Logiweb aspects of lemma bs subset union(bs/r) in pyk

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The predefined "pyk" aspect

define pyk of lemma bs subset union(bs/r) 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 b unicode small s unicode space unicode small s unicode small u unicode small b unicode small s unicode small e unicode small t unicode space unicode small u unicode small n unicode small i unicode small o unicode small n unicode left parenthesis unicode small b unicode small s unicode slash unicode small r unicode right parenthesis unicode end of text end unicode text end text end define

The predefined "tex" aspect

define tex of lemma bs subset union(bs/r) as text unicode start of text unicode capital b unicode capital s unicode small s unicode small u unicode small b unicode small s unicode small e unicode small t unicode end of text end unicode text end text end define

The user defined "the statement aspect" aspect

define statement of lemma bs subset union(bs/r) as system zf infer all metavar var r end metavar indeed all metavar var s end metavar indeed all metavar var big set end metavar indeed quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var big set end metavar end quote endorse not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply metavar var s end metavar zermelo in metavar var big set end metavar imply metavar var s end metavar zermelo in union the set of ph in power metavar var big set end metavar end power such that not0 existential var var t end var zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in metavar var r end metavar end set zermelo is placeholder-var var b end var end set end union end define

The user defined "the proof aspect" aspect

define proof of lemma bs subset union(bs/r) as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var r end metavar indeed all metavar var s end metavar indeed all metavar var big set end metavar indeed quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote endorse not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar infer metavar var s end metavar zermelo in metavar var big set end metavar infer lemma auto member modus probans quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote modus probans quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote modus ponens not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar modus ponens metavar var s end metavar zermelo in metavar var big set end metavar conclude metavar var s end metavar zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var s end metavar end pair end pair zermelo in metavar var r end metavar end set cut lemma equivalence class is subset conclude object var var s end var zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var s end metavar end pair end pair zermelo in metavar var r end metavar end set imply object var var s end var zermelo in metavar var big set end metavar cut lemma subset in power set modus ponens object var var s end var zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var s end metavar end pair end pair zermelo in metavar var r end metavar end set imply object var var s end var zermelo in metavar var big set end metavar conclude the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var s end metavar end pair end pair zermelo in metavar var r end metavar end set zermelo in power metavar var big set end metavar end power cut lemma =reflexivity conclude the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var s end metavar end pair end pair zermelo in metavar var r end metavar end set zermelo is the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var s end metavar end pair end pair zermelo in metavar var r end metavar end set cut prop lemma join conjuncts modus ponens metavar var s end metavar zermelo in metavar var big set end metavar modus ponens the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var s end metavar end pair end pair zermelo in metavar var r end metavar end set zermelo is the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var s end metavar end pair end pair zermelo in metavar var r end metavar end set conclude not0 metavar var s end metavar zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var s end metavar end pair end pair zermelo in metavar var r end metavar end set zermelo is the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var s end metavar end pair end pair zermelo in metavar var r end metavar end set cut 1rule exist intro at existential var var a end var at metavar var s end metavar modus ponens not0 metavar var s end metavar zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var s end metavar end pair end pair zermelo in metavar var r end metavar end set zermelo is the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var s end metavar end pair end pair zermelo in metavar var r end metavar end set conclude not0 existential var var a end var zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in metavar var r end metavar end set zermelo is the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var s end metavar end pair end pair zermelo in metavar var r end metavar end set cut lemma formula2separation modus ponens the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var s end metavar end pair end pair zermelo in metavar var r end metavar end set zermelo in power metavar var big set end metavar end power modus ponens not0 existential var var a end var zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var a end var end pair end pair zermelo in metavar var r end metavar end set zermelo is the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var s end metavar end pair end pair zermelo in metavar var r end metavar end set conclude the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var s end metavar end pair end pair zermelo in metavar var r end metavar end set zermelo in the set of ph in power metavar var big set end metavar end power such that not0 existential var var t end var zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in metavar var r end metavar end set zermelo is placeholder-var var b end var end set cut 1rule exist intro at existential var var j end var at the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var s end metavar end pair end pair zermelo in metavar var r end metavar end set modus ponens metavar var s end metavar zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var s end metavar end pair end pair zermelo in metavar var r end metavar end set conclude metavar var s end metavar zermelo in existential var var j end var cut 1rule exist intro at existential var var j end var at the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var s end metavar end pair end pair zermelo in metavar var r end metavar end set modus ponens the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var s end metavar end pair end pair zermelo in metavar var r end metavar end set zermelo in the set of ph in power metavar var big set end metavar end power such that not0 existential var var t end var zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in metavar var r end metavar end set zermelo is placeholder-var var b end var end set conclude existential var var j end var zermelo in the set of ph in power metavar var big set end metavar end power such that not0 existential var var t end var zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in metavar var r end metavar end set zermelo is placeholder-var var b end var end set cut lemma formula2union modus ponens metavar var s end metavar zermelo in existential var var j end var modus ponens existential var var j end var zermelo in the set of ph in power metavar var big set end metavar end power such that not0 existential var var t end var zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in metavar var r end metavar end set zermelo is placeholder-var var b end var end set conclude metavar var s end metavar zermelo in union the set of ph in power metavar var big set end metavar end power such that not0 existential var var t end var zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in metavar var r end metavar end set zermelo is placeholder-var var b end var end set end union cut all metavar var r end metavar indeed all metavar var s end metavar indeed all metavar var big set end metavar indeed 1rule deduction modus ponens all metavar var r end metavar indeed all metavar var s end metavar indeed all metavar var big set end metavar indeed quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote endorse not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar infer metavar var s end metavar zermelo in metavar var big set end metavar infer metavar var s end metavar zermelo in union the set of ph in power metavar var big set end metavar end power such that not0 existential var var t end var zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in metavar var r end metavar end set zermelo is placeholder-var var b end var end set end union conclude quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var big set end metavar end quote endorse not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply metavar var s end metavar zermelo in metavar var big set end metavar imply metavar var s end metavar zermelo in union the set of ph in power metavar var big set end metavar end power such that not0 existential var var t end var zermelo in metavar var big set end metavar imply not0 the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma existential var var t end var end pair end pair zermelo in metavar var r end metavar end set zermelo is placeholder-var var b end var end set end union end quote state proof state cache var c end expand end define