%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU697^1 : TPTP v9.2.1. Released v3.7.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n031.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 09:36:28 AM UTC 2026
% Result : Theorem 2.57s 1.52s
% Output : Refutation 2.57s
% Verified :
% SZS Type : Refutation
% Derivation depth : 9
% Number of leaves : 3
% Syntax : Number of formulae : 15 ( 7 unt; 0 typ; 0 def)
% Number of atoms : 3279 ( 885 equ; 0 cnn)
% Maximal formula atoms : 216 ( 218 avg)
% Number of connectives : 1682 ( 15 ~; 5 |; 644 &; 157 @)
% ( 0 <=>; 861 =>; 0 <=; 0 <~>)
% Maximal formula depth : 224 ( 104 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 13 ( 13 >; 0 *; 0 +; 0 <<)
% Number of symbols : 294 ( 291 usr; 217 con; 0-4 aty)
% Number of variables : 82 ( 52 ^; 21 !; 9 ?; 82 :)
% Comments :
%------------------------------------------------------------------------------
thf(func_def_0,type,
in: $i > $i > $o ).
thf(func_def_1,type,
exu: ( $i > $o ) > $o ).
thf(func_def_6,type,
setadjoin: $i > $i > $i ).
thf(func_def_8,type,
powerset: $i > $i ).
thf(func_def_10,type,
setunion: $i > $i ).
thf(func_def_19,type,
descr: ( $i > $o ) > $i ).
thf(func_def_21,type,
dsetconstr: $i > ( $i > $o ) > $i ).
thf(func_def_26,type,
prop2set: $o > $i ).
thf(func_def_36,type,
nonempty: $i > $o ).
thf(func_def_69,type,
set2prop: $i > $o ).
thf(func_def_88,type,
subset: $i > $i > $o ).
thf(func_def_89,type,
disjoint: $i > $i > $o ).
thf(func_def_90,type,
setsmeet: $i > $i > $o ).
thf(func_def_114,type,
binunion: $i > $i > $i ).
thf(func_def_122,type,
binintersect: $i > $i > $i ).
thf(func_def_135,type,
regular: $i > $o ).
thf(func_def_136,type,
setminus: $i > $i > $i ).
thf(func_def_147,type,
symdiff: $i > $i > $i ).
thf(func_def_153,type,
iskpair: $i > $o ).
thf(func_def_158,type,
kpair: $i > $i > $i ).
thf(func_def_160,type,
cartprod: $i > $i > $i ).
thf(func_def_177,type,
singleton: $i > $o ).
thf(func_def_179,type,
ex1: $i > ( $i > $o ) > $o ).
thf(func_def_184,type,
atmost1p: $i > $o ).
thf(func_def_185,type,
atleast2p: $i > $o ).
thf(func_def_186,type,
atmost2p: $i > $o ).
thf(func_def_187,type,
upairsetp: $i > $o ).
thf(func_def_191,type,
kfst: $i > $i ).
thf(func_def_203,type,
ksnd: $i > $i ).
thf(func_def_213,type,
breln: $i > $i > $i > $o ).
thf(func_def_214,type,
dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_222,type,
func: $i > $i > $i > $o ).
thf(func_def_223,type,
funcSet: $i > $i > $i ).
thf(func_def_226,type,
ap: $i > $i > $i > $i > $i ).
thf(func_def_232,type,
lam: $i > $i > ( $i > $i ) > $i ).
thf(func_def_269,type,
sK0: $i > $i ).
thf(func_def_273,type,
sK4: ( $i > $i ) > $i > $i > $i ).
thf(func_def_274,type,
sK5: $i > $i > ( $i > $o ) > $i > $i ).
thf(func_def_275,type,
sK6: $i > $i > $i > ( $i > $o ) > $i ).
thf(func_def_276,type,
sK7: $i > $i > ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_277,type,
sK8: $i > ( $i > $o ) > ( $i > $o ) > $i > $i ).
thf(func_def_278,type,
sK9: ( $i > $i ) > $i > $i > $i ).
thf(func_def_279,type,
sK10: $i > $i ).
thf(func_def_280,type,
sK11: ( $i > $o ) > $i ).
thf(func_def_281,type,
sK12: ( $i > $o ) > $i > $i ).
thf(func_def_282,type,
sK13: $i > ( $i > $o ) > $i ).
thf(func_def_283,type,
sK14: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_284,type,
sK15: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_285,type,
sK16: ( $i > $o ) > $i > $i ).
thf(func_def_286,type,
sK17: ( $i > $o ) > $i > $i ).
thf(func_def_287,type,
sK18: ( $i > $o ) > $i > $i ).
thf(func_def_288,type,
sK19: ( $i > $o ) > $i > $i ).
thf(func_def_289,type,
sK20: ( $i > $o ) > $i > $i ).
thf(func_def_290,type,
sK21: ( $i > $o ) > $i > $i ).
thf(func_def_291,type,
sK22: ( $i > $o ) > $i > $i ).
thf(func_def_292,type,
sK23: ( $i > $o ) > $i > $i ).
thf(func_def_293,type,
sK24: $i > $i ).
thf(func_def_294,type,
sK25: $i > $i > $i > $i > $i ).
thf(func_def_295,type,
sK26: $i > $i > $i > $i > $i ).
thf(func_def_296,type,
sK27: $i > $i > $i > $i > $i ).
thf(func_def_297,type,
sK28: $i > $i > $i > $i > $i ).
thf(func_def_298,type,
sK29: $i > $i > $i > $i > $i ).
thf(func_def_299,type,
sK30: $i > $i > $i > $i > $i ).
thf(func_def_300,type,
sK31: $i > $i > $i > $i > $i ).
thf(func_def_301,type,
sK32: $i > ( $i > $o ) > $i ).
thf(func_def_302,type,
sK33: ( $i > $o ) > $i ).
thf(func_def_303,type,
sK34: ( $i > $o ) > $i ).
thf(func_def_304,type,
sK35: ( $i > $o ) > $i ).
thf(func_def_305,type,
sK36: ( $i > $o ) > $i ).
thf(func_def_306,type,
sK37: ( $i > $o ) > $i ).
thf(func_def_307,type,
sK38: ( $i > $o ) > $i > $i ).
thf(func_def_308,type,
sK39: ( $i > $o ) > $i > $i ).
thf(func_def_309,type,
sK40: ( $i > $o ) > $i > $i ).
thf(func_def_310,type,
sK41: ( $i > $o ) > $i > $i ).
thf(func_def_311,type,
sK42: ( $i > $o ) > $i > $i ).
thf(func_def_312,type,
sK43: ( $i > $o ) > $i > $i ).
thf(f2095,plain,
$false,
inference(trivial_inequality_removal,[status(thm)],[f1177]) ).
thf(f1177,plain,
( ( ^ [Y0: $i,Y1: $i,Y2: $i > $i] :
( dpsetconstr @ Y0 @ Y1
@ ^ [Y3: $i,Y4: $i] :
( ( Y2 @ Y3 )
= Y4 ) )
@ sK1
@ sK2
@ ^ [Y0: $i] : ( sK0 @ Y0 ) )
!= ( ^ [Y0: $i,Y1: $i,Y2: $i > $i] :
( dpsetconstr @ Y0 @ Y1
@ ^ [Y3: $i,Y4: $i] :
( ( Y2 @ Y3 )
= Y4 ) )
@ sK1
@ sK2
@ ^ [Y0: $i] : ( sK0 @ Y0 ) ) ),
inference(definition_unfolding,[status(thm)],[f870,f767,f767]) ).
thf(f767,plain,
( lam
= ( ^ [Y0: $i,Y1: $i,Y2: $i > $i] :
( dpsetconstr @ Y0 @ Y1
@ ^ [Y3: $i,Y4: $i] :
( ( Y2 @ Y3 )
= Y4 ) ) ) ),
inference(cnf_transformation,[status(thm)],[f398]) ).
thf(f398,plain,
( lam
= ( ^ [Y0: $i,Y1: $i,Y2: $i > $i] :
( dpsetconstr @ Y0 @ Y1
@ ^ [Y3: $i,Y4: $i] :
( ( Y2 @ Y3 )
= Y4 ) ) ) ),
inference(fool_elimination,[status(thm)],[f397]) ).
thf(f397,plain,
( lam
= ( ^ [X0: $i,X1: $i,X2: $i > $i] :
( dpsetconstr @ X0 @ X1
@ ^ [X3: $i,X4: $i] :
( ( X2 @ X3 )
= X4 ) ) ) ),
inference(rectify,[status(thm)],[f201]) ).
thf(f201,axiom,
( lam
= ( ^ [X3: $i,X4: $i,X12: $i > $i] :
( dpsetconstr @ X3 @ X4
@ ^ [X1: $i,X2: $i] :
( ( X12 @ X1 )
= X2 ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.EApmVaGZ7F/DTF2THF_24917.p',lam) ).
thf(f870,plain,
( ( lam @ sK1 @ sK2
@ ^ [Y0: $i] : ( sK0 @ Y0 ) )
!= ( lam @ sK1 @ sK2
@ ^ [Y0: $i] : ( sK0 @ Y0 ) ) ),
inference(cnf_transformation,[status(thm)],[f660]) ).
thf(f660,plain,
( ( setminusILneg = $true )
& ( exuE3e = $true )
& ( subsetE2 = $true )
& ( singletoninpowunion = $true )
& ( symdiffIneg2 = $true )
& ( subsetE = $true )
& ( binunionLsub = $true )
& ( setunionsingleton1 = $true )
& ( singletonprop = $true )
& ( emptyE1 = $true )
& ( upairsetE = $true )
& ( setukpairIR = $true )
& ( setextAx = $true )
& ( subsetTrans = $true )
& ( ex1I = $true )
& ( dpsetconstrERa = $true )
& ( ubforcartprodlem3 = $true )
& ( funcext = $true )
& ( notsubsetI = $true )
& ( dsetconstrEL = $true )
& ( inCongP = $true )
& ( funcGraphProp1 = $true )
& ( binunionIL = $true )
& ( notdallE = $true )
& ( exuE2 = $true )
& ( setunionE = $true )
& ( cartprodmempaircEq = $true )
& ( singletonsswitch = $true )
& ( eqinunit = $true )
& ( setadjoinE = $true )
& ( funcinfuncset = $true )
& ( kpairp = $true )
& ( setadjoinSub2 = $true )
& ( symdiffIneg1 = $true )
& ( symdiffI1 = $true )
& ( setukpairinjR12 = $true )
& ( cartprodpairin = $true )
& ( subsetI2 = $true )
& ( bs114d = $true )
& ( nonemptyI1 = $true )
& ( cartprodsndpairEq = $true )
& ( funcGraphProp2 = $true )
& ( cartprodfstin = $true )
& ( setukpairinjR11 = $true )
& ( powerset__Cong = $true )
& ( powersetAx = $true )
& ( lam2p = $true )
& ( brelnall2 = $true )
& ( setukpairinjL2 = $true )
& ( subbreln = $true )
& ( upairinpowunion = $true )
& ( cartprodpairmemER = $true )
& ( powersetsubset = $true )
& ( lamProp = $true )
& ( nonemptyImpWitness = $true )
& ( upairsetIL = $true )
& ( ubforcartprodlem1 = $true )
& ( funcextLem = $true )
& ( nonemptyE1 = $true )
& ( subsetI1 = $true )
& ( setadjoinOr = $true )
& ( exu__Cong = $true )
& ( setminusEL = $true )
& ( prop2set2propI = $true )
& ( notinemptyset = $true )
& ( eqbreln = $true )
& ( setunionE2 = $true )
& ( emptysetimpfalse = $true )
& ( binintersectER = $true )
& ( dpsetconstrER = $true )
& ( emptyI = $true )
& ( vacuousDall = $true )
& ( app = $true )
& ( eta1 = $true )
& ( upairsubunion = $true )
& ( cartprodpairmemEL = $true )
& ( quantDeMorgan4 = $true )
& ( symdiffI2 = $true )
& ( ex1E2 = $true )
& ( binunionE = $true )
& ( kfstsingleton = $true )
& ( binintersectSubset2 = $true )
& ( beta1 = $true )
& ( kpairiskpair = $true )
& ( secondinupair = $true )
& ( quantDeMorgan1 = $true )
& ( lamp = $true )
& ( descrp = $true )
& ( dpsetconstrSub = $true )
& ( setminusIRneg = $true )
& ( setoftrueEq = $true )
& ( ksndsingleton = $true )
& ( ksndpairEq = $true )
& ( sepSubset = $true )
& ( setadjoin__Cong = $true )
& ( binintersectSubset4 = $true )
& ( theprop = $true )
& ( setadjoinAx = $true )
& ( notequalI2 = $true )
& ( binunionEcases = $true )
& ( setukpairinjR = $true )
& ( binintersectSubset5 = $true )
& ( funcext2 = $true )
& ( setukpairinjL = $true )
& ( binintersectSubset1 = $true )
& ( setOfPairsIsBReln = $true )
& ( cartprodmempair = $true )
& ( omegaIndAx = $true )
& ( singletoninpowerset = $true )
& ( foundationAx = $true )
& ( upairset2E = $true )
& ( wellorderingAx = $true )
& ( infuncsetfunc = $true )
& ( emptysetE = $true )
& ( sepInPowerset = $true )
& ( binintersectSubset3 = $true )
& ( ( lam @ sK1 @ sK2
@ ^ [Y0: $i] : ( sK0 @ Y0 ) )
!= ( lam @ sK1 @ sK2
@ ^ [Y0: $i] : ( sK0 @ Y0 ) ) )
& ! [X3: $i] :
( ( ( in @ X3 @ sK1 )
!= $true )
| ( ( in @ ( sK0 @ X3 ) @ sK2 )
= $true ) )
& ( dsetconstrER = $true )
& ( funcGraphProp3 = $true )
& ( funcGraphProp4 = $true )
& ( emptysetAx = $true )
& ( inPowerset = $true )
& ( notinsingleton = $true )
& ( upairequniteq = $true )
& ( setunionsingleton2 = $true )
& ( emptyinunitempty = $true )
& ( emptyInPowerset = $true )
& ( setadjoinIR = $true )
& ( ap2apEq2 = $true )
& ( setminusERneg = $true )
& ( powersetE1 = $true )
& ( emptyset__Cong = $true )
& ( setminusLsub = $true )
& ( setukpairIL = $true )
& ( eqimpsubset1 = $true )
& ( noeltsimpempty = $true )
& ( binintersectLsub = $true )
& ( dpsetconstrEL1 = $true )
& ( setbeta = $true )
& ( kfstpairEq = $true )
& ( exuEu = $true )
& ( setadjoinSub = $true )
& ( exuE1 = $true )
& ( symdiffE = $true )
& ( cartprodfstpairEq = $true )
& ( setextsub = $true )
& ( setukpairinjR1 = $true )
& ( descr__Cong = $true )
& ( binintersectRsub = $true )
& ( quantDeMorgan3 = $true )
& ( exuI3 = $true )
& ( setminusSubset2 = $true )
& ( cartprodmempair1 = $true )
& ( nonemptyI = $true )
& ( dpsetconstrI = $true )
& ( omega0Ax = $true )
& ( setunionsingleton = $true )
& ( subPowSU = $true )
& ( cartprodpairsurjEq = $true )
& ( singletonsubset = $true )
& ( ex1I2 = $true )
& ( binunionIR = $true )
& ( emptysetsubset = $true )
& ( dsetconstr__Cong = $true )
& ( quantDeMorgan2 = $true )
& ( binunionRsub = $true )
& ( setunionI = $true )
& ( setminusI = $true )
& ( powersetI1 = $true )
& ( ubforcartprodlem2 = $true )
& ( powersetE = $true )
& ( omegaSAx = $true )
& ( powersetI = $true )
& ( upairsetIR = $true )
& ( setminusELneg = $true )
& ( funcImageSingleton = $true )
& ( notdexE = $true )
& ( setukpairinjR2 = $true )
& ( dpsetconstrEL2 = $true )
& ( ap2p = $true )
& ( apProp = $true )
& ( replAx = $true )
& ( subsetRefl = $true )
& ( setunion__Cong = $true )
& ( singletonsuniq = $true )
& ( kpairsurjEq = $true )
& ( upairset2IR = $true )
& ( setukpairinjL1 = $true )
& ( binintersectI = $true )
& ( disjointsetsI1 = $true )
& ( dsetconstrI = $true )
& ( exuE3u = $true )
& ( setunionAx = $true )
& ( ex1E1 = $true )
& ( subset2powerset = $true )
& ( brelnall1 = $true )
& ( binintersectEL = $true )
& ( eqimpsubset2 = $true )
& ( prop2setI = $true )
& ( ap2apEq1 = $true )
& ( setminusER = $true )
& ( notequalI1 = $true )
& ( exuI2 = $true )
& ( uniqinunit = $true )
& ( cartprodsndin = $true )
& ( in__Cong = $true )
& ( omega__Cong = $true )
& ( subsetemptysetimpeq = $true )
& ( setminusSubset1 = $true )
& ( prop2setE = $true )
& ( emptyinPowerset = $true )
& ( exuI1 = $true )
& ( setext = $true )
& ( setadjoinIL = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1,sK2])],[f658,f659]) ).
thf(f659,plain,
( ? [X0: $i > $i,X1: $i,X2: $i] :
( ( ( lam @ X1 @ X2
@ ^ [Y0: $i] : ( X0 @ Y0 ) )
!= ( lam @ X1 @ X2
@ ^ [Y0: $i] : ( X0 @ Y0 ) ) )
& ! [X3: $i] :
( ( $true
!= ( in @ X3 @ X1 ) )
| ( ( in @ ( X0 @ X3 ) @ X2 )
= $true ) ) )
=> ( ( ( lam @ sK1 @ sK2
@ ^ [Y0: $i] : ( sK0 @ Y0 ) )
!= ( lam @ sK1 @ sK2
@ ^ [Y0: $i] : ( sK0 @ Y0 ) ) )
& ! [X3: $i] :
( ( ( in @ X3 @ sK1 )
!= $true )
| ( ( in @ ( sK0 @ X3 ) @ sK2 )
= $true ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f658,plain,
( ( setminusILneg = $true )
& ( exuE3e = $true )
& ( subsetE2 = $true )
& ( singletoninpowunion = $true )
& ( symdiffIneg2 = $true )
& ( subsetE = $true )
& ( binunionLsub = $true )
& ( setunionsingleton1 = $true )
& ( singletonprop = $true )
& ( emptyE1 = $true )
& ( upairsetE = $true )
& ( setukpairIR = $true )
& ( setextAx = $true )
& ( subsetTrans = $true )
& ( ex1I = $true )
& ( dpsetconstrERa = $true )
& ( ubforcartprodlem3 = $true )
& ( funcext = $true )
& ( notsubsetI = $true )
& ( dsetconstrEL = $true )
& ( inCongP = $true )
& ( funcGraphProp1 = $true )
& ( binunionIL = $true )
& ( notdallE = $true )
& ( exuE2 = $true )
& ( setunionE = $true )
& ( cartprodmempaircEq = $true )
& ( singletonsswitch = $true )
& ( eqinunit = $true )
& ( setadjoinE = $true )
& ( funcinfuncset = $true )
& ( kpairp = $true )
& ( setadjoinSub2 = $true )
& ( symdiffIneg1 = $true )
& ( symdiffI1 = $true )
& ( setukpairinjR12 = $true )
& ( cartprodpairin = $true )
& ( subsetI2 = $true )
& ( bs114d = $true )
& ( nonemptyI1 = $true )
& ( cartprodsndpairEq = $true )
& ( funcGraphProp2 = $true )
& ( cartprodfstin = $true )
& ( setukpairinjR11 = $true )
& ( powerset__Cong = $true )
& ( powersetAx = $true )
& ( lam2p = $true )
& ( brelnall2 = $true )
& ( setukpairinjL2 = $true )
& ( subbreln = $true )
& ( upairinpowunion = $true )
& ( cartprodpairmemER = $true )
& ( powersetsubset = $true )
& ( lamProp = $true )
& ( nonemptyImpWitness = $true )
& ( upairsetIL = $true )
& ( ubforcartprodlem1 = $true )
& ( funcextLem = $true )
& ( nonemptyE1 = $true )
& ( subsetI1 = $true )
& ( setadjoinOr = $true )
& ( exu__Cong = $true )
& ( setminusEL = $true )
& ( prop2set2propI = $true )
& ( notinemptyset = $true )
& ( eqbreln = $true )
& ( setunionE2 = $true )
& ( emptysetimpfalse = $true )
& ( binintersectER = $true )
& ( dpsetconstrER = $true )
& ( emptyI = $true )
& ( vacuousDall = $true )
& ( app = $true )
& ( eta1 = $true )
& ( upairsubunion = $true )
& ( cartprodpairmemEL = $true )
& ( quantDeMorgan4 = $true )
& ( symdiffI2 = $true )
& ( ex1E2 = $true )
& ( binunionE = $true )
& ( kfstsingleton = $true )
& ( binintersectSubset2 = $true )
& ( beta1 = $true )
& ( kpairiskpair = $true )
& ( secondinupair = $true )
& ( quantDeMorgan1 = $true )
& ( lamp = $true )
& ( descrp = $true )
& ( dpsetconstrSub = $true )
& ( setminusIRneg = $true )
& ( setoftrueEq = $true )
& ( ksndsingleton = $true )
& ( ksndpairEq = $true )
& ( sepSubset = $true )
& ( setadjoin__Cong = $true )
& ( binintersectSubset4 = $true )
& ( theprop = $true )
& ( setadjoinAx = $true )
& ( notequalI2 = $true )
& ( binunionEcases = $true )
& ( setukpairinjR = $true )
& ( binintersectSubset5 = $true )
& ( funcext2 = $true )
& ( setukpairinjL = $true )
& ( binintersectSubset1 = $true )
& ( setOfPairsIsBReln = $true )
& ( cartprodmempair = $true )
& ( omegaIndAx = $true )
& ( singletoninpowerset = $true )
& ( foundationAx = $true )
& ( upairset2E = $true )
& ( wellorderingAx = $true )
& ( infuncsetfunc = $true )
& ( emptysetE = $true )
& ( sepInPowerset = $true )
& ( binintersectSubset3 = $true )
& ? [X0: $i > $i,X1: $i,X2: $i] :
( ( ( lam @ X1 @ X2
@ ^ [Y0: $i] : ( X0 @ Y0 ) )
!= ( lam @ X1 @ X2
@ ^ [Y0: $i] : ( X0 @ Y0 ) ) )
& ! [X3: $i] :
( ( $true
!= ( in @ X3 @ X1 ) )
| ( ( in @ ( X0 @ X3 ) @ X2 )
= $true ) ) )
& ( dsetconstrER = $true )
& ( funcGraphProp3 = $true )
& ( funcGraphProp4 = $true )
& ( emptysetAx = $true )
& ( inPowerset = $true )
& ( notinsingleton = $true )
& ( upairequniteq = $true )
& ( setunionsingleton2 = $true )
& ( emptyinunitempty = $true )
& ( emptyInPowerset = $true )
& ( setadjoinIR = $true )
& ( ap2apEq2 = $true )
& ( setminusERneg = $true )
& ( powersetE1 = $true )
& ( emptyset__Cong = $true )
& ( setminusLsub = $true )
& ( setukpairIL = $true )
& ( eqimpsubset1 = $true )
& ( noeltsimpempty = $true )
& ( binintersectLsub = $true )
& ( dpsetconstrEL1 = $true )
& ( setbeta = $true )
& ( kfstpairEq = $true )
& ( exuEu = $true )
& ( setadjoinSub = $true )
& ( exuE1 = $true )
& ( symdiffE = $true )
& ( cartprodfstpairEq = $true )
& ( setextsub = $true )
& ( setukpairinjR1 = $true )
& ( descr__Cong = $true )
& ( binintersectRsub = $true )
& ( quantDeMorgan3 = $true )
& ( exuI3 = $true )
& ( setminusSubset2 = $true )
& ( cartprodmempair1 = $true )
& ( nonemptyI = $true )
& ( dpsetconstrI = $true )
& ( omega0Ax = $true )
& ( setunionsingleton = $true )
& ( subPowSU = $true )
& ( cartprodpairsurjEq = $true )
& ( singletonsubset = $true )
& ( ex1I2 = $true )
& ( binunionIR = $true )
& ( emptysetsubset = $true )
& ( dsetconstr__Cong = $true )
& ( quantDeMorgan2 = $true )
& ( binunionRsub = $true )
& ( setunionI = $true )
& ( setminusI = $true )
& ( powersetI1 = $true )
& ( ubforcartprodlem2 = $true )
& ( powersetE = $true )
& ( omegaSAx = $true )
& ( powersetI = $true )
& ( upairsetIR = $true )
& ( setminusELneg = $true )
& ( funcImageSingleton = $true )
& ( notdexE = $true )
& ( setukpairinjR2 = $true )
& ( dpsetconstrEL2 = $true )
& ( ap2p = $true )
& ( apProp = $true )
& ( replAx = $true )
& ( subsetRefl = $true )
& ( setunion__Cong = $true )
& ( singletonsuniq = $true )
& ( kpairsurjEq = $true )
& ( upairset2IR = $true )
& ( setukpairinjL1 = $true )
& ( binintersectI = $true )
& ( disjointsetsI1 = $true )
& ( dsetconstrI = $true )
& ( exuE3u = $true )
& ( setunionAx = $true )
& ( ex1E1 = $true )
& ( subset2powerset = $true )
& ( brelnall1 = $true )
& ( binintersectEL = $true )
& ( eqimpsubset2 = $true )
& ( prop2setI = $true )
& ( ap2apEq1 = $true )
& ( setminusER = $true )
& ( notequalI1 = $true )
& ( exuI2 = $true )
& ( uniqinunit = $true )
& ( cartprodsndin = $true )
& ( in__Cong = $true )
& ( omega__Cong = $true )
& ( subsetemptysetimpeq = $true )
& ( setminusSubset1 = $true )
& ( prop2setE = $true )
& ( emptyinPowerset = $true )
& ( exuI1 = $true )
& ( setext = $true )
& ( setadjoinIL = $true ) ),
inference(flattening,[status(thm)],[f657]) ).
thf(f657,plain,
( ? [X0: $i > $i,X1: $i,X2: $i] :
( ( ( lam @ X1 @ X2
@ ^ [Y0: $i] : ( X0 @ Y0 ) )
!= ( lam @ X1 @ X2
@ ^ [Y0: $i] : ( X0 @ Y0 ) ) )
& ! [X3: $i] :
( ( $true
!= ( in @ X3 @ X1 ) )
| ( ( in @ ( X0 @ X3 ) @ X2 )
= $true ) ) )
& ( eta1 = $true )
& ( beta1 = $true )
& ( ap2apEq2 = $true )
& ( ap2apEq1 = $true )
& ( funcext2 = $true )
& ( funcext = $true )
& ( eqbreln = $true )
& ( subbreln = $true )
& ( funcGraphProp4 = $true )
& ( funcextLem = $true )
& ( funcGraphProp2 = $true )
& ( funcGraphProp3 = $true )
& ( funcGraphProp1 = $true )
& ( ex1E2 = $true )
& ( brelnall2 = $true )
& ( brelnall1 = $true )
& ( lam2p = $true )
& ( lamp = $true )
& ( lamProp = $true )
& ( funcinfuncset = $true )
& ( ap2p = $true )
& ( infuncsetfunc = $true )
& ( app = $true )
& ( apProp = $true )
& ( funcImageSingleton = $true )
& ( dpsetconstrER = $true )
& ( dpsetconstrEL2 = $true )
& ( dpsetconstrEL1 = $true )
& ( dpsetconstrERa = $true )
& ( setOfPairsIsBReln = $true )
& ( dpsetconstrSub = $true )
& ( dpsetconstrI = $true )
& ( cartprodpairsurjEq = $true )
& ( cartprodsndpairEq = $true )
& ( cartprodfstpairEq = $true )
& ( cartprodmempaircEq = $true )
& ( cartprodpairmemER = $true )
& ( cartprodpairmemEL = $true )
& ( cartprodsndin = $true )
& ( kpairsurjEq = $true )
& ( ksndpairEq = $true )
& ( ksndsingleton = $true )
& ( setukpairinjR = $true )
& ( setukpairinjR2 = $true )
& ( upairequniteq = $true )
& ( setukpairinjR1 = $true )
& ( setukpairinjR12 = $true )
& ( setukpairinjR11 = $true )
& ( setukpairinjL = $true )
& ( setukpairinjL2 = $true )
& ( cartprodfstin = $true )
& ( kfstpairEq = $true )
& ( theprop = $true )
& ( kfstsingleton = $true )
& ( setukpairinjL1 = $true )
& ( singletonsuniq = $true )
& ( ex1I2 = $true )
& ( ex1I = $true )
& ( ex1E1 = $true )
& ( singletonprop = $true )
& ( setunionsingleton = $true )
& ( setunionsingleton2 = $true )
& ( setunionsingleton1 = $true )
& ( setunionE2 = $true )
& ( cartprodmempair = $true )
& ( cartprodmempair1 = $true )
& ( cartprodpairin = $true )
& ( ubforcartprodlem3 = $true )
& ( ubforcartprodlem2 = $true )
& ( ubforcartprodlem1 = $true )
& ( upairinpowunion = $true )
& ( upairsubunion = $true )
& ( upairset2E = $true )
& ( singletoninpowunion = $true )
& ( singletoninpowerset = $true )
& ( singletonsubset = $true )
& ( kpairp = $true )
& ( kpairiskpair = $true )
& ( setukpairIR = $true )
& ( setukpairIL = $true )
& ( secondinupair = $true )
& ( symdiffIneg2 = $true )
& ( symdiffIneg1 = $true )
& ( symdiffI2 = $true )
& ( symdiffI1 = $true )
& ( symdiffE = $true )
& ( setminusSubset1 = $true )
& ( setminusLsub = $true )
& ( setminusIRneg = $true )
& ( setminusILneg = $true )
& ( setminusELneg = $true )
& ( setminusERneg = $true )
& ( setminusSubset2 = $true )
& ( setminusER = $true )
& ( setminusEL = $true )
& ( setminusI = $true )
& ( bs114d = $true )
& ( binintersectSubset1 = $true )
& ( binintersectSubset4 = $true )
& ( binintersectRsub = $true )
& ( disjointsetsI1 = $true )
& ( binintersectER = $true )
& ( binintersectSubset3 = $true )
& ( binintersectSubset2 = $true )
& ( binintersectLsub = $true )
& ( binintersectEL = $true )
& ( binintersectSubset5 = $true )
& ( binintersectI = $true )
& ( binunionRsub = $true )
& ( binunionLsub = $true )
& ( binunionE = $true )
& ( binunionEcases = $true )
& ( binunionIR = $true )
& ( upairset2IR = $true )
& ( binunionIL = $true )
& ( sepSubset = $true )
& ( sepInPowerset = $true )
& ( powersetsubset = $true )
& ( inPowerset = $true )
& ( powersetE1 = $true )
& ( powersetI1 = $true )
& ( subsetemptysetimpeq = $true )
& ( setextsub = $true )
& ( subset2powerset = $true )
& ( setadjoinSub2 = $true )
& ( setadjoinSub = $true )
& ( subsetTrans = $true )
& ( subsetRefl = $true )
& ( notequalI2 = $true )
& ( notequalI1 = $true )
& ( notsubsetI = $true )
& ( subsetE2 = $true )
& ( subsetE = $true )
& ( emptysetsubset = $true )
& ( subsetI2 = $true )
& ( eqimpsubset1 = $true )
& ( eqimpsubset2 = $true )
& ( subsetI1 = $true )
& ( dsetconstr__Cong = $true )
& ( descr__Cong = $true )
& ( exuEu = $true )
& ( omega__Cong = $true )
& ( setunion__Cong = $true )
& ( powerset__Cong = $true )
& ( setadjoin__Cong = $true )
& ( emptyset__Cong = $true )
& ( exu__Cong = $true )
& ( exuE3u = $true )
& ( in__Cong = $true )
& ( inCongP = $true )
& ( exuI2 = $true )
& ( exuI3 = $true )
& ( exuI1 = $true )
& ( notdallE = $true )
& ( notdexE = $true )
& ( prop2set2propI = $true )
& ( prop2setI = $true )
& ( quantDeMorgan4 = $true )
& ( quantDeMorgan3 = $true )
& ( quantDeMorgan2 = $true )
& ( quantDeMorgan1 = $true )
& ( vacuousDall = $true )
& ( emptyE1 = $true )
& ( upairsetIR = $true )
& ( upairsetIL = $true )
& ( upairsetE = $true )
& ( singletonsswitch = $true )
& ( eqinunit = $true )
& ( notinsingleton = $true )
& ( uniqinunit = $true )
& ( nonemptyImpWitness = $true )
& ( exuE2 = $true )
& ( subPowSU = $true )
& ( setunionE = $true )
& ( setunionI = $true )
& ( powersetE = $true )
& ( emptyInPowerset = $true )
& ( emptyinPowerset = $true )
& ( powersetI = $true )
& ( setoftrueEq = $true )
& ( setadjoinOr = $true )
& ( setadjoinE = $true )
& ( setadjoinIR = $true )
& ( emptyinunitempty = $true )
& ( setadjoinIL = $true )
& ( nonemptyI1 = $true )
& ( nonemptyI = $true )
& ( nonemptyE1 = $true )
& ( setbeta = $true )
& ( noeltsimpempty = $true )
& ( emptyI = $true )
& ( setext = $true )
& ( exuE3e = $true )
& ( notinemptyset = $true )
& ( emptysetimpfalse = $true )
& ( emptysetE = $true )
& ( prop2setE = $true )
& ( exuE1 = $true )
& ( dsetconstrER = $true )
& ( dsetconstrEL = $true )
& ( dsetconstrI = $true )
& ( descrp = $true )
& ( wellorderingAx = $true )
& ( foundationAx = $true )
& ( replAx = $true )
& ( omegaIndAx = $true )
& ( omegaSAx = $true )
& ( omega0Ax = $true )
& ( setunionAx = $true )
& ( powersetAx = $true )
& ( setadjoinAx = $true )
& ( emptysetAx = $true )
& ( setextAx = $true ) ),
inference(ennf_transformation,[status(thm)],[f418]) ).
thf(f418,plain,
~ ( ( setextAx = $true )
=> ( ( emptysetAx = $true )
=> ( ( setadjoinAx = $true )
=> ( ( powersetAx = $true )
=> ( ( setunionAx = $true )
=> ( ( omega0Ax = $true )
=> ( ( omegaSAx = $true )
=> ( ( omegaIndAx = $true )
=> ( ( replAx = $true )
=> ( ( foundationAx = $true )
=> ( ( wellorderingAx = $true )
=> ( ( descrp = $true )
=> ( ( dsetconstrI = $true )
=> ( ( dsetconstrEL = $true )
=> ( ( dsetconstrER = $true )
=> ( ( exuE1 = $true )
=> ( ( prop2setE = $true )
=> ( ( emptysetE = $true )
=> ( ( emptysetimpfalse = $true )
=> ( ( notinemptyset = $true )
=> ( ( exuE3e = $true )
=> ( ( setext = $true )
=> ( ( emptyI = $true )
=> ( ( noeltsimpempty = $true )
=> ( ( setbeta = $true )
=> ( ( nonemptyE1 = $true )
=> ( ( nonemptyI = $true )
=> ( ( nonemptyI1 = $true )
=> ( ( setadjoinIL = $true )
=> ( ( emptyinunitempty = $true )
=> ( ( setadjoinIR = $true )
=> ( ( setadjoinE = $true )
=> ( ( setadjoinOr = $true )
=> ( ( setoftrueEq = $true )
=> ( ( powersetI = $true )
=> ( ( emptyinPowerset = $true )
=> ( ( emptyInPowerset = $true )
=> ( ( powersetE = $true )
=> ( ( setunionI = $true )
=> ( ( setunionE = $true )
=> ( ( subPowSU = $true )
=> ( ( exuE2 = $true )
=> ( ( nonemptyImpWitness = $true )
=> ( ( uniqinunit = $true )
=> ( ( notinsingleton = $true )
=> ( ( eqinunit = $true )
=> ( ( singletonsswitch = $true )
=> ( ( upairsetE = $true )
=> ( ( upairsetIL = $true )
=> ( ( upairsetIR = $true )
=> ( ( emptyE1 = $true )
=> ( ( vacuousDall = $true )
=> ( ( quantDeMorgan1 = $true )
=> ( ( quantDeMorgan2 = $true )
=> ( ( quantDeMorgan3 = $true )
=> ( ( quantDeMorgan4 = $true )
=> ( ( prop2setI = $true )
=> ( ( prop2set2propI = $true )
=> ( ( notdexE = $true )
=> ( ( notdallE = $true )
=> ( ( exuI1 = $true )
=> ( ( exuI3 = $true )
=> ( ( exuI2 = $true )
=> ( ( inCongP = $true )
=> ( ( in__Cong = $true )
=> ( ( exuE3u = $true )
=> ( ( exu__Cong = $true )
=> ( ( emptyset__Cong = $true )
=> ( ( setadjoin__Cong = $true )
=> ( ( powerset__Cong = $true )
=> ( ( setunion__Cong = $true )
=> ( ( omega__Cong = $true )
=> ( ( exuEu = $true )
=> ( ( descr__Cong = $true )
=> ( ( dsetconstr__Cong = $true )
=> ( ( subsetI1 = $true )
=> ( ( eqimpsubset2 = $true )
=> ( ( eqimpsubset1 = $true )
=> ( ( subsetI2 = $true )
=> ( ( emptysetsubset = $true )
=> ( ( subsetE = $true )
=> ( ( subsetE2 = $true )
=> ( ( notsubsetI = $true )
=> ( ( notequalI1 = $true )
=> ( ( notequalI2 = $true )
=> ( ( subsetRefl = $true )
=> ( ( subsetTrans = $true )
=> ( ( setadjoinSub = $true )
=> ( ( setadjoinSub2 = $true )
=> ( ( subset2powerset = $true )
=> ( ( setextsub = $true )
=> ( ( subsetemptysetimpeq = $true )
=> ( ( powersetI1 = $true )
=> ( ( powersetE1 = $true )
=> ( ( inPowerset = $true )
=> ( ( powersetsubset = $true )
=> ( ( sepInPowerset = $true )
=> ( ( sepSubset = $true )
=> ( ( binunionIL = $true )
=> ( ( upairset2IR = $true )
=> ( ( binunionIR = $true )
=> ( ( binunionEcases = $true )
=> ( ( binunionE = $true )
=> ( ( binunionLsub = $true )
=> ( ( binunionRsub = $true )
=> ( ( binintersectI = $true )
=> ( ( binintersectSubset5 = $true )
=> ( ( binintersectEL = $true )
=> ( ( binintersectLsub = $true )
=> ( ( binintersectSubset2 = $true )
=> ( ( binintersectSubset3 = $true )
=> ( ( binintersectER = $true )
=> ( ( disjointsetsI1 = $true )
=> ( ( binintersectRsub = $true )
=> ( ( binintersectSubset4 = $true )
=> ( ( binintersectSubset1 = $true )
=> ( ( bs114d = $true )
=> ( ( setminusI = $true )
=> ( ( setminusEL = $true )
=> ( ( setminusER = $true )
=> ( ( setminusSubset2 = $true )
=> ( ( setminusERneg = $true )
=> ( ( setminusELneg = $true )
=> ( ( setminusILneg = $true )
=> ( ( setminusIRneg = $true )
=> ( ( setminusLsub = $true )
=> ( ( setminusSubset1 = $true )
=> ( ( symdiffE = $true )
=> ( ( symdiffI1 = $true )
=> ( ( symdiffI2 = $true )
=> ( ( symdiffIneg1 = $true )
=> ( ( symdiffIneg2 = $true )
=> ( ( secondinupair = $true )
=> ( ( setukpairIL = $true )
=> ( ( setukpairIR = $true )
=> ( ( kpairiskpair = $true )
=> ( ( kpairp = $true )
=> ( ( singletonsubset = $true )
=> ( ( singletoninpowerset = $true )
=> ( ( singletoninpowunion = $true )
=> ( ( upairset2E = $true )
=> ( ( upairsubunion = $true )
=> ( ( upairinpowunion = $true )
=> ( ( ubforcartprodlem1 = $true )
=> ( ( ubforcartprodlem2 = $true )
=> ( ( ubforcartprodlem3 = $true )
=> ( ( cartprodpairin = $true )
=> ( ( cartprodmempair1 = $true )
=> ( ( cartprodmempair = $true )
=> ( ( setunionE2 = $true )
=> ( ( setunionsingleton1 = $true )
=> ( ( setunionsingleton2 = $true )
=> ( ( setunionsingleton = $true )
=> ( ( singletonprop = $true )
=> ( ( ex1E1 = $true )
=> ( ( ex1I = $true )
=> ( ( ex1I2 = $true )
=> ( ( singletonsuniq = $true )
=> ( ( setukpairinjL1 = $true )
=> ( ( kfstsingleton = $true )
=> ( ( theprop = $true )
=> ( ( kfstpairEq = $true )
=> ( ( cartprodfstin = $true )
=> ( ( setukpairinjL2 = $true )
=> ( ( setukpairinjL = $true )
=> ( ( setukpairinjR11 = $true )
=> ( ( setukpairinjR12 = $true )
=> ( ( setukpairinjR1 = $true )
=> ( ( upairequniteq = $true )
=> ( ( setukpairinjR2 = $true )
=> ( ( setukpairinjR = $true )
=> ( ( ksndsingleton = $true )
=> ( ( ksndpairEq = $true )
=> ( ( kpairsurjEq = $true )
=> ( ( cartprodsndin = $true )
=> ( ( cartprodpairmemEL = $true )
=> ( ( cartprodpairmemER = $true )
=> ( ( cartprodmempaircEq = $true )
=> ( ( cartprodfstpairEq = $true )
=> ( ( cartprodsndpairEq = $true )
=> ( ( cartprodpairsurjEq = $true )
=> ( ( dpsetconstrI = $true )
=> ( ( dpsetconstrSub = $true )
=> ( ( setOfPairsIsBReln = $true )
=> ( ( dpsetconstrERa = $true )
=> ( ( dpsetconstrEL1 = $true )
=> ( ( dpsetconstrEL2 = $true )
=> ( ( dpsetconstrER = $true )
=> ( ( funcImageSingleton = $true )
=> ( ( apProp = $true )
=> ( ( app = $true )
=> ( ( infuncsetfunc = $true )
=> ( ( ap2p = $true )
=> ( ( funcinfuncset = $true )
=> ( ( lamProp = $true )
=> ( ( lamp = $true )
=> ( ( lam2p = $true )
=> ( ( brelnall1 = $true )
=> ( ( brelnall2 = $true )
=> ( ( ex1E2 = $true )
=> ( ( funcGraphProp1 = $true )
=> ( ( funcGraphProp3 = $true )
=> ( ( funcGraphProp2 = $true )
=> ( ( funcextLem = $true )
=> ( ( funcGraphProp4 = $true )
=> ( ( subbreln = $true )
=> ( ( eqbreln = $true )
=> ( ( funcext = $true )
=> ( ( funcext2 = $true )
=> ( ( ap2apEq1 = $true )
=> ( ( ap2apEq2 = $true )
=> ( ( beta1 = $true )
=> ( ( eta1 = $true )
=> ! [X0: $i > $i,X1: $i,X2: $i] :
( ! [X3: $i] :
( ( $true
= ( in @ X3 @ X1 ) )
=> ( ( in @ ( X0 @ X3 ) @ X2 )
= $true ) )
=> ( ( lam @ X1 @ X2
@ ^ [Y0: $i] : ( X0 @ Y0 ) )
= ( lam @ X1 @ X2
@ ^ [Y0: $i] : ( X0 @ Y0 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f417]) ).
thf(f417,plain,
~ ( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> ( setunionE2
=> ( setunionsingleton1
=> ( setunionsingleton2
=> ( setunionsingleton
=> ( singletonprop
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> ( setukpairinjL1
=> ( kfstsingleton
=> ( theprop
=> ( kfstpairEq
=> ( cartprodfstin
=> ( setukpairinjL2
=> ( setukpairinjL
=> ( setukpairinjR11
=> ( setukpairinjR12
=> ( setukpairinjR1
=> ( upairequniteq
=> ( setukpairinjR2
=> ( setukpairinjR
=> ( ksndsingleton
=> ( ksndpairEq
=> ( kpairsurjEq
=> ( cartprodsndin
=> ( cartprodpairmemEL
=> ( cartprodpairmemER
=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ( cartprodpairsurjEq
=> ( dpsetconstrI
=> ( dpsetconstrSub
=> ( setOfPairsIsBReln
=> ( dpsetconstrERa
=> ( dpsetconstrEL1
=> ( dpsetconstrEL2
=> ( dpsetconstrER
=> ( funcImageSingleton
=> ( apProp
=> ( app
=> ( infuncsetfunc
=> ( ap2p
=> ( funcinfuncset
=> ( lamProp
=> ( lamp
=> ( lam2p
=> ( brelnall1
=> ( brelnall2
=> ( ex1E2
=> ( funcGraphProp1
=> ( funcGraphProp3
=> ( funcGraphProp2
=> ( funcextLem
=> ( funcGraphProp4
=> ( subbreln
=> ( eqbreln
=> ( funcext
=> ( funcext2
=> ( ap2apEq1
=> ( ap2apEq2
=> ( beta1
=> ( eta1
=> ! [X0: $i > $i,X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( X0 @ X3 ) @ X2 ) )
=> ( ( lam @ X1 @ X2
@ ^ [X4: $i] : ( X0 @ X4 ) )
= ( lam @ X1 @ X2
@ ^ [X5: $i] : ( X0 @ X5 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f221]) ).
thf(f221,negated_conjecture,
~ ( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> ( setunionE2
=> ( setunionsingleton1
=> ( setunionsingleton2
=> ( setunionsingleton
=> ( singletonprop
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> ( setukpairinjL1
=> ( kfstsingleton
=> ( theprop
=> ( kfstpairEq
=> ( cartprodfstin
=> ( setukpairinjL2
=> ( setukpairinjL
=> ( setukpairinjR11
=> ( setukpairinjR12
=> ( setukpairinjR1
=> ( upairequniteq
=> ( setukpairinjR2
=> ( setukpairinjR
=> ( ksndsingleton
=> ( ksndpairEq
=> ( kpairsurjEq
=> ( cartprodsndin
=> ( cartprodpairmemEL
=> ( cartprodpairmemER
=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ( cartprodpairsurjEq
=> ( dpsetconstrI
=> ( dpsetconstrSub
=> ( setOfPairsIsBReln
=> ( dpsetconstrERa
=> ( dpsetconstrEL1
=> ( dpsetconstrEL2
=> ( dpsetconstrER
=> ( funcImageSingleton
=> ( apProp
=> ( app
=> ( infuncsetfunc
=> ( ap2p
=> ( funcinfuncset
=> ( lamProp
=> ( lamp
=> ( lam2p
=> ( brelnall1
=> ( brelnall2
=> ( ex1E2
=> ( funcGraphProp1
=> ( funcGraphProp3
=> ( funcGraphProp2
=> ( funcextLem
=> ( funcGraphProp4
=> ( subbreln
=> ( eqbreln
=> ( funcext
=> ( funcext2
=> ( ap2apEq1
=> ( ap2apEq2
=> ( beta1
=> ( eta1
=> ! [X12: $i > $i,X3: $i,X4: $i] :
( ! [X1: $i] :
( ( in @ X1 @ X3 )
=> ( in @ ( X12 @ X1 ) @ X4 ) )
=> ( ( lam @ X3 @ X4
@ ^ [X1: $i] : ( X12 @ X1 ) )
= ( lam @ X3 @ X4
@ ^ [X1: $i] : ( X12 @ X1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f220]) ).
thf(f220,conjecture,
( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> ( setunionE2
=> ( setunionsingleton1
=> ( setunionsingleton2
=> ( setunionsingleton
=> ( singletonprop
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> ( setukpairinjL1
=> ( kfstsingleton
=> ( theprop
=> ( kfstpairEq
=> ( cartprodfstin
=> ( setukpairinjL2
=> ( setukpairinjL
=> ( setukpairinjR11
=> ( setukpairinjR12
=> ( setukpairinjR1
=> ( upairequniteq
=> ( setukpairinjR2
=> ( setukpairinjR
=> ( ksndsingleton
=> ( ksndpairEq
=> ( kpairsurjEq
=> ( cartprodsndin
=> ( cartprodpairmemEL
=> ( cartprodpairmemER
=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ( cartprodpairsurjEq
=> ( dpsetconstrI
=> ( dpsetconstrSub
=> ( setOfPairsIsBReln
=> ( dpsetconstrERa
=> ( dpsetconstrEL1
=> ( dpsetconstrEL2
=> ( dpsetconstrER
=> ( funcImageSingleton
=> ( apProp
=> ( app
=> ( infuncsetfunc
=> ( ap2p
=> ( funcinfuncset
=> ( lamProp
=> ( lamp
=> ( lam2p
=> ( brelnall1
=> ( brelnall2
=> ( ex1E2
=> ( funcGraphProp1
=> ( funcGraphProp3
=> ( funcGraphProp2
=> ( funcextLem
=> ( funcGraphProp4
=> ( subbreln
=> ( eqbreln
=> ( funcext
=> ( funcext2
=> ( ap2apEq1
=> ( ap2apEq2
=> ( beta1
=> ( eta1
=> ! [X12: $i > $i,X3: $i,X4: $i] :
( ! [X1: $i] :
( ( in @ X1 @ X3 )
=> ( in @ ( X12 @ X1 ) @ X4 ) )
=> ( ( lam @ X3 @ X4
@ ^ [X1: $i] : ( X12 @ X1 ) )
= ( lam @ X3 @ X4
@ ^ [X1: $i] : ( X12 @ X1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.EApmVaGZ7F/DTF2THF_24917.p',lam2lamEq) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SEU697^1 : TPTP v9.2.1. Released v3.7.0.
% 0.12/0.13 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.17/0.34 % Computer : n031.cluster.edu
% 0.17/0.34 % Model : x86_64 x86_64
% 0.17/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.34 % Memory : 8042.1875MB
% 0.17/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.17/0.34 % CPULimit : 300
% 0.17/0.34 % WCLimit : 300
% 0.17/0.34 % DateTime : Tue Feb 24 12:32:18 EST 2026
% 0.17/0.34 % CPUTime :
% 0.17/0.34 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.27/0.46 ---- Original DTF file ---
% 0.27/0.46 thf(spec,logic,$$dhol).
% 0.27/0.46 %------------------------------------------------------------------------------
% 0.27/0.46 % File : SEU697^1 : TPTP v9.2.1. Released v3.7.0.
% 0.27/0.46 % Domain : Set Theory
% 0.27/0.46 % Problem : Functions - Extensionality and Beta Reduction
% 0.27/0.46 % Version : Especial.
% 0.27/0.46 % English : (! A:i.! B:i.! f:i>i.(! x:i.in x A -> in (f x) B) ->
% 0.27/0.46 % lam A B (^ x:i.f x) = lam A B (^ x:i.f x))
% 0.27/0.46
% 0.27/0.46 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.27/0.46 % Source : [Bro08]
% 0.27/0.46 % Names : ZFC199g [Bro08]
% 0.27/0.46
% 0.27/0.46 % Status : Theorem
% 0.27/0.46 % Rating : 0.11 v9.1.0, 0.12 v9.0.0, 0.30 v8.2.0, 0.15 v8.1.0, 0.00 v7.4.0, 0.11 v7.2.0, 0.00 v7.1.0, 0.12 v7.0.0, 0.14 v6.4.0, 0.17 v6.3.0, 0.20 v6.2.0, 0.14 v6.1.0, 0.29 v5.5.0, 0.33 v5.4.0, 0.40 v5.1.0, 0.60 v5.0.0, 0.40 v4.1.0, 0.67 v4.0.1, 1.00 v3.7.0
% 0.27/0.46 % Syntax : Number of formulae : 470 ( 219 unt; 250 typ; 219 def)
% 0.27/0.46 % Number of atoms : 1205 ( 328 equ; 0 cnn)
% 0.27/0.46 % Maximal formula atoms : 216 ( 5 avg)
% 0.27/0.46 % Number of connectives : 2313 ( 47 ~; 7 |; 35 &;1649 @)
% 0.27/0.46 % ( 14 <=>; 561 =>; 0 <=; 0 <~>)
% 0.27/0.46 % Maximal formula depth : 223 ( 2 avg)
% 0.27/0.46 % Number of types : 2 ( 0 usr)
% 0.27/0.46 % Number of type conns : 128 ( 128 >; 0 *; 0 +; 0 <<)
% 0.27/0.46 % Number of symbols : 253 ( 250 usr; 217 con; 0-4 aty)
% 0.27/0.46 % Number of variables : 726 ( 73 ^; 617 !; 36 ?; 726 :)
% 0.27/0.46 % SPC : TH0_THM_EQU_NAR
% 0.27/0.46
% 0.27/0.46 % Comments : http://mathgate.info/detsetitem.php?id=249
% 0.27/0.46 %------------------------------------------------------------------------------
% 0.27/0.46 thf(in_type,type,
% 0.27/0.46 in: $i > $i > $o ).
% 0.27/0.46
% 0.27/0.46 thf(exu_type,type,
% 0.27/0.46 exu: ( $i > $o ) > $o ).
% 0.27/0.46
% 0.27/0.46 thf(exu,definition,
% 0.27/0.46 ( exu
% 0.27/0.46 = ( ^ [Xphi: $i > $o] :
% 0.27/0.46 ? [Xx: $i] :
% 0.27/0.46 ( ( Xphi @ Xx )
% 0.27/0.46 & ! [Xy: $i] :
% 0.27/0.46 ( ( Xphi @ Xy )
% 0.27/0.46 => ( Xx = Xy ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(setextAx_type,type,
% 0.27/0.46 setextAx: $o ).
% 0.27/0.46
% 0.27/0.46 thf(setextAx,definition,
% 0.27/0.46 ( setextAx
% 0.27/0.46 = ( ! [A: $i,B: $i] :
% 0.27/0.46 ( ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 <=> ( in @ Xx @ B ) )
% 0.27/0.46 => ( A = B ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(emptyset_type,type,
% 0.27/0.46 emptyset: $i ).
% 0.27/0.46
% 0.27/0.46 thf(emptysetAx_type,type,
% 0.27/0.46 emptysetAx: $o ).
% 0.27/0.46
% 0.27/0.46 thf(emptysetAx,definition,
% 0.27/0.46 ( emptysetAx
% 0.27/0.46 = ( ! [Xx: $i] :
% 0.27/0.46 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(setadjoin_type,type,
% 0.27/0.46 setadjoin: $i > $i > $i ).
% 0.27/0.46
% 0.27/0.46 thf(setadjoinAx_type,type,
% 0.27/0.46 setadjoinAx: $o ).
% 0.27/0.46
% 0.27/0.46 thf(setadjoinAx,definition,
% 0.27/0.46 ( setadjoinAx
% 0.27/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.27/0.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.27/0.46 <=> ( ( Xy = Xx )
% 0.27/0.46 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(powerset_type,type,
% 0.27/0.46 powerset: $i > $i ).
% 0.27/0.46
% 0.27/0.46 thf(powersetAx_type,type,
% 0.27/0.46 powersetAx: $o ).
% 0.27/0.46
% 0.27/0.46 thf(powersetAx,definition,
% 0.27/0.46 ( powersetAx
% 0.27/0.46 = ( ! [A: $i,B: $i] :
% 0.27/0.46 ( ( in @ B @ ( powerset @ A ) )
% 0.27/0.46 <=> ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ B )
% 0.27/0.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(setunion_type,type,
% 0.27/0.46 setunion: $i > $i ).
% 0.27/0.46
% 0.27/0.46 thf(setunionAx_type,type,
% 0.27/0.46 setunionAx: $o ).
% 0.27/0.46
% 0.27/0.46 thf(setunionAx,definition,
% 0.27/0.46 ( setunionAx
% 0.27/0.46 = ( ! [A: $i,Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ ( setunion @ A ) )
% 0.27/0.46 <=> ? [B: $i] :
% 0.27/0.46 ( ( in @ Xx @ B )
% 0.27/0.46 & ( in @ B @ A ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(omega_type,type,
% 0.27/0.46 omega: $i ).
% 0.27/0.46
% 0.27/0.46 thf(omega0Ax_type,type,
% 0.27/0.46 omega0Ax: $o ).
% 0.27/0.46
% 0.27/0.46 thf(omega0Ax,definition,
% 0.27/0.46 ( omega0Ax
% 0.27/0.46 = ( in @ emptyset @ omega ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(omegaSAx_type,type,
% 0.27/0.46 omegaSAx: $o ).
% 0.27/0.46
% 0.27/0.46 thf(omegaSAx,definition,
% 0.27/0.46 ( omegaSAx
% 0.27/0.46 = ( ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ omega )
% 0.27/0.46 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(omegaIndAx_type,type,
% 0.27/0.46 omegaIndAx: $o ).
% 0.27/0.46
% 0.27/0.46 thf(omegaIndAx,definition,
% 0.27/0.46 ( omegaIndAx
% 0.27/0.46 = ( ! [A: $i] :
% 0.27/0.46 ( ( ( in @ emptyset @ A )
% 0.27/0.46 & ! [Xx: $i] :
% 0.27/0.46 ( ( ( in @ Xx @ omega )
% 0.27/0.46 & ( in @ Xx @ A ) )
% 0.27/0.46 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.27/0.46 => ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ omega )
% 0.27/0.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(replAx_type,type,
% 0.27/0.46 replAx: $o ).
% 0.27/0.46
% 0.27/0.46 thf(replAx,definition,
% 0.27/0.46 ( replAx
% 0.27/0.46 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.27/0.46 ( ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( exu
% 0.27/0.46 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.27/0.46 => ? [B: $i] :
% 0.27/0.46 ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ B )
% 0.27/0.46 <=> ? [Xy: $i] :
% 0.27/0.46 ( ( in @ Xy @ A )
% 0.27/0.46 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(foundationAx_type,type,
% 0.27/0.46 foundationAx: $o ).
% 0.27/0.46
% 0.27/0.46 thf(foundationAx,definition,
% 0.27/0.46 ( foundationAx
% 0.27/0.46 = ( ! [A: $i] :
% 0.27/0.46 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.27/0.46 => ? [B: $i] :
% 0.27/0.46 ( ( in @ B @ A )
% 0.27/0.46 & ~ ? [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ B )
% 0.27/0.46 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(wellorderingAx_type,type,
% 0.27/0.46 wellorderingAx: $o ).
% 0.27/0.46
% 0.27/0.46 thf(wellorderingAx,definition,
% 0.27/0.46 ( wellorderingAx
% 0.27/0.46 = ( ! [A: $i] :
% 0.27/0.46 ? [B: $i] :
% 0.27/0.46 ( ! [C: $i] :
% 0.27/0.46 ( ( in @ C @ B )
% 0.27/0.46 => ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ C )
% 0.27/0.46 => ( in @ Xx @ A ) ) )
% 0.27/0.46 & ! [Xx: $i,Xy: $i] :
% 0.27/0.46 ( ( ( in @ Xx @ A )
% 0.27/0.46 & ( in @ Xy @ A ) )
% 0.27/0.46 => ( ! [C: $i] :
% 0.27/0.46 ( ( in @ C @ B )
% 0.27/0.46 => ( ( in @ Xx @ C )
% 0.27/0.46 <=> ( in @ Xy @ C ) ) )
% 0.27/0.46 => ( Xx = Xy ) ) )
% 0.27/0.46 & ! [C: $i,D: $i] :
% 0.27/0.46 ( ( ( in @ C @ B )
% 0.27/0.46 & ( in @ D @ B ) )
% 0.27/0.46 => ( ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ C )
% 0.27/0.46 => ( in @ Xx @ D ) )
% 0.27/0.46 | ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ D )
% 0.27/0.46 => ( in @ Xx @ C ) ) ) )
% 0.27/0.46 & ! [C: $i] :
% 0.27/0.46 ( ( ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ C )
% 0.27/0.46 => ( in @ Xx @ A ) )
% 0.27/0.46 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.27/0.46 => ? [D: $i,Xx: $i] :
% 0.27/0.46 ( ( in @ D @ B )
% 0.27/0.46 & ( in @ Xx @ C )
% 0.27/0.46 & ~ ? [Xy: $i] :
% 0.27/0.46 ( ( in @ Xy @ D )
% 0.27/0.46 & ( in @ Xy @ C ) )
% 0.27/0.46 & ! [E: $i] :
% 0.27/0.46 ( ( in @ E @ B )
% 0.27/0.46 => ( ! [Xy: $i] :
% 0.27/0.46 ( ( in @ Xy @ E )
% 0.27/0.46 => ( in @ Xy @ D ) )
% 0.27/0.46 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(descr_type,type,
% 0.27/0.46 descr: ( $i > $o ) > $i ).
% 0.27/0.46
% 0.27/0.46 thf(descrp_type,type,
% 0.27/0.46 descrp: $o ).
% 0.27/0.46
% 0.27/0.46 thf(descrp,definition,
% 0.27/0.46 ( descrp
% 0.27/0.46 = ( ! [Xphi: $i > $o] :
% 0.27/0.46 ( ( exu
% 0.27/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.27/0.46 => ( Xphi
% 0.27/0.46 @ ( descr
% 0.27/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(dsetconstr_type,type,
% 0.27/0.46 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.27/0.46
% 0.27/0.46 thf(dsetconstrI_type,type,
% 0.27/0.46 dsetconstrI: $o ).
% 0.27/0.46
% 0.27/0.46 thf(dsetconstrI,definition,
% 0.27/0.46 ( dsetconstrI
% 0.27/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( ( Xphi @ Xx )
% 0.27/0.46 => ( in @ Xx
% 0.27/0.46 @ ( dsetconstr @ A
% 0.27/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(dsetconstrEL_type,type,
% 0.27/0.46 dsetconstrEL: $o ).
% 0.27/0.46
% 0.27/0.46 thf(dsetconstrEL,definition,
% 0.27/0.46 ( dsetconstrEL
% 0.27/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.27/0.46 ( ( in @ Xx
% 0.27/0.46 @ ( dsetconstr @ A
% 0.27/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.27/0.46 => ( in @ Xx @ A ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(dsetconstrER_type,type,
% 0.27/0.46 dsetconstrER: $o ).
% 0.27/0.46
% 0.27/0.46 thf(dsetconstrER,definition,
% 0.27/0.46 ( dsetconstrER
% 0.27/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.27/0.46 ( ( in @ Xx
% 0.27/0.46 @ ( dsetconstr @ A
% 0.27/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.27/0.46 => ( Xphi @ Xx ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(exuE1_type,type,
% 0.27/0.46 exuE1: $o ).
% 0.27/0.46
% 0.27/0.46 thf(exuE1,definition,
% 0.27/0.46 ( exuE1
% 0.27/0.46 = ( ! [Xphi: $i > $o] :
% 0.27/0.46 ( ( exu
% 0.27/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.27/0.46 => ? [Xx: $i] :
% 0.27/0.46 ( ( Xphi @ Xx )
% 0.27/0.46 & ! [Xy: $i] :
% 0.27/0.46 ( ( Xphi @ Xy )
% 0.27/0.46 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(prop2set_type,type,
% 0.27/0.46 prop2set: $o > $i ).
% 0.27/0.46
% 0.27/0.46 thf(prop2setE_type,type,
% 0.27/0.46 prop2setE: $o ).
% 0.27/0.46
% 0.27/0.46 thf(prop2setE,definition,
% 0.27/0.46 ( prop2setE
% 0.27/0.46 = ( ! [Xphi: $o,Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.27/0.46 => Xphi ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(emptysetE_type,type,
% 0.27/0.46 emptysetE: $o ).
% 0.27/0.46
% 0.27/0.46 thf(emptysetE,definition,
% 0.27/0.46 ( emptysetE
% 0.27/0.46 = ( ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ emptyset )
% 0.27/0.46 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(emptysetimpfalse_type,type,
% 0.27/0.46 emptysetimpfalse: $o ).
% 0.27/0.46
% 0.27/0.46 thf(emptysetimpfalse,definition,
% 0.27/0.46 ( emptysetimpfalse
% 0.27/0.46 = ( ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ emptyset )
% 0.27/0.46 => $false ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(notinemptyset_type,type,
% 0.27/0.46 notinemptyset: $o ).
% 0.27/0.46
% 0.27/0.46 thf(notinemptyset,definition,
% 0.27/0.46 ( notinemptyset
% 0.27/0.46 = ( ! [Xx: $i] :
% 0.27/0.46 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(exuE3e_type,type,
% 0.27/0.46 exuE3e: $o ).
% 0.27/0.46
% 0.27/0.46 thf(exuE3e,definition,
% 0.27/0.46 ( exuE3e
% 0.27/0.46 = ( ! [Xphi: $i > $o] :
% 0.27/0.46 ( ( exu
% 0.27/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.27/0.46 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(setext_type,type,
% 0.27/0.46 setext: $o ).
% 0.27/0.46
% 0.27/0.46 thf(setext,definition,
% 0.27/0.46 ( setext
% 0.27/0.46 = ( ! [A: $i,B: $i] :
% 0.27/0.46 ( ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( in @ Xx @ B ) )
% 0.27/0.46 => ( ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ B )
% 0.27/0.46 => ( in @ Xx @ A ) )
% 0.27/0.46 => ( A = B ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(emptyI_type,type,
% 0.27/0.46 emptyI: $o ).
% 0.27/0.46
% 0.27/0.46 thf(emptyI,definition,
% 0.27/0.46 ( emptyI
% 0.27/0.46 = ( ! [A: $i] :
% 0.27/0.46 ( ! [Xx: $i] :
% 0.27/0.46 ~ ( in @ Xx @ A )
% 0.27/0.46 => ( A = emptyset ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(noeltsimpempty_type,type,
% 0.27/0.46 noeltsimpempty: $o ).
% 0.27/0.46
% 0.27/0.46 thf(noeltsimpempty,definition,
% 0.27/0.46 ( noeltsimpempty
% 0.27/0.46 = ( ! [A: $i] :
% 0.27/0.46 ( ! [Xx: $i] :
% 0.27/0.46 ~ ( in @ Xx @ A )
% 0.27/0.46 => ( A = emptyset ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(setbeta_type,type,
% 0.27/0.46 setbeta: $o ).
% 0.27/0.46
% 0.27/0.46 thf(setbeta,definition,
% 0.27/0.46 ( setbeta
% 0.27/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( ( in @ Xx
% 0.27/0.46 @ ( dsetconstr @ A
% 0.27/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.27/0.46 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(nonempty_type,type,
% 0.27/0.46 nonempty: $i > $o ).
% 0.27/0.46
% 0.27/0.46 thf(nonempty,definition,
% 0.27/0.46 ( nonempty
% 0.27/0.46 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(nonemptyE1_type,type,
% 0.27/0.46 nonemptyE1: $o ).
% 0.27/0.46
% 0.27/0.46 thf(nonemptyE1,definition,
% 0.27/0.46 ( nonemptyE1
% 0.27/0.46 = ( ! [A: $i] :
% 0.27/0.46 ( ( nonempty @ A )
% 0.27/0.46 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(nonemptyI_type,type,
% 0.27/0.46 nonemptyI: $o ).
% 0.27/0.46
% 0.27/0.46 thf(nonemptyI,definition,
% 0.27/0.46 ( nonemptyI
% 0.27/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( ( Xphi @ Xx )
% 0.27/0.46 => ( nonempty
% 0.27/0.46 @ ( dsetconstr @ A
% 0.27/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(nonemptyI1_type,type,
% 0.27/0.46 nonemptyI1: $o ).
% 0.27/0.46
% 0.27/0.46 thf(nonemptyI1,definition,
% 0.27/0.46 ( nonemptyI1
% 0.27/0.46 = ( ! [A: $i] :
% 0.27/0.46 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.27/0.46 => ( nonempty @ A ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(setadjoinIL_type,type,
% 0.27/0.46 setadjoinIL: $o ).
% 0.27/0.46
% 0.27/0.46 thf(setadjoinIL,definition,
% 0.27/0.46 ( setadjoinIL
% 0.27/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(emptyinunitempty_type,type,
% 0.27/0.46 emptyinunitempty: $o ).
% 0.27/0.46
% 0.27/0.46 thf(emptyinunitempty,definition,
% 0.27/0.46 ( emptyinunitempty
% 0.27/0.46 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(setadjoinIR_type,type,
% 0.27/0.46 setadjoinIR: $o ).
% 0.27/0.46
% 0.27/0.46 thf(setadjoinIR,definition,
% 0.27/0.46 ( setadjoinIR
% 0.27/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.27/0.46 ( ( in @ Xy @ A )
% 0.27/0.46 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(setadjoinE_type,type,
% 0.27/0.46 setadjoinE: $o ).
% 0.27/0.46
% 0.27/0.46 thf(setadjoinE,definition,
% 0.27/0.46 ( setadjoinE
% 0.27/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.27/0.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.27/0.46 => ! [Xphi: $o] :
% 0.27/0.46 ( ( ( Xy = Xx )
% 0.27/0.46 => Xphi )
% 0.27/0.46 => ( ( ( in @ Xy @ A )
% 0.27/0.46 => Xphi )
% 0.27/0.46 => Xphi ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(setadjoinOr_type,type,
% 0.27/0.46 setadjoinOr: $o ).
% 0.27/0.46
% 0.27/0.46 thf(setadjoinOr,definition,
% 0.27/0.46 ( setadjoinOr
% 0.27/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.27/0.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.27/0.46 => ( ( Xy = Xx )
% 0.27/0.46 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(setoftrueEq_type,type,
% 0.27/0.46 setoftrueEq: $o ).
% 0.27/0.46
% 0.27/0.46 thf(setoftrueEq,definition,
% 0.27/0.46 ( setoftrueEq
% 0.27/0.46 = ( ! [A: $i] :
% 0.27/0.46 ( ( dsetconstr @ A
% 0.27/0.46 @ ^ [Xx: $i] : $true )
% 0.27/0.46 = A ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(powersetI_type,type,
% 0.27/0.46 powersetI: $o ).
% 0.27/0.46
% 0.27/0.46 thf(powersetI,definition,
% 0.27/0.46 ( powersetI
% 0.27/0.46 = ( ! [A: $i,B: $i] :
% 0.27/0.46 ( ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ B )
% 0.27/0.46 => ( in @ Xx @ A ) )
% 0.27/0.46 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(emptyinPowerset_type,type,
% 0.27/0.46 emptyinPowerset: $o ).
% 0.27/0.46
% 0.27/0.46 thf(emptyinPowerset,definition,
% 0.27/0.46 ( emptyinPowerset
% 0.27/0.46 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(emptyInPowerset_type,type,
% 0.27/0.46 emptyInPowerset: $o ).
% 0.27/0.46
% 0.27/0.46 thf(emptyInPowerset,definition,
% 0.27/0.46 ( emptyInPowerset
% 0.27/0.46 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(powersetE_type,type,
% 0.27/0.46 powersetE: $o ).
% 0.27/0.46
% 0.27/0.46 thf(powersetE,definition,
% 0.27/0.46 ( powersetE
% 0.27/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.46 ( ( in @ B @ ( powerset @ A ) )
% 0.27/0.46 => ( ( in @ Xx @ B )
% 0.27/0.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(setunionI_type,type,
% 0.27/0.46 setunionI: $o ).
% 0.27/0.46
% 0.27/0.46 thf(setunionI,definition,
% 0.27/0.46 ( setunionI
% 0.27/0.46 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.27/0.46 ( ( in @ Xx @ B )
% 0.27/0.46 => ( ( in @ B @ A )
% 0.27/0.46 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(setunionE_type,type,
% 0.27/0.46 setunionE: $o ).
% 0.27/0.46
% 0.27/0.46 thf(setunionE,definition,
% 0.27/0.46 ( setunionE
% 0.27/0.46 = ( ! [A: $i,Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ ( setunion @ A ) )
% 0.27/0.46 => ! [Xphi: $o] :
% 0.27/0.46 ( ! [B: $i] :
% 0.27/0.46 ( ( in @ Xx @ B )
% 0.27/0.46 => ( ( in @ B @ A )
% 0.27/0.46 => Xphi ) )
% 0.27/0.46 => Xphi ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(subPowSU_type,type,
% 0.27/0.46 subPowSU: $o ).
% 0.27/0.46
% 0.27/0.46 thf(subPowSU,definition,
% 0.27/0.46 ( subPowSU
% 0.27/0.46 = ( ! [A: $i,Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(exuE2_type,type,
% 0.27/0.46 exuE2: $o ).
% 0.27/0.46
% 0.27/0.46 thf(exuE2,definition,
% 0.27/0.46 ( exuE2
% 0.27/0.46 = ( ! [Xphi: $i > $o] :
% 0.27/0.46 ( ( exu
% 0.27/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.27/0.46 => ? [Xx: $i] :
% 0.27/0.46 ! [Xy: $i] :
% 0.27/0.46 ( ( Xphi @ Xy )
% 0.27/0.46 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(nonemptyImpWitness_type,type,
% 0.27/0.46 nonemptyImpWitness: $o ).
% 0.27/0.46
% 0.27/0.46 thf(nonemptyImpWitness,definition,
% 0.27/0.46 ( nonemptyImpWitness
% 0.27/0.46 = ( ! [A: $i] :
% 0.27/0.46 ( ( nonempty @ A )
% 0.27/0.46 => ? [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 & $true ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(uniqinunit_type,type,
% 0.27/0.46 uniqinunit: $o ).
% 0.27/0.46
% 0.27/0.46 thf(uniqinunit,definition,
% 0.27/0.46 ( uniqinunit
% 0.27/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.27/0.46 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.27/0.46 => ( Xx = Xy ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(notinsingleton_type,type,
% 0.27/0.46 notinsingleton: $o ).
% 0.27/0.46
% 0.27/0.46 thf(notinsingleton,definition,
% 0.27/0.46 ( notinsingleton
% 0.27/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.27/0.46 ( ( Xx != Xy )
% 0.27/0.46 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(eqinunit_type,type,
% 0.27/0.46 eqinunit: $o ).
% 0.27/0.46
% 0.27/0.46 thf(eqinunit,definition,
% 0.27/0.46 ( eqinunit
% 0.27/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.27/0.46 ( ( Xx = Xy )
% 0.27/0.46 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(singletonsswitch_type,type,
% 0.27/0.46 singletonsswitch: $o ).
% 0.27/0.46
% 0.27/0.46 thf(singletonsswitch,definition,
% 0.27/0.46 ( singletonsswitch
% 0.27/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.27/0.46 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.27/0.46 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(upairsetE_type,type,
% 0.27/0.46 upairsetE: $o ).
% 0.27/0.46
% 0.27/0.46 thf(upairsetE,definition,
% 0.27/0.46 ( upairsetE
% 0.27/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.27/0.46 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.27/0.46 => ( ( Xz = Xx )
% 0.27/0.46 | ( Xz = Xy ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(upairsetIL_type,type,
% 0.27/0.46 upairsetIL: $o ).
% 0.27/0.46
% 0.27/0.46 thf(upairsetIL,definition,
% 0.27/0.46 ( upairsetIL
% 0.27/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(upairsetIR_type,type,
% 0.27/0.46 upairsetIR: $o ).
% 0.27/0.46
% 0.27/0.46 thf(upairsetIR,definition,
% 0.27/0.46 ( upairsetIR
% 0.27/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(emptyE1_type,type,
% 0.27/0.46 emptyE1: $o ).
% 0.27/0.46
% 0.27/0.46 thf(emptyE1,definition,
% 0.27/0.46 ( emptyE1
% 0.27/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.27/0.46 ( ? [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 & ( Xphi @ Xx ) )
% 0.27/0.46 => ( ( ( dsetconstr @ A
% 0.27/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.27/0.46 = emptyset )
% 0.27/0.46 => $false ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(vacuousDall_type,type,
% 0.27/0.46 vacuousDall: $o ).
% 0.27/0.46
% 0.27/0.46 thf(vacuousDall,definition,
% 0.27/0.46 ( vacuousDall
% 0.27/0.46 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ emptyset )
% 0.27/0.46 => ( Xphi @ Xx ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(quantDeMorgan1_type,type,
% 0.27/0.46 quantDeMorgan1: $o ).
% 0.27/0.46
% 0.27/0.46 thf(quantDeMorgan1,definition,
% 0.27/0.46 ( quantDeMorgan1
% 0.27/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.27/0.46 ( ~ ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( Xphi @ Xx ) )
% 0.27/0.46 => ? [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(quantDeMorgan2_type,type,
% 0.27/0.46 quantDeMorgan2: $o ).
% 0.27/0.46
% 0.27/0.46 thf(quantDeMorgan2,definition,
% 0.27/0.46 ( quantDeMorgan2
% 0.27/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.27/0.46 ( ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ~ ( Xphi @ Xx ) )
% 0.27/0.46 => ~ ? [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(quantDeMorgan3_type,type,
% 0.27/0.46 quantDeMorgan3: $o ).
% 0.27/0.46
% 0.27/0.46 thf(quantDeMorgan3,definition,
% 0.27/0.46 ( quantDeMorgan3
% 0.27/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.27/0.46 ( ~ ? [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 & ( Xphi @ Xx ) )
% 0.27/0.46 => ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(quantDeMorgan4_type,type,
% 0.27/0.46 quantDeMorgan4: $o ).
% 0.27/0.46
% 0.27/0.46 thf(quantDeMorgan4,definition,
% 0.27/0.46 ( quantDeMorgan4
% 0.27/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.27/0.46 ( ? [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 & ~ ( Xphi @ Xx ) )
% 0.27/0.46 => ~ ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(prop2setI_type,type,
% 0.27/0.46 prop2setI: $o ).
% 0.27/0.46
% 0.27/0.46 thf(prop2setI,definition,
% 0.27/0.46 ( prop2setI
% 0.27/0.46 = ( ! [Xphi: $o] :
% 0.27/0.46 ( Xphi
% 0.27/0.46 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(set2prop_type,type,
% 0.27/0.46 set2prop: $i > $o ).
% 0.27/0.46
% 0.27/0.46 thf(prop2set2propI_type,type,
% 0.27/0.46 prop2set2propI: $o ).
% 0.27/0.46
% 0.27/0.46 thf(prop2set2propI,definition,
% 0.27/0.46 ( prop2set2propI
% 0.27/0.46 = ( ! [Xphi: $o] :
% 0.27/0.46 ( Xphi
% 0.27/0.46 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(notdexE_type,type,
% 0.27/0.46 notdexE: $o ).
% 0.27/0.46
% 0.27/0.46 thf(notdexE,definition,
% 0.27/0.46 ( notdexE
% 0.27/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.27/0.46 ( ~ ? [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 & ( Xphi @ Xx ) )
% 0.27/0.46 => ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(notdallE_type,type,
% 0.27/0.46 notdallE: $o ).
% 0.27/0.46
% 0.27/0.46 thf(notdallE,definition,
% 0.27/0.46 ( notdallE
% 0.27/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.27/0.46 ( ~ ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( Xphi @ Xx ) )
% 0.27/0.46 => ? [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(exuI1_type,type,
% 0.27/0.46 exuI1: $o ).
% 0.27/0.46
% 0.27/0.46 thf(exuI1,definition,
% 0.27/0.46 ( exuI1
% 0.27/0.46 = ( ! [Xphi: $i > $o] :
% 0.27/0.46 ( ? [Xx: $i] :
% 0.27/0.46 ( ( Xphi @ Xx )
% 0.27/0.46 & ! [Xy: $i] :
% 0.27/0.46 ( ( Xphi @ Xy )
% 0.27/0.46 => ( Xx = Xy ) ) )
% 0.27/0.46 => ( exu
% 0.27/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(exuI3_type,type,
% 0.27/0.46 exuI3: $o ).
% 0.27/0.46
% 0.27/0.46 thf(exuI3,definition,
% 0.27/0.46 ( exuI3
% 0.27/0.46 = ( ! [Xphi: $i > $o] :
% 0.27/0.46 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.27/0.46 => ( ! [Xx: $i,Xy: $i] :
% 0.27/0.46 ( ( Xphi @ Xx )
% 0.27/0.46 => ( ( Xphi @ Xy )
% 0.27/0.46 => ( Xx = Xy ) ) )
% 0.27/0.46 => ( exu
% 0.27/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(exuI2_type,type,
% 0.27/0.46 exuI2: $o ).
% 0.27/0.46
% 0.27/0.46 thf(exuI2,definition,
% 0.27/0.46 ( exuI2
% 0.27/0.46 = ( ! [Xphi: $i > $o] :
% 0.27/0.46 ( ? [Xx: $i] :
% 0.27/0.46 ! [Xy: $i] :
% 0.27/0.46 ( ( Xphi @ Xy )
% 0.27/0.46 <=> ( Xy = Xx ) )
% 0.27/0.46 => ( exu
% 0.27/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(inCongP_type,type,
% 0.27/0.46 inCongP: $o ).
% 0.27/0.46
% 0.27/0.46 thf(inCongP,definition,
% 0.27/0.46 ( inCongP
% 0.27/0.46 = ( ! [A: $i,B: $i] :
% 0.27/0.46 ( ( A = B )
% 0.27/0.46 => ! [Xx: $i,Xy: $i] :
% 0.27/0.46 ( ( Xx = Xy )
% 0.27/0.46 => ( ( in @ Xx @ A )
% 0.27/0.46 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(in__Cong_type,type,
% 0.27/0.46 in__Cong: $o ).
% 0.27/0.46
% 0.27/0.46 thf(in__Cong,definition,
% 0.27/0.46 ( in__Cong
% 0.27/0.46 = ( ! [A: $i,B: $i] :
% 0.27/0.46 ( ( A = B )
% 0.27/0.46 => ! [Xx: $i,Xy: $i] :
% 0.27/0.46 ( ( Xx = Xy )
% 0.27/0.46 => ( ( in @ Xx @ A )
% 0.27/0.46 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(exuE3u_type,type,
% 0.27/0.47 exuE3u: $o ).
% 0.27/0.47
% 0.27/0.47 thf(exuE3u,definition,
% 0.27/0.47 ( exuE3u
% 0.27/0.47 = ( ! [Xphi: $i > $o] :
% 0.27/0.47 ( ( exu
% 0.27/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.27/0.47 => ! [Xx: $i,Xy: $i] :
% 0.27/0.47 ( ( Xphi @ Xx )
% 0.27/0.47 => ( ( Xphi @ Xy )
% 0.27/0.47 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(exu__Cong_type,type,
% 0.27/0.47 exu__Cong: $o ).
% 0.27/0.47
% 0.27/0.47 thf(exu__Cong,definition,
% 0.27/0.47 ( exu__Cong
% 0.27/0.47 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.27/0.47 ( ! [Xx: $i,Xy: $i] :
% 0.27/0.47 ( ( Xx = Xy )
% 0.27/0.47 => ( ( Xphi @ Xx )
% 0.27/0.47 <=> ( Xpsi @ Xy ) ) )
% 0.27/0.47 => ( ( exu
% 0.27/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.27/0.47 <=> ( exu
% 0.27/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(emptyset__Cong_type,type,
% 0.27/0.47 emptyset__Cong: $o ).
% 0.27/0.47
% 0.27/0.47 thf(emptyset__Cong,definition,
% 0.27/0.47 ( emptyset__Cong
% 0.27/0.47 = ( emptyset = emptyset ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setadjoin__Cong_type,type,
% 0.27/0.47 setadjoin__Cong: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setadjoin__Cong,definition,
% 0.27/0.47 ( setadjoin__Cong
% 0.27/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.27/0.47 ( ( Xx = Xy )
% 0.27/0.47 => ! [Xz: $i,Xu: $i] :
% 0.27/0.47 ( ( Xz = Xu )
% 0.27/0.47 => ( ( setadjoin @ Xx @ Xz )
% 0.27/0.47 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(powerset__Cong_type,type,
% 0.27/0.47 powerset__Cong: $o ).
% 0.27/0.47
% 0.27/0.47 thf(powerset__Cong,definition,
% 0.27/0.47 ( powerset__Cong
% 0.27/0.47 = ( ! [A: $i,B: $i] :
% 0.27/0.47 ( ( A = B )
% 0.27/0.47 => ( ( powerset @ A )
% 0.27/0.47 = ( powerset @ B ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setunion__Cong_type,type,
% 0.27/0.47 setunion__Cong: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setunion__Cong,definition,
% 0.27/0.47 ( setunion__Cong
% 0.27/0.47 = ( ! [A: $i,B: $i] :
% 0.27/0.47 ( ( A = B )
% 0.27/0.47 => ( ( setunion @ A )
% 0.27/0.47 = ( setunion @ B ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(omega__Cong_type,type,
% 0.27/0.47 omega__Cong: $o ).
% 0.27/0.47
% 0.27/0.47 thf(omega__Cong,definition,
% 0.27/0.47 ( omega__Cong
% 0.27/0.47 = ( omega = omega ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(exuEu_type,type,
% 0.27/0.47 exuEu: $o ).
% 0.27/0.47
% 0.27/0.47 thf(exuEu,definition,
% 0.27/0.47 ( exuEu
% 0.27/0.47 = ( ! [Xphi: $i > $o] :
% 0.27/0.47 ( ( exu
% 0.27/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.27/0.47 => ! [Xx: $i,Xy: $i] :
% 0.27/0.47 ( ( Xphi @ Xx )
% 0.27/0.47 => ( ( Xphi @ Xy )
% 0.27/0.47 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(descr__Cong_type,type,
% 0.27/0.47 descr__Cong: $o ).
% 0.27/0.47
% 0.27/0.47 thf(descr__Cong,definition,
% 0.27/0.47 ( descr__Cong
% 0.27/0.47 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.27/0.47 ( ! [Xx: $i,Xy: $i] :
% 0.27/0.47 ( ( Xx = Xy )
% 0.27/0.47 => ( ( Xphi @ Xx )
% 0.27/0.47 <=> ( Xpsi @ Xy ) ) )
% 0.27/0.47 => ( ( exu
% 0.27/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.27/0.47 => ( ( exu
% 0.27/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.27/0.47 => ( ( descr
% 0.27/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.27/0.47 = ( descr
% 0.27/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(dsetconstr__Cong_type,type,
% 0.27/0.47 dsetconstr__Cong: $o ).
% 0.27/0.47
% 0.27/0.47 thf(dsetconstr__Cong,definition,
% 0.27/0.47 ( dsetconstr__Cong
% 0.27/0.47 = ( ! [A: $i,B: $i] :
% 0.27/0.47 ( ( A = B )
% 0.27/0.47 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.27/0.47 ( ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ! [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ B )
% 0.27/0.47 => ( ( Xx = Xy )
% 0.27/0.47 => ( ( Xphi @ Xx )
% 0.27/0.47 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.27/0.47 => ( ( dsetconstr @ A
% 0.27/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.27/0.47 = ( dsetconstr @ B
% 0.27/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(subset_type,type,
% 0.27/0.47 subset: $i > $i > $o ).
% 0.27/0.47
% 0.27/0.47 thf(disjoint_type,type,
% 0.27/0.47 disjoint: $i > $i > $o ).
% 0.27/0.47
% 0.27/0.47 thf(setsmeet_type,type,
% 0.27/0.47 setsmeet: $i > $i > $o ).
% 0.27/0.47
% 0.27/0.47 thf(subsetI1_type,type,
% 0.27/0.47 subsetI1: $o ).
% 0.27/0.47
% 0.27/0.47 thf(subsetI1,definition,
% 0.27/0.47 ( subsetI1
% 0.27/0.47 = ( ! [A: $i,B: $i] :
% 0.27/0.47 ( ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( in @ Xx @ B ) )
% 0.27/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(eqimpsubset2_type,type,
% 0.27/0.47 eqimpsubset2: $o ).
% 0.27/0.47
% 0.27/0.47 thf(eqimpsubset2,definition,
% 0.27/0.47 ( eqimpsubset2
% 0.27/0.47 = ( ! [A: $i,B: $i] :
% 0.27/0.47 ( ( A = B )
% 0.27/0.47 => ( subset @ B @ A ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(eqimpsubset1_type,type,
% 0.27/0.47 eqimpsubset1: $o ).
% 0.27/0.47
% 0.27/0.47 thf(eqimpsubset1,definition,
% 0.27/0.47 ( eqimpsubset1
% 0.27/0.47 = ( ! [A: $i,B: $i] :
% 0.27/0.47 ( ( A = B )
% 0.27/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(subsetI2_type,type,
% 0.27/0.47 subsetI2: $o ).
% 0.27/0.47
% 0.27/0.47 thf(subsetI2,definition,
% 0.27/0.47 ( subsetI2
% 0.27/0.47 = ( ! [A: $i,B: $i] :
% 0.27/0.47 ( ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( in @ Xx @ B ) )
% 0.27/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(emptysetsubset_type,type,
% 0.27/0.47 emptysetsubset: $o ).
% 0.27/0.47
% 0.27/0.47 thf(emptysetsubset,definition,
% 0.27/0.47 ( emptysetsubset
% 0.27/0.47 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(subsetE_type,type,
% 0.27/0.47 subsetE: $o ).
% 0.27/0.47
% 0.27/0.47 thf(subsetE,definition,
% 0.27/0.47 ( subsetE
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ( subset @ A @ B )
% 0.27/0.47 => ( ( in @ Xx @ A )
% 0.27/0.47 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(subsetE2_type,type,
% 0.27/0.47 subsetE2: $o ).
% 0.27/0.47
% 0.27/0.47 thf(subsetE2,definition,
% 0.27/0.47 ( subsetE2
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ( subset @ A @ B )
% 0.27/0.47 => ( ~ ( in @ Xx @ B )
% 0.27/0.47 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(notsubsetI_type,type,
% 0.27/0.47 notsubsetI: $o ).
% 0.27/0.47
% 0.27/0.47 thf(notsubsetI,definition,
% 0.27/0.47 ( notsubsetI
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( ~ ( in @ Xx @ B )
% 0.27/0.47 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(notequalI1_type,type,
% 0.27/0.47 notequalI1: $o ).
% 0.27/0.47
% 0.27/0.47 thf(notequalI1,definition,
% 0.27/0.47 ( notequalI1
% 0.27/0.47 = ( ! [A: $i,B: $i] :
% 0.27/0.47 ( ~ ( subset @ A @ B )
% 0.27/0.47 => ( A != B ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(notequalI2_type,type,
% 0.27/0.47 notequalI2: $o ).
% 0.27/0.47
% 0.27/0.47 thf(notequalI2,definition,
% 0.27/0.47 ( notequalI2
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( ~ ( in @ Xx @ B )
% 0.27/0.47 => ( A != B ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(subsetRefl_type,type,
% 0.27/0.47 subsetRefl: $o ).
% 0.27/0.47
% 0.27/0.47 thf(subsetRefl,definition,
% 0.27/0.47 ( subsetRefl
% 0.27/0.47 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(subsetTrans_type,type,
% 0.27/0.47 subsetTrans: $o ).
% 0.27/0.47
% 0.27/0.47 thf(subsetTrans,definition,
% 0.27/0.47 ( subsetTrans
% 0.27/0.47 = ( ! [A: $i,B: $i,C: $i] :
% 0.27/0.47 ( ( subset @ A @ B )
% 0.27/0.47 => ( ( subset @ B @ C )
% 0.27/0.47 => ( subset @ A @ C ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setadjoinSub_type,type,
% 0.27/0.47 setadjoinSub: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setadjoinSub,definition,
% 0.27/0.47 ( setadjoinSub
% 0.27/0.47 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setadjoinSub2_type,type,
% 0.27/0.47 setadjoinSub2: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setadjoinSub2,definition,
% 0.27/0.47 ( setadjoinSub2
% 0.27/0.47 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.27/0.47 ( ( subset @ A @ B )
% 0.27/0.47 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(subset2powerset_type,type,
% 0.27/0.47 subset2powerset: $o ).
% 0.27/0.47
% 0.27/0.47 thf(subset2powerset,definition,
% 0.27/0.47 ( subset2powerset
% 0.27/0.47 = ( ! [A: $i,B: $i] :
% 0.27/0.47 ( ( subset @ A @ B )
% 0.27/0.47 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setextsub_type,type,
% 0.27/0.47 setextsub: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setextsub,definition,
% 0.27/0.47 ( setextsub
% 0.27/0.47 = ( ! [A: $i,B: $i] :
% 0.27/0.47 ( ( subset @ A @ B )
% 0.27/0.47 => ( ( subset @ B @ A )
% 0.27/0.47 => ( A = B ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(subsetemptysetimpeq_type,type,
% 0.27/0.47 subsetemptysetimpeq: $o ).
% 0.27/0.47
% 0.27/0.47 thf(subsetemptysetimpeq,definition,
% 0.27/0.47 ( subsetemptysetimpeq
% 0.27/0.47 = ( ! [A: $i] :
% 0.27/0.47 ( ( subset @ A @ emptyset )
% 0.27/0.47 => ( A = emptyset ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(powersetI1_type,type,
% 0.27/0.47 powersetI1: $o ).
% 0.27/0.47
% 0.27/0.47 thf(powersetI1,definition,
% 0.27/0.47 ( powersetI1
% 0.27/0.47 = ( ! [A: $i,B: $i] :
% 0.27/0.47 ( ( subset @ B @ A )
% 0.27/0.47 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(powersetE1_type,type,
% 0.27/0.47 powersetE1: $o ).
% 0.27/0.47
% 0.27/0.47 thf(powersetE1,definition,
% 0.27/0.47 ( powersetE1
% 0.27/0.47 = ( ! [A: $i,B: $i] :
% 0.27/0.47 ( ( in @ B @ ( powerset @ A ) )
% 0.27/0.47 => ( subset @ B @ A ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(inPowerset_type,type,
% 0.27/0.47 inPowerset: $o ).
% 0.27/0.47
% 0.27/0.47 thf(inPowerset,definition,
% 0.27/0.47 ( inPowerset
% 0.27/0.47 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(powersetsubset_type,type,
% 0.27/0.47 powersetsubset: $o ).
% 0.27/0.47
% 0.27/0.47 thf(powersetsubset,definition,
% 0.27/0.47 ( powersetsubset
% 0.27/0.47 = ( ! [A: $i,B: $i] :
% 0.27/0.47 ( ( subset @ A @ B )
% 0.27/0.47 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(sepInPowerset_type,type,
% 0.27/0.47 sepInPowerset: $o ).
% 0.27/0.47
% 0.27/0.47 thf(sepInPowerset,definition,
% 0.27/0.47 ( sepInPowerset
% 0.27/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.27/0.47 ( in
% 0.27/0.47 @ ( dsetconstr @ A
% 0.27/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.27/0.47 @ ( powerset @ A ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(sepSubset_type,type,
% 0.27/0.47 sepSubset: $o ).
% 0.27/0.47
% 0.27/0.47 thf(sepSubset,definition,
% 0.27/0.47 ( sepSubset
% 0.27/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.27/0.47 ( subset
% 0.27/0.47 @ ( dsetconstr @ A
% 0.27/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.27/0.47 @ A ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(binunion_type,type,
% 0.27/0.47 binunion: $i > $i > $i ).
% 0.27/0.47
% 0.27/0.47 thf(binunionIL_type,type,
% 0.27/0.47 binunionIL: $o ).
% 0.27/0.47
% 0.27/0.47 thf(binunionIL,definition,
% 0.27/0.47 ( binunionIL
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(upairset2IR_type,type,
% 0.27/0.47 upairset2IR: $o ).
% 0.27/0.47
% 0.27/0.47 thf(upairset2IR,definition,
% 0.27/0.47 ( upairset2IR
% 0.27/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(binunionIR_type,type,
% 0.27/0.47 binunionIR: $o ).
% 0.27/0.47
% 0.27/0.47 thf(binunionIR,definition,
% 0.27/0.47 ( binunionIR
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ B )
% 0.27/0.47 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(binunionEcases_type,type,
% 0.27/0.47 binunionEcases: $o ).
% 0.27/0.47
% 0.27/0.47 thf(binunionEcases,definition,
% 0.27/0.47 ( binunionEcases
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.27/0.47 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.27/0.47 => ( ( ( in @ Xx @ A )
% 0.27/0.47 => Xphi )
% 0.27/0.47 => ( ( ( in @ Xx @ B )
% 0.27/0.47 => Xphi )
% 0.27/0.47 => Xphi ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(binunionE_type,type,
% 0.27/0.47 binunionE: $o ).
% 0.27/0.47
% 0.27/0.47 thf(binunionE,definition,
% 0.27/0.47 ( binunionE
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.27/0.47 => ( ( in @ Xx @ A )
% 0.27/0.47 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(binunionLsub_type,type,
% 0.27/0.47 binunionLsub: $o ).
% 0.27/0.47
% 0.27/0.47 thf(binunionLsub,definition,
% 0.27/0.47 ( binunionLsub
% 0.27/0.47 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(binunionRsub_type,type,
% 0.27/0.47 binunionRsub: $o ).
% 0.27/0.47
% 0.27/0.47 thf(binunionRsub,definition,
% 0.27/0.47 ( binunionRsub
% 0.27/0.47 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(binintersect_type,type,
% 0.27/0.47 binintersect: $i > $i > $i ).
% 0.27/0.47
% 0.27/0.47 thf(binintersectI_type,type,
% 0.27/0.47 binintersectI: $o ).
% 0.27/0.47
% 0.27/0.47 thf(binintersectI,definition,
% 0.27/0.47 ( binintersectI
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( ( in @ Xx @ B )
% 0.27/0.47 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(binintersectSubset5_type,type,
% 0.27/0.47 binintersectSubset5: $o ).
% 0.27/0.47
% 0.27/0.47 thf(binintersectSubset5,definition,
% 0.27/0.47 ( binintersectSubset5
% 0.27/0.47 = ( ! [A: $i,B: $i,C: $i] :
% 0.27/0.47 ( ( subset @ C @ A )
% 0.27/0.47 => ( ( subset @ C @ B )
% 0.27/0.47 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(binintersectEL_type,type,
% 0.27/0.47 binintersectEL: $o ).
% 0.27/0.47
% 0.27/0.47 thf(binintersectEL,definition,
% 0.27/0.47 ( binintersectEL
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.27/0.47 => ( in @ Xx @ A ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(binintersectLsub_type,type,
% 0.27/0.47 binintersectLsub: $o ).
% 0.27/0.47
% 0.27/0.47 thf(binintersectLsub,definition,
% 0.27/0.47 ( binintersectLsub
% 0.27/0.47 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(binintersectSubset2_type,type,
% 0.27/0.47 binintersectSubset2: $o ).
% 0.27/0.47
% 0.27/0.47 thf(binintersectSubset2,definition,
% 0.27/0.47 ( binintersectSubset2
% 0.27/0.47 = ( ! [A: $i,B: $i] :
% 0.27/0.47 ( ( subset @ A @ B )
% 0.27/0.47 => ( ( binintersect @ A @ B )
% 0.27/0.47 = A ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(binintersectSubset3_type,type,
% 0.27/0.47 binintersectSubset3: $o ).
% 0.27/0.47
% 0.27/0.47 thf(binintersectSubset3,definition,
% 0.27/0.47 ( binintersectSubset3
% 0.27/0.47 = ( ! [A: $i,B: $i] :
% 0.27/0.47 ( ( ( binintersect @ A @ B )
% 0.27/0.47 = B )
% 0.27/0.47 => ( subset @ B @ A ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(binintersectER_type,type,
% 0.27/0.47 binintersectER: $o ).
% 0.27/0.47
% 0.27/0.47 thf(binintersectER,definition,
% 0.27/0.47 ( binintersectER
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.27/0.47 => ( in @ Xx @ B ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(disjointsetsI1_type,type,
% 0.27/0.47 disjointsetsI1: $o ).
% 0.27/0.47
% 0.27/0.47 thf(disjointsetsI1,definition,
% 0.27/0.47 ( disjointsetsI1
% 0.27/0.47 = ( ! [A: $i,B: $i] :
% 0.27/0.47 ( ~ ? [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 & ( in @ Xx @ B ) )
% 0.27/0.47 => ( ( binintersect @ A @ B )
% 0.27/0.47 = emptyset ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(binintersectRsub_type,type,
% 0.27/0.47 binintersectRsub: $o ).
% 0.27/0.47
% 0.27/0.47 thf(binintersectRsub,definition,
% 0.27/0.47 ( binintersectRsub
% 0.27/0.47 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(binintersectSubset4_type,type,
% 0.27/0.47 binintersectSubset4: $o ).
% 0.27/0.47
% 0.27/0.47 thf(binintersectSubset4,definition,
% 0.27/0.47 ( binintersectSubset4
% 0.27/0.47 = ( ! [A: $i,B: $i] :
% 0.27/0.47 ( ( subset @ B @ A )
% 0.27/0.47 => ( ( binintersect @ A @ B )
% 0.27/0.47 = B ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(binintersectSubset1_type,type,
% 0.27/0.47 binintersectSubset1: $o ).
% 0.27/0.47
% 0.27/0.47 thf(binintersectSubset1,definition,
% 0.27/0.47 ( binintersectSubset1
% 0.27/0.47 = ( ! [A: $i,B: $i] :
% 0.27/0.47 ( ( ( binintersect @ A @ B )
% 0.27/0.47 = A )
% 0.27/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(bs114d_type,type,
% 0.27/0.47 bs114d: $o ).
% 0.27/0.47
% 0.27/0.47 thf(bs114d,definition,
% 0.27/0.47 ( bs114d
% 0.27/0.47 = ( ! [A: $i,B: $i,C: $i] :
% 0.27/0.47 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 0.27/0.47 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(regular_type,type,
% 0.27/0.47 regular: $i > $o ).
% 0.27/0.47
% 0.27/0.47 thf(setminus_type,type,
% 0.27/0.47 setminus: $i > $i > $i ).
% 0.27/0.47
% 0.27/0.47 thf(setminusI_type,type,
% 0.27/0.47 setminusI: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setminusI,definition,
% 0.27/0.47 ( setminusI
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( ~ ( in @ Xx @ B )
% 0.27/0.47 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setminusEL_type,type,
% 0.27/0.47 setminusEL: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setminusEL,definition,
% 0.27/0.47 ( setminusEL
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.27/0.47 => ( in @ Xx @ A ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setminusER_type,type,
% 0.27/0.47 setminusER: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setminusER,definition,
% 0.27/0.47 ( setminusER
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.27/0.47 => ~ ( in @ Xx @ B ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setminusSubset2_type,type,
% 0.27/0.47 setminusSubset2: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setminusSubset2,definition,
% 0.27/0.47 ( setminusSubset2
% 0.27/0.47 = ( ! [A: $i,B: $i] :
% 0.27/0.47 ( ( subset @ A @ B )
% 0.27/0.47 => ( ( setminus @ A @ B )
% 0.27/0.47 = emptyset ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setminusERneg_type,type,
% 0.27/0.47 setminusERneg: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setminusERneg,definition,
% 0.27/0.47 ( setminusERneg
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.27/0.47 => ( ( in @ Xx @ A )
% 0.27/0.47 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setminusELneg_type,type,
% 0.27/0.47 setminusELneg: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setminusELneg,definition,
% 0.27/0.47 ( setminusELneg
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.27/0.47 => ( ~ ( in @ Xx @ B )
% 0.27/0.47 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setminusILneg_type,type,
% 0.27/0.47 setminusILneg: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setminusILneg,definition,
% 0.27/0.47 ( setminusILneg
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ~ ( in @ Xx @ A )
% 0.27/0.47 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setminusIRneg_type,type,
% 0.27/0.47 setminusIRneg: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setminusIRneg,definition,
% 0.27/0.47 ( setminusIRneg
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ B )
% 0.27/0.47 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setminusLsub_type,type,
% 0.27/0.47 setminusLsub: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setminusLsub,definition,
% 0.27/0.47 ( setminusLsub
% 0.27/0.47 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setminusSubset1_type,type,
% 0.27/0.47 setminusSubset1: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setminusSubset1,definition,
% 0.27/0.47 ( setminusSubset1
% 0.27/0.47 = ( ! [A: $i,B: $i] :
% 0.27/0.47 ( ( ( setminus @ A @ B )
% 0.27/0.47 = emptyset )
% 0.27/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(symdiff_type,type,
% 0.27/0.47 symdiff: $i > $i > $i ).
% 0.27/0.47
% 0.27/0.47 thf(symdiffE_type,type,
% 0.27/0.47 symdiffE: $o ).
% 0.27/0.47
% 0.27/0.47 thf(symdiffE,definition,
% 0.27/0.47 ( symdiffE
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 0.27/0.47 => ! [Xphi: $o] :
% 0.27/0.47 ( ( ( in @ Xx @ A )
% 0.27/0.47 => ( ~ ( in @ Xx @ B )
% 0.27/0.47 => Xphi ) )
% 0.27/0.47 => ( ( ~ ( in @ Xx @ A )
% 0.27/0.47 => ( ( in @ Xx @ B )
% 0.27/0.47 => Xphi ) )
% 0.27/0.47 => Xphi ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(symdiffI1_type,type,
% 0.27/0.47 symdiffI1: $o ).
% 0.27/0.47
% 0.27/0.47 thf(symdiffI1,definition,
% 0.27/0.47 ( symdiffI1
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( ~ ( in @ Xx @ B )
% 0.27/0.47 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(symdiffI2_type,type,
% 0.27/0.47 symdiffI2: $o ).
% 0.27/0.47
% 0.27/0.47 thf(symdiffI2,definition,
% 0.27/0.47 ( symdiffI2
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ~ ( in @ Xx @ A )
% 0.27/0.47 => ( ( in @ Xx @ B )
% 0.27/0.47 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(symdiffIneg1_type,type,
% 0.27/0.47 symdiffIneg1: $o ).
% 0.27/0.47
% 0.27/0.47 thf(symdiffIneg1,definition,
% 0.27/0.47 ( symdiffIneg1
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( ( in @ Xx @ B )
% 0.27/0.47 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(symdiffIneg2_type,type,
% 0.27/0.47 symdiffIneg2: $o ).
% 0.27/0.47
% 0.27/0.47 thf(symdiffIneg2,definition,
% 0.27/0.47 ( symdiffIneg2
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ~ ( in @ Xx @ A )
% 0.27/0.47 => ( ~ ( in @ Xx @ B )
% 0.27/0.47 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(iskpair_type,type,
% 0.27/0.47 iskpair: $i > $o ).
% 0.27/0.47
% 0.27/0.47 thf(secondinupair_type,type,
% 0.27/0.47 secondinupair: $o ).
% 0.27/0.47
% 0.27/0.47 thf(secondinupair,definition,
% 0.27/0.47 ( secondinupair
% 0.27/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setukpairIL_type,type,
% 0.27/0.47 setukpairIL: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setukpairIL,definition,
% 0.27/0.47 ( setukpairIL
% 0.27/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setukpairIR_type,type,
% 0.27/0.47 setukpairIR: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setukpairIR,definition,
% 0.27/0.47 ( setukpairIR
% 0.27/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(kpairiskpair_type,type,
% 0.27/0.47 kpairiskpair: $o ).
% 0.27/0.47
% 0.27/0.47 thf(kpairiskpair,definition,
% 0.27/0.47 ( kpairiskpair
% 0.27/0.47 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(kpair_type,type,
% 0.27/0.47 kpair: $i > $i > $i ).
% 0.27/0.47
% 0.27/0.47 thf(kpairp_type,type,
% 0.27/0.47 kpairp: $o ).
% 0.27/0.47
% 0.27/0.47 thf(kpairp,definition,
% 0.27/0.47 ( kpairp
% 0.27/0.47 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(cartprod_type,type,
% 0.27/0.47 cartprod: $i > $i > $i ).
% 0.27/0.47
% 0.27/0.47 thf(singletonsubset_type,type,
% 0.27/0.47 singletonsubset: $o ).
% 0.27/0.47
% 0.27/0.47 thf(singletonsubset,definition,
% 0.27/0.47 ( singletonsubset
% 0.27/0.47 = ( ! [A: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(singletoninpowerset_type,type,
% 0.27/0.47 singletoninpowerset: $o ).
% 0.27/0.47
% 0.27/0.47 thf(singletoninpowerset,definition,
% 0.27/0.47 ( singletoninpowerset
% 0.27/0.47 = ( ! [A: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(singletoninpowunion_type,type,
% 0.27/0.47 singletoninpowunion: $o ).
% 0.27/0.47
% 0.27/0.47 thf(singletoninpowunion,definition,
% 0.27/0.47 ( singletoninpowunion
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(upairset2E_type,type,
% 0.27/0.47 upairset2E: $o ).
% 0.27/0.47
% 0.27/0.47 thf(upairset2E,definition,
% 0.27/0.47 ( upairset2E
% 0.27/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.27/0.47 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.27/0.47 => ( ( Xz = Xx )
% 0.27/0.47 | ( Xz = Xy ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(upairsubunion_type,type,
% 0.27/0.47 upairsubunion: $o ).
% 0.27/0.47
% 0.27/0.47 thf(upairsubunion,definition,
% 0.27/0.47 ( upairsubunion
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ! [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ B )
% 0.27/0.47 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(upairinpowunion_type,type,
% 0.27/0.47 upairinpowunion: $o ).
% 0.27/0.47
% 0.27/0.47 thf(upairinpowunion,definition,
% 0.27/0.47 ( upairinpowunion
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ! [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ B )
% 0.27/0.47 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(ubforcartprodlem1_type,type,
% 0.27/0.47 ubforcartprodlem1: $o ).
% 0.27/0.47
% 0.27/0.47 thf(ubforcartprodlem1,definition,
% 0.27/0.47 ( ubforcartprodlem1
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ! [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ B )
% 0.27/0.47 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(ubforcartprodlem2_type,type,
% 0.27/0.47 ubforcartprodlem2: $o ).
% 0.27/0.47
% 0.27/0.47 thf(ubforcartprodlem2,definition,
% 0.27/0.47 ( ubforcartprodlem2
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ! [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ B )
% 0.27/0.47 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(ubforcartprodlem3_type,type,
% 0.27/0.47 ubforcartprodlem3: $o ).
% 0.27/0.47
% 0.27/0.47 thf(ubforcartprodlem3,definition,
% 0.27/0.47 ( ubforcartprodlem3
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ! [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ B )
% 0.27/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(cartprodpairin_type,type,
% 0.27/0.47 cartprodpairin: $o ).
% 0.27/0.47
% 0.27/0.47 thf(cartprodpairin,definition,
% 0.27/0.47 ( cartprodpairin
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ! [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ B )
% 0.27/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(cartprodmempair1_type,type,
% 0.27/0.47 cartprodmempair1: $o ).
% 0.27/0.47
% 0.27/0.47 thf(cartprodmempair1,definition,
% 0.27/0.47 ( cartprodmempair1
% 0.27/0.47 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.27/0.47 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.27/0.47 => ? [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 & ? [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ B )
% 0.27/0.47 & ( Xu
% 0.27/0.47 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(cartprodmempair_type,type,
% 0.27/0.47 cartprodmempair: $o ).
% 0.27/0.47
% 0.27/0.47 thf(cartprodmempair,definition,
% 0.27/0.47 ( cartprodmempair
% 0.27/0.47 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.27/0.47 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.27/0.47 => ( iskpair @ Xu ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setunionE2_type,type,
% 0.27/0.47 setunionE2: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setunionE2,definition,
% 0.27/0.47 ( setunionE2
% 0.27/0.47 = ( ! [A: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ ( setunion @ A ) )
% 0.27/0.47 => ? [X: $i] :
% 0.27/0.47 ( ( in @ X @ A )
% 0.27/0.47 & ( in @ Xx @ X ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setunionsingleton1_type,type,
% 0.27/0.47 setunionsingleton1: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setunionsingleton1,definition,
% 0.27/0.47 ( setunionsingleton1
% 0.27/0.47 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setunionsingleton2_type,type,
% 0.27/0.47 setunionsingleton2: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setunionsingleton2,definition,
% 0.27/0.47 ( setunionsingleton2
% 0.27/0.47 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setunionsingleton_type,type,
% 0.27/0.47 setunionsingleton: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setunionsingleton,definition,
% 0.27/0.47 ( setunionsingleton
% 0.27/0.47 = ( ! [Xx: $i] :
% 0.27/0.47 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 0.27/0.47 = Xx ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(singleton_type,type,
% 0.27/0.47 singleton: $i > $o ).
% 0.27/0.47
% 0.27/0.47 thf(singleton,definition,
% 0.27/0.47 ( singleton
% 0.27/0.47 = ( ^ [A: $i] :
% 0.27/0.47 ? [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 & ( A
% 0.27/0.47 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(singletonprop_type,type,
% 0.27/0.47 singletonprop: $o ).
% 0.27/0.47
% 0.27/0.47 thf(singletonprop,definition,
% 0.27/0.47 ( singletonprop
% 0.27/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.27/0.47 ( ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ! [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ A )
% 0.27/0.47 => ( ( Xphi @ Xx )
% 0.27/0.47 => ( ( Xphi @ Xy )
% 0.27/0.47 => ( Xx = Xy ) ) ) ) )
% 0.27/0.47 => ( ? [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 & ( Xphi @ Xx ) )
% 0.27/0.47 => ( singleton
% 0.27/0.47 @ ( dsetconstr @ A
% 0.27/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(ex1_type,type,
% 0.27/0.47 ex1: $i > ( $i > $o ) > $o ).
% 0.27/0.47
% 0.27/0.47 thf(ex1,definition,
% 0.27/0.47 ( ex1
% 0.27/0.47 = ( ^ [A: $i,Xphi: $i > $o] :
% 0.27/0.47 ( singleton
% 0.27/0.47 @ ( dsetconstr @ A
% 0.27/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(ex1E1_type,type,
% 0.27/0.47 ex1E1: $o ).
% 0.27/0.47
% 0.27/0.47 thf(ex1E1,definition,
% 0.27/0.47 ( ex1E1
% 0.27/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.27/0.47 ( ( ex1 @ A
% 0.27/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.27/0.47 => ? [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(ex1I_type,type,
% 0.27/0.47 ex1I: $o ).
% 0.27/0.47
% 0.27/0.47 thf(ex1I,definition,
% 0.27/0.47 ( ex1I
% 0.27/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( ( Xphi @ Xx )
% 0.27/0.47 => ( ! [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ A )
% 0.27/0.47 => ( ( Xphi @ Xy )
% 0.27/0.47 => ( Xy = Xx ) ) )
% 0.27/0.47 => ( ex1 @ A
% 0.27/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(ex1I2_type,type,
% 0.27/0.47 ex1I2: $o ).
% 0.27/0.47
% 0.27/0.47 thf(ex1I2,definition,
% 0.27/0.47 ( ex1I2
% 0.27/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.27/0.47 ( ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ! [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ A )
% 0.27/0.47 => ( ( Xphi @ Xx )
% 0.27/0.47 => ( ( Xphi @ Xy )
% 0.27/0.47 => ( Xx = Xy ) ) ) ) )
% 0.27/0.47 => ( ? [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 & ( Xphi @ Xx ) )
% 0.27/0.47 => ( ex1 @ A
% 0.27/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(singletonsuniq_type,type,
% 0.27/0.47 singletonsuniq: $o ).
% 0.27/0.47
% 0.27/0.47 thf(singletonsuniq,definition,
% 0.27/0.47 ( singletonsuniq
% 0.27/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.27/0.47 ( ( ( setadjoin @ Xx @ emptyset )
% 0.27/0.47 = ( setadjoin @ Xy @ emptyset ) )
% 0.27/0.47 => ( Xx = Xy ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(atmost1p_type,type,
% 0.27/0.47 atmost1p: $i > $o ).
% 0.27/0.47
% 0.27/0.47 thf(atleast2p_type,type,
% 0.27/0.47 atleast2p: $i > $o ).
% 0.27/0.47
% 0.27/0.47 thf(atmost2p_type,type,
% 0.27/0.47 atmost2p: $i > $o ).
% 0.27/0.47
% 0.27/0.47 thf(upairsetp_type,type,
% 0.27/0.47 upairsetp: $i > $o ).
% 0.27/0.47
% 0.27/0.47 thf(setukpairinjL1_type,type,
% 0.27/0.47 setukpairinjL1: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setukpairinjL1,definition,
% 0.27/0.47 ( setukpairinjL1
% 0.27/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.27/0.47 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 0.27/0.47 => ( Xx = Xz ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(kfstsingleton_type,type,
% 0.27/0.47 kfstsingleton: $o ).
% 0.27/0.47
% 0.27/0.47 thf(kfstsingleton,definition,
% 0.27/0.47 ( kfstsingleton
% 0.27/0.47 = ( ! [Xu: $i] :
% 0.27/0.47 ( ( iskpair @ Xu )
% 0.27/0.47 => ( singleton
% 0.27/0.47 @ ( dsetconstr @ ( setunion @ Xu )
% 0.27/0.47 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(theprop_type,type,
% 0.27/0.47 theprop: $o ).
% 0.27/0.47
% 0.27/0.47 thf(theprop,definition,
% 0.27/0.47 ( theprop
% 0.27/0.47 = ( ! [X: $i] :
% 0.27/0.47 ( ( singleton @ X )
% 0.27/0.47 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(kfst_type,type,
% 0.27/0.47 kfst: $i > $i ).
% 0.27/0.47
% 0.27/0.47 thf(kfstpairEq_type,type,
% 0.27/0.47 kfstpairEq: $o ).
% 0.27/0.47
% 0.27/0.47 thf(kfstpairEq,definition,
% 0.27/0.47 ( kfstpairEq
% 0.27/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.27/0.47 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.27/0.47 = Xx ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(cartprodfstin_type,type,
% 0.27/0.47 cartprodfstin: $o ).
% 0.27/0.47
% 0.27/0.47 thf(cartprodfstin,definition,
% 0.27/0.47 ( cartprodfstin
% 0.27/0.47 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.27/0.47 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.27/0.47 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setukpairinjL2_type,type,
% 0.27/0.47 setukpairinjL2: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setukpairinjL2,definition,
% 0.27/0.47 ( setukpairinjL2
% 0.27/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.27/0.47 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.27/0.47 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.27/0.47 => ( Xx = Xz ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setukpairinjL_type,type,
% 0.27/0.47 setukpairinjL: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setukpairinjL,definition,
% 0.27/0.47 ( setukpairinjL
% 0.27/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.27/0.47 ( ( ( kpair @ Xx @ Xy )
% 0.27/0.47 = ( kpair @ Xz @ Xu ) )
% 0.27/0.47 => ( Xx = Xz ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setukpairinjR11_type,type,
% 0.27/0.47 setukpairinjR11: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setukpairinjR11,definition,
% 0.27/0.47 ( setukpairinjR11
% 0.27/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.27/0.47 ( ( Xx = Xy )
% 0.27/0.47 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.27/0.47 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setukpairinjR12_type,type,
% 0.27/0.47 setukpairinjR12: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setukpairinjR12,definition,
% 0.27/0.47 ( setukpairinjR12
% 0.27/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.27/0.47 ( ( Xx = Xy )
% 0.27/0.47 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.27/0.47 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setukpairinjR1_type,type,
% 0.27/0.47 setukpairinjR1: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setukpairinjR1,definition,
% 0.27/0.47 ( setukpairinjR1
% 0.27/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.27/0.47 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.27/0.47 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.27/0.47 => ( ( Xz = Xu )
% 0.27/0.47 => ( Xy = Xu ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(upairequniteq_type,type,
% 0.27/0.47 upairequniteq: $o ).
% 0.27/0.47
% 0.27/0.47 thf(upairequniteq,definition,
% 0.27/0.47 ( upairequniteq
% 0.27/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.27/0.47 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.27/0.47 = ( setadjoin @ Xz @ emptyset ) )
% 0.27/0.47 => ( Xx = Xy ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setukpairinjR2_type,type,
% 0.27/0.47 setukpairinjR2: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setukpairinjR2,definition,
% 0.27/0.47 ( setukpairinjR2
% 0.27/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.27/0.47 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.27/0.47 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.27/0.47 => ( Xy = Xu ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setukpairinjR_type,type,
% 0.27/0.47 setukpairinjR: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setukpairinjR,definition,
% 0.27/0.47 ( setukpairinjR
% 0.27/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.27/0.47 ( ( ( kpair @ Xx @ Xy )
% 0.27/0.47 = ( kpair @ Xz @ Xu ) )
% 0.27/0.47 => ( Xy = Xu ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(ksndsingleton_type,type,
% 0.27/0.47 ksndsingleton: $o ).
% 0.27/0.47
% 0.27/0.47 thf(ksndsingleton,definition,
% 0.27/0.47 ( ksndsingleton
% 0.27/0.47 = ( ! [Xu: $i] :
% 0.27/0.47 ( ( iskpair @ Xu )
% 0.27/0.47 => ( singleton
% 0.27/0.47 @ ( dsetconstr @ ( setunion @ Xu )
% 0.27/0.47 @ ^ [Xx: $i] :
% 0.27/0.47 ( Xu
% 0.27/0.47 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(ksnd_type,type,
% 0.27/0.47 ksnd: $i > $i ).
% 0.27/0.47
% 0.27/0.47 thf(ksndpairEq_type,type,
% 0.27/0.47 ksndpairEq: $o ).
% 0.27/0.47
% 0.27/0.47 thf(ksndpairEq,definition,
% 0.27/0.47 ( ksndpairEq
% 0.27/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.27/0.47 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.27/0.47 = Xy ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(kpairsurjEq_type,type,
% 0.27/0.47 kpairsurjEq: $o ).
% 0.27/0.47
% 0.27/0.47 thf(kpairsurjEq,definition,
% 0.27/0.47 ( kpairsurjEq
% 0.27/0.47 = ( ! [Xu: $i] :
% 0.27/0.47 ( ( iskpair @ Xu )
% 0.27/0.47 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.27/0.47 = Xu ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(cartprodsndin_type,type,
% 0.27/0.47 cartprodsndin: $o ).
% 0.27/0.47
% 0.27/0.47 thf(cartprodsndin,definition,
% 0.27/0.47 ( cartprodsndin
% 0.27/0.47 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.27/0.47 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.27/0.47 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(cartprodpairmemEL_type,type,
% 0.27/0.47 cartprodpairmemEL: $o ).
% 0.27/0.47
% 0.27/0.47 thf(cartprodpairmemEL,definition,
% 0.27/0.47 ( cartprodpairmemEL
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.27/0.47 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.27/0.47 => ( in @ Xx @ A ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(cartprodpairmemER_type,type,
% 0.27/0.47 cartprodpairmemER: $o ).
% 0.27/0.47
% 0.27/0.47 thf(cartprodpairmemER,definition,
% 0.27/0.47 ( cartprodpairmemER
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.27/0.47 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.27/0.47 => ( in @ Xy @ B ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(cartprodmempaircEq_type,type,
% 0.27/0.47 cartprodmempaircEq: $o ).
% 0.27/0.47
% 0.27/0.47 thf(cartprodmempaircEq,definition,
% 0.27/0.47 ( cartprodmempaircEq
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ! [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ B )
% 0.27/0.47 => ( ( kpair @ Xx @ Xy )
% 0.27/0.47 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(cartprodfstpairEq_type,type,
% 0.27/0.47 cartprodfstpairEq: $o ).
% 0.27/0.47
% 0.27/0.47 thf(cartprodfstpairEq,definition,
% 0.27/0.47 ( cartprodfstpairEq
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ! [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ B )
% 0.27/0.47 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.27/0.47 = Xx ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(cartprodsndpairEq_type,type,
% 0.27/0.47 cartprodsndpairEq: $o ).
% 0.27/0.47
% 0.27/0.47 thf(cartprodsndpairEq,definition,
% 0.27/0.47 ( cartprodsndpairEq
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ! [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ B )
% 0.27/0.47 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.27/0.47 = Xy ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(cartprodpairsurjEq_type,type,
% 0.27/0.47 cartprodpairsurjEq: $o ).
% 0.27/0.47
% 0.27/0.47 thf(cartprodpairsurjEq,definition,
% 0.27/0.47 ( cartprodpairsurjEq
% 0.27/0.47 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.27/0.47 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.27/0.47 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.27/0.47 = Xu ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(breln_type,type,
% 0.27/0.47 breln: $i > $i > $i > $o ).
% 0.27/0.47
% 0.27/0.47 thf(breln,definition,
% 0.27/0.47 ( breln
% 0.27/0.47 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(dpsetconstr_type,type,
% 0.27/0.47 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 0.27/0.47
% 0.27/0.47 thf(dpsetconstrI_type,type,
% 0.27/0.47 dpsetconstrI: $o ).
% 0.27/0.47
% 0.27/0.47 thf(dpsetconstrI,definition,
% 0.27/0.47 ( dpsetconstrI
% 0.27/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ! [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ B )
% 0.27/0.47 => ( ( Xphi @ Xx @ Xy )
% 0.27/0.47 => ( in @ ( kpair @ Xx @ Xy )
% 0.27/0.47 @ ( dpsetconstr @ A @ B
% 0.27/0.47 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(dpsetconstrSub_type,type,
% 0.27/0.47 dpsetconstrSub: $o ).
% 0.27/0.47
% 0.27/0.47 thf(dpsetconstrSub,definition,
% 0.27/0.47 ( dpsetconstrSub
% 0.27/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.27/0.47 ( subset
% 0.27/0.47 @ ( dpsetconstr @ A @ B
% 0.27/0.47 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 0.27/0.47 @ ( cartprod @ A @ B ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setOfPairsIsBReln_type,type,
% 0.27/0.47 setOfPairsIsBReln: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setOfPairsIsBReln,definition,
% 0.27/0.47 ( setOfPairsIsBReln
% 0.27/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.27/0.47 ( breln @ A @ B
% 0.27/0.47 @ ( dpsetconstr @ A @ B
% 0.27/0.47 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(dpsetconstrERa_type,type,
% 0.27/0.47 dpsetconstrERa: $o ).
% 0.27/0.47
% 0.27/0.47 thf(dpsetconstrERa,definition,
% 0.27/0.47 ( dpsetconstrERa
% 0.27/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ! [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ B )
% 0.27/0.47 => ( ( in @ ( kpair @ Xx @ Xy )
% 0.27/0.47 @ ( dpsetconstr @ A @ B
% 0.27/0.47 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.27/0.47 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(dpsetconstrEL1_type,type,
% 0.27/0.47 dpsetconstrEL1: $o ).
% 0.27/0.47
% 0.27/0.47 thf(dpsetconstrEL1,definition,
% 0.27/0.47 ( dpsetconstrEL1
% 0.27/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.27/0.47 ( ( in @ ( kpair @ Xx @ Xy )
% 0.27/0.47 @ ( dpsetconstr @ A @ B
% 0.27/0.47 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.27/0.47 => ( in @ Xx @ A ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(dpsetconstrEL2_type,type,
% 0.27/0.47 dpsetconstrEL2: $o ).
% 0.27/0.47
% 0.27/0.47 thf(dpsetconstrEL2,definition,
% 0.27/0.47 ( dpsetconstrEL2
% 0.27/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.27/0.47 ( ( in @ ( kpair @ Xx @ Xy )
% 0.27/0.47 @ ( dpsetconstr @ A @ B
% 0.27/0.47 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.27/0.47 => ( in @ Xy @ B ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(dpsetconstrER_type,type,
% 0.27/0.47 dpsetconstrER: $o ).
% 0.27/0.47
% 0.27/0.47 thf(dpsetconstrER,definition,
% 0.27/0.47 ( dpsetconstrER
% 0.27/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.27/0.47 ( ( in @ ( kpair @ Xx @ Xy )
% 0.27/0.47 @ ( dpsetconstr @ A @ B
% 0.27/0.47 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.27/0.47 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(func_type,type,
% 0.27/0.47 func: $i > $i > $i > $o ).
% 0.27/0.47
% 0.27/0.47 thf(funcSet_type,type,
% 0.27/0.47 funcSet: $i > $i > $i ).
% 0.27/0.47
% 0.27/0.47 thf(funcImageSingleton_type,type,
% 0.27/0.47 funcImageSingleton: $o ).
% 0.27/0.47
% 0.27/0.47 thf(funcImageSingleton,definition,
% 0.27/0.47 ( funcImageSingleton
% 0.27/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.27/0.47 ( ( func @ A @ B @ Xf )
% 0.27/0.47 => ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( singleton
% 0.27/0.47 @ ( dsetconstr @ B
% 0.27/0.47 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(apProp_type,type,
% 0.27/0.47 apProp: $o ).
% 0.27/0.47
% 0.27/0.47 thf(apProp,definition,
% 0.27/0.47 ( apProp
% 0.27/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.27/0.47 ( ( func @ A @ B @ Xf )
% 0.27/0.47 => ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( in
% 0.27/0.47 @ ( setunion
% 0.27/0.47 @ ( dsetconstr @ B
% 0.27/0.47 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 0.27/0.47 @ B ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(ap_type,type,
% 0.27/0.47 ap: $i > $i > $i > $i > $i ).
% 0.27/0.47
% 0.27/0.47 thf(app_type,type,
% 0.27/0.47 app: $o ).
% 0.27/0.47
% 0.27/0.47 thf(app,definition,
% 0.27/0.47 ( app
% 0.27/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.27/0.47 ( ( func @ A @ B @ Xf )
% 0.27/0.47 => ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(infuncsetfunc_type,type,
% 0.27/0.47 infuncsetfunc: $o ).
% 0.27/0.47
% 0.27/0.47 thf(infuncsetfunc,definition,
% 0.27/0.47 ( infuncsetfunc
% 0.27/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.27/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.27/0.47 => ( func @ A @ B @ Xf ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(ap2p_type,type,
% 0.27/0.47 ap2p: $o ).
% 0.27/0.47
% 0.27/0.47 thf(ap2p,definition,
% 0.27/0.47 ( ap2p
% 0.27/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.27/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.27/0.47 => ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(funcinfuncset_type,type,
% 0.27/0.47 funcinfuncset: $o ).
% 0.27/0.47
% 0.27/0.47 thf(funcinfuncset,definition,
% 0.27/0.47 ( funcinfuncset
% 0.27/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.27/0.47 ( ( func @ A @ B @ Xf )
% 0.27/0.47 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(lamProp_type,type,
% 0.27/0.47 lamProp: $o ).
% 0.27/0.47
% 0.27/0.47 thf(lamProp,definition,
% 0.27/0.47 ( lamProp
% 0.27/0.47 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.27/0.47 ( ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.27/0.47 => ( func @ A @ B
% 0.27/0.47 @ ( dpsetconstr @ A @ B
% 0.27/0.47 @ ^ [Xx: $i,Xy: $i] :
% 0.27/0.47 ( ( Xf @ Xx )
% 0.27/0.47 = Xy ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(lam_type,type,
% 0.27/0.47 lam: $i > $i > ( $i > $i ) > $i ).
% 0.27/0.47
% 0.27/0.47 thf(lam,definition,
% 0.27/0.47 ( lam
% 0.27/0.47 = ( ^ [A: $i,B: $i,Xf: $i > $i] :
% 0.27/0.47 ( dpsetconstr @ A @ B
% 0.27/0.47 @ ^ [Xx: $i,Xy: $i] :
% 0.27/0.47 ( ( Xf @ Xx )
% 0.27/0.47 = Xy ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(lamp_type,type,
% 0.27/0.47 lamp: $o ).
% 0.27/0.47
% 0.27/0.47 thf(lamp,definition,
% 0.27/0.47 ( lamp
% 0.27/0.47 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.27/0.47 ( ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.27/0.47 => ( func @ A @ B
% 0.27/0.47 @ ( lam @ A @ B
% 0.27/0.47 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(lam2p_type,type,
% 0.27/0.47 lam2p: $o ).
% 0.27/0.47
% 0.27/0.47 thf(lam2p,definition,
% 0.27/0.47 ( lam2p
% 0.27/0.47 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.27/0.47 ( ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.27/0.47 => ( in
% 0.27/0.47 @ ( lam @ A @ B
% 0.27/0.47 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.27/0.47 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(brelnall1_type,type,
% 0.27/0.47 brelnall1: $o ).
% 0.27/0.47
% 0.27/0.47 thf(brelnall1,definition,
% 0.27/0.47 ( brelnall1
% 0.27/0.47 = ( ! [A: $i,B: $i,R: $i] :
% 0.27/0.47 ( ( breln @ A @ B @ R )
% 0.27/0.47 => ! [Xphi: $i > $o] :
% 0.27/0.47 ( ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ! [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ B )
% 0.27/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.27/0.47 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.27/0.47 => ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ R )
% 0.27/0.47 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(brelnall2_type,type,
% 0.27/0.47 brelnall2: $o ).
% 0.27/0.47
% 0.27/0.47 thf(brelnall2,definition,
% 0.27/0.47 ( brelnall2
% 0.27/0.47 = ( ! [A: $i,B: $i,R: $i] :
% 0.27/0.47 ( ( breln @ A @ B @ R )
% 0.27/0.47 => ! [Xphi: $i > $o] :
% 0.27/0.47 ( ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ! [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ B )
% 0.27/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.27/0.47 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.27/0.47 => ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ R )
% 0.27/0.47 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(ex1E2_type,type,
% 0.27/0.47 ex1E2: $o ).
% 0.27/0.47
% 0.27/0.47 thf(ex1E2,definition,
% 0.27/0.47 ( ex1E2
% 0.27/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.27/0.47 ( ( ex1 @ A
% 0.27/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.27/0.47 => ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ! [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ A )
% 0.27/0.47 => ( ( Xphi @ Xx )
% 0.27/0.47 => ( ( Xphi @ Xy )
% 0.27/0.47 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(funcGraphProp1_type,type,
% 0.27/0.47 funcGraphProp1: $o ).
% 0.27/0.47
% 0.27/0.47 thf(funcGraphProp1,definition,
% 0.27/0.47 ( funcGraphProp1
% 0.27/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.27/0.47 ( ( func @ A @ B @ Xf )
% 0.27/0.47 => ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(funcGraphProp3_type,type,
% 0.27/0.47 funcGraphProp3: $o ).
% 0.27/0.47
% 0.27/0.47 thf(funcGraphProp3,definition,
% 0.27/0.47 ( funcGraphProp3
% 0.27/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.27/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.27/0.47 => ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(funcGraphProp2_type,type,
% 0.27/0.47 funcGraphProp2: $o ).
% 0.27/0.47
% 0.27/0.47 thf(funcGraphProp2,definition,
% 0.27/0.47 ( funcGraphProp2
% 0.27/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.27/0.47 ( ( func @ A @ B @ Xf )
% 0.27/0.47 => ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ! [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ B )
% 0.27/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.27/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.27/0.47 = Xy ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(funcextLem_type,type,
% 0.27/0.47 funcextLem: $o ).
% 0.27/0.47
% 0.27/0.47 thf(funcextLem,definition,
% 0.27/0.47 ( funcextLem
% 0.27/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.27/0.47 ( ( func @ A @ B @ Xf )
% 0.27/0.47 => ! [Xg: $i] :
% 0.27/0.47 ( ( func @ A @ B @ Xg )
% 0.27/0.47 => ( ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.27/0.47 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.27/0.47 => ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ! [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ B )
% 0.27/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 0.27/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(funcGraphProp4_type,type,
% 0.27/0.47 funcGraphProp4: $o ).
% 0.27/0.47
% 0.27/0.47 thf(funcGraphProp4,definition,
% 0.27/0.47 ( funcGraphProp4
% 0.27/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.27/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.27/0.47 => ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ! [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ B )
% 0.27/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.27/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.27/0.47 = Xy ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(subbreln_type,type,
% 0.27/0.47 subbreln: $o ).
% 0.27/0.47
% 0.27/0.47 thf(subbreln,definition,
% 0.27/0.47 ( subbreln
% 0.27/0.47 = ( ! [A: $i,B: $i,R: $i] :
% 0.27/0.47 ( ( breln @ A @ B @ R )
% 0.27/0.47 => ! [S: $i] :
% 0.27/0.47 ( ( breln @ A @ B @ S )
% 0.27/0.47 => ( ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ! [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ B )
% 0.27/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.27/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.27/0.47 => ( subset @ R @ S ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(eqbreln_type,type,
% 0.27/0.47 eqbreln: $o ).
% 0.27/0.47
% 0.27/0.47 thf(eqbreln,definition,
% 0.27/0.47 ( eqbreln
% 0.27/0.47 = ( ! [A: $i,B: $i,R: $i] :
% 0.27/0.47 ( ( breln @ A @ B @ R )
% 0.27/0.47 => ! [S: $i] :
% 0.27/0.47 ( ( breln @ A @ B @ S )
% 0.27/0.47 => ( ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ! [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ B )
% 0.27/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.27/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.27/0.47 => ( ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ! [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ B )
% 0.27/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.27/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 0.27/0.47 => ( R = S ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(funcext_type,type,
% 0.27/0.47 funcext: $o ).
% 0.27/0.47
% 0.27/0.47 thf(funcext,definition,
% 0.27/0.47 ( funcext
% 0.27/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.27/0.47 ( ( func @ A @ B @ Xf )
% 0.27/0.47 => ! [Xg: $i] :
% 0.27/0.47 ( ( func @ A @ B @ Xg )
% 0.27/0.47 => ( ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.27/0.47 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.27/0.47 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(funcext2_type,type,
% 0.27/0.47 funcext2: $o ).
% 0.27/0.47
% 0.27/0.47 thf(funcext2,definition,
% 0.27/0.47 ( funcext2
% 0.27/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.27/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.27/0.47 => ! [Xg: $i] :
% 0.27/0.47 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 0.27/0.47 => ( ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.27/0.47 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.27/0.47 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(ap2apEq1_type,type,
% 0.27/0.47 ap2apEq1: $o ).
% 0.27/0.47
% 0.27/0.47 thf(ap2apEq1,definition,
% 0.27/0.47 ( ap2apEq1
% 0.27/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.27/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.27/0.47 => ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.27/0.47 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(ap2apEq2_type,type,
% 0.27/0.47 ap2apEq2: $o ).
% 0.27/0.47
% 0.27/0.47 thf(ap2apEq2,definition,
% 0.27/0.47 ( ap2apEq2
% 0.27/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.27/0.47 ( ( func @ A @ B @ Xf )
% 0.27/0.47 => ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.27/0.47 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(beta1_type,type,
% 0.27/0.47 beta1: $o ).
% 0.27/0.47
% 0.27/0.47 thf(beta1,definition,
% 0.27/0.47 ( beta1
% 0.27/0.47 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.27/0.47 ( ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.27/0.47 => ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( ( ap @ A @ B
% 0.27/0.47 @ ( lam @ A @ B
% 0.27/0.47 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.27/0.47 @ Xx )
% 0.27/0.47 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(eta1_type,type,
% 0.27/0.47 eta1: $o ).
% 0.27/0.47
% 0.27/0.47 thf(eta1,definition,
% 0.27/0.47 ( eta1
% 0.27/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.27/0.47 ( ( func @ A @ B @ Xf )
% 0.27/0.47 => ( ( lam @ A @ B
% 0.27/0.47 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.27/0.47 = Xf ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(lam2lamEq,conjecture,
% 0.27/0.47 ( setextAx
% 0.27/0.47 => ( emptysetAx
% 0.27/0.47 => ( setadjoinAx
% 0.27/0.47 => ( powersetAx
% 0.27/0.47 => ( setunionAx
% 0.27/0.47 => ( omega0Ax
% 0.27/0.47 => ( omegaSAx
% 0.27/0.47 => ( omegaIndAx
% 0.27/0.47 => ( replAx
% 0.27/0.47 => ( foundationAx
% 0.27/0.47 => ( wellorderingAx
% 0.27/0.47 => ( descrp
% 0.27/0.47 => ( dsetconstrI
% 0.27/0.47 => ( dsetconstrEL
% 0.27/0.47 => ( dsetconstrER
% 0.27/0.47 => ( exuE1
% 0.27/0.47 => ( prop2setE
% 0.27/0.47 => ( emptysetE
% 0.27/0.47 => ( emptysetimpfalse
% 0.27/0.47 => ( notinemptyset
% 0.27/0.47 => ( exuE3e
% 0.27/0.47 => ( setext
% 0.27/0.47 => ( emptyI
% 0.27/0.47 => ( noeltsimpempty
% 0.27/0.47 => ( setbeta
% 0.27/0.47 => ( nonemptyE1
% 0.27/0.47 => ( nonemptyI
% 0.27/0.47 => ( nonemptyI1
% 0.27/0.47 => ( setadjoinIL
% 0.27/0.47 => ( emptyinunitempty
% 0.27/0.47 => ( setadjoinIR
% 0.27/0.47 => ( setadjoinE
% 0.27/0.47 => ( setadjoinOr
% 0.27/0.47 => ( setoftrueEq
% 0.27/0.47 => ( powersetI
% 0.27/0.47 => ( emptyinPowerset
% 0.27/0.47 => ( emptyInPowerset
% 0.27/0.47 => ( powersetE
% 0.27/0.47 => ( setunionI
% 0.27/0.47 => ( setunionE
% 0.27/0.47 => ( subPowSU
% 0.27/0.47 => ( exuE2
% 0.27/0.47 => ( nonemptyImpWitness
% 0.27/0.47 => ( uniqinunit
% 0.27/0.47 => ( notinsingleton
% 0.27/0.47 => ( eqinunit
% 0.27/0.47 => ( singletonsswitch
% 0.27/0.47 => ( upairsetE
% 0.27/0.47 => ( upairsetIL
% 0.27/0.47 => ( upairsetIR
% 0.27/0.47 => ( emptyE1
% 0.27/0.47 => ( vacuousDall
% 0.27/0.47 => ( quantDeMorgan1
% 0.27/0.47 => ( quantDeMorgan2
% 0.27/0.47 => ( quantDeMorgan3
% 0.27/0.47 => ( quantDeMorgan4
% 0.27/0.47 => ( prop2setI
% 0.27/0.47 => ( prop2set2propI
% 0.27/0.47 => ( notdexE
% 0.27/0.47 => ( notdallE
% 0.27/0.47 => ( exuI1
% 0.27/0.47 => ( exuI3
% 0.27/0.47 => ( exuI2
% 0.27/0.47 => ( inCongP
% 0.27/0.47 => ( in__Cong
% 0.27/0.47 => ( exuE3u
% 0.27/0.47 => ( exu__Cong
% 0.27/0.47 => ( emptyset__Cong
% 0.27/0.47 => ( setadjoin__Cong
% 0.27/0.47 => ( powerset__Cong
% 0.27/0.47 => ( setunion__Cong
% 0.27/0.47 => ( omega__Cong
% 0.27/0.47 => ( exuEu
% 0.27/0.47 => ( descr__Cong
% 0.27/0.47 => ( dsetconstr__Cong
% 0.27/0.47 => ( subsetI1
% 0.27/0.47 => ( eqimpsubset2
% 0.27/0.47 => ( eqimpsubset1
% 0.27/0.47 => ( subsetI2
% 0.27/0.47 => ( emptysetsubset
% 0.27/0.47 => ( subsetE
% 0.27/0.47 => ( subsetE2
% 0.27/0.47 => ( notsubsetI
% 0.27/0.47 => ( notequalI1
% 0.27/0.47 => ( notequalI2
% 0.27/0.47 => ( subsetRefl
% 0.27/0.47 => ( subsetTrans
% 0.27/0.47 => ( setadjoinSub
% 0.27/0.47 => ( setadjoinSub2
% 0.27/0.47 => ( subset2powerset
% 0.27/0.47 => ( setextsub
% 0.27/0.47 => ( subsetemptysetimpeq
% 0.27/0.47 => ( powersetI1
% 0.27/0.47 => ( powersetE1
% 0.27/0.47 => ( inPowerset
% 0.27/0.47 => ( powersetsubset
% 0.27/0.47 => ( sepInPowerset
% 0.27/0.47 => ( sepSubset
% 0.27/0.47 => ( binunionIL
% 0.27/0.47 => ( upairset2IR
% 0.27/0.47 => ( binunionIR
% 0.27/0.47 => ( binunionEcases
% 0.27/0.47 => ( binunionE
% 0.27/0.47 => ( binunionLsub
% 0.27/0.47 => ( binunionRsub
% 0.27/0.47 => ( binintersectI
% 0.27/0.47 => ( binintersectSubset5
% 0.27/0.47 => ( binintersectEL
% 0.27/0.47 => ( binintersectLsub
% 0.27/0.47 => ( binintersectSubset2
% 0.27/0.47 => ( binintersectSubset3
% 0.27/0.47 => ( binintersectER
% 0.27/0.47 => ( disjointsetsI1
% 0.27/0.47 => ( binintersectRsub
% 0.27/0.47 => ( binintersectSubset4
% 0.27/0.47 => ( binintersectSubset1
% 0.27/0.47 => ( bs114d
% 0.27/0.47 => ( setminusI
% 0.27/0.47 => ( setminusEL
% 0.27/0.47 => ( setminusER
% 0.27/0.47 => ( setminusSubset2
% 0.27/0.47 => ( setminusERneg
% 0.27/0.47 => ( setminusELneg
% 0.27/0.47 => ( setminusILneg
% 0.27/0.47 => ( setminusIRneg
% 0.27/0.47 => ( setminusLsub
% 0.27/0.47 => ( setminusSubset1
% 0.27/0.47 => ( symdiffE
% 0.27/0.47 => ( symdiffI1
% 0.27/0.47 => ( symdiffI2
% 0.27/0.47 => ( symdiffIneg1
% 0.27/0.47 => ( symdiffIneg2
% 0.27/0.47 => ( secondinupair
% 0.27/0.47 => ( setukpairIL
% 0.27/0.47 => ( setukpairIR
% 0.27/0.47 => ( kpairiskpair
% 0.27/0.47 => ( kpairp
% 0.27/0.47 => ( singletonsubset
% 0.27/0.47 => ( singletoninpowerset
% 0.27/0.47 => ( singletoninpowunion
% 0.27/0.47 => ( upairset2E
% 0.27/0.47 => ( upairsubunion
% 0.27/0.47 => ( upairinpowunion
% 0.27/0.47 => ( ubforcartprodlem1
% 0.27/0.47 => ( ubforcartprodlem2
% 0.27/0.47 => ( ubforcartprodlem3
% 0.27/0.47 => ( cartprodpairin
% 0.27/0.47 => ( cartprodmempair1
% 0.27/0.47 => ( cartprodmempair
% 0.27/0.47 => ( setunionE2
% 0.27/0.47 => ( setunionsingleton1
% 0.27/0.47 => ( setunionsingleton2
% 0.27/0.47 => ( setunionsingleton
% 0.27/0.47 => ( singletonprop
% 0.27/0.47 => ( ex1E1
% 0.27/0.47 => ( ex1I
% 0.27/0.47 => ( ex1I2
% 0.27/0.47 => ( singletonsuniq
% 0.27/0.47 => ( setukpairinjL1
% 0.27/0.47 => ( kfstsingleton
% 0.27/0.47 => ( theprop
% 0.27/0.47 => ( kfstpairEq
% 0.27/0.47 => ( cartprodfstin
% 0.27/0.47 => ( setukpairinjL2
% 0.27/0.47 => ( setukpairinjL
% 0.27/0.47 => ( setukpairinjR11
% 0.27/0.47 => ( setukpairinjR12
% 0.27/0.47 => ( setukpairinjR1
% 0.27/0.47 => ( upairequniteq
% 0.27/0.47 => ( setukpairinjR2
% 0.27/0.47 => ( setukpairinjR
% 0.27/0.47 => ( ksndsingleton
% 0.27/0.47 => ( ksndpairEq
% 0.27/0.47 => ( kpairsurjEq
% 0.27/0.47 => ( cartprodsndin
% 0.27/0.47 => ( cartprodpairmemEL
% 0.27/0.47 => ( cartprodpairmemER
% 0.27/0.47 => ( cartprodmempaircEq
% 0.27/0.47 => ( cartprodfstpairEq
% 0.27/0.47 => ( cartprodsndpairEq
% 0.27/0.47 => ( cartprodpairsurjEq
% 0.27/0.47 => ( dpsetconstrI
% 0.27/0.47 => ( dpsetconstrSub
% 0.27/0.47 => ( setOfPairsIsBReln
% 0.27/0.47 => ( dpsetconstrERa
% 0.27/0.47 => ( dpsetconstrEL1
% 0.27/0.47 => ( dpsetconstrEL2
% 0.27/0.47 => ( dpsetconstrER
% 0.27/0.47 => ( funcImageSingleton
% 0.27/0.47 => ( apProp
% 0.27/0.47 => ( app
% 0.27/0.47 => ( infuncsetfunc
% 0.27/0.47 => ( ap2p
% 0.27/0.47 => ( funcinfuncset
% 0.27/0.47 => ( lamProp
% 0.27/0.47 => ( lamp
% 0.27/0.47 => ( lam2p
% 0.27/0.47 => ( brelnall1
% 0.27/0.47 => ( brelnall2
% 0.27/0.47 => ( ex1E2
% 0.27/0.47 => ( funcGraphProp1
% 0.27/0.47 => ( funcGraphProp3
% 0.27/0.47 => ( funcGraphProp2
% 0.27/0.47 => ( funcextLem
% 0.27/0.47 => ( funcGraphProp4
% 0.27/0.47 => ( subbreln
% 0.27/0.47 => ( eqbreln
% 0.27/0.47 => ( funcext
% 0.27/0.47 => ( funcext2
% 0.27/0.47 => ( ap2apEq1
% 0.27/0.47 => ( ap2apEq2
% 0.27/0.47 => ( beta1
% 0.27/0.47 => ( eta1
% 0.27/0.47 => ! [A: $i,B: $i,Xf: $i > $i] :
% 0.27/0.47 ( ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.27/0.47 => ( ( lam @ A @ B
% 0.27/0.47 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.27/0.47 = ( lam @ A @ B
% 0.27/0.47 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 %------------------------------------------------------------------------------
% 0.27/0.47 ------------------------
% 0.31/1.40 ---- Embedded in THF ---
% 0.31/1.40 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 0.31/1.40 %%% Generated on Tue Feb 24 12:32:19 EST 2026
% 0.31/1.40 %%% using '$$dhol' embedding, version 1.3.0.
% 0.31/1.40 %%% Logic specification used:
% 0.31/1.40 %%% thf(spec, logic, $$dhol).
% 0.31/1.40
% 0.31/1.40 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.EApmVaGZ7F/DTF2DTF24917.p
% See solution above
% 0.31/1.40 ------------------------
% 0.31/1.44 ---- Cleaned THF ---
% 0.31/1.44 thf(in_type,type,
% 0.31/1.44 in: $i > $i > $o ).
% 0.31/1.44
% 0.31/1.44 thf(exu_type,type,
% 0.31/1.44 exu: ( $i > $o ) > $o ).
% 0.31/1.44
% 0.31/1.44 thf(exu,definition,
% 0.31/1.44 ( exu
% 0.31/1.44 = ( ^ [Xphi: $i > $o] :
% 0.31/1.44 ? [Xx: $i] :
% 0.31/1.44 ( ( Xphi @ Xx )
% 0.31/1.44 & ! [Xy: $i] :
% 0.31/1.44 ( ( Xphi @ Xy )
% 0.31/1.44 => ( Xx = Xy ) ) ) ) ) ).
% 0.31/1.44
% 0.31/1.44 thf(setextAx_type,type,
% 0.31/1.44 setextAx: $o ).
% 0.31/1.44
% 0.31/1.44 thf(setextAx,definition,
% 0.31/1.44 ( setextAx
% 0.31/1.44 = ( ! [A: $i,B: $i] :
% 0.31/1.44 ( ! [Xx: $i] :
% 0.31/1.44 ( ( in @ Xx @ A )
% 0.31/1.44 <=> ( in @ Xx @ B ) )
% 0.31/1.44 => ( A = B ) ) ) ) ).
% 0.31/1.44
% 0.31/1.44 thf(emptyset_type,type,
% 0.31/1.44 emptyset: $i ).
% 0.31/1.44
% 0.31/1.44 thf(emptysetAx_type,type,
% 0.31/1.44 emptysetAx: $o ).
% 0.31/1.44
% 0.31/1.44 thf(emptysetAx,definition,
% 0.31/1.44 ( emptysetAx
% 0.31/1.44 = ( ! [Xx: $i] :
% 0.31/1.44 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.31/1.44
% 0.31/1.44 thf(setadjoin_type,type,
% 0.31/1.44 setadjoin: $i > $i > $i ).
% 0.31/1.44
% 0.31/1.44 thf(setadjoinAx_type,type,
% 0.31/1.44 setadjoinAx: $o ).
% 0.31/1.44
% 0.31/1.44 thf(setadjoinAx,definition,
% 0.31/1.44 ( setadjoinAx
% 0.31/1.44 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.31/1.44 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.31/1.44 <=> ( ( Xy = Xx )
% 0.31/1.44 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.31/1.44
% 0.31/1.44 thf(powerset_type,type,
% 0.31/1.44 powerset: $i > $i ).
% 0.31/1.44
% 0.31/1.44 thf(powersetAx_type,type,
% 0.31/1.44 powersetAx: $o ).
% 0.31/1.44
% 0.31/1.44 thf(powersetAx,definition,
% 0.31/1.44 ( powersetAx
% 0.31/1.44 = ( ! [A: $i,B: $i] :
% 0.31/1.44 ( ( in @ B @ ( powerset @ A ) )
% 0.31/1.44 <=> ! [Xx: $i] :
% 0.31/1.44 ( ( in @ Xx @ B )
% 0.31/1.44 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.31/1.44
% 0.31/1.44 thf(setunion_type,type,
% 0.31/1.44 setunion: $i > $i ).
% 0.31/1.44
% 0.31/1.44 thf(setunionAx_type,type,
% 0.31/1.44 setunionAx: $o ).
% 0.31/1.44
% 0.31/1.44 thf(setunionAx,definition,
% 0.31/1.44 ( setunionAx
% 0.31/1.44 = ( ! [A: $i,Xx: $i] :
% 0.31/1.44 ( ( in @ Xx @ ( setunion @ A ) )
% 0.31/1.44 <=> ? [B: $i] :
% 0.31/1.44 ( ( in @ Xx @ B )
% 0.31/1.44 & ( in @ B @ A ) ) ) ) ) ).
% 0.31/1.44
% 0.31/1.44 thf(omega_type,type,
% 0.31/1.44 omega: $i ).
% 0.31/1.44
% 0.31/1.44 thf(omega0Ax_type,type,
% 0.31/1.44 omega0Ax: $o ).
% 0.31/1.44
% 0.31/1.44 thf(omega0Ax,definition,
% 0.31/1.44 ( omega0Ax
% 0.31/1.44 = ( in @ emptyset @ omega ) ) ).
% 0.31/1.44
% 0.31/1.44 thf(omegaSAx_type,type,
% 0.31/1.44 omegaSAx: $o ).
% 0.31/1.44
% 0.31/1.44 thf(omegaSAx,definition,
% 0.31/1.44 ( omegaSAx
% 0.31/1.44 = ( ! [Xx: $i] :
% 0.31/1.44 ( ( in @ Xx @ omega )
% 0.31/1.44 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.31/1.44
% 0.31/1.44 thf(omegaIndAx_type,type,
% 0.31/1.44 omegaIndAx: $o ).
% 0.31/1.44
% 0.31/1.44 thf(omegaIndAx,definition,
% 0.31/1.44 ( omegaIndAx
% 0.31/1.44 = ( ! [A: $i] :
% 0.31/1.44 ( ( ( in @ emptyset @ A )
% 0.31/1.44 & ! [Xx: $i] :
% 0.31/1.44 ( ( ( in @ Xx @ omega )
% 0.31/1.44 & ( in @ Xx @ A ) )
% 0.31/1.44 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.31/1.44 => ! [Xx: $i] :
% 0.31/1.44 ( ( in @ Xx @ omega )
% 0.31/1.44 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.31/1.44
% 0.31/1.44 thf(replAx_type,type,
% 0.31/1.44 replAx: $o ).
% 0.31/1.44
% 0.31/1.44 thf(replAx,definition,
% 0.31/1.44 ( replAx
% 0.31/1.44 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.31/1.44 ( ! [Xx: $i] :
% 0.31/1.44 ( ( in @ Xx @ A )
% 0.31/1.44 => ( exu
% 0.31/1.44 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.31/1.44 => ? [B: $i] :
% 0.31/1.44 ! [Xx: $i] :
% 0.31/1.44 ( ( in @ Xx @ B )
% 0.31/1.44 <=> ? [Xy: $i] :
% 0.31/1.44 ( ( in @ Xy @ A )
% 0.31/1.44 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.31/1.44
% 0.31/1.44 thf(foundationAx_type,type,
% 0.31/1.44 foundationAx: $o ).
% 0.31/1.44
% 0.31/1.44 thf(foundationAx,definition,
% 0.31/1.44 ( foundationAx
% 0.31/1.44 = ( ! [A: $i] :
% 0.31/1.44 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.31/1.44 => ? [B: $i] :
% 0.31/1.44 ( ( in @ B @ A )
% 0.31/1.44 & ~ ? [Xx: $i] :
% 0.31/1.44 ( ( in @ Xx @ B )
% 0.31/1.44 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.31/1.44
% 0.31/1.44 thf(wellorderingAx_type,type,
% 0.31/1.44 wellorderingAx: $o ).
% 0.31/1.44
% 0.31/1.44 thf(wellorderingAx,definition,
% 0.31/1.44 ( wellorderingAx
% 0.31/1.44 = ( ! [A: $i] :
% 0.31/1.44 ? [B: $i] :
% 0.31/1.44 ( ! [C: $i] :
% 0.31/1.44 ( ( in @ C @ B )
% 0.31/1.44 => ! [Xx: $i] :
% 0.31/1.44 ( ( in @ Xx @ C )
% 0.31/1.44 => ( in @ Xx @ A ) ) )
% 0.31/1.44 & ! [Xx: $i,Xy: $i] :
% 0.31/1.44 ( ( ( in @ Xx @ A )
% 0.31/1.44 & ( in @ Xy @ A ) )
% 0.31/1.44 => ( ! [C: $i] :
% 0.31/1.44 ( ( in @ C @ B )
% 0.31/1.44 => ( ( in @ Xx @ C )
% 0.31/1.44 <=> ( in @ Xy @ C ) ) )
% 0.31/1.44 => ( Xx = Xy ) ) )
% 0.31/1.44 & ! [C: $i,D: $i] :
% 0.31/1.44 ( ( ( in @ C @ B )
% 0.31/1.44 & ( in @ D @ B ) )
% 0.31/1.44 => ( ! [Xx: $i] :
% 0.31/1.44 ( ( in @ Xx @ C )
% 0.31/1.44 => ( in @ Xx @ D ) )
% 0.31/1.44 | ! [Xx: $i] :
% 0.31/1.44 ( ( in @ Xx @ D )
% 0.31/1.44 => ( in @ Xx @ C ) ) ) )
% 0.31/1.44 & ! [C: $i] :
% 0.31/1.44 ( ( ! [Xx: $i] :
% 0.31/1.44 ( ( in @ Xx @ C )
% 0.31/1.44 => ( in @ Xx @ A ) )
% 0.31/1.44 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.31/1.44 => ? [D: $i,Xx: $i] :
% 0.31/1.44 ( ( in @ D @ B )
% 0.31/1.44 & ( in @ Xx @ C )
% 0.31/1.44 & ~ ? [Xy: $i] :
% 0.31/1.44 ( ( in @ Xy @ D )
% 0.31/1.44 & ( in @ Xy @ C ) )
% 0.31/1.44 & ! [E: $i] :
% 0.31/1.44 ( ( in @ E @ B )
% 0.31/1.44 => ( ! [Xy: $i] :
% 0.31/1.44 ( ( in @ Xy @ E )
% 0.31/1.44 => ( in @ Xy @ D ) )
% 0.31/1.44 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.31/1.44
% 0.31/1.44 thf(descr_type,type,
% 0.31/1.44 descr: ( $i > $o ) > $i ).
% 0.31/1.44
% 0.31/1.44 thf(descrp_type,type,
% 0.31/1.44 descrp: $o ).
% 0.31/1.44
% 0.31/1.44 thf(descrp,definition,
% 0.31/1.44 ( descrp
% 0.31/1.44 = ( ! [Xphi: $i > $o] :
% 0.31/1.44 ( ( exu
% 0.31/1.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/1.44 => ( Xphi
% 0.31/1.44 @ ( descr
% 0.31/1.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.31/1.44
% 0.31/1.44 thf(dsetconstr_type,type,
% 0.31/1.44 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.31/1.44
% 0.31/1.44 thf(dsetconstrI_type,type,
% 0.31/1.44 dsetconstrI: $o ).
% 0.31/1.44
% 0.31/1.44 thf(dsetconstrI,definition,
% 0.31/1.44 ( dsetconstrI
% 0.31/1.44 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.31/1.44 ( ( in @ Xx @ A )
% 0.31/1.44 => ( ( Xphi @ Xx )
% 0.31/1.44 => ( in @ Xx
% 0.31/1.44 @ ( dsetconstr @ A
% 0.31/1.44 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.31/1.44
% 0.31/1.44 thf(dsetconstrEL_type,type,
% 0.31/1.44 dsetconstrEL: $o ).
% 0.31/1.44
% 0.31/1.44 thf(dsetconstrEL,definition,
% 0.31/1.44 ( dsetconstrEL
% 0.31/1.44 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.31/1.44 ( ( in @ Xx
% 0.31/1.44 @ ( dsetconstr @ A
% 0.31/1.44 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.31/1.44 => ( in @ Xx @ A ) ) ) ) ).
% 0.31/1.44
% 0.31/1.44 thf(dsetconstrER_type,type,
% 0.31/1.44 dsetconstrER: $o ).
% 0.31/1.44
% 0.31/1.44 thf(dsetconstrER,definition,
% 0.31/1.44 ( dsetconstrER
% 0.31/1.44 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.31/1.44 ( ( in @ Xx
% 0.31/1.44 @ ( dsetconstr @ A
% 0.31/1.44 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.31/1.44 => ( Xphi @ Xx ) ) ) ) ).
% 0.31/1.44
% 0.31/1.44 thf(exuE1_type,type,
% 0.31/1.44 exuE1: $o ).
% 0.31/1.44
% 0.31/1.44 thf(exuE1,definition,
% 0.31/1.44 ( exuE1
% 0.31/1.44 = ( ! [Xphi: $i > $o] :
% 0.31/1.44 ( ( exu
% 0.31/1.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/1.44 => ? [Xx: $i] :
% 0.31/1.44 ( ( Xphi @ Xx )
% 0.31/1.44 & ! [Xy: $i] :
% 0.31/1.44 ( ( Xphi @ Xy )
% 0.31/1.44 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.31/1.44
% 0.31/1.44 thf(prop2set_type,type,
% 0.31/1.44 prop2set: $o > $i ).
% 0.31/1.44
% 0.31/1.44 thf(prop2setE_type,type,
% 0.31/1.44 prop2setE: $o ).
% 0.31/1.44
% 0.31/1.44 thf(prop2setE,definition,
% 0.31/1.44 ( prop2setE
% 0.31/1.44 = ( ! [Xphi: $o,Xx: $i] :
% 0.31/1.44 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.31/1.44 => Xphi ) ) ) ).
% 0.31/1.44
% 0.31/1.44 thf(emptysetE_type,type,
% 0.31/1.44 emptysetE: $o ).
% 0.31/1.44
% 0.31/1.44 thf(emptysetE,definition,
% 0.31/1.44 ( emptysetE
% 0.31/1.44 = ( ! [Xx: $i] :
% 0.31/1.44 ( ( in @ Xx @ emptyset )
% 0.31/1.44 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.31/1.44
% 0.31/1.44 thf(emptysetimpfalse_type,type,
% 0.31/1.44 emptysetimpfalse: $o ).
% 0.31/1.44
% 0.31/1.44 thf(emptysetimpfalse,definition,
% 0.31/1.44 ( emptysetimpfalse
% 0.31/1.44 = ( ! [Xx: $i] :
% 0.31/1.44 ( ( in @ Xx @ emptyset )
% 0.31/1.44 => $false ) ) ) ).
% 0.31/1.44
% 0.31/1.44 thf(notinemptyset_type,type,
% 0.31/1.44 notinemptyset: $o ).
% 0.31/1.44
% 0.31/1.44 thf(notinemptyset,definition,
% 0.31/1.44 ( notinemptyset
% 0.31/1.44 = ( ! [Xx: $i] :
% 0.31/1.44 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.31/1.44
% 0.31/1.44 thf(exuE3e_type,type,
% 0.31/1.44 exuE3e: $o ).
% 0.31/1.44
% 0.31/1.44 thf(exuE3e,definition,
% 0.31/1.44 ( exuE3e
% 0.31/1.44 = ( ! [Xphi: $i > $o] :
% 0.31/1.44 ( ( exu
% 0.31/1.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/1.44 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.31/1.44
% 0.31/1.44 thf(setext_type,type,
% 0.31/1.44 setext: $o ).
% 0.31/1.44
% 0.31/1.44 thf(setext,definition,
% 0.31/1.44 ( setext
% 0.31/1.44 = ( ! [A: $i,B: $i] :
% 0.31/1.44 ( ! [Xx: $i] :
% 0.31/1.44 ( ( in @ Xx @ A )
% 0.31/1.44 => ( in @ Xx @ B ) )
% 0.31/1.44 => ( ! [Xx: $i] :
% 0.31/1.44 ( ( in @ Xx @ B )
% 0.31/1.44 => ( in @ Xx @ A ) )
% 0.31/1.44 => ( A = B ) ) ) ) ) ).
% 0.31/1.44
% 0.31/1.44 thf(emptyI_type,type,
% 0.31/1.44 emptyI: $o ).
% 0.31/1.44
% 0.31/1.44 thf(emptyI,definition,
% 0.31/1.44 ( emptyI
% 0.31/1.44 = ( ! [A: $i] :
% 0.31/1.44 ( ! [Xx: $i] :
% 0.31/1.44 ~ ( in @ Xx @ A )
% 0.31/1.44 => ( A = emptyset ) ) ) ) ).
% 0.31/1.44
% 0.31/1.44 thf(noeltsimpempty_type,type,
% 0.31/1.44 noeltsimpempty: $o ).
% 0.31/1.44
% 0.31/1.44 thf(noeltsimpempty,definition,
% 0.31/1.44 ( noeltsimpempty
% 0.31/1.44 = ( ! [A: $i] :
% 0.31/1.44 ( ! [Xx: $i] :
% 0.31/1.44 ~ ( in @ Xx @ A )
% 0.31/1.44 => ( A = emptyset ) ) ) ) ).
% 0.31/1.44
% 0.31/1.44 thf(setbeta_type,type,
% 0.31/1.44 setbeta: $o ).
% 0.31/1.44
% 0.31/1.44 thf(setbeta,definition,
% 0.31/1.44 ( setbeta
% 0.31/1.44 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.31/1.44 ( ( in @ Xx @ A )
% 0.31/1.44 => ( ( in @ Xx
% 0.31/1.44 @ ( dsetconstr @ A
% 0.31/1.44 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.31/1.44 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/1.44
% 0.31/1.44 thf(nonempty_type,type,
% 0.31/1.44 nonempty: $i > $o ).
% 0.31/1.44
% 0.31/1.44 thf(nonempty,definition,
% 0.31/1.44 ( nonempty
% 0.31/1.44 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.31/1.44
% 0.31/1.44 thf(nonemptyE1_type,type,
% 0.31/1.44 nonemptyE1: $o ).
% 0.31/1.44
% 0.31/1.44 thf(nonemptyE1,definition,
% 0.31/1.44 ( nonemptyE1
% 0.31/1.44 = ( ! [A: $i] :
% 0.31/1.44 ( ( nonempty @ A )
% 0.31/1.44 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.31/1.44
% 0.31/1.44 thf(nonemptyI_type,type,
% 0.31/1.44 nonemptyI: $o ).
% 0.31/1.44
% 0.31/1.44 thf(nonemptyI,definition,
% 0.31/1.44 ( nonemptyI
% 0.31/1.44 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.31/1.44 ( ( in @ Xx @ A )
% 0.31/1.44 => ( ( Xphi @ Xx )
% 0.31/1.44 => ( nonempty
% 0.31/1.44 @ ( dsetconstr @ A
% 0.31/1.44 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.31/1.44
% 0.31/1.44 thf(nonemptyI1_type,type,
% 0.31/1.44 nonemptyI1: $o ).
% 0.31/1.44
% 0.31/1.44 thf(nonemptyI1,definition,
% 0.31/1.44 ( nonemptyI1
% 0.31/1.44 = ( ! [A: $i] :
% 0.31/1.44 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.31/1.44 => ( nonempty @ A ) ) ) ) ).
% 0.31/1.44
% 0.31/1.44 thf(setadjoinIL_type,type,
% 0.31/1.44 setadjoinIL: $o ).
% 0.31/1.44
% 0.31/1.44 thf(setadjoinIL,definition,
% 0.31/1.44 ( setadjoinIL
% 0.31/1.44 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.31/1.44
% 0.31/1.44 thf(emptyinunitempty_type,type,
% 0.31/1.44 emptyinunitempty: $o ).
% 0.31/1.44
% 0.31/1.44 thf(emptyinunitempty,definition,
% 0.31/1.44 ( emptyinunitempty
% 0.31/1.44 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.31/1.44
% 0.31/1.44 thf(setadjoinIR_type,type,
% 0.31/1.44 setadjoinIR: $o ).
% 0.31/1.44
% 0.31/1.44 thf(setadjoinIR,definition,
% 0.31/1.44 ( setadjoinIR
% 0.31/1.44 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.31/1.44 ( ( in @ Xy @ A )
% 0.31/1.44 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.31/1.44
% 0.31/1.44 thf(setadjoinE_type,type,
% 0.31/1.44 setadjoinE: $o ).
% 0.31/1.44
% 0.31/1.44 thf(setadjoinE,definition,
% 0.31/1.44 ( setadjoinE
% 0.31/1.44 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.31/1.44 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.31/1.44 => ! [Xphi: $o] :
% 0.31/1.44 ( ( ( Xy = Xx )
% 0.31/1.44 => Xphi )
% 0.31/1.44 => ( ( ( in @ Xy @ A )
% 0.31/1.44 => Xphi )
% 0.31/1.44 => Xphi ) ) ) ) ) ).
% 0.31/1.44
% 0.31/1.44 thf(setadjoinOr_type,type,
% 0.31/1.44 setadjoinOr: $o ).
% 0.31/1.44
% 0.31/1.44 thf(setadjoinOr,definition,
% 0.31/1.44 ( setadjoinOr
% 0.31/1.44 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.31/1.44 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.31/1.44 => ( ( Xy = Xx )
% 0.31/1.44 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.31/1.44
% 0.31/1.44 thf(setoftrueEq_type,type,
% 0.31/1.44 setoftrueEq: $o ).
% 0.31/1.44
% 0.31/1.44 thf(setoftrueEq,definition,
% 0.31/1.44 ( setoftrueEq
% 0.31/1.44 = ( ! [A: $i] :
% 0.31/1.44 ( ( dsetconstr @ A
% 0.31/1.44 @ ^ [Xx: $i] : $true )
% 0.31/1.44 = A ) ) ) ).
% 0.31/1.44
% 0.31/1.44 thf(powersetI_type,type,
% 0.31/1.44 powersetI: $o ).
% 0.31/1.44
% 0.31/1.44 thf(powersetI,definition,
% 0.31/1.44 ( powersetI
% 0.31/1.44 = ( ! [A: $i,B: $i] :
% 0.31/1.44 ( ! [Xx: $i] :
% 0.31/1.44 ( ( in @ Xx @ B )
% 0.31/1.44 => ( in @ Xx @ A ) )
% 0.31/1.44 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.31/1.44
% 0.31/1.44 thf(emptyinPowerset_type,type,
% 0.31/1.44 emptyinPowerset: $o ).
% 0.31/1.44
% 0.31/1.44 thf(emptyinPowerset,definition,
% 0.31/1.44 ( emptyinPowerset
% 0.31/1.44 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.31/1.44
% 0.31/1.44 thf(emptyInPowerset_type,type,
% 0.31/1.44 emptyInPowerset: $o ).
% 0.31/1.44
% 0.31/1.44 thf(emptyInPowerset,definition,
% 0.31/1.44 ( emptyInPowerset
% 0.31/1.44 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(powersetE_type,type,
% 0.31/1.45 powersetE: $o ).
% 0.31/1.45
% 0.31/1.45 thf(powersetE,definition,
% 0.31/1.45 ( powersetE
% 0.31/1.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/1.45 ( ( in @ B @ ( powerset @ A ) )
% 0.31/1.45 => ( ( in @ Xx @ B )
% 0.31/1.45 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(setunionI_type,type,
% 0.31/1.45 setunionI: $o ).
% 0.31/1.45
% 0.31/1.45 thf(setunionI,definition,
% 0.31/1.45 ( setunionI
% 0.31/1.45 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.31/1.45 ( ( in @ Xx @ B )
% 0.31/1.45 => ( ( in @ B @ A )
% 0.31/1.45 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(setunionE_type,type,
% 0.31/1.45 setunionE: $o ).
% 0.31/1.45
% 0.31/1.45 thf(setunionE,definition,
% 0.31/1.45 ( setunionE
% 0.31/1.45 = ( ! [A: $i,Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ ( setunion @ A ) )
% 0.31/1.45 => ! [Xphi: $o] :
% 0.31/1.45 ( ! [B: $i] :
% 0.31/1.45 ( ( in @ Xx @ B )
% 0.31/1.45 => ( ( in @ B @ A )
% 0.31/1.45 => Xphi ) )
% 0.31/1.45 => Xphi ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(subPowSU_type,type,
% 0.31/1.45 subPowSU: $o ).
% 0.31/1.45
% 0.31/1.45 thf(subPowSU,definition,
% 0.31/1.45 ( subPowSU
% 0.31/1.45 = ( ! [A: $i,Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ A )
% 0.31/1.45 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(exuE2_type,type,
% 0.31/1.45 exuE2: $o ).
% 0.31/1.45
% 0.31/1.45 thf(exuE2,definition,
% 0.31/1.45 ( exuE2
% 0.31/1.45 = ( ! [Xphi: $i > $o] :
% 0.31/1.45 ( ( exu
% 0.31/1.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/1.45 => ? [Xx: $i] :
% 0.31/1.45 ! [Xy: $i] :
% 0.31/1.45 ( ( Xphi @ Xy )
% 0.31/1.45 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(nonemptyImpWitness_type,type,
% 0.31/1.45 nonemptyImpWitness: $o ).
% 0.31/1.45
% 0.31/1.45 thf(nonemptyImpWitness,definition,
% 0.31/1.45 ( nonemptyImpWitness
% 0.31/1.45 = ( ! [A: $i] :
% 0.31/1.45 ( ( nonempty @ A )
% 0.31/1.45 => ? [Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ A )
% 0.31/1.45 & $true ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(uniqinunit_type,type,
% 0.31/1.45 uniqinunit: $o ).
% 0.31/1.45
% 0.31/1.45 thf(uniqinunit,definition,
% 0.31/1.45 ( uniqinunit
% 0.31/1.45 = ( ! [Xx: $i,Xy: $i] :
% 0.31/1.45 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.31/1.45 => ( Xx = Xy ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(notinsingleton_type,type,
% 0.31/1.45 notinsingleton: $o ).
% 0.31/1.45
% 0.31/1.45 thf(notinsingleton,definition,
% 0.31/1.45 ( notinsingleton
% 0.31/1.45 = ( ! [Xx: $i,Xy: $i] :
% 0.31/1.45 ( ( Xx != Xy )
% 0.31/1.45 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(eqinunit_type,type,
% 0.31/1.45 eqinunit: $o ).
% 0.31/1.45
% 0.31/1.45 thf(eqinunit,definition,
% 0.31/1.45 ( eqinunit
% 0.31/1.45 = ( ! [Xx: $i,Xy: $i] :
% 0.31/1.45 ( ( Xx = Xy )
% 0.31/1.45 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(singletonsswitch_type,type,
% 0.31/1.45 singletonsswitch: $o ).
% 0.31/1.45
% 0.31/1.45 thf(singletonsswitch,definition,
% 0.31/1.45 ( singletonsswitch
% 0.31/1.45 = ( ! [Xx: $i,Xy: $i] :
% 0.31/1.45 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.31/1.45 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(upairsetE_type,type,
% 0.31/1.45 upairsetE: $o ).
% 0.31/1.45
% 0.31/1.45 thf(upairsetE,definition,
% 0.31/1.45 ( upairsetE
% 0.31/1.45 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.31/1.45 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.31/1.45 => ( ( Xz = Xx )
% 0.31/1.45 | ( Xz = Xy ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(upairsetIL_type,type,
% 0.31/1.45 upairsetIL: $o ).
% 0.31/1.45
% 0.31/1.45 thf(upairsetIL,definition,
% 0.31/1.45 ( upairsetIL
% 0.31/1.45 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(upairsetIR_type,type,
% 0.31/1.45 upairsetIR: $o ).
% 0.31/1.45
% 0.31/1.45 thf(upairsetIR,definition,
% 0.31/1.45 ( upairsetIR
% 0.31/1.45 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(emptyE1_type,type,
% 0.31/1.45 emptyE1: $o ).
% 0.31/1.45
% 0.31/1.45 thf(emptyE1,definition,
% 0.31/1.45 ( emptyE1
% 0.31/1.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/1.45 ( ? [Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ A )
% 0.31/1.45 & ( Xphi @ Xx ) )
% 0.31/1.45 => ( ( ( dsetconstr @ A
% 0.31/1.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/1.45 = emptyset )
% 0.31/1.45 => $false ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(vacuousDall_type,type,
% 0.31/1.45 vacuousDall: $o ).
% 0.31/1.45
% 0.31/1.45 thf(vacuousDall,definition,
% 0.31/1.45 ( vacuousDall
% 0.31/1.45 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ emptyset )
% 0.31/1.45 => ( Xphi @ Xx ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(quantDeMorgan1_type,type,
% 0.31/1.45 quantDeMorgan1: $o ).
% 0.31/1.45
% 0.31/1.45 thf(quantDeMorgan1,definition,
% 0.31/1.45 ( quantDeMorgan1
% 0.31/1.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/1.45 ( ~ ! [Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ A )
% 0.31/1.45 => ( Xphi @ Xx ) )
% 0.31/1.45 => ? [Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ A )
% 0.31/1.45 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(quantDeMorgan2_type,type,
% 0.31/1.45 quantDeMorgan2: $o ).
% 0.31/1.45
% 0.31/1.45 thf(quantDeMorgan2,definition,
% 0.31/1.45 ( quantDeMorgan2
% 0.31/1.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/1.45 ( ! [Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ A )
% 0.31/1.45 => ~ ( Xphi @ Xx ) )
% 0.31/1.45 => ~ ? [Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ A )
% 0.31/1.45 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(quantDeMorgan3_type,type,
% 0.31/1.45 quantDeMorgan3: $o ).
% 0.31/1.45
% 0.31/1.45 thf(quantDeMorgan3,definition,
% 0.31/1.45 ( quantDeMorgan3
% 0.31/1.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/1.45 ( ~ ? [Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ A )
% 0.31/1.45 & ( Xphi @ Xx ) )
% 0.31/1.45 => ! [Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ A )
% 0.31/1.45 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(quantDeMorgan4_type,type,
% 0.31/1.45 quantDeMorgan4: $o ).
% 0.31/1.45
% 0.31/1.45 thf(quantDeMorgan4,definition,
% 0.31/1.45 ( quantDeMorgan4
% 0.31/1.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/1.45 ( ? [Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ A )
% 0.31/1.45 & ~ ( Xphi @ Xx ) )
% 0.31/1.45 => ~ ! [Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ A )
% 0.31/1.45 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(prop2setI_type,type,
% 0.31/1.45 prop2setI: $o ).
% 0.31/1.45
% 0.31/1.45 thf(prop2setI,definition,
% 0.31/1.45 ( prop2setI
% 0.31/1.45 = ( ! [Xphi: $o] :
% 0.31/1.45 ( Xphi
% 0.31/1.45 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(set2prop_type,type,
% 0.31/1.45 set2prop: $i > $o ).
% 0.31/1.45
% 0.31/1.45 thf(prop2set2propI_type,type,
% 0.31/1.45 prop2set2propI: $o ).
% 0.31/1.45
% 0.31/1.45 thf(prop2set2propI,definition,
% 0.31/1.45 ( prop2set2propI
% 0.31/1.45 = ( ! [Xphi: $o] :
% 0.31/1.45 ( Xphi
% 0.31/1.45 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(notdexE_type,type,
% 0.31/1.45 notdexE: $o ).
% 0.31/1.45
% 0.31/1.45 thf(notdexE,definition,
% 0.31/1.45 ( notdexE
% 0.31/1.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/1.45 ( ~ ? [Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ A )
% 0.31/1.45 & ( Xphi @ Xx ) )
% 0.31/1.45 => ! [Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ A )
% 0.31/1.45 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(notdallE_type,type,
% 0.31/1.45 notdallE: $o ).
% 0.31/1.45
% 0.31/1.45 thf(notdallE,definition,
% 0.31/1.45 ( notdallE
% 0.31/1.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/1.45 ( ~ ! [Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ A )
% 0.31/1.45 => ( Xphi @ Xx ) )
% 0.31/1.45 => ? [Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ A )
% 0.31/1.45 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(exuI1_type,type,
% 0.31/1.45 exuI1: $o ).
% 0.31/1.45
% 0.31/1.45 thf(exuI1,definition,
% 0.31/1.45 ( exuI1
% 0.31/1.45 = ( ! [Xphi: $i > $o] :
% 0.31/1.45 ( ? [Xx: $i] :
% 0.31/1.45 ( ( Xphi @ Xx )
% 0.31/1.45 & ! [Xy: $i] :
% 0.31/1.45 ( ( Xphi @ Xy )
% 0.31/1.45 => ( Xx = Xy ) ) )
% 0.31/1.45 => ( exu
% 0.31/1.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(exuI3_type,type,
% 0.31/1.45 exuI3: $o ).
% 0.31/1.45
% 0.31/1.45 thf(exuI3,definition,
% 0.31/1.45 ( exuI3
% 0.31/1.45 = ( ! [Xphi: $i > $o] :
% 0.31/1.45 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.31/1.45 => ( ! [Xx: $i,Xy: $i] :
% 0.31/1.45 ( ( Xphi @ Xx )
% 0.31/1.45 => ( ( Xphi @ Xy )
% 0.31/1.45 => ( Xx = Xy ) ) )
% 0.31/1.45 => ( exu
% 0.31/1.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(exuI2_type,type,
% 0.31/1.45 exuI2: $o ).
% 0.31/1.45
% 0.31/1.45 thf(exuI2,definition,
% 0.31/1.45 ( exuI2
% 0.31/1.45 = ( ! [Xphi: $i > $o] :
% 0.31/1.45 ( ? [Xx: $i] :
% 0.31/1.45 ! [Xy: $i] :
% 0.31/1.45 ( ( Xphi @ Xy )
% 0.31/1.45 <=> ( Xy = Xx ) )
% 0.31/1.45 => ( exu
% 0.31/1.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(inCongP_type,type,
% 0.31/1.45 inCongP: $o ).
% 0.31/1.45
% 0.31/1.45 thf(inCongP,definition,
% 0.31/1.45 ( inCongP
% 0.31/1.45 = ( ! [A: $i,B: $i] :
% 0.31/1.45 ( ( A = B )
% 0.31/1.45 => ! [Xx: $i,Xy: $i] :
% 0.31/1.45 ( ( Xx = Xy )
% 0.31/1.45 => ( ( in @ Xx @ A )
% 0.31/1.45 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(in__Cong_type,type,
% 0.31/1.45 in__Cong: $o ).
% 0.31/1.45
% 0.31/1.45 thf(in__Cong,definition,
% 0.31/1.45 ( in__Cong
% 0.31/1.45 = ( ! [A: $i,B: $i] :
% 0.31/1.45 ( ( A = B )
% 0.31/1.45 => ! [Xx: $i,Xy: $i] :
% 0.31/1.45 ( ( Xx = Xy )
% 0.31/1.45 => ( ( in @ Xx @ A )
% 0.31/1.45 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(exuE3u_type,type,
% 0.31/1.45 exuE3u: $o ).
% 0.31/1.45
% 0.31/1.45 thf(exuE3u,definition,
% 0.31/1.45 ( exuE3u
% 0.31/1.45 = ( ! [Xphi: $i > $o] :
% 0.31/1.45 ( ( exu
% 0.31/1.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/1.45 => ! [Xx: $i,Xy: $i] :
% 0.31/1.45 ( ( Xphi @ Xx )
% 0.31/1.45 => ( ( Xphi @ Xy )
% 0.31/1.45 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(exu__Cong_type,type,
% 0.31/1.45 exu__Cong: $o ).
% 0.31/1.45
% 0.31/1.45 thf(exu__Cong,definition,
% 0.31/1.45 ( exu__Cong
% 0.31/1.45 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.31/1.45 ( ! [Xx: $i,Xy: $i] :
% 0.31/1.45 ( ( Xx = Xy )
% 0.31/1.45 => ( ( Xphi @ Xx )
% 0.31/1.45 <=> ( Xpsi @ Xy ) ) )
% 0.31/1.45 => ( ( exu
% 0.31/1.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/1.45 <=> ( exu
% 0.31/1.45 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(emptyset__Cong_type,type,
% 0.31/1.45 emptyset__Cong: $o ).
% 0.31/1.45
% 0.31/1.45 thf(emptyset__Cong,definition,
% 0.31/1.45 ( emptyset__Cong
% 0.31/1.45 = ( emptyset = emptyset ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(setadjoin__Cong_type,type,
% 0.31/1.45 setadjoin__Cong: $o ).
% 0.31/1.45
% 0.31/1.45 thf(setadjoin__Cong,definition,
% 0.31/1.45 ( setadjoin__Cong
% 0.31/1.45 = ( ! [Xx: $i,Xy: $i] :
% 0.31/1.45 ( ( Xx = Xy )
% 0.31/1.45 => ! [Xz: $i,Xu: $i] :
% 0.31/1.45 ( ( Xz = Xu )
% 0.31/1.45 => ( ( setadjoin @ Xx @ Xz )
% 0.31/1.45 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(powerset__Cong_type,type,
% 0.31/1.45 powerset__Cong: $o ).
% 0.31/1.45
% 0.31/1.45 thf(powerset__Cong,definition,
% 0.31/1.45 ( powerset__Cong
% 0.31/1.45 = ( ! [A: $i,B: $i] :
% 0.31/1.45 ( ( A = B )
% 0.31/1.45 => ( ( powerset @ A )
% 0.31/1.45 = ( powerset @ B ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(setunion__Cong_type,type,
% 0.31/1.45 setunion__Cong: $o ).
% 0.31/1.45
% 0.31/1.45 thf(setunion__Cong,definition,
% 0.31/1.45 ( setunion__Cong
% 0.31/1.45 = ( ! [A: $i,B: $i] :
% 0.31/1.45 ( ( A = B )
% 0.31/1.45 => ( ( setunion @ A )
% 0.31/1.45 = ( setunion @ B ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(omega__Cong_type,type,
% 0.31/1.45 omega__Cong: $o ).
% 0.31/1.45
% 0.31/1.45 thf(omega__Cong,definition,
% 0.31/1.45 ( omega__Cong
% 0.31/1.45 = ( omega = omega ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(exuEu_type,type,
% 0.31/1.45 exuEu: $o ).
% 0.31/1.45
% 0.31/1.45 thf(exuEu,definition,
% 0.31/1.45 ( exuEu
% 0.31/1.45 = ( ! [Xphi: $i > $o] :
% 0.31/1.45 ( ( exu
% 0.31/1.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/1.45 => ! [Xx: $i,Xy: $i] :
% 0.31/1.45 ( ( Xphi @ Xx )
% 0.31/1.45 => ( ( Xphi @ Xy )
% 0.31/1.45 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(descr__Cong_type,type,
% 0.31/1.45 descr__Cong: $o ).
% 0.31/1.45
% 0.31/1.45 thf(descr__Cong,definition,
% 0.31/1.45 ( descr__Cong
% 0.31/1.45 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.31/1.45 ( ! [Xx: $i,Xy: $i] :
% 0.31/1.45 ( ( Xx = Xy )
% 0.31/1.45 => ( ( Xphi @ Xx )
% 0.31/1.45 <=> ( Xpsi @ Xy ) ) )
% 0.31/1.45 => ( ( exu
% 0.31/1.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/1.45 => ( ( exu
% 0.31/1.45 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.31/1.45 => ( ( descr
% 0.31/1.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/1.45 = ( descr
% 0.31/1.45 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(dsetconstr__Cong_type,type,
% 0.31/1.45 dsetconstr__Cong: $o ).
% 0.31/1.45
% 0.31/1.45 thf(dsetconstr__Cong,definition,
% 0.31/1.45 ( dsetconstr__Cong
% 0.31/1.45 = ( ! [A: $i,B: $i] :
% 0.31/1.45 ( ( A = B )
% 0.31/1.45 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.31/1.45 ( ! [Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ A )
% 0.31/1.45 => ! [Xy: $i] :
% 0.31/1.45 ( ( in @ Xy @ B )
% 0.31/1.45 => ( ( Xx = Xy )
% 0.31/1.45 => ( ( Xphi @ Xx )
% 0.31/1.45 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.31/1.45 => ( ( dsetconstr @ A
% 0.31/1.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/1.45 = ( dsetconstr @ B
% 0.31/1.45 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(subset_type,type,
% 0.31/1.45 subset: $i > $i > $o ).
% 0.31/1.45
% 0.31/1.45 thf(disjoint_type,type,
% 0.31/1.45 disjoint: $i > $i > $o ).
% 0.31/1.45
% 0.31/1.45 thf(setsmeet_type,type,
% 0.31/1.45 setsmeet: $i > $i > $o ).
% 0.31/1.45
% 0.31/1.45 thf(subsetI1_type,type,
% 0.31/1.45 subsetI1: $o ).
% 0.31/1.45
% 0.31/1.45 thf(subsetI1,definition,
% 0.31/1.45 ( subsetI1
% 0.31/1.45 = ( ! [A: $i,B: $i] :
% 0.31/1.45 ( ! [Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ A )
% 0.31/1.45 => ( in @ Xx @ B ) )
% 0.31/1.45 => ( subset @ A @ B ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(eqimpsubset2_type,type,
% 0.31/1.45 eqimpsubset2: $o ).
% 0.31/1.45
% 0.31/1.45 thf(eqimpsubset2,definition,
% 0.31/1.45 ( eqimpsubset2
% 0.31/1.45 = ( ! [A: $i,B: $i] :
% 0.31/1.45 ( ( A = B )
% 0.31/1.45 => ( subset @ B @ A ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(eqimpsubset1_type,type,
% 0.31/1.45 eqimpsubset1: $o ).
% 0.31/1.45
% 0.31/1.45 thf(eqimpsubset1,definition,
% 0.31/1.45 ( eqimpsubset1
% 0.31/1.45 = ( ! [A: $i,B: $i] :
% 0.31/1.45 ( ( A = B )
% 0.31/1.45 => ( subset @ A @ B ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(subsetI2_type,type,
% 0.31/1.45 subsetI2: $o ).
% 0.31/1.45
% 0.31/1.45 thf(subsetI2,definition,
% 0.31/1.45 ( subsetI2
% 0.31/1.45 = ( ! [A: $i,B: $i] :
% 0.31/1.45 ( ! [Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ A )
% 0.31/1.45 => ( in @ Xx @ B ) )
% 0.31/1.45 => ( subset @ A @ B ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(emptysetsubset_type,type,
% 0.31/1.45 emptysetsubset: $o ).
% 0.31/1.45
% 0.31/1.45 thf(emptysetsubset,definition,
% 0.31/1.45 ( emptysetsubset
% 0.31/1.45 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(subsetE_type,type,
% 0.31/1.45 subsetE: $o ).
% 0.31/1.45
% 0.31/1.45 thf(subsetE,definition,
% 0.31/1.45 ( subsetE
% 0.31/1.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/1.45 ( ( subset @ A @ B )
% 0.31/1.45 => ( ( in @ Xx @ A )
% 0.31/1.45 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(subsetE2_type,type,
% 0.31/1.45 subsetE2: $o ).
% 0.31/1.45
% 0.31/1.45 thf(subsetE2,definition,
% 0.31/1.45 ( subsetE2
% 0.31/1.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/1.45 ( ( subset @ A @ B )
% 0.31/1.45 => ( ~ ( in @ Xx @ B )
% 0.31/1.45 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(notsubsetI_type,type,
% 0.31/1.45 notsubsetI: $o ).
% 0.31/1.45
% 0.31/1.45 thf(notsubsetI,definition,
% 0.31/1.45 ( notsubsetI
% 0.31/1.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ A )
% 0.31/1.45 => ( ~ ( in @ Xx @ B )
% 0.31/1.45 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(notequalI1_type,type,
% 0.31/1.45 notequalI1: $o ).
% 0.31/1.45
% 0.31/1.45 thf(notequalI1,definition,
% 0.31/1.45 ( notequalI1
% 0.31/1.45 = ( ! [A: $i,B: $i] :
% 0.31/1.45 ( ~ ( subset @ A @ B )
% 0.31/1.45 => ( A != B ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(notequalI2_type,type,
% 0.31/1.45 notequalI2: $o ).
% 0.31/1.45
% 0.31/1.45 thf(notequalI2,definition,
% 0.31/1.45 ( notequalI2
% 0.31/1.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ A )
% 0.31/1.45 => ( ~ ( in @ Xx @ B )
% 0.31/1.45 => ( A != B ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(subsetRefl_type,type,
% 0.31/1.45 subsetRefl: $o ).
% 0.31/1.45
% 0.31/1.45 thf(subsetRefl,definition,
% 0.31/1.45 ( subsetRefl
% 0.31/1.45 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(subsetTrans_type,type,
% 0.31/1.45 subsetTrans: $o ).
% 0.31/1.45
% 0.31/1.45 thf(subsetTrans,definition,
% 0.31/1.45 ( subsetTrans
% 0.31/1.45 = ( ! [A: $i,B: $i,C: $i] :
% 0.31/1.45 ( ( subset @ A @ B )
% 0.31/1.45 => ( ( subset @ B @ C )
% 0.31/1.45 => ( subset @ A @ C ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(setadjoinSub_type,type,
% 0.31/1.45 setadjoinSub: $o ).
% 0.31/1.45
% 0.31/1.45 thf(setadjoinSub,definition,
% 0.31/1.45 ( setadjoinSub
% 0.31/1.45 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(setadjoinSub2_type,type,
% 0.31/1.45 setadjoinSub2: $o ).
% 0.31/1.45
% 0.31/1.45 thf(setadjoinSub2,definition,
% 0.31/1.45 ( setadjoinSub2
% 0.31/1.45 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.31/1.45 ( ( subset @ A @ B )
% 0.31/1.45 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(subset2powerset_type,type,
% 0.31/1.45 subset2powerset: $o ).
% 0.31/1.45
% 0.31/1.45 thf(subset2powerset,definition,
% 0.31/1.45 ( subset2powerset
% 0.31/1.45 = ( ! [A: $i,B: $i] :
% 0.31/1.45 ( ( subset @ A @ B )
% 0.31/1.45 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(setextsub_type,type,
% 0.31/1.45 setextsub: $o ).
% 0.31/1.45
% 0.31/1.45 thf(setextsub,definition,
% 0.31/1.45 ( setextsub
% 0.31/1.45 = ( ! [A: $i,B: $i] :
% 0.31/1.45 ( ( subset @ A @ B )
% 0.31/1.45 => ( ( subset @ B @ A )
% 0.31/1.45 => ( A = B ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(subsetemptysetimpeq_type,type,
% 0.31/1.45 subsetemptysetimpeq: $o ).
% 0.31/1.45
% 0.31/1.45 thf(subsetemptysetimpeq,definition,
% 0.31/1.45 ( subsetemptysetimpeq
% 0.31/1.45 = ( ! [A: $i] :
% 0.31/1.45 ( ( subset @ A @ emptyset )
% 0.31/1.45 => ( A = emptyset ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(powersetI1_type,type,
% 0.31/1.45 powersetI1: $o ).
% 0.31/1.45
% 0.31/1.45 thf(powersetI1,definition,
% 0.31/1.45 ( powersetI1
% 0.31/1.45 = ( ! [A: $i,B: $i] :
% 0.31/1.45 ( ( subset @ B @ A )
% 0.31/1.45 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(powersetE1_type,type,
% 0.31/1.45 powersetE1: $o ).
% 0.31/1.45
% 0.31/1.45 thf(powersetE1,definition,
% 0.31/1.45 ( powersetE1
% 0.31/1.45 = ( ! [A: $i,B: $i] :
% 0.31/1.45 ( ( in @ B @ ( powerset @ A ) )
% 0.31/1.45 => ( subset @ B @ A ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(inPowerset_type,type,
% 0.31/1.45 inPowerset: $o ).
% 0.31/1.45
% 0.31/1.45 thf(inPowerset,definition,
% 0.31/1.45 ( inPowerset
% 0.31/1.45 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(powersetsubset_type,type,
% 0.31/1.45 powersetsubset: $o ).
% 0.31/1.45
% 0.31/1.45 thf(powersetsubset,definition,
% 0.31/1.45 ( powersetsubset
% 0.31/1.45 = ( ! [A: $i,B: $i] :
% 0.31/1.45 ( ( subset @ A @ B )
% 0.31/1.45 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(sepInPowerset_type,type,
% 0.31/1.45 sepInPowerset: $o ).
% 0.31/1.45
% 0.31/1.45 thf(sepInPowerset,definition,
% 0.31/1.45 ( sepInPowerset
% 0.31/1.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/1.45 ( in
% 0.31/1.45 @ ( dsetconstr @ A
% 0.31/1.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/1.45 @ ( powerset @ A ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(sepSubset_type,type,
% 0.31/1.45 sepSubset: $o ).
% 0.31/1.45
% 0.31/1.45 thf(sepSubset,definition,
% 0.31/1.45 ( sepSubset
% 0.31/1.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/1.45 ( subset
% 0.31/1.45 @ ( dsetconstr @ A
% 0.31/1.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/1.45 @ A ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(binunion_type,type,
% 0.31/1.45 binunion: $i > $i > $i ).
% 0.31/1.45
% 0.31/1.45 thf(binunionIL_type,type,
% 0.31/1.45 binunionIL: $o ).
% 0.31/1.45
% 0.31/1.45 thf(binunionIL,definition,
% 0.31/1.45 ( binunionIL
% 0.31/1.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ A )
% 0.31/1.45 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(upairset2IR_type,type,
% 0.31/1.45 upairset2IR: $o ).
% 0.31/1.45
% 0.31/1.45 thf(upairset2IR,definition,
% 0.31/1.45 ( upairset2IR
% 0.31/1.45 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(binunionIR_type,type,
% 0.31/1.45 binunionIR: $o ).
% 0.31/1.45
% 0.31/1.45 thf(binunionIR,definition,
% 0.31/1.45 ( binunionIR
% 0.31/1.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ B )
% 0.31/1.45 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(binunionEcases_type,type,
% 0.31/1.45 binunionEcases: $o ).
% 0.31/1.45
% 0.31/1.45 thf(binunionEcases,definition,
% 0.31/1.45 ( binunionEcases
% 0.31/1.45 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.31/1.45 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.31/1.45 => ( ( ( in @ Xx @ A )
% 0.31/1.45 => Xphi )
% 0.31/1.45 => ( ( ( in @ Xx @ B )
% 0.31/1.45 => Xphi )
% 0.31/1.45 => Xphi ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(binunionE_type,type,
% 0.31/1.45 binunionE: $o ).
% 0.31/1.45
% 0.31/1.45 thf(binunionE,definition,
% 0.31/1.45 ( binunionE
% 0.31/1.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.31/1.45 => ( ( in @ Xx @ A )
% 0.31/1.45 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(binunionLsub_type,type,
% 0.31/1.45 binunionLsub: $o ).
% 0.31/1.45
% 0.31/1.45 thf(binunionLsub,definition,
% 0.31/1.45 ( binunionLsub
% 0.31/1.45 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(binunionRsub_type,type,
% 0.31/1.45 binunionRsub: $o ).
% 0.31/1.45
% 0.31/1.45 thf(binunionRsub,definition,
% 0.31/1.45 ( binunionRsub
% 0.31/1.45 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(binintersect_type,type,
% 0.31/1.45 binintersect: $i > $i > $i ).
% 0.31/1.45
% 0.31/1.45 thf(binintersectI_type,type,
% 0.31/1.45 binintersectI: $o ).
% 0.31/1.45
% 0.31/1.45 thf(binintersectI,definition,
% 0.31/1.45 ( binintersectI
% 0.31/1.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ A )
% 0.31/1.45 => ( ( in @ Xx @ B )
% 0.31/1.45 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(binintersectSubset5_type,type,
% 0.31/1.45 binintersectSubset5: $o ).
% 0.31/1.45
% 0.31/1.45 thf(binintersectSubset5,definition,
% 0.31/1.45 ( binintersectSubset5
% 0.31/1.45 = ( ! [A: $i,B: $i,C: $i] :
% 0.31/1.45 ( ( subset @ C @ A )
% 0.31/1.45 => ( ( subset @ C @ B )
% 0.31/1.45 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(binintersectEL_type,type,
% 0.31/1.45 binintersectEL: $o ).
% 0.31/1.45
% 0.31/1.45 thf(binintersectEL,definition,
% 0.31/1.45 ( binintersectEL
% 0.31/1.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.31/1.45 => ( in @ Xx @ A ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(binintersectLsub_type,type,
% 0.31/1.45 binintersectLsub: $o ).
% 0.31/1.45
% 0.31/1.45 thf(binintersectLsub,definition,
% 0.31/1.45 ( binintersectLsub
% 0.31/1.45 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(binintersectSubset2_type,type,
% 0.31/1.45 binintersectSubset2: $o ).
% 0.31/1.45
% 0.31/1.45 thf(binintersectSubset2,definition,
% 0.31/1.45 ( binintersectSubset2
% 0.31/1.45 = ( ! [A: $i,B: $i] :
% 0.31/1.45 ( ( subset @ A @ B )
% 0.31/1.45 => ( ( binintersect @ A @ B )
% 0.31/1.45 = A ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(binintersectSubset3_type,type,
% 0.31/1.45 binintersectSubset3: $o ).
% 0.31/1.45
% 0.31/1.45 thf(binintersectSubset3,definition,
% 0.31/1.45 ( binintersectSubset3
% 0.31/1.45 = ( ! [A: $i,B: $i] :
% 0.31/1.45 ( ( ( binintersect @ A @ B )
% 0.31/1.45 = B )
% 0.31/1.45 => ( subset @ B @ A ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(binintersectER_type,type,
% 0.31/1.45 binintersectER: $o ).
% 0.31/1.45
% 0.31/1.45 thf(binintersectER,definition,
% 0.31/1.45 ( binintersectER
% 0.31/1.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.31/1.45 => ( in @ Xx @ B ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(disjointsetsI1_type,type,
% 0.31/1.45 disjointsetsI1: $o ).
% 0.31/1.45
% 0.31/1.45 thf(disjointsetsI1,definition,
% 0.31/1.45 ( disjointsetsI1
% 0.31/1.45 = ( ! [A: $i,B: $i] :
% 0.31/1.45 ( ~ ? [Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ A )
% 0.31/1.45 & ( in @ Xx @ B ) )
% 0.31/1.45 => ( ( binintersect @ A @ B )
% 0.31/1.45 = emptyset ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(binintersectRsub_type,type,
% 0.31/1.45 binintersectRsub: $o ).
% 0.31/1.45
% 0.31/1.45 thf(binintersectRsub,definition,
% 0.31/1.45 ( binintersectRsub
% 0.31/1.45 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(binintersectSubset4_type,type,
% 0.31/1.45 binintersectSubset4: $o ).
% 0.31/1.45
% 0.31/1.45 thf(binintersectSubset4,definition,
% 0.31/1.45 ( binintersectSubset4
% 0.31/1.45 = ( ! [A: $i,B: $i] :
% 0.31/1.45 ( ( subset @ B @ A )
% 0.31/1.45 => ( ( binintersect @ A @ B )
% 0.31/1.45 = B ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(binintersectSubset1_type,type,
% 0.31/1.45 binintersectSubset1: $o ).
% 0.31/1.45
% 0.31/1.45 thf(binintersectSubset1,definition,
% 0.31/1.45 ( binintersectSubset1
% 0.31/1.45 = ( ! [A: $i,B: $i] :
% 0.31/1.45 ( ( ( binintersect @ A @ B )
% 0.31/1.45 = A )
% 0.31/1.45 => ( subset @ A @ B ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(bs114d_type,type,
% 0.31/1.45 bs114d: $o ).
% 0.31/1.45
% 0.31/1.45 thf(bs114d,definition,
% 0.31/1.45 ( bs114d
% 0.31/1.45 = ( ! [A: $i,B: $i,C: $i] :
% 0.31/1.45 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 0.31/1.45 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(regular_type,type,
% 0.31/1.45 regular: $i > $o ).
% 0.31/1.45
% 0.31/1.45 thf(setminus_type,type,
% 0.31/1.45 setminus: $i > $i > $i ).
% 0.31/1.45
% 0.31/1.45 thf(setminusI_type,type,
% 0.31/1.45 setminusI: $o ).
% 0.31/1.45
% 0.31/1.45 thf(setminusI,definition,
% 0.31/1.45 ( setminusI
% 0.31/1.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ A )
% 0.31/1.45 => ( ~ ( in @ Xx @ B )
% 0.31/1.45 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(setminusEL_type,type,
% 0.31/1.45 setminusEL: $o ).
% 0.31/1.45
% 0.31/1.45 thf(setminusEL,definition,
% 0.31/1.45 ( setminusEL
% 0.31/1.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.31/1.45 => ( in @ Xx @ A ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(setminusER_type,type,
% 0.31/1.45 setminusER: $o ).
% 0.31/1.45
% 0.31/1.45 thf(setminusER,definition,
% 0.31/1.45 ( setminusER
% 0.31/1.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.31/1.45 => ~ ( in @ Xx @ B ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(setminusSubset2_type,type,
% 0.31/1.45 setminusSubset2: $o ).
% 0.31/1.45
% 0.31/1.45 thf(setminusSubset2,definition,
% 0.31/1.45 ( setminusSubset2
% 0.31/1.45 = ( ! [A: $i,B: $i] :
% 0.31/1.45 ( ( subset @ A @ B )
% 0.31/1.45 => ( ( setminus @ A @ B )
% 0.31/1.45 = emptyset ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(setminusERneg_type,type,
% 0.31/1.45 setminusERneg: $o ).
% 0.31/1.45
% 0.31/1.45 thf(setminusERneg,definition,
% 0.31/1.45 ( setminusERneg
% 0.31/1.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/1.45 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.31/1.45 => ( ( in @ Xx @ A )
% 0.31/1.45 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(setminusELneg_type,type,
% 0.31/1.45 setminusELneg: $o ).
% 0.31/1.45
% 0.31/1.45 thf(setminusELneg,definition,
% 0.31/1.45 ( setminusELneg
% 0.31/1.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/1.45 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.31/1.45 => ( ~ ( in @ Xx @ B )
% 0.31/1.45 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(setminusILneg_type,type,
% 0.31/1.45 setminusILneg: $o ).
% 0.31/1.45
% 0.31/1.45 thf(setminusILneg,definition,
% 0.31/1.45 ( setminusILneg
% 0.31/1.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/1.45 ( ~ ( in @ Xx @ A )
% 0.31/1.45 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(setminusIRneg_type,type,
% 0.31/1.45 setminusIRneg: $o ).
% 0.31/1.45
% 0.31/1.45 thf(setminusIRneg,definition,
% 0.31/1.45 ( setminusIRneg
% 0.31/1.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ B )
% 0.31/1.45 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(setminusLsub_type,type,
% 0.31/1.45 setminusLsub: $o ).
% 0.31/1.45
% 0.31/1.45 thf(setminusLsub,definition,
% 0.31/1.45 ( setminusLsub
% 0.31/1.45 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(setminusSubset1_type,type,
% 0.31/1.45 setminusSubset1: $o ).
% 0.31/1.45
% 0.31/1.45 thf(setminusSubset1,definition,
% 0.31/1.45 ( setminusSubset1
% 0.31/1.45 = ( ! [A: $i,B: $i] :
% 0.31/1.45 ( ( ( setminus @ A @ B )
% 0.31/1.45 = emptyset )
% 0.31/1.45 => ( subset @ A @ B ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(symdiff_type,type,
% 0.31/1.45 symdiff: $i > $i > $i ).
% 0.31/1.45
% 0.31/1.45 thf(symdiffE_type,type,
% 0.31/1.45 symdiffE: $o ).
% 0.31/1.45
% 0.31/1.45 thf(symdiffE,definition,
% 0.31/1.45 ( symdiffE
% 0.31/1.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 0.31/1.45 => ! [Xphi: $o] :
% 0.31/1.45 ( ( ( in @ Xx @ A )
% 0.31/1.45 => ( ~ ( in @ Xx @ B )
% 0.31/1.45 => Xphi ) )
% 0.31/1.45 => ( ( ~ ( in @ Xx @ A )
% 0.31/1.45 => ( ( in @ Xx @ B )
% 0.31/1.45 => Xphi ) )
% 0.31/1.45 => Xphi ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(symdiffI1_type,type,
% 0.31/1.45 symdiffI1: $o ).
% 0.31/1.45
% 0.31/1.45 thf(symdiffI1,definition,
% 0.31/1.45 ( symdiffI1
% 0.31/1.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ A )
% 0.31/1.45 => ( ~ ( in @ Xx @ B )
% 0.31/1.45 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(symdiffI2_type,type,
% 0.31/1.45 symdiffI2: $o ).
% 0.31/1.45
% 0.31/1.45 thf(symdiffI2,definition,
% 0.31/1.45 ( symdiffI2
% 0.31/1.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/1.45 ( ~ ( in @ Xx @ A )
% 0.31/1.45 => ( ( in @ Xx @ B )
% 0.31/1.45 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(symdiffIneg1_type,type,
% 0.31/1.45 symdiffIneg1: $o ).
% 0.31/1.45
% 0.31/1.45 thf(symdiffIneg1,definition,
% 0.31/1.45 ( symdiffIneg1
% 0.31/1.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ A )
% 0.31/1.45 => ( ( in @ Xx @ B )
% 0.31/1.45 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(symdiffIneg2_type,type,
% 0.31/1.45 symdiffIneg2: $o ).
% 0.31/1.45
% 0.31/1.45 thf(symdiffIneg2,definition,
% 0.31/1.45 ( symdiffIneg2
% 0.31/1.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/1.45 ( ~ ( in @ Xx @ A )
% 0.31/1.45 => ( ~ ( in @ Xx @ B )
% 0.31/1.45 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(iskpair_type,type,
% 0.31/1.45 iskpair: $i > $o ).
% 0.31/1.45
% 0.31/1.45 thf(secondinupair_type,type,
% 0.31/1.45 secondinupair: $o ).
% 0.31/1.45
% 0.31/1.45 thf(secondinupair,definition,
% 0.31/1.45 ( secondinupair
% 0.31/1.45 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(setukpairIL_type,type,
% 0.31/1.45 setukpairIL: $o ).
% 0.31/1.45
% 0.31/1.45 thf(setukpairIL,definition,
% 0.31/1.45 ( setukpairIL
% 0.31/1.45 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(setukpairIR_type,type,
% 0.31/1.45 setukpairIR: $o ).
% 0.31/1.45
% 0.31/1.45 thf(setukpairIR,definition,
% 0.31/1.45 ( setukpairIR
% 0.31/1.45 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(kpairiskpair_type,type,
% 0.31/1.45 kpairiskpair: $o ).
% 0.31/1.45
% 0.31/1.45 thf(kpairiskpair,definition,
% 0.31/1.45 ( kpairiskpair
% 0.31/1.45 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(kpair_type,type,
% 0.31/1.45 kpair: $i > $i > $i ).
% 0.31/1.45
% 0.31/1.45 thf(kpairp_type,type,
% 0.31/1.45 kpairp: $o ).
% 0.31/1.45
% 0.31/1.45 thf(kpairp,definition,
% 0.31/1.45 ( kpairp
% 0.31/1.45 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(cartprod_type,type,
% 0.31/1.45 cartprod: $i > $i > $i ).
% 0.31/1.45
% 0.31/1.45 thf(singletonsubset_type,type,
% 0.31/1.45 singletonsubset: $o ).
% 0.31/1.45
% 0.31/1.45 thf(singletonsubset,definition,
% 0.31/1.45 ( singletonsubset
% 0.31/1.45 = ( ! [A: $i,Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ A )
% 0.31/1.45 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(singletoninpowerset_type,type,
% 0.31/1.45 singletoninpowerset: $o ).
% 0.31/1.45
% 0.31/1.45 thf(singletoninpowerset,definition,
% 0.31/1.45 ( singletoninpowerset
% 0.31/1.45 = ( ! [A: $i,Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ A )
% 0.31/1.45 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(singletoninpowunion_type,type,
% 0.31/1.45 singletoninpowunion: $o ).
% 0.31/1.45
% 0.31/1.45 thf(singletoninpowunion,definition,
% 0.31/1.45 ( singletoninpowunion
% 0.31/1.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ A )
% 0.31/1.45 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(upairset2E_type,type,
% 0.31/1.45 upairset2E: $o ).
% 0.31/1.45
% 0.31/1.45 thf(upairset2E,definition,
% 0.31/1.45 ( upairset2E
% 0.31/1.45 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.31/1.45 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.31/1.45 => ( ( Xz = Xx )
% 0.31/1.45 | ( Xz = Xy ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(upairsubunion_type,type,
% 0.31/1.45 upairsubunion: $o ).
% 0.31/1.45
% 0.31/1.45 thf(upairsubunion,definition,
% 0.31/1.45 ( upairsubunion
% 0.31/1.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ A )
% 0.31/1.45 => ! [Xy: $i] :
% 0.31/1.45 ( ( in @ Xy @ B )
% 0.31/1.45 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(upairinpowunion_type,type,
% 0.31/1.45 upairinpowunion: $o ).
% 0.31/1.45
% 0.31/1.45 thf(upairinpowunion,definition,
% 0.31/1.45 ( upairinpowunion
% 0.31/1.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ A )
% 0.31/1.45 => ! [Xy: $i] :
% 0.31/1.45 ( ( in @ Xy @ B )
% 0.31/1.45 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(ubforcartprodlem1_type,type,
% 0.31/1.45 ubforcartprodlem1: $o ).
% 0.31/1.45
% 0.31/1.45 thf(ubforcartprodlem1,definition,
% 0.31/1.45 ( ubforcartprodlem1
% 0.31/1.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ A )
% 0.31/1.45 => ! [Xy: $i] :
% 0.31/1.45 ( ( in @ Xy @ B )
% 0.31/1.45 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(ubforcartprodlem2_type,type,
% 0.31/1.45 ubforcartprodlem2: $o ).
% 0.31/1.45
% 0.31/1.45 thf(ubforcartprodlem2,definition,
% 0.31/1.45 ( ubforcartprodlem2
% 0.31/1.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ A )
% 0.31/1.45 => ! [Xy: $i] :
% 0.31/1.45 ( ( in @ Xy @ B )
% 0.31/1.45 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(ubforcartprodlem3_type,type,
% 0.31/1.45 ubforcartprodlem3: $o ).
% 0.31/1.45
% 0.31/1.45 thf(ubforcartprodlem3,definition,
% 0.31/1.45 ( ubforcartprodlem3
% 0.31/1.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ A )
% 0.31/1.45 => ! [Xy: $i] :
% 0.31/1.45 ( ( in @ Xy @ B )
% 0.31/1.45 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(cartprodpairin_type,type,
% 0.31/1.45 cartprodpairin: $o ).
% 0.31/1.45
% 0.31/1.45 thf(cartprodpairin,definition,
% 0.31/1.45 ( cartprodpairin
% 0.31/1.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ A )
% 0.31/1.45 => ! [Xy: $i] :
% 0.31/1.45 ( ( in @ Xy @ B )
% 0.31/1.45 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(cartprodmempair1_type,type,
% 0.31/1.45 cartprodmempair1: $o ).
% 0.31/1.45
% 0.31/1.45 thf(cartprodmempair1,definition,
% 0.31/1.45 ( cartprodmempair1
% 0.31/1.45 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.31/1.45 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.31/1.45 => ? [Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ A )
% 0.31/1.45 & ? [Xy: $i] :
% 0.31/1.45 ( ( in @ Xy @ B )
% 0.31/1.45 & ( Xu
% 0.31/1.45 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(cartprodmempair_type,type,
% 0.31/1.45 cartprodmempair: $o ).
% 0.31/1.45
% 0.31/1.45 thf(cartprodmempair,definition,
% 0.31/1.45 ( cartprodmempair
% 0.31/1.45 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.31/1.45 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.31/1.45 => ( iskpair @ Xu ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(setunionE2_type,type,
% 0.31/1.45 setunionE2: $o ).
% 0.31/1.45
% 0.31/1.45 thf(setunionE2,definition,
% 0.31/1.45 ( setunionE2
% 0.31/1.45 = ( ! [A: $i,Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ ( setunion @ A ) )
% 0.31/1.45 => ? [X: $i] :
% 0.31/1.45 ( ( in @ X @ A )
% 0.31/1.45 & ( in @ Xx @ X ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(setunionsingleton1_type,type,
% 0.31/1.45 setunionsingleton1: $o ).
% 0.31/1.45
% 0.31/1.45 thf(setunionsingleton1,definition,
% 0.31/1.45 ( setunionsingleton1
% 0.31/1.45 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(setunionsingleton2_type,type,
% 0.31/1.45 setunionsingleton2: $o ).
% 0.31/1.45
% 0.31/1.45 thf(setunionsingleton2,definition,
% 0.31/1.45 ( setunionsingleton2
% 0.31/1.45 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(setunionsingleton_type,type,
% 0.31/1.45 setunionsingleton: $o ).
% 0.31/1.45
% 0.31/1.45 thf(setunionsingleton,definition,
% 0.31/1.45 ( setunionsingleton
% 0.31/1.45 = ( ! [Xx: $i] :
% 0.31/1.45 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 0.31/1.45 = Xx ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(singleton_type,type,
% 0.31/1.45 singleton: $i > $o ).
% 0.31/1.45
% 0.31/1.45 thf(singleton,definition,
% 0.31/1.45 ( singleton
% 0.31/1.45 = ( ^ [A: $i] :
% 0.31/1.45 ? [Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ A )
% 0.31/1.45 & ( A
% 0.31/1.45 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(singletonprop_type,type,
% 0.31/1.45 singletonprop: $o ).
% 0.31/1.45
% 0.31/1.45 thf(singletonprop,definition,
% 0.31/1.45 ( singletonprop
% 0.31/1.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/1.45 ( ! [Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ A )
% 0.31/1.45 => ! [Xy: $i] :
% 0.31/1.45 ( ( in @ Xy @ A )
% 0.31/1.45 => ( ( Xphi @ Xx )
% 0.31/1.45 => ( ( Xphi @ Xy )
% 0.31/1.45 => ( Xx = Xy ) ) ) ) )
% 0.31/1.45 => ( ? [Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ A )
% 0.31/1.45 & ( Xphi @ Xx ) )
% 0.31/1.45 => ( singleton
% 0.31/1.45 @ ( dsetconstr @ A
% 0.31/1.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(ex1_type,type,
% 0.31/1.45 ex1: $i > ( $i > $o ) > $o ).
% 0.31/1.45
% 0.31/1.45 thf(ex1,definition,
% 0.31/1.45 ( ex1
% 0.31/1.45 = ( ^ [A: $i,Xphi: $i > $o] :
% 0.31/1.45 ( singleton
% 0.31/1.45 @ ( dsetconstr @ A
% 0.31/1.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(ex1E1_type,type,
% 0.31/1.45 ex1E1: $o ).
% 0.31/1.45
% 0.31/1.45 thf(ex1E1,definition,
% 0.31/1.45 ( ex1E1
% 0.31/1.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/1.45 ( ( ex1 @ A
% 0.31/1.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/1.45 => ? [Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ A )
% 0.31/1.45 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(ex1I_type,type,
% 0.31/1.45 ex1I: $o ).
% 0.31/1.45
% 0.31/1.45 thf(ex1I,definition,
% 0.31/1.45 ( ex1I
% 0.31/1.45 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ A )
% 0.31/1.45 => ( ( Xphi @ Xx )
% 0.31/1.45 => ( ! [Xy: $i] :
% 0.31/1.45 ( ( in @ Xy @ A )
% 0.31/1.45 => ( ( Xphi @ Xy )
% 0.31/1.45 => ( Xy = Xx ) ) )
% 0.31/1.45 => ( ex1 @ A
% 0.31/1.45 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(ex1I2_type,type,
% 0.31/1.45 ex1I2: $o ).
% 0.31/1.45
% 0.31/1.45 thf(ex1I2,definition,
% 0.31/1.45 ( ex1I2
% 0.31/1.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/1.45 ( ! [Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ A )
% 0.31/1.45 => ! [Xy: $i] :
% 0.31/1.45 ( ( in @ Xy @ A )
% 0.31/1.45 => ( ( Xphi @ Xx )
% 0.31/1.45 => ( ( Xphi @ Xy )
% 0.31/1.45 => ( Xx = Xy ) ) ) ) )
% 0.31/1.45 => ( ? [Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ A )
% 0.31/1.45 & ( Xphi @ Xx ) )
% 0.31/1.45 => ( ex1 @ A
% 0.31/1.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(singletonsuniq_type,type,
% 0.31/1.45 singletonsuniq: $o ).
% 0.31/1.45
% 0.31/1.45 thf(singletonsuniq,definition,
% 0.31/1.45 ( singletonsuniq
% 0.31/1.45 = ( ! [Xx: $i,Xy: $i] :
% 0.31/1.45 ( ( ( setadjoin @ Xx @ emptyset )
% 0.31/1.45 = ( setadjoin @ Xy @ emptyset ) )
% 0.31/1.45 => ( Xx = Xy ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(atmost1p_type,type,
% 0.31/1.45 atmost1p: $i > $o ).
% 0.31/1.45
% 0.31/1.45 thf(atleast2p_type,type,
% 0.31/1.45 atleast2p: $i > $o ).
% 0.31/1.45
% 0.31/1.45 thf(atmost2p_type,type,
% 0.31/1.45 atmost2p: $i > $o ).
% 0.31/1.45
% 0.31/1.45 thf(upairsetp_type,type,
% 0.31/1.45 upairsetp: $i > $o ).
% 0.31/1.45
% 0.31/1.45 thf(setukpairinjL1_type,type,
% 0.31/1.45 setukpairinjL1: $o ).
% 0.31/1.45
% 0.31/1.45 thf(setukpairinjL1,definition,
% 0.31/1.45 ( setukpairinjL1
% 0.31/1.45 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.31/1.45 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 0.31/1.45 => ( Xx = Xz ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(kfstsingleton_type,type,
% 0.31/1.45 kfstsingleton: $o ).
% 0.31/1.45
% 0.31/1.45 thf(kfstsingleton,definition,
% 0.31/1.45 ( kfstsingleton
% 0.31/1.45 = ( ! [Xu: $i] :
% 0.31/1.45 ( ( iskpair @ Xu )
% 0.31/1.45 => ( singleton
% 0.31/1.45 @ ( dsetconstr @ ( setunion @ Xu )
% 0.31/1.45 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(theprop_type,type,
% 0.31/1.45 theprop: $o ).
% 0.31/1.45
% 0.31/1.45 thf(theprop,definition,
% 0.31/1.45 ( theprop
% 0.31/1.45 = ( ! [X: $i] :
% 0.31/1.45 ( ( singleton @ X )
% 0.31/1.45 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(kfst_type,type,
% 0.31/1.45 kfst: $i > $i ).
% 0.31/1.45
% 0.31/1.45 thf(kfstpairEq_type,type,
% 0.31/1.45 kfstpairEq: $o ).
% 0.31/1.45
% 0.31/1.45 thf(kfstpairEq,definition,
% 0.31/1.45 ( kfstpairEq
% 0.31/1.45 = ( ! [Xx: $i,Xy: $i] :
% 0.31/1.45 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.31/1.45 = Xx ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(cartprodfstin_type,type,
% 0.31/1.45 cartprodfstin: $o ).
% 0.31/1.45
% 0.31/1.45 thf(cartprodfstin,definition,
% 0.31/1.45 ( cartprodfstin
% 0.31/1.45 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.31/1.45 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.31/1.45 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(setukpairinjL2_type,type,
% 0.31/1.45 setukpairinjL2: $o ).
% 0.31/1.45
% 0.31/1.45 thf(setukpairinjL2,definition,
% 0.31/1.45 ( setukpairinjL2
% 0.31/1.45 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.31/1.45 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.31/1.45 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.31/1.45 => ( Xx = Xz ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(setukpairinjL_type,type,
% 0.31/1.45 setukpairinjL: $o ).
% 0.31/1.45
% 0.31/1.45 thf(setukpairinjL,definition,
% 0.31/1.45 ( setukpairinjL
% 0.31/1.45 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.31/1.45 ( ( ( kpair @ Xx @ Xy )
% 0.31/1.45 = ( kpair @ Xz @ Xu ) )
% 0.31/1.45 => ( Xx = Xz ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(setukpairinjR11_type,type,
% 0.31/1.45 setukpairinjR11: $o ).
% 0.31/1.45
% 0.31/1.45 thf(setukpairinjR11,definition,
% 0.31/1.45 ( setukpairinjR11
% 0.31/1.45 = ( ! [Xx: $i,Xy: $i] :
% 0.31/1.45 ( ( Xx = Xy )
% 0.31/1.45 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.31/1.45 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(setukpairinjR12_type,type,
% 0.31/1.45 setukpairinjR12: $o ).
% 0.31/1.45
% 0.31/1.45 thf(setukpairinjR12,definition,
% 0.31/1.45 ( setukpairinjR12
% 0.31/1.45 = ( ! [Xx: $i,Xy: $i] :
% 0.31/1.45 ( ( Xx = Xy )
% 0.31/1.45 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.31/1.45 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(setukpairinjR1_type,type,
% 0.31/1.45 setukpairinjR1: $o ).
% 0.31/1.45
% 0.31/1.45 thf(setukpairinjR1,definition,
% 0.31/1.45 ( setukpairinjR1
% 0.31/1.45 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.31/1.45 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.31/1.45 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.31/1.45 => ( ( Xz = Xu )
% 0.31/1.45 => ( Xy = Xu ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(upairequniteq_type,type,
% 0.31/1.45 upairequniteq: $o ).
% 0.31/1.45
% 0.31/1.45 thf(upairequniteq,definition,
% 0.31/1.45 ( upairequniteq
% 0.31/1.45 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.31/1.45 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.31/1.45 = ( setadjoin @ Xz @ emptyset ) )
% 0.31/1.45 => ( Xx = Xy ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(setukpairinjR2_type,type,
% 0.31/1.45 setukpairinjR2: $o ).
% 0.31/1.45
% 0.31/1.45 thf(setukpairinjR2,definition,
% 0.31/1.45 ( setukpairinjR2
% 0.31/1.45 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.31/1.45 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.31/1.45 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.31/1.45 => ( Xy = Xu ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(setukpairinjR_type,type,
% 0.31/1.45 setukpairinjR: $o ).
% 0.31/1.45
% 0.31/1.45 thf(setukpairinjR,definition,
% 0.31/1.45 ( setukpairinjR
% 0.31/1.45 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.31/1.45 ( ( ( kpair @ Xx @ Xy )
% 0.31/1.45 = ( kpair @ Xz @ Xu ) )
% 0.31/1.45 => ( Xy = Xu ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(ksndsingleton_type,type,
% 0.31/1.45 ksndsingleton: $o ).
% 0.31/1.45
% 0.31/1.45 thf(ksndsingleton,definition,
% 0.31/1.45 ( ksndsingleton
% 0.31/1.45 = ( ! [Xu: $i] :
% 0.31/1.45 ( ( iskpair @ Xu )
% 0.31/1.45 => ( singleton
% 0.31/1.45 @ ( dsetconstr @ ( setunion @ Xu )
% 0.31/1.45 @ ^ [Xx: $i] :
% 0.31/1.45 ( Xu
% 0.31/1.45 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(ksnd_type,type,
% 0.31/1.45 ksnd: $i > $i ).
% 0.31/1.45
% 0.31/1.45 thf(ksndpairEq_type,type,
% 0.31/1.45 ksndpairEq: $o ).
% 0.31/1.45
% 0.31/1.45 thf(ksndpairEq,definition,
% 0.31/1.45 ( ksndpairEq
% 0.31/1.45 = ( ! [Xx: $i,Xy: $i] :
% 0.31/1.45 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.31/1.45 = Xy ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(kpairsurjEq_type,type,
% 0.31/1.45 kpairsurjEq: $o ).
% 0.31/1.45
% 0.31/1.45 thf(kpairsurjEq,definition,
% 0.31/1.45 ( kpairsurjEq
% 0.31/1.45 = ( ! [Xu: $i] :
% 0.31/1.45 ( ( iskpair @ Xu )
% 0.31/1.45 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.31/1.45 = Xu ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(cartprodsndin_type,type,
% 0.31/1.45 cartprodsndin: $o ).
% 0.31/1.45
% 0.31/1.45 thf(cartprodsndin,definition,
% 0.31/1.45 ( cartprodsndin
% 0.31/1.45 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.31/1.45 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.31/1.45 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(cartprodpairmemEL_type,type,
% 0.31/1.45 cartprodpairmemEL: $o ).
% 0.31/1.45
% 0.31/1.45 thf(cartprodpairmemEL,definition,
% 0.31/1.45 ( cartprodpairmemEL
% 0.31/1.45 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.31/1.45 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.31/1.45 => ( in @ Xx @ A ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(cartprodpairmemER_type,type,
% 0.31/1.45 cartprodpairmemER: $o ).
% 0.31/1.45
% 0.31/1.45 thf(cartprodpairmemER,definition,
% 0.31/1.45 ( cartprodpairmemER
% 0.31/1.45 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.31/1.45 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.31/1.45 => ( in @ Xy @ B ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(cartprodmempaircEq_type,type,
% 0.31/1.45 cartprodmempaircEq: $o ).
% 0.31/1.45
% 0.31/1.45 thf(cartprodmempaircEq,definition,
% 0.31/1.45 ( cartprodmempaircEq
% 0.31/1.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ A )
% 0.31/1.45 => ! [Xy: $i] :
% 0.31/1.45 ( ( in @ Xy @ B )
% 0.31/1.45 => ( ( kpair @ Xx @ Xy )
% 0.31/1.45 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(cartprodfstpairEq_type,type,
% 0.31/1.45 cartprodfstpairEq: $o ).
% 0.31/1.45
% 0.31/1.45 thf(cartprodfstpairEq,definition,
% 0.31/1.45 ( cartprodfstpairEq
% 0.31/1.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ A )
% 0.31/1.45 => ! [Xy: $i] :
% 0.31/1.45 ( ( in @ Xy @ B )
% 0.31/1.45 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.31/1.45 = Xx ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(cartprodsndpairEq_type,type,
% 0.31/1.45 cartprodsndpairEq: $o ).
% 0.31/1.45
% 0.31/1.45 thf(cartprodsndpairEq,definition,
% 0.31/1.45 ( cartprodsndpairEq
% 0.31/1.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ A )
% 0.31/1.45 => ! [Xy: $i] :
% 0.31/1.45 ( ( in @ Xy @ B )
% 0.31/1.45 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.31/1.45 = Xy ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(cartprodpairsurjEq_type,type,
% 0.31/1.45 cartprodpairsurjEq: $o ).
% 0.31/1.45
% 0.31/1.45 thf(cartprodpairsurjEq,definition,
% 0.31/1.45 ( cartprodpairsurjEq
% 0.31/1.45 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.31/1.45 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.31/1.45 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.31/1.45 = Xu ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(breln_type,type,
% 0.31/1.45 breln: $i > $i > $i > $o ).
% 0.31/1.45
% 0.31/1.45 thf(breln,definition,
% 0.31/1.45 ( breln
% 0.31/1.45 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(dpsetconstr_type,type,
% 0.31/1.45 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 0.31/1.45
% 0.31/1.45 thf(dpsetconstrI_type,type,
% 0.31/1.45 dpsetconstrI: $o ).
% 0.31/1.45
% 0.31/1.45 thf(dpsetconstrI,definition,
% 0.31/1.45 ( dpsetconstrI
% 0.31/1.45 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ A )
% 0.31/1.45 => ! [Xy: $i] :
% 0.31/1.45 ( ( in @ Xy @ B )
% 0.31/1.45 => ( ( Xphi @ Xx @ Xy )
% 0.31/1.45 => ( in @ ( kpair @ Xx @ Xy )
% 0.31/1.45 @ ( dpsetconstr @ A @ B
% 0.31/1.45 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(dpsetconstrSub_type,type,
% 0.31/1.45 dpsetconstrSub: $o ).
% 0.31/1.45
% 0.31/1.45 thf(dpsetconstrSub,definition,
% 0.31/1.45 ( dpsetconstrSub
% 0.31/1.45 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.31/1.45 ( subset
% 0.31/1.45 @ ( dpsetconstr @ A @ B
% 0.31/1.45 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 0.31/1.45 @ ( cartprod @ A @ B ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(setOfPairsIsBReln_type,type,
% 0.31/1.45 setOfPairsIsBReln: $o ).
% 0.31/1.45
% 0.31/1.45 thf(setOfPairsIsBReln,definition,
% 0.31/1.45 ( setOfPairsIsBReln
% 0.31/1.45 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.31/1.45 ( breln @ A @ B
% 0.31/1.45 @ ( dpsetconstr @ A @ B
% 0.31/1.45 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(dpsetconstrERa_type,type,
% 0.31/1.45 dpsetconstrERa: $o ).
% 0.31/1.45
% 0.31/1.45 thf(dpsetconstrERa,definition,
% 0.31/1.45 ( dpsetconstrERa
% 0.31/1.45 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ A )
% 0.31/1.45 => ! [Xy: $i] :
% 0.31/1.45 ( ( in @ Xy @ B )
% 0.31/1.45 => ( ( in @ ( kpair @ Xx @ Xy )
% 0.31/1.45 @ ( dpsetconstr @ A @ B
% 0.31/1.45 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.31/1.45 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(dpsetconstrEL1_type,type,
% 0.31/1.45 dpsetconstrEL1: $o ).
% 0.31/1.45
% 0.31/1.45 thf(dpsetconstrEL1,definition,
% 0.31/1.45 ( dpsetconstrEL1
% 0.31/1.45 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.31/1.45 ( ( in @ ( kpair @ Xx @ Xy )
% 0.31/1.45 @ ( dpsetconstr @ A @ B
% 0.31/1.45 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.31/1.45 => ( in @ Xx @ A ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(dpsetconstrEL2_type,type,
% 0.31/1.45 dpsetconstrEL2: $o ).
% 0.31/1.45
% 0.31/1.45 thf(dpsetconstrEL2,definition,
% 0.31/1.45 ( dpsetconstrEL2
% 0.31/1.45 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.31/1.45 ( ( in @ ( kpair @ Xx @ Xy )
% 0.31/1.45 @ ( dpsetconstr @ A @ B
% 0.31/1.45 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.31/1.45 => ( in @ Xy @ B ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(dpsetconstrER_type,type,
% 0.31/1.45 dpsetconstrER: $o ).
% 0.31/1.45
% 0.31/1.45 thf(dpsetconstrER,definition,
% 0.31/1.45 ( dpsetconstrER
% 0.31/1.45 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.31/1.45 ( ( in @ ( kpair @ Xx @ Xy )
% 0.31/1.45 @ ( dpsetconstr @ A @ B
% 0.31/1.45 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.31/1.45 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(func_type,type,
% 0.31/1.45 func: $i > $i > $i > $o ).
% 0.31/1.45
% 0.31/1.45 thf(funcSet_type,type,
% 0.31/1.45 funcSet: $i > $i > $i ).
% 0.31/1.45
% 0.31/1.45 thf(funcImageSingleton_type,type,
% 0.31/1.45 funcImageSingleton: $o ).
% 0.31/1.45
% 0.31/1.45 thf(funcImageSingleton,definition,
% 0.31/1.45 ( funcImageSingleton
% 0.31/1.45 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/1.45 ( ( func @ A @ B @ Xf )
% 0.31/1.45 => ! [Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ A )
% 0.31/1.45 => ( singleton
% 0.31/1.45 @ ( dsetconstr @ B
% 0.31/1.45 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(apProp_type,type,
% 0.31/1.45 apProp: $o ).
% 0.31/1.45
% 0.31/1.45 thf(apProp,definition,
% 0.31/1.45 ( apProp
% 0.31/1.45 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/1.45 ( ( func @ A @ B @ Xf )
% 0.31/1.45 => ! [Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ A )
% 0.31/1.45 => ( in
% 0.31/1.45 @ ( setunion
% 0.31/1.45 @ ( dsetconstr @ B
% 0.31/1.45 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 0.31/1.45 @ B ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(ap_type,type,
% 0.31/1.45 ap: $i > $i > $i > $i > $i ).
% 0.31/1.45
% 0.31/1.45 thf(app_type,type,
% 0.31/1.45 app: $o ).
% 0.31/1.45
% 0.31/1.45 thf(app,definition,
% 0.31/1.45 ( app
% 0.31/1.45 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/1.45 ( ( func @ A @ B @ Xf )
% 0.31/1.45 => ! [Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ A )
% 0.31/1.45 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(infuncsetfunc_type,type,
% 0.31/1.45 infuncsetfunc: $o ).
% 0.31/1.45
% 0.31/1.45 thf(infuncsetfunc,definition,
% 0.31/1.45 ( infuncsetfunc
% 0.31/1.45 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/1.45 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.31/1.45 => ( func @ A @ B @ Xf ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(ap2p_type,type,
% 0.31/1.45 ap2p: $o ).
% 0.31/1.45
% 0.31/1.45 thf(ap2p,definition,
% 0.31/1.45 ( ap2p
% 0.31/1.45 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/1.45 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.31/1.45 => ! [Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ A )
% 0.31/1.45 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(funcinfuncset_type,type,
% 0.31/1.45 funcinfuncset: $o ).
% 0.31/1.45
% 0.31/1.45 thf(funcinfuncset,definition,
% 0.31/1.45 ( funcinfuncset
% 0.31/1.45 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/1.45 ( ( func @ A @ B @ Xf )
% 0.31/1.45 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(lamProp_type,type,
% 0.31/1.45 lamProp: $o ).
% 0.31/1.45
% 0.31/1.45 thf(lamProp,definition,
% 0.31/1.45 ( lamProp
% 0.31/1.45 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.31/1.45 ( ! [Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ A )
% 0.31/1.45 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.31/1.45 => ( func @ A @ B
% 0.31/1.45 @ ( dpsetconstr @ A @ B
% 0.31/1.45 @ ^ [Xx: $i,Xy: $i] :
% 0.31/1.45 ( ( Xf @ Xx )
% 0.31/1.45 = Xy ) ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(lam_type,type,
% 0.31/1.45 lam: $i > $i > ( $i > $i ) > $i ).
% 0.31/1.45
% 0.31/1.45 thf(lam,definition,
% 0.31/1.45 ( lam
% 0.31/1.45 = ( ^ [A: $i,B: $i,Xf: $i > $i] :
% 0.31/1.45 ( dpsetconstr @ A @ B
% 0.31/1.45 @ ^ [Xx: $i,Xy: $i] :
% 0.31/1.45 ( ( Xf @ Xx )
% 0.31/1.45 = Xy ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(lamp_type,type,
% 0.31/1.45 lamp: $o ).
% 0.31/1.45
% 0.31/1.45 thf(lamp,definition,
% 0.31/1.45 ( lamp
% 0.31/1.45 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.31/1.45 ( ! [Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ A )
% 0.31/1.45 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.31/1.45 => ( func @ A @ B
% 0.31/1.45 @ ( lam @ A @ B
% 0.31/1.45 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(lam2p_type,type,
% 0.31/1.45 lam2p: $o ).
% 0.31/1.45
% 0.31/1.45 thf(lam2p,definition,
% 0.31/1.45 ( lam2p
% 0.31/1.45 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.31/1.45 ( ! [Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ A )
% 0.31/1.45 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.31/1.45 => ( in
% 0.31/1.45 @ ( lam @ A @ B
% 0.31/1.45 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.31/1.45 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(brelnall1_type,type,
% 0.31/1.45 brelnall1: $o ).
% 0.31/1.45
% 0.31/1.45 thf(brelnall1,definition,
% 0.31/1.45 ( brelnall1
% 0.31/1.45 = ( ! [A: $i,B: $i,R: $i] :
% 0.31/1.45 ( ( breln @ A @ B @ R )
% 0.31/1.45 => ! [Xphi: $i > $o] :
% 0.31/1.45 ( ! [Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ A )
% 0.31/1.45 => ! [Xy: $i] :
% 0.31/1.45 ( ( in @ Xy @ B )
% 0.31/1.45 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.31/1.45 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.31/1.45 => ! [Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ R )
% 0.31/1.45 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(brelnall2_type,type,
% 0.31/1.45 brelnall2: $o ).
% 0.31/1.45
% 0.31/1.45 thf(brelnall2,definition,
% 0.31/1.45 ( brelnall2
% 0.31/1.45 = ( ! [A: $i,B: $i,R: $i] :
% 0.31/1.45 ( ( breln @ A @ B @ R )
% 0.31/1.45 => ! [Xphi: $i > $o] :
% 0.31/1.45 ( ! [Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ A )
% 0.31/1.45 => ! [Xy: $i] :
% 0.31/1.45 ( ( in @ Xy @ B )
% 0.31/1.45 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.31/1.45 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.31/1.45 => ! [Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ R )
% 0.31/1.45 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(ex1E2_type,type,
% 0.31/1.45 ex1E2: $o ).
% 0.31/1.45
% 0.31/1.45 thf(ex1E2,definition,
% 0.31/1.45 ( ex1E2
% 0.31/1.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/1.45 ( ( ex1 @ A
% 0.31/1.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/1.45 => ! [Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ A )
% 0.31/1.45 => ! [Xy: $i] :
% 0.31/1.45 ( ( in @ Xy @ A )
% 0.31/1.45 => ( ( Xphi @ Xx )
% 0.31/1.45 => ( ( Xphi @ Xy )
% 0.31/1.45 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(funcGraphProp1_type,type,
% 0.31/1.45 funcGraphProp1: $o ).
% 0.31/1.45
% 0.31/1.45 thf(funcGraphProp1,definition,
% 0.31/1.45 ( funcGraphProp1
% 0.31/1.45 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/1.45 ( ( func @ A @ B @ Xf )
% 0.31/1.45 => ! [Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ A )
% 0.31/1.45 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(funcGraphProp3_type,type,
% 0.31/1.45 funcGraphProp3: $o ).
% 0.31/1.45
% 0.31/1.45 thf(funcGraphProp3,definition,
% 0.31/1.45 ( funcGraphProp3
% 0.31/1.45 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/1.45 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.31/1.45 => ! [Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ A )
% 0.31/1.45 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(funcGraphProp2_type,type,
% 0.31/1.45 funcGraphProp2: $o ).
% 0.31/1.45
% 0.31/1.45 thf(funcGraphProp2,definition,
% 0.31/1.45 ( funcGraphProp2
% 0.31/1.45 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/1.45 ( ( func @ A @ B @ Xf )
% 0.31/1.45 => ! [Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ A )
% 0.31/1.45 => ! [Xy: $i] :
% 0.31/1.45 ( ( in @ Xy @ B )
% 0.31/1.45 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.31/1.45 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.31/1.45 = Xy ) ) ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(funcextLem_type,type,
% 0.31/1.45 funcextLem: $o ).
% 0.31/1.45
% 0.31/1.45 thf(funcextLem,definition,
% 0.31/1.45 ( funcextLem
% 0.31/1.45 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/1.45 ( ( func @ A @ B @ Xf )
% 0.31/1.45 => ! [Xg: $i] :
% 0.31/1.45 ( ( func @ A @ B @ Xg )
% 0.31/1.45 => ( ! [Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ A )
% 0.31/1.45 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.31/1.45 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.31/1.45 => ! [Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ A )
% 0.31/1.45 => ! [Xy: $i] :
% 0.31/1.45 ( ( in @ Xy @ B )
% 0.31/1.45 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 0.31/1.45 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(funcGraphProp4_type,type,
% 0.31/1.45 funcGraphProp4: $o ).
% 0.31/1.45
% 0.31/1.45 thf(funcGraphProp4,definition,
% 0.31/1.45 ( funcGraphProp4
% 0.31/1.45 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/1.45 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.31/1.45 => ! [Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ A )
% 0.31/1.45 => ! [Xy: $i] :
% 0.31/1.45 ( ( in @ Xy @ B )
% 0.31/1.45 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.31/1.45 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.31/1.45 = Xy ) ) ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(subbreln_type,type,
% 0.31/1.45 subbreln: $o ).
% 0.31/1.45
% 0.31/1.45 thf(subbreln,definition,
% 0.31/1.45 ( subbreln
% 0.31/1.45 = ( ! [A: $i,B: $i,R: $i] :
% 0.31/1.45 ( ( breln @ A @ B @ R )
% 0.31/1.45 => ! [S: $i] :
% 0.31/1.45 ( ( breln @ A @ B @ S )
% 0.31/1.45 => ( ! [Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ A )
% 0.31/1.45 => ! [Xy: $i] :
% 0.31/1.45 ( ( in @ Xy @ B )
% 0.31/1.45 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.31/1.45 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.31/1.45 => ( subset @ R @ S ) ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(eqbreln_type,type,
% 0.31/1.45 eqbreln: $o ).
% 0.31/1.45
% 0.31/1.45 thf(eqbreln,definition,
% 0.31/1.45 ( eqbreln
% 0.31/1.45 = ( ! [A: $i,B: $i,R: $i] :
% 0.31/1.45 ( ( breln @ A @ B @ R )
% 0.31/1.45 => ! [S: $i] :
% 0.31/1.45 ( ( breln @ A @ B @ S )
% 0.31/1.45 => ( ! [Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ A )
% 0.31/1.45 => ! [Xy: $i] :
% 0.31/1.45 ( ( in @ Xy @ B )
% 0.31/1.45 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.31/1.45 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.31/1.45 => ( ! [Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ A )
% 0.31/1.45 => ! [Xy: $i] :
% 0.31/1.45 ( ( in @ Xy @ B )
% 0.31/1.45 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.31/1.45 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 0.31/1.45 => ( R = S ) ) ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(funcext_type,type,
% 0.31/1.45 funcext: $o ).
% 0.31/1.45
% 0.31/1.45 thf(funcext,definition,
% 0.31/1.45 ( funcext
% 0.31/1.45 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/1.45 ( ( func @ A @ B @ Xf )
% 0.31/1.45 => ! [Xg: $i] :
% 0.31/1.45 ( ( func @ A @ B @ Xg )
% 0.31/1.45 => ( ! [Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ A )
% 0.31/1.45 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.31/1.45 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.31/1.45 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(funcext2_type,type,
% 0.31/1.45 funcext2: $o ).
% 0.31/1.45
% 0.31/1.45 thf(funcext2,definition,
% 0.31/1.45 ( funcext2
% 0.31/1.45 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/1.45 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.31/1.45 => ! [Xg: $i] :
% 0.31/1.45 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 0.31/1.45 => ( ! [Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ A )
% 0.31/1.45 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.31/1.45 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.31/1.45 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(ap2apEq1_type,type,
% 0.31/1.45 ap2apEq1: $o ).
% 0.31/1.45
% 0.31/1.45 thf(ap2apEq1,definition,
% 0.31/1.45 ( ap2apEq1
% 0.31/1.45 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/1.45 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.31/1.45 => ! [Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ A )
% 0.31/1.45 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.31/1.45 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(ap2apEq2_type,type,
% 0.31/1.45 ap2apEq2: $o ).
% 0.31/1.45
% 0.31/1.45 thf(ap2apEq2,definition,
% 0.31/1.45 ( ap2apEq2
% 0.31/1.45 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/1.45 ( ( func @ A @ B @ Xf )
% 0.31/1.45 => ! [Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ A )
% 0.31/1.45 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.31/1.45 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(beta1_type,type,
% 0.31/1.45 beta1: $o ).
% 0.31/1.45
% 0.31/1.45 thf(beta1,definition,
% 0.31/1.45 ( beta1
% 0.31/1.45 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.31/1.45 ( ! [Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ A )
% 0.31/1.45 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.31/1.45 => ! [Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ A )
% 0.31/1.45 => ( ( ap @ A @ B
% 0.31/1.45 @ ( lam @ A @ B
% 0.31/1.45 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.31/1.45 @ Xx )
% 0.31/1.45 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(eta1_type,type,
% 0.31/1.45 eta1: $o ).
% 0.31/1.45
% 0.31/1.45 thf(eta1,definition,
% 0.31/1.45 ( eta1
% 0.31/1.45 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/1.45 ( ( func @ A @ B @ Xf )
% 0.31/1.45 => ( ( lam @ A @ B
% 0.31/1.45 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.31/1.45 = Xf ) ) ) ) ).
% 0.31/1.45
% 0.31/1.45 thf(lam2lamEq,conjecture,
% 0.31/1.45 ( setextAx
% 0.31/1.45 => ( emptysetAx
% 0.31/1.45 => ( setadjoinAx
% 0.31/1.45 => ( powersetAx
% 0.31/1.45 => ( setunionAx
% 0.31/1.45 => ( omega0Ax
% 0.31/1.45 => ( omegaSAx
% 0.31/1.45 => ( omegaIndAx
% 0.31/1.45 => ( replAx
% 0.31/1.45 => ( foundationAx
% 0.31/1.45 => ( wellorderingAx
% 0.31/1.45 => ( descrp
% 0.31/1.45 => ( dsetconstrI
% 0.31/1.45 => ( dsetconstrEL
% 0.31/1.45 => ( dsetconstrER
% 0.31/1.45 => ( exuE1
% 0.31/1.45 => ( prop2setE
% 0.31/1.45 => ( emptysetE
% 0.31/1.45 => ( emptysetimpfalse
% 0.31/1.45 => ( notinemptyset
% 0.31/1.45 => ( exuE3e
% 0.31/1.45 => ( setext
% 0.31/1.45 => ( emptyI
% 0.31/1.45 => ( noeltsimpempty
% 0.31/1.45 => ( setbeta
% 0.31/1.45 => ( nonemptyE1
% 0.31/1.45 => ( nonemptyI
% 0.31/1.45 => ( nonemptyI1
% 0.31/1.45 => ( setadjoinIL
% 0.31/1.45 => ( emptyinunitempty
% 0.31/1.45 => ( setadjoinIR
% 0.31/1.45 => ( setadjoinE
% 0.31/1.45 => ( setadjoinOr
% 0.31/1.45 => ( setoftrueEq
% 0.31/1.45 => ( powersetI
% 0.31/1.45 => ( emptyinPowerset
% 0.31/1.45 => ( emptyInPowerset
% 0.31/1.45 => ( powersetE
% 0.31/1.45 => ( setunionI
% 0.31/1.45 => ( setunionE
% 0.31/1.45 => ( subPowSU
% 0.31/1.45 => ( exuE2
% 0.31/1.45 => ( nonemptyImpWitness
% 0.31/1.45 => ( uniqinunit
% 0.31/1.45 => ( notinsingleton
% 0.31/1.45 => ( eqinunit
% 0.31/1.45 => ( singletonsswitch
% 0.31/1.45 => ( upairsetE
% 0.31/1.45 => ( upairsetIL
% 0.31/1.45 => ( upairsetIR
% 0.31/1.45 => ( emptyE1
% 0.31/1.45 => ( vacuousDall
% 0.31/1.45 => ( quantDeMorgan1
% 0.31/1.45 => ( quantDeMorgan2
% 0.31/1.45 => ( quantDeMorgan3
% 0.31/1.45 => ( quantDeMorgan4
% 0.31/1.45 => ( prop2setI
% 0.31/1.45 => ( prop2set2propI
% 0.31/1.45 => ( notdexE
% 0.31/1.45 => ( notdallE
% 0.31/1.45 => ( exuI1
% 0.31/1.45 => ( exuI3
% 0.31/1.45 => ( exuI2
% 0.31/1.45 => ( inCongP
% 0.31/1.45 => ( in__Cong
% 0.31/1.45 => ( exuE3u
% 0.31/1.45 => ( exu__Cong
% 0.31/1.45 => ( emptyset__Cong
% 0.31/1.45 => ( setadjoin__Cong
% 0.31/1.45 => ( powerset__Cong
% 0.31/1.45 => ( setunion__Cong
% 0.31/1.45 => ( omega__Cong
% 0.31/1.45 => ( exuEu
% 0.31/1.45 => ( descr__Cong
% 0.31/1.45 => ( dsetconstr__Cong
% 0.31/1.45 => ( subsetI1
% 0.31/1.45 => ( eqimpsubset2
% 0.31/1.45 => ( eqimpsubset1
% 0.31/1.45 => ( subsetI2
% 0.31/1.45 => ( emptysetsubset
% 0.31/1.45 => ( subsetE
% 0.31/1.45 => ( subsetE2
% 0.31/1.45 => ( notsubsetI
% 0.31/1.45 => ( notequalI1
% 0.31/1.45 => ( notequalI2
% 0.31/1.45 => ( subsetRefl
% 0.31/1.45 => ( subsetTrans
% 0.31/1.45 => ( setadjoinSub
% 0.31/1.45 => ( setadjoinSub2
% 0.31/1.45 => ( subset2powerset
% 0.31/1.45 => ( setextsub
% 0.31/1.45 => ( subsetemptysetimpeq
% 0.31/1.45 => ( powersetI1
% 0.31/1.45 => ( powersetE1
% 0.31/1.45 => ( inPowerset
% 0.31/1.45 => ( powersetsubset
% 0.31/1.45 => ( sepInPowerset
% 0.31/1.45 => ( sepSubset
% 0.31/1.45 => ( binunionIL
% 0.31/1.45 => ( upairset2IR
% 0.31/1.45 => ( binunionIR
% 0.31/1.45 => ( binunionEcases
% 0.31/1.45 => ( binunionE
% 0.31/1.45 => ( binunionLsub
% 0.31/1.45 => ( binunionRsub
% 0.31/1.45 => ( binintersectI
% 0.31/1.45 => ( binintersectSubset5
% 0.31/1.45 => ( binintersectEL
% 0.31/1.45 => ( binintersectLsub
% 0.31/1.45 => ( binintersectSubset2
% 0.31/1.45 => ( binintersectSubset3
% 0.31/1.45 => ( binintersectER
% 0.31/1.45 => ( disjointsetsI1
% 0.31/1.45 => ( binintersectRsub
% 0.31/1.45 => ( binintersectSubset4
% 0.31/1.45 => ( binintersectSubset1
% 0.31/1.45 => ( bs114d
% 0.31/1.45 => ( setminusI
% 0.31/1.45 => ( setminusEL
% 0.31/1.45 => ( setminusER
% 0.31/1.45 => ( setminusSubset2
% 0.31/1.45 => ( setminusERneg
% 0.31/1.45 => ( setminusELneg
% 0.31/1.45 => ( setminusILneg
% 0.31/1.45 => ( setminusIRneg
% 0.31/1.45 => ( setminusLsub
% 0.31/1.45 => ( setminusSubset1
% 0.31/1.45 => ( symdiffE
% 0.31/1.45 => ( symdiffI1
% 0.31/1.45 => ( symdiffI2
% 0.31/1.45 => ( symdiffIneg1
% 0.31/1.45 => ( symdiffIneg2
% 0.31/1.45 => ( secondinupair
% 0.31/1.45 => ( setukpairIL
% 0.31/1.45 => ( setukpairIR
% 0.31/1.45 => ( kpairiskpair
% 0.31/1.45 => ( kpairp
% 0.31/1.45 => ( singletonsubset
% 0.31/1.45 => ( singletoninpowerset
% 0.31/1.45 => ( singletoninpowunion
% 0.31/1.45 => ( upairset2E
% 0.31/1.45 => ( upairsubunion
% 0.31/1.45 => ( upairinpowunion
% 0.31/1.45 => ( ubforcartprodlem1
% 0.31/1.45 => ( ubforcartprodlem2
% 0.31/1.45 => ( ubforcartprodlem3
% 0.31/1.45 => ( cartprodpairin
% 0.31/1.45 => ( cartprodmempair1
% 0.31/1.45 => ( cartprodmempair
% 0.31/1.45 => ( setunionE2
% 0.31/1.45 => ( setunionsingleton1
% 0.31/1.45 => ( setunionsingleton2
% 0.31/1.45 => ( setunionsingleton
% 0.31/1.45 => ( singletonprop
% 0.31/1.45 => ( ex1E1
% 0.31/1.45 => ( ex1I
% 0.31/1.45 => ( ex1I2
% 0.31/1.45 => ( singletonsuniq
% 0.31/1.45 => ( setukpairinjL1
% 0.31/1.45 => ( kfstsingleton
% 0.31/1.45 => ( theprop
% 0.31/1.45 => ( kfstpairEq
% 0.31/1.45 => ( cartprodfstin
% 0.31/1.45 => ( setukpairinjL2
% 0.31/1.45 => ( setukpairinjL
% 0.31/1.45 => ( setukpairinjR11
% 0.31/1.45 => ( setukpairinjR12
% 0.31/1.45 => ( setukpairinjR1
% 0.31/1.45 => ( upairequniteq
% 0.31/1.45 => ( setukpairinjR2
% 0.31/1.45 => ( setukpairinjR
% 0.31/1.45 => ( ksndsingleton
% 0.31/1.45 => ( ksndpairEq
% 0.31/1.45 => ( kpairsurjEq
% 0.31/1.45 => ( cartprodsndin
% 0.31/1.45 => ( cartprodpairmemEL
% 0.31/1.45 => ( cartprodpairmemER
% 0.31/1.45 => ( cartprodmempaircEq
% 0.31/1.45 => ( cartprodfstpairEq
% 0.31/1.45 => ( cartprodsndpairEq
% 0.31/1.45 => ( cartprodpairsurjEq
% 0.31/1.45 => ( dpsetconstrI
% 0.31/1.45 => ( dpsetconstrSub
% 0.31/1.45 => ( setOfPairsIsBReln
% 0.31/1.45 => ( dpsetconstrERa
% 0.31/1.45 => ( dpsetconstrEL1
% 0.31/1.45 => ( dpsetconstrEL2
% 0.31/1.45 => ( dpsetconstrER
% 0.31/1.45 => ( funcImageSingleton
% 0.31/1.45 => ( apProp
% 0.31/1.45 => ( app
% 0.31/1.45 => ( infuncsetfunc
% 0.31/1.45 => ( ap2p
% 0.31/1.45 => ( funcinfuncset
% 0.31/1.45 => ( lamProp
% 0.31/1.45 => ( lamp
% 0.31/1.45 => ( lam2p
% 0.31/1.45 => ( brelnall1
% 0.31/1.45 => ( brelnall2
% 0.31/1.45 => ( ex1E2
% 0.31/1.45 => ( funcGraphProp1
% 0.31/1.45 => ( funcGraphProp3
% 0.31/1.45 => ( funcGraphProp2
% 0.31/1.45 => ( funcextLem
% 0.31/1.45 => ( funcGraphProp4
% 0.31/1.45 => ( subbreln
% 0.31/1.45 => ( eqbreln
% 0.31/1.45 => ( funcext
% 0.31/1.45 => ( funcext2
% 0.31/1.45 => ( ap2apEq1
% 0.31/1.45 => ( ap2apEq2
% 0.31/1.45 => ( beta1
% 0.31/1.45 => ( eta1
% 0.31/1.45 => ! [A: $i,B: $i,Xf: $i > $i] :
% 0.31/1.45 ( ! [Xx: $i] :
% 0.31/1.45 ( ( in @ Xx @ A )
% 0.31/1.45 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.31/1.45 => ( ( lam @ A @ B
% 0.31/1.45 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.31/1.45 = ( lam @ A @ B
% 0.31/1.45 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.31/1.45 ------------------------
% 0.31/1.47 % (25018)dis+1010_1:1_au=on:cbe=off:chr=on:fsr=off:hfsq=on:nm=64:sos=theory:sp=weighted_frequency:i=27:si=on:rtra=on_0 on DTF2THF_24917 for (2999ds/27Mi)
% 0.31/1.47 % (25016)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_24917 for (2999ds/183Mi)
% 0.31/1.48 % (25018)Instruction limit reached!
% 0.31/1.48 % (25018)------------------------------
% 0.31/1.48 % (25018)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.31/1.48 % (25018)Termination reason: Unknown
% 0.31/1.48 % (25018)Termination phase: shuffling
% 0.31/1.48
% 0.31/1.48 % (25018)Memory used [KB]: 1918
% 0.31/1.48 % (25018)Time elapsed: 0.009 s
% 0.31/1.48 % (25018)Instructions burned: 27 (million)
% 0.31/1.48 % (25018)------------------------------
% 0.31/1.48 % (25018)------------------------------
% 0.31/1.48 % (25021)lrs+1002_1:1_au=on:bd=off:e2e=on:sd=2:sos=on:ss=axioms:i=275:si=on:rtra=on_0 on DTF2THF_24917 for (2999ds/275Mi)
% 0.31/1.48 % (25020)lrs+1002_1:128_aac=none:au=on:cnfonf=lazy_not_gen_be_off:sos=all:i=2:si=on:rtra=on_0 on DTF2THF_24917 for (2999ds/2Mi)
% 0.31/1.48 % (25017)lrs+10_1:1_c=on:cnfonf=conj_eager:fd=off:fe=off:kws=frequency:spb=intro:i=4:si=on:rtra=on_0 on DTF2THF_24917 for (2999ds/4Mi)
% 0.31/1.48 % (25019)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_24917 for (2999ds/2Mi)
% 0.31/1.48 % (25022)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_24917 for (2999ds/18Mi)
% 0.31/1.48 % (25019)Instruction limit reached!
% 0.31/1.48 % (25019)------------------------------
% 0.31/1.48 % (25019)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.31/1.48 % (25019)Termination reason: Unknown
% 0.31/1.48 % (25019)Termination phase: shuffling
% 0.31/1.48
% 0.31/1.48 % (25019)Memory used [KB]: 1407
% 0.31/1.48 % (25019)Time elapsed: 0.003 s
% 0.31/1.48 % (25019)Instructions burned: 3 (million)
% 0.31/1.48 % (25020)Instruction limit reached!
% 0.31/1.48 % (25020)------------------------------
% 0.31/1.48 % (25020)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.31/1.48 % (25019)------------------------------
% 0.31/1.48 % (25019)------------------------------
% 0.31/1.48 % (25020)Termination reason: Unknown
% 0.31/1.48 % (25020)Termination phase: shuffling
% 0.31/1.48
% 0.31/1.48 % (25020)Memory used [KB]: 1407
% 0.31/1.48 % (25020)Time elapsed: 0.003 s
% 0.31/1.48 % (25020)Instructions burned: 3 (million)
% 0.31/1.48 % (25020)------------------------------
% 0.31/1.48 % (25020)------------------------------
% 0.31/1.48 % (25017)Instruction limit reached!
% 0.31/1.48 % (25017)------------------------------
% 0.31/1.48 % (25017)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.31/1.48 % (25017)Termination reason: Unknown
% 0.31/1.48 % (25017)Termination phase: shuffling
% 0.31/1.48
% 0.31/1.48 % (25017)Memory used [KB]: 1407
% 0.31/1.48 % (25017)Time elapsed: 0.004 s
% 0.31/1.48 % (25017)Instructions burned: 5 (million)
% 0.31/1.48 % (25017)------------------------------
% 0.31/1.48 % (25017)------------------------------
% 0.31/1.49 % (25023)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_24917 for (2999ds/3Mi)
% 0.31/1.49 % (25022)Instruction limit reached!
% 0.31/1.49 % (25022)------------------------------
% 0.31/1.49 % (25022)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.31/1.49 % (25022)Termination reason: Unknown
% 0.31/1.49 % (25022)Termination phase: shuffling
% 0.31/1.49
% 0.31/1.49 % (25022)Memory used [KB]: 1663
% 0.31/1.49 % (25022)Time elapsed: 0.013 s
% 0.31/1.49 % (25022)Instructions burned: 18 (million)
% 0.31/1.49 % (25022)------------------------------
% 0.31/1.49 % (25022)------------------------------
% 0.31/1.49 % (25023)Instruction limit reached!
% 0.31/1.49 % (25023)------------------------------
% 0.31/1.49 % (25023)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.31/1.49 % (25023)Termination reason: Unknown
% 0.31/1.49 % (25023)Termination phase: shuffling
% 0.31/1.49
% 0.31/1.49 % (25023)Memory used [KB]: 1407
% 0.31/1.49 % (25023)Time elapsed: 0.003 s
% 0.31/1.49 % (25023)Instructions burned: 6 (million)
% 0.31/1.49 % (25023)------------------------------
% 0.31/1.49 % (25023)------------------------------
% 0.31/1.50 % (25024)lrs+1002_1:1_cnfonf=lazy_not_be_gen:hud=14:prag=on:sp=weighted_frequency:tnu=1:i=37:si=on:rtra=on_0 on DTF2THF_24917 for (2999ds/37Mi)
% 2.57/1.50 % (25025)lrs+2_16:1_acc=model:au=on:bd=off:c=on:e2e=on:nm=2:sos=all:i=15:si=on:rtra=on_0 on DTF2THF_24917 for (2999ds/15Mi)
% 2.57/1.50 % (25026)dis+21_1:1_cbe=off:cnfonf=off:fs=off:fsr=off:hud=1:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_24917 for (2999ds/3Mi)
% 2.57/1.50 % (25027)lrs+1002_1:1_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=4203469,65536:i=1041:si=on:rtra=on_0 on DTF2THF_24917 for (2999ds/1041Mi)
% 2.57/1.50 % (25026)Instruction limit reached!
% 2.57/1.50 % (25026)------------------------------
% 2.57/1.50 % (25026)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.57/1.50 % (25026)Termination reason: Unknown
% 2.57/1.50 % (25026)Termination phase: shuffling
% 2.57/1.50
% 2.57/1.50 % (25026)Memory used [KB]: 1407
% 2.57/1.50 % (25026)Time elapsed: 0.004 s
% 2.57/1.50 % (25026)Instructions burned: 3 (million)
% 2.57/1.50 % (25026)------------------------------
% 2.57/1.50 % (25026)------------------------------
% 2.57/1.51 % (25028)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_24917 for (2999ds/7Mi)
% 2.57/1.51 % (25025)Instruction limit reached!
% 2.57/1.51 % (25025)------------------------------
% 2.57/1.51 % (25025)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.57/1.51 % (25025)Termination reason: Unknown
% 2.57/1.51 % (25025)Termination phase: shuffling
% 2.57/1.51
% 2.57/1.51 % (25025)Memory used [KB]: 1663
% 2.57/1.51 % (25025)Time elapsed: 0.012 s
% 2.57/1.51 % (25025)Instructions burned: 16 (million)
% 2.57/1.51 % (25025)------------------------------
% 2.57/1.51 % (25025)------------------------------
% 2.57/1.51 % (25028)Instruction limit reached!
% 2.57/1.51 % (25028)------------------------------
% 2.57/1.51 % (25028)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.57/1.51 % (25028)Termination reason: Unknown
% 2.57/1.51 % (25028)Termination phase: shuffling
% 2.57/1.51
% 2.57/1.51 % (25028)Memory used [KB]: 1535
% 2.57/1.51 % (25028)Time elapsed: 0.006 s
% 2.57/1.51 % (25028)Instructions burned: 7 (million)
% 2.57/1.51 % (25028)------------------------------
% 2.57/1.51 % (25028)------------------------------
% 2.57/1.51 % (25024)Instruction limit reached!
% 2.57/1.51 % (25024)------------------------------
% 2.57/1.51 % (25024)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.57/1.51 % (25024)Termination reason: Unknown
% 2.57/1.51 % (25024)Termination phase: Property scanning
% 2.57/1.51
% 2.57/1.51 % (25024)Memory used [KB]: 2174
% 2.57/1.51 % (25024)Time elapsed: 0.020 s
% 2.57/1.51 % (25024)Instructions burned: 39 (million)
% 2.57/1.51 % (25024)------------------------------
% 2.57/1.51 % (25024)------------------------------
% 2.57/1.52 % (25016)First to succeed.
% 2.57/1.52 % (25029)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_24917 for (2999ds/16Mi)
% 2.57/1.52 % (25016)Refutation found. Thanks to Tanya!
% 2.57/1.52 % SZS status Theorem for DTF2THF_24917
% 2.57/1.52 % SZS output start Proof for DTF2THF_24917
% 2.57/1.52 thf(func_def_0, type, in: $i > $i > $o).
% 2.57/1.52 thf(func_def_1, type, exu: ($i > $o) > $o).
% 2.57/1.52 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 2.57/1.52 thf(func_def_8, type, powerset: $i > $i).
% 2.57/1.52 thf(func_def_10, type, setunion: $i > $i).
% 2.57/1.52 thf(func_def_19, type, descr: ($i > $o) > $i).
% 2.57/1.52 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 2.57/1.52 thf(func_def_26, type, prop2set: $o > $i).
% 2.57/1.52 thf(func_def_36, type, nonempty: $i > $o).
% 2.57/1.52 thf(func_def_69, type, set2prop: $i > $o).
% 2.57/1.52 thf(func_def_88, type, subset: $i > $i > $o).
% 2.57/1.52 thf(func_def_89, type, disjoint: $i > $i > $o).
% 2.57/1.52 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 2.57/1.52 thf(func_def_114, type, binunion: $i > $i > $i).
% 2.57/1.52 thf(func_def_122, type, binintersect: $i > $i > $i).
% 2.57/1.52 thf(func_def_135, type, regular: $i > $o).
% 2.57/1.52 thf(func_def_136, type, setminus: $i > $i > $i).
% 2.57/1.52 thf(func_def_147, type, symdiff: $i > $i > $i).
% 2.57/1.52 thf(func_def_153, type, iskpair: $i > $o).
% 2.57/1.52 thf(func_def_158, type, kpair: $i > $i > $i).
% 2.57/1.52 thf(func_def_160, type, cartprod: $i > $i > $i).
% 2.57/1.52 thf(func_def_177, type, singleton: $i > $o).
% 2.57/1.52 thf(func_def_179, type, ex1: $i > ($i > $o) > $o).
% 2.57/1.52 thf(func_def_184, type, atmost1p: $i > $o).
% 2.57/1.52 thf(func_def_185, type, atleast2p: $i > $o).
% 2.57/1.52 thf(func_def_186, type, atmost2p: $i > $o).
% 2.57/1.52 thf(func_def_187, type, upairsetp: $i > $o).
% 2.57/1.52 thf(func_def_191, type, kfst: $i > $i).
% 2.57/1.52 thf(func_def_203, type, ksnd: $i > $i).
% 2.57/1.52 thf(func_def_213, type, breln: $i > $i > $i > $o).
% 2.57/1.52 thf(func_def_214, type, dpsetconstr: $i > $i > ($i > $i > $o) > $i).
% 2.57/1.52 thf(func_def_222, type, func: $i > $i > $i > $o).
% 2.57/1.52 thf(func_def_223, type, funcSet: $i > $i > $i).
% 2.57/1.52 thf(func_def_226, type, ap: $i > $i > $i > $i > $i).
% 2.57/1.52 thf(func_def_232, type, lam: $i > $i > ($i > $i) > $i).
% 2.57/1.52 thf(func_def_269, type, sK0: $i > $i).
% 2.57/1.52 thf(func_def_273, type, sK4: ($i > $i) > $i > $i > $i).
% 2.57/1.52 thf(func_def_274, type, sK5: $i > $i > ($i > $o) > $i > $i).
% 2.57/1.52 thf(func_def_275, type, sK6: $i > $i > $i > ($i > $o) > $i).
% 2.57/1.52 thf(func_def_276, type, sK7: $i > $i > ($i > $o) > ($i > $o) > $i).
% 2.57/1.52 thf(func_def_277, type, sK8: $i > ($i > $o) > ($i > $o) > $i > $i).
% 2.57/1.52 thf(func_def_278, type, sK9: ($i > $i) > $i > $i > $i).
% 2.57/1.52 thf(func_def_279, type, sK10: $i > $i).
% 2.57/1.52 thf(func_def_280, type, sK11: ($i > $o) > $i).
% 2.57/1.52 thf(func_def_281, type, sK12: ($i > $o) > $i > $i).
% 2.57/1.52 thf(func_def_282, type, sK13: $i > ($i > $o) > $i).
% 2.57/1.52 thf(func_def_283, type, sK14: ($i > $o) > ($i > $o) > $i).
% 2.57/1.52 thf(func_def_284, type, sK15: ($i > $o) > ($i > $o) > $i).
% 2.57/1.52 thf(func_def_285, type, sK16: ($i > $o) > $i > $i).
% 2.57/1.52 thf(func_def_286, type, sK17: ($i > $o) > $i > $i).
% 2.57/1.52 thf(func_def_287, type, sK18: ($i > $o) > $i > $i).
% 2.57/1.52 thf(func_def_288, type, sK19: ($i > $o) > $i > $i).
% 2.57/1.52 thf(func_def_289, type, sK20: ($i > $o) > $i > $i).
% 2.57/1.52 thf(func_def_290, type, sK21: ($i > $o) > $i > $i).
% 2.57/1.52 thf(func_def_291, type, sK22: ($i > $o) > $i > $i).
% 2.57/1.52 thf(func_def_292, type, sK23: ($i > $o) > $i > $i).
% 2.57/1.52 thf(func_def_293, type, sK24: $i > $i).
% 2.57/1.52 thf(func_def_294, type, sK25: $i > $i > $i > $i > $i).
% 2.57/1.52 thf(func_def_295, type, sK26: $i > $i > $i > $i > $i).
% 2.57/1.52 thf(func_def_296, type, sK27: $i > $i > $i > $i > $i).
% 2.57/1.52 thf(func_def_297, type, sK28: $i > $i > $i > $i > $i).
% 2.57/1.52 thf(func_def_298, type, sK29: $i > $i > $i > $i > $i).
% 2.57/1.52 thf(func_def_299, type, sK30: $i > $i > $i > $i > $i).
% 2.57/1.52 thf(func_def_300, type, sK31: $i > $i > $i > $i > $i).
% 2.57/1.52 thf(func_def_301, type, sK32: $i > ($i > $o) > $i).
% 2.57/1.52 thf(func_def_302, type, sK33: ($i > $o) > $i).
% 2.57/1.52 thf(func_def_303, type, sK34: ($i > $o) > $i).
% 2.57/1.52 thf(func_def_304, type, sK35: ($i > $o) > $i).
% 2.57/1.52 thf(func_def_305, type, sK36: ($i > $o) > $i).
% 2.57/1.52 thf(func_def_306, type, sK37: ($i > $o) > $i).
% 2.57/1.52 thf(func_def_307, type, sK38: ($i > $o) > $i > $i).
% 2.57/1.52 thf(func_def_308, type, sK39: ($i > $o) > $i > $i).
% 2.57/1.52 thf(func_def_309, type, sK40: ($i > $o) > $i > $i).
% 2.57/1.52 thf(func_def_310, type, sK41: ($i > $o) > $i > $i).
% 2.57/1.52 thf(func_def_311, type, sK42: ($i > $o) > $i > $i).
% 2.57/1.52 thf(func_def_312, type, sK43: ($i > $o) > $i > $i).
% 2.57/1.52 thf(f2095,plain,(
% 2.57/1.52 $false),
% 2.57/1.52 inference(trivial_inequality_removal,[status(thm)],[f1177])).
% 2.57/1.52 thf(f1177,plain,(
% 2.57/1.52 (((^[Y0 : $i]: ((^[Y1 : $i]: ((^[Y2 : $i > $i]: (dpsetconstr @ Y0 @ Y1 @ (^[Y3 : $i]: ((^[Y4 : $i]: ((Y2 @ Y3) = Y4)))))))))) @ sK1 @ sK2 @ (^[Y0 : $i]: (sK0 @ Y0))) != ((^[Y0 : $i]: ((^[Y1 : $i]: ((^[Y2 : $i > $i]: (dpsetconstr @ Y0 @ Y1 @ (^[Y3 : $i]: ((^[Y4 : $i]: ((Y2 @ Y3) = Y4)))))))))) @ sK1 @ sK2 @ (^[Y0 : $i]: (sK0 @ Y0))))),
% 2.57/1.52 inference(definition_unfolding,[status(thm)],[f870,f767,f767])).
% 2.57/1.52 thf(f767,plain,(
% 2.57/1.52 (lam = (^[Y0 : $i]: ((^[Y1 : $i]: ((^[Y2 : $i > $i]: (dpsetconstr @ Y0 @ Y1 @ (^[Y3 : $i]: ((^[Y4 : $i]: ((Y2 @ Y3) = Y4)))))))))))),
% 2.57/1.52 inference(cnf_transformation,[status(thm)],[f398])).
% 2.57/1.52 thf(f398,plain,(
% 2.57/1.52 (lam = (^[Y0 : $i]: ((^[Y1 : $i]: ((^[Y2 : $i > $i]: (dpsetconstr @ Y0 @ Y1 @ (^[Y3 : $i]: ((^[Y4 : $i]: ((Y2 @ Y3) = Y4)))))))))))),
% 2.57/1.52 inference(fool_elimination,[status(thm)],[f397])).
% 2.57/1.52 thf(f397,plain,(
% 2.57/1.52 (lam = (^[X0 : $i, X1 : $i, X2 : $i > $i] : (dpsetconstr @ X0 @ X1 @ (^[X3 : $i, X4 : $i] : (((X2 @ X3) = X4))))))),
% 2.57/1.52 inference(rectify,[status(thm)],[f201])).
% 2.57/1.52 thf(f201,axiom,(
% 2.57/1.52 (lam = (^[X3 : $i, X4 : $i, X12 : $i > $i] : (dpsetconstr @ X3 @ X4 @ (^[X1 : $i, X2 : $i] : (((X12 @ X1) = X2))))))),
% 2.57/1.52 file('/export/starexec/sandbox2/tmp/tmp.EApmVaGZ7F/DTF2THF_24917.p',lam)).
% 2.57/1.52 thf(f870,plain,(
% 2.57/1.52 ((lam @ sK1 @ sK2 @ (^[Y0 : $i]: (sK0 @ Y0))) != (lam @ sK1 @ sK2 @ (^[Y0 : $i]: (sK0 @ Y0))))),
% 2.57/1.52 inference(cnf_transformation,[status(thm)],[f660])).
% 2.57/1.52 thf(f660,plain,(
% 2.57/1.52 (setminusILneg = $true) & (exuE3e = $true) & (subsetE2 = $true) & (singletoninpowunion = $true) & (symdiffIneg2 = $true) & (subsetE = $true) & (binunionLsub = $true) & (setunionsingleton1 = $true) & (singletonprop = $true) & (emptyE1 = $true) & (upairsetE = $true) & (setukpairIR = $true) & (setextAx = $true) & (subsetTrans = $true) & (ex1I = $true) & (dpsetconstrERa = $true) & (ubforcartprodlem3 = $true) & (funcext = $true) & (notsubsetI = $true) & (dsetconstrEL = $true) & (inCongP = $true) & (funcGraphProp1 = $true) & (binunionIL = $true) & (notdallE = $true) & (exuE2 = $true) & (setunionE = $true) & (cartprodmempaircEq = $true) & (singletonsswitch = $true) & (eqinunit = $true) & (setadjoinE = $true) & (funcinfuncset = $true) & (kpairp = $true) & (setadjoinSub2 = $true) & (symdiffIneg1 = $true) & (symdiffI1 = $true) & (setukpairinjR12 = $true) & (cartprodpairin = $true) & (subsetI2 = $true) & (bs114d = $true) & (nonemptyI1 = $true) & (cartprodsndpairEq = $true) & (funcGraphProp2 = $true) & (cartprodfstin = $true) & (setukpairinjR11 = $true) & (powerset__Cong = $true) & (powersetAx = $true) & (lam2p = $true) & (brelnall2 = $true) & (setukpairinjL2 = $true) & (subbreln = $true) & (upairinpowunion = $true) & (cartprodpairmemER = $true) & (powersetsubset = $true) & (lamProp = $true) & (nonemptyImpWitness = $true) & (upairsetIL = $true) & (ubforcartprodlem1 = $true) & (funcextLem = $true) & (nonemptyE1 = $true) & (subsetI1 = $true) & (setadjoinOr = $true) & (exu__Cong = $true) & (setminusEL = $true) & (prop2set2propI = $true) & (notinemptyset = $true) & (eqbreln = $true) & (setunionE2 = $true) & (emptysetimpfalse = $true) & (binintersectER = $true) & (dpsetconstrER = $true) & (emptyI = $true) & (vacuousDall = $true) & (app = $true) & (eta1 = $true) & (upairsubunion = $true) & (cartprodpairmemEL = $true) & (quantDeMorgan4 = $true) & (symdiffI2 = $true) & (ex1E2 = $true) & (binunionE = $true) & (kfstsingleton = $true) & (binintersectSubset2 = $true) & (beta1 = $true) & (kpairiskpair = $true) & (secondinupair = $true) & (quantDeMorgan1 = $true) & (lamp = $true) & (descrp = $true) & (dpsetconstrSub = $true) & (setminusIRneg = $true) & (setoftrueEq = $true) & (ksndsingleton = $true) & (ksndpairEq = $true) & (sepSubset = $true) & (setadjoin__Cong = $true) & (binintersectSubset4 = $true) & (theprop = $true) & (setadjoinAx = $true) & (notequalI2 = $true) & (binunionEcases = $true) & (setukpairinjR = $true) & (binintersectSubset5 = $true) & (funcext2 = $true) & (setukpairinjL = $true) & (binintersectSubset1 = $true) & (setOfPairsIsBReln = $true) & (cartprodmempair = $true) & (omegaIndAx = $true) & (singletoninpowerset = $true) & (foundationAx = $true) & (upairset2E = $true) & (wellorderingAx = $true) & (infuncsetfunc = $true) & (emptysetE = $true) & (sepInPowerset = $true) & (binintersectSubset3 = $true) & (((lam @ sK1 @ sK2 @ (^[Y0 : $i]: (sK0 @ Y0))) != (lam @ sK1 @ sK2 @ (^[Y0 : $i]: (sK0 @ Y0)))) & ! [X3 : $i] : (((in @ X3 @ sK1) != $true) | ((in @ (sK0 @ X3) @ sK2) = $true))) & (dsetconstrER = $true) & (funcGraphProp3 = $true) & (funcGraphProp4 = $true) & (emptysetAx = $true) & (inPowerset = $true) & (notinsingleton = $true) & (upairequniteq = $true) & (setunionsingleton2 = $true) & (emptyinunitempty = $true) & (emptyInPowerset = $true) & (setadjoinIR = $true) & (ap2apEq2 = $true) & (setminusERneg = $true) & (powersetE1 = $true) & (emptyset__Cong = $true) & (setminusLsub = $true) & (setukpairIL = $true) & (eqimpsubset1 = $true) & (noeltsimpempty = $true) & (binintersectLsub = $true) & (dpsetconstrEL1 = $true) & (setbeta = $true) & (kfstpairEq = $true) & (exuEu = $true) & (setadjoinSub = $true) & (exuE1 = $true) & (symdiffE = $true) & (cartprodfstpairEq = $true) & (setextsub = $true) & (setukpairinjR1 = $true) & (descr__Cong = $true) & (binintersectRsub = $true) & (quantDeMorgan3 = $true) & (exuI3 = $true) & (setminusSubset2 = $true) & (cartprodmempair1 = $true) & (nonemptyI = $true) & (dpsetconstrI = $true) & (omega0Ax = $true) & (setunionsingleton = $true) & (subPowSU = $true) & (cartprodpairsurjEq = $true) & (singletonsubset = $true) & (ex1I2 = $true) & (binunionIR = $true) & (emptysetsubset = $true) & (dsetconstr__Cong = $true) & (quantDeMorgan2 = $true) & (binunionRsub = $true) & (setunionI = $true) & (setminusI = $true) & (powersetI1 = $true) & (ubforcartprodlem2 = $true) & (powersetE = $true) & (omegaSAx = $true) & (powersetI = $true) & (upairsetIR = $true) & (setminusELneg = $true) & (funcImageSingleton = $true) & (notdexE = $true) & (setukpairinjR2 = $true) & (dpsetconstrEL2 = $true) & (ap2p = $true) & (apProp = $true) & (replAx = $true) & (subsetRefl = $true) & (setunion__Cong = $true) & (singletonsuniq = $true) & (kpairsurjEq = $true) & (upairset2IR = $true) & (setukpairinjL1 = $true) & (binintersectI = $true) & (disjointsetsI1 = $true) & (dsetconstrI = $true) & (exuE3u = $true) & (setunionAx = $true) & (ex1E1 = $true) & (subset2powerset = $true) & (brelnall1 = $true) & (binintersectEL = $true) & (eqimpsubset2 = $true) & (prop2setI = $true) & (ap2apEq1 = $true) & (setminusER = $true) & (notequalI1 = $true) & (exuI2 = $true) & (uniqinunit = $true) & (cartprodsndin = $true) & (in__Cong = $true) & (omega__Cong = $true) & (subsetemptysetimpeq = $true) & (setminusSubset1 = $true) & (prop2setE = $true) & (emptyinPowerset = $true) & (exuI1 = $true) & (setext = $true) & (setadjoinIL = $true)),
% 2.57/1.52 inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1,sK2])],[f658,f659])).
% 2.57/1.52 thf(f659,plain,(
% 2.57/1.52 ? [X0 : $i > $i,X1 : $i,X2 : $i] : (((lam @ X1 @ X2 @ (^[Y0 : $i]: (X0 @ Y0))) != (lam @ X1 @ X2 @ (^[Y0 : $i]: (X0 @ Y0)))) & ! [X3 : $i] : (($true != (in @ X3 @ X1)) | ((in @ (X0 @ X3) @ X2) = $true))) => (((lam @ sK1 @ sK2 @ (^[Y0 : $i]: (sK0 @ Y0))) != (lam @ sK1 @ sK2 @ (^[Y0 : $i]: (sK0 @ Y0)))) & ! [X3 : $i] : (((in @ X3 @ sK1) != $true) | ((in @ (sK0 @ X3) @ sK2) = $true)))),
% 2.57/1.52 introduced(definition,[],[choice_axiom])).
% 2.57/1.52 thf(f658,plain,(
% 2.57/1.52 (setminusILneg = $true) & (exuE3e = $true) & (subsetE2 = $true) & (singletoninpowunion = $true) & (symdiffIneg2 = $true) & (subsetE = $true) & (binunionLsub = $true) & (setunionsingleton1 = $true) & (singletonprop = $true) & (emptyE1 = $true) & (upairsetE = $true) & (setukpairIR = $true) & (setextAx = $true) & (subsetTrans = $true) & (ex1I = $true) & (dpsetconstrERa = $true) & (ubforcartprodlem3 = $true) & (funcext = $true) & (notsubsetI = $true) & (dsetconstrEL = $true) & (inCongP = $true) & (funcGraphProp1 = $true) & (binunionIL = $true) & (notdallE = $true) & (exuE2 = $true) & (setunionE = $true) & (cartprodmempaircEq = $true) & (singletonsswitch = $true) & (eqinunit = $true) & (setadjoinE = $true) & (funcinfuncset = $true) & (kpairp = $true) & (setadjoinSub2 = $true) & (symdiffIneg1 = $true) & (symdiffI1 = $true) & (setukpairinjR12 = $true) & (cartprodpairin = $true) & (subsetI2 = $true) & (bs114d = $true) & (nonemptyI1 = $true) & (cartprodsndpairEq = $true) & (funcGraphProp2 = $true) & (cartprodfstin = $true) & (setukpairinjR11 = $true) & (powerset__Cong = $true) & (powersetAx = $true) & (lam2p = $true) & (brelnall2 = $true) & (setukpairinjL2 = $true) & (subbreln = $true) & (upairinpowunion = $true) & (cartprodpairmemER = $true) & (powersetsubset = $true) & (lamProp = $true) & (nonemptyImpWitness = $true) & (upairsetIL = $true) & (ubforcartprodlem1 = $true) & (funcextLem = $true) & (nonemptyE1 = $true) & (subsetI1 = $true) & (setadjoinOr = $true) & (exu__Cong = $true) & (setminusEL = $true) & (prop2set2propI = $true) & (notinemptyset = $true) & (eqbreln = $true) & (setunionE2 = $true) & (emptysetimpfalse = $true) & (binintersectER = $true) & (dpsetconstrER = $true) & (emptyI = $true) & (vacuousDall = $true) & (app = $true) & (eta1 = $true) & (upairsubunion = $true) & (cartprodpairmemEL = $true) & (quantDeMorgan4 = $true) & (symdiffI2 = $true) & (ex1E2 = $true) & (binunionE = $true) & (kfstsingleton = $true) & (binintersectSubset2 = $true) & (beta1 = $true) & (kpairiskpair = $true) & (secondinupair = $true) & (quantDeMorgan1 = $true) & (lamp = $true) & (descrp = $true) & (dpsetconstrSub = $true) & (setminusIRneg = $true) & (setoftrueEq = $true) & (ksndsingleton = $true) & (ksndpairEq = $true) & (sepSubset = $true) & (setadjoin__Cong = $true) & (binintersectSubset4 = $true) & (theprop = $true) & (setadjoinAx = $true) & (notequalI2 = $true) & (binunionEcases = $true) & (setukpairinjR = $true) & (binintersectSubset5 = $true) & (funcext2 = $true) & (setukpairinjL = $true) & (binintersectSubset1 = $true) & (setOfPairsIsBReln = $true) & (cartprodmempair = $true) & (omegaIndAx = $true) & (singletoninpowerset = $true) & (foundationAx = $true) & (upairset2E = $true) & (wellorderingAx = $true) & (infuncsetfunc = $true) & (emptysetE = $true) & (sepInPowerset = $true) & (binintersectSubset3 = $true) & ? [X0 : $i > $i,X1 : $i,X2 : $i] : (((lam @ X1 @ X2 @ (^[Y0 : $i]: (X0 @ Y0))) != (lam @ X1 @ X2 @ (^[Y0 : $i]: (X0 @ Y0)))) & ! [X3 : $i] : (($true != (in @ X3 @ X1)) | ((in @ (X0 @ X3) @ X2) = $true))) & (dsetconstrER = $true) & (funcGraphProp3 = $true) & (funcGraphProp4 = $true) & (emptysetAx = $true) & (inPowerset = $true) & (notinsingleton = $true) & (upairequniteq = $true) & (setunionsingleton2 = $true) & (emptyinunitempty = $true) & (emptyInPowerset = $true) & (setadjoinIR = $true) & (ap2apEq2 = $true) & (setminusERneg = $true) & (powersetE1 = $true) & (emptyset__Cong = $true) & (setminusLsub = $true) & (setukpairIL = $true) & (eqimpsubset1 = $true) & (noeltsimpempty = $true) & (binintersectLsub = $true) & (dpsetconstrEL1 = $true) & (setbeta = $true) & (kfstpairEq = $true) & (exuEu = $true) & (setadjoinSub = $true) & (exuE1 = $true) & (symdiffE = $true) & (cartprodfstpairEq = $true) & (setextsub = $true) & (setukpairinjR1 = $true) & (descr__Cong = $true) & (binintersectRsub = $true) & (quantDeMorgan3 = $true) & (exuI3 = $true) & (setminusSubset2 = $true) & (cartprodmempair1 = $true) & (nonemptyI = $true) & (dpsetconstrI = $true) & (omega0Ax = $true) & (setunionsingleton = $true) & (subPowSU = $true) & (cartprodpairsurjEq = $true) & (singletonsubset = $true) & (ex1I2 = $true) & (binunionIR = $true) & (emptysetsubset = $true) & (dsetconstr__Cong = $true) & (quantDeMorgan2 = $true) & (binunionRsub = $true) & (setunionI = $true) & (setminusI = $true) & (powersetI1 = $true) & (ubforcartprodlem2 = $true) & (powersetE = $true) & (omegaSAx = $true) & (powersetI = $true) & (upairsetIR = $true) & (setminusELneg = $true) & (funcImageSingleton = $true) & (notdexE = $true) & (setukpairinjR2 = $true) & (dpsetconstrEL2 = $true) & (ap2p = $true) & (apProp = $true) & (replAx = $true) & (subsetRefl = $true) & (setunion__Cong = $true) & (singletonsuniq = $true) & (kpairsurjEq = $true) & (upairset2IR = $true) & (setukpairinjL1 = $true) & (binintersectI = $true) & (disjointsetsI1 = $true) & (dsetconstrI = $true) & (exuE3u = $true) & (setunionAx = $true) & (ex1E1 = $true) & (subset2powerset = $true) & (brelnall1 = $true) & (binintersectEL = $true) & (eqimpsubset2 = $true) & (prop2setI = $true) & (ap2apEq1 = $true) & (setminusER = $true) & (notequalI1 = $true) & (exuI2 = $true) & (uniqinunit = $true) & (cartprodsndin = $true) & (in__Cong = $true) & (omega__Cong = $true) & (subsetemptysetimpeq = $true) & (setminusSubset1 = $true) & (prop2setE = $true) & (emptyinPowerset = $true) & (exuI1 = $true) & (setext = $true) & (setadjoinIL = $true)),
% 2.57/1.52 inference(flattening,[status(thm)],[f657])).
% 2.57/1.52 thf(f657,plain,(
% 2.57/1.52 ((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i > $i,X1 : $i,X2 : $i] : (((lam @ X1 @ X2 @ (^[Y0 : $i]: (X0 @ Y0))) != (lam @ X1 @ X2 @ (^[Y0 : $i]: (X0 @ Y0)))) & ! [X3 : $i] : (($true != (in @ X3 @ X1)) | ((in @ (X0 @ X3) @ X2) = $true))) & (eta1 = $true)) & (beta1 = $true)) & (ap2apEq2 = $true)) & (ap2apEq1 = $true)) & (funcext2 = $true)) & (funcext = $true)) & (eqbreln = $true)) & (subbreln = $true)) & (funcGraphProp4 = $true)) & (funcextLem = $true)) & (funcGraphProp2 = $true)) & (funcGraphProp3 = $true)) & (funcGraphProp1 = $true)) & (ex1E2 = $true)) & (brelnall2 = $true)) & (brelnall1 = $true)) & (lam2p = $true)) & (lamp = $true)) & (lamProp = $true)) & (funcinfuncset = $true)) & (ap2p = $true)) & (infuncsetfunc = $true)) & (app = $true)) & (apProp = $true)) & (funcImageSingleton = $true)) & (dpsetconstrER = $true)) & (dpsetconstrEL2 = $true)) & (dpsetconstrEL1 = $true)) & (dpsetconstrERa = $true)) & (setOfPairsIsBReln = $true)) & (dpsetconstrSub = $true)) & (dpsetconstrI = $true)) & (cartprodpairsurjEq = $true)) & (cartprodsndpairEq = $true)) & (cartprodfstpairEq = $true)) & (cartprodmempaircEq = $true)) & (cartprodpairmemER = $true)) & (cartprodpairmemEL = $true)) & (cartprodsndin = $true)) & (kpairsurjEq = $true)) & (ksndpairEq = $true)) & (ksndsingleton = $true)) & (setukpairinjR = $true)) & (setukpairinjR2 = $true)) & (upairequniteq = $true)) & (setukpairinjR1 = $true)) & (setukpairinjR12 = $true)) & (setukpairinjR11 = $true)) & (setukpairinjL = $true)) & (setukpairinjL2 = $true)) & (cartprodfstin = $true)) & (kfstpairEq = $true)) & (theprop = $true)) & (kfstsingleton = $true)) & (setukpairinjL1 = $true)) & (singletonsuniq = $true)) & (ex1I2 = $true)) & (ex1I = $true)) & (ex1E1 = $true)) & (singletonprop = $true)) & (setunionsingleton = $true)) & (setunionsingleton2 = $true)) & (setunionsingleton1 = $true)) & (setunionE2 = $true)) & (cartprodmempair = $true)) & (cartprodmempair1 = $true)) & (cartprodpairin = $true)) & (ubforcartprodlem3 = $true)) & (ubforcartprodlem2 = $true)) & (ubforcartprodlem1 = $true)) & (upairinpowunion = $true)) & (upairsubunion = $true)) & (upairset2E = $true)) & (singletoninpowunion = $true)) & (singletoninpowerset = $true)) & (singletonsubset = $true)) & (kpairp = $true)) & (kpairiskpair = $true)) & (setukpairIR = $true)) & (setukpairIL = $true)) & (secondinupair = $true)) & (symdiffIneg2 = $true)) & (symdiffIneg1 = $true)) & (symdiffI2 = $true)) & (symdiffI1 = $true)) & (symdiffE = $true)) & (setminusSubset1 = $true)) & (setminusLsub = $true)) & (setminusIRneg = $true)) & (setminusILneg = $true)) & (setminusELneg = $true)) & (setminusERneg = $true)) & (setminusSubset2 = $true)) & (setminusER = $true)) & (setminusEL = $true)) & (setminusI = $true)) & (bs114d = $true)) & (binintersectSubset1 = $true)) & (binintersectSubset4 = $true)) & (binintersectRsub = $true)) & (disjointsetsI1 = $true)) & (binintersectER = $true)) & (binintersectSubset3 = $true)) & (binintersectSubset2 = $true)) & (binintersectLsub = $true)) & (binintersectEL = $true)) & (binintersectSubset5 = $true)) & (binintersectI = $true)) & (binunionRsub = $true)) & (binunionLsub = $true)) & (binunionE = $true)) & (binunionEcases = $true)) & (binunionIR = $true)) & (upairset2IR = $true)) & (binunionIL = $true)) & (sepSubset = $true)) & (sepInPowerset = $true)) & (powersetsubset = $true)) & (inPowerset = $true)) & (powersetE1 = $true)) & (powersetI1 = $true)) & (subsetemptysetimpeq = $true)) & (setextsub = $true)) & (subset2powerset = $true)) & (setadjoinSub2 = $true)) & (setadjoinSub = $true)) & (subsetTrans = $true)) & (subsetRefl = $true)) & (notequalI2 = $true)) & (notequalI1 = $true)) & (notsubsetI = $true)) & (subsetE2 = $true)) & (subsetE = $true)) & (emptysetsubset = $true)) & (subsetI2 = $true)) & (eqimpsubset1 = $true)) & (eqimpsubset2 = $true)) & (subsetI1 = $true)) & (dsetconstr__Cong = $true)) & (descr__Cong = $true)) & (exuEu = $true)) & (omega__Cong = $true)) & (setunion__Cong = $true)) & (powerset__Cong = $true)) & (setadjoin__Cong = $true)) & (emptyset__Cong = $true)) & (exu__Cong = $true)) & (exuE3u = $true)) & (in__Cong = $true)) & (inCongP = $true)) & (exuI2 = $true)) & (exuI3 = $true)) & (exuI1 = $true)) & (notdallE = $true)) & (notdexE = $true)) & (prop2set2propI = $true)) & (prop2setI = $true)) & (quantDeMorgan4 = $true)) & (quantDeMorgan3 = $true)) & (quantDeMorgan2 = $true)) & (quantDeMorgan1 = $true)) & (vacuousDall = $true)) & (emptyE1 = $true)) & (upairsetIR = $true)) & (upairsetIL = $true)) & (upairsetE = $true)) & (singletonsswitch = $true)) & (eqinunit = $true)) & (notinsingleton = $true)) & (uniqinunit = $true)) & (nonemptyImpWitness = $true)) & (exuE2 = $true)) & (subPowSU = $true)) & (setunionE = $true)) & (setunionI = $true)) & (powersetE = $true)) & (emptyInPowerset = $true)) & (emptyinPowerset = $true)) & (powersetI = $true)) & (setoftrueEq = $true)) & (setadjoinOr = $true)) & (setadjoinE = $true)) & (setadjoinIR = $true)) & (emptyinunitempty = $true)) & (setadjoinIL = $true)) & (nonemptyI1 = $true)) & (nonemptyI = $true)) & (nonemptyE1 = $true)) & (setbeta = $true)) & (noeltsimpempty = $true)) & (emptyI = $true)) & (setext = $true)) & (exuE3e = $true)) & (notinemptyset = $true)) & (emptysetimpfalse = $true)) & (emptysetE = $true)) & (prop2setE = $true)) & (exuE1 = $true)) & (dsetconstrER = $true)) & (dsetconstrEL = $true)) & (dsetconstrI = $true)) & (descrp = $true)) & (wellorderingAx = $true)) & (foundationAx = $true)) & (replAx = $true)) & (omegaIndAx = $true)) & (omegaSAx = $true)) & (omega0Ax = $true)) & (setunionAx = $true)) & (powersetAx = $true)) & (setadjoinAx = $true)) & (emptysetAx = $true)) & (setextAx = $true)),
% 2.57/1.52 inference(ennf_transformation,[status(thm)],[f418])).
% 2.57/1.52 thf(f418,plain,(
% 2.57/1.52 ~((setextAx = $true) => ((emptysetAx = $true) => ((setadjoinAx = $true) => ((powersetAx = $true) => ((setunionAx = $true) => ((omega0Ax = $true) => ((omegaSAx = $true) => ((omegaIndAx = $true) => ((replAx = $true) => ((foundationAx = $true) => ((wellorderingAx = $true) => ((descrp = $true) => ((dsetconstrI = $true) => ((dsetconstrEL = $true) => ((dsetconstrER = $true) => ((exuE1 = $true) => ((prop2setE = $true) => ((emptysetE = $true) => ((emptysetimpfalse = $true) => ((notinemptyset = $true) => ((exuE3e = $true) => ((setext = $true) => ((emptyI = $true) => ((noeltsimpempty = $true) => ((setbeta = $true) => ((nonemptyE1 = $true) => ((nonemptyI = $true) => ((nonemptyI1 = $true) => ((setadjoinIL = $true) => ((emptyinunitempty = $true) => ((setadjoinIR = $true) => ((setadjoinE = $true) => ((setadjoinOr = $true) => ((setoftrueEq = $true) => ((powersetI = $true) => ((emptyinPowerset = $true) => ((emptyInPowerset = $true) => ((powersetE = $true) => ((setunionI = $true) => ((setunionE = $true) => ((subPowSU = $true) => ((exuE2 = $true) => ((nonemptyImpWitness = $true) => ((uniqinunit = $true) => ((notinsingleton = $true) => ((eqinunit = $true) => ((singletonsswitch = $true) => ((upairsetE = $true) => ((upairsetIL = $true) => ((upairsetIR = $true) => ((emptyE1 = $true) => ((vacuousDall = $true) => ((quantDeMorgan1 = $true) => ((quantDeMorgan2 = $true) => ((quantDeMorgan3 = $true) => ((quantDeMorgan4 = $true) => ((prop2setI = $true) => ((prop2set2propI = $true) => ((notdexE = $true) => ((notdallE = $true) => ((exuI1 = $true) => ((exuI3 = $true) => ((exuI2 = $true) => ((inCongP = $true) => ((in__Cong = $true) => ((exuE3u = $true) => ((exu__Cong = $true) => ((emptyset__Cong = $true) => ((setadjoin__Cong = $true) => ((powerset__Cong = $true) => ((setunion__Cong = $true) => ((omega__Cong = $true) => ((exuEu = $true) => ((descr__Cong = $true) => ((dsetconstr__Cong = $true) => ((subsetI1 = $true) => ((eqimpsubset2 = $true) => ((eqimpsubset1 = $true) => ((subsetI2 = $true) => ((emptysetsubset = $true) => ((subsetE = $true) => ((subsetE2 = $true) => ((notsubsetI = $true) => ((notequalI1 = $true) => ((notequalI2 = $true) => ((subsetRefl = $true) => ((subsetTrans = $true) => ((setadjoinSub = $true) => ((setadjoinSub2 = $true) => ((subset2powerset = $true) => ((setextsub = $true) => ((subsetemptysetimpeq = $true) => ((powersetI1 = $true) => ((powersetE1 = $true) => ((inPowerset = $true) => ((powersetsubset = $true) => ((sepInPowerset = $true) => ((sepSubset = $true) => ((binunionIL = $true) => ((upairset2IR = $true) => ((binunionIR = $true) => ((binunionEcases = $true) => ((binunionE = $true) => ((binunionLsub = $true) => ((binunionRsub = $true) => ((binintersectI = $true) => ((binintersectSubset5 = $true) => ((binintersectEL = $true) => ((binintersectLsub = $true) => ((binintersectSubset2 = $true) => ((binintersectSubset3 = $true) => ((binintersectER = $true) => ((disjointsetsI1 = $true) => ((binintersectRsub = $true) => ((binintersectSubset4 = $true) => ((binintersectSubset1 = $true) => ((bs114d = $true) => ((setminusI = $true) => ((setminusEL = $true) => ((setminusER = $true) => ((setminusSubset2 = $true) => ((setminusERneg = $true) => ((setminusELneg = $true) => ((setminusILneg = $true) => ((setminusIRneg = $true) => ((setminusLsub = $true) => ((setminusSubset1 = $true) => ((symdiffE = $true) => ((symdiffI1 = $true) => ((symdiffI2 = $true) => ((symdiffIneg1 = $true) => ((symdiffIneg2 = $true) => ((secondinupair = $true) => ((setukpairIL = $true) => ((setukpairIR = $true) => ((kpairiskpair = $true) => ((kpairp = $true) => ((singletonsubset = $true) => ((singletoninpowerset = $true) => ((singletoninpowunion = $true) => ((upairset2E = $true) => ((upairsubunion = $true) => ((upairinpowunion = $true) => ((ubforcartprodlem1 = $true) => ((ubforcartprodlem2 = $true) => ((ubforcartprodlem3 = $true) => ((cartprodpairin = $true) => ((cartprodmempair1 = $true) => ((cartprodmempair = $true) => ((setunionE2 = $true) => ((setunionsingleton1 = $true) => ((setunionsingleton2 = $true) => ((setunionsingleton = $true) => ((singletonprop = $true) => ((ex1E1 = $true) => ((ex1I = $true) => ((ex1I2 = $true) => ((singletonsuniq = $true) => ((setukpairinjL1 = $true) => ((kfstsingleton = $true) => ((theprop = $true) => ((kfstpairEq = $true) => ((cartprodfstin = $true) => ((setukpairinjL2 = $true) => ((setukpairinjL = $true) => ((setukpairinjR11 = $true) => ((setukpairinjR12 = $true) => ((setukpairinjR1 = $true) => ((upairequniteq = $true) => ((setukpairinjR2 = $true) => ((setukpairinjR = $true) => ((ksndsingleton = $true) => ((ksndpairEq = $true) => ((kpairsurjEq = $true) => ((cartprodsndin = $true) => ((cartprodpairmemEL = $true) => ((cartprodpairmemER = $true) => ((cartprodmempaircEq = $true) => ((cartprodfstpairEq = $true) => ((cartprodsndpairEq = $true) => ((cartprodpairsurjEq = $true) => ((dpsetconstrI = $true) => ((dpsetconstrSub = $true) => ((setOfPairsIsBReln = $true) => ((dpsetconstrERa = $true) => ((dpsetconstrEL1 = $true) => ((dpsetconstrEL2 = $true) => ((dpsetconstrER = $true) => ((funcImageSingleton = $true) => ((apProp = $true) => ((app = $true) => ((infuncsetfunc = $true) => ((ap2p = $true) => ((funcinfuncset = $true) => ((lamProp = $true) => ((lamp = $true) => ((lam2p = $true) => ((brelnall1 = $true) => ((brelnall2 = $true) => ((ex1E2 = $true) => ((funcGraphProp1 = $true) => ((funcGraphProp3 = $true) => ((funcGraphProp2 = $true) => ((funcextLem = $true) => ((funcGraphProp4 = $true) => ((subbreln = $true) => ((eqbreln = $true) => ((funcext = $true) => ((funcext2 = $true) => ((ap2apEq1 = $true) => ((ap2apEq2 = $true) => ((beta1 = $true) => ((eta1 = $true) => ! [X0 : $i > $i,X1 : $i,X2 : $i] : (! [X3 : $i] : (($true = (in @ X3 @ X1)) => ((in @ (X0 @ X3) @ X2) = $true)) => ((lam @ X1 @ X2 @ (^[Y0 : $i]: (X0 @ Y0))) = (lam @ X1 @ X2 @ (^[Y0 : $i]: (X0 @ Y0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.57/1.52 inference(fool_elimination,[status(thm)],[f417])).
% 2.57/1.52 thf(f417,plain,(
% 2.57/1.52 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => (ubforcartprodlem2 => (ubforcartprodlem3 => (cartprodpairin => (cartprodmempair1 => (cartprodmempair => (setunionE2 => (setunionsingleton1 => (setunionsingleton2 => (setunionsingleton => (singletonprop => (ex1E1 => (ex1I => (ex1I2 => (singletonsuniq => (setukpairinjL1 => (kfstsingleton => (theprop => (kfstpairEq => (cartprodfstin => (setukpairinjL2 => (setukpairinjL => (setukpairinjR11 => (setukpairinjR12 => (setukpairinjR1 => (upairequniteq => (setukpairinjR2 => (setukpairinjR => (ksndsingleton => (ksndpairEq => (kpairsurjEq => (cartprodsndin => (cartprodpairmemEL => (cartprodpairmemER => (cartprodmempaircEq => (cartprodfstpairEq => (cartprodsndpairEq => (cartprodpairsurjEq => (dpsetconstrI => (dpsetconstrSub => (setOfPairsIsBReln => (dpsetconstrERa => (dpsetconstrEL1 => (dpsetconstrEL2 => (dpsetconstrER => (funcImageSingleton => (apProp => (app => (infuncsetfunc => (ap2p => (funcinfuncset => (lamProp => (lamp => (lam2p => (brelnall1 => (brelnall2 => (ex1E2 => (funcGraphProp1 => (funcGraphProp3 => (funcGraphProp2 => (funcextLem => (funcGraphProp4 => (subbreln => (eqbreln => (funcext => (funcext2 => (ap2apEq1 => (ap2apEq2 => (beta1 => (eta1 => ! [X0 : $i > $i,X1 : $i,X2 : $i] : (! [X3 : $i] : ((in @ X3 @ X1) => (in @ (X0 @ X3) @ X2)) => ((lam @ X1 @ X2 @ (^[X4 : $i] : (X0 @ X4))) = (lam @ X1 @ X2 @ (^[X5 : $i] : (X0 @ X5))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.57/1.52 inference(rectify,[status(thm)],[f221])).
% 2.57/1.52 thf(f221,negated_conjecture,(
% 2.57/1.52 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => (ubforcartprodlem2 => (ubforcartprodlem3 => (cartprodpairin => (cartprodmempair1 => (cartprodmempair => (setunionE2 => (setunionsingleton1 => (setunionsingleton2 => (setunionsingleton => (singletonprop => (ex1E1 => (ex1I => (ex1I2 => (singletonsuniq => (setukpairinjL1 => (kfstsingleton => (theprop => (kfstpairEq => (cartprodfstin => (setukpairinjL2 => (setukpairinjL => (setukpairinjR11 => (setukpairinjR12 => (setukpairinjR1 => (upairequniteq => (setukpairinjR2 => (setukpairinjR => (ksndsingleton => (ksndpairEq => (kpairsurjEq => (cartprodsndin => (cartprodpairmemEL => (cartprodpairmemER => (cartprodmempaircEq => (cartprodfstpairEq => (cartprodsndpairEq => (cartprodpairsurjEq => (dpsetconstrI => (dpsetconstrSub => (setOfPairsIsBReln => (dpsetconstrERa => (dpsetconstrEL1 => (dpsetconstrEL2 => (dpsetconstrER => (funcImageSingleton => (apProp => (app => (infuncsetfunc => (ap2p => (funcinfuncset => (lamProp => (lamp => (lam2p => (brelnall1 => (brelnall2 => (ex1E2 => (funcGraphProp1 => (funcGraphProp3 => (funcGraphProp2 => (funcextLem => (funcGraphProp4 => (subbreln => (eqbreln => (funcext => (funcext2 => (ap2apEq1 => (ap2apEq2 => (beta1 => (eta1 => ! [X12 : $i > $i,X3 : $i,X4 : $i] : (! [X1 : $i] : ((in @ X1 @ X3) => (in @ (X12 @ X1) @ X4)) => ((lam @ X3 @ X4 @ (^[X1 : $i] : (X12 @ X1))) = (lam @ X3 @ X4 @ (^[X1 : $i] : (X12 @ X1))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.57/1.52 inference(negated_conjecture,[status(cth)],[f220])).
% 2.57/1.52 thf(f220,conjecture,(
% 2.57/1.52 setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => (ubforcartprodlem2 => (ubforcartprodlem3 => (cartprodpairin => (cartprodmempair1 => (cartprodmempair => (setunionE2 => (setunionsingleton1 => (setunionsingleton2 => (setunionsingleton => (singletonprop => (ex1E1 => (ex1I => (ex1I2 => (singletonsuniq => (setukpairinjL1 => (kfstsingleton => (theprop => (kfstpairEq => (cartprodfstin => (setukpairinjL2 => (setukpairinjL => (setukpairinjR11 => (setukpairinjR12 => (setukpairinjR1 => (upairequniteq => (setukpairinjR2 => (setukpairinjR => (ksndsingleton => (ksndpairEq => (kpairsurjEq => (cartprodsndin => (cartprodpairmemEL => (cartprodpairmemER => (cartprodmempaircEq => (cartprodfstpairEq => (cartprodsndpairEq => (cartprodpairsurjEq => (dpsetconstrI => (dpsetconstrSub => (setOfPairsIsBReln => (dpsetconstrERa => (dpsetconstrEL1 => (dpsetconstrEL2 => (dpsetconstrER => (funcImageSingleton => (apProp => (app => (infuncsetfunc => (ap2p => (funcinfuncset => (lamProp => (lamp => (lam2p => (brelnall1 => (brelnall2 => (ex1E2 => (funcGraphProp1 => (funcGraphProp3 => (funcGraphProp2 => (funcextLem => (funcGraphProp4 => (subbreln => (eqbreln => (funcext => (funcext2 => (ap2apEq1 => (ap2apEq2 => (beta1 => (eta1 => ! [X12 : $i > $i,X3 : $i,X4 : $i] : (! [X1 : $i] : ((in @ X1 @ X3) => (in @ (X12 @ X1) @ X4)) => ((lam @ X3 @ X4 @ (^[X1 : $i] : (X12 @ X1))) = (lam @ X3 @ X4 @ (^[X1 : $i] : (X12 @ X1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.57/1.52 file('/export/starexec/sandbox2/tmp/tmp.EApmVaGZ7F/DTF2THF_24917.p',lam2lamEq)).
% 2.57/1.52 % SZS output end Proof for DTF2THF_24917
% 2.57/1.52 % (25016)------------------------------
% 2.57/1.52 % (25016)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.57/1.52 % (25016)Termination reason: Refutation
% 2.57/1.52
% 2.57/1.52 % (25016)Memory used [KB]: 7419
% 2.57/1.52 % (25016)Time elapsed: 0.048 s
% 2.57/1.52 % (25016)Instructions burned: 158 (million)
% 2.57/1.52 % (25016)------------------------------
% 2.57/1.52 % (25016)------------------------------
% 2.57/1.52 % (25015)Success in time 0.062 s
%------------------------------------------------------------------------------