%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU559^1 : 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 : n025.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 09:35:16 AM UTC 2026
% Result : Theorem 0.30s 1.49s
% Output : Refutation 0.30s
% Verified :
% SZS Type : Refutation
% Derivation depth : 15
% Number of leaves : 3
% Syntax : Number of formulae : 23 ( 13 unt; 0 typ; 0 def)
% Number of atoms : 1280 ( 334 equ; 0 cnn)
% Maximal formula atoms : 78 ( 55 avg)
% Number of connectives : 688 ( 19 ~; 7 |; 230 &; 111 @)
% ( 0 <=>; 317 =>; 0 <=; 0 <~>)
% Maximal formula depth : 84 ( 26 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 187 ( 183 usr; 81 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_103,type,
sP0: $i > $i > $o ).
thf(func_def_104,type,
sP1: $i > $o ).
thf(func_def_105,type,
sP2: $i > $i > $o ).
thf(func_def_106,type,
sP3: $i > $i > $o ).
thf(func_def_109,type,
sK6: $i > $o ).
thf(func_def_110,type,
sK7: $i > $i > $i ).
thf(func_def_111,type,
sK8: $i > $i > $i ).
thf(func_def_112,type,
sK9: $i > $i > $i ).
thf(func_def_113,type,
sK10: $i > $i > $i ).
thf(func_def_114,type,
sK11: $i > $i > $i > $i ).
thf(func_def_115,type,
sK12: $i > $i ).
thf(func_def_116,type,
sK13: $i > $i ).
thf(func_def_117,type,
sK14: $i > $i ).
thf(func_def_118,type,
sK15: $i > $i ).
thf(func_def_119,type,
sK16: $i > $i > $i ).
thf(func_def_120,type,
sK17: $i > $i > $i > $i ).
thf(func_def_121,type,
sK18: $i > $i > $i > $i ).
thf(func_def_122,type,
sK19: $i > $i > $i ).
thf(func_def_123,type,
sK20: $i > $i > $i ).
thf(func_def_125,type,
sK22: $i > $i ).
thf(func_def_126,type,
sK23: $i > $i ).
thf(func_def_127,type,
sK24: $i > $i ).
thf(func_def_128,type,
sK25: $i > $i > $i ).
thf(func_def_132,type,
sK29: $i > $o ).
thf(func_def_133,type,
sK30: ( $i > $o ) > $i ).
thf(func_def_136,type,
sK33: $i > $i > $i ).
thf(func_def_137,type,
sK34: $i > $o ).
thf(func_def_144,type,
sK41: $i > $o ).
thf(func_def_145,type,
sK42: $i > $o ).
thf(func_def_146,type,
sK43: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_147,type,
sK44: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_152,type,
sK49: $i > $o ).
thf(func_def_155,type,
sK52: $i > $o ).
thf(func_def_157,type,
sK54: $i > ( $i > $o ) > $i ).
thf(func_def_159,type,
sK56: $i > $i > $i ).
thf(func_def_160,type,
sK57: $i > $i > $i ).
thf(func_def_163,type,
sK60: ( $i > $o ) > $i ).
thf(func_def_164,type,
sK61: ( $i > $o ) > $i ).
thf(func_def_165,type,
sK62: $i > $o ).
thf(func_def_173,type,
sK70: $i > $i ).
thf(func_def_177,type,
sK74: $i > $o ).
thf(func_def_180,type,
sK77: $i > ( $i > $o ) > $i ).
thf(func_def_181,type,
sK78: $i > $o ).
thf(func_def_187,type,
sK84: ( $i > $o ) > $i ).
thf(func_def_188,type,
sK85: $i > $o ).
thf(func_def_189,type,
sK86: $i > $i ).
thf(func_def_196,type,
sK93: $i > $i > $i ).
thf(func_def_199,type,
sK96: $i > $i > $i ).
thf(func_def_207,type,
sK104: $o > $i > $i > $i ).
thf(func_def_209,type,
sK106: $i > $o ).
thf(func_def_211,type,
sK108: ( $i > $o ) > $i > $i ).
thf(func_def_214,type,
sK111: $i > $i ).
thf(func_def_219,type,
sK116: $i > ( $i > $o ) > $i ).
thf(func_def_220,type,
sK117: $i > $o ).
thf(func_def_223,type,
sK120: $i > $o ).
thf(func_def_230,type,
sK127: $i > $i ).
thf(func_def_232,type,
sK129: $i > ( $i > $o ) > $i ).
thf(func_def_233,type,
sK130: $i > $o ).
thf(func_def_238,type,
sK135: $i > $o ).
thf(func_def_243,type,
sK140: $i > $i > $i ).
thf(func_def_244,type,
sK141: $i > $o ).
thf(func_def_245,type,
sK142: $i > $o ).
thf(func_def_246,type,
sK143: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_247,type,
sK144: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_256,type,
sK153: $i > $i ).
thf(func_def_259,type,
sK156: $i > $i ).
thf(func_def_268,type,
sK165: $i > $o ).
thf(func_def_271,type,
sK168: $i > ( $i > $o ) > $i ).
thf(func_def_274,type,
sK171: $i > $o ).
thf(func_def_276,type,
sK173: $i > ( $i > $o ) > $i ).
thf(func_def_277,type,
sK174: $i > $o ).
thf(func_def_282,type,
sK179: $i > $o ).
thf(func_def_285,type,
sK182: $i > $i ).
thf(func_def_288,type,
sK185: $i > $i > $o ).
thf(func_def_289,type,
sK186: $i > $i ).
thf(func_def_290,type,
sK187: $i > $i ).
thf(func_def_291,type,
sK188: ( $i > $i > $o ) > $i > $i ).
thf(func_def_292,type,
sK189: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_293,type,
sK190: ( $i > $i > $o ) > $i > $i ).
thf(func_def_294,type,
sK191: $i > $i ).
thf(func_def_297,type,
sK194: $i > $o ).
thf(func_def_299,type,
sK196: $i > $o ).
thf(func_def_305,type,
sK202: $i > $o ).
thf(func_def_306,type,
sK203: $i > $o ).
thf(func_def_307,type,
sK204: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_308,type,
sK205: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_311,type,
sK208: $i > $o ).
thf(func_def_312,type,
sK209: $i > $i ).
thf(func_def_313,type,
sK210: ( $i > $o ) > $i ).
thf(func_def_317,type,
sK214: $i > $o ).
thf(func_def_319,type,
sK216: ( $i > $o ) > $i > $i ).
thf(func_def_325,type,
ph222:
!>[X0: $tType] : X0 ).
thf(f1887,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f1881,f1455]) ).
thf(f1455,plain,
( $true
= ( in @ sK223 @ sK46 ) ),
inference(binary_proxy_clausification,[status(thm)],[f1453]) ).
thf(f1453,plain,
( $false
= ( ( in @ sK223 @ sK46 )
=> ( in @ sK223 @ sK45 ) ) ),
inference(beta_eta_normalization,[status(thm)],[f1452]) ).
thf(f1452,plain,
( $false
= ( ^ [Y0: $i] :
( ( in @ Y0 @ sK46 )
=> ( in @ Y0 @ sK45 ) )
@ sK223 ) ),
inference(sigma_clausification,[status(thm)],[f1451]) ).
thf(f1451,plain,
( $true
!= ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ sK46 )
=> ( in @ Y0 @ sK45 ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f1155]) ).
thf(f1155,plain,
( ( ^ [Y0: $i,Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y0 )
=> ( in @ Y2 @ Y1 ) ) )
@ sK46
@ sK45 )
!= $true ),
inference(definition_unfolding,[status(thm)],[f857,f1037]) ).
thf(f1037,plain,
( subset
= ( ^ [Y0: $i,Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y0 )
=> ( in @ Y2 @ Y1 ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f99]) ).
thf(f99,plain,
( subset
= ( ^ [Y0: $i,Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y0 )
=> ( in @ Y2 @ Y1 ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f98]) ).
thf(f98,plain,
( ( ^ [X0: $i,X1: $i] :
! [X2: $i] :
( ( in @ X2 @ X0 )
=> ( in @ X2 @ X1 ) ) )
= subset ),
inference(rectify,[status(thm)],[f78]) ).
thf(f78,axiom,
( ( ^ [X3: $i,X4: $i] :
! [X1: $i] :
( ( in @ X1 @ X3 )
=> ( in @ X1 @ X4 ) ) )
= subset ),
file('/export/starexec/sandbox2/tmp/tmp.zNeidnJOtA/DTF2THF_7736.p',subset) ).
thf(f857,plain,
( $true
!= ( subset @ sK46 @ sK45 ) ),
inference(cnf_transformation,[status(thm)],[f407]) ).
thf(f407,plain,
( ! [X2: $i] :
( ( $true
!= ( in @ X2 @ sK46 ) )
| ( ( in @ X2 @ sK45 )
= $true ) )
& ( $true
!= ( subset @ sK46 @ sK45 ) )
& ( exuE3e = $true )
& ( upairsetIR = $true )
& ( nonemptyI = $true )
& ( exuI1 = $true )
& ( nonemptyI1 = $true )
& ( wellorderingAx = $true )
& ( singletonsswitch = $true )
& ( in__Cong = $true )
& ( emptysetE = $true )
& ( exuI3 = $true )
& ( exu__Cong = $true )
& ( dsetconstr__Cong = $true )
& ( powersetE = $true )
& ( quantDeMorgan1 = $true )
& ( prop2set2propI = $true )
& ( noeltsimpempty = $true )
& ( powersetI = $true )
& ( subPowSU = $true )
& ( dsetconstrI = $true )
& ( omegaIndAx = $true )
& ( powersetAx = $true )
& ( emptyI = $true )
& ( omegaSAx = $true )
& ( setunionI = $true )
& ( emptyinPowerset = $true )
& ( exuE3u = $true )
& ( notdexE = $true )
& ( setadjoin__Cong = $true )
& ( emptyinunitempty = $true )
& ( setunionE = $true )
& ( prop2setE = $true )
& ( emptysetAx = $true )
& ( prop2setI = $true )
& ( setadjoinIR = $true )
& ( descr__Cong = $true )
& ( setadjoinIL = $true )
& ( omega__Cong = $true )
& ( vacuousDall = $true )
& ( exuE2 = $true )
& ( foundationAx = $true )
& ( setoftrueEq = $true )
& ( emptysetimpfalse = $true )
& ( setextAx = $true )
& ( replAx = $true )
& ( setunionAx = $true )
& ( exuEu = $true )
& ( setext = $true )
& ( setunion__Cong = $true )
& ( setadjoinOr = $true )
& ( nonemptyE1 = $true )
& ( quantDeMorgan3 = $true )
& ( dsetconstrER = $true )
& ( notinsingleton = $true )
& ( exuI2 = $true )
& ( inCongP = $true )
& ( exuE1 = $true )
& ( quantDeMorgan2 = $true )
& ( emptyInPowerset = $true )
& ( notdallE = $true )
& ( upairsetE = $true )
& ( emptyE1 = $true )
& ( emptyset__Cong = $true )
& ( dsetconstrEL = $true )
& ( omega0Ax = $true )
& ( setadjoinE = $true )
& ( setadjoinAx = $true )
& ( eqinunit = $true )
& ( powerset__Cong = $true )
& ( upairsetIL = $true )
& ( notinemptyset = $true )
& ( descrp = $true )
& ( uniqinunit = $true )
& ( quantDeMorgan4 = $true )
& ( nonemptyImpWitness = $true )
& ( setbeta = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK45,sK46])],[f329,f406]) ).
thf(f406,plain,
( ? [X0: $i,X1: $i] :
( ! [X2: $i] :
( ( ( in @ X2 @ X1 )
!= $true )
| ( $true
= ( in @ X2 @ X0 ) ) )
& ( ( subset @ X1 @ X0 )
!= $true ) )
=> ( ! [X2: $i] :
( ( $true
!= ( in @ X2 @ sK46 ) )
| ( ( in @ X2 @ sK45 )
= $true ) )
& ( $true
!= ( subset @ sK46 @ sK45 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f329,plain,
( ? [X0: $i,X1: $i] :
( ! [X2: $i] :
( ( ( in @ X2 @ X1 )
!= $true )
| ( $true
= ( in @ X2 @ X0 ) ) )
& ( ( subset @ X1 @ X0 )
!= $true ) )
& ( exuE3e = $true )
& ( upairsetIR = $true )
& ( nonemptyI = $true )
& ( exuI1 = $true )
& ( nonemptyI1 = $true )
& ( wellorderingAx = $true )
& ( singletonsswitch = $true )
& ( in__Cong = $true )
& ( emptysetE = $true )
& ( exuI3 = $true )
& ( exu__Cong = $true )
& ( dsetconstr__Cong = $true )
& ( powersetE = $true )
& ( quantDeMorgan1 = $true )
& ( prop2set2propI = $true )
& ( noeltsimpempty = $true )
& ( powersetI = $true )
& ( subPowSU = $true )
& ( dsetconstrI = $true )
& ( omegaIndAx = $true )
& ( powersetAx = $true )
& ( emptyI = $true )
& ( omegaSAx = $true )
& ( setunionI = $true )
& ( emptyinPowerset = $true )
& ( exuE3u = $true )
& ( notdexE = $true )
& ( setadjoin__Cong = $true )
& ( emptyinunitempty = $true )
& ( setunionE = $true )
& ( prop2setE = $true )
& ( emptysetAx = $true )
& ( prop2setI = $true )
& ( setadjoinIR = $true )
& ( descr__Cong = $true )
& ( setadjoinIL = $true )
& ( omega__Cong = $true )
& ( vacuousDall = $true )
& ( exuE2 = $true )
& ( foundationAx = $true )
& ( setoftrueEq = $true )
& ( emptysetimpfalse = $true )
& ( setextAx = $true )
& ( replAx = $true )
& ( setunionAx = $true )
& ( exuEu = $true )
& ( setext = $true )
& ( setunion__Cong = $true )
& ( setadjoinOr = $true )
& ( nonemptyE1 = $true )
& ( quantDeMorgan3 = $true )
& ( dsetconstrER = $true )
& ( notinsingleton = $true )
& ( exuI2 = $true )
& ( inCongP = $true )
& ( exuE1 = $true )
& ( quantDeMorgan2 = $true )
& ( emptyInPowerset = $true )
& ( notdallE = $true )
& ( upairsetE = $true )
& ( emptyE1 = $true )
& ( emptyset__Cong = $true )
& ( dsetconstrEL = $true )
& ( omega0Ax = $true )
& ( setadjoinE = $true )
& ( setadjoinAx = $true )
& ( eqinunit = $true )
& ( powerset__Cong = $true )
& ( upairsetIL = $true )
& ( notinemptyset = $true )
& ( descrp = $true )
& ( uniqinunit = $true )
& ( quantDeMorgan4 = $true )
& ( nonemptyImpWitness = $true )
& ( setbeta = $true ) ),
inference(flattening,[status(thm)],[f328]) ).
thf(f328,plain,
( ? [X0: $i,X1: $i] :
( ! [X2: $i] :
( ( ( in @ X2 @ X1 )
!= $true )
| ( $true
= ( in @ X2 @ X0 ) ) )
& ( ( subset @ X1 @ X0 )
!= $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)],[f225]) ).
thf(f225,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 )
=> ! [X0: $i,X1: $i] :
( ! [X2: $i] :
( ( ( in @ X2 @ X1 )
= $true )
=> ( $true
= ( in @ X2 @ X0 ) ) )
=> ( ( subset @ X1 @ X0 )
= $true ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f224]) ).
thf(f224,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
=> ! [X0: $i,X1: $i] :
( ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ X0 ) )
=> ( subset @ X1 @ X0 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f80]) ).
thf(f80,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
=> ! [X4: $i,X3: $i] :
( ! [X1: $i] :
( ( in @ X1 @ X3 )
=> ( in @ X1 @ X4 ) )
=> ( subset @ X3 @ X4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f79]) ).
thf(f79,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
=> ! [X4: $i,X3: $i] :
( ! [X1: $i] :
( ( in @ X1 @ X3 )
=> ( in @ X1 @ X4 ) )
=> ( subset @ X3 @ X4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.zNeidnJOtA/DTF2THF_7736.p',subsetI1) ).
thf(f1881,plain,
( $true
!= ( in @ sK223 @ sK46 ) ),
inference(trivial_inequality_removal,[status(thm)],[f1871]) ).
thf(f1871,plain,
( ( $true
!= ( in @ sK223 @ sK46 ) )
| ( $false = $true ) ),
inference(superposition,[status(thm)],[f1454,f858]) ).
thf(f858,plain,
! [X2: $i] :
( ( ( in @ X2 @ sK45 )
= $true )
| ( $true
!= ( in @ X2 @ sK46 ) ) ),
inference(cnf_transformation,[status(thm)],[f407]) ).
thf(f1454,plain,
( $false
= ( in @ sK223 @ sK45 ) ),
inference(binary_proxy_clausification,[status(thm)],[f1453]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : SEU559^1 : TPTP v9.2.1. Released v3.7.0.
% 0.00/0.12 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.14/0.33 % Computer : n025.cluster.edu
% 0.14/0.33 % Model : x86_64 x86_64
% 0.14/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.33 % Memory : 8042.1875MB
% 0.14/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.33 % CPULimit : 300
% 0.14/0.33 % WCLimit : 300
% 0.14/0.33 % DateTime : Tue Feb 24 12:26:57 EST 2026
% 0.14/0.33 % CPUTime :
% 0.14/0.33 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.18/0.45 ---- Original DTF file ---
% 0.18/0.45 thf(spec,logic,$$dhol).
% 0.18/0.45 %------------------------------------------------------------------------------
% 0.18/0.45 % File : SEU559^1 : TPTP v9.2.1. Released v3.7.0.
% 0.18/0.45 % Domain : Set Theory
% 0.18/0.45 % Problem : Preliminary Notions - Relations on Sets - Subsets
% 0.18/0.45 % Version : Especial.
% 0.18/0.45 % English : (! A:i.! B:i.(! x:i.in x A -> in x B) -> subset A B)
% 0.18/0.45
% 0.18/0.45 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.18/0.45 % Source : [Bro08]
% 0.18/0.45 % Names : ZFC061g [Bro08]
% 0.18/0.45
% 0.18/0.45 % Status : Theorem
% 0.18/0.45 % Rating : 0.22 v9.1.0, 0.25 v9.0.0, 0.30 v8.2.0, 0.31 v8.1.0, 0.09 v7.5.0, 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 v5.0.0, 0.40 v4.1.0, 0.67 v3.7.0
% 0.18/0.45 % Syntax : Number of formulae : 169 ( 78 unt; 90 typ; 78 def)
% 0.18/0.45 % Number of atoms : 428 ( 121 equ; 0 cnn)
% 0.18/0.45 % Maximal formula atoms : 78 ( 5 avg)
% 0.18/0.45 % Number of connectives : 694 ( 21 ~; 5 |; 27 &; 430 @)
% 0.18/0.45 % ( 14 <=>; 197 =>; 0 <=; 0 <~>)
% 0.18/0.45 % Maximal formula depth : 83 ( 2 avg)
% 0.18/0.45 % Number of types : 2 ( 0 usr)
% 0.18/0.45 % Number of type conns : 53 ( 53 >; 0 *; 0 +; 0 <<)
% 0.18/0.45 % Number of symbols : 93 ( 90 usr; 79 con; 0-2 aty)
% 0.18/0.45 % Number of variables : 248 ( 30 ^; 190 !; 28 ?; 248 :)
% 0.18/0.45 % SPC : TH0_THM_EQU_NAR
% 0.18/0.45
% 0.18/0.45 % Comments : http://mathgate.info/detsetitem.php?id=150
% 0.18/0.45 %------------------------------------------------------------------------------
% 0.18/0.45 thf(in_type,type,
% 0.18/0.45 in: $i > $i > $o ).
% 0.18/0.45
% 0.18/0.45 thf(exu_type,type,
% 0.18/0.45 exu: ( $i > $o ) > $o ).
% 0.18/0.45
% 0.18/0.45 thf(exu,definition,
% 0.18/0.45 ( exu
% 0.18/0.45 = ( ^ [Xphi: $i > $o] :
% 0.18/0.45 ? [Xx: $i] :
% 0.18/0.45 ( ( Xphi @ Xx )
% 0.18/0.45 & ! [Xy: $i] :
% 0.18/0.45 ( ( Xphi @ Xy )
% 0.18/0.45 => ( Xx = Xy ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 thf(setextAx_type,type,
% 0.18/0.45 setextAx: $o ).
% 0.18/0.45
% 0.18/0.45 thf(setextAx,definition,
% 0.18/0.45 ( setextAx
% 0.18/0.45 = ( ! [A: $i,B: $i] :
% 0.18/0.45 ( ! [Xx: $i] :
% 0.18/0.45 ( ( in @ Xx @ A )
% 0.18/0.45 <=> ( in @ Xx @ B ) )
% 0.18/0.45 => ( A = B ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 thf(emptyset_type,type,
% 0.18/0.45 emptyset: $i ).
% 0.18/0.45
% 0.18/0.45 thf(emptysetAx_type,type,
% 0.18/0.45 emptysetAx: $o ).
% 0.18/0.45
% 0.18/0.45 thf(emptysetAx,definition,
% 0.18/0.45 ( emptysetAx
% 0.18/0.45 = ( ! [Xx: $i] :
% 0.18/0.45 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.18/0.45
% 0.18/0.45 thf(setadjoin_type,type,
% 0.18/0.45 setadjoin: $i > $i > $i ).
% 0.18/0.45
% 0.18/0.45 thf(setadjoinAx_type,type,
% 0.18/0.45 setadjoinAx: $o ).
% 0.18/0.45
% 0.18/0.45 thf(setadjoinAx,definition,
% 0.18/0.45 ( setadjoinAx
% 0.18/0.45 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.18/0.45 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.18/0.45 <=> ( ( Xy = Xx )
% 0.18/0.45 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 thf(powerset_type,type,
% 0.18/0.45 powerset: $i > $i ).
% 0.18/0.45
% 0.18/0.45 thf(powersetAx_type,type,
% 0.18/0.45 powersetAx: $o ).
% 0.18/0.45
% 0.18/0.45 thf(powersetAx,definition,
% 0.18/0.45 ( powersetAx
% 0.18/0.45 = ( ! [A: $i,B: $i] :
% 0.18/0.45 ( ( in @ B @ ( powerset @ A ) )
% 0.18/0.45 <=> ! [Xx: $i] :
% 0.18/0.45 ( ( in @ Xx @ B )
% 0.18/0.45 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 thf(setunion_type,type,
% 0.18/0.45 setunion: $i > $i ).
% 0.18/0.45
% 0.18/0.45 thf(setunionAx_type,type,
% 0.18/0.45 setunionAx: $o ).
% 0.18/0.45
% 0.18/0.45 thf(setunionAx,definition,
% 0.18/0.45 ( setunionAx
% 0.18/0.45 = ( ! [A: $i,Xx: $i] :
% 0.18/0.45 ( ( in @ Xx @ ( setunion @ A ) )
% 0.18/0.45 <=> ? [B: $i] :
% 0.18/0.45 ( ( in @ Xx @ B )
% 0.18/0.45 & ( in @ B @ A ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 thf(omega_type,type,
% 0.18/0.45 omega: $i ).
% 0.18/0.45
% 0.18/0.45 thf(omega0Ax_type,type,
% 0.18/0.45 omega0Ax: $o ).
% 0.18/0.45
% 0.18/0.45 thf(omega0Ax,definition,
% 0.18/0.45 ( omega0Ax
% 0.18/0.45 = ( in @ emptyset @ omega ) ) ).
% 0.18/0.45
% 0.18/0.45 thf(omegaSAx_type,type,
% 0.18/0.45 omegaSAx: $o ).
% 0.18/0.45
% 0.18/0.45 thf(omegaSAx,definition,
% 0.18/0.45 ( omegaSAx
% 0.18/0.45 = ( ! [Xx: $i] :
% 0.18/0.45 ( ( in @ Xx @ omega )
% 0.18/0.45 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 thf(omegaIndAx_type,type,
% 0.18/0.45 omegaIndAx: $o ).
% 0.18/0.45
% 0.18/0.45 thf(omegaIndAx,definition,
% 0.18/0.45 ( omegaIndAx
% 0.18/0.45 = ( ! [A: $i] :
% 0.18/0.45 ( ( ( in @ emptyset @ A )
% 0.18/0.45 & ! [Xx: $i] :
% 0.18/0.45 ( ( ( in @ Xx @ omega )
% 0.18/0.45 & ( in @ Xx @ A ) )
% 0.18/0.45 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.18/0.45 => ! [Xx: $i] :
% 0.18/0.45 ( ( in @ Xx @ omega )
% 0.18/0.45 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 thf(replAx_type,type,
% 0.18/0.45 replAx: $o ).
% 0.18/0.45
% 0.18/0.45 thf(replAx,definition,
% 0.18/0.45 ( replAx
% 0.18/0.45 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.18/0.45 ( ! [Xx: $i] :
% 0.18/0.45 ( ( in @ Xx @ A )
% 0.18/0.45 => ( exu
% 0.18/0.45 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.18/0.45 => ? [B: $i] :
% 0.18/0.45 ! [Xx: $i] :
% 0.18/0.45 ( ( in @ Xx @ B )
% 0.18/0.45 <=> ? [Xy: $i] :
% 0.18/0.45 ( ( in @ Xy @ A )
% 0.18/0.45 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 thf(foundationAx_type,type,
% 0.18/0.45 foundationAx: $o ).
% 0.18/0.45
% 0.18/0.45 thf(foundationAx,definition,
% 0.18/0.45 ( foundationAx
% 0.18/0.45 = ( ! [A: $i] :
% 0.18/0.45 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.18/0.45 => ? [B: $i] :
% 0.18/0.45 ( ( in @ B @ A )
% 0.18/0.45 & ~ ? [Xx: $i] :
% 0.18/0.45 ( ( in @ Xx @ B )
% 0.18/0.45 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 thf(wellorderingAx_type,type,
% 0.18/0.45 wellorderingAx: $o ).
% 0.18/0.45
% 0.18/0.45 thf(wellorderingAx,definition,
% 0.18/0.45 ( wellorderingAx
% 0.18/0.45 = ( ! [A: $i] :
% 0.18/0.45 ? [B: $i] :
% 0.18/0.45 ( ! [C: $i] :
% 0.18/0.45 ( ( in @ C @ B )
% 0.18/0.45 => ! [Xx: $i] :
% 0.18/0.45 ( ( in @ Xx @ C )
% 0.18/0.45 => ( in @ Xx @ A ) ) )
% 0.18/0.45 & ! [Xx: $i,Xy: $i] :
% 0.18/0.45 ( ( ( in @ Xx @ A )
% 0.18/0.45 & ( in @ Xy @ A ) )
% 0.18/0.45 => ( ! [C: $i] :
% 0.18/0.45 ( ( in @ C @ B )
% 0.18/0.45 => ( ( in @ Xx @ C )
% 0.18/0.45 <=> ( in @ Xy @ C ) ) )
% 0.18/0.45 => ( Xx = Xy ) ) )
% 0.18/0.45 & ! [C: $i,D: $i] :
% 0.18/0.45 ( ( ( in @ C @ B )
% 0.18/0.45 & ( in @ D @ B ) )
% 0.18/0.45 => ( ! [Xx: $i] :
% 0.18/0.45 ( ( in @ Xx @ C )
% 0.18/0.45 => ( in @ Xx @ D ) )
% 0.18/0.45 | ! [Xx: $i] :
% 0.18/0.45 ( ( in @ Xx @ D )
% 0.18/0.45 => ( in @ Xx @ C ) ) ) )
% 0.18/0.45 & ! [C: $i] :
% 0.18/0.45 ( ( ! [Xx: $i] :
% 0.18/0.45 ( ( in @ Xx @ C )
% 0.18/0.45 => ( in @ Xx @ A ) )
% 0.18/0.45 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.18/0.45 => ? [D: $i,Xx: $i] :
% 0.18/0.45 ( ( in @ D @ B )
% 0.18/0.45 & ( in @ Xx @ C )
% 0.18/0.45 & ~ ? [Xy: $i] :
% 0.18/0.45 ( ( in @ Xy @ D )
% 0.18/0.45 & ( in @ Xy @ C ) )
% 0.18/0.45 & ! [E: $i] :
% 0.18/0.45 ( ( in @ E @ B )
% 0.18/0.45 => ( ! [Xy: $i] :
% 0.18/0.45 ( ( in @ Xy @ E )
% 0.18/0.45 => ( in @ Xy @ D ) )
% 0.18/0.45 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 thf(descr_type,type,
% 0.18/0.45 descr: ( $i > $o ) > $i ).
% 0.18/0.45
% 0.18/0.45 thf(descrp_type,type,
% 0.18/0.45 descrp: $o ).
% 0.18/0.45
% 0.18/0.45 thf(descrp,definition,
% 0.18/0.45 ( descrp
% 0.18/0.45 = ( ! [Xphi: $i > $o] :
% 0.18/0.45 ( ( exu
% 0.18/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.18/0.45 => ( Xphi
% 0.18/0.45 @ ( descr
% 0.18/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 thf(dsetconstr_type,type,
% 0.18/0.45 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.18/0.45
% 0.18/0.45 thf(dsetconstrI_type,type,
% 0.18/0.45 dsetconstrI: $o ).
% 0.18/0.45
% 0.18/0.45 thf(dsetconstrI,definition,
% 0.18/0.45 ( dsetconstrI
% 0.18/0.45 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.18/0.45 ( ( in @ Xx @ A )
% 0.18/0.45 => ( ( Xphi @ Xx )
% 0.18/0.45 => ( in @ Xx
% 0.18/0.45 @ ( dsetconstr @ A
% 0.18/0.45 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.18/0.45
% 0.18/0.45 thf(dsetconstrEL_type,type,
% 0.18/0.45 dsetconstrEL: $o ).
% 0.18/0.45
% 0.18/0.45 thf(dsetconstrEL,definition,
% 0.18/0.45 ( dsetconstrEL
% 0.29/0.45 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.29/0.45 ( ( in @ Xx
% 0.29/0.45 @ ( dsetconstr @ A
% 0.29/0.45 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.29/0.45 => ( in @ Xx @ A ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(dsetconstrER_type,type,
% 0.29/0.45 dsetconstrER: $o ).
% 0.29/0.45
% 0.29/0.45 thf(dsetconstrER,definition,
% 0.29/0.45 ( dsetconstrER
% 0.29/0.45 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.29/0.45 ( ( in @ Xx
% 0.29/0.45 @ ( dsetconstr @ A
% 0.29/0.45 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.29/0.45 => ( Xphi @ Xx ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(exuE1_type,type,
% 0.29/0.45 exuE1: $o ).
% 0.29/0.45
% 0.29/0.45 thf(exuE1,definition,
% 0.29/0.45 ( exuE1
% 0.29/0.45 = ( ! [Xphi: $i > $o] :
% 0.29/0.45 ( ( exu
% 0.29/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.29/0.45 => ? [Xx: $i] :
% 0.29/0.45 ( ( Xphi @ Xx )
% 0.29/0.45 & ! [Xy: $i] :
% 0.29/0.45 ( ( Xphi @ Xy )
% 0.29/0.45 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(prop2set_type,type,
% 0.29/0.45 prop2set: $o > $i ).
% 0.29/0.45
% 0.29/0.45 thf(prop2setE_type,type,
% 0.29/0.45 prop2setE: $o ).
% 0.29/0.45
% 0.29/0.45 thf(prop2setE,definition,
% 0.29/0.45 ( prop2setE
% 0.29/0.45 = ( ! [Xphi: $o,Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.29/0.45 => Xphi ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(emptysetE_type,type,
% 0.29/0.45 emptysetE: $o ).
% 0.29/0.45
% 0.29/0.45 thf(emptysetE,definition,
% 0.29/0.45 ( emptysetE
% 0.29/0.45 = ( ! [Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ emptyset )
% 0.29/0.45 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(emptysetimpfalse_type,type,
% 0.29/0.45 emptysetimpfalse: $o ).
% 0.29/0.45
% 0.29/0.45 thf(emptysetimpfalse,definition,
% 0.29/0.45 ( emptysetimpfalse
% 0.29/0.45 = ( ! [Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ emptyset )
% 0.29/0.45 => $false ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(notinemptyset_type,type,
% 0.29/0.45 notinemptyset: $o ).
% 0.29/0.45
% 0.29/0.45 thf(notinemptyset,definition,
% 0.29/0.45 ( notinemptyset
% 0.29/0.45 = ( ! [Xx: $i] :
% 0.29/0.45 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(exuE3e_type,type,
% 0.29/0.45 exuE3e: $o ).
% 0.29/0.45
% 0.29/0.45 thf(exuE3e,definition,
% 0.29/0.45 ( exuE3e
% 0.29/0.45 = ( ! [Xphi: $i > $o] :
% 0.29/0.45 ( ( exu
% 0.29/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.29/0.45 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(setext_type,type,
% 0.29/0.45 setext: $o ).
% 0.29/0.45
% 0.29/0.45 thf(setext,definition,
% 0.29/0.45 ( setext
% 0.29/0.45 = ( ! [A: $i,B: $i] :
% 0.29/0.45 ( ! [Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ A )
% 0.29/0.45 => ( in @ Xx @ B ) )
% 0.29/0.45 => ( ! [Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ B )
% 0.29/0.45 => ( in @ Xx @ A ) )
% 0.29/0.45 => ( A = B ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(emptyI_type,type,
% 0.29/0.45 emptyI: $o ).
% 0.29/0.45
% 0.29/0.45 thf(emptyI,definition,
% 0.29/0.45 ( emptyI
% 0.29/0.45 = ( ! [A: $i] :
% 0.29/0.45 ( ! [Xx: $i] :
% 0.29/0.45 ~ ( in @ Xx @ A )
% 0.29/0.45 => ( A = emptyset ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(noeltsimpempty_type,type,
% 0.29/0.45 noeltsimpempty: $o ).
% 0.29/0.45
% 0.29/0.45 thf(noeltsimpempty,definition,
% 0.29/0.45 ( noeltsimpempty
% 0.29/0.45 = ( ! [A: $i] :
% 0.29/0.45 ( ! [Xx: $i] :
% 0.29/0.45 ~ ( in @ Xx @ A )
% 0.29/0.45 => ( A = emptyset ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(setbeta_type,type,
% 0.29/0.45 setbeta: $o ).
% 0.29/0.45
% 0.29/0.45 thf(setbeta,definition,
% 0.29/0.45 ( setbeta
% 0.29/0.45 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ A )
% 0.29/0.45 => ( ( in @ Xx
% 0.29/0.45 @ ( dsetconstr @ A
% 0.29/0.45 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.29/0.45 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(nonempty_type,type,
% 0.29/0.45 nonempty: $i > $o ).
% 0.29/0.45
% 0.29/0.45 thf(nonempty,definition,
% 0.29/0.45 ( nonempty
% 0.29/0.45 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(nonemptyE1_type,type,
% 0.29/0.45 nonemptyE1: $o ).
% 0.29/0.45
% 0.29/0.45 thf(nonemptyE1,definition,
% 0.29/0.45 ( nonemptyE1
% 0.29/0.45 = ( ! [A: $i] :
% 0.29/0.45 ( ( nonempty @ A )
% 0.29/0.45 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(nonemptyI_type,type,
% 0.29/0.45 nonemptyI: $o ).
% 0.29/0.45
% 0.29/0.45 thf(nonemptyI,definition,
% 0.29/0.45 ( nonemptyI
% 0.29/0.45 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ A )
% 0.29/0.45 => ( ( Xphi @ Xx )
% 0.29/0.45 => ( nonempty
% 0.29/0.45 @ ( dsetconstr @ A
% 0.29/0.45 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(nonemptyI1_type,type,
% 0.29/0.45 nonemptyI1: $o ).
% 0.29/0.45
% 0.29/0.45 thf(nonemptyI1,definition,
% 0.29/0.45 ( nonemptyI1
% 0.29/0.45 = ( ! [A: $i] :
% 0.29/0.45 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.29/0.45 => ( nonempty @ A ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(setadjoinIL_type,type,
% 0.29/0.45 setadjoinIL: $o ).
% 0.29/0.45
% 0.29/0.45 thf(setadjoinIL,definition,
% 0.29/0.45 ( setadjoinIL
% 0.29/0.45 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(emptyinunitempty_type,type,
% 0.29/0.45 emptyinunitempty: $o ).
% 0.29/0.45
% 0.29/0.45 thf(emptyinunitempty,definition,
% 0.29/0.45 ( emptyinunitempty
% 0.29/0.45 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(setadjoinIR_type,type,
% 0.29/0.45 setadjoinIR: $o ).
% 0.29/0.45
% 0.29/0.45 thf(setadjoinIR,definition,
% 0.29/0.45 ( setadjoinIR
% 0.29/0.45 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.29/0.45 ( ( in @ Xy @ A )
% 0.29/0.45 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(setadjoinE_type,type,
% 0.29/0.45 setadjoinE: $o ).
% 0.29/0.45
% 0.29/0.45 thf(setadjoinE,definition,
% 0.29/0.45 ( setadjoinE
% 0.29/0.45 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.29/0.45 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.29/0.45 => ! [Xphi: $o] :
% 0.29/0.45 ( ( ( Xy = Xx )
% 0.29/0.45 => Xphi )
% 0.29/0.45 => ( ( ( in @ Xy @ A )
% 0.29/0.45 => Xphi )
% 0.29/0.45 => Xphi ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(setadjoinOr_type,type,
% 0.29/0.45 setadjoinOr: $o ).
% 0.29/0.45
% 0.29/0.45 thf(setadjoinOr,definition,
% 0.29/0.45 ( setadjoinOr
% 0.29/0.45 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.29/0.45 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.29/0.45 => ( ( Xy = Xx )
% 0.29/0.45 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(setoftrueEq_type,type,
% 0.29/0.45 setoftrueEq: $o ).
% 0.29/0.45
% 0.29/0.45 thf(setoftrueEq,definition,
% 0.29/0.45 ( setoftrueEq
% 0.29/0.45 = ( ! [A: $i] :
% 0.29/0.45 ( ( dsetconstr @ A
% 0.29/0.45 @ ^ [Xx: $i] : $true )
% 0.29/0.45 = A ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(powersetI_type,type,
% 0.29/0.45 powersetI: $o ).
% 0.29/0.45
% 0.29/0.45 thf(powersetI,definition,
% 0.29/0.45 ( powersetI
% 0.29/0.45 = ( ! [A: $i,B: $i] :
% 0.29/0.45 ( ! [Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ B )
% 0.29/0.45 => ( in @ Xx @ A ) )
% 0.29/0.45 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(emptyinPowerset_type,type,
% 0.29/0.45 emptyinPowerset: $o ).
% 0.29/0.45
% 0.29/0.45 thf(emptyinPowerset,definition,
% 0.29/0.45 ( emptyinPowerset
% 0.29/0.45 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(emptyInPowerset_type,type,
% 0.29/0.45 emptyInPowerset: $o ).
% 0.29/0.45
% 0.29/0.45 thf(emptyInPowerset,definition,
% 0.29/0.45 ( emptyInPowerset
% 0.29/0.45 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(powersetE_type,type,
% 0.29/0.45 powersetE: $o ).
% 0.29/0.45
% 0.29/0.45 thf(powersetE,definition,
% 0.29/0.45 ( powersetE
% 0.29/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.45 ( ( in @ B @ ( powerset @ A ) )
% 0.29/0.45 => ( ( in @ Xx @ B )
% 0.29/0.45 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(setunionI_type,type,
% 0.29/0.45 setunionI: $o ).
% 0.29/0.45
% 0.29/0.45 thf(setunionI,definition,
% 0.29/0.45 ( setunionI
% 0.29/0.45 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.29/0.45 ( ( in @ Xx @ B )
% 0.29/0.45 => ( ( in @ B @ A )
% 0.29/0.45 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(setunionE_type,type,
% 0.29/0.45 setunionE: $o ).
% 0.29/0.45
% 0.29/0.45 thf(setunionE,definition,
% 0.29/0.45 ( setunionE
% 0.29/0.45 = ( ! [A: $i,Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ ( setunion @ A ) )
% 0.29/0.45 => ! [Xphi: $o] :
% 0.29/0.45 ( ! [B: $i] :
% 0.29/0.45 ( ( in @ Xx @ B )
% 0.29/0.45 => ( ( in @ B @ A )
% 0.29/0.45 => Xphi ) )
% 0.29/0.45 => Xphi ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(subPowSU_type,type,
% 0.29/0.45 subPowSU: $o ).
% 0.29/0.45
% 0.29/0.45 thf(subPowSU,definition,
% 0.29/0.45 ( subPowSU
% 0.29/0.45 = ( ! [A: $i,Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ A )
% 0.29/0.45 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(exuE2_type,type,
% 0.29/0.45 exuE2: $o ).
% 0.29/0.45
% 0.29/0.45 thf(exuE2,definition,
% 0.29/0.45 ( exuE2
% 0.29/0.45 = ( ! [Xphi: $i > $o] :
% 0.29/0.45 ( ( exu
% 0.29/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.29/0.45 => ? [Xx: $i] :
% 0.29/0.45 ! [Xy: $i] :
% 0.29/0.45 ( ( Xphi @ Xy )
% 0.29/0.45 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(nonemptyImpWitness_type,type,
% 0.29/0.45 nonemptyImpWitness: $o ).
% 0.29/0.45
% 0.29/0.45 thf(nonemptyImpWitness,definition,
% 0.29/0.45 ( nonemptyImpWitness
% 0.29/0.45 = ( ! [A: $i] :
% 0.29/0.45 ( ( nonempty @ A )
% 0.29/0.45 => ? [Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ A )
% 0.29/0.45 & $true ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(uniqinunit_type,type,
% 0.29/0.45 uniqinunit: $o ).
% 0.29/0.45
% 0.29/0.45 thf(uniqinunit,definition,
% 0.29/0.45 ( uniqinunit
% 0.29/0.45 = ( ! [Xx: $i,Xy: $i] :
% 0.29/0.45 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.29/0.45 => ( Xx = Xy ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(notinsingleton_type,type,
% 0.29/0.45 notinsingleton: $o ).
% 0.29/0.45
% 0.29/0.45 thf(notinsingleton,definition,
% 0.29/0.45 ( notinsingleton
% 0.29/0.45 = ( ! [Xx: $i,Xy: $i] :
% 0.29/0.45 ( ( Xx != Xy )
% 0.29/0.45 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(eqinunit_type,type,
% 0.29/0.45 eqinunit: $o ).
% 0.29/0.45
% 0.29/0.45 thf(eqinunit,definition,
% 0.29/0.45 ( eqinunit
% 0.29/0.45 = ( ! [Xx: $i,Xy: $i] :
% 0.29/0.45 ( ( Xx = Xy )
% 0.29/0.45 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(singletonsswitch_type,type,
% 0.29/0.45 singletonsswitch: $o ).
% 0.29/0.45
% 0.29/0.45 thf(singletonsswitch,definition,
% 0.29/0.45 ( singletonsswitch
% 0.29/0.45 = ( ! [Xx: $i,Xy: $i] :
% 0.29/0.45 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.29/0.45 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(upairsetE_type,type,
% 0.29/0.45 upairsetE: $o ).
% 0.29/0.45
% 0.29/0.45 thf(upairsetE,definition,
% 0.29/0.45 ( upairsetE
% 0.29/0.45 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.29/0.45 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.29/0.45 => ( ( Xz = Xx )
% 0.29/0.45 | ( Xz = Xy ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(upairsetIL_type,type,
% 0.29/0.45 upairsetIL: $o ).
% 0.29/0.45
% 0.29/0.45 thf(upairsetIL,definition,
% 0.29/0.45 ( upairsetIL
% 0.29/0.45 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(upairsetIR_type,type,
% 0.29/0.45 upairsetIR: $o ).
% 0.29/0.45
% 0.29/0.45 thf(upairsetIR,definition,
% 0.29/0.45 ( upairsetIR
% 0.29/0.45 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(emptyE1_type,type,
% 0.29/0.45 emptyE1: $o ).
% 0.29/0.45
% 0.29/0.45 thf(emptyE1,definition,
% 0.29/0.45 ( emptyE1
% 0.29/0.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.29/0.45 ( ? [Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ A )
% 0.29/0.45 & ( Xphi @ Xx ) )
% 0.29/0.45 => ( ( ( dsetconstr @ A
% 0.29/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.29/0.45 = emptyset )
% 0.29/0.45 => $false ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(vacuousDall_type,type,
% 0.29/0.45 vacuousDall: $o ).
% 0.29/0.45
% 0.29/0.45 thf(vacuousDall,definition,
% 0.29/0.45 ( vacuousDall
% 0.29/0.45 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ emptyset )
% 0.29/0.45 => ( Xphi @ Xx ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(quantDeMorgan1_type,type,
% 0.29/0.45 quantDeMorgan1: $o ).
% 0.29/0.45
% 0.29/0.45 thf(quantDeMorgan1,definition,
% 0.29/0.45 ( quantDeMorgan1
% 0.29/0.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.29/0.45 ( ~ ! [Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ A )
% 0.29/0.45 => ( Xphi @ Xx ) )
% 0.29/0.45 => ? [Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ A )
% 0.29/0.45 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(quantDeMorgan2_type,type,
% 0.29/0.45 quantDeMorgan2: $o ).
% 0.29/0.45
% 0.29/0.45 thf(quantDeMorgan2,definition,
% 0.29/0.45 ( quantDeMorgan2
% 0.29/0.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.29/0.45 ( ! [Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ A )
% 0.29/0.45 => ~ ( Xphi @ Xx ) )
% 0.29/0.45 => ~ ? [Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ A )
% 0.29/0.45 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(quantDeMorgan3_type,type,
% 0.29/0.45 quantDeMorgan3: $o ).
% 0.29/0.45
% 0.29/0.45 thf(quantDeMorgan3,definition,
% 0.29/0.45 ( quantDeMorgan3
% 0.29/0.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.29/0.45 ( ~ ? [Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ A )
% 0.29/0.45 & ( Xphi @ Xx ) )
% 0.29/0.45 => ! [Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ A )
% 0.29/0.45 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(quantDeMorgan4_type,type,
% 0.29/0.45 quantDeMorgan4: $o ).
% 0.29/0.45
% 0.29/0.45 thf(quantDeMorgan4,definition,
% 0.29/0.45 ( quantDeMorgan4
% 0.29/0.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.29/0.45 ( ? [Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ A )
% 0.29/0.45 & ~ ( Xphi @ Xx ) )
% 0.29/0.45 => ~ ! [Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ A )
% 0.29/0.45 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(prop2setI_type,type,
% 0.29/0.45 prop2setI: $o ).
% 0.29/0.45
% 0.29/0.45 thf(prop2setI,definition,
% 0.29/0.45 ( prop2setI
% 0.29/0.45 = ( ! [Xphi: $o] :
% 0.29/0.45 ( Xphi
% 0.29/0.45 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(set2prop_type,type,
% 0.29/0.45 set2prop: $i > $o ).
% 0.29/0.45
% 0.29/0.45 thf(prop2set2propI_type,type,
% 0.29/0.45 prop2set2propI: $o ).
% 0.29/0.45
% 0.29/0.45 thf(prop2set2propI,definition,
% 0.29/0.45 ( prop2set2propI
% 0.29/0.45 = ( ! [Xphi: $o] :
% 0.29/0.45 ( Xphi
% 0.29/0.45 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(notdexE_type,type,
% 0.29/0.45 notdexE: $o ).
% 0.29/0.45
% 0.29/0.45 thf(notdexE,definition,
% 0.29/0.45 ( notdexE
% 0.29/0.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.29/0.45 ( ~ ? [Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ A )
% 0.29/0.45 & ( Xphi @ Xx ) )
% 0.29/0.45 => ! [Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ A )
% 0.29/0.45 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(notdallE_type,type,
% 0.29/0.45 notdallE: $o ).
% 0.29/0.45
% 0.29/0.45 thf(notdallE,definition,
% 0.29/0.45 ( notdallE
% 0.29/0.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.29/0.45 ( ~ ! [Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ A )
% 0.29/0.45 => ( Xphi @ Xx ) )
% 0.29/0.45 => ? [Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ A )
% 0.29/0.45 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(exuI1_type,type,
% 0.29/0.45 exuI1: $o ).
% 0.29/0.45
% 0.29/0.45 thf(exuI1,definition,
% 0.29/0.45 ( exuI1
% 0.29/0.45 = ( ! [Xphi: $i > $o] :
% 0.29/0.45 ( ? [Xx: $i] :
% 0.29/0.45 ( ( Xphi @ Xx )
% 0.29/0.45 & ! [Xy: $i] :
% 0.29/0.45 ( ( Xphi @ Xy )
% 0.29/0.45 => ( Xx = Xy ) ) )
% 0.29/0.45 => ( exu
% 0.29/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(exuI3_type,type,
% 0.29/0.45 exuI3: $o ).
% 0.29/0.45
% 0.29/0.45 thf(exuI3,definition,
% 0.29/0.45 ( exuI3
% 0.29/0.45 = ( ! [Xphi: $i > $o] :
% 0.29/0.45 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.29/0.45 => ( ! [Xx: $i,Xy: $i] :
% 0.29/0.45 ( ( Xphi @ Xx )
% 0.29/0.45 => ( ( Xphi @ Xy )
% 0.29/0.45 => ( Xx = Xy ) ) )
% 0.29/0.45 => ( exu
% 0.29/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(exuI2_type,type,
% 0.29/0.45 exuI2: $o ).
% 0.29/0.45
% 0.29/0.45 thf(exuI2,definition,
% 0.29/0.45 ( exuI2
% 0.29/0.45 = ( ! [Xphi: $i > $o] :
% 0.29/0.45 ( ? [Xx: $i] :
% 0.29/0.45 ! [Xy: $i] :
% 0.29/0.45 ( ( Xphi @ Xy )
% 0.29/0.45 <=> ( Xy = Xx ) )
% 0.29/0.45 => ( exu
% 0.29/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(inCongP_type,type,
% 0.29/0.45 inCongP: $o ).
% 0.29/0.45
% 0.29/0.45 thf(inCongP,definition,
% 0.29/0.45 ( inCongP
% 0.29/0.45 = ( ! [A: $i,B: $i] :
% 0.29/0.45 ( ( A = B )
% 0.29/0.45 => ! [Xx: $i,Xy: $i] :
% 0.29/0.45 ( ( Xx = Xy )
% 0.29/0.45 => ( ( in @ Xx @ A )
% 0.29/0.45 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(in__Cong_type,type,
% 0.29/0.45 in__Cong: $o ).
% 0.29/0.45
% 0.29/0.45 thf(in__Cong,definition,
% 0.29/0.45 ( in__Cong
% 0.29/0.45 = ( ! [A: $i,B: $i] :
% 0.29/0.45 ( ( A = B )
% 0.29/0.45 => ! [Xx: $i,Xy: $i] :
% 0.29/0.45 ( ( Xx = Xy )
% 0.29/0.45 => ( ( in @ Xx @ A )
% 0.29/0.45 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(exuE3u_type,type,
% 0.29/0.45 exuE3u: $o ).
% 0.29/0.45
% 0.29/0.45 thf(exuE3u,definition,
% 0.29/0.45 ( exuE3u
% 0.29/0.45 = ( ! [Xphi: $i > $o] :
% 0.29/0.45 ( ( exu
% 0.29/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.29/0.45 => ! [Xx: $i,Xy: $i] :
% 0.29/0.45 ( ( Xphi @ Xx )
% 0.29/0.45 => ( ( Xphi @ Xy )
% 0.29/0.45 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(exu__Cong_type,type,
% 0.29/0.45 exu__Cong: $o ).
% 0.29/0.45
% 0.29/0.45 thf(exu__Cong,definition,
% 0.29/0.45 ( exu__Cong
% 0.29/0.45 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.29/0.45 ( ! [Xx: $i,Xy: $i] :
% 0.29/0.45 ( ( Xx = Xy )
% 0.29/0.45 => ( ( Xphi @ Xx )
% 0.29/0.45 <=> ( Xpsi @ Xy ) ) )
% 0.29/0.45 => ( ( exu
% 0.29/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.29/0.45 <=> ( exu
% 0.29/0.45 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(emptyset__Cong_type,type,
% 0.29/0.45 emptyset__Cong: $o ).
% 0.29/0.45
% 0.29/0.45 thf(emptyset__Cong,definition,
% 0.29/0.45 ( emptyset__Cong
% 0.29/0.45 = ( emptyset = emptyset ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(setadjoin__Cong_type,type,
% 0.29/0.45 setadjoin__Cong: $o ).
% 0.29/0.45
% 0.29/0.45 thf(setadjoin__Cong,definition,
% 0.29/0.45 ( setadjoin__Cong
% 0.29/0.45 = ( ! [Xx: $i,Xy: $i] :
% 0.29/0.45 ( ( Xx = Xy )
% 0.29/0.45 => ! [Xz: $i,Xu: $i] :
% 0.29/0.45 ( ( Xz = Xu )
% 0.29/0.45 => ( ( setadjoin @ Xx @ Xz )
% 0.29/0.45 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(powerset__Cong_type,type,
% 0.29/0.45 powerset__Cong: $o ).
% 0.29/0.45
% 0.29/0.45 thf(powerset__Cong,definition,
% 0.29/0.45 ( powerset__Cong
% 0.29/0.45 = ( ! [A: $i,B: $i] :
% 0.29/0.45 ( ( A = B )
% 0.29/0.45 => ( ( powerset @ A )
% 0.29/0.45 = ( powerset @ B ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(setunion__Cong_type,type,
% 0.29/0.45 setunion__Cong: $o ).
% 0.29/0.45
% 0.29/0.45 thf(setunion__Cong,definition,
% 0.29/0.45 ( setunion__Cong
% 0.29/0.45 = ( ! [A: $i,B: $i] :
% 0.29/0.45 ( ( A = B )
% 0.29/0.45 => ( ( setunion @ A )
% 0.29/0.45 = ( setunion @ B ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(omega__Cong_type,type,
% 0.29/0.45 omega__Cong: $o ).
% 0.29/0.45
% 0.29/0.45 thf(omega__Cong,definition,
% 0.29/0.45 ( omega__Cong
% 0.29/0.45 = ( omega = omega ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(exuEu_type,type,
% 0.29/0.45 exuEu: $o ).
% 0.29/0.45
% 0.29/0.45 thf(exuEu,definition,
% 0.29/0.45 ( exuEu
% 0.29/0.45 = ( ! [Xphi: $i > $o] :
% 0.29/0.45 ( ( exu
% 0.29/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.29/0.45 => ! [Xx: $i,Xy: $i] :
% 0.29/0.45 ( ( Xphi @ Xx )
% 0.29/0.45 => ( ( Xphi @ Xy )
% 0.29/0.45 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(descr__Cong_type,type,
% 0.29/0.45 descr__Cong: $o ).
% 0.29/0.45
% 0.29/0.45 thf(descr__Cong,definition,
% 0.29/0.45 ( descr__Cong
% 0.29/0.45 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.29/0.45 ( ! [Xx: $i,Xy: $i] :
% 0.29/0.45 ( ( Xx = Xy )
% 0.29/0.45 => ( ( Xphi @ Xx )
% 0.29/0.45 <=> ( Xpsi @ Xy ) ) )
% 0.29/0.45 => ( ( exu
% 0.29/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.29/0.45 => ( ( exu
% 0.29/0.45 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.29/0.45 => ( ( descr
% 0.29/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.29/0.45 = ( descr
% 0.29/0.45 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(dsetconstr__Cong_type,type,
% 0.29/0.45 dsetconstr__Cong: $o ).
% 0.29/0.45
% 0.29/0.45 thf(dsetconstr__Cong,definition,
% 0.29/0.45 ( dsetconstr__Cong
% 0.29/0.45 = ( ! [A: $i,B: $i] :
% 0.29/0.45 ( ( A = B )
% 0.29/0.45 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.29/0.45 ( ! [Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ A )
% 0.29/0.45 => ! [Xy: $i] :
% 0.29/0.45 ( ( in @ Xy @ B )
% 0.29/0.45 => ( ( Xx = Xy )
% 0.29/0.45 => ( ( Xphi @ Xx )
% 0.29/0.45 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.29/0.45 => ( ( dsetconstr @ A
% 0.29/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.29/0.45 = ( dsetconstr @ B
% 0.29/0.45 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(subset_type,type,
% 0.29/0.45 subset: $i > $i > $o ).
% 0.29/0.45
% 0.29/0.45 thf(subset,definition,
% 0.29/0.45 ( subset
% 0.29/0.45 = ( ^ [A: $i,B: $i] :
% 0.29/0.45 ! [Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ A )
% 0.29/0.45 => ( in @ Xx @ B ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(disjoint_type,type,
% 0.29/0.45 disjoint: $i > $i > $o ).
% 0.29/0.45
% 0.29/0.45 thf(setsmeet_type,type,
% 0.29/0.45 setsmeet: $i > $i > $o ).
% 0.29/0.45
% 0.29/0.45 thf(subsetI1,conjecture,
% 0.29/0.45 ( setextAx
% 0.29/0.45 => ( emptysetAx
% 0.29/0.45 => ( setadjoinAx
% 0.29/0.45 => ( powersetAx
% 0.29/0.45 => ( setunionAx
% 0.29/0.45 => ( omega0Ax
% 0.29/0.45 => ( omegaSAx
% 0.29/0.45 => ( omegaIndAx
% 0.29/0.45 => ( replAx
% 0.29/0.45 => ( foundationAx
% 0.29/0.45 => ( wellorderingAx
% 0.29/0.45 => ( descrp
% 0.29/0.45 => ( dsetconstrI
% 0.29/0.45 => ( dsetconstrEL
% 0.29/0.45 => ( dsetconstrER
% 0.29/0.45 => ( exuE1
% 0.29/0.45 => ( prop2setE
% 0.29/0.45 => ( emptysetE
% 0.29/0.45 => ( emptysetimpfalse
% 0.29/0.45 => ( notinemptyset
% 0.29/0.45 => ( exuE3e
% 0.29/0.45 => ( setext
% 0.29/0.45 => ( emptyI
% 0.29/0.45 => ( noeltsimpempty
% 0.29/0.45 => ( setbeta
% 0.29/0.45 => ( nonemptyE1
% 0.29/0.45 => ( nonemptyI
% 0.29/0.45 => ( nonemptyI1
% 0.29/0.45 => ( setadjoinIL
% 0.29/0.45 => ( emptyinunitempty
% 0.29/0.45 => ( setadjoinIR
% 0.29/0.45 => ( setadjoinE
% 0.29/0.45 => ( setadjoinOr
% 0.29/0.45 => ( setoftrueEq
% 0.29/0.45 => ( powersetI
% 0.29/0.45 => ( emptyinPowerset
% 0.29/0.45 => ( emptyInPowerset
% 0.29/0.45 => ( powersetE
% 0.29/0.45 => ( setunionI
% 0.29/0.45 => ( setunionE
% 0.29/0.45 => ( subPowSU
% 0.29/0.45 => ( exuE2
% 0.29/0.45 => ( nonemptyImpWitness
% 0.29/0.45 => ( uniqinunit
% 0.29/0.45 => ( notinsingleton
% 0.29/0.45 => ( eqinunit
% 0.29/0.45 => ( singletonsswitch
% 0.29/0.45 => ( upairsetE
% 0.29/0.45 => ( upairsetIL
% 0.29/0.45 => ( upairsetIR
% 0.29/0.45 => ( emptyE1
% 0.29/0.45 => ( vacuousDall
% 0.29/0.45 => ( quantDeMorgan1
% 0.29/0.45 => ( quantDeMorgan2
% 0.29/0.45 => ( quantDeMorgan3
% 0.29/0.45 => ( quantDeMorgan4
% 0.29/0.45 => ( prop2setI
% 0.29/0.45 => ( prop2set2propI
% 0.29/0.45 => ( notdexE
% 0.29/0.45 => ( notdallE
% 0.29/0.45 => ( exuI1
% 0.29/0.45 => ( exuI3
% 0.29/0.45 => ( exuI2
% 0.29/0.45 => ( inCongP
% 0.29/0.45 => ( in__Cong
% 0.29/0.45 => ( exuE3u
% 0.29/0.45 => ( exu__Cong
% 0.29/0.45 => ( emptyset__Cong
% 0.29/0.45 => ( setadjoin__Cong
% 0.29/0.45 => ( powerset__Cong
% 0.29/0.45 => ( setunion__Cong
% 0.29/0.45 => ( omega__Cong
% 0.29/0.45 => ( exuEu
% 0.29/0.45 => ( descr__Cong
% 0.29/0.45 => ( dsetconstr__Cong
% 0.29/0.45 => ! [A: $i,B: $i] :
% 0.29/0.45 ( ! [Xx: $i] :
% 0.29/0.45 ( ( in @ Xx @ A )
% 0.29/0.45 => ( in @ Xx @ B ) )
% 0.29/0.45 => ( subset @ A @ B ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 %------------------------------------------------------------------------------
% 0.29/0.45 ------------------------
% 0.30/1.40 ---- Embedded in THF ---
% 0.30/1.40 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 0.30/1.40 %%% Generated on Tue Feb 24 12:26:58 EST 2026
% 0.30/1.40 %%% using '$$dhol' embedding, version 1.3.0.
% 0.30/1.40 %%% Logic specification used:
% 0.30/1.40 %%% thf(spec, logic, $$dhol).
% 0.30/1.40
% 0.30/1.40 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.zNeidnJOtA/DTF2DTF7736.p
% See solution above
% 0.30/1.40 ------------------------
% 0.30/1.42 ---- Cleaned THF ---
% 0.30/1.42 thf(in_type,type,
% 0.30/1.42 in: $i > $i > $o ).
% 0.30/1.42
% 0.30/1.42 thf(exu_type,type,
% 0.30/1.42 exu: ( $i > $o ) > $o ).
% 0.30/1.42
% 0.30/1.42 thf(exu,definition,
% 0.30/1.42 ( exu
% 0.30/1.42 = ( ^ [Xphi: $i > $o] :
% 0.30/1.42 ? [Xx: $i] :
% 0.30/1.42 ( ( Xphi @ Xx )
% 0.30/1.42 & ! [Xy: $i] :
% 0.30/1.42 ( ( Xphi @ Xy )
% 0.30/1.42 => ( Xx = Xy ) ) ) ) ) ).
% 0.30/1.42
% 0.30/1.42 thf(setextAx_type,type,
% 0.30/1.42 setextAx: $o ).
% 0.30/1.42
% 0.30/1.42 thf(setextAx,definition,
% 0.30/1.42 ( setextAx
% 0.30/1.42 = ( ! [A: $i,B: $i] :
% 0.30/1.42 ( ! [Xx: $i] :
% 0.30/1.42 ( ( in @ Xx @ A )
% 0.30/1.42 <=> ( in @ Xx @ B ) )
% 0.30/1.42 => ( A = B ) ) ) ) ).
% 0.30/1.42
% 0.30/1.42 thf(emptyset_type,type,
% 0.30/1.42 emptyset: $i ).
% 0.30/1.42
% 0.30/1.42 thf(emptysetAx_type,type,
% 0.30/1.42 emptysetAx: $o ).
% 0.30/1.42
% 0.30/1.42 thf(emptysetAx,definition,
% 0.30/1.42 ( emptysetAx
% 0.30/1.42 = ( ! [Xx: $i] :
% 0.30/1.42 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.30/1.42
% 0.30/1.42 thf(setadjoin_type,type,
% 0.30/1.42 setadjoin: $i > $i > $i ).
% 0.30/1.42
% 0.30/1.42 thf(setadjoinAx_type,type,
% 0.30/1.42 setadjoinAx: $o ).
% 0.30/1.42
% 0.30/1.42 thf(setadjoinAx,definition,
% 0.30/1.42 ( setadjoinAx
% 0.30/1.42 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.30/1.42 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.30/1.42 <=> ( ( Xy = Xx )
% 0.30/1.42 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.30/1.42
% 0.30/1.42 thf(powerset_type,type,
% 0.30/1.42 powerset: $i > $i ).
% 0.30/1.42
% 0.30/1.42 thf(powersetAx_type,type,
% 0.30/1.42 powersetAx: $o ).
% 0.30/1.42
% 0.30/1.42 thf(powersetAx,definition,
% 0.30/1.42 ( powersetAx
% 0.30/1.42 = ( ! [A: $i,B: $i] :
% 0.30/1.42 ( ( in @ B @ ( powerset @ A ) )
% 0.30/1.42 <=> ! [Xx: $i] :
% 0.30/1.42 ( ( in @ Xx @ B )
% 0.30/1.42 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.30/1.42
% 0.30/1.42 thf(setunion_type,type,
% 0.30/1.42 setunion: $i > $i ).
% 0.30/1.42
% 0.30/1.42 thf(setunionAx_type,type,
% 0.30/1.42 setunionAx: $o ).
% 0.30/1.42
% 0.30/1.42 thf(setunionAx,definition,
% 0.30/1.42 ( setunionAx
% 0.30/1.42 = ( ! [A: $i,Xx: $i] :
% 0.30/1.42 ( ( in @ Xx @ ( setunion @ A ) )
% 0.30/1.42 <=> ? [B: $i] :
% 0.30/1.42 ( ( in @ Xx @ B )
% 0.30/1.42 & ( in @ B @ A ) ) ) ) ) ).
% 0.30/1.42
% 0.30/1.42 thf(omega_type,type,
% 0.30/1.42 omega: $i ).
% 0.30/1.42
% 0.30/1.42 thf(omega0Ax_type,type,
% 0.30/1.42 omega0Ax: $o ).
% 0.30/1.42
% 0.30/1.42 thf(omega0Ax,definition,
% 0.30/1.42 ( omega0Ax
% 0.30/1.42 = ( in @ emptyset @ omega ) ) ).
% 0.30/1.42
% 0.30/1.42 thf(omegaSAx_type,type,
% 0.30/1.42 omegaSAx: $o ).
% 0.30/1.42
% 0.30/1.42 thf(omegaSAx,definition,
% 0.30/1.42 ( omegaSAx
% 0.30/1.42 = ( ! [Xx: $i] :
% 0.30/1.42 ( ( in @ Xx @ omega )
% 0.30/1.42 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.30/1.42
% 0.30/1.42 thf(omegaIndAx_type,type,
% 0.30/1.42 omegaIndAx: $o ).
% 0.30/1.42
% 0.30/1.42 thf(omegaIndAx,definition,
% 0.30/1.42 ( omegaIndAx
% 0.30/1.42 = ( ! [A: $i] :
% 0.30/1.42 ( ( ( in @ emptyset @ A )
% 0.30/1.42 & ! [Xx: $i] :
% 0.30/1.42 ( ( ( in @ Xx @ omega )
% 0.30/1.42 & ( in @ Xx @ A ) )
% 0.30/1.42 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.30/1.42 => ! [Xx: $i] :
% 0.30/1.42 ( ( in @ Xx @ omega )
% 0.30/1.42 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.30/1.42
% 0.30/1.42 thf(replAx_type,type,
% 0.30/1.42 replAx: $o ).
% 0.30/1.42
% 0.30/1.42 thf(replAx,definition,
% 0.30/1.42 ( replAx
% 0.30/1.42 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.30/1.42 ( ! [Xx: $i] :
% 0.30/1.42 ( ( in @ Xx @ A )
% 0.30/1.42 => ( exu
% 0.30/1.42 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.30/1.42 => ? [B: $i] :
% 0.30/1.42 ! [Xx: $i] :
% 0.30/1.42 ( ( in @ Xx @ B )
% 0.30/1.42 <=> ? [Xy: $i] :
% 0.30/1.42 ( ( in @ Xy @ A )
% 0.30/1.42 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.30/1.42
% 0.30/1.42 thf(foundationAx_type,type,
% 0.30/1.42 foundationAx: $o ).
% 0.30/1.42
% 0.30/1.42 thf(foundationAx,definition,
% 0.30/1.42 ( foundationAx
% 0.30/1.42 = ( ! [A: $i] :
% 0.30/1.42 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.30/1.42 => ? [B: $i] :
% 0.30/1.42 ( ( in @ B @ A )
% 0.30/1.42 & ~ ? [Xx: $i] :
% 0.30/1.42 ( ( in @ Xx @ B )
% 0.30/1.42 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.30/1.42
% 0.30/1.42 thf(wellorderingAx_type,type,
% 0.30/1.42 wellorderingAx: $o ).
% 0.30/1.42
% 0.30/1.42 thf(wellorderingAx,definition,
% 0.30/1.42 ( wellorderingAx
% 0.30/1.42 = ( ! [A: $i] :
% 0.30/1.42 ? [B: $i] :
% 0.30/1.42 ( ! [C: $i] :
% 0.30/1.42 ( ( in @ C @ B )
% 0.30/1.42 => ! [Xx: $i] :
% 0.30/1.42 ( ( in @ Xx @ C )
% 0.30/1.42 => ( in @ Xx @ A ) ) )
% 0.30/1.42 & ! [Xx: $i,Xy: $i] :
% 0.30/1.42 ( ( ( in @ Xx @ A )
% 0.30/1.42 & ( in @ Xy @ A ) )
% 0.30/1.42 => ( ! [C: $i] :
% 0.30/1.42 ( ( in @ C @ B )
% 0.30/1.42 => ( ( in @ Xx @ C )
% 0.30/1.42 <=> ( in @ Xy @ C ) ) )
% 0.30/1.42 => ( Xx = Xy ) ) )
% 0.30/1.42 & ! [C: $i,D: $i] :
% 0.30/1.42 ( ( ( in @ C @ B )
% 0.30/1.42 & ( in @ D @ B ) )
% 0.30/1.42 => ( ! [Xx: $i] :
% 0.30/1.42 ( ( in @ Xx @ C )
% 0.30/1.42 => ( in @ Xx @ D ) )
% 0.30/1.42 | ! [Xx: $i] :
% 0.30/1.42 ( ( in @ Xx @ D )
% 0.30/1.42 => ( in @ Xx @ C ) ) ) )
% 0.30/1.42 & ! [C: $i] :
% 0.30/1.42 ( ( ! [Xx: $i] :
% 0.30/1.42 ( ( in @ Xx @ C )
% 0.30/1.42 => ( in @ Xx @ A ) )
% 0.30/1.42 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.30/1.42 => ? [D: $i,Xx: $i] :
% 0.30/1.42 ( ( in @ D @ B )
% 0.30/1.42 & ( in @ Xx @ C )
% 0.30/1.42 & ~ ? [Xy: $i] :
% 0.30/1.42 ( ( in @ Xy @ D )
% 0.30/1.42 & ( in @ Xy @ C ) )
% 0.30/1.42 & ! [E: $i] :
% 0.30/1.42 ( ( in @ E @ B )
% 0.30/1.42 => ( ! [Xy: $i] :
% 0.30/1.42 ( ( in @ Xy @ E )
% 0.30/1.42 => ( in @ Xy @ D ) )
% 0.30/1.42 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.30/1.42
% 0.30/1.42 thf(descr_type,type,
% 0.30/1.42 descr: ( $i > $o ) > $i ).
% 0.30/1.42
% 0.30/1.42 thf(descrp_type,type,
% 0.30/1.42 descrp: $o ).
% 0.30/1.42
% 0.30/1.42 thf(descrp,definition,
% 0.30/1.42 ( descrp
% 0.30/1.42 = ( ! [Xphi: $i > $o] :
% 0.30/1.42 ( ( exu
% 0.30/1.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/1.42 => ( Xphi
% 0.30/1.42 @ ( descr
% 0.30/1.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.30/1.42
% 0.30/1.42 thf(dsetconstr_type,type,
% 0.30/1.42 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.30/1.42
% 0.30/1.42 thf(dsetconstrI_type,type,
% 0.30/1.42 dsetconstrI: $o ).
% 0.30/1.42
% 0.30/1.42 thf(dsetconstrI,definition,
% 0.30/1.42 ( dsetconstrI
% 0.30/1.42 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.30/1.42 ( ( in @ Xx @ A )
% 0.30/1.42 => ( ( Xphi @ Xx )
% 0.30/1.42 => ( in @ Xx
% 0.30/1.42 @ ( dsetconstr @ A
% 0.30/1.42 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.30/1.42
% 0.30/1.42 thf(dsetconstrEL_type,type,
% 0.30/1.42 dsetconstrEL: $o ).
% 0.30/1.42
% 0.30/1.42 thf(dsetconstrEL,definition,
% 0.30/1.42 ( dsetconstrEL
% 0.30/1.42 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.30/1.42 ( ( in @ Xx
% 0.30/1.42 @ ( dsetconstr @ A
% 0.30/1.42 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.30/1.42 => ( in @ Xx @ A ) ) ) ) ).
% 0.30/1.42
% 0.30/1.42 thf(dsetconstrER_type,type,
% 0.30/1.42 dsetconstrER: $o ).
% 0.30/1.42
% 0.30/1.42 thf(dsetconstrER,definition,
% 0.30/1.42 ( dsetconstrER
% 0.30/1.42 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.30/1.42 ( ( in @ Xx
% 0.30/1.42 @ ( dsetconstr @ A
% 0.30/1.42 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.30/1.42 => ( Xphi @ Xx ) ) ) ) ).
% 0.30/1.42
% 0.30/1.42 thf(exuE1_type,type,
% 0.30/1.42 exuE1: $o ).
% 0.30/1.42
% 0.30/1.42 thf(exuE1,definition,
% 0.30/1.42 ( exuE1
% 0.30/1.42 = ( ! [Xphi: $i > $o] :
% 0.30/1.42 ( ( exu
% 0.30/1.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/1.42 => ? [Xx: $i] :
% 0.30/1.42 ( ( Xphi @ Xx )
% 0.30/1.42 & ! [Xy: $i] :
% 0.30/1.42 ( ( Xphi @ Xy )
% 0.30/1.42 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.30/1.42
% 0.30/1.42 thf(prop2set_type,type,
% 0.30/1.42 prop2set: $o > $i ).
% 0.30/1.42
% 0.30/1.42 thf(prop2setE_type,type,
% 0.30/1.42 prop2setE: $o ).
% 0.30/1.42
% 0.30/1.42 thf(prop2setE,definition,
% 0.30/1.42 ( prop2setE
% 0.30/1.42 = ( ! [Xphi: $o,Xx: $i] :
% 0.30/1.42 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.30/1.42 => Xphi ) ) ) ).
% 0.30/1.42
% 0.30/1.42 thf(emptysetE_type,type,
% 0.30/1.42 emptysetE: $o ).
% 0.30/1.42
% 0.30/1.42 thf(emptysetE,definition,
% 0.30/1.42 ( emptysetE
% 0.30/1.42 = ( ! [Xx: $i] :
% 0.30/1.42 ( ( in @ Xx @ emptyset )
% 0.30/1.42 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.30/1.42
% 0.30/1.42 thf(emptysetimpfalse_type,type,
% 0.30/1.42 emptysetimpfalse: $o ).
% 0.30/1.42
% 0.30/1.42 thf(emptysetimpfalse,definition,
% 0.30/1.42 ( emptysetimpfalse
% 0.30/1.42 = ( ! [Xx: $i] :
% 0.30/1.42 ( ( in @ Xx @ emptyset )
% 0.30/1.42 => $false ) ) ) ).
% 0.30/1.42
% 0.30/1.42 thf(notinemptyset_type,type,
% 0.30/1.42 notinemptyset: $o ).
% 0.30/1.42
% 0.30/1.42 thf(notinemptyset,definition,
% 0.30/1.42 ( notinemptyset
% 0.30/1.42 = ( ! [Xx: $i] :
% 0.30/1.42 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.30/1.42
% 0.30/1.42 thf(exuE3e_type,type,
% 0.30/1.42 exuE3e: $o ).
% 0.30/1.42
% 0.30/1.42 thf(exuE3e,definition,
% 0.30/1.42 ( exuE3e
% 0.30/1.42 = ( ! [Xphi: $i > $o] :
% 0.30/1.42 ( ( exu
% 0.30/1.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/1.42 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.30/1.42
% 0.30/1.42 thf(setext_type,type,
% 0.30/1.42 setext: $o ).
% 0.30/1.42
% 0.30/1.42 thf(setext,definition,
% 0.30/1.42 ( setext
% 0.30/1.42 = ( ! [A: $i,B: $i] :
% 0.30/1.42 ( ! [Xx: $i] :
% 0.30/1.42 ( ( in @ Xx @ A )
% 0.30/1.42 => ( in @ Xx @ B ) )
% 0.30/1.42 => ( ! [Xx: $i] :
% 0.30/1.42 ( ( in @ Xx @ B )
% 0.30/1.42 => ( in @ Xx @ A ) )
% 0.30/1.42 => ( A = B ) ) ) ) ) ).
% 0.30/1.42
% 0.30/1.42 thf(emptyI_type,type,
% 0.30/1.42 emptyI: $o ).
% 0.30/1.42
% 0.30/1.42 thf(emptyI,definition,
% 0.30/1.42 ( emptyI
% 0.30/1.42 = ( ! [A: $i] :
% 0.30/1.42 ( ! [Xx: $i] :
% 0.30/1.42 ~ ( in @ Xx @ A )
% 0.30/1.42 => ( A = emptyset ) ) ) ) ).
% 0.30/1.42
% 0.30/1.42 thf(noeltsimpempty_type,type,
% 0.30/1.42 noeltsimpempty: $o ).
% 0.30/1.42
% 0.30/1.42 thf(noeltsimpempty,definition,
% 0.30/1.42 ( noeltsimpempty
% 0.30/1.42 = ( ! [A: $i] :
% 0.30/1.42 ( ! [Xx: $i] :
% 0.30/1.42 ~ ( in @ Xx @ A )
% 0.30/1.42 => ( A = emptyset ) ) ) ) ).
% 0.30/1.42
% 0.30/1.42 thf(setbeta_type,type,
% 0.30/1.42 setbeta: $o ).
% 0.30/1.42
% 0.30/1.42 thf(setbeta,definition,
% 0.30/1.42 ( setbeta
% 0.30/1.42 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.30/1.42 ( ( in @ Xx @ A )
% 0.30/1.42 => ( ( in @ Xx
% 0.30/1.42 @ ( dsetconstr @ A
% 0.30/1.42 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.30/1.42 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/1.42
% 0.30/1.42 thf(nonempty_type,type,
% 0.30/1.42 nonempty: $i > $o ).
% 0.30/1.42
% 0.30/1.42 thf(nonempty,definition,
% 0.30/1.42 ( nonempty
% 0.30/1.42 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.30/1.42
% 0.30/1.42 thf(nonemptyE1_type,type,
% 0.30/1.42 nonemptyE1: $o ).
% 0.30/1.42
% 0.30/1.42 thf(nonemptyE1,definition,
% 0.30/1.42 ( nonemptyE1
% 0.30/1.42 = ( ! [A: $i] :
% 0.30/1.42 ( ( nonempty @ A )
% 0.30/1.42 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.30/1.42
% 0.30/1.42 thf(nonemptyI_type,type,
% 0.30/1.42 nonemptyI: $o ).
% 0.30/1.42
% 0.30/1.42 thf(nonemptyI,definition,
% 0.30/1.42 ( nonemptyI
% 0.30/1.42 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.30/1.42 ( ( in @ Xx @ A )
% 0.30/1.42 => ( ( Xphi @ Xx )
% 0.30/1.42 => ( nonempty
% 0.30/1.42 @ ( dsetconstr @ A
% 0.30/1.42 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.30/1.42
% 0.30/1.42 thf(nonemptyI1_type,type,
% 0.30/1.42 nonemptyI1: $o ).
% 0.30/1.42
% 0.30/1.42 thf(nonemptyI1,definition,
% 0.30/1.42 ( nonemptyI1
% 0.30/1.42 = ( ! [A: $i] :
% 0.30/1.42 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.30/1.42 => ( nonempty @ A ) ) ) ) ).
% 0.30/1.42
% 0.30/1.42 thf(setadjoinIL_type,type,
% 0.30/1.42 setadjoinIL: $o ).
% 0.30/1.42
% 0.30/1.42 thf(setadjoinIL,definition,
% 0.30/1.42 ( setadjoinIL
% 0.30/1.42 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.30/1.42
% 0.30/1.42 thf(emptyinunitempty_type,type,
% 0.30/1.42 emptyinunitempty: $o ).
% 0.30/1.42
% 0.30/1.42 thf(emptyinunitempty,definition,
% 0.30/1.42 ( emptyinunitempty
% 0.30/1.42 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.30/1.42
% 0.30/1.42 thf(setadjoinIR_type,type,
% 0.30/1.42 setadjoinIR: $o ).
% 0.30/1.42
% 0.30/1.42 thf(setadjoinIR,definition,
% 0.30/1.42 ( setadjoinIR
% 0.30/1.42 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.30/1.42 ( ( in @ Xy @ A )
% 0.30/1.42 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.30/1.42
% 0.30/1.42 thf(setadjoinE_type,type,
% 0.30/1.42 setadjoinE: $o ).
% 0.30/1.42
% 0.30/1.42 thf(setadjoinE,definition,
% 0.30/1.42 ( setadjoinE
% 0.30/1.42 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.30/1.42 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.30/1.42 => ! [Xphi: $o] :
% 0.30/1.42 ( ( ( Xy = Xx )
% 0.30/1.42 => Xphi )
% 0.30/1.42 => ( ( ( in @ Xy @ A )
% 0.30/1.42 => Xphi )
% 0.30/1.42 => Xphi ) ) ) ) ) ).
% 0.30/1.42
% 0.30/1.42 thf(setadjoinOr_type,type,
% 0.30/1.42 setadjoinOr: $o ).
% 0.30/1.42
% 0.30/1.42 thf(setadjoinOr,definition,
% 0.30/1.42 ( setadjoinOr
% 0.30/1.42 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.30/1.42 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.30/1.42 => ( ( Xy = Xx )
% 0.30/1.42 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.30/1.42
% 0.30/1.42 thf(setoftrueEq_type,type,
% 0.30/1.42 setoftrueEq: $o ).
% 0.30/1.42
% 0.30/1.42 thf(setoftrueEq,definition,
% 0.30/1.42 ( setoftrueEq
% 0.30/1.42 = ( ! [A: $i] :
% 0.30/1.42 ( ( dsetconstr @ A
% 0.30/1.42 @ ^ [Xx: $i] : $true )
% 0.30/1.42 = A ) ) ) ).
% 0.30/1.42
% 0.30/1.42 thf(powersetI_type,type,
% 0.30/1.42 powersetI: $o ).
% 0.30/1.42
% 0.30/1.42 thf(powersetI,definition,
% 0.30/1.42 ( powersetI
% 0.30/1.42 = ( ! [A: $i,B: $i] :
% 0.30/1.42 ( ! [Xx: $i] :
% 0.30/1.42 ( ( in @ Xx @ B )
% 0.30/1.42 => ( in @ Xx @ A ) )
% 0.30/1.42 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.30/1.42
% 0.30/1.42 thf(emptyinPowerset_type,type,
% 0.30/1.42 emptyinPowerset: $o ).
% 0.30/1.42
% 0.30/1.42 thf(emptyinPowerset,definition,
% 0.30/1.42 ( emptyinPowerset
% 0.30/1.42 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.30/1.42
% 0.30/1.42 thf(emptyInPowerset_type,type,
% 0.30/1.42 emptyInPowerset: $o ).
% 0.30/1.42
% 0.30/1.42 thf(emptyInPowerset,definition,
% 0.30/1.42 ( emptyInPowerset
% 0.30/1.42 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.30/1.42
% 0.30/1.42 thf(powersetE_type,type,
% 0.30/1.42 powersetE: $o ).
% 0.30/1.42
% 0.30/1.42 thf(powersetE,definition,
% 0.30/1.42 ( powersetE
% 0.30/1.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/1.42 ( ( in @ B @ ( powerset @ A ) )
% 0.30/1.42 => ( ( in @ Xx @ B )
% 0.30/1.42 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.30/1.42
% 0.30/1.42 thf(setunionI_type,type,
% 0.30/1.42 setunionI: $o ).
% 0.30/1.42
% 0.30/1.42 thf(setunionI,definition,
% 0.30/1.42 ( setunionI
% 0.30/1.42 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.30/1.42 ( ( in @ Xx @ B )
% 0.30/1.42 => ( ( in @ B @ A )
% 0.30/1.42 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.30/1.42
% 0.30/1.42 thf(setunionE_type,type,
% 0.30/1.42 setunionE: $o ).
% 0.30/1.42
% 0.30/1.42 thf(setunionE,definition,
% 0.30/1.42 ( setunionE
% 0.30/1.42 = ( ! [A: $i,Xx: $i] :
% 0.30/1.42 ( ( in @ Xx @ ( setunion @ A ) )
% 0.30/1.42 => ! [Xphi: $o] :
% 0.30/1.42 ( ! [B: $i] :
% 0.30/1.42 ( ( in @ Xx @ B )
% 0.30/1.42 => ( ( in @ B @ A )
% 0.30/1.42 => Xphi ) )
% 0.30/1.42 => Xphi ) ) ) ) ).
% 0.30/1.42
% 0.30/1.42 thf(subPowSU_type,type,
% 0.30/1.42 subPowSU: $o ).
% 0.30/1.42
% 0.30/1.42 thf(subPowSU,definition,
% 0.30/1.42 ( subPowSU
% 0.30/1.42 = ( ! [A: $i,Xx: $i] :
% 0.30/1.42 ( ( in @ Xx @ A )
% 0.30/1.42 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.30/1.42
% 0.30/1.42 thf(exuE2_type,type,
% 0.30/1.42 exuE2: $o ).
% 0.30/1.42
% 0.30/1.42 thf(exuE2,definition,
% 0.30/1.42 ( exuE2
% 0.30/1.42 = ( ! [Xphi: $i > $o] :
% 0.30/1.42 ( ( exu
% 0.30/1.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/1.42 => ? [Xx: $i] :
% 0.30/1.42 ! [Xy: $i] :
% 0.30/1.42 ( ( Xphi @ Xy )
% 0.30/1.42 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.30/1.42
% 0.30/1.42 thf(nonemptyImpWitness_type,type,
% 0.30/1.42 nonemptyImpWitness: $o ).
% 0.30/1.42
% 0.30/1.42 thf(nonemptyImpWitness,definition,
% 0.30/1.42 ( nonemptyImpWitness
% 0.30/1.42 = ( ! [A: $i] :
% 0.30/1.42 ( ( nonempty @ A )
% 0.30/1.42 => ? [Xx: $i] :
% 0.30/1.42 ( ( in @ Xx @ A )
% 0.30/1.42 & $true ) ) ) ) ).
% 0.30/1.42
% 0.30/1.42 thf(uniqinunit_type,type,
% 0.30/1.42 uniqinunit: $o ).
% 0.30/1.42
% 0.30/1.42 thf(uniqinunit,definition,
% 0.30/1.42 ( uniqinunit
% 0.30/1.42 = ( ! [Xx: $i,Xy: $i] :
% 0.30/1.42 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.30/1.42 => ( Xx = Xy ) ) ) ) ).
% 0.30/1.42
% 0.30/1.42 thf(notinsingleton_type,type,
% 0.30/1.42 notinsingleton: $o ).
% 0.30/1.42
% 0.30/1.42 thf(notinsingleton,definition,
% 0.30/1.42 ( notinsingleton
% 0.30/1.42 = ( ! [Xx: $i,Xy: $i] :
% 0.30/1.42 ( ( Xx != Xy )
% 0.30/1.42 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.30/1.42
% 0.30/1.42 thf(eqinunit_type,type,
% 0.30/1.42 eqinunit: $o ).
% 0.30/1.42
% 0.30/1.42 thf(eqinunit,definition,
% 0.30/1.42 ( eqinunit
% 0.30/1.42 = ( ! [Xx: $i,Xy: $i] :
% 0.30/1.42 ( ( Xx = Xy )
% 0.30/1.42 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.30/1.42
% 0.30/1.42 thf(singletonsswitch_type,type,
% 0.30/1.42 singletonsswitch: $o ).
% 0.30/1.42
% 0.30/1.42 thf(singletonsswitch,definition,
% 0.30/1.42 ( singletonsswitch
% 0.30/1.42 = ( ! [Xx: $i,Xy: $i] :
% 0.30/1.42 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.30/1.42 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.30/1.42
% 0.30/1.42 thf(upairsetE_type,type,
% 0.30/1.42 upairsetE: $o ).
% 0.30/1.42
% 0.30/1.42 thf(upairsetE,definition,
% 0.30/1.42 ( upairsetE
% 0.30/1.42 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.30/1.42 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.30/1.42 => ( ( Xz = Xx )
% 0.30/1.42 | ( Xz = Xy ) ) ) ) ) ).
% 0.30/1.42
% 0.30/1.42 thf(upairsetIL_type,type,
% 0.30/1.42 upairsetIL: $o ).
% 0.30/1.42
% 0.30/1.42 thf(upairsetIL,definition,
% 0.30/1.42 ( upairsetIL
% 0.30/1.42 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.30/1.42
% 0.30/1.42 thf(upairsetIR_type,type,
% 0.30/1.42 upairsetIR: $o ).
% 0.30/1.42
% 0.30/1.42 thf(upairsetIR,definition,
% 0.30/1.42 ( upairsetIR
% 0.30/1.42 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.30/1.42
% 0.30/1.42 thf(emptyE1_type,type,
% 0.30/1.42 emptyE1: $o ).
% 0.30/1.42
% 0.30/1.42 thf(emptyE1,definition,
% 0.30/1.42 ( emptyE1
% 0.30/1.42 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/1.42 ( ? [Xx: $i] :
% 0.30/1.42 ( ( in @ Xx @ A )
% 0.30/1.42 & ( Xphi @ Xx ) )
% 0.30/1.42 => ( ( ( dsetconstr @ A
% 0.30/1.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/1.42 = emptyset )
% 0.30/1.42 => $false ) ) ) ) ).
% 0.30/1.42
% 0.30/1.42 thf(vacuousDall_type,type,
% 0.30/1.42 vacuousDall: $o ).
% 0.30/1.42
% 0.30/1.42 thf(vacuousDall,definition,
% 0.30/1.42 ( vacuousDall
% 0.30/1.42 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.30/1.42 ( ( in @ Xx @ emptyset )
% 0.30/1.42 => ( Xphi @ Xx ) ) ) ) ).
% 0.30/1.42
% 0.30/1.42 thf(quantDeMorgan1_type,type,
% 0.30/1.42 quantDeMorgan1: $o ).
% 0.30/1.42
% 0.30/1.42 thf(quantDeMorgan1,definition,
% 0.30/1.42 ( quantDeMorgan1
% 0.30/1.42 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/1.42 ( ~ ! [Xx: $i] :
% 0.30/1.42 ( ( in @ Xx @ A )
% 0.30/1.42 => ( Xphi @ Xx ) )
% 0.30/1.42 => ? [Xx: $i] :
% 0.30/1.42 ( ( in @ Xx @ A )
% 0.30/1.42 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/1.42
% 0.30/1.42 thf(quantDeMorgan2_type,type,
% 0.30/1.42 quantDeMorgan2: $o ).
% 0.30/1.42
% 0.30/1.42 thf(quantDeMorgan2,definition,
% 0.30/1.42 ( quantDeMorgan2
% 0.30/1.42 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/1.42 ( ! [Xx: $i] :
% 0.30/1.42 ( ( in @ Xx @ A )
% 0.30/1.42 => ~ ( Xphi @ Xx ) )
% 0.30/1.42 => ~ ? [Xx: $i] :
% 0.30/1.42 ( ( in @ Xx @ A )
% 0.30/1.42 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/1.42
% 0.30/1.42 thf(quantDeMorgan3_type,type,
% 0.30/1.42 quantDeMorgan3: $o ).
% 0.30/1.42
% 0.30/1.42 thf(quantDeMorgan3,definition,
% 0.30/1.42 ( quantDeMorgan3
% 0.30/1.42 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/1.42 ( ~ ? [Xx: $i] :
% 0.30/1.42 ( ( in @ Xx @ A )
% 0.30/1.42 & ( Xphi @ Xx ) )
% 0.30/1.42 => ! [Xx: $i] :
% 0.30/1.42 ( ( in @ Xx @ A )
% 0.30/1.42 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/1.42
% 0.30/1.42 thf(quantDeMorgan4_type,type,
% 0.30/1.42 quantDeMorgan4: $o ).
% 0.30/1.42
% 0.30/1.42 thf(quantDeMorgan4,definition,
% 0.30/1.42 ( quantDeMorgan4
% 0.30/1.42 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/1.42 ( ? [Xx: $i] :
% 0.30/1.42 ( ( in @ Xx @ A )
% 0.30/1.42 & ~ ( Xphi @ Xx ) )
% 0.30/1.42 => ~ ! [Xx: $i] :
% 0.30/1.42 ( ( in @ Xx @ A )
% 0.30/1.42 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/1.42
% 0.30/1.42 thf(prop2setI_type,type,
% 0.30/1.42 prop2setI: $o ).
% 0.30/1.42
% 0.30/1.42 thf(prop2setI,definition,
% 0.30/1.42 ( prop2setI
% 0.30/1.42 = ( ! [Xphi: $o] :
% 0.30/1.42 ( Xphi
% 0.30/1.42 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.30/1.42
% 0.30/1.42 thf(set2prop_type,type,
% 0.30/1.42 set2prop: $i > $o ).
% 0.30/1.42
% 0.30/1.42 thf(prop2set2propI_type,type,
% 0.30/1.42 prop2set2propI: $o ).
% 0.30/1.42
% 0.30/1.42 thf(prop2set2propI,definition,
% 0.30/1.42 ( prop2set2propI
% 0.30/1.42 = ( ! [Xphi: $o] :
% 0.30/1.42 ( Xphi
% 0.30/1.42 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.30/1.42
% 0.30/1.42 thf(notdexE_type,type,
% 0.30/1.42 notdexE: $o ).
% 0.30/1.42
% 0.30/1.42 thf(notdexE,definition,
% 0.30/1.42 ( notdexE
% 0.30/1.42 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/1.42 ( ~ ? [Xx: $i] :
% 0.30/1.42 ( ( in @ Xx @ A )
% 0.30/1.42 & ( Xphi @ Xx ) )
% 0.30/1.42 => ! [Xx: $i] :
% 0.30/1.42 ( ( in @ Xx @ A )
% 0.30/1.42 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/1.42
% 0.30/1.42 thf(notdallE_type,type,
% 0.30/1.42 notdallE: $o ).
% 0.30/1.42
% 0.30/1.42 thf(notdallE,definition,
% 0.30/1.42 ( notdallE
% 0.30/1.42 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/1.42 ( ~ ! [Xx: $i] :
% 0.30/1.42 ( ( in @ Xx @ A )
% 0.30/1.42 => ( Xphi @ Xx ) )
% 0.30/1.42 => ? [Xx: $i] :
% 0.30/1.42 ( ( in @ Xx @ A )
% 0.30/1.42 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/1.42
% 0.30/1.42 thf(exuI1_type,type,
% 0.30/1.42 exuI1: $o ).
% 0.30/1.42
% 0.30/1.42 thf(exuI1,definition,
% 0.30/1.42 ( exuI1
% 0.30/1.42 = ( ! [Xphi: $i > $o] :
% 0.30/1.42 ( ? [Xx: $i] :
% 0.30/1.42 ( ( Xphi @ Xx )
% 0.30/1.42 & ! [Xy: $i] :
% 0.30/1.42 ( ( Xphi @ Xy )
% 0.30/1.42 => ( Xx = Xy ) ) )
% 0.30/1.42 => ( exu
% 0.30/1.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/1.42
% 0.30/1.42 thf(exuI3_type,type,
% 0.30/1.42 exuI3: $o ).
% 0.30/1.42
% 0.30/1.42 thf(exuI3,definition,
% 0.30/1.42 ( exuI3
% 0.30/1.42 = ( ! [Xphi: $i > $o] :
% 0.30/1.42 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.30/1.42 => ( ! [Xx: $i,Xy: $i] :
% 0.30/1.42 ( ( Xphi @ Xx )
% 0.30/1.42 => ( ( Xphi @ Xy )
% 0.30/1.42 => ( Xx = Xy ) ) )
% 0.30/1.42 => ( exu
% 0.30/1.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.30/1.42
% 0.30/1.42 thf(exuI2_type,type,
% 0.30/1.42 exuI2: $o ).
% 0.30/1.42
% 0.30/1.42 thf(exuI2,definition,
% 0.30/1.42 ( exuI2
% 0.30/1.42 = ( ! [Xphi: $i > $o] :
% 0.30/1.42 ( ? [Xx: $i] :
% 0.30/1.42 ! [Xy: $i] :
% 0.30/1.42 ( ( Xphi @ Xy )
% 0.30/1.42 <=> ( Xy = Xx ) )
% 0.30/1.42 => ( exu
% 0.30/1.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/1.42
% 0.30/1.42 thf(inCongP_type,type,
% 0.30/1.42 inCongP: $o ).
% 0.30/1.42
% 0.30/1.42 thf(inCongP,definition,
% 0.30/1.42 ( inCongP
% 0.30/1.42 = ( ! [A: $i,B: $i] :
% 0.30/1.42 ( ( A = B )
% 0.30/1.42 => ! [Xx: $i,Xy: $i] :
% 0.30/1.42 ( ( Xx = Xy )
% 0.30/1.42 => ( ( in @ Xx @ A )
% 0.30/1.42 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.30/1.42
% 0.30/1.42 thf(in__Cong_type,type,
% 0.30/1.42 in__Cong: $o ).
% 0.30/1.42
% 0.30/1.42 thf(in__Cong,definition,
% 0.30/1.42 ( in__Cong
% 0.30/1.42 = ( ! [A: $i,B: $i] :
% 0.30/1.42 ( ( A = B )
% 0.30/1.42 => ! [Xx: $i,Xy: $i] :
% 0.30/1.42 ( ( Xx = Xy )
% 0.30/1.42 => ( ( in @ Xx @ A )
% 0.30/1.42 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.30/1.42
% 0.30/1.42 thf(exuE3u_type,type,
% 0.30/1.42 exuE3u: $o ).
% 0.30/1.42
% 0.30/1.42 thf(exuE3u,definition,
% 0.30/1.42 ( exuE3u
% 0.30/1.42 = ( ! [Xphi: $i > $o] :
% 0.30/1.42 ( ( exu
% 0.30/1.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/1.42 => ! [Xx: $i,Xy: $i] :
% 0.30/1.42 ( ( Xphi @ Xx )
% 0.30/1.42 => ( ( Xphi @ Xy )
% 0.30/1.42 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.30/1.42
% 0.30/1.42 thf(exu__Cong_type,type,
% 0.30/1.42 exu__Cong: $o ).
% 0.30/1.42
% 0.30/1.42 thf(exu__Cong,definition,
% 0.30/1.42 ( exu__Cong
% 0.30/1.42 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.30/1.42 ( ! [Xx: $i,Xy: $i] :
% 0.30/1.42 ( ( Xx = Xy )
% 0.30/1.42 => ( ( Xphi @ Xx )
% 0.30/1.42 <=> ( Xpsi @ Xy ) ) )
% 0.30/1.42 => ( ( exu
% 0.30/1.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/1.42 <=> ( exu
% 0.30/1.42 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.30/1.42
% 0.30/1.42 thf(emptyset__Cong_type,type,
% 0.30/1.42 emptyset__Cong: $o ).
% 0.30/1.42
% 0.30/1.42 thf(emptyset__Cong,definition,
% 0.30/1.42 ( emptyset__Cong
% 0.30/1.42 = ( emptyset = emptyset ) ) ).
% 0.30/1.42
% 0.30/1.42 thf(setadjoin__Cong_type,type,
% 0.30/1.42 setadjoin__Cong: $o ).
% 0.30/1.42
% 0.30/1.42 thf(setadjoin__Cong,definition,
% 0.30/1.42 ( setadjoin__Cong
% 0.30/1.42 = ( ! [Xx: $i,Xy: $i] :
% 0.30/1.42 ( ( Xx = Xy )
% 0.30/1.42 => ! [Xz: $i,Xu: $i] :
% 0.30/1.42 ( ( Xz = Xu )
% 0.30/1.42 => ( ( setadjoin @ Xx @ Xz )
% 0.30/1.42 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.30/1.42
% 0.30/1.42 thf(powerset__Cong_type,type,
% 0.30/1.42 powerset__Cong: $o ).
% 0.30/1.42
% 0.30/1.42 thf(powerset__Cong,definition,
% 0.30/1.42 ( powerset__Cong
% 0.30/1.42 = ( ! [A: $i,B: $i] :
% 0.30/1.42 ( ( A = B )
% 0.30/1.42 => ( ( powerset @ A )
% 0.30/1.42 = ( powerset @ B ) ) ) ) ) ).
% 0.30/1.42
% 0.30/1.42 thf(setunion__Cong_type,type,
% 0.30/1.42 setunion__Cong: $o ).
% 0.30/1.42
% 0.30/1.42 thf(setunion__Cong,definition,
% 0.30/1.42 ( setunion__Cong
% 0.30/1.42 = ( ! [A: $i,B: $i] :
% 0.30/1.42 ( ( A = B )
% 0.30/1.42 => ( ( setunion @ A )
% 0.30/1.42 = ( setunion @ B ) ) ) ) ) ).
% 0.30/1.42
% 0.30/1.42 thf(omega__Cong_type,type,
% 0.30/1.42 omega__Cong: $o ).
% 0.30/1.42
% 0.30/1.42 thf(omega__Cong,definition,
% 0.30/1.42 ( omega__Cong
% 0.30/1.42 = ( omega = omega ) ) ).
% 0.30/1.42
% 0.30/1.42 thf(exuEu_type,type,
% 0.30/1.42 exuEu: $o ).
% 0.30/1.42
% 0.30/1.42 thf(exuEu,definition,
% 0.30/1.42 ( exuEu
% 0.30/1.42 = ( ! [Xphi: $i > $o] :
% 0.30/1.42 ( ( exu
% 0.30/1.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/1.42 => ! [Xx: $i,Xy: $i] :
% 0.30/1.42 ( ( Xphi @ Xx )
% 0.30/1.42 => ( ( Xphi @ Xy )
% 0.30/1.42 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.30/1.42
% 0.30/1.42 thf(descr__Cong_type,type,
% 0.30/1.42 descr__Cong: $o ).
% 0.30/1.42
% 0.30/1.42 thf(descr__Cong,definition,
% 0.30/1.42 ( descr__Cong
% 0.30/1.42 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.30/1.42 ( ! [Xx: $i,Xy: $i] :
% 0.30/1.42 ( ( Xx = Xy )
% 0.30/1.42 => ( ( Xphi @ Xx )
% 0.30/1.42 <=> ( Xpsi @ Xy ) ) )
% 0.30/1.42 => ( ( exu
% 0.30/1.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/1.42 => ( ( exu
% 0.30/1.42 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.30/1.42 => ( ( descr
% 0.30/1.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/1.42 = ( descr
% 0.30/1.42 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.30/1.42
% 0.30/1.42 thf(dsetconstr__Cong_type,type,
% 0.30/1.42 dsetconstr__Cong: $o ).
% 0.30/1.42
% 0.30/1.42 thf(dsetconstr__Cong,definition,
% 0.30/1.42 ( dsetconstr__Cong
% 0.30/1.42 = ( ! [A: $i,B: $i] :
% 0.30/1.42 ( ( A = B )
% 0.30/1.42 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.30/1.42 ( ! [Xx: $i] :
% 0.30/1.42 ( ( in @ Xx @ A )
% 0.30/1.42 => ! [Xy: $i] :
% 0.30/1.42 ( ( in @ Xy @ B )
% 0.30/1.42 => ( ( Xx = Xy )
% 0.30/1.42 => ( ( Xphi @ Xx )
% 0.30/1.42 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.30/1.42 => ( ( dsetconstr @ A
% 0.30/1.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/1.42 = ( dsetconstr @ B
% 0.30/1.42 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.30/1.42
% 0.30/1.42 thf(subset_type,type,
% 0.30/1.42 subset: $i > $i > $o ).
% 0.30/1.42
% 0.30/1.42 thf(subset,definition,
% 0.30/1.42 ( subset
% 0.30/1.42 = ( ^ [A: $i,B: $i] :
% 0.30/1.42 ! [Xx: $i] :
% 0.30/1.42 ( ( in @ Xx @ A )
% 0.30/1.42 => ( in @ Xx @ B ) ) ) ) ).
% 0.30/1.42
% 0.30/1.42 thf(disjoint_type,type,
% 0.30/1.42 disjoint: $i > $i > $o ).
% 0.30/1.42
% 0.30/1.42 thf(setsmeet_type,type,
% 0.30/1.42 setsmeet: $i > $i > $o ).
% 0.30/1.42
% 0.30/1.42 thf(subsetI1,conjecture,
% 0.30/1.42 ( setextAx
% 0.30/1.42 => ( emptysetAx
% 0.30/1.42 => ( setadjoinAx
% 0.30/1.42 => ( powersetAx
% 0.30/1.42 => ( setunionAx
% 0.30/1.42 => ( omega0Ax
% 0.30/1.42 => ( omegaSAx
% 0.30/1.42 => ( omegaIndAx
% 0.30/1.42 => ( replAx
% 0.30/1.42 => ( foundationAx
% 0.30/1.42 => ( wellorderingAx
% 0.30/1.42 => ( descrp
% 0.30/1.42 => ( dsetconstrI
% 0.30/1.42 => ( dsetconstrEL
% 0.30/1.42 => ( dsetconstrER
% 0.30/1.42 => ( exuE1
% 0.30/1.42 => ( prop2setE
% 0.30/1.42 => ( emptysetE
% 0.30/1.42 => ( emptysetimpfalse
% 0.30/1.42 => ( notinemptyset
% 0.30/1.42 => ( exuE3e
% 0.30/1.42 => ( setext
% 0.30/1.42 => ( emptyI
% 0.30/1.42 => ( noeltsimpempty
% 0.30/1.42 => ( setbeta
% 0.30/1.42 => ( nonemptyE1
% 0.30/1.42 => ( nonemptyI
% 0.30/1.42 => ( nonemptyI1
% 0.30/1.42 => ( setadjoinIL
% 0.30/1.42 => ( emptyinunitempty
% 0.30/1.42 => ( setadjoinIR
% 0.30/1.42 => ( setadjoinE
% 0.30/1.42 => ( setadjoinOr
% 0.30/1.42 => ( setoftrueEq
% 0.30/1.42 => ( powersetI
% 0.30/1.42 => ( emptyinPowerset
% 0.30/1.42 => ( emptyInPowerset
% 0.30/1.42 => ( powersetE
% 0.30/1.42 => ( setunionI
% 0.30/1.42 => ( setunionE
% 0.30/1.42 => ( subPowSU
% 0.30/1.42 => ( exuE2
% 0.30/1.42 => ( nonemptyImpWitness
% 0.30/1.42 => ( uniqinunit
% 0.30/1.42 => ( notinsingleton
% 0.30/1.42 => ( eqinunit
% 0.30/1.42 => ( singletonsswitch
% 0.30/1.42 => ( upairsetE
% 0.30/1.42 => ( upairsetIL
% 0.30/1.42 => ( upairsetIR
% 0.30/1.42 => ( emptyE1
% 0.30/1.42 => ( vacuousDall
% 0.30/1.42 => ( quantDeMorgan1
% 0.30/1.42 => ( quantDeMorgan2
% 0.30/1.42 => ( quantDeMorgan3
% 0.30/1.42 => ( quantDeMorgan4
% 0.30/1.42 => ( prop2setI
% 0.30/1.42 => ( prop2set2propI
% 0.30/1.42 => ( notdexE
% 0.30/1.42 => ( notdallE
% 0.30/1.42 => ( exuI1
% 0.30/1.42 => ( exuI3
% 0.30/1.42 => ( exuI2
% 0.30/1.42 => ( inCongP
% 0.30/1.42 => ( in__Cong
% 0.30/1.42 => ( exuE3u
% 0.30/1.42 => ( exu__Cong
% 0.30/1.42 => ( emptyset__Cong
% 0.30/1.42 => ( setadjoin__Cong
% 0.30/1.42 => ( powerset__Cong
% 0.30/1.42 => ( setunion__Cong
% 0.30/1.42 => ( omega__Cong
% 0.30/1.42 => ( exuEu
% 0.30/1.42 => ( descr__Cong
% 0.30/1.42 => ( dsetconstr__Cong
% 0.30/1.42 => ! [A: $i,B: $i] :
% 0.30/1.42 ( ! [Xx: $i] :
% 0.30/1.42 ( ( in @ Xx @ A )
% 0.30/1.42 => ( in @ Xx @ B ) )
% 0.30/1.42 => ( subset @ A @ B ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/1.42 ------------------------
% 0.30/1.43 % (7961)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_7736 for (2999ds/18Mi)
% 0.30/1.43 % (7955)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_7736 for (2999ds/183Mi)
% 0.30/1.44 % (7960)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_7736 for (2999ds/275Mi)
% 0.30/1.44 % (7961)Instruction limit reached!
% 0.30/1.44 % (7961)------------------------------
% 0.30/1.44 % (7961)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.30/1.44 % (7961)Termination reason: Unknown
% 0.30/1.44 % (7961)Termination phase: Preprocessing 3
% 0.30/1.44
% 0.30/1.44 % (7961)Memory used [KB]: 1535
% 0.30/1.44 % (7961)Time elapsed: 0.007 s
% 0.30/1.44 % (7961)Instructions burned: 18 (million)
% 0.30/1.44 % (7961)------------------------------
% 0.30/1.44 % (7961)------------------------------
% 0.30/1.44 % (7956)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_7736 for (2999ds/4Mi)
% 0.30/1.44 % (7957)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_7736 for (2999ds/27Mi)
% 0.30/1.44 % (7958)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_7736 for (2999ds/2Mi)
% 0.30/1.44 % (7959)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_7736 for (2999ds/2Mi)
% 0.30/1.44 % (7958)Instruction limit reached!
% 0.30/1.44 % (7958)------------------------------
% 0.30/1.44 % (7958)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.30/1.44 % (7958)Termination reason: Unknown
% 0.30/1.44 % (7958)Termination phase: shuffling
% 0.30/1.44
% 0.30/1.44 % (7958)Memory used [KB]: 1151
% 0.30/1.44 % (7958)Time elapsed: 0.003 s
% 0.30/1.44 % (7958)Instructions burned: 2 (million)
% 0.30/1.44 % (7958)------------------------------
% 0.30/1.44 % (7958)------------------------------
% 0.30/1.44 % (7959)Instruction limit reached!
% 0.30/1.44 % (7959)------------------------------
% 0.30/1.44 % (7959)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.30/1.44 % (7959)Termination reason: Unknown
% 0.30/1.44 % (7959)Termination phase: shuffling
% 0.30/1.44
% 0.30/1.44 % (7959)Memory used [KB]: 1151
% 0.30/1.44 % (7959)Time elapsed: 0.003 s
% 0.30/1.44 % (7959)Instructions burned: 2 (million)
% 0.30/1.44 % (7959)------------------------------
% 0.30/1.44 % (7959)------------------------------
% 0.30/1.44 % (7956)Instruction limit reached!
% 0.30/1.44 % (7956)------------------------------
% 0.30/1.44 % (7956)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.30/1.44 % (7956)Termination reason: Unknown
% 0.30/1.44 % (7956)Termination phase: shuffling
% 0.30/1.44
% 0.30/1.44 % (7956)Memory used [KB]: 1151
% 0.30/1.44 % (7956)Time elapsed: 0.004 s
% 0.30/1.44 % (7956)Instructions burned: 4 (million)
% 0.30/1.44 % (7956)------------------------------
% 0.30/1.44 % (7956)------------------------------
% 0.30/1.45 % (7962)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_7736 for (2999ds/3Mi)
% 0.30/1.45 % (7962)Instruction limit reached!
% 0.30/1.45 % (7962)------------------------------
% 0.30/1.45 % (7962)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.30/1.45 % (7962)Termination reason: Unknown
% 0.30/1.45 % (7962)Termination phase: shuffling
% 0.30/1.45
% 0.30/1.45 % (7962)Memory used [KB]: 1151
% 0.30/1.45 % (7962)Time elapsed: 0.003 s
% 0.30/1.45 % (7962)Instructions burned: 5 (million)
% 0.30/1.45 % (7962)------------------------------
% 0.30/1.45 % (7962)------------------------------
% 0.30/1.45 % (7957)Instruction limit reached!
% 0.30/1.45 % (7957)------------------------------
% 0.30/1.45 % (7957)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.30/1.45 % (7957)Termination reason: Unknown
% 0.30/1.45 % (7957)Termination phase: Property scanning
% 0.30/1.45
% 0.30/1.45 % (7957)Memory used [KB]: 1407
% 0.30/1.45 % (7957)Time elapsed: 0.015 s
% 0.30/1.45 % (7957)Instructions burned: 27 (million)
% 0.30/1.45 % (7957)------------------------------
% 0.30/1.45 % (7957)------------------------------
% 0.30/1.46 % (7963)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_7736 for (2999ds/37Mi)
% 0.30/1.46 % (7964)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_7736 for (2999ds/15Mi)
% 0.30/1.46 % (7965)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_7736 for (2999ds/3Mi)
% 0.30/1.46 % (7966)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_7736 for (2999ds/1041Mi)
% 0.30/1.46 % (7965)Instruction limit reached!
% 0.30/1.46 % (7965)------------------------------
% 0.30/1.46 % (7965)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.30/1.46 % (7965)Termination reason: Unknown
% 0.30/1.46 % (7965)Termination phase: shuffling
% 0.30/1.46
% 0.30/1.46 % (7965)Memory used [KB]: 1151
% 0.30/1.46 % (7965)Time elapsed: 0.004 s
% 0.30/1.46 % (7965)Instructions burned: 4 (million)
% 0.30/1.46 % (7965)------------------------------
% 0.30/1.46 % (7965)------------------------------
% 0.30/1.47 % (7964)Instruction limit reached!
% 0.30/1.47 % (7964)------------------------------
% 0.30/1.47 % (7964)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.30/1.47 % (7964)Termination reason: Unknown
% 0.30/1.47 % (7964)Termination phase: Preprocessing 3
% 0.30/1.47
% 0.30/1.47 % (7964)Memory used [KB]: 1535
% 0.30/1.47 % (7964)Time elapsed: 0.011 s
% 0.30/1.47 % (7964)Instructions burned: 16 (million)
% 0.30/1.47 % (7964)------------------------------
% 0.30/1.47 % (7964)------------------------------
% 0.30/1.47 % (7967)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_7736 for (2999ds/7Mi)
% 0.30/1.48 % (7967)Instruction limit reached!
% 0.30/1.48 % (7967)------------------------------
% 0.30/1.48 % (7967)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.30/1.48 % (7967)Termination reason: Unknown
% 0.30/1.48 % (7967)Termination phase: shuffling
% 0.30/1.48
% 0.30/1.48 % (7967)Memory used [KB]: 1279
% 0.30/1.48 % (7967)Time elapsed: 0.006 s
% 0.30/1.48 % (7967)Instructions burned: 8 (million)
% 0.30/1.48 % (7967)------------------------------
% 0.30/1.48 % (7967)------------------------------
% 0.30/1.48 % (7963)Instruction limit reached!
% 0.30/1.48 % (7963)------------------------------
% 0.30/1.48 % (7963)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.30/1.48 % (7963)Termination reason: Unknown
% 0.30/1.48 % (7963)Termination phase: Property scanning
% 0.30/1.48
% 0.30/1.48 % (7963)Memory used [KB]: 1407
% 0.30/1.48 % (7963)Time elapsed: 0.019 s
% 0.30/1.48 % (7963)Instructions burned: 37 (million)
% 0.30/1.48 % (7963)------------------------------
% 0.30/1.48 % (7963)------------------------------
% 0.30/1.48 % (7968)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_7736 for (2999ds/16Mi)
% 0.30/1.49 % (7969)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_7736 for (2999ds/3Mi)
% 0.30/1.49 % (7968)Instruction limit reached!
% 0.30/1.49 % (7968)------------------------------
% 0.30/1.49 % (7968)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.30/1.49 % (7968)Termination reason: Unknown
% 0.30/1.49 % (7968)Termination phase: Property scanning
% 0.30/1.49
% 0.30/1.49 % (7968)Memory used [KB]: 1407
% 0.30/1.49 % (7968)Time elapsed: 0.010 s
% 0.30/1.49 % (7968)Instructions burned: 17 (million)
% 0.30/1.49 % (7968)------------------------------
% 0.30/1.49 % (7968)------------------------------
% 0.30/1.49 % (7969)Instruction limit reached!
% 0.30/1.49 % (7969)------------------------------
% 0.30/1.49 % (7969)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.30/1.49 % (7969)Termination reason: Unknown
% 0.30/1.49 % (7969)Termination phase: shuffling
% 0.30/1.49
% 0.30/1.49 % (7969)Memory used [KB]: 1151
% 0.30/1.49 % (7969)Time elapsed: 0.004 s
% 0.30/1.49 % (7969)Instructions burned: 4 (million)
% 0.30/1.49 % (7969)------------------------------
% 0.30/1.49 % (7969)------------------------------
% 0.30/1.49 % (7960)First to succeed.
% 0.30/1.49 % (7955)Instruction limit reached!
% 0.30/1.49 % (7955)------------------------------
% 0.30/1.49 % (7955)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.30/1.49 % (7970)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_7736 for (2999ds/3Mi)
% 0.30/1.49 % (7955)Termination reason: Unknown
% 0.30/1.49 % (7955)Termination phase: Saturation
% 0.30/1.49
% 0.30/1.49 % (7955)Memory used [KB]: 6780
% 0.30/1.49 % (7955)Time elapsed: 0.060 s
% 0.30/1.49 % (7955)Instructions burned: 184 (million)
% 0.30/1.49 % (7955)------------------------------
% 0.30/1.49 % (7955)------------------------------
% 0.30/1.49 % (7971)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_7736 for (2999ds/7Mi)
% 0.30/1.49 % (7960)Refutation found. Thanks to Tanya!
% 0.30/1.49 % SZS status Theorem for DTF2THF_7736
% 0.30/1.49 % SZS output start Proof for DTF2THF_7736
% 0.30/1.49 thf(func_def_0, type, in: $i > $i > $o).
% 0.30/1.49 thf(func_def_1, type, exu: ($i > $o) > $o).
% 0.30/1.49 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 0.30/1.49 thf(func_def_8, type, powerset: $i > $i).
% 0.30/1.49 thf(func_def_10, type, setunion: $i > $i).
% 0.30/1.49 thf(func_def_19, type, descr: ($i > $o) > $i).
% 0.30/1.49 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 0.30/1.49 thf(func_def_26, type, prop2set: $o > $i).
% 0.30/1.49 thf(func_def_36, type, nonempty: $i > $o).
% 0.30/1.49 thf(func_def_69, type, set2prop: $i > $o).
% 0.30/1.49 thf(func_def_88, type, subset: $i > $i > $o).
% 0.30/1.49 thf(func_def_89, type, disjoint: $i > $i > $o).
% 0.30/1.49 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 0.30/1.49 thf(func_def_103, type, sP0: $i > $i > $o).
% 0.30/1.49 thf(func_def_104, type, sP1: $i > $o).
% 0.30/1.49 thf(func_def_105, type, sP2: $i > $i > $o).
% 0.30/1.49 thf(func_def_106, type, sP3: $i > $i > $o).
% 0.30/1.49 thf(func_def_109, type, sK6: $i > $o).
% 0.30/1.49 thf(func_def_110, type, sK7: $i > $i > $i).
% 0.30/1.49 thf(func_def_111, type, sK8: $i > $i > $i).
% 0.30/1.49 thf(func_def_112, type, sK9: $i > $i > $i).
% 0.30/1.49 thf(func_def_113, type, sK10: $i > $i > $i).
% 0.30/1.49 thf(func_def_114, type, sK11: $i > $i > $i > $i).
% 0.30/1.49 thf(func_def_115, type, sK12: $i > $i).
% 0.30/1.49 thf(func_def_116, type, sK13: $i > $i).
% 0.30/1.49 thf(func_def_117, type, sK14: $i > $i).
% 0.30/1.49 thf(func_def_118, type, sK15: $i > $i).
% 0.30/1.49 thf(func_def_119, type, sK16: $i > $i > $i).
% 0.30/1.49 thf(func_def_120, type, sK17: $i > $i > $i > $i).
% 0.30/1.49 thf(func_def_121, type, sK18: $i > $i > $i > $i).
% 0.30/1.49 thf(func_def_122, type, sK19: $i > $i > $i).
% 0.30/1.49 thf(func_def_123, type, sK20: $i > $i > $i).
% 0.30/1.49 thf(func_def_125, type, sK22: $i > $i).
% 0.30/1.49 thf(func_def_126, type, sK23: $i > $i).
% 0.30/1.49 thf(func_def_127, type, sK24: $i > $i).
% 0.30/1.49 thf(func_def_128, type, sK25: $i > $i > $i).
% 0.30/1.49 thf(func_def_132, type, sK29: $i > $o).
% 0.30/1.49 thf(func_def_133, type, sK30: ($i > $o) > $i).
% 0.30/1.49 thf(func_def_136, type, sK33: $i > $i > $i).
% 0.30/1.49 thf(func_def_137, type, sK34: $i > $o).
% 0.30/1.49 thf(func_def_144, type, sK41: $i > $o).
% 0.30/1.49 thf(func_def_145, type, sK42: $i > $o).
% 0.30/1.49 thf(func_def_146, type, sK43: ($i > $o) > ($i > $o) > $i).
% 0.30/1.49 thf(func_def_147, type, sK44: ($i > $o) > ($i > $o) > $i).
% 0.30/1.49 thf(func_def_152, type, sK49: $i > $o).
% 0.30/1.49 thf(func_def_155, type, sK52: $i > $o).
% 0.30/1.49 thf(func_def_157, type, sK54: $i > ($i > $o) > $i).
% 0.30/1.49 thf(func_def_159, type, sK56: $i > $i > $i).
% 0.30/1.49 thf(func_def_160, type, sK57: $i > $i > $i).
% 0.30/1.49 thf(func_def_163, type, sK60: ($i > $o) > $i).
% 0.30/1.49 thf(func_def_164, type, sK61: ($i > $o) > $i).
% 0.30/1.49 thf(func_def_165, type, sK62: $i > $o).
% 0.30/1.49 thf(func_def_173, type, sK70: $i > $i).
% 0.30/1.49 thf(func_def_177, type, sK74: $i > $o).
% 0.30/1.49 thf(func_def_180, type, sK77: $i > ($i > $o) > $i).
% 0.30/1.49 thf(func_def_181, type, sK78: $i > $o).
% 0.30/1.49 thf(func_def_187, type, sK84: ($i > $o) > $i).
% 0.30/1.49 thf(func_def_188, type, sK85: $i > $o).
% 0.30/1.49 thf(func_def_189, type, sK86: $i > $i).
% 0.30/1.49 thf(func_def_196, type, sK93: $i > $i > $i).
% 0.30/1.49 thf(func_def_199, type, sK96: $i > $i > $i).
% 0.30/1.49 thf(func_def_207, type, sK104: $o > $i > $i > $i).
% 0.30/1.49 thf(func_def_209, type, sK106: $i > $o).
% 0.30/1.49 thf(func_def_211, type, sK108: ($i > $o) > $i > $i).
% 0.30/1.49 thf(func_def_214, type, sK111: $i > $i).
% 0.30/1.49 thf(func_def_219, type, sK116: $i > ($i > $o) > $i).
% 0.30/1.49 thf(func_def_220, type, sK117: $i > $o).
% 0.30/1.49 thf(func_def_223, type, sK120: $i > $o).
% 0.30/1.49 thf(func_def_230, type, sK127: $i > $i).
% 0.30/1.49 thf(func_def_232, type, sK129: $i > ($i > $o) > $i).
% 0.30/1.49 thf(func_def_233, type, sK130: $i > $o).
% 0.30/1.49 thf(func_def_238, type, sK135: $i > $o).
% 0.30/1.49 thf(func_def_243, type, sK140: $i > $i > $i).
% 0.30/1.49 thf(func_def_244, type, sK141: $i > $o).
% 0.30/1.49 thf(func_def_245, type, sK142: $i > $o).
% 0.30/1.49 thf(func_def_246, type, sK143: ($i > $o) > ($i > $o) > $i).
% 0.30/1.49 thf(func_def_247, type, sK144: ($i > $o) > ($i > $o) > $i).
% 0.30/1.49 thf(func_def_256, type, sK153: $i > $i).
% 0.30/1.49 thf(func_def_259, type, sK156: $i > $i).
% 0.30/1.49 thf(func_def_268, type, sK165: $i > $o).
% 0.30/1.49 thf(func_def_271, type, sK168: $i > ($i > $o) > $i).
% 0.30/1.49 thf(func_def_274, type, sK171: $i > $o).
% 0.30/1.49 thf(func_def_276, type, sK173: $i > ($i > $o) > $i).
% 0.30/1.49 thf(func_def_277, type, sK174: $i > $o).
% 0.30/1.49 thf(func_def_282, type, sK179: $i > $o).
% 0.30/1.49 thf(func_def_285, type, sK182: $i > $i).
% 0.30/1.49 thf(func_def_288, type, sK185: $i > $i > $o).
% 0.30/1.49 thf(func_def_289, type, sK186: $i > $i).
% 0.30/1.49 thf(func_def_290, type, sK187: $i > $i).
% 0.30/1.49 thf(func_def_291, type, sK188: ($i > $i > $o) > $i > $i).
% 0.30/1.49 thf(func_def_292, type, sK189: $i > ($i > $i > $o) > $i > $i).
% 0.30/1.49 thf(func_def_293, type, sK190: ($i > $i > $o) > $i > $i).
% 0.30/1.49 thf(func_def_294, type, sK191: $i > $i).
% 0.30/1.49 thf(func_def_297, type, sK194: $i > $o).
% 0.30/1.49 thf(func_def_299, type, sK196: $i > $o).
% 0.30/1.49 thf(func_def_305, type, sK202: $i > $o).
% 0.30/1.49 thf(func_def_306, type, sK203: $i > $o).
% 0.30/1.49 thf(func_def_307, type, sK204: ($i > $o) > ($i > $o) > $i > $i > $i).
% 0.30/1.49 thf(func_def_308, type, sK205: ($i > $o) > ($i > $o) > $i > $i > $i).
% 0.30/1.49 thf(func_def_311, type, sK208: $i > $o).
% 0.30/1.49 thf(func_def_312, type, sK209: $i > $i).
% 0.30/1.49 thf(func_def_313, type, sK210: ($i > $o) > $i).
% 0.30/1.49 thf(func_def_317, type, sK214: $i > $o).
% 0.30/1.49 thf(func_def_319, type, sK216: ($i > $o) > $i > $i).
% 0.30/1.49 thf(func_def_325, type, ph222: !>[X0: $tType]:(X0)).
% 0.30/1.49 thf(f1887,plain,(
% 0.30/1.49 $false),
% 0.30/1.49 inference(subsumption_resolution,[status(thm)],[f1881,f1455])).
% 0.30/1.49 thf(f1455,plain,(
% 0.30/1.49 ($true = (in @ sK223 @ sK46))),
% 0.30/1.49 inference(binary_proxy_clausification,[status(thm)],[f1453])).
% 0.30/1.49 thf(f1453,plain,(
% 0.30/1.49 ($false = ((in @ sK223 @ sK46) => (in @ sK223 @ sK45)))),
% 0.30/1.49 inference(beta_eta_normalization,[status(thm)],[f1452])).
% 0.30/1.49 thf(f1452,plain,(
% 0.30/1.49 ($false = ((^[Y0 : $i]: ((in @ Y0 @ sK46) => (in @ Y0 @ sK45))) @ sK223))),
% 0.30/1.49 inference(sigma_clausification,[status(thm)],[f1451])).
% 0.30/1.49 thf(f1451,plain,(
% 0.30/1.49 ($true != (!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ sK46) => (in @ Y0 @ sK45)))))),
% 0.30/1.49 inference(beta_eta_normalization,[status(thm)],[f1155])).
% 0.30/1.49 thf(f1155,plain,(
% 0.30/1.49 (((^[Y0 : $i]: ((^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: ((in @ Y2 @ Y0) => (in @ Y2 @ Y1))))))) @ sK46 @ sK45) != $true)),
% 0.30/1.49 inference(definition_unfolding,[status(thm)],[f857,f1037])).
% 0.30/1.49 thf(f1037,plain,(
% 0.30/1.49 (subset = (^[Y0 : $i]: ((^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: ((in @ Y2 @ Y0) => (in @ Y2 @ Y1))))))))),
% 0.30/1.49 inference(cnf_transformation,[status(thm)],[f99])).
% 0.30/1.49 thf(f99,plain,(
% 0.30/1.49 (subset = (^[Y0 : $i]: ((^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: ((in @ Y2 @ Y0) => (in @ Y2 @ Y1))))))))),
% 0.30/1.49 inference(fool_elimination,[status(thm)],[f98])).
% 0.30/1.49 thf(f98,plain,(
% 0.30/1.49 ((^[X0 : $i, X1 : $i] : (! [X2 : $i] : ((in @ X2 @ X0) => (in @ X2 @ X1)))) = subset)),
% 0.30/1.49 inference(rectify,[status(thm)],[f78])).
% 0.30/1.49 thf(f78,axiom,(
% 0.30/1.49 ((^[X3 : $i, X4 : $i] : (! [X1 : $i] : ((in @ X1 @ X3) => (in @ X1 @ X4)))) = subset)),
% 0.30/1.49 file('/export/starexec/sandbox2/tmp/tmp.zNeidnJOtA/DTF2THF_7736.p',subset)).
% 0.30/1.49 thf(f857,plain,(
% 0.30/1.49 ($true != (subset @ sK46 @ sK45))),
% 0.30/1.49 inference(cnf_transformation,[status(thm)],[f407])).
% 0.30/1.49 thf(f407,plain,(
% 0.30/1.49 (! [X2 : $i] : (($true != (in @ X2 @ sK46)) | ((in @ X2 @ sK45) = $true)) & ($true != (subset @ sK46 @ sK45))) & (exuE3e = $true) & (upairsetIR = $true) & (nonemptyI = $true) & (exuI1 = $true) & (nonemptyI1 = $true) & (wellorderingAx = $true) & (singletonsswitch = $true) & (in__Cong = $true) & (emptysetE = $true) & (exuI3 = $true) & (exu__Cong = $true) & (dsetconstr__Cong = $true) & (powersetE = $true) & (quantDeMorgan1 = $true) & (prop2set2propI = $true) & (noeltsimpempty = $true) & (powersetI = $true) & (subPowSU = $true) & (dsetconstrI = $true) & (omegaIndAx = $true) & (powersetAx = $true) & (emptyI = $true) & (omegaSAx = $true) & (setunionI = $true) & (emptyinPowerset = $true) & (exuE3u = $true) & (notdexE = $true) & (setadjoin__Cong = $true) & (emptyinunitempty = $true) & (setunionE = $true) & (prop2setE = $true) & (emptysetAx = $true) & (prop2setI = $true) & (setadjoinIR = $true) & (descr__Cong = $true) & (setadjoinIL = $true) & (omega__Cong = $true) & (vacuousDall = $true) & (exuE2 = $true) & (foundationAx = $true) & (setoftrueEq = $true) & (emptysetimpfalse = $true) & (setextAx = $true) & (replAx = $true) & (setunionAx = $true) & (exuEu = $true) & (setext = $true) & (setunion__Cong = $true) & (setadjoinOr = $true) & (nonemptyE1 = $true) & (quantDeMorgan3 = $true) & (dsetconstrER = $true) & (notinsingleton = $true) & (exuI2 = $true) & (inCongP = $true) & (exuE1 = $true) & (quantDeMorgan2 = $true) & (emptyInPowerset = $true) & (notdallE = $true) & (upairsetE = $true) & (emptyE1 = $true) & (emptyset__Cong = $true) & (dsetconstrEL = $true) & (omega0Ax = $true) & (setadjoinE = $true) & (setadjoinAx = $true) & (eqinunit = $true) & (powerset__Cong = $true) & (upairsetIL = $true) & (notinemptyset = $true) & (descrp = $true) & (uniqinunit = $true) & (quantDeMorgan4 = $true) & (nonemptyImpWitness = $true) & (setbeta = $true)),
% 0.30/1.49 inference(skolemisation,[status(esa),new_symbols(skolem,[sK45,sK46])],[f329,f406])).
% 0.30/1.49 thf(f406,plain,(
% 0.30/1.49 ? [X0 : $i,X1 : $i] : (! [X2 : $i] : (((in @ X2 @ X1) != $true) | ($true = (in @ X2 @ X0))) & ((subset @ X1 @ X0) != $true)) => (! [X2 : $i] : (($true != (in @ X2 @ sK46)) | ((in @ X2 @ sK45) = $true)) & ($true != (subset @ sK46 @ sK45)))),
% 0.30/1.49 introduced(definition,[],[choice_axiom])).
% 0.30/1.49 thf(f329,plain,(
% 0.30/1.49 ? [X0 : $i,X1 : $i] : (! [X2 : $i] : (((in @ X2 @ X1) != $true) | ($true = (in @ X2 @ X0))) & ((subset @ X1 @ X0) != $true)) & (exuE3e = $true) & (upairsetIR = $true) & (nonemptyI = $true) & (exuI1 = $true) & (nonemptyI1 = $true) & (wellorderingAx = $true) & (singletonsswitch = $true) & (in__Cong = $true) & (emptysetE = $true) & (exuI3 = $true) & (exu__Cong = $true) & (dsetconstr__Cong = $true) & (powersetE = $true) & (quantDeMorgan1 = $true) & (prop2set2propI = $true) & (noeltsimpempty = $true) & (powersetI = $true) & (subPowSU = $true) & (dsetconstrI = $true) & (omegaIndAx = $true) & (powersetAx = $true) & (emptyI = $true) & (omegaSAx = $true) & (setunionI = $true) & (emptyinPowerset = $true) & (exuE3u = $true) & (notdexE = $true) & (setadjoin__Cong = $true) & (emptyinunitempty = $true) & (setunionE = $true) & (prop2setE = $true) & (emptysetAx = $true) & (prop2setI = $true) & (setadjoinIR = $true) & (descr__Cong = $true) & (setadjoinIL = $true) & (omega__Cong = $true) & (vacuousDall = $true) & (exuE2 = $true) & (foundationAx = $true) & (setoftrueEq = $true) & (emptysetimpfalse = $true) & (setextAx = $true) & (replAx = $true) & (setunionAx = $true) & (exuEu = $true) & (setext = $true) & (setunion__Cong = $true) & (setadjoinOr = $true) & (nonemptyE1 = $true) & (quantDeMorgan3 = $true) & (dsetconstrER = $true) & (notinsingleton = $true) & (exuI2 = $true) & (inCongP = $true) & (exuE1 = $true) & (quantDeMorgan2 = $true) & (emptyInPowerset = $true) & (notdallE = $true) & (upairsetE = $true) & (emptyE1 = $true) & (emptyset__Cong = $true) & (dsetconstrEL = $true) & (omega0Ax = $true) & (setadjoinE = $true) & (setadjoinAx = $true) & (eqinunit = $true) & (powerset__Cong = $true) & (upairsetIL = $true) & (notinemptyset = $true) & (descrp = $true) & (uniqinunit = $true) & (quantDeMorgan4 = $true) & (nonemptyImpWitness = $true) & (setbeta = $true)),
% 0.30/1.49 inference(flattening,[status(thm)],[f328])).
% 0.30/1.49 thf(f328,plain,(
% 0.30/1.49 ((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i,X1 : $i] : (! [X2 : $i] : (((in @ X2 @ X1) != $true) | ($true = (in @ X2 @ X0))) & ((subset @ X1 @ X0) != $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)),
% 0.30/1.49 inference(ennf_transformation,[status(thm)],[f225])).
% 0.30/1.49 thf(f225,plain,(
% 0.30/1.49 ~((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) => ! [X0 : $i,X1 : $i] : (! [X2 : $i] : (((in @ X2 @ X1) = $true) => ($true = (in @ X2 @ X0))) => ((subset @ X1 @ X0) = $true)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 0.30/1.49 inference(fool_elimination,[status(thm)],[f224])).
% 0.30/1.49 thf(f224,plain,(
% 0.30/1.49 ~(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 => ! [X0 : $i,X1 : $i] : (! [X2 : $i] : ((in @ X2 @ X1) => (in @ X2 @ X0)) => (subset @ X1 @ X0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 0.30/1.49 inference(rectify,[status(thm)],[f80])).
% 0.30/1.49 thf(f80,negated_conjecture,(
% 0.30/1.49 ~(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 => ! [X4 : $i,X3 : $i] : (! [X1 : $i] : ((in @ X1 @ X3) => (in @ X1 @ X4)) => (subset @ X3 @ X4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 0.30/1.49 inference(negated_conjecture,[status(cth)],[f79])).
% 0.30/1.49 thf(f79,conjecture,(
% 0.30/1.49 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 => ! [X4 : $i,X3 : $i] : (! [X1 : $i] : ((in @ X1 @ X3) => (in @ X1 @ X4)) => (subset @ X3 @ X4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 0.30/1.49 file('/export/starexec/sandbox2/tmp/tmp.zNeidnJOtA/DTF2THF_7736.p',subsetI1)).
% 0.30/1.49 thf(f1881,plain,(
% 0.30/1.49 ($true != (in @ sK223 @ sK46))),
% 0.30/1.49 inference(trivial_inequality_removal,[status(thm)],[f1871])).
% 0.30/1.49 thf(f1871,plain,(
% 0.30/1.49 ($true != (in @ sK223 @ sK46)) | ($false = $true)),
% 0.30/1.49 inference(superposition,[status(thm)],[f1454,f858])).
% 0.30/1.50 thf(f858,plain,(
% 0.30/1.50 ( ! [X2 : $i] : (((in @ X2 @ sK45) = $true) | ($true != (in @ X2 @ sK46))) )),
% 0.30/1.50 inference(cnf_transformation,[status(thm)],[f407])).
% 0.30/1.50 thf(f1454,plain,(
% 0.30/1.50 ($false = (in @ sK223 @ sK45))),
% 0.30/1.50 inference(binary_proxy_clausification,[status(thm)],[f1453])).
% 0.30/1.50 % SZS output end Proof for DTF2THF_7736
% 0.30/1.50 % (7960)------------------------------
% 0.30/1.50 % (7960)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.30/1.50 % (7960)Termination reason: Refutation
% 0.30/1.50
% 0.30/1.50 % (7960)Memory used [KB]: 7036
% 0.30/1.50 % (7960)Time elapsed: 0.060 s
% 0.30/1.50 % (7960)Instructions burned: 166 (million)
% 0.30/1.50 % (7960)------------------------------
% 0.30/1.50 % (7960)------------------------------
% 0.30/1.50 % (7954)Success in time 0.066 s
%------------------------------------------------------------------------------