%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU622^1 : TPTP v9.2.1. Released v3.7.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n012.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 1.80s 1.23s
% Output : Refutation 1.80s
% Verified :
% SZS Type : Refutation
% Derivation depth : 17
% Number of leaves : 6
% Syntax : Number of formulae : 40 ( 12 unt; 0 typ; 0 def)
% Number of atoms : 2449 ( 651 equ; 0 cnn)
% Maximal formula atoms : 141 ( 61 avg)
% Number of connectives : 1327 ( 54 ~; 32 |; 435 &; 237 @)
% ( 4 <=>; 565 =>; 0 <=; 0 <~>)
% Maximal formula depth : 147 ( 28 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 267 ( 264 usr; 147 con; 0-4 aty)
% Number of variables : 58 ( 0 ^; 40 !; 18 ?; 58 :)
% Comments :
%------------------------------------------------------------------------------
thf(func_def_0,type,
in: $i > $i > $o ).
thf(func_def_1,type,
exu: ( $i > $o ) > $o ).
thf(func_def_6,type,
setadjoin: $i > $i > $i ).
thf(func_def_8,type,
powerset: $i > $i ).
thf(func_def_10,type,
setunion: $i > $i ).
thf(func_def_19,type,
descr: ( $i > $o ) > $i ).
thf(func_def_21,type,
dsetconstr: $i > ( $i > $o ) > $i ).
thf(func_def_26,type,
prop2set: $o > $i ).
thf(func_def_36,type,
nonempty: $i > $o ).
thf(func_def_69,type,
set2prop: $i > $o ).
thf(func_def_88,type,
subset: $i > $i > $o ).
thf(func_def_89,type,
disjoint: $i > $i > $o ).
thf(func_def_90,type,
setsmeet: $i > $i > $o ).
thf(func_def_114,type,
binunion: $i > $i > $i ).
thf(func_def_122,type,
binintersect: $i > $i > $i ).
thf(func_def_135,type,
regular: $i > $o ).
thf(func_def_136,type,
setminus: $i > $i > $i ).
thf(func_def_147,type,
symdiff: $i > $i > $i ).
thf(func_def_153,type,
iskpair: $i > $o ).
thf(func_def_158,type,
kpair: $i > $i > $i ).
thf(func_def_160,type,
cartprod: $i > $i > $i ).
thf(func_def_175,type,
sP0: $o > $i > $i > $i > $o ).
thf(func_def_177,type,
sP2: $i > $i > $o ).
thf(func_def_178,type,
sP3: $i > $o ).
thf(func_def_179,type,
sP4: $i > $i > $o ).
thf(func_def_180,type,
sP5: $i > $i > $o ).
thf(func_def_181,type,
sK6: $i > $o ).
thf(func_def_184,type,
sK9: $i > ( $i > $o ) > $i ).
thf(func_def_185,type,
sK10: $i > $o ).
thf(func_def_195,type,
sK20: ( $i > $o ) > $i ).
thf(func_def_196,type,
sK21: $i > $o ).
thf(func_def_197,type,
sK22: $i > $i ).
thf(func_def_207,type,
sK32: $i > $i ).
thf(func_def_214,type,
sK39: $i > $i > $i ).
thf(func_def_228,type,
sK53: $i > $i > $i ).
thf(func_def_230,type,
sK55: $i > $i > $i ).
thf(func_def_231,type,
sK56: $i > $i > $i ).
thf(func_def_240,type,
sK65: $i > $i ).
thf(func_def_253,type,
sK78: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_254,type,
sK79: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_255,type,
sK80: $i > $o ).
thf(func_def_256,type,
sK81: $i > $o ).
thf(func_def_263,type,
sK88: $i > ( $i > $o ) > $i ).
thf(func_def_264,type,
sK89: $i > $o ).
thf(func_def_272,type,
sK97: $i > $o ).
thf(func_def_280,type,
sK105: $o > $i > $i > $i ).
thf(func_def_284,type,
sK109: $i > $o ).
thf(func_def_286,type,
sK111: ( $i > $o ) > $i ).
thf(func_def_287,type,
sK112: ( $i > $o ) > $i ).
thf(func_def_307,type,
sK132: $i > $i > $i ).
thf(func_def_317,type,
sK142: $i > $o ).
thf(func_def_319,type,
sK144: ( $i > $o ) > $i > $i ).
thf(func_def_337,type,
sK162: $i > $o ).
thf(func_def_339,type,
sK164: ( $i > $o ) > $i > $i ).
thf(func_def_340,type,
sK165: $i > $i > $i ).
thf(func_def_345,type,
sK170: $i > $o ).
thf(func_def_346,type,
sK171: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_347,type,
sK172: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_350,type,
sK175: $i > $o ).
thf(func_def_351,type,
sK176: $i > $o ).
thf(func_def_354,type,
sK179: $i > $o ).
thf(func_def_362,type,
sK187: $i > $i ).
thf(func_def_365,type,
sK190: $i > $i ).
thf(func_def_375,type,
sK200: $i > $o ).
thf(func_def_377,type,
sK202: $i > ( $i > $o ) > $i ).
thf(func_def_382,type,
sK207: $i > $i > $i ).
thf(func_def_386,type,
sK211: $i > $o ).
thf(func_def_387,type,
sK212: ( $i > $o ) > $i ).
thf(func_def_399,type,
sK224: $i > $o ).
thf(func_def_408,type,
sK233: $i > $i > $i ).
thf(func_def_412,type,
sK237: $i > $o ).
thf(func_def_414,type,
sK239: ( $i > $o ) > $i > $i ).
thf(func_def_417,type,
sK242: $i > $i > $o ).
thf(func_def_419,type,
sK244: $i > $i ).
thf(func_def_420,type,
sK245: $i > $i ).
thf(func_def_421,type,
sK246: $i > ( $i > $i > $o ) > $i ).
thf(func_def_422,type,
sK247: $i > ( $i > $i > $o ) > $i ).
thf(func_def_423,type,
sK248: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_442,type,
sK267: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_443,type,
sK268: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_444,type,
sK269: $i > $o ).
thf(func_def_445,type,
sK270: $i > $o ).
thf(func_def_457,type,
sK282: $i > $i ).
thf(func_def_465,type,
sK290: $i > $o ).
thf(func_def_477,type,
sK302: $i > $i > $i ).
thf(func_def_478,type,
sK303: $i > $i > $i ).
thf(func_def_479,type,
sK304: $i > $i > $i ).
thf(func_def_480,type,
sK305: $i > $i > $i ).
thf(func_def_481,type,
sK306: $i > $i > $i ).
thf(func_def_482,type,
sK307: $i > $i > $i > $i ).
thf(func_def_483,type,
sK308: $i > $i ).
thf(func_def_484,type,
sK309: $i > $i ).
thf(func_def_485,type,
sK310: $i > $i ).
thf(func_def_486,type,
sK311: $i > $i ).
thf(func_def_487,type,
sK312: $i > $i > $i > $i ).
thf(func_def_488,type,
sK313: $i > $i > $i > $i ).
thf(func_def_489,type,
sK314: $i > $i > $i ).
thf(func_def_490,type,
sK315: $i > $i > $i ).
thf(func_def_491,type,
sK316: $i > $i > $i ).
thf(func_def_493,type,
sK318: $i > $i ).
thf(func_def_494,type,
sK319: $i > $i ).
thf(func_def_495,type,
sK320: $i > $i ).
thf(func_def_497,type,
sK322: $i > $o ).
thf(func_def_500,type,
sK325: $i > $o ).
thf(func_def_502,type,
sK327: ( $i > $o ) > $i > $i ).
thf(func_def_509,type,
sK334: $i > $i ).
thf(func_def_512,type,
sK337: $i > $o ).
thf(func_def_513,type,
sK338: $i > ( $i > $o ) > $i ).
thf(func_def_514,type,
sK339: $i > $o ).
thf(func_def_527,type,
sK352: $i > $o ).
thf(func_def_529,type,
sK354: $i > $i ).
thf(func_def_531,type,
sK356: $i > $o ).
thf(func_def_534,type,
sK359: $i > $i > $i ).
thf(func_def_537,type,
sK362: $i > $o ).
thf(func_def_540,type,
sK365: $i > $o ).
thf(func_def_542,type,
sK367: ( $i > $o ) > $i ).
thf(func_def_543,type,
sK368: $i > $o ).
thf(func_def_544,type,
sK369: $i > $i ).
thf(func_def_553,type,
ph378:
!>[X0: $tType] : X0 ).
thf(f2527,plain,
$false,
inference(trivial_inequality_removal,[status(thm)],[f2522]) ).
thf(f2522,plain,
$true != $true,
inference(superposition,[status(thm)],[f2064,f1495]) ).
thf(f1495,plain,
( $true
= ( in @ sK155 @ sK156 ) ),
inference(cnf_transformation,[status(thm)],[f872]) ).
thf(f872,plain,
( ( dsetconstrI = $true )
& ( exuE3u = $true )
& ( binunionIR = $true )
& ( symdiffI2 = $true )
& ( setoftrueEq = $true )
& ( powersetAx = $true )
& ( omegaSAx = $true )
& ( symdiffIneg1 = $true )
& ( emptysetimpfalse = $true )
& ( binunionRsub = $true )
& ( subPowSU = $true )
& ( upairsetIL = $true )
& ( sepInPowerset = $true )
& ( binintersectSubset4 = $true )
& ( setadjoinIR = $true )
& ( notsubsetI = $true )
& ( setminusSubset2 = $true )
& ( setunion__Cong = $true )
& ( setadjoinE = $true )
& ( exuEu = $true )
& ( binunionE = $true )
& ( inCongP = $true )
& ( notinsingleton = $true )
& ( eqimpsubset1 = $true )
& ( disjointsetsI1 = $true )
& ( binunionLsub = $true )
& ( quantDeMorgan4 = $true )
& ( inPowerset = $true )
& ( singletonsswitch = $true )
& ( setminusI = $true )
& ( setadjoinOr = $true )
& ( quantDeMorgan1 = $true )
& ( setminusERneg = $true )
& ( nonemptyE1 = $true )
& ( setadjoinSub = $true )
& ( setminusSubset1 = $true )
& ( exuE2 = $true )
& ( setadjoinIL = $true )
& ( binintersectER = $true )
& ( binintersectSubset2 = $true )
& ( setadjoinSub2 = $true )
& ( powersetI = $true )
& ( exuI1 = $true )
& ( emptyI = $true )
& ( notequalI2 = $true )
& ( nonemptyI = $true )
& ( powersetE = $true )
& ( binintersectI = $true )
& ( descr__Cong = $true )
& ( eqinunit = $true )
& ( secondinupair = $true )
& ( subsetE2 = $true )
& ( notdexE = $true )
& ( emptyinunitempty = $true )
& ( wellorderingAx = $true )
& ( exu__Cong = $true )
& ( emptyinPowerset = $true )
& ( subsetI2 = $true )
& ( emptyset__Cong = $true )
& ( dsetconstrEL = $true )
& ( emptysetAx = $true )
& ( setextAx = $true )
& ( setminusEL = $true )
& ( setminusILneg = $true )
& ( omegaIndAx = $true )
& ( emptyInPowerset = $true )
& ( kpairiskpair = $true )
& ( setukpairIL = $true )
& ( singletonsubset = $true )
& ( quantDeMorgan2 = $true )
& ( binintersectSubset5 = $true )
& ( setadjoin__Cong = $true )
& ( exuE3e = $true )
& ( prop2setE = $true )
& ( setunionI = $true )
& ( notequalI1 = $true )
& ( exuI3 = $true )
& ( powersetsubset = $true )
& ( setukpairIR = $true )
& ( upairsetIR = $true )
& ( setbeta = $true )
& ( descrp = $true )
& ( emptyE1 = $true )
& ( subsetemptysetimpeq = $true )
& ( noeltsimpempty = $true )
& ( exuE1 = $true )
& ( nonemptyI1 = $true )
& ( binintersectEL = $true )
& ( subsetTrans = $true )
& ( emptysetE = $true )
& ( upairsetE = $true )
& ( dsetconstrER = $true )
& ( powersetE1 = $true )
& ( notinemptyset = $true )
& ( subsetI1 = $true )
& ( quantDeMorgan3 = $true )
& ( symdiffE = $true )
& ( setadjoinAx = $true )
& ( powerset__Cong = $true )
& ( nonemptyImpWitness = $true )
& ( bs114d = $true )
& ( binintersectSubset3 = $true )
& ( binunionEcases = $true )
& ( dsetconstr__Cong = $true )
& ( symdiffIneg2 = $true )
& ( binintersectLsub = $true )
& ( setminusLsub = $true )
& ( subsetE = $true )
& ( uniqinunit = $true )
& ( exuI2 = $true )
& ( binintersectRsub = $true )
& ( ( in @ ( setadjoin @ sK155 @ emptyset ) @ ( powerset @ sK156 ) )
!= $true )
& ( $true
= ( in @ sK155 @ sK156 ) )
& ( setminusELneg = $true )
& ( setminusER = $true )
& ( kpairp = $true )
& ( omega0Ax = $true )
& ( setext = $true )
& ( prop2set2propI = $true )
& ( emptysetsubset = $true )
& ( subset2powerset = $true )
& ( setminusIRneg = $true )
& ( upairset2IR = $true )
& ( binintersectSubset1 = $true )
& ( subsetRefl = $true )
& ( powersetI1 = $true )
& ( in__Cong = $true )
& ( sepSubset = $true )
& ( foundationAx = $true )
& ( symdiffI1 = $true )
& ( setunionAx = $true )
& ( setunionE = $true )
& ( vacuousDall = $true )
& ( prop2setI = $true )
& ( replAx = $true )
& ( omega__Cong = $true )
& ( notdallE = $true )
& ( setextsub = $true )
& ( eqimpsubset2 = $true )
& ( binunionIL = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK155,sK156])],[f585,f871]) ).
thf(f871,plain,
( ? [X0: $i,X1: $i] :
( ( $true
!= ( in @ ( setadjoin @ X0 @ emptyset ) @ ( powerset @ X1 ) ) )
& ( $true
= ( in @ X0 @ X1 ) ) )
=> ( ( ( in @ ( setadjoin @ sK155 @ emptyset ) @ ( powerset @ sK156 ) )
!= $true )
& ( $true
= ( in @ sK155 @ sK156 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f585,plain,
( ( dsetconstrI = $true )
& ( exuE3u = $true )
& ( binunionIR = $true )
& ( symdiffI2 = $true )
& ( setoftrueEq = $true )
& ( powersetAx = $true )
& ( omegaSAx = $true )
& ( symdiffIneg1 = $true )
& ( emptysetimpfalse = $true )
& ( binunionRsub = $true )
& ( subPowSU = $true )
& ( upairsetIL = $true )
& ( sepInPowerset = $true )
& ( binintersectSubset4 = $true )
& ( setadjoinIR = $true )
& ( notsubsetI = $true )
& ( setminusSubset2 = $true )
& ( setunion__Cong = $true )
& ( setadjoinE = $true )
& ( exuEu = $true )
& ( binunionE = $true )
& ( inCongP = $true )
& ( notinsingleton = $true )
& ( eqimpsubset1 = $true )
& ( disjointsetsI1 = $true )
& ( binunionLsub = $true )
& ( quantDeMorgan4 = $true )
& ( inPowerset = $true )
& ( singletonsswitch = $true )
& ( setminusI = $true )
& ( setadjoinOr = $true )
& ( quantDeMorgan1 = $true )
& ( setminusERneg = $true )
& ( nonemptyE1 = $true )
& ( setadjoinSub = $true )
& ( setminusSubset1 = $true )
& ( exuE2 = $true )
& ( setadjoinIL = $true )
& ( binintersectER = $true )
& ( binintersectSubset2 = $true )
& ( setadjoinSub2 = $true )
& ( powersetI = $true )
& ( exuI1 = $true )
& ( emptyI = $true )
& ( notequalI2 = $true )
& ( nonemptyI = $true )
& ( powersetE = $true )
& ( binintersectI = $true )
& ( descr__Cong = $true )
& ( eqinunit = $true )
& ( secondinupair = $true )
& ( subsetE2 = $true )
& ( notdexE = $true )
& ( emptyinunitempty = $true )
& ( wellorderingAx = $true )
& ( exu__Cong = $true )
& ( emptyinPowerset = $true )
& ( subsetI2 = $true )
& ( emptyset__Cong = $true )
& ( dsetconstrEL = $true )
& ( emptysetAx = $true )
& ( setextAx = $true )
& ( setminusEL = $true )
& ( setminusILneg = $true )
& ( omegaIndAx = $true )
& ( emptyInPowerset = $true )
& ( kpairiskpair = $true )
& ( setukpairIL = $true )
& ( singletonsubset = $true )
& ( quantDeMorgan2 = $true )
& ( binintersectSubset5 = $true )
& ( setadjoin__Cong = $true )
& ( exuE3e = $true )
& ( prop2setE = $true )
& ( setunionI = $true )
& ( notequalI1 = $true )
& ( exuI3 = $true )
& ( powersetsubset = $true )
& ( setukpairIR = $true )
& ( upairsetIR = $true )
& ( setbeta = $true )
& ( descrp = $true )
& ( emptyE1 = $true )
& ( subsetemptysetimpeq = $true )
& ( noeltsimpempty = $true )
& ( exuE1 = $true )
& ( nonemptyI1 = $true )
& ( binintersectEL = $true )
& ( subsetTrans = $true )
& ( emptysetE = $true )
& ( upairsetE = $true )
& ( dsetconstrER = $true )
& ( powersetE1 = $true )
& ( notinemptyset = $true )
& ( subsetI1 = $true )
& ( quantDeMorgan3 = $true )
& ( symdiffE = $true )
& ( setadjoinAx = $true )
& ( powerset__Cong = $true )
& ( nonemptyImpWitness = $true )
& ( bs114d = $true )
& ( binintersectSubset3 = $true )
& ( binunionEcases = $true )
& ( dsetconstr__Cong = $true )
& ( symdiffIneg2 = $true )
& ( binintersectLsub = $true )
& ( setminusLsub = $true )
& ( subsetE = $true )
& ( uniqinunit = $true )
& ( exuI2 = $true )
& ( binintersectRsub = $true )
& ? [X0: $i,X1: $i] :
( ( $true
!= ( in @ ( setadjoin @ X0 @ emptyset ) @ ( powerset @ X1 ) ) )
& ( $true
= ( in @ X0 @ X1 ) ) )
& ( setminusELneg = $true )
& ( setminusER = $true )
& ( kpairp = $true )
& ( omega0Ax = $true )
& ( setext = $true )
& ( prop2set2propI = $true )
& ( emptysetsubset = $true )
& ( subset2powerset = $true )
& ( setminusIRneg = $true )
& ( upairset2IR = $true )
& ( binintersectSubset1 = $true )
& ( subsetRefl = $true )
& ( powersetI1 = $true )
& ( in__Cong = $true )
& ( sepSubset = $true )
& ( foundationAx = $true )
& ( symdiffI1 = $true )
& ( setunionAx = $true )
& ( setunionE = $true )
& ( vacuousDall = $true )
& ( prop2setI = $true )
& ( replAx = $true )
& ( omega__Cong = $true )
& ( notdallE = $true )
& ( setextsub = $true )
& ( eqimpsubset2 = $true )
& ( binunionIL = $true ) ),
inference(flattening,[status(thm)],[f584]) ).
thf(f584,plain,
( ? [X0: $i,X1: $i] :
( ( $true
!= ( in @ ( setadjoin @ X0 @ emptyset ) @ ( powerset @ X1 ) ) )
& ( $true
= ( in @ X0 @ X1 ) ) )
& ( singletonsubset = $true )
& ( kpairp = $true )
& ( kpairiskpair = $true )
& ( setukpairIR = $true )
& ( setukpairIL = $true )
& ( secondinupair = $true )
& ( symdiffIneg2 = $true )
& ( symdiffIneg1 = $true )
& ( symdiffI2 = $true )
& ( symdiffI1 = $true )
& ( symdiffE = $true )
& ( setminusSubset1 = $true )
& ( setminusLsub = $true )
& ( setminusIRneg = $true )
& ( setminusILneg = $true )
& ( setminusELneg = $true )
& ( setminusERneg = $true )
& ( setminusSubset2 = $true )
& ( setminusER = $true )
& ( setminusEL = $true )
& ( setminusI = $true )
& ( bs114d = $true )
& ( binintersectSubset1 = $true )
& ( binintersectSubset4 = $true )
& ( binintersectRsub = $true )
& ( disjointsetsI1 = $true )
& ( binintersectER = $true )
& ( binintersectSubset3 = $true )
& ( binintersectSubset2 = $true )
& ( binintersectLsub = $true )
& ( binintersectEL = $true )
& ( binintersectSubset5 = $true )
& ( binintersectI = $true )
& ( binunionRsub = $true )
& ( binunionLsub = $true )
& ( binunionE = $true )
& ( binunionEcases = $true )
& ( binunionIR = $true )
& ( upairset2IR = $true )
& ( binunionIL = $true )
& ( sepSubset = $true )
& ( sepInPowerset = $true )
& ( powersetsubset = $true )
& ( inPowerset = $true )
& ( powersetE1 = $true )
& ( powersetI1 = $true )
& ( subsetemptysetimpeq = $true )
& ( setextsub = $true )
& ( subset2powerset = $true )
& ( setadjoinSub2 = $true )
& ( setadjoinSub = $true )
& ( subsetTrans = $true )
& ( subsetRefl = $true )
& ( notequalI2 = $true )
& ( notequalI1 = $true )
& ( notsubsetI = $true )
& ( subsetE2 = $true )
& ( subsetE = $true )
& ( emptysetsubset = $true )
& ( subsetI2 = $true )
& ( eqimpsubset1 = $true )
& ( eqimpsubset2 = $true )
& ( subsetI1 = $true )
& ( dsetconstr__Cong = $true )
& ( descr__Cong = $true )
& ( exuEu = $true )
& ( omega__Cong = $true )
& ( setunion__Cong = $true )
& ( powerset__Cong = $true )
& ( setadjoin__Cong = $true )
& ( emptyset__Cong = $true )
& ( exu__Cong = $true )
& ( exuE3u = $true )
& ( in__Cong = $true )
& ( inCongP = $true )
& ( exuI2 = $true )
& ( exuI3 = $true )
& ( exuI1 = $true )
& ( notdallE = $true )
& ( notdexE = $true )
& ( prop2set2propI = $true )
& ( prop2setI = $true )
& ( quantDeMorgan4 = $true )
& ( quantDeMorgan3 = $true )
& ( quantDeMorgan2 = $true )
& ( quantDeMorgan1 = $true )
& ( vacuousDall = $true )
& ( emptyE1 = $true )
& ( upairsetIR = $true )
& ( upairsetIL = $true )
& ( upairsetE = $true )
& ( singletonsswitch = $true )
& ( eqinunit = $true )
& ( notinsingleton = $true )
& ( uniqinunit = $true )
& ( nonemptyImpWitness = $true )
& ( exuE2 = $true )
& ( subPowSU = $true )
& ( setunionE = $true )
& ( setunionI = $true )
& ( powersetE = $true )
& ( emptyInPowerset = $true )
& ( emptyinPowerset = $true )
& ( powersetI = $true )
& ( setoftrueEq = $true )
& ( setadjoinOr = $true )
& ( setadjoinE = $true )
& ( setadjoinIR = $true )
& ( emptyinunitempty = $true )
& ( setadjoinIL = $true )
& ( nonemptyI1 = $true )
& ( nonemptyI = $true )
& ( nonemptyE1 = $true )
& ( setbeta = $true )
& ( noeltsimpempty = $true )
& ( emptyI = $true )
& ( setext = $true )
& ( exuE3e = $true )
& ( notinemptyset = $true )
& ( emptysetimpfalse = $true )
& ( emptysetE = $true )
& ( prop2setE = $true )
& ( exuE1 = $true )
& ( dsetconstrER = $true )
& ( dsetconstrEL = $true )
& ( dsetconstrI = $true )
& ( descrp = $true )
& ( wellorderingAx = $true )
& ( foundationAx = $true )
& ( replAx = $true )
& ( omegaIndAx = $true )
& ( omegaSAx = $true )
& ( omega0Ax = $true )
& ( setunionAx = $true )
& ( powersetAx = $true )
& ( setadjoinAx = $true )
& ( emptysetAx = $true )
& ( setextAx = $true ) ),
inference(ennf_transformation,[status(thm)],[f294]) ).
thf(f294,plain,
~ ( ( setextAx = $true )
=> ( ( emptysetAx = $true )
=> ( ( setadjoinAx = $true )
=> ( ( powersetAx = $true )
=> ( ( setunionAx = $true )
=> ( ( omega0Ax = $true )
=> ( ( omegaSAx = $true )
=> ( ( omegaIndAx = $true )
=> ( ( replAx = $true )
=> ( ( foundationAx = $true )
=> ( ( wellorderingAx = $true )
=> ( ( descrp = $true )
=> ( ( dsetconstrI = $true )
=> ( ( dsetconstrEL = $true )
=> ( ( dsetconstrER = $true )
=> ( ( exuE1 = $true )
=> ( ( prop2setE = $true )
=> ( ( emptysetE = $true )
=> ( ( emptysetimpfalse = $true )
=> ( ( notinemptyset = $true )
=> ( ( exuE3e = $true )
=> ( ( setext = $true )
=> ( ( emptyI = $true )
=> ( ( noeltsimpempty = $true )
=> ( ( setbeta = $true )
=> ( ( nonemptyE1 = $true )
=> ( ( nonemptyI = $true )
=> ( ( nonemptyI1 = $true )
=> ( ( setadjoinIL = $true )
=> ( ( emptyinunitempty = $true )
=> ( ( setadjoinIR = $true )
=> ( ( setadjoinE = $true )
=> ( ( setadjoinOr = $true )
=> ( ( setoftrueEq = $true )
=> ( ( powersetI = $true )
=> ( ( emptyinPowerset = $true )
=> ( ( emptyInPowerset = $true )
=> ( ( powersetE = $true )
=> ( ( setunionI = $true )
=> ( ( setunionE = $true )
=> ( ( subPowSU = $true )
=> ( ( exuE2 = $true )
=> ( ( nonemptyImpWitness = $true )
=> ( ( uniqinunit = $true )
=> ( ( notinsingleton = $true )
=> ( ( eqinunit = $true )
=> ( ( singletonsswitch = $true )
=> ( ( upairsetE = $true )
=> ( ( upairsetIL = $true )
=> ( ( upairsetIR = $true )
=> ( ( emptyE1 = $true )
=> ( ( vacuousDall = $true )
=> ( ( quantDeMorgan1 = $true )
=> ( ( quantDeMorgan2 = $true )
=> ( ( quantDeMorgan3 = $true )
=> ( ( quantDeMorgan4 = $true )
=> ( ( prop2setI = $true )
=> ( ( prop2set2propI = $true )
=> ( ( notdexE = $true )
=> ( ( notdallE = $true )
=> ( ( exuI1 = $true )
=> ( ( exuI3 = $true )
=> ( ( exuI2 = $true )
=> ( ( inCongP = $true )
=> ( ( in__Cong = $true )
=> ( ( exuE3u = $true )
=> ( ( exu__Cong = $true )
=> ( ( emptyset__Cong = $true )
=> ( ( setadjoin__Cong = $true )
=> ( ( powerset__Cong = $true )
=> ( ( setunion__Cong = $true )
=> ( ( omega__Cong = $true )
=> ( ( exuEu = $true )
=> ( ( descr__Cong = $true )
=> ( ( dsetconstr__Cong = $true )
=> ( ( subsetI1 = $true )
=> ( ( eqimpsubset2 = $true )
=> ( ( eqimpsubset1 = $true )
=> ( ( subsetI2 = $true )
=> ( ( emptysetsubset = $true )
=> ( ( subsetE = $true )
=> ( ( subsetE2 = $true )
=> ( ( notsubsetI = $true )
=> ( ( notequalI1 = $true )
=> ( ( notequalI2 = $true )
=> ( ( subsetRefl = $true )
=> ( ( subsetTrans = $true )
=> ( ( setadjoinSub = $true )
=> ( ( setadjoinSub2 = $true )
=> ( ( subset2powerset = $true )
=> ( ( setextsub = $true )
=> ( ( subsetemptysetimpeq = $true )
=> ( ( powersetI1 = $true )
=> ( ( powersetE1 = $true )
=> ( ( inPowerset = $true )
=> ( ( powersetsubset = $true )
=> ( ( sepInPowerset = $true )
=> ( ( sepSubset = $true )
=> ( ( binunionIL = $true )
=> ( ( upairset2IR = $true )
=> ( ( binunionIR = $true )
=> ( ( binunionEcases = $true )
=> ( ( binunionE = $true )
=> ( ( binunionLsub = $true )
=> ( ( binunionRsub = $true )
=> ( ( binintersectI = $true )
=> ( ( binintersectSubset5 = $true )
=> ( ( binintersectEL = $true )
=> ( ( binintersectLsub = $true )
=> ( ( binintersectSubset2 = $true )
=> ( ( binintersectSubset3 = $true )
=> ( ( binintersectER = $true )
=> ( ( disjointsetsI1 = $true )
=> ( ( binintersectRsub = $true )
=> ( ( binintersectSubset4 = $true )
=> ( ( binintersectSubset1 = $true )
=> ( ( bs114d = $true )
=> ( ( setminusI = $true )
=> ( ( setminusEL = $true )
=> ( ( setminusER = $true )
=> ( ( setminusSubset2 = $true )
=> ( ( setminusERneg = $true )
=> ( ( setminusELneg = $true )
=> ( ( setminusILneg = $true )
=> ( ( setminusIRneg = $true )
=> ( ( setminusLsub = $true )
=> ( ( setminusSubset1 = $true )
=> ( ( symdiffE = $true )
=> ( ( symdiffI1 = $true )
=> ( ( symdiffI2 = $true )
=> ( ( symdiffIneg1 = $true )
=> ( ( symdiffIneg2 = $true )
=> ( ( secondinupair = $true )
=> ( ( setukpairIL = $true )
=> ( ( setukpairIR = $true )
=> ( ( kpairiskpair = $true )
=> ( ( kpairp = $true )
=> ( ( singletonsubset = $true )
=> ! [X0: $i,X1: $i] :
( ( $true
= ( in @ X0 @ X1 ) )
=> ( $true
= ( in @ ( setadjoin @ X0 @ emptyset ) @ ( powerset @ X1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f293]) ).
thf(f293,plain,
~ ( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ! [X0: $i,X1: $i] :
( ( in @ X0 @ X1 )
=> ( in @ ( setadjoin @ X0 @ emptyset ) @ ( powerset @ X1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f145]) ).
thf(f145,negated_conjecture,
~ ( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ! [X1: $i,X3: $i] :
( ( in @ X1 @ X3 )
=> ( in @ ( setadjoin @ X1 @ emptyset ) @ ( powerset @ X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f144]) ).
thf(f144,conjecture,
( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ! [X1: $i,X3: $i] :
( ( in @ X1 @ X3 )
=> ( in @ ( setadjoin @ X1 @ emptyset ) @ ( powerset @ X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.P13AcNWfIy/DTF2THF_4603.p',singletoninpowerset) ).
thf(f2064,plain,
( $true
!= ( in @ sK155 @ sK156 ) ),
inference(trivial_inequality_removal,[status(thm)],[f2063]) ).
thf(f2063,plain,
( ( $true != $true )
| ( $true
!= ( in @ sK155 @ sK156 ) ) ),
inference(forward_demodulation,[status(thm)],[f2056,f1539]) ).
thf(f1539,plain,
singletonsubset = $true,
inference(cnf_transformation,[status(thm)],[f872]) ).
thf(f2056,plain,
( ( singletonsubset != $true )
| ( $true
!= ( in @ sK155 @ sK156 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f2049]) ).
thf(f2049,plain,
( ( $true != $true )
| ( $true
!= ( in @ sK155 @ sK156 ) )
| ( singletonsubset != $true ) ),
inference(superposition,[status(thm)],[f2042,f1458]) ).
thf(f1458,plain,
! [X2: $i,X3: $i] :
( ( $true
= ( subset @ ( setadjoin @ X3 @ emptyset ) @ X2 ) )
| ( singletonsubset != $true )
| ( $true
!= ( in @ X3 @ X2 ) ) ),
inference(cnf_transformation,[status(thm)],[f862]) ).
thf(f862,plain,
( ( ( singletonsubset = $true )
| ( ( $true
!= ( subset @ ( setadjoin @ sK149 @ emptyset ) @ sK148 ) )
& ( $true
= ( in @ sK149 @ sK148 ) ) ) )
& ( ! [X2: $i,X3: $i] :
( ( $true
= ( subset @ ( setadjoin @ X3 @ emptyset ) @ X2 ) )
| ( $true
!= ( in @ X3 @ X2 ) ) )
| ( singletonsubset != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK148,sK149])],[f860,f861]) ).
thf(f861,plain,
( ? [X0: $i,X1: $i] :
( ( $true
!= ( subset @ ( setadjoin @ X1 @ emptyset ) @ X0 ) )
& ( $true
= ( in @ X1 @ X0 ) ) )
=> ( ( $true
!= ( subset @ ( setadjoin @ sK149 @ emptyset ) @ sK148 ) )
& ( $true
= ( in @ sK149 @ sK148 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f860,plain,
( ( ( singletonsubset = $true )
| ? [X0: $i,X1: $i] :
( ( $true
!= ( subset @ ( setadjoin @ X1 @ emptyset ) @ X0 ) )
& ( $true
= ( in @ X1 @ X0 ) ) ) )
& ( ! [X2: $i,X3: $i] :
( ( $true
= ( subset @ ( setadjoin @ X3 @ emptyset ) @ X2 ) )
| ( $true
!= ( in @ X3 @ X2 ) ) )
| ( singletonsubset != $true ) ) ),
inference(rectify,[status(thm)],[f859]) ).
thf(f859,plain,
( ( ( singletonsubset = $true )
| ? [X0: $i,X1: $i] :
( ( $true
!= ( subset @ ( setadjoin @ X1 @ emptyset ) @ X0 ) )
& ( $true
= ( in @ X1 @ X0 ) ) ) )
& ( ! [X0: $i,X1: $i] :
( ( $true
= ( subset @ ( setadjoin @ X1 @ emptyset ) @ X0 ) )
| ( $true
!= ( in @ X1 @ X0 ) ) )
| ( singletonsubset != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f523]) ).
thf(f523,plain,
( ( singletonsubset = $true )
<=> ! [X0: $i,X1: $i] :
( ( $true
= ( subset @ ( setadjoin @ X1 @ emptyset ) @ X0 ) )
| ( $true
!= ( in @ X1 @ X0 ) ) ) ),
inference(ennf_transformation,[status(thm)],[f352]) ).
thf(f352,plain,
( ! [X0: $i,X1: $i] :
( ( $true
= ( in @ X1 @ X0 ) )
=> ( $true
= ( subset @ ( setadjoin @ X1 @ emptyset ) @ X0 ) ) )
<=> ( singletonsubset = $true ) ),
inference(fool_elimination,[status(thm)],[f351]) ).
thf(f351,plain,
( singletonsubset
= ( ! [X0: $i,X1: $i] :
( ( in @ X1 @ X0 )
=> ( subset @ ( setadjoin @ X1 @ emptyset ) @ X0 ) ) ) ),
inference(rectify,[status(thm)],[f143]) ).
thf(f143,axiom,
( singletonsubset
= ( ! [X3: $i,X1: $i] :
( ( in @ X1 @ X3 )
=> ( subset @ ( setadjoin @ X1 @ emptyset ) @ X3 ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.P13AcNWfIy/DTF2THF_4603.p',singletonsubset) ).
thf(f2042,plain,
( $true
!= ( subset @ ( setadjoin @ sK155 @ emptyset ) @ sK156 ) ),
inference(trivial_inequality_removal,[status(thm)],[f2041]) ).
thf(f2041,plain,
( ( $true != $true )
| ( $true
!= ( subset @ ( setadjoin @ sK155 @ emptyset ) @ sK156 ) ) ),
inference(forward_demodulation,[status(thm)],[f2036,f1487]) ).
thf(f1487,plain,
subset2powerset = $true,
inference(cnf_transformation,[status(thm)],[f872]) ).
thf(f2036,plain,
( ( subset2powerset != $true )
| ( $true
!= ( subset @ ( setadjoin @ sK155 @ emptyset ) @ sK156 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f2028]) ).
thf(f2028,plain,
( ( subset2powerset != $true )
| ( $true
!= ( subset @ ( setadjoin @ sK155 @ emptyset ) @ sK156 ) )
| ( $true != $true ) ),
inference(superposition,[status(thm)],[f1496,f1420]) ).
thf(f1420,plain,
! [X0: $i,X1: $i] :
( ( $true
= ( in @ X1 @ ( powerset @ X0 ) ) )
| ( $true
!= ( subset @ X1 @ X0 ) )
| ( subset2powerset != $true ) ),
inference(cnf_transformation,[status(thm)],[f814]) ).
thf(f814,plain,
( ( ! [X0: $i,X1: $i] :
( ( $true
= ( in @ X1 @ ( powerset @ X0 ) ) )
| ( $true
!= ( subset @ X1 @ X0 ) ) )
| ( subset2powerset != $true ) )
& ( ( subset2powerset = $true )
| ( ( $true
!= ( in @ sK119 @ ( powerset @ sK118 ) ) )
& ( $true
= ( subset @ sK119 @ sK118 ) ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK118,sK119])],[f812,f813]) ).
thf(f813,plain,
( ? [X2: $i,X3: $i] :
( ( $true
!= ( in @ X3 @ ( powerset @ X2 ) ) )
& ( $true
= ( subset @ X3 @ X2 ) ) )
=> ( ( $true
!= ( in @ sK119 @ ( powerset @ sK118 ) ) )
& ( $true
= ( subset @ sK119 @ sK118 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f812,plain,
( ( ! [X0: $i,X1: $i] :
( ( $true
= ( in @ X1 @ ( powerset @ X0 ) ) )
| ( $true
!= ( subset @ X1 @ X0 ) ) )
| ( subset2powerset != $true ) )
& ( ( subset2powerset = $true )
| ? [X2: $i,X3: $i] :
( ( $true
!= ( in @ X3 @ ( powerset @ X2 ) ) )
& ( $true
= ( subset @ X3 @ X2 ) ) ) ) ),
inference(rectify,[status(thm)],[f811]) ).
thf(f811,plain,
( ( ! [X0: $i,X1: $i] :
( ( $true
= ( in @ X1 @ ( powerset @ X0 ) ) )
| ( $true
!= ( subset @ X1 @ X0 ) ) )
| ( subset2powerset != $true ) )
& ( ( subset2powerset = $true )
| ? [X0: $i,X1: $i] :
( ( $true
!= ( in @ X1 @ ( powerset @ X0 ) ) )
& ( $true
= ( subset @ X1 @ X0 ) ) ) ) ),
inference(nnf_transformation,[status(thm)],[f536]) ).
thf(f536,plain,
( ! [X0: $i,X1: $i] :
( ( $true
= ( in @ X1 @ ( powerset @ X0 ) ) )
| ( $true
!= ( subset @ X1 @ X0 ) ) )
<=> ( subset2powerset = $true ) ),
inference(ennf_transformation,[status(thm)],[f219]) ).
thf(f219,plain,
( ! [X0: $i,X1: $i] :
( ( $true
= ( subset @ X1 @ X0 ) )
=> ( $true
= ( in @ X1 @ ( powerset @ X0 ) ) ) )
<=> ( subset2powerset = $true ) ),
inference(fool_elimination,[status(thm)],[f218]) ).
thf(f218,plain,
! [X0: $i,X1: $i] :
( ( ( subset @ X1 @ X0 )
=> ( in @ X1 @ ( powerset @ X0 ) ) )
= subset2powerset ),
inference(rectify,[status(thm)],[f92]) ).
thf(f92,axiom,
! [X4: $i,X3: $i] :
( ( ( subset @ X3 @ X4 )
=> ( in @ X3 @ ( powerset @ X4 ) ) )
= subset2powerset ),
file('/export/starexec/sandbox/tmp/tmp.P13AcNWfIy/DTF2THF_4603.p',subset2powerset) ).
thf(f1496,plain,
( ( in @ ( setadjoin @ sK155 @ emptyset ) @ ( powerset @ sK156 ) )
!= $true ),
inference(cnf_transformation,[status(thm)],[f872]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.06 % Problem : SEU622^1 : TPTP v9.2.1. Released v3.7.0.
% 0.00/0.07 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.07/0.25 % Computer : n012.cluster.edu
% 0.07/0.25 % Model : x86_64 x86_64
% 0.07/0.25 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.07/0.25 % Memory : 8042.1875MB
% 0.07/0.25 % OS : Linux 3.10.0-693.el7.x86_64
% 0.07/0.25 % CPULimit : 300
% 0.07/0.25 % WCLimit : 300
% 0.07/0.25 % DateTime : Tue Feb 24 12:28:30 EST 2026
% 0.07/0.25 % CPUTime :
% 0.07/0.25 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.07/0.31 ---- Original DTF file ---
% 0.07/0.31 thf(spec,logic,$$dhol).
% 0.07/0.31 %------------------------------------------------------------------------------
% 0.07/0.31 % File : SEU622^1 : TPTP v9.2.1. Released v3.7.0.
% 0.07/0.31 % Domain : Set Theory
% 0.07/0.31 % Problem : Ordered Pairs - Cartesian Products
% 0.07/0.31 % Version : Especial.
% 0.07/0.31 % English : (! A:i.! x:i.in x A -> in (setadjoin x emptyset) (powerset A))
% 0.07/0.31
% 0.07/0.31 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.07/0.31 % Source : [Bro08]
% 0.07/0.31 % Names : ZFC124g [Bro08]
% 0.07/0.31
% 0.07/0.31 % Status : Theorem
% 0.07/0.31 % Rating : 0.11 v9.1.0, 0.12 v9.0.0, 0.20 v8.2.0, 0.31 v8.1.0, 0.18 v7.5.0, 0.14 v7.4.0, 0.22 v7.2.0, 0.12 v7.1.0, 0.38 v7.0.0, 0.29 v6.4.0, 0.33 v6.3.0, 0.40 v6.2.0, 0.29 v6.1.0, 0.43 v5.5.0, 0.50 v5.4.0, 0.40 v5.3.0, 0.60 v5.0.0, 0.40 v4.1.0, 0.67 v3.7.0
% 0.07/0.31 % Syntax : Number of formulae : 305 ( 143 unt; 161 typ; 143 def)
% 0.07/0.31 % Number of atoms : 753 ( 202 equ; 0 cnn)
% 0.07/0.31 % Maximal formula atoms : 140 ( 5 avg)
% 0.07/0.31 % Number of connectives : 1261 ( 47 ~; 6 |; 32 &; 832 @)
% 0.07/0.31 % ( 14 <=>; 330 =>; 0 <=; 0 <~>)
% 0.07/0.31 % Maximal formula depth : 146 ( 2 avg)
% 0.07/0.31 % Number of types : 2 ( 0 usr)
% 0.07/0.31 % Number of type conns : 69 ( 69 >; 0 *; 0 +; 0 <<)
% 0.07/0.31 % Number of symbols : 164 ( 161 usr; 142 con; 0-2 aty)
% 0.07/0.31 % Number of variables : 410 ( 36 ^; 341 !; 33 ?; 410 :)
% 0.07/0.31 % SPC : TH0_THM_EQU_NAR
% 0.07/0.31
% 0.07/0.31 % Comments : http://mathgate.info/detsetitem.php?id=179
% 0.07/0.31 %------------------------------------------------------------------------------
% 0.07/0.31 thf(in_type,type,
% 0.07/0.31 in: $i > $i > $o ).
% 0.07/0.31
% 0.07/0.31 thf(exu_type,type,
% 0.07/0.31 exu: ( $i > $o ) > $o ).
% 0.07/0.31
% 0.07/0.31 thf(exu,definition,
% 0.07/0.31 ( exu
% 0.07/0.31 = ( ^ [Xphi: $i > $o] :
% 0.07/0.31 ? [Xx: $i] :
% 0.07/0.31 ( ( Xphi @ Xx )
% 0.07/0.31 & ! [Xy: $i] :
% 0.07/0.31 ( ( Xphi @ Xy )
% 0.07/0.31 => ( Xx = Xy ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(setextAx_type,type,
% 0.07/0.31 setextAx: $o ).
% 0.07/0.31
% 0.07/0.31 thf(setextAx,definition,
% 0.07/0.31 ( setextAx
% 0.07/0.31 = ( ! [A: $i,B: $i] :
% 0.07/0.31 ( ! [Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ A )
% 0.07/0.31 <=> ( in @ Xx @ B ) )
% 0.07/0.31 => ( A = B ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(emptyset_type,type,
% 0.07/0.31 emptyset: $i ).
% 0.07/0.31
% 0.07/0.31 thf(emptysetAx_type,type,
% 0.07/0.31 emptysetAx: $o ).
% 0.07/0.31
% 0.07/0.31 thf(emptysetAx,definition,
% 0.07/0.31 ( emptysetAx
% 0.07/0.31 = ( ! [Xx: $i] :
% 0.07/0.31 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(setadjoin_type,type,
% 0.07/0.31 setadjoin: $i > $i > $i ).
% 0.07/0.31
% 0.07/0.31 thf(setadjoinAx_type,type,
% 0.07/0.31 setadjoinAx: $o ).
% 0.07/0.31
% 0.07/0.31 thf(setadjoinAx,definition,
% 0.07/0.31 ( setadjoinAx
% 0.07/0.31 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.07/0.31 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.07/0.31 <=> ( ( Xy = Xx )
% 0.07/0.31 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(powerset_type,type,
% 0.07/0.31 powerset: $i > $i ).
% 0.07/0.31
% 0.07/0.31 thf(powersetAx_type,type,
% 0.07/0.31 powersetAx: $o ).
% 0.07/0.31
% 0.07/0.31 thf(powersetAx,definition,
% 0.07/0.31 ( powersetAx
% 0.07/0.31 = ( ! [A: $i,B: $i] :
% 0.07/0.31 ( ( in @ B @ ( powerset @ A ) )
% 0.07/0.31 <=> ! [Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ B )
% 0.07/0.31 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(setunion_type,type,
% 0.07/0.31 setunion: $i > $i ).
% 0.07/0.31
% 0.07/0.31 thf(setunionAx_type,type,
% 0.07/0.31 setunionAx: $o ).
% 0.07/0.31
% 0.07/0.31 thf(setunionAx,definition,
% 0.07/0.31 ( setunionAx
% 0.07/0.31 = ( ! [A: $i,Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ ( setunion @ A ) )
% 0.07/0.31 <=> ? [B: $i] :
% 0.07/0.31 ( ( in @ Xx @ B )
% 0.07/0.31 & ( in @ B @ A ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(omega_type,type,
% 0.07/0.31 omega: $i ).
% 0.07/0.31
% 0.07/0.31 thf(omega0Ax_type,type,
% 0.07/0.31 omega0Ax: $o ).
% 0.07/0.31
% 0.07/0.31 thf(omega0Ax,definition,
% 0.07/0.31 ( omega0Ax
% 0.07/0.31 = ( in @ emptyset @ omega ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(omegaSAx_type,type,
% 0.07/0.31 omegaSAx: $o ).
% 0.07/0.31
% 0.07/0.31 thf(omegaSAx,definition,
% 0.07/0.31 ( omegaSAx
% 0.07/0.31 = ( ! [Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ omega )
% 0.07/0.31 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(omegaIndAx_type,type,
% 0.07/0.31 omegaIndAx: $o ).
% 0.07/0.31
% 0.07/0.31 thf(omegaIndAx,definition,
% 0.07/0.31 ( omegaIndAx
% 0.07/0.31 = ( ! [A: $i] :
% 0.07/0.31 ( ( ( in @ emptyset @ A )
% 0.07/0.31 & ! [Xx: $i] :
% 0.07/0.31 ( ( ( in @ Xx @ omega )
% 0.07/0.31 & ( in @ Xx @ A ) )
% 0.07/0.31 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.07/0.31 => ! [Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ omega )
% 0.07/0.31 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(replAx_type,type,
% 0.07/0.31 replAx: $o ).
% 0.07/0.31
% 0.07/0.31 thf(replAx,definition,
% 0.07/0.31 ( replAx
% 0.07/0.31 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.07/0.31 ( ! [Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ A )
% 0.07/0.31 => ( exu
% 0.07/0.31 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.07/0.31 => ? [B: $i] :
% 0.07/0.31 ! [Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ B )
% 0.07/0.31 <=> ? [Xy: $i] :
% 0.07/0.31 ( ( in @ Xy @ A )
% 0.07/0.31 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(foundationAx_type,type,
% 0.07/0.31 foundationAx: $o ).
% 0.07/0.31
% 0.07/0.31 thf(foundationAx,definition,
% 0.07/0.31 ( foundationAx
% 0.07/0.31 = ( ! [A: $i] :
% 0.07/0.31 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.07/0.31 => ? [B: $i] :
% 0.07/0.31 ( ( in @ B @ A )
% 0.07/0.31 & ~ ? [Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ B )
% 0.07/0.31 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(wellorderingAx_type,type,
% 0.07/0.31 wellorderingAx: $o ).
% 0.07/0.31
% 0.07/0.31 thf(wellorderingAx,definition,
% 0.07/0.31 ( wellorderingAx
% 0.07/0.31 = ( ! [A: $i] :
% 0.07/0.31 ? [B: $i] :
% 0.07/0.31 ( ! [C: $i] :
% 0.07/0.31 ( ( in @ C @ B )
% 0.07/0.31 => ! [Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ C )
% 0.07/0.31 => ( in @ Xx @ A ) ) )
% 0.07/0.31 & ! [Xx: $i,Xy: $i] :
% 0.07/0.31 ( ( ( in @ Xx @ A )
% 0.07/0.31 & ( in @ Xy @ A ) )
% 0.07/0.31 => ( ! [C: $i] :
% 0.07/0.31 ( ( in @ C @ B )
% 0.07/0.31 => ( ( in @ Xx @ C )
% 0.07/0.31 <=> ( in @ Xy @ C ) ) )
% 0.07/0.31 => ( Xx = Xy ) ) )
% 0.07/0.31 & ! [C: $i,D: $i] :
% 0.07/0.31 ( ( ( in @ C @ B )
% 0.07/0.31 & ( in @ D @ B ) )
% 0.07/0.31 => ( ! [Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ C )
% 0.07/0.31 => ( in @ Xx @ D ) )
% 0.07/0.31 | ! [Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ D )
% 0.07/0.31 => ( in @ Xx @ C ) ) ) )
% 0.07/0.31 & ! [C: $i] :
% 0.07/0.31 ( ( ! [Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ C )
% 0.07/0.31 => ( in @ Xx @ A ) )
% 0.07/0.31 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.07/0.31 => ? [D: $i,Xx: $i] :
% 0.07/0.31 ( ( in @ D @ B )
% 0.07/0.31 & ( in @ Xx @ C )
% 0.07/0.31 & ~ ? [Xy: $i] :
% 0.07/0.31 ( ( in @ Xy @ D )
% 0.07/0.31 & ( in @ Xy @ C ) )
% 0.07/0.31 & ! [E: $i] :
% 0.07/0.31 ( ( in @ E @ B )
% 0.07/0.31 => ( ! [Xy: $i] :
% 0.07/0.31 ( ( in @ Xy @ E )
% 0.07/0.31 => ( in @ Xy @ D ) )
% 0.07/0.31 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(descr_type,type,
% 0.07/0.31 descr: ( $i > $o ) > $i ).
% 0.07/0.31
% 0.07/0.31 thf(descrp_type,type,
% 0.07/0.31 descrp: $o ).
% 0.07/0.31
% 0.07/0.31 thf(descrp,definition,
% 0.07/0.31 ( descrp
% 0.07/0.31 = ( ! [Xphi: $i > $o] :
% 0.07/0.31 ( ( exu
% 0.07/0.31 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.07/0.31 => ( Xphi
% 0.07/0.31 @ ( descr
% 0.07/0.31 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(dsetconstr_type,type,
% 0.07/0.31 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.07/0.31
% 0.07/0.31 thf(dsetconstrI_type,type,
% 0.07/0.31 dsetconstrI: $o ).
% 0.07/0.31
% 0.07/0.31 thf(dsetconstrI,definition,
% 0.07/0.31 ( dsetconstrI
% 0.07/0.31 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ A )
% 0.07/0.31 => ( ( Xphi @ Xx )
% 0.07/0.31 => ( in @ Xx
% 0.07/0.31 @ ( dsetconstr @ A
% 0.07/0.31 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(dsetconstrEL_type,type,
% 0.07/0.31 dsetconstrEL: $o ).
% 0.07/0.31
% 0.07/0.31 thf(dsetconstrEL,definition,
% 0.07/0.31 ( dsetconstrEL
% 0.07/0.31 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.07/0.31 ( ( in @ Xx
% 0.07/0.31 @ ( dsetconstr @ A
% 0.07/0.31 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.07/0.31 => ( in @ Xx @ A ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(dsetconstrER_type,type,
% 0.07/0.31 dsetconstrER: $o ).
% 0.07/0.31
% 0.07/0.31 thf(dsetconstrER,definition,
% 0.07/0.31 ( dsetconstrER
% 0.07/0.31 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.07/0.31 ( ( in @ Xx
% 0.07/0.31 @ ( dsetconstr @ A
% 0.07/0.31 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.07/0.31 => ( Xphi @ Xx ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(exuE1_type,type,
% 0.07/0.31 exuE1: $o ).
% 0.07/0.31
% 0.07/0.31 thf(exuE1,definition,
% 0.07/0.31 ( exuE1
% 0.07/0.31 = ( ! [Xphi: $i > $o] :
% 0.07/0.31 ( ( exu
% 0.07/0.31 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.07/0.31 => ? [Xx: $i] :
% 0.07/0.31 ( ( Xphi @ Xx )
% 0.07/0.31 & ! [Xy: $i] :
% 0.07/0.31 ( ( Xphi @ Xy )
% 0.07/0.31 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(prop2set_type,type,
% 0.07/0.31 prop2set: $o > $i ).
% 0.07/0.31
% 0.07/0.31 thf(prop2setE_type,type,
% 0.07/0.31 prop2setE: $o ).
% 0.07/0.31
% 0.07/0.31 thf(prop2setE,definition,
% 0.07/0.31 ( prop2setE
% 0.07/0.31 = ( ! [Xphi: $o,Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.07/0.31 => Xphi ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(emptysetE_type,type,
% 0.07/0.32 emptysetE: $o ).
% 0.07/0.32
% 0.07/0.32 thf(emptysetE,definition,
% 0.07/0.32 ( emptysetE
% 0.07/0.32 = ( ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ emptyset )
% 0.07/0.32 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(emptysetimpfalse_type,type,
% 0.07/0.32 emptysetimpfalse: $o ).
% 0.07/0.32
% 0.07/0.32 thf(emptysetimpfalse,definition,
% 0.07/0.32 ( emptysetimpfalse
% 0.07/0.32 = ( ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ emptyset )
% 0.07/0.32 => $false ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(notinemptyset_type,type,
% 0.07/0.32 notinemptyset: $o ).
% 0.07/0.32
% 0.07/0.32 thf(notinemptyset,definition,
% 0.07/0.32 ( notinemptyset
% 0.07/0.32 = ( ! [Xx: $i] :
% 0.07/0.32 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(exuE3e_type,type,
% 0.07/0.32 exuE3e: $o ).
% 0.07/0.32
% 0.07/0.32 thf(exuE3e,definition,
% 0.07/0.32 ( exuE3e
% 0.07/0.32 = ( ! [Xphi: $i > $o] :
% 0.07/0.32 ( ( exu
% 0.07/0.32 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.07/0.32 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(setext_type,type,
% 0.07/0.32 setext: $o ).
% 0.07/0.32
% 0.07/0.32 thf(setext,definition,
% 0.07/0.32 ( setext
% 0.07/0.32 = ( ! [A: $i,B: $i] :
% 0.07/0.32 ( ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( in @ Xx @ B ) )
% 0.07/0.32 => ( ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ B )
% 0.07/0.32 => ( in @ Xx @ A ) )
% 0.07/0.32 => ( A = B ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(emptyI_type,type,
% 0.07/0.32 emptyI: $o ).
% 0.07/0.32
% 0.07/0.32 thf(emptyI,definition,
% 0.07/0.32 ( emptyI
% 0.07/0.32 = ( ! [A: $i] :
% 0.07/0.32 ( ! [Xx: $i] :
% 0.07/0.32 ~ ( in @ Xx @ A )
% 0.07/0.32 => ( A = emptyset ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(noeltsimpempty_type,type,
% 0.07/0.32 noeltsimpempty: $o ).
% 0.07/0.32
% 0.07/0.32 thf(noeltsimpempty,definition,
% 0.07/0.32 ( noeltsimpempty
% 0.07/0.32 = ( ! [A: $i] :
% 0.07/0.32 ( ! [Xx: $i] :
% 0.07/0.32 ~ ( in @ Xx @ A )
% 0.07/0.32 => ( A = emptyset ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(setbeta_type,type,
% 0.07/0.32 setbeta: $o ).
% 0.07/0.32
% 0.07/0.32 thf(setbeta,definition,
% 0.07/0.32 ( setbeta
% 0.07/0.32 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( ( in @ Xx
% 0.07/0.32 @ ( dsetconstr @ A
% 0.07/0.32 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.07/0.32 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(nonempty_type,type,
% 0.07/0.32 nonempty: $i > $o ).
% 0.07/0.32
% 0.07/0.32 thf(nonempty,definition,
% 0.07/0.32 ( nonempty
% 0.07/0.32 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(nonemptyE1_type,type,
% 0.07/0.32 nonemptyE1: $o ).
% 0.07/0.32
% 0.07/0.32 thf(nonemptyE1,definition,
% 0.07/0.32 ( nonemptyE1
% 0.07/0.32 = ( ! [A: $i] :
% 0.07/0.32 ( ( nonempty @ A )
% 0.07/0.32 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(nonemptyI_type,type,
% 0.07/0.32 nonemptyI: $o ).
% 0.07/0.32
% 0.07/0.32 thf(nonemptyI,definition,
% 0.07/0.32 ( nonemptyI
% 0.07/0.32 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( ( Xphi @ Xx )
% 0.07/0.32 => ( nonempty
% 0.07/0.32 @ ( dsetconstr @ A
% 0.07/0.32 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(nonemptyI1_type,type,
% 0.07/0.32 nonemptyI1: $o ).
% 0.07/0.32
% 0.07/0.32 thf(nonemptyI1,definition,
% 0.07/0.32 ( nonemptyI1
% 0.07/0.32 = ( ! [A: $i] :
% 0.07/0.32 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.07/0.32 => ( nonempty @ A ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(setadjoinIL_type,type,
% 0.07/0.32 setadjoinIL: $o ).
% 0.07/0.32
% 0.07/0.32 thf(setadjoinIL,definition,
% 0.07/0.32 ( setadjoinIL
% 0.07/0.32 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(emptyinunitempty_type,type,
% 0.07/0.32 emptyinunitempty: $o ).
% 0.07/0.32
% 0.07/0.32 thf(emptyinunitempty,definition,
% 0.07/0.32 ( emptyinunitempty
% 0.07/0.32 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(setadjoinIR_type,type,
% 0.07/0.32 setadjoinIR: $o ).
% 0.07/0.32
% 0.07/0.32 thf(setadjoinIR,definition,
% 0.07/0.32 ( setadjoinIR
% 0.07/0.32 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.07/0.32 ( ( in @ Xy @ A )
% 0.07/0.32 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(setadjoinE_type,type,
% 0.07/0.32 setadjoinE: $o ).
% 0.07/0.32
% 0.07/0.32 thf(setadjoinE,definition,
% 0.07/0.32 ( setadjoinE
% 0.07/0.32 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.07/0.32 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.07/0.32 => ! [Xphi: $o] :
% 0.07/0.32 ( ( ( Xy = Xx )
% 0.07/0.32 => Xphi )
% 0.07/0.32 => ( ( ( in @ Xy @ A )
% 0.07/0.32 => Xphi )
% 0.07/0.32 => Xphi ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(setadjoinOr_type,type,
% 0.07/0.32 setadjoinOr: $o ).
% 0.07/0.32
% 0.07/0.32 thf(setadjoinOr,definition,
% 0.07/0.32 ( setadjoinOr
% 0.07/0.32 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.07/0.32 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.07/0.32 => ( ( Xy = Xx )
% 0.07/0.32 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(setoftrueEq_type,type,
% 0.07/0.32 setoftrueEq: $o ).
% 0.07/0.32
% 0.07/0.32 thf(setoftrueEq,definition,
% 0.07/0.32 ( setoftrueEq
% 0.07/0.32 = ( ! [A: $i] :
% 0.07/0.32 ( ( dsetconstr @ A
% 0.07/0.32 @ ^ [Xx: $i] : $true )
% 0.07/0.32 = A ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(powersetI_type,type,
% 0.07/0.32 powersetI: $o ).
% 0.07/0.32
% 0.07/0.32 thf(powersetI,definition,
% 0.07/0.32 ( powersetI
% 0.07/0.32 = ( ! [A: $i,B: $i] :
% 0.07/0.32 ( ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ B )
% 0.07/0.32 => ( in @ Xx @ A ) )
% 0.07/0.32 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(emptyinPowerset_type,type,
% 0.07/0.32 emptyinPowerset: $o ).
% 0.07/0.32
% 0.07/0.32 thf(emptyinPowerset,definition,
% 0.07/0.32 ( emptyinPowerset
% 0.07/0.32 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(emptyInPowerset_type,type,
% 0.07/0.32 emptyInPowerset: $o ).
% 0.07/0.32
% 0.07/0.32 thf(emptyInPowerset,definition,
% 0.07/0.32 ( emptyInPowerset
% 0.07/0.32 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(powersetE_type,type,
% 0.07/0.32 powersetE: $o ).
% 0.07/0.32
% 0.07/0.32 thf(powersetE,definition,
% 0.07/0.32 ( powersetE
% 0.07/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.32 ( ( in @ B @ ( powerset @ A ) )
% 0.07/0.32 => ( ( in @ Xx @ B )
% 0.07/0.32 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(setunionI_type,type,
% 0.07/0.32 setunionI: $o ).
% 0.07/0.32
% 0.07/0.32 thf(setunionI,definition,
% 0.07/0.32 ( setunionI
% 0.07/0.32 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.07/0.32 ( ( in @ Xx @ B )
% 0.07/0.32 => ( ( in @ B @ A )
% 0.07/0.32 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(setunionE_type,type,
% 0.07/0.32 setunionE: $o ).
% 0.07/0.32
% 0.07/0.32 thf(setunionE,definition,
% 0.07/0.32 ( setunionE
% 0.07/0.32 = ( ! [A: $i,Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ ( setunion @ A ) )
% 0.07/0.32 => ! [Xphi: $o] :
% 0.07/0.32 ( ! [B: $i] :
% 0.07/0.32 ( ( in @ Xx @ B )
% 0.07/0.32 => ( ( in @ B @ A )
% 0.07/0.32 => Xphi ) )
% 0.07/0.32 => Xphi ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(subPowSU_type,type,
% 0.07/0.32 subPowSU: $o ).
% 0.07/0.32
% 0.07/0.32 thf(subPowSU,definition,
% 0.07/0.32 ( subPowSU
% 0.07/0.32 = ( ! [A: $i,Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(exuE2_type,type,
% 0.07/0.32 exuE2: $o ).
% 0.07/0.32
% 0.07/0.32 thf(exuE2,definition,
% 0.07/0.32 ( exuE2
% 0.07/0.32 = ( ! [Xphi: $i > $o] :
% 0.07/0.32 ( ( exu
% 0.07/0.32 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.07/0.32 => ? [Xx: $i] :
% 0.07/0.32 ! [Xy: $i] :
% 0.07/0.32 ( ( Xphi @ Xy )
% 0.07/0.32 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(nonemptyImpWitness_type,type,
% 0.07/0.32 nonemptyImpWitness: $o ).
% 0.07/0.32
% 0.07/0.32 thf(nonemptyImpWitness,definition,
% 0.07/0.32 ( nonemptyImpWitness
% 0.07/0.32 = ( ! [A: $i] :
% 0.07/0.32 ( ( nonempty @ A )
% 0.07/0.32 => ? [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 & $true ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(uniqinunit_type,type,
% 0.07/0.32 uniqinunit: $o ).
% 0.07/0.32
% 0.07/0.32 thf(uniqinunit,definition,
% 0.07/0.32 ( uniqinunit
% 0.07/0.32 = ( ! [Xx: $i,Xy: $i] :
% 0.07/0.32 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.07/0.32 => ( Xx = Xy ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(notinsingleton_type,type,
% 0.07/0.32 notinsingleton: $o ).
% 0.07/0.32
% 0.07/0.32 thf(notinsingleton,definition,
% 0.07/0.32 ( notinsingleton
% 0.07/0.32 = ( ! [Xx: $i,Xy: $i] :
% 0.07/0.32 ( ( Xx != Xy )
% 0.07/0.32 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(eqinunit_type,type,
% 0.07/0.32 eqinunit: $o ).
% 0.07/0.32
% 0.07/0.32 thf(eqinunit,definition,
% 0.07/0.32 ( eqinunit
% 0.07/0.32 = ( ! [Xx: $i,Xy: $i] :
% 0.07/0.32 ( ( Xx = Xy )
% 0.07/0.32 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(singletonsswitch_type,type,
% 0.07/0.32 singletonsswitch: $o ).
% 0.07/0.32
% 0.07/0.32 thf(singletonsswitch,definition,
% 0.07/0.32 ( singletonsswitch
% 0.07/0.32 = ( ! [Xx: $i,Xy: $i] :
% 0.07/0.32 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.07/0.32 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(upairsetE_type,type,
% 0.07/0.32 upairsetE: $o ).
% 0.07/0.32
% 0.07/0.32 thf(upairsetE,definition,
% 0.07/0.32 ( upairsetE
% 0.07/0.32 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.07/0.32 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.07/0.32 => ( ( Xz = Xx )
% 0.07/0.32 | ( Xz = Xy ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(upairsetIL_type,type,
% 0.07/0.32 upairsetIL: $o ).
% 0.07/0.32
% 0.07/0.32 thf(upairsetIL,definition,
% 0.07/0.32 ( upairsetIL
% 0.07/0.32 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(upairsetIR_type,type,
% 0.07/0.32 upairsetIR: $o ).
% 0.07/0.32
% 0.07/0.32 thf(upairsetIR,definition,
% 0.07/0.32 ( upairsetIR
% 0.07/0.32 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(emptyE1_type,type,
% 0.07/0.32 emptyE1: $o ).
% 0.07/0.32
% 0.07/0.32 thf(emptyE1,definition,
% 0.07/0.32 ( emptyE1
% 0.07/0.32 = ( ! [A: $i,Xphi: $i > $o] :
% 0.07/0.32 ( ? [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 & ( Xphi @ Xx ) )
% 0.07/0.32 => ( ( ( dsetconstr @ A
% 0.07/0.32 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.07/0.32 = emptyset )
% 0.07/0.32 => $false ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(vacuousDall_type,type,
% 0.07/0.32 vacuousDall: $o ).
% 0.07/0.32
% 0.07/0.32 thf(vacuousDall,definition,
% 0.07/0.32 ( vacuousDall
% 0.07/0.32 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ emptyset )
% 0.07/0.32 => ( Xphi @ Xx ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(quantDeMorgan1_type,type,
% 0.07/0.32 quantDeMorgan1: $o ).
% 0.07/0.32
% 0.07/0.32 thf(quantDeMorgan1,definition,
% 0.07/0.32 ( quantDeMorgan1
% 0.07/0.32 = ( ! [A: $i,Xphi: $i > $o] :
% 0.07/0.32 ( ~ ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( Xphi @ Xx ) )
% 0.07/0.32 => ? [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(quantDeMorgan2_type,type,
% 0.07/0.32 quantDeMorgan2: $o ).
% 0.07/0.32
% 0.07/0.32 thf(quantDeMorgan2,definition,
% 0.07/0.32 ( quantDeMorgan2
% 0.07/0.32 = ( ! [A: $i,Xphi: $i > $o] :
% 0.07/0.32 ( ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ~ ( Xphi @ Xx ) )
% 0.07/0.32 => ~ ? [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(quantDeMorgan3_type,type,
% 0.07/0.32 quantDeMorgan3: $o ).
% 0.07/0.32
% 0.07/0.32 thf(quantDeMorgan3,definition,
% 0.07/0.32 ( quantDeMorgan3
% 0.07/0.32 = ( ! [A: $i,Xphi: $i > $o] :
% 0.07/0.32 ( ~ ? [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 & ( Xphi @ Xx ) )
% 0.07/0.32 => ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(quantDeMorgan4_type,type,
% 0.07/0.32 quantDeMorgan4: $o ).
% 0.07/0.32
% 0.07/0.32 thf(quantDeMorgan4,definition,
% 0.07/0.32 ( quantDeMorgan4
% 0.07/0.32 = ( ! [A: $i,Xphi: $i > $o] :
% 0.07/0.32 ( ? [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 & ~ ( Xphi @ Xx ) )
% 0.07/0.32 => ~ ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(prop2setI_type,type,
% 0.07/0.32 prop2setI: $o ).
% 0.07/0.32
% 0.07/0.32 thf(prop2setI,definition,
% 0.07/0.32 ( prop2setI
% 0.07/0.32 = ( ! [Xphi: $o] :
% 0.07/0.32 ( Xphi
% 0.07/0.32 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(set2prop_type,type,
% 0.07/0.32 set2prop: $i > $o ).
% 0.07/0.32
% 0.07/0.32 thf(prop2set2propI_type,type,
% 0.07/0.32 prop2set2propI: $o ).
% 0.07/0.32
% 0.07/0.32 thf(prop2set2propI,definition,
% 0.07/0.32 ( prop2set2propI
% 0.07/0.32 = ( ! [Xphi: $o] :
% 0.07/0.32 ( Xphi
% 0.07/0.32 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(notdexE_type,type,
% 0.07/0.32 notdexE: $o ).
% 0.07/0.32
% 0.07/0.32 thf(notdexE,definition,
% 0.07/0.32 ( notdexE
% 0.07/0.32 = ( ! [A: $i,Xphi: $i > $o] :
% 0.07/0.32 ( ~ ? [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 & ( Xphi @ Xx ) )
% 0.07/0.32 => ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(notdallE_type,type,
% 0.07/0.32 notdallE: $o ).
% 0.07/0.32
% 0.07/0.32 thf(notdallE,definition,
% 0.07/0.32 ( notdallE
% 0.07/0.32 = ( ! [A: $i,Xphi: $i > $o] :
% 0.07/0.32 ( ~ ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( Xphi @ Xx ) )
% 0.07/0.32 => ? [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(exuI1_type,type,
% 0.07/0.32 exuI1: $o ).
% 0.07/0.32
% 0.07/0.32 thf(exuI1,definition,
% 0.07/0.32 ( exuI1
% 0.07/0.32 = ( ! [Xphi: $i > $o] :
% 0.07/0.32 ( ? [Xx: $i] :
% 0.07/0.32 ( ( Xphi @ Xx )
% 0.07/0.32 & ! [Xy: $i] :
% 0.07/0.32 ( ( Xphi @ Xy )
% 0.07/0.32 => ( Xx = Xy ) ) )
% 0.07/0.32 => ( exu
% 0.07/0.32 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(exuI3_type,type,
% 0.07/0.32 exuI3: $o ).
% 0.07/0.32
% 0.07/0.32 thf(exuI3,definition,
% 0.07/0.32 ( exuI3
% 0.07/0.32 = ( ! [Xphi: $i > $o] :
% 0.07/0.32 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.07/0.32 => ( ! [Xx: $i,Xy: $i] :
% 0.07/0.32 ( ( Xphi @ Xx )
% 0.07/0.32 => ( ( Xphi @ Xy )
% 0.07/0.32 => ( Xx = Xy ) ) )
% 0.07/0.32 => ( exu
% 0.07/0.32 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(exuI2_type,type,
% 0.07/0.32 exuI2: $o ).
% 0.07/0.32
% 0.07/0.32 thf(exuI2,definition,
% 0.07/0.32 ( exuI2
% 0.07/0.32 = ( ! [Xphi: $i > $o] :
% 0.07/0.32 ( ? [Xx: $i] :
% 0.07/0.32 ! [Xy: $i] :
% 0.07/0.32 ( ( Xphi @ Xy )
% 0.07/0.32 <=> ( Xy = Xx ) )
% 0.07/0.32 => ( exu
% 0.07/0.32 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(inCongP_type,type,
% 0.07/0.32 inCongP: $o ).
% 0.07/0.32
% 0.07/0.32 thf(inCongP,definition,
% 0.07/0.32 ( inCongP
% 0.07/0.32 = ( ! [A: $i,B: $i] :
% 0.07/0.32 ( ( A = B )
% 0.07/0.32 => ! [Xx: $i,Xy: $i] :
% 0.07/0.32 ( ( Xx = Xy )
% 0.07/0.32 => ( ( in @ Xx @ A )
% 0.07/0.32 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(in__Cong_type,type,
% 0.07/0.32 in__Cong: $o ).
% 0.07/0.32
% 0.07/0.32 thf(in__Cong,definition,
% 0.07/0.32 ( in__Cong
% 0.07/0.32 = ( ! [A: $i,B: $i] :
% 0.07/0.32 ( ( A = B )
% 0.07/0.32 => ! [Xx: $i,Xy: $i] :
% 0.07/0.32 ( ( Xx = Xy )
% 0.07/0.32 => ( ( in @ Xx @ A )
% 0.07/0.32 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(exuE3u_type,type,
% 0.07/0.32 exuE3u: $o ).
% 0.07/0.32
% 0.07/0.32 thf(exuE3u,definition,
% 0.07/0.32 ( exuE3u
% 0.07/0.32 = ( ! [Xphi: $i > $o] :
% 0.07/0.32 ( ( exu
% 0.07/0.32 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.07/0.32 => ! [Xx: $i,Xy: $i] :
% 0.07/0.32 ( ( Xphi @ Xx )
% 0.07/0.32 => ( ( Xphi @ Xy )
% 0.07/0.32 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(exu__Cong_type,type,
% 0.07/0.32 exu__Cong: $o ).
% 0.07/0.32
% 0.07/0.32 thf(exu__Cong,definition,
% 0.07/0.32 ( exu__Cong
% 0.07/0.32 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.07/0.32 ( ! [Xx: $i,Xy: $i] :
% 0.07/0.32 ( ( Xx = Xy )
% 0.07/0.32 => ( ( Xphi @ Xx )
% 0.07/0.32 <=> ( Xpsi @ Xy ) ) )
% 0.07/0.32 => ( ( exu
% 0.07/0.32 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.07/0.32 <=> ( exu
% 0.07/0.32 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(emptyset__Cong_type,type,
% 0.07/0.32 emptyset__Cong: $o ).
% 0.07/0.32
% 0.07/0.32 thf(emptyset__Cong,definition,
% 0.07/0.32 ( emptyset__Cong
% 0.07/0.32 = ( emptyset = emptyset ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(setadjoin__Cong_type,type,
% 0.07/0.32 setadjoin__Cong: $o ).
% 0.07/0.32
% 0.07/0.32 thf(setadjoin__Cong,definition,
% 0.07/0.32 ( setadjoin__Cong
% 0.07/0.32 = ( ! [Xx: $i,Xy: $i] :
% 0.07/0.32 ( ( Xx = Xy )
% 0.07/0.32 => ! [Xz: $i,Xu: $i] :
% 0.07/0.32 ( ( Xz = Xu )
% 0.07/0.32 => ( ( setadjoin @ Xx @ Xz )
% 0.07/0.32 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(powerset__Cong_type,type,
% 0.07/0.32 powerset__Cong: $o ).
% 0.07/0.32
% 0.07/0.32 thf(powerset__Cong,definition,
% 0.07/0.32 ( powerset__Cong
% 0.07/0.32 = ( ! [A: $i,B: $i] :
% 0.07/0.32 ( ( A = B )
% 0.07/0.32 => ( ( powerset @ A )
% 0.07/0.32 = ( powerset @ B ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(setunion__Cong_type,type,
% 0.07/0.32 setunion__Cong: $o ).
% 0.07/0.32
% 0.07/0.32 thf(setunion__Cong,definition,
% 0.07/0.32 ( setunion__Cong
% 0.07/0.32 = ( ! [A: $i,B: $i] :
% 0.07/0.32 ( ( A = B )
% 0.07/0.32 => ( ( setunion @ A )
% 0.07/0.32 = ( setunion @ B ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(omega__Cong_type,type,
% 0.07/0.32 omega__Cong: $o ).
% 0.07/0.32
% 0.07/0.32 thf(omega__Cong,definition,
% 0.07/0.32 ( omega__Cong
% 0.07/0.32 = ( omega = omega ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(exuEu_type,type,
% 0.07/0.32 exuEu: $o ).
% 0.07/0.32
% 0.07/0.32 thf(exuEu,definition,
% 0.07/0.32 ( exuEu
% 0.07/0.32 = ( ! [Xphi: $i > $o] :
% 0.07/0.32 ( ( exu
% 0.07/0.32 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.07/0.32 => ! [Xx: $i,Xy: $i] :
% 0.07/0.32 ( ( Xphi @ Xx )
% 0.07/0.32 => ( ( Xphi @ Xy )
% 0.07/0.32 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(descr__Cong_type,type,
% 0.07/0.32 descr__Cong: $o ).
% 0.07/0.32
% 0.07/0.32 thf(descr__Cong,definition,
% 0.07/0.32 ( descr__Cong
% 0.07/0.32 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.07/0.32 ( ! [Xx: $i,Xy: $i] :
% 0.07/0.32 ( ( Xx = Xy )
% 0.07/0.32 => ( ( Xphi @ Xx )
% 0.07/0.32 <=> ( Xpsi @ Xy ) ) )
% 0.07/0.32 => ( ( exu
% 0.07/0.32 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.07/0.32 => ( ( exu
% 0.07/0.32 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.07/0.32 => ( ( descr
% 0.07/0.32 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.07/0.32 = ( descr
% 0.07/0.32 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(dsetconstr__Cong_type,type,
% 0.07/0.32 dsetconstr__Cong: $o ).
% 0.07/0.32
% 0.07/0.32 thf(dsetconstr__Cong,definition,
% 0.07/0.32 ( dsetconstr__Cong
% 0.07/0.32 = ( ! [A: $i,B: $i] :
% 0.07/0.32 ( ( A = B )
% 0.07/0.32 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.07/0.32 ( ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ! [Xy: $i] :
% 0.07/0.32 ( ( in @ Xy @ B )
% 0.07/0.32 => ( ( Xx = Xy )
% 0.07/0.32 => ( ( Xphi @ Xx )
% 0.07/0.32 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.07/0.32 => ( ( dsetconstr @ A
% 0.07/0.32 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.07/0.32 = ( dsetconstr @ B
% 0.07/0.32 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(subset_type,type,
% 0.07/0.32 subset: $i > $i > $o ).
% 0.07/0.32
% 0.07/0.32 thf(disjoint_type,type,
% 0.07/0.32 disjoint: $i > $i > $o ).
% 0.07/0.32
% 0.07/0.32 thf(setsmeet_type,type,
% 0.07/0.32 setsmeet: $i > $i > $o ).
% 0.07/0.32
% 0.07/0.32 thf(subsetI1_type,type,
% 0.07/0.32 subsetI1: $o ).
% 0.07/0.32
% 0.07/0.32 thf(subsetI1,definition,
% 0.07/0.32 ( subsetI1
% 0.07/0.32 = ( ! [A: $i,B: $i] :
% 0.07/0.32 ( ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( in @ Xx @ B ) )
% 0.07/0.32 => ( subset @ A @ B ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(eqimpsubset2_type,type,
% 0.07/0.32 eqimpsubset2: $o ).
% 0.07/0.32
% 0.07/0.32 thf(eqimpsubset2,definition,
% 0.07/0.32 ( eqimpsubset2
% 0.07/0.32 = ( ! [A: $i,B: $i] :
% 0.07/0.32 ( ( A = B )
% 0.07/0.32 => ( subset @ B @ A ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(eqimpsubset1_type,type,
% 0.07/0.32 eqimpsubset1: $o ).
% 0.07/0.32
% 0.07/0.32 thf(eqimpsubset1,definition,
% 0.07/0.32 ( eqimpsubset1
% 0.07/0.32 = ( ! [A: $i,B: $i] :
% 0.07/0.32 ( ( A = B )
% 0.07/0.32 => ( subset @ A @ B ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(subsetI2_type,type,
% 0.07/0.32 subsetI2: $o ).
% 0.07/0.32
% 0.07/0.32 thf(subsetI2,definition,
% 0.07/0.32 ( subsetI2
% 0.07/0.32 = ( ! [A: $i,B: $i] :
% 0.07/0.32 ( ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( in @ Xx @ B ) )
% 0.07/0.32 => ( subset @ A @ B ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(emptysetsubset_type,type,
% 0.07/0.32 emptysetsubset: $o ).
% 0.07/0.32
% 0.07/0.32 thf(emptysetsubset,definition,
% 0.07/0.32 ( emptysetsubset
% 0.07/0.32 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(subsetE_type,type,
% 0.07/0.32 subsetE: $o ).
% 0.07/0.32
% 0.07/0.32 thf(subsetE,definition,
% 0.07/0.32 ( subsetE
% 0.07/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.32 ( ( subset @ A @ B )
% 0.07/0.32 => ( ( in @ Xx @ A )
% 0.07/0.32 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(subsetE2_type,type,
% 0.07/0.32 subsetE2: $o ).
% 0.07/0.32
% 0.07/0.32 thf(subsetE2,definition,
% 0.07/0.32 ( subsetE2
% 0.07/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.32 ( ( subset @ A @ B )
% 0.07/0.32 => ( ~ ( in @ Xx @ B )
% 0.07/0.32 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(notsubsetI_type,type,
% 0.07/0.32 notsubsetI: $o ).
% 0.07/0.32
% 0.07/0.32 thf(notsubsetI,definition,
% 0.07/0.32 ( notsubsetI
% 0.07/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( ~ ( in @ Xx @ B )
% 0.07/0.32 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(notequalI1_type,type,
% 0.07/0.32 notequalI1: $o ).
% 0.07/0.32
% 0.07/0.32 thf(notequalI1,definition,
% 0.07/0.32 ( notequalI1
% 0.07/0.32 = ( ! [A: $i,B: $i] :
% 0.07/0.32 ( ~ ( subset @ A @ B )
% 0.07/0.32 => ( A != B ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(notequalI2_type,type,
% 0.07/0.32 notequalI2: $o ).
% 0.07/0.32
% 0.07/0.32 thf(notequalI2,definition,
% 0.07/0.32 ( notequalI2
% 0.07/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( ~ ( in @ Xx @ B )
% 0.07/0.32 => ( A != B ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(subsetRefl_type,type,
% 0.07/0.32 subsetRefl: $o ).
% 0.07/0.32
% 0.07/0.32 thf(subsetRefl,definition,
% 0.07/0.32 ( subsetRefl
% 0.07/0.32 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(subsetTrans_type,type,
% 0.07/0.32 subsetTrans: $o ).
% 0.07/0.32
% 0.07/0.32 thf(subsetTrans,definition,
% 0.07/0.32 ( subsetTrans
% 0.07/0.32 = ( ! [A: $i,B: $i,C: $i] :
% 0.07/0.32 ( ( subset @ A @ B )
% 0.07/0.32 => ( ( subset @ B @ C )
% 0.07/0.32 => ( subset @ A @ C ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(setadjoinSub_type,type,
% 0.07/0.32 setadjoinSub: $o ).
% 0.07/0.32
% 0.07/0.32 thf(setadjoinSub,definition,
% 0.07/0.32 ( setadjoinSub
% 0.07/0.32 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(setadjoinSub2_type,type,
% 0.07/0.32 setadjoinSub2: $o ).
% 0.07/0.32
% 0.07/0.32 thf(setadjoinSub2,definition,
% 0.07/0.32 ( setadjoinSub2
% 0.07/0.32 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.07/0.32 ( ( subset @ A @ B )
% 0.07/0.32 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(subset2powerset_type,type,
% 0.07/0.32 subset2powerset: $o ).
% 0.07/0.32
% 0.07/0.32 thf(subset2powerset,definition,
% 0.07/0.32 ( subset2powerset
% 0.07/0.32 = ( ! [A: $i,B: $i] :
% 0.07/0.32 ( ( subset @ A @ B )
% 0.07/0.32 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(setextsub_type,type,
% 0.07/0.32 setextsub: $o ).
% 0.07/0.32
% 0.07/0.32 thf(setextsub,definition,
% 0.07/0.32 ( setextsub
% 0.07/0.32 = ( ! [A: $i,B: $i] :
% 0.07/0.32 ( ( subset @ A @ B )
% 0.07/0.32 => ( ( subset @ B @ A )
% 0.07/0.32 => ( A = B ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(subsetemptysetimpeq_type,type,
% 0.07/0.32 subsetemptysetimpeq: $o ).
% 0.07/0.32
% 0.07/0.32 thf(subsetemptysetimpeq,definition,
% 0.07/0.32 ( subsetemptysetimpeq
% 0.07/0.32 = ( ! [A: $i] :
% 0.07/0.32 ( ( subset @ A @ emptyset )
% 0.07/0.32 => ( A = emptyset ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(powersetI1_type,type,
% 0.07/0.32 powersetI1: $o ).
% 0.07/0.32
% 0.07/0.32 thf(powersetI1,definition,
% 0.07/0.32 ( powersetI1
% 0.07/0.32 = ( ! [A: $i,B: $i] :
% 0.07/0.32 ( ( subset @ B @ A )
% 0.07/0.32 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(powersetE1_type,type,
% 0.07/0.32 powersetE1: $o ).
% 0.07/0.32
% 0.07/0.32 thf(powersetE1,definition,
% 0.07/0.32 ( powersetE1
% 0.07/0.32 = ( ! [A: $i,B: $i] :
% 0.07/0.32 ( ( in @ B @ ( powerset @ A ) )
% 0.07/0.32 => ( subset @ B @ A ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(inPowerset_type,type,
% 0.07/0.32 inPowerset: $o ).
% 0.07/0.32
% 0.07/0.32 thf(inPowerset,definition,
% 0.07/0.32 ( inPowerset
% 0.07/0.32 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(powersetsubset_type,type,
% 0.07/0.32 powersetsubset: $o ).
% 0.07/0.32
% 0.07/0.32 thf(powersetsubset,definition,
% 0.07/0.32 ( powersetsubset
% 0.07/0.32 = ( ! [A: $i,B: $i] :
% 0.07/0.32 ( ( subset @ A @ B )
% 0.07/0.32 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(sepInPowerset_type,type,
% 0.07/0.32 sepInPowerset: $o ).
% 0.07/0.32
% 0.07/0.32 thf(sepInPowerset,definition,
% 0.07/0.32 ( sepInPowerset
% 0.07/0.32 = ( ! [A: $i,Xphi: $i > $o] :
% 0.07/0.32 ( in
% 0.07/0.32 @ ( dsetconstr @ A
% 0.07/0.32 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.07/0.32 @ ( powerset @ A ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(sepSubset_type,type,
% 0.07/0.32 sepSubset: $o ).
% 0.07/0.32
% 0.07/0.32 thf(sepSubset,definition,
% 0.07/0.32 ( sepSubset
% 0.07/0.32 = ( ! [A: $i,Xphi: $i > $o] :
% 0.07/0.32 ( subset
% 0.07/0.32 @ ( dsetconstr @ A
% 0.07/0.32 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.07/0.32 @ A ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(binunion_type,type,
% 0.07/0.32 binunion: $i > $i > $i ).
% 0.07/0.32
% 0.07/0.32 thf(binunionIL_type,type,
% 0.07/0.32 binunionIL: $o ).
% 0.07/0.32
% 0.07/0.32 thf(binunionIL,definition,
% 0.07/0.32 ( binunionIL
% 0.07/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(upairset2IR_type,type,
% 0.07/0.32 upairset2IR: $o ).
% 0.07/0.32
% 0.07/0.32 thf(upairset2IR,definition,
% 0.07/0.32 ( upairset2IR
% 0.07/0.32 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(binunionIR_type,type,
% 0.07/0.32 binunionIR: $o ).
% 0.07/0.32
% 0.07/0.32 thf(binunionIR,definition,
% 0.07/0.32 ( binunionIR
% 0.07/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ B )
% 0.07/0.32 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(binunionEcases_type,type,
% 0.07/0.32 binunionEcases: $o ).
% 0.07/0.32
% 0.07/0.32 thf(binunionEcases,definition,
% 0.07/0.32 ( binunionEcases
% 0.07/0.32 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.07/0.32 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.07/0.32 => ( ( ( in @ Xx @ A )
% 0.07/0.32 => Xphi )
% 0.07/0.32 => ( ( ( in @ Xx @ B )
% 0.07/0.32 => Xphi )
% 0.07/0.32 => Xphi ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(binunionE_type,type,
% 0.07/0.32 binunionE: $o ).
% 0.07/0.32
% 0.07/0.32 thf(binunionE,definition,
% 0.07/0.32 ( binunionE
% 0.07/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.07/0.32 => ( ( in @ Xx @ A )
% 0.07/0.32 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(binunionLsub_type,type,
% 0.07/0.32 binunionLsub: $o ).
% 0.07/0.32
% 0.07/0.32 thf(binunionLsub,definition,
% 0.07/0.32 ( binunionLsub
% 0.07/0.32 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(binunionRsub_type,type,
% 0.07/0.32 binunionRsub: $o ).
% 0.07/0.32
% 0.07/0.32 thf(binunionRsub,definition,
% 0.07/0.32 ( binunionRsub
% 0.07/0.32 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(binintersect_type,type,
% 0.07/0.32 binintersect: $i > $i > $i ).
% 0.07/0.32
% 0.07/0.32 thf(binintersectI_type,type,
% 0.07/0.32 binintersectI: $o ).
% 0.07/0.32
% 0.07/0.32 thf(binintersectI,definition,
% 0.07/0.32 ( binintersectI
% 0.07/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( ( in @ Xx @ B )
% 0.07/0.32 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(binintersectSubset5_type,type,
% 0.07/0.32 binintersectSubset5: $o ).
% 0.07/0.32
% 0.07/0.32 thf(binintersectSubset5,definition,
% 0.07/0.32 ( binintersectSubset5
% 0.07/0.32 = ( ! [A: $i,B: $i,C: $i] :
% 0.07/0.32 ( ( subset @ C @ A )
% 0.07/0.32 => ( ( subset @ C @ B )
% 0.07/0.32 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(binintersectEL_type,type,
% 0.07/0.32 binintersectEL: $o ).
% 0.07/0.32
% 0.07/0.32 thf(binintersectEL,definition,
% 0.07/0.32 ( binintersectEL
% 0.07/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.07/0.32 => ( in @ Xx @ A ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(binintersectLsub_type,type,
% 0.07/0.32 binintersectLsub: $o ).
% 0.07/0.32
% 0.07/0.32 thf(binintersectLsub,definition,
% 0.07/0.32 ( binintersectLsub
% 0.07/0.32 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(binintersectSubset2_type,type,
% 0.07/0.32 binintersectSubset2: $o ).
% 0.07/0.32
% 0.07/0.32 thf(binintersectSubset2,definition,
% 0.07/0.32 ( binintersectSubset2
% 0.07/0.32 = ( ! [A: $i,B: $i] :
% 0.07/0.32 ( ( subset @ A @ B )
% 0.07/0.32 => ( ( binintersect @ A @ B )
% 0.07/0.32 = A ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(binintersectSubset3_type,type,
% 0.07/0.32 binintersectSubset3: $o ).
% 0.07/0.32
% 0.07/0.32 thf(binintersectSubset3,definition,
% 0.07/0.32 ( binintersectSubset3
% 0.07/0.32 = ( ! [A: $i,B: $i] :
% 0.07/0.32 ( ( ( binintersect @ A @ B )
% 0.07/0.32 = B )
% 0.07/0.32 => ( subset @ B @ A ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(binintersectER_type,type,
% 0.07/0.32 binintersectER: $o ).
% 0.07/0.32
% 0.07/0.32 thf(binintersectER,definition,
% 0.07/0.32 ( binintersectER
% 0.07/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.07/0.32 => ( in @ Xx @ B ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(disjointsetsI1_type,type,
% 0.07/0.32 disjointsetsI1: $o ).
% 0.07/0.32
% 0.07/0.32 thf(disjointsetsI1,definition,
% 0.07/0.32 ( disjointsetsI1
% 0.07/0.32 = ( ! [A: $i,B: $i] :
% 0.07/0.32 ( ~ ? [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 & ( in @ Xx @ B ) )
% 0.07/0.32 => ( ( binintersect @ A @ B )
% 0.07/0.32 = emptyset ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(binintersectRsub_type,type,
% 0.07/0.32 binintersectRsub: $o ).
% 0.07/0.32
% 0.07/0.32 thf(binintersectRsub,definition,
% 0.07/0.32 ( binintersectRsub
% 0.07/0.32 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(binintersectSubset4_type,type,
% 0.07/0.32 binintersectSubset4: $o ).
% 0.07/0.32
% 0.07/0.32 thf(binintersectSubset4,definition,
% 0.07/0.32 ( binintersectSubset4
% 0.07/0.32 = ( ! [A: $i,B: $i] :
% 0.07/0.32 ( ( subset @ B @ A )
% 0.07/0.32 => ( ( binintersect @ A @ B )
% 0.07/0.32 = B ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(binintersectSubset1_type,type,
% 0.07/0.32 binintersectSubset1: $o ).
% 0.07/0.32
% 0.07/0.32 thf(binintersectSubset1,definition,
% 0.07/0.32 ( binintersectSubset1
% 0.07/0.32 = ( ! [A: $i,B: $i] :
% 0.07/0.32 ( ( ( binintersect @ A @ B )
% 0.07/0.32 = A )
% 0.07/0.32 => ( subset @ A @ B ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(bs114d_type,type,
% 0.07/0.32 bs114d: $o ).
% 0.07/0.32
% 0.07/0.32 thf(bs114d,definition,
% 0.07/0.32 ( bs114d
% 0.07/0.32 = ( ! [A: $i,B: $i,C: $i] :
% 0.07/0.32 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 0.07/0.32 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(regular_type,type,
% 0.07/0.32 regular: $i > $o ).
% 0.07/0.32
% 0.07/0.32 thf(setminus_type,type,
% 0.07/0.32 setminus: $i > $i > $i ).
% 0.07/0.32
% 0.07/0.32 thf(setminusI_type,type,
% 0.07/0.32 setminusI: $o ).
% 0.07/0.32
% 0.07/0.32 thf(setminusI,definition,
% 0.07/0.32 ( setminusI
% 0.07/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( ~ ( in @ Xx @ B )
% 0.07/0.32 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(setminusEL_type,type,
% 0.07/0.32 setminusEL: $o ).
% 0.07/0.32
% 0.07/0.32 thf(setminusEL,definition,
% 0.07/0.32 ( setminusEL
% 0.07/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.07/0.32 => ( in @ Xx @ A ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(setminusER_type,type,
% 0.07/0.32 setminusER: $o ).
% 0.07/0.32
% 0.07/0.32 thf(setminusER,definition,
% 0.07/0.32 ( setminusER
% 0.07/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.07/0.32 => ~ ( in @ Xx @ B ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(setminusSubset2_type,type,
% 0.07/0.32 setminusSubset2: $o ).
% 0.07/0.32
% 0.07/0.32 thf(setminusSubset2,definition,
% 0.07/0.32 ( setminusSubset2
% 0.07/0.32 = ( ! [A: $i,B: $i] :
% 0.07/0.32 ( ( subset @ A @ B )
% 0.07/0.32 => ( ( setminus @ A @ B )
% 0.07/0.32 = emptyset ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(setminusERneg_type,type,
% 0.07/0.32 setminusERneg: $o ).
% 0.07/0.32
% 0.07/0.32 thf(setminusERneg,definition,
% 0.07/0.32 ( setminusERneg
% 0.07/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.32 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.07/0.32 => ( ( in @ Xx @ A )
% 0.07/0.32 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(setminusELneg_type,type,
% 0.07/0.32 setminusELneg: $o ).
% 0.07/0.32
% 0.07/0.32 thf(setminusELneg,definition,
% 0.07/0.32 ( setminusELneg
% 0.07/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.32 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.07/0.32 => ( ~ ( in @ Xx @ B )
% 0.07/0.32 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(setminusILneg_type,type,
% 0.07/0.32 setminusILneg: $o ).
% 0.07/0.32
% 0.07/0.32 thf(setminusILneg,definition,
% 0.07/0.32 ( setminusILneg
% 0.07/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.32 ( ~ ( in @ Xx @ A )
% 0.07/0.32 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(setminusIRneg_type,type,
% 0.07/0.32 setminusIRneg: $o ).
% 0.07/0.32
% 0.07/0.32 thf(setminusIRneg,definition,
% 0.07/0.32 ( setminusIRneg
% 0.07/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ B )
% 0.07/0.32 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(setminusLsub_type,type,
% 0.07/0.32 setminusLsub: $o ).
% 0.07/0.32
% 0.07/0.32 thf(setminusLsub,definition,
% 0.07/0.32 ( setminusLsub
% 0.07/0.32 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(setminusSubset1_type,type,
% 0.07/0.32 setminusSubset1: $o ).
% 0.07/0.32
% 0.07/0.32 thf(setminusSubset1,definition,
% 0.07/0.32 ( setminusSubset1
% 0.07/0.32 = ( ! [A: $i,B: $i] :
% 0.07/0.32 ( ( ( setminus @ A @ B )
% 0.07/0.32 = emptyset )
% 0.07/0.32 => ( subset @ A @ B ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(symdiff_type,type,
% 0.07/0.32 symdiff: $i > $i > $i ).
% 0.07/0.32
% 0.07/0.32 thf(symdiffE_type,type,
% 0.07/0.32 symdiffE: $o ).
% 0.07/0.32
% 0.07/0.32 thf(symdiffE,definition,
% 0.07/0.32 ( symdiffE
% 0.07/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 0.07/0.32 => ! [Xphi: $o] :
% 0.07/0.32 ( ( ( in @ Xx @ A )
% 0.07/0.32 => ( ~ ( in @ Xx @ B )
% 0.07/0.32 => Xphi ) )
% 0.07/0.32 => ( ( ~ ( in @ Xx @ A )
% 0.07/0.32 => ( ( in @ Xx @ B )
% 0.07/0.32 => Xphi ) )
% 0.07/0.32 => Xphi ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(symdiffI1_type,type,
% 0.07/0.32 symdiffI1: $o ).
% 0.07/0.32
% 0.07/0.32 thf(symdiffI1,definition,
% 0.07/0.32 ( symdiffI1
% 0.07/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( ~ ( in @ Xx @ B )
% 0.07/0.32 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(symdiffI2_type,type,
% 0.07/0.32 symdiffI2: $o ).
% 0.07/0.32
% 0.07/0.32 thf(symdiffI2,definition,
% 0.07/0.32 ( symdiffI2
% 0.07/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.32 ( ~ ( in @ Xx @ A )
% 0.07/0.32 => ( ( in @ Xx @ B )
% 0.07/0.32 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(symdiffIneg1_type,type,
% 0.07/0.32 symdiffIneg1: $o ).
% 0.07/0.32
% 0.07/0.32 thf(symdiffIneg1,definition,
% 0.07/0.32 ( symdiffIneg1
% 0.07/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( ( in @ Xx @ B )
% 0.07/0.32 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(symdiffIneg2_type,type,
% 0.07/0.32 symdiffIneg2: $o ).
% 0.07/0.32
% 0.07/0.32 thf(symdiffIneg2,definition,
% 0.07/0.32 ( symdiffIneg2
% 0.07/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.32 ( ~ ( in @ Xx @ A )
% 0.07/0.32 => ( ~ ( in @ Xx @ B )
% 0.07/0.32 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(iskpair_type,type,
% 0.07/0.32 iskpair: $i > $o ).
% 0.07/0.32
% 0.07/0.32 thf(iskpair,definition,
% 0.07/0.32 ( iskpair
% 0.07/0.32 = ( ^ [A: $i] :
% 0.07/0.32 ? [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ ( setunion @ A ) )
% 0.07/0.32 & ? [Xy: $i] :
% 0.07/0.32 ( ( in @ Xy @ ( setunion @ A ) )
% 0.07/0.32 & ( A
% 0.07/0.32 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(secondinupair_type,type,
% 0.07/0.32 secondinupair: $o ).
% 0.07/0.32
% 0.07/0.32 thf(secondinupair,definition,
% 0.07/0.32 ( secondinupair
% 0.07/0.32 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(setukpairIL_type,type,
% 0.07/0.32 setukpairIL: $o ).
% 0.07/0.32
% 0.07/0.32 thf(setukpairIL,definition,
% 0.07/0.32 ( setukpairIL
% 0.07/0.32 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(setukpairIR_type,type,
% 0.07/0.32 setukpairIR: $o ).
% 0.07/0.32
% 0.07/0.32 thf(setukpairIR,definition,
% 0.07/0.32 ( setukpairIR
% 0.07/0.32 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(kpairiskpair_type,type,
% 0.07/0.32 kpairiskpair: $o ).
% 0.07/0.32
% 0.07/0.32 thf(kpairiskpair,definition,
% 0.07/0.32 ( kpairiskpair
% 0.07/0.32 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(kpair_type,type,
% 0.07/0.32 kpair: $i > $i > $i ).
% 0.07/0.32
% 0.07/0.32 thf(kpair,definition,
% 0.07/0.32 ( kpair
% 0.07/0.32 = ( ^ [Xx: $i,Xy: $i] : ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(kpairp_type,type,
% 0.07/0.32 kpairp: $o ).
% 0.07/0.32
% 0.07/0.32 thf(kpairp,definition,
% 0.07/0.32 ( kpairp
% 0.07/0.32 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(cartprod_type,type,
% 0.07/0.32 cartprod: $i > $i > $i ).
% 0.07/0.32
% 0.07/0.32 thf(cartprod,definition,
% 0.07/0.32 ( cartprod
% 0.07/0.32 = ( ^ [A: $i,B: $i] :
% 0.07/0.32 ( dsetconstr @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) )
% 0.07/0.32 @ ^ [Xx: $i] :
% 0.07/0.32 ? [Xy: $i] :
% 0.07/0.32 ( ( in @ Xy @ A )
% 0.07/0.32 & ? [Xz: $i] :
% 0.07/0.32 ( ( in @ Xz @ B )
% 0.07/0.32 & ( Xx
% 0.07/0.32 = ( kpair @ Xy @ Xz ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(singletonsubset_type,type,
% 0.07/0.32 singletonsubset: $o ).
% 0.07/0.32
% 0.07/0.32 thf(singletonsubset,definition,
% 0.07/0.32 ( singletonsubset
% 0.07/0.32 = ( ! [A: $i,Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(singletoninpowerset,conjecture,
% 0.07/0.32 ( setextAx
% 0.07/0.32 => ( emptysetAx
% 0.07/0.32 => ( setadjoinAx
% 0.07/0.32 => ( powersetAx
% 0.07/0.32 => ( setunionAx
% 0.07/0.32 => ( omega0Ax
% 0.07/0.32 => ( omegaSAx
% 0.07/0.32 => ( omegaIndAx
% 0.07/0.32 => ( replAx
% 0.07/0.32 => ( foundationAx
% 0.07/0.32 => ( wellorderingAx
% 0.07/0.32 => ( descrp
% 0.07/0.32 => ( dsetconstrI
% 0.07/0.32 => ( dsetconstrEL
% 0.07/0.32 => ( dsetconstrER
% 0.07/0.32 => ( exuE1
% 0.07/0.32 => ( prop2setE
% 0.07/0.32 => ( emptysetE
% 0.07/0.32 => ( emptysetimpfalse
% 0.07/0.32 => ( notinemptyset
% 0.07/0.32 => ( exuE3e
% 0.07/0.32 => ( setext
% 0.07/0.32 => ( emptyI
% 0.07/0.32 => ( noeltsimpempty
% 0.07/0.32 => ( setbeta
% 0.07/0.32 => ( nonemptyE1
% 0.07/0.32 => ( nonemptyI
% 0.07/0.32 => ( nonemptyI1
% 0.07/0.32 => ( setadjoinIL
% 0.07/0.32 => ( emptyinunitempty
% 0.07/0.32 => ( setadjoinIR
% 0.07/0.32 => ( setadjoinE
% 0.07/0.32 => ( setadjoinOr
% 0.07/0.32 => ( setoftrueEq
% 0.07/0.32 => ( powersetI
% 0.07/0.32 => ( emptyinPowerset
% 0.07/0.32 => ( emptyInPowerset
% 0.07/0.32 => ( powersetE
% 0.07/0.32 => ( setunionI
% 0.07/0.32 => ( setunionE
% 0.07/0.32 => ( subPowSU
% 0.07/0.32 => ( exuE2
% 0.07/0.32 => ( nonemptyImpWitness
% 0.07/0.32 => ( uniqinunit
% 0.07/0.32 => ( notinsingleton
% 0.07/0.32 => ( eqinunit
% 0.07/0.32 => ( singletonsswitch
% 0.07/0.32 => ( upairsetE
% 0.07/0.32 => ( upairsetIL
% 0.07/0.32 => ( upairsetIR
% 0.07/0.32 => ( emptyE1
% 0.07/0.32 => ( vacuousDall
% 0.07/0.32 => ( quantDeMorgan1
% 0.07/0.32 => ( quantDeMorgan2
% 0.07/0.32 => ( quantDeMorgan3
% 0.07/0.32 => ( quantDeMorgan4
% 0.07/0.32 => ( prop2setI
% 0.07/0.32 => ( prop2set2propI
% 0.07/0.32 => ( notdexE
% 0.07/0.32 => ( notdallE
% 0.07/0.32 => ( exuI1
% 0.07/0.32 => ( exuI3
% 0.07/0.32 => ( exuI2
% 0.07/0.32 => ( inCongP
% 0.07/0.32 => ( in__Cong
% 0.07/0.32 => ( exuE3u
% 0.07/0.32 => ( exu__Cong
% 0.07/0.32 => ( emptyset__Cong
% 0.07/0.32 => ( setadjoin__Cong
% 0.07/0.32 => ( powerset__Cong
% 0.07/0.32 => ( setunion__Cong
% 0.07/0.32 => ( omega__Cong
% 0.07/0.32 => ( exuEu
% 0.07/0.32 => ( descr__Cong
% 0.07/0.32 => ( dsetconstr__Cong
% 0.07/0.32 => ( subsetI1
% 0.07/0.32 => ( eqimpsubset2
% 0.07/0.32 => ( eqimpsubset1
% 0.07/0.32 => ( subsetI2
% 0.07/0.32 => ( emptysetsubset
% 0.07/0.32 => ( subsetE
% 0.07/0.32 => ( subsetE2
% 0.07/0.32 => ( notsubsetI
% 0.07/0.32 => ( notequalI1
% 0.07/0.32 => ( notequalI2
% 0.07/0.32 => ( subsetRefl
% 0.07/0.32 => ( subsetTrans
% 0.07/0.32 => ( setadjoinSub
% 0.07/0.32 => ( setadjoinSub2
% 0.07/0.32 => ( subset2powerset
% 0.07/0.32 => ( setextsub
% 0.07/0.32 => ( subsetemptysetimpeq
% 0.07/0.32 => ( powersetI1
% 0.07/0.32 => ( powersetE1
% 0.07/0.32 => ( inPowerset
% 0.07/0.32 => ( powersetsubset
% 0.07/0.32 => ( sepInPowerset
% 0.07/0.32 => ( sepSubset
% 0.07/0.32 => ( binunionIL
% 0.07/0.32 => ( upairset2IR
% 0.07/0.32 => ( binunionIR
% 0.07/0.32 => ( binunionEcases
% 0.07/0.32 => ( binunionE
% 0.07/0.32 => ( binunionLsub
% 0.07/0.32 => ( binunionRsub
% 0.07/0.32 => ( binintersectI
% 0.07/0.32 => ( binintersectSubset5
% 0.07/0.32 => ( binintersectEL
% 0.07/0.32 => ( binintersectLsub
% 0.07/0.32 => ( binintersectSubset2
% 0.07/0.32 => ( binintersectSubset3
% 0.07/0.32 => ( binintersectER
% 0.07/0.32 => ( disjointsetsI1
% 0.07/0.32 => ( binintersectRsub
% 0.07/0.32 => ( binintersectSubset4
% 0.07/0.32 => ( binintersectSubset1
% 0.07/0.32 => ( bs114d
% 0.07/0.32 => ( setminusI
% 0.07/0.32 => ( setminusEL
% 0.07/0.32 => ( setminusER
% 0.07/0.32 => ( setminusSubset2
% 0.07/0.32 => ( setminusERneg
% 0.07/0.32 => ( setminusELneg
% 0.07/0.32 => ( setminusILneg
% 0.07/0.32 => ( setminusIRneg
% 0.07/0.32 => ( setminusLsub
% 0.07/0.32 => ( setminusSubset1
% 0.07/0.32 => ( symdiffE
% 0.07/0.32 => ( symdiffI1
% 0.07/0.32 => ( symdiffI2
% 0.07/0.32 => ( symdiffIneg1
% 0.07/0.32 => ( symdiffIneg2
% 0.07/0.32 => ( secondinupair
% 0.07/0.32 => ( setukpairIL
% 0.07/0.32 => ( setukpairIR
% 0.07/0.32 => ( kpairiskpair
% 0.07/0.32 => ( kpairp
% 0.07/0.32 => ( singletonsubset
% 0.07/0.32 => ! [A: $i,Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 %------------------------------------------------------------------------------
% 0.07/0.32 ------------------------
% 1.66/1.06 ---- Embedded in THF ---
% 1.66/1.06 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 1.66/1.06 %%% Generated on Tue Feb 24 12:28:31 EST 2026
% 1.66/1.06 %%% using '$$dhol' embedding, version 1.3.0.
% 1.66/1.06 %%% Logic specification used:
% 1.66/1.06 %%% thf(spec, logic, $$dhol).
% 1.66/1.06
% 1.66/1.06 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.P13AcNWfIy/DTF2DTF4603.p
% See solution above
% 1.66/1.06 ------------------------
% 1.66/1.08 ---- Cleaned THF ---
% 1.66/1.08 thf(in_type,type,
% 1.66/1.08 in: $i > $i > $o ).
% 1.66/1.08
% 1.66/1.08 thf(exu_type,type,
% 1.66/1.08 exu: ( $i > $o ) > $o ).
% 1.66/1.08
% 1.66/1.08 thf(exu,definition,
% 1.66/1.08 ( exu
% 1.66/1.08 = ( ^ [Xphi: $i > $o] :
% 1.66/1.08 ? [Xx: $i] :
% 1.66/1.08 ( ( Xphi @ Xx )
% 1.66/1.08 & ! [Xy: $i] :
% 1.66/1.08 ( ( Xphi @ Xy )
% 1.66/1.08 => ( Xx = Xy ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(setextAx_type,type,
% 1.66/1.08 setextAx: $o ).
% 1.66/1.08
% 1.66/1.08 thf(setextAx,definition,
% 1.66/1.08 ( setextAx
% 1.66/1.08 = ( ! [A: $i,B: $i] :
% 1.66/1.08 ( ! [Xx: $i] :
% 1.66/1.08 ( ( in @ Xx @ A )
% 1.66/1.08 <=> ( in @ Xx @ B ) )
% 1.66/1.08 => ( A = B ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(emptyset_type,type,
% 1.66/1.08 emptyset: $i ).
% 1.66/1.08
% 1.66/1.08 thf(emptysetAx_type,type,
% 1.66/1.08 emptysetAx: $o ).
% 1.66/1.08
% 1.66/1.08 thf(emptysetAx,definition,
% 1.66/1.08 ( emptysetAx
% 1.66/1.08 = ( ! [Xx: $i] :
% 1.66/1.08 ~ ( in @ Xx @ emptyset ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(setadjoin_type,type,
% 1.66/1.08 setadjoin: $i > $i > $i ).
% 1.66/1.08
% 1.66/1.08 thf(setadjoinAx_type,type,
% 1.66/1.08 setadjoinAx: $o ).
% 1.66/1.08
% 1.66/1.08 thf(setadjoinAx,definition,
% 1.66/1.08 ( setadjoinAx
% 1.66/1.08 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 1.66/1.08 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 1.66/1.08 <=> ( ( Xy = Xx )
% 1.66/1.08 | ( in @ Xy @ A ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(powerset_type,type,
% 1.66/1.08 powerset: $i > $i ).
% 1.66/1.08
% 1.66/1.08 thf(powersetAx_type,type,
% 1.66/1.08 powersetAx: $o ).
% 1.66/1.08
% 1.66/1.08 thf(powersetAx,definition,
% 1.66/1.08 ( powersetAx
% 1.66/1.08 = ( ! [A: $i,B: $i] :
% 1.66/1.08 ( ( in @ B @ ( powerset @ A ) )
% 1.66/1.08 <=> ! [Xx: $i] :
% 1.66/1.08 ( ( in @ Xx @ B )
% 1.66/1.08 => ( in @ Xx @ A ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(setunion_type,type,
% 1.66/1.08 setunion: $i > $i ).
% 1.66/1.08
% 1.66/1.08 thf(setunionAx_type,type,
% 1.66/1.08 setunionAx: $o ).
% 1.66/1.08
% 1.66/1.08 thf(setunionAx,definition,
% 1.66/1.08 ( setunionAx
% 1.66/1.08 = ( ! [A: $i,Xx: $i] :
% 1.66/1.08 ( ( in @ Xx @ ( setunion @ A ) )
% 1.66/1.08 <=> ? [B: $i] :
% 1.66/1.08 ( ( in @ Xx @ B )
% 1.66/1.08 & ( in @ B @ A ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(omega_type,type,
% 1.66/1.08 omega: $i ).
% 1.66/1.08
% 1.66/1.08 thf(omega0Ax_type,type,
% 1.66/1.08 omega0Ax: $o ).
% 1.66/1.08
% 1.66/1.08 thf(omega0Ax,definition,
% 1.66/1.08 ( omega0Ax
% 1.66/1.08 = ( in @ emptyset @ omega ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(omegaSAx_type,type,
% 1.66/1.08 omegaSAx: $o ).
% 1.66/1.08
% 1.66/1.08 thf(omegaSAx,definition,
% 1.66/1.08 ( omegaSAx
% 1.66/1.08 = ( ! [Xx: $i] :
% 1.66/1.08 ( ( in @ Xx @ omega )
% 1.66/1.08 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(omegaIndAx_type,type,
% 1.66/1.08 omegaIndAx: $o ).
% 1.66/1.08
% 1.66/1.08 thf(omegaIndAx,definition,
% 1.66/1.08 ( omegaIndAx
% 1.66/1.08 = ( ! [A: $i] :
% 1.66/1.08 ( ( ( in @ emptyset @ A )
% 1.66/1.08 & ! [Xx: $i] :
% 1.66/1.08 ( ( ( in @ Xx @ omega )
% 1.66/1.08 & ( in @ Xx @ A ) )
% 1.66/1.08 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 1.66/1.08 => ! [Xx: $i] :
% 1.66/1.08 ( ( in @ Xx @ omega )
% 1.66/1.08 => ( in @ Xx @ A ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(replAx_type,type,
% 1.66/1.08 replAx: $o ).
% 1.66/1.08
% 1.66/1.08 thf(replAx,definition,
% 1.66/1.08 ( replAx
% 1.66/1.08 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 1.66/1.08 ( ! [Xx: $i] :
% 1.66/1.08 ( ( in @ Xx @ A )
% 1.66/1.08 => ( exu
% 1.66/1.08 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 1.66/1.08 => ? [B: $i] :
% 1.66/1.08 ! [Xx: $i] :
% 1.66/1.08 ( ( in @ Xx @ B )
% 1.66/1.08 <=> ? [Xy: $i] :
% 1.66/1.08 ( ( in @ Xy @ A )
% 1.66/1.08 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(foundationAx_type,type,
% 1.66/1.08 foundationAx: $o ).
% 1.66/1.08
% 1.66/1.08 thf(foundationAx,definition,
% 1.66/1.08 ( foundationAx
% 1.66/1.08 = ( ! [A: $i] :
% 1.66/1.08 ( ? [Xx: $i] : ( in @ Xx @ A )
% 1.66/1.08 => ? [B: $i] :
% 1.66/1.08 ( ( in @ B @ A )
% 1.66/1.08 & ~ ? [Xx: $i] :
% 1.66/1.08 ( ( in @ Xx @ B )
% 1.66/1.08 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(wellorderingAx_type,type,
% 1.66/1.08 wellorderingAx: $o ).
% 1.66/1.08
% 1.66/1.08 thf(wellorderingAx,definition,
% 1.66/1.08 ( wellorderingAx
% 1.66/1.08 = ( ! [A: $i] :
% 1.66/1.08 ? [B: $i] :
% 1.66/1.08 ( ! [C: $i] :
% 1.66/1.08 ( ( in @ C @ B )
% 1.66/1.08 => ! [Xx: $i] :
% 1.66/1.08 ( ( in @ Xx @ C )
% 1.66/1.08 => ( in @ Xx @ A ) ) )
% 1.66/1.08 & ! [Xx: $i,Xy: $i] :
% 1.66/1.08 ( ( ( in @ Xx @ A )
% 1.66/1.08 & ( in @ Xy @ A ) )
% 1.66/1.08 => ( ! [C: $i] :
% 1.66/1.08 ( ( in @ C @ B )
% 1.66/1.08 => ( ( in @ Xx @ C )
% 1.66/1.08 <=> ( in @ Xy @ C ) ) )
% 1.66/1.08 => ( Xx = Xy ) ) )
% 1.66/1.08 & ! [C: $i,D: $i] :
% 1.66/1.08 ( ( ( in @ C @ B )
% 1.66/1.08 & ( in @ D @ B ) )
% 1.66/1.08 => ( ! [Xx: $i] :
% 1.66/1.08 ( ( in @ Xx @ C )
% 1.66/1.08 => ( in @ Xx @ D ) )
% 1.66/1.08 | ! [Xx: $i] :
% 1.66/1.08 ( ( in @ Xx @ D )
% 1.66/1.08 => ( in @ Xx @ C ) ) ) )
% 1.66/1.08 & ! [C: $i] :
% 1.66/1.08 ( ( ! [Xx: $i] :
% 1.66/1.08 ( ( in @ Xx @ C )
% 1.66/1.08 => ( in @ Xx @ A ) )
% 1.66/1.08 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 1.66/1.08 => ? [D: $i,Xx: $i] :
% 1.66/1.08 ( ( in @ D @ B )
% 1.66/1.08 & ( in @ Xx @ C )
% 1.66/1.08 & ~ ? [Xy: $i] :
% 1.66/1.08 ( ( in @ Xy @ D )
% 1.66/1.08 & ( in @ Xy @ C ) )
% 1.66/1.08 & ! [E: $i] :
% 1.66/1.08 ( ( in @ E @ B )
% 1.66/1.08 => ( ! [Xy: $i] :
% 1.66/1.08 ( ( in @ Xy @ E )
% 1.66/1.08 => ( in @ Xy @ D ) )
% 1.66/1.08 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(descr_type,type,
% 1.66/1.08 descr: ( $i > $o ) > $i ).
% 1.66/1.08
% 1.66/1.08 thf(descrp_type,type,
% 1.66/1.08 descrp: $o ).
% 1.66/1.08
% 1.66/1.08 thf(descrp,definition,
% 1.66/1.08 ( descrp
% 1.66/1.08 = ( ! [Xphi: $i > $o] :
% 1.66/1.08 ( ( exu
% 1.66/1.08 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 1.66/1.08 => ( Xphi
% 1.66/1.08 @ ( descr
% 1.66/1.08 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(dsetconstr_type,type,
% 1.66/1.08 dsetconstr: $i > ( $i > $o ) > $i ).
% 1.66/1.08
% 1.66/1.08 thf(dsetconstrI_type,type,
% 1.66/1.08 dsetconstrI: $o ).
% 1.66/1.08
% 1.66/1.08 thf(dsetconstrI,definition,
% 1.66/1.08 ( dsetconstrI
% 1.66/1.08 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 1.66/1.08 ( ( in @ Xx @ A )
% 1.66/1.08 => ( ( Xphi @ Xx )
% 1.66/1.08 => ( in @ Xx
% 1.66/1.08 @ ( dsetconstr @ A
% 1.66/1.08 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(dsetconstrEL_type,type,
% 1.66/1.08 dsetconstrEL: $o ).
% 1.66/1.08
% 1.66/1.08 thf(dsetconstrEL,definition,
% 1.66/1.08 ( dsetconstrEL
% 1.66/1.08 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 1.66/1.08 ( ( in @ Xx
% 1.66/1.08 @ ( dsetconstr @ A
% 1.66/1.08 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 1.66/1.08 => ( in @ Xx @ A ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(dsetconstrER_type,type,
% 1.66/1.08 dsetconstrER: $o ).
% 1.66/1.08
% 1.66/1.08 thf(dsetconstrER,definition,
% 1.66/1.08 ( dsetconstrER
% 1.66/1.08 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 1.66/1.08 ( ( in @ Xx
% 1.66/1.08 @ ( dsetconstr @ A
% 1.66/1.08 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 1.66/1.08 => ( Xphi @ Xx ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(exuE1_type,type,
% 1.66/1.08 exuE1: $o ).
% 1.66/1.08
% 1.66/1.08 thf(exuE1,definition,
% 1.66/1.08 ( exuE1
% 1.66/1.08 = ( ! [Xphi: $i > $o] :
% 1.66/1.08 ( ( exu
% 1.66/1.08 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 1.66/1.08 => ? [Xx: $i] :
% 1.66/1.08 ( ( Xphi @ Xx )
% 1.66/1.08 & ! [Xy: $i] :
% 1.66/1.08 ( ( Xphi @ Xy )
% 1.66/1.08 => ( Xx = Xy ) ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(prop2set_type,type,
% 1.66/1.08 prop2set: $o > $i ).
% 1.66/1.08
% 1.66/1.08 thf(prop2setE_type,type,
% 1.66/1.08 prop2setE: $o ).
% 1.66/1.08
% 1.66/1.08 thf(prop2setE,definition,
% 1.66/1.08 ( prop2setE
% 1.66/1.08 = ( ! [Xphi: $o,Xx: $i] :
% 1.66/1.08 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 1.66/1.08 => Xphi ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(emptysetE_type,type,
% 1.66/1.08 emptysetE: $o ).
% 1.66/1.08
% 1.66/1.08 thf(emptysetE,definition,
% 1.66/1.08 ( emptysetE
% 1.66/1.08 = ( ! [Xx: $i] :
% 1.66/1.08 ( ( in @ Xx @ emptyset )
% 1.66/1.08 => ! [Xphi: $o] : Xphi ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(emptysetimpfalse_type,type,
% 1.66/1.08 emptysetimpfalse: $o ).
% 1.66/1.08
% 1.66/1.08 thf(emptysetimpfalse,definition,
% 1.66/1.08 ( emptysetimpfalse
% 1.66/1.08 = ( ! [Xx: $i] :
% 1.66/1.08 ( ( in @ Xx @ emptyset )
% 1.66/1.08 => $false ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(notinemptyset_type,type,
% 1.66/1.08 notinemptyset: $o ).
% 1.66/1.08
% 1.66/1.08 thf(notinemptyset,definition,
% 1.66/1.08 ( notinemptyset
% 1.66/1.08 = ( ! [Xx: $i] :
% 1.66/1.08 ~ ( in @ Xx @ emptyset ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(exuE3e_type,type,
% 1.66/1.08 exuE3e: $o ).
% 1.66/1.08
% 1.66/1.08 thf(exuE3e,definition,
% 1.66/1.08 ( exuE3e
% 1.66/1.08 = ( ! [Xphi: $i > $o] :
% 1.66/1.08 ( ( exu
% 1.66/1.08 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 1.66/1.08 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(setext_type,type,
% 1.66/1.08 setext: $o ).
% 1.66/1.08
% 1.66/1.08 thf(setext,definition,
% 1.66/1.08 ( setext
% 1.66/1.08 = ( ! [A: $i,B: $i] :
% 1.66/1.08 ( ! [Xx: $i] :
% 1.66/1.08 ( ( in @ Xx @ A )
% 1.66/1.08 => ( in @ Xx @ B ) )
% 1.66/1.08 => ( ! [Xx: $i] :
% 1.66/1.08 ( ( in @ Xx @ B )
% 1.66/1.08 => ( in @ Xx @ A ) )
% 1.66/1.08 => ( A = B ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(emptyI_type,type,
% 1.66/1.08 emptyI: $o ).
% 1.66/1.08
% 1.66/1.08 thf(emptyI,definition,
% 1.66/1.08 ( emptyI
% 1.66/1.08 = ( ! [A: $i] :
% 1.66/1.08 ( ! [Xx: $i] :
% 1.66/1.08 ~ ( in @ Xx @ A )
% 1.66/1.08 => ( A = emptyset ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(noeltsimpempty_type,type,
% 1.66/1.08 noeltsimpempty: $o ).
% 1.66/1.08
% 1.66/1.08 thf(noeltsimpempty,definition,
% 1.66/1.08 ( noeltsimpempty
% 1.66/1.08 = ( ! [A: $i] :
% 1.66/1.08 ( ! [Xx: $i] :
% 1.66/1.08 ~ ( in @ Xx @ A )
% 1.66/1.08 => ( A = emptyset ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(setbeta_type,type,
% 1.66/1.08 setbeta: $o ).
% 1.66/1.08
% 1.66/1.08 thf(setbeta,definition,
% 1.66/1.08 ( setbeta
% 1.66/1.08 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 1.66/1.08 ( ( in @ Xx @ A )
% 1.66/1.08 => ( ( in @ Xx
% 1.66/1.08 @ ( dsetconstr @ A
% 1.66/1.08 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 1.66/1.08 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(nonempty_type,type,
% 1.66/1.08 nonempty: $i > $o ).
% 1.66/1.08
% 1.66/1.08 thf(nonempty,definition,
% 1.66/1.08 ( nonempty
% 1.66/1.08 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(nonemptyE1_type,type,
% 1.66/1.08 nonemptyE1: $o ).
% 1.66/1.08
% 1.66/1.08 thf(nonemptyE1,definition,
% 1.66/1.08 ( nonemptyE1
% 1.66/1.08 = ( ! [A: $i] :
% 1.66/1.08 ( ( nonempty @ A )
% 1.66/1.08 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(nonemptyI_type,type,
% 1.66/1.08 nonemptyI: $o ).
% 1.66/1.08
% 1.66/1.08 thf(nonemptyI,definition,
% 1.66/1.08 ( nonemptyI
% 1.66/1.08 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 1.66/1.08 ( ( in @ Xx @ A )
% 1.66/1.08 => ( ( Xphi @ Xx )
% 1.66/1.08 => ( nonempty
% 1.66/1.08 @ ( dsetconstr @ A
% 1.66/1.08 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(nonemptyI1_type,type,
% 1.66/1.08 nonemptyI1: $o ).
% 1.66/1.08
% 1.66/1.08 thf(nonemptyI1,definition,
% 1.66/1.08 ( nonemptyI1
% 1.66/1.08 = ( ! [A: $i] :
% 1.66/1.08 ( ? [Xx: $i] : ( in @ Xx @ A )
% 1.66/1.08 => ( nonempty @ A ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(setadjoinIL_type,type,
% 1.66/1.08 setadjoinIL: $o ).
% 1.66/1.08
% 1.66/1.08 thf(setadjoinIL,definition,
% 1.66/1.08 ( setadjoinIL
% 1.66/1.08 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(emptyinunitempty_type,type,
% 1.66/1.08 emptyinunitempty: $o ).
% 1.66/1.08
% 1.66/1.08 thf(emptyinunitempty,definition,
% 1.66/1.08 ( emptyinunitempty
% 1.66/1.08 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(setadjoinIR_type,type,
% 1.66/1.08 setadjoinIR: $o ).
% 1.66/1.08
% 1.66/1.08 thf(setadjoinIR,definition,
% 1.66/1.08 ( setadjoinIR
% 1.66/1.08 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 1.66/1.08 ( ( in @ Xy @ A )
% 1.66/1.08 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(setadjoinE_type,type,
% 1.66/1.08 setadjoinE: $o ).
% 1.66/1.08
% 1.66/1.08 thf(setadjoinE,definition,
% 1.66/1.08 ( setadjoinE
% 1.66/1.08 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 1.66/1.08 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 1.66/1.08 => ! [Xphi: $o] :
% 1.66/1.08 ( ( ( Xy = Xx )
% 1.66/1.08 => Xphi )
% 1.66/1.08 => ( ( ( in @ Xy @ A )
% 1.66/1.08 => Xphi )
% 1.66/1.08 => Xphi ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(setadjoinOr_type,type,
% 1.66/1.08 setadjoinOr: $o ).
% 1.66/1.08
% 1.66/1.08 thf(setadjoinOr,definition,
% 1.66/1.08 ( setadjoinOr
% 1.66/1.08 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 1.66/1.08 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 1.66/1.08 => ( ( Xy = Xx )
% 1.66/1.08 | ( in @ Xy @ A ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(setoftrueEq_type,type,
% 1.66/1.08 setoftrueEq: $o ).
% 1.66/1.08
% 1.66/1.08 thf(setoftrueEq,definition,
% 1.66/1.08 ( setoftrueEq
% 1.66/1.08 = ( ! [A: $i] :
% 1.66/1.08 ( ( dsetconstr @ A
% 1.66/1.08 @ ^ [Xx: $i] : $true )
% 1.66/1.08 = A ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(powersetI_type,type,
% 1.66/1.08 powersetI: $o ).
% 1.66/1.08
% 1.66/1.08 thf(powersetI,definition,
% 1.66/1.08 ( powersetI
% 1.66/1.08 = ( ! [A: $i,B: $i] :
% 1.66/1.08 ( ! [Xx: $i] :
% 1.66/1.08 ( ( in @ Xx @ B )
% 1.66/1.08 => ( in @ Xx @ A ) )
% 1.66/1.08 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(emptyinPowerset_type,type,
% 1.66/1.08 emptyinPowerset: $o ).
% 1.66/1.08
% 1.66/1.08 thf(emptyinPowerset,definition,
% 1.66/1.08 ( emptyinPowerset
% 1.66/1.08 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(emptyInPowerset_type,type,
% 1.66/1.08 emptyInPowerset: $o ).
% 1.66/1.08
% 1.66/1.08 thf(emptyInPowerset,definition,
% 1.66/1.08 ( emptyInPowerset
% 1.66/1.08 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(powersetE_type,type,
% 1.66/1.08 powersetE: $o ).
% 1.66/1.08
% 1.66/1.08 thf(powersetE,definition,
% 1.66/1.08 ( powersetE
% 1.66/1.08 = ( ! [A: $i,B: $i,Xx: $i] :
% 1.66/1.08 ( ( in @ B @ ( powerset @ A ) )
% 1.66/1.08 => ( ( in @ Xx @ B )
% 1.66/1.08 => ( in @ Xx @ A ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(setunionI_type,type,
% 1.66/1.08 setunionI: $o ).
% 1.66/1.08
% 1.66/1.08 thf(setunionI,definition,
% 1.66/1.08 ( setunionI
% 1.66/1.08 = ( ! [A: $i,Xx: $i,B: $i] :
% 1.66/1.08 ( ( in @ Xx @ B )
% 1.66/1.08 => ( ( in @ B @ A )
% 1.66/1.08 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(setunionE_type,type,
% 1.66/1.08 setunionE: $o ).
% 1.66/1.08
% 1.66/1.08 thf(setunionE,definition,
% 1.66/1.08 ( setunionE
% 1.66/1.08 = ( ! [A: $i,Xx: $i] :
% 1.66/1.08 ( ( in @ Xx @ ( setunion @ A ) )
% 1.66/1.08 => ! [Xphi: $o] :
% 1.66/1.08 ( ! [B: $i] :
% 1.66/1.08 ( ( in @ Xx @ B )
% 1.66/1.08 => ( ( in @ B @ A )
% 1.66/1.08 => Xphi ) )
% 1.66/1.08 => Xphi ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(subPowSU_type,type,
% 1.66/1.08 subPowSU: $o ).
% 1.66/1.08
% 1.66/1.08 thf(subPowSU,definition,
% 1.66/1.08 ( subPowSU
% 1.66/1.08 = ( ! [A: $i,Xx: $i] :
% 1.66/1.08 ( ( in @ Xx @ A )
% 1.66/1.08 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(exuE2_type,type,
% 1.66/1.08 exuE2: $o ).
% 1.66/1.08
% 1.66/1.08 thf(exuE2,definition,
% 1.66/1.08 ( exuE2
% 1.66/1.08 = ( ! [Xphi: $i > $o] :
% 1.66/1.08 ( ( exu
% 1.66/1.08 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 1.66/1.08 => ? [Xx: $i] :
% 1.66/1.08 ! [Xy: $i] :
% 1.66/1.08 ( ( Xphi @ Xy )
% 1.66/1.08 <=> ( Xy = Xx ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(nonemptyImpWitness_type,type,
% 1.66/1.08 nonemptyImpWitness: $o ).
% 1.66/1.08
% 1.66/1.08 thf(nonemptyImpWitness,definition,
% 1.66/1.08 ( nonemptyImpWitness
% 1.66/1.08 = ( ! [A: $i] :
% 1.66/1.08 ( ( nonempty @ A )
% 1.66/1.08 => ? [Xx: $i] :
% 1.66/1.08 ( ( in @ Xx @ A )
% 1.66/1.08 & $true ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(uniqinunit_type,type,
% 1.66/1.08 uniqinunit: $o ).
% 1.66/1.08
% 1.66/1.08 thf(uniqinunit,definition,
% 1.66/1.08 ( uniqinunit
% 1.66/1.08 = ( ! [Xx: $i,Xy: $i] :
% 1.66/1.08 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 1.66/1.08 => ( Xx = Xy ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(notinsingleton_type,type,
% 1.66/1.08 notinsingleton: $o ).
% 1.66/1.08
% 1.66/1.08 thf(notinsingleton,definition,
% 1.66/1.08 ( notinsingleton
% 1.66/1.08 = ( ! [Xx: $i,Xy: $i] :
% 1.66/1.08 ( ( Xx != Xy )
% 1.66/1.08 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(eqinunit_type,type,
% 1.66/1.08 eqinunit: $o ).
% 1.66/1.08
% 1.66/1.08 thf(eqinunit,definition,
% 1.66/1.08 ( eqinunit
% 1.66/1.08 = ( ! [Xx: $i,Xy: $i] :
% 1.66/1.08 ( ( Xx = Xy )
% 1.66/1.08 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(singletonsswitch_type,type,
% 1.66/1.08 singletonsswitch: $o ).
% 1.66/1.08
% 1.66/1.08 thf(singletonsswitch,definition,
% 1.66/1.08 ( singletonsswitch
% 1.66/1.08 = ( ! [Xx: $i,Xy: $i] :
% 1.66/1.08 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 1.66/1.08 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(upairsetE_type,type,
% 1.66/1.08 upairsetE: $o ).
% 1.66/1.08
% 1.66/1.08 thf(upairsetE,definition,
% 1.66/1.08 ( upairsetE
% 1.66/1.08 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 1.66/1.08 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 1.66/1.08 => ( ( Xz = Xx )
% 1.66/1.08 | ( Xz = Xy ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(upairsetIL_type,type,
% 1.66/1.08 upairsetIL: $o ).
% 1.66/1.08
% 1.66/1.08 thf(upairsetIL,definition,
% 1.66/1.08 ( upairsetIL
% 1.66/1.08 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(upairsetIR_type,type,
% 1.66/1.08 upairsetIR: $o ).
% 1.66/1.08
% 1.66/1.08 thf(upairsetIR,definition,
% 1.66/1.08 ( upairsetIR
% 1.66/1.08 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(emptyE1_type,type,
% 1.66/1.08 emptyE1: $o ).
% 1.66/1.08
% 1.66/1.08 thf(emptyE1,definition,
% 1.66/1.08 ( emptyE1
% 1.66/1.08 = ( ! [A: $i,Xphi: $i > $o] :
% 1.66/1.08 ( ? [Xx: $i] :
% 1.66/1.08 ( ( in @ Xx @ A )
% 1.66/1.08 & ( Xphi @ Xx ) )
% 1.66/1.08 => ( ( ( dsetconstr @ A
% 1.66/1.08 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 1.66/1.08 = emptyset )
% 1.66/1.08 => $false ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(vacuousDall_type,type,
% 1.66/1.08 vacuousDall: $o ).
% 1.66/1.08
% 1.66/1.08 thf(vacuousDall,definition,
% 1.66/1.08 ( vacuousDall
% 1.66/1.08 = ( ! [Xphi: $i > $o,Xx: $i] :
% 1.66/1.08 ( ( in @ Xx @ emptyset )
% 1.66/1.08 => ( Xphi @ Xx ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(quantDeMorgan1_type,type,
% 1.66/1.08 quantDeMorgan1: $o ).
% 1.66/1.08
% 1.66/1.08 thf(quantDeMorgan1,definition,
% 1.66/1.08 ( quantDeMorgan1
% 1.66/1.08 = ( ! [A: $i,Xphi: $i > $o] :
% 1.66/1.08 ( ~ ! [Xx: $i] :
% 1.66/1.08 ( ( in @ Xx @ A )
% 1.66/1.08 => ( Xphi @ Xx ) )
% 1.66/1.08 => ? [Xx: $i] :
% 1.66/1.08 ( ( in @ Xx @ A )
% 1.66/1.08 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(quantDeMorgan2_type,type,
% 1.66/1.08 quantDeMorgan2: $o ).
% 1.66/1.08
% 1.66/1.08 thf(quantDeMorgan2,definition,
% 1.66/1.08 ( quantDeMorgan2
% 1.66/1.08 = ( ! [A: $i,Xphi: $i > $o] :
% 1.66/1.08 ( ! [Xx: $i] :
% 1.66/1.08 ( ( in @ Xx @ A )
% 1.66/1.08 => ~ ( Xphi @ Xx ) )
% 1.66/1.08 => ~ ? [Xx: $i] :
% 1.66/1.08 ( ( in @ Xx @ A )
% 1.66/1.08 & ( Xphi @ Xx ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(quantDeMorgan3_type,type,
% 1.66/1.08 quantDeMorgan3: $o ).
% 1.66/1.08
% 1.66/1.08 thf(quantDeMorgan3,definition,
% 1.66/1.08 ( quantDeMorgan3
% 1.66/1.08 = ( ! [A: $i,Xphi: $i > $o] :
% 1.66/1.08 ( ~ ? [Xx: $i] :
% 1.66/1.08 ( ( in @ Xx @ A )
% 1.66/1.08 & ( Xphi @ Xx ) )
% 1.66/1.08 => ! [Xx: $i] :
% 1.66/1.08 ( ( in @ Xx @ A )
% 1.66/1.08 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(quantDeMorgan4_type,type,
% 1.66/1.08 quantDeMorgan4: $o ).
% 1.66/1.08
% 1.66/1.08 thf(quantDeMorgan4,definition,
% 1.66/1.08 ( quantDeMorgan4
% 1.66/1.08 = ( ! [A: $i,Xphi: $i > $o] :
% 1.66/1.08 ( ? [Xx: $i] :
% 1.66/1.08 ( ( in @ Xx @ A )
% 1.66/1.08 & ~ ( Xphi @ Xx ) )
% 1.66/1.08 => ~ ! [Xx: $i] :
% 1.66/1.08 ( ( in @ Xx @ A )
% 1.66/1.08 => ( Xphi @ Xx ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(prop2setI_type,type,
% 1.66/1.08 prop2setI: $o ).
% 1.66/1.08
% 1.66/1.08 thf(prop2setI,definition,
% 1.66/1.08 ( prop2setI
% 1.66/1.08 = ( ! [Xphi: $o] :
% 1.66/1.08 ( Xphi
% 1.66/1.08 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(set2prop_type,type,
% 1.66/1.08 set2prop: $i > $o ).
% 1.66/1.08
% 1.66/1.08 thf(prop2set2propI_type,type,
% 1.66/1.08 prop2set2propI: $o ).
% 1.66/1.08
% 1.66/1.08 thf(prop2set2propI,definition,
% 1.66/1.08 ( prop2set2propI
% 1.66/1.08 = ( ! [Xphi: $o] :
% 1.66/1.08 ( Xphi
% 1.66/1.08 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(notdexE_type,type,
% 1.66/1.08 notdexE: $o ).
% 1.66/1.08
% 1.66/1.08 thf(notdexE,definition,
% 1.66/1.08 ( notdexE
% 1.66/1.08 = ( ! [A: $i,Xphi: $i > $o] :
% 1.66/1.08 ( ~ ? [Xx: $i] :
% 1.66/1.08 ( ( in @ Xx @ A )
% 1.66/1.08 & ( Xphi @ Xx ) )
% 1.66/1.08 => ! [Xx: $i] :
% 1.66/1.08 ( ( in @ Xx @ A )
% 1.66/1.08 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(notdallE_type,type,
% 1.66/1.08 notdallE: $o ).
% 1.66/1.08
% 1.66/1.08 thf(notdallE,definition,
% 1.66/1.08 ( notdallE
% 1.66/1.08 = ( ! [A: $i,Xphi: $i > $o] :
% 1.66/1.08 ( ~ ! [Xx: $i] :
% 1.66/1.08 ( ( in @ Xx @ A )
% 1.66/1.08 => ( Xphi @ Xx ) )
% 1.66/1.08 => ? [Xx: $i] :
% 1.66/1.08 ( ( in @ Xx @ A )
% 1.66/1.08 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(exuI1_type,type,
% 1.66/1.08 exuI1: $o ).
% 1.66/1.08
% 1.66/1.08 thf(exuI1,definition,
% 1.66/1.08 ( exuI1
% 1.66/1.08 = ( ! [Xphi: $i > $o] :
% 1.66/1.08 ( ? [Xx: $i] :
% 1.66/1.08 ( ( Xphi @ Xx )
% 1.66/1.08 & ! [Xy: $i] :
% 1.66/1.08 ( ( Xphi @ Xy )
% 1.66/1.08 => ( Xx = Xy ) ) )
% 1.66/1.08 => ( exu
% 1.66/1.08 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(exuI3_type,type,
% 1.66/1.08 exuI3: $o ).
% 1.66/1.08
% 1.66/1.08 thf(exuI3,definition,
% 1.66/1.08 ( exuI3
% 1.66/1.08 = ( ! [Xphi: $i > $o] :
% 1.66/1.08 ( ? [Xx: $i] : ( Xphi @ Xx )
% 1.66/1.08 => ( ! [Xx: $i,Xy: $i] :
% 1.66/1.08 ( ( Xphi @ Xx )
% 1.66/1.08 => ( ( Xphi @ Xy )
% 1.66/1.08 => ( Xx = Xy ) ) )
% 1.66/1.08 => ( exu
% 1.66/1.08 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(exuI2_type,type,
% 1.66/1.08 exuI2: $o ).
% 1.66/1.08
% 1.66/1.08 thf(exuI2,definition,
% 1.66/1.08 ( exuI2
% 1.66/1.08 = ( ! [Xphi: $i > $o] :
% 1.66/1.08 ( ? [Xx: $i] :
% 1.66/1.08 ! [Xy: $i] :
% 1.66/1.08 ( ( Xphi @ Xy )
% 1.66/1.08 <=> ( Xy = Xx ) )
% 1.66/1.08 => ( exu
% 1.66/1.08 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(inCongP_type,type,
% 1.66/1.08 inCongP: $o ).
% 1.66/1.08
% 1.66/1.08 thf(inCongP,definition,
% 1.66/1.08 ( inCongP
% 1.66/1.08 = ( ! [A: $i,B: $i] :
% 1.66/1.08 ( ( A = B )
% 1.66/1.08 => ! [Xx: $i,Xy: $i] :
% 1.66/1.08 ( ( Xx = Xy )
% 1.66/1.08 => ( ( in @ Xx @ A )
% 1.66/1.08 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(in__Cong_type,type,
% 1.66/1.08 in__Cong: $o ).
% 1.66/1.08
% 1.66/1.08 thf(in__Cong,definition,
% 1.66/1.08 ( in__Cong
% 1.66/1.08 = ( ! [A: $i,B: $i] :
% 1.66/1.08 ( ( A = B )
% 1.66/1.08 => ! [Xx: $i,Xy: $i] :
% 1.66/1.08 ( ( Xx = Xy )
% 1.66/1.08 => ( ( in @ Xx @ A )
% 1.66/1.08 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(exuE3u_type,type,
% 1.66/1.08 exuE3u: $o ).
% 1.66/1.08
% 1.66/1.08 thf(exuE3u,definition,
% 1.66/1.08 ( exuE3u
% 1.66/1.08 = ( ! [Xphi: $i > $o] :
% 1.66/1.08 ( ( exu
% 1.66/1.08 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 1.66/1.08 => ! [Xx: $i,Xy: $i] :
% 1.66/1.08 ( ( Xphi @ Xx )
% 1.66/1.08 => ( ( Xphi @ Xy )
% 1.66/1.08 => ( Xx = Xy ) ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(exu__Cong_type,type,
% 1.66/1.08 exu__Cong: $o ).
% 1.66/1.08
% 1.66/1.08 thf(exu__Cong,definition,
% 1.66/1.08 ( exu__Cong
% 1.66/1.08 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 1.66/1.08 ( ! [Xx: $i,Xy: $i] :
% 1.66/1.08 ( ( Xx = Xy )
% 1.66/1.08 => ( ( Xphi @ Xx )
% 1.66/1.08 <=> ( Xpsi @ Xy ) ) )
% 1.66/1.08 => ( ( exu
% 1.66/1.08 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 1.66/1.08 <=> ( exu
% 1.66/1.08 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(emptyset__Cong_type,type,
% 1.66/1.08 emptyset__Cong: $o ).
% 1.66/1.08
% 1.66/1.08 thf(emptyset__Cong,definition,
% 1.66/1.08 ( emptyset__Cong
% 1.66/1.08 = ( emptyset = emptyset ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(setadjoin__Cong_type,type,
% 1.66/1.08 setadjoin__Cong: $o ).
% 1.66/1.08
% 1.66/1.08 thf(setadjoin__Cong,definition,
% 1.66/1.08 ( setadjoin__Cong
% 1.66/1.08 = ( ! [Xx: $i,Xy: $i] :
% 1.66/1.08 ( ( Xx = Xy )
% 1.66/1.08 => ! [Xz: $i,Xu: $i] :
% 1.66/1.08 ( ( Xz = Xu )
% 1.66/1.08 => ( ( setadjoin @ Xx @ Xz )
% 1.66/1.08 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(powerset__Cong_type,type,
% 1.66/1.08 powerset__Cong: $o ).
% 1.66/1.08
% 1.66/1.08 thf(powerset__Cong,definition,
% 1.66/1.08 ( powerset__Cong
% 1.66/1.08 = ( ! [A: $i,B: $i] :
% 1.66/1.08 ( ( A = B )
% 1.66/1.08 => ( ( powerset @ A )
% 1.66/1.08 = ( powerset @ B ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(setunion__Cong_type,type,
% 1.66/1.08 setunion__Cong: $o ).
% 1.66/1.08
% 1.66/1.08 thf(setunion__Cong,definition,
% 1.66/1.08 ( setunion__Cong
% 1.66/1.08 = ( ! [A: $i,B: $i] :
% 1.66/1.08 ( ( A = B )
% 1.66/1.08 => ( ( setunion @ A )
% 1.66/1.08 = ( setunion @ B ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(omega__Cong_type,type,
% 1.66/1.08 omega__Cong: $o ).
% 1.66/1.08
% 1.66/1.08 thf(omega__Cong,definition,
% 1.66/1.08 ( omega__Cong
% 1.66/1.08 = ( omega = omega ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(exuEu_type,type,
% 1.66/1.08 exuEu: $o ).
% 1.66/1.08
% 1.66/1.08 thf(exuEu,definition,
% 1.66/1.08 ( exuEu
% 1.66/1.08 = ( ! [Xphi: $i > $o] :
% 1.66/1.08 ( ( exu
% 1.66/1.08 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 1.66/1.08 => ! [Xx: $i,Xy: $i] :
% 1.66/1.08 ( ( Xphi @ Xx )
% 1.66/1.08 => ( ( Xphi @ Xy )
% 1.66/1.08 => ( Xx = Xy ) ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(descr__Cong_type,type,
% 1.66/1.08 descr__Cong: $o ).
% 1.66/1.08
% 1.66/1.08 thf(descr__Cong,definition,
% 1.66/1.08 ( descr__Cong
% 1.66/1.08 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 1.66/1.08 ( ! [Xx: $i,Xy: $i] :
% 1.66/1.08 ( ( Xx = Xy )
% 1.66/1.08 => ( ( Xphi @ Xx )
% 1.66/1.08 <=> ( Xpsi @ Xy ) ) )
% 1.66/1.08 => ( ( exu
% 1.66/1.08 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 1.66/1.08 => ( ( exu
% 1.66/1.08 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 1.66/1.08 => ( ( descr
% 1.66/1.08 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 1.66/1.08 = ( descr
% 1.66/1.08 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(dsetconstr__Cong_type,type,
% 1.66/1.08 dsetconstr__Cong: $o ).
% 1.66/1.08
% 1.66/1.08 thf(dsetconstr__Cong,definition,
% 1.66/1.08 ( dsetconstr__Cong
% 1.66/1.08 = ( ! [A: $i,B: $i] :
% 1.66/1.08 ( ( A = B )
% 1.66/1.08 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 1.66/1.08 ( ! [Xx: $i] :
% 1.66/1.08 ( ( in @ Xx @ A )
% 1.66/1.08 => ! [Xy: $i] :
% 1.66/1.08 ( ( in @ Xy @ B )
% 1.66/1.08 => ( ( Xx = Xy )
% 1.66/1.08 => ( ( Xphi @ Xx )
% 1.66/1.08 <=> ( Xpsi @ Xy ) ) ) ) )
% 1.66/1.08 => ( ( dsetconstr @ A
% 1.66/1.08 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 1.66/1.08 = ( dsetconstr @ B
% 1.66/1.08 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(subset_type,type,
% 1.66/1.08 subset: $i > $i > $o ).
% 1.66/1.08
% 1.66/1.08 thf(disjoint_type,type,
% 1.66/1.08 disjoint: $i > $i > $o ).
% 1.66/1.08
% 1.66/1.08 thf(setsmeet_type,type,
% 1.66/1.08 setsmeet: $i > $i > $o ).
% 1.66/1.08
% 1.66/1.08 thf(subsetI1_type,type,
% 1.66/1.08 subsetI1: $o ).
% 1.66/1.08
% 1.66/1.08 thf(subsetI1,definition,
% 1.66/1.08 ( subsetI1
% 1.66/1.08 = ( ! [A: $i,B: $i] :
% 1.66/1.08 ( ! [Xx: $i] :
% 1.66/1.08 ( ( in @ Xx @ A )
% 1.66/1.08 => ( in @ Xx @ B ) )
% 1.66/1.08 => ( subset @ A @ B ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(eqimpsubset2_type,type,
% 1.66/1.08 eqimpsubset2: $o ).
% 1.66/1.08
% 1.66/1.08 thf(eqimpsubset2,definition,
% 1.66/1.08 ( eqimpsubset2
% 1.66/1.08 = ( ! [A: $i,B: $i] :
% 1.66/1.08 ( ( A = B )
% 1.66/1.08 => ( subset @ B @ A ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(eqimpsubset1_type,type,
% 1.66/1.08 eqimpsubset1: $o ).
% 1.66/1.08
% 1.66/1.08 thf(eqimpsubset1,definition,
% 1.66/1.08 ( eqimpsubset1
% 1.66/1.08 = ( ! [A: $i,B: $i] :
% 1.66/1.08 ( ( A = B )
% 1.66/1.08 => ( subset @ A @ B ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(subsetI2_type,type,
% 1.66/1.08 subsetI2: $o ).
% 1.66/1.08
% 1.66/1.08 thf(subsetI2,definition,
% 1.66/1.08 ( subsetI2
% 1.66/1.08 = ( ! [A: $i,B: $i] :
% 1.66/1.08 ( ! [Xx: $i] :
% 1.66/1.08 ( ( in @ Xx @ A )
% 1.66/1.08 => ( in @ Xx @ B ) )
% 1.66/1.08 => ( subset @ A @ B ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(emptysetsubset_type,type,
% 1.66/1.08 emptysetsubset: $o ).
% 1.66/1.08
% 1.66/1.08 thf(emptysetsubset,definition,
% 1.66/1.08 ( emptysetsubset
% 1.66/1.08 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(subsetE_type,type,
% 1.66/1.08 subsetE: $o ).
% 1.66/1.08
% 1.66/1.08 thf(subsetE,definition,
% 1.66/1.08 ( subsetE
% 1.66/1.08 = ( ! [A: $i,B: $i,Xx: $i] :
% 1.66/1.08 ( ( subset @ A @ B )
% 1.66/1.08 => ( ( in @ Xx @ A )
% 1.66/1.08 => ( in @ Xx @ B ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(subsetE2_type,type,
% 1.66/1.08 subsetE2: $o ).
% 1.66/1.08
% 1.66/1.08 thf(subsetE2,definition,
% 1.66/1.08 ( subsetE2
% 1.66/1.08 = ( ! [A: $i,B: $i,Xx: $i] :
% 1.66/1.08 ( ( subset @ A @ B )
% 1.66/1.08 => ( ~ ( in @ Xx @ B )
% 1.66/1.08 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(notsubsetI_type,type,
% 1.66/1.08 notsubsetI: $o ).
% 1.66/1.08
% 1.66/1.08 thf(notsubsetI,definition,
% 1.66/1.08 ( notsubsetI
% 1.66/1.08 = ( ! [A: $i,B: $i,Xx: $i] :
% 1.66/1.08 ( ( in @ Xx @ A )
% 1.66/1.08 => ( ~ ( in @ Xx @ B )
% 1.66/1.08 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(notequalI1_type,type,
% 1.66/1.08 notequalI1: $o ).
% 1.66/1.08
% 1.66/1.08 thf(notequalI1,definition,
% 1.66/1.08 ( notequalI1
% 1.66/1.08 = ( ! [A: $i,B: $i] :
% 1.66/1.08 ( ~ ( subset @ A @ B )
% 1.66/1.08 => ( A != B ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(notequalI2_type,type,
% 1.66/1.08 notequalI2: $o ).
% 1.66/1.08
% 1.66/1.08 thf(notequalI2,definition,
% 1.66/1.08 ( notequalI2
% 1.66/1.08 = ( ! [A: $i,B: $i,Xx: $i] :
% 1.66/1.08 ( ( in @ Xx @ A )
% 1.66/1.08 => ( ~ ( in @ Xx @ B )
% 1.66/1.08 => ( A != B ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(subsetRefl_type,type,
% 1.66/1.08 subsetRefl: $o ).
% 1.66/1.08
% 1.66/1.08 thf(subsetRefl,definition,
% 1.66/1.08 ( subsetRefl
% 1.66/1.08 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(subsetTrans_type,type,
% 1.66/1.08 subsetTrans: $o ).
% 1.66/1.08
% 1.66/1.08 thf(subsetTrans,definition,
% 1.66/1.08 ( subsetTrans
% 1.66/1.08 = ( ! [A: $i,B: $i,C: $i] :
% 1.66/1.08 ( ( subset @ A @ B )
% 1.66/1.08 => ( ( subset @ B @ C )
% 1.66/1.08 => ( subset @ A @ C ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(setadjoinSub_type,type,
% 1.66/1.08 setadjoinSub: $o ).
% 1.66/1.08
% 1.66/1.08 thf(setadjoinSub,definition,
% 1.66/1.08 ( setadjoinSub
% 1.66/1.08 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(setadjoinSub2_type,type,
% 1.66/1.08 setadjoinSub2: $o ).
% 1.66/1.08
% 1.66/1.08 thf(setadjoinSub2,definition,
% 1.66/1.08 ( setadjoinSub2
% 1.66/1.08 = ( ! [A: $i,Xx: $i,B: $i] :
% 1.66/1.08 ( ( subset @ A @ B )
% 1.66/1.08 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(subset2powerset_type,type,
% 1.66/1.08 subset2powerset: $o ).
% 1.66/1.08
% 1.66/1.08 thf(subset2powerset,definition,
% 1.66/1.08 ( subset2powerset
% 1.66/1.08 = ( ! [A: $i,B: $i] :
% 1.66/1.08 ( ( subset @ A @ B )
% 1.66/1.08 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(setextsub_type,type,
% 1.66/1.08 setextsub: $o ).
% 1.66/1.08
% 1.66/1.08 thf(setextsub,definition,
% 1.66/1.08 ( setextsub
% 1.66/1.08 = ( ! [A: $i,B: $i] :
% 1.66/1.08 ( ( subset @ A @ B )
% 1.66/1.08 => ( ( subset @ B @ A )
% 1.66/1.08 => ( A = B ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(subsetemptysetimpeq_type,type,
% 1.66/1.08 subsetemptysetimpeq: $o ).
% 1.66/1.08
% 1.66/1.08 thf(subsetemptysetimpeq,definition,
% 1.66/1.08 ( subsetemptysetimpeq
% 1.66/1.08 = ( ! [A: $i] :
% 1.66/1.08 ( ( subset @ A @ emptyset )
% 1.66/1.08 => ( A = emptyset ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(powersetI1_type,type,
% 1.66/1.08 powersetI1: $o ).
% 1.66/1.08
% 1.66/1.08 thf(powersetI1,definition,
% 1.66/1.08 ( powersetI1
% 1.66/1.08 = ( ! [A: $i,B: $i] :
% 1.66/1.08 ( ( subset @ B @ A )
% 1.66/1.08 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(powersetE1_type,type,
% 1.66/1.08 powersetE1: $o ).
% 1.66/1.08
% 1.66/1.08 thf(powersetE1,definition,
% 1.66/1.08 ( powersetE1
% 1.66/1.08 = ( ! [A: $i,B: $i] :
% 1.66/1.08 ( ( in @ B @ ( powerset @ A ) )
% 1.66/1.08 => ( subset @ B @ A ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(inPowerset_type,type,
% 1.66/1.08 inPowerset: $o ).
% 1.66/1.08
% 1.66/1.08 thf(inPowerset,definition,
% 1.66/1.08 ( inPowerset
% 1.66/1.08 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(powersetsubset_type,type,
% 1.66/1.08 powersetsubset: $o ).
% 1.66/1.08
% 1.66/1.08 thf(powersetsubset,definition,
% 1.66/1.08 ( powersetsubset
% 1.66/1.08 = ( ! [A: $i,B: $i] :
% 1.66/1.08 ( ( subset @ A @ B )
% 1.66/1.08 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(sepInPowerset_type,type,
% 1.66/1.08 sepInPowerset: $o ).
% 1.66/1.08
% 1.66/1.08 thf(sepInPowerset,definition,
% 1.66/1.08 ( sepInPowerset
% 1.66/1.08 = ( ! [A: $i,Xphi: $i > $o] :
% 1.66/1.08 ( in
% 1.66/1.08 @ ( dsetconstr @ A
% 1.66/1.08 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 1.66/1.08 @ ( powerset @ A ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(sepSubset_type,type,
% 1.66/1.08 sepSubset: $o ).
% 1.66/1.08
% 1.66/1.08 thf(sepSubset,definition,
% 1.66/1.08 ( sepSubset
% 1.66/1.08 = ( ! [A: $i,Xphi: $i > $o] :
% 1.66/1.08 ( subset
% 1.66/1.08 @ ( dsetconstr @ A
% 1.66/1.08 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 1.66/1.08 @ A ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(binunion_type,type,
% 1.66/1.08 binunion: $i > $i > $i ).
% 1.66/1.08
% 1.66/1.08 thf(binunionIL_type,type,
% 1.66/1.08 binunionIL: $o ).
% 1.66/1.08
% 1.66/1.08 thf(binunionIL,definition,
% 1.66/1.08 ( binunionIL
% 1.66/1.08 = ( ! [A: $i,B: $i,Xx: $i] :
% 1.66/1.08 ( ( in @ Xx @ A )
% 1.66/1.08 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(upairset2IR_type,type,
% 1.66/1.08 upairset2IR: $o ).
% 1.66/1.08
% 1.66/1.08 thf(upairset2IR,definition,
% 1.66/1.08 ( upairset2IR
% 1.66/1.08 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(binunionIR_type,type,
% 1.66/1.08 binunionIR: $o ).
% 1.66/1.08
% 1.66/1.08 thf(binunionIR,definition,
% 1.66/1.08 ( binunionIR
% 1.66/1.08 = ( ! [A: $i,B: $i,Xx: $i] :
% 1.66/1.08 ( ( in @ Xx @ B )
% 1.66/1.08 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(binunionEcases_type,type,
% 1.66/1.08 binunionEcases: $o ).
% 1.66/1.08
% 1.66/1.08 thf(binunionEcases,definition,
% 1.66/1.08 ( binunionEcases
% 1.66/1.08 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 1.66/1.08 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 1.66/1.08 => ( ( ( in @ Xx @ A )
% 1.66/1.08 => Xphi )
% 1.66/1.08 => ( ( ( in @ Xx @ B )
% 1.66/1.08 => Xphi )
% 1.66/1.08 => Xphi ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(binunionE_type,type,
% 1.66/1.08 binunionE: $o ).
% 1.66/1.08
% 1.66/1.08 thf(binunionE,definition,
% 1.66/1.08 ( binunionE
% 1.66/1.08 = ( ! [A: $i,B: $i,Xx: $i] :
% 1.66/1.08 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 1.66/1.08 => ( ( in @ Xx @ A )
% 1.66/1.08 | ( in @ Xx @ B ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(binunionLsub_type,type,
% 1.66/1.08 binunionLsub: $o ).
% 1.66/1.08
% 1.66/1.08 thf(binunionLsub,definition,
% 1.66/1.08 ( binunionLsub
% 1.66/1.08 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(binunionRsub_type,type,
% 1.66/1.08 binunionRsub: $o ).
% 1.66/1.08
% 1.66/1.08 thf(binunionRsub,definition,
% 1.66/1.08 ( binunionRsub
% 1.66/1.08 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(binintersect_type,type,
% 1.66/1.08 binintersect: $i > $i > $i ).
% 1.66/1.08
% 1.66/1.08 thf(binintersectI_type,type,
% 1.66/1.08 binintersectI: $o ).
% 1.66/1.08
% 1.66/1.08 thf(binintersectI,definition,
% 1.66/1.08 ( binintersectI
% 1.66/1.08 = ( ! [A: $i,B: $i,Xx: $i] :
% 1.66/1.08 ( ( in @ Xx @ A )
% 1.66/1.08 => ( ( in @ Xx @ B )
% 1.66/1.08 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(binintersectSubset5_type,type,
% 1.66/1.08 binintersectSubset5: $o ).
% 1.66/1.08
% 1.66/1.08 thf(binintersectSubset5,definition,
% 1.66/1.08 ( binintersectSubset5
% 1.66/1.08 = ( ! [A: $i,B: $i,C: $i] :
% 1.66/1.08 ( ( subset @ C @ A )
% 1.66/1.08 => ( ( subset @ C @ B )
% 1.66/1.08 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(binintersectEL_type,type,
% 1.66/1.08 binintersectEL: $o ).
% 1.66/1.08
% 1.66/1.08 thf(binintersectEL,definition,
% 1.66/1.08 ( binintersectEL
% 1.66/1.08 = ( ! [A: $i,B: $i,Xx: $i] :
% 1.66/1.08 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 1.66/1.08 => ( in @ Xx @ A ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(binintersectLsub_type,type,
% 1.66/1.08 binintersectLsub: $o ).
% 1.66/1.08
% 1.66/1.08 thf(binintersectLsub,definition,
% 1.66/1.08 ( binintersectLsub
% 1.66/1.08 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(binintersectSubset2_type,type,
% 1.66/1.08 binintersectSubset2: $o ).
% 1.66/1.08
% 1.66/1.08 thf(binintersectSubset2,definition,
% 1.66/1.08 ( binintersectSubset2
% 1.66/1.08 = ( ! [A: $i,B: $i] :
% 1.66/1.08 ( ( subset @ A @ B )
% 1.66/1.08 => ( ( binintersect @ A @ B )
% 1.66/1.08 = A ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(binintersectSubset3_type,type,
% 1.66/1.08 binintersectSubset3: $o ).
% 1.66/1.08
% 1.66/1.08 thf(binintersectSubset3,definition,
% 1.66/1.08 ( binintersectSubset3
% 1.66/1.08 = ( ! [A: $i,B: $i] :
% 1.66/1.08 ( ( ( binintersect @ A @ B )
% 1.66/1.08 = B )
% 1.66/1.08 => ( subset @ B @ A ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(binintersectER_type,type,
% 1.66/1.08 binintersectER: $o ).
% 1.66/1.08
% 1.66/1.08 thf(binintersectER,definition,
% 1.66/1.08 ( binintersectER
% 1.66/1.08 = ( ! [A: $i,B: $i,Xx: $i] :
% 1.66/1.08 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 1.66/1.08 => ( in @ Xx @ B ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(disjointsetsI1_type,type,
% 1.66/1.08 disjointsetsI1: $o ).
% 1.66/1.08
% 1.66/1.08 thf(disjointsetsI1,definition,
% 1.66/1.08 ( disjointsetsI1
% 1.66/1.08 = ( ! [A: $i,B: $i] :
% 1.66/1.08 ( ~ ? [Xx: $i] :
% 1.66/1.08 ( ( in @ Xx @ A )
% 1.66/1.08 & ( in @ Xx @ B ) )
% 1.66/1.08 => ( ( binintersect @ A @ B )
% 1.66/1.08 = emptyset ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(binintersectRsub_type,type,
% 1.66/1.08 binintersectRsub: $o ).
% 1.66/1.08
% 1.66/1.08 thf(binintersectRsub,definition,
% 1.66/1.08 ( binintersectRsub
% 1.66/1.08 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(binintersectSubset4_type,type,
% 1.66/1.08 binintersectSubset4: $o ).
% 1.66/1.08
% 1.66/1.08 thf(binintersectSubset4,definition,
% 1.66/1.08 ( binintersectSubset4
% 1.66/1.08 = ( ! [A: $i,B: $i] :
% 1.66/1.08 ( ( subset @ B @ A )
% 1.66/1.08 => ( ( binintersect @ A @ B )
% 1.66/1.08 = B ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(binintersectSubset1_type,type,
% 1.66/1.08 binintersectSubset1: $o ).
% 1.66/1.08
% 1.66/1.08 thf(binintersectSubset1,definition,
% 1.66/1.08 ( binintersectSubset1
% 1.66/1.08 = ( ! [A: $i,B: $i] :
% 1.66/1.08 ( ( ( binintersect @ A @ B )
% 1.66/1.08 = A )
% 1.66/1.08 => ( subset @ A @ B ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(bs114d_type,type,
% 1.66/1.08 bs114d: $o ).
% 1.66/1.08
% 1.66/1.08 thf(bs114d,definition,
% 1.66/1.08 ( bs114d
% 1.66/1.08 = ( ! [A: $i,B: $i,C: $i] :
% 1.66/1.08 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 1.66/1.08 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(regular_type,type,
% 1.66/1.08 regular: $i > $o ).
% 1.66/1.08
% 1.66/1.08 thf(setminus_type,type,
% 1.66/1.08 setminus: $i > $i > $i ).
% 1.66/1.08
% 1.66/1.08 thf(setminusI_type,type,
% 1.66/1.08 setminusI: $o ).
% 1.66/1.08
% 1.66/1.08 thf(setminusI,definition,
% 1.66/1.08 ( setminusI
% 1.66/1.08 = ( ! [A: $i,B: $i,Xx: $i] :
% 1.66/1.08 ( ( in @ Xx @ A )
% 1.66/1.08 => ( ~ ( in @ Xx @ B )
% 1.66/1.08 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(setminusEL_type,type,
% 1.66/1.08 setminusEL: $o ).
% 1.66/1.08
% 1.66/1.08 thf(setminusEL,definition,
% 1.66/1.08 ( setminusEL
% 1.66/1.08 = ( ! [A: $i,B: $i,Xx: $i] :
% 1.66/1.08 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 1.66/1.08 => ( in @ Xx @ A ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(setminusER_type,type,
% 1.66/1.08 setminusER: $o ).
% 1.66/1.08
% 1.66/1.08 thf(setminusER,definition,
% 1.66/1.08 ( setminusER
% 1.66/1.08 = ( ! [A: $i,B: $i,Xx: $i] :
% 1.66/1.08 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 1.66/1.08 => ~ ( in @ Xx @ B ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(setminusSubset2_type,type,
% 1.66/1.08 setminusSubset2: $o ).
% 1.66/1.08
% 1.66/1.08 thf(setminusSubset2,definition,
% 1.66/1.08 ( setminusSubset2
% 1.66/1.08 = ( ! [A: $i,B: $i] :
% 1.66/1.08 ( ( subset @ A @ B )
% 1.66/1.08 => ( ( setminus @ A @ B )
% 1.66/1.08 = emptyset ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(setminusERneg_type,type,
% 1.66/1.08 setminusERneg: $o ).
% 1.66/1.08
% 1.66/1.08 thf(setminusERneg,definition,
% 1.66/1.08 ( setminusERneg
% 1.66/1.08 = ( ! [A: $i,B: $i,Xx: $i] :
% 1.66/1.08 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 1.66/1.08 => ( ( in @ Xx @ A )
% 1.66/1.08 => ( in @ Xx @ B ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(setminusELneg_type,type,
% 1.66/1.08 setminusELneg: $o ).
% 1.66/1.08
% 1.66/1.08 thf(setminusELneg,definition,
% 1.66/1.08 ( setminusELneg
% 1.66/1.08 = ( ! [A: $i,B: $i,Xx: $i] :
% 1.66/1.08 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 1.66/1.08 => ( ~ ( in @ Xx @ B )
% 1.66/1.08 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(setminusILneg_type,type,
% 1.66/1.08 setminusILneg: $o ).
% 1.66/1.08
% 1.66/1.08 thf(setminusILneg,definition,
% 1.66/1.08 ( setminusILneg
% 1.66/1.08 = ( ! [A: $i,B: $i,Xx: $i] :
% 1.66/1.08 ( ~ ( in @ Xx @ A )
% 1.66/1.08 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(setminusIRneg_type,type,
% 1.66/1.08 setminusIRneg: $o ).
% 1.66/1.08
% 1.66/1.08 thf(setminusIRneg,definition,
% 1.66/1.08 ( setminusIRneg
% 1.66/1.08 = ( ! [A: $i,B: $i,Xx: $i] :
% 1.66/1.08 ( ( in @ Xx @ B )
% 1.66/1.08 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(setminusLsub_type,type,
% 1.66/1.08 setminusLsub: $o ).
% 1.66/1.08
% 1.66/1.08 thf(setminusLsub,definition,
% 1.66/1.08 ( setminusLsub
% 1.66/1.08 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(setminusSubset1_type,type,
% 1.66/1.08 setminusSubset1: $o ).
% 1.66/1.08
% 1.66/1.08 thf(setminusSubset1,definition,
% 1.66/1.08 ( setminusSubset1
% 1.66/1.08 = ( ! [A: $i,B: $i] :
% 1.66/1.08 ( ( ( setminus @ A @ B )
% 1.66/1.08 = emptyset )
% 1.66/1.08 => ( subset @ A @ B ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(symdiff_type,type,
% 1.66/1.08 symdiff: $i > $i > $i ).
% 1.66/1.08
% 1.66/1.08 thf(symdiffE_type,type,
% 1.66/1.08 symdiffE: $o ).
% 1.66/1.08
% 1.66/1.08 thf(symdiffE,definition,
% 1.66/1.08 ( symdiffE
% 1.66/1.08 = ( ! [A: $i,B: $i,Xx: $i] :
% 1.66/1.08 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 1.66/1.08 => ! [Xphi: $o] :
% 1.66/1.08 ( ( ( in @ Xx @ A )
% 1.66/1.08 => ( ~ ( in @ Xx @ B )
% 1.66/1.08 => Xphi ) )
% 1.66/1.08 => ( ( ~ ( in @ Xx @ A )
% 1.66/1.08 => ( ( in @ Xx @ B )
% 1.66/1.08 => Xphi ) )
% 1.66/1.08 => Xphi ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(symdiffI1_type,type,
% 1.66/1.08 symdiffI1: $o ).
% 1.66/1.08
% 1.66/1.08 thf(symdiffI1,definition,
% 1.66/1.08 ( symdiffI1
% 1.66/1.08 = ( ! [A: $i,B: $i,Xx: $i] :
% 1.66/1.08 ( ( in @ Xx @ A )
% 1.66/1.08 => ( ~ ( in @ Xx @ B )
% 1.66/1.08 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(symdiffI2_type,type,
% 1.66/1.08 symdiffI2: $o ).
% 1.66/1.08
% 1.66/1.08 thf(symdiffI2,definition,
% 1.66/1.08 ( symdiffI2
% 1.66/1.08 = ( ! [A: $i,B: $i,Xx: $i] :
% 1.66/1.08 ( ~ ( in @ Xx @ A )
% 1.66/1.08 => ( ( in @ Xx @ B )
% 1.66/1.08 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(symdiffIneg1_type,type,
% 1.66/1.08 symdiffIneg1: $o ).
% 1.66/1.08
% 1.66/1.08 thf(symdiffIneg1,definition,
% 1.66/1.08 ( symdiffIneg1
% 1.66/1.08 = ( ! [A: $i,B: $i,Xx: $i] :
% 1.66/1.08 ( ( in @ Xx @ A )
% 1.66/1.08 => ( ( in @ Xx @ B )
% 1.66/1.08 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(symdiffIneg2_type,type,
% 1.66/1.08 symdiffIneg2: $o ).
% 1.66/1.08
% 1.66/1.08 thf(symdiffIneg2,definition,
% 1.66/1.08 ( symdiffIneg2
% 1.66/1.08 = ( ! [A: $i,B: $i,Xx: $i] :
% 1.66/1.08 ( ~ ( in @ Xx @ A )
% 1.66/1.08 => ( ~ ( in @ Xx @ B )
% 1.66/1.08 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(iskpair_type,type,
% 1.66/1.08 iskpair: $i > $o ).
% 1.66/1.08
% 1.66/1.08 thf(iskpair,definition,
% 1.66/1.08 ( iskpair
% 1.66/1.08 = ( ^ [A: $i] :
% 1.66/1.08 ? [Xx: $i] :
% 1.66/1.08 ( ( in @ Xx @ ( setunion @ A ) )
% 1.66/1.08 & ? [Xy: $i] :
% 1.66/1.08 ( ( in @ Xy @ ( setunion @ A ) )
% 1.66/1.08 & ( A
% 1.66/1.08 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(secondinupair_type,type,
% 1.66/1.08 secondinupair: $o ).
% 1.66/1.08
% 1.66/1.08 thf(secondinupair,definition,
% 1.66/1.08 ( secondinupair
% 1.66/1.08 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(setukpairIL_type,type,
% 1.66/1.08 setukpairIL: $o ).
% 1.66/1.08
% 1.66/1.08 thf(setukpairIL,definition,
% 1.66/1.08 ( setukpairIL
% 1.66/1.08 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(setukpairIR_type,type,
% 1.66/1.08 setukpairIR: $o ).
% 1.66/1.08
% 1.66/1.08 thf(setukpairIR,definition,
% 1.66/1.08 ( setukpairIR
% 1.66/1.08 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(kpairiskpair_type,type,
% 1.66/1.08 kpairiskpair: $o ).
% 1.66/1.08
% 1.66/1.08 thf(kpairiskpair,definition,
% 1.66/1.08 ( kpairiskpair
% 1.66/1.08 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(kpair_type,type,
% 1.66/1.08 kpair: $i > $i > $i ).
% 1.66/1.08
% 1.66/1.08 thf(kpair,definition,
% 1.66/1.08 ( kpair
% 1.66/1.08 = ( ^ [Xx: $i,Xy: $i] : ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(kpairp_type,type,
% 1.66/1.08 kpairp: $o ).
% 1.66/1.08
% 1.66/1.08 thf(kpairp,definition,
% 1.66/1.08 ( kpairp
% 1.66/1.08 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(cartprod_type,type,
% 1.66/1.08 cartprod: $i > $i > $i ).
% 1.66/1.08
% 1.66/1.08 thf(cartprod,definition,
% 1.66/1.08 ( cartprod
% 1.66/1.08 = ( ^ [A: $i,B: $i] :
% 1.66/1.08 ( dsetconstr @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) )
% 1.66/1.08 @ ^ [Xx: $i] :
% 1.66/1.08 ? [Xy: $i] :
% 1.66/1.08 ( ( in @ Xy @ A )
% 1.66/1.08 & ? [Xz: $i] :
% 1.66/1.08 ( ( in @ Xz @ B )
% 1.66/1.08 & ( Xx
% 1.66/1.08 = ( kpair @ Xy @ Xz ) ) ) ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(singletonsubset_type,type,
% 1.66/1.08 singletonsubset: $o ).
% 1.66/1.08
% 1.66/1.08 thf(singletonsubset,definition,
% 1.66/1.08 ( singletonsubset
% 1.66/1.08 = ( ! [A: $i,Xx: $i] :
% 1.66/1.08 ( ( in @ Xx @ A )
% 1.66/1.08 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 1.66/1.08
% 1.66/1.08 thf(singletoninpowerset,conjecture,
% 1.66/1.08 ( setextAx
% 1.66/1.08 => ( emptysetAx
% 1.66/1.08 => ( setadjoinAx
% 1.66/1.08 => ( powersetAx
% 1.66/1.08 => ( setunionAx
% 1.66/1.08 => ( omega0Ax
% 1.66/1.08 => ( omegaSAx
% 1.66/1.08 => ( omegaIndAx
% 1.66/1.08 => ( replAx
% 1.66/1.08 => ( foundationAx
% 1.66/1.08 => ( wellorderingAx
% 1.66/1.08 => ( descrp
% 1.66/1.08 => ( dsetconstrI
% 1.66/1.08 => ( dsetconstrEL
% 1.66/1.08 => ( dsetconstrER
% 1.66/1.08 => ( exuE1
% 1.66/1.08 => ( prop2setE
% 1.66/1.08 => ( emptysetE
% 1.66/1.08 => ( emptysetimpfalse
% 1.66/1.08 => ( notinemptyset
% 1.66/1.08 => ( exuE3e
% 1.66/1.08 => ( setext
% 1.66/1.08 => ( emptyI
% 1.66/1.08 => ( noeltsimpempty
% 1.66/1.08 => ( setbeta
% 1.66/1.08 => ( nonemptyE1
% 1.66/1.08 => ( nonemptyI
% 1.66/1.08 => ( nonemptyI1
% 1.66/1.08 => ( setadjoinIL
% 1.66/1.08 => ( emptyinunitempty
% 1.66/1.08 => ( setadjoinIR
% 1.66/1.08 => ( setadjoinE
% 1.66/1.08 => ( setadjoinOr
% 1.66/1.08 => ( setoftrueEq
% 1.66/1.08 => ( powersetI
% 1.66/1.08 => ( emptyinPowerset
% 1.66/1.08 => ( emptyInPowerset
% 1.66/1.08 => ( powersetE
% 1.66/1.08 => ( setunionI
% 1.66/1.08 => ( setunionE
% 1.66/1.08 => ( subPowSU
% 1.66/1.08 => ( exuE2
% 1.66/1.08 => ( nonemptyImpWitness
% 1.66/1.08 => ( uniqinunit
% 1.66/1.08 => ( notinsingleton
% 1.66/1.08 => ( eqinunit
% 1.66/1.08 => ( singletonsswitch
% 1.66/1.08 => ( upairsetE
% 1.66/1.08 => ( upairsetIL
% 1.66/1.08 => ( upairsetIR
% 1.66/1.08 => ( emptyE1
% 1.66/1.08 => ( vacuousDall
% 1.66/1.08 => ( quantDeMorgan1
% 1.66/1.08 => ( quantDeMorgan2
% 1.66/1.08 => ( quantDeMorgan3
% 1.66/1.08 => ( quantDeMorgan4
% 1.66/1.08 => ( prop2setI
% 1.66/1.08 => ( prop2set2propI
% 1.66/1.08 => ( notdexE
% 1.66/1.08 => ( notdallE
% 1.66/1.08 => ( exuI1
% 1.66/1.08 => ( exuI3
% 1.66/1.08 => ( exuI2
% 1.66/1.08 => ( inCongP
% 1.66/1.08 => ( in__Cong
% 1.66/1.08 => ( exuE3u
% 1.66/1.08 => ( exu__Cong
% 1.66/1.08 => ( emptyset__Cong
% 1.66/1.08 => ( setadjoin__Cong
% 1.66/1.08 => ( powerset__Cong
% 1.66/1.08 => ( setunion__Cong
% 1.66/1.08 => ( omega__Cong
% 1.66/1.08 => ( exuEu
% 1.66/1.08 => ( descr__Cong
% 1.66/1.08 => ( dsetconstr__Cong
% 1.66/1.08 => ( subsetI1
% 1.66/1.08 => ( eqimpsubset2
% 1.66/1.08 => ( eqimpsubset1
% 1.66/1.08 => ( subsetI2
% 1.66/1.08 => ( emptysetsubset
% 1.66/1.08 => ( subsetE
% 1.66/1.08 => ( subsetE2
% 1.66/1.08 => ( notsubsetI
% 1.66/1.08 => ( notequalI1
% 1.66/1.08 => ( notequalI2
% 1.66/1.08 => ( subsetRefl
% 1.66/1.08 => ( subsetTrans
% 1.66/1.08 => ( setadjoinSub
% 1.66/1.08 => ( setadjoinSub2
% 1.66/1.08 => ( subset2powerset
% 1.66/1.08 => ( setextsub
% 1.66/1.08 => ( subsetemptysetimpeq
% 1.66/1.08 => ( powersetI1
% 1.66/1.08 => ( powersetE1
% 1.66/1.08 => ( inPowerset
% 1.66/1.08 => ( powersetsubset
% 1.66/1.08 => ( sepInPowerset
% 1.66/1.08 => ( sepSubset
% 1.66/1.08 => ( binunionIL
% 1.66/1.08 => ( upairset2IR
% 1.66/1.08 => ( binunionIR
% 1.66/1.08 => ( binunionEcases
% 1.66/1.08 => ( binunionE
% 1.66/1.08 => ( binunionLsub
% 1.66/1.08 => ( binunionRsub
% 1.66/1.08 => ( binintersectI
% 1.66/1.08 => ( binintersectSubset5
% 1.66/1.08 => ( binintersectEL
% 1.66/1.08 => ( binintersectLsub
% 1.66/1.08 => ( binintersectSubset2
% 1.66/1.08 => ( binintersectSubset3
% 1.66/1.08 => ( binintersectER
% 1.66/1.08 => ( disjointsetsI1
% 1.66/1.08 => ( binintersectRsub
% 1.66/1.08 => ( binintersectSubset4
% 1.66/1.08 => ( binintersectSubset1
% 1.66/1.08 => ( bs114d
% 1.66/1.08 => ( setminusI
% 1.66/1.08 => ( setminusEL
% 1.66/1.08 => ( setminusER
% 1.66/1.08 => ( setminusSubset2
% 1.66/1.08 => ( setminusERneg
% 1.66/1.08 => ( setminusELneg
% 1.66/1.08 => ( setminusILneg
% 1.66/1.08 => ( setminusIRneg
% 1.66/1.08 => ( setminusLsub
% 1.66/1.08 => ( setminusSubset1
% 1.66/1.08 => ( symdiffE
% 1.66/1.08 => ( symdiffI1
% 1.66/1.08 => ( symdiffI2
% 1.66/1.08 => ( symdiffIneg1
% 1.66/1.08 => ( symdiffIneg2
% 1.66/1.08 => ( secondinupair
% 1.66/1.08 => ( setukpairIL
% 1.66/1.08 => ( setukpairIR
% 1.66/1.08 => ( kpairiskpair
% 1.66/1.08 => ( kpairp
% 1.66/1.08 => ( singletonsubset
% 1.66/1.08 => ! [A: $i,Xx: $i] :
% 1.66/1.08 ( ( in @ Xx @ A )
% 1.66/1.08 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 1.66/1.08 ------------------------
% 1.66/1.10 % (4799)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_4603 for (2999ds/2Mi)
% 1.66/1.10 % (4796)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_4603 for (2999ds/183Mi)
% 1.66/1.10 % (4802)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_4603 for (2999ds/18Mi)
% 1.66/1.10 % (4799)Instruction limit reached!
% 1.66/1.10 % (4799)------------------------------
% 1.66/1.10 % (4799)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.66/1.10 % (4799)Termination reason: Unknown
% 1.66/1.10 % (4799)Termination phase: shuffling
% 1.66/1.10
% 1.66/1.10 % (4799)Memory used [KB]: 1279
% 1.66/1.10 % (4799)Time elapsed: 0.003 s
% 1.66/1.10 % (4799)Instructions burned: 5 (million)
% 1.66/1.10 % (4799)------------------------------
% 1.66/1.10 % (4799)------------------------------
% 1.66/1.10 % (4800)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_4603 for (2999ds/2Mi)
% 1.66/1.10 % (4797)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_4603 for (2999ds/4Mi)
% 1.66/1.10 % (4800)Instruction limit reached!
% 1.66/1.10 % (4800)------------------------------
% 1.66/1.10 % (4800)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.66/1.10 % (4800)Termination reason: Unknown
% 1.66/1.10 % (4800)Termination phase: shuffling
% 1.66/1.10
% 1.66/1.10 % (4800)Memory used [KB]: 1279
% 1.66/1.10 % (4800)Time elapsed: 0.004 s
% 1.66/1.10 % (4800)Instructions burned: 2 (million)
% 1.66/1.10 % (4800)------------------------------
% 1.66/1.10 % (4800)------------------------------
% 1.66/1.10 % (4797)Instruction limit reached!
% 1.66/1.10 % (4797)------------------------------
% 1.66/1.10 % (4797)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.66/1.10 % (4797)Termination reason: Unknown
% 1.66/1.10 % (4797)Termination phase: shuffling
% 1.66/1.10
% 1.66/1.10 % (4797)Memory used [KB]: 1279
% 1.66/1.10 % (4797)Time elapsed: 0.003 s
% 1.66/1.10 % (4797)Instructions burned: 4 (million)
% 1.66/1.10 % (4797)------------------------------
% 1.66/1.10 % (4797)------------------------------
% 1.66/1.10 % (4801)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_4603 for (2999ds/275Mi)
% 1.66/1.10 % (4798)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_4603 for (2999ds/27Mi)
% 1.66/1.10 % (4802)Instruction limit reached!
% 1.66/1.10 % (4802)------------------------------
% 1.66/1.10 % (4802)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.66/1.10 % (4802)Termination reason: Unknown
% 1.66/1.10 % (4802)Termination phase: Preprocessing 2
% 1.66/1.10
% 1.66/1.10 % (4802)Memory used [KB]: 1535
% 1.66/1.10 % (4802)Time elapsed: 0.009 s
% 1.66/1.10 % (4802)Instructions burned: 20 (million)
% 1.66/1.10 % (4802)------------------------------
% 1.66/1.10 % (4802)------------------------------
% 1.66/1.11 % (4803)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_4603 for (2999ds/3Mi)
% 1.66/1.11 % (4803)Instruction limit reached!
% 1.66/1.11 % (4803)------------------------------
% 1.66/1.11 % (4803)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.66/1.11 % (4803)Termination reason: Unknown
% 1.66/1.11 % (4803)Termination phase: shuffling
% 1.66/1.11
% 1.66/1.11 % (4803)Memory used [KB]: 1279
% 1.66/1.11 % (4803)Time elapsed: 0.003 s
% 1.66/1.11 % (4803)Instructions burned: 6 (million)
% 1.66/1.11 % (4803)------------------------------
% 1.66/1.11 % (4803)------------------------------
% 1.66/1.11 % (4798)Instruction limit reached!
% 1.66/1.11 % (4798)------------------------------
% 1.66/1.11 % (4798)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.66/1.11 % (4798)Termination reason: Unknown
% 1.66/1.11 % (4798)Termination phase: Property scanning
% 1.66/1.11
% 1.66/1.11 % (4798)Memory used [KB]: 1663
% 1.66/1.11 % (4798)Time elapsed: 0.009 s
% 1.66/1.11 % (4798)Instructions burned: 28 (million)
% 1.66/1.11 % (4798)------------------------------
% 1.66/1.11 % (4798)------------------------------
% 1.80/1.11 % (4804)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_4603 for (2999ds/37Mi)
% 1.80/1.11 % (4806)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_4603 for (2999ds/3Mi)
% 1.80/1.12 % (4806)Instruction limit reached!
% 1.80/1.12 % (4806)------------------------------
% 1.80/1.12 % (4806)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.80/1.12 % (4806)Termination reason: Unknown
% 1.80/1.12 % (4806)Termination phase: shuffling
% 1.80/1.12
% 1.80/1.12 % (4806)Memory used [KB]: 1279
% 1.80/1.12 % (4806)Time elapsed: 0.002 s
% 1.80/1.12 % (4806)Instructions burned: 3 (million)
% 1.80/1.12 % (4806)------------------------------
% 1.80/1.12 % (4806)------------------------------
% 1.80/1.12 % (4805)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_4603 for (2999ds/15Mi)
% 1.80/1.12 % (4807)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_4603 for (2999ds/1041Mi)
% 1.80/1.12 % (4808)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_4603 for (2999ds/7Mi)
% 1.80/1.12 % (4808)Instruction limit reached!
% 1.80/1.12 % (4808)------------------------------
% 1.80/1.12 % (4808)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.80/1.12 % (4808)Termination reason: Unknown
% 1.80/1.12 % (4808)Termination phase: shuffling
% 1.80/1.12
% 1.80/1.12 % (4808)Memory used [KB]: 1279
% 1.80/1.12 % (4808)Time elapsed: 0.003 s
% 1.80/1.12 % (4808)Instructions burned: 7 (million)
% 1.80/1.12 % (4808)------------------------------
% 1.80/1.12 % (4808)------------------------------
% 1.80/1.12 % (4804)Instruction limit reached!
% 1.80/1.12 % (4804)------------------------------
% 1.80/1.12 % (4804)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.80/1.12 % (4804)Termination reason: Unknown
% 1.80/1.12 % (4804)Termination phase: Property scanning
% 1.80/1.12
% 1.80/1.12 % (4804)Memory used [KB]: 1663
% 1.80/1.12 % (4804)Time elapsed: 0.013 s
% 1.80/1.12 % (4804)Instructions burned: 39 (million)
% 1.80/1.12 % (4804)------------------------------
% 1.80/1.12 % (4804)------------------------------
% 1.80/1.12 % (4805)Instruction limit reached!
% 1.80/1.12 % (4805)------------------------------
% 1.80/1.12 % (4805)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.80/1.12 % (4805)Termination reason: Unknown
% 1.80/1.12 % (4805)Termination phase: Property scanning
% 1.80/1.12
% 1.80/1.12 % (4805)Memory used [KB]: 1407
% 1.80/1.12 % (4805)Time elapsed: 0.009 s
% 1.80/1.12 % (4805)Instructions burned: 16 (million)
% 1.80/1.12 % (4805)------------------------------
% 1.80/1.12 % (4805)------------------------------
% 1.80/1.13 % (4809)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_4603 for (2999ds/16Mi)
% 1.80/1.13 % (4810)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_4603 for (2999ds/3Mi)
% 1.80/1.13 % (4810)Instruction limit reached!
% 1.80/1.13 % (4810)------------------------------
% 1.80/1.13 % (4810)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.80/1.13 % (4810)Termination reason: Unknown
% 1.80/1.13 % (4810)Termination phase: shuffling
% 1.80/1.13
% 1.80/1.13 % (4810)Memory used [KB]: 1279
% 1.80/1.13 % (4810)Time elapsed: 0.002 s
% 1.80/1.13 % (4810)Instructions burned: 3 (million)
% 1.80/1.13 % (4810)------------------------------
% 1.80/1.13 % (4810)------------------------------
% 1.80/1.13 % (4811)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_4603 for (2999ds/3Mi)
% 1.80/1.13 % (4811)Instruction limit reached!
% 1.80/1.13 % (4811)------------------------------
% 1.80/1.13 % (4811)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.80/1.13 % (4811)Termination reason: Unknown
% 1.80/1.13 % (4811)Termination phase: shuffling
% 1.80/1.13
% 1.80/1.13 % (4811)Memory used [KB]: 1279
% 1.80/1.13 % (4811)Time elapsed: 0.002 s
% 1.80/1.13 % (4811)Instructions burned: 3 (million)
% 1.80/1.13 % (4811)------------------------------
% 1.80/1.13 % (4811)------------------------------
% 1.80/1.14 % (4809)Instruction limit reached!
% 1.80/1.14 % (4809)------------------------------
% 1.80/1.14 % (4809)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.80/1.14 % (4809)Termination reason: Unknown
% 1.80/1.14 % (4809)Termination phase: shuffling
% 1.80/1.14
% 1.80/1.14 % (4809)Memory used [KB]: 1535
% 1.80/1.14 % (4809)Time elapsed: 0.006 s
% 1.80/1.14 % (4809)Instructions burned: 18 (million)
% 1.80/1.14 % (4809)------------------------------
% 1.80/1.14 % (4809)------------------------------
% 1.80/1.14 % (4812)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_4603 for (2999ds/7Mi)
% 1.80/1.14 % (4813)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_4603 for (2999ds/3Mi)
% 1.80/1.14 % (4813)Instruction limit reached!
% 1.80/1.14 % (4813)------------------------------
% 1.80/1.14 % (4813)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.80/1.14 % (4813)Termination reason: Unknown
% 1.80/1.14 % (4813)Termination phase: shuffling
% 1.80/1.14
% 1.80/1.14 % (4813)Memory used [KB]: 1279
% 1.80/1.14 % (4813)Time elapsed: 0.002 s
% 1.80/1.14 % (4813)Instructions burned: 3 (million)
% 1.80/1.14 % (4813)------------------------------
% 1.80/1.14 % (4813)------------------------------
% 1.80/1.15 % (4814)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_4603 for (2999ds/4Mi)
% 1.80/1.15 % (4812)Instruction limit reached!
% 1.80/1.15 % (4812)------------------------------
% 1.80/1.15 % (4812)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.80/1.15 % (4812)Termination reason: Unknown
% 1.80/1.15 % (4812)Termination phase: shuffling
% 1.80/1.15
% 1.80/1.15 % (4812)Memory used [KB]: 1407
% 1.80/1.15 % (4812)Time elapsed: 0.005 s
% 1.80/1.15 % (4812)Instructions burned: 8 (million)
% 1.80/1.15 % (4812)------------------------------
% 1.80/1.15 % (4812)------------------------------
% 1.80/1.15 % (4814)Instruction limit reached!
% 1.80/1.15 % (4814)------------------------------
% 1.80/1.15 % (4814)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.80/1.15 % (4814)Termination reason: Unknown
% 1.80/1.15 % (4814)Termination phase: shuffling
% 1.80/1.15
% 1.80/1.15 % (4814)Memory used [KB]: 1279
% 1.80/1.15 % (4814)Time elapsed: 0.003 s
% 1.80/1.15 % (4814)Instructions burned: 6 (million)
% 1.80/1.15 % (4814)------------------------------
% 1.80/1.15 % (4814)------------------------------
% 1.80/1.15 % (4815)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_4603 for (2999ds/18Mi)
% 1.80/1.15 % (4796)Instruction limit reached!
% 1.80/1.15 % (4796)------------------------------
% 1.80/1.15 % (4796)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.80/1.15 % (4796)Termination reason: Unknown
% 1.80/1.15 % (4796)Termination phase: Saturation
% 1.80/1.15
% 1.80/1.15 % (4796)Memory used [KB]: 7164
% 1.80/1.15 % (4796)Time elapsed: 0.053 s
% 1.80/1.15 % (4796)Instructions burned: 185 (million)
% 1.80/1.15 % (4796)------------------------------
% 1.80/1.15 % (4796)------------------------------
% 1.80/1.15 % (4815)Instruction limit reached!
% 1.80/1.15 % (4815)------------------------------
% 1.80/1.15 % (4815)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.80/1.15 % (4815)Termination reason: Unknown
% 1.80/1.15 % (4815)Termination phase: shuffling
% 1.80/1.15
% 1.80/1.15 % (4815)Memory used [KB]: 1663
% 1.80/1.15 % (4815)Time elapsed: 0.007 s
% 1.80/1.15 % (4815)Instructions burned: 20 (million)
% 1.80/1.15 % (4815)------------------------------
% 1.80/1.15 % (4815)------------------------------
% 1.80/1.15 % (4816)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_4603 for (2999ds/710Mi)
% 1.80/1.16 % (4817)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_4603 for (2999ds/6Mi)
% 1.80/1.16 % (4818)dis+1002_5:1_au=on:bd=off:e2e=on:fde=none:fs=off:fsr=off:sos=on:i=902:si=on:rtra=on_0 on DTF2THF_4603 for (2999ds/902Mi)
% 1.80/1.16 % (4817)Instruction limit reached!
% 1.80/1.16 % (4817)------------------------------
% 1.80/1.16 % (4817)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.80/1.16 % (4817)Termination reason: Unknown
% 1.80/1.16 % (4817)Termination phase: shuffling
% 1.80/1.16
% 1.80/1.16 % (4817)Memory used [KB]: 1279
% 1.80/1.16 % (4817)Time elapsed: 0.003 s
% 1.80/1.16 % (4817)Instructions burned: 6 (million)
% 1.80/1.16 % (4817)------------------------------
% 1.80/1.16 % (4817)------------------------------
% 1.80/1.16 % (4819)dis+21_1:8_apa=on:cnfonf=off:fd=off:fsr=off:hud=0:ins=1:kws=inv_frequency:nwc=10.0:ss=axioms:st=5.0:i=21:si=on:rtra=on_0 on DTF2THF_4603 for (2999ds/21Mi)
% 1.80/1.16 % (4820)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_4603 for (2999ds/5Mi)
% 1.80/1.16 % (4819)Instruction limit reached!
% 1.80/1.16 % (4819)------------------------------
% 1.80/1.16 % (4819)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.80/1.16 % (4819)Termination reason: Unknown
% 1.80/1.16 % (4819)Termination phase: Property scanning
% 1.80/1.16
% 1.80/1.16 % (4819)Memory used [KB]: 1663
% 1.80/1.16 % (4819)Time elapsed: 0.007 s
% 1.80/1.16 % (4819)Instructions burned: 21 (million)
% 1.80/1.16 % (4819)------------------------------
% 1.80/1.16 % (4819)------------------------------
% 1.80/1.16 % (4820)Instruction limit reached!
% 1.80/1.16 % (4820)------------------------------
% 1.80/1.16 % (4820)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.80/1.16 % (4820)Termination reason: Unknown
% 1.80/1.16 % (4820)Termination phase: shuffling
% 1.80/1.16
% 1.80/1.16 % (4820)Memory used [KB]: 1279
% 1.80/1.16 % (4820)Time elapsed: 0.003 s
% 1.80/1.16 % (4820)Instructions burned: 6 (million)
% 1.80/1.16 % (4820)------------------------------
% 1.80/1.16 % (4820)------------------------------
% 1.80/1.17 % (4821)lrs+2_1:1_cnfonf=lazy_not_gen_be_off:cs=on:fe=off:hud=10:inj=on:ins=3:plsq=on:plsqc=1:sd=10:ss=axioms:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_4603 for (2999ds/6Mi)
% 1.80/1.17 % (4827)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_4603 for (2999ds/377Mi)
% 1.80/1.17 % (4821)Instruction limit reached!
% 1.80/1.17 % (4821)------------------------------
% 1.80/1.17 % (4821)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.80/1.17 % (4821)Termination reason: Unknown
% 1.80/1.17 % (4821)Termination phase: shuffling
% 1.80/1.17
% 1.80/1.17 % (4821)Memory used [KB]: 1279
% 1.80/1.17 % (4821)Time elapsed: 0.004 s
% 1.80/1.17 % (4821)Instructions burned: 6 (million)
% 1.80/1.17 % (4821)------------------------------
% 1.80/1.17 % (4821)------------------------------
% 1.80/1.17 % (4828)dis+1010_1:4_atotf=0.2:c=on:cbe=off:cnfonf=lazy_simp:fe=off:ins=2:ntd=on:s2a=on:s2at=5.0:sgt=5:ss=axioms:st=1.5:i=779:si=on:rtra=on_0 on DTF2THF_4603 for (2999ds/779Mi)
% 1.80/1.18 % (4829)lrs+10_1:1_cnfonf=lazy_not_be_gen:ntd=on:sp=const_min:ss=axioms:sup=off:i=19:si=on:rtra=on_0 on DTF2THF_4603 for (2999ds/19Mi)
% 1.80/1.19 % (4829)Instruction limit reached!
% 1.80/1.19 % (4829)------------------------------
% 1.80/1.19 % (4829)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.80/1.19 % (4829)Termination reason: Unknown
% 1.80/1.19 % (4829)Termination phase: Property scanning
% 1.80/1.19
% 1.80/1.19 % (4829)Memory used [KB]: 1663
% 1.80/1.19 % (4829)Time elapsed: 0.007 s
% 1.80/1.19 % (4829)Instructions burned: 22 (million)
% 1.80/1.19 % (4829)------------------------------
% 1.80/1.19 % (4829)------------------------------
% 1.80/1.20 % (4839)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_4603 for (2998ds/879Mi)
% 1.80/1.21 % (4801)Instruction limit reached!
% 1.80/1.21 % (4801)------------------------------
% 1.80/1.21 % (4801)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.80/1.21 % (4801)Termination reason: Unknown
% 1.80/1.21 % (4801)Termination phase: Saturation
% 1.80/1.21
% 1.80/1.21 % (4801)Memory used [KB]: 7931
% 1.80/1.21 % (4801)Time elapsed: 0.113 s
% 1.80/1.21 % (4801)Instructions burned: 275 (million)
% 1.80/1.21 % (4801)------------------------------
% 1.80/1.21 % (4801)------------------------------
% 1.80/1.22 % (4818)First to succeed.
% 1.80/1.23 % (4818)Refutation found. Thanks to Tanya!
% 1.80/1.23 % SZS status Theorem for DTF2THF_4603
% 1.80/1.23 % SZS output start Proof for DTF2THF_4603
% 1.80/1.23 thf(func_def_0, type, in: $i > $i > $o).
% 1.80/1.23 thf(func_def_1, type, exu: ($i > $o) > $o).
% 1.80/1.23 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 1.80/1.23 thf(func_def_8, type, powerset: $i > $i).
% 1.80/1.23 thf(func_def_10, type, setunion: $i > $i).
% 1.80/1.23 thf(func_def_19, type, descr: ($i > $o) > $i).
% 1.80/1.23 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 1.80/1.23 thf(func_def_26, type, prop2set: $o > $i).
% 1.80/1.23 thf(func_def_36, type, nonempty: $i > $o).
% 1.80/1.23 thf(func_def_69, type, set2prop: $i > $o).
% 1.80/1.23 thf(func_def_88, type, subset: $i > $i > $o).
% 1.80/1.23 thf(func_def_89, type, disjoint: $i > $i > $o).
% 1.80/1.23 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 1.80/1.23 thf(func_def_114, type, binunion: $i > $i > $i).
% 1.80/1.23 thf(func_def_122, type, binintersect: $i > $i > $i).
% 1.80/1.23 thf(func_def_135, type, regular: $i > $o).
% 1.80/1.23 thf(func_def_136, type, setminus: $i > $i > $i).
% 1.80/1.23 thf(func_def_147, type, symdiff: $i > $i > $i).
% 1.80/1.23 thf(func_def_153, type, iskpair: $i > $o).
% 1.80/1.23 thf(func_def_158, type, kpair: $i > $i > $i).
% 1.80/1.23 thf(func_def_160, type, cartprod: $i > $i > $i).
% 1.80/1.23 thf(func_def_175, type, sP0: $o > $i > $i > $i > $o).
% 1.80/1.23 thf(func_def_177, type, sP2: $i > $i > $o).
% 1.80/1.23 thf(func_def_178, type, sP3: $i > $o).
% 1.80/1.23 thf(func_def_179, type, sP4: $i > $i > $o).
% 1.80/1.23 thf(func_def_180, type, sP5: $i > $i > $o).
% 1.80/1.23 thf(func_def_181, type, sK6: $i > $o).
% 1.80/1.23 thf(func_def_184, type, sK9: $i > ($i > $o) > $i).
% 1.80/1.23 thf(func_def_185, type, sK10: $i > $o).
% 1.80/1.23 thf(func_def_195, type, sK20: ($i > $o) > $i).
% 1.80/1.23 thf(func_def_196, type, sK21: $i > $o).
% 1.80/1.23 thf(func_def_197, type, sK22: $i > $i).
% 1.80/1.23 thf(func_def_207, type, sK32: $i > $i).
% 1.80/1.23 thf(func_def_214, type, sK39: $i > $i > $i).
% 1.80/1.23 thf(func_def_228, type, sK53: $i > $i > $i).
% 1.80/1.23 thf(func_def_230, type, sK55: $i > $i > $i).
% 1.80/1.23 thf(func_def_231, type, sK56: $i > $i > $i).
% 1.80/1.23 thf(func_def_240, type, sK65: $i > $i).
% 1.80/1.23 thf(func_def_253, type, sK78: ($i > $o) > ($i > $o) > $i).
% 1.80/1.23 thf(func_def_254, type, sK79: ($i > $o) > ($i > $o) > $i).
% 1.80/1.23 thf(func_def_255, type, sK80: $i > $o).
% 1.80/1.23 thf(func_def_256, type, sK81: $i > $o).
% 1.80/1.23 thf(func_def_263, type, sK88: $i > ($i > $o) > $i).
% 1.80/1.23 thf(func_def_264, type, sK89: $i > $o).
% 1.80/1.23 thf(func_def_272, type, sK97: $i > $o).
% 1.80/1.23 thf(func_def_280, type, sK105: $o > $i > $i > $i).
% 1.80/1.23 thf(func_def_284, type, sK109: $i > $o).
% 1.80/1.23 thf(func_def_286, type, sK111: ($i > $o) > $i).
% 1.80/1.23 thf(func_def_287, type, sK112: ($i > $o) > $i).
% 1.80/1.23 thf(func_def_307, type, sK132: $i > $i > $i).
% 1.80/1.23 thf(func_def_317, type, sK142: $i > $o).
% 1.80/1.23 thf(func_def_319, type, sK144: ($i > $o) > $i > $i).
% 1.80/1.23 thf(func_def_337, type, sK162: $i > $o).
% 1.80/1.23 thf(func_def_339, type, sK164: ($i > $o) > $i > $i).
% 1.80/1.23 thf(func_def_340, type, sK165: $i > $i > $i).
% 1.80/1.23 thf(func_def_345, type, sK170: $i > $o).
% 1.80/1.23 thf(func_def_346, type, sK171: ($i > $o) > ($i > $o) > $i > $i > $i).
% 1.80/1.23 thf(func_def_347, type, sK172: ($i > $o) > ($i > $o) > $i > $i > $i).
% 1.80/1.23 thf(func_def_350, type, sK175: $i > $o).
% 1.80/1.23 thf(func_def_351, type, sK176: $i > $o).
% 1.80/1.23 thf(func_def_354, type, sK179: $i > $o).
% 1.80/1.23 thf(func_def_362, type, sK187: $i > $i).
% 1.80/1.23 thf(func_def_365, type, sK190: $i > $i).
% 1.80/1.23 thf(func_def_375, type, sK200: $i > $o).
% 1.80/1.23 thf(func_def_377, type, sK202: $i > ($i > $o) > $i).
% 1.80/1.23 thf(func_def_382, type, sK207: $i > $i > $i).
% 1.80/1.23 thf(func_def_386, type, sK211: $i > $o).
% 1.80/1.23 thf(func_def_387, type, sK212: ($i > $o) > $i).
% 1.80/1.23 thf(func_def_399, type, sK224: $i > $o).
% 1.80/1.23 thf(func_def_408, type, sK233: $i > $i > $i).
% 1.80/1.23 thf(func_def_412, type, sK237: $i > $o).
% 1.80/1.23 thf(func_def_414, type, sK239: ($i > $o) > $i > $i).
% 1.80/1.23 thf(func_def_417, type, sK242: $i > $i > $o).
% 1.80/1.23 thf(func_def_419, type, sK244: $i > $i).
% 1.80/1.23 thf(func_def_420, type, sK245: $i > $i).
% 1.80/1.23 thf(func_def_421, type, sK246: $i > ($i > $i > $o) > $i).
% 1.80/1.23 thf(func_def_422, type, sK247: $i > ($i > $i > $o) > $i).
% 1.80/1.23 thf(func_def_423, type, sK248: $i > $i > ($i > $i > $o) > $i).
% 1.80/1.23 thf(func_def_442, type, sK267: ($i > $o) > ($i > $o) > $i).
% 1.80/1.23 thf(func_def_443, type, sK268: ($i > $o) > ($i > $o) > $i).
% 1.80/1.23 thf(func_def_444, type, sK269: $i > $o).
% 1.80/1.23 thf(func_def_445, type, sK270: $i > $o).
% 1.80/1.23 thf(func_def_457, type, sK282: $i > $i).
% 1.80/1.23 thf(func_def_465, type, sK290: $i > $o).
% 1.80/1.23 thf(func_def_477, type, sK302: $i > $i > $i).
% 1.80/1.23 thf(func_def_478, type, sK303: $i > $i > $i).
% 1.80/1.23 thf(func_def_479, type, sK304: $i > $i > $i).
% 1.80/1.23 thf(func_def_480, type, sK305: $i > $i > $i).
% 1.80/1.23 thf(func_def_481, type, sK306: $i > $i > $i).
% 1.80/1.23 thf(func_def_482, type, sK307: $i > $i > $i > $i).
% 1.80/1.23 thf(func_def_483, type, sK308: $i > $i).
% 1.80/1.23 thf(func_def_484, type, sK309: $i > $i).
% 1.80/1.23 thf(func_def_485, type, sK310: $i > $i).
% 1.80/1.23 thf(func_def_486, type, sK311: $i > $i).
% 1.80/1.23 thf(func_def_487, type, sK312: $i > $i > $i > $i).
% 1.80/1.23 thf(func_def_488, type, sK313: $i > $i > $i > $i).
% 1.80/1.23 thf(func_def_489, type, sK314: $i > $i > $i).
% 1.80/1.23 thf(func_def_490, type, sK315: $i > $i > $i).
% 1.80/1.23 thf(func_def_491, type, sK316: $i > $i > $i).
% 1.80/1.23 thf(func_def_493, type, sK318: $i > $i).
% 1.80/1.23 thf(func_def_494, type, sK319: $i > $i).
% 1.80/1.23 thf(func_def_495, type, sK320: $i > $i).
% 1.80/1.23 thf(func_def_497, type, sK322: $i > $o).
% 1.80/1.23 thf(func_def_500, type, sK325: $i > $o).
% 1.80/1.23 thf(func_def_502, type, sK327: ($i > $o) > $i > $i).
% 1.80/1.23 thf(func_def_509, type, sK334: $i > $i).
% 1.80/1.23 thf(func_def_512, type, sK337: $i > $o).
% 1.80/1.23 thf(func_def_513, type, sK338: $i > ($i > $o) > $i).
% 1.80/1.23 thf(func_def_514, type, sK339: $i > $o).
% 1.80/1.23 thf(func_def_527, type, sK352: $i > $o).
% 1.80/1.23 thf(func_def_529, type, sK354: $i > $i).
% 1.80/1.23 thf(func_def_531, type, sK356: $i > $o).
% 1.80/1.23 thf(func_def_534, type, sK359: $i > $i > $i).
% 1.80/1.23 thf(func_def_537, type, sK362: $i > $o).
% 1.80/1.23 thf(func_def_540, type, sK365: $i > $o).
% 1.80/1.23 thf(func_def_542, type, sK367: ($i > $o) > $i).
% 1.80/1.23 thf(func_def_543, type, sK368: $i > $o).
% 1.80/1.23 thf(func_def_544, type, sK369: $i > $i).
% 1.80/1.23 thf(func_def_553, type, ph378: !>[X0: $tType]:(X0)).
% 1.80/1.23 thf(f2527,plain,(
% 1.80/1.23 $false),
% 1.80/1.23 inference(trivial_inequality_removal,[status(thm)],[f2522])).
% 1.80/1.23 thf(f2522,plain,(
% 1.80/1.23 ($true != $true)),
% 1.80/1.23 inference(superposition,[status(thm)],[f2064,f1495])).
% 1.80/1.23 thf(f1495,plain,(
% 1.80/1.23 ($true = (in @ sK155 @ sK156))),
% 1.80/1.23 inference(cnf_transformation,[status(thm)],[f872])).
% 1.80/1.23 thf(f872,plain,(
% 1.80/1.23 (dsetconstrI = $true) & (exuE3u = $true) & (binunionIR = $true) & (symdiffI2 = $true) & (setoftrueEq = $true) & (powersetAx = $true) & (omegaSAx = $true) & (symdiffIneg1 = $true) & (emptysetimpfalse = $true) & (binunionRsub = $true) & (subPowSU = $true) & (upairsetIL = $true) & (sepInPowerset = $true) & (binintersectSubset4 = $true) & (setadjoinIR = $true) & (notsubsetI = $true) & (setminusSubset2 = $true) & (setunion__Cong = $true) & (setadjoinE = $true) & (exuEu = $true) & (binunionE = $true) & (inCongP = $true) & (notinsingleton = $true) & (eqimpsubset1 = $true) & (disjointsetsI1 = $true) & (binunionLsub = $true) & (quantDeMorgan4 = $true) & (inPowerset = $true) & (singletonsswitch = $true) & (setminusI = $true) & (setadjoinOr = $true) & (quantDeMorgan1 = $true) & (setminusERneg = $true) & (nonemptyE1 = $true) & (setadjoinSub = $true) & (setminusSubset1 = $true) & (exuE2 = $true) & (setadjoinIL = $true) & (binintersectER = $true) & (binintersectSubset2 = $true) & (setadjoinSub2 = $true) & (powersetI = $true) & (exuI1 = $true) & (emptyI = $true) & (notequalI2 = $true) & (nonemptyI = $true) & (powersetE = $true) & (binintersectI = $true) & (descr__Cong = $true) & (eqinunit = $true) & (secondinupair = $true) & (subsetE2 = $true) & (notdexE = $true) & (emptyinunitempty = $true) & (wellorderingAx = $true) & (exu__Cong = $true) & (emptyinPowerset = $true) & (subsetI2 = $true) & (emptyset__Cong = $true) & (dsetconstrEL = $true) & (emptysetAx = $true) & (setextAx = $true) & (setminusEL = $true) & (setminusILneg = $true) & (omegaIndAx = $true) & (emptyInPowerset = $true) & (kpairiskpair = $true) & (setukpairIL = $true) & (singletonsubset = $true) & (quantDeMorgan2 = $true) & (binintersectSubset5 = $true) & (setadjoin__Cong = $true) & (exuE3e = $true) & (prop2setE = $true) & (setunionI = $true) & (notequalI1 = $true) & (exuI3 = $true) & (powersetsubset = $true) & (setukpairIR = $true) & (upairsetIR = $true) & (setbeta = $true) & (descrp = $true) & (emptyE1 = $true) & (subsetemptysetimpeq = $true) & (noeltsimpempty = $true) & (exuE1 = $true) & (nonemptyI1 = $true) & (binintersectEL = $true) & (subsetTrans = $true) & (emptysetE = $true) & (upairsetE = $true) & (dsetconstrER = $true) & (powersetE1 = $true) & (notinemptyset = $true) & (subsetI1 = $true) & (quantDeMorgan3 = $true) & (symdiffE = $true) & (setadjoinAx = $true) & (powerset__Cong = $true) & (nonemptyImpWitness = $true) & (bs114d = $true) & (binintersectSubset3 = $true) & (binunionEcases = $true) & (dsetconstr__Cong = $true) & (symdiffIneg2 = $true) & (binintersectLsub = $true) & (setminusLsub = $true) & (subsetE = $true) & (uniqinunit = $true) & (exuI2 = $true) & (binintersectRsub = $true) & (((in @ (setadjoin @ sK155 @ emptyset) @ (powerset @ sK156)) != $true) & ($true = (in @ sK155 @ sK156))) & (setminusELneg = $true) & (setminusER = $true) & (kpairp = $true) & (omega0Ax = $true) & (setext = $true) & (prop2set2propI = $true) & (emptysetsubset = $true) & (subset2powerset = $true) & (setminusIRneg = $true) & (upairset2IR = $true) & (binintersectSubset1 = $true) & (subsetRefl = $true) & (powersetI1 = $true) & (in__Cong = $true) & (sepSubset = $true) & (foundationAx = $true) & (symdiffI1 = $true) & (setunionAx = $true) & (setunionE = $true) & (vacuousDall = $true) & (prop2setI = $true) & (replAx = $true) & (omega__Cong = $true) & (notdallE = $true) & (setextsub = $true) & (eqimpsubset2 = $true) & (binunionIL = $true)),
% 1.80/1.23 inference(skolemisation,[status(esa),new_symbols(skolem,[sK155,sK156])],[f585,f871])).
% 1.80/1.23 thf(f871,plain,(
% 1.80/1.23 ? [X0 : $i,X1 : $i] : (($true != (in @ (setadjoin @ X0 @ emptyset) @ (powerset @ X1))) & ($true = (in @ X0 @ X1))) => (((in @ (setadjoin @ sK155 @ emptyset) @ (powerset @ sK156)) != $true) & ($true = (in @ sK155 @ sK156)))),
% 1.80/1.23 introduced(definition,[],[choice_axiom])).
% 1.80/1.23 thf(f585,plain,(
% 1.80/1.23 (dsetconstrI = $true) & (exuE3u = $true) & (binunionIR = $true) & (symdiffI2 = $true) & (setoftrueEq = $true) & (powersetAx = $true) & (omegaSAx = $true) & (symdiffIneg1 = $true) & (emptysetimpfalse = $true) & (binunionRsub = $true) & (subPowSU = $true) & (upairsetIL = $true) & (sepInPowerset = $true) & (binintersectSubset4 = $true) & (setadjoinIR = $true) & (notsubsetI = $true) & (setminusSubset2 = $true) & (setunion__Cong = $true) & (setadjoinE = $true) & (exuEu = $true) & (binunionE = $true) & (inCongP = $true) & (notinsingleton = $true) & (eqimpsubset1 = $true) & (disjointsetsI1 = $true) & (binunionLsub = $true) & (quantDeMorgan4 = $true) & (inPowerset = $true) & (singletonsswitch = $true) & (setminusI = $true) & (setadjoinOr = $true) & (quantDeMorgan1 = $true) & (setminusERneg = $true) & (nonemptyE1 = $true) & (setadjoinSub = $true) & (setminusSubset1 = $true) & (exuE2 = $true) & (setadjoinIL = $true) & (binintersectER = $true) & (binintersectSubset2 = $true) & (setadjoinSub2 = $true) & (powersetI = $true) & (exuI1 = $true) & (emptyI = $true) & (notequalI2 = $true) & (nonemptyI = $true) & (powersetE = $true) & (binintersectI = $true) & (descr__Cong = $true) & (eqinunit = $true) & (secondinupair = $true) & (subsetE2 = $true) & (notdexE = $true) & (emptyinunitempty = $true) & (wellorderingAx = $true) & (exu__Cong = $true) & (emptyinPowerset = $true) & (subsetI2 = $true) & (emptyset__Cong = $true) & (dsetconstrEL = $true) & (emptysetAx = $true) & (setextAx = $true) & (setminusEL = $true) & (setminusILneg = $true) & (omegaIndAx = $true) & (emptyInPowerset = $true) & (kpairiskpair = $true) & (setukpairIL = $true) & (singletonsubset = $true) & (quantDeMorgan2 = $true) & (binintersectSubset5 = $true) & (setadjoin__Cong = $true) & (exuE3e = $true) & (prop2setE = $true) & (setunionI = $true) & (notequalI1 = $true) & (exuI3 = $true) & (powersetsubset = $true) & (setukpairIR = $true) & (upairsetIR = $true) & (setbeta = $true) & (descrp = $true) & (emptyE1 = $true) & (subsetemptysetimpeq = $true) & (noeltsimpempty = $true) & (exuE1 = $true) & (nonemptyI1 = $true) & (binintersectEL = $true) & (subsetTrans = $true) & (emptysetE = $true) & (upairsetE = $true) & (dsetconstrER = $true) & (powersetE1 = $true) & (notinemptyset = $true) & (subsetI1 = $true) & (quantDeMorgan3 = $true) & (symdiffE = $true) & (setadjoinAx = $true) & (powerset__Cong = $true) & (nonemptyImpWitness = $true) & (bs114d = $true) & (binintersectSubset3 = $true) & (binunionEcases = $true) & (dsetconstr__Cong = $true) & (symdiffIneg2 = $true) & (binintersectLsub = $true) & (setminusLsub = $true) & (subsetE = $true) & (uniqinunit = $true) & (exuI2 = $true) & (binintersectRsub = $true) & ? [X0 : $i,X1 : $i] : (($true != (in @ (setadjoin @ X0 @ emptyset) @ (powerset @ X1))) & ($true = (in @ X0 @ X1))) & (setminusELneg = $true) & (setminusER = $true) & (kpairp = $true) & (omega0Ax = $true) & (setext = $true) & (prop2set2propI = $true) & (emptysetsubset = $true) & (subset2powerset = $true) & (setminusIRneg = $true) & (upairset2IR = $true) & (binintersectSubset1 = $true) & (subsetRefl = $true) & (powersetI1 = $true) & (in__Cong = $true) & (sepSubset = $true) & (foundationAx = $true) & (symdiffI1 = $true) & (setunionAx = $true) & (setunionE = $true) & (vacuousDall = $true) & (prop2setI = $true) & (replAx = $true) & (omega__Cong = $true) & (notdallE = $true) & (setextsub = $true) & (eqimpsubset2 = $true) & (binunionIL = $true)),
% 1.80/1.23 inference(flattening,[status(thm)],[f584])).
% 1.80/1.23 thf(f584,plain,(
% 1.80/1.23 (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i,X1 : $i] : (($true != (in @ (setadjoin @ X0 @ emptyset) @ (powerset @ X1))) & ($true = (in @ X0 @ X1))) & (singletonsubset = $true)) & (kpairp = $true)) & (kpairiskpair = $true)) & (setukpairIR = $true)) & (setukpairIL = $true)) & (secondinupair = $true)) & (symdiffIneg2 = $true)) & (symdiffIneg1 = $true)) & (symdiffI2 = $true)) & (symdiffI1 = $true)) & (symdiffE = $true)) & (setminusSubset1 = $true)) & (setminusLsub = $true)) & (setminusIRneg = $true)) & (setminusILneg = $true)) & (setminusELneg = $true)) & (setminusERneg = $true)) & (setminusSubset2 = $true)) & (setminusER = $true)) & (setminusEL = $true)) & (setminusI = $true)) & (bs114d = $true)) & (binintersectSubset1 = $true)) & (binintersectSubset4 = $true)) & (binintersectRsub = $true)) & (disjointsetsI1 = $true)) & (binintersectER = $true)) & (binintersectSubset3 = $true)) & (binintersectSubset2 = $true)) & (binintersectLsub = $true)) & (binintersectEL = $true)) & (binintersectSubset5 = $true)) & (binintersectI = $true)) & (binunionRsub = $true)) & (binunionLsub = $true)) & (binunionE = $true)) & (binunionEcases = $true)) & (binunionIR = $true)) & (upairset2IR = $true)) & (binunionIL = $true)) & (sepSubset = $true)) & (sepInPowerset = $true)) & (powersetsubset = $true)) & (inPowerset = $true)) & (powersetE1 = $true)) & (powersetI1 = $true)) & (subsetemptysetimpeq = $true)) & (setextsub = $true)) & (subset2powerset = $true)) & (setadjoinSub2 = $true)) & (setadjoinSub = $true)) & (subsetTrans = $true)) & (subsetRefl = $true)) & (notequalI2 = $true)) & (notequalI1 = $true)) & (notsubsetI = $true)) & (subsetE2 = $true)) & (subsetE = $true)) & (emptysetsubset = $true)) & (subsetI2 = $true)) & (eqimpsubset1 = $true)) & (eqimpsubset2 = $true)) & (subsetI1 = $true)) & (dsetconstr__Cong = $true)) & (descr__Cong = $true)) & (exuEu = $true)) & (omega__Cong = $true)) & (setunion__Cong = $true)) & (powerset__Cong = $true)) & (setadjoin__Cong = $true)) & (emptyset__Cong = $true)) & (exu__Cong = $true)) & (exuE3u = $true)) & (in__Cong = $true)) & (inCongP = $true)) & (exuI2 = $true)) & (exuI3 = $true)) & (exuI1 = $true)) & (notdallE = $true)) & (notdexE = $true)) & (prop2set2propI = $true)) & (prop2setI = $true)) & (quantDeMorgan4 = $true)) & (quantDeMorgan3 = $true)) & (quantDeMorgan2 = $true)) & (quantDeMorgan1 = $true)) & (vacuousDall = $true)) & (emptyE1 = $true)) & (upairsetIR = $true)) & (upairsetIL = $true)) & (upairsetE = $true)) & (singletonsswitch = $true)) & (eqinunit = $true)) & (notinsingleton = $true)) & (uniqinunit = $true)) & (nonemptyImpWitness = $true)) & (exuE2 = $true)) & (subPowSU = $true)) & (setunionE = $true)) & (setunionI = $true)) & (powersetE = $true)) & (emptyInPowerset = $true)) & (emptyinPowerset = $true)) & (powersetI = $true)) & (setoftrueEq = $true)) & (setadjoinOr = $true)) & (setadjoinE = $true)) & (setadjoinIR = $true)) & (emptyinunitempty = $true)) & (setadjoinIL = $true)) & (nonemptyI1 = $true)) & (nonemptyI = $true)) & (nonemptyE1 = $true)) & (setbeta = $true)) & (noeltsimpempty = $true)) & (emptyI = $true)) & (setext = $true)) & (exuE3e = $true)) & (notinemptyset = $true)) & (emptysetimpfalse = $true)) & (emptysetE = $true)) & (prop2setE = $true)) & (exuE1 = $true)) & (dsetconstrER = $true)) & (dsetconstrEL = $true)) & (dsetconstrI = $true)) & (descrp = $true)) & (wellorderingAx = $true)) & (foundationAx = $true)) & (replAx = $true)) & (omegaIndAx = $true)) & (omegaSAx = $true)) & (omega0Ax = $true)) & (setunionAx = $true)) & (powersetAx = $true)) & (setadjoinAx = $true)) & (emptysetAx = $true)) & (setextAx = $true)),
% 1.80/1.23 inference(ennf_transformation,[status(thm)],[f294])).
% 1.80/1.23 thf(f294,plain,(
% 1.80/1.23 ~((setextAx = $true) => ((emptysetAx = $true) => ((setadjoinAx = $true) => ((powersetAx = $true) => ((setunionAx = $true) => ((omega0Ax = $true) => ((omegaSAx = $true) => ((omegaIndAx = $true) => ((replAx = $true) => ((foundationAx = $true) => ((wellorderingAx = $true) => ((descrp = $true) => ((dsetconstrI = $true) => ((dsetconstrEL = $true) => ((dsetconstrER = $true) => ((exuE1 = $true) => ((prop2setE = $true) => ((emptysetE = $true) => ((emptysetimpfalse = $true) => ((notinemptyset = $true) => ((exuE3e = $true) => ((setext = $true) => ((emptyI = $true) => ((noeltsimpempty = $true) => ((setbeta = $true) => ((nonemptyE1 = $true) => ((nonemptyI = $true) => ((nonemptyI1 = $true) => ((setadjoinIL = $true) => ((emptyinunitempty = $true) => ((setadjoinIR = $true) => ((setadjoinE = $true) => ((setadjoinOr = $true) => ((setoftrueEq = $true) => ((powersetI = $true) => ((emptyinPowerset = $true) => ((emptyInPowerset = $true) => ((powersetE = $true) => ((setunionI = $true) => ((setunionE = $true) => ((subPowSU = $true) => ((exuE2 = $true) => ((nonemptyImpWitness = $true) => ((uniqinunit = $true) => ((notinsingleton = $true) => ((eqinunit = $true) => ((singletonsswitch = $true) => ((upairsetE = $true) => ((upairsetIL = $true) => ((upairsetIR = $true) => ((emptyE1 = $true) => ((vacuousDall = $true) => ((quantDeMorgan1 = $true) => ((quantDeMorgan2 = $true) => ((quantDeMorgan3 = $true) => ((quantDeMorgan4 = $true) => ((prop2setI = $true) => ((prop2set2propI = $true) => ((notdexE = $true) => ((notdallE = $true) => ((exuI1 = $true) => ((exuI3 = $true) => ((exuI2 = $true) => ((inCongP = $true) => ((in__Cong = $true) => ((exuE3u = $true) => ((exu__Cong = $true) => ((emptyset__Cong = $true) => ((setadjoin__Cong = $true) => ((powerset__Cong = $true) => ((setunion__Cong = $true) => ((omega__Cong = $true) => ((exuEu = $true) => ((descr__Cong = $true) => ((dsetconstr__Cong = $true) => ((subsetI1 = $true) => ((eqimpsubset2 = $true) => ((eqimpsubset1 = $true) => ((subsetI2 = $true) => ((emptysetsubset = $true) => ((subsetE = $true) => ((subsetE2 = $true) => ((notsubsetI = $true) => ((notequalI1 = $true) => ((notequalI2 = $true) => ((subsetRefl = $true) => ((subsetTrans = $true) => ((setadjoinSub = $true) => ((setadjoinSub2 = $true) => ((subset2powerset = $true) => ((setextsub = $true) => ((subsetemptysetimpeq = $true) => ((powersetI1 = $true) => ((powersetE1 = $true) => ((inPowerset = $true) => ((powersetsubset = $true) => ((sepInPowerset = $true) => ((sepSubset = $true) => ((binunionIL = $true) => ((upairset2IR = $true) => ((binunionIR = $true) => ((binunionEcases = $true) => ((binunionE = $true) => ((binunionLsub = $true) => ((binunionRsub = $true) => ((binintersectI = $true) => ((binintersectSubset5 = $true) => ((binintersectEL = $true) => ((binintersectLsub = $true) => ((binintersectSubset2 = $true) => ((binintersectSubset3 = $true) => ((binintersectER = $true) => ((disjointsetsI1 = $true) => ((binintersectRsub = $true) => ((binintersectSubset4 = $true) => ((binintersectSubset1 = $true) => ((bs114d = $true) => ((setminusI = $true) => ((setminusEL = $true) => ((setminusER = $true) => ((setminusSubset2 = $true) => ((setminusERneg = $true) => ((setminusELneg = $true) => ((setminusILneg = $true) => ((setminusIRneg = $true) => ((setminusLsub = $true) => ((setminusSubset1 = $true) => ((symdiffE = $true) => ((symdiffI1 = $true) => ((symdiffI2 = $true) => ((symdiffIneg1 = $true) => ((symdiffIneg2 = $true) => ((secondinupair = $true) => ((setukpairIL = $true) => ((setukpairIR = $true) => ((kpairiskpair = $true) => ((kpairp = $true) => ((singletonsubset = $true) => ! [X0 : $i,X1 : $i] : (($true = (in @ X0 @ X1)) => ($true = (in @ (setadjoin @ X0 @ emptyset) @ (powerset @ X1))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 1.80/1.23 inference(fool_elimination,[status(thm)],[f293])).
% 1.80/1.23 thf(f293,plain,(
% 1.80/1.23 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => ! [X0 : $i,X1 : $i] : ((in @ X0 @ X1) => (in @ (setadjoin @ X0 @ emptyset) @ (powerset @ X1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 1.80/1.23 inference(rectify,[status(thm)],[f145])).
% 1.80/1.23 thf(f145,negated_conjecture,(
% 1.80/1.23 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => ! [X1 : $i,X3 : $i] : ((in @ X1 @ X3) => (in @ (setadjoin @ X1 @ emptyset) @ (powerset @ X3)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 1.80/1.23 inference(negated_conjecture,[status(cth)],[f144])).
% 1.80/1.23 thf(f144,conjecture,(
% 1.80/1.23 setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => ! [X1 : $i,X3 : $i] : ((in @ X1 @ X3) => (in @ (setadjoin @ X1 @ emptyset) @ (powerset @ X3))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 1.80/1.23 file('/export/starexec/sandbox/tmp/tmp.P13AcNWfIy/DTF2THF_4603.p',singletoninpowerset)).
% 1.80/1.23 thf(f2064,plain,(
% 1.80/1.23 ($true != (in @ sK155 @ sK156))),
% 1.80/1.23 inference(trivial_inequality_removal,[status(thm)],[f2063])).
% 1.80/1.23 thf(f2063,plain,(
% 1.80/1.23 ($true != $true) | ($true != (in @ sK155 @ sK156))),
% 1.80/1.23 inference(forward_demodulation,[status(thm)],[f2056,f1539])).
% 1.80/1.23 thf(f1539,plain,(
% 1.80/1.23 (singletonsubset = $true)),
% 1.80/1.23 inference(cnf_transformation,[status(thm)],[f872])).
% 1.80/1.23 thf(f2056,plain,(
% 1.80/1.23 (singletonsubset != $true) | ($true != (in @ sK155 @ sK156))),
% 1.80/1.23 inference(trivial_inequality_removal,[status(thm)],[f2049])).
% 1.80/1.23 thf(f2049,plain,(
% 1.80/1.23 ($true != $true) | ($true != (in @ sK155 @ sK156)) | (singletonsubset != $true)),
% 1.80/1.23 inference(superposition,[status(thm)],[f2042,f1458])).
% 1.80/1.23 thf(f1458,plain,(
% 1.80/1.23 ( ! [X2 : $i,X3 : $i] : (($true = (subset @ (setadjoin @ X3 @ emptyset) @ X2)) | (singletonsubset != $true) | ($true != (in @ X3 @ X2))) )),
% 1.80/1.23 inference(cnf_transformation,[status(thm)],[f862])).
% 1.80/1.23 thf(f862,plain,(
% 1.80/1.23 ((singletonsubset = $true) | (($true != (subset @ (setadjoin @ sK149 @ emptyset) @ sK148)) & ($true = (in @ sK149 @ sK148)))) & (! [X2 : $i,X3 : $i] : (($true = (subset @ (setadjoin @ X3 @ emptyset) @ X2)) | ($true != (in @ X3 @ X2))) | (singletonsubset != $true))),
% 1.80/1.23 inference(skolemisation,[status(esa),new_symbols(skolem,[sK148,sK149])],[f860,f861])).
% 1.80/1.23 thf(f861,plain,(
% 1.80/1.23 ? [X0 : $i,X1 : $i] : (($true != (subset @ (setadjoin @ X1 @ emptyset) @ X0)) & ($true = (in @ X1 @ X0))) => (($true != (subset @ (setadjoin @ sK149 @ emptyset) @ sK148)) & ($true = (in @ sK149 @ sK148)))),
% 1.80/1.23 introduced(definition,[],[choice_axiom])).
% 1.80/1.23 thf(f860,plain,(
% 1.80/1.23 ((singletonsubset = $true) | ? [X0 : $i,X1 : $i] : (($true != (subset @ (setadjoin @ X1 @ emptyset) @ X0)) & ($true = (in @ X1 @ X0)))) & (! [X2 : $i,X3 : $i] : (($true = (subset @ (setadjoin @ X3 @ emptyset) @ X2)) | ($true != (in @ X3 @ X2))) | (singletonsubset != $true))),
% 1.80/1.23 inference(rectify,[status(thm)],[f859])).
% 1.80/1.23 thf(f859,plain,(
% 1.80/1.23 ((singletonsubset = $true) | ? [X0 : $i,X1 : $i] : (($true != (subset @ (setadjoin @ X1 @ emptyset) @ X0)) & ($true = (in @ X1 @ X0)))) & (! [X0 : $i,X1 : $i] : (($true = (subset @ (setadjoin @ X1 @ emptyset) @ X0)) | ($true != (in @ X1 @ X0))) | (singletonsubset != $true))),
% 1.80/1.23 inference(nnf_transformation,[status(thm)],[f523])).
% 1.80/1.23 thf(f523,plain,(
% 1.80/1.23 (singletonsubset = $true) <=> ! [X0 : $i,X1 : $i] : (($true = (subset @ (setadjoin @ X1 @ emptyset) @ X0)) | ($true != (in @ X1 @ X0)))),
% 1.80/1.23 inference(ennf_transformation,[status(thm)],[f352])).
% 1.80/1.23 thf(f352,plain,(
% 1.80/1.23 ! [X0 : $i,X1 : $i] : (($true = (in @ X1 @ X0)) => ($true = (subset @ (setadjoin @ X1 @ emptyset) @ X0))) <=> (singletonsubset = $true)),
% 1.80/1.23 inference(fool_elimination,[status(thm)],[f351])).
% 1.80/1.23 thf(f351,plain,(
% 1.80/1.23 (singletonsubset = ! [X0 : $i,X1 : $i] : ((in @ X1 @ X0) => (subset @ (setadjoin @ X1 @ emptyset) @ X0)))),
% 1.80/1.23 inference(rectify,[status(thm)],[f143])).
% 1.80/1.23 thf(f143,axiom,(
% 1.80/1.23 (singletonsubset = ! [X3 : $i,X1 : $i] : ((in @ X1 @ X3) => (subset @ (setadjoin @ X1 @ emptyset) @ X3)))),
% 1.80/1.23 file('/export/starexec/sandbox/tmp/tmp.P13AcNWfIy/DTF2THF_4603.p',singletonsubset)).
% 1.80/1.23 thf(f2042,plain,(
% 1.80/1.23 ($true != (subset @ (setadjoin @ sK155 @ emptyset) @ sK156))),
% 1.80/1.23 inference(trivial_inequality_removal,[status(thm)],[f2041])).
% 1.80/1.23 thf(f2041,plain,(
% 1.80/1.23 ($true != $true) | ($true != (subset @ (setadjoin @ sK155 @ emptyset) @ sK156))),
% 1.80/1.23 inference(forward_demodulation,[status(thm)],[f2036,f1487])).
% 1.80/1.23 thf(f1487,plain,(
% 1.80/1.23 (subset2powerset = $true)),
% 1.80/1.23 inference(cnf_transformation,[status(thm)],[f872])).
% 1.80/1.23 thf(f2036,plain,(
% 1.80/1.23 (subset2powerset != $true) | ($true != (subset @ (setadjoin @ sK155 @ emptyset) @ sK156))),
% 1.80/1.23 inference(trivial_inequality_removal,[status(thm)],[f2028])).
% 1.80/1.23 thf(f2028,plain,(
% 1.80/1.23 (subset2powerset != $true) | ($true != (subset @ (setadjoin @ sK155 @ emptyset) @ sK156)) | ($true != $true)),
% 1.80/1.23 inference(superposition,[status(thm)],[f1496,f1420])).
% 1.80/1.23 thf(f1420,plain,(
% 1.80/1.23 ( ! [X0 : $i,X1 : $i] : (($true = (in @ X1 @ (powerset @ X0))) | ($true != (subset @ X1 @ X0)) | (subset2powerset != $true)) )),
% 1.80/1.23 inference(cnf_transformation,[status(thm)],[f814])).
% 1.80/1.23 thf(f814,plain,(
% 1.80/1.23 (! [X0 : $i,X1 : $i] : (($true = (in @ X1 @ (powerset @ X0))) | ($true != (subset @ X1 @ X0))) | (subset2powerset != $true)) & ((subset2powerset = $true) | (($true != (in @ sK119 @ (powerset @ sK118))) & ($true = (subset @ sK119 @ sK118))))),
% 1.80/1.23 inference(skolemisation,[status(esa),new_symbols(skolem,[sK118,sK119])],[f812,f813])).
% 1.80/1.23 thf(f813,plain,(
% 1.80/1.23 ? [X2 : $i,X3 : $i] : (($true != (in @ X3 @ (powerset @ X2))) & ($true = (subset @ X3 @ X2))) => (($true != (in @ sK119 @ (powerset @ sK118))) & ($true = (subset @ sK119 @ sK118)))),
% 1.80/1.23 introduced(definition,[],[choice_axiom])).
% 1.80/1.23 thf(f812,plain,(
% 1.80/1.23 (! [X0 : $i,X1 : $i] : (($true = (in @ X1 @ (powerset @ X0))) | ($true != (subset @ X1 @ X0))) | (subset2powerset != $true)) & ((subset2powerset = $true) | ? [X2 : $i,X3 : $i] : (($true != (in @ X3 @ (powerset @ X2))) & ($true = (subset @ X3 @ X2))))),
% 1.80/1.23 inference(rectify,[status(thm)],[f811])).
% 1.80/1.23 thf(f811,plain,(
% 1.80/1.23 (! [X0 : $i,X1 : $i] : (($true = (in @ X1 @ (powerset @ X0))) | ($true != (subset @ X1 @ X0))) | (subset2powerset != $true)) & ((subset2powerset = $true) | ? [X0 : $i,X1 : $i] : (($true != (in @ X1 @ (powerset @ X0))) & ($true = (subset @ X1 @ X0))))),
% 1.80/1.23 inference(nnf_transformation,[status(thm)],[f536])).
% 1.80/1.23 thf(f536,plain,(
% 1.80/1.23 ! [X0 : $i,X1 : $i] : (($true = (in @ X1 @ (powerset @ X0))) | ($true != (subset @ X1 @ X0))) <=> (subset2powerset = $true)),
% 1.80/1.23 inference(ennf_transformation,[status(thm)],[f219])).
% 1.80/1.23 thf(f219,plain,(
% 1.80/1.23 ! [X0 : $i,X1 : $i] : (($true = (subset @ X1 @ X0)) => ($true = (in @ X1 @ (powerset @ X0)))) <=> (subset2powerset = $true)),
% 1.80/1.23 inference(fool_elimination,[status(thm)],[f218])).
% 1.80/1.23 thf(f218,plain,(
% 1.80/1.23 (! [X0 : $i,X1 : $i] : ((subset @ X1 @ X0) => (in @ X1 @ (powerset @ X0))) = subset2powerset)),
% 1.80/1.23 inference(rectify,[status(thm)],[f92])).
% 1.80/1.23 thf(f92,axiom,(
% 1.80/1.23 (! [X4 : $i,X3 : $i] : ((subset @ X3 @ X4) => (in @ X3 @ (powerset @ X4))) = subset2powerset)),
% 1.80/1.23 file('/export/starexec/sandbox/tmp/tmp.P13AcNWfIy/DTF2THF_4603.p',subset2powerset)).
% 1.80/1.23 thf(f1496,plain,(
% 1.80/1.23 ((in @ (setadjoin @ sK155 @ emptyset) @ (powerset @ sK156)) != $true)),
% 1.80/1.23 inference(cnf_transformation,[status(thm)],[f872])).
% 1.80/1.23 % SZS output end Proof for DTF2THF_4603
% 1.80/1.23 % (4818)------------------------------
% 1.80/1.23 % (4818)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.80/1.23 % (4818)Termination reason: Refutation
% 1.80/1.23
% 1.80/1.23 % (4818)Memory used [KB]: 7803
% 1.80/1.23 % (4818)Time elapsed: 0.092 s
% 1.80/1.23 % (4818)Instructions burned: 179 (million)
% 1.80/1.23 % (4818)------------------------------
% 1.80/1.23 % (4818)------------------------------
% 1.80/1.23 % (4795)Success in time 0.146 s
%------------------------------------------------------------------------------