%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU562^1 : TPTP v9.2.1. Released v3.7.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : 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 09:35:17 AM UTC 2026
% Result : Theorem 2.26s 1.56s
% Output : Refutation 2.26s
% Verified :
% SZS Type : Refutation
% Derivation depth : 16
% Number of leaves : 3
% Syntax : Number of formulae : 24 ( 14 unt; 0 typ; 0 def)
% Number of atoms : 1328 ( 347 equ; 0 cnn)
% Maximal formula atoms : 81 ( 55 avg)
% Number of connectives : 708 ( 18 ~; 7 |; 239 &; 111 @)
% ( 0 <=>; 329 =>; 0 <=; 0 <~>)
% Maximal formula depth : 87 ( 26 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 191 ( 187 usr; 84 con; 0-4 aty)
% ( 4 !!; 0 ??; 0 @@+; 0 @@-)
% Number of variables : 41 ( 15 ^; 20 !; 6 ?; 41 :)
% 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_106,type,
sP0: $i > $i > $o ).
thf(func_def_107,type,
sP1: $i > $o ).
thf(func_def_108,type,
sP2: $i > $i > $o ).
thf(func_def_109,type,
sP3: $i > $i > $o ).
thf(func_def_111,type,
sK5: $i > $o ).
thf(func_def_113,type,
sK7: $i > ( $i > $o ) > $i ).
thf(func_def_118,type,
sK12: $i > $i ).
thf(func_def_120,type,
sK14: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_121,type,
sK15: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_124,type,
sK18: $i > $o ).
thf(func_def_125,type,
sK19: $i > $o ).
thf(func_def_128,type,
sK22: $i > $o ).
thf(func_def_129,type,
sK23: $i > $i ).
thf(func_def_130,type,
sK24: ( $i > $o ) > $i ).
thf(func_def_133,type,
sK27: $i > ( $i > $o ) > $i ).
thf(func_def_134,type,
sK28: $i > $o ).
thf(func_def_136,type,
sK30: $i > $i ).
thf(func_def_141,type,
sK35: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_142,type,
sK36: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_143,type,
sK37: $i > $o ).
thf(func_def_144,type,
sK38: $i > $o ).
thf(func_def_146,type,
sK40: $i > $o ).
thf(func_def_148,type,
sK42: ( $i > $o ) > $i > $i ).
thf(func_def_150,type,
sK44: $i > $o ).
thf(func_def_152,type,
sK46: ( $i > $o ) > $i > $i ).
thf(func_def_156,type,
sK50: $i > $i ).
thf(func_def_159,type,
sK53: ( $i > $o ) > $i > $i ).
thf(func_def_161,type,
sK55: $i > $o ).
thf(func_def_164,type,
sK58: $i > $i ).
thf(func_def_166,type,
sK60: $i > $i ).
thf(func_def_169,type,
sK63: $i > $o ).
thf(func_def_173,type,
sK67: $i > $o ).
thf(func_def_181,type,
sK75: $i > $i > $i ).
thf(func_def_182,type,
sK76: $i > $i > $i ).
thf(func_def_183,type,
sK77: $i > $i > $i ).
thf(func_def_184,type,
sK78: $i > $i > $i ).
thf(func_def_185,type,
sK79: $i > $i > $i ).
thf(func_def_186,type,
sK80: $i > $i > $i ).
thf(func_def_187,type,
sK81: $i > $i > $i > $i ).
thf(func_def_188,type,
sK82: $i > $i ).
thf(func_def_189,type,
sK83: $i > $i ).
thf(func_def_190,type,
sK84: $i > $i ).
thf(func_def_191,type,
sK85: $i > $i ).
thf(func_def_192,type,
sK86: $i > $i > $i > $i ).
thf(func_def_193,type,
sK87: $i > $i > $i > $i ).
thf(func_def_194,type,
sK88: $i > $i > $i ).
thf(func_def_195,type,
sK89: $i > $i > $i ).
thf(func_def_196,type,
sK90: $i > $i > $i ).
thf(func_def_197,type,
sK91: $i > $i ).
thf(func_def_199,type,
sK93: $i > $i ).
thf(func_def_200,type,
sK94: $i > $i ).
thf(func_def_202,type,
sK96: $i > $i ).
thf(func_def_203,type,
sK97: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_204,type,
sK98: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_205,type,
sK99: $i > $o ).
thf(func_def_206,type,
sK100: $i > $o ).
thf(func_def_211,type,
sK105: $i > $i > $i ).
thf(func_def_212,type,
sK106: $i > $i > $i ).
thf(func_def_215,type,
sK109: ( $i > $o ) > $i ).
thf(func_def_216,type,
sK110: ( $i > $o ) > $i ).
thf(func_def_217,type,
sK111: $i > $o ).
thf(func_def_219,type,
sK113: $i > $o ).
thf(func_def_223,type,
sK117: $i > $i ).
thf(func_def_226,type,
sK120: $i > $i > $i ).
thf(func_def_229,type,
sK123: $i > $o ).
thf(func_def_234,type,
sK128: $i > $o ).
thf(func_def_237,type,
sK131: $i > $o ).
thf(func_def_240,type,
sK134: $i > $i > $i ).
thf(func_def_248,type,
sK142: $i > $o ).
thf(func_def_259,type,
sK153: $i > ( $i > $o ) > $i ).
thf(func_def_260,type,
sK154: $i > $o ).
thf(func_def_268,type,
sK162: $i > $i > $o ).
thf(func_def_269,type,
sK163: $i > $i ).
thf(func_def_270,type,
sK164: $i > $i ).
thf(func_def_271,type,
sK165: ( $i > $i > $o ) > $i > $i ).
thf(func_def_272,type,
sK166: ( $i > $i > $o ) > $i > $i ).
thf(func_def_273,type,
sK167: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_274,type,
sK168: ( $i > $o ) > $i ).
thf(func_def_275,type,
sK169: $i > $o ).
thf(func_def_276,type,
sK170: $i > $i ).
thf(func_def_285,type,
sK179: $o > $i > $i > $i ).
thf(func_def_292,type,
sK186: $i > $i > $i ).
thf(func_def_293,type,
sK187: $i > $o ).
thf(func_def_294,type,
sK188: ( $i > $o ) > $i ).
thf(func_def_297,type,
sK191: $i > $o ).
thf(func_def_303,type,
sK197: $i > $o ).
thf(func_def_305,type,
sK199: ( $i > $o ) > $i > $i ).
thf(func_def_311,type,
sK205: $i > $o ).
thf(func_def_314,type,
sK208: $i > ( $i > $o ) > $i ).
thf(func_def_318,type,
sK212: $i > $o ).
thf(func_def_326,type,
sK220: $i > $i > $i ).
thf(func_def_328,type,
sK222: $i > $o ).
thf(func_def_337,type,
ph231:
!>[X0: $tType] : X0 ).
thf(f1832,plain,
$false,
inference(trivial_inequality_removal,[status(thm)],[f1831]) ).
thf(f1831,plain,
$true = $false,
inference(forward_demodulation,[status(thm)],[f1830,f1492]) ).
thf(f1492,plain,
( ( in @ sK230 @ sK69 )
= $false ),
inference(binary_proxy_clausification,[status(thm)],[f1491]) ).
thf(f1491,plain,
( ( ( in @ sK230 @ sK68 )
=> ( in @ sK230 @ sK69 ) )
= $false ),
inference(beta_eta_normalization,[status(thm)],[f1490]) ).
thf(f1490,plain,
( ( ^ [Y0: $i] :
( ( in @ Y0 @ sK68 )
=> ( in @ Y0 @ sK69 ) )
@ sK230 )
= $false ),
inference(sigma_clausification,[status(thm)],[f1489]) ).
thf(f1489,plain,
( $true
!= ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ sK68 )
=> ( in @ Y0 @ sK69 ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f1257]) ).
thf(f1257,plain,
( $true
!= ( ^ [Y0: $i,Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y0 )
=> ( in @ Y2 @ Y1 ) ) )
@ sK68
@ sK69 ) ),
inference(definition_unfolding,[status(thm)],[f850,f839]) ).
thf(f839,plain,
( subset
= ( ^ [Y0: $i,Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y0 )
=> ( in @ Y2 @ Y1 ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f94]) ).
thf(f94,plain,
( subset
= ( ^ [Y0: $i,Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y0 )
=> ( in @ Y2 @ Y1 ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f93]) ).
thf(f93,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/sandbox2/tmp/tmp.vNo7MRU8lP/DTF2THF_20822.p',subset) ).
thf(f850,plain,
( ( subset @ sK68 @ sK69 )
!= $true ),
inference(cnf_transformation,[status(thm)],[f472]) ).
thf(f472,plain,
( ( prop2setE = $true )
& ( setunionAx = $true )
& ( notinemptyset = $true )
& ( wellorderingAx = $true )
& ( noeltsimpempty = $true )
& ( exuEu = $true )
& ( nonemptyI1 = $true )
& ( emptyInPowerset = $true )
& ( setadjoinOr = $true )
& ( emptyI = $true )
& ( setadjoinIR = $true )
& ( eqinunit = $true )
& ( setunionI = $true )
& ( eqimpsubset1 = $true )
& ( dsetconstr__Cong = $true )
& ( setunion__Cong = $true )
& ( exuE2 = $true )
& ( dsetconstrER = $true )
& ( eqimpsubset2 = $true )
& ( notdexE = $true )
& ( setunionE = $true )
& ( prop2setI = $true )
& ( singletonsswitch = $true )
& ( prop2set2propI = $true )
& ( exuI2 = $true )
& ( upairsetIL = $true )
& ( omega__Cong = $true )
& ( quantDeMorgan3 = $true )
& ( exuI3 = $true )
& ( dsetconstrI = $true )
& ( nonemptyI = $true )
& ( powersetAx = $true )
& ( emptysetimpfalse = $true )
& ( quantDeMorgan2 = $true )
& ( nonemptyE1 = $true )
& ( uniqinunit = $true )
& ( upairsetIR = $true )
& ( setextAx = $true )
& ( emptyinPowerset = $true )
& ( setadjoin__Cong = $true )
& ( powerset__Cong = $true )
& ( setoftrueEq = $true )
& ( setext = $true )
& ( exuE3e = $true )
& ( quantDeMorgan4 = $true )
& ( emptyinunitempty = $true )
& ( upairsetE = $true )
& ( exuE3u = $true )
& ( in__Cong = $true )
& ( dsetconstrEL = $true )
& ( replAx = $true )
& ( notdallE = $true )
& ( emptyE1 = $true )
& ( omegaIndAx = $true )
& ( setadjoinIL = $true )
& ( powersetE = $true )
& ( nonemptyImpWitness = $true )
& ( quantDeMorgan1 = $true )
& ( notinsingleton = $true )
& ( exu__Cong = $true )
& ( emptysetAx = $true )
& ( setadjoinAx = $true )
& ( exuI1 = $true )
& ( vacuousDall = $true )
& ( descr__Cong = $true )
& ( setbeta = $true )
& ( foundationAx = $true )
& ( omega0Ax = $true )
& ( subPowSU = $true )
& ( ( subset @ sK68 @ sK69 )
!= $true )
& ! [X2: $i] :
( ( ( in @ X2 @ sK68 )
!= $true )
| ( ( in @ X2 @ sK69 )
= $true ) )
& ( omegaSAx = $true )
& ( powersetI = $true )
& ( descrp = $true )
& ( setadjoinE = $true )
& ( emptysetE = $true )
& ( exuE1 = $true )
& ( emptyset__Cong = $true )
& ( subsetI1 = $true )
& ( inCongP = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK68,sK69])],[f312,f471]) ).
thf(f471,plain,
( ? [X0: $i,X1: $i] :
( ( $true
!= ( subset @ X0 @ X1 ) )
& ! [X2: $i] :
( ( $true
!= ( in @ X2 @ X0 ) )
| ( $true
= ( in @ X2 @ X1 ) ) ) )
=> ( ( ( subset @ sK68 @ sK69 )
!= $true )
& ! [X2: $i] :
( ( ( in @ X2 @ sK68 )
!= $true )
| ( ( in @ X2 @ sK69 )
= $true ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f312,plain,
( ( prop2setE = $true )
& ( setunionAx = $true )
& ( notinemptyset = $true )
& ( wellorderingAx = $true )
& ( noeltsimpempty = $true )
& ( exuEu = $true )
& ( nonemptyI1 = $true )
& ( emptyInPowerset = $true )
& ( setadjoinOr = $true )
& ( emptyI = $true )
& ( setadjoinIR = $true )
& ( eqinunit = $true )
& ( setunionI = $true )
& ( eqimpsubset1 = $true )
& ( dsetconstr__Cong = $true )
& ( setunion__Cong = $true )
& ( exuE2 = $true )
& ( dsetconstrER = $true )
& ( eqimpsubset2 = $true )
& ( notdexE = $true )
& ( setunionE = $true )
& ( prop2setI = $true )
& ( singletonsswitch = $true )
& ( prop2set2propI = $true )
& ( exuI2 = $true )
& ( upairsetIL = $true )
& ( omega__Cong = $true )
& ( quantDeMorgan3 = $true )
& ( exuI3 = $true )
& ( dsetconstrI = $true )
& ( nonemptyI = $true )
& ( powersetAx = $true )
& ( emptysetimpfalse = $true )
& ( quantDeMorgan2 = $true )
& ( nonemptyE1 = $true )
& ( uniqinunit = $true )
& ( upairsetIR = $true )
& ( setextAx = $true )
& ( emptyinPowerset = $true )
& ( setadjoin__Cong = $true )
& ( powerset__Cong = $true )
& ( setoftrueEq = $true )
& ( setext = $true )
& ( exuE3e = $true )
& ( quantDeMorgan4 = $true )
& ( emptyinunitempty = $true )
& ( upairsetE = $true )
& ( exuE3u = $true )
& ( in__Cong = $true )
& ( dsetconstrEL = $true )
& ( replAx = $true )
& ( notdallE = $true )
& ( emptyE1 = $true )
& ( omegaIndAx = $true )
& ( setadjoinIL = $true )
& ( powersetE = $true )
& ( nonemptyImpWitness = $true )
& ( quantDeMorgan1 = $true )
& ( notinsingleton = $true )
& ( exu__Cong = $true )
& ( emptysetAx = $true )
& ( setadjoinAx = $true )
& ( exuI1 = $true )
& ( vacuousDall = $true )
& ( descr__Cong = $true )
& ( setbeta = $true )
& ( foundationAx = $true )
& ( omega0Ax = $true )
& ( subPowSU = $true )
& ? [X0: $i,X1: $i] :
( ( $true
!= ( subset @ X0 @ X1 ) )
& ! [X2: $i] :
( ( $true
!= ( in @ X2 @ X0 ) )
| ( $true
= ( in @ X2 @ X1 ) ) ) )
& ( omegaSAx = $true )
& ( powersetI = $true )
& ( descrp = $true )
& ( setadjoinE = $true )
& ( emptysetE = $true )
& ( exuE1 = $true )
& ( emptyset__Cong = $true )
& ( subsetI1 = $true )
& ( inCongP = $true ) ),
inference(flattening,[status(thm)],[f311]) ).
thf(f311,plain,
( ? [X0: $i,X1: $i] :
( ( $true
!= ( subset @ X0 @ X1 ) )
& ! [X2: $i] :
( ( $true
!= ( in @ X2 @ X0 ) )
| ( $true
= ( in @ X2 @ X1 ) ) ) )
& ( eqimpsubset1 = $true )
& ( eqimpsubset2 = $true )
& ( subsetI1 = $true )
& ( dsetconstr__Cong = $true )
& ( descr__Cong = $true )
& ( exuEu = $true )
& ( omega__Cong = $true )
& ( setunion__Cong = $true )
& ( powerset__Cong = $true )
& ( setadjoin__Cong = $true )
& ( emptyset__Cong = $true )
& ( exu__Cong = $true )
& ( exuE3u = $true )
& ( in__Cong = $true )
& ( inCongP = $true )
& ( exuI2 = $true )
& ( exuI3 = $true )
& ( exuI1 = $true )
& ( notdallE = $true )
& ( notdexE = $true )
& ( prop2set2propI = $true )
& ( prop2setI = $true )
& ( quantDeMorgan4 = $true )
& ( quantDeMorgan3 = $true )
& ( quantDeMorgan2 = $true )
& ( quantDeMorgan1 = $true )
& ( vacuousDall = $true )
& ( emptyE1 = $true )
& ( upairsetIR = $true )
& ( upairsetIL = $true )
& ( upairsetE = $true )
& ( singletonsswitch = $true )
& ( eqinunit = $true )
& ( notinsingleton = $true )
& ( uniqinunit = $true )
& ( nonemptyImpWitness = $true )
& ( exuE2 = $true )
& ( subPowSU = $true )
& ( setunionE = $true )
& ( setunionI = $true )
& ( powersetE = $true )
& ( emptyInPowerset = $true )
& ( emptyinPowerset = $true )
& ( powersetI = $true )
& ( setoftrueEq = $true )
& ( setadjoinOr = $true )
& ( setadjoinE = $true )
& ( setadjoinIR = $true )
& ( emptyinunitempty = $true )
& ( setadjoinIL = $true )
& ( nonemptyI1 = $true )
& ( nonemptyI = $true )
& ( nonemptyE1 = $true )
& ( setbeta = $true )
& ( noeltsimpempty = $true )
& ( emptyI = $true )
& ( setext = $true )
& ( exuE3e = $true )
& ( notinemptyset = $true )
& ( emptysetimpfalse = $true )
& ( emptysetE = $true )
& ( prop2setE = $true )
& ( exuE1 = $true )
& ( dsetconstrER = $true )
& ( dsetconstrEL = $true )
& ( dsetconstrI = $true )
& ( descrp = $true )
& ( wellorderingAx = $true )
& ( foundationAx = $true )
& ( replAx = $true )
& ( omegaIndAx = $true )
& ( omegaSAx = $true )
& ( omega0Ax = $true )
& ( setunionAx = $true )
& ( powersetAx = $true )
& ( setadjoinAx = $true )
& ( emptysetAx = $true )
& ( setextAx = $true ) ),
inference(ennf_transformation,[status(thm)],[f136]) ).
thf(f136,plain,
~ ( ( setextAx = $true )
=> ( ( emptysetAx = $true )
=> ( ( setadjoinAx = $true )
=> ( ( powersetAx = $true )
=> ( ( setunionAx = $true )
=> ( ( omega0Ax = $true )
=> ( ( omegaSAx = $true )
=> ( ( omegaIndAx = $true )
=> ( ( replAx = $true )
=> ( ( foundationAx = $true )
=> ( ( wellorderingAx = $true )
=> ( ( descrp = $true )
=> ( ( dsetconstrI = $true )
=> ( ( dsetconstrEL = $true )
=> ( ( dsetconstrER = $true )
=> ( ( exuE1 = $true )
=> ( ( prop2setE = $true )
=> ( ( emptysetE = $true )
=> ( ( emptysetimpfalse = $true )
=> ( ( notinemptyset = $true )
=> ( ( exuE3e = $true )
=> ( ( setext = $true )
=> ( ( emptyI = $true )
=> ( ( noeltsimpempty = $true )
=> ( ( setbeta = $true )
=> ( ( nonemptyE1 = $true )
=> ( ( nonemptyI = $true )
=> ( ( nonemptyI1 = $true )
=> ( ( setadjoinIL = $true )
=> ( ( emptyinunitempty = $true )
=> ( ( setadjoinIR = $true )
=> ( ( setadjoinE = $true )
=> ( ( setadjoinOr = $true )
=> ( ( setoftrueEq = $true )
=> ( ( powersetI = $true )
=> ( ( emptyinPowerset = $true )
=> ( ( emptyInPowerset = $true )
=> ( ( powersetE = $true )
=> ( ( setunionI = $true )
=> ( ( setunionE = $true )
=> ( ( subPowSU = $true )
=> ( ( exuE2 = $true )
=> ( ( nonemptyImpWitness = $true )
=> ( ( uniqinunit = $true )
=> ( ( notinsingleton = $true )
=> ( ( eqinunit = $true )
=> ( ( singletonsswitch = $true )
=> ( ( upairsetE = $true )
=> ( ( upairsetIL = $true )
=> ( ( upairsetIR = $true )
=> ( ( emptyE1 = $true )
=> ( ( vacuousDall = $true )
=> ( ( quantDeMorgan1 = $true )
=> ( ( quantDeMorgan2 = $true )
=> ( ( quantDeMorgan3 = $true )
=> ( ( quantDeMorgan4 = $true )
=> ( ( prop2setI = $true )
=> ( ( prop2set2propI = $true )
=> ( ( notdexE = $true )
=> ( ( notdallE = $true )
=> ( ( exuI1 = $true )
=> ( ( exuI3 = $true )
=> ( ( exuI2 = $true )
=> ( ( inCongP = $true )
=> ( ( in__Cong = $true )
=> ( ( exuE3u = $true )
=> ( ( exu__Cong = $true )
=> ( ( emptyset__Cong = $true )
=> ( ( setadjoin__Cong = $true )
=> ( ( powerset__Cong = $true )
=> ( ( setunion__Cong = $true )
=> ( ( omega__Cong = $true )
=> ( ( exuEu = $true )
=> ( ( descr__Cong = $true )
=> ( ( dsetconstr__Cong = $true )
=> ( ( subsetI1 = $true )
=> ( ( eqimpsubset2 = $true )
=> ( ( eqimpsubset1 = $true )
=> ! [X0: $i,X1: $i] :
( ! [X2: $i] :
( ( $true
= ( in @ X2 @ X0 ) )
=> ( $true
= ( in @ X2 @ X1 ) ) )
=> ( $true
= ( subset @ X0 @ X1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f135]) ).
thf(f135,plain,
~ ( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ! [X0: $i,X1: $i] :
( ! [X2: $i] :
( ( in @ X2 @ X0 )
=> ( in @ X2 @ X1 ) )
=> ( subset @ X0 @ X1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f83]) ).
thf(f83,negated_conjecture,
~ ( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ! [X3: $i,X4: $i] :
( ! [X1: $i] :
( ( in @ X1 @ X3 )
=> ( in @ X1 @ X4 ) )
=> ( subset @ X3 @ X4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f82]) ).
thf(f82,conjecture,
( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ! [X3: $i,X4: $i] :
( ! [X1: $i] :
( ( in @ X1 @ X3 )
=> ( in @ X1 @ X4 ) )
=> ( subset @ X3 @ X4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.vNo7MRU8lP/DTF2THF_20822.p',subsetI2) ).
thf(f1830,plain,
( $true
= ( in @ sK230 @ sK69 ) ),
inference(trivial_inequality_removal,[status(thm)],[f1821]) ).
thf(f1821,plain,
( ( $true
= ( in @ sK230 @ sK69 ) )
| ( $true != $true ) ),
inference(superposition,[status(thm)],[f849,f1493]) ).
thf(f1493,plain,
( $true
= ( in @ sK230 @ sK68 ) ),
inference(binary_proxy_clausification,[status(thm)],[f1491]) ).
thf(f849,plain,
! [X2: $i] :
( ( ( in @ X2 @ sK68 )
!= $true )
| ( ( in @ X2 @ sK69 )
= $true ) ),
inference(cnf_transformation,[status(thm)],[f472]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SEU562^1 : TPTP v9.2.1. Released v3.7.0.
% 0.12/0.13 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.16/0.34 % Computer : 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/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.30/0.46 ---- Original DTF file ---
% 0.30/0.46 thf(spec,logic,$$dhol).
% 0.30/0.46 %------------------------------------------------------------------------------
% 0.30/0.46 % File : SEU562^1 : TPTP v9.2.1. Released v3.7.0.
% 0.30/0.46 % Domain : Set Theory
% 0.30/0.46 % Problem : Preliminary Notions - Relations on Sets - Subsets
% 0.30/0.46 % Version : Especial.
% 0.30/0.46 % English : (! A:i.! B:i.(! x:i.in x A -> in x B) -> subset A B)
% 0.30/0.46
% 0.30/0.46 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.30/0.46 % Source : [Bro08]
% 0.30/0.46 % Names : ZFC064g [Bro08]
% 0.30/0.46
% 0.30/0.46 % Status : Theorem
% 0.30/0.46 % 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, 0.12 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
% 0.30/0.46 % Syntax : Number of formulae : 175 ( 81 unt; 93 typ; 81 def)
% 0.30/0.46 % Number of atoms : 444 ( 126 equ; 0 cnn)
% 0.30/0.46 % Maximal formula atoms : 81 ( 5 avg)
% 0.30/0.46 % Number of connectives : 711 ( 21 ~; 5 |; 27 &; 440 @)
% 0.30/0.46 % ( 14 <=>; 204 =>; 0 <=; 0 <~>)
% 0.30/0.46 % Maximal formula depth : 86 ( 2 avg)
% 0.30/0.46 % Number of types : 2 ( 0 usr)
% 0.30/0.46 % Number of type conns : 53 ( 53 >; 0 *; 0 +; 0 <<)
% 0.30/0.46 % Number of symbols : 96 ( 93 usr; 82 con; 0-2 aty)
% 0.30/0.46 % Number of variables : 255 ( 30 ^; 197 !; 28 ?; 255 :)
% 0.30/0.46 % SPC : TH0_THM_EQU_NAR
% 0.30/0.46
% 0.30/0.46 % Comments : http://mathgate.info/detsetitem.php?id=151
% 0.30/0.46 %------------------------------------------------------------------------------
% 0.30/0.46 thf(in_type,type,
% 0.30/0.46 in: $i > $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(exu_type,type,
% 0.30/0.46 exu: ( $i > $o ) > $o ).
% 0.30/0.46
% 0.30/0.46 thf(exu,definition,
% 0.30/0.46 ( exu
% 0.30/0.46 = ( ^ [Xphi: $i > $o] :
% 0.30/0.46 ? [Xx: $i] :
% 0.30/0.46 ( ( Xphi @ Xx )
% 0.30/0.46 & ! [Xy: $i] :
% 0.30/0.46 ( ( Xphi @ Xy )
% 0.30/0.46 => ( Xx = Xy ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setextAx_type,type,
% 0.30/0.46 setextAx: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setextAx,definition,
% 0.30/0.46 ( setextAx
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 <=> ( in @ Xx @ B ) )
% 0.30/0.46 => ( A = B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(emptyset_type,type,
% 0.30/0.46 emptyset: $i ).
% 0.30/0.46
% 0.30/0.46 thf(emptysetAx_type,type,
% 0.30/0.46 emptysetAx: $o ).
% 0.30/0.46
% 0.30/0.46 thf(emptysetAx,definition,
% 0.30/0.46 ( emptysetAx
% 0.30/0.46 = ( ! [Xx: $i] :
% 0.30/0.46 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoin_type,type,
% 0.30/0.46 setadjoin: $i > $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinAx_type,type,
% 0.30/0.46 setadjoinAx: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinAx,definition,
% 0.30/0.46 ( setadjoinAx
% 0.30/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.30/0.46 <=> ( ( Xy = Xx )
% 0.30/0.46 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(powerset_type,type,
% 0.30/0.46 powerset: $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(powersetAx_type,type,
% 0.30/0.46 powersetAx: $o ).
% 0.30/0.46
% 0.30/0.46 thf(powersetAx,definition,
% 0.30/0.46 ( powersetAx
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( in @ B @ ( powerset @ A ) )
% 0.30/0.46 <=> ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ B )
% 0.30/0.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setunion_type,type,
% 0.30/0.46 setunion: $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(setunionAx_type,type,
% 0.30/0.46 setunionAx: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setunionAx,definition,
% 0.30/0.46 ( setunionAx
% 0.30/0.46 = ( ! [A: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ ( setunion @ A ) )
% 0.30/0.46 <=> ? [B: $i] :
% 0.30/0.46 ( ( in @ Xx @ B )
% 0.30/0.46 & ( in @ B @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(omega_type,type,
% 0.30/0.46 omega: $i ).
% 0.30/0.46
% 0.30/0.46 thf(omega0Ax_type,type,
% 0.30/0.46 omega0Ax: $o ).
% 0.30/0.46
% 0.30/0.46 thf(omega0Ax,definition,
% 0.30/0.46 ( omega0Ax
% 0.30/0.46 = ( in @ emptyset @ omega ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(omegaSAx_type,type,
% 0.30/0.46 omegaSAx: $o ).
% 0.30/0.46
% 0.30/0.46 thf(omegaSAx,definition,
% 0.30/0.46 ( omegaSAx
% 0.30/0.46 = ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ omega )
% 0.30/0.46 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(omegaIndAx_type,type,
% 0.30/0.46 omegaIndAx: $o ).
% 0.30/0.46
% 0.30/0.46 thf(omegaIndAx,definition,
% 0.30/0.46 ( omegaIndAx
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ( ( ( in @ emptyset @ A )
% 0.30/0.46 & ! [Xx: $i] :
% 0.30/0.46 ( ( ( in @ Xx @ omega )
% 0.30/0.46 & ( in @ Xx @ A ) )
% 0.30/0.46 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ omega )
% 0.30/0.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(replAx_type,type,
% 0.30/0.46 replAx: $o ).
% 0.30/0.46
% 0.30/0.46 thf(replAx,definition,
% 0.30/0.46 ( replAx
% 0.30/0.46 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( exu
% 0.30/0.46 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.30/0.46 => ? [B: $i] :
% 0.30/0.46 ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ B )
% 0.30/0.46 <=> ? [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(foundationAx_type,type,
% 0.30/0.46 foundationAx: $o ).
% 0.30/0.46
% 0.30/0.46 thf(foundationAx,definition,
% 0.30/0.46 ( foundationAx
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.30/0.46 => ? [B: $i] :
% 0.30/0.46 ( ( in @ B @ A )
% 0.30/0.46 & ~ ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ B )
% 0.30/0.46 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(wellorderingAx_type,type,
% 0.30/0.46 wellorderingAx: $o ).
% 0.30/0.46
% 0.30/0.46 thf(wellorderingAx,definition,
% 0.30/0.46 ( wellorderingAx
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ? [B: $i] :
% 0.30/0.46 ( ! [C: $i] :
% 0.30/0.46 ( ( in @ C @ B )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ C )
% 0.30/0.46 => ( in @ Xx @ A ) ) )
% 0.30/0.46 & ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( ( in @ Xx @ A )
% 0.30/0.46 & ( in @ Xy @ A ) )
% 0.30/0.46 => ( ! [C: $i] :
% 0.30/0.46 ( ( in @ C @ B )
% 0.30/0.46 => ( ( in @ Xx @ C )
% 0.30/0.46 <=> ( in @ Xy @ C ) ) )
% 0.30/0.46 => ( Xx = Xy ) ) )
% 0.30/0.46 & ! [C: $i,D: $i] :
% 0.30/0.46 ( ( ( in @ C @ B )
% 0.30/0.46 & ( in @ D @ B ) )
% 0.30/0.46 => ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ C )
% 0.30/0.46 => ( in @ Xx @ D ) )
% 0.30/0.46 | ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ D )
% 0.30/0.46 => ( in @ Xx @ C ) ) ) )
% 0.30/0.46 & ! [C: $i] :
% 0.30/0.46 ( ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ C )
% 0.30/0.46 => ( in @ Xx @ A ) )
% 0.30/0.46 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.30/0.46 => ? [D: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ D @ B )
% 0.30/0.46 & ( in @ Xx @ C )
% 0.30/0.46 & ~ ? [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ D )
% 0.30/0.46 & ( in @ Xy @ C ) )
% 0.30/0.46 & ! [E: $i] :
% 0.30/0.46 ( ( in @ E @ B )
% 0.30/0.46 => ( ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ E )
% 0.30/0.46 => ( in @ Xy @ D ) )
% 0.30/0.46 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(descr_type,type,
% 0.30/0.46 descr: ( $i > $o ) > $i ).
% 0.30/0.46
% 0.30/0.46 thf(descrp_type,type,
% 0.30/0.46 descrp: $o ).
% 0.30/0.46
% 0.30/0.46 thf(descrp,definition,
% 0.30/0.46 ( descrp
% 0.30/0.46 = ( ! [Xphi: $i > $o] :
% 0.30/0.46 ( ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 => ( Xphi
% 0.30/0.46 @ ( descr
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(dsetconstr_type,type,
% 0.30/0.46 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.30/0.46
% 0.30/0.46 thf(dsetconstrI_type,type,
% 0.30/0.46 dsetconstrI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(dsetconstrI,definition,
% 0.30/0.46 ( dsetconstrI
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( Xphi @ Xx )
% 0.30/0.46 => ( in @ Xx
% 0.30/0.46 @ ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(dsetconstrEL_type,type,
% 0.30/0.46 dsetconstrEL: $o ).
% 0.30/0.46
% 0.30/0.46 thf(dsetconstrEL,definition,
% 0.30/0.46 ( dsetconstrEL
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx
% 0.30/0.46 @ ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.30/0.46 => ( in @ Xx @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(dsetconstrER_type,type,
% 0.30/0.46 dsetconstrER: $o ).
% 0.30/0.46
% 0.30/0.46 thf(dsetconstrER,definition,
% 0.30/0.46 ( dsetconstrER
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx
% 0.30/0.46 @ ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.30/0.46 => ( Xphi @ Xx ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(exuE1_type,type,
% 0.30/0.46 exuE1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(exuE1,definition,
% 0.30/0.46 ( exuE1
% 0.30/0.46 = ( ! [Xphi: $i > $o] :
% 0.30/0.46 ( ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 => ? [Xx: $i] :
% 0.30/0.46 ( ( Xphi @ Xx )
% 0.30/0.46 & ! [Xy: $i] :
% 0.30/0.46 ( ( Xphi @ Xy )
% 0.30/0.46 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(prop2set_type,type,
% 0.30/0.46 prop2set: $o > $i ).
% 0.30/0.46
% 0.30/0.46 thf(prop2setE_type,type,
% 0.30/0.46 prop2setE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(prop2setE,definition,
% 0.30/0.46 ( prop2setE
% 0.30/0.46 = ( ! [Xphi: $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.30/0.46 => Xphi ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(emptysetE_type,type,
% 0.30/0.46 emptysetE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(emptysetE,definition,
% 0.30/0.46 ( emptysetE
% 0.30/0.46 = ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ emptyset )
% 0.30/0.46 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(emptysetimpfalse_type,type,
% 0.30/0.46 emptysetimpfalse: $o ).
% 0.30/0.46
% 0.30/0.46 thf(emptysetimpfalse,definition,
% 0.30/0.46 ( emptysetimpfalse
% 0.30/0.46 = ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ emptyset )
% 0.30/0.46 => $false ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(notinemptyset_type,type,
% 0.30/0.46 notinemptyset: $o ).
% 0.30/0.46
% 0.30/0.46 thf(notinemptyset,definition,
% 0.30/0.46 ( notinemptyset
% 0.30/0.46 = ( ! [Xx: $i] :
% 0.30/0.46 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(exuE3e_type,type,
% 0.30/0.46 exuE3e: $o ).
% 0.30/0.46
% 0.30/0.46 thf(exuE3e,definition,
% 0.30/0.46 ( exuE3e
% 0.30/0.46 = ( ! [Xphi: $i > $o] :
% 0.30/0.46 ( ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setext_type,type,
% 0.30/0.46 setext: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setext,definition,
% 0.30/0.46 ( setext
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ Xx @ B ) )
% 0.30/0.46 => ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ B )
% 0.30/0.46 => ( in @ Xx @ A ) )
% 0.30/0.46 => ( A = B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(emptyI_type,type,
% 0.30/0.46 emptyI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(emptyI,definition,
% 0.30/0.46 ( emptyI
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ~ ( in @ Xx @ A )
% 0.30/0.46 => ( A = emptyset ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(noeltsimpempty_type,type,
% 0.30/0.46 noeltsimpempty: $o ).
% 0.30/0.46
% 0.30/0.46 thf(noeltsimpempty,definition,
% 0.30/0.46 ( noeltsimpempty
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ~ ( in @ Xx @ A )
% 0.30/0.46 => ( A = emptyset ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setbeta_type,type,
% 0.30/0.46 setbeta: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setbeta,definition,
% 0.30/0.46 ( setbeta
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx
% 0.30/0.46 @ ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.30/0.46 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(nonempty_type,type,
% 0.30/0.46 nonempty: $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(nonempty,definition,
% 0.30/0.46 ( nonempty
% 0.30/0.46 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(nonemptyE1_type,type,
% 0.30/0.46 nonemptyE1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(nonemptyE1,definition,
% 0.30/0.46 ( nonemptyE1
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ( ( nonempty @ A )
% 0.30/0.46 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(nonemptyI_type,type,
% 0.30/0.46 nonemptyI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(nonemptyI,definition,
% 0.30/0.46 ( nonemptyI
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( Xphi @ Xx )
% 0.30/0.46 => ( nonempty
% 0.30/0.46 @ ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(nonemptyI1_type,type,
% 0.30/0.46 nonemptyI1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(nonemptyI1,definition,
% 0.30/0.46 ( nonemptyI1
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.30/0.46 => ( nonempty @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinIL_type,type,
% 0.30/0.46 setadjoinIL: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinIL,definition,
% 0.30/0.46 ( setadjoinIL
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(emptyinunitempty_type,type,
% 0.30/0.46 emptyinunitempty: $o ).
% 0.30/0.46
% 0.30/0.46 thf(emptyinunitempty,definition,
% 0.30/0.46 ( emptyinunitempty
% 0.30/0.46 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinIR_type,type,
% 0.30/0.46 setadjoinIR: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinIR,definition,
% 0.30/0.46 ( setadjoinIR
% 0.30/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinE_type,type,
% 0.30/0.46 setadjoinE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinE,definition,
% 0.30/0.46 ( setadjoinE
% 0.30/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.30/0.46 => ! [Xphi: $o] :
% 0.30/0.46 ( ( ( Xy = Xx )
% 0.30/0.46 => Xphi )
% 0.30/0.46 => ( ( ( in @ Xy @ A )
% 0.30/0.46 => Xphi )
% 0.30/0.46 => Xphi ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinOr_type,type,
% 0.30/0.46 setadjoinOr: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinOr,definition,
% 0.30/0.46 ( setadjoinOr
% 0.30/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.30/0.46 => ( ( Xy = Xx )
% 0.30/0.46 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setoftrueEq_type,type,
% 0.30/0.46 setoftrueEq: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setoftrueEq,definition,
% 0.30/0.46 ( setoftrueEq
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ( ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xx: $i] : $true )
% 0.30/0.46 = A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(powersetI_type,type,
% 0.30/0.46 powersetI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(powersetI,definition,
% 0.30/0.46 ( powersetI
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ B )
% 0.30/0.46 => ( in @ Xx @ A ) )
% 0.30/0.46 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(emptyinPowerset_type,type,
% 0.30/0.46 emptyinPowerset: $o ).
% 0.30/0.46
% 0.30/0.46 thf(emptyinPowerset,definition,
% 0.30/0.46 ( emptyinPowerset
% 0.30/0.46 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(emptyInPowerset_type,type,
% 0.30/0.46 emptyInPowerset: $o ).
% 0.30/0.46
% 0.30/0.46 thf(emptyInPowerset,definition,
% 0.30/0.46 ( emptyInPowerset
% 0.30/0.46 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(powersetE_type,type,
% 0.30/0.46 powersetE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(powersetE,definition,
% 0.30/0.46 ( powersetE
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ B @ ( powerset @ A ) )
% 0.30/0.46 => ( ( in @ Xx @ B )
% 0.30/0.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setunionI_type,type,
% 0.30/0.46 setunionI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setunionI,definition,
% 0.30/0.46 ( setunionI
% 0.30/0.46 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.30/0.46 ( ( in @ Xx @ B )
% 0.30/0.46 => ( ( in @ B @ A )
% 0.30/0.46 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setunionE_type,type,
% 0.30/0.46 setunionE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setunionE,definition,
% 0.30/0.46 ( setunionE
% 0.30/0.46 = ( ! [A: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ ( setunion @ A ) )
% 0.30/0.46 => ! [Xphi: $o] :
% 0.30/0.46 ( ! [B: $i] :
% 0.30/0.46 ( ( in @ Xx @ B )
% 0.30/0.46 => ( ( in @ B @ A )
% 0.30/0.46 => Xphi ) )
% 0.30/0.46 => Xphi ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(subPowSU_type,type,
% 0.30/0.46 subPowSU: $o ).
% 0.30/0.46
% 0.30/0.46 thf(subPowSU,definition,
% 0.30/0.46 ( subPowSU
% 0.30/0.46 = ( ! [A: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(exuE2_type,type,
% 0.30/0.46 exuE2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(exuE2,definition,
% 0.30/0.46 ( exuE2
% 0.30/0.46 = ( ! [Xphi: $i > $o] :
% 0.30/0.46 ( ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 => ? [Xx: $i] :
% 0.30/0.46 ! [Xy: $i] :
% 0.30/0.46 ( ( Xphi @ Xy )
% 0.30/0.46 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(nonemptyImpWitness_type,type,
% 0.30/0.46 nonemptyImpWitness: $o ).
% 0.30/0.46
% 0.30/0.46 thf(nonemptyImpWitness,definition,
% 0.30/0.46 ( nonemptyImpWitness
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ( ( nonempty @ A )
% 0.30/0.46 => ? [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 & $true ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(uniqinunit_type,type,
% 0.30/0.47 uniqinunit: $o ).
% 0.30/0.47
% 0.30/0.47 thf(uniqinunit,definition,
% 0.30/0.47 ( uniqinunit
% 0.30/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.47 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.30/0.47 => ( Xx = Xy ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(notinsingleton_type,type,
% 0.30/0.47 notinsingleton: $o ).
% 0.30/0.47
% 0.30/0.47 thf(notinsingleton,definition,
% 0.30/0.47 ( notinsingleton
% 0.30/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.47 ( ( Xx != Xy )
% 0.30/0.47 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(eqinunit_type,type,
% 0.30/0.47 eqinunit: $o ).
% 0.30/0.47
% 0.30/0.47 thf(eqinunit,definition,
% 0.30/0.47 ( eqinunit
% 0.30/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.47 ( ( Xx = Xy )
% 0.30/0.47 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(singletonsswitch_type,type,
% 0.30/0.47 singletonsswitch: $o ).
% 0.30/0.47
% 0.30/0.47 thf(singletonsswitch,definition,
% 0.30/0.47 ( singletonsswitch
% 0.30/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.47 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.30/0.47 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(upairsetE_type,type,
% 0.30/0.47 upairsetE: $o ).
% 0.30/0.47
% 0.30/0.47 thf(upairsetE,definition,
% 0.30/0.47 ( upairsetE
% 0.30/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.30/0.47 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.30/0.47 => ( ( Xz = Xx )
% 0.30/0.47 | ( Xz = Xy ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(upairsetIL_type,type,
% 0.30/0.47 upairsetIL: $o ).
% 0.30/0.47
% 0.30/0.47 thf(upairsetIL,definition,
% 0.30/0.47 ( upairsetIL
% 0.30/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(upairsetIR_type,type,
% 0.30/0.47 upairsetIR: $o ).
% 0.30/0.47
% 0.30/0.47 thf(upairsetIR,definition,
% 0.30/0.47 ( upairsetIR
% 0.30/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(emptyE1_type,type,
% 0.30/0.47 emptyE1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(emptyE1,definition,
% 0.30/0.47 ( emptyE1
% 0.30/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.47 ( ? [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 & ( Xphi @ Xx ) )
% 0.30/0.47 => ( ( ( dsetconstr @ A
% 0.30/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.47 = emptyset )
% 0.30/0.47 => $false ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(vacuousDall_type,type,
% 0.30/0.47 vacuousDall: $o ).
% 0.30/0.47
% 0.30/0.47 thf(vacuousDall,definition,
% 0.30/0.47 ( vacuousDall
% 0.30/0.47 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ emptyset )
% 0.30/0.47 => ( Xphi @ Xx ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(quantDeMorgan1_type,type,
% 0.30/0.47 quantDeMorgan1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(quantDeMorgan1,definition,
% 0.30/0.47 ( quantDeMorgan1
% 0.30/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.47 ( ~ ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( Xphi @ Xx ) )
% 0.30/0.47 => ? [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(quantDeMorgan2_type,type,
% 0.30/0.47 quantDeMorgan2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(quantDeMorgan2,definition,
% 0.30/0.47 ( quantDeMorgan2
% 0.30/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.47 ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ~ ( Xphi @ Xx ) )
% 0.30/0.47 => ~ ? [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(quantDeMorgan3_type,type,
% 0.30/0.47 quantDeMorgan3: $o ).
% 0.30/0.47
% 0.30/0.47 thf(quantDeMorgan3,definition,
% 0.30/0.47 ( quantDeMorgan3
% 0.30/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.47 ( ~ ? [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 & ( Xphi @ Xx ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(quantDeMorgan4_type,type,
% 0.30/0.47 quantDeMorgan4: $o ).
% 0.30/0.47
% 0.30/0.47 thf(quantDeMorgan4,definition,
% 0.30/0.47 ( quantDeMorgan4
% 0.30/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.47 ( ? [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 & ~ ( Xphi @ Xx ) )
% 0.30/0.47 => ~ ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(prop2setI_type,type,
% 0.30/0.47 prop2setI: $o ).
% 0.30/0.47
% 0.30/0.47 thf(prop2setI,definition,
% 0.30/0.47 ( prop2setI
% 0.30/0.47 = ( ! [Xphi: $o] :
% 0.30/0.47 ( Xphi
% 0.30/0.47 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(set2prop_type,type,
% 0.30/0.47 set2prop: $i > $o ).
% 0.30/0.47
% 0.30/0.47 thf(prop2set2propI_type,type,
% 0.30/0.47 prop2set2propI: $o ).
% 0.30/0.47
% 0.30/0.47 thf(prop2set2propI,definition,
% 0.30/0.47 ( prop2set2propI
% 0.30/0.47 = ( ! [Xphi: $o] :
% 0.30/0.47 ( Xphi
% 0.30/0.47 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(notdexE_type,type,
% 0.30/0.47 notdexE: $o ).
% 0.30/0.47
% 0.30/0.47 thf(notdexE,definition,
% 0.30/0.47 ( notdexE
% 0.30/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.47 ( ~ ? [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 & ( Xphi @ Xx ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(notdallE_type,type,
% 0.30/0.47 notdallE: $o ).
% 0.30/0.47
% 0.30/0.47 thf(notdallE,definition,
% 0.30/0.47 ( notdallE
% 0.30/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.47 ( ~ ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( Xphi @ Xx ) )
% 0.30/0.47 => ? [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(exuI1_type,type,
% 0.30/0.47 exuI1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(exuI1,definition,
% 0.30/0.47 ( exuI1
% 0.30/0.47 = ( ! [Xphi: $i > $o] :
% 0.30/0.47 ( ? [Xx: $i] :
% 0.30/0.47 ( ( Xphi @ Xx )
% 0.30/0.47 & ! [Xy: $i] :
% 0.30/0.47 ( ( Xphi @ Xy )
% 0.30/0.47 => ( Xx = Xy ) ) )
% 0.30/0.47 => ( exu
% 0.30/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(exuI3_type,type,
% 0.30/0.47 exuI3: $o ).
% 0.30/0.47
% 0.30/0.47 thf(exuI3,definition,
% 0.30/0.47 ( exuI3
% 0.30/0.47 = ( ! [Xphi: $i > $o] :
% 0.30/0.47 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.30/0.47 => ( ! [Xx: $i,Xy: $i] :
% 0.30/0.47 ( ( Xphi @ Xx )
% 0.30/0.47 => ( ( Xphi @ Xy )
% 0.30/0.47 => ( Xx = Xy ) ) )
% 0.30/0.47 => ( exu
% 0.30/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(exuI2_type,type,
% 0.30/0.47 exuI2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(exuI2,definition,
% 0.30/0.47 ( exuI2
% 0.30/0.47 = ( ! [Xphi: $i > $o] :
% 0.30/0.47 ( ? [Xx: $i] :
% 0.30/0.47 ! [Xy: $i] :
% 0.30/0.47 ( ( Xphi @ Xy )
% 0.30/0.47 <=> ( Xy = Xx ) )
% 0.30/0.47 => ( exu
% 0.30/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(inCongP_type,type,
% 0.30/0.47 inCongP: $o ).
% 0.30/0.47
% 0.30/0.47 thf(inCongP,definition,
% 0.30/0.47 ( inCongP
% 0.30/0.47 = ( ! [A: $i,B: $i] :
% 0.30/0.47 ( ( A = B )
% 0.30/0.47 => ! [Xx: $i,Xy: $i] :
% 0.30/0.47 ( ( Xx = Xy )
% 0.30/0.47 => ( ( in @ Xx @ A )
% 0.30/0.47 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(in__Cong_type,type,
% 0.30/0.47 in__Cong: $o ).
% 0.30/0.47
% 0.30/0.47 thf(in__Cong,definition,
% 0.30/0.47 ( in__Cong
% 0.30/0.47 = ( ! [A: $i,B: $i] :
% 0.30/0.47 ( ( A = B )
% 0.30/0.47 => ! [Xx: $i,Xy: $i] :
% 0.30/0.47 ( ( Xx = Xy )
% 0.30/0.47 => ( ( in @ Xx @ A )
% 0.30/0.47 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(exuE3u_type,type,
% 0.30/0.47 exuE3u: $o ).
% 0.30/0.47
% 0.30/0.47 thf(exuE3u,definition,
% 0.30/0.47 ( exuE3u
% 0.30/0.47 = ( ! [Xphi: $i > $o] :
% 0.30/0.47 ( ( exu
% 0.30/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.47 => ! [Xx: $i,Xy: $i] :
% 0.30/0.47 ( ( Xphi @ Xx )
% 0.30/0.47 => ( ( Xphi @ Xy )
% 0.30/0.47 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(exu__Cong_type,type,
% 0.30/0.47 exu__Cong: $o ).
% 0.30/0.47
% 0.30/0.47 thf(exu__Cong,definition,
% 0.30/0.47 ( exu__Cong
% 0.30/0.47 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.30/0.47 ( ! [Xx: $i,Xy: $i] :
% 0.30/0.47 ( ( Xx = Xy )
% 0.30/0.47 => ( ( Xphi @ Xx )
% 0.30/0.47 <=> ( Xpsi @ Xy ) ) )
% 0.30/0.47 => ( ( exu
% 0.30/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.47 <=> ( exu
% 0.30/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(emptyset__Cong_type,type,
% 0.30/0.47 emptyset__Cong: $o ).
% 0.30/0.47
% 0.30/0.47 thf(emptyset__Cong,definition,
% 0.30/0.47 ( emptyset__Cong
% 0.30/0.47 = ( emptyset = emptyset ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(setadjoin__Cong_type,type,
% 0.30/0.47 setadjoin__Cong: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setadjoin__Cong,definition,
% 0.30/0.47 ( setadjoin__Cong
% 0.30/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.47 ( ( Xx = Xy )
% 0.30/0.47 => ! [Xz: $i,Xu: $i] :
% 0.30/0.47 ( ( Xz = Xu )
% 0.30/0.47 => ( ( setadjoin @ Xx @ Xz )
% 0.30/0.47 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(powerset__Cong_type,type,
% 0.30/0.47 powerset__Cong: $o ).
% 0.30/0.47
% 0.30/0.47 thf(powerset__Cong,definition,
% 0.30/0.47 ( powerset__Cong
% 0.30/0.47 = ( ! [A: $i,B: $i] :
% 0.30/0.47 ( ( A = B )
% 0.30/0.47 => ( ( powerset @ A )
% 0.30/0.47 = ( powerset @ B ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(setunion__Cong_type,type,
% 0.30/0.47 setunion__Cong: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setunion__Cong,definition,
% 0.30/0.47 ( setunion__Cong
% 0.30/0.47 = ( ! [A: $i,B: $i] :
% 0.30/0.47 ( ( A = B )
% 0.30/0.47 => ( ( setunion @ A )
% 0.30/0.47 = ( setunion @ B ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(omega__Cong_type,type,
% 0.30/0.47 omega__Cong: $o ).
% 0.30/0.47
% 0.30/0.47 thf(omega__Cong,definition,
% 0.30/0.47 ( omega__Cong
% 0.30/0.47 = ( omega = omega ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(exuEu_type,type,
% 0.30/0.47 exuEu: $o ).
% 0.30/0.47
% 0.30/0.47 thf(exuEu,definition,
% 0.30/0.47 ( exuEu
% 0.30/0.47 = ( ! [Xphi: $i > $o] :
% 0.30/0.47 ( ( exu
% 0.30/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.47 => ! [Xx: $i,Xy: $i] :
% 0.30/0.47 ( ( Xphi @ Xx )
% 0.30/0.47 => ( ( Xphi @ Xy )
% 0.30/0.47 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(descr__Cong_type,type,
% 0.30/0.47 descr__Cong: $o ).
% 0.30/0.47
% 0.30/0.47 thf(descr__Cong,definition,
% 0.30/0.47 ( descr__Cong
% 0.30/0.47 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.30/0.47 ( ! [Xx: $i,Xy: $i] :
% 0.30/0.47 ( ( Xx = Xy )
% 0.30/0.47 => ( ( Xphi @ Xx )
% 0.30/0.47 <=> ( Xpsi @ Xy ) ) )
% 0.30/0.47 => ( ( exu
% 0.30/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.47 => ( ( exu
% 0.30/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.30/0.47 => ( ( descr
% 0.30/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.47 = ( descr
% 0.30/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(dsetconstr__Cong_type,type,
% 0.30/0.47 dsetconstr__Cong: $o ).
% 0.30/0.47
% 0.30/0.47 thf(dsetconstr__Cong,definition,
% 0.30/0.47 ( dsetconstr__Cong
% 0.30/0.47 = ( ! [A: $i,B: $i] :
% 0.30/0.47 ( ( A = B )
% 0.30/0.47 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.30/0.47 ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ B )
% 0.30/0.47 => ( ( Xx = Xy )
% 0.30/0.47 => ( ( Xphi @ Xx )
% 0.30/0.47 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.30/0.47 => ( ( dsetconstr @ A
% 0.30/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.47 = ( dsetconstr @ B
% 0.30/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(subset_type,type,
% 0.30/0.47 subset: $i > $i > $o ).
% 0.30/0.47
% 0.30/0.47 thf(subset,definition,
% 0.30/0.47 ( subset
% 0.30/0.47 = ( ^ [A: $i,B: $i] :
% 0.30/0.47 ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( in @ Xx @ B ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(disjoint_type,type,
% 0.30/0.47 disjoint: $i > $i > $o ).
% 0.30/0.47
% 0.30/0.47 thf(setsmeet_type,type,
% 0.30/0.47 setsmeet: $i > $i > $o ).
% 0.30/0.47
% 0.30/0.47 thf(subsetI1_type,type,
% 0.30/0.47 subsetI1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(subsetI1,definition,
% 0.30/0.47 ( subsetI1
% 0.30/0.47 = ( ! [A: $i,B: $i] :
% 0.30/0.47 ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( in @ Xx @ B ) )
% 0.30/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(eqimpsubset2_type,type,
% 0.30/0.47 eqimpsubset2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(eqimpsubset2,definition,
% 0.30/0.47 ( eqimpsubset2
% 0.30/0.47 = ( ! [A: $i,B: $i] :
% 0.30/0.47 ( ( A = B )
% 0.30/0.47 => ( subset @ B @ A ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(eqimpsubset1_type,type,
% 0.30/0.47 eqimpsubset1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(eqimpsubset1,definition,
% 0.30/0.47 ( eqimpsubset1
% 0.30/0.47 = ( ! [A: $i,B: $i] :
% 0.30/0.47 ( ( A = B )
% 0.30/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(subsetI2,conjecture,
% 0.30/0.47 ( setextAx
% 0.30/0.47 => ( emptysetAx
% 0.30/0.47 => ( setadjoinAx
% 0.30/0.47 => ( powersetAx
% 0.30/0.47 => ( setunionAx
% 0.30/0.47 => ( omega0Ax
% 0.30/0.47 => ( omegaSAx
% 0.30/0.47 => ( omegaIndAx
% 0.30/0.47 => ( replAx
% 0.30/0.47 => ( foundationAx
% 0.30/0.47 => ( wellorderingAx
% 0.30/0.47 => ( descrp
% 0.30/0.47 => ( dsetconstrI
% 0.30/0.47 => ( dsetconstrEL
% 0.30/0.47 => ( dsetconstrER
% 0.30/0.47 => ( exuE1
% 0.30/0.47 => ( prop2setE
% 0.30/0.47 => ( emptysetE
% 0.30/0.47 => ( emptysetimpfalse
% 0.30/0.47 => ( notinemptyset
% 0.30/0.47 => ( exuE3e
% 0.30/0.47 => ( setext
% 0.30/0.47 => ( emptyI
% 0.30/0.47 => ( noeltsimpempty
% 0.30/0.47 => ( setbeta
% 0.30/0.47 => ( nonemptyE1
% 0.30/0.47 => ( nonemptyI
% 0.30/0.47 => ( nonemptyI1
% 0.30/0.47 => ( setadjoinIL
% 0.30/0.47 => ( emptyinunitempty
% 0.30/0.47 => ( setadjoinIR
% 0.30/0.47 => ( setadjoinE
% 0.30/0.47 => ( setadjoinOr
% 0.30/0.47 => ( setoftrueEq
% 0.30/0.47 => ( powersetI
% 0.30/0.47 => ( emptyinPowerset
% 0.30/0.47 => ( emptyInPowerset
% 0.30/0.47 => ( powersetE
% 0.30/0.47 => ( setunionI
% 0.30/0.47 => ( setunionE
% 0.30/0.47 => ( subPowSU
% 0.30/0.47 => ( exuE2
% 0.30/0.47 => ( nonemptyImpWitness
% 0.30/0.47 => ( uniqinunit
% 0.30/0.47 => ( notinsingleton
% 0.30/0.47 => ( eqinunit
% 0.30/0.47 => ( singletonsswitch
% 0.30/0.47 => ( upairsetE
% 0.30/0.47 => ( upairsetIL
% 0.30/0.47 => ( upairsetIR
% 0.30/0.47 => ( emptyE1
% 0.30/0.47 => ( vacuousDall
% 0.30/0.47 => ( quantDeMorgan1
% 0.30/0.47 => ( quantDeMorgan2
% 0.30/0.47 => ( quantDeMorgan3
% 0.30/0.47 => ( quantDeMorgan4
% 0.30/0.47 => ( prop2setI
% 0.30/0.47 => ( prop2set2propI
% 0.30/0.47 => ( notdexE
% 0.30/0.47 => ( notdallE
% 0.30/0.47 => ( exuI1
% 0.30/0.47 => ( exuI3
% 0.30/0.47 => ( exuI2
% 0.30/0.47 => ( inCongP
% 0.30/0.47 => ( in__Cong
% 0.30/0.47 => ( exuE3u
% 0.30/0.47 => ( exu__Cong
% 0.30/0.47 => ( emptyset__Cong
% 0.30/0.47 => ( setadjoin__Cong
% 0.30/0.47 => ( powerset__Cong
% 0.30/0.47 => ( setunion__Cong
% 0.30/0.47 => ( omega__Cong
% 0.30/0.47 => ( exuEu
% 0.30/0.47 => ( descr__Cong
% 0.30/0.47 => ( dsetconstr__Cong
% 0.30/0.47 => ( subsetI1
% 0.30/0.47 => ( eqimpsubset2
% 0.30/0.47 => ( eqimpsubset1
% 0.30/0.47 => ! [A: $i,B: $i] :
% 0.30/0.47 ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( in @ Xx @ B ) )
% 0.30/0.47 => ( subset @ A @ B ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 %------------------------------------------------------------------------------
% 0.30/0.47 ------------------------
% 2.17/1.43 ---- Embedded in THF ---
% 2.17/1.43 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.17/1.43 %%% Generated on Tue Feb 24 12:27:35 EST 2026
% 2.17/1.43 %%% using '$$dhol' embedding, version 1.3.0.
% 2.17/1.43 %%% Logic specification used:
% 2.17/1.43 %%% thf(spec, logic, $$dhol).
% 2.17/1.43
% 2.17/1.43 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.vNo7MRU8lP/DTF2DTF20822.p
% See solution above
% 2.17/1.43 ------------------------
% 2.17/1.45 ---- Cleaned THF ---
% 2.17/1.45 thf(in_type,type,
% 2.17/1.45 in: $i > $i > $o ).
% 2.17/1.45
% 2.17/1.45 thf(exu_type,type,
% 2.17/1.45 exu: ( $i > $o ) > $o ).
% 2.17/1.45
% 2.17/1.45 thf(exu,definition,
% 2.17/1.45 ( exu
% 2.17/1.45 = ( ^ [Xphi: $i > $o] :
% 2.17/1.45 ? [Xx: $i] :
% 2.17/1.45 ( ( Xphi @ Xx )
% 2.17/1.45 & ! [Xy: $i] :
% 2.17/1.45 ( ( Xphi @ Xy )
% 2.17/1.45 => ( Xx = Xy ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(setextAx_type,type,
% 2.17/1.45 setextAx: $o ).
% 2.17/1.45
% 2.17/1.45 thf(setextAx,definition,
% 2.17/1.45 ( setextAx
% 2.17/1.45 = ( ! [A: $i,B: $i] :
% 2.17/1.45 ( ! [Xx: $i] :
% 2.17/1.45 ( ( in @ Xx @ A )
% 2.17/1.45 <=> ( in @ Xx @ B ) )
% 2.17/1.45 => ( A = B ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(emptyset_type,type,
% 2.17/1.45 emptyset: $i ).
% 2.17/1.45
% 2.17/1.45 thf(emptysetAx_type,type,
% 2.17/1.45 emptysetAx: $o ).
% 2.17/1.45
% 2.17/1.45 thf(emptysetAx,definition,
% 2.17/1.45 ( emptysetAx
% 2.17/1.45 = ( ! [Xx: $i] :
% 2.17/1.45 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(setadjoin_type,type,
% 2.17/1.45 setadjoin: $i > $i > $i ).
% 2.17/1.45
% 2.17/1.45 thf(setadjoinAx_type,type,
% 2.17/1.45 setadjoinAx: $o ).
% 2.17/1.45
% 2.17/1.45 thf(setadjoinAx,definition,
% 2.17/1.45 ( setadjoinAx
% 2.17/1.45 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.17/1.45 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.17/1.45 <=> ( ( Xy = Xx )
% 2.17/1.45 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(powerset_type,type,
% 2.17/1.45 powerset: $i > $i ).
% 2.17/1.45
% 2.17/1.45 thf(powersetAx_type,type,
% 2.17/1.45 powersetAx: $o ).
% 2.17/1.45
% 2.17/1.45 thf(powersetAx,definition,
% 2.17/1.45 ( powersetAx
% 2.17/1.45 = ( ! [A: $i,B: $i] :
% 2.17/1.45 ( ( in @ B @ ( powerset @ A ) )
% 2.17/1.45 <=> ! [Xx: $i] :
% 2.17/1.45 ( ( in @ Xx @ B )
% 2.17/1.45 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(setunion_type,type,
% 2.17/1.45 setunion: $i > $i ).
% 2.17/1.45
% 2.17/1.45 thf(setunionAx_type,type,
% 2.17/1.45 setunionAx: $o ).
% 2.17/1.45
% 2.17/1.45 thf(setunionAx,definition,
% 2.17/1.45 ( setunionAx
% 2.17/1.45 = ( ! [A: $i,Xx: $i] :
% 2.17/1.45 ( ( in @ Xx @ ( setunion @ A ) )
% 2.17/1.45 <=> ? [B: $i] :
% 2.17/1.45 ( ( in @ Xx @ B )
% 2.17/1.45 & ( in @ B @ A ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(omega_type,type,
% 2.17/1.45 omega: $i ).
% 2.17/1.45
% 2.17/1.45 thf(omega0Ax_type,type,
% 2.17/1.45 omega0Ax: $o ).
% 2.17/1.45
% 2.17/1.45 thf(omega0Ax,definition,
% 2.17/1.45 ( omega0Ax
% 2.17/1.45 = ( in @ emptyset @ omega ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(omegaSAx_type,type,
% 2.17/1.45 omegaSAx: $o ).
% 2.17/1.45
% 2.17/1.45 thf(omegaSAx,definition,
% 2.17/1.45 ( omegaSAx
% 2.17/1.45 = ( ! [Xx: $i] :
% 2.17/1.45 ( ( in @ Xx @ omega )
% 2.17/1.45 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(omegaIndAx_type,type,
% 2.17/1.45 omegaIndAx: $o ).
% 2.17/1.45
% 2.17/1.45 thf(omegaIndAx,definition,
% 2.17/1.45 ( omegaIndAx
% 2.17/1.45 = ( ! [A: $i] :
% 2.17/1.45 ( ( ( in @ emptyset @ A )
% 2.17/1.45 & ! [Xx: $i] :
% 2.17/1.45 ( ( ( in @ Xx @ omega )
% 2.17/1.45 & ( in @ Xx @ A ) )
% 2.17/1.45 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 2.17/1.45 => ! [Xx: $i] :
% 2.17/1.45 ( ( in @ Xx @ omega )
% 2.17/1.45 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(replAx_type,type,
% 2.17/1.45 replAx: $o ).
% 2.17/1.45
% 2.17/1.45 thf(replAx,definition,
% 2.17/1.45 ( replAx
% 2.17/1.45 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 2.17/1.45 ( ! [Xx: $i] :
% 2.17/1.45 ( ( in @ Xx @ A )
% 2.17/1.45 => ( exu
% 2.17/1.45 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 2.17/1.45 => ? [B: $i] :
% 2.17/1.45 ! [Xx: $i] :
% 2.17/1.45 ( ( in @ Xx @ B )
% 2.17/1.45 <=> ? [Xy: $i] :
% 2.17/1.45 ( ( in @ Xy @ A )
% 2.17/1.45 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(foundationAx_type,type,
% 2.17/1.45 foundationAx: $o ).
% 2.17/1.45
% 2.17/1.45 thf(foundationAx,definition,
% 2.17/1.45 ( foundationAx
% 2.17/1.45 = ( ! [A: $i] :
% 2.17/1.45 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.17/1.45 => ? [B: $i] :
% 2.17/1.45 ( ( in @ B @ A )
% 2.17/1.45 & ~ ? [Xx: $i] :
% 2.17/1.45 ( ( in @ Xx @ B )
% 2.17/1.45 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(wellorderingAx_type,type,
% 2.17/1.45 wellorderingAx: $o ).
% 2.17/1.45
% 2.17/1.45 thf(wellorderingAx,definition,
% 2.17/1.45 ( wellorderingAx
% 2.17/1.45 = ( ! [A: $i] :
% 2.17/1.45 ? [B: $i] :
% 2.17/1.45 ( ! [C: $i] :
% 2.17/1.45 ( ( in @ C @ B )
% 2.17/1.45 => ! [Xx: $i] :
% 2.17/1.45 ( ( in @ Xx @ C )
% 2.17/1.45 => ( in @ Xx @ A ) ) )
% 2.17/1.45 & ! [Xx: $i,Xy: $i] :
% 2.17/1.45 ( ( ( in @ Xx @ A )
% 2.17/1.45 & ( in @ Xy @ A ) )
% 2.17/1.45 => ( ! [C: $i] :
% 2.17/1.45 ( ( in @ C @ B )
% 2.17/1.45 => ( ( in @ Xx @ C )
% 2.17/1.45 <=> ( in @ Xy @ C ) ) )
% 2.17/1.45 => ( Xx = Xy ) ) )
% 2.17/1.45 & ! [C: $i,D: $i] :
% 2.17/1.45 ( ( ( in @ C @ B )
% 2.17/1.45 & ( in @ D @ B ) )
% 2.17/1.45 => ( ! [Xx: $i] :
% 2.17/1.45 ( ( in @ Xx @ C )
% 2.17/1.45 => ( in @ Xx @ D ) )
% 2.17/1.45 | ! [Xx: $i] :
% 2.17/1.45 ( ( in @ Xx @ D )
% 2.17/1.45 => ( in @ Xx @ C ) ) ) )
% 2.17/1.45 & ! [C: $i] :
% 2.17/1.45 ( ( ! [Xx: $i] :
% 2.17/1.45 ( ( in @ Xx @ C )
% 2.17/1.45 => ( in @ Xx @ A ) )
% 2.17/1.45 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 2.17/1.45 => ? [D: $i,Xx: $i] :
% 2.17/1.45 ( ( in @ D @ B )
% 2.17/1.45 & ( in @ Xx @ C )
% 2.17/1.45 & ~ ? [Xy: $i] :
% 2.17/1.45 ( ( in @ Xy @ D )
% 2.17/1.45 & ( in @ Xy @ C ) )
% 2.17/1.45 & ! [E: $i] :
% 2.17/1.45 ( ( in @ E @ B )
% 2.17/1.45 => ( ! [Xy: $i] :
% 2.17/1.45 ( ( in @ Xy @ E )
% 2.17/1.45 => ( in @ Xy @ D ) )
% 2.17/1.45 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(descr_type,type,
% 2.17/1.45 descr: ( $i > $o ) > $i ).
% 2.17/1.45
% 2.17/1.45 thf(descrp_type,type,
% 2.17/1.45 descrp: $o ).
% 2.17/1.45
% 2.17/1.45 thf(descrp,definition,
% 2.17/1.45 ( descrp
% 2.17/1.45 = ( ! [Xphi: $i > $o] :
% 2.17/1.45 ( ( exu
% 2.17/1.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.17/1.45 => ( Xphi
% 2.17/1.45 @ ( descr
% 2.17/1.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(dsetconstr_type,type,
% 2.17/1.45 dsetconstr: $i > ( $i > $o ) > $i ).
% 2.17/1.45
% 2.17/1.45 thf(dsetconstrI_type,type,
% 2.17/1.45 dsetconstrI: $o ).
% 2.17/1.45
% 2.17/1.45 thf(dsetconstrI,definition,
% 2.17/1.45 ( dsetconstrI
% 2.17/1.45 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.17/1.45 ( ( in @ Xx @ A )
% 2.17/1.45 => ( ( Xphi @ Xx )
% 2.17/1.45 => ( in @ Xx
% 2.17/1.45 @ ( dsetconstr @ A
% 2.17/1.45 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(dsetconstrEL_type,type,
% 2.17/1.45 dsetconstrEL: $o ).
% 2.17/1.45
% 2.17/1.45 thf(dsetconstrEL,definition,
% 2.17/1.45 ( dsetconstrEL
% 2.17/1.45 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.17/1.45 ( ( in @ Xx
% 2.17/1.45 @ ( dsetconstr @ A
% 2.17/1.45 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.17/1.45 => ( in @ Xx @ A ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(dsetconstrER_type,type,
% 2.17/1.45 dsetconstrER: $o ).
% 2.17/1.45
% 2.17/1.45 thf(dsetconstrER,definition,
% 2.17/1.45 ( dsetconstrER
% 2.17/1.45 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.17/1.45 ( ( in @ Xx
% 2.17/1.45 @ ( dsetconstr @ A
% 2.17/1.45 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.17/1.45 => ( Xphi @ Xx ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(exuE1_type,type,
% 2.17/1.45 exuE1: $o ).
% 2.17/1.45
% 2.17/1.45 thf(exuE1,definition,
% 2.17/1.45 ( exuE1
% 2.17/1.45 = ( ! [Xphi: $i > $o] :
% 2.17/1.45 ( ( exu
% 2.17/1.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.17/1.45 => ? [Xx: $i] :
% 2.17/1.45 ( ( Xphi @ Xx )
% 2.17/1.45 & ! [Xy: $i] :
% 2.17/1.45 ( ( Xphi @ Xy )
% 2.17/1.45 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(prop2set_type,type,
% 2.17/1.45 prop2set: $o > $i ).
% 2.17/1.45
% 2.17/1.45 thf(prop2setE_type,type,
% 2.17/1.45 prop2setE: $o ).
% 2.17/1.45
% 2.17/1.45 thf(prop2setE,definition,
% 2.17/1.45 ( prop2setE
% 2.17/1.45 = ( ! [Xphi: $o,Xx: $i] :
% 2.17/1.45 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 2.17/1.45 => Xphi ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(emptysetE_type,type,
% 2.17/1.45 emptysetE: $o ).
% 2.17/1.45
% 2.17/1.45 thf(emptysetE,definition,
% 2.17/1.45 ( emptysetE
% 2.17/1.45 = ( ! [Xx: $i] :
% 2.17/1.45 ( ( in @ Xx @ emptyset )
% 2.17/1.45 => ! [Xphi: $o] : Xphi ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(emptysetimpfalse_type,type,
% 2.17/1.45 emptysetimpfalse: $o ).
% 2.17/1.45
% 2.17/1.45 thf(emptysetimpfalse,definition,
% 2.17/1.45 ( emptysetimpfalse
% 2.17/1.45 = ( ! [Xx: $i] :
% 2.17/1.45 ( ( in @ Xx @ emptyset )
% 2.17/1.45 => $false ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(notinemptyset_type,type,
% 2.17/1.45 notinemptyset: $o ).
% 2.17/1.45
% 2.17/1.45 thf(notinemptyset,definition,
% 2.17/1.45 ( notinemptyset
% 2.17/1.45 = ( ! [Xx: $i] :
% 2.17/1.45 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(exuE3e_type,type,
% 2.17/1.45 exuE3e: $o ).
% 2.17/1.45
% 2.17/1.45 thf(exuE3e,definition,
% 2.17/1.45 ( exuE3e
% 2.17/1.45 = ( ! [Xphi: $i > $o] :
% 2.17/1.45 ( ( exu
% 2.17/1.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.17/1.45 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(setext_type,type,
% 2.17/1.45 setext: $o ).
% 2.17/1.45
% 2.17/1.45 thf(setext,definition,
% 2.17/1.45 ( setext
% 2.17/1.45 = ( ! [A: $i,B: $i] :
% 2.17/1.45 ( ! [Xx: $i] :
% 2.17/1.45 ( ( in @ Xx @ A )
% 2.17/1.45 => ( in @ Xx @ B ) )
% 2.17/1.45 => ( ! [Xx: $i] :
% 2.17/1.45 ( ( in @ Xx @ B )
% 2.17/1.45 => ( in @ Xx @ A ) )
% 2.17/1.45 => ( A = B ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(emptyI_type,type,
% 2.17/1.45 emptyI: $o ).
% 2.17/1.45
% 2.17/1.45 thf(emptyI,definition,
% 2.17/1.45 ( emptyI
% 2.17/1.45 = ( ! [A: $i] :
% 2.17/1.45 ( ! [Xx: $i] :
% 2.17/1.45 ~ ( in @ Xx @ A )
% 2.17/1.45 => ( A = emptyset ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(noeltsimpempty_type,type,
% 2.17/1.45 noeltsimpempty: $o ).
% 2.17/1.45
% 2.17/1.45 thf(noeltsimpempty,definition,
% 2.17/1.45 ( noeltsimpempty
% 2.17/1.45 = ( ! [A: $i] :
% 2.17/1.45 ( ! [Xx: $i] :
% 2.17/1.45 ~ ( in @ Xx @ A )
% 2.17/1.45 => ( A = emptyset ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(setbeta_type,type,
% 2.17/1.45 setbeta: $o ).
% 2.17/1.45
% 2.17/1.45 thf(setbeta,definition,
% 2.17/1.45 ( setbeta
% 2.17/1.45 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.17/1.45 ( ( in @ Xx @ A )
% 2.17/1.45 => ( ( in @ Xx
% 2.17/1.45 @ ( dsetconstr @ A
% 2.17/1.45 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.17/1.45 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(nonempty_type,type,
% 2.17/1.45 nonempty: $i > $o ).
% 2.17/1.45
% 2.17/1.45 thf(nonempty,definition,
% 2.17/1.45 ( nonempty
% 2.17/1.45 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(nonemptyE1_type,type,
% 2.17/1.45 nonemptyE1: $o ).
% 2.17/1.45
% 2.17/1.45 thf(nonemptyE1,definition,
% 2.17/1.45 ( nonemptyE1
% 2.17/1.45 = ( ! [A: $i] :
% 2.17/1.45 ( ( nonempty @ A )
% 2.17/1.45 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(nonemptyI_type,type,
% 2.17/1.45 nonemptyI: $o ).
% 2.17/1.45
% 2.17/1.45 thf(nonemptyI,definition,
% 2.17/1.45 ( nonemptyI
% 2.17/1.45 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.17/1.45 ( ( in @ Xx @ A )
% 2.17/1.45 => ( ( Xphi @ Xx )
% 2.17/1.45 => ( nonempty
% 2.17/1.45 @ ( dsetconstr @ A
% 2.17/1.45 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(nonemptyI1_type,type,
% 2.17/1.45 nonemptyI1: $o ).
% 2.17/1.45
% 2.17/1.45 thf(nonemptyI1,definition,
% 2.17/1.45 ( nonemptyI1
% 2.17/1.45 = ( ! [A: $i] :
% 2.17/1.45 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.17/1.45 => ( nonempty @ A ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(setadjoinIL_type,type,
% 2.17/1.45 setadjoinIL: $o ).
% 2.17/1.45
% 2.17/1.45 thf(setadjoinIL,definition,
% 2.17/1.45 ( setadjoinIL
% 2.17/1.45 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(emptyinunitempty_type,type,
% 2.17/1.45 emptyinunitempty: $o ).
% 2.17/1.45
% 2.17/1.45 thf(emptyinunitempty,definition,
% 2.17/1.45 ( emptyinunitempty
% 2.17/1.45 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(setadjoinIR_type,type,
% 2.17/1.45 setadjoinIR: $o ).
% 2.17/1.45
% 2.17/1.45 thf(setadjoinIR,definition,
% 2.17/1.45 ( setadjoinIR
% 2.17/1.45 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.17/1.45 ( ( in @ Xy @ A )
% 2.17/1.45 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(setadjoinE_type,type,
% 2.17/1.45 setadjoinE: $o ).
% 2.17/1.45
% 2.17/1.45 thf(setadjoinE,definition,
% 2.17/1.45 ( setadjoinE
% 2.17/1.45 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.17/1.45 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.17/1.45 => ! [Xphi: $o] :
% 2.17/1.45 ( ( ( Xy = Xx )
% 2.17/1.45 => Xphi )
% 2.17/1.45 => ( ( ( in @ Xy @ A )
% 2.17/1.45 => Xphi )
% 2.17/1.45 => Xphi ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(setadjoinOr_type,type,
% 2.17/1.45 setadjoinOr: $o ).
% 2.17/1.45
% 2.17/1.45 thf(setadjoinOr,definition,
% 2.17/1.45 ( setadjoinOr
% 2.17/1.45 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.17/1.45 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.17/1.45 => ( ( Xy = Xx )
% 2.17/1.45 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(setoftrueEq_type,type,
% 2.17/1.45 setoftrueEq: $o ).
% 2.17/1.45
% 2.17/1.45 thf(setoftrueEq,definition,
% 2.17/1.45 ( setoftrueEq
% 2.17/1.45 = ( ! [A: $i] :
% 2.17/1.45 ( ( dsetconstr @ A
% 2.17/1.45 @ ^ [Xx: $i] : $true )
% 2.17/1.45 = A ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(powersetI_type,type,
% 2.17/1.45 powersetI: $o ).
% 2.17/1.45
% 2.17/1.45 thf(powersetI,definition,
% 2.17/1.45 ( powersetI
% 2.17/1.45 = ( ! [A: $i,B: $i] :
% 2.17/1.45 ( ! [Xx: $i] :
% 2.17/1.45 ( ( in @ Xx @ B )
% 2.17/1.45 => ( in @ Xx @ A ) )
% 2.17/1.45 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(emptyinPowerset_type,type,
% 2.17/1.45 emptyinPowerset: $o ).
% 2.17/1.45
% 2.17/1.45 thf(emptyinPowerset,definition,
% 2.17/1.45 ( emptyinPowerset
% 2.17/1.45 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(emptyInPowerset_type,type,
% 2.17/1.45 emptyInPowerset: $o ).
% 2.17/1.45
% 2.17/1.45 thf(emptyInPowerset,definition,
% 2.17/1.45 ( emptyInPowerset
% 2.17/1.45 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(powersetE_type,type,
% 2.17/1.45 powersetE: $o ).
% 2.17/1.45
% 2.17/1.45 thf(powersetE,definition,
% 2.17/1.45 ( powersetE
% 2.17/1.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.17/1.45 ( ( in @ B @ ( powerset @ A ) )
% 2.17/1.45 => ( ( in @ Xx @ B )
% 2.17/1.45 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(setunionI_type,type,
% 2.17/1.45 setunionI: $o ).
% 2.17/1.45
% 2.17/1.45 thf(setunionI,definition,
% 2.17/1.45 ( setunionI
% 2.17/1.45 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.17/1.45 ( ( in @ Xx @ B )
% 2.17/1.45 => ( ( in @ B @ A )
% 2.17/1.45 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(setunionE_type,type,
% 2.17/1.45 setunionE: $o ).
% 2.17/1.45
% 2.17/1.45 thf(setunionE,definition,
% 2.17/1.45 ( setunionE
% 2.17/1.45 = ( ! [A: $i,Xx: $i] :
% 2.17/1.45 ( ( in @ Xx @ ( setunion @ A ) )
% 2.17/1.45 => ! [Xphi: $o] :
% 2.17/1.45 ( ! [B: $i] :
% 2.17/1.45 ( ( in @ Xx @ B )
% 2.17/1.45 => ( ( in @ B @ A )
% 2.17/1.45 => Xphi ) )
% 2.17/1.45 => Xphi ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(subPowSU_type,type,
% 2.17/1.45 subPowSU: $o ).
% 2.17/1.45
% 2.17/1.45 thf(subPowSU,definition,
% 2.17/1.45 ( subPowSU
% 2.17/1.45 = ( ! [A: $i,Xx: $i] :
% 2.17/1.45 ( ( in @ Xx @ A )
% 2.17/1.45 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(exuE2_type,type,
% 2.17/1.45 exuE2: $o ).
% 2.17/1.45
% 2.17/1.45 thf(exuE2,definition,
% 2.17/1.45 ( exuE2
% 2.17/1.45 = ( ! [Xphi: $i > $o] :
% 2.17/1.45 ( ( exu
% 2.17/1.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.17/1.45 => ? [Xx: $i] :
% 2.17/1.45 ! [Xy: $i] :
% 2.17/1.45 ( ( Xphi @ Xy )
% 2.17/1.45 <=> ( Xy = Xx ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(nonemptyImpWitness_type,type,
% 2.17/1.45 nonemptyImpWitness: $o ).
% 2.17/1.45
% 2.17/1.45 thf(nonemptyImpWitness,definition,
% 2.17/1.45 ( nonemptyImpWitness
% 2.17/1.45 = ( ! [A: $i] :
% 2.17/1.45 ( ( nonempty @ A )
% 2.17/1.45 => ? [Xx: $i] :
% 2.17/1.45 ( ( in @ Xx @ A )
% 2.17/1.45 & $true ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(uniqinunit_type,type,
% 2.17/1.45 uniqinunit: $o ).
% 2.17/1.45
% 2.17/1.45 thf(uniqinunit,definition,
% 2.17/1.45 ( uniqinunit
% 2.17/1.45 = ( ! [Xx: $i,Xy: $i] :
% 2.17/1.45 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.17/1.45 => ( Xx = Xy ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(notinsingleton_type,type,
% 2.17/1.45 notinsingleton: $o ).
% 2.17/1.45
% 2.17/1.45 thf(notinsingleton,definition,
% 2.17/1.45 ( notinsingleton
% 2.17/1.45 = ( ! [Xx: $i,Xy: $i] :
% 2.17/1.45 ( ( Xx != Xy )
% 2.17/1.45 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(eqinunit_type,type,
% 2.17/1.45 eqinunit: $o ).
% 2.17/1.45
% 2.17/1.45 thf(eqinunit,definition,
% 2.17/1.45 ( eqinunit
% 2.17/1.45 = ( ! [Xx: $i,Xy: $i] :
% 2.17/1.45 ( ( Xx = Xy )
% 2.17/1.45 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(singletonsswitch_type,type,
% 2.17/1.45 singletonsswitch: $o ).
% 2.17/1.45
% 2.17/1.45 thf(singletonsswitch,definition,
% 2.17/1.45 ( singletonsswitch
% 2.17/1.45 = ( ! [Xx: $i,Xy: $i] :
% 2.17/1.45 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.17/1.45 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(upairsetE_type,type,
% 2.17/1.45 upairsetE: $o ).
% 2.17/1.45
% 2.17/1.45 thf(upairsetE,definition,
% 2.17/1.45 ( upairsetE
% 2.17/1.45 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.17/1.45 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.17/1.45 => ( ( Xz = Xx )
% 2.17/1.45 | ( Xz = Xy ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(upairsetIL_type,type,
% 2.17/1.45 upairsetIL: $o ).
% 2.17/1.45
% 2.17/1.45 thf(upairsetIL,definition,
% 2.17/1.45 ( upairsetIL
% 2.17/1.45 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(upairsetIR_type,type,
% 2.17/1.45 upairsetIR: $o ).
% 2.17/1.45
% 2.17/1.45 thf(upairsetIR,definition,
% 2.17/1.45 ( upairsetIR
% 2.17/1.45 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(emptyE1_type,type,
% 2.17/1.45 emptyE1: $o ).
% 2.17/1.45
% 2.17/1.45 thf(emptyE1,definition,
% 2.17/1.45 ( emptyE1
% 2.17/1.45 = ( ! [A: $i,Xphi: $i > $o] :
% 2.17/1.45 ( ? [Xx: $i] :
% 2.17/1.45 ( ( in @ Xx @ A )
% 2.17/1.45 & ( Xphi @ Xx ) )
% 2.17/1.45 => ( ( ( dsetconstr @ A
% 2.17/1.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.17/1.45 = emptyset )
% 2.17/1.45 => $false ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(vacuousDall_type,type,
% 2.17/1.45 vacuousDall: $o ).
% 2.17/1.45
% 2.17/1.45 thf(vacuousDall,definition,
% 2.17/1.45 ( vacuousDall
% 2.17/1.45 = ( ! [Xphi: $i > $o,Xx: $i] :
% 2.17/1.45 ( ( in @ Xx @ emptyset )
% 2.17/1.45 => ( Xphi @ Xx ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(quantDeMorgan1_type,type,
% 2.17/1.45 quantDeMorgan1: $o ).
% 2.17/1.45
% 2.17/1.45 thf(quantDeMorgan1,definition,
% 2.17/1.45 ( quantDeMorgan1
% 2.17/1.45 = ( ! [A: $i,Xphi: $i > $o] :
% 2.17/1.45 ( ~ ! [Xx: $i] :
% 2.17/1.45 ( ( in @ Xx @ A )
% 2.17/1.45 => ( Xphi @ Xx ) )
% 2.17/1.45 => ? [Xx: $i] :
% 2.17/1.45 ( ( in @ Xx @ A )
% 2.17/1.45 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(quantDeMorgan2_type,type,
% 2.17/1.45 quantDeMorgan2: $o ).
% 2.17/1.45
% 2.17/1.45 thf(quantDeMorgan2,definition,
% 2.17/1.45 ( quantDeMorgan2
% 2.17/1.45 = ( ! [A: $i,Xphi: $i > $o] :
% 2.17/1.45 ( ! [Xx: $i] :
% 2.17/1.45 ( ( in @ Xx @ A )
% 2.17/1.45 => ~ ( Xphi @ Xx ) )
% 2.17/1.45 => ~ ? [Xx: $i] :
% 2.17/1.45 ( ( in @ Xx @ A )
% 2.17/1.45 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(quantDeMorgan3_type,type,
% 2.17/1.45 quantDeMorgan3: $o ).
% 2.17/1.45
% 2.17/1.45 thf(quantDeMorgan3,definition,
% 2.17/1.45 ( quantDeMorgan3
% 2.17/1.45 = ( ! [A: $i,Xphi: $i > $o] :
% 2.17/1.45 ( ~ ? [Xx: $i] :
% 2.17/1.45 ( ( in @ Xx @ A )
% 2.17/1.45 & ( Xphi @ Xx ) )
% 2.17/1.45 => ! [Xx: $i] :
% 2.17/1.45 ( ( in @ Xx @ A )
% 2.17/1.45 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(quantDeMorgan4_type,type,
% 2.17/1.45 quantDeMorgan4: $o ).
% 2.17/1.45
% 2.17/1.45 thf(quantDeMorgan4,definition,
% 2.17/1.45 ( quantDeMorgan4
% 2.17/1.45 = ( ! [A: $i,Xphi: $i > $o] :
% 2.17/1.45 ( ? [Xx: $i] :
% 2.17/1.45 ( ( in @ Xx @ A )
% 2.17/1.45 & ~ ( Xphi @ Xx ) )
% 2.17/1.45 => ~ ! [Xx: $i] :
% 2.17/1.45 ( ( in @ Xx @ A )
% 2.17/1.45 => ( Xphi @ Xx ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(prop2setI_type,type,
% 2.17/1.45 prop2setI: $o ).
% 2.17/1.45
% 2.17/1.45 thf(prop2setI,definition,
% 2.17/1.45 ( prop2setI
% 2.17/1.45 = ( ! [Xphi: $o] :
% 2.17/1.45 ( Xphi
% 2.17/1.45 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(set2prop_type,type,
% 2.17/1.45 set2prop: $i > $o ).
% 2.17/1.45
% 2.17/1.45 thf(prop2set2propI_type,type,
% 2.17/1.45 prop2set2propI: $o ).
% 2.17/1.45
% 2.17/1.45 thf(prop2set2propI,definition,
% 2.17/1.45 ( prop2set2propI
% 2.17/1.45 = ( ! [Xphi: $o] :
% 2.17/1.45 ( Xphi
% 2.17/1.45 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(notdexE_type,type,
% 2.17/1.45 notdexE: $o ).
% 2.17/1.45
% 2.17/1.45 thf(notdexE,definition,
% 2.17/1.45 ( notdexE
% 2.17/1.45 = ( ! [A: $i,Xphi: $i > $o] :
% 2.17/1.45 ( ~ ? [Xx: $i] :
% 2.17/1.45 ( ( in @ Xx @ A )
% 2.17/1.45 & ( Xphi @ Xx ) )
% 2.17/1.45 => ! [Xx: $i] :
% 2.17/1.45 ( ( in @ Xx @ A )
% 2.17/1.45 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(notdallE_type,type,
% 2.17/1.45 notdallE: $o ).
% 2.17/1.45
% 2.17/1.45 thf(notdallE,definition,
% 2.17/1.45 ( notdallE
% 2.17/1.45 = ( ! [A: $i,Xphi: $i > $o] :
% 2.17/1.45 ( ~ ! [Xx: $i] :
% 2.17/1.45 ( ( in @ Xx @ A )
% 2.17/1.45 => ( Xphi @ Xx ) )
% 2.17/1.45 => ? [Xx: $i] :
% 2.17/1.45 ( ( in @ Xx @ A )
% 2.17/1.45 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(exuI1_type,type,
% 2.17/1.45 exuI1: $o ).
% 2.17/1.45
% 2.17/1.45 thf(exuI1,definition,
% 2.17/1.45 ( exuI1
% 2.17/1.45 = ( ! [Xphi: $i > $o] :
% 2.17/1.45 ( ? [Xx: $i] :
% 2.17/1.45 ( ( Xphi @ Xx )
% 2.17/1.45 & ! [Xy: $i] :
% 2.17/1.45 ( ( Xphi @ Xy )
% 2.17/1.45 => ( Xx = Xy ) ) )
% 2.17/1.45 => ( exu
% 2.17/1.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(exuI3_type,type,
% 2.17/1.45 exuI3: $o ).
% 2.17/1.45
% 2.17/1.45 thf(exuI3,definition,
% 2.17/1.45 ( exuI3
% 2.17/1.45 = ( ! [Xphi: $i > $o] :
% 2.17/1.45 ( ? [Xx: $i] : ( Xphi @ Xx )
% 2.17/1.45 => ( ! [Xx: $i,Xy: $i] :
% 2.17/1.45 ( ( Xphi @ Xx )
% 2.17/1.45 => ( ( Xphi @ Xy )
% 2.17/1.45 => ( Xx = Xy ) ) )
% 2.17/1.45 => ( exu
% 2.17/1.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(exuI2_type,type,
% 2.17/1.45 exuI2: $o ).
% 2.17/1.45
% 2.17/1.45 thf(exuI2,definition,
% 2.17/1.45 ( exuI2
% 2.17/1.45 = ( ! [Xphi: $i > $o] :
% 2.17/1.45 ( ? [Xx: $i] :
% 2.17/1.45 ! [Xy: $i] :
% 2.17/1.45 ( ( Xphi @ Xy )
% 2.17/1.45 <=> ( Xy = Xx ) )
% 2.17/1.45 => ( exu
% 2.17/1.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(inCongP_type,type,
% 2.17/1.45 inCongP: $o ).
% 2.17/1.45
% 2.17/1.45 thf(inCongP,definition,
% 2.17/1.45 ( inCongP
% 2.17/1.45 = ( ! [A: $i,B: $i] :
% 2.17/1.45 ( ( A = B )
% 2.17/1.45 => ! [Xx: $i,Xy: $i] :
% 2.17/1.45 ( ( Xx = Xy )
% 2.17/1.45 => ( ( in @ Xx @ A )
% 2.17/1.45 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(in__Cong_type,type,
% 2.17/1.45 in__Cong: $o ).
% 2.17/1.45
% 2.17/1.45 thf(in__Cong,definition,
% 2.17/1.45 ( in__Cong
% 2.17/1.45 = ( ! [A: $i,B: $i] :
% 2.17/1.45 ( ( A = B )
% 2.17/1.45 => ! [Xx: $i,Xy: $i] :
% 2.17/1.45 ( ( Xx = Xy )
% 2.17/1.45 => ( ( in @ Xx @ A )
% 2.17/1.45 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(exuE3u_type,type,
% 2.17/1.45 exuE3u: $o ).
% 2.17/1.45
% 2.17/1.45 thf(exuE3u,definition,
% 2.17/1.45 ( exuE3u
% 2.17/1.45 = ( ! [Xphi: $i > $o] :
% 2.17/1.45 ( ( exu
% 2.17/1.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.17/1.45 => ! [Xx: $i,Xy: $i] :
% 2.17/1.45 ( ( Xphi @ Xx )
% 2.17/1.45 => ( ( Xphi @ Xy )
% 2.17/1.45 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(exu__Cong_type,type,
% 2.17/1.45 exu__Cong: $o ).
% 2.17/1.45
% 2.17/1.45 thf(exu__Cong,definition,
% 2.17/1.45 ( exu__Cong
% 2.17/1.45 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.17/1.45 ( ! [Xx: $i,Xy: $i] :
% 2.17/1.45 ( ( Xx = Xy )
% 2.17/1.45 => ( ( Xphi @ Xx )
% 2.17/1.45 <=> ( Xpsi @ Xy ) ) )
% 2.17/1.45 => ( ( exu
% 2.17/1.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.17/1.45 <=> ( exu
% 2.17/1.45 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(emptyset__Cong_type,type,
% 2.17/1.45 emptyset__Cong: $o ).
% 2.17/1.45
% 2.17/1.45 thf(emptyset__Cong,definition,
% 2.17/1.45 ( emptyset__Cong
% 2.17/1.45 = ( emptyset = emptyset ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(setadjoin__Cong_type,type,
% 2.17/1.45 setadjoin__Cong: $o ).
% 2.17/1.45
% 2.17/1.45 thf(setadjoin__Cong,definition,
% 2.17/1.45 ( setadjoin__Cong
% 2.17/1.45 = ( ! [Xx: $i,Xy: $i] :
% 2.17/1.45 ( ( Xx = Xy )
% 2.17/1.45 => ! [Xz: $i,Xu: $i] :
% 2.17/1.45 ( ( Xz = Xu )
% 2.17/1.45 => ( ( setadjoin @ Xx @ Xz )
% 2.17/1.45 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(powerset__Cong_type,type,
% 2.17/1.45 powerset__Cong: $o ).
% 2.17/1.45
% 2.17/1.45 thf(powerset__Cong,definition,
% 2.17/1.45 ( powerset__Cong
% 2.17/1.45 = ( ! [A: $i,B: $i] :
% 2.17/1.45 ( ( A = B )
% 2.17/1.45 => ( ( powerset @ A )
% 2.17/1.45 = ( powerset @ B ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(setunion__Cong_type,type,
% 2.17/1.45 setunion__Cong: $o ).
% 2.17/1.45
% 2.17/1.45 thf(setunion__Cong,definition,
% 2.17/1.45 ( setunion__Cong
% 2.17/1.45 = ( ! [A: $i,B: $i] :
% 2.17/1.45 ( ( A = B )
% 2.17/1.45 => ( ( setunion @ A )
% 2.17/1.45 = ( setunion @ B ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(omega__Cong_type,type,
% 2.17/1.45 omega__Cong: $o ).
% 2.17/1.45
% 2.17/1.45 thf(omega__Cong,definition,
% 2.17/1.45 ( omega__Cong
% 2.17/1.45 = ( omega = omega ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(exuEu_type,type,
% 2.17/1.45 exuEu: $o ).
% 2.17/1.45
% 2.17/1.45 thf(exuEu,definition,
% 2.17/1.45 ( exuEu
% 2.17/1.45 = ( ! [Xphi: $i > $o] :
% 2.17/1.45 ( ( exu
% 2.17/1.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.17/1.45 => ! [Xx: $i,Xy: $i] :
% 2.17/1.45 ( ( Xphi @ Xx )
% 2.17/1.45 => ( ( Xphi @ Xy )
% 2.17/1.45 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(descr__Cong_type,type,
% 2.17/1.45 descr__Cong: $o ).
% 2.17/1.45
% 2.17/1.45 thf(descr__Cong,definition,
% 2.17/1.45 ( descr__Cong
% 2.17/1.45 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.17/1.45 ( ! [Xx: $i,Xy: $i] :
% 2.17/1.45 ( ( Xx = Xy )
% 2.17/1.45 => ( ( Xphi @ Xx )
% 2.17/1.45 <=> ( Xpsi @ Xy ) ) )
% 2.17/1.45 => ( ( exu
% 2.17/1.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.17/1.45 => ( ( exu
% 2.17/1.45 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 2.17/1.45 => ( ( descr
% 2.17/1.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.17/1.45 = ( descr
% 2.17/1.45 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(dsetconstr__Cong_type,type,
% 2.17/1.45 dsetconstr__Cong: $o ).
% 2.17/1.45
% 2.17/1.45 thf(dsetconstr__Cong,definition,
% 2.17/1.45 ( dsetconstr__Cong
% 2.17/1.45 = ( ! [A: $i,B: $i] :
% 2.17/1.45 ( ( A = B )
% 2.17/1.45 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.17/1.45 ( ! [Xx: $i] :
% 2.17/1.45 ( ( in @ Xx @ A )
% 2.17/1.45 => ! [Xy: $i] :
% 2.17/1.45 ( ( in @ Xy @ B )
% 2.17/1.45 => ( ( Xx = Xy )
% 2.17/1.45 => ( ( Xphi @ Xx )
% 2.17/1.45 <=> ( Xpsi @ Xy ) ) ) ) )
% 2.17/1.45 => ( ( dsetconstr @ A
% 2.17/1.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.17/1.45 = ( dsetconstr @ B
% 2.17/1.45 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(subset_type,type,
% 2.17/1.45 subset: $i > $i > $o ).
% 2.17/1.45
% 2.17/1.45 thf(subset,definition,
% 2.17/1.45 ( subset
% 2.17/1.45 = ( ^ [A: $i,B: $i] :
% 2.17/1.45 ! [Xx: $i] :
% 2.17/1.45 ( ( in @ Xx @ A )
% 2.17/1.45 => ( in @ Xx @ B ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(disjoint_type,type,
% 2.17/1.45 disjoint: $i > $i > $o ).
% 2.17/1.45
% 2.17/1.45 thf(setsmeet_type,type,
% 2.17/1.45 setsmeet: $i > $i > $o ).
% 2.17/1.45
% 2.17/1.45 thf(subsetI1_type,type,
% 2.17/1.45 subsetI1: $o ).
% 2.17/1.45
% 2.17/1.45 thf(subsetI1,definition,
% 2.17/1.45 ( subsetI1
% 2.17/1.45 = ( ! [A: $i,B: $i] :
% 2.17/1.45 ( ! [Xx: $i] :
% 2.17/1.45 ( ( in @ Xx @ A )
% 2.17/1.45 => ( in @ Xx @ B ) )
% 2.17/1.45 => ( subset @ A @ B ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(eqimpsubset2_type,type,
% 2.17/1.45 eqimpsubset2: $o ).
% 2.17/1.45
% 2.17/1.45 thf(eqimpsubset2,definition,
% 2.17/1.45 ( eqimpsubset2
% 2.17/1.45 = ( ! [A: $i,B: $i] :
% 2.17/1.45 ( ( A = B )
% 2.17/1.45 => ( subset @ B @ A ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(eqimpsubset1_type,type,
% 2.17/1.45 eqimpsubset1: $o ).
% 2.17/1.45
% 2.17/1.45 thf(eqimpsubset1,definition,
% 2.17/1.45 ( eqimpsubset1
% 2.17/1.45 = ( ! [A: $i,B: $i] :
% 2.17/1.45 ( ( A = B )
% 2.17/1.45 => ( subset @ A @ B ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(subsetI2,conjecture,
% 2.17/1.45 ( setextAx
% 2.17/1.45 => ( emptysetAx
% 2.17/1.45 => ( setadjoinAx
% 2.17/1.45 => ( powersetAx
% 2.17/1.45 => ( setunionAx
% 2.17/1.45 => ( omega0Ax
% 2.17/1.45 => ( omegaSAx
% 2.17/1.45 => ( omegaIndAx
% 2.17/1.45 => ( replAx
% 2.17/1.45 => ( foundationAx
% 2.17/1.45 => ( wellorderingAx
% 2.17/1.45 => ( descrp
% 2.17/1.45 => ( dsetconstrI
% 2.17/1.45 => ( dsetconstrEL
% 2.17/1.45 => ( dsetconstrER
% 2.17/1.45 => ( exuE1
% 2.17/1.45 => ( prop2setE
% 2.17/1.45 => ( emptysetE
% 2.17/1.45 => ( emptysetimpfalse
% 2.17/1.45 => ( notinemptyset
% 2.17/1.45 => ( exuE3e
% 2.17/1.45 => ( setext
% 2.17/1.45 => ( emptyI
% 2.17/1.45 => ( noeltsimpempty
% 2.17/1.45 => ( setbeta
% 2.17/1.45 => ( nonemptyE1
% 2.17/1.45 => ( nonemptyI
% 2.17/1.45 => ( nonemptyI1
% 2.17/1.45 => ( setadjoinIL
% 2.17/1.45 => ( emptyinunitempty
% 2.17/1.45 => ( setadjoinIR
% 2.17/1.45 => ( setadjoinE
% 2.17/1.45 => ( setadjoinOr
% 2.17/1.45 => ( setoftrueEq
% 2.17/1.45 => ( powersetI
% 2.17/1.45 => ( emptyinPowerset
% 2.17/1.45 => ( emptyInPowerset
% 2.17/1.45 => ( powersetE
% 2.17/1.45 => ( setunionI
% 2.17/1.45 => ( setunionE
% 2.17/1.45 => ( subPowSU
% 2.17/1.45 => ( exuE2
% 2.17/1.45 => ( nonemptyImpWitness
% 2.17/1.45 => ( uniqinunit
% 2.17/1.45 => ( notinsingleton
% 2.17/1.45 => ( eqinunit
% 2.17/1.45 => ( singletonsswitch
% 2.17/1.45 => ( upairsetE
% 2.17/1.45 => ( upairsetIL
% 2.17/1.45 => ( upairsetIR
% 2.17/1.45 => ( emptyE1
% 2.17/1.45 => ( vacuousDall
% 2.17/1.45 => ( quantDeMorgan1
% 2.17/1.45 => ( quantDeMorgan2
% 2.17/1.45 => ( quantDeMorgan3
% 2.17/1.45 => ( quantDeMorgan4
% 2.17/1.45 => ( prop2setI
% 2.17/1.45 => ( prop2set2propI
% 2.17/1.45 => ( notdexE
% 2.17/1.45 => ( notdallE
% 2.17/1.45 => ( exuI1
% 2.17/1.45 => ( exuI3
% 2.17/1.45 => ( exuI2
% 2.17/1.45 => ( inCongP
% 2.17/1.45 => ( in__Cong
% 2.17/1.45 => ( exuE3u
% 2.17/1.45 => ( exu__Cong
% 2.17/1.45 => ( emptyset__Cong
% 2.17/1.45 => ( setadjoin__Cong
% 2.17/1.45 => ( powerset__Cong
% 2.17/1.45 => ( setunion__Cong
% 2.17/1.45 => ( omega__Cong
% 2.17/1.45 => ( exuEu
% 2.17/1.45 => ( descr__Cong
% 2.17/1.45 => ( dsetconstr__Cong
% 2.17/1.45 => ( subsetI1
% 2.17/1.45 => ( eqimpsubset2
% 2.17/1.45 => ( eqimpsubset1
% 2.17/1.45 => ! [A: $i,B: $i] :
% 2.17/1.45 ( ! [Xx: $i] :
% 2.17/1.45 ( ( in @ Xx @ A )
% 2.17/1.45 => ( in @ Xx @ B ) )
% 2.17/1.45 => ( subset @ A @ B ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.17/1.45 ------------------------
% 2.17/1.46 % (20991)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_20822 for (2999ds/2Mi)
% 2.17/1.46 % (20988)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_20822 for (2999ds/183Mi)
% 2.17/1.46 % (20991)Instruction limit reached!
% 2.17/1.46 % (20991)------------------------------
% 2.17/1.46 % (20991)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.17/1.46 % (20991)Termination reason: Unknown
% 2.17/1.46 % (20991)Termination phase: shuffling
% 2.17/1.46
% 2.17/1.46 % (20991)Memory used [KB]: 1151
% 2.17/1.46 % (20991)Time elapsed: 0.002 s
% 2.17/1.46 % (20991)Instructions burned: 3 (million)
% 2.17/1.46 % (20991)------------------------------
% 2.17/1.46 % (20991)------------------------------
% 2.17/1.46 % (20990)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_20822 for (2999ds/27Mi)
% 2.17/1.46 % (20992)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_20822 for (2999ds/2Mi)
% 2.17/1.46 % (20992)Instruction limit reached!
% 2.17/1.46 % (20992)------------------------------
% 2.17/1.46 % (20992)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.17/1.46 % (20992)Termination reason: Unknown
% 2.17/1.46 % (20992)Termination phase: shuffling
% 2.17/1.46
% 2.17/1.46 % (20992)Memory used [KB]: 1151
% 2.17/1.46 % (20992)Time elapsed: 0.002 s
% 2.17/1.46 % (20992)Instructions burned: 3 (million)
% 2.17/1.46 % (20992)------------------------------
% 2.17/1.46 % (20992)------------------------------
% 2.17/1.47 % (20993)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_20822 for (2999ds/275Mi)
% 2.17/1.47 % (20994)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_20822 for (2999ds/18Mi)
% 2.17/1.47 % (20989)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_20822 for (2999ds/4Mi)
% 2.17/1.47 % (20990)Instruction limit reached!
% 2.17/1.47 % (20990)------------------------------
% 2.17/1.47 % (20990)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.17/1.47 % (20990)Termination reason: Unknown
% 2.17/1.47 % (20990)Termination phase: Function definition elimination
% 2.17/1.47
% 2.17/1.47 % (20990)Memory used [KB]: 1407
% 2.17/1.47 % (20990)Time elapsed: 0.009 s
% 2.17/1.47 % (20990)Instructions burned: 29 (million)
% 2.17/1.47 % (20990)------------------------------
% 2.17/1.47 % (20990)------------------------------
% 2.17/1.47 % (20989)Instruction limit reached!
% 2.17/1.47 % (20989)------------------------------
% 2.17/1.47 % (20989)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.17/1.47 % (20989)Termination reason: Unknown
% 2.17/1.47 % (20989)Termination phase: shuffling
% 2.17/1.47
% 2.17/1.47 % (20989)Memory used [KB]: 1151
% 2.17/1.47 % (20989)Time elapsed: 0.004 s
% 2.17/1.47 % (20989)Instructions burned: 4 (million)
% 2.17/1.47 % (20989)------------------------------
% 2.17/1.47 % (20989)------------------------------
% 2.17/1.47 % (20995)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_20822 for (2999ds/3Mi)
% 2.17/1.47 % (20995)Instruction limit reached!
% 2.17/1.47 % (20995)------------------------------
% 2.17/1.47 % (20995)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.17/1.47 % (20995)Termination reason: Unknown
% 2.17/1.47 % (20995)Termination phase: shuffling
% 2.17/1.47
% 2.17/1.47 % (20995)Memory used [KB]: 1151
% 2.17/1.47 % (20995)Time elapsed: 0.002 s
% 2.17/1.47 % (20995)Instructions burned: 4 (million)
% 2.17/1.47 % (20995)------------------------------
% 2.17/1.47 % (20995)------------------------------
% 2.17/1.47 % (20996)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_20822 for (2999ds/37Mi)
% 2.17/1.48 % (20994)Instruction limit reached!
% 2.17/1.48 % (20994)------------------------------
% 2.17/1.48 % (20994)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.17/1.48 % (20994)Termination reason: Unknown
% 2.17/1.48 % (20994)Termination phase: Preprocessing 3
% 2.17/1.48
% 2.17/1.48 % (20994)Memory used [KB]: 1535
% 2.17/1.48 % (20994)Time elapsed: 0.013 s
% 2.17/1.48 % (20994)Instructions burned: 19 (million)
% 2.17/1.48 % (20994)------------------------------
% 2.17/1.48 % (20994)------------------------------
% 2.17/1.48 % (20997)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_20822 for (2999ds/15Mi)
% 2.17/1.48 % (20999)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_20822 for (2999ds/1041Mi)
% 2.17/1.48 % (20996)Instruction limit reached!
% 2.17/1.48 % (20996)------------------------------
% 2.17/1.48 % (20996)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.17/1.48 % (20996)Termination reason: Unknown
% 2.17/1.48 % (20996)Termination phase: Property scanning
% 2.17/1.48
% 2.17/1.48 % (20996)Memory used [KB]: 1407
% 2.17/1.48 % (20996)Time elapsed: 0.012 s
% 2.17/1.48 % (20996)Instructions burned: 38 (million)
% 2.17/1.48 % (20996)------------------------------
% 2.17/1.48 % (20996)------------------------------
% 2.17/1.49 % (20997)Instruction limit reached!
% 2.17/1.49 % (20997)------------------------------
% 2.17/1.49 % (20997)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.17/1.49 % (20997)Termination reason: Unknown
% 2.17/1.49 % (20997)Termination phase: Preprocessing 3
% 2.17/1.49
% 2.17/1.49 % (20997)Memory used [KB]: 1535
% 2.17/1.49 % (20997)Time elapsed: 0.007 s
% 2.17/1.49 % (20997)Instructions burned: 16 (million)
% 2.17/1.49 % (20997)------------------------------
% 2.17/1.49 % (20997)------------------------------
% 2.17/1.49 % (20998)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_20822 for (2999ds/3Mi)
% 2.17/1.49 % (20998)Instruction limit reached!
% 2.17/1.49 % (20998)------------------------------
% 2.17/1.49 % (20998)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.17/1.49 % (20998)Termination reason: Unknown
% 2.17/1.49 % (20998)Termination phase: shuffling
% 2.17/1.49
% 2.17/1.49 % (20998)Memory used [KB]: 1151
% 2.17/1.49 % (20998)Time elapsed: 0.004 s
% 2.17/1.49 % (20998)Instructions burned: 4 (million)
% 2.17/1.49 % (20998)------------------------------
% 2.17/1.49 % (20998)------------------------------
% 2.17/1.49 % (21001)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_20822 for (2999ds/16Mi)
% 2.17/1.50 % (21002)lrs+21_1:1_au=on:cnfonf=off:fd=preordered:fe=off:fsr=off:hud=11:inj=on:kws=precedence:s2pl=no:sp=weighted_frequency:tgt=full:i=3:si=on:rtra=on_0 on DTF2THF_20822 for (2999ds/3Mi)
% 2.26/1.50 % (21002)Instruction limit reached!
% 2.26/1.50 % (21002)------------------------------
% 2.26/1.50 % (21002)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.26/1.50 % (21002)Termination reason: Unknown
% 2.26/1.50 % (21002)Termination phase: shuffling
% 2.26/1.50
% 2.26/1.50 % (21002)Memory used [KB]: 1151
% 2.26/1.50 % (21002)Time elapsed: 0.003 s
% 2.26/1.50 % (21002)Instructions burned: 5 (million)
% 2.26/1.50 % (21002)------------------------------
% 2.26/1.50 % (21002)------------------------------
% 2.26/1.50 % (21000)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_20822 for (2999ds/7Mi)
% 2.26/1.50 % (21001)Instruction limit reached!
% 2.26/1.50 % (21001)------------------------------
% 2.26/1.50 % (21001)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.26/1.50 % (21001)Termination reason: Unknown
% 2.26/1.50 % (21001)Termination phase: Property scanning
% 2.26/1.50
% 2.26/1.50 % (21001)Memory used [KB]: 1407
% 2.26/1.50 % (21001)Time elapsed: 0.007 s
% 2.26/1.50 % (21001)Instructions burned: 19 (million)
% 2.26/1.50 % (21001)------------------------------
% 2.26/1.50 % (21001)------------------------------
% 2.26/1.50 % (21000)Instruction limit reached!
% 2.26/1.50 % (21000)------------------------------
% 2.26/1.50 % (21000)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.26/1.50 % (21000)Termination reason: Unknown
% 2.26/1.50 % (21000)Termination phase: shuffling
% 2.26/1.50
% 2.26/1.50 % (21000)Memory used [KB]: 1279
% 2.26/1.50 % (21000)Time elapsed: 0.006 s
% 2.26/1.50 % (21000)Instructions burned: 8 (million)
% 2.26/1.50 % (21000)------------------------------
% 2.26/1.50 % (21000)------------------------------
% 2.26/1.51 % (21004)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_20822 for (2999ds/7Mi)
% 2.26/1.51 % (21003)lrs+2_1:1_apa=on:au=on:bd=preordered:cnfonf=off:cs=on:ixr=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_20822 for (2999ds/3Mi)
% 2.26/1.51 % (21003)Instruction limit reached!
% 2.26/1.51 % (21003)------------------------------
% 2.26/1.51 % (21003)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.26/1.51 % (21003)Termination reason: Unknown
% 2.26/1.51 % (21003)Termination phase: shuffling
% 2.26/1.51
% 2.26/1.51 % (21003)Memory used [KB]: 1151
% 2.26/1.51 % (21003)Time elapsed: 0.003 s
% 2.26/1.51 % (21003)Instructions burned: 5 (million)
% 2.26/1.51 % (21003)------------------------------
% 2.26/1.51 % (21003)------------------------------
% 2.26/1.51 % (21004)Instruction limit reached!
% 2.26/1.51 % (21004)------------------------------
% 2.26/1.51 % (21004)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.26/1.51 % (21004)Termination reason: Unknown
% 2.26/1.51 % (21004)Termination phase: shuffling
% 2.26/1.51
% 2.26/1.51 % (21004)Memory used [KB]: 1279
% 2.26/1.51 % (21004)Time elapsed: 0.003 s
% 2.26/1.51 % (21004)Instructions burned: 8 (million)
% 2.26/1.51 % (21004)------------------------------
% 2.26/1.51 % (21004)------------------------------
% 2.26/1.51 % (21005)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_20822 for (2999ds/3Mi)
% 2.26/1.51 % (21005)Instruction limit reached!
% 2.26/1.51 % (21005)------------------------------
% 2.26/1.51 % (21005)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.26/1.51 % (21005)Termination reason: Unknown
% 2.26/1.51 % (21005)Termination phase: shuffling
% 2.26/1.51
% 2.26/1.51 % (21005)Memory used [KB]: 1151
% 2.26/1.51 % (21005)Time elapsed: 0.003 s
% 2.26/1.51 % (21005)Instructions burned: 5 (million)
% 2.26/1.51 % (21005)------------------------------
% 2.26/1.51 % (21005)------------------------------
% 2.26/1.52 % (21008)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_20822 for (2999ds/710Mi)
% 2.26/1.52 % (21006)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_20822 for (2999ds/4Mi)
% 2.26/1.52 % (21007)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_20822 for (2999ds/18Mi)
% 2.26/1.52 % (20988)Instruction limit reached!
% 2.26/1.52 % (20988)------------------------------
% 2.26/1.52 % (20988)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.26/1.52 % (20988)Termination reason: Unknown
% 2.26/1.52 % (20988)Termination phase: Saturation
% 2.26/1.52
% 2.26/1.52 % (20988)Memory used [KB]: 6780
% 2.26/1.52 % (20988)Time elapsed: 0.059 s
% 2.26/1.52 % (20988)Instructions burned: 183 (million)
% 2.26/1.52 % (20988)------------------------------
% 2.26/1.52 % (20988)------------------------------
% 2.26/1.52 % (21009)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_20822 for (2999ds/6Mi)
% 2.26/1.52 % (21006)Instruction limit reached!
% 2.26/1.52 % (21006)------------------------------
% 2.26/1.52 % (21006)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.26/1.52 % (21006)Termination reason: Unknown
% 2.26/1.52 % (21006)Termination phase: shuffling
% 2.26/1.52
% 2.26/1.52 % (21006)Memory used [KB]: 1151
% 2.26/1.52 % (21006)Time elapsed: 0.004 s
% 2.26/1.52 % (21006)Instructions burned: 4 (million)
% 2.26/1.52 % (21006)------------------------------
% 2.26/1.52 % (21006)------------------------------
% 2.26/1.52 % (21009)Instruction limit reached!
% 2.26/1.52 % (21009)------------------------------
% 2.26/1.52 % (21009)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.26/1.52 % (21009)Termination reason: Unknown
% 2.26/1.52 % (21009)Termination phase: shuffling
% 2.26/1.52
% 2.26/1.52 % (21009)Memory used [KB]: 1279
% 2.26/1.52 % (21009)Time elapsed: 0.004 s
% 2.26/1.52 % (21009)Instructions burned: 8 (million)
% 2.26/1.52 % (21009)------------------------------
% 2.26/1.52 % (21009)------------------------------
% 2.26/1.53 % (21007)Instruction limit reached!
% 2.26/1.53 % (21007)------------------------------
% 2.26/1.53 % (21007)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.26/1.53 % (21007)Termination reason: Unknown
% 2.26/1.53 % (21007)Termination phase: Property scanning
% 2.26/1.53
% 2.26/1.53 % (21007)Memory used [KB]: 1407
% 2.26/1.53 % (21007)Time elapsed: 0.007 s
% 2.26/1.53 % (21007)Instructions burned: 18 (million)
% 2.26/1.53 % (21007)------------------------------
% 2.26/1.53 % (21007)------------------------------
% 2.26/1.53 % (21012)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_20822 for (2999ds/5Mi)
% 2.26/1.53 % (21012)Instruction limit reached!
% 2.26/1.53 % (21012)------------------------------
% 2.26/1.53 % (21012)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.26/1.53 % (21012)Termination reason: Unknown
% 2.26/1.53 % (21012)Termination phase: shuffling
% 2.26/1.53
% 2.26/1.53 % (21012)Memory used [KB]: 1151
% 2.26/1.53 % (21012)Time elapsed: 0.003 s
% 2.26/1.53 % (21012)Instructions burned: 5 (million)
% 2.26/1.53 % (21012)------------------------------
% 2.26/1.53 % (21012)------------------------------
% 2.26/1.54 % (21013)lrs+2_1:1_cnfonf=lazy_not_gen_be_off:cs=on:fe=off:hud=10:inj=on:ins=3:plsq=on:plsqc=1:sd=10:ss=axioms:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_20822 for (2999ds/6Mi)
% 2.26/1.54 % (21013)Instruction limit reached!
% 2.26/1.54 % (21013)------------------------------
% 2.26/1.54 % (21013)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.26/1.54 % (21013)Termination reason: Unknown
% 2.26/1.54 % (21013)Termination phase: shuffling
% 2.26/1.54
% 2.26/1.54 % (21013)Memory used [KB]: 1279
% 2.26/1.54 % (21013)Time elapsed: 0.004 s
% 2.26/1.54 % (21013)Instructions burned: 8 (million)
% 2.26/1.54 % (21013)------------------------------
% 2.26/1.54 % (21013)------------------------------
% 2.26/1.54 % (21011)dis+21_1:8_apa=on:cnfonf=off:fd=off:fsr=off:hud=0:ins=1:kws=inv_frequency:nwc=10.0:ss=axioms:st=5.0:i=21:si=on:rtra=on_0 on DTF2THF_20822 for (2999ds/21Mi)
% 2.26/1.54 % (21014)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_20822 for (2999ds/377Mi)
% 2.26/1.55 % (21015)dis+1010_1:4_atotf=0.2:c=on:cbe=off:cnfonf=lazy_simp:fe=off:ins=2:ntd=on:s2a=on:s2at=5.0:sgt=5:ss=axioms:st=1.5:i=779:si=on:rtra=on_0 on DTF2THF_20822 for (2999ds/779Mi)
% 2.26/1.55 % (21011)Instruction limit reached!
% 2.26/1.55 % (21011)------------------------------
% 2.26/1.55 % (21011)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.26/1.55 % (21011)Termination reason: Unknown
% 2.26/1.55 % (21011)Termination phase: SInE selection
% 2.26/1.55
% 2.26/1.55 % (21011)Memory used [KB]: 1407
% 2.26/1.55 % (21011)Time elapsed: 0.013 s
% 2.26/1.55 % (21011)Instructions burned: 22 (million)
% 2.26/1.55 % (21011)------------------------------
% 2.26/1.55 % (21011)------------------------------
% 2.26/1.55 % (21010)dis+1002_5:1_au=on:bd=off:e2e=on:fde=none:fs=off:fsr=off:sos=on:i=902:si=on:rtra=on_0 on DTF2THF_20822 for (2999ds/902Mi)
% 2.26/1.56 % (21008)First to succeed.
% 2.26/1.56 % (21008)Refutation found. Thanks to Tanya!
% 2.26/1.56 % SZS status Theorem for DTF2THF_20822
% 2.26/1.56 % SZS output start Proof for DTF2THF_20822
% 2.26/1.56 thf(func_def_0, type, in: $i > $i > $o).
% 2.26/1.56 thf(func_def_1, type, exu: ($i > $o) > $o).
% 2.26/1.56 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 2.26/1.56 thf(func_def_8, type, powerset: $i > $i).
% 2.26/1.56 thf(func_def_10, type, setunion: $i > $i).
% 2.26/1.56 thf(func_def_19, type, descr: ($i > $o) > $i).
% 2.26/1.56 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 2.26/1.56 thf(func_def_26, type, prop2set: $o > $i).
% 2.26/1.56 thf(func_def_36, type, nonempty: $i > $o).
% 2.26/1.56 thf(func_def_69, type, set2prop: $i > $o).
% 2.26/1.56 thf(func_def_88, type, subset: $i > $i > $o).
% 2.26/1.56 thf(func_def_89, type, disjoint: $i > $i > $o).
% 2.26/1.56 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 2.26/1.56 thf(func_def_106, type, sP0: $i > $i > $o).
% 2.26/1.56 thf(func_def_107, type, sP1: $i > $o).
% 2.26/1.56 thf(func_def_108, type, sP2: $i > $i > $o).
% 2.26/1.56 thf(func_def_109, type, sP3: $i > $i > $o).
% 2.26/1.56 thf(func_def_111, type, sK5: $i > $o).
% 2.26/1.56 thf(func_def_113, type, sK7: $i > ($i > $o) > $i).
% 2.26/1.56 thf(func_def_118, type, sK12: $i > $i).
% 2.26/1.56 thf(func_def_120, type, sK14: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.26/1.56 thf(func_def_121, type, sK15: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.26/1.56 thf(func_def_124, type, sK18: $i > $o).
% 2.26/1.56 thf(func_def_125, type, sK19: $i > $o).
% 2.26/1.56 thf(func_def_128, type, sK22: $i > $o).
% 2.26/1.56 thf(func_def_129, type, sK23: $i > $i).
% 2.26/1.56 thf(func_def_130, type, sK24: ($i > $o) > $i).
% 2.26/1.56 thf(func_def_133, type, sK27: $i > ($i > $o) > $i).
% 2.26/1.56 thf(func_def_134, type, sK28: $i > $o).
% 2.26/1.56 thf(func_def_136, type, sK30: $i > $i).
% 2.26/1.56 thf(func_def_141, type, sK35: ($i > $o) > ($i > $o) > $i).
% 2.26/1.56 thf(func_def_142, type, sK36: ($i > $o) > ($i > $o) > $i).
% 2.26/1.56 thf(func_def_143, type, sK37: $i > $o).
% 2.26/1.56 thf(func_def_144, type, sK38: $i > $o).
% 2.26/1.56 thf(func_def_146, type, sK40: $i > $o).
% 2.26/1.56 thf(func_def_148, type, sK42: ($i > $o) > $i > $i).
% 2.26/1.56 thf(func_def_150, type, sK44: $i > $o).
% 2.26/1.56 thf(func_def_152, type, sK46: ($i > $o) > $i > $i).
% 2.26/1.56 thf(func_def_156, type, sK50: $i > $i).
% 2.26/1.56 thf(func_def_159, type, sK53: ($i > $o) > $i > $i).
% 2.26/1.56 thf(func_def_161, type, sK55: $i > $o).
% 2.26/1.56 thf(func_def_164, type, sK58: $i > $i).
% 2.26/1.56 thf(func_def_166, type, sK60: $i > $i).
% 2.26/1.56 thf(func_def_169, type, sK63: $i > $o).
% 2.26/1.56 thf(func_def_173, type, sK67: $i > $o).
% 2.26/1.56 thf(func_def_181, type, sK75: $i > $i > $i).
% 2.26/1.56 thf(func_def_182, type, sK76: $i > $i > $i).
% 2.26/1.56 thf(func_def_183, type, sK77: $i > $i > $i).
% 2.26/1.56 thf(func_def_184, type, sK78: $i > $i > $i).
% 2.26/1.56 thf(func_def_185, type, sK79: $i > $i > $i).
% 2.26/1.56 thf(func_def_186, type, sK80: $i > $i > $i).
% 2.26/1.56 thf(func_def_187, type, sK81: $i > $i > $i > $i).
% 2.26/1.56 thf(func_def_188, type, sK82: $i > $i).
% 2.26/1.56 thf(func_def_189, type, sK83: $i > $i).
% 2.26/1.56 thf(func_def_190, type, sK84: $i > $i).
% 2.26/1.56 thf(func_def_191, type, sK85: $i > $i).
% 2.26/1.56 thf(func_def_192, type, sK86: $i > $i > $i > $i).
% 2.26/1.56 thf(func_def_193, type, sK87: $i > $i > $i > $i).
% 2.26/1.56 thf(func_def_194, type, sK88: $i > $i > $i).
% 2.26/1.56 thf(func_def_195, type, sK89: $i > $i > $i).
% 2.26/1.56 thf(func_def_196, type, sK90: $i > $i > $i).
% 2.26/1.56 thf(func_def_197, type, sK91: $i > $i).
% 2.26/1.56 thf(func_def_199, type, sK93: $i > $i).
% 2.26/1.56 thf(func_def_200, type, sK94: $i > $i).
% 2.26/1.56 thf(func_def_202, type, sK96: $i > $i).
% 2.26/1.56 thf(func_def_203, type, sK97: ($i > $o) > ($i > $o) > $i).
% 2.26/1.56 thf(func_def_204, type, sK98: ($i > $o) > ($i > $o) > $i).
% 2.26/1.56 thf(func_def_205, type, sK99: $i > $o).
% 2.26/1.56 thf(func_def_206, type, sK100: $i > $o).
% 2.26/1.56 thf(func_def_211, type, sK105: $i > $i > $i).
% 2.26/1.56 thf(func_def_212, type, sK106: $i > $i > $i).
% 2.26/1.56 thf(func_def_215, type, sK109: ($i > $o) > $i).
% 2.26/1.56 thf(func_def_216, type, sK110: ($i > $o) > $i).
% 2.26/1.56 thf(func_def_217, type, sK111: $i > $o).
% 2.26/1.56 thf(func_def_219, type, sK113: $i > $o).
% 2.26/1.56 thf(func_def_223, type, sK117: $i > $i).
% 2.26/1.56 thf(func_def_226, type, sK120: $i > $i > $i).
% 2.26/1.56 thf(func_def_229, type, sK123: $i > $o).
% 2.26/1.56 thf(func_def_234, type, sK128: $i > $o).
% 2.26/1.56 thf(func_def_237, type, sK131: $i > $o).
% 2.26/1.56 thf(func_def_240, type, sK134: $i > $i > $i).
% 2.26/1.56 thf(func_def_248, type, sK142: $i > $o).
% 2.26/1.56 thf(func_def_259, type, sK153: $i > ($i > $o) > $i).
% 2.26/1.56 thf(func_def_260, type, sK154: $i > $o).
% 2.26/1.56 thf(func_def_268, type, sK162: $i > $i > $o).
% 2.26/1.56 thf(func_def_269, type, sK163: $i > $i).
% 2.26/1.56 thf(func_def_270, type, sK164: $i > $i).
% 2.26/1.56 thf(func_def_271, type, sK165: ($i > $i > $o) > $i > $i).
% 2.26/1.56 thf(func_def_272, type, sK166: ($i > $i > $o) > $i > $i).
% 2.26/1.56 thf(func_def_273, type, sK167: $i > ($i > $i > $o) > $i > $i).
% 2.26/1.56 thf(func_def_274, type, sK168: ($i > $o) > $i).
% 2.26/1.56 thf(func_def_275, type, sK169: $i > $o).
% 2.26/1.56 thf(func_def_276, type, sK170: $i > $i).
% 2.26/1.56 thf(func_def_285, type, sK179: $o > $i > $i > $i).
% 2.26/1.56 thf(func_def_292, type, sK186: $i > $i > $i).
% 2.26/1.56 thf(func_def_293, type, sK187: $i > $o).
% 2.26/1.56 thf(func_def_294, type, sK188: ($i > $o) > $i).
% 2.26/1.56 thf(func_def_297, type, sK191: $i > $o).
% 2.26/1.56 thf(func_def_303, type, sK197: $i > $o).
% 2.26/1.56 thf(func_def_305, type, sK199: ($i > $o) > $i > $i).
% 2.26/1.56 thf(func_def_311, type, sK205: $i > $o).
% 2.26/1.56 thf(func_def_314, type, sK208: $i > ($i > $o) > $i).
% 2.26/1.56 thf(func_def_318, type, sK212: $i > $o).
% 2.26/1.56 thf(func_def_326, type, sK220: $i > $i > $i).
% 2.26/1.56 thf(func_def_328, type, sK222: $i > $o).
% 2.26/1.56 thf(func_def_337, type, ph231: !>[X0: $tType]:(X0)).
% 2.26/1.56 thf(f1832,plain,(
% 2.26/1.56 $false),
% 2.26/1.56 inference(trivial_inequality_removal,[status(thm)],[f1831])).
% 2.26/1.56 thf(f1831,plain,(
% 2.26/1.56 ($true = $false)),
% 2.26/1.56 inference(forward_demodulation,[status(thm)],[f1830,f1492])).
% 2.26/1.56 thf(f1492,plain,(
% 2.26/1.56 ((in @ sK230 @ sK69) = $false)),
% 2.26/1.56 inference(binary_proxy_clausification,[status(thm)],[f1491])).
% 2.26/1.56 thf(f1491,plain,(
% 2.26/1.56 (((in @ sK230 @ sK68) => (in @ sK230 @ sK69)) = $false)),
% 2.26/1.56 inference(beta_eta_normalization,[status(thm)],[f1490])).
% 2.26/1.56 thf(f1490,plain,(
% 2.26/1.56 (((^[Y0 : $i]: ((in @ Y0 @ sK68) => (in @ Y0 @ sK69))) @ sK230) = $false)),
% 2.26/1.56 inference(sigma_clausification,[status(thm)],[f1489])).
% 2.26/1.56 thf(f1489,plain,(
% 2.26/1.56 ($true != (!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ sK68) => (in @ Y0 @ sK69)))))),
% 2.26/1.56 inference(beta_eta_normalization,[status(thm)],[f1257])).
% 2.26/1.56 thf(f1257,plain,(
% 2.26/1.56 ($true != ((^[Y0 : $i]: ((^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: ((in @ Y2 @ Y0) => (in @ Y2 @ Y1))))))) @ sK68 @ sK69))),
% 2.26/1.56 inference(definition_unfolding,[status(thm)],[f850,f839])).
% 2.26/1.56 thf(f839,plain,(
% 2.26/1.56 (subset = (^[Y0 : $i]: ((^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: ((in @ Y2 @ Y0) => (in @ Y2 @ Y1))))))))),
% 2.26/1.56 inference(cnf_transformation,[status(thm)],[f94])).
% 2.26/1.56 thf(f94,plain,(
% 2.26/1.56 (subset = (^[Y0 : $i]: ((^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: ((in @ Y2 @ Y0) => (in @ Y2 @ Y1))))))))),
% 2.26/1.56 inference(fool_elimination,[status(thm)],[f93])).
% 2.26/1.56 thf(f93,plain,(
% 2.26/1.56 (subset = (^[X0 : $i, X1 : $i] : (! [X2 : $i] : ((in @ X2 @ X0) => (in @ X2 @ X1)))))),
% 2.26/1.56 inference(rectify,[status(thm)],[f78])).
% 2.26/1.56 thf(f78,axiom,(
% 2.26/1.56 (subset = (^[X3 : $i, X4 : $i] : (! [X1 : $i] : ((in @ X1 @ X3) => (in @ X1 @ X4)))))),
% 2.26/1.56 file('/export/starexec/sandbox2/tmp/tmp.vNo7MRU8lP/DTF2THF_20822.p',subset)).
% 2.26/1.56 thf(f850,plain,(
% 2.26/1.56 ((subset @ sK68 @ sK69) != $true)),
% 2.26/1.56 inference(cnf_transformation,[status(thm)],[f472])).
% 2.26/1.56 thf(f472,plain,(
% 2.26/1.56 (prop2setE = $true) & (setunionAx = $true) & (notinemptyset = $true) & (wellorderingAx = $true) & (noeltsimpempty = $true) & (exuEu = $true) & (nonemptyI1 = $true) & (emptyInPowerset = $true) & (setadjoinOr = $true) & (emptyI = $true) & (setadjoinIR = $true) & (eqinunit = $true) & (setunionI = $true) & (eqimpsubset1 = $true) & (dsetconstr__Cong = $true) & (setunion__Cong = $true) & (exuE2 = $true) & (dsetconstrER = $true) & (eqimpsubset2 = $true) & (notdexE = $true) & (setunionE = $true) & (prop2setI = $true) & (singletonsswitch = $true) & (prop2set2propI = $true) & (exuI2 = $true) & (upairsetIL = $true) & (omega__Cong = $true) & (quantDeMorgan3 = $true) & (exuI3 = $true) & (dsetconstrI = $true) & (nonemptyI = $true) & (powersetAx = $true) & (emptysetimpfalse = $true) & (quantDeMorgan2 = $true) & (nonemptyE1 = $true) & (uniqinunit = $true) & (upairsetIR = $true) & (setextAx = $true) & (emptyinPowerset = $true) & (setadjoin__Cong = $true) & (powerset__Cong = $true) & (setoftrueEq = $true) & (setext = $true) & (exuE3e = $true) & (quantDeMorgan4 = $true) & (emptyinunitempty = $true) & (upairsetE = $true) & (exuE3u = $true) & (in__Cong = $true) & (dsetconstrEL = $true) & (replAx = $true) & (notdallE = $true) & (emptyE1 = $true) & (omegaIndAx = $true) & (setadjoinIL = $true) & (powersetE = $true) & (nonemptyImpWitness = $true) & (quantDeMorgan1 = $true) & (notinsingleton = $true) & (exu__Cong = $true) & (emptysetAx = $true) & (setadjoinAx = $true) & (exuI1 = $true) & (vacuousDall = $true) & (descr__Cong = $true) & (setbeta = $true) & (foundationAx = $true) & (omega0Ax = $true) & (subPowSU = $true) & (((subset @ sK68 @ sK69) != $true) & ! [X2 : $i] : (((in @ X2 @ sK68) != $true) | ((in @ X2 @ sK69) = $true))) & (omegaSAx = $true) & (powersetI = $true) & (descrp = $true) & (setadjoinE = $true) & (emptysetE = $true) & (exuE1 = $true) & (emptyset__Cong = $true) & (subsetI1 = $true) & (inCongP = $true)),
% 2.26/1.56 inference(skolemisation,[status(esa),new_symbols(skolem,[sK68,sK69])],[f312,f471])).
% 2.26/1.56 thf(f471,plain,(
% 2.26/1.56 ? [X0 : $i,X1 : $i] : (($true != (subset @ X0 @ X1)) & ! [X2 : $i] : (($true != (in @ X2 @ X0)) | ($true = (in @ X2 @ X1)))) => (((subset @ sK68 @ sK69) != $true) & ! [X2 : $i] : (((in @ X2 @ sK68) != $true) | ((in @ X2 @ sK69) = $true)))),
% 2.26/1.56 introduced(definition,[],[choice_axiom])).
% 2.26/1.56 thf(f312,plain,(
% 2.26/1.56 (prop2setE = $true) & (setunionAx = $true) & (notinemptyset = $true) & (wellorderingAx = $true) & (noeltsimpempty = $true) & (exuEu = $true) & (nonemptyI1 = $true) & (emptyInPowerset = $true) & (setadjoinOr = $true) & (emptyI = $true) & (setadjoinIR = $true) & (eqinunit = $true) & (setunionI = $true) & (eqimpsubset1 = $true) & (dsetconstr__Cong = $true) & (setunion__Cong = $true) & (exuE2 = $true) & (dsetconstrER = $true) & (eqimpsubset2 = $true) & (notdexE = $true) & (setunionE = $true) & (prop2setI = $true) & (singletonsswitch = $true) & (prop2set2propI = $true) & (exuI2 = $true) & (upairsetIL = $true) & (omega__Cong = $true) & (quantDeMorgan3 = $true) & (exuI3 = $true) & (dsetconstrI = $true) & (nonemptyI = $true) & (powersetAx = $true) & (emptysetimpfalse = $true) & (quantDeMorgan2 = $true) & (nonemptyE1 = $true) & (uniqinunit = $true) & (upairsetIR = $true) & (setextAx = $true) & (emptyinPowerset = $true) & (setadjoin__Cong = $true) & (powerset__Cong = $true) & (setoftrueEq = $true) & (setext = $true) & (exuE3e = $true) & (quantDeMorgan4 = $true) & (emptyinunitempty = $true) & (upairsetE = $true) & (exuE3u = $true) & (in__Cong = $true) & (dsetconstrEL = $true) & (replAx = $true) & (notdallE = $true) & (emptyE1 = $true) & (omegaIndAx = $true) & (setadjoinIL = $true) & (powersetE = $true) & (nonemptyImpWitness = $true) & (quantDeMorgan1 = $true) & (notinsingleton = $true) & (exu__Cong = $true) & (emptysetAx = $true) & (setadjoinAx = $true) & (exuI1 = $true) & (vacuousDall = $true) & (descr__Cong = $true) & (setbeta = $true) & (foundationAx = $true) & (omega0Ax = $true) & (subPowSU = $true) & ? [X0 : $i,X1 : $i] : (($true != (subset @ X0 @ X1)) & ! [X2 : $i] : (($true != (in @ X2 @ X0)) | ($true = (in @ X2 @ X1)))) & (omegaSAx = $true) & (powersetI = $true) & (descrp = $true) & (setadjoinE = $true) & (emptysetE = $true) & (exuE1 = $true) & (emptyset__Cong = $true) & (subsetI1 = $true) & (inCongP = $true)),
% 2.26/1.56 inference(flattening,[status(thm)],[f311])).
% 2.26/1.56 thf(f311,plain,(
% 2.26/1.56 (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i,X1 : $i] : (($true != (subset @ X0 @ X1)) & ! [X2 : $i] : (($true != (in @ X2 @ X0)) | ($true = (in @ X2 @ X1)))) & (eqimpsubset1 = $true)) & (eqimpsubset2 = $true)) & (subsetI1 = $true)) & (dsetconstr__Cong = $true)) & (descr__Cong = $true)) & (exuEu = $true)) & (omega__Cong = $true)) & (setunion__Cong = $true)) & (powerset__Cong = $true)) & (setadjoin__Cong = $true)) & (emptyset__Cong = $true)) & (exu__Cong = $true)) & (exuE3u = $true)) & (in__Cong = $true)) & (inCongP = $true)) & (exuI2 = $true)) & (exuI3 = $true)) & (exuI1 = $true)) & (notdallE = $true)) & (notdexE = $true)) & (prop2set2propI = $true)) & (prop2setI = $true)) & (quantDeMorgan4 = $true)) & (quantDeMorgan3 = $true)) & (quantDeMorgan2 = $true)) & (quantDeMorgan1 = $true)) & (vacuousDall = $true)) & (emptyE1 = $true)) & (upairsetIR = $true)) & (upairsetIL = $true)) & (upairsetE = $true)) & (singletonsswitch = $true)) & (eqinunit = $true)) & (notinsingleton = $true)) & (uniqinunit = $true)) & (nonemptyImpWitness = $true)) & (exuE2 = $true)) & (subPowSU = $true)) & (setunionE = $true)) & (setunionI = $true)) & (powersetE = $true)) & (emptyInPowerset = $true)) & (emptyinPowerset = $true)) & (powersetI = $true)) & (setoftrueEq = $true)) & (setadjoinOr = $true)) & (setadjoinE = $true)) & (setadjoinIR = $true)) & (emptyinunitempty = $true)) & (setadjoinIL = $true)) & (nonemptyI1 = $true)) & (nonemptyI = $true)) & (nonemptyE1 = $true)) & (setbeta = $true)) & (noeltsimpempty = $true)) & (emptyI = $true)) & (setext = $true)) & (exuE3e = $true)) & (notinemptyset = $true)) & (emptysetimpfalse = $true)) & (emptysetE = $true)) & (prop2setE = $true)) & (exuE1 = $true)) & (dsetconstrER = $true)) & (dsetconstrEL = $true)) & (dsetconstrI = $true)) & (descrp = $true)) & (wellorderingAx = $true)) & (foundationAx = $true)) & (replAx = $true)) & (omegaIndAx = $true)) & (omegaSAx = $true)) & (omega0Ax = $true)) & (setunionAx = $true)) & (powersetAx = $true)) & (setadjoinAx = $true)) & (emptysetAx = $true)) & (setextAx = $true)),
% 2.26/1.56 inference(ennf_transformation,[status(thm)],[f136])).
% 2.26/1.56 thf(f136,plain,(
% 2.26/1.56 ~((setextAx = $true) => ((emptysetAx = $true) => ((setadjoinAx = $true) => ((powersetAx = $true) => ((setunionAx = $true) => ((omega0Ax = $true) => ((omegaSAx = $true) => ((omegaIndAx = $true) => ((replAx = $true) => ((foundationAx = $true) => ((wellorderingAx = $true) => ((descrp = $true) => ((dsetconstrI = $true) => ((dsetconstrEL = $true) => ((dsetconstrER = $true) => ((exuE1 = $true) => ((prop2setE = $true) => ((emptysetE = $true) => ((emptysetimpfalse = $true) => ((notinemptyset = $true) => ((exuE3e = $true) => ((setext = $true) => ((emptyI = $true) => ((noeltsimpempty = $true) => ((setbeta = $true) => ((nonemptyE1 = $true) => ((nonemptyI = $true) => ((nonemptyI1 = $true) => ((setadjoinIL = $true) => ((emptyinunitempty = $true) => ((setadjoinIR = $true) => ((setadjoinE = $true) => ((setadjoinOr = $true) => ((setoftrueEq = $true) => ((powersetI = $true) => ((emptyinPowerset = $true) => ((emptyInPowerset = $true) => ((powersetE = $true) => ((setunionI = $true) => ((setunionE = $true) => ((subPowSU = $true) => ((exuE2 = $true) => ((nonemptyImpWitness = $true) => ((uniqinunit = $true) => ((notinsingleton = $true) => ((eqinunit = $true) => ((singletonsswitch = $true) => ((upairsetE = $true) => ((upairsetIL = $true) => ((upairsetIR = $true) => ((emptyE1 = $true) => ((vacuousDall = $true) => ((quantDeMorgan1 = $true) => ((quantDeMorgan2 = $true) => ((quantDeMorgan3 = $true) => ((quantDeMorgan4 = $true) => ((prop2setI = $true) => ((prop2set2propI = $true) => ((notdexE = $true) => ((notdallE = $true) => ((exuI1 = $true) => ((exuI3 = $true) => ((exuI2 = $true) => ((inCongP = $true) => ((in__Cong = $true) => ((exuE3u = $true) => ((exu__Cong = $true) => ((emptyset__Cong = $true) => ((setadjoin__Cong = $true) => ((powerset__Cong = $true) => ((setunion__Cong = $true) => ((omega__Cong = $true) => ((exuEu = $true) => ((descr__Cong = $true) => ((dsetconstr__Cong = $true) => ((subsetI1 = $true) => ((eqimpsubset2 = $true) => ((eqimpsubset1 = $true) => ! [X0 : $i,X1 : $i] : (! [X2 : $i] : (($true = (in @ X2 @ X0)) => ($true = (in @ X2 @ X1))) => ($true = (subset @ X0 @ X1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.26/1.56 inference(fool_elimination,[status(thm)],[f135])).
% 2.26/1.56 thf(f135,plain,(
% 2.26/1.56 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => ! [X0 : $i,X1 : $i] : (! [X2 : $i] : ((in @ X2 @ X0) => (in @ X2 @ X1)) => (subset @ X0 @ X1))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.26/1.56 inference(rectify,[status(thm)],[f83])).
% 2.26/1.56 thf(f83,negated_conjecture,(
% 2.26/1.56 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => ! [X3 : $i,X4 : $i] : (! [X1 : $i] : ((in @ X1 @ X3) => (in @ X1 @ X4)) => (subset @ X3 @ X4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.26/1.56 inference(negated_conjecture,[status(cth)],[f82])).
% 2.26/1.56 thf(f82,conjecture,(
% 2.26/1.56 setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => ! [X3 : $i,X4 : $i] : (! [X1 : $i] : ((in @ X1 @ X3) => (in @ X1 @ X4)) => (subset @ X3 @ X4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.26/1.56 file('/export/starexec/sandbox2/tmp/tmp.vNo7MRU8lP/DTF2THF_20822.p',subsetI2)).
% 2.26/1.56 thf(f1830,plain,(
% 2.26/1.56 ($true = (in @ sK230 @ sK69))),
% 2.26/1.56 inference(trivial_inequality_removal,[status(thm)],[f1821])).
% 2.26/1.56 thf(f1821,plain,(
% 2.26/1.56 ($true = (in @ sK230 @ sK69)) | ($true != $true)),
% 2.26/1.56 inference(superposition,[status(thm)],[f849,f1493])).
% 2.26/1.56 thf(f1493,plain,(
% 2.26/1.56 ($true = (in @ sK230 @ sK68))),
% 2.26/1.56 inference(binary_proxy_clausification,[status(thm)],[f1491])).
% 2.26/1.56 thf(f849,plain,(
% 2.26/1.56 ( ! [X2 : $i] : (((in @ X2 @ sK68) != $true) | ((in @ X2 @ sK69) = $true)) )),
% 2.26/1.56 inference(cnf_transformation,[status(thm)],[f472])).
% 2.26/1.56 % SZS output end Proof for DTF2THF_20822
% 2.26/1.56 % (21008)------------------------------
% 2.26/1.56 % (21008)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.26/1.56 % (21008)Termination reason: Refutation
% 2.26/1.56
% 2.26/1.56 % (21008)Memory used [KB]: 7036
% 2.26/1.56 % (21008)Time elapsed: 0.046 s
% 2.26/1.56 % (21008)Instructions burned: 135 (million)
% 2.26/1.56 % (21008)------------------------------
% 2.26/1.56 % (21008)------------------------------
% 2.26/1.56 % (20987)Success in time 0.099 s
%------------------------------------------------------------------------------