%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU561^1 : TPTP v9.2.1. Released v3.7.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n023.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 09:35:16 AM UTC 2026
% Result : Theorem 2.32s 1.51s
% Output : Refutation 2.32s
% 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 : 1245 ( 334 equ; 0 cnn)
% Maximal formula atoms : 79 ( 62 avg)
% Number of connectives : 629 ( 12 ~; 0 |; 236 &; 57 @)
% ( 0 <=>; 319 =>; 0 <=; 0 <~>)
% Maximal formula depth : 84 ( 30 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 152 ( 148 usr; 83 con; 0-3 aty)
% ( 5 !!; 0 ??; 0 @@+; 0 @@-)
% Number of variables : 32 ( 16 ^; 10 !; 6 ?; 32 :)
% 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_112,type,
sK3: $i > $i ).
thf(func_def_113,type,
sK4: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_114,type,
sK5: $i > ( $i > $o ) > $i ).
thf(func_def_115,type,
sK6: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_116,type,
sK7: ( $i > $o ) > $i > $i ).
thf(func_def_117,type,
sK8: ( $i > $o ) > $i > $i ).
thf(func_def_118,type,
sK9: ( $i > $o ) > $i > $i ).
thf(func_def_119,type,
sK10: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_120,type,
sK11: ( $i > $o ) > $i > $i ).
thf(func_def_121,type,
sK12: ( $i > $o ) > $i > $i ).
thf(func_def_122,type,
sK13: ( $i > $o ) > $i > $i ).
thf(func_def_123,type,
sK14: ( $i > $o ) > $i ).
thf(func_def_124,type,
sK15: $i > ( $i > $o ) > $i ).
thf(func_def_125,type,
sK16: ( $i > $o ) > $i > $i ).
thf(func_def_126,type,
sK17: $i > $i ).
thf(func_def_127,type,
sK18: ( $i > $o ) > $i ).
thf(func_def_128,type,
sK19: ( $i > $o ) > $i ).
thf(func_def_129,type,
sK20: ( $i > $o ) > $i ).
thf(func_def_130,type,
sK21: $o > $i > $i > $i ).
thf(func_def_131,type,
sK22: $i > ( $i > $o ) > $i ).
thf(func_def_132,type,
sK23: $i > $i ).
thf(func_def_133,type,
sK24: $i > $i > $i ).
thf(func_def_134,type,
sK25: $i > $i > $i ).
thf(func_def_135,type,
sK26: $i > $i > $i ).
thf(func_def_136,type,
sK27: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_137,type,
sK28: ( $i > $o ) > $i ).
thf(func_def_138,type,
sK29: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_139,type,
sK30: $i > ( $i > $o ) > $i ).
thf(func_def_140,type,
sK31: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_141,type,
sK32: $i > ( $i > $o ) > $i ).
thf(func_def_142,type,
sK33: ( $i > $o ) > $i ).
thf(func_def_143,type,
sK34: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_144,type,
sK35: $i > ( $i > $o ) > $i ).
thf(func_def_145,type,
sK36: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_146,type,
sK37: ( $i > $o ) > $i > $i ).
thf(func_def_147,type,
sK38: ( $i > $o ) > $i > $i ).
thf(func_def_148,type,
sK39: ( $i > $o ) > $i > $i ).
thf(func_def_149,type,
sK40: $i > ( $i > $o ) > $i ).
thf(func_def_150,type,
sK41: $i > $i ).
thf(func_def_151,type,
sK42: ( $i > $o ) > $i ).
thf(func_def_152,type,
sK43: ( $i > $o ) > $i > $i ).
thf(func_def_153,type,
sK44: ( $i > $o ) > $i ).
thf(func_def_154,type,
sK45: ( $i > $o ) > $i > $i ).
thf(func_def_155,type,
sK46: ( $i > $o ) > $i > $i ).
thf(func_def_156,type,
sK47: $i > $i ).
thf(func_def_157,type,
sK48: $i > $i > $i ).
thf(func_def_158,type,
sK49: $i > $i > $i ).
thf(func_def_159,type,
sK50: $i > $i > $i ).
thf(func_def_160,type,
sK51: $i > $i > $i ).
thf(func_def_161,type,
sK52: $i > $i > $i > $i ).
thf(func_def_162,type,
sK53: ( $i > $o ) > $i > $i ).
thf(func_def_163,type,
sK54: ( $i > $o ) > $i ).
thf(func_def_164,type,
sK55: ( $i > $o ) > $i > $i ).
thf(func_def_165,type,
sK56: ( $i > $o ) > $i ).
thf(func_def_166,type,
sK57: ( $i > $o ) > $i ).
thf(f1216,plain,
$false,
inference(trivial_inequality_removal,[status(thm)],[f1215]) ).
thf(f1215,plain,
$false = $true,
inference(beta_eta_normalization,[status(thm)],[f1214]) ).
thf(f1214,plain,
( $false
= ( ^ [Y0: $i] : $true
@ sK58 ) ),
inference(sigma_clausification,[status(thm)],[f1213]) ).
thf(f1213,plain,
( $true
!= ( !! @ $i
@ ^ [Y0: $i] : $true ) ),
inference(boolean_simplification,[status(thm)],[f1212]) ).
thf(f1212,plain,
( $true
!= ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ sK0 )
=> ( in @ Y0 @ sK0 ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f412]) ).
thf(f412,plain,
( $true
!= ( ^ [Y0: $i,Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y0 )
=> ( in @ Y2 @ Y1 ) ) )
@ sK0
@ sK0 ) ),
inference(definition_unfolding,[status(thm)],[f324,f380,f325]) ).
thf(f325,plain,
sK1 = sK0,
inference(cnf_transformation,[status(thm)],[f243]) ).
thf(f243,plain,
( ( quantDeMorgan4 = $true )
& ( setadjoin__Cong = $true )
& ( singletonsswitch = $true )
& ( omegaSAx = $true )
& ( setunionI = $true )
& ( setadjoinOr = $true )
& ( prop2set2propI = $true )
& ( exuE3u = $true )
& ( nonemptyI = $true )
& ( dsetconstrI = $true )
& ( upairsetIR = $true )
& ( nonemptyI1 = $true )
& ( emptyInPowerset = $true )
& ( notinemptyset = $true )
& ( dsetconstr__Cong = $true )
& ( omega__Cong = $true )
& ( sK1 = sK0 )
& ( $true
!= ( subset @ sK1 @ sK0 ) )
& ( powerset__Cong = $true )
& ( emptyinunitempty = $true )
& ( exuE1 = $true )
& ( descr__Cong = $true )
& ( dsetconstrER = $true )
& ( setadjoinE = $true )
& ( eqinunit = $true )
& ( notinsingleton = $true )
& ( setadjoinAx = $true )
& ( quantDeMorgan1 = $true )
& ( descrp = $true )
& ( quantDeMorgan3 = $true )
& ( exuI3 = $true )
& ( emptyinPowerset = $true )
& ( subsetI1 = $true )
& ( wellorderingAx = $true )
& ( setadjoinIR = $true )
& ( dsetconstrEL = $true )
& ( exuI2 = $true )
& ( setadjoinIL = $true )
& ( vacuousDall = $true )
& ( setextAx = $true )
& ( powersetI = $true )
& ( setunionAx = $true )
& ( setext = $true )
& ( upairsetE = $true )
& ( exuI1 = $true )
& ( replAx = $true )
& ( nonemptyE1 = $true )
& ( prop2setE = $true )
& ( uniqinunit = $true )
& ( quantDeMorgan2 = $true )
& ( in__Cong = $true )
& ( prop2setI = $true )
& ( omega0Ax = $true )
& ( exuEu = $true )
& ( emptysetimpfalse = $true )
& ( emptyI = $true )
& ( emptysetE = $true )
& ( noeltsimpempty = $true )
& ( subPowSU = $true )
& ( notdexE = $true )
& ( exuE2 = $true )
& ( powersetE = $true )
& ( nonemptyImpWitness = $true )
& ( setunion__Cong = $true )
& ( upairsetIL = $true )
& ( foundationAx = $true )
& ( exuE3e = $true )
& ( inCongP = $true )
& ( omegaIndAx = $true )
& ( setoftrueEq = $true )
& ( emptyE1 = $true )
& ( eqimpsubset2 = $true )
& ( notdallE = $true )
& ( powersetAx = $true )
& ( setbeta = $true )
& ( emptysetAx = $true )
& ( setunionE = $true )
& ( emptyset__Cong = $true )
& ( exu__Cong = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1])],[f241,f242]) ).
thf(f242,plain,
( ? [X0: $i,X1: $i] :
( ( X0 = X1 )
& ( ( subset @ X1 @ X0 )
!= $true ) )
=> ( ( sK1 = sK0 )
& ( $true
!= ( subset @ sK1 @ sK0 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f241,plain,
( ( quantDeMorgan4 = $true )
& ( setadjoin__Cong = $true )
& ( singletonsswitch = $true )
& ( omegaSAx = $true )
& ( setunionI = $true )
& ( setadjoinOr = $true )
& ( prop2set2propI = $true )
& ( exuE3u = $true )
& ( nonemptyI = $true )
& ( dsetconstrI = $true )
& ( upairsetIR = $true )
& ( nonemptyI1 = $true )
& ( emptyInPowerset = $true )
& ( notinemptyset = $true )
& ( dsetconstr__Cong = $true )
& ( omega__Cong = $true )
& ? [X0: $i,X1: $i] :
( ( X0 = X1 )
& ( ( subset @ X1 @ X0 )
!= $true ) )
& ( powerset__Cong = $true )
& ( emptyinunitempty = $true )
& ( exuE1 = $true )
& ( descr__Cong = $true )
& ( dsetconstrER = $true )
& ( setadjoinE = $true )
& ( eqinunit = $true )
& ( notinsingleton = $true )
& ( setadjoinAx = $true )
& ( quantDeMorgan1 = $true )
& ( descrp = $true )
& ( quantDeMorgan3 = $true )
& ( exuI3 = $true )
& ( emptyinPowerset = $true )
& ( subsetI1 = $true )
& ( wellorderingAx = $true )
& ( setadjoinIR = $true )
& ( dsetconstrEL = $true )
& ( exuI2 = $true )
& ( setadjoinIL = $true )
& ( vacuousDall = $true )
& ( setextAx = $true )
& ( powersetI = $true )
& ( setunionAx = $true )
& ( setext = $true )
& ( upairsetE = $true )
& ( exuI1 = $true )
& ( replAx = $true )
& ( nonemptyE1 = $true )
& ( prop2setE = $true )
& ( uniqinunit = $true )
& ( quantDeMorgan2 = $true )
& ( in__Cong = $true )
& ( prop2setI = $true )
& ( omega0Ax = $true )
& ( exuEu = $true )
& ( emptysetimpfalse = $true )
& ( emptyI = $true )
& ( emptysetE = $true )
& ( noeltsimpempty = $true )
& ( subPowSU = $true )
& ( notdexE = $true )
& ( exuE2 = $true )
& ( powersetE = $true )
& ( nonemptyImpWitness = $true )
& ( setunion__Cong = $true )
& ( upairsetIL = $true )
& ( foundationAx = $true )
& ( exuE3e = $true )
& ( inCongP = $true )
& ( omegaIndAx = $true )
& ( setoftrueEq = $true )
& ( emptyE1 = $true )
& ( eqimpsubset2 = $true )
& ( notdallE = $true )
& ( powersetAx = $true )
& ( setbeta = $true )
& ( emptysetAx = $true )
& ( setunionE = $true )
& ( emptyset__Cong = $true )
& ( exu__Cong = $true ) ),
inference(flattening,[status(thm)],[f240]) ).
thf(f240,plain,
( ? [X0: $i,X1: $i] :
( ( X0 = X1 )
& ( ( subset @ X1 @ X0 )
!= $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)],[f103]) ).
thf(f103,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 )
=> ! [X0: $i,X1: $i] :
( ( X0 = X1 )
=> ( ( subset @ X1 @ X0 )
= $true ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f102]) ).
thf(f102,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
=> ! [X0: $i,X1: $i] :
( ( X0 = X1 )
=> ( subset @ X1 @ X0 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f82]) ).
thf(f82,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
=> ! [X4: $i,X3: $i] :
( ( X3 = X4 )
=> ( subset @ X3 @ X4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f81]) ).
thf(f81,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
=> ! [X4: $i,X3: $i] :
( ( X3 = X4 )
=> ( subset @ X3 @ X4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.rYUQzxuxZr/DTF2THF_20774.p',eqimpsubset1) ).
thf(f380,plain,
( subset
= ( ^ [Y0: $i,Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y0 )
=> ( in @ Y2 @ Y1 ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f128]) ).
thf(f128,plain,
( subset
= ( ^ [Y0: $i,Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y0 )
=> ( in @ Y2 @ Y1 ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f127]) ).
thf(f127,plain,
( subset
= ( ^ [X0: $i,X1: $i] :
! [X2: $i] :
( ( in @ X2 @ X0 )
=> ( in @ X2 @ X1 ) ) ) ),
inference(rectify,[status(thm)],[f78]) ).
thf(f78,axiom,
( subset
= ( ^ [X3: $i,X4: $i] :
! [X1: $i] :
( ( in @ X1 @ X3 )
=> ( in @ X1 @ X4 ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.rYUQzxuxZr/DTF2THF_20774.p',subset) ).
thf(f324,plain,
( $true
!= ( subset @ sK1 @ sK0 ) ),
inference(cnf_transformation,[status(thm)],[f243]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SEU561^1 : TPTP v9.2.1. Released v3.7.0.
% 0.11/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.16/0.34 % Computer : n023.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 12:27:34 EST 2026
% 0.16/0.34 % CPUTime :
% 0.16/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.29/0.46 ---- Original DTF file ---
% 0.29/0.46 thf(spec,logic,$$dhol).
% 0.29/0.46 %------------------------------------------------------------------------------
% 0.29/0.46 % File : SEU561^1 : TPTP v9.2.1. Released v3.7.0.
% 0.29/0.46 % Domain : Set Theory
% 0.29/0.46 % Problem : Preliminary Notions - Relations on Sets - Subsets
% 0.29/0.46 % Version : Especial.
% 0.29/0.46 % English : (! A:i.! B:i.A = B -> subset A B)
% 0.29/0.46
% 0.29/0.46 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.29/0.46 % Source : [Bro08]
% 0.29/0.46 % Names : ZFC063g [Bro08]
% 0.29/0.46
% 0.29/0.46 % Status : Theorem
% 0.29/0.46 % Rating : 0.22 v9.1.0, 0.25 v9.0.0, 0.30 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 v4.1.0, 0.67 v3.7.0
% 0.29/0.46 % Syntax : Number of formulae : 173 ( 80 unt; 92 typ; 80 def)
% 0.29/0.46 % Number of atoms : 438 ( 125 equ; 0 cnn)
% 0.29/0.46 % Maximal formula atoms : 79 ( 5 avg)
% 0.29/0.46 % Number of connectives : 702 ( 21 ~; 5 |; 27 &; 434 @)
% 0.29/0.46 % ( 14 <=>; 201 =>; 0 <=; 0 <~>)
% 0.29/0.46 % Maximal formula depth : 83 ( 2 avg)
% 0.29/0.46 % Number of types : 2 ( 0 usr)
% 0.29/0.46 % Number of type conns : 53 ( 53 >; 0 *; 0 +; 0 <<)
% 0.29/0.46 % Number of symbols : 95 ( 92 usr; 81 con; 0-2 aty)
% 0.29/0.46 % Number of variables : 252 ( 30 ^; 194 !; 28 ?; 252 :)
% 0.29/0.46 % SPC : TH0_THM_EQU_NAR
% 0.29/0.46
% 0.29/0.46 % Comments : http://mathgate.info/detsetitem.php?id=395
% 0.29/0.46 %------------------------------------------------------------------------------
% 0.29/0.46 thf(in_type,type,
% 0.29/0.46 in: $i > $i > $o ).
% 0.29/0.46
% 0.29/0.46 thf(exu_type,type,
% 0.29/0.46 exu: ( $i > $o ) > $o ).
% 0.29/0.46
% 0.29/0.46 thf(exu,definition,
% 0.29/0.46 ( exu
% 0.29/0.46 = ( ^ [Xphi: $i > $o] :
% 0.29/0.46 ? [Xx: $i] :
% 0.29/0.46 ( ( Xphi @ Xx )
% 0.29/0.46 & ! [Xy: $i] :
% 0.29/0.46 ( ( Xphi @ Xy )
% 0.29/0.46 => ( Xx = Xy ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(setextAx_type,type,
% 0.29/0.46 setextAx: $o ).
% 0.29/0.46
% 0.29/0.46 thf(setextAx,definition,
% 0.29/0.46 ( setextAx
% 0.29/0.46 = ( ! [A: $i,B: $i] :
% 0.29/0.46 ( ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 <=> ( in @ Xx @ B ) )
% 0.29/0.46 => ( A = B ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(emptyset_type,type,
% 0.29/0.46 emptyset: $i ).
% 0.29/0.46
% 0.29/0.46 thf(emptysetAx_type,type,
% 0.29/0.46 emptysetAx: $o ).
% 0.29/0.46
% 0.29/0.46 thf(emptysetAx,definition,
% 0.29/0.46 ( emptysetAx
% 0.29/0.46 = ( ! [Xx: $i] :
% 0.29/0.46 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(setadjoin_type,type,
% 0.29/0.46 setadjoin: $i > $i > $i ).
% 0.29/0.46
% 0.29/0.46 thf(setadjoinAx_type,type,
% 0.29/0.46 setadjoinAx: $o ).
% 0.29/0.46
% 0.29/0.46 thf(setadjoinAx,definition,
% 0.29/0.46 ( setadjoinAx
% 0.29/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.29/0.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.29/0.46 <=> ( ( Xy = Xx )
% 0.29/0.46 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(powerset_type,type,
% 0.29/0.46 powerset: $i > $i ).
% 0.29/0.46
% 0.29/0.46 thf(powersetAx_type,type,
% 0.29/0.46 powersetAx: $o ).
% 0.29/0.46
% 0.29/0.46 thf(powersetAx,definition,
% 0.29/0.46 ( powersetAx
% 0.29/0.46 = ( ! [A: $i,B: $i] :
% 0.29/0.46 ( ( in @ B @ ( powerset @ A ) )
% 0.29/0.46 <=> ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ B )
% 0.29/0.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(setunion_type,type,
% 0.29/0.46 setunion: $i > $i ).
% 0.29/0.46
% 0.29/0.46 thf(setunionAx_type,type,
% 0.29/0.46 setunionAx: $o ).
% 0.29/0.46
% 0.29/0.46 thf(setunionAx,definition,
% 0.29/0.46 ( setunionAx
% 0.29/0.46 = ( ! [A: $i,Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ ( setunion @ A ) )
% 0.29/0.46 <=> ? [B: $i] :
% 0.29/0.46 ( ( in @ Xx @ B )
% 0.29/0.46 & ( in @ B @ A ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(omega_type,type,
% 0.29/0.46 omega: $i ).
% 0.29/0.46
% 0.29/0.46 thf(omega0Ax_type,type,
% 0.29/0.46 omega0Ax: $o ).
% 0.29/0.46
% 0.29/0.46 thf(omega0Ax,definition,
% 0.29/0.46 ( omega0Ax
% 0.29/0.46 = ( in @ emptyset @ omega ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(omegaSAx_type,type,
% 0.29/0.46 omegaSAx: $o ).
% 0.29/0.46
% 0.29/0.46 thf(omegaSAx,definition,
% 0.29/0.46 ( omegaSAx
% 0.29/0.46 = ( ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ omega )
% 0.29/0.46 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(omegaIndAx_type,type,
% 0.29/0.46 omegaIndAx: $o ).
% 0.29/0.46
% 0.29/0.46 thf(omegaIndAx,definition,
% 0.29/0.46 ( omegaIndAx
% 0.29/0.46 = ( ! [A: $i] :
% 0.29/0.46 ( ( ( in @ emptyset @ A )
% 0.29/0.46 & ! [Xx: $i] :
% 0.29/0.46 ( ( ( in @ Xx @ omega )
% 0.29/0.46 & ( in @ Xx @ A ) )
% 0.29/0.46 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.29/0.46 => ! [Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ omega )
% 0.29/0.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(replAx_type,type,
% 0.29/0.46 replAx: $o ).
% 0.29/0.46
% 0.29/0.46 thf(replAx,definition,
% 0.29/0.46 ( replAx
% 0.29/0.46 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.29/0.46 ( ! [Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ A )
% 0.29/0.47 => ( exu
% 0.29/0.47 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.29/0.47 => ? [B: $i] :
% 0.29/0.47 ! [Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ B )
% 0.29/0.47 <=> ? [Xy: $i] :
% 0.29/0.47 ( ( in @ Xy @ A )
% 0.29/0.47 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(foundationAx_type,type,
% 0.29/0.47 foundationAx: $o ).
% 0.29/0.47
% 0.29/0.47 thf(foundationAx,definition,
% 0.29/0.47 ( foundationAx
% 0.29/0.47 = ( ! [A: $i] :
% 0.29/0.47 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.29/0.47 => ? [B: $i] :
% 0.29/0.47 ( ( in @ B @ A )
% 0.29/0.47 & ~ ? [Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ B )
% 0.29/0.47 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(wellorderingAx_type,type,
% 0.29/0.47 wellorderingAx: $o ).
% 0.29/0.47
% 0.29/0.47 thf(wellorderingAx,definition,
% 0.29/0.47 ( wellorderingAx
% 0.29/0.47 = ( ! [A: $i] :
% 0.29/0.47 ? [B: $i] :
% 0.29/0.47 ( ! [C: $i] :
% 0.29/0.47 ( ( in @ C @ B )
% 0.29/0.47 => ! [Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ C )
% 0.29/0.47 => ( in @ Xx @ A ) ) )
% 0.29/0.47 & ! [Xx: $i,Xy: $i] :
% 0.29/0.47 ( ( ( in @ Xx @ A )
% 0.29/0.47 & ( in @ Xy @ A ) )
% 0.29/0.47 => ( ! [C: $i] :
% 0.29/0.47 ( ( in @ C @ B )
% 0.29/0.47 => ( ( in @ Xx @ C )
% 0.29/0.47 <=> ( in @ Xy @ C ) ) )
% 0.29/0.47 => ( Xx = Xy ) ) )
% 0.29/0.47 & ! [C: $i,D: $i] :
% 0.29/0.47 ( ( ( in @ C @ B )
% 0.29/0.47 & ( in @ D @ B ) )
% 0.29/0.47 => ( ! [Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ C )
% 0.29/0.47 => ( in @ Xx @ D ) )
% 0.29/0.47 | ! [Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ D )
% 0.29/0.47 => ( in @ Xx @ C ) ) ) )
% 0.29/0.47 & ! [C: $i] :
% 0.29/0.47 ( ( ! [Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ C )
% 0.29/0.47 => ( in @ Xx @ A ) )
% 0.29/0.47 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.29/0.47 => ? [D: $i,Xx: $i] :
% 0.29/0.47 ( ( in @ D @ B )
% 0.29/0.47 & ( in @ Xx @ C )
% 0.29/0.47 & ~ ? [Xy: $i] :
% 0.29/0.47 ( ( in @ Xy @ D )
% 0.29/0.47 & ( in @ Xy @ C ) )
% 0.29/0.47 & ! [E: $i] :
% 0.29/0.47 ( ( in @ E @ B )
% 0.29/0.47 => ( ! [Xy: $i] :
% 0.29/0.47 ( ( in @ Xy @ E )
% 0.29/0.47 => ( in @ Xy @ D ) )
% 0.29/0.47 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(descr_type,type,
% 0.29/0.47 descr: ( $i > $o ) > $i ).
% 0.29/0.47
% 0.29/0.47 thf(descrp_type,type,
% 0.29/0.47 descrp: $o ).
% 0.29/0.47
% 0.29/0.47 thf(descrp,definition,
% 0.29/0.47 ( descrp
% 0.29/0.47 = ( ! [Xphi: $i > $o] :
% 0.29/0.47 ( ( exu
% 0.29/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.29/0.47 => ( Xphi
% 0.29/0.47 @ ( descr
% 0.29/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(dsetconstr_type,type,
% 0.29/0.47 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.29/0.47
% 0.29/0.47 thf(dsetconstrI_type,type,
% 0.29/0.47 dsetconstrI: $o ).
% 0.29/0.47
% 0.29/0.47 thf(dsetconstrI,definition,
% 0.29/0.47 ( dsetconstrI
% 0.29/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ A )
% 0.29/0.47 => ( ( Xphi @ Xx )
% 0.29/0.47 => ( in @ Xx
% 0.29/0.47 @ ( dsetconstr @ A
% 0.29/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(dsetconstrEL_type,type,
% 0.29/0.47 dsetconstrEL: $o ).
% 0.29/0.47
% 0.29/0.47 thf(dsetconstrEL,definition,
% 0.29/0.47 ( dsetconstrEL
% 0.29/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.29/0.47 ( ( in @ Xx
% 0.29/0.47 @ ( dsetconstr @ A
% 0.29/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.29/0.47 => ( in @ Xx @ A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(dsetconstrER_type,type,
% 0.29/0.47 dsetconstrER: $o ).
% 0.29/0.47
% 0.29/0.47 thf(dsetconstrER,definition,
% 0.29/0.47 ( dsetconstrER
% 0.29/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.29/0.47 ( ( in @ Xx
% 0.29/0.47 @ ( dsetconstr @ A
% 0.29/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.29/0.47 => ( Xphi @ Xx ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(exuE1_type,type,
% 0.29/0.47 exuE1: $o ).
% 0.29/0.47
% 0.29/0.47 thf(exuE1,definition,
% 0.29/0.47 ( exuE1
% 0.29/0.47 = ( ! [Xphi: $i > $o] :
% 0.29/0.47 ( ( exu
% 0.29/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.29/0.47 => ? [Xx: $i] :
% 0.29/0.47 ( ( Xphi @ Xx )
% 0.29/0.47 & ! [Xy: $i] :
% 0.29/0.47 ( ( Xphi @ Xy )
% 0.29/0.47 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(prop2set_type,type,
% 0.29/0.47 prop2set: $o > $i ).
% 0.29/0.47
% 0.29/0.47 thf(prop2setE_type,type,
% 0.29/0.47 prop2setE: $o ).
% 0.29/0.47
% 0.29/0.47 thf(prop2setE,definition,
% 0.29/0.47 ( prop2setE
% 0.29/0.47 = ( ! [Xphi: $o,Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.29/0.47 => Xphi ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(emptysetE_type,type,
% 0.29/0.47 emptysetE: $o ).
% 0.29/0.47
% 0.29/0.47 thf(emptysetE,definition,
% 0.29/0.47 ( emptysetE
% 0.29/0.47 = ( ! [Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ emptyset )
% 0.29/0.47 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(emptysetimpfalse_type,type,
% 0.29/0.47 emptysetimpfalse: $o ).
% 0.29/0.47
% 0.29/0.47 thf(emptysetimpfalse,definition,
% 0.29/0.47 ( emptysetimpfalse
% 0.29/0.47 = ( ! [Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ emptyset )
% 0.29/0.47 => $false ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(notinemptyset_type,type,
% 0.29/0.47 notinemptyset: $o ).
% 0.29/0.47
% 0.29/0.47 thf(notinemptyset,definition,
% 0.29/0.47 ( notinemptyset
% 0.29/0.47 = ( ! [Xx: $i] :
% 0.29/0.47 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(exuE3e_type,type,
% 0.29/0.47 exuE3e: $o ).
% 0.29/0.47
% 0.29/0.47 thf(exuE3e,definition,
% 0.29/0.47 ( exuE3e
% 0.29/0.47 = ( ! [Xphi: $i > $o] :
% 0.29/0.47 ( ( exu
% 0.29/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.29/0.47 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(setext_type,type,
% 0.29/0.47 setext: $o ).
% 0.29/0.47
% 0.29/0.47 thf(setext,definition,
% 0.29/0.47 ( setext
% 0.29/0.47 = ( ! [A: $i,B: $i] :
% 0.29/0.47 ( ! [Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ A )
% 0.29/0.47 => ( in @ Xx @ B ) )
% 0.29/0.47 => ( ! [Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ B )
% 0.29/0.47 => ( in @ Xx @ A ) )
% 0.29/0.47 => ( A = B ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(emptyI_type,type,
% 0.29/0.47 emptyI: $o ).
% 0.29/0.47
% 0.29/0.47 thf(emptyI,definition,
% 0.29/0.47 ( emptyI
% 0.29/0.47 = ( ! [A: $i] :
% 0.29/0.47 ( ! [Xx: $i] :
% 0.29/0.47 ~ ( in @ Xx @ A )
% 0.29/0.47 => ( A = emptyset ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(noeltsimpempty_type,type,
% 0.29/0.47 noeltsimpempty: $o ).
% 0.29/0.47
% 0.29/0.47 thf(noeltsimpempty,definition,
% 0.29/0.47 ( noeltsimpempty
% 0.29/0.47 = ( ! [A: $i] :
% 0.29/0.47 ( ! [Xx: $i] :
% 0.29/0.47 ~ ( in @ Xx @ A )
% 0.29/0.47 => ( A = emptyset ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(setbeta_type,type,
% 0.29/0.47 setbeta: $o ).
% 0.29/0.47
% 0.29/0.47 thf(setbeta,definition,
% 0.29/0.47 ( setbeta
% 0.29/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ A )
% 0.29/0.47 => ( ( in @ Xx
% 0.29/0.47 @ ( dsetconstr @ A
% 0.29/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.29/0.47 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(nonempty_type,type,
% 0.29/0.47 nonempty: $i > $o ).
% 0.29/0.47
% 0.29/0.47 thf(nonempty,definition,
% 0.29/0.47 ( nonempty
% 0.29/0.47 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(nonemptyE1_type,type,
% 0.29/0.47 nonemptyE1: $o ).
% 0.29/0.47
% 0.29/0.47 thf(nonemptyE1,definition,
% 0.29/0.47 ( nonemptyE1
% 0.29/0.47 = ( ! [A: $i] :
% 0.29/0.47 ( ( nonempty @ A )
% 0.29/0.47 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(nonemptyI_type,type,
% 0.29/0.47 nonemptyI: $o ).
% 0.29/0.47
% 0.29/0.47 thf(nonemptyI,definition,
% 0.29/0.47 ( nonemptyI
% 0.29/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ A )
% 0.29/0.47 => ( ( Xphi @ Xx )
% 0.29/0.47 => ( nonempty
% 0.29/0.47 @ ( dsetconstr @ A
% 0.29/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(nonemptyI1_type,type,
% 0.29/0.47 nonemptyI1: $o ).
% 0.29/0.47
% 0.29/0.47 thf(nonemptyI1,definition,
% 0.29/0.47 ( nonemptyI1
% 0.29/0.47 = ( ! [A: $i] :
% 0.29/0.47 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.29/0.47 => ( nonempty @ A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(setadjoinIL_type,type,
% 0.29/0.47 setadjoinIL: $o ).
% 0.29/0.47
% 0.29/0.47 thf(setadjoinIL,definition,
% 0.29/0.47 ( setadjoinIL
% 0.29/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(emptyinunitempty_type,type,
% 0.29/0.47 emptyinunitempty: $o ).
% 0.29/0.47
% 0.29/0.47 thf(emptyinunitempty,definition,
% 0.29/0.47 ( emptyinunitempty
% 0.29/0.47 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(setadjoinIR_type,type,
% 0.29/0.47 setadjoinIR: $o ).
% 0.29/0.47
% 0.29/0.47 thf(setadjoinIR,definition,
% 0.29/0.47 ( setadjoinIR
% 0.29/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.29/0.47 ( ( in @ Xy @ A )
% 0.29/0.47 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(setadjoinE_type,type,
% 0.29/0.47 setadjoinE: $o ).
% 0.29/0.47
% 0.29/0.47 thf(setadjoinE,definition,
% 0.29/0.47 ( setadjoinE
% 0.29/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.29/0.47 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.29/0.47 => ! [Xphi: $o] :
% 0.29/0.47 ( ( ( Xy = Xx )
% 0.29/0.47 => Xphi )
% 0.29/0.47 => ( ( ( in @ Xy @ A )
% 0.29/0.47 => Xphi )
% 0.29/0.47 => Xphi ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(setadjoinOr_type,type,
% 0.29/0.47 setadjoinOr: $o ).
% 0.29/0.47
% 0.29/0.47 thf(setadjoinOr,definition,
% 0.29/0.47 ( setadjoinOr
% 0.29/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.29/0.47 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.29/0.47 => ( ( Xy = Xx )
% 0.29/0.47 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(setoftrueEq_type,type,
% 0.29/0.47 setoftrueEq: $o ).
% 0.29/0.47
% 0.29/0.47 thf(setoftrueEq,definition,
% 0.29/0.47 ( setoftrueEq
% 0.29/0.47 = ( ! [A: $i] :
% 0.29/0.47 ( ( dsetconstr @ A
% 0.29/0.47 @ ^ [Xx: $i] : $true )
% 0.29/0.47 = A ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(powersetI_type,type,
% 0.29/0.47 powersetI: $o ).
% 0.29/0.47
% 0.29/0.47 thf(powersetI,definition,
% 0.29/0.47 ( powersetI
% 0.29/0.47 = ( ! [A: $i,B: $i] :
% 0.29/0.47 ( ! [Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ B )
% 0.29/0.47 => ( in @ Xx @ A ) )
% 0.29/0.47 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(emptyinPowerset_type,type,
% 0.29/0.47 emptyinPowerset: $o ).
% 0.29/0.47
% 0.29/0.47 thf(emptyinPowerset,definition,
% 0.29/0.47 ( emptyinPowerset
% 0.29/0.47 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(emptyInPowerset_type,type,
% 0.29/0.47 emptyInPowerset: $o ).
% 0.29/0.47
% 0.29/0.47 thf(emptyInPowerset,definition,
% 0.29/0.47 ( emptyInPowerset
% 0.29/0.47 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(powersetE_type,type,
% 0.29/0.47 powersetE: $o ).
% 0.29/0.47
% 0.29/0.47 thf(powersetE,definition,
% 0.29/0.47 ( powersetE
% 0.29/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.47 ( ( in @ B @ ( powerset @ A ) )
% 0.29/0.47 => ( ( in @ Xx @ B )
% 0.29/0.47 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(setunionI_type,type,
% 0.29/0.47 setunionI: $o ).
% 0.29/0.47
% 0.29/0.47 thf(setunionI,definition,
% 0.29/0.47 ( setunionI
% 0.29/0.47 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.29/0.47 ( ( in @ Xx @ B )
% 0.29/0.47 => ( ( in @ B @ A )
% 0.29/0.47 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(setunionE_type,type,
% 0.29/0.47 setunionE: $o ).
% 0.29/0.47
% 0.29/0.47 thf(setunionE,definition,
% 0.29/0.47 ( setunionE
% 0.29/0.47 = ( ! [A: $i,Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ ( setunion @ A ) )
% 0.29/0.47 => ! [Xphi: $o] :
% 0.29/0.47 ( ! [B: $i] :
% 0.29/0.47 ( ( in @ Xx @ B )
% 0.29/0.47 => ( ( in @ B @ A )
% 0.29/0.47 => Xphi ) )
% 0.29/0.47 => Xphi ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(subPowSU_type,type,
% 0.29/0.47 subPowSU: $o ).
% 0.29/0.47
% 0.29/0.47 thf(subPowSU,definition,
% 0.29/0.47 ( subPowSU
% 0.29/0.47 = ( ! [A: $i,Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ A )
% 0.29/0.47 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(exuE2_type,type,
% 0.29/0.47 exuE2: $o ).
% 0.29/0.47
% 0.29/0.47 thf(exuE2,definition,
% 0.29/0.47 ( exuE2
% 0.29/0.47 = ( ! [Xphi: $i > $o] :
% 0.29/0.47 ( ( exu
% 0.29/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.29/0.47 => ? [Xx: $i] :
% 0.29/0.47 ! [Xy: $i] :
% 0.29/0.47 ( ( Xphi @ Xy )
% 0.29/0.47 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(nonemptyImpWitness_type,type,
% 0.29/0.47 nonemptyImpWitness: $o ).
% 0.29/0.47
% 0.29/0.47 thf(nonemptyImpWitness,definition,
% 0.29/0.47 ( nonemptyImpWitness
% 0.29/0.47 = ( ! [A: $i] :
% 0.29/0.47 ( ( nonempty @ A )
% 0.29/0.47 => ? [Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ A )
% 0.29/0.47 & $true ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(uniqinunit_type,type,
% 0.29/0.47 uniqinunit: $o ).
% 0.29/0.47
% 0.29/0.47 thf(uniqinunit,definition,
% 0.29/0.47 ( uniqinunit
% 0.29/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.29/0.47 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.29/0.47 => ( Xx = Xy ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(notinsingleton_type,type,
% 0.29/0.47 notinsingleton: $o ).
% 0.29/0.47
% 0.29/0.47 thf(notinsingleton,definition,
% 0.29/0.47 ( notinsingleton
% 0.29/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.29/0.47 ( ( Xx != Xy )
% 0.29/0.47 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(eqinunit_type,type,
% 0.29/0.47 eqinunit: $o ).
% 0.29/0.47
% 0.29/0.47 thf(eqinunit,definition,
% 0.29/0.47 ( eqinunit
% 0.29/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.29/0.47 ( ( Xx = Xy )
% 0.29/0.47 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(singletonsswitch_type,type,
% 0.29/0.47 singletonsswitch: $o ).
% 0.29/0.47
% 0.29/0.47 thf(singletonsswitch,definition,
% 0.29/0.47 ( singletonsswitch
% 0.29/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.29/0.47 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.29/0.47 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(upairsetE_type,type,
% 0.29/0.47 upairsetE: $o ).
% 0.29/0.47
% 0.29/0.47 thf(upairsetE,definition,
% 0.29/0.47 ( upairsetE
% 0.29/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.29/0.47 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.29/0.47 => ( ( Xz = Xx )
% 0.29/0.47 | ( Xz = Xy ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(upairsetIL_type,type,
% 0.29/0.47 upairsetIL: $o ).
% 0.29/0.47
% 0.29/0.47 thf(upairsetIL,definition,
% 0.29/0.47 ( upairsetIL
% 0.29/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(upairsetIR_type,type,
% 0.29/0.47 upairsetIR: $o ).
% 0.29/0.47
% 0.29/0.47 thf(upairsetIR,definition,
% 0.29/0.47 ( upairsetIR
% 0.29/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(emptyE1_type,type,
% 0.29/0.47 emptyE1: $o ).
% 0.29/0.47
% 0.29/0.47 thf(emptyE1,definition,
% 0.29/0.47 ( emptyE1
% 0.29/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.29/0.47 ( ? [Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ A )
% 0.29/0.47 & ( Xphi @ Xx ) )
% 0.29/0.47 => ( ( ( dsetconstr @ A
% 0.29/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.29/0.47 = emptyset )
% 0.29/0.47 => $false ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(vacuousDall_type,type,
% 0.29/0.47 vacuousDall: $o ).
% 0.29/0.47
% 0.29/0.47 thf(vacuousDall,definition,
% 0.29/0.47 ( vacuousDall
% 0.29/0.47 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ emptyset )
% 0.29/0.47 => ( Xphi @ Xx ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(quantDeMorgan1_type,type,
% 0.29/0.47 quantDeMorgan1: $o ).
% 0.29/0.47
% 0.29/0.47 thf(quantDeMorgan1,definition,
% 0.29/0.47 ( quantDeMorgan1
% 0.29/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.29/0.47 ( ~ ! [Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ A )
% 0.29/0.47 => ( Xphi @ Xx ) )
% 0.29/0.47 => ? [Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ A )
% 0.29/0.47 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(quantDeMorgan2_type,type,
% 0.29/0.47 quantDeMorgan2: $o ).
% 0.29/0.47
% 0.29/0.47 thf(quantDeMorgan2,definition,
% 0.29/0.47 ( quantDeMorgan2
% 0.29/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.29/0.47 ( ! [Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ A )
% 0.29/0.47 => ~ ( Xphi @ Xx ) )
% 0.29/0.47 => ~ ? [Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ A )
% 0.29/0.47 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(quantDeMorgan3_type,type,
% 0.29/0.47 quantDeMorgan3: $o ).
% 0.29/0.47
% 0.29/0.47 thf(quantDeMorgan3,definition,
% 0.29/0.47 ( quantDeMorgan3
% 0.29/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.29/0.47 ( ~ ? [Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ A )
% 0.29/0.47 & ( Xphi @ Xx ) )
% 0.29/0.47 => ! [Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ A )
% 0.29/0.47 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(quantDeMorgan4_type,type,
% 0.29/0.47 quantDeMorgan4: $o ).
% 0.29/0.47
% 0.29/0.47 thf(quantDeMorgan4,definition,
% 0.29/0.47 ( quantDeMorgan4
% 0.29/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.29/0.47 ( ? [Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ A )
% 0.29/0.47 & ~ ( Xphi @ Xx ) )
% 0.29/0.47 => ~ ! [Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ A )
% 0.29/0.47 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(prop2setI_type,type,
% 0.29/0.47 prop2setI: $o ).
% 0.29/0.47
% 0.29/0.47 thf(prop2setI,definition,
% 0.29/0.47 ( prop2setI
% 0.29/0.47 = ( ! [Xphi: $o] :
% 0.29/0.47 ( Xphi
% 0.29/0.47 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(set2prop_type,type,
% 0.29/0.47 set2prop: $i > $o ).
% 0.29/0.47
% 0.29/0.47 thf(prop2set2propI_type,type,
% 0.29/0.47 prop2set2propI: $o ).
% 0.29/0.47
% 0.29/0.47 thf(prop2set2propI,definition,
% 0.29/0.47 ( prop2set2propI
% 0.29/0.47 = ( ! [Xphi: $o] :
% 0.29/0.47 ( Xphi
% 0.29/0.47 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(notdexE_type,type,
% 0.29/0.47 notdexE: $o ).
% 0.29/0.47
% 0.29/0.47 thf(notdexE,definition,
% 0.29/0.47 ( notdexE
% 0.29/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.29/0.47 ( ~ ? [Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ A )
% 0.29/0.47 & ( Xphi @ Xx ) )
% 0.29/0.47 => ! [Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ A )
% 0.29/0.47 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(notdallE_type,type,
% 0.29/0.47 notdallE: $o ).
% 0.29/0.47
% 0.29/0.47 thf(notdallE,definition,
% 0.29/0.47 ( notdallE
% 0.29/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.29/0.47 ( ~ ! [Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ A )
% 0.29/0.47 => ( Xphi @ Xx ) )
% 0.29/0.47 => ? [Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ A )
% 0.29/0.47 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(exuI1_type,type,
% 0.29/0.47 exuI1: $o ).
% 0.29/0.47
% 0.29/0.47 thf(exuI1,definition,
% 0.29/0.47 ( exuI1
% 0.29/0.47 = ( ! [Xphi: $i > $o] :
% 0.29/0.47 ( ? [Xx: $i] :
% 0.29/0.47 ( ( Xphi @ Xx )
% 0.29/0.47 & ! [Xy: $i] :
% 0.29/0.47 ( ( Xphi @ Xy )
% 0.29/0.47 => ( Xx = Xy ) ) )
% 0.29/0.47 => ( exu
% 0.29/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(exuI3_type,type,
% 0.29/0.47 exuI3: $o ).
% 0.29/0.47
% 0.29/0.47 thf(exuI3,definition,
% 0.29/0.47 ( exuI3
% 0.29/0.47 = ( ! [Xphi: $i > $o] :
% 0.29/0.47 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.29/0.47 => ( ! [Xx: $i,Xy: $i] :
% 0.29/0.47 ( ( Xphi @ Xx )
% 0.29/0.47 => ( ( Xphi @ Xy )
% 0.29/0.47 => ( Xx = Xy ) ) )
% 0.29/0.47 => ( exu
% 0.29/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(exuI2_type,type,
% 0.29/0.47 exuI2: $o ).
% 0.29/0.47
% 0.29/0.47 thf(exuI2,definition,
% 0.29/0.47 ( exuI2
% 0.29/0.47 = ( ! [Xphi: $i > $o] :
% 0.29/0.47 ( ? [Xx: $i] :
% 0.29/0.47 ! [Xy: $i] :
% 0.29/0.47 ( ( Xphi @ Xy )
% 0.29/0.47 <=> ( Xy = Xx ) )
% 0.29/0.47 => ( exu
% 0.29/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(inCongP_type,type,
% 0.29/0.47 inCongP: $o ).
% 0.29/0.47
% 0.29/0.47 thf(inCongP,definition,
% 0.29/0.47 ( inCongP
% 0.29/0.47 = ( ! [A: $i,B: $i] :
% 0.29/0.47 ( ( A = B )
% 0.29/0.47 => ! [Xx: $i,Xy: $i] :
% 0.29/0.47 ( ( Xx = Xy )
% 0.29/0.47 => ( ( in @ Xx @ A )
% 0.29/0.47 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(in__Cong_type,type,
% 0.29/0.47 in__Cong: $o ).
% 0.29/0.47
% 0.29/0.47 thf(in__Cong,definition,
% 0.29/0.47 ( in__Cong
% 0.29/0.47 = ( ! [A: $i,B: $i] :
% 0.29/0.47 ( ( A = B )
% 0.29/0.47 => ! [Xx: $i,Xy: $i] :
% 0.29/0.47 ( ( Xx = Xy )
% 0.29/0.47 => ( ( in @ Xx @ A )
% 0.29/0.47 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(exuE3u_type,type,
% 0.29/0.47 exuE3u: $o ).
% 0.29/0.47
% 0.29/0.47 thf(exuE3u,definition,
% 0.29/0.47 ( exuE3u
% 0.29/0.47 = ( ! [Xphi: $i > $o] :
% 0.29/0.47 ( ( exu
% 0.29/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.29/0.47 => ! [Xx: $i,Xy: $i] :
% 0.29/0.47 ( ( Xphi @ Xx )
% 0.29/0.47 => ( ( Xphi @ Xy )
% 0.29/0.47 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(exu__Cong_type,type,
% 0.29/0.47 exu__Cong: $o ).
% 0.29/0.47
% 0.29/0.47 thf(exu__Cong,definition,
% 0.29/0.47 ( exu__Cong
% 0.29/0.47 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.29/0.47 ( ! [Xx: $i,Xy: $i] :
% 0.29/0.47 ( ( Xx = Xy )
% 0.29/0.47 => ( ( Xphi @ Xx )
% 0.29/0.47 <=> ( Xpsi @ Xy ) ) )
% 0.29/0.47 => ( ( exu
% 0.29/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.29/0.47 <=> ( exu
% 0.29/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(emptyset__Cong_type,type,
% 0.29/0.47 emptyset__Cong: $o ).
% 0.29/0.47
% 0.29/0.47 thf(emptyset__Cong,definition,
% 0.29/0.47 ( emptyset__Cong
% 0.29/0.47 = ( emptyset = emptyset ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(setadjoin__Cong_type,type,
% 0.29/0.47 setadjoin__Cong: $o ).
% 0.29/0.47
% 0.29/0.47 thf(setadjoin__Cong,definition,
% 0.29/0.47 ( setadjoin__Cong
% 0.29/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.29/0.47 ( ( Xx = Xy )
% 0.29/0.47 => ! [Xz: $i,Xu: $i] :
% 0.29/0.47 ( ( Xz = Xu )
% 0.29/0.47 => ( ( setadjoin @ Xx @ Xz )
% 0.29/0.47 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(powerset__Cong_type,type,
% 0.29/0.47 powerset__Cong: $o ).
% 0.29/0.47
% 0.29/0.47 thf(powerset__Cong,definition,
% 0.29/0.47 ( powerset__Cong
% 0.29/0.47 = ( ! [A: $i,B: $i] :
% 0.29/0.47 ( ( A = B )
% 0.29/0.47 => ( ( powerset @ A )
% 0.29/0.47 = ( powerset @ B ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(setunion__Cong_type,type,
% 0.29/0.47 setunion__Cong: $o ).
% 0.29/0.47
% 0.29/0.47 thf(setunion__Cong,definition,
% 0.29/0.47 ( setunion__Cong
% 0.29/0.47 = ( ! [A: $i,B: $i] :
% 0.29/0.47 ( ( A = B )
% 0.29/0.47 => ( ( setunion @ A )
% 0.29/0.47 = ( setunion @ B ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(omega__Cong_type,type,
% 0.29/0.47 omega__Cong: $o ).
% 0.29/0.47
% 0.29/0.47 thf(omega__Cong,definition,
% 0.29/0.47 ( omega__Cong
% 0.29/0.47 = ( omega = omega ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(exuEu_type,type,
% 0.29/0.47 exuEu: $o ).
% 0.29/0.47
% 0.29/0.47 thf(exuEu,definition,
% 0.29/0.47 ( exuEu
% 0.29/0.47 = ( ! [Xphi: $i > $o] :
% 0.29/0.47 ( ( exu
% 0.29/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.29/0.47 => ! [Xx: $i,Xy: $i] :
% 0.29/0.47 ( ( Xphi @ Xx )
% 0.29/0.47 => ( ( Xphi @ Xy )
% 0.29/0.47 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(descr__Cong_type,type,
% 0.29/0.47 descr__Cong: $o ).
% 0.29/0.47
% 0.29/0.47 thf(descr__Cong,definition,
% 0.29/0.47 ( descr__Cong
% 0.29/0.47 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.29/0.47 ( ! [Xx: $i,Xy: $i] :
% 0.29/0.47 ( ( Xx = Xy )
% 0.29/0.47 => ( ( Xphi @ Xx )
% 0.29/0.47 <=> ( Xpsi @ Xy ) ) )
% 0.29/0.47 => ( ( exu
% 0.29/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.29/0.47 => ( ( exu
% 0.29/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.29/0.47 => ( ( descr
% 0.29/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.29/0.47 = ( descr
% 0.29/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(dsetconstr__Cong_type,type,
% 0.29/0.47 dsetconstr__Cong: $o ).
% 0.29/0.47
% 0.29/0.47 thf(dsetconstr__Cong,definition,
% 0.29/0.47 ( dsetconstr__Cong
% 0.29/0.47 = ( ! [A: $i,B: $i] :
% 0.29/0.47 ( ( A = B )
% 0.29/0.47 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.29/0.47 ( ! [Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ A )
% 0.29/0.47 => ! [Xy: $i] :
% 0.29/0.47 ( ( in @ Xy @ B )
% 0.29/0.47 => ( ( Xx = Xy )
% 0.29/0.47 => ( ( Xphi @ Xx )
% 0.29/0.47 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.29/0.47 => ( ( dsetconstr @ A
% 0.29/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.29/0.47 = ( dsetconstr @ B
% 0.29/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(subset_type,type,
% 0.29/0.47 subset: $i > $i > $o ).
% 0.29/0.47
% 0.29/0.47 thf(subset,definition,
% 0.29/0.47 ( subset
% 0.29/0.47 = ( ^ [A: $i,B: $i] :
% 0.29/0.47 ! [Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ A )
% 0.29/0.47 => ( in @ Xx @ B ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(disjoint_type,type,
% 0.29/0.47 disjoint: $i > $i > $o ).
% 0.29/0.47
% 0.29/0.47 thf(setsmeet_type,type,
% 0.29/0.47 setsmeet: $i > $i > $o ).
% 0.29/0.47
% 0.29/0.47 thf(subsetI1_type,type,
% 0.29/0.47 subsetI1: $o ).
% 0.29/0.47
% 0.29/0.47 thf(subsetI1,definition,
% 0.29/0.47 ( subsetI1
% 0.29/0.47 = ( ! [A: $i,B: $i] :
% 0.29/0.47 ( ! [Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ A )
% 0.29/0.47 => ( in @ Xx @ B ) )
% 0.29/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(eqimpsubset2_type,type,
% 0.29/0.47 eqimpsubset2: $o ).
% 0.29/0.47
% 0.29/0.47 thf(eqimpsubset2,definition,
% 0.29/0.47 ( eqimpsubset2
% 0.29/0.47 = ( ! [A: $i,B: $i] :
% 0.29/0.47 ( ( A = B )
% 0.29/0.47 => ( subset @ B @ A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(eqimpsubset1,conjecture,
% 0.29/0.47 ( setextAx
% 0.29/0.47 => ( emptysetAx
% 0.29/0.47 => ( setadjoinAx
% 0.29/0.47 => ( powersetAx
% 0.29/0.47 => ( setunionAx
% 0.29/0.47 => ( omega0Ax
% 0.29/0.47 => ( omegaSAx
% 0.29/0.47 => ( omegaIndAx
% 0.29/0.47 => ( replAx
% 0.29/0.47 => ( foundationAx
% 0.29/0.47 => ( wellorderingAx
% 0.29/0.47 => ( descrp
% 0.29/0.47 => ( dsetconstrI
% 0.29/0.47 => ( dsetconstrEL
% 0.29/0.47 => ( dsetconstrER
% 0.29/0.47 => ( exuE1
% 0.29/0.47 => ( prop2setE
% 0.29/0.47 => ( emptysetE
% 0.29/0.47 => ( emptysetimpfalse
% 0.29/0.47 => ( notinemptyset
% 0.29/0.47 => ( exuE3e
% 0.29/0.47 => ( setext
% 0.29/0.47 => ( emptyI
% 0.29/0.47 => ( noeltsimpempty
% 0.29/0.47 => ( setbeta
% 0.29/0.47 => ( nonemptyE1
% 0.29/0.47 => ( nonemptyI
% 0.29/0.47 => ( nonemptyI1
% 0.29/0.47 => ( setadjoinIL
% 0.29/0.47 => ( emptyinunitempty
% 0.29/0.47 => ( setadjoinIR
% 0.29/0.47 => ( setadjoinE
% 0.29/0.47 => ( setadjoinOr
% 0.29/0.47 => ( setoftrueEq
% 0.29/0.47 => ( powersetI
% 0.29/0.47 => ( emptyinPowerset
% 0.29/0.47 => ( emptyInPowerset
% 0.29/0.47 => ( powersetE
% 0.29/0.47 => ( setunionI
% 0.29/0.47 => ( setunionE
% 0.29/0.47 => ( subPowSU
% 0.29/0.47 => ( exuE2
% 0.29/0.47 => ( nonemptyImpWitness
% 0.29/0.47 => ( uniqinunit
% 0.29/0.47 => ( notinsingleton
% 0.29/0.47 => ( eqinunit
% 0.29/0.47 => ( singletonsswitch
% 0.29/0.47 => ( upairsetE
% 0.29/0.47 => ( upairsetIL
% 0.29/0.47 => ( upairsetIR
% 0.29/0.47 => ( emptyE1
% 0.29/0.47 => ( vacuousDall
% 0.29/0.47 => ( quantDeMorgan1
% 0.29/0.47 => ( quantDeMorgan2
% 0.29/0.47 => ( quantDeMorgan3
% 0.29/0.47 => ( quantDeMorgan4
% 0.29/0.47 => ( prop2setI
% 0.29/0.47 => ( prop2set2propI
% 0.29/0.47 => ( notdexE
% 0.29/0.47 => ( notdallE
% 0.29/0.47 => ( exuI1
% 0.29/0.47 => ( exuI3
% 0.29/0.47 => ( exuI2
% 0.29/0.47 => ( inCongP
% 0.29/0.47 => ( in__Cong
% 0.29/0.47 => ( exuE3u
% 0.29/0.47 => ( exu__Cong
% 0.29/0.47 => ( emptyset__Cong
% 0.29/0.47 => ( setadjoin__Cong
% 0.29/0.47 => ( powerset__Cong
% 0.29/0.47 => ( setunion__Cong
% 0.29/0.47 => ( omega__Cong
% 0.29/0.47 => ( exuEu
% 0.29/0.47 => ( descr__Cong
% 0.29/0.47 => ( dsetconstr__Cong
% 0.29/0.47 => ( subsetI1
% 0.29/0.47 => ( eqimpsubset2
% 0.29/0.47 => ! [A: $i,B: $i] :
% 0.29/0.47 ( ( A = B )
% 0.29/0.47 => ( subset @ A @ B ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 %------------------------------------------------------------------------------
% 0.29/0.47 ------------------------
% 2.20/1.45 ---- Embedded in THF ---
% 2.20/1.45 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.20/1.45 %%% Generated on Tue Feb 24 12:27:35 EST 2026
% 2.20/1.45 %%% using '$$dhol' embedding, version 1.3.0.
% 2.20/1.45 %%% Logic specification used:
% 2.20/1.45 %%% thf(spec, logic, $$dhol).
% 2.20/1.45
% 2.20/1.45 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.rYUQzxuxZr/DTF2DTF20774.p
% See solution above
% 2.20/1.45 ------------------------
% 2.20/1.46 ---- Cleaned THF ---
% 2.20/1.46 thf(in_type,type,
% 2.20/1.46 in: $i > $i > $o ).
% 2.20/1.46
% 2.20/1.46 thf(exu_type,type,
% 2.20/1.46 exu: ( $i > $o ) > $o ).
% 2.20/1.46
% 2.20/1.46 thf(exu,definition,
% 2.20/1.46 ( exu
% 2.20/1.46 = ( ^ [Xphi: $i > $o] :
% 2.20/1.46 ? [Xx: $i] :
% 2.20/1.46 ( ( Xphi @ Xx )
% 2.20/1.46 & ! [Xy: $i] :
% 2.20/1.46 ( ( Xphi @ Xy )
% 2.20/1.46 => ( Xx = Xy ) ) ) ) ) ).
% 2.20/1.46
% 2.20/1.46 thf(setextAx_type,type,
% 2.20/1.46 setextAx: $o ).
% 2.20/1.46
% 2.20/1.46 thf(setextAx,definition,
% 2.20/1.46 ( setextAx
% 2.20/1.46 = ( ! [A: $i,B: $i] :
% 2.20/1.46 ( ! [Xx: $i] :
% 2.20/1.46 ( ( in @ Xx @ A )
% 2.20/1.46 <=> ( in @ Xx @ B ) )
% 2.20/1.46 => ( A = B ) ) ) ) ).
% 2.20/1.46
% 2.20/1.46 thf(emptyset_type,type,
% 2.20/1.46 emptyset: $i ).
% 2.20/1.46
% 2.20/1.46 thf(emptysetAx_type,type,
% 2.20/1.46 emptysetAx: $o ).
% 2.20/1.46
% 2.20/1.46 thf(emptysetAx,definition,
% 2.20/1.46 ( emptysetAx
% 2.20/1.46 = ( ! [Xx: $i] :
% 2.20/1.46 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.20/1.46
% 2.20/1.46 thf(setadjoin_type,type,
% 2.20/1.46 setadjoin: $i > $i > $i ).
% 2.20/1.46
% 2.20/1.46 thf(setadjoinAx_type,type,
% 2.20/1.46 setadjoinAx: $o ).
% 2.20/1.46
% 2.20/1.46 thf(setadjoinAx,definition,
% 2.20/1.46 ( setadjoinAx
% 2.20/1.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.20/1.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.20/1.46 <=> ( ( Xy = Xx )
% 2.20/1.46 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.20/1.46
% 2.20/1.46 thf(powerset_type,type,
% 2.20/1.46 powerset: $i > $i ).
% 2.20/1.46
% 2.20/1.46 thf(powersetAx_type,type,
% 2.20/1.46 powersetAx: $o ).
% 2.20/1.46
% 2.20/1.46 thf(powersetAx,definition,
% 2.20/1.46 ( powersetAx
% 2.20/1.46 = ( ! [A: $i,B: $i] :
% 2.20/1.46 ( ( in @ B @ ( powerset @ A ) )
% 2.20/1.46 <=> ! [Xx: $i] :
% 2.20/1.46 ( ( in @ Xx @ B )
% 2.20/1.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.20/1.46
% 2.20/1.46 thf(setunion_type,type,
% 2.20/1.46 setunion: $i > $i ).
% 2.20/1.46
% 2.20/1.46 thf(setunionAx_type,type,
% 2.20/1.46 setunionAx: $o ).
% 2.20/1.46
% 2.20/1.46 thf(setunionAx,definition,
% 2.20/1.46 ( setunionAx
% 2.20/1.46 = ( ! [A: $i,Xx: $i] :
% 2.20/1.46 ( ( in @ Xx @ ( setunion @ A ) )
% 2.20/1.46 <=> ? [B: $i] :
% 2.20/1.46 ( ( in @ Xx @ B )
% 2.20/1.46 & ( in @ B @ A ) ) ) ) ) ).
% 2.20/1.46
% 2.20/1.46 thf(omega_type,type,
% 2.20/1.46 omega: $i ).
% 2.20/1.46
% 2.20/1.46 thf(omega0Ax_type,type,
% 2.20/1.46 omega0Ax: $o ).
% 2.20/1.46
% 2.20/1.46 thf(omega0Ax,definition,
% 2.20/1.46 ( omega0Ax
% 2.20/1.46 = ( in @ emptyset @ omega ) ) ).
% 2.20/1.46
% 2.20/1.46 thf(omegaSAx_type,type,
% 2.20/1.46 omegaSAx: $o ).
% 2.20/1.46
% 2.20/1.46 thf(omegaSAx,definition,
% 2.20/1.46 ( omegaSAx
% 2.20/1.46 = ( ! [Xx: $i] :
% 2.20/1.46 ( ( in @ Xx @ omega )
% 2.20/1.46 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 2.20/1.46
% 2.20/1.46 thf(omegaIndAx_type,type,
% 2.20/1.46 omegaIndAx: $o ).
% 2.20/1.46
% 2.20/1.46 thf(omegaIndAx,definition,
% 2.20/1.46 ( omegaIndAx
% 2.20/1.46 = ( ! [A: $i] :
% 2.20/1.46 ( ( ( in @ emptyset @ A )
% 2.20/1.46 & ! [Xx: $i] :
% 2.20/1.46 ( ( ( in @ Xx @ omega )
% 2.20/1.46 & ( in @ Xx @ A ) )
% 2.20/1.46 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 2.20/1.46 => ! [Xx: $i] :
% 2.20/1.46 ( ( in @ Xx @ omega )
% 2.20/1.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.20/1.46
% 2.20/1.46 thf(replAx_type,type,
% 2.20/1.46 replAx: $o ).
% 2.20/1.46
% 2.20/1.46 thf(replAx,definition,
% 2.20/1.46 ( replAx
% 2.20/1.46 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 2.20/1.46 ( ! [Xx: $i] :
% 2.20/1.46 ( ( in @ Xx @ A )
% 2.20/1.46 => ( exu
% 2.20/1.46 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 2.20/1.46 => ? [B: $i] :
% 2.20/1.46 ! [Xx: $i] :
% 2.20/1.46 ( ( in @ Xx @ B )
% 2.20/1.46 <=> ? [Xy: $i] :
% 2.20/1.46 ( ( in @ Xy @ A )
% 2.20/1.46 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 2.20/1.46
% 2.20/1.46 thf(foundationAx_type,type,
% 2.20/1.46 foundationAx: $o ).
% 2.20/1.46
% 2.20/1.46 thf(foundationAx,definition,
% 2.20/1.46 ( foundationAx
% 2.20/1.46 = ( ! [A: $i] :
% 2.20/1.46 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.20/1.46 => ? [B: $i] :
% 2.20/1.46 ( ( in @ B @ A )
% 2.20/1.46 & ~ ? [Xx: $i] :
% 2.20/1.46 ( ( in @ Xx @ B )
% 2.20/1.46 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 2.20/1.46
% 2.20/1.46 thf(wellorderingAx_type,type,
% 2.20/1.46 wellorderingAx: $o ).
% 2.20/1.46
% 2.20/1.46 thf(wellorderingAx,definition,
% 2.20/1.46 ( wellorderingAx
% 2.20/1.46 = ( ! [A: $i] :
% 2.20/1.46 ? [B: $i] :
% 2.20/1.46 ( ! [C: $i] :
% 2.20/1.46 ( ( in @ C @ B )
% 2.20/1.46 => ! [Xx: $i] :
% 2.20/1.46 ( ( in @ Xx @ C )
% 2.20/1.46 => ( in @ Xx @ A ) ) )
% 2.20/1.46 & ! [Xx: $i,Xy: $i] :
% 2.20/1.46 ( ( ( in @ Xx @ A )
% 2.20/1.46 & ( in @ Xy @ A ) )
% 2.20/1.46 => ( ! [C: $i] :
% 2.20/1.46 ( ( in @ C @ B )
% 2.20/1.46 => ( ( in @ Xx @ C )
% 2.20/1.46 <=> ( in @ Xy @ C ) ) )
% 2.20/1.46 => ( Xx = Xy ) ) )
% 2.20/1.46 & ! [C: $i,D: $i] :
% 2.20/1.46 ( ( ( in @ C @ B )
% 2.20/1.46 & ( in @ D @ B ) )
% 2.20/1.46 => ( ! [Xx: $i] :
% 2.20/1.46 ( ( in @ Xx @ C )
% 2.20/1.46 => ( in @ Xx @ D ) )
% 2.20/1.46 | ! [Xx: $i] :
% 2.20/1.46 ( ( in @ Xx @ D )
% 2.20/1.46 => ( in @ Xx @ C ) ) ) )
% 2.20/1.46 & ! [C: $i] :
% 2.20/1.46 ( ( ! [Xx: $i] :
% 2.20/1.46 ( ( in @ Xx @ C )
% 2.20/1.46 => ( in @ Xx @ A ) )
% 2.20/1.46 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 2.20/1.46 => ? [D: $i,Xx: $i] :
% 2.20/1.46 ( ( in @ D @ B )
% 2.20/1.46 & ( in @ Xx @ C )
% 2.20/1.46 & ~ ? [Xy: $i] :
% 2.20/1.46 ( ( in @ Xy @ D )
% 2.20/1.46 & ( in @ Xy @ C ) )
% 2.20/1.46 & ! [E: $i] :
% 2.20/1.46 ( ( in @ E @ B )
% 2.20/1.46 => ( ! [Xy: $i] :
% 2.20/1.46 ( ( in @ Xy @ E )
% 2.20/1.46 => ( in @ Xy @ D ) )
% 2.20/1.46 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 2.20/1.46
% 2.20/1.46 thf(descr_type,type,
% 2.20/1.46 descr: ( $i > $o ) > $i ).
% 2.20/1.46
% 2.20/1.46 thf(descrp_type,type,
% 2.20/1.46 descrp: $o ).
% 2.20/1.46
% 2.20/1.46 thf(descrp,definition,
% 2.20/1.46 ( descrp
% 2.20/1.46 = ( ! [Xphi: $i > $o] :
% 2.20/1.46 ( ( exu
% 2.20/1.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.20/1.46 => ( Xphi
% 2.20/1.46 @ ( descr
% 2.20/1.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.20/1.46
% 2.20/1.46 thf(dsetconstr_type,type,
% 2.20/1.46 dsetconstr: $i > ( $i > $o ) > $i ).
% 2.20/1.46
% 2.20/1.46 thf(dsetconstrI_type,type,
% 2.20/1.46 dsetconstrI: $o ).
% 2.20/1.46
% 2.20/1.46 thf(dsetconstrI,definition,
% 2.20/1.46 ( dsetconstrI
% 2.20/1.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.20/1.46 ( ( in @ Xx @ A )
% 2.20/1.46 => ( ( Xphi @ Xx )
% 2.20/1.46 => ( in @ Xx
% 2.20/1.46 @ ( dsetconstr @ A
% 2.20/1.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.20/1.46
% 2.20/1.46 thf(dsetconstrEL_type,type,
% 2.20/1.46 dsetconstrEL: $o ).
% 2.20/1.46
% 2.20/1.46 thf(dsetconstrEL,definition,
% 2.20/1.46 ( dsetconstrEL
% 2.20/1.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.20/1.46 ( ( in @ Xx
% 2.20/1.46 @ ( dsetconstr @ A
% 2.20/1.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.20/1.46 => ( in @ Xx @ A ) ) ) ) ).
% 2.20/1.46
% 2.20/1.46 thf(dsetconstrER_type,type,
% 2.20/1.46 dsetconstrER: $o ).
% 2.20/1.46
% 2.20/1.46 thf(dsetconstrER,definition,
% 2.20/1.46 ( dsetconstrER
% 2.20/1.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.20/1.46 ( ( in @ Xx
% 2.20/1.46 @ ( dsetconstr @ A
% 2.20/1.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.20/1.46 => ( Xphi @ Xx ) ) ) ) ).
% 2.20/1.46
% 2.20/1.46 thf(exuE1_type,type,
% 2.20/1.46 exuE1: $o ).
% 2.20/1.46
% 2.20/1.46 thf(exuE1,definition,
% 2.20/1.46 ( exuE1
% 2.20/1.46 = ( ! [Xphi: $i > $o] :
% 2.20/1.46 ( ( exu
% 2.20/1.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.20/1.46 => ? [Xx: $i] :
% 2.20/1.46 ( ( Xphi @ Xx )
% 2.20/1.46 & ! [Xy: $i] :
% 2.20/1.46 ( ( Xphi @ Xy )
% 2.20/1.46 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.20/1.46
% 2.20/1.46 thf(prop2set_type,type,
% 2.20/1.46 prop2set: $o > $i ).
% 2.20/1.46
% 2.20/1.46 thf(prop2setE_type,type,
% 2.20/1.46 prop2setE: $o ).
% 2.20/1.46
% 2.20/1.46 thf(prop2setE,definition,
% 2.20/1.46 ( prop2setE
% 2.20/1.46 = ( ! [Xphi: $o,Xx: $i] :
% 2.20/1.46 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 2.20/1.46 => Xphi ) ) ) ).
% 2.20/1.46
% 2.20/1.46 thf(emptysetE_type,type,
% 2.20/1.46 emptysetE: $o ).
% 2.20/1.46
% 2.20/1.46 thf(emptysetE,definition,
% 2.20/1.46 ( emptysetE
% 2.20/1.46 = ( ! [Xx: $i] :
% 2.20/1.46 ( ( in @ Xx @ emptyset )
% 2.20/1.46 => ! [Xphi: $o] : Xphi ) ) ) ).
% 2.20/1.46
% 2.20/1.46 thf(emptysetimpfalse_type,type,
% 2.20/1.46 emptysetimpfalse: $o ).
% 2.20/1.46
% 2.20/1.46 thf(emptysetimpfalse,definition,
% 2.20/1.46 ( emptysetimpfalse
% 2.20/1.46 = ( ! [Xx: $i] :
% 2.20/1.46 ( ( in @ Xx @ emptyset )
% 2.20/1.46 => $false ) ) ) ).
% 2.20/1.46
% 2.20/1.46 thf(notinemptyset_type,type,
% 2.20/1.46 notinemptyset: $o ).
% 2.20/1.46
% 2.20/1.46 thf(notinemptyset,definition,
% 2.20/1.46 ( notinemptyset
% 2.20/1.46 = ( ! [Xx: $i] :
% 2.20/1.46 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.20/1.46
% 2.20/1.46 thf(exuE3e_type,type,
% 2.20/1.46 exuE3e: $o ).
% 2.20/1.46
% 2.20/1.46 thf(exuE3e,definition,
% 2.20/1.46 ( exuE3e
% 2.20/1.46 = ( ! [Xphi: $i > $o] :
% 2.20/1.46 ( ( exu
% 2.20/1.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.20/1.46 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 2.20/1.46
% 2.20/1.46 thf(setext_type,type,
% 2.20/1.46 setext: $o ).
% 2.20/1.46
% 2.20/1.46 thf(setext,definition,
% 2.20/1.46 ( setext
% 2.20/1.46 = ( ! [A: $i,B: $i] :
% 2.20/1.46 ( ! [Xx: $i] :
% 2.20/1.46 ( ( in @ Xx @ A )
% 2.20/1.46 => ( in @ Xx @ B ) )
% 2.20/1.46 => ( ! [Xx: $i] :
% 2.20/1.46 ( ( in @ Xx @ B )
% 2.20/1.46 => ( in @ Xx @ A ) )
% 2.20/1.46 => ( A = B ) ) ) ) ) ).
% 2.20/1.46
% 2.20/1.46 thf(emptyI_type,type,
% 2.20/1.46 emptyI: $o ).
% 2.20/1.46
% 2.20/1.46 thf(emptyI,definition,
% 2.20/1.46 ( emptyI
% 2.20/1.46 = ( ! [A: $i] :
% 2.20/1.46 ( ! [Xx: $i] :
% 2.20/1.46 ~ ( in @ Xx @ A )
% 2.20/1.46 => ( A = emptyset ) ) ) ) ).
% 2.20/1.46
% 2.20/1.46 thf(noeltsimpempty_type,type,
% 2.20/1.46 noeltsimpempty: $o ).
% 2.20/1.46
% 2.20/1.46 thf(noeltsimpempty,definition,
% 2.20/1.46 ( noeltsimpempty
% 2.20/1.46 = ( ! [A: $i] :
% 2.20/1.46 ( ! [Xx: $i] :
% 2.20/1.46 ~ ( in @ Xx @ A )
% 2.20/1.46 => ( A = emptyset ) ) ) ) ).
% 2.20/1.46
% 2.20/1.46 thf(setbeta_type,type,
% 2.20/1.46 setbeta: $o ).
% 2.20/1.46
% 2.20/1.46 thf(setbeta,definition,
% 2.20/1.46 ( setbeta
% 2.20/1.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.20/1.46 ( ( in @ Xx @ A )
% 2.20/1.46 => ( ( in @ Xx
% 2.20/1.46 @ ( dsetconstr @ A
% 2.20/1.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.20/1.46 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 2.20/1.46
% 2.20/1.46 thf(nonempty_type,type,
% 2.20/1.46 nonempty: $i > $o ).
% 2.20/1.46
% 2.20/1.46 thf(nonempty,definition,
% 2.20/1.46 ( nonempty
% 2.20/1.46 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 2.20/1.46
% 2.20/1.46 thf(nonemptyE1_type,type,
% 2.20/1.46 nonemptyE1: $o ).
% 2.20/1.46
% 2.20/1.46 thf(nonemptyE1,definition,
% 2.20/1.46 ( nonemptyE1
% 2.20/1.46 = ( ! [A: $i] :
% 2.20/1.46 ( ( nonempty @ A )
% 2.20/1.46 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 2.20/1.46
% 2.20/1.46 thf(nonemptyI_type,type,
% 2.20/1.46 nonemptyI: $o ).
% 2.20/1.46
% 2.20/1.46 thf(nonemptyI,definition,
% 2.20/1.46 ( nonemptyI
% 2.20/1.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.20/1.46 ( ( in @ Xx @ A )
% 2.20/1.46 => ( ( Xphi @ Xx )
% 2.20/1.46 => ( nonempty
% 2.20/1.46 @ ( dsetconstr @ A
% 2.20/1.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.20/1.46
% 2.20/1.46 thf(nonemptyI1_type,type,
% 2.20/1.46 nonemptyI1: $o ).
% 2.20/1.46
% 2.20/1.46 thf(nonemptyI1,definition,
% 2.20/1.46 ( nonemptyI1
% 2.20/1.46 = ( ! [A: $i] :
% 2.20/1.46 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.20/1.46 => ( nonempty @ A ) ) ) ) ).
% 2.20/1.46
% 2.20/1.46 thf(setadjoinIL_type,type,
% 2.20/1.46 setadjoinIL: $o ).
% 2.20/1.46
% 2.20/1.46 thf(setadjoinIL,definition,
% 2.20/1.46 ( setadjoinIL
% 2.20/1.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 2.20/1.46
% 2.20/1.46 thf(emptyinunitempty_type,type,
% 2.20/1.46 emptyinunitempty: $o ).
% 2.20/1.46
% 2.20/1.46 thf(emptyinunitempty,definition,
% 2.20/1.46 ( emptyinunitempty
% 2.20/1.46 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 2.20/1.46
% 2.20/1.46 thf(setadjoinIR_type,type,
% 2.20/1.46 setadjoinIR: $o ).
% 2.20/1.46
% 2.20/1.46 thf(setadjoinIR,definition,
% 2.20/1.46 ( setadjoinIR
% 2.20/1.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.20/1.46 ( ( in @ Xy @ A )
% 2.20/1.46 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 2.20/1.46
% 2.20/1.46 thf(setadjoinE_type,type,
% 2.20/1.46 setadjoinE: $o ).
% 2.20/1.46
% 2.20/1.46 thf(setadjoinE,definition,
% 2.20/1.46 ( setadjoinE
% 2.20/1.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.20/1.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.20/1.46 => ! [Xphi: $o] :
% 2.20/1.46 ( ( ( Xy = Xx )
% 2.20/1.46 => Xphi )
% 2.20/1.46 => ( ( ( in @ Xy @ A )
% 2.20/1.46 => Xphi )
% 2.20/1.46 => Xphi ) ) ) ) ) ).
% 2.20/1.46
% 2.20/1.46 thf(setadjoinOr_type,type,
% 2.20/1.46 setadjoinOr: $o ).
% 2.20/1.46
% 2.20/1.46 thf(setadjoinOr,definition,
% 2.20/1.46 ( setadjoinOr
% 2.20/1.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.20/1.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.20/1.46 => ( ( Xy = Xx )
% 2.20/1.46 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.20/1.46
% 2.20/1.46 thf(setoftrueEq_type,type,
% 2.20/1.46 setoftrueEq: $o ).
% 2.20/1.46
% 2.20/1.46 thf(setoftrueEq,definition,
% 2.20/1.46 ( setoftrueEq
% 2.20/1.46 = ( ! [A: $i] :
% 2.20/1.46 ( ( dsetconstr @ A
% 2.20/1.46 @ ^ [Xx: $i] : $true )
% 2.20/1.46 = A ) ) ) ).
% 2.20/1.46
% 2.20/1.46 thf(powersetI_type,type,
% 2.20/1.46 powersetI: $o ).
% 2.20/1.46
% 2.20/1.46 thf(powersetI,definition,
% 2.20/1.46 ( powersetI
% 2.20/1.46 = ( ! [A: $i,B: $i] :
% 2.20/1.46 ( ! [Xx: $i] :
% 2.20/1.46 ( ( in @ Xx @ B )
% 2.20/1.46 => ( in @ Xx @ A ) )
% 2.20/1.46 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.20/1.46
% 2.20/1.46 thf(emptyinPowerset_type,type,
% 2.20/1.46 emptyinPowerset: $o ).
% 2.20/1.46
% 2.20/1.46 thf(emptyinPowerset,definition,
% 2.20/1.46 ( emptyinPowerset
% 2.20/1.46 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.20/1.46
% 2.20/1.46 thf(emptyInPowerset_type,type,
% 2.20/1.46 emptyInPowerset: $o ).
% 2.20/1.46
% 2.20/1.46 thf(emptyInPowerset,definition,
% 2.20/1.46 ( emptyInPowerset
% 2.20/1.46 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.20/1.46
% 2.20/1.46 thf(powersetE_type,type,
% 2.20/1.46 powersetE: $o ).
% 2.20/1.46
% 2.20/1.46 thf(powersetE,definition,
% 2.20/1.46 ( powersetE
% 2.20/1.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.20/1.46 ( ( in @ B @ ( powerset @ A ) )
% 2.20/1.46 => ( ( in @ Xx @ B )
% 2.20/1.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.20/1.46
% 2.20/1.46 thf(setunionI_type,type,
% 2.20/1.46 setunionI: $o ).
% 2.20/1.46
% 2.20/1.46 thf(setunionI,definition,
% 2.20/1.46 ( setunionI
% 2.20/1.46 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.20/1.46 ( ( in @ Xx @ B )
% 2.20/1.46 => ( ( in @ B @ A )
% 2.20/1.46 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 2.20/1.46
% 2.20/1.46 thf(setunionE_type,type,
% 2.20/1.46 setunionE: $o ).
% 2.20/1.46
% 2.20/1.46 thf(setunionE,definition,
% 2.20/1.46 ( setunionE
% 2.20/1.46 = ( ! [A: $i,Xx: $i] :
% 2.20/1.46 ( ( in @ Xx @ ( setunion @ A ) )
% 2.20/1.46 => ! [Xphi: $o] :
% 2.20/1.46 ( ! [B: $i] :
% 2.20/1.46 ( ( in @ Xx @ B )
% 2.20/1.46 => ( ( in @ B @ A )
% 2.20/1.46 => Xphi ) )
% 2.20/1.46 => Xphi ) ) ) ) ).
% 2.20/1.46
% 2.20/1.46 thf(subPowSU_type,type,
% 2.20/1.46 subPowSU: $o ).
% 2.20/1.46
% 2.20/1.46 thf(subPowSU,definition,
% 2.20/1.46 ( subPowSU
% 2.20/1.46 = ( ! [A: $i,Xx: $i] :
% 2.20/1.46 ( ( in @ Xx @ A )
% 2.20/1.46 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 2.20/1.46
% 2.20/1.46 thf(exuE2_type,type,
% 2.20/1.46 exuE2: $o ).
% 2.20/1.46
% 2.20/1.46 thf(exuE2,definition,
% 2.20/1.46 ( exuE2
% 2.20/1.46 = ( ! [Xphi: $i > $o] :
% 2.20/1.46 ( ( exu
% 2.20/1.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.20/1.46 => ? [Xx: $i] :
% 2.20/1.46 ! [Xy: $i] :
% 2.20/1.46 ( ( Xphi @ Xy )
% 2.20/1.46 <=> ( Xy = Xx ) ) ) ) ) ).
% 2.20/1.46
% 2.20/1.46 thf(nonemptyImpWitness_type,type,
% 2.20/1.46 nonemptyImpWitness: $o ).
% 2.20/1.46
% 2.20/1.46 thf(nonemptyImpWitness,definition,
% 2.20/1.46 ( nonemptyImpWitness
% 2.20/1.46 = ( ! [A: $i] :
% 2.20/1.46 ( ( nonempty @ A )
% 2.20/1.46 => ? [Xx: $i] :
% 2.20/1.46 ( ( in @ Xx @ A )
% 2.20/1.46 & $true ) ) ) ) ).
% 2.20/1.46
% 2.20/1.46 thf(uniqinunit_type,type,
% 2.20/1.46 uniqinunit: $o ).
% 2.20/1.46
% 2.20/1.46 thf(uniqinunit,definition,
% 2.20/1.46 ( uniqinunit
% 2.20/1.46 = ( ! [Xx: $i,Xy: $i] :
% 2.20/1.46 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.20/1.46 => ( Xx = Xy ) ) ) ) ).
% 2.20/1.46
% 2.20/1.46 thf(notinsingleton_type,type,
% 2.20/1.46 notinsingleton: $o ).
% 2.20/1.46
% 2.20/1.46 thf(notinsingleton,definition,
% 2.20/1.46 ( notinsingleton
% 2.20/1.46 = ( ! [Xx: $i,Xy: $i] :
% 2.20/1.46 ( ( Xx != Xy )
% 2.20/1.46 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.20/1.46
% 2.20/1.46 thf(eqinunit_type,type,
% 2.20/1.46 eqinunit: $o ).
% 2.20/1.46
% 2.20/1.46 thf(eqinunit,definition,
% 2.20/1.46 ( eqinunit
% 2.20/1.46 = ( ! [Xx: $i,Xy: $i] :
% 2.20/1.46 ( ( Xx = Xy )
% 2.20/1.46 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.20/1.46
% 2.20/1.46 thf(singletonsswitch_type,type,
% 2.20/1.46 singletonsswitch: $o ).
% 2.20/1.46
% 2.20/1.46 thf(singletonsswitch,definition,
% 2.20/1.46 ( singletonsswitch
% 2.20/1.46 = ( ! [Xx: $i,Xy: $i] :
% 2.20/1.46 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.20/1.46 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.20/1.46
% 2.20/1.46 thf(upairsetE_type,type,
% 2.20/1.46 upairsetE: $o ).
% 2.20/1.46
% 2.20/1.46 thf(upairsetE,definition,
% 2.20/1.46 ( upairsetE
% 2.20/1.46 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.20/1.46 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.20/1.46 => ( ( Xz = Xx )
% 2.20/1.46 | ( Xz = Xy ) ) ) ) ) ).
% 2.20/1.46
% 2.20/1.46 thf(upairsetIL_type,type,
% 2.20/1.46 upairsetIL: $o ).
% 2.20/1.46
% 2.20/1.46 thf(upairsetIL,definition,
% 2.20/1.46 ( upairsetIL
% 2.20/1.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.20/1.46
% 2.20/1.46 thf(upairsetIR_type,type,
% 2.20/1.46 upairsetIR: $o ).
% 2.20/1.46
% 2.20/1.46 thf(upairsetIR,definition,
% 2.20/1.46 ( upairsetIR
% 2.20/1.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.20/1.46
% 2.20/1.46 thf(emptyE1_type,type,
% 2.20/1.46 emptyE1: $o ).
% 2.20/1.46
% 2.20/1.46 thf(emptyE1,definition,
% 2.20/1.46 ( emptyE1
% 2.20/1.46 = ( ! [A: $i,Xphi: $i > $o] :
% 2.20/1.46 ( ? [Xx: $i] :
% 2.20/1.46 ( ( in @ Xx @ A )
% 2.20/1.46 & ( Xphi @ Xx ) )
% 2.20/1.46 => ( ( ( dsetconstr @ A
% 2.20/1.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.20/1.46 = emptyset )
% 2.20/1.46 => $false ) ) ) ) ).
% 2.20/1.46
% 2.20/1.46 thf(vacuousDall_type,type,
% 2.20/1.46 vacuousDall: $o ).
% 2.20/1.46
% 2.20/1.46 thf(vacuousDall,definition,
% 2.20/1.46 ( vacuousDall
% 2.20/1.46 = ( ! [Xphi: $i > $o,Xx: $i] :
% 2.20/1.46 ( ( in @ Xx @ emptyset )
% 2.20/1.46 => ( Xphi @ Xx ) ) ) ) ).
% 2.20/1.46
% 2.20/1.46 thf(quantDeMorgan1_type,type,
% 2.20/1.46 quantDeMorgan1: $o ).
% 2.20/1.46
% 2.20/1.46 thf(quantDeMorgan1,definition,
% 2.20/1.46 ( quantDeMorgan1
% 2.20/1.46 = ( ! [A: $i,Xphi: $i > $o] :
% 2.20/1.46 ( ~ ! [Xx: $i] :
% 2.20/1.46 ( ( in @ Xx @ A )
% 2.20/1.46 => ( Xphi @ Xx ) )
% 2.20/1.46 => ? [Xx: $i] :
% 2.20/1.46 ( ( in @ Xx @ A )
% 2.20/1.46 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.20/1.46
% 2.20/1.46 thf(quantDeMorgan2_type,type,
% 2.20/1.46 quantDeMorgan2: $o ).
% 2.20/1.47
% 2.20/1.47 thf(quantDeMorgan2,definition,
% 2.20/1.47 ( quantDeMorgan2
% 2.20/1.47 = ( ! [A: $i,Xphi: $i > $o] :
% 2.20/1.47 ( ! [Xx: $i] :
% 2.20/1.47 ( ( in @ Xx @ A )
% 2.20/1.47 => ~ ( Xphi @ Xx ) )
% 2.20/1.47 => ~ ? [Xx: $i] :
% 2.20/1.47 ( ( in @ Xx @ A )
% 2.20/1.47 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.20/1.47
% 2.20/1.47 thf(quantDeMorgan3_type,type,
% 2.20/1.47 quantDeMorgan3: $o ).
% 2.20/1.47
% 2.20/1.47 thf(quantDeMorgan3,definition,
% 2.20/1.47 ( quantDeMorgan3
% 2.20/1.47 = ( ! [A: $i,Xphi: $i > $o] :
% 2.20/1.47 ( ~ ? [Xx: $i] :
% 2.20/1.47 ( ( in @ Xx @ A )
% 2.20/1.47 & ( Xphi @ Xx ) )
% 2.20/1.47 => ! [Xx: $i] :
% 2.20/1.47 ( ( in @ Xx @ A )
% 2.20/1.47 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.20/1.47
% 2.20/1.47 thf(quantDeMorgan4_type,type,
% 2.20/1.47 quantDeMorgan4: $o ).
% 2.20/1.47
% 2.20/1.47 thf(quantDeMorgan4,definition,
% 2.20/1.47 ( quantDeMorgan4
% 2.20/1.47 = ( ! [A: $i,Xphi: $i > $o] :
% 2.20/1.47 ( ? [Xx: $i] :
% 2.20/1.47 ( ( in @ Xx @ A )
% 2.20/1.47 & ~ ( Xphi @ Xx ) )
% 2.20/1.47 => ~ ! [Xx: $i] :
% 2.20/1.47 ( ( in @ Xx @ A )
% 2.20/1.47 => ( Xphi @ Xx ) ) ) ) ) ).
% 2.20/1.47
% 2.20/1.47 thf(prop2setI_type,type,
% 2.20/1.47 prop2setI: $o ).
% 2.20/1.47
% 2.20/1.47 thf(prop2setI,definition,
% 2.20/1.47 ( prop2setI
% 2.20/1.47 = ( ! [Xphi: $o] :
% 2.20/1.47 ( Xphi
% 2.20/1.47 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.20/1.47
% 2.20/1.47 thf(set2prop_type,type,
% 2.20/1.47 set2prop: $i > $o ).
% 2.20/1.47
% 2.20/1.47 thf(prop2set2propI_type,type,
% 2.20/1.47 prop2set2propI: $o ).
% 2.20/1.47
% 2.20/1.47 thf(prop2set2propI,definition,
% 2.20/1.47 ( prop2set2propI
% 2.20/1.47 = ( ! [Xphi: $o] :
% 2.20/1.47 ( Xphi
% 2.20/1.47 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.20/1.47
% 2.20/1.47 thf(notdexE_type,type,
% 2.20/1.47 notdexE: $o ).
% 2.20/1.47
% 2.20/1.47 thf(notdexE,definition,
% 2.20/1.47 ( notdexE
% 2.20/1.47 = ( ! [A: $i,Xphi: $i > $o] :
% 2.20/1.47 ( ~ ? [Xx: $i] :
% 2.20/1.47 ( ( in @ Xx @ A )
% 2.20/1.47 & ( Xphi @ Xx ) )
% 2.20/1.47 => ! [Xx: $i] :
% 2.20/1.47 ( ( in @ Xx @ A )
% 2.20/1.47 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.20/1.47
% 2.20/1.47 thf(notdallE_type,type,
% 2.20/1.47 notdallE: $o ).
% 2.20/1.47
% 2.20/1.47 thf(notdallE,definition,
% 2.20/1.47 ( notdallE
% 2.20/1.47 = ( ! [A: $i,Xphi: $i > $o] :
% 2.20/1.47 ( ~ ! [Xx: $i] :
% 2.20/1.47 ( ( in @ Xx @ A )
% 2.20/1.47 => ( Xphi @ Xx ) )
% 2.20/1.47 => ? [Xx: $i] :
% 2.20/1.47 ( ( in @ Xx @ A )
% 2.20/1.47 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.20/1.47
% 2.20/1.47 thf(exuI1_type,type,
% 2.20/1.47 exuI1: $o ).
% 2.20/1.47
% 2.20/1.47 thf(exuI1,definition,
% 2.20/1.47 ( exuI1
% 2.20/1.47 = ( ! [Xphi: $i > $o] :
% 2.20/1.47 ( ? [Xx: $i] :
% 2.20/1.47 ( ( Xphi @ Xx )
% 2.20/1.47 & ! [Xy: $i] :
% 2.20/1.47 ( ( Xphi @ Xy )
% 2.20/1.47 => ( Xx = Xy ) ) )
% 2.20/1.47 => ( exu
% 2.20/1.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.20/1.47
% 2.20/1.47 thf(exuI3_type,type,
% 2.20/1.47 exuI3: $o ).
% 2.20/1.47
% 2.20/1.47 thf(exuI3,definition,
% 2.20/1.47 ( exuI3
% 2.20/1.47 = ( ! [Xphi: $i > $o] :
% 2.20/1.47 ( ? [Xx: $i] : ( Xphi @ Xx )
% 2.20/1.47 => ( ! [Xx: $i,Xy: $i] :
% 2.20/1.47 ( ( Xphi @ Xx )
% 2.20/1.47 => ( ( Xphi @ Xy )
% 2.20/1.47 => ( Xx = Xy ) ) )
% 2.20/1.47 => ( exu
% 2.20/1.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.20/1.47
% 2.20/1.47 thf(exuI2_type,type,
% 2.20/1.47 exuI2: $o ).
% 2.20/1.47
% 2.20/1.47 thf(exuI2,definition,
% 2.20/1.47 ( exuI2
% 2.20/1.47 = ( ! [Xphi: $i > $o] :
% 2.20/1.47 ( ? [Xx: $i] :
% 2.20/1.47 ! [Xy: $i] :
% 2.20/1.47 ( ( Xphi @ Xy )
% 2.20/1.47 <=> ( Xy = Xx ) )
% 2.20/1.47 => ( exu
% 2.20/1.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.20/1.47
% 2.20/1.47 thf(inCongP_type,type,
% 2.20/1.47 inCongP: $o ).
% 2.20/1.47
% 2.20/1.47 thf(inCongP,definition,
% 2.20/1.47 ( inCongP
% 2.20/1.47 = ( ! [A: $i,B: $i] :
% 2.20/1.47 ( ( A = B )
% 2.20/1.47 => ! [Xx: $i,Xy: $i] :
% 2.20/1.47 ( ( Xx = Xy )
% 2.20/1.47 => ( ( in @ Xx @ A )
% 2.20/1.47 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.20/1.47
% 2.20/1.47 thf(in__Cong_type,type,
% 2.20/1.47 in__Cong: $o ).
% 2.20/1.47
% 2.20/1.47 thf(in__Cong,definition,
% 2.20/1.47 ( in__Cong
% 2.20/1.47 = ( ! [A: $i,B: $i] :
% 2.20/1.47 ( ( A = B )
% 2.20/1.47 => ! [Xx: $i,Xy: $i] :
% 2.20/1.47 ( ( Xx = Xy )
% 2.20/1.47 => ( ( in @ Xx @ A )
% 2.20/1.47 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.20/1.47
% 2.20/1.47 thf(exuE3u_type,type,
% 2.20/1.47 exuE3u: $o ).
% 2.20/1.47
% 2.20/1.47 thf(exuE3u,definition,
% 2.20/1.47 ( exuE3u
% 2.20/1.47 = ( ! [Xphi: $i > $o] :
% 2.20/1.47 ( ( exu
% 2.20/1.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.20/1.47 => ! [Xx: $i,Xy: $i] :
% 2.20/1.47 ( ( Xphi @ Xx )
% 2.20/1.47 => ( ( Xphi @ Xy )
% 2.20/1.47 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.20/1.47
% 2.20/1.47 thf(exu__Cong_type,type,
% 2.20/1.47 exu__Cong: $o ).
% 2.20/1.47
% 2.20/1.47 thf(exu__Cong,definition,
% 2.20/1.47 ( exu__Cong
% 2.20/1.47 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.20/1.47 ( ! [Xx: $i,Xy: $i] :
% 2.20/1.47 ( ( Xx = Xy )
% 2.20/1.47 => ( ( Xphi @ Xx )
% 2.20/1.47 <=> ( Xpsi @ Xy ) ) )
% 2.20/1.47 => ( ( exu
% 2.20/1.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.20/1.47 <=> ( exu
% 2.20/1.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 2.20/1.47
% 2.20/1.47 thf(emptyset__Cong_type,type,
% 2.20/1.47 emptyset__Cong: $o ).
% 2.20/1.47
% 2.20/1.47 thf(emptyset__Cong,definition,
% 2.20/1.47 ( emptyset__Cong
% 2.20/1.47 = ( emptyset = emptyset ) ) ).
% 2.20/1.47
% 2.20/1.47 thf(setadjoin__Cong_type,type,
% 2.20/1.47 setadjoin__Cong: $o ).
% 2.20/1.47
% 2.20/1.47 thf(setadjoin__Cong,definition,
% 2.20/1.47 ( setadjoin__Cong
% 2.20/1.47 = ( ! [Xx: $i,Xy: $i] :
% 2.20/1.47 ( ( Xx = Xy )
% 2.20/1.47 => ! [Xz: $i,Xu: $i] :
% 2.20/1.47 ( ( Xz = Xu )
% 2.20/1.47 => ( ( setadjoin @ Xx @ Xz )
% 2.20/1.47 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 2.20/1.47
% 2.20/1.47 thf(powerset__Cong_type,type,
% 2.20/1.47 powerset__Cong: $o ).
% 2.20/1.47
% 2.20/1.47 thf(powerset__Cong,definition,
% 2.20/1.47 ( powerset__Cong
% 2.20/1.47 = ( ! [A: $i,B: $i] :
% 2.20/1.47 ( ( A = B )
% 2.20/1.47 => ( ( powerset @ A )
% 2.20/1.47 = ( powerset @ B ) ) ) ) ) ).
% 2.20/1.47
% 2.20/1.47 thf(setunion__Cong_type,type,
% 2.20/1.47 setunion__Cong: $o ).
% 2.20/1.47
% 2.20/1.47 thf(setunion__Cong,definition,
% 2.20/1.47 ( setunion__Cong
% 2.20/1.47 = ( ! [A: $i,B: $i] :
% 2.20/1.47 ( ( A = B )
% 2.20/1.47 => ( ( setunion @ A )
% 2.20/1.47 = ( setunion @ B ) ) ) ) ) ).
% 2.20/1.47
% 2.20/1.47 thf(omega__Cong_type,type,
% 2.20/1.47 omega__Cong: $o ).
% 2.20/1.47
% 2.20/1.47 thf(omega__Cong,definition,
% 2.20/1.47 ( omega__Cong
% 2.20/1.47 = ( omega = omega ) ) ).
% 2.20/1.47
% 2.20/1.47 thf(exuEu_type,type,
% 2.20/1.47 exuEu: $o ).
% 2.20/1.47
% 2.20/1.47 thf(exuEu,definition,
% 2.20/1.47 ( exuEu
% 2.20/1.47 = ( ! [Xphi: $i > $o] :
% 2.20/1.47 ( ( exu
% 2.20/1.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.20/1.47 => ! [Xx: $i,Xy: $i] :
% 2.20/1.47 ( ( Xphi @ Xx )
% 2.20/1.47 => ( ( Xphi @ Xy )
% 2.20/1.47 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.20/1.47
% 2.20/1.47 thf(descr__Cong_type,type,
% 2.20/1.47 descr__Cong: $o ).
% 2.20/1.47
% 2.20/1.47 thf(descr__Cong,definition,
% 2.20/1.47 ( descr__Cong
% 2.20/1.47 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.20/1.47 ( ! [Xx: $i,Xy: $i] :
% 2.20/1.47 ( ( Xx = Xy )
% 2.20/1.47 => ( ( Xphi @ Xx )
% 2.20/1.47 <=> ( Xpsi @ Xy ) ) )
% 2.20/1.47 => ( ( exu
% 2.20/1.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.20/1.47 => ( ( exu
% 2.20/1.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 2.20/1.47 => ( ( descr
% 2.20/1.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.20/1.47 = ( descr
% 2.20/1.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 2.20/1.47
% 2.20/1.47 thf(dsetconstr__Cong_type,type,
% 2.20/1.47 dsetconstr__Cong: $o ).
% 2.20/1.47
% 2.20/1.47 thf(dsetconstr__Cong,definition,
% 2.20/1.47 ( dsetconstr__Cong
% 2.20/1.47 = ( ! [A: $i,B: $i] :
% 2.20/1.47 ( ( A = B )
% 2.20/1.47 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.20/1.47 ( ! [Xx: $i] :
% 2.20/1.47 ( ( in @ Xx @ A )
% 2.20/1.47 => ! [Xy: $i] :
% 2.20/1.47 ( ( in @ Xy @ B )
% 2.20/1.47 => ( ( Xx = Xy )
% 2.20/1.47 => ( ( Xphi @ Xx )
% 2.20/1.47 <=> ( Xpsi @ Xy ) ) ) ) )
% 2.20/1.47 => ( ( dsetconstr @ A
% 2.20/1.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.20/1.47 = ( dsetconstr @ B
% 2.20/1.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 2.20/1.47
% 2.20/1.47 thf(subset_type,type,
% 2.20/1.47 subset: $i > $i > $o ).
% 2.20/1.47
% 2.20/1.47 thf(subset,definition,
% 2.20/1.47 ( subset
% 2.20/1.47 = ( ^ [A: $i,B: $i] :
% 2.20/1.47 ! [Xx: $i] :
% 2.20/1.47 ( ( in @ Xx @ A )
% 2.20/1.47 => ( in @ Xx @ B ) ) ) ) ).
% 2.20/1.47
% 2.20/1.47 thf(disjoint_type,type,
% 2.20/1.47 disjoint: $i > $i > $o ).
% 2.20/1.47
% 2.20/1.47 thf(setsmeet_type,type,
% 2.20/1.47 setsmeet: $i > $i > $o ).
% 2.20/1.47
% 2.20/1.47 thf(subsetI1_type,type,
% 2.20/1.47 subsetI1: $o ).
% 2.20/1.47
% 2.20/1.47 thf(subsetI1,definition,
% 2.20/1.47 ( subsetI1
% 2.20/1.47 = ( ! [A: $i,B: $i] :
% 2.20/1.47 ( ! [Xx: $i] :
% 2.20/1.47 ( ( in @ Xx @ A )
% 2.20/1.47 => ( in @ Xx @ B ) )
% 2.20/1.47 => ( subset @ A @ B ) ) ) ) ).
% 2.20/1.47
% 2.20/1.47 thf(eqimpsubset2_type,type,
% 2.20/1.47 eqimpsubset2: $o ).
% 2.20/1.47
% 2.20/1.47 thf(eqimpsubset2,definition,
% 2.20/1.47 ( eqimpsubset2
% 2.20/1.47 = ( ! [A: $i,B: $i] :
% 2.20/1.47 ( ( A = B )
% 2.20/1.47 => ( subset @ B @ A ) ) ) ) ).
% 2.20/1.47
% 2.20/1.47 thf(eqimpsubset1,conjecture,
% 2.20/1.47 ( setextAx
% 2.20/1.47 => ( emptysetAx
% 2.20/1.47 => ( setadjoinAx
% 2.20/1.47 => ( powersetAx
% 2.20/1.47 => ( setunionAx
% 2.20/1.47 => ( omega0Ax
% 2.20/1.47 => ( omegaSAx
% 2.20/1.47 => ( omegaIndAx
% 2.20/1.47 => ( replAx
% 2.20/1.47 => ( foundationAx
% 2.20/1.47 => ( wellorderingAx
% 2.20/1.47 => ( descrp
% 2.20/1.47 => ( dsetconstrI
% 2.20/1.47 => ( dsetconstrEL
% 2.20/1.47 => ( dsetconstrER
% 2.20/1.47 => ( exuE1
% 2.20/1.47 => ( prop2setE
% 2.20/1.47 => ( emptysetE
% 2.20/1.47 => ( emptysetimpfalse
% 2.20/1.47 => ( notinemptyset
% 2.20/1.47 => ( exuE3e
% 2.20/1.47 => ( setext
% 2.20/1.47 => ( emptyI
% 2.20/1.47 => ( noeltsimpempty
% 2.20/1.47 => ( setbeta
% 2.20/1.47 => ( nonemptyE1
% 2.20/1.47 => ( nonemptyI
% 2.20/1.47 => ( nonemptyI1
% 2.20/1.47 => ( setadjoinIL
% 2.20/1.47 => ( emptyinunitempty
% 2.20/1.47 => ( setadjoinIR
% 2.20/1.47 => ( setadjoinE
% 2.20/1.47 => ( setadjoinOr
% 2.20/1.47 => ( setoftrueEq
% 2.20/1.47 => ( powersetI
% 2.20/1.47 => ( emptyinPowerset
% 2.20/1.47 => ( emptyInPowerset
% 2.20/1.47 => ( powersetE
% 2.20/1.47 => ( setunionI
% 2.20/1.47 => ( setunionE
% 2.20/1.47 => ( subPowSU
% 2.20/1.47 => ( exuE2
% 2.20/1.47 => ( nonemptyImpWitness
% 2.20/1.47 => ( uniqinunit
% 2.20/1.47 => ( notinsingleton
% 2.20/1.47 => ( eqinunit
% 2.20/1.47 => ( singletonsswitch
% 2.20/1.47 => ( upairsetE
% 2.20/1.47 => ( upairsetIL
% 2.20/1.47 => ( upairsetIR
% 2.20/1.47 => ( emptyE1
% 2.20/1.47 => ( vacuousDall
% 2.20/1.47 => ( quantDeMorgan1
% 2.20/1.47 => ( quantDeMorgan2
% 2.20/1.47 => ( quantDeMorgan3
% 2.20/1.47 => ( quantDeMorgan4
% 2.20/1.47 => ( prop2setI
% 2.20/1.47 => ( prop2set2propI
% 2.20/1.47 => ( notdexE
% 2.20/1.47 => ( notdallE
% 2.20/1.47 => ( exuI1
% 2.20/1.47 => ( exuI3
% 2.20/1.47 => ( exuI2
% 2.20/1.47 => ( inCongP
% 2.20/1.47 => ( in__Cong
% 2.20/1.47 => ( exuE3u
% 2.20/1.47 => ( exu__Cong
% 2.20/1.47 => ( emptyset__Cong
% 2.20/1.47 => ( setadjoin__Cong
% 2.20/1.47 => ( powerset__Cong
% 2.20/1.47 => ( setunion__Cong
% 2.20/1.47 => ( omega__Cong
% 2.20/1.47 => ( exuEu
% 2.20/1.47 => ( descr__Cong
% 2.20/1.47 => ( dsetconstr__Cong
% 2.20/1.47 => ( subsetI1
% 2.20/1.47 => ( eqimpsubset2
% 2.20/1.47 => ! [A: $i,B: $i] :
% 2.20/1.47 ( ( A = B )
% 2.20/1.47 => ( subset @ A @ B ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.20/1.47 ------------------------
% 2.20/1.48 % (20966)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_20774 for (2999ds/183Mi)
% 2.20/1.49 % (20969)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_20774 for (2999ds/2Mi)
% 2.20/1.49 % (20969)Instruction limit reached!
% 2.20/1.49 % (20969)------------------------------
% 2.20/1.49 % (20969)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.20/1.49 % (20969)Termination reason: Unknown
% 2.20/1.49 % (20969)Termination phase: shuffling
% 2.20/1.49
% 2.20/1.49 % (20969)Memory used [KB]: 1151
% 2.20/1.49 % (20969)Time elapsed: 0.002 s
% 2.20/1.49 % (20969)Instructions burned: 2 (million)
% 2.20/1.49 % (20969)------------------------------
% 2.20/1.49 % (20969)------------------------------
% 2.20/1.49 % (20972)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_20774 for (2999ds/18Mi)
% 2.20/1.49 % (20971)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_20774 for (2999ds/275Mi)
% 2.20/1.49 % (20968)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_20774 for (2999ds/27Mi)
% 2.20/1.49 % (20967)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_20774 for (2999ds/4Mi)
% 2.20/1.49 % (20967)Instruction limit reached!
% 2.20/1.49 % (20967)------------------------------
% 2.20/1.49 % (20967)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.20/1.49 % (20967)Termination reason: Unknown
% 2.20/1.49 % (20967)Termination phase: shuffling
% 2.20/1.49
% 2.20/1.49 % (20967)Memory used [KB]: 1151
% 2.20/1.49 % (20967)Time elapsed: 0.003 s
% 2.20/1.49 % (20967)Instructions burned: 6 (million)
% 2.20/1.49 % (20967)------------------------------
% 2.20/1.49 % (20967)------------------------------
% 2.20/1.49 % (20972)Instruction limit reached!
% 2.20/1.49 % (20972)------------------------------
% 2.20/1.49 % (20972)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.20/1.49 % (20972)Termination reason: Unknown
% 2.20/1.49 % (20972)Termination phase: Preprocessing 3
% 2.20/1.49
% 2.20/1.49 % (20972)Memory used [KB]: 1663
% 2.20/1.49 % (20972)Time elapsed: 0.009 s
% 2.20/1.49 % (20972)Instructions burned: 20 (million)
% 2.20/1.49 % (20972)------------------------------
% 2.20/1.49 % (20972)------------------------------
% 2.32/1.50 % (20970)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_20774 for (2999ds/2Mi)
% 2.32/1.50 % (20970)Instruction limit reached!
% 2.32/1.50 % (20970)------------------------------
% 2.32/1.50 % (20970)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.32/1.50 % (20970)Termination reason: Unknown
% 2.32/1.50 % (20970)Termination phase: shuffling
% 2.32/1.50
% 2.32/1.50 % (20970)Memory used [KB]: 1151
% 2.32/1.50 % (20970)Time elapsed: 0.004 s
% 2.32/1.50 % (20970)Instructions burned: 3 (million)
% 2.32/1.50 % (20970)------------------------------
% 2.32/1.50 % (20970)------------------------------
% 2.32/1.50 % (20973)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_20774 for (2999ds/3Mi)
% 2.32/1.50 % (20973)Instruction limit reached!
% 2.32/1.50 % (20973)------------------------------
% 2.32/1.50 % (20973)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.32/1.50 % (20973)Termination reason: Unknown
% 2.32/1.50 % (20973)Termination phase: shuffling
% 2.32/1.50
% 2.32/1.50 % (20973)Memory used [KB]: 1151
% 2.32/1.50 % (20973)Time elapsed: 0.003 s
% 2.32/1.50 % (20973)Instructions burned: 5 (million)
% 2.32/1.50 % (20973)------------------------------
% 2.32/1.50 % (20973)------------------------------
% 2.32/1.50 % (20968)Instruction limit reached!
% 2.32/1.50 % (20968)------------------------------
% 2.32/1.50 % (20968)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.32/1.50 % (20968)Termination reason: Unknown
% 2.32/1.50 % (20968)Termination phase: Property scanning
% 2.32/1.50
% 2.32/1.50 % (20968)Memory used [KB]: 1407
% 2.32/1.50 % (20968)Time elapsed: 0.014 s
% 2.32/1.50 % (20968)Instructions burned: 27 (million)
% 2.32/1.50 % (20968)------------------------------
% 2.32/1.50 % (20968)------------------------------
% 2.32/1.50 % (20975)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_20774 for (2999ds/15Mi)
% 2.32/1.50 % (20974)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_20774 for (2999ds/37Mi)
% 2.32/1.51 % (20966)First to succeed.
% 2.32/1.51 % (20975)Instruction limit reached!
% 2.32/1.51 % (20975)------------------------------
% 2.32/1.51 % (20975)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.32/1.51 % (20975)Termination reason: Unknown
% 2.32/1.51 % (20975)Termination phase: Preprocessing 3
% 2.32/1.51
% 2.32/1.51 % (20975)Memory used [KB]: 1535
% 2.32/1.51 % (20975)Time elapsed: 0.006 s
% 2.32/1.51 % (20975)Instructions burned: 16 (million)
% 2.32/1.51 % (20975)------------------------------
% 2.32/1.51 % (20975)------------------------------
% 2.32/1.51 % (20966)Refutation found. Thanks to Tanya!
% 2.32/1.51 % SZS status Theorem for DTF2THF_20774
% 2.32/1.51 % SZS output start Proof for DTF2THF_20774
% 2.32/1.51 thf(func_def_0, type, in: $i > $i > $o).
% 2.32/1.51 thf(func_def_1, type, exu: ($i > $o) > $o).
% 2.32/1.51 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 2.32/1.51 thf(func_def_8, type, powerset: $i > $i).
% 2.32/1.51 thf(func_def_10, type, setunion: $i > $i).
% 2.32/1.51 thf(func_def_19, type, descr: ($i > $o) > $i).
% 2.32/1.51 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 2.32/1.51 thf(func_def_26, type, prop2set: $o > $i).
% 2.32/1.51 thf(func_def_36, type, nonempty: $i > $o).
% 2.32/1.51 thf(func_def_69, type, set2prop: $i > $o).
% 2.32/1.51 thf(func_def_88, type, subset: $i > $i > $o).
% 2.32/1.51 thf(func_def_89, type, disjoint: $i > $i > $o).
% 2.32/1.51 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 2.32/1.51 thf(func_def_112, type, sK3: $i > $i).
% 2.32/1.51 thf(func_def_113, type, sK4: ($i > $o) > ($i > $o) > $i).
% 2.32/1.51 thf(func_def_114, type, sK5: $i > ($i > $o) > $i).
% 2.32/1.51 thf(func_def_115, type, sK6: ($i > $o) > ($i > $o) > $i).
% 2.32/1.51 thf(func_def_116, type, sK7: ($i > $o) > $i > $i).
% 2.32/1.51 thf(func_def_117, type, sK8: ($i > $o) > $i > $i).
% 2.32/1.51 thf(func_def_118, type, sK9: ($i > $o) > $i > $i).
% 2.32/1.51 thf(func_def_119, type, sK10: ($i > $o) > ($i > $o) > $i).
% 2.32/1.51 thf(func_def_120, type, sK11: ($i > $o) > $i > $i).
% 2.32/1.51 thf(func_def_121, type, sK12: ($i > $o) > $i > $i).
% 2.32/1.51 thf(func_def_122, type, sK13: ($i > $o) > $i > $i).
% 2.32/1.51 thf(func_def_123, type, sK14: ($i > $o) > $i).
% 2.32/1.51 thf(func_def_124, type, sK15: $i > ($i > $o) > $i).
% 2.32/1.51 thf(func_def_125, type, sK16: ($i > $o) > $i > $i).
% 2.32/1.51 thf(func_def_126, type, sK17: $i > $i).
% 2.32/1.51 thf(func_def_127, type, sK18: ($i > $o) > $i).
% 2.32/1.51 thf(func_def_128, type, sK19: ($i > $o) > $i).
% 2.32/1.51 thf(func_def_129, type, sK20: ($i > $o) > $i).
% 2.32/1.51 thf(func_def_130, type, sK21: $o > $i > $i > $i).
% 2.32/1.51 thf(func_def_131, type, sK22: $i > ($i > $o) > $i).
% 2.32/1.51 thf(func_def_132, type, sK23: $i > $i).
% 2.32/1.51 thf(func_def_133, type, sK24: $i > $i > $i).
% 2.32/1.51 thf(func_def_134, type, sK25: $i > $i > $i).
% 2.32/1.51 thf(func_def_135, type, sK26: $i > $i > $i).
% 2.32/1.51 thf(func_def_136, type, sK27: ($i > $o) > ($i > $o) > $i).
% 2.32/1.51 thf(func_def_137, type, sK28: ($i > $o) > $i).
% 2.32/1.51 thf(func_def_138, type, sK29: ($i > $o) > ($i > $o) > $i).
% 2.32/1.51 thf(func_def_139, type, sK30: $i > ($i > $o) > $i).
% 2.32/1.51 thf(func_def_140, type, sK31: ($i > $o) > ($i > $o) > $i).
% 2.32/1.51 thf(func_def_141, type, sK32: $i > ($i > $o) > $i).
% 2.32/1.51 thf(func_def_142, type, sK33: ($i > $o) > $i).
% 2.32/1.51 thf(func_def_143, type, sK34: ($i > $o) > ($i > $o) > $i).
% 2.32/1.51 thf(func_def_144, type, sK35: $i > ($i > $o) > $i).
% 2.32/1.51 thf(func_def_145, type, sK36: ($i > $o) > ($i > $o) > $i).
% 2.32/1.51 thf(func_def_146, type, sK37: ($i > $o) > $i > $i).
% 2.32/1.51 thf(func_def_147, type, sK38: ($i > $o) > $i > $i).
% 2.32/1.51 thf(func_def_148, type, sK39: ($i > $o) > $i > $i).
% 2.32/1.51 thf(func_def_149, type, sK40: $i > ($i > $o) > $i).
% 2.32/1.51 thf(func_def_150, type, sK41: $i > $i).
% 2.32/1.51 thf(func_def_151, type, sK42: ($i > $o) > $i).
% 2.32/1.51 thf(func_def_152, type, sK43: ($i > $o) > $i > $i).
% 2.32/1.51 thf(func_def_153, type, sK44: ($i > $o) > $i).
% 2.32/1.51 thf(func_def_154, type, sK45: ($i > $o) > $i > $i).
% 2.32/1.51 thf(func_def_155, type, sK46: ($i > $o) > $i > $i).
% 2.32/1.51 thf(func_def_156, type, sK47: $i > $i).
% 2.32/1.51 thf(func_def_157, type, sK48: $i > $i > $i).
% 2.32/1.51 thf(func_def_158, type, sK49: $i > $i > $i).
% 2.32/1.51 thf(func_def_159, type, sK50: $i > $i > $i).
% 2.32/1.51 thf(func_def_160, type, sK51: $i > $i > $i).
% 2.32/1.51 thf(func_def_161, type, sK52: $i > $i > $i > $i).
% 2.32/1.51 thf(func_def_162, type, sK53: ($i > $o) > $i > $i).
% 2.32/1.51 thf(func_def_163, type, sK54: ($i > $o) > $i).
% 2.32/1.51 thf(func_def_164, type, sK55: ($i > $o) > $i > $i).
% 2.32/1.51 thf(func_def_165, type, sK56: ($i > $o) > $i).
% 2.32/1.51 thf(func_def_166, type, sK57: ($i > $o) > $i).
% 2.32/1.51 thf(f1216,plain,(
% 2.32/1.51 $false),
% 2.32/1.51 inference(trivial_inequality_removal,[status(thm)],[f1215])).
% 2.32/1.51 thf(f1215,plain,(
% 2.32/1.51 ($false = $true)),
% 2.32/1.51 inference(beta_eta_normalization,[status(thm)],[f1214])).
% 2.32/1.51 thf(f1214,plain,(
% 2.32/1.51 ($false = ((^[Y0 : $i]: ($true)) @ sK58))),
% 2.32/1.51 inference(sigma_clausification,[status(thm)],[f1213])).
% 2.32/1.51 thf(f1213,plain,(
% 2.32/1.51 ($true != (!! @ $i @ (^[Y0 : $i]: ($true))))),
% 2.32/1.51 inference(boolean_simplification,[status(thm)],[f1212])).
% 2.32/1.51 thf(f1212,plain,(
% 2.32/1.51 ($true != (!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ sK0) => (in @ Y0 @ sK0)))))),
% 2.32/1.51 inference(beta_eta_normalization,[status(thm)],[f412])).
% 2.32/1.51 thf(f412,plain,(
% 2.32/1.51 ($true != ((^[Y0 : $i]: ((^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: ((in @ Y2 @ Y0) => (in @ Y2 @ Y1))))))) @ sK0 @ sK0))),
% 2.32/1.51 inference(definition_unfolding,[status(thm)],[f324,f380,f325])).
% 2.32/1.51 thf(f325,plain,(
% 2.32/1.51 (sK1 = sK0)),
% 2.32/1.51 inference(cnf_transformation,[status(thm)],[f243])).
% 2.32/1.51 thf(f243,plain,(
% 2.32/1.51 (quantDeMorgan4 = $true) & (setadjoin__Cong = $true) & (singletonsswitch = $true) & (omegaSAx = $true) & (setunionI = $true) & (setadjoinOr = $true) & (prop2set2propI = $true) & (exuE3u = $true) & (nonemptyI = $true) & (dsetconstrI = $true) & (upairsetIR = $true) & (nonemptyI1 = $true) & (emptyInPowerset = $true) & (notinemptyset = $true) & (dsetconstr__Cong = $true) & (omega__Cong = $true) & ((sK1 = sK0) & ($true != (subset @ sK1 @ sK0))) & (powerset__Cong = $true) & (emptyinunitempty = $true) & (exuE1 = $true) & (descr__Cong = $true) & (dsetconstrER = $true) & (setadjoinE = $true) & (eqinunit = $true) & (notinsingleton = $true) & (setadjoinAx = $true) & (quantDeMorgan1 = $true) & (descrp = $true) & (quantDeMorgan3 = $true) & (exuI3 = $true) & (emptyinPowerset = $true) & (subsetI1 = $true) & (wellorderingAx = $true) & (setadjoinIR = $true) & (dsetconstrEL = $true) & (exuI2 = $true) & (setadjoinIL = $true) & (vacuousDall = $true) & (setextAx = $true) & (powersetI = $true) & (setunionAx = $true) & (setext = $true) & (upairsetE = $true) & (exuI1 = $true) & (replAx = $true) & (nonemptyE1 = $true) & (prop2setE = $true) & (uniqinunit = $true) & (quantDeMorgan2 = $true) & (in__Cong = $true) & (prop2setI = $true) & (omega0Ax = $true) & (exuEu = $true) & (emptysetimpfalse = $true) & (emptyI = $true) & (emptysetE = $true) & (noeltsimpempty = $true) & (subPowSU = $true) & (notdexE = $true) & (exuE2 = $true) & (powersetE = $true) & (nonemptyImpWitness = $true) & (setunion__Cong = $true) & (upairsetIL = $true) & (foundationAx = $true) & (exuE3e = $true) & (inCongP = $true) & (omegaIndAx = $true) & (setoftrueEq = $true) & (emptyE1 = $true) & (eqimpsubset2 = $true) & (notdallE = $true) & (powersetAx = $true) & (setbeta = $true) & (emptysetAx = $true) & (setunionE = $true) & (emptyset__Cong = $true) & (exu__Cong = $true)),
% 2.32/1.51 inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1])],[f241,f242])).
% 2.32/1.51 thf(f242,plain,(
% 2.32/1.51 ? [X0 : $i,X1 : $i] : ((X0 = X1) & ((subset @ X1 @ X0) != $true)) => ((sK1 = sK0) & ($true != (subset @ sK1 @ sK0)))),
% 2.32/1.51 introduced(definition,[],[choice_axiom])).
% 2.32/1.51 thf(f241,plain,(
% 2.32/1.51 (quantDeMorgan4 = $true) & (setadjoin__Cong = $true) & (singletonsswitch = $true) & (omegaSAx = $true) & (setunionI = $true) & (setadjoinOr = $true) & (prop2set2propI = $true) & (exuE3u = $true) & (nonemptyI = $true) & (dsetconstrI = $true) & (upairsetIR = $true) & (nonemptyI1 = $true) & (emptyInPowerset = $true) & (notinemptyset = $true) & (dsetconstr__Cong = $true) & (omega__Cong = $true) & ? [X0 : $i,X1 : $i] : ((X0 = X1) & ((subset @ X1 @ X0) != $true)) & (powerset__Cong = $true) & (emptyinunitempty = $true) & (exuE1 = $true) & (descr__Cong = $true) & (dsetconstrER = $true) & (setadjoinE = $true) & (eqinunit = $true) & (notinsingleton = $true) & (setadjoinAx = $true) & (quantDeMorgan1 = $true) & (descrp = $true) & (quantDeMorgan3 = $true) & (exuI3 = $true) & (emptyinPowerset = $true) & (subsetI1 = $true) & (wellorderingAx = $true) & (setadjoinIR = $true) & (dsetconstrEL = $true) & (exuI2 = $true) & (setadjoinIL = $true) & (vacuousDall = $true) & (setextAx = $true) & (powersetI = $true) & (setunionAx = $true) & (setext = $true) & (upairsetE = $true) & (exuI1 = $true) & (replAx = $true) & (nonemptyE1 = $true) & (prop2setE = $true) & (uniqinunit = $true) & (quantDeMorgan2 = $true) & (in__Cong = $true) & (prop2setI = $true) & (omega0Ax = $true) & (exuEu = $true) & (emptysetimpfalse = $true) & (emptyI = $true) & (emptysetE = $true) & (noeltsimpempty = $true) & (subPowSU = $true) & (notdexE = $true) & (exuE2 = $true) & (powersetE = $true) & (nonemptyImpWitness = $true) & (setunion__Cong = $true) & (upairsetIL = $true) & (foundationAx = $true) & (exuE3e = $true) & (inCongP = $true) & (omegaIndAx = $true) & (setoftrueEq = $true) & (emptyE1 = $true) & (eqimpsubset2 = $true) & (notdallE = $true) & (powersetAx = $true) & (setbeta = $true) & (emptysetAx = $true) & (setunionE = $true) & (emptyset__Cong = $true) & (exu__Cong = $true)),
% 2.32/1.51 inference(flattening,[status(thm)],[f240])).
% 2.32/1.51 thf(f240,plain,(
% 2.32/1.51 ((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i,X1 : $i] : ((X0 = X1) & ((subset @ X1 @ X0) != $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)),
% 2.32/1.51 inference(ennf_transformation,[status(thm)],[f103])).
% 2.32/1.51 thf(f103,plain,(
% 2.32/1.51 ~((setextAx = $true) => ((emptysetAx = $true) => ((setadjoinAx = $true) => ((powersetAx = $true) => ((setunionAx = $true) => ((omega0Ax = $true) => ((omegaSAx = $true) => ((omegaIndAx = $true) => ((replAx = $true) => ((foundationAx = $true) => ((wellorderingAx = $true) => ((descrp = $true) => ((dsetconstrI = $true) => ((dsetconstrEL = $true) => ((dsetconstrER = $true) => ((exuE1 = $true) => ((prop2setE = $true) => ((emptysetE = $true) => ((emptysetimpfalse = $true) => ((notinemptyset = $true) => ((exuE3e = $true) => ((setext = $true) => ((emptyI = $true) => ((noeltsimpempty = $true) => ((setbeta = $true) => ((nonemptyE1 = $true) => ((nonemptyI = $true) => ((nonemptyI1 = $true) => ((setadjoinIL = $true) => ((emptyinunitempty = $true) => ((setadjoinIR = $true) => ((setadjoinE = $true) => ((setadjoinOr = $true) => ((setoftrueEq = $true) => ((powersetI = $true) => ((emptyinPowerset = $true) => ((emptyInPowerset = $true) => ((powersetE = $true) => ((setunionI = $true) => ((setunionE = $true) => ((subPowSU = $true) => ((exuE2 = $true) => ((nonemptyImpWitness = $true) => ((uniqinunit = $true) => ((notinsingleton = $true) => ((eqinunit = $true) => ((singletonsswitch = $true) => ((upairsetE = $true) => ((upairsetIL = $true) => ((upairsetIR = $true) => ((emptyE1 = $true) => ((vacuousDall = $true) => ((quantDeMorgan1 = $true) => ((quantDeMorgan2 = $true) => ((quantDeMorgan3 = $true) => ((quantDeMorgan4 = $true) => ((prop2setI = $true) => ((prop2set2propI = $true) => ((notdexE = $true) => ((notdallE = $true) => ((exuI1 = $true) => ((exuI3 = $true) => ((exuI2 = $true) => ((inCongP = $true) => ((in__Cong = $true) => ((exuE3u = $true) => ((exu__Cong = $true) => ((emptyset__Cong = $true) => ((setadjoin__Cong = $true) => ((powerset__Cong = $true) => ((setunion__Cong = $true) => ((omega__Cong = $true) => ((exuEu = $true) => ((descr__Cong = $true) => ((dsetconstr__Cong = $true) => ((subsetI1 = $true) => ((eqimpsubset2 = $true) => ! [X0 : $i,X1 : $i] : ((X0 = X1) => ((subset @ X1 @ X0) = $true)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.32/1.51 inference(fool_elimination,[status(thm)],[f102])).
% 2.32/1.51 thf(f102,plain,(
% 2.32/1.51 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => ! [X0 : $i,X1 : $i] : ((X0 = X1) => (subset @ X1 @ X0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.32/1.51 inference(rectify,[status(thm)],[f82])).
% 2.32/1.51 thf(f82,negated_conjecture,(
% 2.32/1.51 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => ! [X4 : $i,X3 : $i] : ((X3 = X4) => (subset @ X3 @ X4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.32/1.51 inference(negated_conjecture,[status(cth)],[f81])).
% 2.32/1.51 thf(f81,conjecture,(
% 2.32/1.51 setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => ! [X4 : $i,X3 : $i] : ((X3 = X4) => (subset @ X3 @ X4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.32/1.51 file('/export/starexec/sandbox/tmp/tmp.rYUQzxuxZr/DTF2THF_20774.p',eqimpsubset1)).
% 2.32/1.51 thf(f380,plain,(
% 2.32/1.51 (subset = (^[Y0 : $i]: ((^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: ((in @ Y2 @ Y0) => (in @ Y2 @ Y1))))))))),
% 2.32/1.51 inference(cnf_transformation,[status(thm)],[f128])).
% 2.32/1.51 thf(f128,plain,(
% 2.32/1.51 (subset = (^[Y0 : $i]: ((^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: ((in @ Y2 @ Y0) => (in @ Y2 @ Y1))))))))),
% 2.32/1.51 inference(fool_elimination,[status(thm)],[f127])).
% 2.32/1.51 thf(f127,plain,(
% 2.32/1.51 (subset = (^[X0 : $i, X1 : $i] : (! [X2 : $i] : ((in @ X2 @ X0) => (in @ X2 @ X1)))))),
% 2.32/1.51 inference(rectify,[status(thm)],[f78])).
% 2.32/1.51 thf(f78,axiom,(
% 2.32/1.51 (subset = (^[X3 : $i, X4 : $i] : (! [X1 : $i] : ((in @ X1 @ X3) => (in @ X1 @ X4)))))),
% 2.32/1.51 file('/export/starexec/sandbox/tmp/tmp.rYUQzxuxZr/DTF2THF_20774.p',subset)).
% 2.32/1.51 thf(f324,plain,(
% 2.32/1.51 ($true != (subset @ sK1 @ sK0))),
% 2.32/1.51 inference(cnf_transformation,[status(thm)],[f243])).
% 2.32/1.51 % SZS output end Proof for DTF2THF_20774
% 2.32/1.51 % (20966)------------------------------
% 2.32/1.51 % (20966)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.32/1.51 % (20966)Termination reason: Refutation
% 2.32/1.51
% 2.32/1.51 % (20966)Memory used [KB]: 6268
% 2.32/1.51 % (20966)Time elapsed: 0.026 s
% 2.32/1.51 % (20966)Instructions burned: 81 (million)
% 2.32/1.51 % (20966)------------------------------
% 2.32/1.51 % (20966)------------------------------
% 2.32/1.51 % (20965)Success in time 0.035 s
%------------------------------------------------------------------------------