Logiweb aspects of lemma reciprocalIsTotal in pyk

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The predefined "pyk" aspect

define pyk of lemma reciprocalIsTotal as text unicode start of text unicode small l unicode small e unicode small m unicode small m unicode small a unicode space unicode small r unicode small e unicode small c unicode small i unicode small p unicode small r unicode small o unicode small c unicode small a unicode small l unicode capital i unicode small s unicode capital t unicode small o unicode small t unicode small a unicode small l unicode end of text end unicode text end text end define

The predefined "tex" aspect

define tex of lemma reciprocalIsTotal as text unicode start of text unicode capital r unicode small e unicode small c unicode small i unicode small p unicode small r unicode small o unicode small c unicode small a unicode small l unicode capital i unicode small s unicode capital t unicode small o unicode small t unicode small a unicode small l unicode end of text end unicode text end text end define

The user defined "the statement aspect" aspect

define statement of lemma reciprocalIsTotal as system Q infer all metavar var fx end metavar indeed for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set end define

The user defined "the proof aspect" aspect

define proof of lemma reciprocalIsTotal as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed [ metavar var fx end metavar ; object var var s1 end var ] = 0 infer object var var s1 end var in0 N infer axiom rationalType conclude 0 in0 Q cut lemma toCartProd modus ponens object var var s1 end var in0 N modus ponens 0 in0 Q conclude zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 0 end pair end pair in0 the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set cut lemma eqReflexivity conclude zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 0 end pair end pair = zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 0 end pair end pair cut prop lemma join conjuncts modus ponens [ metavar var fx end metavar ; object var var s1 end var ] = 0 modus ponens zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 0 end pair end pair = zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 0 end pair end pair conclude not0 [ metavar var fx end metavar ; object var var s1 end var ] = 0 imply not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 0 end pair end pair = zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 0 end pair end pair cut prop lemma weaken or first modus ponens not0 [ metavar var fx end metavar ; object var var s1 end var ] = 0 imply not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 0 end pair end pair = zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 0 end pair end pair conclude not0 not0 not0 [ metavar var fx end metavar ; object var var s1 end var ] = 0 imply not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 0 end pair end pair = zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 1/ [ metavar var fx end metavar ; object var var s1 end var ] end pair end pair imply not0 [ metavar var fx end metavar ; object var var s1 end var ] = 0 imply not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 0 end pair end pair = zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 0 end pair end pair cut pred lemma intro exist at object var var s1 end var modus ponens not0 not0 not0 [ metavar var fx end metavar ; object var var s1 end var ] = 0 imply not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 0 end pair end pair = zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 1/ [ metavar var fx end metavar ; object var var s1 end var ] end pair end pair imply not0 [ metavar var fx end metavar ; object var var s1 end var ] = 0 imply not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 0 end pair end pair = zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 0 end pair end pair conclude not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 0 end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 0 end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair cut lemma formula2separation modus ponens zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 0 end pair end pair in0 the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set modus ponens not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 0 end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 0 end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair conclude zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 0 end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set cut pred lemma intro exist at 0 modus ponens zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 0 end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set conclude not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set cut all metavar var fx end metavar indeed not0 [ metavar var fx end metavar ; object var var s1 end var ] = 0 infer object var var s1 end var in0 N infer axiom seriesType conclude not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fx end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fx end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fx end metavar cut lemma valueType modus ponens object var var s1 end var in0 N modus ponens not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fx end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fx end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fx end metavar conclude [ metavar var fx end metavar ; object var var s1 end var ] in0 Q cut lemma QisClosed(reciprocal) modus ponens not0 [ metavar var fx end metavar ; object var var s1 end var ] = 0 modus ponens [ metavar var fx end metavar ; object var var s1 end var ] in0 Q conclude 1/ [ metavar var fx end metavar ; object var var s1 end var ] in0 Q cut lemma toCartProd modus ponens object var var s1 end var in0 N modus ponens 1/ [ metavar var fx end metavar ; object var var s1 end var ] in0 Q conclude zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 1/ [ metavar var fx end metavar ; object var var s1 end var ] end pair end pair in0 the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set cut lemma eqReflexivity conclude zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 1/ [ metavar var fx end metavar ; object var var s1 end var ] end pair end pair = zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 1/ [ metavar var fx end metavar ; object var var s1 end var ] end pair end pair cut prop lemma join conjuncts modus ponens not0 [ metavar var fx end metavar ; object var var s1 end var ] = 0 modus ponens zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 1/ [ metavar var fx end metavar ; object var var s1 end var ] end pair end pair = zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 1/ [ metavar var fx end metavar ; object var var s1 end var ] end pair end pair conclude not0 not0 [ metavar var fx end metavar ; object var var s1 end var ] = 0 imply not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 1/ [ metavar var fx end metavar ; object var var s1 end var ] end pair end pair = zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 1/ [ metavar var fx end metavar ; object var var s1 end var ] end pair end pair cut prop lemma weaken or second modus ponens not0 not0 [ metavar var fx end metavar ; object var var s1 end var ] = 0 imply not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 1/ [ metavar var fx end metavar ; object var var s1 end var ] end pair end pair = zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 1/ [ metavar var fx end metavar ; object var var s1 end var ] end pair end pair conclude not0 not0 not0 [ metavar var fx end metavar ; object var var s1 end var ] = 0 imply not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 1/ [ metavar var fx end metavar ; object var var s1 end var ] end pair end pair = zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 1/ [ metavar var fx end metavar ; object var var s1 end var ] end pair end pair imply not0 [ metavar var fx end metavar ; object var var s1 end var ] = 0 imply not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 1/ [ metavar var fx end metavar ; object var var s1 end var ] end pair end pair = zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 0 end pair end pair cut pred lemma intro exist at object var var s1 end var modus ponens not0 not0 not0 [ metavar var fx end metavar ; object var var s1 end var ] = 0 imply not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 1/ [ metavar var fx end metavar ; object var var s1 end var ] end pair end pair = zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 1/ [ metavar var fx end metavar ; object var var s1 end var ] end pair end pair imply not0 [ metavar var fx end metavar ; object var var s1 end var ] = 0 imply not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 1/ [ metavar var fx end metavar ; object var var s1 end var ] end pair end pair = zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 0 end pair end pair conclude not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 1/ [ metavar var fx end metavar ; object var var s1 end var ] end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 1/ [ metavar var fx end metavar ; object var var s1 end var ] end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair cut lemma formula2separation modus ponens zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 1/ [ metavar var fx end metavar ; object var var s1 end var ] end pair end pair in0 the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set modus ponens not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 1/ [ metavar var fx end metavar ; object var var s1 end var ] end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 1/ [ metavar var fx end metavar ; object var var s1 end var ] end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair conclude zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 1/ [ metavar var fx end metavar ; object var var s1 end var ] end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set cut pred lemma intro exist at 1/ [ metavar var fx end metavar ; object var var s1 end var ] modus ponens zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 1/ [ metavar var fx end metavar ; object var var s1 end var ] end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set conclude not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set cut all metavar var fx end metavar indeed 1rule deduction modus ponens all metavar var fx end metavar indeed [ metavar var fx end metavar ; object var var s1 end var ] = 0 infer object var var s1 end var in0 N infer not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set conclude [ metavar var fx end metavar ; object var var s1 end var ] = 0 imply for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set cut 1rule deduction modus ponens all metavar var fx end metavar indeed not0 [ metavar var fx end metavar ; object var var s1 end var ] = 0 infer object var var s1 end var in0 N infer not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set conclude not0 [ metavar var fx end metavar ; object var var s1 end var ] = 0 imply for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set cut prop lemma from negations modus ponens [ metavar var fx end metavar ; object var var s1 end var ] = 0 imply for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set modus ponens not0 [ metavar var fx end metavar ; object var var s1 end var ] = 0 imply for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set conclude for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set end quote state proof state cache var c end expand end define