%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU618^1 : 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 : n020.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 09:35:39 AM UTC 2026
% Result : Theorem 2.45s 1.64s
% Output : Refutation 2.45s
% Verified :
% SZS Type : Refutation
% Derivation depth : 20
% Number of leaves : 8
% Syntax : Number of formulae : 50 ( 14 unt; 0 typ; 0 def)
% Number of atoms : 2518 ( 658 equ; 0 cnn)
% Maximal formula atoms : 138 ( 50 avg)
% Number of connectives : 1542 ( 72 ~; 46 |; 421 &; 452 @)
% ( 5 <=>; 546 =>; 0 <=; 0 <~>)
% Maximal formula depth : 146 ( 23 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 264 ( 261 usr; 146 con; 0-4 aty)
% Number of variables : 94 ( 0 ^; 67 !; 27 ?; 94 :)
% 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_168,type,
sP0: $i > $i > $o ).
thf(func_def_169,type,
sP1: $i > $o ).
thf(func_def_170,type,
sP2: $i > $i > $o ).
thf(func_def_171,type,
sP3: $i > $i > $o ).
thf(func_def_173,type,
sP5: $i > $o > $i > $i > $o ).
thf(func_def_205,type,
sK37: $i > $i ).
thf(func_def_206,type,
sK38: $i > $i > $i ).
thf(func_def_217,type,
sK49: $i > $o ).
thf(func_def_252,type,
sK84: $i > $o ).
thf(func_def_255,type,
sK87: $i > $i > $i ).
thf(func_def_268,type,
sK100: $i > $i > $i ).
thf(func_def_272,type,
sK104: $i > $o ).
thf(func_def_275,type,
sK107: $i > ( $i > $o ) > $i ).
thf(func_def_278,type,
sK110: $i > ( $i > $o ) > $i ).
thf(func_def_279,type,
sK111: $i > $o ).
thf(func_def_282,type,
sK114: $i > $i > $i ).
thf(func_def_297,type,
sK129: $i > $i > $i ).
thf(func_def_298,type,
sK130: $i > $i > $i ).
thf(func_def_299,type,
sK131: $i > $i > $i ).
thf(func_def_300,type,
sK132: $i > $i > $i ).
thf(func_def_301,type,
sK133: $i > $i > $i ).
thf(func_def_302,type,
sK134: $i > $i > $i > $i ).
thf(func_def_303,type,
sK135: $i > $i ).
thf(func_def_304,type,
sK136: $i > $i ).
thf(func_def_305,type,
sK137: $i > $i ).
thf(func_def_306,type,
sK138: $i > $i ).
thf(func_def_307,type,
sK139: $i > $i > $i ).
thf(func_def_308,type,
sK140: $i > $i > $i > $i ).
thf(func_def_309,type,
sK141: $i > $i > $i > $i ).
thf(func_def_310,type,
sK142: $i > $i > $i ).
thf(func_def_311,type,
sK143: $i > $i > $i ).
thf(func_def_312,type,
sK144: $i > $i ).
thf(func_def_314,type,
sK146: $i > $i ).
thf(func_def_315,type,
sK147: $i > $i ).
thf(func_def_317,type,
sK149: $i > $o ).
thf(func_def_324,type,
sK156: $i > $o ).
thf(func_def_328,type,
sK160: $i > $i ).
thf(func_def_342,type,
sK174: $i > $o ).
thf(func_def_347,type,
sK179: $i > $i ).
thf(func_def_350,type,
sK182: $i > $o ).
thf(func_def_361,type,
sK193: $i > $i ).
thf(func_def_362,type,
sK194: $i > $i ).
thf(func_def_367,type,
sK199: $o > $i > $i > $i ).
thf(func_def_371,type,
sK203: $i > ( $i > $o ) > $i ).
thf(func_def_372,type,
sK204: $i > $o ).
thf(func_def_375,type,
sK207: $i > $o ).
thf(func_def_376,type,
sK208: $i > $o ).
thf(func_def_377,type,
sK209: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_378,type,
sK210: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_383,type,
sK215: ( $i > $o ) > $i ).
thf(func_def_384,type,
sK216: ( $i > $o ) > $i ).
thf(func_def_385,type,
sK217: $i > $o ).
thf(func_def_387,type,
sK219: $i > ( $i > $o ) > $i ).
thf(func_def_388,type,
sK220: $i > $o ).
thf(func_def_392,type,
sK224: $i > $i > $i ).
thf(func_def_397,type,
sK229: $i > $i > $i ).
thf(func_def_398,type,
sK230: $i > $i > $i ).
thf(func_def_399,type,
sK231: ( $i > $o ) > $i ).
thf(func_def_400,type,
sK232: $i > $o ).
thf(func_def_401,type,
sK233: $i > $i ).
thf(func_def_402,type,
sK234: $i > $i > $i ).
thf(func_def_405,type,
sK237: $i > $o ).
thf(func_def_411,type,
sK243: $i > $o ).
thf(func_def_412,type,
sK244: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_413,type,
sK245: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_414,type,
sK246: $i > $o ).
thf(func_def_415,type,
sK247: $i > $o ).
thf(func_def_419,type,
sK251: $i > $i > $i ).
thf(func_def_425,type,
sK257: $i > $i > $o ).
thf(func_def_426,type,
sK258: $i > $i ).
thf(func_def_427,type,
sK259: $i > $i ).
thf(func_def_428,type,
sK260: ( $i > $i > $o ) > $i > $i ).
thf(func_def_429,type,
sK261: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_430,type,
sK262: ( $i > $i > $o ) > $i > $i ).
thf(func_def_432,type,
sK264: $i > $o ).
thf(func_def_436,type,
sK268: $i > ( $i > $o ) > $i ).
thf(func_def_437,type,
sK269: $i > $o ).
thf(func_def_446,type,
sK278: $i > $i ).
thf(func_def_453,type,
sK285: $i > $o ).
thf(func_def_457,type,
sK289: $i > $o ).
thf(func_def_458,type,
sK290: ( $i > $o ) > $i ).
thf(func_def_459,type,
sK291: $i > $o ).
thf(func_def_473,type,
sK305: $i > $o ).
thf(func_def_476,type,
sK308: $i > ( $i > $o ) > $i ).
thf(func_def_481,type,
sK313: $i > ( $i > $o ) > $i ).
thf(func_def_482,type,
sK314: $i > $o ).
thf(func_def_488,type,
sK320: $i > $i ).
thf(func_def_504,type,
sK336: $i > $o ).
thf(func_def_523,type,
sK355: $i > $o ).
thf(func_def_524,type,
sK356: $i > $o ).
thf(func_def_525,type,
sK357: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_526,type,
sK358: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_528,type,
sK360: $i > ( $i > $o ) > $i ).
thf(func_def_529,type,
sK361: $i > $o ).
thf(func_def_532,type,
sK364: $i > $o ).
thf(func_def_533,type,
sK365: $i > $i ).
thf(func_def_534,type,
sK366: ( $i > $o ) > $i ).
thf(func_def_538,type,
ph370:
!>[X0: $tType] : X0 ).
thf(f3211,plain,
$false,
inference(trivial_inequality_removal,[status(thm)],[f3210]) ).
thf(f3210,plain,
$true != $true,
inference(superposition,[status(thm)],[f3138,f1417]) ).
thf(f1417,plain,
upairsetIR = $true,
inference(cnf_transformation,[status(thm)],[f774]) ).
thf(f774,plain,
( ( setext = $true )
& ( setadjoinOr = $true )
& ( emptysetimpfalse = $true )
& ( quantDeMorgan4 = $true )
& ( setminusI = $true )
& ( wellorderingAx = $true )
& ( sepSubset = $true )
& ( symdiffI1 = $true )
& ( setadjoinIL = $true )
& ( exuI1 = $true )
& ( setadjoin__Cong = $true )
& ( prop2set2propI = $true )
& ( descrp = $true )
& ( powersetsubset = $true )
& ( notinsingleton = $true )
& ( emptyset__Cong = $true )
& ( nonemptyI1 = $true )
& ( descr__Cong = $true )
& ( setminusERneg = $true )
& ( emptyI = $true )
& ( dsetconstrI = $true )
& ( binintersectLsub = $true )
& ( prop2setI = $true )
& ( omegaSAx = $true )
& ( setminusILneg = $true )
& ( emptysetE = $true )
& ( omegaIndAx = $true )
& ( powerset__Cong = $true )
& ( binunionRsub = $true )
& ( uniqinunit = $true )
& ( subsetE2 = $true )
& ( upairsetE = $true )
& ( exuI3 = $true )
& ( inCongP = $true )
& ( setunionAx = $true )
& ( exuEu = $true )
& ( inPowerset = $true )
& ( notsubsetI = $true )
& ( subset2powerset = $true )
& ( setminusER = $true )
& ( eqinunit = $true )
& ( notinemptyset = $true )
& ( setadjoinSub = $true )
& ( emptysetsubset = $true )
& ( setoftrueEq = $true )
& ( setukpairIL = $true )
& ( binunionLsub = $true )
& ( binintersectEL = $true )
& ( notequalI2 = $true )
& ( setminusSubset1 = $true )
& ( nonemptyE1 = $true )
& ( dsetconstrER = $true )
& ( setminusSubset2 = $true )
& ( powersetAx = $true )
& ( in__Cong = $true )
& ( binintersectSubset1 = $true )
& ( setadjoinAx = $true )
& ( symdiffE = $true )
& ( symdiffIneg1 = $true )
& ( binintersectER = $true )
& ( subsetI2 = $true )
& ( setunionE = $true )
& ( bs114d = $true )
& ( upairsetIL = $true )
& ( setadjoinE = $true )
& ( replAx = $true )
& ( setextsub = $true )
& ( ( in @ sK108 @ ( setunion @ ( setadjoin @ ( setadjoin @ sK109 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK109 @ ( setadjoin @ sK108 @ emptyset ) ) @ emptyset ) ) ) )
!= $true )
& ( eqimpsubset2 = $true )
& ( setadjoinSub2 = $true )
& ( subsetI1 = $true )
& ( notdallE = $true )
& ( binunionEcases = $true )
& ( subsetRefl = $true )
& ( setbeta = $true )
& ( quantDeMorgan1 = $true )
& ( powersetI1 = $true )
& ( notdexE = $true )
& ( powersetE = $true )
& ( emptyinunitempty = $true )
& ( upairsetIR = $true )
& ( emptysetAx = $true )
& ( setminusLsub = $true )
& ( binintersectSubset5 = $true )
& ( setextAx = $true )
& ( disjointsetsI1 = $true )
& ( exuE1 = $true )
& ( subPowSU = $true )
& ( eqimpsubset1 = $true )
& ( setminusIRneg = $true )
& ( exuE3u = $true )
& ( prop2setE = $true )
& ( emptyinPowerset = $true )
& ( notequalI1 = $true )
& ( noeltsimpempty = $true )
& ( singletonsswitch = $true )
& ( binintersectSubset4 = $true )
& ( secondinupair = $true )
& ( powersetI = $true )
& ( exuE3e = $true )
& ( setminusEL = $true )
& ( setunion__Cong = $true )
& ( binunionE = $true )
& ( nonemptyI = $true )
& ( nonemptyImpWitness = $true )
& ( exu__Cong = $true )
& ( binintersectSubset3 = $true )
& ( symdiffI2 = $true )
& ( exuI2 = $true )
& ( dsetconstrEL = $true )
& ( binintersectI = $true )
& ( omega0Ax = $true )
& ( omega__Cong = $true )
& ( symdiffIneg2 = $true )
& ( vacuousDall = $true )
& ( dsetconstr__Cong = $true )
& ( upairset2IR = $true )
& ( quantDeMorgan3 = $true )
& ( setunionI = $true )
& ( emptyE1 = $true )
& ( binunionIR = $true )
& ( setminusELneg = $true )
& ( binunionIL = $true )
& ( subsetemptysetimpeq = $true )
& ( powersetE1 = $true )
& ( foundationAx = $true )
& ( binintersectSubset2 = $true )
& ( binintersectRsub = $true )
& ( sepInPowerset = $true )
& ( quantDeMorgan2 = $true )
& ( emptyInPowerset = $true )
& ( subsetE = $true )
& ( subsetTrans = $true )
& ( setadjoinIR = $true )
& ( exuE2 = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK108,sK109])],[f529,f773]) ).
thf(f773,plain,
( ? [X0: $i,X1: $i] :
( ( in @ X0 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X0 @ emptyset ) ) @ emptyset ) ) ) )
!= $true )
=> ( ( in @ sK108 @ ( setunion @ ( setadjoin @ ( setadjoin @ sK109 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK109 @ ( setadjoin @ sK108 @ emptyset ) ) @ emptyset ) ) ) )
!= $true ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f529,plain,
( ( setext = $true )
& ( setadjoinOr = $true )
& ( emptysetimpfalse = $true )
& ( quantDeMorgan4 = $true )
& ( setminusI = $true )
& ( wellorderingAx = $true )
& ( sepSubset = $true )
& ( symdiffI1 = $true )
& ( setadjoinIL = $true )
& ( exuI1 = $true )
& ( setadjoin__Cong = $true )
& ( prop2set2propI = $true )
& ( descrp = $true )
& ( powersetsubset = $true )
& ( notinsingleton = $true )
& ( emptyset__Cong = $true )
& ( nonemptyI1 = $true )
& ( descr__Cong = $true )
& ( setminusERneg = $true )
& ( emptyI = $true )
& ( dsetconstrI = $true )
& ( binintersectLsub = $true )
& ( prop2setI = $true )
& ( omegaSAx = $true )
& ( setminusILneg = $true )
& ( emptysetE = $true )
& ( omegaIndAx = $true )
& ( powerset__Cong = $true )
& ( binunionRsub = $true )
& ( uniqinunit = $true )
& ( subsetE2 = $true )
& ( upairsetE = $true )
& ( exuI3 = $true )
& ( inCongP = $true )
& ( setunionAx = $true )
& ( exuEu = $true )
& ( inPowerset = $true )
& ( notsubsetI = $true )
& ( subset2powerset = $true )
& ( setminusER = $true )
& ( eqinunit = $true )
& ( notinemptyset = $true )
& ( setadjoinSub = $true )
& ( emptysetsubset = $true )
& ( setoftrueEq = $true )
& ( setukpairIL = $true )
& ( binunionLsub = $true )
& ( binintersectEL = $true )
& ( notequalI2 = $true )
& ( setminusSubset1 = $true )
& ( nonemptyE1 = $true )
& ( dsetconstrER = $true )
& ( setminusSubset2 = $true )
& ( powersetAx = $true )
& ( in__Cong = $true )
& ( binintersectSubset1 = $true )
& ( setadjoinAx = $true )
& ( symdiffE = $true )
& ( symdiffIneg1 = $true )
& ( binintersectER = $true )
& ( subsetI2 = $true )
& ( setunionE = $true )
& ( bs114d = $true )
& ( upairsetIL = $true )
& ( setadjoinE = $true )
& ( replAx = $true )
& ( setextsub = $true )
& ? [X0: $i,X1: $i] :
( ( in @ X0 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X0 @ emptyset ) ) @ emptyset ) ) ) )
!= $true )
& ( eqimpsubset2 = $true )
& ( setadjoinSub2 = $true )
& ( subsetI1 = $true )
& ( notdallE = $true )
& ( binunionEcases = $true )
& ( subsetRefl = $true )
& ( setbeta = $true )
& ( quantDeMorgan1 = $true )
& ( powersetI1 = $true )
& ( notdexE = $true )
& ( powersetE = $true )
& ( emptyinunitempty = $true )
& ( upairsetIR = $true )
& ( emptysetAx = $true )
& ( setminusLsub = $true )
& ( binintersectSubset5 = $true )
& ( setextAx = $true )
& ( disjointsetsI1 = $true )
& ( exuE1 = $true )
& ( subPowSU = $true )
& ( eqimpsubset1 = $true )
& ( setminusIRneg = $true )
& ( exuE3u = $true )
& ( prop2setE = $true )
& ( emptyinPowerset = $true )
& ( notequalI1 = $true )
& ( noeltsimpempty = $true )
& ( singletonsswitch = $true )
& ( binintersectSubset4 = $true )
& ( secondinupair = $true )
& ( powersetI = $true )
& ( exuE3e = $true )
& ( setminusEL = $true )
& ( setunion__Cong = $true )
& ( binunionE = $true )
& ( nonemptyI = $true )
& ( nonemptyImpWitness = $true )
& ( exu__Cong = $true )
& ( binintersectSubset3 = $true )
& ( symdiffI2 = $true )
& ( exuI2 = $true )
& ( dsetconstrEL = $true )
& ( binintersectI = $true )
& ( omega0Ax = $true )
& ( omega__Cong = $true )
& ( symdiffIneg2 = $true )
& ( vacuousDall = $true )
& ( dsetconstr__Cong = $true )
& ( upairset2IR = $true )
& ( quantDeMorgan3 = $true )
& ( setunionI = $true )
& ( emptyE1 = $true )
& ( binunionIR = $true )
& ( setminusELneg = $true )
& ( binunionIL = $true )
& ( subsetemptysetimpeq = $true )
& ( powersetE1 = $true )
& ( foundationAx = $true )
& ( binintersectSubset2 = $true )
& ( binintersectRsub = $true )
& ( sepInPowerset = $true )
& ( quantDeMorgan2 = $true )
& ( emptyInPowerset = $true )
& ( subsetE = $true )
& ( subsetTrans = $true )
& ( setadjoinIR = $true )
& ( exuE2 = $true ) ),
inference(flattening,[status(thm)],[f528]) ).
thf(f528,plain,
( ? [X0: $i,X1: $i] :
( ( in @ X0 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X0 @ emptyset ) ) @ emptyset ) ) ) )
!= $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)],[f404]) ).
thf(f404,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 )
=> ! [X0: $i,X1: $i] :
( ( in @ X0 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X0 @ emptyset ) ) @ emptyset ) ) ) )
= $true ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f403]) ).
thf(f403,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
=> ! [X0: $i,X1: $i] : ( in @ X0 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X0 @ emptyset ) ) @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f139]) ).
thf(f139,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
=> ! [X2: $i,X1: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f138]) ).
thf(f138,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
=> ! [X2: $i,X1: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.BCzCW9enGb/DTF2THF_25972.p',setukpairIR) ).
thf(f3138,plain,
upairsetIR != $true,
inference(trivial_inequality_removal,[status(thm)],[f3126]) ).
thf(f3126,plain,
( ( $true != $true )
| ( upairsetIR != $true ) ),
inference(superposition,[status(thm)],[f2188,f1264]) ).
thf(f1264,plain,
! [X2: $i,X3: $i] :
( ( ( in @ X2 @ ( setadjoin @ X3 @ ( setadjoin @ X2 @ emptyset ) ) )
= $true )
| ( upairsetIR != $true ) ),
inference(cnf_transformation,[status(thm)],[f660]) ).
thf(f660,plain,
( ( ( upairsetIR = $true )
| ( ( in @ sK41 @ ( setadjoin @ sK42 @ ( setadjoin @ sK41 @ emptyset ) ) )
!= $true ) )
& ( ! [X2: $i,X3: $i] :
( ( in @ X2 @ ( setadjoin @ X3 @ ( setadjoin @ X2 @ emptyset ) ) )
= $true )
| ( upairsetIR != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK41,sK42])],[f658,f659]) ).
thf(f659,plain,
( ? [X0: $i,X1: $i] :
( ( in @ X0 @ ( setadjoin @ X1 @ ( setadjoin @ X0 @ emptyset ) ) )
!= $true )
=> ( ( in @ sK41 @ ( setadjoin @ sK42 @ ( setadjoin @ sK41 @ emptyset ) ) )
!= $true ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f658,plain,
( ( ( upairsetIR = $true )
| ? [X0: $i,X1: $i] :
( ( in @ X0 @ ( setadjoin @ X1 @ ( setadjoin @ X0 @ emptyset ) ) )
!= $true ) )
& ( ! [X2: $i,X3: $i] :
( ( in @ X2 @ ( setadjoin @ X3 @ ( setadjoin @ X2 @ emptyset ) ) )
= $true )
| ( upairsetIR != $true ) ) ),
inference(rectify,[status(thm)],[f657]) ).
thf(f657,plain,
( ( ( upairsetIR = $true )
| ? [X0: $i,X1: $i] :
( ( in @ X0 @ ( setadjoin @ X1 @ ( setadjoin @ X0 @ emptyset ) ) )
!= $true ) )
& ( ! [X0: $i,X1: $i] :
( ( in @ X0 @ ( setadjoin @ X1 @ ( setadjoin @ X0 @ emptyset ) ) )
= $true )
| ( upairsetIR != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f369]) ).
thf(f369,plain,
( ( upairsetIR = $true )
<=> ! [X0: $i,X1: $i] :
( ( in @ X0 @ ( setadjoin @ X1 @ ( setadjoin @ X0 @ emptyset ) ) )
= $true ) ),
inference(fool_elimination,[status(thm)],[f368]) ).
thf(f368,plain,
( upairsetIR
= ( ! [X0: $i,X1: $i] : ( in @ X0 @ ( setadjoin @ X1 @ ( setadjoin @ X0 @ emptyset ) ) ) ) ),
inference(rectify,[status(thm)],[f52]) ).
thf(f52,axiom,
( upairsetIR
= ( ! [X2: $i,X1: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.BCzCW9enGb/DTF2THF_25972.p',upairsetIR) ).
thf(f2188,plain,
( ( in @ sK108 @ ( setadjoin @ sK109 @ ( setadjoin @ sK108 @ emptyset ) ) )
!= $true ),
inference(trivial_inequality_removal,[status(thm)],[f2187]) ).
thf(f2187,plain,
( ( $true != $true )
| ( ( in @ sK108 @ ( setadjoin @ sK109 @ ( setadjoin @ sK108 @ emptyset ) ) )
!= $true ) ),
inference(forward_demodulation,[status(thm)],[f2078,f1400]) ).
thf(f1400,plain,
secondinupair = $true,
inference(cnf_transformation,[status(thm)],[f774]) ).
thf(f2078,plain,
( ( ( in @ sK108 @ ( setadjoin @ sK109 @ ( setadjoin @ sK108 @ emptyset ) ) )
!= $true )
| ( secondinupair != $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f1995]) ).
thf(f1995,plain,
( ( secondinupair != $true )
| ( $true != $true )
| ( ( in @ sK108 @ ( setadjoin @ sK109 @ ( setadjoin @ sK108 @ emptyset ) ) )
!= $true ) ),
inference(superposition,[status(thm)],[f1989,f1822]) ).
thf(f1822,plain,
! [X2: $i,X3: $i] :
( ( ( in @ X2 @ ( setadjoin @ X3 @ ( setadjoin @ X2 @ emptyset ) ) )
= $true )
| ( secondinupair != $true ) ),
inference(cnf_transformation,[status(thm)],[f1154]) ).
thf(f1154,plain,
( ( ( secondinupair = $true )
| ( ( in @ sK329 @ ( setadjoin @ sK330 @ ( setadjoin @ sK329 @ emptyset ) ) )
!= $true ) )
& ( ! [X2: $i,X3: $i] :
( ( in @ X2 @ ( setadjoin @ X3 @ ( setadjoin @ X2 @ emptyset ) ) )
= $true )
| ( secondinupair != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK329,sK330])],[f1152,f1153]) ).
thf(f1153,plain,
( ? [X0: $i,X1: $i] :
( ( in @ X0 @ ( setadjoin @ X1 @ ( setadjoin @ X0 @ emptyset ) ) )
!= $true )
=> ( ( in @ sK329 @ ( setadjoin @ sK330 @ ( setadjoin @ sK329 @ emptyset ) ) )
!= $true ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1152,plain,
( ( ( secondinupair = $true )
| ? [X0: $i,X1: $i] :
( ( in @ X0 @ ( setadjoin @ X1 @ ( setadjoin @ X0 @ emptyset ) ) )
!= $true ) )
& ( ! [X2: $i,X3: $i] :
( ( in @ X2 @ ( setadjoin @ X3 @ ( setadjoin @ X2 @ emptyset ) ) )
= $true )
| ( secondinupair != $true ) ) ),
inference(rectify,[status(thm)],[f1151]) ).
thf(f1151,plain,
( ( ( secondinupair = $true )
| ? [X0: $i,X1: $i] :
( ( in @ X0 @ ( setadjoin @ X1 @ ( setadjoin @ X0 @ emptyset ) ) )
!= $true ) )
& ( ! [X0: $i,X1: $i] :
( ( in @ X0 @ ( setadjoin @ X1 @ ( setadjoin @ X0 @ emptyset ) ) )
= $true )
| ( secondinupair != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f283]) ).
thf(f283,plain,
( ( secondinupair = $true )
<=> ! [X0: $i,X1: $i] :
( ( in @ X0 @ ( setadjoin @ X1 @ ( setadjoin @ X0 @ emptyset ) ) )
= $true ) ),
inference(fool_elimination,[status(thm)],[f282]) ).
thf(f282,plain,
( secondinupair
= ( ! [X0: $i,X1: $i] : ( in @ X0 @ ( setadjoin @ X1 @ ( setadjoin @ X0 @ emptyset ) ) ) ) ),
inference(rectify,[status(thm)],[f136]) ).
thf(f136,axiom,
( secondinupair
= ( ! [X2: $i,X1: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.BCzCW9enGb/DTF2THF_25972.p',secondinupair) ).
thf(f1989,plain,
! [X0: $i] :
( ( ( in @ X0 @ ( setadjoin @ ( setadjoin @ sK109 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK109 @ ( setadjoin @ sK108 @ emptyset ) ) @ emptyset ) ) )
!= $true )
| ( ( in @ sK108 @ X0 )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f1988]) ).
thf(f1988,plain,
! [X0: $i] :
( ( ( in @ sK108 @ X0 )
!= $true )
| ( ( in @ X0 @ ( setadjoin @ ( setadjoin @ sK109 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK109 @ ( setadjoin @ sK108 @ emptyset ) ) @ emptyset ) ) )
!= $true )
| ( $true != $true ) ),
inference(forward_demodulation,[status(thm)],[f1981,f1379]) ).
thf(f1379,plain,
setunionI = $true,
inference(cnf_transformation,[status(thm)],[f774]) ).
thf(f1981,plain,
! [X0: $i] :
( ( setunionI != $true )
| ( ( in @ X0 @ ( setadjoin @ ( setadjoin @ sK109 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK109 @ ( setadjoin @ sK108 @ emptyset ) ) @ emptyset ) ) )
!= $true )
| ( ( in @ sK108 @ X0 )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f1973]) ).
thf(f1973,plain,
! [X0: $i] :
( ( ( in @ sK108 @ X0 )
!= $true )
| ( $true != $true )
| ( ( in @ X0 @ ( setadjoin @ ( setadjoin @ sK109 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK109 @ ( setadjoin @ sK108 @ emptyset ) ) @ emptyset ) ) )
!= $true )
| ( setunionI != $true ) ),
inference(superposition,[status(thm)],[f1430,f1788]) ).
thf(f1788,plain,
! [X2: $i,X0: $i,X1: $i] :
( ( ( in @ X1 @ ( setunion @ X2 ) )
= $true )
| ( ( in @ X1 @ X0 )
!= $true )
| ( ( in @ X0 @ X2 )
!= $true )
| ( setunionI != $true ) ),
inference(cnf_transformation,[status(thm)],[f1101]) ).
thf(f1101,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X1 @ ( setunion @ X2 ) )
= $true )
| ( ( in @ X0 @ X2 )
!= $true )
| ( ( in @ X1 @ X0 )
!= $true ) )
| ( setunionI != $true ) )
& ( ( setunionI = $true )
| ( ( ( in @ sK301 @ ( setunion @ sK302 ) )
!= $true )
& ( ( in @ sK300 @ sK302 )
= $true )
& ( ( in @ sK301 @ sK300 )
= $true ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK300,sK301,sK302])],[f1099,f1100]) ).
thf(f1100,plain,
( ? [X3: $i,X4: $i,X5: $i] :
( ( ( in @ X4 @ ( setunion @ X5 ) )
!= $true )
& ( ( in @ X3 @ X5 )
= $true )
& ( ( in @ X4 @ X3 )
= $true ) )
=> ( ( ( in @ sK301 @ ( setunion @ sK302 ) )
!= $true )
& ( ( in @ sK300 @ sK302 )
= $true )
& ( ( in @ sK301 @ sK300 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1099,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X1 @ ( setunion @ X2 ) )
= $true )
| ( ( in @ X0 @ X2 )
!= $true )
| ( ( in @ X1 @ X0 )
!= $true ) )
| ( setunionI != $true ) )
& ( ( setunionI = $true )
| ? [X3: $i,X4: $i,X5: $i] :
( ( ( in @ X4 @ ( setunion @ X5 ) )
!= $true )
& ( ( in @ X3 @ X5 )
= $true )
& ( ( in @ X4 @ X3 )
= $true ) ) ) ),
inference(rectify,[status(thm)],[f1098]) ).
thf(f1098,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X1 @ ( setunion @ X2 ) )
= $true )
| ( ( in @ X0 @ X2 )
!= $true )
| ( ( in @ X1 @ X0 )
!= $true ) )
| ( setunionI != $true ) )
& ( ( setunionI = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X1 @ ( setunion @ X2 ) )
!= $true )
& ( ( in @ X0 @ X2 )
= $true )
& ( ( in @ X1 @ X0 )
= $true ) ) ) ),
inference(nnf_transformation,[status(thm)],[f574]) ).
thf(f574,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X1 @ ( setunion @ X2 ) )
= $true )
| ( ( in @ X0 @ X2 )
!= $true )
| ( ( in @ X1 @ X0 )
!= $true ) )
<=> ( setunionI = $true ) ),
inference(flattening,[status(thm)],[f573]) ).
thf(f573,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X1 @ ( setunion @ X2 ) )
= $true )
| ( ( in @ X0 @ X2 )
!= $true )
| ( ( in @ X1 @ X0 )
!= $true ) )
<=> ( setunionI = $true ) ),
inference(ennf_transformation,[status(thm)],[f253]) ).
thf(f253,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X1 @ X0 )
= $true )
=> ( ( ( in @ X0 @ X2 )
= $true )
=> ( ( in @ X1 @ ( setunion @ X2 ) )
= $true ) ) )
<=> ( setunionI = $true ) ),
inference(fool_elimination,[status(thm)],[f252]) ).
thf(f252,plain,
! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X1 @ X0 )
=> ( ( in @ X0 @ X2 )
=> ( in @ X1 @ ( setunion @ X2 ) ) ) )
= setunionI ),
inference(rectify,[status(thm)],[f41]) ).
thf(f41,axiom,
! [X4: $i,X1: $i,X3: $i] :
( ( ( in @ X1 @ X4 )
=> ( ( in @ X4 @ X3 )
=> ( in @ X1 @ ( setunion @ X3 ) ) ) )
= setunionI ),
file('/export/starexec/sandbox/tmp/tmp.BCzCW9enGb/DTF2THF_25972.p',setunionI) ).
thf(f1430,plain,
( ( in @ sK108 @ ( setunion @ ( setadjoin @ ( setadjoin @ sK109 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK109 @ ( setadjoin @ sK108 @ emptyset ) ) @ emptyset ) ) ) )
!= $true ),
inference(cnf_transformation,[status(thm)],[f774]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SEU618^1 : 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
% 0.15/0.34 % Computer : n020.cluster.edu
% 0.15/0.34 % Model : x86_64 x86_64
% 0.15/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.34 % Memory : 8042.1875MB
% 0.15/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.34 % CPULimit : 300
% 0.15/0.34 % WCLimit : 300
% 0.15/0.34 % DateTime : Tue Feb 24 12:28:42 EST 2026
% 0.15/0.34 % CPUTime :
% 0.15/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.29/0.45 ---- Original DTF file ---
% 0.29/0.46 thf(spec,logic,$$dhol).
% 0.29/0.46 %------------------------------------------------------------------------------
% 0.29/0.46 % File : SEU618^1 : TPTP v9.2.1. Released v3.7.0.
% 0.29/0.46 % Domain : Set Theory
% 0.29/0.46 % Problem : Ordered Pairs - Kuratowski Pairs
% 0.29/0.46 % Version : Especial.
% 0.29/0.46 % English : (! x:i.! y:i.in y (setunion (setadjoin (setadjoin x emptyset)
% 0.29/0.46 % (setadjoin (setadjoin x (setadjoin y emptyset)) emptyset))))
% 0.29/0.46
% 0.29/0.46 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.29/0.46 % Source : [Bro08]
% 0.29/0.46 % Names : ZFC120g [Bro08]
% 0.29/0.46
% 0.29/0.46 % Status : Theorem
% 0.29/0.46 % Rating : 0.11 v9.1.0, 0.12 v9.0.0, 0.20 v8.2.0, 0.31 v8.1.0, 0.18 v7.5.0, 0.14 v7.4.0, 0.22 v7.2.0, 0.12 v7.1.0, 0.38 v7.0.0, 0.43 v6.4.0, 0.50 v6.3.0, 0.60 v6.2.0, 0.43 v5.5.0, 0.67 v5.4.0, 0.80 v5.0.0, 1.00 v3.7.0
% 0.29/0.46 % Syntax : Number of formulae : 293 ( 137 unt; 155 typ; 137 def)
% 0.29/0.46 % Number of atoms : 730 ( 195 equ; 0 cnn)
% 0.29/0.46 % Maximal formula atoms : 135 ( 5 avg)
% 0.29/0.46 % Number of connectives : 1204 ( 47 ~; 6 |; 30 &; 783 @)
% 0.29/0.46 % ( 14 <=>; 324 =>; 0 <=; 0 <~>)
% 0.29/0.46 % Maximal formula depth : 145 ( 2 avg)
% 0.29/0.46 % Number of types : 2 ( 0 usr)
% 0.29/0.46 % Number of type conns : 65 ( 65 >; 0 *; 0 +; 0 <<)
% 0.29/0.46 % Number of symbols : 158 ( 155 usr; 138 con; 0-2 aty)
% 0.29/0.46 % Number of variables : 395 ( 31 ^; 333 !; 31 ?; 395 :)
% 0.29/0.46 % SPC : TH0_THM_EQU_NAR
% 0.29/0.46
% 0.29/0.46 % Comments : http://mathgate.info/detsetitem.php?id=173
% 0.29/0.46 %------------------------------------------------------------------------------
% 0.29/0.46 thf(in_type,type,
% 0.29/0.46 in: $i > $i > $o ).
% 0.29/0.46
% 0.29/0.46 thf(exu_type,type,
% 0.29/0.46 exu: ( $i > $o ) > $o ).
% 0.29/0.46
% 0.29/0.46 thf(exu,definition,
% 0.29/0.46 ( exu
% 0.29/0.46 = ( ^ [Xphi: $i > $o] :
% 0.29/0.46 ? [Xx: $i] :
% 0.29/0.46 ( ( Xphi @ Xx )
% 0.29/0.46 & ! [Xy: $i] :
% 0.29/0.46 ( ( Xphi @ Xy )
% 0.29/0.46 => ( Xx = Xy ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(setextAx_type,type,
% 0.29/0.46 setextAx: $o ).
% 0.29/0.46
% 0.29/0.46 thf(setextAx,definition,
% 0.29/0.46 ( setextAx
% 0.29/0.46 = ( ! [A: $i,B: $i] :
% 0.29/0.46 ( ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 <=> ( in @ Xx @ B ) )
% 0.29/0.46 => ( A = B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(emptyset_type,type,
% 0.29/0.46 emptyset: $i ).
% 0.29/0.46
% 0.29/0.46 thf(emptysetAx_type,type,
% 0.29/0.46 emptysetAx: $o ).
% 0.29/0.46
% 0.29/0.46 thf(emptysetAx,definition,
% 0.29/0.46 ( emptysetAx
% 0.29/0.46 = ( ! [Xx: $i] :
% 0.29/0.46 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(setadjoin_type,type,
% 0.29/0.46 setadjoin: $i > $i > $i ).
% 0.29/0.46
% 0.29/0.46 thf(setadjoinAx_type,type,
% 0.29/0.46 setadjoinAx: $o ).
% 0.29/0.46
% 0.29/0.46 thf(setadjoinAx,definition,
% 0.29/0.46 ( setadjoinAx
% 0.29/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.29/0.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.29/0.46 <=> ( ( Xy = Xx )
% 0.29/0.46 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(powerset_type,type,
% 0.29/0.46 powerset: $i > $i ).
% 0.29/0.46
% 0.29/0.46 thf(powersetAx_type,type,
% 0.29/0.46 powersetAx: $o ).
% 0.29/0.46
% 0.29/0.46 thf(powersetAx,definition,
% 0.29/0.46 ( powersetAx
% 0.29/0.46 = ( ! [A: $i,B: $i] :
% 0.29/0.46 ( ( in @ B @ ( powerset @ A ) )
% 0.29/0.46 <=> ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ B )
% 0.29/0.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(setunion_type,type,
% 0.29/0.46 setunion: $i > $i ).
% 0.29/0.46
% 0.29/0.46 thf(setunionAx_type,type,
% 0.29/0.46 setunionAx: $o ).
% 0.29/0.46
% 0.29/0.46 thf(setunionAx,definition,
% 0.29/0.46 ( setunionAx
% 0.29/0.46 = ( ! [A: $i,Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ ( setunion @ A ) )
% 0.29/0.46 <=> ? [B: $i] :
% 0.29/0.46 ( ( in @ Xx @ B )
% 0.29/0.46 & ( in @ B @ A ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(omega_type,type,
% 0.29/0.46 omega: $i ).
% 0.29/0.46
% 0.29/0.46 thf(omega0Ax_type,type,
% 0.29/0.46 omega0Ax: $o ).
% 0.29/0.46
% 0.29/0.46 thf(omega0Ax,definition,
% 0.29/0.46 ( omega0Ax
% 0.29/0.46 = ( in @ emptyset @ omega ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(omegaSAx_type,type,
% 0.29/0.46 omegaSAx: $o ).
% 0.29/0.46
% 0.29/0.46 thf(omegaSAx,definition,
% 0.29/0.46 ( omegaSAx
% 0.29/0.46 = ( ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ omega )
% 0.29/0.46 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(omegaIndAx_type,type,
% 0.29/0.46 omegaIndAx: $o ).
% 0.29/0.46
% 0.29/0.46 thf(omegaIndAx,definition,
% 0.29/0.46 ( omegaIndAx
% 0.29/0.46 = ( ! [A: $i] :
% 0.29/0.46 ( ( ( in @ emptyset @ A )
% 0.29/0.46 & ! [Xx: $i] :
% 0.29/0.46 ( ( ( in @ Xx @ omega )
% 0.29/0.46 & ( in @ Xx @ A ) )
% 0.29/0.46 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.29/0.46 => ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ omega )
% 0.29/0.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(replAx_type,type,
% 0.29/0.46 replAx: $o ).
% 0.29/0.46
% 0.29/0.46 thf(replAx,definition,
% 0.29/0.46 ( replAx
% 0.29/0.46 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.29/0.46 ( ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ( exu
% 0.29/0.46 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.29/0.46 => ? [B: $i] :
% 0.29/0.46 ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ B )
% 0.29/0.46 <=> ? [Xy: $i] :
% 0.29/0.46 ( ( in @ Xy @ A )
% 0.29/0.46 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(foundationAx_type,type,
% 0.29/0.46 foundationAx: $o ).
% 0.29/0.46
% 0.29/0.46 thf(foundationAx,definition,
% 0.29/0.46 ( foundationAx
% 0.29/0.46 = ( ! [A: $i] :
% 0.29/0.46 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.29/0.46 => ? [B: $i] :
% 0.29/0.46 ( ( in @ B @ A )
% 0.29/0.46 & ~ ? [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ B )
% 0.29/0.46 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(wellorderingAx_type,type,
% 0.29/0.46 wellorderingAx: $o ).
% 0.29/0.46
% 0.29/0.46 thf(wellorderingAx,definition,
% 0.29/0.46 ( wellorderingAx
% 0.29/0.46 = ( ! [A: $i] :
% 0.29/0.46 ? [B: $i] :
% 0.29/0.46 ( ! [C: $i] :
% 0.29/0.46 ( ( in @ C @ B )
% 0.29/0.46 => ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ C )
% 0.29/0.46 => ( in @ Xx @ A ) ) )
% 0.29/0.46 & ! [Xx: $i,Xy: $i] :
% 0.29/0.46 ( ( ( in @ Xx @ A )
% 0.29/0.46 & ( in @ Xy @ A ) )
% 0.29/0.46 => ( ! [C: $i] :
% 0.29/0.46 ( ( in @ C @ B )
% 0.29/0.46 => ( ( in @ Xx @ C )
% 0.29/0.46 <=> ( in @ Xy @ C ) ) )
% 0.29/0.46 => ( Xx = Xy ) ) )
% 0.29/0.46 & ! [C: $i,D: $i] :
% 0.29/0.46 ( ( ( in @ C @ B )
% 0.29/0.46 & ( in @ D @ B ) )
% 0.29/0.46 => ( ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ C )
% 0.29/0.46 => ( in @ Xx @ D ) )
% 0.29/0.46 | ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ D )
% 0.29/0.46 => ( in @ Xx @ C ) ) ) )
% 0.29/0.46 & ! [C: $i] :
% 0.29/0.46 ( ( ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ C )
% 0.29/0.46 => ( in @ Xx @ A ) )
% 0.29/0.46 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.29/0.46 => ? [D: $i,Xx: $i] :
% 0.29/0.46 ( ( in @ D @ B )
% 0.29/0.46 & ( in @ Xx @ C )
% 0.29/0.46 & ~ ? [Xy: $i] :
% 0.29/0.46 ( ( in @ Xy @ D )
% 0.29/0.46 & ( in @ Xy @ C ) )
% 0.29/0.46 & ! [E: $i] :
% 0.29/0.46 ( ( in @ E @ B )
% 0.29/0.46 => ( ! [Xy: $i] :
% 0.29/0.46 ( ( in @ Xy @ E )
% 0.29/0.46 => ( in @ Xy @ D ) )
% 0.29/0.46 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(descr_type,type,
% 0.29/0.46 descr: ( $i > $o ) > $i ).
% 0.29/0.46
% 0.29/0.46 thf(descrp_type,type,
% 0.29/0.46 descrp: $o ).
% 0.29/0.46
% 0.29/0.46 thf(descrp,definition,
% 0.29/0.46 ( descrp
% 0.29/0.46 = ( ! [Xphi: $i > $o] :
% 0.29/0.46 ( ( exu
% 0.29/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.29/0.46 => ( Xphi
% 0.29/0.46 @ ( descr
% 0.29/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(dsetconstr_type,type,
% 0.29/0.46 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.29/0.46
% 0.29/0.46 thf(dsetconstrI_type,type,
% 0.29/0.46 dsetconstrI: $o ).
% 0.29/0.46
% 0.29/0.46 thf(dsetconstrI,definition,
% 0.29/0.46 ( dsetconstrI
% 0.29/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ( ( Xphi @ Xx )
% 0.29/0.46 => ( in @ Xx
% 0.29/0.46 @ ( dsetconstr @ A
% 0.29/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(dsetconstrEL_type,type,
% 0.29/0.46 dsetconstrEL: $o ).
% 0.29/0.46
% 0.29/0.46 thf(dsetconstrEL,definition,
% 0.29/0.46 ( dsetconstrEL
% 0.29/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.29/0.46 ( ( in @ Xx
% 0.29/0.46 @ ( dsetconstr @ A
% 0.29/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.29/0.46 => ( in @ Xx @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(dsetconstrER_type,type,
% 0.29/0.46 dsetconstrER: $o ).
% 0.29/0.46
% 0.29/0.46 thf(dsetconstrER,definition,
% 0.29/0.46 ( dsetconstrER
% 0.29/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.29/0.46 ( ( in @ Xx
% 0.29/0.46 @ ( dsetconstr @ A
% 0.29/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.29/0.46 => ( Xphi @ Xx ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(exuE1_type,type,
% 0.29/0.46 exuE1: $o ).
% 0.29/0.46
% 0.29/0.46 thf(exuE1,definition,
% 0.29/0.46 ( exuE1
% 0.29/0.46 = ( ! [Xphi: $i > $o] :
% 0.29/0.46 ( ( exu
% 0.29/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.29/0.46 => ? [Xx: $i] :
% 0.29/0.46 ( ( Xphi @ Xx )
% 0.29/0.46 & ! [Xy: $i] :
% 0.29/0.46 ( ( Xphi @ Xy )
% 0.29/0.46 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(prop2set_type,type,
% 0.29/0.46 prop2set: $o > $i ).
% 0.29/0.46
% 0.29/0.46 thf(prop2setE_type,type,
% 0.29/0.46 prop2setE: $o ).
% 0.29/0.46
% 0.29/0.46 thf(prop2setE,definition,
% 0.29/0.46 ( prop2setE
% 0.29/0.46 = ( ! [Xphi: $o,Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.29/0.46 => Xphi ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(emptysetE_type,type,
% 0.29/0.46 emptysetE: $o ).
% 0.29/0.46
% 0.29/0.46 thf(emptysetE,definition,
% 0.29/0.46 ( emptysetE
% 0.29/0.46 = ( ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ emptyset )
% 0.29/0.46 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(emptysetimpfalse_type,type,
% 0.29/0.46 emptysetimpfalse: $o ).
% 0.29/0.46
% 0.29/0.46 thf(emptysetimpfalse,definition,
% 0.29/0.46 ( emptysetimpfalse
% 0.29/0.46 = ( ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ emptyset )
% 0.29/0.46 => $false ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(notinemptyset_type,type,
% 0.29/0.46 notinemptyset: $o ).
% 0.29/0.46
% 0.29/0.46 thf(notinemptyset,definition,
% 0.29/0.46 ( notinemptyset
% 0.29/0.46 = ( ! [Xx: $i] :
% 0.29/0.46 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(exuE3e_type,type,
% 0.29/0.46 exuE3e: $o ).
% 0.29/0.46
% 0.29/0.46 thf(exuE3e,definition,
% 0.29/0.46 ( exuE3e
% 0.29/0.46 = ( ! [Xphi: $i > $o] :
% 0.29/0.46 ( ( exu
% 0.29/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.29/0.46 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(setext_type,type,
% 0.29/0.46 setext: $o ).
% 0.29/0.46
% 0.29/0.46 thf(setext,definition,
% 0.29/0.46 ( setext
% 0.29/0.46 = ( ! [A: $i,B: $i] :
% 0.29/0.46 ( ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ( in @ Xx @ B ) )
% 0.29/0.46 => ( ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ B )
% 0.29/0.46 => ( in @ Xx @ A ) )
% 0.29/0.46 => ( A = B ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(emptyI_type,type,
% 0.29/0.46 emptyI: $o ).
% 0.29/0.46
% 0.29/0.46 thf(emptyI,definition,
% 0.29/0.46 ( emptyI
% 0.29/0.46 = ( ! [A: $i] :
% 0.29/0.46 ( ! [Xx: $i] :
% 0.29/0.46 ~ ( in @ Xx @ A )
% 0.29/0.46 => ( A = emptyset ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(noeltsimpempty_type,type,
% 0.29/0.46 noeltsimpempty: $o ).
% 0.29/0.46
% 0.29/0.46 thf(noeltsimpempty,definition,
% 0.29/0.46 ( noeltsimpempty
% 0.29/0.46 = ( ! [A: $i] :
% 0.29/0.46 ( ! [Xx: $i] :
% 0.29/0.46 ~ ( in @ Xx @ A )
% 0.29/0.46 => ( A = emptyset ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(setbeta_type,type,
% 0.29/0.46 setbeta: $o ).
% 0.29/0.46
% 0.29/0.46 thf(setbeta,definition,
% 0.29/0.46 ( setbeta
% 0.29/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ( ( in @ Xx
% 0.29/0.46 @ ( dsetconstr @ A
% 0.29/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.29/0.46 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(nonempty_type,type,
% 0.29/0.46 nonempty: $i > $o ).
% 0.29/0.46
% 0.29/0.46 thf(nonempty,definition,
% 0.29/0.46 ( nonempty
% 0.29/0.46 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(nonemptyE1_type,type,
% 0.29/0.46 nonemptyE1: $o ).
% 0.29/0.46
% 0.29/0.46 thf(nonemptyE1,definition,
% 0.29/0.46 ( nonemptyE1
% 0.29/0.46 = ( ! [A: $i] :
% 0.29/0.46 ( ( nonempty @ A )
% 0.29/0.46 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(nonemptyI_type,type,
% 0.29/0.46 nonemptyI: $o ).
% 0.29/0.46
% 0.29/0.46 thf(nonemptyI,definition,
% 0.29/0.46 ( nonemptyI
% 0.29/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ( ( Xphi @ Xx )
% 0.29/0.46 => ( nonempty
% 0.29/0.46 @ ( dsetconstr @ A
% 0.29/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(nonemptyI1_type,type,
% 0.29/0.46 nonemptyI1: $o ).
% 0.29/0.46
% 0.29/0.46 thf(nonemptyI1,definition,
% 0.29/0.46 ( nonemptyI1
% 0.29/0.46 = ( ! [A: $i] :
% 0.29/0.46 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.29/0.46 => ( nonempty @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(setadjoinIL_type,type,
% 0.29/0.46 setadjoinIL: $o ).
% 0.29/0.46
% 0.29/0.46 thf(setadjoinIL,definition,
% 0.29/0.46 ( setadjoinIL
% 0.29/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(emptyinunitempty_type,type,
% 0.29/0.46 emptyinunitempty: $o ).
% 0.29/0.46
% 0.29/0.46 thf(emptyinunitempty,definition,
% 0.29/0.46 ( emptyinunitempty
% 0.29/0.46 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(setadjoinIR_type,type,
% 0.29/0.46 setadjoinIR: $o ).
% 0.29/0.46
% 0.29/0.46 thf(setadjoinIR,definition,
% 0.29/0.46 ( setadjoinIR
% 0.29/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.29/0.46 ( ( in @ Xy @ A )
% 0.29/0.46 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(setadjoinE_type,type,
% 0.29/0.46 setadjoinE: $o ).
% 0.29/0.46
% 0.29/0.46 thf(setadjoinE,definition,
% 0.29/0.46 ( setadjoinE
% 0.29/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.29/0.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.29/0.46 => ! [Xphi: $o] :
% 0.29/0.46 ( ( ( Xy = Xx )
% 0.29/0.46 => Xphi )
% 0.29/0.46 => ( ( ( in @ Xy @ A )
% 0.29/0.46 => Xphi )
% 0.29/0.46 => Xphi ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(setadjoinOr_type,type,
% 0.29/0.46 setadjoinOr: $o ).
% 0.29/0.46
% 0.29/0.46 thf(setadjoinOr,definition,
% 0.29/0.46 ( setadjoinOr
% 0.29/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.29/0.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.29/0.46 => ( ( Xy = Xx )
% 0.29/0.46 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(setoftrueEq_type,type,
% 0.29/0.46 setoftrueEq: $o ).
% 0.29/0.46
% 0.29/0.46 thf(setoftrueEq,definition,
% 0.29/0.46 ( setoftrueEq
% 0.29/0.46 = ( ! [A: $i] :
% 0.29/0.46 ( ( dsetconstr @ A
% 0.29/0.46 @ ^ [Xx: $i] : $true )
% 0.29/0.46 = A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(powersetI_type,type,
% 0.29/0.46 powersetI: $o ).
% 0.29/0.46
% 0.29/0.46 thf(powersetI,definition,
% 0.29/0.46 ( powersetI
% 0.29/0.46 = ( ! [A: $i,B: $i] :
% 0.29/0.46 ( ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ B )
% 0.29/0.46 => ( in @ Xx @ A ) )
% 0.29/0.46 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(emptyinPowerset_type,type,
% 0.29/0.46 emptyinPowerset: $o ).
% 0.29/0.46
% 0.29/0.46 thf(emptyinPowerset,definition,
% 0.29/0.46 ( emptyinPowerset
% 0.29/0.46 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(emptyInPowerset_type,type,
% 0.29/0.46 emptyInPowerset: $o ).
% 0.29/0.46
% 0.29/0.46 thf(emptyInPowerset,definition,
% 0.29/0.46 ( emptyInPowerset
% 0.29/0.46 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(powersetE_type,type,
% 0.29/0.46 powersetE: $o ).
% 0.29/0.46
% 0.29/0.46 thf(powersetE,definition,
% 0.29/0.46 ( powersetE
% 0.29/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.46 ( ( in @ B @ ( powerset @ A ) )
% 0.29/0.46 => ( ( in @ Xx @ B )
% 0.29/0.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(setunionI_type,type,
% 0.29/0.46 setunionI: $o ).
% 0.29/0.46
% 0.29/0.46 thf(setunionI,definition,
% 0.29/0.46 ( setunionI
% 0.29/0.46 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.29/0.46 ( ( in @ Xx @ B )
% 0.29/0.46 => ( ( in @ B @ A )
% 0.29/0.46 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(setunionE_type,type,
% 0.29/0.46 setunionE: $o ).
% 0.29/0.46
% 0.29/0.46 thf(setunionE,definition,
% 0.29/0.46 ( setunionE
% 0.29/0.46 = ( ! [A: $i,Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ ( setunion @ A ) )
% 0.29/0.46 => ! [Xphi: $o] :
% 0.29/0.46 ( ! [B: $i] :
% 0.29/0.46 ( ( in @ Xx @ B )
% 0.29/0.46 => ( ( in @ B @ A )
% 0.29/0.46 => Xphi ) )
% 0.29/0.46 => Xphi ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(subPowSU_type,type,
% 0.29/0.46 subPowSU: $o ).
% 0.29/0.46
% 0.29/0.46 thf(subPowSU,definition,
% 0.29/0.46 ( subPowSU
% 0.29/0.46 = ( ! [A: $i,Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(exuE2_type,type,
% 0.29/0.46 exuE2: $o ).
% 0.29/0.46
% 0.29/0.46 thf(exuE2,definition,
% 0.29/0.46 ( exuE2
% 0.29/0.46 = ( ! [Xphi: $i > $o] :
% 0.29/0.46 ( ( exu
% 0.29/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.29/0.46 => ? [Xx: $i] :
% 0.29/0.46 ! [Xy: $i] :
% 0.29/0.46 ( ( Xphi @ Xy )
% 0.29/0.46 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(nonemptyImpWitness_type,type,
% 0.29/0.46 nonemptyImpWitness: $o ).
% 0.29/0.46
% 0.29/0.46 thf(nonemptyImpWitness,definition,
% 0.29/0.46 ( nonemptyImpWitness
% 0.29/0.46 = ( ! [A: $i] :
% 0.29/0.46 ( ( nonempty @ A )
% 0.29/0.46 => ? [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 & $true ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(uniqinunit_type,type,
% 0.29/0.46 uniqinunit: $o ).
% 0.29/0.46
% 0.29/0.46 thf(uniqinunit,definition,
% 0.29/0.46 ( uniqinunit
% 0.29/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.29/0.46 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.29/0.46 => ( Xx = Xy ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(notinsingleton_type,type,
% 0.29/0.46 notinsingleton: $o ).
% 0.29/0.46
% 0.29/0.46 thf(notinsingleton,definition,
% 0.29/0.46 ( notinsingleton
% 0.29/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.29/0.46 ( ( Xx != Xy )
% 0.29/0.46 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(eqinunit_type,type,
% 0.29/0.46 eqinunit: $o ).
% 0.29/0.46
% 0.29/0.46 thf(eqinunit,definition,
% 0.29/0.46 ( eqinunit
% 0.29/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.29/0.46 ( ( Xx = Xy )
% 0.29/0.46 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(singletonsswitch_type,type,
% 0.29/0.46 singletonsswitch: $o ).
% 0.29/0.46
% 0.29/0.46 thf(singletonsswitch,definition,
% 0.29/0.46 ( singletonsswitch
% 0.29/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.29/0.46 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.29/0.46 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(upairsetE_type,type,
% 0.29/0.46 upairsetE: $o ).
% 0.29/0.46
% 0.29/0.46 thf(upairsetE,definition,
% 0.29/0.46 ( upairsetE
% 0.29/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.29/0.46 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.29/0.46 => ( ( Xz = Xx )
% 0.29/0.46 | ( Xz = Xy ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(upairsetIL_type,type,
% 0.29/0.46 upairsetIL: $o ).
% 0.29/0.46
% 0.29/0.46 thf(upairsetIL,definition,
% 0.29/0.46 ( upairsetIL
% 0.29/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(upairsetIR_type,type,
% 0.29/0.46 upairsetIR: $o ).
% 0.29/0.46
% 0.29/0.46 thf(upairsetIR,definition,
% 0.29/0.46 ( upairsetIR
% 0.29/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(emptyE1_type,type,
% 0.29/0.46 emptyE1: $o ).
% 0.29/0.46
% 0.29/0.46 thf(emptyE1,definition,
% 0.29/0.46 ( emptyE1
% 0.29/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.29/0.46 ( ? [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 & ( Xphi @ Xx ) )
% 0.29/0.46 => ( ( ( dsetconstr @ A
% 0.29/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.29/0.46 = emptyset )
% 0.29/0.46 => $false ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(vacuousDall_type,type,
% 0.29/0.46 vacuousDall: $o ).
% 0.29/0.46
% 0.29/0.46 thf(vacuousDall,definition,
% 0.29/0.46 ( vacuousDall
% 0.29/0.46 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ emptyset )
% 0.29/0.46 => ( Xphi @ Xx ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(quantDeMorgan1_type,type,
% 0.29/0.46 quantDeMorgan1: $o ).
% 0.29/0.46
% 0.29/0.46 thf(quantDeMorgan1,definition,
% 0.29/0.46 ( quantDeMorgan1
% 0.29/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.29/0.46 ( ~ ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ( Xphi @ Xx ) )
% 0.29/0.46 => ? [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(quantDeMorgan2_type,type,
% 0.29/0.46 quantDeMorgan2: $o ).
% 0.29/0.46
% 0.29/0.46 thf(quantDeMorgan2,definition,
% 0.29/0.46 ( quantDeMorgan2
% 0.29/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.29/0.46 ( ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ~ ( Xphi @ Xx ) )
% 0.29/0.46 => ~ ? [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(quantDeMorgan3_type,type,
% 0.29/0.46 quantDeMorgan3: $o ).
% 0.29/0.46
% 0.29/0.46 thf(quantDeMorgan3,definition,
% 0.29/0.46 ( quantDeMorgan3
% 0.29/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.29/0.46 ( ~ ? [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 & ( Xphi @ Xx ) )
% 0.29/0.46 => ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(quantDeMorgan4_type,type,
% 0.29/0.46 quantDeMorgan4: $o ).
% 0.29/0.46
% 0.29/0.46 thf(quantDeMorgan4,definition,
% 0.29/0.46 ( quantDeMorgan4
% 0.29/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.29/0.46 ( ? [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 & ~ ( Xphi @ Xx ) )
% 0.29/0.46 => ~ ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(prop2setI_type,type,
% 0.29/0.46 prop2setI: $o ).
% 0.29/0.46
% 0.29/0.46 thf(prop2setI,definition,
% 0.29/0.46 ( prop2setI
% 0.29/0.46 = ( ! [Xphi: $o] :
% 0.29/0.46 ( Xphi
% 0.29/0.46 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(set2prop_type,type,
% 0.29/0.46 set2prop: $i > $o ).
% 0.29/0.46
% 0.29/0.46 thf(prop2set2propI_type,type,
% 0.29/0.46 prop2set2propI: $o ).
% 0.29/0.46
% 0.29/0.46 thf(prop2set2propI,definition,
% 0.29/0.46 ( prop2set2propI
% 0.29/0.46 = ( ! [Xphi: $o] :
% 0.29/0.46 ( Xphi
% 0.29/0.46 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(notdexE_type,type,
% 0.29/0.46 notdexE: $o ).
% 0.29/0.46
% 0.29/0.46 thf(notdexE,definition,
% 0.29/0.46 ( notdexE
% 0.29/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.29/0.46 ( ~ ? [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 & ( Xphi @ Xx ) )
% 0.29/0.46 => ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(notdallE_type,type,
% 0.29/0.46 notdallE: $o ).
% 0.29/0.46
% 0.29/0.46 thf(notdallE,definition,
% 0.29/0.46 ( notdallE
% 0.29/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.29/0.46 ( ~ ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ( Xphi @ Xx ) )
% 0.29/0.46 => ? [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(exuI1_type,type,
% 0.29/0.46 exuI1: $o ).
% 0.29/0.46
% 0.29/0.46 thf(exuI1,definition,
% 0.29/0.46 ( exuI1
% 0.29/0.46 = ( ! [Xphi: $i > $o] :
% 0.29/0.46 ( ? [Xx: $i] :
% 0.29/0.46 ( ( Xphi @ Xx )
% 0.29/0.46 & ! [Xy: $i] :
% 0.29/0.46 ( ( Xphi @ Xy )
% 0.29/0.46 => ( Xx = Xy ) ) )
% 0.29/0.46 => ( exu
% 0.29/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(exuI3_type,type,
% 0.29/0.46 exuI3: $o ).
% 0.29/0.46
% 0.29/0.46 thf(exuI3,definition,
% 0.29/0.46 ( exuI3
% 0.29/0.46 = ( ! [Xphi: $i > $o] :
% 0.29/0.46 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.29/0.46 => ( ! [Xx: $i,Xy: $i] :
% 0.29/0.46 ( ( Xphi @ Xx )
% 0.29/0.46 => ( ( Xphi @ Xy )
% 0.29/0.46 => ( Xx = Xy ) ) )
% 0.29/0.46 => ( exu
% 0.29/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(exuI2_type,type,
% 0.29/0.46 exuI2: $o ).
% 0.29/0.46
% 0.29/0.46 thf(exuI2,definition,
% 0.29/0.46 ( exuI2
% 0.29/0.46 = ( ! [Xphi: $i > $o] :
% 0.29/0.46 ( ? [Xx: $i] :
% 0.29/0.46 ! [Xy: $i] :
% 0.29/0.46 ( ( Xphi @ Xy )
% 0.29/0.46 <=> ( Xy = Xx ) )
% 0.29/0.46 => ( exu
% 0.29/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(inCongP_type,type,
% 0.29/0.46 inCongP: $o ).
% 0.29/0.46
% 0.29/0.46 thf(inCongP,definition,
% 0.29/0.46 ( inCongP
% 0.29/0.46 = ( ! [A: $i,B: $i] :
% 0.29/0.46 ( ( A = B )
% 0.29/0.46 => ! [Xx: $i,Xy: $i] :
% 0.29/0.46 ( ( Xx = Xy )
% 0.29/0.46 => ( ( in @ Xx @ A )
% 0.29/0.46 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(in__Cong_type,type,
% 0.29/0.46 in__Cong: $o ).
% 0.29/0.46
% 0.29/0.46 thf(in__Cong,definition,
% 0.29/0.46 ( in__Cong
% 0.29/0.46 = ( ! [A: $i,B: $i] :
% 0.29/0.46 ( ( A = B )
% 0.29/0.46 => ! [Xx: $i,Xy: $i] :
% 0.29/0.46 ( ( Xx = Xy )
% 0.29/0.46 => ( ( in @ Xx @ A )
% 0.29/0.46 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(exuE3u_type,type,
% 0.29/0.46 exuE3u: $o ).
% 0.29/0.46
% 0.29/0.46 thf(exuE3u,definition,
% 0.29/0.46 ( exuE3u
% 0.29/0.46 = ( ! [Xphi: $i > $o] :
% 0.29/0.46 ( ( exu
% 0.29/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.29/0.46 => ! [Xx: $i,Xy: $i] :
% 0.29/0.46 ( ( Xphi @ Xx )
% 0.29/0.46 => ( ( Xphi @ Xy )
% 0.29/0.46 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(exu__Cong_type,type,
% 0.29/0.46 exu__Cong: $o ).
% 0.29/0.46
% 0.29/0.46 thf(exu__Cong,definition,
% 0.29/0.46 ( exu__Cong
% 0.29/0.46 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.29/0.46 ( ! [Xx: $i,Xy: $i] :
% 0.29/0.46 ( ( Xx = Xy )
% 0.29/0.46 => ( ( Xphi @ Xx )
% 0.29/0.46 <=> ( Xpsi @ Xy ) ) )
% 0.29/0.46 => ( ( exu
% 0.29/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.29/0.46 <=> ( exu
% 0.29/0.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(emptyset__Cong_type,type,
% 0.29/0.46 emptyset__Cong: $o ).
% 0.29/0.46
% 0.29/0.46 thf(emptyset__Cong,definition,
% 0.29/0.46 ( emptyset__Cong
% 0.29/0.46 = ( emptyset = emptyset ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(setadjoin__Cong_type,type,
% 0.29/0.46 setadjoin__Cong: $o ).
% 0.29/0.46
% 0.29/0.46 thf(setadjoin__Cong,definition,
% 0.29/0.46 ( setadjoin__Cong
% 0.29/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.29/0.46 ( ( Xx = Xy )
% 0.29/0.46 => ! [Xz: $i,Xu: $i] :
% 0.29/0.46 ( ( Xz = Xu )
% 0.29/0.46 => ( ( setadjoin @ Xx @ Xz )
% 0.29/0.46 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(powerset__Cong_type,type,
% 0.29/0.46 powerset__Cong: $o ).
% 0.29/0.46
% 0.29/0.46 thf(powerset__Cong,definition,
% 0.29/0.46 ( powerset__Cong
% 0.29/0.46 = ( ! [A: $i,B: $i] :
% 0.29/0.46 ( ( A = B )
% 0.29/0.46 => ( ( powerset @ A )
% 0.29/0.46 = ( powerset @ B ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(setunion__Cong_type,type,
% 0.29/0.46 setunion__Cong: $o ).
% 0.29/0.46
% 0.29/0.46 thf(setunion__Cong,definition,
% 0.29/0.46 ( setunion__Cong
% 0.29/0.46 = ( ! [A: $i,B: $i] :
% 0.29/0.46 ( ( A = B )
% 0.29/0.46 => ( ( setunion @ A )
% 0.29/0.46 = ( setunion @ B ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(omega__Cong_type,type,
% 0.29/0.46 omega__Cong: $o ).
% 0.29/0.46
% 0.29/0.46 thf(omega__Cong,definition,
% 0.29/0.46 ( omega__Cong
% 0.29/0.46 = ( omega = omega ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(exuEu_type,type,
% 0.29/0.46 exuEu: $o ).
% 0.29/0.46
% 0.29/0.46 thf(exuEu,definition,
% 0.29/0.46 ( exuEu
% 0.29/0.46 = ( ! [Xphi: $i > $o] :
% 0.29/0.46 ( ( exu
% 0.29/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.29/0.46 => ! [Xx: $i,Xy: $i] :
% 0.29/0.46 ( ( Xphi @ Xx )
% 0.29/0.46 => ( ( Xphi @ Xy )
% 0.29/0.46 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(descr__Cong_type,type,
% 0.29/0.46 descr__Cong: $o ).
% 0.29/0.46
% 0.29/0.46 thf(descr__Cong,definition,
% 0.29/0.46 ( descr__Cong
% 0.29/0.46 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.29/0.46 ( ! [Xx: $i,Xy: $i] :
% 0.29/0.46 ( ( Xx = Xy )
% 0.29/0.46 => ( ( Xphi @ Xx )
% 0.29/0.46 <=> ( Xpsi @ Xy ) ) )
% 0.29/0.46 => ( ( exu
% 0.29/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.29/0.46 => ( ( exu
% 0.29/0.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.29/0.46 => ( ( descr
% 0.29/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.29/0.46 = ( descr
% 0.29/0.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(dsetconstr__Cong_type,type,
% 0.29/0.46 dsetconstr__Cong: $o ).
% 0.29/0.46
% 0.29/0.46 thf(dsetconstr__Cong,definition,
% 0.29/0.46 ( dsetconstr__Cong
% 0.29/0.46 = ( ! [A: $i,B: $i] :
% 0.29/0.46 ( ( A = B )
% 0.29/0.46 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.29/0.46 ( ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ! [Xy: $i] :
% 0.29/0.46 ( ( in @ Xy @ B )
% 0.29/0.46 => ( ( Xx = Xy )
% 0.29/0.46 => ( ( Xphi @ Xx )
% 0.29/0.46 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.29/0.46 => ( ( dsetconstr @ A
% 0.29/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.29/0.46 = ( dsetconstr @ B
% 0.29/0.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(subset_type,type,
% 0.29/0.46 subset: $i > $i > $o ).
% 0.29/0.46
% 0.29/0.46 thf(disjoint_type,type,
% 0.29/0.46 disjoint: $i > $i > $o ).
% 0.29/0.46
% 0.29/0.46 thf(setsmeet_type,type,
% 0.29/0.46 setsmeet: $i > $i > $o ).
% 0.29/0.46
% 0.29/0.46 thf(subsetI1_type,type,
% 0.29/0.46 subsetI1: $o ).
% 0.29/0.46
% 0.29/0.46 thf(subsetI1,definition,
% 0.29/0.46 ( subsetI1
% 0.29/0.46 = ( ! [A: $i,B: $i] :
% 0.29/0.46 ( ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ( in @ Xx @ B ) )
% 0.29/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(eqimpsubset2_type,type,
% 0.29/0.46 eqimpsubset2: $o ).
% 0.29/0.46
% 0.29/0.46 thf(eqimpsubset2,definition,
% 0.29/0.46 ( eqimpsubset2
% 0.29/0.46 = ( ! [A: $i,B: $i] :
% 0.29/0.46 ( ( A = B )
% 0.29/0.46 => ( subset @ B @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(eqimpsubset1_type,type,
% 0.29/0.46 eqimpsubset1: $o ).
% 0.29/0.46
% 0.29/0.46 thf(eqimpsubset1,definition,
% 0.29/0.46 ( eqimpsubset1
% 0.29/0.46 = ( ! [A: $i,B: $i] :
% 0.29/0.46 ( ( A = B )
% 0.29/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(subsetI2_type,type,
% 0.29/0.46 subsetI2: $o ).
% 0.29/0.46
% 0.29/0.46 thf(subsetI2,definition,
% 0.29/0.46 ( subsetI2
% 0.29/0.46 = ( ! [A: $i,B: $i] :
% 0.29/0.46 ( ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ( in @ Xx @ B ) )
% 0.29/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(emptysetsubset_type,type,
% 0.29/0.46 emptysetsubset: $o ).
% 0.29/0.46
% 0.29/0.46 thf(emptysetsubset,definition,
% 0.29/0.46 ( emptysetsubset
% 0.29/0.46 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(subsetE_type,type,
% 0.29/0.46 subsetE: $o ).
% 0.29/0.46
% 0.29/0.46 thf(subsetE,definition,
% 0.29/0.46 ( subsetE
% 0.29/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.46 ( ( subset @ A @ B )
% 0.29/0.46 => ( ( in @ Xx @ A )
% 0.29/0.46 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(subsetE2_type,type,
% 0.29/0.46 subsetE2: $o ).
% 0.29/0.46
% 0.29/0.46 thf(subsetE2,definition,
% 0.29/0.46 ( subsetE2
% 0.29/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.46 ( ( subset @ A @ B )
% 0.29/0.46 => ( ~ ( in @ Xx @ B )
% 0.29/0.46 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(notsubsetI_type,type,
% 0.29/0.46 notsubsetI: $o ).
% 0.29/0.46
% 0.29/0.46 thf(notsubsetI,definition,
% 0.29/0.46 ( notsubsetI
% 0.29/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ( ~ ( in @ Xx @ B )
% 0.29/0.46 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(notequalI1_type,type,
% 0.29/0.46 notequalI1: $o ).
% 0.29/0.46
% 0.29/0.46 thf(notequalI1,definition,
% 0.29/0.46 ( notequalI1
% 0.29/0.46 = ( ! [A: $i,B: $i] :
% 0.29/0.46 ( ~ ( subset @ A @ B )
% 0.29/0.46 => ( A != B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(notequalI2_type,type,
% 0.29/0.46 notequalI2: $o ).
% 0.29/0.46
% 0.29/0.46 thf(notequalI2,definition,
% 0.29/0.46 ( notequalI2
% 0.29/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ( ~ ( in @ Xx @ B )
% 0.29/0.46 => ( A != B ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(subsetRefl_type,type,
% 0.29/0.46 subsetRefl: $o ).
% 0.29/0.46
% 0.29/0.46 thf(subsetRefl,definition,
% 0.29/0.46 ( subsetRefl
% 0.29/0.46 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(subsetTrans_type,type,
% 0.29/0.46 subsetTrans: $o ).
% 0.29/0.46
% 0.29/0.46 thf(subsetTrans,definition,
% 0.29/0.46 ( subsetTrans
% 0.29/0.46 = ( ! [A: $i,B: $i,C: $i] :
% 0.29/0.46 ( ( subset @ A @ B )
% 0.29/0.46 => ( ( subset @ B @ C )
% 0.29/0.46 => ( subset @ A @ C ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(setadjoinSub_type,type,
% 0.29/0.46 setadjoinSub: $o ).
% 0.29/0.46
% 0.29/0.46 thf(setadjoinSub,definition,
% 0.29/0.46 ( setadjoinSub
% 0.29/0.46 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(setadjoinSub2_type,type,
% 0.29/0.46 setadjoinSub2: $o ).
% 0.29/0.46
% 0.29/0.46 thf(setadjoinSub2,definition,
% 0.29/0.46 ( setadjoinSub2
% 0.29/0.46 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.29/0.46 ( ( subset @ A @ B )
% 0.29/0.46 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(subset2powerset_type,type,
% 0.29/0.46 subset2powerset: $o ).
% 0.29/0.46
% 0.29/0.46 thf(subset2powerset,definition,
% 0.29/0.46 ( subset2powerset
% 0.29/0.46 = ( ! [A: $i,B: $i] :
% 0.29/0.46 ( ( subset @ A @ B )
% 0.29/0.46 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(setextsub_type,type,
% 0.29/0.46 setextsub: $o ).
% 0.29/0.46
% 0.29/0.46 thf(setextsub,definition,
% 0.29/0.46 ( setextsub
% 0.29/0.46 = ( ! [A: $i,B: $i] :
% 0.29/0.46 ( ( subset @ A @ B )
% 0.29/0.46 => ( ( subset @ B @ A )
% 0.29/0.46 => ( A = B ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(subsetemptysetimpeq_type,type,
% 0.29/0.46 subsetemptysetimpeq: $o ).
% 0.29/0.46
% 0.29/0.46 thf(subsetemptysetimpeq,definition,
% 0.29/0.46 ( subsetemptysetimpeq
% 0.29/0.46 = ( ! [A: $i] :
% 0.29/0.46 ( ( subset @ A @ emptyset )
% 0.29/0.46 => ( A = emptyset ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(powersetI1_type,type,
% 0.29/0.46 powersetI1: $o ).
% 0.29/0.46
% 0.29/0.46 thf(powersetI1,definition,
% 0.29/0.46 ( powersetI1
% 0.29/0.46 = ( ! [A: $i,B: $i] :
% 0.29/0.46 ( ( subset @ B @ A )
% 0.29/0.46 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(powersetE1_type,type,
% 0.29/0.46 powersetE1: $o ).
% 0.29/0.46
% 0.29/0.46 thf(powersetE1,definition,
% 0.29/0.46 ( powersetE1
% 0.29/0.46 = ( ! [A: $i,B: $i] :
% 0.29/0.46 ( ( in @ B @ ( powerset @ A ) )
% 0.29/0.46 => ( subset @ B @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(inPowerset_type,type,
% 0.29/0.46 inPowerset: $o ).
% 0.29/0.46
% 0.29/0.46 thf(inPowerset,definition,
% 0.29/0.46 ( inPowerset
% 0.29/0.46 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(powersetsubset_type,type,
% 0.29/0.46 powersetsubset: $o ).
% 0.29/0.46
% 0.29/0.46 thf(powersetsubset,definition,
% 0.29/0.46 ( powersetsubset
% 0.29/0.46 = ( ! [A: $i,B: $i] :
% 0.29/0.46 ( ( subset @ A @ B )
% 0.29/0.46 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(sepInPowerset_type,type,
% 0.29/0.46 sepInPowerset: $o ).
% 0.29/0.46
% 0.29/0.46 thf(sepInPowerset,definition,
% 0.29/0.46 ( sepInPowerset
% 0.29/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.29/0.46 ( in
% 0.29/0.46 @ ( dsetconstr @ A
% 0.29/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.29/0.46 @ ( powerset @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(sepSubset_type,type,
% 0.29/0.46 sepSubset: $o ).
% 0.29/0.46
% 0.29/0.46 thf(sepSubset,definition,
% 0.29/0.46 ( sepSubset
% 0.29/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.29/0.46 ( subset
% 0.29/0.46 @ ( dsetconstr @ A
% 0.29/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.29/0.46 @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(binunion_type,type,
% 0.29/0.46 binunion: $i > $i > $i ).
% 0.29/0.46
% 0.29/0.46 thf(binunionIL_type,type,
% 0.29/0.46 binunionIL: $o ).
% 0.29/0.46
% 0.29/0.46 thf(binunionIL,definition,
% 0.29/0.46 ( binunionIL
% 0.29/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(upairset2IR_type,type,
% 0.29/0.46 upairset2IR: $o ).
% 0.29/0.46
% 0.29/0.46 thf(upairset2IR,definition,
% 0.29/0.46 ( upairset2IR
% 0.29/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(binunionIR_type,type,
% 0.29/0.46 binunionIR: $o ).
% 0.29/0.46
% 0.29/0.46 thf(binunionIR,definition,
% 0.29/0.46 ( binunionIR
% 0.29/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ B )
% 0.29/0.46 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(binunionEcases_type,type,
% 0.29/0.46 binunionEcases: $o ).
% 0.29/0.46
% 0.29/0.46 thf(binunionEcases,definition,
% 0.29/0.46 ( binunionEcases
% 0.29/0.46 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.29/0.46 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.29/0.46 => ( ( ( in @ Xx @ A )
% 0.29/0.46 => Xphi )
% 0.29/0.46 => ( ( ( in @ Xx @ B )
% 0.29/0.46 => Xphi )
% 0.29/0.46 => Xphi ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(binunionE_type,type,
% 0.29/0.46 binunionE: $o ).
% 0.29/0.46
% 0.29/0.46 thf(binunionE,definition,
% 0.29/0.46 ( binunionE
% 0.29/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.29/0.46 => ( ( in @ Xx @ A )
% 0.29/0.46 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(binunionLsub_type,type,
% 0.29/0.46 binunionLsub: $o ).
% 0.29/0.46
% 0.29/0.46 thf(binunionLsub,definition,
% 0.29/0.46 ( binunionLsub
% 0.29/0.46 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(binunionRsub_type,type,
% 0.29/0.46 binunionRsub: $o ).
% 0.29/0.46
% 0.29/0.46 thf(binunionRsub,definition,
% 0.29/0.46 ( binunionRsub
% 0.29/0.46 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(binintersect_type,type,
% 0.29/0.46 binintersect: $i > $i > $i ).
% 0.29/0.46
% 0.29/0.46 thf(binintersectI_type,type,
% 0.29/0.46 binintersectI: $o ).
% 0.29/0.46
% 0.29/0.46 thf(binintersectI,definition,
% 0.29/0.46 ( binintersectI
% 0.29/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ( ( in @ Xx @ B )
% 0.29/0.46 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(binintersectSubset5_type,type,
% 0.29/0.46 binintersectSubset5: $o ).
% 0.29/0.46
% 0.29/0.46 thf(binintersectSubset5,definition,
% 0.29/0.46 ( binintersectSubset5
% 0.29/0.46 = ( ! [A: $i,B: $i,C: $i] :
% 0.29/0.46 ( ( subset @ C @ A )
% 0.29/0.46 => ( ( subset @ C @ B )
% 0.29/0.46 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(binintersectEL_type,type,
% 0.29/0.46 binintersectEL: $o ).
% 0.29/0.46
% 0.29/0.46 thf(binintersectEL,definition,
% 0.29/0.46 ( binintersectEL
% 0.29/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.29/0.46 => ( in @ Xx @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(binintersectLsub_type,type,
% 0.29/0.46 binintersectLsub: $o ).
% 0.29/0.46
% 0.29/0.46 thf(binintersectLsub,definition,
% 0.29/0.46 ( binintersectLsub
% 0.29/0.46 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(binintersectSubset2_type,type,
% 0.29/0.46 binintersectSubset2: $o ).
% 0.29/0.46
% 0.29/0.46 thf(binintersectSubset2,definition,
% 0.29/0.46 ( binintersectSubset2
% 0.29/0.46 = ( ! [A: $i,B: $i] :
% 0.29/0.46 ( ( subset @ A @ B )
% 0.29/0.46 => ( ( binintersect @ A @ B )
% 0.29/0.46 = A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(binintersectSubset3_type,type,
% 0.29/0.46 binintersectSubset3: $o ).
% 0.29/0.46
% 0.29/0.46 thf(binintersectSubset3,definition,
% 0.29/0.46 ( binintersectSubset3
% 0.29/0.46 = ( ! [A: $i,B: $i] :
% 0.29/0.46 ( ( ( binintersect @ A @ B )
% 0.29/0.46 = B )
% 0.29/0.46 => ( subset @ B @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(binintersectER_type,type,
% 0.29/0.46 binintersectER: $o ).
% 0.29/0.46
% 0.29/0.46 thf(binintersectER,definition,
% 0.29/0.46 ( binintersectER
% 0.29/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.29/0.46 => ( in @ Xx @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(disjointsetsI1_type,type,
% 0.29/0.46 disjointsetsI1: $o ).
% 0.29/0.46
% 0.29/0.46 thf(disjointsetsI1,definition,
% 0.29/0.46 ( disjointsetsI1
% 0.29/0.46 = ( ! [A: $i,B: $i] :
% 0.29/0.46 ( ~ ? [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 & ( in @ Xx @ B ) )
% 0.29/0.46 => ( ( binintersect @ A @ B )
% 0.29/0.46 = emptyset ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(binintersectRsub_type,type,
% 0.29/0.46 binintersectRsub: $o ).
% 0.29/0.46
% 0.29/0.46 thf(binintersectRsub,definition,
% 0.29/0.46 ( binintersectRsub
% 0.29/0.46 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(binintersectSubset4_type,type,
% 0.29/0.46 binintersectSubset4: $o ).
% 0.29/0.46
% 0.29/0.46 thf(binintersectSubset4,definition,
% 0.29/0.46 ( binintersectSubset4
% 0.29/0.46 = ( ! [A: $i,B: $i] :
% 0.29/0.46 ( ( subset @ B @ A )
% 0.29/0.46 => ( ( binintersect @ A @ B )
% 0.29/0.46 = B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(binintersectSubset1_type,type,
% 0.29/0.46 binintersectSubset1: $o ).
% 0.29/0.46
% 0.29/0.46 thf(binintersectSubset1,definition,
% 0.29/0.46 ( binintersectSubset1
% 0.29/0.46 = ( ! [A: $i,B: $i] :
% 0.29/0.46 ( ( ( binintersect @ A @ B )
% 0.29/0.46 = A )
% 0.29/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(bs114d_type,type,
% 0.29/0.46 bs114d: $o ).
% 0.29/0.46
% 0.29/0.46 thf(bs114d,definition,
% 0.29/0.46 ( bs114d
% 0.29/0.46 = ( ! [A: $i,B: $i,C: $i] :
% 0.29/0.46 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 0.29/0.46 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(regular_type,type,
% 0.29/0.46 regular: $i > $o ).
% 0.29/0.46
% 0.29/0.46 thf(setminus_type,type,
% 0.29/0.46 setminus: $i > $i > $i ).
% 0.29/0.46
% 0.29/0.46 thf(setminusI_type,type,
% 0.29/0.46 setminusI: $o ).
% 0.29/0.46
% 0.29/0.46 thf(setminusI,definition,
% 0.29/0.46 ( setminusI
% 0.29/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ( ~ ( in @ Xx @ B )
% 0.29/0.46 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(setminusEL_type,type,
% 0.29/0.46 setminusEL: $o ).
% 0.29/0.46
% 0.29/0.46 thf(setminusEL,definition,
% 0.29/0.46 ( setminusEL
% 0.29/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.29/0.46 => ( in @ Xx @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(setminusER_type,type,
% 0.29/0.46 setminusER: $o ).
% 0.29/0.46
% 0.29/0.46 thf(setminusER,definition,
% 0.29/0.46 ( setminusER
% 0.29/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.29/0.46 => ~ ( in @ Xx @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(setminusSubset2_type,type,
% 0.29/0.46 setminusSubset2: $o ).
% 0.29/0.46
% 0.29/0.46 thf(setminusSubset2,definition,
% 0.29/0.46 ( setminusSubset2
% 0.29/0.46 = ( ! [A: $i,B: $i] :
% 0.29/0.46 ( ( subset @ A @ B )
% 0.29/0.46 => ( ( setminus @ A @ B )
% 0.29/0.46 = emptyset ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(setminusERneg_type,type,
% 0.29/0.46 setminusERneg: $o ).
% 0.29/0.46
% 0.29/0.46 thf(setminusERneg,definition,
% 0.29/0.46 ( setminusERneg
% 0.29/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.46 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.29/0.46 => ( ( in @ Xx @ A )
% 0.29/0.46 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(setminusELneg_type,type,
% 0.29/0.46 setminusELneg: $o ).
% 0.29/0.46
% 0.29/0.46 thf(setminusELneg,definition,
% 0.29/0.46 ( setminusELneg
% 0.29/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.46 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.29/0.46 => ( ~ ( in @ Xx @ B )
% 0.29/0.46 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(setminusILneg_type,type,
% 0.29/0.46 setminusILneg: $o ).
% 0.29/0.46
% 0.29/0.46 thf(setminusILneg,definition,
% 0.29/0.46 ( setminusILneg
% 0.29/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.46 ( ~ ( in @ Xx @ A )
% 0.29/0.46 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(setminusIRneg_type,type,
% 0.29/0.46 setminusIRneg: $o ).
% 0.29/0.46
% 0.29/0.46 thf(setminusIRneg,definition,
% 0.29/0.46 ( setminusIRneg
% 0.29/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ B )
% 0.29/0.46 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(setminusLsub_type,type,
% 0.29/0.46 setminusLsub: $o ).
% 0.29/0.46
% 0.29/0.46 thf(setminusLsub,definition,
% 0.29/0.46 ( setminusLsub
% 0.29/0.46 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(setminusSubset1_type,type,
% 0.29/0.46 setminusSubset1: $o ).
% 0.29/0.46
% 0.29/0.46 thf(setminusSubset1,definition,
% 0.29/0.46 ( setminusSubset1
% 0.29/0.46 = ( ! [A: $i,B: $i] :
% 0.29/0.46 ( ( ( setminus @ A @ B )
% 0.29/0.46 = emptyset )
% 0.29/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(symdiff_type,type,
% 0.29/0.46 symdiff: $i > $i > $i ).
% 0.29/0.46
% 0.29/0.46 thf(symdiffE_type,type,
% 0.29/0.46 symdiffE: $o ).
% 0.29/0.46
% 0.29/0.46 thf(symdiffE,definition,
% 0.29/0.46 ( symdiffE
% 0.29/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 0.29/0.46 => ! [Xphi: $o] :
% 0.29/0.46 ( ( ( in @ Xx @ A )
% 0.29/0.46 => ( ~ ( in @ Xx @ B )
% 0.29/0.46 => Xphi ) )
% 0.29/0.46 => ( ( ~ ( in @ Xx @ A )
% 0.29/0.46 => ( ( in @ Xx @ B )
% 0.29/0.46 => Xphi ) )
% 0.29/0.46 => Xphi ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(symdiffI1_type,type,
% 0.29/0.46 symdiffI1: $o ).
% 0.29/0.46
% 0.29/0.46 thf(symdiffI1,definition,
% 0.29/0.46 ( symdiffI1
% 0.29/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ( ~ ( in @ Xx @ B )
% 0.29/0.46 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(symdiffI2_type,type,
% 0.29/0.46 symdiffI2: $o ).
% 0.29/0.46
% 0.29/0.46 thf(symdiffI2,definition,
% 0.29/0.46 ( symdiffI2
% 0.29/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.46 ( ~ ( in @ Xx @ A )
% 0.29/0.46 => ( ( in @ Xx @ B )
% 0.29/0.46 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(symdiffIneg1_type,type,
% 0.29/0.46 symdiffIneg1: $o ).
% 0.29/0.46
% 0.29/0.46 thf(symdiffIneg1,definition,
% 0.29/0.46 ( symdiffIneg1
% 0.29/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ( ( in @ Xx @ B )
% 0.29/0.46 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(symdiffIneg2_type,type,
% 0.29/0.46 symdiffIneg2: $o ).
% 0.29/0.46
% 0.29/0.46 thf(symdiffIneg2,definition,
% 0.29/0.46 ( symdiffIneg2
% 0.29/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.46 ( ~ ( in @ Xx @ A )
% 0.29/0.46 => ( ~ ( in @ Xx @ B )
% 0.29/0.46 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(iskpair_type,type,
% 0.29/0.46 iskpair: $i > $o ).
% 0.29/0.46
% 0.29/0.46 thf(iskpair,definition,
% 0.29/0.46 ( iskpair
% 0.29/0.46 = ( ^ [A: $i] :
% 0.29/0.46 ? [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ ( setunion @ A ) )
% 0.29/0.46 & ? [Xy: $i] :
% 0.29/0.46 ( ( in @ Xy @ ( setunion @ A ) )
% 0.29/0.46 & ( A
% 0.29/0.46 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(secondinupair_type,type,
% 0.29/0.46 secondinupair: $o ).
% 0.29/0.46
% 0.29/0.46 thf(secondinupair,definition,
% 0.29/0.46 ( secondinupair
% 0.29/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(setukpairIL_type,type,
% 0.29/0.46 setukpairIL: $o ).
% 0.29/0.46
% 0.29/0.46 thf(setukpairIL,definition,
% 0.29/0.46 ( setukpairIL
% 0.29/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(setukpairIR,conjecture,
% 0.29/0.46 ( setextAx
% 0.29/0.46 => ( emptysetAx
% 0.29/0.46 => ( setadjoinAx
% 0.29/0.46 => ( powersetAx
% 0.29/0.46 => ( setunionAx
% 0.29/0.46 => ( omega0Ax
% 0.29/0.46 => ( omegaSAx
% 0.29/0.46 => ( omegaIndAx
% 0.29/0.46 => ( replAx
% 0.29/0.46 => ( foundationAx
% 0.29/0.46 => ( wellorderingAx
% 0.29/0.46 => ( descrp
% 0.29/0.46 => ( dsetconstrI
% 0.29/0.46 => ( dsetconstrEL
% 0.29/0.46 => ( dsetconstrER
% 0.29/0.46 => ( exuE1
% 0.29/0.46 => ( prop2setE
% 0.29/0.46 => ( emptysetE
% 0.29/0.46 => ( emptysetimpfalse
% 0.29/0.46 => ( notinemptyset
% 0.29/0.46 => ( exuE3e
% 0.29/0.46 => ( setext
% 0.29/0.46 => ( emptyI
% 0.29/0.46 => ( noeltsimpempty
% 0.29/0.46 => ( setbeta
% 0.29/0.46 => ( nonemptyE1
% 0.29/0.46 => ( nonemptyI
% 0.29/0.46 => ( nonemptyI1
% 0.29/0.46 => ( setadjoinIL
% 0.29/0.46 => ( emptyinunitempty
% 0.29/0.46 => ( setadjoinIR
% 0.29/0.46 => ( setadjoinE
% 0.29/0.46 => ( setadjoinOr
% 0.29/0.46 => ( setoftrueEq
% 0.29/0.46 => ( powersetI
% 0.29/0.46 => ( emptyinPowerset
% 0.29/0.46 => ( emptyInPowerset
% 0.29/0.46 => ( powersetE
% 0.29/0.46 => ( setunionI
% 0.29/0.46 => ( setunionE
% 0.29/0.46 => ( subPowSU
% 0.29/0.46 => ( exuE2
% 0.29/0.46 => ( nonemptyImpWitness
% 0.29/0.46 => ( uniqinunit
% 0.29/0.46 => ( notinsingleton
% 0.29/0.46 => ( eqinunit
% 0.29/0.46 => ( singletonsswitch
% 0.29/0.46 => ( upairsetE
% 0.29/0.46 => ( upairsetIL
% 0.29/0.46 => ( upairsetIR
% 0.29/0.46 => ( emptyE1
% 0.29/0.46 => ( vacuousDall
% 0.29/0.46 => ( quantDeMorgan1
% 0.29/0.46 => ( quantDeMorgan2
% 0.29/0.46 => ( quantDeMorgan3
% 0.29/0.46 => ( quantDeMorgan4
% 0.29/0.46 => ( prop2setI
% 0.29/0.46 => ( prop2set2propI
% 0.29/0.46 => ( notdexE
% 0.29/0.46 => ( notdallE
% 0.29/0.46 => ( exuI1
% 0.29/0.46 => ( exuI3
% 0.29/0.46 => ( exuI2
% 0.29/0.46 => ( inCongP
% 0.29/0.46 => ( in__Cong
% 0.29/0.46 => ( exuE3u
% 0.29/0.46 => ( exu__Cong
% 0.29/0.46 => ( emptyset__Cong
% 0.29/0.46 => ( setadjoin__Cong
% 0.29/0.46 => ( powerset__Cong
% 0.29/0.46 => ( setunion__Cong
% 0.29/0.46 => ( omega__Cong
% 0.29/0.46 => ( exuEu
% 0.29/0.46 => ( descr__Cong
% 0.29/0.46 => ( dsetconstr__Cong
% 0.29/0.46 => ( subsetI1
% 0.29/0.46 => ( eqimpsubset2
% 0.29/0.46 => ( eqimpsubset1
% 0.29/0.46 => ( subsetI2
% 0.29/0.46 => ( emptysetsubset
% 0.29/0.46 => ( subsetE
% 0.29/0.46 => ( subsetE2
% 0.29/0.46 => ( notsubsetI
% 0.29/0.46 => ( notequalI1
% 0.29/0.46 => ( notequalI2
% 0.29/0.46 => ( subsetRefl
% 0.29/0.46 => ( subsetTrans
% 0.29/0.46 => ( setadjoinSub
% 0.29/0.46 => ( setadjoinSub2
% 0.29/0.46 => ( subset2powerset
% 0.29/0.46 => ( setextsub
% 0.29/0.46 => ( subsetemptysetimpeq
% 0.29/0.46 => ( powersetI1
% 0.29/0.46 => ( powersetE1
% 0.29/0.46 => ( inPowerset
% 0.29/0.46 => ( powersetsubset
% 0.29/0.46 => ( sepInPowerset
% 0.29/0.46 => ( sepSubset
% 0.29/0.46 => ( binunionIL
% 0.29/0.46 => ( upairset2IR
% 0.29/0.46 => ( binunionIR
% 0.29/0.46 => ( binunionEcases
% 0.29/0.46 => ( binunionE
% 0.29/0.46 => ( binunionLsub
% 0.29/0.46 => ( binunionRsub
% 0.29/0.46 => ( binintersectI
% 0.29/0.46 => ( binintersectSubset5
% 0.29/0.46 => ( binintersectEL
% 0.29/0.46 => ( binintersectLsub
% 0.29/0.46 => ( binintersectSubset2
% 0.29/0.46 => ( binintersectSubset3
% 0.29/0.46 => ( binintersectER
% 0.29/0.46 => ( disjointsetsI1
% 0.29/0.46 => ( binintersectRsub
% 0.29/0.46 => ( binintersectSubset4
% 0.29/0.46 => ( binintersectSubset1
% 0.29/0.46 => ( bs114d
% 0.29/0.46 => ( setminusI
% 0.29/0.46 => ( setminusEL
% 0.29/0.46 => ( setminusER
% 0.29/0.46 => ( setminusSubset2
% 0.29/0.46 => ( setminusERneg
% 0.29/0.46 => ( setminusELneg
% 0.29/0.46 => ( setminusILneg
% 0.29/0.46 => ( setminusIRneg
% 0.29/0.46 => ( setminusLsub
% 0.29/0.46 => ( setminusSubset1
% 0.29/0.46 => ( symdiffE
% 0.29/0.46 => ( symdiffI1
% 0.29/0.46 => ( symdiffI2
% 0.29/0.46 => ( symdiffIneg1
% 0.29/0.46 => ( symdiffIneg2
% 0.29/0.46 => ( secondinupair
% 0.29/0.46 => ( setukpairIL
% 0.29/0.46 => ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 %------------------------------------------------------------------------------
% 0.29/0.46 ------------------------
% 2.38/1.43 ---- Embedded in THF ---
% 2.38/1.43 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.38/1.43 %%% Generated on Tue Feb 24 12:28:43 EST 2026
% 2.38/1.43 %%% using '$$dhol' embedding, version 1.3.0.
% 2.38/1.43 %%% Logic specification used:
% 2.38/1.43 %%% thf(spec, logic, $$dhol).
% 2.38/1.43
% 2.38/1.43 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.BCzCW9enGb/DTF2DTF25972.p
% See solution above
% 2.38/1.43 ------------------------
% 2.38/1.46 ---- Cleaned THF ---
% 2.38/1.46 thf(in_type,type,
% 2.38/1.46 in: $i > $i > $o ).
% 2.38/1.46
% 2.38/1.46 thf(exu_type,type,
% 2.38/1.46 exu: ( $i > $o ) > $o ).
% 2.38/1.46
% 2.38/1.46 thf(exu,definition,
% 2.38/1.46 ( exu
% 2.38/1.46 = ( ^ [Xphi: $i > $o] :
% 2.38/1.46 ? [Xx: $i] :
% 2.38/1.46 ( ( Xphi @ Xx )
% 2.38/1.46 & ! [Xy: $i] :
% 2.38/1.46 ( ( Xphi @ Xy )
% 2.38/1.46 => ( Xx = Xy ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(setextAx_type,type,
% 2.38/1.46 setextAx: $o ).
% 2.38/1.46
% 2.38/1.46 thf(setextAx,definition,
% 2.38/1.46 ( setextAx
% 2.38/1.46 = ( ! [A: $i,B: $i] :
% 2.38/1.46 ( ! [Xx: $i] :
% 2.38/1.46 ( ( in @ Xx @ A )
% 2.38/1.46 <=> ( in @ Xx @ B ) )
% 2.38/1.46 => ( A = B ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(emptyset_type,type,
% 2.38/1.46 emptyset: $i ).
% 2.38/1.46
% 2.38/1.46 thf(emptysetAx_type,type,
% 2.38/1.46 emptysetAx: $o ).
% 2.38/1.46
% 2.38/1.46 thf(emptysetAx,definition,
% 2.38/1.46 ( emptysetAx
% 2.38/1.46 = ( ! [Xx: $i] :
% 2.38/1.46 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(setadjoin_type,type,
% 2.38/1.46 setadjoin: $i > $i > $i ).
% 2.38/1.46
% 2.38/1.46 thf(setadjoinAx_type,type,
% 2.38/1.46 setadjoinAx: $o ).
% 2.38/1.46
% 2.38/1.46 thf(setadjoinAx,definition,
% 2.38/1.46 ( setadjoinAx
% 2.38/1.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.38/1.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.38/1.46 <=> ( ( Xy = Xx )
% 2.38/1.46 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(powerset_type,type,
% 2.38/1.46 powerset: $i > $i ).
% 2.38/1.46
% 2.38/1.46 thf(powersetAx_type,type,
% 2.38/1.46 powersetAx: $o ).
% 2.38/1.46
% 2.38/1.46 thf(powersetAx,definition,
% 2.38/1.46 ( powersetAx
% 2.38/1.46 = ( ! [A: $i,B: $i] :
% 2.38/1.46 ( ( in @ B @ ( powerset @ A ) )
% 2.38/1.46 <=> ! [Xx: $i] :
% 2.38/1.46 ( ( in @ Xx @ B )
% 2.38/1.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(setunion_type,type,
% 2.38/1.46 setunion: $i > $i ).
% 2.38/1.46
% 2.38/1.46 thf(setunionAx_type,type,
% 2.38/1.46 setunionAx: $o ).
% 2.38/1.46
% 2.38/1.46 thf(setunionAx,definition,
% 2.38/1.46 ( setunionAx
% 2.38/1.46 = ( ! [A: $i,Xx: $i] :
% 2.38/1.46 ( ( in @ Xx @ ( setunion @ A ) )
% 2.38/1.46 <=> ? [B: $i] :
% 2.38/1.46 ( ( in @ Xx @ B )
% 2.38/1.46 & ( in @ B @ A ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(omega_type,type,
% 2.38/1.46 omega: $i ).
% 2.38/1.46
% 2.38/1.46 thf(omega0Ax_type,type,
% 2.38/1.46 omega0Ax: $o ).
% 2.38/1.46
% 2.38/1.46 thf(omega0Ax,definition,
% 2.38/1.46 ( omega0Ax
% 2.38/1.46 = ( in @ emptyset @ omega ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(omegaSAx_type,type,
% 2.38/1.46 omegaSAx: $o ).
% 2.38/1.46
% 2.38/1.46 thf(omegaSAx,definition,
% 2.38/1.46 ( omegaSAx
% 2.38/1.46 = ( ! [Xx: $i] :
% 2.38/1.46 ( ( in @ Xx @ omega )
% 2.38/1.46 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(omegaIndAx_type,type,
% 2.38/1.46 omegaIndAx: $o ).
% 2.38/1.46
% 2.38/1.46 thf(omegaIndAx,definition,
% 2.38/1.46 ( omegaIndAx
% 2.38/1.46 = ( ! [A: $i] :
% 2.38/1.46 ( ( ( in @ emptyset @ A )
% 2.38/1.46 & ! [Xx: $i] :
% 2.38/1.46 ( ( ( in @ Xx @ omega )
% 2.38/1.46 & ( in @ Xx @ A ) )
% 2.38/1.46 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 2.38/1.46 => ! [Xx: $i] :
% 2.38/1.46 ( ( in @ Xx @ omega )
% 2.38/1.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(replAx_type,type,
% 2.38/1.46 replAx: $o ).
% 2.38/1.46
% 2.38/1.46 thf(replAx,definition,
% 2.38/1.46 ( replAx
% 2.38/1.46 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 2.38/1.46 ( ! [Xx: $i] :
% 2.38/1.46 ( ( in @ Xx @ A )
% 2.38/1.46 => ( exu
% 2.38/1.46 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 2.38/1.46 => ? [B: $i] :
% 2.38/1.46 ! [Xx: $i] :
% 2.38/1.46 ( ( in @ Xx @ B )
% 2.38/1.46 <=> ? [Xy: $i] :
% 2.38/1.46 ( ( in @ Xy @ A )
% 2.38/1.46 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(foundationAx_type,type,
% 2.38/1.46 foundationAx: $o ).
% 2.38/1.46
% 2.38/1.46 thf(foundationAx,definition,
% 2.38/1.46 ( foundationAx
% 2.38/1.46 = ( ! [A: $i] :
% 2.38/1.46 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.38/1.46 => ? [B: $i] :
% 2.38/1.46 ( ( in @ B @ A )
% 2.38/1.46 & ~ ? [Xx: $i] :
% 2.38/1.46 ( ( in @ Xx @ B )
% 2.38/1.46 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(wellorderingAx_type,type,
% 2.38/1.46 wellorderingAx: $o ).
% 2.38/1.46
% 2.38/1.46 thf(wellorderingAx,definition,
% 2.38/1.46 ( wellorderingAx
% 2.38/1.46 = ( ! [A: $i] :
% 2.38/1.46 ? [B: $i] :
% 2.38/1.46 ( ! [C: $i] :
% 2.38/1.46 ( ( in @ C @ B )
% 2.38/1.46 => ! [Xx: $i] :
% 2.38/1.46 ( ( in @ Xx @ C )
% 2.38/1.46 => ( in @ Xx @ A ) ) )
% 2.38/1.46 & ! [Xx: $i,Xy: $i] :
% 2.38/1.46 ( ( ( in @ Xx @ A )
% 2.38/1.46 & ( in @ Xy @ A ) )
% 2.38/1.46 => ( ! [C: $i] :
% 2.38/1.46 ( ( in @ C @ B )
% 2.38/1.46 => ( ( in @ Xx @ C )
% 2.38/1.46 <=> ( in @ Xy @ C ) ) )
% 2.38/1.46 => ( Xx = Xy ) ) )
% 2.38/1.46 & ! [C: $i,D: $i] :
% 2.38/1.46 ( ( ( in @ C @ B )
% 2.38/1.46 & ( in @ D @ B ) )
% 2.38/1.46 => ( ! [Xx: $i] :
% 2.38/1.46 ( ( in @ Xx @ C )
% 2.38/1.46 => ( in @ Xx @ D ) )
% 2.38/1.46 | ! [Xx: $i] :
% 2.38/1.46 ( ( in @ Xx @ D )
% 2.38/1.46 => ( in @ Xx @ C ) ) ) )
% 2.38/1.46 & ! [C: $i] :
% 2.38/1.46 ( ( ! [Xx: $i] :
% 2.38/1.46 ( ( in @ Xx @ C )
% 2.38/1.46 => ( in @ Xx @ A ) )
% 2.38/1.46 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 2.38/1.46 => ? [D: $i,Xx: $i] :
% 2.38/1.46 ( ( in @ D @ B )
% 2.38/1.46 & ( in @ Xx @ C )
% 2.38/1.46 & ~ ? [Xy: $i] :
% 2.38/1.46 ( ( in @ Xy @ D )
% 2.38/1.46 & ( in @ Xy @ C ) )
% 2.38/1.46 & ! [E: $i] :
% 2.38/1.46 ( ( in @ E @ B )
% 2.38/1.46 => ( ! [Xy: $i] :
% 2.38/1.46 ( ( in @ Xy @ E )
% 2.38/1.46 => ( in @ Xy @ D ) )
% 2.38/1.46 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(descr_type,type,
% 2.38/1.46 descr: ( $i > $o ) > $i ).
% 2.38/1.46
% 2.38/1.46 thf(descrp_type,type,
% 2.38/1.46 descrp: $o ).
% 2.38/1.46
% 2.38/1.46 thf(descrp,definition,
% 2.38/1.46 ( descrp
% 2.38/1.46 = ( ! [Xphi: $i > $o] :
% 2.38/1.46 ( ( exu
% 2.38/1.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.38/1.46 => ( Xphi
% 2.38/1.46 @ ( descr
% 2.38/1.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(dsetconstr_type,type,
% 2.38/1.46 dsetconstr: $i > ( $i > $o ) > $i ).
% 2.38/1.46
% 2.38/1.46 thf(dsetconstrI_type,type,
% 2.38/1.46 dsetconstrI: $o ).
% 2.38/1.46
% 2.38/1.46 thf(dsetconstrI,definition,
% 2.38/1.46 ( dsetconstrI
% 2.38/1.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.38/1.46 ( ( in @ Xx @ A )
% 2.38/1.46 => ( ( Xphi @ Xx )
% 2.38/1.46 => ( in @ Xx
% 2.38/1.46 @ ( dsetconstr @ A
% 2.38/1.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(dsetconstrEL_type,type,
% 2.38/1.46 dsetconstrEL: $o ).
% 2.38/1.46
% 2.38/1.46 thf(dsetconstrEL,definition,
% 2.38/1.46 ( dsetconstrEL
% 2.38/1.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.38/1.46 ( ( in @ Xx
% 2.38/1.46 @ ( dsetconstr @ A
% 2.38/1.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.38/1.46 => ( in @ Xx @ A ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(dsetconstrER_type,type,
% 2.38/1.46 dsetconstrER: $o ).
% 2.38/1.46
% 2.38/1.46 thf(dsetconstrER,definition,
% 2.38/1.46 ( dsetconstrER
% 2.38/1.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.38/1.46 ( ( in @ Xx
% 2.38/1.46 @ ( dsetconstr @ A
% 2.38/1.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.38/1.46 => ( Xphi @ Xx ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(exuE1_type,type,
% 2.38/1.46 exuE1: $o ).
% 2.38/1.46
% 2.38/1.46 thf(exuE1,definition,
% 2.38/1.46 ( exuE1
% 2.38/1.46 = ( ! [Xphi: $i > $o] :
% 2.38/1.46 ( ( exu
% 2.38/1.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.38/1.46 => ? [Xx: $i] :
% 2.38/1.46 ( ( Xphi @ Xx )
% 2.38/1.46 & ! [Xy: $i] :
% 2.38/1.46 ( ( Xphi @ Xy )
% 2.38/1.46 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(prop2set_type,type,
% 2.38/1.46 prop2set: $o > $i ).
% 2.38/1.46
% 2.38/1.46 thf(prop2setE_type,type,
% 2.38/1.46 prop2setE: $o ).
% 2.38/1.46
% 2.38/1.46 thf(prop2setE,definition,
% 2.38/1.46 ( prop2setE
% 2.38/1.46 = ( ! [Xphi: $o,Xx: $i] :
% 2.38/1.46 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 2.38/1.46 => Xphi ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(emptysetE_type,type,
% 2.38/1.46 emptysetE: $o ).
% 2.38/1.46
% 2.38/1.46 thf(emptysetE,definition,
% 2.38/1.46 ( emptysetE
% 2.38/1.46 = ( ! [Xx: $i] :
% 2.38/1.46 ( ( in @ Xx @ emptyset )
% 2.38/1.46 => ! [Xphi: $o] : Xphi ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(emptysetimpfalse_type,type,
% 2.38/1.46 emptysetimpfalse: $o ).
% 2.38/1.46
% 2.38/1.46 thf(emptysetimpfalse,definition,
% 2.38/1.46 ( emptysetimpfalse
% 2.38/1.46 = ( ! [Xx: $i] :
% 2.38/1.46 ( ( in @ Xx @ emptyset )
% 2.38/1.46 => $false ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(notinemptyset_type,type,
% 2.38/1.46 notinemptyset: $o ).
% 2.38/1.46
% 2.38/1.46 thf(notinemptyset,definition,
% 2.38/1.46 ( notinemptyset
% 2.38/1.46 = ( ! [Xx: $i] :
% 2.38/1.46 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(exuE3e_type,type,
% 2.38/1.46 exuE3e: $o ).
% 2.38/1.46
% 2.38/1.46 thf(exuE3e,definition,
% 2.38/1.46 ( exuE3e
% 2.38/1.46 = ( ! [Xphi: $i > $o] :
% 2.38/1.46 ( ( exu
% 2.38/1.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.38/1.46 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(setext_type,type,
% 2.38/1.46 setext: $o ).
% 2.38/1.46
% 2.38/1.46 thf(setext,definition,
% 2.38/1.46 ( setext
% 2.38/1.46 = ( ! [A: $i,B: $i] :
% 2.38/1.46 ( ! [Xx: $i] :
% 2.38/1.46 ( ( in @ Xx @ A )
% 2.38/1.46 => ( in @ Xx @ B ) )
% 2.38/1.46 => ( ! [Xx: $i] :
% 2.38/1.46 ( ( in @ Xx @ B )
% 2.38/1.46 => ( in @ Xx @ A ) )
% 2.38/1.46 => ( A = B ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(emptyI_type,type,
% 2.38/1.46 emptyI: $o ).
% 2.38/1.46
% 2.38/1.46 thf(emptyI,definition,
% 2.38/1.46 ( emptyI
% 2.38/1.46 = ( ! [A: $i] :
% 2.38/1.46 ( ! [Xx: $i] :
% 2.38/1.46 ~ ( in @ Xx @ A )
% 2.38/1.46 => ( A = emptyset ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(noeltsimpempty_type,type,
% 2.38/1.46 noeltsimpempty: $o ).
% 2.38/1.46
% 2.38/1.46 thf(noeltsimpempty,definition,
% 2.38/1.46 ( noeltsimpempty
% 2.38/1.46 = ( ! [A: $i] :
% 2.38/1.46 ( ! [Xx: $i] :
% 2.38/1.46 ~ ( in @ Xx @ A )
% 2.38/1.46 => ( A = emptyset ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(setbeta_type,type,
% 2.38/1.46 setbeta: $o ).
% 2.38/1.46
% 2.38/1.46 thf(setbeta,definition,
% 2.38/1.46 ( setbeta
% 2.38/1.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.38/1.46 ( ( in @ Xx @ A )
% 2.38/1.46 => ( ( in @ Xx
% 2.38/1.46 @ ( dsetconstr @ A
% 2.38/1.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.38/1.46 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(nonempty_type,type,
% 2.38/1.46 nonempty: $i > $o ).
% 2.38/1.46
% 2.38/1.46 thf(nonempty,definition,
% 2.38/1.46 ( nonempty
% 2.38/1.46 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(nonemptyE1_type,type,
% 2.38/1.46 nonemptyE1: $o ).
% 2.38/1.46
% 2.38/1.46 thf(nonemptyE1,definition,
% 2.38/1.46 ( nonemptyE1
% 2.38/1.46 = ( ! [A: $i] :
% 2.38/1.46 ( ( nonempty @ A )
% 2.38/1.46 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(nonemptyI_type,type,
% 2.38/1.46 nonemptyI: $o ).
% 2.38/1.46
% 2.38/1.46 thf(nonemptyI,definition,
% 2.38/1.46 ( nonemptyI
% 2.38/1.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.38/1.46 ( ( in @ Xx @ A )
% 2.38/1.46 => ( ( Xphi @ Xx )
% 2.38/1.46 => ( nonempty
% 2.38/1.46 @ ( dsetconstr @ A
% 2.38/1.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(nonemptyI1_type,type,
% 2.38/1.46 nonemptyI1: $o ).
% 2.38/1.46
% 2.38/1.46 thf(nonemptyI1,definition,
% 2.38/1.46 ( nonemptyI1
% 2.38/1.46 = ( ! [A: $i] :
% 2.38/1.46 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.38/1.46 => ( nonempty @ A ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(setadjoinIL_type,type,
% 2.38/1.46 setadjoinIL: $o ).
% 2.38/1.46
% 2.38/1.46 thf(setadjoinIL,definition,
% 2.38/1.46 ( setadjoinIL
% 2.38/1.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(emptyinunitempty_type,type,
% 2.38/1.46 emptyinunitempty: $o ).
% 2.38/1.46
% 2.38/1.46 thf(emptyinunitempty,definition,
% 2.38/1.46 ( emptyinunitempty
% 2.38/1.46 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(setadjoinIR_type,type,
% 2.38/1.46 setadjoinIR: $o ).
% 2.38/1.46
% 2.38/1.46 thf(setadjoinIR,definition,
% 2.38/1.46 ( setadjoinIR
% 2.38/1.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.38/1.46 ( ( in @ Xy @ A )
% 2.38/1.46 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(setadjoinE_type,type,
% 2.38/1.46 setadjoinE: $o ).
% 2.38/1.46
% 2.38/1.46 thf(setadjoinE,definition,
% 2.38/1.46 ( setadjoinE
% 2.38/1.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.38/1.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.38/1.46 => ! [Xphi: $o] :
% 2.38/1.46 ( ( ( Xy = Xx )
% 2.38/1.46 => Xphi )
% 2.38/1.46 => ( ( ( in @ Xy @ A )
% 2.38/1.46 => Xphi )
% 2.38/1.46 => Xphi ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(setadjoinOr_type,type,
% 2.38/1.46 setadjoinOr: $o ).
% 2.38/1.46
% 2.38/1.46 thf(setadjoinOr,definition,
% 2.38/1.46 ( setadjoinOr
% 2.38/1.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.38/1.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.38/1.46 => ( ( Xy = Xx )
% 2.38/1.46 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(setoftrueEq_type,type,
% 2.38/1.46 setoftrueEq: $o ).
% 2.38/1.46
% 2.38/1.46 thf(setoftrueEq,definition,
% 2.38/1.46 ( setoftrueEq
% 2.38/1.46 = ( ! [A: $i] :
% 2.38/1.46 ( ( dsetconstr @ A
% 2.38/1.46 @ ^ [Xx: $i] : $true )
% 2.38/1.46 = A ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(powersetI_type,type,
% 2.38/1.46 powersetI: $o ).
% 2.38/1.46
% 2.38/1.46 thf(powersetI,definition,
% 2.38/1.46 ( powersetI
% 2.38/1.46 = ( ! [A: $i,B: $i] :
% 2.38/1.46 ( ! [Xx: $i] :
% 2.38/1.46 ( ( in @ Xx @ B )
% 2.38/1.46 => ( in @ Xx @ A ) )
% 2.38/1.46 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(emptyinPowerset_type,type,
% 2.38/1.46 emptyinPowerset: $o ).
% 2.38/1.46
% 2.38/1.46 thf(emptyinPowerset,definition,
% 2.38/1.46 ( emptyinPowerset
% 2.38/1.46 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(emptyInPowerset_type,type,
% 2.38/1.46 emptyInPowerset: $o ).
% 2.38/1.46
% 2.38/1.46 thf(emptyInPowerset,definition,
% 2.38/1.46 ( emptyInPowerset
% 2.38/1.46 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(powersetE_type,type,
% 2.38/1.46 powersetE: $o ).
% 2.38/1.46
% 2.38/1.46 thf(powersetE,definition,
% 2.38/1.46 ( powersetE
% 2.38/1.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.38/1.46 ( ( in @ B @ ( powerset @ A ) )
% 2.38/1.46 => ( ( in @ Xx @ B )
% 2.38/1.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(setunionI_type,type,
% 2.38/1.46 setunionI: $o ).
% 2.38/1.46
% 2.38/1.46 thf(setunionI,definition,
% 2.38/1.46 ( setunionI
% 2.38/1.46 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.38/1.46 ( ( in @ Xx @ B )
% 2.38/1.46 => ( ( in @ B @ A )
% 2.38/1.46 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(setunionE_type,type,
% 2.38/1.46 setunionE: $o ).
% 2.38/1.46
% 2.38/1.46 thf(setunionE,definition,
% 2.38/1.46 ( setunionE
% 2.38/1.46 = ( ! [A: $i,Xx: $i] :
% 2.38/1.46 ( ( in @ Xx @ ( setunion @ A ) )
% 2.38/1.46 => ! [Xphi: $o] :
% 2.38/1.46 ( ! [B: $i] :
% 2.38/1.46 ( ( in @ Xx @ B )
% 2.38/1.46 => ( ( in @ B @ A )
% 2.38/1.46 => Xphi ) )
% 2.38/1.46 => Xphi ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(subPowSU_type,type,
% 2.38/1.46 subPowSU: $o ).
% 2.38/1.46
% 2.38/1.46 thf(subPowSU,definition,
% 2.38/1.46 ( subPowSU
% 2.38/1.46 = ( ! [A: $i,Xx: $i] :
% 2.38/1.46 ( ( in @ Xx @ A )
% 2.38/1.46 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(exuE2_type,type,
% 2.38/1.46 exuE2: $o ).
% 2.38/1.46
% 2.38/1.46 thf(exuE2,definition,
% 2.38/1.46 ( exuE2
% 2.38/1.46 = ( ! [Xphi: $i > $o] :
% 2.38/1.46 ( ( exu
% 2.38/1.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.38/1.46 => ? [Xx: $i] :
% 2.38/1.46 ! [Xy: $i] :
% 2.38/1.46 ( ( Xphi @ Xy )
% 2.38/1.46 <=> ( Xy = Xx ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(nonemptyImpWitness_type,type,
% 2.38/1.46 nonemptyImpWitness: $o ).
% 2.38/1.46
% 2.38/1.46 thf(nonemptyImpWitness,definition,
% 2.38/1.46 ( nonemptyImpWitness
% 2.38/1.46 = ( ! [A: $i] :
% 2.38/1.46 ( ( nonempty @ A )
% 2.38/1.46 => ? [Xx: $i] :
% 2.38/1.46 ( ( in @ Xx @ A )
% 2.38/1.46 & $true ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(uniqinunit_type,type,
% 2.38/1.46 uniqinunit: $o ).
% 2.38/1.46
% 2.38/1.46 thf(uniqinunit,definition,
% 2.38/1.46 ( uniqinunit
% 2.38/1.46 = ( ! [Xx: $i,Xy: $i] :
% 2.38/1.46 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.38/1.46 => ( Xx = Xy ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(notinsingleton_type,type,
% 2.38/1.46 notinsingleton: $o ).
% 2.38/1.46
% 2.38/1.46 thf(notinsingleton,definition,
% 2.38/1.46 ( notinsingleton
% 2.38/1.46 = ( ! [Xx: $i,Xy: $i] :
% 2.38/1.46 ( ( Xx != Xy )
% 2.38/1.46 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(eqinunit_type,type,
% 2.38/1.46 eqinunit: $o ).
% 2.38/1.46
% 2.38/1.46 thf(eqinunit,definition,
% 2.38/1.46 ( eqinunit
% 2.38/1.46 = ( ! [Xx: $i,Xy: $i] :
% 2.38/1.46 ( ( Xx = Xy )
% 2.38/1.46 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(singletonsswitch_type,type,
% 2.38/1.46 singletonsswitch: $o ).
% 2.38/1.46
% 2.38/1.46 thf(singletonsswitch,definition,
% 2.38/1.46 ( singletonsswitch
% 2.38/1.46 = ( ! [Xx: $i,Xy: $i] :
% 2.38/1.46 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.38/1.46 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(upairsetE_type,type,
% 2.38/1.46 upairsetE: $o ).
% 2.38/1.46
% 2.38/1.46 thf(upairsetE,definition,
% 2.38/1.46 ( upairsetE
% 2.38/1.46 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.38/1.46 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.38/1.46 => ( ( Xz = Xx )
% 2.38/1.46 | ( Xz = Xy ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(upairsetIL_type,type,
% 2.38/1.46 upairsetIL: $o ).
% 2.38/1.46
% 2.38/1.46 thf(upairsetIL,definition,
% 2.38/1.46 ( upairsetIL
% 2.38/1.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(upairsetIR_type,type,
% 2.38/1.46 upairsetIR: $o ).
% 2.38/1.46
% 2.38/1.46 thf(upairsetIR,definition,
% 2.38/1.46 ( upairsetIR
% 2.38/1.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(emptyE1_type,type,
% 2.38/1.46 emptyE1: $o ).
% 2.38/1.46
% 2.38/1.46 thf(emptyE1,definition,
% 2.38/1.46 ( emptyE1
% 2.38/1.46 = ( ! [A: $i,Xphi: $i > $o] :
% 2.38/1.46 ( ? [Xx: $i] :
% 2.38/1.46 ( ( in @ Xx @ A )
% 2.38/1.46 & ( Xphi @ Xx ) )
% 2.38/1.46 => ( ( ( dsetconstr @ A
% 2.38/1.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.38/1.46 = emptyset )
% 2.38/1.46 => $false ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(vacuousDall_type,type,
% 2.38/1.46 vacuousDall: $o ).
% 2.38/1.46
% 2.38/1.46 thf(vacuousDall,definition,
% 2.38/1.46 ( vacuousDall
% 2.38/1.46 = ( ! [Xphi: $i > $o,Xx: $i] :
% 2.38/1.46 ( ( in @ Xx @ emptyset )
% 2.38/1.46 => ( Xphi @ Xx ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(quantDeMorgan1_type,type,
% 2.38/1.46 quantDeMorgan1: $o ).
% 2.38/1.46
% 2.38/1.46 thf(quantDeMorgan1,definition,
% 2.38/1.46 ( quantDeMorgan1
% 2.38/1.46 = ( ! [A: $i,Xphi: $i > $o] :
% 2.38/1.46 ( ~ ! [Xx: $i] :
% 2.38/1.46 ( ( in @ Xx @ A )
% 2.38/1.46 => ( Xphi @ Xx ) )
% 2.38/1.46 => ? [Xx: $i] :
% 2.38/1.46 ( ( in @ Xx @ A )
% 2.38/1.46 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(quantDeMorgan2_type,type,
% 2.38/1.46 quantDeMorgan2: $o ).
% 2.38/1.46
% 2.38/1.46 thf(quantDeMorgan2,definition,
% 2.38/1.46 ( quantDeMorgan2
% 2.38/1.46 = ( ! [A: $i,Xphi: $i > $o] :
% 2.38/1.46 ( ! [Xx: $i] :
% 2.38/1.46 ( ( in @ Xx @ A )
% 2.38/1.46 => ~ ( Xphi @ Xx ) )
% 2.38/1.46 => ~ ? [Xx: $i] :
% 2.38/1.46 ( ( in @ Xx @ A )
% 2.38/1.46 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(quantDeMorgan3_type,type,
% 2.38/1.46 quantDeMorgan3: $o ).
% 2.38/1.46
% 2.38/1.46 thf(quantDeMorgan3,definition,
% 2.38/1.46 ( quantDeMorgan3
% 2.38/1.46 = ( ! [A: $i,Xphi: $i > $o] :
% 2.38/1.46 ( ~ ? [Xx: $i] :
% 2.38/1.46 ( ( in @ Xx @ A )
% 2.38/1.46 & ( Xphi @ Xx ) )
% 2.38/1.46 => ! [Xx: $i] :
% 2.38/1.46 ( ( in @ Xx @ A )
% 2.38/1.46 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(quantDeMorgan4_type,type,
% 2.38/1.46 quantDeMorgan4: $o ).
% 2.38/1.46
% 2.38/1.46 thf(quantDeMorgan4,definition,
% 2.38/1.46 ( quantDeMorgan4
% 2.38/1.46 = ( ! [A: $i,Xphi: $i > $o] :
% 2.38/1.46 ( ? [Xx: $i] :
% 2.38/1.46 ( ( in @ Xx @ A )
% 2.38/1.46 & ~ ( Xphi @ Xx ) )
% 2.38/1.46 => ~ ! [Xx: $i] :
% 2.38/1.46 ( ( in @ Xx @ A )
% 2.38/1.46 => ( Xphi @ Xx ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(prop2setI_type,type,
% 2.38/1.46 prop2setI: $o ).
% 2.38/1.46
% 2.38/1.46 thf(prop2setI,definition,
% 2.38/1.46 ( prop2setI
% 2.38/1.46 = ( ! [Xphi: $o] :
% 2.38/1.46 ( Xphi
% 2.38/1.46 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(set2prop_type,type,
% 2.38/1.46 set2prop: $i > $o ).
% 2.38/1.46
% 2.38/1.46 thf(prop2set2propI_type,type,
% 2.38/1.46 prop2set2propI: $o ).
% 2.38/1.46
% 2.38/1.46 thf(prop2set2propI,definition,
% 2.38/1.46 ( prop2set2propI
% 2.38/1.46 = ( ! [Xphi: $o] :
% 2.38/1.46 ( Xphi
% 2.38/1.46 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(notdexE_type,type,
% 2.38/1.46 notdexE: $o ).
% 2.38/1.46
% 2.38/1.46 thf(notdexE,definition,
% 2.38/1.46 ( notdexE
% 2.38/1.46 = ( ! [A: $i,Xphi: $i > $o] :
% 2.38/1.46 ( ~ ? [Xx: $i] :
% 2.38/1.46 ( ( in @ Xx @ A )
% 2.38/1.46 & ( Xphi @ Xx ) )
% 2.38/1.46 => ! [Xx: $i] :
% 2.38/1.46 ( ( in @ Xx @ A )
% 2.38/1.46 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(notdallE_type,type,
% 2.38/1.46 notdallE: $o ).
% 2.38/1.46
% 2.38/1.46 thf(notdallE,definition,
% 2.38/1.46 ( notdallE
% 2.38/1.46 = ( ! [A: $i,Xphi: $i > $o] :
% 2.38/1.46 ( ~ ! [Xx: $i] :
% 2.38/1.46 ( ( in @ Xx @ A )
% 2.38/1.46 => ( Xphi @ Xx ) )
% 2.38/1.46 => ? [Xx: $i] :
% 2.38/1.46 ( ( in @ Xx @ A )
% 2.38/1.46 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(exuI1_type,type,
% 2.38/1.46 exuI1: $o ).
% 2.38/1.46
% 2.38/1.46 thf(exuI1,definition,
% 2.38/1.46 ( exuI1
% 2.38/1.46 = ( ! [Xphi: $i > $o] :
% 2.38/1.46 ( ? [Xx: $i] :
% 2.38/1.46 ( ( Xphi @ Xx )
% 2.38/1.46 & ! [Xy: $i] :
% 2.38/1.46 ( ( Xphi @ Xy )
% 2.38/1.46 => ( Xx = Xy ) ) )
% 2.38/1.46 => ( exu
% 2.38/1.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(exuI3_type,type,
% 2.38/1.46 exuI3: $o ).
% 2.38/1.46
% 2.38/1.46 thf(exuI3,definition,
% 2.38/1.46 ( exuI3
% 2.38/1.46 = ( ! [Xphi: $i > $o] :
% 2.38/1.46 ( ? [Xx: $i] : ( Xphi @ Xx )
% 2.38/1.46 => ( ! [Xx: $i,Xy: $i] :
% 2.38/1.46 ( ( Xphi @ Xx )
% 2.38/1.46 => ( ( Xphi @ Xy )
% 2.38/1.46 => ( Xx = Xy ) ) )
% 2.38/1.46 => ( exu
% 2.38/1.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(exuI2_type,type,
% 2.38/1.46 exuI2: $o ).
% 2.38/1.46
% 2.38/1.46 thf(exuI2,definition,
% 2.38/1.46 ( exuI2
% 2.38/1.46 = ( ! [Xphi: $i > $o] :
% 2.38/1.46 ( ? [Xx: $i] :
% 2.38/1.46 ! [Xy: $i] :
% 2.38/1.46 ( ( Xphi @ Xy )
% 2.38/1.46 <=> ( Xy = Xx ) )
% 2.38/1.46 => ( exu
% 2.38/1.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(inCongP_type,type,
% 2.38/1.46 inCongP: $o ).
% 2.38/1.46
% 2.38/1.46 thf(inCongP,definition,
% 2.38/1.46 ( inCongP
% 2.38/1.46 = ( ! [A: $i,B: $i] :
% 2.38/1.46 ( ( A = B )
% 2.38/1.46 => ! [Xx: $i,Xy: $i] :
% 2.38/1.46 ( ( Xx = Xy )
% 2.38/1.46 => ( ( in @ Xx @ A )
% 2.38/1.46 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(in__Cong_type,type,
% 2.38/1.46 in__Cong: $o ).
% 2.38/1.46
% 2.38/1.46 thf(in__Cong,definition,
% 2.38/1.46 ( in__Cong
% 2.38/1.46 = ( ! [A: $i,B: $i] :
% 2.38/1.46 ( ( A = B )
% 2.38/1.46 => ! [Xx: $i,Xy: $i] :
% 2.38/1.46 ( ( Xx = Xy )
% 2.38/1.46 => ( ( in @ Xx @ A )
% 2.38/1.46 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(exuE3u_type,type,
% 2.38/1.46 exuE3u: $o ).
% 2.38/1.46
% 2.38/1.46 thf(exuE3u,definition,
% 2.38/1.46 ( exuE3u
% 2.38/1.46 = ( ! [Xphi: $i > $o] :
% 2.38/1.46 ( ( exu
% 2.38/1.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.38/1.46 => ! [Xx: $i,Xy: $i] :
% 2.38/1.46 ( ( Xphi @ Xx )
% 2.38/1.46 => ( ( Xphi @ Xy )
% 2.38/1.46 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(exu__Cong_type,type,
% 2.38/1.46 exu__Cong: $o ).
% 2.38/1.46
% 2.38/1.46 thf(exu__Cong,definition,
% 2.38/1.46 ( exu__Cong
% 2.38/1.46 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.38/1.46 ( ! [Xx: $i,Xy: $i] :
% 2.38/1.46 ( ( Xx = Xy )
% 2.38/1.46 => ( ( Xphi @ Xx )
% 2.38/1.46 <=> ( Xpsi @ Xy ) ) )
% 2.38/1.46 => ( ( exu
% 2.38/1.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.38/1.46 <=> ( exu
% 2.38/1.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(emptyset__Cong_type,type,
% 2.38/1.46 emptyset__Cong: $o ).
% 2.38/1.46
% 2.38/1.46 thf(emptyset__Cong,definition,
% 2.38/1.46 ( emptyset__Cong
% 2.38/1.46 = ( emptyset = emptyset ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(setadjoin__Cong_type,type,
% 2.38/1.46 setadjoin__Cong: $o ).
% 2.38/1.46
% 2.38/1.46 thf(setadjoin__Cong,definition,
% 2.38/1.46 ( setadjoin__Cong
% 2.38/1.46 = ( ! [Xx: $i,Xy: $i] :
% 2.38/1.46 ( ( Xx = Xy )
% 2.38/1.46 => ! [Xz: $i,Xu: $i] :
% 2.38/1.46 ( ( Xz = Xu )
% 2.38/1.46 => ( ( setadjoin @ Xx @ Xz )
% 2.38/1.46 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(powerset__Cong_type,type,
% 2.38/1.46 powerset__Cong: $o ).
% 2.38/1.46
% 2.38/1.46 thf(powerset__Cong,definition,
% 2.38/1.46 ( powerset__Cong
% 2.38/1.46 = ( ! [A: $i,B: $i] :
% 2.38/1.46 ( ( A = B )
% 2.38/1.46 => ( ( powerset @ A )
% 2.38/1.46 = ( powerset @ B ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(setunion__Cong_type,type,
% 2.38/1.46 setunion__Cong: $o ).
% 2.38/1.46
% 2.38/1.46 thf(setunion__Cong,definition,
% 2.38/1.46 ( setunion__Cong
% 2.38/1.46 = ( ! [A: $i,B: $i] :
% 2.38/1.46 ( ( A = B )
% 2.38/1.46 => ( ( setunion @ A )
% 2.38/1.46 = ( setunion @ B ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(omega__Cong_type,type,
% 2.38/1.46 omega__Cong: $o ).
% 2.38/1.46
% 2.38/1.46 thf(omega__Cong,definition,
% 2.38/1.46 ( omega__Cong
% 2.38/1.46 = ( omega = omega ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(exuEu_type,type,
% 2.38/1.46 exuEu: $o ).
% 2.38/1.46
% 2.38/1.46 thf(exuEu,definition,
% 2.38/1.46 ( exuEu
% 2.38/1.46 = ( ! [Xphi: $i > $o] :
% 2.38/1.46 ( ( exu
% 2.38/1.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.38/1.46 => ! [Xx: $i,Xy: $i] :
% 2.38/1.46 ( ( Xphi @ Xx )
% 2.38/1.46 => ( ( Xphi @ Xy )
% 2.38/1.46 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(descr__Cong_type,type,
% 2.38/1.46 descr__Cong: $o ).
% 2.38/1.46
% 2.38/1.46 thf(descr__Cong,definition,
% 2.38/1.46 ( descr__Cong
% 2.38/1.46 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.38/1.46 ( ! [Xx: $i,Xy: $i] :
% 2.38/1.46 ( ( Xx = Xy )
% 2.38/1.46 => ( ( Xphi @ Xx )
% 2.38/1.46 <=> ( Xpsi @ Xy ) ) )
% 2.38/1.46 => ( ( exu
% 2.38/1.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.38/1.46 => ( ( exu
% 2.38/1.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 2.38/1.46 => ( ( descr
% 2.38/1.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.38/1.46 = ( descr
% 2.38/1.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(dsetconstr__Cong_type,type,
% 2.38/1.46 dsetconstr__Cong: $o ).
% 2.38/1.46
% 2.38/1.46 thf(dsetconstr__Cong,definition,
% 2.38/1.46 ( dsetconstr__Cong
% 2.38/1.46 = ( ! [A: $i,B: $i] :
% 2.38/1.46 ( ( A = B )
% 2.38/1.46 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.38/1.46 ( ! [Xx: $i] :
% 2.38/1.46 ( ( in @ Xx @ A )
% 2.38/1.46 => ! [Xy: $i] :
% 2.38/1.46 ( ( in @ Xy @ B )
% 2.38/1.46 => ( ( Xx = Xy )
% 2.38/1.46 => ( ( Xphi @ Xx )
% 2.38/1.46 <=> ( Xpsi @ Xy ) ) ) ) )
% 2.38/1.46 => ( ( dsetconstr @ A
% 2.38/1.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.38/1.46 = ( dsetconstr @ B
% 2.38/1.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(subset_type,type,
% 2.38/1.46 subset: $i > $i > $o ).
% 2.38/1.46
% 2.38/1.46 thf(disjoint_type,type,
% 2.38/1.46 disjoint: $i > $i > $o ).
% 2.38/1.46
% 2.38/1.46 thf(setsmeet_type,type,
% 2.38/1.46 setsmeet: $i > $i > $o ).
% 2.38/1.46
% 2.38/1.46 thf(subsetI1_type,type,
% 2.38/1.46 subsetI1: $o ).
% 2.38/1.46
% 2.38/1.46 thf(subsetI1,definition,
% 2.38/1.46 ( subsetI1
% 2.38/1.46 = ( ! [A: $i,B: $i] :
% 2.38/1.46 ( ! [Xx: $i] :
% 2.38/1.46 ( ( in @ Xx @ A )
% 2.38/1.46 => ( in @ Xx @ B ) )
% 2.38/1.46 => ( subset @ A @ B ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(eqimpsubset2_type,type,
% 2.38/1.46 eqimpsubset2: $o ).
% 2.38/1.46
% 2.38/1.46 thf(eqimpsubset2,definition,
% 2.38/1.46 ( eqimpsubset2
% 2.38/1.46 = ( ! [A: $i,B: $i] :
% 2.38/1.46 ( ( A = B )
% 2.38/1.46 => ( subset @ B @ A ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(eqimpsubset1_type,type,
% 2.38/1.46 eqimpsubset1: $o ).
% 2.38/1.46
% 2.38/1.46 thf(eqimpsubset1,definition,
% 2.38/1.46 ( eqimpsubset1
% 2.38/1.46 = ( ! [A: $i,B: $i] :
% 2.38/1.46 ( ( A = B )
% 2.38/1.46 => ( subset @ A @ B ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(subsetI2_type,type,
% 2.38/1.46 subsetI2: $o ).
% 2.38/1.46
% 2.38/1.46 thf(subsetI2,definition,
% 2.38/1.46 ( subsetI2
% 2.38/1.46 = ( ! [A: $i,B: $i] :
% 2.38/1.46 ( ! [Xx: $i] :
% 2.38/1.46 ( ( in @ Xx @ A )
% 2.38/1.46 => ( in @ Xx @ B ) )
% 2.38/1.46 => ( subset @ A @ B ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(emptysetsubset_type,type,
% 2.38/1.46 emptysetsubset: $o ).
% 2.38/1.46
% 2.38/1.46 thf(emptysetsubset,definition,
% 2.38/1.46 ( emptysetsubset
% 2.38/1.46 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(subsetE_type,type,
% 2.38/1.46 subsetE: $o ).
% 2.38/1.46
% 2.38/1.46 thf(subsetE,definition,
% 2.38/1.46 ( subsetE
% 2.38/1.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.38/1.46 ( ( subset @ A @ B )
% 2.38/1.46 => ( ( in @ Xx @ A )
% 2.38/1.46 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(subsetE2_type,type,
% 2.38/1.46 subsetE2: $o ).
% 2.38/1.46
% 2.38/1.46 thf(subsetE2,definition,
% 2.38/1.46 ( subsetE2
% 2.38/1.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.38/1.46 ( ( subset @ A @ B )
% 2.38/1.46 => ( ~ ( in @ Xx @ B )
% 2.38/1.46 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(notsubsetI_type,type,
% 2.38/1.46 notsubsetI: $o ).
% 2.38/1.46
% 2.38/1.46 thf(notsubsetI,definition,
% 2.38/1.46 ( notsubsetI
% 2.38/1.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.38/1.46 ( ( in @ Xx @ A )
% 2.38/1.46 => ( ~ ( in @ Xx @ B )
% 2.38/1.46 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(notequalI1_type,type,
% 2.38/1.46 notequalI1: $o ).
% 2.38/1.46
% 2.38/1.46 thf(notequalI1,definition,
% 2.38/1.46 ( notequalI1
% 2.38/1.46 = ( ! [A: $i,B: $i] :
% 2.38/1.46 ( ~ ( subset @ A @ B )
% 2.38/1.46 => ( A != B ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(notequalI2_type,type,
% 2.38/1.46 notequalI2: $o ).
% 2.38/1.46
% 2.38/1.46 thf(notequalI2,definition,
% 2.38/1.46 ( notequalI2
% 2.38/1.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.38/1.46 ( ( in @ Xx @ A )
% 2.38/1.46 => ( ~ ( in @ Xx @ B )
% 2.38/1.46 => ( A != B ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(subsetRefl_type,type,
% 2.38/1.46 subsetRefl: $o ).
% 2.38/1.46
% 2.38/1.46 thf(subsetRefl,definition,
% 2.38/1.46 ( subsetRefl
% 2.38/1.46 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(subsetTrans_type,type,
% 2.38/1.46 subsetTrans: $o ).
% 2.38/1.46
% 2.38/1.46 thf(subsetTrans,definition,
% 2.38/1.46 ( subsetTrans
% 2.38/1.46 = ( ! [A: $i,B: $i,C: $i] :
% 2.38/1.46 ( ( subset @ A @ B )
% 2.38/1.46 => ( ( subset @ B @ C )
% 2.38/1.46 => ( subset @ A @ C ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(setadjoinSub_type,type,
% 2.38/1.46 setadjoinSub: $o ).
% 2.38/1.46
% 2.38/1.46 thf(setadjoinSub,definition,
% 2.38/1.46 ( setadjoinSub
% 2.38/1.46 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(setadjoinSub2_type,type,
% 2.38/1.46 setadjoinSub2: $o ).
% 2.38/1.46
% 2.38/1.46 thf(setadjoinSub2,definition,
% 2.38/1.46 ( setadjoinSub2
% 2.38/1.46 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.38/1.46 ( ( subset @ A @ B )
% 2.38/1.46 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(subset2powerset_type,type,
% 2.38/1.46 subset2powerset: $o ).
% 2.38/1.46
% 2.38/1.46 thf(subset2powerset,definition,
% 2.38/1.46 ( subset2powerset
% 2.38/1.46 = ( ! [A: $i,B: $i] :
% 2.38/1.46 ( ( subset @ A @ B )
% 2.38/1.46 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(setextsub_type,type,
% 2.38/1.46 setextsub: $o ).
% 2.38/1.46
% 2.38/1.46 thf(setextsub,definition,
% 2.38/1.46 ( setextsub
% 2.38/1.46 = ( ! [A: $i,B: $i] :
% 2.38/1.46 ( ( subset @ A @ B )
% 2.38/1.46 => ( ( subset @ B @ A )
% 2.38/1.46 => ( A = B ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(subsetemptysetimpeq_type,type,
% 2.38/1.46 subsetemptysetimpeq: $o ).
% 2.38/1.46
% 2.38/1.46 thf(subsetemptysetimpeq,definition,
% 2.38/1.46 ( subsetemptysetimpeq
% 2.38/1.46 = ( ! [A: $i] :
% 2.38/1.46 ( ( subset @ A @ emptyset )
% 2.38/1.46 => ( A = emptyset ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(powersetI1_type,type,
% 2.38/1.46 powersetI1: $o ).
% 2.38/1.46
% 2.38/1.46 thf(powersetI1,definition,
% 2.38/1.46 ( powersetI1
% 2.38/1.46 = ( ! [A: $i,B: $i] :
% 2.38/1.46 ( ( subset @ B @ A )
% 2.38/1.46 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(powersetE1_type,type,
% 2.38/1.46 powersetE1: $o ).
% 2.38/1.46
% 2.38/1.46 thf(powersetE1,definition,
% 2.38/1.46 ( powersetE1
% 2.38/1.46 = ( ! [A: $i,B: $i] :
% 2.38/1.46 ( ( in @ B @ ( powerset @ A ) )
% 2.38/1.46 => ( subset @ B @ A ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(inPowerset_type,type,
% 2.38/1.46 inPowerset: $o ).
% 2.38/1.46
% 2.38/1.46 thf(inPowerset,definition,
% 2.38/1.46 ( inPowerset
% 2.38/1.46 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(powersetsubset_type,type,
% 2.38/1.46 powersetsubset: $o ).
% 2.38/1.46
% 2.38/1.46 thf(powersetsubset,definition,
% 2.38/1.46 ( powersetsubset
% 2.38/1.46 = ( ! [A: $i,B: $i] :
% 2.38/1.46 ( ( subset @ A @ B )
% 2.38/1.46 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(sepInPowerset_type,type,
% 2.38/1.46 sepInPowerset: $o ).
% 2.38/1.46
% 2.38/1.46 thf(sepInPowerset,definition,
% 2.38/1.46 ( sepInPowerset
% 2.38/1.46 = ( ! [A: $i,Xphi: $i > $o] :
% 2.38/1.46 ( in
% 2.38/1.46 @ ( dsetconstr @ A
% 2.38/1.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.38/1.46 @ ( powerset @ A ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(sepSubset_type,type,
% 2.38/1.46 sepSubset: $o ).
% 2.38/1.46
% 2.38/1.46 thf(sepSubset,definition,
% 2.38/1.46 ( sepSubset
% 2.38/1.46 = ( ! [A: $i,Xphi: $i > $o] :
% 2.38/1.46 ( subset
% 2.38/1.46 @ ( dsetconstr @ A
% 2.38/1.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.38/1.46 @ A ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(binunion_type,type,
% 2.38/1.46 binunion: $i > $i > $i ).
% 2.38/1.46
% 2.38/1.46 thf(binunionIL_type,type,
% 2.38/1.46 binunionIL: $o ).
% 2.38/1.46
% 2.38/1.46 thf(binunionIL,definition,
% 2.38/1.46 ( binunionIL
% 2.38/1.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.38/1.46 ( ( in @ Xx @ A )
% 2.38/1.46 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(upairset2IR_type,type,
% 2.38/1.46 upairset2IR: $o ).
% 2.38/1.46
% 2.38/1.46 thf(upairset2IR,definition,
% 2.38/1.46 ( upairset2IR
% 2.38/1.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(binunionIR_type,type,
% 2.38/1.46 binunionIR: $o ).
% 2.38/1.46
% 2.38/1.46 thf(binunionIR,definition,
% 2.38/1.46 ( binunionIR
% 2.38/1.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.38/1.46 ( ( in @ Xx @ B )
% 2.38/1.46 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(binunionEcases_type,type,
% 2.38/1.46 binunionEcases: $o ).
% 2.38/1.46
% 2.38/1.46 thf(binunionEcases,definition,
% 2.38/1.46 ( binunionEcases
% 2.38/1.46 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 2.38/1.46 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.38/1.46 => ( ( ( in @ Xx @ A )
% 2.38/1.46 => Xphi )
% 2.38/1.46 => ( ( ( in @ Xx @ B )
% 2.38/1.46 => Xphi )
% 2.38/1.46 => Xphi ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(binunionE_type,type,
% 2.38/1.46 binunionE: $o ).
% 2.38/1.46
% 2.38/1.46 thf(binunionE,definition,
% 2.38/1.46 ( binunionE
% 2.38/1.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.38/1.46 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.38/1.46 => ( ( in @ Xx @ A )
% 2.38/1.46 | ( in @ Xx @ B ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(binunionLsub_type,type,
% 2.38/1.46 binunionLsub: $o ).
% 2.38/1.46
% 2.38/1.46 thf(binunionLsub,definition,
% 2.38/1.46 ( binunionLsub
% 2.38/1.46 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(binunionRsub_type,type,
% 2.38/1.46 binunionRsub: $o ).
% 2.38/1.46
% 2.38/1.46 thf(binunionRsub,definition,
% 2.38/1.46 ( binunionRsub
% 2.38/1.46 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(binintersect_type,type,
% 2.38/1.46 binintersect: $i > $i > $i ).
% 2.38/1.46
% 2.38/1.46 thf(binintersectI_type,type,
% 2.38/1.46 binintersectI: $o ).
% 2.38/1.46
% 2.38/1.46 thf(binintersectI,definition,
% 2.38/1.46 ( binintersectI
% 2.38/1.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.38/1.46 ( ( in @ Xx @ A )
% 2.38/1.46 => ( ( in @ Xx @ B )
% 2.38/1.46 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(binintersectSubset5_type,type,
% 2.38/1.46 binintersectSubset5: $o ).
% 2.38/1.46
% 2.38/1.46 thf(binintersectSubset5,definition,
% 2.38/1.46 ( binintersectSubset5
% 2.38/1.46 = ( ! [A: $i,B: $i,C: $i] :
% 2.38/1.46 ( ( subset @ C @ A )
% 2.38/1.46 => ( ( subset @ C @ B )
% 2.38/1.46 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(binintersectEL_type,type,
% 2.38/1.46 binintersectEL: $o ).
% 2.38/1.46
% 2.38/1.46 thf(binintersectEL,definition,
% 2.38/1.46 ( binintersectEL
% 2.38/1.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.38/1.46 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.38/1.46 => ( in @ Xx @ A ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(binintersectLsub_type,type,
% 2.38/1.46 binintersectLsub: $o ).
% 2.38/1.46
% 2.38/1.46 thf(binintersectLsub,definition,
% 2.38/1.46 ( binintersectLsub
% 2.38/1.46 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(binintersectSubset2_type,type,
% 2.38/1.46 binintersectSubset2: $o ).
% 2.38/1.46
% 2.38/1.46 thf(binintersectSubset2,definition,
% 2.38/1.46 ( binintersectSubset2
% 2.38/1.46 = ( ! [A: $i,B: $i] :
% 2.38/1.46 ( ( subset @ A @ B )
% 2.38/1.46 => ( ( binintersect @ A @ B )
% 2.38/1.46 = A ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(binintersectSubset3_type,type,
% 2.38/1.46 binintersectSubset3: $o ).
% 2.38/1.46
% 2.38/1.46 thf(binintersectSubset3,definition,
% 2.38/1.46 ( binintersectSubset3
% 2.38/1.46 = ( ! [A: $i,B: $i] :
% 2.38/1.46 ( ( ( binintersect @ A @ B )
% 2.38/1.46 = B )
% 2.38/1.46 => ( subset @ B @ A ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(binintersectER_type,type,
% 2.38/1.46 binintersectER: $o ).
% 2.38/1.46
% 2.38/1.46 thf(binintersectER,definition,
% 2.38/1.46 ( binintersectER
% 2.38/1.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.38/1.46 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.38/1.46 => ( in @ Xx @ B ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(disjointsetsI1_type,type,
% 2.38/1.46 disjointsetsI1: $o ).
% 2.38/1.46
% 2.38/1.46 thf(disjointsetsI1,definition,
% 2.38/1.46 ( disjointsetsI1
% 2.38/1.46 = ( ! [A: $i,B: $i] :
% 2.38/1.46 ( ~ ? [Xx: $i] :
% 2.38/1.46 ( ( in @ Xx @ A )
% 2.38/1.46 & ( in @ Xx @ B ) )
% 2.38/1.46 => ( ( binintersect @ A @ B )
% 2.38/1.46 = emptyset ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(binintersectRsub_type,type,
% 2.38/1.46 binintersectRsub: $o ).
% 2.38/1.46
% 2.38/1.46 thf(binintersectRsub,definition,
% 2.38/1.46 ( binintersectRsub
% 2.38/1.46 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(binintersectSubset4_type,type,
% 2.38/1.46 binintersectSubset4: $o ).
% 2.38/1.46
% 2.38/1.46 thf(binintersectSubset4,definition,
% 2.38/1.46 ( binintersectSubset4
% 2.38/1.46 = ( ! [A: $i,B: $i] :
% 2.38/1.46 ( ( subset @ B @ A )
% 2.38/1.46 => ( ( binintersect @ A @ B )
% 2.38/1.46 = B ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(binintersectSubset1_type,type,
% 2.38/1.46 binintersectSubset1: $o ).
% 2.38/1.46
% 2.38/1.46 thf(binintersectSubset1,definition,
% 2.38/1.46 ( binintersectSubset1
% 2.38/1.46 = ( ! [A: $i,B: $i] :
% 2.38/1.46 ( ( ( binintersect @ A @ B )
% 2.38/1.46 = A )
% 2.38/1.46 => ( subset @ A @ B ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(bs114d_type,type,
% 2.38/1.46 bs114d: $o ).
% 2.38/1.46
% 2.38/1.46 thf(bs114d,definition,
% 2.38/1.46 ( bs114d
% 2.38/1.46 = ( ! [A: $i,B: $i,C: $i] :
% 2.38/1.46 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 2.38/1.46 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(regular_type,type,
% 2.38/1.46 regular: $i > $o ).
% 2.38/1.46
% 2.38/1.46 thf(setminus_type,type,
% 2.38/1.46 setminus: $i > $i > $i ).
% 2.38/1.46
% 2.38/1.46 thf(setminusI_type,type,
% 2.38/1.46 setminusI: $o ).
% 2.38/1.46
% 2.38/1.46 thf(setminusI,definition,
% 2.38/1.46 ( setminusI
% 2.38/1.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.38/1.46 ( ( in @ Xx @ A )
% 2.38/1.46 => ( ~ ( in @ Xx @ B )
% 2.38/1.46 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(setminusEL_type,type,
% 2.38/1.46 setminusEL: $o ).
% 2.38/1.46
% 2.38/1.46 thf(setminusEL,definition,
% 2.38/1.46 ( setminusEL
% 2.38/1.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.38/1.46 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.38/1.46 => ( in @ Xx @ A ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(setminusER_type,type,
% 2.38/1.46 setminusER: $o ).
% 2.38/1.46
% 2.38/1.46 thf(setminusER,definition,
% 2.38/1.46 ( setminusER
% 2.38/1.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.38/1.46 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.38/1.46 => ~ ( in @ Xx @ B ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(setminusSubset2_type,type,
% 2.38/1.46 setminusSubset2: $o ).
% 2.38/1.46
% 2.38/1.46 thf(setminusSubset2,definition,
% 2.38/1.46 ( setminusSubset2
% 2.38/1.46 = ( ! [A: $i,B: $i] :
% 2.38/1.46 ( ( subset @ A @ B )
% 2.38/1.46 => ( ( setminus @ A @ B )
% 2.38/1.46 = emptyset ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(setminusERneg_type,type,
% 2.38/1.46 setminusERneg: $o ).
% 2.38/1.46
% 2.38/1.46 thf(setminusERneg,definition,
% 2.38/1.46 ( setminusERneg
% 2.38/1.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.38/1.46 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.38/1.46 => ( ( in @ Xx @ A )
% 2.38/1.46 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(setminusELneg_type,type,
% 2.38/1.46 setminusELneg: $o ).
% 2.38/1.46
% 2.38/1.46 thf(setminusELneg,definition,
% 2.38/1.46 ( setminusELneg
% 2.38/1.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.38/1.46 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.38/1.46 => ( ~ ( in @ Xx @ B )
% 2.38/1.46 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(setminusILneg_type,type,
% 2.38/1.46 setminusILneg: $o ).
% 2.38/1.46
% 2.38/1.46 thf(setminusILneg,definition,
% 2.38/1.46 ( setminusILneg
% 2.38/1.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.38/1.46 ( ~ ( in @ Xx @ A )
% 2.38/1.46 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(setminusIRneg_type,type,
% 2.38/1.46 setminusIRneg: $o ).
% 2.38/1.46
% 2.38/1.46 thf(setminusIRneg,definition,
% 2.38/1.46 ( setminusIRneg
% 2.38/1.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.38/1.46 ( ( in @ Xx @ B )
% 2.38/1.46 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(setminusLsub_type,type,
% 2.38/1.46 setminusLsub: $o ).
% 2.38/1.46
% 2.38/1.46 thf(setminusLsub,definition,
% 2.38/1.46 ( setminusLsub
% 2.38/1.46 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(setminusSubset1_type,type,
% 2.38/1.46 setminusSubset1: $o ).
% 2.38/1.46
% 2.38/1.46 thf(setminusSubset1,definition,
% 2.38/1.46 ( setminusSubset1
% 2.38/1.46 = ( ! [A: $i,B: $i] :
% 2.38/1.46 ( ( ( setminus @ A @ B )
% 2.38/1.46 = emptyset )
% 2.38/1.46 => ( subset @ A @ B ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(symdiff_type,type,
% 2.38/1.46 symdiff: $i > $i > $i ).
% 2.38/1.46
% 2.38/1.46 thf(symdiffE_type,type,
% 2.38/1.46 symdiffE: $o ).
% 2.38/1.46
% 2.38/1.46 thf(symdiffE,definition,
% 2.38/1.46 ( symdiffE
% 2.38/1.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.38/1.46 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 2.38/1.46 => ! [Xphi: $o] :
% 2.38/1.46 ( ( ( in @ Xx @ A )
% 2.38/1.46 => ( ~ ( in @ Xx @ B )
% 2.38/1.46 => Xphi ) )
% 2.38/1.46 => ( ( ~ ( in @ Xx @ A )
% 2.38/1.46 => ( ( in @ Xx @ B )
% 2.38/1.46 => Xphi ) )
% 2.38/1.46 => Xphi ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(symdiffI1_type,type,
% 2.38/1.46 symdiffI1: $o ).
% 2.38/1.46
% 2.38/1.46 thf(symdiffI1,definition,
% 2.38/1.46 ( symdiffI1
% 2.38/1.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.38/1.46 ( ( in @ Xx @ A )
% 2.38/1.46 => ( ~ ( in @ Xx @ B )
% 2.38/1.46 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(symdiffI2_type,type,
% 2.38/1.46 symdiffI2: $o ).
% 2.38/1.46
% 2.38/1.46 thf(symdiffI2,definition,
% 2.38/1.46 ( symdiffI2
% 2.38/1.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.38/1.46 ( ~ ( in @ Xx @ A )
% 2.38/1.46 => ( ( in @ Xx @ B )
% 2.38/1.46 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(symdiffIneg1_type,type,
% 2.38/1.46 symdiffIneg1: $o ).
% 2.38/1.46
% 2.38/1.46 thf(symdiffIneg1,definition,
% 2.38/1.46 ( symdiffIneg1
% 2.38/1.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.38/1.46 ( ( in @ Xx @ A )
% 2.38/1.46 => ( ( in @ Xx @ B )
% 2.38/1.46 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(symdiffIneg2_type,type,
% 2.38/1.46 symdiffIneg2: $o ).
% 2.38/1.46
% 2.38/1.46 thf(symdiffIneg2,definition,
% 2.38/1.46 ( symdiffIneg2
% 2.38/1.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.38/1.46 ( ~ ( in @ Xx @ A )
% 2.38/1.46 => ( ~ ( in @ Xx @ B )
% 2.38/1.46 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(iskpair_type,type,
% 2.38/1.46 iskpair: $i > $o ).
% 2.38/1.46
% 2.38/1.46 thf(iskpair,definition,
% 2.38/1.46 ( iskpair
% 2.38/1.46 = ( ^ [A: $i] :
% 2.38/1.46 ? [Xx: $i] :
% 2.38/1.46 ( ( in @ Xx @ ( setunion @ A ) )
% 2.38/1.46 & ? [Xy: $i] :
% 2.38/1.46 ( ( in @ Xy @ ( setunion @ A ) )
% 2.38/1.46 & ( A
% 2.38/1.46 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(secondinupair_type,type,
% 2.38/1.46 secondinupair: $o ).
% 2.38/1.46
% 2.38/1.46 thf(secondinupair,definition,
% 2.38/1.46 ( secondinupair
% 2.38/1.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(setukpairIL_type,type,
% 2.38/1.46 setukpairIL: $o ).
% 2.38/1.46
% 2.38/1.46 thf(setukpairIL,definition,
% 2.38/1.46 ( setukpairIL
% 2.38/1.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.38/1.46
% 2.38/1.46 thf(setukpairIR,conjecture,
% 2.38/1.46 ( setextAx
% 2.38/1.46 => ( emptysetAx
% 2.38/1.46 => ( setadjoinAx
% 2.38/1.46 => ( powersetAx
% 2.38/1.46 => ( setunionAx
% 2.38/1.46 => ( omega0Ax
% 2.38/1.46 => ( omegaSAx
% 2.38/1.46 => ( omegaIndAx
% 2.38/1.46 => ( replAx
% 2.38/1.46 => ( foundationAx
% 2.38/1.46 => ( wellorderingAx
% 2.38/1.46 => ( descrp
% 2.38/1.46 => ( dsetconstrI
% 2.38/1.46 => ( dsetconstrEL
% 2.38/1.46 => ( dsetconstrER
% 2.38/1.46 => ( exuE1
% 2.38/1.46 => ( prop2setE
% 2.38/1.46 => ( emptysetE
% 2.38/1.46 => ( emptysetimpfalse
% 2.38/1.46 => ( notinemptyset
% 2.38/1.46 => ( exuE3e
% 2.38/1.46 => ( setext
% 2.38/1.46 => ( emptyI
% 2.38/1.46 => ( noeltsimpempty
% 2.38/1.46 => ( setbeta
% 2.38/1.46 => ( nonemptyE1
% 2.38/1.46 => ( nonemptyI
% 2.38/1.46 => ( nonemptyI1
% 2.38/1.46 => ( setadjoinIL
% 2.38/1.46 => ( emptyinunitempty
% 2.38/1.46 => ( setadjoinIR
% 2.38/1.46 => ( setadjoinE
% 2.38/1.46 => ( setadjoinOr
% 2.38/1.46 => ( setoftrueEq
% 2.38/1.46 => ( powersetI
% 2.38/1.46 => ( emptyinPowerset
% 2.38/1.46 => ( emptyInPowerset
% 2.38/1.46 => ( powersetE
% 2.38/1.46 => ( setunionI
% 2.38/1.46 => ( setunionE
% 2.38/1.46 => ( subPowSU
% 2.38/1.46 => ( exuE2
% 2.38/1.46 => ( nonemptyImpWitness
% 2.38/1.46 => ( uniqinunit
% 2.38/1.46 => ( notinsingleton
% 2.38/1.46 => ( eqinunit
% 2.38/1.46 => ( singletonsswitch
% 2.38/1.46 => ( upairsetE
% 2.38/1.46 => ( upairsetIL
% 2.38/1.46 => ( upairsetIR
% 2.38/1.46 => ( emptyE1
% 2.38/1.46 => ( vacuousDall
% 2.38/1.46 => ( quantDeMorgan1
% 2.38/1.46 => ( quantDeMorgan2
% 2.38/1.46 => ( quantDeMorgan3
% 2.38/1.46 => ( quantDeMorgan4
% 2.38/1.46 => ( prop2setI
% 2.38/1.46 => ( prop2set2propI
% 2.38/1.46 => ( notdexE
% 2.38/1.46 => ( notdallE
% 2.38/1.46 => ( exuI1
% 2.38/1.46 => ( exuI3
% 2.38/1.46 => ( exuI2
% 2.38/1.46 => ( inCongP
% 2.38/1.46 => ( in__Cong
% 2.38/1.46 => ( exuE3u
% 2.38/1.46 => ( exu__Cong
% 2.38/1.46 => ( emptyset__Cong
% 2.38/1.46 => ( setadjoin__Cong
% 2.38/1.46 => ( powerset__Cong
% 2.38/1.46 => ( setunion__Cong
% 2.38/1.46 => ( omega__Cong
% 2.38/1.46 => ( exuEu
% 2.38/1.46 => ( descr__Cong
% 2.38/1.46 => ( dsetconstr__Cong
% 2.38/1.46 => ( subsetI1
% 2.38/1.46 => ( eqimpsubset2
% 2.38/1.46 => ( eqimpsubset1
% 2.38/1.46 => ( subsetI2
% 2.38/1.46 => ( emptysetsubset
% 2.38/1.46 => ( subsetE
% 2.38/1.46 => ( subsetE2
% 2.38/1.46 => ( notsubsetI
% 2.38/1.46 => ( notequalI1
% 2.38/1.46 => ( notequalI2
% 2.38/1.46 => ( subsetRefl
% 2.38/1.46 => ( subsetTrans
% 2.38/1.46 => ( setadjoinSub
% 2.38/1.46 => ( setadjoinSub2
% 2.38/1.46 => ( subset2powerset
% 2.38/1.47 => ( setextsub
% 2.38/1.47 => ( subsetemptysetimpeq
% 2.38/1.47 => ( powersetI1
% 2.38/1.47 => ( powersetE1
% 2.38/1.47 => ( inPowerset
% 2.38/1.47 => ( powersetsubset
% 2.38/1.47 => ( sepInPowerset
% 2.38/1.47 => ( sepSubset
% 2.38/1.47 => ( binunionIL
% 2.38/1.47 => ( upairset2IR
% 2.38/1.47 => ( binunionIR
% 2.38/1.47 => ( binunionEcases
% 2.38/1.47 => ( binunionE
% 2.38/1.47 => ( binunionLsub
% 2.38/1.47 => ( binunionRsub
% 2.38/1.47 => ( binintersectI
% 2.38/1.47 => ( binintersectSubset5
% 2.38/1.47 => ( binintersectEL
% 2.38/1.47 => ( binintersectLsub
% 2.38/1.47 => ( binintersectSubset2
% 2.38/1.47 => ( binintersectSubset3
% 2.38/1.47 => ( binintersectER
% 2.38/1.47 => ( disjointsetsI1
% 2.38/1.47 => ( binintersectRsub
% 2.38/1.47 => ( binintersectSubset4
% 2.38/1.47 => ( binintersectSubset1
% 2.38/1.47 => ( bs114d
% 2.38/1.47 => ( setminusI
% 2.38/1.47 => ( setminusEL
% 2.38/1.47 => ( setminusER
% 2.38/1.47 => ( setminusSubset2
% 2.38/1.47 => ( setminusERneg
% 2.38/1.47 => ( setminusELneg
% 2.38/1.47 => ( setminusILneg
% 2.38/1.47 => ( setminusIRneg
% 2.38/1.47 => ( setminusLsub
% 2.38/1.47 => ( setminusSubset1
% 2.38/1.47 => ( symdiffE
% 2.38/1.47 => ( symdiffI1
% 2.38/1.47 => ( symdiffI2
% 2.38/1.47 => ( symdiffIneg1
% 2.38/1.47 => ( symdiffIneg2
% 2.38/1.47 => ( secondinupair
% 2.38/1.47 => ( setukpairIL
% 2.38/1.47 => ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.38/1.47 ------------------------
% 2.38/1.48 % (26196)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_25972 for (2999ds/275Mi)
% 2.38/1.48 % (26195)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_25972 for (2999ds/2Mi)
% 2.38/1.48 % (26193)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_25972 for (2999ds/27Mi)
% 2.38/1.48 % (26195)Instruction limit reached!
% 2.38/1.48 % (26195)------------------------------
% 2.38/1.48 % (26195)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.38/1.48 % (26195)Termination reason: Unknown
% 2.38/1.48 % (26195)Termination phase: shuffling
% 2.38/1.48
% 2.38/1.48 % (26195)Memory used [KB]: 1279
% 2.38/1.48 % (26195)Time elapsed: 0.002 s
% 2.38/1.48 % (26195)Instructions burned: 4 (million)
% 2.38/1.48 % (26195)------------------------------
% 2.38/1.48 % (26195)------------------------------
% 2.38/1.49 % (26192)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_25972 for (2999ds/4Mi)
% 2.38/1.49 % (26191)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_25972 for (2999ds/183Mi)
% 2.38/1.49 % (26197)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_25972 for (2999ds/18Mi)
% 2.38/1.49 % (26194)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_25972 for (2999ds/2Mi)
% 2.38/1.49 % (26193)Instruction limit reached!
% 2.38/1.49 % (26193)------------------------------
% 2.38/1.49 % (26193)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.38/1.49 % (26193)Termination reason: Unknown
% 2.38/1.49 % (26193)Termination phase: Property scanning
% 2.38/1.49
% 2.38/1.49 % (26193)Memory used [KB]: 1663
% 2.38/1.49 % (26193)Time elapsed: 0.010 s
% 2.38/1.49 % (26193)Instructions burned: 30 (million)
% 2.38/1.49 % (26193)------------------------------
% 2.38/1.49 % (26193)------------------------------
% 2.38/1.49 % (26194)Instruction limit reached!
% 2.38/1.49 % (26194)------------------------------
% 2.38/1.49 % (26194)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.38/1.49 % (26194)Termination reason: Unknown
% 2.38/1.49 % (26194)Termination phase: shuffling
% 2.38/1.49
% 2.38/1.49 % (26194)Memory used [KB]: 1151
% 2.38/1.49 % (26194)Time elapsed: 0.003 s
% 2.38/1.49 % (26194)Instructions burned: 3 (million)
% 2.38/1.49 % (26194)------------------------------
% 2.38/1.49 % (26194)------------------------------
% 2.38/1.49 % (26192)Instruction limit reached!
% 2.38/1.49 % (26192)------------------------------
% 2.38/1.49 % (26192)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.38/1.49 % (26192)Termination reason: Unknown
% 2.38/1.49 % (26192)Termination phase: shuffling
% 2.38/1.49
% 2.38/1.49 % (26192)Memory used [KB]: 1279
% 2.38/1.49 % (26192)Time elapsed: 0.005 s
% 2.38/1.49 % (26192)Instructions burned: 5 (million)
% 2.38/1.49 % (26192)------------------------------
% 2.38/1.49 % (26192)------------------------------
% 2.38/1.49 % (26203)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_25972 for (2999ds/3Mi)
% 2.38/1.49 % (26203)Instruction limit reached!
% 2.38/1.49 % (26203)------------------------------
% 2.38/1.49 % (26203)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.38/1.49 % (26203)Termination reason: Unknown
% 2.38/1.49 % (26203)Termination phase: shuffling
% 2.38/1.49
% 2.38/1.49 % (26203)Memory used [KB]: 1279
% 2.38/1.49 % (26203)Time elapsed: 0.002 s
% 2.38/1.49 % (26203)Instructions burned: 4 (million)
% 2.38/1.49 % (26203)------------------------------
% 2.38/1.49 % (26203)------------------------------
% 2.45/1.50 % (26197)Instruction limit reached!
% 2.45/1.50 % (26197)------------------------------
% 2.45/1.50 % (26197)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.45/1.50 % (26197)Termination reason: Unknown
% 2.45/1.50 % (26197)Termination phase: Preprocessing 2
% 2.45/1.50
% 2.45/1.50 % (26197)Memory used [KB]: 1535
% 2.45/1.50 % (26197)Time elapsed: 0.013 s
% 2.45/1.50 % (26197)Instructions burned: 19 (million)
% 2.45/1.50 % (26197)------------------------------
% 2.45/1.50 % (26197)------------------------------
% 2.45/1.50 % (26207)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_25972 for (2999ds/37Mi)
% 2.45/1.50 % (26211)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_25972 for (2999ds/1041Mi)
% 2.45/1.50 % (26209)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_25972 for (2999ds/3Mi)
% 2.45/1.50 % (26209)Instruction limit reached!
% 2.45/1.50 % (26209)------------------------------
% 2.45/1.50 % (26209)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.45/1.50 % (26209)Termination reason: Unknown
% 2.45/1.50 % (26209)Termination phase: shuffling
% 2.45/1.50
% 2.45/1.50 % (26209)Memory used [KB]: 1279
% 2.45/1.50 % (26209)Time elapsed: 0.003 s
% 2.45/1.50 % (26209)Instructions burned: 4 (million)
% 2.45/1.50 % (26209)------------------------------
% 2.45/1.50 % (26209)------------------------------
% 2.45/1.51 % (26208)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_25972 for (2999ds/15Mi)
% 2.45/1.51 % (26207)Instruction limit reached!
% 2.45/1.51 % (26207)------------------------------
% 2.45/1.51 % (26207)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.45/1.51 % (26207)Termination reason: Unknown
% 2.45/1.51 % (26207)Termination phase: Property scanning
% 2.45/1.51
% 2.45/1.51 % (26207)Memory used [KB]: 1663
% 2.45/1.51 % (26207)Time elapsed: 0.011 s
% 2.45/1.51 % (26207)Instructions burned: 38 (million)
% 2.45/1.51 % (26207)------------------------------
% 2.45/1.51 % (26207)------------------------------
% 2.45/1.51 % (26217)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_25972 for (2999ds/16Mi)
% 2.45/1.52 % (26208)Instruction limit reached!
% 2.45/1.52 % (26208)------------------------------
% 2.45/1.52 % (26208)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.45/1.52 % (26208)Termination reason: Unknown
% 2.45/1.52 % (26208)Termination phase: Property scanning
% 2.45/1.52
% 2.45/1.52 % (26208)Memory used [KB]: 1407
% 2.45/1.52 % (26208)Time elapsed: 0.010 s
% 2.45/1.52 % (26208)Instructions burned: 15 (million)
% 2.45/1.52 % (26208)------------------------------
% 2.45/1.52 % (26208)------------------------------
% 2.45/1.52 % (26214)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_25972 for (2999ds/7Mi)
% 2.45/1.52 % (26220)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_25972 for (2999ds/3Mi)
% 2.45/1.52 % (26217)Instruction limit reached!
% 2.45/1.52 % (26217)------------------------------
% 2.45/1.52 % (26217)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.45/1.52 % (26220)Instruction limit reached!
% 2.45/1.52 % (26220)------------------------------
% 2.45/1.52 % (26220)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.45/1.52 % (26217)Termination reason: Unknown
% 2.45/1.52 % (26220)Termination reason: Unknown
% 2.45/1.52 % (26217)Termination phase: shuffling
% 2.45/1.52 % (26220)Termination phase: shuffling
% 2.45/1.52
% 2.45/1.52
% 2.45/1.52 % (26217)Memory used [KB]: 1535
% 2.45/1.52 % (26220)Memory used [KB]: 1279
% 2.45/1.52 % (26217)Time elapsed: 0.007 s
% 2.45/1.52 % (26220)Time elapsed: 0.002 s
% 2.45/1.52 % (26217)Instructions burned: 18 (million)
% 2.45/1.52 % (26220)Instructions burned: 4 (million)
% 2.45/1.52 % (26217)------------------------------
% 2.45/1.52 % (26217)------------------------------
% 2.45/1.52 % (26220)------------------------------
% 2.45/1.52 % (26220)------------------------------
% 2.45/1.52 % (26214)Instruction limit reached!
% 2.45/1.52 % (26214)------------------------------
% 2.45/1.52 % (26214)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.45/1.52 % (26214)Termination reason: Unknown
% 2.45/1.52 % (26214)Termination phase: shuffling
% 2.45/1.52
% 2.45/1.52 % (26214)Memory used [KB]: 1279
% 2.45/1.52 % (26214)Time elapsed: 0.006 s
% 2.45/1.52 % (26214)Instructions burned: 7 (million)
% 2.45/1.52 % (26214)------------------------------
% 2.45/1.52 % (26214)------------------------------
% 2.45/1.53 % (26230)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_25972 for (2999ds/3Mi)
% 2.45/1.53 % (26229)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_25972 for (2999ds/7Mi)
% 2.45/1.53 % (26230)Instruction limit reached!
% 2.45/1.53 % (26230)------------------------------
% 2.45/1.53 % (26230)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.45/1.53 % (26230)Termination reason: Unknown
% 2.45/1.53 % (26230)Termination phase: shuffling
% 2.45/1.53
% 2.45/1.53 % (26230)Memory used [KB]: 1279
% 2.45/1.53 % (26230)Time elapsed: 0.002 s
% 2.45/1.53 % (26230)Instructions burned: 3 (million)
% 2.45/1.53 % (26230)------------------------------
% 2.45/1.53 % (26230)------------------------------
% 2.45/1.53 % (26229)Instruction limit reached!
% 2.45/1.53 % (26229)------------------------------
% 2.45/1.53 % (26229)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.45/1.53 % (26229)Termination reason: Unknown
% 2.45/1.53 % (26229)Termination phase: shuffling
% 2.45/1.53
% 2.45/1.53 % (26229)Memory used [KB]: 1279
% 2.45/1.53 % (26229)Time elapsed: 0.004 s
% 2.45/1.53 % (26229)Instructions burned: 8 (million)
% 2.45/1.53 % (26229)------------------------------
% 2.45/1.53 % (26229)------------------------------
% 2.45/1.53 % (26226)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_25972 for (2999ds/3Mi)
% 2.45/1.54 % (26226)Instruction limit reached!
% 2.45/1.54 % (26226)------------------------------
% 2.45/1.54 % (26226)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.45/1.54 % (26226)Termination reason: Unknown
% 2.45/1.54 % (26226)Termination phase: shuffling
% 2.45/1.54
% 2.45/1.54 % (26226)Memory used [KB]: 1279
% 2.45/1.54 % (26226)Time elapsed: 0.004 s
% 2.45/1.54 % (26226)Instructions burned: 4 (million)
% 2.45/1.54 % (26226)------------------------------
% 2.45/1.54 % (26226)------------------------------
% 2.45/1.54 % (26231)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_25972 for (2999ds/4Mi)
% 2.45/1.54 % (26231)Instruction limit reached!
% 2.45/1.54 % (26231)------------------------------
% 2.45/1.54 % (26231)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.45/1.54 % (26231)Termination reason: Unknown
% 2.45/1.54 % (26231)Termination phase: shuffling
% 2.45/1.54
% 2.45/1.54 % (26231)Memory used [KB]: 1279
% 2.45/1.54 % (26231)Time elapsed: 0.003 s
% 2.45/1.54 % (26231)Instructions burned: 6 (million)
% 2.45/1.54 % (26231)------------------------------
% 2.45/1.54 % (26231)------------------------------
% 2.45/1.54 % (26236)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_25972 for (2999ds/18Mi)
% 2.45/1.54 % (26237)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_25972 for (2999ds/710Mi)
% 2.45/1.55 % (26236)Instruction limit reached!
% 2.45/1.55 % (26236)------------------------------
% 2.45/1.55 % (26236)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.45/1.55 % (26236)Termination reason: Unknown
% 2.45/1.55 % (26236)Termination phase: shuffling
% 2.45/1.55
% 2.45/1.55 % (26236)Memory used [KB]: 1535
% 2.45/1.55 % (26236)Time elapsed: 0.006 s
% 2.45/1.55 % (26236)Instructions burned: 18 (million)
% 2.45/1.55 % (26236)------------------------------
% 2.45/1.55 % (26236)------------------------------
% 2.45/1.55 % (26238)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_25972 for (2999ds/6Mi)
% 2.45/1.55 % (26240)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_25972 for (2999ds/902Mi)
% 2.45/1.55 % (26238)Instruction limit reached!
% 2.45/1.55 % (26238)------------------------------
% 2.45/1.55 % (26238)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.45/1.55 % (26238)Termination reason: Unknown
% 2.45/1.55 % (26238)Termination phase: shuffling
% 2.45/1.55
% 2.45/1.55 % (26238)Memory used [KB]: 1279
% 2.45/1.55 % (26238)Time elapsed: 0.004 s
% 2.45/1.55 % (26238)Instructions burned: 7 (million)
% 2.45/1.55 % (26238)------------------------------
% 2.45/1.55 % (26238)------------------------------
% 2.45/1.55 % (26244)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_25972 for (2999ds/21Mi)
% 2.45/1.56 % (26244)Instruction limit reached!
% 2.45/1.56 % (26244)------------------------------
% 2.45/1.56 % (26244)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.45/1.56 % (26244)Termination reason: Unknown
% 2.45/1.56 % (26244)Termination phase: Property scanning
% 2.45/1.56
% 2.45/1.56 % (26244)Memory used [KB]: 1663
% 2.45/1.56 % (26244)Time elapsed: 0.007 s
% 2.45/1.56 % (26244)Instructions burned: 22 (million)
% 2.45/1.56 % (26244)------------------------------
% 2.45/1.56 % (26244)------------------------------
% 2.45/1.56 % (26247)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_25972 for (2999ds/5Mi)
% 2.45/1.56 % (26247)Instruction limit reached!
% 2.45/1.56 % (26247)------------------------------
% 2.45/1.56 % (26247)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.45/1.56 % (26247)Termination reason: Unknown
% 2.45/1.56 % (26247)Termination phase: shuffling
% 2.45/1.56
% 2.45/1.56 % (26247)Memory used [KB]: 1279
% 2.45/1.56 % (26247)Time elapsed: 0.003 s
% 2.45/1.56 % (26247)Instructions burned: 6 (million)
% 2.45/1.56 % (26247)------------------------------
% 2.45/1.56 % (26247)------------------------------
% 2.45/1.57 % (26196)Instruction limit reached!
% 2.45/1.57 % (26196)------------------------------
% 2.45/1.57 % (26196)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.45/1.57 % (26196)Termination reason: Unknown
% 2.45/1.57 % (26196)Termination phase: Saturation
% 2.45/1.57
% 2.45/1.57 % (26196)Memory used [KB]: 7931
% 2.45/1.57 % (26196)Time elapsed: 0.087 s
% 2.45/1.57 % (26196)Instructions burned: 275 (million)
% 2.45/1.57 % (26196)------------------------------
% 2.45/1.57 % (26196)------------------------------
% 2.45/1.57 % (26252)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_25972 for (2999ds/6Mi)
% 2.45/1.57 % (26252)Instruction limit reached!
% 2.45/1.57 % (26252)------------------------------
% 2.45/1.57 % (26252)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.45/1.57 % (26252)Termination reason: Unknown
% 2.45/1.57 % (26252)Termination phase: shuffling
% 2.45/1.57
% 2.45/1.57 % (26252)Memory used [KB]: 1279
% 2.45/1.57 % (26252)Time elapsed: 0.003 s
% 2.45/1.57 % (26252)Instructions burned: 7 (million)
% 2.45/1.57 % (26252)------------------------------
% 2.45/1.57 % (26252)------------------------------
% 2.45/1.57 % (26255)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_25972 for (2999ds/377Mi)
% 2.45/1.58 % (26256)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_25972 for (2999ds/779Mi)
% 2.45/1.58 % (26191)Instruction limit reached!
% 2.45/1.58 % (26191)------------------------------
% 2.45/1.58 % (26191)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.45/1.58 % (26191)Termination reason: Unknown
% 2.45/1.58 % (26191)Termination phase: Saturation
% 2.45/1.58
% 2.45/1.58 % (26191)Memory used [KB]: 7036
% 2.45/1.58 % (26191)Time elapsed: 0.093 s
% 2.45/1.58 % (26191)Instructions burned: 183 (million)
% 2.45/1.58 % (26191)------------------------------
% 2.45/1.58 % (26191)------------------------------
% 2.45/1.58 % (26260)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_25972 for (2999ds/19Mi)
% 2.45/1.59 % (26260)Instruction limit reached!
% 2.45/1.59 % (26260)------------------------------
% 2.45/1.59 % (26260)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.45/1.59 % (26260)Termination reason: Unknown
% 2.45/1.59 % (26260)Termination phase: Property scanning
% 2.45/1.59
% 2.45/1.59 % (26260)Memory used [KB]: 1663
% 2.45/1.59 % (26260)Time elapsed: 0.007 s
% 2.45/1.59 % (26260)Instructions burned: 22 (million)
% 2.45/1.59 % (26260)------------------------------
% 2.45/1.59 % (26260)------------------------------
% 2.45/1.59 % (26263)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_25972 for (2998ds/879Mi)
% 2.45/1.60 % (26264)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_25972 for (2998ds/17Mi)
% 2.45/1.60 % (26264)Instruction limit reached!
% 2.45/1.60 % (26264)------------------------------
% 2.45/1.60 % (26264)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.45/1.60 % (26264)Termination reason: Unknown
% 2.45/1.60 % (26264)Termination phase: shuffling
% 2.45/1.60
% 2.45/1.60 % (26264)Memory used [KB]: 1535
% 2.45/1.60 % (26264)Time elapsed: 0.006 s
% 2.45/1.60 % (26264)Instructions burned: 18 (million)
% 2.45/1.60 % (26264)------------------------------
% 2.45/1.60 % (26264)------------------------------
% 2.45/1.61 % (26272)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_25972 for (2998ds/3Mi)
% 2.45/1.61 % (26272)Instruction limit reached!
% 2.45/1.61 % (26272)------------------------------
% 2.45/1.61 % (26272)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.45/1.61 % (26272)Termination reason: Unknown
% 2.45/1.61 % (26272)Termination phase: shuffling
% 2.45/1.61
% 2.45/1.61 % (26272)Memory used [KB]: 1151
% 2.45/1.61 % (26272)Time elapsed: 0.002 s
% 2.45/1.61 % (26272)Instructions burned: 3 (million)
% 2.45/1.61 % (26272)------------------------------
% 2.45/1.61 % (26272)------------------------------
% 2.45/1.62 % (26275)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_25972 for (2998ds/30Mi)
% 2.45/1.63 % (26275)Instruction limit reached!
% 2.45/1.63 % (26275)------------------------------
% 2.45/1.63 % (26275)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.45/1.63 % (26275)Termination reason: Unknown
% 2.45/1.63 % (26275)Termination phase: Property scanning
% 2.45/1.63
% 2.45/1.63 % (26275)Memory used [KB]: 1663
% 2.45/1.63 % (26275)Time elapsed: 0.009 s
% 2.45/1.63 % (26275)Instructions burned: 31 (million)
% 2.45/1.63 % (26275)------------------------------
% 2.45/1.63 % (26275)------------------------------
% 2.45/1.64 % (26240)First to succeed.
% 2.45/1.64 % (26286)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_25972 for (2998ds/127Mi)
% 2.45/1.64 % (26240)Refutation found. Thanks to Tanya!
% 2.45/1.64 % SZS status Theorem for DTF2THF_25972
% 2.45/1.64 % SZS output start Proof for DTF2THF_25972
% 2.45/1.64 thf(func_def_0, type, in: $i > $i > $o).
% 2.45/1.64 thf(func_def_1, type, exu: ($i > $o) > $o).
% 2.45/1.64 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 2.45/1.64 thf(func_def_8, type, powerset: $i > $i).
% 2.45/1.64 thf(func_def_10, type, setunion: $i > $i).
% 2.45/1.64 thf(func_def_19, type, descr: ($i > $o) > $i).
% 2.45/1.64 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 2.45/1.64 thf(func_def_26, type, prop2set: $o > $i).
% 2.45/1.64 thf(func_def_36, type, nonempty: $i > $o).
% 2.45/1.64 thf(func_def_69, type, set2prop: $i > $o).
% 2.45/1.64 thf(func_def_88, type, subset: $i > $i > $o).
% 2.45/1.64 thf(func_def_89, type, disjoint: $i > $i > $o).
% 2.45/1.64 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 2.45/1.64 thf(func_def_114, type, binunion: $i > $i > $i).
% 2.45/1.64 thf(func_def_122, type, binintersect: $i > $i > $i).
% 2.45/1.64 thf(func_def_135, type, regular: $i > $o).
% 2.45/1.64 thf(func_def_136, type, setminus: $i > $i > $i).
% 2.45/1.64 thf(func_def_147, type, symdiff: $i > $i > $i).
% 2.45/1.64 thf(func_def_153, type, iskpair: $i > $o).
% 2.45/1.64 thf(func_def_168, type, sP0: $i > $i > $o).
% 2.45/1.64 thf(func_def_169, type, sP1: $i > $o).
% 2.45/1.64 thf(func_def_170, type, sP2: $i > $i > $o).
% 2.45/1.64 thf(func_def_171, type, sP3: $i > $i > $o).
% 2.45/1.64 thf(func_def_173, type, sP5: $i > $o > $i > $i > $o).
% 2.45/1.64 thf(func_def_205, type, sK37: $i > $i).
% 2.45/1.64 thf(func_def_206, type, sK38: $i > $i > $i).
% 2.45/1.64 thf(func_def_217, type, sK49: $i > $o).
% 2.45/1.64 thf(func_def_252, type, sK84: $i > $o).
% 2.45/1.64 thf(func_def_255, type, sK87: $i > $i > $i).
% 2.45/1.64 thf(func_def_268, type, sK100: $i > $i > $i).
% 2.45/1.64 thf(func_def_272, type, sK104: $i > $o).
% 2.45/1.64 thf(func_def_275, type, sK107: $i > ($i > $o) > $i).
% 2.45/1.64 thf(func_def_278, type, sK110: $i > ($i > $o) > $i).
% 2.45/1.64 thf(func_def_279, type, sK111: $i > $o).
% 2.45/1.64 thf(func_def_282, type, sK114: $i > $i > $i).
% 2.45/1.64 thf(func_def_297, type, sK129: $i > $i > $i).
% 2.45/1.64 thf(func_def_298, type, sK130: $i > $i > $i).
% 2.45/1.64 thf(func_def_299, type, sK131: $i > $i > $i).
% 2.45/1.64 thf(func_def_300, type, sK132: $i > $i > $i).
% 2.45/1.64 thf(func_def_301, type, sK133: $i > $i > $i).
% 2.45/1.64 thf(func_def_302, type, sK134: $i > $i > $i > $i).
% 2.45/1.64 thf(func_def_303, type, sK135: $i > $i).
% 2.45/1.64 thf(func_def_304, type, sK136: $i > $i).
% 2.45/1.64 thf(func_def_305, type, sK137: $i > $i).
% 2.45/1.64 thf(func_def_306, type, sK138: $i > $i).
% 2.45/1.64 thf(func_def_307, type, sK139: $i > $i > $i).
% 2.45/1.64 thf(func_def_308, type, sK140: $i > $i > $i > $i).
% 2.45/1.64 thf(func_def_309, type, sK141: $i > $i > $i > $i).
% 2.45/1.64 thf(func_def_310, type, sK142: $i > $i > $i).
% 2.45/1.64 thf(func_def_311, type, sK143: $i > $i > $i).
% 2.45/1.64 thf(func_def_312, type, sK144: $i > $i).
% 2.45/1.64 thf(func_def_314, type, sK146: $i > $i).
% 2.45/1.64 thf(func_def_315, type, sK147: $i > $i).
% 2.45/1.64 thf(func_def_317, type, sK149: $i > $o).
% 2.45/1.64 thf(func_def_324, type, sK156: $i > $o).
% 2.45/1.64 thf(func_def_328, type, sK160: $i > $i).
% 2.45/1.64 thf(func_def_342, type, sK174: $i > $o).
% 2.45/1.64 thf(func_def_347, type, sK179: $i > $i).
% 2.45/1.64 thf(func_def_350, type, sK182: $i > $o).
% 2.45/1.64 thf(func_def_361, type, sK193: $i > $i).
% 2.45/1.64 thf(func_def_362, type, sK194: $i > $i).
% 2.45/1.64 thf(func_def_367, type, sK199: $o > $i > $i > $i).
% 2.45/1.64 thf(func_def_371, type, sK203: $i > ($i > $o) > $i).
% 2.45/1.64 thf(func_def_372, type, sK204: $i > $o).
% 2.45/1.64 thf(func_def_375, type, sK207: $i > $o).
% 2.45/1.64 thf(func_def_376, type, sK208: $i > $o).
% 2.45/1.64 thf(func_def_377, type, sK209: ($i > $o) > ($i > $o) > $i).
% 2.45/1.64 thf(func_def_378, type, sK210: ($i > $o) > ($i > $o) > $i).
% 2.45/1.64 thf(func_def_383, type, sK215: ($i > $o) > $i).
% 2.45/1.64 thf(func_def_384, type, sK216: ($i > $o) > $i).
% 2.45/1.64 thf(func_def_385, type, sK217: $i > $o).
% 2.45/1.64 thf(func_def_387, type, sK219: $i > ($i > $o) > $i).
% 2.45/1.64 thf(func_def_388, type, sK220: $i > $o).
% 2.45/1.64 thf(func_def_392, type, sK224: $i > $i > $i).
% 2.45/1.64 thf(func_def_397, type, sK229: $i > $i > $i).
% 2.45/1.64 thf(func_def_398, type, sK230: $i > $i > $i).
% 2.45/1.64 thf(func_def_399, type, sK231: ($i > $o) > $i).
% 2.45/1.64 thf(func_def_400, type, sK232: $i > $o).
% 2.45/1.64 thf(func_def_401, type, sK233: $i > $i).
% 2.45/1.64 thf(func_def_402, type, sK234: $i > $i > $i).
% 2.45/1.64 thf(func_def_405, type, sK237: $i > $o).
% 2.45/1.64 thf(func_def_411, type, sK243: $i > $o).
% 2.45/1.64 thf(func_def_412, type, sK244: ($i > $o) > ($i > $o) > $i).
% 2.45/1.64 thf(func_def_413, type, sK245: ($i > $o) > ($i > $o) > $i).
% 2.45/1.64 thf(func_def_414, type, sK246: $i > $o).
% 2.45/1.64 thf(func_def_415, type, sK247: $i > $o).
% 2.45/1.64 thf(func_def_419, type, sK251: $i > $i > $i).
% 2.45/1.64 thf(func_def_425, type, sK257: $i > $i > $o).
% 2.45/1.64 thf(func_def_426, type, sK258: $i > $i).
% 2.45/1.64 thf(func_def_427, type, sK259: $i > $i).
% 2.45/1.64 thf(func_def_428, type, sK260: ($i > $i > $o) > $i > $i).
% 2.45/1.64 thf(func_def_429, type, sK261: $i > ($i > $i > $o) > $i > $i).
% 2.45/1.64 thf(func_def_430, type, sK262: ($i > $i > $o) > $i > $i).
% 2.45/1.64 thf(func_def_432, type, sK264: $i > $o).
% 2.45/1.64 thf(func_def_436, type, sK268: $i > ($i > $o) > $i).
% 2.45/1.64 thf(func_def_437, type, sK269: $i > $o).
% 2.45/1.64 thf(func_def_446, type, sK278: $i > $i).
% 2.45/1.64 thf(func_def_453, type, sK285: $i > $o).
% 2.45/1.64 thf(func_def_457, type, sK289: $i > $o).
% 2.45/1.64 thf(func_def_458, type, sK290: ($i > $o) > $i).
% 2.45/1.64 thf(func_def_459, type, sK291: $i > $o).
% 2.45/1.64 thf(func_def_473, type, sK305: $i > $o).
% 2.45/1.64 thf(func_def_476, type, sK308: $i > ($i > $o) > $i).
% 2.45/1.64 thf(func_def_481, type, sK313: $i > ($i > $o) > $i).
% 2.45/1.64 thf(func_def_482, type, sK314: $i > $o).
% 2.45/1.64 thf(func_def_488, type, sK320: $i > $i).
% 2.45/1.64 thf(func_def_504, type, sK336: $i > $o).
% 2.45/1.64 thf(func_def_523, type, sK355: $i > $o).
% 2.45/1.64 thf(func_def_524, type, sK356: $i > $o).
% 2.45/1.64 thf(func_def_525, type, sK357: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.45/1.64 thf(func_def_526, type, sK358: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.45/1.64 thf(func_def_528, type, sK360: $i > ($i > $o) > $i).
% 2.45/1.64 thf(func_def_529, type, sK361: $i > $o).
% 2.45/1.64 thf(func_def_532, type, sK364: $i > $o).
% 2.45/1.64 thf(func_def_533, type, sK365: $i > $i).
% 2.45/1.64 thf(func_def_534, type, sK366: ($i > $o) > $i).
% 2.45/1.64 thf(func_def_538, type, ph370: !>[X0: $tType]:(X0)).
% 2.45/1.64 thf(f3211,plain,(
% 2.45/1.64 $false),
% 2.45/1.64 inference(trivial_inequality_removal,[status(thm)],[f3210])).
% 2.45/1.64 thf(f3210,plain,(
% 2.45/1.64 ($true != $true)),
% 2.45/1.64 inference(superposition,[status(thm)],[f3138,f1417])).
% 2.45/1.64 thf(f1417,plain,(
% 2.45/1.64 (upairsetIR = $true)),
% 2.45/1.64 inference(cnf_transformation,[status(thm)],[f774])).
% 2.45/1.64 thf(f774,plain,(
% 2.45/1.64 (setext = $true) & (setadjoinOr = $true) & (emptysetimpfalse = $true) & (quantDeMorgan4 = $true) & (setminusI = $true) & (wellorderingAx = $true) & (sepSubset = $true) & (symdiffI1 = $true) & (setadjoinIL = $true) & (exuI1 = $true) & (setadjoin__Cong = $true) & (prop2set2propI = $true) & (descrp = $true) & (powersetsubset = $true) & (notinsingleton = $true) & (emptyset__Cong = $true) & (nonemptyI1 = $true) & (descr__Cong = $true) & (setminusERneg = $true) & (emptyI = $true) & (dsetconstrI = $true) & (binintersectLsub = $true) & (prop2setI = $true) & (omegaSAx = $true) & (setminusILneg = $true) & (emptysetE = $true) & (omegaIndAx = $true) & (powerset__Cong = $true) & (binunionRsub = $true) & (uniqinunit = $true) & (subsetE2 = $true) & (upairsetE = $true) & (exuI3 = $true) & (inCongP = $true) & (setunionAx = $true) & (exuEu = $true) & (inPowerset = $true) & (notsubsetI = $true) & (subset2powerset = $true) & (setminusER = $true) & (eqinunit = $true) & (notinemptyset = $true) & (setadjoinSub = $true) & (emptysetsubset = $true) & (setoftrueEq = $true) & (setukpairIL = $true) & (binunionLsub = $true) & (binintersectEL = $true) & (notequalI2 = $true) & (setminusSubset1 = $true) & (nonemptyE1 = $true) & (dsetconstrER = $true) & (setminusSubset2 = $true) & (powersetAx = $true) & (in__Cong = $true) & (binintersectSubset1 = $true) & (setadjoinAx = $true) & (symdiffE = $true) & (symdiffIneg1 = $true) & (binintersectER = $true) & (subsetI2 = $true) & (setunionE = $true) & (bs114d = $true) & (upairsetIL = $true) & (setadjoinE = $true) & (replAx = $true) & (setextsub = $true) & ((in @ sK108 @ (setunion @ (setadjoin @ (setadjoin @ sK109 @ emptyset) @ (setadjoin @ (setadjoin @ sK109 @ (setadjoin @ sK108 @ emptyset)) @ emptyset)))) != $true) & (eqimpsubset2 = $true) & (setadjoinSub2 = $true) & (subsetI1 = $true) & (notdallE = $true) & (binunionEcases = $true) & (subsetRefl = $true) & (setbeta = $true) & (quantDeMorgan1 = $true) & (powersetI1 = $true) & (notdexE = $true) & (powersetE = $true) & (emptyinunitempty = $true) & (upairsetIR = $true) & (emptysetAx = $true) & (setminusLsub = $true) & (binintersectSubset5 = $true) & (setextAx = $true) & (disjointsetsI1 = $true) & (exuE1 = $true) & (subPowSU = $true) & (eqimpsubset1 = $true) & (setminusIRneg = $true) & (exuE3u = $true) & (prop2setE = $true) & (emptyinPowerset = $true) & (notequalI1 = $true) & (noeltsimpempty = $true) & (singletonsswitch = $true) & (binintersectSubset4 = $true) & (secondinupair = $true) & (powersetI = $true) & (exuE3e = $true) & (setminusEL = $true) & (setunion__Cong = $true) & (binunionE = $true) & (nonemptyI = $true) & (nonemptyImpWitness = $true) & (exu__Cong = $true) & (binintersectSubset3 = $true) & (symdiffI2 = $true) & (exuI2 = $true) & (dsetconstrEL = $true) & (binintersectI = $true) & (omega0Ax = $true) & (omega__Cong = $true) & (symdiffIneg2 = $true) & (vacuousDall = $true) & (dsetconstr__Cong = $true) & (upairset2IR = $true) & (quantDeMorgan3 = $true) & (setunionI = $true) & (emptyE1 = $true) & (binunionIR = $true) & (setminusELneg = $true) & (binunionIL = $true) & (subsetemptysetimpeq = $true) & (powersetE1 = $true) & (foundationAx = $true) & (binintersectSubset2 = $true) & (binintersectRsub = $true) & (sepInPowerset = $true) & (quantDeMorgan2 = $true) & (emptyInPowerset = $true) & (subsetE = $true) & (subsetTrans = $true) & (setadjoinIR = $true) & (exuE2 = $true)),
% 2.45/1.64 inference(skolemisation,[status(esa),new_symbols(skolem,[sK108,sK109])],[f529,f773])).
% 2.45/1.64 thf(f773,plain,(
% 2.45/1.64 ? [X0 : $i,X1 : $i] : ((in @ X0 @ (setunion @ (setadjoin @ (setadjoin @ X1 @ emptyset) @ (setadjoin @ (setadjoin @ X1 @ (setadjoin @ X0 @ emptyset)) @ emptyset)))) != $true) => ((in @ sK108 @ (setunion @ (setadjoin @ (setadjoin @ sK109 @ emptyset) @ (setadjoin @ (setadjoin @ sK109 @ (setadjoin @ sK108 @ emptyset)) @ emptyset)))) != $true)),
% 2.45/1.64 introduced(definition,[],[choice_axiom])).
% 2.45/1.64 thf(f529,plain,(
% 2.45/1.64 (setext = $true) & (setadjoinOr = $true) & (emptysetimpfalse = $true) & (quantDeMorgan4 = $true) & (setminusI = $true) & (wellorderingAx = $true) & (sepSubset = $true) & (symdiffI1 = $true) & (setadjoinIL = $true) & (exuI1 = $true) & (setadjoin__Cong = $true) & (prop2set2propI = $true) & (descrp = $true) & (powersetsubset = $true) & (notinsingleton = $true) & (emptyset__Cong = $true) & (nonemptyI1 = $true) & (descr__Cong = $true) & (setminusERneg = $true) & (emptyI = $true) & (dsetconstrI = $true) & (binintersectLsub = $true) & (prop2setI = $true) & (omegaSAx = $true) & (setminusILneg = $true) & (emptysetE = $true) & (omegaIndAx = $true) & (powerset__Cong = $true) & (binunionRsub = $true) & (uniqinunit = $true) & (subsetE2 = $true) & (upairsetE = $true) & (exuI3 = $true) & (inCongP = $true) & (setunionAx = $true) & (exuEu = $true) & (inPowerset = $true) & (notsubsetI = $true) & (subset2powerset = $true) & (setminusER = $true) & (eqinunit = $true) & (notinemptyset = $true) & (setadjoinSub = $true) & (emptysetsubset = $true) & (setoftrueEq = $true) & (setukpairIL = $true) & (binunionLsub = $true) & (binintersectEL = $true) & (notequalI2 = $true) & (setminusSubset1 = $true) & (nonemptyE1 = $true) & (dsetconstrER = $true) & (setminusSubset2 = $true) & (powersetAx = $true) & (in__Cong = $true) & (binintersectSubset1 = $true) & (setadjoinAx = $true) & (symdiffE = $true) & (symdiffIneg1 = $true) & (binintersectER = $true) & (subsetI2 = $true) & (setunionE = $true) & (bs114d = $true) & (upairsetIL = $true) & (setadjoinE = $true) & (replAx = $true) & (setextsub = $true) & ? [X0 : $i,X1 : $i] : ((in @ X0 @ (setunion @ (setadjoin @ (setadjoin @ X1 @ emptyset) @ (setadjoin @ (setadjoin @ X1 @ (setadjoin @ X0 @ emptyset)) @ emptyset)))) != $true) & (eqimpsubset2 = $true) & (setadjoinSub2 = $true) & (subsetI1 = $true) & (notdallE = $true) & (binunionEcases = $true) & (subsetRefl = $true) & (setbeta = $true) & (quantDeMorgan1 = $true) & (powersetI1 = $true) & (notdexE = $true) & (powersetE = $true) & (emptyinunitempty = $true) & (upairsetIR = $true) & (emptysetAx = $true) & (setminusLsub = $true) & (binintersectSubset5 = $true) & (setextAx = $true) & (disjointsetsI1 = $true) & (exuE1 = $true) & (subPowSU = $true) & (eqimpsubset1 = $true) & (setminusIRneg = $true) & (exuE3u = $true) & (prop2setE = $true) & (emptyinPowerset = $true) & (notequalI1 = $true) & (noeltsimpempty = $true) & (singletonsswitch = $true) & (binintersectSubset4 = $true) & (secondinupair = $true) & (powersetI = $true) & (exuE3e = $true) & (setminusEL = $true) & (setunion__Cong = $true) & (binunionE = $true) & (nonemptyI = $true) & (nonemptyImpWitness = $true) & (exu__Cong = $true) & (binintersectSubset3 = $true) & (symdiffI2 = $true) & (exuI2 = $true) & (dsetconstrEL = $true) & (binintersectI = $true) & (omega0Ax = $true) & (omega__Cong = $true) & (symdiffIneg2 = $true) & (vacuousDall = $true) & (dsetconstr__Cong = $true) & (upairset2IR = $true) & (quantDeMorgan3 = $true) & (setunionI = $true) & (emptyE1 = $true) & (binunionIR = $true) & (setminusELneg = $true) & (binunionIL = $true) & (subsetemptysetimpeq = $true) & (powersetE1 = $true) & (foundationAx = $true) & (binintersectSubset2 = $true) & (binintersectRsub = $true) & (sepInPowerset = $true) & (quantDeMorgan2 = $true) & (emptyInPowerset = $true) & (subsetE = $true) & (subsetTrans = $true) & (setadjoinIR = $true) & (exuE2 = $true)),
% 2.45/1.64 inference(flattening,[status(thm)],[f528])).
% 2.45/1.64 thf(f528,plain,(
% 2.45/1.64 (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i,X1 : $i] : ((in @ X0 @ (setunion @ (setadjoin @ (setadjoin @ X1 @ emptyset) @ (setadjoin @ (setadjoin @ X1 @ (setadjoin @ X0 @ emptyset)) @ emptyset)))) != $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)),
% 2.45/1.64 inference(ennf_transformation,[status(thm)],[f404])).
% 2.45/1.64 thf(f404,plain,(
% 2.45/1.64 ~((setextAx = $true) => ((emptysetAx = $true) => ((setadjoinAx = $true) => ((powersetAx = $true) => ((setunionAx = $true) => ((omega0Ax = $true) => ((omegaSAx = $true) => ((omegaIndAx = $true) => ((replAx = $true) => ((foundationAx = $true) => ((wellorderingAx = $true) => ((descrp = $true) => ((dsetconstrI = $true) => ((dsetconstrEL = $true) => ((dsetconstrER = $true) => ((exuE1 = $true) => ((prop2setE = $true) => ((emptysetE = $true) => ((emptysetimpfalse = $true) => ((notinemptyset = $true) => ((exuE3e = $true) => ((setext = $true) => ((emptyI = $true) => ((noeltsimpempty = $true) => ((setbeta = $true) => ((nonemptyE1 = $true) => ((nonemptyI = $true) => ((nonemptyI1 = $true) => ((setadjoinIL = $true) => ((emptyinunitempty = $true) => ((setadjoinIR = $true) => ((setadjoinE = $true) => ((setadjoinOr = $true) => ((setoftrueEq = $true) => ((powersetI = $true) => ((emptyinPowerset = $true) => ((emptyInPowerset = $true) => ((powersetE = $true) => ((setunionI = $true) => ((setunionE = $true) => ((subPowSU = $true) => ((exuE2 = $true) => ((nonemptyImpWitness = $true) => ((uniqinunit = $true) => ((notinsingleton = $true) => ((eqinunit = $true) => ((singletonsswitch = $true) => ((upairsetE = $true) => ((upairsetIL = $true) => ((upairsetIR = $true) => ((emptyE1 = $true) => ((vacuousDall = $true) => ((quantDeMorgan1 = $true) => ((quantDeMorgan2 = $true) => ((quantDeMorgan3 = $true) => ((quantDeMorgan4 = $true) => ((prop2setI = $true) => ((prop2set2propI = $true) => ((notdexE = $true) => ((notdallE = $true) => ((exuI1 = $true) => ((exuI3 = $true) => ((exuI2 = $true) => ((inCongP = $true) => ((in__Cong = $true) => ((exuE3u = $true) => ((exu__Cong = $true) => ((emptyset__Cong = $true) => ((setadjoin__Cong = $true) => ((powerset__Cong = $true) => ((setunion__Cong = $true) => ((omega__Cong = $true) => ((exuEu = $true) => ((descr__Cong = $true) => ((dsetconstr__Cong = $true) => ((subsetI1 = $true) => ((eqimpsubset2 = $true) => ((eqimpsubset1 = $true) => ((subsetI2 = $true) => ((emptysetsubset = $true) => ((subsetE = $true) => ((subsetE2 = $true) => ((notsubsetI = $true) => ((notequalI1 = $true) => ((notequalI2 = $true) => ((subsetRefl = $true) => ((subsetTrans = $true) => ((setadjoinSub = $true) => ((setadjoinSub2 = $true) => ((subset2powerset = $true) => ((setextsub = $true) => ((subsetemptysetimpeq = $true) => ((powersetI1 = $true) => ((powersetE1 = $true) => ((inPowerset = $true) => ((powersetsubset = $true) => ((sepInPowerset = $true) => ((sepSubset = $true) => ((binunionIL = $true) => ((upairset2IR = $true) => ((binunionIR = $true) => ((binunionEcases = $true) => ((binunionE = $true) => ((binunionLsub = $true) => ((binunionRsub = $true) => ((binintersectI = $true) => ((binintersectSubset5 = $true) => ((binintersectEL = $true) => ((binintersectLsub = $true) => ((binintersectSubset2 = $true) => ((binintersectSubset3 = $true) => ((binintersectER = $true) => ((disjointsetsI1 = $true) => ((binintersectRsub = $true) => ((binintersectSubset4 = $true) => ((binintersectSubset1 = $true) => ((bs114d = $true) => ((setminusI = $true) => ((setminusEL = $true) => ((setminusER = $true) => ((setminusSubset2 = $true) => ((setminusERneg = $true) => ((setminusELneg = $true) => ((setminusILneg = $true) => ((setminusIRneg = $true) => ((setminusLsub = $true) => ((setminusSubset1 = $true) => ((symdiffE = $true) => ((symdiffI1 = $true) => ((symdiffI2 = $true) => ((symdiffIneg1 = $true) => ((symdiffIneg2 = $true) => ((secondinupair = $true) => ((setukpairIL = $true) => ! [X0 : $i,X1 : $i] : ((in @ X0 @ (setunion @ (setadjoin @ (setadjoin @ X1 @ emptyset) @ (setadjoin @ (setadjoin @ X1 @ (setadjoin @ X0 @ emptyset)) @ emptyset)))) = $true)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.45/1.64 inference(fool_elimination,[status(thm)],[f403])).
% 2.45/1.64 thf(f403,plain,(
% 2.45/1.64 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => ! [X0 : $i,X1 : $i] : (in @ X0 @ (setunion @ (setadjoin @ (setadjoin @ X1 @ emptyset) @ (setadjoin @ (setadjoin @ X1 @ (setadjoin @ X0 @ emptyset)) @ emptyset))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.45/1.64 inference(rectify,[status(thm)],[f139])).
% 2.45/1.64 thf(f139,negated_conjecture,(
% 2.45/1.64 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => ! [X2 : $i,X1 : $i] : (in @ X2 @ (setunion @ (setadjoin @ (setadjoin @ X1 @ emptyset) @ (setadjoin @ (setadjoin @ X1 @ (setadjoin @ X2 @ emptyset)) @ emptyset))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.45/1.64 inference(negated_conjecture,[status(cth)],[f138])).
% 2.45/1.64 thf(f138,conjecture,(
% 2.45/1.64 setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => ! [X2 : $i,X1 : $i] : (in @ X2 @ (setunion @ (setadjoin @ (setadjoin @ X1 @ emptyset) @ (setadjoin @ (setadjoin @ X1 @ (setadjoin @ X2 @ emptyset)) @ emptyset)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.45/1.64 file('/export/starexec/sandbox/tmp/tmp.BCzCW9enGb/DTF2THF_25972.p',setukpairIR)).
% 2.45/1.64 thf(f3138,plain,(
% 2.45/1.64 (upairsetIR != $true)),
% 2.45/1.64 inference(trivial_inequality_removal,[status(thm)],[f3126])).
% 2.45/1.64 thf(f3126,plain,(
% 2.45/1.64 ($true != $true) | (upairsetIR != $true)),
% 2.45/1.64 inference(superposition,[status(thm)],[f2188,f1264])).
% 2.45/1.64 thf(f1264,plain,(
% 2.45/1.64 ( ! [X2 : $i,X3 : $i] : (((in @ X2 @ (setadjoin @ X3 @ (setadjoin @ X2 @ emptyset))) = $true) | (upairsetIR != $true)) )),
% 2.45/1.64 inference(cnf_transformation,[status(thm)],[f660])).
% 2.45/1.64 thf(f660,plain,(
% 2.45/1.64 ((upairsetIR = $true) | ((in @ sK41 @ (setadjoin @ sK42 @ (setadjoin @ sK41 @ emptyset))) != $true)) & (! [X2 : $i,X3 : $i] : ((in @ X2 @ (setadjoin @ X3 @ (setadjoin @ X2 @ emptyset))) = $true) | (upairsetIR != $true))),
% 2.45/1.64 inference(skolemisation,[status(esa),new_symbols(skolem,[sK41,sK42])],[f658,f659])).
% 2.45/1.64 thf(f659,plain,(
% 2.45/1.64 ? [X0 : $i,X1 : $i] : ((in @ X0 @ (setadjoin @ X1 @ (setadjoin @ X0 @ emptyset))) != $true) => ((in @ sK41 @ (setadjoin @ sK42 @ (setadjoin @ sK41 @ emptyset))) != $true)),
% 2.45/1.64 introduced(definition,[],[choice_axiom])).
% 2.45/1.64 thf(f658,plain,(
% 2.45/1.64 ((upairsetIR = $true) | ? [X0 : $i,X1 : $i] : ((in @ X0 @ (setadjoin @ X1 @ (setadjoin @ X0 @ emptyset))) != $true)) & (! [X2 : $i,X3 : $i] : ((in @ X2 @ (setadjoin @ X3 @ (setadjoin @ X2 @ emptyset))) = $true) | (upairsetIR != $true))),
% 2.45/1.64 inference(rectify,[status(thm)],[f657])).
% 2.45/1.64 thf(f657,plain,(
% 2.45/1.64 ((upairsetIR = $true) | ? [X0 : $i,X1 : $i] : ((in @ X0 @ (setadjoin @ X1 @ (setadjoin @ X0 @ emptyset))) != $true)) & (! [X0 : $i,X1 : $i] : ((in @ X0 @ (setadjoin @ X1 @ (setadjoin @ X0 @ emptyset))) = $true) | (upairsetIR != $true))),
% 2.45/1.64 inference(nnf_transformation,[status(thm)],[f369])).
% 2.45/1.64 thf(f369,plain,(
% 2.45/1.64 (upairsetIR = $true) <=> ! [X0 : $i,X1 : $i] : ((in @ X0 @ (setadjoin @ X1 @ (setadjoin @ X0 @ emptyset))) = $true)),
% 2.45/1.64 inference(fool_elimination,[status(thm)],[f368])).
% 2.45/1.64 thf(f368,plain,(
% 2.45/1.64 (upairsetIR = ! [X0 : $i,X1 : $i] : (in @ X0 @ (setadjoin @ X1 @ (setadjoin @ X0 @ emptyset))))),
% 2.45/1.64 inference(rectify,[status(thm)],[f52])).
% 2.45/1.64 thf(f52,axiom,(
% 2.45/1.64 (upairsetIR = ! [X2 : $i,X1 : $i] : (in @ X2 @ (setadjoin @ X1 @ (setadjoin @ X2 @ emptyset))))),
% 2.45/1.64 file('/export/starexec/sandbox/tmp/tmp.BCzCW9enGb/DTF2THF_25972.p',upairsetIR)).
% 2.45/1.64 thf(f2188,plain,(
% 2.45/1.64 ((in @ sK108 @ (setadjoin @ sK109 @ (setadjoin @ sK108 @ emptyset))) != $true)),
% 2.45/1.64 inference(trivial_inequality_removal,[status(thm)],[f2187])).
% 2.45/1.64 thf(f2187,plain,(
% 2.45/1.64 ($true != $true) | ((in @ sK108 @ (setadjoin @ sK109 @ (setadjoin @ sK108 @ emptyset))) != $true)),
% 2.45/1.64 inference(forward_demodulation,[status(thm)],[f2078,f1400])).
% 2.45/1.64 thf(f1400,plain,(
% 2.45/1.64 (secondinupair = $true)),
% 2.45/1.64 inference(cnf_transformation,[status(thm)],[f774])).
% 2.45/1.64 thf(f2078,plain,(
% 2.45/1.64 ((in @ sK108 @ (setadjoin @ sK109 @ (setadjoin @ sK108 @ emptyset))) != $true) | (secondinupair != $true)),
% 2.45/1.64 inference(trivial_inequality_removal,[status(thm)],[f1995])).
% 2.45/1.64 thf(f1995,plain,(
% 2.45/1.64 (secondinupair != $true) | ($true != $true) | ((in @ sK108 @ (setadjoin @ sK109 @ (setadjoin @ sK108 @ emptyset))) != $true)),
% 2.45/1.64 inference(superposition,[status(thm)],[f1989,f1822])).
% 2.45/1.64 thf(f1822,plain,(
% 2.45/1.64 ( ! [X2 : $i,X3 : $i] : (((in @ X2 @ (setadjoin @ X3 @ (setadjoin @ X2 @ emptyset))) = $true) | (secondinupair != $true)) )),
% 2.45/1.64 inference(cnf_transformation,[status(thm)],[f1154])).
% 2.45/1.64 thf(f1154,plain,(
% 2.45/1.64 ((secondinupair = $true) | ((in @ sK329 @ (setadjoin @ sK330 @ (setadjoin @ sK329 @ emptyset))) != $true)) & (! [X2 : $i,X3 : $i] : ((in @ X2 @ (setadjoin @ X3 @ (setadjoin @ X2 @ emptyset))) = $true) | (secondinupair != $true))),
% 2.45/1.64 inference(skolemisation,[status(esa),new_symbols(skolem,[sK329,sK330])],[f1152,f1153])).
% 2.45/1.64 thf(f1153,plain,(
% 2.45/1.64 ? [X0 : $i,X1 : $i] : ((in @ X0 @ (setadjoin @ X1 @ (setadjoin @ X0 @ emptyset))) != $true) => ((in @ sK329 @ (setadjoin @ sK330 @ (setadjoin @ sK329 @ emptyset))) != $true)),
% 2.45/1.64 introduced(definition,[],[choice_axiom])).
% 2.45/1.64 thf(f1152,plain,(
% 2.45/1.64 ((secondinupair = $true) | ? [X0 : $i,X1 : $i] : ((in @ X0 @ (setadjoin @ X1 @ (setadjoin @ X0 @ emptyset))) != $true)) & (! [X2 : $i,X3 : $i] : ((in @ X2 @ (setadjoin @ X3 @ (setadjoin @ X2 @ emptyset))) = $true) | (secondinupair != $true))),
% 2.45/1.64 inference(rectify,[status(thm)],[f1151])).
% 2.45/1.64 thf(f1151,plain,(
% 2.45/1.64 ((secondinupair = $true) | ? [X0 : $i,X1 : $i] : ((in @ X0 @ (setadjoin @ X1 @ (setadjoin @ X0 @ emptyset))) != $true)) & (! [X0 : $i,X1 : $i] : ((in @ X0 @ (setadjoin @ X1 @ (setadjoin @ X0 @ emptyset))) = $true) | (secondinupair != $true))),
% 2.45/1.64 inference(nnf_transformation,[status(thm)],[f283])).
% 2.45/1.64 thf(f283,plain,(
% 2.45/1.64 (secondinupair = $true) <=> ! [X0 : $i,X1 : $i] : ((in @ X0 @ (setadjoin @ X1 @ (setadjoin @ X0 @ emptyset))) = $true)),
% 2.45/1.64 inference(fool_elimination,[status(thm)],[f282])).
% 2.45/1.64 thf(f282,plain,(
% 2.45/1.64 (secondinupair = ! [X0 : $i,X1 : $i] : (in @ X0 @ (setadjoin @ X1 @ (setadjoin @ X0 @ emptyset))))),
% 2.45/1.64 inference(rectify,[status(thm)],[f136])).
% 2.45/1.64 thf(f136,axiom,(
% 2.45/1.64 (secondinupair = ! [X2 : $i,X1 : $i] : (in @ X2 @ (setadjoin @ X1 @ (setadjoin @ X2 @ emptyset))))),
% 2.45/1.64 file('/export/starexec/sandbox/tmp/tmp.BCzCW9enGb/DTF2THF_25972.p',secondinupair)).
% 2.45/1.64 thf(f1989,plain,(
% 2.45/1.64 ( ! [X0 : $i] : (((in @ X0 @ (setadjoin @ (setadjoin @ sK109 @ emptyset) @ (setadjoin @ (setadjoin @ sK109 @ (setadjoin @ sK108 @ emptyset)) @ emptyset))) != $true) | ((in @ sK108 @ X0) != $true)) )),
% 2.45/1.64 inference(trivial_inequality_removal,[status(thm)],[f1988])).
% 2.45/1.64 thf(f1988,plain,(
% 2.45/1.64 ( ! [X0 : $i] : (((in @ sK108 @ X0) != $true) | ((in @ X0 @ (setadjoin @ (setadjoin @ sK109 @ emptyset) @ (setadjoin @ (setadjoin @ sK109 @ (setadjoin @ sK108 @ emptyset)) @ emptyset))) != $true) | ($true != $true)) )),
% 2.45/1.64 inference(forward_demodulation,[status(thm)],[f1981,f1379])).
% 2.45/1.64 thf(f1379,plain,(
% 2.45/1.64 (setunionI = $true)),
% 2.45/1.64 inference(cnf_transformation,[status(thm)],[f774])).
% 2.45/1.64 thf(f1981,plain,(
% 2.45/1.64 ( ! [X0 : $i] : ((setunionI != $true) | ((in @ X0 @ (setadjoin @ (setadjoin @ sK109 @ emptyset) @ (setadjoin @ (setadjoin @ sK109 @ (setadjoin @ sK108 @ emptyset)) @ emptyset))) != $true) | ((in @ sK108 @ X0) != $true)) )),
% 2.45/1.64 inference(trivial_inequality_removal,[status(thm)],[f1973])).
% 2.45/1.64 thf(f1973,plain,(
% 2.45/1.64 ( ! [X0 : $i] : (((in @ sK108 @ X0) != $true) | ($true != $true) | ((in @ X0 @ (setadjoin @ (setadjoin @ sK109 @ emptyset) @ (setadjoin @ (setadjoin @ sK109 @ (setadjoin @ sK108 @ emptyset)) @ emptyset))) != $true) | (setunionI != $true)) )),
% 2.45/1.64 inference(superposition,[status(thm)],[f1430,f1788])).
% 2.45/1.64 thf(f1788,plain,(
% 2.45/1.64 ( ! [X2 : $i,X0 : $i,X1 : $i] : (((in @ X1 @ (setunion @ X2)) = $true) | ((in @ X1 @ X0) != $true) | ((in @ X0 @ X2) != $true) | (setunionI != $true)) )),
% 2.45/1.64 inference(cnf_transformation,[status(thm)],[f1101])).
% 2.45/1.64 thf(f1101,plain,(
% 2.45/1.64 (! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X1 @ (setunion @ X2)) = $true) | ((in @ X0 @ X2) != $true) | ((in @ X1 @ X0) != $true)) | (setunionI != $true)) & ((setunionI = $true) | (((in @ sK301 @ (setunion @ sK302)) != $true) & ((in @ sK300 @ sK302) = $true) & ((in @ sK301 @ sK300) = $true)))),
% 2.45/1.64 inference(skolemisation,[status(esa),new_symbols(skolem,[sK300,sK301,sK302])],[f1099,f1100])).
% 2.45/1.64 thf(f1100,plain,(
% 2.45/1.64 ? [X3 : $i,X4 : $i,X5 : $i] : (((in @ X4 @ (setunion @ X5)) != $true) & ((in @ X3 @ X5) = $true) & ((in @ X4 @ X3) = $true)) => (((in @ sK301 @ (setunion @ sK302)) != $true) & ((in @ sK300 @ sK302) = $true) & ((in @ sK301 @ sK300) = $true))),
% 2.45/1.64 introduced(definition,[],[choice_axiom])).
% 2.45/1.64 thf(f1099,plain,(
% 2.45/1.64 (! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X1 @ (setunion @ X2)) = $true) | ((in @ X0 @ X2) != $true) | ((in @ X1 @ X0) != $true)) | (setunionI != $true)) & ((setunionI = $true) | ? [X3 : $i,X4 : $i,X5 : $i] : (((in @ X4 @ (setunion @ X5)) != $true) & ((in @ X3 @ X5) = $true) & ((in @ X4 @ X3) = $true)))),
% 2.45/1.64 inference(rectify,[status(thm)],[f1098])).
% 2.45/1.64 thf(f1098,plain,(
% 2.45/1.64 (! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X1 @ (setunion @ X2)) = $true) | ((in @ X0 @ X2) != $true) | ((in @ X1 @ X0) != $true)) | (setunionI != $true)) & ((setunionI = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (((in @ X1 @ (setunion @ X2)) != $true) & ((in @ X0 @ X2) = $true) & ((in @ X1 @ X0) = $true)))),
% 2.45/1.64 inference(nnf_transformation,[status(thm)],[f574])).
% 2.45/1.64 thf(f574,plain,(
% 2.45/1.64 ! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X1 @ (setunion @ X2)) = $true) | ((in @ X0 @ X2) != $true) | ((in @ X1 @ X0) != $true)) <=> (setunionI = $true)),
% 2.45/1.64 inference(flattening,[status(thm)],[f573])).
% 2.45/1.64 thf(f573,plain,(
% 2.45/1.64 ! [X0 : $i,X1 : $i,X2 : $i] : ((((in @ X1 @ (setunion @ X2)) = $true) | ((in @ X0 @ X2) != $true)) | ((in @ X1 @ X0) != $true)) <=> (setunionI = $true)),
% 2.45/1.64 inference(ennf_transformation,[status(thm)],[f253])).
% 2.45/1.64 thf(f253,plain,(
% 2.45/1.64 ! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X1 @ X0) = $true) => (((in @ X0 @ X2) = $true) => ((in @ X1 @ (setunion @ X2)) = $true))) <=> (setunionI = $true)),
% 2.45/1.64 inference(fool_elimination,[status(thm)],[f252])).
% 2.45/1.64 thf(f252,plain,(
% 2.45/1.64 (! [X0 : $i,X1 : $i,X2 : $i] : ((in @ X1 @ X0) => ((in @ X0 @ X2) => (in @ X1 @ (setunion @ X2)))) = setunionI)),
% 2.45/1.64 inference(rectify,[status(thm)],[f41])).
% 2.45/1.64 thf(f41,axiom,(
% 2.45/1.64 (! [X4 : $i,X1 : $i,X3 : $i] : ((in @ X1 @ X4) => ((in @ X4 @ X3) => (in @ X1 @ (setunion @ X3)))) = setunionI)),
% 2.45/1.64 file('/export/starexec/sandbox/tmp/tmp.BCzCW9enGb/DTF2THF_25972.p',setunionI)).
% 2.45/1.64 thf(f1430,plain,(
% 2.45/1.64 ((in @ sK108 @ (setunion @ (setadjoin @ (setadjoin @ sK109 @ emptyset) @ (setadjoin @ (setadjoin @ sK109 @ (setadjoin @ sK108 @ emptyset)) @ emptyset)))) != $true)),
% 2.45/1.64 inference(cnf_transformation,[status(thm)],[f774])).
% 2.45/1.64 % SZS output end Proof for DTF2THF_25972
% 2.45/1.64 % (26240)------------------------------
% 2.45/1.64 % (26240)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.45/1.64 % (26240)Termination reason: Refutation
% 2.45/1.64
% 2.45/1.64 % (26240)Memory used [KB]: 7931
% 2.45/1.64 % (26240)Time elapsed: 0.091 s
% 2.45/1.64 % (26240)Instructions burned: 272 (million)
% 2.45/1.64 % (26240)------------------------------
% 2.45/1.64 % (26240)------------------------------
% 2.45/1.64 % (26187)Success in time 0.18 s
%------------------------------------------------------------------------------