%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU641^1 : TPTP v9.2.1. Released v3.7.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n001.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 2.40s 1.64s
% Output : Refutation 2.40s
% Verified :
% SZS Type : Refutation
% Derivation depth : 12
% Number of leaves : 4
% Syntax : Number of formulae : 24 ( 9 unt; 0 typ; 0 def)
% Number of atoms : 2481 ( 676 equ; 0 cnn)
% Maximal formula atoms : 159 ( 103 avg)
% Number of connectives : 1226 ( 19 ~; 8 |; 479 &; 85 @)
% ( 1 <=>; 634 =>; 0 <=; 0 <~>)
% Maximal formula depth : 162 ( 49 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 298 ( 295 usr; 163 con; 0-4 aty)
% Number of variables : 26 ( 0 ^; 17 !; 9 ?; 26 :)
% 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_195,type,
sP0: $i > $o > $i > $i > $o ).
thf(func_def_198,type,
sP3: $i > $i > $o ).
thf(func_def_199,type,
sP4: $i > $o ).
thf(func_def_200,type,
sP5: $i > $i > $o ).
thf(func_def_201,type,
sP6: $i > $i > $o ).
thf(func_def_207,type,
sK12: $i > $o ).
thf(func_def_209,type,
sK14: ( $i > $o ) > $i > $i ).
thf(func_def_216,type,
sK21: $i > $i > $i ).
thf(func_def_217,type,
sK22: $i > $i > $i ).
thf(func_def_222,type,
sK27: $i > $o ).
thf(func_def_225,type,
sK30: $i > $i > $i ).
thf(func_def_234,type,
sK39: $i > $i > $i ).
thf(func_def_235,type,
sK40: $i > $o ).
thf(func_def_236,type,
sK41: ( $i > $o ) > $i ).
thf(func_def_239,type,
sK44: $i > $o ).
thf(func_def_256,type,
sK61: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_257,type,
sK62: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_258,type,
sK63: $i > $o ).
thf(func_def_259,type,
sK64: $i > $o ).
thf(func_def_274,type,
sK79: $i > $i ).
thf(func_def_279,type,
sK84: $i > $o ).
thf(func_def_285,type,
sK90: $i > $i ).
thf(func_def_289,type,
sK94: $i > $o ).
thf(func_def_291,type,
sK96: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_292,type,
sK97: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_295,type,
sK100: $i > $o ).
thf(func_def_296,type,
sK101: $i > $o ).
thf(func_def_297,type,
sK102: $i > $o ).
thf(func_def_298,type,
sK103: $i > $i ).
thf(func_def_299,type,
sK104: ( $i > $o ) > $i ).
thf(func_def_306,type,
sK111: $i > $o ).
thf(func_def_308,type,
sK113: $i > ( $i > $o ) > $i ).
thf(func_def_318,type,
sK123: $i > $o ).
thf(func_def_323,type,
sK128: $i > $o ).
thf(func_def_325,type,
sK130: $i > ( $i > $o ) > $i ).
thf(func_def_328,type,
sK133: $i > $o ).
thf(func_def_330,type,
sK135: $i > ( $i > $o ) > $i ).
thf(func_def_346,type,
sK151: $i > $i > $i ).
thf(func_def_376,type,
sK181: $i > $i > $i > $i ).
thf(func_def_377,type,
sK182: $i > $i > $i > $i ).
thf(func_def_381,type,
sK186: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_382,type,
sK187: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_383,type,
sK188: $i > $o ).
thf(func_def_384,type,
sK189: $i > $o ).
thf(func_def_417,type,
sK222: ( $i > $o ) > $i > $i ).
thf(func_def_419,type,
sK224: $i > $o ).
thf(func_def_424,type,
sK229: $i > $i > $i ).
thf(func_def_428,type,
sK233: $i > $o ).
thf(func_def_430,type,
sK235: ( $i > $o ) > $i > $i ).
thf(func_def_431,type,
sK236: $i > $o ).
thf(func_def_432,type,
sK237: $i > $o ).
thf(func_def_435,type,
sK240: $i > ( $i > $o ) > $i ).
thf(func_def_436,type,
sK241: $i > $i ).
thf(func_def_439,type,
sK244: $i > $i ).
thf(func_def_441,type,
sK246: $i > $o ).
thf(func_def_450,type,
sK255: $o > $i > $i > $i ).
thf(func_def_460,type,
sK265: $i > $o ).
thf(func_def_463,type,
sK268: $i > ( $i > $o ) > $i ).
thf(func_def_464,type,
sK269: $i > ( $i > $o ) > $i ).
thf(func_def_475,type,
sK280: $i > $i > $i ).
thf(func_def_481,type,
sK286: $i > $i > $i ).
thf(func_def_482,type,
sK287: $i > $o ).
thf(func_def_484,type,
sK289: $i > $o ).
thf(func_def_488,type,
sK293: $i > $i > ( $i > $o ) > $i ).
thf(func_def_489,type,
sK294: $i > $o ).
thf(func_def_492,type,
sK297: $i > $i ).
thf(func_def_522,type,
sK327: $i > $o ).
thf(func_def_525,type,
sK330: $i > $i > $o ).
thf(func_def_526,type,
sK331: $i > $i ).
thf(func_def_527,type,
sK332: $i > $i ).
thf(func_def_528,type,
sK333: ( $i > $i > $o ) > $i > $i ).
thf(func_def_529,type,
sK334: ( $i > $i > $o ) > $i > $i ).
thf(func_def_530,type,
sK335: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_534,type,
sK339: ( $i > $o ) > $i ).
thf(func_def_535,type,
sK340: ( $i > $o ) > $i ).
thf(func_def_536,type,
sK341: $i > $o ).
thf(func_def_545,type,
sK350: $i > $o ).
thf(func_def_546,type,
sK351: $i > $i > $i ).
thf(func_def_557,type,
sK362: $i > $i ).
thf(func_def_570,type,
sK375: $i > $i > $i ).
thf(func_def_571,type,
sK376: $i > $i > $i ).
thf(func_def_572,type,
sK377: $i > $i > $i ).
thf(func_def_573,type,
sK378: $i > $i > $i ).
thf(func_def_574,type,
sK379: $i > $i > $i ).
thf(func_def_575,type,
sK380: $i > $i > $i > $i ).
thf(func_def_576,type,
sK381: $i > $i ).
thf(func_def_577,type,
sK382: $i > $i ).
thf(func_def_578,type,
sK383: $i > $i ).
thf(func_def_579,type,
sK384: $i > $i ).
thf(func_def_580,type,
sK385: $i > $i > $i > $i ).
thf(func_def_581,type,
sK386: $i > $i > $i > $i ).
thf(func_def_582,type,
sK387: $i > $i > $i ).
thf(func_def_583,type,
sK388: $i > $i > $i ).
thf(func_def_584,type,
sK389: $i > $i > $i ).
thf(func_def_586,type,
sK391: $i > $i ).
thf(func_def_587,type,
sK392: $i > $i ).
thf(func_def_588,type,
sK393: $i > $i ).
thf(func_def_592,type,
sK397: $i > $i > $i ).
thf(func_def_594,type,
sK399: $i > $i ).
thf(func_def_603,type,
sK408: ( $i > $o ) > $i > $i ).
thf(func_def_605,type,
sK410: $i > $o ).
thf(func_def_607,type,
sK412: $i > $o ).
thf(func_def_610,type,
sK415: $i > ( $i > $o ) > $i ).
thf(func_def_611,type,
sK416: $i > ( $i > $o ) > $i ).
thf(func_def_612,type,
sK417: $i > $o ).
thf(func_def_620,type,
sK425: $i > $o ).
thf(func_def_622,type,
sK427: ( $i > $o ) > $i > $i ).
thf(func_def_634,type,
sK439: ( $i > $o ) > $i ).
thf(func_def_635,type,
sK440: $i > $o ).
thf(func_def_636,type,
sK441: $i > $i ).
thf(func_def_638,type,
ph443:
!>[X0: $tType] : X0 ).
thf(f3196,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f3195,f1616]) ).
thf(f1616,plain,
sK35 != sK36,
inference(cnf_transformation,[status(thm)],[f749]) ).
thf(f749,plain,
( ( symdiffIneg1 = $true )
& ( emptysetsubset = $true )
& ( binintersectI = $true )
& ( setextAx = $true )
& ( setunionE = $true )
& ( dsetconstr__Cong = $true )
& ( in__Cong = $true )
& ( setminusER = $true )
& ( binintersectSubset4 = $true )
& ( upairset2E = $true )
& ( powersetsubset = $true )
& ( emptyInPowerset = $true )
& ( setminusLsub = $true )
& ( setextsub = $true )
& ( sK35 != sK36 )
& ( ( setadjoin @ sK35 @ emptyset )
= ( setadjoin @ sK36 @ emptyset ) )
& ( exuEu = $true )
& ( binintersectLsub = $true )
& ( subsetTrans = $true )
& ( prop2set2propI = $true )
& ( cartprodmempair = $true )
& ( subset2powerset = $true )
& ( binintersectEL = $true )
& ( emptysetAx = $true )
& ( vacuousDall = $true )
& ( setunionsingleton2 = $true )
& ( emptyinPowerset = $true )
& ( cartprodpairin = $true )
& ( quantDeMorgan1 = $true )
& ( exuE2 = $true )
& ( ex1I = $true )
& ( singletoninpowerset = $true )
& ( eqimpsubset2 = $true )
& ( binintersectER = $true )
& ( secondinupair = $true )
& ( cartprodmempair1 = $true )
& ( upairsetE = $true )
& ( notdallE = $true )
& ( notinsingleton = $true )
& ( setminusILneg = $true )
& ( singletonprop = $true )
& ( emptyinunitempty = $true )
& ( symdiffIneg2 = $true )
& ( setadjoinOr = $true )
& ( symdiffI2 = $true )
& ( exuE1 = $true )
& ( setbeta = $true )
& ( setminusIRneg = $true )
& ( setminusEL = $true )
& ( notequalI2 = $true )
& ( emptyI = $true )
& ( ubforcartprodlem1 = $true )
& ( foundationAx = $true )
& ( sepInPowerset = $true )
& ( bs114d = $true )
& ( ubforcartprodlem2 = $true )
& ( powersetE = $true )
& ( setukpairIR = $true )
& ( symdiffE = $true )
& ( sepSubset = $true )
& ( upairinpowunion = $true )
& ( omegaIndAx = $true )
& ( descrp = $true )
& ( subsetI1 = $true )
& ( exuE3u = $true )
& ( ex1I2 = $true )
& ( binunionE = $true )
& ( emptyE1 = $true )
& ( quantDeMorgan3 = $true )
& ( replAx = $true )
& ( subsetE = $true )
& ( exuI2 = $true )
& ( symdiffI1 = $true )
& ( emptysetimpfalse = $true )
& ( setminusERneg = $true )
& ( inPowerset = $true )
& ( powersetI = $true )
& ( subsetI2 = $true )
& ( binunionIL = $true )
& ( emptysetE = $true )
& ( setunionsingleton1 = $true )
& ( prop2setI = $true )
& ( subPowSU = $true )
& ( powersetI1 = $true )
& ( eqimpsubset1 = $true )
& ( setminusI = $true )
& ( setadjoinAx = $true )
& ( powersetAx = $true )
& ( setunionI = $true )
& ( setminusSubset1 = $true )
& ( singletoninpowunion = $true )
& ( omegaSAx = $true )
& ( noeltsimpempty = $true )
& ( nonemptyImpWitness = $true )
& ( subsetemptysetimpeq = $true )
& ( ubforcartprodlem3 = $true )
& ( setadjoinSub2 = $true )
& ( binunionRsub = $true )
& ( dsetconstrI = $true )
& ( exu__Cong = $true )
& ( notsubsetI = $true )
& ( exuI3 = $true )
& ( setadjoinE = $true )
& ( setadjoin__Cong = $true )
& ( setunionsingleton = $true )
& ( wellorderingAx = $true )
& ( setadjoinIL = $true )
& ( setadjoinSub = $true )
& ( binintersectRsub = $true )
& ( setminusELneg = $true )
& ( setminusSubset2 = $true )
& ( singletonsubset = $true )
& ( quantDeMorgan2 = $true )
& ( setukpairIL = $true )
& ( powerset__Cong = $true )
& ( dsetconstrER = $true )
& ( setunion__Cong = $true )
& ( setunionAx = $true )
& ( setadjoinIR = $true )
& ( emptyset__Cong = $true )
& ( nonemptyI = $true )
& ( quantDeMorgan4 = $true )
& ( nonemptyI1 = $true )
& ( omega0Ax = $true )
& ( notinemptyset = $true )
& ( powersetE1 = $true )
& ( kpairiskpair = $true )
& ( binunionLsub = $true )
& ( setext = $true )
& ( ex1E1 = $true )
& ( omega__Cong = $true )
& ( notequalI1 = $true )
& ( kpairp = $true )
& ( subsetE2 = $true )
& ( binintersectSubset2 = $true )
& ( setoftrueEq = $true )
& ( binintersectSubset5 = $true )
& ( upairsetIR = $true )
& ( binintersectSubset1 = $true )
& ( singletonsswitch = $true )
& ( subsetRefl = $true )
& ( exuI1 = $true )
& ( eqinunit = $true )
& ( notdexE = $true )
& ( upairsubunion = $true )
& ( binunionEcases = $true )
& ( upairset2IR = $true )
& ( upairsetIL = $true )
& ( inCongP = $true )
& ( exuE3e = $true )
& ( binunionIR = $true )
& ( uniqinunit = $true )
& ( prop2setE = $true )
& ( binintersectSubset3 = $true )
& ( descr__Cong = $true )
& ( nonemptyE1 = $true )
& ( disjointsetsI1 = $true )
& ( setunionE2 = $true )
& ( dsetconstrEL = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK35,sK36])],[f656,f748]) ).
thf(f748,plain,
( ? [X0: $i,X1: $i] :
( ( X0 != X1 )
& ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X0 @ emptyset ) ) )
=> ( ( sK35 != sK36 )
& ( ( setadjoin @ sK35 @ emptyset )
= ( setadjoin @ sK36 @ emptyset ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f656,plain,
( ( symdiffIneg1 = $true )
& ( emptysetsubset = $true )
& ( binintersectI = $true )
& ( setextAx = $true )
& ( setunionE = $true )
& ( dsetconstr__Cong = $true )
& ( in__Cong = $true )
& ( setminusER = $true )
& ( binintersectSubset4 = $true )
& ( upairset2E = $true )
& ( powersetsubset = $true )
& ( emptyInPowerset = $true )
& ( setminusLsub = $true )
& ( setextsub = $true )
& ? [X0: $i,X1: $i] :
( ( X0 != X1 )
& ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X0 @ emptyset ) ) )
& ( exuEu = $true )
& ( binintersectLsub = $true )
& ( subsetTrans = $true )
& ( prop2set2propI = $true )
& ( cartprodmempair = $true )
& ( subset2powerset = $true )
& ( binintersectEL = $true )
& ( emptysetAx = $true )
& ( vacuousDall = $true )
& ( setunionsingleton2 = $true )
& ( emptyinPowerset = $true )
& ( cartprodpairin = $true )
& ( quantDeMorgan1 = $true )
& ( exuE2 = $true )
& ( ex1I = $true )
& ( singletoninpowerset = $true )
& ( eqimpsubset2 = $true )
& ( binintersectER = $true )
& ( secondinupair = $true )
& ( cartprodmempair1 = $true )
& ( upairsetE = $true )
& ( notdallE = $true )
& ( notinsingleton = $true )
& ( setminusILneg = $true )
& ( singletonprop = $true )
& ( emptyinunitempty = $true )
& ( symdiffIneg2 = $true )
& ( setadjoinOr = $true )
& ( symdiffI2 = $true )
& ( exuE1 = $true )
& ( setbeta = $true )
& ( setminusIRneg = $true )
& ( setminusEL = $true )
& ( notequalI2 = $true )
& ( emptyI = $true )
& ( ubforcartprodlem1 = $true )
& ( foundationAx = $true )
& ( sepInPowerset = $true )
& ( bs114d = $true )
& ( ubforcartprodlem2 = $true )
& ( powersetE = $true )
& ( setukpairIR = $true )
& ( symdiffE = $true )
& ( sepSubset = $true )
& ( upairinpowunion = $true )
& ( omegaIndAx = $true )
& ( descrp = $true )
& ( subsetI1 = $true )
& ( exuE3u = $true )
& ( ex1I2 = $true )
& ( binunionE = $true )
& ( emptyE1 = $true )
& ( quantDeMorgan3 = $true )
& ( replAx = $true )
& ( subsetE = $true )
& ( exuI2 = $true )
& ( symdiffI1 = $true )
& ( emptysetimpfalse = $true )
& ( setminusERneg = $true )
& ( inPowerset = $true )
& ( powersetI = $true )
& ( subsetI2 = $true )
& ( binunionIL = $true )
& ( emptysetE = $true )
& ( setunionsingleton1 = $true )
& ( prop2setI = $true )
& ( subPowSU = $true )
& ( powersetI1 = $true )
& ( eqimpsubset1 = $true )
& ( setminusI = $true )
& ( setadjoinAx = $true )
& ( powersetAx = $true )
& ( setunionI = $true )
& ( setminusSubset1 = $true )
& ( singletoninpowunion = $true )
& ( omegaSAx = $true )
& ( noeltsimpempty = $true )
& ( nonemptyImpWitness = $true )
& ( subsetemptysetimpeq = $true )
& ( ubforcartprodlem3 = $true )
& ( setadjoinSub2 = $true )
& ( binunionRsub = $true )
& ( dsetconstrI = $true )
& ( exu__Cong = $true )
& ( notsubsetI = $true )
& ( exuI3 = $true )
& ( setadjoinE = $true )
& ( setadjoin__Cong = $true )
& ( setunionsingleton = $true )
& ( wellorderingAx = $true )
& ( setadjoinIL = $true )
& ( setadjoinSub = $true )
& ( binintersectRsub = $true )
& ( setminusELneg = $true )
& ( setminusSubset2 = $true )
& ( singletonsubset = $true )
& ( quantDeMorgan2 = $true )
& ( setukpairIL = $true )
& ( powerset__Cong = $true )
& ( dsetconstrER = $true )
& ( setunion__Cong = $true )
& ( setunionAx = $true )
& ( setadjoinIR = $true )
& ( emptyset__Cong = $true )
& ( nonemptyI = $true )
& ( quantDeMorgan4 = $true )
& ( nonemptyI1 = $true )
& ( omega0Ax = $true )
& ( notinemptyset = $true )
& ( powersetE1 = $true )
& ( kpairiskpair = $true )
& ( binunionLsub = $true )
& ( setext = $true )
& ( ex1E1 = $true )
& ( omega__Cong = $true )
& ( notequalI1 = $true )
& ( kpairp = $true )
& ( subsetE2 = $true )
& ( binintersectSubset2 = $true )
& ( setoftrueEq = $true )
& ( binintersectSubset5 = $true )
& ( upairsetIR = $true )
& ( binintersectSubset1 = $true )
& ( singletonsswitch = $true )
& ( subsetRefl = $true )
& ( exuI1 = $true )
& ( eqinunit = $true )
& ( notdexE = $true )
& ( upairsubunion = $true )
& ( binunionEcases = $true )
& ( upairset2IR = $true )
& ( upairsetIL = $true )
& ( inCongP = $true )
& ( exuE3e = $true )
& ( binunionIR = $true )
& ( uniqinunit = $true )
& ( prop2setE = $true )
& ( binintersectSubset3 = $true )
& ( descr__Cong = $true )
& ( nonemptyE1 = $true )
& ( disjointsetsI1 = $true )
& ( setunionE2 = $true )
& ( dsetconstrEL = $true ) ),
inference(flattening,[status(thm)],[f655]) ).
thf(f655,plain,
( ? [X0: $i,X1: $i] :
( ( X0 != X1 )
& ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X0 @ emptyset ) ) )
& ( 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)],[f458]) ).
thf(f458,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 )
=> ! [X0: $i,X1: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X0 @ emptyset ) )
=> ( X0 = X1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f457]) ).
thf(f457,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
=> ! [X0: $i,X1: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X0 @ emptyset ) )
=> ( X0 = X1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f165]) ).
thf(f165,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
=> ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X2 @ emptyset )
= ( setadjoin @ X1 @ emptyset ) )
=> ( X1 = X2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f164]) ).
thf(f164,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
=> ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X2 @ emptyset )
= ( setadjoin @ X1 @ emptyset ) )
=> ( X1 = X2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.DxatUwJPum/DTF2THF_26966.p',singletonsuniq) ).
thf(f3195,plain,
sK35 = sK36,
inference(forward_demodulation,[status(thm)],[f3194,f2815]) ).
thf(f2815,plain,
! [X0: $i] :
( ( setunion @ ( setadjoin @ X0 @ emptyset ) )
= X0 ),
inference(trivial_inequality_removal,[status(thm)],[f2393]) ).
thf(f2393,plain,
! [X0: $i] :
( ( ( setunion @ ( setadjoin @ X0 @ emptyset ) )
= X0 )
| ( $true != $true ) ),
inference(definition_unfolding,[status(thm)],[f1779,f1526]) ).
thf(f1526,plain,
setunionsingleton = $true,
inference(cnf_transformation,[status(thm)],[f749]) ).
thf(f1779,plain,
! [X0: $i] :
( ( ( setunion @ ( setadjoin @ X0 @ emptyset ) )
= X0 )
| ( setunionsingleton != $true ) ),
inference(cnf_transformation,[status(thm)],[f933]) ).
thf(f933,plain,
( ( ! [X0: $i] :
( ( setunion @ ( setadjoin @ X0 @ emptyset ) )
= X0 )
| ( setunionsingleton != $true ) )
& ( ( setunionsingleton = $true )
| ( sK140
!= ( setunion @ ( setadjoin @ sK140 @ emptyset ) ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK140])],[f931,f932]) ).
thf(f932,plain,
( ? [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
!= X1 )
=> ( sK140
!= ( setunion @ ( setadjoin @ sK140 @ emptyset ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f931,plain,
( ( ! [X0: $i] :
( ( setunion @ ( setadjoin @ X0 @ emptyset ) )
= X0 )
| ( setunionsingleton != $true ) )
& ( ( setunionsingleton = $true )
| ? [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
!= X1 ) ) ),
inference(rectify,[status(thm)],[f930]) ).
thf(f930,plain,
( ( ! [X0: $i] :
( ( setunion @ ( setadjoin @ X0 @ emptyset ) )
= X0 )
| ( setunionsingleton != $true ) )
& ( ( setunionsingleton = $true )
| ? [X0: $i] :
( ( setunion @ ( setadjoin @ X0 @ emptyset ) )
!= X0 ) ) ),
inference(nnf_transformation,[status(thm)],[f333]) ).
thf(f333,plain,
( ! [X0: $i] :
( ( setunion @ ( setadjoin @ X0 @ emptyset ) )
= X0 )
<=> ( setunionsingleton = $true ) ),
inference(fool_elimination,[status(thm)],[f332]) ).
thf(f332,plain,
( setunionsingleton
= ( ! [X0: $i] :
( ( setunion @ ( setadjoin @ X0 @ emptyset ) )
= X0 ) ) ),
inference(rectify,[status(thm)],[f157]) ).
thf(f157,axiom,
( setunionsingleton
= ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.DxatUwJPum/DTF2THF_26966.p',setunionsingleton) ).
thf(f3194,plain,
( sK36
= ( setunion @ ( setadjoin @ sK35 @ emptyset ) ) ),
inference(superposition,[status(thm)],[f2815,f1615]) ).
thf(f1615,plain,
( ( setadjoin @ sK35 @ emptyset )
= ( setadjoin @ sK36 @ emptyset ) ),
inference(cnf_transformation,[status(thm)],[f749]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : SEU641^1 : TPTP v9.2.1. Released v3.7.0.
% 0.00/0.12 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.13/0.32 % Computer : n001.cluster.edu
% 0.13/0.32 % Model : x86_64 x86_64
% 0.13/0.32 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.32 % Memory : 8042.1875MB
% 0.13/0.32 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.32 % CPULimit : 300
% 0.13/0.32 % WCLimit : 300
% 0.13/0.32 % DateTime : Tue Feb 24 12:30:20 EST 2026
% 0.13/0.32 % CPUTime :
% 0.13/0.32 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.18/0.43 ---- Original DTF file ---
% 0.18/0.43 thf(spec,logic,$$dhol).
% 0.18/0.43 %------------------------------------------------------------------------------
% 0.18/0.43 % File : SEU641^1 : TPTP v9.2.1. Released v3.7.0.
% 0.18/0.43 % Domain : Set Theory
% 0.18/0.43 % Problem : Ordered Pairs - Singletons
% 0.18/0.43 % Version : Especial.
% 0.18/0.43 % English : (! x:i.! y:i.setadjoin x emptyset = setadjoin y emptyset -> x = y)
% 0.18/0.43
% 0.18/0.43 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.18/0.43 % Source : [Bro08]
% 0.18/0.43 % Names : ZFC143g [Bro08]
% 0.18/0.43
% 0.18/0.43 % Status : Theorem
% 0.18/0.43 % Rating : 0.22 v9.1.0, 0.12 v9.0.0, 0.20 v8.2.0, 0.31 v8.1.0, 0.09 v7.5.0, 0.00 v7.4.0, 0.11 v7.2.0, 0.00 v7.1.0, 0.25 v7.0.0, 0.14 v6.4.0, 0.17 v6.3.0, 0.20 v6.2.0, 0.14 v6.1.0, 0.29 v5.5.0, 0.33 v5.4.0, 0.40 v5.2.0, 0.60 v5.1.0, 0.80 v5.0.0, 0.60 v4.1.0, 0.67 v3.7.0
% 0.18/0.43 % Syntax : Number of formulae : 346 ( 163 unt; 182 typ; 163 def)
% 0.18/0.43 % Number of atoms : 866 ( 231 equ; 0 cnn)
% 0.18/0.43 % Maximal formula atoms : 159 ( 5 avg)
% 0.18/0.43 % Number of connectives : 1498 ( 47 ~; 7 |; 37 &;1008 @)
% 0.18/0.43 % ( 14 <=>; 385 =>; 0 <=; 0 <~>)
% 0.18/0.43 % Maximal formula depth : 161 ( 2 avg)
% 0.18/0.43 % Number of types : 2 ( 0 usr)
% 0.18/0.43 % Number of type conns : 78 ( 78 >; 0 *; 0 +; 0 <<)
% 0.18/0.43 % Number of symbols : 185 ( 182 usr; 161 con; 0-2 aty)
% 0.18/0.43 % Number of variables : 477 ( 41 ^; 398 !; 38 ?; 477 :)
% 0.18/0.43 % SPC : TH0_THM_EQU_NAR
% 0.18/0.43
% 0.18/0.43 % Comments : http://mathgate.info/detsetitem.php?id=197
% 0.18/0.43 %------------------------------------------------------------------------------
% 0.18/0.43 thf(in_type,type,
% 0.18/0.43 in: $i > $i > $o ).
% 0.18/0.43
% 0.18/0.43 thf(exu_type,type,
% 0.18/0.43 exu: ( $i > $o ) > $o ).
% 0.18/0.43
% 0.18/0.43 thf(exu,definition,
% 0.18/0.43 ( exu
% 0.18/0.43 = ( ^ [Xphi: $i > $o] :
% 0.18/0.43 ? [Xx: $i] :
% 0.18/0.43 ( ( Xphi @ Xx )
% 0.18/0.43 & ! [Xy: $i] :
% 0.18/0.43 ( ( Xphi @ Xy )
% 0.18/0.43 => ( Xx = Xy ) ) ) ) ) ).
% 0.18/0.43
% 0.18/0.43 thf(setextAx_type,type,
% 0.18/0.43 setextAx: $o ).
% 0.18/0.43
% 0.18/0.43 thf(setextAx,definition,
% 0.18/0.43 ( setextAx
% 0.18/0.43 = ( ! [A: $i,B: $i] :
% 0.18/0.43 ( ! [Xx: $i] :
% 0.18/0.43 ( ( in @ Xx @ A )
% 0.18/0.43 <=> ( in @ Xx @ B ) )
% 0.18/0.43 => ( A = B ) ) ) ) ).
% 0.18/0.43
% 0.18/0.43 thf(emptyset_type,type,
% 0.18/0.43 emptyset: $i ).
% 0.18/0.43
% 0.18/0.43 thf(emptysetAx_type,type,
% 0.18/0.43 emptysetAx: $o ).
% 0.18/0.43
% 0.18/0.43 thf(emptysetAx,definition,
% 0.18/0.43 ( emptysetAx
% 0.18/0.43 = ( ! [Xx: $i] :
% 0.18/0.43 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.18/0.43
% 0.18/0.43 thf(setadjoin_type,type,
% 0.18/0.43 setadjoin: $i > $i > $i ).
% 0.18/0.43
% 0.18/0.43 thf(setadjoinAx_type,type,
% 0.18/0.43 setadjoinAx: $o ).
% 0.18/0.43
% 0.18/0.43 thf(setadjoinAx,definition,
% 0.18/0.43 ( setadjoinAx
% 0.18/0.43 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.18/0.43 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.18/0.43 <=> ( ( Xy = Xx )
% 0.18/0.43 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.18/0.43
% 0.18/0.43 thf(powerset_type,type,
% 0.18/0.43 powerset: $i > $i ).
% 0.18/0.43
% 0.18/0.43 thf(powersetAx_type,type,
% 0.18/0.43 powersetAx: $o ).
% 0.18/0.43
% 0.18/0.43 thf(powersetAx,definition,
% 0.18/0.43 ( powersetAx
% 0.18/0.43 = ( ! [A: $i,B: $i] :
% 0.18/0.43 ( ( in @ B @ ( powerset @ A ) )
% 0.18/0.43 <=> ! [Xx: $i] :
% 0.18/0.43 ( ( in @ Xx @ B )
% 0.18/0.43 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.18/0.43
% 0.18/0.43 thf(setunion_type,type,
% 0.18/0.43 setunion: $i > $i ).
% 0.18/0.43
% 0.18/0.43 thf(setunionAx_type,type,
% 0.18/0.43 setunionAx: $o ).
% 0.18/0.43
% 0.18/0.43 thf(setunionAx,definition,
% 0.18/0.43 ( setunionAx
% 0.18/0.43 = ( ! [A: $i,Xx: $i] :
% 0.18/0.43 ( ( in @ Xx @ ( setunion @ A ) )
% 0.18/0.43 <=> ? [B: $i] :
% 0.18/0.43 ( ( in @ Xx @ B )
% 0.18/0.43 & ( in @ B @ A ) ) ) ) ) ).
% 0.18/0.43
% 0.18/0.43 thf(omega_type,type,
% 0.18/0.43 omega: $i ).
% 0.18/0.43
% 0.18/0.43 thf(omega0Ax_type,type,
% 0.18/0.43 omega0Ax: $o ).
% 0.18/0.43
% 0.18/0.43 thf(omega0Ax,definition,
% 0.18/0.43 ( omega0Ax
% 0.18/0.43 = ( in @ emptyset @ omega ) ) ).
% 0.18/0.43
% 0.18/0.43 thf(omegaSAx_type,type,
% 0.18/0.43 omegaSAx: $o ).
% 0.18/0.43
% 0.18/0.43 thf(omegaSAx,definition,
% 0.18/0.43 ( omegaSAx
% 0.18/0.43 = ( ! [Xx: $i] :
% 0.18/0.43 ( ( in @ Xx @ omega )
% 0.18/0.43 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.18/0.43
% 0.18/0.43 thf(omegaIndAx_type,type,
% 0.18/0.43 omegaIndAx: $o ).
% 0.18/0.43
% 0.18/0.43 thf(omegaIndAx,definition,
% 0.18/0.43 ( omegaIndAx
% 0.18/0.43 = ( ! [A: $i] :
% 0.18/0.43 ( ( ( in @ emptyset @ A )
% 0.18/0.43 & ! [Xx: $i] :
% 0.18/0.43 ( ( ( in @ Xx @ omega )
% 0.18/0.43 & ( in @ Xx @ A ) )
% 0.18/0.43 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.18/0.43 => ! [Xx: $i] :
% 0.18/0.43 ( ( in @ Xx @ omega )
% 0.18/0.43 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.18/0.43
% 0.18/0.43 thf(replAx_type,type,
% 0.18/0.43 replAx: $o ).
% 0.18/0.43
% 0.18/0.43 thf(replAx,definition,
% 0.18/0.43 ( replAx
% 0.18/0.43 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.18/0.43 ( ! [Xx: $i] :
% 0.18/0.43 ( ( in @ Xx @ A )
% 0.18/0.43 => ( exu
% 0.18/0.43 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.18/0.43 => ? [B: $i] :
% 0.18/0.43 ! [Xx: $i] :
% 0.18/0.43 ( ( in @ Xx @ B )
% 0.18/0.43 <=> ? [Xy: $i] :
% 0.18/0.43 ( ( in @ Xy @ A )
% 0.18/0.43 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.18/0.43
% 0.18/0.43 thf(foundationAx_type,type,
% 0.18/0.43 foundationAx: $o ).
% 0.18/0.43
% 0.18/0.43 thf(foundationAx,definition,
% 0.18/0.43 ( foundationAx
% 0.18/0.43 = ( ! [A: $i] :
% 0.18/0.43 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.18/0.43 => ? [B: $i] :
% 0.18/0.43 ( ( in @ B @ A )
% 0.18/0.43 & ~ ? [Xx: $i] :
% 0.18/0.43 ( ( in @ Xx @ B )
% 0.18/0.43 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.18/0.43
% 0.18/0.43 thf(wellorderingAx_type,type,
% 0.18/0.43 wellorderingAx: $o ).
% 0.18/0.43
% 0.18/0.43 thf(wellorderingAx,definition,
% 0.18/0.43 ( wellorderingAx
% 0.18/0.43 = ( ! [A: $i] :
% 0.18/0.43 ? [B: $i] :
% 0.18/0.43 ( ! [C: $i] :
% 0.18/0.43 ( ( in @ C @ B )
% 0.18/0.43 => ! [Xx: $i] :
% 0.18/0.43 ( ( in @ Xx @ C )
% 0.18/0.43 => ( in @ Xx @ A ) ) )
% 0.18/0.43 & ! [Xx: $i,Xy: $i] :
% 0.18/0.43 ( ( ( in @ Xx @ A )
% 0.18/0.43 & ( in @ Xy @ A ) )
% 0.18/0.43 => ( ! [C: $i] :
% 0.18/0.43 ( ( in @ C @ B )
% 0.18/0.43 => ( ( in @ Xx @ C )
% 0.18/0.43 <=> ( in @ Xy @ C ) ) )
% 0.18/0.43 => ( Xx = Xy ) ) )
% 0.18/0.43 & ! [C: $i,D: $i] :
% 0.18/0.43 ( ( ( in @ C @ B )
% 0.18/0.43 & ( in @ D @ B ) )
% 0.18/0.43 => ( ! [Xx: $i] :
% 0.18/0.43 ( ( in @ Xx @ C )
% 0.18/0.43 => ( in @ Xx @ D ) )
% 0.18/0.43 | ! [Xx: $i] :
% 0.18/0.43 ( ( in @ Xx @ D )
% 0.18/0.43 => ( in @ Xx @ C ) ) ) )
% 0.18/0.43 & ! [C: $i] :
% 0.18/0.43 ( ( ! [Xx: $i] :
% 0.18/0.43 ( ( in @ Xx @ C )
% 0.18/0.43 => ( in @ Xx @ A ) )
% 0.18/0.43 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.18/0.43 => ? [D: $i,Xx: $i] :
% 0.18/0.43 ( ( in @ D @ B )
% 0.18/0.43 & ( in @ Xx @ C )
% 0.18/0.43 & ~ ? [Xy: $i] :
% 0.18/0.43 ( ( in @ Xy @ D )
% 0.18/0.43 & ( in @ Xy @ C ) )
% 0.18/0.43 & ! [E: $i] :
% 0.18/0.43 ( ( in @ E @ B )
% 0.18/0.43 => ( ! [Xy: $i] :
% 0.18/0.43 ( ( in @ Xy @ E )
% 0.18/0.43 => ( in @ Xy @ D ) )
% 0.18/0.43 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.18/0.43
% 0.18/0.43 thf(descr_type,type,
% 0.18/0.43 descr: ( $i > $o ) > $i ).
% 0.18/0.43
% 0.18/0.43 thf(descrp_type,type,
% 0.18/0.43 descrp: $o ).
% 0.18/0.43
% 0.18/0.43 thf(descrp,definition,
% 0.18/0.43 ( descrp
% 0.18/0.43 = ( ! [Xphi: $i > $o] :
% 0.18/0.43 ( ( exu
% 0.18/0.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.18/0.43 => ( Xphi
% 0.18/0.43 @ ( descr
% 0.18/0.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.18/0.43
% 0.18/0.43 thf(dsetconstr_type,type,
% 0.18/0.43 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.18/0.43
% 0.18/0.43 thf(dsetconstrI_type,type,
% 0.18/0.43 dsetconstrI: $o ).
% 0.18/0.43
% 0.18/0.43 thf(dsetconstrI,definition,
% 0.18/0.43 ( dsetconstrI
% 0.18/0.43 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.18/0.43 ( ( in @ Xx @ A )
% 0.18/0.43 => ( ( Xphi @ Xx )
% 0.18/0.43 => ( in @ Xx
% 0.18/0.43 @ ( dsetconstr @ A
% 0.18/0.43 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.18/0.43
% 0.18/0.43 thf(dsetconstrEL_type,type,
% 0.18/0.43 dsetconstrEL: $o ).
% 0.18/0.43
% 0.18/0.43 thf(dsetconstrEL,definition,
% 0.18/0.43 ( dsetconstrEL
% 0.18/0.43 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.18/0.43 ( ( in @ Xx
% 0.18/0.43 @ ( dsetconstr @ A
% 0.18/0.43 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.18/0.43 => ( in @ Xx @ A ) ) ) ) ).
% 0.18/0.43
% 0.18/0.43 thf(dsetconstrER_type,type,
% 0.18/0.43 dsetconstrER: $o ).
% 0.18/0.43
% 0.18/0.43 thf(dsetconstrER,definition,
% 0.18/0.43 ( dsetconstrER
% 0.18/0.43 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.18/0.43 ( ( in @ Xx
% 0.18/0.43 @ ( dsetconstr @ A
% 0.18/0.43 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.18/0.43 => ( Xphi @ Xx ) ) ) ) ).
% 0.18/0.43
% 0.18/0.43 thf(exuE1_type,type,
% 0.18/0.43 exuE1: $o ).
% 0.18/0.43
% 0.18/0.43 thf(exuE1,definition,
% 0.18/0.43 ( exuE1
% 0.18/0.43 = ( ! [Xphi: $i > $o] :
% 0.18/0.43 ( ( exu
% 0.18/0.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.18/0.43 => ? [Xx: $i] :
% 0.18/0.43 ( ( Xphi @ Xx )
% 0.18/0.43 & ! [Xy: $i] :
% 0.18/0.43 ( ( Xphi @ Xy )
% 0.18/0.43 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.18/0.43
% 0.18/0.43 thf(prop2set_type,type,
% 0.18/0.43 prop2set: $o > $i ).
% 0.18/0.43
% 0.18/0.43 thf(prop2setE_type,type,
% 0.18/0.43 prop2setE: $o ).
% 0.18/0.43
% 0.18/0.43 thf(prop2setE,definition,
% 0.18/0.43 ( prop2setE
% 0.18/0.43 = ( ! [Xphi: $o,Xx: $i] :
% 0.18/0.43 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.18/0.43 => Xphi ) ) ) ).
% 0.18/0.43
% 0.18/0.43 thf(emptysetE_type,type,
% 0.18/0.43 emptysetE: $o ).
% 0.18/0.43
% 0.18/0.43 thf(emptysetE,definition,
% 0.18/0.43 ( emptysetE
% 0.18/0.43 = ( ! [Xx: $i] :
% 0.18/0.43 ( ( in @ Xx @ emptyset )
% 0.18/0.43 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.18/0.43
% 0.18/0.43 thf(emptysetimpfalse_type,type,
% 0.18/0.43 emptysetimpfalse: $o ).
% 0.18/0.43
% 0.18/0.43 thf(emptysetimpfalse,definition,
% 0.18/0.43 ( emptysetimpfalse
% 0.18/0.43 = ( ! [Xx: $i] :
% 0.18/0.43 ( ( in @ Xx @ emptyset )
% 0.18/0.43 => $false ) ) ) ).
% 0.18/0.43
% 0.18/0.43 thf(notinemptyset_type,type,
% 0.18/0.43 notinemptyset: $o ).
% 0.18/0.43
% 0.18/0.43 thf(notinemptyset,definition,
% 0.18/0.43 ( notinemptyset
% 0.18/0.43 = ( ! [Xx: $i] :
% 0.18/0.43 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.18/0.43
% 0.18/0.43 thf(exuE3e_type,type,
% 0.18/0.43 exuE3e: $o ).
% 0.18/0.43
% 0.18/0.43 thf(exuE3e,definition,
% 0.18/0.43 ( exuE3e
% 0.18/0.43 = ( ! [Xphi: $i > $o] :
% 0.18/0.43 ( ( exu
% 0.18/0.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.18/0.43 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.18/0.43
% 0.18/0.43 thf(setext_type,type,
% 0.18/0.43 setext: $o ).
% 0.18/0.43
% 0.18/0.43 thf(setext,definition,
% 0.18/0.43 ( setext
% 0.18/0.43 = ( ! [A: $i,B: $i] :
% 0.18/0.43 ( ! [Xx: $i] :
% 0.18/0.43 ( ( in @ Xx @ A )
% 0.18/0.43 => ( in @ Xx @ B ) )
% 0.18/0.43 => ( ! [Xx: $i] :
% 0.18/0.43 ( ( in @ Xx @ B )
% 0.18/0.43 => ( in @ Xx @ A ) )
% 0.18/0.43 => ( A = B ) ) ) ) ) ).
% 0.18/0.43
% 0.18/0.43 thf(emptyI_type,type,
% 0.18/0.43 emptyI: $o ).
% 0.18/0.43
% 0.18/0.43 thf(emptyI,definition,
% 0.18/0.43 ( emptyI
% 0.18/0.43 = ( ! [A: $i] :
% 0.18/0.43 ( ! [Xx: $i] :
% 0.18/0.43 ~ ( in @ Xx @ A )
% 0.18/0.43 => ( A = emptyset ) ) ) ) ).
% 0.18/0.43
% 0.18/0.43 thf(noeltsimpempty_type,type,
% 0.18/0.43 noeltsimpempty: $o ).
% 0.18/0.43
% 0.18/0.43 thf(noeltsimpempty,definition,
% 0.18/0.43 ( noeltsimpempty
% 0.18/0.43 = ( ! [A: $i] :
% 0.18/0.43 ( ! [Xx: $i] :
% 0.18/0.43 ~ ( in @ Xx @ A )
% 0.18/0.43 => ( A = emptyset ) ) ) ) ).
% 0.18/0.43
% 0.18/0.43 thf(setbeta_type,type,
% 0.18/0.43 setbeta: $o ).
% 0.18/0.43
% 0.18/0.43 thf(setbeta,definition,
% 0.18/0.43 ( setbeta
% 0.18/0.43 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.18/0.43 ( ( in @ Xx @ A )
% 0.18/0.43 => ( ( in @ Xx
% 0.18/0.43 @ ( dsetconstr @ A
% 0.18/0.43 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.18/0.43 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.18/0.43
% 0.18/0.43 thf(nonempty_type,type,
% 0.18/0.43 nonempty: $i > $o ).
% 0.18/0.43
% 0.18/0.43 thf(nonempty,definition,
% 0.18/0.43 ( nonempty
% 0.18/0.43 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.18/0.43
% 0.18/0.43 thf(nonemptyE1_type,type,
% 0.18/0.43 nonemptyE1: $o ).
% 0.18/0.43
% 0.18/0.43 thf(nonemptyE1,definition,
% 0.18/0.43 ( nonemptyE1
% 0.18/0.43 = ( ! [A: $i] :
% 0.18/0.43 ( ( nonempty @ A )
% 0.18/0.43 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.18/0.43
% 0.18/0.43 thf(nonemptyI_type,type,
% 0.18/0.43 nonemptyI: $o ).
% 0.18/0.43
% 0.18/0.43 thf(nonemptyI,definition,
% 0.18/0.43 ( nonemptyI
% 0.18/0.43 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.18/0.43 ( ( in @ Xx @ A )
% 0.18/0.43 => ( ( Xphi @ Xx )
% 0.18/0.43 => ( nonempty
% 0.18/0.43 @ ( dsetconstr @ A
% 0.18/0.43 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.18/0.43
% 0.18/0.43 thf(nonemptyI1_type,type,
% 0.18/0.43 nonemptyI1: $o ).
% 0.18/0.43
% 0.18/0.43 thf(nonemptyI1,definition,
% 0.18/0.43 ( nonemptyI1
% 0.18/0.43 = ( ! [A: $i] :
% 0.18/0.43 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.18/0.43 => ( nonempty @ A ) ) ) ) ).
% 0.18/0.43
% 0.18/0.43 thf(setadjoinIL_type,type,
% 0.18/0.43 setadjoinIL: $o ).
% 0.18/0.43
% 0.18/0.43 thf(setadjoinIL,definition,
% 0.18/0.43 ( setadjoinIL
% 0.18/0.43 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.18/0.43
% 0.18/0.43 thf(emptyinunitempty_type,type,
% 0.18/0.43 emptyinunitempty: $o ).
% 0.18/0.43
% 0.18/0.43 thf(emptyinunitempty,definition,
% 0.18/0.43 ( emptyinunitempty
% 0.18/0.43 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.18/0.43
% 0.18/0.43 thf(setadjoinIR_type,type,
% 0.18/0.43 setadjoinIR: $o ).
% 0.18/0.43
% 0.18/0.43 thf(setadjoinIR,definition,
% 0.18/0.43 ( setadjoinIR
% 0.18/0.43 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.18/0.43 ( ( in @ Xy @ A )
% 0.18/0.43 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.18/0.43
% 0.18/0.43 thf(setadjoinE_type,type,
% 0.18/0.43 setadjoinE: $o ).
% 0.18/0.43
% 0.18/0.43 thf(setadjoinE,definition,
% 0.18/0.43 ( setadjoinE
% 0.18/0.43 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.18/0.43 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.18/0.43 => ! [Xphi: $o] :
% 0.18/0.43 ( ( ( Xy = Xx )
% 0.18/0.43 => Xphi )
% 0.18/0.43 => ( ( ( in @ Xy @ A )
% 0.18/0.43 => Xphi )
% 0.18/0.43 => Xphi ) ) ) ) ) ).
% 0.18/0.43
% 0.18/0.43 thf(setadjoinOr_type,type,
% 0.18/0.43 setadjoinOr: $o ).
% 0.18/0.43
% 0.18/0.43 thf(setadjoinOr,definition,
% 0.18/0.43 ( setadjoinOr
% 0.18/0.43 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.18/0.43 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.18/0.43 => ( ( Xy = Xx )
% 0.18/0.43 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.18/0.43
% 0.18/0.43 thf(setoftrueEq_type,type,
% 0.18/0.43 setoftrueEq: $o ).
% 0.18/0.43
% 0.18/0.43 thf(setoftrueEq,definition,
% 0.18/0.43 ( setoftrueEq
% 0.18/0.43 = ( ! [A: $i] :
% 0.18/0.43 ( ( dsetconstr @ A
% 0.18/0.43 @ ^ [Xx: $i] : $true )
% 0.18/0.43 = A ) ) ) ).
% 0.18/0.43
% 0.18/0.43 thf(powersetI_type,type,
% 0.18/0.43 powersetI: $o ).
% 0.18/0.43
% 0.18/0.43 thf(powersetI,definition,
% 0.18/0.43 ( powersetI
% 0.18/0.43 = ( ! [A: $i,B: $i] :
% 0.18/0.43 ( ! [Xx: $i] :
% 0.18/0.43 ( ( in @ Xx @ B )
% 0.18/0.43 => ( in @ Xx @ A ) )
% 0.18/0.43 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.18/0.43
% 0.18/0.43 thf(emptyinPowerset_type,type,
% 0.18/0.43 emptyinPowerset: $o ).
% 0.18/0.43
% 0.18/0.43 thf(emptyinPowerset,definition,
% 0.18/0.43 ( emptyinPowerset
% 0.18/0.43 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.18/0.43
% 0.18/0.43 thf(emptyInPowerset_type,type,
% 0.18/0.43 emptyInPowerset: $o ).
% 0.18/0.43
% 0.18/0.43 thf(emptyInPowerset,definition,
% 0.18/0.43 ( emptyInPowerset
% 0.18/0.43 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.18/0.43
% 0.18/0.43 thf(powersetE_type,type,
% 0.18/0.43 powersetE: $o ).
% 0.18/0.43
% 0.18/0.43 thf(powersetE,definition,
% 0.18/0.43 ( powersetE
% 0.18/0.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.43 ( ( in @ B @ ( powerset @ A ) )
% 0.18/0.43 => ( ( in @ Xx @ B )
% 0.18/0.43 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.18/0.43
% 0.18/0.43 thf(setunionI_type,type,
% 0.18/0.43 setunionI: $o ).
% 0.18/0.43
% 0.18/0.43 thf(setunionI,definition,
% 0.18/0.43 ( setunionI
% 0.18/0.43 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.18/0.43 ( ( in @ Xx @ B )
% 0.18/0.43 => ( ( in @ B @ A )
% 0.18/0.43 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.18/0.43
% 0.18/0.43 thf(setunionE_type,type,
% 0.18/0.43 setunionE: $o ).
% 0.18/0.43
% 0.18/0.43 thf(setunionE,definition,
% 0.18/0.43 ( setunionE
% 0.18/0.43 = ( ! [A: $i,Xx: $i] :
% 0.18/0.43 ( ( in @ Xx @ ( setunion @ A ) )
% 0.18/0.43 => ! [Xphi: $o] :
% 0.18/0.43 ( ! [B: $i] :
% 0.18/0.43 ( ( in @ Xx @ B )
% 0.18/0.43 => ( ( in @ B @ A )
% 0.18/0.43 => Xphi ) )
% 0.18/0.43 => Xphi ) ) ) ) ).
% 0.18/0.43
% 0.18/0.43 thf(subPowSU_type,type,
% 0.18/0.43 subPowSU: $o ).
% 0.18/0.43
% 0.18/0.43 thf(subPowSU,definition,
% 0.18/0.43 ( subPowSU
% 0.18/0.43 = ( ! [A: $i,Xx: $i] :
% 0.18/0.43 ( ( in @ Xx @ A )
% 0.18/0.43 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.18/0.43
% 0.18/0.43 thf(exuE2_type,type,
% 0.18/0.43 exuE2: $o ).
% 0.18/0.43
% 0.18/0.43 thf(exuE2,definition,
% 0.18/0.43 ( exuE2
% 0.18/0.43 = ( ! [Xphi: $i > $o] :
% 0.18/0.43 ( ( exu
% 0.18/0.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.18/0.43 => ? [Xx: $i] :
% 0.18/0.43 ! [Xy: $i] :
% 0.18/0.43 ( ( Xphi @ Xy )
% 0.18/0.43 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.18/0.43
% 0.18/0.43 thf(nonemptyImpWitness_type,type,
% 0.18/0.43 nonemptyImpWitness: $o ).
% 0.18/0.43
% 0.18/0.43 thf(nonemptyImpWitness,definition,
% 0.18/0.43 ( nonemptyImpWitness
% 0.18/0.43 = ( ! [A: $i] :
% 0.18/0.43 ( ( nonempty @ A )
% 0.18/0.44 => ? [Xx: $i] :
% 0.18/0.44 ( ( in @ Xx @ A )
% 0.18/0.44 & $true ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(uniqinunit_type,type,
% 0.18/0.44 uniqinunit: $o ).
% 0.18/0.44
% 0.18/0.44 thf(uniqinunit,definition,
% 0.18/0.44 ( uniqinunit
% 0.18/0.44 = ( ! [Xx: $i,Xy: $i] :
% 0.18/0.44 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.18/0.44 => ( Xx = Xy ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(notinsingleton_type,type,
% 0.18/0.44 notinsingleton: $o ).
% 0.18/0.44
% 0.18/0.44 thf(notinsingleton,definition,
% 0.18/0.44 ( notinsingleton
% 0.18/0.44 = ( ! [Xx: $i,Xy: $i] :
% 0.18/0.44 ( ( Xx != Xy )
% 0.18/0.44 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(eqinunit_type,type,
% 0.18/0.44 eqinunit: $o ).
% 0.18/0.44
% 0.18/0.44 thf(eqinunit,definition,
% 0.18/0.44 ( eqinunit
% 0.18/0.44 = ( ! [Xx: $i,Xy: $i] :
% 0.18/0.44 ( ( Xx = Xy )
% 0.18/0.44 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(singletonsswitch_type,type,
% 0.18/0.44 singletonsswitch: $o ).
% 0.18/0.44
% 0.18/0.44 thf(singletonsswitch,definition,
% 0.18/0.44 ( singletonsswitch
% 0.18/0.44 = ( ! [Xx: $i,Xy: $i] :
% 0.18/0.44 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.18/0.44 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(upairsetE_type,type,
% 0.18/0.44 upairsetE: $o ).
% 0.18/0.44
% 0.18/0.44 thf(upairsetE,definition,
% 0.18/0.44 ( upairsetE
% 0.18/0.44 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.18/0.44 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.18/0.44 => ( ( Xz = Xx )
% 0.18/0.44 | ( Xz = Xy ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(upairsetIL_type,type,
% 0.18/0.44 upairsetIL: $o ).
% 0.18/0.44
% 0.18/0.44 thf(upairsetIL,definition,
% 0.18/0.44 ( upairsetIL
% 0.18/0.44 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(upairsetIR_type,type,
% 0.18/0.44 upairsetIR: $o ).
% 0.18/0.44
% 0.18/0.44 thf(upairsetIR,definition,
% 0.18/0.44 ( upairsetIR
% 0.18/0.44 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(emptyE1_type,type,
% 0.18/0.44 emptyE1: $o ).
% 0.18/0.44
% 0.18/0.44 thf(emptyE1,definition,
% 0.18/0.44 ( emptyE1
% 0.18/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.18/0.44 ( ? [Xx: $i] :
% 0.18/0.44 ( ( in @ Xx @ A )
% 0.18/0.44 & ( Xphi @ Xx ) )
% 0.18/0.44 => ( ( ( dsetconstr @ A
% 0.18/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.18/0.44 = emptyset )
% 0.18/0.44 => $false ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(vacuousDall_type,type,
% 0.18/0.44 vacuousDall: $o ).
% 0.18/0.44
% 0.18/0.44 thf(vacuousDall,definition,
% 0.18/0.44 ( vacuousDall
% 0.18/0.44 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.18/0.44 ( ( in @ Xx @ emptyset )
% 0.18/0.44 => ( Xphi @ Xx ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(quantDeMorgan1_type,type,
% 0.18/0.44 quantDeMorgan1: $o ).
% 0.18/0.44
% 0.18/0.44 thf(quantDeMorgan1,definition,
% 0.18/0.44 ( quantDeMorgan1
% 0.18/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.18/0.44 ( ~ ! [Xx: $i] :
% 0.18/0.44 ( ( in @ Xx @ A )
% 0.18/0.44 => ( Xphi @ Xx ) )
% 0.18/0.44 => ? [Xx: $i] :
% 0.18/0.44 ( ( in @ Xx @ A )
% 0.18/0.44 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(quantDeMorgan2_type,type,
% 0.18/0.44 quantDeMorgan2: $o ).
% 0.18/0.44
% 0.18/0.44 thf(quantDeMorgan2,definition,
% 0.18/0.44 ( quantDeMorgan2
% 0.18/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.18/0.44 ( ! [Xx: $i] :
% 0.18/0.44 ( ( in @ Xx @ A )
% 0.18/0.44 => ~ ( Xphi @ Xx ) )
% 0.18/0.44 => ~ ? [Xx: $i] :
% 0.18/0.44 ( ( in @ Xx @ A )
% 0.18/0.44 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(quantDeMorgan3_type,type,
% 0.18/0.44 quantDeMorgan3: $o ).
% 0.18/0.44
% 0.18/0.44 thf(quantDeMorgan3,definition,
% 0.18/0.44 ( quantDeMorgan3
% 0.18/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.18/0.44 ( ~ ? [Xx: $i] :
% 0.18/0.44 ( ( in @ Xx @ A )
% 0.18/0.44 & ( Xphi @ Xx ) )
% 0.18/0.44 => ! [Xx: $i] :
% 0.18/0.44 ( ( in @ Xx @ A )
% 0.18/0.44 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(quantDeMorgan4_type,type,
% 0.18/0.44 quantDeMorgan4: $o ).
% 0.18/0.44
% 0.18/0.44 thf(quantDeMorgan4,definition,
% 0.18/0.44 ( quantDeMorgan4
% 0.18/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.18/0.44 ( ? [Xx: $i] :
% 0.18/0.44 ( ( in @ Xx @ A )
% 0.18/0.44 & ~ ( Xphi @ Xx ) )
% 0.18/0.44 => ~ ! [Xx: $i] :
% 0.18/0.44 ( ( in @ Xx @ A )
% 0.18/0.44 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(prop2setI_type,type,
% 0.18/0.44 prop2setI: $o ).
% 0.18/0.44
% 0.18/0.44 thf(prop2setI,definition,
% 0.18/0.44 ( prop2setI
% 0.18/0.44 = ( ! [Xphi: $o] :
% 0.18/0.44 ( Xphi
% 0.18/0.44 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(set2prop_type,type,
% 0.18/0.44 set2prop: $i > $o ).
% 0.18/0.44
% 0.18/0.44 thf(prop2set2propI_type,type,
% 0.18/0.44 prop2set2propI: $o ).
% 0.18/0.44
% 0.18/0.44 thf(prop2set2propI,definition,
% 0.18/0.44 ( prop2set2propI
% 0.18/0.44 = ( ! [Xphi: $o] :
% 0.18/0.44 ( Xphi
% 0.18/0.44 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(notdexE_type,type,
% 0.18/0.44 notdexE: $o ).
% 0.18/0.44
% 0.18/0.44 thf(notdexE,definition,
% 0.18/0.44 ( notdexE
% 0.18/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.18/0.44 ( ~ ? [Xx: $i] :
% 0.18/0.44 ( ( in @ Xx @ A )
% 0.18/0.44 & ( Xphi @ Xx ) )
% 0.18/0.44 => ! [Xx: $i] :
% 0.18/0.44 ( ( in @ Xx @ A )
% 0.18/0.44 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(notdallE_type,type,
% 0.18/0.44 notdallE: $o ).
% 0.18/0.44
% 0.18/0.44 thf(notdallE,definition,
% 0.18/0.44 ( notdallE
% 0.18/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.18/0.44 ( ~ ! [Xx: $i] :
% 0.18/0.44 ( ( in @ Xx @ A )
% 0.18/0.44 => ( Xphi @ Xx ) )
% 0.18/0.44 => ? [Xx: $i] :
% 0.18/0.44 ( ( in @ Xx @ A )
% 0.18/0.44 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(exuI1_type,type,
% 0.18/0.44 exuI1: $o ).
% 0.18/0.44
% 0.18/0.44 thf(exuI1,definition,
% 0.18/0.44 ( exuI1
% 0.18/0.44 = ( ! [Xphi: $i > $o] :
% 0.18/0.44 ( ? [Xx: $i] :
% 0.18/0.44 ( ( Xphi @ Xx )
% 0.18/0.44 & ! [Xy: $i] :
% 0.18/0.44 ( ( Xphi @ Xy )
% 0.18/0.44 => ( Xx = Xy ) ) )
% 0.18/0.44 => ( exu
% 0.18/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(exuI3_type,type,
% 0.18/0.44 exuI3: $o ).
% 0.18/0.44
% 0.18/0.44 thf(exuI3,definition,
% 0.18/0.44 ( exuI3
% 0.18/0.44 = ( ! [Xphi: $i > $o] :
% 0.18/0.44 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.18/0.44 => ( ! [Xx: $i,Xy: $i] :
% 0.18/0.44 ( ( Xphi @ Xx )
% 0.18/0.44 => ( ( Xphi @ Xy )
% 0.18/0.44 => ( Xx = Xy ) ) )
% 0.18/0.44 => ( exu
% 0.18/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(exuI2_type,type,
% 0.18/0.44 exuI2: $o ).
% 0.18/0.44
% 0.18/0.44 thf(exuI2,definition,
% 0.18/0.44 ( exuI2
% 0.18/0.44 = ( ! [Xphi: $i > $o] :
% 0.18/0.44 ( ? [Xx: $i] :
% 0.18/0.44 ! [Xy: $i] :
% 0.18/0.44 ( ( Xphi @ Xy )
% 0.18/0.44 <=> ( Xy = Xx ) )
% 0.18/0.44 => ( exu
% 0.18/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(inCongP_type,type,
% 0.18/0.44 inCongP: $o ).
% 0.18/0.44
% 0.18/0.44 thf(inCongP,definition,
% 0.18/0.44 ( inCongP
% 0.18/0.44 = ( ! [A: $i,B: $i] :
% 0.18/0.44 ( ( A = B )
% 0.18/0.44 => ! [Xx: $i,Xy: $i] :
% 0.18/0.44 ( ( Xx = Xy )
% 0.18/0.44 => ( ( in @ Xx @ A )
% 0.18/0.44 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(in__Cong_type,type,
% 0.18/0.44 in__Cong: $o ).
% 0.18/0.44
% 0.18/0.44 thf(in__Cong,definition,
% 0.18/0.44 ( in__Cong
% 0.18/0.44 = ( ! [A: $i,B: $i] :
% 0.18/0.44 ( ( A = B )
% 0.18/0.44 => ! [Xx: $i,Xy: $i] :
% 0.18/0.44 ( ( Xx = Xy )
% 0.18/0.44 => ( ( in @ Xx @ A )
% 0.18/0.44 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(exuE3u_type,type,
% 0.18/0.44 exuE3u: $o ).
% 0.18/0.44
% 0.18/0.44 thf(exuE3u,definition,
% 0.18/0.44 ( exuE3u
% 0.18/0.44 = ( ! [Xphi: $i > $o] :
% 0.18/0.44 ( ( exu
% 0.18/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.18/0.44 => ! [Xx: $i,Xy: $i] :
% 0.18/0.44 ( ( Xphi @ Xx )
% 0.18/0.44 => ( ( Xphi @ Xy )
% 0.18/0.44 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(exu__Cong_type,type,
% 0.18/0.44 exu__Cong: $o ).
% 0.18/0.44
% 0.18/0.44 thf(exu__Cong,definition,
% 0.18/0.44 ( exu__Cong
% 0.18/0.44 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.18/0.44 ( ! [Xx: $i,Xy: $i] :
% 0.18/0.44 ( ( Xx = Xy )
% 0.18/0.44 => ( ( Xphi @ Xx )
% 0.18/0.44 <=> ( Xpsi @ Xy ) ) )
% 0.18/0.44 => ( ( exu
% 0.18/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.18/0.44 <=> ( exu
% 0.18/0.44 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(emptyset__Cong_type,type,
% 0.18/0.44 emptyset__Cong: $o ).
% 0.18/0.44
% 0.18/0.44 thf(emptyset__Cong,definition,
% 0.18/0.44 ( emptyset__Cong
% 0.18/0.44 = ( emptyset = emptyset ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(setadjoin__Cong_type,type,
% 0.18/0.44 setadjoin__Cong: $o ).
% 0.18/0.44
% 0.18/0.44 thf(setadjoin__Cong,definition,
% 0.18/0.44 ( setadjoin__Cong
% 0.18/0.44 = ( ! [Xx: $i,Xy: $i] :
% 0.18/0.44 ( ( Xx = Xy )
% 0.18/0.44 => ! [Xz: $i,Xu: $i] :
% 0.18/0.44 ( ( Xz = Xu )
% 0.18/0.44 => ( ( setadjoin @ Xx @ Xz )
% 0.18/0.44 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(powerset__Cong_type,type,
% 0.18/0.44 powerset__Cong: $o ).
% 0.18/0.44
% 0.18/0.44 thf(powerset__Cong,definition,
% 0.18/0.44 ( powerset__Cong
% 0.18/0.44 = ( ! [A: $i,B: $i] :
% 0.18/0.44 ( ( A = B )
% 0.18/0.44 => ( ( powerset @ A )
% 0.18/0.44 = ( powerset @ B ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(setunion__Cong_type,type,
% 0.18/0.44 setunion__Cong: $o ).
% 0.18/0.44
% 0.18/0.44 thf(setunion__Cong,definition,
% 0.18/0.44 ( setunion__Cong
% 0.18/0.44 = ( ! [A: $i,B: $i] :
% 0.18/0.44 ( ( A = B )
% 0.18/0.44 => ( ( setunion @ A )
% 0.18/0.44 = ( setunion @ B ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(omega__Cong_type,type,
% 0.18/0.44 omega__Cong: $o ).
% 0.18/0.44
% 0.18/0.44 thf(omega__Cong,definition,
% 0.18/0.44 ( omega__Cong
% 0.18/0.44 = ( omega = omega ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(exuEu_type,type,
% 0.18/0.44 exuEu: $o ).
% 0.18/0.44
% 0.18/0.44 thf(exuEu,definition,
% 0.18/0.44 ( exuEu
% 0.18/0.44 = ( ! [Xphi: $i > $o] :
% 0.18/0.44 ( ( exu
% 0.18/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.18/0.44 => ! [Xx: $i,Xy: $i] :
% 0.18/0.44 ( ( Xphi @ Xx )
% 0.18/0.44 => ( ( Xphi @ Xy )
% 0.18/0.44 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(descr__Cong_type,type,
% 0.18/0.44 descr__Cong: $o ).
% 0.18/0.44
% 0.18/0.44 thf(descr__Cong,definition,
% 0.18/0.44 ( descr__Cong
% 0.18/0.44 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.18/0.44 ( ! [Xx: $i,Xy: $i] :
% 0.18/0.44 ( ( Xx = Xy )
% 0.18/0.44 => ( ( Xphi @ Xx )
% 0.18/0.44 <=> ( Xpsi @ Xy ) ) )
% 0.18/0.44 => ( ( exu
% 0.18/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.18/0.44 => ( ( exu
% 0.18/0.44 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.18/0.44 => ( ( descr
% 0.18/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.18/0.44 = ( descr
% 0.18/0.44 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(dsetconstr__Cong_type,type,
% 0.18/0.44 dsetconstr__Cong: $o ).
% 0.18/0.44
% 0.18/0.44 thf(dsetconstr__Cong,definition,
% 0.18/0.44 ( dsetconstr__Cong
% 0.18/0.44 = ( ! [A: $i,B: $i] :
% 0.18/0.44 ( ( A = B )
% 0.18/0.44 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.18/0.44 ( ! [Xx: $i] :
% 0.18/0.44 ( ( in @ Xx @ A )
% 0.18/0.44 => ! [Xy: $i] :
% 0.18/0.44 ( ( in @ Xy @ B )
% 0.18/0.44 => ( ( Xx = Xy )
% 0.18/0.44 => ( ( Xphi @ Xx )
% 0.18/0.44 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.18/0.44 => ( ( dsetconstr @ A
% 0.18/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.18/0.44 = ( dsetconstr @ B
% 0.18/0.44 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(subset_type,type,
% 0.18/0.44 subset: $i > $i > $o ).
% 0.18/0.44
% 0.18/0.44 thf(disjoint_type,type,
% 0.18/0.44 disjoint: $i > $i > $o ).
% 0.18/0.44
% 0.18/0.44 thf(setsmeet_type,type,
% 0.18/0.44 setsmeet: $i > $i > $o ).
% 0.18/0.44
% 0.18/0.44 thf(subsetI1_type,type,
% 0.18/0.44 subsetI1: $o ).
% 0.18/0.44
% 0.18/0.44 thf(subsetI1,definition,
% 0.18/0.44 ( subsetI1
% 0.18/0.44 = ( ! [A: $i,B: $i] :
% 0.18/0.44 ( ! [Xx: $i] :
% 0.18/0.44 ( ( in @ Xx @ A )
% 0.18/0.44 => ( in @ Xx @ B ) )
% 0.18/0.44 => ( subset @ A @ B ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(eqimpsubset2_type,type,
% 0.18/0.44 eqimpsubset2: $o ).
% 0.18/0.44
% 0.18/0.44 thf(eqimpsubset2,definition,
% 0.18/0.44 ( eqimpsubset2
% 0.18/0.44 = ( ! [A: $i,B: $i] :
% 0.18/0.44 ( ( A = B )
% 0.18/0.44 => ( subset @ B @ A ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(eqimpsubset1_type,type,
% 0.18/0.44 eqimpsubset1: $o ).
% 0.18/0.44
% 0.18/0.44 thf(eqimpsubset1,definition,
% 0.18/0.44 ( eqimpsubset1
% 0.18/0.44 = ( ! [A: $i,B: $i] :
% 0.18/0.44 ( ( A = B )
% 0.18/0.44 => ( subset @ A @ B ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(subsetI2_type,type,
% 0.18/0.44 subsetI2: $o ).
% 0.18/0.44
% 0.18/0.44 thf(subsetI2,definition,
% 0.18/0.44 ( subsetI2
% 0.18/0.44 = ( ! [A: $i,B: $i] :
% 0.18/0.44 ( ! [Xx: $i] :
% 0.18/0.44 ( ( in @ Xx @ A )
% 0.18/0.44 => ( in @ Xx @ B ) )
% 0.18/0.44 => ( subset @ A @ B ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(emptysetsubset_type,type,
% 0.18/0.44 emptysetsubset: $o ).
% 0.18/0.44
% 0.18/0.44 thf(emptysetsubset,definition,
% 0.18/0.44 ( emptysetsubset
% 0.18/0.44 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(subsetE_type,type,
% 0.18/0.44 subsetE: $o ).
% 0.18/0.44
% 0.18/0.44 thf(subsetE,definition,
% 0.18/0.44 ( subsetE
% 0.18/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.44 ( ( subset @ A @ B )
% 0.18/0.44 => ( ( in @ Xx @ A )
% 0.18/0.44 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(subsetE2_type,type,
% 0.18/0.44 subsetE2: $o ).
% 0.18/0.44
% 0.18/0.44 thf(subsetE2,definition,
% 0.18/0.44 ( subsetE2
% 0.18/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.44 ( ( subset @ A @ B )
% 0.18/0.44 => ( ~ ( in @ Xx @ B )
% 0.18/0.44 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(notsubsetI_type,type,
% 0.18/0.44 notsubsetI: $o ).
% 0.18/0.44
% 0.18/0.44 thf(notsubsetI,definition,
% 0.18/0.44 ( notsubsetI
% 0.18/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.44 ( ( in @ Xx @ A )
% 0.18/0.44 => ( ~ ( in @ Xx @ B )
% 0.18/0.44 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(notequalI1_type,type,
% 0.18/0.44 notequalI1: $o ).
% 0.18/0.44
% 0.18/0.44 thf(notequalI1,definition,
% 0.18/0.44 ( notequalI1
% 0.18/0.44 = ( ! [A: $i,B: $i] :
% 0.18/0.44 ( ~ ( subset @ A @ B )
% 0.18/0.44 => ( A != B ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(notequalI2_type,type,
% 0.18/0.44 notequalI2: $o ).
% 0.18/0.44
% 0.18/0.44 thf(notequalI2,definition,
% 0.18/0.44 ( notequalI2
% 0.18/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.44 ( ( in @ Xx @ A )
% 0.18/0.44 => ( ~ ( in @ Xx @ B )
% 0.18/0.44 => ( A != B ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(subsetRefl_type,type,
% 0.18/0.44 subsetRefl: $o ).
% 0.18/0.44
% 0.18/0.44 thf(subsetRefl,definition,
% 0.18/0.44 ( subsetRefl
% 0.18/0.44 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(subsetTrans_type,type,
% 0.18/0.44 subsetTrans: $o ).
% 0.18/0.44
% 0.18/0.44 thf(subsetTrans,definition,
% 0.18/0.44 ( subsetTrans
% 0.18/0.44 = ( ! [A: $i,B: $i,C: $i] :
% 0.18/0.44 ( ( subset @ A @ B )
% 0.18/0.44 => ( ( subset @ B @ C )
% 0.18/0.44 => ( subset @ A @ C ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(setadjoinSub_type,type,
% 0.18/0.44 setadjoinSub: $o ).
% 0.18/0.44
% 0.18/0.44 thf(setadjoinSub,definition,
% 0.18/0.44 ( setadjoinSub
% 0.18/0.44 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(setadjoinSub2_type,type,
% 0.18/0.44 setadjoinSub2: $o ).
% 0.18/0.44
% 0.18/0.44 thf(setadjoinSub2,definition,
% 0.18/0.44 ( setadjoinSub2
% 0.18/0.44 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.18/0.44 ( ( subset @ A @ B )
% 0.18/0.44 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(subset2powerset_type,type,
% 0.18/0.44 subset2powerset: $o ).
% 0.18/0.44
% 0.18/0.44 thf(subset2powerset,definition,
% 0.18/0.44 ( subset2powerset
% 0.18/0.44 = ( ! [A: $i,B: $i] :
% 0.18/0.44 ( ( subset @ A @ B )
% 0.18/0.44 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(setextsub_type,type,
% 0.18/0.44 setextsub: $o ).
% 0.18/0.44
% 0.18/0.44 thf(setextsub,definition,
% 0.18/0.44 ( setextsub
% 0.18/0.44 = ( ! [A: $i,B: $i] :
% 0.18/0.44 ( ( subset @ A @ B )
% 0.18/0.44 => ( ( subset @ B @ A )
% 0.18/0.44 => ( A = B ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(subsetemptysetimpeq_type,type,
% 0.18/0.44 subsetemptysetimpeq: $o ).
% 0.18/0.44
% 0.18/0.44 thf(subsetemptysetimpeq,definition,
% 0.18/0.44 ( subsetemptysetimpeq
% 0.18/0.44 = ( ! [A: $i] :
% 0.18/0.44 ( ( subset @ A @ emptyset )
% 0.18/0.44 => ( A = emptyset ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(powersetI1_type,type,
% 0.18/0.44 powersetI1: $o ).
% 0.18/0.44
% 0.18/0.44 thf(powersetI1,definition,
% 0.18/0.44 ( powersetI1
% 0.18/0.44 = ( ! [A: $i,B: $i] :
% 0.18/0.44 ( ( subset @ B @ A )
% 0.18/0.44 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(powersetE1_type,type,
% 0.18/0.44 powersetE1: $o ).
% 0.18/0.44
% 0.18/0.44 thf(powersetE1,definition,
% 0.18/0.44 ( powersetE1
% 0.18/0.44 = ( ! [A: $i,B: $i] :
% 0.18/0.44 ( ( in @ B @ ( powerset @ A ) )
% 0.18/0.44 => ( subset @ B @ A ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(inPowerset_type,type,
% 0.18/0.44 inPowerset: $o ).
% 0.18/0.44
% 0.18/0.44 thf(inPowerset,definition,
% 0.18/0.44 ( inPowerset
% 0.18/0.44 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(powersetsubset_type,type,
% 0.18/0.44 powersetsubset: $o ).
% 0.18/0.44
% 0.18/0.44 thf(powersetsubset,definition,
% 0.18/0.44 ( powersetsubset
% 0.18/0.44 = ( ! [A: $i,B: $i] :
% 0.18/0.44 ( ( subset @ A @ B )
% 0.18/0.44 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(sepInPowerset_type,type,
% 0.18/0.44 sepInPowerset: $o ).
% 0.18/0.44
% 0.18/0.44 thf(sepInPowerset,definition,
% 0.18/0.44 ( sepInPowerset
% 0.18/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.18/0.44 ( in
% 0.18/0.44 @ ( dsetconstr @ A
% 0.18/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.18/0.44 @ ( powerset @ A ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(sepSubset_type,type,
% 0.18/0.44 sepSubset: $o ).
% 0.18/0.44
% 0.18/0.44 thf(sepSubset,definition,
% 0.18/0.44 ( sepSubset
% 0.18/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.18/0.44 ( subset
% 0.18/0.44 @ ( dsetconstr @ A
% 0.18/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.18/0.44 @ A ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(binunion_type,type,
% 0.18/0.44 binunion: $i > $i > $i ).
% 0.18/0.44
% 0.18/0.44 thf(binunionIL_type,type,
% 0.18/0.44 binunionIL: $o ).
% 0.18/0.44
% 0.18/0.44 thf(binunionIL,definition,
% 0.18/0.44 ( binunionIL
% 0.18/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.44 ( ( in @ Xx @ A )
% 0.18/0.44 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(upairset2IR_type,type,
% 0.18/0.44 upairset2IR: $o ).
% 0.18/0.44
% 0.18/0.44 thf(upairset2IR,definition,
% 0.18/0.44 ( upairset2IR
% 0.18/0.44 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(binunionIR_type,type,
% 0.18/0.44 binunionIR: $o ).
% 0.18/0.44
% 0.18/0.44 thf(binunionIR,definition,
% 0.18/0.44 ( binunionIR
% 0.18/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.44 ( ( in @ Xx @ B )
% 0.18/0.44 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(binunionEcases_type,type,
% 0.18/0.44 binunionEcases: $o ).
% 0.18/0.44
% 0.18/0.44 thf(binunionEcases,definition,
% 0.18/0.44 ( binunionEcases
% 0.18/0.44 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.18/0.44 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.18/0.44 => ( ( ( in @ Xx @ A )
% 0.18/0.44 => Xphi )
% 0.18/0.44 => ( ( ( in @ Xx @ B )
% 0.18/0.44 => Xphi )
% 0.18/0.44 => Xphi ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(binunionE_type,type,
% 0.18/0.44 binunionE: $o ).
% 0.18/0.44
% 0.18/0.44 thf(binunionE,definition,
% 0.18/0.44 ( binunionE
% 0.18/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.44 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.18/0.44 => ( ( in @ Xx @ A )
% 0.18/0.44 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(binunionLsub_type,type,
% 0.18/0.44 binunionLsub: $o ).
% 0.18/0.44
% 0.18/0.44 thf(binunionLsub,definition,
% 0.18/0.44 ( binunionLsub
% 0.18/0.44 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(binunionRsub_type,type,
% 0.18/0.44 binunionRsub: $o ).
% 0.18/0.44
% 0.18/0.44 thf(binunionRsub,definition,
% 0.18/0.44 ( binunionRsub
% 0.18/0.44 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(binintersect_type,type,
% 0.18/0.44 binintersect: $i > $i > $i ).
% 0.18/0.44
% 0.18/0.44 thf(binintersectI_type,type,
% 0.18/0.44 binintersectI: $o ).
% 0.18/0.44
% 0.18/0.44 thf(binintersectI,definition,
% 0.18/0.44 ( binintersectI
% 0.18/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.44 ( ( in @ Xx @ A )
% 0.18/0.44 => ( ( in @ Xx @ B )
% 0.18/0.44 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(binintersectSubset5_type,type,
% 0.18/0.44 binintersectSubset5: $o ).
% 0.18/0.44
% 0.18/0.44 thf(binintersectSubset5,definition,
% 0.18/0.44 ( binintersectSubset5
% 0.18/0.44 = ( ! [A: $i,B: $i,C: $i] :
% 0.18/0.44 ( ( subset @ C @ A )
% 0.18/0.44 => ( ( subset @ C @ B )
% 0.18/0.44 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(binintersectEL_type,type,
% 0.18/0.44 binintersectEL: $o ).
% 0.18/0.44
% 0.18/0.44 thf(binintersectEL,definition,
% 0.18/0.44 ( binintersectEL
% 0.18/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.44 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.18/0.44 => ( in @ Xx @ A ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(binintersectLsub_type,type,
% 0.18/0.44 binintersectLsub: $o ).
% 0.18/0.44
% 0.18/0.44 thf(binintersectLsub,definition,
% 0.18/0.44 ( binintersectLsub
% 0.18/0.44 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(binintersectSubset2_type,type,
% 0.18/0.44 binintersectSubset2: $o ).
% 0.18/0.44
% 0.18/0.44 thf(binintersectSubset2,definition,
% 0.18/0.44 ( binintersectSubset2
% 0.18/0.44 = ( ! [A: $i,B: $i] :
% 0.18/0.44 ( ( subset @ A @ B )
% 0.18/0.44 => ( ( binintersect @ A @ B )
% 0.18/0.44 = A ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(binintersectSubset3_type,type,
% 0.18/0.44 binintersectSubset3: $o ).
% 0.18/0.44
% 0.18/0.44 thf(binintersectSubset3,definition,
% 0.18/0.44 ( binintersectSubset3
% 0.18/0.44 = ( ! [A: $i,B: $i] :
% 0.18/0.44 ( ( ( binintersect @ A @ B )
% 0.18/0.44 = B )
% 0.18/0.44 => ( subset @ B @ A ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(binintersectER_type,type,
% 0.18/0.44 binintersectER: $o ).
% 0.18/0.44
% 0.18/0.44 thf(binintersectER,definition,
% 0.18/0.44 ( binintersectER
% 0.18/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.44 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.18/0.44 => ( in @ Xx @ B ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(disjointsetsI1_type,type,
% 0.18/0.44 disjointsetsI1: $o ).
% 0.18/0.44
% 0.18/0.44 thf(disjointsetsI1,definition,
% 0.18/0.44 ( disjointsetsI1
% 0.18/0.44 = ( ! [A: $i,B: $i] :
% 0.18/0.44 ( ~ ? [Xx: $i] :
% 0.18/0.44 ( ( in @ Xx @ A )
% 0.18/0.44 & ( in @ Xx @ B ) )
% 0.18/0.44 => ( ( binintersect @ A @ B )
% 0.18/0.44 = emptyset ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(binintersectRsub_type,type,
% 0.18/0.44 binintersectRsub: $o ).
% 0.18/0.44
% 0.18/0.44 thf(binintersectRsub,definition,
% 0.18/0.44 ( binintersectRsub
% 0.18/0.44 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(binintersectSubset4_type,type,
% 0.18/0.44 binintersectSubset4: $o ).
% 0.18/0.44
% 0.18/0.44 thf(binintersectSubset4,definition,
% 0.18/0.44 ( binintersectSubset4
% 0.18/0.44 = ( ! [A: $i,B: $i] :
% 0.18/0.44 ( ( subset @ B @ A )
% 0.18/0.44 => ( ( binintersect @ A @ B )
% 0.18/0.44 = B ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(binintersectSubset1_type,type,
% 0.18/0.44 binintersectSubset1: $o ).
% 0.18/0.44
% 0.18/0.44 thf(binintersectSubset1,definition,
% 0.18/0.44 ( binintersectSubset1
% 0.18/0.44 = ( ! [A: $i,B: $i] :
% 0.18/0.44 ( ( ( binintersect @ A @ B )
% 0.18/0.44 = A )
% 0.18/0.44 => ( subset @ A @ B ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(bs114d_type,type,
% 0.18/0.44 bs114d: $o ).
% 0.18/0.44
% 0.18/0.44 thf(bs114d,definition,
% 0.18/0.44 ( bs114d
% 0.18/0.44 = ( ! [A: $i,B: $i,C: $i] :
% 0.18/0.44 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 0.18/0.44 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(regular_type,type,
% 0.18/0.44 regular: $i > $o ).
% 0.18/0.44
% 0.18/0.44 thf(setminus_type,type,
% 0.18/0.44 setminus: $i > $i > $i ).
% 0.18/0.44
% 0.18/0.44 thf(setminusI_type,type,
% 0.18/0.44 setminusI: $o ).
% 0.18/0.44
% 0.18/0.44 thf(setminusI,definition,
% 0.18/0.44 ( setminusI
% 0.18/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.44 ( ( in @ Xx @ A )
% 0.18/0.44 => ( ~ ( in @ Xx @ B )
% 0.18/0.44 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(setminusEL_type,type,
% 0.18/0.44 setminusEL: $o ).
% 0.18/0.44
% 0.18/0.44 thf(setminusEL,definition,
% 0.18/0.44 ( setminusEL
% 0.18/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.44 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.18/0.44 => ( in @ Xx @ A ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(setminusER_type,type,
% 0.18/0.44 setminusER: $o ).
% 0.18/0.44
% 0.18/0.44 thf(setminusER,definition,
% 0.18/0.44 ( setminusER
% 0.18/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.44 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.18/0.44 => ~ ( in @ Xx @ B ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(setminusSubset2_type,type,
% 0.18/0.44 setminusSubset2: $o ).
% 0.18/0.44
% 0.18/0.44 thf(setminusSubset2,definition,
% 0.18/0.44 ( setminusSubset2
% 0.18/0.44 = ( ! [A: $i,B: $i] :
% 0.18/0.44 ( ( subset @ A @ B )
% 0.18/0.44 => ( ( setminus @ A @ B )
% 0.18/0.44 = emptyset ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(setminusERneg_type,type,
% 0.18/0.44 setminusERneg: $o ).
% 0.18/0.44
% 0.18/0.44 thf(setminusERneg,definition,
% 0.18/0.44 ( setminusERneg
% 0.18/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.44 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.18/0.44 => ( ( in @ Xx @ A )
% 0.18/0.44 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(setminusELneg_type,type,
% 0.18/0.44 setminusELneg: $o ).
% 0.18/0.44
% 0.18/0.44 thf(setminusELneg,definition,
% 0.18/0.44 ( setminusELneg
% 0.18/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.44 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.18/0.44 => ( ~ ( in @ Xx @ B )
% 0.18/0.44 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(setminusILneg_type,type,
% 0.18/0.44 setminusILneg: $o ).
% 0.18/0.44
% 0.18/0.44 thf(setminusILneg,definition,
% 0.18/0.44 ( setminusILneg
% 0.18/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.44 ( ~ ( in @ Xx @ A )
% 0.18/0.44 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(setminusIRneg_type,type,
% 0.18/0.44 setminusIRneg: $o ).
% 0.18/0.44
% 0.18/0.44 thf(setminusIRneg,definition,
% 0.18/0.44 ( setminusIRneg
% 0.18/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.44 ( ( in @ Xx @ B )
% 0.18/0.44 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(setminusLsub_type,type,
% 0.18/0.44 setminusLsub: $o ).
% 0.18/0.44
% 0.18/0.44 thf(setminusLsub,definition,
% 0.18/0.44 ( setminusLsub
% 0.18/0.44 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(setminusSubset1_type,type,
% 0.18/0.44 setminusSubset1: $o ).
% 0.18/0.44
% 0.18/0.44 thf(setminusSubset1,definition,
% 0.18/0.44 ( setminusSubset1
% 0.18/0.44 = ( ! [A: $i,B: $i] :
% 0.18/0.44 ( ( ( setminus @ A @ B )
% 0.18/0.44 = emptyset )
% 0.18/0.44 => ( subset @ A @ B ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(symdiff_type,type,
% 0.18/0.44 symdiff: $i > $i > $i ).
% 0.18/0.44
% 0.18/0.44 thf(symdiffE_type,type,
% 0.18/0.44 symdiffE: $o ).
% 0.18/0.44
% 0.18/0.44 thf(symdiffE,definition,
% 0.18/0.44 ( symdiffE
% 0.18/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.44 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 0.18/0.44 => ! [Xphi: $o] :
% 0.18/0.44 ( ( ( in @ Xx @ A )
% 0.18/0.44 => ( ~ ( in @ Xx @ B )
% 0.18/0.44 => Xphi ) )
% 0.18/0.44 => ( ( ~ ( in @ Xx @ A )
% 0.18/0.44 => ( ( in @ Xx @ B )
% 0.18/0.44 => Xphi ) )
% 0.18/0.44 => Xphi ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(symdiffI1_type,type,
% 0.18/0.44 symdiffI1: $o ).
% 0.18/0.44
% 0.18/0.44 thf(symdiffI1,definition,
% 0.18/0.44 ( symdiffI1
% 0.18/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.44 ( ( in @ Xx @ A )
% 0.18/0.44 => ( ~ ( in @ Xx @ B )
% 0.18/0.44 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(symdiffI2_type,type,
% 0.18/0.44 symdiffI2: $o ).
% 0.18/0.44
% 0.18/0.44 thf(symdiffI2,definition,
% 0.18/0.44 ( symdiffI2
% 0.18/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.44 ( ~ ( in @ Xx @ A )
% 0.18/0.44 => ( ( in @ Xx @ B )
% 0.18/0.44 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(symdiffIneg1_type,type,
% 0.18/0.44 symdiffIneg1: $o ).
% 0.18/0.44
% 0.18/0.44 thf(symdiffIneg1,definition,
% 0.18/0.44 ( symdiffIneg1
% 0.18/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.44 ( ( in @ Xx @ A )
% 0.18/0.44 => ( ( in @ Xx @ B )
% 0.18/0.44 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(symdiffIneg2_type,type,
% 0.18/0.44 symdiffIneg2: $o ).
% 0.18/0.44
% 0.18/0.44 thf(symdiffIneg2,definition,
% 0.18/0.44 ( symdiffIneg2
% 0.18/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.44 ( ~ ( in @ Xx @ A )
% 0.18/0.44 => ( ~ ( in @ Xx @ B )
% 0.18/0.44 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(iskpair_type,type,
% 0.18/0.44 iskpair: $i > $o ).
% 0.18/0.44
% 0.18/0.44 thf(iskpair,definition,
% 0.18/0.44 ( iskpair
% 0.18/0.44 = ( ^ [A: $i] :
% 0.18/0.44 ? [Xx: $i] :
% 0.18/0.44 ( ( in @ Xx @ ( setunion @ A ) )
% 0.18/0.44 & ? [Xy: $i] :
% 0.18/0.44 ( ( in @ Xy @ ( setunion @ A ) )
% 0.18/0.44 & ( A
% 0.18/0.44 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(secondinupair_type,type,
% 0.18/0.44 secondinupair: $o ).
% 0.18/0.44
% 0.18/0.44 thf(secondinupair,definition,
% 0.18/0.44 ( secondinupair
% 0.18/0.44 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(setukpairIL_type,type,
% 0.18/0.44 setukpairIL: $o ).
% 0.18/0.44
% 0.18/0.44 thf(setukpairIL,definition,
% 0.18/0.44 ( setukpairIL
% 0.18/0.44 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(setukpairIR_type,type,
% 0.18/0.44 setukpairIR: $o ).
% 0.18/0.44
% 0.18/0.44 thf(setukpairIR,definition,
% 0.18/0.44 ( setukpairIR
% 0.18/0.44 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(kpairiskpair_type,type,
% 0.18/0.44 kpairiskpair: $o ).
% 0.18/0.44
% 0.18/0.44 thf(kpairiskpair,definition,
% 0.18/0.44 ( kpairiskpair
% 0.18/0.44 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(kpair_type,type,
% 0.18/0.44 kpair: $i > $i > $i ).
% 0.18/0.44
% 0.18/0.44 thf(kpair,definition,
% 0.18/0.44 ( kpair
% 0.18/0.44 = ( ^ [Xx: $i,Xy: $i] : ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(kpairp_type,type,
% 0.18/0.44 kpairp: $o ).
% 0.18/0.44
% 0.18/0.44 thf(kpairp,definition,
% 0.18/0.44 ( kpairp
% 0.18/0.44 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(cartprod_type,type,
% 0.18/0.44 cartprod: $i > $i > $i ).
% 0.18/0.44
% 0.18/0.44 thf(singletonsubset_type,type,
% 0.18/0.44 singletonsubset: $o ).
% 0.18/0.44
% 0.18/0.44 thf(singletonsubset,definition,
% 0.18/0.44 ( singletonsubset
% 0.18/0.44 = ( ! [A: $i,Xx: $i] :
% 0.18/0.44 ( ( in @ Xx @ A )
% 0.18/0.44 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(singletoninpowerset_type,type,
% 0.18/0.44 singletoninpowerset: $o ).
% 0.18/0.44
% 0.18/0.44 thf(singletoninpowerset,definition,
% 0.18/0.44 ( singletoninpowerset
% 0.18/0.44 = ( ! [A: $i,Xx: $i] :
% 0.18/0.44 ( ( in @ Xx @ A )
% 0.18/0.44 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(singletoninpowunion_type,type,
% 0.18/0.44 singletoninpowunion: $o ).
% 0.18/0.44
% 0.18/0.44 thf(singletoninpowunion,definition,
% 0.18/0.44 ( singletoninpowunion
% 0.18/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.44 ( ( in @ Xx @ A )
% 0.18/0.44 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(upairset2E_type,type,
% 0.18/0.44 upairset2E: $o ).
% 0.18/0.44
% 0.18/0.44 thf(upairset2E,definition,
% 0.18/0.44 ( upairset2E
% 0.18/0.44 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.18/0.44 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.18/0.44 => ( ( Xz = Xx )
% 0.18/0.44 | ( Xz = Xy ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(upairsubunion_type,type,
% 0.18/0.44 upairsubunion: $o ).
% 0.18/0.44
% 0.18/0.44 thf(upairsubunion,definition,
% 0.18/0.44 ( upairsubunion
% 0.18/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.44 ( ( in @ Xx @ A )
% 0.18/0.44 => ! [Xy: $i] :
% 0.18/0.44 ( ( in @ Xy @ B )
% 0.18/0.44 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(upairinpowunion_type,type,
% 0.18/0.44 upairinpowunion: $o ).
% 0.18/0.44
% 0.18/0.44 thf(upairinpowunion,definition,
% 0.18/0.44 ( upairinpowunion
% 0.18/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.44 ( ( in @ Xx @ A )
% 0.18/0.44 => ! [Xy: $i] :
% 0.18/0.44 ( ( in @ Xy @ B )
% 0.18/0.44 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(ubforcartprodlem1_type,type,
% 0.18/0.44 ubforcartprodlem1: $o ).
% 0.18/0.44
% 0.18/0.44 thf(ubforcartprodlem1,definition,
% 0.18/0.44 ( ubforcartprodlem1
% 0.18/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.44 ( ( in @ Xx @ A )
% 0.18/0.44 => ! [Xy: $i] :
% 0.18/0.44 ( ( in @ Xy @ B )
% 0.18/0.44 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(ubforcartprodlem2_type,type,
% 0.18/0.44 ubforcartprodlem2: $o ).
% 0.18/0.44
% 0.18/0.44 thf(ubforcartprodlem2,definition,
% 0.18/0.44 ( ubforcartprodlem2
% 0.18/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.44 ( ( in @ Xx @ A )
% 0.18/0.44 => ! [Xy: $i] :
% 0.18/0.44 ( ( in @ Xy @ B )
% 0.18/0.44 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(ubforcartprodlem3_type,type,
% 0.18/0.44 ubforcartprodlem3: $o ).
% 0.18/0.44
% 0.18/0.44 thf(ubforcartprodlem3,definition,
% 0.18/0.44 ( ubforcartprodlem3
% 0.18/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.44 ( ( in @ Xx @ A )
% 0.18/0.44 => ! [Xy: $i] :
% 0.18/0.44 ( ( in @ Xy @ B )
% 0.18/0.44 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(cartprodpairin_type,type,
% 0.18/0.44 cartprodpairin: $o ).
% 0.18/0.44
% 0.18/0.44 thf(cartprodpairin,definition,
% 0.18/0.44 ( cartprodpairin
% 0.18/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.44 ( ( in @ Xx @ A )
% 0.18/0.44 => ! [Xy: $i] :
% 0.18/0.44 ( ( in @ Xy @ B )
% 0.18/0.44 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(cartprodmempair1_type,type,
% 0.18/0.44 cartprodmempair1: $o ).
% 0.18/0.44
% 0.18/0.44 thf(cartprodmempair1,definition,
% 0.18/0.44 ( cartprodmempair1
% 0.18/0.44 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.18/0.44 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.18/0.44 => ? [Xx: $i] :
% 0.18/0.44 ( ( in @ Xx @ A )
% 0.18/0.44 & ? [Xy: $i] :
% 0.18/0.44 ( ( in @ Xy @ B )
% 0.18/0.44 & ( Xu
% 0.18/0.44 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(cartprodmempair_type,type,
% 0.18/0.44 cartprodmempair: $o ).
% 0.18/0.44
% 0.18/0.44 thf(cartprodmempair,definition,
% 0.18/0.44 ( cartprodmempair
% 0.18/0.44 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.18/0.44 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.18/0.44 => ( iskpair @ Xu ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(setunionE2_type,type,
% 0.18/0.44 setunionE2: $o ).
% 0.18/0.44
% 0.18/0.44 thf(setunionE2,definition,
% 0.18/0.44 ( setunionE2
% 0.18/0.44 = ( ! [A: $i,Xx: $i] :
% 0.18/0.44 ( ( in @ Xx @ ( setunion @ A ) )
% 0.18/0.44 => ? [X: $i] :
% 0.18/0.44 ( ( in @ X @ A )
% 0.18/0.44 & ( in @ Xx @ X ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(setunionsingleton1_type,type,
% 0.18/0.44 setunionsingleton1: $o ).
% 0.18/0.44
% 0.18/0.44 thf(setunionsingleton1,definition,
% 0.18/0.44 ( setunionsingleton1
% 0.18/0.44 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(setunionsingleton2_type,type,
% 0.18/0.44 setunionsingleton2: $o ).
% 0.18/0.44
% 0.18/0.44 thf(setunionsingleton2,definition,
% 0.18/0.44 ( setunionsingleton2
% 0.18/0.44 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(setunionsingleton_type,type,
% 0.18/0.44 setunionsingleton: $o ).
% 0.18/0.44
% 0.18/0.44 thf(setunionsingleton,definition,
% 0.18/0.44 ( setunionsingleton
% 0.18/0.44 = ( ! [Xx: $i] :
% 0.18/0.44 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 0.18/0.44 = Xx ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(singleton_type,type,
% 0.18/0.44 singleton: $i > $o ).
% 0.18/0.44
% 0.18/0.44 thf(singleton,definition,
% 0.18/0.44 ( singleton
% 0.18/0.44 = ( ^ [A: $i] :
% 0.18/0.44 ? [Xx: $i] :
% 0.18/0.44 ( ( in @ Xx @ A )
% 0.18/0.44 & ( A
% 0.18/0.44 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(singletonprop_type,type,
% 0.18/0.44 singletonprop: $o ).
% 0.18/0.44
% 0.18/0.44 thf(singletonprop,definition,
% 0.18/0.44 ( singletonprop
% 0.18/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.18/0.44 ( ! [Xx: $i] :
% 0.18/0.44 ( ( in @ Xx @ A )
% 0.18/0.44 => ! [Xy: $i] :
% 0.18/0.44 ( ( in @ Xy @ A )
% 0.18/0.44 => ( ( Xphi @ Xx )
% 0.18/0.44 => ( ( Xphi @ Xy )
% 0.18/0.44 => ( Xx = Xy ) ) ) ) )
% 0.18/0.44 => ( ? [Xx: $i] :
% 0.18/0.44 ( ( in @ Xx @ A )
% 0.18/0.44 & ( Xphi @ Xx ) )
% 0.18/0.44 => ( singleton
% 0.18/0.44 @ ( dsetconstr @ A
% 0.18/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(ex1_type,type,
% 0.18/0.44 ex1: $i > ( $i > $o ) > $o ).
% 0.18/0.44
% 0.18/0.44 thf(ex1,definition,
% 0.18/0.44 ( ex1
% 0.18/0.44 = ( ^ [A: $i,Xphi: $i > $o] :
% 0.18/0.44 ( singleton
% 0.18/0.44 @ ( dsetconstr @ A
% 0.18/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(ex1E1_type,type,
% 0.18/0.44 ex1E1: $o ).
% 0.18/0.44
% 0.18/0.44 thf(ex1E1,definition,
% 0.18/0.44 ( ex1E1
% 0.18/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.18/0.44 ( ( ex1 @ A
% 0.18/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.18/0.44 => ? [Xx: $i] :
% 0.18/0.44 ( ( in @ Xx @ A )
% 0.18/0.44 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(ex1I_type,type,
% 0.18/0.44 ex1I: $o ).
% 0.18/0.44
% 0.18/0.44 thf(ex1I,definition,
% 0.18/0.44 ( ex1I
% 0.18/0.44 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.18/0.44 ( ( in @ Xx @ A )
% 0.18/0.44 => ( ( Xphi @ Xx )
% 0.18/0.44 => ( ! [Xy: $i] :
% 0.18/0.44 ( ( in @ Xy @ A )
% 0.18/0.44 => ( ( Xphi @ Xy )
% 0.18/0.44 => ( Xy = Xx ) ) )
% 0.18/0.44 => ( ex1 @ A
% 0.18/0.44 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(ex1I2_type,type,
% 0.18/0.44 ex1I2: $o ).
% 0.18/0.44
% 0.18/0.44 thf(ex1I2,definition,
% 0.18/0.44 ( ex1I2
% 0.18/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.18/0.44 ( ! [Xx: $i] :
% 0.18/0.44 ( ( in @ Xx @ A )
% 0.18/0.44 => ! [Xy: $i] :
% 0.18/0.44 ( ( in @ Xy @ A )
% 0.18/0.44 => ( ( Xphi @ Xx )
% 0.18/0.44 => ( ( Xphi @ Xy )
% 0.18/0.44 => ( Xx = Xy ) ) ) ) )
% 0.18/0.44 => ( ? [Xx: $i] :
% 0.18/0.44 ( ( in @ Xx @ A )
% 0.18/0.44 & ( Xphi @ Xx ) )
% 0.18/0.44 => ( ex1 @ A
% 0.18/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 thf(singletonsuniq,conjecture,
% 0.18/0.44 ( setextAx
% 0.18/0.44 => ( emptysetAx
% 0.18/0.44 => ( setadjoinAx
% 0.18/0.44 => ( powersetAx
% 0.18/0.44 => ( setunionAx
% 0.18/0.44 => ( omega0Ax
% 0.18/0.44 => ( omegaSAx
% 0.18/0.44 => ( omegaIndAx
% 0.18/0.44 => ( replAx
% 0.18/0.44 => ( foundationAx
% 0.18/0.44 => ( wellorderingAx
% 0.18/0.44 => ( descrp
% 0.18/0.44 => ( dsetconstrI
% 0.18/0.44 => ( dsetconstrEL
% 0.18/0.44 => ( dsetconstrER
% 0.18/0.44 => ( exuE1
% 0.18/0.44 => ( prop2setE
% 0.18/0.44 => ( emptysetE
% 0.18/0.44 => ( emptysetimpfalse
% 0.18/0.44 => ( notinemptyset
% 0.18/0.44 => ( exuE3e
% 0.18/0.44 => ( setext
% 0.18/0.44 => ( emptyI
% 0.18/0.44 => ( noeltsimpempty
% 0.18/0.44 => ( setbeta
% 0.18/0.44 => ( nonemptyE1
% 0.18/0.44 => ( nonemptyI
% 0.18/0.44 => ( nonemptyI1
% 0.18/0.44 => ( setadjoinIL
% 0.18/0.44 => ( emptyinunitempty
% 0.18/0.44 => ( setadjoinIR
% 0.18/0.44 => ( setadjoinE
% 0.18/0.44 => ( setadjoinOr
% 0.18/0.44 => ( setoftrueEq
% 0.18/0.44 => ( powersetI
% 0.18/0.44 => ( emptyinPowerset
% 0.18/0.44 => ( emptyInPowerset
% 0.18/0.44 => ( powersetE
% 0.18/0.44 => ( setunionI
% 0.18/0.44 => ( setunionE
% 0.18/0.44 => ( subPowSU
% 0.18/0.44 => ( exuE2
% 0.18/0.44 => ( nonemptyImpWitness
% 0.18/0.44 => ( uniqinunit
% 0.18/0.44 => ( notinsingleton
% 0.18/0.44 => ( eqinunit
% 0.18/0.44 => ( singletonsswitch
% 0.18/0.44 => ( upairsetE
% 0.18/0.44 => ( upairsetIL
% 0.18/0.44 => ( upairsetIR
% 0.18/0.44 => ( emptyE1
% 0.18/0.44 => ( vacuousDall
% 0.18/0.44 => ( quantDeMorgan1
% 0.18/0.44 => ( quantDeMorgan2
% 0.18/0.44 => ( quantDeMorgan3
% 0.18/0.44 => ( quantDeMorgan4
% 0.18/0.44 => ( prop2setI
% 0.18/0.44 => ( prop2set2propI
% 0.18/0.44 => ( notdexE
% 0.18/0.44 => ( notdallE
% 0.18/0.44 => ( exuI1
% 0.18/0.44 => ( exuI3
% 0.18/0.44 => ( exuI2
% 0.18/0.44 => ( inCongP
% 0.18/0.44 => ( in__Cong
% 0.18/0.44 => ( exuE3u
% 0.18/0.44 => ( exu__Cong
% 0.18/0.44 => ( emptyset__Cong
% 0.18/0.44 => ( setadjoin__Cong
% 0.18/0.44 => ( powerset__Cong
% 0.18/0.44 => ( setunion__Cong
% 0.18/0.44 => ( omega__Cong
% 0.18/0.44 => ( exuEu
% 0.18/0.44 => ( descr__Cong
% 0.18/0.44 => ( dsetconstr__Cong
% 0.18/0.44 => ( subsetI1
% 0.18/0.44 => ( eqimpsubset2
% 0.18/0.44 => ( eqimpsubset1
% 0.18/0.44 => ( subsetI2
% 0.18/0.44 => ( emptysetsubset
% 0.18/0.44 => ( subsetE
% 0.18/0.44 => ( subsetE2
% 0.18/0.44 => ( notsubsetI
% 0.18/0.44 => ( notequalI1
% 0.18/0.44 => ( notequalI2
% 0.18/0.44 => ( subsetRefl
% 0.18/0.44 => ( subsetTrans
% 0.18/0.44 => ( setadjoinSub
% 0.18/0.44 => ( setadjoinSub2
% 0.18/0.44 => ( subset2powerset
% 0.18/0.44 => ( setextsub
% 0.18/0.44 => ( subsetemptysetimpeq
% 0.18/0.44 => ( powersetI1
% 0.18/0.44 => ( powersetE1
% 0.18/0.44 => ( inPowerset
% 0.18/0.44 => ( powersetsubset
% 0.18/0.44 => ( sepInPowerset
% 0.18/0.44 => ( sepSubset
% 0.18/0.44 => ( binunionIL
% 0.18/0.44 => ( upairset2IR
% 0.18/0.44 => ( binunionIR
% 0.18/0.44 => ( binunionEcases
% 0.18/0.44 => ( binunionE
% 0.18/0.44 => ( binunionLsub
% 0.18/0.44 => ( binunionRsub
% 0.18/0.44 => ( binintersectI
% 0.18/0.44 => ( binintersectSubset5
% 0.18/0.44 => ( binintersectEL
% 0.18/0.44 => ( binintersectLsub
% 0.18/0.44 => ( binintersectSubset2
% 0.18/0.44 => ( binintersectSubset3
% 0.18/0.44 => ( binintersectER
% 0.18/0.44 => ( disjointsetsI1
% 0.18/0.44 => ( binintersectRsub
% 0.18/0.44 => ( binintersectSubset4
% 0.18/0.44 => ( binintersectSubset1
% 0.18/0.44 => ( bs114d
% 0.18/0.44 => ( setminusI
% 0.18/0.44 => ( setminusEL
% 0.18/0.44 => ( setminusER
% 0.18/0.44 => ( setminusSubset2
% 0.18/0.44 => ( setminusERneg
% 0.18/0.44 => ( setminusELneg
% 0.18/0.44 => ( setminusILneg
% 0.18/0.44 => ( setminusIRneg
% 0.18/0.44 => ( setminusLsub
% 0.18/0.44 => ( setminusSubset1
% 0.18/0.44 => ( symdiffE
% 0.18/0.44 => ( symdiffI1
% 0.18/0.44 => ( symdiffI2
% 0.18/0.44 => ( symdiffIneg1
% 0.18/0.44 => ( symdiffIneg2
% 0.18/0.44 => ( secondinupair
% 0.18/0.44 => ( setukpairIL
% 0.18/0.44 => ( setukpairIR
% 0.18/0.44 => ( kpairiskpair
% 0.18/0.44 => ( kpairp
% 0.18/0.44 => ( singletonsubset
% 0.18/0.44 => ( singletoninpowerset
% 0.18/0.44 => ( singletoninpowunion
% 0.18/0.44 => ( upairset2E
% 0.18/0.44 => ( upairsubunion
% 0.18/0.44 => ( upairinpowunion
% 0.18/0.44 => ( ubforcartprodlem1
% 0.18/0.44 => ( ubforcartprodlem2
% 0.18/0.44 => ( ubforcartprodlem3
% 0.18/0.44 => ( cartprodpairin
% 0.18/0.44 => ( cartprodmempair1
% 0.18/0.44 => ( cartprodmempair
% 0.18/0.44 => ( setunionE2
% 0.18/0.44 => ( setunionsingleton1
% 0.18/0.44 => ( setunionsingleton2
% 0.18/0.44 => ( setunionsingleton
% 0.18/0.44 => ( singletonprop
% 0.18/0.44 => ( ex1E1
% 0.18/0.44 => ( ex1I
% 0.18/0.44 => ( ex1I2
% 0.18/0.44 => ! [Xx: $i,Xy: $i] :
% 0.18/0.44 ( ( ( setadjoin @ Xx @ emptyset )
% 0.18/0.44 = ( setadjoin @ Xy @ emptyset ) )
% 0.18/0.44 => ( Xx = Xy ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.18/0.44
% 0.18/0.44 %------------------------------------------------------------------------------
% 0.18/0.44 ------------------------
% 2.28/1.45 ---- Embedded in THF ---
% 2.28/1.45 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.28/1.45 %%% Generated on Tue Feb 24 12:30:21 EST 2026
% 2.28/1.45 %%% using '$$dhol' embedding, version 1.3.0.
% 2.28/1.45 %%% Logic specification used:
% 2.28/1.45 %%% thf(spec, logic, $$dhol).
% 2.28/1.45
% 2.28/1.45 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.DxatUwJPum/DTF2DTF26966.p
% See solution above
% 2.28/1.46 ------------------------
% 2.28/1.47 ---- Cleaned THF ---
% 2.28/1.47 thf(in_type,type,
% 2.28/1.47 in: $i > $i > $o ).
% 2.28/1.47
% 2.28/1.47 thf(exu_type,type,
% 2.28/1.47 exu: ( $i > $o ) > $o ).
% 2.28/1.47
% 2.28/1.47 thf(exu,definition,
% 2.28/1.47 ( exu
% 2.28/1.47 = ( ^ [Xphi: $i > $o] :
% 2.28/1.47 ? [Xx: $i] :
% 2.28/1.47 ( ( Xphi @ Xx )
% 2.28/1.47 & ! [Xy: $i] :
% 2.28/1.47 ( ( Xphi @ Xy )
% 2.28/1.47 => ( Xx = Xy ) ) ) ) ) ).
% 2.28/1.47
% 2.28/1.47 thf(setextAx_type,type,
% 2.28/1.47 setextAx: $o ).
% 2.28/1.47
% 2.28/1.47 thf(setextAx,definition,
% 2.28/1.47 ( setextAx
% 2.28/1.47 = ( ! [A: $i,B: $i] :
% 2.28/1.47 ( ! [Xx: $i] :
% 2.28/1.47 ( ( in @ Xx @ A )
% 2.28/1.47 <=> ( in @ Xx @ B ) )
% 2.28/1.47 => ( A = B ) ) ) ) ).
% 2.28/1.47
% 2.28/1.47 thf(emptyset_type,type,
% 2.28/1.47 emptyset: $i ).
% 2.28/1.47
% 2.28/1.47 thf(emptysetAx_type,type,
% 2.28/1.47 emptysetAx: $o ).
% 2.28/1.47
% 2.28/1.47 thf(emptysetAx,definition,
% 2.28/1.47 ( emptysetAx
% 2.28/1.47 = ( ! [Xx: $i] :
% 2.28/1.47 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.28/1.47
% 2.28/1.47 thf(setadjoin_type,type,
% 2.28/1.47 setadjoin: $i > $i > $i ).
% 2.28/1.47
% 2.28/1.47 thf(setadjoinAx_type,type,
% 2.28/1.47 setadjoinAx: $o ).
% 2.28/1.47
% 2.28/1.47 thf(setadjoinAx,definition,
% 2.28/1.47 ( setadjoinAx
% 2.28/1.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.28/1.47 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.28/1.47 <=> ( ( Xy = Xx )
% 2.28/1.47 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.28/1.47
% 2.28/1.47 thf(powerset_type,type,
% 2.28/1.47 powerset: $i > $i ).
% 2.28/1.47
% 2.28/1.47 thf(powersetAx_type,type,
% 2.28/1.47 powersetAx: $o ).
% 2.28/1.47
% 2.28/1.47 thf(powersetAx,definition,
% 2.28/1.47 ( powersetAx
% 2.28/1.47 = ( ! [A: $i,B: $i] :
% 2.28/1.47 ( ( in @ B @ ( powerset @ A ) )
% 2.28/1.47 <=> ! [Xx: $i] :
% 2.28/1.47 ( ( in @ Xx @ B )
% 2.28/1.47 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.28/1.47
% 2.28/1.47 thf(setunion_type,type,
% 2.28/1.47 setunion: $i > $i ).
% 2.28/1.47
% 2.28/1.47 thf(setunionAx_type,type,
% 2.28/1.47 setunionAx: $o ).
% 2.28/1.47
% 2.28/1.47 thf(setunionAx,definition,
% 2.28/1.47 ( setunionAx
% 2.28/1.47 = ( ! [A: $i,Xx: $i] :
% 2.28/1.47 ( ( in @ Xx @ ( setunion @ A ) )
% 2.28/1.47 <=> ? [B: $i] :
% 2.28/1.47 ( ( in @ Xx @ B )
% 2.28/1.47 & ( in @ B @ A ) ) ) ) ) ).
% 2.28/1.47
% 2.28/1.47 thf(omega_type,type,
% 2.28/1.47 omega: $i ).
% 2.28/1.47
% 2.28/1.47 thf(omega0Ax_type,type,
% 2.28/1.47 omega0Ax: $o ).
% 2.28/1.47
% 2.28/1.47 thf(omega0Ax,definition,
% 2.28/1.47 ( omega0Ax
% 2.28/1.47 = ( in @ emptyset @ omega ) ) ).
% 2.28/1.47
% 2.28/1.47 thf(omegaSAx_type,type,
% 2.28/1.47 omegaSAx: $o ).
% 2.28/1.47
% 2.28/1.47 thf(omegaSAx,definition,
% 2.28/1.47 ( omegaSAx
% 2.28/1.47 = ( ! [Xx: $i] :
% 2.28/1.47 ( ( in @ Xx @ omega )
% 2.28/1.47 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 2.28/1.47
% 2.28/1.47 thf(omegaIndAx_type,type,
% 2.28/1.47 omegaIndAx: $o ).
% 2.28/1.47
% 2.28/1.47 thf(omegaIndAx,definition,
% 2.28/1.47 ( omegaIndAx
% 2.28/1.47 = ( ! [A: $i] :
% 2.28/1.47 ( ( ( in @ emptyset @ A )
% 2.28/1.47 & ! [Xx: $i] :
% 2.28/1.47 ( ( ( in @ Xx @ omega )
% 2.28/1.47 & ( in @ Xx @ A ) )
% 2.28/1.47 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 2.28/1.47 => ! [Xx: $i] :
% 2.28/1.47 ( ( in @ Xx @ omega )
% 2.28/1.47 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.28/1.47
% 2.28/1.47 thf(replAx_type,type,
% 2.28/1.47 replAx: $o ).
% 2.28/1.47
% 2.28/1.47 thf(replAx,definition,
% 2.28/1.47 ( replAx
% 2.28/1.47 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 2.28/1.47 ( ! [Xx: $i] :
% 2.28/1.47 ( ( in @ Xx @ A )
% 2.28/1.47 => ( exu
% 2.28/1.47 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 2.28/1.47 => ? [B: $i] :
% 2.28/1.47 ! [Xx: $i] :
% 2.28/1.47 ( ( in @ Xx @ B )
% 2.28/1.47 <=> ? [Xy: $i] :
% 2.28/1.47 ( ( in @ Xy @ A )
% 2.28/1.47 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 2.28/1.47
% 2.28/1.47 thf(foundationAx_type,type,
% 2.28/1.47 foundationAx: $o ).
% 2.28/1.47
% 2.28/1.47 thf(foundationAx,definition,
% 2.28/1.47 ( foundationAx
% 2.28/1.47 = ( ! [A: $i] :
% 2.28/1.47 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.28/1.47 => ? [B: $i] :
% 2.28/1.47 ( ( in @ B @ A )
% 2.28/1.47 & ~ ? [Xx: $i] :
% 2.28/1.47 ( ( in @ Xx @ B )
% 2.28/1.47 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 2.28/1.47
% 2.28/1.47 thf(wellorderingAx_type,type,
% 2.28/1.47 wellorderingAx: $o ).
% 2.28/1.47
% 2.28/1.47 thf(wellorderingAx,definition,
% 2.28/1.47 ( wellorderingAx
% 2.28/1.47 = ( ! [A: $i] :
% 2.28/1.47 ? [B: $i] :
% 2.28/1.47 ( ! [C: $i] :
% 2.28/1.47 ( ( in @ C @ B )
% 2.28/1.47 => ! [Xx: $i] :
% 2.28/1.47 ( ( in @ Xx @ C )
% 2.28/1.47 => ( in @ Xx @ A ) ) )
% 2.28/1.47 & ! [Xx: $i,Xy: $i] :
% 2.28/1.47 ( ( ( in @ Xx @ A )
% 2.28/1.47 & ( in @ Xy @ A ) )
% 2.28/1.47 => ( ! [C: $i] :
% 2.28/1.47 ( ( in @ C @ B )
% 2.28/1.47 => ( ( in @ Xx @ C )
% 2.28/1.47 <=> ( in @ Xy @ C ) ) )
% 2.28/1.47 => ( Xx = Xy ) ) )
% 2.28/1.47 & ! [C: $i,D: $i] :
% 2.28/1.47 ( ( ( in @ C @ B )
% 2.28/1.47 & ( in @ D @ B ) )
% 2.28/1.47 => ( ! [Xx: $i] :
% 2.28/1.47 ( ( in @ Xx @ C )
% 2.28/1.47 => ( in @ Xx @ D ) )
% 2.28/1.47 | ! [Xx: $i] :
% 2.28/1.47 ( ( in @ Xx @ D )
% 2.28/1.48 => ( in @ Xx @ C ) ) ) )
% 2.28/1.48 & ! [C: $i] :
% 2.28/1.48 ( ( ! [Xx: $i] :
% 2.28/1.48 ( ( in @ Xx @ C )
% 2.28/1.48 => ( in @ Xx @ A ) )
% 2.28/1.48 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 2.28/1.48 => ? [D: $i,Xx: $i] :
% 2.28/1.48 ( ( in @ D @ B )
% 2.28/1.48 & ( in @ Xx @ C )
% 2.28/1.48 & ~ ? [Xy: $i] :
% 2.28/1.48 ( ( in @ Xy @ D )
% 2.28/1.48 & ( in @ Xy @ C ) )
% 2.28/1.48 & ! [E: $i] :
% 2.28/1.48 ( ( in @ E @ B )
% 2.28/1.48 => ( ! [Xy: $i] :
% 2.28/1.48 ( ( in @ Xy @ E )
% 2.28/1.48 => ( in @ Xy @ D ) )
% 2.28/1.48 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(descr_type,type,
% 2.28/1.48 descr: ( $i > $o ) > $i ).
% 2.28/1.48
% 2.28/1.48 thf(descrp_type,type,
% 2.28/1.48 descrp: $o ).
% 2.28/1.48
% 2.28/1.48 thf(descrp,definition,
% 2.28/1.48 ( descrp
% 2.28/1.48 = ( ! [Xphi: $i > $o] :
% 2.28/1.48 ( ( exu
% 2.28/1.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.28/1.48 => ( Xphi
% 2.28/1.48 @ ( descr
% 2.28/1.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(dsetconstr_type,type,
% 2.28/1.48 dsetconstr: $i > ( $i > $o ) > $i ).
% 2.28/1.48
% 2.28/1.48 thf(dsetconstrI_type,type,
% 2.28/1.48 dsetconstrI: $o ).
% 2.28/1.48
% 2.28/1.48 thf(dsetconstrI,definition,
% 2.28/1.48 ( dsetconstrI
% 2.28/1.48 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.28/1.48 ( ( in @ Xx @ A )
% 2.28/1.48 => ( ( Xphi @ Xx )
% 2.28/1.48 => ( in @ Xx
% 2.28/1.48 @ ( dsetconstr @ A
% 2.28/1.48 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(dsetconstrEL_type,type,
% 2.28/1.48 dsetconstrEL: $o ).
% 2.28/1.48
% 2.28/1.48 thf(dsetconstrEL,definition,
% 2.28/1.48 ( dsetconstrEL
% 2.28/1.48 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.28/1.48 ( ( in @ Xx
% 2.28/1.48 @ ( dsetconstr @ A
% 2.28/1.48 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.28/1.48 => ( in @ Xx @ A ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(dsetconstrER_type,type,
% 2.28/1.48 dsetconstrER: $o ).
% 2.28/1.48
% 2.28/1.48 thf(dsetconstrER,definition,
% 2.28/1.48 ( dsetconstrER
% 2.28/1.48 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.28/1.48 ( ( in @ Xx
% 2.28/1.48 @ ( dsetconstr @ A
% 2.28/1.48 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.28/1.48 => ( Xphi @ Xx ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(exuE1_type,type,
% 2.28/1.48 exuE1: $o ).
% 2.28/1.48
% 2.28/1.48 thf(exuE1,definition,
% 2.28/1.48 ( exuE1
% 2.28/1.48 = ( ! [Xphi: $i > $o] :
% 2.28/1.48 ( ( exu
% 2.28/1.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.28/1.48 => ? [Xx: $i] :
% 2.28/1.48 ( ( Xphi @ Xx )
% 2.28/1.48 & ! [Xy: $i] :
% 2.28/1.48 ( ( Xphi @ Xy )
% 2.28/1.48 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(prop2set_type,type,
% 2.28/1.48 prop2set: $o > $i ).
% 2.28/1.48
% 2.28/1.48 thf(prop2setE_type,type,
% 2.28/1.48 prop2setE: $o ).
% 2.28/1.48
% 2.28/1.48 thf(prop2setE,definition,
% 2.28/1.48 ( prop2setE
% 2.28/1.48 = ( ! [Xphi: $o,Xx: $i] :
% 2.28/1.48 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 2.28/1.48 => Xphi ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(emptysetE_type,type,
% 2.28/1.48 emptysetE: $o ).
% 2.28/1.48
% 2.28/1.48 thf(emptysetE,definition,
% 2.28/1.48 ( emptysetE
% 2.28/1.48 = ( ! [Xx: $i] :
% 2.28/1.48 ( ( in @ Xx @ emptyset )
% 2.28/1.48 => ! [Xphi: $o] : Xphi ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(emptysetimpfalse_type,type,
% 2.28/1.48 emptysetimpfalse: $o ).
% 2.28/1.48
% 2.28/1.48 thf(emptysetimpfalse,definition,
% 2.28/1.48 ( emptysetimpfalse
% 2.28/1.48 = ( ! [Xx: $i] :
% 2.28/1.48 ( ( in @ Xx @ emptyset )
% 2.28/1.48 => $false ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(notinemptyset_type,type,
% 2.28/1.48 notinemptyset: $o ).
% 2.28/1.48
% 2.28/1.48 thf(notinemptyset,definition,
% 2.28/1.48 ( notinemptyset
% 2.28/1.48 = ( ! [Xx: $i] :
% 2.28/1.48 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(exuE3e_type,type,
% 2.28/1.48 exuE3e: $o ).
% 2.28/1.48
% 2.28/1.48 thf(exuE3e,definition,
% 2.28/1.48 ( exuE3e
% 2.28/1.48 = ( ! [Xphi: $i > $o] :
% 2.28/1.48 ( ( exu
% 2.28/1.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.28/1.48 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(setext_type,type,
% 2.28/1.48 setext: $o ).
% 2.28/1.48
% 2.28/1.48 thf(setext,definition,
% 2.28/1.48 ( setext
% 2.28/1.48 = ( ! [A: $i,B: $i] :
% 2.28/1.48 ( ! [Xx: $i] :
% 2.28/1.48 ( ( in @ Xx @ A )
% 2.28/1.48 => ( in @ Xx @ B ) )
% 2.28/1.48 => ( ! [Xx: $i] :
% 2.28/1.48 ( ( in @ Xx @ B )
% 2.28/1.48 => ( in @ Xx @ A ) )
% 2.28/1.48 => ( A = B ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(emptyI_type,type,
% 2.28/1.48 emptyI: $o ).
% 2.28/1.48
% 2.28/1.48 thf(emptyI,definition,
% 2.28/1.48 ( emptyI
% 2.28/1.48 = ( ! [A: $i] :
% 2.28/1.48 ( ! [Xx: $i] :
% 2.28/1.48 ~ ( in @ Xx @ A )
% 2.28/1.48 => ( A = emptyset ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(noeltsimpempty_type,type,
% 2.28/1.48 noeltsimpempty: $o ).
% 2.28/1.48
% 2.28/1.48 thf(noeltsimpempty,definition,
% 2.28/1.48 ( noeltsimpempty
% 2.28/1.48 = ( ! [A: $i] :
% 2.28/1.48 ( ! [Xx: $i] :
% 2.28/1.48 ~ ( in @ Xx @ A )
% 2.28/1.48 => ( A = emptyset ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(setbeta_type,type,
% 2.28/1.48 setbeta: $o ).
% 2.28/1.48
% 2.28/1.48 thf(setbeta,definition,
% 2.28/1.48 ( setbeta
% 2.28/1.48 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.28/1.48 ( ( in @ Xx @ A )
% 2.28/1.48 => ( ( in @ Xx
% 2.28/1.48 @ ( dsetconstr @ A
% 2.28/1.48 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.28/1.48 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(nonempty_type,type,
% 2.28/1.48 nonempty: $i > $o ).
% 2.28/1.48
% 2.28/1.48 thf(nonempty,definition,
% 2.28/1.48 ( nonempty
% 2.28/1.48 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(nonemptyE1_type,type,
% 2.28/1.48 nonemptyE1: $o ).
% 2.28/1.48
% 2.28/1.48 thf(nonemptyE1,definition,
% 2.28/1.48 ( nonemptyE1
% 2.28/1.48 = ( ! [A: $i] :
% 2.28/1.48 ( ( nonempty @ A )
% 2.28/1.48 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(nonemptyI_type,type,
% 2.28/1.48 nonemptyI: $o ).
% 2.28/1.48
% 2.28/1.48 thf(nonemptyI,definition,
% 2.28/1.48 ( nonemptyI
% 2.28/1.48 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.28/1.48 ( ( in @ Xx @ A )
% 2.28/1.48 => ( ( Xphi @ Xx )
% 2.28/1.48 => ( nonempty
% 2.28/1.48 @ ( dsetconstr @ A
% 2.28/1.48 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(nonemptyI1_type,type,
% 2.28/1.48 nonemptyI1: $o ).
% 2.28/1.48
% 2.28/1.48 thf(nonemptyI1,definition,
% 2.28/1.48 ( nonemptyI1
% 2.28/1.48 = ( ! [A: $i] :
% 2.28/1.48 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.28/1.48 => ( nonempty @ A ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(setadjoinIL_type,type,
% 2.28/1.48 setadjoinIL: $o ).
% 2.28/1.48
% 2.28/1.48 thf(setadjoinIL,definition,
% 2.28/1.48 ( setadjoinIL
% 2.28/1.48 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(emptyinunitempty_type,type,
% 2.28/1.48 emptyinunitempty: $o ).
% 2.28/1.48
% 2.28/1.48 thf(emptyinunitempty,definition,
% 2.28/1.48 ( emptyinunitempty
% 2.28/1.48 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(setadjoinIR_type,type,
% 2.28/1.48 setadjoinIR: $o ).
% 2.28/1.48
% 2.28/1.48 thf(setadjoinIR,definition,
% 2.28/1.48 ( setadjoinIR
% 2.28/1.48 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.28/1.48 ( ( in @ Xy @ A )
% 2.28/1.48 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(setadjoinE_type,type,
% 2.28/1.48 setadjoinE: $o ).
% 2.28/1.48
% 2.28/1.48 thf(setadjoinE,definition,
% 2.28/1.48 ( setadjoinE
% 2.28/1.48 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.28/1.48 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.28/1.48 => ! [Xphi: $o] :
% 2.28/1.48 ( ( ( Xy = Xx )
% 2.28/1.48 => Xphi )
% 2.28/1.48 => ( ( ( in @ Xy @ A )
% 2.28/1.48 => Xphi )
% 2.28/1.48 => Xphi ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(setadjoinOr_type,type,
% 2.28/1.48 setadjoinOr: $o ).
% 2.28/1.48
% 2.28/1.48 thf(setadjoinOr,definition,
% 2.28/1.48 ( setadjoinOr
% 2.28/1.48 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.28/1.48 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.28/1.48 => ( ( Xy = Xx )
% 2.28/1.48 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(setoftrueEq_type,type,
% 2.28/1.48 setoftrueEq: $o ).
% 2.28/1.48
% 2.28/1.48 thf(setoftrueEq,definition,
% 2.28/1.48 ( setoftrueEq
% 2.28/1.48 = ( ! [A: $i] :
% 2.28/1.48 ( ( dsetconstr @ A
% 2.28/1.48 @ ^ [Xx: $i] : $true )
% 2.28/1.48 = A ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(powersetI_type,type,
% 2.28/1.48 powersetI: $o ).
% 2.28/1.48
% 2.28/1.48 thf(powersetI,definition,
% 2.28/1.48 ( powersetI
% 2.28/1.48 = ( ! [A: $i,B: $i] :
% 2.28/1.48 ( ! [Xx: $i] :
% 2.28/1.48 ( ( in @ Xx @ B )
% 2.28/1.48 => ( in @ Xx @ A ) )
% 2.28/1.48 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(emptyinPowerset_type,type,
% 2.28/1.48 emptyinPowerset: $o ).
% 2.28/1.48
% 2.28/1.48 thf(emptyinPowerset,definition,
% 2.28/1.48 ( emptyinPowerset
% 2.28/1.48 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(emptyInPowerset_type,type,
% 2.28/1.48 emptyInPowerset: $o ).
% 2.28/1.48
% 2.28/1.48 thf(emptyInPowerset,definition,
% 2.28/1.48 ( emptyInPowerset
% 2.28/1.48 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(powersetE_type,type,
% 2.28/1.48 powersetE: $o ).
% 2.28/1.48
% 2.28/1.48 thf(powersetE,definition,
% 2.28/1.48 ( powersetE
% 2.28/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.28/1.48 ( ( in @ B @ ( powerset @ A ) )
% 2.28/1.48 => ( ( in @ Xx @ B )
% 2.28/1.48 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(setunionI_type,type,
% 2.28/1.48 setunionI: $o ).
% 2.28/1.48
% 2.28/1.48 thf(setunionI,definition,
% 2.28/1.48 ( setunionI
% 2.28/1.48 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.28/1.48 ( ( in @ Xx @ B )
% 2.28/1.48 => ( ( in @ B @ A )
% 2.28/1.48 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(setunionE_type,type,
% 2.28/1.48 setunionE: $o ).
% 2.28/1.48
% 2.28/1.48 thf(setunionE,definition,
% 2.28/1.48 ( setunionE
% 2.28/1.48 = ( ! [A: $i,Xx: $i] :
% 2.28/1.48 ( ( in @ Xx @ ( setunion @ A ) )
% 2.28/1.48 => ! [Xphi: $o] :
% 2.28/1.48 ( ! [B: $i] :
% 2.28/1.48 ( ( in @ Xx @ B )
% 2.28/1.48 => ( ( in @ B @ A )
% 2.28/1.48 => Xphi ) )
% 2.28/1.48 => Xphi ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(subPowSU_type,type,
% 2.28/1.48 subPowSU: $o ).
% 2.28/1.48
% 2.28/1.48 thf(subPowSU,definition,
% 2.28/1.48 ( subPowSU
% 2.28/1.48 = ( ! [A: $i,Xx: $i] :
% 2.28/1.48 ( ( in @ Xx @ A )
% 2.28/1.48 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(exuE2_type,type,
% 2.28/1.48 exuE2: $o ).
% 2.28/1.48
% 2.28/1.48 thf(exuE2,definition,
% 2.28/1.48 ( exuE2
% 2.28/1.48 = ( ! [Xphi: $i > $o] :
% 2.28/1.48 ( ( exu
% 2.28/1.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.28/1.48 => ? [Xx: $i] :
% 2.28/1.48 ! [Xy: $i] :
% 2.28/1.48 ( ( Xphi @ Xy )
% 2.28/1.48 <=> ( Xy = Xx ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(nonemptyImpWitness_type,type,
% 2.28/1.48 nonemptyImpWitness: $o ).
% 2.28/1.48
% 2.28/1.48 thf(nonemptyImpWitness,definition,
% 2.28/1.48 ( nonemptyImpWitness
% 2.28/1.48 = ( ! [A: $i] :
% 2.28/1.48 ( ( nonempty @ A )
% 2.28/1.48 => ? [Xx: $i] :
% 2.28/1.48 ( ( in @ Xx @ A )
% 2.28/1.48 & $true ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(uniqinunit_type,type,
% 2.28/1.48 uniqinunit: $o ).
% 2.28/1.48
% 2.28/1.48 thf(uniqinunit,definition,
% 2.28/1.48 ( uniqinunit
% 2.28/1.48 = ( ! [Xx: $i,Xy: $i] :
% 2.28/1.48 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.28/1.48 => ( Xx = Xy ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(notinsingleton_type,type,
% 2.28/1.48 notinsingleton: $o ).
% 2.28/1.48
% 2.28/1.48 thf(notinsingleton,definition,
% 2.28/1.48 ( notinsingleton
% 2.28/1.48 = ( ! [Xx: $i,Xy: $i] :
% 2.28/1.48 ( ( Xx != Xy )
% 2.28/1.48 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(eqinunit_type,type,
% 2.28/1.48 eqinunit: $o ).
% 2.28/1.48
% 2.28/1.48 thf(eqinunit,definition,
% 2.28/1.48 ( eqinunit
% 2.28/1.48 = ( ! [Xx: $i,Xy: $i] :
% 2.28/1.48 ( ( Xx = Xy )
% 2.28/1.48 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(singletonsswitch_type,type,
% 2.28/1.48 singletonsswitch: $o ).
% 2.28/1.48
% 2.28/1.48 thf(singletonsswitch,definition,
% 2.28/1.48 ( singletonsswitch
% 2.28/1.48 = ( ! [Xx: $i,Xy: $i] :
% 2.28/1.48 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.28/1.48 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(upairsetE_type,type,
% 2.28/1.48 upairsetE: $o ).
% 2.28/1.48
% 2.28/1.48 thf(upairsetE,definition,
% 2.28/1.48 ( upairsetE
% 2.28/1.48 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.28/1.48 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.28/1.48 => ( ( Xz = Xx )
% 2.28/1.48 | ( Xz = Xy ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(upairsetIL_type,type,
% 2.28/1.48 upairsetIL: $o ).
% 2.28/1.48
% 2.28/1.48 thf(upairsetIL,definition,
% 2.28/1.48 ( upairsetIL
% 2.28/1.48 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(upairsetIR_type,type,
% 2.28/1.48 upairsetIR: $o ).
% 2.28/1.48
% 2.28/1.48 thf(upairsetIR,definition,
% 2.28/1.48 ( upairsetIR
% 2.28/1.48 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(emptyE1_type,type,
% 2.28/1.48 emptyE1: $o ).
% 2.28/1.48
% 2.28/1.48 thf(emptyE1,definition,
% 2.28/1.48 ( emptyE1
% 2.28/1.48 = ( ! [A: $i,Xphi: $i > $o] :
% 2.28/1.48 ( ? [Xx: $i] :
% 2.28/1.48 ( ( in @ Xx @ A )
% 2.28/1.48 & ( Xphi @ Xx ) )
% 2.28/1.48 => ( ( ( dsetconstr @ A
% 2.28/1.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.28/1.48 = emptyset )
% 2.28/1.48 => $false ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(vacuousDall_type,type,
% 2.28/1.48 vacuousDall: $o ).
% 2.28/1.48
% 2.28/1.48 thf(vacuousDall,definition,
% 2.28/1.48 ( vacuousDall
% 2.28/1.48 = ( ! [Xphi: $i > $o,Xx: $i] :
% 2.28/1.48 ( ( in @ Xx @ emptyset )
% 2.28/1.48 => ( Xphi @ Xx ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(quantDeMorgan1_type,type,
% 2.28/1.48 quantDeMorgan1: $o ).
% 2.28/1.48
% 2.28/1.48 thf(quantDeMorgan1,definition,
% 2.28/1.48 ( quantDeMorgan1
% 2.28/1.48 = ( ! [A: $i,Xphi: $i > $o] :
% 2.28/1.48 ( ~ ! [Xx: $i] :
% 2.28/1.48 ( ( in @ Xx @ A )
% 2.28/1.48 => ( Xphi @ Xx ) )
% 2.28/1.48 => ? [Xx: $i] :
% 2.28/1.48 ( ( in @ Xx @ A )
% 2.28/1.48 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(quantDeMorgan2_type,type,
% 2.28/1.48 quantDeMorgan2: $o ).
% 2.28/1.48
% 2.28/1.48 thf(quantDeMorgan2,definition,
% 2.28/1.48 ( quantDeMorgan2
% 2.28/1.48 = ( ! [A: $i,Xphi: $i > $o] :
% 2.28/1.48 ( ! [Xx: $i] :
% 2.28/1.48 ( ( in @ Xx @ A )
% 2.28/1.48 => ~ ( Xphi @ Xx ) )
% 2.28/1.48 => ~ ? [Xx: $i] :
% 2.28/1.48 ( ( in @ Xx @ A )
% 2.28/1.48 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(quantDeMorgan3_type,type,
% 2.28/1.48 quantDeMorgan3: $o ).
% 2.28/1.48
% 2.28/1.48 thf(quantDeMorgan3,definition,
% 2.28/1.48 ( quantDeMorgan3
% 2.28/1.48 = ( ! [A: $i,Xphi: $i > $o] :
% 2.28/1.48 ( ~ ? [Xx: $i] :
% 2.28/1.48 ( ( in @ Xx @ A )
% 2.28/1.48 & ( Xphi @ Xx ) )
% 2.28/1.48 => ! [Xx: $i] :
% 2.28/1.48 ( ( in @ Xx @ A )
% 2.28/1.48 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(quantDeMorgan4_type,type,
% 2.28/1.48 quantDeMorgan4: $o ).
% 2.28/1.48
% 2.28/1.48 thf(quantDeMorgan4,definition,
% 2.28/1.48 ( quantDeMorgan4
% 2.28/1.48 = ( ! [A: $i,Xphi: $i > $o] :
% 2.28/1.48 ( ? [Xx: $i] :
% 2.28/1.48 ( ( in @ Xx @ A )
% 2.28/1.48 & ~ ( Xphi @ Xx ) )
% 2.28/1.48 => ~ ! [Xx: $i] :
% 2.28/1.48 ( ( in @ Xx @ A )
% 2.28/1.48 => ( Xphi @ Xx ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(prop2setI_type,type,
% 2.28/1.48 prop2setI: $o ).
% 2.28/1.48
% 2.28/1.48 thf(prop2setI,definition,
% 2.28/1.48 ( prop2setI
% 2.28/1.48 = ( ! [Xphi: $o] :
% 2.28/1.48 ( Xphi
% 2.28/1.48 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(set2prop_type,type,
% 2.28/1.48 set2prop: $i > $o ).
% 2.28/1.48
% 2.28/1.48 thf(prop2set2propI_type,type,
% 2.28/1.48 prop2set2propI: $o ).
% 2.28/1.48
% 2.28/1.48 thf(prop2set2propI,definition,
% 2.28/1.48 ( prop2set2propI
% 2.28/1.48 = ( ! [Xphi: $o] :
% 2.28/1.48 ( Xphi
% 2.28/1.48 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(notdexE_type,type,
% 2.28/1.48 notdexE: $o ).
% 2.28/1.48
% 2.28/1.48 thf(notdexE,definition,
% 2.28/1.48 ( notdexE
% 2.28/1.48 = ( ! [A: $i,Xphi: $i > $o] :
% 2.28/1.48 ( ~ ? [Xx: $i] :
% 2.28/1.48 ( ( in @ Xx @ A )
% 2.28/1.48 & ( Xphi @ Xx ) )
% 2.28/1.48 => ! [Xx: $i] :
% 2.28/1.48 ( ( in @ Xx @ A )
% 2.28/1.48 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(notdallE_type,type,
% 2.28/1.48 notdallE: $o ).
% 2.28/1.48
% 2.28/1.48 thf(notdallE,definition,
% 2.28/1.48 ( notdallE
% 2.28/1.48 = ( ! [A: $i,Xphi: $i > $o] :
% 2.28/1.48 ( ~ ! [Xx: $i] :
% 2.28/1.48 ( ( in @ Xx @ A )
% 2.28/1.48 => ( Xphi @ Xx ) )
% 2.28/1.48 => ? [Xx: $i] :
% 2.28/1.48 ( ( in @ Xx @ A )
% 2.28/1.48 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(exuI1_type,type,
% 2.28/1.48 exuI1: $o ).
% 2.28/1.48
% 2.28/1.48 thf(exuI1,definition,
% 2.28/1.48 ( exuI1
% 2.28/1.48 = ( ! [Xphi: $i > $o] :
% 2.28/1.48 ( ? [Xx: $i] :
% 2.28/1.48 ( ( Xphi @ Xx )
% 2.28/1.48 & ! [Xy: $i] :
% 2.28/1.48 ( ( Xphi @ Xy )
% 2.28/1.48 => ( Xx = Xy ) ) )
% 2.28/1.48 => ( exu
% 2.28/1.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(exuI3_type,type,
% 2.28/1.48 exuI3: $o ).
% 2.28/1.48
% 2.28/1.48 thf(exuI3,definition,
% 2.28/1.48 ( exuI3
% 2.28/1.48 = ( ! [Xphi: $i > $o] :
% 2.28/1.48 ( ? [Xx: $i] : ( Xphi @ Xx )
% 2.28/1.48 => ( ! [Xx: $i,Xy: $i] :
% 2.28/1.48 ( ( Xphi @ Xx )
% 2.28/1.48 => ( ( Xphi @ Xy )
% 2.28/1.48 => ( Xx = Xy ) ) )
% 2.28/1.48 => ( exu
% 2.28/1.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(exuI2_type,type,
% 2.28/1.48 exuI2: $o ).
% 2.28/1.48
% 2.28/1.48 thf(exuI2,definition,
% 2.28/1.48 ( exuI2
% 2.28/1.48 = ( ! [Xphi: $i > $o] :
% 2.28/1.48 ( ? [Xx: $i] :
% 2.28/1.48 ! [Xy: $i] :
% 2.28/1.48 ( ( Xphi @ Xy )
% 2.28/1.48 <=> ( Xy = Xx ) )
% 2.28/1.48 => ( exu
% 2.28/1.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(inCongP_type,type,
% 2.28/1.48 inCongP: $o ).
% 2.28/1.48
% 2.28/1.48 thf(inCongP,definition,
% 2.28/1.48 ( inCongP
% 2.28/1.48 = ( ! [A: $i,B: $i] :
% 2.28/1.48 ( ( A = B )
% 2.28/1.48 => ! [Xx: $i,Xy: $i] :
% 2.28/1.48 ( ( Xx = Xy )
% 2.28/1.48 => ( ( in @ Xx @ A )
% 2.28/1.48 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(in__Cong_type,type,
% 2.28/1.48 in__Cong: $o ).
% 2.28/1.48
% 2.28/1.48 thf(in__Cong,definition,
% 2.28/1.48 ( in__Cong
% 2.28/1.48 = ( ! [A: $i,B: $i] :
% 2.28/1.48 ( ( A = B )
% 2.28/1.48 => ! [Xx: $i,Xy: $i] :
% 2.28/1.48 ( ( Xx = Xy )
% 2.28/1.48 => ( ( in @ Xx @ A )
% 2.28/1.48 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(exuE3u_type,type,
% 2.28/1.48 exuE3u: $o ).
% 2.28/1.48
% 2.28/1.48 thf(exuE3u,definition,
% 2.28/1.48 ( exuE3u
% 2.28/1.48 = ( ! [Xphi: $i > $o] :
% 2.28/1.48 ( ( exu
% 2.28/1.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.28/1.48 => ! [Xx: $i,Xy: $i] :
% 2.28/1.48 ( ( Xphi @ Xx )
% 2.28/1.48 => ( ( Xphi @ Xy )
% 2.28/1.48 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(exu__Cong_type,type,
% 2.28/1.48 exu__Cong: $o ).
% 2.28/1.48
% 2.28/1.48 thf(exu__Cong,definition,
% 2.28/1.48 ( exu__Cong
% 2.28/1.48 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.28/1.48 ( ! [Xx: $i,Xy: $i] :
% 2.28/1.48 ( ( Xx = Xy )
% 2.28/1.48 => ( ( Xphi @ Xx )
% 2.28/1.48 <=> ( Xpsi @ Xy ) ) )
% 2.28/1.48 => ( ( exu
% 2.28/1.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.28/1.48 <=> ( exu
% 2.28/1.48 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(emptyset__Cong_type,type,
% 2.28/1.48 emptyset__Cong: $o ).
% 2.28/1.48
% 2.28/1.48 thf(emptyset__Cong,definition,
% 2.28/1.48 ( emptyset__Cong
% 2.28/1.48 = ( emptyset = emptyset ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(setadjoin__Cong_type,type,
% 2.28/1.48 setadjoin__Cong: $o ).
% 2.28/1.48
% 2.28/1.48 thf(setadjoin__Cong,definition,
% 2.28/1.48 ( setadjoin__Cong
% 2.28/1.48 = ( ! [Xx: $i,Xy: $i] :
% 2.28/1.48 ( ( Xx = Xy )
% 2.28/1.48 => ! [Xz: $i,Xu: $i] :
% 2.28/1.48 ( ( Xz = Xu )
% 2.28/1.48 => ( ( setadjoin @ Xx @ Xz )
% 2.28/1.48 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(powerset__Cong_type,type,
% 2.28/1.48 powerset__Cong: $o ).
% 2.28/1.48
% 2.28/1.48 thf(powerset__Cong,definition,
% 2.28/1.48 ( powerset__Cong
% 2.28/1.48 = ( ! [A: $i,B: $i] :
% 2.28/1.48 ( ( A = B )
% 2.28/1.48 => ( ( powerset @ A )
% 2.28/1.48 = ( powerset @ B ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(setunion__Cong_type,type,
% 2.28/1.48 setunion__Cong: $o ).
% 2.28/1.48
% 2.28/1.48 thf(setunion__Cong,definition,
% 2.28/1.48 ( setunion__Cong
% 2.28/1.48 = ( ! [A: $i,B: $i] :
% 2.28/1.48 ( ( A = B )
% 2.28/1.48 => ( ( setunion @ A )
% 2.28/1.48 = ( setunion @ B ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(omega__Cong_type,type,
% 2.28/1.48 omega__Cong: $o ).
% 2.28/1.48
% 2.28/1.48 thf(omega__Cong,definition,
% 2.28/1.48 ( omega__Cong
% 2.28/1.48 = ( omega = omega ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(exuEu_type,type,
% 2.28/1.48 exuEu: $o ).
% 2.28/1.48
% 2.28/1.48 thf(exuEu,definition,
% 2.28/1.48 ( exuEu
% 2.28/1.48 = ( ! [Xphi: $i > $o] :
% 2.28/1.48 ( ( exu
% 2.28/1.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.28/1.48 => ! [Xx: $i,Xy: $i] :
% 2.28/1.48 ( ( Xphi @ Xx )
% 2.28/1.48 => ( ( Xphi @ Xy )
% 2.28/1.48 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(descr__Cong_type,type,
% 2.28/1.48 descr__Cong: $o ).
% 2.28/1.48
% 2.28/1.48 thf(descr__Cong,definition,
% 2.28/1.48 ( descr__Cong
% 2.28/1.48 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.28/1.48 ( ! [Xx: $i,Xy: $i] :
% 2.28/1.48 ( ( Xx = Xy )
% 2.28/1.48 => ( ( Xphi @ Xx )
% 2.28/1.48 <=> ( Xpsi @ Xy ) ) )
% 2.28/1.48 => ( ( exu
% 2.28/1.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.28/1.48 => ( ( exu
% 2.28/1.48 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 2.28/1.48 => ( ( descr
% 2.28/1.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.28/1.48 = ( descr
% 2.28/1.48 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(dsetconstr__Cong_type,type,
% 2.28/1.48 dsetconstr__Cong: $o ).
% 2.28/1.48
% 2.28/1.48 thf(dsetconstr__Cong,definition,
% 2.28/1.48 ( dsetconstr__Cong
% 2.28/1.48 = ( ! [A: $i,B: $i] :
% 2.28/1.48 ( ( A = B )
% 2.28/1.48 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.28/1.48 ( ! [Xx: $i] :
% 2.28/1.48 ( ( in @ Xx @ A )
% 2.28/1.48 => ! [Xy: $i] :
% 2.28/1.48 ( ( in @ Xy @ B )
% 2.28/1.48 => ( ( Xx = Xy )
% 2.28/1.48 => ( ( Xphi @ Xx )
% 2.28/1.48 <=> ( Xpsi @ Xy ) ) ) ) )
% 2.28/1.48 => ( ( dsetconstr @ A
% 2.28/1.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.28/1.48 = ( dsetconstr @ B
% 2.28/1.48 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(subset_type,type,
% 2.28/1.48 subset: $i > $i > $o ).
% 2.28/1.48
% 2.28/1.48 thf(disjoint_type,type,
% 2.28/1.48 disjoint: $i > $i > $o ).
% 2.28/1.48
% 2.28/1.48 thf(setsmeet_type,type,
% 2.28/1.48 setsmeet: $i > $i > $o ).
% 2.28/1.48
% 2.28/1.48 thf(subsetI1_type,type,
% 2.28/1.48 subsetI1: $o ).
% 2.28/1.48
% 2.28/1.48 thf(subsetI1,definition,
% 2.28/1.48 ( subsetI1
% 2.28/1.48 = ( ! [A: $i,B: $i] :
% 2.28/1.48 ( ! [Xx: $i] :
% 2.28/1.48 ( ( in @ Xx @ A )
% 2.28/1.48 => ( in @ Xx @ B ) )
% 2.28/1.48 => ( subset @ A @ B ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(eqimpsubset2_type,type,
% 2.28/1.48 eqimpsubset2: $o ).
% 2.28/1.48
% 2.28/1.48 thf(eqimpsubset2,definition,
% 2.28/1.48 ( eqimpsubset2
% 2.28/1.48 = ( ! [A: $i,B: $i] :
% 2.28/1.48 ( ( A = B )
% 2.28/1.48 => ( subset @ B @ A ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(eqimpsubset1_type,type,
% 2.28/1.48 eqimpsubset1: $o ).
% 2.28/1.48
% 2.28/1.48 thf(eqimpsubset1,definition,
% 2.28/1.48 ( eqimpsubset1
% 2.28/1.48 = ( ! [A: $i,B: $i] :
% 2.28/1.48 ( ( A = B )
% 2.28/1.48 => ( subset @ A @ B ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(subsetI2_type,type,
% 2.28/1.48 subsetI2: $o ).
% 2.28/1.48
% 2.28/1.48 thf(subsetI2,definition,
% 2.28/1.48 ( subsetI2
% 2.28/1.48 = ( ! [A: $i,B: $i] :
% 2.28/1.48 ( ! [Xx: $i] :
% 2.28/1.48 ( ( in @ Xx @ A )
% 2.28/1.48 => ( in @ Xx @ B ) )
% 2.28/1.48 => ( subset @ A @ B ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(emptysetsubset_type,type,
% 2.28/1.48 emptysetsubset: $o ).
% 2.28/1.48
% 2.28/1.48 thf(emptysetsubset,definition,
% 2.28/1.48 ( emptysetsubset
% 2.28/1.48 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(subsetE_type,type,
% 2.28/1.48 subsetE: $o ).
% 2.28/1.48
% 2.28/1.48 thf(subsetE,definition,
% 2.28/1.48 ( subsetE
% 2.28/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.28/1.48 ( ( subset @ A @ B )
% 2.28/1.48 => ( ( in @ Xx @ A )
% 2.28/1.48 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(subsetE2_type,type,
% 2.28/1.48 subsetE2: $o ).
% 2.28/1.48
% 2.28/1.48 thf(subsetE2,definition,
% 2.28/1.48 ( subsetE2
% 2.28/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.28/1.48 ( ( subset @ A @ B )
% 2.28/1.48 => ( ~ ( in @ Xx @ B )
% 2.28/1.48 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(notsubsetI_type,type,
% 2.28/1.48 notsubsetI: $o ).
% 2.28/1.48
% 2.28/1.48 thf(notsubsetI,definition,
% 2.28/1.48 ( notsubsetI
% 2.28/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.28/1.48 ( ( in @ Xx @ A )
% 2.28/1.48 => ( ~ ( in @ Xx @ B )
% 2.28/1.48 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(notequalI1_type,type,
% 2.28/1.48 notequalI1: $o ).
% 2.28/1.48
% 2.28/1.48 thf(notequalI1,definition,
% 2.28/1.48 ( notequalI1
% 2.28/1.48 = ( ! [A: $i,B: $i] :
% 2.28/1.48 ( ~ ( subset @ A @ B )
% 2.28/1.48 => ( A != B ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(notequalI2_type,type,
% 2.28/1.48 notequalI2: $o ).
% 2.28/1.48
% 2.28/1.48 thf(notequalI2,definition,
% 2.28/1.48 ( notequalI2
% 2.28/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.28/1.48 ( ( in @ Xx @ A )
% 2.28/1.48 => ( ~ ( in @ Xx @ B )
% 2.28/1.48 => ( A != B ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(subsetRefl_type,type,
% 2.28/1.48 subsetRefl: $o ).
% 2.28/1.48
% 2.28/1.48 thf(subsetRefl,definition,
% 2.28/1.48 ( subsetRefl
% 2.28/1.48 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(subsetTrans_type,type,
% 2.28/1.48 subsetTrans: $o ).
% 2.28/1.48
% 2.28/1.48 thf(subsetTrans,definition,
% 2.28/1.48 ( subsetTrans
% 2.28/1.48 = ( ! [A: $i,B: $i,C: $i] :
% 2.28/1.48 ( ( subset @ A @ B )
% 2.28/1.48 => ( ( subset @ B @ C )
% 2.28/1.48 => ( subset @ A @ C ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(setadjoinSub_type,type,
% 2.28/1.48 setadjoinSub: $o ).
% 2.28/1.48
% 2.28/1.48 thf(setadjoinSub,definition,
% 2.28/1.48 ( setadjoinSub
% 2.28/1.48 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(setadjoinSub2_type,type,
% 2.28/1.48 setadjoinSub2: $o ).
% 2.28/1.48
% 2.28/1.48 thf(setadjoinSub2,definition,
% 2.28/1.48 ( setadjoinSub2
% 2.28/1.48 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.28/1.48 ( ( subset @ A @ B )
% 2.28/1.48 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(subset2powerset_type,type,
% 2.28/1.48 subset2powerset: $o ).
% 2.28/1.48
% 2.28/1.48 thf(subset2powerset,definition,
% 2.28/1.48 ( subset2powerset
% 2.28/1.48 = ( ! [A: $i,B: $i] :
% 2.28/1.48 ( ( subset @ A @ B )
% 2.28/1.48 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(setextsub_type,type,
% 2.28/1.48 setextsub: $o ).
% 2.28/1.48
% 2.28/1.48 thf(setextsub,definition,
% 2.28/1.48 ( setextsub
% 2.28/1.48 = ( ! [A: $i,B: $i] :
% 2.28/1.48 ( ( subset @ A @ B )
% 2.28/1.48 => ( ( subset @ B @ A )
% 2.28/1.48 => ( A = B ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(subsetemptysetimpeq_type,type,
% 2.28/1.48 subsetemptysetimpeq: $o ).
% 2.28/1.48
% 2.28/1.48 thf(subsetemptysetimpeq,definition,
% 2.28/1.48 ( subsetemptysetimpeq
% 2.28/1.48 = ( ! [A: $i] :
% 2.28/1.48 ( ( subset @ A @ emptyset )
% 2.28/1.48 => ( A = emptyset ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(powersetI1_type,type,
% 2.28/1.48 powersetI1: $o ).
% 2.28/1.48
% 2.28/1.48 thf(powersetI1,definition,
% 2.28/1.48 ( powersetI1
% 2.28/1.48 = ( ! [A: $i,B: $i] :
% 2.28/1.48 ( ( subset @ B @ A )
% 2.28/1.48 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(powersetE1_type,type,
% 2.28/1.48 powersetE1: $o ).
% 2.28/1.48
% 2.28/1.48 thf(powersetE1,definition,
% 2.28/1.48 ( powersetE1
% 2.28/1.48 = ( ! [A: $i,B: $i] :
% 2.28/1.48 ( ( in @ B @ ( powerset @ A ) )
% 2.28/1.48 => ( subset @ B @ A ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(inPowerset_type,type,
% 2.28/1.48 inPowerset: $o ).
% 2.28/1.48
% 2.28/1.48 thf(inPowerset,definition,
% 2.28/1.48 ( inPowerset
% 2.28/1.48 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(powersetsubset_type,type,
% 2.28/1.48 powersetsubset: $o ).
% 2.28/1.48
% 2.28/1.48 thf(powersetsubset,definition,
% 2.28/1.48 ( powersetsubset
% 2.28/1.48 = ( ! [A: $i,B: $i] :
% 2.28/1.48 ( ( subset @ A @ B )
% 2.28/1.48 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(sepInPowerset_type,type,
% 2.28/1.48 sepInPowerset: $o ).
% 2.28/1.48
% 2.28/1.48 thf(sepInPowerset,definition,
% 2.28/1.48 ( sepInPowerset
% 2.28/1.48 = ( ! [A: $i,Xphi: $i > $o] :
% 2.28/1.48 ( in
% 2.28/1.48 @ ( dsetconstr @ A
% 2.28/1.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.28/1.48 @ ( powerset @ A ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(sepSubset_type,type,
% 2.28/1.48 sepSubset: $o ).
% 2.28/1.48
% 2.28/1.48 thf(sepSubset,definition,
% 2.28/1.48 ( sepSubset
% 2.28/1.48 = ( ! [A: $i,Xphi: $i > $o] :
% 2.28/1.48 ( subset
% 2.28/1.48 @ ( dsetconstr @ A
% 2.28/1.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.28/1.48 @ A ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(binunion_type,type,
% 2.28/1.48 binunion: $i > $i > $i ).
% 2.28/1.48
% 2.28/1.48 thf(binunionIL_type,type,
% 2.28/1.48 binunionIL: $o ).
% 2.28/1.48
% 2.28/1.48 thf(binunionIL,definition,
% 2.28/1.48 ( binunionIL
% 2.28/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.28/1.48 ( ( in @ Xx @ A )
% 2.28/1.48 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(upairset2IR_type,type,
% 2.28/1.48 upairset2IR: $o ).
% 2.28/1.48
% 2.28/1.48 thf(upairset2IR,definition,
% 2.28/1.48 ( upairset2IR
% 2.28/1.48 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(binunionIR_type,type,
% 2.28/1.48 binunionIR: $o ).
% 2.28/1.48
% 2.28/1.48 thf(binunionIR,definition,
% 2.28/1.48 ( binunionIR
% 2.28/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.28/1.48 ( ( in @ Xx @ B )
% 2.28/1.48 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(binunionEcases_type,type,
% 2.28/1.48 binunionEcases: $o ).
% 2.28/1.48
% 2.28/1.48 thf(binunionEcases,definition,
% 2.28/1.48 ( binunionEcases
% 2.28/1.48 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 2.28/1.48 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.28/1.48 => ( ( ( in @ Xx @ A )
% 2.28/1.48 => Xphi )
% 2.28/1.48 => ( ( ( in @ Xx @ B )
% 2.28/1.48 => Xphi )
% 2.28/1.48 => Xphi ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(binunionE_type,type,
% 2.28/1.48 binunionE: $o ).
% 2.28/1.48
% 2.28/1.48 thf(binunionE,definition,
% 2.28/1.48 ( binunionE
% 2.28/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.28/1.48 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.28/1.48 => ( ( in @ Xx @ A )
% 2.28/1.48 | ( in @ Xx @ B ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(binunionLsub_type,type,
% 2.28/1.48 binunionLsub: $o ).
% 2.28/1.48
% 2.28/1.48 thf(binunionLsub,definition,
% 2.28/1.48 ( binunionLsub
% 2.28/1.48 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(binunionRsub_type,type,
% 2.28/1.48 binunionRsub: $o ).
% 2.28/1.48
% 2.28/1.48 thf(binunionRsub,definition,
% 2.28/1.48 ( binunionRsub
% 2.28/1.48 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(binintersect_type,type,
% 2.28/1.48 binintersect: $i > $i > $i ).
% 2.28/1.48
% 2.28/1.48 thf(binintersectI_type,type,
% 2.28/1.48 binintersectI: $o ).
% 2.28/1.48
% 2.28/1.48 thf(binintersectI,definition,
% 2.28/1.48 ( binintersectI
% 2.28/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.28/1.48 ( ( in @ Xx @ A )
% 2.28/1.48 => ( ( in @ Xx @ B )
% 2.28/1.48 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(binintersectSubset5_type,type,
% 2.28/1.48 binintersectSubset5: $o ).
% 2.28/1.48
% 2.28/1.48 thf(binintersectSubset5,definition,
% 2.28/1.48 ( binintersectSubset5
% 2.28/1.48 = ( ! [A: $i,B: $i,C: $i] :
% 2.28/1.48 ( ( subset @ C @ A )
% 2.28/1.48 => ( ( subset @ C @ B )
% 2.28/1.48 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(binintersectEL_type,type,
% 2.28/1.48 binintersectEL: $o ).
% 2.28/1.48
% 2.28/1.48 thf(binintersectEL,definition,
% 2.28/1.48 ( binintersectEL
% 2.28/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.28/1.48 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.28/1.48 => ( in @ Xx @ A ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(binintersectLsub_type,type,
% 2.28/1.48 binintersectLsub: $o ).
% 2.28/1.48
% 2.28/1.48 thf(binintersectLsub,definition,
% 2.28/1.48 ( binintersectLsub
% 2.28/1.48 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(binintersectSubset2_type,type,
% 2.28/1.48 binintersectSubset2: $o ).
% 2.28/1.48
% 2.28/1.48 thf(binintersectSubset2,definition,
% 2.28/1.48 ( binintersectSubset2
% 2.28/1.48 = ( ! [A: $i,B: $i] :
% 2.28/1.48 ( ( subset @ A @ B )
% 2.28/1.48 => ( ( binintersect @ A @ B )
% 2.28/1.48 = A ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(binintersectSubset3_type,type,
% 2.28/1.48 binintersectSubset3: $o ).
% 2.28/1.48
% 2.28/1.48 thf(binintersectSubset3,definition,
% 2.28/1.48 ( binintersectSubset3
% 2.28/1.48 = ( ! [A: $i,B: $i] :
% 2.28/1.48 ( ( ( binintersect @ A @ B )
% 2.28/1.48 = B )
% 2.28/1.48 => ( subset @ B @ A ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(binintersectER_type,type,
% 2.28/1.48 binintersectER: $o ).
% 2.28/1.48
% 2.28/1.48 thf(binintersectER,definition,
% 2.28/1.48 ( binintersectER
% 2.28/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.28/1.48 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.28/1.48 => ( in @ Xx @ B ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(disjointsetsI1_type,type,
% 2.28/1.48 disjointsetsI1: $o ).
% 2.28/1.48
% 2.28/1.48 thf(disjointsetsI1,definition,
% 2.28/1.48 ( disjointsetsI1
% 2.28/1.48 = ( ! [A: $i,B: $i] :
% 2.28/1.48 ( ~ ? [Xx: $i] :
% 2.28/1.48 ( ( in @ Xx @ A )
% 2.28/1.48 & ( in @ Xx @ B ) )
% 2.28/1.48 => ( ( binintersect @ A @ B )
% 2.28/1.48 = emptyset ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(binintersectRsub_type,type,
% 2.28/1.48 binintersectRsub: $o ).
% 2.28/1.48
% 2.28/1.48 thf(binintersectRsub,definition,
% 2.28/1.48 ( binintersectRsub
% 2.28/1.48 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(binintersectSubset4_type,type,
% 2.28/1.48 binintersectSubset4: $o ).
% 2.28/1.48
% 2.28/1.48 thf(binintersectSubset4,definition,
% 2.28/1.48 ( binintersectSubset4
% 2.28/1.48 = ( ! [A: $i,B: $i] :
% 2.28/1.48 ( ( subset @ B @ A )
% 2.28/1.48 => ( ( binintersect @ A @ B )
% 2.28/1.48 = B ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(binintersectSubset1_type,type,
% 2.28/1.48 binintersectSubset1: $o ).
% 2.28/1.48
% 2.28/1.48 thf(binintersectSubset1,definition,
% 2.28/1.48 ( binintersectSubset1
% 2.28/1.48 = ( ! [A: $i,B: $i] :
% 2.28/1.48 ( ( ( binintersect @ A @ B )
% 2.28/1.48 = A )
% 2.28/1.48 => ( subset @ A @ B ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(bs114d_type,type,
% 2.28/1.48 bs114d: $o ).
% 2.28/1.48
% 2.28/1.48 thf(bs114d,definition,
% 2.28/1.48 ( bs114d
% 2.28/1.48 = ( ! [A: $i,B: $i,C: $i] :
% 2.28/1.48 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 2.28/1.48 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(regular_type,type,
% 2.28/1.48 regular: $i > $o ).
% 2.28/1.48
% 2.28/1.48 thf(setminus_type,type,
% 2.28/1.48 setminus: $i > $i > $i ).
% 2.28/1.48
% 2.28/1.48 thf(setminusI_type,type,
% 2.28/1.48 setminusI: $o ).
% 2.28/1.48
% 2.28/1.48 thf(setminusI,definition,
% 2.28/1.48 ( setminusI
% 2.28/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.28/1.48 ( ( in @ Xx @ A )
% 2.28/1.48 => ( ~ ( in @ Xx @ B )
% 2.28/1.48 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(setminusEL_type,type,
% 2.28/1.48 setminusEL: $o ).
% 2.28/1.48
% 2.28/1.48 thf(setminusEL,definition,
% 2.28/1.48 ( setminusEL
% 2.28/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.28/1.48 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.28/1.48 => ( in @ Xx @ A ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(setminusER_type,type,
% 2.28/1.48 setminusER: $o ).
% 2.28/1.48
% 2.28/1.48 thf(setminusER,definition,
% 2.28/1.48 ( setminusER
% 2.28/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.28/1.48 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.28/1.48 => ~ ( in @ Xx @ B ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(setminusSubset2_type,type,
% 2.28/1.48 setminusSubset2: $o ).
% 2.28/1.48
% 2.28/1.48 thf(setminusSubset2,definition,
% 2.28/1.48 ( setminusSubset2
% 2.28/1.48 = ( ! [A: $i,B: $i] :
% 2.28/1.48 ( ( subset @ A @ B )
% 2.28/1.48 => ( ( setminus @ A @ B )
% 2.28/1.48 = emptyset ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(setminusERneg_type,type,
% 2.28/1.48 setminusERneg: $o ).
% 2.28/1.48
% 2.28/1.48 thf(setminusERneg,definition,
% 2.28/1.48 ( setminusERneg
% 2.28/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.28/1.48 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.28/1.48 => ( ( in @ Xx @ A )
% 2.28/1.48 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(setminusELneg_type,type,
% 2.28/1.48 setminusELneg: $o ).
% 2.28/1.48
% 2.28/1.48 thf(setminusELneg,definition,
% 2.28/1.48 ( setminusELneg
% 2.28/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.28/1.48 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.28/1.48 => ( ~ ( in @ Xx @ B )
% 2.28/1.48 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(setminusILneg_type,type,
% 2.28/1.48 setminusILneg: $o ).
% 2.28/1.48
% 2.28/1.48 thf(setminusILneg,definition,
% 2.28/1.48 ( setminusILneg
% 2.28/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.28/1.48 ( ~ ( in @ Xx @ A )
% 2.28/1.48 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(setminusIRneg_type,type,
% 2.28/1.48 setminusIRneg: $o ).
% 2.28/1.48
% 2.28/1.48 thf(setminusIRneg,definition,
% 2.28/1.48 ( setminusIRneg
% 2.28/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.28/1.48 ( ( in @ Xx @ B )
% 2.28/1.48 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(setminusLsub_type,type,
% 2.28/1.48 setminusLsub: $o ).
% 2.28/1.48
% 2.28/1.48 thf(setminusLsub,definition,
% 2.28/1.48 ( setminusLsub
% 2.28/1.48 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(setminusSubset1_type,type,
% 2.28/1.48 setminusSubset1: $o ).
% 2.28/1.48
% 2.28/1.48 thf(setminusSubset1,definition,
% 2.28/1.48 ( setminusSubset1
% 2.28/1.48 = ( ! [A: $i,B: $i] :
% 2.28/1.48 ( ( ( setminus @ A @ B )
% 2.28/1.48 = emptyset )
% 2.28/1.48 => ( subset @ A @ B ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(symdiff_type,type,
% 2.28/1.48 symdiff: $i > $i > $i ).
% 2.28/1.48
% 2.28/1.48 thf(symdiffE_type,type,
% 2.28/1.48 symdiffE: $o ).
% 2.28/1.48
% 2.28/1.48 thf(symdiffE,definition,
% 2.28/1.48 ( symdiffE
% 2.28/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.28/1.48 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 2.28/1.48 => ! [Xphi: $o] :
% 2.28/1.48 ( ( ( in @ Xx @ A )
% 2.28/1.48 => ( ~ ( in @ Xx @ B )
% 2.28/1.48 => Xphi ) )
% 2.28/1.48 => ( ( ~ ( in @ Xx @ A )
% 2.28/1.48 => ( ( in @ Xx @ B )
% 2.28/1.48 => Xphi ) )
% 2.28/1.48 => Xphi ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(symdiffI1_type,type,
% 2.28/1.48 symdiffI1: $o ).
% 2.28/1.48
% 2.28/1.48 thf(symdiffI1,definition,
% 2.28/1.48 ( symdiffI1
% 2.28/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.28/1.48 ( ( in @ Xx @ A )
% 2.28/1.48 => ( ~ ( in @ Xx @ B )
% 2.28/1.48 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(symdiffI2_type,type,
% 2.28/1.48 symdiffI2: $o ).
% 2.28/1.48
% 2.28/1.48 thf(symdiffI2,definition,
% 2.28/1.48 ( symdiffI2
% 2.28/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.28/1.48 ( ~ ( in @ Xx @ A )
% 2.28/1.48 => ( ( in @ Xx @ B )
% 2.28/1.48 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(symdiffIneg1_type,type,
% 2.28/1.48 symdiffIneg1: $o ).
% 2.28/1.48
% 2.28/1.48 thf(symdiffIneg1,definition,
% 2.28/1.48 ( symdiffIneg1
% 2.28/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.28/1.48 ( ( in @ Xx @ A )
% 2.28/1.48 => ( ( in @ Xx @ B )
% 2.28/1.48 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(symdiffIneg2_type,type,
% 2.28/1.48 symdiffIneg2: $o ).
% 2.28/1.48
% 2.28/1.48 thf(symdiffIneg2,definition,
% 2.28/1.48 ( symdiffIneg2
% 2.28/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.28/1.48 ( ~ ( in @ Xx @ A )
% 2.28/1.48 => ( ~ ( in @ Xx @ B )
% 2.28/1.48 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(iskpair_type,type,
% 2.28/1.48 iskpair: $i > $o ).
% 2.28/1.48
% 2.28/1.48 thf(iskpair,definition,
% 2.28/1.48 ( iskpair
% 2.28/1.48 = ( ^ [A: $i] :
% 2.28/1.48 ? [Xx: $i] :
% 2.28/1.48 ( ( in @ Xx @ ( setunion @ A ) )
% 2.28/1.48 & ? [Xy: $i] :
% 2.28/1.48 ( ( in @ Xy @ ( setunion @ A ) )
% 2.28/1.48 & ( A
% 2.28/1.48 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(secondinupair_type,type,
% 2.28/1.48 secondinupair: $o ).
% 2.28/1.48
% 2.28/1.48 thf(secondinupair,definition,
% 2.28/1.48 ( secondinupair
% 2.28/1.48 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(setukpairIL_type,type,
% 2.28/1.48 setukpairIL: $o ).
% 2.28/1.48
% 2.28/1.48 thf(setukpairIL,definition,
% 2.28/1.48 ( setukpairIL
% 2.28/1.48 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(setukpairIR_type,type,
% 2.28/1.48 setukpairIR: $o ).
% 2.28/1.48
% 2.28/1.48 thf(setukpairIR,definition,
% 2.28/1.48 ( setukpairIR
% 2.28/1.48 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(kpairiskpair_type,type,
% 2.28/1.48 kpairiskpair: $o ).
% 2.28/1.48
% 2.28/1.48 thf(kpairiskpair,definition,
% 2.28/1.48 ( kpairiskpair
% 2.28/1.48 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(kpair_type,type,
% 2.28/1.48 kpair: $i > $i > $i ).
% 2.28/1.48
% 2.28/1.48 thf(kpair,definition,
% 2.28/1.48 ( kpair
% 2.28/1.48 = ( ^ [Xx: $i,Xy: $i] : ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(kpairp_type,type,
% 2.28/1.48 kpairp: $o ).
% 2.28/1.48
% 2.28/1.48 thf(kpairp,definition,
% 2.28/1.48 ( kpairp
% 2.28/1.48 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(cartprod_type,type,
% 2.28/1.48 cartprod: $i > $i > $i ).
% 2.28/1.48
% 2.28/1.48 thf(singletonsubset_type,type,
% 2.28/1.48 singletonsubset: $o ).
% 2.28/1.48
% 2.28/1.48 thf(singletonsubset,definition,
% 2.28/1.48 ( singletonsubset
% 2.28/1.48 = ( ! [A: $i,Xx: $i] :
% 2.28/1.48 ( ( in @ Xx @ A )
% 2.28/1.48 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(singletoninpowerset_type,type,
% 2.28/1.48 singletoninpowerset: $o ).
% 2.28/1.48
% 2.28/1.48 thf(singletoninpowerset,definition,
% 2.28/1.48 ( singletoninpowerset
% 2.28/1.48 = ( ! [A: $i,Xx: $i] :
% 2.28/1.48 ( ( in @ Xx @ A )
% 2.28/1.48 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(singletoninpowunion_type,type,
% 2.28/1.48 singletoninpowunion: $o ).
% 2.28/1.48
% 2.28/1.48 thf(singletoninpowunion,definition,
% 2.28/1.48 ( singletoninpowunion
% 2.28/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.28/1.48 ( ( in @ Xx @ A )
% 2.28/1.48 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(upairset2E_type,type,
% 2.28/1.48 upairset2E: $o ).
% 2.28/1.48
% 2.28/1.48 thf(upairset2E,definition,
% 2.28/1.48 ( upairset2E
% 2.28/1.48 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.28/1.48 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.28/1.48 => ( ( Xz = Xx )
% 2.28/1.48 | ( Xz = Xy ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(upairsubunion_type,type,
% 2.28/1.48 upairsubunion: $o ).
% 2.28/1.48
% 2.28/1.48 thf(upairsubunion,definition,
% 2.28/1.48 ( upairsubunion
% 2.28/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.28/1.48 ( ( in @ Xx @ A )
% 2.28/1.48 => ! [Xy: $i] :
% 2.28/1.48 ( ( in @ Xy @ B )
% 2.28/1.48 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(upairinpowunion_type,type,
% 2.28/1.48 upairinpowunion: $o ).
% 2.28/1.48
% 2.28/1.48 thf(upairinpowunion,definition,
% 2.28/1.48 ( upairinpowunion
% 2.28/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.28/1.48 ( ( in @ Xx @ A )
% 2.28/1.48 => ! [Xy: $i] :
% 2.28/1.48 ( ( in @ Xy @ B )
% 2.28/1.48 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(ubforcartprodlem1_type,type,
% 2.28/1.48 ubforcartprodlem1: $o ).
% 2.28/1.48
% 2.28/1.48 thf(ubforcartprodlem1,definition,
% 2.28/1.48 ( ubforcartprodlem1
% 2.28/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.28/1.48 ( ( in @ Xx @ A )
% 2.28/1.48 => ! [Xy: $i] :
% 2.28/1.48 ( ( in @ Xy @ B )
% 2.28/1.48 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(ubforcartprodlem2_type,type,
% 2.28/1.48 ubforcartprodlem2: $o ).
% 2.28/1.48
% 2.28/1.48 thf(ubforcartprodlem2,definition,
% 2.28/1.48 ( ubforcartprodlem2
% 2.28/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.28/1.48 ( ( in @ Xx @ A )
% 2.28/1.48 => ! [Xy: $i] :
% 2.28/1.48 ( ( in @ Xy @ B )
% 2.28/1.48 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(ubforcartprodlem3_type,type,
% 2.28/1.48 ubforcartprodlem3: $o ).
% 2.28/1.48
% 2.28/1.48 thf(ubforcartprodlem3,definition,
% 2.28/1.48 ( ubforcartprodlem3
% 2.28/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.28/1.48 ( ( in @ Xx @ A )
% 2.28/1.48 => ! [Xy: $i] :
% 2.28/1.48 ( ( in @ Xy @ B )
% 2.28/1.48 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(cartprodpairin_type,type,
% 2.28/1.48 cartprodpairin: $o ).
% 2.28/1.48
% 2.28/1.48 thf(cartprodpairin,definition,
% 2.28/1.48 ( cartprodpairin
% 2.28/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.28/1.48 ( ( in @ Xx @ A )
% 2.28/1.48 => ! [Xy: $i] :
% 2.28/1.48 ( ( in @ Xy @ B )
% 2.28/1.48 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(cartprodmempair1_type,type,
% 2.28/1.48 cartprodmempair1: $o ).
% 2.28/1.48
% 2.28/1.48 thf(cartprodmempair1,definition,
% 2.28/1.48 ( cartprodmempair1
% 2.28/1.48 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.28/1.48 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.28/1.48 => ? [Xx: $i] :
% 2.28/1.48 ( ( in @ Xx @ A )
% 2.28/1.48 & ? [Xy: $i] :
% 2.28/1.48 ( ( in @ Xy @ B )
% 2.28/1.48 & ( Xu
% 2.28/1.48 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(cartprodmempair_type,type,
% 2.28/1.48 cartprodmempair: $o ).
% 2.28/1.48
% 2.28/1.48 thf(cartprodmempair,definition,
% 2.28/1.48 ( cartprodmempair
% 2.28/1.48 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.28/1.48 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.28/1.48 => ( iskpair @ Xu ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(setunionE2_type,type,
% 2.28/1.48 setunionE2: $o ).
% 2.28/1.48
% 2.28/1.48 thf(setunionE2,definition,
% 2.28/1.48 ( setunionE2
% 2.28/1.48 = ( ! [A: $i,Xx: $i] :
% 2.28/1.48 ( ( in @ Xx @ ( setunion @ A ) )
% 2.28/1.48 => ? [X: $i] :
% 2.28/1.48 ( ( in @ X @ A )
% 2.28/1.48 & ( in @ Xx @ X ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(setunionsingleton1_type,type,
% 2.28/1.48 setunionsingleton1: $o ).
% 2.28/1.48
% 2.28/1.48 thf(setunionsingleton1,definition,
% 2.28/1.48 ( setunionsingleton1
% 2.28/1.48 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(setunionsingleton2_type,type,
% 2.28/1.48 setunionsingleton2: $o ).
% 2.28/1.48
% 2.28/1.48 thf(setunionsingleton2,definition,
% 2.28/1.48 ( setunionsingleton2
% 2.28/1.48 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(setunionsingleton_type,type,
% 2.28/1.48 setunionsingleton: $o ).
% 2.28/1.48
% 2.28/1.48 thf(setunionsingleton,definition,
% 2.28/1.48 ( setunionsingleton
% 2.28/1.48 = ( ! [Xx: $i] :
% 2.28/1.48 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 2.28/1.48 = Xx ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(singleton_type,type,
% 2.28/1.48 singleton: $i > $o ).
% 2.28/1.48
% 2.28/1.48 thf(singleton,definition,
% 2.28/1.48 ( singleton
% 2.28/1.48 = ( ^ [A: $i] :
% 2.28/1.48 ? [Xx: $i] :
% 2.28/1.48 ( ( in @ Xx @ A )
% 2.28/1.48 & ( A
% 2.28/1.48 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(singletonprop_type,type,
% 2.28/1.48 singletonprop: $o ).
% 2.28/1.48
% 2.28/1.48 thf(singletonprop,definition,
% 2.28/1.48 ( singletonprop
% 2.28/1.48 = ( ! [A: $i,Xphi: $i > $o] :
% 2.28/1.48 ( ! [Xx: $i] :
% 2.28/1.48 ( ( in @ Xx @ A )
% 2.28/1.48 => ! [Xy: $i] :
% 2.28/1.48 ( ( in @ Xy @ A )
% 2.28/1.48 => ( ( Xphi @ Xx )
% 2.28/1.48 => ( ( Xphi @ Xy )
% 2.28/1.48 => ( Xx = Xy ) ) ) ) )
% 2.28/1.48 => ( ? [Xx: $i] :
% 2.28/1.48 ( ( in @ Xx @ A )
% 2.28/1.48 & ( Xphi @ Xx ) )
% 2.28/1.48 => ( singleton
% 2.28/1.48 @ ( dsetconstr @ A
% 2.28/1.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(ex1_type,type,
% 2.28/1.48 ex1: $i > ( $i > $o ) > $o ).
% 2.28/1.48
% 2.28/1.48 thf(ex1,definition,
% 2.28/1.48 ( ex1
% 2.28/1.48 = ( ^ [A: $i,Xphi: $i > $o] :
% 2.28/1.48 ( singleton
% 2.28/1.48 @ ( dsetconstr @ A
% 2.28/1.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(ex1E1_type,type,
% 2.28/1.48 ex1E1: $o ).
% 2.28/1.48
% 2.28/1.48 thf(ex1E1,definition,
% 2.28/1.48 ( ex1E1
% 2.28/1.48 = ( ! [A: $i,Xphi: $i > $o] :
% 2.28/1.48 ( ( ex1 @ A
% 2.28/1.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.28/1.48 => ? [Xx: $i] :
% 2.28/1.48 ( ( in @ Xx @ A )
% 2.28/1.48 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(ex1I_type,type,
% 2.28/1.48 ex1I: $o ).
% 2.28/1.48
% 2.28/1.48 thf(ex1I,definition,
% 2.28/1.48 ( ex1I
% 2.28/1.48 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.28/1.48 ( ( in @ Xx @ A )
% 2.28/1.48 => ( ( Xphi @ Xx )
% 2.28/1.48 => ( ! [Xy: $i] :
% 2.28/1.48 ( ( in @ Xy @ A )
% 2.28/1.48 => ( ( Xphi @ Xy )
% 2.28/1.48 => ( Xy = Xx ) ) )
% 2.28/1.48 => ( ex1 @ A
% 2.28/1.48 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(ex1I2_type,type,
% 2.28/1.48 ex1I2: $o ).
% 2.28/1.48
% 2.28/1.48 thf(ex1I2,definition,
% 2.28/1.48 ( ex1I2
% 2.28/1.48 = ( ! [A: $i,Xphi: $i > $o] :
% 2.28/1.48 ( ! [Xx: $i] :
% 2.28/1.48 ( ( in @ Xx @ A )
% 2.28/1.48 => ! [Xy: $i] :
% 2.28/1.48 ( ( in @ Xy @ A )
% 2.28/1.48 => ( ( Xphi @ Xx )
% 2.28/1.48 => ( ( Xphi @ Xy )
% 2.28/1.48 => ( Xx = Xy ) ) ) ) )
% 2.28/1.48 => ( ? [Xx: $i] :
% 2.28/1.48 ( ( in @ Xx @ A )
% 2.28/1.48 & ( Xphi @ Xx ) )
% 2.28/1.48 => ( ex1 @ A
% 2.28/1.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.28/1.48
% 2.28/1.48 thf(singletonsuniq,conjecture,
% 2.28/1.48 ( setextAx
% 2.28/1.48 => ( emptysetAx
% 2.28/1.48 => ( setadjoinAx
% 2.28/1.48 => ( powersetAx
% 2.28/1.48 => ( setunionAx
% 2.28/1.48 => ( omega0Ax
% 2.28/1.48 => ( omegaSAx
% 2.28/1.48 => ( omegaIndAx
% 2.28/1.48 => ( replAx
% 2.28/1.48 => ( foundationAx
% 2.28/1.48 => ( wellorderingAx
% 2.28/1.48 => ( descrp
% 2.28/1.48 => ( dsetconstrI
% 2.28/1.48 => ( dsetconstrEL
% 2.28/1.48 => ( dsetconstrER
% 2.28/1.48 => ( exuE1
% 2.28/1.48 => ( prop2setE
% 2.28/1.48 => ( emptysetE
% 2.28/1.48 => ( emptysetimpfalse
% 2.28/1.48 => ( notinemptyset
% 2.28/1.48 => ( exuE3e
% 2.28/1.48 => ( setext
% 2.28/1.48 => ( emptyI
% 2.28/1.48 => ( noeltsimpempty
% 2.28/1.48 => ( setbeta
% 2.28/1.48 => ( nonemptyE1
% 2.28/1.48 => ( nonemptyI
% 2.28/1.48 => ( nonemptyI1
% 2.28/1.48 => ( setadjoinIL
% 2.28/1.48 => ( emptyinunitempty
% 2.28/1.48 => ( setadjoinIR
% 2.28/1.48 => ( setadjoinE
% 2.28/1.48 => ( setadjoinOr
% 2.28/1.48 => ( setoftrueEq
% 2.28/1.48 => ( powersetI
% 2.28/1.48 => ( emptyinPowerset
% 2.28/1.48 => ( emptyInPowerset
% 2.28/1.48 => ( powersetE
% 2.28/1.48 => ( setunionI
% 2.28/1.48 => ( setunionE
% 2.28/1.48 => ( subPowSU
% 2.28/1.48 => ( exuE2
% 2.28/1.48 => ( nonemptyImpWitness
% 2.28/1.48 => ( uniqinunit
% 2.28/1.48 => ( notinsingleton
% 2.28/1.48 => ( eqinunit
% 2.28/1.48 => ( singletonsswitch
% 2.28/1.48 => ( upairsetE
% 2.28/1.48 => ( upairsetIL
% 2.28/1.48 => ( upairsetIR
% 2.28/1.48 => ( emptyE1
% 2.28/1.48 => ( vacuousDall
% 2.28/1.48 => ( quantDeMorgan1
% 2.28/1.48 => ( quantDeMorgan2
% 2.28/1.48 => ( quantDeMorgan3
% 2.28/1.48 => ( quantDeMorgan4
% 2.28/1.48 => ( prop2setI
% 2.28/1.48 => ( prop2set2propI
% 2.28/1.48 => ( notdexE
% 2.28/1.48 => ( notdallE
% 2.28/1.48 => ( exuI1
% 2.28/1.48 => ( exuI3
% 2.28/1.48 => ( exuI2
% 2.28/1.48 => ( inCongP
% 2.28/1.48 => ( in__Cong
% 2.28/1.48 => ( exuE3u
% 2.28/1.48 => ( exu__Cong
% 2.28/1.48 => ( emptyset__Cong
% 2.28/1.48 => ( setadjoin__Cong
% 2.28/1.48 => ( powerset__Cong
% 2.28/1.48 => ( setunion__Cong
% 2.28/1.48 => ( omega__Cong
% 2.28/1.48 => ( exuEu
% 2.28/1.48 => ( descr__Cong
% 2.28/1.48 => ( dsetconstr__Cong
% 2.28/1.48 => ( subsetI1
% 2.28/1.48 => ( eqimpsubset2
% 2.28/1.48 => ( eqimpsubset1
% 2.28/1.48 => ( subsetI2
% 2.28/1.48 => ( emptysetsubset
% 2.28/1.48 => ( subsetE
% 2.28/1.48 => ( subsetE2
% 2.28/1.48 => ( notsubsetI
% 2.28/1.48 => ( notequalI1
% 2.28/1.48 => ( notequalI2
% 2.28/1.48 => ( subsetRefl
% 2.28/1.48 => ( subsetTrans
% 2.28/1.48 => ( setadjoinSub
% 2.28/1.48 => ( setadjoinSub2
% 2.28/1.48 => ( subset2powerset
% 2.28/1.48 => ( setextsub
% 2.28/1.48 => ( subsetemptysetimpeq
% 2.28/1.48 => ( powersetI1
% 2.28/1.48 => ( powersetE1
% 2.28/1.48 => ( inPowerset
% 2.28/1.48 => ( powersetsubset
% 2.28/1.48 => ( sepInPowerset
% 2.28/1.48 => ( sepSubset
% 2.28/1.48 => ( binunionIL
% 2.28/1.48 => ( upairset2IR
% 2.28/1.48 => ( binunionIR
% 2.28/1.48 => ( binunionEcases
% 2.28/1.48 => ( binunionE
% 2.28/1.48 => ( binunionLsub
% 2.28/1.48 => ( binunionRsub
% 2.28/1.48 => ( binintersectI
% 2.28/1.48 => ( binintersectSubset5
% 2.28/1.48 => ( binintersectEL
% 2.28/1.48 => ( binintersectLsub
% 2.28/1.48 => ( binintersectSubset2
% 2.28/1.48 => ( binintersectSubset3
% 2.28/1.48 => ( binintersectER
% 2.28/1.48 => ( disjointsetsI1
% 2.28/1.48 => ( binintersectRsub
% 2.28/1.48 => ( binintersectSubset4
% 2.28/1.48 => ( binintersectSubset1
% 2.28/1.48 => ( bs114d
% 2.28/1.48 => ( setminusI
% 2.28/1.48 => ( setminusEL
% 2.28/1.48 => ( setminusER
% 2.28/1.48 => ( setminusSubset2
% 2.28/1.48 => ( setminusERneg
% 2.28/1.48 => ( setminusELneg
% 2.28/1.48 => ( setminusILneg
% 2.28/1.48 => ( setminusIRneg
% 2.28/1.48 => ( setminusLsub
% 2.28/1.48 => ( setminusSubset1
% 2.28/1.48 => ( symdiffE
% 2.28/1.48 => ( symdiffI1
% 2.28/1.48 => ( symdiffI2
% 2.28/1.48 => ( symdiffIneg1
% 2.28/1.48 => ( symdiffIneg2
% 2.28/1.48 => ( secondinupair
% 2.28/1.48 => ( setukpairIL
% 2.28/1.48 => ( setukpairIR
% 2.28/1.48 => ( kpairiskpair
% 2.28/1.48 => ( kpairp
% 2.28/1.48 => ( singletonsubset
% 2.28/1.48 => ( singletoninpowerset
% 2.28/1.48 => ( singletoninpowunion
% 2.28/1.48 => ( upairset2E
% 2.28/1.48 => ( upairsubunion
% 2.28/1.48 => ( upairinpowunion
% 2.28/1.48 => ( ubforcartprodlem1
% 2.28/1.48 => ( ubforcartprodlem2
% 2.28/1.48 => ( ubforcartprodlem3
% 2.28/1.48 => ( cartprodpairin
% 2.28/1.48 => ( cartprodmempair1
% 2.28/1.48 => ( cartprodmempair
% 2.28/1.48 => ( setunionE2
% 2.28/1.48 => ( setunionsingleton1
% 2.28/1.48 => ( setunionsingleton2
% 2.28/1.48 => ( setunionsingleton
% 2.28/1.48 => ( singletonprop
% 2.28/1.48 => ( ex1E1
% 2.28/1.48 => ( ex1I
% 2.28/1.48 => ( ex1I2
% 2.28/1.48 => ! [Xx: $i,Xy: $i] :
% 2.28/1.48 ( ( ( setadjoin @ Xx @ emptyset )
% 2.28/1.48 = ( setadjoin @ Xy @ emptyset ) )
% 2.28/1.48 => ( Xx = Xy ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.28/1.48 ------------------------
% 2.28/1.49 % (27067)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_26966 for (2999ds/183Mi)
% 2.40/1.50 % (27071)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_26966 for (2999ds/2Mi)
% 2.40/1.50 % (27070)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_26966 for (2999ds/2Mi)
% 2.40/1.50 % (27068)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_26966 for (2999ds/4Mi)
% 2.40/1.50 % (27069)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_26966 for (2999ds/27Mi)
% 2.40/1.50 % (27072)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_26966 for (2999ds/275Mi)
% 2.40/1.50 % (27073)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_26966 for (2999ds/18Mi)
% 2.40/1.50 % (27071)Instruction limit reached!
% 2.40/1.50 % (27071)------------------------------
% 2.40/1.50 % (27071)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.40/1.50 % (27071)Termination reason: Unknown
% 2.40/1.50 % (27071)Termination phase: shuffling
% 2.40/1.50
% 2.40/1.50 % (27071)Memory used [KB]: 1279
% 2.40/1.50 % (27071)Time elapsed: 0.002 s
% 2.40/1.50 % (27071)Instructions burned: 2 (million)
% 2.40/1.50 % (27071)------------------------------
% 2.40/1.50 % (27071)------------------------------
% 2.40/1.50 % (27070)Instruction limit reached!
% 2.40/1.50 % (27070)------------------------------
% 2.40/1.50 % (27070)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.40/1.50 % (27070)Termination reason: Unknown
% 2.40/1.50 % (27070)Termination phase: shuffling
% 2.40/1.50
% 2.40/1.50 % (27070)Memory used [KB]: 1279
% 2.40/1.50 % (27070)Time elapsed: 0.003 s
% 2.40/1.50 % (27070)Instructions burned: 3 (million)
% 2.40/1.50 % (27070)------------------------------
% 2.40/1.50 % (27070)------------------------------
% 2.40/1.50 % (27068)Instruction limit reached!
% 2.40/1.50 % (27068)------------------------------
% 2.40/1.50 % (27068)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.40/1.50 % (27068)Termination reason: Unknown
% 2.40/1.50 % (27068)Termination phase: shuffling
% 2.40/1.50
% 2.40/1.50 % (27068)Memory used [KB]: 1279
% 2.40/1.50 % (27068)Time elapsed: 0.004 s
% 2.40/1.50 % (27068)Instructions burned: 5 (million)
% 2.40/1.50 % (27068)------------------------------
% 2.40/1.50 % (27068)------------------------------
% 2.40/1.51 % (27073)Instruction limit reached!
% 2.40/1.51 % (27073)------------------------------
% 2.40/1.51 % (27073)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.40/1.51 % (27073)Termination reason: Unknown
% 2.40/1.51 % (27073)Termination phase: Property scanning
% 2.40/1.51
% 2.40/1.51 % (27073)Memory used [KB]: 1535
% 2.40/1.51 % (27073)Time elapsed: 0.011 s
% 2.40/1.51 % (27073)Instructions burned: 19 (million)
% 2.40/1.51 % (27073)------------------------------
% 2.40/1.51 % (27073)------------------------------
% 2.40/1.51 % (27069)Instruction limit reached!
% 2.40/1.51 % (27069)------------------------------
% 2.40/1.51 % (27069)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.40/1.51 % (27069)Termination reason: Unknown
% 2.40/1.51 % (27069)Termination phase: Property scanning
% 2.40/1.51
% 2.40/1.51 % (27069)Memory used [KB]: 1791
% 2.40/1.51 % (27069)Time elapsed: 0.014 s
% 2.40/1.51 % (27069)Instructions burned: 27 (million)
% 2.40/1.51 % (27069)------------------------------
% 2.40/1.51 % (27069)------------------------------
% 2.40/1.52 % (27074)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_26966 for (2999ds/3Mi)
% 2.40/1.52 % (27075)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_26966 for (2999ds/37Mi)
% 2.40/1.52 % (27074)Instruction limit reached!
% 2.40/1.52 % (27074)------------------------------
% 2.40/1.52 % (27074)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.40/1.52 % (27074)Termination reason: Unknown
% 2.40/1.52 % (27074)Termination phase: shuffling
% 2.40/1.52
% 2.40/1.52 % (27074)Memory used [KB]: 1279
% 2.40/1.52 % (27074)Time elapsed: 0.003 s
% 2.40/1.52 % (27074)Instructions burned: 3 (million)
% 2.40/1.52 % (27076)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_26966 for (2999ds/15Mi)
% 2.40/1.52 % (27074)------------------------------
% 2.40/1.52 % (27074)------------------------------
% 2.40/1.53 % (27076)Instruction limit reached!
% 2.40/1.53 % (27076)------------------------------
% 2.40/1.53 % (27076)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.40/1.53 % (27076)Termination reason: Unknown
% 2.40/1.53 % (27076)Termination phase: Property scanning
% 2.40/1.53
% 2.40/1.53 % (27076)Memory used [KB]: 1535
% 2.40/1.53 % (27076)Time elapsed: 0.009 s
% 2.40/1.53 % (27076)Instructions burned: 16 (million)
% 2.40/1.53 % (27076)------------------------------
% 2.40/1.53 % (27076)------------------------------
% 2.40/1.53 % (27077)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_26966 for (2999ds/3Mi)
% 2.40/1.53 % (27077)Instruction limit reached!
% 2.40/1.53 % (27077)------------------------------
% 2.40/1.53 % (27077)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.40/1.53 % (27077)Termination reason: Unknown
% 2.40/1.53 % (27077)Termination phase: shuffling
% 2.40/1.53
% 2.40/1.53 % (27077)Memory used [KB]: 1279
% 2.40/1.53 % (27077)Time elapsed: 0.003 s
% 2.40/1.53 % (27077)Instructions burned: 4 (million)
% 2.40/1.53 % (27077)------------------------------
% 2.40/1.53 % (27077)------------------------------
% 2.40/1.53 % (27078)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_26966 for (2999ds/1041Mi)
% 2.40/1.53 % (27075)Instruction limit reached!
% 2.40/1.53 % (27075)------------------------------
% 2.40/1.53 % (27075)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.40/1.53 % (27075)Termination reason: Unknown
% 2.40/1.53 % (27075)Termination phase: Property scanning
% 2.40/1.53
% 2.40/1.53 % (27075)Memory used [KB]: 1791
% 2.40/1.53 % (27075)Time elapsed: 0.018 s
% 2.40/1.53 % (27075)Instructions burned: 38 (million)
% 2.40/1.53 % (27075)------------------------------
% 2.40/1.53 % (27075)------------------------------
% 2.40/1.54 % (27079)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_26966 for (2999ds/7Mi)
% 2.40/1.54 % (27079)Instruction limit reached!
% 2.40/1.54 % (27079)------------------------------
% 2.40/1.54 % (27079)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.40/1.54 % (27079)Termination reason: Unknown
% 2.40/1.54 % (27079)Termination phase: shuffling
% 2.40/1.54
% 2.40/1.54 % (27079)Memory used [KB]: 1407
% 2.40/1.54 % (27079)Time elapsed: 0.005 s
% 2.40/1.54 % (27079)Instructions burned: 8 (million)
% 2.40/1.54 % (27079)------------------------------
% 2.40/1.54 % (27079)------------------------------
% 2.40/1.54 % (27080)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_26966 for (2999ds/16Mi)
% 2.40/1.54 % (27081)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_26966 for (2999ds/3Mi)
% 2.40/1.54 % (27067)Instruction limit reached!
% 2.40/1.54 % (27067)------------------------------
% 2.40/1.54 % (27067)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.40/1.54 % (27067)Termination reason: Unknown
% 2.40/1.54 % (27067)Termination phase: Saturation
% 2.40/1.54
% 2.40/1.54 % (27067)Memory used [KB]: 7419
% 2.40/1.54 % (27067)Time elapsed: 0.052 s
% 2.40/1.54 % (27067)Instructions burned: 185 (million)
% 2.40/1.54 % (27067)------------------------------
% 2.40/1.54 % (27067)------------------------------
% 2.40/1.55 % (27081)Instruction limit reached!
% 2.40/1.55 % (27081)------------------------------
% 2.40/1.55 % (27081)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.40/1.55 % (27081)Termination reason: Unknown
% 2.40/1.55 % (27081)Termination phase: shuffling
% 2.40/1.55
% 2.40/1.55 % (27081)Memory used [KB]: 1279
% 2.40/1.55 % (27081)Time elapsed: 0.003 s
% 2.40/1.55 % (27081)Instructions burned: 4 (million)
% 2.40/1.55 % (27081)------------------------------
% 2.40/1.55 % (27081)------------------------------
% 2.40/1.55 % (27080)Instruction limit reached!
% 2.40/1.55 % (27080)------------------------------
% 2.40/1.55 % (27080)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.40/1.55 % (27080)Termination reason: Unknown
% 2.40/1.55 % (27080)Termination phase: shuffling
% 2.40/1.55
% 2.40/1.55 % (27080)Memory used [KB]: 1535
% 2.40/1.55 % (27080)Time elapsed: 0.009 s
% 2.40/1.55 % (27080)Instructions burned: 16 (million)
% 2.40/1.55 % (27080)------------------------------
% 2.40/1.55 % (27080)------------------------------
% 2.40/1.55 % (27082)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_26966 for (2999ds/3Mi)
% 2.40/1.55 % (27082)Instruction limit reached!
% 2.40/1.55 % (27082)------------------------------
% 2.40/1.55 % (27082)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.40/1.55 % (27082)Termination reason: Unknown
% 2.40/1.55 % (27082)Termination phase: shuffling
% 2.40/1.55
% 2.40/1.55 % (27082)Memory used [KB]: 1279
% 2.40/1.55 % (27082)Time elapsed: 0.003 s
% 2.40/1.55 % (27082)Instructions burned: 3 (million)
% 2.40/1.55 % (27082)------------------------------
% 2.40/1.55 % (27082)------------------------------
% 2.40/1.55 % (27084)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_26966 for (2999ds/3Mi)
% 2.40/1.56 % (27084)Instruction limit reached!
% 2.40/1.56 % (27084)------------------------------
% 2.40/1.56 % (27084)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.40/1.56 % (27084)Termination reason: Unknown
% 2.40/1.56 % (27084)Termination phase: shuffling
% 2.40/1.56
% 2.40/1.56 % (27084)Memory used [KB]: 1279
% 2.40/1.56 % (27084)Time elapsed: 0.002 s
% 2.40/1.56 % (27084)Instructions burned: 3 (million)
% 2.40/1.56 % (27084)------------------------------
% 2.40/1.56 % (27084)------------------------------
% 2.40/1.56 % (27083)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_26966 for (2999ds/7Mi)
% 2.40/1.56 % (27083)Instruction limit reached!
% 2.40/1.56 % (27083)------------------------------
% 2.40/1.56 % (27083)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.40/1.56 % (27083)Termination reason: Unknown
% 2.40/1.56 % (27083)Termination phase: shuffling
% 2.40/1.56
% 2.40/1.56 % (27083)Memory used [KB]: 1407
% 2.40/1.56 % (27083)Time elapsed: 0.005 s
% 2.40/1.56 % (27083)Instructions burned: 8 (million)
% 2.40/1.56 % (27083)------------------------------
% 2.40/1.56 % (27083)------------------------------
% 2.40/1.56 % (27087)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_26966 for (2999ds/710Mi)
% 2.40/1.57 % (27086)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_26966 for (2999ds/18Mi)
% 2.40/1.57 % (27085)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_26966 for (2999ds/4Mi)
% 2.40/1.57 % (27085)Instruction limit reached!
% 2.40/1.57 % (27085)------------------------------
% 2.40/1.57 % (27085)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.40/1.57 % (27085)Termination reason: Unknown
% 2.40/1.57 % (27085)Termination phase: shuffling
% 2.40/1.57
% 2.40/1.57 % (27085)Memory used [KB]: 1279
% 2.40/1.57 % (27085)Time elapsed: 0.003 s
% 2.40/1.57 % (27085)Instructions burned: 4 (million)
% 2.40/1.57 % (27085)------------------------------
% 2.40/1.57 % (27085)------------------------------
% 2.40/1.57 % (27088)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_26966 for (2999ds/6Mi)
% 2.40/1.57 % (27088)Instruction limit reached!
% 2.40/1.57 % (27088)------------------------------
% 2.40/1.57 % (27088)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.40/1.57 % (27088)Termination reason: Unknown
% 2.40/1.57 % (27088)Termination phase: shuffling
% 2.40/1.57
% 2.40/1.57 % (27088)Memory used [KB]: 1279
% 2.40/1.57 % (27088)Time elapsed: 0.004 s
% 2.40/1.57 % (27088)Instructions burned: 6 (million)
% 2.40/1.57 % (27088)------------------------------
% 2.40/1.57 % (27088)------------------------------
% 2.40/1.57 % (27086)Instruction limit reached!
% 2.40/1.57 % (27086)------------------------------
% 2.40/1.57 % (27086)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.40/1.57 % (27086)Termination reason: Unknown
% 2.40/1.57 % (27086)Termination phase: shuffling
% 2.40/1.57
% 2.40/1.57 % (27086)Memory used [KB]: 1663
% 2.40/1.57 % (27086)Time elapsed: 0.010 s
% 2.40/1.57 % (27086)Instructions burned: 19 (million)
% 2.40/1.57 % (27086)------------------------------
% 2.40/1.57 % (27086)------------------------------
% 2.40/1.58 % (27089)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_26966 for (2999ds/902Mi)
% 2.40/1.58 % (27090)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_26966 for (2999ds/21Mi)
% 2.40/1.59 % (27091)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_26966 for (2999ds/5Mi)
% 2.40/1.59 % (27092)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_26966 for (2999ds/6Mi)
% 2.40/1.59 % (27091)Instruction limit reached!
% 2.40/1.59 % (27091)------------------------------
% 2.40/1.59 % (27091)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.40/1.59 % (27091)Termination reason: Unknown
% 2.40/1.59 % (27091)Termination phase: shuffling
% 2.40/1.59
% 2.40/1.59 % (27091)Memory used [KB]: 1279
% 2.40/1.59 % (27091)Time elapsed: 0.004 s
% 2.40/1.59 % (27091)Instructions burned: 6 (million)
% 2.40/1.59 % (27091)------------------------------
% 2.40/1.59 % (27091)------------------------------
% 2.40/1.59 % (27092)Instruction limit reached!
% 2.40/1.59 % (27092)------------------------------
% 2.40/1.59 % (27092)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.40/1.59 % (27092)Termination reason: Unknown
% 2.40/1.59 % (27092)Termination phase: shuffling
% 2.40/1.59
% 2.40/1.59 % (27092)Memory used [KB]: 1279
% 2.40/1.59 % (27092)Time elapsed: 0.004 s
% 2.40/1.59 % (27092)Instructions burned: 6 (million)
% 2.40/1.59 % (27092)------------------------------
% 2.40/1.59 % (27092)------------------------------
% 2.40/1.59 % (27090)Instruction limit reached!
% 2.40/1.59 % (27090)------------------------------
% 2.40/1.59 % (27090)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.40/1.59 % (27090)Termination reason: Unknown
% 2.40/1.59 % (27090)Termination phase: shuffling
% 2.40/1.59
% 2.40/1.59 % (27090)Memory used [KB]: 1663
% 2.40/1.59 % (27090)Time elapsed: 0.011 s
% 2.40/1.59 % (27090)Instructions burned: 21 (million)
% 2.40/1.59 % (27090)------------------------------
% 2.40/1.59 % (27090)------------------------------
% 2.40/1.61 % (27093)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_26966 for (2998ds/377Mi)
% 2.40/1.61 % (27095)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_26966 for (2998ds/19Mi)
% 2.40/1.61 % (27094)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_26966 for (2998ds/779Mi)
% 2.40/1.62 % (27095)Instruction limit reached!
% 2.40/1.62 % (27095)------------------------------
% 2.40/1.62 % (27095)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.40/1.62 % (27095)Termination reason: Unknown
% 2.40/1.62 % (27095)Termination phase: shuffling
% 2.40/1.62
% 2.40/1.62 % (27095)Memory used [KB]: 1663
% 2.40/1.62 % (27095)Time elapsed: 0.010 s
% 2.40/1.62 % (27095)Instructions burned: 19 (million)
% 2.40/1.62 % (27095)------------------------------
% 2.40/1.62 % (27095)------------------------------
% 2.40/1.63 % (27087)First to succeed.
% 2.40/1.63 % (27072)Also succeeded, but the first one will report.
% 2.40/1.63 % (27096)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_26966 for (2998ds/879Mi)
% 2.40/1.64 % (27087)Refutation found. Thanks to Tanya!
% 2.40/1.64 % SZS status Theorem for DTF2THF_26966
% 2.40/1.64 % SZS output start Proof for DTF2THF_26966
% 2.40/1.64 thf(func_def_0, type, in: $i > $i > $o).
% 2.40/1.64 thf(func_def_1, type, exu: ($i > $o) > $o).
% 2.40/1.64 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 2.40/1.64 thf(func_def_8, type, powerset: $i > $i).
% 2.40/1.64 thf(func_def_10, type, setunion: $i > $i).
% 2.40/1.64 thf(func_def_19, type, descr: ($i > $o) > $i).
% 2.40/1.64 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 2.40/1.64 thf(func_def_26, type, prop2set: $o > $i).
% 2.40/1.64 thf(func_def_36, type, nonempty: $i > $o).
% 2.40/1.64 thf(func_def_69, type, set2prop: $i > $o).
% 2.40/1.64 thf(func_def_88, type, subset: $i > $i > $o).
% 2.40/1.64 thf(func_def_89, type, disjoint: $i > $i > $o).
% 2.40/1.64 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 2.40/1.64 thf(func_def_114, type, binunion: $i > $i > $i).
% 2.40/1.64 thf(func_def_122, type, binintersect: $i > $i > $i).
% 2.40/1.64 thf(func_def_135, type, regular: $i > $o).
% 2.40/1.64 thf(func_def_136, type, setminus: $i > $i > $i).
% 2.40/1.64 thf(func_def_147, type, symdiff: $i > $i > $i).
% 2.40/1.64 thf(func_def_153, type, iskpair: $i > $o).
% 2.40/1.64 thf(func_def_158, type, kpair: $i > $i > $i).
% 2.40/1.64 thf(func_def_160, type, cartprod: $i > $i > $i).
% 2.40/1.64 thf(func_def_177, type, singleton: $i > $o).
% 2.40/1.64 thf(func_def_179, type, ex1: $i > ($i > $o) > $o).
% 2.40/1.64 thf(func_def_195, type, sP0: $i > $o > $i > $i > $o).
% 2.40/1.64 thf(func_def_198, type, sP3: $i > $i > $o).
% 2.40/1.64 thf(func_def_199, type, sP4: $i > $o).
% 2.40/1.64 thf(func_def_200, type, sP5: $i > $i > $o).
% 2.40/1.64 thf(func_def_201, type, sP6: $i > $i > $o).
% 2.40/1.64 thf(func_def_207, type, sK12: $i > $o).
% 2.40/1.64 thf(func_def_209, type, sK14: ($i > $o) > $i > $i).
% 2.40/1.64 thf(func_def_216, type, sK21: $i > $i > $i).
% 2.40/1.64 thf(func_def_217, type, sK22: $i > $i > $i).
% 2.40/1.64 thf(func_def_222, type, sK27: $i > $o).
% 2.40/1.64 thf(func_def_225, type, sK30: $i > $i > $i).
% 2.40/1.64 thf(func_def_234, type, sK39: $i > $i > $i).
% 2.40/1.64 thf(func_def_235, type, sK40: $i > $o).
% 2.40/1.64 thf(func_def_236, type, sK41: ($i > $o) > $i).
% 2.40/1.64 thf(func_def_239, type, sK44: $i > $o).
% 2.40/1.64 thf(func_def_256, type, sK61: ($i > $o) > ($i > $o) > $i).
% 2.40/1.64 thf(func_def_257, type, sK62: ($i > $o) > ($i > $o) > $i).
% 2.40/1.64 thf(func_def_258, type, sK63: $i > $o).
% 2.40/1.64 thf(func_def_259, type, sK64: $i > $o).
% 2.40/1.64 thf(func_def_274, type, sK79: $i > $i).
% 2.40/1.64 thf(func_def_279, type, sK84: $i > $o).
% 2.40/1.64 thf(func_def_285, type, sK90: $i > $i).
% 2.40/1.64 thf(func_def_289, type, sK94: $i > $o).
% 2.40/1.64 thf(func_def_291, type, sK96: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.40/1.64 thf(func_def_292, type, sK97: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.40/1.64 thf(func_def_295, type, sK100: $i > $o).
% 2.40/1.64 thf(func_def_296, type, sK101: $i > $o).
% 2.40/1.64 thf(func_def_297, type, sK102: $i > $o).
% 2.40/1.64 thf(func_def_298, type, sK103: $i > $i).
% 2.40/1.64 thf(func_def_299, type, sK104: ($i > $o) > $i).
% 2.40/1.64 thf(func_def_306, type, sK111: $i > $o).
% 2.40/1.64 thf(func_def_308, type, sK113: $i > ($i > $o) > $i).
% 2.40/1.64 thf(func_def_318, type, sK123: $i > $o).
% 2.40/1.64 thf(func_def_323, type, sK128: $i > $o).
% 2.40/1.64 thf(func_def_325, type, sK130: $i > ($i > $o) > $i).
% 2.40/1.64 thf(func_def_328, type, sK133: $i > $o).
% 2.40/1.64 thf(func_def_330, type, sK135: $i > ($i > $o) > $i).
% 2.40/1.64 thf(func_def_346, type, sK151: $i > $i > $i).
% 2.40/1.64 thf(func_def_376, type, sK181: $i > $i > $i > $i).
% 2.40/1.64 thf(func_def_377, type, sK182: $i > $i > $i > $i).
% 2.40/1.64 thf(func_def_381, type, sK186: ($i > $o) > ($i > $o) > $i).
% 2.40/1.64 thf(func_def_382, type, sK187: ($i > $o) > ($i > $o) > $i).
% 2.40/1.64 thf(func_def_383, type, sK188: $i > $o).
% 2.40/1.64 thf(func_def_384, type, sK189: $i > $o).
% 2.40/1.64 thf(func_def_417, type, sK222: ($i > $o) > $i > $i).
% 2.40/1.64 thf(func_def_419, type, sK224: $i > $o).
% 2.40/1.64 thf(func_def_424, type, sK229: $i > $i > $i).
% 2.40/1.64 thf(func_def_428, type, sK233: $i > $o).
% 2.40/1.64 thf(func_def_430, type, sK235: ($i > $o) > $i > $i).
% 2.40/1.64 thf(func_def_431, type, sK236: $i > $o).
% 2.40/1.64 thf(func_def_432, type, sK237: $i > $o).
% 2.40/1.64 thf(func_def_435, type, sK240: $i > ($i > $o) > $i).
% 2.40/1.64 thf(func_def_436, type, sK241: $i > $i).
% 2.40/1.64 thf(func_def_439, type, sK244: $i > $i).
% 2.40/1.64 thf(func_def_441, type, sK246: $i > $o).
% 2.40/1.64 thf(func_def_450, type, sK255: $o > $i > $i > $i).
% 2.40/1.64 thf(func_def_460, type, sK265: $i > $o).
% 2.40/1.64 thf(func_def_463, type, sK268: $i > ($i > $o) > $i).
% 2.40/1.64 thf(func_def_464, type, sK269: $i > ($i > $o) > $i).
% 2.40/1.64 thf(func_def_475, type, sK280: $i > $i > $i).
% 2.40/1.64 thf(func_def_481, type, sK286: $i > $i > $i).
% 2.40/1.64 thf(func_def_482, type, sK287: $i > $o).
% 2.40/1.64 thf(func_def_484, type, sK289: $i > $o).
% 2.40/1.64 thf(func_def_488, type, sK293: $i > $i > ($i > $o) > $i).
% 2.40/1.64 thf(func_def_489, type, sK294: $i > $o).
% 2.40/1.64 thf(func_def_492, type, sK297: $i > $i).
% 2.40/1.64 thf(func_def_522, type, sK327: $i > $o).
% 2.40/1.64 thf(func_def_525, type, sK330: $i > $i > $o).
% 2.40/1.64 thf(func_def_526, type, sK331: $i > $i).
% 2.40/1.64 thf(func_def_527, type, sK332: $i > $i).
% 2.40/1.64 thf(func_def_528, type, sK333: ($i > $i > $o) > $i > $i).
% 2.40/1.64 thf(func_def_529, type, sK334: ($i > $i > $o) > $i > $i).
% 2.40/1.64 thf(func_def_530, type, sK335: $i > ($i > $i > $o) > $i > $i).
% 2.40/1.64 thf(func_def_534, type, sK339: ($i > $o) > $i).
% 2.40/1.64 thf(func_def_535, type, sK340: ($i > $o) > $i).
% 2.40/1.64 thf(func_def_536, type, sK341: $i > $o).
% 2.40/1.64 thf(func_def_545, type, sK350: $i > $o).
% 2.40/1.64 thf(func_def_546, type, sK351: $i > $i > $i).
% 2.40/1.64 thf(func_def_557, type, sK362: $i > $i).
% 2.40/1.64 thf(func_def_570, type, sK375: $i > $i > $i).
% 2.40/1.64 thf(func_def_571, type, sK376: $i > $i > $i).
% 2.40/1.64 thf(func_def_572, type, sK377: $i > $i > $i).
% 2.40/1.64 thf(func_def_573, type, sK378: $i > $i > $i).
% 2.40/1.64 thf(func_def_574, type, sK379: $i > $i > $i).
% 2.40/1.64 thf(func_def_575, type, sK380: $i > $i > $i > $i).
% 2.40/1.64 thf(func_def_576, type, sK381: $i > $i).
% 2.40/1.64 thf(func_def_577, type, sK382: $i > $i).
% 2.40/1.64 thf(func_def_578, type, sK383: $i > $i).
% 2.40/1.64 thf(func_def_579, type, sK384: $i > $i).
% 2.40/1.64 thf(func_def_580, type, sK385: $i > $i > $i > $i).
% 2.40/1.64 thf(func_def_581, type, sK386: $i > $i > $i > $i).
% 2.40/1.64 thf(func_def_582, type, sK387: $i > $i > $i).
% 2.40/1.64 thf(func_def_583, type, sK388: $i > $i > $i).
% 2.40/1.64 thf(func_def_584, type, sK389: $i > $i > $i).
% 2.40/1.64 thf(func_def_586, type, sK391: $i > $i).
% 2.40/1.64 thf(func_def_587, type, sK392: $i > $i).
% 2.40/1.64 thf(func_def_588, type, sK393: $i > $i).
% 2.40/1.64 thf(func_def_592, type, sK397: $i > $i > $i).
% 2.40/1.64 thf(func_def_594, type, sK399: $i > $i).
% 2.40/1.64 thf(func_def_603, type, sK408: ($i > $o) > $i > $i).
% 2.40/1.64 thf(func_def_605, type, sK410: $i > $o).
% 2.40/1.64 thf(func_def_607, type, sK412: $i > $o).
% 2.40/1.64 thf(func_def_610, type, sK415: $i > ($i > $o) > $i).
% 2.40/1.64 thf(func_def_611, type, sK416: $i > ($i > $o) > $i).
% 2.40/1.64 thf(func_def_612, type, sK417: $i > $o).
% 2.40/1.64 thf(func_def_620, type, sK425: $i > $o).
% 2.40/1.64 thf(func_def_622, type, sK427: ($i > $o) > $i > $i).
% 2.40/1.64 thf(func_def_634, type, sK439: ($i > $o) > $i).
% 2.40/1.64 thf(func_def_635, type, sK440: $i > $o).
% 2.40/1.64 thf(func_def_636, type, sK441: $i > $i).
% 2.40/1.64 thf(func_def_638, type, ph443: !>[X0: $tType]:(X0)).
% 2.40/1.64 thf(f3196,plain,(
% 2.40/1.64 $false),
% 2.40/1.64 inference(subsumption_resolution,[status(thm)],[f3195,f1616])).
% 2.40/1.64 thf(f1616,plain,(
% 2.40/1.64 (sK35 != sK36)),
% 2.40/1.64 inference(cnf_transformation,[status(thm)],[f749])).
% 2.40/1.64 thf(f749,plain,(
% 2.40/1.64 (symdiffIneg1 = $true) & (emptysetsubset = $true) & (binintersectI = $true) & (setextAx = $true) & (setunionE = $true) & (dsetconstr__Cong = $true) & (in__Cong = $true) & (setminusER = $true) & (binintersectSubset4 = $true) & (upairset2E = $true) & (powersetsubset = $true) & (emptyInPowerset = $true) & (setminusLsub = $true) & (setextsub = $true) & ((sK35 != sK36) & ((setadjoin @ sK35 @ emptyset) = (setadjoin @ sK36 @ emptyset))) & (exuEu = $true) & (binintersectLsub = $true) & (subsetTrans = $true) & (prop2set2propI = $true) & (cartprodmempair = $true) & (subset2powerset = $true) & (binintersectEL = $true) & (emptysetAx = $true) & (vacuousDall = $true) & (setunionsingleton2 = $true) & (emptyinPowerset = $true) & (cartprodpairin = $true) & (quantDeMorgan1 = $true) & (exuE2 = $true) & (ex1I = $true) & (singletoninpowerset = $true) & (eqimpsubset2 = $true) & (binintersectER = $true) & (secondinupair = $true) & (cartprodmempair1 = $true) & (upairsetE = $true) & (notdallE = $true) & (notinsingleton = $true) & (setminusILneg = $true) & (singletonprop = $true) & (emptyinunitempty = $true) & (symdiffIneg2 = $true) & (setadjoinOr = $true) & (symdiffI2 = $true) & (exuE1 = $true) & (setbeta = $true) & (setminusIRneg = $true) & (setminusEL = $true) & (notequalI2 = $true) & (emptyI = $true) & (ubforcartprodlem1 = $true) & (foundationAx = $true) & (sepInPowerset = $true) & (bs114d = $true) & (ubforcartprodlem2 = $true) & (powersetE = $true) & (setukpairIR = $true) & (symdiffE = $true) & (sepSubset = $true) & (upairinpowunion = $true) & (omegaIndAx = $true) & (descrp = $true) & (subsetI1 = $true) & (exuE3u = $true) & (ex1I2 = $true) & (binunionE = $true) & (emptyE1 = $true) & (quantDeMorgan3 = $true) & (replAx = $true) & (subsetE = $true) & (exuI2 = $true) & (symdiffI1 = $true) & (emptysetimpfalse = $true) & (setminusERneg = $true) & (inPowerset = $true) & (powersetI = $true) & (subsetI2 = $true) & (binunionIL = $true) & (emptysetE = $true) & (setunionsingleton1 = $true) & (prop2setI = $true) & (subPowSU = $true) & (powersetI1 = $true) & (eqimpsubset1 = $true) & (setminusI = $true) & (setadjoinAx = $true) & (powersetAx = $true) & (setunionI = $true) & (setminusSubset1 = $true) & (singletoninpowunion = $true) & (omegaSAx = $true) & (noeltsimpempty = $true) & (nonemptyImpWitness = $true) & (subsetemptysetimpeq = $true) & (ubforcartprodlem3 = $true) & (setadjoinSub2 = $true) & (binunionRsub = $true) & (dsetconstrI = $true) & (exu__Cong = $true) & (notsubsetI = $true) & (exuI3 = $true) & (setadjoinE = $true) & (setadjoin__Cong = $true) & (setunionsingleton = $true) & (wellorderingAx = $true) & (setadjoinIL = $true) & (setadjoinSub = $true) & (binintersectRsub = $true) & (setminusELneg = $true) & (setminusSubset2 = $true) & (singletonsubset = $true) & (quantDeMorgan2 = $true) & (setukpairIL = $true) & (powerset__Cong = $true) & (dsetconstrER = $true) & (setunion__Cong = $true) & (setunionAx = $true) & (setadjoinIR = $true) & (emptyset__Cong = $true) & (nonemptyI = $true) & (quantDeMorgan4 = $true) & (nonemptyI1 = $true) & (omega0Ax = $true) & (notinemptyset = $true) & (powersetE1 = $true) & (kpairiskpair = $true) & (binunionLsub = $true) & (setext = $true) & (ex1E1 = $true) & (omega__Cong = $true) & (notequalI1 = $true) & (kpairp = $true) & (subsetE2 = $true) & (binintersectSubset2 = $true) & (setoftrueEq = $true) & (binintersectSubset5 = $true) & (upairsetIR = $true) & (binintersectSubset1 = $true) & (singletonsswitch = $true) & (subsetRefl = $true) & (exuI1 = $true) & (eqinunit = $true) & (notdexE = $true) & (upairsubunion = $true) & (binunionEcases = $true) & (upairset2IR = $true) & (upairsetIL = $true) & (inCongP = $true) & (exuE3e = $true) & (binunionIR = $true) & (uniqinunit = $true) & (prop2setE = $true) & (binintersectSubset3 = $true) & (descr__Cong = $true) & (nonemptyE1 = $true) & (disjointsetsI1 = $true) & (setunionE2 = $true) & (dsetconstrEL = $true)),
% 2.40/1.64 inference(skolemisation,[status(esa),new_symbols(skolem,[sK35,sK36])],[f656,f748])).
% 2.40/1.64 thf(f748,plain,(
% 2.40/1.64 ? [X0 : $i,X1 : $i] : ((X0 != X1) & ((setadjoin @ X1 @ emptyset) = (setadjoin @ X0 @ emptyset))) => ((sK35 != sK36) & ((setadjoin @ sK35 @ emptyset) = (setadjoin @ sK36 @ emptyset)))),
% 2.40/1.64 introduced(definition,[],[choice_axiom])).
% 2.40/1.64 thf(f656,plain,(
% 2.40/1.64 (symdiffIneg1 = $true) & (emptysetsubset = $true) & (binintersectI = $true) & (setextAx = $true) & (setunionE = $true) & (dsetconstr__Cong = $true) & (in__Cong = $true) & (setminusER = $true) & (binintersectSubset4 = $true) & (upairset2E = $true) & (powersetsubset = $true) & (emptyInPowerset = $true) & (setminusLsub = $true) & (setextsub = $true) & ? [X0 : $i,X1 : $i] : ((X0 != X1) & ((setadjoin @ X1 @ emptyset) = (setadjoin @ X0 @ emptyset))) & (exuEu = $true) & (binintersectLsub = $true) & (subsetTrans = $true) & (prop2set2propI = $true) & (cartprodmempair = $true) & (subset2powerset = $true) & (binintersectEL = $true) & (emptysetAx = $true) & (vacuousDall = $true) & (setunionsingleton2 = $true) & (emptyinPowerset = $true) & (cartprodpairin = $true) & (quantDeMorgan1 = $true) & (exuE2 = $true) & (ex1I = $true) & (singletoninpowerset = $true) & (eqimpsubset2 = $true) & (binintersectER = $true) & (secondinupair = $true) & (cartprodmempair1 = $true) & (upairsetE = $true) & (notdallE = $true) & (notinsingleton = $true) & (setminusILneg = $true) & (singletonprop = $true) & (emptyinunitempty = $true) & (symdiffIneg2 = $true) & (setadjoinOr = $true) & (symdiffI2 = $true) & (exuE1 = $true) & (setbeta = $true) & (setminusIRneg = $true) & (setminusEL = $true) & (notequalI2 = $true) & (emptyI = $true) & (ubforcartprodlem1 = $true) & (foundationAx = $true) & (sepInPowerset = $true) & (bs114d = $true) & (ubforcartprodlem2 = $true) & (powersetE = $true) & (setukpairIR = $true) & (symdiffE = $true) & (sepSubset = $true) & (upairinpowunion = $true) & (omegaIndAx = $true) & (descrp = $true) & (subsetI1 = $true) & (exuE3u = $true) & (ex1I2 = $true) & (binunionE = $true) & (emptyE1 = $true) & (quantDeMorgan3 = $true) & (replAx = $true) & (subsetE = $true) & (exuI2 = $true) & (symdiffI1 = $true) & (emptysetimpfalse = $true) & (setminusERneg = $true) & (inPowerset = $true) & (powersetI = $true) & (subsetI2 = $true) & (binunionIL = $true) & (emptysetE = $true) & (setunionsingleton1 = $true) & (prop2setI = $true) & (subPowSU = $true) & (powersetI1 = $true) & (eqimpsubset1 = $true) & (setminusI = $true) & (setadjoinAx = $true) & (powersetAx = $true) & (setunionI = $true) & (setminusSubset1 = $true) & (singletoninpowunion = $true) & (omegaSAx = $true) & (noeltsimpempty = $true) & (nonemptyImpWitness = $true) & (subsetemptysetimpeq = $true) & (ubforcartprodlem3 = $true) & (setadjoinSub2 = $true) & (binunionRsub = $true) & (dsetconstrI = $true) & (exu__Cong = $true) & (notsubsetI = $true) & (exuI3 = $true) & (setadjoinE = $true) & (setadjoin__Cong = $true) & (setunionsingleton = $true) & (wellorderingAx = $true) & (setadjoinIL = $true) & (setadjoinSub = $true) & (binintersectRsub = $true) & (setminusELneg = $true) & (setminusSubset2 = $true) & (singletonsubset = $true) & (quantDeMorgan2 = $true) & (setukpairIL = $true) & (powerset__Cong = $true) & (dsetconstrER = $true) & (setunion__Cong = $true) & (setunionAx = $true) & (setadjoinIR = $true) & (emptyset__Cong = $true) & (nonemptyI = $true) & (quantDeMorgan4 = $true) & (nonemptyI1 = $true) & (omega0Ax = $true) & (notinemptyset = $true) & (powersetE1 = $true) & (kpairiskpair = $true) & (binunionLsub = $true) & (setext = $true) & (ex1E1 = $true) & (omega__Cong = $true) & (notequalI1 = $true) & (kpairp = $true) & (subsetE2 = $true) & (binintersectSubset2 = $true) & (setoftrueEq = $true) & (binintersectSubset5 = $true) & (upairsetIR = $true) & (binintersectSubset1 = $true) & (singletonsswitch = $true) & (subsetRefl = $true) & (exuI1 = $true) & (eqinunit = $true) & (notdexE = $true) & (upairsubunion = $true) & (binunionEcases = $true) & (upairset2IR = $true) & (upairsetIL = $true) & (inCongP = $true) & (exuE3e = $true) & (binunionIR = $true) & (uniqinunit = $true) & (prop2setE = $true) & (binintersectSubset3 = $true) & (descr__Cong = $true) & (nonemptyE1 = $true) & (disjointsetsI1 = $true) & (setunionE2 = $true) & (dsetconstrEL = $true)),
% 2.40/1.64 inference(flattening,[status(thm)],[f655])).
% 2.40/1.64 thf(f655,plain,(
% 2.40/1.64 ((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i,X1 : $i] : ((X0 != X1) & ((setadjoin @ X1 @ emptyset) = (setadjoin @ X0 @ emptyset))) & (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)),
% 2.40/1.64 inference(ennf_transformation,[status(thm)],[f458])).
% 2.40/1.64 thf(f458,plain,(
% 2.40/1.64 ~((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) => ! [X0 : $i,X1 : $i] : (((setadjoin @ X1 @ emptyset) = (setadjoin @ X0 @ emptyset)) => (X0 = X1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.40/1.64 inference(fool_elimination,[status(thm)],[f457])).
% 2.40/1.64 thf(f457,plain,(
% 2.40/1.64 ~(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 => ! [X0 : $i,X1 : $i] : (((setadjoin @ X1 @ emptyset) = (setadjoin @ X0 @ emptyset)) => (X0 = X1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.40/1.64 inference(rectify,[status(thm)],[f165])).
% 2.40/1.64 thf(f165,negated_conjecture,(
% 2.40/1.64 ~(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 => ! [X1 : $i,X2 : $i] : (((setadjoin @ X2 @ emptyset) = (setadjoin @ X1 @ emptyset)) => (X1 = X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.40/1.64 inference(negated_conjecture,[status(cth)],[f164])).
% 2.40/1.64 thf(f164,conjecture,(
% 2.40/1.64 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 => ! [X1 : $i,X2 : $i] : (((setadjoin @ X2 @ emptyset) = (setadjoin @ X1 @ emptyset)) => (X1 = X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.40/1.64 file('/export/starexec/sandbox2/tmp/tmp.DxatUwJPum/DTF2THF_26966.p',singletonsuniq)).
% 2.40/1.64 thf(f3195,plain,(
% 2.40/1.64 (sK35 = sK36)),
% 2.40/1.64 inference(forward_demodulation,[status(thm)],[f3194,f2815])).
% 2.40/1.64 thf(f2815,plain,(
% 2.40/1.64 ( ! [X0 : $i] : (((setunion @ (setadjoin @ X0 @ emptyset)) = X0)) )),
% 2.40/1.64 inference(trivial_inequality_removal,[status(thm)],[f2393])).
% 2.40/1.64 thf(f2393,plain,(
% 2.40/1.64 ( ! [X0 : $i] : (((setunion @ (setadjoin @ X0 @ emptyset)) = X0) | ($true != $true)) )),
% 2.40/1.64 inference(definition_unfolding,[status(thm)],[f1779,f1526])).
% 2.40/1.64 thf(f1526,plain,(
% 2.40/1.64 (setunionsingleton = $true)),
% 2.40/1.64 inference(cnf_transformation,[status(thm)],[f749])).
% 2.40/1.64 thf(f1779,plain,(
% 2.40/1.64 ( ! [X0 : $i] : (((setunion @ (setadjoin @ X0 @ emptyset)) = X0) | (setunionsingleton != $true)) )),
% 2.40/1.64 inference(cnf_transformation,[status(thm)],[f933])).
% 2.40/1.64 thf(f933,plain,(
% 2.40/1.64 (! [X0 : $i] : ((setunion @ (setadjoin @ X0 @ emptyset)) = X0) | (setunionsingleton != $true)) & ((setunionsingleton = $true) | (sK140 != (setunion @ (setadjoin @ sK140 @ emptyset))))),
% 2.40/1.64 inference(skolemisation,[status(esa),new_symbols(skolem,[sK140])],[f931,f932])).
% 2.40/1.64 thf(f932,plain,(
% 2.40/1.64 ? [X1 : $i] : ((setunion @ (setadjoin @ X1 @ emptyset)) != X1) => (sK140 != (setunion @ (setadjoin @ sK140 @ emptyset)))),
% 2.40/1.64 introduced(definition,[],[choice_axiom])).
% 2.40/1.64 thf(f931,plain,(
% 2.40/1.64 (! [X0 : $i] : ((setunion @ (setadjoin @ X0 @ emptyset)) = X0) | (setunionsingleton != $true)) & ((setunionsingleton = $true) | ? [X1 : $i] : ((setunion @ (setadjoin @ X1 @ emptyset)) != X1))),
% 2.40/1.64 inference(rectify,[status(thm)],[f930])).
% 2.40/1.64 thf(f930,plain,(
% 2.40/1.64 (! [X0 : $i] : ((setunion @ (setadjoin @ X0 @ emptyset)) = X0) | (setunionsingleton != $true)) & ((setunionsingleton = $true) | ? [X0 : $i] : ((setunion @ (setadjoin @ X0 @ emptyset)) != X0))),
% 2.40/1.64 inference(nnf_transformation,[status(thm)],[f333])).
% 2.40/1.64 thf(f333,plain,(
% 2.40/1.64 ! [X0 : $i] : ((setunion @ (setadjoin @ X0 @ emptyset)) = X0) <=> (setunionsingleton = $true)),
% 2.40/1.64 inference(fool_elimination,[status(thm)],[f332])).
% 2.40/1.64 thf(f332,plain,(
% 2.40/1.64 (setunionsingleton = ! [X0 : $i] : ((setunion @ (setadjoin @ X0 @ emptyset)) = X0))),
% 2.40/1.64 inference(rectify,[status(thm)],[f157])).
% 2.40/1.64 thf(f157,axiom,(
% 2.40/1.64 (setunionsingleton = ! [X1 : $i] : ((setunion @ (setadjoin @ X1 @ emptyset)) = X1))),
% 2.40/1.64 file('/export/starexec/sandbox2/tmp/tmp.DxatUwJPum/DTF2THF_26966.p',setunionsingleton)).
% 2.40/1.64 thf(f3194,plain,(
% 2.40/1.64 (sK36 = (setunion @ (setadjoin @ sK35 @ emptyset)))),
% 2.40/1.64 inference(superposition,[status(thm)],[f2815,f1615])).
% 2.40/1.64 thf(f1615,plain,(
% 2.40/1.64 ((setadjoin @ sK35 @ emptyset) = (setadjoin @ sK36 @ emptyset))),
% 2.40/1.64 inference(cnf_transformation,[status(thm)],[f749])).
% 2.40/1.64 % SZS output end Proof for DTF2THF_26966
% 2.40/1.64 % (27087)------------------------------
% 2.40/1.64 % (27087)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.40/1.64 % (27087)Termination reason: Refutation
% 2.40/1.64
% 2.40/1.64 % (27087)Memory used [KB]: 8187
% 2.40/1.64 % (27087)Time elapsed: 0.070 s
% 2.40/1.64 % (27087)Instructions burned: 218 (million)
% 2.40/1.64 % (27087)------------------------------
% 2.40/1.64 % (27087)------------------------------
% 2.40/1.64 % (27066)Success in time 0.143 s
%------------------------------------------------------------------------------