Logiweb aspects of lemma CPseparationIsRelation in pyk

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The predefined "pyk" aspect

define pyk of lemma CPseparationIsRelation as text unicode start of text unicode small l unicode small e unicode small m unicode small m unicode small a unicode space unicode capital c unicode capital p unicode small s unicode small e unicode small p unicode small a unicode small r unicode small a unicode small t unicode small i unicode small o unicode small n unicode capital i unicode small s unicode capital r unicode small e unicode small l unicode small a 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 CPseparationIsRelation as text unicode start of text unicode capital c unicode capital p unicode small s unicode small e unicode small p unicode small a unicode small r unicode small a unicode small t unicode small i unicode small o unicode small n unicode capital i unicode small s unicode capital r unicode small e unicode small l unicode small a 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 CPseparationIsRelation as system Q infer all metavar var a end metavar indeed all metavar var sx end metavar indeed all metavar var sy end metavar indeed for all objects object var var r1 end var indeed object var var r1 end var in0 the set of ph in the set of ph in power power U( zermelo pair metavar var sx end metavar comma metavar var sy 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 sx end metavar imply not0 object var var op2 end var in0 metavar var sy 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 such that metavar var a end metavar end set imply not0 for all objects object var var op1 end 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 sx end metavar imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end define

The user defined "the proof aspect" aspect

define proof of lemma CPseparationIsRelation as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var a end metavar indeed all metavar var sx end metavar indeed all metavar var sy end metavar indeed object var var s1 end var in0 the set of ph in the set of ph in power power U( zermelo pair metavar var sx end metavar comma metavar var sy 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 sx end metavar imply not0 object var var op2 end var in0 metavar var sy 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 such that metavar var a end metavar end set infer lemma separation2formula(1) modus ponens object var var s1 end var in0 the set of ph in the set of ph in power power U( zermelo pair metavar var sx end metavar comma metavar var sy 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 sx end metavar imply not0 object var var op2 end var in0 metavar var sy 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 such that metavar var a end metavar end set conclude object var var s1 end var in0 the set of ph in power power U( zermelo pair metavar var sx end metavar comma metavar var sy 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 sx end metavar imply not0 object var var op2 end var in0 metavar var sy 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 all metavar var a end metavar indeed all metavar var sx end metavar indeed all metavar var sy end metavar indeed 1rule deduction modus ponens all metavar var a end metavar indeed all metavar var sx end metavar indeed all metavar var sy end metavar indeed object var var s1 end var in0 the set of ph in the set of ph in power power U( zermelo pair metavar var sx end metavar comma metavar var sy 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 sx end metavar imply not0 object var var op2 end var in0 metavar var sy 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 such that metavar var a end metavar end set infer object var var s1 end var in0 the set of ph in power power U( zermelo pair metavar var sx end metavar comma metavar var sy 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 sx end metavar imply not0 object var var op2 end var in0 metavar var sy 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 for all objects object var var s1 end var indeed object var var s1 end var in0 the set of ph in the set of ph in power power U( zermelo pair metavar var sx end metavar comma metavar var sy 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 sx end metavar imply not0 object var var op2 end var in0 metavar var sy 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 such that metavar var a end metavar end set imply object var var s1 end var in0 the set of ph in power power U( zermelo pair metavar var sx end metavar comma metavar var sy 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 sx end metavar imply not0 object var var op2 end var in0 metavar var sy 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 1rule repetition modus ponens for all objects object var var s1 end var indeed object var var s1 end var in0 the set of ph in the set of ph in power power U( zermelo pair metavar var sx end metavar comma metavar var sy 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 sx end metavar imply not0 object var var op2 end var in0 metavar var sy 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 such that metavar var a end metavar end set imply object var var s1 end var in0 the set of ph in power power U( zermelo pair metavar var sx end metavar comma metavar var sy 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 sx end metavar imply not0 object var var op2 end var in0 metavar var sy 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 for all objects object var var s1 end var indeed object var var s1 end var in0 the set of ph in the set of ph in power power U( zermelo pair metavar var sx end metavar comma metavar var sy 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 sx end metavar imply not0 object var var op2 end var in0 metavar var sy 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 such that metavar var a end metavar end set imply object var var s1 end var in0 the set of ph in power power U( zermelo pair metavar var sx end metavar comma metavar var sy 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 sx end metavar imply not0 object var var op2 end var in0 metavar var sy 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 cartProdIsRelation conclude for all objects object var var r1 end var indeed object var var r1 end var in0 the set of ph in power power U( zermelo pair metavar var sx end metavar comma metavar var sy 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 sx end metavar imply not0 object var var op2 end var in0 metavar var sy 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 imply not0 for all objects object var var op1 end 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 sx end metavar imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 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 subsetIsRelation modus ponens for all objects object var var r1 end var indeed object var var r1 end var in0 the set of ph in power power U( zermelo pair metavar var sx end metavar comma metavar var sy 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 sx end metavar imply not0 object var var op2 end var in0 metavar var sy 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 imply not0 for all objects object var var op1 end 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 sx end metavar imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair modus ponens for all objects object var var s1 end var indeed object var var s1 end var in0 the set of ph in the set of ph in power power U( zermelo pair metavar var sx end metavar comma metavar var sy 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 sx end metavar imply not0 object var var op2 end var in0 metavar var sy 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 such that metavar var a end metavar end set imply object var var s1 end var in0 the set of ph in power power U( zermelo pair metavar var sx end metavar comma metavar var sy 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 sx end metavar imply not0 object var var op2 end var in0 metavar var sy 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 for all objects object var var r1 end var indeed object var var r1 end var in0 the set of ph in the set of ph in power power U( zermelo pair metavar var sx end metavar comma metavar var sy 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 sx end metavar imply not0 object var var op2 end var in0 metavar var sy 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 such that metavar var a end metavar end set imply not0 for all objects object var var op1 end 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 sx end metavar imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end quote state proof state cache var c end expand end define