%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU594^1 : TPTP v9.2.1. Released v3.7.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n008.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 09:35:29 AM UTC 2026
% Result : Theorem 2.27s 1.51s
% Output : Refutation 2.27s
% Verified :
% SZS Type : Refutation
% Derivation depth : 11
% Number of leaves : 4
% Syntax : Number of formulae : 23 ( 8 unt; 0 typ; 0 def)
% Number of atoms : 1783 ( 482 equ; 0 cnn)
% Maximal formula atoms : 112 ( 77 avg)
% Number of connectives : 910 ( 19 ~; 8 |; 338 &; 98 @)
% ( 1 <=>; 446 =>; 0 <=; 0 <~>)
% Maximal formula depth : 117 ( 37 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 228 ( 225 usr; 116 con; 0-4 aty)
% Number of variables : 38 ( 0 ^; 26 !; 12 ?; 38 :)
% 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_140,type,
sP0: $i > $i > $o ).
thf(func_def_141,type,
sP1: $i > $o ).
thf(func_def_142,type,
sP2: $i > $i > $o ).
thf(func_def_143,type,
sP3: $i > $i > $o ).
thf(func_def_148,type,
sK8: $i > $o ).
thf(func_def_150,type,
sK10: ( $i > $o ) > $i > $i ).
thf(func_def_157,type,
sK17: $i > $o ).
thf(func_def_159,type,
sK19: $i > $o ).
thf(func_def_165,type,
sK25: $i > $o ).
thf(func_def_166,type,
sK26: $i > $i > $i ).
thf(func_def_175,type,
sK35: $i > $o ).
thf(func_def_176,type,
sK36: $i > $i ).
thf(func_def_177,type,
sK37: ( $i > $o ) > $i ).
thf(func_def_179,type,
sK39: $i > $i ).
thf(func_def_181,type,
sK41: $i > $o ).
thf(func_def_189,type,
sK49: $i > $i > $i ).
thf(func_def_190,type,
sK50: $i > $i > $i ).
thf(func_def_191,type,
sK51: $i > $i > $i ).
thf(func_def_192,type,
sK52: $i > $i > $i ).
thf(func_def_193,type,
sK53: $i > $i > $i > $i ).
thf(func_def_194,type,
sK54: $i > $i ).
thf(func_def_195,type,
sK55: $i > $i ).
thf(func_def_196,type,
sK56: $i > $i ).
thf(func_def_197,type,
sK57: $i > $i ).
thf(func_def_198,type,
sK58: $i > $i > $i ).
thf(func_def_199,type,
sK59: $i > $i > $i > $i ).
thf(func_def_200,type,
sK60: $i > $i > $i > $i ).
thf(func_def_201,type,
sK61: $i > $i > $i ).
thf(func_def_202,type,
sK62: $i > $i > $i ).
thf(func_def_203,type,
sK63: $i > $i ).
thf(func_def_204,type,
sK64: $i > $i > $i ).
thf(func_def_206,type,
sK66: $i > $i ).
thf(func_def_207,type,
sK67: $i > $i ).
thf(func_def_208,type,
sK68: $i > $o ).
thf(func_def_210,type,
sK70: $i > ( $i > $o ) > $i ).
thf(func_def_216,type,
sK76: $i > $i > $i ).
thf(func_def_217,type,
sK77: $i > $i > $o ).
thf(func_def_219,type,
sK79: $i > $i ).
thf(func_def_220,type,
sK80: $i > $i ).
thf(func_def_221,type,
sK81: $i > ( $i > $i > $o ) > $i ).
thf(func_def_222,type,
sK82: $i > ( $i > $i > $o ) > $i ).
thf(func_def_223,type,
sK83: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_231,type,
sK91: $i > $o ).
thf(func_def_233,type,
sK93: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_234,type,
sK94: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_235,type,
sK95: $i > $o ).
thf(func_def_236,type,
sK96: $i > $o ).
thf(func_def_238,type,
sK98: $i > $i ).
thf(func_def_239,type,
sK99: $i > $i ).
thf(func_def_246,type,
sK106: $i > $i > $i ).
thf(func_def_247,type,
sK107: $i > $i ).
thf(func_def_250,type,
sK110: $i > $i ).
thf(func_def_251,type,
sK111: ( $i > $o ) > $i ).
thf(func_def_252,type,
sK112: ( $i > $o ) > $i ).
thf(func_def_253,type,
sK113: $i > $o ).
thf(func_def_258,type,
sK118: $i > $o ).
thf(func_def_264,type,
sK124: $i > $o ).
thf(func_def_271,type,
sK131: $i > ( $i > $o ) > $i ).
thf(func_def_272,type,
sK132: $i > $o ).
thf(func_def_278,type,
sK138: $i > $i > $i ).
thf(func_def_290,type,
sK150: $i > $i > $i ).
thf(func_def_300,type,
sK160: $o > $i > $i > $i ).
thf(func_def_302,type,
sK162: $i > $o ).
thf(func_def_311,type,
sK171: $i > $o ).
thf(func_def_313,type,
sK173: ( $i > $o ) > $i > $i ).
thf(func_def_318,type,
sK178: ( $i > $o ) > $i > $i ).
thf(func_def_320,type,
sK180: $i > $o ).
thf(func_def_327,type,
sK187: $i > $o ).
thf(func_def_331,type,
sK191: $i > $i ).
thf(func_def_354,type,
sK214: $i > $i > $i ).
thf(func_def_355,type,
sK215: $i > $i > $i ).
thf(func_def_364,type,
sK224: $i > $o ).
thf(func_def_367,type,
sK227: $i > ( $i > $o ) > $i ).
thf(func_def_370,type,
sK230: $i > $o ).
thf(func_def_372,type,
sK232: $i > ( $i > $o ) > $i ).
thf(func_def_377,type,
sK237: $i > $o ).
thf(func_def_384,type,
sK244: ( $i > $o ) > $i ).
thf(func_def_385,type,
sK245: $i > $o ).
thf(func_def_386,type,
sK246: $i > $i ).
thf(func_def_387,type,
sK247: $i > $i ).
thf(func_def_394,type,
sK254: $i > $i > $i ).
thf(func_def_404,type,
sK264: $i > $o ).
thf(func_def_408,type,
sK268: $i > $o ).
thf(func_def_409,type,
sK269: $i > $o ).
thf(func_def_410,type,
sK270: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_411,type,
sK271: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_417,type,
sK277: $i > $o ).
thf(func_def_418,type,
sK278: ( $i > $o ) > $i ).
thf(func_def_419,type,
sK279: $i > $o ).
thf(func_def_422,type,
sK282: $i > ( $i > $o ) > $i ).
thf(func_def_423,type,
sK283: $i > $o ).
thf(func_def_428,type,
sK288: $i > $o ).
thf(func_def_429,type,
sK289: $i > $o ).
thf(func_def_430,type,
sK290: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_431,type,
sK291: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_444,type,
ph304:
!>[X0: $tType] : X0 ).
thf(f2460,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f2457,f1260]) ).
thf(f1260,plain,
( $true
!= ( subset @ sK151 @ sK152 ) ),
inference(cnf_transformation,[status(thm)],[f743]) ).
thf(f743,plain,
( ( subsetemptysetimpeq = $true )
& ( foundationAx = $true )
& ( notinemptyset = $true )
& ( powersetAx = $true )
& ( noeltsimpempty = $true )
& ( singletonsswitch = $true )
& ( nonemptyImpWitness = $true )
& ( emptyset__Cong = $true )
& ( prop2set2propI = $true )
& ( emptyinPowerset = $true )
& ( upairsetIL = $true )
& ( exuE3e = $true )
& ( sepSubset = $true )
& ( notequalI1 = $true )
& ( setunionAx = $true )
& ( powersetI1 = $true )
& ( setext = $true )
& ( subsetE = $true )
& ( exu__Cong = $true )
& ( binunionEcases = $true )
& ( exuE1 = $true )
& ( upairsetIR = $true )
& ( wellorderingAx = $true )
& ( binunionRsub = $true )
& ( inCongP = $true )
& ( quantDeMorgan3 = $true )
& ( quantDeMorgan4 = $true )
& ( binintersectSubset2 = $true )
& ( emptyInPowerset = $true )
& ( emptysetAx = $true )
& ( quantDeMorgan1 = $true )
& ( setadjoin__Cong = $true )
& ( omega0Ax = $true )
& ( binintersectEL = $true )
& ( dsetconstr__Cong = $true )
& ( setunionI = $true )
& ( omegaIndAx = $true )
& ( dsetconstrEL = $true )
& ( descrp = $true )
& ( notdallE = $true )
& ( exuI2 = $true )
& ( setextAx = $true )
& ( prop2setE = $true )
& ( eqimpsubset1 = $true )
& ( nonemptyI = $true )
& ( binunionLsub = $true )
& ( setadjoinAx = $true )
& ( setadjoinSub2 = $true )
& ( powersetE1 = $true )
& ( binunionE = $true )
& ( setadjoinIR = $true )
& ( emptysetimpfalse = $true )
& ( notsubsetI = $true )
& ( subsetRefl = $true )
& ( binunionIL = $true )
& ( nonemptyI1 = $true )
& ( subsetI2 = $true )
& ( upairset2IR = $true )
& ( powersetI = $true )
& ( subset2powerset = $true )
& ( eqinunit = $true )
& ( emptysetE = $true )
& ( setunionE = $true )
& ( upairsetE = $true )
& ( powerset__Cong = $true )
& ( setadjoinSub = $true )
& ( emptyE1 = $true )
& ( sepInPowerset = $true )
& ( setbeta = $true )
& ( binintersectLsub = $true )
& ( subsetE2 = $true )
& ( exuI3 = $true )
& ( subsetI1 = $true )
& ( nonemptyE1 = $true )
& ( exuEu = $true )
& ( sK151
= ( binintersect @ sK152 @ sK151 ) )
& ( $true
!= ( subset @ sK151 @ sK152 ) )
& ( inPowerset = $true )
& ( eqimpsubset2 = $true )
& ( subsetTrans = $true )
& ( dsetconstrI = $true )
& ( emptysetsubset = $true )
& ( binunionIR = $true )
& ( omega__Cong = $true )
& ( exuI1 = $true )
& ( powersetsubset = $true )
& ( in__Cong = $true )
& ( powersetE = $true )
& ( exuE3u = $true )
& ( setunion__Cong = $true )
& ( prop2setI = $true )
& ( vacuousDall = $true )
& ( subPowSU = $true )
& ( binintersectI = $true )
& ( exuE2 = $true )
& ( replAx = $true )
& ( notdexE = $true )
& ( emptyI = $true )
& ( uniqinunit = $true )
& ( quantDeMorgan2 = $true )
& ( setadjoinE = $true )
& ( binintersectSubset5 = $true )
& ( setextsub = $true )
& ( setadjoinOr = $true )
& ( dsetconstrER = $true )
& ( omegaSAx = $true )
& ( emptyinunitempty = $true )
& ( notinsingleton = $true )
& ( notequalI2 = $true )
& ( setadjoinIL = $true )
& ( descr__Cong = $true )
& ( setoftrueEq = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK151,sK152])],[f412,f742]) ).
thf(f742,plain,
( ? [X0: $i,X1: $i] :
( ( ( binintersect @ X1 @ X0 )
= X0 )
& ( $true
!= ( subset @ X0 @ X1 ) ) )
=> ( ( sK151
= ( binintersect @ sK152 @ sK151 ) )
& ( $true
!= ( subset @ sK151 @ sK152 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f412,plain,
( ( subsetemptysetimpeq = $true )
& ( foundationAx = $true )
& ( notinemptyset = $true )
& ( powersetAx = $true )
& ( noeltsimpempty = $true )
& ( singletonsswitch = $true )
& ( nonemptyImpWitness = $true )
& ( emptyset__Cong = $true )
& ( prop2set2propI = $true )
& ( emptyinPowerset = $true )
& ( upairsetIL = $true )
& ( exuE3e = $true )
& ( sepSubset = $true )
& ( notequalI1 = $true )
& ( setunionAx = $true )
& ( powersetI1 = $true )
& ( setext = $true )
& ( subsetE = $true )
& ( exu__Cong = $true )
& ( binunionEcases = $true )
& ( exuE1 = $true )
& ( upairsetIR = $true )
& ( wellorderingAx = $true )
& ( binunionRsub = $true )
& ( inCongP = $true )
& ( quantDeMorgan3 = $true )
& ( quantDeMorgan4 = $true )
& ( binintersectSubset2 = $true )
& ( emptyInPowerset = $true )
& ( emptysetAx = $true )
& ( quantDeMorgan1 = $true )
& ( setadjoin__Cong = $true )
& ( omega0Ax = $true )
& ( binintersectEL = $true )
& ( dsetconstr__Cong = $true )
& ( setunionI = $true )
& ( omegaIndAx = $true )
& ( dsetconstrEL = $true )
& ( descrp = $true )
& ( notdallE = $true )
& ( exuI2 = $true )
& ( setextAx = $true )
& ( prop2setE = $true )
& ( eqimpsubset1 = $true )
& ( nonemptyI = $true )
& ( binunionLsub = $true )
& ( setadjoinAx = $true )
& ( setadjoinSub2 = $true )
& ( powersetE1 = $true )
& ( binunionE = $true )
& ( setadjoinIR = $true )
& ( emptysetimpfalse = $true )
& ( notsubsetI = $true )
& ( subsetRefl = $true )
& ( binunionIL = $true )
& ( nonemptyI1 = $true )
& ( subsetI2 = $true )
& ( upairset2IR = $true )
& ( powersetI = $true )
& ( subset2powerset = $true )
& ( eqinunit = $true )
& ( emptysetE = $true )
& ( setunionE = $true )
& ( upairsetE = $true )
& ( powerset__Cong = $true )
& ( setadjoinSub = $true )
& ( emptyE1 = $true )
& ( sepInPowerset = $true )
& ( setbeta = $true )
& ( binintersectLsub = $true )
& ( subsetE2 = $true )
& ( exuI3 = $true )
& ( subsetI1 = $true )
& ( nonemptyE1 = $true )
& ( exuEu = $true )
& ? [X0: $i,X1: $i] :
( ( ( binintersect @ X1 @ X0 )
= X0 )
& ( $true
!= ( subset @ X0 @ X1 ) ) )
& ( inPowerset = $true )
& ( eqimpsubset2 = $true )
& ( subsetTrans = $true )
& ( dsetconstrI = $true )
& ( emptysetsubset = $true )
& ( binunionIR = $true )
& ( omega__Cong = $true )
& ( exuI1 = $true )
& ( powersetsubset = $true )
& ( in__Cong = $true )
& ( powersetE = $true )
& ( exuE3u = $true )
& ( setunion__Cong = $true )
& ( prop2setI = $true )
& ( vacuousDall = $true )
& ( subPowSU = $true )
& ( binintersectI = $true )
& ( exuE2 = $true )
& ( replAx = $true )
& ( notdexE = $true )
& ( emptyI = $true )
& ( uniqinunit = $true )
& ( quantDeMorgan2 = $true )
& ( setadjoinE = $true )
& ( binintersectSubset5 = $true )
& ( setextsub = $true )
& ( setadjoinOr = $true )
& ( dsetconstrER = $true )
& ( omegaSAx = $true )
& ( emptyinunitempty = $true )
& ( notinsingleton = $true )
& ( notequalI2 = $true )
& ( setadjoinIL = $true )
& ( descr__Cong = $true )
& ( setoftrueEq = $true ) ),
inference(flattening,[status(thm)],[f411]) ).
thf(f411,plain,
( ? [X0: $i,X1: $i] :
( ( ( binintersect @ X1 @ X0 )
= X0 )
& ( $true
!= ( subset @ X0 @ X1 ) ) )
& ( 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)],[f134]) ).
thf(f134,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 )
=> ! [X0: $i,X1: $i] :
( ( ( binintersect @ X1 @ X0 )
= X0 )
=> ( $true
= ( subset @ X0 @ X1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f133]) ).
thf(f133,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
=> ! [X0: $i,X1: $i] :
( ( ( binintersect @ X1 @ X0 )
= X0 )
=> ( subset @ X0 @ X1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f115]) ).
thf(f115,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
=> ! [X4: $i,X3: $i] :
( ( ( binintersect @ X3 @ X4 )
= X4 )
=> ( subset @ X4 @ X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f114]) ).
thf(f114,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
=> ! [X4: $i,X3: $i] :
( ( ( binintersect @ X3 @ X4 )
= X4 )
=> ( subset @ X4 @ X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.ncwJItUqjX/DTF2THF_24245.p',binintersectSubset3) ).
thf(f2457,plain,
( $true
= ( subset @ sK151 @ sK152 ) ),
inference(superposition,[status(thm)],[f2010,f1261]) ).
thf(f1261,plain,
( sK151
= ( binintersect @ sK152 @ sK151 ) ),
inference(cnf_transformation,[status(thm)],[f743]) ).
thf(f2010,plain,
! [X0: $i,X1: $i] :
( $true
= ( subset @ ( binintersect @ X0 @ X1 ) @ X0 ) ),
inference(trivial_inequality_removal,[status(thm)],[f1790]) ).
thf(f1790,plain,
! [X0: $i,X1: $i] :
( ( $true
= ( subset @ ( binintersect @ X0 @ X1 ) @ X0 ) )
| ( $true != $true ) ),
inference(definition_unfolding,[status(thm)],[f1395,f1267]) ).
thf(f1267,plain,
binintersectLsub = $true,
inference(cnf_transformation,[status(thm)],[f743]) ).
thf(f1395,plain,
! [X0: $i,X1: $i] :
( ( $true
= ( subset @ ( binintersect @ X0 @ X1 ) @ X0 ) )
| ( binintersectLsub != $true ) ),
inference(cnf_transformation,[status(thm)],[f813]) ).
thf(f813,plain,
( ( ! [X0: $i,X1: $i] :
( $true
= ( subset @ ( binintersect @ X0 @ X1 ) @ X0 ) )
| ( binintersectLsub != $true ) )
& ( ( binintersectLsub = $true )
| ( $true
!= ( subset @ ( binintersect @ sK194 @ sK195 ) @ sK194 ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK194,sK195])],[f811,f812]) ).
thf(f812,plain,
( ? [X2: $i,X3: $i] :
( $true
!= ( subset @ ( binintersect @ X2 @ X3 ) @ X2 ) )
=> ( $true
!= ( subset @ ( binintersect @ sK194 @ sK195 ) @ sK194 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f811,plain,
( ( ! [X0: $i,X1: $i] :
( $true
= ( subset @ ( binintersect @ X0 @ X1 ) @ X0 ) )
| ( binintersectLsub != $true ) )
& ( ( binintersectLsub = $true )
| ? [X2: $i,X3: $i] :
( $true
!= ( subset @ ( binintersect @ X2 @ X3 ) @ X2 ) ) ) ),
inference(rectify,[status(thm)],[f810]) ).
thf(f810,plain,
( ( ! [X0: $i,X1: $i] :
( $true
= ( subset @ ( binintersect @ X0 @ X1 ) @ X0 ) )
| ( binintersectLsub != $true ) )
& ( ( binintersectLsub = $true )
| ? [X0: $i,X1: $i] :
( $true
!= ( subset @ ( binintersect @ X0 @ X1 ) @ X0 ) ) ) ),
inference(nnf_transformation,[status(thm)],[f220]) ).
thf(f220,plain,
( ! [X0: $i,X1: $i] :
( $true
= ( subset @ ( binintersect @ X0 @ X1 ) @ X0 ) )
<=> ( binintersectLsub = $true ) ),
inference(fool_elimination,[status(thm)],[f219]) ).
thf(f219,plain,
! [X0: $i,X1: $i] :
( ( subset @ ( binintersect @ X0 @ X1 ) @ X0 )
= binintersectLsub ),
inference(rectify,[status(thm)],[f112]) ).
thf(f112,axiom,
! [X3: $i,X4: $i] :
( ( subset @ ( binintersect @ X3 @ X4 ) @ X3 )
= binintersectLsub ),
file('/export/starexec/sandbox2/tmp/tmp.ncwJItUqjX/DTF2THF_24245.p',binintersectLsub) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : SEU594^1 : TPTP v9.2.1. Released v3.7.0.
% 0.12/0.12 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.13/0.32 % Computer : n008.cluster.edu
% 0.13/0.32 % Model : x86_64 x86_64
% 0.13/0.32 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.32 % Memory : 8042.1875MB
% 0.13/0.32 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.32 % CPULimit : 300
% 0.13/0.32 % WCLimit : 300
% 0.13/0.32 % DateTime : Tue Feb 24 12:27:54 EST 2026
% 0.13/0.32 % CPUTime :
% 0.13/0.32 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.17/0.44 ---- Original DTF file ---
% 0.17/0.44 thf(spec,logic,$$dhol).
% 0.17/0.44 %------------------------------------------------------------------------------
% 0.17/0.44 % File : SEU594^1 : TPTP v9.2.1. Released v3.7.0.
% 0.17/0.44 % Domain : Set Theory
% 0.17/0.44 % Problem : Preliminary Notions - Ops on Sets - Unions and Intersections
% 0.17/0.44 % Version : Especial.
% 0.17/0.44 % English : (! A:i.! B:i.binintersect A B = B -> subset B A)
% 0.17/0.44
% 0.17/0.44 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.17/0.44 % Source : [Bro08]
% 0.17/0.44 % Names : ZFC096g [Bro08]
% 0.17/0.44
% 0.17/0.44 % Status : Theorem
% 0.17/0.44 % Rating : 0.11 v9.1.0, 0.12 v9.0.0, 0.30 v8.2.0, 0.38 v8.1.0, 0.27 v7.5.0, 0.29 v7.4.0, 0.44 v7.3.0, 0.33 v7.2.0, 0.25 v7.1.0, 0.50 v7.0.0, 0.43 v6.4.0, 0.50 v6.3.0, 0.40 v6.2.0, 0.29 v6.1.0, 0.57 v5.5.0, 0.67 v5.4.0, 0.80 v4.1.0, 1.00 v3.7.0
% 0.17/0.44 % Syntax : Number of formulae : 241 ( 113 unt; 127 typ; 113 def)
% 0.17/0.44 % Number of atoms : 602 ( 164 equ; 0 cnn)
% 0.17/0.44 % Maximal formula atoms : 112 ( 5 avg)
% 0.17/0.44 % Number of connectives : 946 ( 29 ~; 6 |; 27 &; 601 @)
% 0.17/0.44 % ( 14 <=>; 269 =>; 0 <=; 0 <~>)
% 0.17/0.44 % Maximal formula depth : 116 ( 2 avg)
% 0.17/0.44 % Number of types : 2 ( 0 usr)
% 0.17/0.44 % Number of type conns : 59 ( 59 >; 0 *; 0 +; 0 <<)
% 0.17/0.44 % Number of symbols : 130 ( 127 usr; 114 con; 0-2 aty)
% 0.17/0.44 % Number of variables : 331 ( 33 ^; 270 !; 28 ?; 331 :)
% 0.17/0.44 % SPC : TH0_THM_EQU_NAR
% 0.17/0.44
% 0.17/0.44 % Comments : http://mathgate.info/detsetitem.php?id=364
% 0.17/0.44 %------------------------------------------------------------------------------
% 0.17/0.44 thf(in_type,type,
% 0.17/0.44 in: $i > $i > $o ).
% 0.17/0.44
% 0.17/0.44 thf(exu_type,type,
% 0.17/0.44 exu: ( $i > $o ) > $o ).
% 0.17/0.44
% 0.17/0.44 thf(exu,definition,
% 0.17/0.44 ( exu
% 0.17/0.44 = ( ^ [Xphi: $i > $o] :
% 0.17/0.44 ? [Xx: $i] :
% 0.17/0.44 ( ( Xphi @ Xx )
% 0.17/0.44 & ! [Xy: $i] :
% 0.17/0.44 ( ( Xphi @ Xy )
% 0.17/0.44 => ( Xx = Xy ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setextAx_type,type,
% 0.17/0.44 setextAx: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setextAx,definition,
% 0.17/0.44 ( setextAx
% 0.17/0.44 = ( ! [A: $i,B: $i] :
% 0.17/0.44 ( ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 <=> ( in @ Xx @ B ) )
% 0.17/0.44 => ( A = B ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(emptyset_type,type,
% 0.17/0.44 emptyset: $i ).
% 0.17/0.44
% 0.17/0.44 thf(emptysetAx_type,type,
% 0.17/0.44 emptysetAx: $o ).
% 0.17/0.44
% 0.17/0.44 thf(emptysetAx,definition,
% 0.17/0.44 ( emptysetAx
% 0.17/0.44 = ( ! [Xx: $i] :
% 0.17/0.44 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setadjoin_type,type,
% 0.17/0.44 setadjoin: $i > $i > $i ).
% 0.17/0.44
% 0.17/0.44 thf(setadjoinAx_type,type,
% 0.17/0.44 setadjoinAx: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setadjoinAx,definition,
% 0.17/0.44 ( setadjoinAx
% 0.17/0.44 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.17/0.44 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.17/0.44 <=> ( ( Xy = Xx )
% 0.17/0.44 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(powerset_type,type,
% 0.17/0.44 powerset: $i > $i ).
% 0.17/0.44
% 0.17/0.44 thf(powersetAx_type,type,
% 0.17/0.44 powersetAx: $o ).
% 0.17/0.44
% 0.17/0.44 thf(powersetAx,definition,
% 0.17/0.44 ( powersetAx
% 0.17/0.44 = ( ! [A: $i,B: $i] :
% 0.17/0.44 ( ( in @ B @ ( powerset @ A ) )
% 0.17/0.44 <=> ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ B )
% 0.17/0.44 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setunion_type,type,
% 0.17/0.44 setunion: $i > $i ).
% 0.17/0.44
% 0.17/0.44 thf(setunionAx_type,type,
% 0.17/0.44 setunionAx: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setunionAx,definition,
% 0.17/0.44 ( setunionAx
% 0.17/0.44 = ( ! [A: $i,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ ( setunion @ A ) )
% 0.17/0.44 <=> ? [B: $i] :
% 0.17/0.44 ( ( in @ Xx @ B )
% 0.17/0.44 & ( in @ B @ A ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(omega_type,type,
% 0.17/0.44 omega: $i ).
% 0.17/0.44
% 0.17/0.44 thf(omega0Ax_type,type,
% 0.17/0.44 omega0Ax: $o ).
% 0.17/0.44
% 0.17/0.44 thf(omega0Ax,definition,
% 0.17/0.44 ( omega0Ax
% 0.17/0.44 = ( in @ emptyset @ omega ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(omegaSAx_type,type,
% 0.17/0.44 omegaSAx: $o ).
% 0.17/0.44
% 0.17/0.44 thf(omegaSAx,definition,
% 0.17/0.44 ( omegaSAx
% 0.17/0.44 = ( ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ omega )
% 0.17/0.44 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(omegaIndAx_type,type,
% 0.17/0.44 omegaIndAx: $o ).
% 0.17/0.44
% 0.17/0.44 thf(omegaIndAx,definition,
% 0.17/0.44 ( omegaIndAx
% 0.17/0.44 = ( ! [A: $i] :
% 0.17/0.44 ( ( ( in @ emptyset @ A )
% 0.17/0.44 & ! [Xx: $i] :
% 0.17/0.44 ( ( ( in @ Xx @ omega )
% 0.17/0.44 & ( in @ Xx @ A ) )
% 0.17/0.44 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.17/0.44 => ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ omega )
% 0.17/0.44 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(replAx_type,type,
% 0.17/0.44 replAx: $o ).
% 0.17/0.44
% 0.17/0.44 thf(replAx,definition,
% 0.17/0.44 ( replAx
% 0.17/0.44 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.17/0.44 ( ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ( exu
% 0.17/0.44 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.17/0.44 => ? [B: $i] :
% 0.17/0.44 ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ B )
% 0.17/0.44 <=> ? [Xy: $i] :
% 0.17/0.44 ( ( in @ Xy @ A )
% 0.17/0.44 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(foundationAx_type,type,
% 0.17/0.44 foundationAx: $o ).
% 0.17/0.44
% 0.17/0.44 thf(foundationAx,definition,
% 0.17/0.44 ( foundationAx
% 0.17/0.44 = ( ! [A: $i] :
% 0.17/0.44 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.17/0.44 => ? [B: $i] :
% 0.17/0.44 ( ( in @ B @ A )
% 0.17/0.44 & ~ ? [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ B )
% 0.17/0.44 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(wellorderingAx_type,type,
% 0.17/0.44 wellorderingAx: $o ).
% 0.17/0.44
% 0.17/0.44 thf(wellorderingAx,definition,
% 0.17/0.44 ( wellorderingAx
% 0.17/0.44 = ( ! [A: $i] :
% 0.17/0.44 ? [B: $i] :
% 0.17/0.44 ( ! [C: $i] :
% 0.17/0.44 ( ( in @ C @ B )
% 0.17/0.44 => ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ C )
% 0.17/0.44 => ( in @ Xx @ A ) ) )
% 0.17/0.44 & ! [Xx: $i,Xy: $i] :
% 0.17/0.44 ( ( ( in @ Xx @ A )
% 0.17/0.44 & ( in @ Xy @ A ) )
% 0.17/0.44 => ( ! [C: $i] :
% 0.17/0.44 ( ( in @ C @ B )
% 0.17/0.44 => ( ( in @ Xx @ C )
% 0.17/0.44 <=> ( in @ Xy @ C ) ) )
% 0.17/0.44 => ( Xx = Xy ) ) )
% 0.17/0.44 & ! [C: $i,D: $i] :
% 0.17/0.44 ( ( ( in @ C @ B )
% 0.17/0.44 & ( in @ D @ B ) )
% 0.17/0.44 => ( ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ C )
% 0.17/0.44 => ( in @ Xx @ D ) )
% 0.17/0.44 | ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ D )
% 0.17/0.44 => ( in @ Xx @ C ) ) ) )
% 0.17/0.44 & ! [C: $i] :
% 0.17/0.44 ( ( ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ C )
% 0.17/0.44 => ( in @ Xx @ A ) )
% 0.17/0.44 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.17/0.44 => ? [D: $i,Xx: $i] :
% 0.17/0.44 ( ( in @ D @ B )
% 0.17/0.44 & ( in @ Xx @ C )
% 0.17/0.44 & ~ ? [Xy: $i] :
% 0.17/0.44 ( ( in @ Xy @ D )
% 0.17/0.44 & ( in @ Xy @ C ) )
% 0.17/0.44 & ! [E: $i] :
% 0.17/0.44 ( ( in @ E @ B )
% 0.17/0.44 => ( ! [Xy: $i] :
% 0.17/0.44 ( ( in @ Xy @ E )
% 0.17/0.44 => ( in @ Xy @ D ) )
% 0.17/0.44 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(descr_type,type,
% 0.17/0.44 descr: ( $i > $o ) > $i ).
% 0.17/0.44
% 0.17/0.44 thf(descrp_type,type,
% 0.17/0.44 descrp: $o ).
% 0.17/0.44
% 0.17/0.44 thf(descrp,definition,
% 0.17/0.44 ( descrp
% 0.17/0.44 = ( ! [Xphi: $i > $o] :
% 0.17/0.44 ( ( exu
% 0.17/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.44 => ( Xphi
% 0.17/0.44 @ ( descr
% 0.17/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(dsetconstr_type,type,
% 0.17/0.44 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.17/0.44
% 0.17/0.44 thf(dsetconstrI_type,type,
% 0.17/0.44 dsetconstrI: $o ).
% 0.17/0.44
% 0.17/0.44 thf(dsetconstrI,definition,
% 0.17/0.44 ( dsetconstrI
% 0.17/0.44 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ( ( Xphi @ Xx )
% 0.17/0.44 => ( in @ Xx
% 0.17/0.44 @ ( dsetconstr @ A
% 0.17/0.44 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(dsetconstrEL_type,type,
% 0.17/0.44 dsetconstrEL: $o ).
% 0.17/0.44
% 0.17/0.44 thf(dsetconstrEL,definition,
% 0.17/0.44 ( dsetconstrEL
% 0.17/0.44 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx
% 0.17/0.44 @ ( dsetconstr @ A
% 0.17/0.44 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.17/0.44 => ( in @ Xx @ A ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(dsetconstrER_type,type,
% 0.17/0.44 dsetconstrER: $o ).
% 0.17/0.44
% 0.17/0.44 thf(dsetconstrER,definition,
% 0.17/0.44 ( dsetconstrER
% 0.17/0.44 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx
% 0.17/0.44 @ ( dsetconstr @ A
% 0.17/0.44 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.17/0.44 => ( Xphi @ Xx ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(exuE1_type,type,
% 0.17/0.44 exuE1: $o ).
% 0.17/0.44
% 0.17/0.44 thf(exuE1,definition,
% 0.17/0.44 ( exuE1
% 0.17/0.44 = ( ! [Xphi: $i > $o] :
% 0.17/0.44 ( ( exu
% 0.17/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.44 => ? [Xx: $i] :
% 0.17/0.44 ( ( Xphi @ Xx )
% 0.17/0.44 & ! [Xy: $i] :
% 0.17/0.44 ( ( Xphi @ Xy )
% 0.17/0.44 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(prop2set_type,type,
% 0.17/0.44 prop2set: $o > $i ).
% 0.17/0.44
% 0.17/0.44 thf(prop2setE_type,type,
% 0.17/0.44 prop2setE: $o ).
% 0.17/0.44
% 0.17/0.44 thf(prop2setE,definition,
% 0.17/0.44 ( prop2setE
% 0.17/0.44 = ( ! [Xphi: $o,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.17/0.44 => Xphi ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(emptysetE_type,type,
% 0.17/0.44 emptysetE: $o ).
% 0.17/0.44
% 0.17/0.44 thf(emptysetE,definition,
% 0.17/0.44 ( emptysetE
% 0.17/0.44 = ( ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ emptyset )
% 0.17/0.44 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(emptysetimpfalse_type,type,
% 0.17/0.44 emptysetimpfalse: $o ).
% 0.17/0.44
% 0.17/0.44 thf(emptysetimpfalse,definition,
% 0.17/0.44 ( emptysetimpfalse
% 0.17/0.44 = ( ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ emptyset )
% 0.17/0.44 => $false ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(notinemptyset_type,type,
% 0.17/0.44 notinemptyset: $o ).
% 0.17/0.44
% 0.17/0.44 thf(notinemptyset,definition,
% 0.17/0.44 ( notinemptyset
% 0.17/0.44 = ( ! [Xx: $i] :
% 0.17/0.44 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(exuE3e_type,type,
% 0.17/0.44 exuE3e: $o ).
% 0.17/0.44
% 0.17/0.44 thf(exuE3e,definition,
% 0.17/0.44 ( exuE3e
% 0.17/0.44 = ( ! [Xphi: $i > $o] :
% 0.17/0.44 ( ( exu
% 0.17/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.44 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setext_type,type,
% 0.17/0.44 setext: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setext,definition,
% 0.17/0.44 ( setext
% 0.17/0.44 = ( ! [A: $i,B: $i] :
% 0.17/0.44 ( ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ( in @ Xx @ B ) )
% 0.17/0.44 => ( ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ B )
% 0.17/0.44 => ( in @ Xx @ A ) )
% 0.17/0.44 => ( A = B ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(emptyI_type,type,
% 0.17/0.44 emptyI: $o ).
% 0.17/0.44
% 0.17/0.44 thf(emptyI,definition,
% 0.17/0.44 ( emptyI
% 0.17/0.44 = ( ! [A: $i] :
% 0.17/0.44 ( ! [Xx: $i] :
% 0.17/0.44 ~ ( in @ Xx @ A )
% 0.17/0.44 => ( A = emptyset ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(noeltsimpempty_type,type,
% 0.17/0.44 noeltsimpempty: $o ).
% 0.17/0.44
% 0.17/0.44 thf(noeltsimpempty,definition,
% 0.17/0.44 ( noeltsimpempty
% 0.17/0.44 = ( ! [A: $i] :
% 0.17/0.44 ( ! [Xx: $i] :
% 0.17/0.44 ~ ( in @ Xx @ A )
% 0.17/0.44 => ( A = emptyset ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setbeta_type,type,
% 0.17/0.44 setbeta: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setbeta,definition,
% 0.17/0.44 ( setbeta
% 0.17/0.44 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ( ( in @ Xx
% 0.17/0.44 @ ( dsetconstr @ A
% 0.17/0.44 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.17/0.44 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(nonempty_type,type,
% 0.17/0.44 nonempty: $i > $o ).
% 0.17/0.44
% 0.17/0.44 thf(nonempty,definition,
% 0.17/0.44 ( nonempty
% 0.17/0.44 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(nonemptyE1_type,type,
% 0.17/0.44 nonemptyE1: $o ).
% 0.17/0.44
% 0.17/0.44 thf(nonemptyE1,definition,
% 0.17/0.44 ( nonemptyE1
% 0.17/0.44 = ( ! [A: $i] :
% 0.17/0.44 ( ( nonempty @ A )
% 0.17/0.44 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(nonemptyI_type,type,
% 0.17/0.44 nonemptyI: $o ).
% 0.17/0.44
% 0.17/0.44 thf(nonemptyI,definition,
% 0.17/0.44 ( nonemptyI
% 0.17/0.44 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ( ( Xphi @ Xx )
% 0.17/0.44 => ( nonempty
% 0.17/0.44 @ ( dsetconstr @ A
% 0.17/0.44 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(nonemptyI1_type,type,
% 0.17/0.44 nonemptyI1: $o ).
% 0.17/0.44
% 0.17/0.44 thf(nonemptyI1,definition,
% 0.17/0.44 ( nonemptyI1
% 0.17/0.44 = ( ! [A: $i] :
% 0.17/0.44 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.17/0.44 => ( nonempty @ A ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setadjoinIL_type,type,
% 0.17/0.44 setadjoinIL: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setadjoinIL,definition,
% 0.17/0.44 ( setadjoinIL
% 0.17/0.44 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(emptyinunitempty_type,type,
% 0.17/0.44 emptyinunitempty: $o ).
% 0.17/0.44
% 0.17/0.44 thf(emptyinunitempty,definition,
% 0.17/0.44 ( emptyinunitempty
% 0.17/0.44 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setadjoinIR_type,type,
% 0.17/0.44 setadjoinIR: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setadjoinIR,definition,
% 0.17/0.44 ( setadjoinIR
% 0.17/0.44 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.17/0.44 ( ( in @ Xy @ A )
% 0.17/0.44 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setadjoinE_type,type,
% 0.17/0.44 setadjoinE: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setadjoinE,definition,
% 0.17/0.44 ( setadjoinE
% 0.17/0.44 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.17/0.44 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.17/0.44 => ! [Xphi: $o] :
% 0.17/0.44 ( ( ( Xy = Xx )
% 0.17/0.44 => Xphi )
% 0.17/0.44 => ( ( ( in @ Xy @ A )
% 0.17/0.44 => Xphi )
% 0.17/0.44 => Xphi ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setadjoinOr_type,type,
% 0.17/0.44 setadjoinOr: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setadjoinOr,definition,
% 0.17/0.44 ( setadjoinOr
% 0.17/0.44 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.17/0.44 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.17/0.44 => ( ( Xy = Xx )
% 0.17/0.44 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setoftrueEq_type,type,
% 0.17/0.44 setoftrueEq: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setoftrueEq,definition,
% 0.17/0.44 ( setoftrueEq
% 0.17/0.44 = ( ! [A: $i] :
% 0.17/0.44 ( ( dsetconstr @ A
% 0.17/0.44 @ ^ [Xx: $i] : $true )
% 0.17/0.44 = A ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(powersetI_type,type,
% 0.17/0.44 powersetI: $o ).
% 0.17/0.44
% 0.17/0.44 thf(powersetI,definition,
% 0.17/0.44 ( powersetI
% 0.17/0.44 = ( ! [A: $i,B: $i] :
% 0.17/0.44 ( ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ B )
% 0.17/0.44 => ( in @ Xx @ A ) )
% 0.17/0.44 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(emptyinPowerset_type,type,
% 0.17/0.44 emptyinPowerset: $o ).
% 0.17/0.44
% 0.17/0.44 thf(emptyinPowerset,definition,
% 0.17/0.44 ( emptyinPowerset
% 0.17/0.44 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(emptyInPowerset_type,type,
% 0.17/0.44 emptyInPowerset: $o ).
% 0.17/0.44
% 0.17/0.44 thf(emptyInPowerset,definition,
% 0.17/0.44 ( emptyInPowerset
% 0.17/0.44 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(powersetE_type,type,
% 0.17/0.44 powersetE: $o ).
% 0.17/0.44
% 0.17/0.44 thf(powersetE,definition,
% 0.17/0.44 ( powersetE
% 0.17/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.44 ( ( in @ B @ ( powerset @ A ) )
% 0.17/0.44 => ( ( in @ Xx @ B )
% 0.17/0.44 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setunionI_type,type,
% 0.17/0.44 setunionI: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setunionI,definition,
% 0.17/0.44 ( setunionI
% 0.17/0.44 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.17/0.44 ( ( in @ Xx @ B )
% 0.17/0.44 => ( ( in @ B @ A )
% 0.17/0.44 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setunionE_type,type,
% 0.17/0.44 setunionE: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setunionE,definition,
% 0.17/0.44 ( setunionE
% 0.17/0.44 = ( ! [A: $i,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ ( setunion @ A ) )
% 0.17/0.44 => ! [Xphi: $o] :
% 0.17/0.44 ( ! [B: $i] :
% 0.17/0.44 ( ( in @ Xx @ B )
% 0.17/0.44 => ( ( in @ B @ A )
% 0.17/0.44 => Xphi ) )
% 0.17/0.44 => Xphi ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(subPowSU_type,type,
% 0.17/0.44 subPowSU: $o ).
% 0.17/0.44
% 0.17/0.44 thf(subPowSU,definition,
% 0.17/0.44 ( subPowSU
% 0.17/0.44 = ( ! [A: $i,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(exuE2_type,type,
% 0.17/0.44 exuE2: $o ).
% 0.17/0.44
% 0.17/0.44 thf(exuE2,definition,
% 0.17/0.44 ( exuE2
% 0.17/0.44 = ( ! [Xphi: $i > $o] :
% 0.17/0.44 ( ( exu
% 0.17/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.44 => ? [Xx: $i] :
% 0.17/0.44 ! [Xy: $i] :
% 0.17/0.44 ( ( Xphi @ Xy )
% 0.17/0.44 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(nonemptyImpWitness_type,type,
% 0.17/0.44 nonemptyImpWitness: $o ).
% 0.17/0.44
% 0.17/0.44 thf(nonemptyImpWitness,definition,
% 0.17/0.44 ( nonemptyImpWitness
% 0.17/0.44 = ( ! [A: $i] :
% 0.17/0.44 ( ( nonempty @ A )
% 0.17/0.44 => ? [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 & $true ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(uniqinunit_type,type,
% 0.17/0.44 uniqinunit: $o ).
% 0.17/0.44
% 0.17/0.44 thf(uniqinunit,definition,
% 0.17/0.44 ( uniqinunit
% 0.17/0.44 = ( ! [Xx: $i,Xy: $i] :
% 0.17/0.44 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.17/0.44 => ( Xx = Xy ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(notinsingleton_type,type,
% 0.17/0.44 notinsingleton: $o ).
% 0.17/0.44
% 0.17/0.44 thf(notinsingleton,definition,
% 0.17/0.44 ( notinsingleton
% 0.17/0.44 = ( ! [Xx: $i,Xy: $i] :
% 0.17/0.44 ( ( Xx != Xy )
% 0.17/0.44 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(eqinunit_type,type,
% 0.17/0.44 eqinunit: $o ).
% 0.17/0.44
% 0.17/0.44 thf(eqinunit,definition,
% 0.17/0.44 ( eqinunit
% 0.17/0.44 = ( ! [Xx: $i,Xy: $i] :
% 0.17/0.44 ( ( Xx = Xy )
% 0.17/0.44 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(singletonsswitch_type,type,
% 0.17/0.44 singletonsswitch: $o ).
% 0.17/0.44
% 0.17/0.44 thf(singletonsswitch,definition,
% 0.17/0.44 ( singletonsswitch
% 0.17/0.44 = ( ! [Xx: $i,Xy: $i] :
% 0.17/0.44 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.17/0.44 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(upairsetE_type,type,
% 0.17/0.44 upairsetE: $o ).
% 0.17/0.44
% 0.17/0.44 thf(upairsetE,definition,
% 0.17/0.44 ( upairsetE
% 0.17/0.44 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.17/0.44 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.17/0.44 => ( ( Xz = Xx )
% 0.17/0.44 | ( Xz = Xy ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(upairsetIL_type,type,
% 0.17/0.44 upairsetIL: $o ).
% 0.17/0.44
% 0.17/0.44 thf(upairsetIL,definition,
% 0.17/0.44 ( upairsetIL
% 0.17/0.44 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(upairsetIR_type,type,
% 0.17/0.44 upairsetIR: $o ).
% 0.17/0.44
% 0.17/0.44 thf(upairsetIR,definition,
% 0.17/0.44 ( upairsetIR
% 0.17/0.44 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(emptyE1_type,type,
% 0.17/0.44 emptyE1: $o ).
% 0.17/0.44
% 0.17/0.44 thf(emptyE1,definition,
% 0.17/0.44 ( emptyE1
% 0.17/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.17/0.44 ( ? [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 & ( Xphi @ Xx ) )
% 0.17/0.44 => ( ( ( dsetconstr @ A
% 0.17/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.44 = emptyset )
% 0.17/0.44 => $false ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(vacuousDall_type,type,
% 0.17/0.44 vacuousDall: $o ).
% 0.17/0.44
% 0.17/0.44 thf(vacuousDall,definition,
% 0.17/0.44 ( vacuousDall
% 0.17/0.44 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ emptyset )
% 0.17/0.44 => ( Xphi @ Xx ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(quantDeMorgan1_type,type,
% 0.17/0.44 quantDeMorgan1: $o ).
% 0.17/0.44
% 0.17/0.44 thf(quantDeMorgan1,definition,
% 0.17/0.44 ( quantDeMorgan1
% 0.17/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.17/0.44 ( ~ ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ( Xphi @ Xx ) )
% 0.17/0.44 => ? [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(quantDeMorgan2_type,type,
% 0.17/0.44 quantDeMorgan2: $o ).
% 0.17/0.44
% 0.17/0.44 thf(quantDeMorgan2,definition,
% 0.17/0.44 ( quantDeMorgan2
% 0.17/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.17/0.44 ( ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ~ ( Xphi @ Xx ) )
% 0.17/0.44 => ~ ? [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(quantDeMorgan3_type,type,
% 0.17/0.44 quantDeMorgan3: $o ).
% 0.17/0.44
% 0.17/0.44 thf(quantDeMorgan3,definition,
% 0.17/0.44 ( quantDeMorgan3
% 0.17/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.17/0.44 ( ~ ? [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 & ( Xphi @ Xx ) )
% 0.17/0.44 => ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(quantDeMorgan4_type,type,
% 0.17/0.44 quantDeMorgan4: $o ).
% 0.17/0.44
% 0.17/0.44 thf(quantDeMorgan4,definition,
% 0.17/0.44 ( quantDeMorgan4
% 0.17/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.17/0.44 ( ? [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 & ~ ( Xphi @ Xx ) )
% 0.17/0.44 => ~ ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(prop2setI_type,type,
% 0.17/0.44 prop2setI: $o ).
% 0.17/0.44
% 0.17/0.44 thf(prop2setI,definition,
% 0.17/0.44 ( prop2setI
% 0.17/0.44 = ( ! [Xphi: $o] :
% 0.17/0.44 ( Xphi
% 0.17/0.44 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(set2prop_type,type,
% 0.17/0.44 set2prop: $i > $o ).
% 0.17/0.44
% 0.17/0.44 thf(prop2set2propI_type,type,
% 0.17/0.44 prop2set2propI: $o ).
% 0.17/0.44
% 0.17/0.44 thf(prop2set2propI,definition,
% 0.17/0.44 ( prop2set2propI
% 0.17/0.44 = ( ! [Xphi: $o] :
% 0.17/0.44 ( Xphi
% 0.17/0.44 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(notdexE_type,type,
% 0.17/0.44 notdexE: $o ).
% 0.17/0.44
% 0.17/0.44 thf(notdexE,definition,
% 0.17/0.44 ( notdexE
% 0.17/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.17/0.44 ( ~ ? [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 & ( Xphi @ Xx ) )
% 0.17/0.44 => ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(notdallE_type,type,
% 0.17/0.44 notdallE: $o ).
% 0.17/0.44
% 0.17/0.44 thf(notdallE,definition,
% 0.17/0.44 ( notdallE
% 0.17/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.17/0.44 ( ~ ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ( Xphi @ Xx ) )
% 0.17/0.44 => ? [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(exuI1_type,type,
% 0.17/0.44 exuI1: $o ).
% 0.17/0.44
% 0.17/0.44 thf(exuI1,definition,
% 0.17/0.44 ( exuI1
% 0.17/0.44 = ( ! [Xphi: $i > $o] :
% 0.17/0.44 ( ? [Xx: $i] :
% 0.17/0.44 ( ( Xphi @ Xx )
% 0.17/0.44 & ! [Xy: $i] :
% 0.17/0.44 ( ( Xphi @ Xy )
% 0.17/0.44 => ( Xx = Xy ) ) )
% 0.17/0.44 => ( exu
% 0.17/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(exuI3_type,type,
% 0.17/0.44 exuI3: $o ).
% 0.17/0.44
% 0.17/0.44 thf(exuI3,definition,
% 0.17/0.44 ( exuI3
% 0.17/0.44 = ( ! [Xphi: $i > $o] :
% 0.17/0.44 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.17/0.44 => ( ! [Xx: $i,Xy: $i] :
% 0.17/0.44 ( ( Xphi @ Xx )
% 0.17/0.44 => ( ( Xphi @ Xy )
% 0.17/0.44 => ( Xx = Xy ) ) )
% 0.17/0.44 => ( exu
% 0.17/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(exuI2_type,type,
% 0.17/0.44 exuI2: $o ).
% 0.17/0.44
% 0.17/0.44 thf(exuI2,definition,
% 0.17/0.44 ( exuI2
% 0.17/0.44 = ( ! [Xphi: $i > $o] :
% 0.17/0.44 ( ? [Xx: $i] :
% 0.17/0.44 ! [Xy: $i] :
% 0.17/0.44 ( ( Xphi @ Xy )
% 0.17/0.44 <=> ( Xy = Xx ) )
% 0.17/0.44 => ( exu
% 0.17/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(inCongP_type,type,
% 0.17/0.44 inCongP: $o ).
% 0.17/0.44
% 0.17/0.44 thf(inCongP,definition,
% 0.17/0.44 ( inCongP
% 0.17/0.44 = ( ! [A: $i,B: $i] :
% 0.17/0.44 ( ( A = B )
% 0.17/0.44 => ! [Xx: $i,Xy: $i] :
% 0.17/0.44 ( ( Xx = Xy )
% 0.17/0.44 => ( ( in @ Xx @ A )
% 0.17/0.44 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(in__Cong_type,type,
% 0.17/0.44 in__Cong: $o ).
% 0.17/0.44
% 0.17/0.44 thf(in__Cong,definition,
% 0.17/0.44 ( in__Cong
% 0.17/0.44 = ( ! [A: $i,B: $i] :
% 0.17/0.44 ( ( A = B )
% 0.17/0.44 => ! [Xx: $i,Xy: $i] :
% 0.17/0.44 ( ( Xx = Xy )
% 0.17/0.44 => ( ( in @ Xx @ A )
% 0.17/0.44 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(exuE3u_type,type,
% 0.17/0.44 exuE3u: $o ).
% 0.17/0.44
% 0.17/0.44 thf(exuE3u,definition,
% 0.17/0.44 ( exuE3u
% 0.17/0.44 = ( ! [Xphi: $i > $o] :
% 0.17/0.44 ( ( exu
% 0.17/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.44 => ! [Xx: $i,Xy: $i] :
% 0.17/0.44 ( ( Xphi @ Xx )
% 0.17/0.44 => ( ( Xphi @ Xy )
% 0.17/0.44 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(exu__Cong_type,type,
% 0.17/0.44 exu__Cong: $o ).
% 0.17/0.44
% 0.17/0.44 thf(exu__Cong,definition,
% 0.17/0.44 ( exu__Cong
% 0.17/0.44 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.17/0.44 ( ! [Xx: $i,Xy: $i] :
% 0.17/0.44 ( ( Xx = Xy )
% 0.17/0.44 => ( ( Xphi @ Xx )
% 0.17/0.44 <=> ( Xpsi @ Xy ) ) )
% 0.17/0.44 => ( ( exu
% 0.17/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.44 <=> ( exu
% 0.17/0.44 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(emptyset__Cong_type,type,
% 0.17/0.44 emptyset__Cong: $o ).
% 0.17/0.44
% 0.17/0.44 thf(emptyset__Cong,definition,
% 0.17/0.44 ( emptyset__Cong
% 0.17/0.44 = ( emptyset = emptyset ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setadjoin__Cong_type,type,
% 0.17/0.44 setadjoin__Cong: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setadjoin__Cong,definition,
% 0.17/0.44 ( setadjoin__Cong
% 0.17/0.44 = ( ! [Xx: $i,Xy: $i] :
% 0.17/0.44 ( ( Xx = Xy )
% 0.17/0.44 => ! [Xz: $i,Xu: $i] :
% 0.17/0.44 ( ( Xz = Xu )
% 0.17/0.44 => ( ( setadjoin @ Xx @ Xz )
% 0.17/0.44 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(powerset__Cong_type,type,
% 0.17/0.44 powerset__Cong: $o ).
% 0.17/0.44
% 0.17/0.44 thf(powerset__Cong,definition,
% 0.17/0.44 ( powerset__Cong
% 0.17/0.44 = ( ! [A: $i,B: $i] :
% 0.17/0.44 ( ( A = B )
% 0.17/0.44 => ( ( powerset @ A )
% 0.17/0.44 = ( powerset @ B ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setunion__Cong_type,type,
% 0.17/0.44 setunion__Cong: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setunion__Cong,definition,
% 0.17/0.44 ( setunion__Cong
% 0.17/0.44 = ( ! [A: $i,B: $i] :
% 0.17/0.44 ( ( A = B )
% 0.17/0.44 => ( ( setunion @ A )
% 0.17/0.44 = ( setunion @ B ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(omega__Cong_type,type,
% 0.17/0.44 omega__Cong: $o ).
% 0.17/0.44
% 0.17/0.44 thf(omega__Cong,definition,
% 0.17/0.44 ( omega__Cong
% 0.17/0.44 = ( omega = omega ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(exuEu_type,type,
% 0.17/0.44 exuEu: $o ).
% 0.17/0.44
% 0.17/0.44 thf(exuEu,definition,
% 0.17/0.44 ( exuEu
% 0.17/0.44 = ( ! [Xphi: $i > $o] :
% 0.17/0.44 ( ( exu
% 0.17/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.44 => ! [Xx: $i,Xy: $i] :
% 0.17/0.44 ( ( Xphi @ Xx )
% 0.17/0.44 => ( ( Xphi @ Xy )
% 0.17/0.44 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(descr__Cong_type,type,
% 0.17/0.44 descr__Cong: $o ).
% 0.17/0.44
% 0.17/0.44 thf(descr__Cong,definition,
% 0.17/0.44 ( descr__Cong
% 0.17/0.44 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.17/0.44 ( ! [Xx: $i,Xy: $i] :
% 0.17/0.44 ( ( Xx = Xy )
% 0.17/0.44 => ( ( Xphi @ Xx )
% 0.17/0.44 <=> ( Xpsi @ Xy ) ) )
% 0.17/0.44 => ( ( exu
% 0.17/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.44 => ( ( exu
% 0.17/0.44 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.17/0.44 => ( ( descr
% 0.17/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.44 = ( descr
% 0.17/0.44 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(dsetconstr__Cong_type,type,
% 0.17/0.44 dsetconstr__Cong: $o ).
% 0.17/0.44
% 0.17/0.44 thf(dsetconstr__Cong,definition,
% 0.17/0.44 ( dsetconstr__Cong
% 0.17/0.44 = ( ! [A: $i,B: $i] :
% 0.17/0.44 ( ( A = B )
% 0.17/0.44 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.17/0.44 ( ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ! [Xy: $i] :
% 0.17/0.44 ( ( in @ Xy @ B )
% 0.17/0.44 => ( ( Xx = Xy )
% 0.17/0.44 => ( ( Xphi @ Xx )
% 0.17/0.44 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.17/0.44 => ( ( dsetconstr @ A
% 0.17/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.44 = ( dsetconstr @ B
% 0.17/0.44 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(subset_type,type,
% 0.17/0.44 subset: $i > $i > $o ).
% 0.17/0.44
% 0.17/0.44 thf(disjoint_type,type,
% 0.17/0.44 disjoint: $i > $i > $o ).
% 0.17/0.44
% 0.17/0.44 thf(setsmeet_type,type,
% 0.17/0.44 setsmeet: $i > $i > $o ).
% 0.17/0.44
% 0.17/0.44 thf(subsetI1_type,type,
% 0.17/0.44 subsetI1: $o ).
% 0.17/0.44
% 0.17/0.44 thf(subsetI1,definition,
% 0.17/0.44 ( subsetI1
% 0.17/0.44 = ( ! [A: $i,B: $i] :
% 0.17/0.44 ( ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ( in @ Xx @ B ) )
% 0.17/0.44 => ( subset @ A @ B ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(eqimpsubset2_type,type,
% 0.17/0.44 eqimpsubset2: $o ).
% 0.17/0.44
% 0.17/0.44 thf(eqimpsubset2,definition,
% 0.17/0.44 ( eqimpsubset2
% 0.17/0.44 = ( ! [A: $i,B: $i] :
% 0.17/0.44 ( ( A = B )
% 0.17/0.44 => ( subset @ B @ A ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(eqimpsubset1_type,type,
% 0.17/0.44 eqimpsubset1: $o ).
% 0.17/0.44
% 0.17/0.44 thf(eqimpsubset1,definition,
% 0.17/0.44 ( eqimpsubset1
% 0.17/0.44 = ( ! [A: $i,B: $i] :
% 0.17/0.44 ( ( A = B )
% 0.17/0.44 => ( subset @ A @ B ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(subsetI2_type,type,
% 0.17/0.44 subsetI2: $o ).
% 0.17/0.44
% 0.17/0.44 thf(subsetI2,definition,
% 0.17/0.44 ( subsetI2
% 0.17/0.44 = ( ! [A: $i,B: $i] :
% 0.17/0.44 ( ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ( in @ Xx @ B ) )
% 0.17/0.44 => ( subset @ A @ B ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(emptysetsubset_type,type,
% 0.17/0.44 emptysetsubset: $o ).
% 0.17/0.44
% 0.17/0.44 thf(emptysetsubset,definition,
% 0.17/0.44 ( emptysetsubset
% 0.17/0.44 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(subsetE_type,type,
% 0.17/0.44 subsetE: $o ).
% 0.17/0.44
% 0.17/0.44 thf(subsetE,definition,
% 0.17/0.44 ( subsetE
% 0.17/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.44 ( ( subset @ A @ B )
% 0.17/0.44 => ( ( in @ Xx @ A )
% 0.17/0.44 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(subsetE2_type,type,
% 0.17/0.44 subsetE2: $o ).
% 0.17/0.44
% 0.17/0.44 thf(subsetE2,definition,
% 0.17/0.44 ( subsetE2
% 0.17/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.44 ( ( subset @ A @ B )
% 0.17/0.44 => ( ~ ( in @ Xx @ B )
% 0.17/0.44 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(notsubsetI_type,type,
% 0.17/0.44 notsubsetI: $o ).
% 0.17/0.44
% 0.17/0.44 thf(notsubsetI,definition,
% 0.17/0.44 ( notsubsetI
% 0.17/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ( ~ ( in @ Xx @ B )
% 0.17/0.44 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(notequalI1_type,type,
% 0.17/0.44 notequalI1: $o ).
% 0.17/0.44
% 0.17/0.44 thf(notequalI1,definition,
% 0.17/0.44 ( notequalI1
% 0.17/0.44 = ( ! [A: $i,B: $i] :
% 0.17/0.44 ( ~ ( subset @ A @ B )
% 0.17/0.44 => ( A != B ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(notequalI2_type,type,
% 0.17/0.44 notequalI2: $o ).
% 0.17/0.44
% 0.17/0.44 thf(notequalI2,definition,
% 0.17/0.44 ( notequalI2
% 0.17/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ( ~ ( in @ Xx @ B )
% 0.17/0.44 => ( A != B ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(subsetRefl_type,type,
% 0.17/0.44 subsetRefl: $o ).
% 0.17/0.44
% 0.17/0.44 thf(subsetRefl,definition,
% 0.17/0.44 ( subsetRefl
% 0.17/0.44 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(subsetTrans_type,type,
% 0.17/0.44 subsetTrans: $o ).
% 0.17/0.44
% 0.17/0.44 thf(subsetTrans,definition,
% 0.17/0.44 ( subsetTrans
% 0.17/0.44 = ( ! [A: $i,B: $i,C: $i] :
% 0.17/0.44 ( ( subset @ A @ B )
% 0.17/0.44 => ( ( subset @ B @ C )
% 0.17/0.44 => ( subset @ A @ C ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setadjoinSub_type,type,
% 0.17/0.44 setadjoinSub: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setadjoinSub,definition,
% 0.17/0.44 ( setadjoinSub
% 0.17/0.44 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setadjoinSub2_type,type,
% 0.17/0.44 setadjoinSub2: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setadjoinSub2,definition,
% 0.17/0.44 ( setadjoinSub2
% 0.17/0.44 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.17/0.44 ( ( subset @ A @ B )
% 0.17/0.44 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(subset2powerset_type,type,
% 0.17/0.44 subset2powerset: $o ).
% 0.17/0.44
% 0.17/0.44 thf(subset2powerset,definition,
% 0.17/0.44 ( subset2powerset
% 0.17/0.44 = ( ! [A: $i,B: $i] :
% 0.17/0.44 ( ( subset @ A @ B )
% 0.17/0.44 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setextsub_type,type,
% 0.17/0.44 setextsub: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setextsub,definition,
% 0.17/0.44 ( setextsub
% 0.17/0.44 = ( ! [A: $i,B: $i] :
% 0.17/0.44 ( ( subset @ A @ B )
% 0.17/0.44 => ( ( subset @ B @ A )
% 0.17/0.44 => ( A = B ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(subsetemptysetimpeq_type,type,
% 0.17/0.44 subsetemptysetimpeq: $o ).
% 0.17/0.44
% 0.17/0.44 thf(subsetemptysetimpeq,definition,
% 0.17/0.44 ( subsetemptysetimpeq
% 0.17/0.44 = ( ! [A: $i] :
% 0.17/0.44 ( ( subset @ A @ emptyset )
% 0.17/0.44 => ( A = emptyset ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(powersetI1_type,type,
% 0.17/0.44 powersetI1: $o ).
% 0.17/0.44
% 0.17/0.44 thf(powersetI1,definition,
% 0.17/0.44 ( powersetI1
% 0.17/0.44 = ( ! [A: $i,B: $i] :
% 0.17/0.44 ( ( subset @ B @ A )
% 0.17/0.44 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(powersetE1_type,type,
% 0.17/0.44 powersetE1: $o ).
% 0.17/0.44
% 0.17/0.44 thf(powersetE1,definition,
% 0.17/0.44 ( powersetE1
% 0.17/0.44 = ( ! [A: $i,B: $i] :
% 0.17/0.44 ( ( in @ B @ ( powerset @ A ) )
% 0.17/0.44 => ( subset @ B @ A ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(inPowerset_type,type,
% 0.17/0.44 inPowerset: $o ).
% 0.17/0.44
% 0.17/0.44 thf(inPowerset,definition,
% 0.17/0.44 ( inPowerset
% 0.17/0.44 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(powersetsubset_type,type,
% 0.17/0.44 powersetsubset: $o ).
% 0.17/0.44
% 0.17/0.44 thf(powersetsubset,definition,
% 0.17/0.44 ( powersetsubset
% 0.17/0.44 = ( ! [A: $i,B: $i] :
% 0.17/0.44 ( ( subset @ A @ B )
% 0.17/0.44 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(sepInPowerset_type,type,
% 0.17/0.44 sepInPowerset: $o ).
% 0.17/0.44
% 0.17/0.44 thf(sepInPowerset,definition,
% 0.17/0.44 ( sepInPowerset
% 0.17/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.17/0.44 ( in
% 0.17/0.44 @ ( dsetconstr @ A
% 0.17/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.44 @ ( powerset @ A ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(sepSubset_type,type,
% 0.17/0.44 sepSubset: $o ).
% 0.17/0.44
% 0.17/0.44 thf(sepSubset,definition,
% 0.17/0.44 ( sepSubset
% 0.17/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.17/0.44 ( subset
% 0.17/0.44 @ ( dsetconstr @ A
% 0.17/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.44 @ A ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(binunion_type,type,
% 0.17/0.44 binunion: $i > $i > $i ).
% 0.17/0.44
% 0.17/0.44 thf(binunionIL_type,type,
% 0.17/0.44 binunionIL: $o ).
% 0.17/0.44
% 0.17/0.44 thf(binunionIL,definition,
% 0.17/0.44 ( binunionIL
% 0.17/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(upairset2IR_type,type,
% 0.17/0.44 upairset2IR: $o ).
% 0.17/0.44
% 0.17/0.44 thf(upairset2IR,definition,
% 0.17/0.44 ( upairset2IR
% 0.17/0.44 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(binunionIR_type,type,
% 0.17/0.44 binunionIR: $o ).
% 0.17/0.44
% 0.17/0.44 thf(binunionIR,definition,
% 0.17/0.44 ( binunionIR
% 0.17/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ B )
% 0.17/0.44 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(binunionEcases_type,type,
% 0.17/0.44 binunionEcases: $o ).
% 0.17/0.44
% 0.17/0.44 thf(binunionEcases,definition,
% 0.17/0.44 ( binunionEcases
% 0.17/0.44 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.17/0.44 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.17/0.44 => ( ( ( in @ Xx @ A )
% 0.17/0.44 => Xphi )
% 0.17/0.44 => ( ( ( in @ Xx @ B )
% 0.17/0.44 => Xphi )
% 0.17/0.44 => Xphi ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(binunionE_type,type,
% 0.17/0.44 binunionE: $o ).
% 0.17/0.44
% 0.17/0.44 thf(binunionE,definition,
% 0.17/0.44 ( binunionE
% 0.17/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.17/0.44 => ( ( in @ Xx @ A )
% 0.17/0.44 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(binunionLsub_type,type,
% 0.17/0.44 binunionLsub: $o ).
% 0.17/0.44
% 0.17/0.44 thf(binunionLsub,definition,
% 0.17/0.44 ( binunionLsub
% 0.17/0.44 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(binunionRsub_type,type,
% 0.17/0.44 binunionRsub: $o ).
% 0.17/0.44
% 0.17/0.44 thf(binunionRsub,definition,
% 0.17/0.44 ( binunionRsub
% 0.17/0.44 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(binintersect_type,type,
% 0.17/0.44 binintersect: $i > $i > $i ).
% 0.17/0.44
% 0.17/0.44 thf(binintersect,definition,
% 0.17/0.44 ( binintersect
% 0.17/0.44 = ( ^ [A: $i,B: $i] :
% 0.17/0.44 ( dsetconstr @ A
% 0.17/0.44 @ ^ [Xx: $i] : ( in @ Xx @ B ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(binintersectI_type,type,
% 0.17/0.44 binintersectI: $o ).
% 0.17/0.44
% 0.17/0.44 thf(binintersectI,definition,
% 0.17/0.44 ( binintersectI
% 0.17/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ( ( in @ Xx @ B )
% 0.17/0.44 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(binintersectSubset5_type,type,
% 0.17/0.44 binintersectSubset5: $o ).
% 0.17/0.44
% 0.17/0.44 thf(binintersectSubset5,definition,
% 0.17/0.44 ( binintersectSubset5
% 0.17/0.44 = ( ! [A: $i,B: $i,C: $i] :
% 0.17/0.44 ( ( subset @ C @ A )
% 0.17/0.44 => ( ( subset @ C @ B )
% 0.17/0.44 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(binintersectEL_type,type,
% 0.17/0.44 binintersectEL: $o ).
% 0.17/0.44
% 0.17/0.44 thf(binintersectEL,definition,
% 0.17/0.44 ( binintersectEL
% 0.17/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.17/0.44 => ( in @ Xx @ A ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(binintersectLsub_type,type,
% 0.17/0.44 binintersectLsub: $o ).
% 0.17/0.44
% 0.17/0.44 thf(binintersectLsub,definition,
% 0.17/0.44 ( binintersectLsub
% 0.17/0.44 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(binintersectSubset2_type,type,
% 0.17/0.44 binintersectSubset2: $o ).
% 0.17/0.44
% 0.17/0.44 thf(binintersectSubset2,definition,
% 0.17/0.44 ( binintersectSubset2
% 0.17/0.44 = ( ! [A: $i,B: $i] :
% 0.17/0.44 ( ( subset @ A @ B )
% 0.17/0.45 => ( ( binintersect @ A @ B )
% 0.17/0.45 = A ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(binintersectSubset3,conjecture,
% 0.17/0.45 ( setextAx
% 0.17/0.45 => ( emptysetAx
% 0.17/0.45 => ( setadjoinAx
% 0.17/0.45 => ( powersetAx
% 0.17/0.45 => ( setunionAx
% 0.17/0.45 => ( omega0Ax
% 0.17/0.45 => ( omegaSAx
% 0.17/0.45 => ( omegaIndAx
% 0.17/0.45 => ( replAx
% 0.17/0.45 => ( foundationAx
% 0.17/0.45 => ( wellorderingAx
% 0.17/0.45 => ( descrp
% 0.17/0.45 => ( dsetconstrI
% 0.17/0.45 => ( dsetconstrEL
% 0.17/0.45 => ( dsetconstrER
% 0.17/0.45 => ( exuE1
% 0.17/0.45 => ( prop2setE
% 0.17/0.45 => ( emptysetE
% 0.17/0.45 => ( emptysetimpfalse
% 0.17/0.45 => ( notinemptyset
% 0.17/0.45 => ( exuE3e
% 0.17/0.45 => ( setext
% 0.17/0.45 => ( emptyI
% 0.17/0.45 => ( noeltsimpempty
% 0.17/0.45 => ( setbeta
% 0.17/0.45 => ( nonemptyE1
% 0.17/0.45 => ( nonemptyI
% 0.17/0.45 => ( nonemptyI1
% 0.17/0.45 => ( setadjoinIL
% 0.17/0.45 => ( emptyinunitempty
% 0.17/0.45 => ( setadjoinIR
% 0.17/0.45 => ( setadjoinE
% 0.17/0.45 => ( setadjoinOr
% 0.17/0.45 => ( setoftrueEq
% 0.17/0.45 => ( powersetI
% 0.17/0.45 => ( emptyinPowerset
% 0.17/0.45 => ( emptyInPowerset
% 0.17/0.45 => ( powersetE
% 0.17/0.45 => ( setunionI
% 0.17/0.45 => ( setunionE
% 0.17/0.45 => ( subPowSU
% 0.17/0.45 => ( exuE2
% 0.17/0.45 => ( nonemptyImpWitness
% 0.17/0.45 => ( uniqinunit
% 0.17/0.45 => ( notinsingleton
% 0.17/0.45 => ( eqinunit
% 0.17/0.45 => ( singletonsswitch
% 0.17/0.45 => ( upairsetE
% 0.17/0.45 => ( upairsetIL
% 0.17/0.45 => ( upairsetIR
% 0.17/0.45 => ( emptyE1
% 0.17/0.45 => ( vacuousDall
% 0.17/0.45 => ( quantDeMorgan1
% 0.17/0.45 => ( quantDeMorgan2
% 0.17/0.45 => ( quantDeMorgan3
% 0.17/0.45 => ( quantDeMorgan4
% 0.17/0.45 => ( prop2setI
% 0.17/0.45 => ( prop2set2propI
% 0.17/0.45 => ( notdexE
% 0.17/0.45 => ( notdallE
% 0.17/0.45 => ( exuI1
% 0.17/0.45 => ( exuI3
% 0.17/0.45 => ( exuI2
% 0.17/0.45 => ( inCongP
% 0.17/0.45 => ( in__Cong
% 0.17/0.45 => ( exuE3u
% 0.17/0.45 => ( exu__Cong
% 0.17/0.45 => ( emptyset__Cong
% 0.17/0.45 => ( setadjoin__Cong
% 0.17/0.45 => ( powerset__Cong
% 0.17/0.45 => ( setunion__Cong
% 0.17/0.45 => ( omega__Cong
% 0.17/0.45 => ( exuEu
% 0.17/0.45 => ( descr__Cong
% 0.17/0.45 => ( dsetconstr__Cong
% 0.17/0.45 => ( subsetI1
% 0.17/0.45 => ( eqimpsubset2
% 0.17/0.45 => ( eqimpsubset1
% 0.17/0.45 => ( subsetI2
% 0.17/0.45 => ( emptysetsubset
% 0.17/0.45 => ( subsetE
% 0.17/0.45 => ( subsetE2
% 0.17/0.45 => ( notsubsetI
% 0.17/0.45 => ( notequalI1
% 0.17/0.45 => ( notequalI2
% 0.17/0.45 => ( subsetRefl
% 0.17/0.45 => ( subsetTrans
% 0.17/0.45 => ( setadjoinSub
% 0.17/0.45 => ( setadjoinSub2
% 0.17/0.45 => ( subset2powerset
% 0.17/0.45 => ( setextsub
% 0.17/0.45 => ( subsetemptysetimpeq
% 0.17/0.45 => ( powersetI1
% 0.17/0.45 => ( powersetE1
% 0.17/0.45 => ( inPowerset
% 0.17/0.45 => ( powersetsubset
% 0.17/0.45 => ( sepInPowerset
% 0.17/0.45 => ( sepSubset
% 0.17/0.45 => ( binunionIL
% 0.17/0.45 => ( upairset2IR
% 0.17/0.45 => ( binunionIR
% 0.17/0.45 => ( binunionEcases
% 0.17/0.45 => ( binunionE
% 0.17/0.45 => ( binunionLsub
% 0.17/0.45 => ( binunionRsub
% 0.17/0.45 => ( binintersectI
% 0.17/0.45 => ( binintersectSubset5
% 0.17/0.45 => ( binintersectEL
% 0.17/0.45 => ( binintersectLsub
% 0.17/0.45 => ( binintersectSubset2
% 0.17/0.45 => ! [A: $i,B: $i] :
% 0.17/0.45 ( ( ( binintersect @ A @ B )
% 0.17/0.45 = B )
% 0.17/0.45 => ( subset @ B @ A ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 %------------------------------------------------------------------------------
% 0.17/0.45 ------------------------
% 2.21/1.41 ---- Embedded in THF ---
% 2.21/1.41 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.21/1.41 %%% Generated on Tue Feb 24 12:27:55 EST 2026
% 2.21/1.41 %%% using '$$dhol' embedding, version 1.3.0.
% 2.21/1.41 %%% Logic specification used:
% 2.21/1.41 %%% thf(spec, logic, $$dhol).
% 2.21/1.41
% 2.21/1.41 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.ncwJItUqjX/DTF2DTF24245.p
% See solution above
% 2.21/1.41 ------------------------
% 2.21/1.43 ---- Cleaned THF ---
% 2.21/1.43 thf(in_type,type,
% 2.21/1.43 in: $i > $i > $o ).
% 2.21/1.43
% 2.21/1.43 thf(exu_type,type,
% 2.21/1.43 exu: ( $i > $o ) > $o ).
% 2.21/1.43
% 2.21/1.43 thf(exu,definition,
% 2.21/1.43 ( exu
% 2.21/1.43 = ( ^ [Xphi: $i > $o] :
% 2.21/1.43 ? [Xx: $i] :
% 2.21/1.43 ( ( Xphi @ Xx )
% 2.21/1.43 & ! [Xy: $i] :
% 2.21/1.43 ( ( Xphi @ Xy )
% 2.21/1.43 => ( Xx = Xy ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(setextAx_type,type,
% 2.21/1.43 setextAx: $o ).
% 2.21/1.43
% 2.21/1.43 thf(setextAx,definition,
% 2.21/1.43 ( setextAx
% 2.21/1.43 = ( ! [A: $i,B: $i] :
% 2.21/1.43 ( ! [Xx: $i] :
% 2.21/1.43 ( ( in @ Xx @ A )
% 2.21/1.43 <=> ( in @ Xx @ B ) )
% 2.21/1.43 => ( A = B ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(emptyset_type,type,
% 2.21/1.43 emptyset: $i ).
% 2.21/1.43
% 2.21/1.43 thf(emptysetAx_type,type,
% 2.21/1.43 emptysetAx: $o ).
% 2.21/1.43
% 2.21/1.43 thf(emptysetAx,definition,
% 2.21/1.43 ( emptysetAx
% 2.21/1.43 = ( ! [Xx: $i] :
% 2.21/1.43 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(setadjoin_type,type,
% 2.21/1.43 setadjoin: $i > $i > $i ).
% 2.21/1.43
% 2.21/1.43 thf(setadjoinAx_type,type,
% 2.21/1.43 setadjoinAx: $o ).
% 2.21/1.43
% 2.21/1.43 thf(setadjoinAx,definition,
% 2.21/1.43 ( setadjoinAx
% 2.21/1.43 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.21/1.43 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.21/1.43 <=> ( ( Xy = Xx )
% 2.21/1.43 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(powerset_type,type,
% 2.21/1.43 powerset: $i > $i ).
% 2.21/1.43
% 2.21/1.43 thf(powersetAx_type,type,
% 2.21/1.43 powersetAx: $o ).
% 2.21/1.43
% 2.21/1.43 thf(powersetAx,definition,
% 2.21/1.43 ( powersetAx
% 2.21/1.43 = ( ! [A: $i,B: $i] :
% 2.21/1.43 ( ( in @ B @ ( powerset @ A ) )
% 2.21/1.43 <=> ! [Xx: $i] :
% 2.21/1.43 ( ( in @ Xx @ B )
% 2.21/1.43 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(setunion_type,type,
% 2.21/1.43 setunion: $i > $i ).
% 2.21/1.43
% 2.21/1.43 thf(setunionAx_type,type,
% 2.21/1.43 setunionAx: $o ).
% 2.21/1.43
% 2.21/1.43 thf(setunionAx,definition,
% 2.21/1.43 ( setunionAx
% 2.21/1.43 = ( ! [A: $i,Xx: $i] :
% 2.21/1.43 ( ( in @ Xx @ ( setunion @ A ) )
% 2.21/1.43 <=> ? [B: $i] :
% 2.21/1.43 ( ( in @ Xx @ B )
% 2.21/1.43 & ( in @ B @ A ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(omega_type,type,
% 2.21/1.43 omega: $i ).
% 2.21/1.43
% 2.21/1.43 thf(omega0Ax_type,type,
% 2.21/1.43 omega0Ax: $o ).
% 2.21/1.43
% 2.21/1.43 thf(omega0Ax,definition,
% 2.21/1.43 ( omega0Ax
% 2.21/1.43 = ( in @ emptyset @ omega ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(omegaSAx_type,type,
% 2.21/1.43 omegaSAx: $o ).
% 2.21/1.43
% 2.21/1.43 thf(omegaSAx,definition,
% 2.21/1.43 ( omegaSAx
% 2.21/1.43 = ( ! [Xx: $i] :
% 2.21/1.43 ( ( in @ Xx @ omega )
% 2.21/1.43 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(omegaIndAx_type,type,
% 2.21/1.43 omegaIndAx: $o ).
% 2.21/1.43
% 2.21/1.43 thf(omegaIndAx,definition,
% 2.21/1.43 ( omegaIndAx
% 2.21/1.43 = ( ! [A: $i] :
% 2.21/1.43 ( ( ( in @ emptyset @ A )
% 2.21/1.43 & ! [Xx: $i] :
% 2.21/1.43 ( ( ( in @ Xx @ omega )
% 2.21/1.43 & ( in @ Xx @ A ) )
% 2.21/1.43 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 2.21/1.43 => ! [Xx: $i] :
% 2.21/1.43 ( ( in @ Xx @ omega )
% 2.21/1.43 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(replAx_type,type,
% 2.21/1.43 replAx: $o ).
% 2.21/1.43
% 2.21/1.43 thf(replAx,definition,
% 2.21/1.43 ( replAx
% 2.21/1.43 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 2.21/1.43 ( ! [Xx: $i] :
% 2.21/1.43 ( ( in @ Xx @ A )
% 2.21/1.43 => ( exu
% 2.21/1.43 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 2.21/1.43 => ? [B: $i] :
% 2.21/1.43 ! [Xx: $i] :
% 2.21/1.43 ( ( in @ Xx @ B )
% 2.21/1.43 <=> ? [Xy: $i] :
% 2.21/1.43 ( ( in @ Xy @ A )
% 2.21/1.43 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(foundationAx_type,type,
% 2.21/1.43 foundationAx: $o ).
% 2.21/1.43
% 2.21/1.43 thf(foundationAx,definition,
% 2.21/1.43 ( foundationAx
% 2.21/1.43 = ( ! [A: $i] :
% 2.21/1.43 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.21/1.43 => ? [B: $i] :
% 2.21/1.43 ( ( in @ B @ A )
% 2.21/1.43 & ~ ? [Xx: $i] :
% 2.21/1.43 ( ( in @ Xx @ B )
% 2.21/1.43 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(wellorderingAx_type,type,
% 2.21/1.43 wellorderingAx: $o ).
% 2.21/1.43
% 2.21/1.43 thf(wellorderingAx,definition,
% 2.21/1.43 ( wellorderingAx
% 2.21/1.43 = ( ! [A: $i] :
% 2.21/1.43 ? [B: $i] :
% 2.21/1.43 ( ! [C: $i] :
% 2.21/1.43 ( ( in @ C @ B )
% 2.21/1.43 => ! [Xx: $i] :
% 2.21/1.43 ( ( in @ Xx @ C )
% 2.21/1.43 => ( in @ Xx @ A ) ) )
% 2.21/1.43 & ! [Xx: $i,Xy: $i] :
% 2.21/1.43 ( ( ( in @ Xx @ A )
% 2.21/1.43 & ( in @ Xy @ A ) )
% 2.21/1.43 => ( ! [C: $i] :
% 2.21/1.43 ( ( in @ C @ B )
% 2.21/1.43 => ( ( in @ Xx @ C )
% 2.21/1.43 <=> ( in @ Xy @ C ) ) )
% 2.21/1.43 => ( Xx = Xy ) ) )
% 2.21/1.43 & ! [C: $i,D: $i] :
% 2.21/1.43 ( ( ( in @ C @ B )
% 2.21/1.43 & ( in @ D @ B ) )
% 2.21/1.43 => ( ! [Xx: $i] :
% 2.21/1.43 ( ( in @ Xx @ C )
% 2.21/1.43 => ( in @ Xx @ D ) )
% 2.21/1.43 | ! [Xx: $i] :
% 2.21/1.43 ( ( in @ Xx @ D )
% 2.21/1.43 => ( in @ Xx @ C ) ) ) )
% 2.21/1.43 & ! [C: $i] :
% 2.21/1.43 ( ( ! [Xx: $i] :
% 2.21/1.43 ( ( in @ Xx @ C )
% 2.21/1.43 => ( in @ Xx @ A ) )
% 2.21/1.43 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 2.21/1.43 => ? [D: $i,Xx: $i] :
% 2.21/1.43 ( ( in @ D @ B )
% 2.21/1.43 & ( in @ Xx @ C )
% 2.21/1.43 & ~ ? [Xy: $i] :
% 2.21/1.43 ( ( in @ Xy @ D )
% 2.21/1.43 & ( in @ Xy @ C ) )
% 2.21/1.43 & ! [E: $i] :
% 2.21/1.43 ( ( in @ E @ B )
% 2.21/1.43 => ( ! [Xy: $i] :
% 2.21/1.43 ( ( in @ Xy @ E )
% 2.21/1.43 => ( in @ Xy @ D ) )
% 2.21/1.43 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(descr_type,type,
% 2.21/1.43 descr: ( $i > $o ) > $i ).
% 2.21/1.43
% 2.21/1.43 thf(descrp_type,type,
% 2.21/1.43 descrp: $o ).
% 2.21/1.43
% 2.21/1.43 thf(descrp,definition,
% 2.21/1.43 ( descrp
% 2.21/1.43 = ( ! [Xphi: $i > $o] :
% 2.21/1.43 ( ( exu
% 2.21/1.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.21/1.43 => ( Xphi
% 2.21/1.43 @ ( descr
% 2.21/1.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(dsetconstr_type,type,
% 2.21/1.43 dsetconstr: $i > ( $i > $o ) > $i ).
% 2.21/1.43
% 2.21/1.43 thf(dsetconstrI_type,type,
% 2.21/1.43 dsetconstrI: $o ).
% 2.21/1.43
% 2.21/1.43 thf(dsetconstrI,definition,
% 2.21/1.43 ( dsetconstrI
% 2.21/1.43 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.21/1.43 ( ( in @ Xx @ A )
% 2.21/1.43 => ( ( Xphi @ Xx )
% 2.21/1.43 => ( in @ Xx
% 2.21/1.43 @ ( dsetconstr @ A
% 2.21/1.43 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(dsetconstrEL_type,type,
% 2.21/1.43 dsetconstrEL: $o ).
% 2.21/1.43
% 2.21/1.43 thf(dsetconstrEL,definition,
% 2.21/1.43 ( dsetconstrEL
% 2.21/1.43 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.21/1.43 ( ( in @ Xx
% 2.21/1.43 @ ( dsetconstr @ A
% 2.21/1.43 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.21/1.43 => ( in @ Xx @ A ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(dsetconstrER_type,type,
% 2.21/1.43 dsetconstrER: $o ).
% 2.21/1.43
% 2.21/1.43 thf(dsetconstrER,definition,
% 2.21/1.43 ( dsetconstrER
% 2.21/1.43 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.21/1.43 ( ( in @ Xx
% 2.21/1.43 @ ( dsetconstr @ A
% 2.21/1.43 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.21/1.43 => ( Xphi @ Xx ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(exuE1_type,type,
% 2.21/1.43 exuE1: $o ).
% 2.21/1.43
% 2.21/1.43 thf(exuE1,definition,
% 2.21/1.43 ( exuE1
% 2.21/1.43 = ( ! [Xphi: $i > $o] :
% 2.21/1.43 ( ( exu
% 2.21/1.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.21/1.43 => ? [Xx: $i] :
% 2.21/1.43 ( ( Xphi @ Xx )
% 2.21/1.43 & ! [Xy: $i] :
% 2.21/1.43 ( ( Xphi @ Xy )
% 2.21/1.43 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(prop2set_type,type,
% 2.21/1.43 prop2set: $o > $i ).
% 2.21/1.43
% 2.21/1.43 thf(prop2setE_type,type,
% 2.21/1.43 prop2setE: $o ).
% 2.21/1.43
% 2.21/1.43 thf(prop2setE,definition,
% 2.21/1.43 ( prop2setE
% 2.21/1.43 = ( ! [Xphi: $o,Xx: $i] :
% 2.21/1.43 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 2.21/1.43 => Xphi ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(emptysetE_type,type,
% 2.21/1.43 emptysetE: $o ).
% 2.21/1.43
% 2.21/1.43 thf(emptysetE,definition,
% 2.21/1.43 ( emptysetE
% 2.21/1.43 = ( ! [Xx: $i] :
% 2.21/1.43 ( ( in @ Xx @ emptyset )
% 2.21/1.43 => ! [Xphi: $o] : Xphi ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(emptysetimpfalse_type,type,
% 2.21/1.43 emptysetimpfalse: $o ).
% 2.21/1.43
% 2.21/1.43 thf(emptysetimpfalse,definition,
% 2.21/1.43 ( emptysetimpfalse
% 2.21/1.43 = ( ! [Xx: $i] :
% 2.21/1.43 ( ( in @ Xx @ emptyset )
% 2.21/1.43 => $false ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(notinemptyset_type,type,
% 2.21/1.43 notinemptyset: $o ).
% 2.21/1.43
% 2.21/1.43 thf(notinemptyset,definition,
% 2.21/1.43 ( notinemptyset
% 2.21/1.43 = ( ! [Xx: $i] :
% 2.21/1.43 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(exuE3e_type,type,
% 2.21/1.43 exuE3e: $o ).
% 2.21/1.43
% 2.21/1.43 thf(exuE3e,definition,
% 2.21/1.43 ( exuE3e
% 2.21/1.43 = ( ! [Xphi: $i > $o] :
% 2.21/1.43 ( ( exu
% 2.21/1.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.21/1.43 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(setext_type,type,
% 2.21/1.43 setext: $o ).
% 2.21/1.43
% 2.21/1.43 thf(setext,definition,
% 2.21/1.43 ( setext
% 2.21/1.43 = ( ! [A: $i,B: $i] :
% 2.21/1.43 ( ! [Xx: $i] :
% 2.21/1.43 ( ( in @ Xx @ A )
% 2.21/1.43 => ( in @ Xx @ B ) )
% 2.21/1.43 => ( ! [Xx: $i] :
% 2.21/1.43 ( ( in @ Xx @ B )
% 2.21/1.43 => ( in @ Xx @ A ) )
% 2.21/1.43 => ( A = B ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(emptyI_type,type,
% 2.21/1.43 emptyI: $o ).
% 2.21/1.43
% 2.21/1.43 thf(emptyI,definition,
% 2.21/1.43 ( emptyI
% 2.21/1.43 = ( ! [A: $i] :
% 2.21/1.43 ( ! [Xx: $i] :
% 2.21/1.43 ~ ( in @ Xx @ A )
% 2.21/1.43 => ( A = emptyset ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(noeltsimpempty_type,type,
% 2.21/1.43 noeltsimpempty: $o ).
% 2.21/1.43
% 2.21/1.43 thf(noeltsimpempty,definition,
% 2.21/1.43 ( noeltsimpempty
% 2.21/1.43 = ( ! [A: $i] :
% 2.21/1.43 ( ! [Xx: $i] :
% 2.21/1.43 ~ ( in @ Xx @ A )
% 2.21/1.43 => ( A = emptyset ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(setbeta_type,type,
% 2.21/1.43 setbeta: $o ).
% 2.21/1.43
% 2.21/1.43 thf(setbeta,definition,
% 2.21/1.43 ( setbeta
% 2.21/1.43 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.21/1.43 ( ( in @ Xx @ A )
% 2.21/1.43 => ( ( in @ Xx
% 2.21/1.43 @ ( dsetconstr @ A
% 2.21/1.43 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.21/1.43 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(nonempty_type,type,
% 2.21/1.43 nonempty: $i > $o ).
% 2.21/1.43
% 2.21/1.43 thf(nonempty,definition,
% 2.21/1.43 ( nonempty
% 2.21/1.43 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(nonemptyE1_type,type,
% 2.21/1.43 nonemptyE1: $o ).
% 2.21/1.43
% 2.21/1.43 thf(nonemptyE1,definition,
% 2.21/1.43 ( nonemptyE1
% 2.21/1.43 = ( ! [A: $i] :
% 2.21/1.43 ( ( nonempty @ A )
% 2.21/1.43 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(nonemptyI_type,type,
% 2.21/1.43 nonemptyI: $o ).
% 2.21/1.43
% 2.21/1.43 thf(nonemptyI,definition,
% 2.21/1.43 ( nonemptyI
% 2.21/1.43 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.21/1.43 ( ( in @ Xx @ A )
% 2.21/1.43 => ( ( Xphi @ Xx )
% 2.21/1.43 => ( nonempty
% 2.21/1.43 @ ( dsetconstr @ A
% 2.21/1.43 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(nonemptyI1_type,type,
% 2.21/1.43 nonemptyI1: $o ).
% 2.21/1.43
% 2.21/1.43 thf(nonemptyI1,definition,
% 2.21/1.43 ( nonemptyI1
% 2.21/1.43 = ( ! [A: $i] :
% 2.21/1.43 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.21/1.43 => ( nonempty @ A ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(setadjoinIL_type,type,
% 2.21/1.43 setadjoinIL: $o ).
% 2.21/1.43
% 2.21/1.43 thf(setadjoinIL,definition,
% 2.21/1.43 ( setadjoinIL
% 2.21/1.43 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(emptyinunitempty_type,type,
% 2.21/1.43 emptyinunitempty: $o ).
% 2.21/1.43
% 2.21/1.43 thf(emptyinunitempty,definition,
% 2.21/1.43 ( emptyinunitempty
% 2.21/1.43 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(setadjoinIR_type,type,
% 2.21/1.43 setadjoinIR: $o ).
% 2.21/1.43
% 2.21/1.43 thf(setadjoinIR,definition,
% 2.21/1.43 ( setadjoinIR
% 2.21/1.43 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.21/1.43 ( ( in @ Xy @ A )
% 2.21/1.43 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(setadjoinE_type,type,
% 2.21/1.43 setadjoinE: $o ).
% 2.21/1.43
% 2.21/1.43 thf(setadjoinE,definition,
% 2.21/1.43 ( setadjoinE
% 2.21/1.43 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.21/1.43 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.21/1.43 => ! [Xphi: $o] :
% 2.21/1.43 ( ( ( Xy = Xx )
% 2.21/1.43 => Xphi )
% 2.21/1.43 => ( ( ( in @ Xy @ A )
% 2.21/1.43 => Xphi )
% 2.21/1.43 => Xphi ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(setadjoinOr_type,type,
% 2.21/1.43 setadjoinOr: $o ).
% 2.21/1.43
% 2.21/1.43 thf(setadjoinOr,definition,
% 2.21/1.43 ( setadjoinOr
% 2.21/1.43 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.21/1.43 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.21/1.43 => ( ( Xy = Xx )
% 2.21/1.43 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(setoftrueEq_type,type,
% 2.21/1.43 setoftrueEq: $o ).
% 2.21/1.43
% 2.21/1.43 thf(setoftrueEq,definition,
% 2.21/1.43 ( setoftrueEq
% 2.21/1.43 = ( ! [A: $i] :
% 2.21/1.43 ( ( dsetconstr @ A
% 2.21/1.43 @ ^ [Xx: $i] : $true )
% 2.21/1.43 = A ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(powersetI_type,type,
% 2.21/1.43 powersetI: $o ).
% 2.21/1.43
% 2.21/1.43 thf(powersetI,definition,
% 2.21/1.43 ( powersetI
% 2.21/1.43 = ( ! [A: $i,B: $i] :
% 2.21/1.43 ( ! [Xx: $i] :
% 2.21/1.43 ( ( in @ Xx @ B )
% 2.21/1.43 => ( in @ Xx @ A ) )
% 2.21/1.43 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(emptyinPowerset_type,type,
% 2.21/1.43 emptyinPowerset: $o ).
% 2.21/1.43
% 2.21/1.43 thf(emptyinPowerset,definition,
% 2.21/1.43 ( emptyinPowerset
% 2.21/1.43 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(emptyInPowerset_type,type,
% 2.21/1.43 emptyInPowerset: $o ).
% 2.21/1.43
% 2.21/1.43 thf(emptyInPowerset,definition,
% 2.21/1.43 ( emptyInPowerset
% 2.21/1.43 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(powersetE_type,type,
% 2.21/1.43 powersetE: $o ).
% 2.21/1.43
% 2.21/1.43 thf(powersetE,definition,
% 2.21/1.43 ( powersetE
% 2.21/1.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.21/1.43 ( ( in @ B @ ( powerset @ A ) )
% 2.21/1.43 => ( ( in @ Xx @ B )
% 2.21/1.43 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(setunionI_type,type,
% 2.21/1.43 setunionI: $o ).
% 2.21/1.43
% 2.21/1.43 thf(setunionI,definition,
% 2.21/1.43 ( setunionI
% 2.21/1.43 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.21/1.43 ( ( in @ Xx @ B )
% 2.21/1.43 => ( ( in @ B @ A )
% 2.21/1.43 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(setunionE_type,type,
% 2.21/1.43 setunionE: $o ).
% 2.21/1.43
% 2.21/1.43 thf(setunionE,definition,
% 2.21/1.43 ( setunionE
% 2.21/1.43 = ( ! [A: $i,Xx: $i] :
% 2.21/1.43 ( ( in @ Xx @ ( setunion @ A ) )
% 2.21/1.43 => ! [Xphi: $o] :
% 2.21/1.43 ( ! [B: $i] :
% 2.21/1.43 ( ( in @ Xx @ B )
% 2.21/1.43 => ( ( in @ B @ A )
% 2.21/1.43 => Xphi ) )
% 2.21/1.43 => Xphi ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(subPowSU_type,type,
% 2.21/1.43 subPowSU: $o ).
% 2.21/1.43
% 2.21/1.43 thf(subPowSU,definition,
% 2.21/1.43 ( subPowSU
% 2.21/1.43 = ( ! [A: $i,Xx: $i] :
% 2.21/1.43 ( ( in @ Xx @ A )
% 2.21/1.43 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(exuE2_type,type,
% 2.21/1.43 exuE2: $o ).
% 2.21/1.43
% 2.21/1.43 thf(exuE2,definition,
% 2.21/1.43 ( exuE2
% 2.21/1.43 = ( ! [Xphi: $i > $o] :
% 2.21/1.43 ( ( exu
% 2.21/1.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.21/1.43 => ? [Xx: $i] :
% 2.21/1.43 ! [Xy: $i] :
% 2.21/1.43 ( ( Xphi @ Xy )
% 2.21/1.43 <=> ( Xy = Xx ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(nonemptyImpWitness_type,type,
% 2.21/1.43 nonemptyImpWitness: $o ).
% 2.21/1.43
% 2.21/1.43 thf(nonemptyImpWitness,definition,
% 2.21/1.43 ( nonemptyImpWitness
% 2.21/1.43 = ( ! [A: $i] :
% 2.21/1.43 ( ( nonempty @ A )
% 2.21/1.43 => ? [Xx: $i] :
% 2.21/1.43 ( ( in @ Xx @ A )
% 2.21/1.43 & $true ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(uniqinunit_type,type,
% 2.21/1.43 uniqinunit: $o ).
% 2.21/1.43
% 2.21/1.43 thf(uniqinunit,definition,
% 2.21/1.43 ( uniqinunit
% 2.21/1.43 = ( ! [Xx: $i,Xy: $i] :
% 2.21/1.43 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.21/1.43 => ( Xx = Xy ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(notinsingleton_type,type,
% 2.21/1.43 notinsingleton: $o ).
% 2.21/1.43
% 2.21/1.43 thf(notinsingleton,definition,
% 2.21/1.43 ( notinsingleton
% 2.21/1.43 = ( ! [Xx: $i,Xy: $i] :
% 2.21/1.43 ( ( Xx != Xy )
% 2.21/1.43 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(eqinunit_type,type,
% 2.21/1.43 eqinunit: $o ).
% 2.21/1.43
% 2.21/1.43 thf(eqinunit,definition,
% 2.21/1.43 ( eqinunit
% 2.21/1.43 = ( ! [Xx: $i,Xy: $i] :
% 2.21/1.43 ( ( Xx = Xy )
% 2.21/1.43 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(singletonsswitch_type,type,
% 2.21/1.43 singletonsswitch: $o ).
% 2.21/1.43
% 2.21/1.43 thf(singletonsswitch,definition,
% 2.21/1.43 ( singletonsswitch
% 2.21/1.43 = ( ! [Xx: $i,Xy: $i] :
% 2.21/1.43 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.21/1.43 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(upairsetE_type,type,
% 2.21/1.43 upairsetE: $o ).
% 2.21/1.43
% 2.21/1.43 thf(upairsetE,definition,
% 2.21/1.43 ( upairsetE
% 2.21/1.43 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.21/1.43 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.21/1.43 => ( ( Xz = Xx )
% 2.21/1.43 | ( Xz = Xy ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(upairsetIL_type,type,
% 2.21/1.43 upairsetIL: $o ).
% 2.21/1.43
% 2.21/1.43 thf(upairsetIL,definition,
% 2.21/1.43 ( upairsetIL
% 2.21/1.43 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(upairsetIR_type,type,
% 2.21/1.43 upairsetIR: $o ).
% 2.21/1.43
% 2.21/1.43 thf(upairsetIR,definition,
% 2.21/1.43 ( upairsetIR
% 2.21/1.43 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(emptyE1_type,type,
% 2.21/1.43 emptyE1: $o ).
% 2.21/1.43
% 2.21/1.43 thf(emptyE1,definition,
% 2.21/1.43 ( emptyE1
% 2.21/1.43 = ( ! [A: $i,Xphi: $i > $o] :
% 2.21/1.43 ( ? [Xx: $i] :
% 2.21/1.43 ( ( in @ Xx @ A )
% 2.21/1.43 & ( Xphi @ Xx ) )
% 2.21/1.43 => ( ( ( dsetconstr @ A
% 2.21/1.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.21/1.43 = emptyset )
% 2.21/1.43 => $false ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(vacuousDall_type,type,
% 2.21/1.43 vacuousDall: $o ).
% 2.21/1.43
% 2.21/1.43 thf(vacuousDall,definition,
% 2.21/1.43 ( vacuousDall
% 2.21/1.43 = ( ! [Xphi: $i > $o,Xx: $i] :
% 2.21/1.43 ( ( in @ Xx @ emptyset )
% 2.21/1.43 => ( Xphi @ Xx ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(quantDeMorgan1_type,type,
% 2.21/1.43 quantDeMorgan1: $o ).
% 2.21/1.43
% 2.21/1.43 thf(quantDeMorgan1,definition,
% 2.21/1.43 ( quantDeMorgan1
% 2.21/1.43 = ( ! [A: $i,Xphi: $i > $o] :
% 2.21/1.43 ( ~ ! [Xx: $i] :
% 2.21/1.43 ( ( in @ Xx @ A )
% 2.21/1.43 => ( Xphi @ Xx ) )
% 2.21/1.43 => ? [Xx: $i] :
% 2.21/1.43 ( ( in @ Xx @ A )
% 2.21/1.43 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(quantDeMorgan2_type,type,
% 2.21/1.43 quantDeMorgan2: $o ).
% 2.21/1.43
% 2.21/1.43 thf(quantDeMorgan2,definition,
% 2.21/1.43 ( quantDeMorgan2
% 2.21/1.43 = ( ! [A: $i,Xphi: $i > $o] :
% 2.21/1.43 ( ! [Xx: $i] :
% 2.21/1.43 ( ( in @ Xx @ A )
% 2.21/1.43 => ~ ( Xphi @ Xx ) )
% 2.21/1.43 => ~ ? [Xx: $i] :
% 2.21/1.43 ( ( in @ Xx @ A )
% 2.21/1.43 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(quantDeMorgan3_type,type,
% 2.21/1.43 quantDeMorgan3: $o ).
% 2.21/1.43
% 2.21/1.43 thf(quantDeMorgan3,definition,
% 2.21/1.43 ( quantDeMorgan3
% 2.21/1.43 = ( ! [A: $i,Xphi: $i > $o] :
% 2.21/1.43 ( ~ ? [Xx: $i] :
% 2.21/1.43 ( ( in @ Xx @ A )
% 2.21/1.43 & ( Xphi @ Xx ) )
% 2.21/1.43 => ! [Xx: $i] :
% 2.21/1.43 ( ( in @ Xx @ A )
% 2.21/1.43 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(quantDeMorgan4_type,type,
% 2.21/1.43 quantDeMorgan4: $o ).
% 2.21/1.43
% 2.21/1.43 thf(quantDeMorgan4,definition,
% 2.21/1.43 ( quantDeMorgan4
% 2.21/1.43 = ( ! [A: $i,Xphi: $i > $o] :
% 2.21/1.43 ( ? [Xx: $i] :
% 2.21/1.43 ( ( in @ Xx @ A )
% 2.21/1.43 & ~ ( Xphi @ Xx ) )
% 2.21/1.43 => ~ ! [Xx: $i] :
% 2.21/1.43 ( ( in @ Xx @ A )
% 2.21/1.43 => ( Xphi @ Xx ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(prop2setI_type,type,
% 2.21/1.43 prop2setI: $o ).
% 2.21/1.43
% 2.21/1.43 thf(prop2setI,definition,
% 2.21/1.43 ( prop2setI
% 2.21/1.43 = ( ! [Xphi: $o] :
% 2.21/1.43 ( Xphi
% 2.21/1.43 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(set2prop_type,type,
% 2.21/1.43 set2prop: $i > $o ).
% 2.21/1.43
% 2.21/1.43 thf(prop2set2propI_type,type,
% 2.21/1.43 prop2set2propI: $o ).
% 2.21/1.43
% 2.21/1.43 thf(prop2set2propI,definition,
% 2.21/1.43 ( prop2set2propI
% 2.21/1.43 = ( ! [Xphi: $o] :
% 2.21/1.43 ( Xphi
% 2.21/1.43 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(notdexE_type,type,
% 2.21/1.43 notdexE: $o ).
% 2.21/1.43
% 2.21/1.43 thf(notdexE,definition,
% 2.21/1.43 ( notdexE
% 2.21/1.43 = ( ! [A: $i,Xphi: $i > $o] :
% 2.21/1.43 ( ~ ? [Xx: $i] :
% 2.21/1.43 ( ( in @ Xx @ A )
% 2.21/1.43 & ( Xphi @ Xx ) )
% 2.21/1.43 => ! [Xx: $i] :
% 2.21/1.43 ( ( in @ Xx @ A )
% 2.21/1.43 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(notdallE_type,type,
% 2.21/1.43 notdallE: $o ).
% 2.21/1.43
% 2.21/1.43 thf(notdallE,definition,
% 2.21/1.43 ( notdallE
% 2.21/1.43 = ( ! [A: $i,Xphi: $i > $o] :
% 2.21/1.43 ( ~ ! [Xx: $i] :
% 2.21/1.43 ( ( in @ Xx @ A )
% 2.21/1.43 => ( Xphi @ Xx ) )
% 2.21/1.43 => ? [Xx: $i] :
% 2.21/1.43 ( ( in @ Xx @ A )
% 2.21/1.43 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(exuI1_type,type,
% 2.21/1.43 exuI1: $o ).
% 2.21/1.43
% 2.21/1.43 thf(exuI1,definition,
% 2.21/1.43 ( exuI1
% 2.21/1.43 = ( ! [Xphi: $i > $o] :
% 2.21/1.43 ( ? [Xx: $i] :
% 2.21/1.43 ( ( Xphi @ Xx )
% 2.21/1.43 & ! [Xy: $i] :
% 2.21/1.43 ( ( Xphi @ Xy )
% 2.21/1.43 => ( Xx = Xy ) ) )
% 2.21/1.43 => ( exu
% 2.21/1.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(exuI3_type,type,
% 2.21/1.43 exuI3: $o ).
% 2.21/1.43
% 2.21/1.43 thf(exuI3,definition,
% 2.21/1.43 ( exuI3
% 2.21/1.43 = ( ! [Xphi: $i > $o] :
% 2.21/1.43 ( ? [Xx: $i] : ( Xphi @ Xx )
% 2.21/1.43 => ( ! [Xx: $i,Xy: $i] :
% 2.21/1.43 ( ( Xphi @ Xx )
% 2.21/1.43 => ( ( Xphi @ Xy )
% 2.21/1.43 => ( Xx = Xy ) ) )
% 2.21/1.43 => ( exu
% 2.21/1.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(exuI2_type,type,
% 2.21/1.43 exuI2: $o ).
% 2.21/1.43
% 2.21/1.43 thf(exuI2,definition,
% 2.21/1.43 ( exuI2
% 2.21/1.43 = ( ! [Xphi: $i > $o] :
% 2.21/1.43 ( ? [Xx: $i] :
% 2.21/1.43 ! [Xy: $i] :
% 2.21/1.43 ( ( Xphi @ Xy )
% 2.21/1.43 <=> ( Xy = Xx ) )
% 2.21/1.43 => ( exu
% 2.21/1.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(inCongP_type,type,
% 2.21/1.43 inCongP: $o ).
% 2.21/1.43
% 2.21/1.43 thf(inCongP,definition,
% 2.21/1.43 ( inCongP
% 2.21/1.43 = ( ! [A: $i,B: $i] :
% 2.21/1.43 ( ( A = B )
% 2.21/1.43 => ! [Xx: $i,Xy: $i] :
% 2.21/1.43 ( ( Xx = Xy )
% 2.21/1.43 => ( ( in @ Xx @ A )
% 2.21/1.43 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(in__Cong_type,type,
% 2.21/1.43 in__Cong: $o ).
% 2.21/1.43
% 2.21/1.43 thf(in__Cong,definition,
% 2.21/1.43 ( in__Cong
% 2.21/1.43 = ( ! [A: $i,B: $i] :
% 2.21/1.43 ( ( A = B )
% 2.21/1.43 => ! [Xx: $i,Xy: $i] :
% 2.21/1.43 ( ( Xx = Xy )
% 2.21/1.43 => ( ( in @ Xx @ A )
% 2.21/1.43 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(exuE3u_type,type,
% 2.21/1.43 exuE3u: $o ).
% 2.21/1.43
% 2.21/1.43 thf(exuE3u,definition,
% 2.21/1.43 ( exuE3u
% 2.21/1.43 = ( ! [Xphi: $i > $o] :
% 2.21/1.43 ( ( exu
% 2.21/1.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.21/1.43 => ! [Xx: $i,Xy: $i] :
% 2.21/1.43 ( ( Xphi @ Xx )
% 2.21/1.43 => ( ( Xphi @ Xy )
% 2.21/1.43 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(exu__Cong_type,type,
% 2.21/1.43 exu__Cong: $o ).
% 2.21/1.43
% 2.21/1.43 thf(exu__Cong,definition,
% 2.21/1.43 ( exu__Cong
% 2.21/1.43 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.21/1.43 ( ! [Xx: $i,Xy: $i] :
% 2.21/1.43 ( ( Xx = Xy )
% 2.21/1.43 => ( ( Xphi @ Xx )
% 2.21/1.43 <=> ( Xpsi @ Xy ) ) )
% 2.21/1.43 => ( ( exu
% 2.21/1.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.21/1.43 <=> ( exu
% 2.21/1.43 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(emptyset__Cong_type,type,
% 2.21/1.43 emptyset__Cong: $o ).
% 2.21/1.43
% 2.21/1.43 thf(emptyset__Cong,definition,
% 2.21/1.43 ( emptyset__Cong
% 2.21/1.43 = ( emptyset = emptyset ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(setadjoin__Cong_type,type,
% 2.21/1.43 setadjoin__Cong: $o ).
% 2.21/1.43
% 2.21/1.43 thf(setadjoin__Cong,definition,
% 2.21/1.43 ( setadjoin__Cong
% 2.21/1.43 = ( ! [Xx: $i,Xy: $i] :
% 2.21/1.43 ( ( Xx = Xy )
% 2.21/1.43 => ! [Xz: $i,Xu: $i] :
% 2.21/1.43 ( ( Xz = Xu )
% 2.21/1.43 => ( ( setadjoin @ Xx @ Xz )
% 2.21/1.43 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(powerset__Cong_type,type,
% 2.21/1.43 powerset__Cong: $o ).
% 2.21/1.43
% 2.21/1.43 thf(powerset__Cong,definition,
% 2.21/1.43 ( powerset__Cong
% 2.21/1.43 = ( ! [A: $i,B: $i] :
% 2.21/1.43 ( ( A = B )
% 2.21/1.43 => ( ( powerset @ A )
% 2.21/1.43 = ( powerset @ B ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(setunion__Cong_type,type,
% 2.21/1.43 setunion__Cong: $o ).
% 2.21/1.43
% 2.21/1.43 thf(setunion__Cong,definition,
% 2.21/1.43 ( setunion__Cong
% 2.21/1.43 = ( ! [A: $i,B: $i] :
% 2.21/1.43 ( ( A = B )
% 2.21/1.43 => ( ( setunion @ A )
% 2.21/1.43 = ( setunion @ B ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(omega__Cong_type,type,
% 2.21/1.43 omega__Cong: $o ).
% 2.21/1.43
% 2.21/1.43 thf(omega__Cong,definition,
% 2.21/1.43 ( omega__Cong
% 2.21/1.43 = ( omega = omega ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(exuEu_type,type,
% 2.21/1.43 exuEu: $o ).
% 2.21/1.43
% 2.21/1.43 thf(exuEu,definition,
% 2.21/1.43 ( exuEu
% 2.21/1.43 = ( ! [Xphi: $i > $o] :
% 2.21/1.43 ( ( exu
% 2.21/1.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.21/1.43 => ! [Xx: $i,Xy: $i] :
% 2.21/1.43 ( ( Xphi @ Xx )
% 2.21/1.43 => ( ( Xphi @ Xy )
% 2.21/1.43 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(descr__Cong_type,type,
% 2.21/1.43 descr__Cong: $o ).
% 2.21/1.43
% 2.21/1.43 thf(descr__Cong,definition,
% 2.21/1.43 ( descr__Cong
% 2.21/1.43 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.21/1.43 ( ! [Xx: $i,Xy: $i] :
% 2.21/1.43 ( ( Xx = Xy )
% 2.21/1.43 => ( ( Xphi @ Xx )
% 2.21/1.43 <=> ( Xpsi @ Xy ) ) )
% 2.21/1.43 => ( ( exu
% 2.21/1.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.21/1.43 => ( ( exu
% 2.21/1.43 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 2.21/1.43 => ( ( descr
% 2.21/1.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.21/1.43 = ( descr
% 2.21/1.43 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(dsetconstr__Cong_type,type,
% 2.21/1.43 dsetconstr__Cong: $o ).
% 2.21/1.43
% 2.21/1.43 thf(dsetconstr__Cong,definition,
% 2.21/1.43 ( dsetconstr__Cong
% 2.21/1.43 = ( ! [A: $i,B: $i] :
% 2.21/1.43 ( ( A = B )
% 2.21/1.43 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.21/1.43 ( ! [Xx: $i] :
% 2.21/1.43 ( ( in @ Xx @ A )
% 2.21/1.43 => ! [Xy: $i] :
% 2.21/1.43 ( ( in @ Xy @ B )
% 2.21/1.43 => ( ( Xx = Xy )
% 2.21/1.43 => ( ( Xphi @ Xx )
% 2.21/1.43 <=> ( Xpsi @ Xy ) ) ) ) )
% 2.21/1.43 => ( ( dsetconstr @ A
% 2.21/1.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.21/1.43 = ( dsetconstr @ B
% 2.21/1.43 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(subset_type,type,
% 2.21/1.43 subset: $i > $i > $o ).
% 2.21/1.43
% 2.21/1.43 thf(disjoint_type,type,
% 2.21/1.43 disjoint: $i > $i > $o ).
% 2.21/1.43
% 2.21/1.43 thf(setsmeet_type,type,
% 2.21/1.43 setsmeet: $i > $i > $o ).
% 2.21/1.43
% 2.21/1.43 thf(subsetI1_type,type,
% 2.21/1.43 subsetI1: $o ).
% 2.21/1.43
% 2.21/1.43 thf(subsetI1,definition,
% 2.21/1.43 ( subsetI1
% 2.21/1.43 = ( ! [A: $i,B: $i] :
% 2.21/1.43 ( ! [Xx: $i] :
% 2.21/1.43 ( ( in @ Xx @ A )
% 2.21/1.43 => ( in @ Xx @ B ) )
% 2.21/1.43 => ( subset @ A @ B ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(eqimpsubset2_type,type,
% 2.21/1.43 eqimpsubset2: $o ).
% 2.21/1.43
% 2.21/1.43 thf(eqimpsubset2,definition,
% 2.21/1.43 ( eqimpsubset2
% 2.21/1.43 = ( ! [A: $i,B: $i] :
% 2.21/1.43 ( ( A = B )
% 2.21/1.43 => ( subset @ B @ A ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(eqimpsubset1_type,type,
% 2.21/1.43 eqimpsubset1: $o ).
% 2.21/1.43
% 2.21/1.43 thf(eqimpsubset1,definition,
% 2.21/1.43 ( eqimpsubset1
% 2.21/1.43 = ( ! [A: $i,B: $i] :
% 2.21/1.43 ( ( A = B )
% 2.21/1.43 => ( subset @ A @ B ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(subsetI2_type,type,
% 2.21/1.43 subsetI2: $o ).
% 2.21/1.43
% 2.21/1.43 thf(subsetI2,definition,
% 2.21/1.43 ( subsetI2
% 2.21/1.43 = ( ! [A: $i,B: $i] :
% 2.21/1.43 ( ! [Xx: $i] :
% 2.21/1.43 ( ( in @ Xx @ A )
% 2.21/1.43 => ( in @ Xx @ B ) )
% 2.21/1.43 => ( subset @ A @ B ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(emptysetsubset_type,type,
% 2.21/1.43 emptysetsubset: $o ).
% 2.21/1.43
% 2.21/1.43 thf(emptysetsubset,definition,
% 2.21/1.43 ( emptysetsubset
% 2.21/1.43 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(subsetE_type,type,
% 2.21/1.43 subsetE: $o ).
% 2.21/1.43
% 2.21/1.43 thf(subsetE,definition,
% 2.21/1.43 ( subsetE
% 2.21/1.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.21/1.43 ( ( subset @ A @ B )
% 2.21/1.43 => ( ( in @ Xx @ A )
% 2.21/1.43 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(subsetE2_type,type,
% 2.21/1.43 subsetE2: $o ).
% 2.21/1.43
% 2.21/1.43 thf(subsetE2,definition,
% 2.21/1.43 ( subsetE2
% 2.21/1.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.21/1.43 ( ( subset @ A @ B )
% 2.21/1.43 => ( ~ ( in @ Xx @ B )
% 2.21/1.43 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(notsubsetI_type,type,
% 2.21/1.43 notsubsetI: $o ).
% 2.21/1.43
% 2.21/1.43 thf(notsubsetI,definition,
% 2.21/1.43 ( notsubsetI
% 2.21/1.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.21/1.43 ( ( in @ Xx @ A )
% 2.21/1.43 => ( ~ ( in @ Xx @ B )
% 2.21/1.43 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(notequalI1_type,type,
% 2.21/1.43 notequalI1: $o ).
% 2.21/1.43
% 2.21/1.43 thf(notequalI1,definition,
% 2.21/1.43 ( notequalI1
% 2.21/1.43 = ( ! [A: $i,B: $i] :
% 2.21/1.43 ( ~ ( subset @ A @ B )
% 2.21/1.43 => ( A != B ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(notequalI2_type,type,
% 2.21/1.43 notequalI2: $o ).
% 2.21/1.43
% 2.21/1.43 thf(notequalI2,definition,
% 2.21/1.43 ( notequalI2
% 2.21/1.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.21/1.43 ( ( in @ Xx @ A )
% 2.21/1.43 => ( ~ ( in @ Xx @ B )
% 2.21/1.43 => ( A != B ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(subsetRefl_type,type,
% 2.21/1.43 subsetRefl: $o ).
% 2.21/1.43
% 2.21/1.43 thf(subsetRefl,definition,
% 2.21/1.43 ( subsetRefl
% 2.21/1.43 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(subsetTrans_type,type,
% 2.21/1.43 subsetTrans: $o ).
% 2.21/1.43
% 2.21/1.43 thf(subsetTrans,definition,
% 2.21/1.43 ( subsetTrans
% 2.21/1.43 = ( ! [A: $i,B: $i,C: $i] :
% 2.21/1.43 ( ( subset @ A @ B )
% 2.21/1.43 => ( ( subset @ B @ C )
% 2.21/1.43 => ( subset @ A @ C ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(setadjoinSub_type,type,
% 2.21/1.43 setadjoinSub: $o ).
% 2.21/1.43
% 2.21/1.43 thf(setadjoinSub,definition,
% 2.21/1.43 ( setadjoinSub
% 2.21/1.43 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(setadjoinSub2_type,type,
% 2.21/1.43 setadjoinSub2: $o ).
% 2.21/1.43
% 2.21/1.43 thf(setadjoinSub2,definition,
% 2.21/1.43 ( setadjoinSub2
% 2.21/1.43 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.21/1.43 ( ( subset @ A @ B )
% 2.21/1.43 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(subset2powerset_type,type,
% 2.21/1.43 subset2powerset: $o ).
% 2.21/1.43
% 2.21/1.43 thf(subset2powerset,definition,
% 2.21/1.43 ( subset2powerset
% 2.21/1.43 = ( ! [A: $i,B: $i] :
% 2.21/1.43 ( ( subset @ A @ B )
% 2.21/1.43 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(setextsub_type,type,
% 2.21/1.43 setextsub: $o ).
% 2.21/1.43
% 2.21/1.43 thf(setextsub,definition,
% 2.21/1.43 ( setextsub
% 2.21/1.43 = ( ! [A: $i,B: $i] :
% 2.21/1.43 ( ( subset @ A @ B )
% 2.21/1.44 => ( ( subset @ B @ A )
% 2.21/1.44 => ( A = B ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(subsetemptysetimpeq_type,type,
% 2.21/1.44 subsetemptysetimpeq: $o ).
% 2.21/1.44
% 2.21/1.44 thf(subsetemptysetimpeq,definition,
% 2.21/1.44 ( subsetemptysetimpeq
% 2.21/1.44 = ( ! [A: $i] :
% 2.21/1.44 ( ( subset @ A @ emptyset )
% 2.21/1.44 => ( A = emptyset ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(powersetI1_type,type,
% 2.21/1.44 powersetI1: $o ).
% 2.21/1.44
% 2.21/1.44 thf(powersetI1,definition,
% 2.21/1.44 ( powersetI1
% 2.21/1.44 = ( ! [A: $i,B: $i] :
% 2.21/1.44 ( ( subset @ B @ A )
% 2.21/1.44 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(powersetE1_type,type,
% 2.21/1.44 powersetE1: $o ).
% 2.21/1.44
% 2.21/1.44 thf(powersetE1,definition,
% 2.21/1.44 ( powersetE1
% 2.21/1.44 = ( ! [A: $i,B: $i] :
% 2.21/1.44 ( ( in @ B @ ( powerset @ A ) )
% 2.21/1.44 => ( subset @ B @ A ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(inPowerset_type,type,
% 2.21/1.44 inPowerset: $o ).
% 2.21/1.44
% 2.21/1.44 thf(inPowerset,definition,
% 2.21/1.44 ( inPowerset
% 2.21/1.44 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(powersetsubset_type,type,
% 2.21/1.44 powersetsubset: $o ).
% 2.21/1.44
% 2.21/1.44 thf(powersetsubset,definition,
% 2.21/1.44 ( powersetsubset
% 2.21/1.44 = ( ! [A: $i,B: $i] :
% 2.21/1.44 ( ( subset @ A @ B )
% 2.21/1.44 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(sepInPowerset_type,type,
% 2.21/1.44 sepInPowerset: $o ).
% 2.21/1.44
% 2.21/1.44 thf(sepInPowerset,definition,
% 2.21/1.44 ( sepInPowerset
% 2.21/1.44 = ( ! [A: $i,Xphi: $i > $o] :
% 2.21/1.44 ( in
% 2.21/1.44 @ ( dsetconstr @ A
% 2.21/1.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.21/1.44 @ ( powerset @ A ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(sepSubset_type,type,
% 2.21/1.44 sepSubset: $o ).
% 2.21/1.44
% 2.21/1.44 thf(sepSubset,definition,
% 2.21/1.44 ( sepSubset
% 2.21/1.44 = ( ! [A: $i,Xphi: $i > $o] :
% 2.21/1.44 ( subset
% 2.21/1.44 @ ( dsetconstr @ A
% 2.21/1.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.21/1.44 @ A ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(binunion_type,type,
% 2.21/1.44 binunion: $i > $i > $i ).
% 2.21/1.44
% 2.21/1.44 thf(binunionIL_type,type,
% 2.21/1.44 binunionIL: $o ).
% 2.21/1.44
% 2.21/1.44 thf(binunionIL,definition,
% 2.21/1.44 ( binunionIL
% 2.21/1.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ A )
% 2.21/1.44 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(upairset2IR_type,type,
% 2.21/1.44 upairset2IR: $o ).
% 2.21/1.44
% 2.21/1.44 thf(upairset2IR,definition,
% 2.21/1.44 ( upairset2IR
% 2.21/1.44 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(binunionIR_type,type,
% 2.21/1.44 binunionIR: $o ).
% 2.21/1.44
% 2.21/1.44 thf(binunionIR,definition,
% 2.21/1.44 ( binunionIR
% 2.21/1.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ B )
% 2.21/1.44 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(binunionEcases_type,type,
% 2.21/1.44 binunionEcases: $o ).
% 2.21/1.44
% 2.21/1.44 thf(binunionEcases,definition,
% 2.21/1.44 ( binunionEcases
% 2.21/1.44 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 2.21/1.44 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.21/1.44 => ( ( ( in @ Xx @ A )
% 2.21/1.44 => Xphi )
% 2.21/1.44 => ( ( ( in @ Xx @ B )
% 2.21/1.44 => Xphi )
% 2.21/1.44 => Xphi ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(binunionE_type,type,
% 2.21/1.44 binunionE: $o ).
% 2.21/1.44
% 2.21/1.44 thf(binunionE,definition,
% 2.21/1.44 ( binunionE
% 2.21/1.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.21/1.44 => ( ( in @ Xx @ A )
% 2.21/1.44 | ( in @ Xx @ B ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(binunionLsub_type,type,
% 2.21/1.44 binunionLsub: $o ).
% 2.21/1.44
% 2.21/1.44 thf(binunionLsub,definition,
% 2.21/1.44 ( binunionLsub
% 2.21/1.44 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(binunionRsub_type,type,
% 2.21/1.44 binunionRsub: $o ).
% 2.21/1.44
% 2.21/1.44 thf(binunionRsub,definition,
% 2.21/1.44 ( binunionRsub
% 2.21/1.44 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(binintersect_type,type,
% 2.21/1.44 binintersect: $i > $i > $i ).
% 2.21/1.44
% 2.21/1.44 thf(binintersect,definition,
% 2.21/1.44 ( binintersect
% 2.21/1.44 = ( ^ [A: $i,B: $i] :
% 2.21/1.44 ( dsetconstr @ A
% 2.21/1.44 @ ^ [Xx: $i] : ( in @ Xx @ B ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(binintersectI_type,type,
% 2.21/1.44 binintersectI: $o ).
% 2.21/1.44
% 2.21/1.44 thf(binintersectI,definition,
% 2.21/1.44 ( binintersectI
% 2.21/1.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ A )
% 2.21/1.44 => ( ( in @ Xx @ B )
% 2.21/1.44 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(binintersectSubset5_type,type,
% 2.21/1.44 binintersectSubset5: $o ).
% 2.21/1.44
% 2.21/1.44 thf(binintersectSubset5,definition,
% 2.21/1.44 ( binintersectSubset5
% 2.21/1.44 = ( ! [A: $i,B: $i,C: $i] :
% 2.21/1.44 ( ( subset @ C @ A )
% 2.21/1.44 => ( ( subset @ C @ B )
% 2.21/1.44 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(binintersectEL_type,type,
% 2.21/1.44 binintersectEL: $o ).
% 2.21/1.44
% 2.21/1.44 thf(binintersectEL,definition,
% 2.21/1.44 ( binintersectEL
% 2.21/1.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.21/1.44 => ( in @ Xx @ A ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(binintersectLsub_type,type,
% 2.21/1.44 binintersectLsub: $o ).
% 2.21/1.44
% 2.21/1.44 thf(binintersectLsub,definition,
% 2.21/1.44 ( binintersectLsub
% 2.21/1.44 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(binintersectSubset2_type,type,
% 2.21/1.44 binintersectSubset2: $o ).
% 2.21/1.44
% 2.21/1.44 thf(binintersectSubset2,definition,
% 2.21/1.44 ( binintersectSubset2
% 2.21/1.44 = ( ! [A: $i,B: $i] :
% 2.21/1.44 ( ( subset @ A @ B )
% 2.21/1.44 => ( ( binintersect @ A @ B )
% 2.21/1.44 = A ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(binintersectSubset3,conjecture,
% 2.21/1.44 ( setextAx
% 2.21/1.44 => ( emptysetAx
% 2.21/1.44 => ( setadjoinAx
% 2.21/1.44 => ( powersetAx
% 2.21/1.44 => ( setunionAx
% 2.21/1.44 => ( omega0Ax
% 2.21/1.44 => ( omegaSAx
% 2.21/1.44 => ( omegaIndAx
% 2.21/1.44 => ( replAx
% 2.21/1.44 => ( foundationAx
% 2.21/1.44 => ( wellorderingAx
% 2.21/1.44 => ( descrp
% 2.21/1.44 => ( dsetconstrI
% 2.21/1.44 => ( dsetconstrEL
% 2.21/1.44 => ( dsetconstrER
% 2.21/1.44 => ( exuE1
% 2.21/1.44 => ( prop2setE
% 2.21/1.44 => ( emptysetE
% 2.21/1.44 => ( emptysetimpfalse
% 2.21/1.44 => ( notinemptyset
% 2.21/1.44 => ( exuE3e
% 2.21/1.44 => ( setext
% 2.21/1.44 => ( emptyI
% 2.21/1.44 => ( noeltsimpempty
% 2.21/1.44 => ( setbeta
% 2.21/1.44 => ( nonemptyE1
% 2.21/1.44 => ( nonemptyI
% 2.21/1.44 => ( nonemptyI1
% 2.21/1.44 => ( setadjoinIL
% 2.21/1.44 => ( emptyinunitempty
% 2.21/1.44 => ( setadjoinIR
% 2.21/1.44 => ( setadjoinE
% 2.21/1.44 => ( setadjoinOr
% 2.21/1.44 => ( setoftrueEq
% 2.21/1.44 => ( powersetI
% 2.21/1.44 => ( emptyinPowerset
% 2.21/1.44 => ( emptyInPowerset
% 2.21/1.44 => ( powersetE
% 2.21/1.44 => ( setunionI
% 2.21/1.44 => ( setunionE
% 2.21/1.44 => ( subPowSU
% 2.21/1.44 => ( exuE2
% 2.21/1.44 => ( nonemptyImpWitness
% 2.21/1.44 => ( uniqinunit
% 2.21/1.44 => ( notinsingleton
% 2.21/1.44 => ( eqinunit
% 2.21/1.44 => ( singletonsswitch
% 2.21/1.44 => ( upairsetE
% 2.21/1.44 => ( upairsetIL
% 2.21/1.44 => ( upairsetIR
% 2.21/1.44 => ( emptyE1
% 2.21/1.44 => ( vacuousDall
% 2.21/1.44 => ( quantDeMorgan1
% 2.21/1.44 => ( quantDeMorgan2
% 2.21/1.44 => ( quantDeMorgan3
% 2.21/1.44 => ( quantDeMorgan4
% 2.21/1.44 => ( prop2setI
% 2.21/1.44 => ( prop2set2propI
% 2.21/1.44 => ( notdexE
% 2.21/1.44 => ( notdallE
% 2.21/1.44 => ( exuI1
% 2.21/1.44 => ( exuI3
% 2.21/1.44 => ( exuI2
% 2.21/1.44 => ( inCongP
% 2.21/1.44 => ( in__Cong
% 2.21/1.44 => ( exuE3u
% 2.21/1.44 => ( exu__Cong
% 2.21/1.44 => ( emptyset__Cong
% 2.21/1.44 => ( setadjoin__Cong
% 2.21/1.44 => ( powerset__Cong
% 2.21/1.44 => ( setunion__Cong
% 2.21/1.44 => ( omega__Cong
% 2.21/1.44 => ( exuEu
% 2.21/1.44 => ( descr__Cong
% 2.21/1.44 => ( dsetconstr__Cong
% 2.21/1.44 => ( subsetI1
% 2.21/1.44 => ( eqimpsubset2
% 2.21/1.44 => ( eqimpsubset1
% 2.21/1.44 => ( subsetI2
% 2.21/1.44 => ( emptysetsubset
% 2.21/1.44 => ( subsetE
% 2.21/1.44 => ( subsetE2
% 2.21/1.44 => ( notsubsetI
% 2.21/1.44 => ( notequalI1
% 2.21/1.44 => ( notequalI2
% 2.21/1.44 => ( subsetRefl
% 2.21/1.44 => ( subsetTrans
% 2.21/1.44 => ( setadjoinSub
% 2.21/1.44 => ( setadjoinSub2
% 2.21/1.44 => ( subset2powerset
% 2.21/1.44 => ( setextsub
% 2.21/1.44 => ( subsetemptysetimpeq
% 2.21/1.44 => ( powersetI1
% 2.21/1.44 => ( powersetE1
% 2.21/1.44 => ( inPowerset
% 2.21/1.44 => ( powersetsubset
% 2.21/1.44 => ( sepInPowerset
% 2.21/1.44 => ( sepSubset
% 2.21/1.44 => ( binunionIL
% 2.21/1.44 => ( upairset2IR
% 2.21/1.44 => ( binunionIR
% 2.21/1.44 => ( binunionEcases
% 2.21/1.44 => ( binunionE
% 2.21/1.44 => ( binunionLsub
% 2.21/1.44 => ( binunionRsub
% 2.21/1.44 => ( binintersectI
% 2.21/1.44 => ( binintersectSubset5
% 2.21/1.44 => ( binintersectEL
% 2.21/1.44 => ( binintersectLsub
% 2.21/1.44 => ( binintersectSubset2
% 2.21/1.44 => ! [A: $i,B: $i] :
% 2.21/1.44 ( ( ( binintersect @ A @ B )
% 2.21/1.44 = B )
% 2.21/1.44 => ( subset @ B @ A ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.21/1.44 ------------------------
% 2.21/1.45 % (24472)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_24245 for (2999ds/275Mi)
% 2.21/1.45 % (24468)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_24245 for (2999ds/4Mi)
% 2.21/1.45 % (24468)Instruction limit reached!
% 2.21/1.45 % (24468)------------------------------
% 2.21/1.45 % (24468)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.21/1.45 % (24468)Termination reason: Unknown
% 2.21/1.45 % (24468)Termination phase: shuffling
% 2.21/1.45
% 2.21/1.45 % (24468)Memory used [KB]: 1279
% 2.21/1.45 % (24468)Time elapsed: 0.003 s
% 2.21/1.45 % (24468)Instructions burned: 6 (million)
% 2.21/1.45 % (24468)------------------------------
% 2.21/1.45 % (24468)------------------------------
% 2.21/1.46 % (24467)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_24245 for (2999ds/183Mi)
% 2.21/1.46 % (24469)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_24245 for (2999ds/27Mi)
% 2.21/1.46 % (24470)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_24245 for (2999ds/2Mi)
% 2.21/1.46 % (24471)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_24245 for (2999ds/2Mi)
% 2.21/1.46 % (24473)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_24245 for (2999ds/18Mi)
% 2.21/1.46 % (24470)Instruction limit reached!
% 2.21/1.46 % (24470)------------------------------
% 2.21/1.46 % (24470)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.21/1.46 % (24470)Termination reason: Unknown
% 2.21/1.46 % (24470)Termination phase: shuffling
% 2.21/1.46
% 2.21/1.46 % (24470)Memory used [KB]: 1151
% 2.21/1.46 % (24470)Time elapsed: 0.004 s
% 2.21/1.46 % (24470)Instructions burned: 3 (million)
% 2.21/1.46 % (24470)------------------------------
% 2.21/1.46 % (24470)------------------------------
% 2.21/1.46 % (24471)Instruction limit reached!
% 2.21/1.46 % (24471)------------------------------
% 2.21/1.46 % (24471)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.21/1.46 % (24471)Termination reason: Unknown
% 2.21/1.46 % (24471)Termination phase: shuffling
% 2.21/1.46
% 2.21/1.46 % (24471)Memory used [KB]: 1151
% 2.21/1.46 % (24471)Time elapsed: 0.004 s
% 2.21/1.46 % (24471)Instructions burned: 3 (million)
% 2.21/1.46 % (24471)------------------------------
% 2.21/1.46 % (24471)------------------------------
% 2.21/1.47 % (24476)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_24245 for (2999ds/3Mi)
% 2.21/1.47 % (24476)Instruction limit reached!
% 2.21/1.47 % (24476)------------------------------
% 2.21/1.47 % (24476)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.21/1.47 % (24476)Termination reason: Unknown
% 2.21/1.47 % (24476)Termination phase: shuffling
% 2.21/1.47
% 2.21/1.47 % (24476)Memory used [KB]: 1151
% 2.21/1.47 % (24476)Time elapsed: 0.002 s
% 2.21/1.47 % (24476)Instructions burned: 4 (million)
% 2.21/1.47 % (24476)------------------------------
% 2.21/1.47 % (24476)------------------------------
% 2.21/1.47 % (24473)Instruction limit reached!
% 2.21/1.47 % (24473)------------------------------
% 2.21/1.47 % (24473)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.21/1.47 % (24473)Termination reason: Unknown
% 2.21/1.47 % (24473)Termination phase: Preprocessing 3
% 2.21/1.47
% 2.21/1.47 % (24473)Memory used [KB]: 1535
% 2.21/1.47 % (24473)Time elapsed: 0.013 s
% 2.21/1.47 % (24473)Instructions burned: 18 (million)
% 2.21/1.47 % (24473)------------------------------
% 2.21/1.47 % (24473)------------------------------
% 2.21/1.47 % (24469)Instruction limit reached!
% 2.21/1.47 % (24469)------------------------------
% 2.21/1.47 % (24469)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.21/1.47 % (24469)Termination reason: Unknown
% 2.21/1.47 % (24469)Termination phase: Property scanning
% 2.21/1.47
% 2.21/1.47 % (24469)Memory used [KB]: 1535
% 2.21/1.47 % (24469)Time elapsed: 0.016 s
% 2.21/1.47 % (24469)Instructions burned: 27 (million)
% 2.21/1.47 % (24469)------------------------------
% 2.21/1.47 % (24469)------------------------------
% 2.21/1.48 % (24486)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_24245 for (2999ds/3Mi)
% 2.21/1.48 % (24486)Instruction limit reached!
% 2.21/1.48 % (24486)------------------------------
% 2.21/1.48 % (24486)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.21/1.48 % (24486)Termination reason: Unknown
% 2.21/1.48 % (24486)Termination phase: shuffling
% 2.21/1.48
% 2.21/1.48 % (24486)Memory used [KB]: 1151
% 2.21/1.48 % (24486)Time elapsed: 0.002 s
% 2.21/1.48 % (24486)Instructions burned: 4 (million)
% 2.21/1.48 % (24486)------------------------------
% 2.21/1.48 % (24486)------------------------------
% 2.21/1.48 % (24480)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_24245 for (2999ds/37Mi)
% 2.21/1.48 % (24481)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_24245 for (2999ds/15Mi)
% 2.21/1.49 % (24490)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_24245 for (2999ds/16Mi)
% 2.21/1.49 % (24481)Instruction limit reached!
% 2.21/1.49 % (24481)------------------------------
% 2.21/1.49 % (24481)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.21/1.49 % (24481)Termination reason: Unknown
% 2.21/1.49 % (24481)Termination phase: Naming
% 2.21/1.49
% 2.21/1.49 % (24481)Memory used [KB]: 1407
% 2.21/1.49 % (24481)Time elapsed: 0.011 s
% 2.21/1.49 % (24481)Instructions burned: 15 (million)
% 2.21/1.49 % (24481)------------------------------
% 2.21/1.49 % (24481)------------------------------
% 2.21/1.49 % (24488)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_24245 for (2999ds/1041Mi)
% 2.21/1.49 % (24489)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_24245 for (2999ds/7Mi)
% 2.21/1.49 % (24490)Instruction limit reached!
% 2.21/1.49 % (24490)------------------------------
% 2.21/1.49 % (24490)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.21/1.49 % (24490)Termination reason: Unknown
% 2.21/1.49 % (24490)Termination phase: Property scanning
% 2.21/1.49
% 2.21/1.49 % (24490)Memory used [KB]: 1535
% 2.21/1.49 % (24490)Time elapsed: 0.007 s
% 2.21/1.49 % (24490)Instructions burned: 17 (million)
% 2.21/1.49 % (24490)------------------------------
% 2.21/1.49 % (24490)------------------------------
% 2.21/1.50 % (24489)Instruction limit reached!
% 2.21/1.50 % (24489)------------------------------
% 2.21/1.50 % (24489)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.21/1.50 % (24489)Termination reason: Unknown
% 2.21/1.50 % (24489)Termination phase: shuffling
% 2.21/1.50
% 2.21/1.50 % (24489)Memory used [KB]: 1279
% 2.21/1.50 % (24489)Time elapsed: 0.006 s
% 2.21/1.50 % (24489)Instructions burned: 8 (million)
% 2.21/1.50 % (24489)------------------------------
% 2.21/1.50 % (24489)------------------------------
% 2.27/1.50 % (24480)Instruction limit reached!
% 2.27/1.50 % (24480)------------------------------
% 2.27/1.50 % (24480)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.27/1.50 % (24480)Termination reason: Unknown
% 2.27/1.50 % (24480)Termination phase: Property scanning
% 2.27/1.50
% 2.27/1.50 % (24480)Memory used [KB]: 1535
% 2.27/1.50 % (24480)Time elapsed: 0.020 s
% 2.27/1.50 % (24480)Instructions burned: 37 (million)
% 2.27/1.50 % (24480)------------------------------
% 2.27/1.50 % (24480)------------------------------
% 2.27/1.50 % (24503)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_24245 for (2999ds/3Mi)
% 2.27/1.50 % (24503)Instruction limit reached!
% 2.27/1.50 % (24503)------------------------------
% 2.27/1.50 % (24503)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.27/1.50 % (24503)Termination reason: Unknown
% 2.27/1.50 % (24503)Termination phase: shuffling
% 2.27/1.50
% 2.27/1.50 % (24503)Memory used [KB]: 1151
% 2.27/1.50 % (24503)Time elapsed: 0.003 s
% 2.27/1.50 % (24503)Instructions burned: 4 (million)
% 2.27/1.50 % (24503)------------------------------
% 2.27/1.50 % (24503)------------------------------
% 2.27/1.51 % (24499)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_24245 for (2999ds/3Mi)
% 2.27/1.51 % (24499)Instruction limit reached!
% 2.27/1.51 % (24499)------------------------------
% 2.27/1.51 % (24499)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.27/1.51 % (24499)Termination reason: Unknown
% 2.27/1.51 % (24499)Termination phase: shuffling
% 2.27/1.51
% 2.27/1.51 % (24499)Memory used [KB]: 1151
% 2.27/1.51 % (24499)Time elapsed: 0.004 s
% 2.27/1.51 % (24499)Instructions burned: 4 (million)
% 2.27/1.51 % (24499)------------------------------
% 2.27/1.51 % (24499)------------------------------
% 2.27/1.51 % (24472)First to succeed.
% 2.27/1.51 % (24505)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_24245 for (2999ds/7Mi)
% 2.27/1.51 % (24472)Refutation found. Thanks to Tanya!
% 2.27/1.51 % SZS status Theorem for DTF2THF_24245
% 2.27/1.51 % SZS output start Proof for DTF2THF_24245
% 2.27/1.51 thf(func_def_0, type, in: $i > $i > $o).
% 2.27/1.51 thf(func_def_1, type, exu: ($i > $o) > $o).
% 2.27/1.51 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 2.27/1.51 thf(func_def_8, type, powerset: $i > $i).
% 2.27/1.51 thf(func_def_10, type, setunion: $i > $i).
% 2.27/1.51 thf(func_def_19, type, descr: ($i > $o) > $i).
% 2.27/1.51 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 2.27/1.51 thf(func_def_26, type, prop2set: $o > $i).
% 2.27/1.51 thf(func_def_36, type, nonempty: $i > $o).
% 2.27/1.51 thf(func_def_69, type, set2prop: $i > $o).
% 2.27/1.51 thf(func_def_88, type, subset: $i > $i > $o).
% 2.27/1.51 thf(func_def_89, type, disjoint: $i > $i > $o).
% 2.27/1.51 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 2.27/1.51 thf(func_def_114, type, binunion: $i > $i > $i).
% 2.27/1.51 thf(func_def_122, type, binintersect: $i > $i > $i).
% 2.27/1.51 thf(func_def_140, type, sP0: $i > $i > $o).
% 2.27/1.51 thf(func_def_141, type, sP1: $i > $o).
% 2.27/1.51 thf(func_def_142, type, sP2: $i > $i > $o).
% 2.27/1.51 thf(func_def_143, type, sP3: $i > $i > $o).
% 2.27/1.51 thf(func_def_148, type, sK8: $i > $o).
% 2.27/1.51 thf(func_def_150, type, sK10: ($i > $o) > $i > $i).
% 2.27/1.51 thf(func_def_157, type, sK17: $i > $o).
% 2.27/1.51 thf(func_def_159, type, sK19: $i > $o).
% 2.27/1.51 thf(func_def_165, type, sK25: $i > $o).
% 2.27/1.51 thf(func_def_166, type, sK26: $i > $i > $i).
% 2.27/1.51 thf(func_def_175, type, sK35: $i > $o).
% 2.27/1.51 thf(func_def_176, type, sK36: $i > $i).
% 2.27/1.51 thf(func_def_177, type, sK37: ($i > $o) > $i).
% 2.27/1.51 thf(func_def_179, type, sK39: $i > $i).
% 2.27/1.51 thf(func_def_181, type, sK41: $i > $o).
% 2.27/1.51 thf(func_def_189, type, sK49: $i > $i > $i).
% 2.27/1.51 thf(func_def_190, type, sK50: $i > $i > $i).
% 2.27/1.51 thf(func_def_191, type, sK51: $i > $i > $i).
% 2.27/1.51 thf(func_def_192, type, sK52: $i > $i > $i).
% 2.27/1.51 thf(func_def_193, type, sK53: $i > $i > $i > $i).
% 2.27/1.51 thf(func_def_194, type, sK54: $i > $i).
% 2.27/1.51 thf(func_def_195, type, sK55: $i > $i).
% 2.27/1.51 thf(func_def_196, type, sK56: $i > $i).
% 2.27/1.51 thf(func_def_197, type, sK57: $i > $i).
% 2.27/1.51 thf(func_def_198, type, sK58: $i > $i > $i).
% 2.27/1.51 thf(func_def_199, type, sK59: $i > $i > $i > $i).
% 2.27/1.51 thf(func_def_200, type, sK60: $i > $i > $i > $i).
% 2.27/1.51 thf(func_def_201, type, sK61: $i > $i > $i).
% 2.27/1.51 thf(func_def_202, type, sK62: $i > $i > $i).
% 2.27/1.51 thf(func_def_203, type, sK63: $i > $i).
% 2.27/1.51 thf(func_def_204, type, sK64: $i > $i > $i).
% 2.27/1.51 thf(func_def_206, type, sK66: $i > $i).
% 2.27/1.51 thf(func_def_207, type, sK67: $i > $i).
% 2.27/1.51 thf(func_def_208, type, sK68: $i > $o).
% 2.27/1.51 thf(func_def_210, type, sK70: $i > ($i > $o) > $i).
% 2.27/1.51 thf(func_def_216, type, sK76: $i > $i > $i).
% 2.27/1.51 thf(func_def_217, type, sK77: $i > $i > $o).
% 2.27/1.51 thf(func_def_219, type, sK79: $i > $i).
% 2.27/1.51 thf(func_def_220, type, sK80: $i > $i).
% 2.27/1.51 thf(func_def_221, type, sK81: $i > ($i > $i > $o) > $i).
% 2.27/1.51 thf(func_def_222, type, sK82: $i > ($i > $i > $o) > $i).
% 2.27/1.51 thf(func_def_223, type, sK83: $i > $i > ($i > $i > $o) > $i).
% 2.27/1.51 thf(func_def_231, type, sK91: $i > $o).
% 2.27/1.51 thf(func_def_233, type, sK93: ($i > $o) > ($i > $o) > $i).
% 2.27/1.51 thf(func_def_234, type, sK94: ($i > $o) > ($i > $o) > $i).
% 2.27/1.51 thf(func_def_235, type, sK95: $i > $o).
% 2.27/1.51 thf(func_def_236, type, sK96: $i > $o).
% 2.27/1.51 thf(func_def_238, type, sK98: $i > $i).
% 2.27/1.51 thf(func_def_239, type, sK99: $i > $i).
% 2.27/1.51 thf(func_def_246, type, sK106: $i > $i > $i).
% 2.27/1.51 thf(func_def_247, type, sK107: $i > $i).
% 2.27/1.51 thf(func_def_250, type, sK110: $i > $i).
% 2.27/1.51 thf(func_def_251, type, sK111: ($i > $o) > $i).
% 2.27/1.51 thf(func_def_252, type, sK112: ($i > $o) > $i).
% 2.27/1.51 thf(func_def_253, type, sK113: $i > $o).
% 2.27/1.51 thf(func_def_258, type, sK118: $i > $o).
% 2.27/1.51 thf(func_def_264, type, sK124: $i > $o).
% 2.27/1.51 thf(func_def_271, type, sK131: $i > ($i > $o) > $i).
% 2.27/1.51 thf(func_def_272, type, sK132: $i > $o).
% 2.27/1.51 thf(func_def_278, type, sK138: $i > $i > $i).
% 2.27/1.51 thf(func_def_290, type, sK150: $i > $i > $i).
% 2.27/1.51 thf(func_def_300, type, sK160: $o > $i > $i > $i).
% 2.27/1.51 thf(func_def_302, type, sK162: $i > $o).
% 2.27/1.51 thf(func_def_311, type, sK171: $i > $o).
% 2.27/1.51 thf(func_def_313, type, sK173: ($i > $o) > $i > $i).
% 2.27/1.51 thf(func_def_318, type, sK178: ($i > $o) > $i > $i).
% 2.27/1.51 thf(func_def_320, type, sK180: $i > $o).
% 2.27/1.51 thf(func_def_327, type, sK187: $i > $o).
% 2.27/1.51 thf(func_def_331, type, sK191: $i > $i).
% 2.27/1.51 thf(func_def_354, type, sK214: $i > $i > $i).
% 2.27/1.51 thf(func_def_355, type, sK215: $i > $i > $i).
% 2.27/1.51 thf(func_def_364, type, sK224: $i > $o).
% 2.27/1.51 thf(func_def_367, type, sK227: $i > ($i > $o) > $i).
% 2.27/1.51 thf(func_def_370, type, sK230: $i > $o).
% 2.27/1.51 thf(func_def_372, type, sK232: $i > ($i > $o) > $i).
% 2.27/1.51 thf(func_def_377, type, sK237: $i > $o).
% 2.27/1.51 thf(func_def_384, type, sK244: ($i > $o) > $i).
% 2.27/1.51 thf(func_def_385, type, sK245: $i > $o).
% 2.27/1.51 thf(func_def_386, type, sK246: $i > $i).
% 2.27/1.51 thf(func_def_387, type, sK247: $i > $i).
% 2.27/1.51 thf(func_def_394, type, sK254: $i > $i > $i).
% 2.27/1.51 thf(func_def_404, type, sK264: $i > $o).
% 2.27/1.51 thf(func_def_408, type, sK268: $i > $o).
% 2.27/1.51 thf(func_def_409, type, sK269: $i > $o).
% 2.27/1.51 thf(func_def_410, type, sK270: ($i > $o) > ($i > $o) > $i).
% 2.27/1.51 thf(func_def_411, type, sK271: ($i > $o) > ($i > $o) > $i).
% 2.27/1.51 thf(func_def_417, type, sK277: $i > $o).
% 2.27/1.51 thf(func_def_418, type, sK278: ($i > $o) > $i).
% 2.27/1.51 thf(func_def_419, type, sK279: $i > $o).
% 2.27/1.51 thf(func_def_422, type, sK282: $i > ($i > $o) > $i).
% 2.27/1.51 thf(func_def_423, type, sK283: $i > $o).
% 2.27/1.51 thf(func_def_428, type, sK288: $i > $o).
% 2.27/1.51 thf(func_def_429, type, sK289: $i > $o).
% 2.27/1.51 thf(func_def_430, type, sK290: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.27/1.51 thf(func_def_431, type, sK291: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.27/1.51 thf(func_def_444, type, ph304: !>[X0: $tType]:(X0)).
% 2.27/1.51 thf(f2460,plain,(
% 2.27/1.51 $false),
% 2.27/1.51 inference(subsumption_resolution,[status(thm)],[f2457,f1260])).
% 2.27/1.51 thf(f1260,plain,(
% 2.27/1.51 ($true != (subset @ sK151 @ sK152))),
% 2.27/1.51 inference(cnf_transformation,[status(thm)],[f743])).
% 2.27/1.51 thf(f743,plain,(
% 2.27/1.51 (subsetemptysetimpeq = $true) & (foundationAx = $true) & (notinemptyset = $true) & (powersetAx = $true) & (noeltsimpempty = $true) & (singletonsswitch = $true) & (nonemptyImpWitness = $true) & (emptyset__Cong = $true) & (prop2set2propI = $true) & (emptyinPowerset = $true) & (upairsetIL = $true) & (exuE3e = $true) & (sepSubset = $true) & (notequalI1 = $true) & (setunionAx = $true) & (powersetI1 = $true) & (setext = $true) & (subsetE = $true) & (exu__Cong = $true) & (binunionEcases = $true) & (exuE1 = $true) & (upairsetIR = $true) & (wellorderingAx = $true) & (binunionRsub = $true) & (inCongP = $true) & (quantDeMorgan3 = $true) & (quantDeMorgan4 = $true) & (binintersectSubset2 = $true) & (emptyInPowerset = $true) & (emptysetAx = $true) & (quantDeMorgan1 = $true) & (setadjoin__Cong = $true) & (omega0Ax = $true) & (binintersectEL = $true) & (dsetconstr__Cong = $true) & (setunionI = $true) & (omegaIndAx = $true) & (dsetconstrEL = $true) & (descrp = $true) & (notdallE = $true) & (exuI2 = $true) & (setextAx = $true) & (prop2setE = $true) & (eqimpsubset1 = $true) & (nonemptyI = $true) & (binunionLsub = $true) & (setadjoinAx = $true) & (setadjoinSub2 = $true) & (powersetE1 = $true) & (binunionE = $true) & (setadjoinIR = $true) & (emptysetimpfalse = $true) & (notsubsetI = $true) & (subsetRefl = $true) & (binunionIL = $true) & (nonemptyI1 = $true) & (subsetI2 = $true) & (upairset2IR = $true) & (powersetI = $true) & (subset2powerset = $true) & (eqinunit = $true) & (emptysetE = $true) & (setunionE = $true) & (upairsetE = $true) & (powerset__Cong = $true) & (setadjoinSub = $true) & (emptyE1 = $true) & (sepInPowerset = $true) & (setbeta = $true) & (binintersectLsub = $true) & (subsetE2 = $true) & (exuI3 = $true) & (subsetI1 = $true) & (nonemptyE1 = $true) & (exuEu = $true) & ((sK151 = (binintersect @ sK152 @ sK151)) & ($true != (subset @ sK151 @ sK152))) & (inPowerset = $true) & (eqimpsubset2 = $true) & (subsetTrans = $true) & (dsetconstrI = $true) & (emptysetsubset = $true) & (binunionIR = $true) & (omega__Cong = $true) & (exuI1 = $true) & (powersetsubset = $true) & (in__Cong = $true) & (powersetE = $true) & (exuE3u = $true) & (setunion__Cong = $true) & (prop2setI = $true) & (vacuousDall = $true) & (subPowSU = $true) & (binintersectI = $true) & (exuE2 = $true) & (replAx = $true) & (notdexE = $true) & (emptyI = $true) & (uniqinunit = $true) & (quantDeMorgan2 = $true) & (setadjoinE = $true) & (binintersectSubset5 = $true) & (setextsub = $true) & (setadjoinOr = $true) & (dsetconstrER = $true) & (omegaSAx = $true) & (emptyinunitempty = $true) & (notinsingleton = $true) & (notequalI2 = $true) & (setadjoinIL = $true) & (descr__Cong = $true) & (setoftrueEq = $true)),
% 2.27/1.51 inference(skolemisation,[status(esa),new_symbols(skolem,[sK151,sK152])],[f412,f742])).
% 2.27/1.51 thf(f742,plain,(
% 2.27/1.51 ? [X0 : $i,X1 : $i] : (((binintersect @ X1 @ X0) = X0) & ($true != (subset @ X0 @ X1))) => ((sK151 = (binintersect @ sK152 @ sK151)) & ($true != (subset @ sK151 @ sK152)))),
% 2.27/1.51 introduced(definition,[],[choice_axiom])).
% 2.27/1.51 thf(f412,plain,(
% 2.27/1.51 (subsetemptysetimpeq = $true) & (foundationAx = $true) & (notinemptyset = $true) & (powersetAx = $true) & (noeltsimpempty = $true) & (singletonsswitch = $true) & (nonemptyImpWitness = $true) & (emptyset__Cong = $true) & (prop2set2propI = $true) & (emptyinPowerset = $true) & (upairsetIL = $true) & (exuE3e = $true) & (sepSubset = $true) & (notequalI1 = $true) & (setunionAx = $true) & (powersetI1 = $true) & (setext = $true) & (subsetE = $true) & (exu__Cong = $true) & (binunionEcases = $true) & (exuE1 = $true) & (upairsetIR = $true) & (wellorderingAx = $true) & (binunionRsub = $true) & (inCongP = $true) & (quantDeMorgan3 = $true) & (quantDeMorgan4 = $true) & (binintersectSubset2 = $true) & (emptyInPowerset = $true) & (emptysetAx = $true) & (quantDeMorgan1 = $true) & (setadjoin__Cong = $true) & (omega0Ax = $true) & (binintersectEL = $true) & (dsetconstr__Cong = $true) & (setunionI = $true) & (omegaIndAx = $true) & (dsetconstrEL = $true) & (descrp = $true) & (notdallE = $true) & (exuI2 = $true) & (setextAx = $true) & (prop2setE = $true) & (eqimpsubset1 = $true) & (nonemptyI = $true) & (binunionLsub = $true) & (setadjoinAx = $true) & (setadjoinSub2 = $true) & (powersetE1 = $true) & (binunionE = $true) & (setadjoinIR = $true) & (emptysetimpfalse = $true) & (notsubsetI = $true) & (subsetRefl = $true) & (binunionIL = $true) & (nonemptyI1 = $true) & (subsetI2 = $true) & (upairset2IR = $true) & (powersetI = $true) & (subset2powerset = $true) & (eqinunit = $true) & (emptysetE = $true) & (setunionE = $true) & (upairsetE = $true) & (powerset__Cong = $true) & (setadjoinSub = $true) & (emptyE1 = $true) & (sepInPowerset = $true) & (setbeta = $true) & (binintersectLsub = $true) & (subsetE2 = $true) & (exuI3 = $true) & (subsetI1 = $true) & (nonemptyE1 = $true) & (exuEu = $true) & ? [X0 : $i,X1 : $i] : (((binintersect @ X1 @ X0) = X0) & ($true != (subset @ X0 @ X1))) & (inPowerset = $true) & (eqimpsubset2 = $true) & (subsetTrans = $true) & (dsetconstrI = $true) & (emptysetsubset = $true) & (binunionIR = $true) & (omega__Cong = $true) & (exuI1 = $true) & (powersetsubset = $true) & (in__Cong = $true) & (powersetE = $true) & (exuE3u = $true) & (setunion__Cong = $true) & (prop2setI = $true) & (vacuousDall = $true) & (subPowSU = $true) & (binintersectI = $true) & (exuE2 = $true) & (replAx = $true) & (notdexE = $true) & (emptyI = $true) & (uniqinunit = $true) & (quantDeMorgan2 = $true) & (setadjoinE = $true) & (binintersectSubset5 = $true) & (setextsub = $true) & (setadjoinOr = $true) & (dsetconstrER = $true) & (omegaSAx = $true) & (emptyinunitempty = $true) & (notinsingleton = $true) & (notequalI2 = $true) & (setadjoinIL = $true) & (descr__Cong = $true) & (setoftrueEq = $true)),
% 2.27/1.51 inference(flattening,[status(thm)],[f411])).
% 2.27/1.51 thf(f411,plain,(
% 2.27/1.51 (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i,X1 : $i] : (((binintersect @ X1 @ X0) = X0) & ($true != (subset @ X0 @ X1))) & (binintersectSubset2 = $true)) & (binintersectLsub = $true)) & (binintersectEL = $true)) & (binintersectSubset5 = $true)) & (binintersectI = $true)) & (binunionRsub = $true)) & (binunionLsub = $true)) & (binunionE = $true)) & (binunionEcases = $true)) & (binunionIR = $true)) & (upairset2IR = $true)) & (binunionIL = $true)) & (sepSubset = $true)) & (sepInPowerset = $true)) & (powersetsubset = $true)) & (inPowerset = $true)) & (powersetE1 = $true)) & (powersetI1 = $true)) & (subsetemptysetimpeq = $true)) & (setextsub = $true)) & (subset2powerset = $true)) & (setadjoinSub2 = $true)) & (setadjoinSub = $true)) & (subsetTrans = $true)) & (subsetRefl = $true)) & (notequalI2 = $true)) & (notequalI1 = $true)) & (notsubsetI = $true)) & (subsetE2 = $true)) & (subsetE = $true)) & (emptysetsubset = $true)) & (subsetI2 = $true)) & (eqimpsubset1 = $true)) & (eqimpsubset2 = $true)) & (subsetI1 = $true)) & (dsetconstr__Cong = $true)) & (descr__Cong = $true)) & (exuEu = $true)) & (omega__Cong = $true)) & (setunion__Cong = $true)) & (powerset__Cong = $true)) & (setadjoin__Cong = $true)) & (emptyset__Cong = $true)) & (exu__Cong = $true)) & (exuE3u = $true)) & (in__Cong = $true)) & (inCongP = $true)) & (exuI2 = $true)) & (exuI3 = $true)) & (exuI1 = $true)) & (notdallE = $true)) & (notdexE = $true)) & (prop2set2propI = $true)) & (prop2setI = $true)) & (quantDeMorgan4 = $true)) & (quantDeMorgan3 = $true)) & (quantDeMorgan2 = $true)) & (quantDeMorgan1 = $true)) & (vacuousDall = $true)) & (emptyE1 = $true)) & (upairsetIR = $true)) & (upairsetIL = $true)) & (upairsetE = $true)) & (singletonsswitch = $true)) & (eqinunit = $true)) & (notinsingleton = $true)) & (uniqinunit = $true)) & (nonemptyImpWitness = $true)) & (exuE2 = $true)) & (subPowSU = $true)) & (setunionE = $true)) & (setunionI = $true)) & (powersetE = $true)) & (emptyInPowerset = $true)) & (emptyinPowerset = $true)) & (powersetI = $true)) & (setoftrueEq = $true)) & (setadjoinOr = $true)) & (setadjoinE = $true)) & (setadjoinIR = $true)) & (emptyinunitempty = $true)) & (setadjoinIL = $true)) & (nonemptyI1 = $true)) & (nonemptyI = $true)) & (nonemptyE1 = $true)) & (setbeta = $true)) & (noeltsimpempty = $true)) & (emptyI = $true)) & (setext = $true)) & (exuE3e = $true)) & (notinemptyset = $true)) & (emptysetimpfalse = $true)) & (emptysetE = $true)) & (prop2setE = $true)) & (exuE1 = $true)) & (dsetconstrER = $true)) & (dsetconstrEL = $true)) & (dsetconstrI = $true)) & (descrp = $true)) & (wellorderingAx = $true)) & (foundationAx = $true)) & (replAx = $true)) & (omegaIndAx = $true)) & (omegaSAx = $true)) & (omega0Ax = $true)) & (setunionAx = $true)) & (powersetAx = $true)) & (setadjoinAx = $true)) & (emptysetAx = $true)) & (setextAx = $true)),
% 2.27/1.51 inference(ennf_transformation,[status(thm)],[f134])).
% 2.27/1.51 thf(f134,plain,(
% 2.27/1.51 ~((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) => ! [X0 : $i,X1 : $i] : (((binintersect @ X1 @ X0) = X0) => ($true = (subset @ X0 @ X1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.27/1.51 inference(fool_elimination,[status(thm)],[f133])).
% 2.27/1.51 thf(f133,plain,(
% 2.27/1.51 ~(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 => ! [X0 : $i,X1 : $i] : (((binintersect @ X1 @ X0) = X0) => (subset @ X0 @ X1))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.27/1.51 inference(rectify,[status(thm)],[f115])).
% 2.27/1.51 thf(f115,negated_conjecture,(
% 2.27/1.51 ~(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 => ! [X4 : $i,X3 : $i] : (((binintersect @ X3 @ X4) = X4) => (subset @ X4 @ X3))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.27/1.51 inference(negated_conjecture,[status(cth)],[f114])).
% 2.27/1.51 thf(f114,conjecture,(
% 2.27/1.51 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 => ! [X4 : $i,X3 : $i] : (((binintersect @ X3 @ X4) = X4) => (subset @ X4 @ X3)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.27/1.51 file('/export/starexec/sandbox2/tmp/tmp.ncwJItUqjX/DTF2THF_24245.p',binintersectSubset3)).
% 2.27/1.51 thf(f2457,plain,(
% 2.27/1.51 ($true = (subset @ sK151 @ sK152))),
% 2.27/1.51 inference(superposition,[status(thm)],[f2010,f1261])).
% 2.27/1.51 thf(f1261,plain,(
% 2.27/1.51 (sK151 = (binintersect @ sK152 @ sK151))),
% 2.27/1.51 inference(cnf_transformation,[status(thm)],[f743])).
% 2.27/1.51 thf(f2010,plain,(
% 2.27/1.51 ( ! [X0 : $i,X1 : $i] : (($true = (subset @ (binintersect @ X0 @ X1) @ X0))) )),
% 2.27/1.51 inference(trivial_inequality_removal,[status(thm)],[f1790])).
% 2.27/1.51 thf(f1790,plain,(
% 2.27/1.51 ( ! [X0 : $i,X1 : $i] : (($true = (subset @ (binintersect @ X0 @ X1) @ X0)) | ($true != $true)) )),
% 2.27/1.51 inference(definition_unfolding,[status(thm)],[f1395,f1267])).
% 2.27/1.51 thf(f1267,plain,(
% 2.27/1.51 (binintersectLsub = $true)),
% 2.27/1.51 inference(cnf_transformation,[status(thm)],[f743])).
% 2.27/1.51 thf(f1395,plain,(
% 2.27/1.51 ( ! [X0 : $i,X1 : $i] : (($true = (subset @ (binintersect @ X0 @ X1) @ X0)) | (binintersectLsub != $true)) )),
% 2.27/1.51 inference(cnf_transformation,[status(thm)],[f813])).
% 2.27/1.51 thf(f813,plain,(
% 2.27/1.51 (! [X0 : $i,X1 : $i] : ($true = (subset @ (binintersect @ X0 @ X1) @ X0)) | (binintersectLsub != $true)) & ((binintersectLsub = $true) | ($true != (subset @ (binintersect @ sK194 @ sK195) @ sK194)))),
% 2.27/1.51 inference(skolemisation,[status(esa),new_symbols(skolem,[sK194,sK195])],[f811,f812])).
% 2.27/1.51 thf(f812,plain,(
% 2.27/1.51 ? [X2 : $i,X3 : $i] : ($true != (subset @ (binintersect @ X2 @ X3) @ X2)) => ($true != (subset @ (binintersect @ sK194 @ sK195) @ sK194))),
% 2.27/1.51 introduced(definition,[],[choice_axiom])).
% 2.27/1.51 thf(f811,plain,(
% 2.27/1.51 (! [X0 : $i,X1 : $i] : ($true = (subset @ (binintersect @ X0 @ X1) @ X0)) | (binintersectLsub != $true)) & ((binintersectLsub = $true) | ? [X2 : $i,X3 : $i] : ($true != (subset @ (binintersect @ X2 @ X3) @ X2)))),
% 2.27/1.51 inference(rectify,[status(thm)],[f810])).
% 2.27/1.51 thf(f810,plain,(
% 2.27/1.51 (! [X0 : $i,X1 : $i] : ($true = (subset @ (binintersect @ X0 @ X1) @ X0)) | (binintersectLsub != $true)) & ((binintersectLsub = $true) | ? [X0 : $i,X1 : $i] : ($true != (subset @ (binintersect @ X0 @ X1) @ X0)))),
% 2.27/1.51 inference(nnf_transformation,[status(thm)],[f220])).
% 2.27/1.51 thf(f220,plain,(
% 2.27/1.51 ! [X0 : $i,X1 : $i] : ($true = (subset @ (binintersect @ X0 @ X1) @ X0)) <=> (binintersectLsub = $true)),
% 2.27/1.51 inference(fool_elimination,[status(thm)],[f219])).
% 2.27/1.51 thf(f219,plain,(
% 2.27/1.51 (! [X0 : $i,X1 : $i] : (subset @ (binintersect @ X0 @ X1) @ X0) = binintersectLsub)),
% 2.27/1.51 inference(rectify,[status(thm)],[f112])).
% 2.27/1.51 thf(f112,axiom,(
% 2.27/1.51 (! [X3 : $i,X4 : $i] : (subset @ (binintersect @ X3 @ X4) @ X3) = binintersectLsub)),
% 2.27/1.51 file('/export/starexec/sandbox2/tmp/tmp.ncwJItUqjX/DTF2THF_24245.p',binintersectLsub)).
% 2.27/1.51 % SZS output end Proof for DTF2THF_24245
% 2.27/1.51 % (24472)------------------------------
% 2.27/1.51 % (24472)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.27/1.51 % (24472)Termination reason: Refutation
% 2.27/1.51
% 2.27/1.51 % (24472)Memory used [KB]: 7547
% 2.27/1.51 % (24472)Time elapsed: 0.060 s
% 2.27/1.51 % (24472)Instructions burned: 185 (million)
% 2.27/1.51 % (24472)------------------------------
% 2.27/1.51 % (24472)------------------------------
% 2.27/1.51 % (24462)Success in time 0.07 s
%------------------------------------------------------------------------------