Logiweb aspects of lemma reciprocalIsFunction in pyk

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The predefined "pyk" aspect

define pyk of lemma reciprocalIsFunction 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 f unicode small u unicode small n unicode small c unicode small t unicode small i unicode small o unicode small n unicode end of text end unicode text end text end define

The predefined "tex" aspect

define tex of lemma reciprocalIsFunction 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 f unicode small u unicode small n unicode small c unicode small t unicode small i unicode small o unicode small n unicode end of text end unicode text end text end define

The user defined "the statement aspect" aspect

define statement of lemma reciprocalIsFunction as system Q infer all metavar var m end metavar indeed all metavar var m end metavar indeed all metavar var fx end metavar indeed for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 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 imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 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 imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var end define

The user defined "the proof aspect" aspect

define proof of lemma reciprocalIsFunction 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 m end metavar indeed all metavar var fx end metavar indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 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 infer zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 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 infer object var var f1 end var = object var var f3 end var infer 1rule repetition modus ponens zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 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 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 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 lemma separation2formula(2) modus ponens zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 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 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 f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 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 f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 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 1rule repetition modus ponens zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 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 zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 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 lemma separation2formula(2) modus ponens zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 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 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 f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 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 f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 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 all metavar var m end metavar indeed all metavar var m end metavar indeed all metavar var fx end metavar indeed object var var f1 end var = object var var f3 end var infer not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 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 infer not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 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 infer prop lemma first conjunct modus ponens not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 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 conclude not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 cut prop lemma second conjunct modus ponens not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 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 conclude zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 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 cut prop lemma second conjunct modus ponens not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 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 conclude zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 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 cut lemma fromOrderedPair(1) modus ponens zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 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 conclude object var var f1 end var = metavar var m end metavar cut lemma eqSymmetry modus ponens object var var f1 end var = metavar var m end metavar conclude metavar var m end metavar = object var var f1 end var cut lemma fromOrderedPair(1) modus ponens zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 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 conclude object var var f3 end var = metavar var m end metavar cut lemma eqTransitivity4 modus ponens metavar var m end metavar = object var var f1 end var modus ponens object var var f1 end var = object var var f3 end var modus ponens object var var f3 end var = metavar var m end metavar conclude metavar var m end metavar = metavar var m end metavar cut lemma sameSeries modus ponens metavar var m end metavar = metavar var m end metavar conclude [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] cut lemma sameReciprocal modus ponens not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 modus ponens [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] conclude 1/ [ metavar var fx end metavar ; metavar var m end metavar ] = 1/ [ metavar var fx end metavar ; metavar var m end metavar ] cut lemma fromOrderedPair(2) modus ponens zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 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 conclude object var var f2 end var = 1/ [ metavar var fx end metavar ; metavar var m end metavar ] cut lemma fromOrderedPair(2) modus ponens zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 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 conclude object var var f4 end var = 1/ [ metavar var fx end metavar ; metavar var m end metavar ] cut lemma eqSymmetry modus ponens object var var f4 end var = 1/ [ metavar var fx end metavar ; metavar var m end metavar ] conclude 1/ [ metavar var fx end metavar ; metavar var m end metavar ] = object var var f4 end var cut lemma eqTransitivity4 modus ponens object var var f2 end var = 1/ [ metavar var fx end metavar ; metavar var m end metavar ] modus ponens 1/ [ metavar var fx end metavar ; metavar var m end metavar ] = 1/ [ metavar var fx end metavar ; metavar var m end metavar ] modus ponens 1/ [ metavar var fx end metavar ; metavar var m end metavar ] = object var var f4 end var conclude object var var f2 end var = object var var f4 end var cut all metavar var m end metavar indeed all metavar var m end metavar indeed all metavar var fx end metavar indeed object var var f1 end var = object var var f3 end var infer not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 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 infer not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 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 infer prop lemma first conjunct modus ponens not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 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 conclude not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 cut prop lemma second conjunct modus ponens not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 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 conclude zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 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 cut prop lemma first conjunct modus ponens not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 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 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 cut prop lemma second conjunct modus ponens not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 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 f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 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 orderedPairEquality modus ponens zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 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 modus ponens zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 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 modus ponens object var var f1 end var = object var var f3 end var conclude metavar var m end metavar = metavar var m end metavar cut lemma sameSeries modus ponens metavar var m end metavar = metavar var m end metavar conclude [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] cut lemma eqTransitivity modus ponens [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] modus ponens [ metavar var fx end metavar ; metavar var m end metavar ] = 0 conclude [ metavar var fx end metavar ; metavar var m end metavar ] = 0 cut prop lemma from contradiction modus ponens [ metavar var fx end metavar ; metavar var m end metavar ] = 0 modus ponens not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 conclude object var var f2 end var = object var var f4 end var cut all metavar var m end metavar indeed all metavar var m end metavar indeed all metavar var fx end metavar indeed object var var f1 end var = object var var f3 end var infer not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 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 infer not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 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 infer prop lemma first conjunct modus ponens not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 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 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 cut prop lemma second conjunct modus ponens not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 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 f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 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 prop lemma first conjunct modus ponens not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 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 conclude not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 cut prop lemma second conjunct modus ponens not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 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 conclude zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 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 cut lemma orderedPairEquality modus ponens zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 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 modus ponens zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 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 modus ponens object var var f1 end var = object var var f3 end var conclude metavar var m end metavar = metavar var m end metavar cut lemma sameSeries modus ponens metavar var m end metavar = metavar var m end metavar conclude [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] cut lemma eqSymmetry modus ponens [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] conclude [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] cut lemma eqTransitivity modus ponens [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] modus ponens [ metavar var fx end metavar ; metavar var m end metavar ] = 0 conclude [ metavar var fx end metavar ; metavar var m end metavar ] = 0 cut prop lemma from contradiction modus ponens [ metavar var fx end metavar ; metavar var m end metavar ] = 0 modus ponens not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 conclude object var var f2 end var = object var var f4 end var cut all metavar var m end metavar indeed all metavar var m end metavar indeed all metavar var fx end metavar indeed not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 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 infer not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 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 infer prop lemma second conjunct modus ponens not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 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 f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 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 prop lemma second conjunct modus ponens not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 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 f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 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 fromOrderedPair(2) modus ponens zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 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 object var var f2 end var = 0 cut lemma fromOrderedPair(2) modus ponens zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 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 object var var f4 end var = 0 cut lemma eqSymmetry modus ponens object var var f4 end var = 0 conclude 0 = object var var f4 end var cut lemma eqTransitivity modus ponens object var var f2 end var = 0 modus ponens 0 = object var var f4 end var conclude object var var f2 end var = object var var f4 end var cut all metavar var m end metavar indeed all metavar var m end metavar indeed all metavar var fx end metavar indeed object var var f1 end var = object var var f3 end var infer not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 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 f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 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 infer not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 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 f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 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 infer 1rule deduction modus ponens all metavar var m end metavar indeed all metavar var m end metavar indeed all metavar var fx end metavar indeed object var var f1 end var = object var var f3 end var infer not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 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 infer not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 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 infer object var var f2 end var = object var var f4 end var conclude object var var f1 end var = object var var f3 end var imply not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 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 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 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 object var var f2 end var = object var var f4 end var cut 1rule mp modus ponens object var var f1 end var = object var var f3 end var imply not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 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 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 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 object var var f2 end var = object var var f4 end var modus ponens object var var f1 end var = object var var f3 end var conclude not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 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 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 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 object var var f2 end var = object var var f4 end var cut 1rule deduction modus ponens all metavar var m end metavar indeed all metavar var m end metavar indeed all metavar var fx end metavar indeed object var var f1 end var = object var var f3 end var infer not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 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 infer not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 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 infer object var var f2 end var = object var var f4 end var conclude object var var f1 end var = object var var f3 end var imply not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 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 f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 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 imply object var var f2 end var = object var var f4 end var cut 1rule mp modus ponens object var var f1 end var = object var var f3 end var imply not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 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 f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 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 imply object var var f2 end var = object var var f4 end var modus ponens object var var f1 end var = object var var f3 end var conclude not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 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 f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 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 imply object var var f2 end var = object var var f4 end var cut 1rule deduction modus ponens all metavar var m end metavar indeed all metavar var m end metavar indeed all metavar var fx end metavar indeed object var var f1 end var = object var var f3 end var infer not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 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 infer not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 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 infer object var var f2 end var = object var var f4 end var conclude object var var f1 end var = object var var f3 end var imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 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 imply not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 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 object var var f2 end var = object var var f4 end var cut 1rule mp modus ponens object var var f1 end var = object var var f3 end var imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 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 imply not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 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 object var var f2 end var = object var var f4 end var modus ponens object var var f1 end var = object var var f3 end var conclude not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 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 imply not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 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 object var var f2 end var = object var var f4 end var cut 1rule deduction modus ponens all metavar var m end metavar indeed all metavar var m end metavar indeed all metavar var fx end metavar indeed not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 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 infer not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 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 infer object var var f2 end var = object var var f4 end var conclude not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 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 imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 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 imply object var var f2 end var = object var var f4 end var cut prop lemma from two times two disjuncts modus ponens not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 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 f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 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 modus ponens not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 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 f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 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 modus ponens not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 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 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 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 object var var f2 end var = object var var f4 end var modus ponens not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 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 f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 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 imply object var var f2 end var = object var var f4 end var modus ponens not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 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 imply not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 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 object var var f2 end var = object var var f4 end var modus ponens not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 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 imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 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 imply object var var f2 end var = object var var f4 end var conclude object var var f2 end var = object var var f4 end var cut 1rule deduction modus ponens all metavar var m end metavar indeed all metavar var m end metavar indeed all metavar var fx end metavar indeed object var var f1 end var = object var var f3 end var infer not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 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 f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 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 infer not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 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 f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 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 infer object var var f2 end var = object var var f4 end var conclude object var var f1 end var = object var var f3 end var imply not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 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 f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 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 imply not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 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 f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 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 imply object var var f2 end var = object var var f4 end var cut 1rule mp modus ponens object var var f1 end var = object var var f3 end var imply not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 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 f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 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 imply not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 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 f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 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 imply object var var f2 end var = object var var f4 end var modus ponens object var var f1 end var = object var var f3 end var conclude not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 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 f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 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 imply not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 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 f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 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 imply object var var f2 end var = object var var f4 end var cut pred lemma exist mp2 modus ponens not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 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 f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 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 imply not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 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 f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 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 imply object var var f2 end var = object var var f4 end var 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 f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 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 f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 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 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 f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 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 f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 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 object var var f2 end var = object var var f4 end var cut all metavar var m end metavar indeed all metavar var m end metavar indeed all metavar var fx end metavar indeed 1rule deduction modus ponens all metavar var m end metavar indeed all metavar var m end metavar indeed all metavar var fx end metavar indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 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 infer zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 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 infer object var var f1 end var = object var var f3 end var infer object var var f2 end var = object var var f4 end var conclude for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 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 imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 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 imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var end quote state proof state cache var c end expand end define