%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU583^1 : 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 : n011.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 09:35:25 AM UTC 2026
% Result : Theorem 2.45s 1.52s
% Output : Refutation 2.45s
% Verified :
% SZS Type : Refutation
% Derivation depth : 13
% Number of leaves : 3
% Syntax : Number of formulae : 20 ( 12 unt; 0 typ; 0 def)
% Number of atoms : 1576 ( 413 equ; 0 cnn)
% Maximal formula atoms : 101 ( 78 avg)
% Number of connectives : 843 ( 9 ~; 0 |; 297 &; 132 @)
% ( 0 <=>; 397 =>; 0 <=; 0 <~>)
% Maximal formula depth : 107 ( 37 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 200 ( 196 usr; 105 con; 0-4 aty)
% ( 8 !!; 0 ??; 0 @@+; 0 @@-)
% Number of variables : 34 ( 10 ^; 18 !; 6 ?; 34 :)
% 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_136,type,
sK3: $i > $i ).
thf(func_def_137,type,
sK4: $i > $i > $i ).
thf(func_def_138,type,
sK5: ( $i > $o ) > $i > $i ).
thf(func_def_139,type,
sK6: ( $i > $o ) > $i ).
thf(func_def_140,type,
sK7: ( $i > $o ) > $i ).
thf(func_def_141,type,
sK8: $i > $i ).
thf(func_def_142,type,
sK9: ( $i > $o ) > $i ).
thf(func_def_143,type,
sK10: ( $i > $o ) > $i > $i ).
thf(func_def_144,type,
sK11: $i > $i > $i ).
thf(func_def_145,type,
sK12: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_146,type,
sK13: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_147,type,
sK14: $i > ( $i > $o ) > $i ).
thf(func_def_148,type,
sK15: $i > ( $i > $o ) > $i ).
thf(func_def_149,type,
sK16: $i > ( $i > $o ) > $i ).
thf(func_def_150,type,
sK17: $i > ( $i > $o ) > $i ).
thf(func_def_151,type,
sK18: $i > ( $i > $o ) > $i ).
thf(func_def_152,type,
sK19: $i > ( $i > $o ) > $i ).
thf(func_def_153,type,
sK20: $i > ( $i > $o ) > $i ).
thf(func_def_154,type,
sK21: $i > ( $i > $o ) > $i ).
thf(func_def_155,type,
sK22: $i > $i > $i ).
thf(func_def_156,type,
sK23: ( $i > $o ) > $i > $i ).
thf(func_def_157,type,
sK24: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_158,type,
sK25: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_159,type,
sK26: ( $i > $o ) > $i ).
thf(func_def_160,type,
sK27: $i > ( $i > $o ) > $i ).
thf(func_def_161,type,
sK28: $i > ( $i > $o ) > $i ).
thf(func_def_162,type,
sK29: ( $i > $o ) > $i ).
thf(func_def_163,type,
sK30: $i > ( $i > $o ) > $i ).
thf(func_def_164,type,
sK31: $i > ( $i > $o ) > $i ).
thf(func_def_165,type,
sK32: ( $i > $o ) > $i ).
thf(func_def_166,type,
sK33: $i > ( $i > $o ) > $i ).
thf(func_def_167,type,
sK34: $i > ( $i > $o ) > $i ).
thf(func_def_168,type,
sK35: $i > ( $i > $o ) > $i ).
thf(func_def_169,type,
sK36: ( $i > $o ) > $i ).
thf(func_def_170,type,
sK37: ( $i > $o ) > $i ).
thf(func_def_171,type,
sK38: ( $i > $o ) > $i ).
thf(func_def_172,type,
sK39: $i > ( $i > $o ) > $i ).
thf(func_def_173,type,
sK40: $i > ( $i > $o ) > $i ).
thf(func_def_174,type,
sK41: ( $i > $o ) > $i ).
thf(func_def_175,type,
sK42: ( $i > $o ) > $i > $i ).
thf(func_def_176,type,
sK43: $i > ( $i > $o ) > $i ).
thf(func_def_177,type,
sK44: $i > ( $i > $o ) > $i ).
thf(func_def_178,type,
sK45: ( $i > $o ) > $i > $i ).
thf(func_def_179,type,
sK46: $i > ( $i > $i > $o ) > $i ).
thf(func_def_180,type,
sK47: $i > ( $i > $i > $o ) > $i ).
thf(func_def_181,type,
sK48: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_182,type,
sK49: $i > ( $i > $i > $o ) > $i ).
thf(func_def_183,type,
sK50: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_184,type,
sK51: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_185,type,
sK52: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_186,type,
sK53: $i > $i ).
thf(func_def_187,type,
sK54: $i > $i ).
thf(func_def_188,type,
sK55: $i > $i > $i ).
thf(func_def_189,type,
sK56: $i > $i > $i ).
thf(func_def_190,type,
sK57: $i > $i > $i ).
thf(func_def_191,type,
sK58: ( $i > $o ) > $i ).
thf(func_def_192,type,
sK59: ( $i > $o ) > $i > $i ).
thf(func_def_193,type,
sK60: $i > $i > $i ).
thf(func_def_194,type,
sK61: $i > ( $i > $o ) > $i ).
thf(func_def_195,type,
sK62: $i > $i ).
thf(func_def_196,type,
sK63: $i > $i > $i > $i ).
thf(func_def_197,type,
sK64: $i > $i > $i ).
thf(func_def_198,type,
sK65: $i > $i > $i ).
thf(func_def_199,type,
sK66: $i > $i > $i ).
thf(func_def_200,type,
sK67: $o > $i > $i > $i ).
thf(func_def_201,type,
sK68: $i > $i ).
thf(func_def_202,type,
sK69: $i > ( $i > $o ) > $i ).
thf(func_def_203,type,
sK70: ( $i > $o ) > $i > $i ).
thf(func_def_204,type,
sK71: ( $i > $o ) > $i ).
thf(func_def_205,type,
sK72: ( $i > $o ) > $i ).
thf(func_def_206,type,
sK73: ( $i > $o ) > $i ).
thf(func_def_207,type,
sK74: ( $i > $o ) > $i ).
thf(func_def_208,type,
sK75: $i > $i > $i ).
thf(func_def_209,type,
sK76: $i > $i > $i ).
thf(func_def_210,type,
sK77: ( $i > $o ) > $i > $i ).
thf(func_def_211,type,
sK78: $i > ( $i > $o ) > $i ).
thf(func_def_212,type,
sK79: ( $i > $o ) > $i > $i ).
thf(func_def_213,type,
sK80: ( $i > $o ) > $i > ( $i > $o ) > $i > $i ).
thf(func_def_214,type,
sK81: $i > $i > ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_215,type,
sK82: $i > $i ).
thf(f1895,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f454,f1227]) ).
thf(f1227,plain,
! [X2: $i,X1: $i] :
( ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f1226]) ).
thf(f1226,plain,
! [X2: $i,X1: $i] :
( $true
= ( ^ [Y0: $i] : ( in @ Y0 @ ( setadjoin @ X1 @ ( setadjoin @ Y0 @ emptyset ) ) )
@ X2 ) ),
inference(pi_clausification,[status(thm)],[f1225]) ).
thf(f1225,plain,
! [X1: $i] :
( $true
= ( !! @ $i
@ ^ [Y0: $i] : ( in @ Y0 @ ( setadjoin @ X1 @ ( setadjoin @ Y0 @ emptyset ) ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f1224]) ).
thf(f1224,plain,
! [X1: $i] :
( ( ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] : ( in @ Y1 @ ( setadjoin @ Y0 @ ( setadjoin @ Y1 @ emptyset ) ) ) )
@ X1 )
= $true ),
inference(pi_clausification,[status(thm)],[f544]) ).
thf(f544,plain,
( ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] : ( in @ Y1 @ ( setadjoin @ Y0 @ ( setadjoin @ Y1 @ emptyset ) ) ) ) )
= $true ),
inference(definition_unfolding,[status(thm)],[f408,f360]) ).
thf(f360,plain,
( upairsetIR
= ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] : ( in @ Y1 @ ( setadjoin @ Y0 @ ( setadjoin @ Y1 @ emptyset ) ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f235]) ).
thf(f235,plain,
( upairsetIR
= ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] : ( in @ Y1 @ ( setadjoin @ Y0 @ ( setadjoin @ Y1 @ emptyset ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f234]) ).
thf(f234,plain,
( upairsetIR
= ( ! [X0: $i,X1: $i] : ( in @ X0 @ ( setadjoin @ X1 @ ( setadjoin @ X0 @ emptyset ) ) ) ) ),
inference(rectify,[status(thm)],[f52]) ).
thf(f52,axiom,
( upairsetIR
= ( ! [X2: $i,X1: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.KUfZ1qlWTQ/DTF2THF_6747.p',upairsetIR) ).
thf(f408,plain,
upairsetIR = $true,
inference(cnf_transformation,[status(thm)],[f309]) ).
thf(f309,plain,
( ( vacuousDall = $true )
& ( emptyinunitempty = $true )
& ( sepInPowerset = $true )
& ( emptyE1 = $true )
& ( powerset__Cong = $true )
& ( emptyset__Cong = $true )
& ( ( in @ sK1 @ ( setadjoin @ sK0 @ ( setadjoin @ sK1 @ emptyset ) ) )
!= $true )
& ( notinemptyset = $true )
& ( setadjoinSub = $true )
& ( exuE3u = $true )
& ( dsetconstrI = $true )
& ( sepSubset = $true )
& ( nonemptyI1 = $true )
& ( setadjoin__Cong = $true )
& ( descr__Cong = $true )
& ( in__Cong = $true )
& ( setunionAx = $true )
& ( exu__Cong = $true )
& ( omegaSAx = $true )
& ( omegaIndAx = $true )
& ( emptyInPowerset = $true )
& ( nonemptyI = $true )
& ( subsetI2 = $true )
& ( singletonsswitch = $true )
& ( binunionIL = $true )
& ( prop2setI = $true )
& ( setadjoinSub2 = $true )
& ( setbeta = $true )
& ( exuE3e = $true )
& ( subPowSU = $true )
& ( nonemptyImpWitness = $true )
& ( notequalI1 = $true )
& ( notdexE = $true )
& ( powersetI = $true )
& ( emptyinPowerset = $true )
& ( setunionI = $true )
& ( setextsub = $true )
& ( noeltsimpempty = $true )
& ( subsetE2 = $true )
& ( exuI3 = $true )
& ( setadjoinE = $true )
& ( subsetemptysetimpeq = $true )
& ( emptysetsubset = $true )
& ( upairsetIL = $true )
& ( powersetsubset = $true )
& ( setunionE = $true )
& ( powersetAx = $true )
& ( uniqinunit = $true )
& ( omega0Ax = $true )
& ( exuE2 = $true )
& ( nonemptyE1 = $true )
& ( setext = $true )
& ( upairsetIR = $true )
& ( quantDeMorgan1 = $true )
& ( setadjoinIL = $true )
& ( descrp = $true )
& ( setextAx = $true )
& ( setadjoinAx = $true )
& ( powersetI1 = $true )
& ( subsetI1 = $true )
& ( prop2set2propI = $true )
& ( emptysetE = $true )
& ( emptyI = $true )
& ( subsetE = $true )
& ( upairsetE = $true )
& ( setunion__Cong = $true )
& ( notequalI2 = $true )
& ( dsetconstrER = $true )
& ( wellorderingAx = $true )
& ( eqinunit = $true )
& ( powersetE = $true )
& ( subsetRefl = $true )
& ( notinsingleton = $true )
& ( eqimpsubset1 = $true )
& ( inCongP = $true )
& ( omega__Cong = $true )
& ( subset2powerset = $true )
& ( replAx = $true )
& ( notsubsetI = $true )
& ( dsetconstrEL = $true )
& ( setadjoinOr = $true )
& ( exuI2 = $true )
& ( inPowerset = $true )
& ( subsetTrans = $true )
& ( emptysetAx = $true )
& ( quantDeMorgan2 = $true )
& ( exuEu = $true )
& ( notdallE = $true )
& ( setoftrueEq = $true )
& ( foundationAx = $true )
& ( powersetE1 = $true )
& ( setadjoinIR = $true )
& ( exuE1 = $true )
& ( dsetconstr__Cong = $true )
& ( prop2setE = $true )
& ( quantDeMorgan4 = $true )
& ( emptysetimpfalse = $true )
& ( eqimpsubset2 = $true )
& ( exuI1 = $true )
& ( quantDeMorgan3 = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1])],[f307,f308]) ).
thf(f308,plain,
( ? [X0: $i,X1: $i] :
( $true
!= ( in @ X1 @ ( setadjoin @ X0 @ ( setadjoin @ X1 @ emptyset ) ) ) )
=> ( ( in @ sK1 @ ( setadjoin @ sK0 @ ( setadjoin @ sK1 @ emptyset ) ) )
!= $true ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f307,plain,
( ( vacuousDall = $true )
& ( emptyinunitempty = $true )
& ( sepInPowerset = $true )
& ( emptyE1 = $true )
& ( powerset__Cong = $true )
& ( emptyset__Cong = $true )
& ? [X0: $i,X1: $i] :
( $true
!= ( in @ X1 @ ( setadjoin @ X0 @ ( setadjoin @ X1 @ emptyset ) ) ) )
& ( notinemptyset = $true )
& ( setadjoinSub = $true )
& ( exuE3u = $true )
& ( dsetconstrI = $true )
& ( sepSubset = $true )
& ( nonemptyI1 = $true )
& ( setadjoin__Cong = $true )
& ( descr__Cong = $true )
& ( in__Cong = $true )
& ( setunionAx = $true )
& ( exu__Cong = $true )
& ( omegaSAx = $true )
& ( omegaIndAx = $true )
& ( emptyInPowerset = $true )
& ( nonemptyI = $true )
& ( subsetI2 = $true )
& ( singletonsswitch = $true )
& ( binunionIL = $true )
& ( prop2setI = $true )
& ( setadjoinSub2 = $true )
& ( setbeta = $true )
& ( exuE3e = $true )
& ( subPowSU = $true )
& ( nonemptyImpWitness = $true )
& ( notequalI1 = $true )
& ( notdexE = $true )
& ( powersetI = $true )
& ( emptyinPowerset = $true )
& ( setunionI = $true )
& ( setextsub = $true )
& ( noeltsimpempty = $true )
& ( subsetE2 = $true )
& ( exuI3 = $true )
& ( setadjoinE = $true )
& ( subsetemptysetimpeq = $true )
& ( emptysetsubset = $true )
& ( upairsetIL = $true )
& ( powersetsubset = $true )
& ( setunionE = $true )
& ( powersetAx = $true )
& ( uniqinunit = $true )
& ( omega0Ax = $true )
& ( exuE2 = $true )
& ( nonemptyE1 = $true )
& ( setext = $true )
& ( upairsetIR = $true )
& ( quantDeMorgan1 = $true )
& ( setadjoinIL = $true )
& ( descrp = $true )
& ( setextAx = $true )
& ( setadjoinAx = $true )
& ( powersetI1 = $true )
& ( subsetI1 = $true )
& ( prop2set2propI = $true )
& ( emptysetE = $true )
& ( emptyI = $true )
& ( subsetE = $true )
& ( upairsetE = $true )
& ( setunion__Cong = $true )
& ( notequalI2 = $true )
& ( dsetconstrER = $true )
& ( wellorderingAx = $true )
& ( eqinunit = $true )
& ( powersetE = $true )
& ( subsetRefl = $true )
& ( notinsingleton = $true )
& ( eqimpsubset1 = $true )
& ( inCongP = $true )
& ( omega__Cong = $true )
& ( subset2powerset = $true )
& ( replAx = $true )
& ( notsubsetI = $true )
& ( dsetconstrEL = $true )
& ( setadjoinOr = $true )
& ( exuI2 = $true )
& ( inPowerset = $true )
& ( subsetTrans = $true )
& ( emptysetAx = $true )
& ( quantDeMorgan2 = $true )
& ( exuEu = $true )
& ( notdallE = $true )
& ( setoftrueEq = $true )
& ( foundationAx = $true )
& ( powersetE1 = $true )
& ( setadjoinIR = $true )
& ( exuE1 = $true )
& ( dsetconstr__Cong = $true )
& ( prop2setE = $true )
& ( quantDeMorgan4 = $true )
& ( emptysetimpfalse = $true )
& ( eqimpsubset2 = $true )
& ( exuI1 = $true )
& ( quantDeMorgan3 = $true ) ),
inference(flattening,[status(thm)],[f306]) ).
thf(f306,plain,
( ? [X0: $i,X1: $i] :
( $true
!= ( in @ X1 @ ( setadjoin @ X0 @ ( setadjoin @ X1 @ emptyset ) ) ) )
& ( 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)],[f127]) ).
thf(f127,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 )
=> ! [X0: $i,X1: $i] :
( $true
= ( in @ X1 @ ( setadjoin @ X0 @ ( setadjoin @ X1 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f126]) ).
thf(f126,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
=> ! [X0: $i,X1: $i] : ( in @ X1 @ ( setadjoin @ X0 @ ( setadjoin @ X1 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f104]) ).
thf(f104,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
=> ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f103]) ).
thf(f103,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
=> ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.KUfZ1qlWTQ/DTF2THF_6747.p',upairset2IR) ).
thf(f454,plain,
( ( in @ sK1 @ ( setadjoin @ sK0 @ ( setadjoin @ sK1 @ emptyset ) ) )
!= $true ),
inference(cnf_transformation,[status(thm)],[f309]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SEU583^1 : TPTP v9.2.1. Released v3.7.0.
% 0.12/0.13 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.16/0.34 % Computer : n011.cluster.edu
% 0.16/0.34 % Model : x86_64 x86_64
% 0.16/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.34 % Memory : 8042.1875MB
% 0.16/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.34 % CPULimit : 300
% 0.16/0.34 % WCLimit : 300
% 0.16/0.34 % DateTime : Tue Feb 24 12:27:17 EST 2026
% 0.16/0.34 % CPUTime :
% 0.16/0.34 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.30/0.46 ---- Original DTF file ---
% 0.30/0.46 thf(spec,logic,$$dhol).
% 0.30/0.46 %------------------------------------------------------------------------------
% 0.30/0.46 % File : SEU583^1 : TPTP v9.2.1. Released v3.7.0.
% 0.30/0.46 % Domain : Set Theory
% 0.30/0.46 % Problem : Preliminary Notions - Ops on Sets - Unions and Intersections
% 0.30/0.46 % Version : Especial.
% 0.30/0.46 % English : (! x:i.! y:i.in y (setadjoin x (setadjoin y emptyset)))
% 0.30/0.46
% 0.30/0.46 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.30/0.46 % Source : [Bro08]
% 0.30/0.46 % Names : ZFC085g [Bro08]
% 0.30/0.46
% 0.30/0.46 % Status : Theorem
% 0.30/0.46 % Rating : 0.22 v9.1.0, 0.12 v9.0.0, 0.20 v8.2.0, 0.23 v8.1.0, 0.00 v7.4.0, 0.11 v7.2.0, 0.00 v7.1.0, 0.25 v7.0.0, 0.14 v6.4.0, 0.17 v6.3.0, 0.20 v6.2.0, 0.14 v6.1.0, 0.29 v5.5.0, 0.33 v5.4.0, 0.40 v5.1.0, 0.60 v5.0.0, 0.40 v4.1.0, 0.33 v4.0.1, 0.67 v4.0.0, 0.33 v3.7.0
% 0.30/0.46 % Syntax : Number of formulae : 218 ( 102 unt; 115 typ; 102 def)
% 0.30/0.46 % Number of atoms : 545 ( 151 equ; 0 cnn)
% 0.30/0.46 % Maximal formula atoms : 100 ( 5 avg)
% 0.30/0.46 % Number of connectives : 857 ( 29 ~; 5 |; 27 &; 538 @)
% 0.30/0.46 % ( 14 <=>; 244 =>; 0 <=; 0 <~>)
% 0.30/0.46 % Maximal formula depth : 106 ( 2 avg)
% 0.30/0.46 % Number of types : 2 ( 0 usr)
% 0.30/0.46 % Number of type conns : 57 ( 57 >; 0 *; 0 +; 0 <<)
% 0.30/0.46 % Number of symbols : 118 ( 115 usr; 103 con; 0-2 aty)
% 0.30/0.46 % Number of variables : 301 ( 32 ^; 241 !; 28 ?; 301 :)
% 0.30/0.46 % SPC : TH0_THM_EQU_NAR
% 0.30/0.46
% 0.30/0.46 % Comments : http://mathgate.info/detsetitem.php?id=159
% 0.30/0.46 %------------------------------------------------------------------------------
% 0.30/0.46 thf(in_type,type,
% 0.30/0.46 in: $i > $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(exu_type,type,
% 0.30/0.46 exu: ( $i > $o ) > $o ).
% 0.30/0.46
% 0.30/0.46 thf(exu,definition,
% 0.30/0.46 ( exu
% 0.30/0.46 = ( ^ [Xphi: $i > $o] :
% 0.30/0.46 ? [Xx: $i] :
% 0.30/0.46 ( ( Xphi @ Xx )
% 0.30/0.46 & ! [Xy: $i] :
% 0.30/0.46 ( ( Xphi @ Xy )
% 0.30/0.46 => ( Xx = Xy ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setextAx_type,type,
% 0.30/0.46 setextAx: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setextAx,definition,
% 0.30/0.46 ( setextAx
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 <=> ( in @ Xx @ B ) )
% 0.30/0.46 => ( A = B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(emptyset_type,type,
% 0.30/0.46 emptyset: $i ).
% 0.30/0.46
% 0.30/0.46 thf(emptysetAx_type,type,
% 0.30/0.46 emptysetAx: $o ).
% 0.30/0.46
% 0.30/0.46 thf(emptysetAx,definition,
% 0.30/0.46 ( emptysetAx
% 0.30/0.46 = ( ! [Xx: $i] :
% 0.30/0.46 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoin_type,type,
% 0.30/0.46 setadjoin: $i > $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinAx_type,type,
% 0.30/0.46 setadjoinAx: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinAx,definition,
% 0.30/0.46 ( setadjoinAx
% 0.30/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.30/0.46 <=> ( ( Xy = Xx )
% 0.30/0.46 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(powerset_type,type,
% 0.30/0.46 powerset: $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(powersetAx_type,type,
% 0.30/0.46 powersetAx: $o ).
% 0.30/0.46
% 0.30/0.46 thf(powersetAx,definition,
% 0.30/0.46 ( powersetAx
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( in @ B @ ( powerset @ A ) )
% 0.30/0.46 <=> ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ B )
% 0.30/0.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setunion_type,type,
% 0.30/0.46 setunion: $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(setunionAx_type,type,
% 0.30/0.46 setunionAx: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setunionAx,definition,
% 0.30/0.46 ( setunionAx
% 0.30/0.46 = ( ! [A: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ ( setunion @ A ) )
% 0.30/0.46 <=> ? [B: $i] :
% 0.30/0.46 ( ( in @ Xx @ B )
% 0.30/0.46 & ( in @ B @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(omega_type,type,
% 0.30/0.46 omega: $i ).
% 0.30/0.46
% 0.30/0.46 thf(omega0Ax_type,type,
% 0.30/0.46 omega0Ax: $o ).
% 0.30/0.46
% 0.30/0.46 thf(omega0Ax,definition,
% 0.30/0.46 ( omega0Ax
% 0.30/0.46 = ( in @ emptyset @ omega ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(omegaSAx_type,type,
% 0.30/0.46 omegaSAx: $o ).
% 0.30/0.46
% 0.30/0.46 thf(omegaSAx,definition,
% 0.30/0.46 ( omegaSAx
% 0.30/0.46 = ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ omega )
% 0.30/0.46 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(omegaIndAx_type,type,
% 0.30/0.46 omegaIndAx: $o ).
% 0.30/0.46
% 0.30/0.46 thf(omegaIndAx,definition,
% 0.30/0.46 ( omegaIndAx
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ( ( ( in @ emptyset @ A )
% 0.30/0.46 & ! [Xx: $i] :
% 0.30/0.46 ( ( ( in @ Xx @ omega )
% 0.30/0.46 & ( in @ Xx @ A ) )
% 0.30/0.46 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ omega )
% 0.30/0.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(replAx_type,type,
% 0.30/0.46 replAx: $o ).
% 0.30/0.46
% 0.30/0.46 thf(replAx,definition,
% 0.30/0.46 ( replAx
% 0.30/0.46 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( exu
% 0.30/0.46 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.30/0.46 => ? [B: $i] :
% 0.30/0.46 ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ B )
% 0.30/0.46 <=> ? [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(foundationAx_type,type,
% 0.30/0.46 foundationAx: $o ).
% 0.30/0.46
% 0.30/0.46 thf(foundationAx,definition,
% 0.30/0.46 ( foundationAx
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.30/0.46 => ? [B: $i] :
% 0.30/0.46 ( ( in @ B @ A )
% 0.30/0.46 & ~ ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ B )
% 0.30/0.46 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(wellorderingAx_type,type,
% 0.30/0.46 wellorderingAx: $o ).
% 0.30/0.46
% 0.30/0.46 thf(wellorderingAx,definition,
% 0.30/0.46 ( wellorderingAx
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ? [B: $i] :
% 0.30/0.46 ( ! [C: $i] :
% 0.30/0.46 ( ( in @ C @ B )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ C )
% 0.30/0.46 => ( in @ Xx @ A ) ) )
% 0.30/0.46 & ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( ( in @ Xx @ A )
% 0.30/0.46 & ( in @ Xy @ A ) )
% 0.30/0.46 => ( ! [C: $i] :
% 0.30/0.46 ( ( in @ C @ B )
% 0.30/0.46 => ( ( in @ Xx @ C )
% 0.30/0.46 <=> ( in @ Xy @ C ) ) )
% 0.30/0.46 => ( Xx = Xy ) ) )
% 0.30/0.46 & ! [C: $i,D: $i] :
% 0.30/0.46 ( ( ( in @ C @ B )
% 0.30/0.46 & ( in @ D @ B ) )
% 0.30/0.46 => ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ C )
% 0.30/0.46 => ( in @ Xx @ D ) )
% 0.30/0.46 | ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ D )
% 0.30/0.46 => ( in @ Xx @ C ) ) ) )
% 0.30/0.46 & ! [C: $i] :
% 0.30/0.46 ( ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ C )
% 0.30/0.46 => ( in @ Xx @ A ) )
% 0.30/0.46 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.30/0.46 => ? [D: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ D @ B )
% 0.30/0.46 & ( in @ Xx @ C )
% 0.30/0.46 & ~ ? [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ D )
% 0.30/0.46 & ( in @ Xy @ C ) )
% 0.30/0.46 & ! [E: $i] :
% 0.30/0.46 ( ( in @ E @ B )
% 0.30/0.46 => ( ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ E )
% 0.30/0.46 => ( in @ Xy @ D ) )
% 0.30/0.46 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(descr_type,type,
% 0.30/0.46 descr: ( $i > $o ) > $i ).
% 0.30/0.46
% 0.30/0.46 thf(descrp_type,type,
% 0.30/0.46 descrp: $o ).
% 0.30/0.46
% 0.30/0.46 thf(descrp,definition,
% 0.30/0.46 ( descrp
% 0.30/0.46 = ( ! [Xphi: $i > $o] :
% 0.30/0.46 ( ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 => ( Xphi
% 0.30/0.46 @ ( descr
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(dsetconstr_type,type,
% 0.30/0.46 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.30/0.46
% 0.30/0.46 thf(dsetconstrI_type,type,
% 0.30/0.46 dsetconstrI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(dsetconstrI,definition,
% 0.30/0.46 ( dsetconstrI
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( Xphi @ Xx )
% 0.30/0.46 => ( in @ Xx
% 0.30/0.46 @ ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(dsetconstrEL_type,type,
% 0.30/0.46 dsetconstrEL: $o ).
% 0.30/0.46
% 0.30/0.46 thf(dsetconstrEL,definition,
% 0.30/0.46 ( dsetconstrEL
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx
% 0.30/0.46 @ ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.30/0.46 => ( in @ Xx @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(dsetconstrER_type,type,
% 0.30/0.46 dsetconstrER: $o ).
% 0.30/0.46
% 0.30/0.46 thf(dsetconstrER,definition,
% 0.30/0.46 ( dsetconstrER
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx
% 0.30/0.46 @ ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.30/0.46 => ( Xphi @ Xx ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(exuE1_type,type,
% 0.30/0.46 exuE1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(exuE1,definition,
% 0.30/0.46 ( exuE1
% 0.30/0.46 = ( ! [Xphi: $i > $o] :
% 0.30/0.46 ( ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 => ? [Xx: $i] :
% 0.30/0.46 ( ( Xphi @ Xx )
% 0.30/0.46 & ! [Xy: $i] :
% 0.30/0.46 ( ( Xphi @ Xy )
% 0.30/0.46 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(prop2set_type,type,
% 0.30/0.46 prop2set: $o > $i ).
% 0.30/0.46
% 0.30/0.46 thf(prop2setE_type,type,
% 0.30/0.46 prop2setE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(prop2setE,definition,
% 0.30/0.46 ( prop2setE
% 0.30/0.46 = ( ! [Xphi: $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.30/0.46 => Xphi ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(emptysetE_type,type,
% 0.30/0.46 emptysetE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(emptysetE,definition,
% 0.30/0.46 ( emptysetE
% 0.30/0.46 = ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ emptyset )
% 0.30/0.46 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(emptysetimpfalse_type,type,
% 0.30/0.46 emptysetimpfalse: $o ).
% 0.30/0.46
% 0.30/0.46 thf(emptysetimpfalse,definition,
% 0.30/0.46 ( emptysetimpfalse
% 0.30/0.46 = ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ emptyset )
% 0.30/0.46 => $false ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(notinemptyset_type,type,
% 0.30/0.46 notinemptyset: $o ).
% 0.30/0.46
% 0.30/0.46 thf(notinemptyset,definition,
% 0.30/0.46 ( notinemptyset
% 0.30/0.46 = ( ! [Xx: $i] :
% 0.30/0.46 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(exuE3e_type,type,
% 0.30/0.46 exuE3e: $o ).
% 0.30/0.46
% 0.30/0.46 thf(exuE3e,definition,
% 0.30/0.46 ( exuE3e
% 0.30/0.46 = ( ! [Xphi: $i > $o] :
% 0.30/0.46 ( ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setext_type,type,
% 0.30/0.46 setext: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setext,definition,
% 0.30/0.46 ( setext
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ Xx @ B ) )
% 0.30/0.46 => ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ B )
% 0.30/0.46 => ( in @ Xx @ A ) )
% 0.30/0.46 => ( A = B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(emptyI_type,type,
% 0.30/0.46 emptyI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(emptyI,definition,
% 0.30/0.46 ( emptyI
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ~ ( in @ Xx @ A )
% 0.30/0.46 => ( A = emptyset ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(noeltsimpempty_type,type,
% 0.30/0.46 noeltsimpempty: $o ).
% 0.30/0.46
% 0.30/0.46 thf(noeltsimpempty,definition,
% 0.30/0.46 ( noeltsimpempty
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ~ ( in @ Xx @ A )
% 0.30/0.46 => ( A = emptyset ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setbeta_type,type,
% 0.30/0.46 setbeta: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setbeta,definition,
% 0.30/0.46 ( setbeta
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx
% 0.30/0.46 @ ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.30/0.46 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(nonempty_type,type,
% 0.30/0.46 nonempty: $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(nonempty,definition,
% 0.30/0.46 ( nonempty
% 0.30/0.46 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(nonemptyE1_type,type,
% 0.30/0.46 nonemptyE1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(nonemptyE1,definition,
% 0.30/0.46 ( nonemptyE1
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ( ( nonempty @ A )
% 0.30/0.46 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(nonemptyI_type,type,
% 0.30/0.46 nonemptyI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(nonemptyI,definition,
% 0.30/0.46 ( nonemptyI
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( Xphi @ Xx )
% 0.30/0.46 => ( nonempty
% 0.30/0.46 @ ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(nonemptyI1_type,type,
% 0.30/0.46 nonemptyI1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(nonemptyI1,definition,
% 0.30/0.46 ( nonemptyI1
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.30/0.46 => ( nonempty @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinIL_type,type,
% 0.30/0.46 setadjoinIL: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinIL,definition,
% 0.30/0.46 ( setadjoinIL
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(emptyinunitempty_type,type,
% 0.30/0.46 emptyinunitempty: $o ).
% 0.30/0.46
% 0.30/0.46 thf(emptyinunitempty,definition,
% 0.30/0.46 ( emptyinunitempty
% 0.30/0.46 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinIR_type,type,
% 0.30/0.46 setadjoinIR: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinIR,definition,
% 0.30/0.46 ( setadjoinIR
% 0.30/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinE_type,type,
% 0.30/0.46 setadjoinE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinE,definition,
% 0.30/0.46 ( setadjoinE
% 0.30/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.30/0.46 => ! [Xphi: $o] :
% 0.30/0.46 ( ( ( Xy = Xx )
% 0.30/0.46 => Xphi )
% 0.30/0.46 => ( ( ( in @ Xy @ A )
% 0.30/0.46 => Xphi )
% 0.30/0.46 => Xphi ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinOr_type,type,
% 0.30/0.46 setadjoinOr: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinOr,definition,
% 0.30/0.46 ( setadjoinOr
% 0.30/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.30/0.46 => ( ( Xy = Xx )
% 0.30/0.46 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setoftrueEq_type,type,
% 0.30/0.46 setoftrueEq: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setoftrueEq,definition,
% 0.30/0.46 ( setoftrueEq
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ( ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xx: $i] : $true )
% 0.30/0.46 = A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(powersetI_type,type,
% 0.30/0.46 powersetI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(powersetI,definition,
% 0.30/0.46 ( powersetI
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ B )
% 0.30/0.46 => ( in @ Xx @ A ) )
% 0.30/0.46 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(emptyinPowerset_type,type,
% 0.30/0.46 emptyinPowerset: $o ).
% 0.30/0.46
% 0.30/0.46 thf(emptyinPowerset,definition,
% 0.30/0.46 ( emptyinPowerset
% 0.30/0.46 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(emptyInPowerset_type,type,
% 0.30/0.46 emptyInPowerset: $o ).
% 0.30/0.46
% 0.30/0.46 thf(emptyInPowerset,definition,
% 0.30/0.46 ( emptyInPowerset
% 0.30/0.46 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(powersetE_type,type,
% 0.30/0.46 powersetE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(powersetE,definition,
% 0.30/0.46 ( powersetE
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ B @ ( powerset @ A ) )
% 0.30/0.46 => ( ( in @ Xx @ B )
% 0.30/0.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setunionI_type,type,
% 0.30/0.46 setunionI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setunionI,definition,
% 0.30/0.46 ( setunionI
% 0.30/0.46 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.30/0.46 ( ( in @ Xx @ B )
% 0.30/0.46 => ( ( in @ B @ A )
% 0.30/0.46 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setunionE_type,type,
% 0.30/0.46 setunionE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setunionE,definition,
% 0.30/0.46 ( setunionE
% 0.30/0.46 = ( ! [A: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ ( setunion @ A ) )
% 0.30/0.46 => ! [Xphi: $o] :
% 0.30/0.46 ( ! [B: $i] :
% 0.30/0.46 ( ( in @ Xx @ B )
% 0.30/0.46 => ( ( in @ B @ A )
% 0.30/0.46 => Xphi ) )
% 0.30/0.46 => Xphi ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(subPowSU_type,type,
% 0.30/0.46 subPowSU: $o ).
% 0.30/0.46
% 0.30/0.46 thf(subPowSU,definition,
% 0.30/0.46 ( subPowSU
% 0.30/0.46 = ( ! [A: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(exuE2_type,type,
% 0.30/0.46 exuE2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(exuE2,definition,
% 0.30/0.46 ( exuE2
% 0.30/0.46 = ( ! [Xphi: $i > $o] :
% 0.30/0.46 ( ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 => ? [Xx: $i] :
% 0.30/0.46 ! [Xy: $i] :
% 0.30/0.46 ( ( Xphi @ Xy )
% 0.30/0.46 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(nonemptyImpWitness_type,type,
% 0.30/0.46 nonemptyImpWitness: $o ).
% 0.30/0.46
% 0.30/0.46 thf(nonemptyImpWitness,definition,
% 0.30/0.46 ( nonemptyImpWitness
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ( ( nonempty @ A )
% 0.30/0.46 => ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & $true ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(uniqinunit_type,type,
% 0.30/0.46 uniqinunit: $o ).
% 0.30/0.46
% 0.30/0.46 thf(uniqinunit,definition,
% 0.30/0.46 ( uniqinunit
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.30/0.46 => ( Xx = Xy ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(notinsingleton_type,type,
% 0.30/0.46 notinsingleton: $o ).
% 0.30/0.46
% 0.30/0.46 thf(notinsingleton,definition,
% 0.30/0.46 ( notinsingleton
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xx != Xy )
% 0.30/0.46 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(eqinunit_type,type,
% 0.30/0.46 eqinunit: $o ).
% 0.30/0.46
% 0.30/0.46 thf(eqinunit,definition,
% 0.30/0.46 ( eqinunit
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xx = Xy )
% 0.30/0.46 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(singletonsswitch_type,type,
% 0.30/0.46 singletonsswitch: $o ).
% 0.30/0.46
% 0.30/0.46 thf(singletonsswitch,definition,
% 0.30/0.46 ( singletonsswitch
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.30/0.46 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(upairsetE_type,type,
% 0.30/0.46 upairsetE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(upairsetE,definition,
% 0.30/0.46 ( upairsetE
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.30/0.46 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.30/0.46 => ( ( Xz = Xx )
% 0.30/0.46 | ( Xz = Xy ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(upairsetIL_type,type,
% 0.30/0.46 upairsetIL: $o ).
% 0.30/0.46
% 0.30/0.46 thf(upairsetIL,definition,
% 0.30/0.46 ( upairsetIL
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(upairsetIR_type,type,
% 0.30/0.46 upairsetIR: $o ).
% 0.30/0.46
% 0.30/0.46 thf(upairsetIR,definition,
% 0.30/0.46 ( upairsetIR
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(emptyE1_type,type,
% 0.30/0.46 emptyE1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(emptyE1,definition,
% 0.30/0.46 ( emptyE1
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & ( Xphi @ Xx ) )
% 0.30/0.46 => ( ( ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 = emptyset )
% 0.30/0.46 => $false ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(vacuousDall_type,type,
% 0.30/0.46 vacuousDall: $o ).
% 0.30/0.46
% 0.30/0.46 thf(vacuousDall,definition,
% 0.30/0.46 ( vacuousDall
% 0.30/0.46 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ emptyset )
% 0.30/0.46 => ( Xphi @ Xx ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(quantDeMorgan1_type,type,
% 0.30/0.46 quantDeMorgan1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(quantDeMorgan1,definition,
% 0.30/0.46 ( quantDeMorgan1
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( ~ ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( Xphi @ Xx ) )
% 0.30/0.46 => ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(quantDeMorgan2_type,type,
% 0.30/0.46 quantDeMorgan2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(quantDeMorgan2,definition,
% 0.30/0.46 ( quantDeMorgan2
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ~ ( Xphi @ Xx ) )
% 0.30/0.46 => ~ ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(quantDeMorgan3_type,type,
% 0.30/0.46 quantDeMorgan3: $o ).
% 0.30/0.46
% 0.30/0.46 thf(quantDeMorgan3,definition,
% 0.30/0.46 ( quantDeMorgan3
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( ~ ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & ( Xphi @ Xx ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(quantDeMorgan4_type,type,
% 0.30/0.46 quantDeMorgan4: $o ).
% 0.30/0.46
% 0.30/0.46 thf(quantDeMorgan4,definition,
% 0.30/0.46 ( quantDeMorgan4
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & ~ ( Xphi @ Xx ) )
% 0.30/0.46 => ~ ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(prop2setI_type,type,
% 0.30/0.46 prop2setI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(prop2setI,definition,
% 0.30/0.46 ( prop2setI
% 0.30/0.46 = ( ! [Xphi: $o] :
% 0.30/0.46 ( Xphi
% 0.30/0.46 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(set2prop_type,type,
% 0.30/0.46 set2prop: $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(prop2set2propI_type,type,
% 0.30/0.46 prop2set2propI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(prop2set2propI,definition,
% 0.30/0.46 ( prop2set2propI
% 0.30/0.46 = ( ! [Xphi: $o] :
% 0.30/0.46 ( Xphi
% 0.30/0.46 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(notdexE_type,type,
% 0.30/0.46 notdexE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(notdexE,definition,
% 0.30/0.46 ( notdexE
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( ~ ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & ( Xphi @ Xx ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(notdallE_type,type,
% 0.30/0.46 notdallE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(notdallE,definition,
% 0.30/0.46 ( notdallE
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( ~ ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( Xphi @ Xx ) )
% 0.30/0.46 => ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(exuI1_type,type,
% 0.30/0.46 exuI1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(exuI1,definition,
% 0.30/0.46 ( exuI1
% 0.30/0.46 = ( ! [Xphi: $i > $o] :
% 0.30/0.46 ( ? [Xx: $i] :
% 0.30/0.46 ( ( Xphi @ Xx )
% 0.30/0.46 & ! [Xy: $i] :
% 0.30/0.46 ( ( Xphi @ Xy )
% 0.30/0.46 => ( Xx = Xy ) ) )
% 0.30/0.46 => ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(exuI3_type,type,
% 0.30/0.46 exuI3: $o ).
% 0.30/0.46
% 0.30/0.46 thf(exuI3,definition,
% 0.30/0.46 ( exuI3
% 0.30/0.46 = ( ! [Xphi: $i > $o] :
% 0.30/0.46 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.30/0.46 => ( ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xphi @ Xx )
% 0.30/0.46 => ( ( Xphi @ Xy )
% 0.30/0.46 => ( Xx = Xy ) ) )
% 0.30/0.46 => ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(exuI2_type,type,
% 0.30/0.46 exuI2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(exuI2,definition,
% 0.30/0.46 ( exuI2
% 0.30/0.46 = ( ! [Xphi: $i > $o] :
% 0.30/0.46 ( ? [Xx: $i] :
% 0.30/0.46 ! [Xy: $i] :
% 0.30/0.46 ( ( Xphi @ Xy )
% 0.30/0.46 <=> ( Xy = Xx ) )
% 0.30/0.46 => ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(inCongP_type,type,
% 0.30/0.46 inCongP: $o ).
% 0.30/0.46
% 0.30/0.46 thf(inCongP,definition,
% 0.30/0.46 ( inCongP
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( A = B )
% 0.30/0.46 => ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xx = Xy )
% 0.30/0.46 => ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(in__Cong_type,type,
% 0.30/0.46 in__Cong: $o ).
% 0.30/0.46
% 0.30/0.46 thf(in__Cong,definition,
% 0.30/0.46 ( in__Cong
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( A = B )
% 0.30/0.46 => ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xx = Xy )
% 0.30/0.46 => ( ( in @ Xx @ A )
% 0.30/0.46 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(exuE3u_type,type,
% 0.30/0.46 exuE3u: $o ).
% 0.30/0.46
% 0.30/0.46 thf(exuE3u,definition,
% 0.30/0.46 ( exuE3u
% 0.30/0.46 = ( ! [Xphi: $i > $o] :
% 0.30/0.46 ( ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 => ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xphi @ Xx )
% 0.30/0.46 => ( ( Xphi @ Xy )
% 0.30/0.46 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(exu__Cong_type,type,
% 0.30/0.46 exu__Cong: $o ).
% 0.30/0.46
% 0.30/0.46 thf(exu__Cong,definition,
% 0.30/0.46 ( exu__Cong
% 0.30/0.46 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.30/0.46 ( ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xx = Xy )
% 0.30/0.46 => ( ( Xphi @ Xx )
% 0.30/0.46 <=> ( Xpsi @ Xy ) ) )
% 0.30/0.46 => ( ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 <=> ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(emptyset__Cong_type,type,
% 0.30/0.46 emptyset__Cong: $o ).
% 0.30/0.46
% 0.30/0.46 thf(emptyset__Cong,definition,
% 0.30/0.46 ( emptyset__Cong
% 0.30/0.46 = ( emptyset = emptyset ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoin__Cong_type,type,
% 0.30/0.46 setadjoin__Cong: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoin__Cong,definition,
% 0.30/0.46 ( setadjoin__Cong
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xx = Xy )
% 0.30/0.46 => ! [Xz: $i,Xu: $i] :
% 0.30/0.46 ( ( Xz = Xu )
% 0.30/0.46 => ( ( setadjoin @ Xx @ Xz )
% 0.30/0.46 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(powerset__Cong_type,type,
% 0.30/0.46 powerset__Cong: $o ).
% 0.30/0.46
% 0.30/0.46 thf(powerset__Cong,definition,
% 0.30/0.46 ( powerset__Cong
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( A = B )
% 0.30/0.46 => ( ( powerset @ A )
% 0.30/0.46 = ( powerset @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setunion__Cong_type,type,
% 0.30/0.46 setunion__Cong: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setunion__Cong,definition,
% 0.30/0.46 ( setunion__Cong
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( A = B )
% 0.30/0.46 => ( ( setunion @ A )
% 0.30/0.46 = ( setunion @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(omega__Cong_type,type,
% 0.30/0.46 omega__Cong: $o ).
% 0.30/0.46
% 0.30/0.46 thf(omega__Cong,definition,
% 0.30/0.46 ( omega__Cong
% 0.30/0.46 = ( omega = omega ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(exuEu_type,type,
% 0.30/0.46 exuEu: $o ).
% 0.30/0.46
% 0.30/0.46 thf(exuEu,definition,
% 0.30/0.46 ( exuEu
% 0.30/0.46 = ( ! [Xphi: $i > $o] :
% 0.30/0.46 ( ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 => ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xphi @ Xx )
% 0.30/0.46 => ( ( Xphi @ Xy )
% 0.30/0.46 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(descr__Cong_type,type,
% 0.30/0.46 descr__Cong: $o ).
% 0.30/0.46
% 0.30/0.46 thf(descr__Cong,definition,
% 0.30/0.46 ( descr__Cong
% 0.30/0.46 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.30/0.46 ( ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xx = Xy )
% 0.30/0.46 => ( ( Xphi @ Xx )
% 0.30/0.46 <=> ( Xpsi @ Xy ) ) )
% 0.30/0.46 => ( ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 => ( ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.30/0.46 => ( ( descr
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 = ( descr
% 0.30/0.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(dsetconstr__Cong_type,type,
% 0.30/0.46 dsetconstr__Cong: $o ).
% 0.30/0.46
% 0.30/0.46 thf(dsetconstr__Cong,definition,
% 0.30/0.46 ( dsetconstr__Cong
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( A = B )
% 0.30/0.46 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ B )
% 0.30/0.46 => ( ( Xx = Xy )
% 0.30/0.46 => ( ( Xphi @ Xx )
% 0.30/0.46 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.30/0.46 => ( ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 = ( dsetconstr @ B
% 0.30/0.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(subset_type,type,
% 0.30/0.46 subset: $i > $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(disjoint_type,type,
% 0.30/0.46 disjoint: $i > $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(setsmeet_type,type,
% 0.30/0.46 setsmeet: $i > $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(subsetI1_type,type,
% 0.30/0.46 subsetI1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(subsetI1,definition,
% 0.30/0.46 ( subsetI1
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ Xx @ B ) )
% 0.30/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(eqimpsubset2_type,type,
% 0.30/0.46 eqimpsubset2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(eqimpsubset2,definition,
% 0.30/0.46 ( eqimpsubset2
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( A = B )
% 0.30/0.46 => ( subset @ B @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(eqimpsubset1_type,type,
% 0.30/0.46 eqimpsubset1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(eqimpsubset1,definition,
% 0.30/0.46 ( eqimpsubset1
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( A = B )
% 0.30/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(subsetI2_type,type,
% 0.30/0.46 subsetI2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(subsetI2,definition,
% 0.30/0.46 ( subsetI2
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ Xx @ B ) )
% 0.30/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(emptysetsubset_type,type,
% 0.30/0.46 emptysetsubset: $o ).
% 0.30/0.46
% 0.30/0.46 thf(emptysetsubset,definition,
% 0.30/0.46 ( emptysetsubset
% 0.30/0.46 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(subsetE_type,type,
% 0.30/0.46 subsetE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(subsetE,definition,
% 0.30/0.46 ( subsetE
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( subset @ A @ B )
% 0.30/0.46 => ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(subsetE2_type,type,
% 0.30/0.46 subsetE2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(subsetE2,definition,
% 0.30/0.46 ( subsetE2
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( subset @ A @ B )
% 0.30/0.46 => ( ~ ( in @ Xx @ B )
% 0.30/0.46 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(notsubsetI_type,type,
% 0.30/0.46 notsubsetI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(notsubsetI,definition,
% 0.30/0.46 ( notsubsetI
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ~ ( in @ Xx @ B )
% 0.30/0.46 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(notequalI1_type,type,
% 0.30/0.46 notequalI1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(notequalI1,definition,
% 0.30/0.46 ( notequalI1
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ~ ( subset @ A @ B )
% 0.30/0.46 => ( A != B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(notequalI2_type,type,
% 0.30/0.46 notequalI2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(notequalI2,definition,
% 0.30/0.46 ( notequalI2
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ~ ( in @ Xx @ B )
% 0.30/0.46 => ( A != B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(subsetRefl_type,type,
% 0.30/0.46 subsetRefl: $o ).
% 0.30/0.46
% 0.30/0.46 thf(subsetRefl,definition,
% 0.30/0.46 ( subsetRefl
% 0.30/0.46 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(subsetTrans_type,type,
% 0.30/0.46 subsetTrans: $o ).
% 0.30/0.46
% 0.30/0.46 thf(subsetTrans,definition,
% 0.30/0.46 ( subsetTrans
% 0.30/0.46 = ( ! [A: $i,B: $i,C: $i] :
% 0.30/0.46 ( ( subset @ A @ B )
% 0.30/0.46 => ( ( subset @ B @ C )
% 0.30/0.46 => ( subset @ A @ C ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinSub_type,type,
% 0.30/0.46 setadjoinSub: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinSub,definition,
% 0.30/0.46 ( setadjoinSub
% 0.30/0.46 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinSub2_type,type,
% 0.30/0.46 setadjoinSub2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinSub2,definition,
% 0.30/0.46 ( setadjoinSub2
% 0.30/0.46 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.30/0.46 ( ( subset @ A @ B )
% 0.30/0.46 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(subset2powerset_type,type,
% 0.30/0.46 subset2powerset: $o ).
% 0.30/0.46
% 0.30/0.46 thf(subset2powerset,definition,
% 0.30/0.46 ( subset2powerset
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( subset @ A @ B )
% 0.30/0.46 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setextsub_type,type,
% 0.30/0.46 setextsub: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setextsub,definition,
% 0.30/0.46 ( setextsub
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( subset @ A @ B )
% 0.30/0.46 => ( ( subset @ B @ A )
% 0.30/0.46 => ( A = B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(subsetemptysetimpeq_type,type,
% 0.30/0.46 subsetemptysetimpeq: $o ).
% 0.30/0.46
% 0.30/0.46 thf(subsetemptysetimpeq,definition,
% 0.30/0.46 ( subsetemptysetimpeq
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ( ( subset @ A @ emptyset )
% 0.30/0.46 => ( A = emptyset ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(powersetI1_type,type,
% 0.30/0.46 powersetI1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(powersetI1,definition,
% 0.30/0.46 ( powersetI1
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( subset @ B @ A )
% 0.30/0.46 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(powersetE1_type,type,
% 0.30/0.46 powersetE1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(powersetE1,definition,
% 0.30/0.46 ( powersetE1
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( in @ B @ ( powerset @ A ) )
% 0.30/0.46 => ( subset @ B @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(inPowerset_type,type,
% 0.30/0.46 inPowerset: $o ).
% 0.30/0.46
% 0.30/0.46 thf(inPowerset,definition,
% 0.30/0.46 ( inPowerset
% 0.30/0.46 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(powersetsubset_type,type,
% 0.30/0.46 powersetsubset: $o ).
% 0.30/0.46
% 0.30/0.46 thf(powersetsubset,definition,
% 0.30/0.46 ( powersetsubset
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( subset @ A @ B )
% 0.30/0.46 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(sepInPowerset_type,type,
% 0.30/0.46 sepInPowerset: $o ).
% 0.30/0.46
% 0.30/0.46 thf(sepInPowerset,definition,
% 0.30/0.46 ( sepInPowerset
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( in
% 0.30/0.46 @ ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 @ ( powerset @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(sepSubset_type,type,
% 0.30/0.46 sepSubset: $o ).
% 0.30/0.46
% 0.30/0.46 thf(sepSubset,definition,
% 0.30/0.46 ( sepSubset
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( subset
% 0.30/0.46 @ ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binunion_type,type,
% 0.30/0.46 binunion: $i > $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(binunion,definition,
% 0.30/0.46 ( binunion
% 0.30/0.46 = ( ^ [Xx: $i,Xy: $i] : ( setunion @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binunionIL_type,type,
% 0.30/0.46 binunionIL: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binunionIL,definition,
% 0.30/0.46 ( binunionIL
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(upairset2IR,conjecture,
% 0.30/0.46 ( setextAx
% 0.30/0.46 => ( emptysetAx
% 0.30/0.46 => ( setadjoinAx
% 0.30/0.46 => ( powersetAx
% 0.30/0.46 => ( setunionAx
% 0.30/0.46 => ( omega0Ax
% 0.30/0.46 => ( omegaSAx
% 0.30/0.46 => ( omegaIndAx
% 0.30/0.46 => ( replAx
% 0.30/0.46 => ( foundationAx
% 0.30/0.46 => ( wellorderingAx
% 0.30/0.46 => ( descrp
% 0.30/0.46 => ( dsetconstrI
% 0.30/0.46 => ( dsetconstrEL
% 0.30/0.46 => ( dsetconstrER
% 0.30/0.46 => ( exuE1
% 0.30/0.46 => ( prop2setE
% 0.30/0.46 => ( emptysetE
% 0.30/0.46 => ( emptysetimpfalse
% 0.30/0.46 => ( notinemptyset
% 0.30/0.46 => ( exuE3e
% 0.30/0.46 => ( setext
% 0.30/0.46 => ( emptyI
% 0.30/0.46 => ( noeltsimpempty
% 0.30/0.46 => ( setbeta
% 0.30/0.46 => ( nonemptyE1
% 0.30/0.46 => ( nonemptyI
% 0.30/0.46 => ( nonemptyI1
% 0.30/0.46 => ( setadjoinIL
% 0.30/0.46 => ( emptyinunitempty
% 0.30/0.46 => ( setadjoinIR
% 0.30/0.46 => ( setadjoinE
% 0.30/0.46 => ( setadjoinOr
% 0.30/0.46 => ( setoftrueEq
% 0.30/0.46 => ( powersetI
% 0.30/0.46 => ( emptyinPowerset
% 0.30/0.46 => ( emptyInPowerset
% 0.30/0.46 => ( powersetE
% 0.30/0.46 => ( setunionI
% 0.30/0.46 => ( setunionE
% 0.30/0.46 => ( subPowSU
% 0.30/0.46 => ( exuE2
% 0.30/0.46 => ( nonemptyImpWitness
% 0.30/0.46 => ( uniqinunit
% 0.30/0.46 => ( notinsingleton
% 0.30/0.46 => ( eqinunit
% 0.30/0.46 => ( singletonsswitch
% 0.30/0.46 => ( upairsetE
% 0.30/0.46 => ( upairsetIL
% 0.30/0.46 => ( upairsetIR
% 0.30/0.46 => ( emptyE1
% 0.30/0.46 => ( vacuousDall
% 0.30/0.46 => ( quantDeMorgan1
% 0.30/0.46 => ( quantDeMorgan2
% 0.30/0.46 => ( quantDeMorgan3
% 0.30/0.46 => ( quantDeMorgan4
% 0.30/0.46 => ( prop2setI
% 0.30/0.46 => ( prop2set2propI
% 0.30/0.46 => ( notdexE
% 0.30/0.46 => ( notdallE
% 0.30/0.46 => ( exuI1
% 0.30/0.46 => ( exuI3
% 0.30/0.46 => ( exuI2
% 0.30/0.46 => ( inCongP
% 0.30/0.46 => ( in__Cong
% 0.30/0.46 => ( exuE3u
% 0.30/0.46 => ( exu__Cong
% 0.30/0.46 => ( emptyset__Cong
% 0.30/0.46 => ( setadjoin__Cong
% 0.30/0.46 => ( powerset__Cong
% 0.30/0.46 => ( setunion__Cong
% 0.30/0.46 => ( omega__Cong
% 0.30/0.46 => ( exuEu
% 0.30/0.46 => ( descr__Cong
% 0.30/0.46 => ( dsetconstr__Cong
% 0.30/0.46 => ( subsetI1
% 0.30/0.46 => ( eqimpsubset2
% 0.30/0.46 => ( eqimpsubset1
% 0.30/0.46 => ( subsetI2
% 0.30/0.46 => ( emptysetsubset
% 0.30/0.46 => ( subsetE
% 0.30/0.46 => ( subsetE2
% 0.30/0.46 => ( notsubsetI
% 0.30/0.46 => ( notequalI1
% 0.30/0.46 => ( notequalI2
% 0.30/0.46 => ( subsetRefl
% 0.30/0.46 => ( subsetTrans
% 0.30/0.46 => ( setadjoinSub
% 0.30/0.46 => ( setadjoinSub2
% 0.30/0.46 => ( subset2powerset
% 0.30/0.46 => ( setextsub
% 0.30/0.46 => ( subsetemptysetimpeq
% 0.30/0.46 => ( powersetI1
% 0.30/0.46 => ( powersetE1
% 0.30/0.46 => ( inPowerset
% 0.30/0.46 => ( powersetsubset
% 0.30/0.46 => ( sepInPowerset
% 0.30/0.46 => ( sepSubset
% 0.30/0.46 => ( binunionIL
% 0.30/0.46 => ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 %------------------------------------------------------------------------------
% 0.30/0.46 ------------------------
% 2.41/1.41 ---- Embedded in THF ---
% 2.41/1.42 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.41/1.42 %%% Generated on Tue Feb 24 12:27:18 EST 2026
% 2.41/1.42 %%% using '$$dhol' embedding, version 1.3.0.
% 2.41/1.42 %%% Logic specification used:
% 2.41/1.42 %%% thf(spec, logic, $$dhol).
% 2.41/1.42
% 2.41/1.42 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.KUfZ1qlWTQ/DTF2DTF6747.p
% See solution above
% 2.41/1.42 ------------------------
% 2.41/1.44 ---- Cleaned THF ---
% 2.41/1.44 thf(in_type,type,
% 2.41/1.44 in: $i > $i > $o ).
% 2.41/1.44
% 2.41/1.44 thf(exu_type,type,
% 2.41/1.44 exu: ( $i > $o ) > $o ).
% 2.41/1.44
% 2.41/1.44 thf(exu,definition,
% 2.41/1.44 ( exu
% 2.41/1.44 = ( ^ [Xphi: $i > $o] :
% 2.41/1.44 ? [Xx: $i] :
% 2.41/1.44 ( ( Xphi @ Xx )
% 2.41/1.44 & ! [Xy: $i] :
% 2.41/1.44 ( ( Xphi @ Xy )
% 2.41/1.44 => ( Xx = Xy ) ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(setextAx_type,type,
% 2.41/1.44 setextAx: $o ).
% 2.41/1.44
% 2.41/1.44 thf(setextAx,definition,
% 2.41/1.44 ( setextAx
% 2.41/1.44 = ( ! [A: $i,B: $i] :
% 2.41/1.44 ( ! [Xx: $i] :
% 2.41/1.44 ( ( in @ Xx @ A )
% 2.41/1.44 <=> ( in @ Xx @ B ) )
% 2.41/1.44 => ( A = B ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(emptyset_type,type,
% 2.41/1.44 emptyset: $i ).
% 2.41/1.44
% 2.41/1.44 thf(emptysetAx_type,type,
% 2.41/1.44 emptysetAx: $o ).
% 2.41/1.44
% 2.41/1.44 thf(emptysetAx,definition,
% 2.41/1.44 ( emptysetAx
% 2.41/1.44 = ( ! [Xx: $i] :
% 2.41/1.44 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(setadjoin_type,type,
% 2.41/1.44 setadjoin: $i > $i > $i ).
% 2.41/1.44
% 2.41/1.44 thf(setadjoinAx_type,type,
% 2.41/1.44 setadjoinAx: $o ).
% 2.41/1.44
% 2.41/1.44 thf(setadjoinAx,definition,
% 2.41/1.44 ( setadjoinAx
% 2.41/1.44 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.41/1.44 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.41/1.44 <=> ( ( Xy = Xx )
% 2.41/1.44 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(powerset_type,type,
% 2.41/1.44 powerset: $i > $i ).
% 2.41/1.44
% 2.41/1.44 thf(powersetAx_type,type,
% 2.41/1.44 powersetAx: $o ).
% 2.41/1.44
% 2.41/1.44 thf(powersetAx,definition,
% 2.41/1.44 ( powersetAx
% 2.41/1.44 = ( ! [A: $i,B: $i] :
% 2.41/1.44 ( ( in @ B @ ( powerset @ A ) )
% 2.41/1.44 <=> ! [Xx: $i] :
% 2.41/1.44 ( ( in @ Xx @ B )
% 2.41/1.44 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(setunion_type,type,
% 2.41/1.44 setunion: $i > $i ).
% 2.41/1.44
% 2.41/1.44 thf(setunionAx_type,type,
% 2.41/1.44 setunionAx: $o ).
% 2.41/1.44
% 2.41/1.44 thf(setunionAx,definition,
% 2.41/1.44 ( setunionAx
% 2.41/1.44 = ( ! [A: $i,Xx: $i] :
% 2.41/1.44 ( ( in @ Xx @ ( setunion @ A ) )
% 2.41/1.44 <=> ? [B: $i] :
% 2.41/1.44 ( ( in @ Xx @ B )
% 2.41/1.44 & ( in @ B @ A ) ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(omega_type,type,
% 2.41/1.44 omega: $i ).
% 2.41/1.44
% 2.41/1.44 thf(omega0Ax_type,type,
% 2.41/1.44 omega0Ax: $o ).
% 2.41/1.44
% 2.41/1.44 thf(omega0Ax,definition,
% 2.41/1.44 ( omega0Ax
% 2.41/1.44 = ( in @ emptyset @ omega ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(omegaSAx_type,type,
% 2.41/1.44 omegaSAx: $o ).
% 2.41/1.44
% 2.41/1.44 thf(omegaSAx,definition,
% 2.41/1.44 ( omegaSAx
% 2.41/1.44 = ( ! [Xx: $i] :
% 2.41/1.44 ( ( in @ Xx @ omega )
% 2.41/1.44 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(omegaIndAx_type,type,
% 2.41/1.44 omegaIndAx: $o ).
% 2.41/1.44
% 2.41/1.44 thf(omegaIndAx,definition,
% 2.41/1.44 ( omegaIndAx
% 2.41/1.44 = ( ! [A: $i] :
% 2.41/1.44 ( ( ( in @ emptyset @ A )
% 2.41/1.44 & ! [Xx: $i] :
% 2.41/1.44 ( ( ( in @ Xx @ omega )
% 2.41/1.44 & ( in @ Xx @ A ) )
% 2.41/1.44 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 2.41/1.44 => ! [Xx: $i] :
% 2.41/1.44 ( ( in @ Xx @ omega )
% 2.41/1.44 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(replAx_type,type,
% 2.41/1.44 replAx: $o ).
% 2.41/1.44
% 2.41/1.44 thf(replAx,definition,
% 2.41/1.44 ( replAx
% 2.41/1.44 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 2.41/1.44 ( ! [Xx: $i] :
% 2.41/1.44 ( ( in @ Xx @ A )
% 2.41/1.44 => ( exu
% 2.41/1.44 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 2.41/1.44 => ? [B: $i] :
% 2.41/1.44 ! [Xx: $i] :
% 2.41/1.44 ( ( in @ Xx @ B )
% 2.41/1.44 <=> ? [Xy: $i] :
% 2.41/1.44 ( ( in @ Xy @ A )
% 2.41/1.44 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(foundationAx_type,type,
% 2.41/1.44 foundationAx: $o ).
% 2.41/1.44
% 2.41/1.44 thf(foundationAx,definition,
% 2.41/1.44 ( foundationAx
% 2.41/1.44 = ( ! [A: $i] :
% 2.41/1.44 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.41/1.44 => ? [B: $i] :
% 2.41/1.44 ( ( in @ B @ A )
% 2.41/1.44 & ~ ? [Xx: $i] :
% 2.41/1.44 ( ( in @ Xx @ B )
% 2.41/1.44 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(wellorderingAx_type,type,
% 2.41/1.44 wellorderingAx: $o ).
% 2.41/1.44
% 2.41/1.44 thf(wellorderingAx,definition,
% 2.41/1.44 ( wellorderingAx
% 2.41/1.44 = ( ! [A: $i] :
% 2.41/1.44 ? [B: $i] :
% 2.41/1.44 ( ! [C: $i] :
% 2.41/1.44 ( ( in @ C @ B )
% 2.41/1.44 => ! [Xx: $i] :
% 2.41/1.44 ( ( in @ Xx @ C )
% 2.41/1.44 => ( in @ Xx @ A ) ) )
% 2.41/1.44 & ! [Xx: $i,Xy: $i] :
% 2.41/1.44 ( ( ( in @ Xx @ A )
% 2.41/1.44 & ( in @ Xy @ A ) )
% 2.41/1.44 => ( ! [C: $i] :
% 2.41/1.44 ( ( in @ C @ B )
% 2.41/1.44 => ( ( in @ Xx @ C )
% 2.41/1.44 <=> ( in @ Xy @ C ) ) )
% 2.41/1.44 => ( Xx = Xy ) ) )
% 2.41/1.44 & ! [C: $i,D: $i] :
% 2.41/1.44 ( ( ( in @ C @ B )
% 2.41/1.44 & ( in @ D @ B ) )
% 2.41/1.44 => ( ! [Xx: $i] :
% 2.41/1.44 ( ( in @ Xx @ C )
% 2.41/1.44 => ( in @ Xx @ D ) )
% 2.41/1.44 | ! [Xx: $i] :
% 2.41/1.44 ( ( in @ Xx @ D )
% 2.41/1.44 => ( in @ Xx @ C ) ) ) )
% 2.41/1.44 & ! [C: $i] :
% 2.41/1.44 ( ( ! [Xx: $i] :
% 2.41/1.44 ( ( in @ Xx @ C )
% 2.41/1.44 => ( in @ Xx @ A ) )
% 2.41/1.44 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 2.41/1.44 => ? [D: $i,Xx: $i] :
% 2.41/1.44 ( ( in @ D @ B )
% 2.41/1.44 & ( in @ Xx @ C )
% 2.41/1.44 & ~ ? [Xy: $i] :
% 2.41/1.44 ( ( in @ Xy @ D )
% 2.41/1.44 & ( in @ Xy @ C ) )
% 2.41/1.44 & ! [E: $i] :
% 2.41/1.44 ( ( in @ E @ B )
% 2.41/1.44 => ( ! [Xy: $i] :
% 2.41/1.44 ( ( in @ Xy @ E )
% 2.41/1.44 => ( in @ Xy @ D ) )
% 2.41/1.44 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(descr_type,type,
% 2.41/1.44 descr: ( $i > $o ) > $i ).
% 2.41/1.44
% 2.41/1.44 thf(descrp_type,type,
% 2.41/1.44 descrp: $o ).
% 2.41/1.44
% 2.41/1.44 thf(descrp,definition,
% 2.41/1.44 ( descrp
% 2.41/1.44 = ( ! [Xphi: $i > $o] :
% 2.41/1.44 ( ( exu
% 2.41/1.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.41/1.44 => ( Xphi
% 2.41/1.44 @ ( descr
% 2.41/1.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(dsetconstr_type,type,
% 2.41/1.44 dsetconstr: $i > ( $i > $o ) > $i ).
% 2.41/1.44
% 2.41/1.44 thf(dsetconstrI_type,type,
% 2.41/1.44 dsetconstrI: $o ).
% 2.41/1.44
% 2.41/1.44 thf(dsetconstrI,definition,
% 2.41/1.44 ( dsetconstrI
% 2.41/1.44 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.41/1.44 ( ( in @ Xx @ A )
% 2.41/1.44 => ( ( Xphi @ Xx )
% 2.41/1.44 => ( in @ Xx
% 2.41/1.44 @ ( dsetconstr @ A
% 2.41/1.44 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(dsetconstrEL_type,type,
% 2.41/1.44 dsetconstrEL: $o ).
% 2.41/1.44
% 2.41/1.44 thf(dsetconstrEL,definition,
% 2.41/1.44 ( dsetconstrEL
% 2.41/1.44 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.41/1.44 ( ( in @ Xx
% 2.41/1.44 @ ( dsetconstr @ A
% 2.41/1.44 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.41/1.44 => ( in @ Xx @ A ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(dsetconstrER_type,type,
% 2.41/1.44 dsetconstrER: $o ).
% 2.41/1.44
% 2.41/1.44 thf(dsetconstrER,definition,
% 2.41/1.44 ( dsetconstrER
% 2.41/1.44 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.41/1.44 ( ( in @ Xx
% 2.41/1.44 @ ( dsetconstr @ A
% 2.41/1.44 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.41/1.44 => ( Xphi @ Xx ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(exuE1_type,type,
% 2.41/1.44 exuE1: $o ).
% 2.41/1.44
% 2.41/1.44 thf(exuE1,definition,
% 2.41/1.44 ( exuE1
% 2.41/1.44 = ( ! [Xphi: $i > $o] :
% 2.41/1.44 ( ( exu
% 2.41/1.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.41/1.44 => ? [Xx: $i] :
% 2.41/1.44 ( ( Xphi @ Xx )
% 2.41/1.44 & ! [Xy: $i] :
% 2.41/1.44 ( ( Xphi @ Xy )
% 2.41/1.44 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(prop2set_type,type,
% 2.41/1.44 prop2set: $o > $i ).
% 2.41/1.44
% 2.41/1.44 thf(prop2setE_type,type,
% 2.41/1.44 prop2setE: $o ).
% 2.41/1.44
% 2.41/1.44 thf(prop2setE,definition,
% 2.41/1.44 ( prop2setE
% 2.41/1.44 = ( ! [Xphi: $o,Xx: $i] :
% 2.41/1.44 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 2.41/1.44 => Xphi ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(emptysetE_type,type,
% 2.41/1.44 emptysetE: $o ).
% 2.41/1.44
% 2.41/1.44 thf(emptysetE,definition,
% 2.41/1.44 ( emptysetE
% 2.41/1.44 = ( ! [Xx: $i] :
% 2.41/1.44 ( ( in @ Xx @ emptyset )
% 2.41/1.44 => ! [Xphi: $o] : Xphi ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(emptysetimpfalse_type,type,
% 2.41/1.44 emptysetimpfalse: $o ).
% 2.41/1.44
% 2.41/1.44 thf(emptysetimpfalse,definition,
% 2.41/1.44 ( emptysetimpfalse
% 2.41/1.44 = ( ! [Xx: $i] :
% 2.41/1.44 ( ( in @ Xx @ emptyset )
% 2.41/1.44 => $false ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(notinemptyset_type,type,
% 2.41/1.44 notinemptyset: $o ).
% 2.41/1.44
% 2.41/1.44 thf(notinemptyset,definition,
% 2.41/1.44 ( notinemptyset
% 2.41/1.44 = ( ! [Xx: $i] :
% 2.41/1.44 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(exuE3e_type,type,
% 2.41/1.44 exuE3e: $o ).
% 2.41/1.44
% 2.41/1.44 thf(exuE3e,definition,
% 2.41/1.44 ( exuE3e
% 2.41/1.44 = ( ! [Xphi: $i > $o] :
% 2.41/1.44 ( ( exu
% 2.41/1.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.41/1.44 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(setext_type,type,
% 2.41/1.44 setext: $o ).
% 2.41/1.44
% 2.41/1.44 thf(setext,definition,
% 2.41/1.44 ( setext
% 2.41/1.44 = ( ! [A: $i,B: $i] :
% 2.41/1.44 ( ! [Xx: $i] :
% 2.41/1.44 ( ( in @ Xx @ A )
% 2.41/1.44 => ( in @ Xx @ B ) )
% 2.41/1.44 => ( ! [Xx: $i] :
% 2.41/1.44 ( ( in @ Xx @ B )
% 2.41/1.44 => ( in @ Xx @ A ) )
% 2.41/1.44 => ( A = B ) ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(emptyI_type,type,
% 2.41/1.44 emptyI: $o ).
% 2.41/1.44
% 2.41/1.44 thf(emptyI,definition,
% 2.41/1.44 ( emptyI
% 2.41/1.44 = ( ! [A: $i] :
% 2.41/1.44 ( ! [Xx: $i] :
% 2.41/1.44 ~ ( in @ Xx @ A )
% 2.41/1.44 => ( A = emptyset ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(noeltsimpempty_type,type,
% 2.41/1.44 noeltsimpempty: $o ).
% 2.41/1.44
% 2.41/1.44 thf(noeltsimpempty,definition,
% 2.41/1.44 ( noeltsimpempty
% 2.41/1.44 = ( ! [A: $i] :
% 2.41/1.44 ( ! [Xx: $i] :
% 2.41/1.44 ~ ( in @ Xx @ A )
% 2.41/1.44 => ( A = emptyset ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(setbeta_type,type,
% 2.41/1.44 setbeta: $o ).
% 2.41/1.44
% 2.41/1.44 thf(setbeta,definition,
% 2.41/1.44 ( setbeta
% 2.41/1.44 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.41/1.44 ( ( in @ Xx @ A )
% 2.41/1.44 => ( ( in @ Xx
% 2.41/1.44 @ ( dsetconstr @ A
% 2.41/1.44 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.41/1.44 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(nonempty_type,type,
% 2.41/1.44 nonempty: $i > $o ).
% 2.41/1.44
% 2.41/1.44 thf(nonempty,definition,
% 2.41/1.44 ( nonempty
% 2.41/1.44 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(nonemptyE1_type,type,
% 2.41/1.44 nonemptyE1: $o ).
% 2.41/1.44
% 2.41/1.44 thf(nonemptyE1,definition,
% 2.41/1.44 ( nonemptyE1
% 2.41/1.44 = ( ! [A: $i] :
% 2.41/1.44 ( ( nonempty @ A )
% 2.41/1.44 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(nonemptyI_type,type,
% 2.41/1.44 nonemptyI: $o ).
% 2.41/1.44
% 2.41/1.44 thf(nonemptyI,definition,
% 2.41/1.44 ( nonemptyI
% 2.41/1.44 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.41/1.44 ( ( in @ Xx @ A )
% 2.41/1.44 => ( ( Xphi @ Xx )
% 2.41/1.44 => ( nonempty
% 2.41/1.44 @ ( dsetconstr @ A
% 2.41/1.44 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(nonemptyI1_type,type,
% 2.41/1.44 nonemptyI1: $o ).
% 2.41/1.44
% 2.41/1.44 thf(nonemptyI1,definition,
% 2.41/1.44 ( nonemptyI1
% 2.41/1.44 = ( ! [A: $i] :
% 2.41/1.44 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.41/1.44 => ( nonempty @ A ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(setadjoinIL_type,type,
% 2.41/1.44 setadjoinIL: $o ).
% 2.41/1.44
% 2.41/1.44 thf(setadjoinIL,definition,
% 2.41/1.44 ( setadjoinIL
% 2.41/1.44 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(emptyinunitempty_type,type,
% 2.41/1.44 emptyinunitempty: $o ).
% 2.41/1.44
% 2.41/1.44 thf(emptyinunitempty,definition,
% 2.41/1.44 ( emptyinunitempty
% 2.41/1.44 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(setadjoinIR_type,type,
% 2.41/1.44 setadjoinIR: $o ).
% 2.41/1.44
% 2.41/1.44 thf(setadjoinIR,definition,
% 2.41/1.44 ( setadjoinIR
% 2.41/1.44 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.41/1.44 ( ( in @ Xy @ A )
% 2.41/1.44 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(setadjoinE_type,type,
% 2.41/1.44 setadjoinE: $o ).
% 2.41/1.44
% 2.41/1.44 thf(setadjoinE,definition,
% 2.41/1.44 ( setadjoinE
% 2.41/1.44 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.41/1.44 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.41/1.44 => ! [Xphi: $o] :
% 2.41/1.44 ( ( ( Xy = Xx )
% 2.41/1.44 => Xphi )
% 2.41/1.44 => ( ( ( in @ Xy @ A )
% 2.41/1.44 => Xphi )
% 2.41/1.44 => Xphi ) ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(setadjoinOr_type,type,
% 2.41/1.44 setadjoinOr: $o ).
% 2.41/1.44
% 2.41/1.44 thf(setadjoinOr,definition,
% 2.41/1.44 ( setadjoinOr
% 2.41/1.44 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.41/1.44 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.41/1.44 => ( ( Xy = Xx )
% 2.41/1.44 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(setoftrueEq_type,type,
% 2.41/1.44 setoftrueEq: $o ).
% 2.41/1.44
% 2.41/1.44 thf(setoftrueEq,definition,
% 2.41/1.44 ( setoftrueEq
% 2.41/1.44 = ( ! [A: $i] :
% 2.41/1.44 ( ( dsetconstr @ A
% 2.41/1.44 @ ^ [Xx: $i] : $true )
% 2.41/1.44 = A ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(powersetI_type,type,
% 2.41/1.44 powersetI: $o ).
% 2.41/1.44
% 2.41/1.44 thf(powersetI,definition,
% 2.41/1.44 ( powersetI
% 2.41/1.44 = ( ! [A: $i,B: $i] :
% 2.41/1.44 ( ! [Xx: $i] :
% 2.41/1.44 ( ( in @ Xx @ B )
% 2.41/1.44 => ( in @ Xx @ A ) )
% 2.41/1.44 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(emptyinPowerset_type,type,
% 2.41/1.44 emptyinPowerset: $o ).
% 2.41/1.44
% 2.41/1.44 thf(emptyinPowerset,definition,
% 2.41/1.44 ( emptyinPowerset
% 2.41/1.44 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(emptyInPowerset_type,type,
% 2.41/1.44 emptyInPowerset: $o ).
% 2.41/1.44
% 2.41/1.44 thf(emptyInPowerset,definition,
% 2.41/1.44 ( emptyInPowerset
% 2.41/1.44 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(powersetE_type,type,
% 2.41/1.44 powersetE: $o ).
% 2.41/1.44
% 2.41/1.44 thf(powersetE,definition,
% 2.41/1.44 ( powersetE
% 2.41/1.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.41/1.44 ( ( in @ B @ ( powerset @ A ) )
% 2.41/1.44 => ( ( in @ Xx @ B )
% 2.41/1.44 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(setunionI_type,type,
% 2.41/1.44 setunionI: $o ).
% 2.41/1.44
% 2.41/1.44 thf(setunionI,definition,
% 2.41/1.44 ( setunionI
% 2.41/1.44 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.41/1.44 ( ( in @ Xx @ B )
% 2.41/1.44 => ( ( in @ B @ A )
% 2.41/1.44 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(setunionE_type,type,
% 2.41/1.44 setunionE: $o ).
% 2.41/1.44
% 2.41/1.44 thf(setunionE,definition,
% 2.41/1.44 ( setunionE
% 2.41/1.44 = ( ! [A: $i,Xx: $i] :
% 2.41/1.44 ( ( in @ Xx @ ( setunion @ A ) )
% 2.41/1.44 => ! [Xphi: $o] :
% 2.41/1.44 ( ! [B: $i] :
% 2.41/1.44 ( ( in @ Xx @ B )
% 2.41/1.44 => ( ( in @ B @ A )
% 2.41/1.44 => Xphi ) )
% 2.41/1.44 => Xphi ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(subPowSU_type,type,
% 2.41/1.44 subPowSU: $o ).
% 2.41/1.44
% 2.41/1.44 thf(subPowSU,definition,
% 2.41/1.44 ( subPowSU
% 2.41/1.44 = ( ! [A: $i,Xx: $i] :
% 2.41/1.44 ( ( in @ Xx @ A )
% 2.41/1.44 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(exuE2_type,type,
% 2.41/1.44 exuE2: $o ).
% 2.41/1.44
% 2.41/1.44 thf(exuE2,definition,
% 2.41/1.44 ( exuE2
% 2.41/1.44 = ( ! [Xphi: $i > $o] :
% 2.41/1.44 ( ( exu
% 2.41/1.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.41/1.44 => ? [Xx: $i] :
% 2.41/1.44 ! [Xy: $i] :
% 2.41/1.44 ( ( Xphi @ Xy )
% 2.41/1.44 <=> ( Xy = Xx ) ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(nonemptyImpWitness_type,type,
% 2.41/1.44 nonemptyImpWitness: $o ).
% 2.41/1.44
% 2.41/1.44 thf(nonemptyImpWitness,definition,
% 2.41/1.44 ( nonemptyImpWitness
% 2.41/1.44 = ( ! [A: $i] :
% 2.41/1.44 ( ( nonempty @ A )
% 2.41/1.44 => ? [Xx: $i] :
% 2.41/1.44 ( ( in @ Xx @ A )
% 2.41/1.44 & $true ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(uniqinunit_type,type,
% 2.41/1.44 uniqinunit: $o ).
% 2.41/1.44
% 2.41/1.44 thf(uniqinunit,definition,
% 2.41/1.44 ( uniqinunit
% 2.41/1.44 = ( ! [Xx: $i,Xy: $i] :
% 2.41/1.44 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.41/1.44 => ( Xx = Xy ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(notinsingleton_type,type,
% 2.41/1.44 notinsingleton: $o ).
% 2.41/1.44
% 2.41/1.44 thf(notinsingleton,definition,
% 2.41/1.44 ( notinsingleton
% 2.41/1.44 = ( ! [Xx: $i,Xy: $i] :
% 2.41/1.44 ( ( Xx != Xy )
% 2.41/1.44 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(eqinunit_type,type,
% 2.41/1.44 eqinunit: $o ).
% 2.41/1.44
% 2.41/1.44 thf(eqinunit,definition,
% 2.41/1.44 ( eqinunit
% 2.41/1.44 = ( ! [Xx: $i,Xy: $i] :
% 2.41/1.44 ( ( Xx = Xy )
% 2.41/1.44 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(singletonsswitch_type,type,
% 2.41/1.44 singletonsswitch: $o ).
% 2.41/1.44
% 2.41/1.44 thf(singletonsswitch,definition,
% 2.41/1.44 ( singletonsswitch
% 2.41/1.44 = ( ! [Xx: $i,Xy: $i] :
% 2.41/1.44 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.41/1.44 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(upairsetE_type,type,
% 2.41/1.44 upairsetE: $o ).
% 2.41/1.44
% 2.41/1.44 thf(upairsetE,definition,
% 2.41/1.44 ( upairsetE
% 2.41/1.44 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.41/1.44 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.41/1.44 => ( ( Xz = Xx )
% 2.41/1.44 | ( Xz = Xy ) ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(upairsetIL_type,type,
% 2.41/1.44 upairsetIL: $o ).
% 2.41/1.44
% 2.41/1.44 thf(upairsetIL,definition,
% 2.41/1.44 ( upairsetIL
% 2.41/1.44 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(upairsetIR_type,type,
% 2.41/1.44 upairsetIR: $o ).
% 2.41/1.44
% 2.41/1.44 thf(upairsetIR,definition,
% 2.41/1.44 ( upairsetIR
% 2.41/1.44 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(emptyE1_type,type,
% 2.41/1.44 emptyE1: $o ).
% 2.41/1.44
% 2.41/1.44 thf(emptyE1,definition,
% 2.41/1.44 ( emptyE1
% 2.41/1.44 = ( ! [A: $i,Xphi: $i > $o] :
% 2.41/1.44 ( ? [Xx: $i] :
% 2.41/1.44 ( ( in @ Xx @ A )
% 2.41/1.44 & ( Xphi @ Xx ) )
% 2.41/1.44 => ( ( ( dsetconstr @ A
% 2.41/1.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.41/1.44 = emptyset )
% 2.41/1.44 => $false ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(vacuousDall_type,type,
% 2.41/1.44 vacuousDall: $o ).
% 2.41/1.44
% 2.41/1.44 thf(vacuousDall,definition,
% 2.41/1.44 ( vacuousDall
% 2.41/1.44 = ( ! [Xphi: $i > $o,Xx: $i] :
% 2.41/1.44 ( ( in @ Xx @ emptyset )
% 2.41/1.44 => ( Xphi @ Xx ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(quantDeMorgan1_type,type,
% 2.41/1.44 quantDeMorgan1: $o ).
% 2.41/1.44
% 2.41/1.44 thf(quantDeMorgan1,definition,
% 2.41/1.44 ( quantDeMorgan1
% 2.41/1.44 = ( ! [A: $i,Xphi: $i > $o] :
% 2.41/1.44 ( ~ ! [Xx: $i] :
% 2.41/1.44 ( ( in @ Xx @ A )
% 2.41/1.44 => ( Xphi @ Xx ) )
% 2.41/1.44 => ? [Xx: $i] :
% 2.41/1.44 ( ( in @ Xx @ A )
% 2.41/1.44 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(quantDeMorgan2_type,type,
% 2.41/1.44 quantDeMorgan2: $o ).
% 2.41/1.44
% 2.41/1.44 thf(quantDeMorgan2,definition,
% 2.41/1.44 ( quantDeMorgan2
% 2.41/1.44 = ( ! [A: $i,Xphi: $i > $o] :
% 2.41/1.44 ( ! [Xx: $i] :
% 2.41/1.44 ( ( in @ Xx @ A )
% 2.41/1.44 => ~ ( Xphi @ Xx ) )
% 2.41/1.44 => ~ ? [Xx: $i] :
% 2.41/1.44 ( ( in @ Xx @ A )
% 2.41/1.44 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(quantDeMorgan3_type,type,
% 2.41/1.44 quantDeMorgan3: $o ).
% 2.41/1.44
% 2.41/1.44 thf(quantDeMorgan3,definition,
% 2.41/1.44 ( quantDeMorgan3
% 2.41/1.44 = ( ! [A: $i,Xphi: $i > $o] :
% 2.41/1.44 ( ~ ? [Xx: $i] :
% 2.41/1.44 ( ( in @ Xx @ A )
% 2.41/1.44 & ( Xphi @ Xx ) )
% 2.41/1.44 => ! [Xx: $i] :
% 2.41/1.44 ( ( in @ Xx @ A )
% 2.41/1.44 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(quantDeMorgan4_type,type,
% 2.41/1.44 quantDeMorgan4: $o ).
% 2.41/1.44
% 2.41/1.44 thf(quantDeMorgan4,definition,
% 2.41/1.44 ( quantDeMorgan4
% 2.41/1.44 = ( ! [A: $i,Xphi: $i > $o] :
% 2.41/1.44 ( ? [Xx: $i] :
% 2.41/1.44 ( ( in @ Xx @ A )
% 2.41/1.44 & ~ ( Xphi @ Xx ) )
% 2.41/1.44 => ~ ! [Xx: $i] :
% 2.41/1.44 ( ( in @ Xx @ A )
% 2.41/1.44 => ( Xphi @ Xx ) ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(prop2setI_type,type,
% 2.41/1.44 prop2setI: $o ).
% 2.41/1.44
% 2.41/1.44 thf(prop2setI,definition,
% 2.41/1.44 ( prop2setI
% 2.41/1.44 = ( ! [Xphi: $o] :
% 2.41/1.44 ( Xphi
% 2.41/1.44 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(set2prop_type,type,
% 2.41/1.44 set2prop: $i > $o ).
% 2.41/1.44
% 2.41/1.44 thf(prop2set2propI_type,type,
% 2.41/1.44 prop2set2propI: $o ).
% 2.41/1.44
% 2.41/1.44 thf(prop2set2propI,definition,
% 2.41/1.44 ( prop2set2propI
% 2.41/1.44 = ( ! [Xphi: $o] :
% 2.41/1.44 ( Xphi
% 2.41/1.44 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(notdexE_type,type,
% 2.41/1.44 notdexE: $o ).
% 2.41/1.44
% 2.41/1.44 thf(notdexE,definition,
% 2.41/1.44 ( notdexE
% 2.41/1.44 = ( ! [A: $i,Xphi: $i > $o] :
% 2.41/1.44 ( ~ ? [Xx: $i] :
% 2.41/1.44 ( ( in @ Xx @ A )
% 2.41/1.44 & ( Xphi @ Xx ) )
% 2.41/1.44 => ! [Xx: $i] :
% 2.41/1.44 ( ( in @ Xx @ A )
% 2.41/1.44 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(notdallE_type,type,
% 2.41/1.44 notdallE: $o ).
% 2.41/1.44
% 2.41/1.44 thf(notdallE,definition,
% 2.41/1.44 ( notdallE
% 2.41/1.44 = ( ! [A: $i,Xphi: $i > $o] :
% 2.41/1.44 ( ~ ! [Xx: $i] :
% 2.41/1.44 ( ( in @ Xx @ A )
% 2.41/1.44 => ( Xphi @ Xx ) )
% 2.41/1.44 => ? [Xx: $i] :
% 2.41/1.44 ( ( in @ Xx @ A )
% 2.41/1.44 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(exuI1_type,type,
% 2.41/1.44 exuI1: $o ).
% 2.41/1.44
% 2.41/1.44 thf(exuI1,definition,
% 2.41/1.44 ( exuI1
% 2.41/1.44 = ( ! [Xphi: $i > $o] :
% 2.41/1.44 ( ? [Xx: $i] :
% 2.41/1.44 ( ( Xphi @ Xx )
% 2.41/1.44 & ! [Xy: $i] :
% 2.41/1.44 ( ( Xphi @ Xy )
% 2.41/1.44 => ( Xx = Xy ) ) )
% 2.41/1.44 => ( exu
% 2.41/1.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(exuI3_type,type,
% 2.41/1.44 exuI3: $o ).
% 2.41/1.44
% 2.41/1.44 thf(exuI3,definition,
% 2.41/1.44 ( exuI3
% 2.41/1.44 = ( ! [Xphi: $i > $o] :
% 2.41/1.44 ( ? [Xx: $i] : ( Xphi @ Xx )
% 2.41/1.44 => ( ! [Xx: $i,Xy: $i] :
% 2.41/1.44 ( ( Xphi @ Xx )
% 2.41/1.44 => ( ( Xphi @ Xy )
% 2.41/1.44 => ( Xx = Xy ) ) )
% 2.41/1.44 => ( exu
% 2.41/1.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(exuI2_type,type,
% 2.41/1.44 exuI2: $o ).
% 2.41/1.44
% 2.41/1.44 thf(exuI2,definition,
% 2.41/1.44 ( exuI2
% 2.41/1.44 = ( ! [Xphi: $i > $o] :
% 2.41/1.44 ( ? [Xx: $i] :
% 2.41/1.44 ! [Xy: $i] :
% 2.41/1.44 ( ( Xphi @ Xy )
% 2.41/1.44 <=> ( Xy = Xx ) )
% 2.41/1.44 => ( exu
% 2.41/1.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(inCongP_type,type,
% 2.41/1.44 inCongP: $o ).
% 2.41/1.44
% 2.41/1.44 thf(inCongP,definition,
% 2.41/1.44 ( inCongP
% 2.41/1.44 = ( ! [A: $i,B: $i] :
% 2.41/1.44 ( ( A = B )
% 2.41/1.44 => ! [Xx: $i,Xy: $i] :
% 2.41/1.44 ( ( Xx = Xy )
% 2.41/1.44 => ( ( in @ Xx @ A )
% 2.41/1.44 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(in__Cong_type,type,
% 2.41/1.44 in__Cong: $o ).
% 2.41/1.44
% 2.41/1.44 thf(in__Cong,definition,
% 2.41/1.44 ( in__Cong
% 2.41/1.44 = ( ! [A: $i,B: $i] :
% 2.41/1.44 ( ( A = B )
% 2.41/1.44 => ! [Xx: $i,Xy: $i] :
% 2.41/1.44 ( ( Xx = Xy )
% 2.41/1.44 => ( ( in @ Xx @ A )
% 2.41/1.44 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(exuE3u_type,type,
% 2.41/1.44 exuE3u: $o ).
% 2.41/1.44
% 2.41/1.44 thf(exuE3u,definition,
% 2.41/1.44 ( exuE3u
% 2.41/1.44 = ( ! [Xphi: $i > $o] :
% 2.41/1.44 ( ( exu
% 2.41/1.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.41/1.44 => ! [Xx: $i,Xy: $i] :
% 2.41/1.44 ( ( Xphi @ Xx )
% 2.41/1.44 => ( ( Xphi @ Xy )
% 2.41/1.44 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(exu__Cong_type,type,
% 2.41/1.44 exu__Cong: $o ).
% 2.41/1.44
% 2.41/1.44 thf(exu__Cong,definition,
% 2.41/1.44 ( exu__Cong
% 2.41/1.44 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.41/1.44 ( ! [Xx: $i,Xy: $i] :
% 2.41/1.44 ( ( Xx = Xy )
% 2.41/1.44 => ( ( Xphi @ Xx )
% 2.41/1.44 <=> ( Xpsi @ Xy ) ) )
% 2.41/1.44 => ( ( exu
% 2.41/1.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.41/1.44 <=> ( exu
% 2.41/1.44 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(emptyset__Cong_type,type,
% 2.41/1.44 emptyset__Cong: $o ).
% 2.41/1.44
% 2.41/1.44 thf(emptyset__Cong,definition,
% 2.41/1.44 ( emptyset__Cong
% 2.41/1.44 = ( emptyset = emptyset ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(setadjoin__Cong_type,type,
% 2.41/1.44 setadjoin__Cong: $o ).
% 2.41/1.44
% 2.41/1.44 thf(setadjoin__Cong,definition,
% 2.41/1.44 ( setadjoin__Cong
% 2.41/1.44 = ( ! [Xx: $i,Xy: $i] :
% 2.41/1.44 ( ( Xx = Xy )
% 2.41/1.44 => ! [Xz: $i,Xu: $i] :
% 2.41/1.44 ( ( Xz = Xu )
% 2.41/1.44 => ( ( setadjoin @ Xx @ Xz )
% 2.41/1.44 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(powerset__Cong_type,type,
% 2.41/1.44 powerset__Cong: $o ).
% 2.41/1.44
% 2.41/1.44 thf(powerset__Cong,definition,
% 2.41/1.44 ( powerset__Cong
% 2.41/1.44 = ( ! [A: $i,B: $i] :
% 2.41/1.44 ( ( A = B )
% 2.41/1.44 => ( ( powerset @ A )
% 2.41/1.44 = ( powerset @ B ) ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(setunion__Cong_type,type,
% 2.41/1.44 setunion__Cong: $o ).
% 2.41/1.44
% 2.41/1.44 thf(setunion__Cong,definition,
% 2.41/1.44 ( setunion__Cong
% 2.41/1.44 = ( ! [A: $i,B: $i] :
% 2.41/1.44 ( ( A = B )
% 2.41/1.44 => ( ( setunion @ A )
% 2.41/1.44 = ( setunion @ B ) ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(omega__Cong_type,type,
% 2.41/1.44 omega__Cong: $o ).
% 2.41/1.44
% 2.41/1.44 thf(omega__Cong,definition,
% 2.41/1.44 ( omega__Cong
% 2.41/1.44 = ( omega = omega ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(exuEu_type,type,
% 2.41/1.44 exuEu: $o ).
% 2.41/1.44
% 2.41/1.44 thf(exuEu,definition,
% 2.41/1.44 ( exuEu
% 2.41/1.44 = ( ! [Xphi: $i > $o] :
% 2.41/1.44 ( ( exu
% 2.41/1.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.41/1.44 => ! [Xx: $i,Xy: $i] :
% 2.41/1.44 ( ( Xphi @ Xx )
% 2.41/1.44 => ( ( Xphi @ Xy )
% 2.41/1.44 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(descr__Cong_type,type,
% 2.41/1.44 descr__Cong: $o ).
% 2.41/1.44
% 2.41/1.44 thf(descr__Cong,definition,
% 2.41/1.44 ( descr__Cong
% 2.41/1.44 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.41/1.44 ( ! [Xx: $i,Xy: $i] :
% 2.41/1.44 ( ( Xx = Xy )
% 2.41/1.44 => ( ( Xphi @ Xx )
% 2.41/1.44 <=> ( Xpsi @ Xy ) ) )
% 2.41/1.44 => ( ( exu
% 2.41/1.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.41/1.44 => ( ( exu
% 2.41/1.44 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 2.41/1.44 => ( ( descr
% 2.41/1.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.41/1.44 = ( descr
% 2.41/1.44 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(dsetconstr__Cong_type,type,
% 2.41/1.44 dsetconstr__Cong: $o ).
% 2.41/1.44
% 2.41/1.44 thf(dsetconstr__Cong,definition,
% 2.41/1.44 ( dsetconstr__Cong
% 2.41/1.44 = ( ! [A: $i,B: $i] :
% 2.41/1.44 ( ( A = B )
% 2.41/1.44 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.41/1.44 ( ! [Xx: $i] :
% 2.41/1.44 ( ( in @ Xx @ A )
% 2.41/1.44 => ! [Xy: $i] :
% 2.41/1.44 ( ( in @ Xy @ B )
% 2.41/1.44 => ( ( Xx = Xy )
% 2.41/1.44 => ( ( Xphi @ Xx )
% 2.41/1.44 <=> ( Xpsi @ Xy ) ) ) ) )
% 2.41/1.44 => ( ( dsetconstr @ A
% 2.41/1.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.41/1.44 = ( dsetconstr @ B
% 2.41/1.44 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(subset_type,type,
% 2.41/1.44 subset: $i > $i > $o ).
% 2.41/1.44
% 2.41/1.44 thf(disjoint_type,type,
% 2.41/1.44 disjoint: $i > $i > $o ).
% 2.41/1.44
% 2.41/1.44 thf(setsmeet_type,type,
% 2.41/1.44 setsmeet: $i > $i > $o ).
% 2.41/1.44
% 2.41/1.44 thf(subsetI1_type,type,
% 2.41/1.44 subsetI1: $o ).
% 2.41/1.44
% 2.41/1.44 thf(subsetI1,definition,
% 2.41/1.44 ( subsetI1
% 2.41/1.44 = ( ! [A: $i,B: $i] :
% 2.41/1.44 ( ! [Xx: $i] :
% 2.41/1.44 ( ( in @ Xx @ A )
% 2.41/1.44 => ( in @ Xx @ B ) )
% 2.41/1.44 => ( subset @ A @ B ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(eqimpsubset2_type,type,
% 2.41/1.44 eqimpsubset2: $o ).
% 2.41/1.44
% 2.41/1.44 thf(eqimpsubset2,definition,
% 2.41/1.44 ( eqimpsubset2
% 2.41/1.44 = ( ! [A: $i,B: $i] :
% 2.41/1.44 ( ( A = B )
% 2.41/1.44 => ( subset @ B @ A ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(eqimpsubset1_type,type,
% 2.41/1.44 eqimpsubset1: $o ).
% 2.41/1.44
% 2.41/1.44 thf(eqimpsubset1,definition,
% 2.41/1.44 ( eqimpsubset1
% 2.41/1.44 = ( ! [A: $i,B: $i] :
% 2.41/1.44 ( ( A = B )
% 2.41/1.44 => ( subset @ A @ B ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(subsetI2_type,type,
% 2.41/1.44 subsetI2: $o ).
% 2.41/1.44
% 2.41/1.44 thf(subsetI2,definition,
% 2.41/1.44 ( subsetI2
% 2.41/1.44 = ( ! [A: $i,B: $i] :
% 2.41/1.44 ( ! [Xx: $i] :
% 2.41/1.44 ( ( in @ Xx @ A )
% 2.41/1.44 => ( in @ Xx @ B ) )
% 2.41/1.44 => ( subset @ A @ B ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(emptysetsubset_type,type,
% 2.41/1.44 emptysetsubset: $o ).
% 2.41/1.44
% 2.41/1.44 thf(emptysetsubset,definition,
% 2.41/1.44 ( emptysetsubset
% 2.41/1.44 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(subsetE_type,type,
% 2.41/1.44 subsetE: $o ).
% 2.41/1.44
% 2.41/1.44 thf(subsetE,definition,
% 2.41/1.44 ( subsetE
% 2.41/1.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.41/1.44 ( ( subset @ A @ B )
% 2.41/1.44 => ( ( in @ Xx @ A )
% 2.41/1.44 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(subsetE2_type,type,
% 2.41/1.44 subsetE2: $o ).
% 2.41/1.44
% 2.41/1.44 thf(subsetE2,definition,
% 2.41/1.44 ( subsetE2
% 2.41/1.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.41/1.44 ( ( subset @ A @ B )
% 2.41/1.44 => ( ~ ( in @ Xx @ B )
% 2.41/1.44 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(notsubsetI_type,type,
% 2.41/1.44 notsubsetI: $o ).
% 2.41/1.44
% 2.41/1.44 thf(notsubsetI,definition,
% 2.41/1.44 ( notsubsetI
% 2.41/1.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.41/1.44 ( ( in @ Xx @ A )
% 2.41/1.44 => ( ~ ( in @ Xx @ B )
% 2.41/1.44 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(notequalI1_type,type,
% 2.41/1.44 notequalI1: $o ).
% 2.41/1.44
% 2.41/1.44 thf(notequalI1,definition,
% 2.41/1.44 ( notequalI1
% 2.41/1.44 = ( ! [A: $i,B: $i] :
% 2.41/1.44 ( ~ ( subset @ A @ B )
% 2.41/1.44 => ( A != B ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(notequalI2_type,type,
% 2.41/1.44 notequalI2: $o ).
% 2.41/1.44
% 2.41/1.44 thf(notequalI2,definition,
% 2.41/1.44 ( notequalI2
% 2.41/1.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.41/1.44 ( ( in @ Xx @ A )
% 2.41/1.44 => ( ~ ( in @ Xx @ B )
% 2.41/1.44 => ( A != B ) ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(subsetRefl_type,type,
% 2.41/1.44 subsetRefl: $o ).
% 2.41/1.44
% 2.41/1.44 thf(subsetRefl,definition,
% 2.41/1.44 ( subsetRefl
% 2.41/1.44 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(subsetTrans_type,type,
% 2.41/1.44 subsetTrans: $o ).
% 2.41/1.44
% 2.41/1.44 thf(subsetTrans,definition,
% 2.41/1.44 ( subsetTrans
% 2.41/1.44 = ( ! [A: $i,B: $i,C: $i] :
% 2.41/1.44 ( ( subset @ A @ B )
% 2.41/1.44 => ( ( subset @ B @ C )
% 2.41/1.44 => ( subset @ A @ C ) ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(setadjoinSub_type,type,
% 2.41/1.44 setadjoinSub: $o ).
% 2.41/1.44
% 2.41/1.44 thf(setadjoinSub,definition,
% 2.41/1.44 ( setadjoinSub
% 2.41/1.44 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(setadjoinSub2_type,type,
% 2.41/1.44 setadjoinSub2: $o ).
% 2.41/1.44
% 2.41/1.44 thf(setadjoinSub2,definition,
% 2.41/1.44 ( setadjoinSub2
% 2.41/1.44 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.41/1.44 ( ( subset @ A @ B )
% 2.41/1.44 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(subset2powerset_type,type,
% 2.41/1.44 subset2powerset: $o ).
% 2.41/1.44
% 2.41/1.44 thf(subset2powerset,definition,
% 2.41/1.44 ( subset2powerset
% 2.41/1.44 = ( ! [A: $i,B: $i] :
% 2.41/1.44 ( ( subset @ A @ B )
% 2.41/1.44 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(setextsub_type,type,
% 2.41/1.44 setextsub: $o ).
% 2.41/1.44
% 2.41/1.44 thf(setextsub,definition,
% 2.41/1.44 ( setextsub
% 2.41/1.44 = ( ! [A: $i,B: $i] :
% 2.41/1.44 ( ( subset @ A @ B )
% 2.41/1.44 => ( ( subset @ B @ A )
% 2.41/1.44 => ( A = B ) ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(subsetemptysetimpeq_type,type,
% 2.41/1.44 subsetemptysetimpeq: $o ).
% 2.41/1.44
% 2.41/1.44 thf(subsetemptysetimpeq,definition,
% 2.41/1.44 ( subsetemptysetimpeq
% 2.41/1.44 = ( ! [A: $i] :
% 2.41/1.44 ( ( subset @ A @ emptyset )
% 2.41/1.44 => ( A = emptyset ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(powersetI1_type,type,
% 2.41/1.44 powersetI1: $o ).
% 2.41/1.44
% 2.41/1.44 thf(powersetI1,definition,
% 2.41/1.44 ( powersetI1
% 2.41/1.44 = ( ! [A: $i,B: $i] :
% 2.41/1.44 ( ( subset @ B @ A )
% 2.41/1.44 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(powersetE1_type,type,
% 2.41/1.44 powersetE1: $o ).
% 2.41/1.44
% 2.41/1.44 thf(powersetE1,definition,
% 2.41/1.44 ( powersetE1
% 2.41/1.44 = ( ! [A: $i,B: $i] :
% 2.41/1.44 ( ( in @ B @ ( powerset @ A ) )
% 2.41/1.44 => ( subset @ B @ A ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(inPowerset_type,type,
% 2.41/1.44 inPowerset: $o ).
% 2.41/1.44
% 2.41/1.44 thf(inPowerset,definition,
% 2.41/1.44 ( inPowerset
% 2.41/1.44 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(powersetsubset_type,type,
% 2.41/1.44 powersetsubset: $o ).
% 2.41/1.44
% 2.41/1.44 thf(powersetsubset,definition,
% 2.41/1.44 ( powersetsubset
% 2.41/1.44 = ( ! [A: $i,B: $i] :
% 2.41/1.44 ( ( subset @ A @ B )
% 2.41/1.44 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(sepInPowerset_type,type,
% 2.41/1.44 sepInPowerset: $o ).
% 2.41/1.44
% 2.41/1.44 thf(sepInPowerset,definition,
% 2.41/1.44 ( sepInPowerset
% 2.41/1.44 = ( ! [A: $i,Xphi: $i > $o] :
% 2.41/1.44 ( in
% 2.41/1.44 @ ( dsetconstr @ A
% 2.41/1.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.41/1.44 @ ( powerset @ A ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(sepSubset_type,type,
% 2.41/1.44 sepSubset: $o ).
% 2.41/1.44
% 2.41/1.44 thf(sepSubset,definition,
% 2.41/1.44 ( sepSubset
% 2.41/1.44 = ( ! [A: $i,Xphi: $i > $o] :
% 2.41/1.44 ( subset
% 2.41/1.44 @ ( dsetconstr @ A
% 2.41/1.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.41/1.44 @ A ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(binunion_type,type,
% 2.41/1.44 binunion: $i > $i > $i ).
% 2.41/1.44
% 2.41/1.44 thf(binunion,definition,
% 2.41/1.44 ( binunion
% 2.41/1.44 = ( ^ [Xx: $i,Xy: $i] : ( setunion @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(binunionIL_type,type,
% 2.41/1.44 binunionIL: $o ).
% 2.41/1.44
% 2.41/1.44 thf(binunionIL,definition,
% 2.41/1.44 ( binunionIL
% 2.41/1.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.41/1.44 ( ( in @ Xx @ A )
% 2.41/1.44 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.41/1.44
% 2.41/1.44 thf(upairset2IR,conjecture,
% 2.41/1.44 ( setextAx
% 2.41/1.44 => ( emptysetAx
% 2.41/1.44 => ( setadjoinAx
% 2.41/1.44 => ( powersetAx
% 2.41/1.44 => ( setunionAx
% 2.41/1.44 => ( omega0Ax
% 2.41/1.44 => ( omegaSAx
% 2.41/1.44 => ( omegaIndAx
% 2.41/1.44 => ( replAx
% 2.41/1.44 => ( foundationAx
% 2.41/1.44 => ( wellorderingAx
% 2.41/1.44 => ( descrp
% 2.41/1.44 => ( dsetconstrI
% 2.41/1.44 => ( dsetconstrEL
% 2.41/1.44 => ( dsetconstrER
% 2.41/1.44 => ( exuE1
% 2.41/1.44 => ( prop2setE
% 2.41/1.44 => ( emptysetE
% 2.41/1.44 => ( emptysetimpfalse
% 2.41/1.44 => ( notinemptyset
% 2.41/1.44 => ( exuE3e
% 2.41/1.44 => ( setext
% 2.41/1.44 => ( emptyI
% 2.41/1.44 => ( noeltsimpempty
% 2.41/1.44 => ( setbeta
% 2.41/1.44 => ( nonemptyE1
% 2.41/1.44 => ( nonemptyI
% 2.41/1.44 => ( nonemptyI1
% 2.41/1.44 => ( setadjoinIL
% 2.41/1.44 => ( emptyinunitempty
% 2.41/1.44 => ( setadjoinIR
% 2.41/1.44 => ( setadjoinE
% 2.41/1.44 => ( setadjoinOr
% 2.41/1.44 => ( setoftrueEq
% 2.41/1.44 => ( powersetI
% 2.41/1.44 => ( emptyinPowerset
% 2.41/1.44 => ( emptyInPowerset
% 2.41/1.44 => ( powersetE
% 2.41/1.44 => ( setunionI
% 2.41/1.44 => ( setunionE
% 2.41/1.44 => ( subPowSU
% 2.41/1.44 => ( exuE2
% 2.41/1.44 => ( nonemptyImpWitness
% 2.41/1.44 => ( uniqinunit
% 2.41/1.44 => ( notinsingleton
% 2.41/1.44 => ( eqinunit
% 2.41/1.44 => ( singletonsswitch
% 2.41/1.44 => ( upairsetE
% 2.41/1.44 => ( upairsetIL
% 2.41/1.44 => ( upairsetIR
% 2.41/1.44 => ( emptyE1
% 2.41/1.44 => ( vacuousDall
% 2.41/1.44 => ( quantDeMorgan1
% 2.41/1.44 => ( quantDeMorgan2
% 2.41/1.44 => ( quantDeMorgan3
% 2.41/1.44 => ( quantDeMorgan4
% 2.41/1.44 => ( prop2setI
% 2.41/1.44 => ( prop2set2propI
% 2.41/1.44 => ( notdexE
% 2.41/1.44 => ( notdallE
% 2.41/1.44 => ( exuI1
% 2.41/1.44 => ( exuI3
% 2.41/1.44 => ( exuI2
% 2.41/1.44 => ( inCongP
% 2.41/1.44 => ( in__Cong
% 2.41/1.44 => ( exuE3u
% 2.41/1.44 => ( exu__Cong
% 2.41/1.44 => ( emptyset__Cong
% 2.41/1.44 => ( setadjoin__Cong
% 2.41/1.44 => ( powerset__Cong
% 2.41/1.44 => ( setunion__Cong
% 2.41/1.44 => ( omega__Cong
% 2.41/1.44 => ( exuEu
% 2.41/1.44 => ( descr__Cong
% 2.41/1.44 => ( dsetconstr__Cong
% 2.41/1.44 => ( subsetI1
% 2.41/1.44 => ( eqimpsubset2
% 2.41/1.44 => ( eqimpsubset1
% 2.41/1.44 => ( subsetI2
% 2.41/1.44 => ( emptysetsubset
% 2.41/1.44 => ( subsetE
% 2.41/1.44 => ( subsetE2
% 2.41/1.44 => ( notsubsetI
% 2.41/1.44 => ( notequalI1
% 2.41/1.44 => ( notequalI2
% 2.41/1.44 => ( subsetRefl
% 2.41/1.44 => ( subsetTrans
% 2.41/1.44 => ( setadjoinSub
% 2.41/1.44 => ( setadjoinSub2
% 2.41/1.44 => ( subset2powerset
% 2.41/1.44 => ( setextsub
% 2.41/1.44 => ( subsetemptysetimpeq
% 2.41/1.44 => ( powersetI1
% 2.41/1.44 => ( powersetE1
% 2.41/1.44 => ( inPowerset
% 2.41/1.44 => ( powersetsubset
% 2.41/1.44 => ( sepInPowerset
% 2.41/1.44 => ( sepSubset
% 2.41/1.44 => ( binunionIL
% 2.41/1.44 => ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.41/1.44 ------------------------
% 2.41/1.46 % (6970)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_6747 for (2999ds/2Mi)
% 2.41/1.46 % (6971)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_6747 for (2999ds/275Mi)
% 2.41/1.46 % (6970)Instruction limit reached!
% 2.41/1.46 % (6970)------------------------------
% 2.41/1.46 % (6970)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.41/1.46 % (6970)Termination reason: Unknown
% 2.41/1.46 % (6970)Termination phase: shuffling
% 2.41/1.46
% 2.41/1.46 % (6970)Memory used [KB]: 1279
% 2.41/1.46 % (6970)Time elapsed: 0.003 s
% 2.41/1.46 % (6970)Instructions burned: 4 (million)
% 2.41/1.46 % (6970)------------------------------
% 2.41/1.46 % (6970)------------------------------
% 2.41/1.47 % (6966)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_6747 for (2999ds/183Mi)
% 2.41/1.47 % (6967)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_6747 for (2999ds/4Mi)
% 2.41/1.47 % (6969)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_6747 for (2999ds/2Mi)
% 2.41/1.47 % (6968)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_6747 for (2999ds/27Mi)
% 2.41/1.47 % (6972)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_6747 for (2999ds/18Mi)
% 2.41/1.47 % (6969)Instruction limit reached!
% 2.41/1.47 % (6969)------------------------------
% 2.41/1.47 % (6969)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.41/1.47 % (6969)Termination reason: Unknown
% 2.41/1.47 % (6969)Termination phase: shuffling
% 2.41/1.47
% 2.41/1.47 % (6969)Memory used [KB]: 1151
% 2.41/1.47 % (6969)Time elapsed: 0.004 s
% 2.41/1.47 % (6969)Instructions burned: 3 (million)
% 2.41/1.47 % (6969)------------------------------
% 2.41/1.47 % (6969)------------------------------
% 2.41/1.47 % (6967)Instruction limit reached!
% 2.41/1.47 % (6967)------------------------------
% 2.41/1.47 % (6967)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.41/1.47 % (6967)Termination reason: Unknown
% 2.41/1.47 % (6967)Termination phase: shuffling
% 2.41/1.47
% 2.41/1.47 % (6967)Memory used [KB]: 1279
% 2.41/1.47 % (6967)Time elapsed: 0.005 s
% 2.41/1.47 % (6967)Instructions burned: 5 (million)
% 2.41/1.47 % (6967)------------------------------
% 2.41/1.47 % (6967)------------------------------
% 2.41/1.47 % (6973)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_6747 for (2999ds/3Mi)
% 2.41/1.47 % (6973)Instruction limit reached!
% 2.41/1.47 % (6973)------------------------------
% 2.41/1.47 % (6973)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.41/1.47 % (6973)Termination reason: Unknown
% 2.41/1.47 % (6973)Termination phase: shuffling
% 2.41/1.47
% 2.41/1.47 % (6973)Memory used [KB]: 1151
% 2.41/1.47 % (6973)Time elapsed: 0.002 s
% 2.41/1.47 % (6973)Instructions burned: 3 (million)
% 2.41/1.47 % (6973)------------------------------
% 2.41/1.47 % (6973)------------------------------
% 2.41/1.48 % (6972)Instruction limit reached!
% 2.41/1.48 % (6972)------------------------------
% 2.41/1.48 % (6972)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.41/1.48 % (6972)Termination reason: Unknown
% 2.41/1.48 % (6972)Termination phase: Preprocessing 3
% 2.41/1.48
% 2.41/1.48 % (6972)Memory used [KB]: 1535
% 2.41/1.48 % (6972)Time elapsed: 0.013 s
% 2.41/1.48 % (6972)Instructions burned: 18 (million)
% 2.41/1.48 % (6972)------------------------------
% 2.41/1.48 % (6972)------------------------------
% 2.41/1.48 % (6968)Instruction limit reached!
% 2.41/1.48 % (6968)------------------------------
% 2.41/1.48 % (6968)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.41/1.48 % (6968)Termination reason: Unknown
% 2.41/1.48 % (6968)Termination phase: Property scanning
% 2.41/1.48
% 2.41/1.48 % (6968)Memory used [KB]: 1535
% 2.41/1.48 % (6968)Time elapsed: 0.016 s
% 2.41/1.48 % (6968)Instructions burned: 27 (million)
% 2.41/1.48 % (6968)------------------------------
% 2.41/1.48 % (6968)------------------------------
% 2.41/1.48 % (6976)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_6747 for (2999ds/3Mi)
% 2.41/1.48 % (6976)Instruction limit reached!
% 2.41/1.48 % (6976)------------------------------
% 2.41/1.48 % (6976)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.41/1.48 % (6976)Termination reason: Unknown
% 2.41/1.48 % (6976)Termination phase: shuffling
% 2.41/1.48
% 2.41/1.48 % (6976)Memory used [KB]: 1279
% 2.41/1.48 % (6976)Time elapsed: 0.003 s
% 2.41/1.48 % (6976)Instructions burned: 6 (million)
% 2.41/1.48 % (6976)------------------------------
% 2.41/1.48 % (6976)------------------------------
% 2.41/1.49 % (6974)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_6747 for (2999ds/37Mi)
% 2.41/1.49 % (6975)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_6747 for (2999ds/15Mi)
% 2.41/1.49 % (6979)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_6747 for (2999ds/16Mi)
% 2.45/1.50 % (6977)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_6747 for (2999ds/1041Mi)
% 2.45/1.50 % (6975)Instruction limit reached!
% 2.45/1.50 % (6975)------------------------------
% 2.45/1.50 % (6975)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.45/1.50 % (6975)Termination reason: Unknown
% 2.45/1.50 % (6975)Termination phase: Naming
% 2.45/1.50
% 2.45/1.50 % (6975)Memory used [KB]: 1407
% 2.45/1.50 % (6975)Time elapsed: 0.011 s
% 2.45/1.50 % (6975)Instructions burned: 15 (million)
% 2.45/1.50 % (6975)------------------------------
% 2.45/1.50 % (6975)------------------------------
% 2.45/1.50 % (6979)Instruction limit reached!
% 2.45/1.50 % (6979)------------------------------
% 2.45/1.50 % (6979)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.45/1.50 % (6979)Termination reason: Unknown
% 2.45/1.50 % (6979)Termination phase: Property scanning
% 2.45/1.50
% 2.45/1.50 % (6979)Memory used [KB]: 1407
% 2.45/1.50 % (6979)Time elapsed: 0.006 s
% 2.45/1.50 % (6979)Instructions burned: 19 (million)
% 2.45/1.50 % (6979)------------------------------
% 2.45/1.50 % (6979)------------------------------
% 2.45/1.50 % (6978)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_6747 for (2999ds/7Mi)
% 2.45/1.50 % (6978)Instruction limit reached!
% 2.45/1.50 % (6978)------------------------------
% 2.45/1.50 % (6978)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.45/1.50 % (6978)Termination reason: Unknown
% 2.45/1.50 % (6978)Termination phase: shuffling
% 2.45/1.50
% 2.45/1.50 % (6978)Memory used [KB]: 1279
% 2.45/1.50 % (6978)Time elapsed: 0.006 s
% 2.45/1.50 % (6978)Instructions burned: 8 (million)
% 2.45/1.50 % (6978)------------------------------
% 2.45/1.50 % (6978)------------------------------
% 2.45/1.51 % (6974)Instruction limit reached!
% 2.45/1.51 % (6974)------------------------------
% 2.45/1.51 % (6974)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.45/1.51 % (6974)Termination reason: Unknown
% 2.45/1.51 % (6974)Termination phase: Function definition elimination
% 2.45/1.51
% 2.45/1.51 % (6974)Memory used [KB]: 1535
% 2.45/1.51 % (6974)Time elapsed: 0.019 s
% 2.45/1.51 % (6974)Instructions burned: 37 (million)
% 2.45/1.51 % (6974)------------------------------
% 2.45/1.51 % (6974)------------------------------
% 2.45/1.51 % (6981)lrs+2_1:1_apa=on:au=on:bd=preordered:cnfonf=off:cs=on:ixr=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_6747 for (2999ds/3Mi)
% 2.45/1.51 % (6981)Instruction limit reached!
% 2.45/1.51 % (6981)------------------------------
% 2.45/1.51 % (6981)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.45/1.51 % (6981)Termination reason: Unknown
% 2.45/1.51 % (6981)Termination phase: shuffling
% 2.45/1.51
% 2.45/1.51 % (6981)Memory used [KB]: 1151
% 2.45/1.51 % (6981)Time elapsed: 0.002 s
% 2.45/1.51 % (6981)Instructions burned: 4 (million)
% 2.45/1.51 % (6981)------------------------------
% 2.45/1.51 % (6981)------------------------------
% 2.45/1.52 % (6980)lrs+21_1:1_au=on:cnfonf=off:fd=preordered:fe=off:fsr=off:hud=11:inj=on:kws=precedence:s2pl=no:sp=weighted_frequency:tgt=full:i=3:si=on:rtra=on_0 on DTF2THF_6747 for (2999ds/3Mi)
% 2.45/1.52 % (6980)Instruction limit reached!
% 2.45/1.52 % (6980)------------------------------
% 2.45/1.52 % (6980)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.45/1.52 % (6980)Termination reason: Unknown
% 2.45/1.52 % (6980)Termination phase: shuffling
% 2.45/1.52
% 2.45/1.52 % (6980)Memory used [KB]: 1279
% 2.45/1.52 % (6980)Time elapsed: 0.004 s
% 2.45/1.52 % (6980)Instructions burned: 5 (million)
% 2.45/1.52 % (6980)------------------------------
% 2.45/1.52 % (6980)------------------------------
% 2.45/1.52 % (6983)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_6747 for (2999ds/3Mi)
% 2.45/1.52 % (6984)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_6747 for (2999ds/4Mi)
% 2.45/1.52 % (6983)Instruction limit reached!
% 2.45/1.52 % (6983)------------------------------
% 2.45/1.52 % (6983)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.45/1.52 % (6983)Termination reason: Unknown
% 2.45/1.52 % (6983)Termination phase: shuffling
% 2.45/1.52
% 2.45/1.52 % (6983)Memory used [KB]: 1279
% 2.45/1.52 % (6983)Time elapsed: 0.003 s
% 2.45/1.52 % (6983)Instructions burned: 5 (million)
% 2.45/1.52 % (6983)------------------------------
% 2.45/1.52 % (6983)------------------------------
% 2.45/1.52 % (6984)Instruction limit reached!
% 2.45/1.52 % (6984)------------------------------
% 2.45/1.52 % (6984)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.45/1.52 % (6984)Termination reason: Unknown
% 2.45/1.52 % (6984)Termination phase: shuffling
% 2.45/1.52
% 2.45/1.52 % (6984)Memory used [KB]: 1279
% 2.45/1.52 % (6984)Time elapsed: 0.003 s
% 2.45/1.52 % (6984)Instructions burned: 6 (million)
% 2.45/1.52 % (6984)------------------------------
% 2.45/1.52 % (6984)------------------------------
% 2.45/1.52 % (6982)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_6747 for (2999ds/7Mi)
% 2.45/1.52 % (6966)First to succeed.
% 2.45/1.52 % (6966)Refutation found. Thanks to Tanya!
% 2.45/1.52 % SZS status Theorem for DTF2THF_6747
% 2.45/1.52 % SZS output start Proof for DTF2THF_6747
% 2.45/1.52 thf(func_def_0, type, in: $i > $i > $o).
% 2.45/1.52 thf(func_def_1, type, exu: ($i > $o) > $o).
% 2.45/1.52 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 2.45/1.52 thf(func_def_8, type, powerset: $i > $i).
% 2.45/1.52 thf(func_def_10, type, setunion: $i > $i).
% 2.45/1.52 thf(func_def_19, type, descr: ($i > $o) > $i).
% 2.45/1.52 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 2.45/1.52 thf(func_def_26, type, prop2set: $o > $i).
% 2.45/1.52 thf(func_def_36, type, nonempty: $i > $o).
% 2.45/1.52 thf(func_def_69, type, set2prop: $i > $o).
% 2.45/1.52 thf(func_def_88, type, subset: $i > $i > $o).
% 2.45/1.52 thf(func_def_89, type, disjoint: $i > $i > $o).
% 2.45/1.52 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 2.45/1.52 thf(func_def_114, type, binunion: $i > $i > $i).
% 2.45/1.52 thf(func_def_136, type, sK3: $i > $i).
% 2.45/1.52 thf(func_def_137, type, sK4: $i > $i > $i).
% 2.45/1.52 thf(func_def_138, type, sK5: ($i > $o) > $i > $i).
% 2.45/1.52 thf(func_def_139, type, sK6: ($i > $o) > $i).
% 2.45/1.52 thf(func_def_140, type, sK7: ($i > $o) > $i).
% 2.45/1.52 thf(func_def_141, type, sK8: $i > $i).
% 2.45/1.52 thf(func_def_142, type, sK9: ($i > $o) > $i).
% 2.45/1.52 thf(func_def_143, type, sK10: ($i > $o) > $i > $i).
% 2.45/1.52 thf(func_def_144, type, sK11: $i > $i > $i).
% 2.45/1.52 thf(func_def_145, type, sK12: ($i > $o) > ($i > $o) > $i).
% 2.45/1.52 thf(func_def_146, type, sK13: ($i > $o) > ($i > $o) > $i).
% 2.45/1.52 thf(func_def_147, type, sK14: $i > ($i > $o) > $i).
% 2.45/1.52 thf(func_def_148, type, sK15: $i > ($i > $o) > $i).
% 2.45/1.52 thf(func_def_149, type, sK16: $i > ($i > $o) > $i).
% 2.45/1.52 thf(func_def_150, type, sK17: $i > ($i > $o) > $i).
% 2.45/1.52 thf(func_def_151, type, sK18: $i > ($i > $o) > $i).
% 2.45/1.52 thf(func_def_152, type, sK19: $i > ($i > $o) > $i).
% 2.45/1.52 thf(func_def_153, type, sK20: $i > ($i > $o) > $i).
% 2.45/1.52 thf(func_def_154, type, sK21: $i > ($i > $o) > $i).
% 2.45/1.52 thf(func_def_155, type, sK22: $i > $i > $i).
% 2.45/1.52 thf(func_def_156, type, sK23: ($i > $o) > $i > $i).
% 2.45/1.52 thf(func_def_157, type, sK24: ($i > $o) > ($i > $o) > $i).
% 2.45/1.52 thf(func_def_158, type, sK25: ($i > $o) > ($i > $o) > $i).
% 2.45/1.52 thf(func_def_159, type, sK26: ($i > $o) > $i).
% 2.45/1.52 thf(func_def_160, type, sK27: $i > ($i > $o) > $i).
% 2.45/1.52 thf(func_def_161, type, sK28: $i > ($i > $o) > $i).
% 2.45/1.52 thf(func_def_162, type, sK29: ($i > $o) > $i).
% 2.45/1.52 thf(func_def_163, type, sK30: $i > ($i > $o) > $i).
% 2.45/1.52 thf(func_def_164, type, sK31: $i > ($i > $o) > $i).
% 2.45/1.52 thf(func_def_165, type, sK32: ($i > $o) > $i).
% 2.45/1.52 thf(func_def_166, type, sK33: $i > ($i > $o) > $i).
% 2.45/1.52 thf(func_def_167, type, sK34: $i > ($i > $o) > $i).
% 2.45/1.52 thf(func_def_168, type, sK35: $i > ($i > $o) > $i).
% 2.45/1.52 thf(func_def_169, type, sK36: ($i > $o) > $i).
% 2.45/1.52 thf(func_def_170, type, sK37: ($i > $o) > $i).
% 2.45/1.52 thf(func_def_171, type, sK38: ($i > $o) > $i).
% 2.45/1.52 thf(func_def_172, type, sK39: $i > ($i > $o) > $i).
% 2.45/1.52 thf(func_def_173, type, sK40: $i > ($i > $o) > $i).
% 2.45/1.52 thf(func_def_174, type, sK41: ($i > $o) > $i).
% 2.45/1.52 thf(func_def_175, type, sK42: ($i > $o) > $i > $i).
% 2.45/1.52 thf(func_def_176, type, sK43: $i > ($i > $o) > $i).
% 2.45/1.52 thf(func_def_177, type, sK44: $i > ($i > $o) > $i).
% 2.45/1.52 thf(func_def_178, type, sK45: ($i > $o) > $i > $i).
% 2.45/1.52 thf(func_def_179, type, sK46: $i > ($i > $i > $o) > $i).
% 2.45/1.52 thf(func_def_180, type, sK47: $i > ($i > $i > $o) > $i).
% 2.45/1.52 thf(func_def_181, type, sK48: $i > $i > ($i > $i > $o) > $i).
% 2.45/1.52 thf(func_def_182, type, sK49: $i > ($i > $i > $o) > $i).
% 2.45/1.52 thf(func_def_183, type, sK50: $i > $i > ($i > $i > $o) > $i).
% 2.45/1.52 thf(func_def_184, type, sK51: $i > $i > ($i > $i > $o) > $i).
% 2.45/1.52 thf(func_def_185, type, sK52: $i > $i > ($i > $i > $o) > $i).
% 2.45/1.52 thf(func_def_186, type, sK53: $i > $i).
% 2.45/1.52 thf(func_def_187, type, sK54: $i > $i).
% 2.45/1.52 thf(func_def_188, type, sK55: $i > $i > $i).
% 2.45/1.52 thf(func_def_189, type, sK56: $i > $i > $i).
% 2.45/1.52 thf(func_def_190, type, sK57: $i > $i > $i).
% 2.45/1.52 thf(func_def_191, type, sK58: ($i > $o) > $i).
% 2.45/1.52 thf(func_def_192, type, sK59: ($i > $o) > $i > $i).
% 2.45/1.52 thf(func_def_193, type, sK60: $i > $i > $i).
% 2.45/1.52 thf(func_def_194, type, sK61: $i > ($i > $o) > $i).
% 2.45/1.52 thf(func_def_195, type, sK62: $i > $i).
% 2.45/1.52 thf(func_def_196, type, sK63: $i > $i > $i > $i).
% 2.45/1.52 thf(func_def_197, type, sK64: $i > $i > $i).
% 2.45/1.52 thf(func_def_198, type, sK65: $i > $i > $i).
% 2.45/1.52 thf(func_def_199, type, sK66: $i > $i > $i).
% 2.45/1.52 thf(func_def_200, type, sK67: $o > $i > $i > $i).
% 2.45/1.52 thf(func_def_201, type, sK68: $i > $i).
% 2.45/1.52 thf(func_def_202, type, sK69: $i > ($i > $o) > $i).
% 2.45/1.52 thf(func_def_203, type, sK70: ($i > $o) > $i > $i).
% 2.45/1.52 thf(func_def_204, type, sK71: ($i > $o) > $i).
% 2.45/1.52 thf(func_def_205, type, sK72: ($i > $o) > $i).
% 2.45/1.52 thf(func_def_206, type, sK73: ($i > $o) > $i).
% 2.45/1.52 thf(func_def_207, type, sK74: ($i > $o) > $i).
% 2.45/1.52 thf(func_def_208, type, sK75: $i > $i > $i).
% 2.45/1.52 thf(func_def_209, type, sK76: $i > $i > $i).
% 2.45/1.52 thf(func_def_210, type, sK77: ($i > $o) > $i > $i).
% 2.45/1.52 thf(func_def_211, type, sK78: $i > ($i > $o) > $i).
% 2.45/1.52 thf(func_def_212, type, sK79: ($i > $o) > $i > $i).
% 2.45/1.52 thf(func_def_213, type, sK80: ($i > $o) > $i > ($i > $o) > $i > $i).
% 2.45/1.52 thf(func_def_214, type, sK81: $i > $i > ($i > $o) > ($i > $o) > $i).
% 2.45/1.52 thf(func_def_215, type, sK82: $i > $i).
% 2.45/1.52 thf(f1895,plain,(
% 2.45/1.52 $false),
% 2.45/1.52 inference(subsumption_resolution,[status(thm)],[f454,f1227])).
% 2.45/1.52 thf(f1227,plain,(
% 2.45/1.52 ( ! [X2 : $i,X1 : $i] : (((in @ X2 @ (setadjoin @ X1 @ (setadjoin @ X2 @ emptyset))) = $true)) )),
% 2.45/1.52 inference(beta_eta_normalization,[status(thm)],[f1226])).
% 2.45/1.52 thf(f1226,plain,(
% 2.45/1.52 ( ! [X2 : $i,X1 : $i] : (($true = ((^[Y0 : $i]: (in @ Y0 @ (setadjoin @ X1 @ (setadjoin @ Y0 @ emptyset)))) @ X2))) )),
% 2.45/1.52 inference(pi_clausification,[status(thm)],[f1225])).
% 2.45/1.52 thf(f1225,plain,(
% 2.45/1.52 ( ! [X1 : $i] : (($true = (!! @ $i @ (^[Y0 : $i]: (in @ Y0 @ (setadjoin @ X1 @ (setadjoin @ Y0 @ emptyset))))))) )),
% 2.45/1.52 inference(beta_eta_normalization,[status(thm)],[f1224])).
% 2.45/1.52 thf(f1224,plain,(
% 2.45/1.52 ( ! [X1 : $i] : ((((^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (in @ Y1 @ (setadjoin @ Y0 @ (setadjoin @ Y1 @ emptyset)))))) @ X1) = $true)) )),
% 2.45/1.52 inference(pi_clausification,[status(thm)],[f544])).
% 2.45/1.52 thf(f544,plain,(
% 2.45/1.52 ((!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (in @ Y1 @ (setadjoin @ Y0 @ (setadjoin @ Y1 @ emptyset))))))) = $true)),
% 2.45/1.52 inference(definition_unfolding,[status(thm)],[f408,f360])).
% 2.45/1.52 thf(f360,plain,(
% 2.45/1.52 (upairsetIR = (!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (in @ Y1 @ (setadjoin @ Y0 @ (setadjoin @ Y1 @ emptyset))))))))),
% 2.45/1.52 inference(cnf_transformation,[status(thm)],[f235])).
% 2.45/1.52 thf(f235,plain,(
% 2.45/1.52 (upairsetIR = (!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (in @ Y1 @ (setadjoin @ Y0 @ (setadjoin @ Y1 @ emptyset))))))))),
% 2.45/1.52 inference(fool_elimination,[status(thm)],[f234])).
% 2.45/1.52 thf(f234,plain,(
% 2.45/1.52 (upairsetIR = ! [X0 : $i,X1 : $i] : (in @ X0 @ (setadjoin @ X1 @ (setadjoin @ X0 @ emptyset))))),
% 2.45/1.52 inference(rectify,[status(thm)],[f52])).
% 2.45/1.52 thf(f52,axiom,(
% 2.45/1.52 (upairsetIR = ! [X2 : $i,X1 : $i] : (in @ X2 @ (setadjoin @ X1 @ (setadjoin @ X2 @ emptyset))))),
% 2.45/1.52 file('/export/starexec/sandbox2/tmp/tmp.KUfZ1qlWTQ/DTF2THF_6747.p',upairsetIR)).
% 2.45/1.52 thf(f408,plain,(
% 2.45/1.52 (upairsetIR = $true)),
% 2.45/1.52 inference(cnf_transformation,[status(thm)],[f309])).
% 2.45/1.52 thf(f309,plain,(
% 2.45/1.52 (vacuousDall = $true) & (emptyinunitempty = $true) & (sepInPowerset = $true) & (emptyE1 = $true) & (powerset__Cong = $true) & (emptyset__Cong = $true) & ((in @ sK1 @ (setadjoin @ sK0 @ (setadjoin @ sK1 @ emptyset))) != $true) & (notinemptyset = $true) & (setadjoinSub = $true) & (exuE3u = $true) & (dsetconstrI = $true) & (sepSubset = $true) & (nonemptyI1 = $true) & (setadjoin__Cong = $true) & (descr__Cong = $true) & (in__Cong = $true) & (setunionAx = $true) & (exu__Cong = $true) & (omegaSAx = $true) & (omegaIndAx = $true) & (emptyInPowerset = $true) & (nonemptyI = $true) & (subsetI2 = $true) & (singletonsswitch = $true) & (binunionIL = $true) & (prop2setI = $true) & (setadjoinSub2 = $true) & (setbeta = $true) & (exuE3e = $true) & (subPowSU = $true) & (nonemptyImpWitness = $true) & (notequalI1 = $true) & (notdexE = $true) & (powersetI = $true) & (emptyinPowerset = $true) & (setunionI = $true) & (setextsub = $true) & (noeltsimpempty = $true) & (subsetE2 = $true) & (exuI3 = $true) & (setadjoinE = $true) & (subsetemptysetimpeq = $true) & (emptysetsubset = $true) & (upairsetIL = $true) & (powersetsubset = $true) & (setunionE = $true) & (powersetAx = $true) & (uniqinunit = $true) & (omega0Ax = $true) & (exuE2 = $true) & (nonemptyE1 = $true) & (setext = $true) & (upairsetIR = $true) & (quantDeMorgan1 = $true) & (setadjoinIL = $true) & (descrp = $true) & (setextAx = $true) & (setadjoinAx = $true) & (powersetI1 = $true) & (subsetI1 = $true) & (prop2set2propI = $true) & (emptysetE = $true) & (emptyI = $true) & (subsetE = $true) & (upairsetE = $true) & (setunion__Cong = $true) & (notequalI2 = $true) & (dsetconstrER = $true) & (wellorderingAx = $true) & (eqinunit = $true) & (powersetE = $true) & (subsetRefl = $true) & (notinsingleton = $true) & (eqimpsubset1 = $true) & (inCongP = $true) & (omega__Cong = $true) & (subset2powerset = $true) & (replAx = $true) & (notsubsetI = $true) & (dsetconstrEL = $true) & (setadjoinOr = $true) & (exuI2 = $true) & (inPowerset = $true) & (subsetTrans = $true) & (emptysetAx = $true) & (quantDeMorgan2 = $true) & (exuEu = $true) & (notdallE = $true) & (setoftrueEq = $true) & (foundationAx = $true) & (powersetE1 = $true) & (setadjoinIR = $true) & (exuE1 = $true) & (dsetconstr__Cong = $true) & (prop2setE = $true) & (quantDeMorgan4 = $true) & (emptysetimpfalse = $true) & (eqimpsubset2 = $true) & (exuI1 = $true) & (quantDeMorgan3 = $true)),
% 2.45/1.52 inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1])],[f307,f308])).
% 2.45/1.52 thf(f308,plain,(
% 2.45/1.52 ? [X0 : $i,X1 : $i] : ($true != (in @ X1 @ (setadjoin @ X0 @ (setadjoin @ X1 @ emptyset)))) => ((in @ sK1 @ (setadjoin @ sK0 @ (setadjoin @ sK1 @ emptyset))) != $true)),
% 2.45/1.52 introduced(definition,[],[choice_axiom])).
% 2.45/1.52 thf(f307,plain,(
% 2.45/1.52 (vacuousDall = $true) & (emptyinunitempty = $true) & (sepInPowerset = $true) & (emptyE1 = $true) & (powerset__Cong = $true) & (emptyset__Cong = $true) & ? [X0 : $i,X1 : $i] : ($true != (in @ X1 @ (setadjoin @ X0 @ (setadjoin @ X1 @ emptyset)))) & (notinemptyset = $true) & (setadjoinSub = $true) & (exuE3u = $true) & (dsetconstrI = $true) & (sepSubset = $true) & (nonemptyI1 = $true) & (setadjoin__Cong = $true) & (descr__Cong = $true) & (in__Cong = $true) & (setunionAx = $true) & (exu__Cong = $true) & (omegaSAx = $true) & (omegaIndAx = $true) & (emptyInPowerset = $true) & (nonemptyI = $true) & (subsetI2 = $true) & (singletonsswitch = $true) & (binunionIL = $true) & (prop2setI = $true) & (setadjoinSub2 = $true) & (setbeta = $true) & (exuE3e = $true) & (subPowSU = $true) & (nonemptyImpWitness = $true) & (notequalI1 = $true) & (notdexE = $true) & (powersetI = $true) & (emptyinPowerset = $true) & (setunionI = $true) & (setextsub = $true) & (noeltsimpempty = $true) & (subsetE2 = $true) & (exuI3 = $true) & (setadjoinE = $true) & (subsetemptysetimpeq = $true) & (emptysetsubset = $true) & (upairsetIL = $true) & (powersetsubset = $true) & (setunionE = $true) & (powersetAx = $true) & (uniqinunit = $true) & (omega0Ax = $true) & (exuE2 = $true) & (nonemptyE1 = $true) & (setext = $true) & (upairsetIR = $true) & (quantDeMorgan1 = $true) & (setadjoinIL = $true) & (descrp = $true) & (setextAx = $true) & (setadjoinAx = $true) & (powersetI1 = $true) & (subsetI1 = $true) & (prop2set2propI = $true) & (emptysetE = $true) & (emptyI = $true) & (subsetE = $true) & (upairsetE = $true) & (setunion__Cong = $true) & (notequalI2 = $true) & (dsetconstrER = $true) & (wellorderingAx = $true) & (eqinunit = $true) & (powersetE = $true) & (subsetRefl = $true) & (notinsingleton = $true) & (eqimpsubset1 = $true) & (inCongP = $true) & (omega__Cong = $true) & (subset2powerset = $true) & (replAx = $true) & (notsubsetI = $true) & (dsetconstrEL = $true) & (setadjoinOr = $true) & (exuI2 = $true) & (inPowerset = $true) & (subsetTrans = $true) & (emptysetAx = $true) & (quantDeMorgan2 = $true) & (exuEu = $true) & (notdallE = $true) & (setoftrueEq = $true) & (foundationAx = $true) & (powersetE1 = $true) & (setadjoinIR = $true) & (exuE1 = $true) & (dsetconstr__Cong = $true) & (prop2setE = $true) & (quantDeMorgan4 = $true) & (emptysetimpfalse = $true) & (eqimpsubset2 = $true) & (exuI1 = $true) & (quantDeMorgan3 = $true)),
% 2.45/1.52 inference(flattening,[status(thm)],[f306])).
% 2.45/1.52 thf(f306,plain,(
% 2.45/1.52 ((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i,X1 : $i] : ($true != (in @ X1 @ (setadjoin @ X0 @ (setadjoin @ X1 @ emptyset)))) & (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)),
% 2.45/1.52 inference(ennf_transformation,[status(thm)],[f127])).
% 2.45/1.52 thf(f127,plain,(
% 2.45/1.52 ~((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) => ! [X0 : $i,X1 : $i] : ($true = (in @ X1 @ (setadjoin @ X0 @ (setadjoin @ X1 @ emptyset)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.45/1.52 inference(fool_elimination,[status(thm)],[f126])).
% 2.45/1.52 thf(f126,plain,(
% 2.45/1.52 ~(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 => ! [X0 : $i,X1 : $i] : (in @ X1 @ (setadjoin @ X0 @ (setadjoin @ X1 @ emptyset))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.45/1.52 inference(rectify,[status(thm)],[f104])).
% 2.45/1.52 thf(f104,negated_conjecture,(
% 2.45/1.52 ~(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 => ! [X1 : $i,X2 : $i] : (in @ X2 @ (setadjoin @ X1 @ (setadjoin @ X2 @ emptyset))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.45/1.52 inference(negated_conjecture,[status(cth)],[f103])).
% 2.45/1.52 thf(f103,conjecture,(
% 2.45/1.52 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 => ! [X1 : $i,X2 : $i] : (in @ X2 @ (setadjoin @ X1 @ (setadjoin @ X2 @ emptyset)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.45/1.52 file('/export/starexec/sandbox2/tmp/tmp.KUfZ1qlWTQ/DTF2THF_6747.p',upairset2IR)).
% 2.45/1.52 thf(f454,plain,(
% 2.45/1.52 ((in @ sK1 @ (setadjoin @ sK0 @ (setadjoin @ sK1 @ emptyset))) != $true)),
% 2.45/1.52 inference(cnf_transformation,[status(thm)],[f309])).
% 2.45/1.52 % SZS output end Proof for DTF2THF_6747
% 2.45/1.52 % (6966)------------------------------
% 2.45/1.52 % (6966)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.45/1.52 % (6966)Termination reason: Refutation
% 2.45/1.52
% 2.45/1.52 % (6966)Memory used [KB]: 6908
% 2.45/1.52 % (6966)Time elapsed: 0.056 s
% 2.45/1.52 % (6966)Instructions burned: 139 (million)
% 2.45/1.52 % (6966)------------------------------
% 2.45/1.52 % (6966)------------------------------
% 2.45/1.52 % (6965)Success in time 0.068 s
%------------------------------------------------------------------------------