%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU617^1 : TPTP v9.2.1. Released v3.7.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n010.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 09:35:38 AM UTC 2026
% Result : Theorem 2.39s 1.66s
% Output : Refutation 2.39s
% Verified :
% SZS Type : Refutation
% Derivation depth : 19
% Number of leaves : 8
% Syntax : Number of formulae : 41 ( 11 unt; 0 typ; 0 def)
% Number of atoms : 2490 ( 657 equ; 0 cnn)
% Maximal formula atoms : 137 ( 60 avg)
% Number of connectives : 1509 ( 72 ~; 53 |; 428 &; 414 @)
% ( 5 <=>; 537 =>; 0 <=; 0 <~>)
% Maximal formula depth : 145 ( 27 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 261 ( 258 usr; 143 con; 0-4 aty)
% Number of variables : 80 ( 0 ^; 51 !; 29 ?; 80 :)
% 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_167,type,
sP0: $i > $i > $o ).
thf(func_def_168,type,
sP1: $i > $o ).
thf(func_def_169,type,
sP2: $i > $i > $o ).
thf(func_def_170,type,
sP3: $i > $i > $o ).
thf(func_def_171,type,
sP4: $i > $i > $i > $o > $o ).
thf(func_def_175,type,
sK8: $i > $i > $i ).
thf(func_def_176,type,
sK9: $i > $i > $i ).
thf(func_def_177,type,
sK10: $i > $i > $i ).
thf(func_def_178,type,
sK11: $i > $i > $i ).
thf(func_def_179,type,
sK12: $i > $i > $i > $i ).
thf(func_def_180,type,
sK13: $i > $i ).
thf(func_def_181,type,
sK14: $i > $i ).
thf(func_def_182,type,
sK15: $i > $i ).
thf(func_def_183,type,
sK16: $i > $i ).
thf(func_def_184,type,
sK17: $i > $i > $i > $i ).
thf(func_def_185,type,
sK18: $i > $i > $i > $i ).
thf(func_def_186,type,
sK19: $i > $i > $i ).
thf(func_def_187,type,
sK20: $i > $i > $i ).
thf(func_def_188,type,
sK21: $i > $i > $i ).
thf(func_def_190,type,
sK23: $i > $i ).
thf(func_def_191,type,
sK24: $i > $i ).
thf(func_def_192,type,
sK25: $i > $i ).
thf(func_def_193,type,
sK26: $i > $i > $i ).
thf(func_def_194,type,
sK27: $i > $o ).
thf(func_def_196,type,
sK29: $i > ( $i > $o ) > $i ).
thf(func_def_212,type,
sK45: $i > $i > $i ).
thf(func_def_220,type,
sK53: $i > $o ).
thf(func_def_227,type,
sK60: $i > $o ).
thf(func_def_229,type,
sK62: ( $i > $o ) > $i ).
thf(func_def_230,type,
sK63: ( $i > $o ) > $i ).
thf(func_def_233,type,
sK66: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_234,type,
sK67: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_235,type,
sK68: $i > $o ).
thf(func_def_236,type,
sK69: $i > $o ).
thf(func_def_244,type,
sK77: $i > $o ).
thf(func_def_252,type,
sK85: $i > $i ).
thf(func_def_266,type,
sK99: $i > $o ).
thf(func_def_267,type,
sK100: $i > $o ).
thf(func_def_270,type,
sK103: $i > $o ).
thf(func_def_273,type,
sK106: $i > ( $i > $o ) > $i ).
thf(func_def_280,type,
sK113: $i > $o ).
thf(func_def_284,type,
sK117: ( $i > $o ) > $i ).
thf(func_def_285,type,
sK118: $i > $o ).
thf(func_def_286,type,
sK119: $i > $i ).
thf(func_def_290,type,
sK123: $i > $i ).
thf(func_def_292,type,
sK125: ( $i > $o ) > $i > $i ).
thf(func_def_294,type,
sK127: $i > $o ).
thf(func_def_302,type,
sK135: $i > $i > $i ).
thf(func_def_303,type,
sK136: $i > $i > $i ).
thf(func_def_315,type,
sK148: $i > $i ).
thf(func_def_317,type,
sK150: $i > $i ).
thf(func_def_326,type,
sK159: $i > $o ).
thf(func_def_335,type,
sK168: $i > $i > $i ).
thf(func_def_339,type,
sK172: $i > ( $i > $o ) > $i ).
thf(func_def_340,type,
sK173: $i > $o ).
thf(func_def_343,type,
sK176: $i > $o ).
thf(func_def_344,type,
sK177: ( $i > $o ) > $i ).
thf(func_def_353,type,
sK186: $i > $o ).
thf(func_def_362,type,
sK195: $i > $o ).
thf(func_def_365,type,
sK198: ( $i > $o ) > $i > $i ).
thf(func_def_367,type,
sK200: $i > $o ).
thf(func_def_371,type,
sK204: $i > $o ).
thf(func_def_372,type,
sK205: $i > $o ).
thf(func_def_373,type,
sK206: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_374,type,
sK207: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_375,type,
sK208: $i > $i ).
thf(func_def_393,type,
sK226: $i > $o ).
thf(func_def_396,type,
sK229: $i > ( $i > $o ) > $i ).
thf(func_def_399,type,
sK232: $i > $i > $i ).
thf(func_def_406,type,
sK239: $o > $i > $i > $i ).
thf(func_def_426,type,
sK259: $i > $i > $o ).
thf(func_def_427,type,
sK260: $i > $i ).
thf(func_def_428,type,
sK261: $i > $i ).
thf(func_def_429,type,
sK262: ( $i > $i > $o ) > $i > $i ).
thf(func_def_430,type,
sK263: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_431,type,
sK264: ( $i > $i > $o ) > $i > $i ).
thf(func_def_442,type,
sK275: $i > $o ).
thf(func_def_445,type,
sK278: $i > $i > $i ).
thf(func_def_455,type,
sK288: ( $i > $o ) > $i > $i ).
thf(func_def_457,type,
sK290: $i > $o ).
thf(func_def_459,type,
sK292: $i > $o ).
thf(func_def_461,type,
sK294: $i > ( $i > $o ) > $i ).
thf(func_def_462,type,
sK295: $i > $o ).
thf(func_def_463,type,
sK296: $i > $o ).
thf(func_def_464,type,
sK297: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_465,type,
sK298: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_477,type,
sK310: $i > $o ).
thf(func_def_485,type,
sK318: $i > $i > $i ).
thf(func_def_492,type,
sK325: $i > $i > $i ).
thf(func_def_501,type,
sK334: $i > $o ).
thf(func_def_514,type,
sK347: $i > $i > $i ).
thf(func_def_518,type,
sK351: $i > $o ).
thf(func_def_522,type,
sK355: $i > $i ).
thf(func_def_525,type,
sK358: $i > $i ).
thf(func_def_530,type,
sK363: ( $i > $o ) > $i ).
thf(func_def_531,type,
sK364: $i > $o ).
thf(func_def_532,type,
sK365: $i > $i ).
thf(func_def_535,type,
ph368:
!>[X0: $tType] : X0 ).
thf(f3181,plain,
$false,
inference(trivial_inequality_removal,[status(thm)],[f3180]) ).
thf(f3180,plain,
$true != $true,
inference(superposition,[status(thm)],[f3055,f1678]) ).
thf(f1678,plain,
setadjoinIL = $true,
inference(cnf_transformation,[status(thm)],[f1107]) ).
thf(f1107,plain,
( ( emptyInPowerset = $true )
& ( emptysetimpfalse = $true )
& ( subsetTrans = $true )
& ( inPowerset = $true )
& ( descrp = $true )
& ( emptyinPowerset = $true )
& ( descr__Cong = $true )
& ( powersetE1 = $true )
& ( exuEu = $true )
& ( powersetI1 = $true )
& ( binintersectRsub = $true )
& ( notdallE = $true )
& ( binintersectSubset5 = $true )
& ( subsetemptysetimpeq = $true )
& ( setextAx = $true )
& ( emptyinunitempty = $true )
& ( setunion__Cong = $true )
& ( eqinunit = $true )
& ( setunionAx = $true )
& ( quantDeMorgan2 = $true )
& ( setminusLsub = $true )
& ( nonemptyE1 = $true )
& ( exuE1 = $true )
& ( symdiffE = $true )
& ( subsetRefl = $true )
& ( binunionRsub = $true )
& ( replAx = $true )
& ( subset2powerset = $true )
& ( sepSubset = $true )
& ( setadjoinSub = $true )
& ( binintersectER = $true )
& ( dsetconstr__Cong = $true )
& ( exuE3u = $true )
& ( binintersectI = $true )
& ( setminusSubset2 = $true )
& ( exuI1 = $true )
& ( setminusERneg = $true )
& ( setminusER = $true )
& ( emptyE1 = $true )
& ( setminusEL = $true )
& ( setminusSubset1 = $true )
& ( omegaSAx = $true )
& ( subPowSU = $true )
& ( setunionE = $true )
& ( upairsetE = $true )
& ( nonemptyImpWitness = $true )
& ( setminusILneg = $true )
& ( quantDeMorgan4 = $true )
& ( subsetE2 = $true )
& ( emptysetE = $true )
& ( omegaIndAx = $true )
& ( setadjoinIR = $true )
& ( exuE2 = $true )
& ( wellorderingAx = $true )
& ( subsetI1 = $true )
& ( setext = $true )
& ( upairsetIR = $true )
& ( powerset__Cong = $true )
& ( binintersectSubset4 = $true )
& ( subsetE = $true )
& ( vacuousDall = $true )
& ( exu__Cong = $true )
& ( setadjoinOr = $true )
& ( quantDeMorgan3 = $true )
& ( emptyI = $true )
& ( powersetI = $true )
& ( setminusI = $true )
& ( symdiffI1 = $true )
& ( dsetconstrI = $true )
& ( emptysetAx = $true )
& ( binunionIR = $true )
& ( secondinupair = $true )
& ( powersetE = $true )
& ( eqimpsubset2 = $true )
& ( subsetI2 = $true )
& ( binintersectLsub = $true )
& ( setbeta = $true )
& ( prop2setI = $true )
& ( binunionLsub = $true )
& ( binintersectEL = $true )
& ( in__Cong = $true )
& ( setextsub = $true )
& ( powersetsubset = $true )
& ( binintersectSubset1 = $true )
& ( symdiffI2 = $true )
& ( sepInPowerset = $true )
& ( exuI2 = $true )
& ( quantDeMorgan1 = $true )
& ( omega__Cong = $true )
& ( setoftrueEq = $true )
& ( eqimpsubset1 = $true )
& ( notequalI1 = $true )
& ( symdiffIneg1 = $true )
& ( foundationAx = $true )
& ( symdiffIneg2 = $true )
& ( setadjoin__Cong = $true )
& ( powersetAx = $true )
& ( noeltsimpempty = $true )
& ( notdexE = $true )
& ( upairsetIL = $true )
& ( bs114d = $true )
& ( notinsingleton = $true )
& ( emptyset__Cong = $true )
& ( binunionIL = $true )
& ( exuE3e = $true )
& ( notinemptyset = $true )
& ( setadjoinIL = $true )
& ( setunionI = $true )
& ( setminusELneg = $true )
& ( disjointsetsI1 = $true )
& ( inCongP = $true )
& ( exuI3 = $true )
& ( binintersectSubset2 = $true )
& ( binintersectSubset3 = $true )
& ( dsetconstrEL = $true )
& ( $true
!= ( in @ sK305 @ ( setunion @ ( setadjoin @ ( setadjoin @ sK305 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK305 @ ( setadjoin @ sK306 @ emptyset ) ) @ emptyset ) ) ) ) )
& ( setadjoinSub2 = $true )
& ( binunionE = $true )
& ( emptysetsubset = $true )
& ( omega0Ax = $true )
& ( binunionEcases = $true )
& ( prop2setE = $true )
& ( dsetconstrER = $true )
& ( nonemptyI1 = $true )
& ( upairset2IR = $true )
& ( setadjoinE = $true )
& ( prop2set2propI = $true )
& ( setminusIRneg = $true )
& ( uniqinunit = $true )
& ( notequalI2 = $true )
& ( notsubsetI = $true )
& ( nonemptyI = $true )
& ( singletonsswitch = $true )
& ( setadjoinAx = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK305,sK306])],[f457,f1106]) ).
thf(f1106,plain,
( ? [X0: $i,X1: $i] :
( $true
!= ( in @ X0 @ ( setunion @ ( setadjoin @ ( setadjoin @ X0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X0 @ ( setadjoin @ X1 @ emptyset ) ) @ emptyset ) ) ) ) )
=> ( $true
!= ( in @ sK305 @ ( setunion @ ( setadjoin @ ( setadjoin @ sK305 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK305 @ ( setadjoin @ sK306 @ emptyset ) ) @ emptyset ) ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f457,plain,
( ( emptyInPowerset = $true )
& ( emptysetimpfalse = $true )
& ( subsetTrans = $true )
& ( inPowerset = $true )
& ( descrp = $true )
& ( emptyinPowerset = $true )
& ( descr__Cong = $true )
& ( powersetE1 = $true )
& ( exuEu = $true )
& ( powersetI1 = $true )
& ( binintersectRsub = $true )
& ( notdallE = $true )
& ( binintersectSubset5 = $true )
& ( subsetemptysetimpeq = $true )
& ( setextAx = $true )
& ( emptyinunitempty = $true )
& ( setunion__Cong = $true )
& ( eqinunit = $true )
& ( setunionAx = $true )
& ( quantDeMorgan2 = $true )
& ( setminusLsub = $true )
& ( nonemptyE1 = $true )
& ( exuE1 = $true )
& ( symdiffE = $true )
& ( subsetRefl = $true )
& ( binunionRsub = $true )
& ( replAx = $true )
& ( subset2powerset = $true )
& ( sepSubset = $true )
& ( setadjoinSub = $true )
& ( binintersectER = $true )
& ( dsetconstr__Cong = $true )
& ( exuE3u = $true )
& ( binintersectI = $true )
& ( setminusSubset2 = $true )
& ( exuI1 = $true )
& ( setminusERneg = $true )
& ( setminusER = $true )
& ( emptyE1 = $true )
& ( setminusEL = $true )
& ( setminusSubset1 = $true )
& ( omegaSAx = $true )
& ( subPowSU = $true )
& ( setunionE = $true )
& ( upairsetE = $true )
& ( nonemptyImpWitness = $true )
& ( setminusILneg = $true )
& ( quantDeMorgan4 = $true )
& ( subsetE2 = $true )
& ( emptysetE = $true )
& ( omegaIndAx = $true )
& ( setadjoinIR = $true )
& ( exuE2 = $true )
& ( wellorderingAx = $true )
& ( subsetI1 = $true )
& ( setext = $true )
& ( upairsetIR = $true )
& ( powerset__Cong = $true )
& ( binintersectSubset4 = $true )
& ( subsetE = $true )
& ( vacuousDall = $true )
& ( exu__Cong = $true )
& ( setadjoinOr = $true )
& ( quantDeMorgan3 = $true )
& ( emptyI = $true )
& ( powersetI = $true )
& ( setminusI = $true )
& ( symdiffI1 = $true )
& ( dsetconstrI = $true )
& ( emptysetAx = $true )
& ( binunionIR = $true )
& ( secondinupair = $true )
& ( powersetE = $true )
& ( eqimpsubset2 = $true )
& ( subsetI2 = $true )
& ( binintersectLsub = $true )
& ( setbeta = $true )
& ( prop2setI = $true )
& ( binunionLsub = $true )
& ( binintersectEL = $true )
& ( in__Cong = $true )
& ( setextsub = $true )
& ( powersetsubset = $true )
& ( binintersectSubset1 = $true )
& ( symdiffI2 = $true )
& ( sepInPowerset = $true )
& ( exuI2 = $true )
& ( quantDeMorgan1 = $true )
& ( omega__Cong = $true )
& ( setoftrueEq = $true )
& ( eqimpsubset1 = $true )
& ( notequalI1 = $true )
& ( symdiffIneg1 = $true )
& ( foundationAx = $true )
& ( symdiffIneg2 = $true )
& ( setadjoin__Cong = $true )
& ( powersetAx = $true )
& ( noeltsimpempty = $true )
& ( notdexE = $true )
& ( upairsetIL = $true )
& ( bs114d = $true )
& ( notinsingleton = $true )
& ( emptyset__Cong = $true )
& ( binunionIL = $true )
& ( exuE3e = $true )
& ( notinemptyset = $true )
& ( setadjoinIL = $true )
& ( setunionI = $true )
& ( setminusELneg = $true )
& ( disjointsetsI1 = $true )
& ( inCongP = $true )
& ( exuI3 = $true )
& ( binintersectSubset2 = $true )
& ( binintersectSubset3 = $true )
& ( dsetconstrEL = $true )
& ? [X0: $i,X1: $i] :
( $true
!= ( in @ X0 @ ( setunion @ ( setadjoin @ ( setadjoin @ X0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X0 @ ( setadjoin @ X1 @ emptyset ) ) @ emptyset ) ) ) ) )
& ( setadjoinSub2 = $true )
& ( binunionE = $true )
& ( emptysetsubset = $true )
& ( omega0Ax = $true )
& ( binunionEcases = $true )
& ( prop2setE = $true )
& ( dsetconstrER = $true )
& ( nonemptyI1 = $true )
& ( upairset2IR = $true )
& ( setadjoinE = $true )
& ( prop2set2propI = $true )
& ( setminusIRneg = $true )
& ( uniqinunit = $true )
& ( notequalI2 = $true )
& ( notsubsetI = $true )
& ( nonemptyI = $true )
& ( singletonsswitch = $true )
& ( setadjoinAx = $true ) ),
inference(flattening,[status(thm)],[f456]) ).
thf(f456,plain,
( ? [X0: $i,X1: $i] :
( $true
!= ( in @ X0 @ ( setunion @ ( setadjoin @ ( setadjoin @ X0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X0 @ ( setadjoin @ X1 @ emptyset ) ) @ emptyset ) ) ) ) )
& ( 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)],[f326]) ).
thf(f326,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 )
=> ! [X0: $i,X1: $i] :
( $true
= ( in @ X0 @ ( setunion @ ( setadjoin @ ( setadjoin @ X0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X0 @ ( setadjoin @ X1 @ emptyset ) ) @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f325]) ).
thf(f325,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
=> ! [X0: $i,X1: $i] : ( in @ X0 @ ( setunion @ ( setadjoin @ ( setadjoin @ X0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X0 @ ( setadjoin @ X1 @ emptyset ) ) @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f138]) ).
thf(f138,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
=> ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f137]) ).
thf(f137,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
=> ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.q7XDm9XVbs/DTF2THF_21061.p',setukpairIL) ).
thf(f3055,plain,
setadjoinIL != $true,
inference(trivial_inequality_removal,[status(thm)],[f3046]) ).
thf(f3046,plain,
( ( $true != $true )
| ( setadjoinIL != $true ) ),
inference(superposition,[status(thm)],[f2194,f1443]) ).
thf(f1443,plain,
! [X2: $i,X3: $i] :
( ( ( in @ X2 @ ( setadjoin @ X2 @ X3 ) )
= $true )
| ( setadjoinIL != $true ) ),
inference(cnf_transformation,[status(thm)],[f854]) ).
thf(f854,plain,
( ( ( setadjoinIL = $true )
| ( $true
!= ( in @ sK156 @ ( setadjoin @ sK156 @ sK157 ) ) ) )
& ( ! [X2: $i,X3: $i] :
( ( in @ X2 @ ( setadjoin @ X2 @ X3 ) )
= $true )
| ( setadjoinIL != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK156,sK157])],[f852,f853]) ).
thf(f853,plain,
( ? [X0: $i,X1: $i] :
( $true
!= ( in @ X0 @ ( setadjoin @ X0 @ X1 ) ) )
=> ( $true
!= ( in @ sK156 @ ( setadjoin @ sK156 @ sK157 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f852,plain,
( ( ( setadjoinIL = $true )
| ? [X0: $i,X1: $i] :
( $true
!= ( in @ X0 @ ( setadjoin @ X0 @ X1 ) ) ) )
& ( ! [X2: $i,X3: $i] :
( ( in @ X2 @ ( setadjoin @ X2 @ X3 ) )
= $true )
| ( setadjoinIL != $true ) ) ),
inference(rectify,[status(thm)],[f851]) ).
thf(f851,plain,
( ( ( setadjoinIL = $true )
| ? [X0: $i,X1: $i] :
( $true
!= ( in @ X0 @ ( setadjoin @ X0 @ X1 ) ) ) )
& ( ! [X0: $i,X1: $i] :
( $true
= ( in @ X0 @ ( setadjoin @ X0 @ X1 ) ) )
| ( setadjoinIL != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f191]) ).
thf(f191,plain,
( ( setadjoinIL = $true )
<=> ! [X0: $i,X1: $i] :
( $true
= ( in @ X0 @ ( setadjoin @ X0 @ X1 ) ) ) ),
inference(fool_elimination,[status(thm)],[f190]) ).
thf(f190,plain,
( setadjoinIL
= ( ! [X0: $i,X1: $i] : ( in @ X0 @ ( setadjoin @ X0 @ X1 ) ) ) ),
inference(rectify,[status(thm)],[f31]) ).
thf(f31,axiom,
( setadjoinIL
= ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X2 ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.q7XDm9XVbs/DTF2THF_21061.p',setadjoinIL) ).
thf(f2194,plain,
( $true
!= ( in @ sK305 @ ( setadjoin @ sK305 @ emptyset ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f2193]) ).
thf(f2193,plain,
( ( $true != $true )
| ( $true
!= ( in @ sK305 @ ( setadjoin @ sK305 @ emptyset ) ) ) ),
inference(forward_demodulation,[status(thm)],[f2075,f1678]) ).
thf(f2075,plain,
( ( setadjoinIL != $true )
| ( $true
!= ( in @ sK305 @ ( setadjoin @ sK305 @ emptyset ) ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f1990]) ).
thf(f1990,plain,
( ( setadjoinIL != $true )
| ( $true
!= ( in @ sK305 @ ( setadjoin @ sK305 @ emptyset ) ) )
| ( $true != $true ) ),
inference(superposition,[status(thm)],[f1981,f1443]) ).
thf(f1981,plain,
! [X0: $i] :
( ( $true
!= ( in @ X0 @ ( setadjoin @ ( setadjoin @ sK305 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK305 @ ( setadjoin @ sK306 @ emptyset ) ) @ emptyset ) ) ) )
| ( ( in @ sK305 @ X0 )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f1980]) ).
thf(f1980,plain,
! [X0: $i] :
( ( ( in @ sK305 @ X0 )
!= $true )
| ( $true != $true )
| ( $true
!= ( in @ X0 @ ( setadjoin @ ( setadjoin @ sK305 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK305 @ ( setadjoin @ sK306 @ emptyset ) ) @ emptyset ) ) ) ) ),
inference(forward_demodulation,[status(thm)],[f1972,f1766]) ).
thf(f1766,plain,
setunionAx = $true,
inference(cnf_transformation,[status(thm)],[f1107]) ).
thf(f1972,plain,
! [X0: $i] :
( ( setunionAx != $true )
| ( $true
!= ( in @ X0 @ ( setadjoin @ ( setadjoin @ sK305 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK305 @ ( setadjoin @ sK306 @ emptyset ) ) @ emptyset ) ) ) )
| ( ( in @ sK305 @ X0 )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f1963]) ).
thf(f1963,plain,
! [X0: $i] :
( ( setunionAx != $true )
| ( ( in @ sK305 @ X0 )
!= $true )
| ( $true != $true )
| ( $true
!= ( in @ X0 @ ( setadjoin @ ( setadjoin @ sK305 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK305 @ ( setadjoin @ sK306 @ emptyset ) ) @ emptyset ) ) ) ) ),
inference(superposition,[status(thm)],[f1669,f1276]) ).
thf(f1276,plain,
! [X6: $i,X4: $i,X5: $i] :
( ( ( in @ X5 @ ( setunion @ X4 ) )
= $true )
| ( ( in @ X6 @ X4 )
!= $true )
| ( ( in @ X5 @ X6 )
!= $true )
| ( setunionAx != $true ) ),
inference(cnf_transformation,[status(thm)],[f658]) ).
thf(f658,plain,
( ( ( setunionAx = $true )
| ( ( ! [X2: $i] :
( ( ( in @ X2 @ sK42 )
!= $true )
| ( $true
!= ( in @ sK43 @ X2 ) ) )
| ( ( in @ sK43 @ ( setunion @ sK42 ) )
!= $true ) )
& ( ( ( $true
= ( in @ sK44 @ sK42 ) )
& ( $true
= ( in @ sK43 @ sK44 ) ) )
| ( ( in @ sK43 @ ( setunion @ sK42 ) )
= $true ) ) ) )
& ( ! [X4: $i,X5: $i] :
( ( ( ( in @ X5 @ ( setunion @ X4 ) )
= $true )
| ! [X6: $i] :
( ( ( in @ X6 @ X4 )
!= $true )
| ( ( in @ X5 @ X6 )
!= $true ) ) )
& ( ( ( $true
= ( in @ ( sK45 @ X5 @ X4 ) @ X4 ) )
& ( $true
= ( in @ X5 @ ( sK45 @ X5 @ X4 ) ) ) )
| ( ( in @ X5 @ ( setunion @ X4 ) )
!= $true ) ) )
| ( setunionAx != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK42,sK43,sK44,sK45])],[f654,f657,f656,f655]) ).
thf(f655,plain,
( ? [X0: $i,X1: $i] :
( ( ! [X2: $i] :
( ( ( in @ X2 @ X0 )
!= $true )
| ( ( in @ X1 @ X2 )
!= $true ) )
| ( ( in @ X1 @ ( setunion @ X0 ) )
!= $true ) )
& ( ? [X3: $i] :
( ( $true
= ( in @ X3 @ X0 ) )
& ( ( in @ X1 @ X3 )
= $true ) )
| ( ( in @ X1 @ ( setunion @ X0 ) )
= $true ) ) )
=> ( ( ! [X2: $i] :
( ( ( in @ X2 @ sK42 )
!= $true )
| ( $true
!= ( in @ sK43 @ X2 ) ) )
| ( ( in @ sK43 @ ( setunion @ sK42 ) )
!= $true ) )
& ( ? [X3: $i] :
( ( $true
= ( in @ X3 @ sK42 ) )
& ( ( in @ sK43 @ X3 )
= $true ) )
| ( ( in @ sK43 @ ( setunion @ sK42 ) )
= $true ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f656,plain,
( ? [X3: $i] :
( ( $true
= ( in @ X3 @ sK42 ) )
& ( ( in @ sK43 @ X3 )
= $true ) )
=> ( ( $true
= ( in @ sK44 @ sK42 ) )
& ( $true
= ( in @ sK43 @ sK44 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f657,plain,
! [X4: $i,X5: $i] :
( ? [X7: $i] :
( ( ( in @ X7 @ X4 )
= $true )
& ( ( in @ X5 @ X7 )
= $true ) )
=> ( ( $true
= ( in @ ( sK45 @ X5 @ X4 ) @ X4 ) )
& ( $true
= ( in @ X5 @ ( sK45 @ X5 @ X4 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f654,plain,
( ( ( setunionAx = $true )
| ? [X0: $i,X1: $i] :
( ( ! [X2: $i] :
( ( ( in @ X2 @ X0 )
!= $true )
| ( ( in @ X1 @ X2 )
!= $true ) )
| ( ( in @ X1 @ ( setunion @ X0 ) )
!= $true ) )
& ( ? [X3: $i] :
( ( $true
= ( in @ X3 @ X0 ) )
& ( ( in @ X1 @ X3 )
= $true ) )
| ( ( in @ X1 @ ( setunion @ X0 ) )
= $true ) ) ) )
& ( ! [X4: $i,X5: $i] :
( ( ( ( in @ X5 @ ( setunion @ X4 ) )
= $true )
| ! [X6: $i] :
( ( ( in @ X6 @ X4 )
!= $true )
| ( ( in @ X5 @ X6 )
!= $true ) ) )
& ( ? [X7: $i] :
( ( ( in @ X7 @ X4 )
= $true )
& ( ( in @ X5 @ X7 )
= $true ) )
| ( ( in @ X5 @ ( setunion @ X4 ) )
!= $true ) ) )
| ( setunionAx != $true ) ) ),
inference(rectify,[status(thm)],[f653]) ).
thf(f653,plain,
( ( ( setunionAx = $true )
| ? [X0: $i,X1: $i] :
( ( ! [X2: $i] :
( ( ( in @ X2 @ X0 )
!= $true )
| ( ( in @ X1 @ X2 )
!= $true ) )
| ( ( in @ X1 @ ( setunion @ X0 ) )
!= $true ) )
& ( ? [X2: $i] :
( ( ( in @ X2 @ X0 )
= $true )
& ( ( in @ X1 @ X2 )
= $true ) )
| ( ( in @ X1 @ ( setunion @ X0 ) )
= $true ) ) ) )
& ( ! [X0: $i,X1: $i] :
( ( ( ( in @ X1 @ ( setunion @ X0 ) )
= $true )
| ! [X2: $i] :
( ( ( in @ X2 @ X0 )
!= $true )
| ( ( in @ X1 @ X2 )
!= $true ) ) )
& ( ? [X2: $i] :
( ( ( in @ X2 @ X0 )
= $true )
& ( ( in @ X1 @ X2 )
= $true ) )
| ( ( in @ X1 @ ( setunion @ X0 ) )
!= $true ) ) )
| ( setunionAx != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f399]) ).
thf(f399,plain,
( ( setunionAx = $true )
<=> ! [X0: $i,X1: $i] :
( ( ( in @ X1 @ ( setunion @ X0 ) )
= $true )
<=> ? [X2: $i] :
( ( ( in @ X2 @ X0 )
= $true )
& ( ( in @ X1 @ X2 )
= $true ) ) ) ),
inference(fool_elimination,[status(thm)],[f398]) ).
thf(f398,plain,
( setunionAx
= ( ! [X0: $i,X1: $i] :
( ( in @ X1 @ ( setunion @ X0 ) )
<=> ? [X2: $i] :
( ( in @ X1 @ X2 )
& ( in @ X2 @ X0 ) ) ) ) ),
inference(rectify,[status(thm)],[f6]) ).
thf(f6,axiom,
( setunionAx
= ( ! [X3: $i,X1: $i] :
( ( in @ X1 @ ( setunion @ X3 ) )
<=> ? [X4: $i] :
( ( in @ X1 @ X4 )
& ( in @ X4 @ X3 ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.q7XDm9XVbs/DTF2THF_21061.p',setunionAx) ).
thf(f1669,plain,
( $true
!= ( in @ sK305 @ ( setunion @ ( setadjoin @ ( setadjoin @ sK305 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK305 @ ( setadjoin @ sK306 @ emptyset ) ) @ emptyset ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f1107]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : SEU617^1 : TPTP v9.2.1. Released v3.7.0.
% 0.00/0.12 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.12/0.32 % Computer : n010.cluster.edu
% 0.12/0.32 % Model : x86_64 x86_64
% 0.12/0.32 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.32 % Memory : 8042.1875MB
% 0.12/0.32 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.32 % CPULimit : 300
% 0.12/0.32 % WCLimit : 300
% 0.12/0.32 % DateTime : Tue Feb 24 12:33:33 EST 2026
% 0.12/0.32 % CPUTime :
% 0.12/0.32 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.16/0.42 ---- Original DTF file ---
% 0.16/0.42 thf(spec,logic,$$dhol).
% 0.16/0.42 %------------------------------------------------------------------------------
% 0.16/0.42 % File : SEU617^1 : TPTP v9.2.1. Released v3.7.0.
% 0.16/0.42 % Domain : Set Theory
% 0.16/0.42 % Problem : Ordered Pairs - Kuratowski Pairs
% 0.16/0.42 % Version : Especial.
% 0.16/0.42 % English : (! x:i.! y:i.in x (setunion (setadjoin (setadjoin x emptyset)
% 0.16/0.42 % (setadjoin (setadjoin x (setadjoin y emptyset)) emptyset))))
% 0.16/0.42
% 0.16/0.42 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.16/0.42 % Source : [Bro08]
% 0.16/0.42 % Names : ZFC119g [Bro08]
% 0.16/0.42
% 0.16/0.42 % Status : Theorem
% 0.16/0.42 % Rating : 0.11 v9.1.0, 0.12 v9.0.0, 0.20 v8.2.0, 0.31 v8.1.0, 0.18 v7.5.0, 0.14 v7.4.0, 0.22 v7.2.0, 0.12 v7.1.0, 0.38 v7.0.0, 0.43 v6.4.0, 0.50 v6.3.0, 0.60 v6.2.0, 0.43 v5.5.0, 0.67 v5.4.0, 0.80 v4.1.0, 1.00 v3.7.0
% 0.16/0.42 % Syntax : Number of formulae : 291 ( 136 unt; 154 typ; 136 def)
% 0.16/0.42 % Number of atoms : 726 ( 194 equ; 0 cnn)
% 0.16/0.42 % Maximal formula atoms : 134 ( 5 avg)
% 0.16/0.42 % Number of connectives : 1190 ( 47 ~; 6 |; 30 &; 770 @)
% 0.16/0.42 % ( 14 <=>; 323 =>; 0 <=; 0 <~>)
% 0.16/0.42 % Maximal formula depth : 144 ( 2 avg)
% 0.16/0.42 % Number of types : 2 ( 0 usr)
% 0.16/0.42 % Number of type conns : 65 ( 65 >; 0 *; 0 +; 0 <<)
% 0.16/0.42 % Number of symbols : 157 ( 154 usr; 137 con; 0-2 aty)
% 0.16/0.42 % Number of variables : 393 ( 31 ^; 331 !; 31 ?; 393 :)
% 0.16/0.42 % SPC : TH0_THM_EQU_NAR
% 0.16/0.42
% 0.16/0.42 % Comments : http://mathgate.info/detsetitem.php?id=172
% 0.16/0.42 %------------------------------------------------------------------------------
% 0.16/0.42 thf(in_type,type,
% 0.16/0.42 in: $i > $i > $o ).
% 0.16/0.42
% 0.16/0.42 thf(exu_type,type,
% 0.16/0.42 exu: ( $i > $o ) > $o ).
% 0.16/0.42
% 0.16/0.42 thf(exu,definition,
% 0.16/0.42 ( exu
% 0.16/0.42 = ( ^ [Xphi: $i > $o] :
% 0.16/0.42 ? [Xx: $i] :
% 0.16/0.42 ( ( Xphi @ Xx )
% 0.16/0.42 & ! [Xy: $i] :
% 0.16/0.42 ( ( Xphi @ Xy )
% 0.16/0.42 => ( Xx = Xy ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setextAx_type,type,
% 0.16/0.42 setextAx: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setextAx,definition,
% 0.16/0.42 ( setextAx
% 0.16/0.42 = ( ! [A: $i,B: $i] :
% 0.16/0.42 ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 <=> ( in @ Xx @ B ) )
% 0.16/0.42 => ( A = B ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(emptyset_type,type,
% 0.16/0.42 emptyset: $i ).
% 0.16/0.42
% 0.16/0.42 thf(emptysetAx_type,type,
% 0.16/0.42 emptysetAx: $o ).
% 0.16/0.42
% 0.16/0.42 thf(emptysetAx,definition,
% 0.16/0.42 ( emptysetAx
% 0.16/0.42 = ( ! [Xx: $i] :
% 0.16/0.42 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setadjoin_type,type,
% 0.16/0.42 setadjoin: $i > $i > $i ).
% 0.16/0.42
% 0.16/0.42 thf(setadjoinAx_type,type,
% 0.16/0.42 setadjoinAx: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setadjoinAx,definition,
% 0.16/0.42 ( setadjoinAx
% 0.16/0.42 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.16/0.42 <=> ( ( Xy = Xx )
% 0.16/0.42 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(powerset_type,type,
% 0.16/0.42 powerset: $i > $i ).
% 0.16/0.42
% 0.16/0.42 thf(powersetAx_type,type,
% 0.16/0.42 powersetAx: $o ).
% 0.16/0.42
% 0.16/0.42 thf(powersetAx,definition,
% 0.16/0.42 ( powersetAx
% 0.16/0.42 = ( ! [A: $i,B: $i] :
% 0.16/0.42 ( ( in @ B @ ( powerset @ A ) )
% 0.16/0.42 <=> ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ B )
% 0.16/0.42 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setunion_type,type,
% 0.16/0.42 setunion: $i > $i ).
% 0.16/0.42
% 0.16/0.42 thf(setunionAx_type,type,
% 0.16/0.42 setunionAx: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setunionAx,definition,
% 0.16/0.42 ( setunionAx
% 0.16/0.42 = ( ! [A: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ ( setunion @ A ) )
% 0.16/0.42 <=> ? [B: $i] :
% 0.16/0.42 ( ( in @ Xx @ B )
% 0.16/0.42 & ( in @ B @ A ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(omega_type,type,
% 0.16/0.42 omega: $i ).
% 0.16/0.42
% 0.16/0.42 thf(omega0Ax_type,type,
% 0.16/0.42 omega0Ax: $o ).
% 0.16/0.42
% 0.16/0.42 thf(omega0Ax,definition,
% 0.16/0.42 ( omega0Ax
% 0.16/0.42 = ( in @ emptyset @ omega ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(omegaSAx_type,type,
% 0.16/0.42 omegaSAx: $o ).
% 0.16/0.42
% 0.16/0.42 thf(omegaSAx,definition,
% 0.16/0.42 ( omegaSAx
% 0.16/0.42 = ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ omega )
% 0.16/0.42 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(omegaIndAx_type,type,
% 0.16/0.42 omegaIndAx: $o ).
% 0.16/0.42
% 0.16/0.42 thf(omegaIndAx,definition,
% 0.16/0.42 ( omegaIndAx
% 0.16/0.42 = ( ! [A: $i] :
% 0.16/0.42 ( ( ( in @ emptyset @ A )
% 0.16/0.42 & ! [Xx: $i] :
% 0.16/0.42 ( ( ( in @ Xx @ omega )
% 0.16/0.42 & ( in @ Xx @ A ) )
% 0.16/0.42 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ omega )
% 0.16/0.42 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(replAx_type,type,
% 0.16/0.42 replAx: $o ).
% 0.16/0.42
% 0.16/0.42 thf(replAx,definition,
% 0.16/0.42 ( replAx
% 0.16/0.42 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.16/0.42 ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( exu
% 0.16/0.42 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.16/0.42 => ? [B: $i] :
% 0.16/0.42 ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ B )
% 0.16/0.42 <=> ? [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ A )
% 0.16/0.42 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(foundationAx_type,type,
% 0.16/0.42 foundationAx: $o ).
% 0.16/0.42
% 0.16/0.42 thf(foundationAx,definition,
% 0.16/0.42 ( foundationAx
% 0.16/0.42 = ( ! [A: $i] :
% 0.16/0.42 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.16/0.42 => ? [B: $i] :
% 0.16/0.42 ( ( in @ B @ A )
% 0.16/0.42 & ~ ? [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ B )
% 0.16/0.42 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(wellorderingAx_type,type,
% 0.16/0.42 wellorderingAx: $o ).
% 0.16/0.42
% 0.16/0.42 thf(wellorderingAx,definition,
% 0.16/0.42 ( wellorderingAx
% 0.16/0.42 = ( ! [A: $i] :
% 0.16/0.42 ? [B: $i] :
% 0.16/0.42 ( ! [C: $i] :
% 0.16/0.42 ( ( in @ C @ B )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ C )
% 0.16/0.42 => ( in @ Xx @ A ) ) )
% 0.16/0.42 & ! [Xx: $i,Xy: $i] :
% 0.16/0.42 ( ( ( in @ Xx @ A )
% 0.16/0.42 & ( in @ Xy @ A ) )
% 0.16/0.42 => ( ! [C: $i] :
% 0.16/0.42 ( ( in @ C @ B )
% 0.16/0.42 => ( ( in @ Xx @ C )
% 0.16/0.42 <=> ( in @ Xy @ C ) ) )
% 0.16/0.42 => ( Xx = Xy ) ) )
% 0.16/0.42 & ! [C: $i,D: $i] :
% 0.16/0.42 ( ( ( in @ C @ B )
% 0.16/0.42 & ( in @ D @ B ) )
% 0.16/0.42 => ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ C )
% 0.16/0.42 => ( in @ Xx @ D ) )
% 0.16/0.42 | ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ D )
% 0.16/0.42 => ( in @ Xx @ C ) ) ) )
% 0.16/0.42 & ! [C: $i] :
% 0.16/0.42 ( ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ C )
% 0.16/0.42 => ( in @ Xx @ A ) )
% 0.16/0.42 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.16/0.42 => ? [D: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ D @ B )
% 0.16/0.42 & ( in @ Xx @ C )
% 0.16/0.42 & ~ ? [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ D )
% 0.16/0.42 & ( in @ Xy @ C ) )
% 0.16/0.42 & ! [E: $i] :
% 0.16/0.42 ( ( in @ E @ B )
% 0.16/0.42 => ( ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ E )
% 0.16/0.42 => ( in @ Xy @ D ) )
% 0.16/0.42 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(descr_type,type,
% 0.16/0.42 descr: ( $i > $o ) > $i ).
% 0.16/0.42
% 0.16/0.42 thf(descrp_type,type,
% 0.16/0.42 descrp: $o ).
% 0.16/0.42
% 0.16/0.42 thf(descrp,definition,
% 0.16/0.42 ( descrp
% 0.16/0.42 = ( ! [Xphi: $i > $o] :
% 0.16/0.42 ( ( exu
% 0.16/0.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.42 => ( Xphi
% 0.16/0.42 @ ( descr
% 0.16/0.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(dsetconstr_type,type,
% 0.16/0.42 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.16/0.42
% 0.16/0.42 thf(dsetconstrI_type,type,
% 0.16/0.42 dsetconstrI: $o ).
% 0.16/0.42
% 0.16/0.42 thf(dsetconstrI,definition,
% 0.16/0.42 ( dsetconstrI
% 0.16/0.42 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( ( Xphi @ Xx )
% 0.16/0.42 => ( in @ Xx
% 0.16/0.42 @ ( dsetconstr @ A
% 0.16/0.42 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(dsetconstrEL_type,type,
% 0.16/0.42 dsetconstrEL: $o ).
% 0.16/0.42
% 0.16/0.42 thf(dsetconstrEL,definition,
% 0.16/0.42 ( dsetconstrEL
% 0.16/0.42 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx
% 0.16/0.42 @ ( dsetconstr @ A
% 0.16/0.42 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.16/0.42 => ( in @ Xx @ A ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(dsetconstrER_type,type,
% 0.16/0.42 dsetconstrER: $o ).
% 0.16/0.42
% 0.16/0.42 thf(dsetconstrER,definition,
% 0.16/0.42 ( dsetconstrER
% 0.16/0.42 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx
% 0.16/0.42 @ ( dsetconstr @ A
% 0.16/0.42 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.16/0.42 => ( Xphi @ Xx ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(exuE1_type,type,
% 0.16/0.42 exuE1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(exuE1,definition,
% 0.16/0.42 ( exuE1
% 0.16/0.42 = ( ! [Xphi: $i > $o] :
% 0.16/0.42 ( ( exu
% 0.16/0.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.42 => ? [Xx: $i] :
% 0.16/0.42 ( ( Xphi @ Xx )
% 0.16/0.42 & ! [Xy: $i] :
% 0.16/0.42 ( ( Xphi @ Xy )
% 0.16/0.42 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(prop2set_type,type,
% 0.16/0.42 prop2set: $o > $i ).
% 0.16/0.42
% 0.16/0.42 thf(prop2setE_type,type,
% 0.16/0.42 prop2setE: $o ).
% 0.16/0.42
% 0.16/0.42 thf(prop2setE,definition,
% 0.16/0.42 ( prop2setE
% 0.16/0.42 = ( ! [Xphi: $o,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.16/0.42 => Xphi ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(emptysetE_type,type,
% 0.16/0.42 emptysetE: $o ).
% 0.16/0.42
% 0.16/0.42 thf(emptysetE,definition,
% 0.16/0.42 ( emptysetE
% 0.16/0.42 = ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ emptyset )
% 0.16/0.42 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(emptysetimpfalse_type,type,
% 0.16/0.42 emptysetimpfalse: $o ).
% 0.16/0.42
% 0.16/0.42 thf(emptysetimpfalse,definition,
% 0.16/0.42 ( emptysetimpfalse
% 0.16/0.42 = ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ emptyset )
% 0.16/0.42 => $false ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(notinemptyset_type,type,
% 0.16/0.42 notinemptyset: $o ).
% 0.16/0.42
% 0.16/0.42 thf(notinemptyset,definition,
% 0.16/0.42 ( notinemptyset
% 0.16/0.42 = ( ! [Xx: $i] :
% 0.16/0.42 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(exuE3e_type,type,
% 0.16/0.42 exuE3e: $o ).
% 0.16/0.42
% 0.16/0.42 thf(exuE3e,definition,
% 0.16/0.42 ( exuE3e
% 0.16/0.42 = ( ! [Xphi: $i > $o] :
% 0.16/0.42 ( ( exu
% 0.16/0.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.42 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setext_type,type,
% 0.16/0.42 setext: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setext,definition,
% 0.16/0.42 ( setext
% 0.16/0.42 = ( ! [A: $i,B: $i] :
% 0.16/0.42 ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( in @ Xx @ B ) )
% 0.16/0.42 => ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ B )
% 0.16/0.42 => ( in @ Xx @ A ) )
% 0.16/0.42 => ( A = B ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(emptyI_type,type,
% 0.16/0.42 emptyI: $o ).
% 0.16/0.42
% 0.16/0.42 thf(emptyI,definition,
% 0.16/0.42 ( emptyI
% 0.16/0.42 = ( ! [A: $i] :
% 0.16/0.42 ( ! [Xx: $i] :
% 0.16/0.42 ~ ( in @ Xx @ A )
% 0.16/0.42 => ( A = emptyset ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(noeltsimpempty_type,type,
% 0.16/0.42 noeltsimpempty: $o ).
% 0.16/0.42
% 0.16/0.42 thf(noeltsimpempty,definition,
% 0.16/0.42 ( noeltsimpempty
% 0.16/0.42 = ( ! [A: $i] :
% 0.16/0.42 ( ! [Xx: $i] :
% 0.16/0.42 ~ ( in @ Xx @ A )
% 0.16/0.42 => ( A = emptyset ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setbeta_type,type,
% 0.16/0.42 setbeta: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setbeta,definition,
% 0.16/0.42 ( setbeta
% 0.16/0.42 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( ( in @ Xx
% 0.16/0.42 @ ( dsetconstr @ A
% 0.16/0.42 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.16/0.42 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(nonempty_type,type,
% 0.16/0.42 nonempty: $i > $o ).
% 0.16/0.42
% 0.16/0.42 thf(nonempty,definition,
% 0.16/0.42 ( nonempty
% 0.16/0.42 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(nonemptyE1_type,type,
% 0.16/0.42 nonemptyE1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(nonemptyE1,definition,
% 0.16/0.42 ( nonemptyE1
% 0.16/0.42 = ( ! [A: $i] :
% 0.16/0.42 ( ( nonempty @ A )
% 0.16/0.42 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(nonemptyI_type,type,
% 0.16/0.42 nonemptyI: $o ).
% 0.16/0.42
% 0.16/0.42 thf(nonemptyI,definition,
% 0.16/0.42 ( nonemptyI
% 0.16/0.42 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( ( Xphi @ Xx )
% 0.16/0.42 => ( nonempty
% 0.16/0.42 @ ( dsetconstr @ A
% 0.16/0.42 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(nonemptyI1_type,type,
% 0.16/0.42 nonemptyI1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(nonemptyI1,definition,
% 0.16/0.42 ( nonemptyI1
% 0.16/0.42 = ( ! [A: $i] :
% 0.16/0.42 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.16/0.42 => ( nonempty @ A ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setadjoinIL_type,type,
% 0.16/0.42 setadjoinIL: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setadjoinIL,definition,
% 0.16/0.42 ( setadjoinIL
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(emptyinunitempty_type,type,
% 0.16/0.42 emptyinunitempty: $o ).
% 0.16/0.42
% 0.16/0.42 thf(emptyinunitempty,definition,
% 0.16/0.42 ( emptyinunitempty
% 0.16/0.42 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setadjoinIR_type,type,
% 0.16/0.42 setadjoinIR: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setadjoinIR,definition,
% 0.16/0.42 ( setadjoinIR
% 0.16/0.42 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ A )
% 0.16/0.42 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setadjoinE_type,type,
% 0.16/0.42 setadjoinE: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setadjoinE,definition,
% 0.16/0.42 ( setadjoinE
% 0.16/0.42 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.16/0.42 => ! [Xphi: $o] :
% 0.16/0.42 ( ( ( Xy = Xx )
% 0.16/0.42 => Xphi )
% 0.16/0.42 => ( ( ( in @ Xy @ A )
% 0.16/0.42 => Xphi )
% 0.16/0.42 => Xphi ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setadjoinOr_type,type,
% 0.16/0.42 setadjoinOr: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setadjoinOr,definition,
% 0.16/0.42 ( setadjoinOr
% 0.16/0.42 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.16/0.42 => ( ( Xy = Xx )
% 0.16/0.42 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setoftrueEq_type,type,
% 0.16/0.42 setoftrueEq: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setoftrueEq,definition,
% 0.16/0.42 ( setoftrueEq
% 0.16/0.42 = ( ! [A: $i] :
% 0.16/0.42 ( ( dsetconstr @ A
% 0.16/0.42 @ ^ [Xx: $i] : $true )
% 0.16/0.42 = A ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(powersetI_type,type,
% 0.16/0.42 powersetI: $o ).
% 0.16/0.42
% 0.16/0.42 thf(powersetI,definition,
% 0.16/0.42 ( powersetI
% 0.16/0.42 = ( ! [A: $i,B: $i] :
% 0.16/0.42 ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ B )
% 0.16/0.42 => ( in @ Xx @ A ) )
% 0.16/0.42 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(emptyinPowerset_type,type,
% 0.16/0.42 emptyinPowerset: $o ).
% 0.16/0.42
% 0.16/0.42 thf(emptyinPowerset,definition,
% 0.16/0.42 ( emptyinPowerset
% 0.16/0.42 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(emptyInPowerset_type,type,
% 0.16/0.42 emptyInPowerset: $o ).
% 0.16/0.42
% 0.16/0.42 thf(emptyInPowerset,definition,
% 0.16/0.42 ( emptyInPowerset
% 0.16/0.42 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(powersetE_type,type,
% 0.16/0.42 powersetE: $o ).
% 0.16/0.42
% 0.16/0.42 thf(powersetE,definition,
% 0.16/0.42 ( powersetE
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ B @ ( powerset @ A ) )
% 0.16/0.42 => ( ( in @ Xx @ B )
% 0.16/0.42 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setunionI_type,type,
% 0.16/0.42 setunionI: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setunionI,definition,
% 0.16/0.42 ( setunionI
% 0.16/0.42 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.16/0.42 ( ( in @ Xx @ B )
% 0.16/0.42 => ( ( in @ B @ A )
% 0.16/0.42 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setunionE_type,type,
% 0.16/0.42 setunionE: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setunionE,definition,
% 0.16/0.42 ( setunionE
% 0.16/0.42 = ( ! [A: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ ( setunion @ A ) )
% 0.16/0.42 => ! [Xphi: $o] :
% 0.16/0.42 ( ! [B: $i] :
% 0.16/0.42 ( ( in @ Xx @ B )
% 0.16/0.42 => ( ( in @ B @ A )
% 0.16/0.42 => Xphi ) )
% 0.16/0.42 => Xphi ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(subPowSU_type,type,
% 0.16/0.42 subPowSU: $o ).
% 0.16/0.42
% 0.16/0.42 thf(subPowSU,definition,
% 0.16/0.42 ( subPowSU
% 0.16/0.42 = ( ! [A: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(exuE2_type,type,
% 0.16/0.42 exuE2: $o ).
% 0.16/0.42
% 0.16/0.42 thf(exuE2,definition,
% 0.16/0.42 ( exuE2
% 0.16/0.42 = ( ! [Xphi: $i > $o] :
% 0.16/0.42 ( ( exu
% 0.16/0.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.42 => ? [Xx: $i] :
% 0.16/0.42 ! [Xy: $i] :
% 0.16/0.42 ( ( Xphi @ Xy )
% 0.16/0.42 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(nonemptyImpWitness_type,type,
% 0.16/0.42 nonemptyImpWitness: $o ).
% 0.16/0.42
% 0.16/0.42 thf(nonemptyImpWitness,definition,
% 0.16/0.42 ( nonemptyImpWitness
% 0.16/0.42 = ( ! [A: $i] :
% 0.16/0.42 ( ( nonempty @ A )
% 0.16/0.42 => ? [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 & $true ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(uniqinunit_type,type,
% 0.16/0.42 uniqinunit: $o ).
% 0.16/0.42
% 0.16/0.42 thf(uniqinunit,definition,
% 0.16/0.42 ( uniqinunit
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.42 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.16/0.42 => ( Xx = Xy ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(notinsingleton_type,type,
% 0.16/0.42 notinsingleton: $o ).
% 0.16/0.42
% 0.16/0.42 thf(notinsingleton,definition,
% 0.16/0.42 ( notinsingleton
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.42 ( ( Xx != Xy )
% 0.16/0.42 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(eqinunit_type,type,
% 0.16/0.42 eqinunit: $o ).
% 0.16/0.42
% 0.16/0.42 thf(eqinunit,definition,
% 0.16/0.42 ( eqinunit
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.42 ( ( Xx = Xy )
% 0.16/0.42 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(singletonsswitch_type,type,
% 0.16/0.42 singletonsswitch: $o ).
% 0.16/0.42
% 0.16/0.42 thf(singletonsswitch,definition,
% 0.16/0.42 ( singletonsswitch
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.42 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.16/0.42 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(upairsetE_type,type,
% 0.16/0.42 upairsetE: $o ).
% 0.16/0.42
% 0.16/0.42 thf(upairsetE,definition,
% 0.16/0.42 ( upairsetE
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.16/0.42 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.16/0.42 => ( ( Xz = Xx )
% 0.16/0.42 | ( Xz = Xy ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(upairsetIL_type,type,
% 0.16/0.42 upairsetIL: $o ).
% 0.16/0.42
% 0.16/0.42 thf(upairsetIL,definition,
% 0.16/0.42 ( upairsetIL
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(upairsetIR_type,type,
% 0.16/0.42 upairsetIR: $o ).
% 0.16/0.42
% 0.16/0.42 thf(upairsetIR,definition,
% 0.16/0.42 ( upairsetIR
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(emptyE1_type,type,
% 0.16/0.42 emptyE1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(emptyE1,definition,
% 0.16/0.42 ( emptyE1
% 0.16/0.42 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.42 ( ? [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 & ( Xphi @ Xx ) )
% 0.16/0.42 => ( ( ( dsetconstr @ A
% 0.16/0.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.42 = emptyset )
% 0.16/0.42 => $false ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(vacuousDall_type,type,
% 0.16/0.42 vacuousDall: $o ).
% 0.16/0.42
% 0.16/0.42 thf(vacuousDall,definition,
% 0.16/0.42 ( vacuousDall
% 0.16/0.42 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ emptyset )
% 0.16/0.42 => ( Xphi @ Xx ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(quantDeMorgan1_type,type,
% 0.16/0.42 quantDeMorgan1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(quantDeMorgan1,definition,
% 0.16/0.42 ( quantDeMorgan1
% 0.16/0.42 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.42 ( ~ ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( Xphi @ Xx ) )
% 0.16/0.42 => ? [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(quantDeMorgan2_type,type,
% 0.16/0.42 quantDeMorgan2: $o ).
% 0.16/0.42
% 0.16/0.42 thf(quantDeMorgan2,definition,
% 0.16/0.42 ( quantDeMorgan2
% 0.16/0.42 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.42 ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ~ ( Xphi @ Xx ) )
% 0.16/0.42 => ~ ? [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(quantDeMorgan3_type,type,
% 0.16/0.42 quantDeMorgan3: $o ).
% 0.16/0.42
% 0.16/0.42 thf(quantDeMorgan3,definition,
% 0.16/0.42 ( quantDeMorgan3
% 0.16/0.42 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.42 ( ~ ? [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 & ( Xphi @ Xx ) )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(quantDeMorgan4_type,type,
% 0.16/0.42 quantDeMorgan4: $o ).
% 0.16/0.42
% 0.16/0.42 thf(quantDeMorgan4,definition,
% 0.16/0.42 ( quantDeMorgan4
% 0.16/0.42 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.42 ( ? [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 & ~ ( Xphi @ Xx ) )
% 0.16/0.42 => ~ ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(prop2setI_type,type,
% 0.16/0.42 prop2setI: $o ).
% 0.16/0.42
% 0.16/0.42 thf(prop2setI,definition,
% 0.16/0.42 ( prop2setI
% 0.16/0.42 = ( ! [Xphi: $o] :
% 0.16/0.42 ( Xphi
% 0.16/0.42 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(set2prop_type,type,
% 0.16/0.42 set2prop: $i > $o ).
% 0.16/0.42
% 0.16/0.42 thf(prop2set2propI_type,type,
% 0.16/0.42 prop2set2propI: $o ).
% 0.16/0.42
% 0.16/0.42 thf(prop2set2propI,definition,
% 0.16/0.42 ( prop2set2propI
% 0.16/0.42 = ( ! [Xphi: $o] :
% 0.16/0.42 ( Xphi
% 0.16/0.42 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(notdexE_type,type,
% 0.16/0.42 notdexE: $o ).
% 0.16/0.42
% 0.16/0.42 thf(notdexE,definition,
% 0.16/0.42 ( notdexE
% 0.16/0.42 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.42 ( ~ ? [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 & ( Xphi @ Xx ) )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(notdallE_type,type,
% 0.16/0.42 notdallE: $o ).
% 0.16/0.42
% 0.16/0.42 thf(notdallE,definition,
% 0.16/0.42 ( notdallE
% 0.16/0.42 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.42 ( ~ ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( Xphi @ Xx ) )
% 0.16/0.42 => ? [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(exuI1_type,type,
% 0.16/0.42 exuI1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(exuI1,definition,
% 0.16/0.42 ( exuI1
% 0.16/0.42 = ( ! [Xphi: $i > $o] :
% 0.16/0.42 ( ? [Xx: $i] :
% 0.16/0.42 ( ( Xphi @ Xx )
% 0.16/0.42 & ! [Xy: $i] :
% 0.16/0.42 ( ( Xphi @ Xy )
% 0.16/0.42 => ( Xx = Xy ) ) )
% 0.16/0.42 => ( exu
% 0.16/0.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(exuI3_type,type,
% 0.16/0.42 exuI3: $o ).
% 0.16/0.42
% 0.16/0.42 thf(exuI3,definition,
% 0.16/0.42 ( exuI3
% 0.16/0.42 = ( ! [Xphi: $i > $o] :
% 0.16/0.42 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.16/0.42 => ( ! [Xx: $i,Xy: $i] :
% 0.16/0.42 ( ( Xphi @ Xx )
% 0.16/0.42 => ( ( Xphi @ Xy )
% 0.16/0.42 => ( Xx = Xy ) ) )
% 0.16/0.42 => ( exu
% 0.16/0.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(exuI2_type,type,
% 0.16/0.42 exuI2: $o ).
% 0.16/0.42
% 0.16/0.42 thf(exuI2,definition,
% 0.16/0.42 ( exuI2
% 0.16/0.42 = ( ! [Xphi: $i > $o] :
% 0.16/0.42 ( ? [Xx: $i] :
% 0.16/0.42 ! [Xy: $i] :
% 0.16/0.42 ( ( Xphi @ Xy )
% 0.16/0.42 <=> ( Xy = Xx ) )
% 0.16/0.42 => ( exu
% 0.16/0.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(inCongP_type,type,
% 0.16/0.42 inCongP: $o ).
% 0.16/0.42
% 0.16/0.42 thf(inCongP,definition,
% 0.16/0.42 ( inCongP
% 0.16/0.42 = ( ! [A: $i,B: $i] :
% 0.16/0.42 ( ( A = B )
% 0.16/0.42 => ! [Xx: $i,Xy: $i] :
% 0.16/0.42 ( ( Xx = Xy )
% 0.16/0.42 => ( ( in @ Xx @ A )
% 0.16/0.42 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(in__Cong_type,type,
% 0.16/0.42 in__Cong: $o ).
% 0.16/0.42
% 0.16/0.42 thf(in__Cong,definition,
% 0.16/0.42 ( in__Cong
% 0.16/0.42 = ( ! [A: $i,B: $i] :
% 0.16/0.42 ( ( A = B )
% 0.16/0.42 => ! [Xx: $i,Xy: $i] :
% 0.16/0.42 ( ( Xx = Xy )
% 0.16/0.42 => ( ( in @ Xx @ A )
% 0.16/0.42 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(exuE3u_type,type,
% 0.16/0.42 exuE3u: $o ).
% 0.16/0.42
% 0.16/0.42 thf(exuE3u,definition,
% 0.16/0.42 ( exuE3u
% 0.16/0.42 = ( ! [Xphi: $i > $o] :
% 0.16/0.42 ( ( exu
% 0.16/0.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.42 => ! [Xx: $i,Xy: $i] :
% 0.16/0.42 ( ( Xphi @ Xx )
% 0.16/0.42 => ( ( Xphi @ Xy )
% 0.16/0.42 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(exu__Cong_type,type,
% 0.16/0.42 exu__Cong: $o ).
% 0.16/0.42
% 0.16/0.42 thf(exu__Cong,definition,
% 0.16/0.42 ( exu__Cong
% 0.16/0.42 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.16/0.42 ( ! [Xx: $i,Xy: $i] :
% 0.16/0.42 ( ( Xx = Xy )
% 0.16/0.42 => ( ( Xphi @ Xx )
% 0.16/0.42 <=> ( Xpsi @ Xy ) ) )
% 0.16/0.42 => ( ( exu
% 0.16/0.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.42 <=> ( exu
% 0.16/0.42 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(emptyset__Cong_type,type,
% 0.16/0.42 emptyset__Cong: $o ).
% 0.16/0.42
% 0.16/0.42 thf(emptyset__Cong,definition,
% 0.16/0.42 ( emptyset__Cong
% 0.16/0.42 = ( emptyset = emptyset ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setadjoin__Cong_type,type,
% 0.16/0.42 setadjoin__Cong: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setadjoin__Cong,definition,
% 0.16/0.42 ( setadjoin__Cong
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.42 ( ( Xx = Xy )
% 0.16/0.42 => ! [Xz: $i,Xu: $i] :
% 0.16/0.42 ( ( Xz = Xu )
% 0.16/0.42 => ( ( setadjoin @ Xx @ Xz )
% 0.16/0.42 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(powerset__Cong_type,type,
% 0.16/0.42 powerset__Cong: $o ).
% 0.16/0.42
% 0.16/0.42 thf(powerset__Cong,definition,
% 0.16/0.42 ( powerset__Cong
% 0.16/0.42 = ( ! [A: $i,B: $i] :
% 0.16/0.42 ( ( A = B )
% 0.16/0.42 => ( ( powerset @ A )
% 0.16/0.42 = ( powerset @ B ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setunion__Cong_type,type,
% 0.16/0.42 setunion__Cong: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setunion__Cong,definition,
% 0.16/0.42 ( setunion__Cong
% 0.16/0.42 = ( ! [A: $i,B: $i] :
% 0.16/0.42 ( ( A = B )
% 0.16/0.42 => ( ( setunion @ A )
% 0.16/0.42 = ( setunion @ B ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(omega__Cong_type,type,
% 0.16/0.42 omega__Cong: $o ).
% 0.16/0.42
% 0.16/0.42 thf(omega__Cong,definition,
% 0.16/0.42 ( omega__Cong
% 0.16/0.42 = ( omega = omega ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(exuEu_type,type,
% 0.16/0.42 exuEu: $o ).
% 0.16/0.42
% 0.16/0.42 thf(exuEu,definition,
% 0.16/0.42 ( exuEu
% 0.16/0.42 = ( ! [Xphi: $i > $o] :
% 0.16/0.42 ( ( exu
% 0.16/0.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.42 => ! [Xx: $i,Xy: $i] :
% 0.16/0.42 ( ( Xphi @ Xx )
% 0.16/0.42 => ( ( Xphi @ Xy )
% 0.16/0.42 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(descr__Cong_type,type,
% 0.16/0.42 descr__Cong: $o ).
% 0.16/0.42
% 0.16/0.42 thf(descr__Cong,definition,
% 0.16/0.42 ( descr__Cong
% 0.16/0.42 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.16/0.42 ( ! [Xx: $i,Xy: $i] :
% 0.16/0.42 ( ( Xx = Xy )
% 0.16/0.42 => ( ( Xphi @ Xx )
% 0.16/0.42 <=> ( Xpsi @ Xy ) ) )
% 0.16/0.42 => ( ( exu
% 0.16/0.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.42 => ( ( exu
% 0.16/0.42 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.16/0.42 => ( ( descr
% 0.16/0.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.42 = ( descr
% 0.16/0.42 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(dsetconstr__Cong_type,type,
% 0.16/0.42 dsetconstr__Cong: $o ).
% 0.16/0.42
% 0.16/0.42 thf(dsetconstr__Cong,definition,
% 0.16/0.42 ( dsetconstr__Cong
% 0.16/0.42 = ( ! [A: $i,B: $i] :
% 0.16/0.42 ( ( A = B )
% 0.16/0.42 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.16/0.42 ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ B )
% 0.16/0.42 => ( ( Xx = Xy )
% 0.16/0.42 => ( ( Xphi @ Xx )
% 0.16/0.42 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.16/0.42 => ( ( dsetconstr @ A
% 0.16/0.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.42 = ( dsetconstr @ B
% 0.16/0.42 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(subset_type,type,
% 0.16/0.42 subset: $i > $i > $o ).
% 0.16/0.42
% 0.16/0.42 thf(disjoint_type,type,
% 0.16/0.42 disjoint: $i > $i > $o ).
% 0.16/0.42
% 0.16/0.42 thf(setsmeet_type,type,
% 0.16/0.42 setsmeet: $i > $i > $o ).
% 0.16/0.42
% 0.16/0.42 thf(subsetI1_type,type,
% 0.16/0.42 subsetI1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(subsetI1,definition,
% 0.16/0.42 ( subsetI1
% 0.16/0.42 = ( ! [A: $i,B: $i] :
% 0.16/0.42 ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( in @ Xx @ B ) )
% 0.16/0.42 => ( subset @ A @ B ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(eqimpsubset2_type,type,
% 0.16/0.42 eqimpsubset2: $o ).
% 0.16/0.42
% 0.16/0.42 thf(eqimpsubset2,definition,
% 0.16/0.42 ( eqimpsubset2
% 0.16/0.42 = ( ! [A: $i,B: $i] :
% 0.16/0.42 ( ( A = B )
% 0.16/0.42 => ( subset @ B @ A ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(eqimpsubset1_type,type,
% 0.16/0.42 eqimpsubset1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(eqimpsubset1,definition,
% 0.16/0.42 ( eqimpsubset1
% 0.16/0.42 = ( ! [A: $i,B: $i] :
% 0.16/0.42 ( ( A = B )
% 0.16/0.42 => ( subset @ A @ B ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(subsetI2_type,type,
% 0.16/0.42 subsetI2: $o ).
% 0.16/0.42
% 0.16/0.42 thf(subsetI2,definition,
% 0.16/0.42 ( subsetI2
% 0.16/0.42 = ( ! [A: $i,B: $i] :
% 0.16/0.42 ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( in @ Xx @ B ) )
% 0.16/0.42 => ( subset @ A @ B ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(emptysetsubset_type,type,
% 0.16/0.42 emptysetsubset: $o ).
% 0.16/0.42
% 0.16/0.42 thf(emptysetsubset,definition,
% 0.16/0.42 ( emptysetsubset
% 0.16/0.42 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(subsetE_type,type,
% 0.16/0.42 subsetE: $o ).
% 0.16/0.42
% 0.16/0.42 thf(subsetE,definition,
% 0.16/0.42 ( subsetE
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ( subset @ A @ B )
% 0.16/0.42 => ( ( in @ Xx @ A )
% 0.16/0.42 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(subsetE2_type,type,
% 0.16/0.42 subsetE2: $o ).
% 0.16/0.42
% 0.16/0.42 thf(subsetE2,definition,
% 0.16/0.42 ( subsetE2
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ( subset @ A @ B )
% 0.16/0.42 => ( ~ ( in @ Xx @ B )
% 0.16/0.42 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(notsubsetI_type,type,
% 0.16/0.42 notsubsetI: $o ).
% 0.16/0.42
% 0.16/0.42 thf(notsubsetI,definition,
% 0.16/0.42 ( notsubsetI
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( ~ ( in @ Xx @ B )
% 0.16/0.42 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(notequalI1_type,type,
% 0.16/0.42 notequalI1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(notequalI1,definition,
% 0.16/0.42 ( notequalI1
% 0.16/0.42 = ( ! [A: $i,B: $i] :
% 0.16/0.42 ( ~ ( subset @ A @ B )
% 0.16/0.42 => ( A != B ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(notequalI2_type,type,
% 0.16/0.42 notequalI2: $o ).
% 0.16/0.42
% 0.16/0.42 thf(notequalI2,definition,
% 0.16/0.42 ( notequalI2
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( ~ ( in @ Xx @ B )
% 0.16/0.42 => ( A != B ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(subsetRefl_type,type,
% 0.16/0.42 subsetRefl: $o ).
% 0.16/0.42
% 0.16/0.42 thf(subsetRefl,definition,
% 0.16/0.42 ( subsetRefl
% 0.16/0.42 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(subsetTrans_type,type,
% 0.16/0.42 subsetTrans: $o ).
% 0.16/0.42
% 0.16/0.42 thf(subsetTrans,definition,
% 0.16/0.42 ( subsetTrans
% 0.16/0.42 = ( ! [A: $i,B: $i,C: $i] :
% 0.16/0.42 ( ( subset @ A @ B )
% 0.16/0.42 => ( ( subset @ B @ C )
% 0.16/0.42 => ( subset @ A @ C ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setadjoinSub_type,type,
% 0.16/0.42 setadjoinSub: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setadjoinSub,definition,
% 0.16/0.42 ( setadjoinSub
% 0.16/0.42 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setadjoinSub2_type,type,
% 0.16/0.42 setadjoinSub2: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setadjoinSub2,definition,
% 0.16/0.42 ( setadjoinSub2
% 0.16/0.42 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.16/0.42 ( ( subset @ A @ B )
% 0.16/0.42 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(subset2powerset_type,type,
% 0.16/0.42 subset2powerset: $o ).
% 0.16/0.42
% 0.16/0.42 thf(subset2powerset,definition,
% 0.16/0.42 ( subset2powerset
% 0.16/0.42 = ( ! [A: $i,B: $i] :
% 0.16/0.42 ( ( subset @ A @ B )
% 0.16/0.42 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setextsub_type,type,
% 0.16/0.42 setextsub: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setextsub,definition,
% 0.16/0.42 ( setextsub
% 0.16/0.42 = ( ! [A: $i,B: $i] :
% 0.16/0.42 ( ( subset @ A @ B )
% 0.16/0.42 => ( ( subset @ B @ A )
% 0.16/0.42 => ( A = B ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(subsetemptysetimpeq_type,type,
% 0.16/0.42 subsetemptysetimpeq: $o ).
% 0.16/0.42
% 0.16/0.42 thf(subsetemptysetimpeq,definition,
% 0.16/0.42 ( subsetemptysetimpeq
% 0.16/0.42 = ( ! [A: $i] :
% 0.16/0.42 ( ( subset @ A @ emptyset )
% 0.16/0.42 => ( A = emptyset ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(powersetI1_type,type,
% 0.16/0.42 powersetI1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(powersetI1,definition,
% 0.16/0.42 ( powersetI1
% 0.16/0.42 = ( ! [A: $i,B: $i] :
% 0.16/0.42 ( ( subset @ B @ A )
% 0.16/0.42 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(powersetE1_type,type,
% 0.16/0.42 powersetE1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(powersetE1,definition,
% 0.16/0.42 ( powersetE1
% 0.16/0.42 = ( ! [A: $i,B: $i] :
% 0.16/0.42 ( ( in @ B @ ( powerset @ A ) )
% 0.16/0.42 => ( subset @ B @ A ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(inPowerset_type,type,
% 0.16/0.42 inPowerset: $o ).
% 0.16/0.42
% 0.16/0.42 thf(inPowerset,definition,
% 0.16/0.42 ( inPowerset
% 0.16/0.42 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(powersetsubset_type,type,
% 0.16/0.42 powersetsubset: $o ).
% 0.16/0.42
% 0.16/0.42 thf(powersetsubset,definition,
% 0.16/0.42 ( powersetsubset
% 0.16/0.42 = ( ! [A: $i,B: $i] :
% 0.16/0.42 ( ( subset @ A @ B )
% 0.16/0.42 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(sepInPowerset_type,type,
% 0.16/0.42 sepInPowerset: $o ).
% 0.16/0.42
% 0.16/0.42 thf(sepInPowerset,definition,
% 0.16/0.42 ( sepInPowerset
% 0.16/0.42 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.42 ( in
% 0.16/0.42 @ ( dsetconstr @ A
% 0.16/0.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.42 @ ( powerset @ A ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(sepSubset_type,type,
% 0.16/0.42 sepSubset: $o ).
% 0.16/0.42
% 0.16/0.42 thf(sepSubset,definition,
% 0.16/0.42 ( sepSubset
% 0.16/0.42 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.42 ( subset
% 0.16/0.42 @ ( dsetconstr @ A
% 0.16/0.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.42 @ A ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(binunion_type,type,
% 0.16/0.42 binunion: $i > $i > $i ).
% 0.16/0.42
% 0.16/0.42 thf(binunionIL_type,type,
% 0.16/0.42 binunionIL: $o ).
% 0.16/0.42
% 0.16/0.42 thf(binunionIL,definition,
% 0.16/0.42 ( binunionIL
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(upairset2IR_type,type,
% 0.16/0.42 upairset2IR: $o ).
% 0.16/0.42
% 0.16/0.42 thf(upairset2IR,definition,
% 0.16/0.42 ( upairset2IR
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(binunionIR_type,type,
% 0.16/0.42 binunionIR: $o ).
% 0.16/0.42
% 0.16/0.42 thf(binunionIR,definition,
% 0.16/0.42 ( binunionIR
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ B )
% 0.16/0.42 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(binunionEcases_type,type,
% 0.16/0.42 binunionEcases: $o ).
% 0.16/0.42
% 0.16/0.42 thf(binunionEcases,definition,
% 0.16/0.42 ( binunionEcases
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.16/0.42 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.16/0.42 => ( ( ( in @ Xx @ A )
% 0.16/0.42 => Xphi )
% 0.16/0.42 => ( ( ( in @ Xx @ B )
% 0.16/0.42 => Xphi )
% 0.16/0.42 => Xphi ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(binunionE_type,type,
% 0.16/0.42 binunionE: $o ).
% 0.16/0.42
% 0.16/0.42 thf(binunionE,definition,
% 0.16/0.42 ( binunionE
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.16/0.42 => ( ( in @ Xx @ A )
% 0.16/0.42 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(binunionLsub_type,type,
% 0.16/0.42 binunionLsub: $o ).
% 0.16/0.42
% 0.16/0.42 thf(binunionLsub,definition,
% 0.16/0.42 ( binunionLsub
% 0.16/0.42 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(binunionRsub_type,type,
% 0.16/0.42 binunionRsub: $o ).
% 0.16/0.42
% 0.16/0.42 thf(binunionRsub,definition,
% 0.16/0.42 ( binunionRsub
% 0.16/0.42 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(binintersect_type,type,
% 0.16/0.42 binintersect: $i > $i > $i ).
% 0.16/0.42
% 0.16/0.42 thf(binintersectI_type,type,
% 0.16/0.42 binintersectI: $o ).
% 0.16/0.42
% 0.16/0.42 thf(binintersectI,definition,
% 0.16/0.42 ( binintersectI
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( ( in @ Xx @ B )
% 0.16/0.42 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(binintersectSubset5_type,type,
% 0.16/0.42 binintersectSubset5: $o ).
% 0.16/0.42
% 0.16/0.42 thf(binintersectSubset5,definition,
% 0.16/0.42 ( binintersectSubset5
% 0.16/0.42 = ( ! [A: $i,B: $i,C: $i] :
% 0.16/0.42 ( ( subset @ C @ A )
% 0.16/0.42 => ( ( subset @ C @ B )
% 0.16/0.42 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(binintersectEL_type,type,
% 0.16/0.42 binintersectEL: $o ).
% 0.16/0.42
% 0.16/0.42 thf(binintersectEL,definition,
% 0.16/0.42 ( binintersectEL
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.16/0.42 => ( in @ Xx @ A ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(binintersectLsub_type,type,
% 0.16/0.42 binintersectLsub: $o ).
% 0.16/0.42
% 0.16/0.42 thf(binintersectLsub,definition,
% 0.16/0.42 ( binintersectLsub
% 0.16/0.42 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(binintersectSubset2_type,type,
% 0.16/0.42 binintersectSubset2: $o ).
% 0.16/0.42
% 0.16/0.42 thf(binintersectSubset2,definition,
% 0.16/0.42 ( binintersectSubset2
% 0.16/0.42 = ( ! [A: $i,B: $i] :
% 0.16/0.42 ( ( subset @ A @ B )
% 0.16/0.42 => ( ( binintersect @ A @ B )
% 0.16/0.42 = A ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(binintersectSubset3_type,type,
% 0.16/0.42 binintersectSubset3: $o ).
% 0.16/0.42
% 0.16/0.42 thf(binintersectSubset3,definition,
% 0.16/0.42 ( binintersectSubset3
% 0.16/0.42 = ( ! [A: $i,B: $i] :
% 0.16/0.42 ( ( ( binintersect @ A @ B )
% 0.16/0.42 = B )
% 0.16/0.42 => ( subset @ B @ A ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(binintersectER_type,type,
% 0.16/0.42 binintersectER: $o ).
% 0.16/0.42
% 0.16/0.42 thf(binintersectER,definition,
% 0.16/0.42 ( binintersectER
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.16/0.42 => ( in @ Xx @ B ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(disjointsetsI1_type,type,
% 0.16/0.42 disjointsetsI1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(disjointsetsI1,definition,
% 0.16/0.42 ( disjointsetsI1
% 0.16/0.42 = ( ! [A: $i,B: $i] :
% 0.16/0.42 ( ~ ? [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 & ( in @ Xx @ B ) )
% 0.16/0.42 => ( ( binintersect @ A @ B )
% 0.16/0.42 = emptyset ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(binintersectRsub_type,type,
% 0.16/0.42 binintersectRsub: $o ).
% 0.16/0.42
% 0.16/0.42 thf(binintersectRsub,definition,
% 0.16/0.42 ( binintersectRsub
% 0.16/0.42 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(binintersectSubset4_type,type,
% 0.16/0.42 binintersectSubset4: $o ).
% 0.16/0.42
% 0.16/0.42 thf(binintersectSubset4,definition,
% 0.16/0.42 ( binintersectSubset4
% 0.16/0.42 = ( ! [A: $i,B: $i] :
% 0.16/0.42 ( ( subset @ B @ A )
% 0.16/0.42 => ( ( binintersect @ A @ B )
% 0.16/0.42 = B ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(binintersectSubset1_type,type,
% 0.16/0.42 binintersectSubset1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(binintersectSubset1,definition,
% 0.16/0.42 ( binintersectSubset1
% 0.16/0.42 = ( ! [A: $i,B: $i] :
% 0.16/0.42 ( ( ( binintersect @ A @ B )
% 0.16/0.42 = A )
% 0.16/0.42 => ( subset @ A @ B ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(bs114d_type,type,
% 0.16/0.42 bs114d: $o ).
% 0.16/0.42
% 0.16/0.42 thf(bs114d,definition,
% 0.16/0.42 ( bs114d
% 0.16/0.42 = ( ! [A: $i,B: $i,C: $i] :
% 0.16/0.42 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 0.16/0.42 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(regular_type,type,
% 0.16/0.42 regular: $i > $o ).
% 0.16/0.42
% 0.16/0.42 thf(setminus_type,type,
% 0.16/0.42 setminus: $i > $i > $i ).
% 0.16/0.42
% 0.16/0.42 thf(setminusI_type,type,
% 0.16/0.42 setminusI: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setminusI,definition,
% 0.16/0.42 ( setminusI
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( ~ ( in @ Xx @ B )
% 0.16/0.42 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setminusEL_type,type,
% 0.16/0.42 setminusEL: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setminusEL,definition,
% 0.16/0.42 ( setminusEL
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.16/0.42 => ( in @ Xx @ A ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setminusER_type,type,
% 0.16/0.42 setminusER: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setminusER,definition,
% 0.16/0.42 ( setminusER
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.16/0.42 => ~ ( in @ Xx @ B ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setminusSubset2_type,type,
% 0.16/0.42 setminusSubset2: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setminusSubset2,definition,
% 0.16/0.42 ( setminusSubset2
% 0.16/0.42 = ( ! [A: $i,B: $i] :
% 0.16/0.42 ( ( subset @ A @ B )
% 0.16/0.42 => ( ( setminus @ A @ B )
% 0.16/0.42 = emptyset ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setminusERneg_type,type,
% 0.16/0.42 setminusERneg: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setminusERneg,definition,
% 0.16/0.42 ( setminusERneg
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.16/0.42 => ( ( in @ Xx @ A )
% 0.16/0.42 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setminusELneg_type,type,
% 0.16/0.42 setminusELneg: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setminusELneg,definition,
% 0.16/0.42 ( setminusELneg
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.16/0.42 => ( ~ ( in @ Xx @ B )
% 0.16/0.42 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setminusILneg_type,type,
% 0.16/0.42 setminusILneg: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setminusILneg,definition,
% 0.16/0.42 ( setminusILneg
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ~ ( in @ Xx @ A )
% 0.16/0.42 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setminusIRneg_type,type,
% 0.16/0.42 setminusIRneg: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setminusIRneg,definition,
% 0.16/0.42 ( setminusIRneg
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ B )
% 0.16/0.42 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setminusLsub_type,type,
% 0.16/0.42 setminusLsub: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setminusLsub,definition,
% 0.16/0.42 ( setminusLsub
% 0.16/0.42 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setminusSubset1_type,type,
% 0.16/0.42 setminusSubset1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setminusSubset1,definition,
% 0.16/0.42 ( setminusSubset1
% 0.16/0.42 = ( ! [A: $i,B: $i] :
% 0.16/0.42 ( ( ( setminus @ A @ B )
% 0.16/0.42 = emptyset )
% 0.16/0.42 => ( subset @ A @ B ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(symdiff_type,type,
% 0.16/0.42 symdiff: $i > $i > $i ).
% 0.16/0.42
% 0.16/0.42 thf(symdiffE_type,type,
% 0.16/0.42 symdiffE: $o ).
% 0.16/0.42
% 0.16/0.42 thf(symdiffE,definition,
% 0.16/0.42 ( symdiffE
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 0.16/0.42 => ! [Xphi: $o] :
% 0.16/0.42 ( ( ( in @ Xx @ A )
% 0.16/0.42 => ( ~ ( in @ Xx @ B )
% 0.16/0.42 => Xphi ) )
% 0.16/0.42 => ( ( ~ ( in @ Xx @ A )
% 0.16/0.42 => ( ( in @ Xx @ B )
% 0.16/0.42 => Xphi ) )
% 0.16/0.42 => Xphi ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(symdiffI1_type,type,
% 0.16/0.42 symdiffI1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(symdiffI1,definition,
% 0.16/0.42 ( symdiffI1
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( ~ ( in @ Xx @ B )
% 0.16/0.42 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(symdiffI2_type,type,
% 0.16/0.42 symdiffI2: $o ).
% 0.16/0.42
% 0.16/0.42 thf(symdiffI2,definition,
% 0.16/0.42 ( symdiffI2
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ~ ( in @ Xx @ A )
% 0.16/0.42 => ( ( in @ Xx @ B )
% 0.16/0.42 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(symdiffIneg1_type,type,
% 0.16/0.42 symdiffIneg1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(symdiffIneg1,definition,
% 0.16/0.42 ( symdiffIneg1
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( ( in @ Xx @ B )
% 0.16/0.42 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(symdiffIneg2_type,type,
% 0.16/0.42 symdiffIneg2: $o ).
% 0.16/0.42
% 0.16/0.42 thf(symdiffIneg2,definition,
% 0.16/0.42 ( symdiffIneg2
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ~ ( in @ Xx @ A )
% 0.16/0.42 => ( ~ ( in @ Xx @ B )
% 0.16/0.42 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(iskpair_type,type,
% 0.16/0.42 iskpair: $i > $o ).
% 0.16/0.42
% 0.16/0.42 thf(iskpair,definition,
% 0.16/0.42 ( iskpair
% 0.16/0.42 = ( ^ [A: $i] :
% 0.16/0.42 ? [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ ( setunion @ A ) )
% 0.16/0.42 & ? [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ ( setunion @ A ) )
% 0.16/0.42 & ( A
% 0.16/0.42 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(secondinupair_type,type,
% 0.16/0.42 secondinupair: $o ).
% 0.16/0.42
% 0.16/0.42 thf(secondinupair,definition,
% 0.16/0.42 ( secondinupair
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setukpairIL,conjecture,
% 0.16/0.42 ( setextAx
% 0.16/0.42 => ( emptysetAx
% 0.16/0.42 => ( setadjoinAx
% 0.16/0.42 => ( powersetAx
% 0.16/0.42 => ( setunionAx
% 0.16/0.42 => ( omega0Ax
% 0.16/0.42 => ( omegaSAx
% 0.16/0.42 => ( omegaIndAx
% 0.16/0.42 => ( replAx
% 0.16/0.42 => ( foundationAx
% 0.16/0.42 => ( wellorderingAx
% 0.16/0.42 => ( descrp
% 0.16/0.42 => ( dsetconstrI
% 0.16/0.42 => ( dsetconstrEL
% 0.16/0.42 => ( dsetconstrER
% 0.16/0.42 => ( exuE1
% 0.16/0.42 => ( prop2setE
% 0.16/0.42 => ( emptysetE
% 0.16/0.42 => ( emptysetimpfalse
% 0.16/0.42 => ( notinemptyset
% 0.16/0.42 => ( exuE3e
% 0.16/0.42 => ( setext
% 0.16/0.42 => ( emptyI
% 0.16/0.42 => ( noeltsimpempty
% 0.16/0.42 => ( setbeta
% 0.16/0.42 => ( nonemptyE1
% 0.16/0.42 => ( nonemptyI
% 0.16/0.42 => ( nonemptyI1
% 0.16/0.42 => ( setadjoinIL
% 0.16/0.42 => ( emptyinunitempty
% 0.16/0.42 => ( setadjoinIR
% 0.16/0.42 => ( setadjoinE
% 0.16/0.42 => ( setadjoinOr
% 0.16/0.42 => ( setoftrueEq
% 0.16/0.42 => ( powersetI
% 0.16/0.42 => ( emptyinPowerset
% 0.16/0.42 => ( emptyInPowerset
% 0.16/0.42 => ( powersetE
% 0.16/0.42 => ( setunionI
% 0.16/0.42 => ( setunionE
% 0.16/0.42 => ( subPowSU
% 0.16/0.42 => ( exuE2
% 0.16/0.42 => ( nonemptyImpWitness
% 0.16/0.42 => ( uniqinunit
% 0.16/0.42 => ( notinsingleton
% 0.16/0.42 => ( eqinunit
% 0.16/0.42 => ( singletonsswitch
% 0.16/0.42 => ( upairsetE
% 0.16/0.42 => ( upairsetIL
% 0.16/0.42 => ( upairsetIR
% 0.16/0.42 => ( emptyE1
% 0.16/0.42 => ( vacuousDall
% 0.16/0.42 => ( quantDeMorgan1
% 0.16/0.42 => ( quantDeMorgan2
% 0.16/0.42 => ( quantDeMorgan3
% 0.16/0.42 => ( quantDeMorgan4
% 0.16/0.42 => ( prop2setI
% 0.16/0.42 => ( prop2set2propI
% 0.16/0.42 => ( notdexE
% 0.16/0.42 => ( notdallE
% 0.16/0.42 => ( exuI1
% 0.16/0.42 => ( exuI3
% 0.16/0.42 => ( exuI2
% 0.16/0.42 => ( inCongP
% 0.16/0.42 => ( in__Cong
% 0.16/0.42 => ( exuE3u
% 0.16/0.42 => ( exu__Cong
% 0.16/0.42 => ( emptyset__Cong
% 0.16/0.42 => ( setadjoin__Cong
% 0.16/0.42 => ( powerset__Cong
% 0.16/0.42 => ( setunion__Cong
% 0.16/0.42 => ( omega__Cong
% 0.16/0.42 => ( exuEu
% 0.16/0.42 => ( descr__Cong
% 0.16/0.42 => ( dsetconstr__Cong
% 0.16/0.42 => ( subsetI1
% 0.16/0.42 => ( eqimpsubset2
% 0.16/0.42 => ( eqimpsubset1
% 0.16/0.42 => ( subsetI2
% 0.16/0.42 => ( emptysetsubset
% 0.16/0.42 => ( subsetE
% 0.16/0.42 => ( subsetE2
% 0.16/0.42 => ( notsubsetI
% 0.16/0.42 => ( notequalI1
% 0.16/0.42 => ( notequalI2
% 0.16/0.42 => ( subsetRefl
% 0.16/0.42 => ( subsetTrans
% 0.16/0.42 => ( setadjoinSub
% 0.16/0.42 => ( setadjoinSub2
% 0.16/0.42 => ( subset2powerset
% 0.16/0.42 => ( setextsub
% 0.16/0.42 => ( subsetemptysetimpeq
% 0.16/0.42 => ( powersetI1
% 0.16/0.42 => ( powersetE1
% 0.16/0.42 => ( inPowerset
% 0.16/0.42 => ( powersetsubset
% 0.16/0.42 => ( sepInPowerset
% 0.16/0.42 => ( sepSubset
% 0.16/0.42 => ( binunionIL
% 0.16/0.42 => ( upairset2IR
% 0.16/0.42 => ( binunionIR
% 0.16/0.42 => ( binunionEcases
% 0.16/0.42 => ( binunionE
% 0.16/0.42 => ( binunionLsub
% 0.16/0.42 => ( binunionRsub
% 0.16/0.42 => ( binintersectI
% 0.16/0.42 => ( binintersectSubset5
% 0.16/0.42 => ( binintersectEL
% 0.16/0.42 => ( binintersectLsub
% 0.16/0.42 => ( binintersectSubset2
% 0.16/0.42 => ( binintersectSubset3
% 0.16/0.42 => ( binintersectER
% 0.16/0.42 => ( disjointsetsI1
% 0.16/0.42 => ( binintersectRsub
% 0.16/0.42 => ( binintersectSubset4
% 0.16/0.42 => ( binintersectSubset1
% 0.16/0.43 => ( bs114d
% 0.16/0.43 => ( setminusI
% 0.16/0.43 => ( setminusEL
% 0.16/0.43 => ( setminusER
% 0.16/0.43 => ( setminusSubset2
% 0.16/0.43 => ( setminusERneg
% 0.16/0.43 => ( setminusELneg
% 0.16/0.43 => ( setminusILneg
% 0.16/0.43 => ( setminusIRneg
% 0.16/0.43 => ( setminusLsub
% 0.16/0.43 => ( setminusSubset1
% 0.16/0.43 => ( symdiffE
% 0.16/0.43 => ( symdiffI1
% 0.16/0.43 => ( symdiffI2
% 0.16/0.43 => ( symdiffIneg1
% 0.16/0.43 => ( symdiffIneg2
% 0.16/0.43 => ( secondinupair
% 0.16/0.43 => ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 %------------------------------------------------------------------------------
% 0.16/0.43 ------------------------
% 0.25/1.41 ---- Embedded in THF ---
% 0.25/1.41 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 0.25/1.41 %%% Generated on Tue Feb 24 12:33:34 EST 2026
% 0.25/1.41 %%% using '$$dhol' embedding, version 1.3.0.
% 0.25/1.41 %%% Logic specification used:
% 0.25/1.41 %%% thf(spec, logic, $$dhol).
% 0.25/1.41
% 0.25/1.41 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.q7XDm9XVbs/DTF2DTF21061.p
% See solution above
% 0.25/1.41 ------------------------
% 0.25/1.42 ---- Cleaned THF ---
% 0.25/1.42 thf(in_type,type,
% 0.25/1.42 in: $i > $i > $o ).
% 0.25/1.42
% 0.25/1.42 thf(exu_type,type,
% 0.25/1.42 exu: ( $i > $o ) > $o ).
% 0.25/1.42
% 0.25/1.42 thf(exu,definition,
% 0.25/1.42 ( exu
% 0.25/1.42 = ( ^ [Xphi: $i > $o] :
% 0.25/1.42 ? [Xx: $i] :
% 0.25/1.42 ( ( Xphi @ Xx )
% 0.25/1.42 & ! [Xy: $i] :
% 0.25/1.42 ( ( Xphi @ Xy )
% 0.25/1.42 => ( Xx = Xy ) ) ) ) ) ).
% 0.25/1.42
% 0.25/1.42 thf(setextAx_type,type,
% 0.25/1.42 setextAx: $o ).
% 0.25/1.42
% 0.25/1.42 thf(setextAx,definition,
% 0.25/1.42 ( setextAx
% 0.25/1.42 = ( ! [A: $i,B: $i] :
% 0.25/1.42 ( ! [Xx: $i] :
% 0.25/1.42 ( ( in @ Xx @ A )
% 0.25/1.42 <=> ( in @ Xx @ B ) )
% 0.25/1.42 => ( A = B ) ) ) ) ).
% 0.25/1.42
% 0.25/1.42 thf(emptyset_type,type,
% 0.25/1.42 emptyset: $i ).
% 0.25/1.42
% 0.25/1.42 thf(emptysetAx_type,type,
% 0.25/1.42 emptysetAx: $o ).
% 0.25/1.42
% 0.25/1.42 thf(emptysetAx,definition,
% 0.25/1.42 ( emptysetAx
% 0.25/1.42 = ( ! [Xx: $i] :
% 0.25/1.42 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.25/1.42
% 0.25/1.42 thf(setadjoin_type,type,
% 0.25/1.42 setadjoin: $i > $i > $i ).
% 0.25/1.42
% 0.25/1.42 thf(setadjoinAx_type,type,
% 0.25/1.42 setadjoinAx: $o ).
% 0.25/1.42
% 0.25/1.42 thf(setadjoinAx,definition,
% 0.25/1.42 ( setadjoinAx
% 0.25/1.42 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.25/1.42 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.25/1.42 <=> ( ( Xy = Xx )
% 0.25/1.42 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.25/1.42
% 0.25/1.42 thf(powerset_type,type,
% 0.25/1.42 powerset: $i > $i ).
% 0.25/1.42
% 0.25/1.42 thf(powersetAx_type,type,
% 0.25/1.42 powersetAx: $o ).
% 0.25/1.42
% 0.25/1.42 thf(powersetAx,definition,
% 0.25/1.42 ( powersetAx
% 0.25/1.42 = ( ! [A: $i,B: $i] :
% 0.25/1.42 ( ( in @ B @ ( powerset @ A ) )
% 0.25/1.42 <=> ! [Xx: $i] :
% 0.25/1.42 ( ( in @ Xx @ B )
% 0.25/1.42 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.25/1.42
% 0.25/1.42 thf(setunion_type,type,
% 0.25/1.42 setunion: $i > $i ).
% 0.25/1.42
% 0.25/1.42 thf(setunionAx_type,type,
% 0.25/1.42 setunionAx: $o ).
% 0.25/1.42
% 0.25/1.42 thf(setunionAx,definition,
% 0.25/1.42 ( setunionAx
% 0.25/1.42 = ( ! [A: $i,Xx: $i] :
% 0.25/1.42 ( ( in @ Xx @ ( setunion @ A ) )
% 0.25/1.42 <=> ? [B: $i] :
% 0.25/1.42 ( ( in @ Xx @ B )
% 0.25/1.42 & ( in @ B @ A ) ) ) ) ) ).
% 0.25/1.42
% 0.25/1.42 thf(omega_type,type,
% 0.25/1.42 omega: $i ).
% 0.25/1.42
% 0.25/1.42 thf(omega0Ax_type,type,
% 0.25/1.42 omega0Ax: $o ).
% 0.25/1.42
% 0.25/1.42 thf(omega0Ax,definition,
% 0.25/1.42 ( omega0Ax
% 0.25/1.42 = ( in @ emptyset @ omega ) ) ).
% 0.25/1.42
% 0.25/1.42 thf(omegaSAx_type,type,
% 0.25/1.42 omegaSAx: $o ).
% 0.25/1.42
% 0.25/1.42 thf(omegaSAx,definition,
% 0.25/1.42 ( omegaSAx
% 0.25/1.42 = ( ! [Xx: $i] :
% 0.25/1.42 ( ( in @ Xx @ omega )
% 0.25/1.42 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.25/1.42
% 0.25/1.42 thf(omegaIndAx_type,type,
% 0.25/1.42 omegaIndAx: $o ).
% 0.25/1.42
% 0.25/1.42 thf(omegaIndAx,definition,
% 0.25/1.42 ( omegaIndAx
% 0.25/1.42 = ( ! [A: $i] :
% 0.25/1.42 ( ( ( in @ emptyset @ A )
% 0.25/1.42 & ! [Xx: $i] :
% 0.25/1.42 ( ( ( in @ Xx @ omega )
% 0.25/1.42 & ( in @ Xx @ A ) )
% 0.25/1.42 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.25/1.42 => ! [Xx: $i] :
% 0.25/1.42 ( ( in @ Xx @ omega )
% 0.25/1.42 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.25/1.42
% 0.25/1.42 thf(replAx_type,type,
% 0.25/1.42 replAx: $o ).
% 0.25/1.42
% 0.25/1.42 thf(replAx,definition,
% 0.25/1.42 ( replAx
% 0.25/1.42 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.25/1.42 ( ! [Xx: $i] :
% 0.25/1.42 ( ( in @ Xx @ A )
% 0.25/1.42 => ( exu
% 0.25/1.42 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.25/1.42 => ? [B: $i] :
% 0.25/1.42 ! [Xx: $i] :
% 0.25/1.42 ( ( in @ Xx @ B )
% 0.25/1.42 <=> ? [Xy: $i] :
% 0.25/1.42 ( ( in @ Xy @ A )
% 0.25/1.42 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.25/1.42
% 0.25/1.42 thf(foundationAx_type,type,
% 0.25/1.42 foundationAx: $o ).
% 0.25/1.42
% 0.25/1.42 thf(foundationAx,definition,
% 0.25/1.42 ( foundationAx
% 0.25/1.42 = ( ! [A: $i] :
% 0.25/1.42 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.25/1.42 => ? [B: $i] :
% 0.25/1.42 ( ( in @ B @ A )
% 0.25/1.42 & ~ ? [Xx: $i] :
% 0.25/1.42 ( ( in @ Xx @ B )
% 0.25/1.42 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.25/1.42
% 0.25/1.42 thf(wellorderingAx_type,type,
% 0.25/1.42 wellorderingAx: $o ).
% 0.25/1.42
% 0.25/1.42 thf(wellorderingAx,definition,
% 0.25/1.42 ( wellorderingAx
% 0.25/1.42 = ( ! [A: $i] :
% 0.25/1.42 ? [B: $i] :
% 0.25/1.42 ( ! [C: $i] :
% 0.25/1.42 ( ( in @ C @ B )
% 0.25/1.42 => ! [Xx: $i] :
% 0.25/1.42 ( ( in @ Xx @ C )
% 0.25/1.42 => ( in @ Xx @ A ) ) )
% 0.25/1.42 & ! [Xx: $i,Xy: $i] :
% 0.25/1.42 ( ( ( in @ Xx @ A )
% 0.25/1.42 & ( in @ Xy @ A ) )
% 0.25/1.42 => ( ! [C: $i] :
% 0.25/1.42 ( ( in @ C @ B )
% 0.25/1.42 => ( ( in @ Xx @ C )
% 0.25/1.42 <=> ( in @ Xy @ C ) ) )
% 0.25/1.42 => ( Xx = Xy ) ) )
% 0.25/1.42 & ! [C: $i,D: $i] :
% 0.25/1.42 ( ( ( in @ C @ B )
% 0.25/1.42 & ( in @ D @ B ) )
% 0.25/1.42 => ( ! [Xx: $i] :
% 0.25/1.42 ( ( in @ Xx @ C )
% 0.25/1.42 => ( in @ Xx @ D ) )
% 0.25/1.42 | ! [Xx: $i] :
% 0.25/1.42 ( ( in @ Xx @ D )
% 0.25/1.42 => ( in @ Xx @ C ) ) ) )
% 0.25/1.42 & ! [C: $i] :
% 0.25/1.42 ( ( ! [Xx: $i] :
% 0.25/1.42 ( ( in @ Xx @ C )
% 0.25/1.42 => ( in @ Xx @ A ) )
% 0.25/1.42 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.25/1.42 => ? [D: $i,Xx: $i] :
% 0.25/1.42 ( ( in @ D @ B )
% 0.25/1.42 & ( in @ Xx @ C )
% 0.25/1.42 & ~ ? [Xy: $i] :
% 0.25/1.42 ( ( in @ Xy @ D )
% 0.25/1.42 & ( in @ Xy @ C ) )
% 0.25/1.42 & ! [E: $i] :
% 0.25/1.42 ( ( in @ E @ B )
% 0.25/1.42 => ( ! [Xy: $i] :
% 0.25/1.42 ( ( in @ Xy @ E )
% 0.25/1.42 => ( in @ Xy @ D ) )
% 0.25/1.42 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.25/1.42
% 0.25/1.42 thf(descr_type,type,
% 0.25/1.42 descr: ( $i > $o ) > $i ).
% 0.25/1.42
% 0.25/1.42 thf(descrp_type,type,
% 0.25/1.42 descrp: $o ).
% 0.25/1.42
% 0.25/1.42 thf(descrp,definition,
% 0.25/1.42 ( descrp
% 0.25/1.42 = ( ! [Xphi: $i > $o] :
% 0.25/1.42 ( ( exu
% 0.25/1.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.25/1.42 => ( Xphi
% 0.25/1.42 @ ( descr
% 0.25/1.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.25/1.42
% 0.25/1.42 thf(dsetconstr_type,type,
% 0.25/1.42 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.25/1.42
% 0.25/1.42 thf(dsetconstrI_type,type,
% 0.25/1.42 dsetconstrI: $o ).
% 0.25/1.42
% 0.25/1.42 thf(dsetconstrI,definition,
% 0.25/1.42 ( dsetconstrI
% 0.25/1.42 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.25/1.42 ( ( in @ Xx @ A )
% 0.25/1.42 => ( ( Xphi @ Xx )
% 0.25/1.42 => ( in @ Xx
% 0.25/1.42 @ ( dsetconstr @ A
% 0.25/1.42 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.25/1.42
% 0.25/1.42 thf(dsetconstrEL_type,type,
% 0.25/1.42 dsetconstrEL: $o ).
% 0.25/1.42
% 0.25/1.42 thf(dsetconstrEL,definition,
% 0.25/1.42 ( dsetconstrEL
% 0.25/1.42 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.25/1.42 ( ( in @ Xx
% 0.25/1.42 @ ( dsetconstr @ A
% 0.25/1.42 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.25/1.42 => ( in @ Xx @ A ) ) ) ) ).
% 0.25/1.42
% 0.25/1.42 thf(dsetconstrER_type,type,
% 0.25/1.42 dsetconstrER: $o ).
% 0.25/1.42
% 0.25/1.42 thf(dsetconstrER,definition,
% 0.25/1.42 ( dsetconstrER
% 0.25/1.42 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.25/1.42 ( ( in @ Xx
% 0.25/1.42 @ ( dsetconstr @ A
% 0.25/1.42 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.25/1.42 => ( Xphi @ Xx ) ) ) ) ).
% 0.25/1.42
% 0.25/1.42 thf(exuE1_type,type,
% 0.25/1.42 exuE1: $o ).
% 0.25/1.42
% 0.25/1.42 thf(exuE1,definition,
% 0.25/1.42 ( exuE1
% 0.25/1.42 = ( ! [Xphi: $i > $o] :
% 0.25/1.42 ( ( exu
% 0.25/1.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.25/1.42 => ? [Xx: $i] :
% 0.25/1.42 ( ( Xphi @ Xx )
% 0.25/1.42 & ! [Xy: $i] :
% 0.25/1.42 ( ( Xphi @ Xy )
% 0.25/1.42 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.25/1.42
% 0.25/1.42 thf(prop2set_type,type,
% 0.25/1.42 prop2set: $o > $i ).
% 0.25/1.42
% 0.25/1.42 thf(prop2setE_type,type,
% 0.25/1.42 prop2setE: $o ).
% 0.25/1.42
% 0.25/1.42 thf(prop2setE,definition,
% 0.25/1.42 ( prop2setE
% 0.25/1.42 = ( ! [Xphi: $o,Xx: $i] :
% 0.25/1.42 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.25/1.42 => Xphi ) ) ) ).
% 0.25/1.42
% 0.25/1.42 thf(emptysetE_type,type,
% 0.25/1.42 emptysetE: $o ).
% 0.25/1.42
% 0.25/1.42 thf(emptysetE,definition,
% 0.25/1.42 ( emptysetE
% 0.25/1.42 = ( ! [Xx: $i] :
% 0.25/1.42 ( ( in @ Xx @ emptyset )
% 0.25/1.42 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.25/1.42
% 0.25/1.42 thf(emptysetimpfalse_type,type,
% 0.25/1.42 emptysetimpfalse: $o ).
% 0.25/1.42
% 0.25/1.42 thf(emptysetimpfalse,definition,
% 0.25/1.42 ( emptysetimpfalse
% 0.25/1.42 = ( ! [Xx: $i] :
% 0.25/1.42 ( ( in @ Xx @ emptyset )
% 0.25/1.42 => $false ) ) ) ).
% 0.25/1.42
% 0.25/1.42 thf(notinemptyset_type,type,
% 0.25/1.42 notinemptyset: $o ).
% 0.25/1.42
% 0.25/1.42 thf(notinemptyset,definition,
% 0.25/1.42 ( notinemptyset
% 0.25/1.42 = ( ! [Xx: $i] :
% 0.25/1.42 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.25/1.42
% 0.25/1.42 thf(exuE3e_type,type,
% 0.25/1.42 exuE3e: $o ).
% 0.25/1.42
% 0.25/1.42 thf(exuE3e,definition,
% 0.25/1.42 ( exuE3e
% 0.25/1.42 = ( ! [Xphi: $i > $o] :
% 0.25/1.42 ( ( exu
% 0.25/1.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.25/1.42 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.25/1.42
% 0.25/1.42 thf(setext_type,type,
% 0.25/1.42 setext: $o ).
% 0.25/1.42
% 0.25/1.42 thf(setext,definition,
% 0.25/1.42 ( setext
% 0.25/1.42 = ( ! [A: $i,B: $i] :
% 0.25/1.42 ( ! [Xx: $i] :
% 0.25/1.42 ( ( in @ Xx @ A )
% 0.25/1.42 => ( in @ Xx @ B ) )
% 0.25/1.42 => ( ! [Xx: $i] :
% 0.25/1.42 ( ( in @ Xx @ B )
% 0.25/1.42 => ( in @ Xx @ A ) )
% 0.25/1.42 => ( A = B ) ) ) ) ) ).
% 0.25/1.42
% 0.25/1.42 thf(emptyI_type,type,
% 0.25/1.42 emptyI: $o ).
% 0.25/1.42
% 0.25/1.42 thf(emptyI,definition,
% 0.25/1.42 ( emptyI
% 0.25/1.42 = ( ! [A: $i] :
% 0.25/1.42 ( ! [Xx: $i] :
% 0.25/1.42 ~ ( in @ Xx @ A )
% 0.25/1.42 => ( A = emptyset ) ) ) ) ).
% 0.25/1.42
% 0.25/1.42 thf(noeltsimpempty_type,type,
% 0.25/1.42 noeltsimpempty: $o ).
% 0.25/1.42
% 0.25/1.42 thf(noeltsimpempty,definition,
% 0.25/1.42 ( noeltsimpempty
% 0.25/1.42 = ( ! [A: $i] :
% 0.25/1.42 ( ! [Xx: $i] :
% 0.25/1.42 ~ ( in @ Xx @ A )
% 0.25/1.42 => ( A = emptyset ) ) ) ) ).
% 0.25/1.42
% 0.25/1.42 thf(setbeta_type,type,
% 0.25/1.42 setbeta: $o ).
% 0.25/1.42
% 0.25/1.42 thf(setbeta,definition,
% 0.25/1.42 ( setbeta
% 0.25/1.42 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.25/1.42 ( ( in @ Xx @ A )
% 0.25/1.42 => ( ( in @ Xx
% 0.25/1.42 @ ( dsetconstr @ A
% 0.25/1.42 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.25/1.42 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.25/1.42
% 0.25/1.42 thf(nonempty_type,type,
% 0.25/1.42 nonempty: $i > $o ).
% 0.25/1.42
% 0.25/1.42 thf(nonempty,definition,
% 0.25/1.42 ( nonempty
% 0.25/1.42 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.25/1.42
% 0.25/1.42 thf(nonemptyE1_type,type,
% 0.25/1.42 nonemptyE1: $o ).
% 0.25/1.42
% 0.25/1.42 thf(nonemptyE1,definition,
% 0.25/1.42 ( nonemptyE1
% 0.25/1.42 = ( ! [A: $i] :
% 0.25/1.42 ( ( nonempty @ A )
% 0.25/1.42 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.25/1.42
% 0.25/1.42 thf(nonemptyI_type,type,
% 0.25/1.42 nonemptyI: $o ).
% 0.25/1.42
% 0.25/1.42 thf(nonemptyI,definition,
% 0.25/1.42 ( nonemptyI
% 0.25/1.42 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.25/1.42 ( ( in @ Xx @ A )
% 0.25/1.42 => ( ( Xphi @ Xx )
% 0.25/1.42 => ( nonempty
% 0.25/1.42 @ ( dsetconstr @ A
% 0.25/1.42 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.25/1.42
% 0.25/1.42 thf(nonemptyI1_type,type,
% 0.25/1.42 nonemptyI1: $o ).
% 0.25/1.42
% 0.25/1.42 thf(nonemptyI1,definition,
% 0.25/1.42 ( nonemptyI1
% 0.25/1.42 = ( ! [A: $i] :
% 0.25/1.42 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.25/1.42 => ( nonempty @ A ) ) ) ) ).
% 0.25/1.42
% 0.25/1.42 thf(setadjoinIL_type,type,
% 0.25/1.42 setadjoinIL: $o ).
% 0.25/1.42
% 0.25/1.42 thf(setadjoinIL,definition,
% 0.25/1.42 ( setadjoinIL
% 0.25/1.42 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.25/1.42
% 0.25/1.42 thf(emptyinunitempty_type,type,
% 0.25/1.42 emptyinunitempty: $o ).
% 0.25/1.42
% 0.25/1.42 thf(emptyinunitempty,definition,
% 0.25/1.42 ( emptyinunitempty
% 0.25/1.42 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.25/1.42
% 0.25/1.42 thf(setadjoinIR_type,type,
% 0.25/1.42 setadjoinIR: $o ).
% 0.25/1.42
% 0.25/1.42 thf(setadjoinIR,definition,
% 0.25/1.42 ( setadjoinIR
% 0.25/1.42 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.25/1.42 ( ( in @ Xy @ A )
% 0.25/1.42 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.25/1.42
% 0.25/1.42 thf(setadjoinE_type,type,
% 0.25/1.42 setadjoinE: $o ).
% 0.25/1.42
% 0.25/1.42 thf(setadjoinE,definition,
% 0.25/1.42 ( setadjoinE
% 0.25/1.42 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.25/1.42 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.25/1.42 => ! [Xphi: $o] :
% 0.25/1.42 ( ( ( Xy = Xx )
% 0.25/1.42 => Xphi )
% 0.25/1.42 => ( ( ( in @ Xy @ A )
% 0.25/1.42 => Xphi )
% 0.25/1.42 => Xphi ) ) ) ) ) ).
% 0.25/1.42
% 0.25/1.42 thf(setadjoinOr_type,type,
% 0.25/1.42 setadjoinOr: $o ).
% 0.25/1.42
% 0.25/1.42 thf(setadjoinOr,definition,
% 0.25/1.42 ( setadjoinOr
% 0.25/1.42 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.25/1.42 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.25/1.42 => ( ( Xy = Xx )
% 0.25/1.42 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.25/1.42
% 0.25/1.42 thf(setoftrueEq_type,type,
% 0.25/1.42 setoftrueEq: $o ).
% 0.25/1.42
% 0.25/1.42 thf(setoftrueEq,definition,
% 0.25/1.42 ( setoftrueEq
% 0.25/1.42 = ( ! [A: $i] :
% 0.25/1.42 ( ( dsetconstr @ A
% 0.25/1.42 @ ^ [Xx: $i] : $true )
% 0.25/1.42 = A ) ) ) ).
% 0.25/1.42
% 0.25/1.42 thf(powersetI_type,type,
% 0.25/1.42 powersetI: $o ).
% 0.25/1.42
% 0.25/1.42 thf(powersetI,definition,
% 0.25/1.42 ( powersetI
% 0.25/1.42 = ( ! [A: $i,B: $i] :
% 0.25/1.42 ( ! [Xx: $i] :
% 0.25/1.42 ( ( in @ Xx @ B )
% 0.25/1.42 => ( in @ Xx @ A ) )
% 0.25/1.42 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.25/1.42
% 0.25/1.42 thf(emptyinPowerset_type,type,
% 0.25/1.42 emptyinPowerset: $o ).
% 0.25/1.42
% 0.25/1.42 thf(emptyinPowerset,definition,
% 0.25/1.42 ( emptyinPowerset
% 0.25/1.42 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.25/1.42
% 0.25/1.42 thf(emptyInPowerset_type,type,
% 0.25/1.42 emptyInPowerset: $o ).
% 0.25/1.42
% 0.25/1.42 thf(emptyInPowerset,definition,
% 0.25/1.42 ( emptyInPowerset
% 0.25/1.42 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.25/1.42
% 0.25/1.42 thf(powersetE_type,type,
% 0.25/1.42 powersetE: $o ).
% 0.25/1.42
% 0.25/1.42 thf(powersetE,definition,
% 0.25/1.42 ( powersetE
% 0.25/1.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.25/1.42 ( ( in @ B @ ( powerset @ A ) )
% 0.25/1.42 => ( ( in @ Xx @ B )
% 0.25/1.42 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.25/1.42
% 0.25/1.42 thf(setunionI_type,type,
% 0.25/1.42 setunionI: $o ).
% 0.25/1.42
% 0.25/1.42 thf(setunionI,definition,
% 0.25/1.42 ( setunionI
% 0.25/1.42 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.25/1.42 ( ( in @ Xx @ B )
% 0.25/1.42 => ( ( in @ B @ A )
% 0.25/1.42 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.25/1.42
% 0.25/1.42 thf(setunionE_type,type,
% 0.25/1.42 setunionE: $o ).
% 0.25/1.42
% 0.25/1.42 thf(setunionE,definition,
% 0.25/1.42 ( setunionE
% 0.25/1.42 = ( ! [A: $i,Xx: $i] :
% 0.25/1.42 ( ( in @ Xx @ ( setunion @ A ) )
% 0.25/1.42 => ! [Xphi: $o] :
% 0.25/1.42 ( ! [B: $i] :
% 0.25/1.42 ( ( in @ Xx @ B )
% 0.25/1.42 => ( ( in @ B @ A )
% 0.25/1.42 => Xphi ) )
% 0.25/1.42 => Xphi ) ) ) ) ).
% 0.25/1.42
% 0.25/1.42 thf(subPowSU_type,type,
% 0.25/1.42 subPowSU: $o ).
% 0.25/1.42
% 0.25/1.42 thf(subPowSU,definition,
% 0.25/1.42 ( subPowSU
% 0.25/1.42 = ( ! [A: $i,Xx: $i] :
% 0.25/1.42 ( ( in @ Xx @ A )
% 0.25/1.42 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.25/1.42
% 0.25/1.42 thf(exuE2_type,type,
% 0.25/1.42 exuE2: $o ).
% 0.25/1.42
% 0.25/1.42 thf(exuE2,definition,
% 0.25/1.42 ( exuE2
% 0.25/1.42 = ( ! [Xphi: $i > $o] :
% 0.25/1.42 ( ( exu
% 0.25/1.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.25/1.42 => ? [Xx: $i] :
% 0.25/1.42 ! [Xy: $i] :
% 0.25/1.42 ( ( Xphi @ Xy )
% 0.25/1.42 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.25/1.42
% 0.25/1.42 thf(nonemptyImpWitness_type,type,
% 0.25/1.42 nonemptyImpWitness: $o ).
% 0.25/1.42
% 0.25/1.42 thf(nonemptyImpWitness,definition,
% 0.25/1.42 ( nonemptyImpWitness
% 0.25/1.42 = ( ! [A: $i] :
% 0.25/1.42 ( ( nonempty @ A )
% 0.25/1.42 => ? [Xx: $i] :
% 0.25/1.42 ( ( in @ Xx @ A )
% 0.25/1.42 & $true ) ) ) ) ).
% 0.25/1.42
% 0.25/1.42 thf(uniqinunit_type,type,
% 0.25/1.42 uniqinunit: $o ).
% 0.25/1.42
% 0.25/1.42 thf(uniqinunit,definition,
% 0.25/1.42 ( uniqinunit
% 0.25/1.42 = ( ! [Xx: $i,Xy: $i] :
% 0.25/1.42 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.25/1.42 => ( Xx = Xy ) ) ) ) ).
% 0.25/1.42
% 0.25/1.42 thf(notinsingleton_type,type,
% 0.25/1.42 notinsingleton: $o ).
% 0.25/1.42
% 0.25/1.42 thf(notinsingleton,definition,
% 0.25/1.42 ( notinsingleton
% 0.25/1.42 = ( ! [Xx: $i,Xy: $i] :
% 0.25/1.42 ( ( Xx != Xy )
% 0.25/1.42 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.25/1.42
% 0.25/1.42 thf(eqinunit_type,type,
% 0.25/1.42 eqinunit: $o ).
% 0.25/1.42
% 0.25/1.42 thf(eqinunit,definition,
% 0.25/1.42 ( eqinunit
% 0.25/1.42 = ( ! [Xx: $i,Xy: $i] :
% 0.25/1.42 ( ( Xx = Xy )
% 0.25/1.42 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.25/1.42
% 0.25/1.42 thf(singletonsswitch_type,type,
% 0.25/1.42 singletonsswitch: $o ).
% 0.25/1.42
% 0.25/1.42 thf(singletonsswitch,definition,
% 0.25/1.42 ( singletonsswitch
% 0.25/1.42 = ( ! [Xx: $i,Xy: $i] :
% 0.25/1.42 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.25/1.42 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.25/1.42
% 0.25/1.42 thf(upairsetE_type,type,
% 0.25/1.42 upairsetE: $o ).
% 0.25/1.42
% 0.25/1.42 thf(upairsetE,definition,
% 0.25/1.42 ( upairsetE
% 0.25/1.42 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.25/1.42 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.25/1.42 => ( ( Xz = Xx )
% 0.25/1.42 | ( Xz = Xy ) ) ) ) ) ).
% 0.25/1.42
% 0.25/1.42 thf(upairsetIL_type,type,
% 0.25/1.42 upairsetIL: $o ).
% 0.25/1.42
% 0.25/1.42 thf(upairsetIL,definition,
% 0.25/1.42 ( upairsetIL
% 0.25/1.42 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.25/1.42
% 0.25/1.42 thf(upairsetIR_type,type,
% 0.25/1.42 upairsetIR: $o ).
% 0.25/1.42
% 0.25/1.42 thf(upairsetIR,definition,
% 0.25/1.42 ( upairsetIR
% 0.25/1.42 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(emptyE1_type,type,
% 0.25/1.43 emptyE1: $o ).
% 0.25/1.43
% 0.25/1.43 thf(emptyE1,definition,
% 0.25/1.43 ( emptyE1
% 0.25/1.43 = ( ! [A: $i,Xphi: $i > $o] :
% 0.25/1.43 ( ? [Xx: $i] :
% 0.25/1.43 ( ( in @ Xx @ A )
% 0.25/1.43 & ( Xphi @ Xx ) )
% 0.25/1.43 => ( ( ( dsetconstr @ A
% 0.25/1.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.25/1.43 = emptyset )
% 0.25/1.43 => $false ) ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(vacuousDall_type,type,
% 0.25/1.43 vacuousDall: $o ).
% 0.25/1.43
% 0.25/1.43 thf(vacuousDall,definition,
% 0.25/1.43 ( vacuousDall
% 0.25/1.43 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.25/1.43 ( ( in @ Xx @ emptyset )
% 0.25/1.43 => ( Xphi @ Xx ) ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(quantDeMorgan1_type,type,
% 0.25/1.43 quantDeMorgan1: $o ).
% 0.25/1.43
% 0.25/1.43 thf(quantDeMorgan1,definition,
% 0.25/1.43 ( quantDeMorgan1
% 0.25/1.43 = ( ! [A: $i,Xphi: $i > $o] :
% 0.25/1.43 ( ~ ! [Xx: $i] :
% 0.25/1.43 ( ( in @ Xx @ A )
% 0.25/1.43 => ( Xphi @ Xx ) )
% 0.25/1.43 => ? [Xx: $i] :
% 0.25/1.43 ( ( in @ Xx @ A )
% 0.25/1.43 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(quantDeMorgan2_type,type,
% 0.25/1.43 quantDeMorgan2: $o ).
% 0.25/1.43
% 0.25/1.43 thf(quantDeMorgan2,definition,
% 0.25/1.43 ( quantDeMorgan2
% 0.25/1.43 = ( ! [A: $i,Xphi: $i > $o] :
% 0.25/1.43 ( ! [Xx: $i] :
% 0.25/1.43 ( ( in @ Xx @ A )
% 0.25/1.43 => ~ ( Xphi @ Xx ) )
% 0.25/1.43 => ~ ? [Xx: $i] :
% 0.25/1.43 ( ( in @ Xx @ A )
% 0.25/1.43 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(quantDeMorgan3_type,type,
% 0.25/1.43 quantDeMorgan3: $o ).
% 0.25/1.43
% 0.25/1.43 thf(quantDeMorgan3,definition,
% 0.25/1.43 ( quantDeMorgan3
% 0.25/1.43 = ( ! [A: $i,Xphi: $i > $o] :
% 0.25/1.43 ( ~ ? [Xx: $i] :
% 0.25/1.43 ( ( in @ Xx @ A )
% 0.25/1.43 & ( Xphi @ Xx ) )
% 0.25/1.43 => ! [Xx: $i] :
% 0.25/1.43 ( ( in @ Xx @ A )
% 0.25/1.43 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(quantDeMorgan4_type,type,
% 0.25/1.43 quantDeMorgan4: $o ).
% 0.25/1.43
% 0.25/1.43 thf(quantDeMorgan4,definition,
% 0.25/1.43 ( quantDeMorgan4
% 0.25/1.43 = ( ! [A: $i,Xphi: $i > $o] :
% 0.25/1.43 ( ? [Xx: $i] :
% 0.25/1.43 ( ( in @ Xx @ A )
% 0.25/1.43 & ~ ( Xphi @ Xx ) )
% 0.25/1.43 => ~ ! [Xx: $i] :
% 0.25/1.43 ( ( in @ Xx @ A )
% 0.25/1.43 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(prop2setI_type,type,
% 0.25/1.43 prop2setI: $o ).
% 0.25/1.43
% 0.25/1.43 thf(prop2setI,definition,
% 0.25/1.43 ( prop2setI
% 0.25/1.43 = ( ! [Xphi: $o] :
% 0.25/1.43 ( Xphi
% 0.25/1.43 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(set2prop_type,type,
% 0.25/1.43 set2prop: $i > $o ).
% 0.25/1.43
% 0.25/1.43 thf(prop2set2propI_type,type,
% 0.25/1.43 prop2set2propI: $o ).
% 0.25/1.43
% 0.25/1.43 thf(prop2set2propI,definition,
% 0.25/1.43 ( prop2set2propI
% 0.25/1.43 = ( ! [Xphi: $o] :
% 0.25/1.43 ( Xphi
% 0.25/1.43 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(notdexE_type,type,
% 0.25/1.43 notdexE: $o ).
% 0.25/1.43
% 0.25/1.43 thf(notdexE,definition,
% 0.25/1.43 ( notdexE
% 0.25/1.43 = ( ! [A: $i,Xphi: $i > $o] :
% 0.25/1.43 ( ~ ? [Xx: $i] :
% 0.25/1.43 ( ( in @ Xx @ A )
% 0.25/1.43 & ( Xphi @ Xx ) )
% 0.25/1.43 => ! [Xx: $i] :
% 0.25/1.43 ( ( in @ Xx @ A )
% 0.25/1.43 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(notdallE_type,type,
% 0.25/1.43 notdallE: $o ).
% 0.25/1.43
% 0.25/1.43 thf(notdallE,definition,
% 0.25/1.43 ( notdallE
% 0.25/1.43 = ( ! [A: $i,Xphi: $i > $o] :
% 0.25/1.43 ( ~ ! [Xx: $i] :
% 0.25/1.43 ( ( in @ Xx @ A )
% 0.25/1.43 => ( Xphi @ Xx ) )
% 0.25/1.43 => ? [Xx: $i] :
% 0.25/1.43 ( ( in @ Xx @ A )
% 0.25/1.43 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(exuI1_type,type,
% 0.25/1.43 exuI1: $o ).
% 0.25/1.43
% 0.25/1.43 thf(exuI1,definition,
% 0.25/1.43 ( exuI1
% 0.25/1.43 = ( ! [Xphi: $i > $o] :
% 0.25/1.43 ( ? [Xx: $i] :
% 0.25/1.43 ( ( Xphi @ Xx )
% 0.25/1.43 & ! [Xy: $i] :
% 0.25/1.43 ( ( Xphi @ Xy )
% 0.25/1.43 => ( Xx = Xy ) ) )
% 0.25/1.43 => ( exu
% 0.25/1.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(exuI3_type,type,
% 0.25/1.43 exuI3: $o ).
% 0.25/1.43
% 0.25/1.43 thf(exuI3,definition,
% 0.25/1.43 ( exuI3
% 0.25/1.43 = ( ! [Xphi: $i > $o] :
% 0.25/1.43 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.25/1.43 => ( ! [Xx: $i,Xy: $i] :
% 0.25/1.43 ( ( Xphi @ Xx )
% 0.25/1.43 => ( ( Xphi @ Xy )
% 0.25/1.43 => ( Xx = Xy ) ) )
% 0.25/1.43 => ( exu
% 0.25/1.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(exuI2_type,type,
% 0.25/1.43 exuI2: $o ).
% 0.25/1.43
% 0.25/1.43 thf(exuI2,definition,
% 0.25/1.43 ( exuI2
% 0.25/1.43 = ( ! [Xphi: $i > $o] :
% 0.25/1.43 ( ? [Xx: $i] :
% 0.25/1.43 ! [Xy: $i] :
% 0.25/1.43 ( ( Xphi @ Xy )
% 0.25/1.43 <=> ( Xy = Xx ) )
% 0.25/1.43 => ( exu
% 0.25/1.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(inCongP_type,type,
% 0.25/1.43 inCongP: $o ).
% 0.25/1.43
% 0.25/1.43 thf(inCongP,definition,
% 0.25/1.43 ( inCongP
% 0.25/1.43 = ( ! [A: $i,B: $i] :
% 0.25/1.43 ( ( A = B )
% 0.25/1.43 => ! [Xx: $i,Xy: $i] :
% 0.25/1.43 ( ( Xx = Xy )
% 0.25/1.43 => ( ( in @ Xx @ A )
% 0.25/1.43 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(in__Cong_type,type,
% 0.25/1.43 in__Cong: $o ).
% 0.25/1.43
% 0.25/1.43 thf(in__Cong,definition,
% 0.25/1.43 ( in__Cong
% 0.25/1.43 = ( ! [A: $i,B: $i] :
% 0.25/1.43 ( ( A = B )
% 0.25/1.43 => ! [Xx: $i,Xy: $i] :
% 0.25/1.43 ( ( Xx = Xy )
% 0.25/1.43 => ( ( in @ Xx @ A )
% 0.25/1.43 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(exuE3u_type,type,
% 0.25/1.43 exuE3u: $o ).
% 0.25/1.43
% 0.25/1.43 thf(exuE3u,definition,
% 0.25/1.43 ( exuE3u
% 0.25/1.43 = ( ! [Xphi: $i > $o] :
% 0.25/1.43 ( ( exu
% 0.25/1.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.25/1.43 => ! [Xx: $i,Xy: $i] :
% 0.25/1.43 ( ( Xphi @ Xx )
% 0.25/1.43 => ( ( Xphi @ Xy )
% 0.25/1.43 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(exu__Cong_type,type,
% 0.25/1.43 exu__Cong: $o ).
% 0.25/1.43
% 0.25/1.43 thf(exu__Cong,definition,
% 0.25/1.43 ( exu__Cong
% 0.25/1.43 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.25/1.43 ( ! [Xx: $i,Xy: $i] :
% 0.25/1.43 ( ( Xx = Xy )
% 0.25/1.43 => ( ( Xphi @ Xx )
% 0.25/1.43 <=> ( Xpsi @ Xy ) ) )
% 0.25/1.43 => ( ( exu
% 0.25/1.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.25/1.43 <=> ( exu
% 0.25/1.43 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(emptyset__Cong_type,type,
% 0.25/1.43 emptyset__Cong: $o ).
% 0.25/1.43
% 0.25/1.43 thf(emptyset__Cong,definition,
% 0.25/1.43 ( emptyset__Cong
% 0.25/1.43 = ( emptyset = emptyset ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(setadjoin__Cong_type,type,
% 0.25/1.43 setadjoin__Cong: $o ).
% 0.25/1.43
% 0.25/1.43 thf(setadjoin__Cong,definition,
% 0.25/1.43 ( setadjoin__Cong
% 0.25/1.43 = ( ! [Xx: $i,Xy: $i] :
% 0.25/1.43 ( ( Xx = Xy )
% 0.25/1.43 => ! [Xz: $i,Xu: $i] :
% 0.25/1.43 ( ( Xz = Xu )
% 0.25/1.43 => ( ( setadjoin @ Xx @ Xz )
% 0.25/1.43 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(powerset__Cong_type,type,
% 0.25/1.43 powerset__Cong: $o ).
% 0.25/1.43
% 0.25/1.43 thf(powerset__Cong,definition,
% 0.25/1.43 ( powerset__Cong
% 0.25/1.43 = ( ! [A: $i,B: $i] :
% 0.25/1.43 ( ( A = B )
% 0.25/1.43 => ( ( powerset @ A )
% 0.25/1.43 = ( powerset @ B ) ) ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(setunion__Cong_type,type,
% 0.25/1.43 setunion__Cong: $o ).
% 0.25/1.43
% 0.25/1.43 thf(setunion__Cong,definition,
% 0.25/1.43 ( setunion__Cong
% 0.25/1.43 = ( ! [A: $i,B: $i] :
% 0.25/1.43 ( ( A = B )
% 0.25/1.43 => ( ( setunion @ A )
% 0.25/1.43 = ( setunion @ B ) ) ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(omega__Cong_type,type,
% 0.25/1.43 omega__Cong: $o ).
% 0.25/1.43
% 0.25/1.43 thf(omega__Cong,definition,
% 0.25/1.43 ( omega__Cong
% 0.25/1.43 = ( omega = omega ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(exuEu_type,type,
% 0.25/1.43 exuEu: $o ).
% 0.25/1.43
% 0.25/1.43 thf(exuEu,definition,
% 0.25/1.43 ( exuEu
% 0.25/1.43 = ( ! [Xphi: $i > $o] :
% 0.25/1.43 ( ( exu
% 0.25/1.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.25/1.43 => ! [Xx: $i,Xy: $i] :
% 0.25/1.43 ( ( Xphi @ Xx )
% 0.25/1.43 => ( ( Xphi @ Xy )
% 0.25/1.43 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(descr__Cong_type,type,
% 0.25/1.43 descr__Cong: $o ).
% 0.25/1.43
% 0.25/1.43 thf(descr__Cong,definition,
% 0.25/1.43 ( descr__Cong
% 0.25/1.43 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.25/1.43 ( ! [Xx: $i,Xy: $i] :
% 0.25/1.43 ( ( Xx = Xy )
% 0.25/1.43 => ( ( Xphi @ Xx )
% 0.25/1.43 <=> ( Xpsi @ Xy ) ) )
% 0.25/1.43 => ( ( exu
% 0.25/1.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.25/1.43 => ( ( exu
% 0.25/1.43 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.25/1.43 => ( ( descr
% 0.25/1.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.25/1.43 = ( descr
% 0.25/1.43 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(dsetconstr__Cong_type,type,
% 0.25/1.43 dsetconstr__Cong: $o ).
% 0.25/1.43
% 0.25/1.43 thf(dsetconstr__Cong,definition,
% 0.25/1.43 ( dsetconstr__Cong
% 0.25/1.43 = ( ! [A: $i,B: $i] :
% 0.25/1.43 ( ( A = B )
% 0.25/1.43 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.25/1.43 ( ! [Xx: $i] :
% 0.25/1.43 ( ( in @ Xx @ A )
% 0.25/1.43 => ! [Xy: $i] :
% 0.25/1.43 ( ( in @ Xy @ B )
% 0.25/1.43 => ( ( Xx = Xy )
% 0.25/1.43 => ( ( Xphi @ Xx )
% 0.25/1.43 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.25/1.43 => ( ( dsetconstr @ A
% 0.25/1.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.25/1.43 = ( dsetconstr @ B
% 0.25/1.43 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(subset_type,type,
% 0.25/1.43 subset: $i > $i > $o ).
% 0.25/1.43
% 0.25/1.43 thf(disjoint_type,type,
% 0.25/1.43 disjoint: $i > $i > $o ).
% 0.25/1.43
% 0.25/1.43 thf(setsmeet_type,type,
% 0.25/1.43 setsmeet: $i > $i > $o ).
% 0.25/1.43
% 0.25/1.43 thf(subsetI1_type,type,
% 0.25/1.43 subsetI1: $o ).
% 0.25/1.43
% 0.25/1.43 thf(subsetI1,definition,
% 0.25/1.43 ( subsetI1
% 0.25/1.43 = ( ! [A: $i,B: $i] :
% 0.25/1.43 ( ! [Xx: $i] :
% 0.25/1.43 ( ( in @ Xx @ A )
% 0.25/1.43 => ( in @ Xx @ B ) )
% 0.25/1.43 => ( subset @ A @ B ) ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(eqimpsubset2_type,type,
% 0.25/1.43 eqimpsubset2: $o ).
% 0.25/1.43
% 0.25/1.43 thf(eqimpsubset2,definition,
% 0.25/1.43 ( eqimpsubset2
% 0.25/1.43 = ( ! [A: $i,B: $i] :
% 0.25/1.43 ( ( A = B )
% 0.25/1.43 => ( subset @ B @ A ) ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(eqimpsubset1_type,type,
% 0.25/1.43 eqimpsubset1: $o ).
% 0.25/1.43
% 0.25/1.43 thf(eqimpsubset1,definition,
% 0.25/1.43 ( eqimpsubset1
% 0.25/1.43 = ( ! [A: $i,B: $i] :
% 0.25/1.43 ( ( A = B )
% 0.25/1.43 => ( subset @ A @ B ) ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(subsetI2_type,type,
% 0.25/1.43 subsetI2: $o ).
% 0.25/1.43
% 0.25/1.43 thf(subsetI2,definition,
% 0.25/1.43 ( subsetI2
% 0.25/1.43 = ( ! [A: $i,B: $i] :
% 0.25/1.43 ( ! [Xx: $i] :
% 0.25/1.43 ( ( in @ Xx @ A )
% 0.25/1.43 => ( in @ Xx @ B ) )
% 0.25/1.43 => ( subset @ A @ B ) ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(emptysetsubset_type,type,
% 0.25/1.43 emptysetsubset: $o ).
% 0.25/1.43
% 0.25/1.43 thf(emptysetsubset,definition,
% 0.25/1.43 ( emptysetsubset
% 0.25/1.43 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(subsetE_type,type,
% 0.25/1.43 subsetE: $o ).
% 0.25/1.43
% 0.25/1.43 thf(subsetE,definition,
% 0.25/1.43 ( subsetE
% 0.25/1.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.25/1.43 ( ( subset @ A @ B )
% 0.25/1.43 => ( ( in @ Xx @ A )
% 0.25/1.43 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(subsetE2_type,type,
% 0.25/1.43 subsetE2: $o ).
% 0.25/1.43
% 0.25/1.43 thf(subsetE2,definition,
% 0.25/1.43 ( subsetE2
% 0.25/1.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.25/1.43 ( ( subset @ A @ B )
% 0.25/1.43 => ( ~ ( in @ Xx @ B )
% 0.25/1.43 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(notsubsetI_type,type,
% 0.25/1.43 notsubsetI: $o ).
% 0.25/1.43
% 0.25/1.43 thf(notsubsetI,definition,
% 0.25/1.43 ( notsubsetI
% 0.25/1.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.25/1.43 ( ( in @ Xx @ A )
% 0.25/1.43 => ( ~ ( in @ Xx @ B )
% 0.25/1.43 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(notequalI1_type,type,
% 0.25/1.43 notequalI1: $o ).
% 0.25/1.43
% 0.25/1.43 thf(notequalI1,definition,
% 0.25/1.43 ( notequalI1
% 0.25/1.43 = ( ! [A: $i,B: $i] :
% 0.25/1.43 ( ~ ( subset @ A @ B )
% 0.25/1.43 => ( A != B ) ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(notequalI2_type,type,
% 0.25/1.43 notequalI2: $o ).
% 0.25/1.43
% 0.25/1.43 thf(notequalI2,definition,
% 0.25/1.43 ( notequalI2
% 0.25/1.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.25/1.43 ( ( in @ Xx @ A )
% 0.25/1.43 => ( ~ ( in @ Xx @ B )
% 0.25/1.43 => ( A != B ) ) ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(subsetRefl_type,type,
% 0.25/1.43 subsetRefl: $o ).
% 0.25/1.43
% 0.25/1.43 thf(subsetRefl,definition,
% 0.25/1.43 ( subsetRefl
% 0.25/1.43 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(subsetTrans_type,type,
% 0.25/1.43 subsetTrans: $o ).
% 0.25/1.43
% 0.25/1.43 thf(subsetTrans,definition,
% 0.25/1.43 ( subsetTrans
% 0.25/1.43 = ( ! [A: $i,B: $i,C: $i] :
% 0.25/1.43 ( ( subset @ A @ B )
% 0.25/1.43 => ( ( subset @ B @ C )
% 0.25/1.43 => ( subset @ A @ C ) ) ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(setadjoinSub_type,type,
% 0.25/1.43 setadjoinSub: $o ).
% 0.25/1.43
% 0.25/1.43 thf(setadjoinSub,definition,
% 0.25/1.43 ( setadjoinSub
% 0.25/1.43 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(setadjoinSub2_type,type,
% 0.25/1.43 setadjoinSub2: $o ).
% 0.25/1.43
% 0.25/1.43 thf(setadjoinSub2,definition,
% 0.25/1.43 ( setadjoinSub2
% 0.25/1.43 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.25/1.43 ( ( subset @ A @ B )
% 0.25/1.43 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(subset2powerset_type,type,
% 0.25/1.43 subset2powerset: $o ).
% 0.25/1.43
% 0.25/1.43 thf(subset2powerset,definition,
% 0.25/1.43 ( subset2powerset
% 0.25/1.43 = ( ! [A: $i,B: $i] :
% 0.25/1.43 ( ( subset @ A @ B )
% 0.25/1.43 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(setextsub_type,type,
% 0.25/1.43 setextsub: $o ).
% 0.25/1.43
% 0.25/1.43 thf(setextsub,definition,
% 0.25/1.43 ( setextsub
% 0.25/1.43 = ( ! [A: $i,B: $i] :
% 0.25/1.43 ( ( subset @ A @ B )
% 0.25/1.43 => ( ( subset @ B @ A )
% 0.25/1.43 => ( A = B ) ) ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(subsetemptysetimpeq_type,type,
% 0.25/1.43 subsetemptysetimpeq: $o ).
% 0.25/1.43
% 0.25/1.43 thf(subsetemptysetimpeq,definition,
% 0.25/1.43 ( subsetemptysetimpeq
% 0.25/1.43 = ( ! [A: $i] :
% 0.25/1.43 ( ( subset @ A @ emptyset )
% 0.25/1.43 => ( A = emptyset ) ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(powersetI1_type,type,
% 0.25/1.43 powersetI1: $o ).
% 0.25/1.43
% 0.25/1.43 thf(powersetI1,definition,
% 0.25/1.43 ( powersetI1
% 0.25/1.43 = ( ! [A: $i,B: $i] :
% 0.25/1.43 ( ( subset @ B @ A )
% 0.25/1.43 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(powersetE1_type,type,
% 0.25/1.43 powersetE1: $o ).
% 0.25/1.43
% 0.25/1.43 thf(powersetE1,definition,
% 0.25/1.43 ( powersetE1
% 0.25/1.43 = ( ! [A: $i,B: $i] :
% 0.25/1.43 ( ( in @ B @ ( powerset @ A ) )
% 0.25/1.43 => ( subset @ B @ A ) ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(inPowerset_type,type,
% 0.25/1.43 inPowerset: $o ).
% 0.25/1.43
% 0.25/1.43 thf(inPowerset,definition,
% 0.25/1.43 ( inPowerset
% 0.25/1.43 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(powersetsubset_type,type,
% 0.25/1.43 powersetsubset: $o ).
% 0.25/1.43
% 0.25/1.43 thf(powersetsubset,definition,
% 0.25/1.43 ( powersetsubset
% 0.25/1.43 = ( ! [A: $i,B: $i] :
% 0.25/1.43 ( ( subset @ A @ B )
% 0.25/1.43 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(sepInPowerset_type,type,
% 0.25/1.43 sepInPowerset: $o ).
% 0.25/1.43
% 0.25/1.43 thf(sepInPowerset,definition,
% 0.25/1.43 ( sepInPowerset
% 0.25/1.43 = ( ! [A: $i,Xphi: $i > $o] :
% 0.25/1.43 ( in
% 0.25/1.43 @ ( dsetconstr @ A
% 0.25/1.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.25/1.43 @ ( powerset @ A ) ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(sepSubset_type,type,
% 0.25/1.43 sepSubset: $o ).
% 0.25/1.43
% 0.25/1.43 thf(sepSubset,definition,
% 0.25/1.43 ( sepSubset
% 0.25/1.43 = ( ! [A: $i,Xphi: $i > $o] :
% 0.25/1.43 ( subset
% 0.25/1.43 @ ( dsetconstr @ A
% 0.25/1.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.25/1.43 @ A ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(binunion_type,type,
% 0.25/1.43 binunion: $i > $i > $i ).
% 0.25/1.43
% 0.25/1.43 thf(binunionIL_type,type,
% 0.25/1.43 binunionIL: $o ).
% 0.25/1.43
% 0.25/1.43 thf(binunionIL,definition,
% 0.25/1.43 ( binunionIL
% 0.25/1.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.25/1.43 ( ( in @ Xx @ A )
% 0.25/1.43 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(upairset2IR_type,type,
% 0.25/1.43 upairset2IR: $o ).
% 0.25/1.43
% 0.25/1.43 thf(upairset2IR,definition,
% 0.25/1.43 ( upairset2IR
% 0.25/1.43 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(binunionIR_type,type,
% 0.25/1.43 binunionIR: $o ).
% 0.25/1.43
% 0.25/1.43 thf(binunionIR,definition,
% 0.25/1.43 ( binunionIR
% 0.25/1.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.25/1.43 ( ( in @ Xx @ B )
% 0.25/1.43 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(binunionEcases_type,type,
% 0.25/1.43 binunionEcases: $o ).
% 0.25/1.43
% 0.25/1.43 thf(binunionEcases,definition,
% 0.25/1.43 ( binunionEcases
% 0.25/1.43 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.25/1.43 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.25/1.43 => ( ( ( in @ Xx @ A )
% 0.25/1.43 => Xphi )
% 0.25/1.43 => ( ( ( in @ Xx @ B )
% 0.25/1.43 => Xphi )
% 0.25/1.43 => Xphi ) ) ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(binunionE_type,type,
% 0.25/1.43 binunionE: $o ).
% 0.25/1.43
% 0.25/1.43 thf(binunionE,definition,
% 0.25/1.43 ( binunionE
% 0.25/1.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.25/1.43 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.25/1.43 => ( ( in @ Xx @ A )
% 0.25/1.43 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(binunionLsub_type,type,
% 0.25/1.43 binunionLsub: $o ).
% 0.25/1.43
% 0.25/1.43 thf(binunionLsub,definition,
% 0.25/1.43 ( binunionLsub
% 0.25/1.43 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(binunionRsub_type,type,
% 0.25/1.43 binunionRsub: $o ).
% 0.25/1.43
% 0.25/1.43 thf(binunionRsub,definition,
% 0.25/1.43 ( binunionRsub
% 0.25/1.43 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(binintersect_type,type,
% 0.25/1.43 binintersect: $i > $i > $i ).
% 0.25/1.43
% 0.25/1.43 thf(binintersectI_type,type,
% 0.25/1.43 binintersectI: $o ).
% 0.25/1.43
% 0.25/1.43 thf(binintersectI,definition,
% 0.25/1.43 ( binintersectI
% 0.25/1.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.25/1.43 ( ( in @ Xx @ A )
% 0.25/1.43 => ( ( in @ Xx @ B )
% 0.25/1.43 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(binintersectSubset5_type,type,
% 0.25/1.43 binintersectSubset5: $o ).
% 0.25/1.43
% 0.25/1.43 thf(binintersectSubset5,definition,
% 0.25/1.43 ( binintersectSubset5
% 0.25/1.43 = ( ! [A: $i,B: $i,C: $i] :
% 0.25/1.43 ( ( subset @ C @ A )
% 0.25/1.43 => ( ( subset @ C @ B )
% 0.25/1.43 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(binintersectEL_type,type,
% 0.25/1.43 binintersectEL: $o ).
% 0.25/1.43
% 0.25/1.43 thf(binintersectEL,definition,
% 0.25/1.43 ( binintersectEL
% 0.25/1.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.25/1.43 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.25/1.43 => ( in @ Xx @ A ) ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(binintersectLsub_type,type,
% 0.25/1.43 binintersectLsub: $o ).
% 0.25/1.43
% 0.25/1.43 thf(binintersectLsub,definition,
% 0.25/1.43 ( binintersectLsub
% 0.25/1.43 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(binintersectSubset2_type,type,
% 0.25/1.43 binintersectSubset2: $o ).
% 0.25/1.43
% 0.25/1.43 thf(binintersectSubset2,definition,
% 0.25/1.43 ( binintersectSubset2
% 0.25/1.43 = ( ! [A: $i,B: $i] :
% 0.25/1.43 ( ( subset @ A @ B )
% 0.25/1.43 => ( ( binintersect @ A @ B )
% 0.25/1.43 = A ) ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(binintersectSubset3_type,type,
% 0.25/1.43 binintersectSubset3: $o ).
% 0.25/1.43
% 0.25/1.43 thf(binintersectSubset3,definition,
% 0.25/1.43 ( binintersectSubset3
% 0.25/1.43 = ( ! [A: $i,B: $i] :
% 0.25/1.43 ( ( ( binintersect @ A @ B )
% 0.25/1.43 = B )
% 0.25/1.43 => ( subset @ B @ A ) ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(binintersectER_type,type,
% 0.25/1.43 binintersectER: $o ).
% 0.25/1.43
% 0.25/1.43 thf(binintersectER,definition,
% 0.25/1.43 ( binintersectER
% 0.25/1.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.25/1.43 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.25/1.43 => ( in @ Xx @ B ) ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(disjointsetsI1_type,type,
% 0.25/1.43 disjointsetsI1: $o ).
% 0.25/1.43
% 0.25/1.43 thf(disjointsetsI1,definition,
% 0.25/1.43 ( disjointsetsI1
% 0.25/1.43 = ( ! [A: $i,B: $i] :
% 0.25/1.43 ( ~ ? [Xx: $i] :
% 0.25/1.43 ( ( in @ Xx @ A )
% 0.25/1.43 & ( in @ Xx @ B ) )
% 0.25/1.43 => ( ( binintersect @ A @ B )
% 0.25/1.43 = emptyset ) ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(binintersectRsub_type,type,
% 0.25/1.43 binintersectRsub: $o ).
% 0.25/1.43
% 0.25/1.43 thf(binintersectRsub,definition,
% 0.25/1.43 ( binintersectRsub
% 0.25/1.43 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(binintersectSubset4_type,type,
% 0.25/1.43 binintersectSubset4: $o ).
% 0.25/1.43
% 0.25/1.43 thf(binintersectSubset4,definition,
% 0.25/1.43 ( binintersectSubset4
% 0.25/1.43 = ( ! [A: $i,B: $i] :
% 0.25/1.43 ( ( subset @ B @ A )
% 0.25/1.43 => ( ( binintersect @ A @ B )
% 0.25/1.43 = B ) ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(binintersectSubset1_type,type,
% 0.25/1.43 binintersectSubset1: $o ).
% 0.25/1.43
% 0.25/1.43 thf(binintersectSubset1,definition,
% 0.25/1.43 ( binintersectSubset1
% 0.25/1.43 = ( ! [A: $i,B: $i] :
% 0.25/1.43 ( ( ( binintersect @ A @ B )
% 0.25/1.43 = A )
% 0.25/1.43 => ( subset @ A @ B ) ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(bs114d_type,type,
% 0.25/1.43 bs114d: $o ).
% 0.25/1.43
% 0.25/1.43 thf(bs114d,definition,
% 0.25/1.43 ( bs114d
% 0.25/1.43 = ( ! [A: $i,B: $i,C: $i] :
% 0.25/1.43 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 0.25/1.43 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(regular_type,type,
% 0.25/1.43 regular: $i > $o ).
% 0.25/1.43
% 0.25/1.43 thf(setminus_type,type,
% 0.25/1.43 setminus: $i > $i > $i ).
% 0.25/1.43
% 0.25/1.43 thf(setminusI_type,type,
% 0.25/1.43 setminusI: $o ).
% 0.25/1.43
% 0.25/1.43 thf(setminusI,definition,
% 0.25/1.43 ( setminusI
% 0.25/1.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.25/1.43 ( ( in @ Xx @ A )
% 0.25/1.43 => ( ~ ( in @ Xx @ B )
% 0.25/1.43 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(setminusEL_type,type,
% 0.25/1.43 setminusEL: $o ).
% 0.25/1.43
% 0.25/1.43 thf(setminusEL,definition,
% 0.25/1.43 ( setminusEL
% 0.25/1.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.25/1.43 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.25/1.43 => ( in @ Xx @ A ) ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(setminusER_type,type,
% 0.25/1.43 setminusER: $o ).
% 0.25/1.43
% 0.25/1.43 thf(setminusER,definition,
% 0.25/1.43 ( setminusER
% 0.25/1.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.25/1.43 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.25/1.43 => ~ ( in @ Xx @ B ) ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(setminusSubset2_type,type,
% 0.25/1.43 setminusSubset2: $o ).
% 0.25/1.43
% 0.25/1.43 thf(setminusSubset2,definition,
% 0.25/1.43 ( setminusSubset2
% 0.25/1.43 = ( ! [A: $i,B: $i] :
% 0.25/1.43 ( ( subset @ A @ B )
% 0.25/1.43 => ( ( setminus @ A @ B )
% 0.25/1.43 = emptyset ) ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(setminusERneg_type,type,
% 0.25/1.43 setminusERneg: $o ).
% 0.25/1.43
% 0.25/1.43 thf(setminusERneg,definition,
% 0.25/1.43 ( setminusERneg
% 0.25/1.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.25/1.43 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.25/1.43 => ( ( in @ Xx @ A )
% 0.25/1.43 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(setminusELneg_type,type,
% 0.25/1.43 setminusELneg: $o ).
% 0.25/1.43
% 0.25/1.43 thf(setminusELneg,definition,
% 0.25/1.43 ( setminusELneg
% 0.25/1.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.25/1.43 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.25/1.43 => ( ~ ( in @ Xx @ B )
% 0.25/1.43 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(setminusILneg_type,type,
% 0.25/1.43 setminusILneg: $o ).
% 0.25/1.43
% 0.25/1.43 thf(setminusILneg,definition,
% 0.25/1.43 ( setminusILneg
% 0.25/1.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.25/1.43 ( ~ ( in @ Xx @ A )
% 0.25/1.43 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(setminusIRneg_type,type,
% 0.25/1.43 setminusIRneg: $o ).
% 0.25/1.43
% 0.25/1.43 thf(setminusIRneg,definition,
% 0.25/1.43 ( setminusIRneg
% 0.25/1.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.25/1.43 ( ( in @ Xx @ B )
% 0.25/1.43 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(setminusLsub_type,type,
% 0.25/1.43 setminusLsub: $o ).
% 0.25/1.43
% 0.25/1.43 thf(setminusLsub,definition,
% 0.25/1.43 ( setminusLsub
% 0.25/1.43 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(setminusSubset1_type,type,
% 0.25/1.43 setminusSubset1: $o ).
% 0.25/1.43
% 0.25/1.43 thf(setminusSubset1,definition,
% 0.25/1.43 ( setminusSubset1
% 0.25/1.43 = ( ! [A: $i,B: $i] :
% 0.25/1.43 ( ( ( setminus @ A @ B )
% 0.25/1.43 = emptyset )
% 0.25/1.43 => ( subset @ A @ B ) ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(symdiff_type,type,
% 0.25/1.43 symdiff: $i > $i > $i ).
% 0.25/1.43
% 0.25/1.43 thf(symdiffE_type,type,
% 0.25/1.43 symdiffE: $o ).
% 0.25/1.43
% 0.25/1.43 thf(symdiffE,definition,
% 0.25/1.43 ( symdiffE
% 0.25/1.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.25/1.43 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 0.25/1.43 => ! [Xphi: $o] :
% 0.25/1.43 ( ( ( in @ Xx @ A )
% 0.25/1.43 => ( ~ ( in @ Xx @ B )
% 0.25/1.43 => Xphi ) )
% 0.25/1.43 => ( ( ~ ( in @ Xx @ A )
% 0.25/1.43 => ( ( in @ Xx @ B )
% 0.25/1.43 => Xphi ) )
% 0.25/1.43 => Xphi ) ) ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(symdiffI1_type,type,
% 0.25/1.43 symdiffI1: $o ).
% 0.25/1.43
% 0.25/1.43 thf(symdiffI1,definition,
% 0.25/1.43 ( symdiffI1
% 0.25/1.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.25/1.43 ( ( in @ Xx @ A )
% 0.25/1.43 => ( ~ ( in @ Xx @ B )
% 0.25/1.43 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(symdiffI2_type,type,
% 0.25/1.43 symdiffI2: $o ).
% 0.25/1.43
% 0.25/1.43 thf(symdiffI2,definition,
% 0.25/1.43 ( symdiffI2
% 0.25/1.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.25/1.43 ( ~ ( in @ Xx @ A )
% 0.25/1.43 => ( ( in @ Xx @ B )
% 0.25/1.43 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(symdiffIneg1_type,type,
% 0.25/1.43 symdiffIneg1: $o ).
% 0.25/1.43
% 0.25/1.43 thf(symdiffIneg1,definition,
% 0.25/1.43 ( symdiffIneg1
% 0.25/1.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.25/1.43 ( ( in @ Xx @ A )
% 0.25/1.43 => ( ( in @ Xx @ B )
% 0.25/1.43 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(symdiffIneg2_type,type,
% 0.25/1.43 symdiffIneg2: $o ).
% 0.25/1.43
% 0.25/1.43 thf(symdiffIneg2,definition,
% 0.25/1.43 ( symdiffIneg2
% 0.25/1.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.25/1.43 ( ~ ( in @ Xx @ A )
% 0.25/1.43 => ( ~ ( in @ Xx @ B )
% 0.25/1.43 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(iskpair_type,type,
% 0.25/1.43 iskpair: $i > $o ).
% 0.25/1.43
% 0.25/1.43 thf(iskpair,definition,
% 0.25/1.43 ( iskpair
% 0.25/1.43 = ( ^ [A: $i] :
% 0.25/1.43 ? [Xx: $i] :
% 0.25/1.43 ( ( in @ Xx @ ( setunion @ A ) )
% 0.25/1.43 & ? [Xy: $i] :
% 0.25/1.43 ( ( in @ Xy @ ( setunion @ A ) )
% 0.25/1.43 & ( A
% 0.25/1.43 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(secondinupair_type,type,
% 0.25/1.43 secondinupair: $o ).
% 0.25/1.43
% 0.25/1.43 thf(secondinupair,definition,
% 0.25/1.43 ( secondinupair
% 0.25/1.43 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.25/1.43
% 0.25/1.43 thf(setukpairIL,conjecture,
% 0.25/1.43 ( setextAx
% 0.25/1.43 => ( emptysetAx
% 0.25/1.43 => ( setadjoinAx
% 0.25/1.43 => ( powersetAx
% 0.25/1.43 => ( setunionAx
% 0.25/1.43 => ( omega0Ax
% 0.25/1.43 => ( omegaSAx
% 0.25/1.43 => ( omegaIndAx
% 0.25/1.43 => ( replAx
% 0.25/1.43 => ( foundationAx
% 0.25/1.43 => ( wellorderingAx
% 0.25/1.43 => ( descrp
% 0.25/1.43 => ( dsetconstrI
% 0.25/1.43 => ( dsetconstrEL
% 0.25/1.43 => ( dsetconstrER
% 0.25/1.43 => ( exuE1
% 0.25/1.43 => ( prop2setE
% 0.25/1.43 => ( emptysetE
% 0.25/1.43 => ( emptysetimpfalse
% 0.25/1.43 => ( notinemptyset
% 0.25/1.43 => ( exuE3e
% 0.25/1.43 => ( setext
% 0.25/1.43 => ( emptyI
% 0.25/1.43 => ( noeltsimpempty
% 0.25/1.43 => ( setbeta
% 0.25/1.43 => ( nonemptyE1
% 0.25/1.43 => ( nonemptyI
% 0.25/1.43 => ( nonemptyI1
% 0.25/1.43 => ( setadjoinIL
% 0.25/1.43 => ( emptyinunitempty
% 0.25/1.43 => ( setadjoinIR
% 0.25/1.43 => ( setadjoinE
% 0.25/1.43 => ( setadjoinOr
% 0.25/1.43 => ( setoftrueEq
% 0.25/1.43 => ( powersetI
% 0.25/1.43 => ( emptyinPowerset
% 0.25/1.43 => ( emptyInPowerset
% 0.25/1.43 => ( powersetE
% 0.25/1.43 => ( setunionI
% 0.25/1.43 => ( setunionE
% 0.25/1.43 => ( subPowSU
% 0.25/1.43 => ( exuE2
% 0.25/1.43 => ( nonemptyImpWitness
% 0.25/1.43 => ( uniqinunit
% 0.25/1.43 => ( notinsingleton
% 0.25/1.43 => ( eqinunit
% 0.25/1.43 => ( singletonsswitch
% 0.25/1.43 => ( upairsetE
% 0.25/1.43 => ( upairsetIL
% 0.25/1.43 => ( upairsetIR
% 0.25/1.43 => ( emptyE1
% 0.25/1.43 => ( vacuousDall
% 0.25/1.43 => ( quantDeMorgan1
% 0.25/1.43 => ( quantDeMorgan2
% 0.25/1.43 => ( quantDeMorgan3
% 0.25/1.43 => ( quantDeMorgan4
% 0.25/1.43 => ( prop2setI
% 0.25/1.43 => ( prop2set2propI
% 0.25/1.43 => ( notdexE
% 0.25/1.43 => ( notdallE
% 0.25/1.43 => ( exuI1
% 0.25/1.43 => ( exuI3
% 0.25/1.43 => ( exuI2
% 0.25/1.43 => ( inCongP
% 0.25/1.43 => ( in__Cong
% 0.25/1.43 => ( exuE3u
% 0.25/1.43 => ( exu__Cong
% 0.25/1.43 => ( emptyset__Cong
% 0.25/1.43 => ( setadjoin__Cong
% 0.25/1.43 => ( powerset__Cong
% 0.25/1.43 => ( setunion__Cong
% 0.25/1.43 => ( omega__Cong
% 0.25/1.43 => ( exuEu
% 0.25/1.43 => ( descr__Cong
% 0.25/1.43 => ( dsetconstr__Cong
% 0.25/1.43 => ( subsetI1
% 0.25/1.43 => ( eqimpsubset2
% 0.25/1.43 => ( eqimpsubset1
% 0.25/1.43 => ( subsetI2
% 0.25/1.43 => ( emptysetsubset
% 0.25/1.43 => ( subsetE
% 0.25/1.43 => ( subsetE2
% 0.25/1.43 => ( notsubsetI
% 0.25/1.43 => ( notequalI1
% 0.25/1.43 => ( notequalI2
% 0.25/1.43 => ( subsetRefl
% 0.25/1.43 => ( subsetTrans
% 0.25/1.43 => ( setadjoinSub
% 0.25/1.43 => ( setadjoinSub2
% 0.25/1.43 => ( subset2powerset
% 0.25/1.43 => ( setextsub
% 0.25/1.43 => ( subsetemptysetimpeq
% 0.25/1.43 => ( powersetI1
% 0.25/1.43 => ( powersetE1
% 0.25/1.43 => ( inPowerset
% 0.25/1.43 => ( powersetsubset
% 0.25/1.43 => ( sepInPowerset
% 0.25/1.43 => ( sepSubset
% 0.25/1.43 => ( binunionIL
% 0.25/1.43 => ( upairset2IR
% 0.25/1.43 => ( binunionIR
% 0.25/1.43 => ( binunionEcases
% 0.25/1.43 => ( binunionE
% 0.25/1.43 => ( binunionLsub
% 0.25/1.43 => ( binunionRsub
% 0.25/1.43 => ( binintersectI
% 0.25/1.43 => ( binintersectSubset5
% 0.25/1.43 => ( binintersectEL
% 0.25/1.43 => ( binintersectLsub
% 0.25/1.43 => ( binintersectSubset2
% 0.25/1.43 => ( binintersectSubset3
% 0.25/1.43 => ( binintersectER
% 0.25/1.43 => ( disjointsetsI1
% 0.25/1.43 => ( binintersectRsub
% 0.25/1.43 => ( binintersectSubset4
% 0.25/1.43 => ( binintersectSubset1
% 0.25/1.43 => ( bs114d
% 0.25/1.43 => ( setminusI
% 0.25/1.43 => ( setminusEL
% 0.25/1.43 => ( setminusER
% 0.25/1.43 => ( setminusSubset2
% 0.25/1.43 => ( setminusERneg
% 0.25/1.43 => ( setminusELneg
% 0.25/1.43 => ( setminusILneg
% 0.25/1.43 => ( setminusIRneg
% 0.25/1.43 => ( setminusLsub
% 0.25/1.43 => ( setminusSubset1
% 0.25/1.43 => ( symdiffE
% 0.25/1.43 => ( symdiffI1
% 0.25/1.43 => ( symdiffI2
% 0.25/1.43 => ( symdiffIneg1
% 0.25/1.43 => ( symdiffIneg2
% 0.25/1.43 => ( secondinupair
% 0.25/1.43 => ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.25/1.43 ------------------------
% 0.25/1.44 % (21161)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_21061 for (2999ds/183Mi)
% 0.25/1.45 % (21164)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_21061 for (2999ds/2Mi)
% 0.25/1.45 % (21165)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_21061 for (2999ds/2Mi)
% 0.25/1.45 % (21163)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_21061 for (2999ds/27Mi)
% 0.25/1.45 % (21167)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_21061 for (2999ds/18Mi)
% 0.25/1.45 % (21162)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_21061 for (2999ds/4Mi)
% 0.25/1.45 % (21166)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_21061 for (2999ds/275Mi)
% 0.25/1.45 % (21164)Instruction limit reached!
% 0.25/1.45 % (21164)------------------------------
% 0.25/1.45 % (21164)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.25/1.45 % (21164)Termination reason: Unknown
% 0.25/1.45 % (21165)Instruction limit reached!
% 0.25/1.45 % (21165)------------------------------
% 0.25/1.45 % (21165)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.25/1.45 % (21164)Termination phase: shuffling
% 0.25/1.45
% 0.25/1.45 % (21164)Memory used [KB]: 1279
% 0.25/1.45 % (21164)Time elapsed: 0.003 s
% 0.25/1.45 % (21164)Instructions burned: 3 (million)
% 0.25/1.45 % (21164)------------------------------
% 0.25/1.45 % (21164)------------------------------
% 0.25/1.45 % (21165)Termination reason: Unknown
% 0.25/1.45 % (21165)Termination phase: shuffling
% 0.25/1.45
% 0.25/1.45 % (21165)Memory used [KB]: 1279
% 0.25/1.45 % (21165)Time elapsed: 0.003 s
% 0.25/1.45 % (21165)Instructions burned: 4 (million)
% 0.25/1.45 % (21165)------------------------------
% 0.25/1.45 % (21165)------------------------------
% 0.25/1.45 % (21162)Instruction limit reached!
% 0.25/1.45 % (21162)------------------------------
% 0.25/1.45 % (21162)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.25/1.45 % (21162)Termination reason: Unknown
% 0.25/1.45 % (21162)Termination phase: shuffling
% 0.25/1.45
% 0.25/1.45 % (21162)Memory used [KB]: 1279
% 0.25/1.45 % (21162)Time elapsed: 0.004 s
% 0.25/1.45 % (21162)Instructions burned: 5 (million)
% 0.25/1.45 % (21162)------------------------------
% 0.25/1.45 % (21162)------------------------------
% 0.25/1.45 % (21167)Instruction limit reached!
% 0.25/1.45 % (21167)------------------------------
% 0.25/1.45 % (21167)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.25/1.45 % (21167)Termination reason: Unknown
% 0.25/1.45 % (21167)Termination phase: shuffling
% 0.25/1.46
% 0.25/1.46 % (21167)Memory used [KB]: 1407
% 0.25/1.46 % (21167)Time elapsed: 0.010 s
% 0.25/1.46 % (21167)Instructions burned: 18 (million)
% 0.25/1.46 % (21167)------------------------------
% 0.25/1.46 % (21167)------------------------------
% 0.25/1.46 % (21163)Instruction limit reached!
% 0.25/1.46 % (21163)------------------------------
% 0.25/1.46 % (21163)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.25/1.46 % (21163)Termination reason: Unknown
% 0.25/1.46 % (21163)Termination phase: Property scanning
% 0.25/1.46
% 0.25/1.46 % (21163)Memory used [KB]: 1663
% 0.25/1.46 % (21163)Time elapsed: 0.014 s
% 0.25/1.46 % (21163)Instructions burned: 27 (million)
% 0.25/1.46 % (21163)------------------------------
% 0.25/1.46 % (21163)------------------------------
% 0.25/1.46 % (21168)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_21061 for (2999ds/3Mi)
% 0.25/1.46 % (21169)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_21061 for (2999ds/37Mi)
% 0.25/1.46 % (21170)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_21061 for (2999ds/15Mi)
% 0.25/1.47 % (21168)Instruction limit reached!
% 0.25/1.47 % (21168)------------------------------
% 0.25/1.47 % (21168)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.25/1.47 % (21168)Termination reason: Unknown
% 0.25/1.47 % (21168)Termination phase: shuffling
% 0.25/1.47
% 0.25/1.47 % (21168)Memory used [KB]: 1279
% 0.25/1.47 % (21168)Time elapsed: 0.003 s
% 0.25/1.47 % (21168)Instructions burned: 4 (million)
% 0.25/1.47 % (21168)------------------------------
% 0.25/1.47 % (21168)------------------------------
% 0.25/1.47 % (21171)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_21061 for (2999ds/3Mi)
% 0.25/1.47 % (21170)Instruction limit reached!
% 0.25/1.47 % (21170)------------------------------
% 0.25/1.47 % (21170)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.25/1.47 % (21170)Termination reason: Unknown
% 0.25/1.47 % (21170)Termination phase: Property scanning
% 0.25/1.47
% 0.25/1.47 % (21170)Memory used [KB]: 1407
% 0.25/1.47 % (21170)Time elapsed: 0.009 s
% 0.25/1.47 % (21170)Instructions burned: 16 (million)
% 0.25/1.47 % (21170)------------------------------
% 0.25/1.47 % (21170)------------------------------
% 0.25/1.47 % (21171)Instruction limit reached!
% 0.25/1.47 % (21171)------------------------------
% 0.25/1.47 % (21171)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.25/1.47 % (21171)Termination reason: Unknown
% 0.25/1.47 % (21171)Termination phase: shuffling
% 0.25/1.47
% 0.25/1.47 % (21171)Memory used [KB]: 1279
% 0.25/1.47 % (21171)Time elapsed: 0.003 s
% 0.25/1.47 % (21171)Instructions burned: 4 (million)
% 0.25/1.47 % (21171)------------------------------
% 0.25/1.47 % (21171)------------------------------
% 0.25/1.48 % (21172)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_21061 for (2999ds/1041Mi)
% 0.25/1.48 % (21169)Instruction limit reached!
% 0.25/1.48 % (21169)------------------------------
% 0.25/1.48 % (21169)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.25/1.48 % (21169)Termination reason: Unknown
% 0.25/1.48 % (21169)Termination phase: Property scanning
% 0.25/1.48
% 0.25/1.48 % (21169)Memory used [KB]: 1663
% 0.25/1.48 % (21169)Time elapsed: 0.018 s
% 0.25/1.48 % (21169)Instructions burned: 38 (million)
% 0.25/1.48 % (21169)------------------------------
% 0.25/1.48 % (21169)------------------------------
% 0.25/1.48 % (21173)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_21061 for (2999ds/7Mi)
% 0.25/1.48 % (21173)Instruction limit reached!
% 0.25/1.48 % (21173)------------------------------
% 0.25/1.48 % (21173)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.25/1.48 % (21173)Termination reason: Unknown
% 0.25/1.48 % (21173)Termination phase: shuffling
% 0.25/1.48
% 0.25/1.48 % (21173)Memory used [KB]: 1407
% 0.25/1.48 % (21173)Time elapsed: 0.005 s
% 0.25/1.48 % (21173)Instructions burned: 8 (million)
% 0.25/1.48 % (21173)------------------------------
% 0.25/1.48 % (21173)------------------------------
% 0.25/1.49 % (21174)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_21061 for (2999ds/16Mi)
% 0.25/1.49 % (21175)lrs+21_1:1_au=on:cnfonf=off:fd=preordered:fe=off:fsr=off:hud=11:inj=on:kws=precedence:s2pl=no:sp=weighted_frequency:tgt=full:i=3:si=on:rtra=on_0 on DTF2THF_21061 for (2999ds/3Mi)
% 0.25/1.49 % (21175)Instruction limit reached!
% 0.25/1.49 % (21175)------------------------------
% 0.25/1.49 % (21175)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.25/1.49 % (21175)Termination reason: Unknown
% 0.25/1.49 % (21175)Termination phase: shuffling
% 0.25/1.49
% 0.25/1.49 % (21175)Memory used [KB]: 1279
% 0.25/1.49 % (21175)Time elapsed: 0.003 s
% 0.25/1.49 % (21175)Instructions burned: 4 (million)
% 0.25/1.49 % (21175)------------------------------
% 0.25/1.49 % (21175)------------------------------
% 0.25/1.50 % (21174)Instruction limit reached!
% 0.25/1.50 % (21174)------------------------------
% 0.25/1.50 % (21174)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.25/1.50 % (21174)Termination reason: Unknown
% 0.25/1.50 % (21174)Termination phase: shuffling
% 0.25/1.50
% 0.25/1.50 % (21174)Memory used [KB]: 1535
% 0.25/1.50 % (21174)Time elapsed: 0.009 s
% 0.25/1.50 % (21174)Instructions burned: 17 (million)
% 0.25/1.50 % (21174)------------------------------
% 0.25/1.50 % (21174)------------------------------
% 2.39/1.50 % (21176)lrs+2_1:1_apa=on:au=on:bd=preordered:cnfonf=off:cs=on:ixr=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_21061 for (2999ds/3Mi)
% 2.39/1.50 % (21176)Instruction limit reached!
% 2.39/1.50 % (21176)------------------------------
% 2.39/1.50 % (21176)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.39/1.50 % (21176)Termination reason: Unknown
% 2.39/1.50 % (21176)Termination phase: shuffling
% 2.39/1.50
% 2.39/1.50 % (21176)Memory used [KB]: 1279
% 2.39/1.50 % (21176)Time elapsed: 0.003 s
% 2.39/1.50 % (21176)Instructions burned: 4 (million)
% 2.39/1.50 % (21176)------------------------------
% 2.39/1.50 % (21176)------------------------------
% 2.39/1.50 % (21177)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_21061 for (2999ds/7Mi)
% 2.39/1.50 % (21177)Instruction limit reached!
% 2.39/1.50 % (21177)------------------------------
% 2.39/1.50 % (21177)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.39/1.50 % (21177)Termination reason: Unknown
% 2.39/1.50 % (21177)Termination phase: shuffling
% 2.39/1.50
% 2.39/1.50 % (21177)Memory used [KB]: 1279
% 2.39/1.50 % (21177)Time elapsed: 0.005 s
% 2.39/1.50 % (21177)Instructions burned: 8 (million)
% 2.39/1.50 % (21177)------------------------------
% 2.39/1.50 % (21177)------------------------------
% 2.39/1.51 % (21178)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_21061 for (2999ds/3Mi)
% 2.39/1.51 % (21178)Instruction limit reached!
% 2.39/1.51 % (21178)------------------------------
% 2.39/1.51 % (21178)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.39/1.51 % (21178)Termination reason: Unknown
% 2.39/1.51 % (21178)Termination phase: shuffling
% 2.39/1.51
% 2.39/1.51 % (21178)Memory used [KB]: 1279
% 2.39/1.51 % (21178)Time elapsed: 0.003 s
% 2.39/1.51 % (21178)Instructions burned: 4 (million)
% 2.39/1.51 % (21178)------------------------------
% 2.39/1.51 % (21178)------------------------------
% 2.39/1.51 % (21161)Instruction limit reached!
% 2.39/1.51 % (21161)------------------------------
% 2.39/1.51 % (21161)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.39/1.51 % (21161)Termination reason: Unknown
% 2.39/1.51 % (21161)Termination phase: Saturation
% 2.39/1.51
% 2.39/1.51 % (21161)Memory used [KB]: 7036
% 2.39/1.51 % (21161)Time elapsed: 0.069 s
% 2.39/1.51 % (21161)Instructions burned: 183 (million)
% 2.39/1.51 % (21161)------------------------------
% 2.39/1.51 % (21161)------------------------------
% 2.39/1.51 % (21179)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_21061 for (2999ds/4Mi)
% 2.39/1.51 % (21179)Instruction limit reached!
% 2.39/1.51 % (21179)------------------------------
% 2.39/1.51 % (21179)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.39/1.51 % (21179)Termination reason: Unknown
% 2.39/1.51 % (21179)Termination phase: shuffling
% 2.39/1.51
% 2.39/1.51 % (21179)Memory used [KB]: 1279
% 2.39/1.51 % (21179)Time elapsed: 0.003 s
% 2.39/1.51 % (21179)Instructions burned: 4 (million)
% 2.39/1.51 % (21179)------------------------------
% 2.39/1.51 % (21179)------------------------------
% 2.39/1.51 % (21181)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_21061 for (2999ds/710Mi)
% 2.39/1.51 % (21180)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_21061 for (2999ds/18Mi)
% 2.39/1.52 % (21180)Instruction limit reached!
% 2.39/1.52 % (21180)------------------------------
% 2.39/1.52 % (21180)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.39/1.52 % (21180)Termination reason: Unknown
% 2.39/1.52 % (21180)Termination phase: shuffling
% 2.39/1.52
% 2.39/1.52 % (21180)Memory used [KB]: 1535
% 2.39/1.52 % (21180)Time elapsed: 0.010 s
% 2.39/1.52 % (21180)Instructions burned: 19 (million)
% 2.39/1.52 % (21180)------------------------------
% 2.39/1.52 % (21180)------------------------------
% 2.39/1.52 % (21183)dis+1002_5:1_au=on:bd=off:e2e=on:fde=none:fs=off:fsr=off:sos=on:i=902:si=on:rtra=on_0 on DTF2THF_21061 for (2999ds/902Mi)
% 2.39/1.53 % (21184)dis+21_1:8_apa=on:cnfonf=off:fd=off:fsr=off:hud=0:ins=1:kws=inv_frequency:nwc=10.0:ss=axioms:st=5.0:i=21:si=on:rtra=on_0 on DTF2THF_21061 for (2999ds/21Mi)
% 2.39/1.53 % (21182)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_21061 for (2999ds/6Mi)
% 2.39/1.53 % (21182)Instruction limit reached!
% 2.39/1.53 % (21182)------------------------------
% 2.39/1.53 % (21182)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.39/1.53 % (21182)Termination reason: Unknown
% 2.39/1.53 % (21182)Termination phase: shuffling
% 2.39/1.53
% 2.39/1.53 % (21182)Memory used [KB]: 1279
% 2.39/1.53 % (21182)Time elapsed: 0.004 s
% 2.39/1.53 % (21182)Instructions burned: 6 (million)
% 2.39/1.53 % (21182)------------------------------
% 2.39/1.53 % (21182)------------------------------
% 2.39/1.54 % (21185)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_21061 for (2999ds/5Mi)
% 2.39/1.54 % (21184)Instruction limit reached!
% 2.39/1.54 % (21184)------------------------------
% 2.39/1.54 % (21184)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.39/1.54 % (21184)Termination reason: Unknown
% 2.39/1.54 % (21184)Termination phase: Property scanning
% 2.39/1.54
% 2.39/1.54 % (21184)Memory used [KB]: 1663
% 2.39/1.54 % (21184)Time elapsed: 0.011 s
% 2.39/1.54 % (21184)Instructions burned: 22 (million)
% 2.39/1.54 % (21184)------------------------------
% 2.39/1.54 % (21184)------------------------------
% 2.39/1.54 % (21185)Instruction limit reached!
% 2.39/1.54 % (21185)------------------------------
% 2.39/1.54 % (21185)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.39/1.54 % (21185)Termination reason: Unknown
% 2.39/1.54 % (21185)Termination phase: shuffling
% 2.39/1.54
% 2.39/1.54 % (21185)Memory used [KB]: 1279
% 2.39/1.54 % (21185)Time elapsed: 0.004 s
% 2.39/1.54 % (21185)Instructions burned: 6 (million)
% 2.39/1.54 % (21185)------------------------------
% 2.39/1.54 % (21185)------------------------------
% 2.39/1.55 % (21186)lrs+2_1:1_cnfonf=lazy_not_gen_be_off:cs=on:fe=off:hud=10:inj=on:ins=3:plsq=on:plsqc=1:sd=10:ss=axioms:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_21061 for (2999ds/6Mi)
% 2.39/1.55 % (21186)Instruction limit reached!
% 2.39/1.55 % (21186)------------------------------
% 2.39/1.55 % (21186)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.39/1.55 % (21186)Termination reason: Unknown
% 2.39/1.55 % (21186)Termination phase: shuffling
% 2.39/1.55
% 2.39/1.55 % (21186)Memory used [KB]: 1279
% 2.39/1.55 % (21186)Time elapsed: 0.004 s
% 2.39/1.55 % (21186)Instructions burned: 6 (million)
% 2.39/1.55 % (21186)------------------------------
% 2.39/1.55 % (21186)------------------------------
% 2.39/1.55 % (21187)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_21061 for (2998ds/377Mi)
% 2.39/1.56 % (21188)dis+1010_1:4_atotf=0.2:c=on:cbe=off:cnfonf=lazy_simp:fe=off:ins=2:ntd=on:s2a=on:s2at=5.0:sgt=5:ss=axioms:st=1.5:i=779:si=on:rtra=on_0 on DTF2THF_21061 for (2998ds/779Mi)
% 2.39/1.57 % (21189)lrs+10_1:1_cnfonf=lazy_not_be_gen:ntd=on:sp=const_min:ss=axioms:sup=off:i=19:si=on:rtra=on_0 on DTF2THF_21061 for (2998ds/19Mi)
% 2.39/1.58 % (21189)Instruction limit reached!
% 2.39/1.58 % (21189)------------------------------
% 2.39/1.58 % (21189)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.39/1.58 % (21189)Termination reason: Unknown
% 2.39/1.58 % (21189)Termination phase: shuffling
% 2.39/1.58
% 2.39/1.58 % (21189)Memory used [KB]: 1535
% 2.39/1.58 % (21189)Time elapsed: 0.010 s
% 2.39/1.58 % (21189)Instructions burned: 19 (million)
% 2.39/1.58 % (21189)------------------------------
% 2.39/1.58 % (21189)------------------------------
% 2.39/1.58 % (21166)Instruction limit reached!
% 2.39/1.58 % (21166)------------------------------
% 2.39/1.58 % (21166)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.39/1.58 % (21166)Termination reason: Unknown
% 2.39/1.58 % (21166)Termination phase: Saturation
% 2.39/1.58
% 2.39/1.58 % (21166)Memory used [KB]: 7803
% 2.39/1.58 % (21166)Time elapsed: 0.138 s
% 2.39/1.58 % (21166)Instructions burned: 275 (million)
% 2.39/1.58 % (21166)------------------------------
% 2.39/1.58 % (21166)------------------------------
% 2.39/1.59 % (21190)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_21061 for (2998ds/879Mi)
% 2.39/1.60 % (21191)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_21061 for (2998ds/17Mi)
% 2.39/1.61 % (21191)Instruction limit reached!
% 2.39/1.61 % (21191)------------------------------
% 2.39/1.61 % (21191)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.39/1.61 % (21191)Termination reason: Unknown
% 2.39/1.61 % (21191)Termination phase: shuffling
% 2.39/1.61
% 2.39/1.61 % (21191)Memory used [KB]: 1535
% 2.39/1.61 % (21191)Time elapsed: 0.010 s
% 2.39/1.61 % (21191)Instructions burned: 18 (million)
% 2.39/1.61 % (21191)------------------------------
% 2.39/1.61 % (21191)------------------------------
% 2.39/1.63 % (21192)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_21061 for (2998ds/3Mi)
% 2.39/1.63 % (21192)Instruction limit reached!
% 2.39/1.63 % (21192)------------------------------
% 2.39/1.63 % (21192)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.39/1.63 % (21192)Termination reason: Unknown
% 2.39/1.63 % (21192)Termination phase: shuffling
% 2.39/1.63
% 2.39/1.63 % (21192)Memory used [KB]: 1151
% 2.39/1.63 % (21192)Time elapsed: 0.003 s
% 2.39/1.63 % (21192)Instructions burned: 3 (million)
% 2.39/1.63 % (21192)------------------------------
% 2.39/1.63 % (21192)------------------------------
% 2.39/1.65 % (21193)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_21061 for (2998ds/30Mi)
% 2.39/1.66 % (21183)First to succeed.
% 2.39/1.66 % (21193)Instruction limit reached!
% 2.39/1.66 % (21193)------------------------------
% 2.39/1.66 % (21193)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.39/1.66 % (21193)Termination reason: Unknown
% 2.39/1.66 % (21193)Termination phase: Preprocessing 3
% 2.39/1.66
% 2.39/1.66 % (21193)Memory used [KB]: 1663
% 2.39/1.66 % (21193)Time elapsed: 0.010 s
% 2.39/1.66 % (21193)Instructions burned: 33 (million)
% 2.39/1.66 % (21193)------------------------------
% 2.39/1.66 % (21193)------------------------------
% 2.39/1.66 % (21183)Refutation found. Thanks to Tanya!
% 2.39/1.66 % SZS status Theorem for DTF2THF_21061
% 2.39/1.66 % SZS output start Proof for DTF2THF_21061
% 2.39/1.66 thf(func_def_0, type, in: $i > $i > $o).
% 2.39/1.66 thf(func_def_1, type, exu: ($i > $o) > $o).
% 2.39/1.66 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 2.39/1.66 thf(func_def_8, type, powerset: $i > $i).
% 2.39/1.66 thf(func_def_10, type, setunion: $i > $i).
% 2.39/1.66 thf(func_def_19, type, descr: ($i > $o) > $i).
% 2.39/1.66 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 2.39/1.66 thf(func_def_26, type, prop2set: $o > $i).
% 2.39/1.66 thf(func_def_36, type, nonempty: $i > $o).
% 2.39/1.66 thf(func_def_69, type, set2prop: $i > $o).
% 2.39/1.66 thf(func_def_88, type, subset: $i > $i > $o).
% 2.39/1.66 thf(func_def_89, type, disjoint: $i > $i > $o).
% 2.39/1.66 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 2.39/1.66 thf(func_def_114, type, binunion: $i > $i > $i).
% 2.39/1.66 thf(func_def_122, type, binintersect: $i > $i > $i).
% 2.39/1.66 thf(func_def_135, type, regular: $i > $o).
% 2.39/1.66 thf(func_def_136, type, setminus: $i > $i > $i).
% 2.39/1.66 thf(func_def_147, type, symdiff: $i > $i > $i).
% 2.39/1.66 thf(func_def_153, type, iskpair: $i > $o).
% 2.39/1.66 thf(func_def_167, type, sP0: $i > $i > $o).
% 2.39/1.66 thf(func_def_168, type, sP1: $i > $o).
% 2.39/1.66 thf(func_def_169, type, sP2: $i > $i > $o).
% 2.39/1.66 thf(func_def_170, type, sP3: $i > $i > $o).
% 2.39/1.66 thf(func_def_171, type, sP4: $i > $i > $i > $o > $o).
% 2.39/1.66 thf(func_def_175, type, sK8: $i > $i > $i).
% 2.39/1.66 thf(func_def_176, type, sK9: $i > $i > $i).
% 2.39/1.66 thf(func_def_177, type, sK10: $i > $i > $i).
% 2.39/1.66 thf(func_def_178, type, sK11: $i > $i > $i).
% 2.39/1.66 thf(func_def_179, type, sK12: $i > $i > $i > $i).
% 2.39/1.66 thf(func_def_180, type, sK13: $i > $i).
% 2.39/1.66 thf(func_def_181, type, sK14: $i > $i).
% 2.39/1.66 thf(func_def_182, type, sK15: $i > $i).
% 2.39/1.66 thf(func_def_183, type, sK16: $i > $i).
% 2.39/1.66 thf(func_def_184, type, sK17: $i > $i > $i > $i).
% 2.39/1.66 thf(func_def_185, type, sK18: $i > $i > $i > $i).
% 2.39/1.66 thf(func_def_186, type, sK19: $i > $i > $i).
% 2.39/1.66 thf(func_def_187, type, sK20: $i > $i > $i).
% 2.39/1.66 thf(func_def_188, type, sK21: $i > $i > $i).
% 2.39/1.66 thf(func_def_190, type, sK23: $i > $i).
% 2.39/1.66 thf(func_def_191, type, sK24: $i > $i).
% 2.39/1.66 thf(func_def_192, type, sK25: $i > $i).
% 2.39/1.66 thf(func_def_193, type, sK26: $i > $i > $i).
% 2.39/1.66 thf(func_def_194, type, sK27: $i > $o).
% 2.39/1.66 thf(func_def_196, type, sK29: $i > ($i > $o) > $i).
% 2.39/1.66 thf(func_def_212, type, sK45: $i > $i > $i).
% 2.39/1.66 thf(func_def_220, type, sK53: $i > $o).
% 2.39/1.66 thf(func_def_227, type, sK60: $i > $o).
% 2.39/1.66 thf(func_def_229, type, sK62: ($i > $o) > $i).
% 2.39/1.66 thf(func_def_230, type, sK63: ($i > $o) > $i).
% 2.39/1.66 thf(func_def_233, type, sK66: ($i > $o) > ($i > $o) > $i).
% 2.39/1.66 thf(func_def_234, type, sK67: ($i > $o) > ($i > $o) > $i).
% 2.39/1.66 thf(func_def_235, type, sK68: $i > $o).
% 2.39/1.66 thf(func_def_236, type, sK69: $i > $o).
% 2.39/1.66 thf(func_def_244, type, sK77: $i > $o).
% 2.39/1.66 thf(func_def_252, type, sK85: $i > $i).
% 2.39/1.66 thf(func_def_266, type, sK99: $i > $o).
% 2.39/1.66 thf(func_def_267, type, sK100: $i > $o).
% 2.39/1.66 thf(func_def_270, type, sK103: $i > $o).
% 2.39/1.66 thf(func_def_273, type, sK106: $i > ($i > $o) > $i).
% 2.39/1.66 thf(func_def_280, type, sK113: $i > $o).
% 2.39/1.66 thf(func_def_284, type, sK117: ($i > $o) > $i).
% 2.39/1.66 thf(func_def_285, type, sK118: $i > $o).
% 2.39/1.66 thf(func_def_286, type, sK119: $i > $i).
% 2.39/1.66 thf(func_def_290, type, sK123: $i > $i).
% 2.39/1.66 thf(func_def_292, type, sK125: ($i > $o) > $i > $i).
% 2.39/1.66 thf(func_def_294, type, sK127: $i > $o).
% 2.39/1.66 thf(func_def_302, type, sK135: $i > $i > $i).
% 2.39/1.66 thf(func_def_303, type, sK136: $i > $i > $i).
% 2.39/1.66 thf(func_def_315, type, sK148: $i > $i).
% 2.39/1.66 thf(func_def_317, type, sK150: $i > $i).
% 2.39/1.66 thf(func_def_326, type, sK159: $i > $o).
% 2.39/1.66 thf(func_def_335, type, sK168: $i > $i > $i).
% 2.39/1.66 thf(func_def_339, type, sK172: $i > ($i > $o) > $i).
% 2.39/1.66 thf(func_def_340, type, sK173: $i > $o).
% 2.39/1.66 thf(func_def_343, type, sK176: $i > $o).
% 2.39/1.66 thf(func_def_344, type, sK177: ($i > $o) > $i).
% 2.39/1.66 thf(func_def_353, type, sK186: $i > $o).
% 2.39/1.66 thf(func_def_362, type, sK195: $i > $o).
% 2.39/1.66 thf(func_def_365, type, sK198: ($i > $o) > $i > $i).
% 2.39/1.66 thf(func_def_367, type, sK200: $i > $o).
% 2.39/1.66 thf(func_def_371, type, sK204: $i > $o).
% 2.39/1.66 thf(func_def_372, type, sK205: $i > $o).
% 2.39/1.66 thf(func_def_373, type, sK206: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.39/1.66 thf(func_def_374, type, sK207: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.39/1.66 thf(func_def_375, type, sK208: $i > $i).
% 2.39/1.66 thf(func_def_393, type, sK226: $i > $o).
% 2.39/1.66 thf(func_def_396, type, sK229: $i > ($i > $o) > $i).
% 2.39/1.66 thf(func_def_399, type, sK232: $i > $i > $i).
% 2.39/1.66 thf(func_def_406, type, sK239: $o > $i > $i > $i).
% 2.39/1.66 thf(func_def_426, type, sK259: $i > $i > $o).
% 2.39/1.66 thf(func_def_427, type, sK260: $i > $i).
% 2.39/1.66 thf(func_def_428, type, sK261: $i > $i).
% 2.39/1.66 thf(func_def_429, type, sK262: ($i > $i > $o) > $i > $i).
% 2.39/1.66 thf(func_def_430, type, sK263: $i > ($i > $i > $o) > $i > $i).
% 2.39/1.66 thf(func_def_431, type, sK264: ($i > $i > $o) > $i > $i).
% 2.39/1.66 thf(func_def_442, type, sK275: $i > $o).
% 2.39/1.66 thf(func_def_445, type, sK278: $i > $i > $i).
% 2.39/1.66 thf(func_def_455, type, sK288: ($i > $o) > $i > $i).
% 2.39/1.66 thf(func_def_457, type, sK290: $i > $o).
% 2.39/1.66 thf(func_def_459, type, sK292: $i > $o).
% 2.39/1.66 thf(func_def_461, type, sK294: $i > ($i > $o) > $i).
% 2.39/1.66 thf(func_def_462, type, sK295: $i > $o).
% 2.39/1.66 thf(func_def_463, type, sK296: $i > $o).
% 2.39/1.66 thf(func_def_464, type, sK297: ($i > $o) > ($i > $o) > $i).
% 2.39/1.66 thf(func_def_465, type, sK298: ($i > $o) > ($i > $o) > $i).
% 2.39/1.66 thf(func_def_477, type, sK310: $i > $o).
% 2.39/1.66 thf(func_def_485, type, sK318: $i > $i > $i).
% 2.39/1.66 thf(func_def_492, type, sK325: $i > $i > $i).
% 2.39/1.66 thf(func_def_501, type, sK334: $i > $o).
% 2.39/1.66 thf(func_def_514, type, sK347: $i > $i > $i).
% 2.39/1.66 thf(func_def_518, type, sK351: $i > $o).
% 2.39/1.66 thf(func_def_522, type, sK355: $i > $i).
% 2.39/1.66 thf(func_def_525, type, sK358: $i > $i).
% 2.39/1.66 thf(func_def_530, type, sK363: ($i > $o) > $i).
% 2.39/1.66 thf(func_def_531, type, sK364: $i > $o).
% 2.39/1.66 thf(func_def_532, type, sK365: $i > $i).
% 2.39/1.66 thf(func_def_535, type, ph368: !>[X0: $tType]:(X0)).
% 2.39/1.66 thf(f3181,plain,(
% 2.39/1.66 $false),
% 2.39/1.66 inference(trivial_inequality_removal,[status(thm)],[f3180])).
% 2.39/1.66 thf(f3180,plain,(
% 2.39/1.66 ($true != $true)),
% 2.39/1.66 inference(superposition,[status(thm)],[f3055,f1678])).
% 2.39/1.66 thf(f1678,plain,(
% 2.39/1.66 (setadjoinIL = $true)),
% 2.39/1.66 inference(cnf_transformation,[status(thm)],[f1107])).
% 2.39/1.66 thf(f1107,plain,(
% 2.39/1.66 (emptyInPowerset = $true) & (emptysetimpfalse = $true) & (subsetTrans = $true) & (inPowerset = $true) & (descrp = $true) & (emptyinPowerset = $true) & (descr__Cong = $true) & (powersetE1 = $true) & (exuEu = $true) & (powersetI1 = $true) & (binintersectRsub = $true) & (notdallE = $true) & (binintersectSubset5 = $true) & (subsetemptysetimpeq = $true) & (setextAx = $true) & (emptyinunitempty = $true) & (setunion__Cong = $true) & (eqinunit = $true) & (setunionAx = $true) & (quantDeMorgan2 = $true) & (setminusLsub = $true) & (nonemptyE1 = $true) & (exuE1 = $true) & (symdiffE = $true) & (subsetRefl = $true) & (binunionRsub = $true) & (replAx = $true) & (subset2powerset = $true) & (sepSubset = $true) & (setadjoinSub = $true) & (binintersectER = $true) & (dsetconstr__Cong = $true) & (exuE3u = $true) & (binintersectI = $true) & (setminusSubset2 = $true) & (exuI1 = $true) & (setminusERneg = $true) & (setminusER = $true) & (emptyE1 = $true) & (setminusEL = $true) & (setminusSubset1 = $true) & (omegaSAx = $true) & (subPowSU = $true) & (setunionE = $true) & (upairsetE = $true) & (nonemptyImpWitness = $true) & (setminusILneg = $true) & (quantDeMorgan4 = $true) & (subsetE2 = $true) & (emptysetE = $true) & (omegaIndAx = $true) & (setadjoinIR = $true) & (exuE2 = $true) & (wellorderingAx = $true) & (subsetI1 = $true) & (setext = $true) & (upairsetIR = $true) & (powerset__Cong = $true) & (binintersectSubset4 = $true) & (subsetE = $true) & (vacuousDall = $true) & (exu__Cong = $true) & (setadjoinOr = $true) & (quantDeMorgan3 = $true) & (emptyI = $true) & (powersetI = $true) & (setminusI = $true) & (symdiffI1 = $true) & (dsetconstrI = $true) & (emptysetAx = $true) & (binunionIR = $true) & (secondinupair = $true) & (powersetE = $true) & (eqimpsubset2 = $true) & (subsetI2 = $true) & (binintersectLsub = $true) & (setbeta = $true) & (prop2setI = $true) & (binunionLsub = $true) & (binintersectEL = $true) & (in__Cong = $true) & (setextsub = $true) & (powersetsubset = $true) & (binintersectSubset1 = $true) & (symdiffI2 = $true) & (sepInPowerset = $true) & (exuI2 = $true) & (quantDeMorgan1 = $true) & (omega__Cong = $true) & (setoftrueEq = $true) & (eqimpsubset1 = $true) & (notequalI1 = $true) & (symdiffIneg1 = $true) & (foundationAx = $true) & (symdiffIneg2 = $true) & (setadjoin__Cong = $true) & (powersetAx = $true) & (noeltsimpempty = $true) & (notdexE = $true) & (upairsetIL = $true) & (bs114d = $true) & (notinsingleton = $true) & (emptyset__Cong = $true) & (binunionIL = $true) & (exuE3e = $true) & (notinemptyset = $true) & (setadjoinIL = $true) & (setunionI = $true) & (setminusELneg = $true) & (disjointsetsI1 = $true) & (inCongP = $true) & (exuI3 = $true) & (binintersectSubset2 = $true) & (binintersectSubset3 = $true) & (dsetconstrEL = $true) & ($true != (in @ sK305 @ (setunion @ (setadjoin @ (setadjoin @ sK305 @ emptyset) @ (setadjoin @ (setadjoin @ sK305 @ (setadjoin @ sK306 @ emptyset)) @ emptyset))))) & (setadjoinSub2 = $true) & (binunionE = $true) & (emptysetsubset = $true) & (omega0Ax = $true) & (binunionEcases = $true) & (prop2setE = $true) & (dsetconstrER = $true) & (nonemptyI1 = $true) & (upairset2IR = $true) & (setadjoinE = $true) & (prop2set2propI = $true) & (setminusIRneg = $true) & (uniqinunit = $true) & (notequalI2 = $true) & (notsubsetI = $true) & (nonemptyI = $true) & (singletonsswitch = $true) & (setadjoinAx = $true)),
% 2.39/1.66 inference(skolemisation,[status(esa),new_symbols(skolem,[sK305,sK306])],[f457,f1106])).
% 2.39/1.66 thf(f1106,plain,(
% 2.39/1.66 ? [X0 : $i,X1 : $i] : ($true != (in @ X0 @ (setunion @ (setadjoin @ (setadjoin @ X0 @ emptyset) @ (setadjoin @ (setadjoin @ X0 @ (setadjoin @ X1 @ emptyset)) @ emptyset))))) => ($true != (in @ sK305 @ (setunion @ (setadjoin @ (setadjoin @ sK305 @ emptyset) @ (setadjoin @ (setadjoin @ sK305 @ (setadjoin @ sK306 @ emptyset)) @ emptyset)))))),
% 2.39/1.66 introduced(definition,[],[choice_axiom])).
% 2.39/1.66 thf(f457,plain,(
% 2.39/1.66 (emptyInPowerset = $true) & (emptysetimpfalse = $true) & (subsetTrans = $true) & (inPowerset = $true) & (descrp = $true) & (emptyinPowerset = $true) & (descr__Cong = $true) & (powersetE1 = $true) & (exuEu = $true) & (powersetI1 = $true) & (binintersectRsub = $true) & (notdallE = $true) & (binintersectSubset5 = $true) & (subsetemptysetimpeq = $true) & (setextAx = $true) & (emptyinunitempty = $true) & (setunion__Cong = $true) & (eqinunit = $true) & (setunionAx = $true) & (quantDeMorgan2 = $true) & (setminusLsub = $true) & (nonemptyE1 = $true) & (exuE1 = $true) & (symdiffE = $true) & (subsetRefl = $true) & (binunionRsub = $true) & (replAx = $true) & (subset2powerset = $true) & (sepSubset = $true) & (setadjoinSub = $true) & (binintersectER = $true) & (dsetconstr__Cong = $true) & (exuE3u = $true) & (binintersectI = $true) & (setminusSubset2 = $true) & (exuI1 = $true) & (setminusERneg = $true) & (setminusER = $true) & (emptyE1 = $true) & (setminusEL = $true) & (setminusSubset1 = $true) & (omegaSAx = $true) & (subPowSU = $true) & (setunionE = $true) & (upairsetE = $true) & (nonemptyImpWitness = $true) & (setminusILneg = $true) & (quantDeMorgan4 = $true) & (subsetE2 = $true) & (emptysetE = $true) & (omegaIndAx = $true) & (setadjoinIR = $true) & (exuE2 = $true) & (wellorderingAx = $true) & (subsetI1 = $true) & (setext = $true) & (upairsetIR = $true) & (powerset__Cong = $true) & (binintersectSubset4 = $true) & (subsetE = $true) & (vacuousDall = $true) & (exu__Cong = $true) & (setadjoinOr = $true) & (quantDeMorgan3 = $true) & (emptyI = $true) & (powersetI = $true) & (setminusI = $true) & (symdiffI1 = $true) & (dsetconstrI = $true) & (emptysetAx = $true) & (binunionIR = $true) & (secondinupair = $true) & (powersetE = $true) & (eqimpsubset2 = $true) & (subsetI2 = $true) & (binintersectLsub = $true) & (setbeta = $true) & (prop2setI = $true) & (binunionLsub = $true) & (binintersectEL = $true) & (in__Cong = $true) & (setextsub = $true) & (powersetsubset = $true) & (binintersectSubset1 = $true) & (symdiffI2 = $true) & (sepInPowerset = $true) & (exuI2 = $true) & (quantDeMorgan1 = $true) & (omega__Cong = $true) & (setoftrueEq = $true) & (eqimpsubset1 = $true) & (notequalI1 = $true) & (symdiffIneg1 = $true) & (foundationAx = $true) & (symdiffIneg2 = $true) & (setadjoin__Cong = $true) & (powersetAx = $true) & (noeltsimpempty = $true) & (notdexE = $true) & (upairsetIL = $true) & (bs114d = $true) & (notinsingleton = $true) & (emptyset__Cong = $true) & (binunionIL = $true) & (exuE3e = $true) & (notinemptyset = $true) & (setadjoinIL = $true) & (setunionI = $true) & (setminusELneg = $true) & (disjointsetsI1 = $true) & (inCongP = $true) & (exuI3 = $true) & (binintersectSubset2 = $true) & (binintersectSubset3 = $true) & (dsetconstrEL = $true) & ? [X0 : $i,X1 : $i] : ($true != (in @ X0 @ (setunion @ (setadjoin @ (setadjoin @ X0 @ emptyset) @ (setadjoin @ (setadjoin @ X0 @ (setadjoin @ X1 @ emptyset)) @ emptyset))))) & (setadjoinSub2 = $true) & (binunionE = $true) & (emptysetsubset = $true) & (omega0Ax = $true) & (binunionEcases = $true) & (prop2setE = $true) & (dsetconstrER = $true) & (nonemptyI1 = $true) & (upairset2IR = $true) & (setadjoinE = $true) & (prop2set2propI = $true) & (setminusIRneg = $true) & (uniqinunit = $true) & (notequalI2 = $true) & (notsubsetI = $true) & (nonemptyI = $true) & (singletonsswitch = $true) & (setadjoinAx = $true)),
% 2.39/1.66 inference(flattening,[status(thm)],[f456])).
% 2.39/1.66 thf(f456,plain,(
% 2.39/1.66 ((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i,X1 : $i] : ($true != (in @ X0 @ (setunion @ (setadjoin @ (setadjoin @ X0 @ emptyset) @ (setadjoin @ (setadjoin @ X0 @ (setadjoin @ X1 @ emptyset)) @ emptyset))))) & (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)),
% 2.39/1.66 inference(ennf_transformation,[status(thm)],[f326])).
% 2.39/1.66 thf(f326,plain,(
% 2.39/1.66 ~((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) => ! [X0 : $i,X1 : $i] : ($true = (in @ X0 @ (setunion @ (setadjoin @ (setadjoin @ X0 @ emptyset) @ (setadjoin @ (setadjoin @ X0 @ (setadjoin @ X1 @ emptyset)) @ emptyset))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.39/1.66 inference(fool_elimination,[status(thm)],[f325])).
% 2.39/1.66 thf(f325,plain,(
% 2.39/1.66 ~(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 => ! [X0 : $i,X1 : $i] : (in @ X0 @ (setunion @ (setadjoin @ (setadjoin @ X0 @ emptyset) @ (setadjoin @ (setadjoin @ X0 @ (setadjoin @ X1 @ emptyset)) @ emptyset)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.39/1.66 inference(rectify,[status(thm)],[f138])).
% 2.39/1.66 thf(f138,negated_conjecture,(
% 2.39/1.66 ~(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 => ! [X1 : $i,X2 : $i] : (in @ X1 @ (setunion @ (setadjoin @ (setadjoin @ X1 @ emptyset) @ (setadjoin @ (setadjoin @ X1 @ (setadjoin @ X2 @ emptyset)) @ emptyset)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.39/1.66 inference(negated_conjecture,[status(cth)],[f137])).
% 2.39/1.66 thf(f137,conjecture,(
% 2.39/1.66 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 => ! [X1 : $i,X2 : $i] : (in @ X1 @ (setunion @ (setadjoin @ (setadjoin @ X1 @ emptyset) @ (setadjoin @ (setadjoin @ X1 @ (setadjoin @ X2 @ emptyset)) @ emptyset))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.39/1.66 file('/export/starexec/sandbox/tmp/tmp.q7XDm9XVbs/DTF2THF_21061.p',setukpairIL)).
% 2.39/1.66 thf(f3055,plain,(
% 2.39/1.66 (setadjoinIL != $true)),
% 2.39/1.66 inference(trivial_inequality_removal,[status(thm)],[f3046])).
% 2.39/1.66 thf(f3046,plain,(
% 2.39/1.66 ($true != $true) | (setadjoinIL != $true)),
% 2.39/1.66 inference(superposition,[status(thm)],[f2194,f1443])).
% 2.39/1.66 thf(f1443,plain,(
% 2.39/1.66 ( ! [X2 : $i,X3 : $i] : (((in @ X2 @ (setadjoin @ X2 @ X3)) = $true) | (setadjoinIL != $true)) )),
% 2.39/1.66 inference(cnf_transformation,[status(thm)],[f854])).
% 2.39/1.66 thf(f854,plain,(
% 2.39/1.66 ((setadjoinIL = $true) | ($true != (in @ sK156 @ (setadjoin @ sK156 @ sK157)))) & (! [X2 : $i,X3 : $i] : ((in @ X2 @ (setadjoin @ X2 @ X3)) = $true) | (setadjoinIL != $true))),
% 2.39/1.66 inference(skolemisation,[status(esa),new_symbols(skolem,[sK156,sK157])],[f852,f853])).
% 2.39/1.66 thf(f853,plain,(
% 2.39/1.66 ? [X0 : $i,X1 : $i] : ($true != (in @ X0 @ (setadjoin @ X0 @ X1))) => ($true != (in @ sK156 @ (setadjoin @ sK156 @ sK157)))),
% 2.39/1.66 introduced(definition,[],[choice_axiom])).
% 2.39/1.66 thf(f852,plain,(
% 2.39/1.66 ((setadjoinIL = $true) | ? [X0 : $i,X1 : $i] : ($true != (in @ X0 @ (setadjoin @ X0 @ X1)))) & (! [X2 : $i,X3 : $i] : ((in @ X2 @ (setadjoin @ X2 @ X3)) = $true) | (setadjoinIL != $true))),
% 2.39/1.66 inference(rectify,[status(thm)],[f851])).
% 2.39/1.66 thf(f851,plain,(
% 2.39/1.66 ((setadjoinIL = $true) | ? [X0 : $i,X1 : $i] : ($true != (in @ X0 @ (setadjoin @ X0 @ X1)))) & (! [X0 : $i,X1 : $i] : ($true = (in @ X0 @ (setadjoin @ X0 @ X1))) | (setadjoinIL != $true))),
% 2.39/1.66 inference(nnf_transformation,[status(thm)],[f191])).
% 2.39/1.66 thf(f191,plain,(
% 2.39/1.66 (setadjoinIL = $true) <=> ! [X0 : $i,X1 : $i] : ($true = (in @ X0 @ (setadjoin @ X0 @ X1)))),
% 2.39/1.66 inference(fool_elimination,[status(thm)],[f190])).
% 2.39/1.66 thf(f190,plain,(
% 2.39/1.66 (setadjoinIL = ! [X0 : $i,X1 : $i] : (in @ X0 @ (setadjoin @ X0 @ X1)))),
% 2.39/1.66 inference(rectify,[status(thm)],[f31])).
% 2.39/1.66 thf(f31,axiom,(
% 2.39/1.66 (setadjoinIL = ! [X1 : $i,X2 : $i] : (in @ X1 @ (setadjoin @ X1 @ X2)))),
% 2.39/1.66 file('/export/starexec/sandbox/tmp/tmp.q7XDm9XVbs/DTF2THF_21061.p',setadjoinIL)).
% 2.39/1.66 thf(f2194,plain,(
% 2.39/1.66 ($true != (in @ sK305 @ (setadjoin @ sK305 @ emptyset)))),
% 2.39/1.66 inference(trivial_inequality_removal,[status(thm)],[f2193])).
% 2.39/1.66 thf(f2193,plain,(
% 2.39/1.66 ($true != $true) | ($true != (in @ sK305 @ (setadjoin @ sK305 @ emptyset)))),
% 2.39/1.66 inference(forward_demodulation,[status(thm)],[f2075,f1678])).
% 2.39/1.66 thf(f2075,plain,(
% 2.39/1.66 (setadjoinIL != $true) | ($true != (in @ sK305 @ (setadjoin @ sK305 @ emptyset)))),
% 2.39/1.66 inference(trivial_inequality_removal,[status(thm)],[f1990])).
% 2.39/1.66 thf(f1990,plain,(
% 2.39/1.66 (setadjoinIL != $true) | ($true != (in @ sK305 @ (setadjoin @ sK305 @ emptyset))) | ($true != $true)),
% 2.39/1.66 inference(superposition,[status(thm)],[f1981,f1443])).
% 2.39/1.66 thf(f1981,plain,(
% 2.39/1.66 ( ! [X0 : $i] : (($true != (in @ X0 @ (setadjoin @ (setadjoin @ sK305 @ emptyset) @ (setadjoin @ (setadjoin @ sK305 @ (setadjoin @ sK306 @ emptyset)) @ emptyset)))) | ((in @ sK305 @ X0) != $true)) )),
% 2.39/1.66 inference(trivial_inequality_removal,[status(thm)],[f1980])).
% 2.39/1.66 thf(f1980,plain,(
% 2.39/1.66 ( ! [X0 : $i] : (((in @ sK305 @ X0) != $true) | ($true != $true) | ($true != (in @ X0 @ (setadjoin @ (setadjoin @ sK305 @ emptyset) @ (setadjoin @ (setadjoin @ sK305 @ (setadjoin @ sK306 @ emptyset)) @ emptyset))))) )),
% 2.39/1.66 inference(forward_demodulation,[status(thm)],[f1972,f1766])).
% 2.39/1.66 thf(f1766,plain,(
% 2.39/1.66 (setunionAx = $true)),
% 2.39/1.66 inference(cnf_transformation,[status(thm)],[f1107])).
% 2.39/1.66 thf(f1972,plain,(
% 2.39/1.66 ( ! [X0 : $i] : ((setunionAx != $true) | ($true != (in @ X0 @ (setadjoin @ (setadjoin @ sK305 @ emptyset) @ (setadjoin @ (setadjoin @ sK305 @ (setadjoin @ sK306 @ emptyset)) @ emptyset)))) | ((in @ sK305 @ X0) != $true)) )),
% 2.39/1.66 inference(trivial_inequality_removal,[status(thm)],[f1963])).
% 2.39/1.66 thf(f1963,plain,(
% 2.39/1.66 ( ! [X0 : $i] : ((setunionAx != $true) | ((in @ sK305 @ X0) != $true) | ($true != $true) | ($true != (in @ X0 @ (setadjoin @ (setadjoin @ sK305 @ emptyset) @ (setadjoin @ (setadjoin @ sK305 @ (setadjoin @ sK306 @ emptyset)) @ emptyset))))) )),
% 2.39/1.66 inference(superposition,[status(thm)],[f1669,f1276])).
% 2.39/1.66 thf(f1276,plain,(
% 2.39/1.66 ( ! [X6 : $i,X4 : $i,X5 : $i] : (((in @ X5 @ (setunion @ X4)) = $true) | ((in @ X6 @ X4) != $true) | ((in @ X5 @ X6) != $true) | (setunionAx != $true)) )),
% 2.39/1.66 inference(cnf_transformation,[status(thm)],[f658])).
% 2.39/1.66 thf(f658,plain,(
% 2.39/1.66 ((setunionAx = $true) | ((! [X2 : $i] : (((in @ X2 @ sK42) != $true) | ($true != (in @ sK43 @ X2))) | ((in @ sK43 @ (setunion @ sK42)) != $true)) & ((($true = (in @ sK44 @ sK42)) & ($true = (in @ sK43 @ sK44))) | ((in @ sK43 @ (setunion @ sK42)) = $true)))) & (! [X4 : $i,X5 : $i] : ((((in @ X5 @ (setunion @ X4)) = $true) | ! [X6 : $i] : (((in @ X6 @ X4) != $true) | ((in @ X5 @ X6) != $true))) & ((($true = (in @ (sK45 @ X5 @ X4) @ X4)) & ($true = (in @ X5 @ (sK45 @ X5 @ X4)))) | ((in @ X5 @ (setunion @ X4)) != $true))) | (setunionAx != $true))),
% 2.39/1.66 inference(skolemisation,[status(esa),new_symbols(skolem,[sK42,sK43,sK44,sK45])],[f654,f657,f656,f655])).
% 2.39/1.66 thf(f655,plain,(
% 2.39/1.66 ? [X0 : $i,X1 : $i] : ((! [X2 : $i] : (((in @ X2 @ X0) != $true) | ((in @ X1 @ X2) != $true)) | ((in @ X1 @ (setunion @ X0)) != $true)) & (? [X3 : $i] : (($true = (in @ X3 @ X0)) & ((in @ X1 @ X3) = $true)) | ((in @ X1 @ (setunion @ X0)) = $true))) => ((! [X2 : $i] : (((in @ X2 @ sK42) != $true) | ($true != (in @ sK43 @ X2))) | ((in @ sK43 @ (setunion @ sK42)) != $true)) & (? [X3 : $i] : (($true = (in @ X3 @ sK42)) & ((in @ sK43 @ X3) = $true)) | ((in @ sK43 @ (setunion @ sK42)) = $true)))),
% 2.39/1.66 introduced(definition,[],[choice_axiom])).
% 2.39/1.66 thf(f656,plain,(
% 2.39/1.66 ? [X3 : $i] : (($true = (in @ X3 @ sK42)) & ((in @ sK43 @ X3) = $true)) => (($true = (in @ sK44 @ sK42)) & ($true = (in @ sK43 @ sK44)))),
% 2.39/1.66 introduced(definition,[],[choice_axiom])).
% 2.39/1.66 thf(f657,plain,(
% 2.39/1.66 ! [X4 : $i,X5 : $i] : (? [X7 : $i] : (((in @ X7 @ X4) = $true) & ((in @ X5 @ X7) = $true)) => (($true = (in @ (sK45 @ X5 @ X4) @ X4)) & ($true = (in @ X5 @ (sK45 @ X5 @ X4)))))),
% 2.39/1.66 introduced(definition,[],[choice_axiom])).
% 2.39/1.66 thf(f654,plain,(
% 2.39/1.66 ((setunionAx = $true) | ? [X0 : $i,X1 : $i] : ((! [X2 : $i] : (((in @ X2 @ X0) != $true) | ((in @ X1 @ X2) != $true)) | ((in @ X1 @ (setunion @ X0)) != $true)) & (? [X3 : $i] : (($true = (in @ X3 @ X0)) & ((in @ X1 @ X3) = $true)) | ((in @ X1 @ (setunion @ X0)) = $true)))) & (! [X4 : $i,X5 : $i] : ((((in @ X5 @ (setunion @ X4)) = $true) | ! [X6 : $i] : (((in @ X6 @ X4) != $true) | ((in @ X5 @ X6) != $true))) & (? [X7 : $i] : (((in @ X7 @ X4) = $true) & ((in @ X5 @ X7) = $true)) | ((in @ X5 @ (setunion @ X4)) != $true))) | (setunionAx != $true))),
% 2.39/1.66 inference(rectify,[status(thm)],[f653])).
% 2.39/1.66 thf(f653,plain,(
% 2.39/1.66 ((setunionAx = $true) | ? [X0 : $i,X1 : $i] : ((! [X2 : $i] : (((in @ X2 @ X0) != $true) | ((in @ X1 @ X2) != $true)) | ((in @ X1 @ (setunion @ X0)) != $true)) & (? [X2 : $i] : (((in @ X2 @ X0) = $true) & ((in @ X1 @ X2) = $true)) | ((in @ X1 @ (setunion @ X0)) = $true)))) & (! [X0 : $i,X1 : $i] : ((((in @ X1 @ (setunion @ X0)) = $true) | ! [X2 : $i] : (((in @ X2 @ X0) != $true) | ((in @ X1 @ X2) != $true))) & (? [X2 : $i] : (((in @ X2 @ X0) = $true) & ((in @ X1 @ X2) = $true)) | ((in @ X1 @ (setunion @ X0)) != $true))) | (setunionAx != $true))),
% 2.39/1.66 inference(nnf_transformation,[status(thm)],[f399])).
% 2.39/1.66 thf(f399,plain,(
% 2.39/1.66 (setunionAx = $true) <=> ! [X0 : $i,X1 : $i] : (((in @ X1 @ (setunion @ X0)) = $true) <=> ? [X2 : $i] : (((in @ X2 @ X0) = $true) & ((in @ X1 @ X2) = $true)))),
% 2.39/1.66 inference(fool_elimination,[status(thm)],[f398])).
% 2.39/1.66 thf(f398,plain,(
% 2.39/1.66 (setunionAx = ! [X0 : $i,X1 : $i] : ((in @ X1 @ (setunion @ X0)) <=> ? [X2 : $i] : ((in @ X1 @ X2) & (in @ X2 @ X0))))),
% 2.39/1.66 inference(rectify,[status(thm)],[f6])).
% 2.39/1.66 thf(f6,axiom,(
% 2.39/1.66 (setunionAx = ! [X3 : $i,X1 : $i] : ((in @ X1 @ (setunion @ X3)) <=> ? [X4 : $i] : ((in @ X1 @ X4) & (in @ X4 @ X3))))),
% 2.39/1.66 file('/export/starexec/sandbox/tmp/tmp.q7XDm9XVbs/DTF2THF_21061.p',setunionAx)).
% 2.39/1.66 thf(f1669,plain,(
% 2.39/1.66 ($true != (in @ sK305 @ (setunion @ (setadjoin @ (setadjoin @ sK305 @ emptyset) @ (setadjoin @ (setadjoin @ sK305 @ (setadjoin @ sK306 @ emptyset)) @ emptyset)))))),
% 2.39/1.66 inference(cnf_transformation,[status(thm)],[f1107])).
% 2.39/1.66 % SZS output end Proof for DTF2THF_21061
% 2.39/1.66 % (21183)------------------------------
% 2.39/1.66 % (21183)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.39/1.66 % (21183)Termination reason: Refutation
% 2.39/1.66
% 2.39/1.66 % (21183)Memory used [KB]: 8059
% 2.39/1.66 % (21183)Time elapsed: 0.136 s
% 2.39/1.66 % (21183)Instructions burned: 275 (million)
% 2.39/1.66 % (21183)------------------------------
% 2.39/1.66 % (21183)------------------------------
% 2.39/1.66 % (21160)Success in time 0.221 s
%------------------------------------------------------------------------------