%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU642^1 : 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 : n027.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 09:35:53 AM UTC 2026
% Result : Theorem 2.89s 1.93s
% Output : Refutation 2.89s
% Verified :
% SZS Type : Refutation
% Derivation depth : 15
% Number of leaves : 16
% Syntax : Number of formulae : 96 ( 19 unt; 0 typ; 4 def)
% Number of atoms : 3301 ( 901 equ; 0 cnn)
% Maximal formula atoms : 164 ( 34 avg)
% Number of connectives : 2082 ( 145 ~; 131 |; 525 &; 617 @)
% ( 15 <=>; 649 =>; 0 <=; 0 <~>)
% Maximal formula depth : 171 ( 16 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 317 ( 314 usr; 178 con; 0-4 aty)
% Number of variables : 133 ( 0 ^; 91 !; 42 ?; 133 :)
% 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_147,type,
symdiff: $i > $i > $i ).
thf(func_def_153,type,
iskpair: $i > $o ).
thf(func_def_158,type,
kpair: $i > $i > $i ).
thf(func_def_160,type,
cartprod: $i > $i > $i ).
thf(func_def_177,type,
singleton: $i > $o ).
thf(func_def_179,type,
ex1: $i > ( $i > $o ) > $o ).
thf(func_def_184,type,
atmost1p: $i > $o ).
thf(func_def_185,type,
atleast2p: $i > $o ).
thf(func_def_186,type,
atmost2p: $i > $o ).
thf(func_def_187,type,
upairsetp: $i > $o ).
thf(func_def_202,type,
sP2: $i > $o > $i > $i > $o ).
thf(func_def_204,type,
sP4: $i > $i > $o ).
thf(func_def_205,type,
sP5: $i > $o ).
thf(func_def_206,type,
sP6: $i > $i > $o ).
thf(func_def_207,type,
sP7: $i > $i > $o ).
thf(func_def_212,type,
sK12: $i > $o ).
thf(func_def_215,type,
sK15: $i > ( $i > $o ) > $i ).
thf(func_def_234,type,
sK34: $i > $i ).
thf(func_def_243,type,
sK43: $i > $i ).
thf(func_def_244,type,
sK44: $i > $i > ( $i > $o ) > $i ).
thf(func_def_245,type,
sK45: $i > $o ).
thf(func_def_253,type,
sK53: $i > $i > $i ).
thf(func_def_256,type,
sK56: $i > $i > $i ).
thf(func_def_257,type,
sK57: $i > $i > $i ).
thf(func_def_269,type,
sK69: $i > $o ).
thf(func_def_270,type,
sK70: $i > $o ).
thf(func_def_271,type,
sK71: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_272,type,
sK72: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_278,type,
sK78: $i > $i > $i ).
thf(func_def_284,type,
sK84: $i > $i > $i ).
thf(func_def_294,type,
sK94: $i > $o ).
thf(func_def_296,type,
sK96: ( $i > $o ) > $i ).
thf(func_def_297,type,
sK97: ( $i > $o ) > $i ).
thf(func_def_309,type,
sK109: $i > $i > $i ).
thf(func_def_313,type,
sK113: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_314,type,
sK114: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_315,type,
sK115: $i > $o ).
thf(func_def_316,type,
sK116: $i > $o ).
thf(func_def_317,type,
sK117: $i > $o ).
thf(func_def_331,type,
sK131: $i > $i ).
thf(func_def_333,type,
sK133: $i > $o ).
thf(func_def_336,type,
sK136: ( $i > $o ) > $i > $i ).
thf(func_def_338,type,
sK138: $i > $o ).
thf(func_def_358,type,
sK158: $i > ( $i > $o ) > $i ).
thf(func_def_359,type,
sK159: $i > $o ).
thf(func_def_363,type,
sK163: $i > $i ).
thf(func_def_369,type,
sK169: ( $i > $o ) > $i ).
thf(func_def_370,type,
sK170: $i > $o ).
thf(func_def_371,type,
sK171: $i > $i ).
thf(func_def_382,type,
sK182: $i > $o ).
thf(func_def_396,type,
sK196: $i > $o ).
thf(func_def_400,type,
sK200: $i > $o ).
thf(func_def_402,type,
sK202: ( $i > $o ) > $i > $i ).
thf(func_def_403,type,
sK203: ( $i > $o ) > $i > $i ).
thf(func_def_406,type,
sK206: $i > $o ).
thf(func_def_407,type,
sK207: $i > $o ).
thf(func_def_408,type,
sK208: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_409,type,
sK209: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_414,type,
sK214: $i > $i ).
thf(func_def_416,type,
sK216: $i > $i ).
thf(func_def_423,type,
sK223: $o > $i > $i > $i ).
thf(func_def_424,type,
sK224: $i > $i > $i > $i ).
thf(func_def_425,type,
sK225: $i > $i > $i > $i ).
thf(func_def_429,type,
sK229: ( $i > $o ) > $i ).
thf(func_def_430,type,
sK230: $i > $o ).
thf(func_def_431,type,
sK231: $i > $i ).
thf(func_def_432,type,
sK232: $i > $o ).
thf(func_def_436,type,
sK236: $i > $i > $i ).
thf(func_def_456,type,
sK256: $i > $o ).
thf(func_def_458,type,
sK258: ( $i > $o ) > $i > $i ).
thf(func_def_469,type,
sK269: $i > $o ).
thf(func_def_472,type,
sK272: $i > ( $i > $o ) > $i ).
thf(func_def_476,type,
sK276: $i > $o ).
thf(func_def_478,type,
sK278: $i > ( $i > $o ) > $i ).
thf(func_def_479,type,
sK279: $i > $i > $i ).
thf(func_def_489,type,
sK289: $i > $o ).
thf(func_def_512,type,
sK312: ( $i > $o ) > $i ).
thf(func_def_513,type,
sK313: $i > $o ).
thf(func_def_532,type,
sK332: $i > $o ).
thf(func_def_539,type,
sK339: $i > $i > $o ).
thf(func_def_540,type,
sK340: $i > $i ).
thf(func_def_541,type,
sK341: $i > $i ).
thf(func_def_542,type,
sK342: ( $i > $i > $o ) > $i > $i ).
thf(func_def_543,type,
sK343: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_544,type,
sK344: ( $i > $i > $o ) > $i > $i ).
thf(func_def_551,type,
sK351: $i > $o ).
thf(func_def_557,type,
sK357: ( $i > $o ) > $i > $i ).
thf(func_def_559,type,
sK359: $i > $o ).
thf(func_def_564,type,
sK364: $i > $o ).
thf(func_def_568,type,
sK368: $i > $i ).
thf(func_def_575,type,
sK375: $i > $i > $i ).
thf(func_def_576,type,
sK376: ( $i > $o ) > $i > $i ).
thf(func_def_578,type,
sK378: $i > $o ).
thf(func_def_591,type,
sK391: $i > $i > $i ).
thf(func_def_594,type,
sK394: $i > $o ).
thf(func_def_596,type,
sK396: $i > ( $i > $o ) > $i ).
thf(func_def_597,type,
sK397: $i > $o ).
thf(func_def_604,type,
sK404: $i > $o ).
thf(func_def_607,type,
sK407: $i > ( $i > $o ) > $i ).
thf(func_def_608,type,
sK408: $i > ( $i > $o ) > $i ).
thf(func_def_611,type,
sK411: $i > $o ).
thf(func_def_616,type,
sK416: $i > $i > $i ).
thf(func_def_617,type,
sK417: $i > $i > $i ).
thf(func_def_618,type,
sK418: $i > $i > $i ).
thf(func_def_619,type,
sK419: $i > $i > $i ).
thf(func_def_620,type,
sK420: $i > $i > $i > $i ).
thf(func_def_621,type,
sK421: $i > $i ).
thf(func_def_622,type,
sK422: $i > $i ).
thf(func_def_623,type,
sK423: $i > $i ).
thf(func_def_624,type,
sK424: $i > $i ).
thf(func_def_625,type,
sK425: $i > $i > $i > $i ).
thf(func_def_626,type,
sK426: $i > $i > $i > $i ).
thf(func_def_627,type,
sK427: $i > $i > $i ).
thf(func_def_628,type,
sK428: $i > $i > $i ).
thf(func_def_629,type,
sK429: $i > $i > $i ).
thf(func_def_630,type,
sK430: $i > $i ).
thf(func_def_631,type,
sK431: $i > $i > $i ).
thf(func_def_633,type,
sK433: $i > $i ).
thf(func_def_634,type,
sK434: $i > $i ).
thf(func_def_635,type,
sK435: $i > $o ).
thf(func_def_646,type,
ph446:
!>[X0: $tType] : X0 ).
thf(f2907,plain,
$false,
inference(avatar_sat_refutation,[status(thm)],[f2383,f2611,f2768,f2853,f2906]) ).
thf(f2906,plain,
( spl445_7
| ~ spl445_2 ),
inference(avatar_split_clause,[status(thm)],[f2905,f2361,f2598]) ).
thf(f2598,definition,
( spl445_7
<=> ( sK372 = sK370 ) ),
introduced(definition,[new_symbols(naming,[spl445_7])],[avatar_definition]) ).
thf(f2361,definition,
( spl445_2
<=> ( ( setadjoin @ sK372 @ emptyset )
= ( setadjoin @ sK370 @ emptyset ) ) ),
introduced(definition,[new_symbols(naming,[spl445_2])],[avatar_definition]) ).
thf(f2905,plain,
( ( sK372 = sK370 )
| ~ spl445_2 ),
inference(trivial_inequality_removal,[status(thm)],[f2904]) ).
thf(f2904,plain,
( ( sK372 = sK370 )
| ( $true != $true )
| ~ spl445_2 ),
inference(forward_demodulation,[status(thm)],[f2885,f2045]) ).
thf(f2045,plain,
setunionsingleton = $true,
inference(cnf_transformation,[status(thm)],[f1311]) ).
thf(f1311,plain,
( ( notequalI2 = $true )
& ( powersetAx = $true )
& ( setunionsingleton2 = $true )
& ( binintersectLsub = $true )
& ( emptysetAx = $true )
& ( binintersectSubset3 = $true )
& ( foundationAx = $true )
& ( binunionEcases = $true )
& ( vacuousDall = $true )
& ( quantDeMorgan4 = $true )
& ( descr__Cong = $true )
& ( eqinunit = $true )
& ( notequalI1 = $true )
& ( uniqinunit = $true )
& ( notinsingleton = $true )
& ( singletoninpowunion = $true )
& ( symdiffI1 = $true )
& ( subsetE = $true )
& ( powerset__Cong = $true )
& ( dsetconstr__Cong = $true )
& ( emptyinPowerset = $true )
& ( setbeta = $true )
& ( setminusIRneg = $true )
& ( kpairiskpair = $true )
& ( ( in @ ( setadjoin @ sK370 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK372 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK372 @ ( setadjoin @ sK371 @ emptyset ) ) @ emptyset ) ) )
= $true )
& ( sK372 != sK370 )
& ( setminusILneg = $true )
& ( binintersectSubset2 = $true )
& ( dsetconstrEL = $true )
& ( setext = $true )
& ( setminusSubset1 = $true )
& ( bs114d = $true )
& ( in__Cong = $true )
& ( emptyset__Cong = $true )
& ( subsetRefl = $true )
& ( replAx = $true )
& ( upairsetIR = $true )
& ( setunionE = $true )
& ( inPowerset = $true )
& ( setadjoinIL = $true )
& ( subsetI1 = $true )
& ( exu__Cong = $true )
& ( singletonprop = $true )
& ( noeltsimpempty = $true )
& ( setadjoinE = $true )
& ( setunionI = $true )
& ( subsetemptysetimpeq = $true )
& ( powersetI1 = $true )
& ( emptysetimpfalse = $true )
& ( prop2setE = $true )
& ( binintersectI = $true )
& ( kpairp = $true )
& ( setunionAx = $true )
& ( exuI3 = $true )
& ( subsetTrans = $true )
& ( emptyinunitempty = $true )
& ( setunionsingleton = $true )
& ( setukpairIR = $true )
& ( upairset2E = $true )
& ( emptysetsubset = $true )
& ( binintersectER = $true )
& ( ex1I = $true )
& ( emptysetE = $true )
& ( inCongP = $true )
& ( omega__Cong = $true )
& ( binintersectEL = $true )
& ( setadjoinIR = $true )
& ( ex1I2 = $true )
& ( setunionsingleton1 = $true )
& ( setadjoin__Cong = $true )
& ( nonemptyE1 = $true )
& ( setunion__Cong = $true )
& ( upairset2IR = $true )
& ( binintersectSubset1 = $true )
& ( setminusER = $true )
& ( setminusLsub = $true )
& ( exuI1 = $true )
& ( upairsubunion = $true )
& ( powersetE = $true )
& ( binunionRsub = $true )
& ( symdiffIneg1 = $true )
& ( setadjoinOr = $true )
& ( binunionIR = $true )
& ( powersetE1 = $true )
& ( prop2setI = $true )
& ( singletoninpowerset = $true )
& ( nonemptyI = $true )
& ( singletonsubset = $true )
& ( omegaIndAx = $true )
& ( cartprodpairin = $true )
& ( setadjoinSub = $true )
& ( upairsetIL = $true )
& ( sepSubset = $true )
& ( setukpairIL = $true )
& ( emptyInPowerset = $true )
& ( notsubsetI = $true )
& ( subset2powerset = $true )
& ( eqimpsubset2 = $true )
& ( sepInPowerset = $true )
& ( exuI2 = $true )
& ( omegaSAx = $true )
& ( ex1E1 = $true )
& ( emptyI = $true )
& ( powersetI = $true )
& ( setextsub = $true )
& ( exuE3e = $true )
& ( setextAx = $true )
& ( dsetconstrI = $true )
& ( nonemptyImpWitness = $true )
& ( symdiffI2 = $true )
& ( subsetE2 = $true )
& ( subPowSU = $true )
& ( notinemptyset = $true )
& ( subsetI2 = $true )
& ( ubforcartprodlem3 = $true )
& ( ubforcartprodlem1 = $true )
& ( upairsetE = $true )
& ( binintersectSubset4 = $true )
& ( binintersectRsub = $true )
& ( binunionIL = $true )
& ( symdiffE = $true )
& ( omega0Ax = $true )
& ( singletonsswitch = $true )
& ( cartprodmempair = $true )
& ( setunionE2 = $true )
& ( symdiffIneg2 = $true )
& ( singletonsuniq = $true )
& ( setminusERneg = $true )
& ( dsetconstrER = $true )
& ( setadjoinAx = $true )
& ( quantDeMorgan3 = $true )
& ( setminusSubset2 = $true )
& ( wellorderingAx = $true )
& ( notdexE = $true )
& ( setminusI = $true )
& ( binunionE = $true )
& ( upairinpowunion = $true )
& ( ubforcartprodlem2 = $true )
& ( setminusEL = $true )
& ( setminusELneg = $true )
& ( eqimpsubset1 = $true )
& ( prop2set2propI = $true )
& ( quantDeMorgan2 = $true )
& ( setoftrueEq = $true )
& ( emptyE1 = $true )
& ( exuE1 = $true )
& ( powersetsubset = $true )
& ( exuE2 = $true )
& ( setadjoinSub2 = $true )
& ( exuEu = $true )
& ( nonemptyI1 = $true )
& ( binintersectSubset5 = $true )
& ( cartprodmempair1 = $true )
& ( binunionLsub = $true )
& ( notdallE = $true )
& ( descrp = $true )
& ( exuE3u = $true )
& ( quantDeMorgan1 = $true )
& ( disjointsetsI1 = $true )
& ( secondinupair = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK370,sK371,sK372])],[f548,f1310]) ).
thf(f1310,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ( ( in @ ( setadjoin @ X0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X2 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X2 @ ( setadjoin @ X1 @ emptyset ) ) @ emptyset ) ) )
= $true )
& ( X0 != X2 ) )
=> ( ( ( in @ ( setadjoin @ sK370 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK372 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK372 @ ( setadjoin @ sK371 @ emptyset ) ) @ emptyset ) ) )
= $true )
& ( sK372 != sK370 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f548,plain,
( ( notequalI2 = $true )
& ( powersetAx = $true )
& ( setunionsingleton2 = $true )
& ( binintersectLsub = $true )
& ( emptysetAx = $true )
& ( binintersectSubset3 = $true )
& ( foundationAx = $true )
& ( binunionEcases = $true )
& ( vacuousDall = $true )
& ( quantDeMorgan4 = $true )
& ( descr__Cong = $true )
& ( eqinunit = $true )
& ( notequalI1 = $true )
& ( uniqinunit = $true )
& ( notinsingleton = $true )
& ( singletoninpowunion = $true )
& ( symdiffI1 = $true )
& ( subsetE = $true )
& ( powerset__Cong = $true )
& ( dsetconstr__Cong = $true )
& ( emptyinPowerset = $true )
& ( setbeta = $true )
& ( setminusIRneg = $true )
& ( kpairiskpair = $true )
& ? [X0: $i,X1: $i,X2: $i] :
( ( ( in @ ( setadjoin @ X0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X2 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X2 @ ( setadjoin @ X1 @ emptyset ) ) @ emptyset ) ) )
= $true )
& ( X0 != X2 ) )
& ( setminusILneg = $true )
& ( binintersectSubset2 = $true )
& ( dsetconstrEL = $true )
& ( setext = $true )
& ( setminusSubset1 = $true )
& ( bs114d = $true )
& ( in__Cong = $true )
& ( emptyset__Cong = $true )
& ( subsetRefl = $true )
& ( replAx = $true )
& ( upairsetIR = $true )
& ( setunionE = $true )
& ( inPowerset = $true )
& ( setadjoinIL = $true )
& ( subsetI1 = $true )
& ( exu__Cong = $true )
& ( singletonprop = $true )
& ( noeltsimpempty = $true )
& ( setadjoinE = $true )
& ( setunionI = $true )
& ( subsetemptysetimpeq = $true )
& ( powersetI1 = $true )
& ( emptysetimpfalse = $true )
& ( prop2setE = $true )
& ( binintersectI = $true )
& ( kpairp = $true )
& ( setunionAx = $true )
& ( exuI3 = $true )
& ( subsetTrans = $true )
& ( emptyinunitempty = $true )
& ( setunionsingleton = $true )
& ( setukpairIR = $true )
& ( upairset2E = $true )
& ( emptysetsubset = $true )
& ( binintersectER = $true )
& ( ex1I = $true )
& ( emptysetE = $true )
& ( inCongP = $true )
& ( omega__Cong = $true )
& ( binintersectEL = $true )
& ( setadjoinIR = $true )
& ( ex1I2 = $true )
& ( setunionsingleton1 = $true )
& ( setadjoin__Cong = $true )
& ( nonemptyE1 = $true )
& ( setunion__Cong = $true )
& ( upairset2IR = $true )
& ( binintersectSubset1 = $true )
& ( setminusER = $true )
& ( setminusLsub = $true )
& ( exuI1 = $true )
& ( upairsubunion = $true )
& ( powersetE = $true )
& ( binunionRsub = $true )
& ( symdiffIneg1 = $true )
& ( setadjoinOr = $true )
& ( binunionIR = $true )
& ( powersetE1 = $true )
& ( prop2setI = $true )
& ( singletoninpowerset = $true )
& ( nonemptyI = $true )
& ( singletonsubset = $true )
& ( omegaIndAx = $true )
& ( cartprodpairin = $true )
& ( setadjoinSub = $true )
& ( upairsetIL = $true )
& ( sepSubset = $true )
& ( setukpairIL = $true )
& ( emptyInPowerset = $true )
& ( notsubsetI = $true )
& ( subset2powerset = $true )
& ( eqimpsubset2 = $true )
& ( sepInPowerset = $true )
& ( exuI2 = $true )
& ( omegaSAx = $true )
& ( ex1E1 = $true )
& ( emptyI = $true )
& ( powersetI = $true )
& ( setextsub = $true )
& ( exuE3e = $true )
& ( setextAx = $true )
& ( dsetconstrI = $true )
& ( nonemptyImpWitness = $true )
& ( symdiffI2 = $true )
& ( subsetE2 = $true )
& ( subPowSU = $true )
& ( notinemptyset = $true )
& ( subsetI2 = $true )
& ( ubforcartprodlem3 = $true )
& ( ubforcartprodlem1 = $true )
& ( upairsetE = $true )
& ( binintersectSubset4 = $true )
& ( binintersectRsub = $true )
& ( binunionIL = $true )
& ( symdiffE = $true )
& ( omega0Ax = $true )
& ( singletonsswitch = $true )
& ( cartprodmempair = $true )
& ( setunionE2 = $true )
& ( symdiffIneg2 = $true )
& ( singletonsuniq = $true )
& ( setminusERneg = $true )
& ( dsetconstrER = $true )
& ( setadjoinAx = $true )
& ( quantDeMorgan3 = $true )
& ( setminusSubset2 = $true )
& ( wellorderingAx = $true )
& ( notdexE = $true )
& ( setminusI = $true )
& ( binunionE = $true )
& ( upairinpowunion = $true )
& ( ubforcartprodlem2 = $true )
& ( setminusEL = $true )
& ( setminusELneg = $true )
& ( eqimpsubset1 = $true )
& ( prop2set2propI = $true )
& ( quantDeMorgan2 = $true )
& ( setoftrueEq = $true )
& ( emptyE1 = $true )
& ( exuE1 = $true )
& ( powersetsubset = $true )
& ( exuE2 = $true )
& ( setadjoinSub2 = $true )
& ( exuEu = $true )
& ( nonemptyI1 = $true )
& ( binintersectSubset5 = $true )
& ( cartprodmempair1 = $true )
& ( binunionLsub = $true )
& ( notdallE = $true )
& ( descrp = $true )
& ( exuE3u = $true )
& ( quantDeMorgan1 = $true )
& ( disjointsetsI1 = $true )
& ( secondinupair = $true ) ),
inference(flattening,[status(thm)],[f547]) ).
thf(f547,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ( ( in @ ( setadjoin @ X0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X2 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X2 @ ( setadjoin @ X1 @ emptyset ) ) @ emptyset ) ) )
= $true )
& ( X0 != X2 ) )
& ( singletonsuniq = $true )
& ( ex1I2 = $true )
& ( ex1I = $true )
& ( ex1E1 = $true )
& ( singletonprop = $true )
& ( setunionsingleton = $true )
& ( setunionsingleton2 = $true )
& ( setunionsingleton1 = $true )
& ( setunionE2 = $true )
& ( cartprodmempair = $true )
& ( cartprodmempair1 = $true )
& ( cartprodpairin = $true )
& ( ubforcartprodlem3 = $true )
& ( ubforcartprodlem2 = $true )
& ( ubforcartprodlem1 = $true )
& ( upairinpowunion = $true )
& ( upairsubunion = $true )
& ( upairset2E = $true )
& ( singletoninpowunion = $true )
& ( singletoninpowerset = $true )
& ( singletonsubset = $true )
& ( kpairp = $true )
& ( kpairiskpair = $true )
& ( setukpairIR = $true )
& ( setukpairIL = $true )
& ( secondinupair = $true )
& ( symdiffIneg2 = $true )
& ( symdiffIneg1 = $true )
& ( symdiffI2 = $true )
& ( symdiffI1 = $true )
& ( symdiffE = $true )
& ( setminusSubset1 = $true )
& ( setminusLsub = $true )
& ( setminusIRneg = $true )
& ( setminusILneg = $true )
& ( setminusELneg = $true )
& ( setminusERneg = $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)],[f477]) ).
thf(f477,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 )
=> ( ( setminusERneg = $true )
=> ( ( setminusELneg = $true )
=> ( ( setminusILneg = $true )
=> ( ( setminusIRneg = $true )
=> ( ( setminusLsub = $true )
=> ( ( setminusSubset1 = $true )
=> ( ( symdiffE = $true )
=> ( ( symdiffI1 = $true )
=> ( ( symdiffI2 = $true )
=> ( ( symdiffIneg1 = $true )
=> ( ( symdiffIneg2 = $true )
=> ( ( secondinupair = $true )
=> ( ( setukpairIL = $true )
=> ( ( setukpairIR = $true )
=> ( ( kpairiskpair = $true )
=> ( ( kpairp = $true )
=> ( ( singletonsubset = $true )
=> ( ( singletoninpowerset = $true )
=> ( ( singletoninpowunion = $true )
=> ( ( upairset2E = $true )
=> ( ( upairsubunion = $true )
=> ( ( upairinpowunion = $true )
=> ( ( ubforcartprodlem1 = $true )
=> ( ( ubforcartprodlem2 = $true )
=> ( ( ubforcartprodlem3 = $true )
=> ( ( cartprodpairin = $true )
=> ( ( cartprodmempair1 = $true )
=> ( ( cartprodmempair = $true )
=> ( ( setunionE2 = $true )
=> ( ( setunionsingleton1 = $true )
=> ( ( setunionsingleton2 = $true )
=> ( ( setunionsingleton = $true )
=> ( ( singletonprop = $true )
=> ( ( ex1E1 = $true )
=> ( ( ex1I = $true )
=> ( ( ex1I2 = $true )
=> ( ( singletonsuniq = $true )
=> ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ ( setadjoin @ X0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X2 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X2 @ ( setadjoin @ X1 @ emptyset ) ) @ emptyset ) ) )
= $true )
=> ( X0 = X2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f476]) ).
thf(f476,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
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> ( setunionE2
=> ( setunionsingleton1
=> ( setunionsingleton2
=> ( setunionsingleton
=> ( singletonprop
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> ! [X0: $i,X1: $i,X2: $i] :
( ( in @ ( setadjoin @ X0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X2 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X2 @ ( setadjoin @ X1 @ emptyset ) ) @ emptyset ) ) )
=> ( X0 = X2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f166]) ).
thf(f166,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
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> ( setunionE2
=> ( setunionsingleton1
=> ( setunionsingleton2
=> ( setunionsingleton
=> ( singletonprop
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> ! [X8: $i,X2: $i,X1: $i] :
( ( in @ ( setadjoin @ X8 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X8 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f165]) ).
thf(f165,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
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> ( setunionE2
=> ( setunionsingleton1
=> ( setunionsingleton2
=> ( setunionsingleton
=> ( singletonprop
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> ! [X8: $i,X2: $i,X1: $i] :
( ( in @ ( setadjoin @ X8 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X8 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.vWjcIjdJ0D/DTF2THF_13509.p',setukpairinjL1) ).
thf(f2885,plain,
( ( sK372 = sK370 )
| ( setunionsingleton != $true )
| ~ spl445_2 ),
inference(superposition,[status(thm)],[f1597,f2692]) ).
thf(f2692,plain,
( ( sK372
= ( setunion @ ( setadjoin @ sK370 @ emptyset ) ) )
| ~ spl445_2 ),
inference(trivial_inequality_removal,[status(thm)],[f2691]) ).
thf(f2691,plain,
( ( sK372
= ( setunion @ ( setadjoin @ sK370 @ emptyset ) ) )
| ( $true != $true )
| ~ spl445_2 ),
inference(forward_demodulation,[status(thm)],[f2630,f2045]) ).
thf(f2630,plain,
( ( setunionsingleton != $true )
| ( sK372
= ( setunion @ ( setadjoin @ sK370 @ emptyset ) ) )
| ~ spl445_2 ),
inference(superposition,[status(thm)],[f1597,f2363]) ).
thf(f2363,plain,
( ( ( setadjoin @ sK372 @ emptyset )
= ( setadjoin @ sK370 @ emptyset ) )
| ~ spl445_2 ),
inference(avatar_component_clause,[status(thm)],[f2361]) ).
thf(f1597,plain,
! [X0: $i] :
( ( ( setunion @ ( setadjoin @ X0 @ emptyset ) )
= X0 )
| ( setunionsingleton != $true ) ),
inference(cnf_transformation,[status(thm)],[f903]) ).
thf(f903,plain,
( ( ! [X0: $i] :
( ( setunion @ ( setadjoin @ X0 @ emptyset ) )
= X0 )
| ( setunionsingleton != $true ) )
& ( ( setunionsingleton = $true )
| ( ( setunion @ ( setadjoin @ sK124 @ emptyset ) )
!= sK124 ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK124])],[f901,f902]) ).
thf(f902,plain,
( ? [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
!= X1 )
=> ( ( setunion @ ( setadjoin @ sK124 @ emptyset ) )
!= sK124 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f901,plain,
( ( ! [X0: $i] :
( ( setunion @ ( setadjoin @ X0 @ emptyset ) )
= X0 )
| ( setunionsingleton != $true ) )
& ( ( setunionsingleton = $true )
| ? [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
!= X1 ) ) ),
inference(rectify,[status(thm)],[f900]) ).
thf(f900,plain,
( ( ! [X0: $i] :
( ( setunion @ ( setadjoin @ X0 @ emptyset ) )
= X0 )
| ( setunionsingleton != $true ) )
& ( ( setunionsingleton = $true )
| ? [X0: $i] :
( ( setunion @ ( setadjoin @ X0 @ emptyset ) )
!= X0 ) ) ),
inference(nnf_transformation,[status(thm)],[f235]) ).
thf(f235,plain,
( ! [X0: $i] :
( ( setunion @ ( setadjoin @ X0 @ emptyset ) )
= X0 )
<=> ( setunionsingleton = $true ) ),
inference(fool_elimination,[status(thm)],[f234]) ).
thf(f234,plain,
! [X0: $i] :
( ( ( setunion @ ( setadjoin @ X0 @ emptyset ) )
= X0 )
= setunionsingleton ),
inference(rectify,[status(thm)],[f157]) ).
thf(f157,axiom,
! [X1: $i] :
( ( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
= setunionsingleton ),
file('/export/starexec/sandbox/tmp/tmp.vWjcIjdJ0D/DTF2THF_13509.p',setunionsingleton) ).
thf(f2853,plain,
~ spl445_7,
inference(avatar_contradiction_clause,[status(thm)],[f2852]) ).
thf(f2852,plain,
( $false
| ~ spl445_7 ),
inference(trivial_inequality_removal,[status(thm)],[f2851]) ).
thf(f2851,plain,
( ( sK370 != sK370 )
| ~ spl445_7 ),
inference(superposition,[status(thm)],[f2076,f2600]) ).
thf(f2600,plain,
( ( sK372 = sK370 )
| ~ spl445_7 ),
inference(avatar_component_clause,[status(thm)],[f2598]) ).
thf(f2076,plain,
sK372 != sK370,
inference(cnf_transformation,[status(thm)],[f1311]) ).
thf(f2768,plain,
( spl445_4
| ~ spl445_1 ),
inference(avatar_split_clause,[status(thm)],[f2767,f2357,f2371]) ).
thf(f2371,definition,
( spl445_4
<=> ( ( setadjoin @ sK372 @ ( setadjoin @ sK371 @ emptyset ) )
= ( setadjoin @ sK370 @ emptyset ) ) ),
introduced(definition,[new_symbols(naming,[spl445_4])],[avatar_definition]) ).
thf(f2357,definition,
( spl445_1
<=> ( ( in @ ( setadjoin @ sK370 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK372 @ ( setadjoin @ sK371 @ emptyset ) ) @ emptyset ) )
= $true ) ),
introduced(definition,[new_symbols(naming,[spl445_1])],[avatar_definition]) ).
thf(f2767,plain,
( ( ( setadjoin @ sK372 @ ( setadjoin @ sK371 @ emptyset ) )
= ( setadjoin @ sK370 @ emptyset ) )
| ~ spl445_1 ),
inference(trivial_inequality_removal,[status(thm)],[f2766]) ).
thf(f2766,plain,
( ( ( setadjoin @ sK372 @ ( setadjoin @ sK371 @ emptyset ) )
= ( setadjoin @ sK370 @ emptyset ) )
| ( $true != $true )
| ~ spl445_1 ),
inference(forward_demodulation,[status(thm)],[f2742,f2087]) ).
thf(f2087,plain,
notinsingleton = $true,
inference(cnf_transformation,[status(thm)],[f1311]) ).
thf(f2742,plain,
( ( ( setadjoin @ sK372 @ ( setadjoin @ sK371 @ emptyset ) )
= ( setadjoin @ sK370 @ emptyset ) )
| ( notinsingleton != $true )
| ~ spl445_1 ),
inference(trivial_inequality_removal,[status(thm)],[f2736]) ).
thf(f2736,plain,
( ( $true != $true )
| ( notinsingleton != $true )
| ( ( setadjoin @ sK372 @ ( setadjoin @ sK371 @ emptyset ) )
= ( setadjoin @ sK370 @ emptyset ) )
| ~ spl445_1 ),
inference(superposition,[status(thm)],[f1543,f2359]) ).
thf(f2359,plain,
( ( ( in @ ( setadjoin @ sK370 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK372 @ ( setadjoin @ sK371 @ emptyset ) ) @ emptyset ) )
= $true )
| ~ spl445_1 ),
inference(avatar_component_clause,[status(thm)],[f2357]) ).
thf(f1543,plain,
! [X0: $i,X1: $i] :
( ( $true
!= ( in @ X1 @ ( setadjoin @ X0 @ emptyset ) ) )
| ( X0 = X1 )
| ( notinsingleton != $true ) ),
inference(cnf_transformation,[status(thm)],[f831]) ).
thf(f831,plain,
( ( ! [X0: $i,X1: $i] :
( ( $true
!= ( in @ X1 @ ( setadjoin @ X0 @ emptyset ) ) )
| ( X0 = X1 ) )
| ( notinsingleton != $true ) )
& ( ( notinsingleton = $true )
| ( ( $true
= ( in @ sK83 @ ( setadjoin @ sK82 @ emptyset ) ) )
& ( sK82 != sK83 ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK82,sK83])],[f829,f830]) ).
thf(f830,plain,
( ? [X2: $i,X3: $i] :
( ( ( in @ X3 @ ( setadjoin @ X2 @ emptyset ) )
= $true )
& ( X2 != X3 ) )
=> ( ( $true
= ( in @ sK83 @ ( setadjoin @ sK82 @ emptyset ) ) )
& ( sK82 != sK83 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f829,plain,
( ( ! [X0: $i,X1: $i] :
( ( $true
!= ( in @ X1 @ ( setadjoin @ X0 @ emptyset ) ) )
| ( X0 = X1 ) )
| ( notinsingleton != $true ) )
& ( ( notinsingleton = $true )
| ? [X2: $i,X3: $i] :
( ( ( in @ X3 @ ( setadjoin @ X2 @ emptyset ) )
= $true )
& ( X2 != X3 ) ) ) ),
inference(rectify,[status(thm)],[f828]) ).
thf(f828,plain,
( ( ! [X0: $i,X1: $i] :
( ( $true
!= ( in @ X1 @ ( setadjoin @ X0 @ emptyset ) ) )
| ( X0 = X1 ) )
| ( notinsingleton != $true ) )
& ( ( notinsingleton = $true )
| ? [X0: $i,X1: $i] :
( ( $true
= ( in @ X1 @ ( setadjoin @ X0 @ emptyset ) ) )
& ( X0 != X1 ) ) ) ),
inference(nnf_transformation,[status(thm)],[f579]) ).
thf(f579,plain,
( ! [X0: $i,X1: $i] :
( ( $true
!= ( in @ X1 @ ( setadjoin @ X0 @ emptyset ) ) )
| ( X0 = X1 ) )
<=> ( notinsingleton = $true ) ),
inference(ennf_transformation,[status(thm)],[f513]) ).
thf(f513,plain,
( ( notinsingleton = $true )
<=> ! [X0: $i,X1: $i] :
( ( X0 != X1 )
=> ( $true
!= ( in @ X1 @ ( setadjoin @ X0 @ emptyset ) ) ) ) ),
inference(flattening,[status(thm)],[f376]) ).
thf(f376,plain,
( ! [X0: $i,X1: $i] :
( ( X0 != X1 )
=> ( $true
!= ( in @ X1 @ ( setadjoin @ X0 @ emptyset ) ) ) )
<=> ( notinsingleton = $true ) ),
inference(fool_elimination,[status(thm)],[f375]) ).
thf(f375,plain,
! [X0: $i,X1: $i] :
( ( ( X0 != X1 )
=> ~ ( in @ X1 @ ( setadjoin @ X0 @ emptyset ) ) )
= notinsingleton ),
inference(rectify,[status(thm)],[f47]) ).
thf(f47,axiom,
! [X1: $i,X2: $i] :
( ( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
= notinsingleton ),
file('/export/starexec/sandbox/tmp/tmp.vWjcIjdJ0D/DTF2THF_13509.p',notinsingleton) ).
thf(f2611,plain,
( spl445_7
| ~ spl445_4 ),
inference(avatar_split_clause,[status(thm)],[f2610,f2371,f2598]) ).
thf(f2610,plain,
( ( sK372 = sK370 )
| ~ spl445_4 ),
inference(trivial_inequality_removal,[status(thm)],[f2609]) ).
thf(f2609,plain,
( ( sK372 = sK370 )
| ( $true != $true )
| ~ spl445_4 ),
inference(forward_demodulation,[status(thm)],[f2586,f2088]) ).
thf(f2088,plain,
uniqinunit = $true,
inference(cnf_transformation,[status(thm)],[f1311]) ).
thf(f2586,plain,
( ( sK372 = sK370 )
| ( uniqinunit != $true )
| ~ spl445_4 ),
inference(trivial_inequality_removal,[status(thm)],[f2580]) ).
thf(f2580,plain,
( ( uniqinunit != $true )
| ( $true != $true )
| ( sK372 = sK370 )
| ~ spl445_4 ),
inference(superposition,[status(thm)],[f1532,f2539]) ).
thf(f2539,plain,
( ( $true
= ( in @ sK372 @ ( setadjoin @ sK370 @ emptyset ) ) )
| ~ spl445_4 ),
inference(trivial_inequality_removal,[status(thm)],[f2538]) ).
thf(f2538,plain,
( ( $true != $true )
| ( $true
= ( in @ sK372 @ ( setadjoin @ sK370 @ emptyset ) ) )
| ~ spl445_4 ),
inference(forward_demodulation,[status(thm)],[f2521,f2062]) ).
thf(f2062,plain,
setadjoinIL = $true,
inference(cnf_transformation,[status(thm)],[f1311]) ).
thf(f2521,plain,
( ( setadjoinIL != $true )
| ( $true
= ( in @ sK372 @ ( setadjoin @ sK370 @ emptyset ) ) )
| ~ spl445_4 ),
inference(superposition,[status(thm)],[f1459,f2373]) ).
thf(f2373,plain,
( ( ( setadjoin @ sK372 @ ( setadjoin @ sK371 @ emptyset ) )
= ( setadjoin @ sK370 @ emptyset ) )
| ~ spl445_4 ),
inference(avatar_component_clause,[status(thm)],[f2371]) ).
thf(f1459,plain,
! [X0: $i,X1: $i] :
( ( $true
= ( in @ X1 @ ( setadjoin @ X1 @ X0 ) ) )
| ( setadjoinIL != $true ) ),
inference(cnf_transformation,[status(thm)],[f728]) ).
thf(f728,plain,
( ( ! [X0: $i,X1: $i] :
( $true
= ( in @ X1 @ ( setadjoin @ X1 @ X0 ) ) )
| ( setadjoinIL != $true ) )
& ( ( setadjoinIL = $true )
| ( ( in @ sK21 @ ( setadjoin @ sK21 @ sK20 ) )
!= $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK20,sK21])],[f726,f727]) ).
thf(f727,plain,
( ? [X2: $i,X3: $i] :
( $true
!= ( in @ X3 @ ( setadjoin @ X3 @ X2 ) ) )
=> ( ( in @ sK21 @ ( setadjoin @ sK21 @ sK20 ) )
!= $true ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f726,plain,
( ( ! [X0: $i,X1: $i] :
( $true
= ( in @ X1 @ ( setadjoin @ X1 @ X0 ) ) )
| ( setadjoinIL != $true ) )
& ( ( setadjoinIL = $true )
| ? [X2: $i,X3: $i] :
( $true
!= ( in @ X3 @ ( setadjoin @ X3 @ X2 ) ) ) ) ),
inference(rectify,[status(thm)],[f725]) ).
thf(f725,plain,
( ( ! [X0: $i,X1: $i] :
( $true
= ( in @ X1 @ ( setadjoin @ X1 @ X0 ) ) )
| ( setadjoinIL != $true ) )
& ( ( setadjoinIL = $true )
| ? [X0: $i,X1: $i] :
( $true
!= ( in @ X1 @ ( setadjoin @ X1 @ X0 ) ) ) ) ),
inference(nnf_transformation,[status(thm)],[f417]) ).
thf(f417,plain,
( ! [X0: $i,X1: $i] :
( $true
= ( in @ X1 @ ( setadjoin @ X1 @ X0 ) ) )
<=> ( setadjoinIL = $true ) ),
inference(fool_elimination,[status(thm)],[f416]) ).
thf(f416,plain,
! [X0: $i,X1: $i] :
( ( in @ X1 @ ( setadjoin @ X1 @ X0 ) )
= setadjoinIL ),
inference(rectify,[status(thm)],[f31]) ).
thf(f31,axiom,
! [X2: $i,X1: $i] :
( ( in @ X1 @ ( setadjoin @ X1 @ X2 ) )
= setadjoinIL ),
file('/export/starexec/sandbox/tmp/tmp.vWjcIjdJ0D/DTF2THF_13509.p',setadjoinIL) ).
thf(f1532,plain,
! [X2: $i,X3: $i] :
( ( $true
!= ( in @ X2 @ ( setadjoin @ X3 @ emptyset ) ) )
| ( uniqinunit != $true )
| ( X2 = X3 ) ),
inference(cnf_transformation,[status(thm)],[f821]) ).
thf(f821,plain,
( ( ( uniqinunit = $true )
| ( ( ( in @ sK76 @ ( setadjoin @ sK77 @ emptyset ) )
= $true )
& ( sK77 != sK76 ) ) )
& ( ! [X2: $i,X3: $i] :
( ( $true
!= ( in @ X2 @ ( setadjoin @ X3 @ emptyset ) ) )
| ( X2 = X3 ) )
| ( uniqinunit != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK76,sK77])],[f819,f820]) ).
thf(f820,plain,
( ? [X0: $i,X1: $i] :
( ( $true
= ( in @ X0 @ ( setadjoin @ X1 @ emptyset ) ) )
& ( X0 != X1 ) )
=> ( ( ( in @ sK76 @ ( setadjoin @ sK77 @ emptyset ) )
= $true )
& ( sK77 != sK76 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f819,plain,
( ( ( uniqinunit = $true )
| ? [X0: $i,X1: $i] :
( ( $true
= ( in @ X0 @ ( setadjoin @ X1 @ emptyset ) ) )
& ( X0 != X1 ) ) )
& ( ! [X2: $i,X3: $i] :
( ( $true
!= ( in @ X2 @ ( setadjoin @ X3 @ emptyset ) ) )
| ( X2 = X3 ) )
| ( uniqinunit != $true ) ) ),
inference(rectify,[status(thm)],[f818]) ).
thf(f818,plain,
( ( ( uniqinunit = $true )
| ? [X0: $i,X1: $i] :
( ( $true
= ( in @ X0 @ ( setadjoin @ X1 @ emptyset ) ) )
& ( X0 != X1 ) ) )
& ( ! [X0: $i,X1: $i] :
( ( $true
!= ( in @ X0 @ ( setadjoin @ X1 @ emptyset ) ) )
| ( X0 = X1 ) )
| ( uniqinunit != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f682]) ).
thf(f682,plain,
( ( uniqinunit = $true )
<=> ! [X0: $i,X1: $i] :
( ( $true
!= ( in @ X0 @ ( setadjoin @ X1 @ emptyset ) ) )
| ( X0 = X1 ) ) ),
inference(ennf_transformation,[status(thm)],[f319]) ).
thf(f319,plain,
( ! [X0: $i,X1: $i] :
( ( $true
= ( in @ X0 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( X0 = X1 ) )
<=> ( uniqinunit = $true ) ),
inference(fool_elimination,[status(thm)],[f318]) ).
thf(f318,plain,
( uniqinunit
= ( ! [X0: $i,X1: $i] :
( ( in @ X0 @ ( setadjoin @ X1 @ emptyset ) )
=> ( X0 = X1 ) ) ) ),
inference(rectify,[status(thm)],[f46]) ).
thf(f46,axiom,
( uniqinunit
= ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.vWjcIjdJ0D/DTF2THF_13509.p',uniqinunit) ).
thf(f2383,plain,
( spl445_1
| spl445_2 ),
inference(avatar_split_clause,[status(thm)],[f2382,f2361,f2357]) ).
thf(f2382,plain,
( ( ( in @ ( setadjoin @ sK370 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK372 @ ( setadjoin @ sK371 @ emptyset ) ) @ emptyset ) )
= $true )
| ( ( setadjoin @ sK372 @ emptyset )
= ( setadjoin @ sK370 @ emptyset ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f2381]) ).
thf(f2381,plain,
( ( $true != $true )
| ( ( setadjoin @ sK372 @ emptyset )
= ( setadjoin @ sK370 @ emptyset ) )
| ( ( in @ ( setadjoin @ sK370 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK372 @ ( setadjoin @ sK371 @ emptyset ) ) @ emptyset ) )
= $true ) ),
inference(forward_demodulation,[status(thm)],[f2346,f1972]) ).
thf(f1972,plain,
setadjoinAx = $true,
inference(cnf_transformation,[status(thm)],[f1311]) ).
thf(f2346,plain,
( ( ( in @ ( setadjoin @ sK370 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK372 @ ( setadjoin @ sK371 @ emptyset ) ) @ emptyset ) )
= $true )
| ( ( setadjoin @ sK372 @ emptyset )
= ( setadjoin @ sK370 @ emptyset ) )
| ( setadjoinAx != $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f2342]) ).
thf(f2342,plain,
( ( ( in @ ( setadjoin @ sK370 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK372 @ ( setadjoin @ sK371 @ emptyset ) ) @ emptyset ) )
= $true )
| ( setadjoinAx != $true )
| ( ( setadjoin @ sK372 @ emptyset )
= ( setadjoin @ sK370 @ emptyset ) )
| ( $true != $true ) ),
inference(superposition,[status(thm)],[f2220,f2077]) ).
thf(f2077,plain,
( ( in @ ( setadjoin @ sK370 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK372 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK372 @ ( setadjoin @ sK371 @ emptyset ) ) @ emptyset ) ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f1311]) ).
thf(f2220,plain,
! [X2: $i,X0: $i,X1: $i] :
( ( $true
!= ( in @ X0 @ ( setadjoin @ X2 @ X1 ) ) )
| ( $true
= ( in @ X0 @ X1 ) )
| ( X0 = X2 )
| ( setadjoinAx != $true ) ),
inference(cnf_transformation,[status(thm)],[f1432]) ).
thf(f1432,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( $true
= ( in @ X0 @ X1 ) )
| ( X0 = X2 )
| ( $true
!= ( in @ X0 @ ( setadjoin @ X2 @ X1 ) ) ) )
& ( ( $true
= ( in @ X0 @ ( setadjoin @ X2 @ X1 ) ) )
| ( ( $true
!= ( in @ X0 @ X1 ) )
& ( X0 != X2 ) ) ) )
| ( setadjoinAx != $true ) )
& ( ( setadjoinAx = $true )
| ( ( ( $true
!= ( in @ sK438 @ ( setadjoin @ sK440 @ sK439 ) ) )
| ( ( $true
!= ( in @ sK438 @ sK439 ) )
& ( sK438 != sK440 ) ) )
& ( ( $true
= ( in @ sK438 @ ( setadjoin @ sK440 @ sK439 ) ) )
| ( $true
= ( in @ sK438 @ sK439 ) )
| ( sK438 = sK440 ) ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK438,sK439,sK440])],[f1430,f1431]) ).
thf(f1431,plain,
( ? [X3: $i,X4: $i,X5: $i] :
( ( ( $true
!= ( in @ X3 @ ( setadjoin @ X5 @ X4 ) ) )
| ( ( $true
!= ( in @ X3 @ X4 ) )
& ( X3 != X5 ) ) )
& ( ( $true
= ( in @ X3 @ ( setadjoin @ X5 @ X4 ) ) )
| ( $true
= ( in @ X3 @ X4 ) )
| ( X3 = X5 ) ) )
=> ( ( ( $true
!= ( in @ sK438 @ ( setadjoin @ sK440 @ sK439 ) ) )
| ( ( $true
!= ( in @ sK438 @ sK439 ) )
& ( sK438 != sK440 ) ) )
& ( ( $true
= ( in @ sK438 @ ( setadjoin @ sK440 @ sK439 ) ) )
| ( $true
= ( in @ sK438 @ sK439 ) )
| ( sK438 = sK440 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1430,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( $true
= ( in @ X0 @ X1 ) )
| ( X0 = X2 )
| ( $true
!= ( in @ X0 @ ( setadjoin @ X2 @ X1 ) ) ) )
& ( ( $true
= ( in @ X0 @ ( setadjoin @ X2 @ X1 ) ) )
| ( ( $true
!= ( in @ X0 @ X1 ) )
& ( X0 != X2 ) ) ) )
| ( setadjoinAx != $true ) )
& ( ( setadjoinAx = $true )
| ? [X3: $i,X4: $i,X5: $i] :
( ( ( $true
!= ( in @ X3 @ ( setadjoin @ X5 @ X4 ) ) )
| ( ( $true
!= ( in @ X3 @ X4 ) )
& ( X3 != X5 ) ) )
& ( ( $true
= ( in @ X3 @ ( setadjoin @ X5 @ X4 ) ) )
| ( $true
= ( in @ X3 @ X4 ) )
| ( X3 = X5 ) ) ) ) ),
inference(rectify,[status(thm)],[f1429]) ).
thf(f1429,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( $true
= ( in @ X0 @ X1 ) )
| ( X0 = X2 )
| ( $true
!= ( in @ X0 @ ( setadjoin @ X2 @ X1 ) ) ) )
& ( ( $true
= ( in @ X0 @ ( setadjoin @ X2 @ X1 ) ) )
| ( ( $true
!= ( in @ X0 @ X1 ) )
& ( X0 != X2 ) ) ) )
| ( setadjoinAx != $true ) )
& ( ( setadjoinAx = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( ( $true
!= ( in @ X0 @ ( setadjoin @ X2 @ X1 ) ) )
| ( ( $true
!= ( in @ X0 @ X1 ) )
& ( X0 != X2 ) ) )
& ( ( $true
= ( in @ X0 @ ( setadjoin @ X2 @ X1 ) ) )
| ( $true
= ( in @ X0 @ X1 ) )
| ( X0 = X2 ) ) ) ) ),
inference(flattening,[status(thm)],[f1428]) ).
thf(f1428,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( $true
= ( in @ X0 @ X1 ) )
| ( X0 = X2 )
| ( $true
!= ( in @ X0 @ ( setadjoin @ X2 @ X1 ) ) ) )
& ( ( $true
= ( in @ X0 @ ( setadjoin @ X2 @ X1 ) ) )
| ( ( $true
!= ( in @ X0 @ X1 ) )
& ( X0 != X2 ) ) ) )
| ( setadjoinAx != $true ) )
& ( ( setadjoinAx = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( ( $true
!= ( in @ X0 @ ( setadjoin @ X2 @ X1 ) ) )
| ( ( $true
!= ( in @ X0 @ X1 ) )
& ( X0 != X2 ) ) )
& ( ( $true
= ( in @ X0 @ ( setadjoin @ X2 @ X1 ) ) )
| ( $true
= ( in @ X0 @ X1 ) )
| ( X0 = X2 ) ) ) ) ),
inference(nnf_transformation,[status(thm)],[f295]) ).
thf(f295,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( ( $true
= ( in @ X0 @ X1 ) )
| ( X0 = X2 ) )
<=> ( $true
= ( in @ X0 @ ( setadjoin @ X2 @ X1 ) ) ) )
<=> ( setadjoinAx = $true ) ),
inference(fool_elimination,[status(thm)],[f294]) ).
thf(f294,plain,
! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X0 @ ( setadjoin @ X2 @ X1 ) )
<=> ( ( X0 = X2 )
| ( in @ X0 @ X1 ) ) )
= setadjoinAx ),
inference(rectify,[status(thm)],[f4]) ).
thf(f4,axiom,
! [X2: $i,X3: $i,X1: $i] :
( ( ( in @ X2 @ ( setadjoin @ X1 @ X3 ) )
<=> ( ( X1 = X2 )
| ( in @ X2 @ X3 ) ) )
= setadjoinAx ),
file('/export/starexec/sandbox/tmp/tmp.vWjcIjdJ0D/DTF2THF_13509.p',setadjoinAx) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SEU642^1 : TPTP v9.2.1. Released v3.7.0.
% 0.11/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.16/0.34 % Computer : n027.cluster.edu
% 0.16/0.34 % Model : x86_64 x86_64
% 0.16/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.34 % Memory : 8042.1875MB
% 0.16/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.34 % CPULimit : 300
% 0.16/0.34 % WCLimit : 300
% 0.16/0.34 % DateTime : Tue Feb 24 12:30:07 EST 2026
% 0.16/0.34 % CPUTime :
% 0.16/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.21/0.46 ---- Original DTF file ---
% 0.21/0.46 thf(spec,logic,$$dhol).
% 0.21/0.46 %------------------------------------------------------------------------------
% 0.21/0.46 % File : SEU642^1 : TPTP v9.2.1. Released v3.7.0.
% 0.21/0.46 % Domain : Set Theory
% 0.21/0.46 % Problem : Ordered Pairs - Properties of Pairs
% 0.21/0.46 % Version : Especial.
% 0.21/0.46 % English : (! x:i.! y:i.! z:i.in (setadjoin z emptyset) (setadjoin
% 0.21/0.46 % (setadjoin x emptyset) (setadjoin (setadjoin x
% 0.21/0.46 % (setadjoin y emptyset)) emptyset)) -> x = z)
% 0.21/0.46
% 0.21/0.46 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.21/0.46 % Source : [Bro08]
% 0.21/0.46 % Names : ZFC144g [Bro08]
% 0.21/0.46
% 0.21/0.46 % Status : Theorem
% 0.21/0.46 % Rating : 0.22 v9.1.0, 0.12 v9.0.0, 0.20 v8.2.0, 0.31 v8.1.0, 0.27 v7.5.0, 0.29 v7.4.0, 0.56 v7.2.0, 0.50 v7.1.0, 0.62 v7.0.0, 0.71 v6.4.0, 0.83 v6.3.0, 0.80 v6.2.0, 0.86 v5.5.0, 1.00 v5.4.0, 0.80 v4.1.0, 1.00 v3.7.0
% 0.21/0.46 % Syntax : Number of formulae : 352 ( 164 unt; 187 typ; 164 def)
% 0.21/0.46 % Number of atoms : 871 ( 233 equ; 0 cnn)
% 0.21/0.46 % Maximal formula atoms : 160 ( 5 avg)
% 0.21/0.46 % Number of connectives : 1514 ( 47 ~; 7 |; 37 &;1022 @)
% 0.21/0.46 % ( 14 <=>; 387 =>; 0 <=; 0 <~>)
% 0.21/0.46 % Maximal formula depth : 170 ( 2 avg)
% 0.21/0.46 % Number of types : 2 ( 0 usr)
% 0.21/0.46 % Number of type conns : 82 ( 82 >; 0 *; 0 +; 0 <<)
% 0.21/0.46 % Number of symbols : 190 ( 187 usr; 162 con; 0-2 aty)
% 0.21/0.46 % Number of variables : 480 ( 41 ^; 401 !; 38 ?; 480 :)
% 0.21/0.46 % SPC : TH0_THM_EQU_NAR
% 0.21/0.46
% 0.21/0.46 % Comments : http://mathgate.info/detsetitem.php?id=198
% 0.21/0.46 %------------------------------------------------------------------------------
% 0.21/0.46 thf(in_type,type,
% 0.21/0.46 in: $i > $i > $o ).
% 0.21/0.46
% 0.21/0.46 thf(exu_type,type,
% 0.21/0.46 exu: ( $i > $o ) > $o ).
% 0.21/0.46
% 0.21/0.46 thf(exu,definition,
% 0.21/0.46 ( exu
% 0.21/0.46 = ( ^ [Xphi: $i > $o] :
% 0.21/0.46 ? [Xx: $i] :
% 0.21/0.46 ( ( Xphi @ Xx )
% 0.21/0.46 & ! [Xy: $i] :
% 0.21/0.46 ( ( Xphi @ Xy )
% 0.21/0.46 => ( Xx = Xy ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(setextAx_type,type,
% 0.21/0.46 setextAx: $o ).
% 0.21/0.46
% 0.21/0.46 thf(setextAx,definition,
% 0.21/0.46 ( setextAx
% 0.21/0.46 = ( ! [A: $i,B: $i] :
% 0.21/0.46 ( ! [Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ A )
% 0.21/0.46 <=> ( in @ Xx @ B ) )
% 0.21/0.46 => ( A = B ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(emptyset_type,type,
% 0.21/0.46 emptyset: $i ).
% 0.21/0.46
% 0.21/0.46 thf(emptysetAx_type,type,
% 0.21/0.46 emptysetAx: $o ).
% 0.21/0.46
% 0.21/0.46 thf(emptysetAx,definition,
% 0.21/0.46 ( emptysetAx
% 0.21/0.46 = ( ! [Xx: $i] :
% 0.21/0.46 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(setadjoin_type,type,
% 0.21/0.46 setadjoin: $i > $i > $i ).
% 0.21/0.46
% 0.21/0.46 thf(setadjoinAx_type,type,
% 0.21/0.46 setadjoinAx: $o ).
% 0.21/0.46
% 0.21/0.46 thf(setadjoinAx,definition,
% 0.21/0.46 ( setadjoinAx
% 0.21/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.21/0.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.21/0.46 <=> ( ( Xy = Xx )
% 0.21/0.46 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(powerset_type,type,
% 0.21/0.46 powerset: $i > $i ).
% 0.21/0.46
% 0.21/0.46 thf(powersetAx_type,type,
% 0.21/0.46 powersetAx: $o ).
% 0.21/0.46
% 0.21/0.46 thf(powersetAx,definition,
% 0.21/0.46 ( powersetAx
% 0.21/0.46 = ( ! [A: $i,B: $i] :
% 0.21/0.46 ( ( in @ B @ ( powerset @ A ) )
% 0.21/0.46 <=> ! [Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ B )
% 0.21/0.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(setunion_type,type,
% 0.21/0.46 setunion: $i > $i ).
% 0.21/0.46
% 0.21/0.46 thf(setunionAx_type,type,
% 0.21/0.46 setunionAx: $o ).
% 0.21/0.46
% 0.21/0.46 thf(setunionAx,definition,
% 0.21/0.46 ( setunionAx
% 0.21/0.46 = ( ! [A: $i,Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ ( setunion @ A ) )
% 0.21/0.46 <=> ? [B: $i] :
% 0.21/0.46 ( ( in @ Xx @ B )
% 0.21/0.46 & ( in @ B @ A ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(omega_type,type,
% 0.21/0.46 omega: $i ).
% 0.21/0.46
% 0.21/0.46 thf(omega0Ax_type,type,
% 0.21/0.46 omega0Ax: $o ).
% 0.21/0.46
% 0.21/0.46 thf(omega0Ax,definition,
% 0.21/0.46 ( omega0Ax
% 0.21/0.46 = ( in @ emptyset @ omega ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(omegaSAx_type,type,
% 0.21/0.46 omegaSAx: $o ).
% 0.21/0.46
% 0.21/0.46 thf(omegaSAx,definition,
% 0.21/0.46 ( omegaSAx
% 0.21/0.46 = ( ! [Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ omega )
% 0.21/0.46 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(omegaIndAx_type,type,
% 0.21/0.46 omegaIndAx: $o ).
% 0.21/0.46
% 0.21/0.46 thf(omegaIndAx,definition,
% 0.21/0.46 ( omegaIndAx
% 0.21/0.46 = ( ! [A: $i] :
% 0.21/0.46 ( ( ( in @ emptyset @ A )
% 0.21/0.46 & ! [Xx: $i] :
% 0.21/0.46 ( ( ( in @ Xx @ omega )
% 0.21/0.46 & ( in @ Xx @ A ) )
% 0.21/0.46 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.21/0.46 => ! [Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ omega )
% 0.21/0.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(replAx_type,type,
% 0.21/0.46 replAx: $o ).
% 0.21/0.46
% 0.21/0.46 thf(replAx,definition,
% 0.21/0.46 ( replAx
% 0.21/0.46 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.21/0.46 ( ! [Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ A )
% 0.21/0.46 => ( exu
% 0.21/0.46 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.21/0.46 => ? [B: $i] :
% 0.21/0.46 ! [Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ B )
% 0.21/0.46 <=> ? [Xy: $i] :
% 0.21/0.46 ( ( in @ Xy @ A )
% 0.21/0.46 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(foundationAx_type,type,
% 0.21/0.46 foundationAx: $o ).
% 0.21/0.46
% 0.21/0.46 thf(foundationAx,definition,
% 0.21/0.46 ( foundationAx
% 0.21/0.46 = ( ! [A: $i] :
% 0.21/0.46 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.21/0.46 => ? [B: $i] :
% 0.21/0.46 ( ( in @ B @ A )
% 0.21/0.46 & ~ ? [Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ B )
% 0.21/0.46 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(wellorderingAx_type,type,
% 0.21/0.46 wellorderingAx: $o ).
% 0.21/0.46
% 0.21/0.46 thf(wellorderingAx,definition,
% 0.21/0.46 ( wellorderingAx
% 0.21/0.46 = ( ! [A: $i] :
% 0.21/0.46 ? [B: $i] :
% 0.21/0.46 ( ! [C: $i] :
% 0.21/0.46 ( ( in @ C @ B )
% 0.21/0.46 => ! [Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ C )
% 0.21/0.46 => ( in @ Xx @ A ) ) )
% 0.21/0.46 & ! [Xx: $i,Xy: $i] :
% 0.21/0.46 ( ( ( in @ Xx @ A )
% 0.21/0.46 & ( in @ Xy @ A ) )
% 0.21/0.46 => ( ! [C: $i] :
% 0.21/0.46 ( ( in @ C @ B )
% 0.21/0.46 => ( ( in @ Xx @ C )
% 0.21/0.46 <=> ( in @ Xy @ C ) ) )
% 0.21/0.46 => ( Xx = Xy ) ) )
% 0.21/0.46 & ! [C: $i,D: $i] :
% 0.21/0.46 ( ( ( in @ C @ B )
% 0.21/0.46 & ( in @ D @ B ) )
% 0.21/0.46 => ( ! [Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ C )
% 0.21/0.46 => ( in @ Xx @ D ) )
% 0.21/0.46 | ! [Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ D )
% 0.21/0.46 => ( in @ Xx @ C ) ) ) )
% 0.21/0.46 & ! [C: $i] :
% 0.21/0.46 ( ( ! [Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ C )
% 0.21/0.46 => ( in @ Xx @ A ) )
% 0.21/0.46 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.21/0.46 => ? [D: $i,Xx: $i] :
% 0.21/0.46 ( ( in @ D @ B )
% 0.21/0.46 & ( in @ Xx @ C )
% 0.21/0.46 & ~ ? [Xy: $i] :
% 0.21/0.46 ( ( in @ Xy @ D )
% 0.21/0.46 & ( in @ Xy @ C ) )
% 0.21/0.46 & ! [E: $i] :
% 0.21/0.46 ( ( in @ E @ B )
% 0.21/0.46 => ( ! [Xy: $i] :
% 0.21/0.46 ( ( in @ Xy @ E )
% 0.21/0.46 => ( in @ Xy @ D ) )
% 0.21/0.46 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(descr_type,type,
% 0.21/0.46 descr: ( $i > $o ) > $i ).
% 0.21/0.46
% 0.21/0.46 thf(descrp_type,type,
% 0.21/0.46 descrp: $o ).
% 0.21/0.46
% 0.21/0.46 thf(descrp,definition,
% 0.21/0.46 ( descrp
% 0.21/0.46 = ( ! [Xphi: $i > $o] :
% 0.21/0.46 ( ( exu
% 0.21/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.21/0.46 => ( Xphi
% 0.21/0.46 @ ( descr
% 0.21/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(dsetconstr_type,type,
% 0.21/0.46 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.21/0.46
% 0.21/0.46 thf(dsetconstrI_type,type,
% 0.21/0.46 dsetconstrI: $o ).
% 0.21/0.46
% 0.21/0.46 thf(dsetconstrI,definition,
% 0.21/0.46 ( dsetconstrI
% 0.21/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ A )
% 0.21/0.46 => ( ( Xphi @ Xx )
% 0.21/0.46 => ( in @ Xx
% 0.21/0.46 @ ( dsetconstr @ A
% 0.21/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(dsetconstrEL_type,type,
% 0.21/0.46 dsetconstrEL: $o ).
% 0.21/0.46
% 0.21/0.46 thf(dsetconstrEL,definition,
% 0.21/0.46 ( dsetconstrEL
% 0.21/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.21/0.46 ( ( in @ Xx
% 0.21/0.46 @ ( dsetconstr @ A
% 0.21/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.21/0.46 => ( in @ Xx @ A ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(dsetconstrER_type,type,
% 0.21/0.46 dsetconstrER: $o ).
% 0.21/0.46
% 0.21/0.46 thf(dsetconstrER,definition,
% 0.21/0.46 ( dsetconstrER
% 0.21/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.21/0.46 ( ( in @ Xx
% 0.21/0.46 @ ( dsetconstr @ A
% 0.21/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.21/0.46 => ( Xphi @ Xx ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(exuE1_type,type,
% 0.21/0.46 exuE1: $o ).
% 0.21/0.46
% 0.21/0.46 thf(exuE1,definition,
% 0.21/0.46 ( exuE1
% 0.21/0.46 = ( ! [Xphi: $i > $o] :
% 0.21/0.46 ( ( exu
% 0.21/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.21/0.46 => ? [Xx: $i] :
% 0.21/0.46 ( ( Xphi @ Xx )
% 0.21/0.46 & ! [Xy: $i] :
% 0.21/0.46 ( ( Xphi @ Xy )
% 0.21/0.46 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(prop2set_type,type,
% 0.21/0.46 prop2set: $o > $i ).
% 0.21/0.46
% 0.21/0.46 thf(prop2setE_type,type,
% 0.21/0.46 prop2setE: $o ).
% 0.21/0.46
% 0.21/0.46 thf(prop2setE,definition,
% 0.21/0.46 ( prop2setE
% 0.21/0.46 = ( ! [Xphi: $o,Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.21/0.46 => Xphi ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(emptysetE_type,type,
% 0.21/0.46 emptysetE: $o ).
% 0.21/0.46
% 0.21/0.46 thf(emptysetE,definition,
% 0.21/0.46 ( emptysetE
% 0.21/0.46 = ( ! [Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ emptyset )
% 0.21/0.46 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(emptysetimpfalse_type,type,
% 0.21/0.46 emptysetimpfalse: $o ).
% 0.21/0.46
% 0.21/0.46 thf(emptysetimpfalse,definition,
% 0.21/0.46 ( emptysetimpfalse
% 0.21/0.46 = ( ! [Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ emptyset )
% 0.21/0.46 => $false ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(notinemptyset_type,type,
% 0.21/0.46 notinemptyset: $o ).
% 0.21/0.46
% 0.21/0.46 thf(notinemptyset,definition,
% 0.21/0.46 ( notinemptyset
% 0.21/0.46 = ( ! [Xx: $i] :
% 0.21/0.46 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(exuE3e_type,type,
% 0.21/0.46 exuE3e: $o ).
% 0.21/0.46
% 0.21/0.46 thf(exuE3e,definition,
% 0.21/0.46 ( exuE3e
% 0.21/0.46 = ( ! [Xphi: $i > $o] :
% 0.21/0.46 ( ( exu
% 0.21/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.21/0.46 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(setext_type,type,
% 0.21/0.46 setext: $o ).
% 0.21/0.46
% 0.21/0.46 thf(setext,definition,
% 0.21/0.46 ( setext
% 0.21/0.46 = ( ! [A: $i,B: $i] :
% 0.21/0.46 ( ! [Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ A )
% 0.21/0.46 => ( in @ Xx @ B ) )
% 0.21/0.46 => ( ! [Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ B )
% 0.21/0.46 => ( in @ Xx @ A ) )
% 0.21/0.46 => ( A = B ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(emptyI_type,type,
% 0.21/0.46 emptyI: $o ).
% 0.21/0.46
% 0.21/0.46 thf(emptyI,definition,
% 0.21/0.46 ( emptyI
% 0.21/0.46 = ( ! [A: $i] :
% 0.21/0.46 ( ! [Xx: $i] :
% 0.21/0.46 ~ ( in @ Xx @ A )
% 0.21/0.46 => ( A = emptyset ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(noeltsimpempty_type,type,
% 0.21/0.46 noeltsimpempty: $o ).
% 0.21/0.46
% 0.21/0.46 thf(noeltsimpempty,definition,
% 0.21/0.46 ( noeltsimpempty
% 0.21/0.46 = ( ! [A: $i] :
% 0.21/0.46 ( ! [Xx: $i] :
% 0.21/0.46 ~ ( in @ Xx @ A )
% 0.21/0.46 => ( A = emptyset ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(setbeta_type,type,
% 0.21/0.46 setbeta: $o ).
% 0.21/0.46
% 0.21/0.46 thf(setbeta,definition,
% 0.21/0.46 ( setbeta
% 0.21/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ A )
% 0.21/0.46 => ( ( in @ Xx
% 0.21/0.46 @ ( dsetconstr @ A
% 0.21/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.21/0.46 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(nonempty_type,type,
% 0.21/0.46 nonempty: $i > $o ).
% 0.21/0.46
% 0.21/0.46 thf(nonempty,definition,
% 0.21/0.46 ( nonempty
% 0.21/0.46 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(nonemptyE1_type,type,
% 0.21/0.46 nonemptyE1: $o ).
% 0.21/0.46
% 0.21/0.46 thf(nonemptyE1,definition,
% 0.21/0.46 ( nonemptyE1
% 0.21/0.46 = ( ! [A: $i] :
% 0.21/0.46 ( ( nonempty @ A )
% 0.21/0.46 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(nonemptyI_type,type,
% 0.21/0.46 nonemptyI: $o ).
% 0.21/0.46
% 0.21/0.46 thf(nonemptyI,definition,
% 0.21/0.46 ( nonemptyI
% 0.21/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ A )
% 0.21/0.46 => ( ( Xphi @ Xx )
% 0.21/0.46 => ( nonempty
% 0.21/0.46 @ ( dsetconstr @ A
% 0.21/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(nonemptyI1_type,type,
% 0.21/0.46 nonemptyI1: $o ).
% 0.21/0.46
% 0.21/0.46 thf(nonemptyI1,definition,
% 0.21/0.46 ( nonemptyI1
% 0.21/0.46 = ( ! [A: $i] :
% 0.21/0.46 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.21/0.46 => ( nonempty @ A ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(setadjoinIL_type,type,
% 0.21/0.46 setadjoinIL: $o ).
% 0.21/0.46
% 0.21/0.46 thf(setadjoinIL,definition,
% 0.21/0.46 ( setadjoinIL
% 0.21/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(emptyinunitempty_type,type,
% 0.21/0.46 emptyinunitempty: $o ).
% 0.21/0.46
% 0.21/0.46 thf(emptyinunitempty,definition,
% 0.21/0.46 ( emptyinunitempty
% 0.21/0.46 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(setadjoinIR_type,type,
% 0.21/0.46 setadjoinIR: $o ).
% 0.21/0.46
% 0.21/0.46 thf(setadjoinIR,definition,
% 0.21/0.46 ( setadjoinIR
% 0.21/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.21/0.46 ( ( in @ Xy @ A )
% 0.21/0.46 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(setadjoinE_type,type,
% 0.21/0.46 setadjoinE: $o ).
% 0.21/0.46
% 0.21/0.46 thf(setadjoinE,definition,
% 0.21/0.46 ( setadjoinE
% 0.21/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.21/0.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.21/0.46 => ! [Xphi: $o] :
% 0.21/0.46 ( ( ( Xy = Xx )
% 0.21/0.46 => Xphi )
% 0.21/0.46 => ( ( ( in @ Xy @ A )
% 0.21/0.46 => Xphi )
% 0.21/0.46 => Xphi ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(setadjoinOr_type,type,
% 0.21/0.46 setadjoinOr: $o ).
% 0.21/0.46
% 0.21/0.46 thf(setadjoinOr,definition,
% 0.21/0.46 ( setadjoinOr
% 0.21/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.21/0.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.21/0.46 => ( ( Xy = Xx )
% 0.21/0.46 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(setoftrueEq_type,type,
% 0.21/0.46 setoftrueEq: $o ).
% 0.21/0.46
% 0.21/0.46 thf(setoftrueEq,definition,
% 0.21/0.46 ( setoftrueEq
% 0.21/0.46 = ( ! [A: $i] :
% 0.21/0.46 ( ( dsetconstr @ A
% 0.21/0.46 @ ^ [Xx: $i] : $true )
% 0.21/0.46 = A ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(powersetI_type,type,
% 0.21/0.46 powersetI: $o ).
% 0.21/0.46
% 0.21/0.46 thf(powersetI,definition,
% 0.21/0.46 ( powersetI
% 0.21/0.46 = ( ! [A: $i,B: $i] :
% 0.21/0.46 ( ! [Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ B )
% 0.21/0.46 => ( in @ Xx @ A ) )
% 0.21/0.46 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(emptyinPowerset_type,type,
% 0.21/0.46 emptyinPowerset: $o ).
% 0.21/0.46
% 0.21/0.46 thf(emptyinPowerset,definition,
% 0.21/0.46 ( emptyinPowerset
% 0.21/0.46 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(emptyInPowerset_type,type,
% 0.21/0.46 emptyInPowerset: $o ).
% 0.21/0.46
% 0.21/0.46 thf(emptyInPowerset,definition,
% 0.21/0.46 ( emptyInPowerset
% 0.21/0.46 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(powersetE_type,type,
% 0.21/0.46 powersetE: $o ).
% 0.21/0.46
% 0.21/0.46 thf(powersetE,definition,
% 0.21/0.46 ( powersetE
% 0.21/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.21/0.46 ( ( in @ B @ ( powerset @ A ) )
% 0.21/0.46 => ( ( in @ Xx @ B )
% 0.21/0.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(setunionI_type,type,
% 0.21/0.46 setunionI: $o ).
% 0.21/0.46
% 0.21/0.46 thf(setunionI,definition,
% 0.21/0.46 ( setunionI
% 0.21/0.46 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.21/0.46 ( ( in @ Xx @ B )
% 0.21/0.46 => ( ( in @ B @ A )
% 0.21/0.46 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(setunionE_type,type,
% 0.21/0.46 setunionE: $o ).
% 0.21/0.46
% 0.21/0.46 thf(setunionE,definition,
% 0.21/0.46 ( setunionE
% 0.21/0.46 = ( ! [A: $i,Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ ( setunion @ A ) )
% 0.21/0.46 => ! [Xphi: $o] :
% 0.21/0.46 ( ! [B: $i] :
% 0.21/0.46 ( ( in @ Xx @ B )
% 0.21/0.46 => ( ( in @ B @ A )
% 0.21/0.46 => Xphi ) )
% 0.21/0.46 => Xphi ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(subPowSU_type,type,
% 0.21/0.46 subPowSU: $o ).
% 0.21/0.46
% 0.21/0.46 thf(subPowSU,definition,
% 0.21/0.46 ( subPowSU
% 0.21/0.46 = ( ! [A: $i,Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ A )
% 0.21/0.46 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(exuE2_type,type,
% 0.21/0.46 exuE2: $o ).
% 0.21/0.46
% 0.21/0.46 thf(exuE2,definition,
% 0.21/0.46 ( exuE2
% 0.21/0.46 = ( ! [Xphi: $i > $o] :
% 0.21/0.46 ( ( exu
% 0.21/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.21/0.46 => ? [Xx: $i] :
% 0.21/0.46 ! [Xy: $i] :
% 0.21/0.46 ( ( Xphi @ Xy )
% 0.21/0.46 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(nonemptyImpWitness_type,type,
% 0.21/0.46 nonemptyImpWitness: $o ).
% 0.21/0.46
% 0.21/0.46 thf(nonemptyImpWitness,definition,
% 0.21/0.46 ( nonemptyImpWitness
% 0.21/0.46 = ( ! [A: $i] :
% 0.21/0.46 ( ( nonempty @ A )
% 0.21/0.46 => ? [Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ A )
% 0.21/0.46 & $true ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(uniqinunit_type,type,
% 0.21/0.46 uniqinunit: $o ).
% 0.21/0.46
% 0.21/0.46 thf(uniqinunit,definition,
% 0.21/0.46 ( uniqinunit
% 0.21/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.21/0.46 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.21/0.46 => ( Xx = Xy ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(notinsingleton_type,type,
% 0.21/0.46 notinsingleton: $o ).
% 0.21/0.46
% 0.21/0.46 thf(notinsingleton,definition,
% 0.21/0.46 ( notinsingleton
% 0.21/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.21/0.46 ( ( Xx != Xy )
% 0.21/0.46 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(eqinunit_type,type,
% 0.21/0.46 eqinunit: $o ).
% 0.21/0.46
% 0.21/0.46 thf(eqinunit,definition,
% 0.21/0.46 ( eqinunit
% 0.21/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.21/0.46 ( ( Xx = Xy )
% 0.21/0.46 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(singletonsswitch_type,type,
% 0.21/0.46 singletonsswitch: $o ).
% 0.21/0.46
% 0.21/0.46 thf(singletonsswitch,definition,
% 0.21/0.46 ( singletonsswitch
% 0.21/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.21/0.46 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.21/0.46 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(upairsetE_type,type,
% 0.21/0.46 upairsetE: $o ).
% 0.21/0.46
% 0.21/0.46 thf(upairsetE,definition,
% 0.21/0.46 ( upairsetE
% 0.21/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.21/0.46 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.21/0.46 => ( ( Xz = Xx )
% 0.21/0.46 | ( Xz = Xy ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(upairsetIL_type,type,
% 0.21/0.46 upairsetIL: $o ).
% 0.21/0.46
% 0.21/0.46 thf(upairsetIL,definition,
% 0.21/0.46 ( upairsetIL
% 0.21/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(upairsetIR_type,type,
% 0.21/0.46 upairsetIR: $o ).
% 0.21/0.46
% 0.21/0.46 thf(upairsetIR,definition,
% 0.21/0.46 ( upairsetIR
% 0.21/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(emptyE1_type,type,
% 0.21/0.46 emptyE1: $o ).
% 0.21/0.46
% 0.21/0.46 thf(emptyE1,definition,
% 0.21/0.46 ( emptyE1
% 0.21/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.21/0.46 ( ? [Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ A )
% 0.21/0.46 & ( Xphi @ Xx ) )
% 0.21/0.46 => ( ( ( dsetconstr @ A
% 0.21/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.21/0.46 = emptyset )
% 0.21/0.46 => $false ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(vacuousDall_type,type,
% 0.21/0.46 vacuousDall: $o ).
% 0.21/0.46
% 0.21/0.46 thf(vacuousDall,definition,
% 0.21/0.46 ( vacuousDall
% 0.21/0.46 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ emptyset )
% 0.21/0.46 => ( Xphi @ Xx ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(quantDeMorgan1_type,type,
% 0.21/0.46 quantDeMorgan1: $o ).
% 0.21/0.46
% 0.21/0.46 thf(quantDeMorgan1,definition,
% 0.21/0.46 ( quantDeMorgan1
% 0.21/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.21/0.46 ( ~ ! [Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ A )
% 0.21/0.46 => ( Xphi @ Xx ) )
% 0.21/0.46 => ? [Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ A )
% 0.21/0.46 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(quantDeMorgan2_type,type,
% 0.21/0.46 quantDeMorgan2: $o ).
% 0.21/0.46
% 0.21/0.46 thf(quantDeMorgan2,definition,
% 0.21/0.46 ( quantDeMorgan2
% 0.21/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.21/0.46 ( ! [Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ A )
% 0.21/0.46 => ~ ( Xphi @ Xx ) )
% 0.21/0.46 => ~ ? [Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ A )
% 0.21/0.46 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(quantDeMorgan3_type,type,
% 0.21/0.46 quantDeMorgan3: $o ).
% 0.21/0.46
% 0.21/0.46 thf(quantDeMorgan3,definition,
% 0.21/0.46 ( quantDeMorgan3
% 0.21/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.21/0.46 ( ~ ? [Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ A )
% 0.21/0.46 & ( Xphi @ Xx ) )
% 0.21/0.46 => ! [Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ A )
% 0.21/0.46 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(quantDeMorgan4_type,type,
% 0.21/0.46 quantDeMorgan4: $o ).
% 0.21/0.46
% 0.21/0.46 thf(quantDeMorgan4,definition,
% 0.21/0.46 ( quantDeMorgan4
% 0.21/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.21/0.46 ( ? [Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ A )
% 0.21/0.46 & ~ ( Xphi @ Xx ) )
% 0.21/0.46 => ~ ! [Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ A )
% 0.21/0.46 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(prop2setI_type,type,
% 0.21/0.46 prop2setI: $o ).
% 0.21/0.46
% 0.21/0.46 thf(prop2setI,definition,
% 0.21/0.46 ( prop2setI
% 0.21/0.46 = ( ! [Xphi: $o] :
% 0.21/0.46 ( Xphi
% 0.21/0.46 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(set2prop_type,type,
% 0.21/0.46 set2prop: $i > $o ).
% 0.21/0.46
% 0.21/0.46 thf(prop2set2propI_type,type,
% 0.21/0.46 prop2set2propI: $o ).
% 0.21/0.46
% 0.21/0.46 thf(prop2set2propI,definition,
% 0.21/0.46 ( prop2set2propI
% 0.21/0.46 = ( ! [Xphi: $o] :
% 0.21/0.46 ( Xphi
% 0.21/0.46 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(notdexE_type,type,
% 0.21/0.46 notdexE: $o ).
% 0.21/0.46
% 0.21/0.46 thf(notdexE,definition,
% 0.21/0.46 ( notdexE
% 0.21/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.21/0.46 ( ~ ? [Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ A )
% 0.21/0.46 & ( Xphi @ Xx ) )
% 0.21/0.46 => ! [Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ A )
% 0.21/0.46 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(notdallE_type,type,
% 0.21/0.46 notdallE: $o ).
% 0.21/0.46
% 0.21/0.46 thf(notdallE,definition,
% 0.21/0.46 ( notdallE
% 0.21/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.21/0.46 ( ~ ! [Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ A )
% 0.21/0.46 => ( Xphi @ Xx ) )
% 0.21/0.46 => ? [Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ A )
% 0.21/0.46 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(exuI1_type,type,
% 0.21/0.46 exuI1: $o ).
% 0.21/0.46
% 0.21/0.46 thf(exuI1,definition,
% 0.21/0.46 ( exuI1
% 0.21/0.46 = ( ! [Xphi: $i > $o] :
% 0.21/0.46 ( ? [Xx: $i] :
% 0.21/0.46 ( ( Xphi @ Xx )
% 0.21/0.46 & ! [Xy: $i] :
% 0.21/0.46 ( ( Xphi @ Xy )
% 0.21/0.46 => ( Xx = Xy ) ) )
% 0.21/0.46 => ( exu
% 0.21/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(exuI3_type,type,
% 0.21/0.46 exuI3: $o ).
% 0.21/0.46
% 0.21/0.46 thf(exuI3,definition,
% 0.21/0.46 ( exuI3
% 0.21/0.46 = ( ! [Xphi: $i > $o] :
% 0.21/0.46 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.21/0.46 => ( ! [Xx: $i,Xy: $i] :
% 0.21/0.46 ( ( Xphi @ Xx )
% 0.21/0.46 => ( ( Xphi @ Xy )
% 0.21/0.46 => ( Xx = Xy ) ) )
% 0.21/0.46 => ( exu
% 0.21/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(exuI2_type,type,
% 0.21/0.46 exuI2: $o ).
% 0.21/0.46
% 0.21/0.46 thf(exuI2,definition,
% 0.21/0.46 ( exuI2
% 0.21/0.46 = ( ! [Xphi: $i > $o] :
% 0.21/0.46 ( ? [Xx: $i] :
% 0.21/0.46 ! [Xy: $i] :
% 0.21/0.46 ( ( Xphi @ Xy )
% 0.21/0.46 <=> ( Xy = Xx ) )
% 0.21/0.46 => ( exu
% 0.21/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(inCongP_type,type,
% 0.21/0.46 inCongP: $o ).
% 0.21/0.46
% 0.21/0.46 thf(inCongP,definition,
% 0.21/0.46 ( inCongP
% 0.21/0.46 = ( ! [A: $i,B: $i] :
% 0.21/0.46 ( ( A = B )
% 0.21/0.46 => ! [Xx: $i,Xy: $i] :
% 0.21/0.46 ( ( Xx = Xy )
% 0.21/0.46 => ( ( in @ Xx @ A )
% 0.21/0.46 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(in__Cong_type,type,
% 0.21/0.46 in__Cong: $o ).
% 0.21/0.46
% 0.21/0.46 thf(in__Cong,definition,
% 0.21/0.46 ( in__Cong
% 0.21/0.46 = ( ! [A: $i,B: $i] :
% 0.21/0.46 ( ( A = B )
% 0.21/0.46 => ! [Xx: $i,Xy: $i] :
% 0.21/0.46 ( ( Xx = Xy )
% 0.21/0.46 => ( ( in @ Xx @ A )
% 0.21/0.46 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(exuE3u_type,type,
% 0.21/0.46 exuE3u: $o ).
% 0.21/0.46
% 0.21/0.46 thf(exuE3u,definition,
% 0.21/0.46 ( exuE3u
% 0.21/0.46 = ( ! [Xphi: $i > $o] :
% 0.21/0.46 ( ( exu
% 0.21/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.21/0.46 => ! [Xx: $i,Xy: $i] :
% 0.21/0.46 ( ( Xphi @ Xx )
% 0.21/0.46 => ( ( Xphi @ Xy )
% 0.21/0.46 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(exu__Cong_type,type,
% 0.21/0.46 exu__Cong: $o ).
% 0.21/0.46
% 0.21/0.46 thf(exu__Cong,definition,
% 0.21/0.46 ( exu__Cong
% 0.21/0.46 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.21/0.46 ( ! [Xx: $i,Xy: $i] :
% 0.21/0.46 ( ( Xx = Xy )
% 0.21/0.46 => ( ( Xphi @ Xx )
% 0.21/0.46 <=> ( Xpsi @ Xy ) ) )
% 0.21/0.46 => ( ( exu
% 0.21/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.21/0.46 <=> ( exu
% 0.21/0.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(emptyset__Cong_type,type,
% 0.21/0.46 emptyset__Cong: $o ).
% 0.21/0.46
% 0.21/0.46 thf(emptyset__Cong,definition,
% 0.21/0.46 ( emptyset__Cong
% 0.21/0.46 = ( emptyset = emptyset ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(setadjoin__Cong_type,type,
% 0.21/0.46 setadjoin__Cong: $o ).
% 0.21/0.46
% 0.21/0.46 thf(setadjoin__Cong,definition,
% 0.21/0.46 ( setadjoin__Cong
% 0.21/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.21/0.46 ( ( Xx = Xy )
% 0.21/0.46 => ! [Xz: $i,Xu: $i] :
% 0.21/0.46 ( ( Xz = Xu )
% 0.21/0.46 => ( ( setadjoin @ Xx @ Xz )
% 0.21/0.46 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(powerset__Cong_type,type,
% 0.21/0.46 powerset__Cong: $o ).
% 0.21/0.46
% 0.21/0.46 thf(powerset__Cong,definition,
% 0.21/0.46 ( powerset__Cong
% 0.21/0.46 = ( ! [A: $i,B: $i] :
% 0.21/0.46 ( ( A = B )
% 0.21/0.46 => ( ( powerset @ A )
% 0.21/0.46 = ( powerset @ B ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(setunion__Cong_type,type,
% 0.21/0.46 setunion__Cong: $o ).
% 0.21/0.46
% 0.21/0.46 thf(setunion__Cong,definition,
% 0.21/0.46 ( setunion__Cong
% 0.21/0.46 = ( ! [A: $i,B: $i] :
% 0.21/0.46 ( ( A = B )
% 0.21/0.46 => ( ( setunion @ A )
% 0.21/0.46 = ( setunion @ B ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(omega__Cong_type,type,
% 0.21/0.46 omega__Cong: $o ).
% 0.21/0.46
% 0.21/0.46 thf(omega__Cong,definition,
% 0.21/0.46 ( omega__Cong
% 0.21/0.46 = ( omega = omega ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(exuEu_type,type,
% 0.21/0.46 exuEu: $o ).
% 0.21/0.46
% 0.21/0.46 thf(exuEu,definition,
% 0.21/0.46 ( exuEu
% 0.21/0.46 = ( ! [Xphi: $i > $o] :
% 0.21/0.46 ( ( exu
% 0.21/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.21/0.46 => ! [Xx: $i,Xy: $i] :
% 0.21/0.46 ( ( Xphi @ Xx )
% 0.21/0.46 => ( ( Xphi @ Xy )
% 0.21/0.46 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(descr__Cong_type,type,
% 0.21/0.46 descr__Cong: $o ).
% 0.21/0.46
% 0.21/0.46 thf(descr__Cong,definition,
% 0.21/0.46 ( descr__Cong
% 0.21/0.46 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.21/0.46 ( ! [Xx: $i,Xy: $i] :
% 0.21/0.46 ( ( Xx = Xy )
% 0.21/0.46 => ( ( Xphi @ Xx )
% 0.21/0.46 <=> ( Xpsi @ Xy ) ) )
% 0.21/0.46 => ( ( exu
% 0.21/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.21/0.46 => ( ( exu
% 0.21/0.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.21/0.46 => ( ( descr
% 0.21/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.21/0.46 = ( descr
% 0.21/0.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(dsetconstr__Cong_type,type,
% 0.21/0.46 dsetconstr__Cong: $o ).
% 0.21/0.46
% 0.21/0.46 thf(dsetconstr__Cong,definition,
% 0.21/0.46 ( dsetconstr__Cong
% 0.21/0.46 = ( ! [A: $i,B: $i] :
% 0.21/0.46 ( ( A = B )
% 0.21/0.46 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.21/0.46 ( ! [Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ A )
% 0.21/0.46 => ! [Xy: $i] :
% 0.21/0.46 ( ( in @ Xy @ B )
% 0.21/0.46 => ( ( Xx = Xy )
% 0.21/0.46 => ( ( Xphi @ Xx )
% 0.21/0.46 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.21/0.46 => ( ( dsetconstr @ A
% 0.21/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.21/0.46 = ( dsetconstr @ B
% 0.21/0.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(subset_type,type,
% 0.21/0.46 subset: $i > $i > $o ).
% 0.21/0.46
% 0.21/0.46 thf(disjoint_type,type,
% 0.21/0.46 disjoint: $i > $i > $o ).
% 0.21/0.46
% 0.21/0.46 thf(setsmeet_type,type,
% 0.21/0.46 setsmeet: $i > $i > $o ).
% 0.21/0.46
% 0.21/0.46 thf(subsetI1_type,type,
% 0.21/0.46 subsetI1: $o ).
% 0.21/0.46
% 0.21/0.46 thf(subsetI1,definition,
% 0.21/0.46 ( subsetI1
% 0.21/0.46 = ( ! [A: $i,B: $i] :
% 0.21/0.46 ( ! [Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ A )
% 0.21/0.46 => ( in @ Xx @ B ) )
% 0.21/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(eqimpsubset2_type,type,
% 0.21/0.46 eqimpsubset2: $o ).
% 0.21/0.46
% 0.21/0.46 thf(eqimpsubset2,definition,
% 0.21/0.46 ( eqimpsubset2
% 0.21/0.46 = ( ! [A: $i,B: $i] :
% 0.21/0.46 ( ( A = B )
% 0.21/0.46 => ( subset @ B @ A ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(eqimpsubset1_type,type,
% 0.21/0.46 eqimpsubset1: $o ).
% 0.21/0.46
% 0.21/0.46 thf(eqimpsubset1,definition,
% 0.21/0.46 ( eqimpsubset1
% 0.21/0.46 = ( ! [A: $i,B: $i] :
% 0.21/0.46 ( ( A = B )
% 0.21/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(subsetI2_type,type,
% 0.21/0.46 subsetI2: $o ).
% 0.21/0.46
% 0.21/0.46 thf(subsetI2,definition,
% 0.21/0.46 ( subsetI2
% 0.21/0.46 = ( ! [A: $i,B: $i] :
% 0.21/0.46 ( ! [Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ A )
% 0.21/0.46 => ( in @ Xx @ B ) )
% 0.21/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(emptysetsubset_type,type,
% 0.21/0.46 emptysetsubset: $o ).
% 0.21/0.46
% 0.21/0.46 thf(emptysetsubset,definition,
% 0.21/0.46 ( emptysetsubset
% 0.21/0.46 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(subsetE_type,type,
% 0.21/0.46 subsetE: $o ).
% 0.21/0.46
% 0.21/0.46 thf(subsetE,definition,
% 0.21/0.46 ( subsetE
% 0.21/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.21/0.46 ( ( subset @ A @ B )
% 0.21/0.46 => ( ( in @ Xx @ A )
% 0.21/0.46 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(subsetE2_type,type,
% 0.21/0.46 subsetE2: $o ).
% 0.21/0.46
% 0.21/0.46 thf(subsetE2,definition,
% 0.21/0.46 ( subsetE2
% 0.21/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.21/0.46 ( ( subset @ A @ B )
% 0.21/0.46 => ( ~ ( in @ Xx @ B )
% 0.21/0.46 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(notsubsetI_type,type,
% 0.21/0.46 notsubsetI: $o ).
% 0.21/0.46
% 0.21/0.46 thf(notsubsetI,definition,
% 0.21/0.46 ( notsubsetI
% 0.21/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ A )
% 0.21/0.46 => ( ~ ( in @ Xx @ B )
% 0.21/0.46 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(notequalI1_type,type,
% 0.21/0.46 notequalI1: $o ).
% 0.21/0.46
% 0.21/0.46 thf(notequalI1,definition,
% 0.21/0.46 ( notequalI1
% 0.21/0.46 = ( ! [A: $i,B: $i] :
% 0.21/0.46 ( ~ ( subset @ A @ B )
% 0.21/0.46 => ( A != B ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(notequalI2_type,type,
% 0.21/0.46 notequalI2: $o ).
% 0.21/0.46
% 0.21/0.46 thf(notequalI2,definition,
% 0.21/0.46 ( notequalI2
% 0.21/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ A )
% 0.21/0.46 => ( ~ ( in @ Xx @ B )
% 0.21/0.46 => ( A != B ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(subsetRefl_type,type,
% 0.21/0.46 subsetRefl: $o ).
% 0.21/0.46
% 0.21/0.46 thf(subsetRefl,definition,
% 0.21/0.46 ( subsetRefl
% 0.21/0.46 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(subsetTrans_type,type,
% 0.21/0.46 subsetTrans: $o ).
% 0.21/0.46
% 0.21/0.46 thf(subsetTrans,definition,
% 0.21/0.46 ( subsetTrans
% 0.21/0.46 = ( ! [A: $i,B: $i,C: $i] :
% 0.21/0.46 ( ( subset @ A @ B )
% 0.21/0.46 => ( ( subset @ B @ C )
% 0.21/0.46 => ( subset @ A @ C ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(setadjoinSub_type,type,
% 0.21/0.46 setadjoinSub: $o ).
% 0.21/0.46
% 0.21/0.46 thf(setadjoinSub,definition,
% 0.21/0.46 ( setadjoinSub
% 0.21/0.46 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(setadjoinSub2_type,type,
% 0.21/0.46 setadjoinSub2: $o ).
% 0.21/0.46
% 0.21/0.46 thf(setadjoinSub2,definition,
% 0.21/0.46 ( setadjoinSub2
% 0.21/0.46 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.21/0.46 ( ( subset @ A @ B )
% 0.21/0.46 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(subset2powerset_type,type,
% 0.21/0.46 subset2powerset: $o ).
% 0.21/0.46
% 0.21/0.46 thf(subset2powerset,definition,
% 0.21/0.46 ( subset2powerset
% 0.21/0.46 = ( ! [A: $i,B: $i] :
% 0.21/0.46 ( ( subset @ A @ B )
% 0.21/0.46 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(setextsub_type,type,
% 0.21/0.46 setextsub: $o ).
% 0.21/0.46
% 0.21/0.46 thf(setextsub,definition,
% 0.21/0.46 ( setextsub
% 0.21/0.46 = ( ! [A: $i,B: $i] :
% 0.21/0.46 ( ( subset @ A @ B )
% 0.21/0.46 => ( ( subset @ B @ A )
% 0.21/0.46 => ( A = B ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(subsetemptysetimpeq_type,type,
% 0.21/0.46 subsetemptysetimpeq: $o ).
% 0.21/0.46
% 0.21/0.46 thf(subsetemptysetimpeq,definition,
% 0.21/0.46 ( subsetemptysetimpeq
% 0.21/0.46 = ( ! [A: $i] :
% 0.21/0.46 ( ( subset @ A @ emptyset )
% 0.21/0.46 => ( A = emptyset ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(powersetI1_type,type,
% 0.21/0.46 powersetI1: $o ).
% 0.21/0.46
% 0.21/0.46 thf(powersetI1,definition,
% 0.21/0.46 ( powersetI1
% 0.21/0.46 = ( ! [A: $i,B: $i] :
% 0.21/0.46 ( ( subset @ B @ A )
% 0.21/0.46 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(powersetE1_type,type,
% 0.21/0.46 powersetE1: $o ).
% 0.21/0.46
% 0.21/0.46 thf(powersetE1,definition,
% 0.21/0.46 ( powersetE1
% 0.21/0.46 = ( ! [A: $i,B: $i] :
% 0.21/0.46 ( ( in @ B @ ( powerset @ A ) )
% 0.21/0.46 => ( subset @ B @ A ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(inPowerset_type,type,
% 0.21/0.46 inPowerset: $o ).
% 0.21/0.46
% 0.21/0.46 thf(inPowerset,definition,
% 0.21/0.46 ( inPowerset
% 0.21/0.46 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(powersetsubset_type,type,
% 0.21/0.46 powersetsubset: $o ).
% 0.21/0.46
% 0.21/0.46 thf(powersetsubset,definition,
% 0.21/0.46 ( powersetsubset
% 0.21/0.46 = ( ! [A: $i,B: $i] :
% 0.21/0.46 ( ( subset @ A @ B )
% 0.21/0.46 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(sepInPowerset_type,type,
% 0.21/0.46 sepInPowerset: $o ).
% 0.21/0.46
% 0.21/0.46 thf(sepInPowerset,definition,
% 0.21/0.46 ( sepInPowerset
% 0.21/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.21/0.46 ( in
% 0.21/0.46 @ ( dsetconstr @ A
% 0.21/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.21/0.46 @ ( powerset @ A ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(sepSubset_type,type,
% 0.21/0.46 sepSubset: $o ).
% 0.21/0.46
% 0.21/0.46 thf(sepSubset,definition,
% 0.21/0.46 ( sepSubset
% 0.21/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.21/0.46 ( subset
% 0.21/0.46 @ ( dsetconstr @ A
% 0.21/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.21/0.46 @ A ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(binunion_type,type,
% 0.21/0.46 binunion: $i > $i > $i ).
% 0.21/0.46
% 0.21/0.46 thf(binunionIL_type,type,
% 0.21/0.46 binunionIL: $o ).
% 0.21/0.46
% 0.21/0.46 thf(binunionIL,definition,
% 0.21/0.46 ( binunionIL
% 0.21/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ A )
% 0.21/0.46 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(upairset2IR_type,type,
% 0.21/0.46 upairset2IR: $o ).
% 0.21/0.46
% 0.21/0.46 thf(upairset2IR,definition,
% 0.21/0.46 ( upairset2IR
% 0.21/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(binunionIR_type,type,
% 0.21/0.46 binunionIR: $o ).
% 0.21/0.46
% 0.21/0.46 thf(binunionIR,definition,
% 0.21/0.46 ( binunionIR
% 0.21/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ B )
% 0.21/0.46 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(binunionEcases_type,type,
% 0.21/0.46 binunionEcases: $o ).
% 0.21/0.46
% 0.21/0.46 thf(binunionEcases,definition,
% 0.21/0.46 ( binunionEcases
% 0.21/0.46 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.21/0.46 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.21/0.46 => ( ( ( in @ Xx @ A )
% 0.21/0.46 => Xphi )
% 0.21/0.46 => ( ( ( in @ Xx @ B )
% 0.21/0.46 => Xphi )
% 0.21/0.46 => Xphi ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(binunionE_type,type,
% 0.21/0.46 binunionE: $o ).
% 0.21/0.46
% 0.21/0.46 thf(binunionE,definition,
% 0.21/0.46 ( binunionE
% 0.21/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.21/0.46 => ( ( in @ Xx @ A )
% 0.21/0.46 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(binunionLsub_type,type,
% 0.21/0.46 binunionLsub: $o ).
% 0.21/0.46
% 0.21/0.46 thf(binunionLsub,definition,
% 0.21/0.46 ( binunionLsub
% 0.21/0.46 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(binunionRsub_type,type,
% 0.21/0.46 binunionRsub: $o ).
% 0.21/0.46
% 0.21/0.46 thf(binunionRsub,definition,
% 0.21/0.46 ( binunionRsub
% 0.21/0.46 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(binintersect_type,type,
% 0.21/0.46 binintersect: $i > $i > $i ).
% 0.21/0.46
% 0.21/0.46 thf(binintersectI_type,type,
% 0.21/0.46 binintersectI: $o ).
% 0.21/0.46
% 0.21/0.46 thf(binintersectI,definition,
% 0.21/0.46 ( binintersectI
% 0.21/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ A )
% 0.21/0.46 => ( ( in @ Xx @ B )
% 0.21/0.46 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(binintersectSubset5_type,type,
% 0.21/0.46 binintersectSubset5: $o ).
% 0.21/0.46
% 0.21/0.46 thf(binintersectSubset5,definition,
% 0.21/0.46 ( binintersectSubset5
% 0.21/0.46 = ( ! [A: $i,B: $i,C: $i] :
% 0.21/0.46 ( ( subset @ C @ A )
% 0.21/0.46 => ( ( subset @ C @ B )
% 0.21/0.46 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(binintersectEL_type,type,
% 0.21/0.46 binintersectEL: $o ).
% 0.21/0.46
% 0.21/0.46 thf(binintersectEL,definition,
% 0.21/0.46 ( binintersectEL
% 0.21/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.21/0.46 => ( in @ Xx @ A ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(binintersectLsub_type,type,
% 0.21/0.46 binintersectLsub: $o ).
% 0.21/0.46
% 0.21/0.46 thf(binintersectLsub,definition,
% 0.21/0.46 ( binintersectLsub
% 0.21/0.46 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(binintersectSubset2_type,type,
% 0.21/0.46 binintersectSubset2: $o ).
% 0.21/0.46
% 0.21/0.46 thf(binintersectSubset2,definition,
% 0.21/0.46 ( binintersectSubset2
% 0.21/0.46 = ( ! [A: $i,B: $i] :
% 0.21/0.46 ( ( subset @ A @ B )
% 0.21/0.46 => ( ( binintersect @ A @ B )
% 0.21/0.46 = A ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(binintersectSubset3_type,type,
% 0.21/0.46 binintersectSubset3: $o ).
% 0.21/0.46
% 0.21/0.46 thf(binintersectSubset3,definition,
% 0.21/0.46 ( binintersectSubset3
% 0.21/0.46 = ( ! [A: $i,B: $i] :
% 0.21/0.46 ( ( ( binintersect @ A @ B )
% 0.21/0.46 = B )
% 0.21/0.46 => ( subset @ B @ A ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(binintersectER_type,type,
% 0.21/0.46 binintersectER: $o ).
% 0.21/0.46
% 0.21/0.46 thf(binintersectER,definition,
% 0.21/0.46 ( binintersectER
% 0.21/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.21/0.46 => ( in @ Xx @ B ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(disjointsetsI1_type,type,
% 0.21/0.46 disjointsetsI1: $o ).
% 0.21/0.46
% 0.21/0.46 thf(disjointsetsI1,definition,
% 0.21/0.46 ( disjointsetsI1
% 0.21/0.46 = ( ! [A: $i,B: $i] :
% 0.21/0.46 ( ~ ? [Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ A )
% 0.21/0.46 & ( in @ Xx @ B ) )
% 0.21/0.46 => ( ( binintersect @ A @ B )
% 0.21/0.46 = emptyset ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(binintersectRsub_type,type,
% 0.21/0.46 binintersectRsub: $o ).
% 0.21/0.46
% 0.21/0.46 thf(binintersectRsub,definition,
% 0.21/0.46 ( binintersectRsub
% 0.21/0.46 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(binintersectSubset4_type,type,
% 0.21/0.46 binintersectSubset4: $o ).
% 0.21/0.46
% 0.21/0.46 thf(binintersectSubset4,definition,
% 0.21/0.46 ( binintersectSubset4
% 0.21/0.46 = ( ! [A: $i,B: $i] :
% 0.21/0.46 ( ( subset @ B @ A )
% 0.21/0.46 => ( ( binintersect @ A @ B )
% 0.21/0.46 = B ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(binintersectSubset1_type,type,
% 0.21/0.46 binintersectSubset1: $o ).
% 0.21/0.46
% 0.21/0.46 thf(binintersectSubset1,definition,
% 0.21/0.46 ( binintersectSubset1
% 0.21/0.46 = ( ! [A: $i,B: $i] :
% 0.21/0.46 ( ( ( binintersect @ A @ B )
% 0.21/0.46 = A )
% 0.21/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(bs114d_type,type,
% 0.21/0.46 bs114d: $o ).
% 0.21/0.46
% 0.21/0.46 thf(bs114d,definition,
% 0.21/0.46 ( bs114d
% 0.21/0.46 = ( ! [A: $i,B: $i,C: $i] :
% 0.21/0.46 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 0.21/0.46 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(regular_type,type,
% 0.21/0.46 regular: $i > $o ).
% 0.21/0.46
% 0.21/0.46 thf(setminus_type,type,
% 0.21/0.46 setminus: $i > $i > $i ).
% 0.21/0.46
% 0.21/0.46 thf(setminusI_type,type,
% 0.21/0.46 setminusI: $o ).
% 0.21/0.46
% 0.21/0.46 thf(setminusI,definition,
% 0.21/0.46 ( setminusI
% 0.21/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ A )
% 0.21/0.46 => ( ~ ( in @ Xx @ B )
% 0.21/0.46 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(setminusEL_type,type,
% 0.21/0.46 setminusEL: $o ).
% 0.21/0.46
% 0.21/0.46 thf(setminusEL,definition,
% 0.21/0.46 ( setminusEL
% 0.21/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.21/0.46 => ( in @ Xx @ A ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(setminusER_type,type,
% 0.21/0.46 setminusER: $o ).
% 0.21/0.46
% 0.21/0.46 thf(setminusER,definition,
% 0.21/0.46 ( setminusER
% 0.21/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.21/0.46 => ~ ( in @ Xx @ B ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(setminusSubset2_type,type,
% 0.21/0.46 setminusSubset2: $o ).
% 0.21/0.46
% 0.21/0.46 thf(setminusSubset2,definition,
% 0.21/0.46 ( setminusSubset2
% 0.21/0.46 = ( ! [A: $i,B: $i] :
% 0.21/0.46 ( ( subset @ A @ B )
% 0.21/0.46 => ( ( setminus @ A @ B )
% 0.21/0.46 = emptyset ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(setminusERneg_type,type,
% 0.21/0.46 setminusERneg: $o ).
% 0.21/0.46
% 0.21/0.46 thf(setminusERneg,definition,
% 0.21/0.46 ( setminusERneg
% 0.21/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.21/0.46 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.21/0.46 => ( ( in @ Xx @ A )
% 0.21/0.46 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(setminusELneg_type,type,
% 0.21/0.46 setminusELneg: $o ).
% 0.21/0.46
% 0.21/0.46 thf(setminusELneg,definition,
% 0.21/0.46 ( setminusELneg
% 0.21/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.21/0.46 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.21/0.46 => ( ~ ( in @ Xx @ B )
% 0.21/0.46 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(setminusILneg_type,type,
% 0.21/0.46 setminusILneg: $o ).
% 0.21/0.46
% 0.21/0.46 thf(setminusILneg,definition,
% 0.21/0.46 ( setminusILneg
% 0.21/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.21/0.46 ( ~ ( in @ Xx @ A )
% 0.21/0.46 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(setminusIRneg_type,type,
% 0.21/0.46 setminusIRneg: $o ).
% 0.21/0.46
% 0.21/0.46 thf(setminusIRneg,definition,
% 0.21/0.46 ( setminusIRneg
% 0.21/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ B )
% 0.21/0.46 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(setminusLsub_type,type,
% 0.21/0.46 setminusLsub: $o ).
% 0.21/0.46
% 0.21/0.46 thf(setminusLsub,definition,
% 0.21/0.46 ( setminusLsub
% 0.21/0.46 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(setminusSubset1_type,type,
% 0.21/0.46 setminusSubset1: $o ).
% 0.21/0.46
% 0.21/0.46 thf(setminusSubset1,definition,
% 0.21/0.46 ( setminusSubset1
% 0.21/0.46 = ( ! [A: $i,B: $i] :
% 0.21/0.46 ( ( ( setminus @ A @ B )
% 0.21/0.46 = emptyset )
% 0.31/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(symdiff_type,type,
% 0.31/0.46 symdiff: $i > $i > $i ).
% 0.31/0.46
% 0.31/0.46 thf(symdiffE_type,type,
% 0.31/0.46 symdiffE: $o ).
% 0.31/0.46
% 0.31/0.46 thf(symdiffE,definition,
% 0.31/0.46 ( symdiffE
% 0.31/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 0.31/0.46 => ! [Xphi: $o] :
% 0.31/0.46 ( ( ( in @ Xx @ A )
% 0.31/0.46 => ( ~ ( in @ Xx @ B )
% 0.31/0.46 => Xphi ) )
% 0.31/0.46 => ( ( ~ ( in @ Xx @ A )
% 0.31/0.46 => ( ( in @ Xx @ B )
% 0.31/0.46 => Xphi ) )
% 0.31/0.46 => Xphi ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(symdiffI1_type,type,
% 0.31/0.46 symdiffI1: $o ).
% 0.31/0.46
% 0.31/0.46 thf(symdiffI1,definition,
% 0.31/0.46 ( symdiffI1
% 0.31/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ( ~ ( in @ Xx @ B )
% 0.31/0.46 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(symdiffI2_type,type,
% 0.31/0.46 symdiffI2: $o ).
% 0.31/0.46
% 0.31/0.46 thf(symdiffI2,definition,
% 0.31/0.46 ( symdiffI2
% 0.31/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.46 ( ~ ( in @ Xx @ A )
% 0.31/0.46 => ( ( in @ Xx @ B )
% 0.31/0.46 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(symdiffIneg1_type,type,
% 0.31/0.46 symdiffIneg1: $o ).
% 0.31/0.46
% 0.31/0.46 thf(symdiffIneg1,definition,
% 0.31/0.46 ( symdiffIneg1
% 0.31/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ( ( in @ Xx @ B )
% 0.31/0.46 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(symdiffIneg2_type,type,
% 0.31/0.46 symdiffIneg2: $o ).
% 0.31/0.46
% 0.31/0.46 thf(symdiffIneg2,definition,
% 0.31/0.46 ( symdiffIneg2
% 0.31/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.46 ( ~ ( in @ Xx @ A )
% 0.31/0.46 => ( ~ ( in @ Xx @ B )
% 0.31/0.46 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(iskpair_type,type,
% 0.31/0.46 iskpair: $i > $o ).
% 0.31/0.46
% 0.31/0.46 thf(iskpair,definition,
% 0.31/0.46 ( iskpair
% 0.31/0.46 = ( ^ [A: $i] :
% 0.31/0.46 ? [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ ( setunion @ A ) )
% 0.31/0.46 & ? [Xy: $i] :
% 0.31/0.46 ( ( in @ Xy @ ( setunion @ A ) )
% 0.31/0.46 & ( A
% 0.31/0.46 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(secondinupair_type,type,
% 0.31/0.46 secondinupair: $o ).
% 0.31/0.46
% 0.31/0.46 thf(secondinupair,definition,
% 0.31/0.46 ( secondinupair
% 0.31/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(setukpairIL_type,type,
% 0.31/0.46 setukpairIL: $o ).
% 0.31/0.46
% 0.31/0.46 thf(setukpairIL,definition,
% 0.31/0.46 ( setukpairIL
% 0.31/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(setukpairIR_type,type,
% 0.31/0.46 setukpairIR: $o ).
% 0.31/0.46
% 0.31/0.46 thf(setukpairIR,definition,
% 0.31/0.46 ( setukpairIR
% 0.31/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(kpairiskpair_type,type,
% 0.31/0.46 kpairiskpair: $o ).
% 0.31/0.46
% 0.31/0.46 thf(kpairiskpair,definition,
% 0.31/0.46 ( kpairiskpair
% 0.31/0.46 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(kpair_type,type,
% 0.31/0.46 kpair: $i > $i > $i ).
% 0.31/0.46
% 0.31/0.46 thf(kpair,definition,
% 0.31/0.46 ( kpair
% 0.31/0.46 = ( ^ [Xx: $i,Xy: $i] : ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(kpairp_type,type,
% 0.31/0.46 kpairp: $o ).
% 0.31/0.46
% 0.31/0.46 thf(kpairp,definition,
% 0.31/0.46 ( kpairp
% 0.31/0.46 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(cartprod_type,type,
% 0.31/0.46 cartprod: $i > $i > $i ).
% 0.31/0.46
% 0.31/0.46 thf(singletonsubset_type,type,
% 0.31/0.46 singletonsubset: $o ).
% 0.31/0.46
% 0.31/0.46 thf(singletonsubset,definition,
% 0.31/0.46 ( singletonsubset
% 0.31/0.46 = ( ! [A: $i,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(singletoninpowerset_type,type,
% 0.31/0.46 singletoninpowerset: $o ).
% 0.31/0.46
% 0.31/0.46 thf(singletoninpowerset,definition,
% 0.31/0.46 ( singletoninpowerset
% 0.31/0.46 = ( ! [A: $i,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(singletoninpowunion_type,type,
% 0.31/0.46 singletoninpowunion: $o ).
% 0.31/0.46
% 0.31/0.46 thf(singletoninpowunion,definition,
% 0.31/0.46 ( singletoninpowunion
% 0.31/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(upairset2E_type,type,
% 0.31/0.46 upairset2E: $o ).
% 0.31/0.46
% 0.31/0.46 thf(upairset2E,definition,
% 0.31/0.46 ( upairset2E
% 0.31/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.31/0.46 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.31/0.46 => ( ( Xz = Xx )
% 0.31/0.46 | ( Xz = Xy ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(upairsubunion_type,type,
% 0.31/0.46 upairsubunion: $o ).
% 0.31/0.46
% 0.31/0.46 thf(upairsubunion,definition,
% 0.31/0.46 ( upairsubunion
% 0.31/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ! [Xy: $i] :
% 0.31/0.46 ( ( in @ Xy @ B )
% 0.31/0.46 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(upairinpowunion_type,type,
% 0.31/0.46 upairinpowunion: $o ).
% 0.31/0.46
% 0.31/0.46 thf(upairinpowunion,definition,
% 0.31/0.46 ( upairinpowunion
% 0.31/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ! [Xy: $i] :
% 0.31/0.46 ( ( in @ Xy @ B )
% 0.31/0.46 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(ubforcartprodlem1_type,type,
% 0.31/0.46 ubforcartprodlem1: $o ).
% 0.31/0.46
% 0.31/0.46 thf(ubforcartprodlem1,definition,
% 0.31/0.46 ( ubforcartprodlem1
% 0.31/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ! [Xy: $i] :
% 0.31/0.46 ( ( in @ Xy @ B )
% 0.31/0.46 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(ubforcartprodlem2_type,type,
% 0.31/0.46 ubforcartprodlem2: $o ).
% 0.31/0.46
% 0.31/0.46 thf(ubforcartprodlem2,definition,
% 0.31/0.46 ( ubforcartprodlem2
% 0.31/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ! [Xy: $i] :
% 0.31/0.46 ( ( in @ Xy @ B )
% 0.31/0.46 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(ubforcartprodlem3_type,type,
% 0.31/0.46 ubforcartprodlem3: $o ).
% 0.31/0.46
% 0.31/0.46 thf(ubforcartprodlem3,definition,
% 0.31/0.46 ( ubforcartprodlem3
% 0.31/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ! [Xy: $i] :
% 0.31/0.46 ( ( in @ Xy @ B )
% 0.31/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(cartprodpairin_type,type,
% 0.31/0.46 cartprodpairin: $o ).
% 0.31/0.46
% 0.31/0.46 thf(cartprodpairin,definition,
% 0.31/0.46 ( cartprodpairin
% 0.31/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ! [Xy: $i] :
% 0.31/0.46 ( ( in @ Xy @ B )
% 0.31/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(cartprodmempair1_type,type,
% 0.31/0.46 cartprodmempair1: $o ).
% 0.31/0.46
% 0.31/0.46 thf(cartprodmempair1,definition,
% 0.31/0.46 ( cartprodmempair1
% 0.31/0.46 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.31/0.46 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.31/0.46 => ? [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 & ? [Xy: $i] :
% 0.31/0.46 ( ( in @ Xy @ B )
% 0.31/0.46 & ( Xu
% 0.31/0.46 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(cartprodmempair_type,type,
% 0.31/0.46 cartprodmempair: $o ).
% 0.31/0.46
% 0.31/0.46 thf(cartprodmempair,definition,
% 0.31/0.46 ( cartprodmempair
% 0.31/0.46 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.31/0.46 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.31/0.46 => ( iskpair @ Xu ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(setunionE2_type,type,
% 0.31/0.46 setunionE2: $o ).
% 0.31/0.46
% 0.31/0.46 thf(setunionE2,definition,
% 0.31/0.46 ( setunionE2
% 0.31/0.46 = ( ! [A: $i,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ ( setunion @ A ) )
% 0.31/0.46 => ? [X: $i] :
% 0.31/0.46 ( ( in @ X @ A )
% 0.31/0.46 & ( in @ Xx @ X ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(setunionsingleton1_type,type,
% 0.31/0.46 setunionsingleton1: $o ).
% 0.31/0.46
% 0.31/0.46 thf(setunionsingleton1,definition,
% 0.31/0.46 ( setunionsingleton1
% 0.31/0.46 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(setunionsingleton2_type,type,
% 0.31/0.46 setunionsingleton2: $o ).
% 0.31/0.46
% 0.31/0.46 thf(setunionsingleton2,definition,
% 0.31/0.46 ( setunionsingleton2
% 0.31/0.46 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(setunionsingleton_type,type,
% 0.31/0.46 setunionsingleton: $o ).
% 0.31/0.46
% 0.31/0.46 thf(setunionsingleton,definition,
% 0.31/0.46 ( setunionsingleton
% 0.31/0.46 = ( ! [Xx: $i] :
% 0.31/0.46 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 0.31/0.46 = Xx ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(singleton_type,type,
% 0.31/0.46 singleton: $i > $o ).
% 0.31/0.46
% 0.31/0.46 thf(singleton,definition,
% 0.31/0.46 ( singleton
% 0.31/0.46 = ( ^ [A: $i] :
% 0.31/0.46 ? [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 & ( A
% 0.31/0.46 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(singletonprop_type,type,
% 0.31/0.46 singletonprop: $o ).
% 0.31/0.46
% 0.31/0.46 thf(singletonprop,definition,
% 0.31/0.46 ( singletonprop
% 0.31/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/0.46 ( ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ! [Xy: $i] :
% 0.31/0.46 ( ( in @ Xy @ A )
% 0.31/0.46 => ( ( Xphi @ Xx )
% 0.31/0.46 => ( ( Xphi @ Xy )
% 0.31/0.46 => ( Xx = Xy ) ) ) ) )
% 0.31/0.46 => ( ? [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 & ( Xphi @ Xx ) )
% 0.31/0.46 => ( singleton
% 0.31/0.46 @ ( dsetconstr @ A
% 0.31/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(ex1_type,type,
% 0.31/0.46 ex1: $i > ( $i > $o ) > $o ).
% 0.31/0.46
% 0.31/0.46 thf(ex1,definition,
% 0.31/0.46 ( ex1
% 0.31/0.46 = ( ^ [A: $i,Xphi: $i > $o] :
% 0.31/0.46 ( singleton
% 0.31/0.46 @ ( dsetconstr @ A
% 0.31/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(ex1E1_type,type,
% 0.31/0.46 ex1E1: $o ).
% 0.31/0.46
% 0.31/0.46 thf(ex1E1,definition,
% 0.31/0.46 ( ex1E1
% 0.31/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/0.46 ( ( ex1 @ A
% 0.31/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.46 => ? [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(ex1I_type,type,
% 0.31/0.46 ex1I: $o ).
% 0.31/0.46
% 0.31/0.46 thf(ex1I,definition,
% 0.31/0.46 ( ex1I
% 0.31/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ( ( Xphi @ Xx )
% 0.31/0.46 => ( ! [Xy: $i] :
% 0.31/0.46 ( ( in @ Xy @ A )
% 0.31/0.46 => ( ( Xphi @ Xy )
% 0.31/0.46 => ( Xy = Xx ) ) )
% 0.31/0.46 => ( ex1 @ A
% 0.31/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(ex1I2_type,type,
% 0.31/0.46 ex1I2: $o ).
% 0.31/0.46
% 0.31/0.46 thf(ex1I2,definition,
% 0.31/0.46 ( ex1I2
% 0.31/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/0.46 ( ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ! [Xy: $i] :
% 0.31/0.46 ( ( in @ Xy @ A )
% 0.31/0.46 => ( ( Xphi @ Xx )
% 0.31/0.46 => ( ( Xphi @ Xy )
% 0.31/0.46 => ( Xx = Xy ) ) ) ) )
% 0.31/0.46 => ( ? [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 & ( Xphi @ Xx ) )
% 0.31/0.46 => ( ex1 @ A
% 0.31/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(singletonsuniq_type,type,
% 0.31/0.46 singletonsuniq: $o ).
% 0.31/0.46
% 0.31/0.46 thf(singletonsuniq,definition,
% 0.31/0.46 ( singletonsuniq
% 0.31/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.31/0.46 ( ( ( setadjoin @ Xx @ emptyset )
% 0.31/0.46 = ( setadjoin @ Xy @ emptyset ) )
% 0.31/0.46 => ( Xx = Xy ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(atmost1p_type,type,
% 0.31/0.46 atmost1p: $i > $o ).
% 0.31/0.46
% 0.31/0.46 thf(atleast2p_type,type,
% 0.31/0.46 atleast2p: $i > $o ).
% 0.31/0.46
% 0.31/0.46 thf(atmost2p_type,type,
% 0.31/0.46 atmost2p: $i > $o ).
% 0.31/0.46
% 0.31/0.46 thf(upairsetp_type,type,
% 0.31/0.46 upairsetp: $i > $o ).
% 0.31/0.46
% 0.31/0.46 thf(setukpairinjL1,conjecture,
% 0.31/0.46 ( setextAx
% 0.31/0.46 => ( emptysetAx
% 0.31/0.46 => ( setadjoinAx
% 0.31/0.46 => ( powersetAx
% 0.31/0.46 => ( setunionAx
% 0.31/0.46 => ( omega0Ax
% 0.31/0.46 => ( omegaSAx
% 0.31/0.46 => ( omegaIndAx
% 0.31/0.46 => ( replAx
% 0.31/0.46 => ( foundationAx
% 0.31/0.46 => ( wellorderingAx
% 0.31/0.46 => ( descrp
% 0.31/0.46 => ( dsetconstrI
% 0.31/0.46 => ( dsetconstrEL
% 0.31/0.46 => ( dsetconstrER
% 0.31/0.46 => ( exuE1
% 0.31/0.46 => ( prop2setE
% 0.31/0.46 => ( emptysetE
% 0.31/0.46 => ( emptysetimpfalse
% 0.31/0.46 => ( notinemptyset
% 0.31/0.46 => ( exuE3e
% 0.31/0.46 => ( setext
% 0.31/0.46 => ( emptyI
% 0.31/0.46 => ( noeltsimpempty
% 0.31/0.46 => ( setbeta
% 0.31/0.46 => ( nonemptyE1
% 0.31/0.46 => ( nonemptyI
% 0.31/0.46 => ( nonemptyI1
% 0.31/0.46 => ( setadjoinIL
% 0.31/0.46 => ( emptyinunitempty
% 0.31/0.46 => ( setadjoinIR
% 0.31/0.46 => ( setadjoinE
% 0.31/0.46 => ( setadjoinOr
% 0.31/0.46 => ( setoftrueEq
% 0.31/0.46 => ( powersetI
% 0.31/0.46 => ( emptyinPowerset
% 0.31/0.46 => ( emptyInPowerset
% 0.31/0.46 => ( powersetE
% 0.31/0.46 => ( setunionI
% 0.31/0.46 => ( setunionE
% 0.31/0.46 => ( subPowSU
% 0.31/0.46 => ( exuE2
% 0.31/0.46 => ( nonemptyImpWitness
% 0.31/0.46 => ( uniqinunit
% 0.31/0.46 => ( notinsingleton
% 0.31/0.46 => ( eqinunit
% 0.31/0.46 => ( singletonsswitch
% 0.31/0.46 => ( upairsetE
% 0.31/0.46 => ( upairsetIL
% 0.31/0.46 => ( upairsetIR
% 0.31/0.46 => ( emptyE1
% 0.31/0.46 => ( vacuousDall
% 0.31/0.46 => ( quantDeMorgan1
% 0.31/0.46 => ( quantDeMorgan2
% 0.31/0.46 => ( quantDeMorgan3
% 0.31/0.46 => ( quantDeMorgan4
% 0.31/0.46 => ( prop2setI
% 0.31/0.46 => ( prop2set2propI
% 0.31/0.46 => ( notdexE
% 0.31/0.46 => ( notdallE
% 0.31/0.46 => ( exuI1
% 0.31/0.46 => ( exuI3
% 0.31/0.46 => ( exuI2
% 0.31/0.46 => ( inCongP
% 0.31/0.46 => ( in__Cong
% 0.31/0.46 => ( exuE3u
% 0.31/0.46 => ( exu__Cong
% 0.31/0.46 => ( emptyset__Cong
% 0.31/0.46 => ( setadjoin__Cong
% 0.31/0.46 => ( powerset__Cong
% 0.31/0.46 => ( setunion__Cong
% 0.31/0.46 => ( omega__Cong
% 0.31/0.46 => ( exuEu
% 0.31/0.46 => ( descr__Cong
% 0.31/0.46 => ( dsetconstr__Cong
% 0.31/0.46 => ( subsetI1
% 0.31/0.46 => ( eqimpsubset2
% 0.31/0.46 => ( eqimpsubset1
% 0.31/0.46 => ( subsetI2
% 0.31/0.46 => ( emptysetsubset
% 0.31/0.46 => ( subsetE
% 0.31/0.46 => ( subsetE2
% 0.31/0.46 => ( notsubsetI
% 0.31/0.46 => ( notequalI1
% 0.31/0.46 => ( notequalI2
% 0.31/0.46 => ( subsetRefl
% 0.31/0.46 => ( subsetTrans
% 0.31/0.46 => ( setadjoinSub
% 0.31/0.46 => ( setadjoinSub2
% 0.31/0.46 => ( subset2powerset
% 0.31/0.46 => ( setextsub
% 0.31/0.46 => ( subsetemptysetimpeq
% 0.31/0.46 => ( powersetI1
% 0.31/0.46 => ( powersetE1
% 0.31/0.46 => ( inPowerset
% 0.31/0.46 => ( powersetsubset
% 0.31/0.46 => ( sepInPowerset
% 0.31/0.46 => ( sepSubset
% 0.31/0.46 => ( binunionIL
% 0.31/0.46 => ( upairset2IR
% 0.31/0.46 => ( binunionIR
% 0.31/0.46 => ( binunionEcases
% 0.31/0.46 => ( binunionE
% 0.31/0.46 => ( binunionLsub
% 0.31/0.46 => ( binunionRsub
% 0.31/0.46 => ( binintersectI
% 0.31/0.46 => ( binintersectSubset5
% 0.31/0.46 => ( binintersectEL
% 0.31/0.46 => ( binintersectLsub
% 0.31/0.46 => ( binintersectSubset2
% 0.31/0.46 => ( binintersectSubset3
% 0.31/0.46 => ( binintersectER
% 0.31/0.46 => ( disjointsetsI1
% 0.31/0.46 => ( binintersectRsub
% 0.31/0.46 => ( binintersectSubset4
% 0.31/0.46 => ( binintersectSubset1
% 0.31/0.46 => ( bs114d
% 0.31/0.46 => ( setminusI
% 0.31/0.46 => ( setminusEL
% 0.31/0.46 => ( setminusER
% 0.31/0.46 => ( setminusSubset2
% 0.31/0.46 => ( setminusERneg
% 0.31/0.46 => ( setminusELneg
% 0.31/0.46 => ( setminusILneg
% 0.31/0.46 => ( setminusIRneg
% 0.31/0.46 => ( setminusLsub
% 0.31/0.46 => ( setminusSubset1
% 0.31/0.46 => ( symdiffE
% 0.31/0.46 => ( symdiffI1
% 0.31/0.46 => ( symdiffI2
% 0.31/0.46 => ( symdiffIneg1
% 0.31/0.46 => ( symdiffIneg2
% 0.31/0.46 => ( secondinupair
% 0.31/0.46 => ( setukpairIL
% 0.31/0.46 => ( setukpairIR
% 0.31/0.46 => ( kpairiskpair
% 0.31/0.46 => ( kpairp
% 0.31/0.46 => ( singletonsubset
% 0.31/0.46 => ( singletoninpowerset
% 0.31/0.46 => ( singletoninpowunion
% 0.31/0.46 => ( upairset2E
% 0.31/0.46 => ( upairsubunion
% 0.31/0.46 => ( upairinpowunion
% 0.31/0.46 => ( ubforcartprodlem1
% 0.31/0.46 => ( ubforcartprodlem2
% 0.31/0.46 => ( ubforcartprodlem3
% 0.31/0.46 => ( cartprodpairin
% 0.31/0.46 => ( cartprodmempair1
% 0.31/0.46 => ( cartprodmempair
% 0.31/0.46 => ( setunionE2
% 0.31/0.46 => ( setunionsingleton1
% 0.31/0.46 => ( setunionsingleton2
% 0.31/0.46 => ( setunionsingleton
% 0.31/0.46 => ( singletonprop
% 0.31/0.46 => ( ex1E1
% 0.31/0.46 => ( ex1I
% 0.31/0.46 => ( ex1I2
% 0.31/0.46 => ( singletonsuniq
% 0.31/0.46 => ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.31/0.46 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 0.31/0.46 => ( Xx = Xz ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 %------------------------------------------------------------------------------
% 0.31/0.46 ------------------------
% 2.69/1.67 ---- Embedded in THF ---
% 2.69/1.67 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.69/1.67 %%% Generated on Tue Feb 24 12:30:08 EST 2026
% 2.69/1.67 %%% using '$$dhol' embedding, version 1.3.0.
% 2.69/1.67 %%% Logic specification used:
% 2.69/1.67 %%% thf(spec, logic, $$dhol).
% 2.69/1.67
% 2.69/1.67 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.vWjcIjdJ0D/DTF2DTF13509.p
% See solution above
% 2.69/1.67 ------------------------
% 2.86/1.70 ---- Cleaned THF ---
% 2.86/1.70 thf(in_type,type,
% 2.86/1.70 in: $i > $i > $o ).
% 2.86/1.70
% 2.86/1.70 thf(exu_type,type,
% 2.86/1.70 exu: ( $i > $o ) > $o ).
% 2.86/1.70
% 2.86/1.70 thf(exu,definition,
% 2.86/1.70 ( exu
% 2.86/1.70 = ( ^ [Xphi: $i > $o] :
% 2.86/1.70 ? [Xx: $i] :
% 2.86/1.70 ( ( Xphi @ Xx )
% 2.86/1.70 & ! [Xy: $i] :
% 2.86/1.70 ( ( Xphi @ Xy )
% 2.86/1.70 => ( Xx = Xy ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(setextAx_type,type,
% 2.86/1.70 setextAx: $o ).
% 2.86/1.70
% 2.86/1.70 thf(setextAx,definition,
% 2.86/1.70 ( setextAx
% 2.86/1.70 = ( ! [A: $i,B: $i] :
% 2.86/1.70 ( ! [Xx: $i] :
% 2.86/1.70 ( ( in @ Xx @ A )
% 2.86/1.70 <=> ( in @ Xx @ B ) )
% 2.86/1.70 => ( A = B ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(emptyset_type,type,
% 2.86/1.70 emptyset: $i ).
% 2.86/1.70
% 2.86/1.70 thf(emptysetAx_type,type,
% 2.86/1.70 emptysetAx: $o ).
% 2.86/1.70
% 2.86/1.70 thf(emptysetAx,definition,
% 2.86/1.70 ( emptysetAx
% 2.86/1.70 = ( ! [Xx: $i] :
% 2.86/1.70 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(setadjoin_type,type,
% 2.86/1.70 setadjoin: $i > $i > $i ).
% 2.86/1.70
% 2.86/1.70 thf(setadjoinAx_type,type,
% 2.86/1.70 setadjoinAx: $o ).
% 2.86/1.70
% 2.86/1.70 thf(setadjoinAx,definition,
% 2.86/1.70 ( setadjoinAx
% 2.86/1.70 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.86/1.70 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.86/1.70 <=> ( ( Xy = Xx )
% 2.86/1.70 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(powerset_type,type,
% 2.86/1.70 powerset: $i > $i ).
% 2.86/1.70
% 2.86/1.70 thf(powersetAx_type,type,
% 2.86/1.70 powersetAx: $o ).
% 2.86/1.70
% 2.86/1.70 thf(powersetAx,definition,
% 2.86/1.70 ( powersetAx
% 2.86/1.70 = ( ! [A: $i,B: $i] :
% 2.86/1.70 ( ( in @ B @ ( powerset @ A ) )
% 2.86/1.70 <=> ! [Xx: $i] :
% 2.86/1.70 ( ( in @ Xx @ B )
% 2.86/1.70 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(setunion_type,type,
% 2.86/1.70 setunion: $i > $i ).
% 2.86/1.70
% 2.86/1.70 thf(setunionAx_type,type,
% 2.86/1.70 setunionAx: $o ).
% 2.86/1.70
% 2.86/1.70 thf(setunionAx,definition,
% 2.86/1.70 ( setunionAx
% 2.86/1.70 = ( ! [A: $i,Xx: $i] :
% 2.86/1.70 ( ( in @ Xx @ ( setunion @ A ) )
% 2.86/1.70 <=> ? [B: $i] :
% 2.86/1.70 ( ( in @ Xx @ B )
% 2.86/1.70 & ( in @ B @ A ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(omega_type,type,
% 2.86/1.70 omega: $i ).
% 2.86/1.70
% 2.86/1.70 thf(omega0Ax_type,type,
% 2.86/1.70 omega0Ax: $o ).
% 2.86/1.70
% 2.86/1.70 thf(omega0Ax,definition,
% 2.86/1.70 ( omega0Ax
% 2.86/1.70 = ( in @ emptyset @ omega ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(omegaSAx_type,type,
% 2.86/1.70 omegaSAx: $o ).
% 2.86/1.70
% 2.86/1.70 thf(omegaSAx,definition,
% 2.86/1.70 ( omegaSAx
% 2.86/1.70 = ( ! [Xx: $i] :
% 2.86/1.70 ( ( in @ Xx @ omega )
% 2.86/1.70 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(omegaIndAx_type,type,
% 2.86/1.70 omegaIndAx: $o ).
% 2.86/1.70
% 2.86/1.70 thf(omegaIndAx,definition,
% 2.86/1.70 ( omegaIndAx
% 2.86/1.70 = ( ! [A: $i] :
% 2.86/1.70 ( ( ( in @ emptyset @ A )
% 2.86/1.70 & ! [Xx: $i] :
% 2.86/1.70 ( ( ( in @ Xx @ omega )
% 2.86/1.70 & ( in @ Xx @ A ) )
% 2.86/1.70 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 2.86/1.70 => ! [Xx: $i] :
% 2.86/1.70 ( ( in @ Xx @ omega )
% 2.86/1.70 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(replAx_type,type,
% 2.86/1.70 replAx: $o ).
% 2.86/1.70
% 2.86/1.70 thf(replAx,definition,
% 2.86/1.70 ( replAx
% 2.86/1.70 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 2.86/1.70 ( ! [Xx: $i] :
% 2.86/1.70 ( ( in @ Xx @ A )
% 2.86/1.70 => ( exu
% 2.86/1.70 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 2.86/1.70 => ? [B: $i] :
% 2.86/1.70 ! [Xx: $i] :
% 2.86/1.70 ( ( in @ Xx @ B )
% 2.86/1.70 <=> ? [Xy: $i] :
% 2.86/1.70 ( ( in @ Xy @ A )
% 2.86/1.70 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(foundationAx_type,type,
% 2.86/1.70 foundationAx: $o ).
% 2.86/1.70
% 2.86/1.70 thf(foundationAx,definition,
% 2.86/1.70 ( foundationAx
% 2.86/1.70 = ( ! [A: $i] :
% 2.86/1.70 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.86/1.70 => ? [B: $i] :
% 2.86/1.70 ( ( in @ B @ A )
% 2.86/1.70 & ~ ? [Xx: $i] :
% 2.86/1.70 ( ( in @ Xx @ B )
% 2.86/1.70 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(wellorderingAx_type,type,
% 2.86/1.70 wellorderingAx: $o ).
% 2.86/1.70
% 2.86/1.70 thf(wellorderingAx,definition,
% 2.86/1.70 ( wellorderingAx
% 2.86/1.70 = ( ! [A: $i] :
% 2.86/1.70 ? [B: $i] :
% 2.86/1.70 ( ! [C: $i] :
% 2.86/1.70 ( ( in @ C @ B )
% 2.86/1.70 => ! [Xx: $i] :
% 2.86/1.70 ( ( in @ Xx @ C )
% 2.86/1.70 => ( in @ Xx @ A ) ) )
% 2.86/1.70 & ! [Xx: $i,Xy: $i] :
% 2.86/1.70 ( ( ( in @ Xx @ A )
% 2.86/1.70 & ( in @ Xy @ A ) )
% 2.86/1.70 => ( ! [C: $i] :
% 2.86/1.70 ( ( in @ C @ B )
% 2.86/1.70 => ( ( in @ Xx @ C )
% 2.86/1.70 <=> ( in @ Xy @ C ) ) )
% 2.86/1.70 => ( Xx = Xy ) ) )
% 2.86/1.70 & ! [C: $i,D: $i] :
% 2.86/1.70 ( ( ( in @ C @ B )
% 2.86/1.70 & ( in @ D @ B ) )
% 2.86/1.70 => ( ! [Xx: $i] :
% 2.86/1.70 ( ( in @ Xx @ C )
% 2.86/1.70 => ( in @ Xx @ D ) )
% 2.86/1.70 | ! [Xx: $i] :
% 2.86/1.70 ( ( in @ Xx @ D )
% 2.86/1.70 => ( in @ Xx @ C ) ) ) )
% 2.86/1.70 & ! [C: $i] :
% 2.86/1.70 ( ( ! [Xx: $i] :
% 2.86/1.70 ( ( in @ Xx @ C )
% 2.86/1.70 => ( in @ Xx @ A ) )
% 2.86/1.70 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 2.86/1.70 => ? [D: $i,Xx: $i] :
% 2.86/1.70 ( ( in @ D @ B )
% 2.86/1.70 & ( in @ Xx @ C )
% 2.86/1.70 & ~ ? [Xy: $i] :
% 2.86/1.70 ( ( in @ Xy @ D )
% 2.86/1.70 & ( in @ Xy @ C ) )
% 2.86/1.70 & ! [E: $i] :
% 2.86/1.70 ( ( in @ E @ B )
% 2.86/1.70 => ( ! [Xy: $i] :
% 2.86/1.70 ( ( in @ Xy @ E )
% 2.86/1.70 => ( in @ Xy @ D ) )
% 2.86/1.70 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(descr_type,type,
% 2.86/1.70 descr: ( $i > $o ) > $i ).
% 2.86/1.70
% 2.86/1.70 thf(descrp_type,type,
% 2.86/1.70 descrp: $o ).
% 2.86/1.70
% 2.86/1.70 thf(descrp,definition,
% 2.86/1.70 ( descrp
% 2.86/1.70 = ( ! [Xphi: $i > $o] :
% 2.86/1.70 ( ( exu
% 2.86/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.86/1.70 => ( Xphi
% 2.86/1.70 @ ( descr
% 2.86/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(dsetconstr_type,type,
% 2.86/1.70 dsetconstr: $i > ( $i > $o ) > $i ).
% 2.86/1.70
% 2.86/1.70 thf(dsetconstrI_type,type,
% 2.86/1.70 dsetconstrI: $o ).
% 2.86/1.70
% 2.86/1.70 thf(dsetconstrI,definition,
% 2.86/1.70 ( dsetconstrI
% 2.86/1.70 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.86/1.70 ( ( in @ Xx @ A )
% 2.86/1.70 => ( ( Xphi @ Xx )
% 2.86/1.70 => ( in @ Xx
% 2.86/1.70 @ ( dsetconstr @ A
% 2.86/1.70 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(dsetconstrEL_type,type,
% 2.86/1.70 dsetconstrEL: $o ).
% 2.86/1.70
% 2.86/1.70 thf(dsetconstrEL,definition,
% 2.86/1.70 ( dsetconstrEL
% 2.86/1.70 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.86/1.70 ( ( in @ Xx
% 2.86/1.70 @ ( dsetconstr @ A
% 2.86/1.70 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.86/1.70 => ( in @ Xx @ A ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(dsetconstrER_type,type,
% 2.86/1.70 dsetconstrER: $o ).
% 2.86/1.70
% 2.86/1.70 thf(dsetconstrER,definition,
% 2.86/1.70 ( dsetconstrER
% 2.86/1.70 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.86/1.70 ( ( in @ Xx
% 2.86/1.70 @ ( dsetconstr @ A
% 2.86/1.70 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.86/1.70 => ( Xphi @ Xx ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(exuE1_type,type,
% 2.86/1.70 exuE1: $o ).
% 2.86/1.70
% 2.86/1.70 thf(exuE1,definition,
% 2.86/1.70 ( exuE1
% 2.86/1.70 = ( ! [Xphi: $i > $o] :
% 2.86/1.70 ( ( exu
% 2.86/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.86/1.70 => ? [Xx: $i] :
% 2.86/1.70 ( ( Xphi @ Xx )
% 2.86/1.70 & ! [Xy: $i] :
% 2.86/1.70 ( ( Xphi @ Xy )
% 2.86/1.70 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(prop2set_type,type,
% 2.86/1.70 prop2set: $o > $i ).
% 2.86/1.70
% 2.86/1.70 thf(prop2setE_type,type,
% 2.86/1.70 prop2setE: $o ).
% 2.86/1.70
% 2.86/1.70 thf(prop2setE,definition,
% 2.86/1.70 ( prop2setE
% 2.86/1.70 = ( ! [Xphi: $o,Xx: $i] :
% 2.86/1.70 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 2.86/1.70 => Xphi ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(emptysetE_type,type,
% 2.86/1.70 emptysetE: $o ).
% 2.86/1.70
% 2.86/1.70 thf(emptysetE,definition,
% 2.86/1.70 ( emptysetE
% 2.86/1.70 = ( ! [Xx: $i] :
% 2.86/1.70 ( ( in @ Xx @ emptyset )
% 2.86/1.70 => ! [Xphi: $o] : Xphi ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(emptysetimpfalse_type,type,
% 2.86/1.70 emptysetimpfalse: $o ).
% 2.86/1.70
% 2.86/1.70 thf(emptysetimpfalse,definition,
% 2.86/1.70 ( emptysetimpfalse
% 2.86/1.70 = ( ! [Xx: $i] :
% 2.86/1.70 ( ( in @ Xx @ emptyset )
% 2.86/1.70 => $false ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(notinemptyset_type,type,
% 2.86/1.70 notinemptyset: $o ).
% 2.86/1.70
% 2.86/1.70 thf(notinemptyset,definition,
% 2.86/1.70 ( notinemptyset
% 2.86/1.70 = ( ! [Xx: $i] :
% 2.86/1.70 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(exuE3e_type,type,
% 2.86/1.70 exuE3e: $o ).
% 2.86/1.70
% 2.86/1.70 thf(exuE3e,definition,
% 2.86/1.70 ( exuE3e
% 2.86/1.70 = ( ! [Xphi: $i > $o] :
% 2.86/1.70 ( ( exu
% 2.86/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.86/1.70 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(setext_type,type,
% 2.86/1.70 setext: $o ).
% 2.86/1.70
% 2.86/1.70 thf(setext,definition,
% 2.86/1.70 ( setext
% 2.86/1.70 = ( ! [A: $i,B: $i] :
% 2.86/1.70 ( ! [Xx: $i] :
% 2.86/1.70 ( ( in @ Xx @ A )
% 2.86/1.70 => ( in @ Xx @ B ) )
% 2.86/1.70 => ( ! [Xx: $i] :
% 2.86/1.70 ( ( in @ Xx @ B )
% 2.86/1.70 => ( in @ Xx @ A ) )
% 2.86/1.70 => ( A = B ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(emptyI_type,type,
% 2.86/1.70 emptyI: $o ).
% 2.86/1.70
% 2.86/1.70 thf(emptyI,definition,
% 2.86/1.70 ( emptyI
% 2.86/1.70 = ( ! [A: $i] :
% 2.86/1.70 ( ! [Xx: $i] :
% 2.86/1.70 ~ ( in @ Xx @ A )
% 2.86/1.70 => ( A = emptyset ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(noeltsimpempty_type,type,
% 2.86/1.70 noeltsimpempty: $o ).
% 2.86/1.70
% 2.86/1.70 thf(noeltsimpempty,definition,
% 2.86/1.70 ( noeltsimpempty
% 2.86/1.70 = ( ! [A: $i] :
% 2.86/1.70 ( ! [Xx: $i] :
% 2.86/1.70 ~ ( in @ Xx @ A )
% 2.86/1.70 => ( A = emptyset ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(setbeta_type,type,
% 2.86/1.70 setbeta: $o ).
% 2.86/1.70
% 2.86/1.70 thf(setbeta,definition,
% 2.86/1.70 ( setbeta
% 2.86/1.70 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.86/1.70 ( ( in @ Xx @ A )
% 2.86/1.70 => ( ( in @ Xx
% 2.86/1.70 @ ( dsetconstr @ A
% 2.86/1.70 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.86/1.70 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(nonempty_type,type,
% 2.86/1.70 nonempty: $i > $o ).
% 2.86/1.70
% 2.86/1.70 thf(nonempty,definition,
% 2.86/1.70 ( nonempty
% 2.86/1.70 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(nonemptyE1_type,type,
% 2.86/1.70 nonemptyE1: $o ).
% 2.86/1.70
% 2.86/1.70 thf(nonemptyE1,definition,
% 2.86/1.70 ( nonemptyE1
% 2.86/1.70 = ( ! [A: $i] :
% 2.86/1.70 ( ( nonempty @ A )
% 2.86/1.70 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(nonemptyI_type,type,
% 2.86/1.70 nonemptyI: $o ).
% 2.86/1.70
% 2.86/1.70 thf(nonemptyI,definition,
% 2.86/1.70 ( nonemptyI
% 2.86/1.70 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.86/1.70 ( ( in @ Xx @ A )
% 2.86/1.70 => ( ( Xphi @ Xx )
% 2.86/1.70 => ( nonempty
% 2.86/1.70 @ ( dsetconstr @ A
% 2.86/1.70 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(nonemptyI1_type,type,
% 2.86/1.70 nonemptyI1: $o ).
% 2.86/1.70
% 2.86/1.70 thf(nonemptyI1,definition,
% 2.86/1.70 ( nonemptyI1
% 2.86/1.70 = ( ! [A: $i] :
% 2.86/1.70 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.86/1.70 => ( nonempty @ A ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(setadjoinIL_type,type,
% 2.86/1.70 setadjoinIL: $o ).
% 2.86/1.70
% 2.86/1.70 thf(setadjoinIL,definition,
% 2.86/1.70 ( setadjoinIL
% 2.86/1.70 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(emptyinunitempty_type,type,
% 2.86/1.70 emptyinunitempty: $o ).
% 2.86/1.70
% 2.86/1.70 thf(emptyinunitempty,definition,
% 2.86/1.70 ( emptyinunitempty
% 2.86/1.70 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(setadjoinIR_type,type,
% 2.86/1.70 setadjoinIR: $o ).
% 2.86/1.70
% 2.86/1.70 thf(setadjoinIR,definition,
% 2.86/1.70 ( setadjoinIR
% 2.86/1.70 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.86/1.70 ( ( in @ Xy @ A )
% 2.86/1.70 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(setadjoinE_type,type,
% 2.86/1.70 setadjoinE: $o ).
% 2.86/1.70
% 2.86/1.70 thf(setadjoinE,definition,
% 2.86/1.70 ( setadjoinE
% 2.86/1.70 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.86/1.70 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.86/1.70 => ! [Xphi: $o] :
% 2.86/1.70 ( ( ( Xy = Xx )
% 2.86/1.70 => Xphi )
% 2.86/1.70 => ( ( ( in @ Xy @ A )
% 2.86/1.70 => Xphi )
% 2.86/1.70 => Xphi ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(setadjoinOr_type,type,
% 2.86/1.70 setadjoinOr: $o ).
% 2.86/1.70
% 2.86/1.70 thf(setadjoinOr,definition,
% 2.86/1.70 ( setadjoinOr
% 2.86/1.70 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.86/1.70 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.86/1.70 => ( ( Xy = Xx )
% 2.86/1.70 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(setoftrueEq_type,type,
% 2.86/1.70 setoftrueEq: $o ).
% 2.86/1.70
% 2.86/1.70 thf(setoftrueEq,definition,
% 2.86/1.70 ( setoftrueEq
% 2.86/1.70 = ( ! [A: $i] :
% 2.86/1.70 ( ( dsetconstr @ A
% 2.86/1.70 @ ^ [Xx: $i] : $true )
% 2.86/1.70 = A ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(powersetI_type,type,
% 2.86/1.70 powersetI: $o ).
% 2.86/1.70
% 2.86/1.70 thf(powersetI,definition,
% 2.86/1.70 ( powersetI
% 2.86/1.70 = ( ! [A: $i,B: $i] :
% 2.86/1.70 ( ! [Xx: $i] :
% 2.86/1.70 ( ( in @ Xx @ B )
% 2.86/1.70 => ( in @ Xx @ A ) )
% 2.86/1.70 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(emptyinPowerset_type,type,
% 2.86/1.70 emptyinPowerset: $o ).
% 2.86/1.70
% 2.86/1.70 thf(emptyinPowerset,definition,
% 2.86/1.70 ( emptyinPowerset
% 2.86/1.70 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(emptyInPowerset_type,type,
% 2.86/1.70 emptyInPowerset: $o ).
% 2.86/1.70
% 2.86/1.70 thf(emptyInPowerset,definition,
% 2.86/1.70 ( emptyInPowerset
% 2.86/1.70 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(powersetE_type,type,
% 2.86/1.70 powersetE: $o ).
% 2.86/1.70
% 2.86/1.70 thf(powersetE,definition,
% 2.86/1.70 ( powersetE
% 2.86/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.86/1.70 ( ( in @ B @ ( powerset @ A ) )
% 2.86/1.70 => ( ( in @ Xx @ B )
% 2.86/1.70 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(setunionI_type,type,
% 2.86/1.70 setunionI: $o ).
% 2.86/1.70
% 2.86/1.70 thf(setunionI,definition,
% 2.86/1.70 ( setunionI
% 2.86/1.70 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.86/1.70 ( ( in @ Xx @ B )
% 2.86/1.70 => ( ( in @ B @ A )
% 2.86/1.70 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(setunionE_type,type,
% 2.86/1.70 setunionE: $o ).
% 2.86/1.70
% 2.86/1.70 thf(setunionE,definition,
% 2.86/1.70 ( setunionE
% 2.86/1.70 = ( ! [A: $i,Xx: $i] :
% 2.86/1.70 ( ( in @ Xx @ ( setunion @ A ) )
% 2.86/1.70 => ! [Xphi: $o] :
% 2.86/1.70 ( ! [B: $i] :
% 2.86/1.70 ( ( in @ Xx @ B )
% 2.86/1.70 => ( ( in @ B @ A )
% 2.86/1.70 => Xphi ) )
% 2.86/1.70 => Xphi ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(subPowSU_type,type,
% 2.86/1.70 subPowSU: $o ).
% 2.86/1.70
% 2.86/1.70 thf(subPowSU,definition,
% 2.86/1.70 ( subPowSU
% 2.86/1.70 = ( ! [A: $i,Xx: $i] :
% 2.86/1.70 ( ( in @ Xx @ A )
% 2.86/1.70 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(exuE2_type,type,
% 2.86/1.70 exuE2: $o ).
% 2.86/1.70
% 2.86/1.70 thf(exuE2,definition,
% 2.86/1.70 ( exuE2
% 2.86/1.70 = ( ! [Xphi: $i > $o] :
% 2.86/1.70 ( ( exu
% 2.86/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.86/1.70 => ? [Xx: $i] :
% 2.86/1.70 ! [Xy: $i] :
% 2.86/1.70 ( ( Xphi @ Xy )
% 2.86/1.70 <=> ( Xy = Xx ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(nonemptyImpWitness_type,type,
% 2.86/1.70 nonemptyImpWitness: $o ).
% 2.86/1.70
% 2.86/1.70 thf(nonemptyImpWitness,definition,
% 2.86/1.70 ( nonemptyImpWitness
% 2.86/1.70 = ( ! [A: $i] :
% 2.86/1.70 ( ( nonempty @ A )
% 2.86/1.70 => ? [Xx: $i] :
% 2.86/1.70 ( ( in @ Xx @ A )
% 2.86/1.70 & $true ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(uniqinunit_type,type,
% 2.86/1.70 uniqinunit: $o ).
% 2.86/1.70
% 2.86/1.70 thf(uniqinunit,definition,
% 2.86/1.70 ( uniqinunit
% 2.86/1.70 = ( ! [Xx: $i,Xy: $i] :
% 2.86/1.70 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.86/1.70 => ( Xx = Xy ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(notinsingleton_type,type,
% 2.86/1.70 notinsingleton: $o ).
% 2.86/1.70
% 2.86/1.70 thf(notinsingleton,definition,
% 2.86/1.70 ( notinsingleton
% 2.86/1.70 = ( ! [Xx: $i,Xy: $i] :
% 2.86/1.70 ( ( Xx != Xy )
% 2.86/1.70 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(eqinunit_type,type,
% 2.86/1.70 eqinunit: $o ).
% 2.86/1.70
% 2.86/1.70 thf(eqinunit,definition,
% 2.86/1.70 ( eqinunit
% 2.86/1.70 = ( ! [Xx: $i,Xy: $i] :
% 2.86/1.70 ( ( Xx = Xy )
% 2.86/1.70 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(singletonsswitch_type,type,
% 2.86/1.70 singletonsswitch: $o ).
% 2.86/1.70
% 2.86/1.70 thf(singletonsswitch,definition,
% 2.86/1.70 ( singletonsswitch
% 2.86/1.70 = ( ! [Xx: $i,Xy: $i] :
% 2.86/1.70 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.86/1.70 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(upairsetE_type,type,
% 2.86/1.70 upairsetE: $o ).
% 2.86/1.70
% 2.86/1.70 thf(upairsetE,definition,
% 2.86/1.70 ( upairsetE
% 2.86/1.70 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.86/1.70 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.86/1.70 => ( ( Xz = Xx )
% 2.86/1.70 | ( Xz = Xy ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(upairsetIL_type,type,
% 2.86/1.70 upairsetIL: $o ).
% 2.86/1.70
% 2.86/1.70 thf(upairsetIL,definition,
% 2.86/1.70 ( upairsetIL
% 2.86/1.70 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(upairsetIR_type,type,
% 2.86/1.70 upairsetIR: $o ).
% 2.86/1.70
% 2.86/1.70 thf(upairsetIR,definition,
% 2.86/1.70 ( upairsetIR
% 2.86/1.70 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(emptyE1_type,type,
% 2.86/1.70 emptyE1: $o ).
% 2.86/1.70
% 2.86/1.70 thf(emptyE1,definition,
% 2.86/1.70 ( emptyE1
% 2.86/1.70 = ( ! [A: $i,Xphi: $i > $o] :
% 2.86/1.70 ( ? [Xx: $i] :
% 2.86/1.70 ( ( in @ Xx @ A )
% 2.86/1.70 & ( Xphi @ Xx ) )
% 2.86/1.70 => ( ( ( dsetconstr @ A
% 2.86/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.86/1.70 = emptyset )
% 2.86/1.70 => $false ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(vacuousDall_type,type,
% 2.86/1.70 vacuousDall: $o ).
% 2.86/1.70
% 2.86/1.70 thf(vacuousDall,definition,
% 2.86/1.70 ( vacuousDall
% 2.86/1.70 = ( ! [Xphi: $i > $o,Xx: $i] :
% 2.86/1.70 ( ( in @ Xx @ emptyset )
% 2.86/1.70 => ( Xphi @ Xx ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(quantDeMorgan1_type,type,
% 2.86/1.70 quantDeMorgan1: $o ).
% 2.86/1.70
% 2.86/1.70 thf(quantDeMorgan1,definition,
% 2.86/1.70 ( quantDeMorgan1
% 2.86/1.70 = ( ! [A: $i,Xphi: $i > $o] :
% 2.86/1.70 ( ~ ! [Xx: $i] :
% 2.86/1.70 ( ( in @ Xx @ A )
% 2.86/1.70 => ( Xphi @ Xx ) )
% 2.86/1.70 => ? [Xx: $i] :
% 2.86/1.70 ( ( in @ Xx @ A )
% 2.86/1.70 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(quantDeMorgan2_type,type,
% 2.86/1.70 quantDeMorgan2: $o ).
% 2.86/1.70
% 2.86/1.70 thf(quantDeMorgan2,definition,
% 2.86/1.70 ( quantDeMorgan2
% 2.86/1.70 = ( ! [A: $i,Xphi: $i > $o] :
% 2.86/1.70 ( ! [Xx: $i] :
% 2.86/1.70 ( ( in @ Xx @ A )
% 2.86/1.70 => ~ ( Xphi @ Xx ) )
% 2.86/1.70 => ~ ? [Xx: $i] :
% 2.86/1.70 ( ( in @ Xx @ A )
% 2.86/1.70 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(quantDeMorgan3_type,type,
% 2.86/1.70 quantDeMorgan3: $o ).
% 2.86/1.70
% 2.86/1.70 thf(quantDeMorgan3,definition,
% 2.86/1.70 ( quantDeMorgan3
% 2.86/1.70 = ( ! [A: $i,Xphi: $i > $o] :
% 2.86/1.70 ( ~ ? [Xx: $i] :
% 2.86/1.70 ( ( in @ Xx @ A )
% 2.86/1.70 & ( Xphi @ Xx ) )
% 2.86/1.70 => ! [Xx: $i] :
% 2.86/1.70 ( ( in @ Xx @ A )
% 2.86/1.70 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(quantDeMorgan4_type,type,
% 2.86/1.70 quantDeMorgan4: $o ).
% 2.86/1.70
% 2.86/1.70 thf(quantDeMorgan4,definition,
% 2.86/1.70 ( quantDeMorgan4
% 2.86/1.70 = ( ! [A: $i,Xphi: $i > $o] :
% 2.86/1.70 ( ? [Xx: $i] :
% 2.86/1.70 ( ( in @ Xx @ A )
% 2.86/1.70 & ~ ( Xphi @ Xx ) )
% 2.86/1.70 => ~ ! [Xx: $i] :
% 2.86/1.70 ( ( in @ Xx @ A )
% 2.86/1.70 => ( Xphi @ Xx ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(prop2setI_type,type,
% 2.86/1.70 prop2setI: $o ).
% 2.86/1.70
% 2.86/1.70 thf(prop2setI,definition,
% 2.86/1.70 ( prop2setI
% 2.86/1.70 = ( ! [Xphi: $o] :
% 2.86/1.70 ( Xphi
% 2.86/1.70 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(set2prop_type,type,
% 2.86/1.70 set2prop: $i > $o ).
% 2.86/1.70
% 2.86/1.70 thf(prop2set2propI_type,type,
% 2.86/1.70 prop2set2propI: $o ).
% 2.86/1.70
% 2.86/1.70 thf(prop2set2propI,definition,
% 2.86/1.70 ( prop2set2propI
% 2.86/1.70 = ( ! [Xphi: $o] :
% 2.86/1.70 ( Xphi
% 2.86/1.70 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(notdexE_type,type,
% 2.86/1.70 notdexE: $o ).
% 2.86/1.70
% 2.86/1.70 thf(notdexE,definition,
% 2.86/1.70 ( notdexE
% 2.86/1.70 = ( ! [A: $i,Xphi: $i > $o] :
% 2.86/1.70 ( ~ ? [Xx: $i] :
% 2.86/1.70 ( ( in @ Xx @ A )
% 2.86/1.70 & ( Xphi @ Xx ) )
% 2.86/1.70 => ! [Xx: $i] :
% 2.86/1.70 ( ( in @ Xx @ A )
% 2.86/1.70 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(notdallE_type,type,
% 2.86/1.70 notdallE: $o ).
% 2.86/1.70
% 2.86/1.70 thf(notdallE,definition,
% 2.86/1.70 ( notdallE
% 2.86/1.70 = ( ! [A: $i,Xphi: $i > $o] :
% 2.86/1.70 ( ~ ! [Xx: $i] :
% 2.86/1.70 ( ( in @ Xx @ A )
% 2.86/1.70 => ( Xphi @ Xx ) )
% 2.86/1.70 => ? [Xx: $i] :
% 2.86/1.70 ( ( in @ Xx @ A )
% 2.86/1.70 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(exuI1_type,type,
% 2.86/1.70 exuI1: $o ).
% 2.86/1.70
% 2.86/1.70 thf(exuI1,definition,
% 2.86/1.70 ( exuI1
% 2.86/1.70 = ( ! [Xphi: $i > $o] :
% 2.86/1.70 ( ? [Xx: $i] :
% 2.86/1.70 ( ( Xphi @ Xx )
% 2.86/1.70 & ! [Xy: $i] :
% 2.86/1.70 ( ( Xphi @ Xy )
% 2.86/1.70 => ( Xx = Xy ) ) )
% 2.86/1.70 => ( exu
% 2.86/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(exuI3_type,type,
% 2.86/1.70 exuI3: $o ).
% 2.86/1.70
% 2.86/1.70 thf(exuI3,definition,
% 2.86/1.70 ( exuI3
% 2.86/1.70 = ( ! [Xphi: $i > $o] :
% 2.86/1.70 ( ? [Xx: $i] : ( Xphi @ Xx )
% 2.86/1.70 => ( ! [Xx: $i,Xy: $i] :
% 2.86/1.70 ( ( Xphi @ Xx )
% 2.86/1.70 => ( ( Xphi @ Xy )
% 2.86/1.70 => ( Xx = Xy ) ) )
% 2.86/1.70 => ( exu
% 2.86/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(exuI2_type,type,
% 2.86/1.70 exuI2: $o ).
% 2.86/1.70
% 2.86/1.70 thf(exuI2,definition,
% 2.86/1.70 ( exuI2
% 2.86/1.70 = ( ! [Xphi: $i > $o] :
% 2.86/1.70 ( ? [Xx: $i] :
% 2.86/1.70 ! [Xy: $i] :
% 2.86/1.70 ( ( Xphi @ Xy )
% 2.86/1.70 <=> ( Xy = Xx ) )
% 2.86/1.70 => ( exu
% 2.86/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(inCongP_type,type,
% 2.86/1.70 inCongP: $o ).
% 2.86/1.70
% 2.86/1.70 thf(inCongP,definition,
% 2.86/1.70 ( inCongP
% 2.86/1.70 = ( ! [A: $i,B: $i] :
% 2.86/1.70 ( ( A = B )
% 2.86/1.70 => ! [Xx: $i,Xy: $i] :
% 2.86/1.70 ( ( Xx = Xy )
% 2.86/1.70 => ( ( in @ Xx @ A )
% 2.86/1.70 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(in__Cong_type,type,
% 2.86/1.70 in__Cong: $o ).
% 2.86/1.70
% 2.86/1.70 thf(in__Cong,definition,
% 2.86/1.70 ( in__Cong
% 2.86/1.70 = ( ! [A: $i,B: $i] :
% 2.86/1.70 ( ( A = B )
% 2.86/1.70 => ! [Xx: $i,Xy: $i] :
% 2.86/1.70 ( ( Xx = Xy )
% 2.86/1.70 => ( ( in @ Xx @ A )
% 2.86/1.70 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(exuE3u_type,type,
% 2.86/1.70 exuE3u: $o ).
% 2.86/1.70
% 2.86/1.70 thf(exuE3u,definition,
% 2.86/1.70 ( exuE3u
% 2.86/1.70 = ( ! [Xphi: $i > $o] :
% 2.86/1.70 ( ( exu
% 2.86/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.86/1.70 => ! [Xx: $i,Xy: $i] :
% 2.86/1.70 ( ( Xphi @ Xx )
% 2.86/1.70 => ( ( Xphi @ Xy )
% 2.86/1.70 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(exu__Cong_type,type,
% 2.86/1.70 exu__Cong: $o ).
% 2.86/1.70
% 2.86/1.70 thf(exu__Cong,definition,
% 2.86/1.70 ( exu__Cong
% 2.86/1.70 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.86/1.70 ( ! [Xx: $i,Xy: $i] :
% 2.86/1.70 ( ( Xx = Xy )
% 2.86/1.70 => ( ( Xphi @ Xx )
% 2.86/1.70 <=> ( Xpsi @ Xy ) ) )
% 2.86/1.70 => ( ( exu
% 2.86/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.86/1.70 <=> ( exu
% 2.86/1.70 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(emptyset__Cong_type,type,
% 2.86/1.70 emptyset__Cong: $o ).
% 2.86/1.70
% 2.86/1.70 thf(emptyset__Cong,definition,
% 2.86/1.70 ( emptyset__Cong
% 2.86/1.70 = ( emptyset = emptyset ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(setadjoin__Cong_type,type,
% 2.86/1.70 setadjoin__Cong: $o ).
% 2.86/1.70
% 2.86/1.70 thf(setadjoin__Cong,definition,
% 2.86/1.70 ( setadjoin__Cong
% 2.86/1.70 = ( ! [Xx: $i,Xy: $i] :
% 2.86/1.70 ( ( Xx = Xy )
% 2.86/1.70 => ! [Xz: $i,Xu: $i] :
% 2.86/1.70 ( ( Xz = Xu )
% 2.86/1.70 => ( ( setadjoin @ Xx @ Xz )
% 2.86/1.70 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(powerset__Cong_type,type,
% 2.86/1.70 powerset__Cong: $o ).
% 2.86/1.70
% 2.86/1.70 thf(powerset__Cong,definition,
% 2.86/1.70 ( powerset__Cong
% 2.86/1.70 = ( ! [A: $i,B: $i] :
% 2.86/1.70 ( ( A = B )
% 2.86/1.70 => ( ( powerset @ A )
% 2.86/1.70 = ( powerset @ B ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(setunion__Cong_type,type,
% 2.86/1.70 setunion__Cong: $o ).
% 2.86/1.70
% 2.86/1.70 thf(setunion__Cong,definition,
% 2.86/1.70 ( setunion__Cong
% 2.86/1.70 = ( ! [A: $i,B: $i] :
% 2.86/1.70 ( ( A = B )
% 2.86/1.70 => ( ( setunion @ A )
% 2.86/1.70 = ( setunion @ B ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(omega__Cong_type,type,
% 2.86/1.70 omega__Cong: $o ).
% 2.86/1.70
% 2.86/1.70 thf(omega__Cong,definition,
% 2.86/1.70 ( omega__Cong
% 2.86/1.70 = ( omega = omega ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(exuEu_type,type,
% 2.86/1.70 exuEu: $o ).
% 2.86/1.70
% 2.86/1.70 thf(exuEu,definition,
% 2.86/1.70 ( exuEu
% 2.86/1.70 = ( ! [Xphi: $i > $o] :
% 2.86/1.70 ( ( exu
% 2.86/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.86/1.70 => ! [Xx: $i,Xy: $i] :
% 2.86/1.70 ( ( Xphi @ Xx )
% 2.86/1.70 => ( ( Xphi @ Xy )
% 2.86/1.70 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(descr__Cong_type,type,
% 2.86/1.70 descr__Cong: $o ).
% 2.86/1.70
% 2.86/1.70 thf(descr__Cong,definition,
% 2.86/1.70 ( descr__Cong
% 2.86/1.70 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.86/1.70 ( ! [Xx: $i,Xy: $i] :
% 2.86/1.70 ( ( Xx = Xy )
% 2.86/1.70 => ( ( Xphi @ Xx )
% 2.86/1.70 <=> ( Xpsi @ Xy ) ) )
% 2.86/1.70 => ( ( exu
% 2.86/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.86/1.70 => ( ( exu
% 2.86/1.70 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 2.86/1.70 => ( ( descr
% 2.86/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.86/1.70 = ( descr
% 2.86/1.70 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(dsetconstr__Cong_type,type,
% 2.86/1.70 dsetconstr__Cong: $o ).
% 2.86/1.70
% 2.86/1.70 thf(dsetconstr__Cong,definition,
% 2.86/1.70 ( dsetconstr__Cong
% 2.86/1.70 = ( ! [A: $i,B: $i] :
% 2.86/1.70 ( ( A = B )
% 2.86/1.70 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.86/1.70 ( ! [Xx: $i] :
% 2.86/1.70 ( ( in @ Xx @ A )
% 2.86/1.70 => ! [Xy: $i] :
% 2.86/1.70 ( ( in @ Xy @ B )
% 2.86/1.70 => ( ( Xx = Xy )
% 2.86/1.70 => ( ( Xphi @ Xx )
% 2.86/1.70 <=> ( Xpsi @ Xy ) ) ) ) )
% 2.86/1.70 => ( ( dsetconstr @ A
% 2.86/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.86/1.70 = ( dsetconstr @ B
% 2.86/1.70 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(subset_type,type,
% 2.86/1.70 subset: $i > $i > $o ).
% 2.86/1.70
% 2.86/1.70 thf(disjoint_type,type,
% 2.86/1.70 disjoint: $i > $i > $o ).
% 2.86/1.70
% 2.86/1.70 thf(setsmeet_type,type,
% 2.86/1.70 setsmeet: $i > $i > $o ).
% 2.86/1.70
% 2.86/1.70 thf(subsetI1_type,type,
% 2.86/1.70 subsetI1: $o ).
% 2.86/1.70
% 2.86/1.70 thf(subsetI1,definition,
% 2.86/1.70 ( subsetI1
% 2.86/1.70 = ( ! [A: $i,B: $i] :
% 2.86/1.70 ( ! [Xx: $i] :
% 2.86/1.70 ( ( in @ Xx @ A )
% 2.86/1.70 => ( in @ Xx @ B ) )
% 2.86/1.70 => ( subset @ A @ B ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(eqimpsubset2_type,type,
% 2.86/1.70 eqimpsubset2: $o ).
% 2.86/1.70
% 2.86/1.70 thf(eqimpsubset2,definition,
% 2.86/1.70 ( eqimpsubset2
% 2.86/1.70 = ( ! [A: $i,B: $i] :
% 2.86/1.70 ( ( A = B )
% 2.86/1.70 => ( subset @ B @ A ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(eqimpsubset1_type,type,
% 2.86/1.70 eqimpsubset1: $o ).
% 2.86/1.70
% 2.86/1.70 thf(eqimpsubset1,definition,
% 2.86/1.70 ( eqimpsubset1
% 2.86/1.70 = ( ! [A: $i,B: $i] :
% 2.86/1.70 ( ( A = B )
% 2.86/1.70 => ( subset @ A @ B ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(subsetI2_type,type,
% 2.86/1.70 subsetI2: $o ).
% 2.86/1.70
% 2.86/1.70 thf(subsetI2,definition,
% 2.86/1.70 ( subsetI2
% 2.86/1.70 = ( ! [A: $i,B: $i] :
% 2.86/1.70 ( ! [Xx: $i] :
% 2.86/1.70 ( ( in @ Xx @ A )
% 2.86/1.70 => ( in @ Xx @ B ) )
% 2.86/1.70 => ( subset @ A @ B ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(emptysetsubset_type,type,
% 2.86/1.70 emptysetsubset: $o ).
% 2.86/1.70
% 2.86/1.70 thf(emptysetsubset,definition,
% 2.86/1.70 ( emptysetsubset
% 2.86/1.70 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(subsetE_type,type,
% 2.86/1.70 subsetE: $o ).
% 2.86/1.70
% 2.86/1.70 thf(subsetE,definition,
% 2.86/1.70 ( subsetE
% 2.86/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.86/1.70 ( ( subset @ A @ B )
% 2.86/1.70 => ( ( in @ Xx @ A )
% 2.86/1.70 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(subsetE2_type,type,
% 2.86/1.70 subsetE2: $o ).
% 2.86/1.70
% 2.86/1.70 thf(subsetE2,definition,
% 2.86/1.70 ( subsetE2
% 2.86/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.86/1.70 ( ( subset @ A @ B )
% 2.86/1.70 => ( ~ ( in @ Xx @ B )
% 2.86/1.70 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(notsubsetI_type,type,
% 2.86/1.70 notsubsetI: $o ).
% 2.86/1.70
% 2.86/1.70 thf(notsubsetI,definition,
% 2.86/1.70 ( notsubsetI
% 2.86/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.86/1.70 ( ( in @ Xx @ A )
% 2.86/1.70 => ( ~ ( in @ Xx @ B )
% 2.86/1.70 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(notequalI1_type,type,
% 2.86/1.70 notequalI1: $o ).
% 2.86/1.70
% 2.86/1.70 thf(notequalI1,definition,
% 2.86/1.70 ( notequalI1
% 2.86/1.70 = ( ! [A: $i,B: $i] :
% 2.86/1.70 ( ~ ( subset @ A @ B )
% 2.86/1.70 => ( A != B ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(notequalI2_type,type,
% 2.86/1.70 notequalI2: $o ).
% 2.86/1.70
% 2.86/1.70 thf(notequalI2,definition,
% 2.86/1.70 ( notequalI2
% 2.86/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.86/1.70 ( ( in @ Xx @ A )
% 2.86/1.70 => ( ~ ( in @ Xx @ B )
% 2.86/1.70 => ( A != B ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(subsetRefl_type,type,
% 2.86/1.70 subsetRefl: $o ).
% 2.86/1.70
% 2.86/1.70 thf(subsetRefl,definition,
% 2.86/1.70 ( subsetRefl
% 2.86/1.70 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(subsetTrans_type,type,
% 2.86/1.70 subsetTrans: $o ).
% 2.86/1.70
% 2.86/1.70 thf(subsetTrans,definition,
% 2.86/1.70 ( subsetTrans
% 2.86/1.70 = ( ! [A: $i,B: $i,C: $i] :
% 2.86/1.70 ( ( subset @ A @ B )
% 2.86/1.70 => ( ( subset @ B @ C )
% 2.86/1.70 => ( subset @ A @ C ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(setadjoinSub_type,type,
% 2.86/1.70 setadjoinSub: $o ).
% 2.86/1.70
% 2.86/1.70 thf(setadjoinSub,definition,
% 2.86/1.70 ( setadjoinSub
% 2.86/1.70 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(setadjoinSub2_type,type,
% 2.86/1.70 setadjoinSub2: $o ).
% 2.86/1.70
% 2.86/1.70 thf(setadjoinSub2,definition,
% 2.86/1.70 ( setadjoinSub2
% 2.86/1.70 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.86/1.70 ( ( subset @ A @ B )
% 2.86/1.70 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(subset2powerset_type,type,
% 2.86/1.70 subset2powerset: $o ).
% 2.86/1.70
% 2.86/1.70 thf(subset2powerset,definition,
% 2.86/1.70 ( subset2powerset
% 2.86/1.70 = ( ! [A: $i,B: $i] :
% 2.86/1.70 ( ( subset @ A @ B )
% 2.86/1.70 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(setextsub_type,type,
% 2.86/1.70 setextsub: $o ).
% 2.86/1.70
% 2.86/1.70 thf(setextsub,definition,
% 2.86/1.70 ( setextsub
% 2.86/1.70 = ( ! [A: $i,B: $i] :
% 2.86/1.70 ( ( subset @ A @ B )
% 2.86/1.70 => ( ( subset @ B @ A )
% 2.86/1.70 => ( A = B ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(subsetemptysetimpeq_type,type,
% 2.86/1.70 subsetemptysetimpeq: $o ).
% 2.86/1.70
% 2.86/1.70 thf(subsetemptysetimpeq,definition,
% 2.86/1.70 ( subsetemptysetimpeq
% 2.86/1.70 = ( ! [A: $i] :
% 2.86/1.70 ( ( subset @ A @ emptyset )
% 2.86/1.70 => ( A = emptyset ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(powersetI1_type,type,
% 2.86/1.70 powersetI1: $o ).
% 2.86/1.70
% 2.86/1.70 thf(powersetI1,definition,
% 2.86/1.70 ( powersetI1
% 2.86/1.70 = ( ! [A: $i,B: $i] :
% 2.86/1.70 ( ( subset @ B @ A )
% 2.86/1.70 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(powersetE1_type,type,
% 2.86/1.70 powersetE1: $o ).
% 2.86/1.70
% 2.86/1.70 thf(powersetE1,definition,
% 2.86/1.70 ( powersetE1
% 2.86/1.70 = ( ! [A: $i,B: $i] :
% 2.86/1.70 ( ( in @ B @ ( powerset @ A ) )
% 2.86/1.70 => ( subset @ B @ A ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(inPowerset_type,type,
% 2.86/1.70 inPowerset: $o ).
% 2.86/1.70
% 2.86/1.70 thf(inPowerset,definition,
% 2.86/1.70 ( inPowerset
% 2.86/1.70 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(powersetsubset_type,type,
% 2.86/1.70 powersetsubset: $o ).
% 2.86/1.70
% 2.86/1.70 thf(powersetsubset,definition,
% 2.86/1.70 ( powersetsubset
% 2.86/1.70 = ( ! [A: $i,B: $i] :
% 2.86/1.70 ( ( subset @ A @ B )
% 2.86/1.70 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(sepInPowerset_type,type,
% 2.86/1.70 sepInPowerset: $o ).
% 2.86/1.70
% 2.86/1.70 thf(sepInPowerset,definition,
% 2.86/1.70 ( sepInPowerset
% 2.86/1.70 = ( ! [A: $i,Xphi: $i > $o] :
% 2.86/1.70 ( in
% 2.86/1.70 @ ( dsetconstr @ A
% 2.86/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.86/1.70 @ ( powerset @ A ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(sepSubset_type,type,
% 2.86/1.70 sepSubset: $o ).
% 2.86/1.70
% 2.86/1.70 thf(sepSubset,definition,
% 2.86/1.70 ( sepSubset
% 2.86/1.70 = ( ! [A: $i,Xphi: $i > $o] :
% 2.86/1.70 ( subset
% 2.86/1.70 @ ( dsetconstr @ A
% 2.86/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.86/1.70 @ A ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(binunion_type,type,
% 2.86/1.70 binunion: $i > $i > $i ).
% 2.86/1.70
% 2.86/1.70 thf(binunionIL_type,type,
% 2.86/1.70 binunionIL: $o ).
% 2.86/1.70
% 2.86/1.70 thf(binunionIL,definition,
% 2.86/1.70 ( binunionIL
% 2.86/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.86/1.70 ( ( in @ Xx @ A )
% 2.86/1.70 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(upairset2IR_type,type,
% 2.86/1.70 upairset2IR: $o ).
% 2.86/1.70
% 2.86/1.70 thf(upairset2IR,definition,
% 2.86/1.70 ( upairset2IR
% 2.86/1.70 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(binunionIR_type,type,
% 2.86/1.70 binunionIR: $o ).
% 2.86/1.70
% 2.86/1.70 thf(binunionIR,definition,
% 2.86/1.70 ( binunionIR
% 2.86/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.86/1.70 ( ( in @ Xx @ B )
% 2.86/1.70 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(binunionEcases_type,type,
% 2.86/1.70 binunionEcases: $o ).
% 2.86/1.70
% 2.86/1.70 thf(binunionEcases,definition,
% 2.86/1.70 ( binunionEcases
% 2.86/1.70 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 2.86/1.70 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.86/1.70 => ( ( ( in @ Xx @ A )
% 2.86/1.70 => Xphi )
% 2.86/1.70 => ( ( ( in @ Xx @ B )
% 2.86/1.70 => Xphi )
% 2.86/1.70 => Xphi ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(binunionE_type,type,
% 2.86/1.70 binunionE: $o ).
% 2.86/1.70
% 2.86/1.70 thf(binunionE,definition,
% 2.86/1.70 ( binunionE
% 2.86/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.86/1.70 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.86/1.70 => ( ( in @ Xx @ A )
% 2.86/1.70 | ( in @ Xx @ B ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(binunionLsub_type,type,
% 2.86/1.70 binunionLsub: $o ).
% 2.86/1.70
% 2.86/1.70 thf(binunionLsub,definition,
% 2.86/1.70 ( binunionLsub
% 2.86/1.70 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(binunionRsub_type,type,
% 2.86/1.70 binunionRsub: $o ).
% 2.86/1.70
% 2.86/1.70 thf(binunionRsub,definition,
% 2.86/1.70 ( binunionRsub
% 2.86/1.70 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(binintersect_type,type,
% 2.86/1.70 binintersect: $i > $i > $i ).
% 2.86/1.70
% 2.86/1.70 thf(binintersectI_type,type,
% 2.86/1.70 binintersectI: $o ).
% 2.86/1.70
% 2.86/1.70 thf(binintersectI,definition,
% 2.86/1.70 ( binintersectI
% 2.86/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.86/1.70 ( ( in @ Xx @ A )
% 2.86/1.70 => ( ( in @ Xx @ B )
% 2.86/1.70 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(binintersectSubset5_type,type,
% 2.86/1.70 binintersectSubset5: $o ).
% 2.86/1.70
% 2.86/1.70 thf(binintersectSubset5,definition,
% 2.86/1.70 ( binintersectSubset5
% 2.86/1.70 = ( ! [A: $i,B: $i,C: $i] :
% 2.86/1.70 ( ( subset @ C @ A )
% 2.86/1.70 => ( ( subset @ C @ B )
% 2.86/1.70 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(binintersectEL_type,type,
% 2.86/1.70 binintersectEL: $o ).
% 2.86/1.70
% 2.86/1.70 thf(binintersectEL,definition,
% 2.86/1.70 ( binintersectEL
% 2.86/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.86/1.70 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.86/1.70 => ( in @ Xx @ A ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(binintersectLsub_type,type,
% 2.86/1.70 binintersectLsub: $o ).
% 2.86/1.70
% 2.86/1.70 thf(binintersectLsub,definition,
% 2.86/1.70 ( binintersectLsub
% 2.86/1.70 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(binintersectSubset2_type,type,
% 2.86/1.70 binintersectSubset2: $o ).
% 2.86/1.70
% 2.86/1.70 thf(binintersectSubset2,definition,
% 2.86/1.70 ( binintersectSubset2
% 2.86/1.70 = ( ! [A: $i,B: $i] :
% 2.86/1.70 ( ( subset @ A @ B )
% 2.86/1.70 => ( ( binintersect @ A @ B )
% 2.86/1.70 = A ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(binintersectSubset3_type,type,
% 2.86/1.70 binintersectSubset3: $o ).
% 2.86/1.70
% 2.86/1.70 thf(binintersectSubset3,definition,
% 2.86/1.70 ( binintersectSubset3
% 2.86/1.70 = ( ! [A: $i,B: $i] :
% 2.86/1.70 ( ( ( binintersect @ A @ B )
% 2.86/1.70 = B )
% 2.86/1.70 => ( subset @ B @ A ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(binintersectER_type,type,
% 2.86/1.70 binintersectER: $o ).
% 2.86/1.70
% 2.86/1.70 thf(binintersectER,definition,
% 2.86/1.70 ( binintersectER
% 2.86/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.86/1.70 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.86/1.70 => ( in @ Xx @ B ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(disjointsetsI1_type,type,
% 2.86/1.70 disjointsetsI1: $o ).
% 2.86/1.70
% 2.86/1.70 thf(disjointsetsI1,definition,
% 2.86/1.70 ( disjointsetsI1
% 2.86/1.70 = ( ! [A: $i,B: $i] :
% 2.86/1.70 ( ~ ? [Xx: $i] :
% 2.86/1.70 ( ( in @ Xx @ A )
% 2.86/1.70 & ( in @ Xx @ B ) )
% 2.86/1.70 => ( ( binintersect @ A @ B )
% 2.86/1.70 = emptyset ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(binintersectRsub_type,type,
% 2.86/1.70 binintersectRsub: $o ).
% 2.86/1.70
% 2.86/1.70 thf(binintersectRsub,definition,
% 2.86/1.70 ( binintersectRsub
% 2.86/1.70 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(binintersectSubset4_type,type,
% 2.86/1.70 binintersectSubset4: $o ).
% 2.86/1.70
% 2.86/1.70 thf(binintersectSubset4,definition,
% 2.86/1.70 ( binintersectSubset4
% 2.86/1.70 = ( ! [A: $i,B: $i] :
% 2.86/1.70 ( ( subset @ B @ A )
% 2.86/1.70 => ( ( binintersect @ A @ B )
% 2.86/1.70 = B ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(binintersectSubset1_type,type,
% 2.86/1.70 binintersectSubset1: $o ).
% 2.86/1.70
% 2.86/1.70 thf(binintersectSubset1,definition,
% 2.86/1.70 ( binintersectSubset1
% 2.86/1.70 = ( ! [A: $i,B: $i] :
% 2.86/1.70 ( ( ( binintersect @ A @ B )
% 2.86/1.70 = A )
% 2.86/1.70 => ( subset @ A @ B ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(bs114d_type,type,
% 2.86/1.70 bs114d: $o ).
% 2.86/1.70
% 2.86/1.70 thf(bs114d,definition,
% 2.86/1.70 ( bs114d
% 2.86/1.70 = ( ! [A: $i,B: $i,C: $i] :
% 2.86/1.70 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 2.86/1.70 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(regular_type,type,
% 2.86/1.70 regular: $i > $o ).
% 2.86/1.70
% 2.86/1.70 thf(setminus_type,type,
% 2.86/1.70 setminus: $i > $i > $i ).
% 2.86/1.70
% 2.86/1.70 thf(setminusI_type,type,
% 2.86/1.70 setminusI: $o ).
% 2.86/1.70
% 2.86/1.70 thf(setminusI,definition,
% 2.86/1.70 ( setminusI
% 2.86/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.86/1.70 ( ( in @ Xx @ A )
% 2.86/1.70 => ( ~ ( in @ Xx @ B )
% 2.86/1.70 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(setminusEL_type,type,
% 2.86/1.70 setminusEL: $o ).
% 2.86/1.70
% 2.86/1.70 thf(setminusEL,definition,
% 2.86/1.70 ( setminusEL
% 2.86/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.86/1.70 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.86/1.70 => ( in @ Xx @ A ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(setminusER_type,type,
% 2.86/1.70 setminusER: $o ).
% 2.86/1.70
% 2.86/1.70 thf(setminusER,definition,
% 2.86/1.70 ( setminusER
% 2.86/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.86/1.70 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.86/1.70 => ~ ( in @ Xx @ B ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(setminusSubset2_type,type,
% 2.86/1.70 setminusSubset2: $o ).
% 2.86/1.70
% 2.86/1.70 thf(setminusSubset2,definition,
% 2.86/1.70 ( setminusSubset2
% 2.86/1.70 = ( ! [A: $i,B: $i] :
% 2.86/1.70 ( ( subset @ A @ B )
% 2.86/1.70 => ( ( setminus @ A @ B )
% 2.86/1.70 = emptyset ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(setminusERneg_type,type,
% 2.86/1.70 setminusERneg: $o ).
% 2.86/1.70
% 2.86/1.70 thf(setminusERneg,definition,
% 2.86/1.70 ( setminusERneg
% 2.86/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.86/1.70 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.86/1.70 => ( ( in @ Xx @ A )
% 2.86/1.70 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(setminusELneg_type,type,
% 2.86/1.70 setminusELneg: $o ).
% 2.86/1.70
% 2.86/1.70 thf(setminusELneg,definition,
% 2.86/1.70 ( setminusELneg
% 2.86/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.86/1.70 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.86/1.70 => ( ~ ( in @ Xx @ B )
% 2.86/1.70 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(setminusILneg_type,type,
% 2.86/1.70 setminusILneg: $o ).
% 2.86/1.70
% 2.86/1.70 thf(setminusILneg,definition,
% 2.86/1.70 ( setminusILneg
% 2.86/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.86/1.70 ( ~ ( in @ Xx @ A )
% 2.86/1.70 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(setminusIRneg_type,type,
% 2.86/1.70 setminusIRneg: $o ).
% 2.86/1.70
% 2.86/1.70 thf(setminusIRneg,definition,
% 2.86/1.70 ( setminusIRneg
% 2.86/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.86/1.70 ( ( in @ Xx @ B )
% 2.86/1.70 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(setminusLsub_type,type,
% 2.86/1.70 setminusLsub: $o ).
% 2.86/1.70
% 2.86/1.70 thf(setminusLsub,definition,
% 2.86/1.70 ( setminusLsub
% 2.86/1.70 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(setminusSubset1_type,type,
% 2.86/1.70 setminusSubset1: $o ).
% 2.86/1.70
% 2.86/1.70 thf(setminusSubset1,definition,
% 2.86/1.70 ( setminusSubset1
% 2.86/1.70 = ( ! [A: $i,B: $i] :
% 2.86/1.70 ( ( ( setminus @ A @ B )
% 2.86/1.70 = emptyset )
% 2.86/1.70 => ( subset @ A @ B ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(symdiff_type,type,
% 2.86/1.70 symdiff: $i > $i > $i ).
% 2.86/1.70
% 2.86/1.70 thf(symdiffE_type,type,
% 2.86/1.70 symdiffE: $o ).
% 2.86/1.70
% 2.86/1.70 thf(symdiffE,definition,
% 2.86/1.70 ( symdiffE
% 2.86/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.86/1.70 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 2.86/1.70 => ! [Xphi: $o] :
% 2.86/1.70 ( ( ( in @ Xx @ A )
% 2.86/1.70 => ( ~ ( in @ Xx @ B )
% 2.86/1.70 => Xphi ) )
% 2.86/1.70 => ( ( ~ ( in @ Xx @ A )
% 2.86/1.70 => ( ( in @ Xx @ B )
% 2.86/1.70 => Xphi ) )
% 2.86/1.70 => Xphi ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(symdiffI1_type,type,
% 2.86/1.70 symdiffI1: $o ).
% 2.86/1.70
% 2.86/1.70 thf(symdiffI1,definition,
% 2.86/1.70 ( symdiffI1
% 2.86/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.86/1.70 ( ( in @ Xx @ A )
% 2.86/1.70 => ( ~ ( in @ Xx @ B )
% 2.86/1.70 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(symdiffI2_type,type,
% 2.86/1.70 symdiffI2: $o ).
% 2.86/1.70
% 2.86/1.70 thf(symdiffI2,definition,
% 2.86/1.70 ( symdiffI2
% 2.86/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.86/1.70 ( ~ ( in @ Xx @ A )
% 2.86/1.70 => ( ( in @ Xx @ B )
% 2.86/1.70 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(symdiffIneg1_type,type,
% 2.86/1.70 symdiffIneg1: $o ).
% 2.86/1.70
% 2.86/1.70 thf(symdiffIneg1,definition,
% 2.86/1.70 ( symdiffIneg1
% 2.86/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.86/1.70 ( ( in @ Xx @ A )
% 2.86/1.70 => ( ( in @ Xx @ B )
% 2.86/1.70 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(symdiffIneg2_type,type,
% 2.86/1.70 symdiffIneg2: $o ).
% 2.86/1.70
% 2.86/1.70 thf(symdiffIneg2,definition,
% 2.86/1.70 ( symdiffIneg2
% 2.86/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.86/1.70 ( ~ ( in @ Xx @ A )
% 2.86/1.70 => ( ~ ( in @ Xx @ B )
% 2.86/1.70 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(iskpair_type,type,
% 2.86/1.70 iskpair: $i > $o ).
% 2.86/1.70
% 2.86/1.70 thf(iskpair,definition,
% 2.86/1.70 ( iskpair
% 2.86/1.70 = ( ^ [A: $i] :
% 2.86/1.70 ? [Xx: $i] :
% 2.86/1.70 ( ( in @ Xx @ ( setunion @ A ) )
% 2.86/1.70 & ? [Xy: $i] :
% 2.86/1.70 ( ( in @ Xy @ ( setunion @ A ) )
% 2.86/1.70 & ( A
% 2.86/1.70 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(secondinupair_type,type,
% 2.86/1.70 secondinupair: $o ).
% 2.86/1.70
% 2.86/1.70 thf(secondinupair,definition,
% 2.86/1.70 ( secondinupair
% 2.86/1.70 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(setukpairIL_type,type,
% 2.86/1.70 setukpairIL: $o ).
% 2.86/1.70
% 2.86/1.70 thf(setukpairIL,definition,
% 2.86/1.70 ( setukpairIL
% 2.86/1.70 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(setukpairIR_type,type,
% 2.86/1.70 setukpairIR: $o ).
% 2.86/1.70
% 2.86/1.70 thf(setukpairIR,definition,
% 2.86/1.70 ( setukpairIR
% 2.86/1.70 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(kpairiskpair_type,type,
% 2.86/1.70 kpairiskpair: $o ).
% 2.86/1.70
% 2.86/1.70 thf(kpairiskpair,definition,
% 2.86/1.70 ( kpairiskpair
% 2.86/1.70 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(kpair_type,type,
% 2.86/1.70 kpair: $i > $i > $i ).
% 2.86/1.70
% 2.86/1.70 thf(kpair,definition,
% 2.86/1.70 ( kpair
% 2.86/1.70 = ( ^ [Xx: $i,Xy: $i] : ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(kpairp_type,type,
% 2.86/1.70 kpairp: $o ).
% 2.86/1.70
% 2.86/1.70 thf(kpairp,definition,
% 2.86/1.70 ( kpairp
% 2.86/1.70 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(cartprod_type,type,
% 2.86/1.70 cartprod: $i > $i > $i ).
% 2.86/1.70
% 2.86/1.70 thf(singletonsubset_type,type,
% 2.86/1.70 singletonsubset: $o ).
% 2.86/1.70
% 2.86/1.70 thf(singletonsubset,definition,
% 2.86/1.70 ( singletonsubset
% 2.86/1.70 = ( ! [A: $i,Xx: $i] :
% 2.86/1.70 ( ( in @ Xx @ A )
% 2.86/1.70 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(singletoninpowerset_type,type,
% 2.86/1.70 singletoninpowerset: $o ).
% 2.86/1.70
% 2.86/1.70 thf(singletoninpowerset,definition,
% 2.86/1.70 ( singletoninpowerset
% 2.86/1.70 = ( ! [A: $i,Xx: $i] :
% 2.86/1.70 ( ( in @ Xx @ A )
% 2.86/1.70 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(singletoninpowunion_type,type,
% 2.86/1.70 singletoninpowunion: $o ).
% 2.86/1.70
% 2.86/1.70 thf(singletoninpowunion,definition,
% 2.86/1.70 ( singletoninpowunion
% 2.86/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.86/1.70 ( ( in @ Xx @ A )
% 2.86/1.70 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(upairset2E_type,type,
% 2.86/1.70 upairset2E: $o ).
% 2.86/1.70
% 2.86/1.70 thf(upairset2E,definition,
% 2.86/1.70 ( upairset2E
% 2.86/1.70 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.86/1.70 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.86/1.70 => ( ( Xz = Xx )
% 2.86/1.70 | ( Xz = Xy ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(upairsubunion_type,type,
% 2.86/1.70 upairsubunion: $o ).
% 2.86/1.70
% 2.86/1.70 thf(upairsubunion,definition,
% 2.86/1.70 ( upairsubunion
% 2.86/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.86/1.70 ( ( in @ Xx @ A )
% 2.86/1.70 => ! [Xy: $i] :
% 2.86/1.70 ( ( in @ Xy @ B )
% 2.86/1.70 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(upairinpowunion_type,type,
% 2.86/1.70 upairinpowunion: $o ).
% 2.86/1.70
% 2.86/1.70 thf(upairinpowunion,definition,
% 2.86/1.70 ( upairinpowunion
% 2.86/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.86/1.70 ( ( in @ Xx @ A )
% 2.86/1.70 => ! [Xy: $i] :
% 2.86/1.70 ( ( in @ Xy @ B )
% 2.86/1.70 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(ubforcartprodlem1_type,type,
% 2.86/1.70 ubforcartprodlem1: $o ).
% 2.86/1.70
% 2.86/1.70 thf(ubforcartprodlem1,definition,
% 2.86/1.70 ( ubforcartprodlem1
% 2.86/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.86/1.70 ( ( in @ Xx @ A )
% 2.86/1.70 => ! [Xy: $i] :
% 2.86/1.70 ( ( in @ Xy @ B )
% 2.86/1.70 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(ubforcartprodlem2_type,type,
% 2.86/1.70 ubforcartprodlem2: $o ).
% 2.86/1.70
% 2.86/1.70 thf(ubforcartprodlem2,definition,
% 2.86/1.70 ( ubforcartprodlem2
% 2.86/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.86/1.70 ( ( in @ Xx @ A )
% 2.86/1.70 => ! [Xy: $i] :
% 2.86/1.70 ( ( in @ Xy @ B )
% 2.86/1.70 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(ubforcartprodlem3_type,type,
% 2.86/1.70 ubforcartprodlem3: $o ).
% 2.86/1.70
% 2.86/1.70 thf(ubforcartprodlem3,definition,
% 2.86/1.70 ( ubforcartprodlem3
% 2.86/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.86/1.70 ( ( in @ Xx @ A )
% 2.86/1.70 => ! [Xy: $i] :
% 2.86/1.70 ( ( in @ Xy @ B )
% 2.86/1.70 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(cartprodpairin_type,type,
% 2.86/1.70 cartprodpairin: $o ).
% 2.86/1.70
% 2.86/1.70 thf(cartprodpairin,definition,
% 2.86/1.70 ( cartprodpairin
% 2.86/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.86/1.70 ( ( in @ Xx @ A )
% 2.86/1.70 => ! [Xy: $i] :
% 2.86/1.70 ( ( in @ Xy @ B )
% 2.86/1.70 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(cartprodmempair1_type,type,
% 2.86/1.70 cartprodmempair1: $o ).
% 2.86/1.70
% 2.86/1.70 thf(cartprodmempair1,definition,
% 2.86/1.70 ( cartprodmempair1
% 2.86/1.70 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.86/1.70 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.86/1.70 => ? [Xx: $i] :
% 2.86/1.70 ( ( in @ Xx @ A )
% 2.86/1.70 & ? [Xy: $i] :
% 2.86/1.70 ( ( in @ Xy @ B )
% 2.86/1.70 & ( Xu
% 2.86/1.70 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(cartprodmempair_type,type,
% 2.86/1.70 cartprodmempair: $o ).
% 2.86/1.70
% 2.86/1.70 thf(cartprodmempair,definition,
% 2.86/1.70 ( cartprodmempair
% 2.86/1.70 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.86/1.70 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.86/1.70 => ( iskpair @ Xu ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(setunionE2_type,type,
% 2.86/1.70 setunionE2: $o ).
% 2.86/1.70
% 2.86/1.70 thf(setunionE2,definition,
% 2.86/1.70 ( setunionE2
% 2.86/1.70 = ( ! [A: $i,Xx: $i] :
% 2.86/1.70 ( ( in @ Xx @ ( setunion @ A ) )
% 2.86/1.70 => ? [X: $i] :
% 2.86/1.70 ( ( in @ X @ A )
% 2.86/1.70 & ( in @ Xx @ X ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(setunionsingleton1_type,type,
% 2.86/1.70 setunionsingleton1: $o ).
% 2.86/1.70
% 2.86/1.70 thf(setunionsingleton1,definition,
% 2.86/1.70 ( setunionsingleton1
% 2.86/1.70 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(setunionsingleton2_type,type,
% 2.86/1.70 setunionsingleton2: $o ).
% 2.86/1.70
% 2.86/1.70 thf(setunionsingleton2,definition,
% 2.86/1.70 ( setunionsingleton2
% 2.86/1.70 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(setunionsingleton_type,type,
% 2.86/1.70 setunionsingleton: $o ).
% 2.86/1.70
% 2.86/1.70 thf(setunionsingleton,definition,
% 2.86/1.70 ( setunionsingleton
% 2.86/1.70 = ( ! [Xx: $i] :
% 2.86/1.70 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 2.86/1.70 = Xx ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(singleton_type,type,
% 2.86/1.70 singleton: $i > $o ).
% 2.86/1.70
% 2.86/1.70 thf(singleton,definition,
% 2.86/1.70 ( singleton
% 2.86/1.70 = ( ^ [A: $i] :
% 2.86/1.70 ? [Xx: $i] :
% 2.86/1.70 ( ( in @ Xx @ A )
% 2.86/1.70 & ( A
% 2.86/1.70 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(singletonprop_type,type,
% 2.86/1.70 singletonprop: $o ).
% 2.86/1.70
% 2.86/1.70 thf(singletonprop,definition,
% 2.86/1.70 ( singletonprop
% 2.86/1.70 = ( ! [A: $i,Xphi: $i > $o] :
% 2.86/1.70 ( ! [Xx: $i] :
% 2.86/1.70 ( ( in @ Xx @ A )
% 2.86/1.70 => ! [Xy: $i] :
% 2.86/1.70 ( ( in @ Xy @ A )
% 2.86/1.70 => ( ( Xphi @ Xx )
% 2.86/1.70 => ( ( Xphi @ Xy )
% 2.86/1.70 => ( Xx = Xy ) ) ) ) )
% 2.86/1.70 => ( ? [Xx: $i] :
% 2.86/1.70 ( ( in @ Xx @ A )
% 2.86/1.70 & ( Xphi @ Xx ) )
% 2.86/1.70 => ( singleton
% 2.86/1.70 @ ( dsetconstr @ A
% 2.86/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(ex1_type,type,
% 2.86/1.70 ex1: $i > ( $i > $o ) > $o ).
% 2.86/1.70
% 2.86/1.70 thf(ex1,definition,
% 2.86/1.70 ( ex1
% 2.86/1.70 = ( ^ [A: $i,Xphi: $i > $o] :
% 2.86/1.70 ( singleton
% 2.86/1.70 @ ( dsetconstr @ A
% 2.86/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(ex1E1_type,type,
% 2.86/1.70 ex1E1: $o ).
% 2.86/1.70
% 2.86/1.70 thf(ex1E1,definition,
% 2.86/1.70 ( ex1E1
% 2.86/1.70 = ( ! [A: $i,Xphi: $i > $o] :
% 2.86/1.70 ( ( ex1 @ A
% 2.86/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.86/1.70 => ? [Xx: $i] :
% 2.86/1.70 ( ( in @ Xx @ A )
% 2.86/1.70 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(ex1I_type,type,
% 2.86/1.70 ex1I: $o ).
% 2.86/1.70
% 2.86/1.70 thf(ex1I,definition,
% 2.86/1.70 ( ex1I
% 2.86/1.70 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.86/1.70 ( ( in @ Xx @ A )
% 2.86/1.70 => ( ( Xphi @ Xx )
% 2.86/1.70 => ( ! [Xy: $i] :
% 2.86/1.70 ( ( in @ Xy @ A )
% 2.86/1.70 => ( ( Xphi @ Xy )
% 2.86/1.70 => ( Xy = Xx ) ) )
% 2.86/1.70 => ( ex1 @ A
% 2.86/1.70 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(ex1I2_type,type,
% 2.86/1.70 ex1I2: $o ).
% 2.86/1.70
% 2.86/1.70 thf(ex1I2,definition,
% 2.86/1.70 ( ex1I2
% 2.86/1.70 = ( ! [A: $i,Xphi: $i > $o] :
% 2.86/1.70 ( ! [Xx: $i] :
% 2.86/1.70 ( ( in @ Xx @ A )
% 2.86/1.70 => ! [Xy: $i] :
% 2.86/1.70 ( ( in @ Xy @ A )
% 2.86/1.70 => ( ( Xphi @ Xx )
% 2.86/1.70 => ( ( Xphi @ Xy )
% 2.86/1.70 => ( Xx = Xy ) ) ) ) )
% 2.86/1.70 => ( ? [Xx: $i] :
% 2.86/1.70 ( ( in @ Xx @ A )
% 2.86/1.70 & ( Xphi @ Xx ) )
% 2.86/1.70 => ( ex1 @ A
% 2.86/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(singletonsuniq_type,type,
% 2.86/1.70 singletonsuniq: $o ).
% 2.86/1.70
% 2.86/1.70 thf(singletonsuniq,definition,
% 2.86/1.70 ( singletonsuniq
% 2.86/1.70 = ( ! [Xx: $i,Xy: $i] :
% 2.86/1.70 ( ( ( setadjoin @ Xx @ emptyset )
% 2.86/1.70 = ( setadjoin @ Xy @ emptyset ) )
% 2.86/1.70 => ( Xx = Xy ) ) ) ) ).
% 2.86/1.70
% 2.86/1.70 thf(atmost1p_type,type,
% 2.86/1.70 atmost1p: $i > $o ).
% 2.86/1.70
% 2.86/1.70 thf(atleast2p_type,type,
% 2.86/1.70 atleast2p: $i > $o ).
% 2.86/1.70
% 2.86/1.70 thf(atmost2p_type,type,
% 2.86/1.70 atmost2p: $i > $o ).
% 2.86/1.70
% 2.86/1.70 thf(upairsetp_type,type,
% 2.86/1.70 upairsetp: $i > $o ).
% 2.86/1.70
% 2.86/1.70 thf(setukpairinjL1,conjecture,
% 2.86/1.70 ( setextAx
% 2.86/1.70 => ( emptysetAx
% 2.86/1.70 => ( setadjoinAx
% 2.86/1.70 => ( powersetAx
% 2.86/1.70 => ( setunionAx
% 2.86/1.70 => ( omega0Ax
% 2.86/1.70 => ( omegaSAx
% 2.86/1.70 => ( omegaIndAx
% 2.86/1.70 => ( replAx
% 2.86/1.70 => ( foundationAx
% 2.86/1.70 => ( wellorderingAx
% 2.86/1.70 => ( descrp
% 2.86/1.70 => ( dsetconstrI
% 2.86/1.70 => ( dsetconstrEL
% 2.86/1.70 => ( dsetconstrER
% 2.86/1.70 => ( exuE1
% 2.86/1.70 => ( prop2setE
% 2.86/1.70 => ( emptysetE
% 2.86/1.70 => ( emptysetimpfalse
% 2.86/1.70 => ( notinemptyset
% 2.86/1.70 => ( exuE3e
% 2.86/1.70 => ( setext
% 2.86/1.70 => ( emptyI
% 2.86/1.70 => ( noeltsimpempty
% 2.86/1.70 => ( setbeta
% 2.86/1.70 => ( nonemptyE1
% 2.86/1.70 => ( nonemptyI
% 2.86/1.70 => ( nonemptyI1
% 2.86/1.70 => ( setadjoinIL
% 2.86/1.70 => ( emptyinunitempty
% 2.86/1.70 => ( setadjoinIR
% 2.86/1.70 => ( setadjoinE
% 2.86/1.70 => ( setadjoinOr
% 2.86/1.70 => ( setoftrueEq
% 2.86/1.70 => ( powersetI
% 2.86/1.70 => ( emptyinPowerset
% 2.86/1.70 => ( emptyInPowerset
% 2.86/1.70 => ( powersetE
% 2.86/1.70 => ( setunionI
% 2.86/1.70 => ( setunionE
% 2.86/1.70 => ( subPowSU
% 2.86/1.70 => ( exuE2
% 2.86/1.70 => ( nonemptyImpWitness
% 2.86/1.70 => ( uniqinunit
% 2.86/1.70 => ( notinsingleton
% 2.86/1.70 => ( eqinunit
% 2.86/1.70 => ( singletonsswitch
% 2.86/1.70 => ( upairsetE
% 2.86/1.70 => ( upairsetIL
% 2.86/1.70 => ( upairsetIR
% 2.86/1.70 => ( emptyE1
% 2.86/1.70 => ( vacuousDall
% 2.86/1.70 => ( quantDeMorgan1
% 2.86/1.70 => ( quantDeMorgan2
% 2.86/1.70 => ( quantDeMorgan3
% 2.86/1.70 => ( quantDeMorgan4
% 2.86/1.70 => ( prop2setI
% 2.86/1.70 => ( prop2set2propI
% 2.86/1.70 => ( notdexE
% 2.86/1.70 => ( notdallE
% 2.86/1.70 => ( exuI1
% 2.86/1.70 => ( exuI3
% 2.86/1.70 => ( exuI2
% 2.86/1.70 => ( inCongP
% 2.86/1.70 => ( in__Cong
% 2.86/1.70 => ( exuE3u
% 2.86/1.70 => ( exu__Cong
% 2.86/1.70 => ( emptyset__Cong
% 2.86/1.70 => ( setadjoin__Cong
% 2.86/1.70 => ( powerset__Cong
% 2.86/1.70 => ( setunion__Cong
% 2.86/1.70 => ( omega__Cong
% 2.86/1.70 => ( exuEu
% 2.86/1.70 => ( descr__Cong
% 2.86/1.70 => ( dsetconstr__Cong
% 2.86/1.70 => ( subsetI1
% 2.86/1.70 => ( eqimpsubset2
% 2.86/1.70 => ( eqimpsubset1
% 2.86/1.70 => ( subsetI2
% 2.86/1.70 => ( emptysetsubset
% 2.86/1.70 => ( subsetE
% 2.86/1.70 => ( subsetE2
% 2.86/1.71 => ( notsubsetI
% 2.86/1.71 => ( notequalI1
% 2.86/1.71 => ( notequalI2
% 2.86/1.71 => ( subsetRefl
% 2.86/1.71 => ( subsetTrans
% 2.86/1.71 => ( setadjoinSub
% 2.86/1.71 => ( setadjoinSub2
% 2.86/1.71 => ( subset2powerset
% 2.86/1.71 => ( setextsub
% 2.86/1.71 => ( subsetemptysetimpeq
% 2.86/1.71 => ( powersetI1
% 2.86/1.71 => ( powersetE1
% 2.86/1.71 => ( inPowerset
% 2.86/1.71 => ( powersetsubset
% 2.86/1.71 => ( sepInPowerset
% 2.86/1.71 => ( sepSubset
% 2.86/1.71 => ( binunionIL
% 2.86/1.71 => ( upairset2IR
% 2.86/1.71 => ( binunionIR
% 2.86/1.71 => ( binunionEcases
% 2.86/1.71 => ( binunionE
% 2.86/1.71 => ( binunionLsub
% 2.86/1.71 => ( binunionRsub
% 2.86/1.71 => ( binintersectI
% 2.86/1.71 => ( binintersectSubset5
% 2.86/1.71 => ( binintersectEL
% 2.86/1.71 => ( binintersectLsub
% 2.86/1.71 => ( binintersectSubset2
% 2.86/1.71 => ( binintersectSubset3
% 2.86/1.71 => ( binintersectER
% 2.86/1.71 => ( disjointsetsI1
% 2.86/1.71 => ( binintersectRsub
% 2.86/1.71 => ( binintersectSubset4
% 2.86/1.71 => ( binintersectSubset1
% 2.86/1.71 => ( bs114d
% 2.86/1.71 => ( setminusI
% 2.86/1.71 => ( setminusEL
% 2.86/1.71 => ( setminusER
% 2.86/1.71 => ( setminusSubset2
% 2.86/1.71 => ( setminusERneg
% 2.86/1.71 => ( setminusELneg
% 2.86/1.71 => ( setminusILneg
% 2.86/1.71 => ( setminusIRneg
% 2.86/1.71 => ( setminusLsub
% 2.86/1.71 => ( setminusSubset1
% 2.86/1.71 => ( symdiffE
% 2.86/1.71 => ( symdiffI1
% 2.86/1.71 => ( symdiffI2
% 2.86/1.71 => ( symdiffIneg1
% 2.86/1.71 => ( symdiffIneg2
% 2.86/1.71 => ( secondinupair
% 2.86/1.71 => ( setukpairIL
% 2.86/1.71 => ( setukpairIR
% 2.86/1.71 => ( kpairiskpair
% 2.86/1.71 => ( kpairp
% 2.86/1.71 => ( singletonsubset
% 2.86/1.71 => ( singletoninpowerset
% 2.86/1.71 => ( singletoninpowunion
% 2.86/1.71 => ( upairset2E
% 2.86/1.71 => ( upairsubunion
% 2.86/1.71 => ( upairinpowunion
% 2.86/1.71 => ( ubforcartprodlem1
% 2.86/1.71 => ( ubforcartprodlem2
% 2.86/1.71 => ( ubforcartprodlem3
% 2.86/1.71 => ( cartprodpairin
% 2.86/1.71 => ( cartprodmempair1
% 2.86/1.71 => ( cartprodmempair
% 2.86/1.71 => ( setunionE2
% 2.86/1.71 => ( setunionsingleton1
% 2.86/1.71 => ( setunionsingleton2
% 2.86/1.71 => ( setunionsingleton
% 2.86/1.71 => ( singletonprop
% 2.86/1.71 => ( ex1E1
% 2.86/1.71 => ( ex1I
% 2.86/1.71 => ( ex1I2
% 2.86/1.71 => ( singletonsuniq
% 2.86/1.71 => ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.86/1.71 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 2.86/1.71 => ( Xx = Xz ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.86/1.71 ------------------------
% 2.86/1.73 % (13609)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_13509 for (2999ds/27Mi)
% 2.86/1.73 % (13610)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_13509 for (2999ds/2Mi)
% 2.86/1.73 % (13608)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_13509 for (2999ds/4Mi)
% 2.86/1.73 % (13607)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_13509 for (2999ds/183Mi)
% 2.86/1.73 % (13611)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_13509 for (2999ds/2Mi)
% 2.86/1.73 % (13612)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_13509 for (2999ds/275Mi)
% 2.86/1.73 % (13613)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_13509 for (2999ds/18Mi)
% 2.86/1.73 % (13610)Instruction limit reached!
% 2.86/1.73 % (13610)------------------------------
% 2.86/1.73 % (13610)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.86/1.73 % (13610)Termination reason: Unknown
% 2.86/1.73 % (13610)Termination phase: shuffling
% 2.86/1.73 % (13611)Instruction limit reached!
% 2.86/1.73 % (13611)------------------------------
% 2.86/1.73 % (13611)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.86/1.73 % (13611)Termination reason: Unknown
% 2.86/1.73 % (13611)Termination phase: shuffling
% 2.86/1.73
% 2.86/1.73 % (13611)Memory used [KB]: 1279
% 2.86/1.73 % (13611)Time elapsed: 0.004 s
% 2.86/1.73 % (13611)Instructions burned: 4 (million)
% 2.86/1.73 % (13611)------------------------------
% 2.86/1.73 % (13611)------------------------------
% 2.86/1.73
% 2.86/1.73 % (13610)Memory used [KB]: 1279
% 2.86/1.73 % (13610)Time elapsed: 0.003 s
% 2.86/1.73 % (13610)Instructions burned: 3 (million)
% 2.86/1.73 % (13610)------------------------------
% 2.86/1.73 % (13610)------------------------------
% 2.86/1.74 % (13608)Instruction limit reached!
% 2.86/1.74 % (13608)------------------------------
% 2.86/1.74 % (13608)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.86/1.74 % (13608)Termination reason: Unknown
% 2.86/1.74 % (13608)Termination phase: shuffling
% 2.86/1.74
% 2.86/1.74 % (13608)Memory used [KB]: 1279
% 2.86/1.74 % (13608)Time elapsed: 0.004 s
% 2.86/1.74 % (13608)Instructions burned: 5 (million)
% 2.86/1.74 % (13608)------------------------------
% 2.86/1.74 % (13608)------------------------------
% 2.86/1.74 % (13613)Instruction limit reached!
% 2.86/1.74 % (13613)------------------------------
% 2.86/1.74 % (13613)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.86/1.74 % (13613)Termination reason: Unknown
% 2.86/1.74 % (13613)Termination phase: Property scanning
% 2.86/1.74
% 2.86/1.74 % (13613)Memory used [KB]: 1535
% 2.86/1.74 % (13613)Time elapsed: 0.011 s
% 2.86/1.74 % (13613)Instructions burned: 18 (million)
% 2.86/1.74 % (13613)------------------------------
% 2.86/1.74 % (13613)------------------------------
% 2.86/1.75 % (13609)Instruction limit reached!
% 2.86/1.75 % (13609)------------------------------
% 2.86/1.75 % (13609)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.86/1.75 % (13609)Termination reason: Unknown
% 2.86/1.75 % (13609)Termination phase: Property scanning
% 2.86/1.75
% 2.86/1.75 % (13609)Memory used [KB]: 1791
% 2.86/1.75 % (13609)Time elapsed: 0.014 s
% 2.86/1.75 % (13609)Instructions burned: 27 (million)
% 2.86/1.75 % (13609)------------------------------
% 2.86/1.75 % (13609)------------------------------
% 2.86/1.75 % (13615)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_13509 for (2999ds/37Mi)
% 2.86/1.75 % (13614)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_13509 for (2999ds/3Mi)
% 2.86/1.75 % (13616)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_13509 for (2999ds/15Mi)
% 2.86/1.75 % (13614)Instruction limit reached!
% 2.86/1.75 % (13614)------------------------------
% 2.86/1.75 % (13614)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.86/1.75 % (13614)Termination reason: Unknown
% 2.86/1.75 % (13614)Termination phase: shuffling
% 2.86/1.75
% 2.86/1.75 % (13614)Memory used [KB]: 1279
% 2.86/1.75 % (13614)Time elapsed: 0.004 s
% 2.86/1.75 % (13614)Instructions burned: 5 (million)
% 2.86/1.75 % (13614)------------------------------
% 2.86/1.75 % (13614)------------------------------
% 2.86/1.76 % (13617)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_13509 for (2999ds/3Mi)
% 2.86/1.76 % (13616)Instruction limit reached!
% 2.86/1.76 % (13616)------------------------------
% 2.86/1.76 % (13616)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.86/1.76 % (13616)Termination reason: Unknown
% 2.86/1.76 % (13616)Termination phase: Property scanning
% 2.86/1.76
% 2.86/1.76 % (13616)Memory used [KB]: 1535
% 2.86/1.76 % (13616)Time elapsed: 0.009 s
% 2.86/1.76 % (13616)Instructions burned: 15 (million)
% 2.86/1.76 % (13616)------------------------------
% 2.86/1.76 % (13616)------------------------------
% 2.86/1.76 % (13617)Instruction limit reached!
% 2.86/1.76 % (13617)------------------------------
% 2.86/1.76 % (13617)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.86/1.76 % (13617)Termination reason: Unknown
% 2.86/1.76 % (13617)Termination phase: shuffling
% 2.86/1.76
% 2.86/1.76 % (13617)Memory used [KB]: 1279
% 2.86/1.76 % (13617)Time elapsed: 0.003 s
% 2.86/1.76 % (13617)Instructions burned: 3 (million)
% 2.86/1.76 % (13617)------------------------------
% 2.86/1.76 % (13617)------------------------------
% 2.86/1.76 % (13618)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_13509 for (2999ds/1041Mi)
% 2.86/1.77 % (13615)Instruction limit reached!
% 2.86/1.77 % (13615)------------------------------
% 2.86/1.77 % (13615)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.86/1.77 % (13615)Termination reason: Unknown
% 2.86/1.77 % (13615)Termination phase: Property scanning
% 2.86/1.77
% 2.86/1.77 % (13615)Memory used [KB]: 1791
% 2.86/1.77 % (13615)Time elapsed: 0.019 s
% 2.86/1.77 % (13615)Instructions burned: 39 (million)
% 2.86/1.77 % (13615)------------------------------
% 2.86/1.77 % (13615)------------------------------
% 2.86/1.77 % (13619)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_13509 for (2999ds/7Mi)
% 2.86/1.77 % (13619)Instruction limit reached!
% 2.86/1.77 % (13619)------------------------------
% 2.86/1.77 % (13619)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.86/1.77 % (13619)Termination reason: Unknown
% 2.86/1.77 % (13619)Termination phase: shuffling
% 2.86/1.77
% 2.86/1.77 % (13619)Memory used [KB]: 1407
% 2.86/1.77 % (13619)Time elapsed: 0.006 s
% 2.86/1.77 % (13619)Instructions burned: 8 (million)
% 2.86/1.77 % (13619)------------------------------
% 2.86/1.77 % (13619)------------------------------
% 2.86/1.78 % (13620)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_13509 for (2999ds/16Mi)
% 2.86/1.78 % (13621)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_13509 for (2999ds/3Mi)
% 2.86/1.78 % (13621)Instruction limit reached!
% 2.86/1.78 % (13621)------------------------------
% 2.86/1.78 % (13621)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.86/1.78 % (13621)Termination reason: Unknown
% 2.86/1.78 % (13621)Termination phase: shuffling
% 2.86/1.78
% 2.86/1.78 % (13621)Memory used [KB]: 1279
% 2.86/1.78 % (13621)Time elapsed: 0.003 s
% 2.86/1.78 % (13621)Instructions burned: 3 (million)
% 2.86/1.78 % (13621)------------------------------
% 2.86/1.78 % (13621)------------------------------
% 2.86/1.78 % (13620)Instruction limit reached!
% 2.86/1.78 % (13620)------------------------------
% 2.86/1.78 % (13620)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.86/1.78 % (13620)Termination reason: Unknown
% 2.86/1.78 % (13620)Termination phase: shuffling
% 2.86/1.78
% 2.86/1.78 % (13620)Memory used [KB]: 1535
% 2.86/1.78 % (13620)Time elapsed: 0.009 s
% 2.86/1.78 % (13620)Instructions burned: 16 (million)
% 2.86/1.78 % (13620)------------------------------
% 2.86/1.78 % (13620)------------------------------
% 2.86/1.78 % (13622)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_13509 for (2999ds/3Mi)
% 2.86/1.79 % (13622)Instruction limit reached!
% 2.86/1.79 % (13622)------------------------------
% 2.86/1.79 % (13622)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.86/1.79 % (13622)Termination reason: Unknown
% 2.86/1.79 % (13622)Termination phase: shuffling
% 2.86/1.79
% 2.86/1.79 % (13622)Memory used [KB]: 1279
% 2.86/1.79 % (13622)Time elapsed: 0.003 s
% 2.86/1.79 % (13622)Instructions burned: 3 (million)
% 2.86/1.79 % (13622)------------------------------
% 2.86/1.79 % (13622)------------------------------
% 2.86/1.79 % (13623)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_13509 for (2999ds/7Mi)
% 2.86/1.79 % (13624)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_13509 for (2999ds/3Mi)
% 2.86/1.79 % (13623)Instruction limit reached!
% 2.86/1.79 % (13623)------------------------------
% 2.86/1.79 % (13623)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.86/1.79 % (13623)Termination reason: Unknown
% 2.86/1.79 % (13623)Termination phase: shuffling
% 2.86/1.79
% 2.86/1.79 % (13623)Memory used [KB]: 1407
% 2.86/1.79 % (13623)Time elapsed: 0.005 s
% 2.86/1.79 % (13623)Instructions burned: 7 (million)
% 2.86/1.79 % (13623)------------------------------
% 2.86/1.79 % (13623)------------------------------
% 2.86/1.79 % (13624)Instruction limit reached!
% 2.86/1.79 % (13624)------------------------------
% 2.86/1.79 % (13624)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.86/1.79 % (13624)Termination reason: Unknown
% 2.86/1.79 % (13624)Termination phase: shuffling
% 2.86/1.79
% 2.86/1.79 % (13624)Memory used [KB]: 1279
% 2.86/1.79 % (13624)Time elapsed: 0.003 s
% 2.86/1.79 % (13624)Instructions burned: 4 (million)
% 2.86/1.79 % (13624)------------------------------
% 2.86/1.79 % (13624)------------------------------
% 2.89/1.80 % (13625)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_13509 for (2999ds/4Mi)
% 2.89/1.80 % (13625)Instruction limit reached!
% 2.89/1.80 % (13625)------------------------------
% 2.89/1.80 % (13625)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.89/1.80 % (13625)Termination reason: Unknown
% 2.89/1.80 % (13625)Termination phase: shuffling
% 2.89/1.80
% 2.89/1.80 % (13625)Memory used [KB]: 1279
% 2.89/1.80 % (13625)Time elapsed: 0.004 s
% 2.89/1.80 % (13625)Instructions burned: 5 (million)
% 2.89/1.80 % (13625)------------------------------
% 2.89/1.80 % (13625)------------------------------
% 2.89/1.80 % (13626)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_13509 for (2999ds/18Mi)
% 2.89/1.81 % (13627)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_13509 for (2999ds/710Mi)
% 2.89/1.81 % (13628)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_13509 for (2999ds/6Mi)
% 2.89/1.81 % (13626)Instruction limit reached!
% 2.89/1.81 % (13626)------------------------------
% 2.89/1.81 % (13626)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.89/1.81 % (13626)Termination reason: Unknown
% 2.89/1.81 % (13626)Termination phase: shuffling
% 2.89/1.81
% 2.89/1.81 % (13626)Memory used [KB]: 1663
% 2.89/1.81 % (13626)Time elapsed: 0.010 s
% 2.89/1.81 % (13626)Instructions burned: 18 (million)
% 2.89/1.81 % (13626)------------------------------
% 2.89/1.81 % (13626)------------------------------
% 2.89/1.81 % (13628)Instruction limit reached!
% 2.89/1.81 % (13628)------------------------------
% 2.89/1.81 % (13628)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.89/1.81 % (13628)Termination reason: Unknown
% 2.89/1.81 % (13628)Termination phase: shuffling
% 2.89/1.81
% 2.89/1.81 % (13628)Memory used [KB]: 1279
% 2.89/1.81 % (13628)Time elapsed: 0.004 s
% 2.89/1.81 % (13628)Instructions burned: 6 (million)
% 2.89/1.81 % (13628)------------------------------
% 2.89/1.81 % (13628)------------------------------
% 2.89/1.82 % (13607)Instruction limit reached!
% 2.89/1.82 % (13607)------------------------------
% 2.89/1.82 % (13607)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.89/1.82 % (13607)Termination reason: Unknown
% 2.89/1.82 % (13607)Termination phase: Saturation
% 2.89/1.82
% 2.89/1.82 % (13607)Memory used [KB]: 7419
% 2.89/1.82 % (13607)Time elapsed: 0.085 s
% 2.89/1.82 % (13607)Instructions burned: 183 (million)
% 2.89/1.82 % (13607)------------------------------
% 2.89/1.82 % (13607)------------------------------
% 2.89/1.82 % (13629)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_13509 for (2999ds/902Mi)
% 2.89/1.83 % (13630)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_13509 for (2999ds/21Mi)
% 2.89/1.83 % (13631)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_13509 for (2999ds/5Mi)
% 2.89/1.83 % (13631)Instruction limit reached!
% 2.89/1.83 % (13631)------------------------------
% 2.89/1.83 % (13631)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.89/1.83 % (13631)Termination reason: Unknown
% 2.89/1.83 % (13631)Termination phase: shuffling
% 2.89/1.83
% 2.89/1.83 % (13631)Memory used [KB]: 1279
% 2.89/1.83 % (13631)Time elapsed: 0.004 s
% 2.89/1.83 % (13631)Instructions burned: 5 (million)
% 2.89/1.83 % (13631)------------------------------
% 2.89/1.83 % (13631)------------------------------
% 2.89/1.83 % (13632)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_13509 for (2999ds/6Mi)
% 2.89/1.84 % (13632)Instruction limit reached!
% 2.89/1.84 % (13632)------------------------------
% 2.89/1.84 % (13632)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.89/1.84 % (13632)Termination reason: Unknown
% 2.89/1.84 % (13632)Termination phase: shuffling
% 2.89/1.84
% 2.89/1.84 % (13632)Memory used [KB]: 1407
% 2.89/1.84 % (13632)Time elapsed: 0.005 s
% 2.89/1.84 % (13632)Instructions burned: 6 (million)
% 2.89/1.84 % (13632)------------------------------
% 2.89/1.84 % (13632)------------------------------
% 2.89/1.84 % (13630)Instruction limit reached!
% 2.89/1.84 % (13630)------------------------------
% 2.89/1.84 % (13630)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.89/1.84 % (13630)Termination reason: Unknown
% 2.89/1.84 % (13630)Termination phase: shuffling
% 2.89/1.84
% 2.89/1.84 % (13630)Memory used [KB]: 1791
% 2.89/1.84 % (13630)Time elapsed: 0.012 s
% 2.89/1.84 % (13630)Instructions burned: 22 (million)
% 2.89/1.84 % (13630)------------------------------
% 2.89/1.84 % (13630)------------------------------
% 2.89/1.85 % (13633)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_13509 for (2998ds/377Mi)
% 2.89/1.85 % (13634)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_13509 for (2998ds/779Mi)
% 2.89/1.85 % (13635)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_13509 for (2998ds/19Mi)
% 2.89/1.86 % (13635)Instruction limit reached!
% 2.89/1.86 % (13635)------------------------------
% 2.89/1.86 % (13635)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.89/1.86 % (13635)Termination reason: Unknown
% 2.89/1.86 % (13635)Termination phase: shuffling
% 2.89/1.86
% 2.89/1.86 % (13635)Memory used [KB]: 1663
% 2.89/1.86 % (13635)Time elapsed: 0.010 s
% 2.89/1.86 % (13635)Instructions burned: 19 (million)
% 2.89/1.86 % (13635)------------------------------
% 2.89/1.86 % (13635)------------------------------
% 2.89/1.87 % (13612)Instruction limit reached!
% 2.89/1.87 % (13612)------------------------------
% 2.89/1.87 % (13612)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.89/1.87 % (13612)Termination reason: Unknown
% 2.89/1.87 % (13612)Termination phase: Saturation
% 2.89/1.87
% 2.89/1.87 % (13612)Memory used [KB]: 8187
% 2.89/1.87 % (13612)Time elapsed: 0.140 s
% 2.89/1.87 % (13612)Instructions burned: 276 (million)
% 2.89/1.87 % (13612)------------------------------
% 2.89/1.87 % (13612)------------------------------
% 2.89/1.88 % (13636)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_13509 for (2998ds/879Mi)
% 2.89/1.89 % (13637)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_13509 for (2998ds/17Mi)
% 2.89/1.90 % (13637)Instruction limit reached!
% 2.89/1.90 % (13637)------------------------------
% 2.89/1.90 % (13637)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.89/1.90 % (13637)Termination reason: Unknown
% 2.89/1.90 % (13637)Termination phase: shuffling
% 2.89/1.90
% 2.89/1.90 % (13637)Memory used [KB]: 1535
% 2.89/1.90 % (13637)Time elapsed: 0.010 s
% 2.89/1.90 % (13637)Instructions burned: 17 (million)
% 2.89/1.90 % (13637)------------------------------
% 2.89/1.90 % (13637)------------------------------
% 2.89/1.91 % (13638)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_13509 for (2998ds/3Mi)
% 2.89/1.92 % (13638)Instruction limit reached!
% 2.89/1.92 % (13638)------------------------------
% 2.89/1.92 % (13638)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.89/1.92 % (13638)Termination reason: Unknown
% 2.89/1.92 % (13638)Termination phase: shuffling
% 2.89/1.92
% 2.89/1.92 % (13638)Memory used [KB]: 1279
% 2.89/1.92 % (13638)Time elapsed: 0.004 s
% 2.89/1.92 % (13638)Instructions burned: 4 (million)
% 2.89/1.92 % (13638)------------------------------
% 2.89/1.92 % (13638)------------------------------
% 2.89/1.93 % (13629)First to succeed.
% 2.89/1.93 % (13629)Refutation found. Thanks to Tanya!
% 2.89/1.93 % SZS status Theorem for DTF2THF_13509
% 2.89/1.93 % SZS output start Proof for DTF2THF_13509
% 2.89/1.93 thf(func_def_0, type, in: $i > $i > $o).
% 2.89/1.93 thf(func_def_1, type, exu: ($i > $o) > $o).
% 2.89/1.93 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 2.89/1.93 thf(func_def_8, type, powerset: $i > $i).
% 2.89/1.93 thf(func_def_10, type, setunion: $i > $i).
% 2.89/1.93 thf(func_def_19, type, descr: ($i > $o) > $i).
% 2.89/1.93 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 2.89/1.93 thf(func_def_26, type, prop2set: $o > $i).
% 2.89/1.93 thf(func_def_36, type, nonempty: $i > $o).
% 2.89/1.93 thf(func_def_69, type, set2prop: $i > $o).
% 2.89/1.93 thf(func_def_88, type, subset: $i > $i > $o).
% 2.89/1.93 thf(func_def_89, type, disjoint: $i > $i > $o).
% 2.89/1.93 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 2.89/1.93 thf(func_def_114, type, binunion: $i > $i > $i).
% 2.89/1.93 thf(func_def_122, type, binintersect: $i > $i > $i).
% 2.89/1.93 thf(func_def_135, type, regular: $i > $o).
% 2.89/1.93 thf(func_def_136, type, setminus: $i > $i > $i).
% 2.89/1.93 thf(func_def_147, type, symdiff: $i > $i > $i).
% 2.89/1.93 thf(func_def_153, type, iskpair: $i > $o).
% 2.89/1.93 thf(func_def_158, type, kpair: $i > $i > $i).
% 2.89/1.93 thf(func_def_160, type, cartprod: $i > $i > $i).
% 2.89/1.93 thf(func_def_177, type, singleton: $i > $o).
% 2.89/1.93 thf(func_def_179, type, ex1: $i > ($i > $o) > $o).
% 2.89/1.93 thf(func_def_184, type, atmost1p: $i > $o).
% 2.89/1.93 thf(func_def_185, type, atleast2p: $i > $o).
% 2.89/1.93 thf(func_def_186, type, atmost2p: $i > $o).
% 2.89/1.93 thf(func_def_187, type, upairsetp: $i > $o).
% 2.89/1.93 thf(func_def_202, type, sP2: $i > $o > $i > $i > $o).
% 2.89/1.93 thf(func_def_204, type, sP4: $i > $i > $o).
% 2.89/1.93 thf(func_def_205, type, sP5: $i > $o).
% 2.89/1.93 thf(func_def_206, type, sP6: $i > $i > $o).
% 2.89/1.93 thf(func_def_207, type, sP7: $i > $i > $o).
% 2.89/1.93 thf(func_def_212, type, sK12: $i > $o).
% 2.89/1.93 thf(func_def_215, type, sK15: $i > ($i > $o) > $i).
% 2.89/1.93 thf(func_def_234, type, sK34: $i > $i).
% 2.89/1.93 thf(func_def_243, type, sK43: $i > $i).
% 2.89/1.93 thf(func_def_244, type, sK44: $i > $i > ($i > $o) > $i).
% 2.89/1.93 thf(func_def_245, type, sK45: $i > $o).
% 2.89/1.93 thf(func_def_253, type, sK53: $i > $i > $i).
% 2.89/1.93 thf(func_def_256, type, sK56: $i > $i > $i).
% 2.89/1.93 thf(func_def_257, type, sK57: $i > $i > $i).
% 2.89/1.93 thf(func_def_269, type, sK69: $i > $o).
% 2.89/1.93 thf(func_def_270, type, sK70: $i > $o).
% 2.89/1.93 thf(func_def_271, type, sK71: ($i > $o) > ($i > $o) > $i).
% 2.89/1.93 thf(func_def_272, type, sK72: ($i > $o) > ($i > $o) > $i).
% 2.89/1.93 thf(func_def_278, type, sK78: $i > $i > $i).
% 2.89/1.93 thf(func_def_284, type, sK84: $i > $i > $i).
% 2.89/1.93 thf(func_def_294, type, sK94: $i > $o).
% 2.89/1.93 thf(func_def_296, type, sK96: ($i > $o) > $i).
% 2.89/1.93 thf(func_def_297, type, sK97: ($i > $o) > $i).
% 2.89/1.93 thf(func_def_309, type, sK109: $i > $i > $i).
% 2.89/1.93 thf(func_def_313, type, sK113: ($i > $o) > ($i > $o) > $i).
% 2.89/1.93 thf(func_def_314, type, sK114: ($i > $o) > ($i > $o) > $i).
% 2.89/1.93 thf(func_def_315, type, sK115: $i > $o).
% 2.89/1.93 thf(func_def_316, type, sK116: $i > $o).
% 2.89/1.93 thf(func_def_317, type, sK117: $i > $o).
% 2.89/1.93 thf(func_def_331, type, sK131: $i > $i).
% 2.89/1.93 thf(func_def_333, type, sK133: $i > $o).
% 2.89/1.93 thf(func_def_336, type, sK136: ($i > $o) > $i > $i).
% 2.89/1.93 thf(func_def_338, type, sK138: $i > $o).
% 2.89/1.93 thf(func_def_358, type, sK158: $i > ($i > $o) > $i).
% 2.89/1.93 thf(func_def_359, type, sK159: $i > $o).
% 2.89/1.93 thf(func_def_363, type, sK163: $i > $i).
% 2.89/1.93 thf(func_def_369, type, sK169: ($i > $o) > $i).
% 2.89/1.93 thf(func_def_370, type, sK170: $i > $o).
% 2.89/1.93 thf(func_def_371, type, sK171: $i > $i).
% 2.89/1.93 thf(func_def_382, type, sK182: $i > $o).
% 2.89/1.93 thf(func_def_396, type, sK196: $i > $o).
% 2.89/1.93 thf(func_def_400, type, sK200: $i > $o).
% 2.89/1.93 thf(func_def_402, type, sK202: ($i > $o) > $i > $i).
% 2.89/1.93 thf(func_def_403, type, sK203: ($i > $o) > $i > $i).
% 2.89/1.93 thf(func_def_406, type, sK206: $i > $o).
% 2.89/1.93 thf(func_def_407, type, sK207: $i > $o).
% 2.89/1.93 thf(func_def_408, type, sK208: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.89/1.93 thf(func_def_409, type, sK209: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.89/1.93 thf(func_def_414, type, sK214: $i > $i).
% 2.89/1.93 thf(func_def_416, type, sK216: $i > $i).
% 2.89/1.93 thf(func_def_423, type, sK223: $o > $i > $i > $i).
% 2.89/1.93 thf(func_def_424, type, sK224: $i > $i > $i > $i).
% 2.89/1.93 thf(func_def_425, type, sK225: $i > $i > $i > $i).
% 2.89/1.93 thf(func_def_429, type, sK229: ($i > $o) > $i).
% 2.89/1.93 thf(func_def_430, type, sK230: $i > $o).
% 2.89/1.93 thf(func_def_431, type, sK231: $i > $i).
% 2.89/1.93 thf(func_def_432, type, sK232: $i > $o).
% 2.89/1.93 thf(func_def_436, type, sK236: $i > $i > $i).
% 2.89/1.93 thf(func_def_456, type, sK256: $i > $o).
% 2.89/1.93 thf(func_def_458, type, sK258: ($i > $o) > $i > $i).
% 2.89/1.93 thf(func_def_469, type, sK269: $i > $o).
% 2.89/1.93 thf(func_def_472, type, sK272: $i > ($i > $o) > $i).
% 2.89/1.93 thf(func_def_476, type, sK276: $i > $o).
% 2.89/1.93 thf(func_def_478, type, sK278: $i > ($i > $o) > $i).
% 2.89/1.93 thf(func_def_479, type, sK279: $i > $i > $i).
% 2.89/1.93 thf(func_def_489, type, sK289: $i > $o).
% 2.89/1.93 thf(func_def_512, type, sK312: ($i > $o) > $i).
% 2.89/1.93 thf(func_def_513, type, sK313: $i > $o).
% 2.89/1.93 thf(func_def_532, type, sK332: $i > $o).
% 2.89/1.93 thf(func_def_539, type, sK339: $i > $i > $o).
% 2.89/1.93 thf(func_def_540, type, sK340: $i > $i).
% 2.89/1.93 thf(func_def_541, type, sK341: $i > $i).
% 2.89/1.93 thf(func_def_542, type, sK342: ($i > $i > $o) > $i > $i).
% 2.89/1.93 thf(func_def_543, type, sK343: $i > ($i > $i > $o) > $i > $i).
% 2.89/1.93 thf(func_def_544, type, sK344: ($i > $i > $o) > $i > $i).
% 2.89/1.93 thf(func_def_551, type, sK351: $i > $o).
% 2.89/1.93 thf(func_def_557, type, sK357: ($i > $o) > $i > $i).
% 2.89/1.93 thf(func_def_559, type, sK359: $i > $o).
% 2.89/1.93 thf(func_def_564, type, sK364: $i > $o).
% 2.89/1.93 thf(func_def_568, type, sK368: $i > $i).
% 2.89/1.93 thf(func_def_575, type, sK375: $i > $i > $i).
% 2.89/1.93 thf(func_def_576, type, sK376: ($i > $o) > $i > $i).
% 2.89/1.93 thf(func_def_578, type, sK378: $i > $o).
% 2.89/1.93 thf(func_def_591, type, sK391: $i > $i > $i).
% 2.89/1.93 thf(func_def_594, type, sK394: $i > $o).
% 2.89/1.93 thf(func_def_596, type, sK396: $i > ($i > $o) > $i).
% 2.89/1.93 thf(func_def_597, type, sK397: $i > $o).
% 2.89/1.93 thf(func_def_604, type, sK404: $i > $o).
% 2.89/1.93 thf(func_def_607, type, sK407: $i > ($i > $o) > $i).
% 2.89/1.93 thf(func_def_608, type, sK408: $i > ($i > $o) > $i).
% 2.89/1.93 thf(func_def_611, type, sK411: $i > $o).
% 2.89/1.93 thf(func_def_616, type, sK416: $i > $i > $i).
% 2.89/1.93 thf(func_def_617, type, sK417: $i > $i > $i).
% 2.89/1.93 thf(func_def_618, type, sK418: $i > $i > $i).
% 2.89/1.93 thf(func_def_619, type, sK419: $i > $i > $i).
% 2.89/1.93 thf(func_def_620, type, sK420: $i > $i > $i > $i).
% 2.89/1.93 thf(func_def_621, type, sK421: $i > $i).
% 2.89/1.93 thf(func_def_622, type, sK422: $i > $i).
% 2.89/1.93 thf(func_def_623, type, sK423: $i > $i).
% 2.89/1.93 thf(func_def_624, type, sK424: $i > $i).
% 2.89/1.93 thf(func_def_625, type, sK425: $i > $i > $i > $i).
% 2.89/1.93 thf(func_def_626, type, sK426: $i > $i > $i > $i).
% 2.89/1.93 thf(func_def_627, type, sK427: $i > $i > $i).
% 2.89/1.93 thf(func_def_628, type, sK428: $i > $i > $i).
% 2.89/1.93 thf(func_def_629, type, sK429: $i > $i > $i).
% 2.89/1.93 thf(func_def_630, type, sK430: $i > $i).
% 2.89/1.93 thf(func_def_631, type, sK431: $i > $i > $i).
% 2.89/1.93 thf(func_def_633, type, sK433: $i > $i).
% 2.89/1.93 thf(func_def_634, type, sK434: $i > $i).
% 2.89/1.93 thf(func_def_635, type, sK435: $i > $o).
% 2.89/1.93 thf(func_def_646, type, ph446: !>[X0: $tType]:(X0)).
% 2.89/1.93 thf(f2907,plain,(
% 2.89/1.93 $false),
% 2.89/1.93 inference(avatar_sat_refutation,[status(thm)],[f2383,f2611,f2768,f2853,f2906])).
% 2.89/1.93 thf(f2906,plain,(
% 2.89/1.93 spl445_7 | ~spl445_2),
% 2.89/1.93 inference(avatar_split_clause,[status(thm)],[f2905,f2361,f2598])).
% 2.89/1.93 thf(f2598,definition,(
% 2.89/1.93 spl445_7 <=> (sK372 = sK370)),
% 2.89/1.93 introduced(definition,[new_symbols(naming,[spl445_7])],[avatar_definition])).
% 2.89/1.93 thf(f2361,definition,(
% 2.89/1.93 spl445_2 <=> ((setadjoin @ sK372 @ emptyset) = (setadjoin @ sK370 @ emptyset))),
% 2.89/1.93 introduced(definition,[new_symbols(naming,[spl445_2])],[avatar_definition])).
% 2.89/1.93 thf(f2905,plain,(
% 2.89/1.93 (sK372 = sK370) | ~spl445_2),
% 2.89/1.93 inference(trivial_inequality_removal,[status(thm)],[f2904])).
% 2.89/1.93 thf(f2904,plain,(
% 2.89/1.93 (sK372 = sK370) | ($true != $true) | ~spl445_2),
% 2.89/1.93 inference(forward_demodulation,[status(thm)],[f2885,f2045])).
% 2.89/1.93 thf(f2045,plain,(
% 2.89/1.93 (setunionsingleton = $true)),
% 2.89/1.93 inference(cnf_transformation,[status(thm)],[f1311])).
% 2.89/1.93 thf(f1311,plain,(
% 2.89/1.93 (notequalI2 = $true) & (powersetAx = $true) & (setunionsingleton2 = $true) & (binintersectLsub = $true) & (emptysetAx = $true) & (binintersectSubset3 = $true) & (foundationAx = $true) & (binunionEcases = $true) & (vacuousDall = $true) & (quantDeMorgan4 = $true) & (descr__Cong = $true) & (eqinunit = $true) & (notequalI1 = $true) & (uniqinunit = $true) & (notinsingleton = $true) & (singletoninpowunion = $true) & (symdiffI1 = $true) & (subsetE = $true) & (powerset__Cong = $true) & (dsetconstr__Cong = $true) & (emptyinPowerset = $true) & (setbeta = $true) & (setminusIRneg = $true) & (kpairiskpair = $true) & (((in @ (setadjoin @ sK370 @ emptyset) @ (setadjoin @ (setadjoin @ sK372 @ emptyset) @ (setadjoin @ (setadjoin @ sK372 @ (setadjoin @ sK371 @ emptyset)) @ emptyset))) = $true) & (sK372 != sK370)) & (setminusILneg = $true) & (binintersectSubset2 = $true) & (dsetconstrEL = $true) & (setext = $true) & (setminusSubset1 = $true) & (bs114d = $true) & (in__Cong = $true) & (emptyset__Cong = $true) & (subsetRefl = $true) & (replAx = $true) & (upairsetIR = $true) & (setunionE = $true) & (inPowerset = $true) & (setadjoinIL = $true) & (subsetI1 = $true) & (exu__Cong = $true) & (singletonprop = $true) & (noeltsimpempty = $true) & (setadjoinE = $true) & (setunionI = $true) & (subsetemptysetimpeq = $true) & (powersetI1 = $true) & (emptysetimpfalse = $true) & (prop2setE = $true) & (binintersectI = $true) & (kpairp = $true) & (setunionAx = $true) & (exuI3 = $true) & (subsetTrans = $true) & (emptyinunitempty = $true) & (setunionsingleton = $true) & (setukpairIR = $true) & (upairset2E = $true) & (emptysetsubset = $true) & (binintersectER = $true) & (ex1I = $true) & (emptysetE = $true) & (inCongP = $true) & (omega__Cong = $true) & (binintersectEL = $true) & (setadjoinIR = $true) & (ex1I2 = $true) & (setunionsingleton1 = $true) & (setadjoin__Cong = $true) & (nonemptyE1 = $true) & (setunion__Cong = $true) & (upairset2IR = $true) & (binintersectSubset1 = $true) & (setminusER = $true) & (setminusLsub = $true) & (exuI1 = $true) & (upairsubunion = $true) & (powersetE = $true) & (binunionRsub = $true) & (symdiffIneg1 = $true) & (setadjoinOr = $true) & (binunionIR = $true) & (powersetE1 = $true) & (prop2setI = $true) & (singletoninpowerset = $true) & (nonemptyI = $true) & (singletonsubset = $true) & (omegaIndAx = $true) & (cartprodpairin = $true) & (setadjoinSub = $true) & (upairsetIL = $true) & (sepSubset = $true) & (setukpairIL = $true) & (emptyInPowerset = $true) & (notsubsetI = $true) & (subset2powerset = $true) & (eqimpsubset2 = $true) & (sepInPowerset = $true) & (exuI2 = $true) & (omegaSAx = $true) & (ex1E1 = $true) & (emptyI = $true) & (powersetI = $true) & (setextsub = $true) & (exuE3e = $true) & (setextAx = $true) & (dsetconstrI = $true) & (nonemptyImpWitness = $true) & (symdiffI2 = $true) & (subsetE2 = $true) & (subPowSU = $true) & (notinemptyset = $true) & (subsetI2 = $true) & (ubforcartprodlem3 = $true) & (ubforcartprodlem1 = $true) & (upairsetE = $true) & (binintersectSubset4 = $true) & (binintersectRsub = $true) & (binunionIL = $true) & (symdiffE = $true) & (omega0Ax = $true) & (singletonsswitch = $true) & (cartprodmempair = $true) & (setunionE2 = $true) & (symdiffIneg2 = $true) & (singletonsuniq = $true) & (setminusERneg = $true) & (dsetconstrER = $true) & (setadjoinAx = $true) & (quantDeMorgan3 = $true) & (setminusSubset2 = $true) & (wellorderingAx = $true) & (notdexE = $true) & (setminusI = $true) & (binunionE = $true) & (upairinpowunion = $true) & (ubforcartprodlem2 = $true) & (setminusEL = $true) & (setminusELneg = $true) & (eqimpsubset1 = $true) & (prop2set2propI = $true) & (quantDeMorgan2 = $true) & (setoftrueEq = $true) & (emptyE1 = $true) & (exuE1 = $true) & (powersetsubset = $true) & (exuE2 = $true) & (setadjoinSub2 = $true) & (exuEu = $true) & (nonemptyI1 = $true) & (binintersectSubset5 = $true) & (cartprodmempair1 = $true) & (binunionLsub = $true) & (notdallE = $true) & (descrp = $true) & (exuE3u = $true) & (quantDeMorgan1 = $true) & (disjointsetsI1 = $true) & (secondinupair = $true)),
% 2.89/1.93 inference(skolemisation,[status(esa),new_symbols(skolem,[sK370,sK371,sK372])],[f548,f1310])).
% 2.89/1.93 thf(f1310,plain,(
% 2.89/1.93 ? [X0 : $i,X1 : $i,X2 : $i] : (((in @ (setadjoin @ X0 @ emptyset) @ (setadjoin @ (setadjoin @ X2 @ emptyset) @ (setadjoin @ (setadjoin @ X2 @ (setadjoin @ X1 @ emptyset)) @ emptyset))) = $true) & (X0 != X2)) => (((in @ (setadjoin @ sK370 @ emptyset) @ (setadjoin @ (setadjoin @ sK372 @ emptyset) @ (setadjoin @ (setadjoin @ sK372 @ (setadjoin @ sK371 @ emptyset)) @ emptyset))) = $true) & (sK372 != sK370))),
% 2.89/1.93 introduced(definition,[],[choice_axiom])).
% 2.89/1.93 thf(f548,plain,(
% 2.89/1.93 (notequalI2 = $true) & (powersetAx = $true) & (setunionsingleton2 = $true) & (binintersectLsub = $true) & (emptysetAx = $true) & (binintersectSubset3 = $true) & (foundationAx = $true) & (binunionEcases = $true) & (vacuousDall = $true) & (quantDeMorgan4 = $true) & (descr__Cong = $true) & (eqinunit = $true) & (notequalI1 = $true) & (uniqinunit = $true) & (notinsingleton = $true) & (singletoninpowunion = $true) & (symdiffI1 = $true) & (subsetE = $true) & (powerset__Cong = $true) & (dsetconstr__Cong = $true) & (emptyinPowerset = $true) & (setbeta = $true) & (setminusIRneg = $true) & (kpairiskpair = $true) & ? [X0 : $i,X1 : $i,X2 : $i] : (((in @ (setadjoin @ X0 @ emptyset) @ (setadjoin @ (setadjoin @ X2 @ emptyset) @ (setadjoin @ (setadjoin @ X2 @ (setadjoin @ X1 @ emptyset)) @ emptyset))) = $true) & (X0 != X2)) & (setminusILneg = $true) & (binintersectSubset2 = $true) & (dsetconstrEL = $true) & (setext = $true) & (setminusSubset1 = $true) & (bs114d = $true) & (in__Cong = $true) & (emptyset__Cong = $true) & (subsetRefl = $true) & (replAx = $true) & (upairsetIR = $true) & (setunionE = $true) & (inPowerset = $true) & (setadjoinIL = $true) & (subsetI1 = $true) & (exu__Cong = $true) & (singletonprop = $true) & (noeltsimpempty = $true) & (setadjoinE = $true) & (setunionI = $true) & (subsetemptysetimpeq = $true) & (powersetI1 = $true) & (emptysetimpfalse = $true) & (prop2setE = $true) & (binintersectI = $true) & (kpairp = $true) & (setunionAx = $true) & (exuI3 = $true) & (subsetTrans = $true) & (emptyinunitempty = $true) & (setunionsingleton = $true) & (setukpairIR = $true) & (upairset2E = $true) & (emptysetsubset = $true) & (binintersectER = $true) & (ex1I = $true) & (emptysetE = $true) & (inCongP = $true) & (omega__Cong = $true) & (binintersectEL = $true) & (setadjoinIR = $true) & (ex1I2 = $true) & (setunionsingleton1 = $true) & (setadjoin__Cong = $true) & (nonemptyE1 = $true) & (setunion__Cong = $true) & (upairset2IR = $true) & (binintersectSubset1 = $true) & (setminusER = $true) & (setminusLsub = $true) & (exuI1 = $true) & (upairsubunion = $true) & (powersetE = $true) & (binunionRsub = $true) & (symdiffIneg1 = $true) & (setadjoinOr = $true) & (binunionIR = $true) & (powersetE1 = $true) & (prop2setI = $true) & (singletoninpowerset = $true) & (nonemptyI = $true) & (singletonsubset = $true) & (omegaIndAx = $true) & (cartprodpairin = $true) & (setadjoinSub = $true) & (upairsetIL = $true) & (sepSubset = $true) & (setukpairIL = $true) & (emptyInPowerset = $true) & (notsubsetI = $true) & (subset2powerset = $true) & (eqimpsubset2 = $true) & (sepInPowerset = $true) & (exuI2 = $true) & (omegaSAx = $true) & (ex1E1 = $true) & (emptyI = $true) & (powersetI = $true) & (setextsub = $true) & (exuE3e = $true) & (setextAx = $true) & (dsetconstrI = $true) & (nonemptyImpWitness = $true) & (symdiffI2 = $true) & (subsetE2 = $true) & (subPowSU = $true) & (notinemptyset = $true) & (subsetI2 = $true) & (ubforcartprodlem3 = $true) & (ubforcartprodlem1 = $true) & (upairsetE = $true) & (binintersectSubset4 = $true) & (binintersectRsub = $true) & (binunionIL = $true) & (symdiffE = $true) & (omega0Ax = $true) & (singletonsswitch = $true) & (cartprodmempair = $true) & (setunionE2 = $true) & (symdiffIneg2 = $true) & (singletonsuniq = $true) & (setminusERneg = $true) & (dsetconstrER = $true) & (setadjoinAx = $true) & (quantDeMorgan3 = $true) & (setminusSubset2 = $true) & (wellorderingAx = $true) & (notdexE = $true) & (setminusI = $true) & (binunionE = $true) & (upairinpowunion = $true) & (ubforcartprodlem2 = $true) & (setminusEL = $true) & (setminusELneg = $true) & (eqimpsubset1 = $true) & (prop2set2propI = $true) & (quantDeMorgan2 = $true) & (setoftrueEq = $true) & (emptyE1 = $true) & (exuE1 = $true) & (powersetsubset = $true) & (exuE2 = $true) & (setadjoinSub2 = $true) & (exuEu = $true) & (nonemptyI1 = $true) & (binintersectSubset5 = $true) & (cartprodmempair1 = $true) & (binunionLsub = $true) & (notdallE = $true) & (descrp = $true) & (exuE3u = $true) & (quantDeMorgan1 = $true) & (disjointsetsI1 = $true) & (secondinupair = $true)),
% 2.89/1.93 inference(flattening,[status(thm)],[f547])).
% 2.89/1.93 thf(f547,plain,(
% 2.89/1.93 (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i,X1 : $i,X2 : $i] : (((in @ (setadjoin @ X0 @ emptyset) @ (setadjoin @ (setadjoin @ X2 @ emptyset) @ (setadjoin @ (setadjoin @ X2 @ (setadjoin @ X1 @ emptyset)) @ emptyset))) = $true) & (X0 != X2)) & (singletonsuniq = $true)) & (ex1I2 = $true)) & (ex1I = $true)) & (ex1E1 = $true)) & (singletonprop = $true)) & (setunionsingleton = $true)) & (setunionsingleton2 = $true)) & (setunionsingleton1 = $true)) & (setunionE2 = $true)) & (cartprodmempair = $true)) & (cartprodmempair1 = $true)) & (cartprodpairin = $true)) & (ubforcartprodlem3 = $true)) & (ubforcartprodlem2 = $true)) & (ubforcartprodlem1 = $true)) & (upairinpowunion = $true)) & (upairsubunion = $true)) & (upairset2E = $true)) & (singletoninpowunion = $true)) & (singletoninpowerset = $true)) & (singletonsubset = $true)) & (kpairp = $true)) & (kpairiskpair = $true)) & (setukpairIR = $true)) & (setukpairIL = $true)) & (secondinupair = $true)) & (symdiffIneg2 = $true)) & (symdiffIneg1 = $true)) & (symdiffI2 = $true)) & (symdiffI1 = $true)) & (symdiffE = $true)) & (setminusSubset1 = $true)) & (setminusLsub = $true)) & (setminusIRneg = $true)) & (setminusILneg = $true)) & (setminusELneg = $true)) & (setminusERneg = $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)),
% 2.89/1.93 inference(ennf_transformation,[status(thm)],[f477])).
% 2.89/1.93 thf(f477,plain,(
% 2.89/1.93 ~((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) => ((setminusERneg = $true) => ((setminusELneg = $true) => ((setminusILneg = $true) => ((setminusIRneg = $true) => ((setminusLsub = $true) => ((setminusSubset1 = $true) => ((symdiffE = $true) => ((symdiffI1 = $true) => ((symdiffI2 = $true) => ((symdiffIneg1 = $true) => ((symdiffIneg2 = $true) => ((secondinupair = $true) => ((setukpairIL = $true) => ((setukpairIR = $true) => ((kpairiskpair = $true) => ((kpairp = $true) => ((singletonsubset = $true) => ((singletoninpowerset = $true) => ((singletoninpowunion = $true) => ((upairset2E = $true) => ((upairsubunion = $true) => ((upairinpowunion = $true) => ((ubforcartprodlem1 = $true) => ((ubforcartprodlem2 = $true) => ((ubforcartprodlem3 = $true) => ((cartprodpairin = $true) => ((cartprodmempair1 = $true) => ((cartprodmempair = $true) => ((setunionE2 = $true) => ((setunionsingleton1 = $true) => ((setunionsingleton2 = $true) => ((setunionsingleton = $true) => ((singletonprop = $true) => ((ex1E1 = $true) => ((ex1I = $true) => ((ex1I2 = $true) => ((singletonsuniq = $true) => ! [X0 : $i,X1 : $i,X2 : $i] : (((in @ (setadjoin @ X0 @ emptyset) @ (setadjoin @ (setadjoin @ X2 @ emptyset) @ (setadjoin @ (setadjoin @ X2 @ (setadjoin @ X1 @ emptyset)) @ emptyset))) = $true) => (X0 = X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.89/1.93 inference(fool_elimination,[status(thm)],[f476])).
% 2.89/1.93 thf(f476,plain,(
% 2.89/1.93 ~(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 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => (ubforcartprodlem2 => (ubforcartprodlem3 => (cartprodpairin => (cartprodmempair1 => (cartprodmempair => (setunionE2 => (setunionsingleton1 => (setunionsingleton2 => (setunionsingleton => (singletonprop => (ex1E1 => (ex1I => (ex1I2 => (singletonsuniq => ! [X0 : $i,X1 : $i,X2 : $i] : ((in @ (setadjoin @ X0 @ emptyset) @ (setadjoin @ (setadjoin @ X2 @ emptyset) @ (setadjoin @ (setadjoin @ X2 @ (setadjoin @ X1 @ emptyset)) @ emptyset))) => (X0 = X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.89/1.93 inference(rectify,[status(thm)],[f166])).
% 2.89/1.93 thf(f166,negated_conjecture,(
% 2.89/1.93 ~(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 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => (ubforcartprodlem2 => (ubforcartprodlem3 => (cartprodpairin => (cartprodmempair1 => (cartprodmempair => (setunionE2 => (setunionsingleton1 => (setunionsingleton2 => (setunionsingleton => (singletonprop => (ex1E1 => (ex1I => (ex1I2 => (singletonsuniq => ! [X8 : $i,X2 : $i,X1 : $i] : ((in @ (setadjoin @ X8 @ emptyset) @ (setadjoin @ (setadjoin @ X1 @ emptyset) @ (setadjoin @ (setadjoin @ X1 @ (setadjoin @ X2 @ emptyset)) @ emptyset))) => (X1 = X8))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.89/1.93 inference(negated_conjecture,[status(cth)],[f165])).
% 2.89/1.93 thf(f165,conjecture,(
% 2.89/1.93 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 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => (ubforcartprodlem2 => (ubforcartprodlem3 => (cartprodpairin => (cartprodmempair1 => (cartprodmempair => (setunionE2 => (setunionsingleton1 => (setunionsingleton2 => (setunionsingleton => (singletonprop => (ex1E1 => (ex1I => (ex1I2 => (singletonsuniq => ! [X8 : $i,X2 : $i,X1 : $i] : ((in @ (setadjoin @ X8 @ emptyset) @ (setadjoin @ (setadjoin @ X1 @ emptyset) @ (setadjoin @ (setadjoin @ X1 @ (setadjoin @ X2 @ emptyset)) @ emptyset))) => (X1 = X8)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.89/1.93 file('/export/starexec/sandbox/tmp/tmp.vWjcIjdJ0D/DTF2THF_13509.p',setukpairinjL1)).
% 2.89/1.93 thf(f2885,plain,(
% 2.89/1.93 (sK372 = sK370) | (setunionsingleton != $true) | ~spl445_2),
% 2.89/1.93 inference(superposition,[status(thm)],[f1597,f2692])).
% 2.89/1.93 thf(f2692,plain,(
% 2.89/1.93 (sK372 = (setunion @ (setadjoin @ sK370 @ emptyset))) | ~spl445_2),
% 2.89/1.93 inference(trivial_inequality_removal,[status(thm)],[f2691])).
% 2.89/1.93 thf(f2691,plain,(
% 2.89/1.93 (sK372 = (setunion @ (setadjoin @ sK370 @ emptyset))) | ($true != $true) | ~spl445_2),
% 2.89/1.93 inference(forward_demodulation,[status(thm)],[f2630,f2045])).
% 2.89/1.93 thf(f2630,plain,(
% 2.89/1.93 (setunionsingleton != $true) | (sK372 = (setunion @ (setadjoin @ sK370 @ emptyset))) | ~spl445_2),
% 2.89/1.93 inference(superposition,[status(thm)],[f1597,f2363])).
% 2.89/1.93 thf(f2363,plain,(
% 2.89/1.93 ((setadjoin @ sK372 @ emptyset) = (setadjoin @ sK370 @ emptyset)) | ~spl445_2),
% 2.89/1.93 inference(avatar_component_clause,[status(thm)],[f2361])).
% 2.89/1.93 thf(f1597,plain,(
% 2.89/1.93 ( ! [X0 : $i] : (((setunion @ (setadjoin @ X0 @ emptyset)) = X0) | (setunionsingleton != $true)) )),
% 2.89/1.93 inference(cnf_transformation,[status(thm)],[f903])).
% 2.89/1.93 thf(f903,plain,(
% 2.89/1.93 (! [X0 : $i] : ((setunion @ (setadjoin @ X0 @ emptyset)) = X0) | (setunionsingleton != $true)) & ((setunionsingleton = $true) | ((setunion @ (setadjoin @ sK124 @ emptyset)) != sK124))),
% 2.89/1.93 inference(skolemisation,[status(esa),new_symbols(skolem,[sK124])],[f901,f902])).
% 2.89/1.93 thf(f902,plain,(
% 2.89/1.93 ? [X1 : $i] : ((setunion @ (setadjoin @ X1 @ emptyset)) != X1) => ((setunion @ (setadjoin @ sK124 @ emptyset)) != sK124)),
% 2.89/1.93 introduced(definition,[],[choice_axiom])).
% 2.89/1.93 thf(f901,plain,(
% 2.89/1.93 (! [X0 : $i] : ((setunion @ (setadjoin @ X0 @ emptyset)) = X0) | (setunionsingleton != $true)) & ((setunionsingleton = $true) | ? [X1 : $i] : ((setunion @ (setadjoin @ X1 @ emptyset)) != X1))),
% 2.89/1.93 inference(rectify,[status(thm)],[f900])).
% 2.89/1.93 thf(f900,plain,(
% 2.89/1.93 (! [X0 : $i] : ((setunion @ (setadjoin @ X0 @ emptyset)) = X0) | (setunionsingleton != $true)) & ((setunionsingleton = $true) | ? [X0 : $i] : ((setunion @ (setadjoin @ X0 @ emptyset)) != X0))),
% 2.89/1.93 inference(nnf_transformation,[status(thm)],[f235])).
% 2.89/1.93 thf(f235,plain,(
% 2.89/1.93 ! [X0 : $i] : ((setunion @ (setadjoin @ X0 @ emptyset)) = X0) <=> (setunionsingleton = $true)),
% 2.89/1.93 inference(fool_elimination,[status(thm)],[f234])).
% 2.89/1.93 thf(f234,plain,(
% 2.89/1.93 (! [X0 : $i] : ((setunion @ (setadjoin @ X0 @ emptyset)) = X0) = setunionsingleton)),
% 2.89/1.93 inference(rectify,[status(thm)],[f157])).
% 2.89/1.93 thf(f157,axiom,(
% 2.89/1.93 (! [X1 : $i] : ((setunion @ (setadjoin @ X1 @ emptyset)) = X1) = setunionsingleton)),
% 2.89/1.93 file('/export/starexec/sandbox/tmp/tmp.vWjcIjdJ0D/DTF2THF_13509.p',setunionsingleton)).
% 2.89/1.93 thf(f2853,plain,(
% 2.89/1.93 ~spl445_7),
% 2.89/1.93 inference(avatar_contradiction_clause,[status(thm)],[f2852])).
% 2.89/1.93 thf(f2852,plain,(
% 2.89/1.93 $false | ~spl445_7),
% 2.89/1.93 inference(trivial_inequality_removal,[status(thm)],[f2851])).
% 2.89/1.93 thf(f2851,plain,(
% 2.89/1.93 (sK370 != sK370) | ~spl445_7),
% 2.89/1.93 inference(superposition,[status(thm)],[f2076,f2600])).
% 2.89/1.93 thf(f2600,plain,(
% 2.89/1.93 (sK372 = sK370) | ~spl445_7),
% 2.89/1.93 inference(avatar_component_clause,[status(thm)],[f2598])).
% 2.89/1.93 thf(f2076,plain,(
% 2.89/1.93 (sK372 != sK370)),
% 2.89/1.93 inference(cnf_transformation,[status(thm)],[f1311])).
% 2.89/1.93 thf(f2768,plain,(
% 2.89/1.93 spl445_4 | ~spl445_1),
% 2.89/1.93 inference(avatar_split_clause,[status(thm)],[f2767,f2357,f2371])).
% 2.89/1.93 thf(f2371,definition,(
% 2.89/1.93 spl445_4 <=> ((setadjoin @ sK372 @ (setadjoin @ sK371 @ emptyset)) = (setadjoin @ sK370 @ emptyset))),
% 2.89/1.93 introduced(definition,[new_symbols(naming,[spl445_4])],[avatar_definition])).
% 2.89/1.93 thf(f2357,definition,(
% 2.89/1.93 spl445_1 <=> ((in @ (setadjoin @ sK370 @ emptyset) @ (setadjoin @ (setadjoin @ sK372 @ (setadjoin @ sK371 @ emptyset)) @ emptyset)) = $true)),
% 2.89/1.93 introduced(definition,[new_symbols(naming,[spl445_1])],[avatar_definition])).
% 2.89/1.93 thf(f2767,plain,(
% 2.89/1.93 ((setadjoin @ sK372 @ (setadjoin @ sK371 @ emptyset)) = (setadjoin @ sK370 @ emptyset)) | ~spl445_1),
% 2.89/1.93 inference(trivial_inequality_removal,[status(thm)],[f2766])).
% 2.89/1.93 thf(f2766,plain,(
% 2.89/1.93 ((setadjoin @ sK372 @ (setadjoin @ sK371 @ emptyset)) = (setadjoin @ sK370 @ emptyset)) | ($true != $true) | ~spl445_1),
% 2.89/1.93 inference(forward_demodulation,[status(thm)],[f2742,f2087])).
% 2.89/1.93 thf(f2087,plain,(
% 2.89/1.93 (notinsingleton = $true)),
% 2.89/1.93 inference(cnf_transformation,[status(thm)],[f1311])).
% 2.89/1.93 thf(f2742,plain,(
% 2.89/1.93 ((setadjoin @ sK372 @ (setadjoin @ sK371 @ emptyset)) = (setadjoin @ sK370 @ emptyset)) | (notinsingleton != $true) | ~spl445_1),
% 2.89/1.93 inference(trivial_inequality_removal,[status(thm)],[f2736])).
% 2.89/1.93 thf(f2736,plain,(
% 2.89/1.93 ($true != $true) | (notinsingleton != $true) | ((setadjoin @ sK372 @ (setadjoin @ sK371 @ emptyset)) = (setadjoin @ sK370 @ emptyset)) | ~spl445_1),
% 2.89/1.93 inference(superposition,[status(thm)],[f1543,f2359])).
% 2.89/1.93 thf(f2359,plain,(
% 2.89/1.93 ((in @ (setadjoin @ sK370 @ emptyset) @ (setadjoin @ (setadjoin @ sK372 @ (setadjoin @ sK371 @ emptyset)) @ emptyset)) = $true) | ~spl445_1),
% 2.89/1.93 inference(avatar_component_clause,[status(thm)],[f2357])).
% 2.89/1.93 thf(f1543,plain,(
% 2.89/1.93 ( ! [X0 : $i,X1 : $i] : (($true != (in @ X1 @ (setadjoin @ X0 @ emptyset))) | (X0 = X1) | (notinsingleton != $true)) )),
% 2.89/1.93 inference(cnf_transformation,[status(thm)],[f831])).
% 2.89/1.93 thf(f831,plain,(
% 2.89/1.93 (! [X0 : $i,X1 : $i] : (($true != (in @ X1 @ (setadjoin @ X0 @ emptyset))) | (X0 = X1)) | (notinsingleton != $true)) & ((notinsingleton = $true) | (($true = (in @ sK83 @ (setadjoin @ sK82 @ emptyset))) & (sK82 != sK83)))),
% 2.89/1.93 inference(skolemisation,[status(esa),new_symbols(skolem,[sK82,sK83])],[f829,f830])).
% 2.89/1.93 thf(f830,plain,(
% 2.89/1.93 ? [X2 : $i,X3 : $i] : (((in @ X3 @ (setadjoin @ X2 @ emptyset)) = $true) & (X2 != X3)) => (($true = (in @ sK83 @ (setadjoin @ sK82 @ emptyset))) & (sK82 != sK83))),
% 2.89/1.93 introduced(definition,[],[choice_axiom])).
% 2.89/1.93 thf(f829,plain,(
% 2.89/1.93 (! [X0 : $i,X1 : $i] : (($true != (in @ X1 @ (setadjoin @ X0 @ emptyset))) | (X0 = X1)) | (notinsingleton != $true)) & ((notinsingleton = $true) | ? [X2 : $i,X3 : $i] : (((in @ X3 @ (setadjoin @ X2 @ emptyset)) = $true) & (X2 != X3)))),
% 2.89/1.93 inference(rectify,[status(thm)],[f828])).
% 2.89/1.93 thf(f828,plain,(
% 2.89/1.93 (! [X0 : $i,X1 : $i] : (($true != (in @ X1 @ (setadjoin @ X0 @ emptyset))) | (X0 = X1)) | (notinsingleton != $true)) & ((notinsingleton = $true) | ? [X0 : $i,X1 : $i] : (($true = (in @ X1 @ (setadjoin @ X0 @ emptyset))) & (X0 != X1)))),
% 2.89/1.93 inference(nnf_transformation,[status(thm)],[f579])).
% 2.89/1.93 thf(f579,plain,(
% 2.89/1.93 ! [X0 : $i,X1 : $i] : (($true != (in @ X1 @ (setadjoin @ X0 @ emptyset))) | (X0 = X1)) <=> (notinsingleton = $true)),
% 2.89/1.93 inference(ennf_transformation,[status(thm)],[f513])).
% 2.89/1.93 thf(f513,plain,(
% 2.89/1.93 (notinsingleton = $true) <=> ! [X0 : $i,X1 : $i] : ((X0 != X1) => ($true != (in @ X1 @ (setadjoin @ X0 @ emptyset))))),
% 2.89/1.93 inference(flattening,[status(thm)],[f376])).
% 2.89/1.93 thf(f376,plain,(
% 2.89/1.93 ! [X0 : $i,X1 : $i] : ((X0 != X1) => ~($true = (in @ X1 @ (setadjoin @ X0 @ emptyset)))) <=> (notinsingleton = $true)),
% 2.89/1.93 inference(fool_elimination,[status(thm)],[f375])).
% 2.89/1.93 thf(f375,plain,(
% 2.89/1.93 (! [X0 : $i,X1 : $i] : ((X0 != X1) => ~(in @ X1 @ (setadjoin @ X0 @ emptyset))) = notinsingleton)),
% 2.89/1.93 inference(rectify,[status(thm)],[f47])).
% 2.89/1.93 thf(f47,axiom,(
% 2.89/1.93 (! [X1 : $i,X2 : $i] : ((X1 != X2) => ~(in @ X2 @ (setadjoin @ X1 @ emptyset))) = notinsingleton)),
% 2.89/1.93 file('/export/starexec/sandbox/tmp/tmp.vWjcIjdJ0D/DTF2THF_13509.p',notinsingleton)).
% 2.89/1.93 thf(f2611,plain,(
% 2.89/1.93 spl445_7 | ~spl445_4),
% 2.89/1.93 inference(avatar_split_clause,[status(thm)],[f2610,f2371,f2598])).
% 2.89/1.93 thf(f2610,plain,(
% 2.89/1.93 (sK372 = sK370) | ~spl445_4),
% 2.89/1.93 inference(trivial_inequality_removal,[status(thm)],[f2609])).
% 2.89/1.93 thf(f2609,plain,(
% 2.89/1.93 (sK372 = sK370) | ($true != $true) | ~spl445_4),
% 2.89/1.93 inference(forward_demodulation,[status(thm)],[f2586,f2088])).
% 2.89/1.93 thf(f2088,plain,(
% 2.89/1.93 (uniqinunit = $true)),
% 2.89/1.93 inference(cnf_transformation,[status(thm)],[f1311])).
% 2.89/1.93 thf(f2586,plain,(
% 2.89/1.93 (sK372 = sK370) | (uniqinunit != $true) | ~spl445_4),
% 2.89/1.93 inference(trivial_inequality_removal,[status(thm)],[f2580])).
% 2.89/1.93 thf(f2580,plain,(
% 2.89/1.93 (uniqinunit != $true) | ($true != $true) | (sK372 = sK370) | ~spl445_4),
% 2.89/1.93 inference(superposition,[status(thm)],[f1532,f2539])).
% 2.89/1.93 thf(f2539,plain,(
% 2.89/1.93 ($true = (in @ sK372 @ (setadjoin @ sK370 @ emptyset))) | ~spl445_4),
% 2.89/1.93 inference(trivial_inequality_removal,[status(thm)],[f2538])).
% 2.89/1.93 thf(f2538,plain,(
% 2.89/1.93 ($true != $true) | ($true = (in @ sK372 @ (setadjoin @ sK370 @ emptyset))) | ~spl445_4),
% 2.89/1.93 inference(forward_demodulation,[status(thm)],[f2521,f2062])).
% 2.89/1.93 thf(f2062,plain,(
% 2.89/1.93 (setadjoinIL = $true)),
% 2.89/1.93 inference(cnf_transformation,[status(thm)],[f1311])).
% 2.89/1.93 thf(f2521,plain,(
% 2.89/1.93 (setadjoinIL != $true) | ($true = (in @ sK372 @ (setadjoin @ sK370 @ emptyset))) | ~spl445_4),
% 2.89/1.93 inference(superposition,[status(thm)],[f1459,f2373])).
% 2.89/1.93 thf(f2373,plain,(
% 2.89/1.93 ((setadjoin @ sK372 @ (setadjoin @ sK371 @ emptyset)) = (setadjoin @ sK370 @ emptyset)) | ~spl445_4),
% 2.89/1.93 inference(avatar_component_clause,[status(thm)],[f2371])).
% 2.89/1.93 thf(f1459,plain,(
% 2.89/1.93 ( ! [X0 : $i,X1 : $i] : (($true = (in @ X1 @ (setadjoin @ X1 @ X0))) | (setadjoinIL != $true)) )),
% 2.89/1.93 inference(cnf_transformation,[status(thm)],[f728])).
% 2.89/1.93 thf(f728,plain,(
% 2.89/1.93 (! [X0 : $i,X1 : $i] : ($true = (in @ X1 @ (setadjoin @ X1 @ X0))) | (setadjoinIL != $true)) & ((setadjoinIL = $true) | ((in @ sK21 @ (setadjoin @ sK21 @ sK20)) != $true))),
% 2.89/1.93 inference(skolemisation,[status(esa),new_symbols(skolem,[sK20,sK21])],[f726,f727])).
% 2.89/1.93 thf(f727,plain,(
% 2.89/1.93 ? [X2 : $i,X3 : $i] : ($true != (in @ X3 @ (setadjoin @ X3 @ X2))) => ((in @ sK21 @ (setadjoin @ sK21 @ sK20)) != $true)),
% 2.89/1.93 introduced(definition,[],[choice_axiom])).
% 2.89/1.93 thf(f726,plain,(
% 2.89/1.93 (! [X0 : $i,X1 : $i] : ($true = (in @ X1 @ (setadjoin @ X1 @ X0))) | (setadjoinIL != $true)) & ((setadjoinIL = $true) | ? [X2 : $i,X3 : $i] : ($true != (in @ X3 @ (setadjoin @ X3 @ X2))))),
% 2.89/1.93 inference(rectify,[status(thm)],[f725])).
% 2.89/1.93 thf(f725,plain,(
% 2.89/1.93 (! [X0 : $i,X1 : $i] : ($true = (in @ X1 @ (setadjoin @ X1 @ X0))) | (setadjoinIL != $true)) & ((setadjoinIL = $true) | ? [X0 : $i,X1 : $i] : ($true != (in @ X1 @ (setadjoin @ X1 @ X0))))),
% 2.89/1.93 inference(nnf_transformation,[status(thm)],[f417])).
% 2.89/1.93 thf(f417,plain,(
% 2.89/1.93 ! [X0 : $i,X1 : $i] : ($true = (in @ X1 @ (setadjoin @ X1 @ X0))) <=> (setadjoinIL = $true)),
% 2.89/1.93 inference(fool_elimination,[status(thm)],[f416])).
% 2.89/1.93 thf(f416,plain,(
% 2.89/1.93 (! [X0 : $i,X1 : $i] : (in @ X1 @ (setadjoin @ X1 @ X0)) = setadjoinIL)),
% 2.89/1.93 inference(rectify,[status(thm)],[f31])).
% 2.89/1.93 thf(f31,axiom,(
% 2.89/1.93 (! [X2 : $i,X1 : $i] : (in @ X1 @ (setadjoin @ X1 @ X2)) = setadjoinIL)),
% 2.89/1.93 file('/export/starexec/sandbox/tmp/tmp.vWjcIjdJ0D/DTF2THF_13509.p',setadjoinIL)).
% 2.89/1.93 thf(f1532,plain,(
% 2.89/1.93 ( ! [X2 : $i,X3 : $i] : (($true != (in @ X2 @ (setadjoin @ X3 @ emptyset))) | (uniqinunit != $true) | (X2 = X3)) )),
% 2.89/1.93 inference(cnf_transformation,[status(thm)],[f821])).
% 2.89/1.93 thf(f821,plain,(
% 2.89/1.93 ((uniqinunit = $true) | (((in @ sK76 @ (setadjoin @ sK77 @ emptyset)) = $true) & (sK77 != sK76))) & (! [X2 : $i,X3 : $i] : (($true != (in @ X2 @ (setadjoin @ X3 @ emptyset))) | (X2 = X3)) | (uniqinunit != $true))),
% 2.89/1.93 inference(skolemisation,[status(esa),new_symbols(skolem,[sK76,sK77])],[f819,f820])).
% 2.89/1.93 thf(f820,plain,(
% 2.89/1.93 ? [X0 : $i,X1 : $i] : (($true = (in @ X0 @ (setadjoin @ X1 @ emptyset))) & (X0 != X1)) => (((in @ sK76 @ (setadjoin @ sK77 @ emptyset)) = $true) & (sK77 != sK76))),
% 2.89/1.93 introduced(definition,[],[choice_axiom])).
% 2.89/1.93 thf(f819,plain,(
% 2.89/1.93 ((uniqinunit = $true) | ? [X0 : $i,X1 : $i] : (($true = (in @ X0 @ (setadjoin @ X1 @ emptyset))) & (X0 != X1))) & (! [X2 : $i,X3 : $i] : (($true != (in @ X2 @ (setadjoin @ X3 @ emptyset))) | (X2 = X3)) | (uniqinunit != $true))),
% 2.89/1.93 inference(rectify,[status(thm)],[f818])).
% 2.89/1.93 thf(f818,plain,(
% 2.89/1.93 ((uniqinunit = $true) | ? [X0 : $i,X1 : $i] : (($true = (in @ X0 @ (setadjoin @ X1 @ emptyset))) & (X0 != X1))) & (! [X0 : $i,X1 : $i] : (($true != (in @ X0 @ (setadjoin @ X1 @ emptyset))) | (X0 = X1)) | (uniqinunit != $true))),
% 2.89/1.93 inference(nnf_transformation,[status(thm)],[f682])).
% 2.89/1.93 thf(f682,plain,(
% 2.89/1.93 (uniqinunit = $true) <=> ! [X0 : $i,X1 : $i] : (($true != (in @ X0 @ (setadjoin @ X1 @ emptyset))) | (X0 = X1))),
% 2.89/1.93 inference(ennf_transformation,[status(thm)],[f319])).
% 2.89/1.93 thf(f319,plain,(
% 2.89/1.93 ! [X0 : $i,X1 : $i] : (($true = (in @ X0 @ (setadjoin @ X1 @ emptyset))) => (X0 = X1)) <=> (uniqinunit = $true)),
% 2.89/1.93 inference(fool_elimination,[status(thm)],[f318])).
% 2.89/1.93 thf(f318,plain,(
% 2.89/1.93 (uniqinunit = ! [X0 : $i,X1 : $i] : ((in @ X0 @ (setadjoin @ X1 @ emptyset)) => (X0 = X1)))),
% 2.89/1.93 inference(rectify,[status(thm)],[f46])).
% 2.89/1.93 thf(f46,axiom,(
% 2.89/1.93 (uniqinunit = ! [X1 : $i,X2 : $i] : ((in @ X1 @ (setadjoin @ X2 @ emptyset)) => (X1 = X2)))),
% 2.89/1.93 file('/export/starexec/sandbox/tmp/tmp.vWjcIjdJ0D/DTF2THF_13509.p',uniqinunit)).
% 2.89/1.93 thf(f2383,plain,(
% 2.89/1.93 spl445_1 | spl445_2),
% 2.89/1.93 inference(avatar_split_clause,[status(thm)],[f2382,f2361,f2357])).
% 2.89/1.93 thf(f2382,plain,(
% 2.89/1.93 ((in @ (setadjoin @ sK370 @ emptyset) @ (setadjoin @ (setadjoin @ sK372 @ (setadjoin @ sK371 @ emptyset)) @ emptyset)) = $true) | ((setadjoin @ sK372 @ emptyset) = (setadjoin @ sK370 @ emptyset))),
% 2.89/1.93 inference(trivial_inequality_removal,[status(thm)],[f2381])).
% 2.89/1.93 thf(f2381,plain,(
% 2.89/1.93 ($true != $true) | ((setadjoin @ sK372 @ emptyset) = (setadjoin @ sK370 @ emptyset)) | ((in @ (setadjoin @ sK370 @ emptyset) @ (setadjoin @ (setadjoin @ sK372 @ (setadjoin @ sK371 @ emptyset)) @ emptyset)) = $true)),
% 2.89/1.93 inference(forward_demodulation,[status(thm)],[f2346,f1972])).
% 2.89/1.93 thf(f1972,plain,(
% 2.89/1.93 (setadjoinAx = $true)),
% 2.89/1.93 inference(cnf_transformation,[status(thm)],[f1311])).
% 2.89/1.93 thf(f2346,plain,(
% 2.89/1.93 ((in @ (setadjoin @ sK370 @ emptyset) @ (setadjoin @ (setadjoin @ sK372 @ (setadjoin @ sK371 @ emptyset)) @ emptyset)) = $true) | ((setadjoin @ sK372 @ emptyset) = (setadjoin @ sK370 @ emptyset)) | (setadjoinAx != $true)),
% 2.89/1.93 inference(trivial_inequality_removal,[status(thm)],[f2342])).
% 2.89/1.93 thf(f2342,plain,(
% 2.89/1.93 ((in @ (setadjoin @ sK370 @ emptyset) @ (setadjoin @ (setadjoin @ sK372 @ (setadjoin @ sK371 @ emptyset)) @ emptyset)) = $true) | (setadjoinAx != $true) | ((setadjoin @ sK372 @ emptyset) = (setadjoin @ sK370 @ emptyset)) | ($true != $true)),
% 2.89/1.93 inference(superposition,[status(thm)],[f2220,f2077])).
% 2.89/1.93 thf(f2077,plain,(
% 2.89/1.93 ((in @ (setadjoin @ sK370 @ emptyset) @ (setadjoin @ (setadjoin @ sK372 @ emptyset) @ (setadjoin @ (setadjoin @ sK372 @ (setadjoin @ sK371 @ emptyset)) @ emptyset))) = $true)),
% 2.89/1.93 inference(cnf_transformation,[status(thm)],[f1311])).
% 2.89/1.93 thf(f2220,plain,(
% 2.89/1.93 ( ! [X2 : $i,X0 : $i,X1 : $i] : (($true != (in @ X0 @ (setadjoin @ X2 @ X1))) | ($true = (in @ X0 @ X1)) | (X0 = X2) | (setadjoinAx != $true)) )),
% 2.89/1.93 inference(cnf_transformation,[status(thm)],[f1432])).
% 2.89/1.93 thf(f1432,plain,(
% 2.89/1.93 (! [X0 : $i,X1 : $i,X2 : $i] : ((($true = (in @ X0 @ X1)) | (X0 = X2) | ($true != (in @ X0 @ (setadjoin @ X2 @ X1)))) & (($true = (in @ X0 @ (setadjoin @ X2 @ X1))) | (($true != (in @ X0 @ X1)) & (X0 != X2)))) | (setadjoinAx != $true)) & ((setadjoinAx = $true) | ((($true != (in @ sK438 @ (setadjoin @ sK440 @ sK439))) | (($true != (in @ sK438 @ sK439)) & (sK438 != sK440))) & (($true = (in @ sK438 @ (setadjoin @ sK440 @ sK439))) | ($true = (in @ sK438 @ sK439)) | (sK438 = sK440))))),
% 2.89/1.93 inference(skolemisation,[status(esa),new_symbols(skolem,[sK438,sK439,sK440])],[f1430,f1431])).
% 2.89/1.93 thf(f1431,plain,(
% 2.89/1.93 ? [X3 : $i,X4 : $i,X5 : $i] : ((($true != (in @ X3 @ (setadjoin @ X5 @ X4))) | (($true != (in @ X3 @ X4)) & (X3 != X5))) & (($true = (in @ X3 @ (setadjoin @ X5 @ X4))) | ($true = (in @ X3 @ X4)) | (X3 = X5))) => ((($true != (in @ sK438 @ (setadjoin @ sK440 @ sK439))) | (($true != (in @ sK438 @ sK439)) & (sK438 != sK440))) & (($true = (in @ sK438 @ (setadjoin @ sK440 @ sK439))) | ($true = (in @ sK438 @ sK439)) | (sK438 = sK440)))),
% 2.89/1.93 introduced(definition,[],[choice_axiom])).
% 2.89/1.93 thf(f1430,plain,(
% 2.89/1.93 (! [X0 : $i,X1 : $i,X2 : $i] : ((($true = (in @ X0 @ X1)) | (X0 = X2) | ($true != (in @ X0 @ (setadjoin @ X2 @ X1)))) & (($true = (in @ X0 @ (setadjoin @ X2 @ X1))) | (($true != (in @ X0 @ X1)) & (X0 != X2)))) | (setadjoinAx != $true)) & ((setadjoinAx = $true) | ? [X3 : $i,X4 : $i,X5 : $i] : ((($true != (in @ X3 @ (setadjoin @ X5 @ X4))) | (($true != (in @ X3 @ X4)) & (X3 != X5))) & (($true = (in @ X3 @ (setadjoin @ X5 @ X4))) | ($true = (in @ X3 @ X4)) | (X3 = X5))))),
% 2.89/1.93 inference(rectify,[status(thm)],[f1429])).
% 2.89/1.93 thf(f1429,plain,(
% 2.89/1.93 (! [X0 : $i,X1 : $i,X2 : $i] : ((($true = (in @ X0 @ X1)) | (X0 = X2) | ($true != (in @ X0 @ (setadjoin @ X2 @ X1)))) & (($true = (in @ X0 @ (setadjoin @ X2 @ X1))) | (($true != (in @ X0 @ X1)) & (X0 != X2)))) | (setadjoinAx != $true)) & ((setadjoinAx = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : ((($true != (in @ X0 @ (setadjoin @ X2 @ X1))) | (($true != (in @ X0 @ X1)) & (X0 != X2))) & (($true = (in @ X0 @ (setadjoin @ X2 @ X1))) | ($true = (in @ X0 @ X1)) | (X0 = X2))))),
% 2.89/1.93 inference(flattening,[status(thm)],[f1428])).
% 2.89/1.93 thf(f1428,plain,(
% 2.89/1.93 (! [X0 : $i,X1 : $i,X2 : $i] : (((($true = (in @ X0 @ X1)) | (X0 = X2)) | ($true != (in @ X0 @ (setadjoin @ X2 @ X1)))) & (($true = (in @ X0 @ (setadjoin @ X2 @ X1))) | (($true != (in @ X0 @ X1)) & (X0 != X2)))) | (setadjoinAx != $true)) & ((setadjoinAx = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : ((($true != (in @ X0 @ (setadjoin @ X2 @ X1))) | (($true != (in @ X0 @ X1)) & (X0 != X2))) & (($true = (in @ X0 @ (setadjoin @ X2 @ X1))) | (($true = (in @ X0 @ X1)) | (X0 = X2)))))),
% 2.89/1.93 inference(nnf_transformation,[status(thm)],[f295])).
% 2.89/1.93 thf(f295,plain,(
% 2.89/1.93 ! [X0 : $i,X1 : $i,X2 : $i] : ((($true = (in @ X0 @ X1)) | (X0 = X2)) <=> ($true = (in @ X0 @ (setadjoin @ X2 @ X1)))) <=> (setadjoinAx = $true)),
% 2.89/1.93 inference(fool_elimination,[status(thm)],[f294])).
% 2.89/1.93 thf(f294,plain,(
% 2.89/1.93 (! [X0 : $i,X1 : $i,X2 : $i] : ((in @ X0 @ (setadjoin @ X2 @ X1)) <=> ((X0 = X2) | (in @ X0 @ X1))) = setadjoinAx)),
% 2.89/1.93 inference(rectify,[status(thm)],[f4])).
% 2.89/1.93 thf(f4,axiom,(
% 2.89/1.93 (! [X2 : $i,X3 : $i,X1 : $i] : ((in @ X2 @ (setadjoin @ X1 @ X3)) <=> ((X1 = X2) | (in @ X2 @ X3))) = setadjoinAx)),
% 2.89/1.93 file('/export/starexec/sandbox/tmp/tmp.vWjcIjdJ0D/DTF2THF_13509.p',setadjoinAx)).
% 2.89/1.93 % SZS output end Proof for DTF2THF_13509
% 2.89/1.93 % (13629)------------------------------
% 2.89/1.93 % (13629)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.89/1.93 % (13629)Termination reason: Refutation
% 2.89/1.93
% 2.89/1.93 % (13629)Memory used [KB]: 8315
% 2.89/1.93 % (13629)Time elapsed: 0.116 s
% 2.89/1.93 % (13629)Instructions burned: 222 (million)
% 2.89/1.93 % (13629)------------------------------
% 2.89/1.93 % (13629)------------------------------
% 2.89/1.93 % (13606)Success in time 0.209 s
%------------------------------------------------------------------------------