Logiweb aspects of lemma fromCartProd in pyk

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The predefined "pyk" aspect

define pyk of lemma fromCartProd 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 f unicode small r unicode small o unicode small m unicode capital c unicode small a unicode small r unicode small t unicode capital p unicode small r unicode small o unicode small d unicode end of text end unicode text end text end define

The predefined "tex" aspect

define tex of lemma fromCartProd as text unicode start of text unicode capital f unicode small r unicode small o unicode small m unicode capital c unicode small a unicode small r unicode small t unicode capital p unicode small r unicode small o unicode small d unicode end of text end unicode text end text end define

The user defined "the statement aspect" aspect

define statement of lemma fromCartProd as system Q infer all metavar var sx end metavar indeed all metavar var sx1 end metavar indeed all metavar var sy end metavar indeed all metavar var sy1 end metavar indeed zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair in0 the set of ph in power power U( zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sx1 end metavar imply not0 object var var op2 end var in0 metavar var sy1 end metavar imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set infer not0 metavar var sx end metavar in0 metavar var sx1 end metavar imply not0 metavar var sy end metavar in0 metavar var sy1 end metavar end define

The user defined "the proof aspect" aspect

define proof of lemma fromCartProd as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var sx end metavar indeed all metavar var sx1 end metavar indeed all metavar var sy end metavar indeed all metavar var sy1 end metavar indeed zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair in0 the set of ph in power power U( zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sx1 end metavar imply not0 object var var op2 end var in0 metavar var sy1 end metavar imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set infer 1rule repetition modus ponens zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair in0 the set of ph in power power U( zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sx1 end metavar imply not0 object var var op2 end var in0 metavar var sy1 end metavar imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set conclude zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair in0 the set of ph in power power U( zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sx1 end metavar imply not0 object var var op2 end var in0 metavar var sy1 end metavar imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var 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 separation2formula modus ponens zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair in0 the set of ph in power power U( zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sx1 end metavar imply not0 object var var op2 end var in0 metavar var sy1 end metavar imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set conclude not0 zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair in0 power power U( zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair ) end power end power imply not0 not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sx1 end metavar imply not0 object var var op2 end var in0 metavar var sy1 end metavar imply not0 zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair cut prop lemma second conjunct modus ponens not0 zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair in0 power power U( zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair ) end power end power imply not0 not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sx1 end metavar imply not0 object var var op2 end var in0 metavar var sy1 end metavar imply not0 zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair conclude not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sx1 end metavar imply not0 object var var op2 end var in0 metavar var sy1 end metavar imply not0 zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair cut 1rule repetition modus ponens not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sx1 end metavar imply not0 object var var op2 end var in0 metavar var sy1 end metavar imply not0 zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair conclude not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sx1 end metavar imply not0 object var var op2 end var in0 metavar var sy1 end metavar imply not0 zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair cut all metavar var sx end metavar indeed all metavar var sx1 end metavar indeed all metavar var sy end metavar indeed all metavar var sy1 end metavar indeed not0 not0 object var var op1 end var in0 metavar var sx1 end metavar imply not0 object var var op2 end var in0 metavar var sy1 end metavar imply not0 zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair infer prop lemma first conjunct modus ponens not0 not0 object var var op1 end var in0 metavar var sx1 end metavar imply not0 object var var op2 end var in0 metavar var sy1 end metavar imply not0 zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair conclude not0 object var var op1 end var in0 metavar var sx1 end metavar imply not0 object var var op2 end var in0 metavar var sy1 end metavar cut prop lemma first conjunct modus ponens not0 object var var op1 end var in0 metavar var sx1 end metavar imply not0 object var var op2 end var in0 metavar var sy1 end metavar conclude object var var op1 end var in0 metavar var sx1 end metavar cut prop lemma second conjunct modus ponens not0 object var var op1 end var in0 metavar var sx1 end metavar imply not0 object var var op2 end var in0 metavar var sy1 end metavar conclude object var var op2 end var in0 metavar var sy1 end metavar cut prop lemma second conjunct modus ponens not0 not0 object var var op1 end var in0 metavar var sx1 end metavar imply not0 object var var op2 end var in0 metavar var sy1 end metavar imply not0 zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair conclude zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair cut lemma fromOrderedPair(1) modus ponens zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair conclude metavar var sx end metavar = object var var op1 end var cut lemma sameMember(2) modus ponens metavar var sx end metavar = object var var op1 end var modus ponens object var var op1 end var in0 metavar var sx1 end metavar conclude metavar var sx end metavar in0 metavar var sx1 end metavar cut lemma fromOrderedPair(2) modus ponens zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair conclude metavar var sy end metavar = object var var op2 end var cut lemma sameMember(2) modus ponens metavar var sy end metavar = object var var op2 end var modus ponens object var var op2 end var in0 metavar var sy1 end metavar conclude metavar var sy end metavar in0 metavar var sy1 end metavar cut prop lemma join conjuncts modus ponens metavar var sx end metavar in0 metavar var sx1 end metavar modus ponens metavar var sy end metavar in0 metavar var sy1 end metavar conclude not0 metavar var sx end metavar in0 metavar var sx1 end metavar imply not0 metavar var sy end metavar in0 metavar var sy1 end metavar cut 1rule deduction modus ponens all metavar var sx end metavar indeed all metavar var sx1 end metavar indeed all metavar var sy end metavar indeed all metavar var sy1 end metavar indeed not0 not0 object var var op1 end var in0 metavar var sx1 end metavar imply not0 object var var op2 end var in0 metavar var sy1 end metavar imply not0 zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair infer not0 metavar var sx end metavar in0 metavar var sx1 end metavar imply not0 metavar var sy end metavar in0 metavar var sy1 end metavar conclude not0 not0 object var var op1 end var in0 metavar var sx1 end metavar imply not0 object var var op2 end var in0 metavar var sy1 end metavar imply not0 zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 metavar var sx end metavar in0 metavar var sx1 end metavar imply not0 metavar var sy end metavar in0 metavar var sy1 end metavar cut pred lemma 2exist mp modus ponens not0 not0 object var var op1 end var in0 metavar var sx1 end metavar imply not0 object var var op2 end var in0 metavar var sy1 end metavar imply not0 zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 metavar var sx end metavar in0 metavar var sx1 end metavar imply not0 metavar var sy end metavar in0 metavar var sy1 end metavar modus ponens not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sx1 end metavar imply not0 object var var op2 end var in0 metavar var sy1 end metavar imply not0 zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair conclude not0 metavar var sx end metavar in0 metavar var sx1 end metavar imply not0 metavar var sy end metavar in0 metavar var sy1 end metavar end quote state proof state cache var c end expand end define