%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU602^1 : 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 : n003.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 09:35:32 AM UTC 2026
% Result : Theorem 2.30s 1.78s
% Output : Refutation 2.30s
% Verified :
% SZS Type : Refutation
% Derivation depth : 19
% Number of leaves : 4
% Syntax : Number of formulae : 33 ( 23 unt; 0 typ; 0 def)
% Number of atoms : 1944 ( 510 equ; 0 cnn)
% Maximal formula atoms : 120 ( 58 avg)
% Number of connectives : 1099 ( 15 ~; 2 |; 359 &; 216 @)
% ( 0 <=>; 489 =>; 0 <=; 0 <~>)
% Maximal formula depth : 127 ( 28 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 20 ( 20 >; 0 *; 0 +; 0 <<)
% Number of symbols : 188 ( 184 usr; 124 con; 0-4 aty)
% ( 18 !!; 0 ??; 0 @@+; 0 @@-)
% Number of variables : 84 ( 42 ^; 33 !; 9 ?; 84 :)
% Comments :
%------------------------------------------------------------------------------
thf(func_def_0,type,
in: $i > $i > $o ).
thf(func_def_1,type,
exu: ( $i > $o ) > $o ).
thf(func_def_6,type,
setadjoin: $i > $i > $i ).
thf(func_def_8,type,
powerset: $i > $i ).
thf(func_def_10,type,
setunion: $i > $i ).
thf(func_def_19,type,
descr: ( $i > $o ) > $i ).
thf(func_def_21,type,
dsetconstr: $i > ( $i > $o ) > $i ).
thf(func_def_26,type,
prop2set: $o > $i ).
thf(func_def_36,type,
nonempty: $i > $o ).
thf(func_def_69,type,
set2prop: $i > $o ).
thf(func_def_88,type,
subset: $i > $i > $o ).
thf(func_def_89,type,
disjoint: $i > $i > $o ).
thf(func_def_90,type,
setsmeet: $i > $i > $o ).
thf(func_def_114,type,
binunion: $i > $i > $i ).
thf(func_def_122,type,
binintersect: $i > $i > $i ).
thf(func_def_135,type,
regular: $i > $o ).
thf(func_def_136,type,
setminus: $i > $i > $i ).
thf(func_def_159,type,
sK4: ( $i > $o ) > $i > $i ).
thf(func_def_160,type,
sK5: $i > $i > $i ).
thf(func_def_161,type,
sK6: $i > $i > $i ).
thf(func_def_162,type,
sK7: ( $i > $o ) > $i > $i ).
thf(func_def_163,type,
sK8: $i > $i ).
thf(func_def_164,type,
sK9: $i > $i ).
thf(func_def_165,type,
sK10: $i > $i > $i > $i ).
thf(func_def_166,type,
sK11: $i > $i > $i ).
thf(func_def_167,type,
sK12: $i > $i > $i ).
thf(func_def_168,type,
sK13: $i > $i > $i ).
thf(func_def_169,type,
sK14: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_170,type,
sK15: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_171,type,
sK16: $i > $i ).
thf(func_def_172,type,
sK17: $i > ( $i > $o ) > $i ).
thf(func_def_173,type,
sK18: ( $i > $o ) > $i > ( $i > $o ) > $i > $i ).
thf(func_def_174,type,
sK19: $i > $i > ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_175,type,
sK20: $i > ( $i > $o ) > $i ).
thf(func_def_176,type,
sK21: ( $i > $o ) > $i ).
thf(func_def_177,type,
sK22: $i > $i > $i ).
thf(func_def_178,type,
sK23: ( $i > $o ) > $i ).
thf(func_def_179,type,
sK24: ( $i > $o ) > $i ).
thf(func_def_180,type,
sK25: ( $i > $o ) > $i ).
thf(func_def_181,type,
sK26: $i > $i ).
thf(func_def_182,type,
sK27: $i > ( $i > $i > $o ) > $i ).
thf(func_def_183,type,
sK28: $i > ( $i > $i > $o ) > $i ).
thf(func_def_184,type,
sK29: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_185,type,
sK30: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_186,type,
sK31: $i > ( $i > $o ) > $i ).
thf(func_def_187,type,
sK32: $i > $i > $i ).
thf(func_def_188,type,
sK33: $i > $i > $i ).
thf(func_def_189,type,
sK34: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_190,type,
sK35: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_191,type,
sK36: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_192,type,
sK37: $i > $i ).
thf(func_def_193,type,
sK38: $i > ( $i > $o ) > $i ).
thf(func_def_194,type,
sK39: $i > $i > $i ).
thf(func_def_195,type,
sK40: $i > $o > $i > $i ).
thf(func_def_196,type,
sK41: $i > $i ).
thf(func_def_197,type,
sK42: $i > $i > $i ).
thf(func_def_198,type,
sK43: $i > $i > $i ).
thf(func_def_199,type,
sK44: $i > $i > $i ).
thf(func_def_200,type,
sK45: ( $i > $o ) > $i > $i ).
thf(func_def_201,type,
sK46: ( $i > $o ) > $i > $i ).
thf(func_def_202,type,
sK47: $i > $i ).
thf(func_def_203,type,
sK48: ( $i > $o ) > $i ).
thf(func_def_204,type,
ph49:
!>[X0: $tType] : X0 ).
thf(f3069,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f3055,f534]) ).
thf(f534,plain,
( $true
!= ( in @ sK2 @ sK1 ) ),
inference(cnf_transformation,[status(thm)],[f366]) ).
thf(f366,plain,
( ( setunionAx = $true )
& ( setbeta = $true )
& ( powerset__Cong = $true )
& ( nonemptyImpWitness = $true )
& ( exuE1 = $true )
& ( upairsetE = $true )
& ( dsetconstrER = $true )
& ( emptysetsubset = $true )
& ( wellorderingAx = $true )
& ( uniqinunit = $true )
& ( emptysetE = $true )
& ( setoftrueEq = $true )
& ( binintersectI = $true )
& ( powersetAx = $true )
& ( emptysetimpfalse = $true )
& ( emptysetAx = $true )
& ( binunionIL = $true )
& ( setadjoinAx = $true )
& ( setadjoinSub = $true )
& ( binintersectSubset4 = $true )
& ( omegaSAx = $true )
& ( binintersectSubset5 = $true )
& ( foundationAx = $true )
& ( binintersectSubset2 = $true )
& ( exuEu = $true )
& ( binunionEcases = $true )
& ( subsetRefl = $true )
& ( setunion__Cong = $true )
& ( binintersectSubset1 = $true )
& ( setadjoinIR = $true )
& ( in__Cong = $true )
& ( subsetE = $true )
& ( binunionIR = $true )
& ( binintersectRsub = $true )
& ( prop2set2propI = $true )
& ( exuE3u = $true )
& ( setadjoinSub2 = $true )
& ( subset2powerset = $true )
& ( eqinunit = $true )
& ( exuI1 = $true )
& ( subsetI2 = $true )
& ( setadjoinIL = $true )
& ( powersetE1 = $true )
& ( subsetTrans = $true )
& ( binintersectSubset3 = $true )
& ( sepSubset = $true )
& ( setextsub = $true )
& ( subPowSU = $true )
& ( eqimpsubset2 = $true )
& ( omegaIndAx = $true )
& ( quantDeMorgan1 = $true )
& ( inPowerset = $true )
& ( emptyinunitempty = $true )
& ( exu__Cong = $true )
& ( setadjoin__Cong = $true )
& ( nonemptyE1 = $true )
& ( dsetconstrEL = $true )
& ( subsetemptysetimpeq = $true )
& ( noeltsimpempty = $true )
& ( dsetconstrI = $true )
& ( notdallE = $true )
& ( $true
!= ( in @ sK2 @ sK1 ) )
& ( $true
= ( in @ sK2 @ ( setminus @ sK1 @ sK0 ) ) )
& ( notequalI1 = $true )
& ( setadjoinOr = $true )
& ( prop2setE = $true )
& ( upairsetIL = $true )
& ( prop2setI = $true )
& ( eqimpsubset1 = $true )
& ( notinsingleton = $true )
& ( descrp = $true )
& ( binunionLsub = $true )
& ( dsetconstr__Cong = $true )
& ( subsetE2 = $true )
& ( disjointsetsI1 = $true )
& ( setextAx = $true )
& ( subsetI1 = $true )
& ( powersetsubset = $true )
& ( binintersectLsub = $true )
& ( upairsetIR = $true )
& ( nonemptyI1 = $true )
& ( sepInPowerset = $true )
& ( omega0Ax = $true )
& ( exuI3 = $true )
& ( notsubsetI = $true )
& ( notinemptyset = $true )
& ( emptyE1 = $true )
& ( emptyinPowerset = $true )
& ( descr__Cong = $true )
& ( powersetE = $true )
& ( bs114d = $true )
& ( quantDeMorgan4 = $true )
& ( exuE2 = $true )
& ( binunionRsub = $true )
& ( quantDeMorgan2 = $true )
& ( omega__Cong = $true )
& ( inCongP = $true )
& ( vacuousDall = $true )
& ( quantDeMorgan3 = $true )
& ( binintersectEL = $true )
& ( binunionE = $true )
& ( replAx = $true )
& ( singletonsswitch = $true )
& ( setunionE = $true )
& ( binintersectER = $true )
& ( notequalI2 = $true )
& ( notdexE = $true )
& ( exuE3e = $true )
& ( emptyI = $true )
& ( exuI2 = $true )
& ( powersetI = $true )
& ( emptyset__Cong = $true )
& ( powersetI1 = $true )
& ( setadjoinE = $true )
& ( emptyInPowerset = $true )
& ( nonemptyI = $true )
& ( setext = $true )
& ( upairset2IR = $true )
& ( setminusI = $true )
& ( setunionI = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1,sK2])],[f364,f365]) ).
thf(f365,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ( $true
!= ( in @ X2 @ X1 ) )
& ( $true
= ( in @ X2 @ ( setminus @ X1 @ X0 ) ) ) )
=> ( ( $true
!= ( in @ sK2 @ sK1 ) )
& ( $true
= ( in @ sK2 @ ( setminus @ sK1 @ sK0 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f364,plain,
( ( setunionAx = $true )
& ( setbeta = $true )
& ( powerset__Cong = $true )
& ( nonemptyImpWitness = $true )
& ( exuE1 = $true )
& ( upairsetE = $true )
& ( dsetconstrER = $true )
& ( emptysetsubset = $true )
& ( wellorderingAx = $true )
& ( uniqinunit = $true )
& ( emptysetE = $true )
& ( setoftrueEq = $true )
& ( binintersectI = $true )
& ( powersetAx = $true )
& ( emptysetimpfalse = $true )
& ( emptysetAx = $true )
& ( binunionIL = $true )
& ( setadjoinAx = $true )
& ( setadjoinSub = $true )
& ( binintersectSubset4 = $true )
& ( omegaSAx = $true )
& ( binintersectSubset5 = $true )
& ( foundationAx = $true )
& ( binintersectSubset2 = $true )
& ( exuEu = $true )
& ( binunionEcases = $true )
& ( subsetRefl = $true )
& ( setunion__Cong = $true )
& ( binintersectSubset1 = $true )
& ( setadjoinIR = $true )
& ( in__Cong = $true )
& ( subsetE = $true )
& ( binunionIR = $true )
& ( binintersectRsub = $true )
& ( prop2set2propI = $true )
& ( exuE3u = $true )
& ( setadjoinSub2 = $true )
& ( subset2powerset = $true )
& ( eqinunit = $true )
& ( exuI1 = $true )
& ( subsetI2 = $true )
& ( setadjoinIL = $true )
& ( powersetE1 = $true )
& ( subsetTrans = $true )
& ( binintersectSubset3 = $true )
& ( sepSubset = $true )
& ( setextsub = $true )
& ( subPowSU = $true )
& ( eqimpsubset2 = $true )
& ( omegaIndAx = $true )
& ( quantDeMorgan1 = $true )
& ( inPowerset = $true )
& ( emptyinunitempty = $true )
& ( exu__Cong = $true )
& ( setadjoin__Cong = $true )
& ( nonemptyE1 = $true )
& ( dsetconstrEL = $true )
& ( subsetemptysetimpeq = $true )
& ( noeltsimpempty = $true )
& ( dsetconstrI = $true )
& ( notdallE = $true )
& ? [X0: $i,X1: $i,X2: $i] :
( ( $true
!= ( in @ X2 @ X1 ) )
& ( $true
= ( in @ X2 @ ( setminus @ X1 @ X0 ) ) ) )
& ( notequalI1 = $true )
& ( setadjoinOr = $true )
& ( prop2setE = $true )
& ( upairsetIL = $true )
& ( prop2setI = $true )
& ( eqimpsubset1 = $true )
& ( notinsingleton = $true )
& ( descrp = $true )
& ( binunionLsub = $true )
& ( dsetconstr__Cong = $true )
& ( subsetE2 = $true )
& ( disjointsetsI1 = $true )
& ( setextAx = $true )
& ( subsetI1 = $true )
& ( powersetsubset = $true )
& ( binintersectLsub = $true )
& ( upairsetIR = $true )
& ( nonemptyI1 = $true )
& ( sepInPowerset = $true )
& ( omega0Ax = $true )
& ( exuI3 = $true )
& ( notsubsetI = $true )
& ( notinemptyset = $true )
& ( emptyE1 = $true )
& ( emptyinPowerset = $true )
& ( descr__Cong = $true )
& ( powersetE = $true )
& ( bs114d = $true )
& ( quantDeMorgan4 = $true )
& ( exuE2 = $true )
& ( binunionRsub = $true )
& ( quantDeMorgan2 = $true )
& ( omega__Cong = $true )
& ( inCongP = $true )
& ( vacuousDall = $true )
& ( quantDeMorgan3 = $true )
& ( binintersectEL = $true )
& ( binunionE = $true )
& ( replAx = $true )
& ( singletonsswitch = $true )
& ( setunionE = $true )
& ( binintersectER = $true )
& ( notequalI2 = $true )
& ( notdexE = $true )
& ( exuE3e = $true )
& ( emptyI = $true )
& ( exuI2 = $true )
& ( powersetI = $true )
& ( emptyset__Cong = $true )
& ( powersetI1 = $true )
& ( setadjoinE = $true )
& ( emptyInPowerset = $true )
& ( nonemptyI = $true )
& ( setext = $true )
& ( upairset2IR = $true )
& ( setminusI = $true )
& ( setunionI = $true ) ),
inference(flattening,[status(thm)],[f363]) ).
thf(f363,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ( $true
!= ( in @ X2 @ X1 ) )
& ( $true
= ( in @ X2 @ ( setminus @ X1 @ X0 ) ) ) )
& ( setminusI = $true )
& ( bs114d = $true )
& ( binintersectSubset1 = $true )
& ( binintersectSubset4 = $true )
& ( binintersectRsub = $true )
& ( disjointsetsI1 = $true )
& ( binintersectER = $true )
& ( binintersectSubset3 = $true )
& ( binintersectSubset2 = $true )
& ( binintersectLsub = $true )
& ( binintersectEL = $true )
& ( binintersectSubset5 = $true )
& ( binintersectI = $true )
& ( binunionRsub = $true )
& ( binunionLsub = $true )
& ( binunionE = $true )
& ( binunionEcases = $true )
& ( binunionIR = $true )
& ( upairset2IR = $true )
& ( binunionIL = $true )
& ( sepSubset = $true )
& ( sepInPowerset = $true )
& ( powersetsubset = $true )
& ( inPowerset = $true )
& ( powersetE1 = $true )
& ( powersetI1 = $true )
& ( subsetemptysetimpeq = $true )
& ( setextsub = $true )
& ( subset2powerset = $true )
& ( setadjoinSub2 = $true )
& ( setadjoinSub = $true )
& ( subsetTrans = $true )
& ( subsetRefl = $true )
& ( notequalI2 = $true )
& ( notequalI1 = $true )
& ( notsubsetI = $true )
& ( subsetE2 = $true )
& ( subsetE = $true )
& ( emptysetsubset = $true )
& ( subsetI2 = $true )
& ( eqimpsubset1 = $true )
& ( eqimpsubset2 = $true )
& ( subsetI1 = $true )
& ( dsetconstr__Cong = $true )
& ( descr__Cong = $true )
& ( exuEu = $true )
& ( omega__Cong = $true )
& ( setunion__Cong = $true )
& ( powerset__Cong = $true )
& ( setadjoin__Cong = $true )
& ( emptyset__Cong = $true )
& ( exu__Cong = $true )
& ( exuE3u = $true )
& ( in__Cong = $true )
& ( inCongP = $true )
& ( exuI2 = $true )
& ( exuI3 = $true )
& ( exuI1 = $true )
& ( notdallE = $true )
& ( notdexE = $true )
& ( prop2set2propI = $true )
& ( prop2setI = $true )
& ( quantDeMorgan4 = $true )
& ( quantDeMorgan3 = $true )
& ( quantDeMorgan2 = $true )
& ( quantDeMorgan1 = $true )
& ( vacuousDall = $true )
& ( emptyE1 = $true )
& ( upairsetIR = $true )
& ( upairsetIL = $true )
& ( upairsetE = $true )
& ( singletonsswitch = $true )
& ( eqinunit = $true )
& ( notinsingleton = $true )
& ( uniqinunit = $true )
& ( nonemptyImpWitness = $true )
& ( exuE2 = $true )
& ( subPowSU = $true )
& ( setunionE = $true )
& ( setunionI = $true )
& ( powersetE = $true )
& ( emptyInPowerset = $true )
& ( emptyinPowerset = $true )
& ( powersetI = $true )
& ( setoftrueEq = $true )
& ( setadjoinOr = $true )
& ( setadjoinE = $true )
& ( setadjoinIR = $true )
& ( emptyinunitempty = $true )
& ( setadjoinIL = $true )
& ( nonemptyI1 = $true )
& ( nonemptyI = $true )
& ( nonemptyE1 = $true )
& ( setbeta = $true )
& ( noeltsimpempty = $true )
& ( emptyI = $true )
& ( setext = $true )
& ( exuE3e = $true )
& ( notinemptyset = $true )
& ( emptysetimpfalse = $true )
& ( emptysetE = $true )
& ( prop2setE = $true )
& ( exuE1 = $true )
& ( dsetconstrER = $true )
& ( dsetconstrEL = $true )
& ( dsetconstrI = $true )
& ( descrp = $true )
& ( wellorderingAx = $true )
& ( foundationAx = $true )
& ( replAx = $true )
& ( omegaIndAx = $true )
& ( omegaSAx = $true )
& ( omega0Ax = $true )
& ( setunionAx = $true )
& ( powersetAx = $true )
& ( setadjoinAx = $true )
& ( emptysetAx = $true )
& ( setextAx = $true ) ),
inference(ennf_transformation,[status(thm)],[f155]) ).
thf(f155,plain,
~ ( ( setextAx = $true )
=> ( ( emptysetAx = $true )
=> ( ( setadjoinAx = $true )
=> ( ( powersetAx = $true )
=> ( ( setunionAx = $true )
=> ( ( omega0Ax = $true )
=> ( ( omegaSAx = $true )
=> ( ( omegaIndAx = $true )
=> ( ( replAx = $true )
=> ( ( foundationAx = $true )
=> ( ( wellorderingAx = $true )
=> ( ( descrp = $true )
=> ( ( dsetconstrI = $true )
=> ( ( dsetconstrEL = $true )
=> ( ( dsetconstrER = $true )
=> ( ( exuE1 = $true )
=> ( ( prop2setE = $true )
=> ( ( emptysetE = $true )
=> ( ( emptysetimpfalse = $true )
=> ( ( notinemptyset = $true )
=> ( ( exuE3e = $true )
=> ( ( setext = $true )
=> ( ( emptyI = $true )
=> ( ( noeltsimpempty = $true )
=> ( ( setbeta = $true )
=> ( ( nonemptyE1 = $true )
=> ( ( nonemptyI = $true )
=> ( ( nonemptyI1 = $true )
=> ( ( setadjoinIL = $true )
=> ( ( emptyinunitempty = $true )
=> ( ( setadjoinIR = $true )
=> ( ( setadjoinE = $true )
=> ( ( setadjoinOr = $true )
=> ( ( setoftrueEq = $true )
=> ( ( powersetI = $true )
=> ( ( emptyinPowerset = $true )
=> ( ( emptyInPowerset = $true )
=> ( ( powersetE = $true )
=> ( ( setunionI = $true )
=> ( ( setunionE = $true )
=> ( ( subPowSU = $true )
=> ( ( exuE2 = $true )
=> ( ( nonemptyImpWitness = $true )
=> ( ( uniqinunit = $true )
=> ( ( notinsingleton = $true )
=> ( ( eqinunit = $true )
=> ( ( singletonsswitch = $true )
=> ( ( upairsetE = $true )
=> ( ( upairsetIL = $true )
=> ( ( upairsetIR = $true )
=> ( ( emptyE1 = $true )
=> ( ( vacuousDall = $true )
=> ( ( quantDeMorgan1 = $true )
=> ( ( quantDeMorgan2 = $true )
=> ( ( quantDeMorgan3 = $true )
=> ( ( quantDeMorgan4 = $true )
=> ( ( prop2setI = $true )
=> ( ( prop2set2propI = $true )
=> ( ( notdexE = $true )
=> ( ( notdallE = $true )
=> ( ( exuI1 = $true )
=> ( ( exuI3 = $true )
=> ( ( exuI2 = $true )
=> ( ( inCongP = $true )
=> ( ( in__Cong = $true )
=> ( ( exuE3u = $true )
=> ( ( exu__Cong = $true )
=> ( ( emptyset__Cong = $true )
=> ( ( setadjoin__Cong = $true )
=> ( ( powerset__Cong = $true )
=> ( ( setunion__Cong = $true )
=> ( ( omega__Cong = $true )
=> ( ( exuEu = $true )
=> ( ( descr__Cong = $true )
=> ( ( dsetconstr__Cong = $true )
=> ( ( subsetI1 = $true )
=> ( ( eqimpsubset2 = $true )
=> ( ( eqimpsubset1 = $true )
=> ( ( subsetI2 = $true )
=> ( ( emptysetsubset = $true )
=> ( ( subsetE = $true )
=> ( ( subsetE2 = $true )
=> ( ( notsubsetI = $true )
=> ( ( notequalI1 = $true )
=> ( ( notequalI2 = $true )
=> ( ( subsetRefl = $true )
=> ( ( subsetTrans = $true )
=> ( ( setadjoinSub = $true )
=> ( ( setadjoinSub2 = $true )
=> ( ( subset2powerset = $true )
=> ( ( setextsub = $true )
=> ( ( subsetemptysetimpeq = $true )
=> ( ( powersetI1 = $true )
=> ( ( powersetE1 = $true )
=> ( ( inPowerset = $true )
=> ( ( powersetsubset = $true )
=> ( ( sepInPowerset = $true )
=> ( ( sepSubset = $true )
=> ( ( binunionIL = $true )
=> ( ( upairset2IR = $true )
=> ( ( binunionIR = $true )
=> ( ( binunionEcases = $true )
=> ( ( binunionE = $true )
=> ( ( binunionLsub = $true )
=> ( ( binunionRsub = $true )
=> ( ( binintersectI = $true )
=> ( ( binintersectSubset5 = $true )
=> ( ( binintersectEL = $true )
=> ( ( binintersectLsub = $true )
=> ( ( binintersectSubset2 = $true )
=> ( ( binintersectSubset3 = $true )
=> ( ( binintersectER = $true )
=> ( ( disjointsetsI1 = $true )
=> ( ( binintersectRsub = $true )
=> ( ( binintersectSubset4 = $true )
=> ( ( binintersectSubset1 = $true )
=> ( ( bs114d = $true )
=> ( ( setminusI = $true )
=> ! [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X2 @ ( setminus @ X1 @ X0 ) ) )
=> ( $true
= ( in @ X2 @ X1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f154]) ).
thf(f154,plain,
~ ( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ! [X0: $i,X1: $i,X2: $i] :
( ( in @ X2 @ ( setminus @ X1 @ X0 ) )
=> ( in @ X2 @ X1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f123]) ).
thf(f123,negated_conjecture,
~ ( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ! [X4: $i,X3: $i,X1: $i] :
( ( in @ X1 @ ( setminus @ X3 @ X4 ) )
=> ( in @ X1 @ X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f122]) ).
thf(f122,conjecture,
( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ! [X4: $i,X3: $i,X1: $i] :
( ( in @ X1 @ ( setminus @ X3 @ X4 ) )
=> ( in @ X1 @ X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.U8znWXlJ6W/DTF2THF_14115.p',setminusEL) ).
thf(f3055,plain,
( $true
= ( in @ sK2 @ sK1 ) ),
inference(trivial_inequality_removal,[status(thm)],[f3045]) ).
thf(f3045,plain,
( ( $true
= ( in @ sK2 @ sK1 ) )
| ( $true = $false ) ),
inference(superposition,[status(thm)],[f1811,f1498]) ).
thf(f1498,plain,
! [X2: $i > $o,X3: $i,X1: $i] :
( ( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
= $false )
| ( $true
= ( in @ X3 @ X1 ) ) ),
inference(binary_proxy_clausification,[status(thm)],[f1497]) ).
thf(f1497,plain,
! [X2: $i > $o,X3: $i,X1: $i] :
( ( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f1496]) ).
thf(f1496,plain,
! [X2: $i > $o,X3: $i,X1: $i] :
( $true
= ( ^ [Y0: $i] :
( ( in @ Y0 @ ( dsetconstr @ X1 @ X2 ) )
=> ( in @ Y0 @ X1 ) )
@ X3 ) ),
inference(pi_clausification,[status(thm)],[f1495]) ).
thf(f1495,plain,
! [X2: $i > $o,X1: $i] :
( ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ ( dsetconstr @ X1 @ X2 ) )
=> ( in @ Y0 @ X1 ) ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f1494]) ).
thf(f1494,plain,
! [X2: $i > $o,X1: $i] :
( ( ^ [Y0: $i > $o] :
( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ ( dsetconstr @ X1 @ Y0 ) )
=> ( in @ Y1 @ X1 ) ) )
@ X2 )
= $true ),
inference(pi_clausification,[status(thm)],[f1493]) ).
thf(f1493,plain,
! [X1: $i] :
( $true
= ( !! @ ( $i > $o )
@ ^ [Y0: $i > $o] :
( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ ( dsetconstr @ X1 @ Y0 ) )
=> ( in @ Y1 @ X1 ) ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f1492]) ).
thf(f1492,plain,
! [X1: $i] :
( ( ^ [Y0: $i] :
( !! @ ( $i > $o )
@ ^ [Y1: $i > $o] :
( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y2 @ ( dsetconstr @ Y0 @ Y1 ) )
=> ( in @ Y2 @ Y0 ) ) ) )
@ X1 )
= $true ),
inference(pi_clausification,[status(thm)],[f1491]) ).
thf(f1491,plain,
( $true
= ( !! @ $i
@ ^ [Y0: $i] :
( !! @ ( $i > $o )
@ ^ [Y1: $i > $o] :
( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y2 @ ( dsetconstr @ Y0 @ Y1 ) )
=> ( in @ Y2 @ Y0 ) ) ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f655]) ).
thf(f655,plain,
( $true
= ( !! @ $i
@ ^ [Y0: $i] :
( !! @ ( $i > $o )
@ ^ [Y1: $i > $o] :
( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y2
@ ( dsetconstr @ Y0
@ ^ [Y3: $i] : ( Y1 @ Y3 ) ) )
=> ( in @ Y2 @ Y0 ) ) ) ) ) ),
inference(definition_unfolding,[status(thm)],[f539,f445]) ).
thf(f445,plain,
( dsetconstrEL
= ( !! @ $i
@ ^ [Y0: $i] :
( !! @ ( $i > $o )
@ ^ [Y1: $i > $o] :
( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y2
@ ( dsetconstr @ Y0
@ ^ [Y3: $i] : ( Y1 @ Y3 ) ) )
=> ( in @ Y2 @ Y0 ) ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f348]) ).
thf(f348,plain,
( dsetconstrEL
= ( !! @ $i
@ ^ [Y0: $i] :
( !! @ ( $i > $o )
@ ^ [Y1: $i > $o] :
( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y2
@ ( dsetconstr @ Y0
@ ^ [Y3: $i] : ( Y1 @ Y3 ) ) )
=> ( in @ Y2 @ Y0 ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f347]) ).
thf(f347,plain,
! [X0: $i,X1: $i > $o,X2: $i] :
( ( ( in @ X0
@ ( dsetconstr @ X2
@ ^ [X3: $i] : ( X1 @ X3 ) ) )
=> ( in @ X0 @ X2 ) )
= dsetconstrEL ),
inference(rectify,[status(thm)],[f15]) ).
thf(f15,axiom,
! [X1: $i,X0: $i > $o,X3: $i] :
( ( ( in @ X1
@ ( dsetconstr @ X3
@ ^ [X2: $i] : ( X0 @ X2 ) ) )
=> ( in @ X1 @ X3 ) )
= dsetconstrEL ),
file('/export/starexec/sandbox2/tmp/tmp.U8znWXlJ6W/DTF2THF_14115.p',dsetconstrEL) ).
thf(f539,plain,
dsetconstrEL = $true,
inference(cnf_transformation,[status(thm)],[f366]) ).
thf(f1811,plain,
( ( in @ sK2
@ ( dsetconstr @ sK1
@ ^ [Y0: $i] :
~ ( in @ Y0 @ sK0 ) ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f660]) ).
thf(f660,plain,
( $true
= ( in @ sK2
@ ( ^ [Y0: $i,Y1: $i] :
( dsetconstr @ Y0
@ ^ [Y2: $i] :
~ ( in @ Y2 @ Y1 ) )
@ sK1
@ sK0 ) ) ),
inference(definition_unfolding,[status(thm)],[f533,f454]) ).
thf(f454,plain,
( setminus
= ( ^ [Y0: $i,Y1: $i] :
( dsetconstr @ Y0
@ ^ [Y2: $i] :
~ ( in @ Y2 @ Y1 ) ) ) ),
inference(cnf_transformation,[status(thm)],[f282]) ).
thf(f282,plain,
( setminus
= ( ^ [Y0: $i,Y1: $i] :
( dsetconstr @ Y0
@ ^ [Y2: $i] :
~ ( in @ Y2 @ Y1 ) ) ) ),
inference(fool_elimination,[status(thm)],[f281]) ).
thf(f281,plain,
( setminus
= ( ^ [X0: $i,X1: $i] :
( dsetconstr @ X0
@ ^ [X2: $i] :
~ ( in @ X2 @ X1 ) ) ) ),
inference(rectify,[status(thm)],[f120]) ).
thf(f120,axiom,
( setminus
= ( ^ [X3: $i,X4: $i] :
( dsetconstr @ X3
@ ^ [X1: $i] :
~ ( in @ X1 @ X4 ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.U8znWXlJ6W/DTF2THF_14115.p',setminus) ).
thf(f533,plain,
( $true
= ( in @ sK2 @ ( setminus @ sK1 @ sK0 ) ) ),
inference(cnf_transformation,[status(thm)],[f366]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.10 % Problem : SEU602^1 : TPTP v9.2.1. Released v3.7.0.
% 0.00/0.12 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.13/0.32 % Computer : n003.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 12:27:55 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
% 0.16/0.43 ---- Original DTF file ---
% 0.16/0.43 thf(spec,logic,$$dhol).
% 0.16/0.43 %------------------------------------------------------------------------------
% 0.16/0.43 % File : SEU602^1 : TPTP v9.2.1. Released v3.7.0.
% 0.16/0.43 % Domain : Set Theory
% 0.16/0.43 % Problem : Preliminary Notions - Operations on Sets - Set Difference
% 0.16/0.43 % Version : Especial.
% 0.16/0.43 % English : (! A:i.! B:i.! x:i.in x (setminus A B) -> in x A)
% 0.16/0.43
% 0.16/0.43 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.16/0.43 % Source : [Bro08]
% 0.16/0.43 % Names : ZFC104g [Bro08]
% 0.16/0.43
% 0.16/0.43 % Status : Theorem
% 0.16/0.43 % Rating : 0.22 v9.1.0, 0.25 v9.0.0, 0.30 v8.2.0, 0.38 v8.1.0, 0.36 v7.5.0, 0.14 v7.4.0, 0.22 v7.2.0, 0.12 v7.1.0, 0.38 v7.0.0, 0.29 v6.4.0, 0.33 v6.3.0, 0.40 v6.2.0, 0.43 v5.5.0, 0.50 v5.4.0, 0.60 v4.1.0, 1.00 v3.7.0
% 0.16/0.43 % Syntax : Number of formulae : 259 ( 121 unt; 137 typ; 121 def)
% 0.16/0.43 % Number of atoms : 642 ( 176 equ; 0 cnn)
% 0.16/0.43 % Maximal formula atoms : 120 ( 5 avg)
% 0.16/0.43 % Number of connectives : 1013 ( 32 ~; 6 |; 28 &; 649 @)
% 0.16/0.43 % ( 14 <=>; 284 =>; 0 <=; 0 <~>)
% 0.16/0.43 % Maximal formula depth : 126 ( 2 avg)
% 0.16/0.43 % Number of types : 2 ( 0 usr)
% 0.16/0.43 % Number of type conns : 62 ( 62 >; 0 *; 0 +; 0 <<)
% 0.16/0.43 % Number of symbols : 140 ( 137 usr; 122 con; 0-2 aty)
% 0.16/0.43 % Number of variables : 352 ( 33 ^; 290 !; 29 ?; 352 :)
% 0.16/0.43 % SPC : TH0_THM_EQU_NAR
% 0.16/0.43
% 0.16/0.43 % Comments : http://mathgate.info/detsetitem.php?id=270
% 0.16/0.43 %------------------------------------------------------------------------------
% 0.16/0.43 thf(in_type,type,
% 0.16/0.43 in: $i > $i > $o ).
% 0.16/0.43
% 0.16/0.43 thf(exu_type,type,
% 0.16/0.43 exu: ( $i > $o ) > $o ).
% 0.16/0.43
% 0.16/0.43 thf(exu,definition,
% 0.16/0.43 ( exu
% 0.16/0.43 = ( ^ [Xphi: $i > $o] :
% 0.16/0.43 ? [Xx: $i] :
% 0.16/0.43 ( ( Xphi @ Xx )
% 0.16/0.43 & ! [Xy: $i] :
% 0.16/0.43 ( ( Xphi @ Xy )
% 0.16/0.43 => ( Xx = Xy ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setextAx_type,type,
% 0.16/0.43 setextAx: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setextAx,definition,
% 0.16/0.43 ( setextAx
% 0.16/0.43 = ( ! [A: $i,B: $i] :
% 0.16/0.43 ( ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 <=> ( in @ Xx @ B ) )
% 0.16/0.43 => ( A = B ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(emptyset_type,type,
% 0.16/0.43 emptyset: $i ).
% 0.16/0.43
% 0.16/0.43 thf(emptysetAx_type,type,
% 0.16/0.43 emptysetAx: $o ).
% 0.16/0.43
% 0.16/0.43 thf(emptysetAx,definition,
% 0.16/0.43 ( emptysetAx
% 0.16/0.43 = ( ! [Xx: $i] :
% 0.16/0.43 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setadjoin_type,type,
% 0.16/0.43 setadjoin: $i > $i > $i ).
% 0.16/0.43
% 0.16/0.43 thf(setadjoinAx_type,type,
% 0.16/0.43 setadjoinAx: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setadjoinAx,definition,
% 0.16/0.43 ( setadjoinAx
% 0.16/0.43 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.16/0.43 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.16/0.43 <=> ( ( Xy = Xx )
% 0.16/0.43 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(powerset_type,type,
% 0.16/0.43 powerset: $i > $i ).
% 0.16/0.43
% 0.16/0.43 thf(powersetAx_type,type,
% 0.16/0.43 powersetAx: $o ).
% 0.16/0.43
% 0.16/0.43 thf(powersetAx,definition,
% 0.16/0.43 ( powersetAx
% 0.16/0.43 = ( ! [A: $i,B: $i] :
% 0.16/0.43 ( ( in @ B @ ( powerset @ A ) )
% 0.16/0.43 <=> ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ B )
% 0.16/0.43 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setunion_type,type,
% 0.16/0.43 setunion: $i > $i ).
% 0.16/0.43
% 0.16/0.43 thf(setunionAx_type,type,
% 0.16/0.43 setunionAx: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setunionAx,definition,
% 0.16/0.43 ( setunionAx
% 0.16/0.43 = ( ! [A: $i,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ ( setunion @ A ) )
% 0.16/0.43 <=> ? [B: $i] :
% 0.16/0.43 ( ( in @ Xx @ B )
% 0.16/0.43 & ( in @ B @ A ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(omega_type,type,
% 0.16/0.43 omega: $i ).
% 0.16/0.43
% 0.16/0.43 thf(omega0Ax_type,type,
% 0.16/0.43 omega0Ax: $o ).
% 0.16/0.43
% 0.16/0.43 thf(omega0Ax,definition,
% 0.16/0.43 ( omega0Ax
% 0.16/0.43 = ( in @ emptyset @ omega ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(omegaSAx_type,type,
% 0.16/0.43 omegaSAx: $o ).
% 0.16/0.43
% 0.16/0.43 thf(omegaSAx,definition,
% 0.16/0.43 ( omegaSAx
% 0.16/0.43 = ( ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ omega )
% 0.16/0.43 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(omegaIndAx_type,type,
% 0.16/0.43 omegaIndAx: $o ).
% 0.16/0.43
% 0.16/0.43 thf(omegaIndAx,definition,
% 0.16/0.43 ( omegaIndAx
% 0.16/0.43 = ( ! [A: $i] :
% 0.16/0.43 ( ( ( in @ emptyset @ A )
% 0.16/0.43 & ! [Xx: $i] :
% 0.16/0.43 ( ( ( in @ Xx @ omega )
% 0.16/0.43 & ( in @ Xx @ A ) )
% 0.16/0.43 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.16/0.43 => ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ omega )
% 0.16/0.43 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(replAx_type,type,
% 0.16/0.43 replAx: $o ).
% 0.16/0.43
% 0.16/0.43 thf(replAx,definition,
% 0.16/0.43 ( replAx
% 0.16/0.43 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.16/0.43 ( ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ( exu
% 0.16/0.43 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.16/0.43 => ? [B: $i] :
% 0.16/0.43 ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ B )
% 0.16/0.43 <=> ? [Xy: $i] :
% 0.16/0.43 ( ( in @ Xy @ A )
% 0.16/0.43 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(foundationAx_type,type,
% 0.16/0.43 foundationAx: $o ).
% 0.16/0.43
% 0.16/0.43 thf(foundationAx,definition,
% 0.16/0.43 ( foundationAx
% 0.16/0.43 = ( ! [A: $i] :
% 0.16/0.43 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.16/0.43 => ? [B: $i] :
% 0.16/0.43 ( ( in @ B @ A )
% 0.16/0.43 & ~ ? [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ B )
% 0.16/0.43 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(wellorderingAx_type,type,
% 0.16/0.43 wellorderingAx: $o ).
% 0.16/0.43
% 0.16/0.43 thf(wellorderingAx,definition,
% 0.16/0.43 ( wellorderingAx
% 0.16/0.43 = ( ! [A: $i] :
% 0.16/0.43 ? [B: $i] :
% 0.16/0.43 ( ! [C: $i] :
% 0.16/0.43 ( ( in @ C @ B )
% 0.16/0.43 => ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ C )
% 0.16/0.43 => ( in @ Xx @ A ) ) )
% 0.16/0.43 & ! [Xx: $i,Xy: $i] :
% 0.16/0.43 ( ( ( in @ Xx @ A )
% 0.16/0.43 & ( in @ Xy @ A ) )
% 0.16/0.43 => ( ! [C: $i] :
% 0.16/0.43 ( ( in @ C @ B )
% 0.16/0.43 => ( ( in @ Xx @ C )
% 0.16/0.43 <=> ( in @ Xy @ C ) ) )
% 0.16/0.43 => ( Xx = Xy ) ) )
% 0.16/0.43 & ! [C: $i,D: $i] :
% 0.16/0.43 ( ( ( in @ C @ B )
% 0.16/0.43 & ( in @ D @ B ) )
% 0.16/0.43 => ( ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ C )
% 0.16/0.43 => ( in @ Xx @ D ) )
% 0.16/0.43 | ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ D )
% 0.16/0.43 => ( in @ Xx @ C ) ) ) )
% 0.16/0.43 & ! [C: $i] :
% 0.16/0.43 ( ( ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ C )
% 0.16/0.43 => ( in @ Xx @ A ) )
% 0.16/0.43 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.16/0.43 => ? [D: $i,Xx: $i] :
% 0.16/0.43 ( ( in @ D @ B )
% 0.16/0.43 & ( in @ Xx @ C )
% 0.16/0.43 & ~ ? [Xy: $i] :
% 0.16/0.43 ( ( in @ Xy @ D )
% 0.16/0.43 & ( in @ Xy @ C ) )
% 0.16/0.43 & ! [E: $i] :
% 0.16/0.43 ( ( in @ E @ B )
% 0.16/0.43 => ( ! [Xy: $i] :
% 0.16/0.43 ( ( in @ Xy @ E )
% 0.16/0.43 => ( in @ Xy @ D ) )
% 0.16/0.43 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(descr_type,type,
% 0.16/0.43 descr: ( $i > $o ) > $i ).
% 0.16/0.43
% 0.16/0.43 thf(descrp_type,type,
% 0.16/0.43 descrp: $o ).
% 0.16/0.43
% 0.16/0.43 thf(descrp,definition,
% 0.16/0.43 ( descrp
% 0.16/0.43 = ( ! [Xphi: $i > $o] :
% 0.16/0.43 ( ( exu
% 0.16/0.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.43 => ( Xphi
% 0.16/0.43 @ ( descr
% 0.16/0.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(dsetconstr_type,type,
% 0.16/0.43 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.16/0.43
% 0.16/0.43 thf(dsetconstrI_type,type,
% 0.16/0.43 dsetconstrI: $o ).
% 0.16/0.43
% 0.16/0.43 thf(dsetconstrI,definition,
% 0.16/0.43 ( dsetconstrI
% 0.16/0.43 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ( ( Xphi @ Xx )
% 0.16/0.43 => ( in @ Xx
% 0.16/0.43 @ ( dsetconstr @ A
% 0.16/0.43 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(dsetconstrEL_type,type,
% 0.16/0.43 dsetconstrEL: $o ).
% 0.16/0.43
% 0.16/0.43 thf(dsetconstrEL,definition,
% 0.16/0.43 ( dsetconstrEL
% 0.16/0.43 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx
% 0.16/0.43 @ ( dsetconstr @ A
% 0.16/0.43 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.16/0.43 => ( in @ Xx @ A ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(dsetconstrER_type,type,
% 0.16/0.43 dsetconstrER: $o ).
% 0.16/0.43
% 0.16/0.43 thf(dsetconstrER,definition,
% 0.16/0.43 ( dsetconstrER
% 0.16/0.43 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx
% 0.16/0.43 @ ( dsetconstr @ A
% 0.16/0.43 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.16/0.43 => ( Xphi @ Xx ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(exuE1_type,type,
% 0.16/0.43 exuE1: $o ).
% 0.16/0.43
% 0.16/0.43 thf(exuE1,definition,
% 0.16/0.43 ( exuE1
% 0.16/0.43 = ( ! [Xphi: $i > $o] :
% 0.16/0.43 ( ( exu
% 0.16/0.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.43 => ? [Xx: $i] :
% 0.16/0.43 ( ( Xphi @ Xx )
% 0.16/0.43 & ! [Xy: $i] :
% 0.16/0.43 ( ( Xphi @ Xy )
% 0.16/0.43 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(prop2set_type,type,
% 0.16/0.43 prop2set: $o > $i ).
% 0.16/0.43
% 0.16/0.43 thf(prop2setE_type,type,
% 0.16/0.43 prop2setE: $o ).
% 0.16/0.43
% 0.16/0.43 thf(prop2setE,definition,
% 0.16/0.43 ( prop2setE
% 0.16/0.43 = ( ! [Xphi: $o,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.16/0.43 => Xphi ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(emptysetE_type,type,
% 0.16/0.43 emptysetE: $o ).
% 0.16/0.43
% 0.16/0.43 thf(emptysetE,definition,
% 0.16/0.43 ( emptysetE
% 0.16/0.43 = ( ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ emptyset )
% 0.16/0.43 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(emptysetimpfalse_type,type,
% 0.16/0.43 emptysetimpfalse: $o ).
% 0.16/0.43
% 0.16/0.43 thf(emptysetimpfalse,definition,
% 0.16/0.43 ( emptysetimpfalse
% 0.16/0.43 = ( ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ emptyset )
% 0.16/0.43 => $false ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(notinemptyset_type,type,
% 0.16/0.43 notinemptyset: $o ).
% 0.16/0.43
% 0.16/0.43 thf(notinemptyset,definition,
% 0.16/0.43 ( notinemptyset
% 0.16/0.43 = ( ! [Xx: $i] :
% 0.16/0.43 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(exuE3e_type,type,
% 0.16/0.43 exuE3e: $o ).
% 0.16/0.43
% 0.16/0.43 thf(exuE3e,definition,
% 0.16/0.43 ( exuE3e
% 0.16/0.43 = ( ! [Xphi: $i > $o] :
% 0.16/0.43 ( ( exu
% 0.16/0.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.43 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setext_type,type,
% 0.16/0.43 setext: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setext,definition,
% 0.16/0.43 ( setext
% 0.16/0.43 = ( ! [A: $i,B: $i] :
% 0.16/0.43 ( ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ( in @ Xx @ B ) )
% 0.16/0.43 => ( ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ B )
% 0.16/0.43 => ( in @ Xx @ A ) )
% 0.16/0.43 => ( A = B ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(emptyI_type,type,
% 0.16/0.43 emptyI: $o ).
% 0.16/0.43
% 0.16/0.43 thf(emptyI,definition,
% 0.16/0.43 ( emptyI
% 0.16/0.43 = ( ! [A: $i] :
% 0.16/0.43 ( ! [Xx: $i] :
% 0.16/0.43 ~ ( in @ Xx @ A )
% 0.16/0.43 => ( A = emptyset ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(noeltsimpempty_type,type,
% 0.16/0.43 noeltsimpempty: $o ).
% 0.16/0.43
% 0.16/0.43 thf(noeltsimpempty,definition,
% 0.16/0.43 ( noeltsimpempty
% 0.16/0.43 = ( ! [A: $i] :
% 0.16/0.43 ( ! [Xx: $i] :
% 0.16/0.43 ~ ( in @ Xx @ A )
% 0.16/0.43 => ( A = emptyset ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setbeta_type,type,
% 0.16/0.43 setbeta: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setbeta,definition,
% 0.16/0.43 ( setbeta
% 0.16/0.43 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ( ( in @ Xx
% 0.16/0.43 @ ( dsetconstr @ A
% 0.16/0.43 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.16/0.43 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(nonempty_type,type,
% 0.16/0.43 nonempty: $i > $o ).
% 0.16/0.43
% 0.16/0.43 thf(nonempty,definition,
% 0.16/0.43 ( nonempty
% 0.16/0.43 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(nonemptyE1_type,type,
% 0.16/0.43 nonemptyE1: $o ).
% 0.16/0.43
% 0.16/0.43 thf(nonemptyE1,definition,
% 0.16/0.43 ( nonemptyE1
% 0.16/0.43 = ( ! [A: $i] :
% 0.16/0.43 ( ( nonempty @ A )
% 0.16/0.43 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(nonemptyI_type,type,
% 0.16/0.43 nonemptyI: $o ).
% 0.16/0.43
% 0.16/0.43 thf(nonemptyI,definition,
% 0.16/0.43 ( nonemptyI
% 0.16/0.43 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ( ( Xphi @ Xx )
% 0.16/0.43 => ( nonempty
% 0.16/0.43 @ ( dsetconstr @ A
% 0.16/0.43 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(nonemptyI1_type,type,
% 0.16/0.43 nonemptyI1: $o ).
% 0.16/0.43
% 0.16/0.43 thf(nonemptyI1,definition,
% 0.16/0.43 ( nonemptyI1
% 0.16/0.43 = ( ! [A: $i] :
% 0.16/0.43 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.16/0.43 => ( nonempty @ A ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setadjoinIL_type,type,
% 0.16/0.43 setadjoinIL: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setadjoinIL,definition,
% 0.16/0.43 ( setadjoinIL
% 0.16/0.43 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(emptyinunitempty_type,type,
% 0.16/0.43 emptyinunitempty: $o ).
% 0.16/0.43
% 0.16/0.43 thf(emptyinunitempty,definition,
% 0.16/0.43 ( emptyinunitempty
% 0.16/0.43 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setadjoinIR_type,type,
% 0.16/0.43 setadjoinIR: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setadjoinIR,definition,
% 0.16/0.43 ( setadjoinIR
% 0.16/0.43 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.16/0.43 ( ( in @ Xy @ A )
% 0.16/0.43 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setadjoinE_type,type,
% 0.16/0.43 setadjoinE: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setadjoinE,definition,
% 0.16/0.43 ( setadjoinE
% 0.16/0.43 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.16/0.43 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.16/0.43 => ! [Xphi: $o] :
% 0.16/0.43 ( ( ( Xy = Xx )
% 0.16/0.43 => Xphi )
% 0.16/0.43 => ( ( ( in @ Xy @ A )
% 0.16/0.43 => Xphi )
% 0.16/0.43 => Xphi ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setadjoinOr_type,type,
% 0.16/0.43 setadjoinOr: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setadjoinOr,definition,
% 0.16/0.43 ( setadjoinOr
% 0.16/0.43 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.16/0.43 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.16/0.43 => ( ( Xy = Xx )
% 0.16/0.43 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setoftrueEq_type,type,
% 0.16/0.43 setoftrueEq: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setoftrueEq,definition,
% 0.16/0.43 ( setoftrueEq
% 0.16/0.43 = ( ! [A: $i] :
% 0.16/0.43 ( ( dsetconstr @ A
% 0.16/0.43 @ ^ [Xx: $i] : $true )
% 0.16/0.43 = A ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(powersetI_type,type,
% 0.16/0.43 powersetI: $o ).
% 0.16/0.43
% 0.16/0.43 thf(powersetI,definition,
% 0.16/0.43 ( powersetI
% 0.16/0.43 = ( ! [A: $i,B: $i] :
% 0.16/0.43 ( ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ B )
% 0.16/0.43 => ( in @ Xx @ A ) )
% 0.16/0.43 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(emptyinPowerset_type,type,
% 0.16/0.43 emptyinPowerset: $o ).
% 0.16/0.43
% 0.16/0.43 thf(emptyinPowerset,definition,
% 0.16/0.43 ( emptyinPowerset
% 0.16/0.43 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(emptyInPowerset_type,type,
% 0.16/0.43 emptyInPowerset: $o ).
% 0.16/0.43
% 0.16/0.43 thf(emptyInPowerset,definition,
% 0.16/0.43 ( emptyInPowerset
% 0.16/0.43 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(powersetE_type,type,
% 0.16/0.43 powersetE: $o ).
% 0.16/0.43
% 0.16/0.43 thf(powersetE,definition,
% 0.16/0.43 ( powersetE
% 0.16/0.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.43 ( ( in @ B @ ( powerset @ A ) )
% 0.16/0.43 => ( ( in @ Xx @ B )
% 0.16/0.43 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setunionI_type,type,
% 0.16/0.43 setunionI: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setunionI,definition,
% 0.16/0.43 ( setunionI
% 0.16/0.43 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.16/0.43 ( ( in @ Xx @ B )
% 0.16/0.43 => ( ( in @ B @ A )
% 0.16/0.43 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setunionE_type,type,
% 0.16/0.44 setunionE: $o ).
% 0.16/0.44
% 0.16/0.44 thf(setunionE,definition,
% 0.16/0.44 ( setunionE
% 0.16/0.44 = ( ! [A: $i,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ ( setunion @ A ) )
% 0.16/0.44 => ! [Xphi: $o] :
% 0.16/0.44 ( ! [B: $i] :
% 0.16/0.44 ( ( in @ Xx @ B )
% 0.16/0.44 => ( ( in @ B @ A )
% 0.16/0.44 => Xphi ) )
% 0.16/0.44 => Xphi ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(subPowSU_type,type,
% 0.16/0.44 subPowSU: $o ).
% 0.16/0.44
% 0.16/0.44 thf(subPowSU,definition,
% 0.16/0.44 ( subPowSU
% 0.16/0.44 = ( ! [A: $i,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(exuE2_type,type,
% 0.16/0.44 exuE2: $o ).
% 0.16/0.44
% 0.16/0.44 thf(exuE2,definition,
% 0.16/0.44 ( exuE2
% 0.16/0.44 = ( ! [Xphi: $i > $o] :
% 0.16/0.44 ( ( exu
% 0.16/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.44 => ? [Xx: $i] :
% 0.16/0.44 ! [Xy: $i] :
% 0.16/0.44 ( ( Xphi @ Xy )
% 0.16/0.44 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(nonemptyImpWitness_type,type,
% 0.16/0.44 nonemptyImpWitness: $o ).
% 0.16/0.44
% 0.16/0.44 thf(nonemptyImpWitness,definition,
% 0.16/0.44 ( nonemptyImpWitness
% 0.16/0.44 = ( ! [A: $i] :
% 0.16/0.44 ( ( nonempty @ A )
% 0.16/0.44 => ? [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 & $true ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(uniqinunit_type,type,
% 0.16/0.44 uniqinunit: $o ).
% 0.16/0.44
% 0.16/0.44 thf(uniqinunit,definition,
% 0.16/0.44 ( uniqinunit
% 0.16/0.44 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.44 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.16/0.44 => ( Xx = Xy ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(notinsingleton_type,type,
% 0.16/0.44 notinsingleton: $o ).
% 0.16/0.44
% 0.16/0.44 thf(notinsingleton,definition,
% 0.16/0.44 ( notinsingleton
% 0.16/0.44 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.44 ( ( Xx != Xy )
% 0.16/0.44 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(eqinunit_type,type,
% 0.16/0.44 eqinunit: $o ).
% 0.16/0.44
% 0.16/0.44 thf(eqinunit,definition,
% 0.16/0.44 ( eqinunit
% 0.16/0.44 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.44 ( ( Xx = Xy )
% 0.16/0.44 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(singletonsswitch_type,type,
% 0.16/0.44 singletonsswitch: $o ).
% 0.16/0.44
% 0.16/0.44 thf(singletonsswitch,definition,
% 0.16/0.44 ( singletonsswitch
% 0.16/0.44 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.44 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.16/0.44 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(upairsetE_type,type,
% 0.16/0.44 upairsetE: $o ).
% 0.16/0.44
% 0.16/0.44 thf(upairsetE,definition,
% 0.16/0.44 ( upairsetE
% 0.16/0.44 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.16/0.44 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.16/0.44 => ( ( Xz = Xx )
% 0.16/0.44 | ( Xz = Xy ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(upairsetIL_type,type,
% 0.16/0.44 upairsetIL: $o ).
% 0.16/0.44
% 0.16/0.44 thf(upairsetIL,definition,
% 0.16/0.44 ( upairsetIL
% 0.16/0.44 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(upairsetIR_type,type,
% 0.16/0.44 upairsetIR: $o ).
% 0.16/0.44
% 0.16/0.44 thf(upairsetIR,definition,
% 0.16/0.44 ( upairsetIR
% 0.16/0.44 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(emptyE1_type,type,
% 0.16/0.44 emptyE1: $o ).
% 0.16/0.44
% 0.16/0.44 thf(emptyE1,definition,
% 0.16/0.44 ( emptyE1
% 0.16/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.44 ( ? [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 & ( Xphi @ Xx ) )
% 0.16/0.44 => ( ( ( dsetconstr @ A
% 0.16/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.44 = emptyset )
% 0.16/0.44 => $false ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(vacuousDall_type,type,
% 0.16/0.44 vacuousDall: $o ).
% 0.16/0.44
% 0.16/0.44 thf(vacuousDall,definition,
% 0.16/0.44 ( vacuousDall
% 0.16/0.44 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ emptyset )
% 0.16/0.44 => ( Xphi @ Xx ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(quantDeMorgan1_type,type,
% 0.16/0.44 quantDeMorgan1: $o ).
% 0.16/0.44
% 0.16/0.44 thf(quantDeMorgan1,definition,
% 0.16/0.44 ( quantDeMorgan1
% 0.16/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.44 ( ~ ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( Xphi @ Xx ) )
% 0.16/0.44 => ? [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(quantDeMorgan2_type,type,
% 0.16/0.44 quantDeMorgan2: $o ).
% 0.16/0.44
% 0.16/0.44 thf(quantDeMorgan2,definition,
% 0.16/0.44 ( quantDeMorgan2
% 0.16/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.44 ( ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ~ ( Xphi @ Xx ) )
% 0.16/0.44 => ~ ? [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(quantDeMorgan3_type,type,
% 0.16/0.44 quantDeMorgan3: $o ).
% 0.16/0.44
% 0.16/0.44 thf(quantDeMorgan3,definition,
% 0.16/0.44 ( quantDeMorgan3
% 0.16/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.44 ( ~ ? [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 & ( Xphi @ Xx ) )
% 0.16/0.44 => ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(quantDeMorgan4_type,type,
% 0.16/0.44 quantDeMorgan4: $o ).
% 0.16/0.44
% 0.16/0.44 thf(quantDeMorgan4,definition,
% 0.16/0.44 ( quantDeMorgan4
% 0.16/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.44 ( ? [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 & ~ ( Xphi @ Xx ) )
% 0.16/0.44 => ~ ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(prop2setI_type,type,
% 0.16/0.44 prop2setI: $o ).
% 0.16/0.44
% 0.16/0.44 thf(prop2setI,definition,
% 0.16/0.44 ( prop2setI
% 0.16/0.44 = ( ! [Xphi: $o] :
% 0.16/0.44 ( Xphi
% 0.16/0.44 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(set2prop_type,type,
% 0.16/0.44 set2prop: $i > $o ).
% 0.16/0.44
% 0.16/0.44 thf(prop2set2propI_type,type,
% 0.16/0.44 prop2set2propI: $o ).
% 0.16/0.44
% 0.16/0.44 thf(prop2set2propI,definition,
% 0.16/0.44 ( prop2set2propI
% 0.16/0.44 = ( ! [Xphi: $o] :
% 0.16/0.44 ( Xphi
% 0.16/0.44 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(notdexE_type,type,
% 0.16/0.44 notdexE: $o ).
% 0.16/0.44
% 0.16/0.44 thf(notdexE,definition,
% 0.16/0.44 ( notdexE
% 0.16/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.44 ( ~ ? [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 & ( Xphi @ Xx ) )
% 0.16/0.44 => ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(notdallE_type,type,
% 0.16/0.44 notdallE: $o ).
% 0.16/0.44
% 0.16/0.44 thf(notdallE,definition,
% 0.16/0.44 ( notdallE
% 0.16/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.44 ( ~ ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( Xphi @ Xx ) )
% 0.16/0.44 => ? [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(exuI1_type,type,
% 0.16/0.44 exuI1: $o ).
% 0.16/0.44
% 0.16/0.44 thf(exuI1,definition,
% 0.16/0.44 ( exuI1
% 0.16/0.44 = ( ! [Xphi: $i > $o] :
% 0.16/0.44 ( ? [Xx: $i] :
% 0.16/0.44 ( ( Xphi @ Xx )
% 0.16/0.44 & ! [Xy: $i] :
% 0.16/0.44 ( ( Xphi @ Xy )
% 0.16/0.44 => ( Xx = Xy ) ) )
% 0.16/0.44 => ( exu
% 0.16/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(exuI3_type,type,
% 0.16/0.44 exuI3: $o ).
% 0.16/0.44
% 0.16/0.44 thf(exuI3,definition,
% 0.16/0.44 ( exuI3
% 0.16/0.44 = ( ! [Xphi: $i > $o] :
% 0.16/0.44 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.16/0.44 => ( ! [Xx: $i,Xy: $i] :
% 0.16/0.44 ( ( Xphi @ Xx )
% 0.16/0.44 => ( ( Xphi @ Xy )
% 0.16/0.44 => ( Xx = Xy ) ) )
% 0.16/0.44 => ( exu
% 0.16/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(exuI2_type,type,
% 0.16/0.44 exuI2: $o ).
% 0.16/0.44
% 0.16/0.44 thf(exuI2,definition,
% 0.16/0.44 ( exuI2
% 0.16/0.44 = ( ! [Xphi: $i > $o] :
% 0.16/0.44 ( ? [Xx: $i] :
% 0.16/0.44 ! [Xy: $i] :
% 0.16/0.44 ( ( Xphi @ Xy )
% 0.16/0.44 <=> ( Xy = Xx ) )
% 0.16/0.44 => ( exu
% 0.16/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(inCongP_type,type,
% 0.16/0.44 inCongP: $o ).
% 0.16/0.44
% 0.16/0.44 thf(inCongP,definition,
% 0.16/0.44 ( inCongP
% 0.16/0.44 = ( ! [A: $i,B: $i] :
% 0.16/0.44 ( ( A = B )
% 0.16/0.44 => ! [Xx: $i,Xy: $i] :
% 0.16/0.44 ( ( Xx = Xy )
% 0.16/0.44 => ( ( in @ Xx @ A )
% 0.16/0.44 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(in__Cong_type,type,
% 0.16/0.44 in__Cong: $o ).
% 0.16/0.44
% 0.16/0.44 thf(in__Cong,definition,
% 0.16/0.44 ( in__Cong
% 0.16/0.44 = ( ! [A: $i,B: $i] :
% 0.16/0.44 ( ( A = B )
% 0.16/0.44 => ! [Xx: $i,Xy: $i] :
% 0.16/0.44 ( ( Xx = Xy )
% 0.16/0.44 => ( ( in @ Xx @ A )
% 0.16/0.44 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(exuE3u_type,type,
% 0.16/0.44 exuE3u: $o ).
% 0.16/0.44
% 0.16/0.44 thf(exuE3u,definition,
% 0.16/0.44 ( exuE3u
% 0.16/0.44 = ( ! [Xphi: $i > $o] :
% 0.16/0.44 ( ( exu
% 0.16/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.44 => ! [Xx: $i,Xy: $i] :
% 0.16/0.44 ( ( Xphi @ Xx )
% 0.16/0.44 => ( ( Xphi @ Xy )
% 0.16/0.44 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(exu__Cong_type,type,
% 0.16/0.44 exu__Cong: $o ).
% 0.16/0.44
% 0.16/0.44 thf(exu__Cong,definition,
% 0.16/0.44 ( exu__Cong
% 0.16/0.44 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.16/0.44 ( ! [Xx: $i,Xy: $i] :
% 0.16/0.44 ( ( Xx = Xy )
% 0.16/0.44 => ( ( Xphi @ Xx )
% 0.16/0.44 <=> ( Xpsi @ Xy ) ) )
% 0.16/0.44 => ( ( exu
% 0.16/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.44 <=> ( exu
% 0.16/0.44 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(emptyset__Cong_type,type,
% 0.16/0.44 emptyset__Cong: $o ).
% 0.16/0.44
% 0.16/0.44 thf(emptyset__Cong,definition,
% 0.16/0.44 ( emptyset__Cong
% 0.16/0.44 = ( emptyset = emptyset ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(setadjoin__Cong_type,type,
% 0.16/0.44 setadjoin__Cong: $o ).
% 0.16/0.44
% 0.16/0.44 thf(setadjoin__Cong,definition,
% 0.16/0.44 ( setadjoin__Cong
% 0.16/0.44 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.44 ( ( Xx = Xy )
% 0.16/0.44 => ! [Xz: $i,Xu: $i] :
% 0.16/0.44 ( ( Xz = Xu )
% 0.16/0.44 => ( ( setadjoin @ Xx @ Xz )
% 0.16/0.44 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(powerset__Cong_type,type,
% 0.16/0.44 powerset__Cong: $o ).
% 0.16/0.44
% 0.16/0.44 thf(powerset__Cong,definition,
% 0.16/0.44 ( powerset__Cong
% 0.16/0.44 = ( ! [A: $i,B: $i] :
% 0.16/0.44 ( ( A = B )
% 0.16/0.44 => ( ( powerset @ A )
% 0.16/0.44 = ( powerset @ B ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(setunion__Cong_type,type,
% 0.16/0.44 setunion__Cong: $o ).
% 0.16/0.44
% 0.16/0.44 thf(setunion__Cong,definition,
% 0.16/0.44 ( setunion__Cong
% 0.16/0.44 = ( ! [A: $i,B: $i] :
% 0.16/0.44 ( ( A = B )
% 0.16/0.44 => ( ( setunion @ A )
% 0.16/0.44 = ( setunion @ B ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(omega__Cong_type,type,
% 0.16/0.44 omega__Cong: $o ).
% 0.16/0.44
% 0.16/0.44 thf(omega__Cong,definition,
% 0.16/0.44 ( omega__Cong
% 0.16/0.44 = ( omega = omega ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(exuEu_type,type,
% 0.16/0.44 exuEu: $o ).
% 0.16/0.44
% 0.16/0.44 thf(exuEu,definition,
% 0.16/0.44 ( exuEu
% 0.16/0.44 = ( ! [Xphi: $i > $o] :
% 0.16/0.44 ( ( exu
% 0.16/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.44 => ! [Xx: $i,Xy: $i] :
% 0.16/0.44 ( ( Xphi @ Xx )
% 0.16/0.44 => ( ( Xphi @ Xy )
% 0.16/0.44 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(descr__Cong_type,type,
% 0.16/0.44 descr__Cong: $o ).
% 0.16/0.44
% 0.16/0.44 thf(descr__Cong,definition,
% 0.16/0.44 ( descr__Cong
% 0.16/0.44 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.16/0.44 ( ! [Xx: $i,Xy: $i] :
% 0.16/0.44 ( ( Xx = Xy )
% 0.16/0.44 => ( ( Xphi @ Xx )
% 0.16/0.44 <=> ( Xpsi @ Xy ) ) )
% 0.16/0.44 => ( ( exu
% 0.16/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.44 => ( ( exu
% 0.16/0.44 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.16/0.44 => ( ( descr
% 0.16/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.44 = ( descr
% 0.16/0.44 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(dsetconstr__Cong_type,type,
% 0.16/0.44 dsetconstr__Cong: $o ).
% 0.16/0.44
% 0.16/0.44 thf(dsetconstr__Cong,definition,
% 0.16/0.44 ( dsetconstr__Cong
% 0.16/0.44 = ( ! [A: $i,B: $i] :
% 0.16/0.44 ( ( A = B )
% 0.16/0.44 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.16/0.44 ( ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ! [Xy: $i] :
% 0.16/0.44 ( ( in @ Xy @ B )
% 0.16/0.44 => ( ( Xx = Xy )
% 0.16/0.44 => ( ( Xphi @ Xx )
% 0.16/0.44 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.16/0.44 => ( ( dsetconstr @ A
% 0.16/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.44 = ( dsetconstr @ B
% 0.16/0.44 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(subset_type,type,
% 0.16/0.44 subset: $i > $i > $o ).
% 0.16/0.44
% 0.16/0.44 thf(disjoint_type,type,
% 0.16/0.44 disjoint: $i > $i > $o ).
% 0.16/0.44
% 0.16/0.44 thf(setsmeet_type,type,
% 0.16/0.44 setsmeet: $i > $i > $o ).
% 0.16/0.44
% 0.16/0.44 thf(subsetI1_type,type,
% 0.16/0.44 subsetI1: $o ).
% 0.16/0.44
% 0.16/0.44 thf(subsetI1,definition,
% 0.16/0.44 ( subsetI1
% 0.16/0.44 = ( ! [A: $i,B: $i] :
% 0.16/0.44 ( ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( in @ Xx @ B ) )
% 0.16/0.44 => ( subset @ A @ B ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(eqimpsubset2_type,type,
% 0.16/0.44 eqimpsubset2: $o ).
% 0.16/0.44
% 0.16/0.44 thf(eqimpsubset2,definition,
% 0.16/0.44 ( eqimpsubset2
% 0.16/0.44 = ( ! [A: $i,B: $i] :
% 0.16/0.44 ( ( A = B )
% 0.16/0.44 => ( subset @ B @ A ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(eqimpsubset1_type,type,
% 0.16/0.44 eqimpsubset1: $o ).
% 0.16/0.44
% 0.16/0.44 thf(eqimpsubset1,definition,
% 0.16/0.44 ( eqimpsubset1
% 0.16/0.44 = ( ! [A: $i,B: $i] :
% 0.16/0.44 ( ( A = B )
% 0.16/0.44 => ( subset @ A @ B ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(subsetI2_type,type,
% 0.16/0.44 subsetI2: $o ).
% 0.16/0.44
% 0.16/0.44 thf(subsetI2,definition,
% 0.16/0.44 ( subsetI2
% 0.16/0.44 = ( ! [A: $i,B: $i] :
% 0.16/0.44 ( ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( in @ Xx @ B ) )
% 0.16/0.44 => ( subset @ A @ B ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(emptysetsubset_type,type,
% 0.16/0.44 emptysetsubset: $o ).
% 0.16/0.44
% 0.16/0.44 thf(emptysetsubset,definition,
% 0.16/0.44 ( emptysetsubset
% 0.16/0.44 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(subsetE_type,type,
% 0.16/0.44 subsetE: $o ).
% 0.16/0.44
% 0.16/0.44 thf(subsetE,definition,
% 0.16/0.44 ( subsetE
% 0.16/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.44 ( ( subset @ A @ B )
% 0.16/0.44 => ( ( in @ Xx @ A )
% 0.16/0.44 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(subsetE2_type,type,
% 0.16/0.44 subsetE2: $o ).
% 0.16/0.44
% 0.16/0.44 thf(subsetE2,definition,
% 0.16/0.44 ( subsetE2
% 0.16/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.44 ( ( subset @ A @ B )
% 0.16/0.44 => ( ~ ( in @ Xx @ B )
% 0.16/0.44 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(notsubsetI_type,type,
% 0.16/0.44 notsubsetI: $o ).
% 0.16/0.44
% 0.16/0.44 thf(notsubsetI,definition,
% 0.16/0.44 ( notsubsetI
% 0.16/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( ~ ( in @ Xx @ B )
% 0.16/0.44 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(notequalI1_type,type,
% 0.16/0.44 notequalI1: $o ).
% 0.16/0.44
% 0.16/0.44 thf(notequalI1,definition,
% 0.16/0.44 ( notequalI1
% 0.16/0.44 = ( ! [A: $i,B: $i] :
% 0.16/0.44 ( ~ ( subset @ A @ B )
% 0.16/0.44 => ( A != B ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(notequalI2_type,type,
% 0.16/0.44 notequalI2: $o ).
% 0.16/0.44
% 0.16/0.44 thf(notequalI2,definition,
% 0.16/0.44 ( notequalI2
% 0.16/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( ~ ( in @ Xx @ B )
% 0.16/0.44 => ( A != B ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(subsetRefl_type,type,
% 0.16/0.44 subsetRefl: $o ).
% 0.16/0.44
% 0.16/0.44 thf(subsetRefl,definition,
% 0.16/0.44 ( subsetRefl
% 0.16/0.44 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(subsetTrans_type,type,
% 0.16/0.44 subsetTrans: $o ).
% 0.16/0.44
% 0.16/0.44 thf(subsetTrans,definition,
% 0.16/0.44 ( subsetTrans
% 0.16/0.44 = ( ! [A: $i,B: $i,C: $i] :
% 0.16/0.44 ( ( subset @ A @ B )
% 0.16/0.44 => ( ( subset @ B @ C )
% 0.16/0.44 => ( subset @ A @ C ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(setadjoinSub_type,type,
% 0.16/0.44 setadjoinSub: $o ).
% 0.16/0.44
% 0.16/0.44 thf(setadjoinSub,definition,
% 0.16/0.44 ( setadjoinSub
% 0.16/0.44 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(setadjoinSub2_type,type,
% 0.16/0.44 setadjoinSub2: $o ).
% 0.16/0.44
% 0.16/0.44 thf(setadjoinSub2,definition,
% 0.16/0.44 ( setadjoinSub2
% 0.16/0.44 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.16/0.44 ( ( subset @ A @ B )
% 0.16/0.44 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(subset2powerset_type,type,
% 0.16/0.44 subset2powerset: $o ).
% 0.16/0.44
% 0.16/0.44 thf(subset2powerset,definition,
% 0.16/0.44 ( subset2powerset
% 0.16/0.44 = ( ! [A: $i,B: $i] :
% 0.16/0.44 ( ( subset @ A @ B )
% 0.16/0.44 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(setextsub_type,type,
% 0.16/0.44 setextsub: $o ).
% 0.16/0.44
% 0.16/0.44 thf(setextsub,definition,
% 0.16/0.44 ( setextsub
% 0.16/0.44 = ( ! [A: $i,B: $i] :
% 0.16/0.44 ( ( subset @ A @ B )
% 0.16/0.44 => ( ( subset @ B @ A )
% 0.16/0.44 => ( A = B ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(subsetemptysetimpeq_type,type,
% 0.16/0.44 subsetemptysetimpeq: $o ).
% 0.16/0.44
% 0.16/0.44 thf(subsetemptysetimpeq,definition,
% 0.16/0.44 ( subsetemptysetimpeq
% 0.16/0.44 = ( ! [A: $i] :
% 0.16/0.44 ( ( subset @ A @ emptyset )
% 0.16/0.44 => ( A = emptyset ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(powersetI1_type,type,
% 0.16/0.44 powersetI1: $o ).
% 0.16/0.44
% 0.16/0.44 thf(powersetI1,definition,
% 0.16/0.44 ( powersetI1
% 0.16/0.44 = ( ! [A: $i,B: $i] :
% 0.16/0.44 ( ( subset @ B @ A )
% 0.16/0.44 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(powersetE1_type,type,
% 0.16/0.44 powersetE1: $o ).
% 0.16/0.44
% 0.16/0.44 thf(powersetE1,definition,
% 0.16/0.44 ( powersetE1
% 0.16/0.44 = ( ! [A: $i,B: $i] :
% 0.16/0.44 ( ( in @ B @ ( powerset @ A ) )
% 0.16/0.44 => ( subset @ B @ A ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(inPowerset_type,type,
% 0.16/0.44 inPowerset: $o ).
% 0.16/0.44
% 0.16/0.44 thf(inPowerset,definition,
% 0.16/0.44 ( inPowerset
% 0.16/0.44 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(powersetsubset_type,type,
% 0.16/0.44 powersetsubset: $o ).
% 0.16/0.44
% 0.16/0.44 thf(powersetsubset,definition,
% 0.16/0.44 ( powersetsubset
% 0.16/0.44 = ( ! [A: $i,B: $i] :
% 0.16/0.44 ( ( subset @ A @ B )
% 0.16/0.44 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(sepInPowerset_type,type,
% 0.16/0.44 sepInPowerset: $o ).
% 0.16/0.44
% 0.16/0.44 thf(sepInPowerset,definition,
% 0.16/0.44 ( sepInPowerset
% 0.16/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.44 ( in
% 0.16/0.44 @ ( dsetconstr @ A
% 0.16/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.44 @ ( powerset @ A ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(sepSubset_type,type,
% 0.16/0.44 sepSubset: $o ).
% 0.16/0.44
% 0.16/0.44 thf(sepSubset,definition,
% 0.16/0.44 ( sepSubset
% 0.16/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.44 ( subset
% 0.16/0.44 @ ( dsetconstr @ A
% 0.16/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.44 @ A ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(binunion_type,type,
% 0.16/0.44 binunion: $i > $i > $i ).
% 0.16/0.44
% 0.16/0.44 thf(binunionIL_type,type,
% 0.16/0.44 binunionIL: $o ).
% 0.16/0.44
% 0.16/0.44 thf(binunionIL,definition,
% 0.16/0.44 ( binunionIL
% 0.16/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(upairset2IR_type,type,
% 0.16/0.44 upairset2IR: $o ).
% 0.16/0.44
% 0.16/0.44 thf(upairset2IR,definition,
% 0.16/0.44 ( upairset2IR
% 0.16/0.44 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(binunionIR_type,type,
% 0.16/0.44 binunionIR: $o ).
% 0.16/0.44
% 0.16/0.44 thf(binunionIR,definition,
% 0.16/0.44 ( binunionIR
% 0.16/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ B )
% 0.16/0.44 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(binunionEcases_type,type,
% 0.16/0.44 binunionEcases: $o ).
% 0.16/0.44
% 0.16/0.44 thf(binunionEcases,definition,
% 0.16/0.44 ( binunionEcases
% 0.16/0.44 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.16/0.44 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.16/0.44 => ( ( ( in @ Xx @ A )
% 0.16/0.44 => Xphi )
% 0.16/0.44 => ( ( ( in @ Xx @ B )
% 0.16/0.44 => Xphi )
% 0.16/0.44 => Xphi ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(binunionE_type,type,
% 0.16/0.44 binunionE: $o ).
% 0.16/0.44
% 0.16/0.44 thf(binunionE,definition,
% 0.16/0.44 ( binunionE
% 0.16/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.16/0.44 => ( ( in @ Xx @ A )
% 0.16/0.44 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(binunionLsub_type,type,
% 0.16/0.44 binunionLsub: $o ).
% 0.16/0.44
% 0.16/0.44 thf(binunionLsub,definition,
% 0.16/0.44 ( binunionLsub
% 0.16/0.44 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(binunionRsub_type,type,
% 0.16/0.44 binunionRsub: $o ).
% 0.16/0.44
% 0.16/0.44 thf(binunionRsub,definition,
% 0.16/0.44 ( binunionRsub
% 0.16/0.44 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(binintersect_type,type,
% 0.16/0.44 binintersect: $i > $i > $i ).
% 0.16/0.44
% 0.16/0.44 thf(binintersectI_type,type,
% 0.16/0.44 binintersectI: $o ).
% 0.16/0.44
% 0.16/0.44 thf(binintersectI,definition,
% 0.16/0.44 ( binintersectI
% 0.16/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( ( in @ Xx @ B )
% 0.16/0.44 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(binintersectSubset5_type,type,
% 0.16/0.44 binintersectSubset5: $o ).
% 0.16/0.44
% 0.16/0.44 thf(binintersectSubset5,definition,
% 0.16/0.44 ( binintersectSubset5
% 0.16/0.44 = ( ! [A: $i,B: $i,C: $i] :
% 0.16/0.44 ( ( subset @ C @ A )
% 0.16/0.44 => ( ( subset @ C @ B )
% 0.16/0.44 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(binintersectEL_type,type,
% 0.16/0.44 binintersectEL: $o ).
% 0.16/0.44
% 0.16/0.44 thf(binintersectEL,definition,
% 0.16/0.44 ( binintersectEL
% 0.16/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.16/0.44 => ( in @ Xx @ A ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(binintersectLsub_type,type,
% 0.16/0.44 binintersectLsub: $o ).
% 0.16/0.44
% 0.16/0.44 thf(binintersectLsub,definition,
% 0.16/0.44 ( binintersectLsub
% 0.16/0.44 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(binintersectSubset2_type,type,
% 0.16/0.44 binintersectSubset2: $o ).
% 0.16/0.44
% 0.16/0.44 thf(binintersectSubset2,definition,
% 0.16/0.44 ( binintersectSubset2
% 0.16/0.44 = ( ! [A: $i,B: $i] :
% 0.16/0.44 ( ( subset @ A @ B )
% 0.16/0.44 => ( ( binintersect @ A @ B )
% 0.16/0.44 = A ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(binintersectSubset3_type,type,
% 0.16/0.44 binintersectSubset3: $o ).
% 0.16/0.44
% 0.16/0.44 thf(binintersectSubset3,definition,
% 0.16/0.44 ( binintersectSubset3
% 0.16/0.44 = ( ! [A: $i,B: $i] :
% 0.16/0.44 ( ( ( binintersect @ A @ B )
% 0.16/0.44 = B )
% 0.16/0.44 => ( subset @ B @ A ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(binintersectER_type,type,
% 0.16/0.44 binintersectER: $o ).
% 0.16/0.44
% 0.16/0.44 thf(binintersectER,definition,
% 0.16/0.44 ( binintersectER
% 0.16/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.16/0.44 => ( in @ Xx @ B ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(disjointsetsI1_type,type,
% 0.16/0.44 disjointsetsI1: $o ).
% 0.16/0.44
% 0.16/0.44 thf(disjointsetsI1,definition,
% 0.16/0.44 ( disjointsetsI1
% 0.16/0.44 = ( ! [A: $i,B: $i] :
% 0.16/0.44 ( ~ ? [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 & ( in @ Xx @ B ) )
% 0.16/0.44 => ( ( binintersect @ A @ B )
% 0.16/0.44 = emptyset ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(binintersectRsub_type,type,
% 0.16/0.44 binintersectRsub: $o ).
% 0.16/0.44
% 0.16/0.44 thf(binintersectRsub,definition,
% 0.16/0.44 ( binintersectRsub
% 0.16/0.44 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(binintersectSubset4_type,type,
% 0.16/0.44 binintersectSubset4: $o ).
% 0.16/0.44
% 0.16/0.44 thf(binintersectSubset4,definition,
% 0.16/0.44 ( binintersectSubset4
% 0.16/0.44 = ( ! [A: $i,B: $i] :
% 0.16/0.44 ( ( subset @ B @ A )
% 0.16/0.44 => ( ( binintersect @ A @ B )
% 0.16/0.44 = B ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(binintersectSubset1_type,type,
% 0.16/0.44 binintersectSubset1: $o ).
% 0.16/0.44
% 0.16/0.44 thf(binintersectSubset1,definition,
% 0.16/0.44 ( binintersectSubset1
% 0.16/0.44 = ( ! [A: $i,B: $i] :
% 0.16/0.44 ( ( ( binintersect @ A @ B )
% 0.16/0.44 = A )
% 0.16/0.44 => ( subset @ A @ B ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(bs114d_type,type,
% 0.16/0.44 bs114d: $o ).
% 0.16/0.44
% 0.16/0.44 thf(bs114d,definition,
% 0.16/0.44 ( bs114d
% 0.16/0.44 = ( ! [A: $i,B: $i,C: $i] :
% 0.16/0.44 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 0.16/0.44 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(regular_type,type,
% 0.16/0.44 regular: $i > $o ).
% 0.16/0.44
% 0.16/0.44 thf(setminus_type,type,
% 0.16/0.44 setminus: $i > $i > $i ).
% 0.16/0.44
% 0.16/0.44 thf(setminus,definition,
% 0.16/0.44 ( setminus
% 0.16/0.44 = ( ^ [A: $i,B: $i] :
% 0.16/0.44 ( dsetconstr @ A
% 0.16/0.44 @ ^ [Xx: $i] :
% 0.16/0.44 ~ ( in @ Xx @ B ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(setminusI_type,type,
% 0.16/0.44 setminusI: $o ).
% 0.16/0.44
% 0.16/0.44 thf(setminusI,definition,
% 0.16/0.44 ( setminusI
% 0.16/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( ~ ( in @ Xx @ B )
% 0.16/0.44 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(setminusEL,conjecture,
% 0.16/0.44 ( setextAx
% 0.16/0.44 => ( emptysetAx
% 0.16/0.44 => ( setadjoinAx
% 0.16/0.44 => ( powersetAx
% 0.16/0.44 => ( setunionAx
% 0.16/0.44 => ( omega0Ax
% 0.16/0.44 => ( omegaSAx
% 0.16/0.44 => ( omegaIndAx
% 0.16/0.44 => ( replAx
% 0.16/0.44 => ( foundationAx
% 0.16/0.44 => ( wellorderingAx
% 0.16/0.44 => ( descrp
% 0.16/0.44 => ( dsetconstrI
% 0.16/0.44 => ( dsetconstrEL
% 0.16/0.44 => ( dsetconstrER
% 0.16/0.44 => ( exuE1
% 0.16/0.44 => ( prop2setE
% 0.16/0.44 => ( emptysetE
% 0.16/0.44 => ( emptysetimpfalse
% 0.16/0.44 => ( notinemptyset
% 0.16/0.44 => ( exuE3e
% 0.16/0.44 => ( setext
% 0.16/0.44 => ( emptyI
% 0.16/0.44 => ( noeltsimpempty
% 0.16/0.44 => ( setbeta
% 0.16/0.44 => ( nonemptyE1
% 0.16/0.44 => ( nonemptyI
% 0.16/0.44 => ( nonemptyI1
% 0.16/0.44 => ( setadjoinIL
% 0.16/0.44 => ( emptyinunitempty
% 0.16/0.44 => ( setadjoinIR
% 0.16/0.44 => ( setadjoinE
% 0.16/0.44 => ( setadjoinOr
% 0.16/0.44 => ( setoftrueEq
% 0.16/0.44 => ( powersetI
% 0.16/0.44 => ( emptyinPowerset
% 0.16/0.44 => ( emptyInPowerset
% 0.16/0.44 => ( powersetE
% 0.16/0.44 => ( setunionI
% 0.16/0.44 => ( setunionE
% 0.16/0.44 => ( subPowSU
% 0.16/0.44 => ( exuE2
% 0.16/0.44 => ( nonemptyImpWitness
% 0.16/0.44 => ( uniqinunit
% 0.16/0.44 => ( notinsingleton
% 0.16/0.44 => ( eqinunit
% 0.16/0.44 => ( singletonsswitch
% 0.16/0.44 => ( upairsetE
% 0.16/0.44 => ( upairsetIL
% 0.16/0.44 => ( upairsetIR
% 0.16/0.44 => ( emptyE1
% 0.16/0.44 => ( vacuousDall
% 0.16/0.44 => ( quantDeMorgan1
% 0.16/0.44 => ( quantDeMorgan2
% 0.16/0.44 => ( quantDeMorgan3
% 0.16/0.44 => ( quantDeMorgan4
% 0.16/0.44 => ( prop2setI
% 0.16/0.44 => ( prop2set2propI
% 0.16/0.44 => ( notdexE
% 0.16/0.44 => ( notdallE
% 0.16/0.44 => ( exuI1
% 0.16/0.44 => ( exuI3
% 0.16/0.44 => ( exuI2
% 0.16/0.44 => ( inCongP
% 0.16/0.44 => ( in__Cong
% 0.16/0.44 => ( exuE3u
% 0.16/0.44 => ( exu__Cong
% 0.16/0.44 => ( emptyset__Cong
% 0.16/0.44 => ( setadjoin__Cong
% 0.16/0.44 => ( powerset__Cong
% 0.16/0.44 => ( setunion__Cong
% 0.16/0.44 => ( omega__Cong
% 0.16/0.44 => ( exuEu
% 0.16/0.44 => ( descr__Cong
% 0.16/0.44 => ( dsetconstr__Cong
% 0.16/0.44 => ( subsetI1
% 0.16/0.44 => ( eqimpsubset2
% 0.16/0.44 => ( eqimpsubset1
% 0.16/0.44 => ( subsetI2
% 0.16/0.44 => ( emptysetsubset
% 0.16/0.44 => ( subsetE
% 0.16/0.44 => ( subsetE2
% 0.16/0.44 => ( notsubsetI
% 0.16/0.44 => ( notequalI1
% 0.16/0.44 => ( notequalI2
% 0.16/0.44 => ( subsetRefl
% 0.16/0.44 => ( subsetTrans
% 0.16/0.44 => ( setadjoinSub
% 0.16/0.44 => ( setadjoinSub2
% 0.16/0.44 => ( subset2powerset
% 0.16/0.44 => ( setextsub
% 0.16/0.44 => ( subsetemptysetimpeq
% 0.16/0.44 => ( powersetI1
% 0.16/0.44 => ( powersetE1
% 0.16/0.44 => ( inPowerset
% 0.16/0.44 => ( powersetsubset
% 0.16/0.44 => ( sepInPowerset
% 0.16/0.44 => ( sepSubset
% 0.16/0.44 => ( binunionIL
% 0.16/0.44 => ( upairset2IR
% 0.16/0.44 => ( binunionIR
% 0.16/0.44 => ( binunionEcases
% 0.16/0.44 => ( binunionE
% 0.16/0.44 => ( binunionLsub
% 0.16/0.44 => ( binunionRsub
% 0.16/0.44 => ( binintersectI
% 0.16/0.44 => ( binintersectSubset5
% 0.16/0.44 => ( binintersectEL
% 0.16/0.44 => ( binintersectLsub
% 0.16/0.44 => ( binintersectSubset2
% 0.16/0.44 => ( binintersectSubset3
% 0.16/0.44 => ( binintersectER
% 0.16/0.44 => ( disjointsetsI1
% 0.16/0.44 => ( binintersectRsub
% 0.16/0.44 => ( binintersectSubset4
% 0.16/0.44 => ( binintersectSubset1
% 0.16/0.44 => ( bs114d
% 0.16/0.44 => ( setminusI
% 0.16/0.44 => ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.16/0.44 => ( in @ Xx @ A ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 %------------------------------------------------------------------------------
% 0.16/0.44 ------------------------
% 2.05/1.38 ---- Embedded in THF ---
% 2.05/1.38 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.05/1.38 %%% Generated on Tue Feb 24 12:27:56 EST 2026
% 2.05/1.38 %%% using '$$dhol' embedding, version 1.3.0.
% 2.05/1.38 %%% Logic specification used:
% 2.05/1.38 %%% thf(spec, logic, $$dhol).
% 2.05/1.38
% 2.05/1.38 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.U8znWXlJ6W/DTF2DTF14115.p
% See solution above
% 2.05/1.38 ------------------------
% 2.28/1.40 ---- Cleaned THF ---
% 2.28/1.40 thf(in_type,type,
% 2.28/1.40 in: $i > $i > $o ).
% 2.28/1.40
% 2.28/1.40 thf(exu_type,type,
% 2.28/1.40 exu: ( $i > $o ) > $o ).
% 2.28/1.40
% 2.28/1.40 thf(exu,definition,
% 2.28/1.40 ( exu
% 2.28/1.40 = ( ^ [Xphi: $i > $o] :
% 2.28/1.40 ? [Xx: $i] :
% 2.28/1.40 ( ( Xphi @ Xx )
% 2.28/1.40 & ! [Xy: $i] :
% 2.28/1.40 ( ( Xphi @ Xy )
% 2.28/1.40 => ( Xx = Xy ) ) ) ) ) ).
% 2.28/1.40
% 2.28/1.40 thf(setextAx_type,type,
% 2.28/1.40 setextAx: $o ).
% 2.28/1.40
% 2.28/1.40 thf(setextAx,definition,
% 2.28/1.40 ( setextAx
% 2.28/1.40 = ( ! [A: $i,B: $i] :
% 2.28/1.40 ( ! [Xx: $i] :
% 2.28/1.40 ( ( in @ Xx @ A )
% 2.28/1.40 <=> ( in @ Xx @ B ) )
% 2.28/1.40 => ( A = B ) ) ) ) ).
% 2.28/1.40
% 2.28/1.40 thf(emptyset_type,type,
% 2.28/1.40 emptyset: $i ).
% 2.28/1.40
% 2.28/1.40 thf(emptysetAx_type,type,
% 2.28/1.40 emptysetAx: $o ).
% 2.28/1.40
% 2.28/1.40 thf(emptysetAx,definition,
% 2.28/1.40 ( emptysetAx
% 2.28/1.40 = ( ! [Xx: $i] :
% 2.28/1.40 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.28/1.40
% 2.28/1.40 thf(setadjoin_type,type,
% 2.28/1.40 setadjoin: $i > $i > $i ).
% 2.28/1.40
% 2.28/1.40 thf(setadjoinAx_type,type,
% 2.28/1.40 setadjoinAx: $o ).
% 2.28/1.40
% 2.28/1.40 thf(setadjoinAx,definition,
% 2.28/1.40 ( setadjoinAx
% 2.28/1.40 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.28/1.40 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.28/1.40 <=> ( ( Xy = Xx )
% 2.28/1.40 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.28/1.40
% 2.28/1.40 thf(powerset_type,type,
% 2.28/1.40 powerset: $i > $i ).
% 2.28/1.40
% 2.28/1.40 thf(powersetAx_type,type,
% 2.28/1.40 powersetAx: $o ).
% 2.28/1.40
% 2.28/1.40 thf(powersetAx,definition,
% 2.28/1.40 ( powersetAx
% 2.28/1.40 = ( ! [A: $i,B: $i] :
% 2.28/1.40 ( ( in @ B @ ( powerset @ A ) )
% 2.28/1.40 <=> ! [Xx: $i] :
% 2.28/1.40 ( ( in @ Xx @ B )
% 2.28/1.40 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.28/1.40
% 2.28/1.40 thf(setunion_type,type,
% 2.28/1.40 setunion: $i > $i ).
% 2.28/1.40
% 2.28/1.40 thf(setunionAx_type,type,
% 2.28/1.40 setunionAx: $o ).
% 2.28/1.40
% 2.28/1.40 thf(setunionAx,definition,
% 2.28/1.40 ( setunionAx
% 2.28/1.40 = ( ! [A: $i,Xx: $i] :
% 2.28/1.40 ( ( in @ Xx @ ( setunion @ A ) )
% 2.28/1.40 <=> ? [B: $i] :
% 2.28/1.40 ( ( in @ Xx @ B )
% 2.28/1.40 & ( in @ B @ A ) ) ) ) ) ).
% 2.28/1.40
% 2.28/1.40 thf(omega_type,type,
% 2.28/1.40 omega: $i ).
% 2.28/1.40
% 2.28/1.40 thf(omega0Ax_type,type,
% 2.28/1.40 omega0Ax: $o ).
% 2.28/1.40
% 2.28/1.40 thf(omega0Ax,definition,
% 2.28/1.40 ( omega0Ax
% 2.28/1.40 = ( in @ emptyset @ omega ) ) ).
% 2.28/1.40
% 2.28/1.40 thf(omegaSAx_type,type,
% 2.28/1.40 omegaSAx: $o ).
% 2.28/1.40
% 2.28/1.40 thf(omegaSAx,definition,
% 2.28/1.40 ( omegaSAx
% 2.28/1.40 = ( ! [Xx: $i] :
% 2.28/1.40 ( ( in @ Xx @ omega )
% 2.28/1.40 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 2.28/1.40
% 2.28/1.40 thf(omegaIndAx_type,type,
% 2.28/1.40 omegaIndAx: $o ).
% 2.28/1.40
% 2.28/1.40 thf(omegaIndAx,definition,
% 2.28/1.40 ( omegaIndAx
% 2.28/1.40 = ( ! [A: $i] :
% 2.28/1.40 ( ( ( in @ emptyset @ A )
% 2.28/1.40 & ! [Xx: $i] :
% 2.28/1.40 ( ( ( in @ Xx @ omega )
% 2.28/1.40 & ( in @ Xx @ A ) )
% 2.28/1.40 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 2.28/1.40 => ! [Xx: $i] :
% 2.28/1.40 ( ( in @ Xx @ omega )
% 2.28/1.40 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.28/1.40
% 2.28/1.40 thf(replAx_type,type,
% 2.28/1.40 replAx: $o ).
% 2.28/1.40
% 2.28/1.40 thf(replAx,definition,
% 2.28/1.40 ( replAx
% 2.28/1.40 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 2.28/1.40 ( ! [Xx: $i] :
% 2.28/1.40 ( ( in @ Xx @ A )
% 2.28/1.40 => ( exu
% 2.28/1.40 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 2.28/1.40 => ? [B: $i] :
% 2.28/1.40 ! [Xx: $i] :
% 2.28/1.40 ( ( in @ Xx @ B )
% 2.28/1.40 <=> ? [Xy: $i] :
% 2.28/1.40 ( ( in @ Xy @ A )
% 2.28/1.40 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 2.28/1.40
% 2.28/1.40 thf(foundationAx_type,type,
% 2.28/1.40 foundationAx: $o ).
% 2.28/1.40
% 2.28/1.40 thf(foundationAx,definition,
% 2.28/1.40 ( foundationAx
% 2.28/1.40 = ( ! [A: $i] :
% 2.28/1.40 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.28/1.40 => ? [B: $i] :
% 2.28/1.40 ( ( in @ B @ A )
% 2.28/1.40 & ~ ? [Xx: $i] :
% 2.28/1.40 ( ( in @ Xx @ B )
% 2.28/1.40 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 2.28/1.40
% 2.28/1.40 thf(wellorderingAx_type,type,
% 2.28/1.40 wellorderingAx: $o ).
% 2.28/1.40
% 2.28/1.40 thf(wellorderingAx,definition,
% 2.28/1.40 ( wellorderingAx
% 2.28/1.40 = ( ! [A: $i] :
% 2.28/1.40 ? [B: $i] :
% 2.28/1.40 ( ! [C: $i] :
% 2.28/1.40 ( ( in @ C @ B )
% 2.28/1.40 => ! [Xx: $i] :
% 2.28/1.40 ( ( in @ Xx @ C )
% 2.28/1.40 => ( in @ Xx @ A ) ) )
% 2.28/1.40 & ! [Xx: $i,Xy: $i] :
% 2.28/1.40 ( ( ( in @ Xx @ A )
% 2.28/1.40 & ( in @ Xy @ A ) )
% 2.28/1.40 => ( ! [C: $i] :
% 2.28/1.40 ( ( in @ C @ B )
% 2.28/1.40 => ( ( in @ Xx @ C )
% 2.28/1.40 <=> ( in @ Xy @ C ) ) )
% 2.28/1.40 => ( Xx = Xy ) ) )
% 2.28/1.40 & ! [C: $i,D: $i] :
% 2.28/1.40 ( ( ( in @ C @ B )
% 2.28/1.40 & ( in @ D @ B ) )
% 2.28/1.40 => ( ! [Xx: $i] :
% 2.28/1.40 ( ( in @ Xx @ C )
% 2.28/1.40 => ( in @ Xx @ D ) )
% 2.28/1.40 | ! [Xx: $i] :
% 2.28/1.40 ( ( in @ Xx @ D )
% 2.28/1.40 => ( in @ Xx @ C ) ) ) )
% 2.28/1.40 & ! [C: $i] :
% 2.28/1.40 ( ( ! [Xx: $i] :
% 2.28/1.40 ( ( in @ Xx @ C )
% 2.28/1.40 => ( in @ Xx @ A ) )
% 2.28/1.40 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 2.28/1.40 => ? [D: $i,Xx: $i] :
% 2.28/1.40 ( ( in @ D @ B )
% 2.28/1.40 & ( in @ Xx @ C )
% 2.28/1.40 & ~ ? [Xy: $i] :
% 2.28/1.40 ( ( in @ Xy @ D )
% 2.28/1.40 & ( in @ Xy @ C ) )
% 2.28/1.40 & ! [E: $i] :
% 2.28/1.40 ( ( in @ E @ B )
% 2.28/1.40 => ( ! [Xy: $i] :
% 2.28/1.40 ( ( in @ Xy @ E )
% 2.28/1.40 => ( in @ Xy @ D ) )
% 2.28/1.40 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 2.28/1.40
% 2.28/1.40 thf(descr_type,type,
% 2.28/1.40 descr: ( $i > $o ) > $i ).
% 2.28/1.40
% 2.28/1.40 thf(descrp_type,type,
% 2.28/1.40 descrp: $o ).
% 2.28/1.40
% 2.28/1.40 thf(descrp,definition,
% 2.28/1.40 ( descrp
% 2.28/1.40 = ( ! [Xphi: $i > $o] :
% 2.28/1.40 ( ( exu
% 2.28/1.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.28/1.40 => ( Xphi
% 2.28/1.40 @ ( descr
% 2.28/1.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.28/1.40
% 2.28/1.40 thf(dsetconstr_type,type,
% 2.28/1.40 dsetconstr: $i > ( $i > $o ) > $i ).
% 2.28/1.40
% 2.28/1.40 thf(dsetconstrI_type,type,
% 2.28/1.40 dsetconstrI: $o ).
% 2.28/1.40
% 2.28/1.40 thf(dsetconstrI,definition,
% 2.28/1.40 ( dsetconstrI
% 2.28/1.40 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.28/1.40 ( ( in @ Xx @ A )
% 2.28/1.40 => ( ( Xphi @ Xx )
% 2.28/1.40 => ( in @ Xx
% 2.28/1.40 @ ( dsetconstr @ A
% 2.28/1.40 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.28/1.40
% 2.28/1.40 thf(dsetconstrEL_type,type,
% 2.28/1.40 dsetconstrEL: $o ).
% 2.28/1.40
% 2.28/1.40 thf(dsetconstrEL,definition,
% 2.28/1.40 ( dsetconstrEL
% 2.28/1.40 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.28/1.40 ( ( in @ Xx
% 2.28/1.40 @ ( dsetconstr @ A
% 2.28/1.40 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.28/1.40 => ( in @ Xx @ A ) ) ) ) ).
% 2.28/1.40
% 2.28/1.40 thf(dsetconstrER_type,type,
% 2.28/1.40 dsetconstrER: $o ).
% 2.28/1.40
% 2.28/1.40 thf(dsetconstrER,definition,
% 2.28/1.40 ( dsetconstrER
% 2.28/1.40 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.28/1.40 ( ( in @ Xx
% 2.28/1.40 @ ( dsetconstr @ A
% 2.28/1.40 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.28/1.40 => ( Xphi @ Xx ) ) ) ) ).
% 2.28/1.40
% 2.28/1.40 thf(exuE1_type,type,
% 2.28/1.40 exuE1: $o ).
% 2.28/1.40
% 2.28/1.40 thf(exuE1,definition,
% 2.28/1.40 ( exuE1
% 2.28/1.40 = ( ! [Xphi: $i > $o] :
% 2.28/1.40 ( ( exu
% 2.28/1.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.28/1.40 => ? [Xx: $i] :
% 2.28/1.40 ( ( Xphi @ Xx )
% 2.28/1.40 & ! [Xy: $i] :
% 2.28/1.40 ( ( Xphi @ Xy )
% 2.28/1.40 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.28/1.40
% 2.28/1.40 thf(prop2set_type,type,
% 2.28/1.40 prop2set: $o > $i ).
% 2.28/1.40
% 2.28/1.40 thf(prop2setE_type,type,
% 2.28/1.40 prop2setE: $o ).
% 2.28/1.40
% 2.28/1.40 thf(prop2setE,definition,
% 2.28/1.40 ( prop2setE
% 2.28/1.40 = ( ! [Xphi: $o,Xx: $i] :
% 2.28/1.40 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 2.28/1.40 => Xphi ) ) ) ).
% 2.28/1.40
% 2.28/1.40 thf(emptysetE_type,type,
% 2.28/1.40 emptysetE: $o ).
% 2.28/1.40
% 2.28/1.40 thf(emptysetE,definition,
% 2.28/1.40 ( emptysetE
% 2.28/1.40 = ( ! [Xx: $i] :
% 2.28/1.40 ( ( in @ Xx @ emptyset )
% 2.28/1.40 => ! [Xphi: $o] : Xphi ) ) ) ).
% 2.28/1.40
% 2.28/1.40 thf(emptysetimpfalse_type,type,
% 2.28/1.40 emptysetimpfalse: $o ).
% 2.28/1.40
% 2.28/1.40 thf(emptysetimpfalse,definition,
% 2.28/1.40 ( emptysetimpfalse
% 2.28/1.40 = ( ! [Xx: $i] :
% 2.28/1.40 ( ( in @ Xx @ emptyset )
% 2.28/1.40 => $false ) ) ) ).
% 2.28/1.40
% 2.28/1.40 thf(notinemptyset_type,type,
% 2.28/1.40 notinemptyset: $o ).
% 2.28/1.40
% 2.28/1.40 thf(notinemptyset,definition,
% 2.28/1.40 ( notinemptyset
% 2.28/1.40 = ( ! [Xx: $i] :
% 2.28/1.40 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.28/1.40
% 2.28/1.40 thf(exuE3e_type,type,
% 2.28/1.40 exuE3e: $o ).
% 2.28/1.40
% 2.28/1.40 thf(exuE3e,definition,
% 2.28/1.40 ( exuE3e
% 2.28/1.40 = ( ! [Xphi: $i > $o] :
% 2.28/1.40 ( ( exu
% 2.28/1.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.28/1.40 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 2.28/1.40
% 2.28/1.40 thf(setext_type,type,
% 2.28/1.40 setext: $o ).
% 2.28/1.40
% 2.28/1.40 thf(setext,definition,
% 2.28/1.40 ( setext
% 2.28/1.40 = ( ! [A: $i,B: $i] :
% 2.28/1.40 ( ! [Xx: $i] :
% 2.28/1.40 ( ( in @ Xx @ A )
% 2.28/1.40 => ( in @ Xx @ B ) )
% 2.28/1.40 => ( ! [Xx: $i] :
% 2.28/1.40 ( ( in @ Xx @ B )
% 2.28/1.40 => ( in @ Xx @ A ) )
% 2.28/1.40 => ( A = B ) ) ) ) ) ).
% 2.28/1.40
% 2.28/1.40 thf(emptyI_type,type,
% 2.28/1.40 emptyI: $o ).
% 2.28/1.40
% 2.28/1.40 thf(emptyI,definition,
% 2.28/1.40 ( emptyI
% 2.28/1.40 = ( ! [A: $i] :
% 2.28/1.40 ( ! [Xx: $i] :
% 2.28/1.40 ~ ( in @ Xx @ A )
% 2.28/1.40 => ( A = emptyset ) ) ) ) ).
% 2.28/1.40
% 2.28/1.40 thf(noeltsimpempty_type,type,
% 2.28/1.40 noeltsimpempty: $o ).
% 2.28/1.40
% 2.28/1.40 thf(noeltsimpempty,definition,
% 2.28/1.40 ( noeltsimpempty
% 2.28/1.40 = ( ! [A: $i] :
% 2.28/1.40 ( ! [Xx: $i] :
% 2.28/1.40 ~ ( in @ Xx @ A )
% 2.28/1.40 => ( A = emptyset ) ) ) ) ).
% 2.28/1.40
% 2.28/1.40 thf(setbeta_type,type,
% 2.28/1.40 setbeta: $o ).
% 2.28/1.40
% 2.28/1.40 thf(setbeta,definition,
% 2.28/1.40 ( setbeta
% 2.28/1.40 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.28/1.40 ( ( in @ Xx @ A )
% 2.28/1.40 => ( ( in @ Xx
% 2.28/1.40 @ ( dsetconstr @ A
% 2.28/1.40 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.28/1.40 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 2.28/1.40
% 2.28/1.40 thf(nonempty_type,type,
% 2.28/1.40 nonempty: $i > $o ).
% 2.28/1.40
% 2.28/1.40 thf(nonempty,definition,
% 2.28/1.40 ( nonempty
% 2.28/1.40 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 2.28/1.40
% 2.28/1.40 thf(nonemptyE1_type,type,
% 2.28/1.40 nonemptyE1: $o ).
% 2.28/1.40
% 2.28/1.40 thf(nonemptyE1,definition,
% 2.28/1.40 ( nonemptyE1
% 2.28/1.40 = ( ! [A: $i] :
% 2.28/1.40 ( ( nonempty @ A )
% 2.28/1.40 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 2.28/1.40
% 2.28/1.40 thf(nonemptyI_type,type,
% 2.28/1.40 nonemptyI: $o ).
% 2.28/1.40
% 2.28/1.40 thf(nonemptyI,definition,
% 2.28/1.40 ( nonemptyI
% 2.28/1.40 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.28/1.40 ( ( in @ Xx @ A )
% 2.28/1.40 => ( ( Xphi @ Xx )
% 2.28/1.40 => ( nonempty
% 2.28/1.40 @ ( dsetconstr @ A
% 2.28/1.40 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.28/1.40
% 2.28/1.40 thf(nonemptyI1_type,type,
% 2.28/1.40 nonemptyI1: $o ).
% 2.28/1.40
% 2.28/1.40 thf(nonemptyI1,definition,
% 2.28/1.40 ( nonemptyI1
% 2.28/1.40 = ( ! [A: $i] :
% 2.28/1.40 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.28/1.40 => ( nonempty @ A ) ) ) ) ).
% 2.28/1.40
% 2.28/1.40 thf(setadjoinIL_type,type,
% 2.28/1.40 setadjoinIL: $o ).
% 2.28/1.40
% 2.28/1.40 thf(setadjoinIL,definition,
% 2.28/1.40 ( setadjoinIL
% 2.28/1.40 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 2.28/1.40
% 2.28/1.40 thf(emptyinunitempty_type,type,
% 2.28/1.40 emptyinunitempty: $o ).
% 2.28/1.40
% 2.28/1.40 thf(emptyinunitempty,definition,
% 2.28/1.40 ( emptyinunitempty
% 2.28/1.40 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 2.28/1.40
% 2.28/1.40 thf(setadjoinIR_type,type,
% 2.28/1.40 setadjoinIR: $o ).
% 2.28/1.40
% 2.28/1.40 thf(setadjoinIR,definition,
% 2.28/1.40 ( setadjoinIR
% 2.28/1.40 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.28/1.40 ( ( in @ Xy @ A )
% 2.28/1.40 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 2.28/1.40
% 2.28/1.40 thf(setadjoinE_type,type,
% 2.28/1.40 setadjoinE: $o ).
% 2.28/1.40
% 2.28/1.40 thf(setadjoinE,definition,
% 2.28/1.40 ( setadjoinE
% 2.28/1.40 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.28/1.40 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.28/1.40 => ! [Xphi: $o] :
% 2.28/1.40 ( ( ( Xy = Xx )
% 2.28/1.40 => Xphi )
% 2.28/1.40 => ( ( ( in @ Xy @ A )
% 2.28/1.40 => Xphi )
% 2.28/1.40 => Xphi ) ) ) ) ) ).
% 2.28/1.40
% 2.28/1.40 thf(setadjoinOr_type,type,
% 2.28/1.40 setadjoinOr: $o ).
% 2.28/1.40
% 2.28/1.40 thf(setadjoinOr,definition,
% 2.28/1.40 ( setadjoinOr
% 2.28/1.40 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.28/1.40 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.28/1.40 => ( ( Xy = Xx )
% 2.28/1.40 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.28/1.40
% 2.28/1.40 thf(setoftrueEq_type,type,
% 2.28/1.40 setoftrueEq: $o ).
% 2.28/1.40
% 2.28/1.40 thf(setoftrueEq,definition,
% 2.28/1.40 ( setoftrueEq
% 2.28/1.40 = ( ! [A: $i] :
% 2.28/1.40 ( ( dsetconstr @ A
% 2.28/1.40 @ ^ [Xx: $i] : $true )
% 2.28/1.40 = A ) ) ) ).
% 2.28/1.40
% 2.28/1.40 thf(powersetI_type,type,
% 2.28/1.40 powersetI: $o ).
% 2.28/1.40
% 2.28/1.40 thf(powersetI,definition,
% 2.28/1.40 ( powersetI
% 2.28/1.40 = ( ! [A: $i,B: $i] :
% 2.28/1.40 ( ! [Xx: $i] :
% 2.28/1.40 ( ( in @ Xx @ B )
% 2.28/1.40 => ( in @ Xx @ A ) )
% 2.28/1.40 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.28/1.40
% 2.28/1.40 thf(emptyinPowerset_type,type,
% 2.28/1.40 emptyinPowerset: $o ).
% 2.28/1.40
% 2.28/1.40 thf(emptyinPowerset,definition,
% 2.28/1.40 ( emptyinPowerset
% 2.28/1.40 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.28/1.40
% 2.28/1.40 thf(emptyInPowerset_type,type,
% 2.28/1.40 emptyInPowerset: $o ).
% 2.28/1.40
% 2.28/1.40 thf(emptyInPowerset,definition,
% 2.28/1.40 ( emptyInPowerset
% 2.28/1.40 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.28/1.40
% 2.28/1.40 thf(powersetE_type,type,
% 2.28/1.40 powersetE: $o ).
% 2.28/1.40
% 2.28/1.40 thf(powersetE,definition,
% 2.28/1.40 ( powersetE
% 2.28/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.28/1.40 ( ( in @ B @ ( powerset @ A ) )
% 2.28/1.40 => ( ( in @ Xx @ B )
% 2.28/1.40 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.28/1.40
% 2.28/1.40 thf(setunionI_type,type,
% 2.28/1.40 setunionI: $o ).
% 2.28/1.40
% 2.28/1.40 thf(setunionI,definition,
% 2.28/1.40 ( setunionI
% 2.28/1.40 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.28/1.40 ( ( in @ Xx @ B )
% 2.28/1.40 => ( ( in @ B @ A )
% 2.28/1.40 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 2.28/1.40
% 2.28/1.40 thf(setunionE_type,type,
% 2.28/1.40 setunionE: $o ).
% 2.28/1.40
% 2.28/1.40 thf(setunionE,definition,
% 2.28/1.40 ( setunionE
% 2.28/1.40 = ( ! [A: $i,Xx: $i] :
% 2.28/1.40 ( ( in @ Xx @ ( setunion @ A ) )
% 2.28/1.40 => ! [Xphi: $o] :
% 2.28/1.40 ( ! [B: $i] :
% 2.28/1.40 ( ( in @ Xx @ B )
% 2.28/1.40 => ( ( in @ B @ A )
% 2.28/1.40 => Xphi ) )
% 2.28/1.40 => Xphi ) ) ) ) ).
% 2.28/1.40
% 2.28/1.40 thf(subPowSU_type,type,
% 2.28/1.40 subPowSU: $o ).
% 2.28/1.40
% 2.28/1.40 thf(subPowSU,definition,
% 2.28/1.40 ( subPowSU
% 2.28/1.40 = ( ! [A: $i,Xx: $i] :
% 2.28/1.40 ( ( in @ Xx @ A )
% 2.28/1.40 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 2.28/1.41
% 2.28/1.41 thf(exuE2_type,type,
% 2.28/1.41 exuE2: $o ).
% 2.28/1.41
% 2.28/1.41 thf(exuE2,definition,
% 2.28/1.41 ( exuE2
% 2.28/1.41 = ( ! [Xphi: $i > $o] :
% 2.28/1.41 ( ( exu
% 2.28/1.41 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.28/1.41 => ? [Xx: $i] :
% 2.28/1.41 ! [Xy: $i] :
% 2.28/1.41 ( ( Xphi @ Xy )
% 2.28/1.41 <=> ( Xy = Xx ) ) ) ) ) ).
% 2.28/1.41
% 2.28/1.41 thf(nonemptyImpWitness_type,type,
% 2.28/1.41 nonemptyImpWitness: $o ).
% 2.28/1.41
% 2.28/1.41 thf(nonemptyImpWitness,definition,
% 2.28/1.41 ( nonemptyImpWitness
% 2.28/1.41 = ( ! [A: $i] :
% 2.28/1.41 ( ( nonempty @ A )
% 2.28/1.41 => ? [Xx: $i] :
% 2.28/1.41 ( ( in @ Xx @ A )
% 2.28/1.41 & $true ) ) ) ) ).
% 2.28/1.41
% 2.28/1.41 thf(uniqinunit_type,type,
% 2.28/1.41 uniqinunit: $o ).
% 2.28/1.41
% 2.28/1.41 thf(uniqinunit,definition,
% 2.28/1.41 ( uniqinunit
% 2.28/1.41 = ( ! [Xx: $i,Xy: $i] :
% 2.28/1.41 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.28/1.41 => ( Xx = Xy ) ) ) ) ).
% 2.28/1.41
% 2.28/1.41 thf(notinsingleton_type,type,
% 2.28/1.41 notinsingleton: $o ).
% 2.28/1.41
% 2.28/1.41 thf(notinsingleton,definition,
% 2.28/1.41 ( notinsingleton
% 2.28/1.41 = ( ! [Xx: $i,Xy: $i] :
% 2.28/1.41 ( ( Xx != Xy )
% 2.28/1.41 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.28/1.41
% 2.28/1.41 thf(eqinunit_type,type,
% 2.28/1.41 eqinunit: $o ).
% 2.28/1.41
% 2.28/1.41 thf(eqinunit,definition,
% 2.28/1.41 ( eqinunit
% 2.28/1.41 = ( ! [Xx: $i,Xy: $i] :
% 2.28/1.41 ( ( Xx = Xy )
% 2.28/1.41 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.28/1.41
% 2.28/1.41 thf(singletonsswitch_type,type,
% 2.28/1.41 singletonsswitch: $o ).
% 2.28/1.41
% 2.28/1.41 thf(singletonsswitch,definition,
% 2.28/1.41 ( singletonsswitch
% 2.28/1.41 = ( ! [Xx: $i,Xy: $i] :
% 2.28/1.41 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.28/1.41 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.28/1.41
% 2.28/1.41 thf(upairsetE_type,type,
% 2.28/1.41 upairsetE: $o ).
% 2.28/1.41
% 2.28/1.41 thf(upairsetE,definition,
% 2.28/1.41 ( upairsetE
% 2.28/1.41 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.28/1.41 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.28/1.41 => ( ( Xz = Xx )
% 2.28/1.41 | ( Xz = Xy ) ) ) ) ) ).
% 2.28/1.41
% 2.28/1.41 thf(upairsetIL_type,type,
% 2.28/1.41 upairsetIL: $o ).
% 2.28/1.41
% 2.28/1.41 thf(upairsetIL,definition,
% 2.28/1.41 ( upairsetIL
% 2.28/1.41 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.28/1.41
% 2.28/1.41 thf(upairsetIR_type,type,
% 2.28/1.41 upairsetIR: $o ).
% 2.28/1.41
% 2.28/1.41 thf(upairsetIR,definition,
% 2.28/1.41 ( upairsetIR
% 2.28/1.41 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.28/1.41
% 2.28/1.41 thf(emptyE1_type,type,
% 2.28/1.41 emptyE1: $o ).
% 2.28/1.41
% 2.28/1.41 thf(emptyE1,definition,
% 2.28/1.41 ( emptyE1
% 2.28/1.41 = ( ! [A: $i,Xphi: $i > $o] :
% 2.28/1.41 ( ? [Xx: $i] :
% 2.28/1.41 ( ( in @ Xx @ A )
% 2.28/1.41 & ( Xphi @ Xx ) )
% 2.28/1.41 => ( ( ( dsetconstr @ A
% 2.28/1.41 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.28/1.41 = emptyset )
% 2.28/1.41 => $false ) ) ) ) ).
% 2.28/1.41
% 2.28/1.41 thf(vacuousDall_type,type,
% 2.28/1.41 vacuousDall: $o ).
% 2.28/1.41
% 2.28/1.41 thf(vacuousDall,definition,
% 2.28/1.41 ( vacuousDall
% 2.28/1.41 = ( ! [Xphi: $i > $o,Xx: $i] :
% 2.28/1.41 ( ( in @ Xx @ emptyset )
% 2.28/1.41 => ( Xphi @ Xx ) ) ) ) ).
% 2.28/1.41
% 2.28/1.41 thf(quantDeMorgan1_type,type,
% 2.28/1.41 quantDeMorgan1: $o ).
% 2.28/1.41
% 2.28/1.41 thf(quantDeMorgan1,definition,
% 2.28/1.41 ( quantDeMorgan1
% 2.28/1.41 = ( ! [A: $i,Xphi: $i > $o] :
% 2.28/1.41 ( ~ ! [Xx: $i] :
% 2.28/1.41 ( ( in @ Xx @ A )
% 2.28/1.41 => ( Xphi @ Xx ) )
% 2.28/1.41 => ? [Xx: $i] :
% 2.28/1.41 ( ( in @ Xx @ A )
% 2.28/1.41 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.28/1.41
% 2.28/1.41 thf(quantDeMorgan2_type,type,
% 2.28/1.41 quantDeMorgan2: $o ).
% 2.28/1.41
% 2.28/1.41 thf(quantDeMorgan2,definition,
% 2.28/1.41 ( quantDeMorgan2
% 2.28/1.41 = ( ! [A: $i,Xphi: $i > $o] :
% 2.28/1.41 ( ! [Xx: $i] :
% 2.28/1.41 ( ( in @ Xx @ A )
% 2.28/1.41 => ~ ( Xphi @ Xx ) )
% 2.28/1.41 => ~ ? [Xx: $i] :
% 2.28/1.41 ( ( in @ Xx @ A )
% 2.28/1.41 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.28/1.41
% 2.28/1.41 thf(quantDeMorgan3_type,type,
% 2.28/1.41 quantDeMorgan3: $o ).
% 2.28/1.41
% 2.28/1.41 thf(quantDeMorgan3,definition,
% 2.28/1.41 ( quantDeMorgan3
% 2.28/1.41 = ( ! [A: $i,Xphi: $i > $o] :
% 2.28/1.41 ( ~ ? [Xx: $i] :
% 2.28/1.41 ( ( in @ Xx @ A )
% 2.28/1.41 & ( Xphi @ Xx ) )
% 2.28/1.41 => ! [Xx: $i] :
% 2.28/1.41 ( ( in @ Xx @ A )
% 2.28/1.41 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.28/1.41
% 2.28/1.41 thf(quantDeMorgan4_type,type,
% 2.28/1.41 quantDeMorgan4: $o ).
% 2.28/1.41
% 2.28/1.41 thf(quantDeMorgan4,definition,
% 2.28/1.41 ( quantDeMorgan4
% 2.28/1.41 = ( ! [A: $i,Xphi: $i > $o] :
% 2.28/1.41 ( ? [Xx: $i] :
% 2.28/1.41 ( ( in @ Xx @ A )
% 2.28/1.41 & ~ ( Xphi @ Xx ) )
% 2.28/1.41 => ~ ! [Xx: $i] :
% 2.28/1.41 ( ( in @ Xx @ A )
% 2.28/1.41 => ( Xphi @ Xx ) ) ) ) ) ).
% 2.28/1.41
% 2.28/1.41 thf(prop2setI_type,type,
% 2.28/1.41 prop2setI: $o ).
% 2.28/1.41
% 2.28/1.41 thf(prop2setI,definition,
% 2.28/1.41 ( prop2setI
% 2.28/1.41 = ( ! [Xphi: $o] :
% 2.28/1.41 ( Xphi
% 2.28/1.41 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.28/1.41
% 2.28/1.41 thf(set2prop_type,type,
% 2.28/1.41 set2prop: $i > $o ).
% 2.28/1.41
% 2.28/1.41 thf(prop2set2propI_type,type,
% 2.28/1.41 prop2set2propI: $o ).
% 2.28/1.41
% 2.28/1.41 thf(prop2set2propI,definition,
% 2.28/1.41 ( prop2set2propI
% 2.28/1.41 = ( ! [Xphi: $o] :
% 2.28/1.41 ( Xphi
% 2.28/1.41 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.28/1.41
% 2.28/1.41 thf(notdexE_type,type,
% 2.28/1.41 notdexE: $o ).
% 2.28/1.41
% 2.28/1.41 thf(notdexE,definition,
% 2.28/1.41 ( notdexE
% 2.28/1.41 = ( ! [A: $i,Xphi: $i > $o] :
% 2.28/1.41 ( ~ ? [Xx: $i] :
% 2.28/1.41 ( ( in @ Xx @ A )
% 2.28/1.41 & ( Xphi @ Xx ) )
% 2.28/1.41 => ! [Xx: $i] :
% 2.28/1.41 ( ( in @ Xx @ A )
% 2.28/1.41 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.28/1.41
% 2.28/1.41 thf(notdallE_type,type,
% 2.28/1.41 notdallE: $o ).
% 2.28/1.41
% 2.28/1.41 thf(notdallE,definition,
% 2.28/1.41 ( notdallE
% 2.28/1.41 = ( ! [A: $i,Xphi: $i > $o] :
% 2.28/1.41 ( ~ ! [Xx: $i] :
% 2.28/1.41 ( ( in @ Xx @ A )
% 2.28/1.41 => ( Xphi @ Xx ) )
% 2.28/1.41 => ? [Xx: $i] :
% 2.28/1.41 ( ( in @ Xx @ A )
% 2.28/1.41 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.28/1.41
% 2.28/1.41 thf(exuI1_type,type,
% 2.28/1.41 exuI1: $o ).
% 2.28/1.41
% 2.28/1.41 thf(exuI1,definition,
% 2.28/1.41 ( exuI1
% 2.28/1.41 = ( ! [Xphi: $i > $o] :
% 2.28/1.41 ( ? [Xx: $i] :
% 2.28/1.41 ( ( Xphi @ Xx )
% 2.28/1.41 & ! [Xy: $i] :
% 2.28/1.41 ( ( Xphi @ Xy )
% 2.28/1.41 => ( Xx = Xy ) ) )
% 2.28/1.41 => ( exu
% 2.28/1.41 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.28/1.41
% 2.28/1.41 thf(exuI3_type,type,
% 2.28/1.41 exuI3: $o ).
% 2.28/1.41
% 2.28/1.41 thf(exuI3,definition,
% 2.28/1.41 ( exuI3
% 2.28/1.41 = ( ! [Xphi: $i > $o] :
% 2.28/1.41 ( ? [Xx: $i] : ( Xphi @ Xx )
% 2.28/1.41 => ( ! [Xx: $i,Xy: $i] :
% 2.28/1.41 ( ( Xphi @ Xx )
% 2.28/1.41 => ( ( Xphi @ Xy )
% 2.28/1.41 => ( Xx = Xy ) ) )
% 2.28/1.41 => ( exu
% 2.28/1.41 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.28/1.41
% 2.28/1.41 thf(exuI2_type,type,
% 2.28/1.41 exuI2: $o ).
% 2.28/1.41
% 2.28/1.41 thf(exuI2,definition,
% 2.28/1.41 ( exuI2
% 2.28/1.41 = ( ! [Xphi: $i > $o] :
% 2.28/1.41 ( ? [Xx: $i] :
% 2.28/1.41 ! [Xy: $i] :
% 2.28/1.41 ( ( Xphi @ Xy )
% 2.28/1.41 <=> ( Xy = Xx ) )
% 2.28/1.41 => ( exu
% 2.28/1.41 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.28/1.41
% 2.28/1.41 thf(inCongP_type,type,
% 2.28/1.41 inCongP: $o ).
% 2.28/1.41
% 2.28/1.41 thf(inCongP,definition,
% 2.28/1.41 ( inCongP
% 2.28/1.41 = ( ! [A: $i,B: $i] :
% 2.28/1.41 ( ( A = B )
% 2.28/1.41 => ! [Xx: $i,Xy: $i] :
% 2.28/1.41 ( ( Xx = Xy )
% 2.28/1.41 => ( ( in @ Xx @ A )
% 2.28/1.41 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.28/1.41
% 2.28/1.41 thf(in__Cong_type,type,
% 2.28/1.41 in__Cong: $o ).
% 2.28/1.41
% 2.28/1.41 thf(in__Cong,definition,
% 2.28/1.41 ( in__Cong
% 2.28/1.41 = ( ! [A: $i,B: $i] :
% 2.28/1.41 ( ( A = B )
% 2.28/1.41 => ! [Xx: $i,Xy: $i] :
% 2.28/1.41 ( ( Xx = Xy )
% 2.28/1.41 => ( ( in @ Xx @ A )
% 2.28/1.41 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.28/1.41
% 2.28/1.41 thf(exuE3u_type,type,
% 2.28/1.41 exuE3u: $o ).
% 2.28/1.41
% 2.28/1.41 thf(exuE3u,definition,
% 2.28/1.41 ( exuE3u
% 2.28/1.41 = ( ! [Xphi: $i > $o] :
% 2.28/1.41 ( ( exu
% 2.28/1.41 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.28/1.41 => ! [Xx: $i,Xy: $i] :
% 2.28/1.41 ( ( Xphi @ Xx )
% 2.28/1.41 => ( ( Xphi @ Xy )
% 2.28/1.41 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.28/1.41
% 2.28/1.41 thf(exu__Cong_type,type,
% 2.28/1.41 exu__Cong: $o ).
% 2.28/1.41
% 2.28/1.41 thf(exu__Cong,definition,
% 2.28/1.41 ( exu__Cong
% 2.28/1.41 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.28/1.41 ( ! [Xx: $i,Xy: $i] :
% 2.28/1.41 ( ( Xx = Xy )
% 2.28/1.41 => ( ( Xphi @ Xx )
% 2.28/1.41 <=> ( Xpsi @ Xy ) ) )
% 2.28/1.41 => ( ( exu
% 2.28/1.41 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.28/1.41 <=> ( exu
% 2.28/1.41 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 2.28/1.41
% 2.28/1.41 thf(emptyset__Cong_type,type,
% 2.28/1.41 emptyset__Cong: $o ).
% 2.28/1.41
% 2.28/1.41 thf(emptyset__Cong,definition,
% 2.28/1.41 ( emptyset__Cong
% 2.28/1.41 = ( emptyset = emptyset ) ) ).
% 2.28/1.41
% 2.28/1.41 thf(setadjoin__Cong_type,type,
% 2.28/1.41 setadjoin__Cong: $o ).
% 2.28/1.41
% 2.28/1.41 thf(setadjoin__Cong,definition,
% 2.28/1.41 ( setadjoin__Cong
% 2.28/1.41 = ( ! [Xx: $i,Xy: $i] :
% 2.28/1.41 ( ( Xx = Xy )
% 2.28/1.41 => ! [Xz: $i,Xu: $i] :
% 2.28/1.41 ( ( Xz = Xu )
% 2.28/1.41 => ( ( setadjoin @ Xx @ Xz )
% 2.28/1.41 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 2.28/1.41
% 2.28/1.41 thf(powerset__Cong_type,type,
% 2.28/1.41 powerset__Cong: $o ).
% 2.28/1.41
% 2.28/1.41 thf(powerset__Cong,definition,
% 2.28/1.41 ( powerset__Cong
% 2.28/1.41 = ( ! [A: $i,B: $i] :
% 2.28/1.41 ( ( A = B )
% 2.28/1.41 => ( ( powerset @ A )
% 2.28/1.41 = ( powerset @ B ) ) ) ) ) ).
% 2.28/1.41
% 2.28/1.41 thf(setunion__Cong_type,type,
% 2.28/1.41 setunion__Cong: $o ).
% 2.28/1.41
% 2.28/1.41 thf(setunion__Cong,definition,
% 2.28/1.41 ( setunion__Cong
% 2.28/1.41 = ( ! [A: $i,B: $i] :
% 2.28/1.41 ( ( A = B )
% 2.28/1.41 => ( ( setunion @ A )
% 2.28/1.41 = ( setunion @ B ) ) ) ) ) ).
% 2.28/1.41
% 2.28/1.41 thf(omega__Cong_type,type,
% 2.28/1.41 omega__Cong: $o ).
% 2.28/1.41
% 2.28/1.41 thf(omega__Cong,definition,
% 2.28/1.41 ( omega__Cong
% 2.28/1.41 = ( omega = omega ) ) ).
% 2.28/1.41
% 2.28/1.41 thf(exuEu_type,type,
% 2.28/1.41 exuEu: $o ).
% 2.28/1.41
% 2.28/1.41 thf(exuEu,definition,
% 2.28/1.41 ( exuEu
% 2.28/1.41 = ( ! [Xphi: $i > $o] :
% 2.28/1.41 ( ( exu
% 2.28/1.41 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.28/1.41 => ! [Xx: $i,Xy: $i] :
% 2.28/1.41 ( ( Xphi @ Xx )
% 2.28/1.41 => ( ( Xphi @ Xy )
% 2.28/1.41 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.28/1.41
% 2.28/1.41 thf(descr__Cong_type,type,
% 2.28/1.41 descr__Cong: $o ).
% 2.28/1.41
% 2.28/1.41 thf(descr__Cong,definition,
% 2.28/1.41 ( descr__Cong
% 2.28/1.41 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.28/1.41 ( ! [Xx: $i,Xy: $i] :
% 2.28/1.41 ( ( Xx = Xy )
% 2.28/1.41 => ( ( Xphi @ Xx )
% 2.28/1.41 <=> ( Xpsi @ Xy ) ) )
% 2.28/1.41 => ( ( exu
% 2.28/1.41 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.28/1.41 => ( ( exu
% 2.28/1.41 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 2.28/1.41 => ( ( descr
% 2.28/1.41 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.28/1.41 = ( descr
% 2.28/1.41 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 2.28/1.41
% 2.28/1.41 thf(dsetconstr__Cong_type,type,
% 2.28/1.41 dsetconstr__Cong: $o ).
% 2.28/1.41
% 2.28/1.41 thf(dsetconstr__Cong,definition,
% 2.28/1.41 ( dsetconstr__Cong
% 2.28/1.41 = ( ! [A: $i,B: $i] :
% 2.28/1.41 ( ( A = B )
% 2.28/1.41 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.28/1.41 ( ! [Xx: $i] :
% 2.28/1.41 ( ( in @ Xx @ A )
% 2.28/1.41 => ! [Xy: $i] :
% 2.28/1.41 ( ( in @ Xy @ B )
% 2.28/1.41 => ( ( Xx = Xy )
% 2.28/1.41 => ( ( Xphi @ Xx )
% 2.28/1.41 <=> ( Xpsi @ Xy ) ) ) ) )
% 2.28/1.41 => ( ( dsetconstr @ A
% 2.28/1.41 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.28/1.41 = ( dsetconstr @ B
% 2.28/1.41 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 2.28/1.41
% 2.28/1.41 thf(subset_type,type,
% 2.28/1.41 subset: $i > $i > $o ).
% 2.28/1.41
% 2.28/1.41 thf(disjoint_type,type,
% 2.28/1.41 disjoint: $i > $i > $o ).
% 2.28/1.41
% 2.28/1.41 thf(setsmeet_type,type,
% 2.28/1.41 setsmeet: $i > $i > $o ).
% 2.28/1.41
% 2.28/1.41 thf(subsetI1_type,type,
% 2.28/1.41 subsetI1: $o ).
% 2.28/1.41
% 2.28/1.41 thf(subsetI1,definition,
% 2.28/1.41 ( subsetI1
% 2.28/1.41 = ( ! [A: $i,B: $i] :
% 2.28/1.41 ( ! [Xx: $i] :
% 2.28/1.41 ( ( in @ Xx @ A )
% 2.28/1.41 => ( in @ Xx @ B ) )
% 2.28/1.41 => ( subset @ A @ B ) ) ) ) ).
% 2.28/1.41
% 2.28/1.41 thf(eqimpsubset2_type,type,
% 2.28/1.41 eqimpsubset2: $o ).
% 2.28/1.41
% 2.28/1.41 thf(eqimpsubset2,definition,
% 2.28/1.41 ( eqimpsubset2
% 2.28/1.41 = ( ! [A: $i,B: $i] :
% 2.28/1.41 ( ( A = B )
% 2.28/1.41 => ( subset @ B @ A ) ) ) ) ).
% 2.28/1.41
% 2.28/1.41 thf(eqimpsubset1_type,type,
% 2.28/1.41 eqimpsubset1: $o ).
% 2.28/1.41
% 2.28/1.41 thf(eqimpsubset1,definition,
% 2.28/1.41 ( eqimpsubset1
% 2.28/1.41 = ( ! [A: $i,B: $i] :
% 2.28/1.41 ( ( A = B )
% 2.28/1.41 => ( subset @ A @ B ) ) ) ) ).
% 2.28/1.41
% 2.28/1.41 thf(subsetI2_type,type,
% 2.28/1.41 subsetI2: $o ).
% 2.28/1.41
% 2.28/1.41 thf(subsetI2,definition,
% 2.28/1.41 ( subsetI2
% 2.28/1.41 = ( ! [A: $i,B: $i] :
% 2.28/1.41 ( ! [Xx: $i] :
% 2.28/1.41 ( ( in @ Xx @ A )
% 2.28/1.41 => ( in @ Xx @ B ) )
% 2.28/1.41 => ( subset @ A @ B ) ) ) ) ).
% 2.28/1.41
% 2.28/1.41 thf(emptysetsubset_type,type,
% 2.28/1.41 emptysetsubset: $o ).
% 2.28/1.41
% 2.28/1.41 thf(emptysetsubset,definition,
% 2.28/1.41 ( emptysetsubset
% 2.28/1.41 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 2.28/1.41
% 2.28/1.41 thf(subsetE_type,type,
% 2.28/1.41 subsetE: $o ).
% 2.28/1.41
% 2.28/1.41 thf(subsetE,definition,
% 2.28/1.41 ( subsetE
% 2.28/1.41 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.28/1.41 ( ( subset @ A @ B )
% 2.28/1.41 => ( ( in @ Xx @ A )
% 2.28/1.41 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.28/1.41
% 2.28/1.41 thf(subsetE2_type,type,
% 2.28/1.41 subsetE2: $o ).
% 2.28/1.41
% 2.28/1.41 thf(subsetE2,definition,
% 2.28/1.41 ( subsetE2
% 2.28/1.41 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.28/1.41 ( ( subset @ A @ B )
% 2.28/1.41 => ( ~ ( in @ Xx @ B )
% 2.28/1.41 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.28/1.41
% 2.28/1.41 thf(notsubsetI_type,type,
% 2.28/1.41 notsubsetI: $o ).
% 2.28/1.41
% 2.28/1.41 thf(notsubsetI,definition,
% 2.28/1.41 ( notsubsetI
% 2.28/1.41 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.28/1.41 ( ( in @ Xx @ A )
% 2.28/1.41 => ( ~ ( in @ Xx @ B )
% 2.28/1.41 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 2.28/1.41
% 2.28/1.41 thf(notequalI1_type,type,
% 2.28/1.41 notequalI1: $o ).
% 2.28/1.41
% 2.28/1.41 thf(notequalI1,definition,
% 2.28/1.41 ( notequalI1
% 2.28/1.41 = ( ! [A: $i,B: $i] :
% 2.28/1.41 ( ~ ( subset @ A @ B )
% 2.28/1.41 => ( A != B ) ) ) ) ).
% 2.28/1.41
% 2.28/1.41 thf(notequalI2_type,type,
% 2.28/1.41 notequalI2: $o ).
% 2.28/1.41
% 2.28/1.41 thf(notequalI2,definition,
% 2.28/1.41 ( notequalI2
% 2.28/1.41 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.28/1.41 ( ( in @ Xx @ A )
% 2.28/1.41 => ( ~ ( in @ Xx @ B )
% 2.28/1.41 => ( A != B ) ) ) ) ) ).
% 2.28/1.41
% 2.28/1.41 thf(subsetRefl_type,type,
% 2.28/1.41 subsetRefl: $o ).
% 2.28/1.41
% 2.28/1.41 thf(subsetRefl,definition,
% 2.28/1.41 ( subsetRefl
% 2.28/1.41 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 2.28/1.41
% 2.28/1.41 thf(subsetTrans_type,type,
% 2.28/1.41 subsetTrans: $o ).
% 2.28/1.41
% 2.28/1.41 thf(subsetTrans,definition,
% 2.28/1.41 ( subsetTrans
% 2.28/1.41 = ( ! [A: $i,B: $i,C: $i] :
% 2.28/1.41 ( ( subset @ A @ B )
% 2.28/1.41 => ( ( subset @ B @ C )
% 2.28/1.41 => ( subset @ A @ C ) ) ) ) ) ).
% 2.28/1.41
% 2.28/1.41 thf(setadjoinSub_type,type,
% 2.28/1.41 setadjoinSub: $o ).
% 2.28/1.41
% 2.28/1.41 thf(setadjoinSub,definition,
% 2.28/1.41 ( setadjoinSub
% 2.28/1.41 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 2.28/1.41
% 2.28/1.41 thf(setadjoinSub2_type,type,
% 2.28/1.41 setadjoinSub2: $o ).
% 2.28/1.41
% 2.28/1.41 thf(setadjoinSub2,definition,
% 2.28/1.41 ( setadjoinSub2
% 2.28/1.41 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.28/1.41 ( ( subset @ A @ B )
% 2.28/1.41 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 2.28/1.41
% 2.28/1.41 thf(subset2powerset_type,type,
% 2.28/1.41 subset2powerset: $o ).
% 2.28/1.41
% 2.28/1.41 thf(subset2powerset,definition,
% 2.28/1.41 ( subset2powerset
% 2.28/1.41 = ( ! [A: $i,B: $i] :
% 2.28/1.41 ( ( subset @ A @ B )
% 2.28/1.41 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 2.28/1.41
% 2.28/1.41 thf(setextsub_type,type,
% 2.28/1.41 setextsub: $o ).
% 2.28/1.41
% 2.28/1.41 thf(setextsub,definition,
% 2.28/1.41 ( setextsub
% 2.28/1.41 = ( ! [A: $i,B: $i] :
% 2.28/1.41 ( ( subset @ A @ B )
% 2.28/1.41 => ( ( subset @ B @ A )
% 2.28/1.41 => ( A = B ) ) ) ) ) ).
% 2.28/1.41
% 2.28/1.41 thf(subsetemptysetimpeq_type,type,
% 2.28/1.41 subsetemptysetimpeq: $o ).
% 2.28/1.41
% 2.28/1.41 thf(subsetemptysetimpeq,definition,
% 2.28/1.41 ( subsetemptysetimpeq
% 2.28/1.41 = ( ! [A: $i] :
% 2.28/1.41 ( ( subset @ A @ emptyset )
% 2.28/1.41 => ( A = emptyset ) ) ) ) ).
% 2.28/1.41
% 2.28/1.41 thf(powersetI1_type,type,
% 2.28/1.41 powersetI1: $o ).
% 2.28/1.41
% 2.28/1.41 thf(powersetI1,definition,
% 2.28/1.41 ( powersetI1
% 2.28/1.41 = ( ! [A: $i,B: $i] :
% 2.28/1.41 ( ( subset @ B @ A )
% 2.28/1.41 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.28/1.41
% 2.28/1.41 thf(powersetE1_type,type,
% 2.28/1.41 powersetE1: $o ).
% 2.28/1.41
% 2.28/1.41 thf(powersetE1,definition,
% 2.28/1.41 ( powersetE1
% 2.28/1.41 = ( ! [A: $i,B: $i] :
% 2.28/1.41 ( ( in @ B @ ( powerset @ A ) )
% 2.28/1.41 => ( subset @ B @ A ) ) ) ) ).
% 2.28/1.41
% 2.28/1.41 thf(inPowerset_type,type,
% 2.28/1.41 inPowerset: $o ).
% 2.28/1.41
% 2.28/1.41 thf(inPowerset,definition,
% 2.28/1.41 ( inPowerset
% 2.28/1.41 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 2.28/1.41
% 2.28/1.41 thf(powersetsubset_type,type,
% 2.28/1.41 powersetsubset: $o ).
% 2.28/1.41
% 2.28/1.41 thf(powersetsubset,definition,
% 2.28/1.41 ( powersetsubset
% 2.28/1.41 = ( ! [A: $i,B: $i] :
% 2.28/1.41 ( ( subset @ A @ B )
% 2.28/1.41 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 2.28/1.41
% 2.28/1.41 thf(sepInPowerset_type,type,
% 2.28/1.41 sepInPowerset: $o ).
% 2.28/1.41
% 2.28/1.41 thf(sepInPowerset,definition,
% 2.28/1.41 ( sepInPowerset
% 2.28/1.41 = ( ! [A: $i,Xphi: $i > $o] :
% 2.28/1.41 ( in
% 2.28/1.41 @ ( dsetconstr @ A
% 2.28/1.41 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.28/1.41 @ ( powerset @ A ) ) ) ) ).
% 2.28/1.41
% 2.28/1.41 thf(sepSubset_type,type,
% 2.28/1.41 sepSubset: $o ).
% 2.28/1.41
% 2.28/1.41 thf(sepSubset,definition,
% 2.28/1.41 ( sepSubset
% 2.28/1.41 = ( ! [A: $i,Xphi: $i > $o] :
% 2.28/1.41 ( subset
% 2.28/1.41 @ ( dsetconstr @ A
% 2.28/1.41 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.28/1.41 @ A ) ) ) ).
% 2.28/1.41
% 2.28/1.41 thf(binunion_type,type,
% 2.28/1.41 binunion: $i > $i > $i ).
% 2.28/1.41
% 2.28/1.41 thf(binunionIL_type,type,
% 2.28/1.41 binunionIL: $o ).
% 2.28/1.41
% 2.28/1.41 thf(binunionIL,definition,
% 2.28/1.41 ( binunionIL
% 2.28/1.41 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.28/1.41 ( ( in @ Xx @ A )
% 2.28/1.41 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.28/1.41
% 2.28/1.41 thf(upairset2IR_type,type,
% 2.28/1.41 upairset2IR: $o ).
% 2.28/1.41
% 2.28/1.41 thf(upairset2IR,definition,
% 2.28/1.41 ( upairset2IR
% 2.28/1.41 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.28/1.41
% 2.28/1.41 thf(binunionIR_type,type,
% 2.28/1.41 binunionIR: $o ).
% 2.28/1.41
% 2.28/1.41 thf(binunionIR,definition,
% 2.28/1.41 ( binunionIR
% 2.28/1.41 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.28/1.41 ( ( in @ Xx @ B )
% 2.28/1.41 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.28/1.41
% 2.28/1.41 thf(binunionEcases_type,type,
% 2.28/1.41 binunionEcases: $o ).
% 2.28/1.41
% 2.28/1.41 thf(binunionEcases,definition,
% 2.28/1.41 ( binunionEcases
% 2.28/1.41 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 2.28/1.41 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.28/1.41 => ( ( ( in @ Xx @ A )
% 2.28/1.41 => Xphi )
% 2.28/1.41 => ( ( ( in @ Xx @ B )
% 2.28/1.41 => Xphi )
% 2.28/1.41 => Xphi ) ) ) ) ) ).
% 2.28/1.41
% 2.28/1.41 thf(binunionE_type,type,
% 2.28/1.41 binunionE: $o ).
% 2.28/1.41
% 2.28/1.41 thf(binunionE,definition,
% 2.28/1.41 ( binunionE
% 2.28/1.41 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.28/1.41 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.28/1.41 => ( ( in @ Xx @ A )
% 2.28/1.41 | ( in @ Xx @ B ) ) ) ) ) ).
% 2.28/1.41
% 2.28/1.41 thf(binunionLsub_type,type,
% 2.28/1.41 binunionLsub: $o ).
% 2.28/1.41
% 2.28/1.41 thf(binunionLsub,definition,
% 2.28/1.41 ( binunionLsub
% 2.28/1.41 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 2.28/1.41
% 2.28/1.41 thf(binunionRsub_type,type,
% 2.28/1.41 binunionRsub: $o ).
% 2.28/1.41
% 2.28/1.41 thf(binunionRsub,definition,
% 2.28/1.41 ( binunionRsub
% 2.28/1.41 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 2.28/1.41
% 2.28/1.41 thf(binintersect_type,type,
% 2.28/1.41 binintersect: $i > $i > $i ).
% 2.28/1.41
% 2.28/1.41 thf(binintersectI_type,type,
% 2.28/1.41 binintersectI: $o ).
% 2.28/1.41
% 2.28/1.41 thf(binintersectI,definition,
% 2.28/1.41 ( binintersectI
% 2.28/1.41 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.28/1.41 ( ( in @ Xx @ A )
% 2.28/1.41 => ( ( in @ Xx @ B )
% 2.28/1.41 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.28/1.41
% 2.28/1.41 thf(binintersectSubset5_type,type,
% 2.28/1.41 binintersectSubset5: $o ).
% 2.28/1.41
% 2.28/1.41 thf(binintersectSubset5,definition,
% 2.28/1.41 ( binintersectSubset5
% 2.28/1.41 = ( ! [A: $i,B: $i,C: $i] :
% 2.28/1.41 ( ( subset @ C @ A )
% 2.28/1.41 => ( ( subset @ C @ B )
% 2.28/1.41 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.28/1.41
% 2.28/1.41 thf(binintersectEL_type,type,
% 2.28/1.41 binintersectEL: $o ).
% 2.28/1.41
% 2.28/1.41 thf(binintersectEL,definition,
% 2.28/1.41 ( binintersectEL
% 2.28/1.41 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.28/1.41 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.28/1.41 => ( in @ Xx @ A ) ) ) ) ).
% 2.28/1.41
% 2.28/1.41 thf(binintersectLsub_type,type,
% 2.28/1.41 binintersectLsub: $o ).
% 2.28/1.41
% 2.28/1.41 thf(binintersectLsub,definition,
% 2.28/1.41 ( binintersectLsub
% 2.28/1.41 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 2.28/1.41
% 2.28/1.41 thf(binintersectSubset2_type,type,
% 2.28/1.41 binintersectSubset2: $o ).
% 2.28/1.41
% 2.28/1.41 thf(binintersectSubset2,definition,
% 2.28/1.41 ( binintersectSubset2
% 2.28/1.41 = ( ! [A: $i,B: $i] :
% 2.28/1.41 ( ( subset @ A @ B )
% 2.28/1.41 => ( ( binintersect @ A @ B )
% 2.28/1.41 = A ) ) ) ) ).
% 2.28/1.41
% 2.28/1.41 thf(binintersectSubset3_type,type,
% 2.28/1.41 binintersectSubset3: $o ).
% 2.28/1.41
% 2.28/1.41 thf(binintersectSubset3,definition,
% 2.28/1.41 ( binintersectSubset3
% 2.28/1.41 = ( ! [A: $i,B: $i] :
% 2.28/1.41 ( ( ( binintersect @ A @ B )
% 2.28/1.41 = B )
% 2.28/1.41 => ( subset @ B @ A ) ) ) ) ).
% 2.28/1.41
% 2.28/1.41 thf(binintersectER_type,type,
% 2.28/1.41 binintersectER: $o ).
% 2.28/1.41
% 2.28/1.41 thf(binintersectER,definition,
% 2.28/1.41 ( binintersectER
% 2.28/1.41 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.28/1.41 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.28/1.41 => ( in @ Xx @ B ) ) ) ) ).
% 2.28/1.41
% 2.28/1.41 thf(disjointsetsI1_type,type,
% 2.28/1.41 disjointsetsI1: $o ).
% 2.28/1.41
% 2.28/1.41 thf(disjointsetsI1,definition,
% 2.28/1.41 ( disjointsetsI1
% 2.28/1.41 = ( ! [A: $i,B: $i] :
% 2.28/1.41 ( ~ ? [Xx: $i] :
% 2.28/1.41 ( ( in @ Xx @ A )
% 2.28/1.41 & ( in @ Xx @ B ) )
% 2.28/1.41 => ( ( binintersect @ A @ B )
% 2.28/1.41 = emptyset ) ) ) ) ).
% 2.28/1.41
% 2.28/1.41 thf(binintersectRsub_type,type,
% 2.28/1.41 binintersectRsub: $o ).
% 2.28/1.41
% 2.28/1.41 thf(binintersectRsub,definition,
% 2.28/1.41 ( binintersectRsub
% 2.28/1.41 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 2.28/1.41
% 2.28/1.41 thf(binintersectSubset4_type,type,
% 2.28/1.41 binintersectSubset4: $o ).
% 2.28/1.41
% 2.28/1.41 thf(binintersectSubset4,definition,
% 2.28/1.41 ( binintersectSubset4
% 2.28/1.41 = ( ! [A: $i,B: $i] :
% 2.28/1.41 ( ( subset @ B @ A )
% 2.28/1.41 => ( ( binintersect @ A @ B )
% 2.28/1.41 = B ) ) ) ) ).
% 2.28/1.41
% 2.28/1.41 thf(binintersectSubset1_type,type,
% 2.28/1.41 binintersectSubset1: $o ).
% 2.28/1.41
% 2.28/1.41 thf(binintersectSubset1,definition,
% 2.28/1.41 ( binintersectSubset1
% 2.28/1.41 = ( ! [A: $i,B: $i] :
% 2.28/1.41 ( ( ( binintersect @ A @ B )
% 2.28/1.41 = A )
% 2.28/1.41 => ( subset @ A @ B ) ) ) ) ).
% 2.28/1.41
% 2.28/1.41 thf(bs114d_type,type,
% 2.28/1.41 bs114d: $o ).
% 2.28/1.41
% 2.28/1.41 thf(bs114d,definition,
% 2.28/1.41 ( bs114d
% 2.28/1.41 = ( ! [A: $i,B: $i,C: $i] :
% 2.28/1.41 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 2.28/1.41 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 2.28/1.41
% 2.28/1.41 thf(regular_type,type,
% 2.28/1.41 regular: $i > $o ).
% 2.28/1.41
% 2.28/1.41 thf(setminus_type,type,
% 2.28/1.41 setminus: $i > $i > $i ).
% 2.28/1.41
% 2.28/1.41 thf(setminus,definition,
% 2.28/1.41 ( setminus
% 2.28/1.41 = ( ^ [A: $i,B: $i] :
% 2.28/1.41 ( dsetconstr @ A
% 2.28/1.41 @ ^ [Xx: $i] :
% 2.28/1.41 ~ ( in @ Xx @ B ) ) ) ) ).
% 2.28/1.41
% 2.28/1.41 thf(setminusI_type,type,
% 2.28/1.41 setminusI: $o ).
% 2.28/1.41
% 2.28/1.41 thf(setminusI,definition,
% 2.28/1.41 ( setminusI
% 2.28/1.41 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.28/1.41 ( ( in @ Xx @ A )
% 2.28/1.41 => ( ~ ( in @ Xx @ B )
% 2.28/1.41 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 2.28/1.41
% 2.28/1.41 thf(setminusEL,conjecture,
% 2.28/1.41 ( setextAx
% 2.28/1.41 => ( emptysetAx
% 2.28/1.41 => ( setadjoinAx
% 2.28/1.41 => ( powersetAx
% 2.28/1.41 => ( setunionAx
% 2.28/1.41 => ( omega0Ax
% 2.28/1.41 => ( omegaSAx
% 2.28/1.41 => ( omegaIndAx
% 2.28/1.41 => ( replAx
% 2.28/1.41 => ( foundationAx
% 2.28/1.41 => ( wellorderingAx
% 2.28/1.41 => ( descrp
% 2.28/1.41 => ( dsetconstrI
% 2.28/1.41 => ( dsetconstrEL
% 2.28/1.41 => ( dsetconstrER
% 2.28/1.41 => ( exuE1
% 2.28/1.41 => ( prop2setE
% 2.28/1.41 => ( emptysetE
% 2.28/1.41 => ( emptysetimpfalse
% 2.28/1.41 => ( notinemptyset
% 2.28/1.41 => ( exuE3e
% 2.28/1.41 => ( setext
% 2.28/1.41 => ( emptyI
% 2.28/1.41 => ( noeltsimpempty
% 2.28/1.41 => ( setbeta
% 2.28/1.41 => ( nonemptyE1
% 2.28/1.41 => ( nonemptyI
% 2.28/1.41 => ( nonemptyI1
% 2.28/1.41 => ( setadjoinIL
% 2.28/1.41 => ( emptyinunitempty
% 2.28/1.41 => ( setadjoinIR
% 2.28/1.41 => ( setadjoinE
% 2.28/1.41 => ( setadjoinOr
% 2.28/1.41 => ( setoftrueEq
% 2.28/1.41 => ( powersetI
% 2.28/1.41 => ( emptyinPowerset
% 2.28/1.41 => ( emptyInPowerset
% 2.28/1.41 => ( powersetE
% 2.28/1.41 => ( setunionI
% 2.28/1.41 => ( setunionE
% 2.28/1.41 => ( subPowSU
% 2.28/1.41 => ( exuE2
% 2.28/1.41 => ( nonemptyImpWitness
% 2.28/1.41 => ( uniqinunit
% 2.28/1.41 => ( notinsingleton
% 2.28/1.41 => ( eqinunit
% 2.28/1.41 => ( singletonsswitch
% 2.28/1.41 => ( upairsetE
% 2.28/1.41 => ( upairsetIL
% 2.28/1.41 => ( upairsetIR
% 2.28/1.41 => ( emptyE1
% 2.28/1.41 => ( vacuousDall
% 2.28/1.41 => ( quantDeMorgan1
% 2.28/1.41 => ( quantDeMorgan2
% 2.28/1.41 => ( quantDeMorgan3
% 2.28/1.41 => ( quantDeMorgan4
% 2.28/1.41 => ( prop2setI
% 2.28/1.41 => ( prop2set2propI
% 2.28/1.41 => ( notdexE
% 2.28/1.41 => ( notdallE
% 2.28/1.41 => ( exuI1
% 2.28/1.41 => ( exuI3
% 2.28/1.41 => ( exuI2
% 2.28/1.41 => ( inCongP
% 2.28/1.41 => ( in__Cong
% 2.28/1.41 => ( exuE3u
% 2.28/1.41 => ( exu__Cong
% 2.28/1.41 => ( emptyset__Cong
% 2.28/1.41 => ( setadjoin__Cong
% 2.28/1.41 => ( powerset__Cong
% 2.28/1.41 => ( setunion__Cong
% 2.28/1.41 => ( omega__Cong
% 2.28/1.41 => ( exuEu
% 2.28/1.41 => ( descr__Cong
% 2.28/1.41 => ( dsetconstr__Cong
% 2.28/1.41 => ( subsetI1
% 2.28/1.41 => ( eqimpsubset2
% 2.28/1.41 => ( eqimpsubset1
% 2.28/1.41 => ( subsetI2
% 2.28/1.41 => ( emptysetsubset
% 2.28/1.41 => ( subsetE
% 2.28/1.41 => ( subsetE2
% 2.28/1.41 => ( notsubsetI
% 2.28/1.41 => ( notequalI1
% 2.28/1.41 => ( notequalI2
% 2.28/1.41 => ( subsetRefl
% 2.28/1.41 => ( subsetTrans
% 2.28/1.41 => ( setadjoinSub
% 2.28/1.41 => ( setadjoinSub2
% 2.28/1.41 => ( subset2powerset
% 2.28/1.41 => ( setextsub
% 2.28/1.41 => ( subsetemptysetimpeq
% 2.28/1.41 => ( powersetI1
% 2.28/1.41 => ( powersetE1
% 2.28/1.41 => ( inPowerset
% 2.28/1.41 => ( powersetsubset
% 2.28/1.41 => ( sepInPowerset
% 2.28/1.41 => ( sepSubset
% 2.28/1.41 => ( binunionIL
% 2.28/1.41 => ( upairset2IR
% 2.28/1.41 => ( binunionIR
% 2.28/1.41 => ( binunionEcases
% 2.28/1.41 => ( binunionE
% 2.28/1.41 => ( binunionLsub
% 2.28/1.41 => ( binunionRsub
% 2.28/1.41 => ( binintersectI
% 2.28/1.41 => ( binintersectSubset5
% 2.28/1.41 => ( binintersectEL
% 2.28/1.41 => ( binintersectLsub
% 2.28/1.41 => ( binintersectSubset2
% 2.28/1.41 => ( binintersectSubset3
% 2.28/1.41 => ( binintersectER
% 2.28/1.41 => ( disjointsetsI1
% 2.28/1.41 => ( binintersectRsub
% 2.28/1.41 => ( binintersectSubset4
% 2.28/1.41 => ( binintersectSubset1
% 2.28/1.41 => ( bs114d
% 2.28/1.41 => ( setminusI
% 2.28/1.41 => ! [A: $i,B: $i,Xx: $i] :
% 2.28/1.41 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.28/1.41 => ( in @ Xx @ A ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.28/1.41 ------------------------
% 2.28/1.42 % (14216)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_14115 for (2999ds/183Mi)
% 2.28/1.43 % (14220)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_14115 for (2999ds/2Mi)
% 2.28/1.43 % (14218)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_14115 for (2999ds/27Mi)
% 2.28/1.43 % (14219)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_14115 for (2999ds/2Mi)
% 2.28/1.43 % (14221)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_14115 for (2999ds/275Mi)
% 2.28/1.43 % (14222)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_14115 for (2999ds/18Mi)
% 2.28/1.43 % (14220)Instruction limit reached!
% 2.28/1.43 % (14220)------------------------------
% 2.28/1.43 % (14220)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.28/1.43 % (14217)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_14115 for (2999ds/4Mi)
% 2.28/1.43 % (14220)Termination reason: Unknown
% 2.28/1.43 % (14220)Termination phase: shuffling
% 2.28/1.43
% 2.28/1.43 % (14220)Memory used [KB]: 1151
% 2.28/1.43 % (14220)Time elapsed: 0.002 s
% 2.28/1.43 % (14220)Instructions burned: 2 (million)
% 2.28/1.43 % (14220)------------------------------
% 2.28/1.43 % (14220)------------------------------
% 2.28/1.43 % (14219)Instruction limit reached!
% 2.28/1.43 % (14219)------------------------------
% 2.28/1.43 % (14219)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.28/1.43 % (14219)Termination reason: Unknown
% 2.28/1.43 % (14219)Termination phase: shuffling
% 2.28/1.43
% 2.28/1.43 % (14219)Memory used [KB]: 1279
% 2.28/1.43 % (14219)Time elapsed: 0.003 s
% 2.28/1.43 % (14219)Instructions burned: 3 (million)
% 2.28/1.43 % (14219)------------------------------
% 2.28/1.43 % (14219)------------------------------
% 2.28/1.43 % (14217)Instruction limit reached!
% 2.28/1.43 % (14217)------------------------------
% 2.28/1.43 % (14217)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.28/1.43 % (14217)Termination reason: Unknown
% 2.28/1.43 % (14217)Termination phase: shuffling
% 2.28/1.43
% 2.28/1.43 % (14217)Memory used [KB]: 1279
% 2.28/1.43 % (14217)Time elapsed: 0.004 s
% 2.28/1.43 % (14217)Instructions burned: 5 (million)
% 2.28/1.43 % (14217)------------------------------
% 2.28/1.43 % (14217)------------------------------
% 2.28/1.44 % (14222)Instruction limit reached!
% 2.28/1.44 % (14222)------------------------------
% 2.28/1.44 % (14222)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.28/1.44 % (14222)Termination reason: Unknown
% 2.28/1.44 % (14222)Termination phase: Preprocessing 3
% 2.28/1.44
% 2.28/1.44 % (14222)Memory used [KB]: 1535
% 2.28/1.44 % (14222)Time elapsed: 0.011 s
% 2.28/1.44 % (14222)Instructions burned: 19 (million)
% 2.28/1.44 % (14222)------------------------------
% 2.28/1.44 % (14222)------------------------------
% 2.28/1.44 % (14218)Instruction limit reached!
% 2.28/1.44 % (14218)------------------------------
% 2.28/1.44 % (14218)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.28/1.44 % (14218)Termination reason: Unknown
% 2.28/1.44 % (14218)Termination phase: Preprocessing 3
% 2.28/1.44
% 2.28/1.44 % (14218)Memory used [KB]: 1535
% 2.28/1.44 % (14218)Time elapsed: 0.014 s
% 2.28/1.44 % (14218)Instructions burned: 28 (million)
% 2.28/1.44 % (14218)------------------------------
% 2.28/1.44 % (14218)------------------------------
% 2.28/1.45 % (14224)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_14115 for (2999ds/3Mi)
% 2.28/1.45 % (14225)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_14115 for (2999ds/37Mi)
% 2.28/1.45 % (14224)Instruction limit reached!
% 2.28/1.45 % (14224)------------------------------
% 2.28/1.45 % (14224)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.28/1.45 % (14224)Termination reason: Unknown
% 2.28/1.45 % (14224)Termination phase: shuffling
% 2.28/1.45
% 2.28/1.45 % (14224)Memory used [KB]: 1151
% 2.28/1.45 % (14224)Time elapsed: 0.003 s
% 2.28/1.45 % (14224)Instructions burned: 3 (million)
% 2.28/1.45 % (14224)------------------------------
% 2.28/1.45 % (14224)------------------------------
% 2.28/1.45 % (14226)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_14115 for (2999ds/15Mi)
% 2.28/1.46 % (14227)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_14115 for (2999ds/3Mi)
% 2.28/1.46 % (14226)Instruction limit reached!
% 2.28/1.46 % (14226)------------------------------
% 2.28/1.46 % (14226)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.28/1.46 % (14226)Termination reason: Unknown
% 2.28/1.46 % (14226)Termination phase: Preprocessing 2
% 2.28/1.46
% 2.28/1.46 % (14226)Memory used [KB]: 1407
% 2.28/1.46 % (14226)Time elapsed: 0.009 s
% 2.28/1.46 % (14226)Instructions burned: 16 (million)
% 2.28/1.46 % (14226)------------------------------
% 2.28/1.46 % (14226)------------------------------
% 2.28/1.46 % (14227)Instruction limit reached!
% 2.28/1.46 % (14227)------------------------------
% 2.28/1.46 % (14227)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.28/1.46 % (14227)Termination reason: Unknown
% 2.28/1.46 % (14227)Termination phase: shuffling
% 2.28/1.46
% 2.28/1.46 % (14227)Memory used [KB]: 1151
% 2.28/1.46 % (14227)Time elapsed: 0.003 s
% 2.28/1.46 % (14227)Instructions burned: 3 (million)
% 2.28/1.46 % (14227)------------------------------
% 2.28/1.46 % (14227)------------------------------
% 2.28/1.46 % (14228)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_14115 for (2999ds/1041Mi)
% 2.28/1.46 % (14225)Instruction limit reached!
% 2.28/1.46 % (14225)------------------------------
% 2.28/1.46 % (14225)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.28/1.46 % (14225)Termination reason: Unknown
% 2.28/1.46 % (14225)Termination phase: Property scanning
% 2.28/1.46
% 2.28/1.46 % (14225)Memory used [KB]: 1663
% 2.28/1.46 % (14225)Time elapsed: 0.018 s
% 2.28/1.46 % (14225)Instructions burned: 38 (million)
% 2.28/1.46 % (14225)------------------------------
% 2.28/1.46 % (14225)------------------------------
% 2.28/1.46 % (14229)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_14115 for (2999ds/7Mi)
% 2.28/1.47 % (14229)Instruction limit reached!
% 2.28/1.47 % (14229)------------------------------
% 2.28/1.47 % (14229)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.28/1.47 % (14229)Termination reason: Unknown
% 2.28/1.47 % (14229)Termination phase: shuffling
% 2.28/1.47
% 2.28/1.47 % (14229)Memory used [KB]: 1279
% 2.28/1.47 % (14229)Time elapsed: 0.005 s
% 2.28/1.47 % (14229)Instructions burned: 8 (million)
% 2.28/1.47 % (14229)------------------------------
% 2.28/1.47 % (14229)------------------------------
% 2.28/1.47 % (14230)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_14115 for (2999ds/16Mi)
% 2.28/1.47 % (14231)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_14115 for (2999ds/3Mi)
% 2.28/1.47 % (14231)Instruction limit reached!
% 2.28/1.47 % (14231)------------------------------
% 2.28/1.47 % (14231)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.28/1.47 % (14231)Termination reason: Unknown
% 2.28/1.47 % (14231)Termination phase: shuffling
% 2.28/1.47
% 2.28/1.47 % (14231)Memory used [KB]: 1279
% 2.28/1.47 % (14231)Time elapsed: 0.003 s
% 2.28/1.47 % (14231)Instructions burned: 4 (million)
% 2.28/1.47 % (14231)------------------------------
% 2.28/1.47 % (14231)------------------------------
% 2.28/1.48 % (14216)Instruction limit reached!
% 2.28/1.48 % (14216)------------------------------
% 2.28/1.48 % (14216)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.28/1.48 % (14216)Termination reason: Unknown
% 2.28/1.48 % (14216)Termination phase: Saturation
% 2.28/1.48
% 2.28/1.48 % (14216)Memory used [KB]: 6908
% 2.28/1.48 % (14216)Time elapsed: 0.054 s
% 2.28/1.48 % (14216)Instructions burned: 184 (million)
% 2.28/1.48 % (14216)------------------------------
% 2.28/1.48 % (14216)------------------------------
% 2.28/1.48 % (14232)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_14115 for (2999ds/3Mi)
% 2.28/1.48 % (14230)Instruction limit reached!
% 2.28/1.48 % (14230)------------------------------
% 2.28/1.48 % (14230)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.28/1.48 % (14230)Termination reason: Unknown
% 2.28/1.48 % (14230)Termination phase: shuffling
% 2.28/1.48
% 2.28/1.48 % (14230)Memory used [KB]: 1535
% 2.28/1.48 % (14230)Time elapsed: 0.009 s
% 2.28/1.48 % (14230)Instructions burned: 17 (million)
% 2.28/1.48 % (14230)------------------------------
% 2.28/1.48 % (14230)------------------------------
% 2.28/1.48 % (14232)Instruction limit reached!
% 2.28/1.48 % (14232)------------------------------
% 2.28/1.48 % (14232)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.28/1.48 % (14232)Termination reason: Unknown
% 2.28/1.48 % (14232)Termination phase: shuffling
% 2.28/1.48
% 2.28/1.48 % (14232)Memory used [KB]: 1151
% 2.28/1.48 % (14232)Time elapsed: 0.003 s
% 2.28/1.48 % (14232)Instructions burned: 3 (million)
% 2.28/1.48 % (14232)------------------------------
% 2.28/1.48 % (14232)------------------------------
% 2.28/1.48 % (14233)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_14115 for (2999ds/7Mi)
% 2.28/1.49 % (14235)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_14115 for (2999ds/4Mi)
% 2.28/1.49 % (14233)Instruction limit reached!
% 2.28/1.49 % (14233)------------------------------
% 2.28/1.49 % (14233)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.28/1.49 % (14233)Termination reason: Unknown
% 2.28/1.49 % (14235)Instruction limit reached!
% 2.28/1.49 % (14235)------------------------------
% 2.28/1.49 % (14235)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.28/1.49 % (14235)Termination reason: Unknown
% 2.28/1.49 % (14235)Termination phase: shuffling
% 2.28/1.49
% 2.28/1.49 % (14235)Memory used [KB]: 1279
% 2.28/1.49 % (14235)Time elapsed: 0.003 s
% 2.28/1.49 % (14235)Instructions burned: 7 (million)
% 2.28/1.49 % (14235)------------------------------
% 2.28/1.49 % (14235)------------------------------
% 2.28/1.49 % (14233)Termination phase: shuffling
% 2.28/1.49
% 2.28/1.49 % (14233)Memory used [KB]: 1279
% 2.28/1.49 % (14233)Time elapsed: 0.005 s
% 2.28/1.49 % (14233)Instructions burned: 8 (million)
% 2.28/1.49 % (14233)------------------------------
% 2.28/1.49 % (14233)------------------------------
% 2.28/1.49 % (14234)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_14115 for (2999ds/3Mi)
% 2.28/1.49 % (14234)Instruction limit reached!
% 2.28/1.49 % (14234)------------------------------
% 2.28/1.49 % (14234)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.28/1.49 % (14234)Termination reason: Unknown
% 2.28/1.49 % (14234)Termination phase: shuffling
% 2.28/1.49
% 2.28/1.49 % (14234)Memory used [KB]: 1279
% 2.28/1.49 % (14234)Time elapsed: 0.003 s
% 2.28/1.49 % (14234)Instructions burned: 4 (million)
% 2.28/1.49 % (14234)------------------------------
% 2.28/1.49 % (14234)------------------------------
% 2.30/1.50 % (14236)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_14115 for (2999ds/18Mi)
% 2.30/1.50 % (14238)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_14115 for (2999ds/6Mi)
% 2.30/1.50 % (14237)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_14115 for (2999ds/710Mi)
% 2.30/1.50 % (14238)Instruction limit reached!
% 2.30/1.50 % (14238)------------------------------
% 2.30/1.50 % (14238)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.30/1.50 % (14238)Termination reason: Unknown
% 2.30/1.50 % (14238)Termination phase: shuffling
% 2.30/1.50
% 2.30/1.50 % (14238)Memory used [KB]: 1279
% 2.30/1.50 % (14238)Time elapsed: 0.003 s
% 2.30/1.50 % (14238)Instructions burned: 7 (million)
% 2.30/1.50 % (14238)------------------------------
% 2.30/1.50 % (14238)------------------------------
% 2.30/1.50 % (14239)dis+1002_5:1_au=on:bd=off:e2e=on:fde=none:fs=off:fsr=off:sos=on:i=902:si=on:rtra=on_0 on DTF2THF_14115 for (2999ds/902Mi)
% 2.30/1.51 % (14236)Instruction limit reached!
% 2.30/1.51 % (14236)------------------------------
% 2.30/1.51 % (14236)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.30/1.51 % (14236)Termination reason: Unknown
% 2.30/1.51 % (14236)Termination phase: Property scanning
% 2.30/1.51
% 2.30/1.51 % (14236)Memory used [KB]: 1535
% 2.30/1.51 % (14236)Time elapsed: 0.010 s
% 2.30/1.51 % (14236)Instructions burned: 19 (million)
% 2.30/1.51 % (14236)------------------------------
% 2.30/1.51 % (14236)------------------------------
% 2.30/1.51 % (14241)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_14115 for (2999ds/5Mi)
% 2.30/1.51 % (14240)dis+21_1:8_apa=on:cnfonf=off:fd=off:fsr=off:hud=0:ins=1:kws=inv_frequency:nwc=10.0:ss=axioms:st=5.0:i=21:si=on:rtra=on_0 on DTF2THF_14115 for (2999ds/21Mi)
% 2.30/1.51 % (14241)Instruction limit reached!
% 2.30/1.51 % (14241)------------------------------
% 2.30/1.51 % (14241)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.30/1.51 % (14241)Termination reason: Unknown
% 2.30/1.51 % (14241)Termination phase: shuffling
% 2.30/1.51
% 2.30/1.51 % (14241)Memory used [KB]: 1279
% 2.30/1.51 % (14241)Time elapsed: 0.003 s
% 2.30/1.51 % (14241)Instructions burned: 7 (million)
% 2.30/1.51 % (14241)------------------------------
% 2.30/1.51 % (14241)------------------------------
% 2.30/1.52 % (14240)Instruction limit reached!
% 2.30/1.52 % (14240)------------------------------
% 2.30/1.52 % (14240)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.30/1.52 % (14240)Termination reason: Unknown
% 2.30/1.52 % (14240)Termination phase: Property scanning
% 2.30/1.52
% 2.30/1.52 % (14240)Memory used [KB]: 1535
% 2.30/1.52 % (14240)Time elapsed: 0.011 s
% 2.30/1.52 % (14240)Instructions burned: 21 (million)
% 2.30/1.52 % (14240)------------------------------
% 2.30/1.52 % (14240)------------------------------
% 2.30/1.52 % (14243)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_14115 for (2999ds/377Mi)
% 2.30/1.52 % (14242)lrs+2_1:1_cnfonf=lazy_not_gen_be_off:cs=on:fe=off:hud=10:inj=on:ins=3:plsq=on:plsqc=1:sd=10:ss=axioms:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_14115 for (2999ds/6Mi)
% 2.30/1.52 % (14242)Instruction limit reached!
% 2.30/1.52 % (14242)------------------------------
% 2.30/1.52 % (14242)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.30/1.52 % (14242)Termination reason: Unknown
% 2.30/1.52 % (14242)Termination phase: shuffling
% 2.30/1.52
% 2.30/1.52 % (14242)Memory used [KB]: 1279
% 2.30/1.52 % (14242)Time elapsed: 0.004 s
% 2.30/1.52 % (14242)Instructions burned: 6 (million)
% 2.30/1.52 % (14242)------------------------------
% 2.30/1.52 % (14242)------------------------------
% 2.30/1.54 % (14244)dis+1010_1:4_atotf=0.2:c=on:cbe=off:cnfonf=lazy_simp:fe=off:ins=2:ntd=on:s2a=on:s2at=5.0:sgt=5:ss=axioms:st=1.5:i=779:si=on:rtra=on_0 on DTF2THF_14115 for (2999ds/779Mi)
% 2.30/1.54 % (14245)lrs+10_1:1_cnfonf=lazy_not_be_gen:ntd=on:sp=const_min:ss=axioms:sup=off:i=19:si=on:rtra=on_0 on DTF2THF_14115 for (2999ds/19Mi)
% 2.30/1.55 % (14245)Instruction limit reached!
% 2.30/1.55 % (14245)------------------------------
% 2.30/1.55 % (14245)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.30/1.55 % (14245)Termination reason: Unknown
% 2.30/1.55 % (14245)Termination phase: Property scanning
% 2.30/1.55
% 2.30/1.55 % (14245)Memory used [KB]: 1535
% 2.30/1.55 % (14245)Time elapsed: 0.011 s
% 2.30/1.55 % (14245)Instructions burned: 21 (million)
% 2.30/1.55 % (14245)------------------------------
% 2.30/1.55 % (14245)------------------------------
% 2.30/1.57 % (14221)Instruction limit reached!
% 2.30/1.57 % (14221)------------------------------
% 2.30/1.57 % (14221)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.30/1.57 % (14221)Termination reason: Unknown
% 2.30/1.57 % (14221)Termination phase: Saturation
% 2.30/1.57
% 2.30/1.57 % (14221)Memory used [KB]: 7675
% 2.30/1.57 % (14221)Time elapsed: 0.137 s
% 2.30/1.57 % (14221)Instructions burned: 276 (million)
% 2.30/1.57 % (14221)------------------------------
% 2.30/1.57 % (14221)------------------------------
% 2.30/1.57 % (14246)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_14115 for (2998ds/879Mi)
% 2.30/1.58 % (14247)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_14115 for (2998ds/17Mi)
% 2.30/1.59 % (14247)Instruction limit reached!
% 2.30/1.59 % (14247)------------------------------
% 2.30/1.59 % (14247)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.30/1.59 % (14247)Termination reason: Unknown
% 2.30/1.59 % (14247)Termination phase: shuffling
% 2.30/1.59
% 2.30/1.59 % (14247)Memory used [KB]: 1535
% 2.30/1.59 % (14247)Time elapsed: 0.009 s
% 2.30/1.59 % (14247)Instructions burned: 17 (million)
% 2.30/1.59 % (14247)------------------------------
% 2.30/1.59 % (14247)------------------------------
% 2.30/1.61 % (14248)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_14115 for (2998ds/3Mi)
% 2.30/1.61 % (14248)Instruction limit reached!
% 2.30/1.61 % (14248)------------------------------
% 2.30/1.61 % (14248)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.30/1.61 % (14248)Termination reason: Unknown
% 2.30/1.61 % (14248)Termination phase: shuffling
% 2.30/1.61
% 2.30/1.61 % (14248)Memory used [KB]: 1151
% 2.30/1.61 % (14248)Time elapsed: 0.003 s
% 2.30/1.61 % (14248)Instructions burned: 3 (million)
% 2.30/1.61 % (14248)------------------------------
% 2.30/1.61 % (14248)------------------------------
% 2.30/1.62 % (14249)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_14115 for (2998ds/30Mi)
% 2.30/1.63 % (14249)Instruction limit reached!
% 2.30/1.63 % (14249)------------------------------
% 2.30/1.63 % (14249)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.30/1.63 % (14249)Termination reason: Unknown
% 2.30/1.63 % (14249)Termination phase: Property scanning
% 2.30/1.63
% 2.30/1.63 % (14249)Memory used [KB]: 1535
% 2.30/1.63 % (14249)Time elapsed: 0.010 s
% 2.30/1.63 % (14249)Instructions burned: 32 (million)
% 2.30/1.63 % (14249)------------------------------
% 2.30/1.63 % (14249)------------------------------
% 2.30/1.64 % (14250)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_14115 for (2998ds/127Mi)
% 2.30/1.65 % (14243)Instruction limit reached!
% 2.30/1.65 % (14243)------------------------------
% 2.30/1.65 % (14243)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.30/1.65 % (14243)Termination reason: Unknown
% 2.30/1.65 % (14243)Termination phase: Saturation
% 2.30/1.65
% 2.30/1.65 % (14243)Memory used [KB]: 7291
% 2.30/1.65 % (14243)Time elapsed: 0.131 s
% 2.30/1.65 % (14243)Instructions burned: 378 (million)
% 2.30/1.65 % (14243)------------------------------
% 2.30/1.65 % (14243)------------------------------
% 2.30/1.67 % (14251)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_14115 for (2997ds/100Mi)
% 2.30/1.70 % (14250)Instruction limit reached!
% 2.30/1.70 % (14250)------------------------------
% 2.30/1.70 % (14250)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.30/1.70 % (14250)Termination reason: Unknown
% 2.30/1.70 % (14250)Termination phase: Saturation
% 2.30/1.70
% 2.30/1.70 % (14250)Memory used [KB]: 6780
% 2.30/1.70 % (14250)Time elapsed: 0.058 s
% 2.30/1.70 % (14250)Instructions burned: 127 (million)
% 2.30/1.70 % (14250)------------------------------
% 2.30/1.70 % (14250)------------------------------
% 2.30/1.71 % (14251)Instruction limit reached!
% 2.30/1.71 % (14251)------------------------------
% 2.30/1.71 % (14251)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.30/1.71 % (14251)Termination reason: Unknown
% 2.30/1.71 % (14251)Termination phase: Saturation
% 2.30/1.71
% 2.30/1.71 % (14251)Memory used [KB]: 6396
% 2.30/1.71 % (14251)Time elapsed: 0.030 s
% 2.30/1.71 % (14251)Instructions burned: 101 (million)
% 2.30/1.71 % (14251)------------------------------
% 2.30/1.71 % (14251)------------------------------
% 2.30/1.71 % (14252)dis+10_1:1_anc=none:cnfonf=lazy_gen:fd=preordered:fe=off:hud=10:ins=3:ixr=off:nwc=5.0:plsq=on:plsqc=1:plsqr=32,1:sp=const_frequency:uhcvi=on:i=3:si=on:rtra=on_0 on DTF2THF_14115 for (2997ds/3Mi)
% 2.30/1.72 % (14252)Instruction limit reached!
% 2.30/1.72 % (14252)------------------------------
% 2.30/1.72 % (14252)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.30/1.72 % (14252)Termination reason: Unknown
% 2.30/1.72 % (14252)Termination phase: shuffling
% 2.30/1.72
% 2.30/1.72 % (14252)Memory used [KB]: 1151
% 2.30/1.72 % (14252)Time elapsed: 0.003 s
% 2.30/1.72 % (14252)Instructions burned: 3 (million)
% 2.30/1.72 % (14252)------------------------------
% 2.30/1.72 % (14252)------------------------------
% 2.30/1.72 % (14253)lrs+10_8:1_au=on:avsq=on:e2e=on:ins=3:s2a=on:s2at=3.0:ss=axioms:i=20:si=on:rtra=on_0 on DTF2THF_14115 for (2996ds/20Mi)
% 2.30/1.73 % (14253)Instruction limit reached!
% 2.30/1.73 % (14253)------------------------------
% 2.30/1.73 % (14253)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.30/1.73 % (14253)Termination reason: Unknown
% 2.30/1.73 % (14253)Termination phase: Preprocessing 1
% 2.30/1.73
% 2.30/1.73 % (14253)Memory used [KB]: 1407
% 2.30/1.73 % (14253)Time elapsed: 0.007 s
% 2.30/1.73 % (14253)Instructions burned: 22 (million)
% 2.30/1.73 % (14253)------------------------------
% 2.30/1.73 % (14253)------------------------------
% 2.30/1.74 % (14254)dis+1002_1:1_cbe=off:hud=5:nm=4:plsq=on:plsqr=7,1:prag=on:sp=const_max:tnu=1:i=86:si=on:rtra=on_0 on DTF2THF_14115 for (2996ds/86Mi)
% 2.30/1.75 % (14255)lrs+1010_1:1_au=on:cbe=off:nm=2:ntd=on:sd=2:ss=axioms:st=5.0:i=107:si=on:rtra=on_0 on DTF2THF_14115 for (2996ds/107Mi)
% 2.30/1.76 % (14254)Instruction limit reached!
% 2.30/1.76 % (14254)------------------------------
% 2.30/1.76 % (14254)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.30/1.76 % (14254)Termination reason: Unknown
% 2.30/1.76 % (14254)Termination phase: Saturation
% 2.30/1.76
% 2.30/1.76 % (14254)Memory used [KB]: 6396
% 2.30/1.76 % (14254)Time elapsed: 0.024 s
% 2.30/1.76 % (14254)Instructions burned: 88 (million)
% 2.30/1.76 % (14254)------------------------------
% 2.30/1.76 % (14254)------------------------------
% 2.30/1.77 % (14256)lrs+2_1:1024_cnfonf=lazy_gen:fe=off:hud=15:plsq=on:plsqc=1:plsqr=32,1:i=39:si=on:rtra=on_0 on DTF2THF_14115 for (2996ds/39Mi)
% 2.30/1.78 % (14246)First to succeed.
% 2.30/1.78 % (14246)Refutation found. Thanks to Tanya!
% 2.30/1.78 % SZS status Theorem for DTF2THF_14115
% 2.30/1.78 % SZS output start Proof for DTF2THF_14115
% 2.30/1.78 thf(func_def_0, type, in: $i > $i > $o).
% 2.30/1.78 thf(func_def_1, type, exu: ($i > $o) > $o).
% 2.30/1.78 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 2.30/1.78 thf(func_def_8, type, powerset: $i > $i).
% 2.30/1.78 thf(func_def_10, type, setunion: $i > $i).
% 2.30/1.78 thf(func_def_19, type, descr: ($i > $o) > $i).
% 2.30/1.78 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 2.30/1.78 thf(func_def_26, type, prop2set: $o > $i).
% 2.30/1.78 thf(func_def_36, type, nonempty: $i > $o).
% 2.30/1.78 thf(func_def_69, type, set2prop: $i > $o).
% 2.30/1.78 thf(func_def_88, type, subset: $i > $i > $o).
% 2.30/1.78 thf(func_def_89, type, disjoint: $i > $i > $o).
% 2.30/1.78 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 2.30/1.78 thf(func_def_114, type, binunion: $i > $i > $i).
% 2.30/1.78 thf(func_def_122, type, binintersect: $i > $i > $i).
% 2.30/1.78 thf(func_def_135, type, regular: $i > $o).
% 2.30/1.78 thf(func_def_136, type, setminus: $i > $i > $i).
% 2.30/1.78 thf(func_def_159, type, sK4: ($i > $o) > $i > $i).
% 2.30/1.78 thf(func_def_160, type, sK5: $i > $i > $i).
% 2.30/1.78 thf(func_def_161, type, sK6: $i > $i > $i).
% 2.30/1.78 thf(func_def_162, type, sK7: ($i > $o) > $i > $i).
% 2.30/1.78 thf(func_def_163, type, sK8: $i > $i).
% 2.30/1.78 thf(func_def_164, type, sK9: $i > $i).
% 2.30/1.78 thf(func_def_165, type, sK10: $i > $i > $i > $i).
% 2.30/1.78 thf(func_def_166, type, sK11: $i > $i > $i).
% 2.30/1.78 thf(func_def_167, type, sK12: $i > $i > $i).
% 2.30/1.78 thf(func_def_168, type, sK13: $i > $i > $i).
% 2.30/1.78 thf(func_def_169, type, sK14: ($i > $o) > ($i > $o) > $i).
% 2.30/1.78 thf(func_def_170, type, sK15: ($i > $o) > ($i > $o) > $i).
% 2.30/1.78 thf(func_def_171, type, sK16: $i > $i).
% 2.30/1.78 thf(func_def_172, type, sK17: $i > ($i > $o) > $i).
% 2.30/1.78 thf(func_def_173, type, sK18: ($i > $o) > $i > ($i > $o) > $i > $i).
% 2.30/1.78 thf(func_def_174, type, sK19: $i > $i > ($i > $o) > ($i > $o) > $i).
% 2.30/1.78 thf(func_def_175, type, sK20: $i > ($i > $o) > $i).
% 2.30/1.78 thf(func_def_176, type, sK21: ($i > $o) > $i).
% 2.30/1.78 thf(func_def_177, type, sK22: $i > $i > $i).
% 2.30/1.78 thf(func_def_178, type, sK23: ($i > $o) > $i).
% 2.30/1.78 thf(func_def_179, type, sK24: ($i > $o) > $i).
% 2.30/1.78 thf(func_def_180, type, sK25: ($i > $o) > $i).
% 2.30/1.78 thf(func_def_181, type, sK26: $i > $i).
% 2.30/1.78 thf(func_def_182, type, sK27: $i > ($i > $i > $o) > $i).
% 2.30/1.78 thf(func_def_183, type, sK28: $i > ($i > $i > $o) > $i).
% 2.30/1.78 thf(func_def_184, type, sK29: $i > $i > ($i > $i > $o) > $i).
% 2.30/1.78 thf(func_def_185, type, sK30: $i > $i > ($i > $i > $o) > $i).
% 2.30/1.78 thf(func_def_186, type, sK31: $i > ($i > $o) > $i).
% 2.30/1.78 thf(func_def_187, type, sK32: $i > $i > $i).
% 2.30/1.78 thf(func_def_188, type, sK33: $i > $i > $i).
% 2.30/1.78 thf(func_def_189, type, sK34: ($i > $o) > ($i > $o) > $i).
% 2.30/1.78 thf(func_def_190, type, sK35: ($i > $o) > ($i > $o) > $i).
% 2.30/1.78 thf(func_def_191, type, sK36: ($i > $o) > ($i > $o) > $i).
% 2.30/1.78 thf(func_def_192, type, sK37: $i > $i).
% 2.30/1.78 thf(func_def_193, type, sK38: $i > ($i > $o) > $i).
% 2.30/1.78 thf(func_def_194, type, sK39: $i > $i > $i).
% 2.30/1.78 thf(func_def_195, type, sK40: $i > $o > $i > $i).
% 2.30/1.78 thf(func_def_196, type, sK41: $i > $i).
% 2.30/1.78 thf(func_def_197, type, sK42: $i > $i > $i).
% 2.30/1.78 thf(func_def_198, type, sK43: $i > $i > $i).
% 2.30/1.78 thf(func_def_199, type, sK44: $i > $i > $i).
% 2.30/1.78 thf(func_def_200, type, sK45: ($i > $o) > $i > $i).
% 2.30/1.78 thf(func_def_201, type, sK46: ($i > $o) > $i > $i).
% 2.30/1.78 thf(func_def_202, type, sK47: $i > $i).
% 2.30/1.78 thf(func_def_203, type, sK48: ($i > $o) > $i).
% 2.30/1.78 thf(func_def_204, type, ph49: !>[X0: $tType]:(X0)).
% 2.30/1.78 thf(f3069,plain,(
% 2.30/1.78 $false),
% 2.30/1.78 inference(subsumption_resolution,[status(thm)],[f3055,f534])).
% 2.30/1.78 thf(f534,plain,(
% 2.30/1.78 ($true != (in @ sK2 @ sK1))),
% 2.30/1.78 inference(cnf_transformation,[status(thm)],[f366])).
% 2.30/1.78 thf(f366,plain,(
% 2.30/1.78 (setunionAx = $true) & (setbeta = $true) & (powerset__Cong = $true) & (nonemptyImpWitness = $true) & (exuE1 = $true) & (upairsetE = $true) & (dsetconstrER = $true) & (emptysetsubset = $true) & (wellorderingAx = $true) & (uniqinunit = $true) & (emptysetE = $true) & (setoftrueEq = $true) & (binintersectI = $true) & (powersetAx = $true) & (emptysetimpfalse = $true) & (emptysetAx = $true) & (binunionIL = $true) & (setadjoinAx = $true) & (setadjoinSub = $true) & (binintersectSubset4 = $true) & (omegaSAx = $true) & (binintersectSubset5 = $true) & (foundationAx = $true) & (binintersectSubset2 = $true) & (exuEu = $true) & (binunionEcases = $true) & (subsetRefl = $true) & (setunion__Cong = $true) & (binintersectSubset1 = $true) & (setadjoinIR = $true) & (in__Cong = $true) & (subsetE = $true) & (binunionIR = $true) & (binintersectRsub = $true) & (prop2set2propI = $true) & (exuE3u = $true) & (setadjoinSub2 = $true) & (subset2powerset = $true) & (eqinunit = $true) & (exuI1 = $true) & (subsetI2 = $true) & (setadjoinIL = $true) & (powersetE1 = $true) & (subsetTrans = $true) & (binintersectSubset3 = $true) & (sepSubset = $true) & (setextsub = $true) & (subPowSU = $true) & (eqimpsubset2 = $true) & (omegaIndAx = $true) & (quantDeMorgan1 = $true) & (inPowerset = $true) & (emptyinunitempty = $true) & (exu__Cong = $true) & (setadjoin__Cong = $true) & (nonemptyE1 = $true) & (dsetconstrEL = $true) & (subsetemptysetimpeq = $true) & (noeltsimpempty = $true) & (dsetconstrI = $true) & (notdallE = $true) & (($true != (in @ sK2 @ sK1)) & ($true = (in @ sK2 @ (setminus @ sK1 @ sK0)))) & (notequalI1 = $true) & (setadjoinOr = $true) & (prop2setE = $true) & (upairsetIL = $true) & (prop2setI = $true) & (eqimpsubset1 = $true) & (notinsingleton = $true) & (descrp = $true) & (binunionLsub = $true) & (dsetconstr__Cong = $true) & (subsetE2 = $true) & (disjointsetsI1 = $true) & (setextAx = $true) & (subsetI1 = $true) & (powersetsubset = $true) & (binintersectLsub = $true) & (upairsetIR = $true) & (nonemptyI1 = $true) & (sepInPowerset = $true) & (omega0Ax = $true) & (exuI3 = $true) & (notsubsetI = $true) & (notinemptyset = $true) & (emptyE1 = $true) & (emptyinPowerset = $true) & (descr__Cong = $true) & (powersetE = $true) & (bs114d = $true) & (quantDeMorgan4 = $true) & (exuE2 = $true) & (binunionRsub = $true) & (quantDeMorgan2 = $true) & (omega__Cong = $true) & (inCongP = $true) & (vacuousDall = $true) & (quantDeMorgan3 = $true) & (binintersectEL = $true) & (binunionE = $true) & (replAx = $true) & (singletonsswitch = $true) & (setunionE = $true) & (binintersectER = $true) & (notequalI2 = $true) & (notdexE = $true) & (exuE3e = $true) & (emptyI = $true) & (exuI2 = $true) & (powersetI = $true) & (emptyset__Cong = $true) & (powersetI1 = $true) & (setadjoinE = $true) & (emptyInPowerset = $true) & (nonemptyI = $true) & (setext = $true) & (upairset2IR = $true) & (setminusI = $true) & (setunionI = $true)),
% 2.30/1.78 inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1,sK2])],[f364,f365])).
% 2.30/1.78 thf(f365,plain,(
% 2.30/1.78 ? [X0 : $i,X1 : $i,X2 : $i] : (($true != (in @ X2 @ X1)) & ($true = (in @ X2 @ (setminus @ X1 @ X0)))) => (($true != (in @ sK2 @ sK1)) & ($true = (in @ sK2 @ (setminus @ sK1 @ sK0))))),
% 2.30/1.78 introduced(definition,[],[choice_axiom])).
% 2.30/1.78 thf(f364,plain,(
% 2.30/1.78 (setunionAx = $true) & (setbeta = $true) & (powerset__Cong = $true) & (nonemptyImpWitness = $true) & (exuE1 = $true) & (upairsetE = $true) & (dsetconstrER = $true) & (emptysetsubset = $true) & (wellorderingAx = $true) & (uniqinunit = $true) & (emptysetE = $true) & (setoftrueEq = $true) & (binintersectI = $true) & (powersetAx = $true) & (emptysetimpfalse = $true) & (emptysetAx = $true) & (binunionIL = $true) & (setadjoinAx = $true) & (setadjoinSub = $true) & (binintersectSubset4 = $true) & (omegaSAx = $true) & (binintersectSubset5 = $true) & (foundationAx = $true) & (binintersectSubset2 = $true) & (exuEu = $true) & (binunionEcases = $true) & (subsetRefl = $true) & (setunion__Cong = $true) & (binintersectSubset1 = $true) & (setadjoinIR = $true) & (in__Cong = $true) & (subsetE = $true) & (binunionIR = $true) & (binintersectRsub = $true) & (prop2set2propI = $true) & (exuE3u = $true) & (setadjoinSub2 = $true) & (subset2powerset = $true) & (eqinunit = $true) & (exuI1 = $true) & (subsetI2 = $true) & (setadjoinIL = $true) & (powersetE1 = $true) & (subsetTrans = $true) & (binintersectSubset3 = $true) & (sepSubset = $true) & (setextsub = $true) & (subPowSU = $true) & (eqimpsubset2 = $true) & (omegaIndAx = $true) & (quantDeMorgan1 = $true) & (inPowerset = $true) & (emptyinunitempty = $true) & (exu__Cong = $true) & (setadjoin__Cong = $true) & (nonemptyE1 = $true) & (dsetconstrEL = $true) & (subsetemptysetimpeq = $true) & (noeltsimpempty = $true) & (dsetconstrI = $true) & (notdallE = $true) & ? [X0 : $i,X1 : $i,X2 : $i] : (($true != (in @ X2 @ X1)) & ($true = (in @ X2 @ (setminus @ X1 @ X0)))) & (notequalI1 = $true) & (setadjoinOr = $true) & (prop2setE = $true) & (upairsetIL = $true) & (prop2setI = $true) & (eqimpsubset1 = $true) & (notinsingleton = $true) & (descrp = $true) & (binunionLsub = $true) & (dsetconstr__Cong = $true) & (subsetE2 = $true) & (disjointsetsI1 = $true) & (setextAx = $true) & (subsetI1 = $true) & (powersetsubset = $true) & (binintersectLsub = $true) & (upairsetIR = $true) & (nonemptyI1 = $true) & (sepInPowerset = $true) & (omega0Ax = $true) & (exuI3 = $true) & (notsubsetI = $true) & (notinemptyset = $true) & (emptyE1 = $true) & (emptyinPowerset = $true) & (descr__Cong = $true) & (powersetE = $true) & (bs114d = $true) & (quantDeMorgan4 = $true) & (exuE2 = $true) & (binunionRsub = $true) & (quantDeMorgan2 = $true) & (omega__Cong = $true) & (inCongP = $true) & (vacuousDall = $true) & (quantDeMorgan3 = $true) & (binintersectEL = $true) & (binunionE = $true) & (replAx = $true) & (singletonsswitch = $true) & (setunionE = $true) & (binintersectER = $true) & (notequalI2 = $true) & (notdexE = $true) & (exuE3e = $true) & (emptyI = $true) & (exuI2 = $true) & (powersetI = $true) & (emptyset__Cong = $true) & (powersetI1 = $true) & (setadjoinE = $true) & (emptyInPowerset = $true) & (nonemptyI = $true) & (setext = $true) & (upairset2IR = $true) & (setminusI = $true) & (setunionI = $true)),
% 2.30/1.78 inference(flattening,[status(thm)],[f363])).
% 2.30/1.78 thf(f363,plain,(
% 2.30/1.78 (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i,X1 : $i,X2 : $i] : (($true != (in @ X2 @ X1)) & ($true = (in @ X2 @ (setminus @ X1 @ X0)))) & (setminusI = $true)) & (bs114d = $true)) & (binintersectSubset1 = $true)) & (binintersectSubset4 = $true)) & (binintersectRsub = $true)) & (disjointsetsI1 = $true)) & (binintersectER = $true)) & (binintersectSubset3 = $true)) & (binintersectSubset2 = $true)) & (binintersectLsub = $true)) & (binintersectEL = $true)) & (binintersectSubset5 = $true)) & (binintersectI = $true)) & (binunionRsub = $true)) & (binunionLsub = $true)) & (binunionE = $true)) & (binunionEcases = $true)) & (binunionIR = $true)) & (upairset2IR = $true)) & (binunionIL = $true)) & (sepSubset = $true)) & (sepInPowerset = $true)) & (powersetsubset = $true)) & (inPowerset = $true)) & (powersetE1 = $true)) & (powersetI1 = $true)) & (subsetemptysetimpeq = $true)) & (setextsub = $true)) & (subset2powerset = $true)) & (setadjoinSub2 = $true)) & (setadjoinSub = $true)) & (subsetTrans = $true)) & (subsetRefl = $true)) & (notequalI2 = $true)) & (notequalI1 = $true)) & (notsubsetI = $true)) & (subsetE2 = $true)) & (subsetE = $true)) & (emptysetsubset = $true)) & (subsetI2 = $true)) & (eqimpsubset1 = $true)) & (eqimpsubset2 = $true)) & (subsetI1 = $true)) & (dsetconstr__Cong = $true)) & (descr__Cong = $true)) & (exuEu = $true)) & (omega__Cong = $true)) & (setunion__Cong = $true)) & (powerset__Cong = $true)) & (setadjoin__Cong = $true)) & (emptyset__Cong = $true)) & (exu__Cong = $true)) & (exuE3u = $true)) & (in__Cong = $true)) & (inCongP = $true)) & (exuI2 = $true)) & (exuI3 = $true)) & (exuI1 = $true)) & (notdallE = $true)) & (notdexE = $true)) & (prop2set2propI = $true)) & (prop2setI = $true)) & (quantDeMorgan4 = $true)) & (quantDeMorgan3 = $true)) & (quantDeMorgan2 = $true)) & (quantDeMorgan1 = $true)) & (vacuousDall = $true)) & (emptyE1 = $true)) & (upairsetIR = $true)) & (upairsetIL = $true)) & (upairsetE = $true)) & (singletonsswitch = $true)) & (eqinunit = $true)) & (notinsingleton = $true)) & (uniqinunit = $true)) & (nonemptyImpWitness = $true)) & (exuE2 = $true)) & (subPowSU = $true)) & (setunionE = $true)) & (setunionI = $true)) & (powersetE = $true)) & (emptyInPowerset = $true)) & (emptyinPowerset = $true)) & (powersetI = $true)) & (setoftrueEq = $true)) & (setadjoinOr = $true)) & (setadjoinE = $true)) & (setadjoinIR = $true)) & (emptyinunitempty = $true)) & (setadjoinIL = $true)) & (nonemptyI1 = $true)) & (nonemptyI = $true)) & (nonemptyE1 = $true)) & (setbeta = $true)) & (noeltsimpempty = $true)) & (emptyI = $true)) & (setext = $true)) & (exuE3e = $true)) & (notinemptyset = $true)) & (emptysetimpfalse = $true)) & (emptysetE = $true)) & (prop2setE = $true)) & (exuE1 = $true)) & (dsetconstrER = $true)) & (dsetconstrEL = $true)) & (dsetconstrI = $true)) & (descrp = $true)) & (wellorderingAx = $true)) & (foundationAx = $true)) & (replAx = $true)) & (omegaIndAx = $true)) & (omegaSAx = $true)) & (omega0Ax = $true)) & (setunionAx = $true)) & (powersetAx = $true)) & (setadjoinAx = $true)) & (emptysetAx = $true)) & (setextAx = $true)),
% 2.30/1.78 inference(ennf_transformation,[status(thm)],[f155])).
% 2.30/1.78 thf(f155,plain,(
% 2.30/1.78 ~((setextAx = $true) => ((emptysetAx = $true) => ((setadjoinAx = $true) => ((powersetAx = $true) => ((setunionAx = $true) => ((omega0Ax = $true) => ((omegaSAx = $true) => ((omegaIndAx = $true) => ((replAx = $true) => ((foundationAx = $true) => ((wellorderingAx = $true) => ((descrp = $true) => ((dsetconstrI = $true) => ((dsetconstrEL = $true) => ((dsetconstrER = $true) => ((exuE1 = $true) => ((prop2setE = $true) => ((emptysetE = $true) => ((emptysetimpfalse = $true) => ((notinemptyset = $true) => ((exuE3e = $true) => ((setext = $true) => ((emptyI = $true) => ((noeltsimpempty = $true) => ((setbeta = $true) => ((nonemptyE1 = $true) => ((nonemptyI = $true) => ((nonemptyI1 = $true) => ((setadjoinIL = $true) => ((emptyinunitempty = $true) => ((setadjoinIR = $true) => ((setadjoinE = $true) => ((setadjoinOr = $true) => ((setoftrueEq = $true) => ((powersetI = $true) => ((emptyinPowerset = $true) => ((emptyInPowerset = $true) => ((powersetE = $true) => ((setunionI = $true) => ((setunionE = $true) => ((subPowSU = $true) => ((exuE2 = $true) => ((nonemptyImpWitness = $true) => ((uniqinunit = $true) => ((notinsingleton = $true) => ((eqinunit = $true) => ((singletonsswitch = $true) => ((upairsetE = $true) => ((upairsetIL = $true) => ((upairsetIR = $true) => ((emptyE1 = $true) => ((vacuousDall = $true) => ((quantDeMorgan1 = $true) => ((quantDeMorgan2 = $true) => ((quantDeMorgan3 = $true) => ((quantDeMorgan4 = $true) => ((prop2setI = $true) => ((prop2set2propI = $true) => ((notdexE = $true) => ((notdallE = $true) => ((exuI1 = $true) => ((exuI3 = $true) => ((exuI2 = $true) => ((inCongP = $true) => ((in__Cong = $true) => ((exuE3u = $true) => ((exu__Cong = $true) => ((emptyset__Cong = $true) => ((setadjoin__Cong = $true) => ((powerset__Cong = $true) => ((setunion__Cong = $true) => ((omega__Cong = $true) => ((exuEu = $true) => ((descr__Cong = $true) => ((dsetconstr__Cong = $true) => ((subsetI1 = $true) => ((eqimpsubset2 = $true) => ((eqimpsubset1 = $true) => ((subsetI2 = $true) => ((emptysetsubset = $true) => ((subsetE = $true) => ((subsetE2 = $true) => ((notsubsetI = $true) => ((notequalI1 = $true) => ((notequalI2 = $true) => ((subsetRefl = $true) => ((subsetTrans = $true) => ((setadjoinSub = $true) => ((setadjoinSub2 = $true) => ((subset2powerset = $true) => ((setextsub = $true) => ((subsetemptysetimpeq = $true) => ((powersetI1 = $true) => ((powersetE1 = $true) => ((inPowerset = $true) => ((powersetsubset = $true) => ((sepInPowerset = $true) => ((sepSubset = $true) => ((binunionIL = $true) => ((upairset2IR = $true) => ((binunionIR = $true) => ((binunionEcases = $true) => ((binunionE = $true) => ((binunionLsub = $true) => ((binunionRsub = $true) => ((binintersectI = $true) => ((binintersectSubset5 = $true) => ((binintersectEL = $true) => ((binintersectLsub = $true) => ((binintersectSubset2 = $true) => ((binintersectSubset3 = $true) => ((binintersectER = $true) => ((disjointsetsI1 = $true) => ((binintersectRsub = $true) => ((binintersectSubset4 = $true) => ((binintersectSubset1 = $true) => ((bs114d = $true) => ((setminusI = $true) => ! [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X2 @ (setminus @ X1 @ X0))) => ($true = (in @ X2 @ X1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.30/1.78 inference(fool_elimination,[status(thm)],[f154])).
% 2.30/1.78 thf(f154,plain,(
% 2.30/1.78 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => ! [X0 : $i,X1 : $i,X2 : $i] : ((in @ X2 @ (setminus @ X1 @ X0)) => (in @ X2 @ X1))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.30/1.78 inference(rectify,[status(thm)],[f123])).
% 2.30/1.78 thf(f123,negated_conjecture,(
% 2.30/1.78 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => ! [X4 : $i,X3 : $i,X1 : $i] : ((in @ X1 @ (setminus @ X3 @ X4)) => (in @ X1 @ X3))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.30/1.78 inference(negated_conjecture,[status(cth)],[f122])).
% 2.30/1.78 thf(f122,conjecture,(
% 2.30/1.78 setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => ! [X4 : $i,X3 : $i,X1 : $i] : ((in @ X1 @ (setminus @ X3 @ X4)) => (in @ X1 @ X3)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.30/1.78 file('/export/starexec/sandbox2/tmp/tmp.U8znWXlJ6W/DTF2THF_14115.p',setminusEL)).
% 2.30/1.78 thf(f3055,plain,(
% 2.30/1.78 ($true = (in @ sK2 @ sK1))),
% 2.30/1.78 inference(trivial_inequality_removal,[status(thm)],[f3045])).
% 2.30/1.78 thf(f3045,plain,(
% 2.30/1.78 ($true = (in @ sK2 @ sK1)) | ($true = $false)),
% 2.30/1.78 inference(superposition,[status(thm)],[f1811,f1498])).
% 2.30/1.78 thf(f1498,plain,(
% 2.30/1.78 ( ! [X2 : $i > $o,X3 : $i,X1 : $i] : (((in @ X3 @ (dsetconstr @ X1 @ X2)) = $false) | ($true = (in @ X3 @ X1))) )),
% 2.30/1.78 inference(binary_proxy_clausification,[status(thm)],[f1497])).
% 2.30/1.78 thf(f1497,plain,(
% 2.30/1.78 ( ! [X2 : $i > $o,X3 : $i,X1 : $i] : ((((in @ X3 @ (dsetconstr @ X1 @ X2)) => (in @ X3 @ X1)) = $true)) )),
% 2.30/1.78 inference(beta_eta_normalization,[status(thm)],[f1496])).
% 2.30/1.78 thf(f1496,plain,(
% 2.30/1.78 ( ! [X2 : $i > $o,X3 : $i,X1 : $i] : (($true = ((^[Y0 : $i]: ((in @ Y0 @ (dsetconstr @ X1 @ X2)) => (in @ Y0 @ X1))) @ X3))) )),
% 2.30/1.78 inference(pi_clausification,[status(thm)],[f1495])).
% 2.30/1.78 thf(f1495,plain,(
% 2.30/1.78 ( ! [X2 : $i > $o,X1 : $i] : (((!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ (dsetconstr @ X1 @ X2)) => (in @ Y0 @ X1)))) = $true)) )),
% 2.30/1.78 inference(beta_eta_normalization,[status(thm)],[f1494])).
% 2.30/1.78 thf(f1494,plain,(
% 2.30/1.78 ( ! [X2 : $i > $o,X1 : $i] : ((((^[Y0 : $i > $o]: (!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ (dsetconstr @ X1 @ Y0)) => (in @ Y1 @ X1))))) @ X2) = $true)) )),
% 2.30/1.78 inference(pi_clausification,[status(thm)],[f1493])).
% 2.30/1.78 thf(f1493,plain,(
% 2.30/1.78 ( ! [X1 : $i] : (($true = (!! @ ($i > $o) @ (^[Y0 : $i > $o]: (!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ (dsetconstr @ X1 @ Y0)) => (in @ Y1 @ X1)))))))) )),
% 2.30/1.78 inference(beta_eta_normalization,[status(thm)],[f1492])).
% 2.30/1.78 thf(f1492,plain,(
% 2.30/1.78 ( ! [X1 : $i] : ((((^[Y0 : $i]: (!! @ ($i > $o) @ (^[Y1 : $i > $o]: (!! @ $i @ (^[Y2 : $i]: ((in @ Y2 @ (dsetconstr @ Y0 @ Y1)) => (in @ Y2 @ Y0))))))) @ X1) = $true)) )),
% 2.30/1.78 inference(pi_clausification,[status(thm)],[f1491])).
% 2.30/1.78 thf(f1491,plain,(
% 2.30/1.78 ($true = (!! @ $i @ (^[Y0 : $i]: (!! @ ($i > $o) @ (^[Y1 : $i > $o]: (!! @ $i @ (^[Y2 : $i]: ((in @ Y2 @ (dsetconstr @ Y0 @ Y1)) => (in @ Y2 @ Y0)))))))))),
% 2.30/1.78 inference(beta_eta_normalization,[status(thm)],[f655])).
% 2.30/1.78 thf(f655,plain,(
% 2.30/1.78 ($true = (!! @ $i @ (^[Y0 : $i]: (!! @ ($i > $o) @ (^[Y1 : $i > $o]: (!! @ $i @ (^[Y2 : $i]: ((in @ Y2 @ (dsetconstr @ Y0 @ (^[Y3 : $i]: (Y1 @ Y3)))) => (in @ Y2 @ Y0)))))))))),
% 2.30/1.78 inference(definition_unfolding,[status(thm)],[f539,f445])).
% 2.30/1.78 thf(f445,plain,(
% 2.30/1.78 (dsetconstrEL = (!! @ $i @ (^[Y0 : $i]: (!! @ ($i > $o) @ (^[Y1 : $i > $o]: (!! @ $i @ (^[Y2 : $i]: ((in @ Y2 @ (dsetconstr @ Y0 @ (^[Y3 : $i]: (Y1 @ Y3)))) => (in @ Y2 @ Y0)))))))))),
% 2.30/1.78 inference(cnf_transformation,[status(thm)],[f348])).
% 2.30/1.78 thf(f348,plain,(
% 2.30/1.78 (dsetconstrEL = (!! @ $i @ (^[Y0 : $i]: (!! @ ($i > $o) @ (^[Y1 : $i > $o]: (!! @ $i @ (^[Y2 : $i]: ((in @ Y2 @ (dsetconstr @ Y0 @ (^[Y3 : $i]: (Y1 @ Y3)))) => (in @ Y2 @ Y0)))))))))),
% 2.30/1.78 inference(fool_elimination,[status(thm)],[f347])).
% 2.30/1.78 thf(f347,plain,(
% 2.30/1.78 (! [X0 : $i,X1 : $i > $o,X2 : $i] : ((in @ X0 @ (dsetconstr @ X2 @ (^[X3 : $i] : (X1 @ X3)))) => (in @ X0 @ X2)) = dsetconstrEL)),
% 2.30/1.78 inference(rectify,[status(thm)],[f15])).
% 2.30/1.78 thf(f15,axiom,(
% 2.30/1.78 (! [X1 : $i,X0 : $i > $o,X3 : $i] : ((in @ X1 @ (dsetconstr @ X3 @ (^[X2 : $i] : (X0 @ X2)))) => (in @ X1 @ X3)) = dsetconstrEL)),
% 2.30/1.78 file('/export/starexec/sandbox2/tmp/tmp.U8znWXlJ6W/DTF2THF_14115.p',dsetconstrEL)).
% 2.30/1.78 thf(f539,plain,(
% 2.30/1.78 (dsetconstrEL = $true)),
% 2.30/1.78 inference(cnf_transformation,[status(thm)],[f366])).
% 2.30/1.78 thf(f1811,plain,(
% 2.30/1.78 ((in @ sK2 @ (dsetconstr @ sK1 @ (^[Y0 : $i]: (~ (in @ Y0 @ sK0))))) = $true)),
% 2.30/1.78 inference(beta_eta_normalization,[status(thm)],[f660])).
% 2.30/1.78 thf(f660,plain,(
% 2.30/1.78 ($true = (in @ sK2 @ ((^[Y0 : $i]: ((^[Y1 : $i]: (dsetconstr @ Y0 @ (^[Y2 : $i]: (~ (in @ Y2 @ Y1))))))) @ sK1 @ sK0)))),
% 2.30/1.78 inference(definition_unfolding,[status(thm)],[f533,f454])).
% 2.30/1.78 thf(f454,plain,(
% 2.30/1.78 (setminus = (^[Y0 : $i]: ((^[Y1 : $i]: (dsetconstr @ Y0 @ (^[Y2 : $i]: (~ (in @ Y2 @ Y1))))))))),
% 2.30/1.78 inference(cnf_transformation,[status(thm)],[f282])).
% 2.30/1.78 thf(f282,plain,(
% 2.30/1.78 (setminus = (^[Y0 : $i]: ((^[Y1 : $i]: (dsetconstr @ Y0 @ (^[Y2 : $i]: (~ (in @ Y2 @ Y1))))))))),
% 2.30/1.78 inference(fool_elimination,[status(thm)],[f281])).
% 2.30/1.78 thf(f281,plain,(
% 2.30/1.78 (setminus = (^[X0 : $i, X1 : $i] : (dsetconstr @ X0 @ (^[X2 : $i] : (~(in @ X2 @ X1))))))),
% 2.30/1.78 inference(rectify,[status(thm)],[f120])).
% 2.30/1.78 thf(f120,axiom,(
% 2.30/1.78 (setminus = (^[X3 : $i, X4 : $i] : (dsetconstr @ X3 @ (^[X1 : $i] : (~(in @ X1 @ X4))))))),
% 2.30/1.78 file('/export/starexec/sandbox2/tmp/tmp.U8znWXlJ6W/DTF2THF_14115.p',setminus)).
% 2.30/1.78 thf(f533,plain,(
% 2.30/1.78 ($true = (in @ sK2 @ (setminus @ sK1 @ sK0)))),
% 2.30/1.78 inference(cnf_transformation,[status(thm)],[f366])).
% 2.30/1.78 % SZS output end Proof for DTF2THF_14115
% 2.30/1.78 % (14246)------------------------------
% 2.30/1.78 % (14246)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.30/1.78 % (14246)Termination reason: Refutation
% 2.30/1.78
% 2.30/1.78 % (14246)Memory used [KB]: 7419
% 2.30/1.78 % (14246)Time elapsed: 0.235 s
% 2.30/1.78 % (14246)Instructions burned: 423 (million)
% 2.30/1.78 % (14246)------------------------------
% 2.30/1.78 % (14246)------------------------------
% 2.30/1.78 % (14215)Success in time 0.375 s
%------------------------------------------------------------------------------