%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU537^1 : 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 : n003.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 09:35:09 AM UTC 2026
% Result : Theorem 0.23s 1.48s
% Output : Refutation 0.23s
% Verified :
% SZS Type : Refutation
% Derivation depth : 11
% Number of leaves : 3
% Syntax : Number of formulae : 16 ( 4 unt; 0 typ; 0 def)
% Number of atoms : 1084 ( 304 equ; 0 cnn)
% Maximal formula atoms : 57 ( 67 avg)
% Number of connectives : 563 ( 29 ~; 7 |; 176 &; 74 @)
% ( 0 <=>; 277 =>; 0 <=; 0 <~>)
% Maximal formula depth : 63 ( 31 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 8 ( 8 >; 0 *; 0 +; 0 <<)
% Number of symbols : 128 ( 125 usr; 57 con; 0-3 aty)
% Number of variables : 37 ( 0 ^; 21 !; 16 ?; 37 :)
% 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_78,type,
sP1: $i > $i > $o ).
thf(func_def_79,type,
sP2: $i > $o ).
thf(func_def_80,type,
sP3: $i > $i > $o ).
thf(func_def_81,type,
sP4: $i > $i > $o ).
thf(func_def_82,type,
sK5: $i > $i ).
thf(func_def_86,type,
sK9: $i > $i > $i ).
thf(func_def_90,type,
sK13: $i > $i ).
thf(func_def_94,type,
sK17: $i > $i > $o ).
thf(func_def_96,type,
sK19: $i > $i ).
thf(func_def_97,type,
sK20: $i > $i ).
thf(func_def_98,type,
sK21: $i > ( $i > $i > $o ) > $i ).
thf(func_def_99,type,
sK22: $i > ( $i > $i > $o ) > $i ).
thf(func_def_100,type,
sK23: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_103,type,
sK26: $i > $o ).
thf(func_def_107,type,
sK30: $i > $o ).
thf(func_def_110,type,
sK33: $i > ( $i > $o ) > $i ).
thf(func_def_113,type,
sK36: ( $i > $o ) > $i ).
thf(func_def_114,type,
sK37: $i > $o ).
thf(func_def_118,type,
sK41: $i > $i > $i ).
thf(func_def_124,type,
sK47: $i > $i > $i ).
thf(func_def_125,type,
sK48: $i > $i > $i ).
thf(func_def_133,type,
sK56: $i > $i ).
thf(func_def_134,type,
sK57: $i > $o ).
thf(func_def_143,type,
sK66: $i > $i > $i ).
thf(func_def_152,type,
sK75: $i > $o ).
thf(func_def_160,type,
sK83: $i > $i ).
thf(func_def_163,type,
sK86: $i > $i ).
thf(func_def_166,type,
sK89: $i > $i ).
thf(func_def_172,type,
sK95: $o > $i > $i > $i ).
thf(func_def_175,type,
sK98: $i > $o ).
thf(func_def_179,type,
sK102: $i > $o ).
thf(func_def_181,type,
sK104: $i > $i ).
thf(func_def_185,type,
sK108: $i > $o ).
thf(func_def_188,type,
sK111: $i > $i > $i ).
thf(func_def_189,type,
sK112: $i > $i > $i ).
thf(func_def_190,type,
sK113: $i > $i > $i ).
thf(func_def_191,type,
sK114: $i > $i > $i ).
thf(func_def_192,type,
sK115: $i > $i > $i > $i ).
thf(func_def_193,type,
sK116: $i > $i ).
thf(func_def_194,type,
sK117: $i > $i ).
thf(func_def_195,type,
sK118: $i > $i ).
thf(func_def_196,type,
sK119: $i > $i ).
thf(func_def_197,type,
sK120: $i > $i > $i ).
thf(func_def_198,type,
sK121: $i > $i > $i > $i ).
thf(func_def_199,type,
sK122: $i > $i > $i > $i ).
thf(func_def_200,type,
sK123: $i > $i > $i ).
thf(func_def_201,type,
sK124: $i > $i > $i ).
thf(func_def_203,type,
sK126: $i > $i ).
thf(func_def_204,type,
sK127: $i > $i ).
thf(func_def_205,type,
sK128: $i > $i ).
thf(func_def_206,type,
sK129: $i > $i > $i ).
thf(func_def_212,type,
sK135: ( $i > $o ) > $i ).
thf(func_def_213,type,
sK136: $i > $o ).
thf(func_def_214,type,
sK137: $i > $i ).
thf(func_def_216,type,
sK139: $i > $o ).
thf(func_def_220,type,
sK143: $i > $o ).
thf(func_def_225,type,
sK148: $i > $o ).
thf(func_def_226,type,
sK149: $i > $i ).
thf(func_def_227,type,
sK150: ( $i > $o ) > $i ).
thf(func_def_230,type,
sK153: $i > $i > $i ).
thf(func_def_232,type,
ph155:
!>[X0: $tType] : X0 ).
thf(f1179,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f1159,f646]) ).
thf(f646,plain,
( ( sK57 @ sK59 )
= $true ),
inference(cnf_transformation,[status(thm)],[f349]) ).
thf(f349,plain,
( ( subPowSU = $true )
& ( upairsetIR = $true )
& ! [X2: $i] :
( ( ( in @ X2 @ sK58 )
!= $true )
| ( ( sK57 @ X2 )
!= $true ) )
& ( ( sK57 @ sK59 )
= $true )
& ( ( in @ sK59 @ sK58 )
= $true )
& ( exuE2 = $true )
& ( nonemptyE1 = $true )
& ( powersetE = $true )
& ( descrp = $true )
& ( emptyInPowerset = $true )
& ( quantDeMorgan1 = $true )
& ( emptysetE = $true )
& ( setbeta = $true )
& ( omegaSAx = $true )
& ( setext = $true )
& ( dsetconstrEL = $true )
& ( noeltsimpempty = $true )
& ( notinemptyset = $true )
& ( prop2setE = $true )
& ( vacuousDall = $true )
& ( setadjoinOr = $true )
& ( wellorderingAx = $true )
& ( singletonsswitch = $true )
& ( replAx = $true )
& ( setadjoinE = $true )
& ( eqinunit = $true )
& ( emptysetimpfalse = $true )
& ( omega0Ax = $true )
& ( setunionI = $true )
& ( exuE3e = $true )
& ( powersetAx = $true )
& ( setadjoinIL = $true )
& ( setadjoinAx = $true )
& ( dsetconstrI = $true )
& ( setextAx = $true )
& ( emptysetAx = $true )
& ( nonemptyI1 = $true )
& ( emptyE1 = $true )
& ( omegaIndAx = $true )
& ( exuE1 = $true )
& ( nonemptyI = $true )
& ( powersetI = $true )
& ( setunionE = $true )
& ( foundationAx = $true )
& ( setadjoinIR = $true )
& ( setunionAx = $true )
& ( emptyinPowerset = $true )
& ( upairsetIL = $true )
& ( emptyinunitempty = $true )
& ( upairsetE = $true )
& ( notinsingleton = $true )
& ( nonemptyImpWitness = $true )
& ( emptyI = $true )
& ( uniqinunit = $true )
& ( dsetconstrER = $true )
& ( setoftrueEq = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK57,sK58,sK59])],[f228,f348,f347]) ).
thf(f347,plain,
( ? [X0: $i > $o,X1: $i] :
( ! [X2: $i] :
( ( ( in @ X2 @ X1 )
!= $true )
| ( ( X0 @ X2 )
!= $true ) )
& ? [X3: $i] :
( ( ( X0 @ X3 )
= $true )
& ( ( in @ X3 @ X1 )
= $true ) ) )
=> ( ! [X2: $i] :
( ( ( in @ X2 @ sK58 )
!= $true )
| ( ( sK57 @ X2 )
!= $true ) )
& ? [X3: $i] :
( ( ( sK57 @ X3 )
= $true )
& ( ( in @ X3 @ sK58 )
= $true ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f348,plain,
( ? [X3: $i] :
( ( ( sK57 @ X3 )
= $true )
& ( ( in @ X3 @ sK58 )
= $true ) )
=> ( ( ( sK57 @ sK59 )
= $true )
& ( ( in @ sK59 @ sK58 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f228,plain,
( ( subPowSU = $true )
& ( upairsetIR = $true )
& ? [X0: $i > $o,X1: $i] :
( ! [X2: $i] :
( ( ( in @ X2 @ X1 )
!= $true )
| ( ( X0 @ X2 )
!= $true ) )
& ? [X3: $i] :
( ( ( X0 @ X3 )
= $true )
& ( ( in @ X3 @ X1 )
= $true ) ) )
& ( exuE2 = $true )
& ( nonemptyE1 = $true )
& ( powersetE = $true )
& ( descrp = $true )
& ( emptyInPowerset = $true )
& ( quantDeMorgan1 = $true )
& ( emptysetE = $true )
& ( setbeta = $true )
& ( omegaSAx = $true )
& ( setext = $true )
& ( dsetconstrEL = $true )
& ( noeltsimpempty = $true )
& ( notinemptyset = $true )
& ( prop2setE = $true )
& ( vacuousDall = $true )
& ( setadjoinOr = $true )
& ( wellorderingAx = $true )
& ( singletonsswitch = $true )
& ( replAx = $true )
& ( setadjoinE = $true )
& ( eqinunit = $true )
& ( emptysetimpfalse = $true )
& ( omega0Ax = $true )
& ( setunionI = $true )
& ( exuE3e = $true )
& ( powersetAx = $true )
& ( setadjoinIL = $true )
& ( setadjoinAx = $true )
& ( dsetconstrI = $true )
& ( setextAx = $true )
& ( emptysetAx = $true )
& ( nonemptyI1 = $true )
& ( emptyE1 = $true )
& ( omegaIndAx = $true )
& ( exuE1 = $true )
& ( nonemptyI = $true )
& ( powersetI = $true )
& ( setunionE = $true )
& ( foundationAx = $true )
& ( setadjoinIR = $true )
& ( setunionAx = $true )
& ( emptyinPowerset = $true )
& ( upairsetIL = $true )
& ( emptyinunitempty = $true )
& ( upairsetE = $true )
& ( notinsingleton = $true )
& ( nonemptyImpWitness = $true )
& ( emptyI = $true )
& ( uniqinunit = $true )
& ( dsetconstrER = $true )
& ( setoftrueEq = $true ) ),
inference(flattening,[status(thm)],[f227]) ).
thf(f227,plain,
( ? [X0: $i > $o,X1: $i] :
( ! [X2: $i] :
( ( ( in @ X2 @ X1 )
!= $true )
| ( ( X0 @ X2 )
!= $true ) )
& ? [X3: $i] :
( ( ( X0 @ X3 )
= $true )
& ( ( in @ X3 @ X1 )
= $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)],[f169]) ).
thf(f169,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 )
=> ! [X0: $i > $o,X1: $i] :
( ! [X2: $i] :
( ( ( in @ X2 @ X1 )
= $true )
=> ( ( X0 @ X2 )
!= $true ) )
=> ~ ? [X3: $i] :
( ( ( X0 @ X3 )
= $true )
& ( ( in @ X3 @ X1 )
= $true ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(flattening,[status(thm)],[f79]) ).
thf(f79,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 )
=> ! [X0: $i > $o,X1: $i] :
( ! [X2: $i] :
( ( ( in @ X2 @ X1 )
= $true )
=> ( ( X0 @ X2 )
!= $true ) )
=> ~ ? [X3: $i] :
( ( ( X0 @ X3 )
= $true )
& ( ( in @ X3 @ X1 )
= $true ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f78]) ).
thf(f78,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
=> ! [X0: $i > $o,X1: $i] :
( ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ~ ( X0 @ X2 ) )
=> ~ ? [X3: $i] :
( ( in @ X3 @ X1 )
& ( X0 @ X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f58]) ).
thf(f58,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
=> ! [X0: $i > $o,X3: $i] :
( ! [X1: $i] :
( ( in @ X1 @ X3 )
=> ~ ( X0 @ X1 ) )
=> ~ ? [X1: $i] :
( ( in @ X1 @ X3 )
& ( X0 @ X1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f57]) ).
thf(f57,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
=> ! [X0: $i > $o,X3: $i] :
( ! [X1: $i] :
( ( in @ X1 @ X3 )
=> ~ ( X0 @ X1 ) )
=> ~ ? [X1: $i] :
( ( in @ X1 @ X3 )
& ( X0 @ X1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.6gNdrgnz6L/DTF2THF_7087.p',quantDeMorgan2) ).
thf(f1159,plain,
( ( sK57 @ sK59 )
!= $true ),
inference(trivial_inequality_removal,[status(thm)],[f1146]) ).
thf(f1146,plain,
( ( ( sK57 @ sK59 )
!= $true )
| ( $true != $true ) ),
inference(superposition,[status(thm)],[f647,f645]) ).
thf(f645,plain,
( ( in @ sK59 @ sK58 )
= $true ),
inference(cnf_transformation,[status(thm)],[f349]) ).
thf(f647,plain,
! [X2: $i] :
( ( ( in @ X2 @ sK58 )
!= $true )
| ( ( sK57 @ X2 )
!= $true ) ),
inference(cnf_transformation,[status(thm)],[f349]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.09 % Problem : SEU537^1 : TPTP v9.2.1. Released v3.7.0.
% 0.00/0.09 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.11/0.29 % Computer : n003.cluster.edu
% 0.11/0.29 % Model : x86_64 x86_64
% 0.11/0.29 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.29 % Memory : 8042.1875MB
% 0.11/0.29 % OS : Linux 3.10.0-693.el7.x86_64
% 0.11/0.29 % CPULimit : 300
% 0.11/0.29 % WCLimit : 300
% 0.11/0.29 % DateTime : Tue Feb 24 12:25:55 EST 2026
% 0.11/0.29 % CPUTime :
% 0.11/0.29 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.19/0.38 ---- Original DTF file ---
% 0.19/0.38 thf(spec,logic,$$dhol).
% 0.19/0.38 %------------------------------------------------------------------------------
% 0.19/0.38 % File : SEU537^1 : TPTP v9.2.1. Released v3.7.0.
% 0.19/0.38 % Domain : Set Theory
% 0.19/0.38 % Problem : Preliminary Notions - Bounded Quantifier Laws
% 0.19/0.38 % Version : Especial.
% 0.19/0.38 % English : (! A:i.! phi:i>o.(! x:i.in x A -> ~(phi x)) ->
% 0.19/0.38 % ~(? x:i.in x A & phi x))
% 0.19/0.38
% 0.19/0.38 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.19/0.38 % Source : [Bro08]
% 0.19/0.38 % Names : ZFC039g [Bro08]
% 0.19/0.38
% 0.19/0.38 % Status : Theorem
% 0.19/0.38 % Rating : 0.11 v9.1.0, 0.12 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, 0.17 v5.4.0, 0.20 v5.3.0, 0.40 v5.1.0, 0.60 v5.0.0, 0.40 v4.1.0, 0.67 v4.0.0, 0.33 v3.7.0
% 0.19/0.38 % Syntax : Number of formulae : 121 ( 56 unt; 64 typ; 56 def)
% 0.19/0.38 % Number of atoms : 307 ( 75 equ; 0 cnn)
% 0.19/0.38 % Maximal formula atoms : 55 ( 5 avg)
% 0.19/0.38 % Number of connectives : 507 ( 13 ~; 5 |; 22 &; 330 @)
% 0.19/0.38 % ( 8 <=>; 129 =>; 0 <=; 0 <~>)
% 0.19/0.38 % Maximal formula depth : 62 ( 2 avg)
% 0.19/0.38 % Number of types : 2 ( 0 usr)
% 0.19/0.38 % Number of type conns : 31 ( 31 >; 0 *; 0 +; 0 <<)
% 0.19/0.38 % Number of symbols : 67 ( 64 usr; 57 con; 0-2 aty)
% 0.19/0.38 % Number of variables : 167 ( 17 ^; 129 !; 21 ?; 167 :)
% 0.19/0.38 % SPC : TH0_THM_EQU_NAR
% 0.19/0.38
% 0.19/0.38 % Comments : http://mathgate.info/detsetitem.php?id=450
% 0.19/0.38 %------------------------------------------------------------------------------
% 0.19/0.38 thf(in_type,type,
% 0.19/0.38 in: $i > $i > $o ).
% 0.19/0.38
% 0.19/0.38 thf(exu_type,type,
% 0.19/0.38 exu: ( $i > $o ) > $o ).
% 0.19/0.38
% 0.19/0.38 thf(exu,definition,
% 0.19/0.38 ( exu
% 0.19/0.38 = ( ^ [Xphi: $i > $o] :
% 0.19/0.38 ? [Xx: $i] :
% 0.19/0.38 ( ( Xphi @ Xx )
% 0.19/0.38 & ! [Xy: $i] :
% 0.19/0.38 ( ( Xphi @ Xy )
% 0.19/0.38 => ( Xx = Xy ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(setextAx_type,type,
% 0.19/0.38 setextAx: $o ).
% 0.19/0.38
% 0.19/0.38 thf(setextAx,definition,
% 0.19/0.38 ( setextAx
% 0.19/0.38 = ( ! [A: $i,B: $i] :
% 0.19/0.38 ( ! [Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ A )
% 0.19/0.38 <=> ( in @ Xx @ B ) )
% 0.19/0.38 => ( A = B ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(emptyset_type,type,
% 0.19/0.38 emptyset: $i ).
% 0.19/0.38
% 0.19/0.38 thf(emptysetAx_type,type,
% 0.19/0.38 emptysetAx: $o ).
% 0.19/0.38
% 0.19/0.38 thf(emptysetAx,definition,
% 0.19/0.38 ( emptysetAx
% 0.19/0.38 = ( ! [Xx: $i] :
% 0.19/0.38 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(setadjoin_type,type,
% 0.19/0.38 setadjoin: $i > $i > $i ).
% 0.19/0.38
% 0.19/0.38 thf(setadjoinAx_type,type,
% 0.19/0.38 setadjoinAx: $o ).
% 0.19/0.38
% 0.19/0.38 thf(setadjoinAx,definition,
% 0.19/0.38 ( setadjoinAx
% 0.19/0.38 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.19/0.38 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.19/0.38 <=> ( ( Xy = Xx )
% 0.19/0.38 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(powerset_type,type,
% 0.19/0.38 powerset: $i > $i ).
% 0.19/0.38
% 0.19/0.38 thf(powersetAx_type,type,
% 0.19/0.38 powersetAx: $o ).
% 0.19/0.38
% 0.19/0.38 thf(powersetAx,definition,
% 0.19/0.38 ( powersetAx
% 0.19/0.38 = ( ! [A: $i,B: $i] :
% 0.19/0.38 ( ( in @ B @ ( powerset @ A ) )
% 0.19/0.38 <=> ! [Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ B )
% 0.19/0.38 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(setunion_type,type,
% 0.19/0.38 setunion: $i > $i ).
% 0.19/0.38
% 0.19/0.38 thf(setunionAx_type,type,
% 0.19/0.38 setunionAx: $o ).
% 0.19/0.38
% 0.19/0.38 thf(setunionAx,definition,
% 0.19/0.38 ( setunionAx
% 0.19/0.38 = ( ! [A: $i,Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ ( setunion @ A ) )
% 0.19/0.38 <=> ? [B: $i] :
% 0.19/0.38 ( ( in @ Xx @ B )
% 0.19/0.38 & ( in @ B @ A ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(omega_type,type,
% 0.19/0.38 omega: $i ).
% 0.19/0.38
% 0.19/0.38 thf(omega0Ax_type,type,
% 0.19/0.38 omega0Ax: $o ).
% 0.19/0.38
% 0.19/0.38 thf(omega0Ax,definition,
% 0.19/0.38 ( omega0Ax
% 0.19/0.38 = ( in @ emptyset @ omega ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(omegaSAx_type,type,
% 0.19/0.38 omegaSAx: $o ).
% 0.19/0.38
% 0.19/0.38 thf(omegaSAx,definition,
% 0.19/0.38 ( omegaSAx
% 0.19/0.38 = ( ! [Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ omega )
% 0.19/0.38 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(omegaIndAx_type,type,
% 0.19/0.38 omegaIndAx: $o ).
% 0.19/0.38
% 0.19/0.38 thf(omegaIndAx,definition,
% 0.19/0.38 ( omegaIndAx
% 0.19/0.38 = ( ! [A: $i] :
% 0.19/0.38 ( ( ( in @ emptyset @ A )
% 0.19/0.38 & ! [Xx: $i] :
% 0.19/0.38 ( ( ( in @ Xx @ omega )
% 0.19/0.38 & ( in @ Xx @ A ) )
% 0.19/0.38 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.19/0.38 => ! [Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ omega )
% 0.19/0.38 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(replAx_type,type,
% 0.19/0.38 replAx: $o ).
% 0.19/0.38
% 0.19/0.38 thf(replAx,definition,
% 0.19/0.38 ( replAx
% 0.19/0.38 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.19/0.38 ( ! [Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ A )
% 0.19/0.38 => ( exu
% 0.19/0.38 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.19/0.38 => ? [B: $i] :
% 0.19/0.38 ! [Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ B )
% 0.19/0.38 <=> ? [Xy: $i] :
% 0.19/0.38 ( ( in @ Xy @ A )
% 0.19/0.38 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(foundationAx_type,type,
% 0.19/0.38 foundationAx: $o ).
% 0.19/0.38
% 0.19/0.38 thf(foundationAx,definition,
% 0.19/0.38 ( foundationAx
% 0.19/0.38 = ( ! [A: $i] :
% 0.19/0.38 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.19/0.38 => ? [B: $i] :
% 0.19/0.38 ( ( in @ B @ A )
% 0.19/0.38 & ~ ? [Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ B )
% 0.19/0.38 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(wellorderingAx_type,type,
% 0.19/0.38 wellorderingAx: $o ).
% 0.19/0.38
% 0.19/0.38 thf(wellorderingAx,definition,
% 0.19/0.38 ( wellorderingAx
% 0.19/0.38 = ( ! [A: $i] :
% 0.19/0.38 ? [B: $i] :
% 0.19/0.38 ( ! [C: $i] :
% 0.19/0.38 ( ( in @ C @ B )
% 0.19/0.38 => ! [Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ C )
% 0.19/0.38 => ( in @ Xx @ A ) ) )
% 0.19/0.38 & ! [Xx: $i,Xy: $i] :
% 0.19/0.38 ( ( ( in @ Xx @ A )
% 0.19/0.38 & ( in @ Xy @ A ) )
% 0.19/0.38 => ( ! [C: $i] :
% 0.19/0.38 ( ( in @ C @ B )
% 0.19/0.38 => ( ( in @ Xx @ C )
% 0.19/0.38 <=> ( in @ Xy @ C ) ) )
% 0.19/0.38 => ( Xx = Xy ) ) )
% 0.19/0.38 & ! [C: $i,D: $i] :
% 0.19/0.38 ( ( ( in @ C @ B )
% 0.19/0.38 & ( in @ D @ B ) )
% 0.19/0.38 => ( ! [Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ C )
% 0.19/0.38 => ( in @ Xx @ D ) )
% 0.19/0.38 | ! [Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ D )
% 0.19/0.38 => ( in @ Xx @ C ) ) ) )
% 0.19/0.38 & ! [C: $i] :
% 0.19/0.38 ( ( ! [Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ C )
% 0.19/0.38 => ( in @ Xx @ A ) )
% 0.19/0.38 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.19/0.38 => ? [D: $i,Xx: $i] :
% 0.19/0.38 ( ( in @ D @ B )
% 0.19/0.38 & ( in @ Xx @ C )
% 0.19/0.38 & ~ ? [Xy: $i] :
% 0.19/0.38 ( ( in @ Xy @ D )
% 0.19/0.38 & ( in @ Xy @ C ) )
% 0.19/0.38 & ! [E: $i] :
% 0.19/0.38 ( ( in @ E @ B )
% 0.19/0.38 => ( ! [Xy: $i] :
% 0.19/0.38 ( ( in @ Xy @ E )
% 0.19/0.38 => ( in @ Xy @ D ) )
% 0.19/0.38 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(descr_type,type,
% 0.19/0.38 descr: ( $i > $o ) > $i ).
% 0.19/0.38
% 0.19/0.38 thf(descrp_type,type,
% 0.19/0.38 descrp: $o ).
% 0.19/0.38
% 0.19/0.38 thf(descrp,definition,
% 0.19/0.38 ( descrp
% 0.19/0.38 = ( ! [Xphi: $i > $o] :
% 0.19/0.38 ( ( exu
% 0.19/0.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.38 => ( Xphi
% 0.19/0.38 @ ( descr
% 0.19/0.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(dsetconstr_type,type,
% 0.19/0.38 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.19/0.38
% 0.19/0.38 thf(dsetconstrI_type,type,
% 0.19/0.38 dsetconstrI: $o ).
% 0.19/0.38
% 0.19/0.38 thf(dsetconstrI,definition,
% 0.19/0.38 ( dsetconstrI
% 0.19/0.38 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ A )
% 0.19/0.38 => ( ( Xphi @ Xx )
% 0.19/0.38 => ( in @ Xx
% 0.19/0.38 @ ( dsetconstr @ A
% 0.19/0.38 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(dsetconstrEL_type,type,
% 0.19/0.38 dsetconstrEL: $o ).
% 0.19/0.38
% 0.19/0.38 thf(dsetconstrEL,definition,
% 0.19/0.38 ( dsetconstrEL
% 0.19/0.38 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.38 ( ( in @ Xx
% 0.19/0.38 @ ( dsetconstr @ A
% 0.19/0.38 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.19/0.38 => ( in @ Xx @ A ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(dsetconstrER_type,type,
% 0.19/0.38 dsetconstrER: $o ).
% 0.19/0.38
% 0.19/0.38 thf(dsetconstrER,definition,
% 0.19/0.38 ( dsetconstrER
% 0.19/0.38 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.38 ( ( in @ Xx
% 0.19/0.38 @ ( dsetconstr @ A
% 0.19/0.38 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.19/0.38 => ( Xphi @ Xx ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(exuE1_type,type,
% 0.19/0.38 exuE1: $o ).
% 0.19/0.38
% 0.19/0.38 thf(exuE1,definition,
% 0.19/0.38 ( exuE1
% 0.19/0.38 = ( ! [Xphi: $i > $o] :
% 0.19/0.38 ( ( exu
% 0.19/0.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.38 => ? [Xx: $i] :
% 0.19/0.38 ( ( Xphi @ Xx )
% 0.19/0.38 & ! [Xy: $i] :
% 0.19/0.38 ( ( Xphi @ Xy )
% 0.19/0.38 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(prop2set_type,type,
% 0.19/0.38 prop2set: $o > $i ).
% 0.19/0.38
% 0.19/0.38 thf(prop2set,definition,
% 0.19/0.38 ( prop2set
% 0.19/0.38 = ( ^ [Xphi: $o] :
% 0.19/0.38 ( dsetconstr @ ( powerset @ emptyset )
% 0.19/0.38 @ ^ [Xx: $i] : Xphi ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(prop2setE_type,type,
% 0.19/0.38 prop2setE: $o ).
% 0.19/0.38
% 0.19/0.38 thf(prop2setE,definition,
% 0.19/0.38 ( prop2setE
% 0.19/0.38 = ( ! [Xphi: $o,Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.19/0.38 => Xphi ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(emptysetE_type,type,
% 0.19/0.38 emptysetE: $o ).
% 0.19/0.38
% 0.19/0.38 thf(emptysetE,definition,
% 0.19/0.38 ( emptysetE
% 0.19/0.38 = ( ! [Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ emptyset )
% 0.19/0.38 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(emptysetimpfalse_type,type,
% 0.19/0.38 emptysetimpfalse: $o ).
% 0.19/0.38
% 0.19/0.38 thf(emptysetimpfalse,definition,
% 0.19/0.38 ( emptysetimpfalse
% 0.19/0.38 = ( ! [Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ emptyset )
% 0.19/0.38 => $false ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(notinemptyset_type,type,
% 0.19/0.38 notinemptyset: $o ).
% 0.19/0.38
% 0.19/0.38 thf(notinemptyset,definition,
% 0.19/0.38 ( notinemptyset
% 0.19/0.38 = ( ! [Xx: $i] :
% 0.19/0.38 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(exuE3e_type,type,
% 0.19/0.38 exuE3e: $o ).
% 0.19/0.38
% 0.19/0.38 thf(exuE3e,definition,
% 0.19/0.38 ( exuE3e
% 0.19/0.38 = ( ! [Xphi: $i > $o] :
% 0.19/0.38 ( ( exu
% 0.19/0.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.38 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(setext_type,type,
% 0.19/0.38 setext: $o ).
% 0.19/0.38
% 0.19/0.38 thf(setext,definition,
% 0.19/0.38 ( setext
% 0.19/0.38 = ( ! [A: $i,B: $i] :
% 0.19/0.38 ( ! [Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ A )
% 0.19/0.38 => ( in @ Xx @ B ) )
% 0.19/0.38 => ( ! [Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ B )
% 0.19/0.38 => ( in @ Xx @ A ) )
% 0.19/0.38 => ( A = B ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(emptyI_type,type,
% 0.19/0.38 emptyI: $o ).
% 0.19/0.38
% 0.19/0.38 thf(emptyI,definition,
% 0.19/0.38 ( emptyI
% 0.19/0.38 = ( ! [A: $i] :
% 0.19/0.38 ( ! [Xx: $i] :
% 0.19/0.38 ~ ( in @ Xx @ A )
% 0.19/0.38 => ( A = emptyset ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(noeltsimpempty_type,type,
% 0.19/0.38 noeltsimpempty: $o ).
% 0.19/0.38
% 0.19/0.38 thf(noeltsimpempty,definition,
% 0.19/0.38 ( noeltsimpempty
% 0.19/0.38 = ( ! [A: $i] :
% 0.19/0.38 ( ! [Xx: $i] :
% 0.19/0.38 ~ ( in @ Xx @ A )
% 0.19/0.38 => ( A = emptyset ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(setbeta_type,type,
% 0.19/0.38 setbeta: $o ).
% 0.19/0.38
% 0.19/0.38 thf(setbeta,definition,
% 0.19/0.38 ( setbeta
% 0.19/0.38 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ A )
% 0.19/0.38 => ( ( in @ Xx
% 0.19/0.38 @ ( dsetconstr @ A
% 0.19/0.38 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.19/0.38 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(nonempty_type,type,
% 0.19/0.38 nonempty: $i > $o ).
% 0.19/0.38
% 0.19/0.38 thf(nonempty,definition,
% 0.19/0.38 ( nonempty
% 0.19/0.38 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(nonemptyE1_type,type,
% 0.19/0.38 nonemptyE1: $o ).
% 0.19/0.38
% 0.19/0.38 thf(nonemptyE1,definition,
% 0.19/0.38 ( nonemptyE1
% 0.19/0.38 = ( ! [A: $i] :
% 0.19/0.38 ( ( nonempty @ A )
% 0.19/0.38 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(nonemptyI_type,type,
% 0.19/0.38 nonemptyI: $o ).
% 0.19/0.38
% 0.19/0.38 thf(nonemptyI,definition,
% 0.19/0.38 ( nonemptyI
% 0.19/0.38 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ A )
% 0.19/0.38 => ( ( Xphi @ Xx )
% 0.19/0.38 => ( nonempty
% 0.19/0.38 @ ( dsetconstr @ A
% 0.19/0.38 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(nonemptyI1_type,type,
% 0.19/0.38 nonemptyI1: $o ).
% 0.19/0.38
% 0.19/0.38 thf(nonemptyI1,definition,
% 0.19/0.38 ( nonemptyI1
% 0.19/0.38 = ( ! [A: $i] :
% 0.19/0.38 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.19/0.38 => ( nonempty @ A ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(setadjoinIL_type,type,
% 0.19/0.38 setadjoinIL: $o ).
% 0.19/0.38
% 0.19/0.38 thf(setadjoinIL,definition,
% 0.19/0.38 ( setadjoinIL
% 0.19/0.38 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(emptyinunitempty_type,type,
% 0.19/0.38 emptyinunitempty: $o ).
% 0.19/0.38
% 0.19/0.38 thf(emptyinunitempty,definition,
% 0.19/0.38 ( emptyinunitempty
% 0.19/0.38 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(setadjoinIR_type,type,
% 0.19/0.38 setadjoinIR: $o ).
% 0.19/0.38
% 0.19/0.38 thf(setadjoinIR,definition,
% 0.19/0.38 ( setadjoinIR
% 0.19/0.38 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.19/0.38 ( ( in @ Xy @ A )
% 0.19/0.38 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(setadjoinE_type,type,
% 0.19/0.38 setadjoinE: $o ).
% 0.19/0.38
% 0.19/0.38 thf(setadjoinE,definition,
% 0.19/0.38 ( setadjoinE
% 0.19/0.38 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.19/0.38 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.19/0.38 => ! [Xphi: $o] :
% 0.19/0.38 ( ( ( Xy = Xx )
% 0.19/0.38 => Xphi )
% 0.19/0.38 => ( ( ( in @ Xy @ A )
% 0.19/0.38 => Xphi )
% 0.19/0.38 => Xphi ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(setadjoinOr_type,type,
% 0.19/0.38 setadjoinOr: $o ).
% 0.19/0.38
% 0.19/0.38 thf(setadjoinOr,definition,
% 0.19/0.38 ( setadjoinOr
% 0.19/0.38 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.19/0.38 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.19/0.38 => ( ( Xy = Xx )
% 0.19/0.38 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(setoftrueEq_type,type,
% 0.19/0.38 setoftrueEq: $o ).
% 0.19/0.38
% 0.19/0.38 thf(setoftrueEq,definition,
% 0.19/0.38 ( setoftrueEq
% 0.19/0.38 = ( ! [A: $i] :
% 0.19/0.38 ( ( dsetconstr @ A
% 0.19/0.38 @ ^ [Xx: $i] : $true )
% 0.19/0.38 = A ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(powersetI_type,type,
% 0.19/0.38 powersetI: $o ).
% 0.19/0.38
% 0.19/0.38 thf(powersetI,definition,
% 0.19/0.38 ( powersetI
% 0.19/0.38 = ( ! [A: $i,B: $i] :
% 0.19/0.38 ( ! [Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ B )
% 0.19/0.38 => ( in @ Xx @ A ) )
% 0.19/0.38 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(emptyinPowerset_type,type,
% 0.19/0.38 emptyinPowerset: $o ).
% 0.19/0.38
% 0.19/0.38 thf(emptyinPowerset,definition,
% 0.19/0.38 ( emptyinPowerset
% 0.19/0.38 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(emptyInPowerset_type,type,
% 0.19/0.38 emptyInPowerset: $o ).
% 0.19/0.38
% 0.19/0.38 thf(emptyInPowerset,definition,
% 0.19/0.38 ( emptyInPowerset
% 0.19/0.38 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(powersetE_type,type,
% 0.19/0.38 powersetE: $o ).
% 0.19/0.38
% 0.19/0.38 thf(powersetE,definition,
% 0.19/0.38 ( powersetE
% 0.19/0.38 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.38 ( ( in @ B @ ( powerset @ A ) )
% 0.19/0.38 => ( ( in @ Xx @ B )
% 0.19/0.38 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(setunionI_type,type,
% 0.19/0.38 setunionI: $o ).
% 0.19/0.38
% 0.19/0.38 thf(setunionI,definition,
% 0.19/0.38 ( setunionI
% 0.19/0.38 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.19/0.38 ( ( in @ Xx @ B )
% 0.19/0.38 => ( ( in @ B @ A )
% 0.19/0.38 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(setunionE_type,type,
% 0.19/0.38 setunionE: $o ).
% 0.19/0.38
% 0.19/0.38 thf(setunionE,definition,
% 0.19/0.38 ( setunionE
% 0.19/0.38 = ( ! [A: $i,Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ ( setunion @ A ) )
% 0.19/0.38 => ! [Xphi: $o] :
% 0.19/0.38 ( ! [B: $i] :
% 0.19/0.38 ( ( in @ Xx @ B )
% 0.19/0.38 => ( ( in @ B @ A )
% 0.19/0.38 => Xphi ) )
% 0.19/0.38 => Xphi ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(subPowSU_type,type,
% 0.19/0.38 subPowSU: $o ).
% 0.19/0.38
% 0.19/0.38 thf(subPowSU,definition,
% 0.19/0.38 ( subPowSU
% 0.19/0.38 = ( ! [A: $i,Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ A )
% 0.19/0.38 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(exuE2_type,type,
% 0.19/0.38 exuE2: $o ).
% 0.19/0.38
% 0.19/0.38 thf(exuE2,definition,
% 0.19/0.38 ( exuE2
% 0.19/0.38 = ( ! [Xphi: $i > $o] :
% 0.19/0.38 ( ( exu
% 0.19/0.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.38 => ? [Xx: $i] :
% 0.19/0.38 ! [Xy: $i] :
% 0.19/0.38 ( ( Xphi @ Xy )
% 0.19/0.38 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(nonemptyImpWitness_type,type,
% 0.19/0.38 nonemptyImpWitness: $o ).
% 0.19/0.38
% 0.19/0.38 thf(nonemptyImpWitness,definition,
% 0.19/0.38 ( nonemptyImpWitness
% 0.19/0.38 = ( ! [A: $i] :
% 0.19/0.38 ( ( nonempty @ A )
% 0.19/0.38 => ? [Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ A )
% 0.19/0.38 & $true ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(uniqinunit_type,type,
% 0.19/0.38 uniqinunit: $o ).
% 0.19/0.38
% 0.19/0.38 thf(uniqinunit,definition,
% 0.19/0.38 ( uniqinunit
% 0.19/0.38 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.38 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.19/0.38 => ( Xx = Xy ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(notinsingleton_type,type,
% 0.19/0.38 notinsingleton: $o ).
% 0.19/0.38
% 0.19/0.38 thf(notinsingleton,definition,
% 0.19/0.38 ( notinsingleton
% 0.19/0.38 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.38 ( ( Xx != Xy )
% 0.19/0.38 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(eqinunit_type,type,
% 0.19/0.38 eqinunit: $o ).
% 0.19/0.38
% 0.19/0.38 thf(eqinunit,definition,
% 0.19/0.38 ( eqinunit
% 0.19/0.38 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.38 ( ( Xx = Xy )
% 0.19/0.38 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(singletonsswitch_type,type,
% 0.19/0.38 singletonsswitch: $o ).
% 0.19/0.38
% 0.19/0.38 thf(singletonsswitch,definition,
% 0.19/0.38 ( singletonsswitch
% 0.19/0.38 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.38 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.19/0.38 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(upairsetE_type,type,
% 0.19/0.38 upairsetE: $o ).
% 0.19/0.38
% 0.19/0.38 thf(upairsetE,definition,
% 0.19/0.38 ( upairsetE
% 0.19/0.38 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.19/0.38 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.19/0.38 => ( ( Xz = Xx )
% 0.19/0.38 | ( Xz = Xy ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(upairsetIL_type,type,
% 0.19/0.38 upairsetIL: $o ).
% 0.19/0.38
% 0.19/0.38 thf(upairsetIL,definition,
% 0.19/0.38 ( upairsetIL
% 0.19/0.38 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(upairsetIR_type,type,
% 0.19/0.38 upairsetIR: $o ).
% 0.19/0.38
% 0.19/0.38 thf(upairsetIR,definition,
% 0.19/0.38 ( upairsetIR
% 0.19/0.38 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(emptyE1_type,type,
% 0.19/0.38 emptyE1: $o ).
% 0.19/0.38
% 0.19/0.38 thf(emptyE1,definition,
% 0.19/0.38 ( emptyE1
% 0.19/0.38 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.38 ( ? [Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ A )
% 0.19/0.38 & ( Xphi @ Xx ) )
% 0.19/0.38 => ( ( ( dsetconstr @ A
% 0.19/0.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.38 = emptyset )
% 0.19/0.38 => $false ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(vacuousDall_type,type,
% 0.19/0.38 vacuousDall: $o ).
% 0.19/0.38
% 0.19/0.38 thf(vacuousDall,definition,
% 0.19/0.38 ( vacuousDall
% 0.19/0.38 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ emptyset )
% 0.19/0.38 => ( Xphi @ Xx ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(quantDeMorgan1_type,type,
% 0.19/0.38 quantDeMorgan1: $o ).
% 0.19/0.38
% 0.19/0.38 thf(quantDeMorgan1,definition,
% 0.19/0.38 ( quantDeMorgan1
% 0.19/0.38 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.38 ( ~ ! [Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ A )
% 0.19/0.38 => ( Xphi @ Xx ) )
% 0.19/0.38 => ? [Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ A )
% 0.19/0.38 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(quantDeMorgan2,conjecture,
% 0.19/0.38 ( setextAx
% 0.19/0.38 => ( emptysetAx
% 0.19/0.38 => ( setadjoinAx
% 0.19/0.38 => ( powersetAx
% 0.19/0.38 => ( setunionAx
% 0.19/0.38 => ( omega0Ax
% 0.19/0.38 => ( omegaSAx
% 0.19/0.38 => ( omegaIndAx
% 0.19/0.38 => ( replAx
% 0.19/0.38 => ( foundationAx
% 0.19/0.38 => ( wellorderingAx
% 0.19/0.38 => ( descrp
% 0.19/0.38 => ( dsetconstrI
% 0.19/0.38 => ( dsetconstrEL
% 0.19/0.38 => ( dsetconstrER
% 0.19/0.38 => ( exuE1
% 0.19/0.38 => ( prop2setE
% 0.19/0.38 => ( emptysetE
% 0.19/0.38 => ( emptysetimpfalse
% 0.19/0.38 => ( notinemptyset
% 0.19/0.38 => ( exuE3e
% 0.19/0.38 => ( setext
% 0.19/0.38 => ( emptyI
% 0.19/0.38 => ( noeltsimpempty
% 0.19/0.38 => ( setbeta
% 0.19/0.38 => ( nonemptyE1
% 0.19/0.38 => ( nonemptyI
% 0.19/0.38 => ( nonemptyI1
% 0.19/0.38 => ( setadjoinIL
% 0.19/0.38 => ( emptyinunitempty
% 0.19/0.38 => ( setadjoinIR
% 0.19/0.38 => ( setadjoinE
% 0.19/0.38 => ( setadjoinOr
% 0.19/0.38 => ( setoftrueEq
% 0.19/0.38 => ( powersetI
% 0.19/0.38 => ( emptyinPowerset
% 0.19/0.38 => ( emptyInPowerset
% 0.19/0.38 => ( powersetE
% 0.19/0.38 => ( setunionI
% 0.19/0.38 => ( setunionE
% 0.19/0.38 => ( subPowSU
% 0.19/0.38 => ( exuE2
% 0.19/0.38 => ( nonemptyImpWitness
% 0.19/0.38 => ( uniqinunit
% 0.19/0.38 => ( notinsingleton
% 0.19/0.38 => ( eqinunit
% 0.19/0.38 => ( singletonsswitch
% 0.19/0.38 => ( upairsetE
% 0.19/0.38 => ( upairsetIL
% 0.19/0.38 => ( upairsetIR
% 0.19/0.38 => ( emptyE1
% 0.19/0.38 => ( vacuousDall
% 0.19/0.38 => ( quantDeMorgan1
% 0.19/0.38 => ! [A: $i,Xphi: $i > $o] :
% 0.19/0.38 ( ! [Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ A )
% 0.19/0.38 => ~ ( Xphi @ Xx ) )
% 0.19/0.38 => ~ ? [Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ A )
% 0.19/0.38 & ( Xphi @ Xx ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 %------------------------------------------------------------------------------
% 0.19/0.38 ------------------------
% 0.23/1.40 ---- Embedded in THF ---
% 0.23/1.40 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 0.23/1.40 %%% Generated on Tue Feb 24 12:25:56 EST 2026
% 0.23/1.40 %%% using '$$dhol' embedding, version 1.3.0.
% 0.23/1.40 %%% Logic specification used:
% 0.23/1.40 %%% thf(spec, logic, $$dhol).
% 0.23/1.40
% 0.23/1.40 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.6gNdrgnz6L/DTF2DTF7087.p
% See solution above
% 0.23/1.40 ------------------------
% 0.23/1.42 ---- Cleaned THF ---
% 0.23/1.42 thf(in_type,type,
% 0.23/1.42 in: $i > $i > $o ).
% 0.23/1.42
% 0.23/1.42 thf(exu_type,type,
% 0.23/1.42 exu: ( $i > $o ) > $o ).
% 0.23/1.42
% 0.23/1.42 thf(exu,definition,
% 0.23/1.42 ( exu
% 0.23/1.42 = ( ^ [Xphi: $i > $o] :
% 0.23/1.42 ? [Xx: $i] :
% 0.23/1.42 ( ( Xphi @ Xx )
% 0.23/1.42 & ! [Xy: $i] :
% 0.23/1.42 ( ( Xphi @ Xy )
% 0.23/1.42 => ( Xx = Xy ) ) ) ) ) ).
% 0.23/1.42
% 0.23/1.42 thf(setextAx_type,type,
% 0.23/1.42 setextAx: $o ).
% 0.23/1.42
% 0.23/1.42 thf(setextAx,definition,
% 0.23/1.42 ( setextAx
% 0.23/1.42 = ( ! [A: $i,B: $i] :
% 0.23/1.42 ( ! [Xx: $i] :
% 0.23/1.42 ( ( in @ Xx @ A )
% 0.23/1.42 <=> ( in @ Xx @ B ) )
% 0.23/1.42 => ( A = B ) ) ) ) ).
% 0.23/1.42
% 0.23/1.42 thf(emptyset_type,type,
% 0.23/1.42 emptyset: $i ).
% 0.23/1.42
% 0.23/1.42 thf(emptysetAx_type,type,
% 0.23/1.42 emptysetAx: $o ).
% 0.23/1.42
% 0.23/1.42 thf(emptysetAx,definition,
% 0.23/1.42 ( emptysetAx
% 0.23/1.42 = ( ! [Xx: $i] :
% 0.23/1.42 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.23/1.42
% 0.23/1.42 thf(setadjoin_type,type,
% 0.23/1.42 setadjoin: $i > $i > $i ).
% 0.23/1.42
% 0.23/1.42 thf(setadjoinAx_type,type,
% 0.23/1.42 setadjoinAx: $o ).
% 0.23/1.42
% 0.23/1.42 thf(setadjoinAx,definition,
% 0.23/1.42 ( setadjoinAx
% 0.23/1.42 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.23/1.42 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.23/1.42 <=> ( ( Xy = Xx )
% 0.23/1.42 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.23/1.42
% 0.23/1.42 thf(powerset_type,type,
% 0.23/1.42 powerset: $i > $i ).
% 0.23/1.42
% 0.23/1.42 thf(powersetAx_type,type,
% 0.23/1.42 powersetAx: $o ).
% 0.23/1.42
% 0.23/1.42 thf(powersetAx,definition,
% 0.23/1.42 ( powersetAx
% 0.23/1.42 = ( ! [A: $i,B: $i] :
% 0.23/1.42 ( ( in @ B @ ( powerset @ A ) )
% 0.23/1.42 <=> ! [Xx: $i] :
% 0.23/1.42 ( ( in @ Xx @ B )
% 0.23/1.42 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.23/1.42
% 0.23/1.42 thf(setunion_type,type,
% 0.23/1.42 setunion: $i > $i ).
% 0.23/1.42
% 0.23/1.42 thf(setunionAx_type,type,
% 0.23/1.42 setunionAx: $o ).
% 0.23/1.42
% 0.23/1.42 thf(setunionAx,definition,
% 0.23/1.42 ( setunionAx
% 0.23/1.42 = ( ! [A: $i,Xx: $i] :
% 0.23/1.42 ( ( in @ Xx @ ( setunion @ A ) )
% 0.23/1.42 <=> ? [B: $i] :
% 0.23/1.42 ( ( in @ Xx @ B )
% 0.23/1.42 & ( in @ B @ A ) ) ) ) ) ).
% 0.23/1.42
% 0.23/1.42 thf(omega_type,type,
% 0.23/1.42 omega: $i ).
% 0.23/1.42
% 0.23/1.42 thf(omega0Ax_type,type,
% 0.23/1.42 omega0Ax: $o ).
% 0.23/1.42
% 0.23/1.42 thf(omega0Ax,definition,
% 0.23/1.42 ( omega0Ax
% 0.23/1.42 = ( in @ emptyset @ omega ) ) ).
% 0.23/1.42
% 0.23/1.42 thf(omegaSAx_type,type,
% 0.23/1.42 omegaSAx: $o ).
% 0.23/1.42
% 0.23/1.42 thf(omegaSAx,definition,
% 0.23/1.42 ( omegaSAx
% 0.23/1.42 = ( ! [Xx: $i] :
% 0.23/1.42 ( ( in @ Xx @ omega )
% 0.23/1.42 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.23/1.42
% 0.23/1.42 thf(omegaIndAx_type,type,
% 0.23/1.42 omegaIndAx: $o ).
% 0.23/1.42
% 0.23/1.42 thf(omegaIndAx,definition,
% 0.23/1.42 ( omegaIndAx
% 0.23/1.42 = ( ! [A: $i] :
% 0.23/1.42 ( ( ( in @ emptyset @ A )
% 0.23/1.42 & ! [Xx: $i] :
% 0.23/1.42 ( ( ( in @ Xx @ omega )
% 0.23/1.42 & ( in @ Xx @ A ) )
% 0.23/1.42 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.23/1.42 => ! [Xx: $i] :
% 0.23/1.42 ( ( in @ Xx @ omega )
% 0.23/1.42 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.23/1.42
% 0.23/1.42 thf(replAx_type,type,
% 0.23/1.42 replAx: $o ).
% 0.23/1.42
% 0.23/1.42 thf(replAx,definition,
% 0.23/1.42 ( replAx
% 0.23/1.42 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.23/1.42 ( ! [Xx: $i] :
% 0.23/1.42 ( ( in @ Xx @ A )
% 0.23/1.42 => ( exu
% 0.23/1.42 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.23/1.42 => ? [B: $i] :
% 0.23/1.42 ! [Xx: $i] :
% 0.23/1.42 ( ( in @ Xx @ B )
% 0.23/1.42 <=> ? [Xy: $i] :
% 0.23/1.42 ( ( in @ Xy @ A )
% 0.23/1.42 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.23/1.42
% 0.23/1.42 thf(foundationAx_type,type,
% 0.23/1.42 foundationAx: $o ).
% 0.23/1.42
% 0.23/1.42 thf(foundationAx,definition,
% 0.23/1.42 ( foundationAx
% 0.23/1.42 = ( ! [A: $i] :
% 0.23/1.42 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.23/1.42 => ? [B: $i] :
% 0.23/1.42 ( ( in @ B @ A )
% 0.23/1.42 & ~ ? [Xx: $i] :
% 0.23/1.42 ( ( in @ Xx @ B )
% 0.23/1.42 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.23/1.42
% 0.23/1.42 thf(wellorderingAx_type,type,
% 0.23/1.42 wellorderingAx: $o ).
% 0.23/1.42
% 0.23/1.42 thf(wellorderingAx,definition,
% 0.23/1.42 ( wellorderingAx
% 0.23/1.42 = ( ! [A: $i] :
% 0.23/1.42 ? [B: $i] :
% 0.23/1.42 ( ! [C: $i] :
% 0.23/1.42 ( ( in @ C @ B )
% 0.23/1.42 => ! [Xx: $i] :
% 0.23/1.42 ( ( in @ Xx @ C )
% 0.23/1.42 => ( in @ Xx @ A ) ) )
% 0.23/1.42 & ! [Xx: $i,Xy: $i] :
% 0.23/1.42 ( ( ( in @ Xx @ A )
% 0.23/1.42 & ( in @ Xy @ A ) )
% 0.23/1.42 => ( ! [C: $i] :
% 0.23/1.42 ( ( in @ C @ B )
% 0.23/1.42 => ( ( in @ Xx @ C )
% 0.23/1.42 <=> ( in @ Xy @ C ) ) )
% 0.23/1.42 => ( Xx = Xy ) ) )
% 0.23/1.42 & ! [C: $i,D: $i] :
% 0.23/1.42 ( ( ( in @ C @ B )
% 0.23/1.42 & ( in @ D @ B ) )
% 0.23/1.42 => ( ! [Xx: $i] :
% 0.23/1.42 ( ( in @ Xx @ C )
% 0.23/1.42 => ( in @ Xx @ D ) )
% 0.23/1.42 | ! [Xx: $i] :
% 0.23/1.42 ( ( in @ Xx @ D )
% 0.23/1.42 => ( in @ Xx @ C ) ) ) )
% 0.23/1.42 & ! [C: $i] :
% 0.23/1.42 ( ( ! [Xx: $i] :
% 0.23/1.42 ( ( in @ Xx @ C )
% 0.23/1.42 => ( in @ Xx @ A ) )
% 0.23/1.42 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.23/1.42 => ? [D: $i,Xx: $i] :
% 0.23/1.42 ( ( in @ D @ B )
% 0.23/1.42 & ( in @ Xx @ C )
% 0.23/1.42 & ~ ? [Xy: $i] :
% 0.23/1.42 ( ( in @ Xy @ D )
% 0.23/1.42 & ( in @ Xy @ C ) )
% 0.23/1.42 & ! [E: $i] :
% 0.23/1.42 ( ( in @ E @ B )
% 0.23/1.42 => ( ! [Xy: $i] :
% 0.23/1.42 ( ( in @ Xy @ E )
% 0.23/1.42 => ( in @ Xy @ D ) )
% 0.23/1.42 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.23/1.42
% 0.23/1.42 thf(descr_type,type,
% 0.23/1.42 descr: ( $i > $o ) > $i ).
% 0.23/1.42
% 0.23/1.42 thf(descrp_type,type,
% 0.23/1.42 descrp: $o ).
% 0.23/1.42
% 0.23/1.42 thf(descrp,definition,
% 0.23/1.42 ( descrp
% 0.23/1.42 = ( ! [Xphi: $i > $o] :
% 0.23/1.42 ( ( exu
% 0.23/1.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.23/1.42 => ( Xphi
% 0.23/1.42 @ ( descr
% 0.23/1.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.23/1.42
% 0.23/1.42 thf(dsetconstr_type,type,
% 0.23/1.42 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.23/1.42
% 0.23/1.42 thf(dsetconstrI_type,type,
% 0.23/1.42 dsetconstrI: $o ).
% 0.23/1.42
% 0.23/1.42 thf(dsetconstrI,definition,
% 0.23/1.42 ( dsetconstrI
% 0.23/1.42 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.23/1.42 ( ( in @ Xx @ A )
% 0.23/1.42 => ( ( Xphi @ Xx )
% 0.23/1.42 => ( in @ Xx
% 0.23/1.42 @ ( dsetconstr @ A
% 0.23/1.42 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.23/1.42
% 0.23/1.42 thf(dsetconstrEL_type,type,
% 0.23/1.42 dsetconstrEL: $o ).
% 0.23/1.42
% 0.23/1.42 thf(dsetconstrEL,definition,
% 0.23/1.42 ( dsetconstrEL
% 0.23/1.42 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.23/1.42 ( ( in @ Xx
% 0.23/1.42 @ ( dsetconstr @ A
% 0.23/1.42 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.23/1.42 => ( in @ Xx @ A ) ) ) ) ).
% 0.23/1.42
% 0.23/1.42 thf(dsetconstrER_type,type,
% 0.23/1.42 dsetconstrER: $o ).
% 0.23/1.42
% 0.23/1.42 thf(dsetconstrER,definition,
% 0.23/1.42 ( dsetconstrER
% 0.23/1.42 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.23/1.42 ( ( in @ Xx
% 0.23/1.42 @ ( dsetconstr @ A
% 0.23/1.42 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.23/1.42 => ( Xphi @ Xx ) ) ) ) ).
% 0.23/1.42
% 0.23/1.42 thf(exuE1_type,type,
% 0.23/1.42 exuE1: $o ).
% 0.23/1.42
% 0.23/1.42 thf(exuE1,definition,
% 0.23/1.42 ( exuE1
% 0.23/1.42 = ( ! [Xphi: $i > $o] :
% 0.23/1.42 ( ( exu
% 0.23/1.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.23/1.42 => ? [Xx: $i] :
% 0.23/1.42 ( ( Xphi @ Xx )
% 0.23/1.42 & ! [Xy: $i] :
% 0.23/1.42 ( ( Xphi @ Xy )
% 0.23/1.42 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.23/1.42
% 0.23/1.42 thf(prop2set_type,type,
% 0.23/1.42 prop2set: $o > $i ).
% 0.23/1.42
% 0.23/1.42 thf(prop2set,definition,
% 0.23/1.42 ( prop2set
% 0.23/1.42 = ( ^ [Xphi: $o] :
% 0.23/1.42 ( dsetconstr @ ( powerset @ emptyset )
% 0.23/1.42 @ ^ [Xx: $i] : Xphi ) ) ) ).
% 0.23/1.42
% 0.23/1.42 thf(prop2setE_type,type,
% 0.23/1.42 prop2setE: $o ).
% 0.23/1.42
% 0.23/1.42 thf(prop2setE,definition,
% 0.23/1.42 ( prop2setE
% 0.23/1.42 = ( ! [Xphi: $o,Xx: $i] :
% 0.23/1.42 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.23/1.42 => Xphi ) ) ) ).
% 0.23/1.42
% 0.23/1.42 thf(emptysetE_type,type,
% 0.23/1.42 emptysetE: $o ).
% 0.23/1.42
% 0.23/1.42 thf(emptysetE,definition,
% 0.23/1.42 ( emptysetE
% 0.23/1.42 = ( ! [Xx: $i] :
% 0.23/1.42 ( ( in @ Xx @ emptyset )
% 0.23/1.42 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.23/1.42
% 0.23/1.42 thf(emptysetimpfalse_type,type,
% 0.23/1.42 emptysetimpfalse: $o ).
% 0.23/1.42
% 0.23/1.42 thf(emptysetimpfalse,definition,
% 0.23/1.42 ( emptysetimpfalse
% 0.23/1.42 = ( ! [Xx: $i] :
% 0.23/1.42 ( ( in @ Xx @ emptyset )
% 0.23/1.42 => $false ) ) ) ).
% 0.23/1.42
% 0.23/1.42 thf(notinemptyset_type,type,
% 0.23/1.42 notinemptyset: $o ).
% 0.23/1.42
% 0.23/1.42 thf(notinemptyset,definition,
% 0.23/1.42 ( notinemptyset
% 0.23/1.42 = ( ! [Xx: $i] :
% 0.23/1.42 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.23/1.42
% 0.23/1.42 thf(exuE3e_type,type,
% 0.23/1.42 exuE3e: $o ).
% 0.23/1.42
% 0.23/1.42 thf(exuE3e,definition,
% 0.23/1.42 ( exuE3e
% 0.23/1.42 = ( ! [Xphi: $i > $o] :
% 0.23/1.42 ( ( exu
% 0.23/1.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.23/1.42 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.23/1.42
% 0.23/1.42 thf(setext_type,type,
% 0.23/1.42 setext: $o ).
% 0.23/1.42
% 0.23/1.42 thf(setext,definition,
% 0.23/1.42 ( setext
% 0.23/1.42 = ( ! [A: $i,B: $i] :
% 0.23/1.42 ( ! [Xx: $i] :
% 0.23/1.42 ( ( in @ Xx @ A )
% 0.23/1.42 => ( in @ Xx @ B ) )
% 0.23/1.42 => ( ! [Xx: $i] :
% 0.23/1.42 ( ( in @ Xx @ B )
% 0.23/1.42 => ( in @ Xx @ A ) )
% 0.23/1.42 => ( A = B ) ) ) ) ) ).
% 0.23/1.42
% 0.23/1.42 thf(emptyI_type,type,
% 0.23/1.42 emptyI: $o ).
% 0.23/1.42
% 0.23/1.42 thf(emptyI,definition,
% 0.23/1.42 ( emptyI
% 0.23/1.42 = ( ! [A: $i] :
% 0.23/1.42 ( ! [Xx: $i] :
% 0.23/1.42 ~ ( in @ Xx @ A )
% 0.23/1.42 => ( A = emptyset ) ) ) ) ).
% 0.23/1.42
% 0.23/1.42 thf(noeltsimpempty_type,type,
% 0.23/1.42 noeltsimpempty: $o ).
% 0.23/1.42
% 0.23/1.42 thf(noeltsimpempty,definition,
% 0.23/1.42 ( noeltsimpempty
% 0.23/1.42 = ( ! [A: $i] :
% 0.23/1.42 ( ! [Xx: $i] :
% 0.23/1.42 ~ ( in @ Xx @ A )
% 0.23/1.42 => ( A = emptyset ) ) ) ) ).
% 0.23/1.42
% 0.23/1.42 thf(setbeta_type,type,
% 0.23/1.42 setbeta: $o ).
% 0.23/1.42
% 0.23/1.42 thf(setbeta,definition,
% 0.23/1.42 ( setbeta
% 0.23/1.42 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.23/1.42 ( ( in @ Xx @ A )
% 0.23/1.42 => ( ( in @ Xx
% 0.23/1.42 @ ( dsetconstr @ A
% 0.23/1.42 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.23/1.42 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.23/1.42
% 0.23/1.42 thf(nonempty_type,type,
% 0.23/1.42 nonempty: $i > $o ).
% 0.23/1.42
% 0.23/1.42 thf(nonempty,definition,
% 0.23/1.42 ( nonempty
% 0.23/1.42 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.23/1.42
% 0.23/1.42 thf(nonemptyE1_type,type,
% 0.23/1.42 nonemptyE1: $o ).
% 0.23/1.42
% 0.23/1.42 thf(nonemptyE1,definition,
% 0.23/1.42 ( nonemptyE1
% 0.23/1.42 = ( ! [A: $i] :
% 0.23/1.42 ( ( nonempty @ A )
% 0.23/1.42 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.23/1.42
% 0.23/1.42 thf(nonemptyI_type,type,
% 0.23/1.42 nonemptyI: $o ).
% 0.23/1.42
% 0.23/1.42 thf(nonemptyI,definition,
% 0.23/1.42 ( nonemptyI
% 0.23/1.42 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.23/1.42 ( ( in @ Xx @ A )
% 0.23/1.42 => ( ( Xphi @ Xx )
% 0.23/1.42 => ( nonempty
% 0.23/1.42 @ ( dsetconstr @ A
% 0.23/1.42 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.23/1.42
% 0.23/1.42 thf(nonemptyI1_type,type,
% 0.23/1.42 nonemptyI1: $o ).
% 0.23/1.42
% 0.23/1.42 thf(nonemptyI1,definition,
% 0.23/1.42 ( nonemptyI1
% 0.23/1.42 = ( ! [A: $i] :
% 0.23/1.42 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.23/1.42 => ( nonempty @ A ) ) ) ) ).
% 0.23/1.42
% 0.23/1.42 thf(setadjoinIL_type,type,
% 0.23/1.42 setadjoinIL: $o ).
% 0.23/1.42
% 0.23/1.42 thf(setadjoinIL,definition,
% 0.23/1.42 ( setadjoinIL
% 0.23/1.42 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.23/1.42
% 0.23/1.42 thf(emptyinunitempty_type,type,
% 0.23/1.42 emptyinunitempty: $o ).
% 0.23/1.42
% 0.23/1.42 thf(emptyinunitempty,definition,
% 0.23/1.42 ( emptyinunitempty
% 0.23/1.42 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.23/1.42
% 0.23/1.42 thf(setadjoinIR_type,type,
% 0.23/1.42 setadjoinIR: $o ).
% 0.23/1.42
% 0.23/1.42 thf(setadjoinIR,definition,
% 0.23/1.42 ( setadjoinIR
% 0.23/1.42 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.23/1.42 ( ( in @ Xy @ A )
% 0.23/1.42 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.23/1.42
% 0.23/1.42 thf(setadjoinE_type,type,
% 0.23/1.42 setadjoinE: $o ).
% 0.23/1.42
% 0.23/1.42 thf(setadjoinE,definition,
% 0.23/1.42 ( setadjoinE
% 0.23/1.42 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.23/1.42 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.23/1.42 => ! [Xphi: $o] :
% 0.23/1.42 ( ( ( Xy = Xx )
% 0.23/1.42 => Xphi )
% 0.23/1.42 => ( ( ( in @ Xy @ A )
% 0.23/1.42 => Xphi )
% 0.23/1.42 => Xphi ) ) ) ) ) ).
% 0.23/1.42
% 0.23/1.42 thf(setadjoinOr_type,type,
% 0.23/1.42 setadjoinOr: $o ).
% 0.23/1.42
% 0.23/1.42 thf(setadjoinOr,definition,
% 0.23/1.42 ( setadjoinOr
% 0.23/1.42 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.23/1.42 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.23/1.42 => ( ( Xy = Xx )
% 0.23/1.42 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.23/1.42
% 0.23/1.42 thf(setoftrueEq_type,type,
% 0.23/1.42 setoftrueEq: $o ).
% 0.23/1.42
% 0.23/1.42 thf(setoftrueEq,definition,
% 0.23/1.42 ( setoftrueEq
% 0.23/1.42 = ( ! [A: $i] :
% 0.23/1.42 ( ( dsetconstr @ A
% 0.23/1.42 @ ^ [Xx: $i] : $true )
% 0.23/1.42 = A ) ) ) ).
% 0.23/1.42
% 0.23/1.42 thf(powersetI_type,type,
% 0.23/1.42 powersetI: $o ).
% 0.23/1.42
% 0.23/1.42 thf(powersetI,definition,
% 0.23/1.42 ( powersetI
% 0.23/1.42 = ( ! [A: $i,B: $i] :
% 0.23/1.42 ( ! [Xx: $i] :
% 0.23/1.42 ( ( in @ Xx @ B )
% 0.23/1.42 => ( in @ Xx @ A ) )
% 0.23/1.42 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.23/1.42
% 0.23/1.42 thf(emptyinPowerset_type,type,
% 0.23/1.42 emptyinPowerset: $o ).
% 0.23/1.42
% 0.23/1.42 thf(emptyinPowerset,definition,
% 0.23/1.42 ( emptyinPowerset
% 0.23/1.42 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.23/1.42
% 0.23/1.42 thf(emptyInPowerset_type,type,
% 0.23/1.42 emptyInPowerset: $o ).
% 0.23/1.42
% 0.23/1.42 thf(emptyInPowerset,definition,
% 0.23/1.42 ( emptyInPowerset
% 0.23/1.42 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.23/1.42
% 0.23/1.42 thf(powersetE_type,type,
% 0.23/1.42 powersetE: $o ).
% 0.23/1.42
% 0.23/1.42 thf(powersetE,definition,
% 0.23/1.42 ( powersetE
% 0.23/1.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.23/1.42 ( ( in @ B @ ( powerset @ A ) )
% 0.23/1.42 => ( ( in @ Xx @ B )
% 0.23/1.42 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.23/1.42
% 0.23/1.42 thf(setunionI_type,type,
% 0.23/1.42 setunionI: $o ).
% 0.23/1.42
% 0.23/1.42 thf(setunionI,definition,
% 0.23/1.42 ( setunionI
% 0.23/1.42 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.23/1.42 ( ( in @ Xx @ B )
% 0.23/1.42 => ( ( in @ B @ A )
% 0.23/1.42 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.23/1.42
% 0.23/1.42 thf(setunionE_type,type,
% 0.23/1.42 setunionE: $o ).
% 0.23/1.42
% 0.23/1.42 thf(setunionE,definition,
% 0.23/1.42 ( setunionE
% 0.23/1.42 = ( ! [A: $i,Xx: $i] :
% 0.23/1.42 ( ( in @ Xx @ ( setunion @ A ) )
% 0.23/1.42 => ! [Xphi: $o] :
% 0.23/1.42 ( ! [B: $i] :
% 0.23/1.42 ( ( in @ Xx @ B )
% 0.23/1.42 => ( ( in @ B @ A )
% 0.23/1.42 => Xphi ) )
% 0.23/1.42 => Xphi ) ) ) ) ).
% 0.23/1.42
% 0.23/1.42 thf(subPowSU_type,type,
% 0.23/1.42 subPowSU: $o ).
% 0.23/1.42
% 0.23/1.42 thf(subPowSU,definition,
% 0.23/1.42 ( subPowSU
% 0.23/1.42 = ( ! [A: $i,Xx: $i] :
% 0.23/1.42 ( ( in @ Xx @ A )
% 0.23/1.42 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.23/1.42
% 0.23/1.42 thf(exuE2_type,type,
% 0.23/1.42 exuE2: $o ).
% 0.23/1.42
% 0.23/1.42 thf(exuE2,definition,
% 0.23/1.42 ( exuE2
% 0.23/1.42 = ( ! [Xphi: $i > $o] :
% 0.23/1.42 ( ( exu
% 0.23/1.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.23/1.42 => ? [Xx: $i] :
% 0.23/1.42 ! [Xy: $i] :
% 0.23/1.42 ( ( Xphi @ Xy )
% 0.23/1.42 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.23/1.42
% 0.23/1.42 thf(nonemptyImpWitness_type,type,
% 0.23/1.42 nonemptyImpWitness: $o ).
% 0.23/1.42
% 0.23/1.42 thf(nonemptyImpWitness,definition,
% 0.23/1.42 ( nonemptyImpWitness
% 0.23/1.42 = ( ! [A: $i] :
% 0.23/1.42 ( ( nonempty @ A )
% 0.23/1.42 => ? [Xx: $i] :
% 0.23/1.42 ( ( in @ Xx @ A )
% 0.23/1.42 & $true ) ) ) ) ).
% 0.23/1.42
% 0.23/1.42 thf(uniqinunit_type,type,
% 0.23/1.42 uniqinunit: $o ).
% 0.23/1.42
% 0.23/1.42 thf(uniqinunit,definition,
% 0.23/1.42 ( uniqinunit
% 0.23/1.42 = ( ! [Xx: $i,Xy: $i] :
% 0.23/1.42 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.23/1.42 => ( Xx = Xy ) ) ) ) ).
% 0.23/1.42
% 0.23/1.42 thf(notinsingleton_type,type,
% 0.23/1.42 notinsingleton: $o ).
% 0.23/1.42
% 0.23/1.42 thf(notinsingleton,definition,
% 0.23/1.42 ( notinsingleton
% 0.23/1.42 = ( ! [Xx: $i,Xy: $i] :
% 0.23/1.42 ( ( Xx != Xy )
% 0.23/1.42 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.23/1.42
% 0.23/1.42 thf(eqinunit_type,type,
% 0.23/1.42 eqinunit: $o ).
% 0.23/1.42
% 0.23/1.42 thf(eqinunit,definition,
% 0.23/1.42 ( eqinunit
% 0.23/1.42 = ( ! [Xx: $i,Xy: $i] :
% 0.23/1.42 ( ( Xx = Xy )
% 0.23/1.42 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.23/1.42
% 0.23/1.42 thf(singletonsswitch_type,type,
% 0.23/1.42 singletonsswitch: $o ).
% 0.23/1.42
% 0.23/1.42 thf(singletonsswitch,definition,
% 0.23/1.42 ( singletonsswitch
% 0.23/1.42 = ( ! [Xx: $i,Xy: $i] :
% 0.23/1.42 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.23/1.42 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.23/1.42
% 0.23/1.42 thf(upairsetE_type,type,
% 0.23/1.42 upairsetE: $o ).
% 0.23/1.42
% 0.23/1.42 thf(upairsetE,definition,
% 0.23/1.42 ( upairsetE
% 0.23/1.42 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.23/1.42 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.23/1.42 => ( ( Xz = Xx )
% 0.23/1.42 | ( Xz = Xy ) ) ) ) ) ).
% 0.23/1.42
% 0.23/1.42 thf(upairsetIL_type,type,
% 0.23/1.42 upairsetIL: $o ).
% 0.23/1.42
% 0.23/1.42 thf(upairsetIL,definition,
% 0.23/1.42 ( upairsetIL
% 0.23/1.42 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.23/1.42
% 0.23/1.42 thf(upairsetIR_type,type,
% 0.23/1.42 upairsetIR: $o ).
% 0.23/1.42
% 0.23/1.42 thf(upairsetIR,definition,
% 0.23/1.42 ( upairsetIR
% 0.23/1.42 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.23/1.42
% 0.23/1.42 thf(emptyE1_type,type,
% 0.23/1.42 emptyE1: $o ).
% 0.23/1.42
% 0.23/1.42 thf(emptyE1,definition,
% 0.23/1.42 ( emptyE1
% 0.23/1.42 = ( ! [A: $i,Xphi: $i > $o] :
% 0.23/1.42 ( ? [Xx: $i] :
% 0.23/1.42 ( ( in @ Xx @ A )
% 0.23/1.42 & ( Xphi @ Xx ) )
% 0.23/1.42 => ( ( ( dsetconstr @ A
% 0.23/1.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.23/1.42 = emptyset )
% 0.23/1.42 => $false ) ) ) ) ).
% 0.23/1.42
% 0.23/1.42 thf(vacuousDall_type,type,
% 0.23/1.42 vacuousDall: $o ).
% 0.23/1.42
% 0.23/1.42 thf(vacuousDall,definition,
% 0.23/1.42 ( vacuousDall
% 0.23/1.42 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.23/1.42 ( ( in @ Xx @ emptyset )
% 0.23/1.42 => ( Xphi @ Xx ) ) ) ) ).
% 0.23/1.42
% 0.23/1.42 thf(quantDeMorgan1_type,type,
% 0.23/1.42 quantDeMorgan1: $o ).
% 0.23/1.42
% 0.23/1.42 thf(quantDeMorgan1,definition,
% 0.23/1.42 ( quantDeMorgan1
% 0.23/1.42 = ( ! [A: $i,Xphi: $i > $o] :
% 0.23/1.42 ( ~ ! [Xx: $i] :
% 0.23/1.42 ( ( in @ Xx @ A )
% 0.23/1.42 => ( Xphi @ Xx ) )
% 0.23/1.42 => ? [Xx: $i] :
% 0.23/1.42 ( ( in @ Xx @ A )
% 0.23/1.42 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.23/1.42
% 0.23/1.42 thf(quantDeMorgan2,conjecture,
% 0.23/1.42 ( setextAx
% 0.23/1.42 => ( emptysetAx
% 0.23/1.42 => ( setadjoinAx
% 0.23/1.42 => ( powersetAx
% 0.23/1.42 => ( setunionAx
% 0.23/1.42 => ( omega0Ax
% 0.23/1.42 => ( omegaSAx
% 0.23/1.42 => ( omegaIndAx
% 0.23/1.42 => ( replAx
% 0.23/1.42 => ( foundationAx
% 0.23/1.42 => ( wellorderingAx
% 0.23/1.42 => ( descrp
% 0.23/1.42 => ( dsetconstrI
% 0.23/1.42 => ( dsetconstrEL
% 0.23/1.42 => ( dsetconstrER
% 0.23/1.42 => ( exuE1
% 0.23/1.42 => ( prop2setE
% 0.23/1.42 => ( emptysetE
% 0.23/1.42 => ( emptysetimpfalse
% 0.23/1.42 => ( notinemptyset
% 0.23/1.42 => ( exuE3e
% 0.23/1.42 => ( setext
% 0.23/1.42 => ( emptyI
% 0.23/1.42 => ( noeltsimpempty
% 0.23/1.42 => ( setbeta
% 0.23/1.42 => ( nonemptyE1
% 0.23/1.42 => ( nonemptyI
% 0.23/1.42 => ( nonemptyI1
% 0.23/1.42 => ( setadjoinIL
% 0.23/1.42 => ( emptyinunitempty
% 0.23/1.42 => ( setadjoinIR
% 0.23/1.42 => ( setadjoinE
% 0.23/1.42 => ( setadjoinOr
% 0.23/1.42 => ( setoftrueEq
% 0.23/1.42 => ( powersetI
% 0.23/1.42 => ( emptyinPowerset
% 0.23/1.42 => ( emptyInPowerset
% 0.23/1.42 => ( powersetE
% 0.23/1.42 => ( setunionI
% 0.23/1.42 => ( setunionE
% 0.23/1.42 => ( subPowSU
% 0.23/1.42 => ( exuE2
% 0.23/1.42 => ( nonemptyImpWitness
% 0.23/1.42 => ( uniqinunit
% 0.23/1.42 => ( notinsingleton
% 0.23/1.42 => ( eqinunit
% 0.23/1.42 => ( singletonsswitch
% 0.23/1.42 => ( upairsetE
% 0.23/1.42 => ( upairsetIL
% 0.23/1.42 => ( upairsetIR
% 0.23/1.42 => ( emptyE1
% 0.23/1.42 => ( vacuousDall
% 0.23/1.42 => ( quantDeMorgan1
% 0.23/1.42 => ! [A: $i,Xphi: $i > $o] :
% 0.23/1.42 ( ! [Xx: $i] :
% 0.23/1.42 ( ( in @ Xx @ A )
% 0.23/1.42 => ~ ( Xphi @ Xx ) )
% 0.23/1.42 => ~ ? [Xx: $i] :
% 0.23/1.42 ( ( in @ Xx @ A )
% 0.23/1.42 & ( Xphi @ Xx ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.23/1.42 ------------------------
% 0.23/1.44 % (7190)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_7087 for (2999ds/275Mi)
% 0.23/1.44 % (7188)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_7087 for (2999ds/2Mi)
% 0.23/1.44 % (7191)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_7087 for (2999ds/18Mi)
% 0.23/1.44 % (7185)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_7087 for (2999ds/183Mi)
% 0.23/1.44 % (7189)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_7087 for (2999ds/2Mi)
% 0.23/1.44 % (7187)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_7087 for (2999ds/27Mi)
% 0.23/1.44 % (7186)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_7087 for (2999ds/4Mi)
% 0.23/1.44 % (7188)Instruction limit reached!
% 0.23/1.44 % (7188)------------------------------
% 0.23/1.44 % (7188)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.23/1.44 % (7188)Termination reason: Unknown
% 0.23/1.44 % (7189)Instruction limit reached!
% 0.23/1.44 % (7189)------------------------------
% 0.23/1.44 % (7189)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.23/1.44 % (7188)Termination phase: shuffling
% 0.23/1.44
% 0.23/1.44 % (7188)Memory used [KB]: 1023
% 0.23/1.44 % (7188)Time elapsed: 0.003 s
% 0.23/1.44 % (7188)Instructions burned: 3 (million)
% 0.23/1.44 % (7188)------------------------------
% 0.23/1.44 % (7188)------------------------------
% 0.23/1.44 % (7189)Termination reason: Unknown
% 0.23/1.44 % (7189)Termination phase: shuffling
% 0.23/1.44
% 0.23/1.44 % (7189)Memory used [KB]: 1023
% 0.23/1.44 % (7189)Time elapsed: 0.003 s
% 0.23/1.44 % (7189)Instructions burned: 3 (million)
% 0.23/1.44 % (7189)------------------------------
% 0.23/1.44 % (7189)------------------------------
% 0.23/1.44 % (7186)Instruction limit reached!
% 0.23/1.44 % (7186)------------------------------
% 0.23/1.44 % (7186)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.23/1.44 % (7186)Termination reason: Unknown
% 0.23/1.44 % (7186)Termination phase: shuffling
% 0.23/1.44
% 0.23/1.44 % (7186)Memory used [KB]: 1151
% 0.23/1.44 % (7186)Time elapsed: 0.004 s
% 0.23/1.44 % (7186)Instructions burned: 4 (million)
% 0.23/1.44 % (7186)------------------------------
% 0.23/1.44 % (7186)------------------------------
% 0.23/1.45 % (7191)Instruction limit reached!
% 0.23/1.45 % (7191)------------------------------
% 0.23/1.45 % (7191)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.23/1.45 % (7191)Termination reason: Unknown
% 0.23/1.45 % (7191)Termination phase: Preprocessing 3
% 0.23/1.45
% 0.23/1.45 % (7191)Memory used [KB]: 1663
% 0.23/1.45 % (7191)Time elapsed: 0.011 s
% 0.23/1.45 % (7191)Instructions burned: 19 (million)
% 0.23/1.45 % (7191)------------------------------
% 0.23/1.45 % (7191)------------------------------
% 0.23/1.45 % (7187)Instruction limit reached!
% 0.23/1.45 % (7187)------------------------------
% 0.23/1.45 % (7187)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.23/1.45 % (7187)Termination reason: Unknown
% 0.23/1.45 % (7187)Termination phase: Saturation
% 0.23/1.45
% 0.23/1.45 % (7187)Memory used [KB]: 5756
% 0.23/1.45 % (7187)Time elapsed: 0.014 s
% 0.23/1.45 % (7187)Instructions burned: 27 (million)
% 0.23/1.45 % (7187)------------------------------
% 0.23/1.45 % (7187)------------------------------
% 0.23/1.46 % (7192)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_7087 for (2999ds/3Mi)
% 0.23/1.46 % (7193)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_7087 for (2999ds/37Mi)
% 0.23/1.46 % (7194)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_7087 for (2999ds/15Mi)
% 0.23/1.46 % (7192)Instruction limit reached!
% 0.23/1.46 % (7192)------------------------------
% 0.23/1.46 % (7192)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.23/1.46 % (7192)Termination reason: Unknown
% 0.23/1.46 % (7192)Termination phase: shuffling
% 0.23/1.46
% 0.23/1.46 % (7192)Memory used [KB]: 1151
% 0.23/1.46 % (7192)Time elapsed: 0.003 s
% 0.23/1.46 % (7192)Instructions burned: 3 (million)
% 0.23/1.46 % (7192)------------------------------
% 0.23/1.46 % (7192)------------------------------
% 0.23/1.47 % (7194)Instruction limit reached!
% 0.23/1.47 % (7194)------------------------------
% 0.23/1.47 % (7194)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.23/1.47 % (7194)Termination reason: Unknown
% 0.23/1.47 % (7194)Termination phase: Preprocessing 3
% 0.23/1.47 % (7195)dis+21_1:1_cbe=off:cnfonf=off:fs=off:fsr=off:hud=1:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_7087 for (2999ds/3Mi)
% 0.23/1.47
% 0.23/1.47 % (7194)Memory used [KB]: 1407
% 0.23/1.47 % (7194)Time elapsed: 0.009 s
% 0.23/1.47 % (7194)Instructions burned: 16 (million)
% 0.23/1.47 % (7194)------------------------------
% 0.23/1.47 % (7194)------------------------------
% 0.23/1.47 % (7195)Instruction limit reached!
% 0.23/1.47 % (7195)------------------------------
% 0.23/1.47 % (7195)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.23/1.47 % (7195)Termination reason: Unknown
% 0.23/1.47 % (7195)Termination phase: shuffling
% 0.23/1.47
% 0.23/1.47 % (7195)Memory used [KB]: 1151
% 0.23/1.47 % (7195)Time elapsed: 0.003 s
% 0.23/1.47 % (7195)Instructions burned: 4 (million)
% 0.23/1.47 % (7195)------------------------------
% 0.23/1.47 % (7195)------------------------------
% 0.23/1.47 % (7196)lrs+1002_1:1_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=4203469,65536:i=1041:si=on:rtra=on_0 on DTF2THF_7087 for (2999ds/1041Mi)
% 0.23/1.47 % (7193)Instruction limit reached!
% 0.23/1.47 % (7193)------------------------------
% 0.23/1.47 % (7193)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.23/1.47 % (7193)Termination reason: Unknown
% 0.23/1.47 % (7193)Termination phase: Saturation
% 0.23/1.47
% 0.23/1.47 % (7193)Memory used [KB]: 5756
% 0.23/1.47 % (7193)Time elapsed: 0.018 s
% 0.23/1.47 % (7193)Instructions burned: 39 (million)
% 0.23/1.47 % (7193)------------------------------
% 0.23/1.47 % (7193)------------------------------
% 0.23/1.47 % (7197)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_7087 for (2999ds/7Mi)
% 0.23/1.48 % (7197)Instruction limit reached!
% 0.23/1.48 % (7197)------------------------------
% 0.23/1.48 % (7197)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.23/1.48 % (7197)Termination reason: Unknown
% 0.23/1.48 % (7197)Termination phase: shuffling
% 0.23/1.48
% 0.23/1.48 % (7197)Memory used [KB]: 1151
% 0.23/1.48 % (7197)Time elapsed: 0.005 s
% 0.23/1.48 % (7197)Instructions burned: 8 (million)
% 0.23/1.48 % (7197)------------------------------
% 0.23/1.48 % (7197)------------------------------
% 0.23/1.48 % (7190)First to succeed.
% 0.23/1.48 % (7198)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_7087 for (2999ds/16Mi)
% 0.23/1.48 % (7185)Also succeeded, but the first one will report.
% 0.23/1.48 % (7190)Refutation found. Thanks to Tanya!
% 0.23/1.48 % SZS status Theorem for DTF2THF_7087
% 0.23/1.48 % SZS output start Proof for DTF2THF_7087
% 0.23/1.48 thf(func_def_0, type, in: $i > $i > $o).
% 0.23/1.48 thf(func_def_1, type, exu: ($i > $o) > $o).
% 0.23/1.48 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 0.23/1.48 thf(func_def_8, type, powerset: $i > $i).
% 0.23/1.48 thf(func_def_10, type, setunion: $i > $i).
% 0.23/1.48 thf(func_def_19, type, descr: ($i > $o) > $i).
% 0.23/1.48 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 0.23/1.48 thf(func_def_26, type, prop2set: $o > $i).
% 0.23/1.48 thf(func_def_36, type, nonempty: $i > $o).
% 0.23/1.48 thf(func_def_78, type, sP1: $i > $i > $o).
% 0.23/1.48 thf(func_def_79, type, sP2: $i > $o).
% 0.23/1.48 thf(func_def_80, type, sP3: $i > $i > $o).
% 0.23/1.48 thf(func_def_81, type, sP4: $i > $i > $o).
% 0.23/1.48 thf(func_def_82, type, sK5: $i > $i).
% 0.23/1.48 thf(func_def_86, type, sK9: $i > $i > $i).
% 0.23/1.48 thf(func_def_90, type, sK13: $i > $i).
% 0.23/1.48 thf(func_def_94, type, sK17: $i > $i > $o).
% 0.23/1.48 thf(func_def_96, type, sK19: $i > $i).
% 0.23/1.48 thf(func_def_97, type, sK20: $i > $i).
% 0.23/1.48 thf(func_def_98, type, sK21: $i > ($i > $i > $o) > $i).
% 0.23/1.48 thf(func_def_99, type, sK22: $i > ($i > $i > $o) > $i).
% 0.23/1.48 thf(func_def_100, type, sK23: $i > $i > ($i > $i > $o) > $i).
% 0.23/1.48 thf(func_def_103, type, sK26: $i > $o).
% 0.23/1.48 thf(func_def_107, type, sK30: $i > $o).
% 0.23/1.48 thf(func_def_110, type, sK33: $i > ($i > $o) > $i).
% 0.23/1.48 thf(func_def_113, type, sK36: ($i > $o) > $i).
% 0.23/1.48 thf(func_def_114, type, sK37: $i > $o).
% 0.23/1.48 thf(func_def_118, type, sK41: $i > $i > $i).
% 0.23/1.48 thf(func_def_124, type, sK47: $i > $i > $i).
% 0.23/1.48 thf(func_def_125, type, sK48: $i > $i > $i).
% 0.23/1.48 thf(func_def_133, type, sK56: $i > $i).
% 0.23/1.48 thf(func_def_134, type, sK57: $i > $o).
% 0.23/1.48 thf(func_def_143, type, sK66: $i > $i > $i).
% 0.23/1.48 thf(func_def_152, type, sK75: $i > $o).
% 0.23/1.48 thf(func_def_160, type, sK83: $i > $i).
% 0.23/1.48 thf(func_def_163, type, sK86: $i > $i).
% 0.23/1.48 thf(func_def_166, type, sK89: $i > $i).
% 0.23/1.48 thf(func_def_172, type, sK95: $o > $i > $i > $i).
% 0.23/1.48 thf(func_def_175, type, sK98: $i > $o).
% 0.23/1.48 thf(func_def_179, type, sK102: $i > $o).
% 0.23/1.48 thf(func_def_181, type, sK104: $i > $i).
% 0.23/1.48 thf(func_def_185, type, sK108: $i > $o).
% 0.23/1.48 thf(func_def_188, type, sK111: $i > $i > $i).
% 0.23/1.48 thf(func_def_189, type, sK112: $i > $i > $i).
% 0.23/1.48 thf(func_def_190, type, sK113: $i > $i > $i).
% 0.23/1.48 thf(func_def_191, type, sK114: $i > $i > $i).
% 0.23/1.48 thf(func_def_192, type, sK115: $i > $i > $i > $i).
% 0.23/1.48 thf(func_def_193, type, sK116: $i > $i).
% 0.23/1.48 thf(func_def_194, type, sK117: $i > $i).
% 0.23/1.48 thf(func_def_195, type, sK118: $i > $i).
% 0.23/1.48 thf(func_def_196, type, sK119: $i > $i).
% 0.23/1.48 thf(func_def_197, type, sK120: $i > $i > $i).
% 0.23/1.48 thf(func_def_198, type, sK121: $i > $i > $i > $i).
% 0.23/1.48 thf(func_def_199, type, sK122: $i > $i > $i > $i).
% 0.23/1.48 thf(func_def_200, type, sK123: $i > $i > $i).
% 0.23/1.48 thf(func_def_201, type, sK124: $i > $i > $i).
% 0.23/1.48 thf(func_def_203, type, sK126: $i > $i).
% 0.23/1.48 thf(func_def_204, type, sK127: $i > $i).
% 0.23/1.48 thf(func_def_205, type, sK128: $i > $i).
% 0.23/1.48 thf(func_def_206, type, sK129: $i > $i > $i).
% 0.23/1.48 thf(func_def_212, type, sK135: ($i > $o) > $i).
% 0.23/1.48 thf(func_def_213, type, sK136: $i > $o).
% 0.23/1.48 thf(func_def_214, type, sK137: $i > $i).
% 0.23/1.48 thf(func_def_216, type, sK139: $i > $o).
% 0.23/1.48 thf(func_def_220, type, sK143: $i > $o).
% 0.23/1.48 thf(func_def_225, type, sK148: $i > $o).
% 0.23/1.48 thf(func_def_226, type, sK149: $i > $i).
% 0.23/1.48 thf(func_def_227, type, sK150: ($i > $o) > $i).
% 0.23/1.48 thf(func_def_230, type, sK153: $i > $i > $i).
% 0.23/1.48 thf(func_def_232, type, ph155: !>[X0: $tType]:(X0)).
% 0.23/1.48 thf(f1179,plain,(
% 0.23/1.48 $false),
% 0.23/1.48 inference(subsumption_resolution,[status(thm)],[f1159,f646])).
% 0.23/1.48 thf(f646,plain,(
% 0.23/1.48 ((sK57 @ sK59) = $true)),
% 0.23/1.48 inference(cnf_transformation,[status(thm)],[f349])).
% 0.23/1.48 thf(f349,plain,(
% 0.23/1.48 (subPowSU = $true) & (upairsetIR = $true) & (! [X2 : $i] : (((in @ X2 @ sK58) != $true) | ((sK57 @ X2) != $true)) & (((sK57 @ sK59) = $true) & ((in @ sK59 @ sK58) = $true))) & (exuE2 = $true) & (nonemptyE1 = $true) & (powersetE = $true) & (descrp = $true) & (emptyInPowerset = $true) & (quantDeMorgan1 = $true) & (emptysetE = $true) & (setbeta = $true) & (omegaSAx = $true) & (setext = $true) & (dsetconstrEL = $true) & (noeltsimpempty = $true) & (notinemptyset = $true) & (prop2setE = $true) & (vacuousDall = $true) & (setadjoinOr = $true) & (wellorderingAx = $true) & (singletonsswitch = $true) & (replAx = $true) & (setadjoinE = $true) & (eqinunit = $true) & (emptysetimpfalse = $true) & (omega0Ax = $true) & (setunionI = $true) & (exuE3e = $true) & (powersetAx = $true) & (setadjoinIL = $true) & (setadjoinAx = $true) & (dsetconstrI = $true) & (setextAx = $true) & (emptysetAx = $true) & (nonemptyI1 = $true) & (emptyE1 = $true) & (omegaIndAx = $true) & (exuE1 = $true) & (nonemptyI = $true) & (powersetI = $true) & (setunionE = $true) & (foundationAx = $true) & (setadjoinIR = $true) & (setunionAx = $true) & (emptyinPowerset = $true) & (upairsetIL = $true) & (emptyinunitempty = $true) & (upairsetE = $true) & (notinsingleton = $true) & (nonemptyImpWitness = $true) & (emptyI = $true) & (uniqinunit = $true) & (dsetconstrER = $true) & (setoftrueEq = $true)),
% 0.23/1.48 inference(skolemisation,[status(esa),new_symbols(skolem,[sK57,sK58,sK59])],[f228,f348,f347])).
% 0.23/1.48 thf(f347,plain,(
% 0.23/1.48 ? [X0 : $i > $o,X1 : $i] : (! [X2 : $i] : (((in @ X2 @ X1) != $true) | ((X0 @ X2) != $true)) & ? [X3 : $i] : (((X0 @ X3) = $true) & ((in @ X3 @ X1) = $true))) => (! [X2 : $i] : (((in @ X2 @ sK58) != $true) | ((sK57 @ X2) != $true)) & ? [X3 : $i] : (((sK57 @ X3) = $true) & ((in @ X3 @ sK58) = $true)))),
% 0.23/1.48 introduced(definition,[],[choice_axiom])).
% 0.23/1.48 thf(f348,plain,(
% 0.23/1.48 ? [X3 : $i] : (((sK57 @ X3) = $true) & ((in @ X3 @ sK58) = $true)) => (((sK57 @ sK59) = $true) & ((in @ sK59 @ sK58) = $true))),
% 0.23/1.48 introduced(definition,[],[choice_axiom])).
% 0.23/1.48 thf(f228,plain,(
% 0.23/1.48 (subPowSU = $true) & (upairsetIR = $true) & ? [X0 : $i > $o,X1 : $i] : (! [X2 : $i] : (((in @ X2 @ X1) != $true) | ((X0 @ X2) != $true)) & ? [X3 : $i] : (((X0 @ X3) = $true) & ((in @ X3 @ X1) = $true))) & (exuE2 = $true) & (nonemptyE1 = $true) & (powersetE = $true) & (descrp = $true) & (emptyInPowerset = $true) & (quantDeMorgan1 = $true) & (emptysetE = $true) & (setbeta = $true) & (omegaSAx = $true) & (setext = $true) & (dsetconstrEL = $true) & (noeltsimpempty = $true) & (notinemptyset = $true) & (prop2setE = $true) & (vacuousDall = $true) & (setadjoinOr = $true) & (wellorderingAx = $true) & (singletonsswitch = $true) & (replAx = $true) & (setadjoinE = $true) & (eqinunit = $true) & (emptysetimpfalse = $true) & (omega0Ax = $true) & (setunionI = $true) & (exuE3e = $true) & (powersetAx = $true) & (setadjoinIL = $true) & (setadjoinAx = $true) & (dsetconstrI = $true) & (setextAx = $true) & (emptysetAx = $true) & (nonemptyI1 = $true) & (emptyE1 = $true) & (omegaIndAx = $true) & (exuE1 = $true) & (nonemptyI = $true) & (powersetI = $true) & (setunionE = $true) & (foundationAx = $true) & (setadjoinIR = $true) & (setunionAx = $true) & (emptyinPowerset = $true) & (upairsetIL = $true) & (emptyinunitempty = $true) & (upairsetE = $true) & (notinsingleton = $true) & (nonemptyImpWitness = $true) & (emptyI = $true) & (uniqinunit = $true) & (dsetconstrER = $true) & (setoftrueEq = $true)),
% 0.23/1.48 inference(flattening,[status(thm)],[f227])).
% 0.23/1.48 thf(f227,plain,(
% 0.23/1.48 ((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i > $o,X1 : $i] : (! [X2 : $i] : (((in @ X2 @ X1) != $true) | ((X0 @ X2) != $true)) & ? [X3 : $i] : (((X0 @ X3) = $true) & ((in @ X3 @ X1) = $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)),
% 0.23/1.48 inference(ennf_transformation,[status(thm)],[f169])).
% 0.23/1.48 thf(f169,plain,(
% 0.23/1.48 ~((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) => ! [X0 : $i > $o,X1 : $i] : (! [X2 : $i] : (((in @ X2 @ X1) = $true) => ((X0 @ X2) != $true)) => ~? [X3 : $i] : (((X0 @ X3) = $true) & ((in @ X3 @ X1) = $true))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 0.23/1.48 inference(flattening,[status(thm)],[f79])).
% 0.23/1.48 thf(f79,plain,(
% 0.23/1.48 ~((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) => ! [X0 : $i > $o,X1 : $i] : (! [X2 : $i] : (((in @ X2 @ X1) = $true) => ~((X0 @ X2) = $true)) => ~? [X3 : $i] : (((X0 @ X3) = $true) & ((in @ X3 @ X1) = $true))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 0.23/1.48 inference(fool_elimination,[status(thm)],[f78])).
% 0.23/1.48 thf(f78,plain,(
% 0.23/1.48 ~(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 => ! [X0 : $i > $o,X1 : $i] : (! [X2 : $i] : ((in @ X2 @ X1) => ~(X0 @ X2)) => ~? [X3 : $i] : ((in @ X3 @ X1) & (X0 @ X3))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 0.23/1.48 inference(rectify,[status(thm)],[f58])).
% 0.23/1.48 thf(f58,negated_conjecture,(
% 0.23/1.48 ~(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 => ! [X0 : $i > $o,X3 : $i] : (! [X1 : $i] : ((in @ X1 @ X3) => ~(X0 @ X1)) => ~? [X1 : $i] : ((in @ X1 @ X3) & (X0 @ X1))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 0.23/1.48 inference(negated_conjecture,[status(cth)],[f57])).
% 0.23/1.48 thf(f57,conjecture,(
% 0.23/1.48 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 => ! [X0 : $i > $o,X3 : $i] : (! [X1 : $i] : ((in @ X1 @ X3) => ~(X0 @ X1)) => ~? [X1 : $i] : ((in @ X1 @ X3) & (X0 @ X1)))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 0.23/1.48 file('/export/starexec/sandbox/tmp/tmp.6gNdrgnz6L/DTF2THF_7087.p',quantDeMorgan2)).
% 0.23/1.48 thf(f1159,plain,(
% 0.23/1.48 ((sK57 @ sK59) != $true)),
% 0.23/1.48 inference(trivial_inequality_removal,[status(thm)],[f1146])).
% 0.23/1.48 thf(f1146,plain,(
% 0.23/1.48 ((sK57 @ sK59) != $true) | ($true != $true)),
% 0.23/1.48 inference(superposition,[status(thm)],[f647,f645])).
% 0.23/1.48 thf(f645,plain,(
% 0.23/1.48 ((in @ sK59 @ sK58) = $true)),
% 0.23/1.48 inference(cnf_transformation,[status(thm)],[f349])).
% 0.23/1.48 thf(f647,plain,(
% 0.23/1.48 ( ! [X2 : $i] : (((in @ X2 @ sK58) != $true) | ((sK57 @ X2) != $true)) )),
% 0.23/1.48 inference(cnf_transformation,[status(thm)],[f349])).
% 0.23/1.48 % SZS output end Proof for DTF2THF_7087
% 0.23/1.48 % (7190)------------------------------
% 0.23/1.48 % (7190)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.23/1.48 % (7190)Termination reason: Refutation
% 0.23/1.48
% 0.23/1.48 % (7190)Memory used [KB]: 6524
% 0.23/1.48 % (7190)Time elapsed: 0.043 s
% 0.23/1.48 % (7190)Instructions burned: 80 (million)
% 0.23/1.48 % (7190)------------------------------
% 0.23/1.48 % (7190)------------------------------
% 0.23/1.48 % (7184)Success in time 0.047 s
%------------------------------------------------------------------------------