%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU592^1 : 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 : n016.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 09:35:28 AM UTC 2026
% Result : Theorem 2.16s 1.45s
% Output : Refutation 2.16s
% Verified :
% SZS Type : Refutation
% Derivation depth : 14
% Number of leaves : 4
% Syntax : Number of formulae : 27 ( 19 unt; 0 typ; 0 def)
% Number of atoms : 1719 ( 456 equ; 0 cnn)
% Maximal formula atoms : 109 ( 63 avg)
% Number of connectives : 915 ( 11 ~; 0 |; 324 &; 137 @)
% ( 0 <=>; 433 =>; 0 <=; 0 <~>)
% Maximal formula depth : 116 ( 31 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 15 ( 15 >; 0 *; 0 +; 0 <<)
% Number of symbols : 209 ( 205 usr; 113 con; 0-4 aty)
% ( 10 !!; 0 ??; 0 @@+; 0 @@-)
% Number of variables : 57 ( 33 ^; 18 !; 6 ?; 57 :)
% 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_146,type,
sK3: $i > $i ).
thf(func_def_147,type,
sK4: $i > $i > $i ).
thf(func_def_148,type,
sK5: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_149,type,
sK6: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_150,type,
sK7: $i > ( $i > $o ) > $i ).
thf(func_def_151,type,
sK8: $i > ( $i > $o ) > $i ).
thf(func_def_152,type,
sK9: $i > ( $i > $o ) > $i ).
thf(func_def_153,type,
sK10: $i > ( $i > $o ) > $i ).
thf(func_def_154,type,
sK11: $i > ( $i > $o ) > $i ).
thf(func_def_155,type,
sK12: $i > ( $i > $o ) > $i ).
thf(func_def_156,type,
sK13: $i > ( $i > $o ) > $i ).
thf(func_def_157,type,
sK14: $i > $i ).
thf(func_def_158,type,
sK15: $i > $i > $i > $i ).
thf(func_def_159,type,
sK16: $i > $i > $i ).
thf(func_def_160,type,
sK17: $i > $i > $i ).
thf(func_def_161,type,
sK18: $i > $i > $i ).
thf(func_def_162,type,
sK19: $i > $i > $i ).
thf(func_def_163,type,
sK20: $i > $i > $i ).
thf(func_def_164,type,
sK21: $i > $i > $i ).
thf(func_def_165,type,
sK22: ( $i > $o ) > $i > $i ).
thf(func_def_166,type,
sK23: ( $i > $o ) > $i ).
thf(func_def_167,type,
sK24: ( $i > $o ) > $i ).
thf(func_def_168,type,
sK25: ( $i > $o ) > $i ).
thf(func_def_169,type,
sK26: ( $i > $o ) > $i > $i ).
thf(func_def_170,type,
sK27: ( $i > $o ) > $i ).
thf(func_def_171,type,
sK28: ( $i > $o ) > $i ).
thf(func_def_172,type,
sK29: ( $i > $o ) > $i ).
thf(func_def_173,type,
sK30: ( $i > $o ) > $i ).
thf(func_def_174,type,
sK31: ( $i > $o ) > $i ).
thf(func_def_175,type,
sK32: $i > ( $i > $o ) > $i ).
thf(func_def_176,type,
sK33: $o > $i > $i > $i ).
thf(func_def_177,type,
sK34: $i > ( $i > $o ) > $i ).
thf(func_def_178,type,
sK35: $i > $i ).
thf(func_def_179,type,
sK36: $i > $i ).
thf(func_def_180,type,
sK37: $i > $i ).
thf(func_def_181,type,
sK38: ( $i > $o ) > $i > $i ).
thf(func_def_182,type,
sK39: ( $i > $o ) > $i ).
thf(func_def_183,type,
sK40: ( $i > $o ) > $i ).
thf(func_def_184,type,
sK41: $i > ( $i > $o ) > $i ).
thf(func_def_185,type,
sK42: $i > $i > $i ).
thf(func_def_186,type,
sK43: $i > $i > $i ).
thf(func_def_187,type,
sK44: $i > $i > $i ).
thf(func_def_188,type,
sK45: ( $i > $o ) > $i > ( $i > $o ) > $i > $i ).
thf(func_def_189,type,
sK46: $i > $i > ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_190,type,
sK47: ( $i > $o ) > $i > $i ).
thf(func_def_191,type,
sK48: ( $i > $o ) > $i > $i ).
thf(func_def_192,type,
sK49: $i > $i > $i ).
thf(func_def_193,type,
sK50: ( $i > $o ) > $i > $i ).
thf(func_def_194,type,
sK51: $i > $i > $i ).
thf(func_def_195,type,
sK52: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_196,type,
sK53: ( $i > $o ) > $i ).
thf(func_def_197,type,
sK54: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_198,type,
sK55: ( $i > $o ) > $i > $i ).
thf(func_def_199,type,
sK56: ( $i > $o ) > $i > $i ).
thf(func_def_200,type,
sK57: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_201,type,
sK58: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_202,type,
sK59: ( $i > $o ) > $i > $i ).
thf(func_def_203,type,
sK60: ( $i > $o ) > $i ).
thf(func_def_204,type,
sK61: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_205,type,
sK62: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_206,type,
sK63: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_207,type,
sK64: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_208,type,
sK65: ( $i > $o ) > $i ).
thf(func_def_209,type,
sK66: ( $i > $o ) > $i > $i ).
thf(func_def_210,type,
sK67: $i > ( $i > $o ) > $i ).
thf(func_def_211,type,
sK68: $i > ( $i > $i > $o ) > $i ).
thf(func_def_212,type,
sK69: $i > ( $i > $i > $o ) > $i ).
thf(func_def_213,type,
sK70: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_214,type,
sK71: $i > ( $i > $i > $o ) > $i ).
thf(func_def_215,type,
sK72: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_216,type,
sK73: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_217,type,
sK74: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_218,type,
sK75: $i > $i ).
thf(func_def_219,type,
sK76: ( $i > $o ) > $i > $i ).
thf(func_def_220,type,
sK77: ( $i > $o ) > $i ).
thf(func_def_221,type,
sK78: ( $i > $o ) > $i > $i ).
thf(func_def_222,type,
sK79: ( $i > $o ) > $i > $i ).
thf(func_def_223,type,
sK80: $i > $i > $i ).
thf(func_def_224,type,
sK81: ( $i > $o ) > $i > $i ).
thf(func_def_225,type,
sK82: $i > $i ).
thf(f2041,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f1486,f1962]) ).
thf(f1962,plain,
! [X2: $i,X1: $i > $o] :
( ( subset @ ( dsetconstr @ X2 @ X1 ) @ X2 )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f1961]) ).
thf(f1961,plain,
! [X2: $i,X1: $i > $o] :
( ( ^ [Y0: $i] : ( subset @ ( dsetconstr @ Y0 @ X1 ) @ Y0 )
@ X2 )
= $true ),
inference(pi_clausification,[status(thm)],[f1960]) ).
thf(f1960,plain,
! [X1: $i > $o] :
( ( !! @ $i
@ ^ [Y0: $i] : ( subset @ ( dsetconstr @ Y0 @ X1 ) @ Y0 ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f1959]) ).
thf(f1959,plain,
! [X1: $i > $o] :
( ( ^ [Y0: $i > $o] :
( !! @ $i
@ ^ [Y1: $i] : ( subset @ ( dsetconstr @ Y1 @ Y0 ) @ Y1 ) )
@ X1 )
= $true ),
inference(pi_clausification,[status(thm)],[f1958]) ).
thf(f1958,plain,
( ( !! @ ( $i > $o )
@ ^ [Y0: $i > $o] :
( !! @ $i
@ ^ [Y1: $i] : ( subset @ ( dsetconstr @ Y1 @ Y0 ) @ Y1 ) ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f583]) ).
thf(f583,plain,
( ( !! @ ( $i > $o )
@ ^ [Y0: $i > $o] :
( !! @ $i
@ ^ [Y1: $i] :
( subset
@ ( dsetconstr @ Y1
@ ^ [Y2: $i] : ( Y0 @ Y2 ) )
@ Y1 ) ) )
= $true ),
inference(definition_unfolding,[status(thm)],[f533,f360]) ).
thf(f360,plain,
( sepSubset
= ( !! @ ( $i > $o )
@ ^ [Y0: $i > $o] :
( !! @ $i
@ ^ [Y1: $i] :
( subset
@ ( dsetconstr @ Y1
@ ^ [Y2: $i] : ( Y0 @ Y2 ) )
@ Y1 ) ) ) ),
inference(cnf_transformation,[status(thm)],[f297]) ).
thf(f297,plain,
( sepSubset
= ( !! @ ( $i > $o )
@ ^ [Y0: $i > $o] :
( !! @ $i
@ ^ [Y1: $i] :
( subset
@ ( dsetconstr @ Y1
@ ^ [Y2: $i] : ( Y0 @ Y2 ) )
@ Y1 ) ) ) ),
inference(fool_elimination,[status(thm)],[f296]) ).
thf(f296,plain,
! [X0: $i,X1: $i > $o] :
( ( subset
@ ( dsetconstr @ X0
@ ^ [X2: $i] : ( X1 @ X2 ) )
@ X0 )
= sepSubset ),
inference(rectify,[status(thm)],[f100]) ).
thf(f100,axiom,
! [X3: $i,X0: $i > $o] :
( ( subset
@ ( dsetconstr @ X3
@ ^ [X1: $i] : ( X0 @ X1 ) )
@ X3 )
= sepSubset ),
file('/export/starexec/sandbox2/tmp/tmp.Z1Gw6mJcyS/DTF2THF_11947.p',sepSubset) ).
thf(f533,plain,
sepSubset = $true,
inference(cnf_transformation,[status(thm)],[f336]) ).
thf(f336,plain,
( ( notdexE = $true )
& ( setunionAx = $true )
& ( binunionLsub = $true )
& ( binunionEcases = $true )
& ( quantDeMorgan3 = $true )
& ( nonemptyE1 = $true )
& ( setadjoinIL = $true )
& ( wellorderingAx = $true )
& ( setunion__Cong = $true )
& ( upairsetE = $true )
& ( emptysetAx = $true )
& ( emptysetE = $true )
& ( sepSubset = $true )
& ( descrp = $true )
& ( subsetRefl = $true )
& ( setbeta = $true )
& ( setext = $true )
& ( emptyset__Cong = $true )
& ( dsetconstr__Cong = $true )
& ( eqimpsubset1 = $true )
& ( notsubsetI = $true )
& ( omegaIndAx = $true )
& ( setoftrueEq = $true )
& ( nonemptyI = $true )
& ( notinsingleton = $true )
& ( emptyE1 = $true )
& ( binunionIR = $true )
& ( dsetconstrER = $true )
& ( noeltsimpempty = $true )
& ( setextAx = $true )
& ( powersetAx = $true )
& ( emptyI = $true )
& ( exuE2 = $true )
& ( setadjoinSub = $true )
& ( exuEu = $true )
& ( upairsetIR = $true )
& ( setadjoinOr = $true )
& ( emptysetsubset = $true )
& ( setunionE = $true )
& ( uniqinunit = $true )
& ( exuI3 = $true )
& ( emptyinPowerset = $true )
& ( powerset__Cong = $true )
& ( foundationAx = $true )
& ( powersetI1 = $true )
& ( descr__Cong = $true )
& ( emptyInPowerset = $true )
& ( setextsub = $true )
& ( notequalI2 = $true )
& ( notequalI1 = $true )
& ( setunionI = $true )
& ( dsetconstrI = $true )
& ( subsetemptysetimpeq = $true )
& ( upairsetIL = $true )
& ( subPowSU = $true )
& ( exuE1 = $true )
& ( prop2setI = $true )
& ( exuI2 = $true )
& ( powersetE1 = $true )
& ( nonemptyImpWitness = $true )
& ( nonemptyI1 = $true )
& ( inCongP = $true )
& ( omega__Cong = $true )
& ( subset2powerset = $true )
& ( notinemptyset = $true )
& ( setadjoinAx = $true )
& ( emptyinunitempty = $true )
& ( binunionE = $true )
& ( exuI1 = $true )
& ( upairset2IR = $true )
& ( setadjoinSub2 = $true )
& ( quantDeMorgan4 = $true )
& ( singletonsswitch = $true )
& ( binunionIL = $true )
& ( exuE3u = $true )
& ( subsetTrans = $true )
& ( sepInPowerset = $true )
& ( eqimpsubset2 = $true )
& ( vacuousDall = $true )
& ( in__Cong = $true )
& ( binintersectSubset5 = $true )
& ( powersetsubset = $true )
& ( subsetI2 = $true )
& ( subsetE = $true )
& ( notdallE = $true )
& ( setadjoinE = $true )
& ( emptysetimpfalse = $true )
& ( omega0Ax = $true )
& ( inPowerset = $true )
& ( powersetI = $true )
& ( subsetE2 = $true )
& ( replAx = $true )
& ( eqinunit = $true )
& ( setadjoinIR = $true )
& ( prop2setE = $true )
& ( exu__Cong = $true )
& ( setadjoin__Cong = $true )
& ( omegaSAx = $true )
& ( powersetE = $true )
& ( quantDeMorgan2 = $true )
& ( binintersectI = $true )
& ( dsetconstrEL = $true )
& ( quantDeMorgan1 = $true )
& ( binintersectEL = $true )
& ( binunionRsub = $true )
& ( prop2set2propI = $true )
& ( subsetI1 = $true )
& ( ( subset @ ( binintersect @ sK1 @ sK0 ) @ sK1 )
!= $true )
& ( exuE3e = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1])],[f334,f335]) ).
thf(f335,plain,
( ? [X0: $i,X1: $i] :
( $true
!= ( subset @ ( binintersect @ X1 @ X0 ) @ X1 ) )
=> ( ( subset @ ( binintersect @ sK1 @ sK0 ) @ sK1 )
!= $true ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f334,plain,
( ( notdexE = $true )
& ( setunionAx = $true )
& ( binunionLsub = $true )
& ( binunionEcases = $true )
& ( quantDeMorgan3 = $true )
& ( nonemptyE1 = $true )
& ( setadjoinIL = $true )
& ( wellorderingAx = $true )
& ( setunion__Cong = $true )
& ( upairsetE = $true )
& ( emptysetAx = $true )
& ( emptysetE = $true )
& ( sepSubset = $true )
& ( descrp = $true )
& ( subsetRefl = $true )
& ( setbeta = $true )
& ( setext = $true )
& ( emptyset__Cong = $true )
& ( dsetconstr__Cong = $true )
& ( eqimpsubset1 = $true )
& ( notsubsetI = $true )
& ( omegaIndAx = $true )
& ( setoftrueEq = $true )
& ( nonemptyI = $true )
& ( notinsingleton = $true )
& ( emptyE1 = $true )
& ( binunionIR = $true )
& ( dsetconstrER = $true )
& ( noeltsimpempty = $true )
& ( setextAx = $true )
& ( powersetAx = $true )
& ( emptyI = $true )
& ( exuE2 = $true )
& ( setadjoinSub = $true )
& ( exuEu = $true )
& ( upairsetIR = $true )
& ( setadjoinOr = $true )
& ( emptysetsubset = $true )
& ( setunionE = $true )
& ( uniqinunit = $true )
& ( exuI3 = $true )
& ( emptyinPowerset = $true )
& ( powerset__Cong = $true )
& ( foundationAx = $true )
& ( powersetI1 = $true )
& ( descr__Cong = $true )
& ( emptyInPowerset = $true )
& ( setextsub = $true )
& ( notequalI2 = $true )
& ( notequalI1 = $true )
& ( setunionI = $true )
& ( dsetconstrI = $true )
& ( subsetemptysetimpeq = $true )
& ( upairsetIL = $true )
& ( subPowSU = $true )
& ( exuE1 = $true )
& ( prop2setI = $true )
& ( exuI2 = $true )
& ( powersetE1 = $true )
& ( nonemptyImpWitness = $true )
& ( nonemptyI1 = $true )
& ( inCongP = $true )
& ( omega__Cong = $true )
& ( subset2powerset = $true )
& ( notinemptyset = $true )
& ( setadjoinAx = $true )
& ( emptyinunitempty = $true )
& ( binunionE = $true )
& ( exuI1 = $true )
& ( upairset2IR = $true )
& ( setadjoinSub2 = $true )
& ( quantDeMorgan4 = $true )
& ( singletonsswitch = $true )
& ( binunionIL = $true )
& ( exuE3u = $true )
& ( subsetTrans = $true )
& ( sepInPowerset = $true )
& ( eqimpsubset2 = $true )
& ( vacuousDall = $true )
& ( in__Cong = $true )
& ( binintersectSubset5 = $true )
& ( powersetsubset = $true )
& ( subsetI2 = $true )
& ( subsetE = $true )
& ( notdallE = $true )
& ( setadjoinE = $true )
& ( emptysetimpfalse = $true )
& ( omega0Ax = $true )
& ( inPowerset = $true )
& ( powersetI = $true )
& ( subsetE2 = $true )
& ( replAx = $true )
& ( eqinunit = $true )
& ( setadjoinIR = $true )
& ( prop2setE = $true )
& ( exu__Cong = $true )
& ( setadjoin__Cong = $true )
& ( omegaSAx = $true )
& ( powersetE = $true )
& ( quantDeMorgan2 = $true )
& ( binintersectI = $true )
& ( dsetconstrEL = $true )
& ( quantDeMorgan1 = $true )
& ( binintersectEL = $true )
& ( binunionRsub = $true )
& ( prop2set2propI = $true )
& ( subsetI1 = $true )
& ? [X0: $i,X1: $i] :
( $true
!= ( subset @ ( binintersect @ X1 @ X0 ) @ X1 ) )
& ( exuE3e = $true ) ),
inference(flattening,[status(thm)],[f333]) ).
thf(f333,plain,
( ? [X0: $i,X1: $i] :
( $true
!= ( subset @ ( binintersect @ X1 @ X0 ) @ X1 ) )
& ( 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)],[f229]) ).
thf(f229,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 )
=> ! [X0: $i,X1: $i] :
( $true
= ( subset @ ( binintersect @ X1 @ X0 ) @ X1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f228]) ).
thf(f228,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
=> ! [X0: $i,X1: $i] : ( subset @ ( binintersect @ X1 @ X0 ) @ X1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f113]) ).
thf(f113,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
=> ! [X4: $i,X3: $i] : ( subset @ ( binintersect @ X3 @ X4 ) @ X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f112]) ).
thf(f112,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
=> ! [X4: $i,X3: $i] : ( subset @ ( binintersect @ X3 @ X4 ) @ X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.Z1Gw6mJcyS/DTF2THF_11947.p',binintersectLsub) ).
thf(f1486,plain,
( ( subset
@ ( dsetconstr @ sK1
@ ^ [Y0: $i] : ( in @ Y0 @ sK0 ) )
@ sK1 )
!= $true ),
inference(beta_eta_normalization,[status(thm)],[f670]) ).
thf(f670,plain,
( ( subset
@ ( ^ [Y0: $i,Y1: $i] :
( dsetconstr @ Y0
@ ^ [Y2: $i] : ( in @ Y2 @ Y1 ) )
@ sK1
@ sK0 )
@ sK1 )
!= $true ),
inference(definition_unfolding,[status(thm)],[f438,f552]) ).
thf(f552,plain,
( binintersect
= ( ^ [Y0: $i,Y1: $i] :
( dsetconstr @ Y0
@ ^ [Y2: $i] : ( in @ Y2 @ Y1 ) ) ) ),
inference(cnf_transformation,[status(thm)],[f241]) ).
thf(f241,plain,
( binintersect
= ( ^ [Y0: $i,Y1: $i] :
( dsetconstr @ Y0
@ ^ [Y2: $i] : ( in @ Y2 @ Y1 ) ) ) ),
inference(fool_elimination,[status(thm)],[f240]) ).
thf(f240,plain,
( ( ^ [X0: $i,X1: $i] :
( dsetconstr @ X0
@ ^ [X2: $i] : ( in @ X2 @ X1 ) ) )
= binintersect ),
inference(rectify,[status(thm)],[f108]) ).
thf(f108,axiom,
( ( ^ [X3: $i,X4: $i] :
( dsetconstr @ X3
@ ^ [X1: $i] : ( in @ X1 @ X4 ) ) )
= binintersect ),
file('/export/starexec/sandbox2/tmp/tmp.Z1Gw6mJcyS/DTF2THF_11947.p',binintersect) ).
thf(f438,plain,
( ( subset @ ( binintersect @ sK1 @ sK0 ) @ sK1 )
!= $true ),
inference(cnf_transformation,[status(thm)],[f336]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : SEU592^1 : 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
% 0.13/0.30 % Computer : n016.cluster.edu
% 0.13/0.30 % Model : x86_64 x86_64
% 0.13/0.30 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.30 % Memory : 8042.1875MB
% 0.13/0.30 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.30 % CPULimit : 300
% 0.13/0.30 % WCLimit : 300
% 0.13/0.30 % DateTime : Tue Feb 24 12:28:21 EST 2026
% 0.13/0.31 % CPUTime :
% 0.13/0.31 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.16/0.42 ---- Original DTF file ---
% 0.16/0.42 thf(spec,logic,$$dhol).
% 0.16/0.42 %------------------------------------------------------------------------------
% 0.16/0.42 % File : SEU592^1 : TPTP v9.2.1. Released v3.7.0.
% 0.16/0.42 % Domain : Set Theory
% 0.16/0.42 % Problem : Preliminary Notions - Ops on Sets - Unions and Intersections
% 0.16/0.42 % Version : Especial.
% 0.16/0.42 % English : (! A:i.! B:i.subset (binintersect A B) A)
% 0.16/0.42
% 0.16/0.42 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.16/0.42 % Source : [Bro08]
% 0.16/0.42 % Names : ZFC094g [Bro08]
% 0.16/0.42
% 0.16/0.42 % Status : Theorem
% 0.16/0.42 % Rating : 0.22 v9.1.0, 0.25 v9.0.0, 0.30 v8.2.0, 0.38 v8.1.0, 0.27 v7.5.0, 0.14 v7.4.0, 0.22 v7.2.0, 0.12 v7.1.0, 0.38 v7.0.0, 0.29 v6.4.0, 0.33 v6.3.0, 0.40 v6.2.0, 0.29 v6.1.0, 0.43 v5.5.0, 0.50 v5.4.0, 0.60 v4.1.0, 1.00 v3.7.0
% 0.16/0.42 % Syntax : Number of formulae : 237 ( 111 unt; 125 typ; 111 def)
% 0.16/0.42 % Number of atoms : 592 ( 160 equ; 0 cnn)
% 0.16/0.42 % Maximal formula atoms : 109 ( 5 avg)
% 0.16/0.42 % Number of connectives : 934 ( 29 ~; 6 |; 27 &; 593 @)
% 0.16/0.42 % ( 14 <=>; 265 =>; 0 <=; 0 <~>)
% 0.16/0.42 % Maximal formula depth : 115 ( 2 avg)
% 0.16/0.42 % Number of types : 2 ( 0 usr)
% 0.16/0.42 % Number of type conns : 59 ( 59 >; 0 *; 0 +; 0 <<)
% 0.16/0.42 % Number of symbols : 128 ( 125 usr; 112 con; 0-2 aty)
% 0.16/0.42 % Number of variables : 327 ( 33 ^; 266 !; 28 ?; 327 :)
% 0.16/0.42 % SPC : TH0_THM_EQU_NAR
% 0.16/0.42
% 0.16/0.42 % Comments : http://mathgate.info/detsetitem.php?id=292
% 0.16/0.42 %------------------------------------------------------------------------------
% 0.16/0.42 thf(in_type,type,
% 0.16/0.42 in: $i > $i > $o ).
% 0.16/0.42
% 0.16/0.42 thf(exu_type,type,
% 0.16/0.42 exu: ( $i > $o ) > $o ).
% 0.16/0.42
% 0.16/0.42 thf(exu,definition,
% 0.16/0.42 ( exu
% 0.16/0.42 = ( ^ [Xphi: $i > $o] :
% 0.16/0.42 ? [Xx: $i] :
% 0.16/0.42 ( ( Xphi @ Xx )
% 0.16/0.42 & ! [Xy: $i] :
% 0.16/0.42 ( ( Xphi @ Xy )
% 0.16/0.42 => ( Xx = Xy ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setextAx_type,type,
% 0.16/0.42 setextAx: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setextAx,definition,
% 0.16/0.42 ( setextAx
% 0.16/0.42 = ( ! [A: $i,B: $i] :
% 0.16/0.42 ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 <=> ( in @ Xx @ B ) )
% 0.16/0.42 => ( A = B ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(emptyset_type,type,
% 0.16/0.42 emptyset: $i ).
% 0.16/0.42
% 0.16/0.42 thf(emptysetAx_type,type,
% 0.16/0.42 emptysetAx: $o ).
% 0.16/0.42
% 0.16/0.42 thf(emptysetAx,definition,
% 0.16/0.42 ( emptysetAx
% 0.16/0.42 = ( ! [Xx: $i] :
% 0.16/0.42 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setadjoin_type,type,
% 0.16/0.42 setadjoin: $i > $i > $i ).
% 0.16/0.42
% 0.16/0.42 thf(setadjoinAx_type,type,
% 0.16/0.42 setadjoinAx: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setadjoinAx,definition,
% 0.16/0.42 ( setadjoinAx
% 0.16/0.42 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.16/0.42 <=> ( ( Xy = Xx )
% 0.16/0.42 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(powerset_type,type,
% 0.16/0.42 powerset: $i > $i ).
% 0.16/0.42
% 0.16/0.42 thf(powersetAx_type,type,
% 0.16/0.42 powersetAx: $o ).
% 0.16/0.42
% 0.16/0.42 thf(powersetAx,definition,
% 0.16/0.42 ( powersetAx
% 0.16/0.42 = ( ! [A: $i,B: $i] :
% 0.16/0.42 ( ( in @ B @ ( powerset @ A ) )
% 0.16/0.42 <=> ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ B )
% 0.16/0.42 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setunion_type,type,
% 0.16/0.42 setunion: $i > $i ).
% 0.16/0.42
% 0.16/0.42 thf(setunionAx_type,type,
% 0.16/0.42 setunionAx: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setunionAx,definition,
% 0.16/0.42 ( setunionAx
% 0.16/0.42 = ( ! [A: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ ( setunion @ A ) )
% 0.16/0.42 <=> ? [B: $i] :
% 0.16/0.42 ( ( in @ Xx @ B )
% 0.16/0.42 & ( in @ B @ A ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(omega_type,type,
% 0.16/0.42 omega: $i ).
% 0.16/0.42
% 0.16/0.42 thf(omega0Ax_type,type,
% 0.16/0.42 omega0Ax: $o ).
% 0.16/0.42
% 0.16/0.42 thf(omega0Ax,definition,
% 0.16/0.42 ( omega0Ax
% 0.16/0.42 = ( in @ emptyset @ omega ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(omegaSAx_type,type,
% 0.16/0.42 omegaSAx: $o ).
% 0.16/0.42
% 0.16/0.42 thf(omegaSAx,definition,
% 0.16/0.42 ( omegaSAx
% 0.16/0.42 = ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ omega )
% 0.16/0.42 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(omegaIndAx_type,type,
% 0.16/0.42 omegaIndAx: $o ).
% 0.16/0.42
% 0.16/0.42 thf(omegaIndAx,definition,
% 0.16/0.42 ( omegaIndAx
% 0.16/0.42 = ( ! [A: $i] :
% 0.16/0.42 ( ( ( in @ emptyset @ A )
% 0.16/0.42 & ! [Xx: $i] :
% 0.16/0.42 ( ( ( in @ Xx @ omega )
% 0.16/0.42 & ( in @ Xx @ A ) )
% 0.16/0.42 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ omega )
% 0.16/0.42 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(replAx_type,type,
% 0.16/0.42 replAx: $o ).
% 0.16/0.42
% 0.16/0.42 thf(replAx,definition,
% 0.16/0.42 ( replAx
% 0.16/0.42 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.16/0.42 ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( exu
% 0.16/0.42 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.16/0.42 => ? [B: $i] :
% 0.16/0.42 ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ B )
% 0.16/0.42 <=> ? [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ A )
% 0.16/0.42 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(foundationAx_type,type,
% 0.16/0.42 foundationAx: $o ).
% 0.16/0.42
% 0.16/0.42 thf(foundationAx,definition,
% 0.16/0.42 ( foundationAx
% 0.16/0.42 = ( ! [A: $i] :
% 0.16/0.42 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.16/0.42 => ? [B: $i] :
% 0.16/0.42 ( ( in @ B @ A )
% 0.16/0.42 & ~ ? [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ B )
% 0.16/0.42 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(wellorderingAx_type,type,
% 0.16/0.42 wellorderingAx: $o ).
% 0.16/0.42
% 0.16/0.42 thf(wellorderingAx,definition,
% 0.16/0.42 ( wellorderingAx
% 0.16/0.42 = ( ! [A: $i] :
% 0.16/0.42 ? [B: $i] :
% 0.16/0.42 ( ! [C: $i] :
% 0.16/0.42 ( ( in @ C @ B )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ C )
% 0.16/0.42 => ( in @ Xx @ A ) ) )
% 0.16/0.42 & ! [Xx: $i,Xy: $i] :
% 0.16/0.42 ( ( ( in @ Xx @ A )
% 0.16/0.42 & ( in @ Xy @ A ) )
% 0.16/0.42 => ( ! [C: $i] :
% 0.16/0.42 ( ( in @ C @ B )
% 0.16/0.42 => ( ( in @ Xx @ C )
% 0.16/0.42 <=> ( in @ Xy @ C ) ) )
% 0.16/0.42 => ( Xx = Xy ) ) )
% 0.16/0.42 & ! [C: $i,D: $i] :
% 0.16/0.42 ( ( ( in @ C @ B )
% 0.16/0.42 & ( in @ D @ B ) )
% 0.16/0.42 => ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ C )
% 0.16/0.42 => ( in @ Xx @ D ) )
% 0.16/0.42 | ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ D )
% 0.16/0.42 => ( in @ Xx @ C ) ) ) )
% 0.16/0.42 & ! [C: $i] :
% 0.16/0.42 ( ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ C )
% 0.16/0.42 => ( in @ Xx @ A ) )
% 0.16/0.42 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.16/0.42 => ? [D: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ D @ B )
% 0.16/0.42 & ( in @ Xx @ C )
% 0.16/0.42 & ~ ? [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ D )
% 0.16/0.42 & ( in @ Xy @ C ) )
% 0.16/0.42 & ! [E: $i] :
% 0.16/0.42 ( ( in @ E @ B )
% 0.16/0.42 => ( ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ E )
% 0.16/0.42 => ( in @ Xy @ D ) )
% 0.16/0.42 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(descr_type,type,
% 0.16/0.42 descr: ( $i > $o ) > $i ).
% 0.16/0.42
% 0.16/0.42 thf(descrp_type,type,
% 0.16/0.42 descrp: $o ).
% 0.16/0.42
% 0.16/0.42 thf(descrp,definition,
% 0.16/0.42 ( descrp
% 0.16/0.42 = ( ! [Xphi: $i > $o] :
% 0.16/0.42 ( ( exu
% 0.16/0.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.42 => ( Xphi
% 0.16/0.42 @ ( descr
% 0.16/0.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(dsetconstr_type,type,
% 0.16/0.42 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.16/0.42
% 0.16/0.42 thf(dsetconstrI_type,type,
% 0.16/0.42 dsetconstrI: $o ).
% 0.16/0.42
% 0.16/0.42 thf(dsetconstrI,definition,
% 0.16/0.42 ( dsetconstrI
% 0.16/0.42 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( ( Xphi @ Xx )
% 0.16/0.42 => ( in @ Xx
% 0.16/0.42 @ ( dsetconstr @ A
% 0.16/0.42 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(dsetconstrEL_type,type,
% 0.16/0.42 dsetconstrEL: $o ).
% 0.16/0.42
% 0.16/0.42 thf(dsetconstrEL,definition,
% 0.16/0.42 ( dsetconstrEL
% 0.16/0.42 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx
% 0.16/0.42 @ ( dsetconstr @ A
% 0.16/0.42 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.16/0.42 => ( in @ Xx @ A ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(dsetconstrER_type,type,
% 0.16/0.42 dsetconstrER: $o ).
% 0.16/0.42
% 0.16/0.42 thf(dsetconstrER,definition,
% 0.16/0.42 ( dsetconstrER
% 0.16/0.42 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx
% 0.16/0.42 @ ( dsetconstr @ A
% 0.16/0.42 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.16/0.42 => ( Xphi @ Xx ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(exuE1_type,type,
% 0.16/0.42 exuE1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(exuE1,definition,
% 0.16/0.42 ( exuE1
% 0.16/0.42 = ( ! [Xphi: $i > $o] :
% 0.16/0.42 ( ( exu
% 0.16/0.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.42 => ? [Xx: $i] :
% 0.16/0.42 ( ( Xphi @ Xx )
% 0.16/0.42 & ! [Xy: $i] :
% 0.16/0.42 ( ( Xphi @ Xy )
% 0.16/0.42 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(prop2set_type,type,
% 0.16/0.42 prop2set: $o > $i ).
% 0.16/0.42
% 0.16/0.42 thf(prop2setE_type,type,
% 0.16/0.42 prop2setE: $o ).
% 0.16/0.42
% 0.16/0.42 thf(prop2setE,definition,
% 0.16/0.42 ( prop2setE
% 0.16/0.42 = ( ! [Xphi: $o,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.16/0.42 => Xphi ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(emptysetE_type,type,
% 0.16/0.42 emptysetE: $o ).
% 0.16/0.42
% 0.16/0.42 thf(emptysetE,definition,
% 0.16/0.42 ( emptysetE
% 0.16/0.42 = ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ emptyset )
% 0.16/0.42 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(emptysetimpfalse_type,type,
% 0.16/0.42 emptysetimpfalse: $o ).
% 0.16/0.42
% 0.16/0.42 thf(emptysetimpfalse,definition,
% 0.16/0.42 ( emptysetimpfalse
% 0.16/0.42 = ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ emptyset )
% 0.16/0.42 => $false ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(notinemptyset_type,type,
% 0.16/0.42 notinemptyset: $o ).
% 0.16/0.42
% 0.16/0.42 thf(notinemptyset,definition,
% 0.16/0.42 ( notinemptyset
% 0.16/0.42 = ( ! [Xx: $i] :
% 0.16/0.42 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(exuE3e_type,type,
% 0.16/0.42 exuE3e: $o ).
% 0.16/0.42
% 0.16/0.42 thf(exuE3e,definition,
% 0.16/0.42 ( exuE3e
% 0.16/0.42 = ( ! [Xphi: $i > $o] :
% 0.16/0.42 ( ( exu
% 0.16/0.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.42 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setext_type,type,
% 0.16/0.42 setext: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setext,definition,
% 0.16/0.42 ( setext
% 0.16/0.42 = ( ! [A: $i,B: $i] :
% 0.16/0.42 ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( in @ Xx @ B ) )
% 0.16/0.42 => ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ B )
% 0.16/0.42 => ( in @ Xx @ A ) )
% 0.16/0.42 => ( A = B ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(emptyI_type,type,
% 0.16/0.42 emptyI: $o ).
% 0.16/0.42
% 0.16/0.42 thf(emptyI,definition,
% 0.16/0.42 ( emptyI
% 0.16/0.42 = ( ! [A: $i] :
% 0.16/0.42 ( ! [Xx: $i] :
% 0.16/0.42 ~ ( in @ Xx @ A )
% 0.16/0.42 => ( A = emptyset ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(noeltsimpempty_type,type,
% 0.16/0.42 noeltsimpempty: $o ).
% 0.16/0.42
% 0.16/0.42 thf(noeltsimpempty,definition,
% 0.16/0.42 ( noeltsimpempty
% 0.16/0.42 = ( ! [A: $i] :
% 0.16/0.42 ( ! [Xx: $i] :
% 0.16/0.42 ~ ( in @ Xx @ A )
% 0.16/0.42 => ( A = emptyset ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setbeta_type,type,
% 0.16/0.42 setbeta: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setbeta,definition,
% 0.16/0.42 ( setbeta
% 0.16/0.42 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( ( in @ Xx
% 0.16/0.42 @ ( dsetconstr @ A
% 0.16/0.42 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.16/0.42 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(nonempty_type,type,
% 0.16/0.42 nonempty: $i > $o ).
% 0.16/0.42
% 0.16/0.42 thf(nonempty,definition,
% 0.16/0.42 ( nonempty
% 0.16/0.42 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(nonemptyE1_type,type,
% 0.16/0.42 nonemptyE1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(nonemptyE1,definition,
% 0.16/0.42 ( nonemptyE1
% 0.16/0.42 = ( ! [A: $i] :
% 0.16/0.42 ( ( nonempty @ A )
% 0.16/0.42 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(nonemptyI_type,type,
% 0.16/0.42 nonemptyI: $o ).
% 0.16/0.42
% 0.16/0.42 thf(nonemptyI,definition,
% 0.16/0.42 ( nonemptyI
% 0.16/0.42 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( ( Xphi @ Xx )
% 0.16/0.42 => ( nonempty
% 0.16/0.42 @ ( dsetconstr @ A
% 0.16/0.42 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(nonemptyI1_type,type,
% 0.16/0.42 nonemptyI1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(nonemptyI1,definition,
% 0.16/0.42 ( nonemptyI1
% 0.16/0.42 = ( ! [A: $i] :
% 0.16/0.42 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.16/0.42 => ( nonempty @ A ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setadjoinIL_type,type,
% 0.16/0.42 setadjoinIL: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setadjoinIL,definition,
% 0.16/0.42 ( setadjoinIL
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(emptyinunitempty_type,type,
% 0.16/0.42 emptyinunitempty: $o ).
% 0.16/0.42
% 0.16/0.42 thf(emptyinunitempty,definition,
% 0.16/0.42 ( emptyinunitempty
% 0.16/0.42 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setadjoinIR_type,type,
% 0.16/0.42 setadjoinIR: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setadjoinIR,definition,
% 0.16/0.42 ( setadjoinIR
% 0.16/0.42 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ A )
% 0.16/0.42 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setadjoinE_type,type,
% 0.16/0.42 setadjoinE: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setadjoinE,definition,
% 0.16/0.42 ( setadjoinE
% 0.16/0.42 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.16/0.42 => ! [Xphi: $o] :
% 0.16/0.42 ( ( ( Xy = Xx )
% 0.16/0.42 => Xphi )
% 0.16/0.42 => ( ( ( in @ Xy @ A )
% 0.16/0.42 => Xphi )
% 0.16/0.42 => Xphi ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setadjoinOr_type,type,
% 0.16/0.42 setadjoinOr: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setadjoinOr,definition,
% 0.16/0.42 ( setadjoinOr
% 0.16/0.42 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.16/0.42 => ( ( Xy = Xx )
% 0.16/0.42 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setoftrueEq_type,type,
% 0.16/0.42 setoftrueEq: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setoftrueEq,definition,
% 0.16/0.42 ( setoftrueEq
% 0.16/0.42 = ( ! [A: $i] :
% 0.16/0.42 ( ( dsetconstr @ A
% 0.16/0.42 @ ^ [Xx: $i] : $true )
% 0.16/0.42 = A ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(powersetI_type,type,
% 0.16/0.42 powersetI: $o ).
% 0.16/0.42
% 0.16/0.42 thf(powersetI,definition,
% 0.16/0.42 ( powersetI
% 0.16/0.42 = ( ! [A: $i,B: $i] :
% 0.16/0.42 ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ B )
% 0.16/0.42 => ( in @ Xx @ A ) )
% 0.16/0.42 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(emptyinPowerset_type,type,
% 0.16/0.42 emptyinPowerset: $o ).
% 0.16/0.42
% 0.16/0.42 thf(emptyinPowerset,definition,
% 0.16/0.42 ( emptyinPowerset
% 0.16/0.42 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(emptyInPowerset_type,type,
% 0.16/0.42 emptyInPowerset: $o ).
% 0.16/0.42
% 0.16/0.42 thf(emptyInPowerset,definition,
% 0.16/0.42 ( emptyInPowerset
% 0.16/0.42 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(powersetE_type,type,
% 0.16/0.42 powersetE: $o ).
% 0.16/0.42
% 0.16/0.42 thf(powersetE,definition,
% 0.16/0.42 ( powersetE
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ B @ ( powerset @ A ) )
% 0.16/0.42 => ( ( in @ Xx @ B )
% 0.16/0.42 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setunionI_type,type,
% 0.16/0.42 setunionI: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setunionI,definition,
% 0.16/0.42 ( setunionI
% 0.16/0.42 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.16/0.42 ( ( in @ Xx @ B )
% 0.16/0.42 => ( ( in @ B @ A )
% 0.16/0.42 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setunionE_type,type,
% 0.16/0.42 setunionE: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setunionE,definition,
% 0.16/0.42 ( setunionE
% 0.16/0.42 = ( ! [A: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ ( setunion @ A ) )
% 0.16/0.42 => ! [Xphi: $o] :
% 0.16/0.42 ( ! [B: $i] :
% 0.16/0.42 ( ( in @ Xx @ B )
% 0.16/0.42 => ( ( in @ B @ A )
% 0.16/0.42 => Xphi ) )
% 0.16/0.42 => Xphi ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(subPowSU_type,type,
% 0.16/0.42 subPowSU: $o ).
% 0.16/0.42
% 0.16/0.42 thf(subPowSU,definition,
% 0.16/0.42 ( subPowSU
% 0.16/0.42 = ( ! [A: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(exuE2_type,type,
% 0.16/0.42 exuE2: $o ).
% 0.16/0.42
% 0.16/0.42 thf(exuE2,definition,
% 0.16/0.42 ( exuE2
% 0.16/0.42 = ( ! [Xphi: $i > $o] :
% 0.16/0.42 ( ( exu
% 0.16/0.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.42 => ? [Xx: $i] :
% 0.16/0.42 ! [Xy: $i] :
% 0.16/0.42 ( ( Xphi @ Xy )
% 0.16/0.42 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(nonemptyImpWitness_type,type,
% 0.16/0.42 nonemptyImpWitness: $o ).
% 0.16/0.42
% 0.16/0.42 thf(nonemptyImpWitness,definition,
% 0.16/0.42 ( nonemptyImpWitness
% 0.16/0.42 = ( ! [A: $i] :
% 0.16/0.42 ( ( nonempty @ A )
% 0.16/0.42 => ? [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 & $true ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(uniqinunit_type,type,
% 0.16/0.42 uniqinunit: $o ).
% 0.16/0.42
% 0.16/0.42 thf(uniqinunit,definition,
% 0.16/0.42 ( uniqinunit
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.42 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.16/0.42 => ( Xx = Xy ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(notinsingleton_type,type,
% 0.16/0.42 notinsingleton: $o ).
% 0.16/0.42
% 0.16/0.42 thf(notinsingleton,definition,
% 0.16/0.42 ( notinsingleton
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.42 ( ( Xx != Xy )
% 0.16/0.42 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(eqinunit_type,type,
% 0.16/0.42 eqinunit: $o ).
% 0.16/0.42
% 0.16/0.42 thf(eqinunit,definition,
% 0.16/0.42 ( eqinunit
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.42 ( ( Xx = Xy )
% 0.16/0.42 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(singletonsswitch_type,type,
% 0.16/0.42 singletonsswitch: $o ).
% 0.16/0.42
% 0.16/0.42 thf(singletonsswitch,definition,
% 0.16/0.42 ( singletonsswitch
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.42 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.16/0.42 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(upairsetE_type,type,
% 0.16/0.42 upairsetE: $o ).
% 0.16/0.42
% 0.16/0.42 thf(upairsetE,definition,
% 0.16/0.42 ( upairsetE
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.16/0.42 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.16/0.42 => ( ( Xz = Xx )
% 0.16/0.42 | ( Xz = Xy ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(upairsetIL_type,type,
% 0.16/0.42 upairsetIL: $o ).
% 0.16/0.42
% 0.16/0.42 thf(upairsetIL,definition,
% 0.16/0.42 ( upairsetIL
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(upairsetIR_type,type,
% 0.16/0.42 upairsetIR: $o ).
% 0.16/0.42
% 0.16/0.42 thf(upairsetIR,definition,
% 0.16/0.42 ( upairsetIR
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(emptyE1_type,type,
% 0.16/0.42 emptyE1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(emptyE1,definition,
% 0.16/0.42 ( emptyE1
% 0.16/0.42 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.42 ( ? [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 & ( Xphi @ Xx ) )
% 0.16/0.42 => ( ( ( dsetconstr @ A
% 0.16/0.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.42 = emptyset )
% 0.16/0.42 => $false ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(vacuousDall_type,type,
% 0.16/0.42 vacuousDall: $o ).
% 0.16/0.42
% 0.16/0.42 thf(vacuousDall,definition,
% 0.16/0.42 ( vacuousDall
% 0.16/0.42 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ emptyset )
% 0.16/0.42 => ( Xphi @ Xx ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(quantDeMorgan1_type,type,
% 0.16/0.42 quantDeMorgan1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(quantDeMorgan1,definition,
% 0.16/0.42 ( quantDeMorgan1
% 0.16/0.42 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.42 ( ~ ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( Xphi @ Xx ) )
% 0.16/0.42 => ? [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(quantDeMorgan2_type,type,
% 0.16/0.42 quantDeMorgan2: $o ).
% 0.16/0.42
% 0.16/0.42 thf(quantDeMorgan2,definition,
% 0.16/0.42 ( quantDeMorgan2
% 0.16/0.42 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.42 ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ~ ( Xphi @ Xx ) )
% 0.16/0.42 => ~ ? [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(quantDeMorgan3_type,type,
% 0.16/0.42 quantDeMorgan3: $o ).
% 0.16/0.42
% 0.16/0.42 thf(quantDeMorgan3,definition,
% 0.16/0.42 ( quantDeMorgan3
% 0.16/0.42 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.42 ( ~ ? [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 & ( Xphi @ Xx ) )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(quantDeMorgan4_type,type,
% 0.16/0.42 quantDeMorgan4: $o ).
% 0.16/0.42
% 0.16/0.42 thf(quantDeMorgan4,definition,
% 0.16/0.42 ( quantDeMorgan4
% 0.16/0.42 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.42 ( ? [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 & ~ ( Xphi @ Xx ) )
% 0.16/0.42 => ~ ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(prop2setI_type,type,
% 0.16/0.42 prop2setI: $o ).
% 0.16/0.42
% 0.16/0.42 thf(prop2setI,definition,
% 0.16/0.42 ( prop2setI
% 0.16/0.42 = ( ! [Xphi: $o] :
% 0.16/0.42 ( Xphi
% 0.16/0.42 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(set2prop_type,type,
% 0.16/0.42 set2prop: $i > $o ).
% 0.16/0.42
% 0.16/0.42 thf(prop2set2propI_type,type,
% 0.16/0.42 prop2set2propI: $o ).
% 0.16/0.42
% 0.16/0.42 thf(prop2set2propI,definition,
% 0.16/0.42 ( prop2set2propI
% 0.16/0.42 = ( ! [Xphi: $o] :
% 0.16/0.42 ( Xphi
% 0.16/0.42 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(notdexE_type,type,
% 0.16/0.42 notdexE: $o ).
% 0.16/0.42
% 0.16/0.42 thf(notdexE,definition,
% 0.16/0.42 ( notdexE
% 0.16/0.42 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.42 ( ~ ? [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 & ( Xphi @ Xx ) )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(notdallE_type,type,
% 0.16/0.42 notdallE: $o ).
% 0.16/0.42
% 0.16/0.42 thf(notdallE,definition,
% 0.16/0.42 ( notdallE
% 0.16/0.42 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.42 ( ~ ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( Xphi @ Xx ) )
% 0.16/0.42 => ? [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(exuI1_type,type,
% 0.16/0.42 exuI1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(exuI1,definition,
% 0.16/0.42 ( exuI1
% 0.16/0.42 = ( ! [Xphi: $i > $o] :
% 0.16/0.42 ( ? [Xx: $i] :
% 0.16/0.42 ( ( Xphi @ Xx )
% 0.16/0.42 & ! [Xy: $i] :
% 0.16/0.42 ( ( Xphi @ Xy )
% 0.16/0.42 => ( Xx = Xy ) ) )
% 0.16/0.42 => ( exu
% 0.16/0.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(exuI3_type,type,
% 0.16/0.42 exuI3: $o ).
% 0.16/0.42
% 0.16/0.42 thf(exuI3,definition,
% 0.16/0.42 ( exuI3
% 0.16/0.42 = ( ! [Xphi: $i > $o] :
% 0.16/0.42 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.16/0.42 => ( ! [Xx: $i,Xy: $i] :
% 0.16/0.42 ( ( Xphi @ Xx )
% 0.16/0.42 => ( ( Xphi @ Xy )
% 0.16/0.42 => ( Xx = Xy ) ) )
% 0.16/0.42 => ( exu
% 0.16/0.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(exuI2_type,type,
% 0.16/0.42 exuI2: $o ).
% 0.16/0.42
% 0.16/0.42 thf(exuI2,definition,
% 0.16/0.42 ( exuI2
% 0.16/0.42 = ( ! [Xphi: $i > $o] :
% 0.16/0.42 ( ? [Xx: $i] :
% 0.16/0.42 ! [Xy: $i] :
% 0.16/0.42 ( ( Xphi @ Xy )
% 0.16/0.42 <=> ( Xy = Xx ) )
% 0.16/0.42 => ( exu
% 0.16/0.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(inCongP_type,type,
% 0.16/0.42 inCongP: $o ).
% 0.16/0.42
% 0.16/0.42 thf(inCongP,definition,
% 0.16/0.42 ( inCongP
% 0.16/0.42 = ( ! [A: $i,B: $i] :
% 0.16/0.42 ( ( A = B )
% 0.16/0.42 => ! [Xx: $i,Xy: $i] :
% 0.16/0.42 ( ( Xx = Xy )
% 0.16/0.42 => ( ( in @ Xx @ A )
% 0.16/0.42 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(in__Cong_type,type,
% 0.16/0.42 in__Cong: $o ).
% 0.16/0.42
% 0.16/0.42 thf(in__Cong,definition,
% 0.16/0.42 ( in__Cong
% 0.16/0.42 = ( ! [A: $i,B: $i] :
% 0.16/0.42 ( ( A = B )
% 0.16/0.42 => ! [Xx: $i,Xy: $i] :
% 0.16/0.42 ( ( Xx = Xy )
% 0.16/0.42 => ( ( in @ Xx @ A )
% 0.16/0.42 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(exuE3u_type,type,
% 0.16/0.42 exuE3u: $o ).
% 0.16/0.42
% 0.16/0.42 thf(exuE3u,definition,
% 0.16/0.42 ( exuE3u
% 0.16/0.42 = ( ! [Xphi: $i > $o] :
% 0.16/0.42 ( ( exu
% 0.16/0.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.42 => ! [Xx: $i,Xy: $i] :
% 0.16/0.42 ( ( Xphi @ Xx )
% 0.16/0.42 => ( ( Xphi @ Xy )
% 0.16/0.42 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(exu__Cong_type,type,
% 0.16/0.42 exu__Cong: $o ).
% 0.16/0.42
% 0.16/0.42 thf(exu__Cong,definition,
% 0.16/0.42 ( exu__Cong
% 0.16/0.42 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.16/0.42 ( ! [Xx: $i,Xy: $i] :
% 0.16/0.42 ( ( Xx = Xy )
% 0.16/0.42 => ( ( Xphi @ Xx )
% 0.16/0.42 <=> ( Xpsi @ Xy ) ) )
% 0.16/0.42 => ( ( exu
% 0.16/0.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.42 <=> ( exu
% 0.16/0.42 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(emptyset__Cong_type,type,
% 0.16/0.42 emptyset__Cong: $o ).
% 0.16/0.42
% 0.16/0.42 thf(emptyset__Cong,definition,
% 0.16/0.42 ( emptyset__Cong
% 0.16/0.42 = ( emptyset = emptyset ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setadjoin__Cong_type,type,
% 0.16/0.42 setadjoin__Cong: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setadjoin__Cong,definition,
% 0.16/0.42 ( setadjoin__Cong
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.42 ( ( Xx = Xy )
% 0.16/0.42 => ! [Xz: $i,Xu: $i] :
% 0.16/0.42 ( ( Xz = Xu )
% 0.16/0.42 => ( ( setadjoin @ Xx @ Xz )
% 0.16/0.42 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(powerset__Cong_type,type,
% 0.16/0.42 powerset__Cong: $o ).
% 0.16/0.42
% 0.16/0.42 thf(powerset__Cong,definition,
% 0.16/0.42 ( powerset__Cong
% 0.16/0.42 = ( ! [A: $i,B: $i] :
% 0.16/0.42 ( ( A = B )
% 0.16/0.42 => ( ( powerset @ A )
% 0.16/0.42 = ( powerset @ B ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setunion__Cong_type,type,
% 0.16/0.42 setunion__Cong: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setunion__Cong,definition,
% 0.16/0.42 ( setunion__Cong
% 0.16/0.42 = ( ! [A: $i,B: $i] :
% 0.16/0.42 ( ( A = B )
% 0.16/0.42 => ( ( setunion @ A )
% 0.16/0.42 = ( setunion @ B ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(omega__Cong_type,type,
% 0.16/0.42 omega__Cong: $o ).
% 0.16/0.42
% 0.16/0.42 thf(omega__Cong,definition,
% 0.16/0.42 ( omega__Cong
% 0.16/0.42 = ( omega = omega ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(exuEu_type,type,
% 0.16/0.42 exuEu: $o ).
% 0.16/0.42
% 0.16/0.42 thf(exuEu,definition,
% 0.16/0.42 ( exuEu
% 0.16/0.42 = ( ! [Xphi: $i > $o] :
% 0.16/0.42 ( ( exu
% 0.16/0.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.42 => ! [Xx: $i,Xy: $i] :
% 0.16/0.42 ( ( Xphi @ Xx )
% 0.16/0.42 => ( ( Xphi @ Xy )
% 0.16/0.42 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(descr__Cong_type,type,
% 0.16/0.42 descr__Cong: $o ).
% 0.16/0.42
% 0.16/0.42 thf(descr__Cong,definition,
% 0.16/0.42 ( descr__Cong
% 0.16/0.42 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.16/0.42 ( ! [Xx: $i,Xy: $i] :
% 0.16/0.42 ( ( Xx = Xy )
% 0.16/0.42 => ( ( Xphi @ Xx )
% 0.16/0.42 <=> ( Xpsi @ Xy ) ) )
% 0.16/0.42 => ( ( exu
% 0.16/0.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.42 => ( ( exu
% 0.16/0.42 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.16/0.42 => ( ( descr
% 0.16/0.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.42 = ( descr
% 0.16/0.42 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(dsetconstr__Cong_type,type,
% 0.16/0.42 dsetconstr__Cong: $o ).
% 0.16/0.42
% 0.16/0.42 thf(dsetconstr__Cong,definition,
% 0.16/0.42 ( dsetconstr__Cong
% 0.16/0.42 = ( ! [A: $i,B: $i] :
% 0.16/0.42 ( ( A = B )
% 0.16/0.42 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.16/0.42 ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ B )
% 0.16/0.42 => ( ( Xx = Xy )
% 0.16/0.42 => ( ( Xphi @ Xx )
% 0.16/0.42 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.16/0.42 => ( ( dsetconstr @ A
% 0.16/0.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.42 = ( dsetconstr @ B
% 0.16/0.42 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(subset_type,type,
% 0.16/0.42 subset: $i > $i > $o ).
% 0.16/0.42
% 0.16/0.42 thf(disjoint_type,type,
% 0.16/0.42 disjoint: $i > $i > $o ).
% 0.16/0.42
% 0.16/0.42 thf(setsmeet_type,type,
% 0.16/0.42 setsmeet: $i > $i > $o ).
% 0.16/0.42
% 0.16/0.42 thf(subsetI1_type,type,
% 0.16/0.42 subsetI1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(subsetI1,definition,
% 0.16/0.42 ( subsetI1
% 0.16/0.42 = ( ! [A: $i,B: $i] :
% 0.16/0.42 ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( in @ Xx @ B ) )
% 0.16/0.42 => ( subset @ A @ B ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(eqimpsubset2_type,type,
% 0.16/0.42 eqimpsubset2: $o ).
% 0.16/0.42
% 0.16/0.42 thf(eqimpsubset2,definition,
% 0.16/0.42 ( eqimpsubset2
% 0.16/0.42 = ( ! [A: $i,B: $i] :
% 0.16/0.42 ( ( A = B )
% 0.16/0.42 => ( subset @ B @ A ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(eqimpsubset1_type,type,
% 0.16/0.42 eqimpsubset1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(eqimpsubset1,definition,
% 0.16/0.42 ( eqimpsubset1
% 0.16/0.42 = ( ! [A: $i,B: $i] :
% 0.16/0.42 ( ( A = B )
% 0.16/0.42 => ( subset @ A @ B ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(subsetI2_type,type,
% 0.16/0.42 subsetI2: $o ).
% 0.16/0.42
% 0.16/0.42 thf(subsetI2,definition,
% 0.16/0.42 ( subsetI2
% 0.16/0.42 = ( ! [A: $i,B: $i] :
% 0.16/0.42 ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( in @ Xx @ B ) )
% 0.16/0.42 => ( subset @ A @ B ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(emptysetsubset_type,type,
% 0.16/0.42 emptysetsubset: $o ).
% 0.16/0.42
% 0.16/0.42 thf(emptysetsubset,definition,
% 0.16/0.42 ( emptysetsubset
% 0.16/0.42 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(subsetE_type,type,
% 0.16/0.42 subsetE: $o ).
% 0.16/0.42
% 0.16/0.42 thf(subsetE,definition,
% 0.16/0.42 ( subsetE
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ( subset @ A @ B )
% 0.16/0.42 => ( ( in @ Xx @ A )
% 0.16/0.42 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(subsetE2_type,type,
% 0.16/0.42 subsetE2: $o ).
% 0.16/0.42
% 0.16/0.42 thf(subsetE2,definition,
% 0.16/0.42 ( subsetE2
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ( subset @ A @ B )
% 0.16/0.42 => ( ~ ( in @ Xx @ B )
% 0.16/0.42 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(notsubsetI_type,type,
% 0.16/0.42 notsubsetI: $o ).
% 0.16/0.42
% 0.16/0.42 thf(notsubsetI,definition,
% 0.16/0.42 ( notsubsetI
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( ~ ( in @ Xx @ B )
% 0.16/0.42 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(notequalI1_type,type,
% 0.16/0.42 notequalI1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(notequalI1,definition,
% 0.16/0.42 ( notequalI1
% 0.16/0.42 = ( ! [A: $i,B: $i] :
% 0.16/0.42 ( ~ ( subset @ A @ B )
% 0.16/0.42 => ( A != B ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(notequalI2_type,type,
% 0.16/0.42 notequalI2: $o ).
% 0.16/0.42
% 0.16/0.42 thf(notequalI2,definition,
% 0.16/0.42 ( notequalI2
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( ~ ( in @ Xx @ B )
% 0.16/0.42 => ( A != B ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(subsetRefl_type,type,
% 0.16/0.42 subsetRefl: $o ).
% 0.16/0.42
% 0.16/0.42 thf(subsetRefl,definition,
% 0.16/0.42 ( subsetRefl
% 0.16/0.42 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(subsetTrans_type,type,
% 0.16/0.42 subsetTrans: $o ).
% 0.16/0.42
% 0.16/0.42 thf(subsetTrans,definition,
% 0.16/0.42 ( subsetTrans
% 0.16/0.42 = ( ! [A: $i,B: $i,C: $i] :
% 0.16/0.42 ( ( subset @ A @ B )
% 0.16/0.42 => ( ( subset @ B @ C )
% 0.16/0.42 => ( subset @ A @ C ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setadjoinSub_type,type,
% 0.16/0.42 setadjoinSub: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setadjoinSub,definition,
% 0.16/0.42 ( setadjoinSub
% 0.16/0.42 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setadjoinSub2_type,type,
% 0.16/0.42 setadjoinSub2: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setadjoinSub2,definition,
% 0.16/0.42 ( setadjoinSub2
% 0.16/0.42 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.16/0.42 ( ( subset @ A @ B )
% 0.16/0.42 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(subset2powerset_type,type,
% 0.16/0.42 subset2powerset: $o ).
% 0.16/0.42
% 0.16/0.42 thf(subset2powerset,definition,
% 0.16/0.42 ( subset2powerset
% 0.16/0.42 = ( ! [A: $i,B: $i] :
% 0.16/0.42 ( ( subset @ A @ B )
% 0.16/0.42 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setextsub_type,type,
% 0.16/0.42 setextsub: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setextsub,definition,
% 0.16/0.42 ( setextsub
% 0.16/0.42 = ( ! [A: $i,B: $i] :
% 0.16/0.42 ( ( subset @ A @ B )
% 0.16/0.42 => ( ( subset @ B @ A )
% 0.16/0.42 => ( A = B ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(subsetemptysetimpeq_type,type,
% 0.16/0.42 subsetemptysetimpeq: $o ).
% 0.16/0.42
% 0.16/0.42 thf(subsetemptysetimpeq,definition,
% 0.16/0.42 ( subsetemptysetimpeq
% 0.16/0.42 = ( ! [A: $i] :
% 0.16/0.42 ( ( subset @ A @ emptyset )
% 0.16/0.42 => ( A = emptyset ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(powersetI1_type,type,
% 0.16/0.42 powersetI1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(powersetI1,definition,
% 0.16/0.42 ( powersetI1
% 0.16/0.42 = ( ! [A: $i,B: $i] :
% 0.16/0.42 ( ( subset @ B @ A )
% 0.16/0.42 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(powersetE1_type,type,
% 0.16/0.42 powersetE1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(powersetE1,definition,
% 0.16/0.42 ( powersetE1
% 0.16/0.42 = ( ! [A: $i,B: $i] :
% 0.16/0.42 ( ( in @ B @ ( powerset @ A ) )
% 0.16/0.42 => ( subset @ B @ A ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(inPowerset_type,type,
% 0.16/0.42 inPowerset: $o ).
% 0.16/0.42
% 0.16/0.42 thf(inPowerset,definition,
% 0.16/0.42 ( inPowerset
% 0.16/0.42 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(powersetsubset_type,type,
% 0.16/0.42 powersetsubset: $o ).
% 0.16/0.42
% 0.16/0.42 thf(powersetsubset,definition,
% 0.16/0.42 ( powersetsubset
% 0.16/0.42 = ( ! [A: $i,B: $i] :
% 0.16/0.42 ( ( subset @ A @ B )
% 0.16/0.42 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(sepInPowerset_type,type,
% 0.16/0.42 sepInPowerset: $o ).
% 0.16/0.42
% 0.16/0.42 thf(sepInPowerset,definition,
% 0.16/0.42 ( sepInPowerset
% 0.16/0.42 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.42 ( in
% 0.16/0.42 @ ( dsetconstr @ A
% 0.16/0.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.42 @ ( powerset @ A ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(sepSubset_type,type,
% 0.16/0.42 sepSubset: $o ).
% 0.16/0.42
% 0.16/0.42 thf(sepSubset,definition,
% 0.16/0.42 ( sepSubset
% 0.16/0.42 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.42 ( subset
% 0.16/0.42 @ ( dsetconstr @ A
% 0.16/0.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.42 @ A ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(binunion_type,type,
% 0.16/0.42 binunion: $i > $i > $i ).
% 0.16/0.42
% 0.16/0.42 thf(binunionIL_type,type,
% 0.16/0.42 binunionIL: $o ).
% 0.16/0.42
% 0.16/0.42 thf(binunionIL,definition,
% 0.16/0.42 ( binunionIL
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(upairset2IR_type,type,
% 0.16/0.42 upairset2IR: $o ).
% 0.16/0.42
% 0.16/0.42 thf(upairset2IR,definition,
% 0.16/0.42 ( upairset2IR
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(binunionIR_type,type,
% 0.16/0.42 binunionIR: $o ).
% 0.16/0.42
% 0.16/0.42 thf(binunionIR,definition,
% 0.16/0.42 ( binunionIR
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ B )
% 0.16/0.42 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(binunionEcases_type,type,
% 0.16/0.42 binunionEcases: $o ).
% 0.16/0.42
% 0.16/0.42 thf(binunionEcases,definition,
% 0.16/0.42 ( binunionEcases
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.16/0.42 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.16/0.42 => ( ( ( in @ Xx @ A )
% 0.16/0.42 => Xphi )
% 0.16/0.42 => ( ( ( in @ Xx @ B )
% 0.16/0.42 => Xphi )
% 0.16/0.42 => Xphi ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(binunionE_type,type,
% 0.16/0.42 binunionE: $o ).
% 0.16/0.42
% 0.16/0.42 thf(binunionE,definition,
% 0.16/0.42 ( binunionE
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.16/0.42 => ( ( in @ Xx @ A )
% 0.16/0.42 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(binunionLsub_type,type,
% 0.16/0.42 binunionLsub: $o ).
% 0.16/0.42
% 0.16/0.42 thf(binunionLsub,definition,
% 0.16/0.42 ( binunionLsub
% 0.16/0.42 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(binunionRsub_type,type,
% 0.16/0.42 binunionRsub: $o ).
% 0.16/0.42
% 0.16/0.42 thf(binunionRsub,definition,
% 0.16/0.42 ( binunionRsub
% 0.16/0.42 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(binintersect_type,type,
% 0.16/0.42 binintersect: $i > $i > $i ).
% 0.16/0.42
% 0.16/0.42 thf(binintersect,definition,
% 0.16/0.42 ( binintersect
% 0.16/0.42 = ( ^ [A: $i,B: $i] :
% 0.16/0.42 ( dsetconstr @ A
% 0.16/0.42 @ ^ [Xx: $i] : ( in @ Xx @ B ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(binintersectI_type,type,
% 0.16/0.42 binintersectI: $o ).
% 0.16/0.42
% 0.16/0.42 thf(binintersectI,definition,
% 0.16/0.42 ( binintersectI
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( ( in @ Xx @ B )
% 0.16/0.42 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(binintersectSubset5_type,type,
% 0.16/0.42 binintersectSubset5: $o ).
% 0.16/0.42
% 0.16/0.42 thf(binintersectSubset5,definition,
% 0.16/0.42 ( binintersectSubset5
% 0.16/0.42 = ( ! [A: $i,B: $i,C: $i] :
% 0.16/0.42 ( ( subset @ C @ A )
% 0.16/0.42 => ( ( subset @ C @ B )
% 0.16/0.42 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(binintersectEL_type,type,
% 0.16/0.42 binintersectEL: $o ).
% 0.16/0.42
% 0.16/0.42 thf(binintersectEL,definition,
% 0.16/0.42 ( binintersectEL
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.16/0.42 => ( in @ Xx @ A ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(binintersectLsub,conjecture,
% 0.16/0.42 ( setextAx
% 0.16/0.42 => ( emptysetAx
% 0.16/0.42 => ( setadjoinAx
% 0.16/0.42 => ( powersetAx
% 0.16/0.42 => ( setunionAx
% 0.16/0.42 => ( omega0Ax
% 0.16/0.42 => ( omegaSAx
% 0.16/0.42 => ( omegaIndAx
% 0.16/0.42 => ( replAx
% 0.16/0.42 => ( foundationAx
% 0.16/0.42 => ( wellorderingAx
% 0.16/0.42 => ( descrp
% 0.16/0.42 => ( dsetconstrI
% 0.16/0.42 => ( dsetconstrEL
% 0.16/0.42 => ( dsetconstrER
% 0.16/0.42 => ( exuE1
% 0.16/0.42 => ( prop2setE
% 0.16/0.42 => ( emptysetE
% 0.16/0.42 => ( emptysetimpfalse
% 0.16/0.42 => ( notinemptyset
% 0.16/0.42 => ( exuE3e
% 0.16/0.42 => ( setext
% 0.16/0.42 => ( emptyI
% 0.16/0.42 => ( noeltsimpempty
% 0.16/0.42 => ( setbeta
% 0.16/0.42 => ( nonemptyE1
% 0.16/0.42 => ( nonemptyI
% 0.16/0.42 => ( nonemptyI1
% 0.16/0.42 => ( setadjoinIL
% 0.16/0.42 => ( emptyinunitempty
% 0.16/0.42 => ( setadjoinIR
% 0.16/0.42 => ( setadjoinE
% 0.16/0.42 => ( setadjoinOr
% 0.16/0.42 => ( setoftrueEq
% 0.16/0.42 => ( powersetI
% 0.16/0.42 => ( emptyinPowerset
% 0.16/0.42 => ( emptyInPowerset
% 0.16/0.42 => ( powersetE
% 0.16/0.42 => ( setunionI
% 0.16/0.42 => ( setunionE
% 0.16/0.42 => ( subPowSU
% 0.16/0.42 => ( exuE2
% 0.16/0.42 => ( nonemptyImpWitness
% 0.16/0.42 => ( uniqinunit
% 0.16/0.42 => ( notinsingleton
% 0.16/0.42 => ( eqinunit
% 0.16/0.42 => ( singletonsswitch
% 0.16/0.42 => ( upairsetE
% 0.16/0.42 => ( upairsetIL
% 0.16/0.42 => ( upairsetIR
% 0.16/0.42 => ( emptyE1
% 0.16/0.42 => ( vacuousDall
% 0.16/0.42 => ( quantDeMorgan1
% 0.16/0.42 => ( quantDeMorgan2
% 0.16/0.42 => ( quantDeMorgan3
% 0.16/0.42 => ( quantDeMorgan4
% 0.16/0.42 => ( prop2setI
% 0.16/0.42 => ( prop2set2propI
% 0.16/0.42 => ( notdexE
% 0.16/0.42 => ( notdallE
% 0.16/0.42 => ( exuI1
% 0.16/0.42 => ( exuI3
% 0.16/0.42 => ( exuI2
% 0.16/0.42 => ( inCongP
% 0.16/0.42 => ( in__Cong
% 0.16/0.42 => ( exuE3u
% 0.16/0.42 => ( exu__Cong
% 0.16/0.42 => ( emptyset__Cong
% 0.16/0.42 => ( setadjoin__Cong
% 0.16/0.42 => ( powerset__Cong
% 0.16/0.42 => ( setunion__Cong
% 0.16/0.42 => ( omega__Cong
% 0.16/0.42 => ( exuEu
% 0.16/0.42 => ( descr__Cong
% 0.16/0.42 => ( dsetconstr__Cong
% 0.16/0.42 => ( subsetI1
% 0.16/0.42 => ( eqimpsubset2
% 0.16/0.42 => ( eqimpsubset1
% 0.16/0.42 => ( subsetI2
% 0.16/0.42 => ( emptysetsubset
% 0.16/0.42 => ( subsetE
% 0.16/0.42 => ( subsetE2
% 0.16/0.42 => ( notsubsetI
% 0.16/0.42 => ( notequalI1
% 0.16/0.42 => ( notequalI2
% 0.16/0.42 => ( subsetRefl
% 0.16/0.42 => ( subsetTrans
% 0.16/0.42 => ( setadjoinSub
% 0.16/0.42 => ( setadjoinSub2
% 0.16/0.42 => ( subset2powerset
% 0.16/0.42 => ( setextsub
% 0.16/0.42 => ( subsetemptysetimpeq
% 0.16/0.42 => ( powersetI1
% 0.16/0.42 => ( powersetE1
% 0.16/0.42 => ( inPowerset
% 0.16/0.42 => ( powersetsubset
% 0.16/0.42 => ( sepInPowerset
% 0.16/0.42 => ( sepSubset
% 0.16/0.42 => ( binunionIL
% 0.16/0.42 => ( upairset2IR
% 0.16/0.42 => ( binunionIR
% 0.16/0.42 => ( binunionEcases
% 0.16/0.42 => ( binunionE
% 0.16/0.42 => ( binunionLsub
% 0.16/0.42 => ( binunionRsub
% 0.16/0.42 => ( binintersectI
% 0.16/0.42 => ( binintersectSubset5
% 0.16/0.42 => ( binintersectEL
% 0.16/0.42 => ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 %------------------------------------------------------------------------------
% 0.16/0.42 ------------------------
% 1.98/1.36 ---- Embedded in THF ---
% 1.98/1.36 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 1.98/1.36 %%% Generated on Tue Feb 24 12:28:22 EST 2026
% 1.98/1.36 %%% using '$$dhol' embedding, version 1.3.0.
% 1.98/1.36 %%% Logic specification used:
% 1.98/1.36 %%% thf(spec, logic, $$dhol).
% 1.98/1.36
% 1.98/1.36 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.Z1Gw6mJcyS/DTF2DTF11947.p
% See solution above
% 1.98/1.36 ------------------------
% 1.98/1.38 ---- Cleaned THF ---
% 1.98/1.38 thf(in_type,type,
% 1.98/1.38 in: $i > $i > $o ).
% 1.98/1.38
% 1.98/1.38 thf(exu_type,type,
% 1.98/1.38 exu: ( $i > $o ) > $o ).
% 1.98/1.38
% 1.98/1.38 thf(exu,definition,
% 1.98/1.38 ( exu
% 1.98/1.38 = ( ^ [Xphi: $i > $o] :
% 1.98/1.38 ? [Xx: $i] :
% 1.98/1.38 ( ( Xphi @ Xx )
% 1.98/1.38 & ! [Xy: $i] :
% 1.98/1.38 ( ( Xphi @ Xy )
% 1.98/1.38 => ( Xx = Xy ) ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(setextAx_type,type,
% 1.98/1.38 setextAx: $o ).
% 1.98/1.38
% 1.98/1.38 thf(setextAx,definition,
% 1.98/1.38 ( setextAx
% 1.98/1.38 = ( ! [A: $i,B: $i] :
% 1.98/1.38 ( ! [Xx: $i] :
% 1.98/1.38 ( ( in @ Xx @ A )
% 1.98/1.38 <=> ( in @ Xx @ B ) )
% 1.98/1.38 => ( A = B ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(emptyset_type,type,
% 1.98/1.38 emptyset: $i ).
% 1.98/1.38
% 1.98/1.38 thf(emptysetAx_type,type,
% 1.98/1.38 emptysetAx: $o ).
% 1.98/1.38
% 1.98/1.38 thf(emptysetAx,definition,
% 1.98/1.38 ( emptysetAx
% 1.98/1.38 = ( ! [Xx: $i] :
% 1.98/1.38 ~ ( in @ Xx @ emptyset ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(setadjoin_type,type,
% 1.98/1.38 setadjoin: $i > $i > $i ).
% 1.98/1.38
% 1.98/1.38 thf(setadjoinAx_type,type,
% 1.98/1.38 setadjoinAx: $o ).
% 1.98/1.38
% 1.98/1.38 thf(setadjoinAx,definition,
% 1.98/1.38 ( setadjoinAx
% 1.98/1.38 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 1.98/1.38 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 1.98/1.38 <=> ( ( Xy = Xx )
% 1.98/1.38 | ( in @ Xy @ A ) ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(powerset_type,type,
% 1.98/1.38 powerset: $i > $i ).
% 1.98/1.38
% 1.98/1.38 thf(powersetAx_type,type,
% 1.98/1.38 powersetAx: $o ).
% 1.98/1.38
% 1.98/1.38 thf(powersetAx,definition,
% 1.98/1.38 ( powersetAx
% 1.98/1.38 = ( ! [A: $i,B: $i] :
% 1.98/1.38 ( ( in @ B @ ( powerset @ A ) )
% 1.98/1.38 <=> ! [Xx: $i] :
% 1.98/1.38 ( ( in @ Xx @ B )
% 1.98/1.38 => ( in @ Xx @ A ) ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(setunion_type,type,
% 1.98/1.38 setunion: $i > $i ).
% 1.98/1.38
% 1.98/1.38 thf(setunionAx_type,type,
% 1.98/1.38 setunionAx: $o ).
% 1.98/1.38
% 1.98/1.38 thf(setunionAx,definition,
% 1.98/1.38 ( setunionAx
% 1.98/1.38 = ( ! [A: $i,Xx: $i] :
% 1.98/1.38 ( ( in @ Xx @ ( setunion @ A ) )
% 1.98/1.38 <=> ? [B: $i] :
% 1.98/1.38 ( ( in @ Xx @ B )
% 1.98/1.38 & ( in @ B @ A ) ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(omega_type,type,
% 1.98/1.38 omega: $i ).
% 1.98/1.38
% 1.98/1.38 thf(omega0Ax_type,type,
% 1.98/1.38 omega0Ax: $o ).
% 1.98/1.38
% 1.98/1.38 thf(omega0Ax,definition,
% 1.98/1.38 ( omega0Ax
% 1.98/1.38 = ( in @ emptyset @ omega ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(omegaSAx_type,type,
% 1.98/1.38 omegaSAx: $o ).
% 1.98/1.38
% 1.98/1.38 thf(omegaSAx,definition,
% 1.98/1.38 ( omegaSAx
% 1.98/1.38 = ( ! [Xx: $i] :
% 1.98/1.38 ( ( in @ Xx @ omega )
% 1.98/1.38 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(omegaIndAx_type,type,
% 1.98/1.38 omegaIndAx: $o ).
% 1.98/1.38
% 1.98/1.38 thf(omegaIndAx,definition,
% 1.98/1.38 ( omegaIndAx
% 1.98/1.38 = ( ! [A: $i] :
% 1.98/1.38 ( ( ( in @ emptyset @ A )
% 1.98/1.38 & ! [Xx: $i] :
% 1.98/1.38 ( ( ( in @ Xx @ omega )
% 1.98/1.38 & ( in @ Xx @ A ) )
% 1.98/1.38 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 1.98/1.38 => ! [Xx: $i] :
% 1.98/1.38 ( ( in @ Xx @ omega )
% 1.98/1.38 => ( in @ Xx @ A ) ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(replAx_type,type,
% 1.98/1.38 replAx: $o ).
% 1.98/1.38
% 1.98/1.38 thf(replAx,definition,
% 1.98/1.38 ( replAx
% 1.98/1.38 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 1.98/1.38 ( ! [Xx: $i] :
% 1.98/1.38 ( ( in @ Xx @ A )
% 1.98/1.38 => ( exu
% 1.98/1.38 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 1.98/1.38 => ? [B: $i] :
% 1.98/1.38 ! [Xx: $i] :
% 1.98/1.38 ( ( in @ Xx @ B )
% 1.98/1.38 <=> ? [Xy: $i] :
% 1.98/1.38 ( ( in @ Xy @ A )
% 1.98/1.38 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(foundationAx_type,type,
% 1.98/1.38 foundationAx: $o ).
% 1.98/1.38
% 1.98/1.38 thf(foundationAx,definition,
% 1.98/1.38 ( foundationAx
% 1.98/1.38 = ( ! [A: $i] :
% 1.98/1.38 ( ? [Xx: $i] : ( in @ Xx @ A )
% 1.98/1.38 => ? [B: $i] :
% 1.98/1.38 ( ( in @ B @ A )
% 1.98/1.38 & ~ ? [Xx: $i] :
% 1.98/1.38 ( ( in @ Xx @ B )
% 1.98/1.38 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(wellorderingAx_type,type,
% 1.98/1.38 wellorderingAx: $o ).
% 1.98/1.38
% 1.98/1.38 thf(wellorderingAx,definition,
% 1.98/1.38 ( wellorderingAx
% 1.98/1.38 = ( ! [A: $i] :
% 1.98/1.38 ? [B: $i] :
% 1.98/1.38 ( ! [C: $i] :
% 1.98/1.38 ( ( in @ C @ B )
% 1.98/1.38 => ! [Xx: $i] :
% 1.98/1.38 ( ( in @ Xx @ C )
% 1.98/1.38 => ( in @ Xx @ A ) ) )
% 1.98/1.38 & ! [Xx: $i,Xy: $i] :
% 1.98/1.38 ( ( ( in @ Xx @ A )
% 1.98/1.38 & ( in @ Xy @ A ) )
% 1.98/1.38 => ( ! [C: $i] :
% 1.98/1.38 ( ( in @ C @ B )
% 1.98/1.38 => ( ( in @ Xx @ C )
% 1.98/1.38 <=> ( in @ Xy @ C ) ) )
% 1.98/1.38 => ( Xx = Xy ) ) )
% 1.98/1.38 & ! [C: $i,D: $i] :
% 1.98/1.38 ( ( ( in @ C @ B )
% 1.98/1.38 & ( in @ D @ B ) )
% 1.98/1.38 => ( ! [Xx: $i] :
% 1.98/1.38 ( ( in @ Xx @ C )
% 1.98/1.38 => ( in @ Xx @ D ) )
% 1.98/1.38 | ! [Xx: $i] :
% 1.98/1.38 ( ( in @ Xx @ D )
% 1.98/1.38 => ( in @ Xx @ C ) ) ) )
% 1.98/1.38 & ! [C: $i] :
% 1.98/1.38 ( ( ! [Xx: $i] :
% 1.98/1.38 ( ( in @ Xx @ C )
% 1.98/1.38 => ( in @ Xx @ A ) )
% 1.98/1.38 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 1.98/1.38 => ? [D: $i,Xx: $i] :
% 1.98/1.38 ( ( in @ D @ B )
% 1.98/1.38 & ( in @ Xx @ C )
% 1.98/1.38 & ~ ? [Xy: $i] :
% 1.98/1.38 ( ( in @ Xy @ D )
% 1.98/1.38 & ( in @ Xy @ C ) )
% 1.98/1.38 & ! [E: $i] :
% 1.98/1.38 ( ( in @ E @ B )
% 1.98/1.38 => ( ! [Xy: $i] :
% 1.98/1.38 ( ( in @ Xy @ E )
% 1.98/1.38 => ( in @ Xy @ D ) )
% 1.98/1.38 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(descr_type,type,
% 1.98/1.38 descr: ( $i > $o ) > $i ).
% 1.98/1.38
% 1.98/1.38 thf(descrp_type,type,
% 1.98/1.38 descrp: $o ).
% 1.98/1.38
% 1.98/1.38 thf(descrp,definition,
% 1.98/1.38 ( descrp
% 1.98/1.38 = ( ! [Xphi: $i > $o] :
% 1.98/1.38 ( ( exu
% 1.98/1.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 1.98/1.38 => ( Xphi
% 1.98/1.38 @ ( descr
% 1.98/1.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(dsetconstr_type,type,
% 1.98/1.38 dsetconstr: $i > ( $i > $o ) > $i ).
% 1.98/1.38
% 1.98/1.38 thf(dsetconstrI_type,type,
% 1.98/1.38 dsetconstrI: $o ).
% 1.98/1.38
% 1.98/1.38 thf(dsetconstrI,definition,
% 1.98/1.38 ( dsetconstrI
% 1.98/1.38 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 1.98/1.38 ( ( in @ Xx @ A )
% 1.98/1.38 => ( ( Xphi @ Xx )
% 1.98/1.38 => ( in @ Xx
% 1.98/1.38 @ ( dsetconstr @ A
% 1.98/1.38 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(dsetconstrEL_type,type,
% 1.98/1.38 dsetconstrEL: $o ).
% 1.98/1.38
% 1.98/1.38 thf(dsetconstrEL,definition,
% 1.98/1.38 ( dsetconstrEL
% 1.98/1.38 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 1.98/1.38 ( ( in @ Xx
% 1.98/1.38 @ ( dsetconstr @ A
% 1.98/1.38 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 1.98/1.38 => ( in @ Xx @ A ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(dsetconstrER_type,type,
% 1.98/1.38 dsetconstrER: $o ).
% 1.98/1.38
% 1.98/1.38 thf(dsetconstrER,definition,
% 1.98/1.38 ( dsetconstrER
% 1.98/1.38 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 1.98/1.38 ( ( in @ Xx
% 1.98/1.38 @ ( dsetconstr @ A
% 1.98/1.38 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 1.98/1.38 => ( Xphi @ Xx ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(exuE1_type,type,
% 1.98/1.38 exuE1: $o ).
% 1.98/1.38
% 1.98/1.38 thf(exuE1,definition,
% 1.98/1.38 ( exuE1
% 1.98/1.38 = ( ! [Xphi: $i > $o] :
% 1.98/1.38 ( ( exu
% 1.98/1.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 1.98/1.38 => ? [Xx: $i] :
% 1.98/1.38 ( ( Xphi @ Xx )
% 1.98/1.38 & ! [Xy: $i] :
% 1.98/1.38 ( ( Xphi @ Xy )
% 1.98/1.38 => ( Xx = Xy ) ) ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(prop2set_type,type,
% 1.98/1.38 prop2set: $o > $i ).
% 1.98/1.38
% 1.98/1.38 thf(prop2setE_type,type,
% 1.98/1.38 prop2setE: $o ).
% 1.98/1.38
% 1.98/1.38 thf(prop2setE,definition,
% 1.98/1.38 ( prop2setE
% 1.98/1.38 = ( ! [Xphi: $o,Xx: $i] :
% 1.98/1.38 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 1.98/1.38 => Xphi ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(emptysetE_type,type,
% 1.98/1.38 emptysetE: $o ).
% 1.98/1.38
% 1.98/1.38 thf(emptysetE,definition,
% 1.98/1.38 ( emptysetE
% 1.98/1.38 = ( ! [Xx: $i] :
% 1.98/1.38 ( ( in @ Xx @ emptyset )
% 1.98/1.38 => ! [Xphi: $o] : Xphi ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(emptysetimpfalse_type,type,
% 1.98/1.38 emptysetimpfalse: $o ).
% 1.98/1.38
% 1.98/1.38 thf(emptysetimpfalse,definition,
% 1.98/1.38 ( emptysetimpfalse
% 1.98/1.38 = ( ! [Xx: $i] :
% 1.98/1.38 ( ( in @ Xx @ emptyset )
% 1.98/1.38 => $false ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(notinemptyset_type,type,
% 1.98/1.38 notinemptyset: $o ).
% 1.98/1.38
% 1.98/1.38 thf(notinemptyset,definition,
% 1.98/1.38 ( notinemptyset
% 1.98/1.38 = ( ! [Xx: $i] :
% 1.98/1.38 ~ ( in @ Xx @ emptyset ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(exuE3e_type,type,
% 1.98/1.38 exuE3e: $o ).
% 1.98/1.38
% 1.98/1.38 thf(exuE3e,definition,
% 1.98/1.38 ( exuE3e
% 1.98/1.38 = ( ! [Xphi: $i > $o] :
% 1.98/1.38 ( ( exu
% 1.98/1.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 1.98/1.38 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(setext_type,type,
% 1.98/1.38 setext: $o ).
% 1.98/1.38
% 1.98/1.38 thf(setext,definition,
% 1.98/1.38 ( setext
% 1.98/1.38 = ( ! [A: $i,B: $i] :
% 1.98/1.38 ( ! [Xx: $i] :
% 1.98/1.38 ( ( in @ Xx @ A )
% 1.98/1.38 => ( in @ Xx @ B ) )
% 1.98/1.38 => ( ! [Xx: $i] :
% 1.98/1.38 ( ( in @ Xx @ B )
% 1.98/1.38 => ( in @ Xx @ A ) )
% 1.98/1.38 => ( A = B ) ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(emptyI_type,type,
% 1.98/1.38 emptyI: $o ).
% 1.98/1.38
% 1.98/1.38 thf(emptyI,definition,
% 1.98/1.38 ( emptyI
% 1.98/1.38 = ( ! [A: $i] :
% 1.98/1.38 ( ! [Xx: $i] :
% 1.98/1.38 ~ ( in @ Xx @ A )
% 1.98/1.38 => ( A = emptyset ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(noeltsimpempty_type,type,
% 1.98/1.38 noeltsimpempty: $o ).
% 1.98/1.38
% 1.98/1.38 thf(noeltsimpempty,definition,
% 1.98/1.38 ( noeltsimpempty
% 1.98/1.38 = ( ! [A: $i] :
% 1.98/1.38 ( ! [Xx: $i] :
% 1.98/1.38 ~ ( in @ Xx @ A )
% 1.98/1.38 => ( A = emptyset ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(setbeta_type,type,
% 1.98/1.38 setbeta: $o ).
% 1.98/1.38
% 1.98/1.38 thf(setbeta,definition,
% 1.98/1.38 ( setbeta
% 1.98/1.38 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 1.98/1.38 ( ( in @ Xx @ A )
% 1.98/1.38 => ( ( in @ Xx
% 1.98/1.38 @ ( dsetconstr @ A
% 1.98/1.38 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 1.98/1.38 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(nonempty_type,type,
% 1.98/1.38 nonempty: $i > $o ).
% 1.98/1.38
% 1.98/1.38 thf(nonempty,definition,
% 1.98/1.38 ( nonempty
% 1.98/1.38 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(nonemptyE1_type,type,
% 1.98/1.38 nonemptyE1: $o ).
% 1.98/1.38
% 1.98/1.38 thf(nonemptyE1,definition,
% 1.98/1.38 ( nonemptyE1
% 1.98/1.38 = ( ! [A: $i] :
% 1.98/1.38 ( ( nonempty @ A )
% 1.98/1.38 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(nonemptyI_type,type,
% 1.98/1.38 nonemptyI: $o ).
% 1.98/1.38
% 1.98/1.38 thf(nonemptyI,definition,
% 1.98/1.38 ( nonemptyI
% 1.98/1.38 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 1.98/1.38 ( ( in @ Xx @ A )
% 1.98/1.38 => ( ( Xphi @ Xx )
% 1.98/1.38 => ( nonempty
% 1.98/1.38 @ ( dsetconstr @ A
% 1.98/1.38 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(nonemptyI1_type,type,
% 1.98/1.38 nonemptyI1: $o ).
% 1.98/1.38
% 1.98/1.38 thf(nonemptyI1,definition,
% 1.98/1.38 ( nonemptyI1
% 1.98/1.38 = ( ! [A: $i] :
% 1.98/1.38 ( ? [Xx: $i] : ( in @ Xx @ A )
% 1.98/1.38 => ( nonempty @ A ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(setadjoinIL_type,type,
% 1.98/1.38 setadjoinIL: $o ).
% 1.98/1.38
% 1.98/1.38 thf(setadjoinIL,definition,
% 1.98/1.38 ( setadjoinIL
% 1.98/1.38 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(emptyinunitempty_type,type,
% 1.98/1.38 emptyinunitempty: $o ).
% 1.98/1.38
% 1.98/1.38 thf(emptyinunitempty,definition,
% 1.98/1.38 ( emptyinunitempty
% 1.98/1.38 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(setadjoinIR_type,type,
% 1.98/1.38 setadjoinIR: $o ).
% 1.98/1.38
% 1.98/1.38 thf(setadjoinIR,definition,
% 1.98/1.38 ( setadjoinIR
% 1.98/1.38 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 1.98/1.38 ( ( in @ Xy @ A )
% 1.98/1.38 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(setadjoinE_type,type,
% 1.98/1.38 setadjoinE: $o ).
% 1.98/1.38
% 1.98/1.38 thf(setadjoinE,definition,
% 1.98/1.38 ( setadjoinE
% 1.98/1.38 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 1.98/1.38 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 1.98/1.38 => ! [Xphi: $o] :
% 1.98/1.38 ( ( ( Xy = Xx )
% 1.98/1.38 => Xphi )
% 1.98/1.38 => ( ( ( in @ Xy @ A )
% 1.98/1.38 => Xphi )
% 1.98/1.38 => Xphi ) ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(setadjoinOr_type,type,
% 1.98/1.38 setadjoinOr: $o ).
% 1.98/1.38
% 1.98/1.38 thf(setadjoinOr,definition,
% 1.98/1.38 ( setadjoinOr
% 1.98/1.38 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 1.98/1.38 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 1.98/1.38 => ( ( Xy = Xx )
% 1.98/1.38 | ( in @ Xy @ A ) ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(setoftrueEq_type,type,
% 1.98/1.38 setoftrueEq: $o ).
% 1.98/1.38
% 1.98/1.38 thf(setoftrueEq,definition,
% 1.98/1.38 ( setoftrueEq
% 1.98/1.38 = ( ! [A: $i] :
% 1.98/1.38 ( ( dsetconstr @ A
% 1.98/1.38 @ ^ [Xx: $i] : $true )
% 1.98/1.38 = A ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(powersetI_type,type,
% 1.98/1.38 powersetI: $o ).
% 1.98/1.38
% 1.98/1.38 thf(powersetI,definition,
% 1.98/1.38 ( powersetI
% 1.98/1.38 = ( ! [A: $i,B: $i] :
% 1.98/1.38 ( ! [Xx: $i] :
% 1.98/1.38 ( ( in @ Xx @ B )
% 1.98/1.38 => ( in @ Xx @ A ) )
% 1.98/1.38 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(emptyinPowerset_type,type,
% 1.98/1.38 emptyinPowerset: $o ).
% 1.98/1.38
% 1.98/1.38 thf(emptyinPowerset,definition,
% 1.98/1.38 ( emptyinPowerset
% 1.98/1.38 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(emptyInPowerset_type,type,
% 1.98/1.38 emptyInPowerset: $o ).
% 1.98/1.38
% 1.98/1.38 thf(emptyInPowerset,definition,
% 1.98/1.38 ( emptyInPowerset
% 1.98/1.38 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(powersetE_type,type,
% 1.98/1.38 powersetE: $o ).
% 1.98/1.38
% 1.98/1.38 thf(powersetE,definition,
% 1.98/1.38 ( powersetE
% 1.98/1.38 = ( ! [A: $i,B: $i,Xx: $i] :
% 1.98/1.38 ( ( in @ B @ ( powerset @ A ) )
% 1.98/1.38 => ( ( in @ Xx @ B )
% 1.98/1.38 => ( in @ Xx @ A ) ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(setunionI_type,type,
% 1.98/1.38 setunionI: $o ).
% 1.98/1.38
% 1.98/1.38 thf(setunionI,definition,
% 1.98/1.38 ( setunionI
% 1.98/1.38 = ( ! [A: $i,Xx: $i,B: $i] :
% 1.98/1.38 ( ( in @ Xx @ B )
% 1.98/1.38 => ( ( in @ B @ A )
% 1.98/1.38 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(setunionE_type,type,
% 1.98/1.38 setunionE: $o ).
% 1.98/1.38
% 1.98/1.38 thf(setunionE,definition,
% 1.98/1.38 ( setunionE
% 1.98/1.38 = ( ! [A: $i,Xx: $i] :
% 1.98/1.38 ( ( in @ Xx @ ( setunion @ A ) )
% 1.98/1.38 => ! [Xphi: $o] :
% 1.98/1.38 ( ! [B: $i] :
% 1.98/1.38 ( ( in @ Xx @ B )
% 1.98/1.38 => ( ( in @ B @ A )
% 1.98/1.38 => Xphi ) )
% 1.98/1.38 => Xphi ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(subPowSU_type,type,
% 1.98/1.38 subPowSU: $o ).
% 1.98/1.38
% 1.98/1.38 thf(subPowSU,definition,
% 1.98/1.38 ( subPowSU
% 1.98/1.38 = ( ! [A: $i,Xx: $i] :
% 1.98/1.38 ( ( in @ Xx @ A )
% 1.98/1.38 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(exuE2_type,type,
% 1.98/1.38 exuE2: $o ).
% 1.98/1.38
% 1.98/1.38 thf(exuE2,definition,
% 1.98/1.38 ( exuE2
% 1.98/1.38 = ( ! [Xphi: $i > $o] :
% 1.98/1.38 ( ( exu
% 1.98/1.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 1.98/1.38 => ? [Xx: $i] :
% 1.98/1.38 ! [Xy: $i] :
% 1.98/1.38 ( ( Xphi @ Xy )
% 1.98/1.38 <=> ( Xy = Xx ) ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(nonemptyImpWitness_type,type,
% 1.98/1.38 nonemptyImpWitness: $o ).
% 1.98/1.38
% 1.98/1.38 thf(nonemptyImpWitness,definition,
% 1.98/1.38 ( nonemptyImpWitness
% 1.98/1.38 = ( ! [A: $i] :
% 1.98/1.38 ( ( nonempty @ A )
% 1.98/1.38 => ? [Xx: $i] :
% 1.98/1.38 ( ( in @ Xx @ A )
% 1.98/1.38 & $true ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(uniqinunit_type,type,
% 1.98/1.38 uniqinunit: $o ).
% 1.98/1.38
% 1.98/1.38 thf(uniqinunit,definition,
% 1.98/1.38 ( uniqinunit
% 1.98/1.38 = ( ! [Xx: $i,Xy: $i] :
% 1.98/1.38 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 1.98/1.38 => ( Xx = Xy ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(notinsingleton_type,type,
% 1.98/1.38 notinsingleton: $o ).
% 1.98/1.38
% 1.98/1.38 thf(notinsingleton,definition,
% 1.98/1.38 ( notinsingleton
% 1.98/1.38 = ( ! [Xx: $i,Xy: $i] :
% 1.98/1.38 ( ( Xx != Xy )
% 1.98/1.38 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(eqinunit_type,type,
% 1.98/1.38 eqinunit: $o ).
% 1.98/1.38
% 1.98/1.38 thf(eqinunit,definition,
% 1.98/1.38 ( eqinunit
% 1.98/1.38 = ( ! [Xx: $i,Xy: $i] :
% 1.98/1.38 ( ( Xx = Xy )
% 1.98/1.38 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(singletonsswitch_type,type,
% 1.98/1.38 singletonsswitch: $o ).
% 1.98/1.38
% 1.98/1.38 thf(singletonsswitch,definition,
% 1.98/1.38 ( singletonsswitch
% 1.98/1.38 = ( ! [Xx: $i,Xy: $i] :
% 1.98/1.38 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 1.98/1.38 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(upairsetE_type,type,
% 1.98/1.38 upairsetE: $o ).
% 1.98/1.38
% 1.98/1.38 thf(upairsetE,definition,
% 1.98/1.38 ( upairsetE
% 1.98/1.38 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 1.98/1.38 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 1.98/1.38 => ( ( Xz = Xx )
% 1.98/1.38 | ( Xz = Xy ) ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(upairsetIL_type,type,
% 1.98/1.38 upairsetIL: $o ).
% 1.98/1.38
% 1.98/1.38 thf(upairsetIL,definition,
% 1.98/1.38 ( upairsetIL
% 1.98/1.38 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(upairsetIR_type,type,
% 1.98/1.38 upairsetIR: $o ).
% 1.98/1.38
% 1.98/1.38 thf(upairsetIR,definition,
% 1.98/1.38 ( upairsetIR
% 1.98/1.38 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(emptyE1_type,type,
% 1.98/1.38 emptyE1: $o ).
% 1.98/1.38
% 1.98/1.38 thf(emptyE1,definition,
% 1.98/1.38 ( emptyE1
% 1.98/1.38 = ( ! [A: $i,Xphi: $i > $o] :
% 1.98/1.38 ( ? [Xx: $i] :
% 1.98/1.38 ( ( in @ Xx @ A )
% 1.98/1.38 & ( Xphi @ Xx ) )
% 1.98/1.38 => ( ( ( dsetconstr @ A
% 1.98/1.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 1.98/1.38 = emptyset )
% 1.98/1.38 => $false ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(vacuousDall_type,type,
% 1.98/1.38 vacuousDall: $o ).
% 1.98/1.38
% 1.98/1.38 thf(vacuousDall,definition,
% 1.98/1.38 ( vacuousDall
% 1.98/1.38 = ( ! [Xphi: $i > $o,Xx: $i] :
% 1.98/1.38 ( ( in @ Xx @ emptyset )
% 1.98/1.38 => ( Xphi @ Xx ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(quantDeMorgan1_type,type,
% 1.98/1.38 quantDeMorgan1: $o ).
% 1.98/1.38
% 1.98/1.38 thf(quantDeMorgan1,definition,
% 1.98/1.38 ( quantDeMorgan1
% 1.98/1.38 = ( ! [A: $i,Xphi: $i > $o] :
% 1.98/1.38 ( ~ ! [Xx: $i] :
% 1.98/1.38 ( ( in @ Xx @ A )
% 1.98/1.38 => ( Xphi @ Xx ) )
% 1.98/1.38 => ? [Xx: $i] :
% 1.98/1.38 ( ( in @ Xx @ A )
% 1.98/1.38 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(quantDeMorgan2_type,type,
% 1.98/1.38 quantDeMorgan2: $o ).
% 1.98/1.38
% 1.98/1.38 thf(quantDeMorgan2,definition,
% 1.98/1.38 ( quantDeMorgan2
% 1.98/1.38 = ( ! [A: $i,Xphi: $i > $o] :
% 1.98/1.38 ( ! [Xx: $i] :
% 1.98/1.38 ( ( in @ Xx @ A )
% 1.98/1.38 => ~ ( Xphi @ Xx ) )
% 1.98/1.38 => ~ ? [Xx: $i] :
% 1.98/1.38 ( ( in @ Xx @ A )
% 1.98/1.38 & ( Xphi @ Xx ) ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(quantDeMorgan3_type,type,
% 1.98/1.38 quantDeMorgan3: $o ).
% 1.98/1.38
% 1.98/1.38 thf(quantDeMorgan3,definition,
% 1.98/1.38 ( quantDeMorgan3
% 1.98/1.38 = ( ! [A: $i,Xphi: $i > $o] :
% 1.98/1.38 ( ~ ? [Xx: $i] :
% 1.98/1.38 ( ( in @ Xx @ A )
% 1.98/1.38 & ( Xphi @ Xx ) )
% 1.98/1.38 => ! [Xx: $i] :
% 1.98/1.38 ( ( in @ Xx @ A )
% 1.98/1.38 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(quantDeMorgan4_type,type,
% 1.98/1.38 quantDeMorgan4: $o ).
% 1.98/1.38
% 1.98/1.38 thf(quantDeMorgan4,definition,
% 1.98/1.38 ( quantDeMorgan4
% 1.98/1.38 = ( ! [A: $i,Xphi: $i > $o] :
% 1.98/1.38 ( ? [Xx: $i] :
% 1.98/1.38 ( ( in @ Xx @ A )
% 1.98/1.38 & ~ ( Xphi @ Xx ) )
% 1.98/1.38 => ~ ! [Xx: $i] :
% 1.98/1.38 ( ( in @ Xx @ A )
% 1.98/1.38 => ( Xphi @ Xx ) ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(prop2setI_type,type,
% 1.98/1.38 prop2setI: $o ).
% 1.98/1.38
% 1.98/1.38 thf(prop2setI,definition,
% 1.98/1.38 ( prop2setI
% 1.98/1.38 = ( ! [Xphi: $o] :
% 1.98/1.38 ( Xphi
% 1.98/1.38 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(set2prop_type,type,
% 1.98/1.38 set2prop: $i > $o ).
% 1.98/1.38
% 1.98/1.38 thf(prop2set2propI_type,type,
% 1.98/1.38 prop2set2propI: $o ).
% 1.98/1.38
% 1.98/1.38 thf(prop2set2propI,definition,
% 1.98/1.38 ( prop2set2propI
% 1.98/1.38 = ( ! [Xphi: $o] :
% 1.98/1.38 ( Xphi
% 1.98/1.38 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(notdexE_type,type,
% 1.98/1.38 notdexE: $o ).
% 1.98/1.38
% 1.98/1.38 thf(notdexE,definition,
% 1.98/1.38 ( notdexE
% 1.98/1.38 = ( ! [A: $i,Xphi: $i > $o] :
% 1.98/1.38 ( ~ ? [Xx: $i] :
% 1.98/1.38 ( ( in @ Xx @ A )
% 1.98/1.38 & ( Xphi @ Xx ) )
% 1.98/1.38 => ! [Xx: $i] :
% 1.98/1.38 ( ( in @ Xx @ A )
% 1.98/1.38 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(notdallE_type,type,
% 1.98/1.38 notdallE: $o ).
% 1.98/1.38
% 1.98/1.38 thf(notdallE,definition,
% 1.98/1.38 ( notdallE
% 1.98/1.38 = ( ! [A: $i,Xphi: $i > $o] :
% 1.98/1.38 ( ~ ! [Xx: $i] :
% 1.98/1.38 ( ( in @ Xx @ A )
% 1.98/1.38 => ( Xphi @ Xx ) )
% 1.98/1.38 => ? [Xx: $i] :
% 1.98/1.38 ( ( in @ Xx @ A )
% 1.98/1.38 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(exuI1_type,type,
% 1.98/1.38 exuI1: $o ).
% 1.98/1.38
% 1.98/1.38 thf(exuI1,definition,
% 1.98/1.38 ( exuI1
% 1.98/1.38 = ( ! [Xphi: $i > $o] :
% 1.98/1.38 ( ? [Xx: $i] :
% 1.98/1.38 ( ( Xphi @ Xx )
% 1.98/1.38 & ! [Xy: $i] :
% 1.98/1.38 ( ( Xphi @ Xy )
% 1.98/1.38 => ( Xx = Xy ) ) )
% 1.98/1.38 => ( exu
% 1.98/1.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(exuI3_type,type,
% 1.98/1.38 exuI3: $o ).
% 1.98/1.38
% 1.98/1.38 thf(exuI3,definition,
% 1.98/1.38 ( exuI3
% 1.98/1.38 = ( ! [Xphi: $i > $o] :
% 1.98/1.38 ( ? [Xx: $i] : ( Xphi @ Xx )
% 1.98/1.38 => ( ! [Xx: $i,Xy: $i] :
% 1.98/1.38 ( ( Xphi @ Xx )
% 1.98/1.38 => ( ( Xphi @ Xy )
% 1.98/1.38 => ( Xx = Xy ) ) )
% 1.98/1.38 => ( exu
% 1.98/1.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(exuI2_type,type,
% 1.98/1.38 exuI2: $o ).
% 1.98/1.38
% 1.98/1.38 thf(exuI2,definition,
% 1.98/1.38 ( exuI2
% 1.98/1.38 = ( ! [Xphi: $i > $o] :
% 1.98/1.38 ( ? [Xx: $i] :
% 1.98/1.38 ! [Xy: $i] :
% 1.98/1.38 ( ( Xphi @ Xy )
% 1.98/1.38 <=> ( Xy = Xx ) )
% 1.98/1.38 => ( exu
% 1.98/1.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(inCongP_type,type,
% 1.98/1.38 inCongP: $o ).
% 1.98/1.38
% 1.98/1.38 thf(inCongP,definition,
% 1.98/1.38 ( inCongP
% 1.98/1.38 = ( ! [A: $i,B: $i] :
% 1.98/1.38 ( ( A = B )
% 1.98/1.38 => ! [Xx: $i,Xy: $i] :
% 1.98/1.38 ( ( Xx = Xy )
% 1.98/1.38 => ( ( in @ Xx @ A )
% 1.98/1.38 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(in__Cong_type,type,
% 1.98/1.38 in__Cong: $o ).
% 1.98/1.38
% 1.98/1.38 thf(in__Cong,definition,
% 1.98/1.38 ( in__Cong
% 1.98/1.38 = ( ! [A: $i,B: $i] :
% 1.98/1.38 ( ( A = B )
% 1.98/1.38 => ! [Xx: $i,Xy: $i] :
% 1.98/1.38 ( ( Xx = Xy )
% 1.98/1.38 => ( ( in @ Xx @ A )
% 1.98/1.38 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(exuE3u_type,type,
% 1.98/1.38 exuE3u: $o ).
% 1.98/1.38
% 1.98/1.38 thf(exuE3u,definition,
% 1.98/1.38 ( exuE3u
% 1.98/1.38 = ( ! [Xphi: $i > $o] :
% 1.98/1.38 ( ( exu
% 1.98/1.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 1.98/1.38 => ! [Xx: $i,Xy: $i] :
% 1.98/1.38 ( ( Xphi @ Xx )
% 1.98/1.38 => ( ( Xphi @ Xy )
% 1.98/1.38 => ( Xx = Xy ) ) ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(exu__Cong_type,type,
% 1.98/1.38 exu__Cong: $o ).
% 1.98/1.38
% 1.98/1.38 thf(exu__Cong,definition,
% 1.98/1.38 ( exu__Cong
% 1.98/1.38 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 1.98/1.38 ( ! [Xx: $i,Xy: $i] :
% 1.98/1.38 ( ( Xx = Xy )
% 1.98/1.38 => ( ( Xphi @ Xx )
% 1.98/1.38 <=> ( Xpsi @ Xy ) ) )
% 1.98/1.38 => ( ( exu
% 1.98/1.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 1.98/1.38 <=> ( exu
% 1.98/1.38 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(emptyset__Cong_type,type,
% 1.98/1.38 emptyset__Cong: $o ).
% 1.98/1.38
% 1.98/1.38 thf(emptyset__Cong,definition,
% 1.98/1.38 ( emptyset__Cong
% 1.98/1.38 = ( emptyset = emptyset ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(setadjoin__Cong_type,type,
% 1.98/1.38 setadjoin__Cong: $o ).
% 1.98/1.38
% 1.98/1.38 thf(setadjoin__Cong,definition,
% 1.98/1.38 ( setadjoin__Cong
% 1.98/1.38 = ( ! [Xx: $i,Xy: $i] :
% 1.98/1.38 ( ( Xx = Xy )
% 1.98/1.38 => ! [Xz: $i,Xu: $i] :
% 1.98/1.38 ( ( Xz = Xu )
% 1.98/1.38 => ( ( setadjoin @ Xx @ Xz )
% 1.98/1.38 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(powerset__Cong_type,type,
% 1.98/1.38 powerset__Cong: $o ).
% 1.98/1.38
% 1.98/1.38 thf(powerset__Cong,definition,
% 1.98/1.38 ( powerset__Cong
% 1.98/1.38 = ( ! [A: $i,B: $i] :
% 1.98/1.38 ( ( A = B )
% 1.98/1.38 => ( ( powerset @ A )
% 1.98/1.38 = ( powerset @ B ) ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(setunion__Cong_type,type,
% 1.98/1.38 setunion__Cong: $o ).
% 1.98/1.38
% 1.98/1.38 thf(setunion__Cong,definition,
% 1.98/1.38 ( setunion__Cong
% 1.98/1.38 = ( ! [A: $i,B: $i] :
% 1.98/1.38 ( ( A = B )
% 1.98/1.38 => ( ( setunion @ A )
% 1.98/1.38 = ( setunion @ B ) ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(omega__Cong_type,type,
% 1.98/1.38 omega__Cong: $o ).
% 1.98/1.38
% 1.98/1.38 thf(omega__Cong,definition,
% 1.98/1.38 ( omega__Cong
% 1.98/1.38 = ( omega = omega ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(exuEu_type,type,
% 1.98/1.38 exuEu: $o ).
% 1.98/1.38
% 1.98/1.38 thf(exuEu,definition,
% 1.98/1.38 ( exuEu
% 1.98/1.38 = ( ! [Xphi: $i > $o] :
% 1.98/1.38 ( ( exu
% 1.98/1.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 1.98/1.38 => ! [Xx: $i,Xy: $i] :
% 1.98/1.38 ( ( Xphi @ Xx )
% 1.98/1.38 => ( ( Xphi @ Xy )
% 1.98/1.38 => ( Xx = Xy ) ) ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(descr__Cong_type,type,
% 1.98/1.38 descr__Cong: $o ).
% 1.98/1.38
% 1.98/1.38 thf(descr__Cong,definition,
% 1.98/1.38 ( descr__Cong
% 1.98/1.38 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 1.98/1.38 ( ! [Xx: $i,Xy: $i] :
% 1.98/1.38 ( ( Xx = Xy )
% 1.98/1.38 => ( ( Xphi @ Xx )
% 1.98/1.38 <=> ( Xpsi @ Xy ) ) )
% 1.98/1.38 => ( ( exu
% 1.98/1.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 1.98/1.38 => ( ( exu
% 1.98/1.38 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 1.98/1.38 => ( ( descr
% 1.98/1.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 1.98/1.38 = ( descr
% 1.98/1.38 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(dsetconstr__Cong_type,type,
% 1.98/1.38 dsetconstr__Cong: $o ).
% 1.98/1.38
% 1.98/1.38 thf(dsetconstr__Cong,definition,
% 1.98/1.38 ( dsetconstr__Cong
% 1.98/1.38 = ( ! [A: $i,B: $i] :
% 1.98/1.38 ( ( A = B )
% 1.98/1.38 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 1.98/1.38 ( ! [Xx: $i] :
% 1.98/1.38 ( ( in @ Xx @ A )
% 1.98/1.38 => ! [Xy: $i] :
% 1.98/1.38 ( ( in @ Xy @ B )
% 1.98/1.38 => ( ( Xx = Xy )
% 1.98/1.38 => ( ( Xphi @ Xx )
% 1.98/1.38 <=> ( Xpsi @ Xy ) ) ) ) )
% 1.98/1.38 => ( ( dsetconstr @ A
% 1.98/1.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 1.98/1.38 = ( dsetconstr @ B
% 1.98/1.38 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(subset_type,type,
% 1.98/1.38 subset: $i > $i > $o ).
% 1.98/1.38
% 1.98/1.38 thf(disjoint_type,type,
% 1.98/1.38 disjoint: $i > $i > $o ).
% 1.98/1.38
% 1.98/1.38 thf(setsmeet_type,type,
% 1.98/1.38 setsmeet: $i > $i > $o ).
% 1.98/1.38
% 1.98/1.38 thf(subsetI1_type,type,
% 1.98/1.38 subsetI1: $o ).
% 1.98/1.38
% 1.98/1.38 thf(subsetI1,definition,
% 1.98/1.38 ( subsetI1
% 1.98/1.38 = ( ! [A: $i,B: $i] :
% 1.98/1.38 ( ! [Xx: $i] :
% 1.98/1.38 ( ( in @ Xx @ A )
% 1.98/1.38 => ( in @ Xx @ B ) )
% 1.98/1.38 => ( subset @ A @ B ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(eqimpsubset2_type,type,
% 1.98/1.38 eqimpsubset2: $o ).
% 1.98/1.38
% 1.98/1.38 thf(eqimpsubset2,definition,
% 1.98/1.38 ( eqimpsubset2
% 1.98/1.38 = ( ! [A: $i,B: $i] :
% 1.98/1.38 ( ( A = B )
% 1.98/1.38 => ( subset @ B @ A ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(eqimpsubset1_type,type,
% 1.98/1.38 eqimpsubset1: $o ).
% 1.98/1.38
% 1.98/1.38 thf(eqimpsubset1,definition,
% 1.98/1.38 ( eqimpsubset1
% 1.98/1.38 = ( ! [A: $i,B: $i] :
% 1.98/1.38 ( ( A = B )
% 1.98/1.38 => ( subset @ A @ B ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(subsetI2_type,type,
% 1.98/1.38 subsetI2: $o ).
% 1.98/1.38
% 1.98/1.38 thf(subsetI2,definition,
% 1.98/1.38 ( subsetI2
% 1.98/1.38 = ( ! [A: $i,B: $i] :
% 1.98/1.38 ( ! [Xx: $i] :
% 1.98/1.38 ( ( in @ Xx @ A )
% 1.98/1.38 => ( in @ Xx @ B ) )
% 1.98/1.38 => ( subset @ A @ B ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(emptysetsubset_type,type,
% 1.98/1.38 emptysetsubset: $o ).
% 1.98/1.38
% 1.98/1.38 thf(emptysetsubset,definition,
% 1.98/1.38 ( emptysetsubset
% 1.98/1.38 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(subsetE_type,type,
% 1.98/1.38 subsetE: $o ).
% 1.98/1.38
% 1.98/1.38 thf(subsetE,definition,
% 1.98/1.38 ( subsetE
% 1.98/1.38 = ( ! [A: $i,B: $i,Xx: $i] :
% 1.98/1.38 ( ( subset @ A @ B )
% 1.98/1.38 => ( ( in @ Xx @ A )
% 1.98/1.38 => ( in @ Xx @ B ) ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(subsetE2_type,type,
% 1.98/1.38 subsetE2: $o ).
% 1.98/1.38
% 1.98/1.38 thf(subsetE2,definition,
% 1.98/1.38 ( subsetE2
% 1.98/1.38 = ( ! [A: $i,B: $i,Xx: $i] :
% 1.98/1.38 ( ( subset @ A @ B )
% 1.98/1.38 => ( ~ ( in @ Xx @ B )
% 1.98/1.38 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(notsubsetI_type,type,
% 1.98/1.38 notsubsetI: $o ).
% 1.98/1.38
% 1.98/1.38 thf(notsubsetI,definition,
% 1.98/1.38 ( notsubsetI
% 1.98/1.38 = ( ! [A: $i,B: $i,Xx: $i] :
% 1.98/1.38 ( ( in @ Xx @ A )
% 1.98/1.38 => ( ~ ( in @ Xx @ B )
% 1.98/1.38 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(notequalI1_type,type,
% 1.98/1.38 notequalI1: $o ).
% 1.98/1.38
% 1.98/1.38 thf(notequalI1,definition,
% 1.98/1.38 ( notequalI1
% 1.98/1.38 = ( ! [A: $i,B: $i] :
% 1.98/1.38 ( ~ ( subset @ A @ B )
% 1.98/1.38 => ( A != B ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(notequalI2_type,type,
% 1.98/1.38 notequalI2: $o ).
% 1.98/1.38
% 1.98/1.38 thf(notequalI2,definition,
% 1.98/1.38 ( notequalI2
% 1.98/1.38 = ( ! [A: $i,B: $i,Xx: $i] :
% 1.98/1.38 ( ( in @ Xx @ A )
% 1.98/1.38 => ( ~ ( in @ Xx @ B )
% 1.98/1.38 => ( A != B ) ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(subsetRefl_type,type,
% 1.98/1.38 subsetRefl: $o ).
% 1.98/1.38
% 1.98/1.38 thf(subsetRefl,definition,
% 1.98/1.38 ( subsetRefl
% 1.98/1.38 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(subsetTrans_type,type,
% 1.98/1.38 subsetTrans: $o ).
% 1.98/1.38
% 1.98/1.38 thf(subsetTrans,definition,
% 1.98/1.38 ( subsetTrans
% 1.98/1.38 = ( ! [A: $i,B: $i,C: $i] :
% 1.98/1.38 ( ( subset @ A @ B )
% 1.98/1.38 => ( ( subset @ B @ C )
% 1.98/1.38 => ( subset @ A @ C ) ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(setadjoinSub_type,type,
% 1.98/1.38 setadjoinSub: $o ).
% 1.98/1.38
% 1.98/1.38 thf(setadjoinSub,definition,
% 1.98/1.38 ( setadjoinSub
% 1.98/1.38 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(setadjoinSub2_type,type,
% 1.98/1.38 setadjoinSub2: $o ).
% 1.98/1.38
% 1.98/1.38 thf(setadjoinSub2,definition,
% 1.98/1.38 ( setadjoinSub2
% 1.98/1.38 = ( ! [A: $i,Xx: $i,B: $i] :
% 1.98/1.38 ( ( subset @ A @ B )
% 1.98/1.38 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(subset2powerset_type,type,
% 1.98/1.38 subset2powerset: $o ).
% 1.98/1.38
% 1.98/1.38 thf(subset2powerset,definition,
% 1.98/1.38 ( subset2powerset
% 1.98/1.38 = ( ! [A: $i,B: $i] :
% 1.98/1.38 ( ( subset @ A @ B )
% 1.98/1.38 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(setextsub_type,type,
% 1.98/1.38 setextsub: $o ).
% 1.98/1.38
% 1.98/1.38 thf(setextsub,definition,
% 1.98/1.38 ( setextsub
% 1.98/1.38 = ( ! [A: $i,B: $i] :
% 1.98/1.38 ( ( subset @ A @ B )
% 1.98/1.38 => ( ( subset @ B @ A )
% 1.98/1.38 => ( A = B ) ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(subsetemptysetimpeq_type,type,
% 1.98/1.38 subsetemptysetimpeq: $o ).
% 1.98/1.38
% 1.98/1.38 thf(subsetemptysetimpeq,definition,
% 1.98/1.38 ( subsetemptysetimpeq
% 1.98/1.38 = ( ! [A: $i] :
% 1.98/1.38 ( ( subset @ A @ emptyset )
% 1.98/1.38 => ( A = emptyset ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(powersetI1_type,type,
% 1.98/1.38 powersetI1: $o ).
% 1.98/1.38
% 1.98/1.38 thf(powersetI1,definition,
% 1.98/1.38 ( powersetI1
% 1.98/1.38 = ( ! [A: $i,B: $i] :
% 1.98/1.38 ( ( subset @ B @ A )
% 1.98/1.38 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(powersetE1_type,type,
% 1.98/1.38 powersetE1: $o ).
% 1.98/1.38
% 1.98/1.38 thf(powersetE1,definition,
% 1.98/1.38 ( powersetE1
% 1.98/1.38 = ( ! [A: $i,B: $i] :
% 1.98/1.38 ( ( in @ B @ ( powerset @ A ) )
% 1.98/1.38 => ( subset @ B @ A ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(inPowerset_type,type,
% 1.98/1.38 inPowerset: $o ).
% 1.98/1.38
% 1.98/1.38 thf(inPowerset,definition,
% 1.98/1.38 ( inPowerset
% 1.98/1.38 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(powersetsubset_type,type,
% 1.98/1.38 powersetsubset: $o ).
% 1.98/1.38
% 1.98/1.38 thf(powersetsubset,definition,
% 1.98/1.38 ( powersetsubset
% 1.98/1.38 = ( ! [A: $i,B: $i] :
% 1.98/1.38 ( ( subset @ A @ B )
% 1.98/1.38 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(sepInPowerset_type,type,
% 1.98/1.38 sepInPowerset: $o ).
% 1.98/1.38
% 1.98/1.38 thf(sepInPowerset,definition,
% 1.98/1.38 ( sepInPowerset
% 1.98/1.38 = ( ! [A: $i,Xphi: $i > $o] :
% 1.98/1.38 ( in
% 1.98/1.38 @ ( dsetconstr @ A
% 1.98/1.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 1.98/1.38 @ ( powerset @ A ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(sepSubset_type,type,
% 1.98/1.38 sepSubset: $o ).
% 1.98/1.38
% 1.98/1.38 thf(sepSubset,definition,
% 1.98/1.38 ( sepSubset
% 1.98/1.38 = ( ! [A: $i,Xphi: $i > $o] :
% 1.98/1.38 ( subset
% 1.98/1.38 @ ( dsetconstr @ A
% 1.98/1.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 1.98/1.38 @ A ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(binunion_type,type,
% 1.98/1.38 binunion: $i > $i > $i ).
% 1.98/1.38
% 1.98/1.38 thf(binunionIL_type,type,
% 1.98/1.38 binunionIL: $o ).
% 1.98/1.38
% 1.98/1.38 thf(binunionIL,definition,
% 1.98/1.38 ( binunionIL
% 1.98/1.38 = ( ! [A: $i,B: $i,Xx: $i] :
% 1.98/1.38 ( ( in @ Xx @ A )
% 1.98/1.38 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(upairset2IR_type,type,
% 1.98/1.38 upairset2IR: $o ).
% 1.98/1.38
% 1.98/1.38 thf(upairset2IR,definition,
% 1.98/1.38 ( upairset2IR
% 1.98/1.38 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(binunionIR_type,type,
% 1.98/1.38 binunionIR: $o ).
% 1.98/1.38
% 1.98/1.38 thf(binunionIR,definition,
% 1.98/1.38 ( binunionIR
% 1.98/1.38 = ( ! [A: $i,B: $i,Xx: $i] :
% 1.98/1.38 ( ( in @ Xx @ B )
% 1.98/1.38 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(binunionEcases_type,type,
% 1.98/1.38 binunionEcases: $o ).
% 1.98/1.38
% 1.98/1.38 thf(binunionEcases,definition,
% 1.98/1.38 ( binunionEcases
% 1.98/1.38 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 1.98/1.38 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 1.98/1.38 => ( ( ( in @ Xx @ A )
% 1.98/1.38 => Xphi )
% 1.98/1.38 => ( ( ( in @ Xx @ B )
% 1.98/1.38 => Xphi )
% 1.98/1.38 => Xphi ) ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(binunionE_type,type,
% 1.98/1.38 binunionE: $o ).
% 1.98/1.38
% 1.98/1.38 thf(binunionE,definition,
% 1.98/1.38 ( binunionE
% 1.98/1.38 = ( ! [A: $i,B: $i,Xx: $i] :
% 1.98/1.38 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 1.98/1.38 => ( ( in @ Xx @ A )
% 1.98/1.38 | ( in @ Xx @ B ) ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(binunionLsub_type,type,
% 1.98/1.38 binunionLsub: $o ).
% 1.98/1.38
% 1.98/1.38 thf(binunionLsub,definition,
% 1.98/1.38 ( binunionLsub
% 1.98/1.38 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(binunionRsub_type,type,
% 1.98/1.38 binunionRsub: $o ).
% 1.98/1.38
% 1.98/1.38 thf(binunionRsub,definition,
% 1.98/1.38 ( binunionRsub
% 1.98/1.38 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(binintersect_type,type,
% 1.98/1.38 binintersect: $i > $i > $i ).
% 1.98/1.38
% 1.98/1.38 thf(binintersect,definition,
% 1.98/1.38 ( binintersect
% 1.98/1.38 = ( ^ [A: $i,B: $i] :
% 1.98/1.38 ( dsetconstr @ A
% 1.98/1.38 @ ^ [Xx: $i] : ( in @ Xx @ B ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(binintersectI_type,type,
% 1.98/1.38 binintersectI: $o ).
% 1.98/1.38
% 1.98/1.38 thf(binintersectI,definition,
% 1.98/1.38 ( binintersectI
% 1.98/1.38 = ( ! [A: $i,B: $i,Xx: $i] :
% 1.98/1.38 ( ( in @ Xx @ A )
% 1.98/1.38 => ( ( in @ Xx @ B )
% 1.98/1.38 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(binintersectSubset5_type,type,
% 1.98/1.38 binintersectSubset5: $o ).
% 1.98/1.38
% 1.98/1.38 thf(binintersectSubset5,definition,
% 1.98/1.38 ( binintersectSubset5
% 1.98/1.38 = ( ! [A: $i,B: $i,C: $i] :
% 1.98/1.38 ( ( subset @ C @ A )
% 1.98/1.38 => ( ( subset @ C @ B )
% 1.98/1.38 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(binintersectEL_type,type,
% 1.98/1.38 binintersectEL: $o ).
% 1.98/1.38
% 1.98/1.38 thf(binintersectEL,definition,
% 1.98/1.38 ( binintersectEL
% 1.98/1.38 = ( ! [A: $i,B: $i,Xx: $i] :
% 1.98/1.38 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 1.98/1.38 => ( in @ Xx @ A ) ) ) ) ).
% 1.98/1.38
% 1.98/1.38 thf(binintersectLsub,conjecture,
% 1.98/1.38 ( setextAx
% 1.98/1.38 => ( emptysetAx
% 1.98/1.38 => ( setadjoinAx
% 1.98/1.38 => ( powersetAx
% 1.98/1.38 => ( setunionAx
% 1.98/1.38 => ( omega0Ax
% 1.98/1.38 => ( omegaSAx
% 1.98/1.38 => ( omegaIndAx
% 1.98/1.38 => ( replAx
% 1.98/1.38 => ( foundationAx
% 1.98/1.38 => ( wellorderingAx
% 1.98/1.38 => ( descrp
% 1.98/1.38 => ( dsetconstrI
% 1.98/1.38 => ( dsetconstrEL
% 1.98/1.38 => ( dsetconstrER
% 1.98/1.38 => ( exuE1
% 1.98/1.38 => ( prop2setE
% 1.98/1.38 => ( emptysetE
% 1.98/1.38 => ( emptysetimpfalse
% 1.98/1.38 => ( notinemptyset
% 1.98/1.38 => ( exuE3e
% 1.98/1.38 => ( setext
% 1.98/1.38 => ( emptyI
% 1.98/1.38 => ( noeltsimpempty
% 1.98/1.38 => ( setbeta
% 1.98/1.38 => ( nonemptyE1
% 1.98/1.38 => ( nonemptyI
% 1.98/1.38 => ( nonemptyI1
% 1.98/1.38 => ( setadjoinIL
% 1.98/1.38 => ( emptyinunitempty
% 1.98/1.38 => ( setadjoinIR
% 1.98/1.38 => ( setadjoinE
% 1.98/1.38 => ( setadjoinOr
% 1.98/1.38 => ( setoftrueEq
% 1.98/1.38 => ( powersetI
% 1.98/1.38 => ( emptyinPowerset
% 1.98/1.38 => ( emptyInPowerset
% 1.98/1.38 => ( powersetE
% 1.98/1.38 => ( setunionI
% 1.98/1.38 => ( setunionE
% 1.98/1.38 => ( subPowSU
% 1.98/1.38 => ( exuE2
% 1.98/1.38 => ( nonemptyImpWitness
% 1.98/1.38 => ( uniqinunit
% 1.98/1.38 => ( notinsingleton
% 1.98/1.38 => ( eqinunit
% 1.98/1.38 => ( singletonsswitch
% 1.98/1.38 => ( upairsetE
% 1.98/1.38 => ( upairsetIL
% 1.98/1.38 => ( upairsetIR
% 1.98/1.38 => ( emptyE1
% 1.98/1.38 => ( vacuousDall
% 1.98/1.38 => ( quantDeMorgan1
% 1.98/1.38 => ( quantDeMorgan2
% 1.98/1.38 => ( quantDeMorgan3
% 1.98/1.38 => ( quantDeMorgan4
% 1.98/1.38 => ( prop2setI
% 1.98/1.38 => ( prop2set2propI
% 1.98/1.38 => ( notdexE
% 1.98/1.38 => ( notdallE
% 1.98/1.38 => ( exuI1
% 1.98/1.38 => ( exuI3
% 1.98/1.38 => ( exuI2
% 1.98/1.38 => ( inCongP
% 1.98/1.38 => ( in__Cong
% 1.98/1.38 => ( exuE3u
% 1.98/1.38 => ( exu__Cong
% 1.98/1.38 => ( emptyset__Cong
% 1.98/1.38 => ( setadjoin__Cong
% 1.98/1.38 => ( powerset__Cong
% 1.98/1.38 => ( setunion__Cong
% 1.98/1.38 => ( omega__Cong
% 1.98/1.38 => ( exuEu
% 1.98/1.38 => ( descr__Cong
% 1.98/1.38 => ( dsetconstr__Cong
% 1.98/1.38 => ( subsetI1
% 1.98/1.38 => ( eqimpsubset2
% 1.98/1.38 => ( eqimpsubset1
% 1.98/1.38 => ( subsetI2
% 1.98/1.38 => ( emptysetsubset
% 1.98/1.38 => ( subsetE
% 1.98/1.38 => ( subsetE2
% 1.98/1.38 => ( notsubsetI
% 1.98/1.38 => ( notequalI1
% 1.98/1.38 => ( notequalI2
% 1.98/1.38 => ( subsetRefl
% 1.98/1.38 => ( subsetTrans
% 1.98/1.38 => ( setadjoinSub
% 1.98/1.38 => ( setadjoinSub2
% 1.98/1.38 => ( subset2powerset
% 1.98/1.38 => ( setextsub
% 1.98/1.38 => ( subsetemptysetimpeq
% 1.98/1.38 => ( powersetI1
% 1.98/1.38 => ( powersetE1
% 1.98/1.38 => ( inPowerset
% 1.98/1.38 => ( powersetsubset
% 1.98/1.38 => ( sepInPowerset
% 1.98/1.38 => ( sepSubset
% 1.98/1.38 => ( binunionIL
% 1.98/1.38 => ( upairset2IR
% 1.98/1.38 => ( binunionIR
% 1.98/1.38 => ( binunionEcases
% 1.98/1.38 => ( binunionE
% 1.98/1.38 => ( binunionLsub
% 1.98/1.38 => ( binunionRsub
% 1.98/1.38 => ( binintersectI
% 1.98/1.38 => ( binintersectSubset5
% 1.98/1.38 => ( binintersectEL
% 1.98/1.38 => ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 1.98/1.38 ------------------------
% 2.16/1.40 % (12139)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_11947 for (2999ds/183Mi)
% 2.16/1.40 % (12145)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_11947 for (2999ds/18Mi)
% 2.16/1.40 % (12141)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_11947 for (2999ds/27Mi)
% 2.16/1.40 % (12144)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_11947 for (2999ds/275Mi)
% 2.16/1.41 % (12145)Instruction limit reached!
% 2.16/1.41 % (12145)------------------------------
% 2.16/1.41 % (12145)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.16/1.41 % (12145)Termination reason: Unknown
% 2.16/1.41 % (12145)Termination phase: Preprocessing 3
% 2.16/1.41
% 2.16/1.41 % (12145)Memory used [KB]: 1663
% 2.16/1.41 % (12145)Time elapsed: 0.011 s
% 2.16/1.41 % (12145)Instructions burned: 20 (million)
% 2.16/1.41 % (12145)------------------------------
% 2.16/1.41 % (12145)------------------------------
% 2.16/1.41 % (12140)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_11947 for (2999ds/4Mi)
% 2.16/1.41 % (12143)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_11947 for (2999ds/2Mi)
% 2.16/1.41 % (12143)Instruction limit reached!
% 2.16/1.41 % (12143)------------------------------
% 2.16/1.41 % (12143)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.16/1.41 % (12143)Termination reason: Unknown
% 2.16/1.41 % (12143)Termination phase: shuffling
% 2.16/1.41
% 2.16/1.41 % (12143)Memory used [KB]: 1151
% 2.16/1.41 % (12143)Time elapsed: 0.003 s
% 2.16/1.41 % (12143)Instructions burned: 3 (million)
% 2.16/1.41 % (12143)------------------------------
% 2.16/1.41 % (12143)------------------------------
% 2.16/1.42 % (12140)Instruction limit reached!
% 2.16/1.42 % (12140)------------------------------
% 2.16/1.42 % (12140)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.16/1.42 % (12140)Termination reason: Unknown
% 2.16/1.42 % (12140)Termination phase: shuffling
% 2.16/1.42
% 2.16/1.42 % (12140)Memory used [KB]: 1151
% 2.16/1.42 % (12140)Time elapsed: 0.004 s
% 2.16/1.42 % (12140)Instructions burned: 4 (million)
% 2.16/1.42 % (12140)------------------------------
% 2.16/1.42 % (12140)------------------------------
% 2.16/1.42 % (12141)Instruction limit reached!
% 2.16/1.42 % (12141)------------------------------
% 2.16/1.42 % (12141)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.16/1.42 % (12142)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_11947 for (2999ds/2Mi)
% 2.16/1.42 % (12141)Termination reason: Unknown
% 2.16/1.42 % (12141)Termination phase: Property scanning
% 2.16/1.42
% 2.16/1.42 % (12141)Memory used [KB]: 1535
% 2.16/1.42 % (12141)Time elapsed: 0.015 s
% 2.16/1.42 % (12141)Instructions burned: 27 (million)
% 2.16/1.42 % (12141)------------------------------
% 2.16/1.42 % (12141)------------------------------
% 2.16/1.42 % (12142)Instruction limit reached!
% 2.16/1.42 % (12142)------------------------------
% 2.16/1.42 % (12142)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.16/1.42 % (12142)Termination reason: Unknown
% 2.16/1.42 % (12142)Termination phase: shuffling
% 2.16/1.42
% 2.16/1.42 % (12142)Memory used [KB]: 1151
% 2.16/1.42 % (12142)Time elapsed: 0.004 s
% 2.16/1.42 % (12142)Instructions burned: 2 (million)
% 2.16/1.42 % (12142)------------------------------
% 2.16/1.42 % (12142)------------------------------
% 2.16/1.43 % (12151)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_11947 for (2999ds/3Mi)
% 2.16/1.43 % (12151)Instruction limit reached!
% 2.16/1.43 % (12151)------------------------------
% 2.16/1.43 % (12151)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.16/1.43 % (12151)Termination reason: Unknown
% 2.16/1.43 % (12151)Termination phase: shuffling
% 2.16/1.43
% 2.16/1.43 % (12151)Memory used [KB]: 1279
% 2.16/1.43 % (12151)Time elapsed: 0.003 s
% 2.16/1.43 % (12151)Instructions burned: 5 (million)
% 2.16/1.43 % (12151)------------------------------
% 2.16/1.43 % (12151)------------------------------
% 2.16/1.43 % (12153)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_11947 for (2999ds/15Mi)
% 2.16/1.43 % (12155)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_11947 for (2999ds/3Mi)
% 2.16/1.44 % (12155)Instruction limit reached!
% 2.16/1.44 % (12155)------------------------------
% 2.16/1.44 % (12155)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.16/1.44 % (12155)Termination reason: Unknown
% 2.16/1.44 % (12155)Termination phase: shuffling
% 2.16/1.44
% 2.16/1.44 % (12155)Memory used [KB]: 1151
% 2.16/1.44 % (12155)Time elapsed: 0.003 s
% 2.16/1.44 % (12155)Instructions burned: 4 (million)
% 2.16/1.44 % (12155)------------------------------
% 2.16/1.44 % (12155)------------------------------
% 2.16/1.44 % (12159)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_11947 for (2999ds/7Mi)
% 2.16/1.44 % (12159)Instruction limit reached!
% 2.16/1.44 % (12159)------------------------------
% 2.16/1.44 % (12159)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.16/1.44 % (12159)Termination reason: Unknown
% 2.16/1.44 % (12159)Termination phase: shuffling
% 2.16/1.44
% 2.16/1.44 % (12159)Memory used [KB]: 1279
% 2.16/1.44 % (12159)Time elapsed: 0.003 s
% 2.16/1.44 % (12159)Instructions burned: 8 (million)
% 2.16/1.44 % (12159)------------------------------
% 2.16/1.44 % (12159)------------------------------
% 2.16/1.44 % (12156)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_11947 for (2999ds/1041Mi)
% 2.16/1.44 % (12152)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_11947 for (2999ds/37Mi)
% 2.16/1.44 % (12153)Instruction limit reached!
% 2.16/1.44 % (12153)------------------------------
% 2.16/1.44 % (12153)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.16/1.44 % (12153)Termination reason: Unknown
% 2.16/1.44 % (12153)Termination phase: Naming
% 2.16/1.44
% 2.16/1.44 % (12153)Memory used [KB]: 1535
% 2.16/1.44 % (12153)Time elapsed: 0.010 s
% 2.16/1.44 % (12153)Instructions burned: 16 (million)
% 2.16/1.44 % (12153)------------------------------
% 2.16/1.44 % (12153)------------------------------
% 2.16/1.44 % (12139)First to succeed.
% 2.16/1.45 % (12139)Refutation found. Thanks to Tanya!
% 2.16/1.45 % SZS status Theorem for DTF2THF_11947
% 2.16/1.45 % SZS output start Proof for DTF2THF_11947
% 2.16/1.45 thf(func_def_0, type, in: $i > $i > $o).
% 2.16/1.45 thf(func_def_1, type, exu: ($i > $o) > $o).
% 2.16/1.45 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 2.16/1.45 thf(func_def_8, type, powerset: $i > $i).
% 2.16/1.45 thf(func_def_10, type, setunion: $i > $i).
% 2.16/1.45 thf(func_def_19, type, descr: ($i > $o) > $i).
% 2.16/1.45 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 2.16/1.45 thf(func_def_26, type, prop2set: $o > $i).
% 2.16/1.45 thf(func_def_36, type, nonempty: $i > $o).
% 2.16/1.45 thf(func_def_69, type, set2prop: $i > $o).
% 2.16/1.45 thf(func_def_88, type, subset: $i > $i > $o).
% 2.16/1.45 thf(func_def_89, type, disjoint: $i > $i > $o).
% 2.16/1.45 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 2.16/1.45 thf(func_def_114, type, binunion: $i > $i > $i).
% 2.16/1.45 thf(func_def_122, type, binintersect: $i > $i > $i).
% 2.16/1.45 thf(func_def_146, type, sK3: $i > $i).
% 2.16/1.45 thf(func_def_147, type, sK4: $i > $i > $i).
% 2.16/1.45 thf(func_def_148, type, sK5: ($i > $o) > ($i > $o) > $i).
% 2.16/1.45 thf(func_def_149, type, sK6: ($i > $o) > ($i > $o) > $i).
% 2.16/1.45 thf(func_def_150, type, sK7: $i > ($i > $o) > $i).
% 2.16/1.45 thf(func_def_151, type, sK8: $i > ($i > $o) > $i).
% 2.16/1.45 thf(func_def_152, type, sK9: $i > ($i > $o) > $i).
% 2.16/1.45 thf(func_def_153, type, sK10: $i > ($i > $o) > $i).
% 2.16/1.45 thf(func_def_154, type, sK11: $i > ($i > $o) > $i).
% 2.16/1.45 thf(func_def_155, type, sK12: $i > ($i > $o) > $i).
% 2.16/1.45 thf(func_def_156, type, sK13: $i > ($i > $o) > $i).
% 2.16/1.45 thf(func_def_157, type, sK14: $i > $i).
% 2.16/1.45 thf(func_def_158, type, sK15: $i > $i > $i > $i).
% 2.16/1.45 thf(func_def_159, type, sK16: $i > $i > $i).
% 2.16/1.45 thf(func_def_160, type, sK17: $i > $i > $i).
% 2.16/1.45 thf(func_def_161, type, sK18: $i > $i > $i).
% 2.16/1.45 thf(func_def_162, type, sK19: $i > $i > $i).
% 2.16/1.45 thf(func_def_163, type, sK20: $i > $i > $i).
% 2.16/1.45 thf(func_def_164, type, sK21: $i > $i > $i).
% 2.16/1.45 thf(func_def_165, type, sK22: ($i > $o) > $i > $i).
% 2.16/1.45 thf(func_def_166, type, sK23: ($i > $o) > $i).
% 2.16/1.45 thf(func_def_167, type, sK24: ($i > $o) > $i).
% 2.16/1.45 thf(func_def_168, type, sK25: ($i > $o) > $i).
% 2.16/1.45 thf(func_def_169, type, sK26: ($i > $o) > $i > $i).
% 2.16/1.45 thf(func_def_170, type, sK27: ($i > $o) > $i).
% 2.16/1.45 thf(func_def_171, type, sK28: ($i > $o) > $i).
% 2.16/1.45 thf(func_def_172, type, sK29: ($i > $o) > $i).
% 2.16/1.45 thf(func_def_173, type, sK30: ($i > $o) > $i).
% 2.16/1.45 thf(func_def_174, type, sK31: ($i > $o) > $i).
% 2.16/1.45 thf(func_def_175, type, sK32: $i > ($i > $o) > $i).
% 2.16/1.45 thf(func_def_176, type, sK33: $o > $i > $i > $i).
% 2.16/1.45 thf(func_def_177, type, sK34: $i > ($i > $o) > $i).
% 2.16/1.45 thf(func_def_178, type, sK35: $i > $i).
% 2.16/1.45 thf(func_def_179, type, sK36: $i > $i).
% 2.16/1.45 thf(func_def_180, type, sK37: $i > $i).
% 2.16/1.45 thf(func_def_181, type, sK38: ($i > $o) > $i > $i).
% 2.16/1.45 thf(func_def_182, type, sK39: ($i > $o) > $i).
% 2.16/1.45 thf(func_def_183, type, sK40: ($i > $o) > $i).
% 2.16/1.45 thf(func_def_184, type, sK41: $i > ($i > $o) > $i).
% 2.16/1.45 thf(func_def_185, type, sK42: $i > $i > $i).
% 2.16/1.45 thf(func_def_186, type, sK43: $i > $i > $i).
% 2.16/1.45 thf(func_def_187, type, sK44: $i > $i > $i).
% 2.16/1.45 thf(func_def_188, type, sK45: ($i > $o) > $i > ($i > $o) > $i > $i).
% 2.16/1.45 thf(func_def_189, type, sK46: $i > $i > ($i > $o) > ($i > $o) > $i).
% 2.16/1.45 thf(func_def_190, type, sK47: ($i > $o) > $i > $i).
% 2.16/1.45 thf(func_def_191, type, sK48: ($i > $o) > $i > $i).
% 2.16/1.45 thf(func_def_192, type, sK49: $i > $i > $i).
% 2.16/1.45 thf(func_def_193, type, sK50: ($i > $o) > $i > $i).
% 2.16/1.45 thf(func_def_194, type, sK51: $i > $i > $i).
% 2.16/1.45 thf(func_def_195, type, sK52: ($i > $o) > ($i > $o) > $i).
% 2.16/1.45 thf(func_def_196, type, sK53: ($i > $o) > $i).
% 2.16/1.45 thf(func_def_197, type, sK54: ($i > $o) > ($i > $o) > $i).
% 2.16/1.45 thf(func_def_198, type, sK55: ($i > $o) > $i > $i).
% 2.16/1.45 thf(func_def_199, type, sK56: ($i > $o) > $i > $i).
% 2.16/1.45 thf(func_def_200, type, sK57: ($i > $o) > ($i > $o) > $i).
% 2.16/1.45 thf(func_def_201, type, sK58: ($i > $o) > ($i > $o) > $i).
% 2.16/1.45 thf(func_def_202, type, sK59: ($i > $o) > $i > $i).
% 2.16/1.45 thf(func_def_203, type, sK60: ($i > $o) > $i).
% 2.16/1.45 thf(func_def_204, type, sK61: ($i > $o) > ($i > $o) > $i).
% 2.16/1.45 thf(func_def_205, type, sK62: ($i > $o) > ($i > $o) > $i).
% 2.16/1.45 thf(func_def_206, type, sK63: ($i > $o) > ($i > $o) > $i).
% 2.16/1.45 thf(func_def_207, type, sK64: ($i > $o) > ($i > $o) > $i).
% 2.16/1.45 thf(func_def_208, type, sK65: ($i > $o) > $i).
% 2.16/1.45 thf(func_def_209, type, sK66: ($i > $o) > $i > $i).
% 2.16/1.45 thf(func_def_210, type, sK67: $i > ($i > $o) > $i).
% 2.16/1.45 thf(func_def_211, type, sK68: $i > ($i > $i > $o) > $i).
% 2.16/1.45 thf(func_def_212, type, sK69: $i > ($i > $i > $o) > $i).
% 2.16/1.45 thf(func_def_213, type, sK70: $i > $i > ($i > $i > $o) > $i).
% 2.16/1.45 thf(func_def_214, type, sK71: $i > ($i > $i > $o) > $i).
% 2.16/1.45 thf(func_def_215, type, sK72: $i > ($i > $i > $o) > $i > $i).
% 2.16/1.45 thf(func_def_216, type, sK73: $i > $i > ($i > $i > $o) > $i).
% 2.16/1.45 thf(func_def_217, type, sK74: $i > $i > ($i > $i > $o) > $i).
% 2.16/1.45 thf(func_def_218, type, sK75: $i > $i).
% 2.16/1.45 thf(func_def_219, type, sK76: ($i > $o) > $i > $i).
% 2.16/1.45 thf(func_def_220, type, sK77: ($i > $o) > $i).
% 2.16/1.45 thf(func_def_221, type, sK78: ($i > $o) > $i > $i).
% 2.16/1.45 thf(func_def_222, type, sK79: ($i > $o) > $i > $i).
% 2.16/1.45 thf(func_def_223, type, sK80: $i > $i > $i).
% 2.16/1.45 thf(func_def_224, type, sK81: ($i > $o) > $i > $i).
% 2.16/1.45 thf(func_def_225, type, sK82: $i > $i).
% 2.16/1.45 thf(f2041,plain,(
% 2.16/1.45 $false),
% 2.16/1.45 inference(subsumption_resolution,[status(thm)],[f1486,f1962])).
% 2.16/1.45 thf(f1962,plain,(
% 2.16/1.45 ( ! [X2 : $i,X1 : $i > $o] : (((subset @ (dsetconstr @ X2 @ X1) @ X2) = $true)) )),
% 2.16/1.45 inference(beta_eta_normalization,[status(thm)],[f1961])).
% 2.16/1.45 thf(f1961,plain,(
% 2.16/1.45 ( ! [X2 : $i,X1 : $i > $o] : ((((^[Y0 : $i]: (subset @ (dsetconstr @ Y0 @ X1) @ Y0)) @ X2) = $true)) )),
% 2.16/1.45 inference(pi_clausification,[status(thm)],[f1960])).
% 2.16/1.45 thf(f1960,plain,(
% 2.16/1.45 ( ! [X1 : $i > $o] : (((!! @ $i @ (^[Y0 : $i]: (subset @ (dsetconstr @ Y0 @ X1) @ Y0))) = $true)) )),
% 2.16/1.45 inference(beta_eta_normalization,[status(thm)],[f1959])).
% 2.16/1.45 thf(f1959,plain,(
% 2.16/1.45 ( ! [X1 : $i > $o] : ((((^[Y0 : $i > $o]: (!! @ $i @ (^[Y1 : $i]: (subset @ (dsetconstr @ Y1 @ Y0) @ Y1)))) @ X1) = $true)) )),
% 2.16/1.45 inference(pi_clausification,[status(thm)],[f1958])).
% 2.16/1.45 thf(f1958,plain,(
% 2.16/1.45 ((!! @ ($i > $o) @ (^[Y0 : $i > $o]: (!! @ $i @ (^[Y1 : $i]: (subset @ (dsetconstr @ Y1 @ Y0) @ Y1))))) = $true)),
% 2.16/1.45 inference(beta_eta_normalization,[status(thm)],[f583])).
% 2.16/1.45 thf(f583,plain,(
% 2.16/1.45 ((!! @ ($i > $o) @ (^[Y0 : $i > $o]: (!! @ $i @ (^[Y1 : $i]: (subset @ (dsetconstr @ Y1 @ (^[Y2 : $i]: (Y0 @ Y2))) @ Y1))))) = $true)),
% 2.16/1.45 inference(definition_unfolding,[status(thm)],[f533,f360])).
% 2.16/1.45 thf(f360,plain,(
% 2.16/1.45 (sepSubset = (!! @ ($i > $o) @ (^[Y0 : $i > $o]: (!! @ $i @ (^[Y1 : $i]: (subset @ (dsetconstr @ Y1 @ (^[Y2 : $i]: (Y0 @ Y2))) @ Y1))))))),
% 2.16/1.45 inference(cnf_transformation,[status(thm)],[f297])).
% 2.16/1.45 thf(f297,plain,(
% 2.16/1.45 (sepSubset = (!! @ ($i > $o) @ (^[Y0 : $i > $o]: (!! @ $i @ (^[Y1 : $i]: (subset @ (dsetconstr @ Y1 @ (^[Y2 : $i]: (Y0 @ Y2))) @ Y1))))))),
% 2.16/1.45 inference(fool_elimination,[status(thm)],[f296])).
% 2.16/1.45 thf(f296,plain,(
% 2.16/1.45 (! [X0 : $i,X1 : $i > $o] : (subset @ (dsetconstr @ X0 @ (^[X2 : $i] : (X1 @ X2))) @ X0) = sepSubset)),
% 2.16/1.45 inference(rectify,[status(thm)],[f100])).
% 2.16/1.45 thf(f100,axiom,(
% 2.16/1.45 (! [X3 : $i,X0 : $i > $o] : (subset @ (dsetconstr @ X3 @ (^[X1 : $i] : (X0 @ X1))) @ X3) = sepSubset)),
% 2.16/1.45 file('/export/starexec/sandbox2/tmp/tmp.Z1Gw6mJcyS/DTF2THF_11947.p',sepSubset)).
% 2.16/1.45 thf(f533,plain,(
% 2.16/1.45 (sepSubset = $true)),
% 2.16/1.45 inference(cnf_transformation,[status(thm)],[f336])).
% 2.16/1.45 thf(f336,plain,(
% 2.16/1.45 (notdexE = $true) & (setunionAx = $true) & (binunionLsub = $true) & (binunionEcases = $true) & (quantDeMorgan3 = $true) & (nonemptyE1 = $true) & (setadjoinIL = $true) & (wellorderingAx = $true) & (setunion__Cong = $true) & (upairsetE = $true) & (emptysetAx = $true) & (emptysetE = $true) & (sepSubset = $true) & (descrp = $true) & (subsetRefl = $true) & (setbeta = $true) & (setext = $true) & (emptyset__Cong = $true) & (dsetconstr__Cong = $true) & (eqimpsubset1 = $true) & (notsubsetI = $true) & (omegaIndAx = $true) & (setoftrueEq = $true) & (nonemptyI = $true) & (notinsingleton = $true) & (emptyE1 = $true) & (binunionIR = $true) & (dsetconstrER = $true) & (noeltsimpempty = $true) & (setextAx = $true) & (powersetAx = $true) & (emptyI = $true) & (exuE2 = $true) & (setadjoinSub = $true) & (exuEu = $true) & (upairsetIR = $true) & (setadjoinOr = $true) & (emptysetsubset = $true) & (setunionE = $true) & (uniqinunit = $true) & (exuI3 = $true) & (emptyinPowerset = $true) & (powerset__Cong = $true) & (foundationAx = $true) & (powersetI1 = $true) & (descr__Cong = $true) & (emptyInPowerset = $true) & (setextsub = $true) & (notequalI2 = $true) & (notequalI1 = $true) & (setunionI = $true) & (dsetconstrI = $true) & (subsetemptysetimpeq = $true) & (upairsetIL = $true) & (subPowSU = $true) & (exuE1 = $true) & (prop2setI = $true) & (exuI2 = $true) & (powersetE1 = $true) & (nonemptyImpWitness = $true) & (nonemptyI1 = $true) & (inCongP = $true) & (omega__Cong = $true) & (subset2powerset = $true) & (notinemptyset = $true) & (setadjoinAx = $true) & (emptyinunitempty = $true) & (binunionE = $true) & (exuI1 = $true) & (upairset2IR = $true) & (setadjoinSub2 = $true) & (quantDeMorgan4 = $true) & (singletonsswitch = $true) & (binunionIL = $true) & (exuE3u = $true) & (subsetTrans = $true) & (sepInPowerset = $true) & (eqimpsubset2 = $true) & (vacuousDall = $true) & (in__Cong = $true) & (binintersectSubset5 = $true) & (powersetsubset = $true) & (subsetI2 = $true) & (subsetE = $true) & (notdallE = $true) & (setadjoinE = $true) & (emptysetimpfalse = $true) & (omega0Ax = $true) & (inPowerset = $true) & (powersetI = $true) & (subsetE2 = $true) & (replAx = $true) & (eqinunit = $true) & (setadjoinIR = $true) & (prop2setE = $true) & (exu__Cong = $true) & (setadjoin__Cong = $true) & (omegaSAx = $true) & (powersetE = $true) & (quantDeMorgan2 = $true) & (binintersectI = $true) & (dsetconstrEL = $true) & (quantDeMorgan1 = $true) & (binintersectEL = $true) & (binunionRsub = $true) & (prop2set2propI = $true) & (subsetI1 = $true) & ((subset @ (binintersect @ sK1 @ sK0) @ sK1) != $true) & (exuE3e = $true)),
% 2.16/1.45 inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1])],[f334,f335])).
% 2.16/1.45 thf(f335,plain,(
% 2.16/1.45 ? [X0 : $i,X1 : $i] : ($true != (subset @ (binintersect @ X1 @ X0) @ X1)) => ((subset @ (binintersect @ sK1 @ sK0) @ sK1) != $true)),
% 2.16/1.45 introduced(definition,[],[choice_axiom])).
% 2.16/1.45 thf(f334,plain,(
% 2.16/1.45 (notdexE = $true) & (setunionAx = $true) & (binunionLsub = $true) & (binunionEcases = $true) & (quantDeMorgan3 = $true) & (nonemptyE1 = $true) & (setadjoinIL = $true) & (wellorderingAx = $true) & (setunion__Cong = $true) & (upairsetE = $true) & (emptysetAx = $true) & (emptysetE = $true) & (sepSubset = $true) & (descrp = $true) & (subsetRefl = $true) & (setbeta = $true) & (setext = $true) & (emptyset__Cong = $true) & (dsetconstr__Cong = $true) & (eqimpsubset1 = $true) & (notsubsetI = $true) & (omegaIndAx = $true) & (setoftrueEq = $true) & (nonemptyI = $true) & (notinsingleton = $true) & (emptyE1 = $true) & (binunionIR = $true) & (dsetconstrER = $true) & (noeltsimpempty = $true) & (setextAx = $true) & (powersetAx = $true) & (emptyI = $true) & (exuE2 = $true) & (setadjoinSub = $true) & (exuEu = $true) & (upairsetIR = $true) & (setadjoinOr = $true) & (emptysetsubset = $true) & (setunionE = $true) & (uniqinunit = $true) & (exuI3 = $true) & (emptyinPowerset = $true) & (powerset__Cong = $true) & (foundationAx = $true) & (powersetI1 = $true) & (descr__Cong = $true) & (emptyInPowerset = $true) & (setextsub = $true) & (notequalI2 = $true) & (notequalI1 = $true) & (setunionI = $true) & (dsetconstrI = $true) & (subsetemptysetimpeq = $true) & (upairsetIL = $true) & (subPowSU = $true) & (exuE1 = $true) & (prop2setI = $true) & (exuI2 = $true) & (powersetE1 = $true) & (nonemptyImpWitness = $true) & (nonemptyI1 = $true) & (inCongP = $true) & (omega__Cong = $true) & (subset2powerset = $true) & (notinemptyset = $true) & (setadjoinAx = $true) & (emptyinunitempty = $true) & (binunionE = $true) & (exuI1 = $true) & (upairset2IR = $true) & (setadjoinSub2 = $true) & (quantDeMorgan4 = $true) & (singletonsswitch = $true) & (binunionIL = $true) & (exuE3u = $true) & (subsetTrans = $true) & (sepInPowerset = $true) & (eqimpsubset2 = $true) & (vacuousDall = $true) & (in__Cong = $true) & (binintersectSubset5 = $true) & (powersetsubset = $true) & (subsetI2 = $true) & (subsetE = $true) & (notdallE = $true) & (setadjoinE = $true) & (emptysetimpfalse = $true) & (omega0Ax = $true) & (inPowerset = $true) & (powersetI = $true) & (subsetE2 = $true) & (replAx = $true) & (eqinunit = $true) & (setadjoinIR = $true) & (prop2setE = $true) & (exu__Cong = $true) & (setadjoin__Cong = $true) & (omegaSAx = $true) & (powersetE = $true) & (quantDeMorgan2 = $true) & (binintersectI = $true) & (dsetconstrEL = $true) & (quantDeMorgan1 = $true) & (binintersectEL = $true) & (binunionRsub = $true) & (prop2set2propI = $true) & (subsetI1 = $true) & ? [X0 : $i,X1 : $i] : ($true != (subset @ (binintersect @ X1 @ X0) @ X1)) & (exuE3e = $true)),
% 2.16/1.45 inference(flattening,[status(thm)],[f333])).
% 2.16/1.45 thf(f333,plain,(
% 2.16/1.45 (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i,X1 : $i] : ($true != (subset @ (binintersect @ X1 @ X0) @ X1)) & (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)),
% 2.16/1.45 inference(ennf_transformation,[status(thm)],[f229])).
% 2.16/1.45 thf(f229,plain,(
% 2.16/1.45 ~((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) => ! [X0 : $i,X1 : $i] : ($true = (subset @ (binintersect @ X1 @ X0) @ X1))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.16/1.45 inference(fool_elimination,[status(thm)],[f228])).
% 2.16/1.45 thf(f228,plain,(
% 2.16/1.45 ~(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 => ! [X0 : $i,X1 : $i] : (subset @ (binintersect @ X1 @ X0) @ X1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.16/1.45 inference(rectify,[status(thm)],[f113])).
% 2.16/1.45 thf(f113,negated_conjecture,(
% 2.16/1.45 ~(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 => ! [X4 : $i,X3 : $i] : (subset @ (binintersect @ X3 @ X4) @ X3)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.16/1.45 inference(negated_conjecture,[status(cth)],[f112])).
% 2.16/1.45 thf(f112,conjecture,(
% 2.16/1.45 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 => ! [X4 : $i,X3 : $i] : (subset @ (binintersect @ X3 @ X4) @ X3))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.16/1.45 file('/export/starexec/sandbox2/tmp/tmp.Z1Gw6mJcyS/DTF2THF_11947.p',binintersectLsub)).
% 2.16/1.45 thf(f1486,plain,(
% 2.16/1.45 ((subset @ (dsetconstr @ sK1 @ (^[Y0 : $i]: (in @ Y0 @ sK0))) @ sK1) != $true)),
% 2.16/1.45 inference(beta_eta_normalization,[status(thm)],[f670])).
% 2.16/1.45 thf(f670,plain,(
% 2.16/1.45 ((subset @ ((^[Y0 : $i]: ((^[Y1 : $i]: (dsetconstr @ Y0 @ (^[Y2 : $i]: (in @ Y2 @ Y1)))))) @ sK1 @ sK0) @ sK1) != $true)),
% 2.16/1.45 inference(definition_unfolding,[status(thm)],[f438,f552])).
% 2.16/1.45 thf(f552,plain,(
% 2.16/1.45 (binintersect = (^[Y0 : $i]: ((^[Y1 : $i]: (dsetconstr @ Y0 @ (^[Y2 : $i]: (in @ Y2 @ Y1)))))))),
% 2.16/1.45 inference(cnf_transformation,[status(thm)],[f241])).
% 2.16/1.45 thf(f241,plain,(
% 2.16/1.45 (binintersect = (^[Y0 : $i]: ((^[Y1 : $i]: (dsetconstr @ Y0 @ (^[Y2 : $i]: (in @ Y2 @ Y1)))))))),
% 2.16/1.45 inference(fool_elimination,[status(thm)],[f240])).
% 2.16/1.45 thf(f240,plain,(
% 2.16/1.45 ((^[X0 : $i, X1 : $i] : (dsetconstr @ X0 @ (^[X2 : $i] : (in @ X2 @ X1)))) = binintersect)),
% 2.16/1.45 inference(rectify,[status(thm)],[f108])).
% 2.16/1.45 thf(f108,axiom,(
% 2.16/1.45 ((^[X3 : $i, X4 : $i] : (dsetconstr @ X3 @ (^[X1 : $i] : (in @ X1 @ X4)))) = binintersect)),
% 2.16/1.45 file('/export/starexec/sandbox2/tmp/tmp.Z1Gw6mJcyS/DTF2THF_11947.p',binintersect)).
% 2.16/1.45 thf(f438,plain,(
% 2.16/1.45 ((subset @ (binintersect @ sK1 @ sK0) @ sK1) != $true)),
% 2.16/1.45 inference(cnf_transformation,[status(thm)],[f336])).
% 2.16/1.45 % SZS output end Proof for DTF2THF_11947
% 2.16/1.45 % (12139)------------------------------
% 2.16/1.45 % (12139)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.16/1.45 % (12139)Termination reason: Refutation
% 2.16/1.45
% 2.16/1.45 % (12139)Memory used [KB]: 6908
% 2.16/1.45 % (12139)Time elapsed: 0.050 s
% 2.16/1.45 % (12139)Instructions burned: 139 (million)
% 2.16/1.45 % (12139)------------------------------
% 2.16/1.45 % (12139)------------------------------
% 2.16/1.45 % (12137)Success in time 0.057 s
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