%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU650^1 : TPTP v9.2.1. Released v3.7.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n029.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 09:35:57 AM UTC 2026
% Result : Theorem 2.65s 1.65s
% Output : Refutation 2.65s
% Verified :
% SZS Type : Refutation
% Derivation depth : 12
% Number of leaves : 4
% Syntax : Number of formulae : 26 ( 9 unt; 0 typ; 0 def)
% Number of atoms : 2710 ( 730 equ; 0 cnn)
% Maximal formula atoms : 168 ( 104 avg)
% Number of connectives : 1491 ( 28 ~; 15 |; 511 &; 262 @)
% ( 2 <=>; 673 =>; 0 <=; 0 <~>)
% Maximal formula depth : 171 ( 49 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 313 ( 310 usr; 173 con; 0-4 aty)
% Number of variables : 40 ( 0 ^; 28 !; 12 ?; 40 :)
% Comments :
%------------------------------------------------------------------------------
thf(func_def_0,type,
in: $i > $i > $o ).
thf(func_def_1,type,
exu: ( $i > $o ) > $o ).
thf(func_def_6,type,
setadjoin: $i > $i > $i ).
thf(func_def_8,type,
powerset: $i > $i ).
thf(func_def_10,type,
setunion: $i > $i ).
thf(func_def_19,type,
descr: ( $i > $o ) > $i ).
thf(func_def_21,type,
dsetconstr: $i > ( $i > $o ) > $i ).
thf(func_def_26,type,
prop2set: $o > $i ).
thf(func_def_36,type,
nonempty: $i > $o ).
thf(func_def_69,type,
set2prop: $i > $o ).
thf(func_def_88,type,
subset: $i > $i > $o ).
thf(func_def_89,type,
disjoint: $i > $i > $o ).
thf(func_def_90,type,
setsmeet: $i > $i > $o ).
thf(func_def_114,type,
binunion: $i > $i > $i ).
thf(func_def_122,type,
binintersect: $i > $i > $i ).
thf(func_def_135,type,
regular: $i > $o ).
thf(func_def_136,type,
setminus: $i > $i > $i ).
thf(func_def_147,type,
symdiff: $i > $i > $i ).
thf(func_def_153,type,
iskpair: $i > $o ).
thf(func_def_158,type,
kpair: $i > $i > $i ).
thf(func_def_160,type,
cartprod: $i > $i > $i ).
thf(func_def_177,type,
singleton: $i > $o ).
thf(func_def_179,type,
ex1: $i > ( $i > $o ) > $o ).
thf(func_def_184,type,
atmost1p: $i > $o ).
thf(func_def_185,type,
atleast2p: $i > $o ).
thf(func_def_186,type,
atmost2p: $i > $o ).
thf(func_def_187,type,
upairsetp: $i > $o ).
thf(func_def_191,type,
kfst: $i > $i ).
thf(func_def_209,type,
sP0: $i > $i > $i > $o > $o ).
thf(func_def_211,type,
sP2: $i > $i > $o ).
thf(func_def_212,type,
sP3: $i > $o ).
thf(func_def_213,type,
sP4: $i > $i > $o ).
thf(func_def_214,type,
sP5: $i > $i > $o ).
thf(func_def_219,type,
sK10: $i > $o ).
thf(func_def_220,type,
sK11: $i > $o ).
thf(func_def_221,type,
sK12: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_222,type,
sK13: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_227,type,
sK18: $i > $o ).
thf(func_def_244,type,
sK35: ( $i > $o ) > $i ).
thf(func_def_245,type,
sK36: ( $i > $o ) > $i ).
thf(func_def_246,type,
sK37: $i > $o ).
thf(func_def_248,type,
sK39: $i > $o ).
thf(func_def_260,type,
sK51: ( $i > $o ) > $i > $i ).
thf(func_def_262,type,
sK53: $i > $o ).
thf(func_def_268,type,
sK59: $i > $i > $i ).
thf(func_def_269,type,
sK60: $i > $i > $i ).
thf(func_def_275,type,
sK66: $i > $i > $i ).
thf(func_def_294,type,
sK85: $i > $o ).
thf(func_def_297,type,
sK88: $i > $o ).
thf(func_def_303,type,
sK94: $i > $i ).
thf(func_def_305,type,
sK96: $i > $i > $i ).
thf(func_def_333,type,
sK124: $i > $i > $i ).
thf(func_def_334,type,
sK125: $i > $o ).
thf(func_def_340,type,
sK131: $i > $o ).
thf(func_def_348,type,
sK139: $i > $i ).
thf(func_def_357,type,
sK148: $i > ( $i > $o ) > $i ).
thf(func_def_358,type,
sK149: $i > $o ).
thf(func_def_362,type,
sK153: $i > $o ).
thf(func_def_368,type,
sK159: $i > $i > $i ).
thf(func_def_371,type,
sK162: $i > $o ).
thf(func_def_374,type,
sK165: $i > ( $i > $o ) > $i ).
thf(func_def_375,type,
sK166: $i > $o ).
thf(func_def_383,type,
sK174: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_384,type,
sK175: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_387,type,
sK178: $i > $o ).
thf(func_def_388,type,
sK179: $i > $o ).
thf(func_def_404,type,
sK195: $i > $o ).
thf(func_def_409,type,
sK200: $i > $o ).
thf(func_def_411,type,
sK202: ( $i > $o ) > $i > $i ).
thf(func_def_421,type,
sK212: $i > $o ).
thf(func_def_423,type,
sK214: $i > ( $i > $o ) > $i ).
thf(func_def_425,type,
sK216: $i > $i ).
thf(func_def_428,type,
sK219: $i > $i ).
thf(func_def_433,type,
sK224: $i > $i > $i ).
thf(func_def_445,type,
sK236: $i > $i > $i > $i ).
thf(func_def_446,type,
sK237: $i > $i > $i > $i ).
thf(func_def_448,type,
sK239: $i > $i > $o ).
thf(func_def_449,type,
sK240: $i > $i ).
thf(func_def_450,type,
sK241: $i > $i ).
thf(func_def_451,type,
sK242: ( $i > $i > $o ) > $i > $i ).
thf(func_def_452,type,
sK243: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_453,type,
sK244: ( $i > $i > $o ) > $i > $i ).
thf(func_def_457,type,
sK248: $i > $i ).
thf(func_def_458,type,
sK249: $i > $i > $i ).
thf(func_def_459,type,
sK250: $i > $i > $i ).
thf(func_def_460,type,
sK251: $i > $i > $i ).
thf(func_def_461,type,
sK252: $i > $i > $i ).
thf(func_def_462,type,
sK253: $i > $i > $i > $i ).
thf(func_def_463,type,
sK254: $i > $i ).
thf(func_def_464,type,
sK255: $i > $i ).
thf(func_def_465,type,
sK256: $i > $i ).
thf(func_def_466,type,
sK257: $i > $i ).
thf(func_def_467,type,
sK258: $i > $i > $i > $i ).
thf(func_def_468,type,
sK259: $i > $i > $i > $i ).
thf(func_def_469,type,
sK260: $i > $i > $i ).
thf(func_def_470,type,
sK261: $i > $i > $i ).
thf(func_def_471,type,
sK262: $i > $i > $i ).
thf(func_def_472,type,
sK263: $i > $i ).
thf(func_def_473,type,
sK264: $i > $i > $i ).
thf(func_def_475,type,
sK266: $i > $i ).
thf(func_def_476,type,
sK267: $i > $i ).
thf(func_def_500,type,
sK291: $i > $o ).
thf(func_def_503,type,
sK294: $i > ( $i > $o ) > $i ).
thf(func_def_504,type,
sK295: $i > ( $i > $o ) > $i ).
thf(func_def_511,type,
sK302: $i > $o ).
thf(func_def_512,type,
sK303: $i > $i ).
thf(func_def_513,type,
sK304: ( $i > $o ) > $i ).
thf(func_def_514,type,
sK305: $i > $i > $i ).
thf(func_def_518,type,
sK309: $i > ( $i > $o ) > $i ).
thf(func_def_519,type,
sK310: $i > $o ).
thf(func_def_532,type,
sK323: $i > ( $i > $o ) > $i ).
thf(func_def_533,type,
sK324: $i > $o ).
thf(func_def_554,type,
sK345: $i > $o ).
thf(func_def_555,type,
sK346: ( $i > $o ) > $i > $i > $i ).
thf(func_def_556,type,
sK347: $i > $i > $i ).
thf(func_def_559,type,
sK350: ( $i > $o ) > $i ).
thf(func_def_560,type,
sK351: $i > $o ).
thf(func_def_561,type,
sK352: $i > $i ).
thf(func_def_563,type,
sK354: $o > $i > $i > $i ).
thf(func_def_567,type,
sK358: $i > $o ).
thf(func_def_584,type,
sK375: ( $i > $o ) > $i > $i ).
thf(func_def_586,type,
sK377: $i > $o ).
thf(func_def_594,type,
sK385: ( $i > $o ) > $i ).
thf(func_def_595,type,
sK386: $i > $o ).
thf(func_def_598,type,
sK389: $i > $i ).
thf(func_def_599,type,
sK390: $i > $i ).
thf(func_def_602,type,
sK393: $i > ( $i > $o ) > $i ).
thf(func_def_603,type,
sK394: $i > $o ).
thf(func_def_605,type,
sK396: $i > $o ).
thf(func_def_612,type,
sK403: $i > $i > $i ).
thf(func_def_634,type,
sK425: $i > $o ).
thf(func_def_637,type,
sK428: $i > ( $i > $o ) > $i ).
thf(func_def_638,type,
sK429: $i > ( $i > $o ) > $i ).
thf(func_def_641,type,
sK432: $i > $o ).
thf(func_def_657,type,
sK448: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_658,type,
sK449: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_659,type,
sK450: $i > $o ).
thf(func_def_660,type,
sK451: $i > $o ).
thf(func_def_673,type,
ph464:
!>[X0: $tType] : X0 ).
thf(f3197,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f3196,f2966]) ).
thf(f2966,plain,
! [X1: $i] :
( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X1 @ ( setadjoin @ X1 @ emptyset ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f2898]) ).
thf(f2898,plain,
! [X1: $i] :
( ( $true != $true )
| ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X1 @ ( setadjoin @ X1 @ emptyset ) ) ) ),
inference(equality_resolution,[status(thm)],[f2498]) ).
thf(f2498,plain,
! [X0: $i,X1: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X1 @ ( setadjoin @ X0 @ emptyset ) ) )
| ( X0 != X1 )
| ( $true != $true ) ),
inference(definition_unfolding,[status(thm)],[f1695,f2085]) ).
thf(f2085,plain,
setukpairinjR11 = $true,
inference(cnf_transformation,[status(thm)],[f1265]) ).
thf(f1265,plain,
( ( cartprodmempair1 = $true )
& ( setunionsingleton1 = $true )
& ( omega__Cong = $true )
& ( binintersectSubset2 = $true )
& ( quantDeMorgan3 = $true )
& ( binintersectRsub = $true )
& ( prop2setI = $true )
& ( binintersectI = $true )
& ( dsetconstrI = $true )
& ( singletoninpowunion = $true )
& ( setunionsingleton = $true )
& ( upairsetIR = $true )
& ( cartprodfstin = $true )
& ( symdiffIneg1 = $true )
& ( binintersectLsub = $true )
& ( subsetRefl = $true )
& ( binunionIL = $true )
& ( inPowerset = $true )
& ( emptyinPowerset = $true )
& ( setminusELneg = $true )
& ( bs114d = $true )
& ( prop2set2propI = $true )
& ( eqimpsubset2 = $true )
& ( upairinpowunion = $true )
& ( singletoninpowerset = $true )
& ( notequalI1 = $true )
& ( kpairp = $true )
& ( setukpairIL = $true )
& ( cartprodmempair = $true )
& ( eqinunit = $true )
& ( setoftrueEq = $true )
& ( setminusLsub = $true )
& ( uniqinunit = $true )
& ( binunionEcases = $true )
& ( powersetE1 = $true )
& ( binintersectSubset5 = $true )
& ( setukpairinjR11 = $true )
& ( symdiffI1 = $true )
& ( descrp = $true )
& ( emptyInPowerset = $true )
& ( setukpairIR = $true )
& ( emptyset__Cong = $true )
& ( powersetI1 = $true )
& ( setukpairinjL2 = $true )
& ( exuI3 = $true )
& ( ubforcartprodlem3 = $true )
& ( setadjoinAx = $true )
& ( foundationAx = $true )
& ( sepSubset = $true )
& ( ex1E1 = $true )
& ( notinsingleton = $true )
& ( powersetE = $true )
& ( setminusIRneg = $true )
& ( setextsub = $true )
& ( powerset__Cong = $true )
& ( upairsetIL = $true )
& ( in__Cong = $true )
& ( symdiffI2 = $true )
& ( upairset2E = $true )
& ( sepInPowerset = $true )
& ( setadjoinSub = $true )
& ( exuE2 = $true )
& ( ubforcartprodlem1 = $true )
& ( exuI1 = $true )
& ( binintersectSubset4 = $true )
& ( binintersectEL = $true )
& ( setminusER = $true )
& ( ex1I2 = $true )
& ( ( setadjoin @ ( setadjoin @ sK327 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK327 @ ( setadjoin @ sK328 @ emptyset ) ) @ emptyset ) )
!= ( setadjoin @ ( setadjoin @ sK327 @ emptyset ) @ emptyset ) )
& ( sK327 = sK328 )
& ( setadjoin__Cong = $true )
& ( subsetI1 = $true )
& ( exuEu = $true )
& ( binunionLsub = $true )
& ( quantDeMorgan2 = $true )
& ( dsetconstrER = $true )
& ( setminusEL = $true )
& ( setukpairinjL1 = $true )
& ( subsetTrans = $true )
& ( quantDeMorgan4 = $true )
& ( notdallE = $true )
& ( wellorderingAx = $true )
& ( setukpairinjL = $true )
& ( upairset2IR = $true )
& ( setunionI = $true )
& ( kfstsingleton = $true )
& ( setext = $true )
& ( subset2powerset = $true )
& ( setminusSubset2 = $true )
& ( setadjoinSub2 = $true )
& ( subsetE = $true )
& ( powersetI = $true )
& ( singletonprop = $true )
& ( setminusERneg = $true )
& ( omegaIndAx = $true )
& ( setadjoinE = $true )
& ( exu__Cong = $true )
& ( emptyI = $true )
& ( nonemptyE1 = $true )
& ( kpairiskpair = $true )
& ( upairsetE = $true )
& ( powersetAx = $true )
& ( descr__Cong = $true )
& ( setminusI = $true )
& ( secondinupair = $true )
& ( notdexE = $true )
& ( disjointsetsI1 = $true )
& ( setunion__Cong = $true )
& ( setunionsingleton2 = $true )
& ( setunionE = $true )
& ( setminusSubset1 = $true )
& ( emptyinunitempty = $true )
& ( powersetsubset = $true )
& ( emptysetAx = $true )
& ( omega0Ax = $true )
& ( emptyE1 = $true )
& ( setbeta = $true )
& ( exuE3e = $true )
& ( emptysetsubset = $true )
& ( emptysetimpfalse = $true )
& ( exuE1 = $true )
& ( setextAx = $true )
& ( singletonsswitch = $true )
& ( replAx = $true )
& ( notsubsetI = $true )
& ( singletonsubset = $true )
& ( kfstpairEq = $true )
& ( setadjoinIR = $true )
& ( cartprodpairin = $true )
& ( upairsubunion = $true )
& ( subsetemptysetimpeq = $true )
& ( subsetI2 = $true )
& ( singletonsuniq = $true )
& ( omegaSAx = $true )
& ( noeltsimpempty = $true )
& ( vacuousDall = $true )
& ( notinemptyset = $true )
& ( setadjoinOr = $true )
& ( binintersectSubset3 = $true )
& ( setadjoinIL = $true )
& ( ubforcartprodlem2 = $true )
& ( symdiffIneg2 = $true )
& ( theprop = $true )
& ( dsetconstr__Cong = $true )
& ( exuE3u = $true )
& ( setunionAx = $true )
& ( quantDeMorgan1 = $true )
& ( setunionE2 = $true )
& ( nonemptyI = $true )
& ( binintersectER = $true )
& ( binunionIR = $true )
& ( symdiffE = $true )
& ( prop2setE = $true )
& ( binunionE = $true )
& ( emptysetE = $true )
& ( nonemptyImpWitness = $true )
& ( eqimpsubset1 = $true )
& ( inCongP = $true )
& ( notequalI2 = $true )
& ( subsetE2 = $true )
& ( exuI2 = $true )
& ( subPowSU = $true )
& ( binunionRsub = $true )
& ( ex1I = $true )
& ( binintersectSubset1 = $true )
& ( dsetconstrEL = $true )
& ( nonemptyI1 = $true )
& ( setminusILneg = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK327,sK328])],[f687,f1264]) ).
thf(f1264,plain,
( ? [X0: $i,X1: $i] :
( ( ( setadjoin @ ( setadjoin @ X0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X0 @ ( setadjoin @ X1 @ emptyset ) ) @ emptyset ) )
!= ( setadjoin @ ( setadjoin @ X0 @ emptyset ) @ emptyset ) )
& ( X0 = X1 ) )
=> ( ( ( setadjoin @ ( setadjoin @ sK327 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK327 @ ( setadjoin @ sK328 @ emptyset ) ) @ emptyset ) )
!= ( setadjoin @ ( setadjoin @ sK327 @ emptyset ) @ emptyset ) )
& ( sK327 = sK328 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f687,plain,
( ( cartprodmempair1 = $true )
& ( setunionsingleton1 = $true )
& ( omega__Cong = $true )
& ( binintersectSubset2 = $true )
& ( quantDeMorgan3 = $true )
& ( binintersectRsub = $true )
& ( prop2setI = $true )
& ( binintersectI = $true )
& ( dsetconstrI = $true )
& ( singletoninpowunion = $true )
& ( setunionsingleton = $true )
& ( upairsetIR = $true )
& ( cartprodfstin = $true )
& ( symdiffIneg1 = $true )
& ( binintersectLsub = $true )
& ( subsetRefl = $true )
& ( binunionIL = $true )
& ( inPowerset = $true )
& ( emptyinPowerset = $true )
& ( setminusELneg = $true )
& ( bs114d = $true )
& ( prop2set2propI = $true )
& ( eqimpsubset2 = $true )
& ( upairinpowunion = $true )
& ( singletoninpowerset = $true )
& ( notequalI1 = $true )
& ( kpairp = $true )
& ( setukpairIL = $true )
& ( cartprodmempair = $true )
& ( eqinunit = $true )
& ( setoftrueEq = $true )
& ( setminusLsub = $true )
& ( uniqinunit = $true )
& ( binunionEcases = $true )
& ( powersetE1 = $true )
& ( binintersectSubset5 = $true )
& ( setukpairinjR11 = $true )
& ( symdiffI1 = $true )
& ( descrp = $true )
& ( emptyInPowerset = $true )
& ( setukpairIR = $true )
& ( emptyset__Cong = $true )
& ( powersetI1 = $true )
& ( setukpairinjL2 = $true )
& ( exuI3 = $true )
& ( ubforcartprodlem3 = $true )
& ( setadjoinAx = $true )
& ( foundationAx = $true )
& ( sepSubset = $true )
& ( ex1E1 = $true )
& ( notinsingleton = $true )
& ( powersetE = $true )
& ( setminusIRneg = $true )
& ( setextsub = $true )
& ( powerset__Cong = $true )
& ( upairsetIL = $true )
& ( in__Cong = $true )
& ( symdiffI2 = $true )
& ( upairset2E = $true )
& ( sepInPowerset = $true )
& ( setadjoinSub = $true )
& ( exuE2 = $true )
& ( ubforcartprodlem1 = $true )
& ( exuI1 = $true )
& ( binintersectSubset4 = $true )
& ( binintersectEL = $true )
& ( setminusER = $true )
& ( ex1I2 = $true )
& ? [X0: $i,X1: $i] :
( ( ( setadjoin @ ( setadjoin @ X0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X0 @ ( setadjoin @ X1 @ emptyset ) ) @ emptyset ) )
!= ( setadjoin @ ( setadjoin @ X0 @ emptyset ) @ emptyset ) )
& ( X0 = X1 ) )
& ( setadjoin__Cong = $true )
& ( subsetI1 = $true )
& ( exuEu = $true )
& ( binunionLsub = $true )
& ( quantDeMorgan2 = $true )
& ( dsetconstrER = $true )
& ( setminusEL = $true )
& ( setukpairinjL1 = $true )
& ( subsetTrans = $true )
& ( quantDeMorgan4 = $true )
& ( notdallE = $true )
& ( wellorderingAx = $true )
& ( setukpairinjL = $true )
& ( upairset2IR = $true )
& ( setunionI = $true )
& ( kfstsingleton = $true )
& ( setext = $true )
& ( subset2powerset = $true )
& ( setminusSubset2 = $true )
& ( setadjoinSub2 = $true )
& ( subsetE = $true )
& ( powersetI = $true )
& ( singletonprop = $true )
& ( setminusERneg = $true )
& ( omegaIndAx = $true )
& ( setadjoinE = $true )
& ( exu__Cong = $true )
& ( emptyI = $true )
& ( nonemptyE1 = $true )
& ( kpairiskpair = $true )
& ( upairsetE = $true )
& ( powersetAx = $true )
& ( descr__Cong = $true )
& ( setminusI = $true )
& ( secondinupair = $true )
& ( notdexE = $true )
& ( disjointsetsI1 = $true )
& ( setunion__Cong = $true )
& ( setunionsingleton2 = $true )
& ( setunionE = $true )
& ( setminusSubset1 = $true )
& ( emptyinunitempty = $true )
& ( powersetsubset = $true )
& ( emptysetAx = $true )
& ( omega0Ax = $true )
& ( emptyE1 = $true )
& ( setbeta = $true )
& ( exuE3e = $true )
& ( emptysetsubset = $true )
& ( emptysetimpfalse = $true )
& ( exuE1 = $true )
& ( setextAx = $true )
& ( singletonsswitch = $true )
& ( replAx = $true )
& ( notsubsetI = $true )
& ( singletonsubset = $true )
& ( kfstpairEq = $true )
& ( setadjoinIR = $true )
& ( cartprodpairin = $true )
& ( upairsubunion = $true )
& ( subsetemptysetimpeq = $true )
& ( subsetI2 = $true )
& ( singletonsuniq = $true )
& ( omegaSAx = $true )
& ( noeltsimpempty = $true )
& ( vacuousDall = $true )
& ( notinemptyset = $true )
& ( setadjoinOr = $true )
& ( binintersectSubset3 = $true )
& ( setadjoinIL = $true )
& ( ubforcartprodlem2 = $true )
& ( symdiffIneg2 = $true )
& ( theprop = $true )
& ( dsetconstr__Cong = $true )
& ( exuE3u = $true )
& ( setunionAx = $true )
& ( quantDeMorgan1 = $true )
& ( setunionE2 = $true )
& ( nonemptyI = $true )
& ( binintersectER = $true )
& ( binunionIR = $true )
& ( symdiffE = $true )
& ( prop2setE = $true )
& ( binunionE = $true )
& ( emptysetE = $true )
& ( nonemptyImpWitness = $true )
& ( eqimpsubset1 = $true )
& ( inCongP = $true )
& ( notequalI2 = $true )
& ( subsetE2 = $true )
& ( exuI2 = $true )
& ( subPowSU = $true )
& ( binunionRsub = $true )
& ( ex1I = $true )
& ( binintersectSubset1 = $true )
& ( dsetconstrEL = $true )
& ( nonemptyI1 = $true )
& ( setminusILneg = $true ) ),
inference(flattening,[status(thm)],[f686]) ).
thf(f686,plain,
( ? [X0: $i,X1: $i] :
( ( ( setadjoin @ ( setadjoin @ X0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X0 @ ( setadjoin @ X1 @ emptyset ) ) @ emptyset ) )
!= ( setadjoin @ ( setadjoin @ X0 @ emptyset ) @ emptyset ) )
& ( X0 = X1 ) )
& ( setukpairinjR11 = $true )
& ( setukpairinjL = $true )
& ( setukpairinjL2 = $true )
& ( cartprodfstin = $true )
& ( kfstpairEq = $true )
& ( theprop = $true )
& ( kfstsingleton = $true )
& ( setukpairinjL1 = $true )
& ( singletonsuniq = $true )
& ( ex1I2 = $true )
& ( ex1I = $true )
& ( ex1E1 = $true )
& ( singletonprop = $true )
& ( setunionsingleton = $true )
& ( setunionsingleton2 = $true )
& ( setunionsingleton1 = $true )
& ( setunionE2 = $true )
& ( cartprodmempair = $true )
& ( cartprodmempair1 = $true )
& ( cartprodpairin = $true )
& ( ubforcartprodlem3 = $true )
& ( ubforcartprodlem2 = $true )
& ( ubforcartprodlem1 = $true )
& ( upairinpowunion = $true )
& ( upairsubunion = $true )
& ( upairset2E = $true )
& ( singletoninpowunion = $true )
& ( singletoninpowerset = $true )
& ( singletonsubset = $true )
& ( kpairp = $true )
& ( kpairiskpair = $true )
& ( setukpairIR = $true )
& ( setukpairIL = $true )
& ( secondinupair = $true )
& ( symdiffIneg2 = $true )
& ( symdiffIneg1 = $true )
& ( symdiffI2 = $true )
& ( symdiffI1 = $true )
& ( symdiffE = $true )
& ( setminusSubset1 = $true )
& ( setminusLsub = $true )
& ( setminusIRneg = $true )
& ( setminusILneg = $true )
& ( setminusELneg = $true )
& ( setminusERneg = $true )
& ( setminusSubset2 = $true )
& ( setminusER = $true )
& ( setminusEL = $true )
& ( setminusI = $true )
& ( bs114d = $true )
& ( binintersectSubset1 = $true )
& ( binintersectSubset4 = $true )
& ( binintersectRsub = $true )
& ( disjointsetsI1 = $true )
& ( binintersectER = $true )
& ( binintersectSubset3 = $true )
& ( binintersectSubset2 = $true )
& ( binintersectLsub = $true )
& ( binintersectEL = $true )
& ( binintersectSubset5 = $true )
& ( binintersectI = $true )
& ( binunionRsub = $true )
& ( binunionLsub = $true )
& ( binunionE = $true )
& ( binunionEcases = $true )
& ( binunionIR = $true )
& ( upairset2IR = $true )
& ( binunionIL = $true )
& ( sepSubset = $true )
& ( sepInPowerset = $true )
& ( powersetsubset = $true )
& ( inPowerset = $true )
& ( powersetE1 = $true )
& ( powersetI1 = $true )
& ( subsetemptysetimpeq = $true )
& ( setextsub = $true )
& ( subset2powerset = $true )
& ( setadjoinSub2 = $true )
& ( setadjoinSub = $true )
& ( subsetTrans = $true )
& ( subsetRefl = $true )
& ( notequalI2 = $true )
& ( notequalI1 = $true )
& ( notsubsetI = $true )
& ( subsetE2 = $true )
& ( subsetE = $true )
& ( emptysetsubset = $true )
& ( subsetI2 = $true )
& ( eqimpsubset1 = $true )
& ( eqimpsubset2 = $true )
& ( subsetI1 = $true )
& ( dsetconstr__Cong = $true )
& ( descr__Cong = $true )
& ( exuEu = $true )
& ( omega__Cong = $true )
& ( setunion__Cong = $true )
& ( powerset__Cong = $true )
& ( setadjoin__Cong = $true )
& ( emptyset__Cong = $true )
& ( exu__Cong = $true )
& ( exuE3u = $true )
& ( in__Cong = $true )
& ( inCongP = $true )
& ( exuI2 = $true )
& ( exuI3 = $true )
& ( exuI1 = $true )
& ( notdallE = $true )
& ( notdexE = $true )
& ( prop2set2propI = $true )
& ( prop2setI = $true )
& ( quantDeMorgan4 = $true )
& ( quantDeMorgan3 = $true )
& ( quantDeMorgan2 = $true )
& ( quantDeMorgan1 = $true )
& ( vacuousDall = $true )
& ( emptyE1 = $true )
& ( upairsetIR = $true )
& ( upairsetIL = $true )
& ( upairsetE = $true )
& ( singletonsswitch = $true )
& ( eqinunit = $true )
& ( notinsingleton = $true )
& ( uniqinunit = $true )
& ( nonemptyImpWitness = $true )
& ( exuE2 = $true )
& ( subPowSU = $true )
& ( setunionE = $true )
& ( setunionI = $true )
& ( powersetE = $true )
& ( emptyInPowerset = $true )
& ( emptyinPowerset = $true )
& ( powersetI = $true )
& ( setoftrueEq = $true )
& ( setadjoinOr = $true )
& ( setadjoinE = $true )
& ( setadjoinIR = $true )
& ( emptyinunitempty = $true )
& ( setadjoinIL = $true )
& ( nonemptyI1 = $true )
& ( nonemptyI = $true )
& ( nonemptyE1 = $true )
& ( setbeta = $true )
& ( noeltsimpempty = $true )
& ( emptyI = $true )
& ( setext = $true )
& ( exuE3e = $true )
& ( notinemptyset = $true )
& ( emptysetimpfalse = $true )
& ( emptysetE = $true )
& ( prop2setE = $true )
& ( exuE1 = $true )
& ( dsetconstrER = $true )
& ( dsetconstrEL = $true )
& ( dsetconstrI = $true )
& ( descrp = $true )
& ( wellorderingAx = $true )
& ( foundationAx = $true )
& ( replAx = $true )
& ( omegaIndAx = $true )
& ( omegaSAx = $true )
& ( omega0Ax = $true )
& ( setunionAx = $true )
& ( powersetAx = $true )
& ( setadjoinAx = $true )
& ( emptysetAx = $true )
& ( setextAx = $true ) ),
inference(ennf_transformation,[status(thm)],[f504]) ).
thf(f504,plain,
~ ( ( setextAx = $true )
=> ( ( emptysetAx = $true )
=> ( ( setadjoinAx = $true )
=> ( ( powersetAx = $true )
=> ( ( setunionAx = $true )
=> ( ( omega0Ax = $true )
=> ( ( omegaSAx = $true )
=> ( ( omegaIndAx = $true )
=> ( ( replAx = $true )
=> ( ( foundationAx = $true )
=> ( ( wellorderingAx = $true )
=> ( ( descrp = $true )
=> ( ( dsetconstrI = $true )
=> ( ( dsetconstrEL = $true )
=> ( ( dsetconstrER = $true )
=> ( ( exuE1 = $true )
=> ( ( prop2setE = $true )
=> ( ( emptysetE = $true )
=> ( ( emptysetimpfalse = $true )
=> ( ( notinemptyset = $true )
=> ( ( exuE3e = $true )
=> ( ( setext = $true )
=> ( ( emptyI = $true )
=> ( ( noeltsimpempty = $true )
=> ( ( setbeta = $true )
=> ( ( nonemptyE1 = $true )
=> ( ( nonemptyI = $true )
=> ( ( nonemptyI1 = $true )
=> ( ( setadjoinIL = $true )
=> ( ( emptyinunitempty = $true )
=> ( ( setadjoinIR = $true )
=> ( ( setadjoinE = $true )
=> ( ( setadjoinOr = $true )
=> ( ( setoftrueEq = $true )
=> ( ( powersetI = $true )
=> ( ( emptyinPowerset = $true )
=> ( ( emptyInPowerset = $true )
=> ( ( powersetE = $true )
=> ( ( setunionI = $true )
=> ( ( setunionE = $true )
=> ( ( subPowSU = $true )
=> ( ( exuE2 = $true )
=> ( ( nonemptyImpWitness = $true )
=> ( ( uniqinunit = $true )
=> ( ( notinsingleton = $true )
=> ( ( eqinunit = $true )
=> ( ( singletonsswitch = $true )
=> ( ( upairsetE = $true )
=> ( ( upairsetIL = $true )
=> ( ( upairsetIR = $true )
=> ( ( emptyE1 = $true )
=> ( ( vacuousDall = $true )
=> ( ( quantDeMorgan1 = $true )
=> ( ( quantDeMorgan2 = $true )
=> ( ( quantDeMorgan3 = $true )
=> ( ( quantDeMorgan4 = $true )
=> ( ( prop2setI = $true )
=> ( ( prop2set2propI = $true )
=> ( ( notdexE = $true )
=> ( ( notdallE = $true )
=> ( ( exuI1 = $true )
=> ( ( exuI3 = $true )
=> ( ( exuI2 = $true )
=> ( ( inCongP = $true )
=> ( ( in__Cong = $true )
=> ( ( exuE3u = $true )
=> ( ( exu__Cong = $true )
=> ( ( emptyset__Cong = $true )
=> ( ( setadjoin__Cong = $true )
=> ( ( powerset__Cong = $true )
=> ( ( setunion__Cong = $true )
=> ( ( omega__Cong = $true )
=> ( ( exuEu = $true )
=> ( ( descr__Cong = $true )
=> ( ( dsetconstr__Cong = $true )
=> ( ( subsetI1 = $true )
=> ( ( eqimpsubset2 = $true )
=> ( ( eqimpsubset1 = $true )
=> ( ( subsetI2 = $true )
=> ( ( emptysetsubset = $true )
=> ( ( subsetE = $true )
=> ( ( subsetE2 = $true )
=> ( ( notsubsetI = $true )
=> ( ( notequalI1 = $true )
=> ( ( notequalI2 = $true )
=> ( ( subsetRefl = $true )
=> ( ( subsetTrans = $true )
=> ( ( setadjoinSub = $true )
=> ( ( setadjoinSub2 = $true )
=> ( ( subset2powerset = $true )
=> ( ( setextsub = $true )
=> ( ( subsetemptysetimpeq = $true )
=> ( ( powersetI1 = $true )
=> ( ( powersetE1 = $true )
=> ( ( inPowerset = $true )
=> ( ( powersetsubset = $true )
=> ( ( sepInPowerset = $true )
=> ( ( sepSubset = $true )
=> ( ( binunionIL = $true )
=> ( ( upairset2IR = $true )
=> ( ( binunionIR = $true )
=> ( ( binunionEcases = $true )
=> ( ( binunionE = $true )
=> ( ( binunionLsub = $true )
=> ( ( binunionRsub = $true )
=> ( ( binintersectI = $true )
=> ( ( binintersectSubset5 = $true )
=> ( ( binintersectEL = $true )
=> ( ( binintersectLsub = $true )
=> ( ( binintersectSubset2 = $true )
=> ( ( binintersectSubset3 = $true )
=> ( ( binintersectER = $true )
=> ( ( disjointsetsI1 = $true )
=> ( ( binintersectRsub = $true )
=> ( ( binintersectSubset4 = $true )
=> ( ( binintersectSubset1 = $true )
=> ( ( bs114d = $true )
=> ( ( setminusI = $true )
=> ( ( setminusEL = $true )
=> ( ( setminusER = $true )
=> ( ( setminusSubset2 = $true )
=> ( ( setminusERneg = $true )
=> ( ( setminusELneg = $true )
=> ( ( setminusILneg = $true )
=> ( ( setminusIRneg = $true )
=> ( ( setminusLsub = $true )
=> ( ( setminusSubset1 = $true )
=> ( ( symdiffE = $true )
=> ( ( symdiffI1 = $true )
=> ( ( symdiffI2 = $true )
=> ( ( symdiffIneg1 = $true )
=> ( ( symdiffIneg2 = $true )
=> ( ( secondinupair = $true )
=> ( ( setukpairIL = $true )
=> ( ( setukpairIR = $true )
=> ( ( kpairiskpair = $true )
=> ( ( kpairp = $true )
=> ( ( singletonsubset = $true )
=> ( ( singletoninpowerset = $true )
=> ( ( singletoninpowunion = $true )
=> ( ( upairset2E = $true )
=> ( ( upairsubunion = $true )
=> ( ( upairinpowunion = $true )
=> ( ( ubforcartprodlem1 = $true )
=> ( ( ubforcartprodlem2 = $true )
=> ( ( ubforcartprodlem3 = $true )
=> ( ( cartprodpairin = $true )
=> ( ( cartprodmempair1 = $true )
=> ( ( cartprodmempair = $true )
=> ( ( setunionE2 = $true )
=> ( ( setunionsingleton1 = $true )
=> ( ( setunionsingleton2 = $true )
=> ( ( setunionsingleton = $true )
=> ( ( singletonprop = $true )
=> ( ( ex1E1 = $true )
=> ( ( ex1I = $true )
=> ( ( ex1I2 = $true )
=> ( ( singletonsuniq = $true )
=> ( ( setukpairinjL1 = $true )
=> ( ( kfstsingleton = $true )
=> ( ( theprop = $true )
=> ( ( kfstpairEq = $true )
=> ( ( cartprodfstin = $true )
=> ( ( setukpairinjL2 = $true )
=> ( ( setukpairinjL = $true )
=> ( ( setukpairinjR11 = $true )
=> ! [X0: $i,X1: $i] :
( ( X0 = X1 )
=> ( ( setadjoin @ ( setadjoin @ X0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X0 @ ( setadjoin @ X1 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X0 @ emptyset ) @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f503]) ).
thf(f503,plain,
~ ( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> ( setunionE2
=> ( setunionsingleton1
=> ( setunionsingleton2
=> ( setunionsingleton
=> ( singletonprop
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> ( setukpairinjL1
=> ( kfstsingleton
=> ( theprop
=> ( kfstpairEq
=> ( cartprodfstin
=> ( setukpairinjL2
=> ( setukpairinjL
=> ( setukpairinjR11
=> ! [X0: $i,X1: $i] :
( ( X0 = X1 )
=> ( ( setadjoin @ ( setadjoin @ X0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X0 @ ( setadjoin @ X1 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X0 @ emptyset ) @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f174]) ).
thf(f174,negated_conjecture,
~ ( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> ( setunionE2
=> ( setunionsingleton1
=> ( setunionsingleton2
=> ( setunionsingleton
=> ( singletonprop
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> ( setukpairinjL1
=> ( kfstsingleton
=> ( theprop
=> ( kfstpairEq
=> ( cartprodfstin
=> ( setukpairinjL2
=> ( setukpairinjL
=> ( setukpairinjR11
=> ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f173]) ).
thf(f173,conjecture,
( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> ( setunionE2
=> ( setunionsingleton1
=> ( setunionsingleton2
=> ( setunionsingleton
=> ( singletonprop
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> ( setukpairinjL1
=> ( kfstsingleton
=> ( theprop
=> ( kfstpairEq
=> ( cartprodfstin
=> ( setukpairinjL2
=> ( setukpairinjL
=> ( setukpairinjR11
=> ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.mID8C0AdqD/DTF2THF_11959.p',setukpairinjR12) ).
thf(f1695,plain,
! [X0: $i,X1: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X1 @ ( setadjoin @ X0 @ emptyset ) ) )
| ( X0 != X1 )
| ( setukpairinjR11 != $true ) ),
inference(cnf_transformation,[status(thm)],[f965]) ).
thf(f965,plain,
( ( ! [X0: $i,X1: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X1 @ ( setadjoin @ X0 @ emptyset ) ) )
| ( X0 != X1 ) )
| ( setukpairinjR11 != $true ) )
& ( ( setukpairinjR11 = $true )
| ( ( ( setadjoin @ sK145 @ ( setadjoin @ sK144 @ emptyset ) )
!= ( setadjoin @ sK145 @ emptyset ) )
& ( sK144 = sK145 ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK144,sK145])],[f963,f964]) ).
thf(f964,plain,
( ? [X2: $i,X3: $i] :
( ( ( setadjoin @ X3 @ emptyset )
!= ( setadjoin @ X3 @ ( setadjoin @ X2 @ emptyset ) ) )
& ( X2 = X3 ) )
=> ( ( ( setadjoin @ sK145 @ ( setadjoin @ sK144 @ emptyset ) )
!= ( setadjoin @ sK145 @ emptyset ) )
& ( sK144 = sK145 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f963,plain,
( ( ! [X0: $i,X1: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X1 @ ( setadjoin @ X0 @ emptyset ) ) )
| ( X0 != X1 ) )
| ( setukpairinjR11 != $true ) )
& ( ( setukpairinjR11 = $true )
| ? [X2: $i,X3: $i] :
( ( ( setadjoin @ X3 @ emptyset )
!= ( setadjoin @ X3 @ ( setadjoin @ X2 @ emptyset ) ) )
& ( X2 = X3 ) ) ) ),
inference(rectify,[status(thm)],[f962]) ).
thf(f962,plain,
( ( ! [X0: $i,X1: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X1 @ ( setadjoin @ X0 @ emptyset ) ) )
| ( X0 != X1 ) )
| ( setukpairinjR11 != $true ) )
& ( ( setukpairinjR11 = $true )
| ? [X0: $i,X1: $i] :
( ( ( setadjoin @ X1 @ emptyset )
!= ( setadjoin @ X1 @ ( setadjoin @ X0 @ emptyset ) ) )
& ( X0 = X1 ) ) ) ),
inference(nnf_transformation,[status(thm)],[f597]) ).
thf(f597,plain,
( ! [X0: $i,X1: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X1 @ ( setadjoin @ X0 @ emptyset ) ) )
| ( X0 != X1 ) )
<=> ( setukpairinjR11 = $true ) ),
inference(ennf_transformation,[status(thm)],[f378]) ).
thf(f378,plain,
( ( setukpairinjR11 = $true )
<=> ! [X0: $i,X1: $i] :
( ( X0 = X1 )
=> ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X1 @ ( setadjoin @ X0 @ emptyset ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f377]) ).
thf(f377,plain,
! [X0: $i,X1: $i] :
( ( ( X0 = X1 )
=> ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X1 @ ( setadjoin @ X0 @ emptyset ) ) ) )
= setukpairinjR11 ),
inference(rectify,[status(thm)],[f172]) ).
thf(f172,axiom,
! [X2: $i,X1: $i] :
( ( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) ) )
= setukpairinjR11 ),
file('/export/starexec/sandbox/tmp/tmp.mID8C0AdqD/DTF2THF_11959.p',setukpairinjR11) ).
thf(f3196,plain,
( ( setadjoin @ ( setadjoin @ sK328 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK328 @ emptyset ) @ emptyset ) )
!= ( setadjoin @ ( setadjoin @ sK328 @ emptyset ) @ emptyset ) ),
inference(forward_demodulation,[status(thm)],[f2713,f2966]) ).
thf(f2713,plain,
( ( setadjoin @ ( setadjoin @ sK328 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK328 @ ( setadjoin @ sK328 @ emptyset ) ) @ emptyset ) )
!= ( setadjoin @ ( setadjoin @ sK328 @ emptyset ) @ emptyset ) ),
inference(definition_unfolding,[status(thm)],[f2053,f2052,f2052,f2052]) ).
thf(f2052,plain,
sK327 = sK328,
inference(cnf_transformation,[status(thm)],[f1265]) ).
thf(f2053,plain,
( ( setadjoin @ ( setadjoin @ sK327 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK327 @ ( setadjoin @ sK328 @ emptyset ) ) @ emptyset ) )
!= ( setadjoin @ ( setadjoin @ sK327 @ emptyset ) @ emptyset ) ),
inference(cnf_transformation,[status(thm)],[f1265]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SEU650^1 : TPTP v9.2.1. Released v3.7.0.
% 0.12/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.16/0.34 % Computer : n029.cluster.edu
% 0.16/0.34 % Model : x86_64 x86_64
% 0.16/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.34 % Memory : 8042.1875MB
% 0.16/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.34 % CPULimit : 300
% 0.16/0.34 % WCLimit : 300
% 0.16/0.34 % DateTime : Tue Feb 24 12:29:29 EST 2026
% 0.16/0.34 % CPUTime :
% 0.16/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.31/0.46 ---- Original DTF file ---
% 0.31/0.46 thf(spec,logic,$$dhol).
% 0.31/0.46 %------------------------------------------------------------------------------
% 0.31/0.46 % File : SEU650^1 : TPTP v9.2.1. Released v3.7.0.
% 0.31/0.46 % Domain : Set Theory
% 0.31/0.46 % Problem : Ordered Pairs - Properties of Pairs
% 0.31/0.46 % Version : Especial.
% 0.31/0.46 % English : (! x:i.! y:i.x = y -> setadjoin (setadjoin x emptyset) (setadjoin
% 0.31/0.46 % (setadjoin x (setadjoin y emptyset)) emptyset) = setadjoin
% 0.31/0.46 % (setadjoin x emptyset) emptyset)
% 0.31/0.46
% 0.31/0.46 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.31/0.46 % Source : [Bro08]
% 0.31/0.46 % Names : ZFC152g [Bro08]
% 0.31/0.46
% 0.31/0.46 % Status : Theorem
% 0.31/0.46 % Rating : 0.11 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, 0.29 v6.1.0, 0.43 v5.5.0, 0.50 v5.4.0, 0.60 v5.2.0, 0.80 v5.0.0, 0.60 v4.1.0, 1.00 v3.7.0
% 0.31/0.46 % Syntax : Number of formulae : 369 ( 172 unt; 196 typ; 172 def)
% 0.31/0.46 % Number of atoms : 911 ( 250 equ; 0 cnn)
% 0.31/0.46 % Maximal formula atoms : 168 ( 5 avg)
% 0.31/0.46 % Number of connectives : 1596 ( 47 ~; 7 |; 37 &;1089 @)
% 0.31/0.46 % ( 14 <=>; 402 =>; 0 <=; 0 <~>)
% 0.31/0.46 % Maximal formula depth : 170 ( 2 avg)
% 0.31/0.46 % Number of types : 2 ( 0 usr)
% 0.31/0.46 % Number of type conns : 83 ( 83 >; 0 *; 0 +; 0 <<)
% 0.31/0.46 % Number of symbols : 199 ( 196 usr; 170 con; 0-2 aty)
% 0.31/0.46 % Number of variables : 500 ( 42 ^; 420 !; 38 ?; 500 :)
% 0.31/0.46 % SPC : TH0_THM_EQU_NAR
% 0.31/0.46
% 0.31/0.46 % Comments : http://mathgate.info/detsetitem.php?id=206
% 0.31/0.46 %------------------------------------------------------------------------------
% 0.31/0.46 thf(in_type,type,
% 0.31/0.46 in: $i > $i > $o ).
% 0.31/0.46
% 0.31/0.46 thf(exu_type,type,
% 0.31/0.46 exu: ( $i > $o ) > $o ).
% 0.31/0.46
% 0.31/0.46 thf(exu,definition,
% 0.31/0.46 ( exu
% 0.31/0.46 = ( ^ [Xphi: $i > $o] :
% 0.31/0.46 ? [Xx: $i] :
% 0.31/0.46 ( ( Xphi @ Xx )
% 0.31/0.46 & ! [Xy: $i] :
% 0.31/0.46 ( ( Xphi @ Xy )
% 0.31/0.46 => ( Xx = Xy ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(setextAx_type,type,
% 0.31/0.46 setextAx: $o ).
% 0.31/0.46
% 0.31/0.46 thf(setextAx,definition,
% 0.31/0.46 ( setextAx
% 0.31/0.46 = ( ! [A: $i,B: $i] :
% 0.31/0.46 ( ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 <=> ( in @ Xx @ B ) )
% 0.31/0.46 => ( A = B ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(emptyset_type,type,
% 0.31/0.46 emptyset: $i ).
% 0.31/0.46
% 0.31/0.46 thf(emptysetAx_type,type,
% 0.31/0.46 emptysetAx: $o ).
% 0.31/0.46
% 0.31/0.46 thf(emptysetAx,definition,
% 0.31/0.46 ( emptysetAx
% 0.31/0.46 = ( ! [Xx: $i] :
% 0.31/0.46 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(setadjoin_type,type,
% 0.31/0.46 setadjoin: $i > $i > $i ).
% 0.31/0.46
% 0.31/0.46 thf(setadjoinAx_type,type,
% 0.31/0.46 setadjoinAx: $o ).
% 0.31/0.46
% 0.31/0.46 thf(setadjoinAx,definition,
% 0.31/0.46 ( setadjoinAx
% 0.31/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.31/0.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.31/0.46 <=> ( ( Xy = Xx )
% 0.31/0.46 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(powerset_type,type,
% 0.31/0.46 powerset: $i > $i ).
% 0.31/0.46
% 0.31/0.46 thf(powersetAx_type,type,
% 0.31/0.46 powersetAx: $o ).
% 0.31/0.46
% 0.31/0.46 thf(powersetAx,definition,
% 0.31/0.46 ( powersetAx
% 0.31/0.46 = ( ! [A: $i,B: $i] :
% 0.31/0.46 ( ( in @ B @ ( powerset @ A ) )
% 0.31/0.46 <=> ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ B )
% 0.31/0.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(setunion_type,type,
% 0.31/0.46 setunion: $i > $i ).
% 0.31/0.46
% 0.31/0.46 thf(setunionAx_type,type,
% 0.31/0.46 setunionAx: $o ).
% 0.31/0.46
% 0.31/0.46 thf(setunionAx,definition,
% 0.31/0.46 ( setunionAx
% 0.31/0.46 = ( ! [A: $i,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ ( setunion @ A ) )
% 0.31/0.46 <=> ? [B: $i] :
% 0.31/0.46 ( ( in @ Xx @ B )
% 0.31/0.46 & ( in @ B @ A ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(omega_type,type,
% 0.31/0.46 omega: $i ).
% 0.31/0.46
% 0.31/0.46 thf(omega0Ax_type,type,
% 0.31/0.46 omega0Ax: $o ).
% 0.31/0.46
% 0.31/0.46 thf(omega0Ax,definition,
% 0.31/0.46 ( omega0Ax
% 0.31/0.46 = ( in @ emptyset @ omega ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(omegaSAx_type,type,
% 0.31/0.46 omegaSAx: $o ).
% 0.31/0.46
% 0.31/0.46 thf(omegaSAx,definition,
% 0.31/0.46 ( omegaSAx
% 0.31/0.46 = ( ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ omega )
% 0.31/0.46 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(omegaIndAx_type,type,
% 0.31/0.46 omegaIndAx: $o ).
% 0.31/0.46
% 0.31/0.46 thf(omegaIndAx,definition,
% 0.31/0.46 ( omegaIndAx
% 0.31/0.46 = ( ! [A: $i] :
% 0.31/0.46 ( ( ( in @ emptyset @ A )
% 0.31/0.46 & ! [Xx: $i] :
% 0.31/0.46 ( ( ( in @ Xx @ omega )
% 0.31/0.46 & ( in @ Xx @ A ) )
% 0.31/0.46 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.31/0.46 => ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ omega )
% 0.31/0.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(replAx_type,type,
% 0.31/0.46 replAx: $o ).
% 0.31/0.46
% 0.31/0.46 thf(replAx,definition,
% 0.31/0.46 ( replAx
% 0.31/0.46 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.31/0.46 ( ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ( exu
% 0.31/0.46 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.31/0.46 => ? [B: $i] :
% 0.31/0.46 ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ B )
% 0.31/0.46 <=> ? [Xy: $i] :
% 0.31/0.46 ( ( in @ Xy @ A )
% 0.31/0.46 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(foundationAx_type,type,
% 0.31/0.46 foundationAx: $o ).
% 0.31/0.46
% 0.31/0.46 thf(foundationAx,definition,
% 0.31/0.46 ( foundationAx
% 0.31/0.46 = ( ! [A: $i] :
% 0.31/0.46 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.31/0.46 => ? [B: $i] :
% 0.31/0.46 ( ( in @ B @ A )
% 0.31/0.46 & ~ ? [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ B )
% 0.31/0.46 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(wellorderingAx_type,type,
% 0.31/0.46 wellorderingAx: $o ).
% 0.31/0.46
% 0.31/0.46 thf(wellorderingAx,definition,
% 0.31/0.46 ( wellorderingAx
% 0.31/0.46 = ( ! [A: $i] :
% 0.31/0.46 ? [B: $i] :
% 0.31/0.46 ( ! [C: $i] :
% 0.31/0.46 ( ( in @ C @ B )
% 0.31/0.46 => ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ C )
% 0.31/0.46 => ( in @ Xx @ A ) ) )
% 0.31/0.46 & ! [Xx: $i,Xy: $i] :
% 0.31/0.46 ( ( ( in @ Xx @ A )
% 0.31/0.46 & ( in @ Xy @ A ) )
% 0.31/0.46 => ( ! [C: $i] :
% 0.31/0.46 ( ( in @ C @ B )
% 0.31/0.46 => ( ( in @ Xx @ C )
% 0.31/0.46 <=> ( in @ Xy @ C ) ) )
% 0.31/0.46 => ( Xx = Xy ) ) )
% 0.31/0.46 & ! [C: $i,D: $i] :
% 0.31/0.46 ( ( ( in @ C @ B )
% 0.31/0.46 & ( in @ D @ B ) )
% 0.31/0.46 => ( ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ C )
% 0.31/0.46 => ( in @ Xx @ D ) )
% 0.31/0.46 | ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ D )
% 0.31/0.46 => ( in @ Xx @ C ) ) ) )
% 0.31/0.46 & ! [C: $i] :
% 0.31/0.46 ( ( ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ C )
% 0.31/0.46 => ( in @ Xx @ A ) )
% 0.31/0.46 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.31/0.46 => ? [D: $i,Xx: $i] :
% 0.31/0.46 ( ( in @ D @ B )
% 0.31/0.46 & ( in @ Xx @ C )
% 0.31/0.46 & ~ ? [Xy: $i] :
% 0.31/0.46 ( ( in @ Xy @ D )
% 0.31/0.46 & ( in @ Xy @ C ) )
% 0.31/0.46 & ! [E: $i] :
% 0.31/0.46 ( ( in @ E @ B )
% 0.31/0.46 => ( ! [Xy: $i] :
% 0.31/0.46 ( ( in @ Xy @ E )
% 0.31/0.46 => ( in @ Xy @ D ) )
% 0.31/0.46 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(descr_type,type,
% 0.31/0.46 descr: ( $i > $o ) > $i ).
% 0.31/0.46
% 0.31/0.46 thf(descrp_type,type,
% 0.31/0.46 descrp: $o ).
% 0.31/0.46
% 0.31/0.46 thf(descrp,definition,
% 0.31/0.46 ( descrp
% 0.31/0.46 = ( ! [Xphi: $i > $o] :
% 0.31/0.46 ( ( exu
% 0.31/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.46 => ( Xphi
% 0.31/0.46 @ ( descr
% 0.31/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(dsetconstr_type,type,
% 0.31/0.46 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.31/0.46
% 0.31/0.46 thf(dsetconstrI_type,type,
% 0.31/0.46 dsetconstrI: $o ).
% 0.31/0.46
% 0.31/0.46 thf(dsetconstrI,definition,
% 0.31/0.46 ( dsetconstrI
% 0.31/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ( ( Xphi @ Xx )
% 0.31/0.46 => ( in @ Xx
% 0.31/0.46 @ ( dsetconstr @ A
% 0.31/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(dsetconstrEL_type,type,
% 0.31/0.46 dsetconstrEL: $o ).
% 0.31/0.46
% 0.31/0.46 thf(dsetconstrEL,definition,
% 0.31/0.46 ( dsetconstrEL
% 0.31/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx
% 0.31/0.46 @ ( dsetconstr @ A
% 0.31/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.31/0.46 => ( in @ Xx @ A ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(dsetconstrER_type,type,
% 0.31/0.46 dsetconstrER: $o ).
% 0.31/0.46
% 0.31/0.46 thf(dsetconstrER,definition,
% 0.31/0.46 ( dsetconstrER
% 0.31/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx
% 0.31/0.46 @ ( dsetconstr @ A
% 0.31/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.31/0.46 => ( Xphi @ Xx ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(exuE1_type,type,
% 0.31/0.46 exuE1: $o ).
% 0.31/0.46
% 0.31/0.46 thf(exuE1,definition,
% 0.31/0.46 ( exuE1
% 0.31/0.46 = ( ! [Xphi: $i > $o] :
% 0.31/0.46 ( ( exu
% 0.31/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.46 => ? [Xx: $i] :
% 0.31/0.46 ( ( Xphi @ Xx )
% 0.31/0.46 & ! [Xy: $i] :
% 0.31/0.46 ( ( Xphi @ Xy )
% 0.31/0.46 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(prop2set_type,type,
% 0.31/0.46 prop2set: $o > $i ).
% 0.31/0.46
% 0.31/0.46 thf(prop2setE_type,type,
% 0.31/0.46 prop2setE: $o ).
% 0.31/0.46
% 0.31/0.46 thf(prop2setE,definition,
% 0.31/0.47 ( prop2setE
% 0.31/0.47 = ( ! [Xphi: $o,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.31/0.47 => Xphi ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(emptysetE_type,type,
% 0.31/0.47 emptysetE: $o ).
% 0.31/0.47
% 0.31/0.47 thf(emptysetE,definition,
% 0.31/0.47 ( emptysetE
% 0.31/0.47 = ( ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ emptyset )
% 0.31/0.47 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(emptysetimpfalse_type,type,
% 0.31/0.47 emptysetimpfalse: $o ).
% 0.31/0.47
% 0.31/0.47 thf(emptysetimpfalse,definition,
% 0.31/0.47 ( emptysetimpfalse
% 0.31/0.47 = ( ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ emptyset )
% 0.31/0.47 => $false ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(notinemptyset_type,type,
% 0.31/0.47 notinemptyset: $o ).
% 0.31/0.47
% 0.31/0.47 thf(notinemptyset,definition,
% 0.31/0.47 ( notinemptyset
% 0.31/0.47 = ( ! [Xx: $i] :
% 0.31/0.47 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(exuE3e_type,type,
% 0.31/0.47 exuE3e: $o ).
% 0.31/0.47
% 0.31/0.47 thf(exuE3e,definition,
% 0.31/0.47 ( exuE3e
% 0.31/0.47 = ( ! [Xphi: $i > $o] :
% 0.31/0.47 ( ( exu
% 0.31/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.47 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setext_type,type,
% 0.31/0.47 setext: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setext,definition,
% 0.31/0.47 ( setext
% 0.31/0.47 = ( ! [A: $i,B: $i] :
% 0.31/0.47 ( ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( in @ Xx @ B ) )
% 0.31/0.47 => ( ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ B )
% 0.31/0.47 => ( in @ Xx @ A ) )
% 0.31/0.47 => ( A = B ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(emptyI_type,type,
% 0.31/0.47 emptyI: $o ).
% 0.31/0.47
% 0.31/0.47 thf(emptyI,definition,
% 0.31/0.47 ( emptyI
% 0.31/0.47 = ( ! [A: $i] :
% 0.31/0.47 ( ! [Xx: $i] :
% 0.31/0.47 ~ ( in @ Xx @ A )
% 0.31/0.47 => ( A = emptyset ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(noeltsimpempty_type,type,
% 0.31/0.47 noeltsimpempty: $o ).
% 0.31/0.47
% 0.31/0.47 thf(noeltsimpempty,definition,
% 0.31/0.47 ( noeltsimpempty
% 0.31/0.47 = ( ! [A: $i] :
% 0.31/0.47 ( ! [Xx: $i] :
% 0.31/0.47 ~ ( in @ Xx @ A )
% 0.31/0.47 => ( A = emptyset ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setbeta_type,type,
% 0.31/0.47 setbeta: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setbeta,definition,
% 0.31/0.47 ( setbeta
% 0.31/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( ( in @ Xx
% 0.31/0.47 @ ( dsetconstr @ A
% 0.31/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.31/0.47 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(nonempty_type,type,
% 0.31/0.47 nonempty: $i > $o ).
% 0.31/0.47
% 0.31/0.47 thf(nonempty,definition,
% 0.31/0.47 ( nonempty
% 0.31/0.47 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(nonemptyE1_type,type,
% 0.31/0.47 nonemptyE1: $o ).
% 0.31/0.47
% 0.31/0.47 thf(nonemptyE1,definition,
% 0.31/0.47 ( nonemptyE1
% 0.31/0.47 = ( ! [A: $i] :
% 0.31/0.47 ( ( nonempty @ A )
% 0.31/0.47 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(nonemptyI_type,type,
% 0.31/0.47 nonemptyI: $o ).
% 0.31/0.47
% 0.31/0.47 thf(nonemptyI,definition,
% 0.31/0.47 ( nonemptyI
% 0.31/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( ( Xphi @ Xx )
% 0.31/0.47 => ( nonempty
% 0.31/0.47 @ ( dsetconstr @ A
% 0.31/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(nonemptyI1_type,type,
% 0.31/0.47 nonemptyI1: $o ).
% 0.31/0.47
% 0.31/0.47 thf(nonemptyI1,definition,
% 0.31/0.47 ( nonemptyI1
% 0.31/0.47 = ( ! [A: $i] :
% 0.31/0.47 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.31/0.47 => ( nonempty @ A ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setadjoinIL_type,type,
% 0.31/0.47 setadjoinIL: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setadjoinIL,definition,
% 0.31/0.47 ( setadjoinIL
% 0.31/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(emptyinunitempty_type,type,
% 0.31/0.47 emptyinunitempty: $o ).
% 0.31/0.47
% 0.31/0.47 thf(emptyinunitempty,definition,
% 0.31/0.47 ( emptyinunitempty
% 0.31/0.47 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setadjoinIR_type,type,
% 0.31/0.47 setadjoinIR: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setadjoinIR,definition,
% 0.31/0.47 ( setadjoinIR
% 0.31/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.31/0.47 ( ( in @ Xy @ A )
% 0.31/0.47 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setadjoinE_type,type,
% 0.31/0.47 setadjoinE: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setadjoinE,definition,
% 0.31/0.47 ( setadjoinE
% 0.31/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.31/0.47 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.31/0.47 => ! [Xphi: $o] :
% 0.31/0.47 ( ( ( Xy = Xx )
% 0.31/0.47 => Xphi )
% 0.31/0.47 => ( ( ( in @ Xy @ A )
% 0.31/0.47 => Xphi )
% 0.31/0.47 => Xphi ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setadjoinOr_type,type,
% 0.31/0.47 setadjoinOr: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setadjoinOr,definition,
% 0.31/0.47 ( setadjoinOr
% 0.31/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.31/0.47 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.31/0.47 => ( ( Xy = Xx )
% 0.31/0.47 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setoftrueEq_type,type,
% 0.31/0.47 setoftrueEq: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setoftrueEq,definition,
% 0.31/0.47 ( setoftrueEq
% 0.31/0.47 = ( ! [A: $i] :
% 0.31/0.47 ( ( dsetconstr @ A
% 0.31/0.47 @ ^ [Xx: $i] : $true )
% 0.31/0.47 = A ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(powersetI_type,type,
% 0.31/0.47 powersetI: $o ).
% 0.31/0.47
% 0.31/0.47 thf(powersetI,definition,
% 0.31/0.47 ( powersetI
% 0.31/0.47 = ( ! [A: $i,B: $i] :
% 0.31/0.47 ( ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ B )
% 0.31/0.47 => ( in @ Xx @ A ) )
% 0.31/0.47 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(emptyinPowerset_type,type,
% 0.31/0.47 emptyinPowerset: $o ).
% 0.31/0.47
% 0.31/0.47 thf(emptyinPowerset,definition,
% 0.31/0.47 ( emptyinPowerset
% 0.31/0.47 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(emptyInPowerset_type,type,
% 0.31/0.47 emptyInPowerset: $o ).
% 0.31/0.47
% 0.31/0.47 thf(emptyInPowerset,definition,
% 0.31/0.47 ( emptyInPowerset
% 0.31/0.47 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(powersetE_type,type,
% 0.31/0.47 powersetE: $o ).
% 0.31/0.47
% 0.31/0.47 thf(powersetE,definition,
% 0.31/0.47 ( powersetE
% 0.31/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.47 ( ( in @ B @ ( powerset @ A ) )
% 0.31/0.47 => ( ( in @ Xx @ B )
% 0.31/0.47 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setunionI_type,type,
% 0.31/0.47 setunionI: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setunionI,definition,
% 0.31/0.47 ( setunionI
% 0.31/0.47 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.31/0.47 ( ( in @ Xx @ B )
% 0.31/0.47 => ( ( in @ B @ A )
% 0.31/0.47 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setunionE_type,type,
% 0.31/0.47 setunionE: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setunionE,definition,
% 0.31/0.47 ( setunionE
% 0.31/0.47 = ( ! [A: $i,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ ( setunion @ A ) )
% 0.31/0.47 => ! [Xphi: $o] :
% 0.31/0.47 ( ! [B: $i] :
% 0.31/0.47 ( ( in @ Xx @ B )
% 0.31/0.47 => ( ( in @ B @ A )
% 0.31/0.47 => Xphi ) )
% 0.31/0.47 => Xphi ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(subPowSU_type,type,
% 0.31/0.47 subPowSU: $o ).
% 0.31/0.47
% 0.31/0.47 thf(subPowSU,definition,
% 0.31/0.47 ( subPowSU
% 0.31/0.47 = ( ! [A: $i,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(exuE2_type,type,
% 0.31/0.47 exuE2: $o ).
% 0.31/0.47
% 0.31/0.47 thf(exuE2,definition,
% 0.31/0.47 ( exuE2
% 0.31/0.47 = ( ! [Xphi: $i > $o] :
% 0.31/0.47 ( ( exu
% 0.31/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.47 => ? [Xx: $i] :
% 0.31/0.47 ! [Xy: $i] :
% 0.31/0.47 ( ( Xphi @ Xy )
% 0.31/0.47 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(nonemptyImpWitness_type,type,
% 0.31/0.47 nonemptyImpWitness: $o ).
% 0.31/0.47
% 0.31/0.47 thf(nonemptyImpWitness,definition,
% 0.31/0.47 ( nonemptyImpWitness
% 0.31/0.47 = ( ! [A: $i] :
% 0.31/0.47 ( ( nonempty @ A )
% 0.31/0.47 => ? [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 & $true ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(uniqinunit_type,type,
% 0.31/0.47 uniqinunit: $o ).
% 0.31/0.47
% 0.31/0.47 thf(uniqinunit,definition,
% 0.31/0.47 ( uniqinunit
% 0.31/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.31/0.47 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.31/0.47 => ( Xx = Xy ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(notinsingleton_type,type,
% 0.31/0.47 notinsingleton: $o ).
% 0.31/0.47
% 0.31/0.47 thf(notinsingleton,definition,
% 0.31/0.47 ( notinsingleton
% 0.31/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.31/0.47 ( ( Xx != Xy )
% 0.31/0.47 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(eqinunit_type,type,
% 0.31/0.47 eqinunit: $o ).
% 0.31/0.47
% 0.31/0.47 thf(eqinunit,definition,
% 0.31/0.47 ( eqinunit
% 0.31/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.31/0.47 ( ( Xx = Xy )
% 0.31/0.47 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(singletonsswitch_type,type,
% 0.31/0.47 singletonsswitch: $o ).
% 0.31/0.47
% 0.31/0.47 thf(singletonsswitch,definition,
% 0.31/0.47 ( singletonsswitch
% 0.31/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.31/0.47 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.31/0.47 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(upairsetE_type,type,
% 0.31/0.47 upairsetE: $o ).
% 0.31/0.47
% 0.31/0.47 thf(upairsetE,definition,
% 0.31/0.47 ( upairsetE
% 0.31/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.31/0.47 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.31/0.47 => ( ( Xz = Xx )
% 0.31/0.47 | ( Xz = Xy ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(upairsetIL_type,type,
% 0.31/0.47 upairsetIL: $o ).
% 0.31/0.47
% 0.31/0.47 thf(upairsetIL,definition,
% 0.31/0.47 ( upairsetIL
% 0.31/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(upairsetIR_type,type,
% 0.31/0.47 upairsetIR: $o ).
% 0.31/0.47
% 0.31/0.47 thf(upairsetIR,definition,
% 0.31/0.47 ( upairsetIR
% 0.31/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(emptyE1_type,type,
% 0.31/0.47 emptyE1: $o ).
% 0.31/0.47
% 0.31/0.47 thf(emptyE1,definition,
% 0.31/0.47 ( emptyE1
% 0.31/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/0.47 ( ? [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 & ( Xphi @ Xx ) )
% 0.31/0.47 => ( ( ( dsetconstr @ A
% 0.31/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.47 = emptyset )
% 0.31/0.47 => $false ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(vacuousDall_type,type,
% 0.31/0.47 vacuousDall: $o ).
% 0.31/0.47
% 0.31/0.47 thf(vacuousDall,definition,
% 0.31/0.47 ( vacuousDall
% 0.31/0.47 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ emptyset )
% 0.31/0.47 => ( Xphi @ Xx ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(quantDeMorgan1_type,type,
% 0.31/0.47 quantDeMorgan1: $o ).
% 0.31/0.47
% 0.31/0.47 thf(quantDeMorgan1,definition,
% 0.31/0.47 ( quantDeMorgan1
% 0.31/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/0.47 ( ~ ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( Xphi @ Xx ) )
% 0.31/0.47 => ? [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(quantDeMorgan2_type,type,
% 0.31/0.47 quantDeMorgan2: $o ).
% 0.31/0.47
% 0.31/0.47 thf(quantDeMorgan2,definition,
% 0.31/0.47 ( quantDeMorgan2
% 0.31/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/0.47 ( ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ~ ( Xphi @ Xx ) )
% 0.31/0.47 => ~ ? [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(quantDeMorgan3_type,type,
% 0.31/0.47 quantDeMorgan3: $o ).
% 0.31/0.47
% 0.31/0.47 thf(quantDeMorgan3,definition,
% 0.31/0.47 ( quantDeMorgan3
% 0.31/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/0.47 ( ~ ? [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 & ( Xphi @ Xx ) )
% 0.31/0.47 => ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(quantDeMorgan4_type,type,
% 0.31/0.47 quantDeMorgan4: $o ).
% 0.31/0.47
% 0.31/0.47 thf(quantDeMorgan4,definition,
% 0.31/0.47 ( quantDeMorgan4
% 0.31/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/0.47 ( ? [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 & ~ ( Xphi @ Xx ) )
% 0.31/0.47 => ~ ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(prop2setI_type,type,
% 0.31/0.47 prop2setI: $o ).
% 0.31/0.47
% 0.31/0.47 thf(prop2setI,definition,
% 0.31/0.47 ( prop2setI
% 0.31/0.47 = ( ! [Xphi: $o] :
% 0.31/0.47 ( Xphi
% 0.31/0.47 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(set2prop_type,type,
% 0.31/0.47 set2prop: $i > $o ).
% 0.31/0.47
% 0.31/0.47 thf(prop2set2propI_type,type,
% 0.31/0.47 prop2set2propI: $o ).
% 0.31/0.47
% 0.31/0.47 thf(prop2set2propI,definition,
% 0.31/0.47 ( prop2set2propI
% 0.31/0.47 = ( ! [Xphi: $o] :
% 0.31/0.47 ( Xphi
% 0.31/0.47 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(notdexE_type,type,
% 0.31/0.47 notdexE: $o ).
% 0.31/0.47
% 0.31/0.47 thf(notdexE,definition,
% 0.31/0.47 ( notdexE
% 0.31/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/0.47 ( ~ ? [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 & ( Xphi @ Xx ) )
% 0.31/0.47 => ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(notdallE_type,type,
% 0.31/0.47 notdallE: $o ).
% 0.31/0.47
% 0.31/0.47 thf(notdallE,definition,
% 0.31/0.47 ( notdallE
% 0.31/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/0.47 ( ~ ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( Xphi @ Xx ) )
% 0.31/0.47 => ? [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(exuI1_type,type,
% 0.31/0.47 exuI1: $o ).
% 0.31/0.47
% 0.31/0.47 thf(exuI1,definition,
% 0.31/0.47 ( exuI1
% 0.31/0.47 = ( ! [Xphi: $i > $o] :
% 0.31/0.47 ( ? [Xx: $i] :
% 0.31/0.47 ( ( Xphi @ Xx )
% 0.31/0.47 & ! [Xy: $i] :
% 0.31/0.47 ( ( Xphi @ Xy )
% 0.31/0.47 => ( Xx = Xy ) ) )
% 0.31/0.47 => ( exu
% 0.31/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(exuI3_type,type,
% 0.31/0.47 exuI3: $o ).
% 0.31/0.47
% 0.31/0.47 thf(exuI3,definition,
% 0.31/0.47 ( exuI3
% 0.31/0.47 = ( ! [Xphi: $i > $o] :
% 0.31/0.47 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.31/0.47 => ( ! [Xx: $i,Xy: $i] :
% 0.31/0.47 ( ( Xphi @ Xx )
% 0.31/0.47 => ( ( Xphi @ Xy )
% 0.31/0.47 => ( Xx = Xy ) ) )
% 0.31/0.47 => ( exu
% 0.31/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(exuI2_type,type,
% 0.31/0.47 exuI2: $o ).
% 0.31/0.47
% 0.31/0.47 thf(exuI2,definition,
% 0.31/0.47 ( exuI2
% 0.31/0.47 = ( ! [Xphi: $i > $o] :
% 0.31/0.47 ( ? [Xx: $i] :
% 0.31/0.47 ! [Xy: $i] :
% 0.31/0.47 ( ( Xphi @ Xy )
% 0.31/0.47 <=> ( Xy = Xx ) )
% 0.31/0.47 => ( exu
% 0.31/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(inCongP_type,type,
% 0.31/0.47 inCongP: $o ).
% 0.31/0.47
% 0.31/0.47 thf(inCongP,definition,
% 0.31/0.47 ( inCongP
% 0.31/0.47 = ( ! [A: $i,B: $i] :
% 0.31/0.47 ( ( A = B )
% 0.31/0.47 => ! [Xx: $i,Xy: $i] :
% 0.31/0.47 ( ( Xx = Xy )
% 0.31/0.47 => ( ( in @ Xx @ A )
% 0.31/0.47 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(in__Cong_type,type,
% 0.31/0.47 in__Cong: $o ).
% 0.31/0.47
% 0.31/0.47 thf(in__Cong,definition,
% 0.31/0.47 ( in__Cong
% 0.31/0.47 = ( ! [A: $i,B: $i] :
% 0.31/0.47 ( ( A = B )
% 0.31/0.47 => ! [Xx: $i,Xy: $i] :
% 0.31/0.47 ( ( Xx = Xy )
% 0.31/0.47 => ( ( in @ Xx @ A )
% 0.31/0.47 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(exuE3u_type,type,
% 0.31/0.47 exuE3u: $o ).
% 0.31/0.47
% 0.31/0.47 thf(exuE3u,definition,
% 0.31/0.47 ( exuE3u
% 0.31/0.47 = ( ! [Xphi: $i > $o] :
% 0.31/0.47 ( ( exu
% 0.31/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.47 => ! [Xx: $i,Xy: $i] :
% 0.31/0.47 ( ( Xphi @ Xx )
% 0.31/0.47 => ( ( Xphi @ Xy )
% 0.31/0.47 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(exu__Cong_type,type,
% 0.31/0.47 exu__Cong: $o ).
% 0.31/0.47
% 0.31/0.47 thf(exu__Cong,definition,
% 0.31/0.47 ( exu__Cong
% 0.31/0.47 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.31/0.47 ( ! [Xx: $i,Xy: $i] :
% 0.31/0.47 ( ( Xx = Xy )
% 0.31/0.47 => ( ( Xphi @ Xx )
% 0.31/0.47 <=> ( Xpsi @ Xy ) ) )
% 0.31/0.47 => ( ( exu
% 0.31/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.47 <=> ( exu
% 0.31/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(emptyset__Cong_type,type,
% 0.31/0.47 emptyset__Cong: $o ).
% 0.31/0.47
% 0.31/0.47 thf(emptyset__Cong,definition,
% 0.31/0.47 ( emptyset__Cong
% 0.31/0.47 = ( emptyset = emptyset ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setadjoin__Cong_type,type,
% 0.31/0.47 setadjoin__Cong: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setadjoin__Cong,definition,
% 0.31/0.47 ( setadjoin__Cong
% 0.31/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.31/0.47 ( ( Xx = Xy )
% 0.31/0.47 => ! [Xz: $i,Xu: $i] :
% 0.31/0.47 ( ( Xz = Xu )
% 0.31/0.47 => ( ( setadjoin @ Xx @ Xz )
% 0.31/0.47 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(powerset__Cong_type,type,
% 0.31/0.47 powerset__Cong: $o ).
% 0.31/0.47
% 0.31/0.47 thf(powerset__Cong,definition,
% 0.31/0.47 ( powerset__Cong
% 0.31/0.47 = ( ! [A: $i,B: $i] :
% 0.31/0.47 ( ( A = B )
% 0.31/0.47 => ( ( powerset @ A )
% 0.31/0.47 = ( powerset @ B ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setunion__Cong_type,type,
% 0.31/0.47 setunion__Cong: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setunion__Cong,definition,
% 0.31/0.47 ( setunion__Cong
% 0.31/0.47 = ( ! [A: $i,B: $i] :
% 0.31/0.47 ( ( A = B )
% 0.31/0.47 => ( ( setunion @ A )
% 0.31/0.47 = ( setunion @ B ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(omega__Cong_type,type,
% 0.31/0.47 omega__Cong: $o ).
% 0.31/0.47
% 0.31/0.47 thf(omega__Cong,definition,
% 0.31/0.47 ( omega__Cong
% 0.31/0.47 = ( omega = omega ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(exuEu_type,type,
% 0.31/0.47 exuEu: $o ).
% 0.31/0.47
% 0.31/0.47 thf(exuEu,definition,
% 0.31/0.47 ( exuEu
% 0.31/0.47 = ( ! [Xphi: $i > $o] :
% 0.31/0.47 ( ( exu
% 0.31/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.47 => ! [Xx: $i,Xy: $i] :
% 0.31/0.47 ( ( Xphi @ Xx )
% 0.31/0.47 => ( ( Xphi @ Xy )
% 0.31/0.47 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(descr__Cong_type,type,
% 0.31/0.47 descr__Cong: $o ).
% 0.31/0.47
% 0.31/0.47 thf(descr__Cong,definition,
% 0.31/0.47 ( descr__Cong
% 0.31/0.47 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.31/0.47 ( ! [Xx: $i,Xy: $i] :
% 0.31/0.47 ( ( Xx = Xy )
% 0.31/0.47 => ( ( Xphi @ Xx )
% 0.31/0.47 <=> ( Xpsi @ Xy ) ) )
% 0.31/0.47 => ( ( exu
% 0.31/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.47 => ( ( exu
% 0.31/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.31/0.47 => ( ( descr
% 0.31/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.47 = ( descr
% 0.31/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(dsetconstr__Cong_type,type,
% 0.31/0.47 dsetconstr__Cong: $o ).
% 0.31/0.47
% 0.31/0.47 thf(dsetconstr__Cong,definition,
% 0.31/0.47 ( dsetconstr__Cong
% 0.31/0.47 = ( ! [A: $i,B: $i] :
% 0.31/0.47 ( ( A = B )
% 0.31/0.47 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.31/0.47 ( ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ! [Xy: $i] :
% 0.31/0.47 ( ( in @ Xy @ B )
% 0.31/0.47 => ( ( Xx = Xy )
% 0.31/0.47 => ( ( Xphi @ Xx )
% 0.31/0.47 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.31/0.47 => ( ( dsetconstr @ A
% 0.31/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.47 = ( dsetconstr @ B
% 0.31/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(subset_type,type,
% 0.31/0.47 subset: $i > $i > $o ).
% 0.31/0.47
% 0.31/0.47 thf(disjoint_type,type,
% 0.31/0.47 disjoint: $i > $i > $o ).
% 0.31/0.47
% 0.31/0.47 thf(setsmeet_type,type,
% 0.31/0.47 setsmeet: $i > $i > $o ).
% 0.31/0.47
% 0.31/0.47 thf(subsetI1_type,type,
% 0.31/0.47 subsetI1: $o ).
% 0.31/0.47
% 0.31/0.47 thf(subsetI1,definition,
% 0.31/0.47 ( subsetI1
% 0.31/0.47 = ( ! [A: $i,B: $i] :
% 0.31/0.47 ( ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( in @ Xx @ B ) )
% 0.31/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(eqimpsubset2_type,type,
% 0.31/0.47 eqimpsubset2: $o ).
% 0.31/0.47
% 0.31/0.47 thf(eqimpsubset2,definition,
% 0.31/0.47 ( eqimpsubset2
% 0.31/0.47 = ( ! [A: $i,B: $i] :
% 0.31/0.47 ( ( A = B )
% 0.31/0.47 => ( subset @ B @ A ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(eqimpsubset1_type,type,
% 0.31/0.47 eqimpsubset1: $o ).
% 0.31/0.47
% 0.31/0.47 thf(eqimpsubset1,definition,
% 0.31/0.47 ( eqimpsubset1
% 0.31/0.47 = ( ! [A: $i,B: $i] :
% 0.31/0.47 ( ( A = B )
% 0.31/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(subsetI2_type,type,
% 0.31/0.47 subsetI2: $o ).
% 0.31/0.47
% 0.31/0.47 thf(subsetI2,definition,
% 0.31/0.47 ( subsetI2
% 0.31/0.47 = ( ! [A: $i,B: $i] :
% 0.31/0.47 ( ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( in @ Xx @ B ) )
% 0.31/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(emptysetsubset_type,type,
% 0.31/0.47 emptysetsubset: $o ).
% 0.31/0.47
% 0.31/0.47 thf(emptysetsubset,definition,
% 0.31/0.47 ( emptysetsubset
% 0.31/0.47 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(subsetE_type,type,
% 0.31/0.47 subsetE: $o ).
% 0.31/0.47
% 0.31/0.47 thf(subsetE,definition,
% 0.31/0.47 ( subsetE
% 0.31/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.47 ( ( subset @ A @ B )
% 0.31/0.47 => ( ( in @ Xx @ A )
% 0.31/0.47 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(subsetE2_type,type,
% 0.31/0.47 subsetE2: $o ).
% 0.31/0.47
% 0.31/0.47 thf(subsetE2,definition,
% 0.31/0.47 ( subsetE2
% 0.31/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.47 ( ( subset @ A @ B )
% 0.31/0.47 => ( ~ ( in @ Xx @ B )
% 0.31/0.47 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(notsubsetI_type,type,
% 0.31/0.47 notsubsetI: $o ).
% 0.31/0.47
% 0.31/0.47 thf(notsubsetI,definition,
% 0.31/0.47 ( notsubsetI
% 0.31/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( ~ ( in @ Xx @ B )
% 0.31/0.47 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(notequalI1_type,type,
% 0.31/0.47 notequalI1: $o ).
% 0.31/0.47
% 0.31/0.47 thf(notequalI1,definition,
% 0.31/0.47 ( notequalI1
% 0.31/0.47 = ( ! [A: $i,B: $i] :
% 0.31/0.47 ( ~ ( subset @ A @ B )
% 0.31/0.47 => ( A != B ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(notequalI2_type,type,
% 0.31/0.47 notequalI2: $o ).
% 0.31/0.47
% 0.31/0.47 thf(notequalI2,definition,
% 0.31/0.47 ( notequalI2
% 0.31/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( ~ ( in @ Xx @ B )
% 0.31/0.47 => ( A != B ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(subsetRefl_type,type,
% 0.31/0.47 subsetRefl: $o ).
% 0.31/0.47
% 0.31/0.47 thf(subsetRefl,definition,
% 0.31/0.47 ( subsetRefl
% 0.31/0.47 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(subsetTrans_type,type,
% 0.31/0.47 subsetTrans: $o ).
% 0.31/0.47
% 0.31/0.47 thf(subsetTrans,definition,
% 0.31/0.47 ( subsetTrans
% 0.31/0.47 = ( ! [A: $i,B: $i,C: $i] :
% 0.31/0.47 ( ( subset @ A @ B )
% 0.31/0.47 => ( ( subset @ B @ C )
% 0.31/0.47 => ( subset @ A @ C ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setadjoinSub_type,type,
% 0.31/0.47 setadjoinSub: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setadjoinSub,definition,
% 0.31/0.47 ( setadjoinSub
% 0.31/0.47 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setadjoinSub2_type,type,
% 0.31/0.47 setadjoinSub2: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setadjoinSub2,definition,
% 0.31/0.47 ( setadjoinSub2
% 0.31/0.47 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.31/0.47 ( ( subset @ A @ B )
% 0.31/0.47 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(subset2powerset_type,type,
% 0.31/0.47 subset2powerset: $o ).
% 0.31/0.47
% 0.31/0.47 thf(subset2powerset,definition,
% 0.31/0.47 ( subset2powerset
% 0.31/0.47 = ( ! [A: $i,B: $i] :
% 0.31/0.47 ( ( subset @ A @ B )
% 0.31/0.47 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setextsub_type,type,
% 0.31/0.47 setextsub: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setextsub,definition,
% 0.31/0.47 ( setextsub
% 0.31/0.47 = ( ! [A: $i,B: $i] :
% 0.31/0.47 ( ( subset @ A @ B )
% 0.31/0.47 => ( ( subset @ B @ A )
% 0.31/0.47 => ( A = B ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(subsetemptysetimpeq_type,type,
% 0.31/0.47 subsetemptysetimpeq: $o ).
% 0.31/0.47
% 0.31/0.47 thf(subsetemptysetimpeq,definition,
% 0.31/0.47 ( subsetemptysetimpeq
% 0.31/0.47 = ( ! [A: $i] :
% 0.31/0.47 ( ( subset @ A @ emptyset )
% 0.31/0.47 => ( A = emptyset ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(powersetI1_type,type,
% 0.31/0.47 powersetI1: $o ).
% 0.31/0.47
% 0.31/0.47 thf(powersetI1,definition,
% 0.31/0.47 ( powersetI1
% 0.31/0.47 = ( ! [A: $i,B: $i] :
% 0.31/0.47 ( ( subset @ B @ A )
% 0.31/0.47 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(powersetE1_type,type,
% 0.31/0.47 powersetE1: $o ).
% 0.31/0.47
% 0.31/0.47 thf(powersetE1,definition,
% 0.31/0.47 ( powersetE1
% 0.31/0.47 = ( ! [A: $i,B: $i] :
% 0.31/0.47 ( ( in @ B @ ( powerset @ A ) )
% 0.31/0.47 => ( subset @ B @ A ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(inPowerset_type,type,
% 0.31/0.47 inPowerset: $o ).
% 0.31/0.47
% 0.31/0.47 thf(inPowerset,definition,
% 0.31/0.47 ( inPowerset
% 0.31/0.47 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(powersetsubset_type,type,
% 0.31/0.47 powersetsubset: $o ).
% 0.31/0.47
% 0.31/0.47 thf(powersetsubset,definition,
% 0.31/0.47 ( powersetsubset
% 0.31/0.47 = ( ! [A: $i,B: $i] :
% 0.31/0.47 ( ( subset @ A @ B )
% 0.31/0.47 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(sepInPowerset_type,type,
% 0.31/0.47 sepInPowerset: $o ).
% 0.31/0.47
% 0.31/0.47 thf(sepInPowerset,definition,
% 0.31/0.47 ( sepInPowerset
% 0.31/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/0.47 ( in
% 0.31/0.47 @ ( dsetconstr @ A
% 0.31/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.47 @ ( powerset @ A ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(sepSubset_type,type,
% 0.31/0.47 sepSubset: $o ).
% 0.31/0.47
% 0.31/0.47 thf(sepSubset,definition,
% 0.31/0.47 ( sepSubset
% 0.31/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/0.47 ( subset
% 0.31/0.47 @ ( dsetconstr @ A
% 0.31/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.47 @ A ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(binunion_type,type,
% 0.31/0.47 binunion: $i > $i > $i ).
% 0.31/0.47
% 0.31/0.47 thf(binunionIL_type,type,
% 0.31/0.47 binunionIL: $o ).
% 0.31/0.47
% 0.31/0.47 thf(binunionIL,definition,
% 0.31/0.47 ( binunionIL
% 0.31/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(upairset2IR_type,type,
% 0.31/0.47 upairset2IR: $o ).
% 0.31/0.47
% 0.31/0.47 thf(upairset2IR,definition,
% 0.31/0.47 ( upairset2IR
% 0.31/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(binunionIR_type,type,
% 0.31/0.47 binunionIR: $o ).
% 0.31/0.47
% 0.31/0.47 thf(binunionIR,definition,
% 0.31/0.47 ( binunionIR
% 0.31/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ B )
% 0.31/0.47 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(binunionEcases_type,type,
% 0.31/0.47 binunionEcases: $o ).
% 0.31/0.47
% 0.31/0.47 thf(binunionEcases,definition,
% 0.31/0.47 ( binunionEcases
% 0.31/0.47 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.31/0.47 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.31/0.47 => ( ( ( in @ Xx @ A )
% 0.31/0.47 => Xphi )
% 0.31/0.47 => ( ( ( in @ Xx @ B )
% 0.31/0.47 => Xphi )
% 0.31/0.47 => Xphi ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(binunionE_type,type,
% 0.31/0.47 binunionE: $o ).
% 0.31/0.47
% 0.31/0.47 thf(binunionE,definition,
% 0.31/0.47 ( binunionE
% 0.31/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.31/0.47 => ( ( in @ Xx @ A )
% 0.31/0.47 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(binunionLsub_type,type,
% 0.31/0.47 binunionLsub: $o ).
% 0.31/0.47
% 0.31/0.47 thf(binunionLsub,definition,
% 0.31/0.47 ( binunionLsub
% 0.31/0.47 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(binunionRsub_type,type,
% 0.31/0.47 binunionRsub: $o ).
% 0.31/0.47
% 0.31/0.47 thf(binunionRsub,definition,
% 0.31/0.47 ( binunionRsub
% 0.31/0.47 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(binintersect_type,type,
% 0.31/0.47 binintersect: $i > $i > $i ).
% 0.31/0.47
% 0.31/0.47 thf(binintersectI_type,type,
% 0.31/0.47 binintersectI: $o ).
% 0.31/0.47
% 0.31/0.47 thf(binintersectI,definition,
% 0.31/0.47 ( binintersectI
% 0.31/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( ( in @ Xx @ B )
% 0.31/0.47 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(binintersectSubset5_type,type,
% 0.31/0.47 binintersectSubset5: $o ).
% 0.31/0.47
% 0.31/0.47 thf(binintersectSubset5,definition,
% 0.31/0.47 ( binintersectSubset5
% 0.31/0.47 = ( ! [A: $i,B: $i,C: $i] :
% 0.31/0.47 ( ( subset @ C @ A )
% 0.31/0.47 => ( ( subset @ C @ B )
% 0.31/0.47 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(binintersectEL_type,type,
% 0.31/0.47 binintersectEL: $o ).
% 0.31/0.47
% 0.31/0.47 thf(binintersectEL,definition,
% 0.31/0.47 ( binintersectEL
% 0.31/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.31/0.47 => ( in @ Xx @ A ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(binintersectLsub_type,type,
% 0.31/0.47 binintersectLsub: $o ).
% 0.31/0.47
% 0.31/0.47 thf(binintersectLsub,definition,
% 0.31/0.47 ( binintersectLsub
% 0.31/0.47 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(binintersectSubset2_type,type,
% 0.31/0.47 binintersectSubset2: $o ).
% 0.31/0.47
% 0.31/0.47 thf(binintersectSubset2,definition,
% 0.31/0.47 ( binintersectSubset2
% 0.31/0.47 = ( ! [A: $i,B: $i] :
% 0.31/0.47 ( ( subset @ A @ B )
% 0.31/0.47 => ( ( binintersect @ A @ B )
% 0.31/0.47 = A ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(binintersectSubset3_type,type,
% 0.31/0.47 binintersectSubset3: $o ).
% 0.31/0.47
% 0.31/0.47 thf(binintersectSubset3,definition,
% 0.31/0.47 ( binintersectSubset3
% 0.31/0.47 = ( ! [A: $i,B: $i] :
% 0.31/0.47 ( ( ( binintersect @ A @ B )
% 0.31/0.47 = B )
% 0.31/0.47 => ( subset @ B @ A ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(binintersectER_type,type,
% 0.31/0.47 binintersectER: $o ).
% 0.31/0.47
% 0.31/0.47 thf(binintersectER,definition,
% 0.31/0.47 ( binintersectER
% 0.31/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.31/0.47 => ( in @ Xx @ B ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(disjointsetsI1_type,type,
% 0.31/0.47 disjointsetsI1: $o ).
% 0.31/0.47
% 0.31/0.47 thf(disjointsetsI1,definition,
% 0.31/0.47 ( disjointsetsI1
% 0.31/0.47 = ( ! [A: $i,B: $i] :
% 0.31/0.47 ( ~ ? [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 & ( in @ Xx @ B ) )
% 0.31/0.47 => ( ( binintersect @ A @ B )
% 0.31/0.47 = emptyset ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(binintersectRsub_type,type,
% 0.31/0.47 binintersectRsub: $o ).
% 0.31/0.47
% 0.31/0.47 thf(binintersectRsub,definition,
% 0.31/0.47 ( binintersectRsub
% 0.31/0.47 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(binintersectSubset4_type,type,
% 0.31/0.47 binintersectSubset4: $o ).
% 0.31/0.47
% 0.31/0.47 thf(binintersectSubset4,definition,
% 0.31/0.47 ( binintersectSubset4
% 0.31/0.47 = ( ! [A: $i,B: $i] :
% 0.31/0.47 ( ( subset @ B @ A )
% 0.31/0.47 => ( ( binintersect @ A @ B )
% 0.31/0.47 = B ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(binintersectSubset1_type,type,
% 0.31/0.47 binintersectSubset1: $o ).
% 0.31/0.47
% 0.31/0.47 thf(binintersectSubset1,definition,
% 0.31/0.47 ( binintersectSubset1
% 0.31/0.47 = ( ! [A: $i,B: $i] :
% 0.31/0.47 ( ( ( binintersect @ A @ B )
% 0.31/0.47 = A )
% 0.31/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(bs114d_type,type,
% 0.31/0.47 bs114d: $o ).
% 0.31/0.47
% 0.31/0.47 thf(bs114d,definition,
% 0.31/0.47 ( bs114d
% 0.31/0.47 = ( ! [A: $i,B: $i,C: $i] :
% 0.31/0.47 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 0.31/0.47 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(regular_type,type,
% 0.31/0.47 regular: $i > $o ).
% 0.31/0.47
% 0.31/0.47 thf(setminus_type,type,
% 0.31/0.47 setminus: $i > $i > $i ).
% 0.31/0.47
% 0.31/0.47 thf(setminusI_type,type,
% 0.31/0.47 setminusI: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setminusI,definition,
% 0.31/0.47 ( setminusI
% 0.31/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( ~ ( in @ Xx @ B )
% 0.31/0.47 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setminusEL_type,type,
% 0.31/0.47 setminusEL: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setminusEL,definition,
% 0.31/0.47 ( setminusEL
% 0.31/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.31/0.47 => ( in @ Xx @ A ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setminusER_type,type,
% 0.31/0.47 setminusER: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setminusER,definition,
% 0.31/0.47 ( setminusER
% 0.31/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.31/0.47 => ~ ( in @ Xx @ B ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setminusSubset2_type,type,
% 0.31/0.47 setminusSubset2: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setminusSubset2,definition,
% 0.31/0.47 ( setminusSubset2
% 0.31/0.47 = ( ! [A: $i,B: $i] :
% 0.31/0.47 ( ( subset @ A @ B )
% 0.31/0.47 => ( ( setminus @ A @ B )
% 0.31/0.47 = emptyset ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setminusERneg_type,type,
% 0.31/0.47 setminusERneg: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setminusERneg,definition,
% 0.31/0.47 ( setminusERneg
% 0.31/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.47 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.31/0.47 => ( ( in @ Xx @ A )
% 0.31/0.47 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setminusELneg_type,type,
% 0.31/0.47 setminusELneg: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setminusELneg,definition,
% 0.31/0.47 ( setminusELneg
% 0.31/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.47 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.31/0.47 => ( ~ ( in @ Xx @ B )
% 0.31/0.47 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setminusILneg_type,type,
% 0.31/0.47 setminusILneg: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setminusILneg,definition,
% 0.31/0.47 ( setminusILneg
% 0.31/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.47 ( ~ ( in @ Xx @ A )
% 0.31/0.47 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setminusIRneg_type,type,
% 0.31/0.47 setminusIRneg: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setminusIRneg,definition,
% 0.31/0.47 ( setminusIRneg
% 0.31/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ B )
% 0.31/0.47 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setminusLsub_type,type,
% 0.31/0.47 setminusLsub: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setminusLsub,definition,
% 0.31/0.47 ( setminusLsub
% 0.31/0.47 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setminusSubset1_type,type,
% 0.31/0.47 setminusSubset1: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setminusSubset1,definition,
% 0.31/0.47 ( setminusSubset1
% 0.31/0.47 = ( ! [A: $i,B: $i] :
% 0.31/0.47 ( ( ( setminus @ A @ B )
% 0.31/0.47 = emptyset )
% 0.31/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(symdiff_type,type,
% 0.31/0.47 symdiff: $i > $i > $i ).
% 0.31/0.47
% 0.31/0.47 thf(symdiffE_type,type,
% 0.31/0.47 symdiffE: $o ).
% 0.31/0.47
% 0.31/0.47 thf(symdiffE,definition,
% 0.31/0.47 ( symdiffE
% 0.31/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 0.31/0.47 => ! [Xphi: $o] :
% 0.31/0.47 ( ( ( in @ Xx @ A )
% 0.31/0.47 => ( ~ ( in @ Xx @ B )
% 0.31/0.47 => Xphi ) )
% 0.31/0.47 => ( ( ~ ( in @ Xx @ A )
% 0.31/0.47 => ( ( in @ Xx @ B )
% 0.31/0.47 => Xphi ) )
% 0.31/0.47 => Xphi ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(symdiffI1_type,type,
% 0.31/0.47 symdiffI1: $o ).
% 0.31/0.47
% 0.31/0.47 thf(symdiffI1,definition,
% 0.31/0.47 ( symdiffI1
% 0.31/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( ~ ( in @ Xx @ B )
% 0.31/0.47 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(symdiffI2_type,type,
% 0.31/0.47 symdiffI2: $o ).
% 0.31/0.47
% 0.31/0.47 thf(symdiffI2,definition,
% 0.31/0.47 ( symdiffI2
% 0.31/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.47 ( ~ ( in @ Xx @ A )
% 0.31/0.47 => ( ( in @ Xx @ B )
% 0.31/0.47 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(symdiffIneg1_type,type,
% 0.31/0.47 symdiffIneg1: $o ).
% 0.31/0.47
% 0.31/0.47 thf(symdiffIneg1,definition,
% 0.31/0.47 ( symdiffIneg1
% 0.31/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( ( in @ Xx @ B )
% 0.31/0.47 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(symdiffIneg2_type,type,
% 0.31/0.47 symdiffIneg2: $o ).
% 0.31/0.47
% 0.31/0.47 thf(symdiffIneg2,definition,
% 0.31/0.47 ( symdiffIneg2
% 0.31/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.47 ( ~ ( in @ Xx @ A )
% 0.31/0.47 => ( ~ ( in @ Xx @ B )
% 0.31/0.47 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(iskpair_type,type,
% 0.31/0.47 iskpair: $i > $o ).
% 0.31/0.47
% 0.31/0.47 thf(iskpair,definition,
% 0.31/0.47 ( iskpair
% 0.31/0.47 = ( ^ [A: $i] :
% 0.31/0.47 ? [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ ( setunion @ A ) )
% 0.31/0.47 & ? [Xy: $i] :
% 0.31/0.47 ( ( in @ Xy @ ( setunion @ A ) )
% 0.31/0.47 & ( A
% 0.31/0.47 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(secondinupair_type,type,
% 0.31/0.47 secondinupair: $o ).
% 0.31/0.47
% 0.31/0.47 thf(secondinupair,definition,
% 0.31/0.47 ( secondinupair
% 0.31/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setukpairIL_type,type,
% 0.31/0.47 setukpairIL: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setukpairIL,definition,
% 0.31/0.47 ( setukpairIL
% 0.31/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setukpairIR_type,type,
% 0.31/0.47 setukpairIR: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setukpairIR,definition,
% 0.31/0.47 ( setukpairIR
% 0.31/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(kpairiskpair_type,type,
% 0.31/0.47 kpairiskpair: $o ).
% 0.31/0.47
% 0.31/0.47 thf(kpairiskpair,definition,
% 0.31/0.47 ( kpairiskpair
% 0.31/0.47 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(kpair_type,type,
% 0.31/0.47 kpair: $i > $i > $i ).
% 0.31/0.47
% 0.31/0.47 thf(kpair,definition,
% 0.31/0.47 ( kpair
% 0.31/0.47 = ( ^ [Xx: $i,Xy: $i] : ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(kpairp_type,type,
% 0.31/0.47 kpairp: $o ).
% 0.31/0.47
% 0.31/0.47 thf(kpairp,definition,
% 0.31/0.47 ( kpairp
% 0.31/0.47 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(cartprod_type,type,
% 0.31/0.47 cartprod: $i > $i > $i ).
% 0.31/0.47
% 0.31/0.47 thf(singletonsubset_type,type,
% 0.31/0.47 singletonsubset: $o ).
% 0.31/0.47
% 0.31/0.47 thf(singletonsubset,definition,
% 0.31/0.47 ( singletonsubset
% 0.31/0.47 = ( ! [A: $i,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(singletoninpowerset_type,type,
% 0.31/0.47 singletoninpowerset: $o ).
% 0.31/0.47
% 0.31/0.47 thf(singletoninpowerset,definition,
% 0.31/0.47 ( singletoninpowerset
% 0.31/0.47 = ( ! [A: $i,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(singletoninpowunion_type,type,
% 0.31/0.47 singletoninpowunion: $o ).
% 0.31/0.47
% 0.31/0.47 thf(singletoninpowunion,definition,
% 0.31/0.47 ( singletoninpowunion
% 0.31/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(upairset2E_type,type,
% 0.31/0.47 upairset2E: $o ).
% 0.31/0.47
% 0.31/0.47 thf(upairset2E,definition,
% 0.31/0.47 ( upairset2E
% 0.31/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.31/0.47 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.31/0.47 => ( ( Xz = Xx )
% 0.31/0.47 | ( Xz = Xy ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(upairsubunion_type,type,
% 0.31/0.47 upairsubunion: $o ).
% 0.31/0.47
% 0.31/0.47 thf(upairsubunion,definition,
% 0.31/0.47 ( upairsubunion
% 0.31/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ! [Xy: $i] :
% 0.31/0.47 ( ( in @ Xy @ B )
% 0.31/0.47 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(upairinpowunion_type,type,
% 0.31/0.47 upairinpowunion: $o ).
% 0.31/0.47
% 0.31/0.47 thf(upairinpowunion,definition,
% 0.31/0.47 ( upairinpowunion
% 0.31/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ! [Xy: $i] :
% 0.31/0.47 ( ( in @ Xy @ B )
% 0.31/0.47 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(ubforcartprodlem1_type,type,
% 0.31/0.47 ubforcartprodlem1: $o ).
% 0.31/0.47
% 0.31/0.47 thf(ubforcartprodlem1,definition,
% 0.31/0.47 ( ubforcartprodlem1
% 0.31/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ! [Xy: $i] :
% 0.31/0.47 ( ( in @ Xy @ B )
% 0.31/0.47 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(ubforcartprodlem2_type,type,
% 0.31/0.47 ubforcartprodlem2: $o ).
% 0.31/0.47
% 0.31/0.47 thf(ubforcartprodlem2,definition,
% 0.31/0.47 ( ubforcartprodlem2
% 0.31/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ! [Xy: $i] :
% 0.31/0.47 ( ( in @ Xy @ B )
% 0.31/0.47 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(ubforcartprodlem3_type,type,
% 0.31/0.47 ubforcartprodlem3: $o ).
% 0.31/0.47
% 0.31/0.47 thf(ubforcartprodlem3,definition,
% 0.31/0.47 ( ubforcartprodlem3
% 0.31/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ! [Xy: $i] :
% 0.31/0.47 ( ( in @ Xy @ B )
% 0.31/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(cartprodpairin_type,type,
% 0.31/0.47 cartprodpairin: $o ).
% 0.31/0.47
% 0.31/0.47 thf(cartprodpairin,definition,
% 0.31/0.47 ( cartprodpairin
% 0.31/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ! [Xy: $i] :
% 0.31/0.47 ( ( in @ Xy @ B )
% 0.31/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(cartprodmempair1_type,type,
% 0.31/0.47 cartprodmempair1: $o ).
% 0.31/0.47
% 0.31/0.47 thf(cartprodmempair1,definition,
% 0.31/0.47 ( cartprodmempair1
% 0.31/0.47 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.31/0.47 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.31/0.47 => ? [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 & ? [Xy: $i] :
% 0.31/0.47 ( ( in @ Xy @ B )
% 0.31/0.47 & ( Xu
% 0.31/0.47 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(cartprodmempair_type,type,
% 0.31/0.47 cartprodmempair: $o ).
% 0.31/0.47
% 0.31/0.47 thf(cartprodmempair,definition,
% 0.31/0.47 ( cartprodmempair
% 0.31/0.47 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.31/0.47 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.31/0.47 => ( iskpair @ Xu ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setunionE2_type,type,
% 0.31/0.47 setunionE2: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setunionE2,definition,
% 0.31/0.47 ( setunionE2
% 0.31/0.47 = ( ! [A: $i,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ ( setunion @ A ) )
% 0.31/0.47 => ? [X: $i] :
% 0.31/0.47 ( ( in @ X @ A )
% 0.31/0.47 & ( in @ Xx @ X ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setunionsingleton1_type,type,
% 0.31/0.47 setunionsingleton1: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setunionsingleton1,definition,
% 0.31/0.47 ( setunionsingleton1
% 0.31/0.47 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setunionsingleton2_type,type,
% 0.31/0.47 setunionsingleton2: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setunionsingleton2,definition,
% 0.31/0.47 ( setunionsingleton2
% 0.31/0.47 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setunionsingleton_type,type,
% 0.31/0.47 setunionsingleton: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setunionsingleton,definition,
% 0.31/0.47 ( setunionsingleton
% 0.31/0.47 = ( ! [Xx: $i] :
% 0.31/0.47 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 0.31/0.47 = Xx ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(singleton_type,type,
% 0.31/0.47 singleton: $i > $o ).
% 0.31/0.47
% 0.31/0.47 thf(singleton,definition,
% 0.31/0.47 ( singleton
% 0.31/0.47 = ( ^ [A: $i] :
% 0.31/0.47 ? [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 & ( A
% 0.31/0.47 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(singletonprop_type,type,
% 0.31/0.47 singletonprop: $o ).
% 0.31/0.47
% 0.31/0.47 thf(singletonprop,definition,
% 0.31/0.47 ( singletonprop
% 0.31/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/0.47 ( ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ! [Xy: $i] :
% 0.31/0.47 ( ( in @ Xy @ A )
% 0.31/0.47 => ( ( Xphi @ Xx )
% 0.31/0.47 => ( ( Xphi @ Xy )
% 0.31/0.47 => ( Xx = Xy ) ) ) ) )
% 0.31/0.47 => ( ? [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 & ( Xphi @ Xx ) )
% 0.31/0.47 => ( singleton
% 0.31/0.47 @ ( dsetconstr @ A
% 0.31/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(ex1_type,type,
% 0.31/0.47 ex1: $i > ( $i > $o ) > $o ).
% 0.31/0.47
% 0.31/0.47 thf(ex1,definition,
% 0.31/0.47 ( ex1
% 0.31/0.47 = ( ^ [A: $i,Xphi: $i > $o] :
% 0.31/0.47 ( singleton
% 0.31/0.47 @ ( dsetconstr @ A
% 0.31/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(ex1E1_type,type,
% 0.31/0.47 ex1E1: $o ).
% 0.31/0.47
% 0.31/0.47 thf(ex1E1,definition,
% 0.31/0.47 ( ex1E1
% 0.31/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/0.47 ( ( ex1 @ A
% 0.31/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.47 => ? [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(ex1I_type,type,
% 0.31/0.47 ex1I: $o ).
% 0.31/0.47
% 0.31/0.47 thf(ex1I,definition,
% 0.31/0.47 ( ex1I
% 0.31/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( ( Xphi @ Xx )
% 0.31/0.47 => ( ! [Xy: $i] :
% 0.31/0.47 ( ( in @ Xy @ A )
% 0.31/0.47 => ( ( Xphi @ Xy )
% 0.31/0.47 => ( Xy = Xx ) ) )
% 0.31/0.47 => ( ex1 @ A
% 0.31/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(ex1I2_type,type,
% 0.31/0.47 ex1I2: $o ).
% 0.31/0.47
% 0.31/0.47 thf(ex1I2,definition,
% 0.31/0.47 ( ex1I2
% 0.31/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/0.47 ( ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ! [Xy: $i] :
% 0.31/0.47 ( ( in @ Xy @ A )
% 0.31/0.47 => ( ( Xphi @ Xx )
% 0.31/0.47 => ( ( Xphi @ Xy )
% 0.31/0.47 => ( Xx = Xy ) ) ) ) )
% 0.31/0.47 => ( ? [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 & ( Xphi @ Xx ) )
% 0.31/0.47 => ( ex1 @ A
% 0.31/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(singletonsuniq_type,type,
% 0.31/0.47 singletonsuniq: $o ).
% 0.31/0.47
% 0.31/0.47 thf(singletonsuniq,definition,
% 0.31/0.47 ( singletonsuniq
% 0.31/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.31/0.47 ( ( ( setadjoin @ Xx @ emptyset )
% 0.31/0.47 = ( setadjoin @ Xy @ emptyset ) )
% 0.31/0.47 => ( Xx = Xy ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(atmost1p_type,type,
% 0.31/0.47 atmost1p: $i > $o ).
% 0.31/0.47
% 0.31/0.47 thf(atleast2p_type,type,
% 0.31/0.47 atleast2p: $i > $o ).
% 0.31/0.47
% 0.31/0.47 thf(atmost2p_type,type,
% 0.31/0.47 atmost2p: $i > $o ).
% 0.31/0.47
% 0.31/0.47 thf(upairsetp_type,type,
% 0.31/0.47 upairsetp: $i > $o ).
% 0.31/0.47
% 0.31/0.47 thf(setukpairinjL1_type,type,
% 0.31/0.47 setukpairinjL1: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setukpairinjL1,definition,
% 0.31/0.47 ( setukpairinjL1
% 0.31/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.31/0.47 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 0.31/0.47 => ( Xx = Xz ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(kfstsingleton_type,type,
% 0.31/0.47 kfstsingleton: $o ).
% 0.31/0.47
% 0.31/0.47 thf(kfstsingleton,definition,
% 0.31/0.47 ( kfstsingleton
% 0.31/0.47 = ( ! [Xu: $i] :
% 0.31/0.47 ( ( iskpair @ Xu )
% 0.31/0.47 => ( singleton
% 0.31/0.47 @ ( dsetconstr @ ( setunion @ Xu )
% 0.31/0.47 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(theprop_type,type,
% 0.31/0.47 theprop: $o ).
% 0.31/0.47
% 0.31/0.47 thf(theprop,definition,
% 0.31/0.47 ( theprop
% 0.31/0.47 = ( ! [X: $i] :
% 0.31/0.47 ( ( singleton @ X )
% 0.31/0.47 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(kfst_type,type,
% 0.31/0.47 kfst: $i > $i ).
% 0.31/0.47
% 0.31/0.47 thf(kfstpairEq_type,type,
% 0.31/0.47 kfstpairEq: $o ).
% 0.31/0.47
% 0.31/0.47 thf(kfstpairEq,definition,
% 0.31/0.47 ( kfstpairEq
% 0.31/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.31/0.47 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.31/0.47 = Xx ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(cartprodfstin_type,type,
% 0.31/0.47 cartprodfstin: $o ).
% 0.31/0.47
% 0.31/0.47 thf(cartprodfstin,definition,
% 0.31/0.47 ( cartprodfstin
% 0.31/0.47 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.31/0.47 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.31/0.47 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setukpairinjL2_type,type,
% 0.31/0.47 setukpairinjL2: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setukpairinjL2,definition,
% 0.31/0.47 ( setukpairinjL2
% 0.31/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.31/0.47 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.31/0.47 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.31/0.47 => ( Xx = Xz ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setukpairinjL_type,type,
% 0.31/0.47 setukpairinjL: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setukpairinjL,definition,
% 0.31/0.47 ( setukpairinjL
% 0.31/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.31/0.47 ( ( ( kpair @ Xx @ Xy )
% 0.31/0.47 = ( kpair @ Xz @ Xu ) )
% 0.31/0.47 => ( Xx = Xz ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setukpairinjR11_type,type,
% 0.31/0.47 setukpairinjR11: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setukpairinjR11,definition,
% 0.31/0.47 ( setukpairinjR11
% 0.31/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.31/0.47 ( ( Xx = Xy )
% 0.31/0.47 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.31/0.47 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setukpairinjR12,conjecture,
% 0.31/0.47 ( setextAx
% 0.31/0.47 => ( emptysetAx
% 0.31/0.47 => ( setadjoinAx
% 0.31/0.47 => ( powersetAx
% 0.31/0.47 => ( setunionAx
% 0.31/0.47 => ( omega0Ax
% 0.31/0.47 => ( omegaSAx
% 0.31/0.47 => ( omegaIndAx
% 0.31/0.47 => ( replAx
% 0.31/0.47 => ( foundationAx
% 0.31/0.47 => ( wellorderingAx
% 0.31/0.47 => ( descrp
% 0.31/0.47 => ( dsetconstrI
% 0.31/0.47 => ( dsetconstrEL
% 0.31/0.47 => ( dsetconstrER
% 0.31/0.47 => ( exuE1
% 0.31/0.47 => ( prop2setE
% 0.31/0.47 => ( emptysetE
% 0.31/0.47 => ( emptysetimpfalse
% 0.31/0.47 => ( notinemptyset
% 0.31/0.47 => ( exuE3e
% 0.31/0.47 => ( setext
% 0.31/0.47 => ( emptyI
% 0.31/0.47 => ( noeltsimpempty
% 0.31/0.47 => ( setbeta
% 0.31/0.47 => ( nonemptyE1
% 0.31/0.47 => ( nonemptyI
% 0.31/0.47 => ( nonemptyI1
% 0.31/0.47 => ( setadjoinIL
% 0.31/0.47 => ( emptyinunitempty
% 0.31/0.47 => ( setadjoinIR
% 0.31/0.47 => ( setadjoinE
% 0.31/0.47 => ( setadjoinOr
% 0.31/0.47 => ( setoftrueEq
% 0.31/0.47 => ( powersetI
% 0.31/0.47 => ( emptyinPowerset
% 0.31/0.47 => ( emptyInPowerset
% 0.31/0.47 => ( powersetE
% 0.31/0.47 => ( setunionI
% 0.31/0.47 => ( setunionE
% 0.31/0.47 => ( subPowSU
% 0.31/0.47 => ( exuE2
% 0.31/0.47 => ( nonemptyImpWitness
% 0.31/0.47 => ( uniqinunit
% 0.31/0.47 => ( notinsingleton
% 0.31/0.47 => ( eqinunit
% 0.31/0.47 => ( singletonsswitch
% 0.31/0.47 => ( upairsetE
% 0.31/0.47 => ( upairsetIL
% 0.31/0.47 => ( upairsetIR
% 0.31/0.47 => ( emptyE1
% 0.31/0.47 => ( vacuousDall
% 0.31/0.47 => ( quantDeMorgan1
% 0.31/0.47 => ( quantDeMorgan2
% 0.31/0.47 => ( quantDeMorgan3
% 0.31/0.47 => ( quantDeMorgan4
% 0.31/0.47 => ( prop2setI
% 0.31/0.47 => ( prop2set2propI
% 0.31/0.47 => ( notdexE
% 0.31/0.47 => ( notdallE
% 0.31/0.47 => ( exuI1
% 0.31/0.47 => ( exuI3
% 0.31/0.47 => ( exuI2
% 0.31/0.47 => ( inCongP
% 0.31/0.47 => ( in__Cong
% 0.31/0.47 => ( exuE3u
% 0.31/0.47 => ( exu__Cong
% 0.31/0.47 => ( emptyset__Cong
% 0.31/0.47 => ( setadjoin__Cong
% 0.31/0.47 => ( powerset__Cong
% 0.31/0.47 => ( setunion__Cong
% 0.31/0.47 => ( omega__Cong
% 0.31/0.47 => ( exuEu
% 0.31/0.47 => ( descr__Cong
% 0.31/0.47 => ( dsetconstr__Cong
% 0.31/0.47 => ( subsetI1
% 0.31/0.47 => ( eqimpsubset2
% 0.31/0.47 => ( eqimpsubset1
% 0.31/0.47 => ( subsetI2
% 0.31/0.47 => ( emptysetsubset
% 0.31/0.47 => ( subsetE
% 0.31/0.47 => ( subsetE2
% 0.31/0.47 => ( notsubsetI
% 0.31/0.47 => ( notequalI1
% 0.31/0.47 => ( notequalI2
% 0.31/0.47 => ( subsetRefl
% 0.31/0.47 => ( subsetTrans
% 0.31/0.47 => ( setadjoinSub
% 0.31/0.47 => ( setadjoinSub2
% 0.31/0.47 => ( subset2powerset
% 0.31/0.47 => ( setextsub
% 0.31/0.47 => ( subsetemptysetimpeq
% 0.31/0.47 => ( powersetI1
% 0.31/0.47 => ( powersetE1
% 0.31/0.47 => ( inPowerset
% 0.31/0.47 => ( powersetsubset
% 0.31/0.47 => ( sepInPowerset
% 0.31/0.47 => ( sepSubset
% 0.31/0.47 => ( binunionIL
% 0.31/0.47 => ( upairset2IR
% 0.31/0.47 => ( binunionIR
% 0.31/0.47 => ( binunionEcases
% 0.31/0.47 => ( binunionE
% 0.31/0.47 => ( binunionLsub
% 0.31/0.47 => ( binunionRsub
% 0.31/0.47 => ( binintersectI
% 0.31/0.47 => ( binintersectSubset5
% 0.31/0.47 => ( binintersectEL
% 0.31/0.47 => ( binintersectLsub
% 0.31/0.47 => ( binintersectSubset2
% 0.31/0.47 => ( binintersectSubset3
% 0.31/0.47 => ( binintersectER
% 0.31/0.47 => ( disjointsetsI1
% 0.31/0.47 => ( binintersectRsub
% 0.31/0.47 => ( binintersectSubset4
% 0.31/0.47 => ( binintersectSubset1
% 0.31/0.47 => ( bs114d
% 0.31/0.47 => ( setminusI
% 0.31/0.47 => ( setminusEL
% 0.31/0.47 => ( setminusER
% 0.31/0.47 => ( setminusSubset2
% 0.31/0.47 => ( setminusERneg
% 0.31/0.47 => ( setminusELneg
% 0.31/0.47 => ( setminusILneg
% 0.31/0.47 => ( setminusIRneg
% 0.31/0.47 => ( setminusLsub
% 0.31/0.47 => ( setminusSubset1
% 0.31/0.47 => ( symdiffE
% 0.31/0.47 => ( symdiffI1
% 0.31/0.47 => ( symdiffI2
% 0.31/0.47 => ( symdiffIneg1
% 0.31/0.47 => ( symdiffIneg2
% 0.31/0.47 => ( secondinupair
% 0.31/0.47 => ( setukpairIL
% 0.31/0.47 => ( setukpairIR
% 0.31/0.47 => ( kpairiskpair
% 0.31/0.47 => ( kpairp
% 0.31/0.47 => ( singletonsubset
% 0.31/0.47 => ( singletoninpowerset
% 0.31/0.47 => ( singletoninpowunion
% 0.31/0.47 => ( upairset2E
% 0.31/0.47 => ( upairsubunion
% 0.31/0.47 => ( upairinpowunion
% 0.31/0.47 => ( ubforcartprodlem1
% 0.31/0.47 => ( ubforcartprodlem2
% 0.31/0.47 => ( ubforcartprodlem3
% 0.31/0.47 => ( cartprodpairin
% 0.31/0.47 => ( cartprodmempair1
% 0.31/0.47 => ( cartprodmempair
% 0.31/0.47 => ( setunionE2
% 0.31/0.47 => ( setunionsingleton1
% 0.31/0.47 => ( setunionsingleton2
% 0.31/0.47 => ( setunionsingleton
% 0.31/0.47 => ( singletonprop
% 0.31/0.47 => ( ex1E1
% 0.31/0.47 => ( ex1I
% 0.31/0.47 => ( ex1I2
% 0.31/0.47 => ( singletonsuniq
% 0.31/0.47 => ( setukpairinjL1
% 0.31/0.47 => ( kfstsingleton
% 0.31/0.47 => ( theprop
% 0.31/0.47 => ( kfstpairEq
% 0.31/0.47 => ( cartprodfstin
% 0.31/0.47 => ( setukpairinjL2
% 0.31/0.47 => ( setukpairinjL
% 0.31/0.47 => ( setukpairinjR11
% 0.31/0.47 => ! [Xx: $i,Xy: $i] :
% 0.31/0.47 ( ( Xx = Xy )
% 0.31/0.47 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.31/0.47 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 %------------------------------------------------------------------------------
% 0.31/0.47 ------------------------
% 2.41/1.47 ---- Embedded in THF ---
% 2.41/1.47 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.41/1.47 %%% Generated on Tue Feb 24 12:29:30 EST 2026
% 2.41/1.47 %%% using '$$dhol' embedding, version 1.3.0.
% 2.41/1.47 %%% Logic specification used:
% 2.41/1.47 %%% thf(spec, logic, $$dhol).
% 2.41/1.47
% 2.41/1.47 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.mID8C0AdqD/DTF2DTF11959.p
% See solution above
% 2.41/1.47 ------------------------
% 2.61/1.51 ---- Cleaned THF ---
% 2.61/1.51 thf(in_type,type,
% 2.61/1.51 in: $i > $i > $o ).
% 2.61/1.51
% 2.61/1.51 thf(exu_type,type,
% 2.61/1.51 exu: ( $i > $o ) > $o ).
% 2.61/1.51
% 2.61/1.51 thf(exu,definition,
% 2.61/1.51 ( exu
% 2.61/1.51 = ( ^ [Xphi: $i > $o] :
% 2.61/1.51 ? [Xx: $i] :
% 2.61/1.51 ( ( Xphi @ Xx )
% 2.61/1.51 & ! [Xy: $i] :
% 2.61/1.51 ( ( Xphi @ Xy )
% 2.61/1.51 => ( Xx = Xy ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(setextAx_type,type,
% 2.61/1.51 setextAx: $o ).
% 2.61/1.51
% 2.61/1.51 thf(setextAx,definition,
% 2.61/1.51 ( setextAx
% 2.61/1.51 = ( ! [A: $i,B: $i] :
% 2.61/1.51 ( ! [Xx: $i] :
% 2.61/1.51 ( ( in @ Xx @ A )
% 2.61/1.51 <=> ( in @ Xx @ B ) )
% 2.61/1.51 => ( A = B ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(emptyset_type,type,
% 2.61/1.51 emptyset: $i ).
% 2.61/1.51
% 2.61/1.51 thf(emptysetAx_type,type,
% 2.61/1.51 emptysetAx: $o ).
% 2.61/1.51
% 2.61/1.51 thf(emptysetAx,definition,
% 2.61/1.51 ( emptysetAx
% 2.61/1.51 = ( ! [Xx: $i] :
% 2.61/1.51 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(setadjoin_type,type,
% 2.61/1.51 setadjoin: $i > $i > $i ).
% 2.61/1.51
% 2.61/1.51 thf(setadjoinAx_type,type,
% 2.61/1.51 setadjoinAx: $o ).
% 2.61/1.51
% 2.61/1.51 thf(setadjoinAx,definition,
% 2.61/1.51 ( setadjoinAx
% 2.61/1.51 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.61/1.51 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.61/1.51 <=> ( ( Xy = Xx )
% 2.61/1.51 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(powerset_type,type,
% 2.61/1.51 powerset: $i > $i ).
% 2.61/1.51
% 2.61/1.51 thf(powersetAx_type,type,
% 2.61/1.51 powersetAx: $o ).
% 2.61/1.51
% 2.61/1.51 thf(powersetAx,definition,
% 2.61/1.51 ( powersetAx
% 2.61/1.51 = ( ! [A: $i,B: $i] :
% 2.61/1.51 ( ( in @ B @ ( powerset @ A ) )
% 2.61/1.51 <=> ! [Xx: $i] :
% 2.61/1.51 ( ( in @ Xx @ B )
% 2.61/1.51 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(setunion_type,type,
% 2.61/1.51 setunion: $i > $i ).
% 2.61/1.51
% 2.61/1.51 thf(setunionAx_type,type,
% 2.61/1.51 setunionAx: $o ).
% 2.61/1.51
% 2.61/1.51 thf(setunionAx,definition,
% 2.61/1.51 ( setunionAx
% 2.61/1.51 = ( ! [A: $i,Xx: $i] :
% 2.61/1.51 ( ( in @ Xx @ ( setunion @ A ) )
% 2.61/1.51 <=> ? [B: $i] :
% 2.61/1.51 ( ( in @ Xx @ B )
% 2.61/1.51 & ( in @ B @ A ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(omega_type,type,
% 2.61/1.51 omega: $i ).
% 2.61/1.51
% 2.61/1.51 thf(omega0Ax_type,type,
% 2.61/1.51 omega0Ax: $o ).
% 2.61/1.51
% 2.61/1.51 thf(omega0Ax,definition,
% 2.61/1.51 ( omega0Ax
% 2.61/1.51 = ( in @ emptyset @ omega ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(omegaSAx_type,type,
% 2.61/1.51 omegaSAx: $o ).
% 2.61/1.51
% 2.61/1.51 thf(omegaSAx,definition,
% 2.61/1.51 ( omegaSAx
% 2.61/1.51 = ( ! [Xx: $i] :
% 2.61/1.51 ( ( in @ Xx @ omega )
% 2.61/1.51 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(omegaIndAx_type,type,
% 2.61/1.51 omegaIndAx: $o ).
% 2.61/1.51
% 2.61/1.51 thf(omegaIndAx,definition,
% 2.61/1.51 ( omegaIndAx
% 2.61/1.51 = ( ! [A: $i] :
% 2.61/1.51 ( ( ( in @ emptyset @ A )
% 2.61/1.51 & ! [Xx: $i] :
% 2.61/1.51 ( ( ( in @ Xx @ omega )
% 2.61/1.51 & ( in @ Xx @ A ) )
% 2.61/1.51 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 2.61/1.51 => ! [Xx: $i] :
% 2.61/1.51 ( ( in @ Xx @ omega )
% 2.61/1.51 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(replAx_type,type,
% 2.61/1.51 replAx: $o ).
% 2.61/1.51
% 2.61/1.51 thf(replAx,definition,
% 2.61/1.51 ( replAx
% 2.61/1.51 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 2.61/1.51 ( ! [Xx: $i] :
% 2.61/1.51 ( ( in @ Xx @ A )
% 2.61/1.51 => ( exu
% 2.61/1.51 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 2.61/1.51 => ? [B: $i] :
% 2.61/1.51 ! [Xx: $i] :
% 2.61/1.51 ( ( in @ Xx @ B )
% 2.61/1.51 <=> ? [Xy: $i] :
% 2.61/1.51 ( ( in @ Xy @ A )
% 2.61/1.51 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(foundationAx_type,type,
% 2.61/1.51 foundationAx: $o ).
% 2.61/1.51
% 2.61/1.51 thf(foundationAx,definition,
% 2.61/1.51 ( foundationAx
% 2.61/1.51 = ( ! [A: $i] :
% 2.61/1.51 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.61/1.51 => ? [B: $i] :
% 2.61/1.51 ( ( in @ B @ A )
% 2.61/1.51 & ~ ? [Xx: $i] :
% 2.61/1.51 ( ( in @ Xx @ B )
% 2.61/1.51 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(wellorderingAx_type,type,
% 2.61/1.51 wellorderingAx: $o ).
% 2.61/1.51
% 2.61/1.51 thf(wellorderingAx,definition,
% 2.61/1.51 ( wellorderingAx
% 2.61/1.51 = ( ! [A: $i] :
% 2.61/1.51 ? [B: $i] :
% 2.61/1.51 ( ! [C: $i] :
% 2.61/1.51 ( ( in @ C @ B )
% 2.61/1.51 => ! [Xx: $i] :
% 2.61/1.51 ( ( in @ Xx @ C )
% 2.61/1.51 => ( in @ Xx @ A ) ) )
% 2.61/1.51 & ! [Xx: $i,Xy: $i] :
% 2.61/1.51 ( ( ( in @ Xx @ A )
% 2.61/1.51 & ( in @ Xy @ A ) )
% 2.61/1.51 => ( ! [C: $i] :
% 2.61/1.51 ( ( in @ C @ B )
% 2.61/1.51 => ( ( in @ Xx @ C )
% 2.61/1.51 <=> ( in @ Xy @ C ) ) )
% 2.61/1.51 => ( Xx = Xy ) ) )
% 2.61/1.51 & ! [C: $i,D: $i] :
% 2.61/1.51 ( ( ( in @ C @ B )
% 2.61/1.51 & ( in @ D @ B ) )
% 2.61/1.51 => ( ! [Xx: $i] :
% 2.61/1.51 ( ( in @ Xx @ C )
% 2.61/1.51 => ( in @ Xx @ D ) )
% 2.61/1.51 | ! [Xx: $i] :
% 2.61/1.51 ( ( in @ Xx @ D )
% 2.61/1.51 => ( in @ Xx @ C ) ) ) )
% 2.61/1.51 & ! [C: $i] :
% 2.61/1.51 ( ( ! [Xx: $i] :
% 2.61/1.51 ( ( in @ Xx @ C )
% 2.61/1.51 => ( in @ Xx @ A ) )
% 2.61/1.51 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 2.61/1.51 => ? [D: $i,Xx: $i] :
% 2.61/1.51 ( ( in @ D @ B )
% 2.61/1.51 & ( in @ Xx @ C )
% 2.61/1.51 & ~ ? [Xy: $i] :
% 2.61/1.51 ( ( in @ Xy @ D )
% 2.61/1.51 & ( in @ Xy @ C ) )
% 2.61/1.51 & ! [E: $i] :
% 2.61/1.51 ( ( in @ E @ B )
% 2.61/1.51 => ( ! [Xy: $i] :
% 2.61/1.51 ( ( in @ Xy @ E )
% 2.61/1.51 => ( in @ Xy @ D ) )
% 2.61/1.51 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(descr_type,type,
% 2.61/1.51 descr: ( $i > $o ) > $i ).
% 2.61/1.51
% 2.61/1.51 thf(descrp_type,type,
% 2.61/1.51 descrp: $o ).
% 2.61/1.51
% 2.61/1.51 thf(descrp,definition,
% 2.61/1.51 ( descrp
% 2.61/1.51 = ( ! [Xphi: $i > $o] :
% 2.61/1.51 ( ( exu
% 2.61/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.61/1.51 => ( Xphi
% 2.61/1.51 @ ( descr
% 2.61/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(dsetconstr_type,type,
% 2.61/1.51 dsetconstr: $i > ( $i > $o ) > $i ).
% 2.61/1.51
% 2.61/1.51 thf(dsetconstrI_type,type,
% 2.61/1.51 dsetconstrI: $o ).
% 2.61/1.51
% 2.61/1.51 thf(dsetconstrI,definition,
% 2.61/1.51 ( dsetconstrI
% 2.61/1.51 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.61/1.51 ( ( in @ Xx @ A )
% 2.61/1.51 => ( ( Xphi @ Xx )
% 2.61/1.51 => ( in @ Xx
% 2.61/1.51 @ ( dsetconstr @ A
% 2.61/1.51 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(dsetconstrEL_type,type,
% 2.61/1.51 dsetconstrEL: $o ).
% 2.61/1.51
% 2.61/1.51 thf(dsetconstrEL,definition,
% 2.61/1.51 ( dsetconstrEL
% 2.61/1.51 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.61/1.51 ( ( in @ Xx
% 2.61/1.51 @ ( dsetconstr @ A
% 2.61/1.51 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.61/1.51 => ( in @ Xx @ A ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(dsetconstrER_type,type,
% 2.61/1.51 dsetconstrER: $o ).
% 2.61/1.51
% 2.61/1.51 thf(dsetconstrER,definition,
% 2.61/1.51 ( dsetconstrER
% 2.61/1.51 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.61/1.51 ( ( in @ Xx
% 2.61/1.51 @ ( dsetconstr @ A
% 2.61/1.51 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.61/1.51 => ( Xphi @ Xx ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(exuE1_type,type,
% 2.61/1.51 exuE1: $o ).
% 2.61/1.51
% 2.61/1.51 thf(exuE1,definition,
% 2.61/1.51 ( exuE1
% 2.61/1.51 = ( ! [Xphi: $i > $o] :
% 2.61/1.51 ( ( exu
% 2.61/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.61/1.51 => ? [Xx: $i] :
% 2.61/1.51 ( ( Xphi @ Xx )
% 2.61/1.51 & ! [Xy: $i] :
% 2.61/1.51 ( ( Xphi @ Xy )
% 2.61/1.51 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(prop2set_type,type,
% 2.61/1.51 prop2set: $o > $i ).
% 2.61/1.51
% 2.61/1.51 thf(prop2setE_type,type,
% 2.61/1.51 prop2setE: $o ).
% 2.61/1.51
% 2.61/1.51 thf(prop2setE,definition,
% 2.61/1.51 ( prop2setE
% 2.61/1.51 = ( ! [Xphi: $o,Xx: $i] :
% 2.61/1.51 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 2.61/1.51 => Xphi ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(emptysetE_type,type,
% 2.61/1.51 emptysetE: $o ).
% 2.61/1.51
% 2.61/1.51 thf(emptysetE,definition,
% 2.61/1.51 ( emptysetE
% 2.61/1.51 = ( ! [Xx: $i] :
% 2.61/1.51 ( ( in @ Xx @ emptyset )
% 2.61/1.51 => ! [Xphi: $o] : Xphi ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(emptysetimpfalse_type,type,
% 2.61/1.51 emptysetimpfalse: $o ).
% 2.61/1.51
% 2.61/1.51 thf(emptysetimpfalse,definition,
% 2.61/1.51 ( emptysetimpfalse
% 2.61/1.51 = ( ! [Xx: $i] :
% 2.61/1.51 ( ( in @ Xx @ emptyset )
% 2.61/1.51 => $false ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(notinemptyset_type,type,
% 2.61/1.51 notinemptyset: $o ).
% 2.61/1.51
% 2.61/1.51 thf(notinemptyset,definition,
% 2.61/1.51 ( notinemptyset
% 2.61/1.51 = ( ! [Xx: $i] :
% 2.61/1.51 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(exuE3e_type,type,
% 2.61/1.51 exuE3e: $o ).
% 2.61/1.51
% 2.61/1.51 thf(exuE3e,definition,
% 2.61/1.51 ( exuE3e
% 2.61/1.51 = ( ! [Xphi: $i > $o] :
% 2.61/1.51 ( ( exu
% 2.61/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.61/1.51 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(setext_type,type,
% 2.61/1.51 setext: $o ).
% 2.61/1.51
% 2.61/1.51 thf(setext,definition,
% 2.61/1.51 ( setext
% 2.61/1.51 = ( ! [A: $i,B: $i] :
% 2.61/1.51 ( ! [Xx: $i] :
% 2.61/1.51 ( ( in @ Xx @ A )
% 2.61/1.51 => ( in @ Xx @ B ) )
% 2.61/1.51 => ( ! [Xx: $i] :
% 2.61/1.51 ( ( in @ Xx @ B )
% 2.61/1.51 => ( in @ Xx @ A ) )
% 2.61/1.51 => ( A = B ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(emptyI_type,type,
% 2.61/1.51 emptyI: $o ).
% 2.61/1.51
% 2.61/1.51 thf(emptyI,definition,
% 2.61/1.51 ( emptyI
% 2.61/1.51 = ( ! [A: $i] :
% 2.61/1.51 ( ! [Xx: $i] :
% 2.61/1.51 ~ ( in @ Xx @ A )
% 2.61/1.51 => ( A = emptyset ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(noeltsimpempty_type,type,
% 2.61/1.51 noeltsimpempty: $o ).
% 2.61/1.51
% 2.61/1.51 thf(noeltsimpempty,definition,
% 2.61/1.51 ( noeltsimpempty
% 2.61/1.51 = ( ! [A: $i] :
% 2.61/1.51 ( ! [Xx: $i] :
% 2.61/1.51 ~ ( in @ Xx @ A )
% 2.61/1.51 => ( A = emptyset ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(setbeta_type,type,
% 2.61/1.51 setbeta: $o ).
% 2.61/1.51
% 2.61/1.51 thf(setbeta,definition,
% 2.61/1.51 ( setbeta
% 2.61/1.51 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.61/1.51 ( ( in @ Xx @ A )
% 2.61/1.51 => ( ( in @ Xx
% 2.61/1.51 @ ( dsetconstr @ A
% 2.61/1.51 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.61/1.51 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(nonempty_type,type,
% 2.61/1.51 nonempty: $i > $o ).
% 2.61/1.51
% 2.61/1.51 thf(nonempty,definition,
% 2.61/1.51 ( nonempty
% 2.61/1.51 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(nonemptyE1_type,type,
% 2.61/1.51 nonemptyE1: $o ).
% 2.61/1.51
% 2.61/1.51 thf(nonemptyE1,definition,
% 2.61/1.51 ( nonemptyE1
% 2.61/1.51 = ( ! [A: $i] :
% 2.61/1.51 ( ( nonempty @ A )
% 2.61/1.51 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(nonemptyI_type,type,
% 2.61/1.51 nonemptyI: $o ).
% 2.61/1.51
% 2.61/1.51 thf(nonemptyI,definition,
% 2.61/1.51 ( nonemptyI
% 2.61/1.51 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.61/1.51 ( ( in @ Xx @ A )
% 2.61/1.51 => ( ( Xphi @ Xx )
% 2.61/1.51 => ( nonempty
% 2.61/1.51 @ ( dsetconstr @ A
% 2.61/1.51 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(nonemptyI1_type,type,
% 2.61/1.51 nonemptyI1: $o ).
% 2.61/1.51
% 2.61/1.51 thf(nonemptyI1,definition,
% 2.61/1.51 ( nonemptyI1
% 2.61/1.51 = ( ! [A: $i] :
% 2.61/1.51 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.61/1.51 => ( nonempty @ A ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(setadjoinIL_type,type,
% 2.61/1.51 setadjoinIL: $o ).
% 2.61/1.51
% 2.61/1.51 thf(setadjoinIL,definition,
% 2.61/1.51 ( setadjoinIL
% 2.61/1.51 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(emptyinunitempty_type,type,
% 2.61/1.51 emptyinunitempty: $o ).
% 2.61/1.51
% 2.61/1.51 thf(emptyinunitempty,definition,
% 2.61/1.51 ( emptyinunitempty
% 2.61/1.51 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(setadjoinIR_type,type,
% 2.61/1.51 setadjoinIR: $o ).
% 2.61/1.51
% 2.61/1.51 thf(setadjoinIR,definition,
% 2.61/1.51 ( setadjoinIR
% 2.61/1.51 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.61/1.51 ( ( in @ Xy @ A )
% 2.61/1.51 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(setadjoinE_type,type,
% 2.61/1.51 setadjoinE: $o ).
% 2.61/1.51
% 2.61/1.51 thf(setadjoinE,definition,
% 2.61/1.51 ( setadjoinE
% 2.61/1.51 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.61/1.51 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.61/1.51 => ! [Xphi: $o] :
% 2.61/1.51 ( ( ( Xy = Xx )
% 2.61/1.51 => Xphi )
% 2.61/1.51 => ( ( ( in @ Xy @ A )
% 2.61/1.51 => Xphi )
% 2.61/1.51 => Xphi ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(setadjoinOr_type,type,
% 2.61/1.51 setadjoinOr: $o ).
% 2.61/1.51
% 2.61/1.51 thf(setadjoinOr,definition,
% 2.61/1.51 ( setadjoinOr
% 2.61/1.51 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.61/1.51 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.61/1.51 => ( ( Xy = Xx )
% 2.61/1.51 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(setoftrueEq_type,type,
% 2.61/1.51 setoftrueEq: $o ).
% 2.61/1.51
% 2.61/1.51 thf(setoftrueEq,definition,
% 2.61/1.51 ( setoftrueEq
% 2.61/1.51 = ( ! [A: $i] :
% 2.61/1.51 ( ( dsetconstr @ A
% 2.61/1.51 @ ^ [Xx: $i] : $true )
% 2.61/1.51 = A ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(powersetI_type,type,
% 2.61/1.51 powersetI: $o ).
% 2.61/1.51
% 2.61/1.51 thf(powersetI,definition,
% 2.61/1.51 ( powersetI
% 2.61/1.51 = ( ! [A: $i,B: $i] :
% 2.61/1.51 ( ! [Xx: $i] :
% 2.61/1.51 ( ( in @ Xx @ B )
% 2.61/1.51 => ( in @ Xx @ A ) )
% 2.61/1.51 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(emptyinPowerset_type,type,
% 2.61/1.51 emptyinPowerset: $o ).
% 2.61/1.51
% 2.61/1.51 thf(emptyinPowerset,definition,
% 2.61/1.51 ( emptyinPowerset
% 2.61/1.51 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(emptyInPowerset_type,type,
% 2.61/1.51 emptyInPowerset: $o ).
% 2.61/1.51
% 2.61/1.51 thf(emptyInPowerset,definition,
% 2.61/1.51 ( emptyInPowerset
% 2.61/1.51 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(powersetE_type,type,
% 2.61/1.51 powersetE: $o ).
% 2.61/1.51
% 2.61/1.51 thf(powersetE,definition,
% 2.61/1.51 ( powersetE
% 2.61/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.51 ( ( in @ B @ ( powerset @ A ) )
% 2.61/1.51 => ( ( in @ Xx @ B )
% 2.61/1.51 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(setunionI_type,type,
% 2.61/1.51 setunionI: $o ).
% 2.61/1.51
% 2.61/1.51 thf(setunionI,definition,
% 2.61/1.51 ( setunionI
% 2.61/1.51 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.61/1.51 ( ( in @ Xx @ B )
% 2.61/1.51 => ( ( in @ B @ A )
% 2.61/1.51 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(setunionE_type,type,
% 2.61/1.51 setunionE: $o ).
% 2.61/1.51
% 2.61/1.51 thf(setunionE,definition,
% 2.61/1.51 ( setunionE
% 2.61/1.51 = ( ! [A: $i,Xx: $i] :
% 2.61/1.51 ( ( in @ Xx @ ( setunion @ A ) )
% 2.61/1.51 => ! [Xphi: $o] :
% 2.61/1.51 ( ! [B: $i] :
% 2.61/1.51 ( ( in @ Xx @ B )
% 2.61/1.51 => ( ( in @ B @ A )
% 2.61/1.51 => Xphi ) )
% 2.61/1.51 => Xphi ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(subPowSU_type,type,
% 2.61/1.51 subPowSU: $o ).
% 2.61/1.51
% 2.61/1.51 thf(subPowSU,definition,
% 2.61/1.51 ( subPowSU
% 2.61/1.51 = ( ! [A: $i,Xx: $i] :
% 2.61/1.51 ( ( in @ Xx @ A )
% 2.61/1.51 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(exuE2_type,type,
% 2.61/1.51 exuE2: $o ).
% 2.61/1.51
% 2.61/1.51 thf(exuE2,definition,
% 2.61/1.51 ( exuE2
% 2.61/1.51 = ( ! [Xphi: $i > $o] :
% 2.61/1.51 ( ( exu
% 2.61/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.61/1.51 => ? [Xx: $i] :
% 2.61/1.51 ! [Xy: $i] :
% 2.61/1.51 ( ( Xphi @ Xy )
% 2.61/1.51 <=> ( Xy = Xx ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(nonemptyImpWitness_type,type,
% 2.61/1.51 nonemptyImpWitness: $o ).
% 2.61/1.51
% 2.61/1.51 thf(nonemptyImpWitness,definition,
% 2.61/1.51 ( nonemptyImpWitness
% 2.61/1.51 = ( ! [A: $i] :
% 2.61/1.51 ( ( nonempty @ A )
% 2.61/1.51 => ? [Xx: $i] :
% 2.61/1.51 ( ( in @ Xx @ A )
% 2.61/1.51 & $true ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(uniqinunit_type,type,
% 2.61/1.51 uniqinunit: $o ).
% 2.61/1.51
% 2.61/1.51 thf(uniqinunit,definition,
% 2.61/1.51 ( uniqinunit
% 2.61/1.51 = ( ! [Xx: $i,Xy: $i] :
% 2.61/1.51 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.61/1.51 => ( Xx = Xy ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(notinsingleton_type,type,
% 2.61/1.51 notinsingleton: $o ).
% 2.61/1.51
% 2.61/1.51 thf(notinsingleton,definition,
% 2.61/1.51 ( notinsingleton
% 2.61/1.51 = ( ! [Xx: $i,Xy: $i] :
% 2.61/1.51 ( ( Xx != Xy )
% 2.61/1.51 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(eqinunit_type,type,
% 2.61/1.51 eqinunit: $o ).
% 2.61/1.51
% 2.61/1.51 thf(eqinunit,definition,
% 2.61/1.51 ( eqinunit
% 2.61/1.51 = ( ! [Xx: $i,Xy: $i] :
% 2.61/1.51 ( ( Xx = Xy )
% 2.61/1.51 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(singletonsswitch_type,type,
% 2.61/1.51 singletonsswitch: $o ).
% 2.61/1.51
% 2.61/1.51 thf(singletonsswitch,definition,
% 2.61/1.51 ( singletonsswitch
% 2.61/1.51 = ( ! [Xx: $i,Xy: $i] :
% 2.61/1.51 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.61/1.51 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(upairsetE_type,type,
% 2.61/1.51 upairsetE: $o ).
% 2.61/1.51
% 2.61/1.51 thf(upairsetE,definition,
% 2.61/1.51 ( upairsetE
% 2.61/1.51 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.61/1.51 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.61/1.51 => ( ( Xz = Xx )
% 2.61/1.51 | ( Xz = Xy ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(upairsetIL_type,type,
% 2.61/1.51 upairsetIL: $o ).
% 2.61/1.51
% 2.61/1.51 thf(upairsetIL,definition,
% 2.61/1.51 ( upairsetIL
% 2.61/1.51 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(upairsetIR_type,type,
% 2.61/1.51 upairsetIR: $o ).
% 2.61/1.51
% 2.61/1.51 thf(upairsetIR,definition,
% 2.61/1.51 ( upairsetIR
% 2.61/1.51 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(emptyE1_type,type,
% 2.61/1.51 emptyE1: $o ).
% 2.61/1.51
% 2.61/1.51 thf(emptyE1,definition,
% 2.61/1.51 ( emptyE1
% 2.61/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 2.61/1.51 ( ? [Xx: $i] :
% 2.61/1.51 ( ( in @ Xx @ A )
% 2.61/1.51 & ( Xphi @ Xx ) )
% 2.61/1.51 => ( ( ( dsetconstr @ A
% 2.61/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.61/1.51 = emptyset )
% 2.61/1.51 => $false ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(vacuousDall_type,type,
% 2.61/1.51 vacuousDall: $o ).
% 2.61/1.51
% 2.61/1.51 thf(vacuousDall,definition,
% 2.61/1.51 ( vacuousDall
% 2.61/1.51 = ( ! [Xphi: $i > $o,Xx: $i] :
% 2.61/1.51 ( ( in @ Xx @ emptyset )
% 2.61/1.51 => ( Xphi @ Xx ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(quantDeMorgan1_type,type,
% 2.61/1.51 quantDeMorgan1: $o ).
% 2.61/1.51
% 2.61/1.51 thf(quantDeMorgan1,definition,
% 2.61/1.51 ( quantDeMorgan1
% 2.61/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 2.61/1.51 ( ~ ! [Xx: $i] :
% 2.61/1.51 ( ( in @ Xx @ A )
% 2.61/1.51 => ( Xphi @ Xx ) )
% 2.61/1.51 => ? [Xx: $i] :
% 2.61/1.51 ( ( in @ Xx @ A )
% 2.61/1.51 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(quantDeMorgan2_type,type,
% 2.61/1.51 quantDeMorgan2: $o ).
% 2.61/1.51
% 2.61/1.51 thf(quantDeMorgan2,definition,
% 2.61/1.51 ( quantDeMorgan2
% 2.61/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 2.61/1.51 ( ! [Xx: $i] :
% 2.61/1.51 ( ( in @ Xx @ A )
% 2.61/1.51 => ~ ( Xphi @ Xx ) )
% 2.61/1.51 => ~ ? [Xx: $i] :
% 2.61/1.51 ( ( in @ Xx @ A )
% 2.61/1.51 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(quantDeMorgan3_type,type,
% 2.61/1.51 quantDeMorgan3: $o ).
% 2.61/1.51
% 2.61/1.51 thf(quantDeMorgan3,definition,
% 2.61/1.51 ( quantDeMorgan3
% 2.61/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 2.61/1.51 ( ~ ? [Xx: $i] :
% 2.61/1.51 ( ( in @ Xx @ A )
% 2.61/1.51 & ( Xphi @ Xx ) )
% 2.61/1.51 => ! [Xx: $i] :
% 2.61/1.51 ( ( in @ Xx @ A )
% 2.61/1.51 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(quantDeMorgan4_type,type,
% 2.61/1.51 quantDeMorgan4: $o ).
% 2.61/1.51
% 2.61/1.51 thf(quantDeMorgan4,definition,
% 2.61/1.51 ( quantDeMorgan4
% 2.61/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 2.61/1.51 ( ? [Xx: $i] :
% 2.61/1.51 ( ( in @ Xx @ A )
% 2.61/1.51 & ~ ( Xphi @ Xx ) )
% 2.61/1.51 => ~ ! [Xx: $i] :
% 2.61/1.51 ( ( in @ Xx @ A )
% 2.61/1.51 => ( Xphi @ Xx ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(prop2setI_type,type,
% 2.61/1.51 prop2setI: $o ).
% 2.61/1.51
% 2.61/1.51 thf(prop2setI,definition,
% 2.61/1.51 ( prop2setI
% 2.61/1.51 = ( ! [Xphi: $o] :
% 2.61/1.51 ( Xphi
% 2.61/1.51 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(set2prop_type,type,
% 2.61/1.51 set2prop: $i > $o ).
% 2.61/1.51
% 2.61/1.51 thf(prop2set2propI_type,type,
% 2.61/1.51 prop2set2propI: $o ).
% 2.61/1.51
% 2.61/1.51 thf(prop2set2propI,definition,
% 2.61/1.51 ( prop2set2propI
% 2.61/1.51 = ( ! [Xphi: $o] :
% 2.61/1.51 ( Xphi
% 2.61/1.51 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(notdexE_type,type,
% 2.61/1.51 notdexE: $o ).
% 2.61/1.51
% 2.61/1.51 thf(notdexE,definition,
% 2.61/1.51 ( notdexE
% 2.61/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 2.61/1.51 ( ~ ? [Xx: $i] :
% 2.61/1.51 ( ( in @ Xx @ A )
% 2.61/1.51 & ( Xphi @ Xx ) )
% 2.61/1.51 => ! [Xx: $i] :
% 2.61/1.51 ( ( in @ Xx @ A )
% 2.61/1.51 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(notdallE_type,type,
% 2.61/1.51 notdallE: $o ).
% 2.61/1.51
% 2.61/1.51 thf(notdallE,definition,
% 2.61/1.51 ( notdallE
% 2.61/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 2.61/1.51 ( ~ ! [Xx: $i] :
% 2.61/1.51 ( ( in @ Xx @ A )
% 2.61/1.51 => ( Xphi @ Xx ) )
% 2.61/1.51 => ? [Xx: $i] :
% 2.61/1.51 ( ( in @ Xx @ A )
% 2.61/1.51 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(exuI1_type,type,
% 2.61/1.51 exuI1: $o ).
% 2.61/1.51
% 2.61/1.51 thf(exuI1,definition,
% 2.61/1.51 ( exuI1
% 2.61/1.51 = ( ! [Xphi: $i > $o] :
% 2.61/1.51 ( ? [Xx: $i] :
% 2.61/1.51 ( ( Xphi @ Xx )
% 2.61/1.51 & ! [Xy: $i] :
% 2.61/1.51 ( ( Xphi @ Xy )
% 2.61/1.51 => ( Xx = Xy ) ) )
% 2.61/1.51 => ( exu
% 2.61/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(exuI3_type,type,
% 2.61/1.51 exuI3: $o ).
% 2.61/1.51
% 2.61/1.51 thf(exuI3,definition,
% 2.61/1.51 ( exuI3
% 2.61/1.51 = ( ! [Xphi: $i > $o] :
% 2.61/1.51 ( ? [Xx: $i] : ( Xphi @ Xx )
% 2.61/1.51 => ( ! [Xx: $i,Xy: $i] :
% 2.61/1.51 ( ( Xphi @ Xx )
% 2.61/1.51 => ( ( Xphi @ Xy )
% 2.61/1.51 => ( Xx = Xy ) ) )
% 2.61/1.51 => ( exu
% 2.61/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(exuI2_type,type,
% 2.61/1.51 exuI2: $o ).
% 2.61/1.51
% 2.61/1.51 thf(exuI2,definition,
% 2.61/1.51 ( exuI2
% 2.61/1.51 = ( ! [Xphi: $i > $o] :
% 2.61/1.51 ( ? [Xx: $i] :
% 2.61/1.51 ! [Xy: $i] :
% 2.61/1.51 ( ( Xphi @ Xy )
% 2.61/1.51 <=> ( Xy = Xx ) )
% 2.61/1.51 => ( exu
% 2.61/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(inCongP_type,type,
% 2.61/1.51 inCongP: $o ).
% 2.61/1.51
% 2.61/1.51 thf(inCongP,definition,
% 2.61/1.51 ( inCongP
% 2.61/1.51 = ( ! [A: $i,B: $i] :
% 2.61/1.51 ( ( A = B )
% 2.61/1.51 => ! [Xx: $i,Xy: $i] :
% 2.61/1.51 ( ( Xx = Xy )
% 2.61/1.51 => ( ( in @ Xx @ A )
% 2.61/1.51 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(in__Cong_type,type,
% 2.61/1.51 in__Cong: $o ).
% 2.61/1.51
% 2.61/1.51 thf(in__Cong,definition,
% 2.61/1.51 ( in__Cong
% 2.61/1.51 = ( ! [A: $i,B: $i] :
% 2.61/1.51 ( ( A = B )
% 2.61/1.51 => ! [Xx: $i,Xy: $i] :
% 2.61/1.51 ( ( Xx = Xy )
% 2.61/1.51 => ( ( in @ Xx @ A )
% 2.61/1.51 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(exuE3u_type,type,
% 2.61/1.51 exuE3u: $o ).
% 2.61/1.51
% 2.61/1.51 thf(exuE3u,definition,
% 2.61/1.51 ( exuE3u
% 2.61/1.51 = ( ! [Xphi: $i > $o] :
% 2.61/1.51 ( ( exu
% 2.61/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.61/1.51 => ! [Xx: $i,Xy: $i] :
% 2.61/1.51 ( ( Xphi @ Xx )
% 2.61/1.51 => ( ( Xphi @ Xy )
% 2.61/1.51 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(exu__Cong_type,type,
% 2.61/1.51 exu__Cong: $o ).
% 2.61/1.51
% 2.61/1.51 thf(exu__Cong,definition,
% 2.61/1.51 ( exu__Cong
% 2.61/1.51 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.61/1.51 ( ! [Xx: $i,Xy: $i] :
% 2.61/1.51 ( ( Xx = Xy )
% 2.61/1.51 => ( ( Xphi @ Xx )
% 2.61/1.51 <=> ( Xpsi @ Xy ) ) )
% 2.61/1.51 => ( ( exu
% 2.61/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.61/1.51 <=> ( exu
% 2.61/1.51 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(emptyset__Cong_type,type,
% 2.61/1.51 emptyset__Cong: $o ).
% 2.61/1.51
% 2.61/1.51 thf(emptyset__Cong,definition,
% 2.61/1.51 ( emptyset__Cong
% 2.61/1.51 = ( emptyset = emptyset ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(setadjoin__Cong_type,type,
% 2.61/1.51 setadjoin__Cong: $o ).
% 2.61/1.51
% 2.61/1.51 thf(setadjoin__Cong,definition,
% 2.61/1.51 ( setadjoin__Cong
% 2.61/1.51 = ( ! [Xx: $i,Xy: $i] :
% 2.61/1.51 ( ( Xx = Xy )
% 2.61/1.51 => ! [Xz: $i,Xu: $i] :
% 2.61/1.51 ( ( Xz = Xu )
% 2.61/1.51 => ( ( setadjoin @ Xx @ Xz )
% 2.61/1.51 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(powerset__Cong_type,type,
% 2.61/1.51 powerset__Cong: $o ).
% 2.61/1.51
% 2.61/1.51 thf(powerset__Cong,definition,
% 2.61/1.51 ( powerset__Cong
% 2.61/1.51 = ( ! [A: $i,B: $i] :
% 2.61/1.51 ( ( A = B )
% 2.61/1.51 => ( ( powerset @ A )
% 2.61/1.51 = ( powerset @ B ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(setunion__Cong_type,type,
% 2.61/1.51 setunion__Cong: $o ).
% 2.61/1.51
% 2.61/1.51 thf(setunion__Cong,definition,
% 2.61/1.51 ( setunion__Cong
% 2.61/1.51 = ( ! [A: $i,B: $i] :
% 2.61/1.51 ( ( A = B )
% 2.61/1.51 => ( ( setunion @ A )
% 2.61/1.51 = ( setunion @ B ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(omega__Cong_type,type,
% 2.61/1.51 omega__Cong: $o ).
% 2.61/1.51
% 2.61/1.51 thf(omega__Cong,definition,
% 2.61/1.51 ( omega__Cong
% 2.61/1.51 = ( omega = omega ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(exuEu_type,type,
% 2.61/1.51 exuEu: $o ).
% 2.61/1.51
% 2.61/1.51 thf(exuEu,definition,
% 2.61/1.51 ( exuEu
% 2.61/1.51 = ( ! [Xphi: $i > $o] :
% 2.61/1.51 ( ( exu
% 2.61/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.61/1.51 => ! [Xx: $i,Xy: $i] :
% 2.61/1.51 ( ( Xphi @ Xx )
% 2.61/1.51 => ( ( Xphi @ Xy )
% 2.61/1.51 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(descr__Cong_type,type,
% 2.61/1.51 descr__Cong: $o ).
% 2.61/1.51
% 2.61/1.51 thf(descr__Cong,definition,
% 2.61/1.51 ( descr__Cong
% 2.61/1.51 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.61/1.51 ( ! [Xx: $i,Xy: $i] :
% 2.61/1.51 ( ( Xx = Xy )
% 2.61/1.51 => ( ( Xphi @ Xx )
% 2.61/1.51 <=> ( Xpsi @ Xy ) ) )
% 2.61/1.51 => ( ( exu
% 2.61/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.61/1.51 => ( ( exu
% 2.61/1.51 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 2.61/1.51 => ( ( descr
% 2.61/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.61/1.51 = ( descr
% 2.61/1.51 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(dsetconstr__Cong_type,type,
% 2.61/1.51 dsetconstr__Cong: $o ).
% 2.61/1.51
% 2.61/1.51 thf(dsetconstr__Cong,definition,
% 2.61/1.51 ( dsetconstr__Cong
% 2.61/1.51 = ( ! [A: $i,B: $i] :
% 2.61/1.51 ( ( A = B )
% 2.61/1.51 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.61/1.51 ( ! [Xx: $i] :
% 2.61/1.51 ( ( in @ Xx @ A )
% 2.61/1.51 => ! [Xy: $i] :
% 2.61/1.51 ( ( in @ Xy @ B )
% 2.61/1.51 => ( ( Xx = Xy )
% 2.61/1.51 => ( ( Xphi @ Xx )
% 2.61/1.51 <=> ( Xpsi @ Xy ) ) ) ) )
% 2.61/1.51 => ( ( dsetconstr @ A
% 2.61/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.61/1.51 = ( dsetconstr @ B
% 2.61/1.51 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(subset_type,type,
% 2.61/1.51 subset: $i > $i > $o ).
% 2.61/1.51
% 2.61/1.51 thf(disjoint_type,type,
% 2.61/1.51 disjoint: $i > $i > $o ).
% 2.61/1.51
% 2.61/1.51 thf(setsmeet_type,type,
% 2.61/1.51 setsmeet: $i > $i > $o ).
% 2.61/1.51
% 2.61/1.51 thf(subsetI1_type,type,
% 2.61/1.51 subsetI1: $o ).
% 2.61/1.51
% 2.61/1.51 thf(subsetI1,definition,
% 2.61/1.51 ( subsetI1
% 2.61/1.51 = ( ! [A: $i,B: $i] :
% 2.61/1.51 ( ! [Xx: $i] :
% 2.61/1.51 ( ( in @ Xx @ A )
% 2.61/1.51 => ( in @ Xx @ B ) )
% 2.61/1.51 => ( subset @ A @ B ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(eqimpsubset2_type,type,
% 2.61/1.51 eqimpsubset2: $o ).
% 2.61/1.51
% 2.61/1.51 thf(eqimpsubset2,definition,
% 2.61/1.51 ( eqimpsubset2
% 2.61/1.51 = ( ! [A: $i,B: $i] :
% 2.61/1.51 ( ( A = B )
% 2.61/1.51 => ( subset @ B @ A ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(eqimpsubset1_type,type,
% 2.61/1.51 eqimpsubset1: $o ).
% 2.61/1.51
% 2.61/1.51 thf(eqimpsubset1,definition,
% 2.61/1.51 ( eqimpsubset1
% 2.61/1.51 = ( ! [A: $i,B: $i] :
% 2.61/1.51 ( ( A = B )
% 2.61/1.51 => ( subset @ A @ B ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(subsetI2_type,type,
% 2.61/1.51 subsetI2: $o ).
% 2.61/1.51
% 2.61/1.51 thf(subsetI2,definition,
% 2.61/1.51 ( subsetI2
% 2.61/1.51 = ( ! [A: $i,B: $i] :
% 2.61/1.51 ( ! [Xx: $i] :
% 2.61/1.51 ( ( in @ Xx @ A )
% 2.61/1.51 => ( in @ Xx @ B ) )
% 2.61/1.51 => ( subset @ A @ B ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(emptysetsubset_type,type,
% 2.61/1.51 emptysetsubset: $o ).
% 2.61/1.51
% 2.61/1.51 thf(emptysetsubset,definition,
% 2.61/1.51 ( emptysetsubset
% 2.61/1.51 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(subsetE_type,type,
% 2.61/1.51 subsetE: $o ).
% 2.61/1.51
% 2.61/1.51 thf(subsetE,definition,
% 2.61/1.51 ( subsetE
% 2.61/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.51 ( ( subset @ A @ B )
% 2.61/1.51 => ( ( in @ Xx @ A )
% 2.61/1.51 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(subsetE2_type,type,
% 2.61/1.51 subsetE2: $o ).
% 2.61/1.51
% 2.61/1.51 thf(subsetE2,definition,
% 2.61/1.51 ( subsetE2
% 2.61/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.51 ( ( subset @ A @ B )
% 2.61/1.51 => ( ~ ( in @ Xx @ B )
% 2.61/1.51 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(notsubsetI_type,type,
% 2.61/1.51 notsubsetI: $o ).
% 2.61/1.51
% 2.61/1.51 thf(notsubsetI,definition,
% 2.61/1.51 ( notsubsetI
% 2.61/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.51 ( ( in @ Xx @ A )
% 2.61/1.51 => ( ~ ( in @ Xx @ B )
% 2.61/1.51 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(notequalI1_type,type,
% 2.61/1.51 notequalI1: $o ).
% 2.61/1.51
% 2.61/1.51 thf(notequalI1,definition,
% 2.61/1.51 ( notequalI1
% 2.61/1.51 = ( ! [A: $i,B: $i] :
% 2.61/1.51 ( ~ ( subset @ A @ B )
% 2.61/1.51 => ( A != B ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(notequalI2_type,type,
% 2.61/1.51 notequalI2: $o ).
% 2.61/1.51
% 2.61/1.51 thf(notequalI2,definition,
% 2.61/1.51 ( notequalI2
% 2.61/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.51 ( ( in @ Xx @ A )
% 2.61/1.51 => ( ~ ( in @ Xx @ B )
% 2.61/1.51 => ( A != B ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(subsetRefl_type,type,
% 2.61/1.51 subsetRefl: $o ).
% 2.61/1.51
% 2.61/1.51 thf(subsetRefl,definition,
% 2.61/1.51 ( subsetRefl
% 2.61/1.51 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(subsetTrans_type,type,
% 2.61/1.51 subsetTrans: $o ).
% 2.61/1.51
% 2.61/1.51 thf(subsetTrans,definition,
% 2.61/1.51 ( subsetTrans
% 2.61/1.51 = ( ! [A: $i,B: $i,C: $i] :
% 2.61/1.51 ( ( subset @ A @ B )
% 2.61/1.51 => ( ( subset @ B @ C )
% 2.61/1.51 => ( subset @ A @ C ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(setadjoinSub_type,type,
% 2.61/1.51 setadjoinSub: $o ).
% 2.61/1.51
% 2.61/1.51 thf(setadjoinSub,definition,
% 2.61/1.51 ( setadjoinSub
% 2.61/1.51 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(setadjoinSub2_type,type,
% 2.61/1.51 setadjoinSub2: $o ).
% 2.61/1.51
% 2.61/1.51 thf(setadjoinSub2,definition,
% 2.61/1.51 ( setadjoinSub2
% 2.61/1.51 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.61/1.51 ( ( subset @ A @ B )
% 2.61/1.51 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(subset2powerset_type,type,
% 2.61/1.51 subset2powerset: $o ).
% 2.61/1.51
% 2.61/1.51 thf(subset2powerset,definition,
% 2.61/1.51 ( subset2powerset
% 2.61/1.51 = ( ! [A: $i,B: $i] :
% 2.61/1.51 ( ( subset @ A @ B )
% 2.61/1.51 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(setextsub_type,type,
% 2.61/1.51 setextsub: $o ).
% 2.61/1.51
% 2.61/1.51 thf(setextsub,definition,
% 2.61/1.51 ( setextsub
% 2.61/1.51 = ( ! [A: $i,B: $i] :
% 2.61/1.51 ( ( subset @ A @ B )
% 2.61/1.51 => ( ( subset @ B @ A )
% 2.61/1.51 => ( A = B ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(subsetemptysetimpeq_type,type,
% 2.61/1.51 subsetemptysetimpeq: $o ).
% 2.61/1.51
% 2.61/1.51 thf(subsetemptysetimpeq,definition,
% 2.61/1.51 ( subsetemptysetimpeq
% 2.61/1.51 = ( ! [A: $i] :
% 2.61/1.51 ( ( subset @ A @ emptyset )
% 2.61/1.51 => ( A = emptyset ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(powersetI1_type,type,
% 2.61/1.51 powersetI1: $o ).
% 2.61/1.51
% 2.61/1.51 thf(powersetI1,definition,
% 2.61/1.51 ( powersetI1
% 2.61/1.51 = ( ! [A: $i,B: $i] :
% 2.61/1.51 ( ( subset @ B @ A )
% 2.61/1.51 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(powersetE1_type,type,
% 2.61/1.51 powersetE1: $o ).
% 2.61/1.51
% 2.61/1.51 thf(powersetE1,definition,
% 2.61/1.51 ( powersetE1
% 2.61/1.51 = ( ! [A: $i,B: $i] :
% 2.61/1.51 ( ( in @ B @ ( powerset @ A ) )
% 2.61/1.51 => ( subset @ B @ A ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(inPowerset_type,type,
% 2.61/1.51 inPowerset: $o ).
% 2.61/1.51
% 2.61/1.51 thf(inPowerset,definition,
% 2.61/1.51 ( inPowerset
% 2.61/1.51 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(powersetsubset_type,type,
% 2.61/1.51 powersetsubset: $o ).
% 2.61/1.51
% 2.61/1.51 thf(powersetsubset,definition,
% 2.61/1.51 ( powersetsubset
% 2.61/1.51 = ( ! [A: $i,B: $i] :
% 2.61/1.51 ( ( subset @ A @ B )
% 2.61/1.51 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(sepInPowerset_type,type,
% 2.61/1.51 sepInPowerset: $o ).
% 2.61/1.51
% 2.61/1.51 thf(sepInPowerset,definition,
% 2.61/1.51 ( sepInPowerset
% 2.61/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 2.61/1.51 ( in
% 2.61/1.51 @ ( dsetconstr @ A
% 2.61/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.61/1.51 @ ( powerset @ A ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(sepSubset_type,type,
% 2.61/1.51 sepSubset: $o ).
% 2.61/1.51
% 2.61/1.51 thf(sepSubset,definition,
% 2.61/1.51 ( sepSubset
% 2.61/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 2.61/1.51 ( subset
% 2.61/1.51 @ ( dsetconstr @ A
% 2.61/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.61/1.51 @ A ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(binunion_type,type,
% 2.61/1.51 binunion: $i > $i > $i ).
% 2.61/1.51
% 2.61/1.51 thf(binunionIL_type,type,
% 2.61/1.51 binunionIL: $o ).
% 2.61/1.51
% 2.61/1.51 thf(binunionIL,definition,
% 2.61/1.51 ( binunionIL
% 2.61/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.51 ( ( in @ Xx @ A )
% 2.61/1.51 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(upairset2IR_type,type,
% 2.61/1.51 upairset2IR: $o ).
% 2.61/1.51
% 2.61/1.51 thf(upairset2IR,definition,
% 2.61/1.51 ( upairset2IR
% 2.61/1.51 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(binunionIR_type,type,
% 2.61/1.51 binunionIR: $o ).
% 2.61/1.51
% 2.61/1.51 thf(binunionIR,definition,
% 2.61/1.51 ( binunionIR
% 2.61/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.51 ( ( in @ Xx @ B )
% 2.61/1.51 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(binunionEcases_type,type,
% 2.61/1.51 binunionEcases: $o ).
% 2.61/1.51
% 2.61/1.51 thf(binunionEcases,definition,
% 2.61/1.51 ( binunionEcases
% 2.61/1.51 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 2.61/1.51 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.61/1.51 => ( ( ( in @ Xx @ A )
% 2.61/1.51 => Xphi )
% 2.61/1.51 => ( ( ( in @ Xx @ B )
% 2.61/1.51 => Xphi )
% 2.61/1.51 => Xphi ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(binunionE_type,type,
% 2.61/1.51 binunionE: $o ).
% 2.61/1.51
% 2.61/1.51 thf(binunionE,definition,
% 2.61/1.51 ( binunionE
% 2.61/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.51 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.61/1.51 => ( ( in @ Xx @ A )
% 2.61/1.51 | ( in @ Xx @ B ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(binunionLsub_type,type,
% 2.61/1.51 binunionLsub: $o ).
% 2.61/1.51
% 2.61/1.51 thf(binunionLsub,definition,
% 2.61/1.51 ( binunionLsub
% 2.61/1.51 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(binunionRsub_type,type,
% 2.61/1.51 binunionRsub: $o ).
% 2.61/1.51
% 2.61/1.51 thf(binunionRsub,definition,
% 2.61/1.51 ( binunionRsub
% 2.61/1.51 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(binintersect_type,type,
% 2.61/1.51 binintersect: $i > $i > $i ).
% 2.61/1.51
% 2.61/1.51 thf(binintersectI_type,type,
% 2.61/1.51 binintersectI: $o ).
% 2.61/1.51
% 2.61/1.51 thf(binintersectI,definition,
% 2.61/1.51 ( binintersectI
% 2.61/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.51 ( ( in @ Xx @ A )
% 2.61/1.51 => ( ( in @ Xx @ B )
% 2.61/1.51 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(binintersectSubset5_type,type,
% 2.61/1.51 binintersectSubset5: $o ).
% 2.61/1.51
% 2.61/1.51 thf(binintersectSubset5,definition,
% 2.61/1.51 ( binintersectSubset5
% 2.61/1.51 = ( ! [A: $i,B: $i,C: $i] :
% 2.61/1.51 ( ( subset @ C @ A )
% 2.61/1.51 => ( ( subset @ C @ B )
% 2.61/1.51 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(binintersectEL_type,type,
% 2.61/1.51 binintersectEL: $o ).
% 2.61/1.51
% 2.61/1.51 thf(binintersectEL,definition,
% 2.61/1.51 ( binintersectEL
% 2.61/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.51 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.61/1.51 => ( in @ Xx @ A ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(binintersectLsub_type,type,
% 2.61/1.51 binintersectLsub: $o ).
% 2.61/1.51
% 2.61/1.51 thf(binintersectLsub,definition,
% 2.61/1.51 ( binintersectLsub
% 2.61/1.51 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(binintersectSubset2_type,type,
% 2.61/1.51 binintersectSubset2: $o ).
% 2.61/1.51
% 2.61/1.51 thf(binintersectSubset2,definition,
% 2.61/1.51 ( binintersectSubset2
% 2.61/1.51 = ( ! [A: $i,B: $i] :
% 2.61/1.51 ( ( subset @ A @ B )
% 2.61/1.51 => ( ( binintersect @ A @ B )
% 2.61/1.51 = A ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(binintersectSubset3_type,type,
% 2.61/1.51 binintersectSubset3: $o ).
% 2.61/1.51
% 2.61/1.51 thf(binintersectSubset3,definition,
% 2.61/1.51 ( binintersectSubset3
% 2.61/1.51 = ( ! [A: $i,B: $i] :
% 2.61/1.51 ( ( ( binintersect @ A @ B )
% 2.61/1.51 = B )
% 2.61/1.51 => ( subset @ B @ A ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(binintersectER_type,type,
% 2.61/1.51 binintersectER: $o ).
% 2.61/1.51
% 2.61/1.51 thf(binintersectER,definition,
% 2.61/1.51 ( binintersectER
% 2.61/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.51 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.61/1.51 => ( in @ Xx @ B ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(disjointsetsI1_type,type,
% 2.61/1.51 disjointsetsI1: $o ).
% 2.61/1.51
% 2.61/1.51 thf(disjointsetsI1,definition,
% 2.61/1.51 ( disjointsetsI1
% 2.61/1.51 = ( ! [A: $i,B: $i] :
% 2.61/1.51 ( ~ ? [Xx: $i] :
% 2.61/1.51 ( ( in @ Xx @ A )
% 2.61/1.51 & ( in @ Xx @ B ) )
% 2.61/1.51 => ( ( binintersect @ A @ B )
% 2.61/1.51 = emptyset ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(binintersectRsub_type,type,
% 2.61/1.51 binintersectRsub: $o ).
% 2.61/1.51
% 2.61/1.51 thf(binintersectRsub,definition,
% 2.61/1.51 ( binintersectRsub
% 2.61/1.51 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(binintersectSubset4_type,type,
% 2.61/1.51 binintersectSubset4: $o ).
% 2.61/1.51
% 2.61/1.51 thf(binintersectSubset4,definition,
% 2.61/1.51 ( binintersectSubset4
% 2.61/1.51 = ( ! [A: $i,B: $i] :
% 2.61/1.51 ( ( subset @ B @ A )
% 2.61/1.51 => ( ( binintersect @ A @ B )
% 2.61/1.51 = B ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(binintersectSubset1_type,type,
% 2.61/1.51 binintersectSubset1: $o ).
% 2.61/1.51
% 2.61/1.51 thf(binintersectSubset1,definition,
% 2.61/1.51 ( binintersectSubset1
% 2.61/1.51 = ( ! [A: $i,B: $i] :
% 2.61/1.51 ( ( ( binintersect @ A @ B )
% 2.61/1.51 = A )
% 2.61/1.51 => ( subset @ A @ B ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(bs114d_type,type,
% 2.61/1.51 bs114d: $o ).
% 2.61/1.51
% 2.61/1.51 thf(bs114d,definition,
% 2.61/1.51 ( bs114d
% 2.61/1.51 = ( ! [A: $i,B: $i,C: $i] :
% 2.61/1.51 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 2.61/1.51 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(regular_type,type,
% 2.61/1.51 regular: $i > $o ).
% 2.61/1.51
% 2.61/1.51 thf(setminus_type,type,
% 2.61/1.51 setminus: $i > $i > $i ).
% 2.61/1.51
% 2.61/1.51 thf(setminusI_type,type,
% 2.61/1.51 setminusI: $o ).
% 2.61/1.51
% 2.61/1.51 thf(setminusI,definition,
% 2.61/1.51 ( setminusI
% 2.61/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.51 ( ( in @ Xx @ A )
% 2.61/1.51 => ( ~ ( in @ Xx @ B )
% 2.61/1.51 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(setminusEL_type,type,
% 2.61/1.51 setminusEL: $o ).
% 2.61/1.51
% 2.61/1.51 thf(setminusEL,definition,
% 2.61/1.51 ( setminusEL
% 2.61/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.51 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.61/1.51 => ( in @ Xx @ A ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(setminusER_type,type,
% 2.61/1.51 setminusER: $o ).
% 2.61/1.51
% 2.61/1.51 thf(setminusER,definition,
% 2.61/1.51 ( setminusER
% 2.61/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.51 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.61/1.51 => ~ ( in @ Xx @ B ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(setminusSubset2_type,type,
% 2.61/1.51 setminusSubset2: $o ).
% 2.61/1.51
% 2.61/1.51 thf(setminusSubset2,definition,
% 2.61/1.51 ( setminusSubset2
% 2.61/1.51 = ( ! [A: $i,B: $i] :
% 2.61/1.51 ( ( subset @ A @ B )
% 2.61/1.51 => ( ( setminus @ A @ B )
% 2.61/1.51 = emptyset ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(setminusERneg_type,type,
% 2.61/1.51 setminusERneg: $o ).
% 2.61/1.51
% 2.61/1.51 thf(setminusERneg,definition,
% 2.61/1.51 ( setminusERneg
% 2.61/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.51 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.61/1.51 => ( ( in @ Xx @ A )
% 2.61/1.51 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(setminusELneg_type,type,
% 2.61/1.51 setminusELneg: $o ).
% 2.61/1.51
% 2.61/1.51 thf(setminusELneg,definition,
% 2.61/1.51 ( setminusELneg
% 2.61/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.51 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.61/1.51 => ( ~ ( in @ Xx @ B )
% 2.61/1.51 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(setminusILneg_type,type,
% 2.61/1.51 setminusILneg: $o ).
% 2.61/1.51
% 2.61/1.51 thf(setminusILneg,definition,
% 2.61/1.51 ( setminusILneg
% 2.61/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.51 ( ~ ( in @ Xx @ A )
% 2.61/1.51 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(setminusIRneg_type,type,
% 2.61/1.51 setminusIRneg: $o ).
% 2.61/1.51
% 2.61/1.51 thf(setminusIRneg,definition,
% 2.61/1.51 ( setminusIRneg
% 2.61/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.51 ( ( in @ Xx @ B )
% 2.61/1.51 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(setminusLsub_type,type,
% 2.61/1.51 setminusLsub: $o ).
% 2.61/1.51
% 2.61/1.51 thf(setminusLsub,definition,
% 2.61/1.51 ( setminusLsub
% 2.61/1.51 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(setminusSubset1_type,type,
% 2.61/1.51 setminusSubset1: $o ).
% 2.61/1.51
% 2.61/1.51 thf(setminusSubset1,definition,
% 2.61/1.51 ( setminusSubset1
% 2.61/1.51 = ( ! [A: $i,B: $i] :
% 2.61/1.51 ( ( ( setminus @ A @ B )
% 2.61/1.51 = emptyset )
% 2.61/1.51 => ( subset @ A @ B ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(symdiff_type,type,
% 2.61/1.51 symdiff: $i > $i > $i ).
% 2.61/1.51
% 2.61/1.51 thf(symdiffE_type,type,
% 2.61/1.51 symdiffE: $o ).
% 2.61/1.51
% 2.61/1.51 thf(symdiffE,definition,
% 2.61/1.51 ( symdiffE
% 2.61/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.51 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 2.61/1.51 => ! [Xphi: $o] :
% 2.61/1.51 ( ( ( in @ Xx @ A )
% 2.61/1.51 => ( ~ ( in @ Xx @ B )
% 2.61/1.51 => Xphi ) )
% 2.61/1.51 => ( ( ~ ( in @ Xx @ A )
% 2.61/1.51 => ( ( in @ Xx @ B )
% 2.61/1.51 => Xphi ) )
% 2.61/1.51 => Xphi ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(symdiffI1_type,type,
% 2.61/1.51 symdiffI1: $o ).
% 2.61/1.51
% 2.61/1.51 thf(symdiffI1,definition,
% 2.61/1.51 ( symdiffI1
% 2.61/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.51 ( ( in @ Xx @ A )
% 2.61/1.51 => ( ~ ( in @ Xx @ B )
% 2.61/1.51 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(symdiffI2_type,type,
% 2.61/1.51 symdiffI2: $o ).
% 2.61/1.51
% 2.61/1.51 thf(symdiffI2,definition,
% 2.61/1.51 ( symdiffI2
% 2.61/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.51 ( ~ ( in @ Xx @ A )
% 2.61/1.51 => ( ( in @ Xx @ B )
% 2.61/1.51 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(symdiffIneg1_type,type,
% 2.61/1.51 symdiffIneg1: $o ).
% 2.61/1.51
% 2.61/1.51 thf(symdiffIneg1,definition,
% 2.61/1.51 ( symdiffIneg1
% 2.61/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.51 ( ( in @ Xx @ A )
% 2.61/1.51 => ( ( in @ Xx @ B )
% 2.61/1.51 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(symdiffIneg2_type,type,
% 2.61/1.51 symdiffIneg2: $o ).
% 2.61/1.51
% 2.61/1.51 thf(symdiffIneg2,definition,
% 2.61/1.51 ( symdiffIneg2
% 2.61/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.51 ( ~ ( in @ Xx @ A )
% 2.61/1.51 => ( ~ ( in @ Xx @ B )
% 2.61/1.51 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(iskpair_type,type,
% 2.61/1.51 iskpair: $i > $o ).
% 2.61/1.51
% 2.61/1.51 thf(iskpair,definition,
% 2.61/1.51 ( iskpair
% 2.61/1.51 = ( ^ [A: $i] :
% 2.61/1.51 ? [Xx: $i] :
% 2.61/1.51 ( ( in @ Xx @ ( setunion @ A ) )
% 2.61/1.51 & ? [Xy: $i] :
% 2.61/1.51 ( ( in @ Xy @ ( setunion @ A ) )
% 2.61/1.51 & ( A
% 2.61/1.51 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(secondinupair_type,type,
% 2.61/1.51 secondinupair: $o ).
% 2.61/1.51
% 2.61/1.51 thf(secondinupair,definition,
% 2.61/1.51 ( secondinupair
% 2.61/1.51 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(setukpairIL_type,type,
% 2.61/1.51 setukpairIL: $o ).
% 2.61/1.51
% 2.61/1.51 thf(setukpairIL,definition,
% 2.61/1.51 ( setukpairIL
% 2.61/1.51 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(setukpairIR_type,type,
% 2.61/1.51 setukpairIR: $o ).
% 2.61/1.51
% 2.61/1.51 thf(setukpairIR,definition,
% 2.61/1.51 ( setukpairIR
% 2.61/1.51 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(kpairiskpair_type,type,
% 2.61/1.51 kpairiskpair: $o ).
% 2.61/1.51
% 2.61/1.51 thf(kpairiskpair,definition,
% 2.61/1.51 ( kpairiskpair
% 2.61/1.51 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(kpair_type,type,
% 2.61/1.51 kpair: $i > $i > $i ).
% 2.61/1.51
% 2.61/1.51 thf(kpair,definition,
% 2.61/1.51 ( kpair
% 2.61/1.51 = ( ^ [Xx: $i,Xy: $i] : ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(kpairp_type,type,
% 2.61/1.51 kpairp: $o ).
% 2.61/1.51
% 2.61/1.51 thf(kpairp,definition,
% 2.61/1.51 ( kpairp
% 2.61/1.51 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(cartprod_type,type,
% 2.61/1.51 cartprod: $i > $i > $i ).
% 2.61/1.51
% 2.61/1.51 thf(singletonsubset_type,type,
% 2.61/1.51 singletonsubset: $o ).
% 2.61/1.51
% 2.61/1.51 thf(singletonsubset,definition,
% 2.61/1.51 ( singletonsubset
% 2.61/1.51 = ( ! [A: $i,Xx: $i] :
% 2.61/1.51 ( ( in @ Xx @ A )
% 2.61/1.51 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(singletoninpowerset_type,type,
% 2.61/1.51 singletoninpowerset: $o ).
% 2.61/1.51
% 2.61/1.51 thf(singletoninpowerset,definition,
% 2.61/1.51 ( singletoninpowerset
% 2.61/1.51 = ( ! [A: $i,Xx: $i] :
% 2.61/1.51 ( ( in @ Xx @ A )
% 2.61/1.51 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(singletoninpowunion_type,type,
% 2.61/1.51 singletoninpowunion: $o ).
% 2.61/1.51
% 2.61/1.51 thf(singletoninpowunion,definition,
% 2.61/1.51 ( singletoninpowunion
% 2.61/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.51 ( ( in @ Xx @ A )
% 2.61/1.51 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(upairset2E_type,type,
% 2.61/1.51 upairset2E: $o ).
% 2.61/1.51
% 2.61/1.51 thf(upairset2E,definition,
% 2.61/1.51 ( upairset2E
% 2.61/1.51 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.61/1.51 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.61/1.51 => ( ( Xz = Xx )
% 2.61/1.51 | ( Xz = Xy ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(upairsubunion_type,type,
% 2.61/1.51 upairsubunion: $o ).
% 2.61/1.51
% 2.61/1.51 thf(upairsubunion,definition,
% 2.61/1.51 ( upairsubunion
% 2.61/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.51 ( ( in @ Xx @ A )
% 2.61/1.51 => ! [Xy: $i] :
% 2.61/1.51 ( ( in @ Xy @ B )
% 2.61/1.51 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(upairinpowunion_type,type,
% 2.61/1.51 upairinpowunion: $o ).
% 2.61/1.51
% 2.61/1.51 thf(upairinpowunion,definition,
% 2.61/1.51 ( upairinpowunion
% 2.61/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.51 ( ( in @ Xx @ A )
% 2.61/1.51 => ! [Xy: $i] :
% 2.61/1.51 ( ( in @ Xy @ B )
% 2.61/1.51 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(ubforcartprodlem1_type,type,
% 2.61/1.51 ubforcartprodlem1: $o ).
% 2.61/1.51
% 2.61/1.51 thf(ubforcartprodlem1,definition,
% 2.61/1.51 ( ubforcartprodlem1
% 2.61/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.51 ( ( in @ Xx @ A )
% 2.61/1.51 => ! [Xy: $i] :
% 2.61/1.51 ( ( in @ Xy @ B )
% 2.61/1.51 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(ubforcartprodlem2_type,type,
% 2.61/1.51 ubforcartprodlem2: $o ).
% 2.61/1.51
% 2.61/1.51 thf(ubforcartprodlem2,definition,
% 2.61/1.51 ( ubforcartprodlem2
% 2.61/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.51 ( ( in @ Xx @ A )
% 2.61/1.51 => ! [Xy: $i] :
% 2.61/1.51 ( ( in @ Xy @ B )
% 2.61/1.51 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(ubforcartprodlem3_type,type,
% 2.61/1.51 ubforcartprodlem3: $o ).
% 2.61/1.51
% 2.61/1.51 thf(ubforcartprodlem3,definition,
% 2.61/1.51 ( ubforcartprodlem3
% 2.61/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.51 ( ( in @ Xx @ A )
% 2.61/1.51 => ! [Xy: $i] :
% 2.61/1.51 ( ( in @ Xy @ B )
% 2.61/1.51 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(cartprodpairin_type,type,
% 2.61/1.51 cartprodpairin: $o ).
% 2.61/1.51
% 2.61/1.51 thf(cartprodpairin,definition,
% 2.61/1.51 ( cartprodpairin
% 2.61/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.51 ( ( in @ Xx @ A )
% 2.61/1.51 => ! [Xy: $i] :
% 2.61/1.51 ( ( in @ Xy @ B )
% 2.61/1.51 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(cartprodmempair1_type,type,
% 2.61/1.51 cartprodmempair1: $o ).
% 2.61/1.51
% 2.61/1.51 thf(cartprodmempair1,definition,
% 2.61/1.51 ( cartprodmempair1
% 2.61/1.51 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.61/1.51 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.61/1.51 => ? [Xx: $i] :
% 2.61/1.51 ( ( in @ Xx @ A )
% 2.61/1.51 & ? [Xy: $i] :
% 2.61/1.51 ( ( in @ Xy @ B )
% 2.61/1.51 & ( Xu
% 2.61/1.51 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(cartprodmempair_type,type,
% 2.61/1.51 cartprodmempair: $o ).
% 2.61/1.51
% 2.61/1.51 thf(cartprodmempair,definition,
% 2.61/1.51 ( cartprodmempair
% 2.61/1.51 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.61/1.51 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.61/1.51 => ( iskpair @ Xu ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(setunionE2_type,type,
% 2.61/1.51 setunionE2: $o ).
% 2.61/1.51
% 2.61/1.51 thf(setunionE2,definition,
% 2.61/1.51 ( setunionE2
% 2.61/1.51 = ( ! [A: $i,Xx: $i] :
% 2.61/1.51 ( ( in @ Xx @ ( setunion @ A ) )
% 2.61/1.51 => ? [X: $i] :
% 2.61/1.51 ( ( in @ X @ A )
% 2.61/1.51 & ( in @ Xx @ X ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(setunionsingleton1_type,type,
% 2.61/1.51 setunionsingleton1: $o ).
% 2.61/1.51
% 2.61/1.51 thf(setunionsingleton1,definition,
% 2.61/1.51 ( setunionsingleton1
% 2.61/1.51 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(setunionsingleton2_type,type,
% 2.61/1.51 setunionsingleton2: $o ).
% 2.61/1.51
% 2.61/1.51 thf(setunionsingleton2,definition,
% 2.61/1.51 ( setunionsingleton2
% 2.61/1.51 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(setunionsingleton_type,type,
% 2.61/1.51 setunionsingleton: $o ).
% 2.61/1.51
% 2.61/1.51 thf(setunionsingleton,definition,
% 2.61/1.51 ( setunionsingleton
% 2.61/1.51 = ( ! [Xx: $i] :
% 2.61/1.51 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 2.61/1.51 = Xx ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(singleton_type,type,
% 2.61/1.51 singleton: $i > $o ).
% 2.61/1.51
% 2.61/1.51 thf(singleton,definition,
% 2.61/1.51 ( singleton
% 2.61/1.51 = ( ^ [A: $i] :
% 2.61/1.51 ? [Xx: $i] :
% 2.61/1.51 ( ( in @ Xx @ A )
% 2.61/1.51 & ( A
% 2.61/1.51 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(singletonprop_type,type,
% 2.61/1.51 singletonprop: $o ).
% 2.61/1.51
% 2.61/1.51 thf(singletonprop,definition,
% 2.61/1.51 ( singletonprop
% 2.61/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 2.61/1.51 ( ! [Xx: $i] :
% 2.61/1.51 ( ( in @ Xx @ A )
% 2.61/1.51 => ! [Xy: $i] :
% 2.61/1.51 ( ( in @ Xy @ A )
% 2.61/1.51 => ( ( Xphi @ Xx )
% 2.61/1.51 => ( ( Xphi @ Xy )
% 2.61/1.51 => ( Xx = Xy ) ) ) ) )
% 2.61/1.51 => ( ? [Xx: $i] :
% 2.61/1.51 ( ( in @ Xx @ A )
% 2.61/1.51 & ( Xphi @ Xx ) )
% 2.61/1.51 => ( singleton
% 2.61/1.51 @ ( dsetconstr @ A
% 2.61/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(ex1_type,type,
% 2.61/1.51 ex1: $i > ( $i > $o ) > $o ).
% 2.61/1.51
% 2.61/1.51 thf(ex1,definition,
% 2.61/1.51 ( ex1
% 2.61/1.51 = ( ^ [A: $i,Xphi: $i > $o] :
% 2.61/1.51 ( singleton
% 2.61/1.51 @ ( dsetconstr @ A
% 2.61/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(ex1E1_type,type,
% 2.61/1.51 ex1E1: $o ).
% 2.61/1.51
% 2.61/1.51 thf(ex1E1,definition,
% 2.61/1.51 ( ex1E1
% 2.61/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 2.61/1.51 ( ( ex1 @ A
% 2.61/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.61/1.51 => ? [Xx: $i] :
% 2.61/1.51 ( ( in @ Xx @ A )
% 2.61/1.51 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(ex1I_type,type,
% 2.61/1.51 ex1I: $o ).
% 2.61/1.51
% 2.61/1.51 thf(ex1I,definition,
% 2.61/1.51 ( ex1I
% 2.61/1.51 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.61/1.51 ( ( in @ Xx @ A )
% 2.61/1.51 => ( ( Xphi @ Xx )
% 2.61/1.51 => ( ! [Xy: $i] :
% 2.61/1.51 ( ( in @ Xy @ A )
% 2.61/1.51 => ( ( Xphi @ Xy )
% 2.61/1.51 => ( Xy = Xx ) ) )
% 2.61/1.51 => ( ex1 @ A
% 2.61/1.51 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(ex1I2_type,type,
% 2.61/1.51 ex1I2: $o ).
% 2.61/1.51
% 2.61/1.51 thf(ex1I2,definition,
% 2.61/1.51 ( ex1I2
% 2.61/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 2.61/1.51 ( ! [Xx: $i] :
% 2.61/1.51 ( ( in @ Xx @ A )
% 2.61/1.51 => ! [Xy: $i] :
% 2.61/1.51 ( ( in @ Xy @ A )
% 2.61/1.51 => ( ( Xphi @ Xx )
% 2.61/1.51 => ( ( Xphi @ Xy )
% 2.61/1.51 => ( Xx = Xy ) ) ) ) )
% 2.61/1.51 => ( ? [Xx: $i] :
% 2.61/1.51 ( ( in @ Xx @ A )
% 2.61/1.51 & ( Xphi @ Xx ) )
% 2.61/1.51 => ( ex1 @ A
% 2.61/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(singletonsuniq_type,type,
% 2.61/1.51 singletonsuniq: $o ).
% 2.61/1.51
% 2.61/1.51 thf(singletonsuniq,definition,
% 2.61/1.51 ( singletonsuniq
% 2.61/1.51 = ( ! [Xx: $i,Xy: $i] :
% 2.61/1.51 ( ( ( setadjoin @ Xx @ emptyset )
% 2.61/1.51 = ( setadjoin @ Xy @ emptyset ) )
% 2.61/1.51 => ( Xx = Xy ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(atmost1p_type,type,
% 2.61/1.51 atmost1p: $i > $o ).
% 2.61/1.51
% 2.61/1.51 thf(atleast2p_type,type,
% 2.61/1.51 atleast2p: $i > $o ).
% 2.61/1.51
% 2.61/1.51 thf(atmost2p_type,type,
% 2.61/1.51 atmost2p: $i > $o ).
% 2.61/1.51
% 2.61/1.51 thf(upairsetp_type,type,
% 2.61/1.51 upairsetp: $i > $o ).
% 2.61/1.51
% 2.61/1.51 thf(setukpairinjL1_type,type,
% 2.61/1.51 setukpairinjL1: $o ).
% 2.61/1.51
% 2.61/1.51 thf(setukpairinjL1,definition,
% 2.61/1.51 ( setukpairinjL1
% 2.61/1.51 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.61/1.51 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 2.61/1.51 => ( Xx = Xz ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(kfstsingleton_type,type,
% 2.61/1.51 kfstsingleton: $o ).
% 2.61/1.51
% 2.61/1.51 thf(kfstsingleton,definition,
% 2.61/1.51 ( kfstsingleton
% 2.61/1.51 = ( ! [Xu: $i] :
% 2.61/1.51 ( ( iskpair @ Xu )
% 2.61/1.51 => ( singleton
% 2.61/1.51 @ ( dsetconstr @ ( setunion @ Xu )
% 2.61/1.51 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(theprop_type,type,
% 2.61/1.51 theprop: $o ).
% 2.61/1.51
% 2.61/1.51 thf(theprop,definition,
% 2.61/1.51 ( theprop
% 2.61/1.51 = ( ! [X: $i] :
% 2.61/1.51 ( ( singleton @ X )
% 2.61/1.51 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(kfst_type,type,
% 2.61/1.51 kfst: $i > $i ).
% 2.61/1.51
% 2.61/1.51 thf(kfstpairEq_type,type,
% 2.61/1.51 kfstpairEq: $o ).
% 2.61/1.51
% 2.61/1.51 thf(kfstpairEq,definition,
% 2.61/1.51 ( kfstpairEq
% 2.61/1.51 = ( ! [Xx: $i,Xy: $i] :
% 2.61/1.51 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 2.61/1.51 = Xx ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(cartprodfstin_type,type,
% 2.61/1.51 cartprodfstin: $o ).
% 2.61/1.51
% 2.61/1.51 thf(cartprodfstin,definition,
% 2.61/1.51 ( cartprodfstin
% 2.61/1.51 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.61/1.51 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.61/1.51 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(setukpairinjL2_type,type,
% 2.61/1.51 setukpairinjL2: $o ).
% 2.61/1.51
% 2.61/1.51 thf(setukpairinjL2,definition,
% 2.61/1.51 ( setukpairinjL2
% 2.61/1.51 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.61/1.51 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.61/1.51 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.61/1.51 => ( Xx = Xz ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(setukpairinjL_type,type,
% 2.61/1.51 setukpairinjL: $o ).
% 2.61/1.51
% 2.61/1.51 thf(setukpairinjL,definition,
% 2.61/1.51 ( setukpairinjL
% 2.61/1.51 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.61/1.51 ( ( ( kpair @ Xx @ Xy )
% 2.61/1.51 = ( kpair @ Xz @ Xu ) )
% 2.61/1.51 => ( Xx = Xz ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(setukpairinjR11_type,type,
% 2.61/1.51 setukpairinjR11: $o ).
% 2.61/1.51
% 2.61/1.51 thf(setukpairinjR11,definition,
% 2.61/1.51 ( setukpairinjR11
% 2.61/1.51 = ( ! [Xx: $i,Xy: $i] :
% 2.61/1.51 ( ( Xx = Xy )
% 2.61/1.51 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.61/1.51 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.61/1.51
% 2.61/1.51 thf(setukpairinjR12,conjecture,
% 2.61/1.51 ( setextAx
% 2.61/1.51 => ( emptysetAx
% 2.61/1.51 => ( setadjoinAx
% 2.61/1.51 => ( powersetAx
% 2.61/1.51 => ( setunionAx
% 2.61/1.51 => ( omega0Ax
% 2.61/1.51 => ( omegaSAx
% 2.61/1.51 => ( omegaIndAx
% 2.61/1.51 => ( replAx
% 2.61/1.51 => ( foundationAx
% 2.61/1.51 => ( wellorderingAx
% 2.61/1.51 => ( descrp
% 2.61/1.51 => ( dsetconstrI
% 2.61/1.51 => ( dsetconstrEL
% 2.61/1.51 => ( dsetconstrER
% 2.61/1.51 => ( exuE1
% 2.61/1.51 => ( prop2setE
% 2.61/1.51 => ( emptysetE
% 2.61/1.51 => ( emptysetimpfalse
% 2.61/1.51 => ( notinemptyset
% 2.61/1.51 => ( exuE3e
% 2.61/1.51 => ( setext
% 2.61/1.51 => ( emptyI
% 2.61/1.51 => ( noeltsimpempty
% 2.61/1.51 => ( setbeta
% 2.61/1.51 => ( nonemptyE1
% 2.61/1.51 => ( nonemptyI
% 2.61/1.51 => ( nonemptyI1
% 2.61/1.51 => ( setadjoinIL
% 2.61/1.51 => ( emptyinunitempty
% 2.61/1.51 => ( setadjoinIR
% 2.61/1.51 => ( setadjoinE
% 2.61/1.51 => ( setadjoinOr
% 2.61/1.51 => ( setoftrueEq
% 2.61/1.51 => ( powersetI
% 2.61/1.51 => ( emptyinPowerset
% 2.61/1.51 => ( emptyInPowerset
% 2.61/1.51 => ( powersetE
% 2.61/1.51 => ( setunionI
% 2.61/1.51 => ( setunionE
% 2.61/1.51 => ( subPowSU
% 2.61/1.51 => ( exuE2
% 2.61/1.51 => ( nonemptyImpWitness
% 2.61/1.51 => ( uniqinunit
% 2.61/1.51 => ( notinsingleton
% 2.61/1.51 => ( eqinunit
% 2.61/1.51 => ( singletonsswitch
% 2.61/1.51 => ( upairsetE
% 2.61/1.51 => ( upairsetIL
% 2.61/1.51 => ( upairsetIR
% 2.61/1.51 => ( emptyE1
% 2.61/1.51 => ( vacuousDall
% 2.61/1.51 => ( quantDeMorgan1
% 2.61/1.51 => ( quantDeMorgan2
% 2.61/1.51 => ( quantDeMorgan3
% 2.61/1.51 => ( quantDeMorgan4
% 2.61/1.51 => ( prop2setI
% 2.61/1.51 => ( prop2set2propI
% 2.61/1.51 => ( notdexE
% 2.61/1.51 => ( notdallE
% 2.61/1.51 => ( exuI1
% 2.61/1.51 => ( exuI3
% 2.61/1.51 => ( exuI2
% 2.61/1.51 => ( inCongP
% 2.61/1.51 => ( in__Cong
% 2.61/1.51 => ( exuE3u
% 2.61/1.51 => ( exu__Cong
% 2.61/1.51 => ( emptyset__Cong
% 2.61/1.51 => ( setadjoin__Cong
% 2.61/1.51 => ( powerset__Cong
% 2.61/1.51 => ( setunion__Cong
% 2.61/1.51 => ( omega__Cong
% 2.61/1.51 => ( exuEu
% 2.61/1.51 => ( descr__Cong
% 2.61/1.51 => ( dsetconstr__Cong
% 2.61/1.51 => ( subsetI1
% 2.61/1.51 => ( eqimpsubset2
% 2.61/1.51 => ( eqimpsubset1
% 2.61/1.51 => ( subsetI2
% 2.61/1.51 => ( emptysetsubset
% 2.61/1.51 => ( subsetE
% 2.61/1.51 => ( subsetE2
% 2.61/1.51 => ( notsubsetI
% 2.61/1.51 => ( notequalI1
% 2.61/1.51 => ( notequalI2
% 2.61/1.51 => ( subsetRefl
% 2.61/1.51 => ( subsetTrans
% 2.61/1.51 => ( setadjoinSub
% 2.61/1.51 => ( setadjoinSub2
% 2.61/1.51 => ( subset2powerset
% 2.61/1.51 => ( setextsub
% 2.61/1.51 => ( subsetemptysetimpeq
% 2.61/1.51 => ( powersetI1
% 2.61/1.51 => ( powersetE1
% 2.61/1.51 => ( inPowerset
% 2.61/1.51 => ( powersetsubset
% 2.61/1.51 => ( sepInPowerset
% 2.61/1.51 => ( sepSubset
% 2.61/1.51 => ( binunionIL
% 2.61/1.51 => ( upairset2IR
% 2.61/1.51 => ( binunionIR
% 2.61/1.51 => ( binunionEcases
% 2.61/1.51 => ( binunionE
% 2.61/1.51 => ( binunionLsub
% 2.61/1.51 => ( binunionRsub
% 2.61/1.51 => ( binintersectI
% 2.61/1.51 => ( binintersectSubset5
% 2.61/1.51 => ( binintersectEL
% 2.61/1.51 => ( binintersectLsub
% 2.61/1.51 => ( binintersectSubset2
% 2.61/1.51 => ( binintersectSubset3
% 2.61/1.51 => ( binintersectER
% 2.61/1.51 => ( disjointsetsI1
% 2.61/1.51 => ( binintersectRsub
% 2.61/1.51 => ( binintersectSubset4
% 2.61/1.51 => ( binintersectSubset1
% 2.61/1.51 => ( bs114d
% 2.61/1.51 => ( setminusI
% 2.61/1.51 => ( setminusEL
% 2.61/1.51 => ( setminusER
% 2.61/1.51 => ( setminusSubset2
% 2.61/1.51 => ( setminusERneg
% 2.61/1.51 => ( setminusELneg
% 2.61/1.51 => ( setminusILneg
% 2.61/1.51 => ( setminusIRneg
% 2.61/1.51 => ( setminusLsub
% 2.61/1.51 => ( setminusSubset1
% 2.61/1.51 => ( symdiffE
% 2.61/1.51 => ( symdiffI1
% 2.61/1.51 => ( symdiffI2
% 2.61/1.51 => ( symdiffIneg1
% 2.61/1.51 => ( symdiffIneg2
% 2.61/1.51 => ( secondinupair
% 2.61/1.51 => ( setukpairIL
% 2.61/1.51 => ( setukpairIR
% 2.61/1.51 => ( kpairiskpair
% 2.61/1.51 => ( kpairp
% 2.61/1.51 => ( singletonsubset
% 2.61/1.51 => ( singletoninpowerset
% 2.61/1.51 => ( singletoninpowunion
% 2.61/1.51 => ( upairset2E
% 2.61/1.51 => ( upairsubunion
% 2.61/1.51 => ( upairinpowunion
% 2.61/1.51 => ( ubforcartprodlem1
% 2.61/1.51 => ( ubforcartprodlem2
% 2.61/1.51 => ( ubforcartprodlem3
% 2.61/1.51 => ( cartprodpairin
% 2.61/1.51 => ( cartprodmempair1
% 2.61/1.51 => ( cartprodmempair
% 2.61/1.51 => ( setunionE2
% 2.61/1.51 => ( setunionsingleton1
% 2.61/1.51 => ( setunionsingleton2
% 2.61/1.51 => ( setunionsingleton
% 2.61/1.51 => ( singletonprop
% 2.61/1.51 => ( ex1E1
% 2.61/1.51 => ( ex1I
% 2.61/1.51 => ( ex1I2
% 2.61/1.51 => ( singletonsuniq
% 2.61/1.51 => ( setukpairinjL1
% 2.61/1.51 => ( kfstsingleton
% 2.61/1.51 => ( theprop
% 2.61/1.51 => ( kfstpairEq
% 2.61/1.51 => ( cartprodfstin
% 2.61/1.51 => ( setukpairinjL2
% 2.61/1.51 => ( setukpairinjL
% 2.61/1.51 => ( setukpairinjR11
% 2.61/1.51 => ! [Xx: $i,Xy: $i] :
% 2.61/1.51 ( ( Xx = Xy )
% 2.61/1.51 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.61/1.51 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.61/1.51 ------------------------
% 2.61/1.54 % (12293)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_11959 for (2999ds/2Mi)
% 2.61/1.54 % (12292)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_11959 for (2999ds/27Mi)
% 2.61/1.54 % (12293)Instruction limit reached!
% 2.61/1.54 % (12293)------------------------------
% 2.61/1.54 % (12293)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.54 % (12293)Termination reason: Unknown
% 2.61/1.54 % (12293)Termination phase: shuffling
% 2.61/1.54
% 2.61/1.54 % (12293)Memory used [KB]: 1279
% 2.61/1.54 % (12293)Time elapsed: 0.002 s
% 2.61/1.54 % (12293)Instructions burned: 2 (million)
% 2.61/1.54 % (12293)------------------------------
% 2.61/1.54 % (12293)------------------------------
% 2.61/1.54 % (12296)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_11959 for (2999ds/18Mi)
% 2.61/1.54 % (12290)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_11959 for (2999ds/183Mi)
% 2.61/1.54 % (12291)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_11959 for (2999ds/4Mi)
% 2.61/1.54 % (12294)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_11959 for (2999ds/2Mi)
% 2.61/1.54 % (12295)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_11959 for (2999ds/275Mi)
% 2.61/1.54 % (12292)Instruction limit reached!
% 2.61/1.54 % (12292)------------------------------
% 2.61/1.54 % (12292)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.54 % (12292)Termination reason: Unknown
% 2.61/1.54 % (12292)Termination phase: Property scanning
% 2.61/1.54
% 2.61/1.54 % (12292)Memory used [KB]: 1791
% 2.61/1.54 % (12292)Time elapsed: 0.010 s
% 2.61/1.54 % (12292)Instructions burned: 28 (million)
% 2.61/1.54 % (12292)------------------------------
% 2.61/1.54 % (12292)------------------------------
% 2.61/1.54 % (12294)Instruction limit reached!
% 2.61/1.54 % (12294)------------------------------
% 2.61/1.54 % (12294)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.54 % (12294)Termination reason: Unknown
% 2.61/1.54 % (12294)Termination phase: shuffling
% 2.61/1.54
% 2.61/1.54 % (12294)Memory used [KB]: 1279
% 2.61/1.54 % (12294)Time elapsed: 0.003 s
% 2.61/1.54 % (12294)Instructions burned: 3 (million)
% 2.61/1.54 % (12294)------------------------------
% 2.61/1.54 % (12294)------------------------------
% 2.61/1.55 % (12291)Instruction limit reached!
% 2.61/1.55 % (12291)------------------------------
% 2.61/1.55 % (12291)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.55 % (12291)Termination reason: Unknown
% 2.61/1.55 % (12291)Termination phase: shuffling
% 2.61/1.55
% 2.61/1.55 % (12291)Memory used [KB]: 1279
% 2.61/1.55 % (12291)Time elapsed: 0.004 s
% 2.61/1.55 % (12291)Instructions burned: 5 (million)
% 2.61/1.55 % (12291)------------------------------
% 2.61/1.55 % (12291)------------------------------
% 2.61/1.55 % (12297)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_11959 for (2999ds/3Mi)
% 2.61/1.55 % (12297)Instruction limit reached!
% 2.61/1.55 % (12297)------------------------------
% 2.61/1.55 % (12297)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.55 % (12297)Termination reason: Unknown
% 2.61/1.55 % (12297)Termination phase: shuffling
% 2.61/1.55
% 2.61/1.55 % (12297)Memory used [KB]: 1279
% 2.61/1.55 % (12297)Time elapsed: 0.002 s
% 2.61/1.55 % (12297)Instructions burned: 3 (million)
% 2.61/1.55 % (12297)------------------------------
% 2.61/1.55 % (12297)------------------------------
% 2.61/1.55 % (12296)Instruction limit reached!
% 2.61/1.55 % (12296)------------------------------
% 2.61/1.55 % (12296)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.55 % (12296)Termination reason: Unknown
% 2.61/1.55 % (12296)Termination phase: Property scanning
% 2.61/1.55
% 2.61/1.55 % (12296)Memory used [KB]: 1535
% 2.61/1.55 % (12296)Time elapsed: 0.013 s
% 2.61/1.55 % (12296)Instructions burned: 18 (million)
% 2.61/1.55 % (12296)------------------------------
% 2.61/1.55 % (12296)------------------------------
% 2.61/1.55 % (12298)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_11959 for (2999ds/37Mi)
% 2.61/1.56 % (12301)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_11959 for (2999ds/1041Mi)
% 2.61/1.56 % (12299)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_11959 for (2999ds/15Mi)
% 2.61/1.56 % (12300)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_11959 for (2999ds/3Mi)
% 2.61/1.56 % (12302)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_11959 for (2999ds/7Mi)
% 2.61/1.56 % (12300)Instruction limit reached!
% 2.61/1.56 % (12300)------------------------------
% 2.61/1.56 % (12300)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.56 % (12300)Termination reason: Unknown
% 2.61/1.56 % (12300)Termination phase: shuffling
% 2.61/1.56
% 2.61/1.56 % (12300)Memory used [KB]: 1279
% 2.61/1.56 % (12300)Time elapsed: 0.004 s
% 2.61/1.56 % (12300)Instructions burned: 4 (million)
% 2.61/1.56 % (12300)------------------------------
% 2.61/1.56 % (12300)------------------------------
% 2.61/1.57 % (12298)Instruction limit reached!
% 2.61/1.57 % (12298)------------------------------
% 2.61/1.57 % (12298)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.57 % (12298)Termination reason: Unknown
% 2.61/1.57 % (12298)Termination phase: Property scanning
% 2.61/1.57
% 2.61/1.57 % (12298)Memory used [KB]: 1791
% 2.61/1.57 % (12298)Time elapsed: 0.012 s
% 2.61/1.57 % (12298)Instructions burned: 40 (million)
% 2.61/1.57 % (12298)------------------------------
% 2.61/1.57 % (12298)------------------------------
% 2.61/1.57 % (12302)Instruction limit reached!
% 2.61/1.57 % (12302)------------------------------
% 2.61/1.57 % (12302)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.57 % (12302)Termination reason: Unknown
% 2.61/1.57 % (12302)Termination phase: shuffling
% 2.61/1.57
% 2.61/1.57 % (12302)Memory used [KB]: 1407
% 2.61/1.57 % (12302)Time elapsed: 0.004 s
% 2.61/1.57 % (12302)Instructions burned: 7 (million)
% 2.61/1.57 % (12302)------------------------------
% 2.61/1.57 % (12302)------------------------------
% 2.61/1.57 % (12299)Instruction limit reached!
% 2.61/1.57 % (12299)------------------------------
% 2.61/1.57 % (12299)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.57 % (12299)Termination reason: Unknown
% 2.61/1.57 % (12299)Termination phase: shuffling
% 2.61/1.57
% 2.61/1.57 % (12299)Memory used [KB]: 1535
% 2.61/1.57 % (12299)Time elapsed: 0.011 s
% 2.61/1.57 % (12299)Instructions burned: 15 (million)
% 2.61/1.57 % (12299)------------------------------
% 2.61/1.57 % (12299)------------------------------
% 2.61/1.58 % (12304)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_11959 for (2999ds/3Mi)
% 2.61/1.58 % (12304)Instruction limit reached!
% 2.61/1.58 % (12304)------------------------------
% 2.61/1.58 % (12304)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.58 % (12304)Termination reason: Unknown
% 2.61/1.58 % (12304)Termination phase: shuffling
% 2.61/1.58
% 2.61/1.58 % (12304)Memory used [KB]: 1279
% 2.61/1.58 % (12304)Time elapsed: 0.002 s
% 2.61/1.58 % (12304)Instructions burned: 4 (million)
% 2.61/1.58 % (12304)------------------------------
% 2.61/1.58 % (12304)------------------------------
% 2.61/1.58 % (12305)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_11959 for (2999ds/3Mi)
% 2.61/1.58 % (12305)Instruction limit reached!
% 2.61/1.58 % (12305)------------------------------
% 2.61/1.58 % (12305)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.58 % (12305)Termination reason: Unknown
% 2.61/1.58 % (12305)Termination phase: shuffling
% 2.61/1.58
% 2.61/1.58 % (12305)Memory used [KB]: 1407
% 2.61/1.58 % (12305)Time elapsed: 0.003 s
% 2.61/1.58 % (12305)Instructions burned: 5 (million)
% 2.61/1.58 % (12305)------------------------------
% 2.61/1.58 % (12305)------------------------------
% 2.61/1.58 % (12303)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_11959 for (2999ds/16Mi)
% 2.61/1.59 % (12306)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_11959 for (2999ds/7Mi)
% 2.61/1.59 % (12307)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_11959 for (2999ds/3Mi)
% 2.61/1.59 % (12306)Instruction limit reached!
% 2.61/1.59 % (12306)------------------------------
% 2.61/1.59 % (12306)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.59 % (12306)Termination reason: Unknown
% 2.61/1.59 % (12306)Termination phase: shuffling
% 2.61/1.59
% 2.61/1.59 % (12306)Memory used [KB]: 1407
% 2.61/1.59 % (12306)Time elapsed: 0.004 s
% 2.61/1.59 % (12306)Instructions burned: 7 (million)
% 2.61/1.59 % (12306)------------------------------
% 2.61/1.59 % (12306)------------------------------
% 2.61/1.59 % (12307)Instruction limit reached!
% 2.61/1.59 % (12307)------------------------------
% 2.61/1.59 % (12307)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.59 % (12307)Termination reason: Unknown
% 2.61/1.59 % (12307)Termination phase: shuffling
% 2.61/1.59
% 2.61/1.59 % (12307)Memory used [KB]: 1279
% 2.61/1.59 % (12307)Time elapsed: 0.002 s
% 2.61/1.59 % (12307)Instructions burned: 4 (million)
% 2.61/1.59 % (12307)------------------------------
% 2.61/1.59 % (12307)------------------------------
% 2.61/1.59 % (12308)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_11959 for (2999ds/4Mi)
% 2.61/1.59 % (12303)Instruction limit reached!
% 2.61/1.59 % (12303)------------------------------
% 2.61/1.59 % (12303)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.59 % (12303)Termination reason: Unknown
% 2.61/1.59 % (12303)Termination phase: shuffling
% 2.61/1.59
% 2.61/1.59 % (12303)Memory used [KB]: 1663
% 2.61/1.59 % (12303)Time elapsed: 0.010 s
% 2.61/1.59 % (12303)Instructions burned: 16 (million)
% 2.61/1.59 % (12303)------------------------------
% 2.61/1.59 % (12303)------------------------------
% 2.61/1.59 % (12308)Instruction limit reached!
% 2.61/1.59 % (12308)------------------------------
% 2.61/1.59 % (12308)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.59 % (12308)Termination reason: Unknown
% 2.61/1.59 % (12308)Termination phase: shuffling
% 2.61/1.59
% 2.61/1.59 % (12308)Memory used [KB]: 1279
% 2.61/1.59 % (12308)Time elapsed: 0.003 s
% 2.61/1.59 % (12308)Instructions burned: 4 (million)
% 2.61/1.59 % (12308)------------------------------
% 2.61/1.59 % (12308)------------------------------
% 2.65/1.60 % (12309)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_11959 for (2999ds/18Mi)
% 2.65/1.60 % (12310)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_11959 for (2999ds/710Mi)
% 2.65/1.60 % (12290)Instruction limit reached!
% 2.65/1.60 % (12290)------------------------------
% 2.65/1.60 % (12290)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.60 % (12290)Termination reason: Unknown
% 2.65/1.60 % (12290)Termination phase: Saturation
% 2.65/1.60
% 2.65/1.60 % (12290)Memory used [KB]: 7419
% 2.65/1.60 % (12290)Time elapsed: 0.060 s
% 2.65/1.60 % (12290)Instructions burned: 184 (million)
% 2.65/1.60 % (12290)------------------------------
% 2.65/1.60 % (12290)------------------------------
% 2.65/1.60 % (12312)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_11959 for (2999ds/902Mi)
% 2.65/1.60 % (12309)Instruction limit reached!
% 2.65/1.60 % (12309)------------------------------
% 2.65/1.60 % (12309)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.60 % (12309)Termination reason: Unknown
% 2.65/1.60 % (12309)Termination phase: shuffling
% 2.65/1.60
% 2.65/1.60 % (12309)Memory used [KB]: 1663
% 2.65/1.60 % (12309)Time elapsed: 0.006 s
% 2.65/1.60 % (12309)Instructions burned: 18 (million)
% 2.65/1.60 % (12309)------------------------------
% 2.65/1.60 % (12309)------------------------------
% 2.65/1.60 % (12311)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_11959 for (2999ds/6Mi)
% 2.65/1.61 % (12311)Instruction limit reached!
% 2.65/1.61 % (12311)------------------------------
% 2.65/1.61 % (12311)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.61 % (12311)Termination reason: Unknown
% 2.65/1.61 % (12311)Termination phase: shuffling
% 2.65/1.61
% 2.65/1.61 % (12311)Memory used [KB]: 1407
% 2.65/1.61 % (12311)Time elapsed: 0.004 s
% 2.65/1.61 % (12311)Instructions burned: 6 (million)
% 2.65/1.61 % (12311)------------------------------
% 2.65/1.61 % (12311)------------------------------
% 2.65/1.61 % (12313)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_11959 for (2999ds/21Mi)
% 2.65/1.61 % (12314)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_11959 for (2999ds/5Mi)
% 2.65/1.61 % (12314)Instruction limit reached!
% 2.65/1.61 % (12314)------------------------------
% 2.65/1.61 % (12314)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.61 % (12314)Termination reason: Unknown
% 2.65/1.61 % (12314)Termination phase: shuffling
% 2.65/1.61
% 2.65/1.61 % (12314)Memory used [KB]: 1407
% 2.65/1.61 % (12314)Time elapsed: 0.003 s
% 2.65/1.61 % (12314)Instructions burned: 7 (million)
% 2.65/1.61 % (12314)------------------------------
% 2.65/1.61 % (12314)------------------------------
% 2.65/1.62 % (12313)Instruction limit reached!
% 2.65/1.62 % (12313)------------------------------
% 2.65/1.62 % (12313)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.62 % (12313)Termination reason: Unknown
% 2.65/1.62 % (12313)Termination phase: shuffling
% 2.65/1.62
% 2.65/1.62 % (12313)Memory used [KB]: 1663
% 2.65/1.62 % (12313)Time elapsed: 0.007 s
% 2.65/1.62 % (12313)Instructions burned: 21 (million)
% 2.65/1.62 % (12313)------------------------------
% 2.65/1.62 % (12313)------------------------------
% 2.65/1.62 % (12315)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_11959 for (2999ds/6Mi)
% 2.65/1.62 % (12315)Instruction limit reached!
% 2.65/1.62 % (12315)------------------------------
% 2.65/1.62 % (12315)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.62 % (12315)Termination reason: Unknown
% 2.65/1.62 % (12315)Termination phase: shuffling
% 2.65/1.62
% 2.65/1.62 % (12315)Memory used [KB]: 1407
% 2.65/1.62 % (12315)Time elapsed: 0.003 s
% 2.65/1.62 % (12315)Instructions burned: 7 (million)
% 2.65/1.62 % (12315)------------------------------
% 2.65/1.62 % (12315)------------------------------
% 2.65/1.62 % (12316)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_11959 for (2999ds/377Mi)
% 2.65/1.63 % (12317)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_11959 for (2999ds/779Mi)
% 2.65/1.63 % (12318)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_11959 for (2999ds/19Mi)
% 2.65/1.64 % (12318)Instruction limit reached!
% 2.65/1.64 % (12318)------------------------------
% 2.65/1.64 % (12318)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.64 % (12318)Termination reason: Unknown
% 2.65/1.64 % (12318)Termination phase: shuffling
% 2.65/1.64
% 2.65/1.64 % (12318)Memory used [KB]: 1663
% 2.65/1.64 % (12318)Time elapsed: 0.008 s
% 2.65/1.64 % (12318)Instructions burned: 22 (million)
% 2.65/1.64 % (12318)------------------------------
% 2.65/1.64 % (12318)------------------------------
% 2.65/1.64 % (12295)First to succeed.
% 2.65/1.65 % (12319)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_11959 for (2998ds/879Mi)
% 2.65/1.65 % (12295)Refutation found. Thanks to Tanya!
% 2.65/1.65 % SZS status Theorem for DTF2THF_11959
% 2.65/1.65 % SZS output start Proof for DTF2THF_11959
% 2.65/1.65 thf(func_def_0, type, in: $i > $i > $o).
% 2.65/1.65 thf(func_def_1, type, exu: ($i > $o) > $o).
% 2.65/1.65 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 2.65/1.65 thf(func_def_8, type, powerset: $i > $i).
% 2.65/1.65 thf(func_def_10, type, setunion: $i > $i).
% 2.65/1.65 thf(func_def_19, type, descr: ($i > $o) > $i).
% 2.65/1.65 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 2.65/1.65 thf(func_def_26, type, prop2set: $o > $i).
% 2.65/1.65 thf(func_def_36, type, nonempty: $i > $o).
% 2.65/1.65 thf(func_def_69, type, set2prop: $i > $o).
% 2.65/1.65 thf(func_def_88, type, subset: $i > $i > $o).
% 2.65/1.65 thf(func_def_89, type, disjoint: $i > $i > $o).
% 2.65/1.65 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 2.65/1.65 thf(func_def_114, type, binunion: $i > $i > $i).
% 2.65/1.65 thf(func_def_122, type, binintersect: $i > $i > $i).
% 2.65/1.65 thf(func_def_135, type, regular: $i > $o).
% 2.65/1.65 thf(func_def_136, type, setminus: $i > $i > $i).
% 2.65/1.65 thf(func_def_147, type, symdiff: $i > $i > $i).
% 2.65/1.65 thf(func_def_153, type, iskpair: $i > $o).
% 2.65/1.65 thf(func_def_158, type, kpair: $i > $i > $i).
% 2.65/1.65 thf(func_def_160, type, cartprod: $i > $i > $i).
% 2.65/1.65 thf(func_def_177, type, singleton: $i > $o).
% 2.65/1.65 thf(func_def_179, type, ex1: $i > ($i > $o) > $o).
% 2.65/1.65 thf(func_def_184, type, atmost1p: $i > $o).
% 2.65/1.65 thf(func_def_185, type, atleast2p: $i > $o).
% 2.65/1.65 thf(func_def_186, type, atmost2p: $i > $o).
% 2.65/1.65 thf(func_def_187, type, upairsetp: $i > $o).
% 2.65/1.65 thf(func_def_191, type, kfst: $i > $i).
% 2.65/1.65 thf(func_def_209, type, sP0: $i > $i > $i > $o > $o).
% 2.65/1.65 thf(func_def_211, type, sP2: $i > $i > $o).
% 2.65/1.65 thf(func_def_212, type, sP3: $i > $o).
% 2.65/1.65 thf(func_def_213, type, sP4: $i > $i > $o).
% 2.65/1.65 thf(func_def_214, type, sP5: $i > $i > $o).
% 2.65/1.65 thf(func_def_219, type, sK10: $i > $o).
% 2.65/1.65 thf(func_def_220, type, sK11: $i > $o).
% 2.65/1.65 thf(func_def_221, type, sK12: ($i > $o) > ($i > $o) > $i).
% 2.65/1.65 thf(func_def_222, type, sK13: ($i > $o) > ($i > $o) > $i).
% 2.65/1.65 thf(func_def_227, type, sK18: $i > $o).
% 2.65/1.65 thf(func_def_244, type, sK35: ($i > $o) > $i).
% 2.65/1.65 thf(func_def_245, type, sK36: ($i > $o) > $i).
% 2.65/1.65 thf(func_def_246, type, sK37: $i > $o).
% 2.65/1.65 thf(func_def_248, type, sK39: $i > $o).
% 2.65/1.65 thf(func_def_260, type, sK51: ($i > $o) > $i > $i).
% 2.65/1.65 thf(func_def_262, type, sK53: $i > $o).
% 2.65/1.65 thf(func_def_268, type, sK59: $i > $i > $i).
% 2.65/1.65 thf(func_def_269, type, sK60: $i > $i > $i).
% 2.65/1.65 thf(func_def_275, type, sK66: $i > $i > $i).
% 2.65/1.65 thf(func_def_294, type, sK85: $i > $o).
% 2.65/1.65 thf(func_def_297, type, sK88: $i > $o).
% 2.65/1.65 thf(func_def_303, type, sK94: $i > $i).
% 2.65/1.65 thf(func_def_305, type, sK96: $i > $i > $i).
% 2.65/1.65 thf(func_def_333, type, sK124: $i > $i > $i).
% 2.65/1.65 thf(func_def_334, type, sK125: $i > $o).
% 2.65/1.65 thf(func_def_340, type, sK131: $i > $o).
% 2.65/1.65 thf(func_def_348, type, sK139: $i > $i).
% 2.65/1.65 thf(func_def_357, type, sK148: $i > ($i > $o) > $i).
% 2.65/1.65 thf(func_def_358, type, sK149: $i > $o).
% 2.65/1.65 thf(func_def_362, type, sK153: $i > $o).
% 2.65/1.65 thf(func_def_368, type, sK159: $i > $i > $i).
% 2.65/1.65 thf(func_def_371, type, sK162: $i > $o).
% 2.65/1.65 thf(func_def_374, type, sK165: $i > ($i > $o) > $i).
% 2.65/1.65 thf(func_def_375, type, sK166: $i > $o).
% 2.65/1.65 thf(func_def_383, type, sK174: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.65/1.65 thf(func_def_384, type, sK175: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.65/1.65 thf(func_def_387, type, sK178: $i > $o).
% 2.65/1.65 thf(func_def_388, type, sK179: $i > $o).
% 2.65/1.65 thf(func_def_404, type, sK195: $i > $o).
% 2.65/1.65 thf(func_def_409, type, sK200: $i > $o).
% 2.65/1.65 thf(func_def_411, type, sK202: ($i > $o) > $i > $i).
% 2.65/1.65 thf(func_def_421, type, sK212: $i > $o).
% 2.65/1.65 thf(func_def_423, type, sK214: $i > ($i > $o) > $i).
% 2.65/1.65 thf(func_def_425, type, sK216: $i > $i).
% 2.65/1.65 thf(func_def_428, type, sK219: $i > $i).
% 2.65/1.65 thf(func_def_433, type, sK224: $i > $i > $i).
% 2.65/1.65 thf(func_def_445, type, sK236: $i > $i > $i > $i).
% 2.65/1.65 thf(func_def_446, type, sK237: $i > $i > $i > $i).
% 2.65/1.65 thf(func_def_448, type, sK239: $i > $i > $o).
% 2.65/1.65 thf(func_def_449, type, sK240: $i > $i).
% 2.65/1.65 thf(func_def_450, type, sK241: $i > $i).
% 2.65/1.65 thf(func_def_451, type, sK242: ($i > $i > $o) > $i > $i).
% 2.65/1.65 thf(func_def_452, type, sK243: $i > ($i > $i > $o) > $i > $i).
% 2.65/1.65 thf(func_def_453, type, sK244: ($i > $i > $o) > $i > $i).
% 2.65/1.65 thf(func_def_457, type, sK248: $i > $i).
% 2.65/1.65 thf(func_def_458, type, sK249: $i > $i > $i).
% 2.65/1.65 thf(func_def_459, type, sK250: $i > $i > $i).
% 2.65/1.65 thf(func_def_460, type, sK251: $i > $i > $i).
% 2.65/1.65 thf(func_def_461, type, sK252: $i > $i > $i).
% 2.65/1.65 thf(func_def_462, type, sK253: $i > $i > $i > $i).
% 2.65/1.65 thf(func_def_463, type, sK254: $i > $i).
% 2.65/1.65 thf(func_def_464, type, sK255: $i > $i).
% 2.65/1.65 thf(func_def_465, type, sK256: $i > $i).
% 2.65/1.65 thf(func_def_466, type, sK257: $i > $i).
% 2.65/1.65 thf(func_def_467, type, sK258: $i > $i > $i > $i).
% 2.65/1.65 thf(func_def_468, type, sK259: $i > $i > $i > $i).
% 2.65/1.65 thf(func_def_469, type, sK260: $i > $i > $i).
% 2.65/1.65 thf(func_def_470, type, sK261: $i > $i > $i).
% 2.65/1.65 thf(func_def_471, type, sK262: $i > $i > $i).
% 2.65/1.65 thf(func_def_472, type, sK263: $i > $i).
% 2.65/1.65 thf(func_def_473, type, sK264: $i > $i > $i).
% 2.65/1.65 thf(func_def_475, type, sK266: $i > $i).
% 2.65/1.65 thf(func_def_476, type, sK267: $i > $i).
% 2.65/1.65 thf(func_def_500, type, sK291: $i > $o).
% 2.65/1.65 thf(func_def_503, type, sK294: $i > ($i > $o) > $i).
% 2.65/1.65 thf(func_def_504, type, sK295: $i > ($i > $o) > $i).
% 2.65/1.65 thf(func_def_511, type, sK302: $i > $o).
% 2.65/1.65 thf(func_def_512, type, sK303: $i > $i).
% 2.65/1.65 thf(func_def_513, type, sK304: ($i > $o) > $i).
% 2.65/1.65 thf(func_def_514, type, sK305: $i > $i > $i).
% 2.65/1.65 thf(func_def_518, type, sK309: $i > ($i > $o) > $i).
% 2.65/1.65 thf(func_def_519, type, sK310: $i > $o).
% 2.65/1.65 thf(func_def_532, type, sK323: $i > ($i > $o) > $i).
% 2.65/1.65 thf(func_def_533, type, sK324: $i > $o).
% 2.65/1.65 thf(func_def_554, type, sK345: $i > $o).
% 2.65/1.65 thf(func_def_555, type, sK346: ($i > $o) > $i > $i > $i).
% 2.65/1.65 thf(func_def_556, type, sK347: $i > $i > $i).
% 2.65/1.65 thf(func_def_559, type, sK350: ($i > $o) > $i).
% 2.65/1.65 thf(func_def_560, type, sK351: $i > $o).
% 2.65/1.65 thf(func_def_561, type, sK352: $i > $i).
% 2.65/1.65 thf(func_def_563, type, sK354: $o > $i > $i > $i).
% 2.65/1.65 thf(func_def_567, type, sK358: $i > $o).
% 2.65/1.65 thf(func_def_584, type, sK375: ($i > $o) > $i > $i).
% 2.65/1.65 thf(func_def_586, type, sK377: $i > $o).
% 2.65/1.65 thf(func_def_594, type, sK385: ($i > $o) > $i).
% 2.65/1.65 thf(func_def_595, type, sK386: $i > $o).
% 2.65/1.65 thf(func_def_598, type, sK389: $i > $i).
% 2.65/1.65 thf(func_def_599, type, sK390: $i > $i).
% 2.65/1.65 thf(func_def_602, type, sK393: $i > ($i > $o) > $i).
% 2.65/1.65 thf(func_def_603, type, sK394: $i > $o).
% 2.65/1.65 thf(func_def_605, type, sK396: $i > $o).
% 2.65/1.65 thf(func_def_612, type, sK403: $i > $i > $i).
% 2.65/1.65 thf(func_def_634, type, sK425: $i > $o).
% 2.65/1.65 thf(func_def_637, type, sK428: $i > ($i > $o) > $i).
% 2.65/1.65 thf(func_def_638, type, sK429: $i > ($i > $o) > $i).
% 2.65/1.65 thf(func_def_641, type, sK432: $i > $o).
% 2.65/1.65 thf(func_def_657, type, sK448: ($i > $o) > ($i > $o) > $i).
% 2.65/1.65 thf(func_def_658, type, sK449: ($i > $o) > ($i > $o) > $i).
% 2.65/1.65 thf(func_def_659, type, sK450: $i > $o).
% 2.65/1.65 thf(func_def_660, type, sK451: $i > $o).
% 2.65/1.65 thf(func_def_673, type, ph464: !>[X0: $tType]:(X0)).
% 2.65/1.65 thf(f3197,plain,(
% 2.65/1.65 $false),
% 2.65/1.65 inference(subsumption_resolution,[status(thm)],[f3196,f2966])).
% 2.65/1.65 thf(f2966,plain,(
% 2.65/1.65 ( ! [X1 : $i] : (((setadjoin @ X1 @ emptyset) = (setadjoin @ X1 @ (setadjoin @ X1 @ emptyset)))) )),
% 2.65/1.65 inference(trivial_inequality_removal,[status(thm)],[f2898])).
% 2.65/1.65 thf(f2898,plain,(
% 2.65/1.65 ( ! [X1 : $i] : (($true != $true) | ((setadjoin @ X1 @ emptyset) = (setadjoin @ X1 @ (setadjoin @ X1 @ emptyset)))) )),
% 2.65/1.65 inference(equality_resolution,[status(thm)],[f2498])).
% 2.65/1.65 thf(f2498,plain,(
% 2.65/1.65 ( ! [X0 : $i,X1 : $i] : (((setadjoin @ X1 @ emptyset) = (setadjoin @ X1 @ (setadjoin @ X0 @ emptyset))) | (X0 != X1) | ($true != $true)) )),
% 2.65/1.65 inference(definition_unfolding,[status(thm)],[f1695,f2085])).
% 2.65/1.65 thf(f2085,plain,(
% 2.65/1.65 (setukpairinjR11 = $true)),
% 2.65/1.65 inference(cnf_transformation,[status(thm)],[f1265])).
% 2.65/1.65 thf(f1265,plain,(
% 2.65/1.65 (cartprodmempair1 = $true) & (setunionsingleton1 = $true) & (omega__Cong = $true) & (binintersectSubset2 = $true) & (quantDeMorgan3 = $true) & (binintersectRsub = $true) & (prop2setI = $true) & (binintersectI = $true) & (dsetconstrI = $true) & (singletoninpowunion = $true) & (setunionsingleton = $true) & (upairsetIR = $true) & (cartprodfstin = $true) & (symdiffIneg1 = $true) & (binintersectLsub = $true) & (subsetRefl = $true) & (binunionIL = $true) & (inPowerset = $true) & (emptyinPowerset = $true) & (setminusELneg = $true) & (bs114d = $true) & (prop2set2propI = $true) & (eqimpsubset2 = $true) & (upairinpowunion = $true) & (singletoninpowerset = $true) & (notequalI1 = $true) & (kpairp = $true) & (setukpairIL = $true) & (cartprodmempair = $true) & (eqinunit = $true) & (setoftrueEq = $true) & (setminusLsub = $true) & (uniqinunit = $true) & (binunionEcases = $true) & (powersetE1 = $true) & (binintersectSubset5 = $true) & (setukpairinjR11 = $true) & (symdiffI1 = $true) & (descrp = $true) & (emptyInPowerset = $true) & (setukpairIR = $true) & (emptyset__Cong = $true) & (powersetI1 = $true) & (setukpairinjL2 = $true) & (exuI3 = $true) & (ubforcartprodlem3 = $true) & (setadjoinAx = $true) & (foundationAx = $true) & (sepSubset = $true) & (ex1E1 = $true) & (notinsingleton = $true) & (powersetE = $true) & (setminusIRneg = $true) & (setextsub = $true) & (powerset__Cong = $true) & (upairsetIL = $true) & (in__Cong = $true) & (symdiffI2 = $true) & (upairset2E = $true) & (sepInPowerset = $true) & (setadjoinSub = $true) & (exuE2 = $true) & (ubforcartprodlem1 = $true) & (exuI1 = $true) & (binintersectSubset4 = $true) & (binintersectEL = $true) & (setminusER = $true) & (ex1I2 = $true) & (((setadjoin @ (setadjoin @ sK327 @ emptyset) @ (setadjoin @ (setadjoin @ sK327 @ (setadjoin @ sK328 @ emptyset)) @ emptyset)) != (setadjoin @ (setadjoin @ sK327 @ emptyset) @ emptyset)) & (sK327 = sK328)) & (setadjoin__Cong = $true) & (subsetI1 = $true) & (exuEu = $true) & (binunionLsub = $true) & (quantDeMorgan2 = $true) & (dsetconstrER = $true) & (setminusEL = $true) & (setukpairinjL1 = $true) & (subsetTrans = $true) & (quantDeMorgan4 = $true) & (notdallE = $true) & (wellorderingAx = $true) & (setukpairinjL = $true) & (upairset2IR = $true) & (setunionI = $true) & (kfstsingleton = $true) & (setext = $true) & (subset2powerset = $true) & (setminusSubset2 = $true) & (setadjoinSub2 = $true) & (subsetE = $true) & (powersetI = $true) & (singletonprop = $true) & (setminusERneg = $true) & (omegaIndAx = $true) & (setadjoinE = $true) & (exu__Cong = $true) & (emptyI = $true) & (nonemptyE1 = $true) & (kpairiskpair = $true) & (upairsetE = $true) & (powersetAx = $true) & (descr__Cong = $true) & (setminusI = $true) & (secondinupair = $true) & (notdexE = $true) & (disjointsetsI1 = $true) & (setunion__Cong = $true) & (setunionsingleton2 = $true) & (setunionE = $true) & (setminusSubset1 = $true) & (emptyinunitempty = $true) & (powersetsubset = $true) & (emptysetAx = $true) & (omega0Ax = $true) & (emptyE1 = $true) & (setbeta = $true) & (exuE3e = $true) & (emptysetsubset = $true) & (emptysetimpfalse = $true) & (exuE1 = $true) & (setextAx = $true) & (singletonsswitch = $true) & (replAx = $true) & (notsubsetI = $true) & (singletonsubset = $true) & (kfstpairEq = $true) & (setadjoinIR = $true) & (cartprodpairin = $true) & (upairsubunion = $true) & (subsetemptysetimpeq = $true) & (subsetI2 = $true) & (singletonsuniq = $true) & (omegaSAx = $true) & (noeltsimpempty = $true) & (vacuousDall = $true) & (notinemptyset = $true) & (setadjoinOr = $true) & (binintersectSubset3 = $true) & (setadjoinIL = $true) & (ubforcartprodlem2 = $true) & (symdiffIneg2 = $true) & (theprop = $true) & (dsetconstr__Cong = $true) & (exuE3u = $true) & (setunionAx = $true) & (quantDeMorgan1 = $true) & (setunionE2 = $true) & (nonemptyI = $true) & (binintersectER = $true) & (binunionIR = $true) & (symdiffE = $true) & (prop2setE = $true) & (binunionE = $true) & (emptysetE = $true) & (nonemptyImpWitness = $true) & (eqimpsubset1 = $true) & (inCongP = $true) & (notequalI2 = $true) & (subsetE2 = $true) & (exuI2 = $true) & (subPowSU = $true) & (binunionRsub = $true) & (ex1I = $true) & (binintersectSubset1 = $true) & (dsetconstrEL = $true) & (nonemptyI1 = $true) & (setminusILneg = $true)),
% 2.65/1.65 inference(skolemisation,[status(esa),new_symbols(skolem,[sK327,sK328])],[f687,f1264])).
% 2.65/1.65 thf(f1264,plain,(
% 2.65/1.65 ? [X0 : $i,X1 : $i] : (((setadjoin @ (setadjoin @ X0 @ emptyset) @ (setadjoin @ (setadjoin @ X0 @ (setadjoin @ X1 @ emptyset)) @ emptyset)) != (setadjoin @ (setadjoin @ X0 @ emptyset) @ emptyset)) & (X0 = X1)) => (((setadjoin @ (setadjoin @ sK327 @ emptyset) @ (setadjoin @ (setadjoin @ sK327 @ (setadjoin @ sK328 @ emptyset)) @ emptyset)) != (setadjoin @ (setadjoin @ sK327 @ emptyset) @ emptyset)) & (sK327 = sK328))),
% 2.65/1.65 introduced(definition,[],[choice_axiom])).
% 2.65/1.65 thf(f687,plain,(
% 2.65/1.65 (cartprodmempair1 = $true) & (setunionsingleton1 = $true) & (omega__Cong = $true) & (binintersectSubset2 = $true) & (quantDeMorgan3 = $true) & (binintersectRsub = $true) & (prop2setI = $true) & (binintersectI = $true) & (dsetconstrI = $true) & (singletoninpowunion = $true) & (setunionsingleton = $true) & (upairsetIR = $true) & (cartprodfstin = $true) & (symdiffIneg1 = $true) & (binintersectLsub = $true) & (subsetRefl = $true) & (binunionIL = $true) & (inPowerset = $true) & (emptyinPowerset = $true) & (setminusELneg = $true) & (bs114d = $true) & (prop2set2propI = $true) & (eqimpsubset2 = $true) & (upairinpowunion = $true) & (singletoninpowerset = $true) & (notequalI1 = $true) & (kpairp = $true) & (setukpairIL = $true) & (cartprodmempair = $true) & (eqinunit = $true) & (setoftrueEq = $true) & (setminusLsub = $true) & (uniqinunit = $true) & (binunionEcases = $true) & (powersetE1 = $true) & (binintersectSubset5 = $true) & (setukpairinjR11 = $true) & (symdiffI1 = $true) & (descrp = $true) & (emptyInPowerset = $true) & (setukpairIR = $true) & (emptyset__Cong = $true) & (powersetI1 = $true) & (setukpairinjL2 = $true) & (exuI3 = $true) & (ubforcartprodlem3 = $true) & (setadjoinAx = $true) & (foundationAx = $true) & (sepSubset = $true) & (ex1E1 = $true) & (notinsingleton = $true) & (powersetE = $true) & (setminusIRneg = $true) & (setextsub = $true) & (powerset__Cong = $true) & (upairsetIL = $true) & (in__Cong = $true) & (symdiffI2 = $true) & (upairset2E = $true) & (sepInPowerset = $true) & (setadjoinSub = $true) & (exuE2 = $true) & (ubforcartprodlem1 = $true) & (exuI1 = $true) & (binintersectSubset4 = $true) & (binintersectEL = $true) & (setminusER = $true) & (ex1I2 = $true) & ? [X0 : $i,X1 : $i] : (((setadjoin @ (setadjoin @ X0 @ emptyset) @ (setadjoin @ (setadjoin @ X0 @ (setadjoin @ X1 @ emptyset)) @ emptyset)) != (setadjoin @ (setadjoin @ X0 @ emptyset) @ emptyset)) & (X0 = X1)) & (setadjoin__Cong = $true) & (subsetI1 = $true) & (exuEu = $true) & (binunionLsub = $true) & (quantDeMorgan2 = $true) & (dsetconstrER = $true) & (setminusEL = $true) & (setukpairinjL1 = $true) & (subsetTrans = $true) & (quantDeMorgan4 = $true) & (notdallE = $true) & (wellorderingAx = $true) & (setukpairinjL = $true) & (upairset2IR = $true) & (setunionI = $true) & (kfstsingleton = $true) & (setext = $true) & (subset2powerset = $true) & (setminusSubset2 = $true) & (setadjoinSub2 = $true) & (subsetE = $true) & (powersetI = $true) & (singletonprop = $true) & (setminusERneg = $true) & (omegaIndAx = $true) & (setadjoinE = $true) & (exu__Cong = $true) & (emptyI = $true) & (nonemptyE1 = $true) & (kpairiskpair = $true) & (upairsetE = $true) & (powersetAx = $true) & (descr__Cong = $true) & (setminusI = $true) & (secondinupair = $true) & (notdexE = $true) & (disjointsetsI1 = $true) & (setunion__Cong = $true) & (setunionsingleton2 = $true) & (setunionE = $true) & (setminusSubset1 = $true) & (emptyinunitempty = $true) & (powersetsubset = $true) & (emptysetAx = $true) & (omega0Ax = $true) & (emptyE1 = $true) & (setbeta = $true) & (exuE3e = $true) & (emptysetsubset = $true) & (emptysetimpfalse = $true) & (exuE1 = $true) & (setextAx = $true) & (singletonsswitch = $true) & (replAx = $true) & (notsubsetI = $true) & (singletonsubset = $true) & (kfstpairEq = $true) & (setadjoinIR = $true) & (cartprodpairin = $true) & (upairsubunion = $true) & (subsetemptysetimpeq = $true) & (subsetI2 = $true) & (singletonsuniq = $true) & (omegaSAx = $true) & (noeltsimpempty = $true) & (vacuousDall = $true) & (notinemptyset = $true) & (setadjoinOr = $true) & (binintersectSubset3 = $true) & (setadjoinIL = $true) & (ubforcartprodlem2 = $true) & (symdiffIneg2 = $true) & (theprop = $true) & (dsetconstr__Cong = $true) & (exuE3u = $true) & (setunionAx = $true) & (quantDeMorgan1 = $true) & (setunionE2 = $true) & (nonemptyI = $true) & (binintersectER = $true) & (binunionIR = $true) & (symdiffE = $true) & (prop2setE = $true) & (binunionE = $true) & (emptysetE = $true) & (nonemptyImpWitness = $true) & (eqimpsubset1 = $true) & (inCongP = $true) & (notequalI2 = $true) & (subsetE2 = $true) & (exuI2 = $true) & (subPowSU = $true) & (binunionRsub = $true) & (ex1I = $true) & (binintersectSubset1 = $true) & (dsetconstrEL = $true) & (nonemptyI1 = $true) & (setminusILneg = $true)),
% 2.65/1.65 inference(flattening,[status(thm)],[f686])).
% 2.65/1.65 thf(f686,plain,(
% 2.65/1.65 (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i,X1 : $i] : (((setadjoin @ (setadjoin @ X0 @ emptyset) @ (setadjoin @ (setadjoin @ X0 @ (setadjoin @ X1 @ emptyset)) @ emptyset)) != (setadjoin @ (setadjoin @ X0 @ emptyset) @ emptyset)) & (X0 = X1)) & (setukpairinjR11 = $true)) & (setukpairinjL = $true)) & (setukpairinjL2 = $true)) & (cartprodfstin = $true)) & (kfstpairEq = $true)) & (theprop = $true)) & (kfstsingleton = $true)) & (setukpairinjL1 = $true)) & (singletonsuniq = $true)) & (ex1I2 = $true)) & (ex1I = $true)) & (ex1E1 = $true)) & (singletonprop = $true)) & (setunionsingleton = $true)) & (setunionsingleton2 = $true)) & (setunionsingleton1 = $true)) & (setunionE2 = $true)) & (cartprodmempair = $true)) & (cartprodmempair1 = $true)) & (cartprodpairin = $true)) & (ubforcartprodlem3 = $true)) & (ubforcartprodlem2 = $true)) & (ubforcartprodlem1 = $true)) & (upairinpowunion = $true)) & (upairsubunion = $true)) & (upairset2E = $true)) & (singletoninpowunion = $true)) & (singletoninpowerset = $true)) & (singletonsubset = $true)) & (kpairp = $true)) & (kpairiskpair = $true)) & (setukpairIR = $true)) & (setukpairIL = $true)) & (secondinupair = $true)) & (symdiffIneg2 = $true)) & (symdiffIneg1 = $true)) & (symdiffI2 = $true)) & (symdiffI1 = $true)) & (symdiffE = $true)) & (setminusSubset1 = $true)) & (setminusLsub = $true)) & (setminusIRneg = $true)) & (setminusILneg = $true)) & (setminusELneg = $true)) & (setminusERneg = $true)) & (setminusSubset2 = $true)) & (setminusER = $true)) & (setminusEL = $true)) & (setminusI = $true)) & (bs114d = $true)) & (binintersectSubset1 = $true)) & (binintersectSubset4 = $true)) & (binintersectRsub = $true)) & (disjointsetsI1 = $true)) & (binintersectER = $true)) & (binintersectSubset3 = $true)) & (binintersectSubset2 = $true)) & (binintersectLsub = $true)) & (binintersectEL = $true)) & (binintersectSubset5 = $true)) & (binintersectI = $true)) & (binunionRsub = $true)) & (binunionLsub = $true)) & (binunionE = $true)) & (binunionEcases = $true)) & (binunionIR = $true)) & (upairset2IR = $true)) & (binunionIL = $true)) & (sepSubset = $true)) & (sepInPowerset = $true)) & (powersetsubset = $true)) & (inPowerset = $true)) & (powersetE1 = $true)) & (powersetI1 = $true)) & (subsetemptysetimpeq = $true)) & (setextsub = $true)) & (subset2powerset = $true)) & (setadjoinSub2 = $true)) & (setadjoinSub = $true)) & (subsetTrans = $true)) & (subsetRefl = $true)) & (notequalI2 = $true)) & (notequalI1 = $true)) & (notsubsetI = $true)) & (subsetE2 = $true)) & (subsetE = $true)) & (emptysetsubset = $true)) & (subsetI2 = $true)) & (eqimpsubset1 = $true)) & (eqimpsubset2 = $true)) & (subsetI1 = $true)) & (dsetconstr__Cong = $true)) & (descr__Cong = $true)) & (exuEu = $true)) & (omega__Cong = $true)) & (setunion__Cong = $true)) & (powerset__Cong = $true)) & (setadjoin__Cong = $true)) & (emptyset__Cong = $true)) & (exu__Cong = $true)) & (exuE3u = $true)) & (in__Cong = $true)) & (inCongP = $true)) & (exuI2 = $true)) & (exuI3 = $true)) & (exuI1 = $true)) & (notdallE = $true)) & (notdexE = $true)) & (prop2set2propI = $true)) & (prop2setI = $true)) & (quantDeMorgan4 = $true)) & (quantDeMorgan3 = $true)) & (quantDeMorgan2 = $true)) & (quantDeMorgan1 = $true)) & (vacuousDall = $true)) & (emptyE1 = $true)) & (upairsetIR = $true)) & (upairsetIL = $true)) & (upairsetE = $true)) & (singletonsswitch = $true)) & (eqinunit = $true)) & (notinsingleton = $true)) & (uniqinunit = $true)) & (nonemptyImpWitness = $true)) & (exuE2 = $true)) & (subPowSU = $true)) & (setunionE = $true)) & (setunionI = $true)) & (powersetE = $true)) & (emptyInPowerset = $true)) & (emptyinPowerset = $true)) & (powersetI = $true)) & (setoftrueEq = $true)) & (setadjoinOr = $true)) & (setadjoinE = $true)) & (setadjoinIR = $true)) & (emptyinunitempty = $true)) & (setadjoinIL = $true)) & (nonemptyI1 = $true)) & (nonemptyI = $true)) & (nonemptyE1 = $true)) & (setbeta = $true)) & (noeltsimpempty = $true)) & (emptyI = $true)) & (setext = $true)) & (exuE3e = $true)) & (notinemptyset = $true)) & (emptysetimpfalse = $true)) & (emptysetE = $true)) & (prop2setE = $true)) & (exuE1 = $true)) & (dsetconstrER = $true)) & (dsetconstrEL = $true)) & (dsetconstrI = $true)) & (descrp = $true)) & (wellorderingAx = $true)) & (foundationAx = $true)) & (replAx = $true)) & (omegaIndAx = $true)) & (omegaSAx = $true)) & (omega0Ax = $true)) & (setunionAx = $true)) & (powersetAx = $true)) & (setadjoinAx = $true)) & (emptysetAx = $true)) & (setextAx = $true)),
% 2.65/1.65 inference(ennf_transformation,[status(thm)],[f504])).
% 2.65/1.65 thf(f504,plain,(
% 2.65/1.65 ~((setextAx = $true) => ((emptysetAx = $true) => ((setadjoinAx = $true) => ((powersetAx = $true) => ((setunionAx = $true) => ((omega0Ax = $true) => ((omegaSAx = $true) => ((omegaIndAx = $true) => ((replAx = $true) => ((foundationAx = $true) => ((wellorderingAx = $true) => ((descrp = $true) => ((dsetconstrI = $true) => ((dsetconstrEL = $true) => ((dsetconstrER = $true) => ((exuE1 = $true) => ((prop2setE = $true) => ((emptysetE = $true) => ((emptysetimpfalse = $true) => ((notinemptyset = $true) => ((exuE3e = $true) => ((setext = $true) => ((emptyI = $true) => ((noeltsimpempty = $true) => ((setbeta = $true) => ((nonemptyE1 = $true) => ((nonemptyI = $true) => ((nonemptyI1 = $true) => ((setadjoinIL = $true) => ((emptyinunitempty = $true) => ((setadjoinIR = $true) => ((setadjoinE = $true) => ((setadjoinOr = $true) => ((setoftrueEq = $true) => ((powersetI = $true) => ((emptyinPowerset = $true) => ((emptyInPowerset = $true) => ((powersetE = $true) => ((setunionI = $true) => ((setunionE = $true) => ((subPowSU = $true) => ((exuE2 = $true) => ((nonemptyImpWitness = $true) => ((uniqinunit = $true) => ((notinsingleton = $true) => ((eqinunit = $true) => ((singletonsswitch = $true) => ((upairsetE = $true) => ((upairsetIL = $true) => ((upairsetIR = $true) => ((emptyE1 = $true) => ((vacuousDall = $true) => ((quantDeMorgan1 = $true) => ((quantDeMorgan2 = $true) => ((quantDeMorgan3 = $true) => ((quantDeMorgan4 = $true) => ((prop2setI = $true) => ((prop2set2propI = $true) => ((notdexE = $true) => ((notdallE = $true) => ((exuI1 = $true) => ((exuI3 = $true) => ((exuI2 = $true) => ((inCongP = $true) => ((in__Cong = $true) => ((exuE3u = $true) => ((exu__Cong = $true) => ((emptyset__Cong = $true) => ((setadjoin__Cong = $true) => ((powerset__Cong = $true) => ((setunion__Cong = $true) => ((omega__Cong = $true) => ((exuEu = $true) => ((descr__Cong = $true) => ((dsetconstr__Cong = $true) => ((subsetI1 = $true) => ((eqimpsubset2 = $true) => ((eqimpsubset1 = $true) => ((subsetI2 = $true) => ((emptysetsubset = $true) => ((subsetE = $true) => ((subsetE2 = $true) => ((notsubsetI = $true) => ((notequalI1 = $true) => ((notequalI2 = $true) => ((subsetRefl = $true) => ((subsetTrans = $true) => ((setadjoinSub = $true) => ((setadjoinSub2 = $true) => ((subset2powerset = $true) => ((setextsub = $true) => ((subsetemptysetimpeq = $true) => ((powersetI1 = $true) => ((powersetE1 = $true) => ((inPowerset = $true) => ((powersetsubset = $true) => ((sepInPowerset = $true) => ((sepSubset = $true) => ((binunionIL = $true) => ((upairset2IR = $true) => ((binunionIR = $true) => ((binunionEcases = $true) => ((binunionE = $true) => ((binunionLsub = $true) => ((binunionRsub = $true) => ((binintersectI = $true) => ((binintersectSubset5 = $true) => ((binintersectEL = $true) => ((binintersectLsub = $true) => ((binintersectSubset2 = $true) => ((binintersectSubset3 = $true) => ((binintersectER = $true) => ((disjointsetsI1 = $true) => ((binintersectRsub = $true) => ((binintersectSubset4 = $true) => ((binintersectSubset1 = $true) => ((bs114d = $true) => ((setminusI = $true) => ((setminusEL = $true) => ((setminusER = $true) => ((setminusSubset2 = $true) => ((setminusERneg = $true) => ((setminusELneg = $true) => ((setminusILneg = $true) => ((setminusIRneg = $true) => ((setminusLsub = $true) => ((setminusSubset1 = $true) => ((symdiffE = $true) => ((symdiffI1 = $true) => ((symdiffI2 = $true) => ((symdiffIneg1 = $true) => ((symdiffIneg2 = $true) => ((secondinupair = $true) => ((setukpairIL = $true) => ((setukpairIR = $true) => ((kpairiskpair = $true) => ((kpairp = $true) => ((singletonsubset = $true) => ((singletoninpowerset = $true) => ((singletoninpowunion = $true) => ((upairset2E = $true) => ((upairsubunion = $true) => ((upairinpowunion = $true) => ((ubforcartprodlem1 = $true) => ((ubforcartprodlem2 = $true) => ((ubforcartprodlem3 = $true) => ((cartprodpairin = $true) => ((cartprodmempair1 = $true) => ((cartprodmempair = $true) => ((setunionE2 = $true) => ((setunionsingleton1 = $true) => ((setunionsingleton2 = $true) => ((setunionsingleton = $true) => ((singletonprop = $true) => ((ex1E1 = $true) => ((ex1I = $true) => ((ex1I2 = $true) => ((singletonsuniq = $true) => ((setukpairinjL1 = $true) => ((kfstsingleton = $true) => ((theprop = $true) => ((kfstpairEq = $true) => ((cartprodfstin = $true) => ((setukpairinjL2 = $true) => ((setukpairinjL = $true) => ((setukpairinjR11 = $true) => ! [X0 : $i,X1 : $i] : ((X0 = X1) => ((setadjoin @ (setadjoin @ X0 @ emptyset) @ (setadjoin @ (setadjoin @ X0 @ (setadjoin @ X1 @ emptyset)) @ emptyset)) = (setadjoin @ (setadjoin @ X0 @ emptyset) @ emptyset)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.65/1.65 inference(fool_elimination,[status(thm)],[f503])).
% 2.65/1.65 thf(f503,plain,(
% 2.65/1.65 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => (ubforcartprodlem2 => (ubforcartprodlem3 => (cartprodpairin => (cartprodmempair1 => (cartprodmempair => (setunionE2 => (setunionsingleton1 => (setunionsingleton2 => (setunionsingleton => (singletonprop => (ex1E1 => (ex1I => (ex1I2 => (singletonsuniq => (setukpairinjL1 => (kfstsingleton => (theprop => (kfstpairEq => (cartprodfstin => (setukpairinjL2 => (setukpairinjL => (setukpairinjR11 => ! [X0 : $i,X1 : $i] : ((X0 = X1) => ((setadjoin @ (setadjoin @ X0 @ emptyset) @ (setadjoin @ (setadjoin @ X0 @ (setadjoin @ X1 @ emptyset)) @ emptyset)) = (setadjoin @ (setadjoin @ X0 @ emptyset) @ emptyset)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.65/1.65 inference(rectify,[status(thm)],[f174])).
% 2.65/1.65 thf(f174,negated_conjecture,(
% 2.65/1.65 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => (ubforcartprodlem2 => (ubforcartprodlem3 => (cartprodpairin => (cartprodmempair1 => (cartprodmempair => (setunionE2 => (setunionsingleton1 => (setunionsingleton2 => (setunionsingleton => (singletonprop => (ex1E1 => (ex1I => (ex1I2 => (singletonsuniq => (setukpairinjL1 => (kfstsingleton => (theprop => (kfstpairEq => (cartprodfstin => (setukpairinjL2 => (setukpairinjL => (setukpairinjR11 => ! [X1 : $i,X2 : $i] : ((X1 = X2) => ((setadjoin @ (setadjoin @ X1 @ emptyset) @ (setadjoin @ (setadjoin @ X1 @ (setadjoin @ X2 @ emptyset)) @ emptyset)) = (setadjoin @ (setadjoin @ X1 @ emptyset) @ emptyset)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.65/1.65 inference(negated_conjecture,[status(cth)],[f173])).
% 2.65/1.65 thf(f173,conjecture,(
% 2.65/1.65 setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => (ubforcartprodlem2 => (ubforcartprodlem3 => (cartprodpairin => (cartprodmempair1 => (cartprodmempair => (setunionE2 => (setunionsingleton1 => (setunionsingleton2 => (setunionsingleton => (singletonprop => (ex1E1 => (ex1I => (ex1I2 => (singletonsuniq => (setukpairinjL1 => (kfstsingleton => (theprop => (kfstpairEq => (cartprodfstin => (setukpairinjL2 => (setukpairinjL => (setukpairinjR11 => ! [X1 : $i,X2 : $i] : ((X1 = X2) => ((setadjoin @ (setadjoin @ X1 @ emptyset) @ (setadjoin @ (setadjoin @ X1 @ (setadjoin @ X2 @ emptyset)) @ emptyset)) = (setadjoin @ (setadjoin @ X1 @ emptyset) @ emptyset))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.65/1.65 file('/export/starexec/sandbox/tmp/tmp.mID8C0AdqD/DTF2THF_11959.p',setukpairinjR12)).
% 2.65/1.65 thf(f1695,plain,(
% 2.65/1.65 ( ! [X0 : $i,X1 : $i] : (((setadjoin @ X1 @ emptyset) = (setadjoin @ X1 @ (setadjoin @ X0 @ emptyset))) | (X0 != X1) | (setukpairinjR11 != $true)) )),
% 2.65/1.65 inference(cnf_transformation,[status(thm)],[f965])).
% 2.65/1.65 thf(f965,plain,(
% 2.65/1.65 (! [X0 : $i,X1 : $i] : (((setadjoin @ X1 @ emptyset) = (setadjoin @ X1 @ (setadjoin @ X0 @ emptyset))) | (X0 != X1)) | (setukpairinjR11 != $true)) & ((setukpairinjR11 = $true) | (((setadjoin @ sK145 @ (setadjoin @ sK144 @ emptyset)) != (setadjoin @ sK145 @ emptyset)) & (sK144 = sK145)))),
% 2.65/1.65 inference(skolemisation,[status(esa),new_symbols(skolem,[sK144,sK145])],[f963,f964])).
% 2.65/1.65 thf(f964,plain,(
% 2.65/1.65 ? [X2 : $i,X3 : $i] : (((setadjoin @ X3 @ emptyset) != (setadjoin @ X3 @ (setadjoin @ X2 @ emptyset))) & (X2 = X3)) => (((setadjoin @ sK145 @ (setadjoin @ sK144 @ emptyset)) != (setadjoin @ sK145 @ emptyset)) & (sK144 = sK145))),
% 2.65/1.65 introduced(definition,[],[choice_axiom])).
% 2.65/1.65 thf(f963,plain,(
% 2.65/1.65 (! [X0 : $i,X1 : $i] : (((setadjoin @ X1 @ emptyset) = (setadjoin @ X1 @ (setadjoin @ X0 @ emptyset))) | (X0 != X1)) | (setukpairinjR11 != $true)) & ((setukpairinjR11 = $true) | ? [X2 : $i,X3 : $i] : (((setadjoin @ X3 @ emptyset) != (setadjoin @ X3 @ (setadjoin @ X2 @ emptyset))) & (X2 = X3)))),
% 2.65/1.65 inference(rectify,[status(thm)],[f962])).
% 2.65/1.65 thf(f962,plain,(
% 2.65/1.65 (! [X0 : $i,X1 : $i] : (((setadjoin @ X1 @ emptyset) = (setadjoin @ X1 @ (setadjoin @ X0 @ emptyset))) | (X0 != X1)) | (setukpairinjR11 != $true)) & ((setukpairinjR11 = $true) | ? [X0 : $i,X1 : $i] : (((setadjoin @ X1 @ emptyset) != (setadjoin @ X1 @ (setadjoin @ X0 @ emptyset))) & (X0 = X1)))),
% 2.65/1.65 inference(nnf_transformation,[status(thm)],[f597])).
% 2.65/1.65 thf(f597,plain,(
% 2.65/1.65 ! [X0 : $i,X1 : $i] : (((setadjoin @ X1 @ emptyset) = (setadjoin @ X1 @ (setadjoin @ X0 @ emptyset))) | (X0 != X1)) <=> (setukpairinjR11 = $true)),
% 2.65/1.65 inference(ennf_transformation,[status(thm)],[f378])).
% 2.65/1.65 thf(f378,plain,(
% 2.65/1.65 (setukpairinjR11 = $true) <=> ! [X0 : $i,X1 : $i] : ((X0 = X1) => ((setadjoin @ X1 @ emptyset) = (setadjoin @ X1 @ (setadjoin @ X0 @ emptyset))))),
% 2.65/1.65 inference(fool_elimination,[status(thm)],[f377])).
% 2.65/1.65 thf(f377,plain,(
% 2.65/1.65 (! [X0 : $i,X1 : $i] : ((X0 = X1) => ((setadjoin @ X1 @ emptyset) = (setadjoin @ X1 @ (setadjoin @ X0 @ emptyset)))) = setukpairinjR11)),
% 2.65/1.65 inference(rectify,[status(thm)],[f172])).
% 2.65/1.65 thf(f172,axiom,(
% 2.65/1.65 (! [X2 : $i,X1 : $i] : ((X1 = X2) => ((setadjoin @ X1 @ emptyset) = (setadjoin @ X1 @ (setadjoin @ X2 @ emptyset)))) = setukpairinjR11)),
% 2.65/1.65 file('/export/starexec/sandbox/tmp/tmp.mID8C0AdqD/DTF2THF_11959.p',setukpairinjR11)).
% 2.65/1.65 thf(f3196,plain,(
% 2.65/1.65 ((setadjoin @ (setadjoin @ sK328 @ emptyset) @ (setadjoin @ (setadjoin @ sK328 @ emptyset) @ emptyset)) != (setadjoin @ (setadjoin @ sK328 @ emptyset) @ emptyset))),
% 2.65/1.65 inference(forward_demodulation,[status(thm)],[f2713,f2966])).
% 2.65/1.65 thf(f2713,plain,(
% 2.65/1.65 ((setadjoin @ (setadjoin @ sK328 @ emptyset) @ (setadjoin @ (setadjoin @ sK328 @ (setadjoin @ sK328 @ emptyset)) @ emptyset)) != (setadjoin @ (setadjoin @ sK328 @ emptyset) @ emptyset))),
% 2.65/1.65 inference(definition_unfolding,[status(thm)],[f2053,f2052,f2052,f2052])).
% 2.65/1.65 thf(f2052,plain,(
% 2.65/1.65 (sK327 = sK328)),
% 2.65/1.65 inference(cnf_transformation,[status(thm)],[f1265])).
% 2.65/1.65 thf(f2053,plain,(
% 2.65/1.65 ((setadjoin @ (setadjoin @ sK327 @ emptyset) @ (setadjoin @ (setadjoin @ sK327 @ (setadjoin @ sK328 @ emptyset)) @ emptyset)) != (setadjoin @ (setadjoin @ sK327 @ emptyset) @ emptyset))),
% 2.65/1.65 inference(cnf_transformation,[status(thm)],[f1265])).
% 2.65/1.65 % SZS output end Proof for DTF2THF_11959
% 2.65/1.65 % (12295)------------------------------
% 2.65/1.65 % (12295)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.65 % (12295)Termination reason: Refutation
% 2.65/1.65
% 2.65/1.65 % (12295)Memory used [KB]: 8059
% 2.65/1.65 % (12295)Time elapsed: 0.104 s
% 2.65/1.65 % (12295)Instructions burned: 179 (million)
% 2.65/1.65 % (12295)------------------------------
% 2.65/1.65 % (12295)------------------------------
% 2.65/1.65 % (12289)Success in time 0.121 s
%------------------------------------------------------------------------------