%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU605^1 : TPTP v9.2.1. Released v3.7.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n022.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 09:35:33 AM UTC 2026
% Result : Theorem 2.63s 1.84s
% Output : Refutation 2.63s
% Verified :
% SZS Type : Refutation
% Derivation depth : 15
% Number of leaves : 6
% Syntax : Number of formulae : 40 ( 9 unt; 0 typ; 2 def)
% Number of atoms : 2550 ( 700 equ; 0 cnn)
% Maximal formula atoms : 124 ( 63 avg)
% Number of connectives : 1344 ( 61 ~; 34 |; 386 &; 232 @)
% ( 6 <=>; 625 =>; 0 <=; 0 <~>)
% Maximal formula depth : 131 ( 29 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 246 ( 243 usr; 131 con; 0-4 aty)
% Number of variables : 63 ( 0 ^; 45 !; 18 ?; 63 :)
% Comments :
%------------------------------------------------------------------------------
thf(func_def_0,type,
in: $i > $i > $o ).
thf(func_def_1,type,
exu: ( $i > $o ) > $o ).
thf(func_def_6,type,
setadjoin: $i > $i > $i ).
thf(func_def_8,type,
powerset: $i > $i ).
thf(func_def_10,type,
setunion: $i > $i ).
thf(func_def_19,type,
descr: ( $i > $o ) > $i ).
thf(func_def_21,type,
dsetconstr: $i > ( $i > $o ) > $i ).
thf(func_def_26,type,
prop2set: $o > $i ).
thf(func_def_36,type,
nonempty: $i > $o ).
thf(func_def_69,type,
set2prop: $i > $o ).
thf(func_def_88,type,
subset: $i > $i > $o ).
thf(func_def_89,type,
disjoint: $i > $i > $o ).
thf(func_def_90,type,
setsmeet: $i > $i > $o ).
thf(func_def_114,type,
binunion: $i > $i > $i ).
thf(func_def_122,type,
binintersect: $i > $i > $i ).
thf(func_def_135,type,
regular: $i > $o ).
thf(func_def_136,type,
setminus: $i > $i > $i ).
thf(func_def_154,type,
sP1: $i > $i > $o ).
thf(func_def_155,type,
sP2: $i > $o ).
thf(func_def_156,type,
sP3: $i > $i > $o ).
thf(func_def_157,type,
sP4: $i > $i > $o ).
thf(func_def_161,type,
sK8: $i > $o ).
thf(func_def_162,type,
sK9: $i > $i ).
thf(func_def_163,type,
sK10: ( $i > $o ) > $i ).
thf(func_def_168,type,
sK15: $i > $i > $i ).
thf(func_def_169,type,
sK16: $i > $i > $i ).
thf(func_def_173,type,
sK20: $i > ( $i > $o ) > $i ).
thf(func_def_174,type,
sK21: $i > $o ).
thf(func_def_186,type,
sK33: $i > ( $i > $o ) > $i ).
thf(func_def_187,type,
sK34: $i > $o ).
thf(func_def_190,type,
sK37: $i > $i > $i ).
thf(func_def_193,type,
sK40: $i > $i ).
thf(func_def_205,type,
sK52: $i > $o ).
thf(func_def_206,type,
sK53: $i > $o ).
thf(func_def_207,type,
sK54: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_208,type,
sK55: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_211,type,
sK58: $i > $o ).
thf(func_def_215,type,
sK62: $i > $o ).
thf(func_def_224,type,
sK71: $i > $o ).
thf(func_def_227,type,
sK74: $i > ( $i > $o ) > $i ).
thf(func_def_232,type,
sK79: $i > $o ).
thf(func_def_233,type,
sK80: $i > $o ).
thf(func_def_234,type,
sK81: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_235,type,
sK82: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_242,type,
sK89: $i > $i ).
thf(func_def_244,type,
sK91: $i > $i ).
thf(func_def_269,type,
sK116: $i > ( $i > $o ) > $i ).
thf(func_def_270,type,
sK117: $i > $o ).
thf(func_def_277,type,
sK124: ( $i > $o ) > $i > $i ).
thf(func_def_279,type,
sK126: $i > $o ).
thf(func_def_285,type,
sK132: $i > $i ).
thf(func_def_288,type,
sK135: $i > $o ).
thf(func_def_296,type,
sK143: $i > $i > $i ).
thf(func_def_304,type,
sK151: $i > $o ).
thf(func_def_307,type,
sK154: $i > $o ).
thf(func_def_310,type,
sK157: $i > $o ).
thf(func_def_312,type,
sK159: $i > ( $i > $o ) > $i ).
thf(func_def_313,type,
sK160: $o > $i > $i > $i ).
thf(func_def_320,type,
sK167: $i > $o ).
thf(func_def_327,type,
sK174: $i > $o ).
thf(func_def_332,type,
sK179: $i > $o ).
thf(func_def_336,type,
sK183: $i > $i ).
thf(func_def_342,type,
sK189: $i > $i ).
thf(func_def_345,type,
sK192: ( $i > $o ) > $i ).
thf(func_def_346,type,
sK193: $i > $o ).
thf(func_def_351,type,
sK198: $i > $o ).
thf(func_def_356,type,
sK203: $i > ( $i > $o ) > $i ).
thf(func_def_357,type,
sK204: $i > $o ).
thf(func_def_361,type,
sK208: $i > $o ).
thf(func_def_363,type,
sK210: $i > ( $i > $o ) > $i ).
thf(func_def_373,type,
sK220: $i > $i > $i ).
thf(func_def_376,type,
sK223: $i > $i > $o ).
thf(func_def_378,type,
sK225: $i > $i ).
thf(func_def_379,type,
sK226: $i > $i ).
thf(func_def_380,type,
sK227: $i > ( $i > $i > $o ) > $i ).
thf(func_def_381,type,
sK228: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_382,type,
sK229: $i > ( $i > $i > $o ) > $i ).
thf(func_def_386,type,
sK233: $i > $i ).
thf(func_def_395,type,
sK242: $i > $i > $i ).
thf(func_def_403,type,
sK250: $i > $o ).
thf(func_def_404,type,
sK251: $i > $i ).
thf(func_def_405,type,
sK252: ( $i > $o ) > $i ).
thf(func_def_411,type,
sK258: $i > $o ).
thf(func_def_415,type,
sK262: $i > $o ).
thf(func_def_421,type,
sK268: $i > $i > $i ).
thf(func_def_424,type,
sK271: $i > $i > $i ).
thf(func_def_425,type,
sK272: $i > $i > $i ).
thf(func_def_426,type,
sK273: $i > $i > $i ).
thf(func_def_427,type,
sK274: $i > $i > $i ).
thf(func_def_428,type,
sK275: $i > $i > $i ).
thf(func_def_429,type,
sK276: $i > $i > $i > $i ).
thf(func_def_430,type,
sK277: $i > $i ).
thf(func_def_431,type,
sK278: $i > $i ).
thf(func_def_432,type,
sK279: $i > $i ).
thf(func_def_433,type,
sK280: $i > $i ).
thf(func_def_434,type,
sK281: $i > $i > $i > $i ).
thf(func_def_435,type,
sK282: $i > $i > $i > $i ).
thf(func_def_436,type,
sK283: $i > $i > $i ).
thf(func_def_437,type,
sK284: $i > $i > $i ).
thf(func_def_438,type,
sK285: $i > $i > $i ).
thf(func_def_439,type,
sK286: $i > $i ).
thf(func_def_441,type,
sK288: $i > $i ).
thf(func_def_442,type,
sK289: $i > $i ).
thf(func_def_444,type,
sK291: $i > $o ).
thf(func_def_445,type,
sK292: $i > $o ).
thf(func_def_446,type,
sK293: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_447,type,
sK294: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_462,type,
sK309: $i > $o ).
thf(func_def_468,type,
sK315: $i > $i > $i ).
thf(func_def_474,type,
sK321: ( $i > $o ) > $i ).
thf(func_def_475,type,
sK322: ( $i > $o ) > $i ).
thf(func_def_476,type,
sK323: $i > $o ).
thf(func_def_485,type,
sK332: $i > $i > $i ).
thf(func_def_487,type,
ph334:
!>[X0: $tType] : X0 ).
thf(f1878,plain,
$false,
inference(avatar_sat_refutation,[status(thm)],[f1839,f1861,f1875]) ).
thf(f1875,plain,
~ spl333_3,
inference(avatar_contradiction_clause,[status(thm)],[f1874]) ).
thf(f1874,plain,
( $false
| ~ spl333_3 ),
inference(trivial_inequality_removal,[status(thm)],[f1871]) ).
thf(f1871,plain,
( ( $true != $true )
| ~ spl333_3 ),
inference(superposition,[status(thm)],[f1190,f1834]) ).
thf(f1834,plain,
( ( $true
= ( in @ sK5 @ sK6 ) )
| ~ spl333_3 ),
inference(avatar_component_clause,[status(thm)],[f1832]) ).
thf(f1832,definition,
( spl333_3
<=> ( $true
= ( in @ sK5 @ sK6 ) ) ),
introduced(definition,[new_symbols(naming,[spl333_3])],[avatar_definition]) ).
thf(f1190,plain,
( $true
!= ( in @ sK5 @ sK6 ) ),
inference(cnf_transformation,[status(thm)],[f530]) ).
thf(f530,plain,
( ( exuE3u = $true )
& ( subsetTrans = $true )
& ( subset2powerset = $true )
& ( omega0Ax = $true )
& ( setunionI = $true )
& ( descr__Cong = $true )
& ( binintersectSubset3 = $true )
& ( notequalI2 = $true )
& ( quantDeMorgan1 = $true )
& ( setadjoinIL = $true )
& ( exuEu = $true )
& ( omegaIndAx = $true )
& ( quantDeMorgan2 = $true )
& ( binunionRsub = $true )
& ( quantDeMorgan3 = $true )
& ( notdexE = $true )
& ( wellorderingAx = $true )
& ( inCongP = $true )
& ( singletonsswitch = $true )
& ( subsetI2 = $true )
& ( setminusEL = $true )
& ( omega__Cong = $true )
& ( setminusSubset2 = $true )
& ( exuE3e = $true )
& ( quantDeMorgan4 = $true )
& ( exuI2 = $true )
& ( $true
!= ( in @ sK5 @ ( setminus @ sK7 @ sK6 ) ) )
& ( ( in @ sK5 @ sK7 )
= $true )
& ( $true
!= ( in @ sK5 @ sK6 ) )
& ( nonemptyI = $true )
& ( setadjoinOr = $true )
& ( dsetconstr__Cong = $true )
& ( inPowerset = $true )
& ( emptyInPowerset = $true )
& ( binunionLsub = $true )
& ( setext = $true )
& ( in__Cong = $true )
& ( descrp = $true )
& ( setbeta = $true )
& ( dsetconstrER = $true )
& ( noeltsimpempty = $true )
& ( subsetE2 = $true )
& ( powerset__Cong = $true )
& ( binintersectSubset1 = $true )
& ( binintersectSubset5 = $true )
& ( upairsetE = $true )
& ( setoftrueEq = $true )
& ( emptyinunitempty = $true )
& ( subsetemptysetimpeq = $true )
& ( subsetI1 = $true )
& ( emptysetsubset = $true )
& ( setextsub = $true )
& ( powersetE1 = $true )
& ( powersetAx = $true )
& ( nonemptyE1 = $true )
& ( subsetRefl = $true )
& ( powersetI1 = $true )
& ( binintersectRsub = $true )
& ( notinemptyset = $true )
& ( exu__Cong = $true )
& ( exuE1 = $true )
& ( binintersectSubset4 = $true )
& ( binunionE = $true )
& ( emptysetE = $true )
& ( emptysetimpfalse = $true )
& ( exuI3 = $true )
& ( setminusER = $true )
& ( sepInPowerset = $true )
& ( powersetI = $true )
& ( sepSubset = $true )
& ( powersetsubset = $true )
& ( powersetE = $true )
& ( setunionE = $true )
& ( binintersectI = $true )
& ( binintersectLsub = $true )
& ( emptysetAx = $true )
& ( setadjoinIR = $true )
& ( nonemptyI1 = $true )
& ( emptyI = $true )
& ( prop2set2propI = $true )
& ( setadjoinSub2 = $true )
& ( notequalI1 = $true )
& ( prop2setI = $true )
& ( binintersectEL = $true )
& ( setadjoinE = $true )
& ( vacuousDall = $true )
& ( notinsingleton = $true )
& ( binintersectER = $true )
& ( notdallE = $true )
& ( emptyinPowerset = $true )
& ( eqimpsubset2 = $true )
& ( eqimpsubset1 = $true )
& ( setadjoin__Cong = $true )
& ( replAx = $true )
& ( uniqinunit = $true )
& ( setunion__Cong = $true )
& ( dsetconstrI = $true )
& ( setadjoinSub = $true )
& ( notsubsetI = $true )
& ( emptyE1 = $true )
& ( subPowSU = $true )
& ( upairsetIL = $true )
& ( binunionIR = $true )
& ( bs114d = $true )
& ( binunionEcases = $true )
& ( binintersectSubset2 = $true )
& ( omegaSAx = $true )
& ( exuI1 = $true )
& ( setminusI = $true )
& ( eqinunit = $true )
& ( upairset2IR = $true )
& ( disjointsetsI1 = $true )
& ( subsetE = $true )
& ( upairsetIR = $true )
& ( foundationAx = $true )
& ( exuE2 = $true )
& ( dsetconstrEL = $true )
& ( setunionAx = $true )
& ( binunionIL = $true )
& ( setextAx = $true )
& ( prop2setE = $true )
& ( setadjoinAx = $true )
& ( nonemptyImpWitness = $true )
& ( emptyset__Cong = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK5,sK6,sK7])],[f507,f529]) ).
thf(f529,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X0 @ ( setminus @ X2 @ X1 ) )
!= $true )
& ( $true
= ( in @ X0 @ X2 ) )
& ( $true
!= ( in @ X0 @ X1 ) ) )
=> ( ( $true
!= ( in @ sK5 @ ( setminus @ sK7 @ sK6 ) ) )
& ( ( in @ sK5 @ sK7 )
= $true )
& ( $true
!= ( in @ sK5 @ sK6 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f507,plain,
( ( exuE3u = $true )
& ( subsetTrans = $true )
& ( subset2powerset = $true )
& ( omega0Ax = $true )
& ( setunionI = $true )
& ( descr__Cong = $true )
& ( binintersectSubset3 = $true )
& ( notequalI2 = $true )
& ( quantDeMorgan1 = $true )
& ( setadjoinIL = $true )
& ( exuEu = $true )
& ( omegaIndAx = $true )
& ( quantDeMorgan2 = $true )
& ( binunionRsub = $true )
& ( quantDeMorgan3 = $true )
& ( notdexE = $true )
& ( wellorderingAx = $true )
& ( inCongP = $true )
& ( singletonsswitch = $true )
& ( subsetI2 = $true )
& ( setminusEL = $true )
& ( omega__Cong = $true )
& ( setminusSubset2 = $true )
& ( exuE3e = $true )
& ( quantDeMorgan4 = $true )
& ( exuI2 = $true )
& ? [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X0 @ ( setminus @ X2 @ X1 ) )
!= $true )
& ( $true
= ( in @ X0 @ X2 ) )
& ( $true
!= ( in @ X0 @ X1 ) ) )
& ( nonemptyI = $true )
& ( setadjoinOr = $true )
& ( dsetconstr__Cong = $true )
& ( inPowerset = $true )
& ( emptyInPowerset = $true )
& ( binunionLsub = $true )
& ( setext = $true )
& ( in__Cong = $true )
& ( descrp = $true )
& ( setbeta = $true )
& ( dsetconstrER = $true )
& ( noeltsimpempty = $true )
& ( subsetE2 = $true )
& ( powerset__Cong = $true )
& ( binintersectSubset1 = $true )
& ( binintersectSubset5 = $true )
& ( upairsetE = $true )
& ( setoftrueEq = $true )
& ( emptyinunitempty = $true )
& ( subsetemptysetimpeq = $true )
& ( subsetI1 = $true )
& ( emptysetsubset = $true )
& ( setextsub = $true )
& ( powersetE1 = $true )
& ( powersetAx = $true )
& ( nonemptyE1 = $true )
& ( subsetRefl = $true )
& ( powersetI1 = $true )
& ( binintersectRsub = $true )
& ( notinemptyset = $true )
& ( exu__Cong = $true )
& ( exuE1 = $true )
& ( binintersectSubset4 = $true )
& ( binunionE = $true )
& ( emptysetE = $true )
& ( emptysetimpfalse = $true )
& ( exuI3 = $true )
& ( setminusER = $true )
& ( sepInPowerset = $true )
& ( powersetI = $true )
& ( sepSubset = $true )
& ( powersetsubset = $true )
& ( powersetE = $true )
& ( setunionE = $true )
& ( binintersectI = $true )
& ( binintersectLsub = $true )
& ( emptysetAx = $true )
& ( setadjoinIR = $true )
& ( nonemptyI1 = $true )
& ( emptyI = $true )
& ( prop2set2propI = $true )
& ( setadjoinSub2 = $true )
& ( notequalI1 = $true )
& ( prop2setI = $true )
& ( binintersectEL = $true )
& ( setadjoinE = $true )
& ( vacuousDall = $true )
& ( notinsingleton = $true )
& ( binintersectER = $true )
& ( notdallE = $true )
& ( emptyinPowerset = $true )
& ( eqimpsubset2 = $true )
& ( eqimpsubset1 = $true )
& ( setadjoin__Cong = $true )
& ( replAx = $true )
& ( uniqinunit = $true )
& ( setunion__Cong = $true )
& ( dsetconstrI = $true )
& ( setadjoinSub = $true )
& ( notsubsetI = $true )
& ( emptyE1 = $true )
& ( subPowSU = $true )
& ( upairsetIL = $true )
& ( binunionIR = $true )
& ( bs114d = $true )
& ( binunionEcases = $true )
& ( binintersectSubset2 = $true )
& ( omegaSAx = $true )
& ( exuI1 = $true )
& ( setminusI = $true )
& ( eqinunit = $true )
& ( upairset2IR = $true )
& ( disjointsetsI1 = $true )
& ( subsetE = $true )
& ( upairsetIR = $true )
& ( foundationAx = $true )
& ( exuE2 = $true )
& ( dsetconstrEL = $true )
& ( setunionAx = $true )
& ( binunionIL = $true )
& ( setextAx = $true )
& ( prop2setE = $true )
& ( setadjoinAx = $true )
& ( nonemptyImpWitness = $true )
& ( emptyset__Cong = $true ) ),
inference(flattening,[status(thm)],[f506]) ).
thf(f506,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ( $true
!= ( in @ X0 @ X1 ) )
& ( $true
= ( in @ X0 @ X2 ) )
& ( ( in @ X0 @ ( setminus @ X2 @ X1 ) )
!= $true ) )
& ( setminusSubset2 = $true )
& ( setminusER = $true )
& ( setminusEL = $true )
& ( setminusI = $true )
& ( bs114d = $true )
& ( binintersectSubset1 = $true )
& ( binintersectSubset4 = $true )
& ( binintersectRsub = $true )
& ( disjointsetsI1 = $true )
& ( binintersectER = $true )
& ( binintersectSubset3 = $true )
& ( binintersectSubset2 = $true )
& ( binintersectLsub = $true )
& ( binintersectEL = $true )
& ( binintersectSubset5 = $true )
& ( binintersectI = $true )
& ( binunionRsub = $true )
& ( binunionLsub = $true )
& ( binunionE = $true )
& ( binunionEcases = $true )
& ( binunionIR = $true )
& ( upairset2IR = $true )
& ( binunionIL = $true )
& ( sepSubset = $true )
& ( sepInPowerset = $true )
& ( powersetsubset = $true )
& ( inPowerset = $true )
& ( powersetE1 = $true )
& ( powersetI1 = $true )
& ( subsetemptysetimpeq = $true )
& ( setextsub = $true )
& ( subset2powerset = $true )
& ( setadjoinSub2 = $true )
& ( setadjoinSub = $true )
& ( subsetTrans = $true )
& ( subsetRefl = $true )
& ( notequalI2 = $true )
& ( notequalI1 = $true )
& ( notsubsetI = $true )
& ( subsetE2 = $true )
& ( subsetE = $true )
& ( emptysetsubset = $true )
& ( subsetI2 = $true )
& ( eqimpsubset1 = $true )
& ( eqimpsubset2 = $true )
& ( subsetI1 = $true )
& ( dsetconstr__Cong = $true )
& ( descr__Cong = $true )
& ( exuEu = $true )
& ( omega__Cong = $true )
& ( setunion__Cong = $true )
& ( powerset__Cong = $true )
& ( setadjoin__Cong = $true )
& ( emptyset__Cong = $true )
& ( exu__Cong = $true )
& ( exuE3u = $true )
& ( in__Cong = $true )
& ( inCongP = $true )
& ( exuI2 = $true )
& ( exuI3 = $true )
& ( exuI1 = $true )
& ( notdallE = $true )
& ( notdexE = $true )
& ( prop2set2propI = $true )
& ( prop2setI = $true )
& ( quantDeMorgan4 = $true )
& ( quantDeMorgan3 = $true )
& ( quantDeMorgan2 = $true )
& ( quantDeMorgan1 = $true )
& ( vacuousDall = $true )
& ( emptyE1 = $true )
& ( upairsetIR = $true )
& ( upairsetIL = $true )
& ( upairsetE = $true )
& ( singletonsswitch = $true )
& ( eqinunit = $true )
& ( notinsingleton = $true )
& ( uniqinunit = $true )
& ( nonemptyImpWitness = $true )
& ( exuE2 = $true )
& ( subPowSU = $true )
& ( setunionE = $true )
& ( setunionI = $true )
& ( powersetE = $true )
& ( emptyInPowerset = $true )
& ( emptyinPowerset = $true )
& ( powersetI = $true )
& ( setoftrueEq = $true )
& ( setadjoinOr = $true )
& ( setadjoinE = $true )
& ( setadjoinIR = $true )
& ( emptyinunitempty = $true )
& ( setadjoinIL = $true )
& ( nonemptyI1 = $true )
& ( nonemptyI = $true )
& ( nonemptyE1 = $true )
& ( setbeta = $true )
& ( noeltsimpempty = $true )
& ( emptyI = $true )
& ( setext = $true )
& ( exuE3e = $true )
& ( notinemptyset = $true )
& ( emptysetimpfalse = $true )
& ( emptysetE = $true )
& ( prop2setE = $true )
& ( exuE1 = $true )
& ( dsetconstrER = $true )
& ( dsetconstrEL = $true )
& ( dsetconstrI = $true )
& ( descrp = $true )
& ( wellorderingAx = $true )
& ( foundationAx = $true )
& ( replAx = $true )
& ( omegaIndAx = $true )
& ( omegaSAx = $true )
& ( omega0Ax = $true )
& ( setunionAx = $true )
& ( powersetAx = $true )
& ( setadjoinAx = $true )
& ( emptysetAx = $true )
& ( setextAx = $true ) ),
inference(ennf_transformation,[status(thm)],[f370]) ).
thf(f370,plain,
~ ( ( setextAx = $true )
=> ( ( emptysetAx = $true )
=> ( ( setadjoinAx = $true )
=> ( ( powersetAx = $true )
=> ( ( setunionAx = $true )
=> ( ( omega0Ax = $true )
=> ( ( omegaSAx = $true )
=> ( ( omegaIndAx = $true )
=> ( ( replAx = $true )
=> ( ( foundationAx = $true )
=> ( ( wellorderingAx = $true )
=> ( ( descrp = $true )
=> ( ( dsetconstrI = $true )
=> ( ( dsetconstrEL = $true )
=> ( ( dsetconstrER = $true )
=> ( ( exuE1 = $true )
=> ( ( prop2setE = $true )
=> ( ( emptysetE = $true )
=> ( ( emptysetimpfalse = $true )
=> ( ( notinemptyset = $true )
=> ( ( exuE3e = $true )
=> ( ( setext = $true )
=> ( ( emptyI = $true )
=> ( ( noeltsimpempty = $true )
=> ( ( setbeta = $true )
=> ( ( nonemptyE1 = $true )
=> ( ( nonemptyI = $true )
=> ( ( nonemptyI1 = $true )
=> ( ( setadjoinIL = $true )
=> ( ( emptyinunitempty = $true )
=> ( ( setadjoinIR = $true )
=> ( ( setadjoinE = $true )
=> ( ( setadjoinOr = $true )
=> ( ( setoftrueEq = $true )
=> ( ( powersetI = $true )
=> ( ( emptyinPowerset = $true )
=> ( ( emptyInPowerset = $true )
=> ( ( powersetE = $true )
=> ( ( setunionI = $true )
=> ( ( setunionE = $true )
=> ( ( subPowSU = $true )
=> ( ( exuE2 = $true )
=> ( ( nonemptyImpWitness = $true )
=> ( ( uniqinunit = $true )
=> ( ( notinsingleton = $true )
=> ( ( eqinunit = $true )
=> ( ( singletonsswitch = $true )
=> ( ( upairsetE = $true )
=> ( ( upairsetIL = $true )
=> ( ( upairsetIR = $true )
=> ( ( emptyE1 = $true )
=> ( ( vacuousDall = $true )
=> ( ( quantDeMorgan1 = $true )
=> ( ( quantDeMorgan2 = $true )
=> ( ( quantDeMorgan3 = $true )
=> ( ( quantDeMorgan4 = $true )
=> ( ( prop2setI = $true )
=> ( ( prop2set2propI = $true )
=> ( ( notdexE = $true )
=> ( ( notdallE = $true )
=> ( ( exuI1 = $true )
=> ( ( exuI3 = $true )
=> ( ( exuI2 = $true )
=> ( ( inCongP = $true )
=> ( ( in__Cong = $true )
=> ( ( exuE3u = $true )
=> ( ( exu__Cong = $true )
=> ( ( emptyset__Cong = $true )
=> ( ( setadjoin__Cong = $true )
=> ( ( powerset__Cong = $true )
=> ( ( setunion__Cong = $true )
=> ( ( omega__Cong = $true )
=> ( ( exuEu = $true )
=> ( ( descr__Cong = $true )
=> ( ( dsetconstr__Cong = $true )
=> ( ( subsetI1 = $true )
=> ( ( eqimpsubset2 = $true )
=> ( ( eqimpsubset1 = $true )
=> ( ( subsetI2 = $true )
=> ( ( emptysetsubset = $true )
=> ( ( subsetE = $true )
=> ( ( subsetE2 = $true )
=> ( ( notsubsetI = $true )
=> ( ( notequalI1 = $true )
=> ( ( notequalI2 = $true )
=> ( ( subsetRefl = $true )
=> ( ( subsetTrans = $true )
=> ( ( setadjoinSub = $true )
=> ( ( setadjoinSub2 = $true )
=> ( ( subset2powerset = $true )
=> ( ( setextsub = $true )
=> ( ( subsetemptysetimpeq = $true )
=> ( ( powersetI1 = $true )
=> ( ( powersetE1 = $true )
=> ( ( inPowerset = $true )
=> ( ( powersetsubset = $true )
=> ( ( sepInPowerset = $true )
=> ( ( sepSubset = $true )
=> ( ( binunionIL = $true )
=> ( ( upairset2IR = $true )
=> ( ( binunionIR = $true )
=> ( ( binunionEcases = $true )
=> ( ( binunionE = $true )
=> ( ( binunionLsub = $true )
=> ( ( binunionRsub = $true )
=> ( ( binintersectI = $true )
=> ( ( binintersectSubset5 = $true )
=> ( ( binintersectEL = $true )
=> ( ( binintersectLsub = $true )
=> ( ( binintersectSubset2 = $true )
=> ( ( binintersectSubset3 = $true )
=> ( ( binintersectER = $true )
=> ( ( disjointsetsI1 = $true )
=> ( ( binintersectRsub = $true )
=> ( ( binintersectSubset4 = $true )
=> ( ( binintersectSubset1 = $true )
=> ( ( bs114d = $true )
=> ( ( setminusI = $true )
=> ( ( setminusEL = $true )
=> ( ( setminusER = $true )
=> ( ( setminusSubset2 = $true )
=> ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X0 @ ( setminus @ X2 @ X1 ) )
!= $true )
=> ( ( $true
= ( in @ X0 @ X2 ) )
=> ( $true
= ( in @ X0 @ X1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(flattening,[status(thm)],[f136]) ).
thf(f136,plain,
~ ( ( setextAx = $true )
=> ( ( emptysetAx = $true )
=> ( ( setadjoinAx = $true )
=> ( ( powersetAx = $true )
=> ( ( setunionAx = $true )
=> ( ( omega0Ax = $true )
=> ( ( omegaSAx = $true )
=> ( ( omegaIndAx = $true )
=> ( ( replAx = $true )
=> ( ( foundationAx = $true )
=> ( ( wellorderingAx = $true )
=> ( ( descrp = $true )
=> ( ( dsetconstrI = $true )
=> ( ( dsetconstrEL = $true )
=> ( ( dsetconstrER = $true )
=> ( ( exuE1 = $true )
=> ( ( prop2setE = $true )
=> ( ( emptysetE = $true )
=> ( ( emptysetimpfalse = $true )
=> ( ( notinemptyset = $true )
=> ( ( exuE3e = $true )
=> ( ( setext = $true )
=> ( ( emptyI = $true )
=> ( ( noeltsimpempty = $true )
=> ( ( setbeta = $true )
=> ( ( nonemptyE1 = $true )
=> ( ( nonemptyI = $true )
=> ( ( nonemptyI1 = $true )
=> ( ( setadjoinIL = $true )
=> ( ( emptyinunitempty = $true )
=> ( ( setadjoinIR = $true )
=> ( ( setadjoinE = $true )
=> ( ( setadjoinOr = $true )
=> ( ( setoftrueEq = $true )
=> ( ( powersetI = $true )
=> ( ( emptyinPowerset = $true )
=> ( ( emptyInPowerset = $true )
=> ( ( powersetE = $true )
=> ( ( setunionI = $true )
=> ( ( setunionE = $true )
=> ( ( subPowSU = $true )
=> ( ( exuE2 = $true )
=> ( ( nonemptyImpWitness = $true )
=> ( ( uniqinunit = $true )
=> ( ( notinsingleton = $true )
=> ( ( eqinunit = $true )
=> ( ( singletonsswitch = $true )
=> ( ( upairsetE = $true )
=> ( ( upairsetIL = $true )
=> ( ( upairsetIR = $true )
=> ( ( emptyE1 = $true )
=> ( ( vacuousDall = $true )
=> ( ( quantDeMorgan1 = $true )
=> ( ( quantDeMorgan2 = $true )
=> ( ( quantDeMorgan3 = $true )
=> ( ( quantDeMorgan4 = $true )
=> ( ( prop2setI = $true )
=> ( ( prop2set2propI = $true )
=> ( ( notdexE = $true )
=> ( ( notdallE = $true )
=> ( ( exuI1 = $true )
=> ( ( exuI3 = $true )
=> ( ( exuI2 = $true )
=> ( ( inCongP = $true )
=> ( ( in__Cong = $true )
=> ( ( exuE3u = $true )
=> ( ( exu__Cong = $true )
=> ( ( emptyset__Cong = $true )
=> ( ( setadjoin__Cong = $true )
=> ( ( powerset__Cong = $true )
=> ( ( setunion__Cong = $true )
=> ( ( omega__Cong = $true )
=> ( ( exuEu = $true )
=> ( ( descr__Cong = $true )
=> ( ( dsetconstr__Cong = $true )
=> ( ( subsetI1 = $true )
=> ( ( eqimpsubset2 = $true )
=> ( ( eqimpsubset1 = $true )
=> ( ( subsetI2 = $true )
=> ( ( emptysetsubset = $true )
=> ( ( subsetE = $true )
=> ( ( subsetE2 = $true )
=> ( ( notsubsetI = $true )
=> ( ( notequalI1 = $true )
=> ( ( notequalI2 = $true )
=> ( ( subsetRefl = $true )
=> ( ( subsetTrans = $true )
=> ( ( setadjoinSub = $true )
=> ( ( setadjoinSub2 = $true )
=> ( ( subset2powerset = $true )
=> ( ( setextsub = $true )
=> ( ( subsetemptysetimpeq = $true )
=> ( ( powersetI1 = $true )
=> ( ( powersetE1 = $true )
=> ( ( inPowerset = $true )
=> ( ( powersetsubset = $true )
=> ( ( sepInPowerset = $true )
=> ( ( sepSubset = $true )
=> ( ( binunionIL = $true )
=> ( ( upairset2IR = $true )
=> ( ( binunionIR = $true )
=> ( ( binunionEcases = $true )
=> ( ( binunionE = $true )
=> ( ( binunionLsub = $true )
=> ( ( binunionRsub = $true )
=> ( ( binintersectI = $true )
=> ( ( binintersectSubset5 = $true )
=> ( ( binintersectEL = $true )
=> ( ( binintersectLsub = $true )
=> ( ( binintersectSubset2 = $true )
=> ( ( binintersectSubset3 = $true )
=> ( ( binintersectER = $true )
=> ( ( disjointsetsI1 = $true )
=> ( ( binintersectRsub = $true )
=> ( ( binintersectSubset4 = $true )
=> ( ( binintersectSubset1 = $true )
=> ( ( bs114d = $true )
=> ( ( setminusI = $true )
=> ( ( setminusEL = $true )
=> ( ( setminusER = $true )
=> ( ( setminusSubset2 = $true )
=> ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X0 @ ( setminus @ X2 @ X1 ) )
!= $true )
=> ( ( $true
= ( in @ X0 @ X2 ) )
=> ( $true
= ( in @ X0 @ X1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f135]) ).
thf(f135,plain,
~ ( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ! [X0: $i,X1: $i,X2: $i] :
( ~ ( in @ X0 @ ( setminus @ X2 @ X1 ) )
=> ( ( in @ X0 @ X2 )
=> ( in @ X0 @ X1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f125]) ).
thf(f125,negated_conjecture,
~ ( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ! [X1: $i,X4: $i,X3: $i] :
( ~ ( in @ X1 @ ( setminus @ X3 @ X4 ) )
=> ( ( in @ X1 @ X3 )
=> ( in @ X1 @ X4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f124]) ).
thf(f124,conjecture,
( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ! [X1: $i,X4: $i,X3: $i] :
( ~ ( in @ X1 @ ( setminus @ X3 @ X4 ) )
=> ( ( in @ X1 @ X3 )
=> ( in @ X1 @ X4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.FHMJW0nBVa/DTF2THF_8470.p',setminusERneg) ).
thf(f1861,plain,
spl333_4,
inference(avatar_contradiction_clause,[status(thm)],[f1860]) ).
thf(f1860,plain,
( $false
| spl333_4 ),
inference(trivial_inequality_removal,[status(thm)],[f1854]) ).
thf(f1854,plain,
( ( $true != $true )
| spl333_4 ),
inference(superposition,[status(thm)],[f1838,f1191]) ).
thf(f1191,plain,
( ( in @ sK5 @ sK7 )
= $true ),
inference(cnf_transformation,[status(thm)],[f530]) ).
thf(f1838,plain,
( ( ( in @ sK5 @ sK7 )
!= $true )
| spl333_4 ),
inference(avatar_component_clause,[status(thm)],[f1836]) ).
thf(f1836,definition,
( spl333_4
<=> ( ( in @ sK5 @ sK7 )
= $true ) ),
introduced(definition,[new_symbols(naming,[spl333_4])],[avatar_definition]) ).
thf(f1839,plain,
( spl333_3
| ~ spl333_4 ),
inference(avatar_split_clause,[status(thm)],[f1830,f1836,f1832]) ).
thf(f1830,plain,
( ( ( in @ sK5 @ sK7 )
!= $true )
| ( $true
= ( in @ sK5 @ sK6 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f1829]) ).
thf(f1829,plain,
( ( $true != $true )
| ( $true
= ( in @ sK5 @ sK6 ) )
| ( ( in @ sK5 @ sK7 )
!= $true ) ),
inference(forward_demodulation,[status(thm)],[f1808,f1110]) ).
thf(f1110,plain,
setminusI = $true,
inference(cnf_transformation,[status(thm)],[f530]) ).
thf(f1808,plain,
( ( setminusI != $true )
| ( $true
= ( in @ sK5 @ sK6 ) )
| ( ( in @ sK5 @ sK7 )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f1802]) ).
thf(f1802,plain,
( ( $true
= ( in @ sK5 @ sK6 ) )
| ( setminusI != $true )
| ( ( in @ sK5 @ sK7 )
!= $true )
| ( $true != $true ) ),
inference(superposition,[status(thm)],[f1192,f1411]) ).
thf(f1411,plain,
! [X2: $i,X0: $i,X1: $i] :
( ( $true
= ( in @ X2 @ ( setminus @ X0 @ X1 ) ) )
| ( ( in @ X2 @ X0 )
!= $true )
| ( ( in @ X2 @ X1 )
= $true )
| ( setminusI != $true ) ),
inference(cnf_transformation,[status(thm)],[f768]) ).
thf(f768,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X2 @ X1 )
= $true )
| ( $true
= ( in @ X2 @ ( setminus @ X0 @ X1 ) ) )
| ( ( in @ X2 @ X0 )
!= $true ) )
| ( setminusI != $true ) )
& ( ( setminusI = $true )
| ( ( $true
!= ( in @ sK149 @ sK148 ) )
& ( ( in @ sK149 @ ( setminus @ sK147 @ sK148 ) )
!= $true )
& ( $true
= ( in @ sK149 @ sK147 ) ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK147,sK148,sK149])],[f766,f767]) ).
thf(f767,plain,
( ? [X3: $i,X4: $i,X5: $i] :
( ( ( in @ X5 @ X4 )
!= $true )
& ( ( in @ X5 @ ( setminus @ X3 @ X4 ) )
!= $true )
& ( $true
= ( in @ X5 @ X3 ) ) )
=> ( ( $true
!= ( in @ sK149 @ sK148 ) )
& ( ( in @ sK149 @ ( setminus @ sK147 @ sK148 ) )
!= $true )
& ( $true
= ( in @ sK149 @ sK147 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f766,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X2 @ X1 )
= $true )
| ( $true
= ( in @ X2 @ ( setminus @ X0 @ X1 ) ) )
| ( ( in @ X2 @ X0 )
!= $true ) )
| ( setminusI != $true ) )
& ( ( setminusI = $true )
| ? [X3: $i,X4: $i,X5: $i] :
( ( ( in @ X5 @ X4 )
!= $true )
& ( ( in @ X5 @ ( setminus @ X3 @ X4 ) )
!= $true )
& ( $true
= ( in @ X5 @ X3 ) ) ) ) ),
inference(rectify,[status(thm)],[f765]) ).
thf(f765,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X2 @ X1 )
= $true )
| ( $true
= ( in @ X2 @ ( setminus @ X0 @ X1 ) ) )
| ( ( in @ X2 @ X0 )
!= $true ) )
| ( setminusI != $true ) )
& ( ( setminusI = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X2 @ X1 )
!= $true )
& ( $true
!= ( in @ X2 @ ( setminus @ X0 @ X1 ) ) )
& ( ( in @ X2 @ X0 )
= $true ) ) ) ),
inference(nnf_transformation,[status(thm)],[f408]) ).
thf(f408,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X2 @ X1 )
= $true )
| ( $true
= ( in @ X2 @ ( setminus @ X0 @ X1 ) ) )
| ( ( in @ X2 @ X0 )
!= $true ) )
<=> ( setminusI = $true ) ),
inference(flattening,[status(thm)],[f407]) ).
thf(f407,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X2 @ ( setminus @ X0 @ X1 ) ) )
| ( ( in @ X2 @ X1 )
= $true )
| ( ( in @ X2 @ X0 )
!= $true ) )
<=> ( setminusI = $true ) ),
inference(ennf_transformation,[status(thm)],[f392]) ).
thf(f392,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X2 @ X0 )
= $true )
=> ( ( ( in @ X2 @ X1 )
!= $true )
=> ( $true
= ( in @ X2 @ ( setminus @ X0 @ X1 ) ) ) ) )
<=> ( setminusI = $true ) ),
inference(flattening,[status(thm)],[f310]) ).
thf(f310,plain,
( ( setminusI = $true )
<=> ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X2 @ X0 )
= $true )
=> ( ( ( in @ X2 @ X1 )
!= $true )
=> ( $true
= ( in @ X2 @ ( setminus @ X0 @ X1 ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f309]) ).
thf(f309,plain,
( setminusI
= ( ! [X0: $i,X1: $i,X2: $i] :
( ( in @ X2 @ X0 )
=> ( ~ ( in @ X2 @ X1 )
=> ( in @ X2 @ ( setminus @ X0 @ X1 ) ) ) ) ) ),
inference(rectify,[status(thm)],[f120]) ).
thf(f120,axiom,
( setminusI
= ( ! [X3: $i,X4: $i,X1: $i] :
( ( in @ X1 @ X3 )
=> ( ~ ( in @ X1 @ X4 )
=> ( in @ X1 @ ( setminus @ X3 @ X4 ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.FHMJW0nBVa/DTF2THF_8470.p',setminusI) ).
thf(f1192,plain,
( $true
!= ( in @ sK5 @ ( setminus @ sK7 @ sK6 ) ) ),
inference(cnf_transformation,[status(thm)],[f530]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.13 % Problem : SEU605^1 : TPTP v9.2.1. Released v3.7.0.
% 0.11/0.14 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.17/0.35 % Computer : n022.cluster.edu
% 0.17/0.35 % Model : x86_64 x86_64
% 0.17/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.35 % Memory : 8042.1875MB
% 0.17/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.17/0.35 % CPULimit : 300
% 0.17/0.35 % WCLimit : 300
% 0.17/0.35 % DateTime : Tue Feb 24 12:28:21 EST 2026
% 0.17/0.35 % CPUTime :
% 0.17/0.35 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.19/0.47 ---- Original DTF file ---
% 0.19/0.47 thf(spec,logic,$$dhol).
% 0.19/0.47 %------------------------------------------------------------------------------
% 0.19/0.47 % File : SEU605^1 : TPTP v9.2.1. Released v3.7.0.
% 0.19/0.47 % Domain : Set Theory
% 0.19/0.47 % Problem : Preliminary Notions - Operations on Sets - Set Difference
% 0.19/0.47 % Version : Especial.
% 0.19/0.47 % English : (! A:i.! B:i.! x:i.~(in x (setminus A B)) -> in x A -> in x B)
% 0.19/0.47
% 0.19/0.47 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.19/0.47 % Source : [Bro08]
% 0.19/0.47 % Names : ZFC107g [Bro08]
% 0.19/0.47
% 0.19/0.47 % Status : Theorem
% 0.19/0.47 % Rating : 0.11 v9.1.0, 0.12 v9.0.0, 0.20 v8.2.0, 0.23 v8.1.0, 0.09 v7.5.0, 0.14 v7.4.0, 0.22 v7.2.0, 0.12 v7.1.0, 0.38 v7.0.0, 0.29 v6.4.0, 0.33 v6.3.0, 0.40 v6.2.0, 0.29 v6.1.0, 0.43 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
% 0.19/0.47 % Syntax : Number of formulae : 264 ( 123 unt; 140 typ; 123 def)
% 0.19/0.47 % Number of atoms : 656 ( 179 equ; 0 cnn)
% 0.19/0.47 % Maximal formula atoms : 124 ( 5 avg)
% 0.19/0.47 % Number of connectives : 1035 ( 33 ~; 6 |; 28 &; 663 @)
% 0.19/0.47 % ( 14 <=>; 291 =>; 0 <=; 0 <~>)
% 0.19/0.47 % Maximal formula depth : 130 ( 2 avg)
% 0.19/0.47 % Number of types : 2 ( 0 usr)
% 0.19/0.47 % Number of type conns : 62 ( 62 >; 0 *; 0 +; 0 <<)
% 0.19/0.47 % Number of symbols : 143 ( 140 usr; 125 con; 0-2 aty)
% 0.19/0.47 % Number of variables : 357 ( 30 ^; 298 !; 29 ?; 357 :)
% 0.19/0.47 % SPC : TH0_THM_EQU_NAR
% 0.19/0.47
% 0.19/0.47 % Comments : http://mathgate.info/detsetitem.php?id=460
% 0.19/0.47 %------------------------------------------------------------------------------
% 0.19/0.47 thf(in_type,type,
% 0.19/0.47 in: $i > $i > $o ).
% 0.19/0.47
% 0.19/0.47 thf(exu_type,type,
% 0.19/0.47 exu: ( $i > $o ) > $o ).
% 0.19/0.47
% 0.19/0.47 thf(exu,definition,
% 0.19/0.47 ( exu
% 0.19/0.47 = ( ^ [Xphi: $i > $o] :
% 0.19/0.47 ? [Xx: $i] :
% 0.19/0.47 ( ( Xphi @ Xx )
% 0.19/0.47 & ! [Xy: $i] :
% 0.19/0.47 ( ( Xphi @ Xy )
% 0.19/0.47 => ( Xx = Xy ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(setextAx_type,type,
% 0.19/0.47 setextAx: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setextAx,definition,
% 0.19/0.47 ( setextAx
% 0.19/0.47 = ( ! [A: $i,B: $i] :
% 0.19/0.47 ( ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 <=> ( in @ Xx @ B ) )
% 0.19/0.47 => ( A = B ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(emptyset_type,type,
% 0.19/0.47 emptyset: $i ).
% 0.19/0.47
% 0.19/0.47 thf(emptysetAx_type,type,
% 0.19/0.47 emptysetAx: $o ).
% 0.19/0.47
% 0.19/0.47 thf(emptysetAx,definition,
% 0.19/0.47 ( emptysetAx
% 0.19/0.47 = ( ! [Xx: $i] :
% 0.19/0.47 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(setadjoin_type,type,
% 0.19/0.47 setadjoin: $i > $i > $i ).
% 0.19/0.47
% 0.19/0.47 thf(setadjoinAx_type,type,
% 0.19/0.47 setadjoinAx: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setadjoinAx,definition,
% 0.19/0.47 ( setadjoinAx
% 0.19/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.19/0.47 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.19/0.47 <=> ( ( Xy = Xx )
% 0.19/0.47 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(powerset_type,type,
% 0.19/0.47 powerset: $i > $i ).
% 0.19/0.47
% 0.19/0.47 thf(powersetAx_type,type,
% 0.19/0.47 powersetAx: $o ).
% 0.19/0.47
% 0.19/0.47 thf(powersetAx,definition,
% 0.19/0.47 ( powersetAx
% 0.19/0.47 = ( ! [A: $i,B: $i] :
% 0.19/0.47 ( ( in @ B @ ( powerset @ A ) )
% 0.19/0.47 <=> ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ B )
% 0.19/0.47 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(setunion_type,type,
% 0.19/0.47 setunion: $i > $i ).
% 0.19/0.47
% 0.19/0.47 thf(setunionAx_type,type,
% 0.19/0.47 setunionAx: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setunionAx,definition,
% 0.19/0.47 ( setunionAx
% 0.19/0.47 = ( ! [A: $i,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ ( setunion @ A ) )
% 0.19/0.47 <=> ? [B: $i] :
% 0.19/0.47 ( ( in @ Xx @ B )
% 0.19/0.47 & ( in @ B @ A ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(omega_type,type,
% 0.19/0.47 omega: $i ).
% 0.19/0.47
% 0.19/0.47 thf(omega0Ax_type,type,
% 0.19/0.47 omega0Ax: $o ).
% 0.19/0.47
% 0.19/0.47 thf(omega0Ax,definition,
% 0.19/0.47 ( omega0Ax
% 0.19/0.47 = ( in @ emptyset @ omega ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(omegaSAx_type,type,
% 0.19/0.47 omegaSAx: $o ).
% 0.19/0.47
% 0.19/0.47 thf(omegaSAx,definition,
% 0.19/0.47 ( omegaSAx
% 0.19/0.47 = ( ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ omega )
% 0.19/0.47 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(omegaIndAx_type,type,
% 0.19/0.47 omegaIndAx: $o ).
% 0.19/0.47
% 0.19/0.47 thf(omegaIndAx,definition,
% 0.19/0.47 ( omegaIndAx
% 0.19/0.47 = ( ! [A: $i] :
% 0.19/0.47 ( ( ( in @ emptyset @ A )
% 0.19/0.47 & ! [Xx: $i] :
% 0.19/0.47 ( ( ( in @ Xx @ omega )
% 0.19/0.47 & ( in @ Xx @ A ) )
% 0.19/0.47 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.19/0.47 => ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ omega )
% 0.19/0.47 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(replAx_type,type,
% 0.19/0.47 replAx: $o ).
% 0.19/0.47
% 0.19/0.47 thf(replAx,definition,
% 0.19/0.47 ( replAx
% 0.19/0.47 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.19/0.47 ( ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ( exu
% 0.19/0.47 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.19/0.47 => ? [B: $i] :
% 0.19/0.47 ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ B )
% 0.19/0.47 <=> ? [Xy: $i] :
% 0.19/0.47 ( ( in @ Xy @ A )
% 0.19/0.47 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(foundationAx_type,type,
% 0.19/0.47 foundationAx: $o ).
% 0.19/0.47
% 0.19/0.47 thf(foundationAx,definition,
% 0.19/0.47 ( foundationAx
% 0.19/0.47 = ( ! [A: $i] :
% 0.19/0.47 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.19/0.47 => ? [B: $i] :
% 0.19/0.47 ( ( in @ B @ A )
% 0.19/0.47 & ~ ? [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ B )
% 0.19/0.47 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(wellorderingAx_type,type,
% 0.19/0.47 wellorderingAx: $o ).
% 0.19/0.47
% 0.19/0.47 thf(wellorderingAx,definition,
% 0.19/0.47 ( wellorderingAx
% 0.19/0.47 = ( ! [A: $i] :
% 0.19/0.47 ? [B: $i] :
% 0.19/0.47 ( ! [C: $i] :
% 0.19/0.47 ( ( in @ C @ B )
% 0.19/0.47 => ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ C )
% 0.19/0.47 => ( in @ Xx @ A ) ) )
% 0.19/0.47 & ! [Xx: $i,Xy: $i] :
% 0.19/0.47 ( ( ( in @ Xx @ A )
% 0.19/0.47 & ( in @ Xy @ A ) )
% 0.19/0.47 => ( ! [C: $i] :
% 0.19/0.47 ( ( in @ C @ B )
% 0.19/0.47 => ( ( in @ Xx @ C )
% 0.19/0.47 <=> ( in @ Xy @ C ) ) )
% 0.19/0.47 => ( Xx = Xy ) ) )
% 0.19/0.47 & ! [C: $i,D: $i] :
% 0.19/0.47 ( ( ( in @ C @ B )
% 0.19/0.47 & ( in @ D @ B ) )
% 0.19/0.47 => ( ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ C )
% 0.19/0.47 => ( in @ Xx @ D ) )
% 0.19/0.47 | ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ D )
% 0.19/0.47 => ( in @ Xx @ C ) ) ) )
% 0.19/0.47 & ! [C: $i] :
% 0.19/0.47 ( ( ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ C )
% 0.19/0.47 => ( in @ Xx @ A ) )
% 0.19/0.47 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.19/0.47 => ? [D: $i,Xx: $i] :
% 0.19/0.47 ( ( in @ D @ B )
% 0.19/0.47 & ( in @ Xx @ C )
% 0.19/0.47 & ~ ? [Xy: $i] :
% 0.19/0.47 ( ( in @ Xy @ D )
% 0.19/0.47 & ( in @ Xy @ C ) )
% 0.19/0.47 & ! [E: $i] :
% 0.19/0.47 ( ( in @ E @ B )
% 0.19/0.47 => ( ! [Xy: $i] :
% 0.19/0.47 ( ( in @ Xy @ E )
% 0.19/0.47 => ( in @ Xy @ D ) )
% 0.19/0.47 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(descr_type,type,
% 0.19/0.47 descr: ( $i > $o ) > $i ).
% 0.19/0.47
% 0.19/0.47 thf(descrp_type,type,
% 0.19/0.47 descrp: $o ).
% 0.19/0.47
% 0.19/0.47 thf(descrp,definition,
% 0.19/0.47 ( descrp
% 0.19/0.47 = ( ! [Xphi: $i > $o] :
% 0.19/0.47 ( ( exu
% 0.19/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.47 => ( Xphi
% 0.19/0.47 @ ( descr
% 0.19/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(dsetconstr_type,type,
% 0.19/0.47 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.19/0.47
% 0.19/0.47 thf(dsetconstrI_type,type,
% 0.19/0.47 dsetconstrI: $o ).
% 0.19/0.47
% 0.19/0.47 thf(dsetconstrI,definition,
% 0.19/0.47 ( dsetconstrI
% 0.19/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ( ( Xphi @ Xx )
% 0.19/0.47 => ( in @ Xx
% 0.19/0.47 @ ( dsetconstr @ A
% 0.19/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(dsetconstrEL_type,type,
% 0.19/0.47 dsetconstrEL: $o ).
% 0.19/0.47
% 0.19/0.47 thf(dsetconstrEL,definition,
% 0.19/0.47 ( dsetconstrEL
% 0.19/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx
% 0.19/0.47 @ ( dsetconstr @ A
% 0.19/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.19/0.47 => ( in @ Xx @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(dsetconstrER_type,type,
% 0.19/0.47 dsetconstrER: $o ).
% 0.19/0.47
% 0.19/0.47 thf(dsetconstrER,definition,
% 0.19/0.47 ( dsetconstrER
% 0.19/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx
% 0.19/0.47 @ ( dsetconstr @ A
% 0.19/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.19/0.47 => ( Xphi @ Xx ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(exuE1_type,type,
% 0.19/0.47 exuE1: $o ).
% 0.19/0.47
% 0.19/0.47 thf(exuE1,definition,
% 0.19/0.47 ( exuE1
% 0.19/0.47 = ( ! [Xphi: $i > $o] :
% 0.19/0.47 ( ( exu
% 0.19/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.47 => ? [Xx: $i] :
% 0.19/0.47 ( ( Xphi @ Xx )
% 0.19/0.47 & ! [Xy: $i] :
% 0.19/0.47 ( ( Xphi @ Xy )
% 0.19/0.47 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(prop2set_type,type,
% 0.19/0.47 prop2set: $o > $i ).
% 0.19/0.47
% 0.19/0.47 thf(prop2setE_type,type,
% 0.19/0.47 prop2setE: $o ).
% 0.19/0.47
% 0.19/0.47 thf(prop2setE,definition,
% 0.19/0.47 ( prop2setE
% 0.19/0.47 = ( ! [Xphi: $o,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.19/0.47 => Xphi ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(emptysetE_type,type,
% 0.19/0.47 emptysetE: $o ).
% 0.19/0.47
% 0.19/0.47 thf(emptysetE,definition,
% 0.19/0.47 ( emptysetE
% 0.19/0.47 = ( ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ emptyset )
% 0.19/0.47 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(emptysetimpfalse_type,type,
% 0.19/0.47 emptysetimpfalse: $o ).
% 0.19/0.47
% 0.19/0.47 thf(emptysetimpfalse,definition,
% 0.19/0.47 ( emptysetimpfalse
% 0.19/0.47 = ( ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ emptyset )
% 0.19/0.47 => $false ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(notinemptyset_type,type,
% 0.19/0.47 notinemptyset: $o ).
% 0.19/0.47
% 0.19/0.47 thf(notinemptyset,definition,
% 0.19/0.47 ( notinemptyset
% 0.19/0.47 = ( ! [Xx: $i] :
% 0.19/0.47 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(exuE3e_type,type,
% 0.19/0.47 exuE3e: $o ).
% 0.19/0.47
% 0.19/0.47 thf(exuE3e,definition,
% 0.19/0.47 ( exuE3e
% 0.19/0.47 = ( ! [Xphi: $i > $o] :
% 0.19/0.47 ( ( exu
% 0.19/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.47 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(setext_type,type,
% 0.19/0.47 setext: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setext,definition,
% 0.19/0.47 ( setext
% 0.19/0.47 = ( ! [A: $i,B: $i] :
% 0.19/0.47 ( ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ( in @ Xx @ B ) )
% 0.19/0.47 => ( ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ B )
% 0.19/0.47 => ( in @ Xx @ A ) )
% 0.19/0.47 => ( A = B ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(emptyI_type,type,
% 0.19/0.47 emptyI: $o ).
% 0.19/0.47
% 0.19/0.47 thf(emptyI,definition,
% 0.19/0.47 ( emptyI
% 0.19/0.47 = ( ! [A: $i] :
% 0.19/0.47 ( ! [Xx: $i] :
% 0.19/0.47 ~ ( in @ Xx @ A )
% 0.19/0.47 => ( A = emptyset ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(noeltsimpempty_type,type,
% 0.19/0.47 noeltsimpempty: $o ).
% 0.19/0.47
% 0.19/0.47 thf(noeltsimpempty,definition,
% 0.19/0.47 ( noeltsimpempty
% 0.19/0.47 = ( ! [A: $i] :
% 0.19/0.47 ( ! [Xx: $i] :
% 0.19/0.47 ~ ( in @ Xx @ A )
% 0.19/0.47 => ( A = emptyset ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(setbeta_type,type,
% 0.19/0.47 setbeta: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setbeta,definition,
% 0.19/0.47 ( setbeta
% 0.19/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ( ( in @ Xx
% 0.19/0.47 @ ( dsetconstr @ A
% 0.19/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.19/0.47 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(nonempty_type,type,
% 0.19/0.47 nonempty: $i > $o ).
% 0.19/0.47
% 0.19/0.47 thf(nonempty,definition,
% 0.19/0.47 ( nonempty
% 0.19/0.47 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(nonemptyE1_type,type,
% 0.19/0.47 nonemptyE1: $o ).
% 0.19/0.47
% 0.19/0.47 thf(nonemptyE1,definition,
% 0.19/0.47 ( nonemptyE1
% 0.19/0.47 = ( ! [A: $i] :
% 0.19/0.47 ( ( nonempty @ A )
% 0.19/0.47 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(nonemptyI_type,type,
% 0.19/0.47 nonemptyI: $o ).
% 0.19/0.47
% 0.19/0.47 thf(nonemptyI,definition,
% 0.19/0.47 ( nonemptyI
% 0.19/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ( ( Xphi @ Xx )
% 0.19/0.47 => ( nonempty
% 0.19/0.47 @ ( dsetconstr @ A
% 0.19/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(nonemptyI1_type,type,
% 0.19/0.47 nonemptyI1: $o ).
% 0.19/0.47
% 0.19/0.47 thf(nonemptyI1,definition,
% 0.19/0.47 ( nonemptyI1
% 0.19/0.47 = ( ! [A: $i] :
% 0.19/0.47 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.19/0.47 => ( nonempty @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(setadjoinIL_type,type,
% 0.19/0.47 setadjoinIL: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setadjoinIL,definition,
% 0.19/0.47 ( setadjoinIL
% 0.19/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(emptyinunitempty_type,type,
% 0.19/0.47 emptyinunitempty: $o ).
% 0.19/0.47
% 0.19/0.47 thf(emptyinunitempty,definition,
% 0.19/0.47 ( emptyinunitempty
% 0.19/0.47 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(setadjoinIR_type,type,
% 0.19/0.47 setadjoinIR: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setadjoinIR,definition,
% 0.19/0.47 ( setadjoinIR
% 0.19/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.19/0.47 ( ( in @ Xy @ A )
% 0.19/0.47 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(setadjoinE_type,type,
% 0.19/0.47 setadjoinE: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setadjoinE,definition,
% 0.19/0.47 ( setadjoinE
% 0.19/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.19/0.47 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.19/0.47 => ! [Xphi: $o] :
% 0.19/0.47 ( ( ( Xy = Xx )
% 0.19/0.47 => Xphi )
% 0.19/0.47 => ( ( ( in @ Xy @ A )
% 0.19/0.47 => Xphi )
% 0.19/0.47 => Xphi ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(setadjoinOr_type,type,
% 0.19/0.47 setadjoinOr: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setadjoinOr,definition,
% 0.19/0.47 ( setadjoinOr
% 0.19/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.19/0.47 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.19/0.47 => ( ( Xy = Xx )
% 0.19/0.47 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(setoftrueEq_type,type,
% 0.19/0.47 setoftrueEq: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setoftrueEq,definition,
% 0.19/0.47 ( setoftrueEq
% 0.19/0.47 = ( ! [A: $i] :
% 0.19/0.47 ( ( dsetconstr @ A
% 0.19/0.47 @ ^ [Xx: $i] : $true )
% 0.19/0.47 = A ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(powersetI_type,type,
% 0.19/0.47 powersetI: $o ).
% 0.19/0.47
% 0.19/0.47 thf(powersetI,definition,
% 0.19/0.47 ( powersetI
% 0.19/0.47 = ( ! [A: $i,B: $i] :
% 0.19/0.47 ( ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ B )
% 0.19/0.47 => ( in @ Xx @ A ) )
% 0.19/0.47 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(emptyinPowerset_type,type,
% 0.19/0.47 emptyinPowerset: $o ).
% 0.19/0.47
% 0.19/0.47 thf(emptyinPowerset,definition,
% 0.19/0.47 ( emptyinPowerset
% 0.19/0.47 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(emptyInPowerset_type,type,
% 0.19/0.47 emptyInPowerset: $o ).
% 0.19/0.47
% 0.19/0.47 thf(emptyInPowerset,definition,
% 0.19/0.47 ( emptyInPowerset
% 0.19/0.47 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(powersetE_type,type,
% 0.19/0.47 powersetE: $o ).
% 0.19/0.47
% 0.19/0.47 thf(powersetE,definition,
% 0.19/0.47 ( powersetE
% 0.19/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.47 ( ( in @ B @ ( powerset @ A ) )
% 0.19/0.47 => ( ( in @ Xx @ B )
% 0.19/0.47 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(setunionI_type,type,
% 0.19/0.47 setunionI: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setunionI,definition,
% 0.19/0.47 ( setunionI
% 0.19/0.47 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.19/0.47 ( ( in @ Xx @ B )
% 0.19/0.47 => ( ( in @ B @ A )
% 0.19/0.47 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(setunionE_type,type,
% 0.19/0.47 setunionE: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setunionE,definition,
% 0.19/0.47 ( setunionE
% 0.19/0.47 = ( ! [A: $i,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ ( setunion @ A ) )
% 0.19/0.47 => ! [Xphi: $o] :
% 0.19/0.47 ( ! [B: $i] :
% 0.19/0.47 ( ( in @ Xx @ B )
% 0.19/0.47 => ( ( in @ B @ A )
% 0.19/0.47 => Xphi ) )
% 0.19/0.47 => Xphi ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(subPowSU_type,type,
% 0.19/0.47 subPowSU: $o ).
% 0.19/0.47
% 0.19/0.47 thf(subPowSU,definition,
% 0.19/0.47 ( subPowSU
% 0.19/0.47 = ( ! [A: $i,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(exuE2_type,type,
% 0.19/0.47 exuE2: $o ).
% 0.19/0.47
% 0.19/0.47 thf(exuE2,definition,
% 0.19/0.47 ( exuE2
% 0.19/0.47 = ( ! [Xphi: $i > $o] :
% 0.19/0.47 ( ( exu
% 0.19/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.47 => ? [Xx: $i] :
% 0.19/0.47 ! [Xy: $i] :
% 0.19/0.47 ( ( Xphi @ Xy )
% 0.19/0.47 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(nonemptyImpWitness_type,type,
% 0.19/0.47 nonemptyImpWitness: $o ).
% 0.19/0.47
% 0.19/0.47 thf(nonemptyImpWitness,definition,
% 0.19/0.47 ( nonemptyImpWitness
% 0.19/0.47 = ( ! [A: $i] :
% 0.19/0.47 ( ( nonempty @ A )
% 0.19/0.47 => ? [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 & $true ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(uniqinunit_type,type,
% 0.19/0.47 uniqinunit: $o ).
% 0.19/0.47
% 0.19/0.47 thf(uniqinunit,definition,
% 0.19/0.47 ( uniqinunit
% 0.19/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.47 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.19/0.47 => ( Xx = Xy ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(notinsingleton_type,type,
% 0.19/0.47 notinsingleton: $o ).
% 0.19/0.47
% 0.19/0.47 thf(notinsingleton,definition,
% 0.19/0.47 ( notinsingleton
% 0.19/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.47 ( ( Xx != Xy )
% 0.19/0.47 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(eqinunit_type,type,
% 0.19/0.47 eqinunit: $o ).
% 0.19/0.47
% 0.19/0.47 thf(eqinunit,definition,
% 0.19/0.47 ( eqinunit
% 0.19/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.47 ( ( Xx = Xy )
% 0.19/0.47 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(singletonsswitch_type,type,
% 0.19/0.47 singletonsswitch: $o ).
% 0.19/0.47
% 0.19/0.47 thf(singletonsswitch,definition,
% 0.19/0.47 ( singletonsswitch
% 0.19/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.47 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.19/0.47 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(upairsetE_type,type,
% 0.19/0.47 upairsetE: $o ).
% 0.19/0.47
% 0.19/0.47 thf(upairsetE,definition,
% 0.19/0.47 ( upairsetE
% 0.19/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.19/0.47 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.19/0.47 => ( ( Xz = Xx )
% 0.19/0.47 | ( Xz = Xy ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(upairsetIL_type,type,
% 0.19/0.47 upairsetIL: $o ).
% 0.19/0.47
% 0.19/0.47 thf(upairsetIL,definition,
% 0.19/0.47 ( upairsetIL
% 0.19/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(upairsetIR_type,type,
% 0.19/0.47 upairsetIR: $o ).
% 0.19/0.47
% 0.19/0.47 thf(upairsetIR,definition,
% 0.19/0.47 ( upairsetIR
% 0.19/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(emptyE1_type,type,
% 0.19/0.47 emptyE1: $o ).
% 0.19/0.47
% 0.19/0.47 thf(emptyE1,definition,
% 0.19/0.47 ( emptyE1
% 0.19/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.47 ( ? [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 & ( Xphi @ Xx ) )
% 0.19/0.47 => ( ( ( dsetconstr @ A
% 0.19/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.47 = emptyset )
% 0.19/0.47 => $false ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(vacuousDall_type,type,
% 0.19/0.47 vacuousDall: $o ).
% 0.19/0.47
% 0.19/0.47 thf(vacuousDall,definition,
% 0.19/0.47 ( vacuousDall
% 0.19/0.47 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ emptyset )
% 0.19/0.47 => ( Xphi @ Xx ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(quantDeMorgan1_type,type,
% 0.19/0.47 quantDeMorgan1: $o ).
% 0.19/0.47
% 0.19/0.47 thf(quantDeMorgan1,definition,
% 0.19/0.47 ( quantDeMorgan1
% 0.19/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.47 ( ~ ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ( Xphi @ Xx ) )
% 0.19/0.47 => ? [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(quantDeMorgan2_type,type,
% 0.19/0.47 quantDeMorgan2: $o ).
% 0.19/0.47
% 0.19/0.47 thf(quantDeMorgan2,definition,
% 0.19/0.47 ( quantDeMorgan2
% 0.19/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.47 ( ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ~ ( Xphi @ Xx ) )
% 0.19/0.47 => ~ ? [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(quantDeMorgan3_type,type,
% 0.19/0.47 quantDeMorgan3: $o ).
% 0.19/0.47
% 0.19/0.47 thf(quantDeMorgan3,definition,
% 0.19/0.47 ( quantDeMorgan3
% 0.19/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.47 ( ~ ? [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 & ( Xphi @ Xx ) )
% 0.19/0.47 => ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(quantDeMorgan4_type,type,
% 0.19/0.47 quantDeMorgan4: $o ).
% 0.19/0.47
% 0.19/0.47 thf(quantDeMorgan4,definition,
% 0.19/0.47 ( quantDeMorgan4
% 0.19/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.47 ( ? [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 & ~ ( Xphi @ Xx ) )
% 0.19/0.47 => ~ ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(prop2setI_type,type,
% 0.19/0.47 prop2setI: $o ).
% 0.19/0.47
% 0.19/0.47 thf(prop2setI,definition,
% 0.19/0.47 ( prop2setI
% 0.19/0.47 = ( ! [Xphi: $o] :
% 0.19/0.47 ( Xphi
% 0.19/0.47 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(set2prop_type,type,
% 0.19/0.47 set2prop: $i > $o ).
% 0.19/0.47
% 0.19/0.47 thf(prop2set2propI_type,type,
% 0.19/0.47 prop2set2propI: $o ).
% 0.19/0.47
% 0.19/0.47 thf(prop2set2propI,definition,
% 0.19/0.47 ( prop2set2propI
% 0.19/0.47 = ( ! [Xphi: $o] :
% 0.19/0.47 ( Xphi
% 0.19/0.47 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(notdexE_type,type,
% 0.19/0.47 notdexE: $o ).
% 0.19/0.47
% 0.19/0.47 thf(notdexE,definition,
% 0.19/0.47 ( notdexE
% 0.19/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.47 ( ~ ? [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 & ( Xphi @ Xx ) )
% 0.19/0.47 => ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(notdallE_type,type,
% 0.19/0.47 notdallE: $o ).
% 0.19/0.47
% 0.19/0.47 thf(notdallE,definition,
% 0.19/0.47 ( notdallE
% 0.19/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.47 ( ~ ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ( Xphi @ Xx ) )
% 0.19/0.47 => ? [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(exuI1_type,type,
% 0.19/0.47 exuI1: $o ).
% 0.19/0.47
% 0.19/0.47 thf(exuI1,definition,
% 0.19/0.47 ( exuI1
% 0.19/0.47 = ( ! [Xphi: $i > $o] :
% 0.19/0.47 ( ? [Xx: $i] :
% 0.19/0.47 ( ( Xphi @ Xx )
% 0.19/0.47 & ! [Xy: $i] :
% 0.19/0.47 ( ( Xphi @ Xy )
% 0.19/0.47 => ( Xx = Xy ) ) )
% 0.19/0.47 => ( exu
% 0.19/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(exuI3_type,type,
% 0.19/0.47 exuI3: $o ).
% 0.19/0.47
% 0.19/0.47 thf(exuI3,definition,
% 0.19/0.47 ( exuI3
% 0.19/0.47 = ( ! [Xphi: $i > $o] :
% 0.19/0.47 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.19/0.47 => ( ! [Xx: $i,Xy: $i] :
% 0.19/0.47 ( ( Xphi @ Xx )
% 0.19/0.47 => ( ( Xphi @ Xy )
% 0.19/0.47 => ( Xx = Xy ) ) )
% 0.19/0.47 => ( exu
% 0.19/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(exuI2_type,type,
% 0.19/0.47 exuI2: $o ).
% 0.19/0.47
% 0.19/0.47 thf(exuI2,definition,
% 0.19/0.47 ( exuI2
% 0.19/0.47 = ( ! [Xphi: $i > $o] :
% 0.19/0.47 ( ? [Xx: $i] :
% 0.19/0.47 ! [Xy: $i] :
% 0.19/0.47 ( ( Xphi @ Xy )
% 0.19/0.47 <=> ( Xy = Xx ) )
% 0.19/0.47 => ( exu
% 0.19/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(inCongP_type,type,
% 0.19/0.47 inCongP: $o ).
% 0.19/0.47
% 0.19/0.47 thf(inCongP,definition,
% 0.19/0.47 ( inCongP
% 0.19/0.47 = ( ! [A: $i,B: $i] :
% 0.19/0.47 ( ( A = B )
% 0.19/0.47 => ! [Xx: $i,Xy: $i] :
% 0.19/0.47 ( ( Xx = Xy )
% 0.19/0.47 => ( ( in @ Xx @ A )
% 0.19/0.47 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(in__Cong_type,type,
% 0.19/0.47 in__Cong: $o ).
% 0.19/0.47
% 0.19/0.47 thf(in__Cong,definition,
% 0.19/0.47 ( in__Cong
% 0.19/0.47 = ( ! [A: $i,B: $i] :
% 0.19/0.47 ( ( A = B )
% 0.19/0.47 => ! [Xx: $i,Xy: $i] :
% 0.19/0.47 ( ( Xx = Xy )
% 0.19/0.47 => ( ( in @ Xx @ A )
% 0.19/0.47 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(exuE3u_type,type,
% 0.19/0.47 exuE3u: $o ).
% 0.19/0.47
% 0.19/0.47 thf(exuE3u,definition,
% 0.19/0.47 ( exuE3u
% 0.19/0.47 = ( ! [Xphi: $i > $o] :
% 0.19/0.47 ( ( exu
% 0.19/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.47 => ! [Xx: $i,Xy: $i] :
% 0.19/0.47 ( ( Xphi @ Xx )
% 0.19/0.47 => ( ( Xphi @ Xy )
% 0.19/0.47 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(exu__Cong_type,type,
% 0.19/0.47 exu__Cong: $o ).
% 0.19/0.47
% 0.19/0.47 thf(exu__Cong,definition,
% 0.19/0.47 ( exu__Cong
% 0.19/0.47 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.19/0.47 ( ! [Xx: $i,Xy: $i] :
% 0.19/0.47 ( ( Xx = Xy )
% 0.19/0.47 => ( ( Xphi @ Xx )
% 0.19/0.47 <=> ( Xpsi @ Xy ) ) )
% 0.19/0.47 => ( ( exu
% 0.19/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.47 <=> ( exu
% 0.19/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(emptyset__Cong_type,type,
% 0.19/0.47 emptyset__Cong: $o ).
% 0.19/0.47
% 0.19/0.47 thf(emptyset__Cong,definition,
% 0.19/0.47 ( emptyset__Cong
% 0.19/0.47 = ( emptyset = emptyset ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(setadjoin__Cong_type,type,
% 0.19/0.47 setadjoin__Cong: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setadjoin__Cong,definition,
% 0.19/0.47 ( setadjoin__Cong
% 0.19/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.47 ( ( Xx = Xy )
% 0.30/0.47 => ! [Xz: $i,Xu: $i] :
% 0.30/0.47 ( ( Xz = Xu )
% 0.30/0.47 => ( ( setadjoin @ Xx @ Xz )
% 0.30/0.47 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(powerset__Cong_type,type,
% 0.30/0.47 powerset__Cong: $o ).
% 0.30/0.47
% 0.30/0.47 thf(powerset__Cong,definition,
% 0.30/0.47 ( powerset__Cong
% 0.30/0.47 = ( ! [A: $i,B: $i] :
% 0.30/0.47 ( ( A = B )
% 0.30/0.47 => ( ( powerset @ A )
% 0.30/0.47 = ( powerset @ B ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(setunion__Cong_type,type,
% 0.30/0.47 setunion__Cong: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setunion__Cong,definition,
% 0.30/0.47 ( setunion__Cong
% 0.30/0.47 = ( ! [A: $i,B: $i] :
% 0.30/0.47 ( ( A = B )
% 0.30/0.47 => ( ( setunion @ A )
% 0.30/0.47 = ( setunion @ B ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(omega__Cong_type,type,
% 0.30/0.47 omega__Cong: $o ).
% 0.30/0.47
% 0.30/0.47 thf(omega__Cong,definition,
% 0.30/0.47 ( omega__Cong
% 0.30/0.47 = ( omega = omega ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(exuEu_type,type,
% 0.30/0.47 exuEu: $o ).
% 0.30/0.47
% 0.30/0.47 thf(exuEu,definition,
% 0.30/0.47 ( exuEu
% 0.30/0.47 = ( ! [Xphi: $i > $o] :
% 0.30/0.47 ( ( exu
% 0.30/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.47 => ! [Xx: $i,Xy: $i] :
% 0.30/0.47 ( ( Xphi @ Xx )
% 0.30/0.47 => ( ( Xphi @ Xy )
% 0.30/0.47 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(descr__Cong_type,type,
% 0.30/0.47 descr__Cong: $o ).
% 0.30/0.47
% 0.30/0.47 thf(descr__Cong,definition,
% 0.30/0.47 ( descr__Cong
% 0.30/0.47 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.30/0.47 ( ! [Xx: $i,Xy: $i] :
% 0.30/0.47 ( ( Xx = Xy )
% 0.30/0.47 => ( ( Xphi @ Xx )
% 0.30/0.47 <=> ( Xpsi @ Xy ) ) )
% 0.30/0.47 => ( ( exu
% 0.30/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.47 => ( ( exu
% 0.30/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.30/0.47 => ( ( descr
% 0.30/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.47 = ( descr
% 0.30/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(dsetconstr__Cong_type,type,
% 0.30/0.47 dsetconstr__Cong: $o ).
% 0.30/0.47
% 0.30/0.47 thf(dsetconstr__Cong,definition,
% 0.30/0.47 ( dsetconstr__Cong
% 0.30/0.47 = ( ! [A: $i,B: $i] :
% 0.30/0.47 ( ( A = B )
% 0.30/0.47 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.30/0.47 ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ B )
% 0.30/0.47 => ( ( Xx = Xy )
% 0.30/0.47 => ( ( Xphi @ Xx )
% 0.30/0.47 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.30/0.47 => ( ( dsetconstr @ A
% 0.30/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.47 = ( dsetconstr @ B
% 0.30/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(subset_type,type,
% 0.30/0.47 subset: $i > $i > $o ).
% 0.30/0.47
% 0.30/0.47 thf(disjoint_type,type,
% 0.30/0.47 disjoint: $i > $i > $o ).
% 0.30/0.47
% 0.30/0.47 thf(setsmeet_type,type,
% 0.30/0.47 setsmeet: $i > $i > $o ).
% 0.30/0.47
% 0.30/0.47 thf(subsetI1_type,type,
% 0.30/0.47 subsetI1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(subsetI1,definition,
% 0.30/0.47 ( subsetI1
% 0.30/0.47 = ( ! [A: $i,B: $i] :
% 0.30/0.47 ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( in @ Xx @ B ) )
% 0.30/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(eqimpsubset2_type,type,
% 0.30/0.47 eqimpsubset2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(eqimpsubset2,definition,
% 0.30/0.47 ( eqimpsubset2
% 0.30/0.47 = ( ! [A: $i,B: $i] :
% 0.30/0.47 ( ( A = B )
% 0.30/0.47 => ( subset @ B @ A ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(eqimpsubset1_type,type,
% 0.30/0.47 eqimpsubset1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(eqimpsubset1,definition,
% 0.30/0.47 ( eqimpsubset1
% 0.30/0.47 = ( ! [A: $i,B: $i] :
% 0.30/0.47 ( ( A = B )
% 0.30/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(subsetI2_type,type,
% 0.30/0.47 subsetI2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(subsetI2,definition,
% 0.30/0.47 ( subsetI2
% 0.30/0.47 = ( ! [A: $i,B: $i] :
% 0.30/0.47 ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( in @ Xx @ B ) )
% 0.30/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(emptysetsubset_type,type,
% 0.30/0.47 emptysetsubset: $o ).
% 0.30/0.47
% 0.30/0.47 thf(emptysetsubset,definition,
% 0.30/0.47 ( emptysetsubset
% 0.30/0.47 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(subsetE_type,type,
% 0.30/0.47 subsetE: $o ).
% 0.30/0.47
% 0.30/0.47 thf(subsetE,definition,
% 0.30/0.47 ( subsetE
% 0.30/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.47 ( ( subset @ A @ B )
% 0.30/0.47 => ( ( in @ Xx @ A )
% 0.30/0.47 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(subsetE2_type,type,
% 0.30/0.47 subsetE2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(subsetE2,definition,
% 0.30/0.47 ( subsetE2
% 0.30/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.47 ( ( subset @ A @ B )
% 0.30/0.47 => ( ~ ( in @ Xx @ B )
% 0.30/0.47 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(notsubsetI_type,type,
% 0.30/0.47 notsubsetI: $o ).
% 0.30/0.47
% 0.30/0.47 thf(notsubsetI,definition,
% 0.30/0.47 ( notsubsetI
% 0.30/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ~ ( in @ Xx @ B )
% 0.30/0.47 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(notequalI1_type,type,
% 0.30/0.47 notequalI1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(notequalI1,definition,
% 0.30/0.47 ( notequalI1
% 0.30/0.47 = ( ! [A: $i,B: $i] :
% 0.30/0.47 ( ~ ( subset @ A @ B )
% 0.30/0.47 => ( A != B ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(notequalI2_type,type,
% 0.30/0.47 notequalI2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(notequalI2,definition,
% 0.30/0.47 ( notequalI2
% 0.30/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ~ ( in @ Xx @ B )
% 0.30/0.47 => ( A != B ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(subsetRefl_type,type,
% 0.30/0.47 subsetRefl: $o ).
% 0.30/0.47
% 0.30/0.47 thf(subsetRefl,definition,
% 0.30/0.47 ( subsetRefl
% 0.30/0.47 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(subsetTrans_type,type,
% 0.30/0.47 subsetTrans: $o ).
% 0.30/0.47
% 0.30/0.47 thf(subsetTrans,definition,
% 0.30/0.47 ( subsetTrans
% 0.30/0.47 = ( ! [A: $i,B: $i,C: $i] :
% 0.30/0.47 ( ( subset @ A @ B )
% 0.30/0.47 => ( ( subset @ B @ C )
% 0.30/0.47 => ( subset @ A @ C ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(setadjoinSub_type,type,
% 0.30/0.47 setadjoinSub: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setadjoinSub,definition,
% 0.30/0.47 ( setadjoinSub
% 0.30/0.47 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(setadjoinSub2_type,type,
% 0.30/0.47 setadjoinSub2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setadjoinSub2,definition,
% 0.30/0.47 ( setadjoinSub2
% 0.30/0.47 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.30/0.47 ( ( subset @ A @ B )
% 0.30/0.47 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(subset2powerset_type,type,
% 0.30/0.47 subset2powerset: $o ).
% 0.30/0.47
% 0.30/0.47 thf(subset2powerset,definition,
% 0.30/0.47 ( subset2powerset
% 0.30/0.47 = ( ! [A: $i,B: $i] :
% 0.30/0.47 ( ( subset @ A @ B )
% 0.30/0.47 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(setextsub_type,type,
% 0.30/0.47 setextsub: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setextsub,definition,
% 0.30/0.47 ( setextsub
% 0.30/0.47 = ( ! [A: $i,B: $i] :
% 0.30/0.47 ( ( subset @ A @ B )
% 0.30/0.47 => ( ( subset @ B @ A )
% 0.30/0.47 => ( A = B ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(subsetemptysetimpeq_type,type,
% 0.30/0.47 subsetemptysetimpeq: $o ).
% 0.30/0.47
% 0.30/0.47 thf(subsetemptysetimpeq,definition,
% 0.30/0.47 ( subsetemptysetimpeq
% 0.30/0.47 = ( ! [A: $i] :
% 0.30/0.47 ( ( subset @ A @ emptyset )
% 0.30/0.47 => ( A = emptyset ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(powersetI1_type,type,
% 0.30/0.47 powersetI1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(powersetI1,definition,
% 0.30/0.47 ( powersetI1
% 0.30/0.47 = ( ! [A: $i,B: $i] :
% 0.30/0.47 ( ( subset @ B @ A )
% 0.30/0.47 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(powersetE1_type,type,
% 0.30/0.47 powersetE1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(powersetE1,definition,
% 0.30/0.47 ( powersetE1
% 0.30/0.47 = ( ! [A: $i,B: $i] :
% 0.30/0.47 ( ( in @ B @ ( powerset @ A ) )
% 0.30/0.47 => ( subset @ B @ A ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(inPowerset_type,type,
% 0.30/0.47 inPowerset: $o ).
% 0.30/0.47
% 0.30/0.47 thf(inPowerset,definition,
% 0.30/0.47 ( inPowerset
% 0.30/0.47 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(powersetsubset_type,type,
% 0.30/0.47 powersetsubset: $o ).
% 0.30/0.47
% 0.30/0.47 thf(powersetsubset,definition,
% 0.30/0.47 ( powersetsubset
% 0.30/0.47 = ( ! [A: $i,B: $i] :
% 0.30/0.47 ( ( subset @ A @ B )
% 0.30/0.47 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(sepInPowerset_type,type,
% 0.30/0.47 sepInPowerset: $o ).
% 0.30/0.47
% 0.30/0.47 thf(sepInPowerset,definition,
% 0.30/0.47 ( sepInPowerset
% 0.30/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.47 ( in
% 0.30/0.47 @ ( dsetconstr @ A
% 0.30/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.47 @ ( powerset @ A ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(sepSubset_type,type,
% 0.30/0.47 sepSubset: $o ).
% 0.30/0.47
% 0.30/0.47 thf(sepSubset,definition,
% 0.30/0.47 ( sepSubset
% 0.30/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.47 ( subset
% 0.30/0.47 @ ( dsetconstr @ A
% 0.30/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.47 @ A ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(binunion_type,type,
% 0.30/0.47 binunion: $i > $i > $i ).
% 0.30/0.47
% 0.30/0.47 thf(binunionIL_type,type,
% 0.30/0.47 binunionIL: $o ).
% 0.30/0.47
% 0.30/0.47 thf(binunionIL,definition,
% 0.30/0.47 ( binunionIL
% 0.30/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(upairset2IR_type,type,
% 0.30/0.47 upairset2IR: $o ).
% 0.30/0.47
% 0.30/0.47 thf(upairset2IR,definition,
% 0.30/0.47 ( upairset2IR
% 0.30/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(binunionIR_type,type,
% 0.30/0.47 binunionIR: $o ).
% 0.30/0.47
% 0.30/0.47 thf(binunionIR,definition,
% 0.30/0.47 ( binunionIR
% 0.30/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ B )
% 0.30/0.47 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(binunionEcases_type,type,
% 0.30/0.47 binunionEcases: $o ).
% 0.30/0.47
% 0.30/0.47 thf(binunionEcases,definition,
% 0.30/0.47 ( binunionEcases
% 0.30/0.47 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.30/0.47 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.30/0.47 => ( ( ( in @ Xx @ A )
% 0.30/0.47 => Xphi )
% 0.30/0.47 => ( ( ( in @ Xx @ B )
% 0.30/0.47 => Xphi )
% 0.30/0.47 => Xphi ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(binunionE_type,type,
% 0.30/0.47 binunionE: $o ).
% 0.30/0.47
% 0.30/0.47 thf(binunionE,definition,
% 0.30/0.47 ( binunionE
% 0.30/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.30/0.47 => ( ( in @ Xx @ A )
% 0.30/0.47 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(binunionLsub_type,type,
% 0.30/0.47 binunionLsub: $o ).
% 0.30/0.47
% 0.30/0.47 thf(binunionLsub,definition,
% 0.30/0.47 ( binunionLsub
% 0.30/0.47 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(binunionRsub_type,type,
% 0.30/0.47 binunionRsub: $o ).
% 0.30/0.47
% 0.30/0.47 thf(binunionRsub,definition,
% 0.30/0.47 ( binunionRsub
% 0.30/0.47 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(binintersect_type,type,
% 0.30/0.47 binintersect: $i > $i > $i ).
% 0.30/0.47
% 0.30/0.47 thf(binintersectI_type,type,
% 0.30/0.47 binintersectI: $o ).
% 0.30/0.47
% 0.30/0.47 thf(binintersectI,definition,
% 0.30/0.47 ( binintersectI
% 0.30/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( in @ Xx @ B )
% 0.30/0.47 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(binintersectSubset5_type,type,
% 0.30/0.47 binintersectSubset5: $o ).
% 0.30/0.47
% 0.30/0.47 thf(binintersectSubset5,definition,
% 0.30/0.47 ( binintersectSubset5
% 0.30/0.47 = ( ! [A: $i,B: $i,C: $i] :
% 0.30/0.47 ( ( subset @ C @ A )
% 0.30/0.47 => ( ( subset @ C @ B )
% 0.30/0.47 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(binintersectEL_type,type,
% 0.30/0.47 binintersectEL: $o ).
% 0.30/0.47
% 0.30/0.47 thf(binintersectEL,definition,
% 0.30/0.47 ( binintersectEL
% 0.30/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.30/0.47 => ( in @ Xx @ A ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(binintersectLsub_type,type,
% 0.30/0.47 binintersectLsub: $o ).
% 0.30/0.47
% 0.30/0.47 thf(binintersectLsub,definition,
% 0.30/0.47 ( binintersectLsub
% 0.30/0.47 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(binintersectSubset2_type,type,
% 0.30/0.47 binintersectSubset2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(binintersectSubset2,definition,
% 0.30/0.47 ( binintersectSubset2
% 0.30/0.47 = ( ! [A: $i,B: $i] :
% 0.30/0.47 ( ( subset @ A @ B )
% 0.30/0.47 => ( ( binintersect @ A @ B )
% 0.30/0.47 = A ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(binintersectSubset3_type,type,
% 0.30/0.47 binintersectSubset3: $o ).
% 0.30/0.47
% 0.30/0.47 thf(binintersectSubset3,definition,
% 0.30/0.47 ( binintersectSubset3
% 0.30/0.47 = ( ! [A: $i,B: $i] :
% 0.30/0.47 ( ( ( binintersect @ A @ B )
% 0.30/0.47 = B )
% 0.30/0.47 => ( subset @ B @ A ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(binintersectER_type,type,
% 0.30/0.47 binintersectER: $o ).
% 0.30/0.47
% 0.30/0.47 thf(binintersectER,definition,
% 0.30/0.47 ( binintersectER
% 0.30/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.30/0.47 => ( in @ Xx @ B ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(disjointsetsI1_type,type,
% 0.30/0.47 disjointsetsI1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(disjointsetsI1,definition,
% 0.30/0.47 ( disjointsetsI1
% 0.30/0.47 = ( ! [A: $i,B: $i] :
% 0.30/0.47 ( ~ ? [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 & ( in @ Xx @ B ) )
% 0.30/0.47 => ( ( binintersect @ A @ B )
% 0.30/0.47 = emptyset ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(binintersectRsub_type,type,
% 0.30/0.47 binintersectRsub: $o ).
% 0.30/0.47
% 0.30/0.47 thf(binintersectRsub,definition,
% 0.30/0.47 ( binintersectRsub
% 0.30/0.47 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(binintersectSubset4_type,type,
% 0.30/0.47 binintersectSubset4: $o ).
% 0.30/0.47
% 0.30/0.47 thf(binintersectSubset4,definition,
% 0.30/0.47 ( binintersectSubset4
% 0.30/0.47 = ( ! [A: $i,B: $i] :
% 0.30/0.47 ( ( subset @ B @ A )
% 0.30/0.47 => ( ( binintersect @ A @ B )
% 0.30/0.47 = B ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(binintersectSubset1_type,type,
% 0.30/0.47 binintersectSubset1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(binintersectSubset1,definition,
% 0.30/0.47 ( binintersectSubset1
% 0.30/0.47 = ( ! [A: $i,B: $i] :
% 0.30/0.47 ( ( ( binintersect @ A @ B )
% 0.30/0.47 = A )
% 0.30/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(bs114d_type,type,
% 0.30/0.47 bs114d: $o ).
% 0.30/0.47
% 0.30/0.47 thf(bs114d,definition,
% 0.30/0.47 ( bs114d
% 0.30/0.47 = ( ! [A: $i,B: $i,C: $i] :
% 0.30/0.47 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 0.30/0.47 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(regular_type,type,
% 0.30/0.47 regular: $i > $o ).
% 0.30/0.47
% 0.30/0.47 thf(setminus_type,type,
% 0.30/0.47 setminus: $i > $i > $i ).
% 0.30/0.47
% 0.30/0.47 thf(setminusI_type,type,
% 0.30/0.47 setminusI: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setminusI,definition,
% 0.30/0.47 ( setminusI
% 0.30/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ~ ( in @ Xx @ B )
% 0.30/0.47 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(setminusEL_type,type,
% 0.30/0.47 setminusEL: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setminusEL,definition,
% 0.30/0.47 ( setminusEL
% 0.30/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.30/0.47 => ( in @ Xx @ A ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(setminusER_type,type,
% 0.30/0.47 setminusER: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setminusER,definition,
% 0.30/0.47 ( setminusER
% 0.30/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.30/0.47 => ~ ( in @ Xx @ B ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(setminusSubset2_type,type,
% 0.30/0.47 setminusSubset2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setminusSubset2,definition,
% 0.30/0.47 ( setminusSubset2
% 0.30/0.47 = ( ! [A: $i,B: $i] :
% 0.30/0.47 ( ( subset @ A @ B )
% 0.30/0.47 => ( ( setminus @ A @ B )
% 0.30/0.47 = emptyset ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(setminusERneg,conjecture,
% 0.30/0.47 ( setextAx
% 0.30/0.47 => ( emptysetAx
% 0.30/0.47 => ( setadjoinAx
% 0.30/0.47 => ( powersetAx
% 0.30/0.47 => ( setunionAx
% 0.30/0.47 => ( omega0Ax
% 0.30/0.47 => ( omegaSAx
% 0.30/0.47 => ( omegaIndAx
% 0.30/0.47 => ( replAx
% 0.30/0.47 => ( foundationAx
% 0.30/0.47 => ( wellorderingAx
% 0.30/0.47 => ( descrp
% 0.30/0.47 => ( dsetconstrI
% 0.30/0.47 => ( dsetconstrEL
% 0.30/0.47 => ( dsetconstrER
% 0.30/0.47 => ( exuE1
% 0.30/0.47 => ( prop2setE
% 0.30/0.47 => ( emptysetE
% 0.30/0.47 => ( emptysetimpfalse
% 0.30/0.47 => ( notinemptyset
% 0.30/0.47 => ( exuE3e
% 0.30/0.47 => ( setext
% 0.30/0.47 => ( emptyI
% 0.30/0.47 => ( noeltsimpempty
% 0.30/0.47 => ( setbeta
% 0.30/0.47 => ( nonemptyE1
% 0.30/0.47 => ( nonemptyI
% 0.30/0.47 => ( nonemptyI1
% 0.30/0.47 => ( setadjoinIL
% 0.30/0.47 => ( emptyinunitempty
% 0.30/0.47 => ( setadjoinIR
% 0.30/0.47 => ( setadjoinE
% 0.30/0.47 => ( setadjoinOr
% 0.30/0.47 => ( setoftrueEq
% 0.30/0.47 => ( powersetI
% 0.30/0.47 => ( emptyinPowerset
% 0.30/0.47 => ( emptyInPowerset
% 0.30/0.47 => ( powersetE
% 0.30/0.47 => ( setunionI
% 0.30/0.47 => ( setunionE
% 0.30/0.47 => ( subPowSU
% 0.30/0.47 => ( exuE2
% 0.30/0.47 => ( nonemptyImpWitness
% 0.30/0.47 => ( uniqinunit
% 0.30/0.47 => ( notinsingleton
% 0.30/0.47 => ( eqinunit
% 0.30/0.47 => ( singletonsswitch
% 0.30/0.47 => ( upairsetE
% 0.30/0.47 => ( upairsetIL
% 0.30/0.47 => ( upairsetIR
% 0.30/0.47 => ( emptyE1
% 0.30/0.47 => ( vacuousDall
% 0.30/0.47 => ( quantDeMorgan1
% 0.30/0.47 => ( quantDeMorgan2
% 0.30/0.47 => ( quantDeMorgan3
% 0.30/0.47 => ( quantDeMorgan4
% 0.30/0.47 => ( prop2setI
% 0.30/0.47 => ( prop2set2propI
% 0.30/0.47 => ( notdexE
% 0.30/0.47 => ( notdallE
% 0.30/0.47 => ( exuI1
% 0.30/0.47 => ( exuI3
% 0.30/0.47 => ( exuI2
% 0.30/0.47 => ( inCongP
% 0.30/0.47 => ( in__Cong
% 0.30/0.47 => ( exuE3u
% 0.30/0.47 => ( exu__Cong
% 0.30/0.47 => ( emptyset__Cong
% 0.30/0.47 => ( setadjoin__Cong
% 0.30/0.47 => ( powerset__Cong
% 0.30/0.47 => ( setunion__Cong
% 0.30/0.47 => ( omega__Cong
% 0.30/0.47 => ( exuEu
% 0.30/0.47 => ( descr__Cong
% 0.30/0.47 => ( dsetconstr__Cong
% 0.30/0.47 => ( subsetI1
% 0.30/0.47 => ( eqimpsubset2
% 0.30/0.47 => ( eqimpsubset1
% 0.30/0.47 => ( subsetI2
% 0.30/0.47 => ( emptysetsubset
% 0.30/0.47 => ( subsetE
% 0.30/0.47 => ( subsetE2
% 0.30/0.47 => ( notsubsetI
% 0.30/0.47 => ( notequalI1
% 0.30/0.47 => ( notequalI2
% 0.30/0.47 => ( subsetRefl
% 0.30/0.47 => ( subsetTrans
% 0.30/0.47 => ( setadjoinSub
% 0.30/0.47 => ( setadjoinSub2
% 0.30/0.47 => ( subset2powerset
% 0.30/0.47 => ( setextsub
% 0.30/0.47 => ( subsetemptysetimpeq
% 0.30/0.47 => ( powersetI1
% 0.30/0.47 => ( powersetE1
% 0.30/0.47 => ( inPowerset
% 0.30/0.47 => ( powersetsubset
% 0.30/0.47 => ( sepInPowerset
% 0.30/0.47 => ( sepSubset
% 0.30/0.47 => ( binunionIL
% 0.30/0.47 => ( upairset2IR
% 0.30/0.47 => ( binunionIR
% 0.30/0.47 => ( binunionEcases
% 0.30/0.47 => ( binunionE
% 0.30/0.47 => ( binunionLsub
% 0.30/0.47 => ( binunionRsub
% 0.30/0.47 => ( binintersectI
% 0.30/0.47 => ( binintersectSubset5
% 0.30/0.47 => ( binintersectEL
% 0.30/0.47 => ( binintersectLsub
% 0.30/0.47 => ( binintersectSubset2
% 0.30/0.47 => ( binintersectSubset3
% 0.30/0.47 => ( binintersectER
% 0.30/0.47 => ( disjointsetsI1
% 0.30/0.47 => ( binintersectRsub
% 0.30/0.47 => ( binintersectSubset4
% 0.30/0.47 => ( binintersectSubset1
% 0.30/0.47 => ( bs114d
% 0.30/0.47 => ( setminusI
% 0.30/0.47 => ( setminusEL
% 0.30/0.47 => ( setminusER
% 0.30/0.47 => ( setminusSubset2
% 0.30/0.47 => ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.47 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.30/0.47 => ( ( in @ Xx @ A )
% 0.30/0.47 => ( in @ Xx @ B ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 %------------------------------------------------------------------------------
% 0.30/0.47 ------------------------
% 2.56/1.63 ---- Embedded in THF ---
% 2.56/1.63 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.56/1.63 %%% Generated on Tue Feb 24 12:28:22 EST 2026
% 2.56/1.63 %%% using '$$dhol' embedding, version 1.3.0.
% 2.56/1.63 %%% Logic specification used:
% 2.56/1.63 %%% thf(spec, logic, $$dhol).
% 2.56/1.63
% 2.56/1.63 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.FHMJW0nBVa/DTF2DTF8470.p
% See solution above
% 2.56/1.64 ------------------------
% 2.56/1.66 ---- Cleaned THF ---
% 2.56/1.66 thf(in_type,type,
% 2.56/1.66 in: $i > $i > $o ).
% 2.56/1.66
% 2.56/1.66 thf(exu_type,type,
% 2.56/1.66 exu: ( $i > $o ) > $o ).
% 2.56/1.66
% 2.56/1.66 thf(exu,definition,
% 2.56/1.66 ( exu
% 2.56/1.66 = ( ^ [Xphi: $i > $o] :
% 2.56/1.66 ? [Xx: $i] :
% 2.56/1.66 ( ( Xphi @ Xx )
% 2.56/1.66 & ! [Xy: $i] :
% 2.56/1.66 ( ( Xphi @ Xy )
% 2.56/1.66 => ( Xx = Xy ) ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(setextAx_type,type,
% 2.56/1.66 setextAx: $o ).
% 2.56/1.66
% 2.56/1.66 thf(setextAx,definition,
% 2.56/1.66 ( setextAx
% 2.56/1.66 = ( ! [A: $i,B: $i] :
% 2.56/1.66 ( ! [Xx: $i] :
% 2.56/1.66 ( ( in @ Xx @ A )
% 2.56/1.66 <=> ( in @ Xx @ B ) )
% 2.56/1.66 => ( A = B ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(emptyset_type,type,
% 2.56/1.66 emptyset: $i ).
% 2.56/1.66
% 2.56/1.66 thf(emptysetAx_type,type,
% 2.56/1.66 emptysetAx: $o ).
% 2.56/1.66
% 2.56/1.66 thf(emptysetAx,definition,
% 2.56/1.66 ( emptysetAx
% 2.56/1.66 = ( ! [Xx: $i] :
% 2.56/1.66 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(setadjoin_type,type,
% 2.56/1.66 setadjoin: $i > $i > $i ).
% 2.56/1.66
% 2.56/1.66 thf(setadjoinAx_type,type,
% 2.56/1.66 setadjoinAx: $o ).
% 2.56/1.66
% 2.56/1.66 thf(setadjoinAx,definition,
% 2.56/1.66 ( setadjoinAx
% 2.56/1.66 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.56/1.66 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.56/1.66 <=> ( ( Xy = Xx )
% 2.56/1.66 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(powerset_type,type,
% 2.56/1.66 powerset: $i > $i ).
% 2.56/1.66
% 2.56/1.66 thf(powersetAx_type,type,
% 2.56/1.66 powersetAx: $o ).
% 2.56/1.66
% 2.56/1.66 thf(powersetAx,definition,
% 2.56/1.66 ( powersetAx
% 2.56/1.66 = ( ! [A: $i,B: $i] :
% 2.56/1.66 ( ( in @ B @ ( powerset @ A ) )
% 2.56/1.66 <=> ! [Xx: $i] :
% 2.56/1.66 ( ( in @ Xx @ B )
% 2.56/1.66 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(setunion_type,type,
% 2.56/1.66 setunion: $i > $i ).
% 2.56/1.66
% 2.56/1.66 thf(setunionAx_type,type,
% 2.56/1.66 setunionAx: $o ).
% 2.56/1.66
% 2.56/1.66 thf(setunionAx,definition,
% 2.56/1.66 ( setunionAx
% 2.56/1.66 = ( ! [A: $i,Xx: $i] :
% 2.56/1.66 ( ( in @ Xx @ ( setunion @ A ) )
% 2.56/1.66 <=> ? [B: $i] :
% 2.56/1.66 ( ( in @ Xx @ B )
% 2.56/1.66 & ( in @ B @ A ) ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(omega_type,type,
% 2.56/1.66 omega: $i ).
% 2.56/1.66
% 2.56/1.66 thf(omega0Ax_type,type,
% 2.56/1.66 omega0Ax: $o ).
% 2.56/1.66
% 2.56/1.66 thf(omega0Ax,definition,
% 2.56/1.66 ( omega0Ax
% 2.56/1.66 = ( in @ emptyset @ omega ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(omegaSAx_type,type,
% 2.56/1.66 omegaSAx: $o ).
% 2.56/1.66
% 2.56/1.66 thf(omegaSAx,definition,
% 2.56/1.66 ( omegaSAx
% 2.56/1.66 = ( ! [Xx: $i] :
% 2.56/1.66 ( ( in @ Xx @ omega )
% 2.56/1.66 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(omegaIndAx_type,type,
% 2.56/1.66 omegaIndAx: $o ).
% 2.56/1.66
% 2.56/1.66 thf(omegaIndAx,definition,
% 2.56/1.66 ( omegaIndAx
% 2.56/1.66 = ( ! [A: $i] :
% 2.56/1.66 ( ( ( in @ emptyset @ A )
% 2.56/1.66 & ! [Xx: $i] :
% 2.56/1.66 ( ( ( in @ Xx @ omega )
% 2.56/1.66 & ( in @ Xx @ A ) )
% 2.56/1.66 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 2.56/1.66 => ! [Xx: $i] :
% 2.56/1.66 ( ( in @ Xx @ omega )
% 2.56/1.66 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(replAx_type,type,
% 2.56/1.66 replAx: $o ).
% 2.56/1.66
% 2.56/1.66 thf(replAx,definition,
% 2.56/1.66 ( replAx
% 2.56/1.66 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 2.56/1.66 ( ! [Xx: $i] :
% 2.56/1.66 ( ( in @ Xx @ A )
% 2.56/1.66 => ( exu
% 2.56/1.66 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 2.56/1.66 => ? [B: $i] :
% 2.56/1.66 ! [Xx: $i] :
% 2.56/1.66 ( ( in @ Xx @ B )
% 2.56/1.66 <=> ? [Xy: $i] :
% 2.56/1.66 ( ( in @ Xy @ A )
% 2.56/1.66 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(foundationAx_type,type,
% 2.56/1.66 foundationAx: $o ).
% 2.56/1.66
% 2.56/1.66 thf(foundationAx,definition,
% 2.56/1.66 ( foundationAx
% 2.56/1.66 = ( ! [A: $i] :
% 2.56/1.66 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.56/1.66 => ? [B: $i] :
% 2.56/1.66 ( ( in @ B @ A )
% 2.56/1.66 & ~ ? [Xx: $i] :
% 2.56/1.66 ( ( in @ Xx @ B )
% 2.56/1.66 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(wellorderingAx_type,type,
% 2.56/1.66 wellorderingAx: $o ).
% 2.56/1.66
% 2.56/1.66 thf(wellorderingAx,definition,
% 2.56/1.66 ( wellorderingAx
% 2.56/1.66 = ( ! [A: $i] :
% 2.56/1.66 ? [B: $i] :
% 2.56/1.66 ( ! [C: $i] :
% 2.56/1.66 ( ( in @ C @ B )
% 2.56/1.66 => ! [Xx: $i] :
% 2.56/1.66 ( ( in @ Xx @ C )
% 2.56/1.66 => ( in @ Xx @ A ) ) )
% 2.56/1.66 & ! [Xx: $i,Xy: $i] :
% 2.56/1.66 ( ( ( in @ Xx @ A )
% 2.56/1.66 & ( in @ Xy @ A ) )
% 2.56/1.66 => ( ! [C: $i] :
% 2.56/1.66 ( ( in @ C @ B )
% 2.56/1.66 => ( ( in @ Xx @ C )
% 2.56/1.66 <=> ( in @ Xy @ C ) ) )
% 2.56/1.66 => ( Xx = Xy ) ) )
% 2.56/1.66 & ! [C: $i,D: $i] :
% 2.56/1.66 ( ( ( in @ C @ B )
% 2.56/1.66 & ( in @ D @ B ) )
% 2.56/1.66 => ( ! [Xx: $i] :
% 2.56/1.66 ( ( in @ Xx @ C )
% 2.56/1.66 => ( in @ Xx @ D ) )
% 2.56/1.66 | ! [Xx: $i] :
% 2.56/1.66 ( ( in @ Xx @ D )
% 2.56/1.66 => ( in @ Xx @ C ) ) ) )
% 2.56/1.66 & ! [C: $i] :
% 2.56/1.66 ( ( ! [Xx: $i] :
% 2.56/1.66 ( ( in @ Xx @ C )
% 2.56/1.66 => ( in @ Xx @ A ) )
% 2.56/1.66 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 2.56/1.66 => ? [D: $i,Xx: $i] :
% 2.56/1.66 ( ( in @ D @ B )
% 2.56/1.66 & ( in @ Xx @ C )
% 2.56/1.66 & ~ ? [Xy: $i] :
% 2.56/1.66 ( ( in @ Xy @ D )
% 2.56/1.66 & ( in @ Xy @ C ) )
% 2.56/1.66 & ! [E: $i] :
% 2.56/1.66 ( ( in @ E @ B )
% 2.56/1.66 => ( ! [Xy: $i] :
% 2.56/1.66 ( ( in @ Xy @ E )
% 2.56/1.66 => ( in @ Xy @ D ) )
% 2.56/1.66 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(descr_type,type,
% 2.56/1.66 descr: ( $i > $o ) > $i ).
% 2.56/1.66
% 2.56/1.66 thf(descrp_type,type,
% 2.56/1.66 descrp: $o ).
% 2.56/1.66
% 2.56/1.66 thf(descrp,definition,
% 2.56/1.66 ( descrp
% 2.56/1.66 = ( ! [Xphi: $i > $o] :
% 2.56/1.66 ( ( exu
% 2.56/1.66 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.56/1.66 => ( Xphi
% 2.56/1.66 @ ( descr
% 2.56/1.66 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(dsetconstr_type,type,
% 2.56/1.66 dsetconstr: $i > ( $i > $o ) > $i ).
% 2.56/1.66
% 2.56/1.66 thf(dsetconstrI_type,type,
% 2.56/1.66 dsetconstrI: $o ).
% 2.56/1.66
% 2.56/1.66 thf(dsetconstrI,definition,
% 2.56/1.66 ( dsetconstrI
% 2.56/1.66 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.56/1.66 ( ( in @ Xx @ A )
% 2.56/1.66 => ( ( Xphi @ Xx )
% 2.56/1.66 => ( in @ Xx
% 2.56/1.66 @ ( dsetconstr @ A
% 2.56/1.66 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(dsetconstrEL_type,type,
% 2.56/1.66 dsetconstrEL: $o ).
% 2.56/1.66
% 2.56/1.66 thf(dsetconstrEL,definition,
% 2.56/1.66 ( dsetconstrEL
% 2.56/1.66 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.56/1.66 ( ( in @ Xx
% 2.56/1.66 @ ( dsetconstr @ A
% 2.56/1.66 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.56/1.66 => ( in @ Xx @ A ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(dsetconstrER_type,type,
% 2.56/1.66 dsetconstrER: $o ).
% 2.56/1.66
% 2.56/1.66 thf(dsetconstrER,definition,
% 2.56/1.66 ( dsetconstrER
% 2.56/1.66 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.56/1.66 ( ( in @ Xx
% 2.56/1.66 @ ( dsetconstr @ A
% 2.56/1.66 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.56/1.66 => ( Xphi @ Xx ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(exuE1_type,type,
% 2.56/1.66 exuE1: $o ).
% 2.56/1.66
% 2.56/1.66 thf(exuE1,definition,
% 2.56/1.66 ( exuE1
% 2.56/1.66 = ( ! [Xphi: $i > $o] :
% 2.56/1.66 ( ( exu
% 2.56/1.66 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.56/1.66 => ? [Xx: $i] :
% 2.56/1.66 ( ( Xphi @ Xx )
% 2.56/1.66 & ! [Xy: $i] :
% 2.56/1.66 ( ( Xphi @ Xy )
% 2.56/1.66 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(prop2set_type,type,
% 2.56/1.66 prop2set: $o > $i ).
% 2.56/1.66
% 2.56/1.66 thf(prop2setE_type,type,
% 2.56/1.66 prop2setE: $o ).
% 2.56/1.66
% 2.56/1.66 thf(prop2setE,definition,
% 2.56/1.66 ( prop2setE
% 2.56/1.66 = ( ! [Xphi: $o,Xx: $i] :
% 2.56/1.66 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 2.56/1.66 => Xphi ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(emptysetE_type,type,
% 2.56/1.66 emptysetE: $o ).
% 2.56/1.66
% 2.56/1.66 thf(emptysetE,definition,
% 2.56/1.66 ( emptysetE
% 2.56/1.66 = ( ! [Xx: $i] :
% 2.56/1.66 ( ( in @ Xx @ emptyset )
% 2.56/1.66 => ! [Xphi: $o] : Xphi ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(emptysetimpfalse_type,type,
% 2.56/1.66 emptysetimpfalse: $o ).
% 2.56/1.66
% 2.56/1.66 thf(emptysetimpfalse,definition,
% 2.56/1.66 ( emptysetimpfalse
% 2.56/1.66 = ( ! [Xx: $i] :
% 2.56/1.66 ( ( in @ Xx @ emptyset )
% 2.56/1.66 => $false ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(notinemptyset_type,type,
% 2.56/1.66 notinemptyset: $o ).
% 2.56/1.66
% 2.56/1.66 thf(notinemptyset,definition,
% 2.56/1.66 ( notinemptyset
% 2.56/1.66 = ( ! [Xx: $i] :
% 2.56/1.66 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(exuE3e_type,type,
% 2.56/1.66 exuE3e: $o ).
% 2.56/1.66
% 2.56/1.66 thf(exuE3e,definition,
% 2.56/1.66 ( exuE3e
% 2.56/1.66 = ( ! [Xphi: $i > $o] :
% 2.56/1.66 ( ( exu
% 2.56/1.66 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.56/1.66 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(setext_type,type,
% 2.56/1.66 setext: $o ).
% 2.56/1.66
% 2.56/1.66 thf(setext,definition,
% 2.56/1.66 ( setext
% 2.56/1.66 = ( ! [A: $i,B: $i] :
% 2.56/1.66 ( ! [Xx: $i] :
% 2.56/1.66 ( ( in @ Xx @ A )
% 2.56/1.66 => ( in @ Xx @ B ) )
% 2.56/1.66 => ( ! [Xx: $i] :
% 2.56/1.66 ( ( in @ Xx @ B )
% 2.56/1.66 => ( in @ Xx @ A ) )
% 2.56/1.66 => ( A = B ) ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(emptyI_type,type,
% 2.56/1.66 emptyI: $o ).
% 2.56/1.66
% 2.56/1.66 thf(emptyI,definition,
% 2.56/1.66 ( emptyI
% 2.56/1.66 = ( ! [A: $i] :
% 2.56/1.66 ( ! [Xx: $i] :
% 2.56/1.66 ~ ( in @ Xx @ A )
% 2.56/1.66 => ( A = emptyset ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(noeltsimpempty_type,type,
% 2.56/1.66 noeltsimpempty: $o ).
% 2.56/1.66
% 2.56/1.66 thf(noeltsimpempty,definition,
% 2.56/1.66 ( noeltsimpempty
% 2.56/1.66 = ( ! [A: $i] :
% 2.56/1.66 ( ! [Xx: $i] :
% 2.56/1.66 ~ ( in @ Xx @ A )
% 2.56/1.66 => ( A = emptyset ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(setbeta_type,type,
% 2.56/1.66 setbeta: $o ).
% 2.56/1.66
% 2.56/1.66 thf(setbeta,definition,
% 2.56/1.66 ( setbeta
% 2.56/1.66 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.56/1.66 ( ( in @ Xx @ A )
% 2.56/1.66 => ( ( in @ Xx
% 2.56/1.66 @ ( dsetconstr @ A
% 2.56/1.66 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.56/1.66 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(nonempty_type,type,
% 2.56/1.66 nonempty: $i > $o ).
% 2.56/1.66
% 2.56/1.66 thf(nonempty,definition,
% 2.56/1.66 ( nonempty
% 2.56/1.66 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(nonemptyE1_type,type,
% 2.56/1.66 nonemptyE1: $o ).
% 2.56/1.66
% 2.56/1.66 thf(nonemptyE1,definition,
% 2.56/1.66 ( nonemptyE1
% 2.56/1.66 = ( ! [A: $i] :
% 2.56/1.66 ( ( nonempty @ A )
% 2.56/1.66 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(nonemptyI_type,type,
% 2.56/1.66 nonemptyI: $o ).
% 2.56/1.66
% 2.56/1.66 thf(nonemptyI,definition,
% 2.56/1.66 ( nonemptyI
% 2.56/1.66 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.56/1.66 ( ( in @ Xx @ A )
% 2.56/1.66 => ( ( Xphi @ Xx )
% 2.56/1.66 => ( nonempty
% 2.56/1.66 @ ( dsetconstr @ A
% 2.56/1.66 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(nonemptyI1_type,type,
% 2.56/1.66 nonemptyI1: $o ).
% 2.56/1.66
% 2.56/1.66 thf(nonemptyI1,definition,
% 2.56/1.66 ( nonemptyI1
% 2.56/1.66 = ( ! [A: $i] :
% 2.56/1.66 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.56/1.66 => ( nonempty @ A ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(setadjoinIL_type,type,
% 2.56/1.66 setadjoinIL: $o ).
% 2.56/1.66
% 2.56/1.66 thf(setadjoinIL,definition,
% 2.56/1.66 ( setadjoinIL
% 2.56/1.66 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(emptyinunitempty_type,type,
% 2.56/1.66 emptyinunitempty: $o ).
% 2.56/1.66
% 2.56/1.66 thf(emptyinunitempty,definition,
% 2.56/1.66 ( emptyinunitempty
% 2.56/1.66 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(setadjoinIR_type,type,
% 2.56/1.66 setadjoinIR: $o ).
% 2.56/1.66
% 2.56/1.66 thf(setadjoinIR,definition,
% 2.56/1.66 ( setadjoinIR
% 2.56/1.66 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.56/1.66 ( ( in @ Xy @ A )
% 2.56/1.66 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(setadjoinE_type,type,
% 2.56/1.66 setadjoinE: $o ).
% 2.56/1.66
% 2.56/1.66 thf(setadjoinE,definition,
% 2.56/1.66 ( setadjoinE
% 2.56/1.66 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.56/1.66 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.56/1.66 => ! [Xphi: $o] :
% 2.56/1.66 ( ( ( Xy = Xx )
% 2.56/1.66 => Xphi )
% 2.56/1.66 => ( ( ( in @ Xy @ A )
% 2.56/1.66 => Xphi )
% 2.56/1.66 => Xphi ) ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(setadjoinOr_type,type,
% 2.56/1.66 setadjoinOr: $o ).
% 2.56/1.66
% 2.56/1.66 thf(setadjoinOr,definition,
% 2.56/1.66 ( setadjoinOr
% 2.56/1.66 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.56/1.66 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.56/1.66 => ( ( Xy = Xx )
% 2.56/1.66 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(setoftrueEq_type,type,
% 2.56/1.66 setoftrueEq: $o ).
% 2.56/1.66
% 2.56/1.66 thf(setoftrueEq,definition,
% 2.56/1.66 ( setoftrueEq
% 2.56/1.66 = ( ! [A: $i] :
% 2.56/1.66 ( ( dsetconstr @ A
% 2.56/1.66 @ ^ [Xx: $i] : $true )
% 2.56/1.66 = A ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(powersetI_type,type,
% 2.56/1.66 powersetI: $o ).
% 2.56/1.66
% 2.56/1.66 thf(powersetI,definition,
% 2.56/1.66 ( powersetI
% 2.56/1.66 = ( ! [A: $i,B: $i] :
% 2.56/1.66 ( ! [Xx: $i] :
% 2.56/1.66 ( ( in @ Xx @ B )
% 2.56/1.66 => ( in @ Xx @ A ) )
% 2.56/1.66 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(emptyinPowerset_type,type,
% 2.56/1.66 emptyinPowerset: $o ).
% 2.56/1.66
% 2.56/1.66 thf(emptyinPowerset,definition,
% 2.56/1.66 ( emptyinPowerset
% 2.56/1.66 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(emptyInPowerset_type,type,
% 2.56/1.66 emptyInPowerset: $o ).
% 2.56/1.66
% 2.56/1.66 thf(emptyInPowerset,definition,
% 2.56/1.66 ( emptyInPowerset
% 2.56/1.66 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(powersetE_type,type,
% 2.56/1.66 powersetE: $o ).
% 2.56/1.66
% 2.56/1.66 thf(powersetE,definition,
% 2.56/1.66 ( powersetE
% 2.56/1.66 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.56/1.66 ( ( in @ B @ ( powerset @ A ) )
% 2.56/1.66 => ( ( in @ Xx @ B )
% 2.56/1.66 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(setunionI_type,type,
% 2.56/1.66 setunionI: $o ).
% 2.56/1.66
% 2.56/1.66 thf(setunionI,definition,
% 2.56/1.66 ( setunionI
% 2.56/1.66 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.56/1.66 ( ( in @ Xx @ B )
% 2.56/1.66 => ( ( in @ B @ A )
% 2.56/1.66 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(setunionE_type,type,
% 2.56/1.66 setunionE: $o ).
% 2.56/1.66
% 2.56/1.66 thf(setunionE,definition,
% 2.56/1.66 ( setunionE
% 2.56/1.66 = ( ! [A: $i,Xx: $i] :
% 2.56/1.66 ( ( in @ Xx @ ( setunion @ A ) )
% 2.56/1.66 => ! [Xphi: $o] :
% 2.56/1.66 ( ! [B: $i] :
% 2.56/1.66 ( ( in @ Xx @ B )
% 2.56/1.66 => ( ( in @ B @ A )
% 2.56/1.66 => Xphi ) )
% 2.56/1.66 => Xphi ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(subPowSU_type,type,
% 2.56/1.66 subPowSU: $o ).
% 2.56/1.66
% 2.56/1.66 thf(subPowSU,definition,
% 2.56/1.66 ( subPowSU
% 2.56/1.66 = ( ! [A: $i,Xx: $i] :
% 2.56/1.66 ( ( in @ Xx @ A )
% 2.56/1.66 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(exuE2_type,type,
% 2.56/1.66 exuE2: $o ).
% 2.56/1.66
% 2.56/1.66 thf(exuE2,definition,
% 2.56/1.66 ( exuE2
% 2.56/1.66 = ( ! [Xphi: $i > $o] :
% 2.56/1.66 ( ( exu
% 2.56/1.66 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.56/1.66 => ? [Xx: $i] :
% 2.56/1.66 ! [Xy: $i] :
% 2.56/1.66 ( ( Xphi @ Xy )
% 2.56/1.66 <=> ( Xy = Xx ) ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(nonemptyImpWitness_type,type,
% 2.56/1.66 nonemptyImpWitness: $o ).
% 2.56/1.66
% 2.56/1.66 thf(nonemptyImpWitness,definition,
% 2.56/1.66 ( nonemptyImpWitness
% 2.56/1.66 = ( ! [A: $i] :
% 2.56/1.66 ( ( nonempty @ A )
% 2.56/1.66 => ? [Xx: $i] :
% 2.56/1.66 ( ( in @ Xx @ A )
% 2.56/1.66 & $true ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(uniqinunit_type,type,
% 2.56/1.66 uniqinunit: $o ).
% 2.56/1.66
% 2.56/1.66 thf(uniqinunit,definition,
% 2.56/1.66 ( uniqinunit
% 2.56/1.66 = ( ! [Xx: $i,Xy: $i] :
% 2.56/1.66 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.56/1.66 => ( Xx = Xy ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(notinsingleton_type,type,
% 2.56/1.66 notinsingleton: $o ).
% 2.56/1.66
% 2.56/1.66 thf(notinsingleton,definition,
% 2.56/1.66 ( notinsingleton
% 2.56/1.66 = ( ! [Xx: $i,Xy: $i] :
% 2.56/1.66 ( ( Xx != Xy )
% 2.56/1.66 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(eqinunit_type,type,
% 2.56/1.66 eqinunit: $o ).
% 2.56/1.66
% 2.56/1.66 thf(eqinunit,definition,
% 2.56/1.66 ( eqinunit
% 2.56/1.66 = ( ! [Xx: $i,Xy: $i] :
% 2.56/1.66 ( ( Xx = Xy )
% 2.56/1.66 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(singletonsswitch_type,type,
% 2.56/1.66 singletonsswitch: $o ).
% 2.56/1.66
% 2.56/1.66 thf(singletonsswitch,definition,
% 2.56/1.66 ( singletonsswitch
% 2.56/1.66 = ( ! [Xx: $i,Xy: $i] :
% 2.56/1.66 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.56/1.66 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(upairsetE_type,type,
% 2.56/1.66 upairsetE: $o ).
% 2.56/1.66
% 2.56/1.66 thf(upairsetE,definition,
% 2.56/1.66 ( upairsetE
% 2.56/1.66 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.56/1.66 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.56/1.66 => ( ( Xz = Xx )
% 2.56/1.66 | ( Xz = Xy ) ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(upairsetIL_type,type,
% 2.56/1.66 upairsetIL: $o ).
% 2.56/1.66
% 2.56/1.66 thf(upairsetIL,definition,
% 2.56/1.66 ( upairsetIL
% 2.56/1.66 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(upairsetIR_type,type,
% 2.56/1.66 upairsetIR: $o ).
% 2.56/1.66
% 2.56/1.66 thf(upairsetIR,definition,
% 2.56/1.66 ( upairsetIR
% 2.56/1.66 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(emptyE1_type,type,
% 2.56/1.66 emptyE1: $o ).
% 2.56/1.66
% 2.56/1.66 thf(emptyE1,definition,
% 2.56/1.66 ( emptyE1
% 2.56/1.66 = ( ! [A: $i,Xphi: $i > $o] :
% 2.56/1.66 ( ? [Xx: $i] :
% 2.56/1.66 ( ( in @ Xx @ A )
% 2.56/1.66 & ( Xphi @ Xx ) )
% 2.56/1.66 => ( ( ( dsetconstr @ A
% 2.56/1.66 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.56/1.66 = emptyset )
% 2.56/1.66 => $false ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(vacuousDall_type,type,
% 2.56/1.66 vacuousDall: $o ).
% 2.56/1.66
% 2.56/1.66 thf(vacuousDall,definition,
% 2.56/1.66 ( vacuousDall
% 2.56/1.66 = ( ! [Xphi: $i > $o,Xx: $i] :
% 2.56/1.66 ( ( in @ Xx @ emptyset )
% 2.56/1.66 => ( Xphi @ Xx ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(quantDeMorgan1_type,type,
% 2.56/1.66 quantDeMorgan1: $o ).
% 2.56/1.66
% 2.56/1.66 thf(quantDeMorgan1,definition,
% 2.56/1.66 ( quantDeMorgan1
% 2.56/1.66 = ( ! [A: $i,Xphi: $i > $o] :
% 2.56/1.66 ( ~ ! [Xx: $i] :
% 2.56/1.66 ( ( in @ Xx @ A )
% 2.56/1.66 => ( Xphi @ Xx ) )
% 2.56/1.66 => ? [Xx: $i] :
% 2.56/1.66 ( ( in @ Xx @ A )
% 2.56/1.66 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(quantDeMorgan2_type,type,
% 2.56/1.66 quantDeMorgan2: $o ).
% 2.56/1.66
% 2.56/1.66 thf(quantDeMorgan2,definition,
% 2.56/1.66 ( quantDeMorgan2
% 2.56/1.66 = ( ! [A: $i,Xphi: $i > $o] :
% 2.56/1.66 ( ! [Xx: $i] :
% 2.56/1.66 ( ( in @ Xx @ A )
% 2.56/1.66 => ~ ( Xphi @ Xx ) )
% 2.56/1.66 => ~ ? [Xx: $i] :
% 2.56/1.66 ( ( in @ Xx @ A )
% 2.56/1.66 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(quantDeMorgan3_type,type,
% 2.56/1.66 quantDeMorgan3: $o ).
% 2.56/1.66
% 2.56/1.66 thf(quantDeMorgan3,definition,
% 2.56/1.66 ( quantDeMorgan3
% 2.56/1.66 = ( ! [A: $i,Xphi: $i > $o] :
% 2.56/1.66 ( ~ ? [Xx: $i] :
% 2.56/1.66 ( ( in @ Xx @ A )
% 2.56/1.66 & ( Xphi @ Xx ) )
% 2.56/1.66 => ! [Xx: $i] :
% 2.56/1.66 ( ( in @ Xx @ A )
% 2.56/1.66 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(quantDeMorgan4_type,type,
% 2.56/1.66 quantDeMorgan4: $o ).
% 2.56/1.66
% 2.56/1.66 thf(quantDeMorgan4,definition,
% 2.56/1.66 ( quantDeMorgan4
% 2.56/1.66 = ( ! [A: $i,Xphi: $i > $o] :
% 2.56/1.66 ( ? [Xx: $i] :
% 2.56/1.66 ( ( in @ Xx @ A )
% 2.56/1.66 & ~ ( Xphi @ Xx ) )
% 2.56/1.66 => ~ ! [Xx: $i] :
% 2.56/1.66 ( ( in @ Xx @ A )
% 2.56/1.66 => ( Xphi @ Xx ) ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(prop2setI_type,type,
% 2.56/1.66 prop2setI: $o ).
% 2.56/1.66
% 2.56/1.66 thf(prop2setI,definition,
% 2.56/1.66 ( prop2setI
% 2.56/1.66 = ( ! [Xphi: $o] :
% 2.56/1.66 ( Xphi
% 2.56/1.66 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(set2prop_type,type,
% 2.56/1.66 set2prop: $i > $o ).
% 2.56/1.66
% 2.56/1.66 thf(prop2set2propI_type,type,
% 2.56/1.66 prop2set2propI: $o ).
% 2.56/1.66
% 2.56/1.66 thf(prop2set2propI,definition,
% 2.56/1.66 ( prop2set2propI
% 2.56/1.66 = ( ! [Xphi: $o] :
% 2.56/1.66 ( Xphi
% 2.56/1.66 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(notdexE_type,type,
% 2.56/1.66 notdexE: $o ).
% 2.56/1.66
% 2.56/1.66 thf(notdexE,definition,
% 2.56/1.66 ( notdexE
% 2.56/1.66 = ( ! [A: $i,Xphi: $i > $o] :
% 2.56/1.66 ( ~ ? [Xx: $i] :
% 2.56/1.66 ( ( in @ Xx @ A )
% 2.56/1.66 & ( Xphi @ Xx ) )
% 2.56/1.66 => ! [Xx: $i] :
% 2.56/1.66 ( ( in @ Xx @ A )
% 2.56/1.66 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(notdallE_type,type,
% 2.56/1.66 notdallE: $o ).
% 2.56/1.66
% 2.56/1.66 thf(notdallE,definition,
% 2.56/1.66 ( notdallE
% 2.56/1.66 = ( ! [A: $i,Xphi: $i > $o] :
% 2.56/1.66 ( ~ ! [Xx: $i] :
% 2.56/1.66 ( ( in @ Xx @ A )
% 2.56/1.66 => ( Xphi @ Xx ) )
% 2.56/1.66 => ? [Xx: $i] :
% 2.56/1.66 ( ( in @ Xx @ A )
% 2.56/1.66 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(exuI1_type,type,
% 2.56/1.66 exuI1: $o ).
% 2.56/1.66
% 2.56/1.66 thf(exuI1,definition,
% 2.56/1.66 ( exuI1
% 2.56/1.66 = ( ! [Xphi: $i > $o] :
% 2.56/1.66 ( ? [Xx: $i] :
% 2.56/1.66 ( ( Xphi @ Xx )
% 2.56/1.66 & ! [Xy: $i] :
% 2.56/1.66 ( ( Xphi @ Xy )
% 2.56/1.66 => ( Xx = Xy ) ) )
% 2.56/1.66 => ( exu
% 2.56/1.66 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(exuI3_type,type,
% 2.56/1.66 exuI3: $o ).
% 2.56/1.66
% 2.56/1.66 thf(exuI3,definition,
% 2.56/1.66 ( exuI3
% 2.56/1.66 = ( ! [Xphi: $i > $o] :
% 2.56/1.66 ( ? [Xx: $i] : ( Xphi @ Xx )
% 2.56/1.66 => ( ! [Xx: $i,Xy: $i] :
% 2.56/1.66 ( ( Xphi @ Xx )
% 2.56/1.66 => ( ( Xphi @ Xy )
% 2.56/1.66 => ( Xx = Xy ) ) )
% 2.56/1.66 => ( exu
% 2.56/1.66 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(exuI2_type,type,
% 2.56/1.66 exuI2: $o ).
% 2.56/1.66
% 2.56/1.66 thf(exuI2,definition,
% 2.56/1.66 ( exuI2
% 2.56/1.66 = ( ! [Xphi: $i > $o] :
% 2.56/1.66 ( ? [Xx: $i] :
% 2.56/1.66 ! [Xy: $i] :
% 2.56/1.66 ( ( Xphi @ Xy )
% 2.56/1.66 <=> ( Xy = Xx ) )
% 2.56/1.66 => ( exu
% 2.56/1.66 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(inCongP_type,type,
% 2.56/1.66 inCongP: $o ).
% 2.56/1.66
% 2.56/1.66 thf(inCongP,definition,
% 2.56/1.66 ( inCongP
% 2.56/1.66 = ( ! [A: $i,B: $i] :
% 2.56/1.66 ( ( A = B )
% 2.56/1.66 => ! [Xx: $i,Xy: $i] :
% 2.56/1.66 ( ( Xx = Xy )
% 2.56/1.66 => ( ( in @ Xx @ A )
% 2.56/1.66 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(in__Cong_type,type,
% 2.56/1.66 in__Cong: $o ).
% 2.56/1.66
% 2.56/1.66 thf(in__Cong,definition,
% 2.56/1.66 ( in__Cong
% 2.56/1.66 = ( ! [A: $i,B: $i] :
% 2.56/1.66 ( ( A = B )
% 2.56/1.66 => ! [Xx: $i,Xy: $i] :
% 2.56/1.66 ( ( Xx = Xy )
% 2.56/1.66 => ( ( in @ Xx @ A )
% 2.56/1.66 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(exuE3u_type,type,
% 2.56/1.66 exuE3u: $o ).
% 2.56/1.66
% 2.56/1.66 thf(exuE3u,definition,
% 2.56/1.66 ( exuE3u
% 2.56/1.66 = ( ! [Xphi: $i > $o] :
% 2.56/1.66 ( ( exu
% 2.56/1.66 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.56/1.66 => ! [Xx: $i,Xy: $i] :
% 2.56/1.66 ( ( Xphi @ Xx )
% 2.56/1.66 => ( ( Xphi @ Xy )
% 2.56/1.66 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(exu__Cong_type,type,
% 2.56/1.66 exu__Cong: $o ).
% 2.56/1.66
% 2.56/1.66 thf(exu__Cong,definition,
% 2.56/1.66 ( exu__Cong
% 2.56/1.66 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.56/1.66 ( ! [Xx: $i,Xy: $i] :
% 2.56/1.66 ( ( Xx = Xy )
% 2.56/1.66 => ( ( Xphi @ Xx )
% 2.56/1.66 <=> ( Xpsi @ Xy ) ) )
% 2.56/1.66 => ( ( exu
% 2.56/1.66 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.56/1.66 <=> ( exu
% 2.56/1.66 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(emptyset__Cong_type,type,
% 2.56/1.66 emptyset__Cong: $o ).
% 2.56/1.66
% 2.56/1.66 thf(emptyset__Cong,definition,
% 2.56/1.66 ( emptyset__Cong
% 2.56/1.66 = ( emptyset = emptyset ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(setadjoin__Cong_type,type,
% 2.56/1.66 setadjoin__Cong: $o ).
% 2.56/1.66
% 2.56/1.66 thf(setadjoin__Cong,definition,
% 2.56/1.66 ( setadjoin__Cong
% 2.56/1.66 = ( ! [Xx: $i,Xy: $i] :
% 2.56/1.66 ( ( Xx = Xy )
% 2.56/1.66 => ! [Xz: $i,Xu: $i] :
% 2.56/1.66 ( ( Xz = Xu )
% 2.56/1.66 => ( ( setadjoin @ Xx @ Xz )
% 2.56/1.66 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(powerset__Cong_type,type,
% 2.56/1.66 powerset__Cong: $o ).
% 2.56/1.66
% 2.56/1.66 thf(powerset__Cong,definition,
% 2.56/1.66 ( powerset__Cong
% 2.56/1.66 = ( ! [A: $i,B: $i] :
% 2.56/1.66 ( ( A = B )
% 2.56/1.66 => ( ( powerset @ A )
% 2.56/1.66 = ( powerset @ B ) ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(setunion__Cong_type,type,
% 2.56/1.66 setunion__Cong: $o ).
% 2.56/1.66
% 2.56/1.66 thf(setunion__Cong,definition,
% 2.56/1.66 ( setunion__Cong
% 2.56/1.66 = ( ! [A: $i,B: $i] :
% 2.56/1.66 ( ( A = B )
% 2.56/1.66 => ( ( setunion @ A )
% 2.56/1.66 = ( setunion @ B ) ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(omega__Cong_type,type,
% 2.56/1.66 omega__Cong: $o ).
% 2.56/1.66
% 2.56/1.66 thf(omega__Cong,definition,
% 2.56/1.66 ( omega__Cong
% 2.56/1.66 = ( omega = omega ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(exuEu_type,type,
% 2.56/1.66 exuEu: $o ).
% 2.56/1.66
% 2.56/1.66 thf(exuEu,definition,
% 2.56/1.66 ( exuEu
% 2.56/1.66 = ( ! [Xphi: $i > $o] :
% 2.56/1.66 ( ( exu
% 2.56/1.66 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.56/1.66 => ! [Xx: $i,Xy: $i] :
% 2.56/1.66 ( ( Xphi @ Xx )
% 2.56/1.66 => ( ( Xphi @ Xy )
% 2.56/1.66 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(descr__Cong_type,type,
% 2.56/1.66 descr__Cong: $o ).
% 2.56/1.66
% 2.56/1.66 thf(descr__Cong,definition,
% 2.56/1.66 ( descr__Cong
% 2.56/1.66 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.56/1.66 ( ! [Xx: $i,Xy: $i] :
% 2.56/1.66 ( ( Xx = Xy )
% 2.56/1.66 => ( ( Xphi @ Xx )
% 2.56/1.66 <=> ( Xpsi @ Xy ) ) )
% 2.56/1.66 => ( ( exu
% 2.56/1.66 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.56/1.66 => ( ( exu
% 2.56/1.66 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 2.56/1.66 => ( ( descr
% 2.56/1.66 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.56/1.66 = ( descr
% 2.56/1.66 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(dsetconstr__Cong_type,type,
% 2.56/1.66 dsetconstr__Cong: $o ).
% 2.56/1.66
% 2.56/1.66 thf(dsetconstr__Cong,definition,
% 2.56/1.66 ( dsetconstr__Cong
% 2.56/1.66 = ( ! [A: $i,B: $i] :
% 2.56/1.66 ( ( A = B )
% 2.56/1.66 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.56/1.66 ( ! [Xx: $i] :
% 2.56/1.66 ( ( in @ Xx @ A )
% 2.56/1.66 => ! [Xy: $i] :
% 2.56/1.66 ( ( in @ Xy @ B )
% 2.56/1.66 => ( ( Xx = Xy )
% 2.56/1.66 => ( ( Xphi @ Xx )
% 2.56/1.66 <=> ( Xpsi @ Xy ) ) ) ) )
% 2.56/1.66 => ( ( dsetconstr @ A
% 2.56/1.66 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.56/1.66 = ( dsetconstr @ B
% 2.56/1.66 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(subset_type,type,
% 2.56/1.66 subset: $i > $i > $o ).
% 2.56/1.66
% 2.56/1.66 thf(disjoint_type,type,
% 2.56/1.66 disjoint: $i > $i > $o ).
% 2.56/1.66
% 2.56/1.66 thf(setsmeet_type,type,
% 2.56/1.66 setsmeet: $i > $i > $o ).
% 2.56/1.66
% 2.56/1.66 thf(subsetI1_type,type,
% 2.56/1.66 subsetI1: $o ).
% 2.56/1.66
% 2.56/1.66 thf(subsetI1,definition,
% 2.56/1.66 ( subsetI1
% 2.56/1.66 = ( ! [A: $i,B: $i] :
% 2.56/1.66 ( ! [Xx: $i] :
% 2.56/1.66 ( ( in @ Xx @ A )
% 2.56/1.66 => ( in @ Xx @ B ) )
% 2.56/1.66 => ( subset @ A @ B ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(eqimpsubset2_type,type,
% 2.56/1.66 eqimpsubset2: $o ).
% 2.56/1.66
% 2.56/1.66 thf(eqimpsubset2,definition,
% 2.56/1.66 ( eqimpsubset2
% 2.56/1.66 = ( ! [A: $i,B: $i] :
% 2.56/1.66 ( ( A = B )
% 2.56/1.66 => ( subset @ B @ A ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(eqimpsubset1_type,type,
% 2.56/1.66 eqimpsubset1: $o ).
% 2.56/1.66
% 2.56/1.66 thf(eqimpsubset1,definition,
% 2.56/1.66 ( eqimpsubset1
% 2.56/1.66 = ( ! [A: $i,B: $i] :
% 2.56/1.66 ( ( A = B )
% 2.56/1.66 => ( subset @ A @ B ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(subsetI2_type,type,
% 2.56/1.66 subsetI2: $o ).
% 2.56/1.66
% 2.56/1.66 thf(subsetI2,definition,
% 2.56/1.66 ( subsetI2
% 2.56/1.66 = ( ! [A: $i,B: $i] :
% 2.56/1.66 ( ! [Xx: $i] :
% 2.56/1.66 ( ( in @ Xx @ A )
% 2.56/1.66 => ( in @ Xx @ B ) )
% 2.56/1.66 => ( subset @ A @ B ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(emptysetsubset_type,type,
% 2.56/1.66 emptysetsubset: $o ).
% 2.56/1.66
% 2.56/1.66 thf(emptysetsubset,definition,
% 2.56/1.66 ( emptysetsubset
% 2.56/1.66 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(subsetE_type,type,
% 2.56/1.66 subsetE: $o ).
% 2.56/1.66
% 2.56/1.66 thf(subsetE,definition,
% 2.56/1.66 ( subsetE
% 2.56/1.66 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.56/1.66 ( ( subset @ A @ B )
% 2.56/1.66 => ( ( in @ Xx @ A )
% 2.56/1.66 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(subsetE2_type,type,
% 2.56/1.66 subsetE2: $o ).
% 2.56/1.66
% 2.56/1.66 thf(subsetE2,definition,
% 2.56/1.66 ( subsetE2
% 2.56/1.66 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.56/1.66 ( ( subset @ A @ B )
% 2.56/1.66 => ( ~ ( in @ Xx @ B )
% 2.56/1.66 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(notsubsetI_type,type,
% 2.56/1.66 notsubsetI: $o ).
% 2.56/1.66
% 2.56/1.66 thf(notsubsetI,definition,
% 2.56/1.66 ( notsubsetI
% 2.56/1.66 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.56/1.66 ( ( in @ Xx @ A )
% 2.56/1.66 => ( ~ ( in @ Xx @ B )
% 2.56/1.66 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(notequalI1_type,type,
% 2.56/1.66 notequalI1: $o ).
% 2.56/1.66
% 2.56/1.66 thf(notequalI1,definition,
% 2.56/1.66 ( notequalI1
% 2.56/1.66 = ( ! [A: $i,B: $i] :
% 2.56/1.66 ( ~ ( subset @ A @ B )
% 2.56/1.66 => ( A != B ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(notequalI2_type,type,
% 2.56/1.66 notequalI2: $o ).
% 2.56/1.66
% 2.56/1.66 thf(notequalI2,definition,
% 2.56/1.66 ( notequalI2
% 2.56/1.66 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.56/1.66 ( ( in @ Xx @ A )
% 2.56/1.66 => ( ~ ( in @ Xx @ B )
% 2.56/1.66 => ( A != B ) ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(subsetRefl_type,type,
% 2.56/1.66 subsetRefl: $o ).
% 2.56/1.66
% 2.56/1.66 thf(subsetRefl,definition,
% 2.56/1.66 ( subsetRefl
% 2.56/1.66 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(subsetTrans_type,type,
% 2.56/1.66 subsetTrans: $o ).
% 2.56/1.66
% 2.56/1.66 thf(subsetTrans,definition,
% 2.56/1.66 ( subsetTrans
% 2.56/1.66 = ( ! [A: $i,B: $i,C: $i] :
% 2.56/1.66 ( ( subset @ A @ B )
% 2.56/1.66 => ( ( subset @ B @ C )
% 2.56/1.66 => ( subset @ A @ C ) ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(setadjoinSub_type,type,
% 2.56/1.66 setadjoinSub: $o ).
% 2.56/1.66
% 2.56/1.66 thf(setadjoinSub,definition,
% 2.56/1.66 ( setadjoinSub
% 2.56/1.66 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(setadjoinSub2_type,type,
% 2.56/1.66 setadjoinSub2: $o ).
% 2.56/1.66
% 2.56/1.66 thf(setadjoinSub2,definition,
% 2.56/1.66 ( setadjoinSub2
% 2.56/1.66 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.56/1.66 ( ( subset @ A @ B )
% 2.56/1.66 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(subset2powerset_type,type,
% 2.56/1.66 subset2powerset: $o ).
% 2.56/1.66
% 2.56/1.66 thf(subset2powerset,definition,
% 2.56/1.66 ( subset2powerset
% 2.56/1.66 = ( ! [A: $i,B: $i] :
% 2.56/1.66 ( ( subset @ A @ B )
% 2.56/1.66 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(setextsub_type,type,
% 2.56/1.66 setextsub: $o ).
% 2.56/1.66
% 2.56/1.66 thf(setextsub,definition,
% 2.56/1.66 ( setextsub
% 2.56/1.66 = ( ! [A: $i,B: $i] :
% 2.56/1.66 ( ( subset @ A @ B )
% 2.56/1.66 => ( ( subset @ B @ A )
% 2.56/1.66 => ( A = B ) ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(subsetemptysetimpeq_type,type,
% 2.56/1.66 subsetemptysetimpeq: $o ).
% 2.56/1.66
% 2.56/1.66 thf(subsetemptysetimpeq,definition,
% 2.56/1.66 ( subsetemptysetimpeq
% 2.56/1.66 = ( ! [A: $i] :
% 2.56/1.66 ( ( subset @ A @ emptyset )
% 2.56/1.66 => ( A = emptyset ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(powersetI1_type,type,
% 2.56/1.66 powersetI1: $o ).
% 2.56/1.66
% 2.56/1.66 thf(powersetI1,definition,
% 2.56/1.66 ( powersetI1
% 2.56/1.66 = ( ! [A: $i,B: $i] :
% 2.56/1.66 ( ( subset @ B @ A )
% 2.56/1.66 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(powersetE1_type,type,
% 2.56/1.66 powersetE1: $o ).
% 2.56/1.66
% 2.56/1.66 thf(powersetE1,definition,
% 2.56/1.66 ( powersetE1
% 2.56/1.66 = ( ! [A: $i,B: $i] :
% 2.56/1.66 ( ( in @ B @ ( powerset @ A ) )
% 2.56/1.66 => ( subset @ B @ A ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(inPowerset_type,type,
% 2.56/1.66 inPowerset: $o ).
% 2.56/1.66
% 2.56/1.66 thf(inPowerset,definition,
% 2.56/1.66 ( inPowerset
% 2.56/1.66 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(powersetsubset_type,type,
% 2.56/1.66 powersetsubset: $o ).
% 2.56/1.66
% 2.56/1.66 thf(powersetsubset,definition,
% 2.56/1.66 ( powersetsubset
% 2.56/1.66 = ( ! [A: $i,B: $i] :
% 2.56/1.66 ( ( subset @ A @ B )
% 2.56/1.66 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(sepInPowerset_type,type,
% 2.56/1.66 sepInPowerset: $o ).
% 2.56/1.66
% 2.56/1.66 thf(sepInPowerset,definition,
% 2.56/1.66 ( sepInPowerset
% 2.56/1.66 = ( ! [A: $i,Xphi: $i > $o] :
% 2.56/1.66 ( in
% 2.56/1.66 @ ( dsetconstr @ A
% 2.56/1.66 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.56/1.66 @ ( powerset @ A ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(sepSubset_type,type,
% 2.56/1.66 sepSubset: $o ).
% 2.56/1.66
% 2.56/1.66 thf(sepSubset,definition,
% 2.56/1.66 ( sepSubset
% 2.56/1.66 = ( ! [A: $i,Xphi: $i > $o] :
% 2.56/1.66 ( subset
% 2.56/1.66 @ ( dsetconstr @ A
% 2.56/1.66 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.56/1.66 @ A ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(binunion_type,type,
% 2.56/1.66 binunion: $i > $i > $i ).
% 2.56/1.66
% 2.56/1.66 thf(binunionIL_type,type,
% 2.56/1.66 binunionIL: $o ).
% 2.56/1.66
% 2.56/1.66 thf(binunionIL,definition,
% 2.56/1.66 ( binunionIL
% 2.56/1.66 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.56/1.66 ( ( in @ Xx @ A )
% 2.56/1.66 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(upairset2IR_type,type,
% 2.56/1.66 upairset2IR: $o ).
% 2.56/1.66
% 2.56/1.66 thf(upairset2IR,definition,
% 2.56/1.66 ( upairset2IR
% 2.56/1.66 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(binunionIR_type,type,
% 2.56/1.66 binunionIR: $o ).
% 2.56/1.66
% 2.56/1.66 thf(binunionIR,definition,
% 2.56/1.66 ( binunionIR
% 2.56/1.66 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.56/1.66 ( ( in @ Xx @ B )
% 2.56/1.66 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(binunionEcases_type,type,
% 2.56/1.66 binunionEcases: $o ).
% 2.56/1.66
% 2.56/1.66 thf(binunionEcases,definition,
% 2.56/1.66 ( binunionEcases
% 2.56/1.66 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 2.56/1.66 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.56/1.66 => ( ( ( in @ Xx @ A )
% 2.56/1.66 => Xphi )
% 2.56/1.66 => ( ( ( in @ Xx @ B )
% 2.56/1.66 => Xphi )
% 2.56/1.66 => Xphi ) ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(binunionE_type,type,
% 2.56/1.66 binunionE: $o ).
% 2.56/1.66
% 2.56/1.66 thf(binunionE,definition,
% 2.56/1.66 ( binunionE
% 2.56/1.66 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.56/1.66 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.56/1.66 => ( ( in @ Xx @ A )
% 2.56/1.66 | ( in @ Xx @ B ) ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(binunionLsub_type,type,
% 2.56/1.66 binunionLsub: $o ).
% 2.56/1.66
% 2.56/1.66 thf(binunionLsub,definition,
% 2.56/1.66 ( binunionLsub
% 2.56/1.66 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(binunionRsub_type,type,
% 2.56/1.66 binunionRsub: $o ).
% 2.56/1.66
% 2.56/1.66 thf(binunionRsub,definition,
% 2.56/1.66 ( binunionRsub
% 2.56/1.66 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(binintersect_type,type,
% 2.56/1.66 binintersect: $i > $i > $i ).
% 2.56/1.66
% 2.56/1.66 thf(binintersectI_type,type,
% 2.56/1.66 binintersectI: $o ).
% 2.56/1.66
% 2.56/1.66 thf(binintersectI,definition,
% 2.56/1.66 ( binintersectI
% 2.56/1.66 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.56/1.66 ( ( in @ Xx @ A )
% 2.56/1.66 => ( ( in @ Xx @ B )
% 2.56/1.66 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(binintersectSubset5_type,type,
% 2.56/1.66 binintersectSubset5: $o ).
% 2.56/1.66
% 2.56/1.66 thf(binintersectSubset5,definition,
% 2.56/1.66 ( binintersectSubset5
% 2.56/1.66 = ( ! [A: $i,B: $i,C: $i] :
% 2.56/1.66 ( ( subset @ C @ A )
% 2.56/1.66 => ( ( subset @ C @ B )
% 2.56/1.66 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(binintersectEL_type,type,
% 2.56/1.66 binintersectEL: $o ).
% 2.56/1.66
% 2.56/1.66 thf(binintersectEL,definition,
% 2.56/1.66 ( binintersectEL
% 2.56/1.66 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.56/1.66 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.56/1.66 => ( in @ Xx @ A ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(binintersectLsub_type,type,
% 2.56/1.66 binintersectLsub: $o ).
% 2.56/1.66
% 2.56/1.66 thf(binintersectLsub,definition,
% 2.56/1.66 ( binintersectLsub
% 2.56/1.66 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(binintersectSubset2_type,type,
% 2.56/1.66 binintersectSubset2: $o ).
% 2.56/1.66
% 2.56/1.66 thf(binintersectSubset2,definition,
% 2.56/1.66 ( binintersectSubset2
% 2.56/1.66 = ( ! [A: $i,B: $i] :
% 2.56/1.66 ( ( subset @ A @ B )
% 2.56/1.66 => ( ( binintersect @ A @ B )
% 2.56/1.66 = A ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(binintersectSubset3_type,type,
% 2.56/1.66 binintersectSubset3: $o ).
% 2.56/1.66
% 2.56/1.66 thf(binintersectSubset3,definition,
% 2.56/1.66 ( binintersectSubset3
% 2.56/1.66 = ( ! [A: $i,B: $i] :
% 2.56/1.66 ( ( ( binintersect @ A @ B )
% 2.56/1.66 = B )
% 2.56/1.66 => ( subset @ B @ A ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(binintersectER_type,type,
% 2.56/1.66 binintersectER: $o ).
% 2.56/1.66
% 2.56/1.66 thf(binintersectER,definition,
% 2.56/1.66 ( binintersectER
% 2.56/1.66 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.56/1.66 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.56/1.66 => ( in @ Xx @ B ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(disjointsetsI1_type,type,
% 2.56/1.66 disjointsetsI1: $o ).
% 2.56/1.66
% 2.56/1.66 thf(disjointsetsI1,definition,
% 2.56/1.66 ( disjointsetsI1
% 2.56/1.66 = ( ! [A: $i,B: $i] :
% 2.56/1.66 ( ~ ? [Xx: $i] :
% 2.56/1.66 ( ( in @ Xx @ A )
% 2.56/1.66 & ( in @ Xx @ B ) )
% 2.56/1.66 => ( ( binintersect @ A @ B )
% 2.56/1.66 = emptyset ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(binintersectRsub_type,type,
% 2.56/1.66 binintersectRsub: $o ).
% 2.56/1.66
% 2.56/1.66 thf(binintersectRsub,definition,
% 2.56/1.66 ( binintersectRsub
% 2.56/1.66 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(binintersectSubset4_type,type,
% 2.56/1.66 binintersectSubset4: $o ).
% 2.56/1.66
% 2.56/1.66 thf(binintersectSubset4,definition,
% 2.56/1.66 ( binintersectSubset4
% 2.56/1.66 = ( ! [A: $i,B: $i] :
% 2.56/1.66 ( ( subset @ B @ A )
% 2.56/1.66 => ( ( binintersect @ A @ B )
% 2.56/1.66 = B ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(binintersectSubset1_type,type,
% 2.56/1.66 binintersectSubset1: $o ).
% 2.56/1.66
% 2.56/1.66 thf(binintersectSubset1,definition,
% 2.56/1.66 ( binintersectSubset1
% 2.56/1.66 = ( ! [A: $i,B: $i] :
% 2.56/1.66 ( ( ( binintersect @ A @ B )
% 2.56/1.66 = A )
% 2.56/1.66 => ( subset @ A @ B ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(bs114d_type,type,
% 2.56/1.66 bs114d: $o ).
% 2.56/1.66
% 2.56/1.66 thf(bs114d,definition,
% 2.56/1.66 ( bs114d
% 2.56/1.66 = ( ! [A: $i,B: $i,C: $i] :
% 2.56/1.66 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 2.56/1.66 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(regular_type,type,
% 2.56/1.66 regular: $i > $o ).
% 2.56/1.66
% 2.56/1.66 thf(setminus_type,type,
% 2.56/1.66 setminus: $i > $i > $i ).
% 2.56/1.66
% 2.56/1.66 thf(setminusI_type,type,
% 2.56/1.66 setminusI: $o ).
% 2.56/1.66
% 2.56/1.66 thf(setminusI,definition,
% 2.56/1.66 ( setminusI
% 2.56/1.66 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.56/1.66 ( ( in @ Xx @ A )
% 2.56/1.66 => ( ~ ( in @ Xx @ B )
% 2.56/1.66 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(setminusEL_type,type,
% 2.56/1.66 setminusEL: $o ).
% 2.56/1.66
% 2.56/1.66 thf(setminusEL,definition,
% 2.56/1.66 ( setminusEL
% 2.56/1.66 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.56/1.66 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.56/1.66 => ( in @ Xx @ A ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(setminusER_type,type,
% 2.56/1.66 setminusER: $o ).
% 2.56/1.66
% 2.56/1.66 thf(setminusER,definition,
% 2.56/1.66 ( setminusER
% 2.56/1.66 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.56/1.66 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.56/1.66 => ~ ( in @ Xx @ B ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(setminusSubset2_type,type,
% 2.56/1.66 setminusSubset2: $o ).
% 2.56/1.66
% 2.56/1.66 thf(setminusSubset2,definition,
% 2.56/1.66 ( setminusSubset2
% 2.56/1.66 = ( ! [A: $i,B: $i] :
% 2.56/1.66 ( ( subset @ A @ B )
% 2.56/1.66 => ( ( setminus @ A @ B )
% 2.56/1.66 = emptyset ) ) ) ) ).
% 2.56/1.66
% 2.56/1.66 thf(setminusERneg,conjecture,
% 2.56/1.66 ( setextAx
% 2.56/1.66 => ( emptysetAx
% 2.56/1.66 => ( setadjoinAx
% 2.56/1.66 => ( powersetAx
% 2.56/1.66 => ( setunionAx
% 2.56/1.66 => ( omega0Ax
% 2.56/1.66 => ( omegaSAx
% 2.56/1.66 => ( omegaIndAx
% 2.56/1.66 => ( replAx
% 2.56/1.66 => ( foundationAx
% 2.56/1.66 => ( wellorderingAx
% 2.56/1.66 => ( descrp
% 2.56/1.66 => ( dsetconstrI
% 2.56/1.66 => ( dsetconstrEL
% 2.56/1.66 => ( dsetconstrER
% 2.56/1.66 => ( exuE1
% 2.56/1.66 => ( prop2setE
% 2.56/1.66 => ( emptysetE
% 2.56/1.66 => ( emptysetimpfalse
% 2.56/1.66 => ( notinemptyset
% 2.56/1.66 => ( exuE3e
% 2.56/1.66 => ( setext
% 2.56/1.66 => ( emptyI
% 2.56/1.66 => ( noeltsimpempty
% 2.56/1.66 => ( setbeta
% 2.56/1.66 => ( nonemptyE1
% 2.56/1.66 => ( nonemptyI
% 2.56/1.66 => ( nonemptyI1
% 2.56/1.66 => ( setadjoinIL
% 2.56/1.66 => ( emptyinunitempty
% 2.56/1.66 => ( setadjoinIR
% 2.56/1.66 => ( setadjoinE
% 2.56/1.66 => ( setadjoinOr
% 2.56/1.66 => ( setoftrueEq
% 2.56/1.66 => ( powersetI
% 2.56/1.66 => ( emptyinPowerset
% 2.56/1.66 => ( emptyInPowerset
% 2.56/1.66 => ( powersetE
% 2.56/1.66 => ( setunionI
% 2.56/1.66 => ( setunionE
% 2.56/1.66 => ( subPowSU
% 2.56/1.66 => ( exuE2
% 2.56/1.66 => ( nonemptyImpWitness
% 2.56/1.66 => ( uniqinunit
% 2.56/1.66 => ( notinsingleton
% 2.56/1.66 => ( eqinunit
% 2.56/1.66 => ( singletonsswitch
% 2.56/1.66 => ( upairsetE
% 2.56/1.66 => ( upairsetIL
% 2.56/1.66 => ( upairsetIR
% 2.56/1.66 => ( emptyE1
% 2.56/1.66 => ( vacuousDall
% 2.56/1.66 => ( quantDeMorgan1
% 2.56/1.66 => ( quantDeMorgan2
% 2.56/1.66 => ( quantDeMorgan3
% 2.56/1.66 => ( quantDeMorgan4
% 2.56/1.66 => ( prop2setI
% 2.56/1.66 => ( prop2set2propI
% 2.56/1.66 => ( notdexE
% 2.56/1.66 => ( notdallE
% 2.56/1.66 => ( exuI1
% 2.56/1.66 => ( exuI3
% 2.56/1.66 => ( exuI2
% 2.56/1.66 => ( inCongP
% 2.56/1.66 => ( in__Cong
% 2.56/1.66 => ( exuE3u
% 2.56/1.66 => ( exu__Cong
% 2.56/1.66 => ( emptyset__Cong
% 2.56/1.66 => ( setadjoin__Cong
% 2.56/1.66 => ( powerset__Cong
% 2.56/1.67 => ( setunion__Cong
% 2.56/1.67 => ( omega__Cong
% 2.56/1.67 => ( exuEu
% 2.56/1.67 => ( descr__Cong
% 2.56/1.67 => ( dsetconstr__Cong
% 2.56/1.67 => ( subsetI1
% 2.56/1.67 => ( eqimpsubset2
% 2.56/1.67 => ( eqimpsubset1
% 2.56/1.67 => ( subsetI2
% 2.56/1.67 => ( emptysetsubset
% 2.56/1.67 => ( subsetE
% 2.56/1.67 => ( subsetE2
% 2.56/1.67 => ( notsubsetI
% 2.56/1.67 => ( notequalI1
% 2.56/1.67 => ( notequalI2
% 2.56/1.67 => ( subsetRefl
% 2.56/1.67 => ( subsetTrans
% 2.56/1.67 => ( setadjoinSub
% 2.56/1.67 => ( setadjoinSub2
% 2.56/1.67 => ( subset2powerset
% 2.56/1.67 => ( setextsub
% 2.56/1.67 => ( subsetemptysetimpeq
% 2.56/1.67 => ( powersetI1
% 2.56/1.67 => ( powersetE1
% 2.56/1.67 => ( inPowerset
% 2.56/1.67 => ( powersetsubset
% 2.56/1.67 => ( sepInPowerset
% 2.56/1.67 => ( sepSubset
% 2.56/1.67 => ( binunionIL
% 2.56/1.67 => ( upairset2IR
% 2.56/1.67 => ( binunionIR
% 2.56/1.67 => ( binunionEcases
% 2.56/1.67 => ( binunionE
% 2.56/1.67 => ( binunionLsub
% 2.56/1.67 => ( binunionRsub
% 2.56/1.67 => ( binintersectI
% 2.56/1.67 => ( binintersectSubset5
% 2.56/1.67 => ( binintersectEL
% 2.56/1.67 => ( binintersectLsub
% 2.56/1.67 => ( binintersectSubset2
% 2.56/1.67 => ( binintersectSubset3
% 2.56/1.67 => ( binintersectER
% 2.56/1.67 => ( disjointsetsI1
% 2.56/1.67 => ( binintersectRsub
% 2.56/1.67 => ( binintersectSubset4
% 2.56/1.67 => ( binintersectSubset1
% 2.56/1.67 => ( bs114d
% 2.56/1.67 => ( setminusI
% 2.56/1.67 => ( setminusEL
% 2.56/1.67 => ( setminusER
% 2.56/1.67 => ( setminusSubset2
% 2.56/1.67 => ! [A: $i,B: $i,Xx: $i] :
% 2.56/1.67 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.56/1.67 => ( ( in @ Xx @ A )
% 2.56/1.67 => ( in @ Xx @ B ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.56/1.67 ------------------------
% 2.56/1.69 % (8573)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_8470 for (2999ds/275Mi)
% 2.56/1.69 % (8574)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_8470 for (2999ds/18Mi)
% 2.56/1.69 % (8571)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_8470 for (2999ds/2Mi)
% 2.56/1.69 % (8570)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_8470 for (2999ds/27Mi)
% 2.56/1.69 % (8569)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_8470 for (2999ds/4Mi)
% 2.56/1.69 % (8568)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_8470 for (2999ds/183Mi)
% 2.56/1.69 % (8572)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_8470 for (2999ds/2Mi)
% 2.56/1.69 % (8571)Instruction limit reached!
% 2.56/1.69 % (8571)------------------------------
% 2.56/1.69 % (8571)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.69 % (8571)Termination reason: Unknown
% 2.56/1.69 % (8572)Instruction limit reached!
% 2.56/1.69 % (8572)------------------------------
% 2.56/1.69 % (8572)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.69 % (8571)Termination phase: shuffling
% 2.56/1.69
% 2.56/1.69 % (8571)Memory used [KB]: 1151
% 2.56/1.69 % (8571)Time elapsed: 0.003 s
% 2.56/1.69 % (8571)Instructions burned: 2 (million)
% 2.56/1.69 % (8571)------------------------------
% 2.56/1.69 % (8571)------------------------------
% 2.56/1.69 % (8572)Termination reason: Unknown
% 2.56/1.69 % (8572)Termination phase: shuffling
% 2.56/1.69
% 2.56/1.69 % (8572)Memory used [KB]: 1151
% 2.56/1.69 % (8572)Time elapsed: 0.003 s
% 2.56/1.69 % (8572)Instructions burned: 2 (million)
% 2.56/1.69 % (8572)------------------------------
% 2.56/1.69 % (8572)------------------------------
% 2.56/1.69 % (8569)Instruction limit reached!
% 2.56/1.69 % (8569)------------------------------
% 2.56/1.69 % (8569)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.69 % (8569)Termination reason: Unknown
% 2.56/1.69 % (8569)Termination phase: shuffling
% 2.56/1.69
% 2.56/1.69 % (8569)Memory used [KB]: 1279
% 2.56/1.69 % (8569)Time elapsed: 0.004 s
% 2.56/1.69 % (8569)Instructions burned: 4 (million)
% 2.56/1.69 % (8569)------------------------------
% 2.56/1.69 % (8569)------------------------------
% 2.63/1.70 % (8574)Instruction limit reached!
% 2.63/1.70 % (8574)------------------------------
% 2.63/1.70 % (8574)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.63/1.70 % (8574)Termination reason: Unknown
% 2.63/1.70 % (8574)Termination phase: Preprocessing 3
% 2.63/1.70
% 2.63/1.70 % (8574)Memory used [KB]: 1535
% 2.63/1.70 % (8574)Time elapsed: 0.011 s
% 2.63/1.70 % (8574)Instructions burned: 19 (million)
% 2.63/1.70 % (8574)------------------------------
% 2.63/1.70 % (8574)------------------------------
% 2.63/1.70 % (8570)Instruction limit reached!
% 2.63/1.70 % (8570)------------------------------
% 2.63/1.70 % (8570)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.63/1.70 % (8570)Termination reason: Unknown
% 2.63/1.70 % (8570)Termination phase: Preprocessing 2
% 2.63/1.70
% 2.63/1.70 % (8570)Memory used [KB]: 1535
% 2.63/1.70 % (8570)Time elapsed: 0.015 s
% 2.63/1.70 % (8570)Instructions burned: 29 (million)
% 2.63/1.70 % (8570)------------------------------
% 2.63/1.70 % (8570)------------------------------
% 2.63/1.71 % (8576)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_8470 for (2999ds/37Mi)
% 2.63/1.71 % (8575)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_8470 for (2999ds/3Mi)
% 2.63/1.71 % (8577)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_8470 for (2999ds/15Mi)
% 2.63/1.71 % (8575)Instruction limit reached!
% 2.63/1.71 % (8575)------------------------------
% 2.63/1.71 % (8575)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.63/1.71 % (8575)Termination reason: Unknown
% 2.63/1.71 % (8575)Termination phase: shuffling
% 2.63/1.71
% 2.63/1.71 % (8575)Memory used [KB]: 1151
% 2.63/1.71 % (8575)Time elapsed: 0.003 s
% 2.63/1.71 % (8575)Instructions burned: 3 (million)
% 2.63/1.71 % (8575)------------------------------
% 2.63/1.71 % (8575)------------------------------
% 2.63/1.71 % (8577)Instruction limit reached!
% 2.63/1.71 % (8577)------------------------------
% 2.63/1.71 % (8577)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.63/1.71 % (8577)Termination reason: Unknown
% 2.63/1.71 % (8577)Termination phase: Preprocessing 2
% 2.63/1.71 % (8578)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_8470 for (2999ds/3Mi)
% 2.63/1.71
% 2.63/1.71 % (8577)Memory used [KB]: 1407
% 2.63/1.71 % (8577)Time elapsed: 0.009 s
% 2.63/1.71 % (8577)Instructions burned: 16 (million)
% 2.63/1.71 % (8577)------------------------------
% 2.63/1.71 % (8577)------------------------------
% 2.63/1.72 % (8578)Instruction limit reached!
% 2.63/1.72 % (8578)------------------------------
% 2.63/1.72 % (8578)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.63/1.72 % (8578)Termination reason: Unknown
% 2.63/1.72 % (8578)Termination phase: shuffling
% 2.63/1.72
% 2.63/1.72 % (8578)Memory used [KB]: 1151
% 2.63/1.72 % (8578)Time elapsed: 0.003 s
% 2.63/1.72 % (8578)Instructions burned: 3 (million)
% 2.63/1.72 % (8578)------------------------------
% 2.63/1.72 % (8578)------------------------------
% 2.63/1.72 % (8579)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_8470 for (2999ds/1041Mi)
% 2.63/1.72 % (8576)Instruction limit reached!
% 2.63/1.72 % (8576)------------------------------
% 2.63/1.72 % (8576)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.63/1.72 % (8576)Termination reason: Unknown
% 2.63/1.72 % (8576)Termination phase: Property scanning
% 2.63/1.72
% 2.63/1.72 % (8576)Memory used [KB]: 1663
% 2.63/1.72 % (8576)Time elapsed: 0.018 s
% 2.63/1.72 % (8576)Instructions burned: 37 (million)
% 2.63/1.72 % (8576)------------------------------
% 2.63/1.72 % (8576)------------------------------
% 2.63/1.72 % (8580)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_8470 for (2999ds/7Mi)
% 2.63/1.73 % (8580)Instruction limit reached!
% 2.63/1.73 % (8580)------------------------------
% 2.63/1.73 % (8580)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.63/1.73 % (8580)Termination reason: Unknown
% 2.63/1.73 % (8580)Termination phase: shuffling
% 2.63/1.73
% 2.63/1.73 % (8580)Memory used [KB]: 1279
% 2.63/1.73 % (8580)Time elapsed: 0.005 s
% 2.63/1.73 % (8580)Instructions burned: 8 (million)
% 2.63/1.73 % (8580)------------------------------
% 2.63/1.73 % (8580)------------------------------
% 2.63/1.73 % (8581)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_8470 for (2999ds/16Mi)
% 2.63/1.73 % (8582)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_8470 for (2999ds/3Mi)
% 2.63/1.73 % (8582)Instruction limit reached!
% 2.63/1.73 % (8582)------------------------------
% 2.63/1.73 % (8582)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.63/1.73 % (8582)Termination reason: Unknown
% 2.63/1.73 % (8582)Termination phase: shuffling
% 2.63/1.73
% 2.63/1.73 % (8582)Memory used [KB]: 1151
% 2.63/1.73 % (8582)Time elapsed: 0.003 s
% 2.63/1.73 % (8582)Instructions burned: 3 (million)
% 2.63/1.73 % (8582)------------------------------
% 2.63/1.73 % (8582)------------------------------
% 2.63/1.74 % (8581)Instruction limit reached!
% 2.63/1.74 % (8581)------------------------------
% 2.63/1.74 % (8581)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.63/1.74 % (8581)Termination reason: Unknown
% 2.63/1.74 % (8581)Termination phase: shuffling
% 2.63/1.74
% 2.63/1.74 % (8581)Memory used [KB]: 1535
% 2.63/1.74 % (8581)Time elapsed: 0.009 s
% 2.63/1.74 % (8581)Instructions burned: 17 (million)
% 2.63/1.74 % (8581)------------------------------
% 2.63/1.74 % (8581)------------------------------
% 2.63/1.74 % (8583)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_8470 for (2999ds/3Mi)
% 2.63/1.74 % (8583)Instruction limit reached!
% 2.63/1.74 % (8583)------------------------------
% 2.63/1.74 % (8583)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.63/1.74 % (8583)Termination reason: Unknown
% 2.63/1.74 % (8583)Termination phase: shuffling
% 2.63/1.74
% 2.63/1.74 % (8583)Memory used [KB]: 1151
% 2.63/1.74 % (8583)Time elapsed: 0.003 s
% 2.63/1.74 % (8583)Instructions burned: 4 (million)
% 2.63/1.74 % (8583)------------------------------
% 2.63/1.74 % (8583)------------------------------
% 2.63/1.74 % (8584)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_8470 for (2999ds/7Mi)
% 2.63/1.75 % (8584)Instruction limit reached!
% 2.63/1.75 % (8584)------------------------------
% 2.63/1.75 % (8584)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.63/1.75 % (8584)Termination reason: Unknown
% 2.63/1.75 % (8584)Termination phase: shuffling
% 2.63/1.75
% 2.63/1.75 % (8584)Memory used [KB]: 1279
% 2.63/1.75 % (8584)Time elapsed: 0.005 s
% 2.63/1.75 % (8584)Instructions burned: 8 (million)
% 2.63/1.75 % (8584)------------------------------
% 2.63/1.75 % (8584)------------------------------
% 2.63/1.75 % (8585)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_8470 for (2999ds/3Mi)
% 2.63/1.75 % (8585)Instruction limit reached!
% 2.63/1.75 % (8585)------------------------------
% 2.63/1.75 % (8585)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.63/1.75 % (8585)Termination reason: Unknown
% 2.63/1.75 % (8585)Termination phase: shuffling
% 2.63/1.75
% 2.63/1.75 % (8585)Memory used [KB]: 1279
% 2.63/1.75 % (8585)Time elapsed: 0.003 s
% 2.63/1.75 % (8585)Instructions burned: 4 (million)
% 2.63/1.75 % (8585)------------------------------
% 2.63/1.75 % (8585)------------------------------
% 2.63/1.75 % (8586)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_8470 for (2999ds/4Mi)
% 2.63/1.76 % (8586)Instruction limit reached!
% 2.63/1.76 % (8586)------------------------------
% 2.63/1.76 % (8586)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.63/1.76 % (8586)Termination reason: Unknown
% 2.63/1.76 % (8586)Termination phase: shuffling
% 2.63/1.76
% 2.63/1.76 % (8586)Memory used [KB]: 1151
% 2.63/1.76 % (8586)Time elapsed: 0.003 s
% 2.63/1.76 % (8586)Instructions burned: 4 (million)
% 2.63/1.76 % (8586)------------------------------
% 2.63/1.76 % (8586)------------------------------
% 2.63/1.76 % (8587)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_8470 for (2999ds/18Mi)
% 2.63/1.76 % (8588)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_8470 for (2999ds/710Mi)
% 2.63/1.76 % (8587)Instruction limit reached!
% 2.63/1.76 % (8587)------------------------------
% 2.63/1.76 % (8587)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.63/1.76 % (8587)Termination reason: Unknown
% 2.63/1.76 % (8587)Termination phase: Property scanning
% 2.63/1.76
% 2.63/1.76 % (8587)Memory used [KB]: 1535
% 2.63/1.76 % (8587)Time elapsed: 0.010 s
% 2.63/1.76 % (8587)Instructions burned: 19 (million)
% 2.63/1.76 % (8587)------------------------------
% 2.63/1.76 % (8587)------------------------------
% 2.63/1.77 % (8589)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_8470 for (2999ds/6Mi)
% 2.63/1.77 % (8589)Instruction limit reached!
% 2.63/1.77 % (8589)------------------------------
% 2.63/1.77 % (8589)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.63/1.77 % (8589)Termination reason: Unknown
% 2.63/1.77 % (8589)Termination phase: shuffling
% 2.63/1.77
% 2.63/1.77 % (8589)Memory used [KB]: 1279
% 2.63/1.77 % (8589)Time elapsed: 0.004 s
% 2.63/1.77 % (8589)Instructions burned: 6 (million)
% 2.63/1.77 % (8589)------------------------------
% 2.63/1.77 % (8589)------------------------------
% 2.63/1.77 % (8590)dis+1002_5:1_au=on:bd=off:e2e=on:fde=none:fs=off:fsr=off:sos=on:i=902:si=on:rtra=on_0 on DTF2THF_8470 for (2999ds/902Mi)
% 2.63/1.77 % (8568)Instruction limit reached!
% 2.63/1.77 % (8568)------------------------------
% 2.63/1.77 % (8568)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.63/1.77 % (8568)Termination reason: Unknown
% 2.63/1.77 % (8568)Termination phase: Saturation
% 2.63/1.77
% 2.63/1.77 % (8568)Memory used [KB]: 6908
% 2.63/1.77 % (8568)Time elapsed: 0.088 s
% 2.63/1.77 % (8568)Instructions burned: 183 (million)
% 2.63/1.77 % (8568)------------------------------
% 2.63/1.77 % (8568)------------------------------
% 2.63/1.78 % (8591)dis+21_1:8_apa=on:cnfonf=off:fd=off:fsr=off:hud=0:ins=1:kws=inv_frequency:nwc=10.0:ss=axioms:st=5.0:i=21:si=on:rtra=on_0 on DTF2THF_8470 for (2999ds/21Mi)
% 2.63/1.78 % (8592)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_8470 for (2999ds/5Mi)
% 2.63/1.79 % (8592)Instruction limit reached!
% 2.63/1.79 % (8592)------------------------------
% 2.63/1.79 % (8592)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.63/1.79 % (8592)Termination reason: Unknown
% 2.63/1.79 % (8592)Termination phase: shuffling
% 2.63/1.79
% 2.63/1.79 % (8592)Memory used [KB]: 1279
% 2.63/1.79 % (8592)Time elapsed: 0.004 s
% 2.63/1.79 % (8592)Instructions burned: 6 (million)
% 2.63/1.79 % (8592)------------------------------
% 2.63/1.79 % (8592)------------------------------
% 2.63/1.79 % (8591)Instruction limit reached!
% 2.63/1.79 % (8591)------------------------------
% 2.63/1.79 % (8591)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.63/1.79 % (8591)Termination reason: Unknown
% 2.63/1.79 % (8591)Termination phase: Property scanning
% 2.63/1.79
% 2.63/1.79 % (8591)Memory used [KB]: 1535
% 2.63/1.79 % (8591)Time elapsed: 0.011 s
% 2.63/1.79 % (8591)Instructions burned: 21 (million)
% 2.63/1.79 % (8591)------------------------------
% 2.63/1.79 % (8591)------------------------------
% 2.63/1.79 % (8593)lrs+2_1:1_cnfonf=lazy_not_gen_be_off:cs=on:fe=off:hud=10:inj=on:ins=3:plsq=on:plsqc=1:sd=10:ss=axioms:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_8470 for (2998ds/6Mi)
% 2.63/1.79 % (8593)Instruction limit reached!
% 2.63/1.79 % (8593)------------------------------
% 2.63/1.79 % (8593)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.63/1.79 % (8593)Termination reason: Unknown
% 2.63/1.79 % (8593)Termination phase: shuffling
% 2.63/1.79
% 2.63/1.79 % (8593)Memory used [KB]: 1279
% 2.63/1.79 % (8593)Time elapsed: 0.004 s
% 2.63/1.79 % (8593)Instructions burned: 6 (million)
% 2.63/1.79 % (8593)------------------------------
% 2.63/1.79 % (8593)------------------------------
% 2.63/1.80 % (8594)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_8470 for (2998ds/377Mi)
% 2.63/1.81 % (8595)dis+1010_1:4_atotf=0.2:c=on:cbe=off:cnfonf=lazy_simp:fe=off:ins=2:ntd=on:s2a=on:s2at=5.0:sgt=5:ss=axioms:st=1.5:i=779:si=on:rtra=on_0 on DTF2THF_8470 for (2998ds/779Mi)
% 2.63/1.81 % (8596)lrs+10_1:1_cnfonf=lazy_not_be_gen:ntd=on:sp=const_min:ss=axioms:sup=off:i=19:si=on:rtra=on_0 on DTF2THF_8470 for (2998ds/19Mi)
% 2.63/1.82 % (8596)Instruction limit reached!
% 2.63/1.82 % (8596)------------------------------
% 2.63/1.82 % (8596)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.63/1.82 % (8596)Termination reason: Unknown
% 2.63/1.82 % (8596)Termination phase: Property scanning
% 2.63/1.82
% 2.63/1.82 % (8596)Memory used [KB]: 1535
% 2.63/1.82 % (8596)Time elapsed: 0.010 s
% 2.63/1.82 % (8596)Instructions burned: 19 (million)
% 2.63/1.82 % (8596)------------------------------
% 2.63/1.82 % (8596)------------------------------
% 2.63/1.83 % (8573)Instruction limit reached!
% 2.63/1.83 % (8573)------------------------------
% 2.63/1.83 % (8573)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.63/1.83 % (8573)Termination reason: Unknown
% 2.63/1.83 % (8573)Termination phase: Saturation
% 2.63/1.83
% 2.63/1.83 % (8573)Memory used [KB]: 7803
% 2.63/1.83 % (8573)Time elapsed: 0.139 s
% 2.63/1.83 % (8573)Instructions burned: 275 (million)
% 2.63/1.83 % (8573)------------------------------
% 2.63/1.83 % (8573)------------------------------
% 2.63/1.83 % (8590)First to succeed.
% 2.63/1.83 % (8597)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_8470 for (2998ds/879Mi)
% 2.63/1.84 % (8590)Refutation found. Thanks to Tanya!
% 2.63/1.84 % SZS status Theorem for DTF2THF_8470
% 2.63/1.84 % SZS output start Proof for DTF2THF_8470
% 2.63/1.84 thf(func_def_0, type, in: $i > $i > $o).
% 2.63/1.84 thf(func_def_1, type, exu: ($i > $o) > $o).
% 2.63/1.84 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 2.63/1.84 thf(func_def_8, type, powerset: $i > $i).
% 2.63/1.84 thf(func_def_10, type, setunion: $i > $i).
% 2.63/1.84 thf(func_def_19, type, descr: ($i > $o) > $i).
% 2.63/1.84 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 2.63/1.84 thf(func_def_26, type, prop2set: $o > $i).
% 2.63/1.84 thf(func_def_36, type, nonempty: $i > $o).
% 2.63/1.84 thf(func_def_69, type, set2prop: $i > $o).
% 2.63/1.84 thf(func_def_88, type, subset: $i > $i > $o).
% 2.63/1.84 thf(func_def_89, type, disjoint: $i > $i > $o).
% 2.63/1.84 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 2.63/1.84 thf(func_def_114, type, binunion: $i > $i > $i).
% 2.63/1.84 thf(func_def_122, type, binintersect: $i > $i > $i).
% 2.63/1.84 thf(func_def_135, type, regular: $i > $o).
% 2.63/1.84 thf(func_def_136, type, setminus: $i > $i > $i).
% 2.63/1.84 thf(func_def_154, type, sP1: $i > $i > $o).
% 2.63/1.84 thf(func_def_155, type, sP2: $i > $o).
% 2.63/1.84 thf(func_def_156, type, sP3: $i > $i > $o).
% 2.63/1.84 thf(func_def_157, type, sP4: $i > $i > $o).
% 2.63/1.84 thf(func_def_161, type, sK8: $i > $o).
% 2.63/1.84 thf(func_def_162, type, sK9: $i > $i).
% 2.63/1.84 thf(func_def_163, type, sK10: ($i > $o) > $i).
% 2.63/1.84 thf(func_def_168, type, sK15: $i > $i > $i).
% 2.63/1.84 thf(func_def_169, type, sK16: $i > $i > $i).
% 2.63/1.84 thf(func_def_173, type, sK20: $i > ($i > $o) > $i).
% 2.63/1.84 thf(func_def_174, type, sK21: $i > $o).
% 2.63/1.84 thf(func_def_186, type, sK33: $i > ($i > $o) > $i).
% 2.63/1.84 thf(func_def_187, type, sK34: $i > $o).
% 2.63/1.84 thf(func_def_190, type, sK37: $i > $i > $i).
% 2.63/1.84 thf(func_def_193, type, sK40: $i > $i).
% 2.63/1.84 thf(func_def_205, type, sK52: $i > $o).
% 2.63/1.84 thf(func_def_206, type, sK53: $i > $o).
% 2.63/1.84 thf(func_def_207, type, sK54: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.63/1.84 thf(func_def_208, type, sK55: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.63/1.84 thf(func_def_211, type, sK58: $i > $o).
% 2.63/1.84 thf(func_def_215, type, sK62: $i > $o).
% 2.63/1.84 thf(func_def_224, type, sK71: $i > $o).
% 2.63/1.84 thf(func_def_227, type, sK74: $i > ($i > $o) > $i).
% 2.63/1.84 thf(func_def_232, type, sK79: $i > $o).
% 2.63/1.84 thf(func_def_233, type, sK80: $i > $o).
% 2.63/1.84 thf(func_def_234, type, sK81: ($i > $o) > ($i > $o) > $i).
% 2.63/1.84 thf(func_def_235, type, sK82: ($i > $o) > ($i > $o) > $i).
% 2.63/1.84 thf(func_def_242, type, sK89: $i > $i).
% 2.63/1.84 thf(func_def_244, type, sK91: $i > $i).
% 2.63/1.84 thf(func_def_269, type, sK116: $i > ($i > $o) > $i).
% 2.63/1.84 thf(func_def_270, type, sK117: $i > $o).
% 2.63/1.84 thf(func_def_277, type, sK124: ($i > $o) > $i > $i).
% 2.63/1.84 thf(func_def_279, type, sK126: $i > $o).
% 2.63/1.84 thf(func_def_285, type, sK132: $i > $i).
% 2.63/1.84 thf(func_def_288, type, sK135: $i > $o).
% 2.63/1.84 thf(func_def_296, type, sK143: $i > $i > $i).
% 2.63/1.84 thf(func_def_304, type, sK151: $i > $o).
% 2.63/1.84 thf(func_def_307, type, sK154: $i > $o).
% 2.63/1.84 thf(func_def_310, type, sK157: $i > $o).
% 2.63/1.84 thf(func_def_312, type, sK159: $i > ($i > $o) > $i).
% 2.63/1.84 thf(func_def_313, type, sK160: $o > $i > $i > $i).
% 2.63/1.84 thf(func_def_320, type, sK167: $i > $o).
% 2.63/1.84 thf(func_def_327, type, sK174: $i > $o).
% 2.63/1.84 thf(func_def_332, type, sK179: $i > $o).
% 2.63/1.84 thf(func_def_336, type, sK183: $i > $i).
% 2.63/1.84 thf(func_def_342, type, sK189: $i > $i).
% 2.63/1.84 thf(func_def_345, type, sK192: ($i > $o) > $i).
% 2.63/1.84 thf(func_def_346, type, sK193: $i > $o).
% 2.63/1.84 thf(func_def_351, type, sK198: $i > $o).
% 2.63/1.84 thf(func_def_356, type, sK203: $i > ($i > $o) > $i).
% 2.63/1.84 thf(func_def_357, type, sK204: $i > $o).
% 2.63/1.84 thf(func_def_361, type, sK208: $i > $o).
% 2.63/1.84 thf(func_def_363, type, sK210: $i > ($i > $o) > $i).
% 2.63/1.84 thf(func_def_373, type, sK220: $i > $i > $i).
% 2.63/1.84 thf(func_def_376, type, sK223: $i > $i > $o).
% 2.63/1.84 thf(func_def_378, type, sK225: $i > $i).
% 2.63/1.84 thf(func_def_379, type, sK226: $i > $i).
% 2.63/1.84 thf(func_def_380, type, sK227: $i > ($i > $i > $o) > $i).
% 2.63/1.84 thf(func_def_381, type, sK228: $i > $i > ($i > $i > $o) > $i).
% 2.63/1.84 thf(func_def_382, type, sK229: $i > ($i > $i > $o) > $i).
% 2.63/1.84 thf(func_def_386, type, sK233: $i > $i).
% 2.63/1.84 thf(func_def_395, type, sK242: $i > $i > $i).
% 2.63/1.84 thf(func_def_403, type, sK250: $i > $o).
% 2.63/1.84 thf(func_def_404, type, sK251: $i > $i).
% 2.63/1.84 thf(func_def_405, type, sK252: ($i > $o) > $i).
% 2.63/1.84 thf(func_def_411, type, sK258: $i > $o).
% 2.63/1.84 thf(func_def_415, type, sK262: $i > $o).
% 2.63/1.84 thf(func_def_421, type, sK268: $i > $i > $i).
% 2.63/1.84 thf(func_def_424, type, sK271: $i > $i > $i).
% 2.63/1.84 thf(func_def_425, type, sK272: $i > $i > $i).
% 2.63/1.84 thf(func_def_426, type, sK273: $i > $i > $i).
% 2.63/1.84 thf(func_def_427, type, sK274: $i > $i > $i).
% 2.63/1.84 thf(func_def_428, type, sK275: $i > $i > $i).
% 2.63/1.84 thf(func_def_429, type, sK276: $i > $i > $i > $i).
% 2.63/1.84 thf(func_def_430, type, sK277: $i > $i).
% 2.63/1.84 thf(func_def_431, type, sK278: $i > $i).
% 2.63/1.84 thf(func_def_432, type, sK279: $i > $i).
% 2.63/1.84 thf(func_def_433, type, sK280: $i > $i).
% 2.63/1.84 thf(func_def_434, type, sK281: $i > $i > $i > $i).
% 2.63/1.84 thf(func_def_435, type, sK282: $i > $i > $i > $i).
% 2.63/1.84 thf(func_def_436, type, sK283: $i > $i > $i).
% 2.63/1.84 thf(func_def_437, type, sK284: $i > $i > $i).
% 2.63/1.84 thf(func_def_438, type, sK285: $i > $i > $i).
% 2.63/1.84 thf(func_def_439, type, sK286: $i > $i).
% 2.63/1.84 thf(func_def_441, type, sK288: $i > $i).
% 2.63/1.84 thf(func_def_442, type, sK289: $i > $i).
% 2.63/1.84 thf(func_def_444, type, sK291: $i > $o).
% 2.63/1.84 thf(func_def_445, type, sK292: $i > $o).
% 2.63/1.84 thf(func_def_446, type, sK293: ($i > $o) > ($i > $o) > $i).
% 2.63/1.84 thf(func_def_447, type, sK294: ($i > $o) > ($i > $o) > $i).
% 2.63/1.84 thf(func_def_462, type, sK309: $i > $o).
% 2.63/1.84 thf(func_def_468, type, sK315: $i > $i > $i).
% 2.63/1.84 thf(func_def_474, type, sK321: ($i > $o) > $i).
% 2.63/1.84 thf(func_def_475, type, sK322: ($i > $o) > $i).
% 2.63/1.84 thf(func_def_476, type, sK323: $i > $o).
% 2.63/1.84 thf(func_def_485, type, sK332: $i > $i > $i).
% 2.63/1.84 thf(func_def_487, type, ph334: !>[X0: $tType]:(X0)).
% 2.63/1.84 thf(f1878,plain,(
% 2.63/1.84 $false),
% 2.63/1.84 inference(avatar_sat_refutation,[status(thm)],[f1839,f1861,f1875])).
% 2.63/1.84 thf(f1875,plain,(
% 2.63/1.84 ~spl333_3),
% 2.63/1.84 inference(avatar_contradiction_clause,[status(thm)],[f1874])).
% 2.63/1.84 thf(f1874,plain,(
% 2.63/1.84 $false | ~spl333_3),
% 2.63/1.84 inference(trivial_inequality_removal,[status(thm)],[f1871])).
% 2.63/1.84 thf(f1871,plain,(
% 2.63/1.84 ($true != $true) | ~spl333_3),
% 2.63/1.84 inference(superposition,[status(thm)],[f1190,f1834])).
% 2.63/1.84 thf(f1834,plain,(
% 2.63/1.84 ($true = (in @ sK5 @ sK6)) | ~spl333_3),
% 2.63/1.84 inference(avatar_component_clause,[status(thm)],[f1832])).
% 2.63/1.84 thf(f1832,definition,(
% 2.63/1.84 spl333_3 <=> ($true = (in @ sK5 @ sK6))),
% 2.63/1.84 introduced(definition,[new_symbols(naming,[spl333_3])],[avatar_definition])).
% 2.63/1.84 thf(f1190,plain,(
% 2.63/1.84 ($true != (in @ sK5 @ sK6))),
% 2.63/1.84 inference(cnf_transformation,[status(thm)],[f530])).
% 2.63/1.84 thf(f530,plain,(
% 2.63/1.84 (exuE3u = $true) & (subsetTrans = $true) & (subset2powerset = $true) & (omega0Ax = $true) & (setunionI = $true) & (descr__Cong = $true) & (binintersectSubset3 = $true) & (notequalI2 = $true) & (quantDeMorgan1 = $true) & (setadjoinIL = $true) & (exuEu = $true) & (omegaIndAx = $true) & (quantDeMorgan2 = $true) & (binunionRsub = $true) & (quantDeMorgan3 = $true) & (notdexE = $true) & (wellorderingAx = $true) & (inCongP = $true) & (singletonsswitch = $true) & (subsetI2 = $true) & (setminusEL = $true) & (omega__Cong = $true) & (setminusSubset2 = $true) & (exuE3e = $true) & (quantDeMorgan4 = $true) & (exuI2 = $true) & (($true != (in @ sK5 @ (setminus @ sK7 @ sK6))) & ((in @ sK5 @ sK7) = $true) & ($true != (in @ sK5 @ sK6))) & (nonemptyI = $true) & (setadjoinOr = $true) & (dsetconstr__Cong = $true) & (inPowerset = $true) & (emptyInPowerset = $true) & (binunionLsub = $true) & (setext = $true) & (in__Cong = $true) & (descrp = $true) & (setbeta = $true) & (dsetconstrER = $true) & (noeltsimpempty = $true) & (subsetE2 = $true) & (powerset__Cong = $true) & (binintersectSubset1 = $true) & (binintersectSubset5 = $true) & (upairsetE = $true) & (setoftrueEq = $true) & (emptyinunitempty = $true) & (subsetemptysetimpeq = $true) & (subsetI1 = $true) & (emptysetsubset = $true) & (setextsub = $true) & (powersetE1 = $true) & (powersetAx = $true) & (nonemptyE1 = $true) & (subsetRefl = $true) & (powersetI1 = $true) & (binintersectRsub = $true) & (notinemptyset = $true) & (exu__Cong = $true) & (exuE1 = $true) & (binintersectSubset4 = $true) & (binunionE = $true) & (emptysetE = $true) & (emptysetimpfalse = $true) & (exuI3 = $true) & (setminusER = $true) & (sepInPowerset = $true) & (powersetI = $true) & (sepSubset = $true) & (powersetsubset = $true) & (powersetE = $true) & (setunionE = $true) & (binintersectI = $true) & (binintersectLsub = $true) & (emptysetAx = $true) & (setadjoinIR = $true) & (nonemptyI1 = $true) & (emptyI = $true) & (prop2set2propI = $true) & (setadjoinSub2 = $true) & (notequalI1 = $true) & (prop2setI = $true) & (binintersectEL = $true) & (setadjoinE = $true) & (vacuousDall = $true) & (notinsingleton = $true) & (binintersectER = $true) & (notdallE = $true) & (emptyinPowerset = $true) & (eqimpsubset2 = $true) & (eqimpsubset1 = $true) & (setadjoin__Cong = $true) & (replAx = $true) & (uniqinunit = $true) & (setunion__Cong = $true) & (dsetconstrI = $true) & (setadjoinSub = $true) & (notsubsetI = $true) & (emptyE1 = $true) & (subPowSU = $true) & (upairsetIL = $true) & (binunionIR = $true) & (bs114d = $true) & (binunionEcases = $true) & (binintersectSubset2 = $true) & (omegaSAx = $true) & (exuI1 = $true) & (setminusI = $true) & (eqinunit = $true) & (upairset2IR = $true) & (disjointsetsI1 = $true) & (subsetE = $true) & (upairsetIR = $true) & (foundationAx = $true) & (exuE2 = $true) & (dsetconstrEL = $true) & (setunionAx = $true) & (binunionIL = $true) & (setextAx = $true) & (prop2setE = $true) & (setadjoinAx = $true) & (nonemptyImpWitness = $true) & (emptyset__Cong = $true)),
% 2.63/1.84 inference(skolemisation,[status(esa),new_symbols(skolem,[sK5,sK6,sK7])],[f507,f529])).
% 2.63/1.84 thf(f529,plain,(
% 2.63/1.84 ? [X0 : $i,X1 : $i,X2 : $i] : (((in @ X0 @ (setminus @ X2 @ X1)) != $true) & ($true = (in @ X0 @ X2)) & ($true != (in @ X0 @ X1))) => (($true != (in @ sK5 @ (setminus @ sK7 @ sK6))) & ((in @ sK5 @ sK7) = $true) & ($true != (in @ sK5 @ sK6)))),
% 2.63/1.84 introduced(definition,[],[choice_axiom])).
% 2.63/1.84 thf(f507,plain,(
% 2.63/1.84 (exuE3u = $true) & (subsetTrans = $true) & (subset2powerset = $true) & (omega0Ax = $true) & (setunionI = $true) & (descr__Cong = $true) & (binintersectSubset3 = $true) & (notequalI2 = $true) & (quantDeMorgan1 = $true) & (setadjoinIL = $true) & (exuEu = $true) & (omegaIndAx = $true) & (quantDeMorgan2 = $true) & (binunionRsub = $true) & (quantDeMorgan3 = $true) & (notdexE = $true) & (wellorderingAx = $true) & (inCongP = $true) & (singletonsswitch = $true) & (subsetI2 = $true) & (setminusEL = $true) & (omega__Cong = $true) & (setminusSubset2 = $true) & (exuE3e = $true) & (quantDeMorgan4 = $true) & (exuI2 = $true) & ? [X0 : $i,X1 : $i,X2 : $i] : (((in @ X0 @ (setminus @ X2 @ X1)) != $true) & ($true = (in @ X0 @ X2)) & ($true != (in @ X0 @ X1))) & (nonemptyI = $true) & (setadjoinOr = $true) & (dsetconstr__Cong = $true) & (inPowerset = $true) & (emptyInPowerset = $true) & (binunionLsub = $true) & (setext = $true) & (in__Cong = $true) & (descrp = $true) & (setbeta = $true) & (dsetconstrER = $true) & (noeltsimpempty = $true) & (subsetE2 = $true) & (powerset__Cong = $true) & (binintersectSubset1 = $true) & (binintersectSubset5 = $true) & (upairsetE = $true) & (setoftrueEq = $true) & (emptyinunitempty = $true) & (subsetemptysetimpeq = $true) & (subsetI1 = $true) & (emptysetsubset = $true) & (setextsub = $true) & (powersetE1 = $true) & (powersetAx = $true) & (nonemptyE1 = $true) & (subsetRefl = $true) & (powersetI1 = $true) & (binintersectRsub = $true) & (notinemptyset = $true) & (exu__Cong = $true) & (exuE1 = $true) & (binintersectSubset4 = $true) & (binunionE = $true) & (emptysetE = $true) & (emptysetimpfalse = $true) & (exuI3 = $true) & (setminusER = $true) & (sepInPowerset = $true) & (powersetI = $true) & (sepSubset = $true) & (powersetsubset = $true) & (powersetE = $true) & (setunionE = $true) & (binintersectI = $true) & (binintersectLsub = $true) & (emptysetAx = $true) & (setadjoinIR = $true) & (nonemptyI1 = $true) & (emptyI = $true) & (prop2set2propI = $true) & (setadjoinSub2 = $true) & (notequalI1 = $true) & (prop2setI = $true) & (binintersectEL = $true) & (setadjoinE = $true) & (vacuousDall = $true) & (notinsingleton = $true) & (binintersectER = $true) & (notdallE = $true) & (emptyinPowerset = $true) & (eqimpsubset2 = $true) & (eqimpsubset1 = $true) & (setadjoin__Cong = $true) & (replAx = $true) & (uniqinunit = $true) & (setunion__Cong = $true) & (dsetconstrI = $true) & (setadjoinSub = $true) & (notsubsetI = $true) & (emptyE1 = $true) & (subPowSU = $true) & (upairsetIL = $true) & (binunionIR = $true) & (bs114d = $true) & (binunionEcases = $true) & (binintersectSubset2 = $true) & (omegaSAx = $true) & (exuI1 = $true) & (setminusI = $true) & (eqinunit = $true) & (upairset2IR = $true) & (disjointsetsI1 = $true) & (subsetE = $true) & (upairsetIR = $true) & (foundationAx = $true) & (exuE2 = $true) & (dsetconstrEL = $true) & (setunionAx = $true) & (binunionIL = $true) & (setextAx = $true) & (prop2setE = $true) & (setadjoinAx = $true) & (nonemptyImpWitness = $true) & (emptyset__Cong = $true)),
% 2.63/1.84 inference(flattening,[status(thm)],[f506])).
% 2.63/1.84 thf(f506,plain,(
% 2.63/1.84 ((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i,X1 : $i,X2 : $i] : ((($true != (in @ X0 @ X1)) & ($true = (in @ X0 @ X2))) & ((in @ X0 @ (setminus @ X2 @ X1)) != $true)) & (setminusSubset2 = $true)) & (setminusER = $true)) & (setminusEL = $true)) & (setminusI = $true)) & (bs114d = $true)) & (binintersectSubset1 = $true)) & (binintersectSubset4 = $true)) & (binintersectRsub = $true)) & (disjointsetsI1 = $true)) & (binintersectER = $true)) & (binintersectSubset3 = $true)) & (binintersectSubset2 = $true)) & (binintersectLsub = $true)) & (binintersectEL = $true)) & (binintersectSubset5 = $true)) & (binintersectI = $true)) & (binunionRsub = $true)) & (binunionLsub = $true)) & (binunionE = $true)) & (binunionEcases = $true)) & (binunionIR = $true)) & (upairset2IR = $true)) & (binunionIL = $true)) & (sepSubset = $true)) & (sepInPowerset = $true)) & (powersetsubset = $true)) & (inPowerset = $true)) & (powersetE1 = $true)) & (powersetI1 = $true)) & (subsetemptysetimpeq = $true)) & (setextsub = $true)) & (subset2powerset = $true)) & (setadjoinSub2 = $true)) & (setadjoinSub = $true)) & (subsetTrans = $true)) & (subsetRefl = $true)) & (notequalI2 = $true)) & (notequalI1 = $true)) & (notsubsetI = $true)) & (subsetE2 = $true)) & (subsetE = $true)) & (emptysetsubset = $true)) & (subsetI2 = $true)) & (eqimpsubset1 = $true)) & (eqimpsubset2 = $true)) & (subsetI1 = $true)) & (dsetconstr__Cong = $true)) & (descr__Cong = $true)) & (exuEu = $true)) & (omega__Cong = $true)) & (setunion__Cong = $true)) & (powerset__Cong = $true)) & (setadjoin__Cong = $true)) & (emptyset__Cong = $true)) & (exu__Cong = $true)) & (exuE3u = $true)) & (in__Cong = $true)) & (inCongP = $true)) & (exuI2 = $true)) & (exuI3 = $true)) & (exuI1 = $true)) & (notdallE = $true)) & (notdexE = $true)) & (prop2set2propI = $true)) & (prop2setI = $true)) & (quantDeMorgan4 = $true)) & (quantDeMorgan3 = $true)) & (quantDeMorgan2 = $true)) & (quantDeMorgan1 = $true)) & (vacuousDall = $true)) & (emptyE1 = $true)) & (upairsetIR = $true)) & (upairsetIL = $true)) & (upairsetE = $true)) & (singletonsswitch = $true)) & (eqinunit = $true)) & (notinsingleton = $true)) & (uniqinunit = $true)) & (nonemptyImpWitness = $true)) & (exuE2 = $true)) & (subPowSU = $true)) & (setunionE = $true)) & (setunionI = $true)) & (powersetE = $true)) & (emptyInPowerset = $true)) & (emptyinPowerset = $true)) & (powersetI = $true)) & (setoftrueEq = $true)) & (setadjoinOr = $true)) & (setadjoinE = $true)) & (setadjoinIR = $true)) & (emptyinunitempty = $true)) & (setadjoinIL = $true)) & (nonemptyI1 = $true)) & (nonemptyI = $true)) & (nonemptyE1 = $true)) & (setbeta = $true)) & (noeltsimpempty = $true)) & (emptyI = $true)) & (setext = $true)) & (exuE3e = $true)) & (notinemptyset = $true)) & (emptysetimpfalse = $true)) & (emptysetE = $true)) & (prop2setE = $true)) & (exuE1 = $true)) & (dsetconstrER = $true)) & (dsetconstrEL = $true)) & (dsetconstrI = $true)) & (descrp = $true)) & (wellorderingAx = $true)) & (foundationAx = $true)) & (replAx = $true)) & (omegaIndAx = $true)) & (omegaSAx = $true)) & (omega0Ax = $true)) & (setunionAx = $true)) & (powersetAx = $true)) & (setadjoinAx = $true)) & (emptysetAx = $true)) & (setextAx = $true)),
% 2.63/1.84 inference(ennf_transformation,[status(thm)],[f370])).
% 2.63/1.84 thf(f370,plain,(
% 2.63/1.84 ~((setextAx = $true) => ((emptysetAx = $true) => ((setadjoinAx = $true) => ((powersetAx = $true) => ((setunionAx = $true) => ((omega0Ax = $true) => ((omegaSAx = $true) => ((omegaIndAx = $true) => ((replAx = $true) => ((foundationAx = $true) => ((wellorderingAx = $true) => ((descrp = $true) => ((dsetconstrI = $true) => ((dsetconstrEL = $true) => ((dsetconstrER = $true) => ((exuE1 = $true) => ((prop2setE = $true) => ((emptysetE = $true) => ((emptysetimpfalse = $true) => ((notinemptyset = $true) => ((exuE3e = $true) => ((setext = $true) => ((emptyI = $true) => ((noeltsimpempty = $true) => ((setbeta = $true) => ((nonemptyE1 = $true) => ((nonemptyI = $true) => ((nonemptyI1 = $true) => ((setadjoinIL = $true) => ((emptyinunitempty = $true) => ((setadjoinIR = $true) => ((setadjoinE = $true) => ((setadjoinOr = $true) => ((setoftrueEq = $true) => ((powersetI = $true) => ((emptyinPowerset = $true) => ((emptyInPowerset = $true) => ((powersetE = $true) => ((setunionI = $true) => ((setunionE = $true) => ((subPowSU = $true) => ((exuE2 = $true) => ((nonemptyImpWitness = $true) => ((uniqinunit = $true) => ((notinsingleton = $true) => ((eqinunit = $true) => ((singletonsswitch = $true) => ((upairsetE = $true) => ((upairsetIL = $true) => ((upairsetIR = $true) => ((emptyE1 = $true) => ((vacuousDall = $true) => ((quantDeMorgan1 = $true) => ((quantDeMorgan2 = $true) => ((quantDeMorgan3 = $true) => ((quantDeMorgan4 = $true) => ((prop2setI = $true) => ((prop2set2propI = $true) => ((notdexE = $true) => ((notdallE = $true) => ((exuI1 = $true) => ((exuI3 = $true) => ((exuI2 = $true) => ((inCongP = $true) => ((in__Cong = $true) => ((exuE3u = $true) => ((exu__Cong = $true) => ((emptyset__Cong = $true) => ((setadjoin__Cong = $true) => ((powerset__Cong = $true) => ((setunion__Cong = $true) => ((omega__Cong = $true) => ((exuEu = $true) => ((descr__Cong = $true) => ((dsetconstr__Cong = $true) => ((subsetI1 = $true) => ((eqimpsubset2 = $true) => ((eqimpsubset1 = $true) => ((subsetI2 = $true) => ((emptysetsubset = $true) => ((subsetE = $true) => ((subsetE2 = $true) => ((notsubsetI = $true) => ((notequalI1 = $true) => ((notequalI2 = $true) => ((subsetRefl = $true) => ((subsetTrans = $true) => ((setadjoinSub = $true) => ((setadjoinSub2 = $true) => ((subset2powerset = $true) => ((setextsub = $true) => ((subsetemptysetimpeq = $true) => ((powersetI1 = $true) => ((powersetE1 = $true) => ((inPowerset = $true) => ((powersetsubset = $true) => ((sepInPowerset = $true) => ((sepSubset = $true) => ((binunionIL = $true) => ((upairset2IR = $true) => ((binunionIR = $true) => ((binunionEcases = $true) => ((binunionE = $true) => ((binunionLsub = $true) => ((binunionRsub = $true) => ((binintersectI = $true) => ((binintersectSubset5 = $true) => ((binintersectEL = $true) => ((binintersectLsub = $true) => ((binintersectSubset2 = $true) => ((binintersectSubset3 = $true) => ((binintersectER = $true) => ((disjointsetsI1 = $true) => ((binintersectRsub = $true) => ((binintersectSubset4 = $true) => ((binintersectSubset1 = $true) => ((bs114d = $true) => ((setminusI = $true) => ((setminusEL = $true) => ((setminusER = $true) => ((setminusSubset2 = $true) => ! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X0 @ (setminus @ X2 @ X1)) != $true) => (($true = (in @ X0 @ X2)) => ($true = (in @ X0 @ X1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.63/1.84 inference(flattening,[status(thm)],[f136])).
% 2.63/1.84 thf(f136,plain,(
% 2.63/1.84 ~((setextAx = $true) => ((emptysetAx = $true) => ((setadjoinAx = $true) => ((powersetAx = $true) => ((setunionAx = $true) => ((omega0Ax = $true) => ((omegaSAx = $true) => ((omegaIndAx = $true) => ((replAx = $true) => ((foundationAx = $true) => ((wellorderingAx = $true) => ((descrp = $true) => ((dsetconstrI = $true) => ((dsetconstrEL = $true) => ((dsetconstrER = $true) => ((exuE1 = $true) => ((prop2setE = $true) => ((emptysetE = $true) => ((emptysetimpfalse = $true) => ((notinemptyset = $true) => ((exuE3e = $true) => ((setext = $true) => ((emptyI = $true) => ((noeltsimpempty = $true) => ((setbeta = $true) => ((nonemptyE1 = $true) => ((nonemptyI = $true) => ((nonemptyI1 = $true) => ((setadjoinIL = $true) => ((emptyinunitempty = $true) => ((setadjoinIR = $true) => ((setadjoinE = $true) => ((setadjoinOr = $true) => ((setoftrueEq = $true) => ((powersetI = $true) => ((emptyinPowerset = $true) => ((emptyInPowerset = $true) => ((powersetE = $true) => ((setunionI = $true) => ((setunionE = $true) => ((subPowSU = $true) => ((exuE2 = $true) => ((nonemptyImpWitness = $true) => ((uniqinunit = $true) => ((notinsingleton = $true) => ((eqinunit = $true) => ((singletonsswitch = $true) => ((upairsetE = $true) => ((upairsetIL = $true) => ((upairsetIR = $true) => ((emptyE1 = $true) => ((vacuousDall = $true) => ((quantDeMorgan1 = $true) => ((quantDeMorgan2 = $true) => ((quantDeMorgan3 = $true) => ((quantDeMorgan4 = $true) => ((prop2setI = $true) => ((prop2set2propI = $true) => ((notdexE = $true) => ((notdallE = $true) => ((exuI1 = $true) => ((exuI3 = $true) => ((exuI2 = $true) => ((inCongP = $true) => ((in__Cong = $true) => ((exuE3u = $true) => ((exu__Cong = $true) => ((emptyset__Cong = $true) => ((setadjoin__Cong = $true) => ((powerset__Cong = $true) => ((setunion__Cong = $true) => ((omega__Cong = $true) => ((exuEu = $true) => ((descr__Cong = $true) => ((dsetconstr__Cong = $true) => ((subsetI1 = $true) => ((eqimpsubset2 = $true) => ((eqimpsubset1 = $true) => ((subsetI2 = $true) => ((emptysetsubset = $true) => ((subsetE = $true) => ((subsetE2 = $true) => ((notsubsetI = $true) => ((notequalI1 = $true) => ((notequalI2 = $true) => ((subsetRefl = $true) => ((subsetTrans = $true) => ((setadjoinSub = $true) => ((setadjoinSub2 = $true) => ((subset2powerset = $true) => ((setextsub = $true) => ((subsetemptysetimpeq = $true) => ((powersetI1 = $true) => ((powersetE1 = $true) => ((inPowerset = $true) => ((powersetsubset = $true) => ((sepInPowerset = $true) => ((sepSubset = $true) => ((binunionIL = $true) => ((upairset2IR = $true) => ((binunionIR = $true) => ((binunionEcases = $true) => ((binunionE = $true) => ((binunionLsub = $true) => ((binunionRsub = $true) => ((binintersectI = $true) => ((binintersectSubset5 = $true) => ((binintersectEL = $true) => ((binintersectLsub = $true) => ((binintersectSubset2 = $true) => ((binintersectSubset3 = $true) => ((binintersectER = $true) => ((disjointsetsI1 = $true) => ((binintersectRsub = $true) => ((binintersectSubset4 = $true) => ((binintersectSubset1 = $true) => ((bs114d = $true) => ((setminusI = $true) => ((setminusEL = $true) => ((setminusER = $true) => ((setminusSubset2 = $true) => ! [X0 : $i,X1 : $i,X2 : $i] : (~((in @ X0 @ (setminus @ X2 @ X1)) = $true) => (($true = (in @ X0 @ X2)) => ($true = (in @ X0 @ X1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.63/1.84 inference(fool_elimination,[status(thm)],[f135])).
% 2.63/1.84 thf(f135,plain,(
% 2.63/1.84 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => ! [X0 : $i,X1 : $i,X2 : $i] : (~(in @ X0 @ (setminus @ X2 @ X1)) => ((in @ X0 @ X2) => (in @ X0 @ X1))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.63/1.84 inference(rectify,[status(thm)],[f125])).
% 2.63/1.84 thf(f125,negated_conjecture,(
% 2.63/1.84 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => ! [X1 : $i,X4 : $i,X3 : $i] : (~(in @ X1 @ (setminus @ X3 @ X4)) => ((in @ X1 @ X3) => (in @ X1 @ X4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.63/1.84 inference(negated_conjecture,[status(cth)],[f124])).
% 2.63/1.84 thf(f124,conjecture,(
% 2.63/1.84 setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => ! [X1 : $i,X4 : $i,X3 : $i] : (~(in @ X1 @ (setminus @ X3 @ X4)) => ((in @ X1 @ X3) => (in @ X1 @ X4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.63/1.84 file('/export/starexec/sandbox/tmp/tmp.FHMJW0nBVa/DTF2THF_8470.p',setminusERneg)).
% 2.63/1.84 thf(f1861,plain,(
% 2.63/1.84 spl333_4),
% 2.63/1.84 inference(avatar_contradiction_clause,[status(thm)],[f1860])).
% 2.63/1.84 thf(f1860,plain,(
% 2.63/1.84 $false | spl333_4),
% 2.63/1.84 inference(trivial_inequality_removal,[status(thm)],[f1854])).
% 2.63/1.84 thf(f1854,plain,(
% 2.63/1.84 ($true != $true) | spl333_4),
% 2.63/1.84 inference(superposition,[status(thm)],[f1838,f1191])).
% 2.63/1.84 thf(f1191,plain,(
% 2.63/1.84 ((in @ sK5 @ sK7) = $true)),
% 2.63/1.84 inference(cnf_transformation,[status(thm)],[f530])).
% 2.63/1.84 thf(f1838,plain,(
% 2.63/1.84 ((in @ sK5 @ sK7) != $true) | spl333_4),
% 2.63/1.84 inference(avatar_component_clause,[status(thm)],[f1836])).
% 2.63/1.84 thf(f1836,definition,(
% 2.63/1.84 spl333_4 <=> ((in @ sK5 @ sK7) = $true)),
% 2.63/1.84 introduced(definition,[new_symbols(naming,[spl333_4])],[avatar_definition])).
% 2.63/1.84 thf(f1839,plain,(
% 2.63/1.84 spl333_3 | ~spl333_4),
% 2.63/1.84 inference(avatar_split_clause,[status(thm)],[f1830,f1836,f1832])).
% 2.63/1.84 thf(f1830,plain,(
% 2.63/1.84 ((in @ sK5 @ sK7) != $true) | ($true = (in @ sK5 @ sK6))),
% 2.63/1.84 inference(trivial_inequality_removal,[status(thm)],[f1829])).
% 2.63/1.84 thf(f1829,plain,(
% 2.63/1.84 ($true != $true) | ($true = (in @ sK5 @ sK6)) | ((in @ sK5 @ sK7) != $true)),
% 2.63/1.84 inference(forward_demodulation,[status(thm)],[f1808,f1110])).
% 2.63/1.84 thf(f1110,plain,(
% 2.63/1.84 (setminusI = $true)),
% 2.63/1.84 inference(cnf_transformation,[status(thm)],[f530])).
% 2.63/1.84 thf(f1808,plain,(
% 2.63/1.84 (setminusI != $true) | ($true = (in @ sK5 @ sK6)) | ((in @ sK5 @ sK7) != $true)),
% 2.63/1.84 inference(trivial_inequality_removal,[status(thm)],[f1802])).
% 2.63/1.84 thf(f1802,plain,(
% 2.63/1.84 ($true = (in @ sK5 @ sK6)) | (setminusI != $true) | ((in @ sK5 @ sK7) != $true) | ($true != $true)),
% 2.63/1.84 inference(superposition,[status(thm)],[f1192,f1411])).
% 2.63/1.84 thf(f1411,plain,(
% 2.63/1.84 ( ! [X2 : $i,X0 : $i,X1 : $i] : (($true = (in @ X2 @ (setminus @ X0 @ X1))) | ((in @ X2 @ X0) != $true) | ((in @ X2 @ X1) = $true) | (setminusI != $true)) )),
% 2.63/1.84 inference(cnf_transformation,[status(thm)],[f768])).
% 2.63/1.84 thf(f768,plain,(
% 2.63/1.84 (! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X2 @ X1) = $true) | ($true = (in @ X2 @ (setminus @ X0 @ X1))) | ((in @ X2 @ X0) != $true)) | (setminusI != $true)) & ((setminusI = $true) | (($true != (in @ sK149 @ sK148)) & ((in @ sK149 @ (setminus @ sK147 @ sK148)) != $true) & ($true = (in @ sK149 @ sK147))))),
% 2.63/1.84 inference(skolemisation,[status(esa),new_symbols(skolem,[sK147,sK148,sK149])],[f766,f767])).
% 2.63/1.84 thf(f767,plain,(
% 2.63/1.84 ? [X3 : $i,X4 : $i,X5 : $i] : (((in @ X5 @ X4) != $true) & ((in @ X5 @ (setminus @ X3 @ X4)) != $true) & ($true = (in @ X5 @ X3))) => (($true != (in @ sK149 @ sK148)) & ((in @ sK149 @ (setminus @ sK147 @ sK148)) != $true) & ($true = (in @ sK149 @ sK147)))),
% 2.63/1.84 introduced(definition,[],[choice_axiom])).
% 2.63/1.84 thf(f766,plain,(
% 2.63/1.84 (! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X2 @ X1) = $true) | ($true = (in @ X2 @ (setminus @ X0 @ X1))) | ((in @ X2 @ X0) != $true)) | (setminusI != $true)) & ((setminusI = $true) | ? [X3 : $i,X4 : $i,X5 : $i] : (((in @ X5 @ X4) != $true) & ((in @ X5 @ (setminus @ X3 @ X4)) != $true) & ($true = (in @ X5 @ X3))))),
% 2.63/1.84 inference(rectify,[status(thm)],[f765])).
% 2.63/1.84 thf(f765,plain,(
% 2.63/1.84 (! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X2 @ X1) = $true) | ($true = (in @ X2 @ (setminus @ X0 @ X1))) | ((in @ X2 @ X0) != $true)) | (setminusI != $true)) & ((setminusI = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (((in @ X2 @ X1) != $true) & ($true != (in @ X2 @ (setminus @ X0 @ X1))) & ((in @ X2 @ X0) = $true)))),
% 2.63/1.84 inference(nnf_transformation,[status(thm)],[f408])).
% 2.63/1.84 thf(f408,plain,(
% 2.63/1.84 ! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X2 @ X1) = $true) | ($true = (in @ X2 @ (setminus @ X0 @ X1))) | ((in @ X2 @ X0) != $true)) <=> (setminusI = $true)),
% 2.63/1.84 inference(flattening,[status(thm)],[f407])).
% 2.63/1.84 thf(f407,plain,(
% 2.63/1.84 ! [X0 : $i,X1 : $i,X2 : $i] : ((($true = (in @ X2 @ (setminus @ X0 @ X1))) | ((in @ X2 @ X1) = $true)) | ((in @ X2 @ X0) != $true)) <=> (setminusI = $true)),
% 2.63/1.84 inference(ennf_transformation,[status(thm)],[f392])).
% 2.63/1.84 thf(f392,plain,(
% 2.63/1.84 ! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X2 @ X0) = $true) => (((in @ X2 @ X1) != $true) => ($true = (in @ X2 @ (setminus @ X0 @ X1))))) <=> (setminusI = $true)),
% 2.63/1.84 inference(flattening,[status(thm)],[f310])).
% 2.63/1.84 thf(f310,plain,(
% 2.63/1.84 (setminusI = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X2 @ X0) = $true) => (~((in @ X2 @ X1) = $true) => ($true = (in @ X2 @ (setminus @ X0 @ X1)))))),
% 2.63/1.84 inference(fool_elimination,[status(thm)],[f309])).
% 2.63/1.84 thf(f309,plain,(
% 2.63/1.84 (setminusI = ! [X0 : $i,X1 : $i,X2 : $i] : ((in @ X2 @ X0) => (~(in @ X2 @ X1) => (in @ X2 @ (setminus @ X0 @ X1)))))),
% 2.63/1.84 inference(rectify,[status(thm)],[f120])).
% 2.63/1.84 thf(f120,axiom,(
% 2.63/1.84 (setminusI = ! [X3 : $i,X4 : $i,X1 : $i] : ((in @ X1 @ X3) => (~(in @ X1 @ X4) => (in @ X1 @ (setminus @ X3 @ X4)))))),
% 2.63/1.84 file('/export/starexec/sandbox/tmp/tmp.FHMJW0nBVa/DTF2THF_8470.p',setminusI)).
% 2.63/1.84 thf(f1192,plain,(
% 2.63/1.84 ($true != (in @ sK5 @ (setminus @ sK7 @ sK6)))),
% 2.63/1.84 inference(cnf_transformation,[status(thm)],[f530])).
% 2.63/1.84 % SZS output end Proof for DTF2THF_8470
% 2.63/1.84 % (8590)------------------------------
% 2.63/1.84 % (8590)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.63/1.84 % (8590)Termination reason: Refutation
% 2.63/1.84
% 2.63/1.84 % (8590)Memory used [KB]: 7419
% 2.63/1.84 % (8590)Time elapsed: 0.067 s
% 2.63/1.84 % (8590)Instructions burned: 126 (million)
% 2.63/1.84 % (8590)------------------------------
% 2.63/1.84 % (8590)------------------------------
% 2.63/1.84 % (8567)Success in time 0.161 s
%------------------------------------------------------------------------------