%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU626^1 : TPTP v9.2.1. Released v3.7.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n001.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 09:35:42 AM UTC 2026
% Result : Theorem 2.41s 1.56s
% Output : Refutation 2.41s
% 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 : 2669 ( 700 equ; 0 cnn)
% Maximal formula atoms : 146 ( 59 avg)
% Number of connectives : 1608 ( 70 ~; 42 |; 461 &; 441 @)
% ( 4 <=>; 590 =>; 0 <=; 0 <~>)
% Maximal formula depth : 155 ( 27 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 275 ( 272 usr; 155 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_180,type,
sP1: $i > $i > $i > $o > $o ).
thf(func_def_181,type,
sP2: $i > $i > $o ).
thf(func_def_182,type,
sP3: $i > $o ).
thf(func_def_183,type,
sP4: $i > $i > $o ).
thf(func_def_184,type,
sP5: $i > $i > $o ).
thf(func_def_192,type,
sK13: $i > $o ).
thf(func_def_220,type,
sK41: $i > $o ).
thf(func_def_228,type,
sK49: $i > $i ).
thf(func_def_242,type,
sK63: $i > ( $i > $o ) > $i ).
thf(func_def_243,type,
sK64: $i > $o ).
thf(func_def_248,type,
sK69: $i > $o ).
thf(func_def_250,type,
sK71: ( $i > $o ) > $i ).
thf(func_def_251,type,
sK72: ( $i > $o ) > $i ).
thf(func_def_254,type,
sK75: $i > $i > $i ).
thf(func_def_258,type,
sK79: $i > $o ).
thf(func_def_260,type,
sK81: $i > $i ).
thf(func_def_262,type,
sK83: $i > $i ).
thf(func_def_264,type,
sK85: $i > $i > $o ).
thf(func_def_266,type,
sK87: $i > $i ).
thf(func_def_267,type,
sK88: $i > $i ).
thf(func_def_268,type,
sK89: $i > ( $i > $i > $o ) > $i ).
thf(func_def_269,type,
sK90: $i > ( $i > $i > $o ) > $i ).
thf(func_def_270,type,
sK91: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_273,type,
sK94: $i > $i > $i ).
thf(func_def_274,type,
sK95: $i > $i > $i ).
thf(func_def_292,type,
sK113: $i > $i > $i ).
thf(func_def_293,type,
sK114: $i > $i > $i ).
thf(func_def_294,type,
sK115: $i > $i > $i ).
thf(func_def_295,type,
sK116: $i > $i > $i ).
thf(func_def_296,type,
sK117: $i > $i > $i ).
thf(func_def_297,type,
sK118: $i > $i > $i > $i ).
thf(func_def_298,type,
sK119: $i > $i ).
thf(func_def_299,type,
sK120: $i > $i ).
thf(func_def_300,type,
sK121: $i > $i ).
thf(func_def_301,type,
sK122: $i > $i ).
thf(func_def_302,type,
sK123: $i > $i > $i > $i ).
thf(func_def_303,type,
sK124: $i > $i > $i > $i ).
thf(func_def_304,type,
sK125: $i > $i > $i ).
thf(func_def_305,type,
sK126: $i > $i > $i ).
thf(func_def_306,type,
sK127: $i > $i > $i ).
thf(func_def_308,type,
sK129: $i > $i ).
thf(func_def_309,type,
sK130: $i > $i ).
thf(func_def_310,type,
sK131: $i > $i ).
thf(func_def_317,type,
sK138: $i > $i ).
thf(func_def_341,type,
sK162: $i > $i > $i ).
thf(func_def_361,type,
sK182: $i > $o ).
thf(func_def_363,type,
sK184: ( $i > $o ) > $i > $i ).
thf(func_def_364,type,
sK185: ( $i > $o ) > $i ).
thf(func_def_365,type,
sK186: $i > $o ).
thf(func_def_368,type,
sK189: $i > $o ).
thf(func_def_372,type,
sK193: $i > $o ).
thf(func_def_377,type,
sK198: $i > $i > $i ).
thf(func_def_384,type,
sK205: $i > $i > $i ).
thf(func_def_385,type,
sK206: $i > $o ).
thf(func_def_387,type,
sK208: $i > $o ).
thf(func_def_397,type,
sK218: $i > $i ).
thf(func_def_399,type,
sK220: $i > $o ).
thf(func_def_400,type,
sK221: $i > $o ).
thf(func_def_401,type,
sK222: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_402,type,
sK223: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_409,type,
sK230: $i > $i > $i ).
thf(func_def_419,type,
sK240: ( $i > $o ) > $i ).
thf(func_def_420,type,
sK241: $i > $o ).
thf(func_def_421,type,
sK242: $i > $i ).
thf(func_def_422,type,
sK243: $i > $o ).
thf(func_def_428,type,
sK249: $i > ( $i > $o ) > $i ).
thf(func_def_429,type,
sK250: $i > $o ).
thf(func_def_434,type,
sK255: $i > $o ).
thf(func_def_435,type,
sK256: $i > $o ).
thf(func_def_436,type,
sK257: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_437,type,
sK258: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_441,type,
sK262: ( $i > $o ) > $i > $i ).
thf(func_def_443,type,
sK264: $i > $o ).
thf(func_def_459,type,
sK280: ( $i > $o ) > $i > $i ).
thf(func_def_461,type,
sK282: $i > $o ).
thf(func_def_464,type,
sK285: $i > $o ).
thf(func_def_470,type,
sK291: $i > $i ).
thf(func_def_473,type,
sK294: $i > $o ).
thf(func_def_479,type,
sK300: $i > $i > $i ).
thf(func_def_487,type,
sK308: $i > $o ).
thf(func_def_489,type,
sK310: $i > ( $i > $o ) > $i ).
thf(func_def_491,type,
sK312: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_492,type,
sK313: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_493,type,
sK314: $i > $o ).
thf(func_def_494,type,
sK315: $i > $o ).
thf(func_def_503,type,
sK324: $i > $o ).
thf(func_def_506,type,
sK327: $i > $o ).
thf(func_def_507,type,
sK328: $i > $i ).
thf(func_def_508,type,
sK329: ( $i > $o ) > $i ).
thf(func_def_519,type,
sK340: $i > ( $i > $o ) > $i ).
thf(func_def_520,type,
sK341: $i > $o ).
thf(func_def_526,type,
sK347: $i > $o ).
thf(func_def_528,type,
sK349: $i > ( $i > $o ) > $i ).
thf(func_def_529,type,
sK350: $i > $o ).
thf(func_def_536,type,
sK357: $i > $i > $i ).
thf(func_def_541,type,
sK362: $i > $i ).
thf(func_def_545,type,
sK366: $o > $i > $i > $i ).
thf(func_def_571,type,
ph392:
!>[X0: $tType] : X0 ).
thf(f2589,plain,
$false,
inference(trivial_inequality_removal,[status(thm)],[f2584]) ).
thf(f2584,plain,
$true != $true,
inference(superposition,[status(thm)],[f2120,f1623]) ).
thf(f1623,plain,
( ( in @ sK166 @ sK165 )
= $true ),
inference(cnf_transformation,[status(thm)],[f893]) ).
thf(f893,plain,
( ( setadjoin__Cong = $true )
& ( quantDeMorgan1 = $true )
& ( upairsubunion = $true )
& ( upairsetIL = $true )
& ( omegaIndAx = $true )
& ( setminusILneg = $true )
& ( prop2setI = $true )
& ( omega0Ax = $true )
& ( emptyinPowerset = $true )
& ( notequalI1 = $true )
& ( setadjoinAx = $true )
& ( subsetI2 = $true )
& ( dsetconstrEL = $true )
& ( exuI2 = $true )
& ( setminusIRneg = $true )
& ( emptysetE = $true )
& ( subsetE2 = $true )
& ( subset2powerset = $true )
& ( powerset__Cong = $true )
& ( setbeta = $true )
& ( emptysetsubset = $true )
& ( powersetE1 = $true )
& ( subsetemptysetimpeq = $true )
& ( binintersectSubset2 = $true )
& ( exuE3u = $true )
& ( setminusELneg = $true )
& ( singletonsswitch = $true )
& ( setukpairIL = $true )
& ( dsetconstrER = $true )
& ( powersetI = $true )
& ( setminusSubset1 = $true )
& ( binintersectLsub = $true )
& ( setextAx = $true )
& ( setminusEL = $true )
& ( powersetAx = $true )
& ( prop2set2propI = $true )
& ( bs114d = $true )
& ( omegaSAx = $true )
& ( setadjoinE = $true )
& ( notdexE = $true )
& ( binintersectRsub = $true )
& ( exuEu = $true )
& ( ( in @ sK164 @ sK163 )
= $true )
& ( ( in @ sK166 @ sK165 )
= $true )
& ( ( in @ ( setadjoin @ sK164 @ ( setadjoin @ sK166 @ emptyset ) ) @ ( powerset @ ( binunion @ sK163 @ sK165 ) ) )
!= $true )
& ( emptysetAx = $true )
& ( replAx = $true )
& ( inPowerset = $true )
& ( setextsub = $true )
& ( foundationAx = $true )
& ( exuI1 = $true )
& ( singletoninpowerset = $true )
& ( quantDeMorgan2 = $true )
& ( upairset2IR = $true )
& ( setminusER = $true )
& ( eqinunit = $true )
& ( setunionI = $true )
& ( emptysetimpfalse = $true )
& ( subsetE = $true )
& ( dsetconstr__Cong = $true )
& ( vacuousDall = $true )
& ( binunionIL = $true )
& ( singletoninpowunion = $true )
& ( setadjoinIL = $true )
& ( binunionE = $true )
& ( emptyInPowerset = $true )
& ( setminusSubset2 = $true )
& ( setadjoinSub2 = $true )
& ( kpairiskpair = $true )
& ( powersetI1 = $true )
& ( emptyE1 = $true )
& ( symdiffIneg1 = $true )
& ( setminusERneg = $true )
& ( setunionAx = $true )
& ( binunionEcases = $true )
& ( kpairp = $true )
& ( nonemptyI = $true )
& ( emptyinunitempty = $true )
& ( setminusLsub = $true )
& ( setadjoinIR = $true )
& ( descr__Cong = $true )
& ( notdallE = $true )
& ( sepSubset = $true )
& ( setadjoinOr = $true )
& ( notinemptyset = $true )
& ( setunionE = $true )
& ( emptyset__Cong = $true )
& ( exuI3 = $true )
& ( eqimpsubset2 = $true )
& ( symdiffIneg2 = $true )
& ( setminusI = $true )
& ( nonemptyImpWitness = $true )
& ( nonemptyI1 = $true )
& ( powersetE = $true )
& ( powersetsubset = $true )
& ( symdiffE = $true )
& ( exuE3e = $true )
& ( binintersectI = $true )
& ( symdiffI1 = $true )
& ( inCongP = $true )
& ( exuE2 = $true )
& ( binunionIR = $true )
& ( nonemptyE1 = $true )
& ( subPowSU = $true )
& ( binunionLsub = $true )
& ( quantDeMorgan3 = $true )
& ( sepInPowerset = $true )
& ( binintersectSubset5 = $true )
& ( exuE1 = $true )
& ( subsetRefl = $true )
& ( upairsetE = $true )
& ( exu__Cong = $true )
& ( dsetconstrI = $true )
& ( subsetTrans = $true )
& ( binintersectEL = $true )
& ( binintersectER = $true )
& ( secondinupair = $true )
& ( emptyI = $true )
& ( singletonsubset = $true )
& ( eqimpsubset1 = $true )
& ( noeltsimpempty = $true )
& ( omega__Cong = $true )
& ( upairset2E = $true )
& ( prop2setE = $true )
& ( descrp = $true )
& ( wellorderingAx = $true )
& ( notsubsetI = $true )
& ( binunionRsub = $true )
& ( setext = $true )
& ( in__Cong = $true )
& ( uniqinunit = $true )
& ( quantDeMorgan4 = $true )
& ( disjointsetsI1 = $true )
& ( symdiffI2 = $true )
& ( setukpairIR = $true )
& ( upairsetIR = $true )
& ( notinsingleton = $true )
& ( binintersectSubset1 = $true )
& ( setadjoinSub = $true )
& ( setunion__Cong = $true )
& ( setoftrueEq = $true )
& ( binintersectSubset3 = $true )
& ( subsetI1 = $true )
& ( notequalI2 = $true )
& ( binintersectSubset4 = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK163,sK164,sK165,sK166])],[f523,f892,f891]) ).
thf(f891,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X1 @ X0 ) )
& ? [X3: $i] :
( ( ( in @ X3 @ X2 )
= $true )
& ( ( in @ ( setadjoin @ X1 @ ( setadjoin @ X3 @ emptyset ) ) @ ( powerset @ ( binunion @ X0 @ X2 ) ) )
!= $true ) ) )
=> ( ( ( in @ sK164 @ sK163 )
= $true )
& ? [X3: $i] :
( ( $true
= ( in @ X3 @ sK165 ) )
& ( $true
!= ( in @ ( setadjoin @ sK164 @ ( setadjoin @ X3 @ emptyset ) ) @ ( powerset @ ( binunion @ sK163 @ sK165 ) ) ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f892,plain,
( ? [X3: $i] :
( ( $true
= ( in @ X3 @ sK165 ) )
& ( $true
!= ( in @ ( setadjoin @ sK164 @ ( setadjoin @ X3 @ emptyset ) ) @ ( powerset @ ( binunion @ sK163 @ sK165 ) ) ) ) )
=> ( ( ( in @ sK166 @ sK165 )
= $true )
& ( ( in @ ( setadjoin @ sK164 @ ( setadjoin @ sK166 @ emptyset ) ) @ ( powerset @ ( binunion @ sK163 @ sK165 ) ) )
!= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f523,plain,
( ( setadjoin__Cong = $true )
& ( quantDeMorgan1 = $true )
& ( upairsubunion = $true )
& ( upairsetIL = $true )
& ( omegaIndAx = $true )
& ( setminusILneg = $true )
& ( prop2setI = $true )
& ( omega0Ax = $true )
& ( emptyinPowerset = $true )
& ( notequalI1 = $true )
& ( setadjoinAx = $true )
& ( subsetI2 = $true )
& ( dsetconstrEL = $true )
& ( exuI2 = $true )
& ( setminusIRneg = $true )
& ( emptysetE = $true )
& ( subsetE2 = $true )
& ( subset2powerset = $true )
& ( powerset__Cong = $true )
& ( setbeta = $true )
& ( emptysetsubset = $true )
& ( powersetE1 = $true )
& ( subsetemptysetimpeq = $true )
& ( binintersectSubset2 = $true )
& ( exuE3u = $true )
& ( setminusELneg = $true )
& ( singletonsswitch = $true )
& ( setukpairIL = $true )
& ( dsetconstrER = $true )
& ( powersetI = $true )
& ( setminusSubset1 = $true )
& ( binintersectLsub = $true )
& ( setextAx = $true )
& ( setminusEL = $true )
& ( powersetAx = $true )
& ( prop2set2propI = $true )
& ( bs114d = $true )
& ( omegaSAx = $true )
& ( setadjoinE = $true )
& ( notdexE = $true )
& ( binintersectRsub = $true )
& ( exuEu = $true )
& ? [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X1 @ X0 ) )
& ? [X3: $i] :
( ( ( in @ X3 @ X2 )
= $true )
& ( ( in @ ( setadjoin @ X1 @ ( setadjoin @ X3 @ emptyset ) ) @ ( powerset @ ( binunion @ X0 @ X2 ) ) )
!= $true ) ) )
& ( emptysetAx = $true )
& ( replAx = $true )
& ( inPowerset = $true )
& ( setextsub = $true )
& ( foundationAx = $true )
& ( exuI1 = $true )
& ( singletoninpowerset = $true )
& ( quantDeMorgan2 = $true )
& ( upairset2IR = $true )
& ( setminusER = $true )
& ( eqinunit = $true )
& ( setunionI = $true )
& ( emptysetimpfalse = $true )
& ( subsetE = $true )
& ( dsetconstr__Cong = $true )
& ( vacuousDall = $true )
& ( binunionIL = $true )
& ( singletoninpowunion = $true )
& ( setadjoinIL = $true )
& ( binunionE = $true )
& ( emptyInPowerset = $true )
& ( setminusSubset2 = $true )
& ( setadjoinSub2 = $true )
& ( kpairiskpair = $true )
& ( powersetI1 = $true )
& ( emptyE1 = $true )
& ( symdiffIneg1 = $true )
& ( setminusERneg = $true )
& ( setunionAx = $true )
& ( binunionEcases = $true )
& ( kpairp = $true )
& ( nonemptyI = $true )
& ( emptyinunitempty = $true )
& ( setminusLsub = $true )
& ( setadjoinIR = $true )
& ( descr__Cong = $true )
& ( notdallE = $true )
& ( sepSubset = $true )
& ( setadjoinOr = $true )
& ( notinemptyset = $true )
& ( setunionE = $true )
& ( emptyset__Cong = $true )
& ( exuI3 = $true )
& ( eqimpsubset2 = $true )
& ( symdiffIneg2 = $true )
& ( setminusI = $true )
& ( nonemptyImpWitness = $true )
& ( nonemptyI1 = $true )
& ( powersetE = $true )
& ( powersetsubset = $true )
& ( symdiffE = $true )
& ( exuE3e = $true )
& ( binintersectI = $true )
& ( symdiffI1 = $true )
& ( inCongP = $true )
& ( exuE2 = $true )
& ( binunionIR = $true )
& ( nonemptyE1 = $true )
& ( subPowSU = $true )
& ( binunionLsub = $true )
& ( quantDeMorgan3 = $true )
& ( sepInPowerset = $true )
& ( binintersectSubset5 = $true )
& ( exuE1 = $true )
& ( subsetRefl = $true )
& ( upairsetE = $true )
& ( exu__Cong = $true )
& ( dsetconstrI = $true )
& ( subsetTrans = $true )
& ( binintersectEL = $true )
& ( binintersectER = $true )
& ( secondinupair = $true )
& ( emptyI = $true )
& ( singletonsubset = $true )
& ( eqimpsubset1 = $true )
& ( noeltsimpempty = $true )
& ( omega__Cong = $true )
& ( upairset2E = $true )
& ( prop2setE = $true )
& ( descrp = $true )
& ( wellorderingAx = $true )
& ( notsubsetI = $true )
& ( binunionRsub = $true )
& ( setext = $true )
& ( in__Cong = $true )
& ( uniqinunit = $true )
& ( quantDeMorgan4 = $true )
& ( disjointsetsI1 = $true )
& ( symdiffI2 = $true )
& ( setukpairIR = $true )
& ( upairsetIR = $true )
& ( notinsingleton = $true )
& ( binintersectSubset1 = $true )
& ( setadjoinSub = $true )
& ( setunion__Cong = $true )
& ( setoftrueEq = $true )
& ( binintersectSubset3 = $true )
& ( subsetI1 = $true )
& ( notequalI2 = $true )
& ( binintersectSubset4 = $true ) ),
inference(flattening,[status(thm)],[f522]) ).
thf(f522,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X1 @ X0 ) )
& ? [X3: $i] :
( ( ( in @ X3 @ X2 )
= $true )
& ( ( in @ ( setadjoin @ X1 @ ( setadjoin @ X3 @ emptyset ) ) @ ( powerset @ ( binunion @ X0 @ X2 ) ) )
!= $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)],[f296]) ).
thf(f296,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 )
=> ! [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X1 @ X0 ) )
=> ! [X3: $i] :
( ( ( in @ X3 @ X2 )
= $true )
=> ( ( in @ ( setadjoin @ X1 @ ( setadjoin @ X3 @ emptyset ) ) @ ( powerset @ ( binunion @ X0 @ X2 ) ) )
= $true ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f295]) ).
thf(f295,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
=> ! [X0: $i,X1: $i,X2: $i] :
( ( in @ X1 @ X0 )
=> ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ ( setadjoin @ X1 @ ( setadjoin @ X3 @ emptyset ) ) @ ( powerset @ ( binunion @ X0 @ X2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f149]) ).
thf(f149,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
=> ! [X3: $i,X1: $i,X4: $i] :
( ( in @ X1 @ X3 )
=> ! [X2: $i] :
( ( in @ X2 @ X4 )
=> ( in @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ ( powerset @ ( binunion @ X3 @ X4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f148]) ).
thf(f148,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
=> ! [X3: $i,X1: $i,X4: $i] :
( ( in @ X1 @ X3 )
=> ! [X2: $i] :
( ( in @ X2 @ X4 )
=> ( in @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ ( powerset @ ( binunion @ X3 @ X4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.DbdYpcF5sg/DTF2THF_22176.p',upairinpowunion) ).
thf(f2120,plain,
( ( in @ sK166 @ sK165 )
!= $true ),
inference(trivial_inequality_removal,[status(thm)],[f2119]) ).
thf(f2119,plain,
( ( $true != $true )
| ( ( in @ sK166 @ sK165 )
!= $true ) ),
inference(forward_demodulation,[status(thm)],[f2118,f1664]) ).
thf(f1664,plain,
upairsubunion = $true,
inference(cnf_transformation,[status(thm)],[f893]) ).
thf(f2118,plain,
( ( ( in @ sK166 @ sK165 )
!= $true )
| ( upairsubunion != $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f2117]) ).
thf(f2117,plain,
( ( $true != $true )
| ( upairsubunion != $true )
| ( ( in @ sK166 @ sK165 )
!= $true ) ),
inference(forward_demodulation,[status(thm)],[f2116,f1624]) ).
thf(f1624,plain,
( ( in @ sK164 @ sK163 )
= $true ),
inference(cnf_transformation,[status(thm)],[f893]) ).
thf(f2116,plain,
( ( ( in @ sK166 @ sK165 )
!= $true )
| ( ( in @ sK164 @ sK163 )
!= $true )
| ( upairsubunion != $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f2107]) ).
thf(f2107,plain,
( ( upairsubunion != $true )
| ( $true != $true )
| ( ( in @ sK164 @ sK163 )
!= $true )
| ( ( in @ sK166 @ sK165 )
!= $true ) ),
inference(superposition,[status(thm)],[f2098,f1514]) ).
thf(f1514,plain,
! [X6: $i,X7: $i,X4: $i,X5: $i] :
( ( ( subset @ ( setadjoin @ X5 @ ( setadjoin @ X7 @ emptyset ) ) @ ( binunion @ X4 @ X6 ) )
= $true )
| ( upairsubunion != $true )
| ( ( in @ X5 @ X4 )
!= $true )
| ( $true
!= ( in @ X7 @ X6 ) ) ),
inference(cnf_transformation,[status(thm)],[f885]) ).
thf(f885,plain,
( ( ( upairsubunion = $true )
| ( ( ( in @ sK159 @ sK158 )
= $true )
& ( ( subset @ ( setadjoin @ sK157 @ ( setadjoin @ sK159 @ emptyset ) ) @ ( binunion @ sK156 @ sK158 ) )
!= $true )
& ( $true
= ( in @ sK157 @ sK156 ) ) ) )
& ( ! [X4: $i,X5: $i,X6: $i] :
( ! [X7: $i] :
( ( $true
!= ( in @ X7 @ X6 ) )
| ( ( subset @ ( setadjoin @ X5 @ ( setadjoin @ X7 @ emptyset ) ) @ ( binunion @ X4 @ X6 ) )
= $true ) )
| ( ( in @ X5 @ X4 )
!= $true ) )
| ( upairsubunion != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK156,sK157,sK158,sK159])],[f882,f884,f883]) ).
thf(f883,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ? [X3: $i] :
( ( ( in @ X3 @ X2 )
= $true )
& ( ( subset @ ( setadjoin @ X1 @ ( setadjoin @ X3 @ emptyset ) ) @ ( binunion @ X0 @ X2 ) )
!= $true ) )
& ( $true
= ( in @ X1 @ X0 ) ) )
=> ( ? [X3: $i] :
( ( $true
= ( in @ X3 @ sK158 ) )
& ( ( subset @ ( setadjoin @ sK157 @ ( setadjoin @ X3 @ emptyset ) ) @ ( binunion @ sK156 @ sK158 ) )
!= $true ) )
& ( $true
= ( in @ sK157 @ sK156 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f884,plain,
( ? [X3: $i] :
( ( $true
= ( in @ X3 @ sK158 ) )
& ( ( subset @ ( setadjoin @ sK157 @ ( setadjoin @ X3 @ emptyset ) ) @ ( binunion @ sK156 @ sK158 ) )
!= $true ) )
=> ( ( ( in @ sK159 @ sK158 )
= $true )
& ( ( subset @ ( setadjoin @ sK157 @ ( setadjoin @ sK159 @ emptyset ) ) @ ( binunion @ sK156 @ sK158 ) )
!= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f882,plain,
( ( ( upairsubunion = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ? [X3: $i] :
( ( ( in @ X3 @ X2 )
= $true )
& ( ( subset @ ( setadjoin @ X1 @ ( setadjoin @ X3 @ emptyset ) ) @ ( binunion @ X0 @ X2 ) )
!= $true ) )
& ( $true
= ( in @ X1 @ X0 ) ) ) )
& ( ! [X4: $i,X5: $i,X6: $i] :
( ! [X7: $i] :
( ( $true
!= ( in @ X7 @ X6 ) )
| ( ( subset @ ( setadjoin @ X5 @ ( setadjoin @ X7 @ emptyset ) ) @ ( binunion @ X4 @ X6 ) )
= $true ) )
| ( ( in @ X5 @ X4 )
!= $true ) )
| ( upairsubunion != $true ) ) ),
inference(rectify,[status(thm)],[f881]) ).
thf(f881,plain,
( ( ( upairsubunion = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ? [X3: $i] :
( ( ( in @ X3 @ X2 )
= $true )
& ( ( subset @ ( setadjoin @ X1 @ ( setadjoin @ X3 @ emptyset ) ) @ ( binunion @ X0 @ X2 ) )
!= $true ) )
& ( $true
= ( in @ X1 @ X0 ) ) ) )
& ( ! [X0: $i,X1: $i,X2: $i] :
( ! [X3: $i] :
( ( ( in @ X3 @ X2 )
!= $true )
| ( ( subset @ ( setadjoin @ X1 @ ( setadjoin @ X3 @ emptyset ) ) @ ( binunion @ X0 @ X2 ) )
= $true ) )
| ( $true
!= ( in @ X1 @ X0 ) ) )
| ( upairsubunion != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f521]) ).
thf(f521,plain,
( ( upairsubunion = $true )
<=> ! [X0: $i,X1: $i,X2: $i] :
( ! [X3: $i] :
( ( ( in @ X3 @ X2 )
!= $true )
| ( ( subset @ ( setadjoin @ X1 @ ( setadjoin @ X3 @ emptyset ) ) @ ( binunion @ X0 @ X2 ) )
= $true ) )
| ( $true
!= ( in @ X1 @ X0 ) ) ) ),
inference(ennf_transformation,[status(thm)],[f322]) ).
thf(f322,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X1 @ X0 ) )
=> ! [X3: $i] :
( ( ( in @ X3 @ X2 )
= $true )
=> ( ( subset @ ( setadjoin @ X1 @ ( setadjoin @ X3 @ emptyset ) ) @ ( binunion @ X0 @ X2 ) )
= $true ) ) )
<=> ( upairsubunion = $true ) ),
inference(fool_elimination,[status(thm)],[f321]) ).
thf(f321,plain,
( upairsubunion
= ( ! [X0: $i,X1: $i,X2: $i] :
( ( in @ X1 @ X0 )
=> ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( subset @ ( setadjoin @ X1 @ ( setadjoin @ X3 @ emptyset ) ) @ ( binunion @ X0 @ X2 ) ) ) ) ) ),
inference(rectify,[status(thm)],[f147]) ).
thf(f147,axiom,
( upairsubunion
= ( ! [X3: $i,X1: $i,X4: $i] :
( ( in @ X1 @ X3 )
=> ! [X2: $i] :
( ( in @ X2 @ X4 )
=> ( subset @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ ( binunion @ X3 @ X4 ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.DbdYpcF5sg/DTF2THF_22176.p',upairsubunion) ).
thf(f2098,plain,
( ( subset @ ( setadjoin @ sK164 @ ( setadjoin @ sK166 @ emptyset ) ) @ ( binunion @ sK163 @ sK165 ) )
!= $true ),
inference(trivial_inequality_removal,[status(thm)],[f2097]) ).
thf(f2097,plain,
( ( ( subset @ ( setadjoin @ sK164 @ ( setadjoin @ sK166 @ emptyset ) ) @ ( binunion @ sK163 @ sK165 ) )
!= $true )
| ( $true != $true ) ),
inference(forward_demodulation,[status(thm)],[f2096,f1649]) ).
thf(f1649,plain,
subset2powerset = $true,
inference(cnf_transformation,[status(thm)],[f893]) ).
thf(f2096,plain,
( ( subset2powerset != $true )
| ( ( subset @ ( setadjoin @ sK164 @ ( setadjoin @ sK166 @ emptyset ) ) @ ( binunion @ sK163 @ sK165 ) )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f2085]) ).
thf(f2085,plain,
( ( subset2powerset != $true )
| ( ( subset @ ( setadjoin @ sK164 @ ( setadjoin @ sK166 @ emptyset ) ) @ ( binunion @ sK163 @ sK165 ) )
!= $true )
| ( $true != $true ) ),
inference(superposition,[status(thm)],[f1622,f1759]) ).
thf(f1759,plain,
! [X0: $i,X1: $i] :
( ( ( in @ X0 @ ( powerset @ X1 ) )
= $true )
| ( subset2powerset != $true )
| ( ( subset @ X0 @ X1 )
!= $true ) ),
inference(cnf_transformation,[status(thm)],[f1011]) ).
thf(f1011,plain,
( ( ! [X0: $i,X1: $i] :
( ( ( in @ X0 @ ( powerset @ X1 ) )
= $true )
| ( ( subset @ X0 @ X1 )
!= $true ) )
| ( subset2powerset != $true ) )
& ( ( subset2powerset = $true )
| ( ( $true
!= ( in @ sK231 @ ( powerset @ sK232 ) ) )
& ( ( subset @ sK231 @ sK232 )
= $true ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK231,sK232])],[f1009,f1010]) ).
thf(f1010,plain,
( ? [X2: $i,X3: $i] :
( ( ( in @ X2 @ ( powerset @ X3 ) )
!= $true )
& ( $true
= ( subset @ X2 @ X3 ) ) )
=> ( ( $true
!= ( in @ sK231 @ ( powerset @ sK232 ) ) )
& ( ( subset @ sK231 @ sK232 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1009,plain,
( ( ! [X0: $i,X1: $i] :
( ( ( in @ X0 @ ( powerset @ X1 ) )
= $true )
| ( ( subset @ X0 @ X1 )
!= $true ) )
| ( subset2powerset != $true ) )
& ( ( subset2powerset = $true )
| ? [X2: $i,X3: $i] :
( ( ( in @ X2 @ ( powerset @ X3 ) )
!= $true )
& ( $true
= ( subset @ X2 @ X3 ) ) ) ) ),
inference(rectify,[status(thm)],[f1008]) ).
thf(f1008,plain,
( ( ! [X0: $i,X1: $i] :
( ( ( in @ X0 @ ( powerset @ X1 ) )
= $true )
| ( ( subset @ X0 @ X1 )
!= $true ) )
| ( subset2powerset != $true ) )
& ( ( subset2powerset = $true )
| ? [X0: $i,X1: $i] :
( ( ( in @ X0 @ ( powerset @ X1 ) )
!= $true )
& ( ( subset @ X0 @ X1 )
= $true ) ) ) ),
inference(nnf_transformation,[status(thm)],[f566]) ).
thf(f566,plain,
( ! [X0: $i,X1: $i] :
( ( ( in @ X0 @ ( powerset @ X1 ) )
= $true )
| ( ( subset @ X0 @ X1 )
!= $true ) )
<=> ( subset2powerset = $true ) ),
inference(ennf_transformation,[status(thm)],[f188]) ).
thf(f188,plain,
( ( subset2powerset = $true )
<=> ! [X0: $i,X1: $i] :
( ( ( subset @ X0 @ X1 )
= $true )
=> ( ( in @ X0 @ ( powerset @ X1 ) )
= $true ) ) ),
inference(fool_elimination,[status(thm)],[f187]) ).
thf(f187,plain,
( subset2powerset
= ( ! [X0: $i,X1: $i] :
( ( subset @ X0 @ X1 )
=> ( in @ X0 @ ( powerset @ X1 ) ) ) ) ),
inference(rectify,[status(thm)],[f92]) ).
thf(f92,axiom,
( subset2powerset
= ( ! [X3: $i,X4: $i] :
( ( subset @ X3 @ X4 )
=> ( in @ X3 @ ( powerset @ X4 ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.DbdYpcF5sg/DTF2THF_22176.p',subset2powerset) ).
thf(f1622,plain,
( ( in @ ( setadjoin @ sK164 @ ( setadjoin @ sK166 @ emptyset ) ) @ ( powerset @ ( binunion @ sK163 @ sK165 ) ) )
!= $true ),
inference(cnf_transformation,[status(thm)],[f893]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SEU626^1 : TPTP v9.2.1. Released v3.7.0.
% 0.12/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.17/0.34 % Computer : n001.cluster.edu
% 0.17/0.34 % Model : x86_64 x86_64
% 0.17/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.34 % Memory : 8042.1875MB
% 0.17/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.17/0.34 % CPULimit : 300
% 0.17/0.34 % WCLimit : 300
% 0.17/0.34 % DateTime : Tue Feb 24 12:29:20 EST 2026
% 0.17/0.34 % CPUTime :
% 0.17/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.29/0.46 ---- Original DTF file ---
% 0.29/0.47 thf(spec,logic,$$dhol).
% 0.29/0.47 %------------------------------------------------------------------------------
% 0.29/0.47 % File : SEU626^1 : TPTP v9.2.1. Released v3.7.0.
% 0.29/0.47 % Domain : Set Theory
% 0.29/0.47 % Problem : Ordered Pairs - Cartesian Products
% 0.29/0.47 % Version : Especial.
% 0.29/0.47 % English : (! A:i.! B:i.! x:i.in x A -> (! y:i.in y B -> in (setadjoin x
% 0.29/0.47 % (setadjoin y emptyset)) (powerset (binunion A B))))
% 0.29/0.47
% 0.29/0.47 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.29/0.47 % Source : [Bro08]
% 0.29/0.47 % Names : ZFC128g [Bro08]
% 0.29/0.47
% 0.29/0.47 % Status : Theorem
% 0.29/0.47 % Rating : 0.11 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, 0.22 v7.2.0, 0.12 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 v5.5.0, 0.50 v5.4.0, 0.60 v4.1.0, 1.00 v3.7.0
% 0.29/0.47 % Syntax : Number of formulae : 313 ( 147 unt; 165 typ; 147 def)
% 0.29/0.47 % Number of atoms : 776 ( 208 equ; 0 cnn)
% 0.29/0.47 % Maximal formula atoms : 145 ( 5 avg)
% 0.29/0.47 % Number of connectives : 1312 ( 47 ~; 7 |; 32 &; 872 @)
% 0.29/0.47 % ( 14 <=>; 340 =>; 0 <=; 0 <~>)
% 0.29/0.47 % Maximal formula depth : 154 ( 2 avg)
% 0.29/0.47 % Number of types : 2 ( 0 usr)
% 0.29/0.47 % Number of type conns : 69 ( 69 >; 0 *; 0 +; 0 <<)
% 0.29/0.47 % Number of symbols : 168 ( 165 usr; 146 con; 0-2 aty)
% 0.29/0.47 % Number of variables : 424 ( 36 ^; 355 !; 33 ?; 424 :)
% 0.29/0.47 % SPC : TH0_THM_EQU_NAR
% 0.29/0.47
% 0.29/0.47 % Comments : http://mathgate.info/detsetitem.php?id=183
% 0.29/0.47 %------------------------------------------------------------------------------
% 0.29/0.47 thf(in_type,type,
% 0.29/0.47 in: $i > $i > $o ).
% 0.29/0.47
% 0.29/0.47 thf(exu_type,type,
% 0.29/0.47 exu: ( $i > $o ) > $o ).
% 0.29/0.47
% 0.29/0.47 thf(exu,definition,
% 0.29/0.47 ( exu
% 0.29/0.47 = ( ^ [Xphi: $i > $o] :
% 0.29/0.47 ? [Xx: $i] :
% 0.29/0.47 ( ( Xphi @ Xx )
% 0.29/0.47 & ! [Xy: $i] :
% 0.29/0.47 ( ( Xphi @ Xy )
% 0.29/0.47 => ( Xx = Xy ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(setextAx_type,type,
% 0.29/0.47 setextAx: $o ).
% 0.29/0.47
% 0.29/0.47 thf(setextAx,definition,
% 0.29/0.47 ( setextAx
% 0.29/0.47 = ( ! [A: $i,B: $i] :
% 0.29/0.47 ( ! [Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ A )
% 0.29/0.47 <=> ( in @ Xx @ B ) )
% 0.29/0.47 => ( A = B ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(emptyset_type,type,
% 0.29/0.47 emptyset: $i ).
% 0.29/0.47
% 0.29/0.47 thf(emptysetAx_type,type,
% 0.29/0.47 emptysetAx: $o ).
% 0.29/0.47
% 0.29/0.47 thf(emptysetAx,definition,
% 0.29/0.47 ( emptysetAx
% 0.29/0.47 = ( ! [Xx: $i] :
% 0.29/0.47 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(setadjoin_type,type,
% 0.29/0.47 setadjoin: $i > $i > $i ).
% 0.29/0.47
% 0.29/0.47 thf(setadjoinAx_type,type,
% 0.29/0.47 setadjoinAx: $o ).
% 0.29/0.47
% 0.29/0.47 thf(setadjoinAx,definition,
% 0.29/0.47 ( setadjoinAx
% 0.29/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.29/0.47 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.29/0.47 <=> ( ( Xy = Xx )
% 0.29/0.47 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(powerset_type,type,
% 0.29/0.47 powerset: $i > $i ).
% 0.29/0.47
% 0.29/0.47 thf(powersetAx_type,type,
% 0.29/0.47 powersetAx: $o ).
% 0.29/0.47
% 0.29/0.47 thf(powersetAx,definition,
% 0.29/0.47 ( powersetAx
% 0.29/0.47 = ( ! [A: $i,B: $i] :
% 0.29/0.47 ( ( in @ B @ ( powerset @ A ) )
% 0.29/0.47 <=> ! [Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ B )
% 0.29/0.47 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(setunion_type,type,
% 0.29/0.47 setunion: $i > $i ).
% 0.29/0.47
% 0.29/0.47 thf(setunionAx_type,type,
% 0.29/0.47 setunionAx: $o ).
% 0.29/0.47
% 0.29/0.47 thf(setunionAx,definition,
% 0.29/0.47 ( setunionAx
% 0.29/0.47 = ( ! [A: $i,Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ ( setunion @ A ) )
% 0.29/0.47 <=> ? [B: $i] :
% 0.29/0.47 ( ( in @ Xx @ B )
% 0.29/0.47 & ( in @ B @ A ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(omega_type,type,
% 0.29/0.47 omega: $i ).
% 0.29/0.47
% 0.29/0.47 thf(omega0Ax_type,type,
% 0.29/0.47 omega0Ax: $o ).
% 0.29/0.47
% 0.29/0.47 thf(omega0Ax,definition,
% 0.29/0.47 ( omega0Ax
% 0.29/0.47 = ( in @ emptyset @ omega ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(omegaSAx_type,type,
% 0.29/0.47 omegaSAx: $o ).
% 0.29/0.47
% 0.29/0.47 thf(omegaSAx,definition,
% 0.29/0.47 ( omegaSAx
% 0.29/0.47 = ( ! [Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ omega )
% 0.29/0.47 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(omegaIndAx_type,type,
% 0.29/0.47 omegaIndAx: $o ).
% 0.29/0.47
% 0.29/0.47 thf(omegaIndAx,definition,
% 0.29/0.47 ( omegaIndAx
% 0.29/0.47 = ( ! [A: $i] :
% 0.29/0.47 ( ( ( in @ emptyset @ A )
% 0.29/0.47 & ! [Xx: $i] :
% 0.29/0.47 ( ( ( in @ Xx @ omega )
% 0.29/0.47 & ( in @ Xx @ A ) )
% 0.29/0.47 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.29/0.47 => ! [Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ omega )
% 0.29/0.47 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(replAx_type,type,
% 0.29/0.47 replAx: $o ).
% 0.29/0.47
% 0.29/0.47 thf(replAx,definition,
% 0.29/0.47 ( replAx
% 0.29/0.47 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.29/0.47 ( ! [Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ A )
% 0.29/0.47 => ( exu
% 0.29/0.47 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.29/0.47 => ? [B: $i] :
% 0.29/0.47 ! [Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ B )
% 0.29/0.47 <=> ? [Xy: $i] :
% 0.29/0.47 ( ( in @ Xy @ A )
% 0.29/0.47 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(foundationAx_type,type,
% 0.29/0.47 foundationAx: $o ).
% 0.29/0.47
% 0.29/0.47 thf(foundationAx,definition,
% 0.29/0.47 ( foundationAx
% 0.29/0.47 = ( ! [A: $i] :
% 0.29/0.47 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.29/0.47 => ? [B: $i] :
% 0.29/0.47 ( ( in @ B @ A )
% 0.29/0.47 & ~ ? [Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ B )
% 0.29/0.47 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(wellorderingAx_type,type,
% 0.29/0.47 wellorderingAx: $o ).
% 0.29/0.47
% 0.29/0.47 thf(wellorderingAx,definition,
% 0.29/0.47 ( wellorderingAx
% 0.29/0.47 = ( ! [A: $i] :
% 0.29/0.47 ? [B: $i] :
% 0.29/0.47 ( ! [C: $i] :
% 0.29/0.47 ( ( in @ C @ B )
% 0.29/0.47 => ! [Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ C )
% 0.29/0.47 => ( in @ Xx @ A ) ) )
% 0.29/0.47 & ! [Xx: $i,Xy: $i] :
% 0.29/0.47 ( ( ( in @ Xx @ A )
% 0.29/0.47 & ( in @ Xy @ A ) )
% 0.29/0.47 => ( ! [C: $i] :
% 0.29/0.47 ( ( in @ C @ B )
% 0.29/0.47 => ( ( in @ Xx @ C )
% 0.29/0.47 <=> ( in @ Xy @ C ) ) )
% 0.29/0.47 => ( Xx = Xy ) ) )
% 0.29/0.47 & ! [C: $i,D: $i] :
% 0.29/0.47 ( ( ( in @ C @ B )
% 0.29/0.47 & ( in @ D @ B ) )
% 0.29/0.47 => ( ! [Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ C )
% 0.29/0.47 => ( in @ Xx @ D ) )
% 0.29/0.47 | ! [Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ D )
% 0.29/0.47 => ( in @ Xx @ C ) ) ) )
% 0.29/0.47 & ! [C: $i] :
% 0.29/0.47 ( ( ! [Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ C )
% 0.29/0.47 => ( in @ Xx @ A ) )
% 0.29/0.47 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.29/0.47 => ? [D: $i,Xx: $i] :
% 0.29/0.47 ( ( in @ D @ B )
% 0.29/0.47 & ( in @ Xx @ C )
% 0.29/0.47 & ~ ? [Xy: $i] :
% 0.29/0.47 ( ( in @ Xy @ D )
% 0.29/0.47 & ( in @ Xy @ C ) )
% 0.29/0.47 & ! [E: $i] :
% 0.29/0.47 ( ( in @ E @ B )
% 0.29/0.47 => ( ! [Xy: $i] :
% 0.29/0.47 ( ( in @ Xy @ E )
% 0.29/0.47 => ( in @ Xy @ D ) )
% 0.29/0.47 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(descr_type,type,
% 0.29/0.47 descr: ( $i > $o ) > $i ).
% 0.29/0.47
% 0.29/0.47 thf(descrp_type,type,
% 0.29/0.47 descrp: $o ).
% 0.29/0.47
% 0.29/0.47 thf(descrp,definition,
% 0.29/0.47 ( descrp
% 0.29/0.47 = ( ! [Xphi: $i > $o] :
% 0.29/0.47 ( ( exu
% 0.29/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.29/0.47 => ( Xphi
% 0.29/0.47 @ ( descr
% 0.29/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(dsetconstr_type,type,
% 0.29/0.47 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.29/0.47
% 0.29/0.47 thf(dsetconstrI_type,type,
% 0.29/0.47 dsetconstrI: $o ).
% 0.29/0.47
% 0.29/0.47 thf(dsetconstrI,definition,
% 0.29/0.47 ( dsetconstrI
% 0.29/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ A )
% 0.29/0.47 => ( ( Xphi @ Xx )
% 0.29/0.47 => ( in @ Xx
% 0.29/0.47 @ ( dsetconstr @ A
% 0.29/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(dsetconstrEL_type,type,
% 0.29/0.47 dsetconstrEL: $o ).
% 0.29/0.47
% 0.29/0.47 thf(dsetconstrEL,definition,
% 0.29/0.47 ( dsetconstrEL
% 0.29/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.29/0.47 ( ( in @ Xx
% 0.29/0.47 @ ( dsetconstr @ A
% 0.29/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.29/0.47 => ( in @ Xx @ A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(dsetconstrER_type,type,
% 0.29/0.47 dsetconstrER: $o ).
% 0.29/0.47
% 0.29/0.47 thf(dsetconstrER,definition,
% 0.29/0.47 ( dsetconstrER
% 0.29/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.29/0.47 ( ( in @ Xx
% 0.29/0.47 @ ( dsetconstr @ A
% 0.29/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.29/0.47 => ( Xphi @ Xx ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(exuE1_type,type,
% 0.29/0.47 exuE1: $o ).
% 0.29/0.47
% 0.29/0.47 thf(exuE1,definition,
% 0.29/0.47 ( exuE1
% 0.29/0.47 = ( ! [Xphi: $i > $o] :
% 0.29/0.47 ( ( exu
% 0.29/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.29/0.47 => ? [Xx: $i] :
% 0.29/0.47 ( ( Xphi @ Xx )
% 0.29/0.47 & ! [Xy: $i] :
% 0.29/0.47 ( ( Xphi @ Xy )
% 0.29/0.47 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(prop2set_type,type,
% 0.29/0.47 prop2set: $o > $i ).
% 0.29/0.47
% 0.29/0.47 thf(prop2setE_type,type,
% 0.29/0.47 prop2setE: $o ).
% 0.29/0.47
% 0.29/0.47 thf(prop2setE,definition,
% 0.29/0.47 ( prop2setE
% 0.29/0.47 = ( ! [Xphi: $o,Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.29/0.47 => Xphi ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(emptysetE_type,type,
% 0.29/0.47 emptysetE: $o ).
% 0.29/0.47
% 0.29/0.47 thf(emptysetE,definition,
% 0.29/0.47 ( emptysetE
% 0.29/0.47 = ( ! [Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ emptyset )
% 0.29/0.47 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(emptysetimpfalse_type,type,
% 0.29/0.47 emptysetimpfalse: $o ).
% 0.29/0.47
% 0.29/0.47 thf(emptysetimpfalse,definition,
% 0.29/0.47 ( emptysetimpfalse
% 0.29/0.47 = ( ! [Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ emptyset )
% 0.29/0.47 => $false ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(notinemptyset_type,type,
% 0.29/0.47 notinemptyset: $o ).
% 0.29/0.47
% 0.29/0.47 thf(notinemptyset,definition,
% 0.29/0.47 ( notinemptyset
% 0.29/0.47 = ( ! [Xx: $i] :
% 0.29/0.47 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(exuE3e_type,type,
% 0.29/0.47 exuE3e: $o ).
% 0.29/0.47
% 0.29/0.47 thf(exuE3e,definition,
% 0.29/0.47 ( exuE3e
% 0.29/0.47 = ( ! [Xphi: $i > $o] :
% 0.29/0.47 ( ( exu
% 0.29/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.29/0.47 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(setext_type,type,
% 0.29/0.47 setext: $o ).
% 0.29/0.47
% 0.29/0.47 thf(setext,definition,
% 0.29/0.47 ( setext
% 0.29/0.47 = ( ! [A: $i,B: $i] :
% 0.29/0.47 ( ! [Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ A )
% 0.29/0.47 => ( in @ Xx @ B ) )
% 0.29/0.47 => ( ! [Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ B )
% 0.29/0.47 => ( in @ Xx @ A ) )
% 0.29/0.47 => ( A = B ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(emptyI_type,type,
% 0.29/0.47 emptyI: $o ).
% 0.29/0.47
% 0.29/0.47 thf(emptyI,definition,
% 0.29/0.47 ( emptyI
% 0.29/0.47 = ( ! [A: $i] :
% 0.29/0.47 ( ! [Xx: $i] :
% 0.29/0.47 ~ ( in @ Xx @ A )
% 0.29/0.47 => ( A = emptyset ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(noeltsimpempty_type,type,
% 0.29/0.47 noeltsimpempty: $o ).
% 0.29/0.47
% 0.29/0.47 thf(noeltsimpempty,definition,
% 0.29/0.47 ( noeltsimpempty
% 0.29/0.47 = ( ! [A: $i] :
% 0.29/0.47 ( ! [Xx: $i] :
% 0.29/0.47 ~ ( in @ Xx @ A )
% 0.29/0.47 => ( A = emptyset ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(setbeta_type,type,
% 0.29/0.47 setbeta: $o ).
% 0.29/0.47
% 0.29/0.47 thf(setbeta,definition,
% 0.29/0.47 ( setbeta
% 0.29/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ A )
% 0.29/0.47 => ( ( in @ Xx
% 0.29/0.47 @ ( dsetconstr @ A
% 0.29/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.29/0.47 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(nonempty_type,type,
% 0.29/0.47 nonempty: $i > $o ).
% 0.29/0.47
% 0.29/0.47 thf(nonempty,definition,
% 0.29/0.47 ( nonempty
% 0.29/0.47 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(nonemptyE1_type,type,
% 0.29/0.47 nonemptyE1: $o ).
% 0.29/0.47
% 0.29/0.47 thf(nonemptyE1,definition,
% 0.29/0.47 ( nonemptyE1
% 0.29/0.47 = ( ! [A: $i] :
% 0.29/0.47 ( ( nonempty @ A )
% 0.29/0.47 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(nonemptyI_type,type,
% 0.29/0.47 nonemptyI: $o ).
% 0.29/0.47
% 0.29/0.47 thf(nonemptyI,definition,
% 0.29/0.47 ( nonemptyI
% 0.29/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ A )
% 0.29/0.47 => ( ( Xphi @ Xx )
% 0.29/0.47 => ( nonempty
% 0.29/0.47 @ ( dsetconstr @ A
% 0.29/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(nonemptyI1_type,type,
% 0.29/0.47 nonemptyI1: $o ).
% 0.29/0.47
% 0.29/0.47 thf(nonemptyI1,definition,
% 0.29/0.47 ( nonemptyI1
% 0.29/0.47 = ( ! [A: $i] :
% 0.29/0.47 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.29/0.47 => ( nonempty @ A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(setadjoinIL_type,type,
% 0.29/0.47 setadjoinIL: $o ).
% 0.29/0.47
% 0.29/0.47 thf(setadjoinIL,definition,
% 0.29/0.47 ( setadjoinIL
% 0.29/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(emptyinunitempty_type,type,
% 0.29/0.47 emptyinunitempty: $o ).
% 0.29/0.47
% 0.29/0.47 thf(emptyinunitempty,definition,
% 0.29/0.47 ( emptyinunitempty
% 0.29/0.47 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(setadjoinIR_type,type,
% 0.29/0.47 setadjoinIR: $o ).
% 0.29/0.47
% 0.29/0.47 thf(setadjoinIR,definition,
% 0.29/0.47 ( setadjoinIR
% 0.29/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.29/0.47 ( ( in @ Xy @ A )
% 0.29/0.47 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(setadjoinE_type,type,
% 0.29/0.47 setadjoinE: $o ).
% 0.29/0.47
% 0.29/0.47 thf(setadjoinE,definition,
% 0.29/0.47 ( setadjoinE
% 0.29/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.29/0.47 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.29/0.47 => ! [Xphi: $o] :
% 0.29/0.47 ( ( ( Xy = Xx )
% 0.29/0.47 => Xphi )
% 0.29/0.47 => ( ( ( in @ Xy @ A )
% 0.29/0.47 => Xphi )
% 0.29/0.47 => Xphi ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(setadjoinOr_type,type,
% 0.29/0.47 setadjoinOr: $o ).
% 0.29/0.47
% 0.29/0.47 thf(setadjoinOr,definition,
% 0.29/0.47 ( setadjoinOr
% 0.29/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.29/0.47 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.29/0.47 => ( ( Xy = Xx )
% 0.29/0.47 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(setoftrueEq_type,type,
% 0.29/0.47 setoftrueEq: $o ).
% 0.29/0.47
% 0.29/0.47 thf(setoftrueEq,definition,
% 0.29/0.47 ( setoftrueEq
% 0.29/0.47 = ( ! [A: $i] :
% 0.29/0.47 ( ( dsetconstr @ A
% 0.29/0.47 @ ^ [Xx: $i] : $true )
% 0.29/0.47 = A ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(powersetI_type,type,
% 0.29/0.47 powersetI: $o ).
% 0.29/0.47
% 0.29/0.47 thf(powersetI,definition,
% 0.29/0.47 ( powersetI
% 0.29/0.47 = ( ! [A: $i,B: $i] :
% 0.29/0.47 ( ! [Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ B )
% 0.29/0.47 => ( in @ Xx @ A ) )
% 0.29/0.47 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(emptyinPowerset_type,type,
% 0.29/0.47 emptyinPowerset: $o ).
% 0.29/0.47
% 0.29/0.47 thf(emptyinPowerset,definition,
% 0.29/0.47 ( emptyinPowerset
% 0.29/0.47 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(emptyInPowerset_type,type,
% 0.29/0.47 emptyInPowerset: $o ).
% 0.29/0.47
% 0.29/0.47 thf(emptyInPowerset,definition,
% 0.29/0.47 ( emptyInPowerset
% 0.29/0.47 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(powersetE_type,type,
% 0.29/0.47 powersetE: $o ).
% 0.29/0.47
% 0.29/0.47 thf(powersetE,definition,
% 0.29/0.47 ( powersetE
% 0.29/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.47 ( ( in @ B @ ( powerset @ A ) )
% 0.29/0.47 => ( ( in @ Xx @ B )
% 0.29/0.47 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(setunionI_type,type,
% 0.29/0.47 setunionI: $o ).
% 0.29/0.47
% 0.29/0.47 thf(setunionI,definition,
% 0.29/0.47 ( setunionI
% 0.29/0.47 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.29/0.47 ( ( in @ Xx @ B )
% 0.29/0.47 => ( ( in @ B @ A )
% 0.29/0.47 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(setunionE_type,type,
% 0.29/0.47 setunionE: $o ).
% 0.29/0.47
% 0.29/0.47 thf(setunionE,definition,
% 0.29/0.47 ( setunionE
% 0.29/0.47 = ( ! [A: $i,Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ ( setunion @ A ) )
% 0.29/0.47 => ! [Xphi: $o] :
% 0.29/0.47 ( ! [B: $i] :
% 0.29/0.47 ( ( in @ Xx @ B )
% 0.29/0.47 => ( ( in @ B @ A )
% 0.29/0.47 => Xphi ) )
% 0.29/0.47 => Xphi ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(subPowSU_type,type,
% 0.29/0.47 subPowSU: $o ).
% 0.29/0.47
% 0.29/0.47 thf(subPowSU,definition,
% 0.29/0.47 ( subPowSU
% 0.29/0.47 = ( ! [A: $i,Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ A )
% 0.29/0.47 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(exuE2_type,type,
% 0.29/0.47 exuE2: $o ).
% 0.29/0.47
% 0.29/0.47 thf(exuE2,definition,
% 0.29/0.47 ( exuE2
% 0.29/0.47 = ( ! [Xphi: $i > $o] :
% 0.29/0.47 ( ( exu
% 0.29/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.29/0.47 => ? [Xx: $i] :
% 0.29/0.47 ! [Xy: $i] :
% 0.29/0.47 ( ( Xphi @ Xy )
% 0.29/0.47 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(nonemptyImpWitness_type,type,
% 0.29/0.47 nonemptyImpWitness: $o ).
% 0.29/0.47
% 0.29/0.47 thf(nonemptyImpWitness,definition,
% 0.29/0.47 ( nonemptyImpWitness
% 0.29/0.47 = ( ! [A: $i] :
% 0.29/0.47 ( ( nonempty @ A )
% 0.29/0.47 => ? [Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ A )
% 0.29/0.47 & $true ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(uniqinunit_type,type,
% 0.29/0.47 uniqinunit: $o ).
% 0.29/0.47
% 0.29/0.47 thf(uniqinunit,definition,
% 0.29/0.47 ( uniqinunit
% 0.29/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.29/0.47 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.29/0.47 => ( Xx = Xy ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(notinsingleton_type,type,
% 0.29/0.47 notinsingleton: $o ).
% 0.29/0.47
% 0.29/0.47 thf(notinsingleton,definition,
% 0.29/0.47 ( notinsingleton
% 0.29/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.29/0.47 ( ( Xx != Xy )
% 0.29/0.47 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(eqinunit_type,type,
% 0.29/0.47 eqinunit: $o ).
% 0.29/0.47
% 0.29/0.47 thf(eqinunit,definition,
% 0.29/0.47 ( eqinunit
% 0.29/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.29/0.47 ( ( Xx = Xy )
% 0.29/0.47 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(singletonsswitch_type,type,
% 0.29/0.47 singletonsswitch: $o ).
% 0.29/0.47
% 0.29/0.47 thf(singletonsswitch,definition,
% 0.29/0.47 ( singletonsswitch
% 0.29/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.29/0.47 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.29/0.47 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(upairsetE_type,type,
% 0.29/0.47 upairsetE: $o ).
% 0.29/0.47
% 0.29/0.47 thf(upairsetE,definition,
% 0.29/0.47 ( upairsetE
% 0.29/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.29/0.47 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.29/0.47 => ( ( Xz = Xx )
% 0.29/0.47 | ( Xz = Xy ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(upairsetIL_type,type,
% 0.29/0.47 upairsetIL: $o ).
% 0.29/0.47
% 0.29/0.47 thf(upairsetIL,definition,
% 0.29/0.47 ( upairsetIL
% 0.29/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(upairsetIR_type,type,
% 0.29/0.47 upairsetIR: $o ).
% 0.29/0.47
% 0.29/0.47 thf(upairsetIR,definition,
% 0.29/0.47 ( upairsetIR
% 0.29/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(emptyE1_type,type,
% 0.29/0.47 emptyE1: $o ).
% 0.29/0.47
% 0.29/0.47 thf(emptyE1,definition,
% 0.29/0.47 ( emptyE1
% 0.29/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.29/0.47 ( ? [Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ A )
% 0.29/0.47 & ( Xphi @ Xx ) )
% 0.29/0.47 => ( ( ( dsetconstr @ A
% 0.29/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.29/0.47 = emptyset )
% 0.29/0.47 => $false ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(vacuousDall_type,type,
% 0.29/0.47 vacuousDall: $o ).
% 0.29/0.47
% 0.29/0.47 thf(vacuousDall,definition,
% 0.29/0.47 ( vacuousDall
% 0.29/0.47 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ emptyset )
% 0.29/0.47 => ( Xphi @ Xx ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(quantDeMorgan1_type,type,
% 0.29/0.47 quantDeMorgan1: $o ).
% 0.29/0.47
% 0.29/0.47 thf(quantDeMorgan1,definition,
% 0.29/0.47 ( quantDeMorgan1
% 0.29/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.29/0.47 ( ~ ! [Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ A )
% 0.29/0.47 => ( Xphi @ Xx ) )
% 0.29/0.47 => ? [Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ A )
% 0.29/0.47 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(quantDeMorgan2_type,type,
% 0.29/0.47 quantDeMorgan2: $o ).
% 0.29/0.47
% 0.29/0.47 thf(quantDeMorgan2,definition,
% 0.29/0.47 ( quantDeMorgan2
% 0.29/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.29/0.47 ( ! [Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ A )
% 0.29/0.47 => ~ ( Xphi @ Xx ) )
% 0.29/0.47 => ~ ? [Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ A )
% 0.29/0.47 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(quantDeMorgan3_type,type,
% 0.29/0.47 quantDeMorgan3: $o ).
% 0.29/0.47
% 0.29/0.47 thf(quantDeMorgan3,definition,
% 0.29/0.47 ( quantDeMorgan3
% 0.29/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.29/0.47 ( ~ ? [Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ A )
% 0.29/0.47 & ( Xphi @ Xx ) )
% 0.29/0.47 => ! [Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ A )
% 0.29/0.47 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(quantDeMorgan4_type,type,
% 0.29/0.47 quantDeMorgan4: $o ).
% 0.29/0.47
% 0.29/0.47 thf(quantDeMorgan4,definition,
% 0.29/0.47 ( quantDeMorgan4
% 0.29/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.29/0.47 ( ? [Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ A )
% 0.29/0.47 & ~ ( Xphi @ Xx ) )
% 0.29/0.47 => ~ ! [Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ A )
% 0.29/0.47 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(prop2setI_type,type,
% 0.29/0.47 prop2setI: $o ).
% 0.29/0.47
% 0.29/0.47 thf(prop2setI,definition,
% 0.29/0.47 ( prop2setI
% 0.29/0.47 = ( ! [Xphi: $o] :
% 0.29/0.47 ( Xphi
% 0.29/0.47 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(set2prop_type,type,
% 0.29/0.47 set2prop: $i > $o ).
% 0.29/0.47
% 0.29/0.47 thf(prop2set2propI_type,type,
% 0.29/0.47 prop2set2propI: $o ).
% 0.29/0.47
% 0.29/0.47 thf(prop2set2propI,definition,
% 0.29/0.47 ( prop2set2propI
% 0.29/0.47 = ( ! [Xphi: $o] :
% 0.29/0.47 ( Xphi
% 0.29/0.47 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(notdexE_type,type,
% 0.29/0.47 notdexE: $o ).
% 0.29/0.47
% 0.29/0.47 thf(notdexE,definition,
% 0.29/0.47 ( notdexE
% 0.29/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.29/0.47 ( ~ ? [Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ A )
% 0.29/0.47 & ( Xphi @ Xx ) )
% 0.29/0.47 => ! [Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ A )
% 0.29/0.47 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(notdallE_type,type,
% 0.29/0.47 notdallE: $o ).
% 0.29/0.47
% 0.29/0.47 thf(notdallE,definition,
% 0.29/0.47 ( notdallE
% 0.29/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.29/0.47 ( ~ ! [Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ A )
% 0.29/0.47 => ( Xphi @ Xx ) )
% 0.29/0.47 => ? [Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ A )
% 0.29/0.47 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(exuI1_type,type,
% 0.29/0.47 exuI1: $o ).
% 0.29/0.47
% 0.29/0.47 thf(exuI1,definition,
% 0.29/0.47 ( exuI1
% 0.29/0.47 = ( ! [Xphi: $i > $o] :
% 0.29/0.47 ( ? [Xx: $i] :
% 0.29/0.47 ( ( Xphi @ Xx )
% 0.29/0.47 & ! [Xy: $i] :
% 0.29/0.47 ( ( Xphi @ Xy )
% 0.29/0.47 => ( Xx = Xy ) ) )
% 0.29/0.47 => ( exu
% 0.29/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(exuI3_type,type,
% 0.29/0.47 exuI3: $o ).
% 0.29/0.47
% 0.29/0.47 thf(exuI3,definition,
% 0.29/0.47 ( exuI3
% 0.29/0.47 = ( ! [Xphi: $i > $o] :
% 0.29/0.47 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.29/0.47 => ( ! [Xx: $i,Xy: $i] :
% 0.29/0.47 ( ( Xphi @ Xx )
% 0.29/0.47 => ( ( Xphi @ Xy )
% 0.29/0.47 => ( Xx = Xy ) ) )
% 0.29/0.47 => ( exu
% 0.29/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(exuI2_type,type,
% 0.29/0.47 exuI2: $o ).
% 0.29/0.47
% 0.29/0.47 thf(exuI2,definition,
% 0.29/0.47 ( exuI2
% 0.29/0.47 = ( ! [Xphi: $i > $o] :
% 0.29/0.47 ( ? [Xx: $i] :
% 0.29/0.47 ! [Xy: $i] :
% 0.29/0.47 ( ( Xphi @ Xy )
% 0.29/0.47 <=> ( Xy = Xx ) )
% 0.29/0.47 => ( exu
% 0.29/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(inCongP_type,type,
% 0.29/0.47 inCongP: $o ).
% 0.29/0.47
% 0.29/0.47 thf(inCongP,definition,
% 0.29/0.47 ( inCongP
% 0.29/0.47 = ( ! [A: $i,B: $i] :
% 0.29/0.47 ( ( A = B )
% 0.29/0.47 => ! [Xx: $i,Xy: $i] :
% 0.29/0.47 ( ( Xx = Xy )
% 0.29/0.47 => ( ( in @ Xx @ A )
% 0.29/0.47 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(in__Cong_type,type,
% 0.29/0.47 in__Cong: $o ).
% 0.29/0.47
% 0.29/0.47 thf(in__Cong,definition,
% 0.29/0.47 ( in__Cong
% 0.29/0.47 = ( ! [A: $i,B: $i] :
% 0.29/0.47 ( ( A = B )
% 0.29/0.47 => ! [Xx: $i,Xy: $i] :
% 0.29/0.47 ( ( Xx = Xy )
% 0.29/0.47 => ( ( in @ Xx @ A )
% 0.29/0.47 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(exuE3u_type,type,
% 0.29/0.47 exuE3u: $o ).
% 0.29/0.47
% 0.29/0.47 thf(exuE3u,definition,
% 0.29/0.47 ( exuE3u
% 0.29/0.47 = ( ! [Xphi: $i > $o] :
% 0.29/0.47 ( ( exu
% 0.29/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.29/0.47 => ! [Xx: $i,Xy: $i] :
% 0.29/0.47 ( ( Xphi @ Xx )
% 0.29/0.47 => ( ( Xphi @ Xy )
% 0.29/0.47 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(exu__Cong_type,type,
% 0.29/0.47 exu__Cong: $o ).
% 0.29/0.47
% 0.29/0.47 thf(exu__Cong,definition,
% 0.29/0.47 ( exu__Cong
% 0.29/0.47 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.29/0.47 ( ! [Xx: $i,Xy: $i] :
% 0.29/0.47 ( ( Xx = Xy )
% 0.29/0.47 => ( ( Xphi @ Xx )
% 0.29/0.47 <=> ( Xpsi @ Xy ) ) )
% 0.29/0.47 => ( ( exu
% 0.29/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.29/0.47 <=> ( exu
% 0.29/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(emptyset__Cong_type,type,
% 0.29/0.47 emptyset__Cong: $o ).
% 0.29/0.47
% 0.29/0.47 thf(emptyset__Cong,definition,
% 0.29/0.47 ( emptyset__Cong
% 0.29/0.47 = ( emptyset = emptyset ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(setadjoin__Cong_type,type,
% 0.29/0.47 setadjoin__Cong: $o ).
% 0.29/0.47
% 0.29/0.47 thf(setadjoin__Cong,definition,
% 0.29/0.47 ( setadjoin__Cong
% 0.29/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.29/0.47 ( ( Xx = Xy )
% 0.29/0.47 => ! [Xz: $i,Xu: $i] :
% 0.29/0.47 ( ( Xz = Xu )
% 0.29/0.47 => ( ( setadjoin @ Xx @ Xz )
% 0.29/0.47 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(powerset__Cong_type,type,
% 0.29/0.47 powerset__Cong: $o ).
% 0.29/0.47
% 0.29/0.47 thf(powerset__Cong,definition,
% 0.29/0.47 ( powerset__Cong
% 0.29/0.47 = ( ! [A: $i,B: $i] :
% 0.29/0.47 ( ( A = B )
% 0.29/0.47 => ( ( powerset @ A )
% 0.29/0.47 = ( powerset @ B ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(setunion__Cong_type,type,
% 0.29/0.47 setunion__Cong: $o ).
% 0.29/0.47
% 0.29/0.47 thf(setunion__Cong,definition,
% 0.29/0.47 ( setunion__Cong
% 0.29/0.47 = ( ! [A: $i,B: $i] :
% 0.29/0.47 ( ( A = B )
% 0.29/0.47 => ( ( setunion @ A )
% 0.29/0.47 = ( setunion @ B ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(omega__Cong_type,type,
% 0.29/0.47 omega__Cong: $o ).
% 0.29/0.47
% 0.29/0.47 thf(omega__Cong,definition,
% 0.29/0.47 ( omega__Cong
% 0.29/0.47 = ( omega = omega ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(exuEu_type,type,
% 0.29/0.47 exuEu: $o ).
% 0.29/0.47
% 0.29/0.47 thf(exuEu,definition,
% 0.29/0.47 ( exuEu
% 0.29/0.47 = ( ! [Xphi: $i > $o] :
% 0.29/0.47 ( ( exu
% 0.29/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.29/0.47 => ! [Xx: $i,Xy: $i] :
% 0.29/0.47 ( ( Xphi @ Xx )
% 0.29/0.47 => ( ( Xphi @ Xy )
% 0.29/0.47 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(descr__Cong_type,type,
% 0.29/0.47 descr__Cong: $o ).
% 0.29/0.47
% 0.29/0.47 thf(descr__Cong,definition,
% 0.29/0.47 ( descr__Cong
% 0.29/0.47 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.29/0.47 ( ! [Xx: $i,Xy: $i] :
% 0.29/0.47 ( ( Xx = Xy )
% 0.29/0.47 => ( ( Xphi @ Xx )
% 0.29/0.47 <=> ( Xpsi @ Xy ) ) )
% 0.29/0.47 => ( ( exu
% 0.29/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.29/0.47 => ( ( exu
% 0.29/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.29/0.47 => ( ( descr
% 0.29/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.29/0.47 = ( descr
% 0.29/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(dsetconstr__Cong_type,type,
% 0.29/0.47 dsetconstr__Cong: $o ).
% 0.29/0.47
% 0.29/0.47 thf(dsetconstr__Cong,definition,
% 0.29/0.47 ( dsetconstr__Cong
% 0.29/0.47 = ( ! [A: $i,B: $i] :
% 0.29/0.47 ( ( A = B )
% 0.29/0.47 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.29/0.47 ( ! [Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ A )
% 0.29/0.47 => ! [Xy: $i] :
% 0.29/0.47 ( ( in @ Xy @ B )
% 0.29/0.47 => ( ( Xx = Xy )
% 0.29/0.47 => ( ( Xphi @ Xx )
% 0.29/0.47 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.29/0.47 => ( ( dsetconstr @ A
% 0.29/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.29/0.47 = ( dsetconstr @ B
% 0.29/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(subset_type,type,
% 0.29/0.47 subset: $i > $i > $o ).
% 0.29/0.47
% 0.29/0.47 thf(disjoint_type,type,
% 0.29/0.47 disjoint: $i > $i > $o ).
% 0.29/0.47
% 0.29/0.47 thf(setsmeet_type,type,
% 0.29/0.47 setsmeet: $i > $i > $o ).
% 0.29/0.47
% 0.29/0.47 thf(subsetI1_type,type,
% 0.29/0.47 subsetI1: $o ).
% 0.29/0.47
% 0.29/0.47 thf(subsetI1,definition,
% 0.29/0.47 ( subsetI1
% 0.29/0.47 = ( ! [A: $i,B: $i] :
% 0.29/0.47 ( ! [Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ A )
% 0.29/0.47 => ( in @ Xx @ B ) )
% 0.29/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(eqimpsubset2_type,type,
% 0.29/0.47 eqimpsubset2: $o ).
% 0.29/0.47
% 0.29/0.47 thf(eqimpsubset2,definition,
% 0.29/0.47 ( eqimpsubset2
% 0.29/0.47 = ( ! [A: $i,B: $i] :
% 0.29/0.47 ( ( A = B )
% 0.29/0.47 => ( subset @ B @ A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(eqimpsubset1_type,type,
% 0.29/0.47 eqimpsubset1: $o ).
% 0.29/0.47
% 0.29/0.47 thf(eqimpsubset1,definition,
% 0.29/0.47 ( eqimpsubset1
% 0.29/0.47 = ( ! [A: $i,B: $i] :
% 0.29/0.47 ( ( A = B )
% 0.29/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(subsetI2_type,type,
% 0.29/0.47 subsetI2: $o ).
% 0.29/0.47
% 0.29/0.47 thf(subsetI2,definition,
% 0.29/0.47 ( subsetI2
% 0.29/0.47 = ( ! [A: $i,B: $i] :
% 0.29/0.47 ( ! [Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ A )
% 0.29/0.47 => ( in @ Xx @ B ) )
% 0.29/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(emptysetsubset_type,type,
% 0.29/0.47 emptysetsubset: $o ).
% 0.29/0.47
% 0.29/0.47 thf(emptysetsubset,definition,
% 0.29/0.47 ( emptysetsubset
% 0.29/0.47 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(subsetE_type,type,
% 0.29/0.47 subsetE: $o ).
% 0.29/0.47
% 0.29/0.47 thf(subsetE,definition,
% 0.29/0.47 ( subsetE
% 0.29/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.47 ( ( subset @ A @ B )
% 0.29/0.47 => ( ( in @ Xx @ A )
% 0.29/0.47 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(subsetE2_type,type,
% 0.29/0.47 subsetE2: $o ).
% 0.29/0.47
% 0.29/0.47 thf(subsetE2,definition,
% 0.29/0.47 ( subsetE2
% 0.29/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.47 ( ( subset @ A @ B )
% 0.29/0.47 => ( ~ ( in @ Xx @ B )
% 0.29/0.47 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(notsubsetI_type,type,
% 0.29/0.47 notsubsetI: $o ).
% 0.29/0.47
% 0.29/0.47 thf(notsubsetI,definition,
% 0.29/0.47 ( notsubsetI
% 0.29/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ A )
% 0.29/0.47 => ( ~ ( in @ Xx @ B )
% 0.29/0.47 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(notequalI1_type,type,
% 0.29/0.47 notequalI1: $o ).
% 0.29/0.47
% 0.29/0.47 thf(notequalI1,definition,
% 0.29/0.47 ( notequalI1
% 0.29/0.47 = ( ! [A: $i,B: $i] :
% 0.29/0.47 ( ~ ( subset @ A @ B )
% 0.29/0.47 => ( A != B ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(notequalI2_type,type,
% 0.29/0.47 notequalI2: $o ).
% 0.29/0.47
% 0.29/0.47 thf(notequalI2,definition,
% 0.29/0.47 ( notequalI2
% 0.29/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ A )
% 0.29/0.47 => ( ~ ( in @ Xx @ B )
% 0.29/0.47 => ( A != B ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(subsetRefl_type,type,
% 0.29/0.47 subsetRefl: $o ).
% 0.29/0.47
% 0.29/0.47 thf(subsetRefl,definition,
% 0.29/0.47 ( subsetRefl
% 0.29/0.47 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(subsetTrans_type,type,
% 0.29/0.47 subsetTrans: $o ).
% 0.29/0.47
% 0.29/0.47 thf(subsetTrans,definition,
% 0.29/0.47 ( subsetTrans
% 0.29/0.47 = ( ! [A: $i,B: $i,C: $i] :
% 0.29/0.47 ( ( subset @ A @ B )
% 0.29/0.47 => ( ( subset @ B @ C )
% 0.29/0.47 => ( subset @ A @ C ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(setadjoinSub_type,type,
% 0.29/0.47 setadjoinSub: $o ).
% 0.29/0.47
% 0.29/0.47 thf(setadjoinSub,definition,
% 0.29/0.47 ( setadjoinSub
% 0.29/0.47 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(setadjoinSub2_type,type,
% 0.29/0.47 setadjoinSub2: $o ).
% 0.29/0.47
% 0.29/0.47 thf(setadjoinSub2,definition,
% 0.29/0.47 ( setadjoinSub2
% 0.29/0.47 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.29/0.47 ( ( subset @ A @ B )
% 0.29/0.47 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(subset2powerset_type,type,
% 0.29/0.47 subset2powerset: $o ).
% 0.29/0.47
% 0.29/0.47 thf(subset2powerset,definition,
% 0.29/0.47 ( subset2powerset
% 0.29/0.47 = ( ! [A: $i,B: $i] :
% 0.29/0.47 ( ( subset @ A @ B )
% 0.29/0.47 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(setextsub_type,type,
% 0.29/0.47 setextsub: $o ).
% 0.29/0.47
% 0.29/0.47 thf(setextsub,definition,
% 0.29/0.47 ( setextsub
% 0.29/0.47 = ( ! [A: $i,B: $i] :
% 0.29/0.47 ( ( subset @ A @ B )
% 0.29/0.47 => ( ( subset @ B @ A )
% 0.29/0.47 => ( A = B ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(subsetemptysetimpeq_type,type,
% 0.29/0.47 subsetemptysetimpeq: $o ).
% 0.29/0.47
% 0.29/0.47 thf(subsetemptysetimpeq,definition,
% 0.29/0.47 ( subsetemptysetimpeq
% 0.29/0.47 = ( ! [A: $i] :
% 0.29/0.47 ( ( subset @ A @ emptyset )
% 0.29/0.47 => ( A = emptyset ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(powersetI1_type,type,
% 0.29/0.47 powersetI1: $o ).
% 0.29/0.47
% 0.29/0.47 thf(powersetI1,definition,
% 0.29/0.47 ( powersetI1
% 0.29/0.47 = ( ! [A: $i,B: $i] :
% 0.29/0.47 ( ( subset @ B @ A )
% 0.29/0.47 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(powersetE1_type,type,
% 0.29/0.47 powersetE1: $o ).
% 0.29/0.47
% 0.29/0.47 thf(powersetE1,definition,
% 0.29/0.47 ( powersetE1
% 0.29/0.47 = ( ! [A: $i,B: $i] :
% 0.29/0.47 ( ( in @ B @ ( powerset @ A ) )
% 0.29/0.47 => ( subset @ B @ A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(inPowerset_type,type,
% 0.29/0.47 inPowerset: $o ).
% 0.29/0.47
% 0.29/0.47 thf(inPowerset,definition,
% 0.29/0.47 ( inPowerset
% 0.29/0.47 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(powersetsubset_type,type,
% 0.29/0.47 powersetsubset: $o ).
% 0.29/0.47
% 0.29/0.47 thf(powersetsubset,definition,
% 0.29/0.47 ( powersetsubset
% 0.29/0.47 = ( ! [A: $i,B: $i] :
% 0.29/0.47 ( ( subset @ A @ B )
% 0.29/0.47 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(sepInPowerset_type,type,
% 0.29/0.47 sepInPowerset: $o ).
% 0.29/0.47
% 0.29/0.47 thf(sepInPowerset,definition,
% 0.29/0.47 ( sepInPowerset
% 0.29/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.29/0.47 ( in
% 0.29/0.47 @ ( dsetconstr @ A
% 0.29/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.29/0.47 @ ( powerset @ A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(sepSubset_type,type,
% 0.29/0.47 sepSubset: $o ).
% 0.29/0.47
% 0.29/0.47 thf(sepSubset,definition,
% 0.29/0.47 ( sepSubset
% 0.29/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.29/0.47 ( subset
% 0.29/0.47 @ ( dsetconstr @ A
% 0.29/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.29/0.47 @ A ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(binunion_type,type,
% 0.29/0.47 binunion: $i > $i > $i ).
% 0.29/0.47
% 0.29/0.47 thf(binunionIL_type,type,
% 0.29/0.47 binunionIL: $o ).
% 0.29/0.47
% 0.29/0.47 thf(binunionIL,definition,
% 0.29/0.47 ( binunionIL
% 0.29/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ A )
% 0.29/0.47 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(upairset2IR_type,type,
% 0.29/0.47 upairset2IR: $o ).
% 0.29/0.47
% 0.29/0.47 thf(upairset2IR,definition,
% 0.29/0.47 ( upairset2IR
% 0.29/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(binunionIR_type,type,
% 0.29/0.47 binunionIR: $o ).
% 0.29/0.47
% 0.29/0.47 thf(binunionIR,definition,
% 0.29/0.47 ( binunionIR
% 0.29/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ B )
% 0.29/0.47 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(binunionEcases_type,type,
% 0.29/0.47 binunionEcases: $o ).
% 0.29/0.47
% 0.29/0.47 thf(binunionEcases,definition,
% 0.29/0.47 ( binunionEcases
% 0.29/0.47 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.29/0.47 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.29/0.47 => ( ( ( in @ Xx @ A )
% 0.29/0.47 => Xphi )
% 0.29/0.47 => ( ( ( in @ Xx @ B )
% 0.29/0.47 => Xphi )
% 0.29/0.47 => Xphi ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(binunionE_type,type,
% 0.29/0.47 binunionE: $o ).
% 0.29/0.47
% 0.29/0.47 thf(binunionE,definition,
% 0.29/0.47 ( binunionE
% 0.29/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.29/0.47 => ( ( in @ Xx @ A )
% 0.29/0.47 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(binunionLsub_type,type,
% 0.29/0.47 binunionLsub: $o ).
% 0.29/0.47
% 0.29/0.47 thf(binunionLsub,definition,
% 0.29/0.47 ( binunionLsub
% 0.29/0.47 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(binunionRsub_type,type,
% 0.29/0.47 binunionRsub: $o ).
% 0.29/0.47
% 0.29/0.47 thf(binunionRsub,definition,
% 0.29/0.47 ( binunionRsub
% 0.29/0.47 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(binintersect_type,type,
% 0.29/0.47 binintersect: $i > $i > $i ).
% 0.29/0.47
% 0.29/0.47 thf(binintersectI_type,type,
% 0.29/0.47 binintersectI: $o ).
% 0.29/0.47
% 0.29/0.47 thf(binintersectI,definition,
% 0.29/0.47 ( binintersectI
% 0.29/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ A )
% 0.29/0.47 => ( ( in @ Xx @ B )
% 0.29/0.47 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(binintersectSubset5_type,type,
% 0.29/0.47 binintersectSubset5: $o ).
% 0.29/0.47
% 0.29/0.47 thf(binintersectSubset5,definition,
% 0.29/0.47 ( binintersectSubset5
% 0.29/0.47 = ( ! [A: $i,B: $i,C: $i] :
% 0.29/0.47 ( ( subset @ C @ A )
% 0.29/0.47 => ( ( subset @ C @ B )
% 0.29/0.47 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(binintersectEL_type,type,
% 0.29/0.47 binintersectEL: $o ).
% 0.29/0.47
% 0.29/0.47 thf(binintersectEL,definition,
% 0.29/0.47 ( binintersectEL
% 0.29/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.29/0.47 => ( in @ Xx @ A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(binintersectLsub_type,type,
% 0.29/0.47 binintersectLsub: $o ).
% 0.29/0.47
% 0.29/0.47 thf(binintersectLsub,definition,
% 0.29/0.47 ( binintersectLsub
% 0.29/0.47 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(binintersectSubset2_type,type,
% 0.29/0.47 binintersectSubset2: $o ).
% 0.29/0.47
% 0.29/0.47 thf(binintersectSubset2,definition,
% 0.29/0.47 ( binintersectSubset2
% 0.29/0.47 = ( ! [A: $i,B: $i] :
% 0.29/0.47 ( ( subset @ A @ B )
% 0.29/0.47 => ( ( binintersect @ A @ B )
% 0.29/0.47 = A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(binintersectSubset3_type,type,
% 0.29/0.47 binintersectSubset3: $o ).
% 0.29/0.47
% 0.29/0.47 thf(binintersectSubset3,definition,
% 0.29/0.47 ( binintersectSubset3
% 0.29/0.47 = ( ! [A: $i,B: $i] :
% 0.29/0.47 ( ( ( binintersect @ A @ B )
% 0.29/0.47 = B )
% 0.29/0.47 => ( subset @ B @ A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(binintersectER_type,type,
% 0.29/0.47 binintersectER: $o ).
% 0.29/0.47
% 0.29/0.47 thf(binintersectER,definition,
% 0.29/0.47 ( binintersectER
% 0.29/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.29/0.47 => ( in @ Xx @ B ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(disjointsetsI1_type,type,
% 0.29/0.47 disjointsetsI1: $o ).
% 0.29/0.47
% 0.29/0.47 thf(disjointsetsI1,definition,
% 0.29/0.47 ( disjointsetsI1
% 0.29/0.47 = ( ! [A: $i,B: $i] :
% 0.29/0.47 ( ~ ? [Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ A )
% 0.29/0.47 & ( in @ Xx @ B ) )
% 0.29/0.47 => ( ( binintersect @ A @ B )
% 0.29/0.47 = emptyset ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(binintersectRsub_type,type,
% 0.29/0.47 binintersectRsub: $o ).
% 0.29/0.47
% 0.29/0.47 thf(binintersectRsub,definition,
% 0.29/0.47 ( binintersectRsub
% 0.29/0.47 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(binintersectSubset4_type,type,
% 0.29/0.47 binintersectSubset4: $o ).
% 0.29/0.47
% 0.29/0.47 thf(binintersectSubset4,definition,
% 0.29/0.47 ( binintersectSubset4
% 0.29/0.47 = ( ! [A: $i,B: $i] :
% 0.29/0.47 ( ( subset @ B @ A )
% 0.29/0.47 => ( ( binintersect @ A @ B )
% 0.29/0.47 = B ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(binintersectSubset1_type,type,
% 0.29/0.47 binintersectSubset1: $o ).
% 0.29/0.47
% 0.29/0.47 thf(binintersectSubset1,definition,
% 0.29/0.47 ( binintersectSubset1
% 0.29/0.47 = ( ! [A: $i,B: $i] :
% 0.29/0.47 ( ( ( binintersect @ A @ B )
% 0.29/0.47 = A )
% 0.29/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(bs114d_type,type,
% 0.29/0.47 bs114d: $o ).
% 0.29/0.47
% 0.29/0.47 thf(bs114d,definition,
% 0.29/0.47 ( bs114d
% 0.29/0.47 = ( ! [A: $i,B: $i,C: $i] :
% 0.29/0.47 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 0.29/0.47 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(regular_type,type,
% 0.29/0.47 regular: $i > $o ).
% 0.29/0.47
% 0.29/0.47 thf(setminus_type,type,
% 0.29/0.47 setminus: $i > $i > $i ).
% 0.29/0.47
% 0.29/0.47 thf(setminusI_type,type,
% 0.29/0.47 setminusI: $o ).
% 0.29/0.47
% 0.29/0.47 thf(setminusI,definition,
% 0.29/0.47 ( setminusI
% 0.29/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ A )
% 0.29/0.47 => ( ~ ( in @ Xx @ B )
% 0.29/0.47 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(setminusEL_type,type,
% 0.29/0.47 setminusEL: $o ).
% 0.29/0.47
% 0.29/0.47 thf(setminusEL,definition,
% 0.29/0.47 ( setminusEL
% 0.29/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.29/0.47 => ( in @ Xx @ A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(setminusER_type,type,
% 0.29/0.47 setminusER: $o ).
% 0.29/0.47
% 0.29/0.47 thf(setminusER,definition,
% 0.29/0.47 ( setminusER
% 0.29/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.29/0.47 => ~ ( in @ Xx @ B ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(setminusSubset2_type,type,
% 0.29/0.47 setminusSubset2: $o ).
% 0.29/0.47
% 0.29/0.47 thf(setminusSubset2,definition,
% 0.29/0.47 ( setminusSubset2
% 0.29/0.47 = ( ! [A: $i,B: $i] :
% 0.29/0.47 ( ( subset @ A @ B )
% 0.29/0.47 => ( ( setminus @ A @ B )
% 0.29/0.47 = emptyset ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(setminusERneg_type,type,
% 0.29/0.47 setminusERneg: $o ).
% 0.29/0.47
% 0.29/0.47 thf(setminusERneg,definition,
% 0.29/0.47 ( setminusERneg
% 0.29/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.47 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.29/0.47 => ( ( in @ Xx @ A )
% 0.29/0.47 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(setminusELneg_type,type,
% 0.29/0.47 setminusELneg: $o ).
% 0.29/0.47
% 0.29/0.47 thf(setminusELneg,definition,
% 0.29/0.47 ( setminusELneg
% 0.29/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.47 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.29/0.47 => ( ~ ( in @ Xx @ B )
% 0.29/0.47 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(setminusILneg_type,type,
% 0.29/0.47 setminusILneg: $o ).
% 0.29/0.47
% 0.29/0.47 thf(setminusILneg,definition,
% 0.29/0.47 ( setminusILneg
% 0.29/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.47 ( ~ ( in @ Xx @ A )
% 0.29/0.47 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(setminusIRneg_type,type,
% 0.29/0.47 setminusIRneg: $o ).
% 0.29/0.47
% 0.29/0.47 thf(setminusIRneg,definition,
% 0.29/0.47 ( setminusIRneg
% 0.29/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ B )
% 0.29/0.47 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(setminusLsub_type,type,
% 0.29/0.47 setminusLsub: $o ).
% 0.29/0.47
% 0.29/0.47 thf(setminusLsub,definition,
% 0.29/0.47 ( setminusLsub
% 0.29/0.47 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(setminusSubset1_type,type,
% 0.29/0.47 setminusSubset1: $o ).
% 0.29/0.47
% 0.29/0.47 thf(setminusSubset1,definition,
% 0.29/0.47 ( setminusSubset1
% 0.29/0.47 = ( ! [A: $i,B: $i] :
% 0.29/0.47 ( ( ( setminus @ A @ B )
% 0.29/0.47 = emptyset )
% 0.29/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(symdiff_type,type,
% 0.29/0.47 symdiff: $i > $i > $i ).
% 0.29/0.47
% 0.29/0.47 thf(symdiffE_type,type,
% 0.29/0.47 symdiffE: $o ).
% 0.29/0.47
% 0.29/0.47 thf(symdiffE,definition,
% 0.29/0.47 ( symdiffE
% 0.29/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 0.29/0.47 => ! [Xphi: $o] :
% 0.29/0.47 ( ( ( in @ Xx @ A )
% 0.29/0.47 => ( ~ ( in @ Xx @ B )
% 0.29/0.47 => Xphi ) )
% 0.29/0.47 => ( ( ~ ( in @ Xx @ A )
% 0.29/0.47 => ( ( in @ Xx @ B )
% 0.29/0.47 => Xphi ) )
% 0.29/0.47 => Xphi ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(symdiffI1_type,type,
% 0.29/0.47 symdiffI1: $o ).
% 0.29/0.47
% 0.29/0.47 thf(symdiffI1,definition,
% 0.29/0.47 ( symdiffI1
% 0.29/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ A )
% 0.29/0.47 => ( ~ ( in @ Xx @ B )
% 0.29/0.47 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(symdiffI2_type,type,
% 0.29/0.47 symdiffI2: $o ).
% 0.29/0.47
% 0.29/0.47 thf(symdiffI2,definition,
% 0.29/0.47 ( symdiffI2
% 0.29/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.47 ( ~ ( in @ Xx @ A )
% 0.29/0.47 => ( ( in @ Xx @ B )
% 0.29/0.47 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(symdiffIneg1_type,type,
% 0.29/0.47 symdiffIneg1: $o ).
% 0.29/0.47
% 0.29/0.47 thf(symdiffIneg1,definition,
% 0.29/0.47 ( symdiffIneg1
% 0.29/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ A )
% 0.29/0.47 => ( ( in @ Xx @ B )
% 0.29/0.47 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(symdiffIneg2_type,type,
% 0.29/0.47 symdiffIneg2: $o ).
% 0.29/0.47
% 0.29/0.47 thf(symdiffIneg2,definition,
% 0.29/0.47 ( symdiffIneg2
% 0.29/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.47 ( ~ ( in @ Xx @ A )
% 0.29/0.47 => ( ~ ( in @ Xx @ B )
% 0.29/0.47 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(iskpair_type,type,
% 0.29/0.47 iskpair: $i > $o ).
% 0.29/0.47
% 0.29/0.47 thf(iskpair,definition,
% 0.29/0.47 ( iskpair
% 0.29/0.47 = ( ^ [A: $i] :
% 0.29/0.47 ? [Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ ( setunion @ A ) )
% 0.29/0.47 & ? [Xy: $i] :
% 0.29/0.47 ( ( in @ Xy @ ( setunion @ A ) )
% 0.29/0.47 & ( A
% 0.29/0.47 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(secondinupair_type,type,
% 0.29/0.47 secondinupair: $o ).
% 0.29/0.47
% 0.29/0.47 thf(secondinupair,definition,
% 0.29/0.47 ( secondinupair
% 0.29/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(setukpairIL_type,type,
% 0.29/0.47 setukpairIL: $o ).
% 0.29/0.47
% 0.29/0.47 thf(setukpairIL,definition,
% 0.29/0.47 ( setukpairIL
% 0.29/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(setukpairIR_type,type,
% 0.29/0.47 setukpairIR: $o ).
% 0.29/0.47
% 0.29/0.47 thf(setukpairIR,definition,
% 0.29/0.47 ( setukpairIR
% 0.29/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(kpairiskpair_type,type,
% 0.29/0.47 kpairiskpair: $o ).
% 0.29/0.47
% 0.29/0.47 thf(kpairiskpair,definition,
% 0.29/0.47 ( kpairiskpair
% 0.29/0.47 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(kpair_type,type,
% 0.29/0.47 kpair: $i > $i > $i ).
% 0.29/0.47
% 0.29/0.47 thf(kpair,definition,
% 0.29/0.47 ( kpair
% 0.29/0.47 = ( ^ [Xx: $i,Xy: $i] : ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(kpairp_type,type,
% 0.29/0.47 kpairp: $o ).
% 0.29/0.47
% 0.29/0.47 thf(kpairp,definition,
% 0.29/0.47 ( kpairp
% 0.29/0.47 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(cartprod_type,type,
% 0.29/0.47 cartprod: $i > $i > $i ).
% 0.29/0.47
% 0.29/0.47 thf(cartprod,definition,
% 0.29/0.47 ( cartprod
% 0.29/0.47 = ( ^ [A: $i,B: $i] :
% 0.29/0.47 ( dsetconstr @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) )
% 0.29/0.47 @ ^ [Xx: $i] :
% 0.29/0.47 ? [Xy: $i] :
% 0.29/0.47 ( ( in @ Xy @ A )
% 0.29/0.47 & ? [Xz: $i] :
% 0.29/0.47 ( ( in @ Xz @ B )
% 0.29/0.47 & ( Xx
% 0.29/0.47 = ( kpair @ Xy @ Xz ) ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(singletonsubset_type,type,
% 0.29/0.47 singletonsubset: $o ).
% 0.29/0.47
% 0.29/0.47 thf(singletonsubset,definition,
% 0.29/0.47 ( singletonsubset
% 0.29/0.47 = ( ! [A: $i,Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ A )
% 0.29/0.47 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(singletoninpowerset_type,type,
% 0.29/0.47 singletoninpowerset: $o ).
% 0.29/0.47
% 0.29/0.47 thf(singletoninpowerset,definition,
% 0.29/0.47 ( singletoninpowerset
% 0.29/0.47 = ( ! [A: $i,Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ A )
% 0.29/0.47 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(singletoninpowunion_type,type,
% 0.29/0.47 singletoninpowunion: $o ).
% 0.29/0.47
% 0.29/0.47 thf(singletoninpowunion,definition,
% 0.29/0.47 ( singletoninpowunion
% 0.29/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ A )
% 0.29/0.47 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(upairset2E_type,type,
% 0.29/0.47 upairset2E: $o ).
% 0.29/0.47
% 0.29/0.47 thf(upairset2E,definition,
% 0.29/0.47 ( upairset2E
% 0.29/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.29/0.47 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.29/0.47 => ( ( Xz = Xx )
% 0.29/0.47 | ( Xz = Xy ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(upairsubunion_type,type,
% 0.29/0.47 upairsubunion: $o ).
% 0.29/0.47
% 0.29/0.47 thf(upairsubunion,definition,
% 0.29/0.47 ( upairsubunion
% 0.29/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ A )
% 0.29/0.47 => ! [Xy: $i] :
% 0.29/0.47 ( ( in @ Xy @ B )
% 0.29/0.47 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(upairinpowunion,conjecture,
% 0.29/0.47 ( setextAx
% 0.29/0.47 => ( emptysetAx
% 0.29/0.47 => ( setadjoinAx
% 0.29/0.47 => ( powersetAx
% 0.29/0.47 => ( setunionAx
% 0.29/0.47 => ( omega0Ax
% 0.29/0.47 => ( omegaSAx
% 0.29/0.47 => ( omegaIndAx
% 0.29/0.47 => ( replAx
% 0.29/0.47 => ( foundationAx
% 0.29/0.47 => ( wellorderingAx
% 0.29/0.47 => ( descrp
% 0.29/0.47 => ( dsetconstrI
% 0.29/0.47 => ( dsetconstrEL
% 0.29/0.47 => ( dsetconstrER
% 0.29/0.47 => ( exuE1
% 0.29/0.47 => ( prop2setE
% 0.29/0.47 => ( emptysetE
% 0.29/0.47 => ( emptysetimpfalse
% 0.29/0.47 => ( notinemptyset
% 0.29/0.47 => ( exuE3e
% 0.29/0.47 => ( setext
% 0.29/0.47 => ( emptyI
% 0.29/0.47 => ( noeltsimpempty
% 0.29/0.47 => ( setbeta
% 0.29/0.47 => ( nonemptyE1
% 0.29/0.47 => ( nonemptyI
% 0.29/0.47 => ( nonemptyI1
% 0.29/0.47 => ( setadjoinIL
% 0.29/0.47 => ( emptyinunitempty
% 0.29/0.47 => ( setadjoinIR
% 0.29/0.47 => ( setadjoinE
% 0.29/0.47 => ( setadjoinOr
% 0.29/0.47 => ( setoftrueEq
% 0.29/0.47 => ( powersetI
% 0.29/0.47 => ( emptyinPowerset
% 0.29/0.47 => ( emptyInPowerset
% 0.29/0.47 => ( powersetE
% 0.29/0.47 => ( setunionI
% 0.29/0.47 => ( setunionE
% 0.29/0.47 => ( subPowSU
% 0.29/0.47 => ( exuE2
% 0.29/0.47 => ( nonemptyImpWitness
% 0.29/0.47 => ( uniqinunit
% 0.29/0.47 => ( notinsingleton
% 0.29/0.47 => ( eqinunit
% 0.29/0.47 => ( singletonsswitch
% 0.29/0.47 => ( upairsetE
% 0.29/0.47 => ( upairsetIL
% 0.29/0.47 => ( upairsetIR
% 0.29/0.47 => ( emptyE1
% 0.29/0.47 => ( vacuousDall
% 0.29/0.47 => ( quantDeMorgan1
% 0.29/0.47 => ( quantDeMorgan2
% 0.29/0.47 => ( quantDeMorgan3
% 0.29/0.47 => ( quantDeMorgan4
% 0.29/0.47 => ( prop2setI
% 0.29/0.47 => ( prop2set2propI
% 0.29/0.47 => ( notdexE
% 0.29/0.47 => ( notdallE
% 0.29/0.47 => ( exuI1
% 0.29/0.47 => ( exuI3
% 0.29/0.47 => ( exuI2
% 0.29/0.47 => ( inCongP
% 0.29/0.47 => ( in__Cong
% 0.29/0.47 => ( exuE3u
% 0.29/0.47 => ( exu__Cong
% 0.29/0.47 => ( emptyset__Cong
% 0.29/0.47 => ( setadjoin__Cong
% 0.29/0.47 => ( powerset__Cong
% 0.29/0.47 => ( setunion__Cong
% 0.29/0.47 => ( omega__Cong
% 0.29/0.47 => ( exuEu
% 0.29/0.47 => ( descr__Cong
% 0.29/0.47 => ( dsetconstr__Cong
% 0.29/0.47 => ( subsetI1
% 0.29/0.47 => ( eqimpsubset2
% 0.29/0.47 => ( eqimpsubset1
% 0.29/0.47 => ( subsetI2
% 0.29/0.47 => ( emptysetsubset
% 0.29/0.47 => ( subsetE
% 0.29/0.47 => ( subsetE2
% 0.29/0.47 => ( notsubsetI
% 0.29/0.47 => ( notequalI1
% 0.29/0.47 => ( notequalI2
% 0.29/0.47 => ( subsetRefl
% 0.29/0.47 => ( subsetTrans
% 0.29/0.47 => ( setadjoinSub
% 0.29/0.47 => ( setadjoinSub2
% 0.29/0.47 => ( subset2powerset
% 0.29/0.47 => ( setextsub
% 0.29/0.47 => ( subsetemptysetimpeq
% 0.29/0.47 => ( powersetI1
% 0.29/0.47 => ( powersetE1
% 0.29/0.47 => ( inPowerset
% 0.29/0.47 => ( powersetsubset
% 0.29/0.47 => ( sepInPowerset
% 0.29/0.47 => ( sepSubset
% 0.29/0.47 => ( binunionIL
% 0.29/0.47 => ( upairset2IR
% 0.29/0.47 => ( binunionIR
% 0.29/0.47 => ( binunionEcases
% 0.29/0.47 => ( binunionE
% 0.29/0.47 => ( binunionLsub
% 0.29/0.47 => ( binunionRsub
% 0.29/0.47 => ( binintersectI
% 0.29/0.47 => ( binintersectSubset5
% 0.29/0.47 => ( binintersectEL
% 0.29/0.47 => ( binintersectLsub
% 0.29/0.47 => ( binintersectSubset2
% 0.29/0.47 => ( binintersectSubset3
% 0.29/0.47 => ( binintersectER
% 0.29/0.47 => ( disjointsetsI1
% 0.29/0.47 => ( binintersectRsub
% 0.29/0.47 => ( binintersectSubset4
% 0.29/0.47 => ( binintersectSubset1
% 0.29/0.47 => ( bs114d
% 0.29/0.47 => ( setminusI
% 0.29/0.47 => ( setminusEL
% 0.29/0.47 => ( setminusER
% 0.29/0.47 => ( setminusSubset2
% 0.29/0.47 => ( setminusERneg
% 0.29/0.47 => ( setminusELneg
% 0.29/0.47 => ( setminusILneg
% 0.29/0.47 => ( setminusIRneg
% 0.29/0.47 => ( setminusLsub
% 0.29/0.47 => ( setminusSubset1
% 0.29/0.47 => ( symdiffE
% 0.29/0.47 => ( symdiffI1
% 0.29/0.47 => ( symdiffI2
% 0.29/0.47 => ( symdiffIneg1
% 0.29/0.47 => ( symdiffIneg2
% 0.29/0.47 => ( secondinupair
% 0.29/0.47 => ( setukpairIL
% 0.29/0.47 => ( setukpairIR
% 0.29/0.47 => ( kpairiskpair
% 0.29/0.47 => ( kpairp
% 0.29/0.47 => ( singletonsubset
% 0.29/0.47 => ( singletoninpowerset
% 0.29/0.47 => ( singletoninpowunion
% 0.29/0.47 => ( upairset2E
% 0.29/0.47 => ( upairsubunion
% 0.29/0.47 => ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ A )
% 0.29/0.47 => ! [Xy: $i] :
% 0.29/0.47 ( ( in @ Xy @ B )
% 0.29/0.47 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 %------------------------------------------------------------------------------
% 0.29/0.47 ------------------------
% 2.16/1.38 ---- Embedded in THF ---
% 2.16/1.38 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.16/1.38 %%% Generated on Tue Feb 24 12:29:21 EST 2026
% 2.16/1.38 %%% using '$$dhol' embedding, version 1.3.0.
% 2.16/1.38 %%% Logic specification used:
% 2.16/1.38 %%% thf(spec, logic, $$dhol).
% 2.16/1.38
% 2.16/1.38 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.DbdYpcF5sg/DTF2DTF22176.p
% See solution above
% 2.16/1.38 ------------------------
% 2.16/1.40 ---- Cleaned THF ---
% 2.41/1.40 thf(in_type,type,
% 2.41/1.40 in: $i > $i > $o ).
% 2.41/1.40
% 2.41/1.40 thf(exu_type,type,
% 2.41/1.40 exu: ( $i > $o ) > $o ).
% 2.41/1.40
% 2.41/1.40 thf(exu,definition,
% 2.41/1.40 ( exu
% 2.41/1.40 = ( ^ [Xphi: $i > $o] :
% 2.41/1.40 ? [Xx: $i] :
% 2.41/1.40 ( ( Xphi @ Xx )
% 2.41/1.40 & ! [Xy: $i] :
% 2.41/1.40 ( ( Xphi @ Xy )
% 2.41/1.40 => ( Xx = Xy ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(setextAx_type,type,
% 2.41/1.40 setextAx: $o ).
% 2.41/1.40
% 2.41/1.40 thf(setextAx,definition,
% 2.41/1.40 ( setextAx
% 2.41/1.40 = ( ! [A: $i,B: $i] :
% 2.41/1.40 ( ! [Xx: $i] :
% 2.41/1.40 ( ( in @ Xx @ A )
% 2.41/1.40 <=> ( in @ Xx @ B ) )
% 2.41/1.40 => ( A = B ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(emptyset_type,type,
% 2.41/1.40 emptyset: $i ).
% 2.41/1.40
% 2.41/1.40 thf(emptysetAx_type,type,
% 2.41/1.40 emptysetAx: $o ).
% 2.41/1.40
% 2.41/1.40 thf(emptysetAx,definition,
% 2.41/1.40 ( emptysetAx
% 2.41/1.40 = ( ! [Xx: $i] :
% 2.41/1.40 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(setadjoin_type,type,
% 2.41/1.40 setadjoin: $i > $i > $i ).
% 2.41/1.40
% 2.41/1.40 thf(setadjoinAx_type,type,
% 2.41/1.40 setadjoinAx: $o ).
% 2.41/1.40
% 2.41/1.40 thf(setadjoinAx,definition,
% 2.41/1.40 ( setadjoinAx
% 2.41/1.40 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.41/1.40 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.41/1.40 <=> ( ( Xy = Xx )
% 2.41/1.40 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(powerset_type,type,
% 2.41/1.40 powerset: $i > $i ).
% 2.41/1.40
% 2.41/1.40 thf(powersetAx_type,type,
% 2.41/1.40 powersetAx: $o ).
% 2.41/1.40
% 2.41/1.40 thf(powersetAx,definition,
% 2.41/1.40 ( powersetAx
% 2.41/1.40 = ( ! [A: $i,B: $i] :
% 2.41/1.40 ( ( in @ B @ ( powerset @ A ) )
% 2.41/1.40 <=> ! [Xx: $i] :
% 2.41/1.40 ( ( in @ Xx @ B )
% 2.41/1.40 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(setunion_type,type,
% 2.41/1.40 setunion: $i > $i ).
% 2.41/1.40
% 2.41/1.40 thf(setunionAx_type,type,
% 2.41/1.40 setunionAx: $o ).
% 2.41/1.40
% 2.41/1.40 thf(setunionAx,definition,
% 2.41/1.40 ( setunionAx
% 2.41/1.40 = ( ! [A: $i,Xx: $i] :
% 2.41/1.40 ( ( in @ Xx @ ( setunion @ A ) )
% 2.41/1.40 <=> ? [B: $i] :
% 2.41/1.40 ( ( in @ Xx @ B )
% 2.41/1.40 & ( in @ B @ A ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(omega_type,type,
% 2.41/1.40 omega: $i ).
% 2.41/1.40
% 2.41/1.40 thf(omega0Ax_type,type,
% 2.41/1.40 omega0Ax: $o ).
% 2.41/1.40
% 2.41/1.40 thf(omega0Ax,definition,
% 2.41/1.40 ( omega0Ax
% 2.41/1.40 = ( in @ emptyset @ omega ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(omegaSAx_type,type,
% 2.41/1.40 omegaSAx: $o ).
% 2.41/1.40
% 2.41/1.40 thf(omegaSAx,definition,
% 2.41/1.40 ( omegaSAx
% 2.41/1.40 = ( ! [Xx: $i] :
% 2.41/1.40 ( ( in @ Xx @ omega )
% 2.41/1.40 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(omegaIndAx_type,type,
% 2.41/1.40 omegaIndAx: $o ).
% 2.41/1.40
% 2.41/1.40 thf(omegaIndAx,definition,
% 2.41/1.40 ( omegaIndAx
% 2.41/1.40 = ( ! [A: $i] :
% 2.41/1.40 ( ( ( in @ emptyset @ A )
% 2.41/1.40 & ! [Xx: $i] :
% 2.41/1.40 ( ( ( in @ Xx @ omega )
% 2.41/1.40 & ( in @ Xx @ A ) )
% 2.41/1.40 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 2.41/1.40 => ! [Xx: $i] :
% 2.41/1.40 ( ( in @ Xx @ omega )
% 2.41/1.40 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(replAx_type,type,
% 2.41/1.40 replAx: $o ).
% 2.41/1.40
% 2.41/1.40 thf(replAx,definition,
% 2.41/1.40 ( replAx
% 2.41/1.40 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 2.41/1.40 ( ! [Xx: $i] :
% 2.41/1.40 ( ( in @ Xx @ A )
% 2.41/1.40 => ( exu
% 2.41/1.40 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 2.41/1.40 => ? [B: $i] :
% 2.41/1.40 ! [Xx: $i] :
% 2.41/1.40 ( ( in @ Xx @ B )
% 2.41/1.40 <=> ? [Xy: $i] :
% 2.41/1.40 ( ( in @ Xy @ A )
% 2.41/1.40 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(foundationAx_type,type,
% 2.41/1.40 foundationAx: $o ).
% 2.41/1.40
% 2.41/1.40 thf(foundationAx,definition,
% 2.41/1.40 ( foundationAx
% 2.41/1.40 = ( ! [A: $i] :
% 2.41/1.40 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.41/1.40 => ? [B: $i] :
% 2.41/1.40 ( ( in @ B @ A )
% 2.41/1.40 & ~ ? [Xx: $i] :
% 2.41/1.40 ( ( in @ Xx @ B )
% 2.41/1.40 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(wellorderingAx_type,type,
% 2.41/1.40 wellorderingAx: $o ).
% 2.41/1.40
% 2.41/1.40 thf(wellorderingAx,definition,
% 2.41/1.40 ( wellorderingAx
% 2.41/1.40 = ( ! [A: $i] :
% 2.41/1.40 ? [B: $i] :
% 2.41/1.40 ( ! [C: $i] :
% 2.41/1.40 ( ( in @ C @ B )
% 2.41/1.40 => ! [Xx: $i] :
% 2.41/1.40 ( ( in @ Xx @ C )
% 2.41/1.40 => ( in @ Xx @ A ) ) )
% 2.41/1.40 & ! [Xx: $i,Xy: $i] :
% 2.41/1.40 ( ( ( in @ Xx @ A )
% 2.41/1.40 & ( in @ Xy @ A ) )
% 2.41/1.40 => ( ! [C: $i] :
% 2.41/1.40 ( ( in @ C @ B )
% 2.41/1.40 => ( ( in @ Xx @ C )
% 2.41/1.40 <=> ( in @ Xy @ C ) ) )
% 2.41/1.40 => ( Xx = Xy ) ) )
% 2.41/1.40 & ! [C: $i,D: $i] :
% 2.41/1.40 ( ( ( in @ C @ B )
% 2.41/1.40 & ( in @ D @ B ) )
% 2.41/1.40 => ( ! [Xx: $i] :
% 2.41/1.40 ( ( in @ Xx @ C )
% 2.41/1.40 => ( in @ Xx @ D ) )
% 2.41/1.40 | ! [Xx: $i] :
% 2.41/1.40 ( ( in @ Xx @ D )
% 2.41/1.40 => ( in @ Xx @ C ) ) ) )
% 2.41/1.40 & ! [C: $i] :
% 2.41/1.40 ( ( ! [Xx: $i] :
% 2.41/1.40 ( ( in @ Xx @ C )
% 2.41/1.40 => ( in @ Xx @ A ) )
% 2.41/1.40 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 2.41/1.40 => ? [D: $i,Xx: $i] :
% 2.41/1.40 ( ( in @ D @ B )
% 2.41/1.40 & ( in @ Xx @ C )
% 2.41/1.40 & ~ ? [Xy: $i] :
% 2.41/1.40 ( ( in @ Xy @ D )
% 2.41/1.40 & ( in @ Xy @ C ) )
% 2.41/1.40 & ! [E: $i] :
% 2.41/1.40 ( ( in @ E @ B )
% 2.41/1.40 => ( ! [Xy: $i] :
% 2.41/1.40 ( ( in @ Xy @ E )
% 2.41/1.40 => ( in @ Xy @ D ) )
% 2.41/1.40 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(descr_type,type,
% 2.41/1.40 descr: ( $i > $o ) > $i ).
% 2.41/1.40
% 2.41/1.40 thf(descrp_type,type,
% 2.41/1.40 descrp: $o ).
% 2.41/1.40
% 2.41/1.40 thf(descrp,definition,
% 2.41/1.40 ( descrp
% 2.41/1.40 = ( ! [Xphi: $i > $o] :
% 2.41/1.40 ( ( exu
% 2.41/1.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.41/1.40 => ( Xphi
% 2.41/1.40 @ ( descr
% 2.41/1.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(dsetconstr_type,type,
% 2.41/1.40 dsetconstr: $i > ( $i > $o ) > $i ).
% 2.41/1.40
% 2.41/1.40 thf(dsetconstrI_type,type,
% 2.41/1.40 dsetconstrI: $o ).
% 2.41/1.40
% 2.41/1.40 thf(dsetconstrI,definition,
% 2.41/1.40 ( dsetconstrI
% 2.41/1.40 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.41/1.40 ( ( in @ Xx @ A )
% 2.41/1.40 => ( ( Xphi @ Xx )
% 2.41/1.40 => ( in @ Xx
% 2.41/1.40 @ ( dsetconstr @ A
% 2.41/1.40 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(dsetconstrEL_type,type,
% 2.41/1.40 dsetconstrEL: $o ).
% 2.41/1.40
% 2.41/1.40 thf(dsetconstrEL,definition,
% 2.41/1.40 ( dsetconstrEL
% 2.41/1.40 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.41/1.40 ( ( in @ Xx
% 2.41/1.40 @ ( dsetconstr @ A
% 2.41/1.40 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.41/1.40 => ( in @ Xx @ A ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(dsetconstrER_type,type,
% 2.41/1.40 dsetconstrER: $o ).
% 2.41/1.40
% 2.41/1.40 thf(dsetconstrER,definition,
% 2.41/1.40 ( dsetconstrER
% 2.41/1.40 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.41/1.40 ( ( in @ Xx
% 2.41/1.40 @ ( dsetconstr @ A
% 2.41/1.40 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.41/1.40 => ( Xphi @ Xx ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(exuE1_type,type,
% 2.41/1.40 exuE1: $o ).
% 2.41/1.40
% 2.41/1.40 thf(exuE1,definition,
% 2.41/1.40 ( exuE1
% 2.41/1.40 = ( ! [Xphi: $i > $o] :
% 2.41/1.40 ( ( exu
% 2.41/1.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.41/1.40 => ? [Xx: $i] :
% 2.41/1.40 ( ( Xphi @ Xx )
% 2.41/1.40 & ! [Xy: $i] :
% 2.41/1.40 ( ( Xphi @ Xy )
% 2.41/1.40 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(prop2set_type,type,
% 2.41/1.40 prop2set: $o > $i ).
% 2.41/1.40
% 2.41/1.40 thf(prop2setE_type,type,
% 2.41/1.40 prop2setE: $o ).
% 2.41/1.40
% 2.41/1.40 thf(prop2setE,definition,
% 2.41/1.40 ( prop2setE
% 2.41/1.40 = ( ! [Xphi: $o,Xx: $i] :
% 2.41/1.40 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 2.41/1.40 => Xphi ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(emptysetE_type,type,
% 2.41/1.40 emptysetE: $o ).
% 2.41/1.40
% 2.41/1.40 thf(emptysetE,definition,
% 2.41/1.40 ( emptysetE
% 2.41/1.40 = ( ! [Xx: $i] :
% 2.41/1.40 ( ( in @ Xx @ emptyset )
% 2.41/1.40 => ! [Xphi: $o] : Xphi ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(emptysetimpfalse_type,type,
% 2.41/1.40 emptysetimpfalse: $o ).
% 2.41/1.40
% 2.41/1.40 thf(emptysetimpfalse,definition,
% 2.41/1.40 ( emptysetimpfalse
% 2.41/1.40 = ( ! [Xx: $i] :
% 2.41/1.40 ( ( in @ Xx @ emptyset )
% 2.41/1.40 => $false ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(notinemptyset_type,type,
% 2.41/1.40 notinemptyset: $o ).
% 2.41/1.40
% 2.41/1.40 thf(notinemptyset,definition,
% 2.41/1.40 ( notinemptyset
% 2.41/1.40 = ( ! [Xx: $i] :
% 2.41/1.40 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(exuE3e_type,type,
% 2.41/1.40 exuE3e: $o ).
% 2.41/1.40
% 2.41/1.40 thf(exuE3e,definition,
% 2.41/1.40 ( exuE3e
% 2.41/1.40 = ( ! [Xphi: $i > $o] :
% 2.41/1.40 ( ( exu
% 2.41/1.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.41/1.40 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(setext_type,type,
% 2.41/1.40 setext: $o ).
% 2.41/1.40
% 2.41/1.40 thf(setext,definition,
% 2.41/1.40 ( setext
% 2.41/1.40 = ( ! [A: $i,B: $i] :
% 2.41/1.40 ( ! [Xx: $i] :
% 2.41/1.40 ( ( in @ Xx @ A )
% 2.41/1.40 => ( in @ Xx @ B ) )
% 2.41/1.40 => ( ! [Xx: $i] :
% 2.41/1.40 ( ( in @ Xx @ B )
% 2.41/1.40 => ( in @ Xx @ A ) )
% 2.41/1.40 => ( A = B ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(emptyI_type,type,
% 2.41/1.40 emptyI: $o ).
% 2.41/1.40
% 2.41/1.40 thf(emptyI,definition,
% 2.41/1.40 ( emptyI
% 2.41/1.40 = ( ! [A: $i] :
% 2.41/1.40 ( ! [Xx: $i] :
% 2.41/1.40 ~ ( in @ Xx @ A )
% 2.41/1.40 => ( A = emptyset ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(noeltsimpempty_type,type,
% 2.41/1.40 noeltsimpempty: $o ).
% 2.41/1.40
% 2.41/1.40 thf(noeltsimpempty,definition,
% 2.41/1.40 ( noeltsimpempty
% 2.41/1.40 = ( ! [A: $i] :
% 2.41/1.40 ( ! [Xx: $i] :
% 2.41/1.40 ~ ( in @ Xx @ A )
% 2.41/1.40 => ( A = emptyset ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(setbeta_type,type,
% 2.41/1.40 setbeta: $o ).
% 2.41/1.40
% 2.41/1.40 thf(setbeta,definition,
% 2.41/1.40 ( setbeta
% 2.41/1.40 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.41/1.40 ( ( in @ Xx @ A )
% 2.41/1.40 => ( ( in @ Xx
% 2.41/1.40 @ ( dsetconstr @ A
% 2.41/1.40 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.41/1.40 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(nonempty_type,type,
% 2.41/1.40 nonempty: $i > $o ).
% 2.41/1.40
% 2.41/1.40 thf(nonempty,definition,
% 2.41/1.40 ( nonempty
% 2.41/1.40 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(nonemptyE1_type,type,
% 2.41/1.40 nonemptyE1: $o ).
% 2.41/1.40
% 2.41/1.40 thf(nonemptyE1,definition,
% 2.41/1.40 ( nonemptyE1
% 2.41/1.40 = ( ! [A: $i] :
% 2.41/1.40 ( ( nonempty @ A )
% 2.41/1.40 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(nonemptyI_type,type,
% 2.41/1.40 nonemptyI: $o ).
% 2.41/1.40
% 2.41/1.40 thf(nonemptyI,definition,
% 2.41/1.40 ( nonemptyI
% 2.41/1.40 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.41/1.40 ( ( in @ Xx @ A )
% 2.41/1.40 => ( ( Xphi @ Xx )
% 2.41/1.40 => ( nonempty
% 2.41/1.40 @ ( dsetconstr @ A
% 2.41/1.40 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(nonemptyI1_type,type,
% 2.41/1.40 nonemptyI1: $o ).
% 2.41/1.40
% 2.41/1.40 thf(nonemptyI1,definition,
% 2.41/1.40 ( nonemptyI1
% 2.41/1.40 = ( ! [A: $i] :
% 2.41/1.40 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.41/1.40 => ( nonempty @ A ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(setadjoinIL_type,type,
% 2.41/1.40 setadjoinIL: $o ).
% 2.41/1.40
% 2.41/1.40 thf(setadjoinIL,definition,
% 2.41/1.40 ( setadjoinIL
% 2.41/1.40 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(emptyinunitempty_type,type,
% 2.41/1.40 emptyinunitempty: $o ).
% 2.41/1.40
% 2.41/1.40 thf(emptyinunitempty,definition,
% 2.41/1.40 ( emptyinunitempty
% 2.41/1.40 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(setadjoinIR_type,type,
% 2.41/1.40 setadjoinIR: $o ).
% 2.41/1.40
% 2.41/1.40 thf(setadjoinIR,definition,
% 2.41/1.40 ( setadjoinIR
% 2.41/1.40 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.41/1.40 ( ( in @ Xy @ A )
% 2.41/1.40 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(setadjoinE_type,type,
% 2.41/1.40 setadjoinE: $o ).
% 2.41/1.40
% 2.41/1.40 thf(setadjoinE,definition,
% 2.41/1.40 ( setadjoinE
% 2.41/1.40 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.41/1.40 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.41/1.40 => ! [Xphi: $o] :
% 2.41/1.40 ( ( ( Xy = Xx )
% 2.41/1.40 => Xphi )
% 2.41/1.40 => ( ( ( in @ Xy @ A )
% 2.41/1.40 => Xphi )
% 2.41/1.40 => Xphi ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(setadjoinOr_type,type,
% 2.41/1.40 setadjoinOr: $o ).
% 2.41/1.40
% 2.41/1.40 thf(setadjoinOr,definition,
% 2.41/1.40 ( setadjoinOr
% 2.41/1.40 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.41/1.40 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.41/1.40 => ( ( Xy = Xx )
% 2.41/1.40 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(setoftrueEq_type,type,
% 2.41/1.40 setoftrueEq: $o ).
% 2.41/1.40
% 2.41/1.40 thf(setoftrueEq,definition,
% 2.41/1.40 ( setoftrueEq
% 2.41/1.40 = ( ! [A: $i] :
% 2.41/1.40 ( ( dsetconstr @ A
% 2.41/1.40 @ ^ [Xx: $i] : $true )
% 2.41/1.40 = A ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(powersetI_type,type,
% 2.41/1.40 powersetI: $o ).
% 2.41/1.40
% 2.41/1.40 thf(powersetI,definition,
% 2.41/1.40 ( powersetI
% 2.41/1.40 = ( ! [A: $i,B: $i] :
% 2.41/1.40 ( ! [Xx: $i] :
% 2.41/1.40 ( ( in @ Xx @ B )
% 2.41/1.40 => ( in @ Xx @ A ) )
% 2.41/1.40 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(emptyinPowerset_type,type,
% 2.41/1.40 emptyinPowerset: $o ).
% 2.41/1.40
% 2.41/1.40 thf(emptyinPowerset,definition,
% 2.41/1.40 ( emptyinPowerset
% 2.41/1.40 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(emptyInPowerset_type,type,
% 2.41/1.40 emptyInPowerset: $o ).
% 2.41/1.40
% 2.41/1.40 thf(emptyInPowerset,definition,
% 2.41/1.40 ( emptyInPowerset
% 2.41/1.40 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(powersetE_type,type,
% 2.41/1.40 powersetE: $o ).
% 2.41/1.40
% 2.41/1.40 thf(powersetE,definition,
% 2.41/1.40 ( powersetE
% 2.41/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.41/1.40 ( ( in @ B @ ( powerset @ A ) )
% 2.41/1.40 => ( ( in @ Xx @ B )
% 2.41/1.40 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(setunionI_type,type,
% 2.41/1.40 setunionI: $o ).
% 2.41/1.40
% 2.41/1.40 thf(setunionI,definition,
% 2.41/1.40 ( setunionI
% 2.41/1.40 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.41/1.40 ( ( in @ Xx @ B )
% 2.41/1.40 => ( ( in @ B @ A )
% 2.41/1.40 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(setunionE_type,type,
% 2.41/1.40 setunionE: $o ).
% 2.41/1.40
% 2.41/1.40 thf(setunionE,definition,
% 2.41/1.40 ( setunionE
% 2.41/1.40 = ( ! [A: $i,Xx: $i] :
% 2.41/1.40 ( ( in @ Xx @ ( setunion @ A ) )
% 2.41/1.40 => ! [Xphi: $o] :
% 2.41/1.40 ( ! [B: $i] :
% 2.41/1.40 ( ( in @ Xx @ B )
% 2.41/1.40 => ( ( in @ B @ A )
% 2.41/1.40 => Xphi ) )
% 2.41/1.40 => Xphi ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(subPowSU_type,type,
% 2.41/1.40 subPowSU: $o ).
% 2.41/1.40
% 2.41/1.40 thf(subPowSU,definition,
% 2.41/1.40 ( subPowSU
% 2.41/1.40 = ( ! [A: $i,Xx: $i] :
% 2.41/1.40 ( ( in @ Xx @ A )
% 2.41/1.40 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(exuE2_type,type,
% 2.41/1.40 exuE2: $o ).
% 2.41/1.40
% 2.41/1.40 thf(exuE2,definition,
% 2.41/1.40 ( exuE2
% 2.41/1.40 = ( ! [Xphi: $i > $o] :
% 2.41/1.40 ( ( exu
% 2.41/1.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.41/1.40 => ? [Xx: $i] :
% 2.41/1.40 ! [Xy: $i] :
% 2.41/1.40 ( ( Xphi @ Xy )
% 2.41/1.40 <=> ( Xy = Xx ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(nonemptyImpWitness_type,type,
% 2.41/1.40 nonemptyImpWitness: $o ).
% 2.41/1.40
% 2.41/1.40 thf(nonemptyImpWitness,definition,
% 2.41/1.40 ( nonemptyImpWitness
% 2.41/1.40 = ( ! [A: $i] :
% 2.41/1.40 ( ( nonempty @ A )
% 2.41/1.40 => ? [Xx: $i] :
% 2.41/1.40 ( ( in @ Xx @ A )
% 2.41/1.40 & $true ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(uniqinunit_type,type,
% 2.41/1.40 uniqinunit: $o ).
% 2.41/1.40
% 2.41/1.40 thf(uniqinunit,definition,
% 2.41/1.40 ( uniqinunit
% 2.41/1.40 = ( ! [Xx: $i,Xy: $i] :
% 2.41/1.40 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.41/1.40 => ( Xx = Xy ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(notinsingleton_type,type,
% 2.41/1.40 notinsingleton: $o ).
% 2.41/1.40
% 2.41/1.40 thf(notinsingleton,definition,
% 2.41/1.40 ( notinsingleton
% 2.41/1.40 = ( ! [Xx: $i,Xy: $i] :
% 2.41/1.40 ( ( Xx != Xy )
% 2.41/1.40 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(eqinunit_type,type,
% 2.41/1.40 eqinunit: $o ).
% 2.41/1.40
% 2.41/1.40 thf(eqinunit,definition,
% 2.41/1.40 ( eqinunit
% 2.41/1.40 = ( ! [Xx: $i,Xy: $i] :
% 2.41/1.40 ( ( Xx = Xy )
% 2.41/1.40 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(singletonsswitch_type,type,
% 2.41/1.40 singletonsswitch: $o ).
% 2.41/1.40
% 2.41/1.40 thf(singletonsswitch,definition,
% 2.41/1.40 ( singletonsswitch
% 2.41/1.40 = ( ! [Xx: $i,Xy: $i] :
% 2.41/1.40 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.41/1.40 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(upairsetE_type,type,
% 2.41/1.40 upairsetE: $o ).
% 2.41/1.40
% 2.41/1.40 thf(upairsetE,definition,
% 2.41/1.40 ( upairsetE
% 2.41/1.40 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.41/1.40 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.41/1.40 => ( ( Xz = Xx )
% 2.41/1.40 | ( Xz = Xy ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(upairsetIL_type,type,
% 2.41/1.40 upairsetIL: $o ).
% 2.41/1.40
% 2.41/1.40 thf(upairsetIL,definition,
% 2.41/1.40 ( upairsetIL
% 2.41/1.40 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(upairsetIR_type,type,
% 2.41/1.40 upairsetIR: $o ).
% 2.41/1.40
% 2.41/1.40 thf(upairsetIR,definition,
% 2.41/1.40 ( upairsetIR
% 2.41/1.40 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(emptyE1_type,type,
% 2.41/1.40 emptyE1: $o ).
% 2.41/1.40
% 2.41/1.40 thf(emptyE1,definition,
% 2.41/1.40 ( emptyE1
% 2.41/1.40 = ( ! [A: $i,Xphi: $i > $o] :
% 2.41/1.40 ( ? [Xx: $i] :
% 2.41/1.40 ( ( in @ Xx @ A )
% 2.41/1.40 & ( Xphi @ Xx ) )
% 2.41/1.40 => ( ( ( dsetconstr @ A
% 2.41/1.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.41/1.40 = emptyset )
% 2.41/1.40 => $false ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(vacuousDall_type,type,
% 2.41/1.40 vacuousDall: $o ).
% 2.41/1.40
% 2.41/1.40 thf(vacuousDall,definition,
% 2.41/1.40 ( vacuousDall
% 2.41/1.40 = ( ! [Xphi: $i > $o,Xx: $i] :
% 2.41/1.40 ( ( in @ Xx @ emptyset )
% 2.41/1.40 => ( Xphi @ Xx ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(quantDeMorgan1_type,type,
% 2.41/1.40 quantDeMorgan1: $o ).
% 2.41/1.40
% 2.41/1.40 thf(quantDeMorgan1,definition,
% 2.41/1.40 ( quantDeMorgan1
% 2.41/1.40 = ( ! [A: $i,Xphi: $i > $o] :
% 2.41/1.40 ( ~ ! [Xx: $i] :
% 2.41/1.40 ( ( in @ Xx @ A )
% 2.41/1.40 => ( Xphi @ Xx ) )
% 2.41/1.40 => ? [Xx: $i] :
% 2.41/1.40 ( ( in @ Xx @ A )
% 2.41/1.40 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(quantDeMorgan2_type,type,
% 2.41/1.40 quantDeMorgan2: $o ).
% 2.41/1.40
% 2.41/1.40 thf(quantDeMorgan2,definition,
% 2.41/1.40 ( quantDeMorgan2
% 2.41/1.40 = ( ! [A: $i,Xphi: $i > $o] :
% 2.41/1.40 ( ! [Xx: $i] :
% 2.41/1.40 ( ( in @ Xx @ A )
% 2.41/1.40 => ~ ( Xphi @ Xx ) )
% 2.41/1.40 => ~ ? [Xx: $i] :
% 2.41/1.40 ( ( in @ Xx @ A )
% 2.41/1.40 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(quantDeMorgan3_type,type,
% 2.41/1.40 quantDeMorgan3: $o ).
% 2.41/1.40
% 2.41/1.40 thf(quantDeMorgan3,definition,
% 2.41/1.40 ( quantDeMorgan3
% 2.41/1.40 = ( ! [A: $i,Xphi: $i > $o] :
% 2.41/1.40 ( ~ ? [Xx: $i] :
% 2.41/1.40 ( ( in @ Xx @ A )
% 2.41/1.40 & ( Xphi @ Xx ) )
% 2.41/1.40 => ! [Xx: $i] :
% 2.41/1.40 ( ( in @ Xx @ A )
% 2.41/1.40 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(quantDeMorgan4_type,type,
% 2.41/1.40 quantDeMorgan4: $o ).
% 2.41/1.40
% 2.41/1.40 thf(quantDeMorgan4,definition,
% 2.41/1.40 ( quantDeMorgan4
% 2.41/1.40 = ( ! [A: $i,Xphi: $i > $o] :
% 2.41/1.40 ( ? [Xx: $i] :
% 2.41/1.40 ( ( in @ Xx @ A )
% 2.41/1.40 & ~ ( Xphi @ Xx ) )
% 2.41/1.40 => ~ ! [Xx: $i] :
% 2.41/1.40 ( ( in @ Xx @ A )
% 2.41/1.40 => ( Xphi @ Xx ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(prop2setI_type,type,
% 2.41/1.40 prop2setI: $o ).
% 2.41/1.40
% 2.41/1.40 thf(prop2setI,definition,
% 2.41/1.40 ( prop2setI
% 2.41/1.40 = ( ! [Xphi: $o] :
% 2.41/1.40 ( Xphi
% 2.41/1.40 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(set2prop_type,type,
% 2.41/1.40 set2prop: $i > $o ).
% 2.41/1.40
% 2.41/1.40 thf(prop2set2propI_type,type,
% 2.41/1.40 prop2set2propI: $o ).
% 2.41/1.40
% 2.41/1.40 thf(prop2set2propI,definition,
% 2.41/1.40 ( prop2set2propI
% 2.41/1.40 = ( ! [Xphi: $o] :
% 2.41/1.40 ( Xphi
% 2.41/1.40 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(notdexE_type,type,
% 2.41/1.40 notdexE: $o ).
% 2.41/1.40
% 2.41/1.40 thf(notdexE,definition,
% 2.41/1.40 ( notdexE
% 2.41/1.40 = ( ! [A: $i,Xphi: $i > $o] :
% 2.41/1.40 ( ~ ? [Xx: $i] :
% 2.41/1.40 ( ( in @ Xx @ A )
% 2.41/1.40 & ( Xphi @ Xx ) )
% 2.41/1.40 => ! [Xx: $i] :
% 2.41/1.40 ( ( in @ Xx @ A )
% 2.41/1.40 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(notdallE_type,type,
% 2.41/1.40 notdallE: $o ).
% 2.41/1.40
% 2.41/1.40 thf(notdallE,definition,
% 2.41/1.40 ( notdallE
% 2.41/1.40 = ( ! [A: $i,Xphi: $i > $o] :
% 2.41/1.40 ( ~ ! [Xx: $i] :
% 2.41/1.40 ( ( in @ Xx @ A )
% 2.41/1.40 => ( Xphi @ Xx ) )
% 2.41/1.40 => ? [Xx: $i] :
% 2.41/1.40 ( ( in @ Xx @ A )
% 2.41/1.40 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(exuI1_type,type,
% 2.41/1.40 exuI1: $o ).
% 2.41/1.40
% 2.41/1.40 thf(exuI1,definition,
% 2.41/1.40 ( exuI1
% 2.41/1.40 = ( ! [Xphi: $i > $o] :
% 2.41/1.40 ( ? [Xx: $i] :
% 2.41/1.40 ( ( Xphi @ Xx )
% 2.41/1.40 & ! [Xy: $i] :
% 2.41/1.40 ( ( Xphi @ Xy )
% 2.41/1.40 => ( Xx = Xy ) ) )
% 2.41/1.40 => ( exu
% 2.41/1.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(exuI3_type,type,
% 2.41/1.40 exuI3: $o ).
% 2.41/1.40
% 2.41/1.40 thf(exuI3,definition,
% 2.41/1.40 ( exuI3
% 2.41/1.40 = ( ! [Xphi: $i > $o] :
% 2.41/1.40 ( ? [Xx: $i] : ( Xphi @ Xx )
% 2.41/1.40 => ( ! [Xx: $i,Xy: $i] :
% 2.41/1.40 ( ( Xphi @ Xx )
% 2.41/1.40 => ( ( Xphi @ Xy )
% 2.41/1.40 => ( Xx = Xy ) ) )
% 2.41/1.40 => ( exu
% 2.41/1.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(exuI2_type,type,
% 2.41/1.40 exuI2: $o ).
% 2.41/1.40
% 2.41/1.40 thf(exuI2,definition,
% 2.41/1.40 ( exuI2
% 2.41/1.40 = ( ! [Xphi: $i > $o] :
% 2.41/1.40 ( ? [Xx: $i] :
% 2.41/1.40 ! [Xy: $i] :
% 2.41/1.40 ( ( Xphi @ Xy )
% 2.41/1.40 <=> ( Xy = Xx ) )
% 2.41/1.40 => ( exu
% 2.41/1.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(inCongP_type,type,
% 2.41/1.40 inCongP: $o ).
% 2.41/1.40
% 2.41/1.40 thf(inCongP,definition,
% 2.41/1.40 ( inCongP
% 2.41/1.40 = ( ! [A: $i,B: $i] :
% 2.41/1.40 ( ( A = B )
% 2.41/1.40 => ! [Xx: $i,Xy: $i] :
% 2.41/1.40 ( ( Xx = Xy )
% 2.41/1.40 => ( ( in @ Xx @ A )
% 2.41/1.40 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(in__Cong_type,type,
% 2.41/1.40 in__Cong: $o ).
% 2.41/1.40
% 2.41/1.40 thf(in__Cong,definition,
% 2.41/1.40 ( in__Cong
% 2.41/1.40 = ( ! [A: $i,B: $i] :
% 2.41/1.40 ( ( A = B )
% 2.41/1.40 => ! [Xx: $i,Xy: $i] :
% 2.41/1.40 ( ( Xx = Xy )
% 2.41/1.40 => ( ( in @ Xx @ A )
% 2.41/1.40 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(exuE3u_type,type,
% 2.41/1.40 exuE3u: $o ).
% 2.41/1.40
% 2.41/1.40 thf(exuE3u,definition,
% 2.41/1.40 ( exuE3u
% 2.41/1.40 = ( ! [Xphi: $i > $o] :
% 2.41/1.40 ( ( exu
% 2.41/1.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.41/1.40 => ! [Xx: $i,Xy: $i] :
% 2.41/1.40 ( ( Xphi @ Xx )
% 2.41/1.40 => ( ( Xphi @ Xy )
% 2.41/1.40 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(exu__Cong_type,type,
% 2.41/1.40 exu__Cong: $o ).
% 2.41/1.40
% 2.41/1.40 thf(exu__Cong,definition,
% 2.41/1.40 ( exu__Cong
% 2.41/1.40 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.41/1.40 ( ! [Xx: $i,Xy: $i] :
% 2.41/1.40 ( ( Xx = Xy )
% 2.41/1.40 => ( ( Xphi @ Xx )
% 2.41/1.40 <=> ( Xpsi @ Xy ) ) )
% 2.41/1.40 => ( ( exu
% 2.41/1.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.41/1.40 <=> ( exu
% 2.41/1.40 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(emptyset__Cong_type,type,
% 2.41/1.40 emptyset__Cong: $o ).
% 2.41/1.40
% 2.41/1.40 thf(emptyset__Cong,definition,
% 2.41/1.40 ( emptyset__Cong
% 2.41/1.40 = ( emptyset = emptyset ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(setadjoin__Cong_type,type,
% 2.41/1.40 setadjoin__Cong: $o ).
% 2.41/1.40
% 2.41/1.40 thf(setadjoin__Cong,definition,
% 2.41/1.40 ( setadjoin__Cong
% 2.41/1.40 = ( ! [Xx: $i,Xy: $i] :
% 2.41/1.40 ( ( Xx = Xy )
% 2.41/1.40 => ! [Xz: $i,Xu: $i] :
% 2.41/1.40 ( ( Xz = Xu )
% 2.41/1.40 => ( ( setadjoin @ Xx @ Xz )
% 2.41/1.40 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(powerset__Cong_type,type,
% 2.41/1.40 powerset__Cong: $o ).
% 2.41/1.40
% 2.41/1.40 thf(powerset__Cong,definition,
% 2.41/1.40 ( powerset__Cong
% 2.41/1.40 = ( ! [A: $i,B: $i] :
% 2.41/1.40 ( ( A = B )
% 2.41/1.40 => ( ( powerset @ A )
% 2.41/1.40 = ( powerset @ B ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(setunion__Cong_type,type,
% 2.41/1.40 setunion__Cong: $o ).
% 2.41/1.40
% 2.41/1.40 thf(setunion__Cong,definition,
% 2.41/1.40 ( setunion__Cong
% 2.41/1.40 = ( ! [A: $i,B: $i] :
% 2.41/1.40 ( ( A = B )
% 2.41/1.40 => ( ( setunion @ A )
% 2.41/1.40 = ( setunion @ B ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(omega__Cong_type,type,
% 2.41/1.40 omega__Cong: $o ).
% 2.41/1.40
% 2.41/1.40 thf(omega__Cong,definition,
% 2.41/1.40 ( omega__Cong
% 2.41/1.40 = ( omega = omega ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(exuEu_type,type,
% 2.41/1.40 exuEu: $o ).
% 2.41/1.40
% 2.41/1.40 thf(exuEu,definition,
% 2.41/1.40 ( exuEu
% 2.41/1.40 = ( ! [Xphi: $i > $o] :
% 2.41/1.40 ( ( exu
% 2.41/1.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.41/1.40 => ! [Xx: $i,Xy: $i] :
% 2.41/1.40 ( ( Xphi @ Xx )
% 2.41/1.40 => ( ( Xphi @ Xy )
% 2.41/1.40 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(descr__Cong_type,type,
% 2.41/1.40 descr__Cong: $o ).
% 2.41/1.40
% 2.41/1.40 thf(descr__Cong,definition,
% 2.41/1.40 ( descr__Cong
% 2.41/1.40 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.41/1.40 ( ! [Xx: $i,Xy: $i] :
% 2.41/1.40 ( ( Xx = Xy )
% 2.41/1.40 => ( ( Xphi @ Xx )
% 2.41/1.40 <=> ( Xpsi @ Xy ) ) )
% 2.41/1.40 => ( ( exu
% 2.41/1.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.41/1.40 => ( ( exu
% 2.41/1.40 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 2.41/1.40 => ( ( descr
% 2.41/1.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.41/1.40 = ( descr
% 2.41/1.40 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(dsetconstr__Cong_type,type,
% 2.41/1.40 dsetconstr__Cong: $o ).
% 2.41/1.40
% 2.41/1.40 thf(dsetconstr__Cong,definition,
% 2.41/1.40 ( dsetconstr__Cong
% 2.41/1.40 = ( ! [A: $i,B: $i] :
% 2.41/1.40 ( ( A = B )
% 2.41/1.40 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.41/1.40 ( ! [Xx: $i] :
% 2.41/1.40 ( ( in @ Xx @ A )
% 2.41/1.40 => ! [Xy: $i] :
% 2.41/1.40 ( ( in @ Xy @ B )
% 2.41/1.40 => ( ( Xx = Xy )
% 2.41/1.40 => ( ( Xphi @ Xx )
% 2.41/1.40 <=> ( Xpsi @ Xy ) ) ) ) )
% 2.41/1.40 => ( ( dsetconstr @ A
% 2.41/1.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.41/1.40 = ( dsetconstr @ B
% 2.41/1.40 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(subset_type,type,
% 2.41/1.40 subset: $i > $i > $o ).
% 2.41/1.40
% 2.41/1.40 thf(disjoint_type,type,
% 2.41/1.40 disjoint: $i > $i > $o ).
% 2.41/1.40
% 2.41/1.40 thf(setsmeet_type,type,
% 2.41/1.40 setsmeet: $i > $i > $o ).
% 2.41/1.40
% 2.41/1.40 thf(subsetI1_type,type,
% 2.41/1.40 subsetI1: $o ).
% 2.41/1.40
% 2.41/1.40 thf(subsetI1,definition,
% 2.41/1.40 ( subsetI1
% 2.41/1.40 = ( ! [A: $i,B: $i] :
% 2.41/1.40 ( ! [Xx: $i] :
% 2.41/1.40 ( ( in @ Xx @ A )
% 2.41/1.40 => ( in @ Xx @ B ) )
% 2.41/1.40 => ( subset @ A @ B ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(eqimpsubset2_type,type,
% 2.41/1.40 eqimpsubset2: $o ).
% 2.41/1.40
% 2.41/1.40 thf(eqimpsubset2,definition,
% 2.41/1.40 ( eqimpsubset2
% 2.41/1.40 = ( ! [A: $i,B: $i] :
% 2.41/1.40 ( ( A = B )
% 2.41/1.40 => ( subset @ B @ A ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(eqimpsubset1_type,type,
% 2.41/1.40 eqimpsubset1: $o ).
% 2.41/1.40
% 2.41/1.40 thf(eqimpsubset1,definition,
% 2.41/1.40 ( eqimpsubset1
% 2.41/1.40 = ( ! [A: $i,B: $i] :
% 2.41/1.40 ( ( A = B )
% 2.41/1.40 => ( subset @ A @ B ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(subsetI2_type,type,
% 2.41/1.40 subsetI2: $o ).
% 2.41/1.40
% 2.41/1.40 thf(subsetI2,definition,
% 2.41/1.40 ( subsetI2
% 2.41/1.40 = ( ! [A: $i,B: $i] :
% 2.41/1.40 ( ! [Xx: $i] :
% 2.41/1.40 ( ( in @ Xx @ A )
% 2.41/1.40 => ( in @ Xx @ B ) )
% 2.41/1.40 => ( subset @ A @ B ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(emptysetsubset_type,type,
% 2.41/1.40 emptysetsubset: $o ).
% 2.41/1.40
% 2.41/1.40 thf(emptysetsubset,definition,
% 2.41/1.40 ( emptysetsubset
% 2.41/1.40 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(subsetE_type,type,
% 2.41/1.40 subsetE: $o ).
% 2.41/1.40
% 2.41/1.40 thf(subsetE,definition,
% 2.41/1.40 ( subsetE
% 2.41/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.41/1.40 ( ( subset @ A @ B )
% 2.41/1.40 => ( ( in @ Xx @ A )
% 2.41/1.40 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(subsetE2_type,type,
% 2.41/1.40 subsetE2: $o ).
% 2.41/1.40
% 2.41/1.40 thf(subsetE2,definition,
% 2.41/1.40 ( subsetE2
% 2.41/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.41/1.40 ( ( subset @ A @ B )
% 2.41/1.40 => ( ~ ( in @ Xx @ B )
% 2.41/1.40 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(notsubsetI_type,type,
% 2.41/1.40 notsubsetI: $o ).
% 2.41/1.40
% 2.41/1.40 thf(notsubsetI,definition,
% 2.41/1.40 ( notsubsetI
% 2.41/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.41/1.40 ( ( in @ Xx @ A )
% 2.41/1.40 => ( ~ ( in @ Xx @ B )
% 2.41/1.40 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(notequalI1_type,type,
% 2.41/1.40 notequalI1: $o ).
% 2.41/1.40
% 2.41/1.40 thf(notequalI1,definition,
% 2.41/1.40 ( notequalI1
% 2.41/1.40 = ( ! [A: $i,B: $i] :
% 2.41/1.40 ( ~ ( subset @ A @ B )
% 2.41/1.40 => ( A != B ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(notequalI2_type,type,
% 2.41/1.40 notequalI2: $o ).
% 2.41/1.40
% 2.41/1.40 thf(notequalI2,definition,
% 2.41/1.40 ( notequalI2
% 2.41/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.41/1.40 ( ( in @ Xx @ A )
% 2.41/1.40 => ( ~ ( in @ Xx @ B )
% 2.41/1.40 => ( A != B ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(subsetRefl_type,type,
% 2.41/1.40 subsetRefl: $o ).
% 2.41/1.40
% 2.41/1.40 thf(subsetRefl,definition,
% 2.41/1.40 ( subsetRefl
% 2.41/1.40 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(subsetTrans_type,type,
% 2.41/1.40 subsetTrans: $o ).
% 2.41/1.40
% 2.41/1.40 thf(subsetTrans,definition,
% 2.41/1.40 ( subsetTrans
% 2.41/1.40 = ( ! [A: $i,B: $i,C: $i] :
% 2.41/1.40 ( ( subset @ A @ B )
% 2.41/1.40 => ( ( subset @ B @ C )
% 2.41/1.40 => ( subset @ A @ C ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(setadjoinSub_type,type,
% 2.41/1.40 setadjoinSub: $o ).
% 2.41/1.40
% 2.41/1.40 thf(setadjoinSub,definition,
% 2.41/1.40 ( setadjoinSub
% 2.41/1.40 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(setadjoinSub2_type,type,
% 2.41/1.40 setadjoinSub2: $o ).
% 2.41/1.40
% 2.41/1.40 thf(setadjoinSub2,definition,
% 2.41/1.40 ( setadjoinSub2
% 2.41/1.40 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.41/1.40 ( ( subset @ A @ B )
% 2.41/1.40 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(subset2powerset_type,type,
% 2.41/1.40 subset2powerset: $o ).
% 2.41/1.40
% 2.41/1.40 thf(subset2powerset,definition,
% 2.41/1.40 ( subset2powerset
% 2.41/1.40 = ( ! [A: $i,B: $i] :
% 2.41/1.40 ( ( subset @ A @ B )
% 2.41/1.40 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(setextsub_type,type,
% 2.41/1.40 setextsub: $o ).
% 2.41/1.40
% 2.41/1.40 thf(setextsub,definition,
% 2.41/1.40 ( setextsub
% 2.41/1.40 = ( ! [A: $i,B: $i] :
% 2.41/1.40 ( ( subset @ A @ B )
% 2.41/1.40 => ( ( subset @ B @ A )
% 2.41/1.40 => ( A = B ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(subsetemptysetimpeq_type,type,
% 2.41/1.40 subsetemptysetimpeq: $o ).
% 2.41/1.40
% 2.41/1.40 thf(subsetemptysetimpeq,definition,
% 2.41/1.40 ( subsetemptysetimpeq
% 2.41/1.40 = ( ! [A: $i] :
% 2.41/1.40 ( ( subset @ A @ emptyset )
% 2.41/1.40 => ( A = emptyset ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(powersetI1_type,type,
% 2.41/1.40 powersetI1: $o ).
% 2.41/1.40
% 2.41/1.40 thf(powersetI1,definition,
% 2.41/1.40 ( powersetI1
% 2.41/1.40 = ( ! [A: $i,B: $i] :
% 2.41/1.40 ( ( subset @ B @ A )
% 2.41/1.40 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(powersetE1_type,type,
% 2.41/1.40 powersetE1: $o ).
% 2.41/1.40
% 2.41/1.40 thf(powersetE1,definition,
% 2.41/1.40 ( powersetE1
% 2.41/1.40 = ( ! [A: $i,B: $i] :
% 2.41/1.40 ( ( in @ B @ ( powerset @ A ) )
% 2.41/1.40 => ( subset @ B @ A ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(inPowerset_type,type,
% 2.41/1.40 inPowerset: $o ).
% 2.41/1.40
% 2.41/1.40 thf(inPowerset,definition,
% 2.41/1.40 ( inPowerset
% 2.41/1.40 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(powersetsubset_type,type,
% 2.41/1.40 powersetsubset: $o ).
% 2.41/1.40
% 2.41/1.40 thf(powersetsubset,definition,
% 2.41/1.40 ( powersetsubset
% 2.41/1.40 = ( ! [A: $i,B: $i] :
% 2.41/1.40 ( ( subset @ A @ B )
% 2.41/1.40 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(sepInPowerset_type,type,
% 2.41/1.40 sepInPowerset: $o ).
% 2.41/1.40
% 2.41/1.40 thf(sepInPowerset,definition,
% 2.41/1.40 ( sepInPowerset
% 2.41/1.40 = ( ! [A: $i,Xphi: $i > $o] :
% 2.41/1.40 ( in
% 2.41/1.40 @ ( dsetconstr @ A
% 2.41/1.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.41/1.40 @ ( powerset @ A ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(sepSubset_type,type,
% 2.41/1.40 sepSubset: $o ).
% 2.41/1.40
% 2.41/1.40 thf(sepSubset,definition,
% 2.41/1.40 ( sepSubset
% 2.41/1.40 = ( ! [A: $i,Xphi: $i > $o] :
% 2.41/1.40 ( subset
% 2.41/1.40 @ ( dsetconstr @ A
% 2.41/1.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.41/1.40 @ A ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(binunion_type,type,
% 2.41/1.40 binunion: $i > $i > $i ).
% 2.41/1.40
% 2.41/1.40 thf(binunionIL_type,type,
% 2.41/1.40 binunionIL: $o ).
% 2.41/1.40
% 2.41/1.40 thf(binunionIL,definition,
% 2.41/1.40 ( binunionIL
% 2.41/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.41/1.40 ( ( in @ Xx @ A )
% 2.41/1.40 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(upairset2IR_type,type,
% 2.41/1.40 upairset2IR: $o ).
% 2.41/1.40
% 2.41/1.40 thf(upairset2IR,definition,
% 2.41/1.40 ( upairset2IR
% 2.41/1.40 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(binunionIR_type,type,
% 2.41/1.40 binunionIR: $o ).
% 2.41/1.40
% 2.41/1.40 thf(binunionIR,definition,
% 2.41/1.40 ( binunionIR
% 2.41/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.41/1.40 ( ( in @ Xx @ B )
% 2.41/1.40 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(binunionEcases_type,type,
% 2.41/1.40 binunionEcases: $o ).
% 2.41/1.40
% 2.41/1.40 thf(binunionEcases,definition,
% 2.41/1.40 ( binunionEcases
% 2.41/1.40 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 2.41/1.40 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.41/1.40 => ( ( ( in @ Xx @ A )
% 2.41/1.40 => Xphi )
% 2.41/1.40 => ( ( ( in @ Xx @ B )
% 2.41/1.40 => Xphi )
% 2.41/1.40 => Xphi ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(binunionE_type,type,
% 2.41/1.40 binunionE: $o ).
% 2.41/1.40
% 2.41/1.40 thf(binunionE,definition,
% 2.41/1.40 ( binunionE
% 2.41/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.41/1.40 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.41/1.40 => ( ( in @ Xx @ A )
% 2.41/1.40 | ( in @ Xx @ B ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(binunionLsub_type,type,
% 2.41/1.40 binunionLsub: $o ).
% 2.41/1.40
% 2.41/1.40 thf(binunionLsub,definition,
% 2.41/1.40 ( binunionLsub
% 2.41/1.40 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(binunionRsub_type,type,
% 2.41/1.40 binunionRsub: $o ).
% 2.41/1.40
% 2.41/1.40 thf(binunionRsub,definition,
% 2.41/1.40 ( binunionRsub
% 2.41/1.40 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(binintersect_type,type,
% 2.41/1.40 binintersect: $i > $i > $i ).
% 2.41/1.40
% 2.41/1.40 thf(binintersectI_type,type,
% 2.41/1.40 binintersectI: $o ).
% 2.41/1.40
% 2.41/1.40 thf(binintersectI,definition,
% 2.41/1.40 ( binintersectI
% 2.41/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.41/1.40 ( ( in @ Xx @ A )
% 2.41/1.40 => ( ( in @ Xx @ B )
% 2.41/1.40 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(binintersectSubset5_type,type,
% 2.41/1.40 binintersectSubset5: $o ).
% 2.41/1.40
% 2.41/1.40 thf(binintersectSubset5,definition,
% 2.41/1.40 ( binintersectSubset5
% 2.41/1.40 = ( ! [A: $i,B: $i,C: $i] :
% 2.41/1.40 ( ( subset @ C @ A )
% 2.41/1.40 => ( ( subset @ C @ B )
% 2.41/1.40 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(binintersectEL_type,type,
% 2.41/1.40 binintersectEL: $o ).
% 2.41/1.40
% 2.41/1.40 thf(binintersectEL,definition,
% 2.41/1.40 ( binintersectEL
% 2.41/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.41/1.40 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.41/1.40 => ( in @ Xx @ A ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(binintersectLsub_type,type,
% 2.41/1.40 binintersectLsub: $o ).
% 2.41/1.40
% 2.41/1.40 thf(binintersectLsub,definition,
% 2.41/1.40 ( binintersectLsub
% 2.41/1.40 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(binintersectSubset2_type,type,
% 2.41/1.40 binintersectSubset2: $o ).
% 2.41/1.40
% 2.41/1.40 thf(binintersectSubset2,definition,
% 2.41/1.40 ( binintersectSubset2
% 2.41/1.40 = ( ! [A: $i,B: $i] :
% 2.41/1.40 ( ( subset @ A @ B )
% 2.41/1.40 => ( ( binintersect @ A @ B )
% 2.41/1.40 = A ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(binintersectSubset3_type,type,
% 2.41/1.40 binintersectSubset3: $o ).
% 2.41/1.40
% 2.41/1.40 thf(binintersectSubset3,definition,
% 2.41/1.40 ( binintersectSubset3
% 2.41/1.40 = ( ! [A: $i,B: $i] :
% 2.41/1.40 ( ( ( binintersect @ A @ B )
% 2.41/1.40 = B )
% 2.41/1.40 => ( subset @ B @ A ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(binintersectER_type,type,
% 2.41/1.40 binintersectER: $o ).
% 2.41/1.40
% 2.41/1.40 thf(binintersectER,definition,
% 2.41/1.40 ( binintersectER
% 2.41/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.41/1.40 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.41/1.40 => ( in @ Xx @ B ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(disjointsetsI1_type,type,
% 2.41/1.40 disjointsetsI1: $o ).
% 2.41/1.40
% 2.41/1.40 thf(disjointsetsI1,definition,
% 2.41/1.40 ( disjointsetsI1
% 2.41/1.40 = ( ! [A: $i,B: $i] :
% 2.41/1.40 ( ~ ? [Xx: $i] :
% 2.41/1.40 ( ( in @ Xx @ A )
% 2.41/1.40 & ( in @ Xx @ B ) )
% 2.41/1.40 => ( ( binintersect @ A @ B )
% 2.41/1.40 = emptyset ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(binintersectRsub_type,type,
% 2.41/1.40 binintersectRsub: $o ).
% 2.41/1.40
% 2.41/1.40 thf(binintersectRsub,definition,
% 2.41/1.40 ( binintersectRsub
% 2.41/1.40 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(binintersectSubset4_type,type,
% 2.41/1.40 binintersectSubset4: $o ).
% 2.41/1.40
% 2.41/1.40 thf(binintersectSubset4,definition,
% 2.41/1.40 ( binintersectSubset4
% 2.41/1.40 = ( ! [A: $i,B: $i] :
% 2.41/1.40 ( ( subset @ B @ A )
% 2.41/1.40 => ( ( binintersect @ A @ B )
% 2.41/1.40 = B ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(binintersectSubset1_type,type,
% 2.41/1.40 binintersectSubset1: $o ).
% 2.41/1.40
% 2.41/1.40 thf(binintersectSubset1,definition,
% 2.41/1.40 ( binintersectSubset1
% 2.41/1.40 = ( ! [A: $i,B: $i] :
% 2.41/1.40 ( ( ( binintersect @ A @ B )
% 2.41/1.40 = A )
% 2.41/1.40 => ( subset @ A @ B ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(bs114d_type,type,
% 2.41/1.40 bs114d: $o ).
% 2.41/1.40
% 2.41/1.40 thf(bs114d,definition,
% 2.41/1.40 ( bs114d
% 2.41/1.40 = ( ! [A: $i,B: $i,C: $i] :
% 2.41/1.40 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 2.41/1.40 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(regular_type,type,
% 2.41/1.40 regular: $i > $o ).
% 2.41/1.40
% 2.41/1.40 thf(setminus_type,type,
% 2.41/1.40 setminus: $i > $i > $i ).
% 2.41/1.40
% 2.41/1.40 thf(setminusI_type,type,
% 2.41/1.40 setminusI: $o ).
% 2.41/1.40
% 2.41/1.40 thf(setminusI,definition,
% 2.41/1.40 ( setminusI
% 2.41/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.41/1.40 ( ( in @ Xx @ A )
% 2.41/1.40 => ( ~ ( in @ Xx @ B )
% 2.41/1.40 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(setminusEL_type,type,
% 2.41/1.40 setminusEL: $o ).
% 2.41/1.40
% 2.41/1.40 thf(setminusEL,definition,
% 2.41/1.40 ( setminusEL
% 2.41/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.41/1.40 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.41/1.40 => ( in @ Xx @ A ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(setminusER_type,type,
% 2.41/1.40 setminusER: $o ).
% 2.41/1.40
% 2.41/1.40 thf(setminusER,definition,
% 2.41/1.40 ( setminusER
% 2.41/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.41/1.40 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.41/1.40 => ~ ( in @ Xx @ B ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(setminusSubset2_type,type,
% 2.41/1.40 setminusSubset2: $o ).
% 2.41/1.40
% 2.41/1.40 thf(setminusSubset2,definition,
% 2.41/1.40 ( setminusSubset2
% 2.41/1.40 = ( ! [A: $i,B: $i] :
% 2.41/1.40 ( ( subset @ A @ B )
% 2.41/1.40 => ( ( setminus @ A @ B )
% 2.41/1.40 = emptyset ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(setminusERneg_type,type,
% 2.41/1.40 setminusERneg: $o ).
% 2.41/1.40
% 2.41/1.40 thf(setminusERneg,definition,
% 2.41/1.40 ( setminusERneg
% 2.41/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.41/1.40 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.41/1.40 => ( ( in @ Xx @ A )
% 2.41/1.40 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(setminusELneg_type,type,
% 2.41/1.40 setminusELneg: $o ).
% 2.41/1.40
% 2.41/1.40 thf(setminusELneg,definition,
% 2.41/1.40 ( setminusELneg
% 2.41/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.41/1.40 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.41/1.40 => ( ~ ( in @ Xx @ B )
% 2.41/1.40 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(setminusILneg_type,type,
% 2.41/1.40 setminusILneg: $o ).
% 2.41/1.40
% 2.41/1.40 thf(setminusILneg,definition,
% 2.41/1.40 ( setminusILneg
% 2.41/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.41/1.40 ( ~ ( in @ Xx @ A )
% 2.41/1.40 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(setminusIRneg_type,type,
% 2.41/1.40 setminusIRneg: $o ).
% 2.41/1.40
% 2.41/1.40 thf(setminusIRneg,definition,
% 2.41/1.40 ( setminusIRneg
% 2.41/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.41/1.40 ( ( in @ Xx @ B )
% 2.41/1.40 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(setminusLsub_type,type,
% 2.41/1.40 setminusLsub: $o ).
% 2.41/1.40
% 2.41/1.40 thf(setminusLsub,definition,
% 2.41/1.40 ( setminusLsub
% 2.41/1.40 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(setminusSubset1_type,type,
% 2.41/1.40 setminusSubset1: $o ).
% 2.41/1.40
% 2.41/1.40 thf(setminusSubset1,definition,
% 2.41/1.40 ( setminusSubset1
% 2.41/1.40 = ( ! [A: $i,B: $i] :
% 2.41/1.40 ( ( ( setminus @ A @ B )
% 2.41/1.40 = emptyset )
% 2.41/1.40 => ( subset @ A @ B ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(symdiff_type,type,
% 2.41/1.40 symdiff: $i > $i > $i ).
% 2.41/1.40
% 2.41/1.40 thf(symdiffE_type,type,
% 2.41/1.40 symdiffE: $o ).
% 2.41/1.40
% 2.41/1.40 thf(symdiffE,definition,
% 2.41/1.40 ( symdiffE
% 2.41/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.41/1.40 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 2.41/1.40 => ! [Xphi: $o] :
% 2.41/1.40 ( ( ( in @ Xx @ A )
% 2.41/1.40 => ( ~ ( in @ Xx @ B )
% 2.41/1.40 => Xphi ) )
% 2.41/1.40 => ( ( ~ ( in @ Xx @ A )
% 2.41/1.40 => ( ( in @ Xx @ B )
% 2.41/1.40 => Xphi ) )
% 2.41/1.40 => Xphi ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(symdiffI1_type,type,
% 2.41/1.40 symdiffI1: $o ).
% 2.41/1.40
% 2.41/1.40 thf(symdiffI1,definition,
% 2.41/1.40 ( symdiffI1
% 2.41/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.41/1.40 ( ( in @ Xx @ A )
% 2.41/1.40 => ( ~ ( in @ Xx @ B )
% 2.41/1.40 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(symdiffI2_type,type,
% 2.41/1.40 symdiffI2: $o ).
% 2.41/1.40
% 2.41/1.40 thf(symdiffI2,definition,
% 2.41/1.40 ( symdiffI2
% 2.41/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.41/1.40 ( ~ ( in @ Xx @ A )
% 2.41/1.40 => ( ( in @ Xx @ B )
% 2.41/1.40 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(symdiffIneg1_type,type,
% 2.41/1.40 symdiffIneg1: $o ).
% 2.41/1.40
% 2.41/1.40 thf(symdiffIneg1,definition,
% 2.41/1.40 ( symdiffIneg1
% 2.41/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.41/1.40 ( ( in @ Xx @ A )
% 2.41/1.40 => ( ( in @ Xx @ B )
% 2.41/1.40 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(symdiffIneg2_type,type,
% 2.41/1.40 symdiffIneg2: $o ).
% 2.41/1.40
% 2.41/1.40 thf(symdiffIneg2,definition,
% 2.41/1.40 ( symdiffIneg2
% 2.41/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.41/1.40 ( ~ ( in @ Xx @ A )
% 2.41/1.40 => ( ~ ( in @ Xx @ B )
% 2.41/1.40 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(iskpair_type,type,
% 2.41/1.40 iskpair: $i > $o ).
% 2.41/1.40
% 2.41/1.40 thf(iskpair,definition,
% 2.41/1.40 ( iskpair
% 2.41/1.40 = ( ^ [A: $i] :
% 2.41/1.40 ? [Xx: $i] :
% 2.41/1.40 ( ( in @ Xx @ ( setunion @ A ) )
% 2.41/1.40 & ? [Xy: $i] :
% 2.41/1.40 ( ( in @ Xy @ ( setunion @ A ) )
% 2.41/1.40 & ( A
% 2.41/1.40 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(secondinupair_type,type,
% 2.41/1.40 secondinupair: $o ).
% 2.41/1.40
% 2.41/1.40 thf(secondinupair,definition,
% 2.41/1.40 ( secondinupair
% 2.41/1.40 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(setukpairIL_type,type,
% 2.41/1.40 setukpairIL: $o ).
% 2.41/1.40
% 2.41/1.40 thf(setukpairIL,definition,
% 2.41/1.40 ( setukpairIL
% 2.41/1.40 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(setukpairIR_type,type,
% 2.41/1.40 setukpairIR: $o ).
% 2.41/1.40
% 2.41/1.40 thf(setukpairIR,definition,
% 2.41/1.40 ( setukpairIR
% 2.41/1.40 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(kpairiskpair_type,type,
% 2.41/1.40 kpairiskpair: $o ).
% 2.41/1.40
% 2.41/1.40 thf(kpairiskpair,definition,
% 2.41/1.40 ( kpairiskpair
% 2.41/1.40 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(kpair_type,type,
% 2.41/1.40 kpair: $i > $i > $i ).
% 2.41/1.40
% 2.41/1.40 thf(kpair,definition,
% 2.41/1.40 ( kpair
% 2.41/1.40 = ( ^ [Xx: $i,Xy: $i] : ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(kpairp_type,type,
% 2.41/1.40 kpairp: $o ).
% 2.41/1.40
% 2.41/1.40 thf(kpairp,definition,
% 2.41/1.40 ( kpairp
% 2.41/1.40 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(cartprod_type,type,
% 2.41/1.40 cartprod: $i > $i > $i ).
% 2.41/1.40
% 2.41/1.40 thf(cartprod,definition,
% 2.41/1.40 ( cartprod
% 2.41/1.40 = ( ^ [A: $i,B: $i] :
% 2.41/1.40 ( dsetconstr @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) )
% 2.41/1.40 @ ^ [Xx: $i] :
% 2.41/1.40 ? [Xy: $i] :
% 2.41/1.40 ( ( in @ Xy @ A )
% 2.41/1.40 & ? [Xz: $i] :
% 2.41/1.40 ( ( in @ Xz @ B )
% 2.41/1.40 & ( Xx
% 2.41/1.40 = ( kpair @ Xy @ Xz ) ) ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(singletonsubset_type,type,
% 2.41/1.40 singletonsubset: $o ).
% 2.41/1.40
% 2.41/1.40 thf(singletonsubset,definition,
% 2.41/1.40 ( singletonsubset
% 2.41/1.40 = ( ! [A: $i,Xx: $i] :
% 2.41/1.40 ( ( in @ Xx @ A )
% 2.41/1.40 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(singletoninpowerset_type,type,
% 2.41/1.40 singletoninpowerset: $o ).
% 2.41/1.40
% 2.41/1.40 thf(singletoninpowerset,definition,
% 2.41/1.40 ( singletoninpowerset
% 2.41/1.40 = ( ! [A: $i,Xx: $i] :
% 2.41/1.40 ( ( in @ Xx @ A )
% 2.41/1.40 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(singletoninpowunion_type,type,
% 2.41/1.40 singletoninpowunion: $o ).
% 2.41/1.40
% 2.41/1.40 thf(singletoninpowunion,definition,
% 2.41/1.40 ( singletoninpowunion
% 2.41/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.41/1.40 ( ( in @ Xx @ A )
% 2.41/1.40 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(upairset2E_type,type,
% 2.41/1.40 upairset2E: $o ).
% 2.41/1.40
% 2.41/1.40 thf(upairset2E,definition,
% 2.41/1.40 ( upairset2E
% 2.41/1.40 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.41/1.40 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.41/1.40 => ( ( Xz = Xx )
% 2.41/1.40 | ( Xz = Xy ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(upairsubunion_type,type,
% 2.41/1.40 upairsubunion: $o ).
% 2.41/1.40
% 2.41/1.40 thf(upairsubunion,definition,
% 2.41/1.40 ( upairsubunion
% 2.41/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.41/1.40 ( ( in @ Xx @ A )
% 2.41/1.40 => ! [Xy: $i] :
% 2.41/1.40 ( ( in @ Xy @ B )
% 2.41/1.40 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.41/1.40
% 2.41/1.40 thf(upairinpowunion,conjecture,
% 2.41/1.40 ( setextAx
% 2.41/1.40 => ( emptysetAx
% 2.41/1.40 => ( setadjoinAx
% 2.41/1.40 => ( powersetAx
% 2.41/1.40 => ( setunionAx
% 2.41/1.40 => ( omega0Ax
% 2.41/1.40 => ( omegaSAx
% 2.41/1.40 => ( omegaIndAx
% 2.41/1.40 => ( replAx
% 2.41/1.40 => ( foundationAx
% 2.41/1.40 => ( wellorderingAx
% 2.41/1.40 => ( descrp
% 2.41/1.40 => ( dsetconstrI
% 2.41/1.40 => ( dsetconstrEL
% 2.41/1.40 => ( dsetconstrER
% 2.41/1.40 => ( exuE1
% 2.41/1.40 => ( prop2setE
% 2.41/1.40 => ( emptysetE
% 2.41/1.40 => ( emptysetimpfalse
% 2.41/1.40 => ( notinemptyset
% 2.41/1.40 => ( exuE3e
% 2.41/1.40 => ( setext
% 2.41/1.40 => ( emptyI
% 2.41/1.40 => ( noeltsimpempty
% 2.41/1.40 => ( setbeta
% 2.41/1.40 => ( nonemptyE1
% 2.41/1.40 => ( nonemptyI
% 2.41/1.40 => ( nonemptyI1
% 2.41/1.40 => ( setadjoinIL
% 2.41/1.40 => ( emptyinunitempty
% 2.41/1.40 => ( setadjoinIR
% 2.41/1.40 => ( setadjoinE
% 2.41/1.40 => ( setadjoinOr
% 2.41/1.40 => ( setoftrueEq
% 2.41/1.40 => ( powersetI
% 2.41/1.40 => ( emptyinPowerset
% 2.41/1.40 => ( emptyInPowerset
% 2.41/1.40 => ( powersetE
% 2.41/1.40 => ( setunionI
% 2.41/1.40 => ( setunionE
% 2.41/1.40 => ( subPowSU
% 2.41/1.40 => ( exuE2
% 2.41/1.40 => ( nonemptyImpWitness
% 2.41/1.40 => ( uniqinunit
% 2.41/1.40 => ( notinsingleton
% 2.41/1.40 => ( eqinunit
% 2.41/1.40 => ( singletonsswitch
% 2.41/1.40 => ( upairsetE
% 2.41/1.40 => ( upairsetIL
% 2.41/1.40 => ( upairsetIR
% 2.41/1.40 => ( emptyE1
% 2.41/1.40 => ( vacuousDall
% 2.41/1.40 => ( quantDeMorgan1
% 2.41/1.40 => ( quantDeMorgan2
% 2.41/1.40 => ( quantDeMorgan3
% 2.41/1.40 => ( quantDeMorgan4
% 2.41/1.40 => ( prop2setI
% 2.41/1.40 => ( prop2set2propI
% 2.41/1.40 => ( notdexE
% 2.41/1.40 => ( notdallE
% 2.41/1.40 => ( exuI1
% 2.41/1.40 => ( exuI3
% 2.41/1.40 => ( exuI2
% 2.41/1.40 => ( inCongP
% 2.41/1.40 => ( in__Cong
% 2.41/1.40 => ( exuE3u
% 2.41/1.40 => ( exu__Cong
% 2.41/1.40 => ( emptyset__Cong
% 2.41/1.40 => ( setadjoin__Cong
% 2.41/1.40 => ( powerset__Cong
% 2.41/1.40 => ( setunion__Cong
% 2.41/1.40 => ( omega__Cong
% 2.41/1.40 => ( exuEu
% 2.41/1.40 => ( descr__Cong
% 2.41/1.40 => ( dsetconstr__Cong
% 2.41/1.40 => ( subsetI1
% 2.41/1.40 => ( eqimpsubset2
% 2.41/1.40 => ( eqimpsubset1
% 2.41/1.40 => ( subsetI2
% 2.41/1.40 => ( emptysetsubset
% 2.41/1.40 => ( subsetE
% 2.41/1.40 => ( subsetE2
% 2.41/1.40 => ( notsubsetI
% 2.41/1.40 => ( notequalI1
% 2.41/1.40 => ( notequalI2
% 2.41/1.40 => ( subsetRefl
% 2.41/1.40 => ( subsetTrans
% 2.41/1.40 => ( setadjoinSub
% 2.41/1.40 => ( setadjoinSub2
% 2.41/1.40 => ( subset2powerset
% 2.41/1.40 => ( setextsub
% 2.41/1.40 => ( subsetemptysetimpeq
% 2.41/1.40 => ( powersetI1
% 2.41/1.40 => ( powersetE1
% 2.41/1.40 => ( inPowerset
% 2.41/1.40 => ( powersetsubset
% 2.41/1.40 => ( sepInPowerset
% 2.41/1.40 => ( sepSubset
% 2.41/1.40 => ( binunionIL
% 2.41/1.40 => ( upairset2IR
% 2.41/1.40 => ( binunionIR
% 2.41/1.40 => ( binunionEcases
% 2.41/1.40 => ( binunionE
% 2.41/1.40 => ( binunionLsub
% 2.41/1.40 => ( binunionRsub
% 2.41/1.40 => ( binintersectI
% 2.41/1.40 => ( binintersectSubset5
% 2.41/1.40 => ( binintersectEL
% 2.41/1.40 => ( binintersectLsub
% 2.41/1.40 => ( binintersectSubset2
% 2.41/1.40 => ( binintersectSubset3
% 2.41/1.40 => ( binintersectER
% 2.41/1.40 => ( disjointsetsI1
% 2.41/1.40 => ( binintersectRsub
% 2.41/1.40 => ( binintersectSubset4
% 2.41/1.40 => ( binintersectSubset1
% 2.41/1.40 => ( bs114d
% 2.41/1.40 => ( setminusI
% 2.41/1.40 => ( setminusEL
% 2.41/1.40 => ( setminusER
% 2.41/1.40 => ( setminusSubset2
% 2.41/1.40 => ( setminusERneg
% 2.41/1.40 => ( setminusELneg
% 2.41/1.40 => ( setminusILneg
% 2.41/1.40 => ( setminusIRneg
% 2.41/1.40 => ( setminusLsub
% 2.41/1.40 => ( setminusSubset1
% 2.41/1.40 => ( symdiffE
% 2.41/1.40 => ( symdiffI1
% 2.41/1.40 => ( symdiffI2
% 2.41/1.40 => ( symdiffIneg1
% 2.41/1.40 => ( symdiffIneg2
% 2.41/1.40 => ( secondinupair
% 2.41/1.40 => ( setukpairIL
% 2.41/1.40 => ( setukpairIR
% 2.41/1.40 => ( kpairiskpair
% 2.41/1.40 => ( kpairp
% 2.41/1.40 => ( singletonsubset
% 2.41/1.40 => ( singletoninpowerset
% 2.41/1.40 => ( singletoninpowunion
% 2.41/1.40 => ( upairset2E
% 2.41/1.40 => ( upairsubunion
% 2.41/1.40 => ! [A: $i,B: $i,Xx: $i] :
% 2.41/1.40 ( ( in @ Xx @ A )
% 2.41/1.40 => ! [Xy: $i] :
% 2.41/1.40 ( ( in @ Xy @ B )
% 2.41/1.40 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.41/1.40 ------------------------
% 2.41/1.42 % (22463)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_22176 for (2999ds/4Mi)
% 2.41/1.42 % (22462)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_22176 for (2999ds/183Mi)
% 2.41/1.42 % (22463)Instruction limit reached!
% 2.41/1.42 % (22463)------------------------------
% 2.41/1.42 % (22463)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.41/1.42 % (22463)Termination reason: Unknown
% 2.41/1.42 % (22463)Termination phase: shuffling
% 2.41/1.42
% 2.41/1.42 % (22463)Memory used [KB]: 1279
% 2.41/1.42 % (22463)Time elapsed: 0.003 s
% 2.41/1.42 % (22463)Instructions burned: 5 (million)
% 2.41/1.42 % (22463)------------------------------
% 2.41/1.42 % (22463)------------------------------
% 2.41/1.43 % (22464)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_22176 for (2999ds/27Mi)
% 2.41/1.43 % (22465)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_22176 for (2999ds/2Mi)
% 2.41/1.43 % (22466)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_22176 for (2999ds/2Mi)
% 2.41/1.43 % (22467)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_22176 for (2999ds/275Mi)
% 2.41/1.43 % (22468)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_22176 for (2999ds/18Mi)
% 2.41/1.43 % (22465)Instruction limit reached!
% 2.41/1.43 % (22465)------------------------------
% 2.41/1.43 % (22465)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.41/1.43 % (22465)Termination reason: Unknown
% 2.41/1.43 % (22465)Termination phase: shuffling
% 2.41/1.43
% 2.41/1.43 % (22465)Memory used [KB]: 1279
% 2.41/1.43 % (22465)Time elapsed: 0.004 s
% 2.41/1.43 % (22465)Instructions burned: 3 (million)
% 2.41/1.43 % (22465)------------------------------
% 2.41/1.43 % (22465)------------------------------
% 2.41/1.43 % (22466)Instruction limit reached!
% 2.41/1.43 % (22466)------------------------------
% 2.41/1.43 % (22466)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.41/1.43 % (22466)Termination reason: Unknown
% 2.41/1.43 % (22466)Termination phase: shuffling
% 2.41/1.43
% 2.41/1.43 % (22466)Memory used [KB]: 1279
% 2.41/1.43 % (22466)Time elapsed: 0.004 s
% 2.41/1.43 % (22466)Instructions burned: 3 (million)
% 2.41/1.43 % (22466)------------------------------
% 2.41/1.43 % (22466)------------------------------
% 2.41/1.43 % (22469)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_22176 for (2999ds/3Mi)
% 2.41/1.43 % (22469)Instruction limit reached!
% 2.41/1.43 % (22469)------------------------------
% 2.41/1.43 % (22469)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.41/1.43 % (22469)Termination reason: Unknown
% 2.41/1.43 % (22469)Termination phase: shuffling
% 2.41/1.43
% 2.41/1.43 % (22469)Memory used [KB]: 1279
% 2.41/1.43 % (22469)Time elapsed: 0.002 s
% 2.41/1.43 % (22469)Instructions burned: 3 (million)
% 2.41/1.43 % (22469)------------------------------
% 2.41/1.43 % (22469)------------------------------
% 2.41/1.44 % (22468)Instruction limit reached!
% 2.41/1.44 % (22468)------------------------------
% 2.41/1.44 % (22468)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.41/1.44 % (22468)Termination reason: Unknown
% 2.41/1.44 % (22468)Termination phase: Preprocessing 1
% 2.41/1.44
% 2.41/1.44 % (22468)Memory used [KB]: 1535
% 2.41/1.44 % (22468)Time elapsed: 0.013 s
% 2.41/1.44 % (22468)Instructions burned: 19 (million)
% 2.41/1.44 % (22468)------------------------------
% 2.41/1.44 % (22468)------------------------------
% 2.41/1.44 % (22472)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_22176 for (2999ds/3Mi)
% 2.41/1.44 % (22464)Instruction limit reached!
% 2.41/1.44 % (22464)------------------------------
% 2.41/1.44 % (22464)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.41/1.44 % (22464)Termination reason: Unknown
% 2.41/1.44 % (22464)Termination phase: Property scanning
% 2.41/1.44
% 2.41/1.44 % (22464)Memory used [KB]: 1791
% 2.41/1.44 % (22464)Time elapsed: 0.016 s
% 2.41/1.44 % (22464)Instructions burned: 27 (million)
% 2.41/1.44 % (22464)------------------------------
% 2.41/1.44 % (22464)------------------------------
% 2.41/1.44 % (22472)Instruction limit reached!
% 2.41/1.44 % (22472)------------------------------
% 2.41/1.44 % (22472)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.41/1.44 % (22472)Termination reason: Unknown
% 2.41/1.44 % (22472)Termination phase: shuffling
% 2.41/1.44
% 2.41/1.44 % (22472)Memory used [KB]: 1279
% 2.41/1.44 % (22472)Time elapsed: 0.002 s
% 2.41/1.44 % (22472)Instructions burned: 4 (million)
% 2.41/1.44 % (22472)------------------------------
% 2.41/1.44 % (22472)------------------------------
% 2.41/1.45 % (22470)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_22176 for (2999ds/37Mi)
% 2.41/1.45 % (22471)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_22176 for (2999ds/15Mi)
% 2.41/1.45 % (22475)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_22176 for (2999ds/16Mi)
% 2.41/1.46 % (22471)Instruction limit reached!
% 2.41/1.46 % (22471)------------------------------
% 2.41/1.46 % (22471)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.41/1.46 % (22471)Termination reason: Unknown
% 2.41/1.46 % (22471)Termination phase: Property scanning
% 2.41/1.46
% 2.41/1.46 % (22471)Memory used [KB]: 1535
% 2.41/1.46 % (22471)Time elapsed: 0.011 s
% 2.41/1.46 % (22471)Instructions burned: 15 (million)
% 2.41/1.46 % (22471)------------------------------
% 2.41/1.46 % (22471)------------------------------
% 2.41/1.46 % (22475)Instruction limit reached!
% 2.41/1.46 % (22475)------------------------------
% 2.41/1.46 % (22475)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.41/1.46 % (22475)Termination reason: Unknown
% 2.41/1.46 % (22475)Termination phase: shuffling
% 2.41/1.46
% 2.41/1.46 % (22475)Memory used [KB]: 1663
% 2.41/1.46 % (22475)Time elapsed: 0.006 s
% 2.41/1.46 % (22475)Instructions burned: 19 (million)
% 2.41/1.46 % (22475)------------------------------
% 2.41/1.46 % (22475)------------------------------
% 2.41/1.46 % (22473)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_22176 for (2999ds/1041Mi)
% 2.41/1.46 % (22474)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_22176 for (2999ds/7Mi)
% 2.41/1.46 % (22474)Instruction limit reached!
% 2.41/1.46 % (22474)------------------------------
% 2.41/1.46 % (22474)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.41/1.46 % (22474)Termination reason: Unknown
% 2.41/1.46 % (22474)Termination phase: shuffling
% 2.41/1.46
% 2.41/1.46 % (22474)Memory used [KB]: 1407
% 2.41/1.46 % (22474)Time elapsed: 0.006 s
% 2.41/1.46 % (22474)Instructions burned: 8 (million)
% 2.41/1.46 % (22474)------------------------------
% 2.41/1.46 % (22474)------------------------------
% 2.41/1.47 % (22470)Instruction limit reached!
% 2.41/1.47 % (22470)------------------------------
% 2.41/1.47 % (22470)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.41/1.47 % (22470)Termination reason: Unknown
% 2.41/1.47 % (22470)Termination phase: Property scanning
% 2.41/1.47
% 2.41/1.47 % (22470)Memory used [KB]: 1663
% 2.41/1.47 % (22470)Time elapsed: 0.020 s
% 2.41/1.47 % (22470)Instructions burned: 37 (million)
% 2.41/1.47 % (22470)------------------------------
% 2.41/1.47 % (22470)------------------------------
% 2.41/1.47 % (22477)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_22176 for (2999ds/3Mi)
% 2.41/1.47 % (22477)Instruction limit reached!
% 2.41/1.47 % (22477)------------------------------
% 2.41/1.47 % (22477)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.41/1.47 % (22477)Termination reason: Unknown
% 2.41/1.47 % (22477)Termination phase: shuffling
% 2.41/1.47
% 2.41/1.47 % (22477)Memory used [KB]: 1279
% 2.41/1.47 % (22477)Time elapsed: 0.002 s
% 2.41/1.47 % (22477)Instructions burned: 3 (million)
% 2.41/1.47 % (22477)------------------------------
% 2.41/1.47 % (22477)------------------------------
% 2.41/1.47 % (22476)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_22176 for (2999ds/3Mi)
% 2.41/1.48 % (22462)Instruction limit reached!
% 2.41/1.48 % (22462)------------------------------
% 2.41/1.48 % (22462)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.41/1.48 % (22462)Termination reason: Unknown
% 2.41/1.48 % (22462)Termination phase: Saturation
% 2.41/1.48
% 2.41/1.48 % (22462)Memory used [KB]: 7164
% 2.41/1.48 % (22462)Time elapsed: 0.056 s
% 2.41/1.48 % (22462)Instructions burned: 183 (million)
% 2.41/1.48 % (22462)------------------------------
% 2.41/1.48 % (22462)------------------------------
% 2.41/1.48 % (22476)Instruction limit reached!
% 2.41/1.48 % (22476)------------------------------
% 2.41/1.48 % (22476)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.41/1.48 % (22476)Termination reason: Unknown
% 2.41/1.48 % (22476)Termination phase: shuffling
% 2.41/1.48
% 2.41/1.48 % (22476)Memory used [KB]: 1279
% 2.41/1.48 % (22476)Time elapsed: 0.004 s
% 2.41/1.48 % (22476)Instructions burned: 4 (million)
% 2.41/1.48 % (22476)------------------------------
% 2.41/1.48 % (22476)------------------------------
% 2.41/1.48 % (22480)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_22176 for (2999ds/4Mi)
% 2.41/1.48 % (22480)Instruction limit reached!
% 2.41/1.48 % (22480)------------------------------
% 2.41/1.48 % (22480)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.41/1.48 % (22480)Termination reason: Unknown
% 2.41/1.48 % (22480)Termination phase: shuffling
% 2.41/1.48
% 2.41/1.48 % (22480)Memory used [KB]: 1279
% 2.41/1.48 % (22480)Time elapsed: 0.002 s
% 2.41/1.48 % (22480)Instructions burned: 4 (million)
% 2.41/1.48 % (22480)------------------------------
% 2.41/1.48 % (22480)------------------------------
% 2.41/1.48 % (22478)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_22176 for (2999ds/7Mi)
% 2.41/1.48 % (22479)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_22176 for (2999ds/3Mi)
% 2.41/1.48 % (22479)Instruction limit reached!
% 2.41/1.48 % (22479)------------------------------
% 2.41/1.48 % (22479)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.41/1.48 % (22479)Termination reason: Unknown
% 2.41/1.48 % (22479)Termination phase: shuffling
% 2.41/1.48
% 2.41/1.48 % (22479)Memory used [KB]: 1279
% 2.41/1.48 % (22479)Time elapsed: 0.004 s
% 2.41/1.48 % (22479)Instructions burned: 6 (million)
% 2.41/1.48 % (22479)------------------------------
% 2.41/1.48 % (22479)------------------------------
% 2.41/1.49 % (22478)Instruction limit reached!
% 2.41/1.49 % (22478)------------------------------
% 2.41/1.49 % (22478)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.41/1.49 % (22478)Termination reason: Unknown
% 2.41/1.49 % (22478)Termination phase: shuffling
% 2.41/1.49
% 2.41/1.49 % (22478)Memory used [KB]: 1407
% 2.41/1.49 % (22478)Time elapsed: 0.006 s
% 2.41/1.49 % (22478)Instructions burned: 8 (million)
% 2.41/1.49 % (22478)------------------------------
% 2.41/1.49 % (22478)------------------------------
% 2.41/1.49 % (22483)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_22176 for (2999ds/6Mi)
% 2.41/1.49 % (22483)Instruction limit reached!
% 2.41/1.49 % (22483)------------------------------
% 2.41/1.49 % (22483)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.41/1.49 % (22483)Termination reason: Unknown
% 2.41/1.49 % (22483)Termination phase: shuffling
% 2.41/1.49
% 2.41/1.49 % (22483)Memory used [KB]: 1407
% 2.41/1.49 % (22483)Time elapsed: 0.003 s
% 2.41/1.49 % (22483)Instructions burned: 7 (million)
% 2.41/1.49 % (22483)------------------------------
% 2.41/1.49 % (22483)------------------------------
% 2.41/1.49 % (22481)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_22176 for (2999ds/18Mi)
% 2.41/1.49 % (22482)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_22176 for (2999ds/710Mi)
% 2.41/1.49 % (22484)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_22176 for (2999ds/902Mi)
% 2.41/1.50 % (22486)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_22176 for (2999ds/5Mi)
% 2.41/1.50 % (22486)Instruction limit reached!
% 2.41/1.50 % (22486)------------------------------
% 2.41/1.50 % (22486)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.41/1.50 % (22486)Termination reason: Unknown
% 2.41/1.50 % (22486)Termination phase: shuffling
% 2.41/1.50
% 2.41/1.50 % (22486)Memory used [KB]: 1407
% 2.41/1.50 % (22486)Time elapsed: 0.003 s
% 2.41/1.50 % (22486)Instructions burned: 7 (million)
% 2.41/1.50 % (22486)------------------------------
% 2.41/1.50 % (22486)------------------------------
% 2.41/1.50 % (22485)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_22176 for (2999ds/21Mi)
% 2.41/1.50 % (22481)Instruction limit reached!
% 2.41/1.50 % (22481)------------------------------
% 2.41/1.50 % (22481)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.41/1.50 % (22481)Termination reason: Unknown
% 2.41/1.50 % (22481)Termination phase: shuffling
% 2.41/1.50
% 2.41/1.50 % (22481)Memory used [KB]: 1663
% 2.41/1.50 % (22481)Time elapsed: 0.012 s
% 2.41/1.50 % (22481)Instructions burned: 19 (million)
% 2.41/1.50 % (22481)------------------------------
% 2.41/1.50 % (22481)------------------------------
% 2.41/1.51 % (22487)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_22176 for (2999ds/6Mi)
% 2.41/1.51 % (22487)Instruction limit reached!
% 2.41/1.51 % (22487)------------------------------
% 2.41/1.51 % (22487)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.41/1.51 % (22487)Termination reason: Unknown
% 2.41/1.51 % (22487)Termination phase: shuffling
% 2.41/1.51
% 2.41/1.51 % (22487)Memory used [KB]: 1407
% 2.41/1.51 % (22487)Time elapsed: 0.003 s
% 2.41/1.51 % (22487)Instructions burned: 7 (million)
% 2.41/1.51 % (22487)------------------------------
% 2.41/1.51 % (22487)------------------------------
% 2.41/1.51 % (22485)Instruction limit reached!
% 2.41/1.51 % (22485)------------------------------
% 2.41/1.51 % (22485)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.41/1.51 % (22485)Termination reason: Unknown
% 2.41/1.51 % (22485)Termination phase: shuffling
% 2.41/1.51
% 2.41/1.51 % (22485)Memory used [KB]: 1791
% 2.41/1.51 % (22485)Time elapsed: 0.013 s
% 2.41/1.51 % (22485)Instructions burned: 21 (million)
% 2.41/1.51 % (22485)------------------------------
% 2.41/1.51 % (22485)------------------------------
% 2.41/1.52 % (22489)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_22176 for (2999ds/779Mi)
% 2.41/1.52 % (22488)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_22176 for (2999ds/377Mi)
% 2.41/1.53 % (22490)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_22176 for (2998ds/19Mi)
% 2.41/1.53 % (22490)Instruction limit reached!
% 2.41/1.53 % (22490)------------------------------
% 2.41/1.53 % (22490)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.41/1.53 % (22490)Termination reason: Unknown
% 2.41/1.53 % (22490)Termination phase: shuffling
% 2.41/1.53
% 2.41/1.53 % (22490)Memory used [KB]: 1663
% 2.41/1.53 % (22490)Time elapsed: 0.008 s
% 2.41/1.53 % (22490)Instructions burned: 21 (million)
% 2.41/1.53 % (22490)------------------------------
% 2.41/1.53 % (22490)------------------------------
% 2.41/1.54 % (22467)Instruction limit reached!
% 2.41/1.54 % (22467)------------------------------
% 2.41/1.54 % (22467)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.41/1.54 % (22467)Termination reason: Unknown
% 2.41/1.54 % (22467)Termination phase: Saturation
% 2.41/1.54
% 2.41/1.54 % (22467)Memory used [KB]: 7931
% 2.41/1.54 % (22467)Time elapsed: 0.115 s
% 2.41/1.54 % (22467)Instructions burned: 275 (million)
% 2.41/1.54 % (22467)------------------------------
% 2.41/1.54 % (22467)------------------------------
% 2.41/1.54 % (22491)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_22176 for (2998ds/879Mi)
% 2.41/1.55 % (22492)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_22176 for (2998ds/17Mi)
% 2.41/1.56 % (22492)Instruction limit reached!
% 2.41/1.56 % (22492)------------------------------
% 2.41/1.56 % (22492)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.41/1.56 % (22492)Termination reason: Unknown
% 2.41/1.56 % (22492)Termination phase: shuffling
% 2.41/1.56
% 2.41/1.56 % (22492)Memory used [KB]: 1535
% 2.41/1.56 % (22492)Time elapsed: 0.006 s
% 2.41/1.56 % (22492)Instructions burned: 18 (million)
% 2.41/1.56 % (22492)------------------------------
% 2.41/1.56 % (22492)------------------------------
% 2.41/1.56 % (22484)First to succeed.
% 2.41/1.56 % (22484)Refutation found. Thanks to Tanya!
% 2.41/1.56 % SZS status Theorem for DTF2THF_22176
% 2.41/1.56 % SZS output start Proof for DTF2THF_22176
% 2.41/1.56 thf(func_def_0, type, in: $i > $i > $o).
% 2.41/1.56 thf(func_def_1, type, exu: ($i > $o) > $o).
% 2.41/1.56 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 2.41/1.56 thf(func_def_8, type, powerset: $i > $i).
% 2.41/1.56 thf(func_def_10, type, setunion: $i > $i).
% 2.41/1.56 thf(func_def_19, type, descr: ($i > $o) > $i).
% 2.41/1.56 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 2.41/1.56 thf(func_def_26, type, prop2set: $o > $i).
% 2.41/1.56 thf(func_def_36, type, nonempty: $i > $o).
% 2.41/1.56 thf(func_def_69, type, set2prop: $i > $o).
% 2.41/1.56 thf(func_def_88, type, subset: $i > $i > $o).
% 2.41/1.56 thf(func_def_89, type, disjoint: $i > $i > $o).
% 2.41/1.56 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 2.41/1.56 thf(func_def_114, type, binunion: $i > $i > $i).
% 2.41/1.56 thf(func_def_122, type, binintersect: $i > $i > $i).
% 2.41/1.56 thf(func_def_135, type, regular: $i > $o).
% 2.41/1.56 thf(func_def_136, type, setminus: $i > $i > $i).
% 2.41/1.56 thf(func_def_147, type, symdiff: $i > $i > $i).
% 2.41/1.56 thf(func_def_153, type, iskpair: $i > $o).
% 2.41/1.56 thf(func_def_158, type, kpair: $i > $i > $i).
% 2.41/1.56 thf(func_def_160, type, cartprod: $i > $i > $i).
% 2.41/1.56 thf(func_def_180, type, sP1: $i > $i > $i > $o > $o).
% 2.41/1.56 thf(func_def_181, type, sP2: $i > $i > $o).
% 2.41/1.56 thf(func_def_182, type, sP3: $i > $o).
% 2.41/1.56 thf(func_def_183, type, sP4: $i > $i > $o).
% 2.41/1.56 thf(func_def_184, type, sP5: $i > $i > $o).
% 2.41/1.56 thf(func_def_192, type, sK13: $i > $o).
% 2.41/1.56 thf(func_def_220, type, sK41: $i > $o).
% 2.41/1.56 thf(func_def_228, type, sK49: $i > $i).
% 2.41/1.56 thf(func_def_242, type, sK63: $i > ($i > $o) > $i).
% 2.41/1.56 thf(func_def_243, type, sK64: $i > $o).
% 2.41/1.56 thf(func_def_248, type, sK69: $i > $o).
% 2.41/1.56 thf(func_def_250, type, sK71: ($i > $o) > $i).
% 2.41/1.56 thf(func_def_251, type, sK72: ($i > $o) > $i).
% 2.41/1.56 thf(func_def_254, type, sK75: $i > $i > $i).
% 2.41/1.56 thf(func_def_258, type, sK79: $i > $o).
% 2.41/1.56 thf(func_def_260, type, sK81: $i > $i).
% 2.41/1.56 thf(func_def_262, type, sK83: $i > $i).
% 2.41/1.56 thf(func_def_264, type, sK85: $i > $i > $o).
% 2.41/1.56 thf(func_def_266, type, sK87: $i > $i).
% 2.41/1.56 thf(func_def_267, type, sK88: $i > $i).
% 2.41/1.56 thf(func_def_268, type, sK89: $i > ($i > $i > $o) > $i).
% 2.41/1.56 thf(func_def_269, type, sK90: $i > ($i > $i > $o) > $i).
% 2.41/1.56 thf(func_def_270, type, sK91: $i > $i > ($i > $i > $o) > $i).
% 2.41/1.56 thf(func_def_273, type, sK94: $i > $i > $i).
% 2.41/1.56 thf(func_def_274, type, sK95: $i > $i > $i).
% 2.41/1.56 thf(func_def_292, type, sK113: $i > $i > $i).
% 2.41/1.56 thf(func_def_293, type, sK114: $i > $i > $i).
% 2.41/1.56 thf(func_def_294, type, sK115: $i > $i > $i).
% 2.41/1.56 thf(func_def_295, type, sK116: $i > $i > $i).
% 2.41/1.56 thf(func_def_296, type, sK117: $i > $i > $i).
% 2.41/1.56 thf(func_def_297, type, sK118: $i > $i > $i > $i).
% 2.41/1.56 thf(func_def_298, type, sK119: $i > $i).
% 2.41/1.56 thf(func_def_299, type, sK120: $i > $i).
% 2.41/1.56 thf(func_def_300, type, sK121: $i > $i).
% 2.41/1.56 thf(func_def_301, type, sK122: $i > $i).
% 2.41/1.56 thf(func_def_302, type, sK123: $i > $i > $i > $i).
% 2.41/1.56 thf(func_def_303, type, sK124: $i > $i > $i > $i).
% 2.41/1.56 thf(func_def_304, type, sK125: $i > $i > $i).
% 2.41/1.56 thf(func_def_305, type, sK126: $i > $i > $i).
% 2.41/1.56 thf(func_def_306, type, sK127: $i > $i > $i).
% 2.41/1.56 thf(func_def_308, type, sK129: $i > $i).
% 2.41/1.56 thf(func_def_309, type, sK130: $i > $i).
% 2.41/1.56 thf(func_def_310, type, sK131: $i > $i).
% 2.41/1.56 thf(func_def_317, type, sK138: $i > $i).
% 2.41/1.56 thf(func_def_341, type, sK162: $i > $i > $i).
% 2.41/1.56 thf(func_def_361, type, sK182: $i > $o).
% 2.41/1.56 thf(func_def_363, type, sK184: ($i > $o) > $i > $i).
% 2.41/1.56 thf(func_def_364, type, sK185: ($i > $o) > $i).
% 2.41/1.56 thf(func_def_365, type, sK186: $i > $o).
% 2.41/1.56 thf(func_def_368, type, sK189: $i > $o).
% 2.41/1.56 thf(func_def_372, type, sK193: $i > $o).
% 2.41/1.56 thf(func_def_377, type, sK198: $i > $i > $i).
% 2.41/1.56 thf(func_def_384, type, sK205: $i > $i > $i).
% 2.41/1.56 thf(func_def_385, type, sK206: $i > $o).
% 2.41/1.56 thf(func_def_387, type, sK208: $i > $o).
% 2.41/1.56 thf(func_def_397, type, sK218: $i > $i).
% 2.41/1.56 thf(func_def_399, type, sK220: $i > $o).
% 2.41/1.56 thf(func_def_400, type, sK221: $i > $o).
% 2.41/1.56 thf(func_def_401, type, sK222: ($i > $o) > ($i > $o) > $i).
% 2.41/1.56 thf(func_def_402, type, sK223: ($i > $o) > ($i > $o) > $i).
% 2.41/1.56 thf(func_def_409, type, sK230: $i > $i > $i).
% 2.41/1.56 thf(func_def_419, type, sK240: ($i > $o) > $i).
% 2.41/1.56 thf(func_def_420, type, sK241: $i > $o).
% 2.41/1.56 thf(func_def_421, type, sK242: $i > $i).
% 2.41/1.56 thf(func_def_422, type, sK243: $i > $o).
% 2.41/1.56 thf(func_def_428, type, sK249: $i > ($i > $o) > $i).
% 2.41/1.56 thf(func_def_429, type, sK250: $i > $o).
% 2.41/1.56 thf(func_def_434, type, sK255: $i > $o).
% 2.41/1.56 thf(func_def_435, type, sK256: $i > $o).
% 2.41/1.56 thf(func_def_436, type, sK257: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.41/1.56 thf(func_def_437, type, sK258: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.41/1.56 thf(func_def_441, type, sK262: ($i > $o) > $i > $i).
% 2.41/1.56 thf(func_def_443, type, sK264: $i > $o).
% 2.41/1.56 thf(func_def_459, type, sK280: ($i > $o) > $i > $i).
% 2.41/1.56 thf(func_def_461, type, sK282: $i > $o).
% 2.41/1.56 thf(func_def_464, type, sK285: $i > $o).
% 2.41/1.56 thf(func_def_470, type, sK291: $i > $i).
% 2.41/1.56 thf(func_def_473, type, sK294: $i > $o).
% 2.41/1.56 thf(func_def_479, type, sK300: $i > $i > $i).
% 2.41/1.56 thf(func_def_487, type, sK308: $i > $o).
% 2.41/1.56 thf(func_def_489, type, sK310: $i > ($i > $o) > $i).
% 2.41/1.56 thf(func_def_491, type, sK312: ($i > $o) > ($i > $o) > $i).
% 2.41/1.56 thf(func_def_492, type, sK313: ($i > $o) > ($i > $o) > $i).
% 2.41/1.56 thf(func_def_493, type, sK314: $i > $o).
% 2.41/1.56 thf(func_def_494, type, sK315: $i > $o).
% 2.41/1.56 thf(func_def_503, type, sK324: $i > $o).
% 2.41/1.56 thf(func_def_506, type, sK327: $i > $o).
% 2.41/1.56 thf(func_def_507, type, sK328: $i > $i).
% 2.41/1.56 thf(func_def_508, type, sK329: ($i > $o) > $i).
% 2.41/1.56 thf(func_def_519, type, sK340: $i > ($i > $o) > $i).
% 2.41/1.56 thf(func_def_520, type, sK341: $i > $o).
% 2.41/1.56 thf(func_def_526, type, sK347: $i > $o).
% 2.41/1.56 thf(func_def_528, type, sK349: $i > ($i > $o) > $i).
% 2.41/1.56 thf(func_def_529, type, sK350: $i > $o).
% 2.41/1.56 thf(func_def_536, type, sK357: $i > $i > $i).
% 2.41/1.56 thf(func_def_541, type, sK362: $i > $i).
% 2.41/1.56 thf(func_def_545, type, sK366: $o > $i > $i > $i).
% 2.41/1.56 thf(func_def_571, type, ph392: !>[X0: $tType]:(X0)).
% 2.41/1.56 thf(f2589,plain,(
% 2.41/1.56 $false),
% 2.41/1.56 inference(trivial_inequality_removal,[status(thm)],[f2584])).
% 2.41/1.56 thf(f2584,plain,(
% 2.41/1.56 ($true != $true)),
% 2.41/1.56 inference(superposition,[status(thm)],[f2120,f1623])).
% 2.41/1.56 thf(f1623,plain,(
% 2.41/1.56 ((in @ sK166 @ sK165) = $true)),
% 2.41/1.56 inference(cnf_transformation,[status(thm)],[f893])).
% 2.41/1.56 thf(f893,plain,(
% 2.41/1.56 (setadjoin__Cong = $true) & (quantDeMorgan1 = $true) & (upairsubunion = $true) & (upairsetIL = $true) & (omegaIndAx = $true) & (setminusILneg = $true) & (prop2setI = $true) & (omega0Ax = $true) & (emptyinPowerset = $true) & (notequalI1 = $true) & (setadjoinAx = $true) & (subsetI2 = $true) & (dsetconstrEL = $true) & (exuI2 = $true) & (setminusIRneg = $true) & (emptysetE = $true) & (subsetE2 = $true) & (subset2powerset = $true) & (powerset__Cong = $true) & (setbeta = $true) & (emptysetsubset = $true) & (powersetE1 = $true) & (subsetemptysetimpeq = $true) & (binintersectSubset2 = $true) & (exuE3u = $true) & (setminusELneg = $true) & (singletonsswitch = $true) & (setukpairIL = $true) & (dsetconstrER = $true) & (powersetI = $true) & (setminusSubset1 = $true) & (binintersectLsub = $true) & (setextAx = $true) & (setminusEL = $true) & (powersetAx = $true) & (prop2set2propI = $true) & (bs114d = $true) & (omegaSAx = $true) & (setadjoinE = $true) & (notdexE = $true) & (binintersectRsub = $true) & (exuEu = $true) & (((in @ sK164 @ sK163) = $true) & (((in @ sK166 @ sK165) = $true) & ((in @ (setadjoin @ sK164 @ (setadjoin @ sK166 @ emptyset)) @ (powerset @ (binunion @ sK163 @ sK165))) != $true))) & (emptysetAx = $true) & (replAx = $true) & (inPowerset = $true) & (setextsub = $true) & (foundationAx = $true) & (exuI1 = $true) & (singletoninpowerset = $true) & (quantDeMorgan2 = $true) & (upairset2IR = $true) & (setminusER = $true) & (eqinunit = $true) & (setunionI = $true) & (emptysetimpfalse = $true) & (subsetE = $true) & (dsetconstr__Cong = $true) & (vacuousDall = $true) & (binunionIL = $true) & (singletoninpowunion = $true) & (setadjoinIL = $true) & (binunionE = $true) & (emptyInPowerset = $true) & (setminusSubset2 = $true) & (setadjoinSub2 = $true) & (kpairiskpair = $true) & (powersetI1 = $true) & (emptyE1 = $true) & (symdiffIneg1 = $true) & (setminusERneg = $true) & (setunionAx = $true) & (binunionEcases = $true) & (kpairp = $true) & (nonemptyI = $true) & (emptyinunitempty = $true) & (setminusLsub = $true) & (setadjoinIR = $true) & (descr__Cong = $true) & (notdallE = $true) & (sepSubset = $true) & (setadjoinOr = $true) & (notinemptyset = $true) & (setunionE = $true) & (emptyset__Cong = $true) & (exuI3 = $true) & (eqimpsubset2 = $true) & (symdiffIneg2 = $true) & (setminusI = $true) & (nonemptyImpWitness = $true) & (nonemptyI1 = $true) & (powersetE = $true) & (powersetsubset = $true) & (symdiffE = $true) & (exuE3e = $true) & (binintersectI = $true) & (symdiffI1 = $true) & (inCongP = $true) & (exuE2 = $true) & (binunionIR = $true) & (nonemptyE1 = $true) & (subPowSU = $true) & (binunionLsub = $true) & (quantDeMorgan3 = $true) & (sepInPowerset = $true) & (binintersectSubset5 = $true) & (exuE1 = $true) & (subsetRefl = $true) & (upairsetE = $true) & (exu__Cong = $true) & (dsetconstrI = $true) & (subsetTrans = $true) & (binintersectEL = $true) & (binintersectER = $true) & (secondinupair = $true) & (emptyI = $true) & (singletonsubset = $true) & (eqimpsubset1 = $true) & (noeltsimpempty = $true) & (omega__Cong = $true) & (upairset2E = $true) & (prop2setE = $true) & (descrp = $true) & (wellorderingAx = $true) & (notsubsetI = $true) & (binunionRsub = $true) & (setext = $true) & (in__Cong = $true) & (uniqinunit = $true) & (quantDeMorgan4 = $true) & (disjointsetsI1 = $true) & (symdiffI2 = $true) & (setukpairIR = $true) & (upairsetIR = $true) & (notinsingleton = $true) & (binintersectSubset1 = $true) & (setadjoinSub = $true) & (setunion__Cong = $true) & (setoftrueEq = $true) & (binintersectSubset3 = $true) & (subsetI1 = $true) & (notequalI2 = $true) & (binintersectSubset4 = $true)),
% 2.41/1.56 inference(skolemisation,[status(esa),new_symbols(skolem,[sK163,sK164,sK165,sK166])],[f523,f892,f891])).
% 2.41/1.56 thf(f891,plain,(
% 2.41/1.56 ? [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X1 @ X0)) & ? [X3 : $i] : (((in @ X3 @ X2) = $true) & ((in @ (setadjoin @ X1 @ (setadjoin @ X3 @ emptyset)) @ (powerset @ (binunion @ X0 @ X2))) != $true))) => (((in @ sK164 @ sK163) = $true) & ? [X3 : $i] : (($true = (in @ X3 @ sK165)) & ($true != (in @ (setadjoin @ sK164 @ (setadjoin @ X3 @ emptyset)) @ (powerset @ (binunion @ sK163 @ sK165))))))),
% 2.41/1.56 introduced(definition,[],[choice_axiom])).
% 2.41/1.56 thf(f892,plain,(
% 2.41/1.56 ? [X3 : $i] : (($true = (in @ X3 @ sK165)) & ($true != (in @ (setadjoin @ sK164 @ (setadjoin @ X3 @ emptyset)) @ (powerset @ (binunion @ sK163 @ sK165))))) => (((in @ sK166 @ sK165) = $true) & ((in @ (setadjoin @ sK164 @ (setadjoin @ sK166 @ emptyset)) @ (powerset @ (binunion @ sK163 @ sK165))) != $true))),
% 2.41/1.56 introduced(definition,[],[choice_axiom])).
% 2.41/1.56 thf(f523,plain,(
% 2.41/1.56 (setadjoin__Cong = $true) & (quantDeMorgan1 = $true) & (upairsubunion = $true) & (upairsetIL = $true) & (omegaIndAx = $true) & (setminusILneg = $true) & (prop2setI = $true) & (omega0Ax = $true) & (emptyinPowerset = $true) & (notequalI1 = $true) & (setadjoinAx = $true) & (subsetI2 = $true) & (dsetconstrEL = $true) & (exuI2 = $true) & (setminusIRneg = $true) & (emptysetE = $true) & (subsetE2 = $true) & (subset2powerset = $true) & (powerset__Cong = $true) & (setbeta = $true) & (emptysetsubset = $true) & (powersetE1 = $true) & (subsetemptysetimpeq = $true) & (binintersectSubset2 = $true) & (exuE3u = $true) & (setminusELneg = $true) & (singletonsswitch = $true) & (setukpairIL = $true) & (dsetconstrER = $true) & (powersetI = $true) & (setminusSubset1 = $true) & (binintersectLsub = $true) & (setextAx = $true) & (setminusEL = $true) & (powersetAx = $true) & (prop2set2propI = $true) & (bs114d = $true) & (omegaSAx = $true) & (setadjoinE = $true) & (notdexE = $true) & (binintersectRsub = $true) & (exuEu = $true) & ? [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X1 @ X0)) & ? [X3 : $i] : (((in @ X3 @ X2) = $true) & ((in @ (setadjoin @ X1 @ (setadjoin @ X3 @ emptyset)) @ (powerset @ (binunion @ X0 @ X2))) != $true))) & (emptysetAx = $true) & (replAx = $true) & (inPowerset = $true) & (setextsub = $true) & (foundationAx = $true) & (exuI1 = $true) & (singletoninpowerset = $true) & (quantDeMorgan2 = $true) & (upairset2IR = $true) & (setminusER = $true) & (eqinunit = $true) & (setunionI = $true) & (emptysetimpfalse = $true) & (subsetE = $true) & (dsetconstr__Cong = $true) & (vacuousDall = $true) & (binunionIL = $true) & (singletoninpowunion = $true) & (setadjoinIL = $true) & (binunionE = $true) & (emptyInPowerset = $true) & (setminusSubset2 = $true) & (setadjoinSub2 = $true) & (kpairiskpair = $true) & (powersetI1 = $true) & (emptyE1 = $true) & (symdiffIneg1 = $true) & (setminusERneg = $true) & (setunionAx = $true) & (binunionEcases = $true) & (kpairp = $true) & (nonemptyI = $true) & (emptyinunitempty = $true) & (setminusLsub = $true) & (setadjoinIR = $true) & (descr__Cong = $true) & (notdallE = $true) & (sepSubset = $true) & (setadjoinOr = $true) & (notinemptyset = $true) & (setunionE = $true) & (emptyset__Cong = $true) & (exuI3 = $true) & (eqimpsubset2 = $true) & (symdiffIneg2 = $true) & (setminusI = $true) & (nonemptyImpWitness = $true) & (nonemptyI1 = $true) & (powersetE = $true) & (powersetsubset = $true) & (symdiffE = $true) & (exuE3e = $true) & (binintersectI = $true) & (symdiffI1 = $true) & (inCongP = $true) & (exuE2 = $true) & (binunionIR = $true) & (nonemptyE1 = $true) & (subPowSU = $true) & (binunionLsub = $true) & (quantDeMorgan3 = $true) & (sepInPowerset = $true) & (binintersectSubset5 = $true) & (exuE1 = $true) & (subsetRefl = $true) & (upairsetE = $true) & (exu__Cong = $true) & (dsetconstrI = $true) & (subsetTrans = $true) & (binintersectEL = $true) & (binintersectER = $true) & (secondinupair = $true) & (emptyI = $true) & (singletonsubset = $true) & (eqimpsubset1 = $true) & (noeltsimpempty = $true) & (omega__Cong = $true) & (upairset2E = $true) & (prop2setE = $true) & (descrp = $true) & (wellorderingAx = $true) & (notsubsetI = $true) & (binunionRsub = $true) & (setext = $true) & (in__Cong = $true) & (uniqinunit = $true) & (quantDeMorgan4 = $true) & (disjointsetsI1 = $true) & (symdiffI2 = $true) & (setukpairIR = $true) & (upairsetIR = $true) & (notinsingleton = $true) & (binintersectSubset1 = $true) & (setadjoinSub = $true) & (setunion__Cong = $true) & (setoftrueEq = $true) & (binintersectSubset3 = $true) & (subsetI1 = $true) & (notequalI2 = $true) & (binintersectSubset4 = $true)),
% 2.41/1.56 inference(flattening,[status(thm)],[f522])).
% 2.41/1.56 thf(f522,plain,(
% 2.41/1.56 (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X1 @ X0)) & ? [X3 : $i] : (((in @ X3 @ X2) = $true) & ((in @ (setadjoin @ X1 @ (setadjoin @ X3 @ emptyset)) @ (powerset @ (binunion @ X0 @ X2))) != $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)),
% 2.41/1.56 inference(ennf_transformation,[status(thm)],[f296])).
% 2.41/1.56 thf(f296,plain,(
% 2.41/1.56 ~((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) => ! [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X1 @ X0)) => ! [X3 : $i] : (((in @ X3 @ X2) = $true) => ((in @ (setadjoin @ X1 @ (setadjoin @ X3 @ emptyset)) @ (powerset @ (binunion @ X0 @ X2))) = $true)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.41/1.56 inference(fool_elimination,[status(thm)],[f295])).
% 2.41/1.56 thf(f295,plain,(
% 2.41/1.56 ~(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 => ! [X0 : $i,X1 : $i,X2 : $i] : ((in @ X1 @ X0) => ! [X3 : $i] : ((in @ X3 @ X2) => (in @ (setadjoin @ X1 @ (setadjoin @ X3 @ emptyset)) @ (powerset @ (binunion @ X0 @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.41/1.56 inference(rectify,[status(thm)],[f149])).
% 2.41/1.56 thf(f149,negated_conjecture,(
% 2.41/1.56 ~(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 => ! [X3 : $i,X1 : $i,X4 : $i] : ((in @ X1 @ X3) => ! [X2 : $i] : ((in @ X2 @ X4) => (in @ (setadjoin @ X1 @ (setadjoin @ X2 @ emptyset)) @ (powerset @ (binunion @ X3 @ X4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.41/1.56 inference(negated_conjecture,[status(cth)],[f148])).
% 2.41/1.56 thf(f148,conjecture,(
% 2.41/1.56 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 => ! [X3 : $i,X1 : $i,X4 : $i] : ((in @ X1 @ X3) => ! [X2 : $i] : ((in @ X2 @ X4) => (in @ (setadjoin @ X1 @ (setadjoin @ X2 @ emptyset)) @ (powerset @ (binunion @ X3 @ X4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.41/1.56 file('/export/starexec/sandbox/tmp/tmp.DbdYpcF5sg/DTF2THF_22176.p',upairinpowunion)).
% 2.41/1.56 thf(f2120,plain,(
% 2.41/1.56 ((in @ sK166 @ sK165) != $true)),
% 2.41/1.56 inference(trivial_inequality_removal,[status(thm)],[f2119])).
% 2.41/1.56 thf(f2119,plain,(
% 2.41/1.56 ($true != $true) | ((in @ sK166 @ sK165) != $true)),
% 2.41/1.56 inference(forward_demodulation,[status(thm)],[f2118,f1664])).
% 2.41/1.56 thf(f1664,plain,(
% 2.41/1.56 (upairsubunion = $true)),
% 2.41/1.56 inference(cnf_transformation,[status(thm)],[f893])).
% 2.41/1.56 thf(f2118,plain,(
% 2.41/1.56 ((in @ sK166 @ sK165) != $true) | (upairsubunion != $true)),
% 2.41/1.56 inference(trivial_inequality_removal,[status(thm)],[f2117])).
% 2.41/1.56 thf(f2117,plain,(
% 2.41/1.56 ($true != $true) | (upairsubunion != $true) | ((in @ sK166 @ sK165) != $true)),
% 2.41/1.56 inference(forward_demodulation,[status(thm)],[f2116,f1624])).
% 2.41/1.56 thf(f1624,plain,(
% 2.41/1.56 ((in @ sK164 @ sK163) = $true)),
% 2.41/1.56 inference(cnf_transformation,[status(thm)],[f893])).
% 2.41/1.56 thf(f2116,plain,(
% 2.41/1.56 ((in @ sK166 @ sK165) != $true) | ((in @ sK164 @ sK163) != $true) | (upairsubunion != $true)),
% 2.41/1.56 inference(trivial_inequality_removal,[status(thm)],[f2107])).
% 2.41/1.56 thf(f2107,plain,(
% 2.41/1.56 (upairsubunion != $true) | ($true != $true) | ((in @ sK164 @ sK163) != $true) | ((in @ sK166 @ sK165) != $true)),
% 2.41/1.56 inference(superposition,[status(thm)],[f2098,f1514])).
% 2.41/1.56 thf(f1514,plain,(
% 2.41/1.56 ( ! [X6 : $i,X7 : $i,X4 : $i,X5 : $i] : (((subset @ (setadjoin @ X5 @ (setadjoin @ X7 @ emptyset)) @ (binunion @ X4 @ X6)) = $true) | (upairsubunion != $true) | ((in @ X5 @ X4) != $true) | ($true != (in @ X7 @ X6))) )),
% 2.41/1.56 inference(cnf_transformation,[status(thm)],[f885])).
% 2.41/1.56 thf(f885,plain,(
% 2.41/1.56 ((upairsubunion = $true) | ((((in @ sK159 @ sK158) = $true) & ((subset @ (setadjoin @ sK157 @ (setadjoin @ sK159 @ emptyset)) @ (binunion @ sK156 @ sK158)) != $true)) & ($true = (in @ sK157 @ sK156)))) & (! [X4 : $i,X5 : $i,X6 : $i] : (! [X7 : $i] : (($true != (in @ X7 @ X6)) | ((subset @ (setadjoin @ X5 @ (setadjoin @ X7 @ emptyset)) @ (binunion @ X4 @ X6)) = $true)) | ((in @ X5 @ X4) != $true)) | (upairsubunion != $true))),
% 2.41/1.56 inference(skolemisation,[status(esa),new_symbols(skolem,[sK156,sK157,sK158,sK159])],[f882,f884,f883])).
% 2.41/1.56 thf(f883,plain,(
% 2.41/1.56 ? [X0 : $i,X1 : $i,X2 : $i] : (? [X3 : $i] : (((in @ X3 @ X2) = $true) & ((subset @ (setadjoin @ X1 @ (setadjoin @ X3 @ emptyset)) @ (binunion @ X0 @ X2)) != $true)) & ($true = (in @ X1 @ X0))) => (? [X3 : $i] : (($true = (in @ X3 @ sK158)) & ((subset @ (setadjoin @ sK157 @ (setadjoin @ X3 @ emptyset)) @ (binunion @ sK156 @ sK158)) != $true)) & ($true = (in @ sK157 @ sK156)))),
% 2.41/1.56 introduced(definition,[],[choice_axiom])).
% 2.41/1.56 thf(f884,plain,(
% 2.41/1.56 ? [X3 : $i] : (($true = (in @ X3 @ sK158)) & ((subset @ (setadjoin @ sK157 @ (setadjoin @ X3 @ emptyset)) @ (binunion @ sK156 @ sK158)) != $true)) => (((in @ sK159 @ sK158) = $true) & ((subset @ (setadjoin @ sK157 @ (setadjoin @ sK159 @ emptyset)) @ (binunion @ sK156 @ sK158)) != $true))),
% 2.41/1.56 introduced(definition,[],[choice_axiom])).
% 2.41/1.56 thf(f882,plain,(
% 2.41/1.56 ((upairsubunion = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (? [X3 : $i] : (((in @ X3 @ X2) = $true) & ((subset @ (setadjoin @ X1 @ (setadjoin @ X3 @ emptyset)) @ (binunion @ X0 @ X2)) != $true)) & ($true = (in @ X1 @ X0)))) & (! [X4 : $i,X5 : $i,X6 : $i] : (! [X7 : $i] : (($true != (in @ X7 @ X6)) | ((subset @ (setadjoin @ X5 @ (setadjoin @ X7 @ emptyset)) @ (binunion @ X4 @ X6)) = $true)) | ((in @ X5 @ X4) != $true)) | (upairsubunion != $true))),
% 2.41/1.56 inference(rectify,[status(thm)],[f881])).
% 2.41/1.56 thf(f881,plain,(
% 2.41/1.56 ((upairsubunion = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (? [X3 : $i] : (((in @ X3 @ X2) = $true) & ((subset @ (setadjoin @ X1 @ (setadjoin @ X3 @ emptyset)) @ (binunion @ X0 @ X2)) != $true)) & ($true = (in @ X1 @ X0)))) & (! [X0 : $i,X1 : $i,X2 : $i] : (! [X3 : $i] : (((in @ X3 @ X2) != $true) | ((subset @ (setadjoin @ X1 @ (setadjoin @ X3 @ emptyset)) @ (binunion @ X0 @ X2)) = $true)) | ($true != (in @ X1 @ X0))) | (upairsubunion != $true))),
% 2.41/1.56 inference(nnf_transformation,[status(thm)],[f521])).
% 2.41/1.56 thf(f521,plain,(
% 2.41/1.56 (upairsubunion = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i] : (! [X3 : $i] : (((in @ X3 @ X2) != $true) | ((subset @ (setadjoin @ X1 @ (setadjoin @ X3 @ emptyset)) @ (binunion @ X0 @ X2)) = $true)) | ($true != (in @ X1 @ X0)))),
% 2.41/1.56 inference(ennf_transformation,[status(thm)],[f322])).
% 2.41/1.56 thf(f322,plain,(
% 2.41/1.56 ! [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X1 @ X0)) => ! [X3 : $i] : (((in @ X3 @ X2) = $true) => ((subset @ (setadjoin @ X1 @ (setadjoin @ X3 @ emptyset)) @ (binunion @ X0 @ X2)) = $true))) <=> (upairsubunion = $true)),
% 2.41/1.56 inference(fool_elimination,[status(thm)],[f321])).
% 2.41/1.56 thf(f321,plain,(
% 2.41/1.56 (upairsubunion = ! [X0 : $i,X1 : $i,X2 : $i] : ((in @ X1 @ X0) => ! [X3 : $i] : ((in @ X3 @ X2) => (subset @ (setadjoin @ X1 @ (setadjoin @ X3 @ emptyset)) @ (binunion @ X0 @ X2)))))),
% 2.41/1.56 inference(rectify,[status(thm)],[f147])).
% 2.41/1.56 thf(f147,axiom,(
% 2.41/1.56 (upairsubunion = ! [X3 : $i,X1 : $i,X4 : $i] : ((in @ X1 @ X3) => ! [X2 : $i] : ((in @ X2 @ X4) => (subset @ (setadjoin @ X1 @ (setadjoin @ X2 @ emptyset)) @ (binunion @ X3 @ X4)))))),
% 2.41/1.56 file('/export/starexec/sandbox/tmp/tmp.DbdYpcF5sg/DTF2THF_22176.p',upairsubunion)).
% 2.41/1.56 thf(f2098,plain,(
% 2.41/1.56 ((subset @ (setadjoin @ sK164 @ (setadjoin @ sK166 @ emptyset)) @ (binunion @ sK163 @ sK165)) != $true)),
% 2.41/1.56 inference(trivial_inequality_removal,[status(thm)],[f2097])).
% 2.41/1.56 thf(f2097,plain,(
% 2.41/1.56 ((subset @ (setadjoin @ sK164 @ (setadjoin @ sK166 @ emptyset)) @ (binunion @ sK163 @ sK165)) != $true) | ($true != $true)),
% 2.41/1.56 inference(forward_demodulation,[status(thm)],[f2096,f1649])).
% 2.41/1.56 thf(f1649,plain,(
% 2.41/1.56 (subset2powerset = $true)),
% 2.41/1.56 inference(cnf_transformation,[status(thm)],[f893])).
% 2.41/1.56 thf(f2096,plain,(
% 2.41/1.56 (subset2powerset != $true) | ((subset @ (setadjoin @ sK164 @ (setadjoin @ sK166 @ emptyset)) @ (binunion @ sK163 @ sK165)) != $true)),
% 2.41/1.56 inference(trivial_inequality_removal,[status(thm)],[f2085])).
% 2.41/1.56 thf(f2085,plain,(
% 2.41/1.56 (subset2powerset != $true) | ((subset @ (setadjoin @ sK164 @ (setadjoin @ sK166 @ emptyset)) @ (binunion @ sK163 @ sK165)) != $true) | ($true != $true)),
% 2.41/1.56 inference(superposition,[status(thm)],[f1622,f1759])).
% 2.41/1.56 thf(f1759,plain,(
% 2.41/1.56 ( ! [X0 : $i,X1 : $i] : (((in @ X0 @ (powerset @ X1)) = $true) | (subset2powerset != $true) | ((subset @ X0 @ X1) != $true)) )),
% 2.41/1.56 inference(cnf_transformation,[status(thm)],[f1011])).
% 2.41/1.56 thf(f1011,plain,(
% 2.41/1.56 (! [X0 : $i,X1 : $i] : (((in @ X0 @ (powerset @ X1)) = $true) | ((subset @ X0 @ X1) != $true)) | (subset2powerset != $true)) & ((subset2powerset = $true) | (($true != (in @ sK231 @ (powerset @ sK232))) & ((subset @ sK231 @ sK232) = $true)))),
% 2.41/1.56 inference(skolemisation,[status(esa),new_symbols(skolem,[sK231,sK232])],[f1009,f1010])).
% 2.41/1.56 thf(f1010,plain,(
% 2.41/1.56 ? [X2 : $i,X3 : $i] : (((in @ X2 @ (powerset @ X3)) != $true) & ($true = (subset @ X2 @ X3))) => (($true != (in @ sK231 @ (powerset @ sK232))) & ((subset @ sK231 @ sK232) = $true))),
% 2.41/1.56 introduced(definition,[],[choice_axiom])).
% 2.41/1.56 thf(f1009,plain,(
% 2.41/1.56 (! [X0 : $i,X1 : $i] : (((in @ X0 @ (powerset @ X1)) = $true) | ((subset @ X0 @ X1) != $true)) | (subset2powerset != $true)) & ((subset2powerset = $true) | ? [X2 : $i,X3 : $i] : (((in @ X2 @ (powerset @ X3)) != $true) & ($true = (subset @ X2 @ X3))))),
% 2.41/1.56 inference(rectify,[status(thm)],[f1008])).
% 2.41/1.56 thf(f1008,plain,(
% 2.41/1.56 (! [X0 : $i,X1 : $i] : (((in @ X0 @ (powerset @ X1)) = $true) | ((subset @ X0 @ X1) != $true)) | (subset2powerset != $true)) & ((subset2powerset = $true) | ? [X0 : $i,X1 : $i] : (((in @ X0 @ (powerset @ X1)) != $true) & ((subset @ X0 @ X1) = $true)))),
% 2.41/1.56 inference(nnf_transformation,[status(thm)],[f566])).
% 2.41/1.56 thf(f566,plain,(
% 2.41/1.56 ! [X0 : $i,X1 : $i] : (((in @ X0 @ (powerset @ X1)) = $true) | ((subset @ X0 @ X1) != $true)) <=> (subset2powerset = $true)),
% 2.41/1.56 inference(ennf_transformation,[status(thm)],[f188])).
% 2.41/1.56 thf(f188,plain,(
% 2.41/1.56 (subset2powerset = $true) <=> ! [X0 : $i,X1 : $i] : (((subset @ X0 @ X1) = $true) => ((in @ X0 @ (powerset @ X1)) = $true))),
% 2.41/1.56 inference(fool_elimination,[status(thm)],[f187])).
% 2.41/1.56 thf(f187,plain,(
% 2.41/1.56 (subset2powerset = ! [X0 : $i,X1 : $i] : ((subset @ X0 @ X1) => (in @ X0 @ (powerset @ X1))))),
% 2.41/1.56 inference(rectify,[status(thm)],[f92])).
% 2.41/1.56 thf(f92,axiom,(
% 2.41/1.56 (subset2powerset = ! [X3 : $i,X4 : $i] : ((subset @ X3 @ X4) => (in @ X3 @ (powerset @ X4))))),
% 2.41/1.56 file('/export/starexec/sandbox/tmp/tmp.DbdYpcF5sg/DTF2THF_22176.p',subset2powerset)).
% 2.41/1.56 thf(f1622,plain,(
% 2.41/1.56 ((in @ (setadjoin @ sK164 @ (setadjoin @ sK166 @ emptyset)) @ (powerset @ (binunion @ sK163 @ sK165))) != $true)),
% 2.41/1.56 inference(cnf_transformation,[status(thm)],[f893])).
% 2.41/1.56 % SZS output end Proof for DTF2THF_22176
% 2.41/1.56 % (22484)------------------------------
% 2.41/1.56 % (22484)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.41/1.56 % (22484)Termination reason: Refutation
% 2.41/1.56
% 2.41/1.56 % (22484)Memory used [KB]: 7931
% 2.41/1.56 % (22484)Time elapsed: 0.066 s
% 2.41/1.56 % (22484)Instructions burned: 194 (million)
% 2.41/1.56 % (22484)------------------------------
% 2.41/1.56 % (22484)------------------------------
% 2.41/1.56 % (22461)Success in time 0.147 s
%------------------------------------------------------------------------------