%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU628^1 : TPTP v9.2.1. Released v3.7.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n008.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 09:35:43 AM UTC 2026
% Result : Theorem 2.53s 1.64s
% Output : Refutation 2.53s
% Verified :
% SZS Type : Refutation
% Derivation depth : 19
% Number of leaves : 8
% Syntax : Number of formulae : 45 ( 13 unt; 0 typ; 0 def)
% Number of atoms : 2774 ( 708 equ; 0 cnn)
% Maximal formula atoms : 150 ( 61 avg)
% Number of connectives : 1839 ( 70 ~; 42 |; 467 &; 658 @)
% ( 4 <=>; 598 =>; 0 <=; 0 <~>)
% Maximal formula depth : 160 ( 28 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 277 ( 274 usr; 157 con; 0-4 aty)
% Number of variables : 98 ( 0 ^; 64 !; 34 ?; 98 :)
% Comments :
%------------------------------------------------------------------------------
thf(func_def_0,type,
in: $i > $i > $o ).
thf(func_def_1,type,
exu: ( $i > $o ) > $o ).
thf(func_def_6,type,
setadjoin: $i > $i > $i ).
thf(func_def_8,type,
powerset: $i > $i ).
thf(func_def_10,type,
setunion: $i > $i ).
thf(func_def_19,type,
descr: ( $i > $o ) > $i ).
thf(func_def_21,type,
dsetconstr: $i > ( $i > $o ) > $i ).
thf(func_def_26,type,
prop2set: $o > $i ).
thf(func_def_36,type,
nonempty: $i > $o ).
thf(func_def_69,type,
set2prop: $i > $o ).
thf(func_def_88,type,
subset: $i > $i > $o ).
thf(func_def_89,type,
disjoint: $i > $i > $o ).
thf(func_def_90,type,
setsmeet: $i > $i > $o ).
thf(func_def_114,type,
binunion: $i > $i > $i ).
thf(func_def_122,type,
binintersect: $i > $i > $i ).
thf(func_def_135,type,
regular: $i > $o ).
thf(func_def_136,type,
setminus: $i > $i > $i ).
thf(func_def_147,type,
symdiff: $i > $i > $i ).
thf(func_def_153,type,
iskpair: $i > $o ).
thf(func_def_158,type,
kpair: $i > $i > $i ).
thf(func_def_160,type,
cartprod: $i > $i > $i ).
thf(func_def_182,type,
sP1: $i > $i > $o ).
thf(func_def_183,type,
sP2: $i > $o ).
thf(func_def_184,type,
sP3: $i > $i > $o ).
thf(func_def_185,type,
sP4: $i > $i > $o ).
thf(func_def_186,type,
sP5: $o > $i > $i > $i > $o ).
thf(func_def_196,type,
sK15: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_197,type,
sK16: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_198,type,
sK17: $i > $o ).
thf(func_def_199,type,
sK18: $i > $o ).
thf(func_def_204,type,
sK23: $i > $o ).
thf(func_def_208,type,
sK27: $i > $i > $o ).
thf(func_def_209,type,
sK28: $i > $i ).
thf(func_def_210,type,
sK29: $i > $i ).
thf(func_def_211,type,
sK30: ( $i > $i > $o ) > $i > $i ).
thf(func_def_212,type,
sK31: ( $i > $i > $o ) > $i > $i ).
thf(func_def_213,type,
sK32: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_254,type,
sK73: $i > $i ).
thf(func_def_255,type,
sK74: $i > $o ).
thf(func_def_257,type,
sK76: $i > ( $i > $o ) > $i ).
thf(func_def_262,type,
sK81: $i > $o ).
thf(func_def_272,type,
sK91: $i > $i ).
thf(func_def_274,type,
sK93: $i > $o ).
thf(func_def_280,type,
sK99: $i > $i > $i ).
thf(func_def_288,type,
sK107: $i > $o ).
thf(func_def_289,type,
sK108: $i > $o ).
thf(func_def_290,type,
sK109: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_291,type,
sK110: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_299,type,
sK118: ( $i > $o ) > $i > $i ).
thf(func_def_301,type,
sK120: $i > $o ).
thf(func_def_307,type,
sK126: $i > $o ).
thf(func_def_309,type,
sK128: ( $i > $o ) > $i > $i ).
thf(func_def_314,type,
sK133: $i > $i ).
thf(func_def_318,type,
sK137: $i > $i > $i ).
thf(func_def_334,type,
sK153: $i > ( $i > $o ) > $i ).
thf(func_def_335,type,
sK154: $i > $o ).
thf(func_def_338,type,
sK157: $i > $o ).
thf(func_def_341,type,
sK160: $i > $o ).
thf(func_def_343,type,
sK162: $i > $i ).
thf(func_def_347,type,
sK166: $i > $o ).
thf(func_def_358,type,
sK177: $i > $i > $i ).
thf(func_def_362,type,
sK181: $i > $i > $i ).
thf(func_def_363,type,
sK182: $i > $i > $i ).
thf(func_def_372,type,
sK191: $i > $i > $i ).
thf(func_def_375,type,
sK194: $i > $i > $i ).
thf(func_def_376,type,
sK195: $i > $i > $i ).
thf(func_def_377,type,
sK196: $i > $i > $i ).
thf(func_def_378,type,
sK197: $i > $i > $i ).
thf(func_def_379,type,
sK198: $i > $i > $i ).
thf(func_def_380,type,
sK199: $i > $i > $i > $i ).
thf(func_def_381,type,
sK200: $i > $i ).
thf(func_def_382,type,
sK201: $i > $i ).
thf(func_def_383,type,
sK202: $i > $i ).
thf(func_def_384,type,
sK203: $i > $i ).
thf(func_def_385,type,
sK204: $i > $i > $i > $i ).
thf(func_def_386,type,
sK205: $i > $i > $i > $i ).
thf(func_def_387,type,
sK206: $i > $i > $i ).
thf(func_def_388,type,
sK207: $i > $i > $i ).
thf(func_def_389,type,
sK208: $i > $i > $i ).
thf(func_def_391,type,
sK210: $i > $i ).
thf(func_def_392,type,
sK211: $i > $i ).
thf(func_def_393,type,
sK212: $i > $i ).
thf(func_def_401,type,
sK220: $i > $i > $i ).
thf(func_def_402,type,
sK221: $i > $o ).
thf(func_def_407,type,
sK226: $i > $o ).
thf(func_def_409,type,
sK228: ( $i > $o ) > $i ).
thf(func_def_410,type,
sK229: ( $i > $o ) > $i ).
thf(func_def_414,type,
sK233: $i > $o ).
thf(func_def_416,type,
sK235: $i > $o ).
thf(func_def_419,type,
sK238: $i > ( $i > $o ) > $i ).
thf(func_def_431,type,
sK250: $i > $o ).
thf(func_def_432,type,
sK251: $i > $i ).
thf(func_def_433,type,
sK252: ( $i > $o ) > $i ).
thf(func_def_437,type,
sK256: $i > $o ).
thf(func_def_449,type,
sK268: $i > $o ).
thf(func_def_455,type,
sK274: $i > $o ).
thf(func_def_467,type,
sK286: ( $i > $o ) > $i > $i ).
thf(func_def_469,type,
sK288: $i > $o ).
thf(func_def_490,type,
sK309: $i > $o ).
thf(func_def_491,type,
sK310: ( $i > $o ) > $i ).
thf(func_def_492,type,
sK311: $i > $o ).
thf(func_def_493,type,
sK312: $i > $i ).
thf(func_def_494,type,
sK313: ( $i > $o ) > $i ).
thf(func_def_498,type,
sK317: $o > $i > $i > $i ).
thf(func_def_501,type,
sK320: $i > $i > $i ).
thf(func_def_516,type,
sK335: ( $i > $o ) > $i > $i ).
thf(func_def_518,type,
sK337: $i > $o ).
thf(func_def_525,type,
sK344: $i > $i > $i ).
thf(func_def_541,type,
sK360: $i > $o ).
thf(func_def_543,type,
sK362: ( $i > $o ) > $i > $i ).
thf(func_def_549,type,
sK368: $i > $i ).
thf(func_def_556,type,
sK375: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_557,type,
sK376: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_560,type,
sK379: $i > $o ).
thf(func_def_561,type,
sK380: $i > $o ).
thf(func_def_570,type,
sK389: $i > $o ).
thf(func_def_577,type,
sK396: $i > $i ).
thf(func_def_579,type,
sK398: $i > $i ).
thf(func_def_581,type,
ph400:
!>[X0: $tType] : X0 ).
thf(f2185,plain,
$false,
inference(trivial_inequality_removal,[status(thm)],[f2179]) ).
thf(f2179,plain,
$true != $true,
inference(superposition,[status(thm)],[f2150,f1569]) ).
thf(f1569,plain,
( $true
= ( in @ sK188 @ sK185 ) ),
inference(cnf_transformation,[status(thm)],[f948]) ).
thf(f948,plain,
( ( powersetsubset = $true )
& ( omega0Ax = $true )
& ( inPowerset = $true )
& ( eqimpsubset2 = $true )
& ( nonemptyI1 = $true )
& ( notdallE = $true )
& ( symdiffIneg2 = $true )
& ( setukpairIL = $true )
& ( powersetE = $true )
& ( bs114d = $true )
& ( binintersectI = $true )
& ( exuE2 = $true )
& ( dsetconstrER = $true )
& ( exuE1 = $true )
& ( eqimpsubset1 = $true )
& ( in__Cong = $true )
& ( binunionLsub = $true )
& ( prop2set2propI = $true )
& ( exuI3 = $true )
& ( prop2setI = $true )
& ( upairsubunion = $true )
& ( binintersectEL = $true )
& ( setminusI = $true )
& ( nonemptyI = $true )
& ( subsetTrans = $true )
& ( quantDeMorgan2 = $true )
& ( quantDeMorgan4 = $true )
& ( binunionRsub = $true )
& ( nonemptyE1 = $true )
& ( setadjoin__Cong = $true )
& ( emptysetimpfalse = $true )
& ( setbeta = $true )
& ( emptyset__Cong = $true )
& ( binunionIL = $true )
& ( setunionI = $true )
& ( kpairp = $true )
& ( noeltsimpempty = $true )
& ( notsubsetI = $true )
& ( setadjoinSub2 = $true )
& ( dsetconstrI = $true )
& ( notequalI2 = $true )
& ( upairsetE = $true )
& ( notinemptyset = $true )
& ( emptysetsubset = $true )
& ( subPowSU = $true )
& ( setminusERneg = $true )
& ( nonemptyImpWitness = $true )
& ( replAx = $true )
& ( omega__Cong = $true )
& ( powerset__Cong = $true )
& ( setminusSubset1 = $true )
& ( sepInPowerset = $true )
& ( foundationAx = $true )
& ( emptyI = $true )
& ( subsetI1 = $true )
& ( kpairiskpair = $true )
& ( exuE3e = $true )
& ( notinsingleton = $true )
& ( binintersectLsub = $true )
& ( emptyinPowerset = $true )
& ( powersetE1 = $true )
& ( wellorderingAx = $true )
& ( inCongP = $true )
& ( binintersectER = $true )
& ( descrp = $true )
& ( setunionAx = $true )
& ( setminusLsub = $true )
& ( dsetconstrEL = $true )
& ( setunionE = $true )
& ( setadjoinIL = $true )
& ( emptyinunitempty = $true )
& ( omegaIndAx = $true )
& ( upairsetIL = $true )
& ( setminusILneg = $true )
& ( symdiffI2 = $true )
& ( subsetE = $true )
& ( exu__Cong = $true )
& ( symdiffIneg1 = $true )
& ( binintersectRsub = $true )
& ( descr__Cong = $true )
& ( setoftrueEq = $true )
& ( subsetemptysetimpeq = $true )
& ( symdiffE = $true )
& ( singletoninpowerset = $true )
& ( eqinunit = $true )
& ( upairset2E = $true )
& ( upairset2IR = $true )
& ( exuI2 = $true )
& ( setminusELneg = $true )
& ( binintersectSubset3 = $true )
& ( subsetE2 = $true )
& ( powersetAx = $true )
& ( upairsetIR = $true )
& ( setext = $true )
& ( subsetRefl = $true )
& ( emptyE1 = $true )
& ( emptysetE = $true )
& ( dsetconstr__Cong = $true )
& ( powersetI = $true )
& ( singletonsswitch = $true )
& ( notdexE = $true )
& ( binunionE = $true )
& ( setadjoinSub = $true )
& ( emptyInPowerset = $true )
& ( binintersectSubset2 = $true )
& ( binunionEcases = $true )
& ( secondinupair = $true )
& ( binintersectSubset1 = $true )
& ( omegaSAx = $true )
& ( setukpairIR = $true )
& ( sepSubset = $true )
& ( exuE3u = $true )
& ( ubforcartprodlem1 = $true )
& ( quantDeMorgan3 = $true )
& ( binunionIR = $true )
& ( setminusIRneg = $true )
& ( emptysetAx = $true )
& ( subset2powerset = $true )
& ( binintersectSubset5 = $true )
& ( setunion__Cong = $true )
& ( vacuousDall = $true )
& ( exuI1 = $true )
& ( setadjoinIR = $true )
& ( singletonsubset = $true )
& ( setextAx = $true )
& ( exuEu = $true )
& ( setextsub = $true )
& ( subsetI2 = $true )
& ( setminusER = $true )
& ( powersetI1 = $true )
& ( disjointsetsI1 = $true )
& ( uniqinunit = $true )
& ( setadjoinAx = $true )
& ( setadjoinOr = $true )
& ( symdiffI1 = $true )
& ( $true
= ( in @ sK187 @ sK186 ) )
& ( $true
= ( in @ sK188 @ sK185 ) )
& ( $true
!= ( in @ ( setadjoin @ ( setadjoin @ sK187 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK187 @ ( setadjoin @ sK188 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ sK186 @ sK185 ) ) ) ) )
& ( notequalI1 = $true )
& ( setminusSubset2 = $true )
& ( singletoninpowunion = $true )
& ( quantDeMorgan1 = $true )
& ( binintersectSubset4 = $true )
& ( prop2setE = $true )
& ( upairinpowunion = $true )
& ( setadjoinE = $true )
& ( setminusEL = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK185,sK186,sK187,sK188])],[f609,f947,f946]) ).
thf(f946,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X2 @ X1 ) )
& ? [X3: $i] :
( ( ( in @ X3 @ X0 )
= $true )
& ( ( in @ ( setadjoin @ ( setadjoin @ X2 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X2 @ ( setadjoin @ X3 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X0 ) ) ) )
!= $true ) ) )
=> ( ( $true
= ( in @ sK187 @ sK186 ) )
& ? [X3: $i] :
( ( ( in @ X3 @ sK185 )
= $true )
& ( ( in @ ( setadjoin @ ( setadjoin @ sK187 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK187 @ ( setadjoin @ X3 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ sK186 @ sK185 ) ) ) )
!= $true ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f947,plain,
( ? [X3: $i] :
( ( ( in @ X3 @ sK185 )
= $true )
& ( ( in @ ( setadjoin @ ( setadjoin @ sK187 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK187 @ ( setadjoin @ X3 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ sK186 @ sK185 ) ) ) )
!= $true ) )
=> ( ( $true
= ( in @ sK188 @ sK185 ) )
& ( $true
!= ( in @ ( setadjoin @ ( setadjoin @ sK187 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK187 @ ( setadjoin @ sK188 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ sK186 @ sK185 ) ) ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f609,plain,
( ( powersetsubset = $true )
& ( omega0Ax = $true )
& ( inPowerset = $true )
& ( eqimpsubset2 = $true )
& ( nonemptyI1 = $true )
& ( notdallE = $true )
& ( symdiffIneg2 = $true )
& ( setukpairIL = $true )
& ( powersetE = $true )
& ( bs114d = $true )
& ( binintersectI = $true )
& ( exuE2 = $true )
& ( dsetconstrER = $true )
& ( exuE1 = $true )
& ( eqimpsubset1 = $true )
& ( in__Cong = $true )
& ( binunionLsub = $true )
& ( prop2set2propI = $true )
& ( exuI3 = $true )
& ( prop2setI = $true )
& ( upairsubunion = $true )
& ( binintersectEL = $true )
& ( setminusI = $true )
& ( nonemptyI = $true )
& ( subsetTrans = $true )
& ( quantDeMorgan2 = $true )
& ( quantDeMorgan4 = $true )
& ( binunionRsub = $true )
& ( nonemptyE1 = $true )
& ( setadjoin__Cong = $true )
& ( emptysetimpfalse = $true )
& ( setbeta = $true )
& ( emptyset__Cong = $true )
& ( binunionIL = $true )
& ( setunionI = $true )
& ( kpairp = $true )
& ( noeltsimpempty = $true )
& ( notsubsetI = $true )
& ( setadjoinSub2 = $true )
& ( dsetconstrI = $true )
& ( notequalI2 = $true )
& ( upairsetE = $true )
& ( notinemptyset = $true )
& ( emptysetsubset = $true )
& ( subPowSU = $true )
& ( setminusERneg = $true )
& ( nonemptyImpWitness = $true )
& ( replAx = $true )
& ( omega__Cong = $true )
& ( powerset__Cong = $true )
& ( setminusSubset1 = $true )
& ( sepInPowerset = $true )
& ( foundationAx = $true )
& ( emptyI = $true )
& ( subsetI1 = $true )
& ( kpairiskpair = $true )
& ( exuE3e = $true )
& ( notinsingleton = $true )
& ( binintersectLsub = $true )
& ( emptyinPowerset = $true )
& ( powersetE1 = $true )
& ( wellorderingAx = $true )
& ( inCongP = $true )
& ( binintersectER = $true )
& ( descrp = $true )
& ( setunionAx = $true )
& ( setminusLsub = $true )
& ( dsetconstrEL = $true )
& ( setunionE = $true )
& ( setadjoinIL = $true )
& ( emptyinunitempty = $true )
& ( omegaIndAx = $true )
& ( upairsetIL = $true )
& ( setminusILneg = $true )
& ( symdiffI2 = $true )
& ( subsetE = $true )
& ( exu__Cong = $true )
& ( symdiffIneg1 = $true )
& ( binintersectRsub = $true )
& ( descr__Cong = $true )
& ( setoftrueEq = $true )
& ( subsetemptysetimpeq = $true )
& ( symdiffE = $true )
& ( singletoninpowerset = $true )
& ( eqinunit = $true )
& ( upairset2E = $true )
& ( upairset2IR = $true )
& ( exuI2 = $true )
& ( setminusELneg = $true )
& ( binintersectSubset3 = $true )
& ( subsetE2 = $true )
& ( powersetAx = $true )
& ( upairsetIR = $true )
& ( setext = $true )
& ( subsetRefl = $true )
& ( emptyE1 = $true )
& ( emptysetE = $true )
& ( dsetconstr__Cong = $true )
& ( powersetI = $true )
& ( singletonsswitch = $true )
& ( notdexE = $true )
& ( binunionE = $true )
& ( setadjoinSub = $true )
& ( emptyInPowerset = $true )
& ( binintersectSubset2 = $true )
& ( binunionEcases = $true )
& ( secondinupair = $true )
& ( binintersectSubset1 = $true )
& ( omegaSAx = $true )
& ( setukpairIR = $true )
& ( sepSubset = $true )
& ( exuE3u = $true )
& ( ubforcartprodlem1 = $true )
& ( quantDeMorgan3 = $true )
& ( binunionIR = $true )
& ( setminusIRneg = $true )
& ( emptysetAx = $true )
& ( subset2powerset = $true )
& ( binintersectSubset5 = $true )
& ( setunion__Cong = $true )
& ( vacuousDall = $true )
& ( exuI1 = $true )
& ( setadjoinIR = $true )
& ( singletonsubset = $true )
& ( setextAx = $true )
& ( exuEu = $true )
& ( setextsub = $true )
& ( subsetI2 = $true )
& ( setminusER = $true )
& ( powersetI1 = $true )
& ( disjointsetsI1 = $true )
& ( uniqinunit = $true )
& ( setadjoinAx = $true )
& ( setadjoinOr = $true )
& ( symdiffI1 = $true )
& ? [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X2 @ X1 ) )
& ? [X3: $i] :
( ( ( in @ X3 @ X0 )
= $true )
& ( ( in @ ( setadjoin @ ( setadjoin @ X2 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X2 @ ( setadjoin @ X3 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X0 ) ) ) )
!= $true ) ) )
& ( notequalI1 = $true )
& ( setminusSubset2 = $true )
& ( singletoninpowunion = $true )
& ( quantDeMorgan1 = $true )
& ( binintersectSubset4 = $true )
& ( prop2setE = $true )
& ( upairinpowunion = $true )
& ( setadjoinE = $true )
& ( setminusEL = $true ) ),
inference(flattening,[status(thm)],[f608]) ).
thf(f608,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X2 @ X1 ) )
& ? [X3: $i] :
( ( ( in @ X3 @ X0 )
= $true )
& ( ( in @ ( setadjoin @ ( setadjoin @ X2 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X2 @ ( setadjoin @ X3 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X0 ) ) ) )
!= $true ) ) )
& ( ubforcartprodlem1 = $true )
& ( upairinpowunion = $true )
& ( upairsubunion = $true )
& ( upairset2E = $true )
& ( singletoninpowunion = $true )
& ( singletoninpowerset = $true )
& ( singletonsubset = $true )
& ( kpairp = $true )
& ( kpairiskpair = $true )
& ( setukpairIR = $true )
& ( setukpairIL = $true )
& ( secondinupair = $true )
& ( symdiffIneg2 = $true )
& ( symdiffIneg1 = $true )
& ( symdiffI2 = $true )
& ( symdiffI1 = $true )
& ( symdiffE = $true )
& ( setminusSubset1 = $true )
& ( setminusLsub = $true )
& ( setminusIRneg = $true )
& ( setminusILneg = $true )
& ( setminusELneg = $true )
& ( setminusERneg = $true )
& ( setminusSubset2 = $true )
& ( setminusER = $true )
& ( setminusEL = $true )
& ( setminusI = $true )
& ( bs114d = $true )
& ( binintersectSubset1 = $true )
& ( binintersectSubset4 = $true )
& ( binintersectRsub = $true )
& ( disjointsetsI1 = $true )
& ( binintersectER = $true )
& ( binintersectSubset3 = $true )
& ( binintersectSubset2 = $true )
& ( binintersectLsub = $true )
& ( binintersectEL = $true )
& ( binintersectSubset5 = $true )
& ( binintersectI = $true )
& ( binunionRsub = $true )
& ( binunionLsub = $true )
& ( binunionE = $true )
& ( binunionEcases = $true )
& ( binunionIR = $true )
& ( upairset2IR = $true )
& ( binunionIL = $true )
& ( sepSubset = $true )
& ( sepInPowerset = $true )
& ( powersetsubset = $true )
& ( inPowerset = $true )
& ( powersetE1 = $true )
& ( powersetI1 = $true )
& ( subsetemptysetimpeq = $true )
& ( setextsub = $true )
& ( subset2powerset = $true )
& ( setadjoinSub2 = $true )
& ( setadjoinSub = $true )
& ( subsetTrans = $true )
& ( subsetRefl = $true )
& ( notequalI2 = $true )
& ( notequalI1 = $true )
& ( notsubsetI = $true )
& ( subsetE2 = $true )
& ( subsetE = $true )
& ( emptysetsubset = $true )
& ( subsetI2 = $true )
& ( eqimpsubset1 = $true )
& ( eqimpsubset2 = $true )
& ( subsetI1 = $true )
& ( dsetconstr__Cong = $true )
& ( descr__Cong = $true )
& ( exuEu = $true )
& ( omega__Cong = $true )
& ( setunion__Cong = $true )
& ( powerset__Cong = $true )
& ( setadjoin__Cong = $true )
& ( emptyset__Cong = $true )
& ( exu__Cong = $true )
& ( exuE3u = $true )
& ( in__Cong = $true )
& ( inCongP = $true )
& ( exuI2 = $true )
& ( exuI3 = $true )
& ( exuI1 = $true )
& ( notdallE = $true )
& ( notdexE = $true )
& ( prop2set2propI = $true )
& ( prop2setI = $true )
& ( quantDeMorgan4 = $true )
& ( quantDeMorgan3 = $true )
& ( quantDeMorgan2 = $true )
& ( quantDeMorgan1 = $true )
& ( vacuousDall = $true )
& ( emptyE1 = $true )
& ( upairsetIR = $true )
& ( upairsetIL = $true )
& ( upairsetE = $true )
& ( singletonsswitch = $true )
& ( eqinunit = $true )
& ( notinsingleton = $true )
& ( uniqinunit = $true )
& ( nonemptyImpWitness = $true )
& ( exuE2 = $true )
& ( subPowSU = $true )
& ( setunionE = $true )
& ( setunionI = $true )
& ( powersetE = $true )
& ( emptyInPowerset = $true )
& ( emptyinPowerset = $true )
& ( powersetI = $true )
& ( setoftrueEq = $true )
& ( setadjoinOr = $true )
& ( setadjoinE = $true )
& ( setadjoinIR = $true )
& ( emptyinunitempty = $true )
& ( setadjoinIL = $true )
& ( nonemptyI1 = $true )
& ( nonemptyI = $true )
& ( nonemptyE1 = $true )
& ( setbeta = $true )
& ( noeltsimpempty = $true )
& ( emptyI = $true )
& ( setext = $true )
& ( exuE3e = $true )
& ( notinemptyset = $true )
& ( emptysetimpfalse = $true )
& ( emptysetE = $true )
& ( prop2setE = $true )
& ( exuE1 = $true )
& ( dsetconstrER = $true )
& ( dsetconstrEL = $true )
& ( dsetconstrI = $true )
& ( descrp = $true )
& ( wellorderingAx = $true )
& ( foundationAx = $true )
& ( replAx = $true )
& ( omegaIndAx = $true )
& ( omegaSAx = $true )
& ( omega0Ax = $true )
& ( setunionAx = $true )
& ( powersetAx = $true )
& ( setadjoinAx = $true )
& ( emptysetAx = $true )
& ( setextAx = $true ) ),
inference(ennf_transformation,[status(thm)],[f274]) ).
thf(f274,plain,
~ ( ( setextAx = $true )
=> ( ( emptysetAx = $true )
=> ( ( setadjoinAx = $true )
=> ( ( powersetAx = $true )
=> ( ( setunionAx = $true )
=> ( ( omega0Ax = $true )
=> ( ( omegaSAx = $true )
=> ( ( omegaIndAx = $true )
=> ( ( replAx = $true )
=> ( ( foundationAx = $true )
=> ( ( wellorderingAx = $true )
=> ( ( descrp = $true )
=> ( ( dsetconstrI = $true )
=> ( ( dsetconstrEL = $true )
=> ( ( dsetconstrER = $true )
=> ( ( exuE1 = $true )
=> ( ( prop2setE = $true )
=> ( ( emptysetE = $true )
=> ( ( emptysetimpfalse = $true )
=> ( ( notinemptyset = $true )
=> ( ( exuE3e = $true )
=> ( ( setext = $true )
=> ( ( emptyI = $true )
=> ( ( noeltsimpempty = $true )
=> ( ( setbeta = $true )
=> ( ( nonemptyE1 = $true )
=> ( ( nonemptyI = $true )
=> ( ( nonemptyI1 = $true )
=> ( ( setadjoinIL = $true )
=> ( ( emptyinunitempty = $true )
=> ( ( setadjoinIR = $true )
=> ( ( setadjoinE = $true )
=> ( ( setadjoinOr = $true )
=> ( ( setoftrueEq = $true )
=> ( ( powersetI = $true )
=> ( ( emptyinPowerset = $true )
=> ( ( emptyInPowerset = $true )
=> ( ( powersetE = $true )
=> ( ( setunionI = $true )
=> ( ( setunionE = $true )
=> ( ( subPowSU = $true )
=> ( ( exuE2 = $true )
=> ( ( nonemptyImpWitness = $true )
=> ( ( uniqinunit = $true )
=> ( ( notinsingleton = $true )
=> ( ( eqinunit = $true )
=> ( ( singletonsswitch = $true )
=> ( ( upairsetE = $true )
=> ( ( upairsetIL = $true )
=> ( ( upairsetIR = $true )
=> ( ( emptyE1 = $true )
=> ( ( vacuousDall = $true )
=> ( ( quantDeMorgan1 = $true )
=> ( ( quantDeMorgan2 = $true )
=> ( ( quantDeMorgan3 = $true )
=> ( ( quantDeMorgan4 = $true )
=> ( ( prop2setI = $true )
=> ( ( prop2set2propI = $true )
=> ( ( notdexE = $true )
=> ( ( notdallE = $true )
=> ( ( exuI1 = $true )
=> ( ( exuI3 = $true )
=> ( ( exuI2 = $true )
=> ( ( inCongP = $true )
=> ( ( in__Cong = $true )
=> ( ( exuE3u = $true )
=> ( ( exu__Cong = $true )
=> ( ( emptyset__Cong = $true )
=> ( ( setadjoin__Cong = $true )
=> ( ( powerset__Cong = $true )
=> ( ( setunion__Cong = $true )
=> ( ( omega__Cong = $true )
=> ( ( exuEu = $true )
=> ( ( descr__Cong = $true )
=> ( ( dsetconstr__Cong = $true )
=> ( ( subsetI1 = $true )
=> ( ( eqimpsubset2 = $true )
=> ( ( eqimpsubset1 = $true )
=> ( ( subsetI2 = $true )
=> ( ( emptysetsubset = $true )
=> ( ( subsetE = $true )
=> ( ( subsetE2 = $true )
=> ( ( notsubsetI = $true )
=> ( ( notequalI1 = $true )
=> ( ( notequalI2 = $true )
=> ( ( subsetRefl = $true )
=> ( ( subsetTrans = $true )
=> ( ( setadjoinSub = $true )
=> ( ( setadjoinSub2 = $true )
=> ( ( subset2powerset = $true )
=> ( ( setextsub = $true )
=> ( ( subsetemptysetimpeq = $true )
=> ( ( powersetI1 = $true )
=> ( ( powersetE1 = $true )
=> ( ( inPowerset = $true )
=> ( ( powersetsubset = $true )
=> ( ( sepInPowerset = $true )
=> ( ( sepSubset = $true )
=> ( ( binunionIL = $true )
=> ( ( upairset2IR = $true )
=> ( ( binunionIR = $true )
=> ( ( binunionEcases = $true )
=> ( ( binunionE = $true )
=> ( ( binunionLsub = $true )
=> ( ( binunionRsub = $true )
=> ( ( binintersectI = $true )
=> ( ( binintersectSubset5 = $true )
=> ( ( binintersectEL = $true )
=> ( ( binintersectLsub = $true )
=> ( ( binintersectSubset2 = $true )
=> ( ( binintersectSubset3 = $true )
=> ( ( binintersectER = $true )
=> ( ( disjointsetsI1 = $true )
=> ( ( binintersectRsub = $true )
=> ( ( binintersectSubset4 = $true )
=> ( ( binintersectSubset1 = $true )
=> ( ( bs114d = $true )
=> ( ( setminusI = $true )
=> ( ( setminusEL = $true )
=> ( ( setminusER = $true )
=> ( ( setminusSubset2 = $true )
=> ( ( setminusERneg = $true )
=> ( ( setminusELneg = $true )
=> ( ( setminusILneg = $true )
=> ( ( setminusIRneg = $true )
=> ( ( setminusLsub = $true )
=> ( ( setminusSubset1 = $true )
=> ( ( symdiffE = $true )
=> ( ( symdiffI1 = $true )
=> ( ( symdiffI2 = $true )
=> ( ( symdiffIneg1 = $true )
=> ( ( symdiffIneg2 = $true )
=> ( ( secondinupair = $true )
=> ( ( setukpairIL = $true )
=> ( ( setukpairIR = $true )
=> ( ( kpairiskpair = $true )
=> ( ( kpairp = $true )
=> ( ( singletonsubset = $true )
=> ( ( singletoninpowerset = $true )
=> ( ( singletoninpowunion = $true )
=> ( ( upairset2E = $true )
=> ( ( upairsubunion = $true )
=> ( ( upairinpowunion = $true )
=> ( ( ubforcartprodlem1 = $true )
=> ! [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X2 @ X1 ) )
=> ! [X3: $i] :
( ( ( in @ X3 @ X0 )
= $true )
=> ( ( in @ ( setadjoin @ ( setadjoin @ X2 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X2 @ ( setadjoin @ X3 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X0 ) ) ) )
= $true ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f273]) ).
thf(f273,plain,
~ ( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ! [X0: $i,X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ! [X3: $i] :
( ( in @ X3 @ X0 )
=> ( in @ ( setadjoin @ ( setadjoin @ X2 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X2 @ ( setadjoin @ X3 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X0 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f151]) ).
thf(f151,negated_conjecture,
~ ( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ! [X4: $i,X3: $i,X1: $i] :
( ( in @ X1 @ X3 )
=> ! [X2: $i] :
( ( in @ X2 @ X4 )
=> ( in @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X3 @ X4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f150]) ).
thf(f150,conjecture,
( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ! [X4: $i,X3: $i,X1: $i] :
( ( in @ X1 @ X3 )
=> ! [X2: $i] :
( ( in @ X2 @ X4 )
=> ( in @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X3 @ X4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.yC96zsFX7x/DTF2THF_27586.p',ubforcartprodlem2) ).
thf(f2150,plain,
( $true
!= ( in @ sK188 @ sK185 ) ),
inference(trivial_inequality_removal,[status(thm)],[f2149]) ).
thf(f2149,plain,
( ( $true
!= ( in @ sK188 @ sK185 ) )
| ( $true != $true ) ),
inference(forward_demodulation,[status(thm)],[f2148,f1593]) ).
thf(f1593,plain,
ubforcartprodlem1 = $true,
inference(cnf_transformation,[status(thm)],[f948]) ).
thf(f2148,plain,
( ( ubforcartprodlem1 != $true )
| ( $true
!= ( in @ sK188 @ sK185 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f2147]) ).
thf(f2147,plain,
( ( $true != $true )
| ( $true
!= ( in @ sK188 @ sK185 ) )
| ( ubforcartprodlem1 != $true ) ),
inference(forward_demodulation,[status(thm)],[f2143,f1570]) ).
thf(f1570,plain,
( $true
= ( in @ sK187 @ sK186 ) ),
inference(cnf_transformation,[status(thm)],[f948]) ).
thf(f2143,plain,
( ( $true
!= ( in @ sK187 @ sK186 ) )
| ( ubforcartprodlem1 != $true )
| ( $true
!= ( in @ sK188 @ sK185 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f2135]) ).
thf(f2135,plain,
( ( $true
!= ( in @ sK188 @ sK185 ) )
| ( ubforcartprodlem1 != $true )
| ( $true != $true )
| ( $true
!= ( in @ sK187 @ sK186 ) ) ),
inference(superposition,[status(thm)],[f2132,f1445]) ).
thf(f1445,plain,
! [X2: $i,X3: $i,X0: $i,X1: $i] :
( ( ( subset @ ( setadjoin @ ( setadjoin @ X0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X0 @ ( setadjoin @ X3 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) )
= $true )
| ( ubforcartprodlem1 != $true )
| ( ( in @ X3 @ X2 )
!= $true )
| ( $true
!= ( in @ X0 @ X1 ) ) ),
inference(cnf_transformation,[status(thm)],[f808]) ).
thf(f808,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ! [X3: $i] :
( ( ( in @ X3 @ X2 )
!= $true )
| ( ( subset @ ( setadjoin @ ( setadjoin @ X0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X0 @ ( setadjoin @ X3 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) )
= $true ) )
| ( $true
!= ( in @ X0 @ X1 ) ) )
| ( ubforcartprodlem1 != $true ) )
& ( ( ubforcartprodlem1 = $true )
| ( ( $true
= ( in @ sK105 @ sK104 ) )
& ( $true
!= ( subset @ ( setadjoin @ ( setadjoin @ sK102 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK102 @ ( setadjoin @ sK105 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ sK103 @ sK104 ) ) ) )
& ( $true
= ( in @ sK102 @ sK103 ) ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK102,sK103,sK104,sK105])],[f805,f807,f806]) ).
thf(f806,plain,
( ? [X4: $i,X5: $i,X6: $i] :
( ? [X7: $i] :
( ( ( in @ X7 @ X6 )
= $true )
& ( $true
!= ( subset @ ( setadjoin @ ( setadjoin @ X4 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X4 @ ( setadjoin @ X7 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X5 @ X6 ) ) ) ) )
& ( $true
= ( in @ X4 @ X5 ) ) )
=> ( ? [X7: $i] :
( ( $true
= ( in @ X7 @ sK104 ) )
& ( $true
!= ( subset @ ( setadjoin @ ( setadjoin @ sK102 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK102 @ ( setadjoin @ X7 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ sK103 @ sK104 ) ) ) ) )
& ( $true
= ( in @ sK102 @ sK103 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f807,plain,
( ? [X7: $i] :
( ( $true
= ( in @ X7 @ sK104 ) )
& ( $true
!= ( subset @ ( setadjoin @ ( setadjoin @ sK102 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK102 @ ( setadjoin @ X7 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ sK103 @ sK104 ) ) ) ) )
=> ( ( $true
= ( in @ sK105 @ sK104 ) )
& ( $true
!= ( subset @ ( setadjoin @ ( setadjoin @ sK102 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK102 @ ( setadjoin @ sK105 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ sK103 @ sK104 ) ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f805,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ! [X3: $i] :
( ( ( in @ X3 @ X2 )
!= $true )
| ( ( subset @ ( setadjoin @ ( setadjoin @ X0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X0 @ ( setadjoin @ X3 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) )
= $true ) )
| ( $true
!= ( in @ X0 @ X1 ) ) )
| ( ubforcartprodlem1 != $true ) )
& ( ( ubforcartprodlem1 = $true )
| ? [X4: $i,X5: $i,X6: $i] :
( ? [X7: $i] :
( ( ( in @ X7 @ X6 )
= $true )
& ( $true
!= ( subset @ ( setadjoin @ ( setadjoin @ X4 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X4 @ ( setadjoin @ X7 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X5 @ X6 ) ) ) ) )
& ( $true
= ( in @ X4 @ X5 ) ) ) ) ),
inference(rectify,[status(thm)],[f804]) ).
thf(f804,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ! [X3: $i] :
( ( ( in @ X3 @ X2 )
!= $true )
| ( ( subset @ ( setadjoin @ ( setadjoin @ X0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X0 @ ( setadjoin @ X3 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) )
= $true ) )
| ( $true
!= ( in @ X0 @ X1 ) ) )
| ( ubforcartprodlem1 != $true ) )
& ( ( ubforcartprodlem1 = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ? [X3: $i] :
( ( ( in @ X3 @ X2 )
= $true )
& ( ( subset @ ( setadjoin @ ( setadjoin @ X0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X0 @ ( setadjoin @ X3 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) )
!= $true ) )
& ( $true
= ( in @ X0 @ X1 ) ) ) ) ),
inference(nnf_transformation,[status(thm)],[f558]) ).
thf(f558,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ! [X3: $i] :
( ( ( in @ X3 @ X2 )
!= $true )
| ( ( subset @ ( setadjoin @ ( setadjoin @ X0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X0 @ ( setadjoin @ X3 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) )
= $true ) )
| ( $true
!= ( in @ X0 @ X1 ) ) )
<=> ( ubforcartprodlem1 = $true ) ),
inference(ennf_transformation,[status(thm)],[f232]) ).
thf(f232,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X0 @ X1 ) )
=> ! [X3: $i] :
( ( ( in @ X3 @ X2 )
= $true )
=> ( ( subset @ ( setadjoin @ ( setadjoin @ X0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X0 @ ( setadjoin @ X3 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) )
= $true ) ) )
<=> ( ubforcartprodlem1 = $true ) ),
inference(fool_elimination,[status(thm)],[f231]) ).
thf(f231,plain,
( ubforcartprodlem1
= ( ! [X0: $i,X1: $i,X2: $i] :
( ( in @ X0 @ X1 )
=> ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X0 @ ( setadjoin @ X3 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f149]) ).
thf(f149,axiom,
( ubforcartprodlem1
= ( ! [X1: $i,X3: $i,X4: $i] :
( ( in @ X1 @ X3 )
=> ! [X2: $i] :
( ( in @ X2 @ X4 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X3 @ X4 ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.yC96zsFX7x/DTF2THF_27586.p',ubforcartprodlem1) ).
thf(f2132,plain,
( $true
!= ( subset @ ( setadjoin @ ( setadjoin @ sK187 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK187 @ ( setadjoin @ sK188 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ sK186 @ sK185 ) ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f2131]) ).
thf(f2131,plain,
( ( $true != $true )
| ( $true
!= ( subset @ ( setadjoin @ ( setadjoin @ sK187 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK187 @ ( setadjoin @ sK188 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ sK186 @ sK185 ) ) ) ) ),
inference(forward_demodulation,[status(thm)],[f2120,f1576]) ).
thf(f1576,plain,
powersetI1 = $true,
inference(cnf_transformation,[status(thm)],[f948]) ).
thf(f2120,plain,
( ( powersetI1 != $true )
| ( $true
!= ( subset @ ( setadjoin @ ( setadjoin @ sK187 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK187 @ ( setadjoin @ sK188 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ sK186 @ sK185 ) ) ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f2114]) ).
thf(f2114,plain,
( ( $true != $true )
| ( powersetI1 != $true )
| ( $true
!= ( subset @ ( setadjoin @ ( setadjoin @ sK187 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK187 @ ( setadjoin @ sK188 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ sK186 @ sK185 ) ) ) ) ),
inference(superposition,[status(thm)],[f1568,f1352]) ).
thf(f1352,plain,
! [X0: $i,X1: $i] :
( ( ( in @ X0 @ ( powerset @ X1 ) )
= $true )
| ( $true
!= ( subset @ X0 @ X1 ) )
| ( powersetI1 != $true ) ),
inference(cnf_transformation,[status(thm)],[f701]) ).
thf(f701,plain,
( ( ! [X0: $i,X1: $i] :
( ( $true
!= ( subset @ X0 @ X1 ) )
| ( ( in @ X0 @ ( powerset @ X1 ) )
= $true ) )
| ( powersetI1 != $true ) )
& ( ( powersetI1 = $true )
| ( ( $true
= ( subset @ sK37 @ sK38 ) )
& ( $true
!= ( in @ sK37 @ ( powerset @ sK38 ) ) ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK37,sK38])],[f699,f700]) ).
thf(f700,plain,
( ? [X2: $i,X3: $i] :
( ( ( subset @ X2 @ X3 )
= $true )
& ( ( in @ X2 @ ( powerset @ X3 ) )
!= $true ) )
=> ( ( $true
= ( subset @ sK37 @ sK38 ) )
& ( $true
!= ( in @ sK37 @ ( powerset @ sK38 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f699,plain,
( ( ! [X0: $i,X1: $i] :
( ( $true
!= ( subset @ X0 @ X1 ) )
| ( ( in @ X0 @ ( powerset @ X1 ) )
= $true ) )
| ( powersetI1 != $true ) )
& ( ( powersetI1 = $true )
| ? [X2: $i,X3: $i] :
( ( ( subset @ X2 @ X3 )
= $true )
& ( ( in @ X2 @ ( powerset @ X3 ) )
!= $true ) ) ) ),
inference(rectify,[status(thm)],[f698]) ).
thf(f698,plain,
( ( ! [X0: $i,X1: $i] :
( ( $true
!= ( subset @ X0 @ X1 ) )
| ( ( in @ X0 @ ( powerset @ X1 ) )
= $true ) )
| ( powersetI1 != $true ) )
& ( ( powersetI1 = $true )
| ? [X0: $i,X1: $i] :
( ( $true
= ( subset @ X0 @ X1 ) )
& ( ( in @ X0 @ ( powerset @ X1 ) )
!= $true ) ) ) ),
inference(nnf_transformation,[status(thm)],[f616]) ).
thf(f616,plain,
( ! [X0: $i,X1: $i] :
( ( $true
!= ( subset @ X0 @ X1 ) )
| ( ( in @ X0 @ ( powerset @ X1 ) )
= $true ) )
<=> ( powersetI1 = $true ) ),
inference(ennf_transformation,[status(thm)],[f158]) ).
thf(f158,plain,
( ! [X0: $i,X1: $i] :
( ( $true
= ( subset @ X0 @ X1 ) )
=> ( ( in @ X0 @ ( powerset @ X1 ) )
= $true ) )
<=> ( powersetI1 = $true ) ),
inference(fool_elimination,[status(thm)],[f157]) ).
thf(f157,plain,
( powersetI1
= ( ! [X0: $i,X1: $i] :
( ( subset @ X0 @ X1 )
=> ( in @ X0 @ ( powerset @ X1 ) ) ) ) ),
inference(rectify,[status(thm)],[f95]) ).
thf(f95,axiom,
( powersetI1
= ( ! [X4: $i,X3: $i] :
( ( subset @ X4 @ X3 )
=> ( in @ X4 @ ( powerset @ X3 ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.yC96zsFX7x/DTF2THF_27586.p',powersetI1) ).
thf(f1568,plain,
( $true
!= ( in @ ( setadjoin @ ( setadjoin @ sK187 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK187 @ ( setadjoin @ sK188 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ sK186 @ sK185 ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f948]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : SEU628^1 : TPTP v9.2.1. Released v3.7.0.
% 0.11/0.12 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.16/0.33 % Computer : n008.cluster.edu
% 0.16/0.33 % Model : x86_64 x86_64
% 0.16/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.33 % Memory : 8042.1875MB
% 0.16/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.33 % CPULimit : 300
% 0.16/0.33 % WCLimit : 300
% 0.16/0.33 % DateTime : Tue Feb 24 12:28:54 EST 2026
% 0.16/0.34 % CPUTime :
% 0.16/0.34 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.28/0.45 ---- Original DTF file ---
% 0.28/0.45 thf(spec,logic,$$dhol).
% 0.28/0.45 %------------------------------------------------------------------------------
% 0.28/0.45 % File : SEU628^1 : TPTP v9.2.1. Released v3.7.0.
% 0.28/0.45 % Domain : Set Theory
% 0.28/0.45 % Problem : Ordered Pairs - Cartesian Products
% 0.28/0.45 % Version : Especial.
% 0.28/0.45 % English : (! A:i.! B:i.! x:i.in x A -> (! y:i.in y B -> in (setadjoin
% 0.28/0.45 % (setadjoin x emptyset) (setadjoin (setadjoin x
% 0.28/0.45 % (setadjoin y emptyset)) emptyset)) (powerset (powerset
% 0.28/0.45 % (binunion A B)))))
% 0.28/0.45
% 0.28/0.45 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.28/0.45 % Source : [Bro08]
% 0.28/0.45 % Names : ZFC130g [Bro08]
% 0.28/0.45
% 0.28/0.45 % Status : Theorem
% 0.28/0.45 % Rating : 0.22 v9.1.0, 0.25 v9.0.0, 0.20 v8.2.0, 0.31 v8.1.0, 0.18 v7.5.0, 0.14 v7.4.0, 0.33 v7.2.0, 0.25 v7.1.0, 0.38 v7.0.0, 0.29 v6.4.0, 0.33 v6.3.0, 0.40 v6.2.0, 0.29 v6.1.0, 0.43 v6.0.0, 0.57 v5.5.0, 0.50 v5.4.0, 0.60 v4.1.0, 1.00 v3.7.0
% 0.28/0.45 % Syntax : Number of formulae : 317 ( 149 unt; 167 typ; 149 def)
% 0.28/0.45 % Number of atoms : 788 ( 210 equ; 0 cnn)
% 0.28/0.45 % Maximal formula atoms : 147 ( 5 avg)
% 0.28/0.46 % Number of connectives : 1357 ( 47 ~; 7 |; 32 &; 911 @)
% 0.28/0.46 % ( 14 <=>; 346 =>; 0 <=; 0 <~>)
% 0.28/0.46 % Maximal formula depth : 159 ( 2 avg)
% 0.28/0.46 % Number of types : 2 ( 0 usr)
% 0.28/0.46 % Number of type conns : 69 ( 69 >; 0 *; 0 +; 0 <<)
% 0.28/0.46 % Number of symbols : 170 ( 167 usr; 148 con; 0-2 aty)
% 0.28/0.46 % Number of variables : 432 ( 36 ^; 363 !; 33 ?; 432 :)
% 0.28/0.46 % SPC : TH0_THM_EQU_NAR
% 0.28/0.46
% 0.28/0.46 % Comments : http://mathgate.info/detsetitem.php?id=185
% 0.28/0.46 %------------------------------------------------------------------------------
% 0.28/0.46 thf(in_type,type,
% 0.28/0.46 in: $i > $i > $o ).
% 0.28/0.46
% 0.28/0.46 thf(exu_type,type,
% 0.28/0.46 exu: ( $i > $o ) > $o ).
% 0.28/0.46
% 0.28/0.46 thf(exu,definition,
% 0.28/0.46 ( exu
% 0.28/0.46 = ( ^ [Xphi: $i > $o] :
% 0.28/0.46 ? [Xx: $i] :
% 0.28/0.46 ( ( Xphi @ Xx )
% 0.28/0.46 & ! [Xy: $i] :
% 0.28/0.46 ( ( Xphi @ Xy )
% 0.28/0.46 => ( Xx = Xy ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(setextAx_type,type,
% 0.28/0.46 setextAx: $o ).
% 0.28/0.46
% 0.28/0.46 thf(setextAx,definition,
% 0.28/0.46 ( setextAx
% 0.28/0.46 = ( ! [A: $i,B: $i] :
% 0.28/0.46 ( ! [Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ A )
% 0.28/0.46 <=> ( in @ Xx @ B ) )
% 0.28/0.46 => ( A = B ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(emptyset_type,type,
% 0.28/0.46 emptyset: $i ).
% 0.28/0.46
% 0.28/0.46 thf(emptysetAx_type,type,
% 0.28/0.46 emptysetAx: $o ).
% 0.28/0.46
% 0.28/0.46 thf(emptysetAx,definition,
% 0.28/0.46 ( emptysetAx
% 0.28/0.46 = ( ! [Xx: $i] :
% 0.28/0.46 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(setadjoin_type,type,
% 0.28/0.46 setadjoin: $i > $i > $i ).
% 0.28/0.46
% 0.28/0.46 thf(setadjoinAx_type,type,
% 0.28/0.46 setadjoinAx: $o ).
% 0.28/0.46
% 0.28/0.46 thf(setadjoinAx,definition,
% 0.28/0.46 ( setadjoinAx
% 0.28/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.28/0.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.28/0.46 <=> ( ( Xy = Xx )
% 0.28/0.46 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(powerset_type,type,
% 0.28/0.46 powerset: $i > $i ).
% 0.28/0.46
% 0.28/0.46 thf(powersetAx_type,type,
% 0.28/0.46 powersetAx: $o ).
% 0.28/0.46
% 0.28/0.46 thf(powersetAx,definition,
% 0.28/0.46 ( powersetAx
% 0.28/0.46 = ( ! [A: $i,B: $i] :
% 0.28/0.46 ( ( in @ B @ ( powerset @ A ) )
% 0.28/0.46 <=> ! [Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ B )
% 0.28/0.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(setunion_type,type,
% 0.28/0.46 setunion: $i > $i ).
% 0.28/0.46
% 0.28/0.46 thf(setunionAx_type,type,
% 0.28/0.46 setunionAx: $o ).
% 0.28/0.46
% 0.28/0.46 thf(setunionAx,definition,
% 0.28/0.46 ( setunionAx
% 0.28/0.46 = ( ! [A: $i,Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ ( setunion @ A ) )
% 0.28/0.46 <=> ? [B: $i] :
% 0.28/0.46 ( ( in @ Xx @ B )
% 0.28/0.46 & ( in @ B @ A ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(omega_type,type,
% 0.28/0.46 omega: $i ).
% 0.28/0.46
% 0.28/0.46 thf(omega0Ax_type,type,
% 0.28/0.46 omega0Ax: $o ).
% 0.28/0.46
% 0.28/0.46 thf(omega0Ax,definition,
% 0.28/0.46 ( omega0Ax
% 0.28/0.46 = ( in @ emptyset @ omega ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(omegaSAx_type,type,
% 0.28/0.46 omegaSAx: $o ).
% 0.28/0.46
% 0.28/0.46 thf(omegaSAx,definition,
% 0.28/0.46 ( omegaSAx
% 0.28/0.46 = ( ! [Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ omega )
% 0.28/0.46 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(omegaIndAx_type,type,
% 0.28/0.46 omegaIndAx: $o ).
% 0.28/0.46
% 0.28/0.46 thf(omegaIndAx,definition,
% 0.28/0.46 ( omegaIndAx
% 0.28/0.46 = ( ! [A: $i] :
% 0.28/0.46 ( ( ( in @ emptyset @ A )
% 0.28/0.46 & ! [Xx: $i] :
% 0.28/0.46 ( ( ( in @ Xx @ omega )
% 0.28/0.46 & ( in @ Xx @ A ) )
% 0.28/0.46 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.28/0.46 => ! [Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ omega )
% 0.28/0.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(replAx_type,type,
% 0.28/0.46 replAx: $o ).
% 0.28/0.46
% 0.28/0.46 thf(replAx,definition,
% 0.28/0.46 ( replAx
% 0.28/0.46 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.28/0.46 ( ! [Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ A )
% 0.28/0.46 => ( exu
% 0.28/0.46 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.28/0.46 => ? [B: $i] :
% 0.28/0.46 ! [Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ B )
% 0.28/0.46 <=> ? [Xy: $i] :
% 0.28/0.46 ( ( in @ Xy @ A )
% 0.28/0.46 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(foundationAx_type,type,
% 0.28/0.46 foundationAx: $o ).
% 0.28/0.46
% 0.28/0.46 thf(foundationAx,definition,
% 0.28/0.46 ( foundationAx
% 0.28/0.46 = ( ! [A: $i] :
% 0.28/0.46 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.28/0.46 => ? [B: $i] :
% 0.28/0.46 ( ( in @ B @ A )
% 0.28/0.46 & ~ ? [Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ B )
% 0.28/0.46 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(wellorderingAx_type,type,
% 0.28/0.46 wellorderingAx: $o ).
% 0.28/0.46
% 0.28/0.46 thf(wellorderingAx,definition,
% 0.28/0.46 ( wellorderingAx
% 0.28/0.46 = ( ! [A: $i] :
% 0.28/0.46 ? [B: $i] :
% 0.28/0.46 ( ! [C: $i] :
% 0.28/0.46 ( ( in @ C @ B )
% 0.28/0.46 => ! [Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ C )
% 0.28/0.46 => ( in @ Xx @ A ) ) )
% 0.28/0.46 & ! [Xx: $i,Xy: $i] :
% 0.28/0.46 ( ( ( in @ Xx @ A )
% 0.28/0.46 & ( in @ Xy @ A ) )
% 0.28/0.46 => ( ! [C: $i] :
% 0.28/0.46 ( ( in @ C @ B )
% 0.28/0.46 => ( ( in @ Xx @ C )
% 0.28/0.46 <=> ( in @ Xy @ C ) ) )
% 0.28/0.46 => ( Xx = Xy ) ) )
% 0.28/0.46 & ! [C: $i,D: $i] :
% 0.28/0.46 ( ( ( in @ C @ B )
% 0.28/0.46 & ( in @ D @ B ) )
% 0.28/0.46 => ( ! [Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ C )
% 0.28/0.46 => ( in @ Xx @ D ) )
% 0.28/0.46 | ! [Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ D )
% 0.28/0.46 => ( in @ Xx @ C ) ) ) )
% 0.28/0.46 & ! [C: $i] :
% 0.28/0.46 ( ( ! [Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ C )
% 0.28/0.46 => ( in @ Xx @ A ) )
% 0.28/0.46 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.28/0.46 => ? [D: $i,Xx: $i] :
% 0.28/0.46 ( ( in @ D @ B )
% 0.28/0.46 & ( in @ Xx @ C )
% 0.28/0.46 & ~ ? [Xy: $i] :
% 0.28/0.46 ( ( in @ Xy @ D )
% 0.28/0.46 & ( in @ Xy @ C ) )
% 0.28/0.46 & ! [E: $i] :
% 0.28/0.46 ( ( in @ E @ B )
% 0.28/0.46 => ( ! [Xy: $i] :
% 0.28/0.46 ( ( in @ Xy @ E )
% 0.28/0.46 => ( in @ Xy @ D ) )
% 0.28/0.46 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(descr_type,type,
% 0.28/0.46 descr: ( $i > $o ) > $i ).
% 0.28/0.46
% 0.28/0.46 thf(descrp_type,type,
% 0.28/0.46 descrp: $o ).
% 0.28/0.46
% 0.28/0.46 thf(descrp,definition,
% 0.28/0.46 ( descrp
% 0.28/0.46 = ( ! [Xphi: $i > $o] :
% 0.28/0.46 ( ( exu
% 0.28/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.28/0.46 => ( Xphi
% 0.28/0.46 @ ( descr
% 0.28/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(dsetconstr_type,type,
% 0.28/0.46 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.28/0.46
% 0.28/0.46 thf(dsetconstrI_type,type,
% 0.28/0.46 dsetconstrI: $o ).
% 0.28/0.46
% 0.28/0.46 thf(dsetconstrI,definition,
% 0.28/0.46 ( dsetconstrI
% 0.28/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ A )
% 0.28/0.46 => ( ( Xphi @ Xx )
% 0.28/0.46 => ( in @ Xx
% 0.28/0.46 @ ( dsetconstr @ A
% 0.28/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(dsetconstrEL_type,type,
% 0.28/0.46 dsetconstrEL: $o ).
% 0.28/0.46
% 0.28/0.46 thf(dsetconstrEL,definition,
% 0.28/0.46 ( dsetconstrEL
% 0.28/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.28/0.46 ( ( in @ Xx
% 0.28/0.46 @ ( dsetconstr @ A
% 0.28/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.28/0.46 => ( in @ Xx @ A ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(dsetconstrER_type,type,
% 0.28/0.46 dsetconstrER: $o ).
% 0.28/0.46
% 0.28/0.46 thf(dsetconstrER,definition,
% 0.28/0.46 ( dsetconstrER
% 0.28/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.28/0.46 ( ( in @ Xx
% 0.28/0.46 @ ( dsetconstr @ A
% 0.28/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.28/0.46 => ( Xphi @ Xx ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(exuE1_type,type,
% 0.28/0.46 exuE1: $o ).
% 0.28/0.46
% 0.28/0.46 thf(exuE1,definition,
% 0.28/0.46 ( exuE1
% 0.28/0.46 = ( ! [Xphi: $i > $o] :
% 0.28/0.46 ( ( exu
% 0.28/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.28/0.46 => ? [Xx: $i] :
% 0.28/0.46 ( ( Xphi @ Xx )
% 0.28/0.46 & ! [Xy: $i] :
% 0.28/0.46 ( ( Xphi @ Xy )
% 0.28/0.46 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(prop2set_type,type,
% 0.28/0.46 prop2set: $o > $i ).
% 0.28/0.46
% 0.28/0.46 thf(prop2setE_type,type,
% 0.28/0.46 prop2setE: $o ).
% 0.28/0.46
% 0.28/0.46 thf(prop2setE,definition,
% 0.28/0.46 ( prop2setE
% 0.28/0.46 = ( ! [Xphi: $o,Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.28/0.46 => Xphi ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(emptysetE_type,type,
% 0.28/0.46 emptysetE: $o ).
% 0.28/0.46
% 0.28/0.46 thf(emptysetE,definition,
% 0.28/0.46 ( emptysetE
% 0.28/0.46 = ( ! [Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ emptyset )
% 0.28/0.46 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(emptysetimpfalse_type,type,
% 0.28/0.46 emptysetimpfalse: $o ).
% 0.28/0.46
% 0.28/0.46 thf(emptysetimpfalse,definition,
% 0.28/0.46 ( emptysetimpfalse
% 0.28/0.46 = ( ! [Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ emptyset )
% 0.28/0.46 => $false ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(notinemptyset_type,type,
% 0.28/0.46 notinemptyset: $o ).
% 0.28/0.46
% 0.28/0.46 thf(notinemptyset,definition,
% 0.28/0.46 ( notinemptyset
% 0.28/0.46 = ( ! [Xx: $i] :
% 0.28/0.46 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(exuE3e_type,type,
% 0.28/0.46 exuE3e: $o ).
% 0.28/0.46
% 0.28/0.46 thf(exuE3e,definition,
% 0.28/0.46 ( exuE3e
% 0.28/0.46 = ( ! [Xphi: $i > $o] :
% 0.28/0.46 ( ( exu
% 0.28/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.28/0.46 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(setext_type,type,
% 0.28/0.46 setext: $o ).
% 0.28/0.46
% 0.28/0.46 thf(setext,definition,
% 0.28/0.46 ( setext
% 0.28/0.46 = ( ! [A: $i,B: $i] :
% 0.28/0.46 ( ! [Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ A )
% 0.28/0.46 => ( in @ Xx @ B ) )
% 0.28/0.46 => ( ! [Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ B )
% 0.28/0.46 => ( in @ Xx @ A ) )
% 0.28/0.46 => ( A = B ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(emptyI_type,type,
% 0.28/0.46 emptyI: $o ).
% 0.28/0.46
% 0.28/0.46 thf(emptyI,definition,
% 0.28/0.46 ( emptyI
% 0.28/0.46 = ( ! [A: $i] :
% 0.28/0.46 ( ! [Xx: $i] :
% 0.28/0.46 ~ ( in @ Xx @ A )
% 0.28/0.46 => ( A = emptyset ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(noeltsimpempty_type,type,
% 0.28/0.46 noeltsimpempty: $o ).
% 0.28/0.46
% 0.28/0.46 thf(noeltsimpempty,definition,
% 0.28/0.46 ( noeltsimpempty
% 0.28/0.46 = ( ! [A: $i] :
% 0.28/0.46 ( ! [Xx: $i] :
% 0.28/0.46 ~ ( in @ Xx @ A )
% 0.28/0.46 => ( A = emptyset ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(setbeta_type,type,
% 0.28/0.46 setbeta: $o ).
% 0.28/0.46
% 0.28/0.46 thf(setbeta,definition,
% 0.28/0.46 ( setbeta
% 0.28/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ A )
% 0.28/0.46 => ( ( in @ Xx
% 0.28/0.46 @ ( dsetconstr @ A
% 0.28/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.28/0.46 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(nonempty_type,type,
% 0.28/0.46 nonempty: $i > $o ).
% 0.28/0.46
% 0.28/0.46 thf(nonempty,definition,
% 0.28/0.46 ( nonempty
% 0.28/0.46 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(nonemptyE1_type,type,
% 0.28/0.46 nonemptyE1: $o ).
% 0.28/0.46
% 0.28/0.46 thf(nonemptyE1,definition,
% 0.28/0.46 ( nonemptyE1
% 0.28/0.46 = ( ! [A: $i] :
% 0.28/0.46 ( ( nonempty @ A )
% 0.28/0.46 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(nonemptyI_type,type,
% 0.28/0.46 nonemptyI: $o ).
% 0.28/0.46
% 0.28/0.46 thf(nonemptyI,definition,
% 0.28/0.46 ( nonemptyI
% 0.28/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ A )
% 0.28/0.46 => ( ( Xphi @ Xx )
% 0.28/0.46 => ( nonempty
% 0.28/0.46 @ ( dsetconstr @ A
% 0.28/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(nonemptyI1_type,type,
% 0.28/0.46 nonemptyI1: $o ).
% 0.28/0.46
% 0.28/0.46 thf(nonemptyI1,definition,
% 0.28/0.46 ( nonemptyI1
% 0.28/0.46 = ( ! [A: $i] :
% 0.28/0.46 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.28/0.46 => ( nonempty @ A ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(setadjoinIL_type,type,
% 0.28/0.46 setadjoinIL: $o ).
% 0.28/0.46
% 0.28/0.46 thf(setadjoinIL,definition,
% 0.28/0.46 ( setadjoinIL
% 0.28/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(emptyinunitempty_type,type,
% 0.28/0.46 emptyinunitempty: $o ).
% 0.28/0.46
% 0.28/0.46 thf(emptyinunitempty,definition,
% 0.28/0.46 ( emptyinunitempty
% 0.28/0.46 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(setadjoinIR_type,type,
% 0.28/0.46 setadjoinIR: $o ).
% 0.28/0.46
% 0.28/0.46 thf(setadjoinIR,definition,
% 0.28/0.46 ( setadjoinIR
% 0.28/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.28/0.46 ( ( in @ Xy @ A )
% 0.28/0.46 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(setadjoinE_type,type,
% 0.28/0.46 setadjoinE: $o ).
% 0.28/0.46
% 0.28/0.46 thf(setadjoinE,definition,
% 0.28/0.46 ( setadjoinE
% 0.28/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.28/0.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.28/0.46 => ! [Xphi: $o] :
% 0.28/0.46 ( ( ( Xy = Xx )
% 0.28/0.46 => Xphi )
% 0.28/0.46 => ( ( ( in @ Xy @ A )
% 0.28/0.46 => Xphi )
% 0.28/0.46 => Xphi ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(setadjoinOr_type,type,
% 0.28/0.46 setadjoinOr: $o ).
% 0.28/0.46
% 0.28/0.46 thf(setadjoinOr,definition,
% 0.28/0.46 ( setadjoinOr
% 0.28/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.28/0.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.28/0.46 => ( ( Xy = Xx )
% 0.28/0.46 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(setoftrueEq_type,type,
% 0.28/0.46 setoftrueEq: $o ).
% 0.28/0.46
% 0.28/0.46 thf(setoftrueEq,definition,
% 0.28/0.46 ( setoftrueEq
% 0.28/0.46 = ( ! [A: $i] :
% 0.28/0.46 ( ( dsetconstr @ A
% 0.28/0.46 @ ^ [Xx: $i] : $true )
% 0.28/0.46 = A ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(powersetI_type,type,
% 0.28/0.46 powersetI: $o ).
% 0.28/0.46
% 0.28/0.46 thf(powersetI,definition,
% 0.28/0.46 ( powersetI
% 0.28/0.46 = ( ! [A: $i,B: $i] :
% 0.28/0.46 ( ! [Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ B )
% 0.28/0.46 => ( in @ Xx @ A ) )
% 0.28/0.46 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(emptyinPowerset_type,type,
% 0.28/0.46 emptyinPowerset: $o ).
% 0.28/0.46
% 0.28/0.46 thf(emptyinPowerset,definition,
% 0.28/0.46 ( emptyinPowerset
% 0.28/0.46 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(emptyInPowerset_type,type,
% 0.28/0.46 emptyInPowerset: $o ).
% 0.28/0.46
% 0.28/0.46 thf(emptyInPowerset,definition,
% 0.28/0.46 ( emptyInPowerset
% 0.28/0.46 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(powersetE_type,type,
% 0.28/0.46 powersetE: $o ).
% 0.28/0.46
% 0.28/0.46 thf(powersetE,definition,
% 0.28/0.46 ( powersetE
% 0.28/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.46 ( ( in @ B @ ( powerset @ A ) )
% 0.28/0.46 => ( ( in @ Xx @ B )
% 0.28/0.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(setunionI_type,type,
% 0.28/0.46 setunionI: $o ).
% 0.28/0.46
% 0.28/0.46 thf(setunionI,definition,
% 0.28/0.46 ( setunionI
% 0.28/0.46 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.28/0.46 ( ( in @ Xx @ B )
% 0.28/0.46 => ( ( in @ B @ A )
% 0.28/0.46 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(setunionE_type,type,
% 0.28/0.46 setunionE: $o ).
% 0.28/0.46
% 0.28/0.46 thf(setunionE,definition,
% 0.28/0.46 ( setunionE
% 0.28/0.46 = ( ! [A: $i,Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ ( setunion @ A ) )
% 0.28/0.46 => ! [Xphi: $o] :
% 0.28/0.46 ( ! [B: $i] :
% 0.28/0.46 ( ( in @ Xx @ B )
% 0.28/0.46 => ( ( in @ B @ A )
% 0.28/0.46 => Xphi ) )
% 0.28/0.46 => Xphi ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(subPowSU_type,type,
% 0.28/0.46 subPowSU: $o ).
% 0.28/0.46
% 0.28/0.46 thf(subPowSU,definition,
% 0.28/0.46 ( subPowSU
% 0.28/0.46 = ( ! [A: $i,Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ A )
% 0.28/0.46 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(exuE2_type,type,
% 0.28/0.46 exuE2: $o ).
% 0.28/0.46
% 0.28/0.46 thf(exuE2,definition,
% 0.28/0.46 ( exuE2
% 0.28/0.46 = ( ! [Xphi: $i > $o] :
% 0.28/0.46 ( ( exu
% 0.28/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.28/0.46 => ? [Xx: $i] :
% 0.28/0.46 ! [Xy: $i] :
% 0.28/0.46 ( ( Xphi @ Xy )
% 0.28/0.46 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(nonemptyImpWitness_type,type,
% 0.28/0.46 nonemptyImpWitness: $o ).
% 0.28/0.46
% 0.28/0.46 thf(nonemptyImpWitness,definition,
% 0.28/0.46 ( nonemptyImpWitness
% 0.28/0.46 = ( ! [A: $i] :
% 0.28/0.46 ( ( nonempty @ A )
% 0.28/0.46 => ? [Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ A )
% 0.28/0.46 & $true ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(uniqinunit_type,type,
% 0.28/0.46 uniqinunit: $o ).
% 0.28/0.46
% 0.28/0.46 thf(uniqinunit,definition,
% 0.28/0.46 ( uniqinunit
% 0.28/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.28/0.46 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.28/0.46 => ( Xx = Xy ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(notinsingleton_type,type,
% 0.28/0.46 notinsingleton: $o ).
% 0.28/0.46
% 0.28/0.46 thf(notinsingleton,definition,
% 0.28/0.46 ( notinsingleton
% 0.28/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.28/0.46 ( ( Xx != Xy )
% 0.28/0.46 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(eqinunit_type,type,
% 0.28/0.46 eqinunit: $o ).
% 0.28/0.46
% 0.28/0.46 thf(eqinunit,definition,
% 0.28/0.46 ( eqinunit
% 0.28/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.28/0.46 ( ( Xx = Xy )
% 0.28/0.46 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(singletonsswitch_type,type,
% 0.28/0.46 singletonsswitch: $o ).
% 0.28/0.46
% 0.28/0.46 thf(singletonsswitch,definition,
% 0.28/0.46 ( singletonsswitch
% 0.28/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.28/0.46 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.28/0.46 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(upairsetE_type,type,
% 0.28/0.46 upairsetE: $o ).
% 0.28/0.46
% 0.28/0.46 thf(upairsetE,definition,
% 0.28/0.46 ( upairsetE
% 0.28/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.28/0.46 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.28/0.46 => ( ( Xz = Xx )
% 0.28/0.46 | ( Xz = Xy ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(upairsetIL_type,type,
% 0.28/0.46 upairsetIL: $o ).
% 0.28/0.46
% 0.28/0.46 thf(upairsetIL,definition,
% 0.28/0.46 ( upairsetIL
% 0.28/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(upairsetIR_type,type,
% 0.28/0.46 upairsetIR: $o ).
% 0.28/0.46
% 0.28/0.46 thf(upairsetIR,definition,
% 0.28/0.46 ( upairsetIR
% 0.28/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(emptyE1_type,type,
% 0.28/0.46 emptyE1: $o ).
% 0.28/0.46
% 0.28/0.46 thf(emptyE1,definition,
% 0.28/0.46 ( emptyE1
% 0.28/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.28/0.46 ( ? [Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ A )
% 0.28/0.46 & ( Xphi @ Xx ) )
% 0.28/0.46 => ( ( ( dsetconstr @ A
% 0.28/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.28/0.46 = emptyset )
% 0.28/0.46 => $false ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(vacuousDall_type,type,
% 0.28/0.46 vacuousDall: $o ).
% 0.28/0.46
% 0.28/0.46 thf(vacuousDall,definition,
% 0.28/0.46 ( vacuousDall
% 0.28/0.46 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ emptyset )
% 0.28/0.46 => ( Xphi @ Xx ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(quantDeMorgan1_type,type,
% 0.28/0.46 quantDeMorgan1: $o ).
% 0.28/0.46
% 0.28/0.46 thf(quantDeMorgan1,definition,
% 0.28/0.46 ( quantDeMorgan1
% 0.28/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.28/0.46 ( ~ ! [Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ A )
% 0.28/0.46 => ( Xphi @ Xx ) )
% 0.28/0.46 => ? [Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ A )
% 0.28/0.46 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(quantDeMorgan2_type,type,
% 0.28/0.46 quantDeMorgan2: $o ).
% 0.28/0.46
% 0.28/0.46 thf(quantDeMorgan2,definition,
% 0.28/0.46 ( quantDeMorgan2
% 0.28/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.28/0.46 ( ! [Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ A )
% 0.28/0.46 => ~ ( Xphi @ Xx ) )
% 0.28/0.46 => ~ ? [Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ A )
% 0.28/0.46 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(quantDeMorgan3_type,type,
% 0.28/0.46 quantDeMorgan3: $o ).
% 0.28/0.46
% 0.28/0.46 thf(quantDeMorgan3,definition,
% 0.28/0.46 ( quantDeMorgan3
% 0.28/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.28/0.46 ( ~ ? [Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ A )
% 0.28/0.46 & ( Xphi @ Xx ) )
% 0.28/0.46 => ! [Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ A )
% 0.28/0.46 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(quantDeMorgan4_type,type,
% 0.28/0.46 quantDeMorgan4: $o ).
% 0.28/0.46
% 0.28/0.46 thf(quantDeMorgan4,definition,
% 0.28/0.46 ( quantDeMorgan4
% 0.28/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.28/0.46 ( ? [Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ A )
% 0.28/0.46 & ~ ( Xphi @ Xx ) )
% 0.28/0.46 => ~ ! [Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ A )
% 0.28/0.46 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(prop2setI_type,type,
% 0.28/0.46 prop2setI: $o ).
% 0.28/0.46
% 0.28/0.46 thf(prop2setI,definition,
% 0.28/0.46 ( prop2setI
% 0.28/0.46 = ( ! [Xphi: $o] :
% 0.28/0.46 ( Xphi
% 0.28/0.46 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(set2prop_type,type,
% 0.28/0.46 set2prop: $i > $o ).
% 0.28/0.46
% 0.28/0.46 thf(prop2set2propI_type,type,
% 0.28/0.46 prop2set2propI: $o ).
% 0.28/0.46
% 0.28/0.46 thf(prop2set2propI,definition,
% 0.28/0.46 ( prop2set2propI
% 0.28/0.46 = ( ! [Xphi: $o] :
% 0.28/0.46 ( Xphi
% 0.28/0.46 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(notdexE_type,type,
% 0.28/0.46 notdexE: $o ).
% 0.28/0.46
% 0.28/0.46 thf(notdexE,definition,
% 0.28/0.46 ( notdexE
% 0.28/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.28/0.46 ( ~ ? [Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ A )
% 0.28/0.46 & ( Xphi @ Xx ) )
% 0.28/0.46 => ! [Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ A )
% 0.28/0.46 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(notdallE_type,type,
% 0.28/0.46 notdallE: $o ).
% 0.28/0.46
% 0.28/0.46 thf(notdallE,definition,
% 0.28/0.46 ( notdallE
% 0.28/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.28/0.46 ( ~ ! [Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ A )
% 0.28/0.46 => ( Xphi @ Xx ) )
% 0.28/0.46 => ? [Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ A )
% 0.28/0.46 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(exuI1_type,type,
% 0.28/0.46 exuI1: $o ).
% 0.28/0.46
% 0.28/0.46 thf(exuI1,definition,
% 0.28/0.46 ( exuI1
% 0.28/0.46 = ( ! [Xphi: $i > $o] :
% 0.28/0.46 ( ? [Xx: $i] :
% 0.28/0.46 ( ( Xphi @ Xx )
% 0.28/0.46 & ! [Xy: $i] :
% 0.28/0.46 ( ( Xphi @ Xy )
% 0.28/0.46 => ( Xx = Xy ) ) )
% 0.28/0.46 => ( exu
% 0.28/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(exuI3_type,type,
% 0.28/0.46 exuI3: $o ).
% 0.28/0.46
% 0.28/0.46 thf(exuI3,definition,
% 0.28/0.46 ( exuI3
% 0.28/0.46 = ( ! [Xphi: $i > $o] :
% 0.28/0.46 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.28/0.46 => ( ! [Xx: $i,Xy: $i] :
% 0.28/0.46 ( ( Xphi @ Xx )
% 0.28/0.46 => ( ( Xphi @ Xy )
% 0.28/0.46 => ( Xx = Xy ) ) )
% 0.28/0.46 => ( exu
% 0.28/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(exuI2_type,type,
% 0.28/0.46 exuI2: $o ).
% 0.28/0.46
% 0.28/0.46 thf(exuI2,definition,
% 0.28/0.46 ( exuI2
% 0.28/0.46 = ( ! [Xphi: $i > $o] :
% 0.28/0.46 ( ? [Xx: $i] :
% 0.28/0.46 ! [Xy: $i] :
% 0.28/0.46 ( ( Xphi @ Xy )
% 0.28/0.46 <=> ( Xy = Xx ) )
% 0.28/0.46 => ( exu
% 0.28/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(inCongP_type,type,
% 0.28/0.46 inCongP: $o ).
% 0.28/0.46
% 0.28/0.46 thf(inCongP,definition,
% 0.28/0.46 ( inCongP
% 0.28/0.46 = ( ! [A: $i,B: $i] :
% 0.28/0.46 ( ( A = B )
% 0.28/0.46 => ! [Xx: $i,Xy: $i] :
% 0.28/0.46 ( ( Xx = Xy )
% 0.28/0.46 => ( ( in @ Xx @ A )
% 0.28/0.46 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(in__Cong_type,type,
% 0.28/0.46 in__Cong: $o ).
% 0.28/0.46
% 0.28/0.46 thf(in__Cong,definition,
% 0.28/0.46 ( in__Cong
% 0.28/0.46 = ( ! [A: $i,B: $i] :
% 0.28/0.46 ( ( A = B )
% 0.28/0.46 => ! [Xx: $i,Xy: $i] :
% 0.28/0.46 ( ( Xx = Xy )
% 0.28/0.46 => ( ( in @ Xx @ A )
% 0.28/0.46 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(exuE3u_type,type,
% 0.28/0.46 exuE3u: $o ).
% 0.28/0.46
% 0.28/0.46 thf(exuE3u,definition,
% 0.28/0.46 ( exuE3u
% 0.28/0.46 = ( ! [Xphi: $i > $o] :
% 0.28/0.46 ( ( exu
% 0.28/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.28/0.46 => ! [Xx: $i,Xy: $i] :
% 0.28/0.46 ( ( Xphi @ Xx )
% 0.28/0.46 => ( ( Xphi @ Xy )
% 0.28/0.46 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(exu__Cong_type,type,
% 0.28/0.46 exu__Cong: $o ).
% 0.28/0.46
% 0.28/0.46 thf(exu__Cong,definition,
% 0.28/0.46 ( exu__Cong
% 0.28/0.46 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.28/0.46 ( ! [Xx: $i,Xy: $i] :
% 0.28/0.46 ( ( Xx = Xy )
% 0.28/0.46 => ( ( Xphi @ Xx )
% 0.28/0.46 <=> ( Xpsi @ Xy ) ) )
% 0.28/0.46 => ( ( exu
% 0.28/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.28/0.46 <=> ( exu
% 0.28/0.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(emptyset__Cong_type,type,
% 0.28/0.46 emptyset__Cong: $o ).
% 0.28/0.46
% 0.28/0.46 thf(emptyset__Cong,definition,
% 0.28/0.46 ( emptyset__Cong
% 0.28/0.46 = ( emptyset = emptyset ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(setadjoin__Cong_type,type,
% 0.28/0.46 setadjoin__Cong: $o ).
% 0.28/0.46
% 0.28/0.46 thf(setadjoin__Cong,definition,
% 0.28/0.46 ( setadjoin__Cong
% 0.28/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.28/0.46 ( ( Xx = Xy )
% 0.28/0.46 => ! [Xz: $i,Xu: $i] :
% 0.28/0.46 ( ( Xz = Xu )
% 0.28/0.46 => ( ( setadjoin @ Xx @ Xz )
% 0.28/0.46 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(powerset__Cong_type,type,
% 0.28/0.46 powerset__Cong: $o ).
% 0.28/0.46
% 0.28/0.46 thf(powerset__Cong,definition,
% 0.28/0.46 ( powerset__Cong
% 0.28/0.46 = ( ! [A: $i,B: $i] :
% 0.28/0.46 ( ( A = B )
% 0.28/0.46 => ( ( powerset @ A )
% 0.28/0.46 = ( powerset @ B ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(setunion__Cong_type,type,
% 0.28/0.46 setunion__Cong: $o ).
% 0.28/0.46
% 0.28/0.46 thf(setunion__Cong,definition,
% 0.28/0.46 ( setunion__Cong
% 0.28/0.46 = ( ! [A: $i,B: $i] :
% 0.28/0.46 ( ( A = B )
% 0.28/0.46 => ( ( setunion @ A )
% 0.28/0.46 = ( setunion @ B ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(omega__Cong_type,type,
% 0.28/0.46 omega__Cong: $o ).
% 0.28/0.46
% 0.28/0.46 thf(omega__Cong,definition,
% 0.28/0.46 ( omega__Cong
% 0.28/0.46 = ( omega = omega ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(exuEu_type,type,
% 0.28/0.46 exuEu: $o ).
% 0.28/0.46
% 0.28/0.46 thf(exuEu,definition,
% 0.28/0.46 ( exuEu
% 0.28/0.46 = ( ! [Xphi: $i > $o] :
% 0.28/0.46 ( ( exu
% 0.28/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.28/0.46 => ! [Xx: $i,Xy: $i] :
% 0.28/0.46 ( ( Xphi @ Xx )
% 0.28/0.46 => ( ( Xphi @ Xy )
% 0.28/0.46 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(descr__Cong_type,type,
% 0.28/0.46 descr__Cong: $o ).
% 0.28/0.46
% 0.28/0.46 thf(descr__Cong,definition,
% 0.28/0.46 ( descr__Cong
% 0.28/0.46 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.28/0.46 ( ! [Xx: $i,Xy: $i] :
% 0.28/0.46 ( ( Xx = Xy )
% 0.28/0.46 => ( ( Xphi @ Xx )
% 0.28/0.46 <=> ( Xpsi @ Xy ) ) )
% 0.28/0.46 => ( ( exu
% 0.28/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.28/0.46 => ( ( exu
% 0.28/0.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.28/0.46 => ( ( descr
% 0.28/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.28/0.46 = ( descr
% 0.28/0.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(dsetconstr__Cong_type,type,
% 0.28/0.46 dsetconstr__Cong: $o ).
% 0.28/0.46
% 0.28/0.46 thf(dsetconstr__Cong,definition,
% 0.28/0.46 ( dsetconstr__Cong
% 0.28/0.46 = ( ! [A: $i,B: $i] :
% 0.28/0.46 ( ( A = B )
% 0.28/0.46 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.28/0.46 ( ! [Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ A )
% 0.28/0.46 => ! [Xy: $i] :
% 0.28/0.46 ( ( in @ Xy @ B )
% 0.28/0.46 => ( ( Xx = Xy )
% 0.28/0.46 => ( ( Xphi @ Xx )
% 0.28/0.46 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.28/0.46 => ( ( dsetconstr @ A
% 0.28/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.28/0.46 = ( dsetconstr @ B
% 0.28/0.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(subset_type,type,
% 0.28/0.46 subset: $i > $i > $o ).
% 0.28/0.46
% 0.28/0.46 thf(disjoint_type,type,
% 0.28/0.46 disjoint: $i > $i > $o ).
% 0.28/0.46
% 0.28/0.46 thf(setsmeet_type,type,
% 0.28/0.46 setsmeet: $i > $i > $o ).
% 0.28/0.46
% 0.28/0.46 thf(subsetI1_type,type,
% 0.28/0.46 subsetI1: $o ).
% 0.28/0.46
% 0.28/0.46 thf(subsetI1,definition,
% 0.28/0.46 ( subsetI1
% 0.28/0.46 = ( ! [A: $i,B: $i] :
% 0.28/0.46 ( ! [Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ A )
% 0.28/0.46 => ( in @ Xx @ B ) )
% 0.28/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(eqimpsubset2_type,type,
% 0.28/0.46 eqimpsubset2: $o ).
% 0.28/0.46
% 0.28/0.46 thf(eqimpsubset2,definition,
% 0.28/0.46 ( eqimpsubset2
% 0.28/0.46 = ( ! [A: $i,B: $i] :
% 0.28/0.46 ( ( A = B )
% 0.28/0.46 => ( subset @ B @ A ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(eqimpsubset1_type,type,
% 0.28/0.46 eqimpsubset1: $o ).
% 0.28/0.46
% 0.28/0.46 thf(eqimpsubset1,definition,
% 0.28/0.46 ( eqimpsubset1
% 0.28/0.46 = ( ! [A: $i,B: $i] :
% 0.28/0.46 ( ( A = B )
% 0.28/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(subsetI2_type,type,
% 0.28/0.46 subsetI2: $o ).
% 0.28/0.46
% 0.28/0.46 thf(subsetI2,definition,
% 0.28/0.46 ( subsetI2
% 0.28/0.46 = ( ! [A: $i,B: $i] :
% 0.28/0.46 ( ! [Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ A )
% 0.28/0.46 => ( in @ Xx @ B ) )
% 0.28/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(emptysetsubset_type,type,
% 0.28/0.46 emptysetsubset: $o ).
% 0.28/0.46
% 0.28/0.46 thf(emptysetsubset,definition,
% 0.28/0.46 ( emptysetsubset
% 0.28/0.46 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(subsetE_type,type,
% 0.28/0.46 subsetE: $o ).
% 0.28/0.46
% 0.28/0.46 thf(subsetE,definition,
% 0.28/0.46 ( subsetE
% 0.28/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.46 ( ( subset @ A @ B )
% 0.28/0.46 => ( ( in @ Xx @ A )
% 0.28/0.46 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(subsetE2_type,type,
% 0.28/0.46 subsetE2: $o ).
% 0.28/0.46
% 0.28/0.46 thf(subsetE2,definition,
% 0.28/0.46 ( subsetE2
% 0.28/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.46 ( ( subset @ A @ B )
% 0.28/0.46 => ( ~ ( in @ Xx @ B )
% 0.28/0.46 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(notsubsetI_type,type,
% 0.28/0.46 notsubsetI: $o ).
% 0.28/0.46
% 0.28/0.46 thf(notsubsetI,definition,
% 0.28/0.46 ( notsubsetI
% 0.28/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ A )
% 0.28/0.46 => ( ~ ( in @ Xx @ B )
% 0.28/0.46 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(notequalI1_type,type,
% 0.28/0.46 notequalI1: $o ).
% 0.28/0.46
% 0.28/0.46 thf(notequalI1,definition,
% 0.28/0.46 ( notequalI1
% 0.28/0.46 = ( ! [A: $i,B: $i] :
% 0.28/0.46 ( ~ ( subset @ A @ B )
% 0.28/0.46 => ( A != B ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(notequalI2_type,type,
% 0.28/0.46 notequalI2: $o ).
% 0.28/0.46
% 0.28/0.46 thf(notequalI2,definition,
% 0.28/0.46 ( notequalI2
% 0.28/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ A )
% 0.28/0.46 => ( ~ ( in @ Xx @ B )
% 0.28/0.46 => ( A != B ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(subsetRefl_type,type,
% 0.28/0.46 subsetRefl: $o ).
% 0.28/0.46
% 0.28/0.46 thf(subsetRefl,definition,
% 0.28/0.46 ( subsetRefl
% 0.28/0.46 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(subsetTrans_type,type,
% 0.28/0.46 subsetTrans: $o ).
% 0.28/0.46
% 0.28/0.46 thf(subsetTrans,definition,
% 0.28/0.46 ( subsetTrans
% 0.28/0.46 = ( ! [A: $i,B: $i,C: $i] :
% 0.28/0.46 ( ( subset @ A @ B )
% 0.28/0.46 => ( ( subset @ B @ C )
% 0.28/0.46 => ( subset @ A @ C ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(setadjoinSub_type,type,
% 0.28/0.46 setadjoinSub: $o ).
% 0.28/0.46
% 0.28/0.46 thf(setadjoinSub,definition,
% 0.28/0.46 ( setadjoinSub
% 0.28/0.46 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(setadjoinSub2_type,type,
% 0.28/0.46 setadjoinSub2: $o ).
% 0.28/0.46
% 0.28/0.46 thf(setadjoinSub2,definition,
% 0.28/0.46 ( setadjoinSub2
% 0.28/0.46 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.28/0.46 ( ( subset @ A @ B )
% 0.28/0.46 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(subset2powerset_type,type,
% 0.28/0.46 subset2powerset: $o ).
% 0.28/0.46
% 0.28/0.46 thf(subset2powerset,definition,
% 0.28/0.46 ( subset2powerset
% 0.28/0.46 = ( ! [A: $i,B: $i] :
% 0.28/0.46 ( ( subset @ A @ B )
% 0.28/0.46 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(setextsub_type,type,
% 0.28/0.46 setextsub: $o ).
% 0.28/0.46
% 0.28/0.46 thf(setextsub,definition,
% 0.28/0.46 ( setextsub
% 0.28/0.46 = ( ! [A: $i,B: $i] :
% 0.28/0.46 ( ( subset @ A @ B )
% 0.28/0.46 => ( ( subset @ B @ A )
% 0.28/0.46 => ( A = B ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(subsetemptysetimpeq_type,type,
% 0.28/0.46 subsetemptysetimpeq: $o ).
% 0.28/0.46
% 0.28/0.46 thf(subsetemptysetimpeq,definition,
% 0.28/0.46 ( subsetemptysetimpeq
% 0.28/0.46 = ( ! [A: $i] :
% 0.28/0.46 ( ( subset @ A @ emptyset )
% 0.28/0.46 => ( A = emptyset ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(powersetI1_type,type,
% 0.28/0.46 powersetI1: $o ).
% 0.28/0.46
% 0.28/0.46 thf(powersetI1,definition,
% 0.28/0.46 ( powersetI1
% 0.28/0.46 = ( ! [A: $i,B: $i] :
% 0.28/0.46 ( ( subset @ B @ A )
% 0.28/0.46 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(powersetE1_type,type,
% 0.28/0.46 powersetE1: $o ).
% 0.28/0.46
% 0.28/0.46 thf(powersetE1,definition,
% 0.28/0.46 ( powersetE1
% 0.28/0.46 = ( ! [A: $i,B: $i] :
% 0.28/0.46 ( ( in @ B @ ( powerset @ A ) )
% 0.28/0.46 => ( subset @ B @ A ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(inPowerset_type,type,
% 0.28/0.46 inPowerset: $o ).
% 0.28/0.46
% 0.28/0.46 thf(inPowerset,definition,
% 0.28/0.46 ( inPowerset
% 0.28/0.46 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(powersetsubset_type,type,
% 0.28/0.46 powersetsubset: $o ).
% 0.28/0.46
% 0.28/0.46 thf(powersetsubset,definition,
% 0.28/0.46 ( powersetsubset
% 0.28/0.46 = ( ! [A: $i,B: $i] :
% 0.28/0.46 ( ( subset @ A @ B )
% 0.28/0.46 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(sepInPowerset_type,type,
% 0.28/0.46 sepInPowerset: $o ).
% 0.28/0.46
% 0.28/0.46 thf(sepInPowerset,definition,
% 0.28/0.46 ( sepInPowerset
% 0.28/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.28/0.46 ( in
% 0.28/0.46 @ ( dsetconstr @ A
% 0.28/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.28/0.46 @ ( powerset @ A ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(sepSubset_type,type,
% 0.28/0.46 sepSubset: $o ).
% 0.28/0.46
% 0.28/0.46 thf(sepSubset,definition,
% 0.28/0.46 ( sepSubset
% 0.28/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.28/0.46 ( subset
% 0.28/0.46 @ ( dsetconstr @ A
% 0.28/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.28/0.46 @ A ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(binunion_type,type,
% 0.28/0.46 binunion: $i > $i > $i ).
% 0.28/0.46
% 0.28/0.46 thf(binunionIL_type,type,
% 0.28/0.46 binunionIL: $o ).
% 0.28/0.46
% 0.28/0.46 thf(binunionIL,definition,
% 0.28/0.46 ( binunionIL
% 0.28/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ A )
% 0.28/0.46 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(upairset2IR_type,type,
% 0.28/0.46 upairset2IR: $o ).
% 0.28/0.46
% 0.28/0.46 thf(upairset2IR,definition,
% 0.28/0.46 ( upairset2IR
% 0.28/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(binunionIR_type,type,
% 0.28/0.46 binunionIR: $o ).
% 0.28/0.46
% 0.28/0.46 thf(binunionIR,definition,
% 0.28/0.46 ( binunionIR
% 0.28/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ B )
% 0.28/0.46 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(binunionEcases_type,type,
% 0.28/0.46 binunionEcases: $o ).
% 0.28/0.46
% 0.28/0.46 thf(binunionEcases,definition,
% 0.28/0.46 ( binunionEcases
% 0.28/0.46 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.28/0.46 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.28/0.46 => ( ( ( in @ Xx @ A )
% 0.28/0.46 => Xphi )
% 0.28/0.46 => ( ( ( in @ Xx @ B )
% 0.28/0.46 => Xphi )
% 0.28/0.46 => Xphi ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(binunionE_type,type,
% 0.28/0.46 binunionE: $o ).
% 0.28/0.46
% 0.28/0.46 thf(binunionE,definition,
% 0.28/0.46 ( binunionE
% 0.28/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.28/0.46 => ( ( in @ Xx @ A )
% 0.28/0.46 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(binunionLsub_type,type,
% 0.28/0.46 binunionLsub: $o ).
% 0.28/0.46
% 0.28/0.46 thf(binunionLsub,definition,
% 0.28/0.46 ( binunionLsub
% 0.28/0.46 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(binunionRsub_type,type,
% 0.28/0.46 binunionRsub: $o ).
% 0.28/0.46
% 0.28/0.46 thf(binunionRsub,definition,
% 0.28/0.46 ( binunionRsub
% 0.28/0.46 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(binintersect_type,type,
% 0.28/0.46 binintersect: $i > $i > $i ).
% 0.28/0.46
% 0.28/0.46 thf(binintersectI_type,type,
% 0.28/0.46 binintersectI: $o ).
% 0.28/0.46
% 0.28/0.46 thf(binintersectI,definition,
% 0.28/0.46 ( binintersectI
% 0.28/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ A )
% 0.28/0.46 => ( ( in @ Xx @ B )
% 0.28/0.46 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(binintersectSubset5_type,type,
% 0.28/0.46 binintersectSubset5: $o ).
% 0.28/0.46
% 0.28/0.46 thf(binintersectSubset5,definition,
% 0.28/0.46 ( binintersectSubset5
% 0.28/0.46 = ( ! [A: $i,B: $i,C: $i] :
% 0.28/0.46 ( ( subset @ C @ A )
% 0.28/0.46 => ( ( subset @ C @ B )
% 0.28/0.46 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(binintersectEL_type,type,
% 0.28/0.46 binintersectEL: $o ).
% 0.28/0.46
% 0.28/0.46 thf(binintersectEL,definition,
% 0.28/0.46 ( binintersectEL
% 0.28/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.28/0.46 => ( in @ Xx @ A ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(binintersectLsub_type,type,
% 0.28/0.46 binintersectLsub: $o ).
% 0.28/0.46
% 0.28/0.46 thf(binintersectLsub,definition,
% 0.28/0.46 ( binintersectLsub
% 0.28/0.46 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(binintersectSubset2_type,type,
% 0.28/0.46 binintersectSubset2: $o ).
% 0.28/0.46
% 0.28/0.46 thf(binintersectSubset2,definition,
% 0.28/0.46 ( binintersectSubset2
% 0.28/0.46 = ( ! [A: $i,B: $i] :
% 0.28/0.46 ( ( subset @ A @ B )
% 0.28/0.46 => ( ( binintersect @ A @ B )
% 0.28/0.46 = A ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(binintersectSubset3_type,type,
% 0.28/0.46 binintersectSubset3: $o ).
% 0.28/0.46
% 0.28/0.46 thf(binintersectSubset3,definition,
% 0.28/0.46 ( binintersectSubset3
% 0.28/0.46 = ( ! [A: $i,B: $i] :
% 0.28/0.46 ( ( ( binintersect @ A @ B )
% 0.28/0.46 = B )
% 0.28/0.46 => ( subset @ B @ A ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(binintersectER_type,type,
% 0.28/0.46 binintersectER: $o ).
% 0.28/0.46
% 0.28/0.46 thf(binintersectER,definition,
% 0.28/0.46 ( binintersectER
% 0.28/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.28/0.46 => ( in @ Xx @ B ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(disjointsetsI1_type,type,
% 0.28/0.46 disjointsetsI1: $o ).
% 0.28/0.46
% 0.28/0.46 thf(disjointsetsI1,definition,
% 0.28/0.46 ( disjointsetsI1
% 0.28/0.46 = ( ! [A: $i,B: $i] :
% 0.28/0.46 ( ~ ? [Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ A )
% 0.28/0.46 & ( in @ Xx @ B ) )
% 0.28/0.46 => ( ( binintersect @ A @ B )
% 0.28/0.46 = emptyset ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(binintersectRsub_type,type,
% 0.28/0.46 binintersectRsub: $o ).
% 0.28/0.46
% 0.28/0.46 thf(binintersectRsub,definition,
% 0.28/0.46 ( binintersectRsub
% 0.28/0.46 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(binintersectSubset4_type,type,
% 0.28/0.46 binintersectSubset4: $o ).
% 0.28/0.46
% 0.28/0.46 thf(binintersectSubset4,definition,
% 0.28/0.46 ( binintersectSubset4
% 0.28/0.46 = ( ! [A: $i,B: $i] :
% 0.28/0.46 ( ( subset @ B @ A )
% 0.28/0.46 => ( ( binintersect @ A @ B )
% 0.28/0.46 = B ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(binintersectSubset1_type,type,
% 0.28/0.46 binintersectSubset1: $o ).
% 0.28/0.46
% 0.28/0.46 thf(binintersectSubset1,definition,
% 0.28/0.46 ( binintersectSubset1
% 0.28/0.46 = ( ! [A: $i,B: $i] :
% 0.28/0.46 ( ( ( binintersect @ A @ B )
% 0.28/0.46 = A )
% 0.28/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(bs114d_type,type,
% 0.28/0.46 bs114d: $o ).
% 0.28/0.46
% 0.28/0.46 thf(bs114d,definition,
% 0.28/0.46 ( bs114d
% 0.28/0.46 = ( ! [A: $i,B: $i,C: $i] :
% 0.28/0.46 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 0.28/0.46 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(regular_type,type,
% 0.28/0.46 regular: $i > $o ).
% 0.28/0.46
% 0.28/0.46 thf(setminus_type,type,
% 0.28/0.46 setminus: $i > $i > $i ).
% 0.28/0.46
% 0.28/0.46 thf(setminusI_type,type,
% 0.28/0.46 setminusI: $o ).
% 0.28/0.46
% 0.28/0.46 thf(setminusI,definition,
% 0.28/0.46 ( setminusI
% 0.28/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ A )
% 0.28/0.46 => ( ~ ( in @ Xx @ B )
% 0.28/0.46 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(setminusEL_type,type,
% 0.28/0.46 setminusEL: $o ).
% 0.28/0.46
% 0.28/0.46 thf(setminusEL,definition,
% 0.28/0.46 ( setminusEL
% 0.28/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.28/0.46 => ( in @ Xx @ A ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(setminusER_type,type,
% 0.28/0.46 setminusER: $o ).
% 0.28/0.46
% 0.28/0.46 thf(setminusER,definition,
% 0.28/0.46 ( setminusER
% 0.28/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.28/0.46 => ~ ( in @ Xx @ B ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(setminusSubset2_type,type,
% 0.28/0.46 setminusSubset2: $o ).
% 0.28/0.46
% 0.28/0.46 thf(setminusSubset2,definition,
% 0.28/0.46 ( setminusSubset2
% 0.28/0.46 = ( ! [A: $i,B: $i] :
% 0.28/0.46 ( ( subset @ A @ B )
% 0.28/0.46 => ( ( setminus @ A @ B )
% 0.28/0.46 = emptyset ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(setminusERneg_type,type,
% 0.28/0.46 setminusERneg: $o ).
% 0.28/0.46
% 0.28/0.46 thf(setminusERneg,definition,
% 0.28/0.46 ( setminusERneg
% 0.28/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.46 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.28/0.46 => ( ( in @ Xx @ A )
% 0.28/0.46 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(setminusELneg_type,type,
% 0.28/0.46 setminusELneg: $o ).
% 0.28/0.46
% 0.28/0.46 thf(setminusELneg,definition,
% 0.28/0.46 ( setminusELneg
% 0.28/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.46 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.28/0.46 => ( ~ ( in @ Xx @ B )
% 0.28/0.46 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(setminusILneg_type,type,
% 0.28/0.46 setminusILneg: $o ).
% 0.28/0.46
% 0.28/0.46 thf(setminusILneg,definition,
% 0.28/0.46 ( setminusILneg
% 0.28/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.46 ( ~ ( in @ Xx @ A )
% 0.28/0.46 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(setminusIRneg_type,type,
% 0.28/0.46 setminusIRneg: $o ).
% 0.28/0.46
% 0.28/0.46 thf(setminusIRneg,definition,
% 0.28/0.46 ( setminusIRneg
% 0.28/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ B )
% 0.28/0.46 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(setminusLsub_type,type,
% 0.28/0.46 setminusLsub: $o ).
% 0.28/0.46
% 0.28/0.46 thf(setminusLsub,definition,
% 0.28/0.46 ( setminusLsub
% 0.28/0.46 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(setminusSubset1_type,type,
% 0.28/0.46 setminusSubset1: $o ).
% 0.28/0.46
% 0.28/0.46 thf(setminusSubset1,definition,
% 0.28/0.46 ( setminusSubset1
% 0.28/0.46 = ( ! [A: $i,B: $i] :
% 0.28/0.46 ( ( ( setminus @ A @ B )
% 0.28/0.46 = emptyset )
% 0.28/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(symdiff_type,type,
% 0.28/0.46 symdiff: $i > $i > $i ).
% 0.28/0.46
% 0.28/0.46 thf(symdiffE_type,type,
% 0.28/0.46 symdiffE: $o ).
% 0.28/0.46
% 0.28/0.46 thf(symdiffE,definition,
% 0.28/0.46 ( symdiffE
% 0.28/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 0.28/0.46 => ! [Xphi: $o] :
% 0.28/0.46 ( ( ( in @ Xx @ A )
% 0.28/0.46 => ( ~ ( in @ Xx @ B )
% 0.28/0.46 => Xphi ) )
% 0.28/0.46 => ( ( ~ ( in @ Xx @ A )
% 0.28/0.46 => ( ( in @ Xx @ B )
% 0.28/0.46 => Xphi ) )
% 0.28/0.46 => Xphi ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(symdiffI1_type,type,
% 0.28/0.46 symdiffI1: $o ).
% 0.28/0.46
% 0.28/0.46 thf(symdiffI1,definition,
% 0.28/0.46 ( symdiffI1
% 0.28/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ A )
% 0.28/0.46 => ( ~ ( in @ Xx @ B )
% 0.28/0.46 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(symdiffI2_type,type,
% 0.28/0.46 symdiffI2: $o ).
% 0.28/0.46
% 0.28/0.46 thf(symdiffI2,definition,
% 0.28/0.46 ( symdiffI2
% 0.28/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.46 ( ~ ( in @ Xx @ A )
% 0.28/0.46 => ( ( in @ Xx @ B )
% 0.28/0.46 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(symdiffIneg1_type,type,
% 0.28/0.46 symdiffIneg1: $o ).
% 0.28/0.46
% 0.28/0.46 thf(symdiffIneg1,definition,
% 0.28/0.46 ( symdiffIneg1
% 0.28/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ A )
% 0.28/0.46 => ( ( in @ Xx @ B )
% 0.28/0.46 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(symdiffIneg2_type,type,
% 0.28/0.46 symdiffIneg2: $o ).
% 0.28/0.46
% 0.28/0.46 thf(symdiffIneg2,definition,
% 0.28/0.46 ( symdiffIneg2
% 0.28/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.46 ( ~ ( in @ Xx @ A )
% 0.28/0.46 => ( ~ ( in @ Xx @ B )
% 0.28/0.46 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(iskpair_type,type,
% 0.28/0.46 iskpair: $i > $o ).
% 0.28/0.46
% 0.28/0.46 thf(iskpair,definition,
% 0.28/0.46 ( iskpair
% 0.28/0.46 = ( ^ [A: $i] :
% 0.28/0.46 ? [Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ ( setunion @ A ) )
% 0.28/0.46 & ? [Xy: $i] :
% 0.28/0.46 ( ( in @ Xy @ ( setunion @ A ) )
% 0.28/0.46 & ( A
% 0.28/0.46 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(secondinupair_type,type,
% 0.28/0.46 secondinupair: $o ).
% 0.28/0.46
% 0.28/0.46 thf(secondinupair,definition,
% 0.28/0.46 ( secondinupair
% 0.28/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(setukpairIL_type,type,
% 0.28/0.46 setukpairIL: $o ).
% 0.28/0.46
% 0.28/0.46 thf(setukpairIL,definition,
% 0.28/0.46 ( setukpairIL
% 0.28/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(setukpairIR_type,type,
% 0.28/0.46 setukpairIR: $o ).
% 0.28/0.46
% 0.28/0.46 thf(setukpairIR,definition,
% 0.28/0.46 ( setukpairIR
% 0.28/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(kpairiskpair_type,type,
% 0.28/0.46 kpairiskpair: $o ).
% 0.28/0.46
% 0.28/0.46 thf(kpairiskpair,definition,
% 0.28/0.46 ( kpairiskpair
% 0.28/0.46 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(kpair_type,type,
% 0.28/0.46 kpair: $i > $i > $i ).
% 0.28/0.46
% 0.28/0.46 thf(kpair,definition,
% 0.28/0.46 ( kpair
% 0.28/0.46 = ( ^ [Xx: $i,Xy: $i] : ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(kpairp_type,type,
% 0.28/0.46 kpairp: $o ).
% 0.28/0.46
% 0.28/0.46 thf(kpairp,definition,
% 0.28/0.46 ( kpairp
% 0.28/0.46 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(cartprod_type,type,
% 0.28/0.46 cartprod: $i > $i > $i ).
% 0.28/0.46
% 0.28/0.46 thf(cartprod,definition,
% 0.28/0.46 ( cartprod
% 0.28/0.46 = ( ^ [A: $i,B: $i] :
% 0.28/0.46 ( dsetconstr @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) )
% 0.28/0.46 @ ^ [Xx: $i] :
% 0.28/0.46 ? [Xy: $i] :
% 0.28/0.46 ( ( in @ Xy @ A )
% 0.28/0.46 & ? [Xz: $i] :
% 0.28/0.46 ( ( in @ Xz @ B )
% 0.28/0.46 & ( Xx
% 0.28/0.46 = ( kpair @ Xy @ Xz ) ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(singletonsubset_type,type,
% 0.28/0.46 singletonsubset: $o ).
% 0.28/0.46
% 0.28/0.46 thf(singletonsubset,definition,
% 0.28/0.46 ( singletonsubset
% 0.28/0.46 = ( ! [A: $i,Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ A )
% 0.28/0.46 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(singletoninpowerset_type,type,
% 0.28/0.46 singletoninpowerset: $o ).
% 0.28/0.46
% 0.28/0.46 thf(singletoninpowerset,definition,
% 0.28/0.46 ( singletoninpowerset
% 0.28/0.46 = ( ! [A: $i,Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ A )
% 0.28/0.46 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(singletoninpowunion_type,type,
% 0.28/0.46 singletoninpowunion: $o ).
% 0.28/0.46
% 0.28/0.46 thf(singletoninpowunion,definition,
% 0.28/0.46 ( singletoninpowunion
% 0.28/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ A )
% 0.28/0.46 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(upairset2E_type,type,
% 0.28/0.46 upairset2E: $o ).
% 0.28/0.46
% 0.28/0.46 thf(upairset2E,definition,
% 0.28/0.46 ( upairset2E
% 0.28/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.28/0.46 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.28/0.46 => ( ( Xz = Xx )
% 0.28/0.46 | ( Xz = Xy ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(upairsubunion_type,type,
% 0.28/0.46 upairsubunion: $o ).
% 0.28/0.46
% 0.28/0.46 thf(upairsubunion,definition,
% 0.28/0.46 ( upairsubunion
% 0.28/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ A )
% 0.28/0.46 => ! [Xy: $i] :
% 0.28/0.46 ( ( in @ Xy @ B )
% 0.28/0.46 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(upairinpowunion_type,type,
% 0.28/0.46 upairinpowunion: $o ).
% 0.28/0.46
% 0.28/0.46 thf(upairinpowunion,definition,
% 0.28/0.46 ( upairinpowunion
% 0.28/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ A )
% 0.28/0.46 => ! [Xy: $i] :
% 0.28/0.46 ( ( in @ Xy @ B )
% 0.28/0.46 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(ubforcartprodlem1_type,type,
% 0.28/0.46 ubforcartprodlem1: $o ).
% 0.28/0.46
% 0.28/0.46 thf(ubforcartprodlem1,definition,
% 0.28/0.46 ( ubforcartprodlem1
% 0.28/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ A )
% 0.28/0.46 => ! [Xy: $i] :
% 0.28/0.46 ( ( in @ Xy @ B )
% 0.28/0.46 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(ubforcartprodlem2,conjecture,
% 0.28/0.46 ( setextAx
% 0.28/0.46 => ( emptysetAx
% 0.28/0.46 => ( setadjoinAx
% 0.28/0.46 => ( powersetAx
% 0.28/0.46 => ( setunionAx
% 0.28/0.46 => ( omega0Ax
% 0.28/0.46 => ( omegaSAx
% 0.28/0.46 => ( omegaIndAx
% 0.28/0.46 => ( replAx
% 0.28/0.46 => ( foundationAx
% 0.28/0.46 => ( wellorderingAx
% 0.28/0.46 => ( descrp
% 0.28/0.46 => ( dsetconstrI
% 0.28/0.46 => ( dsetconstrEL
% 0.28/0.46 => ( dsetconstrER
% 0.28/0.46 => ( exuE1
% 0.28/0.46 => ( prop2setE
% 0.28/0.46 => ( emptysetE
% 0.28/0.46 => ( emptysetimpfalse
% 0.28/0.46 => ( notinemptyset
% 0.28/0.46 => ( exuE3e
% 0.28/0.46 => ( setext
% 0.28/0.46 => ( emptyI
% 0.28/0.46 => ( noeltsimpempty
% 0.28/0.46 => ( setbeta
% 0.28/0.46 => ( nonemptyE1
% 0.28/0.46 => ( nonemptyI
% 0.28/0.46 => ( nonemptyI1
% 0.28/0.46 => ( setadjoinIL
% 0.28/0.46 => ( emptyinunitempty
% 0.28/0.46 => ( setadjoinIR
% 0.28/0.46 => ( setadjoinE
% 0.28/0.46 => ( setadjoinOr
% 0.28/0.46 => ( setoftrueEq
% 0.28/0.46 => ( powersetI
% 0.28/0.46 => ( emptyinPowerset
% 0.28/0.46 => ( emptyInPowerset
% 0.28/0.46 => ( powersetE
% 0.28/0.46 => ( setunionI
% 0.28/0.46 => ( setunionE
% 0.28/0.46 => ( subPowSU
% 0.28/0.46 => ( exuE2
% 0.28/0.46 => ( nonemptyImpWitness
% 0.28/0.46 => ( uniqinunit
% 0.28/0.46 => ( notinsingleton
% 0.28/0.46 => ( eqinunit
% 0.28/0.46 => ( singletonsswitch
% 0.28/0.46 => ( upairsetE
% 0.28/0.46 => ( upairsetIL
% 0.28/0.46 => ( upairsetIR
% 0.28/0.46 => ( emptyE1
% 0.28/0.46 => ( vacuousDall
% 0.28/0.46 => ( quantDeMorgan1
% 0.28/0.46 => ( quantDeMorgan2
% 0.28/0.46 => ( quantDeMorgan3
% 0.28/0.46 => ( quantDeMorgan4
% 0.28/0.46 => ( prop2setI
% 0.28/0.46 => ( prop2set2propI
% 0.28/0.46 => ( notdexE
% 0.28/0.46 => ( notdallE
% 0.28/0.46 => ( exuI1
% 0.28/0.46 => ( exuI3
% 0.28/0.46 => ( exuI2
% 0.28/0.46 => ( inCongP
% 0.28/0.46 => ( in__Cong
% 0.28/0.46 => ( exuE3u
% 0.28/0.46 => ( exu__Cong
% 0.28/0.46 => ( emptyset__Cong
% 0.28/0.46 => ( setadjoin__Cong
% 0.28/0.46 => ( powerset__Cong
% 0.28/0.46 => ( setunion__Cong
% 0.28/0.46 => ( omega__Cong
% 0.28/0.46 => ( exuEu
% 0.28/0.46 => ( descr__Cong
% 0.28/0.46 => ( dsetconstr__Cong
% 0.28/0.46 => ( subsetI1
% 0.28/0.46 => ( eqimpsubset2
% 0.28/0.46 => ( eqimpsubset1
% 0.28/0.46 => ( subsetI2
% 0.28/0.46 => ( emptysetsubset
% 0.28/0.46 => ( subsetE
% 0.28/0.46 => ( subsetE2
% 0.28/0.46 => ( notsubsetI
% 0.28/0.46 => ( notequalI1
% 0.28/0.46 => ( notequalI2
% 0.28/0.46 => ( subsetRefl
% 0.28/0.46 => ( subsetTrans
% 0.28/0.46 => ( setadjoinSub
% 0.28/0.46 => ( setadjoinSub2
% 0.28/0.46 => ( subset2powerset
% 0.28/0.46 => ( setextsub
% 0.28/0.46 => ( subsetemptysetimpeq
% 0.28/0.46 => ( powersetI1
% 0.28/0.46 => ( powersetE1
% 0.28/0.46 => ( inPowerset
% 0.28/0.46 => ( powersetsubset
% 0.28/0.46 => ( sepInPowerset
% 0.28/0.46 => ( sepSubset
% 0.28/0.46 => ( binunionIL
% 0.28/0.46 => ( upairset2IR
% 0.28/0.46 => ( binunionIR
% 0.28/0.46 => ( binunionEcases
% 0.28/0.46 => ( binunionE
% 0.28/0.46 => ( binunionLsub
% 0.28/0.46 => ( binunionRsub
% 0.28/0.46 => ( binintersectI
% 0.28/0.46 => ( binintersectSubset5
% 0.28/0.46 => ( binintersectEL
% 0.28/0.46 => ( binintersectLsub
% 0.28/0.46 => ( binintersectSubset2
% 0.28/0.46 => ( binintersectSubset3
% 0.28/0.46 => ( binintersectER
% 0.28/0.46 => ( disjointsetsI1
% 0.28/0.46 => ( binintersectRsub
% 0.28/0.46 => ( binintersectSubset4
% 0.28/0.46 => ( binintersectSubset1
% 0.28/0.46 => ( bs114d
% 0.28/0.46 => ( setminusI
% 0.28/0.46 => ( setminusEL
% 0.28/0.46 => ( setminusER
% 0.28/0.46 => ( setminusSubset2
% 0.28/0.46 => ( setminusERneg
% 0.28/0.46 => ( setminusELneg
% 0.28/0.46 => ( setminusILneg
% 0.28/0.46 => ( setminusIRneg
% 0.28/0.46 => ( setminusLsub
% 0.28/0.46 => ( setminusSubset1
% 0.28/0.46 => ( symdiffE
% 0.28/0.46 => ( symdiffI1
% 0.28/0.46 => ( symdiffI2
% 0.28/0.46 => ( symdiffIneg1
% 0.28/0.46 => ( symdiffIneg2
% 0.28/0.46 => ( secondinupair
% 0.28/0.46 => ( setukpairIL
% 0.28/0.46 => ( setukpairIR
% 0.28/0.46 => ( kpairiskpair
% 0.28/0.46 => ( kpairp
% 0.28/0.46 => ( singletonsubset
% 0.28/0.46 => ( singletoninpowerset
% 0.28/0.46 => ( singletoninpowunion
% 0.28/0.46 => ( upairset2E
% 0.28/0.46 => ( upairsubunion
% 0.28/0.46 => ( upairinpowunion
% 0.28/0.46 => ( ubforcartprodlem1
% 0.28/0.46 => ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ A )
% 0.28/0.46 => ! [Xy: $i] :
% 0.28/0.46 ( ( in @ Xy @ B )
% 0.28/0.46 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 %------------------------------------------------------------------------------
% 0.28/0.46 ------------------------
% 2.36/1.45 ---- Embedded in THF ---
% 2.36/1.45 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.36/1.45 %%% Generated on Tue Feb 24 12:28:55 EST 2026
% 2.36/1.45 %%% using '$$dhol' embedding, version 1.3.0.
% 2.36/1.45 %%% Logic specification used:
% 2.36/1.45 %%% thf(spec, logic, $$dhol).
% 2.36/1.45
% 2.36/1.45 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.yC96zsFX7x/DTF2DTF27586.p
% See solution above
% 2.36/1.45 ------------------------
% 2.36/1.49 ---- Cleaned THF ---
% 2.36/1.49 thf(in_type,type,
% 2.36/1.49 in: $i > $i > $o ).
% 2.36/1.49
% 2.36/1.49 thf(exu_type,type,
% 2.36/1.49 exu: ( $i > $o ) > $o ).
% 2.36/1.49
% 2.36/1.49 thf(exu,definition,
% 2.36/1.49 ( exu
% 2.36/1.49 = ( ^ [Xphi: $i > $o] :
% 2.36/1.49 ? [Xx: $i] :
% 2.36/1.49 ( ( Xphi @ Xx )
% 2.36/1.49 & ! [Xy: $i] :
% 2.36/1.49 ( ( Xphi @ Xy )
% 2.36/1.49 => ( Xx = Xy ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(setextAx_type,type,
% 2.36/1.49 setextAx: $o ).
% 2.36/1.49
% 2.36/1.49 thf(setextAx,definition,
% 2.36/1.49 ( setextAx
% 2.36/1.49 = ( ! [A: $i,B: $i] :
% 2.36/1.49 ( ! [Xx: $i] :
% 2.36/1.49 ( ( in @ Xx @ A )
% 2.36/1.49 <=> ( in @ Xx @ B ) )
% 2.36/1.49 => ( A = B ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(emptyset_type,type,
% 2.36/1.49 emptyset: $i ).
% 2.36/1.49
% 2.36/1.49 thf(emptysetAx_type,type,
% 2.36/1.49 emptysetAx: $o ).
% 2.36/1.49
% 2.36/1.49 thf(emptysetAx,definition,
% 2.36/1.49 ( emptysetAx
% 2.36/1.49 = ( ! [Xx: $i] :
% 2.36/1.49 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(setadjoin_type,type,
% 2.36/1.49 setadjoin: $i > $i > $i ).
% 2.36/1.49
% 2.36/1.49 thf(setadjoinAx_type,type,
% 2.36/1.49 setadjoinAx: $o ).
% 2.36/1.49
% 2.36/1.49 thf(setadjoinAx,definition,
% 2.36/1.49 ( setadjoinAx
% 2.36/1.49 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.36/1.49 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.36/1.49 <=> ( ( Xy = Xx )
% 2.36/1.49 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(powerset_type,type,
% 2.36/1.49 powerset: $i > $i ).
% 2.36/1.49
% 2.36/1.49 thf(powersetAx_type,type,
% 2.36/1.49 powersetAx: $o ).
% 2.36/1.49
% 2.36/1.49 thf(powersetAx,definition,
% 2.36/1.49 ( powersetAx
% 2.36/1.49 = ( ! [A: $i,B: $i] :
% 2.36/1.49 ( ( in @ B @ ( powerset @ A ) )
% 2.36/1.49 <=> ! [Xx: $i] :
% 2.36/1.49 ( ( in @ Xx @ B )
% 2.36/1.49 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(setunion_type,type,
% 2.36/1.49 setunion: $i > $i ).
% 2.36/1.49
% 2.36/1.49 thf(setunionAx_type,type,
% 2.36/1.49 setunionAx: $o ).
% 2.36/1.49
% 2.36/1.49 thf(setunionAx,definition,
% 2.36/1.49 ( setunionAx
% 2.36/1.49 = ( ! [A: $i,Xx: $i] :
% 2.36/1.49 ( ( in @ Xx @ ( setunion @ A ) )
% 2.36/1.49 <=> ? [B: $i] :
% 2.36/1.49 ( ( in @ Xx @ B )
% 2.36/1.49 & ( in @ B @ A ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(omega_type,type,
% 2.36/1.49 omega: $i ).
% 2.36/1.49
% 2.36/1.49 thf(omega0Ax_type,type,
% 2.36/1.49 omega0Ax: $o ).
% 2.36/1.49
% 2.36/1.49 thf(omega0Ax,definition,
% 2.36/1.49 ( omega0Ax
% 2.36/1.49 = ( in @ emptyset @ omega ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(omegaSAx_type,type,
% 2.36/1.49 omegaSAx: $o ).
% 2.36/1.49
% 2.36/1.49 thf(omegaSAx,definition,
% 2.36/1.49 ( omegaSAx
% 2.36/1.49 = ( ! [Xx: $i] :
% 2.36/1.49 ( ( in @ Xx @ omega )
% 2.36/1.49 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(omegaIndAx_type,type,
% 2.36/1.49 omegaIndAx: $o ).
% 2.36/1.49
% 2.36/1.49 thf(omegaIndAx,definition,
% 2.36/1.49 ( omegaIndAx
% 2.36/1.49 = ( ! [A: $i] :
% 2.36/1.49 ( ( ( in @ emptyset @ A )
% 2.36/1.49 & ! [Xx: $i] :
% 2.36/1.49 ( ( ( in @ Xx @ omega )
% 2.36/1.49 & ( in @ Xx @ A ) )
% 2.36/1.49 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 2.36/1.49 => ! [Xx: $i] :
% 2.36/1.49 ( ( in @ Xx @ omega )
% 2.36/1.49 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(replAx_type,type,
% 2.36/1.49 replAx: $o ).
% 2.36/1.49
% 2.36/1.49 thf(replAx,definition,
% 2.36/1.49 ( replAx
% 2.36/1.49 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 2.36/1.49 ( ! [Xx: $i] :
% 2.36/1.49 ( ( in @ Xx @ A )
% 2.36/1.49 => ( exu
% 2.36/1.49 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 2.36/1.49 => ? [B: $i] :
% 2.36/1.49 ! [Xx: $i] :
% 2.36/1.49 ( ( in @ Xx @ B )
% 2.36/1.49 <=> ? [Xy: $i] :
% 2.36/1.49 ( ( in @ Xy @ A )
% 2.36/1.49 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(foundationAx_type,type,
% 2.36/1.49 foundationAx: $o ).
% 2.36/1.49
% 2.36/1.49 thf(foundationAx,definition,
% 2.36/1.49 ( foundationAx
% 2.36/1.49 = ( ! [A: $i] :
% 2.36/1.49 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.36/1.49 => ? [B: $i] :
% 2.36/1.49 ( ( in @ B @ A )
% 2.36/1.49 & ~ ? [Xx: $i] :
% 2.36/1.49 ( ( in @ Xx @ B )
% 2.36/1.49 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(wellorderingAx_type,type,
% 2.36/1.49 wellorderingAx: $o ).
% 2.36/1.49
% 2.36/1.49 thf(wellorderingAx,definition,
% 2.36/1.49 ( wellorderingAx
% 2.36/1.49 = ( ! [A: $i] :
% 2.36/1.49 ? [B: $i] :
% 2.36/1.49 ( ! [C: $i] :
% 2.36/1.49 ( ( in @ C @ B )
% 2.36/1.49 => ! [Xx: $i] :
% 2.36/1.49 ( ( in @ Xx @ C )
% 2.36/1.49 => ( in @ Xx @ A ) ) )
% 2.36/1.49 & ! [Xx: $i,Xy: $i] :
% 2.36/1.49 ( ( ( in @ Xx @ A )
% 2.36/1.49 & ( in @ Xy @ A ) )
% 2.36/1.49 => ( ! [C: $i] :
% 2.36/1.49 ( ( in @ C @ B )
% 2.36/1.49 => ( ( in @ Xx @ C )
% 2.36/1.49 <=> ( in @ Xy @ C ) ) )
% 2.36/1.49 => ( Xx = Xy ) ) )
% 2.36/1.49 & ! [C: $i,D: $i] :
% 2.36/1.49 ( ( ( in @ C @ B )
% 2.36/1.49 & ( in @ D @ B ) )
% 2.36/1.49 => ( ! [Xx: $i] :
% 2.36/1.49 ( ( in @ Xx @ C )
% 2.36/1.49 => ( in @ Xx @ D ) )
% 2.36/1.49 | ! [Xx: $i] :
% 2.36/1.49 ( ( in @ Xx @ D )
% 2.36/1.49 => ( in @ Xx @ C ) ) ) )
% 2.36/1.49 & ! [C: $i] :
% 2.36/1.49 ( ( ! [Xx: $i] :
% 2.36/1.49 ( ( in @ Xx @ C )
% 2.36/1.49 => ( in @ Xx @ A ) )
% 2.36/1.49 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 2.36/1.49 => ? [D: $i,Xx: $i] :
% 2.36/1.49 ( ( in @ D @ B )
% 2.36/1.49 & ( in @ Xx @ C )
% 2.36/1.49 & ~ ? [Xy: $i] :
% 2.36/1.49 ( ( in @ Xy @ D )
% 2.36/1.49 & ( in @ Xy @ C ) )
% 2.36/1.49 & ! [E: $i] :
% 2.36/1.49 ( ( in @ E @ B )
% 2.36/1.49 => ( ! [Xy: $i] :
% 2.36/1.49 ( ( in @ Xy @ E )
% 2.36/1.49 => ( in @ Xy @ D ) )
% 2.36/1.49 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(descr_type,type,
% 2.36/1.49 descr: ( $i > $o ) > $i ).
% 2.36/1.49
% 2.36/1.49 thf(descrp_type,type,
% 2.36/1.49 descrp: $o ).
% 2.36/1.49
% 2.36/1.49 thf(descrp,definition,
% 2.36/1.49 ( descrp
% 2.36/1.49 = ( ! [Xphi: $i > $o] :
% 2.36/1.49 ( ( exu
% 2.36/1.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.36/1.49 => ( Xphi
% 2.36/1.49 @ ( descr
% 2.36/1.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(dsetconstr_type,type,
% 2.36/1.49 dsetconstr: $i > ( $i > $o ) > $i ).
% 2.36/1.49
% 2.36/1.49 thf(dsetconstrI_type,type,
% 2.36/1.49 dsetconstrI: $o ).
% 2.36/1.49
% 2.36/1.49 thf(dsetconstrI,definition,
% 2.36/1.49 ( dsetconstrI
% 2.36/1.49 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.36/1.49 ( ( in @ Xx @ A )
% 2.36/1.49 => ( ( Xphi @ Xx )
% 2.36/1.49 => ( in @ Xx
% 2.36/1.49 @ ( dsetconstr @ A
% 2.36/1.49 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(dsetconstrEL_type,type,
% 2.36/1.49 dsetconstrEL: $o ).
% 2.36/1.49
% 2.36/1.49 thf(dsetconstrEL,definition,
% 2.36/1.49 ( dsetconstrEL
% 2.36/1.49 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.36/1.49 ( ( in @ Xx
% 2.36/1.49 @ ( dsetconstr @ A
% 2.36/1.49 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.36/1.49 => ( in @ Xx @ A ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(dsetconstrER_type,type,
% 2.36/1.49 dsetconstrER: $o ).
% 2.36/1.49
% 2.36/1.49 thf(dsetconstrER,definition,
% 2.36/1.49 ( dsetconstrER
% 2.36/1.49 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.36/1.49 ( ( in @ Xx
% 2.36/1.49 @ ( dsetconstr @ A
% 2.36/1.49 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.36/1.49 => ( Xphi @ Xx ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(exuE1_type,type,
% 2.36/1.49 exuE1: $o ).
% 2.36/1.49
% 2.36/1.49 thf(exuE1,definition,
% 2.36/1.49 ( exuE1
% 2.36/1.49 = ( ! [Xphi: $i > $o] :
% 2.36/1.49 ( ( exu
% 2.36/1.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.36/1.49 => ? [Xx: $i] :
% 2.36/1.49 ( ( Xphi @ Xx )
% 2.36/1.49 & ! [Xy: $i] :
% 2.36/1.49 ( ( Xphi @ Xy )
% 2.36/1.49 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(prop2set_type,type,
% 2.36/1.49 prop2set: $o > $i ).
% 2.36/1.49
% 2.36/1.49 thf(prop2setE_type,type,
% 2.36/1.49 prop2setE: $o ).
% 2.36/1.49
% 2.36/1.49 thf(prop2setE,definition,
% 2.36/1.49 ( prop2setE
% 2.36/1.49 = ( ! [Xphi: $o,Xx: $i] :
% 2.36/1.49 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 2.36/1.49 => Xphi ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(emptysetE_type,type,
% 2.36/1.49 emptysetE: $o ).
% 2.36/1.49
% 2.36/1.49 thf(emptysetE,definition,
% 2.36/1.49 ( emptysetE
% 2.36/1.49 = ( ! [Xx: $i] :
% 2.36/1.49 ( ( in @ Xx @ emptyset )
% 2.36/1.49 => ! [Xphi: $o] : Xphi ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(emptysetimpfalse_type,type,
% 2.36/1.49 emptysetimpfalse: $o ).
% 2.36/1.49
% 2.36/1.49 thf(emptysetimpfalse,definition,
% 2.36/1.49 ( emptysetimpfalse
% 2.36/1.49 = ( ! [Xx: $i] :
% 2.36/1.49 ( ( in @ Xx @ emptyset )
% 2.36/1.49 => $false ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(notinemptyset_type,type,
% 2.36/1.49 notinemptyset: $o ).
% 2.36/1.49
% 2.36/1.49 thf(notinemptyset,definition,
% 2.36/1.49 ( notinemptyset
% 2.36/1.49 = ( ! [Xx: $i] :
% 2.36/1.49 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(exuE3e_type,type,
% 2.36/1.49 exuE3e: $o ).
% 2.36/1.49
% 2.36/1.49 thf(exuE3e,definition,
% 2.36/1.49 ( exuE3e
% 2.36/1.49 = ( ! [Xphi: $i > $o] :
% 2.36/1.49 ( ( exu
% 2.36/1.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.36/1.49 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(setext_type,type,
% 2.36/1.49 setext: $o ).
% 2.36/1.49
% 2.36/1.49 thf(setext,definition,
% 2.36/1.49 ( setext
% 2.36/1.49 = ( ! [A: $i,B: $i] :
% 2.36/1.49 ( ! [Xx: $i] :
% 2.36/1.49 ( ( in @ Xx @ A )
% 2.36/1.49 => ( in @ Xx @ B ) )
% 2.36/1.49 => ( ! [Xx: $i] :
% 2.36/1.49 ( ( in @ Xx @ B )
% 2.36/1.49 => ( in @ Xx @ A ) )
% 2.36/1.49 => ( A = B ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(emptyI_type,type,
% 2.36/1.49 emptyI: $o ).
% 2.36/1.49
% 2.36/1.49 thf(emptyI,definition,
% 2.36/1.49 ( emptyI
% 2.36/1.49 = ( ! [A: $i] :
% 2.36/1.49 ( ! [Xx: $i] :
% 2.36/1.49 ~ ( in @ Xx @ A )
% 2.36/1.49 => ( A = emptyset ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(noeltsimpempty_type,type,
% 2.36/1.49 noeltsimpempty: $o ).
% 2.36/1.49
% 2.36/1.49 thf(noeltsimpempty,definition,
% 2.36/1.49 ( noeltsimpempty
% 2.36/1.49 = ( ! [A: $i] :
% 2.36/1.49 ( ! [Xx: $i] :
% 2.36/1.49 ~ ( in @ Xx @ A )
% 2.36/1.49 => ( A = emptyset ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(setbeta_type,type,
% 2.36/1.49 setbeta: $o ).
% 2.36/1.49
% 2.36/1.49 thf(setbeta,definition,
% 2.36/1.49 ( setbeta
% 2.36/1.49 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.36/1.49 ( ( in @ Xx @ A )
% 2.36/1.49 => ( ( in @ Xx
% 2.36/1.49 @ ( dsetconstr @ A
% 2.36/1.49 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.36/1.49 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(nonempty_type,type,
% 2.36/1.49 nonempty: $i > $o ).
% 2.36/1.49
% 2.36/1.49 thf(nonempty,definition,
% 2.36/1.49 ( nonempty
% 2.36/1.49 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(nonemptyE1_type,type,
% 2.36/1.49 nonemptyE1: $o ).
% 2.36/1.49
% 2.36/1.49 thf(nonemptyE1,definition,
% 2.36/1.49 ( nonemptyE1
% 2.36/1.49 = ( ! [A: $i] :
% 2.36/1.49 ( ( nonempty @ A )
% 2.36/1.49 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(nonemptyI_type,type,
% 2.36/1.49 nonemptyI: $o ).
% 2.36/1.49
% 2.36/1.49 thf(nonemptyI,definition,
% 2.36/1.49 ( nonemptyI
% 2.36/1.49 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.36/1.49 ( ( in @ Xx @ A )
% 2.36/1.49 => ( ( Xphi @ Xx )
% 2.36/1.49 => ( nonempty
% 2.36/1.49 @ ( dsetconstr @ A
% 2.36/1.49 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(nonemptyI1_type,type,
% 2.36/1.49 nonemptyI1: $o ).
% 2.36/1.49
% 2.36/1.49 thf(nonemptyI1,definition,
% 2.36/1.49 ( nonemptyI1
% 2.36/1.49 = ( ! [A: $i] :
% 2.36/1.49 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.36/1.49 => ( nonempty @ A ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(setadjoinIL_type,type,
% 2.36/1.49 setadjoinIL: $o ).
% 2.36/1.49
% 2.36/1.49 thf(setadjoinIL,definition,
% 2.36/1.49 ( setadjoinIL
% 2.36/1.49 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(emptyinunitempty_type,type,
% 2.36/1.49 emptyinunitempty: $o ).
% 2.36/1.49
% 2.36/1.49 thf(emptyinunitempty,definition,
% 2.36/1.49 ( emptyinunitempty
% 2.36/1.49 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(setadjoinIR_type,type,
% 2.36/1.49 setadjoinIR: $o ).
% 2.36/1.49
% 2.36/1.49 thf(setadjoinIR,definition,
% 2.36/1.49 ( setadjoinIR
% 2.36/1.49 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.36/1.49 ( ( in @ Xy @ A )
% 2.36/1.49 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(setadjoinE_type,type,
% 2.36/1.49 setadjoinE: $o ).
% 2.36/1.49
% 2.36/1.49 thf(setadjoinE,definition,
% 2.36/1.49 ( setadjoinE
% 2.36/1.49 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.36/1.49 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.36/1.49 => ! [Xphi: $o] :
% 2.36/1.49 ( ( ( Xy = Xx )
% 2.36/1.49 => Xphi )
% 2.36/1.49 => ( ( ( in @ Xy @ A )
% 2.36/1.49 => Xphi )
% 2.36/1.49 => Xphi ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(setadjoinOr_type,type,
% 2.36/1.49 setadjoinOr: $o ).
% 2.36/1.49
% 2.36/1.49 thf(setadjoinOr,definition,
% 2.36/1.49 ( setadjoinOr
% 2.36/1.49 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.36/1.49 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.36/1.49 => ( ( Xy = Xx )
% 2.36/1.49 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(setoftrueEq_type,type,
% 2.36/1.49 setoftrueEq: $o ).
% 2.36/1.49
% 2.36/1.49 thf(setoftrueEq,definition,
% 2.36/1.49 ( setoftrueEq
% 2.36/1.49 = ( ! [A: $i] :
% 2.36/1.49 ( ( dsetconstr @ A
% 2.36/1.49 @ ^ [Xx: $i] : $true )
% 2.36/1.49 = A ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(powersetI_type,type,
% 2.36/1.49 powersetI: $o ).
% 2.36/1.49
% 2.36/1.49 thf(powersetI,definition,
% 2.36/1.49 ( powersetI
% 2.36/1.49 = ( ! [A: $i,B: $i] :
% 2.36/1.49 ( ! [Xx: $i] :
% 2.36/1.49 ( ( in @ Xx @ B )
% 2.36/1.49 => ( in @ Xx @ A ) )
% 2.36/1.49 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(emptyinPowerset_type,type,
% 2.36/1.49 emptyinPowerset: $o ).
% 2.36/1.49
% 2.36/1.49 thf(emptyinPowerset,definition,
% 2.36/1.49 ( emptyinPowerset
% 2.36/1.49 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(emptyInPowerset_type,type,
% 2.36/1.49 emptyInPowerset: $o ).
% 2.36/1.49
% 2.36/1.49 thf(emptyInPowerset,definition,
% 2.36/1.49 ( emptyInPowerset
% 2.36/1.49 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(powersetE_type,type,
% 2.36/1.49 powersetE: $o ).
% 2.36/1.49
% 2.36/1.49 thf(powersetE,definition,
% 2.36/1.49 ( powersetE
% 2.36/1.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.36/1.49 ( ( in @ B @ ( powerset @ A ) )
% 2.36/1.49 => ( ( in @ Xx @ B )
% 2.36/1.49 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(setunionI_type,type,
% 2.36/1.49 setunionI: $o ).
% 2.36/1.49
% 2.36/1.49 thf(setunionI,definition,
% 2.36/1.49 ( setunionI
% 2.36/1.49 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.36/1.49 ( ( in @ Xx @ B )
% 2.36/1.49 => ( ( in @ B @ A )
% 2.36/1.49 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(setunionE_type,type,
% 2.36/1.49 setunionE: $o ).
% 2.36/1.49
% 2.36/1.49 thf(setunionE,definition,
% 2.36/1.49 ( setunionE
% 2.36/1.49 = ( ! [A: $i,Xx: $i] :
% 2.36/1.49 ( ( in @ Xx @ ( setunion @ A ) )
% 2.36/1.49 => ! [Xphi: $o] :
% 2.36/1.49 ( ! [B: $i] :
% 2.36/1.49 ( ( in @ Xx @ B )
% 2.36/1.49 => ( ( in @ B @ A )
% 2.36/1.49 => Xphi ) )
% 2.36/1.49 => Xphi ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(subPowSU_type,type,
% 2.36/1.49 subPowSU: $o ).
% 2.36/1.49
% 2.36/1.49 thf(subPowSU,definition,
% 2.36/1.49 ( subPowSU
% 2.36/1.49 = ( ! [A: $i,Xx: $i] :
% 2.36/1.49 ( ( in @ Xx @ A )
% 2.36/1.49 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(exuE2_type,type,
% 2.36/1.49 exuE2: $o ).
% 2.36/1.49
% 2.36/1.49 thf(exuE2,definition,
% 2.36/1.49 ( exuE2
% 2.36/1.49 = ( ! [Xphi: $i > $o] :
% 2.36/1.49 ( ( exu
% 2.36/1.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.36/1.49 => ? [Xx: $i] :
% 2.36/1.49 ! [Xy: $i] :
% 2.36/1.49 ( ( Xphi @ Xy )
% 2.36/1.49 <=> ( Xy = Xx ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(nonemptyImpWitness_type,type,
% 2.36/1.49 nonemptyImpWitness: $o ).
% 2.36/1.49
% 2.36/1.49 thf(nonemptyImpWitness,definition,
% 2.36/1.49 ( nonemptyImpWitness
% 2.36/1.49 = ( ! [A: $i] :
% 2.36/1.49 ( ( nonempty @ A )
% 2.36/1.49 => ? [Xx: $i] :
% 2.36/1.49 ( ( in @ Xx @ A )
% 2.36/1.49 & $true ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(uniqinunit_type,type,
% 2.36/1.49 uniqinunit: $o ).
% 2.36/1.49
% 2.36/1.49 thf(uniqinunit,definition,
% 2.36/1.49 ( uniqinunit
% 2.36/1.49 = ( ! [Xx: $i,Xy: $i] :
% 2.36/1.49 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.36/1.49 => ( Xx = Xy ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(notinsingleton_type,type,
% 2.36/1.49 notinsingleton: $o ).
% 2.36/1.49
% 2.36/1.49 thf(notinsingleton,definition,
% 2.36/1.49 ( notinsingleton
% 2.36/1.49 = ( ! [Xx: $i,Xy: $i] :
% 2.36/1.49 ( ( Xx != Xy )
% 2.36/1.49 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(eqinunit_type,type,
% 2.36/1.49 eqinunit: $o ).
% 2.36/1.49
% 2.36/1.49 thf(eqinunit,definition,
% 2.36/1.49 ( eqinunit
% 2.36/1.49 = ( ! [Xx: $i,Xy: $i] :
% 2.36/1.49 ( ( Xx = Xy )
% 2.36/1.49 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(singletonsswitch_type,type,
% 2.36/1.49 singletonsswitch: $o ).
% 2.36/1.49
% 2.36/1.49 thf(singletonsswitch,definition,
% 2.36/1.49 ( singletonsswitch
% 2.36/1.49 = ( ! [Xx: $i,Xy: $i] :
% 2.36/1.49 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.36/1.49 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(upairsetE_type,type,
% 2.36/1.49 upairsetE: $o ).
% 2.36/1.49
% 2.36/1.49 thf(upairsetE,definition,
% 2.36/1.49 ( upairsetE
% 2.36/1.49 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.36/1.49 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.36/1.49 => ( ( Xz = Xx )
% 2.36/1.49 | ( Xz = Xy ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(upairsetIL_type,type,
% 2.36/1.49 upairsetIL: $o ).
% 2.36/1.49
% 2.36/1.49 thf(upairsetIL,definition,
% 2.36/1.49 ( upairsetIL
% 2.36/1.49 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(upairsetIR_type,type,
% 2.36/1.49 upairsetIR: $o ).
% 2.36/1.49
% 2.36/1.49 thf(upairsetIR,definition,
% 2.36/1.49 ( upairsetIR
% 2.36/1.49 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(emptyE1_type,type,
% 2.36/1.49 emptyE1: $o ).
% 2.36/1.49
% 2.36/1.49 thf(emptyE1,definition,
% 2.36/1.49 ( emptyE1
% 2.36/1.49 = ( ! [A: $i,Xphi: $i > $o] :
% 2.36/1.49 ( ? [Xx: $i] :
% 2.36/1.49 ( ( in @ Xx @ A )
% 2.36/1.49 & ( Xphi @ Xx ) )
% 2.36/1.49 => ( ( ( dsetconstr @ A
% 2.36/1.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.36/1.49 = emptyset )
% 2.36/1.49 => $false ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(vacuousDall_type,type,
% 2.36/1.49 vacuousDall: $o ).
% 2.36/1.49
% 2.36/1.49 thf(vacuousDall,definition,
% 2.36/1.49 ( vacuousDall
% 2.36/1.49 = ( ! [Xphi: $i > $o,Xx: $i] :
% 2.36/1.49 ( ( in @ Xx @ emptyset )
% 2.36/1.49 => ( Xphi @ Xx ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(quantDeMorgan1_type,type,
% 2.36/1.49 quantDeMorgan1: $o ).
% 2.36/1.49
% 2.36/1.49 thf(quantDeMorgan1,definition,
% 2.36/1.49 ( quantDeMorgan1
% 2.36/1.49 = ( ! [A: $i,Xphi: $i > $o] :
% 2.36/1.49 ( ~ ! [Xx: $i] :
% 2.36/1.49 ( ( in @ Xx @ A )
% 2.36/1.49 => ( Xphi @ Xx ) )
% 2.36/1.49 => ? [Xx: $i] :
% 2.36/1.49 ( ( in @ Xx @ A )
% 2.36/1.49 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(quantDeMorgan2_type,type,
% 2.36/1.49 quantDeMorgan2: $o ).
% 2.36/1.49
% 2.36/1.49 thf(quantDeMorgan2,definition,
% 2.36/1.49 ( quantDeMorgan2
% 2.36/1.49 = ( ! [A: $i,Xphi: $i > $o] :
% 2.36/1.49 ( ! [Xx: $i] :
% 2.36/1.49 ( ( in @ Xx @ A )
% 2.36/1.49 => ~ ( Xphi @ Xx ) )
% 2.36/1.49 => ~ ? [Xx: $i] :
% 2.36/1.49 ( ( in @ Xx @ A )
% 2.36/1.49 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(quantDeMorgan3_type,type,
% 2.36/1.49 quantDeMorgan3: $o ).
% 2.36/1.49
% 2.36/1.49 thf(quantDeMorgan3,definition,
% 2.36/1.49 ( quantDeMorgan3
% 2.36/1.49 = ( ! [A: $i,Xphi: $i > $o] :
% 2.36/1.49 ( ~ ? [Xx: $i] :
% 2.36/1.49 ( ( in @ Xx @ A )
% 2.36/1.49 & ( Xphi @ Xx ) )
% 2.36/1.49 => ! [Xx: $i] :
% 2.36/1.49 ( ( in @ Xx @ A )
% 2.36/1.49 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(quantDeMorgan4_type,type,
% 2.36/1.49 quantDeMorgan4: $o ).
% 2.36/1.49
% 2.36/1.49 thf(quantDeMorgan4,definition,
% 2.36/1.49 ( quantDeMorgan4
% 2.36/1.49 = ( ! [A: $i,Xphi: $i > $o] :
% 2.36/1.49 ( ? [Xx: $i] :
% 2.36/1.49 ( ( in @ Xx @ A )
% 2.36/1.49 & ~ ( Xphi @ Xx ) )
% 2.36/1.49 => ~ ! [Xx: $i] :
% 2.36/1.49 ( ( in @ Xx @ A )
% 2.36/1.49 => ( Xphi @ Xx ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(prop2setI_type,type,
% 2.36/1.49 prop2setI: $o ).
% 2.36/1.49
% 2.36/1.49 thf(prop2setI,definition,
% 2.36/1.49 ( prop2setI
% 2.36/1.49 = ( ! [Xphi: $o] :
% 2.36/1.49 ( Xphi
% 2.36/1.49 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(set2prop_type,type,
% 2.36/1.49 set2prop: $i > $o ).
% 2.36/1.49
% 2.36/1.49 thf(prop2set2propI_type,type,
% 2.36/1.49 prop2set2propI: $o ).
% 2.36/1.49
% 2.36/1.49 thf(prop2set2propI,definition,
% 2.36/1.49 ( prop2set2propI
% 2.36/1.49 = ( ! [Xphi: $o] :
% 2.36/1.49 ( Xphi
% 2.36/1.49 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(notdexE_type,type,
% 2.36/1.49 notdexE: $o ).
% 2.36/1.49
% 2.36/1.49 thf(notdexE,definition,
% 2.36/1.49 ( notdexE
% 2.36/1.49 = ( ! [A: $i,Xphi: $i > $o] :
% 2.36/1.49 ( ~ ? [Xx: $i] :
% 2.36/1.49 ( ( in @ Xx @ A )
% 2.36/1.49 & ( Xphi @ Xx ) )
% 2.36/1.49 => ! [Xx: $i] :
% 2.36/1.49 ( ( in @ Xx @ A )
% 2.36/1.49 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(notdallE_type,type,
% 2.36/1.49 notdallE: $o ).
% 2.36/1.49
% 2.36/1.49 thf(notdallE,definition,
% 2.36/1.49 ( notdallE
% 2.36/1.49 = ( ! [A: $i,Xphi: $i > $o] :
% 2.36/1.49 ( ~ ! [Xx: $i] :
% 2.36/1.49 ( ( in @ Xx @ A )
% 2.36/1.49 => ( Xphi @ Xx ) )
% 2.36/1.49 => ? [Xx: $i] :
% 2.36/1.49 ( ( in @ Xx @ A )
% 2.36/1.49 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(exuI1_type,type,
% 2.36/1.49 exuI1: $o ).
% 2.36/1.49
% 2.36/1.49 thf(exuI1,definition,
% 2.36/1.49 ( exuI1
% 2.36/1.49 = ( ! [Xphi: $i > $o] :
% 2.36/1.49 ( ? [Xx: $i] :
% 2.36/1.49 ( ( Xphi @ Xx )
% 2.36/1.49 & ! [Xy: $i] :
% 2.36/1.49 ( ( Xphi @ Xy )
% 2.36/1.49 => ( Xx = Xy ) ) )
% 2.36/1.49 => ( exu
% 2.36/1.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(exuI3_type,type,
% 2.36/1.49 exuI3: $o ).
% 2.36/1.49
% 2.36/1.49 thf(exuI3,definition,
% 2.36/1.49 ( exuI3
% 2.36/1.49 = ( ! [Xphi: $i > $o] :
% 2.36/1.49 ( ? [Xx: $i] : ( Xphi @ Xx )
% 2.36/1.49 => ( ! [Xx: $i,Xy: $i] :
% 2.36/1.49 ( ( Xphi @ Xx )
% 2.36/1.49 => ( ( Xphi @ Xy )
% 2.36/1.49 => ( Xx = Xy ) ) )
% 2.36/1.49 => ( exu
% 2.36/1.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(exuI2_type,type,
% 2.36/1.49 exuI2: $o ).
% 2.36/1.49
% 2.36/1.49 thf(exuI2,definition,
% 2.36/1.49 ( exuI2
% 2.36/1.49 = ( ! [Xphi: $i > $o] :
% 2.36/1.49 ( ? [Xx: $i] :
% 2.36/1.49 ! [Xy: $i] :
% 2.36/1.49 ( ( Xphi @ Xy )
% 2.36/1.49 <=> ( Xy = Xx ) )
% 2.36/1.49 => ( exu
% 2.36/1.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(inCongP_type,type,
% 2.36/1.49 inCongP: $o ).
% 2.36/1.49
% 2.36/1.49 thf(inCongP,definition,
% 2.36/1.49 ( inCongP
% 2.36/1.49 = ( ! [A: $i,B: $i] :
% 2.36/1.49 ( ( A = B )
% 2.36/1.49 => ! [Xx: $i,Xy: $i] :
% 2.36/1.49 ( ( Xx = Xy )
% 2.36/1.49 => ( ( in @ Xx @ A )
% 2.36/1.49 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(in__Cong_type,type,
% 2.36/1.49 in__Cong: $o ).
% 2.36/1.49
% 2.36/1.49 thf(in__Cong,definition,
% 2.36/1.49 ( in__Cong
% 2.36/1.49 = ( ! [A: $i,B: $i] :
% 2.36/1.49 ( ( A = B )
% 2.36/1.49 => ! [Xx: $i,Xy: $i] :
% 2.36/1.49 ( ( Xx = Xy )
% 2.36/1.49 => ( ( in @ Xx @ A )
% 2.36/1.49 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(exuE3u_type,type,
% 2.36/1.49 exuE3u: $o ).
% 2.36/1.49
% 2.36/1.49 thf(exuE3u,definition,
% 2.36/1.49 ( exuE3u
% 2.36/1.49 = ( ! [Xphi: $i > $o] :
% 2.36/1.49 ( ( exu
% 2.36/1.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.36/1.49 => ! [Xx: $i,Xy: $i] :
% 2.36/1.49 ( ( Xphi @ Xx )
% 2.36/1.49 => ( ( Xphi @ Xy )
% 2.36/1.49 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(exu__Cong_type,type,
% 2.36/1.49 exu__Cong: $o ).
% 2.36/1.49
% 2.36/1.49 thf(exu__Cong,definition,
% 2.36/1.49 ( exu__Cong
% 2.36/1.49 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.36/1.49 ( ! [Xx: $i,Xy: $i] :
% 2.36/1.49 ( ( Xx = Xy )
% 2.36/1.49 => ( ( Xphi @ Xx )
% 2.36/1.49 <=> ( Xpsi @ Xy ) ) )
% 2.36/1.49 => ( ( exu
% 2.36/1.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.36/1.49 <=> ( exu
% 2.36/1.49 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(emptyset__Cong_type,type,
% 2.36/1.49 emptyset__Cong: $o ).
% 2.36/1.49
% 2.36/1.49 thf(emptyset__Cong,definition,
% 2.36/1.49 ( emptyset__Cong
% 2.36/1.49 = ( emptyset = emptyset ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(setadjoin__Cong_type,type,
% 2.36/1.49 setadjoin__Cong: $o ).
% 2.36/1.49
% 2.36/1.49 thf(setadjoin__Cong,definition,
% 2.36/1.49 ( setadjoin__Cong
% 2.36/1.49 = ( ! [Xx: $i,Xy: $i] :
% 2.36/1.49 ( ( Xx = Xy )
% 2.36/1.49 => ! [Xz: $i,Xu: $i] :
% 2.36/1.49 ( ( Xz = Xu )
% 2.36/1.49 => ( ( setadjoin @ Xx @ Xz )
% 2.36/1.49 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(powerset__Cong_type,type,
% 2.36/1.49 powerset__Cong: $o ).
% 2.36/1.49
% 2.36/1.49 thf(powerset__Cong,definition,
% 2.36/1.49 ( powerset__Cong
% 2.36/1.49 = ( ! [A: $i,B: $i] :
% 2.36/1.49 ( ( A = B )
% 2.36/1.49 => ( ( powerset @ A )
% 2.36/1.49 = ( powerset @ B ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(setunion__Cong_type,type,
% 2.36/1.49 setunion__Cong: $o ).
% 2.36/1.49
% 2.36/1.49 thf(setunion__Cong,definition,
% 2.36/1.49 ( setunion__Cong
% 2.36/1.49 = ( ! [A: $i,B: $i] :
% 2.36/1.49 ( ( A = B )
% 2.36/1.49 => ( ( setunion @ A )
% 2.36/1.49 = ( setunion @ B ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(omega__Cong_type,type,
% 2.36/1.49 omega__Cong: $o ).
% 2.36/1.49
% 2.36/1.49 thf(omega__Cong,definition,
% 2.36/1.49 ( omega__Cong
% 2.36/1.49 = ( omega = omega ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(exuEu_type,type,
% 2.36/1.49 exuEu: $o ).
% 2.36/1.49
% 2.36/1.49 thf(exuEu,definition,
% 2.36/1.49 ( exuEu
% 2.36/1.49 = ( ! [Xphi: $i > $o] :
% 2.36/1.49 ( ( exu
% 2.36/1.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.36/1.49 => ! [Xx: $i,Xy: $i] :
% 2.36/1.49 ( ( Xphi @ Xx )
% 2.36/1.49 => ( ( Xphi @ Xy )
% 2.36/1.49 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(descr__Cong_type,type,
% 2.36/1.49 descr__Cong: $o ).
% 2.36/1.49
% 2.36/1.49 thf(descr__Cong,definition,
% 2.36/1.49 ( descr__Cong
% 2.36/1.49 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.36/1.49 ( ! [Xx: $i,Xy: $i] :
% 2.36/1.49 ( ( Xx = Xy )
% 2.36/1.49 => ( ( Xphi @ Xx )
% 2.36/1.49 <=> ( Xpsi @ Xy ) ) )
% 2.36/1.49 => ( ( exu
% 2.36/1.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.36/1.49 => ( ( exu
% 2.36/1.49 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 2.36/1.49 => ( ( descr
% 2.36/1.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.36/1.49 = ( descr
% 2.36/1.49 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(dsetconstr__Cong_type,type,
% 2.36/1.49 dsetconstr__Cong: $o ).
% 2.36/1.49
% 2.36/1.49 thf(dsetconstr__Cong,definition,
% 2.36/1.49 ( dsetconstr__Cong
% 2.36/1.49 = ( ! [A: $i,B: $i] :
% 2.36/1.49 ( ( A = B )
% 2.36/1.49 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.36/1.49 ( ! [Xx: $i] :
% 2.36/1.49 ( ( in @ Xx @ A )
% 2.36/1.49 => ! [Xy: $i] :
% 2.36/1.49 ( ( in @ Xy @ B )
% 2.36/1.49 => ( ( Xx = Xy )
% 2.36/1.49 => ( ( Xphi @ Xx )
% 2.36/1.49 <=> ( Xpsi @ Xy ) ) ) ) )
% 2.36/1.49 => ( ( dsetconstr @ A
% 2.36/1.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.36/1.49 = ( dsetconstr @ B
% 2.36/1.49 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(subset_type,type,
% 2.36/1.49 subset: $i > $i > $o ).
% 2.36/1.49
% 2.36/1.49 thf(disjoint_type,type,
% 2.36/1.49 disjoint: $i > $i > $o ).
% 2.36/1.49
% 2.36/1.49 thf(setsmeet_type,type,
% 2.36/1.49 setsmeet: $i > $i > $o ).
% 2.36/1.49
% 2.36/1.49 thf(subsetI1_type,type,
% 2.36/1.49 subsetI1: $o ).
% 2.36/1.49
% 2.36/1.49 thf(subsetI1,definition,
% 2.36/1.49 ( subsetI1
% 2.36/1.49 = ( ! [A: $i,B: $i] :
% 2.36/1.49 ( ! [Xx: $i] :
% 2.36/1.49 ( ( in @ Xx @ A )
% 2.36/1.49 => ( in @ Xx @ B ) )
% 2.36/1.49 => ( subset @ A @ B ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(eqimpsubset2_type,type,
% 2.36/1.49 eqimpsubset2: $o ).
% 2.36/1.49
% 2.36/1.49 thf(eqimpsubset2,definition,
% 2.36/1.49 ( eqimpsubset2
% 2.36/1.49 = ( ! [A: $i,B: $i] :
% 2.36/1.49 ( ( A = B )
% 2.36/1.49 => ( subset @ B @ A ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(eqimpsubset1_type,type,
% 2.36/1.49 eqimpsubset1: $o ).
% 2.36/1.49
% 2.36/1.49 thf(eqimpsubset1,definition,
% 2.36/1.49 ( eqimpsubset1
% 2.36/1.49 = ( ! [A: $i,B: $i] :
% 2.36/1.49 ( ( A = B )
% 2.36/1.49 => ( subset @ A @ B ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(subsetI2_type,type,
% 2.36/1.49 subsetI2: $o ).
% 2.36/1.49
% 2.36/1.49 thf(subsetI2,definition,
% 2.36/1.49 ( subsetI2
% 2.36/1.49 = ( ! [A: $i,B: $i] :
% 2.36/1.49 ( ! [Xx: $i] :
% 2.36/1.49 ( ( in @ Xx @ A )
% 2.36/1.49 => ( in @ Xx @ B ) )
% 2.36/1.49 => ( subset @ A @ B ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(emptysetsubset_type,type,
% 2.36/1.49 emptysetsubset: $o ).
% 2.36/1.49
% 2.36/1.49 thf(emptysetsubset,definition,
% 2.36/1.49 ( emptysetsubset
% 2.36/1.49 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(subsetE_type,type,
% 2.36/1.49 subsetE: $o ).
% 2.36/1.49
% 2.36/1.49 thf(subsetE,definition,
% 2.36/1.49 ( subsetE
% 2.36/1.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.36/1.49 ( ( subset @ A @ B )
% 2.36/1.49 => ( ( in @ Xx @ A )
% 2.36/1.49 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(subsetE2_type,type,
% 2.36/1.49 subsetE2: $o ).
% 2.36/1.49
% 2.36/1.49 thf(subsetE2,definition,
% 2.36/1.49 ( subsetE2
% 2.36/1.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.36/1.49 ( ( subset @ A @ B )
% 2.36/1.49 => ( ~ ( in @ Xx @ B )
% 2.36/1.49 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(notsubsetI_type,type,
% 2.36/1.49 notsubsetI: $o ).
% 2.36/1.49
% 2.36/1.49 thf(notsubsetI,definition,
% 2.36/1.49 ( notsubsetI
% 2.36/1.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.36/1.49 ( ( in @ Xx @ A )
% 2.36/1.49 => ( ~ ( in @ Xx @ B )
% 2.36/1.49 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(notequalI1_type,type,
% 2.36/1.49 notequalI1: $o ).
% 2.36/1.49
% 2.36/1.49 thf(notequalI1,definition,
% 2.36/1.49 ( notequalI1
% 2.36/1.49 = ( ! [A: $i,B: $i] :
% 2.36/1.49 ( ~ ( subset @ A @ B )
% 2.36/1.49 => ( A != B ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(notequalI2_type,type,
% 2.36/1.49 notequalI2: $o ).
% 2.36/1.49
% 2.36/1.49 thf(notequalI2,definition,
% 2.36/1.49 ( notequalI2
% 2.36/1.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.36/1.49 ( ( in @ Xx @ A )
% 2.36/1.49 => ( ~ ( in @ Xx @ B )
% 2.36/1.49 => ( A != B ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(subsetRefl_type,type,
% 2.36/1.49 subsetRefl: $o ).
% 2.36/1.49
% 2.36/1.49 thf(subsetRefl,definition,
% 2.36/1.49 ( subsetRefl
% 2.36/1.49 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(subsetTrans_type,type,
% 2.36/1.49 subsetTrans: $o ).
% 2.36/1.49
% 2.36/1.49 thf(subsetTrans,definition,
% 2.36/1.49 ( subsetTrans
% 2.36/1.49 = ( ! [A: $i,B: $i,C: $i] :
% 2.36/1.49 ( ( subset @ A @ B )
% 2.36/1.49 => ( ( subset @ B @ C )
% 2.36/1.49 => ( subset @ A @ C ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(setadjoinSub_type,type,
% 2.36/1.49 setadjoinSub: $o ).
% 2.36/1.49
% 2.36/1.49 thf(setadjoinSub,definition,
% 2.36/1.49 ( setadjoinSub
% 2.36/1.49 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(setadjoinSub2_type,type,
% 2.36/1.49 setadjoinSub2: $o ).
% 2.36/1.49
% 2.36/1.49 thf(setadjoinSub2,definition,
% 2.36/1.49 ( setadjoinSub2
% 2.36/1.49 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.36/1.49 ( ( subset @ A @ B )
% 2.36/1.49 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(subset2powerset_type,type,
% 2.36/1.49 subset2powerset: $o ).
% 2.36/1.49
% 2.36/1.49 thf(subset2powerset,definition,
% 2.36/1.49 ( subset2powerset
% 2.36/1.49 = ( ! [A: $i,B: $i] :
% 2.36/1.49 ( ( subset @ A @ B )
% 2.36/1.49 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(setextsub_type,type,
% 2.36/1.49 setextsub: $o ).
% 2.36/1.49
% 2.36/1.49 thf(setextsub,definition,
% 2.36/1.49 ( setextsub
% 2.36/1.49 = ( ! [A: $i,B: $i] :
% 2.36/1.49 ( ( subset @ A @ B )
% 2.36/1.49 => ( ( subset @ B @ A )
% 2.36/1.49 => ( A = B ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(subsetemptysetimpeq_type,type,
% 2.36/1.49 subsetemptysetimpeq: $o ).
% 2.36/1.49
% 2.36/1.49 thf(subsetemptysetimpeq,definition,
% 2.36/1.49 ( subsetemptysetimpeq
% 2.36/1.49 = ( ! [A: $i] :
% 2.36/1.49 ( ( subset @ A @ emptyset )
% 2.36/1.49 => ( A = emptyset ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(powersetI1_type,type,
% 2.36/1.49 powersetI1: $o ).
% 2.36/1.49
% 2.36/1.49 thf(powersetI1,definition,
% 2.36/1.49 ( powersetI1
% 2.36/1.49 = ( ! [A: $i,B: $i] :
% 2.36/1.49 ( ( subset @ B @ A )
% 2.36/1.49 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(powersetE1_type,type,
% 2.36/1.49 powersetE1: $o ).
% 2.36/1.49
% 2.36/1.49 thf(powersetE1,definition,
% 2.36/1.49 ( powersetE1
% 2.36/1.49 = ( ! [A: $i,B: $i] :
% 2.36/1.49 ( ( in @ B @ ( powerset @ A ) )
% 2.36/1.49 => ( subset @ B @ A ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(inPowerset_type,type,
% 2.36/1.49 inPowerset: $o ).
% 2.36/1.49
% 2.36/1.49 thf(inPowerset,definition,
% 2.36/1.49 ( inPowerset
% 2.36/1.49 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(powersetsubset_type,type,
% 2.36/1.49 powersetsubset: $o ).
% 2.36/1.49
% 2.36/1.49 thf(powersetsubset,definition,
% 2.36/1.49 ( powersetsubset
% 2.36/1.49 = ( ! [A: $i,B: $i] :
% 2.36/1.49 ( ( subset @ A @ B )
% 2.36/1.49 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(sepInPowerset_type,type,
% 2.36/1.49 sepInPowerset: $o ).
% 2.36/1.49
% 2.36/1.49 thf(sepInPowerset,definition,
% 2.36/1.49 ( sepInPowerset
% 2.36/1.49 = ( ! [A: $i,Xphi: $i > $o] :
% 2.36/1.49 ( in
% 2.36/1.49 @ ( dsetconstr @ A
% 2.36/1.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.36/1.49 @ ( powerset @ A ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(sepSubset_type,type,
% 2.36/1.49 sepSubset: $o ).
% 2.36/1.49
% 2.36/1.49 thf(sepSubset,definition,
% 2.36/1.49 ( sepSubset
% 2.36/1.49 = ( ! [A: $i,Xphi: $i > $o] :
% 2.36/1.49 ( subset
% 2.36/1.49 @ ( dsetconstr @ A
% 2.36/1.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.36/1.49 @ A ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(binunion_type,type,
% 2.36/1.49 binunion: $i > $i > $i ).
% 2.36/1.49
% 2.36/1.49 thf(binunionIL_type,type,
% 2.36/1.49 binunionIL: $o ).
% 2.36/1.49
% 2.36/1.49 thf(binunionIL,definition,
% 2.36/1.49 ( binunionIL
% 2.36/1.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.36/1.49 ( ( in @ Xx @ A )
% 2.36/1.49 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(upairset2IR_type,type,
% 2.36/1.49 upairset2IR: $o ).
% 2.36/1.49
% 2.36/1.49 thf(upairset2IR,definition,
% 2.36/1.49 ( upairset2IR
% 2.36/1.49 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(binunionIR_type,type,
% 2.36/1.49 binunionIR: $o ).
% 2.36/1.49
% 2.36/1.49 thf(binunionIR,definition,
% 2.36/1.49 ( binunionIR
% 2.36/1.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.36/1.49 ( ( in @ Xx @ B )
% 2.36/1.49 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(binunionEcases_type,type,
% 2.36/1.49 binunionEcases: $o ).
% 2.36/1.49
% 2.36/1.49 thf(binunionEcases,definition,
% 2.36/1.49 ( binunionEcases
% 2.36/1.49 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 2.36/1.49 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.36/1.49 => ( ( ( in @ Xx @ A )
% 2.36/1.49 => Xphi )
% 2.36/1.49 => ( ( ( in @ Xx @ B )
% 2.36/1.49 => Xphi )
% 2.36/1.49 => Xphi ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(binunionE_type,type,
% 2.36/1.49 binunionE: $o ).
% 2.36/1.49
% 2.36/1.49 thf(binunionE,definition,
% 2.36/1.49 ( binunionE
% 2.36/1.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.36/1.49 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.36/1.49 => ( ( in @ Xx @ A )
% 2.36/1.49 | ( in @ Xx @ B ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(binunionLsub_type,type,
% 2.36/1.49 binunionLsub: $o ).
% 2.36/1.49
% 2.36/1.49 thf(binunionLsub,definition,
% 2.36/1.49 ( binunionLsub
% 2.36/1.49 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(binunionRsub_type,type,
% 2.36/1.49 binunionRsub: $o ).
% 2.36/1.49
% 2.36/1.49 thf(binunionRsub,definition,
% 2.36/1.49 ( binunionRsub
% 2.36/1.49 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(binintersect_type,type,
% 2.36/1.49 binintersect: $i > $i > $i ).
% 2.36/1.49
% 2.36/1.49 thf(binintersectI_type,type,
% 2.36/1.49 binintersectI: $o ).
% 2.36/1.49
% 2.36/1.49 thf(binintersectI,definition,
% 2.36/1.49 ( binintersectI
% 2.36/1.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.36/1.49 ( ( in @ Xx @ A )
% 2.36/1.49 => ( ( in @ Xx @ B )
% 2.36/1.49 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(binintersectSubset5_type,type,
% 2.36/1.49 binintersectSubset5: $o ).
% 2.36/1.49
% 2.36/1.49 thf(binintersectSubset5,definition,
% 2.36/1.49 ( binintersectSubset5
% 2.36/1.49 = ( ! [A: $i,B: $i,C: $i] :
% 2.36/1.49 ( ( subset @ C @ A )
% 2.36/1.49 => ( ( subset @ C @ B )
% 2.36/1.49 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(binintersectEL_type,type,
% 2.36/1.49 binintersectEL: $o ).
% 2.36/1.49
% 2.36/1.49 thf(binintersectEL,definition,
% 2.36/1.49 ( binintersectEL
% 2.36/1.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.36/1.49 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.36/1.49 => ( in @ Xx @ A ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(binintersectLsub_type,type,
% 2.36/1.49 binintersectLsub: $o ).
% 2.36/1.49
% 2.36/1.49 thf(binintersectLsub,definition,
% 2.36/1.49 ( binintersectLsub
% 2.36/1.49 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(binintersectSubset2_type,type,
% 2.36/1.49 binintersectSubset2: $o ).
% 2.36/1.49
% 2.36/1.49 thf(binintersectSubset2,definition,
% 2.36/1.49 ( binintersectSubset2
% 2.36/1.49 = ( ! [A: $i,B: $i] :
% 2.36/1.49 ( ( subset @ A @ B )
% 2.36/1.49 => ( ( binintersect @ A @ B )
% 2.36/1.49 = A ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(binintersectSubset3_type,type,
% 2.36/1.49 binintersectSubset3: $o ).
% 2.36/1.49
% 2.36/1.49 thf(binintersectSubset3,definition,
% 2.36/1.49 ( binintersectSubset3
% 2.36/1.49 = ( ! [A: $i,B: $i] :
% 2.36/1.49 ( ( ( binintersect @ A @ B )
% 2.36/1.49 = B )
% 2.36/1.49 => ( subset @ B @ A ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(binintersectER_type,type,
% 2.36/1.49 binintersectER: $o ).
% 2.36/1.49
% 2.36/1.49 thf(binintersectER,definition,
% 2.36/1.49 ( binintersectER
% 2.36/1.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.36/1.49 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.36/1.49 => ( in @ Xx @ B ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(disjointsetsI1_type,type,
% 2.36/1.49 disjointsetsI1: $o ).
% 2.36/1.49
% 2.36/1.49 thf(disjointsetsI1,definition,
% 2.36/1.49 ( disjointsetsI1
% 2.36/1.49 = ( ! [A: $i,B: $i] :
% 2.36/1.49 ( ~ ? [Xx: $i] :
% 2.36/1.49 ( ( in @ Xx @ A )
% 2.36/1.49 & ( in @ Xx @ B ) )
% 2.36/1.49 => ( ( binintersect @ A @ B )
% 2.36/1.49 = emptyset ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(binintersectRsub_type,type,
% 2.36/1.49 binintersectRsub: $o ).
% 2.36/1.49
% 2.36/1.49 thf(binintersectRsub,definition,
% 2.36/1.49 ( binintersectRsub
% 2.36/1.49 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(binintersectSubset4_type,type,
% 2.36/1.49 binintersectSubset4: $o ).
% 2.36/1.49
% 2.36/1.49 thf(binintersectSubset4,definition,
% 2.36/1.49 ( binintersectSubset4
% 2.36/1.49 = ( ! [A: $i,B: $i] :
% 2.36/1.49 ( ( subset @ B @ A )
% 2.36/1.49 => ( ( binintersect @ A @ B )
% 2.36/1.49 = B ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(binintersectSubset1_type,type,
% 2.36/1.49 binintersectSubset1: $o ).
% 2.36/1.49
% 2.36/1.49 thf(binintersectSubset1,definition,
% 2.36/1.49 ( binintersectSubset1
% 2.36/1.49 = ( ! [A: $i,B: $i] :
% 2.36/1.49 ( ( ( binintersect @ A @ B )
% 2.36/1.49 = A )
% 2.36/1.49 => ( subset @ A @ B ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(bs114d_type,type,
% 2.36/1.49 bs114d: $o ).
% 2.36/1.49
% 2.36/1.49 thf(bs114d,definition,
% 2.36/1.49 ( bs114d
% 2.36/1.49 = ( ! [A: $i,B: $i,C: $i] :
% 2.36/1.49 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 2.36/1.49 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(regular_type,type,
% 2.36/1.49 regular: $i > $o ).
% 2.36/1.49
% 2.36/1.49 thf(setminus_type,type,
% 2.36/1.49 setminus: $i > $i > $i ).
% 2.36/1.49
% 2.36/1.49 thf(setminusI_type,type,
% 2.36/1.49 setminusI: $o ).
% 2.36/1.49
% 2.36/1.49 thf(setminusI,definition,
% 2.36/1.49 ( setminusI
% 2.36/1.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.36/1.49 ( ( in @ Xx @ A )
% 2.36/1.49 => ( ~ ( in @ Xx @ B )
% 2.36/1.49 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(setminusEL_type,type,
% 2.36/1.49 setminusEL: $o ).
% 2.36/1.49
% 2.36/1.49 thf(setminusEL,definition,
% 2.36/1.49 ( setminusEL
% 2.36/1.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.36/1.49 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.36/1.49 => ( in @ Xx @ A ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(setminusER_type,type,
% 2.36/1.49 setminusER: $o ).
% 2.36/1.49
% 2.36/1.49 thf(setminusER,definition,
% 2.36/1.49 ( setminusER
% 2.36/1.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.36/1.49 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.36/1.49 => ~ ( in @ Xx @ B ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(setminusSubset2_type,type,
% 2.36/1.49 setminusSubset2: $o ).
% 2.36/1.49
% 2.36/1.49 thf(setminusSubset2,definition,
% 2.36/1.49 ( setminusSubset2
% 2.36/1.49 = ( ! [A: $i,B: $i] :
% 2.36/1.49 ( ( subset @ A @ B )
% 2.36/1.49 => ( ( setminus @ A @ B )
% 2.36/1.49 = emptyset ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(setminusERneg_type,type,
% 2.36/1.49 setminusERneg: $o ).
% 2.36/1.49
% 2.36/1.49 thf(setminusERneg,definition,
% 2.36/1.49 ( setminusERneg
% 2.36/1.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.36/1.49 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.36/1.49 => ( ( in @ Xx @ A )
% 2.36/1.49 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(setminusELneg_type,type,
% 2.36/1.49 setminusELneg: $o ).
% 2.36/1.49
% 2.36/1.49 thf(setminusELneg,definition,
% 2.36/1.49 ( setminusELneg
% 2.36/1.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.36/1.49 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.36/1.49 => ( ~ ( in @ Xx @ B )
% 2.36/1.49 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(setminusILneg_type,type,
% 2.36/1.49 setminusILneg: $o ).
% 2.36/1.49
% 2.36/1.49 thf(setminusILneg,definition,
% 2.36/1.49 ( setminusILneg
% 2.36/1.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.36/1.49 ( ~ ( in @ Xx @ A )
% 2.36/1.49 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(setminusIRneg_type,type,
% 2.36/1.49 setminusIRneg: $o ).
% 2.36/1.49
% 2.36/1.49 thf(setminusIRneg,definition,
% 2.36/1.49 ( setminusIRneg
% 2.36/1.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.36/1.49 ( ( in @ Xx @ B )
% 2.36/1.49 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(setminusLsub_type,type,
% 2.36/1.49 setminusLsub: $o ).
% 2.36/1.49
% 2.36/1.49 thf(setminusLsub,definition,
% 2.36/1.49 ( setminusLsub
% 2.36/1.49 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(setminusSubset1_type,type,
% 2.36/1.49 setminusSubset1: $o ).
% 2.36/1.49
% 2.36/1.49 thf(setminusSubset1,definition,
% 2.36/1.49 ( setminusSubset1
% 2.36/1.49 = ( ! [A: $i,B: $i] :
% 2.36/1.49 ( ( ( setminus @ A @ B )
% 2.36/1.49 = emptyset )
% 2.36/1.49 => ( subset @ A @ B ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(symdiff_type,type,
% 2.36/1.49 symdiff: $i > $i > $i ).
% 2.36/1.49
% 2.36/1.49 thf(symdiffE_type,type,
% 2.36/1.49 symdiffE: $o ).
% 2.36/1.49
% 2.36/1.49 thf(symdiffE,definition,
% 2.36/1.49 ( symdiffE
% 2.36/1.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.36/1.49 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 2.36/1.49 => ! [Xphi: $o] :
% 2.36/1.49 ( ( ( in @ Xx @ A )
% 2.36/1.49 => ( ~ ( in @ Xx @ B )
% 2.36/1.49 => Xphi ) )
% 2.36/1.49 => ( ( ~ ( in @ Xx @ A )
% 2.36/1.49 => ( ( in @ Xx @ B )
% 2.36/1.49 => Xphi ) )
% 2.36/1.49 => Xphi ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(symdiffI1_type,type,
% 2.36/1.49 symdiffI1: $o ).
% 2.36/1.49
% 2.36/1.49 thf(symdiffI1,definition,
% 2.36/1.49 ( symdiffI1
% 2.36/1.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.36/1.49 ( ( in @ Xx @ A )
% 2.36/1.49 => ( ~ ( in @ Xx @ B )
% 2.36/1.49 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(symdiffI2_type,type,
% 2.36/1.49 symdiffI2: $o ).
% 2.36/1.49
% 2.36/1.49 thf(symdiffI2,definition,
% 2.36/1.49 ( symdiffI2
% 2.36/1.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.36/1.49 ( ~ ( in @ Xx @ A )
% 2.36/1.49 => ( ( in @ Xx @ B )
% 2.36/1.49 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(symdiffIneg1_type,type,
% 2.36/1.49 symdiffIneg1: $o ).
% 2.36/1.49
% 2.36/1.49 thf(symdiffIneg1,definition,
% 2.36/1.49 ( symdiffIneg1
% 2.36/1.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.36/1.49 ( ( in @ Xx @ A )
% 2.36/1.49 => ( ( in @ Xx @ B )
% 2.36/1.49 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(symdiffIneg2_type,type,
% 2.36/1.49 symdiffIneg2: $o ).
% 2.36/1.49
% 2.36/1.49 thf(symdiffIneg2,definition,
% 2.36/1.49 ( symdiffIneg2
% 2.36/1.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.36/1.49 ( ~ ( in @ Xx @ A )
% 2.36/1.49 => ( ~ ( in @ Xx @ B )
% 2.36/1.49 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(iskpair_type,type,
% 2.36/1.49 iskpair: $i > $o ).
% 2.36/1.49
% 2.36/1.49 thf(iskpair,definition,
% 2.36/1.49 ( iskpair
% 2.36/1.49 = ( ^ [A: $i] :
% 2.36/1.49 ? [Xx: $i] :
% 2.36/1.49 ( ( in @ Xx @ ( setunion @ A ) )
% 2.36/1.49 & ? [Xy: $i] :
% 2.36/1.49 ( ( in @ Xy @ ( setunion @ A ) )
% 2.36/1.49 & ( A
% 2.36/1.49 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(secondinupair_type,type,
% 2.36/1.49 secondinupair: $o ).
% 2.36/1.49
% 2.36/1.49 thf(secondinupair,definition,
% 2.36/1.49 ( secondinupair
% 2.36/1.49 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(setukpairIL_type,type,
% 2.36/1.49 setukpairIL: $o ).
% 2.36/1.49
% 2.36/1.49 thf(setukpairIL,definition,
% 2.36/1.49 ( setukpairIL
% 2.36/1.49 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(setukpairIR_type,type,
% 2.36/1.49 setukpairIR: $o ).
% 2.36/1.49
% 2.36/1.49 thf(setukpairIR,definition,
% 2.36/1.49 ( setukpairIR
% 2.36/1.49 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(kpairiskpair_type,type,
% 2.36/1.49 kpairiskpair: $o ).
% 2.36/1.49
% 2.36/1.49 thf(kpairiskpair,definition,
% 2.36/1.49 ( kpairiskpair
% 2.36/1.49 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(kpair_type,type,
% 2.36/1.49 kpair: $i > $i > $i ).
% 2.36/1.49
% 2.36/1.49 thf(kpair,definition,
% 2.36/1.49 ( kpair
% 2.36/1.49 = ( ^ [Xx: $i,Xy: $i] : ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(kpairp_type,type,
% 2.36/1.49 kpairp: $o ).
% 2.36/1.49
% 2.36/1.49 thf(kpairp,definition,
% 2.36/1.49 ( kpairp
% 2.36/1.49 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(cartprod_type,type,
% 2.36/1.49 cartprod: $i > $i > $i ).
% 2.36/1.49
% 2.36/1.49 thf(cartprod,definition,
% 2.36/1.49 ( cartprod
% 2.36/1.49 = ( ^ [A: $i,B: $i] :
% 2.36/1.49 ( dsetconstr @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) )
% 2.36/1.49 @ ^ [Xx: $i] :
% 2.36/1.49 ? [Xy: $i] :
% 2.36/1.49 ( ( in @ Xy @ A )
% 2.36/1.49 & ? [Xz: $i] :
% 2.36/1.49 ( ( in @ Xz @ B )
% 2.36/1.49 & ( Xx
% 2.36/1.49 = ( kpair @ Xy @ Xz ) ) ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(singletonsubset_type,type,
% 2.36/1.49 singletonsubset: $o ).
% 2.36/1.49
% 2.36/1.49 thf(singletonsubset,definition,
% 2.36/1.49 ( singletonsubset
% 2.36/1.49 = ( ! [A: $i,Xx: $i] :
% 2.36/1.49 ( ( in @ Xx @ A )
% 2.36/1.49 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(singletoninpowerset_type,type,
% 2.36/1.49 singletoninpowerset: $o ).
% 2.36/1.49
% 2.36/1.49 thf(singletoninpowerset,definition,
% 2.36/1.49 ( singletoninpowerset
% 2.36/1.49 = ( ! [A: $i,Xx: $i] :
% 2.36/1.49 ( ( in @ Xx @ A )
% 2.36/1.49 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(singletoninpowunion_type,type,
% 2.36/1.49 singletoninpowunion: $o ).
% 2.36/1.49
% 2.36/1.49 thf(singletoninpowunion,definition,
% 2.36/1.49 ( singletoninpowunion
% 2.36/1.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.36/1.49 ( ( in @ Xx @ A )
% 2.36/1.49 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(upairset2E_type,type,
% 2.36/1.49 upairset2E: $o ).
% 2.36/1.49
% 2.36/1.49 thf(upairset2E,definition,
% 2.36/1.49 ( upairset2E
% 2.36/1.49 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.36/1.49 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.36/1.49 => ( ( Xz = Xx )
% 2.36/1.49 | ( Xz = Xy ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(upairsubunion_type,type,
% 2.36/1.49 upairsubunion: $o ).
% 2.36/1.49
% 2.36/1.49 thf(upairsubunion,definition,
% 2.36/1.49 ( upairsubunion
% 2.36/1.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.36/1.49 ( ( in @ Xx @ A )
% 2.36/1.49 => ! [Xy: $i] :
% 2.36/1.49 ( ( in @ Xy @ B )
% 2.36/1.49 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(upairinpowunion_type,type,
% 2.36/1.49 upairinpowunion: $o ).
% 2.36/1.49
% 2.36/1.49 thf(upairinpowunion,definition,
% 2.36/1.49 ( upairinpowunion
% 2.36/1.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.36/1.49 ( ( in @ Xx @ A )
% 2.36/1.49 => ! [Xy: $i] :
% 2.36/1.49 ( ( in @ Xy @ B )
% 2.36/1.49 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(ubforcartprodlem1_type,type,
% 2.36/1.49 ubforcartprodlem1: $o ).
% 2.36/1.49
% 2.36/1.49 thf(ubforcartprodlem1,definition,
% 2.36/1.49 ( ubforcartprodlem1
% 2.36/1.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.36/1.49 ( ( in @ Xx @ A )
% 2.36/1.49 => ! [Xy: $i] :
% 2.36/1.49 ( ( in @ Xy @ B )
% 2.36/1.49 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.36/1.49
% 2.36/1.49 thf(ubforcartprodlem2,conjecture,
% 2.36/1.49 ( setextAx
% 2.36/1.49 => ( emptysetAx
% 2.36/1.49 => ( setadjoinAx
% 2.36/1.49 => ( powersetAx
% 2.36/1.49 => ( setunionAx
% 2.36/1.49 => ( omega0Ax
% 2.36/1.49 => ( omegaSAx
% 2.36/1.49 => ( omegaIndAx
% 2.36/1.49 => ( replAx
% 2.36/1.49 => ( foundationAx
% 2.36/1.49 => ( wellorderingAx
% 2.36/1.49 => ( descrp
% 2.36/1.49 => ( dsetconstrI
% 2.36/1.49 => ( dsetconstrEL
% 2.36/1.49 => ( dsetconstrER
% 2.36/1.49 => ( exuE1
% 2.36/1.49 => ( prop2setE
% 2.36/1.49 => ( emptysetE
% 2.36/1.49 => ( emptysetimpfalse
% 2.36/1.49 => ( notinemptyset
% 2.36/1.49 => ( exuE3e
% 2.36/1.49 => ( setext
% 2.36/1.49 => ( emptyI
% 2.36/1.49 => ( noeltsimpempty
% 2.36/1.49 => ( setbeta
% 2.36/1.49 => ( nonemptyE1
% 2.36/1.49 => ( nonemptyI
% 2.36/1.49 => ( nonemptyI1
% 2.36/1.49 => ( setadjoinIL
% 2.36/1.49 => ( emptyinunitempty
% 2.36/1.49 => ( setadjoinIR
% 2.36/1.49 => ( setadjoinE
% 2.36/1.49 => ( setadjoinOr
% 2.36/1.49 => ( setoftrueEq
% 2.36/1.49 => ( powersetI
% 2.36/1.49 => ( emptyinPowerset
% 2.36/1.49 => ( emptyInPowerset
% 2.36/1.49 => ( powersetE
% 2.36/1.49 => ( setunionI
% 2.36/1.49 => ( setunionE
% 2.36/1.49 => ( subPowSU
% 2.36/1.49 => ( exuE2
% 2.36/1.49 => ( nonemptyImpWitness
% 2.36/1.49 => ( uniqinunit
% 2.36/1.49 => ( notinsingleton
% 2.36/1.49 => ( eqinunit
% 2.36/1.49 => ( singletonsswitch
% 2.36/1.49 => ( upairsetE
% 2.36/1.49 => ( upairsetIL
% 2.36/1.49 => ( upairsetIR
% 2.36/1.49 => ( emptyE1
% 2.36/1.49 => ( vacuousDall
% 2.36/1.49 => ( quantDeMorgan1
% 2.36/1.49 => ( quantDeMorgan2
% 2.36/1.49 => ( quantDeMorgan3
% 2.36/1.49 => ( quantDeMorgan4
% 2.36/1.49 => ( prop2setI
% 2.36/1.49 => ( prop2set2propI
% 2.36/1.49 => ( notdexE
% 2.36/1.49 => ( notdallE
% 2.36/1.49 => ( exuI1
% 2.36/1.49 => ( exuI3
% 2.36/1.49 => ( exuI2
% 2.36/1.49 => ( inCongP
% 2.36/1.49 => ( in__Cong
% 2.36/1.49 => ( exuE3u
% 2.36/1.49 => ( exu__Cong
% 2.36/1.49 => ( emptyset__Cong
% 2.36/1.49 => ( setadjoin__Cong
% 2.36/1.49 => ( powerset__Cong
% 2.36/1.49 => ( setunion__Cong
% 2.36/1.49 => ( omega__Cong
% 2.36/1.49 => ( exuEu
% 2.36/1.49 => ( descr__Cong
% 2.36/1.49 => ( dsetconstr__Cong
% 2.36/1.49 => ( subsetI1
% 2.36/1.49 => ( eqimpsubset2
% 2.36/1.49 => ( eqimpsubset1
% 2.36/1.49 => ( subsetI2
% 2.36/1.49 => ( emptysetsubset
% 2.36/1.49 => ( subsetE
% 2.36/1.49 => ( subsetE2
% 2.36/1.49 => ( notsubsetI
% 2.36/1.49 => ( notequalI1
% 2.36/1.49 => ( notequalI2
% 2.36/1.49 => ( subsetRefl
% 2.36/1.49 => ( subsetTrans
% 2.36/1.49 => ( setadjoinSub
% 2.36/1.49 => ( setadjoinSub2
% 2.36/1.49 => ( subset2powerset
% 2.36/1.49 => ( setextsub
% 2.36/1.49 => ( subsetemptysetimpeq
% 2.36/1.49 => ( powersetI1
% 2.36/1.49 => ( powersetE1
% 2.36/1.49 => ( inPowerset
% 2.36/1.49 => ( powersetsubset
% 2.36/1.49 => ( sepInPowerset
% 2.36/1.49 => ( sepSubset
% 2.36/1.49 => ( binunionIL
% 2.36/1.49 => ( upairset2IR
% 2.36/1.49 => ( binunionIR
% 2.36/1.49 => ( binunionEcases
% 2.36/1.49 => ( binunionE
% 2.36/1.49 => ( binunionLsub
% 2.36/1.49 => ( binunionRsub
% 2.36/1.49 => ( binintersectI
% 2.36/1.49 => ( binintersectSubset5
% 2.36/1.49 => ( binintersectEL
% 2.36/1.49 => ( binintersectLsub
% 2.36/1.49 => ( binintersectSubset2
% 2.36/1.49 => ( binintersectSubset3
% 2.36/1.49 => ( binintersectER
% 2.36/1.49 => ( disjointsetsI1
% 2.36/1.49 => ( binintersectRsub
% 2.36/1.49 => ( binintersectSubset4
% 2.36/1.49 => ( binintersectSubset1
% 2.36/1.49 => ( bs114d
% 2.36/1.49 => ( setminusI
% 2.36/1.49 => ( setminusEL
% 2.36/1.49 => ( setminusER
% 2.36/1.49 => ( setminusSubset2
% 2.36/1.49 => ( setminusERneg
% 2.36/1.49 => ( setminusELneg
% 2.36/1.49 => ( setminusILneg
% 2.36/1.49 => ( setminusIRneg
% 2.36/1.49 => ( setminusLsub
% 2.36/1.49 => ( setminusSubset1
% 2.36/1.49 => ( symdiffE
% 2.36/1.49 => ( symdiffI1
% 2.36/1.49 => ( symdiffI2
% 2.36/1.49 => ( symdiffIneg1
% 2.36/1.49 => ( symdiffIneg2
% 2.36/1.49 => ( secondinupair
% 2.36/1.49 => ( setukpairIL
% 2.36/1.49 => ( setukpairIR
% 2.36/1.49 => ( kpairiskpair
% 2.36/1.49 => ( kpairp
% 2.36/1.49 => ( singletonsubset
% 2.36/1.49 => ( singletoninpowerset
% 2.36/1.49 => ( singletoninpowunion
% 2.36/1.49 => ( upairset2E
% 2.36/1.49 => ( upairsubunion
% 2.36/1.49 => ( upairinpowunion
% 2.36/1.49 => ( ubforcartprodlem1
% 2.36/1.49 => ! [A: $i,B: $i,Xx: $i] :
% 2.36/1.49 ( ( in @ Xx @ A )
% 2.36/1.49 => ! [Xy: $i] :
% 2.36/1.49 ( ( in @ Xy @ B )
% 2.36/1.49 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.36/1.49 ------------------------
% 2.53/1.51 % (27798)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_27586 for (2999ds/2Mi)
% 2.53/1.51 % (27796)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_27586 for (2999ds/4Mi)
% 2.53/1.51 % (27798)Instruction limit reached!
% 2.53/1.51 % (27798)------------------------------
% 2.53/1.51 % (27798)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.53/1.51 % (27798)Termination reason: Unknown
% 2.53/1.51 % (27798)Termination phase: shuffling
% 2.53/1.51
% 2.53/1.51 % (27798)Memory used [KB]: 1279
% 2.53/1.51 % (27798)Time elapsed: 0.002 s
% 2.53/1.51 % (27798)Instructions burned: 3 (million)
% 2.53/1.51 % (27798)------------------------------
% 2.53/1.51 % (27798)------------------------------
% 2.53/1.51 % (27796)Instruction limit reached!
% 2.53/1.51 % (27796)------------------------------
% 2.53/1.51 % (27796)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.53/1.51 % (27796)Termination reason: Unknown
% 2.53/1.51 % (27796)Termination phase: shuffling
% 2.53/1.51
% 2.53/1.51 % (27796)Memory used [KB]: 1279
% 2.53/1.51 % (27796)Time elapsed: 0.003 s
% 2.53/1.51 % (27796)Instructions burned: 6 (million)
% 2.53/1.51 % (27796)------------------------------
% 2.53/1.51 % (27796)------------------------------
% 2.53/1.51 % (27797)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_27586 for (2999ds/27Mi)
% 2.53/1.51 % (27799)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_27586 for (2999ds/2Mi)
% 2.53/1.51 % (27801)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_27586 for (2999ds/18Mi)
% 2.53/1.51 % (27800)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_27586 for (2999ds/275Mi)
% 2.53/1.51 % (27795)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_27586 for (2999ds/183Mi)
% 2.53/1.51 % (27799)Instruction limit reached!
% 2.53/1.51 % (27799)------------------------------
% 2.53/1.51 % (27799)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.53/1.51 % (27799)Termination reason: Unknown
% 2.53/1.51 % (27799)Termination phase: shuffling
% 2.53/1.51
% 2.53/1.51 % (27799)Memory used [KB]: 1279
% 2.53/1.51 % (27799)Time elapsed: 0.004 s
% 2.53/1.51 % (27799)Instructions burned: 3 (million)
% 2.53/1.51 % (27799)------------------------------
% 2.53/1.51 % (27799)------------------------------
% 2.53/1.52 % (27802)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_27586 for (2999ds/3Mi)
% 2.53/1.52 % (27803)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_27586 for (2999ds/37Mi)
% 2.53/1.52 % (27802)Instruction limit reached!
% 2.53/1.52 % (27802)------------------------------
% 2.53/1.52 % (27802)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.53/1.52 % (27802)Termination reason: Unknown
% 2.53/1.52 % (27802)Termination phase: shuffling
% 2.53/1.52
% 2.53/1.52 % (27802)Memory used [KB]: 1279
% 2.53/1.52 % (27802)Time elapsed: 0.003 s
% 2.53/1.52 % (27802)Instructions burned: 4 (million)
% 2.53/1.52 % (27802)------------------------------
% 2.53/1.52 % (27802)------------------------------
% 2.53/1.52 % (27801)Instruction limit reached!
% 2.53/1.52 % (27801)------------------------------
% 2.53/1.52 % (27801)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.53/1.52 % (27801)Termination reason: Unknown
% 2.53/1.52 % (27801)Termination phase: Property scanning
% 2.53/1.52
% 2.53/1.52 % (27801)Memory used [KB]: 1535
% 2.53/1.52 % (27801)Time elapsed: 0.013 s
% 2.53/1.52 % (27801)Instructions burned: 19 (million)
% 2.53/1.52 % (27801)------------------------------
% 2.53/1.52 % (27801)------------------------------
% 2.53/1.53 % (27797)Instruction limit reached!
% 2.53/1.53 % (27797)------------------------------
% 2.53/1.53 % (27797)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.53/1.53 % (27797)Termination reason: Unknown
% 2.53/1.53 % (27797)Termination phase: Property scanning
% 2.53/1.53
% 2.53/1.53 % (27797)Memory used [KB]: 1791
% 2.53/1.53 % (27797)Time elapsed: 0.016 s
% 2.53/1.53 % (27797)Instructions burned: 27 (million)
% 2.53/1.53 % (27797)------------------------------
% 2.53/1.53 % (27797)------------------------------
% 2.53/1.53 % (27803)Instruction limit reached!
% 2.53/1.53 % (27803)------------------------------
% 2.53/1.53 % (27803)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.53/1.53 % (27803)Termination reason: Unknown
% 2.53/1.53 % (27803)Termination phase: Property scanning
% 2.53/1.53
% 2.53/1.53 % (27803)Memory used [KB]: 1663
% 2.53/1.53 % (27803)Time elapsed: 0.013 s
% 2.53/1.53 % (27803)Instructions burned: 40 (million)
% 2.53/1.53 % (27803)------------------------------
% 2.53/1.53 % (27803)------------------------------
% 2.53/1.53 % (27805)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_27586 for (2999ds/3Mi)
% 2.53/1.53 % (27805)Instruction limit reached!
% 2.53/1.53 % (27805)------------------------------
% 2.53/1.53 % (27805)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.53/1.53 % (27805)Termination reason: Unknown
% 2.53/1.53 % (27805)Termination phase: shuffling
% 2.53/1.53
% 2.53/1.53 % (27805)Memory used [KB]: 1279
% 2.53/1.53 % (27805)Time elapsed: 0.002 s
% 2.53/1.53 % (27805)Instructions burned: 3 (million)
% 2.53/1.53 % (27805)------------------------------
% 2.53/1.53 % (27805)------------------------------
% 2.53/1.53 % (27804)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_27586 for (2999ds/15Mi)
% 2.53/1.54 % (27808)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_27586 for (2999ds/16Mi)
% 2.53/1.54 % (27809)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_27586 for (2999ds/3Mi)
% 2.53/1.54 % (27804)Instruction limit reached!
% 2.53/1.54 % (27804)------------------------------
% 2.53/1.54 % (27804)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.53/1.54 % (27804)Termination reason: Unknown
% 2.53/1.54 % (27804)Termination phase: Property scanning
% 2.53/1.54
% 2.53/1.54 % (27804)Memory used [KB]: 1535
% 2.53/1.54 % (27804)Time elapsed: 0.011 s
% 2.53/1.54 % (27804)Instructions burned: 16 (million)
% 2.53/1.54 % (27804)------------------------------
% 2.53/1.54 % (27804)------------------------------
% 2.53/1.54 % (27806)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_27586 for (2999ds/1041Mi)
% 2.53/1.54 % (27809)Instruction limit reached!
% 2.53/1.54 % (27809)------------------------------
% 2.53/1.54 % (27809)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.53/1.54 % (27809)Termination reason: Unknown
% 2.53/1.54 % (27809)Termination phase: shuffling
% 2.53/1.54
% 2.53/1.54 % (27809)Memory used [KB]: 1279
% 2.53/1.54 % (27809)Time elapsed: 0.002 s
% 2.53/1.54 % (27809)Instructions burned: 3 (million)
% 2.53/1.54 % (27809)------------------------------
% 2.53/1.54 % (27809)------------------------------
% 2.53/1.54 % (27808)Instruction limit reached!
% 2.53/1.54 % (27808)------------------------------
% 2.53/1.54 % (27808)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.53/1.54 % (27808)Termination reason: Unknown
% 2.53/1.54 % (27808)Termination phase: shuffling
% 2.53/1.54
% 2.53/1.54 % (27808)Memory used [KB]: 1663
% 2.53/1.54 % (27808)Time elapsed: 0.006 s
% 2.53/1.54 % (27808)Instructions burned: 17 (million)
% 2.53/1.54 % (27808)------------------------------
% 2.53/1.54 % (27808)------------------------------
% 2.53/1.54 % (27807)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_27586 for (2999ds/7Mi)
% 2.53/1.55 % (27807)Instruction limit reached!
% 2.53/1.55 % (27807)------------------------------
% 2.53/1.55 % (27807)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.53/1.55 % (27807)Termination reason: Unknown
% 2.53/1.55 % (27807)Termination phase: shuffling
% 2.53/1.55
% 2.53/1.55 % (27807)Memory used [KB]: 1407
% 2.53/1.55 % (27807)Time elapsed: 0.006 s
% 2.53/1.55 % (27807)Instructions burned: 8 (million)
% 2.53/1.55 % (27807)------------------------------
% 2.53/1.55 % (27807)------------------------------
% 2.53/1.55 % (27811)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_27586 for (2999ds/7Mi)
% 2.53/1.55 % (27810)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_27586 for (2999ds/3Mi)
% 2.53/1.55 % (27810)Instruction limit reached!
% 2.53/1.55 % (27810)------------------------------
% 2.53/1.55 % (27810)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.53/1.55 % (27810)Termination reason: Unknown
% 2.53/1.55 % (27810)Termination phase: shuffling
% 2.53/1.55
% 2.53/1.55 % (27810)Memory used [KB]: 1279
% 2.53/1.55 % (27810)Time elapsed: 0.002 s
% 2.53/1.55 % (27810)Instructions burned: 4 (million)
% 2.53/1.55 % (27810)------------------------------
% 2.53/1.55 % (27810)------------------------------
% 2.53/1.55 % (27811)Instruction limit reached!
% 2.53/1.55 % (27811)------------------------------
% 2.53/1.55 % (27811)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.53/1.55 % (27811)Termination reason: Unknown
% 2.53/1.55 % (27811)Termination phase: shuffling
% 2.53/1.55
% 2.53/1.55 % (27811)Memory used [KB]: 1407
% 2.53/1.55 % (27811)Time elapsed: 0.003 s
% 2.53/1.55 % (27811)Instructions burned: 7 (million)
% 2.53/1.55 % (27811)------------------------------
% 2.53/1.55 % (27811)------------------------------
% 2.53/1.55 % (27812)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_27586 for (2999ds/3Mi)
% 2.53/1.56 % (27812)Instruction limit reached!
% 2.53/1.56 % (27812)------------------------------
% 2.53/1.56 % (27812)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.53/1.56 % (27812)Termination reason: Unknown
% 2.53/1.56 % (27812)Termination phase: shuffling
% 2.53/1.56
% 2.53/1.56 % (27812)Memory used [KB]: 1279
% 2.53/1.56 % (27812)Time elapsed: 0.003 s
% 2.53/1.56 % (27812)Instructions burned: 4 (million)
% 2.53/1.56 % (27812)------------------------------
% 2.53/1.56 % (27812)------------------------------
% 2.53/1.56 % (27815)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_27586 for (2999ds/710Mi)
% 2.53/1.56 % (27814)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_27586 for (2999ds/18Mi)
% 2.53/1.57 % (27816)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_27586 for (2999ds/6Mi)
% 2.53/1.57 % (27813)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_27586 for (2999ds/4Mi)
% 2.53/1.57 % (27813)Instruction limit reached!
% 2.53/1.57 % (27813)------------------------------
% 2.53/1.57 % (27813)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.53/1.57 % (27813)Termination reason: Unknown
% 2.53/1.57 % (27813)Termination phase: shuffling
% 2.53/1.57
% 2.53/1.57 % (27813)Memory used [KB]: 1279
% 2.53/1.57 % (27813)Time elapsed: 0.004 s
% 2.53/1.57 % (27813)Instructions burned: 4 (million)
% 2.53/1.57 % (27813)------------------------------
% 2.53/1.57 % (27813)------------------------------
% 2.53/1.57 % (27816)Instruction limit reached!
% 2.53/1.57 % (27816)------------------------------
% 2.53/1.57 % (27816)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.53/1.57 % (27816)Termination reason: Unknown
% 2.53/1.57 % (27816)Termination phase: shuffling
% 2.53/1.57
% 2.53/1.57 % (27816)Memory used [KB]: 1407
% 2.53/1.57 % (27816)Time elapsed: 0.004 s
% 2.53/1.57 % (27816)Instructions burned: 7 (million)
% 2.53/1.57 % (27816)------------------------------
% 2.53/1.57 % (27816)------------------------------
% 2.53/1.57 % (27814)Instruction limit reached!
% 2.53/1.57 % (27814)------------------------------
% 2.53/1.57 % (27814)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.53/1.57 % (27814)Termination reason: Unknown
% 2.53/1.57 % (27814)Termination phase: shuffling
% 2.53/1.57
% 2.53/1.57 % (27814)Memory used [KB]: 1663
% 2.53/1.57 % (27814)Time elapsed: 0.007 s
% 2.53/1.57 % (27814)Instructions burned: 21 (million)
% 2.53/1.57 % (27814)------------------------------
% 2.53/1.57 % (27814)------------------------------
% 2.53/1.58 % (27818)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_27586 for (2999ds/21Mi)
% 2.53/1.58 % (27819)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_27586 for (2999ds/5Mi)
% 2.53/1.58 % (27819)Instruction limit reached!
% 2.53/1.58 % (27819)------------------------------
% 2.53/1.58 % (27819)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.53/1.58 % (27819)Termination reason: Unknown
% 2.53/1.58 % (27819)Termination phase: shuffling
% 2.53/1.58
% 2.53/1.58 % (27819)Memory used [KB]: 1407
% 2.53/1.58 % (27819)Time elapsed: 0.004 s
% 2.53/1.58 % (27819)Instructions burned: 7 (million)
% 2.53/1.58 % (27819)------------------------------
% 2.53/1.58 % (27819)------------------------------
% 2.53/1.58 % (27795)Instruction limit reached!
% 2.53/1.58 % (27795)------------------------------
% 2.53/1.58 % (27795)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.53/1.58 % (27795)Termination reason: Unknown
% 2.53/1.58 % (27795)Termination phase: Saturation
% 2.53/1.58
% 2.53/1.58 % (27795)Memory used [KB]: 7164
% 2.53/1.58 % (27795)Time elapsed: 0.071 s
% 2.53/1.58 % (27795)Instructions burned: 185 (million)
% 2.53/1.58 % (27795)------------------------------
% 2.53/1.58 % (27795)------------------------------
% 2.53/1.58 % (27818)Instruction limit reached!
% 2.53/1.58 % (27818)------------------------------
% 2.53/1.58 % (27818)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.53/1.58 % (27818)Termination reason: Unknown
% 2.53/1.58 % (27818)Termination phase: Property scanning
% 2.53/1.58
% 2.53/1.58 % (27818)Memory used [KB]: 1791
% 2.53/1.58 % (27818)Time elapsed: 0.008 s
% 2.53/1.58 % (27818)Instructions burned: 23 (million)
% 2.53/1.58 % (27818)------------------------------
% 2.53/1.58 % (27818)------------------------------
% 2.53/1.59 % (27817)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_27586 for (2999ds/902Mi)
% 2.53/1.59 % (27820)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_27586 for (2999ds/6Mi)
% 2.53/1.59 % (27821)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_27586 for (2999ds/377Mi)
% 2.53/1.59 % (27820)Instruction limit reached!
% 2.53/1.59 % (27820)------------------------------
% 2.53/1.59 % (27820)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.53/1.59 % (27820)Termination reason: Unknown
% 2.53/1.59 % (27820)Termination phase: shuffling
% 2.53/1.59
% 2.53/1.59 % (27820)Memory used [KB]: 1407
% 2.53/1.59 % (27820)Time elapsed: 0.004 s
% 2.53/1.59 % (27820)Instructions burned: 8 (million)
% 2.53/1.59 % (27820)------------------------------
% 2.53/1.59 % (27820)------------------------------
% 2.53/1.59 % (27822)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_27586 for (2999ds/779Mi)
% 2.53/1.60 % (27823)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_27586 for (2999ds/19Mi)
% 2.53/1.61 % (27823)Instruction limit reached!
% 2.53/1.61 % (27823)------------------------------
% 2.53/1.61 % (27823)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.53/1.61 % (27823)Termination reason: Unknown
% 2.53/1.61 % (27823)Termination phase: shuffling
% 2.53/1.61
% 2.53/1.61 % (27823)Memory used [KB]: 1663
% 2.53/1.61 % (27823)Time elapsed: 0.008 s
% 2.53/1.61 % (27823)Instructions burned: 21 (million)
% 2.53/1.61 % (27823)------------------------------
% 2.53/1.61 % (27823)------------------------------
% 2.53/1.62 % (27800)Instruction limit reached!
% 2.53/1.62 % (27800)------------------------------
% 2.53/1.62 % (27800)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.53/1.62 % (27800)Termination reason: Unknown
% 2.53/1.62 % (27800)Termination phase: Saturation
% 2.53/1.62
% 2.53/1.62 % (27800)Memory used [KB]: 8059
% 2.53/1.62 % (27800)Time elapsed: 0.106 s
% 2.53/1.62 % (27800)Instructions burned: 275 (million)
% 2.53/1.62 % (27800)------------------------------
% 2.53/1.62 % (27800)------------------------------
% 2.53/1.62 % (27824)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_27586 for (2998ds/879Mi)
% 2.53/1.63 % (27825)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_27586 for (2998ds/17Mi)
% 2.53/1.63 % (27825)Instruction limit reached!
% 2.53/1.63 % (27825)------------------------------
% 2.53/1.63 % (27825)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.53/1.63 % (27825)Termination reason: Unknown
% 2.53/1.63 % (27825)Termination phase: shuffling
% 2.53/1.63
% 2.53/1.63 % (27825)Memory used [KB]: 1663
% 2.53/1.63 % (27825)Time elapsed: 0.007 s
% 2.53/1.63 % (27825)Instructions burned: 17 (million)
% 2.53/1.63 % (27825)------------------------------
% 2.53/1.63 % (27825)------------------------------
% 2.53/1.64 % (27817)First to succeed.
% 2.53/1.64 % (27817)Refutation found. Thanks to Tanya!
% 2.53/1.64 % SZS status Theorem for DTF2THF_27586
% 2.53/1.64 % SZS output start Proof for DTF2THF_27586
% 2.53/1.64 thf(func_def_0, type, in: $i > $i > $o).
% 2.53/1.64 thf(func_def_1, type, exu: ($i > $o) > $o).
% 2.53/1.64 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 2.53/1.64 thf(func_def_8, type, powerset: $i > $i).
% 2.53/1.64 thf(func_def_10, type, setunion: $i > $i).
% 2.53/1.64 thf(func_def_19, type, descr: ($i > $o) > $i).
% 2.53/1.64 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 2.53/1.64 thf(func_def_26, type, prop2set: $o > $i).
% 2.53/1.64 thf(func_def_36, type, nonempty: $i > $o).
% 2.53/1.64 thf(func_def_69, type, set2prop: $i > $o).
% 2.53/1.64 thf(func_def_88, type, subset: $i > $i > $o).
% 2.53/1.64 thf(func_def_89, type, disjoint: $i > $i > $o).
% 2.53/1.64 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 2.53/1.64 thf(func_def_114, type, binunion: $i > $i > $i).
% 2.53/1.64 thf(func_def_122, type, binintersect: $i > $i > $i).
% 2.53/1.64 thf(func_def_135, type, regular: $i > $o).
% 2.53/1.64 thf(func_def_136, type, setminus: $i > $i > $i).
% 2.53/1.64 thf(func_def_147, type, symdiff: $i > $i > $i).
% 2.53/1.64 thf(func_def_153, type, iskpair: $i > $o).
% 2.53/1.64 thf(func_def_158, type, kpair: $i > $i > $i).
% 2.53/1.64 thf(func_def_160, type, cartprod: $i > $i > $i).
% 2.53/1.64 thf(func_def_182, type, sP1: $i > $i > $o).
% 2.53/1.64 thf(func_def_183, type, sP2: $i > $o).
% 2.53/1.64 thf(func_def_184, type, sP3: $i > $i > $o).
% 2.53/1.64 thf(func_def_185, type, sP4: $i > $i > $o).
% 2.53/1.64 thf(func_def_186, type, sP5: $o > $i > $i > $i > $o).
% 2.53/1.64 thf(func_def_196, type, sK15: ($i > $o) > ($i > $o) > $i).
% 2.53/1.64 thf(func_def_197, type, sK16: ($i > $o) > ($i > $o) > $i).
% 2.53/1.64 thf(func_def_198, type, sK17: $i > $o).
% 2.53/1.64 thf(func_def_199, type, sK18: $i > $o).
% 2.53/1.64 thf(func_def_204, type, sK23: $i > $o).
% 2.53/1.64 thf(func_def_208, type, sK27: $i > $i > $o).
% 2.53/1.64 thf(func_def_209, type, sK28: $i > $i).
% 2.53/1.64 thf(func_def_210, type, sK29: $i > $i).
% 2.53/1.64 thf(func_def_211, type, sK30: ($i > $i > $o) > $i > $i).
% 2.53/1.64 thf(func_def_212, type, sK31: ($i > $i > $o) > $i > $i).
% 2.53/1.64 thf(func_def_213, type, sK32: $i > ($i > $i > $o) > $i > $i).
% 2.53/1.64 thf(func_def_254, type, sK73: $i > $i).
% 2.53/1.64 thf(func_def_255, type, sK74: $i > $o).
% 2.53/1.64 thf(func_def_257, type, sK76: $i > ($i > $o) > $i).
% 2.53/1.64 thf(func_def_262, type, sK81: $i > $o).
% 2.53/1.64 thf(func_def_272, type, sK91: $i > $i).
% 2.53/1.64 thf(func_def_274, type, sK93: $i > $o).
% 2.53/1.64 thf(func_def_280, type, sK99: $i > $i > $i).
% 2.53/1.64 thf(func_def_288, type, sK107: $i > $o).
% 2.53/1.64 thf(func_def_289, type, sK108: $i > $o).
% 2.53/1.64 thf(func_def_290, type, sK109: ($i > $o) > ($i > $o) > $i).
% 2.53/1.64 thf(func_def_291, type, sK110: ($i > $o) > ($i > $o) > $i).
% 2.53/1.64 thf(func_def_299, type, sK118: ($i > $o) > $i > $i).
% 2.53/1.64 thf(func_def_301, type, sK120: $i > $o).
% 2.53/1.64 thf(func_def_307, type, sK126: $i > $o).
% 2.53/1.64 thf(func_def_309, type, sK128: ($i > $o) > $i > $i).
% 2.53/1.64 thf(func_def_314, type, sK133: $i > $i).
% 2.53/1.64 thf(func_def_318, type, sK137: $i > $i > $i).
% 2.53/1.64 thf(func_def_334, type, sK153: $i > ($i > $o) > $i).
% 2.53/1.64 thf(func_def_335, type, sK154: $i > $o).
% 2.53/1.64 thf(func_def_338, type, sK157: $i > $o).
% 2.53/1.64 thf(func_def_341, type, sK160: $i > $o).
% 2.53/1.64 thf(func_def_343, type, sK162: $i > $i).
% 2.53/1.64 thf(func_def_347, type, sK166: $i > $o).
% 2.53/1.64 thf(func_def_358, type, sK177: $i > $i > $i).
% 2.53/1.64 thf(func_def_362, type, sK181: $i > $i > $i).
% 2.53/1.64 thf(func_def_363, type, sK182: $i > $i > $i).
% 2.53/1.64 thf(func_def_372, type, sK191: $i > $i > $i).
% 2.53/1.64 thf(func_def_375, type, sK194: $i > $i > $i).
% 2.53/1.64 thf(func_def_376, type, sK195: $i > $i > $i).
% 2.53/1.64 thf(func_def_377, type, sK196: $i > $i > $i).
% 2.53/1.64 thf(func_def_378, type, sK197: $i > $i > $i).
% 2.53/1.64 thf(func_def_379, type, sK198: $i > $i > $i).
% 2.53/1.64 thf(func_def_380, type, sK199: $i > $i > $i > $i).
% 2.53/1.64 thf(func_def_381, type, sK200: $i > $i).
% 2.53/1.64 thf(func_def_382, type, sK201: $i > $i).
% 2.53/1.64 thf(func_def_383, type, sK202: $i > $i).
% 2.53/1.64 thf(func_def_384, type, sK203: $i > $i).
% 2.53/1.64 thf(func_def_385, type, sK204: $i > $i > $i > $i).
% 2.53/1.64 thf(func_def_386, type, sK205: $i > $i > $i > $i).
% 2.53/1.64 thf(func_def_387, type, sK206: $i > $i > $i).
% 2.53/1.64 thf(func_def_388, type, sK207: $i > $i > $i).
% 2.53/1.64 thf(func_def_389, type, sK208: $i > $i > $i).
% 2.53/1.64 thf(func_def_391, type, sK210: $i > $i).
% 2.53/1.64 thf(func_def_392, type, sK211: $i > $i).
% 2.53/1.64 thf(func_def_393, type, sK212: $i > $i).
% 2.53/1.64 thf(func_def_401, type, sK220: $i > $i > $i).
% 2.53/1.64 thf(func_def_402, type, sK221: $i > $o).
% 2.53/1.64 thf(func_def_407, type, sK226: $i > $o).
% 2.53/1.64 thf(func_def_409, type, sK228: ($i > $o) > $i).
% 2.53/1.64 thf(func_def_410, type, sK229: ($i > $o) > $i).
% 2.53/1.64 thf(func_def_414, type, sK233: $i > $o).
% 2.53/1.64 thf(func_def_416, type, sK235: $i > $o).
% 2.53/1.64 thf(func_def_419, type, sK238: $i > ($i > $o) > $i).
% 2.53/1.64 thf(func_def_431, type, sK250: $i > $o).
% 2.53/1.64 thf(func_def_432, type, sK251: $i > $i).
% 2.53/1.64 thf(func_def_433, type, sK252: ($i > $o) > $i).
% 2.53/1.64 thf(func_def_437, type, sK256: $i > $o).
% 2.53/1.64 thf(func_def_449, type, sK268: $i > $o).
% 2.53/1.64 thf(func_def_455, type, sK274: $i > $o).
% 2.53/1.64 thf(func_def_467, type, sK286: ($i > $o) > $i > $i).
% 2.53/1.64 thf(func_def_469, type, sK288: $i > $o).
% 2.53/1.64 thf(func_def_490, type, sK309: $i > $o).
% 2.53/1.64 thf(func_def_491, type, sK310: ($i > $o) > $i).
% 2.53/1.64 thf(func_def_492, type, sK311: $i > $o).
% 2.53/1.64 thf(func_def_493, type, sK312: $i > $i).
% 2.53/1.64 thf(func_def_494, type, sK313: ($i > $o) > $i).
% 2.53/1.64 thf(func_def_498, type, sK317: $o > $i > $i > $i).
% 2.53/1.64 thf(func_def_501, type, sK320: $i > $i > $i).
% 2.53/1.64 thf(func_def_516, type, sK335: ($i > $o) > $i > $i).
% 2.53/1.64 thf(func_def_518, type, sK337: $i > $o).
% 2.53/1.64 thf(func_def_525, type, sK344: $i > $i > $i).
% 2.53/1.64 thf(func_def_541, type, sK360: $i > $o).
% 2.53/1.64 thf(func_def_543, type, sK362: ($i > $o) > $i > $i).
% 2.53/1.64 thf(func_def_549, type, sK368: $i > $i).
% 2.53/1.64 thf(func_def_556, type, sK375: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.53/1.64 thf(func_def_557, type, sK376: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.53/1.64 thf(func_def_560, type, sK379: $i > $o).
% 2.53/1.64 thf(func_def_561, type, sK380: $i > $o).
% 2.53/1.64 thf(func_def_570, type, sK389: $i > $o).
% 2.53/1.64 thf(func_def_577, type, sK396: $i > $i).
% 2.53/1.64 thf(func_def_579, type, sK398: $i > $i).
% 2.53/1.64 thf(func_def_581, type, ph400: !>[X0: $tType]:(X0)).
% 2.53/1.64 thf(f2185,plain,(
% 2.53/1.64 $false),
% 2.53/1.64 inference(trivial_inequality_removal,[status(thm)],[f2179])).
% 2.53/1.64 thf(f2179,plain,(
% 2.53/1.64 ($true != $true)),
% 2.53/1.64 inference(superposition,[status(thm)],[f2150,f1569])).
% 2.53/1.64 thf(f1569,plain,(
% 2.53/1.64 ($true = (in @ sK188 @ sK185))),
% 2.53/1.64 inference(cnf_transformation,[status(thm)],[f948])).
% 2.53/1.64 thf(f948,plain,(
% 2.53/1.64 (powersetsubset = $true) & (omega0Ax = $true) & (inPowerset = $true) & (eqimpsubset2 = $true) & (nonemptyI1 = $true) & (notdallE = $true) & (symdiffIneg2 = $true) & (setukpairIL = $true) & (powersetE = $true) & (bs114d = $true) & (binintersectI = $true) & (exuE2 = $true) & (dsetconstrER = $true) & (exuE1 = $true) & (eqimpsubset1 = $true) & (in__Cong = $true) & (binunionLsub = $true) & (prop2set2propI = $true) & (exuI3 = $true) & (prop2setI = $true) & (upairsubunion = $true) & (binintersectEL = $true) & (setminusI = $true) & (nonemptyI = $true) & (subsetTrans = $true) & (quantDeMorgan2 = $true) & (quantDeMorgan4 = $true) & (binunionRsub = $true) & (nonemptyE1 = $true) & (setadjoin__Cong = $true) & (emptysetimpfalse = $true) & (setbeta = $true) & (emptyset__Cong = $true) & (binunionIL = $true) & (setunionI = $true) & (kpairp = $true) & (noeltsimpempty = $true) & (notsubsetI = $true) & (setadjoinSub2 = $true) & (dsetconstrI = $true) & (notequalI2 = $true) & (upairsetE = $true) & (notinemptyset = $true) & (emptysetsubset = $true) & (subPowSU = $true) & (setminusERneg = $true) & (nonemptyImpWitness = $true) & (replAx = $true) & (omega__Cong = $true) & (powerset__Cong = $true) & (setminusSubset1 = $true) & (sepInPowerset = $true) & (foundationAx = $true) & (emptyI = $true) & (subsetI1 = $true) & (kpairiskpair = $true) & (exuE3e = $true) & (notinsingleton = $true) & (binintersectLsub = $true) & (emptyinPowerset = $true) & (powersetE1 = $true) & (wellorderingAx = $true) & (inCongP = $true) & (binintersectER = $true) & (descrp = $true) & (setunionAx = $true) & (setminusLsub = $true) & (dsetconstrEL = $true) & (setunionE = $true) & (setadjoinIL = $true) & (emptyinunitempty = $true) & (omegaIndAx = $true) & (upairsetIL = $true) & (setminusILneg = $true) & (symdiffI2 = $true) & (subsetE = $true) & (exu__Cong = $true) & (symdiffIneg1 = $true) & (binintersectRsub = $true) & (descr__Cong = $true) & (setoftrueEq = $true) & (subsetemptysetimpeq = $true) & (symdiffE = $true) & (singletoninpowerset = $true) & (eqinunit = $true) & (upairset2E = $true) & (upairset2IR = $true) & (exuI2 = $true) & (setminusELneg = $true) & (binintersectSubset3 = $true) & (subsetE2 = $true) & (powersetAx = $true) & (upairsetIR = $true) & (setext = $true) & (subsetRefl = $true) & (emptyE1 = $true) & (emptysetE = $true) & (dsetconstr__Cong = $true) & (powersetI = $true) & (singletonsswitch = $true) & (notdexE = $true) & (binunionE = $true) & (setadjoinSub = $true) & (emptyInPowerset = $true) & (binintersectSubset2 = $true) & (binunionEcases = $true) & (secondinupair = $true) & (binintersectSubset1 = $true) & (omegaSAx = $true) & (setukpairIR = $true) & (sepSubset = $true) & (exuE3u = $true) & (ubforcartprodlem1 = $true) & (quantDeMorgan3 = $true) & (binunionIR = $true) & (setminusIRneg = $true) & (emptysetAx = $true) & (subset2powerset = $true) & (binintersectSubset5 = $true) & (setunion__Cong = $true) & (vacuousDall = $true) & (exuI1 = $true) & (setadjoinIR = $true) & (singletonsubset = $true) & (setextAx = $true) & (exuEu = $true) & (setextsub = $true) & (subsetI2 = $true) & (setminusER = $true) & (powersetI1 = $true) & (disjointsetsI1 = $true) & (uniqinunit = $true) & (setadjoinAx = $true) & (setadjoinOr = $true) & (symdiffI1 = $true) & (($true = (in @ sK187 @ sK186)) & (($true = (in @ sK188 @ sK185)) & ($true != (in @ (setadjoin @ (setadjoin @ sK187 @ emptyset) @ (setadjoin @ (setadjoin @ sK187 @ (setadjoin @ sK188 @ emptyset)) @ emptyset)) @ (powerset @ (powerset @ (binunion @ sK186 @ sK185))))))) & (notequalI1 = $true) & (setminusSubset2 = $true) & (singletoninpowunion = $true) & (quantDeMorgan1 = $true) & (binintersectSubset4 = $true) & (prop2setE = $true) & (upairinpowunion = $true) & (setadjoinE = $true) & (setminusEL = $true)),
% 2.53/1.64 inference(skolemisation,[status(esa),new_symbols(skolem,[sK185,sK186,sK187,sK188])],[f609,f947,f946])).
% 2.53/1.64 thf(f946,plain,(
% 2.53/1.64 ? [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X2 @ X1)) & ? [X3 : $i] : (((in @ X3 @ X0) = $true) & ((in @ (setadjoin @ (setadjoin @ X2 @ emptyset) @ (setadjoin @ (setadjoin @ X2 @ (setadjoin @ X3 @ emptyset)) @ emptyset)) @ (powerset @ (powerset @ (binunion @ X1 @ X0)))) != $true))) => (($true = (in @ sK187 @ sK186)) & ? [X3 : $i] : (((in @ X3 @ sK185) = $true) & ((in @ (setadjoin @ (setadjoin @ sK187 @ emptyset) @ (setadjoin @ (setadjoin @ sK187 @ (setadjoin @ X3 @ emptyset)) @ emptyset)) @ (powerset @ (powerset @ (binunion @ sK186 @ sK185)))) != $true)))),
% 2.53/1.64 introduced(definition,[],[choice_axiom])).
% 2.53/1.64 thf(f947,plain,(
% 2.53/1.64 ? [X3 : $i] : (((in @ X3 @ sK185) = $true) & ((in @ (setadjoin @ (setadjoin @ sK187 @ emptyset) @ (setadjoin @ (setadjoin @ sK187 @ (setadjoin @ X3 @ emptyset)) @ emptyset)) @ (powerset @ (powerset @ (binunion @ sK186 @ sK185)))) != $true)) => (($true = (in @ sK188 @ sK185)) & ($true != (in @ (setadjoin @ (setadjoin @ sK187 @ emptyset) @ (setadjoin @ (setadjoin @ sK187 @ (setadjoin @ sK188 @ emptyset)) @ emptyset)) @ (powerset @ (powerset @ (binunion @ sK186 @ sK185))))))),
% 2.53/1.64 introduced(definition,[],[choice_axiom])).
% 2.53/1.64 thf(f609,plain,(
% 2.53/1.64 (powersetsubset = $true) & (omega0Ax = $true) & (inPowerset = $true) & (eqimpsubset2 = $true) & (nonemptyI1 = $true) & (notdallE = $true) & (symdiffIneg2 = $true) & (setukpairIL = $true) & (powersetE = $true) & (bs114d = $true) & (binintersectI = $true) & (exuE2 = $true) & (dsetconstrER = $true) & (exuE1 = $true) & (eqimpsubset1 = $true) & (in__Cong = $true) & (binunionLsub = $true) & (prop2set2propI = $true) & (exuI3 = $true) & (prop2setI = $true) & (upairsubunion = $true) & (binintersectEL = $true) & (setminusI = $true) & (nonemptyI = $true) & (subsetTrans = $true) & (quantDeMorgan2 = $true) & (quantDeMorgan4 = $true) & (binunionRsub = $true) & (nonemptyE1 = $true) & (setadjoin__Cong = $true) & (emptysetimpfalse = $true) & (setbeta = $true) & (emptyset__Cong = $true) & (binunionIL = $true) & (setunionI = $true) & (kpairp = $true) & (noeltsimpempty = $true) & (notsubsetI = $true) & (setadjoinSub2 = $true) & (dsetconstrI = $true) & (notequalI2 = $true) & (upairsetE = $true) & (notinemptyset = $true) & (emptysetsubset = $true) & (subPowSU = $true) & (setminusERneg = $true) & (nonemptyImpWitness = $true) & (replAx = $true) & (omega__Cong = $true) & (powerset__Cong = $true) & (setminusSubset1 = $true) & (sepInPowerset = $true) & (foundationAx = $true) & (emptyI = $true) & (subsetI1 = $true) & (kpairiskpair = $true) & (exuE3e = $true) & (notinsingleton = $true) & (binintersectLsub = $true) & (emptyinPowerset = $true) & (powersetE1 = $true) & (wellorderingAx = $true) & (inCongP = $true) & (binintersectER = $true) & (descrp = $true) & (setunionAx = $true) & (setminusLsub = $true) & (dsetconstrEL = $true) & (setunionE = $true) & (setadjoinIL = $true) & (emptyinunitempty = $true) & (omegaIndAx = $true) & (upairsetIL = $true) & (setminusILneg = $true) & (symdiffI2 = $true) & (subsetE = $true) & (exu__Cong = $true) & (symdiffIneg1 = $true) & (binintersectRsub = $true) & (descr__Cong = $true) & (setoftrueEq = $true) & (subsetemptysetimpeq = $true) & (symdiffE = $true) & (singletoninpowerset = $true) & (eqinunit = $true) & (upairset2E = $true) & (upairset2IR = $true) & (exuI2 = $true) & (setminusELneg = $true) & (binintersectSubset3 = $true) & (subsetE2 = $true) & (powersetAx = $true) & (upairsetIR = $true) & (setext = $true) & (subsetRefl = $true) & (emptyE1 = $true) & (emptysetE = $true) & (dsetconstr__Cong = $true) & (powersetI = $true) & (singletonsswitch = $true) & (notdexE = $true) & (binunionE = $true) & (setadjoinSub = $true) & (emptyInPowerset = $true) & (binintersectSubset2 = $true) & (binunionEcases = $true) & (secondinupair = $true) & (binintersectSubset1 = $true) & (omegaSAx = $true) & (setukpairIR = $true) & (sepSubset = $true) & (exuE3u = $true) & (ubforcartprodlem1 = $true) & (quantDeMorgan3 = $true) & (binunionIR = $true) & (setminusIRneg = $true) & (emptysetAx = $true) & (subset2powerset = $true) & (binintersectSubset5 = $true) & (setunion__Cong = $true) & (vacuousDall = $true) & (exuI1 = $true) & (setadjoinIR = $true) & (singletonsubset = $true) & (setextAx = $true) & (exuEu = $true) & (setextsub = $true) & (subsetI2 = $true) & (setminusER = $true) & (powersetI1 = $true) & (disjointsetsI1 = $true) & (uniqinunit = $true) & (setadjoinAx = $true) & (setadjoinOr = $true) & (symdiffI1 = $true) & ? [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X2 @ X1)) & ? [X3 : $i] : (((in @ X3 @ X0) = $true) & ((in @ (setadjoin @ (setadjoin @ X2 @ emptyset) @ (setadjoin @ (setadjoin @ X2 @ (setadjoin @ X3 @ emptyset)) @ emptyset)) @ (powerset @ (powerset @ (binunion @ X1 @ X0)))) != $true))) & (notequalI1 = $true) & (setminusSubset2 = $true) & (singletoninpowunion = $true) & (quantDeMorgan1 = $true) & (binintersectSubset4 = $true) & (prop2setE = $true) & (upairinpowunion = $true) & (setadjoinE = $true) & (setminusEL = $true)),
% 2.53/1.64 inference(flattening,[status(thm)],[f608])).
% 2.53/1.64 thf(f608,plain,(
% 2.53/1.64 (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X2 @ X1)) & ? [X3 : $i] : (((in @ X3 @ X0) = $true) & ((in @ (setadjoin @ (setadjoin @ X2 @ emptyset) @ (setadjoin @ (setadjoin @ X2 @ (setadjoin @ X3 @ emptyset)) @ emptyset)) @ (powerset @ (powerset @ (binunion @ X1 @ X0)))) != $true))) & (ubforcartprodlem1 = $true)) & (upairinpowunion = $true)) & (upairsubunion = $true)) & (upairset2E = $true)) & (singletoninpowunion = $true)) & (singletoninpowerset = $true)) & (singletonsubset = $true)) & (kpairp = $true)) & (kpairiskpair = $true)) & (setukpairIR = $true)) & (setukpairIL = $true)) & (secondinupair = $true)) & (symdiffIneg2 = $true)) & (symdiffIneg1 = $true)) & (symdiffI2 = $true)) & (symdiffI1 = $true)) & (symdiffE = $true)) & (setminusSubset1 = $true)) & (setminusLsub = $true)) & (setminusIRneg = $true)) & (setminusILneg = $true)) & (setminusELneg = $true)) & (setminusERneg = $true)) & (setminusSubset2 = $true)) & (setminusER = $true)) & (setminusEL = $true)) & (setminusI = $true)) & (bs114d = $true)) & (binintersectSubset1 = $true)) & (binintersectSubset4 = $true)) & (binintersectRsub = $true)) & (disjointsetsI1 = $true)) & (binintersectER = $true)) & (binintersectSubset3 = $true)) & (binintersectSubset2 = $true)) & (binintersectLsub = $true)) & (binintersectEL = $true)) & (binintersectSubset5 = $true)) & (binintersectI = $true)) & (binunionRsub = $true)) & (binunionLsub = $true)) & (binunionE = $true)) & (binunionEcases = $true)) & (binunionIR = $true)) & (upairset2IR = $true)) & (binunionIL = $true)) & (sepSubset = $true)) & (sepInPowerset = $true)) & (powersetsubset = $true)) & (inPowerset = $true)) & (powersetE1 = $true)) & (powersetI1 = $true)) & (subsetemptysetimpeq = $true)) & (setextsub = $true)) & (subset2powerset = $true)) & (setadjoinSub2 = $true)) & (setadjoinSub = $true)) & (subsetTrans = $true)) & (subsetRefl = $true)) & (notequalI2 = $true)) & (notequalI1 = $true)) & (notsubsetI = $true)) & (subsetE2 = $true)) & (subsetE = $true)) & (emptysetsubset = $true)) & (subsetI2 = $true)) & (eqimpsubset1 = $true)) & (eqimpsubset2 = $true)) & (subsetI1 = $true)) & (dsetconstr__Cong = $true)) & (descr__Cong = $true)) & (exuEu = $true)) & (omega__Cong = $true)) & (setunion__Cong = $true)) & (powerset__Cong = $true)) & (setadjoin__Cong = $true)) & (emptyset__Cong = $true)) & (exu__Cong = $true)) & (exuE3u = $true)) & (in__Cong = $true)) & (inCongP = $true)) & (exuI2 = $true)) & (exuI3 = $true)) & (exuI1 = $true)) & (notdallE = $true)) & (notdexE = $true)) & (prop2set2propI = $true)) & (prop2setI = $true)) & (quantDeMorgan4 = $true)) & (quantDeMorgan3 = $true)) & (quantDeMorgan2 = $true)) & (quantDeMorgan1 = $true)) & (vacuousDall = $true)) & (emptyE1 = $true)) & (upairsetIR = $true)) & (upairsetIL = $true)) & (upairsetE = $true)) & (singletonsswitch = $true)) & (eqinunit = $true)) & (notinsingleton = $true)) & (uniqinunit = $true)) & (nonemptyImpWitness = $true)) & (exuE2 = $true)) & (subPowSU = $true)) & (setunionE = $true)) & (setunionI = $true)) & (powersetE = $true)) & (emptyInPowerset = $true)) & (emptyinPowerset = $true)) & (powersetI = $true)) & (setoftrueEq = $true)) & (setadjoinOr = $true)) & (setadjoinE = $true)) & (setadjoinIR = $true)) & (emptyinunitempty = $true)) & (setadjoinIL = $true)) & (nonemptyI1 = $true)) & (nonemptyI = $true)) & (nonemptyE1 = $true)) & (setbeta = $true)) & (noeltsimpempty = $true)) & (emptyI = $true)) & (setext = $true)) & (exuE3e = $true)) & (notinemptyset = $true)) & (emptysetimpfalse = $true)) & (emptysetE = $true)) & (prop2setE = $true)) & (exuE1 = $true)) & (dsetconstrER = $true)) & (dsetconstrEL = $true)) & (dsetconstrI = $true)) & (descrp = $true)) & (wellorderingAx = $true)) & (foundationAx = $true)) & (replAx = $true)) & (omegaIndAx = $true)) & (omegaSAx = $true)) & (omega0Ax = $true)) & (setunionAx = $true)) & (powersetAx = $true)) & (setadjoinAx = $true)) & (emptysetAx = $true)) & (setextAx = $true)),
% 2.53/1.64 inference(ennf_transformation,[status(thm)],[f274])).
% 2.53/1.64 thf(f274,plain,(
% 2.53/1.64 ~((setextAx = $true) => ((emptysetAx = $true) => ((setadjoinAx = $true) => ((powersetAx = $true) => ((setunionAx = $true) => ((omega0Ax = $true) => ((omegaSAx = $true) => ((omegaIndAx = $true) => ((replAx = $true) => ((foundationAx = $true) => ((wellorderingAx = $true) => ((descrp = $true) => ((dsetconstrI = $true) => ((dsetconstrEL = $true) => ((dsetconstrER = $true) => ((exuE1 = $true) => ((prop2setE = $true) => ((emptysetE = $true) => ((emptysetimpfalse = $true) => ((notinemptyset = $true) => ((exuE3e = $true) => ((setext = $true) => ((emptyI = $true) => ((noeltsimpempty = $true) => ((setbeta = $true) => ((nonemptyE1 = $true) => ((nonemptyI = $true) => ((nonemptyI1 = $true) => ((setadjoinIL = $true) => ((emptyinunitempty = $true) => ((setadjoinIR = $true) => ((setadjoinE = $true) => ((setadjoinOr = $true) => ((setoftrueEq = $true) => ((powersetI = $true) => ((emptyinPowerset = $true) => ((emptyInPowerset = $true) => ((powersetE = $true) => ((setunionI = $true) => ((setunionE = $true) => ((subPowSU = $true) => ((exuE2 = $true) => ((nonemptyImpWitness = $true) => ((uniqinunit = $true) => ((notinsingleton = $true) => ((eqinunit = $true) => ((singletonsswitch = $true) => ((upairsetE = $true) => ((upairsetIL = $true) => ((upairsetIR = $true) => ((emptyE1 = $true) => ((vacuousDall = $true) => ((quantDeMorgan1 = $true) => ((quantDeMorgan2 = $true) => ((quantDeMorgan3 = $true) => ((quantDeMorgan4 = $true) => ((prop2setI = $true) => ((prop2set2propI = $true) => ((notdexE = $true) => ((notdallE = $true) => ((exuI1 = $true) => ((exuI3 = $true) => ((exuI2 = $true) => ((inCongP = $true) => ((in__Cong = $true) => ((exuE3u = $true) => ((exu__Cong = $true) => ((emptyset__Cong = $true) => ((setadjoin__Cong = $true) => ((powerset__Cong = $true) => ((setunion__Cong = $true) => ((omega__Cong = $true) => ((exuEu = $true) => ((descr__Cong = $true) => ((dsetconstr__Cong = $true) => ((subsetI1 = $true) => ((eqimpsubset2 = $true) => ((eqimpsubset1 = $true) => ((subsetI2 = $true) => ((emptysetsubset = $true) => ((subsetE = $true) => ((subsetE2 = $true) => ((notsubsetI = $true) => ((notequalI1 = $true) => ((notequalI2 = $true) => ((subsetRefl = $true) => ((subsetTrans = $true) => ((setadjoinSub = $true) => ((setadjoinSub2 = $true) => ((subset2powerset = $true) => ((setextsub = $true) => ((subsetemptysetimpeq = $true) => ((powersetI1 = $true) => ((powersetE1 = $true) => ((inPowerset = $true) => ((powersetsubset = $true) => ((sepInPowerset = $true) => ((sepSubset = $true) => ((binunionIL = $true) => ((upairset2IR = $true) => ((binunionIR = $true) => ((binunionEcases = $true) => ((binunionE = $true) => ((binunionLsub = $true) => ((binunionRsub = $true) => ((binintersectI = $true) => ((binintersectSubset5 = $true) => ((binintersectEL = $true) => ((binintersectLsub = $true) => ((binintersectSubset2 = $true) => ((binintersectSubset3 = $true) => ((binintersectER = $true) => ((disjointsetsI1 = $true) => ((binintersectRsub = $true) => ((binintersectSubset4 = $true) => ((binintersectSubset1 = $true) => ((bs114d = $true) => ((setminusI = $true) => ((setminusEL = $true) => ((setminusER = $true) => ((setminusSubset2 = $true) => ((setminusERneg = $true) => ((setminusELneg = $true) => ((setminusILneg = $true) => ((setminusIRneg = $true) => ((setminusLsub = $true) => ((setminusSubset1 = $true) => ((symdiffE = $true) => ((symdiffI1 = $true) => ((symdiffI2 = $true) => ((symdiffIneg1 = $true) => ((symdiffIneg2 = $true) => ((secondinupair = $true) => ((setukpairIL = $true) => ((setukpairIR = $true) => ((kpairiskpair = $true) => ((kpairp = $true) => ((singletonsubset = $true) => ((singletoninpowerset = $true) => ((singletoninpowunion = $true) => ((upairset2E = $true) => ((upairsubunion = $true) => ((upairinpowunion = $true) => ((ubforcartprodlem1 = $true) => ! [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X2 @ X1)) => ! [X3 : $i] : (((in @ X3 @ X0) = $true) => ((in @ (setadjoin @ (setadjoin @ X2 @ emptyset) @ (setadjoin @ (setadjoin @ X2 @ (setadjoin @ X3 @ emptyset)) @ emptyset)) @ (powerset @ (powerset @ (binunion @ X1 @ X0)))) = $true)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.53/1.64 inference(fool_elimination,[status(thm)],[f273])).
% 2.53/1.64 thf(f273,plain,(
% 2.53/1.64 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => ! [X0 : $i,X1 : $i,X2 : $i] : ((in @ X2 @ X1) => ! [X3 : $i] : ((in @ X3 @ X0) => (in @ (setadjoin @ (setadjoin @ X2 @ emptyset) @ (setadjoin @ (setadjoin @ X2 @ (setadjoin @ X3 @ emptyset)) @ emptyset)) @ (powerset @ (powerset @ (binunion @ X1 @ X0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.53/1.64 inference(rectify,[status(thm)],[f151])).
% 2.53/1.64 thf(f151,negated_conjecture,(
% 2.53/1.64 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => ! [X4 : $i,X3 : $i,X1 : $i] : ((in @ X1 @ X3) => ! [X2 : $i] : ((in @ X2 @ X4) => (in @ (setadjoin @ (setadjoin @ X1 @ emptyset) @ (setadjoin @ (setadjoin @ X1 @ (setadjoin @ X2 @ emptyset)) @ emptyset)) @ (powerset @ (powerset @ (binunion @ X3 @ X4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.53/1.64 inference(negated_conjecture,[status(cth)],[f150])).
% 2.53/1.64 thf(f150,conjecture,(
% 2.53/1.64 setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => ! [X4 : $i,X3 : $i,X1 : $i] : ((in @ X1 @ X3) => ! [X2 : $i] : ((in @ X2 @ X4) => (in @ (setadjoin @ (setadjoin @ X1 @ emptyset) @ (setadjoin @ (setadjoin @ X1 @ (setadjoin @ X2 @ emptyset)) @ emptyset)) @ (powerset @ (powerset @ (binunion @ X3 @ X4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.53/1.64 file('/export/starexec/sandbox2/tmp/tmp.yC96zsFX7x/DTF2THF_27586.p',ubforcartprodlem2)).
% 2.53/1.64 thf(f2150,plain,(
% 2.53/1.64 ($true != (in @ sK188 @ sK185))),
% 2.53/1.64 inference(trivial_inequality_removal,[status(thm)],[f2149])).
% 2.53/1.64 thf(f2149,plain,(
% 2.53/1.64 ($true != (in @ sK188 @ sK185)) | ($true != $true)),
% 2.53/1.64 inference(forward_demodulation,[status(thm)],[f2148,f1593])).
% 2.53/1.64 thf(f1593,plain,(
% 2.53/1.64 (ubforcartprodlem1 = $true)),
% 2.53/1.64 inference(cnf_transformation,[status(thm)],[f948])).
% 2.53/1.64 thf(f2148,plain,(
% 2.53/1.64 (ubforcartprodlem1 != $true) | ($true != (in @ sK188 @ sK185))),
% 2.53/1.64 inference(trivial_inequality_removal,[status(thm)],[f2147])).
% 2.53/1.64 thf(f2147,plain,(
% 2.53/1.64 ($true != $true) | ($true != (in @ sK188 @ sK185)) | (ubforcartprodlem1 != $true)),
% 2.53/1.64 inference(forward_demodulation,[status(thm)],[f2143,f1570])).
% 2.53/1.64 thf(f1570,plain,(
% 2.53/1.64 ($true = (in @ sK187 @ sK186))),
% 2.53/1.64 inference(cnf_transformation,[status(thm)],[f948])).
% 2.53/1.64 thf(f2143,plain,(
% 2.53/1.64 ($true != (in @ sK187 @ sK186)) | (ubforcartprodlem1 != $true) | ($true != (in @ sK188 @ sK185))),
% 2.53/1.64 inference(trivial_inequality_removal,[status(thm)],[f2135])).
% 2.53/1.64 thf(f2135,plain,(
% 2.53/1.64 ($true != (in @ sK188 @ sK185)) | (ubforcartprodlem1 != $true) | ($true != $true) | ($true != (in @ sK187 @ sK186))),
% 2.53/1.64 inference(superposition,[status(thm)],[f2132,f1445])).
% 2.53/1.64 thf(f1445,plain,(
% 2.53/1.64 ( ! [X2 : $i,X3 : $i,X0 : $i,X1 : $i] : (((subset @ (setadjoin @ (setadjoin @ X0 @ emptyset) @ (setadjoin @ (setadjoin @ X0 @ (setadjoin @ X3 @ emptyset)) @ emptyset)) @ (powerset @ (binunion @ X1 @ X2))) = $true) | (ubforcartprodlem1 != $true) | ((in @ X3 @ X2) != $true) | ($true != (in @ X0 @ X1))) )),
% 2.53/1.64 inference(cnf_transformation,[status(thm)],[f808])).
% 2.53/1.64 thf(f808,plain,(
% 2.53/1.64 (! [X0 : $i,X1 : $i,X2 : $i] : (! [X3 : $i] : (((in @ X3 @ X2) != $true) | ((subset @ (setadjoin @ (setadjoin @ X0 @ emptyset) @ (setadjoin @ (setadjoin @ X0 @ (setadjoin @ X3 @ emptyset)) @ emptyset)) @ (powerset @ (binunion @ X1 @ X2))) = $true)) | ($true != (in @ X0 @ X1))) | (ubforcartprodlem1 != $true)) & ((ubforcartprodlem1 = $true) | ((($true = (in @ sK105 @ sK104)) & ($true != (subset @ (setadjoin @ (setadjoin @ sK102 @ emptyset) @ (setadjoin @ (setadjoin @ sK102 @ (setadjoin @ sK105 @ emptyset)) @ emptyset)) @ (powerset @ (binunion @ sK103 @ sK104))))) & ($true = (in @ sK102 @ sK103))))),
% 2.53/1.64 inference(skolemisation,[status(esa),new_symbols(skolem,[sK102,sK103,sK104,sK105])],[f805,f807,f806])).
% 2.53/1.64 thf(f806,plain,(
% 2.53/1.64 ? [X4 : $i,X5 : $i,X6 : $i] : (? [X7 : $i] : (((in @ X7 @ X6) = $true) & ($true != (subset @ (setadjoin @ (setadjoin @ X4 @ emptyset) @ (setadjoin @ (setadjoin @ X4 @ (setadjoin @ X7 @ emptyset)) @ emptyset)) @ (powerset @ (binunion @ X5 @ X6))))) & ($true = (in @ X4 @ X5))) => (? [X7 : $i] : (($true = (in @ X7 @ sK104)) & ($true != (subset @ (setadjoin @ (setadjoin @ sK102 @ emptyset) @ (setadjoin @ (setadjoin @ sK102 @ (setadjoin @ X7 @ emptyset)) @ emptyset)) @ (powerset @ (binunion @ sK103 @ sK104))))) & ($true = (in @ sK102 @ sK103)))),
% 2.53/1.64 introduced(definition,[],[choice_axiom])).
% 2.53/1.64 thf(f807,plain,(
% 2.53/1.64 ? [X7 : $i] : (($true = (in @ X7 @ sK104)) & ($true != (subset @ (setadjoin @ (setadjoin @ sK102 @ emptyset) @ (setadjoin @ (setadjoin @ sK102 @ (setadjoin @ X7 @ emptyset)) @ emptyset)) @ (powerset @ (binunion @ sK103 @ sK104))))) => (($true = (in @ sK105 @ sK104)) & ($true != (subset @ (setadjoin @ (setadjoin @ sK102 @ emptyset) @ (setadjoin @ (setadjoin @ sK102 @ (setadjoin @ sK105 @ emptyset)) @ emptyset)) @ (powerset @ (binunion @ sK103 @ sK104)))))),
% 2.53/1.64 introduced(definition,[],[choice_axiom])).
% 2.53/1.64 thf(f805,plain,(
% 2.53/1.64 (! [X0 : $i,X1 : $i,X2 : $i] : (! [X3 : $i] : (((in @ X3 @ X2) != $true) | ((subset @ (setadjoin @ (setadjoin @ X0 @ emptyset) @ (setadjoin @ (setadjoin @ X0 @ (setadjoin @ X3 @ emptyset)) @ emptyset)) @ (powerset @ (binunion @ X1 @ X2))) = $true)) | ($true != (in @ X0 @ X1))) | (ubforcartprodlem1 != $true)) & ((ubforcartprodlem1 = $true) | ? [X4 : $i,X5 : $i,X6 : $i] : (? [X7 : $i] : (((in @ X7 @ X6) = $true) & ($true != (subset @ (setadjoin @ (setadjoin @ X4 @ emptyset) @ (setadjoin @ (setadjoin @ X4 @ (setadjoin @ X7 @ emptyset)) @ emptyset)) @ (powerset @ (binunion @ X5 @ X6))))) & ($true = (in @ X4 @ X5))))),
% 2.53/1.64 inference(rectify,[status(thm)],[f804])).
% 2.53/1.64 thf(f804,plain,(
% 2.53/1.64 (! [X0 : $i,X1 : $i,X2 : $i] : (! [X3 : $i] : (((in @ X3 @ X2) != $true) | ((subset @ (setadjoin @ (setadjoin @ X0 @ emptyset) @ (setadjoin @ (setadjoin @ X0 @ (setadjoin @ X3 @ emptyset)) @ emptyset)) @ (powerset @ (binunion @ X1 @ X2))) = $true)) | ($true != (in @ X0 @ X1))) | (ubforcartprodlem1 != $true)) & ((ubforcartprodlem1 = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (? [X3 : $i] : (((in @ X3 @ X2) = $true) & ((subset @ (setadjoin @ (setadjoin @ X0 @ emptyset) @ (setadjoin @ (setadjoin @ X0 @ (setadjoin @ X3 @ emptyset)) @ emptyset)) @ (powerset @ (binunion @ X1 @ X2))) != $true)) & ($true = (in @ X0 @ X1))))),
% 2.53/1.64 inference(nnf_transformation,[status(thm)],[f558])).
% 2.53/1.64 thf(f558,plain,(
% 2.53/1.64 ! [X0 : $i,X1 : $i,X2 : $i] : (! [X3 : $i] : (((in @ X3 @ X2) != $true) | ((subset @ (setadjoin @ (setadjoin @ X0 @ emptyset) @ (setadjoin @ (setadjoin @ X0 @ (setadjoin @ X3 @ emptyset)) @ emptyset)) @ (powerset @ (binunion @ X1 @ X2))) = $true)) | ($true != (in @ X0 @ X1))) <=> (ubforcartprodlem1 = $true)),
% 2.53/1.64 inference(ennf_transformation,[status(thm)],[f232])).
% 2.53/1.64 thf(f232,plain,(
% 2.53/1.64 ! [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X0 @ X1)) => ! [X3 : $i] : (((in @ X3 @ X2) = $true) => ((subset @ (setadjoin @ (setadjoin @ X0 @ emptyset) @ (setadjoin @ (setadjoin @ X0 @ (setadjoin @ X3 @ emptyset)) @ emptyset)) @ (powerset @ (binunion @ X1 @ X2))) = $true))) <=> (ubforcartprodlem1 = $true)),
% 2.53/1.64 inference(fool_elimination,[status(thm)],[f231])).
% 2.53/1.64 thf(f231,plain,(
% 2.53/1.64 (ubforcartprodlem1 = ! [X0 : $i,X1 : $i,X2 : $i] : ((in @ X0 @ X1) => ! [X3 : $i] : ((in @ X3 @ X2) => (subset @ (setadjoin @ (setadjoin @ X0 @ emptyset) @ (setadjoin @ (setadjoin @ X0 @ (setadjoin @ X3 @ emptyset)) @ emptyset)) @ (powerset @ (binunion @ X1 @ X2))))))),
% 2.53/1.64 inference(rectify,[status(thm)],[f149])).
% 2.53/1.64 thf(f149,axiom,(
% 2.53/1.64 (ubforcartprodlem1 = ! [X1 : $i,X3 : $i,X4 : $i] : ((in @ X1 @ X3) => ! [X2 : $i] : ((in @ X2 @ X4) => (subset @ (setadjoin @ (setadjoin @ X1 @ emptyset) @ (setadjoin @ (setadjoin @ X1 @ (setadjoin @ X2 @ emptyset)) @ emptyset)) @ (powerset @ (binunion @ X3 @ X4))))))),
% 2.53/1.64 file('/export/starexec/sandbox2/tmp/tmp.yC96zsFX7x/DTF2THF_27586.p',ubforcartprodlem1)).
% 2.53/1.64 thf(f2132,plain,(
% 2.53/1.64 ($true != (subset @ (setadjoin @ (setadjoin @ sK187 @ emptyset) @ (setadjoin @ (setadjoin @ sK187 @ (setadjoin @ sK188 @ emptyset)) @ emptyset)) @ (powerset @ (binunion @ sK186 @ sK185))))),
% 2.53/1.64 inference(trivial_inequality_removal,[status(thm)],[f2131])).
% 2.53/1.64 thf(f2131,plain,(
% 2.53/1.64 ($true != $true) | ($true != (subset @ (setadjoin @ (setadjoin @ sK187 @ emptyset) @ (setadjoin @ (setadjoin @ sK187 @ (setadjoin @ sK188 @ emptyset)) @ emptyset)) @ (powerset @ (binunion @ sK186 @ sK185))))),
% 2.53/1.64 inference(forward_demodulation,[status(thm)],[f2120,f1576])).
% 2.53/1.64 thf(f1576,plain,(
% 2.53/1.64 (powersetI1 = $true)),
% 2.53/1.64 inference(cnf_transformation,[status(thm)],[f948])).
% 2.53/1.64 thf(f2120,plain,(
% 2.53/1.64 (powersetI1 != $true) | ($true != (subset @ (setadjoin @ (setadjoin @ sK187 @ emptyset) @ (setadjoin @ (setadjoin @ sK187 @ (setadjoin @ sK188 @ emptyset)) @ emptyset)) @ (powerset @ (binunion @ sK186 @ sK185))))),
% 2.53/1.64 inference(trivial_inequality_removal,[status(thm)],[f2114])).
% 2.53/1.64 thf(f2114,plain,(
% 2.53/1.64 ($true != $true) | (powersetI1 != $true) | ($true != (subset @ (setadjoin @ (setadjoin @ sK187 @ emptyset) @ (setadjoin @ (setadjoin @ sK187 @ (setadjoin @ sK188 @ emptyset)) @ emptyset)) @ (powerset @ (binunion @ sK186 @ sK185))))),
% 2.53/1.64 inference(superposition,[status(thm)],[f1568,f1352])).
% 2.53/1.64 thf(f1352,plain,(
% 2.53/1.64 ( ! [X0 : $i,X1 : $i] : (((in @ X0 @ (powerset @ X1)) = $true) | ($true != (subset @ X0 @ X1)) | (powersetI1 != $true)) )),
% 2.53/1.64 inference(cnf_transformation,[status(thm)],[f701])).
% 2.53/1.64 thf(f701,plain,(
% 2.53/1.64 (! [X0 : $i,X1 : $i] : (($true != (subset @ X0 @ X1)) | ((in @ X0 @ (powerset @ X1)) = $true)) | (powersetI1 != $true)) & ((powersetI1 = $true) | (($true = (subset @ sK37 @ sK38)) & ($true != (in @ sK37 @ (powerset @ sK38)))))),
% 2.53/1.64 inference(skolemisation,[status(esa),new_symbols(skolem,[sK37,sK38])],[f699,f700])).
% 2.53/1.64 thf(f700,plain,(
% 2.53/1.64 ? [X2 : $i,X3 : $i] : (((subset @ X2 @ X3) = $true) & ((in @ X2 @ (powerset @ X3)) != $true)) => (($true = (subset @ sK37 @ sK38)) & ($true != (in @ sK37 @ (powerset @ sK38))))),
% 2.53/1.64 introduced(definition,[],[choice_axiom])).
% 2.53/1.64 thf(f699,plain,(
% 2.53/1.64 (! [X0 : $i,X1 : $i] : (($true != (subset @ X0 @ X1)) | ((in @ X0 @ (powerset @ X1)) = $true)) | (powersetI1 != $true)) & ((powersetI1 = $true) | ? [X2 : $i,X3 : $i] : (((subset @ X2 @ X3) = $true) & ((in @ X2 @ (powerset @ X3)) != $true)))),
% 2.53/1.64 inference(rectify,[status(thm)],[f698])).
% 2.53/1.64 thf(f698,plain,(
% 2.53/1.64 (! [X0 : $i,X1 : $i] : (($true != (subset @ X0 @ X1)) | ((in @ X0 @ (powerset @ X1)) = $true)) | (powersetI1 != $true)) & ((powersetI1 = $true) | ? [X0 : $i,X1 : $i] : (($true = (subset @ X0 @ X1)) & ((in @ X0 @ (powerset @ X1)) != $true)))),
% 2.53/1.64 inference(nnf_transformation,[status(thm)],[f616])).
% 2.53/1.64 thf(f616,plain,(
% 2.53/1.64 ! [X0 : $i,X1 : $i] : (($true != (subset @ X0 @ X1)) | ((in @ X0 @ (powerset @ X1)) = $true)) <=> (powersetI1 = $true)),
% 2.53/1.64 inference(ennf_transformation,[status(thm)],[f158])).
% 2.53/1.64 thf(f158,plain,(
% 2.53/1.64 ! [X0 : $i,X1 : $i] : (($true = (subset @ X0 @ X1)) => ((in @ X0 @ (powerset @ X1)) = $true)) <=> (powersetI1 = $true)),
% 2.53/1.64 inference(fool_elimination,[status(thm)],[f157])).
% 2.53/1.64 thf(f157,plain,(
% 2.53/1.64 (powersetI1 = ! [X0 : $i,X1 : $i] : ((subset @ X0 @ X1) => (in @ X0 @ (powerset @ X1))))),
% 2.53/1.64 inference(rectify,[status(thm)],[f95])).
% 2.53/1.64 thf(f95,axiom,(
% 2.53/1.64 (powersetI1 = ! [X4 : $i,X3 : $i] : ((subset @ X4 @ X3) => (in @ X4 @ (powerset @ X3))))),
% 2.53/1.64 file('/export/starexec/sandbox2/tmp/tmp.yC96zsFX7x/DTF2THF_27586.p',powersetI1)).
% 2.53/1.64 thf(f1568,plain,(
% 2.53/1.64 ($true != (in @ (setadjoin @ (setadjoin @ sK187 @ emptyset) @ (setadjoin @ (setadjoin @ sK187 @ (setadjoin @ sK188 @ emptyset)) @ emptyset)) @ (powerset @ (powerset @ (binunion @ sK186 @ sK185)))))),
% 2.53/1.64 inference(cnf_transformation,[status(thm)],[f948])).
% 2.53/1.64 % SZS output end Proof for DTF2THF_27586
% 2.53/1.64 % (27817)------------------------------
% 2.53/1.64 % (27817)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.53/1.64 % (27817)Termination reason: Refutation
% 2.53/1.64
% 2.53/1.64 % (27817)Memory used [KB]: 7803
% 2.53/1.64 % (27817)Time elapsed: 0.057 s
% 2.53/1.64 % (27817)Instructions burned: 159 (million)
% 2.53/1.64 % (27817)------------------------------
% 2.53/1.64 % (27817)------------------------------
% 2.53/1.64 % (27794)Success in time 0.14 s
%------------------------------------------------------------------------------