%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU620^1 : TPTP v9.2.1. Released v3.7.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n013.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 09:35:40 AM UTC 2026
% Result : Theorem 2.53s 1.55s
% Output : Refutation 2.53s
% Verified :
% SZS Type : Refutation
% Derivation depth : 10
% Number of leaves : 5
% Syntax : Number of formulae : 27 ( 12 unt; 0 typ; 0 def)
% Number of atoms : 2224 ( 581 equ; 0 cnn)
% Maximal formula atoms : 137 ( 82 avg)
% Number of connectives : 1235 ( 21 ~; 8 |; 411 &; 248 @)
% ( 1 <=>; 546 =>; 0 <=; 0 <~>)
% Maximal formula depth : 143 ( 39 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 262 ( 259 usr; 143 con; 0-4 aty)
% Number of variables : 48 ( 10 ^; 26 !; 12 ?; 48 :)
% Comments :
%------------------------------------------------------------------------------
thf(func_def_0,type,
in: $i > $i > $o ).
thf(func_def_1,type,
exu: ( $i > $o ) > $o ).
thf(func_def_6,type,
setadjoin: $i > $i > $i ).
thf(func_def_8,type,
powerset: $i > $i ).
thf(func_def_10,type,
setunion: $i > $i ).
thf(func_def_19,type,
descr: ( $i > $o ) > $i ).
thf(func_def_21,type,
dsetconstr: $i > ( $i > $o ) > $i ).
thf(func_def_26,type,
prop2set: $o > $i ).
thf(func_def_36,type,
nonempty: $i > $o ).
thf(func_def_69,type,
set2prop: $i > $o ).
thf(func_def_88,type,
subset: $i > $i > $o ).
thf(func_def_89,type,
disjoint: $i > $i > $o ).
thf(func_def_90,type,
setsmeet: $i > $i > $o ).
thf(func_def_114,type,
binunion: $i > $i > $i ).
thf(func_def_122,type,
binintersect: $i > $i > $i ).
thf(func_def_135,type,
regular: $i > $o ).
thf(func_def_136,type,
setminus: $i > $i > $i ).
thf(func_def_147,type,
symdiff: $i > $i > $i ).
thf(func_def_153,type,
iskpair: $i > $o ).
thf(func_def_158,type,
kpair: $i > $i > $i ).
thf(func_def_171,type,
sP0: $i > $i > $o ).
thf(func_def_172,type,
sP1: $i > $o ).
thf(func_def_173,type,
sP2: $i > $i > $o ).
thf(func_def_174,type,
sP3: $i > $i > $o ).
thf(func_def_176,type,
sP5: $i > $o > $i > $i > $o ).
thf(func_def_185,type,
sK14: $i > $i > $i ).
thf(func_def_189,type,
sK18: $i > $o ).
thf(func_def_194,type,
sK23: $i > $o ).
thf(func_def_199,type,
sK28: $i > $o ).
thf(func_def_200,type,
sK29: ( $i > $o ) > $i ).
thf(func_def_205,type,
sK34: $i > $i > $i ).
thf(func_def_209,type,
sK38: $i > $i ).
thf(func_def_210,type,
sK39: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_211,type,
sK40: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_212,type,
sK41: $i > $o ).
thf(func_def_213,type,
sK42: $i > $o ).
thf(func_def_214,type,
sK43: $i > $o ).
thf(func_def_216,type,
sK45: $i > ( $i > $o ) > $i ).
thf(func_def_219,type,
sK48: $i > $o ).
thf(func_def_220,type,
sK49: ( $i > $o ) > $i > $i ).
thf(func_def_222,type,
sK51: $i > $o ).
thf(func_def_236,type,
sK65: $i > $o ).
thf(func_def_244,type,
sK73: $i > $i > $i ).
thf(func_def_245,type,
sK74: $i > $i > $i ).
thf(func_def_246,type,
sK75: $i > $i > $i ).
thf(func_def_247,type,
sK76: $i > $i > $i ).
thf(func_def_248,type,
sK77: $i > $i > $i ).
thf(func_def_249,type,
sK78: $i > $i > $i > $i ).
thf(func_def_250,type,
sK79: $i > $i ).
thf(func_def_251,type,
sK80: $i > $i ).
thf(func_def_252,type,
sK81: $i > $i ).
thf(func_def_253,type,
sK82: $i > $i ).
thf(func_def_254,type,
sK83: $i > $i > $i ).
thf(func_def_255,type,
sK84: $i > $i > $i > $i ).
thf(func_def_256,type,
sK85: $i > $i > $i > $i ).
thf(func_def_257,type,
sK86: $i > $i > $i ).
thf(func_def_258,type,
sK87: $i > $i > $i ).
thf(func_def_260,type,
sK89: $i > $i ).
thf(func_def_261,type,
sK90: $i > $i ).
thf(func_def_262,type,
sK91: $i > $i ).
thf(func_def_268,type,
sK97: ( $i > $o ) > $i ).
thf(func_def_269,type,
sK98: $i > $o ).
thf(func_def_270,type,
sK99: $i > $i ).
thf(func_def_271,type,
sK100: $i > $o ).
thf(func_def_280,type,
sK109: $i > $i > $i ).
thf(func_def_288,type,
sK117: $i > ( $i > $o ) > $i ).
thf(func_def_289,type,
sK118: $i > $o ).
thf(func_def_294,type,
sK123: $i > $o ).
thf(func_def_308,type,
sK137: $i > $i > $i ).
thf(func_def_311,type,
sK140: $i > $i > $i ).
thf(func_def_314,type,
sK143: $i > $i > $o ).
thf(func_def_316,type,
sK145: $i > $i ).
thf(func_def_317,type,
sK146: $i > $i ).
thf(func_def_318,type,
sK147: $i > ( $i > $i > $o ) > $i ).
thf(func_def_319,type,
sK148: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_320,type,
sK149: $i > ( $i > $i > $o ) > $i ).
thf(func_def_322,type,
sK151: $i > $i ).
thf(func_def_324,type,
sK153: $i > $i ).
thf(func_def_331,type,
sK160: $i > $i ).
thf(func_def_332,type,
sK161: ( $i > $o ) > $i > $i ).
thf(func_def_334,type,
sK163: $i > $o ).
thf(func_def_340,type,
sK169: $i > $o ).
thf(func_def_359,type,
sK188: $i > $i > $i ).
thf(func_def_365,type,
sK194: $i > $o ).
thf(func_def_368,type,
sK197: $i > ( $i > $o ) > $i ).
thf(func_def_379,type,
sK208: $i > $i ).
thf(func_def_380,type,
sK209: $i > $o ).
thf(func_def_385,type,
sK214: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_386,type,
sK215: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_387,type,
sK216: $i > $o ).
thf(func_def_388,type,
sK217: $i > $o ).
thf(func_def_395,type,
sK224: $i > $o ).
thf(func_def_398,type,
sK227: $i > $i > $i ).
thf(func_def_416,type,
sK245: $o > $i > $i > $i ).
thf(func_def_421,type,
sK250: $i > $i ).
thf(func_def_431,type,
sK260: ( $i > $o ) > $i ).
thf(func_def_432,type,
sK261: ( $i > $o ) > $i ).
thf(func_def_433,type,
sK262: $i > $o ).
thf(func_def_440,type,
sK269: $i > $o ).
thf(func_def_459,type,
sK288: ( $i > $o ) > $i ).
thf(func_def_460,type,
sK289: $i > $o ).
thf(func_def_461,type,
sK290: $i > $i ).
thf(func_def_462,type,
sK291: ( $i > $o ) > $i > $i ).
thf(func_def_464,type,
sK293: $i > $o ).
thf(func_def_467,type,
sK296: $i > $o ).
thf(func_def_469,type,
sK298: ( $i > $o ) > $i > $i ).
thf(func_def_477,type,
sK306: $i > $i ).
thf(func_def_478,type,
sK307: ( $i > $o ) > $i > $i ).
thf(func_def_480,type,
sK309: $i > $o ).
thf(func_def_492,type,
sK321: $i > $o ).
thf(func_def_493,type,
sK322: $i > $o ).
thf(func_def_494,type,
sK323: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_495,type,
sK324: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_505,type,
sK334: $i > $o ).
thf(func_def_509,type,
sK338: $i > $o ).
thf(func_def_515,type,
sK344: $i > $i > $i ).
thf(func_def_516,type,
sK345: $i > $i > $i ).
thf(func_def_544,type,
ph373:
!>[X0: $tType] : X0 ).
thf(f2620,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f2545,f2558]) ).
thf(f2558,plain,
! [X2: $i,X3: $i] :
( ( iskpair @ ( setadjoin @ ( setadjoin @ X2 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X2 @ ( setadjoin @ X3 @ emptyset ) ) @ emptyset ) ) )
= $true ),
inference(trivial_inequality_removal,[status(thm)],[f1936]) ).
thf(f1936,plain,
! [X2: $i,X3: $i] :
( ( ( iskpair @ ( setadjoin @ ( setadjoin @ X2 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X2 @ ( setadjoin @ X3 @ emptyset ) ) @ emptyset ) ) )
= $true )
| ( $true != $true ) ),
inference(definition_unfolding,[status(thm)],[f1277,f1680]) ).
thf(f1680,plain,
kpairiskpair = $true,
inference(cnf_transformation,[status(thm)],[f1003]) ).
thf(f1003,plain,
( ( uniqinunit = $true )
& ( omega0Ax = $true )
& ( binintersectSubset3 = $true )
& ( emptyI = $true )
& ( exuE3e = $true )
& ( setminusERneg = $true )
& ( notinsingleton = $true )
& ( emptysetimpfalse = $true )
& ( foundationAx = $true )
& ( setukpairIL = $true )
& ( exuEu = $true )
& ( binunionE = $true )
& ( setminusEL = $true )
& ( powersetsubset = $true )
& ( singletonsswitch = $true )
& ( symdiffI2 = $true )
& ( setextsub = $true )
& ( symdiffE = $true )
& ( eqimpsubset1 = $true )
& ( binunionRsub = $true )
& ( setadjoinIL = $true )
& ( subsetemptysetimpeq = $true )
& ( setadjoinSub2 = $true )
& ( binintersectI = $true )
& ( binintersectSubset5 = $true )
& ( sepSubset = $true )
& ( quantDeMorgan1 = $true )
& ( kpairiskpair = $true )
& ( setunionE = $true )
& ( binunionIL = $true )
& ( exuI1 = $true )
& ( setunion__Cong = $true )
& ( powersetE = $true )
& ( setukpairIR = $true )
& ( noeltsimpempty = $true )
& ( vacuousDall = $true )
& ( binintersectRsub = $true )
& ( setadjoinE = $true )
& ( subsetE2 = $true )
& ( inPowerset = $true )
& ( upairsetIR = $true )
& ( dsetconstrI = $true )
& ( descr__Cong = $true )
& ( notequalI2 = $true )
& ( binunionEcases = $true )
& ( binintersectLsub = $true )
& ( subsetRefl = $true )
& ( emptyinPowerset = $true )
& ( powerset__Cong = $true )
& ( setadjoinSub = $true )
& ( upairset2IR = $true )
& ( nonemptyI = $true )
& ( setunionAx = $true )
& ( emptyinunitempty = $true )
& ( notdallE = $true )
& ( setminusSubset1 = $true )
& ( emptysetE = $true )
& ( emptysetsubset = $true )
& ( exuE3u = $true )
& ( prop2set2propI = $true )
& ( in__Cong = $true )
& ( sepInPowerset = $true )
& ( binunionLsub = $true )
& ( setoftrueEq = $true )
& ( emptysetAx = $true )
& ( wellorderingAx = $true )
& ( subPowSU = $true )
& ( binintersectSubset4 = $true )
& ( powersetAx = $true )
& ( emptyset__Cong = $true )
& ( setbeta = $true )
& ( secondinupair = $true )
& ( prop2setE = $true )
& ( exuE1 = $true )
& ( dsetconstrEL = $true )
& ( nonemptyImpWitness = $true )
& ( subsetTrans = $true )
& ( setminusELneg = $true )
& ( binintersectEL = $true )
& ( subset2powerset = $true )
& ( setext = $true )
& ( dsetconstr__Cong = $true )
& ( notequalI1 = $true )
& ( quantDeMorgan2 = $true )
& ( nonemptyI1 = $true )
& ( exu__Cong = $true )
& ( subsetI2 = $true )
& ( setminusER = $true )
& ( disjointsetsI1 = $true )
& ( nonemptyE1 = $true )
& ( omegaIndAx = $true )
& ( replAx = $true )
& ( prop2setI = $true )
& ( dsetconstrER = $true )
& ( binintersectER = $true )
& ( setadjoin__Cong = $true )
& ( symdiffI1 = $true )
& ( ( iskpair @ ( kpair @ sK237 @ sK238 ) )
!= $true )
& ( setminusI = $true )
& ( setextAx = $true )
& ( eqinunit = $true )
& ( symdiffIneg2 = $true )
& ( quantDeMorgan3 = $true )
& ( setadjoinIR = $true )
& ( powersetE1 = $true )
& ( upairsetE = $true )
& ( exuE2 = $true )
& ( notdexE = $true )
& ( descrp = $true )
& ( binunionIR = $true )
& ( subsetI1 = $true )
& ( subsetE = $true )
& ( symdiffIneg1 = $true )
& ( inCongP = $true )
& ( emptyE1 = $true )
& ( notinemptyset = $true )
& ( setunionI = $true )
& ( eqimpsubset2 = $true )
& ( binintersectSubset2 = $true )
& ( setadjoinAx = $true )
& ( omega__Cong = $true )
& ( setminusSubset2 = $true )
& ( powersetI = $true )
& ( setminusLsub = $true )
& ( binintersectSubset1 = $true )
& ( quantDeMorgan4 = $true )
& ( exuI3 = $true )
& ( bs114d = $true )
& ( exuI2 = $true )
& ( emptyInPowerset = $true )
& ( powersetI1 = $true )
& ( notsubsetI = $true )
& ( setadjoinOr = $true )
& ( setminusIRneg = $true )
& ( upairsetIL = $true )
& ( omegaSAx = $true )
& ( setminusILneg = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK237,sK238])],[f596,f1002]) ).
thf(f1002,plain,
( ? [X0: $i,X1: $i] :
( ( iskpair @ ( kpair @ X0 @ X1 ) )
!= $true )
=> ( ( iskpair @ ( kpair @ sK237 @ sK238 ) )
!= $true ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f596,plain,
( ( uniqinunit = $true )
& ( omega0Ax = $true )
& ( binintersectSubset3 = $true )
& ( emptyI = $true )
& ( exuE3e = $true )
& ( setminusERneg = $true )
& ( notinsingleton = $true )
& ( emptysetimpfalse = $true )
& ( foundationAx = $true )
& ( setukpairIL = $true )
& ( exuEu = $true )
& ( binunionE = $true )
& ( setminusEL = $true )
& ( powersetsubset = $true )
& ( singletonsswitch = $true )
& ( symdiffI2 = $true )
& ( setextsub = $true )
& ( symdiffE = $true )
& ( eqimpsubset1 = $true )
& ( binunionRsub = $true )
& ( setadjoinIL = $true )
& ( subsetemptysetimpeq = $true )
& ( setadjoinSub2 = $true )
& ( binintersectI = $true )
& ( binintersectSubset5 = $true )
& ( sepSubset = $true )
& ( quantDeMorgan1 = $true )
& ( kpairiskpair = $true )
& ( setunionE = $true )
& ( binunionIL = $true )
& ( exuI1 = $true )
& ( setunion__Cong = $true )
& ( powersetE = $true )
& ( setukpairIR = $true )
& ( noeltsimpempty = $true )
& ( vacuousDall = $true )
& ( binintersectRsub = $true )
& ( setadjoinE = $true )
& ( subsetE2 = $true )
& ( inPowerset = $true )
& ( upairsetIR = $true )
& ( dsetconstrI = $true )
& ( descr__Cong = $true )
& ( notequalI2 = $true )
& ( binunionEcases = $true )
& ( binintersectLsub = $true )
& ( subsetRefl = $true )
& ( emptyinPowerset = $true )
& ( powerset__Cong = $true )
& ( setadjoinSub = $true )
& ( upairset2IR = $true )
& ( nonemptyI = $true )
& ( setunionAx = $true )
& ( emptyinunitempty = $true )
& ( notdallE = $true )
& ( setminusSubset1 = $true )
& ( emptysetE = $true )
& ( emptysetsubset = $true )
& ( exuE3u = $true )
& ( prop2set2propI = $true )
& ( in__Cong = $true )
& ( sepInPowerset = $true )
& ( binunionLsub = $true )
& ( setoftrueEq = $true )
& ( emptysetAx = $true )
& ( wellorderingAx = $true )
& ( subPowSU = $true )
& ( binintersectSubset4 = $true )
& ( powersetAx = $true )
& ( emptyset__Cong = $true )
& ( setbeta = $true )
& ( secondinupair = $true )
& ( prop2setE = $true )
& ( exuE1 = $true )
& ( dsetconstrEL = $true )
& ( nonemptyImpWitness = $true )
& ( subsetTrans = $true )
& ( setminusELneg = $true )
& ( binintersectEL = $true )
& ( subset2powerset = $true )
& ( setext = $true )
& ( dsetconstr__Cong = $true )
& ( notequalI1 = $true )
& ( quantDeMorgan2 = $true )
& ( nonemptyI1 = $true )
& ( exu__Cong = $true )
& ( subsetI2 = $true )
& ( setminusER = $true )
& ( disjointsetsI1 = $true )
& ( nonemptyE1 = $true )
& ( omegaIndAx = $true )
& ( replAx = $true )
& ( prop2setI = $true )
& ( dsetconstrER = $true )
& ( binintersectER = $true )
& ( setadjoin__Cong = $true )
& ( symdiffI1 = $true )
& ? [X0: $i,X1: $i] :
( ( iskpair @ ( kpair @ X0 @ X1 ) )
!= $true )
& ( setminusI = $true )
& ( setextAx = $true )
& ( eqinunit = $true )
& ( symdiffIneg2 = $true )
& ( quantDeMorgan3 = $true )
& ( setadjoinIR = $true )
& ( powersetE1 = $true )
& ( upairsetE = $true )
& ( exuE2 = $true )
& ( notdexE = $true )
& ( descrp = $true )
& ( binunionIR = $true )
& ( subsetI1 = $true )
& ( subsetE = $true )
& ( symdiffIneg1 = $true )
& ( inCongP = $true )
& ( emptyE1 = $true )
& ( notinemptyset = $true )
& ( setunionI = $true )
& ( eqimpsubset2 = $true )
& ( binintersectSubset2 = $true )
& ( setadjoinAx = $true )
& ( omega__Cong = $true )
& ( setminusSubset2 = $true )
& ( powersetI = $true )
& ( setminusLsub = $true )
& ( binintersectSubset1 = $true )
& ( quantDeMorgan4 = $true )
& ( exuI3 = $true )
& ( bs114d = $true )
& ( exuI2 = $true )
& ( emptyInPowerset = $true )
& ( powersetI1 = $true )
& ( notsubsetI = $true )
& ( setadjoinOr = $true )
& ( setminusIRneg = $true )
& ( upairsetIL = $true )
& ( omegaSAx = $true )
& ( setminusILneg = $true ) ),
inference(flattening,[status(thm)],[f595]) ).
thf(f595,plain,
( ? [X0: $i,X1: $i] :
( ( iskpair @ ( kpair @ X0 @ X1 ) )
!= $true )
& ( kpairiskpair = $true )
& ( setukpairIR = $true )
& ( setukpairIL = $true )
& ( secondinupair = $true )
& ( symdiffIneg2 = $true )
& ( symdiffIneg1 = $true )
& ( symdiffI2 = $true )
& ( symdiffI1 = $true )
& ( symdiffE = $true )
& ( setminusSubset1 = $true )
& ( setminusLsub = $true )
& ( setminusIRneg = $true )
& ( setminusILneg = $true )
& ( setminusELneg = $true )
& ( setminusERneg = $true )
& ( setminusSubset2 = $true )
& ( setminusER = $true )
& ( setminusEL = $true )
& ( setminusI = $true )
& ( bs114d = $true )
& ( binintersectSubset1 = $true )
& ( binintersectSubset4 = $true )
& ( binintersectRsub = $true )
& ( disjointsetsI1 = $true )
& ( binintersectER = $true )
& ( binintersectSubset3 = $true )
& ( binintersectSubset2 = $true )
& ( binintersectLsub = $true )
& ( binintersectEL = $true )
& ( binintersectSubset5 = $true )
& ( binintersectI = $true )
& ( binunionRsub = $true )
& ( binunionLsub = $true )
& ( binunionE = $true )
& ( binunionEcases = $true )
& ( binunionIR = $true )
& ( upairset2IR = $true )
& ( binunionIL = $true )
& ( sepSubset = $true )
& ( sepInPowerset = $true )
& ( powersetsubset = $true )
& ( inPowerset = $true )
& ( powersetE1 = $true )
& ( powersetI1 = $true )
& ( subsetemptysetimpeq = $true )
& ( setextsub = $true )
& ( subset2powerset = $true )
& ( setadjoinSub2 = $true )
& ( setadjoinSub = $true )
& ( subsetTrans = $true )
& ( subsetRefl = $true )
& ( notequalI2 = $true )
& ( notequalI1 = $true )
& ( notsubsetI = $true )
& ( subsetE2 = $true )
& ( subsetE = $true )
& ( emptysetsubset = $true )
& ( subsetI2 = $true )
& ( eqimpsubset1 = $true )
& ( eqimpsubset2 = $true )
& ( subsetI1 = $true )
& ( dsetconstr__Cong = $true )
& ( descr__Cong = $true )
& ( exuEu = $true )
& ( omega__Cong = $true )
& ( setunion__Cong = $true )
& ( powerset__Cong = $true )
& ( setadjoin__Cong = $true )
& ( emptyset__Cong = $true )
& ( exu__Cong = $true )
& ( exuE3u = $true )
& ( in__Cong = $true )
& ( inCongP = $true )
& ( exuI2 = $true )
& ( exuI3 = $true )
& ( exuI1 = $true )
& ( notdallE = $true )
& ( notdexE = $true )
& ( prop2set2propI = $true )
& ( prop2setI = $true )
& ( quantDeMorgan4 = $true )
& ( quantDeMorgan3 = $true )
& ( quantDeMorgan2 = $true )
& ( quantDeMorgan1 = $true )
& ( vacuousDall = $true )
& ( emptyE1 = $true )
& ( upairsetIR = $true )
& ( upairsetIL = $true )
& ( upairsetE = $true )
& ( singletonsswitch = $true )
& ( eqinunit = $true )
& ( notinsingleton = $true )
& ( uniqinunit = $true )
& ( nonemptyImpWitness = $true )
& ( exuE2 = $true )
& ( subPowSU = $true )
& ( setunionE = $true )
& ( setunionI = $true )
& ( powersetE = $true )
& ( emptyInPowerset = $true )
& ( emptyinPowerset = $true )
& ( powersetI = $true )
& ( setoftrueEq = $true )
& ( setadjoinOr = $true )
& ( setadjoinE = $true )
& ( setadjoinIR = $true )
& ( emptyinunitempty = $true )
& ( setadjoinIL = $true )
& ( nonemptyI1 = $true )
& ( nonemptyI = $true )
& ( nonemptyE1 = $true )
& ( setbeta = $true )
& ( noeltsimpempty = $true )
& ( emptyI = $true )
& ( setext = $true )
& ( exuE3e = $true )
& ( notinemptyset = $true )
& ( emptysetimpfalse = $true )
& ( emptysetE = $true )
& ( prop2setE = $true )
& ( exuE1 = $true )
& ( dsetconstrER = $true )
& ( dsetconstrEL = $true )
& ( dsetconstrI = $true )
& ( descrp = $true )
& ( wellorderingAx = $true )
& ( foundationAx = $true )
& ( replAx = $true )
& ( omegaIndAx = $true )
& ( omegaSAx = $true )
& ( omega0Ax = $true )
& ( setunionAx = $true )
& ( powersetAx = $true )
& ( setadjoinAx = $true )
& ( emptysetAx = $true )
& ( setextAx = $true ) ),
inference(ennf_transformation,[status(thm)],[f358]) ).
thf(f358,plain,
~ ( ( setextAx = $true )
=> ( ( emptysetAx = $true )
=> ( ( setadjoinAx = $true )
=> ( ( powersetAx = $true )
=> ( ( setunionAx = $true )
=> ( ( omega0Ax = $true )
=> ( ( omegaSAx = $true )
=> ( ( omegaIndAx = $true )
=> ( ( replAx = $true )
=> ( ( foundationAx = $true )
=> ( ( wellorderingAx = $true )
=> ( ( descrp = $true )
=> ( ( dsetconstrI = $true )
=> ( ( dsetconstrEL = $true )
=> ( ( dsetconstrER = $true )
=> ( ( exuE1 = $true )
=> ( ( prop2setE = $true )
=> ( ( emptysetE = $true )
=> ( ( emptysetimpfalse = $true )
=> ( ( notinemptyset = $true )
=> ( ( exuE3e = $true )
=> ( ( setext = $true )
=> ( ( emptyI = $true )
=> ( ( noeltsimpempty = $true )
=> ( ( setbeta = $true )
=> ( ( nonemptyE1 = $true )
=> ( ( nonemptyI = $true )
=> ( ( nonemptyI1 = $true )
=> ( ( setadjoinIL = $true )
=> ( ( emptyinunitempty = $true )
=> ( ( setadjoinIR = $true )
=> ( ( setadjoinE = $true )
=> ( ( setadjoinOr = $true )
=> ( ( setoftrueEq = $true )
=> ( ( powersetI = $true )
=> ( ( emptyinPowerset = $true )
=> ( ( emptyInPowerset = $true )
=> ( ( powersetE = $true )
=> ( ( setunionI = $true )
=> ( ( setunionE = $true )
=> ( ( subPowSU = $true )
=> ( ( exuE2 = $true )
=> ( ( nonemptyImpWitness = $true )
=> ( ( uniqinunit = $true )
=> ( ( notinsingleton = $true )
=> ( ( eqinunit = $true )
=> ( ( singletonsswitch = $true )
=> ( ( upairsetE = $true )
=> ( ( upairsetIL = $true )
=> ( ( upairsetIR = $true )
=> ( ( emptyE1 = $true )
=> ( ( vacuousDall = $true )
=> ( ( quantDeMorgan1 = $true )
=> ( ( quantDeMorgan2 = $true )
=> ( ( quantDeMorgan3 = $true )
=> ( ( quantDeMorgan4 = $true )
=> ( ( prop2setI = $true )
=> ( ( prop2set2propI = $true )
=> ( ( notdexE = $true )
=> ( ( notdallE = $true )
=> ( ( exuI1 = $true )
=> ( ( exuI3 = $true )
=> ( ( exuI2 = $true )
=> ( ( inCongP = $true )
=> ( ( in__Cong = $true )
=> ( ( exuE3u = $true )
=> ( ( exu__Cong = $true )
=> ( ( emptyset__Cong = $true )
=> ( ( setadjoin__Cong = $true )
=> ( ( powerset__Cong = $true )
=> ( ( setunion__Cong = $true )
=> ( ( omega__Cong = $true )
=> ( ( exuEu = $true )
=> ( ( descr__Cong = $true )
=> ( ( dsetconstr__Cong = $true )
=> ( ( subsetI1 = $true )
=> ( ( eqimpsubset2 = $true )
=> ( ( eqimpsubset1 = $true )
=> ( ( subsetI2 = $true )
=> ( ( emptysetsubset = $true )
=> ( ( subsetE = $true )
=> ( ( subsetE2 = $true )
=> ( ( notsubsetI = $true )
=> ( ( notequalI1 = $true )
=> ( ( notequalI2 = $true )
=> ( ( subsetRefl = $true )
=> ( ( subsetTrans = $true )
=> ( ( setadjoinSub = $true )
=> ( ( setadjoinSub2 = $true )
=> ( ( subset2powerset = $true )
=> ( ( setextsub = $true )
=> ( ( subsetemptysetimpeq = $true )
=> ( ( powersetI1 = $true )
=> ( ( powersetE1 = $true )
=> ( ( inPowerset = $true )
=> ( ( powersetsubset = $true )
=> ( ( sepInPowerset = $true )
=> ( ( sepSubset = $true )
=> ( ( binunionIL = $true )
=> ( ( upairset2IR = $true )
=> ( ( binunionIR = $true )
=> ( ( binunionEcases = $true )
=> ( ( binunionE = $true )
=> ( ( binunionLsub = $true )
=> ( ( binunionRsub = $true )
=> ( ( binintersectI = $true )
=> ( ( binintersectSubset5 = $true )
=> ( ( binintersectEL = $true )
=> ( ( binintersectLsub = $true )
=> ( ( binintersectSubset2 = $true )
=> ( ( binintersectSubset3 = $true )
=> ( ( binintersectER = $true )
=> ( ( disjointsetsI1 = $true )
=> ( ( binintersectRsub = $true )
=> ( ( binintersectSubset4 = $true )
=> ( ( binintersectSubset1 = $true )
=> ( ( bs114d = $true )
=> ( ( setminusI = $true )
=> ( ( setminusEL = $true )
=> ( ( setminusER = $true )
=> ( ( setminusSubset2 = $true )
=> ( ( setminusERneg = $true )
=> ( ( setminusELneg = $true )
=> ( ( setminusILneg = $true )
=> ( ( setminusIRneg = $true )
=> ( ( setminusLsub = $true )
=> ( ( setminusSubset1 = $true )
=> ( ( symdiffE = $true )
=> ( ( symdiffI1 = $true )
=> ( ( symdiffI2 = $true )
=> ( ( symdiffIneg1 = $true )
=> ( ( symdiffIneg2 = $true )
=> ( ( secondinupair = $true )
=> ( ( setukpairIL = $true )
=> ( ( setukpairIR = $true )
=> ( ( kpairiskpair = $true )
=> ! [X0: $i,X1: $i] :
( ( iskpair @ ( kpair @ X0 @ X1 ) )
= $true ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f357]) ).
thf(f357,plain,
~ ( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ! [X0: $i,X1: $i] : ( iskpair @ ( kpair @ X0 @ X1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f142]) ).
thf(f142,negated_conjecture,
~ ( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f141]) ).
thf(f141,conjecture,
( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.0dbwHN3pEv/DTF2THF_22343.p',kpairp) ).
thf(f1277,plain,
! [X2: $i,X3: $i] :
( ( ( iskpair @ ( setadjoin @ ( setadjoin @ X2 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X2 @ ( setadjoin @ X3 @ emptyset ) ) @ emptyset ) ) )
= $true )
| ( kpairiskpair != $true ) ),
inference(cnf_transformation,[status(thm)],[f663]) ).
thf(f663,plain,
( ( ( kpairiskpair = $true )
| ( ( iskpair @ ( setadjoin @ ( setadjoin @ sK35 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK35 @ ( setadjoin @ sK36 @ emptyset ) ) @ emptyset ) ) )
!= $true ) )
& ( ! [X2: $i,X3: $i] :
( ( iskpair @ ( setadjoin @ ( setadjoin @ X2 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X2 @ ( setadjoin @ X3 @ emptyset ) ) @ emptyset ) ) )
= $true )
| ( kpairiskpair != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK35,sK36])],[f661,f662]) ).
thf(f662,plain,
( ? [X0: $i,X1: $i] :
( ( iskpair @ ( setadjoin @ ( setadjoin @ X0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X0 @ ( setadjoin @ X1 @ emptyset ) ) @ emptyset ) ) )
!= $true )
=> ( ( iskpair @ ( setadjoin @ ( setadjoin @ sK35 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK35 @ ( setadjoin @ sK36 @ emptyset ) ) @ emptyset ) ) )
!= $true ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f661,plain,
( ( ( kpairiskpair = $true )
| ? [X0: $i,X1: $i] :
( ( iskpair @ ( setadjoin @ ( setadjoin @ X0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X0 @ ( setadjoin @ X1 @ emptyset ) ) @ emptyset ) ) )
!= $true ) )
& ( ! [X2: $i,X3: $i] :
( ( iskpair @ ( setadjoin @ ( setadjoin @ X2 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X2 @ ( setadjoin @ X3 @ emptyset ) ) @ emptyset ) ) )
= $true )
| ( kpairiskpair != $true ) ) ),
inference(rectify,[status(thm)],[f660]) ).
thf(f660,plain,
( ( ( kpairiskpair = $true )
| ? [X0: $i,X1: $i] :
( ( iskpair @ ( setadjoin @ ( setadjoin @ X0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X0 @ ( setadjoin @ X1 @ emptyset ) ) @ emptyset ) ) )
!= $true ) )
& ( ! [X0: $i,X1: $i] :
( ( iskpair @ ( setadjoin @ ( setadjoin @ X0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X0 @ ( setadjoin @ X1 @ emptyset ) ) @ emptyset ) ) )
= $true )
| ( kpairiskpair != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f292]) ).
thf(f292,plain,
( ( kpairiskpair = $true )
<=> ! [X0: $i,X1: $i] :
( ( iskpair @ ( setadjoin @ ( setadjoin @ X0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X0 @ ( setadjoin @ X1 @ emptyset ) ) @ emptyset ) ) )
= $true ) ),
inference(fool_elimination,[status(thm)],[f291]) ).
thf(f291,plain,
! [X0: $i,X1: $i] :
( ( iskpair @ ( setadjoin @ ( setadjoin @ X0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X0 @ ( setadjoin @ X1 @ emptyset ) ) @ emptyset ) ) )
= kpairiskpair ),
inference(rectify,[status(thm)],[f139]) ).
thf(f139,axiom,
! [X1: $i,X2: $i] :
( ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
= kpairiskpair ),
file('/export/starexec/sandbox2/tmp/tmp.0dbwHN3pEv/DTF2THF_22343.p',kpairiskpair) ).
thf(f2545,plain,
( $true
!= ( iskpair @ ( setadjoin @ ( setadjoin @ sK237 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK237 @ ( setadjoin @ sK238 @ emptyset ) ) @ emptyset ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f2190]) ).
thf(f2190,plain,
( ( iskpair
@ ( ^ [Y0: $i,Y1: $i] : ( setadjoin @ ( setadjoin @ Y0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ Y0 @ ( setadjoin @ Y1 @ emptyset ) ) @ emptyset ) )
@ sK237
@ sK238 ) )
!= $true ),
inference(definition_unfolding,[status(thm)],[f1610,f1326]) ).
thf(f1326,plain,
( kpair
= ( ^ [Y0: $i,Y1: $i] : ( setadjoin @ ( setadjoin @ Y0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ Y0 @ ( setadjoin @ Y1 @ emptyset ) ) @ emptyset ) ) ) ),
inference(cnf_transformation,[status(thm)],[f266]) ).
thf(f266,plain,
( kpair
= ( ^ [Y0: $i,Y1: $i] : ( setadjoin @ ( setadjoin @ Y0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ Y0 @ ( setadjoin @ Y1 @ emptyset ) ) @ emptyset ) ) ) ),
inference(fool_elimination,[status(thm)],[f265]) ).
thf(f265,plain,
( ( ^ [X0: $i,X1: $i] : ( setadjoin @ ( setadjoin @ X0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X0 @ ( setadjoin @ X1 @ emptyset ) ) @ emptyset ) ) )
= kpair ),
inference(rectify,[status(thm)],[f140]) ).
thf(f140,axiom,
( ( ^ [X1: $i,X2: $i] : ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
= kpair ),
file('/export/starexec/sandbox2/tmp/tmp.0dbwHN3pEv/DTF2THF_22343.p',kpair) ).
thf(f1610,plain,
( ( iskpair @ ( kpair @ sK237 @ sK238 ) )
!= $true ),
inference(cnf_transformation,[status(thm)],[f1003]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SEU620^1 : TPTP v9.2.1. Released v3.7.0.
% 0.12/0.13 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.17/0.34 % Computer : n013.cluster.edu
% 0.17/0.34 % Model : x86_64 x86_64
% 0.17/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.34 % Memory : 8042.1875MB
% 0.17/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.17/0.34 % CPULimit : 300
% 0.17/0.34 % WCLimit : 300
% 0.17/0.34 % DateTime : Tue Feb 24 12:28:23 EST 2026
% 0.17/0.34 % CPUTime :
% 0.17/0.34 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.19/0.46 ---- Original DTF file ---
% 0.19/0.46 thf(spec,logic,$$dhol).
% 0.19/0.46 %------------------------------------------------------------------------------
% 0.19/0.46 % File : SEU620^1 : TPTP v9.2.1. Released v3.7.0.
% 0.19/0.46 % Domain : Set Theory
% 0.19/0.46 % Problem : Ordered Pairs - Kuratowski Pairs
% 0.19/0.46 % Version : Especial.
% 0.19/0.46 % English : (! x:i.! y:i.iskpair (kpair x y))
% 0.19/0.46
% 0.19/0.46 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.19/0.46 % Source : [Bro08]
% 0.19/0.46 % Names : ZFC122g [Bro08]
% 0.19/0.46
% 0.19/0.46 % Status : Theorem
% 0.19/0.46 % Rating : 0.11 v9.1.0, 0.12 v9.0.0, 0.20 v8.2.0, 0.23 v8.1.0, 0.09 v7.5.0, 0.14 v7.4.0, 0.33 v7.2.0, 0.25 v7.0.0, 0.29 v6.4.0, 0.33 v6.3.0, 0.40 v6.2.0, 0.29 v6.1.0, 0.43 v5.5.0, 0.33 v5.4.0, 0.60 v4.1.0, 0.67 v4.0.1, 1.00 v4.0.0, 0.67 v3.7.0
% 0.19/0.46 % Syntax : Number of formulae : 299 ( 140 unt; 158 typ; 140 def)
% 0.19/0.46 % Number of atoms : 739 ( 198 equ; 0 cnn)
% 0.19/0.46 % Maximal formula atoms : 137 ( 5 avg)
% 0.19/0.46 % Number of connectives : 1230 ( 47 ~; 6 |; 30 &; 807 @)
% 0.19/0.46 % ( 14 <=>; 326 =>; 0 <=; 0 <~>)
% 0.19/0.46 % Maximal formula depth : 142 ( 2 avg)
% 0.19/0.46 % Number of types : 2 ( 0 usr)
% 0.19/0.46 % Number of type conns : 67 ( 67 >; 0 *; 0 +; 0 <<)
% 0.19/0.46 % Number of symbols : 161 ( 158 usr; 140 con; 0-2 aty)
% 0.19/0.46 % Number of variables : 401 ( 33 ^; 337 !; 31 ?; 401 :)
% 0.19/0.46 % SPC : TH0_THM_EQU_NAR
% 0.19/0.46
% 0.19/0.46 % Comments : http://mathgate.info/detsetitem.php?id=176
% 0.19/0.46 %------------------------------------------------------------------------------
% 0.19/0.46 thf(in_type,type,
% 0.19/0.46 in: $i > $i > $o ).
% 0.19/0.46
% 0.19/0.46 thf(exu_type,type,
% 0.19/0.46 exu: ( $i > $o ) > $o ).
% 0.19/0.46
% 0.19/0.46 thf(exu,definition,
% 0.19/0.46 ( exu
% 0.19/0.46 = ( ^ [Xphi: $i > $o] :
% 0.19/0.46 ? [Xx: $i] :
% 0.19/0.46 ( ( Xphi @ Xx )
% 0.19/0.46 & ! [Xy: $i] :
% 0.19/0.46 ( ( Xphi @ Xy )
% 0.19/0.46 => ( Xx = Xy ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setextAx_type,type,
% 0.19/0.46 setextAx: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setextAx,definition,
% 0.19/0.46 ( setextAx
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 <=> ( in @ Xx @ B ) )
% 0.19/0.46 => ( A = B ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(emptyset_type,type,
% 0.19/0.46 emptyset: $i ).
% 0.19/0.46
% 0.19/0.46 thf(emptysetAx_type,type,
% 0.19/0.46 emptysetAx: $o ).
% 0.19/0.46
% 0.19/0.46 thf(emptysetAx,definition,
% 0.19/0.46 ( emptysetAx
% 0.19/0.46 = ( ! [Xx: $i] :
% 0.19/0.46 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setadjoin_type,type,
% 0.19/0.46 setadjoin: $i > $i > $i ).
% 0.19/0.46
% 0.19/0.46 thf(setadjoinAx_type,type,
% 0.19/0.46 setadjoinAx: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setadjoinAx,definition,
% 0.19/0.46 ( setadjoinAx
% 0.19/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.19/0.46 <=> ( ( Xy = Xx )
% 0.19/0.46 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(powerset_type,type,
% 0.19/0.46 powerset: $i > $i ).
% 0.19/0.46
% 0.19/0.46 thf(powersetAx_type,type,
% 0.19/0.46 powersetAx: $o ).
% 0.19/0.46
% 0.19/0.46 thf(powersetAx,definition,
% 0.19/0.46 ( powersetAx
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ( in @ B @ ( powerset @ A ) )
% 0.19/0.46 <=> ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ B )
% 0.19/0.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setunion_type,type,
% 0.19/0.46 setunion: $i > $i ).
% 0.19/0.46
% 0.19/0.46 thf(setunionAx_type,type,
% 0.19/0.46 setunionAx: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setunionAx,definition,
% 0.19/0.46 ( setunionAx
% 0.19/0.46 = ( ! [A: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ ( setunion @ A ) )
% 0.19/0.46 <=> ? [B: $i] :
% 0.19/0.46 ( ( in @ Xx @ B )
% 0.19/0.46 & ( in @ B @ A ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(omega_type,type,
% 0.19/0.46 omega: $i ).
% 0.19/0.46
% 0.19/0.46 thf(omega0Ax_type,type,
% 0.19/0.46 omega0Ax: $o ).
% 0.19/0.46
% 0.19/0.46 thf(omega0Ax,definition,
% 0.19/0.46 ( omega0Ax
% 0.19/0.46 = ( in @ emptyset @ omega ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(omegaSAx_type,type,
% 0.19/0.46 omegaSAx: $o ).
% 0.19/0.46
% 0.19/0.46 thf(omegaSAx,definition,
% 0.19/0.46 ( omegaSAx
% 0.19/0.46 = ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ omega )
% 0.19/0.46 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(omegaIndAx_type,type,
% 0.19/0.46 omegaIndAx: $o ).
% 0.19/0.46
% 0.19/0.46 thf(omegaIndAx,definition,
% 0.19/0.46 ( omegaIndAx
% 0.19/0.46 = ( ! [A: $i] :
% 0.19/0.46 ( ( ( in @ emptyset @ A )
% 0.19/0.46 & ! [Xx: $i] :
% 0.19/0.46 ( ( ( in @ Xx @ omega )
% 0.19/0.46 & ( in @ Xx @ A ) )
% 0.19/0.46 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.19/0.46 => ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ omega )
% 0.19/0.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(replAx_type,type,
% 0.19/0.46 replAx: $o ).
% 0.19/0.46
% 0.19/0.46 thf(replAx,definition,
% 0.19/0.46 ( replAx
% 0.19/0.46 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.19/0.46 ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( exu
% 0.19/0.46 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.19/0.46 => ? [B: $i] :
% 0.19/0.46 ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ B )
% 0.19/0.46 <=> ? [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ A )
% 0.19/0.46 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(foundationAx_type,type,
% 0.19/0.46 foundationAx: $o ).
% 0.19/0.46
% 0.19/0.46 thf(foundationAx,definition,
% 0.19/0.46 ( foundationAx
% 0.19/0.46 = ( ! [A: $i] :
% 0.19/0.46 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.19/0.46 => ? [B: $i] :
% 0.19/0.46 ( ( in @ B @ A )
% 0.19/0.46 & ~ ? [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ B )
% 0.19/0.46 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(wellorderingAx_type,type,
% 0.19/0.46 wellorderingAx: $o ).
% 0.19/0.46
% 0.19/0.46 thf(wellorderingAx,definition,
% 0.19/0.46 ( wellorderingAx
% 0.19/0.46 = ( ! [A: $i] :
% 0.19/0.46 ? [B: $i] :
% 0.19/0.46 ( ! [C: $i] :
% 0.19/0.46 ( ( in @ C @ B )
% 0.19/0.46 => ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ C )
% 0.19/0.46 => ( in @ Xx @ A ) ) )
% 0.19/0.46 & ! [Xx: $i,Xy: $i] :
% 0.19/0.46 ( ( ( in @ Xx @ A )
% 0.19/0.46 & ( in @ Xy @ A ) )
% 0.19/0.46 => ( ! [C: $i] :
% 0.19/0.46 ( ( in @ C @ B )
% 0.19/0.46 => ( ( in @ Xx @ C )
% 0.19/0.46 <=> ( in @ Xy @ C ) ) )
% 0.19/0.46 => ( Xx = Xy ) ) )
% 0.19/0.46 & ! [C: $i,D: $i] :
% 0.19/0.46 ( ( ( in @ C @ B )
% 0.19/0.46 & ( in @ D @ B ) )
% 0.19/0.46 => ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ C )
% 0.19/0.46 => ( in @ Xx @ D ) )
% 0.19/0.46 | ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ D )
% 0.19/0.46 => ( in @ Xx @ C ) ) ) )
% 0.19/0.46 & ! [C: $i] :
% 0.19/0.46 ( ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ C )
% 0.19/0.46 => ( in @ Xx @ A ) )
% 0.19/0.46 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.19/0.46 => ? [D: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ D @ B )
% 0.19/0.46 & ( in @ Xx @ C )
% 0.19/0.46 & ~ ? [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ D )
% 0.19/0.46 & ( in @ Xy @ C ) )
% 0.19/0.46 & ! [E: $i] :
% 0.19/0.46 ( ( in @ E @ B )
% 0.19/0.46 => ( ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ E )
% 0.19/0.46 => ( in @ Xy @ D ) )
% 0.19/0.46 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(descr_type,type,
% 0.19/0.46 descr: ( $i > $o ) > $i ).
% 0.19/0.46
% 0.19/0.46 thf(descrp_type,type,
% 0.19/0.46 descrp: $o ).
% 0.19/0.46
% 0.19/0.46 thf(descrp,definition,
% 0.19/0.46 ( descrp
% 0.19/0.46 = ( ! [Xphi: $i > $o] :
% 0.19/0.46 ( ( exu
% 0.19/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.46 => ( Xphi
% 0.19/0.46 @ ( descr
% 0.19/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(dsetconstr_type,type,
% 0.19/0.46 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.19/0.46
% 0.19/0.46 thf(dsetconstrI_type,type,
% 0.19/0.46 dsetconstrI: $o ).
% 0.19/0.46
% 0.19/0.46 thf(dsetconstrI,definition,
% 0.19/0.46 ( dsetconstrI
% 0.19/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( ( Xphi @ Xx )
% 0.19/0.46 => ( in @ Xx
% 0.19/0.46 @ ( dsetconstr @ A
% 0.19/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(dsetconstrEL_type,type,
% 0.19/0.46 dsetconstrEL: $o ).
% 0.19/0.46
% 0.19/0.46 thf(dsetconstrEL,definition,
% 0.19/0.46 ( dsetconstrEL
% 0.19/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx
% 0.19/0.46 @ ( dsetconstr @ A
% 0.19/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.19/0.46 => ( in @ Xx @ A ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(dsetconstrER_type,type,
% 0.19/0.46 dsetconstrER: $o ).
% 0.19/0.46
% 0.19/0.46 thf(dsetconstrER,definition,
% 0.19/0.46 ( dsetconstrER
% 0.19/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx
% 0.19/0.46 @ ( dsetconstr @ A
% 0.19/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.19/0.46 => ( Xphi @ Xx ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(exuE1_type,type,
% 0.19/0.46 exuE1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(exuE1,definition,
% 0.19/0.46 ( exuE1
% 0.19/0.46 = ( ! [Xphi: $i > $o] :
% 0.19/0.46 ( ( exu
% 0.19/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.46 => ? [Xx: $i] :
% 0.19/0.46 ( ( Xphi @ Xx )
% 0.19/0.46 & ! [Xy: $i] :
% 0.19/0.46 ( ( Xphi @ Xy )
% 0.19/0.46 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(prop2set_type,type,
% 0.19/0.46 prop2set: $o > $i ).
% 0.19/0.46
% 0.19/0.46 thf(prop2setE_type,type,
% 0.19/0.46 prop2setE: $o ).
% 0.19/0.46
% 0.19/0.46 thf(prop2setE,definition,
% 0.19/0.46 ( prop2setE
% 0.19/0.46 = ( ! [Xphi: $o,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.19/0.46 => Xphi ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(emptysetE_type,type,
% 0.19/0.46 emptysetE: $o ).
% 0.19/0.46
% 0.19/0.46 thf(emptysetE,definition,
% 0.19/0.46 ( emptysetE
% 0.19/0.46 = ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ emptyset )
% 0.19/0.46 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(emptysetimpfalse_type,type,
% 0.19/0.46 emptysetimpfalse: $o ).
% 0.19/0.46
% 0.19/0.46 thf(emptysetimpfalse,definition,
% 0.19/0.46 ( emptysetimpfalse
% 0.19/0.46 = ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ emptyset )
% 0.19/0.46 => $false ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(notinemptyset_type,type,
% 0.19/0.46 notinemptyset: $o ).
% 0.19/0.46
% 0.19/0.46 thf(notinemptyset,definition,
% 0.19/0.46 ( notinemptyset
% 0.19/0.46 = ( ! [Xx: $i] :
% 0.19/0.46 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(exuE3e_type,type,
% 0.19/0.46 exuE3e: $o ).
% 0.19/0.46
% 0.19/0.46 thf(exuE3e,definition,
% 0.19/0.46 ( exuE3e
% 0.19/0.46 = ( ! [Xphi: $i > $o] :
% 0.19/0.46 ( ( exu
% 0.19/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.46 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setext_type,type,
% 0.19/0.46 setext: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setext,definition,
% 0.19/0.46 ( setext
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( in @ Xx @ B ) )
% 0.19/0.46 => ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ B )
% 0.19/0.46 => ( in @ Xx @ A ) )
% 0.19/0.46 => ( A = B ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(emptyI_type,type,
% 0.19/0.46 emptyI: $o ).
% 0.19/0.46
% 0.19/0.46 thf(emptyI,definition,
% 0.19/0.46 ( emptyI
% 0.19/0.46 = ( ! [A: $i] :
% 0.19/0.46 ( ! [Xx: $i] :
% 0.19/0.46 ~ ( in @ Xx @ A )
% 0.19/0.46 => ( A = emptyset ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(noeltsimpempty_type,type,
% 0.19/0.46 noeltsimpempty: $o ).
% 0.19/0.46
% 0.19/0.46 thf(noeltsimpempty,definition,
% 0.19/0.46 ( noeltsimpempty
% 0.19/0.46 = ( ! [A: $i] :
% 0.19/0.46 ( ! [Xx: $i] :
% 0.19/0.46 ~ ( in @ Xx @ A )
% 0.19/0.46 => ( A = emptyset ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setbeta_type,type,
% 0.19/0.46 setbeta: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setbeta,definition,
% 0.19/0.46 ( setbeta
% 0.19/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( ( in @ Xx
% 0.19/0.46 @ ( dsetconstr @ A
% 0.19/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.19/0.46 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(nonempty_type,type,
% 0.19/0.46 nonempty: $i > $o ).
% 0.19/0.46
% 0.19/0.46 thf(nonempty,definition,
% 0.19/0.46 ( nonempty
% 0.19/0.46 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(nonemptyE1_type,type,
% 0.19/0.46 nonemptyE1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(nonemptyE1,definition,
% 0.19/0.46 ( nonemptyE1
% 0.19/0.46 = ( ! [A: $i] :
% 0.19/0.46 ( ( nonempty @ A )
% 0.19/0.46 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(nonemptyI_type,type,
% 0.19/0.46 nonemptyI: $o ).
% 0.19/0.46
% 0.19/0.46 thf(nonemptyI,definition,
% 0.19/0.46 ( nonemptyI
% 0.19/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( ( Xphi @ Xx )
% 0.19/0.46 => ( nonempty
% 0.19/0.46 @ ( dsetconstr @ A
% 0.19/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(nonemptyI1_type,type,
% 0.19/0.46 nonemptyI1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(nonemptyI1,definition,
% 0.19/0.46 ( nonemptyI1
% 0.19/0.46 = ( ! [A: $i] :
% 0.19/0.46 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.19/0.46 => ( nonempty @ A ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setadjoinIL_type,type,
% 0.19/0.46 setadjoinIL: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setadjoinIL,definition,
% 0.19/0.46 ( setadjoinIL
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(emptyinunitempty_type,type,
% 0.19/0.46 emptyinunitempty: $o ).
% 0.19/0.46
% 0.19/0.46 thf(emptyinunitempty,definition,
% 0.19/0.46 ( emptyinunitempty
% 0.19/0.46 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setadjoinIR_type,type,
% 0.19/0.46 setadjoinIR: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setadjoinIR,definition,
% 0.19/0.46 ( setadjoinIR
% 0.19/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ A )
% 0.19/0.46 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setadjoinE_type,type,
% 0.19/0.46 setadjoinE: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setadjoinE,definition,
% 0.19/0.46 ( setadjoinE
% 0.19/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.19/0.46 => ! [Xphi: $o] :
% 0.19/0.46 ( ( ( Xy = Xx )
% 0.19/0.46 => Xphi )
% 0.19/0.46 => ( ( ( in @ Xy @ A )
% 0.19/0.46 => Xphi )
% 0.19/0.46 => Xphi ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setadjoinOr_type,type,
% 0.19/0.46 setadjoinOr: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setadjoinOr,definition,
% 0.19/0.46 ( setadjoinOr
% 0.19/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.19/0.46 => ( ( Xy = Xx )
% 0.19/0.46 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setoftrueEq_type,type,
% 0.19/0.46 setoftrueEq: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setoftrueEq,definition,
% 0.19/0.46 ( setoftrueEq
% 0.19/0.46 = ( ! [A: $i] :
% 0.19/0.46 ( ( dsetconstr @ A
% 0.19/0.46 @ ^ [Xx: $i] : $true )
% 0.19/0.46 = A ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(powersetI_type,type,
% 0.19/0.46 powersetI: $o ).
% 0.19/0.46
% 0.19/0.46 thf(powersetI,definition,
% 0.19/0.46 ( powersetI
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ B )
% 0.19/0.46 => ( in @ Xx @ A ) )
% 0.19/0.46 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(emptyinPowerset_type,type,
% 0.19/0.46 emptyinPowerset: $o ).
% 0.19/0.46
% 0.19/0.46 thf(emptyinPowerset,definition,
% 0.19/0.46 ( emptyinPowerset
% 0.19/0.46 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(emptyInPowerset_type,type,
% 0.19/0.46 emptyInPowerset: $o ).
% 0.19/0.46
% 0.19/0.46 thf(emptyInPowerset,definition,
% 0.19/0.46 ( emptyInPowerset
% 0.19/0.46 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(powersetE_type,type,
% 0.19/0.46 powersetE: $o ).
% 0.19/0.46
% 0.19/0.46 thf(powersetE,definition,
% 0.19/0.46 ( powersetE
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ B @ ( powerset @ A ) )
% 0.19/0.46 => ( ( in @ Xx @ B )
% 0.19/0.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setunionI_type,type,
% 0.19/0.46 setunionI: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setunionI,definition,
% 0.19/0.46 ( setunionI
% 0.19/0.46 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.19/0.46 ( ( in @ Xx @ B )
% 0.19/0.46 => ( ( in @ B @ A )
% 0.19/0.46 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setunionE_type,type,
% 0.19/0.46 setunionE: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setunionE,definition,
% 0.19/0.46 ( setunionE
% 0.19/0.46 = ( ! [A: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ ( setunion @ A ) )
% 0.19/0.46 => ! [Xphi: $o] :
% 0.19/0.46 ( ! [B: $i] :
% 0.19/0.46 ( ( in @ Xx @ B )
% 0.19/0.46 => ( ( in @ B @ A )
% 0.19/0.46 => Xphi ) )
% 0.19/0.46 => Xphi ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(subPowSU_type,type,
% 0.19/0.46 subPowSU: $o ).
% 0.19/0.46
% 0.19/0.46 thf(subPowSU,definition,
% 0.19/0.46 ( subPowSU
% 0.19/0.46 = ( ! [A: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(exuE2_type,type,
% 0.19/0.46 exuE2: $o ).
% 0.19/0.46
% 0.19/0.46 thf(exuE2,definition,
% 0.19/0.46 ( exuE2
% 0.19/0.46 = ( ! [Xphi: $i > $o] :
% 0.19/0.46 ( ( exu
% 0.19/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.46 => ? [Xx: $i] :
% 0.19/0.46 ! [Xy: $i] :
% 0.19/0.46 ( ( Xphi @ Xy )
% 0.19/0.46 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(nonemptyImpWitness_type,type,
% 0.19/0.46 nonemptyImpWitness: $o ).
% 0.19/0.46
% 0.19/0.46 thf(nonemptyImpWitness,definition,
% 0.19/0.46 ( nonemptyImpWitness
% 0.19/0.46 = ( ! [A: $i] :
% 0.19/0.46 ( ( nonempty @ A )
% 0.19/0.46 => ? [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 & $true ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(uniqinunit_type,type,
% 0.19/0.46 uniqinunit: $o ).
% 0.19/0.46
% 0.19/0.46 thf(uniqinunit,definition,
% 0.19/0.46 ( uniqinunit
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.46 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.19/0.46 => ( Xx = Xy ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(notinsingleton_type,type,
% 0.19/0.46 notinsingleton: $o ).
% 0.19/0.46
% 0.19/0.46 thf(notinsingleton,definition,
% 0.19/0.46 ( notinsingleton
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.46 ( ( Xx != Xy )
% 0.19/0.46 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(eqinunit_type,type,
% 0.19/0.46 eqinunit: $o ).
% 0.19/0.46
% 0.19/0.46 thf(eqinunit,definition,
% 0.19/0.46 ( eqinunit
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.46 ( ( Xx = Xy )
% 0.19/0.46 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(singletonsswitch_type,type,
% 0.19/0.46 singletonsswitch: $o ).
% 0.19/0.46
% 0.19/0.46 thf(singletonsswitch,definition,
% 0.19/0.46 ( singletonsswitch
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.46 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.19/0.46 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(upairsetE_type,type,
% 0.19/0.46 upairsetE: $o ).
% 0.19/0.46
% 0.19/0.46 thf(upairsetE,definition,
% 0.19/0.46 ( upairsetE
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.19/0.46 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.19/0.46 => ( ( Xz = Xx )
% 0.19/0.46 | ( Xz = Xy ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(upairsetIL_type,type,
% 0.19/0.46 upairsetIL: $o ).
% 0.19/0.46
% 0.19/0.46 thf(upairsetIL,definition,
% 0.19/0.46 ( upairsetIL
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(upairsetIR_type,type,
% 0.19/0.46 upairsetIR: $o ).
% 0.19/0.46
% 0.19/0.46 thf(upairsetIR,definition,
% 0.19/0.46 ( upairsetIR
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(emptyE1_type,type,
% 0.19/0.46 emptyE1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(emptyE1,definition,
% 0.19/0.46 ( emptyE1
% 0.19/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.46 ( ? [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 & ( Xphi @ Xx ) )
% 0.19/0.46 => ( ( ( dsetconstr @ A
% 0.19/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.46 = emptyset )
% 0.19/0.46 => $false ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(vacuousDall_type,type,
% 0.19/0.46 vacuousDall: $o ).
% 0.19/0.46
% 0.19/0.46 thf(vacuousDall,definition,
% 0.19/0.46 ( vacuousDall
% 0.19/0.46 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ emptyset )
% 0.19/0.46 => ( Xphi @ Xx ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(quantDeMorgan1_type,type,
% 0.19/0.46 quantDeMorgan1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(quantDeMorgan1,definition,
% 0.19/0.46 ( quantDeMorgan1
% 0.19/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.46 ( ~ ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( Xphi @ Xx ) )
% 0.19/0.46 => ? [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(quantDeMorgan2_type,type,
% 0.19/0.46 quantDeMorgan2: $o ).
% 0.19/0.46
% 0.19/0.46 thf(quantDeMorgan2,definition,
% 0.19/0.46 ( quantDeMorgan2
% 0.19/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.46 ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ~ ( Xphi @ Xx ) )
% 0.19/0.46 => ~ ? [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(quantDeMorgan3_type,type,
% 0.19/0.46 quantDeMorgan3: $o ).
% 0.19/0.46
% 0.19/0.46 thf(quantDeMorgan3,definition,
% 0.19/0.46 ( quantDeMorgan3
% 0.19/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.46 ( ~ ? [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 & ( Xphi @ Xx ) )
% 0.19/0.46 => ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(quantDeMorgan4_type,type,
% 0.19/0.46 quantDeMorgan4: $o ).
% 0.19/0.46
% 0.19/0.46 thf(quantDeMorgan4,definition,
% 0.19/0.46 ( quantDeMorgan4
% 0.19/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.46 ( ? [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 & ~ ( Xphi @ Xx ) )
% 0.19/0.46 => ~ ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(prop2setI_type,type,
% 0.19/0.46 prop2setI: $o ).
% 0.19/0.46
% 0.19/0.46 thf(prop2setI,definition,
% 0.19/0.46 ( prop2setI
% 0.19/0.46 = ( ! [Xphi: $o] :
% 0.19/0.46 ( Xphi
% 0.19/0.46 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(set2prop_type,type,
% 0.19/0.46 set2prop: $i > $o ).
% 0.19/0.46
% 0.19/0.46 thf(prop2set2propI_type,type,
% 0.19/0.46 prop2set2propI: $o ).
% 0.19/0.46
% 0.19/0.46 thf(prop2set2propI,definition,
% 0.19/0.46 ( prop2set2propI
% 0.19/0.46 = ( ! [Xphi: $o] :
% 0.19/0.46 ( Xphi
% 0.19/0.46 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(notdexE_type,type,
% 0.19/0.46 notdexE: $o ).
% 0.19/0.46
% 0.19/0.46 thf(notdexE,definition,
% 0.19/0.46 ( notdexE
% 0.19/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.46 ( ~ ? [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 & ( Xphi @ Xx ) )
% 0.19/0.46 => ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(notdallE_type,type,
% 0.19/0.46 notdallE: $o ).
% 0.19/0.46
% 0.19/0.46 thf(notdallE,definition,
% 0.19/0.46 ( notdallE
% 0.19/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.46 ( ~ ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( Xphi @ Xx ) )
% 0.19/0.46 => ? [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(exuI1_type,type,
% 0.19/0.46 exuI1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(exuI1,definition,
% 0.19/0.46 ( exuI1
% 0.19/0.46 = ( ! [Xphi: $i > $o] :
% 0.19/0.46 ( ? [Xx: $i] :
% 0.19/0.46 ( ( Xphi @ Xx )
% 0.19/0.46 & ! [Xy: $i] :
% 0.19/0.46 ( ( Xphi @ Xy )
% 0.19/0.46 => ( Xx = Xy ) ) )
% 0.19/0.46 => ( exu
% 0.19/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(exuI3_type,type,
% 0.19/0.46 exuI3: $o ).
% 0.19/0.46
% 0.19/0.46 thf(exuI3,definition,
% 0.19/0.46 ( exuI3
% 0.19/0.46 = ( ! [Xphi: $i > $o] :
% 0.19/0.46 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.19/0.46 => ( ! [Xx: $i,Xy: $i] :
% 0.19/0.46 ( ( Xphi @ Xx )
% 0.19/0.46 => ( ( Xphi @ Xy )
% 0.19/0.46 => ( Xx = Xy ) ) )
% 0.19/0.46 => ( exu
% 0.19/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(exuI2_type,type,
% 0.19/0.46 exuI2: $o ).
% 0.19/0.46
% 0.19/0.46 thf(exuI2,definition,
% 0.19/0.46 ( exuI2
% 0.19/0.46 = ( ! [Xphi: $i > $o] :
% 0.19/0.46 ( ? [Xx: $i] :
% 0.19/0.46 ! [Xy: $i] :
% 0.19/0.46 ( ( Xphi @ Xy )
% 0.19/0.46 <=> ( Xy = Xx ) )
% 0.19/0.46 => ( exu
% 0.19/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(inCongP_type,type,
% 0.19/0.46 inCongP: $o ).
% 0.19/0.46
% 0.19/0.46 thf(inCongP,definition,
% 0.19/0.46 ( inCongP
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ( A = B )
% 0.19/0.46 => ! [Xx: $i,Xy: $i] :
% 0.19/0.46 ( ( Xx = Xy )
% 0.19/0.46 => ( ( in @ Xx @ A )
% 0.19/0.46 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(in__Cong_type,type,
% 0.19/0.46 in__Cong: $o ).
% 0.19/0.46
% 0.19/0.46 thf(in__Cong,definition,
% 0.19/0.46 ( in__Cong
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ( A = B )
% 0.19/0.46 => ! [Xx: $i,Xy: $i] :
% 0.19/0.46 ( ( Xx = Xy )
% 0.19/0.46 => ( ( in @ Xx @ A )
% 0.19/0.46 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(exuE3u_type,type,
% 0.19/0.46 exuE3u: $o ).
% 0.19/0.46
% 0.19/0.46 thf(exuE3u,definition,
% 0.19/0.46 ( exuE3u
% 0.19/0.46 = ( ! [Xphi: $i > $o] :
% 0.19/0.46 ( ( exu
% 0.19/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.46 => ! [Xx: $i,Xy: $i] :
% 0.19/0.46 ( ( Xphi @ Xx )
% 0.19/0.46 => ( ( Xphi @ Xy )
% 0.19/0.46 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(exu__Cong_type,type,
% 0.19/0.46 exu__Cong: $o ).
% 0.19/0.46
% 0.19/0.46 thf(exu__Cong,definition,
% 0.19/0.46 ( exu__Cong
% 0.19/0.46 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.19/0.46 ( ! [Xx: $i,Xy: $i] :
% 0.19/0.46 ( ( Xx = Xy )
% 0.19/0.46 => ( ( Xphi @ Xx )
% 0.19/0.46 <=> ( Xpsi @ Xy ) ) )
% 0.19/0.46 => ( ( exu
% 0.19/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.46 <=> ( exu
% 0.19/0.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(emptyset__Cong_type,type,
% 0.19/0.46 emptyset__Cong: $o ).
% 0.19/0.46
% 0.19/0.46 thf(emptyset__Cong,definition,
% 0.19/0.46 ( emptyset__Cong
% 0.19/0.46 = ( emptyset = emptyset ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setadjoin__Cong_type,type,
% 0.19/0.46 setadjoin__Cong: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setadjoin__Cong,definition,
% 0.19/0.46 ( setadjoin__Cong
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.46 ( ( Xx = Xy )
% 0.19/0.46 => ! [Xz: $i,Xu: $i] :
% 0.19/0.46 ( ( Xz = Xu )
% 0.19/0.46 => ( ( setadjoin @ Xx @ Xz )
% 0.19/0.46 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(powerset__Cong_type,type,
% 0.19/0.46 powerset__Cong: $o ).
% 0.19/0.46
% 0.19/0.46 thf(powerset__Cong,definition,
% 0.19/0.46 ( powerset__Cong
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ( A = B )
% 0.19/0.46 => ( ( powerset @ A )
% 0.19/0.46 = ( powerset @ B ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setunion__Cong_type,type,
% 0.19/0.46 setunion__Cong: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setunion__Cong,definition,
% 0.19/0.46 ( setunion__Cong
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ( A = B )
% 0.19/0.46 => ( ( setunion @ A )
% 0.19/0.46 = ( setunion @ B ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(omega__Cong_type,type,
% 0.19/0.46 omega__Cong: $o ).
% 0.19/0.46
% 0.19/0.46 thf(omega__Cong,definition,
% 0.19/0.46 ( omega__Cong
% 0.19/0.46 = ( omega = omega ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(exuEu_type,type,
% 0.19/0.46 exuEu: $o ).
% 0.19/0.46
% 0.19/0.46 thf(exuEu,definition,
% 0.19/0.46 ( exuEu
% 0.19/0.46 = ( ! [Xphi: $i > $o] :
% 0.19/0.46 ( ( exu
% 0.19/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.46 => ! [Xx: $i,Xy: $i] :
% 0.19/0.46 ( ( Xphi @ Xx )
% 0.19/0.46 => ( ( Xphi @ Xy )
% 0.19/0.46 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(descr__Cong_type,type,
% 0.19/0.46 descr__Cong: $o ).
% 0.19/0.46
% 0.19/0.46 thf(descr__Cong,definition,
% 0.19/0.46 ( descr__Cong
% 0.19/0.46 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.19/0.46 ( ! [Xx: $i,Xy: $i] :
% 0.19/0.46 ( ( Xx = Xy )
% 0.19/0.46 => ( ( Xphi @ Xx )
% 0.19/0.46 <=> ( Xpsi @ Xy ) ) )
% 0.19/0.46 => ( ( exu
% 0.19/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.46 => ( ( exu
% 0.19/0.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.19/0.46 => ( ( descr
% 0.19/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.46 = ( descr
% 0.19/0.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(dsetconstr__Cong_type,type,
% 0.19/0.46 dsetconstr__Cong: $o ).
% 0.19/0.46
% 0.19/0.46 thf(dsetconstr__Cong,definition,
% 0.19/0.46 ( dsetconstr__Cong
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ( A = B )
% 0.19/0.46 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.19/0.46 ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ B )
% 0.19/0.46 => ( ( Xx = Xy )
% 0.19/0.46 => ( ( Xphi @ Xx )
% 0.19/0.46 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.19/0.46 => ( ( dsetconstr @ A
% 0.19/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.46 = ( dsetconstr @ B
% 0.19/0.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(subset_type,type,
% 0.19/0.46 subset: $i > $i > $o ).
% 0.19/0.46
% 0.19/0.46 thf(disjoint_type,type,
% 0.19/0.46 disjoint: $i > $i > $o ).
% 0.19/0.46
% 0.19/0.46 thf(setsmeet_type,type,
% 0.19/0.46 setsmeet: $i > $i > $o ).
% 0.19/0.46
% 0.19/0.46 thf(subsetI1_type,type,
% 0.19/0.46 subsetI1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(subsetI1,definition,
% 0.19/0.46 ( subsetI1
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( in @ Xx @ B ) )
% 0.19/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(eqimpsubset2_type,type,
% 0.19/0.46 eqimpsubset2: $o ).
% 0.19/0.46
% 0.19/0.46 thf(eqimpsubset2,definition,
% 0.19/0.46 ( eqimpsubset2
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ( A = B )
% 0.19/0.46 => ( subset @ B @ A ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(eqimpsubset1_type,type,
% 0.19/0.46 eqimpsubset1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(eqimpsubset1,definition,
% 0.19/0.46 ( eqimpsubset1
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ( A = B )
% 0.19/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(subsetI2_type,type,
% 0.19/0.46 subsetI2: $o ).
% 0.19/0.46
% 0.19/0.46 thf(subsetI2,definition,
% 0.19/0.46 ( subsetI2
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( in @ Xx @ B ) )
% 0.19/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(emptysetsubset_type,type,
% 0.19/0.46 emptysetsubset: $o ).
% 0.19/0.46
% 0.19/0.46 thf(emptysetsubset,definition,
% 0.19/0.46 ( emptysetsubset
% 0.19/0.46 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(subsetE_type,type,
% 0.19/0.46 subsetE: $o ).
% 0.19/0.46
% 0.19/0.46 thf(subsetE,definition,
% 0.19/0.46 ( subsetE
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( subset @ A @ B )
% 0.19/0.46 => ( ( in @ Xx @ A )
% 0.19/0.46 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(subsetE2_type,type,
% 0.19/0.46 subsetE2: $o ).
% 0.19/0.46
% 0.19/0.46 thf(subsetE2,definition,
% 0.19/0.46 ( subsetE2
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( subset @ A @ B )
% 0.19/0.46 => ( ~ ( in @ Xx @ B )
% 0.19/0.46 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(notsubsetI_type,type,
% 0.19/0.46 notsubsetI: $o ).
% 0.19/0.46
% 0.19/0.46 thf(notsubsetI,definition,
% 0.19/0.46 ( notsubsetI
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( ~ ( in @ Xx @ B )
% 0.19/0.46 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(notequalI1_type,type,
% 0.19/0.46 notequalI1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(notequalI1,definition,
% 0.19/0.46 ( notequalI1
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ~ ( subset @ A @ B )
% 0.19/0.46 => ( A != B ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(notequalI2_type,type,
% 0.19/0.46 notequalI2: $o ).
% 0.19/0.46
% 0.19/0.46 thf(notequalI2,definition,
% 0.19/0.46 ( notequalI2
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( ~ ( in @ Xx @ B )
% 0.19/0.46 => ( A != B ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(subsetRefl_type,type,
% 0.19/0.46 subsetRefl: $o ).
% 0.19/0.46
% 0.19/0.46 thf(subsetRefl,definition,
% 0.19/0.46 ( subsetRefl
% 0.19/0.46 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(subsetTrans_type,type,
% 0.19/0.46 subsetTrans: $o ).
% 0.19/0.46
% 0.19/0.46 thf(subsetTrans,definition,
% 0.19/0.46 ( subsetTrans
% 0.19/0.46 = ( ! [A: $i,B: $i,C: $i] :
% 0.19/0.46 ( ( subset @ A @ B )
% 0.19/0.46 => ( ( subset @ B @ C )
% 0.19/0.46 => ( subset @ A @ C ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setadjoinSub_type,type,
% 0.19/0.46 setadjoinSub: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setadjoinSub,definition,
% 0.19/0.46 ( setadjoinSub
% 0.19/0.46 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setadjoinSub2_type,type,
% 0.19/0.46 setadjoinSub2: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setadjoinSub2,definition,
% 0.19/0.46 ( setadjoinSub2
% 0.19/0.46 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.19/0.46 ( ( subset @ A @ B )
% 0.19/0.46 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(subset2powerset_type,type,
% 0.19/0.46 subset2powerset: $o ).
% 0.19/0.46
% 0.19/0.46 thf(subset2powerset,definition,
% 0.19/0.46 ( subset2powerset
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ( subset @ A @ B )
% 0.19/0.46 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setextsub_type,type,
% 0.19/0.46 setextsub: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setextsub,definition,
% 0.19/0.46 ( setextsub
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ( subset @ A @ B )
% 0.19/0.46 => ( ( subset @ B @ A )
% 0.19/0.46 => ( A = B ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(subsetemptysetimpeq_type,type,
% 0.19/0.46 subsetemptysetimpeq: $o ).
% 0.19/0.46
% 0.19/0.46 thf(subsetemptysetimpeq,definition,
% 0.19/0.46 ( subsetemptysetimpeq
% 0.19/0.46 = ( ! [A: $i] :
% 0.19/0.46 ( ( subset @ A @ emptyset )
% 0.19/0.46 => ( A = emptyset ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(powersetI1_type,type,
% 0.19/0.46 powersetI1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(powersetI1,definition,
% 0.19/0.46 ( powersetI1
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ( subset @ B @ A )
% 0.19/0.46 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(powersetE1_type,type,
% 0.19/0.46 powersetE1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(powersetE1,definition,
% 0.19/0.46 ( powersetE1
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ( in @ B @ ( powerset @ A ) )
% 0.19/0.46 => ( subset @ B @ A ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(inPowerset_type,type,
% 0.19/0.46 inPowerset: $o ).
% 0.19/0.46
% 0.19/0.46 thf(inPowerset,definition,
% 0.19/0.46 ( inPowerset
% 0.19/0.46 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(powersetsubset_type,type,
% 0.19/0.46 powersetsubset: $o ).
% 0.19/0.46
% 0.19/0.46 thf(powersetsubset,definition,
% 0.19/0.46 ( powersetsubset
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ( subset @ A @ B )
% 0.19/0.46 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(sepInPowerset_type,type,
% 0.19/0.46 sepInPowerset: $o ).
% 0.19/0.46
% 0.19/0.46 thf(sepInPowerset,definition,
% 0.19/0.46 ( sepInPowerset
% 0.19/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.46 ( in
% 0.19/0.46 @ ( dsetconstr @ A
% 0.19/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.46 @ ( powerset @ A ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(sepSubset_type,type,
% 0.19/0.46 sepSubset: $o ).
% 0.19/0.46
% 0.19/0.46 thf(sepSubset,definition,
% 0.19/0.46 ( sepSubset
% 0.19/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.46 ( subset
% 0.19/0.46 @ ( dsetconstr @ A
% 0.19/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.46 @ A ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(binunion_type,type,
% 0.19/0.46 binunion: $i > $i > $i ).
% 0.19/0.46
% 0.19/0.46 thf(binunionIL_type,type,
% 0.19/0.46 binunionIL: $o ).
% 0.19/0.46
% 0.19/0.46 thf(binunionIL,definition,
% 0.19/0.46 ( binunionIL
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(upairset2IR_type,type,
% 0.19/0.46 upairset2IR: $o ).
% 0.19/0.46
% 0.19/0.46 thf(upairset2IR,definition,
% 0.19/0.46 ( upairset2IR
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(binunionIR_type,type,
% 0.19/0.46 binunionIR: $o ).
% 0.19/0.46
% 0.19/0.46 thf(binunionIR,definition,
% 0.19/0.46 ( binunionIR
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ B )
% 0.19/0.46 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(binunionEcases_type,type,
% 0.19/0.46 binunionEcases: $o ).
% 0.19/0.46
% 0.19/0.46 thf(binunionEcases,definition,
% 0.19/0.46 ( binunionEcases
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.19/0.46 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.19/0.46 => ( ( ( in @ Xx @ A )
% 0.19/0.46 => Xphi )
% 0.19/0.46 => ( ( ( in @ Xx @ B )
% 0.19/0.46 => Xphi )
% 0.19/0.46 => Xphi ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(binunionE_type,type,
% 0.19/0.46 binunionE: $o ).
% 0.19/0.46
% 0.19/0.46 thf(binunionE,definition,
% 0.19/0.46 ( binunionE
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.19/0.46 => ( ( in @ Xx @ A )
% 0.19/0.46 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(binunionLsub_type,type,
% 0.19/0.46 binunionLsub: $o ).
% 0.19/0.46
% 0.19/0.46 thf(binunionLsub,definition,
% 0.19/0.46 ( binunionLsub
% 0.19/0.46 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(binunionRsub_type,type,
% 0.19/0.46 binunionRsub: $o ).
% 0.19/0.46
% 0.19/0.46 thf(binunionRsub,definition,
% 0.19/0.46 ( binunionRsub
% 0.19/0.46 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(binintersect_type,type,
% 0.19/0.46 binintersect: $i > $i > $i ).
% 0.19/0.46
% 0.19/0.46 thf(binintersectI_type,type,
% 0.19/0.46 binintersectI: $o ).
% 0.19/0.46
% 0.19/0.46 thf(binintersectI,definition,
% 0.19/0.46 ( binintersectI
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( ( in @ Xx @ B )
% 0.19/0.46 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(binintersectSubset5_type,type,
% 0.19/0.46 binintersectSubset5: $o ).
% 0.19/0.46
% 0.19/0.46 thf(binintersectSubset5,definition,
% 0.19/0.46 ( binintersectSubset5
% 0.19/0.46 = ( ! [A: $i,B: $i,C: $i] :
% 0.19/0.46 ( ( subset @ C @ A )
% 0.19/0.46 => ( ( subset @ C @ B )
% 0.19/0.46 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(binintersectEL_type,type,
% 0.19/0.46 binintersectEL: $o ).
% 0.19/0.46
% 0.19/0.46 thf(binintersectEL,definition,
% 0.19/0.46 ( binintersectEL
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.19/0.46 => ( in @ Xx @ A ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(binintersectLsub_type,type,
% 0.19/0.46 binintersectLsub: $o ).
% 0.19/0.46
% 0.19/0.46 thf(binintersectLsub,definition,
% 0.19/0.46 ( binintersectLsub
% 0.19/0.46 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(binintersectSubset2_type,type,
% 0.19/0.46 binintersectSubset2: $o ).
% 0.19/0.46
% 0.19/0.46 thf(binintersectSubset2,definition,
% 0.19/0.46 ( binintersectSubset2
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ( subset @ A @ B )
% 0.19/0.46 => ( ( binintersect @ A @ B )
% 0.19/0.46 = A ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(binintersectSubset3_type,type,
% 0.19/0.46 binintersectSubset3: $o ).
% 0.19/0.46
% 0.19/0.46 thf(binintersectSubset3,definition,
% 0.19/0.46 ( binintersectSubset3
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ( ( binintersect @ A @ B )
% 0.19/0.46 = B )
% 0.19/0.46 => ( subset @ B @ A ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(binintersectER_type,type,
% 0.19/0.46 binintersectER: $o ).
% 0.19/0.46
% 0.19/0.46 thf(binintersectER,definition,
% 0.19/0.46 ( binintersectER
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.19/0.46 => ( in @ Xx @ B ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(disjointsetsI1_type,type,
% 0.19/0.46 disjointsetsI1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(disjointsetsI1,definition,
% 0.19/0.46 ( disjointsetsI1
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ~ ? [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 & ( in @ Xx @ B ) )
% 0.19/0.46 => ( ( binintersect @ A @ B )
% 0.19/0.46 = emptyset ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(binintersectRsub_type,type,
% 0.19/0.46 binintersectRsub: $o ).
% 0.19/0.46
% 0.19/0.46 thf(binintersectRsub,definition,
% 0.19/0.46 ( binintersectRsub
% 0.19/0.46 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(binintersectSubset4_type,type,
% 0.19/0.46 binintersectSubset4: $o ).
% 0.19/0.46
% 0.19/0.46 thf(binintersectSubset4,definition,
% 0.19/0.46 ( binintersectSubset4
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ( subset @ B @ A )
% 0.19/0.46 => ( ( binintersect @ A @ B )
% 0.19/0.46 = B ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(binintersectSubset1_type,type,
% 0.19/0.46 binintersectSubset1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(binintersectSubset1,definition,
% 0.19/0.46 ( binintersectSubset1
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ( ( binintersect @ A @ B )
% 0.19/0.46 = A )
% 0.19/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(bs114d_type,type,
% 0.19/0.46 bs114d: $o ).
% 0.19/0.46
% 0.19/0.46 thf(bs114d,definition,
% 0.19/0.46 ( bs114d
% 0.19/0.46 = ( ! [A: $i,B: $i,C: $i] :
% 0.19/0.46 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 0.19/0.46 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(regular_type,type,
% 0.19/0.46 regular: $i > $o ).
% 0.19/0.46
% 0.19/0.46 thf(setminus_type,type,
% 0.19/0.46 setminus: $i > $i > $i ).
% 0.19/0.46
% 0.19/0.46 thf(setminusI_type,type,
% 0.19/0.46 setminusI: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setminusI,definition,
% 0.19/0.46 ( setminusI
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( ~ ( in @ Xx @ B )
% 0.19/0.46 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setminusEL_type,type,
% 0.19/0.46 setminusEL: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setminusEL,definition,
% 0.19/0.46 ( setminusEL
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.19/0.46 => ( in @ Xx @ A ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setminusER_type,type,
% 0.19/0.46 setminusER: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setminusER,definition,
% 0.19/0.46 ( setminusER
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.19/0.46 => ~ ( in @ Xx @ B ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setminusSubset2_type,type,
% 0.19/0.46 setminusSubset2: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setminusSubset2,definition,
% 0.19/0.46 ( setminusSubset2
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ( subset @ A @ B )
% 0.19/0.46 => ( ( setminus @ A @ B )
% 0.19/0.46 = emptyset ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setminusERneg_type,type,
% 0.19/0.46 setminusERneg: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setminusERneg,definition,
% 0.19/0.46 ( setminusERneg
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.19/0.46 => ( ( in @ Xx @ A )
% 0.19/0.46 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setminusELneg_type,type,
% 0.19/0.46 setminusELneg: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setminusELneg,definition,
% 0.19/0.46 ( setminusELneg
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.19/0.46 => ( ~ ( in @ Xx @ B )
% 0.19/0.46 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setminusILneg_type,type,
% 0.19/0.46 setminusILneg: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setminusILneg,definition,
% 0.19/0.46 ( setminusILneg
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ~ ( in @ Xx @ A )
% 0.19/0.46 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setminusIRneg_type,type,
% 0.19/0.46 setminusIRneg: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setminusIRneg,definition,
% 0.19/0.46 ( setminusIRneg
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ B )
% 0.19/0.46 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setminusLsub_type,type,
% 0.19/0.46 setminusLsub: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setminusLsub,definition,
% 0.19/0.46 ( setminusLsub
% 0.19/0.46 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setminusSubset1_type,type,
% 0.19/0.46 setminusSubset1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setminusSubset1,definition,
% 0.19/0.46 ( setminusSubset1
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ( ( setminus @ A @ B )
% 0.19/0.46 = emptyset )
% 0.19/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(symdiff_type,type,
% 0.19/0.46 symdiff: $i > $i > $i ).
% 0.19/0.46
% 0.19/0.46 thf(symdiffE_type,type,
% 0.19/0.46 symdiffE: $o ).
% 0.19/0.46
% 0.19/0.46 thf(symdiffE,definition,
% 0.19/0.46 ( symdiffE
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 0.19/0.46 => ! [Xphi: $o] :
% 0.19/0.46 ( ( ( in @ Xx @ A )
% 0.19/0.46 => ( ~ ( in @ Xx @ B )
% 0.19/0.46 => Xphi ) )
% 0.19/0.46 => ( ( ~ ( in @ Xx @ A )
% 0.19/0.46 => ( ( in @ Xx @ B )
% 0.19/0.46 => Xphi ) )
% 0.19/0.46 => Xphi ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(symdiffI1_type,type,
% 0.19/0.46 symdiffI1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(symdiffI1,definition,
% 0.19/0.46 ( symdiffI1
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( ~ ( in @ Xx @ B )
% 0.19/0.46 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(symdiffI2_type,type,
% 0.19/0.46 symdiffI2: $o ).
% 0.19/0.46
% 0.19/0.46 thf(symdiffI2,definition,
% 0.19/0.46 ( symdiffI2
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ~ ( in @ Xx @ A )
% 0.19/0.46 => ( ( in @ Xx @ B )
% 0.19/0.46 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(symdiffIneg1_type,type,
% 0.19/0.46 symdiffIneg1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(symdiffIneg1,definition,
% 0.19/0.46 ( symdiffIneg1
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( ( in @ Xx @ B )
% 0.19/0.46 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(symdiffIneg2_type,type,
% 0.19/0.46 symdiffIneg2: $o ).
% 0.19/0.46
% 0.19/0.46 thf(symdiffIneg2,definition,
% 0.19/0.46 ( symdiffIneg2
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ~ ( in @ Xx @ A )
% 0.19/0.46 => ( ~ ( in @ Xx @ B )
% 0.19/0.46 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(iskpair_type,type,
% 0.19/0.46 iskpair: $i > $o ).
% 0.19/0.46
% 0.19/0.46 thf(iskpair,definition,
% 0.19/0.46 ( iskpair
% 0.19/0.46 = ( ^ [A: $i] :
% 0.19/0.46 ? [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ ( setunion @ A ) )
% 0.19/0.46 & ? [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ ( setunion @ A ) )
% 0.19/0.46 & ( A
% 0.19/0.46 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(secondinupair_type,type,
% 0.19/0.46 secondinupair: $o ).
% 0.19/0.46
% 0.19/0.46 thf(secondinupair,definition,
% 0.19/0.46 ( secondinupair
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setukpairIL_type,type,
% 0.19/0.47 setukpairIL: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setukpairIL,definition,
% 0.19/0.47 ( setukpairIL
% 0.19/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(setukpairIR_type,type,
% 0.19/0.47 setukpairIR: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setukpairIR,definition,
% 0.19/0.47 ( setukpairIR
% 0.19/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(kpairiskpair_type,type,
% 0.19/0.47 kpairiskpair: $o ).
% 0.19/0.47
% 0.19/0.47 thf(kpairiskpair,definition,
% 0.19/0.47 ( kpairiskpair
% 0.19/0.47 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(kpair_type,type,
% 0.19/0.47 kpair: $i > $i > $i ).
% 0.19/0.47
% 0.19/0.47 thf(kpair,definition,
% 0.19/0.47 ( kpair
% 0.19/0.47 = ( ^ [Xx: $i,Xy: $i] : ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(kpairp,conjecture,
% 0.19/0.47 ( setextAx
% 0.19/0.47 => ( emptysetAx
% 0.19/0.47 => ( setadjoinAx
% 0.19/0.47 => ( powersetAx
% 0.19/0.47 => ( setunionAx
% 0.19/0.47 => ( omega0Ax
% 0.19/0.47 => ( omegaSAx
% 0.19/0.47 => ( omegaIndAx
% 0.19/0.47 => ( replAx
% 0.19/0.47 => ( foundationAx
% 0.19/0.47 => ( wellorderingAx
% 0.19/0.47 => ( descrp
% 0.19/0.47 => ( dsetconstrI
% 0.19/0.47 => ( dsetconstrEL
% 0.19/0.47 => ( dsetconstrER
% 0.19/0.47 => ( exuE1
% 0.19/0.47 => ( prop2setE
% 0.19/0.47 => ( emptysetE
% 0.19/0.47 => ( emptysetimpfalse
% 0.19/0.47 => ( notinemptyset
% 0.19/0.47 => ( exuE3e
% 0.19/0.47 => ( setext
% 0.19/0.47 => ( emptyI
% 0.19/0.47 => ( noeltsimpempty
% 0.19/0.47 => ( setbeta
% 0.19/0.47 => ( nonemptyE1
% 0.19/0.47 => ( nonemptyI
% 0.19/0.47 => ( nonemptyI1
% 0.19/0.47 => ( setadjoinIL
% 0.19/0.47 => ( emptyinunitempty
% 0.19/0.47 => ( setadjoinIR
% 0.19/0.47 => ( setadjoinE
% 0.19/0.47 => ( setadjoinOr
% 0.19/0.47 => ( setoftrueEq
% 0.19/0.47 => ( powersetI
% 0.19/0.47 => ( emptyinPowerset
% 0.19/0.47 => ( emptyInPowerset
% 0.19/0.47 => ( powersetE
% 0.19/0.47 => ( setunionI
% 0.19/0.47 => ( setunionE
% 0.19/0.47 => ( subPowSU
% 0.19/0.47 => ( exuE2
% 0.19/0.47 => ( nonemptyImpWitness
% 0.19/0.47 => ( uniqinunit
% 0.19/0.47 => ( notinsingleton
% 0.19/0.47 => ( eqinunit
% 0.19/0.47 => ( singletonsswitch
% 0.19/0.47 => ( upairsetE
% 0.19/0.47 => ( upairsetIL
% 0.19/0.47 => ( upairsetIR
% 0.19/0.47 => ( emptyE1
% 0.19/0.47 => ( vacuousDall
% 0.19/0.47 => ( quantDeMorgan1
% 0.19/0.47 => ( quantDeMorgan2
% 0.19/0.47 => ( quantDeMorgan3
% 0.19/0.47 => ( quantDeMorgan4
% 0.19/0.47 => ( prop2setI
% 0.19/0.47 => ( prop2set2propI
% 0.19/0.47 => ( notdexE
% 0.19/0.47 => ( notdallE
% 0.19/0.47 => ( exuI1
% 0.19/0.47 => ( exuI3
% 0.19/0.47 => ( exuI2
% 0.19/0.47 => ( inCongP
% 0.19/0.47 => ( in__Cong
% 0.19/0.47 => ( exuE3u
% 0.19/0.47 => ( exu__Cong
% 0.19/0.47 => ( emptyset__Cong
% 0.19/0.47 => ( setadjoin__Cong
% 0.19/0.47 => ( powerset__Cong
% 0.19/0.47 => ( setunion__Cong
% 0.19/0.47 => ( omega__Cong
% 0.19/0.47 => ( exuEu
% 0.19/0.47 => ( descr__Cong
% 0.19/0.47 => ( dsetconstr__Cong
% 0.19/0.47 => ( subsetI1
% 0.19/0.47 => ( eqimpsubset2
% 0.19/0.47 => ( eqimpsubset1
% 0.19/0.47 => ( subsetI2
% 0.19/0.47 => ( emptysetsubset
% 0.19/0.47 => ( subsetE
% 0.19/0.47 => ( subsetE2
% 0.19/0.47 => ( notsubsetI
% 0.19/0.47 => ( notequalI1
% 0.19/0.47 => ( notequalI2
% 0.19/0.47 => ( subsetRefl
% 0.19/0.47 => ( subsetTrans
% 0.19/0.47 => ( setadjoinSub
% 0.19/0.47 => ( setadjoinSub2
% 0.19/0.47 => ( subset2powerset
% 0.19/0.47 => ( setextsub
% 0.19/0.47 => ( subsetemptysetimpeq
% 0.19/0.47 => ( powersetI1
% 0.19/0.47 => ( powersetE1
% 0.19/0.47 => ( inPowerset
% 0.19/0.47 => ( powersetsubset
% 0.19/0.47 => ( sepInPowerset
% 0.19/0.47 => ( sepSubset
% 0.19/0.47 => ( binunionIL
% 0.19/0.47 => ( upairset2IR
% 0.19/0.47 => ( binunionIR
% 0.19/0.47 => ( binunionEcases
% 0.19/0.47 => ( binunionE
% 0.19/0.47 => ( binunionLsub
% 0.19/0.47 => ( binunionRsub
% 0.19/0.47 => ( binintersectI
% 0.19/0.47 => ( binintersectSubset5
% 0.19/0.47 => ( binintersectEL
% 0.19/0.47 => ( binintersectLsub
% 0.19/0.47 => ( binintersectSubset2
% 0.19/0.47 => ( binintersectSubset3
% 0.19/0.47 => ( binintersectER
% 0.19/0.47 => ( disjointsetsI1
% 0.19/0.47 => ( binintersectRsub
% 0.19/0.47 => ( binintersectSubset4
% 0.19/0.47 => ( binintersectSubset1
% 0.19/0.47 => ( bs114d
% 0.19/0.47 => ( setminusI
% 0.19/0.47 => ( setminusEL
% 0.19/0.47 => ( setminusER
% 0.19/0.47 => ( setminusSubset2
% 0.19/0.47 => ( setminusERneg
% 0.19/0.47 => ( setminusELneg
% 0.19/0.47 => ( setminusILneg
% 0.19/0.47 => ( setminusIRneg
% 0.19/0.47 => ( setminusLsub
% 0.19/0.47 => ( setminusSubset1
% 0.19/0.47 => ( symdiffE
% 0.19/0.47 => ( symdiffI1
% 0.19/0.47 => ( symdiffI2
% 0.19/0.47 => ( symdiffIneg1
% 0.19/0.47 => ( symdiffIneg2
% 0.19/0.47 => ( secondinupair
% 0.19/0.47 => ( setukpairIL
% 0.19/0.47 => ( setukpairIR
% 0.19/0.47 => ( kpairiskpair
% 0.19/0.47 => ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 %------------------------------------------------------------------------------
% 0.19/0.47 ------------------------
% 2.37/1.45 ---- Embedded in THF ---
% 2.37/1.45 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.37/1.45 %%% Generated on Tue Feb 24 12:28:24 EST 2026
% 2.37/1.45 %%% using '$$dhol' embedding, version 1.3.0.
% 2.37/1.45 %%% Logic specification used:
% 2.37/1.45 %%% thf(spec, logic, $$dhol).
% 2.37/1.45
% 2.37/1.45 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.0dbwHN3pEv/DTF2DTF22343.p
% See solution above
% 2.37/1.45 ------------------------
% 2.37/1.47 ---- Cleaned THF ---
% 2.37/1.47 thf(in_type,type,
% 2.37/1.47 in: $i > $i > $o ).
% 2.37/1.47
% 2.37/1.47 thf(exu_type,type,
% 2.37/1.47 exu: ( $i > $o ) > $o ).
% 2.37/1.47
% 2.37/1.47 thf(exu,definition,
% 2.37/1.47 ( exu
% 2.37/1.47 = ( ^ [Xphi: $i > $o] :
% 2.37/1.47 ? [Xx: $i] :
% 2.37/1.47 ( ( Xphi @ Xx )
% 2.37/1.47 & ! [Xy: $i] :
% 2.37/1.47 ( ( Xphi @ Xy )
% 2.37/1.47 => ( Xx = Xy ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(setextAx_type,type,
% 2.37/1.47 setextAx: $o ).
% 2.37/1.47
% 2.37/1.47 thf(setextAx,definition,
% 2.37/1.47 ( setextAx
% 2.37/1.47 = ( ! [A: $i,B: $i] :
% 2.37/1.47 ( ! [Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ A )
% 2.37/1.47 <=> ( in @ Xx @ B ) )
% 2.37/1.47 => ( A = B ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(emptyset_type,type,
% 2.37/1.47 emptyset: $i ).
% 2.37/1.47
% 2.37/1.47 thf(emptysetAx_type,type,
% 2.37/1.47 emptysetAx: $o ).
% 2.37/1.47
% 2.37/1.47 thf(emptysetAx,definition,
% 2.37/1.47 ( emptysetAx
% 2.37/1.47 = ( ! [Xx: $i] :
% 2.37/1.47 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(setadjoin_type,type,
% 2.37/1.47 setadjoin: $i > $i > $i ).
% 2.37/1.47
% 2.37/1.47 thf(setadjoinAx_type,type,
% 2.37/1.47 setadjoinAx: $o ).
% 2.37/1.47
% 2.37/1.47 thf(setadjoinAx,definition,
% 2.37/1.47 ( setadjoinAx
% 2.37/1.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.37/1.47 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.37/1.47 <=> ( ( Xy = Xx )
% 2.37/1.47 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(powerset_type,type,
% 2.37/1.47 powerset: $i > $i ).
% 2.37/1.47
% 2.37/1.47 thf(powersetAx_type,type,
% 2.37/1.47 powersetAx: $o ).
% 2.37/1.47
% 2.37/1.47 thf(powersetAx,definition,
% 2.37/1.47 ( powersetAx
% 2.37/1.47 = ( ! [A: $i,B: $i] :
% 2.37/1.47 ( ( in @ B @ ( powerset @ A ) )
% 2.37/1.47 <=> ! [Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ B )
% 2.37/1.47 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(setunion_type,type,
% 2.37/1.47 setunion: $i > $i ).
% 2.37/1.47
% 2.37/1.47 thf(setunionAx_type,type,
% 2.37/1.47 setunionAx: $o ).
% 2.37/1.47
% 2.37/1.47 thf(setunionAx,definition,
% 2.37/1.47 ( setunionAx
% 2.37/1.47 = ( ! [A: $i,Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ ( setunion @ A ) )
% 2.37/1.47 <=> ? [B: $i] :
% 2.37/1.47 ( ( in @ Xx @ B )
% 2.37/1.47 & ( in @ B @ A ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(omega_type,type,
% 2.37/1.47 omega: $i ).
% 2.37/1.47
% 2.37/1.47 thf(omega0Ax_type,type,
% 2.37/1.47 omega0Ax: $o ).
% 2.37/1.47
% 2.37/1.47 thf(omega0Ax,definition,
% 2.37/1.47 ( omega0Ax
% 2.37/1.47 = ( in @ emptyset @ omega ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(omegaSAx_type,type,
% 2.37/1.47 omegaSAx: $o ).
% 2.37/1.47
% 2.37/1.47 thf(omegaSAx,definition,
% 2.37/1.47 ( omegaSAx
% 2.37/1.47 = ( ! [Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ omega )
% 2.37/1.47 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(omegaIndAx_type,type,
% 2.37/1.47 omegaIndAx: $o ).
% 2.37/1.47
% 2.37/1.47 thf(omegaIndAx,definition,
% 2.37/1.47 ( omegaIndAx
% 2.37/1.47 = ( ! [A: $i] :
% 2.37/1.47 ( ( ( in @ emptyset @ A )
% 2.37/1.47 & ! [Xx: $i] :
% 2.37/1.47 ( ( ( in @ Xx @ omega )
% 2.37/1.47 & ( in @ Xx @ A ) )
% 2.37/1.47 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 2.37/1.47 => ! [Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ omega )
% 2.37/1.47 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(replAx_type,type,
% 2.37/1.47 replAx: $o ).
% 2.37/1.47
% 2.37/1.47 thf(replAx,definition,
% 2.37/1.47 ( replAx
% 2.37/1.47 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 2.37/1.47 ( ! [Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ A )
% 2.37/1.47 => ( exu
% 2.37/1.47 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 2.37/1.47 => ? [B: $i] :
% 2.37/1.47 ! [Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ B )
% 2.37/1.47 <=> ? [Xy: $i] :
% 2.37/1.47 ( ( in @ Xy @ A )
% 2.37/1.47 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(foundationAx_type,type,
% 2.37/1.47 foundationAx: $o ).
% 2.37/1.47
% 2.37/1.47 thf(foundationAx,definition,
% 2.37/1.47 ( foundationAx
% 2.37/1.47 = ( ! [A: $i] :
% 2.37/1.47 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.37/1.47 => ? [B: $i] :
% 2.37/1.47 ( ( in @ B @ A )
% 2.37/1.47 & ~ ? [Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ B )
% 2.37/1.47 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(wellorderingAx_type,type,
% 2.37/1.47 wellorderingAx: $o ).
% 2.37/1.47
% 2.37/1.47 thf(wellorderingAx,definition,
% 2.37/1.47 ( wellorderingAx
% 2.37/1.47 = ( ! [A: $i] :
% 2.37/1.47 ? [B: $i] :
% 2.37/1.47 ( ! [C: $i] :
% 2.37/1.47 ( ( in @ C @ B )
% 2.37/1.47 => ! [Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ C )
% 2.37/1.47 => ( in @ Xx @ A ) ) )
% 2.37/1.47 & ! [Xx: $i,Xy: $i] :
% 2.37/1.47 ( ( ( in @ Xx @ A )
% 2.37/1.47 & ( in @ Xy @ A ) )
% 2.37/1.47 => ( ! [C: $i] :
% 2.37/1.47 ( ( in @ C @ B )
% 2.37/1.47 => ( ( in @ Xx @ C )
% 2.37/1.47 <=> ( in @ Xy @ C ) ) )
% 2.37/1.47 => ( Xx = Xy ) ) )
% 2.37/1.47 & ! [C: $i,D: $i] :
% 2.37/1.47 ( ( ( in @ C @ B )
% 2.37/1.47 & ( in @ D @ B ) )
% 2.37/1.47 => ( ! [Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ C )
% 2.37/1.47 => ( in @ Xx @ D ) )
% 2.37/1.47 | ! [Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ D )
% 2.37/1.47 => ( in @ Xx @ C ) ) ) )
% 2.37/1.47 & ! [C: $i] :
% 2.37/1.47 ( ( ! [Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ C )
% 2.37/1.47 => ( in @ Xx @ A ) )
% 2.37/1.47 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 2.37/1.47 => ? [D: $i,Xx: $i] :
% 2.37/1.47 ( ( in @ D @ B )
% 2.37/1.47 & ( in @ Xx @ C )
% 2.37/1.47 & ~ ? [Xy: $i] :
% 2.37/1.47 ( ( in @ Xy @ D )
% 2.37/1.47 & ( in @ Xy @ C ) )
% 2.37/1.47 & ! [E: $i] :
% 2.37/1.47 ( ( in @ E @ B )
% 2.37/1.47 => ( ! [Xy: $i] :
% 2.37/1.47 ( ( in @ Xy @ E )
% 2.37/1.47 => ( in @ Xy @ D ) )
% 2.37/1.47 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(descr_type,type,
% 2.37/1.47 descr: ( $i > $o ) > $i ).
% 2.37/1.47
% 2.37/1.47 thf(descrp_type,type,
% 2.37/1.47 descrp: $o ).
% 2.37/1.47
% 2.37/1.47 thf(descrp,definition,
% 2.37/1.47 ( descrp
% 2.37/1.47 = ( ! [Xphi: $i > $o] :
% 2.37/1.47 ( ( exu
% 2.37/1.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.37/1.47 => ( Xphi
% 2.37/1.47 @ ( descr
% 2.37/1.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(dsetconstr_type,type,
% 2.37/1.47 dsetconstr: $i > ( $i > $o ) > $i ).
% 2.37/1.47
% 2.37/1.47 thf(dsetconstrI_type,type,
% 2.37/1.47 dsetconstrI: $o ).
% 2.37/1.47
% 2.37/1.47 thf(dsetconstrI,definition,
% 2.37/1.47 ( dsetconstrI
% 2.37/1.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ A )
% 2.37/1.47 => ( ( Xphi @ Xx )
% 2.37/1.47 => ( in @ Xx
% 2.37/1.47 @ ( dsetconstr @ A
% 2.37/1.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(dsetconstrEL_type,type,
% 2.37/1.47 dsetconstrEL: $o ).
% 2.37/1.47
% 2.37/1.47 thf(dsetconstrEL,definition,
% 2.37/1.47 ( dsetconstrEL
% 2.37/1.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.37/1.47 ( ( in @ Xx
% 2.37/1.47 @ ( dsetconstr @ A
% 2.37/1.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.37/1.47 => ( in @ Xx @ A ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(dsetconstrER_type,type,
% 2.37/1.47 dsetconstrER: $o ).
% 2.37/1.47
% 2.37/1.47 thf(dsetconstrER,definition,
% 2.37/1.47 ( dsetconstrER
% 2.37/1.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.37/1.47 ( ( in @ Xx
% 2.37/1.47 @ ( dsetconstr @ A
% 2.37/1.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.37/1.47 => ( Xphi @ Xx ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(exuE1_type,type,
% 2.37/1.47 exuE1: $o ).
% 2.37/1.47
% 2.37/1.47 thf(exuE1,definition,
% 2.37/1.47 ( exuE1
% 2.37/1.47 = ( ! [Xphi: $i > $o] :
% 2.37/1.47 ( ( exu
% 2.37/1.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.37/1.47 => ? [Xx: $i] :
% 2.37/1.47 ( ( Xphi @ Xx )
% 2.37/1.47 & ! [Xy: $i] :
% 2.37/1.47 ( ( Xphi @ Xy )
% 2.37/1.47 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(prop2set_type,type,
% 2.37/1.47 prop2set: $o > $i ).
% 2.37/1.47
% 2.37/1.47 thf(prop2setE_type,type,
% 2.37/1.47 prop2setE: $o ).
% 2.37/1.47
% 2.37/1.47 thf(prop2setE,definition,
% 2.37/1.47 ( prop2setE
% 2.37/1.47 = ( ! [Xphi: $o,Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 2.37/1.47 => Xphi ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(emptysetE_type,type,
% 2.37/1.47 emptysetE: $o ).
% 2.37/1.47
% 2.37/1.47 thf(emptysetE,definition,
% 2.37/1.47 ( emptysetE
% 2.37/1.47 = ( ! [Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ emptyset )
% 2.37/1.47 => ! [Xphi: $o] : Xphi ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(emptysetimpfalse_type,type,
% 2.37/1.47 emptysetimpfalse: $o ).
% 2.37/1.47
% 2.37/1.47 thf(emptysetimpfalse,definition,
% 2.37/1.47 ( emptysetimpfalse
% 2.37/1.47 = ( ! [Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ emptyset )
% 2.37/1.47 => $false ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(notinemptyset_type,type,
% 2.37/1.47 notinemptyset: $o ).
% 2.37/1.47
% 2.37/1.47 thf(notinemptyset,definition,
% 2.37/1.47 ( notinemptyset
% 2.37/1.47 = ( ! [Xx: $i] :
% 2.37/1.47 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(exuE3e_type,type,
% 2.37/1.47 exuE3e: $o ).
% 2.37/1.47
% 2.37/1.47 thf(exuE3e,definition,
% 2.37/1.47 ( exuE3e
% 2.37/1.47 = ( ! [Xphi: $i > $o] :
% 2.37/1.47 ( ( exu
% 2.37/1.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.37/1.47 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(setext_type,type,
% 2.37/1.47 setext: $o ).
% 2.37/1.47
% 2.37/1.47 thf(setext,definition,
% 2.37/1.47 ( setext
% 2.37/1.47 = ( ! [A: $i,B: $i] :
% 2.37/1.47 ( ! [Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ A )
% 2.37/1.47 => ( in @ Xx @ B ) )
% 2.37/1.47 => ( ! [Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ B )
% 2.37/1.47 => ( in @ Xx @ A ) )
% 2.37/1.47 => ( A = B ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(emptyI_type,type,
% 2.37/1.47 emptyI: $o ).
% 2.37/1.47
% 2.37/1.47 thf(emptyI,definition,
% 2.37/1.47 ( emptyI
% 2.37/1.47 = ( ! [A: $i] :
% 2.37/1.47 ( ! [Xx: $i] :
% 2.37/1.47 ~ ( in @ Xx @ A )
% 2.37/1.47 => ( A = emptyset ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(noeltsimpempty_type,type,
% 2.37/1.47 noeltsimpempty: $o ).
% 2.37/1.47
% 2.37/1.47 thf(noeltsimpempty,definition,
% 2.37/1.47 ( noeltsimpempty
% 2.37/1.47 = ( ! [A: $i] :
% 2.37/1.47 ( ! [Xx: $i] :
% 2.37/1.47 ~ ( in @ Xx @ A )
% 2.37/1.47 => ( A = emptyset ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(setbeta_type,type,
% 2.37/1.47 setbeta: $o ).
% 2.37/1.47
% 2.37/1.47 thf(setbeta,definition,
% 2.37/1.47 ( setbeta
% 2.37/1.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ A )
% 2.37/1.47 => ( ( in @ Xx
% 2.37/1.47 @ ( dsetconstr @ A
% 2.37/1.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.37/1.47 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(nonempty_type,type,
% 2.37/1.47 nonempty: $i > $o ).
% 2.37/1.47
% 2.37/1.47 thf(nonempty,definition,
% 2.37/1.47 ( nonempty
% 2.37/1.47 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(nonemptyE1_type,type,
% 2.37/1.47 nonemptyE1: $o ).
% 2.37/1.47
% 2.37/1.47 thf(nonemptyE1,definition,
% 2.37/1.47 ( nonemptyE1
% 2.37/1.47 = ( ! [A: $i] :
% 2.37/1.47 ( ( nonempty @ A )
% 2.37/1.47 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(nonemptyI_type,type,
% 2.37/1.47 nonemptyI: $o ).
% 2.37/1.47
% 2.37/1.47 thf(nonemptyI,definition,
% 2.37/1.47 ( nonemptyI
% 2.37/1.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ A )
% 2.37/1.47 => ( ( Xphi @ Xx )
% 2.37/1.47 => ( nonempty
% 2.37/1.47 @ ( dsetconstr @ A
% 2.37/1.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(nonemptyI1_type,type,
% 2.37/1.47 nonemptyI1: $o ).
% 2.37/1.47
% 2.37/1.47 thf(nonemptyI1,definition,
% 2.37/1.47 ( nonemptyI1
% 2.37/1.47 = ( ! [A: $i] :
% 2.37/1.47 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.37/1.47 => ( nonempty @ A ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(setadjoinIL_type,type,
% 2.37/1.47 setadjoinIL: $o ).
% 2.37/1.47
% 2.37/1.47 thf(setadjoinIL,definition,
% 2.37/1.47 ( setadjoinIL
% 2.37/1.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(emptyinunitempty_type,type,
% 2.37/1.47 emptyinunitempty: $o ).
% 2.37/1.47
% 2.37/1.47 thf(emptyinunitempty,definition,
% 2.37/1.47 ( emptyinunitempty
% 2.37/1.47 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(setadjoinIR_type,type,
% 2.37/1.47 setadjoinIR: $o ).
% 2.37/1.47
% 2.37/1.47 thf(setadjoinIR,definition,
% 2.37/1.47 ( setadjoinIR
% 2.37/1.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.37/1.47 ( ( in @ Xy @ A )
% 2.37/1.47 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(setadjoinE_type,type,
% 2.37/1.47 setadjoinE: $o ).
% 2.37/1.47
% 2.37/1.47 thf(setadjoinE,definition,
% 2.37/1.47 ( setadjoinE
% 2.37/1.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.37/1.47 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.37/1.47 => ! [Xphi: $o] :
% 2.37/1.47 ( ( ( Xy = Xx )
% 2.37/1.47 => Xphi )
% 2.37/1.47 => ( ( ( in @ Xy @ A )
% 2.37/1.47 => Xphi )
% 2.37/1.47 => Xphi ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(setadjoinOr_type,type,
% 2.37/1.47 setadjoinOr: $o ).
% 2.37/1.47
% 2.37/1.47 thf(setadjoinOr,definition,
% 2.37/1.47 ( setadjoinOr
% 2.37/1.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.37/1.47 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.37/1.47 => ( ( Xy = Xx )
% 2.37/1.47 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(setoftrueEq_type,type,
% 2.37/1.47 setoftrueEq: $o ).
% 2.37/1.47
% 2.37/1.47 thf(setoftrueEq,definition,
% 2.37/1.47 ( setoftrueEq
% 2.37/1.47 = ( ! [A: $i] :
% 2.37/1.47 ( ( dsetconstr @ A
% 2.37/1.47 @ ^ [Xx: $i] : $true )
% 2.37/1.47 = A ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(powersetI_type,type,
% 2.37/1.47 powersetI: $o ).
% 2.37/1.47
% 2.37/1.47 thf(powersetI,definition,
% 2.37/1.47 ( powersetI
% 2.37/1.47 = ( ! [A: $i,B: $i] :
% 2.37/1.47 ( ! [Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ B )
% 2.37/1.47 => ( in @ Xx @ A ) )
% 2.37/1.47 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(emptyinPowerset_type,type,
% 2.37/1.47 emptyinPowerset: $o ).
% 2.37/1.47
% 2.37/1.47 thf(emptyinPowerset,definition,
% 2.37/1.47 ( emptyinPowerset
% 2.37/1.47 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(emptyInPowerset_type,type,
% 2.37/1.47 emptyInPowerset: $o ).
% 2.37/1.47
% 2.37/1.47 thf(emptyInPowerset,definition,
% 2.37/1.47 ( emptyInPowerset
% 2.37/1.47 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(powersetE_type,type,
% 2.37/1.47 powersetE: $o ).
% 2.37/1.47
% 2.37/1.47 thf(powersetE,definition,
% 2.37/1.47 ( powersetE
% 2.37/1.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.37/1.47 ( ( in @ B @ ( powerset @ A ) )
% 2.37/1.47 => ( ( in @ Xx @ B )
% 2.37/1.47 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(setunionI_type,type,
% 2.37/1.47 setunionI: $o ).
% 2.37/1.47
% 2.37/1.47 thf(setunionI,definition,
% 2.37/1.47 ( setunionI
% 2.37/1.47 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.37/1.47 ( ( in @ Xx @ B )
% 2.37/1.47 => ( ( in @ B @ A )
% 2.37/1.47 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(setunionE_type,type,
% 2.37/1.47 setunionE: $o ).
% 2.37/1.47
% 2.37/1.47 thf(setunionE,definition,
% 2.37/1.47 ( setunionE
% 2.37/1.47 = ( ! [A: $i,Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ ( setunion @ A ) )
% 2.37/1.47 => ! [Xphi: $o] :
% 2.37/1.47 ( ! [B: $i] :
% 2.37/1.47 ( ( in @ Xx @ B )
% 2.37/1.47 => ( ( in @ B @ A )
% 2.37/1.47 => Xphi ) )
% 2.37/1.47 => Xphi ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(subPowSU_type,type,
% 2.37/1.47 subPowSU: $o ).
% 2.37/1.47
% 2.37/1.47 thf(subPowSU,definition,
% 2.37/1.47 ( subPowSU
% 2.37/1.47 = ( ! [A: $i,Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ A )
% 2.37/1.47 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(exuE2_type,type,
% 2.37/1.47 exuE2: $o ).
% 2.37/1.47
% 2.37/1.47 thf(exuE2,definition,
% 2.37/1.47 ( exuE2
% 2.37/1.47 = ( ! [Xphi: $i > $o] :
% 2.37/1.47 ( ( exu
% 2.37/1.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.37/1.47 => ? [Xx: $i] :
% 2.37/1.47 ! [Xy: $i] :
% 2.37/1.47 ( ( Xphi @ Xy )
% 2.37/1.47 <=> ( Xy = Xx ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(nonemptyImpWitness_type,type,
% 2.37/1.47 nonemptyImpWitness: $o ).
% 2.37/1.47
% 2.37/1.47 thf(nonemptyImpWitness,definition,
% 2.37/1.47 ( nonemptyImpWitness
% 2.37/1.47 = ( ! [A: $i] :
% 2.37/1.47 ( ( nonempty @ A )
% 2.37/1.47 => ? [Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ A )
% 2.37/1.47 & $true ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(uniqinunit_type,type,
% 2.37/1.47 uniqinunit: $o ).
% 2.37/1.47
% 2.37/1.47 thf(uniqinunit,definition,
% 2.37/1.47 ( uniqinunit
% 2.37/1.47 = ( ! [Xx: $i,Xy: $i] :
% 2.37/1.47 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.37/1.47 => ( Xx = Xy ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(notinsingleton_type,type,
% 2.37/1.47 notinsingleton: $o ).
% 2.37/1.47
% 2.37/1.47 thf(notinsingleton,definition,
% 2.37/1.47 ( notinsingleton
% 2.37/1.47 = ( ! [Xx: $i,Xy: $i] :
% 2.37/1.47 ( ( Xx != Xy )
% 2.37/1.47 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(eqinunit_type,type,
% 2.37/1.47 eqinunit: $o ).
% 2.37/1.47
% 2.37/1.47 thf(eqinunit,definition,
% 2.37/1.47 ( eqinunit
% 2.37/1.47 = ( ! [Xx: $i,Xy: $i] :
% 2.37/1.47 ( ( Xx = Xy )
% 2.37/1.47 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(singletonsswitch_type,type,
% 2.37/1.47 singletonsswitch: $o ).
% 2.37/1.47
% 2.37/1.47 thf(singletonsswitch,definition,
% 2.37/1.47 ( singletonsswitch
% 2.37/1.47 = ( ! [Xx: $i,Xy: $i] :
% 2.37/1.47 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.37/1.47 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(upairsetE_type,type,
% 2.37/1.47 upairsetE: $o ).
% 2.37/1.47
% 2.37/1.47 thf(upairsetE,definition,
% 2.37/1.47 ( upairsetE
% 2.37/1.47 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.37/1.47 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.37/1.47 => ( ( Xz = Xx )
% 2.37/1.47 | ( Xz = Xy ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(upairsetIL_type,type,
% 2.37/1.47 upairsetIL: $o ).
% 2.37/1.47
% 2.37/1.47 thf(upairsetIL,definition,
% 2.37/1.47 ( upairsetIL
% 2.37/1.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(upairsetIR_type,type,
% 2.37/1.47 upairsetIR: $o ).
% 2.37/1.47
% 2.37/1.47 thf(upairsetIR,definition,
% 2.37/1.47 ( upairsetIR
% 2.37/1.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(emptyE1_type,type,
% 2.37/1.47 emptyE1: $o ).
% 2.37/1.47
% 2.37/1.47 thf(emptyE1,definition,
% 2.37/1.47 ( emptyE1
% 2.37/1.47 = ( ! [A: $i,Xphi: $i > $o] :
% 2.37/1.47 ( ? [Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ A )
% 2.37/1.47 & ( Xphi @ Xx ) )
% 2.37/1.47 => ( ( ( dsetconstr @ A
% 2.37/1.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.37/1.47 = emptyset )
% 2.37/1.47 => $false ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(vacuousDall_type,type,
% 2.37/1.47 vacuousDall: $o ).
% 2.37/1.47
% 2.37/1.47 thf(vacuousDall,definition,
% 2.37/1.47 ( vacuousDall
% 2.37/1.47 = ( ! [Xphi: $i > $o,Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ emptyset )
% 2.37/1.47 => ( Xphi @ Xx ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(quantDeMorgan1_type,type,
% 2.37/1.47 quantDeMorgan1: $o ).
% 2.37/1.47
% 2.37/1.47 thf(quantDeMorgan1,definition,
% 2.37/1.47 ( quantDeMorgan1
% 2.37/1.47 = ( ! [A: $i,Xphi: $i > $o] :
% 2.37/1.47 ( ~ ! [Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ A )
% 2.37/1.47 => ( Xphi @ Xx ) )
% 2.37/1.47 => ? [Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ A )
% 2.37/1.47 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(quantDeMorgan2_type,type,
% 2.37/1.47 quantDeMorgan2: $o ).
% 2.37/1.47
% 2.37/1.47 thf(quantDeMorgan2,definition,
% 2.37/1.47 ( quantDeMorgan2
% 2.37/1.47 = ( ! [A: $i,Xphi: $i > $o] :
% 2.37/1.47 ( ! [Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ A )
% 2.37/1.47 => ~ ( Xphi @ Xx ) )
% 2.37/1.47 => ~ ? [Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ A )
% 2.37/1.47 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(quantDeMorgan3_type,type,
% 2.37/1.47 quantDeMorgan3: $o ).
% 2.37/1.47
% 2.37/1.47 thf(quantDeMorgan3,definition,
% 2.37/1.47 ( quantDeMorgan3
% 2.37/1.47 = ( ! [A: $i,Xphi: $i > $o] :
% 2.37/1.47 ( ~ ? [Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ A )
% 2.37/1.47 & ( Xphi @ Xx ) )
% 2.37/1.47 => ! [Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ A )
% 2.37/1.47 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(quantDeMorgan4_type,type,
% 2.37/1.47 quantDeMorgan4: $o ).
% 2.37/1.47
% 2.37/1.47 thf(quantDeMorgan4,definition,
% 2.37/1.47 ( quantDeMorgan4
% 2.37/1.47 = ( ! [A: $i,Xphi: $i > $o] :
% 2.37/1.47 ( ? [Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ A )
% 2.37/1.47 & ~ ( Xphi @ Xx ) )
% 2.37/1.47 => ~ ! [Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ A )
% 2.37/1.47 => ( Xphi @ Xx ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(prop2setI_type,type,
% 2.37/1.47 prop2setI: $o ).
% 2.37/1.47
% 2.37/1.47 thf(prop2setI,definition,
% 2.37/1.47 ( prop2setI
% 2.37/1.47 = ( ! [Xphi: $o] :
% 2.37/1.47 ( Xphi
% 2.37/1.47 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(set2prop_type,type,
% 2.37/1.47 set2prop: $i > $o ).
% 2.37/1.47
% 2.37/1.47 thf(prop2set2propI_type,type,
% 2.37/1.47 prop2set2propI: $o ).
% 2.37/1.47
% 2.37/1.47 thf(prop2set2propI,definition,
% 2.37/1.47 ( prop2set2propI
% 2.37/1.47 = ( ! [Xphi: $o] :
% 2.37/1.47 ( Xphi
% 2.37/1.47 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(notdexE_type,type,
% 2.37/1.47 notdexE: $o ).
% 2.37/1.47
% 2.37/1.47 thf(notdexE,definition,
% 2.37/1.47 ( notdexE
% 2.37/1.47 = ( ! [A: $i,Xphi: $i > $o] :
% 2.37/1.47 ( ~ ? [Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ A )
% 2.37/1.47 & ( Xphi @ Xx ) )
% 2.37/1.47 => ! [Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ A )
% 2.37/1.47 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(notdallE_type,type,
% 2.37/1.47 notdallE: $o ).
% 2.37/1.47
% 2.37/1.47 thf(notdallE,definition,
% 2.37/1.47 ( notdallE
% 2.37/1.47 = ( ! [A: $i,Xphi: $i > $o] :
% 2.37/1.47 ( ~ ! [Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ A )
% 2.37/1.47 => ( Xphi @ Xx ) )
% 2.37/1.47 => ? [Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ A )
% 2.37/1.47 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(exuI1_type,type,
% 2.37/1.47 exuI1: $o ).
% 2.37/1.47
% 2.37/1.47 thf(exuI1,definition,
% 2.37/1.47 ( exuI1
% 2.37/1.47 = ( ! [Xphi: $i > $o] :
% 2.37/1.47 ( ? [Xx: $i] :
% 2.37/1.47 ( ( Xphi @ Xx )
% 2.37/1.47 & ! [Xy: $i] :
% 2.37/1.47 ( ( Xphi @ Xy )
% 2.37/1.47 => ( Xx = Xy ) ) )
% 2.37/1.47 => ( exu
% 2.37/1.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(exuI3_type,type,
% 2.37/1.47 exuI3: $o ).
% 2.37/1.47
% 2.37/1.47 thf(exuI3,definition,
% 2.37/1.47 ( exuI3
% 2.37/1.47 = ( ! [Xphi: $i > $o] :
% 2.37/1.47 ( ? [Xx: $i] : ( Xphi @ Xx )
% 2.37/1.47 => ( ! [Xx: $i,Xy: $i] :
% 2.37/1.47 ( ( Xphi @ Xx )
% 2.37/1.47 => ( ( Xphi @ Xy )
% 2.37/1.47 => ( Xx = Xy ) ) )
% 2.37/1.47 => ( exu
% 2.37/1.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(exuI2_type,type,
% 2.37/1.47 exuI2: $o ).
% 2.37/1.47
% 2.37/1.47 thf(exuI2,definition,
% 2.37/1.47 ( exuI2
% 2.37/1.47 = ( ! [Xphi: $i > $o] :
% 2.37/1.47 ( ? [Xx: $i] :
% 2.37/1.47 ! [Xy: $i] :
% 2.37/1.47 ( ( Xphi @ Xy )
% 2.37/1.47 <=> ( Xy = Xx ) )
% 2.37/1.47 => ( exu
% 2.37/1.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(inCongP_type,type,
% 2.37/1.47 inCongP: $o ).
% 2.37/1.47
% 2.37/1.47 thf(inCongP,definition,
% 2.37/1.47 ( inCongP
% 2.37/1.47 = ( ! [A: $i,B: $i] :
% 2.37/1.47 ( ( A = B )
% 2.37/1.47 => ! [Xx: $i,Xy: $i] :
% 2.37/1.47 ( ( Xx = Xy )
% 2.37/1.47 => ( ( in @ Xx @ A )
% 2.37/1.47 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(in__Cong_type,type,
% 2.37/1.47 in__Cong: $o ).
% 2.37/1.47
% 2.37/1.47 thf(in__Cong,definition,
% 2.37/1.47 ( in__Cong
% 2.37/1.47 = ( ! [A: $i,B: $i] :
% 2.37/1.47 ( ( A = B )
% 2.37/1.47 => ! [Xx: $i,Xy: $i] :
% 2.37/1.47 ( ( Xx = Xy )
% 2.37/1.47 => ( ( in @ Xx @ A )
% 2.37/1.47 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(exuE3u_type,type,
% 2.37/1.47 exuE3u: $o ).
% 2.37/1.47
% 2.37/1.47 thf(exuE3u,definition,
% 2.37/1.47 ( exuE3u
% 2.37/1.47 = ( ! [Xphi: $i > $o] :
% 2.37/1.47 ( ( exu
% 2.37/1.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.37/1.47 => ! [Xx: $i,Xy: $i] :
% 2.37/1.47 ( ( Xphi @ Xx )
% 2.37/1.47 => ( ( Xphi @ Xy )
% 2.37/1.47 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(exu__Cong_type,type,
% 2.37/1.47 exu__Cong: $o ).
% 2.37/1.47
% 2.37/1.47 thf(exu__Cong,definition,
% 2.37/1.47 ( exu__Cong
% 2.37/1.47 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.37/1.47 ( ! [Xx: $i,Xy: $i] :
% 2.37/1.47 ( ( Xx = Xy )
% 2.37/1.47 => ( ( Xphi @ Xx )
% 2.37/1.47 <=> ( Xpsi @ Xy ) ) )
% 2.37/1.47 => ( ( exu
% 2.37/1.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.37/1.47 <=> ( exu
% 2.37/1.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(emptyset__Cong_type,type,
% 2.37/1.47 emptyset__Cong: $o ).
% 2.37/1.47
% 2.37/1.47 thf(emptyset__Cong,definition,
% 2.37/1.47 ( emptyset__Cong
% 2.37/1.47 = ( emptyset = emptyset ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(setadjoin__Cong_type,type,
% 2.37/1.47 setadjoin__Cong: $o ).
% 2.37/1.47
% 2.37/1.47 thf(setadjoin__Cong,definition,
% 2.37/1.47 ( setadjoin__Cong
% 2.37/1.47 = ( ! [Xx: $i,Xy: $i] :
% 2.37/1.47 ( ( Xx = Xy )
% 2.37/1.47 => ! [Xz: $i,Xu: $i] :
% 2.37/1.47 ( ( Xz = Xu )
% 2.37/1.47 => ( ( setadjoin @ Xx @ Xz )
% 2.37/1.47 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(powerset__Cong_type,type,
% 2.37/1.47 powerset__Cong: $o ).
% 2.37/1.47
% 2.37/1.47 thf(powerset__Cong,definition,
% 2.37/1.47 ( powerset__Cong
% 2.37/1.47 = ( ! [A: $i,B: $i] :
% 2.37/1.47 ( ( A = B )
% 2.37/1.47 => ( ( powerset @ A )
% 2.37/1.47 = ( powerset @ B ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(setunion__Cong_type,type,
% 2.37/1.47 setunion__Cong: $o ).
% 2.37/1.47
% 2.37/1.47 thf(setunion__Cong,definition,
% 2.37/1.47 ( setunion__Cong
% 2.37/1.47 = ( ! [A: $i,B: $i] :
% 2.37/1.47 ( ( A = B )
% 2.37/1.47 => ( ( setunion @ A )
% 2.37/1.47 = ( setunion @ B ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(omega__Cong_type,type,
% 2.37/1.47 omega__Cong: $o ).
% 2.37/1.47
% 2.37/1.47 thf(omega__Cong,definition,
% 2.37/1.47 ( omega__Cong
% 2.37/1.47 = ( omega = omega ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(exuEu_type,type,
% 2.37/1.47 exuEu: $o ).
% 2.37/1.47
% 2.37/1.47 thf(exuEu,definition,
% 2.37/1.47 ( exuEu
% 2.37/1.47 = ( ! [Xphi: $i > $o] :
% 2.37/1.47 ( ( exu
% 2.37/1.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.37/1.47 => ! [Xx: $i,Xy: $i] :
% 2.37/1.47 ( ( Xphi @ Xx )
% 2.37/1.47 => ( ( Xphi @ Xy )
% 2.37/1.47 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(descr__Cong_type,type,
% 2.37/1.47 descr__Cong: $o ).
% 2.37/1.47
% 2.37/1.47 thf(descr__Cong,definition,
% 2.37/1.47 ( descr__Cong
% 2.37/1.47 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.37/1.47 ( ! [Xx: $i,Xy: $i] :
% 2.37/1.47 ( ( Xx = Xy )
% 2.37/1.47 => ( ( Xphi @ Xx )
% 2.37/1.47 <=> ( Xpsi @ Xy ) ) )
% 2.37/1.47 => ( ( exu
% 2.37/1.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.37/1.47 => ( ( exu
% 2.37/1.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 2.37/1.47 => ( ( descr
% 2.37/1.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.37/1.47 = ( descr
% 2.37/1.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(dsetconstr__Cong_type,type,
% 2.37/1.47 dsetconstr__Cong: $o ).
% 2.37/1.47
% 2.37/1.47 thf(dsetconstr__Cong,definition,
% 2.37/1.47 ( dsetconstr__Cong
% 2.37/1.47 = ( ! [A: $i,B: $i] :
% 2.37/1.47 ( ( A = B )
% 2.37/1.47 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.37/1.47 ( ! [Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ A )
% 2.37/1.47 => ! [Xy: $i] :
% 2.37/1.47 ( ( in @ Xy @ B )
% 2.37/1.47 => ( ( Xx = Xy )
% 2.37/1.47 => ( ( Xphi @ Xx )
% 2.37/1.47 <=> ( Xpsi @ Xy ) ) ) ) )
% 2.37/1.47 => ( ( dsetconstr @ A
% 2.37/1.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.37/1.47 = ( dsetconstr @ B
% 2.37/1.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(subset_type,type,
% 2.37/1.47 subset: $i > $i > $o ).
% 2.37/1.47
% 2.37/1.47 thf(disjoint_type,type,
% 2.37/1.47 disjoint: $i > $i > $o ).
% 2.37/1.47
% 2.37/1.47 thf(setsmeet_type,type,
% 2.37/1.47 setsmeet: $i > $i > $o ).
% 2.37/1.47
% 2.37/1.47 thf(subsetI1_type,type,
% 2.37/1.47 subsetI1: $o ).
% 2.37/1.47
% 2.37/1.47 thf(subsetI1,definition,
% 2.37/1.47 ( subsetI1
% 2.37/1.47 = ( ! [A: $i,B: $i] :
% 2.37/1.47 ( ! [Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ A )
% 2.37/1.47 => ( in @ Xx @ B ) )
% 2.37/1.47 => ( subset @ A @ B ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(eqimpsubset2_type,type,
% 2.37/1.47 eqimpsubset2: $o ).
% 2.37/1.47
% 2.37/1.47 thf(eqimpsubset2,definition,
% 2.37/1.47 ( eqimpsubset2
% 2.37/1.47 = ( ! [A: $i,B: $i] :
% 2.37/1.47 ( ( A = B )
% 2.37/1.47 => ( subset @ B @ A ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(eqimpsubset1_type,type,
% 2.37/1.47 eqimpsubset1: $o ).
% 2.37/1.47
% 2.37/1.47 thf(eqimpsubset1,definition,
% 2.37/1.47 ( eqimpsubset1
% 2.37/1.47 = ( ! [A: $i,B: $i] :
% 2.37/1.47 ( ( A = B )
% 2.37/1.47 => ( subset @ A @ B ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(subsetI2_type,type,
% 2.37/1.47 subsetI2: $o ).
% 2.37/1.47
% 2.37/1.47 thf(subsetI2,definition,
% 2.37/1.47 ( subsetI2
% 2.37/1.47 = ( ! [A: $i,B: $i] :
% 2.37/1.47 ( ! [Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ A )
% 2.37/1.47 => ( in @ Xx @ B ) )
% 2.37/1.47 => ( subset @ A @ B ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(emptysetsubset_type,type,
% 2.37/1.47 emptysetsubset: $o ).
% 2.37/1.47
% 2.37/1.47 thf(emptysetsubset,definition,
% 2.37/1.47 ( emptysetsubset
% 2.37/1.47 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(subsetE_type,type,
% 2.37/1.47 subsetE: $o ).
% 2.37/1.47
% 2.37/1.47 thf(subsetE,definition,
% 2.37/1.47 ( subsetE
% 2.37/1.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.37/1.47 ( ( subset @ A @ B )
% 2.37/1.47 => ( ( in @ Xx @ A )
% 2.37/1.47 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(subsetE2_type,type,
% 2.37/1.47 subsetE2: $o ).
% 2.37/1.47
% 2.37/1.47 thf(subsetE2,definition,
% 2.37/1.47 ( subsetE2
% 2.37/1.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.37/1.47 ( ( subset @ A @ B )
% 2.37/1.47 => ( ~ ( in @ Xx @ B )
% 2.37/1.47 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(notsubsetI_type,type,
% 2.37/1.47 notsubsetI: $o ).
% 2.37/1.47
% 2.37/1.47 thf(notsubsetI,definition,
% 2.37/1.47 ( notsubsetI
% 2.37/1.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ A )
% 2.37/1.47 => ( ~ ( in @ Xx @ B )
% 2.37/1.47 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(notequalI1_type,type,
% 2.37/1.47 notequalI1: $o ).
% 2.37/1.47
% 2.37/1.47 thf(notequalI1,definition,
% 2.37/1.47 ( notequalI1
% 2.37/1.47 = ( ! [A: $i,B: $i] :
% 2.37/1.47 ( ~ ( subset @ A @ B )
% 2.37/1.47 => ( A != B ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(notequalI2_type,type,
% 2.37/1.47 notequalI2: $o ).
% 2.37/1.47
% 2.37/1.47 thf(notequalI2,definition,
% 2.37/1.47 ( notequalI2
% 2.37/1.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ A )
% 2.37/1.47 => ( ~ ( in @ Xx @ B )
% 2.37/1.47 => ( A != B ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(subsetRefl_type,type,
% 2.37/1.47 subsetRefl: $o ).
% 2.37/1.47
% 2.37/1.47 thf(subsetRefl,definition,
% 2.37/1.47 ( subsetRefl
% 2.37/1.47 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(subsetTrans_type,type,
% 2.37/1.47 subsetTrans: $o ).
% 2.37/1.47
% 2.37/1.47 thf(subsetTrans,definition,
% 2.37/1.47 ( subsetTrans
% 2.37/1.47 = ( ! [A: $i,B: $i,C: $i] :
% 2.37/1.47 ( ( subset @ A @ B )
% 2.37/1.47 => ( ( subset @ B @ C )
% 2.37/1.47 => ( subset @ A @ C ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(setadjoinSub_type,type,
% 2.37/1.47 setadjoinSub: $o ).
% 2.37/1.47
% 2.37/1.47 thf(setadjoinSub,definition,
% 2.37/1.47 ( setadjoinSub
% 2.37/1.47 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(setadjoinSub2_type,type,
% 2.37/1.47 setadjoinSub2: $o ).
% 2.37/1.47
% 2.37/1.47 thf(setadjoinSub2,definition,
% 2.37/1.47 ( setadjoinSub2
% 2.37/1.47 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.37/1.47 ( ( subset @ A @ B )
% 2.37/1.47 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(subset2powerset_type,type,
% 2.37/1.47 subset2powerset: $o ).
% 2.37/1.47
% 2.37/1.47 thf(subset2powerset,definition,
% 2.37/1.47 ( subset2powerset
% 2.37/1.47 = ( ! [A: $i,B: $i] :
% 2.37/1.47 ( ( subset @ A @ B )
% 2.37/1.47 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(setextsub_type,type,
% 2.37/1.47 setextsub: $o ).
% 2.37/1.47
% 2.37/1.47 thf(setextsub,definition,
% 2.37/1.47 ( setextsub
% 2.37/1.47 = ( ! [A: $i,B: $i] :
% 2.37/1.47 ( ( subset @ A @ B )
% 2.37/1.47 => ( ( subset @ B @ A )
% 2.37/1.47 => ( A = B ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(subsetemptysetimpeq_type,type,
% 2.37/1.47 subsetemptysetimpeq: $o ).
% 2.37/1.47
% 2.37/1.47 thf(subsetemptysetimpeq,definition,
% 2.37/1.47 ( subsetemptysetimpeq
% 2.37/1.47 = ( ! [A: $i] :
% 2.37/1.47 ( ( subset @ A @ emptyset )
% 2.37/1.47 => ( A = emptyset ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(powersetI1_type,type,
% 2.37/1.47 powersetI1: $o ).
% 2.37/1.47
% 2.37/1.47 thf(powersetI1,definition,
% 2.37/1.47 ( powersetI1
% 2.37/1.47 = ( ! [A: $i,B: $i] :
% 2.37/1.47 ( ( subset @ B @ A )
% 2.37/1.47 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(powersetE1_type,type,
% 2.37/1.47 powersetE1: $o ).
% 2.37/1.47
% 2.37/1.47 thf(powersetE1,definition,
% 2.37/1.47 ( powersetE1
% 2.37/1.47 = ( ! [A: $i,B: $i] :
% 2.37/1.47 ( ( in @ B @ ( powerset @ A ) )
% 2.37/1.47 => ( subset @ B @ A ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(inPowerset_type,type,
% 2.37/1.47 inPowerset: $o ).
% 2.37/1.47
% 2.37/1.47 thf(inPowerset,definition,
% 2.37/1.47 ( inPowerset
% 2.37/1.47 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(powersetsubset_type,type,
% 2.37/1.47 powersetsubset: $o ).
% 2.37/1.47
% 2.37/1.47 thf(powersetsubset,definition,
% 2.37/1.47 ( powersetsubset
% 2.37/1.47 = ( ! [A: $i,B: $i] :
% 2.37/1.47 ( ( subset @ A @ B )
% 2.37/1.47 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(sepInPowerset_type,type,
% 2.37/1.47 sepInPowerset: $o ).
% 2.37/1.47
% 2.37/1.47 thf(sepInPowerset,definition,
% 2.37/1.47 ( sepInPowerset
% 2.37/1.47 = ( ! [A: $i,Xphi: $i > $o] :
% 2.37/1.47 ( in
% 2.37/1.47 @ ( dsetconstr @ A
% 2.37/1.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.37/1.47 @ ( powerset @ A ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(sepSubset_type,type,
% 2.37/1.47 sepSubset: $o ).
% 2.37/1.47
% 2.37/1.47 thf(sepSubset,definition,
% 2.37/1.47 ( sepSubset
% 2.37/1.47 = ( ! [A: $i,Xphi: $i > $o] :
% 2.37/1.47 ( subset
% 2.37/1.47 @ ( dsetconstr @ A
% 2.37/1.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.37/1.47 @ A ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(binunion_type,type,
% 2.37/1.47 binunion: $i > $i > $i ).
% 2.37/1.47
% 2.37/1.47 thf(binunionIL_type,type,
% 2.37/1.47 binunionIL: $o ).
% 2.37/1.47
% 2.37/1.47 thf(binunionIL,definition,
% 2.37/1.47 ( binunionIL
% 2.37/1.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ A )
% 2.37/1.47 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(upairset2IR_type,type,
% 2.37/1.47 upairset2IR: $o ).
% 2.37/1.47
% 2.37/1.47 thf(upairset2IR,definition,
% 2.37/1.47 ( upairset2IR
% 2.37/1.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(binunionIR_type,type,
% 2.37/1.47 binunionIR: $o ).
% 2.37/1.47
% 2.37/1.47 thf(binunionIR,definition,
% 2.37/1.47 ( binunionIR
% 2.37/1.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ B )
% 2.37/1.47 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(binunionEcases_type,type,
% 2.37/1.47 binunionEcases: $o ).
% 2.37/1.47
% 2.37/1.47 thf(binunionEcases,definition,
% 2.37/1.47 ( binunionEcases
% 2.37/1.47 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 2.37/1.47 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.37/1.47 => ( ( ( in @ Xx @ A )
% 2.37/1.47 => Xphi )
% 2.37/1.47 => ( ( ( in @ Xx @ B )
% 2.37/1.47 => Xphi )
% 2.37/1.47 => Xphi ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(binunionE_type,type,
% 2.37/1.47 binunionE: $o ).
% 2.37/1.47
% 2.37/1.47 thf(binunionE,definition,
% 2.37/1.47 ( binunionE
% 2.37/1.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.37/1.47 => ( ( in @ Xx @ A )
% 2.37/1.47 | ( in @ Xx @ B ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(binunionLsub_type,type,
% 2.37/1.47 binunionLsub: $o ).
% 2.37/1.47
% 2.37/1.47 thf(binunionLsub,definition,
% 2.37/1.47 ( binunionLsub
% 2.37/1.47 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(binunionRsub_type,type,
% 2.37/1.47 binunionRsub: $o ).
% 2.37/1.47
% 2.37/1.47 thf(binunionRsub,definition,
% 2.37/1.47 ( binunionRsub
% 2.37/1.47 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(binintersect_type,type,
% 2.37/1.47 binintersect: $i > $i > $i ).
% 2.37/1.47
% 2.37/1.47 thf(binintersectI_type,type,
% 2.37/1.47 binintersectI: $o ).
% 2.37/1.47
% 2.37/1.47 thf(binintersectI,definition,
% 2.37/1.47 ( binintersectI
% 2.37/1.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ A )
% 2.37/1.47 => ( ( in @ Xx @ B )
% 2.37/1.47 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(binintersectSubset5_type,type,
% 2.37/1.47 binintersectSubset5: $o ).
% 2.37/1.47
% 2.37/1.47 thf(binintersectSubset5,definition,
% 2.37/1.47 ( binintersectSubset5
% 2.37/1.47 = ( ! [A: $i,B: $i,C: $i] :
% 2.37/1.47 ( ( subset @ C @ A )
% 2.37/1.47 => ( ( subset @ C @ B )
% 2.37/1.47 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(binintersectEL_type,type,
% 2.37/1.47 binintersectEL: $o ).
% 2.37/1.47
% 2.37/1.47 thf(binintersectEL,definition,
% 2.37/1.47 ( binintersectEL
% 2.37/1.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.37/1.47 => ( in @ Xx @ A ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(binintersectLsub_type,type,
% 2.37/1.47 binintersectLsub: $o ).
% 2.37/1.47
% 2.37/1.47 thf(binintersectLsub,definition,
% 2.37/1.47 ( binintersectLsub
% 2.37/1.47 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(binintersectSubset2_type,type,
% 2.37/1.47 binintersectSubset2: $o ).
% 2.37/1.47
% 2.37/1.47 thf(binintersectSubset2,definition,
% 2.37/1.47 ( binintersectSubset2
% 2.37/1.47 = ( ! [A: $i,B: $i] :
% 2.37/1.47 ( ( subset @ A @ B )
% 2.37/1.47 => ( ( binintersect @ A @ B )
% 2.37/1.47 = A ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(binintersectSubset3_type,type,
% 2.37/1.47 binintersectSubset3: $o ).
% 2.37/1.47
% 2.37/1.47 thf(binintersectSubset3,definition,
% 2.37/1.47 ( binintersectSubset3
% 2.37/1.47 = ( ! [A: $i,B: $i] :
% 2.37/1.47 ( ( ( binintersect @ A @ B )
% 2.37/1.47 = B )
% 2.37/1.47 => ( subset @ B @ A ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(binintersectER_type,type,
% 2.37/1.47 binintersectER: $o ).
% 2.37/1.47
% 2.37/1.47 thf(binintersectER,definition,
% 2.37/1.47 ( binintersectER
% 2.37/1.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.37/1.47 => ( in @ Xx @ B ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(disjointsetsI1_type,type,
% 2.37/1.47 disjointsetsI1: $o ).
% 2.37/1.47
% 2.37/1.47 thf(disjointsetsI1,definition,
% 2.37/1.47 ( disjointsetsI1
% 2.37/1.47 = ( ! [A: $i,B: $i] :
% 2.37/1.47 ( ~ ? [Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ A )
% 2.37/1.47 & ( in @ Xx @ B ) )
% 2.37/1.47 => ( ( binintersect @ A @ B )
% 2.37/1.47 = emptyset ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(binintersectRsub_type,type,
% 2.37/1.47 binintersectRsub: $o ).
% 2.37/1.47
% 2.37/1.47 thf(binintersectRsub,definition,
% 2.37/1.47 ( binintersectRsub
% 2.37/1.47 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(binintersectSubset4_type,type,
% 2.37/1.47 binintersectSubset4: $o ).
% 2.37/1.47
% 2.37/1.47 thf(binintersectSubset4,definition,
% 2.37/1.47 ( binintersectSubset4
% 2.37/1.47 = ( ! [A: $i,B: $i] :
% 2.37/1.47 ( ( subset @ B @ A )
% 2.37/1.47 => ( ( binintersect @ A @ B )
% 2.37/1.47 = B ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(binintersectSubset1_type,type,
% 2.37/1.47 binintersectSubset1: $o ).
% 2.37/1.47
% 2.37/1.47 thf(binintersectSubset1,definition,
% 2.37/1.47 ( binintersectSubset1
% 2.37/1.47 = ( ! [A: $i,B: $i] :
% 2.37/1.47 ( ( ( binintersect @ A @ B )
% 2.37/1.47 = A )
% 2.37/1.47 => ( subset @ A @ B ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(bs114d_type,type,
% 2.37/1.47 bs114d: $o ).
% 2.37/1.47
% 2.37/1.47 thf(bs114d,definition,
% 2.37/1.47 ( bs114d
% 2.37/1.47 = ( ! [A: $i,B: $i,C: $i] :
% 2.37/1.47 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 2.37/1.47 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(regular_type,type,
% 2.37/1.47 regular: $i > $o ).
% 2.37/1.47
% 2.37/1.47 thf(setminus_type,type,
% 2.37/1.47 setminus: $i > $i > $i ).
% 2.37/1.47
% 2.37/1.47 thf(setminusI_type,type,
% 2.37/1.47 setminusI: $o ).
% 2.37/1.47
% 2.37/1.47 thf(setminusI,definition,
% 2.37/1.47 ( setminusI
% 2.37/1.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ A )
% 2.37/1.47 => ( ~ ( in @ Xx @ B )
% 2.37/1.47 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(setminusEL_type,type,
% 2.37/1.47 setminusEL: $o ).
% 2.37/1.47
% 2.37/1.47 thf(setminusEL,definition,
% 2.37/1.47 ( setminusEL
% 2.37/1.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.37/1.47 => ( in @ Xx @ A ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(setminusER_type,type,
% 2.37/1.47 setminusER: $o ).
% 2.37/1.47
% 2.37/1.47 thf(setminusER,definition,
% 2.37/1.47 ( setminusER
% 2.37/1.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.37/1.47 => ~ ( in @ Xx @ B ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(setminusSubset2_type,type,
% 2.37/1.47 setminusSubset2: $o ).
% 2.37/1.47
% 2.37/1.47 thf(setminusSubset2,definition,
% 2.37/1.47 ( setminusSubset2
% 2.37/1.47 = ( ! [A: $i,B: $i] :
% 2.37/1.47 ( ( subset @ A @ B )
% 2.37/1.47 => ( ( setminus @ A @ B )
% 2.37/1.47 = emptyset ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(setminusERneg_type,type,
% 2.37/1.47 setminusERneg: $o ).
% 2.37/1.47
% 2.37/1.47 thf(setminusERneg,definition,
% 2.37/1.47 ( setminusERneg
% 2.37/1.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.37/1.47 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.37/1.47 => ( ( in @ Xx @ A )
% 2.37/1.47 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(setminusELneg_type,type,
% 2.37/1.47 setminusELneg: $o ).
% 2.37/1.47
% 2.37/1.47 thf(setminusELneg,definition,
% 2.37/1.47 ( setminusELneg
% 2.37/1.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.37/1.47 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.37/1.47 => ( ~ ( in @ Xx @ B )
% 2.37/1.47 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(setminusILneg_type,type,
% 2.37/1.47 setminusILneg: $o ).
% 2.37/1.47
% 2.37/1.47 thf(setminusILneg,definition,
% 2.37/1.47 ( setminusILneg
% 2.37/1.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.37/1.47 ( ~ ( in @ Xx @ A )
% 2.37/1.47 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(setminusIRneg_type,type,
% 2.37/1.47 setminusIRneg: $o ).
% 2.37/1.47
% 2.37/1.47 thf(setminusIRneg,definition,
% 2.37/1.47 ( setminusIRneg
% 2.37/1.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ B )
% 2.37/1.47 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(setminusLsub_type,type,
% 2.37/1.47 setminusLsub: $o ).
% 2.37/1.47
% 2.37/1.47 thf(setminusLsub,definition,
% 2.37/1.47 ( setminusLsub
% 2.37/1.47 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(setminusSubset1_type,type,
% 2.37/1.47 setminusSubset1: $o ).
% 2.37/1.47
% 2.37/1.47 thf(setminusSubset1,definition,
% 2.37/1.47 ( setminusSubset1
% 2.37/1.47 = ( ! [A: $i,B: $i] :
% 2.37/1.47 ( ( ( setminus @ A @ B )
% 2.37/1.47 = emptyset )
% 2.37/1.47 => ( subset @ A @ B ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(symdiff_type,type,
% 2.37/1.47 symdiff: $i > $i > $i ).
% 2.37/1.47
% 2.37/1.47 thf(symdiffE_type,type,
% 2.37/1.47 symdiffE: $o ).
% 2.37/1.47
% 2.37/1.47 thf(symdiffE,definition,
% 2.37/1.47 ( symdiffE
% 2.37/1.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 2.37/1.47 => ! [Xphi: $o] :
% 2.37/1.47 ( ( ( in @ Xx @ A )
% 2.37/1.47 => ( ~ ( in @ Xx @ B )
% 2.37/1.47 => Xphi ) )
% 2.37/1.47 => ( ( ~ ( in @ Xx @ A )
% 2.37/1.47 => ( ( in @ Xx @ B )
% 2.37/1.47 => Xphi ) )
% 2.37/1.47 => Xphi ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(symdiffI1_type,type,
% 2.37/1.47 symdiffI1: $o ).
% 2.37/1.47
% 2.37/1.47 thf(symdiffI1,definition,
% 2.37/1.47 ( symdiffI1
% 2.37/1.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ A )
% 2.37/1.47 => ( ~ ( in @ Xx @ B )
% 2.37/1.47 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(symdiffI2_type,type,
% 2.37/1.47 symdiffI2: $o ).
% 2.37/1.47
% 2.37/1.47 thf(symdiffI2,definition,
% 2.37/1.47 ( symdiffI2
% 2.37/1.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.37/1.47 ( ~ ( in @ Xx @ A )
% 2.37/1.47 => ( ( in @ Xx @ B )
% 2.37/1.47 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(symdiffIneg1_type,type,
% 2.37/1.47 symdiffIneg1: $o ).
% 2.37/1.47
% 2.37/1.47 thf(symdiffIneg1,definition,
% 2.37/1.47 ( symdiffIneg1
% 2.37/1.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ A )
% 2.37/1.47 => ( ( in @ Xx @ B )
% 2.37/1.47 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(symdiffIneg2_type,type,
% 2.37/1.47 symdiffIneg2: $o ).
% 2.37/1.47
% 2.37/1.47 thf(symdiffIneg2,definition,
% 2.37/1.47 ( symdiffIneg2
% 2.37/1.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.37/1.47 ( ~ ( in @ Xx @ A )
% 2.37/1.47 => ( ~ ( in @ Xx @ B )
% 2.37/1.47 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(iskpair_type,type,
% 2.37/1.47 iskpair: $i > $o ).
% 2.37/1.47
% 2.37/1.47 thf(iskpair,definition,
% 2.37/1.47 ( iskpair
% 2.37/1.47 = ( ^ [A: $i] :
% 2.37/1.47 ? [Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ ( setunion @ A ) )
% 2.37/1.47 & ? [Xy: $i] :
% 2.37/1.47 ( ( in @ Xy @ ( setunion @ A ) )
% 2.37/1.47 & ( A
% 2.37/1.47 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(secondinupair_type,type,
% 2.37/1.47 secondinupair: $o ).
% 2.37/1.47
% 2.37/1.47 thf(secondinupair,definition,
% 2.37/1.47 ( secondinupair
% 2.37/1.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(setukpairIL_type,type,
% 2.37/1.47 setukpairIL: $o ).
% 2.37/1.47
% 2.37/1.47 thf(setukpairIL,definition,
% 2.37/1.47 ( setukpairIL
% 2.37/1.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(setukpairIR_type,type,
% 2.37/1.47 setukpairIR: $o ).
% 2.37/1.47
% 2.37/1.47 thf(setukpairIR,definition,
% 2.37/1.47 ( setukpairIR
% 2.37/1.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(kpairiskpair_type,type,
% 2.37/1.47 kpairiskpair: $o ).
% 2.37/1.47
% 2.37/1.47 thf(kpairiskpair,definition,
% 2.37/1.47 ( kpairiskpair
% 2.37/1.47 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(kpair_type,type,
% 2.37/1.47 kpair: $i > $i > $i ).
% 2.37/1.47
% 2.37/1.47 thf(kpair,definition,
% 2.37/1.47 ( kpair
% 2.37/1.47 = ( ^ [Xx: $i,Xy: $i] : ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(kpairp,conjecture,
% 2.37/1.47 ( setextAx
% 2.37/1.47 => ( emptysetAx
% 2.37/1.47 => ( setadjoinAx
% 2.37/1.47 => ( powersetAx
% 2.37/1.47 => ( setunionAx
% 2.37/1.47 => ( omega0Ax
% 2.37/1.47 => ( omegaSAx
% 2.37/1.47 => ( omegaIndAx
% 2.37/1.47 => ( replAx
% 2.37/1.47 => ( foundationAx
% 2.37/1.47 => ( wellorderingAx
% 2.37/1.47 => ( descrp
% 2.37/1.47 => ( dsetconstrI
% 2.37/1.47 => ( dsetconstrEL
% 2.37/1.47 => ( dsetconstrER
% 2.37/1.47 => ( exuE1
% 2.37/1.47 => ( prop2setE
% 2.37/1.47 => ( emptysetE
% 2.37/1.47 => ( emptysetimpfalse
% 2.37/1.47 => ( notinemptyset
% 2.37/1.47 => ( exuE3e
% 2.37/1.47 => ( setext
% 2.37/1.47 => ( emptyI
% 2.37/1.47 => ( noeltsimpempty
% 2.37/1.47 => ( setbeta
% 2.37/1.47 => ( nonemptyE1
% 2.37/1.47 => ( nonemptyI
% 2.37/1.47 => ( nonemptyI1
% 2.37/1.47 => ( setadjoinIL
% 2.37/1.47 => ( emptyinunitempty
% 2.37/1.47 => ( setadjoinIR
% 2.37/1.47 => ( setadjoinE
% 2.37/1.47 => ( setadjoinOr
% 2.37/1.47 => ( setoftrueEq
% 2.37/1.47 => ( powersetI
% 2.37/1.47 => ( emptyinPowerset
% 2.37/1.47 => ( emptyInPowerset
% 2.37/1.47 => ( powersetE
% 2.37/1.47 => ( setunionI
% 2.37/1.47 => ( setunionE
% 2.37/1.47 => ( subPowSU
% 2.37/1.47 => ( exuE2
% 2.37/1.47 => ( nonemptyImpWitness
% 2.37/1.47 => ( uniqinunit
% 2.37/1.47 => ( notinsingleton
% 2.37/1.47 => ( eqinunit
% 2.37/1.47 => ( singletonsswitch
% 2.37/1.47 => ( upairsetE
% 2.37/1.47 => ( upairsetIL
% 2.37/1.47 => ( upairsetIR
% 2.37/1.47 => ( emptyE1
% 2.37/1.47 => ( vacuousDall
% 2.37/1.47 => ( quantDeMorgan1
% 2.37/1.47 => ( quantDeMorgan2
% 2.37/1.47 => ( quantDeMorgan3
% 2.37/1.47 => ( quantDeMorgan4
% 2.37/1.47 => ( prop2setI
% 2.37/1.47 => ( prop2set2propI
% 2.37/1.47 => ( notdexE
% 2.37/1.47 => ( notdallE
% 2.37/1.47 => ( exuI1
% 2.37/1.47 => ( exuI3
% 2.37/1.47 => ( exuI2
% 2.37/1.47 => ( inCongP
% 2.37/1.47 => ( in__Cong
% 2.37/1.47 => ( exuE3u
% 2.37/1.47 => ( exu__Cong
% 2.37/1.47 => ( emptyset__Cong
% 2.37/1.47 => ( setadjoin__Cong
% 2.37/1.47 => ( powerset__Cong
% 2.37/1.47 => ( setunion__Cong
% 2.37/1.47 => ( omega__Cong
% 2.37/1.47 => ( exuEu
% 2.37/1.47 => ( descr__Cong
% 2.37/1.47 => ( dsetconstr__Cong
% 2.37/1.47 => ( subsetI1
% 2.37/1.47 => ( eqimpsubset2
% 2.37/1.47 => ( eqimpsubset1
% 2.37/1.47 => ( subsetI2
% 2.37/1.47 => ( emptysetsubset
% 2.37/1.48 => ( subsetE
% 2.37/1.48 => ( subsetE2
% 2.37/1.48 => ( notsubsetI
% 2.37/1.48 => ( notequalI1
% 2.37/1.48 => ( notequalI2
% 2.37/1.48 => ( subsetRefl
% 2.37/1.48 => ( subsetTrans
% 2.37/1.48 => ( setadjoinSub
% 2.37/1.48 => ( setadjoinSub2
% 2.37/1.48 => ( subset2powerset
% 2.37/1.48 => ( setextsub
% 2.37/1.48 => ( subsetemptysetimpeq
% 2.37/1.48 => ( powersetI1
% 2.37/1.48 => ( powersetE1
% 2.37/1.48 => ( inPowerset
% 2.37/1.48 => ( powersetsubset
% 2.37/1.48 => ( sepInPowerset
% 2.37/1.48 => ( sepSubset
% 2.37/1.48 => ( binunionIL
% 2.37/1.48 => ( upairset2IR
% 2.37/1.48 => ( binunionIR
% 2.37/1.48 => ( binunionEcases
% 2.37/1.48 => ( binunionE
% 2.37/1.48 => ( binunionLsub
% 2.37/1.48 => ( binunionRsub
% 2.37/1.48 => ( binintersectI
% 2.37/1.48 => ( binintersectSubset5
% 2.37/1.48 => ( binintersectEL
% 2.37/1.48 => ( binintersectLsub
% 2.37/1.48 => ( binintersectSubset2
% 2.37/1.48 => ( binintersectSubset3
% 2.37/1.48 => ( binintersectER
% 2.37/1.48 => ( disjointsetsI1
% 2.37/1.48 => ( binintersectRsub
% 2.37/1.48 => ( binintersectSubset4
% 2.37/1.48 => ( binintersectSubset1
% 2.37/1.48 => ( bs114d
% 2.37/1.48 => ( setminusI
% 2.37/1.48 => ( setminusEL
% 2.37/1.48 => ( setminusER
% 2.37/1.48 => ( setminusSubset2
% 2.37/1.48 => ( setminusERneg
% 2.37/1.48 => ( setminusELneg
% 2.37/1.48 => ( setminusILneg
% 2.37/1.48 => ( setminusIRneg
% 2.37/1.48 => ( setminusLsub
% 2.37/1.48 => ( setminusSubset1
% 2.37/1.48 => ( symdiffE
% 2.37/1.48 => ( symdiffI1
% 2.37/1.48 => ( symdiffI2
% 2.37/1.48 => ( symdiffIneg1
% 2.37/1.48 => ( symdiffIneg2
% 2.37/1.48 => ( secondinupair
% 2.37/1.48 => ( setukpairIL
% 2.37/1.48 => ( setukpairIR
% 2.37/1.48 => ( kpairiskpair
% 2.37/1.48 => ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.37/1.48 ------------------------
% 2.37/1.49 % (22524)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_22343 for (2999ds/183Mi)
% 2.37/1.49 % (22526)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_22343 for (2999ds/27Mi)
% 2.37/1.49 % (22527)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_22343 for (2999ds/2Mi)
% 2.37/1.49 % (22525)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_22343 for (2999ds/4Mi)
% 2.37/1.49 % (22528)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_22343 for (2999ds/2Mi)
% 2.37/1.50 % (22527)Instruction limit reached!
% 2.37/1.50 % (22527)------------------------------
% 2.37/1.50 % (22527)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.37/1.50 % (22527)Termination reason: Unknown
% 2.37/1.50 % (22528)Instruction limit reached!
% 2.37/1.50 % (22528)------------------------------
% 2.37/1.50 % (22528)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.37/1.50 % (22527)Termination phase: shuffling
% 2.37/1.50
% 2.37/1.50 % (22527)Memory used [KB]: 1279
% 2.37/1.50 % (22527)Time elapsed: 0.003 s
% 2.37/1.50 % (22527)Instructions burned: 3 (million)
% 2.37/1.50 % (22527)------------------------------
% 2.37/1.50 % (22527)------------------------------
% 2.37/1.50 % (22528)Termination reason: Unknown
% 2.37/1.50 % (22528)Termination phase: shuffling
% 2.37/1.50
% 2.37/1.50 % (22529)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_22343 for (2999ds/275Mi)
% 2.37/1.50 % (22528)Memory used [KB]: 1279
% 2.37/1.50 % (22528)Time elapsed: 0.003 s
% 2.37/1.50 % (22528)Instructions burned: 2 (million)
% 2.37/1.50 % (22528)------------------------------
% 2.37/1.50 % (22528)------------------------------
% 2.37/1.50 % (22525)Instruction limit reached!
% 2.37/1.50 % (22525)------------------------------
% 2.37/1.50 % (22525)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.37/1.50 % (22525)Termination reason: Unknown
% 2.37/1.50 % (22525)Termination phase: shuffling
% 2.37/1.50
% 2.37/1.50 % (22525)Memory used [KB]: 1279
% 2.37/1.50 % (22525)Time elapsed: 0.004 s
% 2.37/1.50 % (22525)Instructions burned: 5 (million)
% 2.37/1.50 % (22525)------------------------------
% 2.37/1.50 % (22525)------------------------------
% 2.53/1.50 % (22526)Instruction limit reached!
% 2.53/1.50 % (22526)------------------------------
% 2.53/1.50 % (22526)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.53/1.50 % (22526)Termination reason: Unknown
% 2.53/1.50 % (22526)Termination phase: Property scanning
% 2.53/1.50
% 2.53/1.50 % (22526)Memory used [KB]: 1663
% 2.53/1.50 % (22526)Time elapsed: 0.010 s
% 2.53/1.50 % (22526)Instructions burned: 28 (million)
% 2.53/1.50 % (22526)------------------------------
% 2.53/1.50 % (22526)------------------------------
% 2.53/1.50 % (22530)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_22343 for (2999ds/18Mi)
% 2.53/1.51 % (22532)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_22343 for (2999ds/3Mi)
% 2.53/1.51 % (22532)Instruction limit reached!
% 2.53/1.51 % (22532)------------------------------
% 2.53/1.51 % (22532)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.53/1.51 % (22532)Termination reason: Unknown
% 2.53/1.51 % (22532)Termination phase: shuffling
% 2.53/1.51
% 2.53/1.51 % (22532)Memory used [KB]: 1279
% 2.53/1.51 % (22532)Time elapsed: 0.003 s
% 2.53/1.51 % (22532)Instructions burned: 3 (million)
% 2.53/1.51 % (22532)------------------------------
% 2.53/1.51 % (22532)------------------------------
% 2.53/1.51 % (22534)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_22343 for (2999ds/15Mi)
% 2.53/1.52 % (22530)Instruction limit reached!
% 2.53/1.52 % (22530)------------------------------
% 2.53/1.52 % (22530)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.53/1.52 % (22530)Termination reason: Unknown
% 2.53/1.52 % (22530)Termination phase: Preprocessing 1
% 2.53/1.52
% 2.53/1.52 % (22530)Memory used [KB]: 1407
% 2.53/1.52 % (22530)Time elapsed: 0.012 s
% 2.53/1.52 % (22530)Instructions burned: 18 (million)
% 2.53/1.52 % (22530)------------------------------
% 2.53/1.52 % (22530)------------------------------
% 2.53/1.52 % (22533)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_22343 for (2999ds/37Mi)
% 2.53/1.52 % (22534)Instruction limit reached!
% 2.53/1.52 % (22534)------------------------------
% 2.53/1.52 % (22534)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.53/1.52 % (22534)Termination reason: Unknown
% 2.53/1.52 % (22534)Termination phase: Property scanning
% 2.53/1.52
% 2.53/1.52 % (22534)Memory used [KB]: 1407
% 2.53/1.52 % (22534)Time elapsed: 0.010 s
% 2.53/1.52 % (22534)Instructions burned: 15 (million)
% 2.53/1.52 % (22534)------------------------------
% 2.53/1.52 % (22534)------------------------------
% 2.53/1.53 % (22536)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_22343 for (2999ds/1041Mi)
% 2.53/1.53 % (22535)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_22343 for (2999ds/3Mi)
% 2.53/1.53 % (22535)Instruction limit reached!
% 2.53/1.53 % (22535)------------------------------
% 2.53/1.53 % (22535)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.53/1.53 % (22535)Termination reason: Unknown
% 2.53/1.53 % (22535)Termination phase: shuffling
% 2.53/1.53
% 2.53/1.53 % (22535)Memory used [KB]: 1279
% 2.53/1.53 % (22535)Time elapsed: 0.003 s
% 2.53/1.53 % (22535)Instructions burned: 3 (million)
% 2.53/1.53 % (22535)------------------------------
% 2.53/1.53 % (22535)------------------------------
% 2.53/1.53 % (22537)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_22343 for (2999ds/7Mi)
% 2.53/1.53 % (22533)Instruction limit reached!
% 2.53/1.53 % (22533)------------------------------
% 2.53/1.53 % (22533)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.53/1.53 % (22533)Termination reason: Unknown
% 2.53/1.53 % (22533)Termination phase: Property scanning
% 2.53/1.53
% 2.53/1.53 % (22533)Memory used [KB]: 1663
% 2.53/1.53 % (22533)Time elapsed: 0.018 s
% 2.53/1.53 % (22533)Instructions burned: 38 (million)
% 2.53/1.53 % (22533)------------------------------
% 2.53/1.53 % (22533)------------------------------
% 2.53/1.54 % (22537)Instruction limit reached!
% 2.53/1.54 % (22537)------------------------------
% 2.53/1.54 % (22537)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.53/1.54 % (22537)Termination reason: Unknown
% 2.53/1.54 % (22537)Termination phase: shuffling
% 2.53/1.54
% 2.53/1.54 % (22537)Memory used [KB]: 1407
% 2.53/1.54 % (22537)Time elapsed: 0.006 s
% 2.53/1.54 % (22537)Instructions burned: 8 (million)
% 2.53/1.54 % (22537)------------------------------
% 2.53/1.54 % (22537)------------------------------
% 2.53/1.54 % (22539)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_22343 for (2999ds/16Mi)
% 2.53/1.54 % (22540)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_22343 for (2999ds/3Mi)
% 2.53/1.55 % (22540)Instruction limit reached!
% 2.53/1.55 % (22540)------------------------------
% 2.53/1.55 % (22540)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.53/1.55 % (22540)Termination reason: Unknown
% 2.53/1.55 % (22540)Termination phase: shuffling
% 2.53/1.55
% 2.53/1.55 % (22540)Memory used [KB]: 1279
% 2.53/1.55 % (22540)Time elapsed: 0.003 s
% 2.53/1.55 % (22540)Instructions burned: 3 (million)
% 2.53/1.55 % (22540)------------------------------
% 2.53/1.55 % (22540)------------------------------
% 2.53/1.55 % (22539)Instruction limit reached!
% 2.53/1.55 % (22539)------------------------------
% 2.53/1.55 % (22539)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.53/1.55 % (22539)Termination reason: Unknown
% 2.53/1.55 % (22539)Termination phase: shuffling
% 2.53/1.55
% 2.53/1.55 % (22539)Memory used [KB]: 1535
% 2.53/1.55 % (22539)Time elapsed: 0.009 s
% 2.53/1.55 % (22539)Instructions burned: 16 (million)
% 2.53/1.55 % (22539)------------------------------
% 2.53/1.55 % (22539)------------------------------
% 2.53/1.55 % (22524)Instruction limit reached!
% 2.53/1.55 % (22524)------------------------------
% 2.53/1.55 % (22524)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.53/1.55 % (22524)Termination reason: Unknown
% 2.53/1.55 % (22524)Termination phase: Saturation
% 2.53/1.55
% 2.53/1.55 % (22524)Memory used [KB]: 7164
% 2.53/1.55 % (22524)Time elapsed: 0.063 s
% 2.53/1.55 % (22524)Instructions burned: 184 (million)
% 2.53/1.55 % (22524)------------------------------
% 2.53/1.55 % (22524)------------------------------
% 2.53/1.55 % (22529)First to succeed.
% 2.53/1.55 % (22541)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_22343 for (2999ds/3Mi)
% 2.53/1.55 % (22529)Refutation found. Thanks to Tanya!
% 2.53/1.55 % SZS status Theorem for DTF2THF_22343
% 2.53/1.55 % SZS output start Proof for DTF2THF_22343
% 2.53/1.55 thf(func_def_0, type, in: $i > $i > $o).
% 2.53/1.55 thf(func_def_1, type, exu: ($i > $o) > $o).
% 2.53/1.55 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 2.53/1.55 thf(func_def_8, type, powerset: $i > $i).
% 2.53/1.55 thf(func_def_10, type, setunion: $i > $i).
% 2.53/1.55 thf(func_def_19, type, descr: ($i > $o) > $i).
% 2.53/1.55 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 2.53/1.55 thf(func_def_26, type, prop2set: $o > $i).
% 2.53/1.55 thf(func_def_36, type, nonempty: $i > $o).
% 2.53/1.55 thf(func_def_69, type, set2prop: $i > $o).
% 2.53/1.55 thf(func_def_88, type, subset: $i > $i > $o).
% 2.53/1.55 thf(func_def_89, type, disjoint: $i > $i > $o).
% 2.53/1.55 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 2.53/1.55 thf(func_def_114, type, binunion: $i > $i > $i).
% 2.53/1.55 thf(func_def_122, type, binintersect: $i > $i > $i).
% 2.53/1.55 thf(func_def_135, type, regular: $i > $o).
% 2.53/1.55 thf(func_def_136, type, setminus: $i > $i > $i).
% 2.53/1.55 thf(func_def_147, type, symdiff: $i > $i > $i).
% 2.53/1.55 thf(func_def_153, type, iskpair: $i > $o).
% 2.53/1.55 thf(func_def_158, type, kpair: $i > $i > $i).
% 2.53/1.55 thf(func_def_171, type, sP0: $i > $i > $o).
% 2.53/1.55 thf(func_def_172, type, sP1: $i > $o).
% 2.53/1.55 thf(func_def_173, type, sP2: $i > $i > $o).
% 2.53/1.55 thf(func_def_174, type, sP3: $i > $i > $o).
% 2.53/1.55 thf(func_def_176, type, sP5: $i > $o > $i > $i > $o).
% 2.53/1.55 thf(func_def_185, type, sK14: $i > $i > $i).
% 2.53/1.55 thf(func_def_189, type, sK18: $i > $o).
% 2.53/1.55 thf(func_def_194, type, sK23: $i > $o).
% 2.53/1.55 thf(func_def_199, type, sK28: $i > $o).
% 2.53/1.55 thf(func_def_200, type, sK29: ($i > $o) > $i).
% 2.53/1.55 thf(func_def_205, type, sK34: $i > $i > $i).
% 2.53/1.55 thf(func_def_209, type, sK38: $i > $i).
% 2.53/1.55 thf(func_def_210, type, sK39: ($i > $o) > ($i > $o) > $i).
% 2.53/1.55 thf(func_def_211, type, sK40: ($i > $o) > ($i > $o) > $i).
% 2.53/1.55 thf(func_def_212, type, sK41: $i > $o).
% 2.53/1.55 thf(func_def_213, type, sK42: $i > $o).
% 2.53/1.55 thf(func_def_214, type, sK43: $i > $o).
% 2.53/1.55 thf(func_def_216, type, sK45: $i > ($i > $o) > $i).
% 2.53/1.55 thf(func_def_219, type, sK48: $i > $o).
% 2.53/1.55 thf(func_def_220, type, sK49: ($i > $o) > $i > $i).
% 2.53/1.55 thf(func_def_222, type, sK51: $i > $o).
% 2.53/1.55 thf(func_def_236, type, sK65: $i > $o).
% 2.53/1.55 thf(func_def_244, type, sK73: $i > $i > $i).
% 2.53/1.55 thf(func_def_245, type, sK74: $i > $i > $i).
% 2.53/1.55 thf(func_def_246, type, sK75: $i > $i > $i).
% 2.53/1.55 thf(func_def_247, type, sK76: $i > $i > $i).
% 2.53/1.55 thf(func_def_248, type, sK77: $i > $i > $i).
% 2.53/1.55 thf(func_def_249, type, sK78: $i > $i > $i > $i).
% 2.53/1.55 thf(func_def_250, type, sK79: $i > $i).
% 2.53/1.55 thf(func_def_251, type, sK80: $i > $i).
% 2.53/1.55 thf(func_def_252, type, sK81: $i > $i).
% 2.53/1.55 thf(func_def_253, type, sK82: $i > $i).
% 2.53/1.55 thf(func_def_254, type, sK83: $i > $i > $i).
% 2.53/1.55 thf(func_def_255, type, sK84: $i > $i > $i > $i).
% 2.53/1.55 thf(func_def_256, type, sK85: $i > $i > $i > $i).
% 2.53/1.55 thf(func_def_257, type, sK86: $i > $i > $i).
% 2.53/1.55 thf(func_def_258, type, sK87: $i > $i > $i).
% 2.53/1.55 thf(func_def_260, type, sK89: $i > $i).
% 2.53/1.55 thf(func_def_261, type, sK90: $i > $i).
% 2.53/1.55 thf(func_def_262, type, sK91: $i > $i).
% 2.53/1.55 thf(func_def_268, type, sK97: ($i > $o) > $i).
% 2.53/1.55 thf(func_def_269, type, sK98: $i > $o).
% 2.53/1.55 thf(func_def_270, type, sK99: $i > $i).
% 2.53/1.55 thf(func_def_271, type, sK100: $i > $o).
% 2.53/1.55 thf(func_def_280, type, sK109: $i > $i > $i).
% 2.53/1.55 thf(func_def_288, type, sK117: $i > ($i > $o) > $i).
% 2.53/1.55 thf(func_def_289, type, sK118: $i > $o).
% 2.53/1.55 thf(func_def_294, type, sK123: $i > $o).
% 2.53/1.55 thf(func_def_308, type, sK137: $i > $i > $i).
% 2.53/1.55 thf(func_def_311, type, sK140: $i > $i > $i).
% 2.53/1.55 thf(func_def_314, type, sK143: $i > $i > $o).
% 2.53/1.55 thf(func_def_316, type, sK145: $i > $i).
% 2.53/1.55 thf(func_def_317, type, sK146: $i > $i).
% 2.53/1.55 thf(func_def_318, type, sK147: $i > ($i > $i > $o) > $i).
% 2.53/1.55 thf(func_def_319, type, sK148: $i > $i > ($i > $i > $o) > $i).
% 2.53/1.55 thf(func_def_320, type, sK149: $i > ($i > $i > $o) > $i).
% 2.53/1.55 thf(func_def_322, type, sK151: $i > $i).
% 2.53/1.55 thf(func_def_324, type, sK153: $i > $i).
% 2.53/1.55 thf(func_def_331, type, sK160: $i > $i).
% 2.53/1.55 thf(func_def_332, type, sK161: ($i > $o) > $i > $i).
% 2.53/1.55 thf(func_def_334, type, sK163: $i > $o).
% 2.53/1.55 thf(func_def_340, type, sK169: $i > $o).
% 2.53/1.55 thf(func_def_359, type, sK188: $i > $i > $i).
% 2.53/1.55 thf(func_def_365, type, sK194: $i > $o).
% 2.53/1.55 thf(func_def_368, type, sK197: $i > ($i > $o) > $i).
% 2.53/1.55 thf(func_def_379, type, sK208: $i > $i).
% 2.53/1.55 thf(func_def_380, type, sK209: $i > $o).
% 2.53/1.55 thf(func_def_385, type, sK214: ($i > $o) > ($i > $o) > $i).
% 2.53/1.55 thf(func_def_386, type, sK215: ($i > $o) > ($i > $o) > $i).
% 2.53/1.55 thf(func_def_387, type, sK216: $i > $o).
% 2.53/1.55 thf(func_def_388, type, sK217: $i > $o).
% 2.53/1.55 thf(func_def_395, type, sK224: $i > $o).
% 2.53/1.55 thf(func_def_398, type, sK227: $i > $i > $i).
% 2.53/1.55 thf(func_def_416, type, sK245: $o > $i > $i > $i).
% 2.53/1.55 thf(func_def_421, type, sK250: $i > $i).
% 2.53/1.55 thf(func_def_431, type, sK260: ($i > $o) > $i).
% 2.53/1.55 thf(func_def_432, type, sK261: ($i > $o) > $i).
% 2.53/1.55 thf(func_def_433, type, sK262: $i > $o).
% 2.53/1.55 thf(func_def_440, type, sK269: $i > $o).
% 2.53/1.55 thf(func_def_459, type, sK288: ($i > $o) > $i).
% 2.53/1.55 thf(func_def_460, type, sK289: $i > $o).
% 2.53/1.55 thf(func_def_461, type, sK290: $i > $i).
% 2.53/1.55 thf(func_def_462, type, sK291: ($i > $o) > $i > $i).
% 2.53/1.55 thf(func_def_464, type, sK293: $i > $o).
% 2.53/1.55 thf(func_def_467, type, sK296: $i > $o).
% 2.53/1.55 thf(func_def_469, type, sK298: ($i > $o) > $i > $i).
% 2.53/1.55 thf(func_def_477, type, sK306: $i > $i).
% 2.53/1.55 thf(func_def_478, type, sK307: ($i > $o) > $i > $i).
% 2.53/1.55 thf(func_def_480, type, sK309: $i > $o).
% 2.53/1.55 thf(func_def_492, type, sK321: $i > $o).
% 2.53/1.55 thf(func_def_493, type, sK322: $i > $o).
% 2.53/1.55 thf(func_def_494, type, sK323: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.53/1.55 thf(func_def_495, type, sK324: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.53/1.55 thf(func_def_505, type, sK334: $i > $o).
% 2.53/1.55 thf(func_def_509, type, sK338: $i > $o).
% 2.53/1.55 thf(func_def_515, type, sK344: $i > $i > $i).
% 2.53/1.55 thf(func_def_516, type, sK345: $i > $i > $i).
% 2.53/1.55 thf(func_def_544, type, ph373: !>[X0: $tType]:(X0)).
% 2.53/1.55 thf(f2620,plain,(
% 2.53/1.55 $false),
% 2.53/1.55 inference(subsumption_resolution,[status(thm)],[f2545,f2558])).
% 2.53/1.55 thf(f2558,plain,(
% 2.53/1.55 ( ! [X2 : $i,X3 : $i] : (((iskpair @ (setadjoin @ (setadjoin @ X2 @ emptyset) @ (setadjoin @ (setadjoin @ X2 @ (setadjoin @ X3 @ emptyset)) @ emptyset))) = $true)) )),
% 2.53/1.55 inference(trivial_inequality_removal,[status(thm)],[f1936])).
% 2.53/1.55 thf(f1936,plain,(
% 2.53/1.55 ( ! [X2 : $i,X3 : $i] : (((iskpair @ (setadjoin @ (setadjoin @ X2 @ emptyset) @ (setadjoin @ (setadjoin @ X2 @ (setadjoin @ X3 @ emptyset)) @ emptyset))) = $true) | ($true != $true)) )),
% 2.53/1.55 inference(definition_unfolding,[status(thm)],[f1277,f1680])).
% 2.53/1.55 thf(f1680,plain,(
% 2.53/1.55 (kpairiskpair = $true)),
% 2.53/1.55 inference(cnf_transformation,[status(thm)],[f1003])).
% 2.53/1.55 thf(f1003,plain,(
% 2.53/1.55 (uniqinunit = $true) & (omega0Ax = $true) & (binintersectSubset3 = $true) & (emptyI = $true) & (exuE3e = $true) & (setminusERneg = $true) & (notinsingleton = $true) & (emptysetimpfalse = $true) & (foundationAx = $true) & (setukpairIL = $true) & (exuEu = $true) & (binunionE = $true) & (setminusEL = $true) & (powersetsubset = $true) & (singletonsswitch = $true) & (symdiffI2 = $true) & (setextsub = $true) & (symdiffE = $true) & (eqimpsubset1 = $true) & (binunionRsub = $true) & (setadjoinIL = $true) & (subsetemptysetimpeq = $true) & (setadjoinSub2 = $true) & (binintersectI = $true) & (binintersectSubset5 = $true) & (sepSubset = $true) & (quantDeMorgan1 = $true) & (kpairiskpair = $true) & (setunionE = $true) & (binunionIL = $true) & (exuI1 = $true) & (setunion__Cong = $true) & (powersetE = $true) & (setukpairIR = $true) & (noeltsimpempty = $true) & (vacuousDall = $true) & (binintersectRsub = $true) & (setadjoinE = $true) & (subsetE2 = $true) & (inPowerset = $true) & (upairsetIR = $true) & (dsetconstrI = $true) & (descr__Cong = $true) & (notequalI2 = $true) & (binunionEcases = $true) & (binintersectLsub = $true) & (subsetRefl = $true) & (emptyinPowerset = $true) & (powerset__Cong = $true) & (setadjoinSub = $true) & (upairset2IR = $true) & (nonemptyI = $true) & (setunionAx = $true) & (emptyinunitempty = $true) & (notdallE = $true) & (setminusSubset1 = $true) & (emptysetE = $true) & (emptysetsubset = $true) & (exuE3u = $true) & (prop2set2propI = $true) & (in__Cong = $true) & (sepInPowerset = $true) & (binunionLsub = $true) & (setoftrueEq = $true) & (emptysetAx = $true) & (wellorderingAx = $true) & (subPowSU = $true) & (binintersectSubset4 = $true) & (powersetAx = $true) & (emptyset__Cong = $true) & (setbeta = $true) & (secondinupair = $true) & (prop2setE = $true) & (exuE1 = $true) & (dsetconstrEL = $true) & (nonemptyImpWitness = $true) & (subsetTrans = $true) & (setminusELneg = $true) & (binintersectEL = $true) & (subset2powerset = $true) & (setext = $true) & (dsetconstr__Cong = $true) & (notequalI1 = $true) & (quantDeMorgan2 = $true) & (nonemptyI1 = $true) & (exu__Cong = $true) & (subsetI2 = $true) & (setminusER = $true) & (disjointsetsI1 = $true) & (nonemptyE1 = $true) & (omegaIndAx = $true) & (replAx = $true) & (prop2setI = $true) & (dsetconstrER = $true) & (binintersectER = $true) & (setadjoin__Cong = $true) & (symdiffI1 = $true) & ((iskpair @ (kpair @ sK237 @ sK238)) != $true) & (setminusI = $true) & (setextAx = $true) & (eqinunit = $true) & (symdiffIneg2 = $true) & (quantDeMorgan3 = $true) & (setadjoinIR = $true) & (powersetE1 = $true) & (upairsetE = $true) & (exuE2 = $true) & (notdexE = $true) & (descrp = $true) & (binunionIR = $true) & (subsetI1 = $true) & (subsetE = $true) & (symdiffIneg1 = $true) & (inCongP = $true) & (emptyE1 = $true) & (notinemptyset = $true) & (setunionI = $true) & (eqimpsubset2 = $true) & (binintersectSubset2 = $true) & (setadjoinAx = $true) & (omega__Cong = $true) & (setminusSubset2 = $true) & (powersetI = $true) & (setminusLsub = $true) & (binintersectSubset1 = $true) & (quantDeMorgan4 = $true) & (exuI3 = $true) & (bs114d = $true) & (exuI2 = $true) & (emptyInPowerset = $true) & (powersetI1 = $true) & (notsubsetI = $true) & (setadjoinOr = $true) & (setminusIRneg = $true) & (upairsetIL = $true) & (omegaSAx = $true) & (setminusILneg = $true)),
% 2.53/1.55 inference(skolemisation,[status(esa),new_symbols(skolem,[sK237,sK238])],[f596,f1002])).
% 2.53/1.55 thf(f1002,plain,(
% 2.53/1.55 ? [X0 : $i,X1 : $i] : ((iskpair @ (kpair @ X0 @ X1)) != $true) => ((iskpair @ (kpair @ sK237 @ sK238)) != $true)),
% 2.53/1.55 introduced(definition,[],[choice_axiom])).
% 2.53/1.55 thf(f596,plain,(
% 2.53/1.55 (uniqinunit = $true) & (omega0Ax = $true) & (binintersectSubset3 = $true) & (emptyI = $true) & (exuE3e = $true) & (setminusERneg = $true) & (notinsingleton = $true) & (emptysetimpfalse = $true) & (foundationAx = $true) & (setukpairIL = $true) & (exuEu = $true) & (binunionE = $true) & (setminusEL = $true) & (powersetsubset = $true) & (singletonsswitch = $true) & (symdiffI2 = $true) & (setextsub = $true) & (symdiffE = $true) & (eqimpsubset1 = $true) & (binunionRsub = $true) & (setadjoinIL = $true) & (subsetemptysetimpeq = $true) & (setadjoinSub2 = $true) & (binintersectI = $true) & (binintersectSubset5 = $true) & (sepSubset = $true) & (quantDeMorgan1 = $true) & (kpairiskpair = $true) & (setunionE = $true) & (binunionIL = $true) & (exuI1 = $true) & (setunion__Cong = $true) & (powersetE = $true) & (setukpairIR = $true) & (noeltsimpempty = $true) & (vacuousDall = $true) & (binintersectRsub = $true) & (setadjoinE = $true) & (subsetE2 = $true) & (inPowerset = $true) & (upairsetIR = $true) & (dsetconstrI = $true) & (descr__Cong = $true) & (notequalI2 = $true) & (binunionEcases = $true) & (binintersectLsub = $true) & (subsetRefl = $true) & (emptyinPowerset = $true) & (powerset__Cong = $true) & (setadjoinSub = $true) & (upairset2IR = $true) & (nonemptyI = $true) & (setunionAx = $true) & (emptyinunitempty = $true) & (notdallE = $true) & (setminusSubset1 = $true) & (emptysetE = $true) & (emptysetsubset = $true) & (exuE3u = $true) & (prop2set2propI = $true) & (in__Cong = $true) & (sepInPowerset = $true) & (binunionLsub = $true) & (setoftrueEq = $true) & (emptysetAx = $true) & (wellorderingAx = $true) & (subPowSU = $true) & (binintersectSubset4 = $true) & (powersetAx = $true) & (emptyset__Cong = $true) & (setbeta = $true) & (secondinupair = $true) & (prop2setE = $true) & (exuE1 = $true) & (dsetconstrEL = $true) & (nonemptyImpWitness = $true) & (subsetTrans = $true) & (setminusELneg = $true) & (binintersectEL = $true) & (subset2powerset = $true) & (setext = $true) & (dsetconstr__Cong = $true) & (notequalI1 = $true) & (quantDeMorgan2 = $true) & (nonemptyI1 = $true) & (exu__Cong = $true) & (subsetI2 = $true) & (setminusER = $true) & (disjointsetsI1 = $true) & (nonemptyE1 = $true) & (omegaIndAx = $true) & (replAx = $true) & (prop2setI = $true) & (dsetconstrER = $true) & (binintersectER = $true) & (setadjoin__Cong = $true) & (symdiffI1 = $true) & ? [X0 : $i,X1 : $i] : ((iskpair @ (kpair @ X0 @ X1)) != $true) & (setminusI = $true) & (setextAx = $true) & (eqinunit = $true) & (symdiffIneg2 = $true) & (quantDeMorgan3 = $true) & (setadjoinIR = $true) & (powersetE1 = $true) & (upairsetE = $true) & (exuE2 = $true) & (notdexE = $true) & (descrp = $true) & (binunionIR = $true) & (subsetI1 = $true) & (subsetE = $true) & (symdiffIneg1 = $true) & (inCongP = $true) & (emptyE1 = $true) & (notinemptyset = $true) & (setunionI = $true) & (eqimpsubset2 = $true) & (binintersectSubset2 = $true) & (setadjoinAx = $true) & (omega__Cong = $true) & (setminusSubset2 = $true) & (powersetI = $true) & (setminusLsub = $true) & (binintersectSubset1 = $true) & (quantDeMorgan4 = $true) & (exuI3 = $true) & (bs114d = $true) & (exuI2 = $true) & (emptyInPowerset = $true) & (powersetI1 = $true) & (notsubsetI = $true) & (setadjoinOr = $true) & (setminusIRneg = $true) & (upairsetIL = $true) & (omegaSAx = $true) & (setminusILneg = $true)),
% 2.53/1.55 inference(flattening,[status(thm)],[f595])).
% 2.53/1.55 thf(f595,plain,(
% 2.53/1.55 (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i,X1 : $i] : ((iskpair @ (kpair @ X0 @ X1)) != $true) & (kpairiskpair = $true)) & (setukpairIR = $true)) & (setukpairIL = $true)) & (secondinupair = $true)) & (symdiffIneg2 = $true)) & (symdiffIneg1 = $true)) & (symdiffI2 = $true)) & (symdiffI1 = $true)) & (symdiffE = $true)) & (setminusSubset1 = $true)) & (setminusLsub = $true)) & (setminusIRneg = $true)) & (setminusILneg = $true)) & (setminusELneg = $true)) & (setminusERneg = $true)) & (setminusSubset2 = $true)) & (setminusER = $true)) & (setminusEL = $true)) & (setminusI = $true)) & (bs114d = $true)) & (binintersectSubset1 = $true)) & (binintersectSubset4 = $true)) & (binintersectRsub = $true)) & (disjointsetsI1 = $true)) & (binintersectER = $true)) & (binintersectSubset3 = $true)) & (binintersectSubset2 = $true)) & (binintersectLsub = $true)) & (binintersectEL = $true)) & (binintersectSubset5 = $true)) & (binintersectI = $true)) & (binunionRsub = $true)) & (binunionLsub = $true)) & (binunionE = $true)) & (binunionEcases = $true)) & (binunionIR = $true)) & (upairset2IR = $true)) & (binunionIL = $true)) & (sepSubset = $true)) & (sepInPowerset = $true)) & (powersetsubset = $true)) & (inPowerset = $true)) & (powersetE1 = $true)) & (powersetI1 = $true)) & (subsetemptysetimpeq = $true)) & (setextsub = $true)) & (subset2powerset = $true)) & (setadjoinSub2 = $true)) & (setadjoinSub = $true)) & (subsetTrans = $true)) & (subsetRefl = $true)) & (notequalI2 = $true)) & (notequalI1 = $true)) & (notsubsetI = $true)) & (subsetE2 = $true)) & (subsetE = $true)) & (emptysetsubset = $true)) & (subsetI2 = $true)) & (eqimpsubset1 = $true)) & (eqimpsubset2 = $true)) & (subsetI1 = $true)) & (dsetconstr__Cong = $true)) & (descr__Cong = $true)) & (exuEu = $true)) & (omega__Cong = $true)) & (setunion__Cong = $true)) & (powerset__Cong = $true)) & (setadjoin__Cong = $true)) & (emptyset__Cong = $true)) & (exu__Cong = $true)) & (exuE3u = $true)) & (in__Cong = $true)) & (inCongP = $true)) & (exuI2 = $true)) & (exuI3 = $true)) & (exuI1 = $true)) & (notdallE = $true)) & (notdexE = $true)) & (prop2set2propI = $true)) & (prop2setI = $true)) & (quantDeMorgan4 = $true)) & (quantDeMorgan3 = $true)) & (quantDeMorgan2 = $true)) & (quantDeMorgan1 = $true)) & (vacuousDall = $true)) & (emptyE1 = $true)) & (upairsetIR = $true)) & (upairsetIL = $true)) & (upairsetE = $true)) & (singletonsswitch = $true)) & (eqinunit = $true)) & (notinsingleton = $true)) & (uniqinunit = $true)) & (nonemptyImpWitness = $true)) & (exuE2 = $true)) & (subPowSU = $true)) & (setunionE = $true)) & (setunionI = $true)) & (powersetE = $true)) & (emptyInPowerset = $true)) & (emptyinPowerset = $true)) & (powersetI = $true)) & (setoftrueEq = $true)) & (setadjoinOr = $true)) & (setadjoinE = $true)) & (setadjoinIR = $true)) & (emptyinunitempty = $true)) & (setadjoinIL = $true)) & (nonemptyI1 = $true)) & (nonemptyI = $true)) & (nonemptyE1 = $true)) & (setbeta = $true)) & (noeltsimpempty = $true)) & (emptyI = $true)) & (setext = $true)) & (exuE3e = $true)) & (notinemptyset = $true)) & (emptysetimpfalse = $true)) & (emptysetE = $true)) & (prop2setE = $true)) & (exuE1 = $true)) & (dsetconstrER = $true)) & (dsetconstrEL = $true)) & (dsetconstrI = $true)) & (descrp = $true)) & (wellorderingAx = $true)) & (foundationAx = $true)) & (replAx = $true)) & (omegaIndAx = $true)) & (omegaSAx = $true)) & (omega0Ax = $true)) & (setunionAx = $true)) & (powersetAx = $true)) & (setadjoinAx = $true)) & (emptysetAx = $true)) & (setextAx = $true)),
% 2.53/1.55 inference(ennf_transformation,[status(thm)],[f358])).
% 2.53/1.55 thf(f358,plain,(
% 2.53/1.55 ~((setextAx = $true) => ((emptysetAx = $true) => ((setadjoinAx = $true) => ((powersetAx = $true) => ((setunionAx = $true) => ((omega0Ax = $true) => ((omegaSAx = $true) => ((omegaIndAx = $true) => ((replAx = $true) => ((foundationAx = $true) => ((wellorderingAx = $true) => ((descrp = $true) => ((dsetconstrI = $true) => ((dsetconstrEL = $true) => ((dsetconstrER = $true) => ((exuE1 = $true) => ((prop2setE = $true) => ((emptysetE = $true) => ((emptysetimpfalse = $true) => ((notinemptyset = $true) => ((exuE3e = $true) => ((setext = $true) => ((emptyI = $true) => ((noeltsimpempty = $true) => ((setbeta = $true) => ((nonemptyE1 = $true) => ((nonemptyI = $true) => ((nonemptyI1 = $true) => ((setadjoinIL = $true) => ((emptyinunitempty = $true) => ((setadjoinIR = $true) => ((setadjoinE = $true) => ((setadjoinOr = $true) => ((setoftrueEq = $true) => ((powersetI = $true) => ((emptyinPowerset = $true) => ((emptyInPowerset = $true) => ((powersetE = $true) => ((setunionI = $true) => ((setunionE = $true) => ((subPowSU = $true) => ((exuE2 = $true) => ((nonemptyImpWitness = $true) => ((uniqinunit = $true) => ((notinsingleton = $true) => ((eqinunit = $true) => ((singletonsswitch = $true) => ((upairsetE = $true) => ((upairsetIL = $true) => ((upairsetIR = $true) => ((emptyE1 = $true) => ((vacuousDall = $true) => ((quantDeMorgan1 = $true) => ((quantDeMorgan2 = $true) => ((quantDeMorgan3 = $true) => ((quantDeMorgan4 = $true) => ((prop2setI = $true) => ((prop2set2propI = $true) => ((notdexE = $true) => ((notdallE = $true) => ((exuI1 = $true) => ((exuI3 = $true) => ((exuI2 = $true) => ((inCongP = $true) => ((in__Cong = $true) => ((exuE3u = $true) => ((exu__Cong = $true) => ((emptyset__Cong = $true) => ((setadjoin__Cong = $true) => ((powerset__Cong = $true) => ((setunion__Cong = $true) => ((omega__Cong = $true) => ((exuEu = $true) => ((descr__Cong = $true) => ((dsetconstr__Cong = $true) => ((subsetI1 = $true) => ((eqimpsubset2 = $true) => ((eqimpsubset1 = $true) => ((subsetI2 = $true) => ((emptysetsubset = $true) => ((subsetE = $true) => ((subsetE2 = $true) => ((notsubsetI = $true) => ((notequalI1 = $true) => ((notequalI2 = $true) => ((subsetRefl = $true) => ((subsetTrans = $true) => ((setadjoinSub = $true) => ((setadjoinSub2 = $true) => ((subset2powerset = $true) => ((setextsub = $true) => ((subsetemptysetimpeq = $true) => ((powersetI1 = $true) => ((powersetE1 = $true) => ((inPowerset = $true) => ((powersetsubset = $true) => ((sepInPowerset = $true) => ((sepSubset = $true) => ((binunionIL = $true) => ((upairset2IR = $true) => ((binunionIR = $true) => ((binunionEcases = $true) => ((binunionE = $true) => ((binunionLsub = $true) => ((binunionRsub = $true) => ((binintersectI = $true) => ((binintersectSubset5 = $true) => ((binintersectEL = $true) => ((binintersectLsub = $true) => ((binintersectSubset2 = $true) => ((binintersectSubset3 = $true) => ((binintersectER = $true) => ((disjointsetsI1 = $true) => ((binintersectRsub = $true) => ((binintersectSubset4 = $true) => ((binintersectSubset1 = $true) => ((bs114d = $true) => ((setminusI = $true) => ((setminusEL = $true) => ((setminusER = $true) => ((setminusSubset2 = $true) => ((setminusERneg = $true) => ((setminusELneg = $true) => ((setminusILneg = $true) => ((setminusIRneg = $true) => ((setminusLsub = $true) => ((setminusSubset1 = $true) => ((symdiffE = $true) => ((symdiffI1 = $true) => ((symdiffI2 = $true) => ((symdiffIneg1 = $true) => ((symdiffIneg2 = $true) => ((secondinupair = $true) => ((setukpairIL = $true) => ((setukpairIR = $true) => ((kpairiskpair = $true) => ! [X0 : $i,X1 : $i] : ((iskpair @ (kpair @ X0 @ X1)) = $true)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.53/1.55 inference(fool_elimination,[status(thm)],[f357])).
% 2.53/1.55 thf(f357,plain,(
% 2.53/1.55 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => ! [X0 : $i,X1 : $i] : (iskpair @ (kpair @ X0 @ X1))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.53/1.55 inference(rectify,[status(thm)],[f142])).
% 2.53/1.55 thf(f142,negated_conjecture,(
% 2.53/1.55 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => ! [X1 : $i,X2 : $i] : (iskpair @ (kpair @ X1 @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.53/1.55 inference(negated_conjecture,[status(cth)],[f141])).
% 2.53/1.55 thf(f141,conjecture,(
% 2.53/1.55 setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => ! [X1 : $i,X2 : $i] : (iskpair @ (kpair @ X1 @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.53/1.55 file('/export/starexec/sandbox2/tmp/tmp.0dbwHN3pEv/DTF2THF_22343.p',kpairp)).
% 2.53/1.55 thf(f1277,plain,(
% 2.53/1.55 ( ! [X2 : $i,X3 : $i] : (((iskpair @ (setadjoin @ (setadjoin @ X2 @ emptyset) @ (setadjoin @ (setadjoin @ X2 @ (setadjoin @ X3 @ emptyset)) @ emptyset))) = $true) | (kpairiskpair != $true)) )),
% 2.53/1.55 inference(cnf_transformation,[status(thm)],[f663])).
% 2.53/1.55 thf(f663,plain,(
% 2.53/1.55 ((kpairiskpair = $true) | ((iskpair @ (setadjoin @ (setadjoin @ sK35 @ emptyset) @ (setadjoin @ (setadjoin @ sK35 @ (setadjoin @ sK36 @ emptyset)) @ emptyset))) != $true)) & (! [X2 : $i,X3 : $i] : ((iskpair @ (setadjoin @ (setadjoin @ X2 @ emptyset) @ (setadjoin @ (setadjoin @ X2 @ (setadjoin @ X3 @ emptyset)) @ emptyset))) = $true) | (kpairiskpair != $true))),
% 2.53/1.55 inference(skolemisation,[status(esa),new_symbols(skolem,[sK35,sK36])],[f661,f662])).
% 2.53/1.55 thf(f662,plain,(
% 2.53/1.55 ? [X0 : $i,X1 : $i] : ((iskpair @ (setadjoin @ (setadjoin @ X0 @ emptyset) @ (setadjoin @ (setadjoin @ X0 @ (setadjoin @ X1 @ emptyset)) @ emptyset))) != $true) => ((iskpair @ (setadjoin @ (setadjoin @ sK35 @ emptyset) @ (setadjoin @ (setadjoin @ sK35 @ (setadjoin @ sK36 @ emptyset)) @ emptyset))) != $true)),
% 2.53/1.55 introduced(definition,[],[choice_axiom])).
% 2.53/1.55 thf(f661,plain,(
% 2.53/1.55 ((kpairiskpair = $true) | ? [X0 : $i,X1 : $i] : ((iskpair @ (setadjoin @ (setadjoin @ X0 @ emptyset) @ (setadjoin @ (setadjoin @ X0 @ (setadjoin @ X1 @ emptyset)) @ emptyset))) != $true)) & (! [X2 : $i,X3 : $i] : ((iskpair @ (setadjoin @ (setadjoin @ X2 @ emptyset) @ (setadjoin @ (setadjoin @ X2 @ (setadjoin @ X3 @ emptyset)) @ emptyset))) = $true) | (kpairiskpair != $true))),
% 2.53/1.55 inference(rectify,[status(thm)],[f660])).
% 2.53/1.55 thf(f660,plain,(
% 2.53/1.55 ((kpairiskpair = $true) | ? [X0 : $i,X1 : $i] : ((iskpair @ (setadjoin @ (setadjoin @ X0 @ emptyset) @ (setadjoin @ (setadjoin @ X0 @ (setadjoin @ X1 @ emptyset)) @ emptyset))) != $true)) & (! [X0 : $i,X1 : $i] : ((iskpair @ (setadjoin @ (setadjoin @ X0 @ emptyset) @ (setadjoin @ (setadjoin @ X0 @ (setadjoin @ X1 @ emptyset)) @ emptyset))) = $true) | (kpairiskpair != $true))),
% 2.53/1.55 inference(nnf_transformation,[status(thm)],[f292])).
% 2.53/1.55 thf(f292,plain,(
% 2.53/1.55 (kpairiskpair = $true) <=> ! [X0 : $i,X1 : $i] : ((iskpair @ (setadjoin @ (setadjoin @ X0 @ emptyset) @ (setadjoin @ (setadjoin @ X0 @ (setadjoin @ X1 @ emptyset)) @ emptyset))) = $true)),
% 2.53/1.55 inference(fool_elimination,[status(thm)],[f291])).
% 2.53/1.55 thf(f291,plain,(
% 2.53/1.55 (! [X0 : $i,X1 : $i] : (iskpair @ (setadjoin @ (setadjoin @ X0 @ emptyset) @ (setadjoin @ (setadjoin @ X0 @ (setadjoin @ X1 @ emptyset)) @ emptyset))) = kpairiskpair)),
% 2.53/1.55 inference(rectify,[status(thm)],[f139])).
% 2.53/1.55 thf(f139,axiom,(
% 2.53/1.55 (! [X1 : $i,X2 : $i] : (iskpair @ (setadjoin @ (setadjoin @ X1 @ emptyset) @ (setadjoin @ (setadjoin @ X1 @ (setadjoin @ X2 @ emptyset)) @ emptyset))) = kpairiskpair)),
% 2.53/1.55 file('/export/starexec/sandbox2/tmp/tmp.0dbwHN3pEv/DTF2THF_22343.p',kpairiskpair)).
% 2.53/1.55 thf(f2545,plain,(
% 2.53/1.55 ($true != (iskpair @ (setadjoin @ (setadjoin @ sK237 @ emptyset) @ (setadjoin @ (setadjoin @ sK237 @ (setadjoin @ sK238 @ emptyset)) @ emptyset))))),
% 2.53/1.55 inference(beta_eta_normalization,[status(thm)],[f2190])).
% 2.53/1.55 thf(f2190,plain,(
% 2.53/1.55 ((iskpair @ ((^[Y0 : $i]: ((^[Y1 : $i]: (setadjoin @ (setadjoin @ Y0 @ emptyset) @ (setadjoin @ (setadjoin @ Y0 @ (setadjoin @ Y1 @ emptyset)) @ emptyset))))) @ sK237 @ sK238)) != $true)),
% 2.53/1.55 inference(definition_unfolding,[status(thm)],[f1610,f1326])).
% 2.53/1.55 thf(f1326,plain,(
% 2.53/1.55 (kpair = (^[Y0 : $i]: ((^[Y1 : $i]: (setadjoin @ (setadjoin @ Y0 @ emptyset) @ (setadjoin @ (setadjoin @ Y0 @ (setadjoin @ Y1 @ emptyset)) @ emptyset))))))),
% 2.53/1.55 inference(cnf_transformation,[status(thm)],[f266])).
% 2.53/1.55 thf(f266,plain,(
% 2.53/1.55 (kpair = (^[Y0 : $i]: ((^[Y1 : $i]: (setadjoin @ (setadjoin @ Y0 @ emptyset) @ (setadjoin @ (setadjoin @ Y0 @ (setadjoin @ Y1 @ emptyset)) @ emptyset))))))),
% 2.53/1.55 inference(fool_elimination,[status(thm)],[f265])).
% 2.53/1.55 thf(f265,plain,(
% 2.53/1.55 ((^[X0 : $i, X1 : $i] : (setadjoin @ (setadjoin @ X0 @ emptyset) @ (setadjoin @ (setadjoin @ X0 @ (setadjoin @ X1 @ emptyset)) @ emptyset))) = kpair)),
% 2.53/1.55 inference(rectify,[status(thm)],[f140])).
% 2.53/1.55 thf(f140,axiom,(
% 2.53/1.55 ((^[X1 : $i, X2 : $i] : (setadjoin @ (setadjoin @ X1 @ emptyset) @ (setadjoin @ (setadjoin @ X1 @ (setadjoin @ X2 @ emptyset)) @ emptyset))) = kpair)),
% 2.53/1.55 file('/export/starexec/sandbox2/tmp/tmp.0dbwHN3pEv/DTF2THF_22343.p',kpair)).
% 2.53/1.55 thf(f1610,plain,(
% 2.53/1.55 ((iskpair @ (kpair @ sK237 @ sK238)) != $true)),
% 2.53/1.55 inference(cnf_transformation,[status(thm)],[f1003])).
% 2.53/1.55 % SZS output end Proof for DTF2THF_22343
% 2.53/1.55 % (22529)------------------------------
% 2.53/1.55 % (22529)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.53/1.55 % (22529)Termination reason: Refutation
% 2.53/1.55
% 2.53/1.55 % (22529)Memory used [KB]: 7419
% 2.53/1.55 % (22529)Time elapsed: 0.059 s
% 2.53/1.55 % (22529)Instructions burned: 106 (million)
% 2.53/1.55 % (22529)------------------------------
% 2.53/1.55 % (22529)------------------------------
% 2.53/1.55 % (22522)Success in time 0.074 s
%------------------------------------------------------------------------------