%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU671^1 : 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 : n015.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 09:36:05 AM UTC 2026
% Result : Theorem 2.99s 1.99s
% Output : Refutation 2.99s
% Verified :
% SZS Type : Refutation
% Derivation depth : 15
% Number of leaves : 12
% Syntax : Number of formulae : 69 ( 18 unt; 0 typ; 3 def)
% Number of atoms : 3545 ( 946 equ; 0 cnn)
% Maximal formula atoms : 189 ( 51 avg)
% Number of connectives : 2287 ( 101 ~; 82 |; 604 &; 718 @)
% ( 10 <=>; 772 =>; 0 <=; 0 <~>)
% Maximal formula depth : 201 ( 25 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 88 ( 88 >; 0 *; 0 +; 0 <<)
% Number of symbols : 358 ( 355 usr; 208 con; 0-4 aty)
% Number of variables : 326 ( 104 ^; 160 !; 62 ?; 326 :)
% 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_203,type,
ksnd: $i > $i ).
thf(func_def_213,type,
breln: $i > $i > $i > $o ).
thf(func_def_214,type,
dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_235,type,
sP0: $o > $i > $i > $i > $o ).
thf(func_def_238,type,
sP3: $i > $i > $o ).
thf(func_def_239,type,
sP4: $i > $o ).
thf(func_def_240,type,
sP5: $i > $i > $o ).
thf(func_def_241,type,
sP6: $i > $i > $o ).
thf(func_def_255,type,
sK20: $i > $o ).
thf(func_def_282,type,
sK47: $i > $o ).
thf(func_def_297,type,
sK62: $i > $i > $i ).
thf(func_def_298,type,
sK63: $i > $i > $i ).
thf(func_def_301,type,
sK66: $i > $o ).
thf(func_def_308,type,
sK73: $i > $i > $i ).
thf(func_def_317,type,
sK82: $i > $i > $o ).
thf(func_def_329,type,
sK94: $i > $i > $o ).
thf(func_def_331,type,
sK96: $i > $o ).
thf(func_def_334,type,
sK99: $i > $o ).
thf(func_def_337,type,
sK102: $i > ( $i > $o ) > $i ).
thf(func_def_338,type,
sK103: $i > ( $i > $o ) > $i > $i ).
thf(func_def_340,type,
sK105: $i > $o ).
thf(func_def_347,type,
sK112: $i > $i > $i > $i ).
thf(func_def_348,type,
sK113: $i > $i > $i > $i ).
thf(func_def_351,type,
sK116: $i > $i > $o ).
thf(func_def_365,type,
sK130: $i > $o ).
thf(func_def_367,type,
sK132: ( $i > $o ) > $i > $i ).
thf(func_def_368,type,
sK133: ( $i > $o ) > $i > $i ).
thf(func_def_385,type,
sK150: $i > $o ).
thf(func_def_387,type,
sK152: ( $i > $o ) > $i > $i ).
thf(func_def_389,type,
sK154: $i > $i ).
thf(func_def_391,type,
sK156: ( $i > $o ) > $i ).
thf(func_def_392,type,
sK157: ( $i > $o ) > $i ).
thf(func_def_393,type,
sK158: $i > $o ).
thf(func_def_398,type,
sK163: $i > $i ).
thf(func_def_406,type,
sK171: ( $i > $o ) > $i ).
thf(func_def_407,type,
sK172: $i > $o ).
thf(func_def_408,type,
sK173: $i > $i ).
thf(func_def_424,type,
sK189: $i > $i > $o ).
thf(func_def_425,type,
sK190: $i > $i ).
thf(func_def_426,type,
sK191: $i > $i ).
thf(func_def_427,type,
sK192: ( $i > $i > $o ) > $i > $i ).
thf(func_def_428,type,
sK193: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_429,type,
sK194: ( $i > $i > $o ) > $i > $i ).
thf(func_def_436,type,
sK201: $i > ( $i > $o ) > $i ).
thf(func_def_437,type,
sK202: $i > $o ).
thf(func_def_439,type,
sK204: ( $i > $o ) > $i > $i ).
thf(func_def_441,type,
sK206: $i > $o ).
thf(func_def_451,type,
sK216: $i > $i > $i ).
thf(func_def_459,type,
sK224: $i > $i ).
thf(func_def_462,type,
sK227: $i > $i > $i ).
thf(func_def_472,type,
sK237: $i > $i > $o ).
thf(func_def_490,type,
sK255: $i > $o ).
thf(func_def_492,type,
sK257: ( $i > $o ) > $i > $i ).
thf(func_def_494,type,
sK259: $i > $o ).
thf(func_def_501,type,
sK266: $i > $i > $i ).
thf(func_def_505,type,
sK270: $i > $o ).
thf(func_def_507,type,
sK272: ( $i > $o ) > $i > $i ).
thf(func_def_510,type,
sK275: $i > $o ).
thf(func_def_525,type,
sK290: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_526,type,
sK291: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_527,type,
sK292: $i > $o ).
thf(func_def_528,type,
sK293: $i > $o ).
thf(func_def_532,type,
sK297: $i > $i > $i ).
thf(func_def_543,type,
sK308: $i > $o ).
thf(func_def_544,type,
sK309: $i > $o ).
thf(func_def_552,type,
sK317: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_553,type,
sK318: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_556,type,
sK321: $i > $o ).
thf(func_def_557,type,
sK322: $i > $o ).
thf(func_def_558,type,
sK323: $i > $o ).
thf(func_def_562,type,
sK327: $i > $i ).
thf(func_def_570,type,
sK335: $i > ( $i > $o ) > $i ).
thf(func_def_571,type,
sK336: $i > $o ).
thf(func_def_575,type,
sK340: $i > $i > $i ).
thf(func_def_578,type,
sK343: $i > $i ).
thf(func_def_579,type,
sK344: $i > $i ).
thf(func_def_593,type,
sK358: $i > $i > $o ).
thf(func_def_596,type,
sK361: $i > $i > $i ).
thf(func_def_597,type,
sK362: $i > $i > $i ).
thf(func_def_598,type,
sK363: $i > $i > $i ).
thf(func_def_599,type,
sK364: $i > $i > $i ).
thf(func_def_600,type,
sK365: $i > $i > $i > $i ).
thf(func_def_601,type,
sK366: $i > $i ).
thf(func_def_602,type,
sK367: $i > $i ).
thf(func_def_603,type,
sK368: $i > $i ).
thf(func_def_604,type,
sK369: $i > $i ).
thf(func_def_605,type,
sK370: $i > $i > $i > $i ).
thf(func_def_606,type,
sK371: $i > $i > $i > $i ).
thf(func_def_607,type,
sK372: $i > $i > $i ).
thf(func_def_608,type,
sK373: $i > $i > $i ).
thf(func_def_609,type,
sK374: $i > $i > $i ).
thf(func_def_611,type,
sK376: $i > $i ).
thf(func_def_612,type,
sK377: $i > $i ).
thf(func_def_613,type,
sK378: $i > $i ).
thf(func_def_614,type,
sK379: $i > $i > $i ).
thf(func_def_615,type,
sK380: $i > $o ).
thf(func_def_626,type,
sK391: $i > $o ).
thf(func_def_629,type,
sK394: $i > ( $i > $o ) > $i ).
thf(func_def_630,type,
sK395: $i > ( $i > $o ) > $i ).
thf(func_def_643,type,
sK408: ( $i > $o ) > $i ).
thf(func_def_644,type,
sK409: $i > $o ).
thf(func_def_645,type,
sK410: $i > $i ).
thf(func_def_651,type,
sK416: $i > $i > $o ).
thf(func_def_664,type,
sK429: $i > ( $i > $o ) > $i ).
thf(func_def_665,type,
sK430: $i > $o ).
thf(func_def_667,type,
sK432: $i > $o ).
thf(func_def_668,type,
sK433: $i > $o ).
thf(func_def_669,type,
sK434: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_670,type,
sK435: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_673,type,
sK438: $i > $i > $i ).
thf(func_def_676,type,
sK441: $i > $i ).
thf(func_def_687,type,
sK452: $i > $i > $i ).
thf(func_def_700,type,
sK465: $o > $i > $i > $i ).
thf(func_def_704,type,
sK469: ( $i > $o ) > $i ).
thf(func_def_705,type,
sK470: $i > $o ).
thf(func_def_731,type,
sK496: $i > $o ).
thf(func_def_743,type,
sK508: $i > $i > $o ).
thf(func_def_754,type,
sK519: $i > $o ).
thf(func_def_758,type,
sK523: $i > $o ).
thf(func_def_761,type,
sK526: $i > ( $i > $o ) > $i ).
thf(func_def_776,type,
ph541:
!>[X0: $tType] : X0 ).
thf(f2747,plain,
$false,
inference(avatar_sat_refutation,[status(thm)],[f2741,f2742,f2743,f2746]) ).
thf(f2746,plain,
~ spl540_1,
inference(avatar_contradiction_clause,[status(thm)],[f2745]) ).
thf(f2745,plain,
( $false
| ~ spl540_1 ),
inference(trivial_inequality_removal,[status(thm)],[f2744]) ).
thf(f2744,plain,
( ( $true != $true )
| ~ spl540_1 ),
inference(superposition,[status(thm)],[f1953,f2732]) ).
thf(f2732,plain,
( ( ( sK116 @ sK118 @ sK115 )
= $true )
| ~ spl540_1 ),
inference(avatar_component_clause,[status(thm)],[f2730]) ).
thf(f2730,definition,
( spl540_1
<=> ( ( sK116 @ sK118 @ sK115 )
= $true ) ),
introduced(definition,[new_symbols(naming,[spl540_1])],[avatar_definition]) ).
thf(f1953,plain,
( ( sK116 @ sK118 @ sK115 )
!= $true ),
inference(cnf_transformation,[status(thm)],[f988]) ).
thf(f988,plain,
( ( prop2setI = $true )
& ( ubforcartprodlem3 = $true )
& ( emptyE1 = $true )
& ( emptyinunitempty = $true )
& ( inCongP = $true )
& ( exu__Cong = $true )
& ( setadjoinAx = $true )
& ( subset2powerset = $true )
& ( setunionI = $true )
& ( binintersectSubset5 = $true )
& ( setukpairinjR12 = $true )
& ( binintersectEL = $true )
& ( setadjoinIL = $true )
& ( setukpairinjR1 = $true )
& ( binintersectSubset2 = $true )
& ( emptysetAx = $true )
& ( upairequniteq = $true )
& ( prop2set2propI = $true )
& ( singletoninpowunion = $true )
& ( setadjoin__Cong = $true )
& ( setukpairinjL2 = $true )
& ( in__Cong = $true )
& ( upairsetIR = $true )
& ( setextAx = $true )
& ( cartprodsndpairEq = $true )
& ( quantDeMorgan3 = $true )
& ( singletonprop = $true )
& ( setminusELneg = $true )
& ( binintersectER = $true )
& ( emptysetimpfalse = $true )
& ( quantDeMorgan1 = $true )
& ( kpairiskpair = $true )
& ( binintersectSubset3 = $true )
& ( setminusSubset2 = $true )
& ( symdiffI1 = $true )
& ( eqimpsubset2 = $true )
& ( nonemptyI1 = $true )
& ( vacuousDall = $true )
& ( singletonsubset = $true )
& ( powerset__Cong = $true )
& ( setukpairinjR11 = $true )
& ( setext = $true )
& ( symdiffE = $true )
& ( setadjoinSub2 = $true )
& ( subsetemptysetimpeq = $true )
& ( binunionLsub = $true )
& ( emptyset__Cong = $true )
& ( subsetE = $true )
& ( setukpairinjR = $true )
& ( dpsetconstrERa = $true )
& ( dsetconstrER = $true )
& ( setukpairinjL1 = $true )
& ( setunionsingleton2 = $true )
& ( setadjoinIR = $true )
& ( ( in @ ( kpair @ sK118 @ sK115 )
@ ( dpsetconstr @ sK117 @ sK114
@ ^ [Y0: $i,Y1: $i] : ( sK116 @ Y0 @ Y1 ) ) )
= $true )
& ( ( sK116 @ sK118 @ sK115 )
!= $true )
& ( sepInPowerset = $true )
& ( setukpairIR = $true )
& ( disjointsetsI1 = $true )
& ( setminusEL = $true )
& ( cartprodpairin = $true )
& ( notequalI2 = $true )
& ( setextsub = $true )
& ( setminusERneg = $true )
& ( binintersectLsub = $true )
& ( subsetTrans = $true )
& ( exuI1 = $true )
& ( setOfPairsIsBReln = $true )
& ( setadjoinOr = $true )
& ( quantDeMorgan2 = $true )
& ( notsubsetI = $true )
& ( notequalI1 = $true )
& ( subPowSU = $true )
& ( binintersectRsub = $true )
& ( kfstpairEq = $true )
& ( setminusILneg = $true )
& ( cartprodfstin = $true )
& ( nonemptyImpWitness = $true )
& ( binunionRsub = $true )
& ( dsetconstrEL = $true )
& ( notinemptyset = $true )
& ( powersetI1 = $true )
& ( secondinupair = $true )
& ( dpsetconstrEL1 = $true )
& ( wellorderingAx = $true )
& ( setunion__Cong = $true )
& ( upairsetE = $true )
& ( binunionIL = $true )
& ( setukpairinjR2 = $true )
& ( powersetE = $true )
& ( exuE3e = $true )
& ( nonemptyI = $true )
& ( setoftrueEq = $true )
& ( replAx = $true )
& ( ubforcartprodlem2 = $true )
& ( dpsetconstrEL2 = $true )
& ( ksndsingleton = $true )
& ( upairset2IR = $true )
& ( ex1E1 = $true )
& ( notdallE = $true )
& ( upairsetIL = $true )
& ( cartprodmempair1 = $true )
& ( setadjoinSub = $true )
& ( upairinpowunion = $true )
& ( cartprodsndin = $true )
& ( setadjoinE = $true )
& ( eqinunit = $true )
& ( dpsetconstrSub = $true )
& ( omega0Ax = $true )
& ( binunionEcases = $true )
& ( upairset2E = $true )
& ( notdexE = $true )
& ( dpsetconstrI = $true )
& ( ksndpairEq = $true )
& ( cartprodpairsurjEq = $true )
& ( omegaSAx = $true )
& ( omegaIndAx = $true )
& ( powersetAx = $true )
& ( exuE3u = $true )
& ( inPowerset = $true )
& ( emptysetE = $true )
& ( exuE2 = $true )
& ( dsetconstrI = $true )
& ( descrp = $true )
& ( descr__Cong = $true )
& ( cartprodmempaircEq = $true )
& ( emptyinPowerset = $true )
& ( binintersectSubset1 = $true )
& ( dsetconstr__Cong = $true )
& ( setunionsingleton = $true )
& ( binunionE = $true )
& ( ex1I = $true )
& ( setminusIRneg = $true )
& ( symdiffI2 = $true )
& ( kfstsingleton = $true )
& ( bs114d = $true )
& ( subsetRefl = $true )
& ( cartprodpairmemEL = $true )
& ( setminusLsub = $true )
& ( subsetE2 = $true )
& ( setunionsingleton1 = $true )
& ( sepSubset = $true )
& ( notinsingleton = $true )
& ( powersetsubset = $true )
& ( setbeta = $true )
& ( exuI2 = $true )
& ( setminusER = $true )
& ( kpairsurjEq = $true )
& ( singletoninpowerset = $true )
& ( setukpairinjL = $true )
& ( exuI3 = $true )
& ( symdiffIneg2 = $true )
& ( cartprodfstpairEq = $true )
& ( setunionAx = $true )
& ( singletonsswitch = $true )
& ( exuEu = $true )
& ( uniqinunit = $true )
& ( theprop = $true )
& ( setunionE2 = $true )
& ( ex1I2 = $true )
& ( noeltsimpempty = $true )
& ( ubforcartprodlem1 = $true )
& ( symdiffIneg1 = $true )
& ( kpairp = $true )
& ( cartprodmempair = $true )
& ( eqimpsubset1 = $true )
& ( setminusI = $true )
& ( binintersectI = $true )
& ( omega__Cong = $true )
& ( binunionIR = $true )
& ( cartprodpairmemER = $true )
& ( exuE1 = $true )
& ( emptyI = $true )
& ( powersetI = $true )
& ( singletonsuniq = $true )
& ( subsetI2 = $true )
& ( setminusSubset1 = $true )
& ( emptysetsubset = $true )
& ( emptyInPowerset = $true )
& ( nonemptyE1 = $true )
& ( upairsubunion = $true )
& ( setunionE = $true )
& ( foundationAx = $true )
& ( quantDeMorgan4 = $true )
& ( binintersectSubset4 = $true )
& ( powersetE1 = $true )
& ( prop2setE = $true )
& ( subsetI1 = $true )
& ( setukpairIL = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK114,sK115,sK116,sK117,sK118])],[f668,f987]) ).
thf(f987,plain,
( ? [X0: $i,X1: $i,X2: $i > $i > $o,X3: $i,X4: $i] :
( ( ( in @ ( kpair @ X4 @ X1 )
@ ( dpsetconstr @ X3 @ X0
@ ^ [Y0: $i,Y1: $i] : ( X2 @ Y0 @ Y1 ) ) )
= $true )
& ( ( X2 @ X4 @ X1 )
!= $true ) )
=> ( ( ( in @ ( kpair @ sK118 @ sK115 )
@ ( dpsetconstr @ sK117 @ sK114
@ ^ [Y0: $i,Y1: $i] : ( sK116 @ Y0 @ Y1 ) ) )
= $true )
& ( ( sK116 @ sK118 @ sK115 )
!= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f668,plain,
( ( prop2setI = $true )
& ( ubforcartprodlem3 = $true )
& ( emptyE1 = $true )
& ( emptyinunitempty = $true )
& ( inCongP = $true )
& ( exu__Cong = $true )
& ( setadjoinAx = $true )
& ( subset2powerset = $true )
& ( setunionI = $true )
& ( binintersectSubset5 = $true )
& ( setukpairinjR12 = $true )
& ( binintersectEL = $true )
& ( setadjoinIL = $true )
& ( setukpairinjR1 = $true )
& ( binintersectSubset2 = $true )
& ( emptysetAx = $true )
& ( upairequniteq = $true )
& ( prop2set2propI = $true )
& ( singletoninpowunion = $true )
& ( setadjoin__Cong = $true )
& ( setukpairinjL2 = $true )
& ( in__Cong = $true )
& ( upairsetIR = $true )
& ( setextAx = $true )
& ( cartprodsndpairEq = $true )
& ( quantDeMorgan3 = $true )
& ( singletonprop = $true )
& ( setminusELneg = $true )
& ( binintersectER = $true )
& ( emptysetimpfalse = $true )
& ( quantDeMorgan1 = $true )
& ( kpairiskpair = $true )
& ( binintersectSubset3 = $true )
& ( setminusSubset2 = $true )
& ( symdiffI1 = $true )
& ( eqimpsubset2 = $true )
& ( nonemptyI1 = $true )
& ( vacuousDall = $true )
& ( singletonsubset = $true )
& ( powerset__Cong = $true )
& ( setukpairinjR11 = $true )
& ( setext = $true )
& ( symdiffE = $true )
& ( setadjoinSub2 = $true )
& ( subsetemptysetimpeq = $true )
& ( binunionLsub = $true )
& ( emptyset__Cong = $true )
& ( subsetE = $true )
& ( setukpairinjR = $true )
& ( dpsetconstrERa = $true )
& ( dsetconstrER = $true )
& ( setukpairinjL1 = $true )
& ( setunionsingleton2 = $true )
& ( setadjoinIR = $true )
& ? [X0: $i,X1: $i,X2: $i > $i > $o,X3: $i,X4: $i] :
( ( ( in @ ( kpair @ X4 @ X1 )
@ ( dpsetconstr @ X3 @ X0
@ ^ [Y0: $i,Y1: $i] : ( X2 @ Y0 @ Y1 ) ) )
= $true )
& ( ( X2 @ X4 @ X1 )
!= $true ) )
& ( sepInPowerset = $true )
& ( setukpairIR = $true )
& ( disjointsetsI1 = $true )
& ( setminusEL = $true )
& ( cartprodpairin = $true )
& ( notequalI2 = $true )
& ( setextsub = $true )
& ( setminusERneg = $true )
& ( binintersectLsub = $true )
& ( subsetTrans = $true )
& ( exuI1 = $true )
& ( setOfPairsIsBReln = $true )
& ( setadjoinOr = $true )
& ( quantDeMorgan2 = $true )
& ( notsubsetI = $true )
& ( notequalI1 = $true )
& ( subPowSU = $true )
& ( binintersectRsub = $true )
& ( kfstpairEq = $true )
& ( setminusILneg = $true )
& ( cartprodfstin = $true )
& ( nonemptyImpWitness = $true )
& ( binunionRsub = $true )
& ( dsetconstrEL = $true )
& ( notinemptyset = $true )
& ( powersetI1 = $true )
& ( secondinupair = $true )
& ( dpsetconstrEL1 = $true )
& ( wellorderingAx = $true )
& ( setunion__Cong = $true )
& ( upairsetE = $true )
& ( binunionIL = $true )
& ( setukpairinjR2 = $true )
& ( powersetE = $true )
& ( exuE3e = $true )
& ( nonemptyI = $true )
& ( setoftrueEq = $true )
& ( replAx = $true )
& ( ubforcartprodlem2 = $true )
& ( dpsetconstrEL2 = $true )
& ( ksndsingleton = $true )
& ( upairset2IR = $true )
& ( ex1E1 = $true )
& ( notdallE = $true )
& ( upairsetIL = $true )
& ( cartprodmempair1 = $true )
& ( setadjoinSub = $true )
& ( upairinpowunion = $true )
& ( cartprodsndin = $true )
& ( setadjoinE = $true )
& ( eqinunit = $true )
& ( dpsetconstrSub = $true )
& ( omega0Ax = $true )
& ( binunionEcases = $true )
& ( upairset2E = $true )
& ( notdexE = $true )
& ( dpsetconstrI = $true )
& ( ksndpairEq = $true )
& ( cartprodpairsurjEq = $true )
& ( omegaSAx = $true )
& ( omegaIndAx = $true )
& ( powersetAx = $true )
& ( exuE3u = $true )
& ( inPowerset = $true )
& ( emptysetE = $true )
& ( exuE2 = $true )
& ( dsetconstrI = $true )
& ( descrp = $true )
& ( descr__Cong = $true )
& ( cartprodmempaircEq = $true )
& ( emptyinPowerset = $true )
& ( binintersectSubset1 = $true )
& ( dsetconstr__Cong = $true )
& ( setunionsingleton = $true )
& ( binunionE = $true )
& ( ex1I = $true )
& ( setminusIRneg = $true )
& ( symdiffI2 = $true )
& ( kfstsingleton = $true )
& ( bs114d = $true )
& ( subsetRefl = $true )
& ( cartprodpairmemEL = $true )
& ( setminusLsub = $true )
& ( subsetE2 = $true )
& ( setunionsingleton1 = $true )
& ( sepSubset = $true )
& ( notinsingleton = $true )
& ( powersetsubset = $true )
& ( setbeta = $true )
& ( exuI2 = $true )
& ( setminusER = $true )
& ( kpairsurjEq = $true )
& ( singletoninpowerset = $true )
& ( setukpairinjL = $true )
& ( exuI3 = $true )
& ( symdiffIneg2 = $true )
& ( cartprodfstpairEq = $true )
& ( setunionAx = $true )
& ( singletonsswitch = $true )
& ( exuEu = $true )
& ( uniqinunit = $true )
& ( theprop = $true )
& ( setunionE2 = $true )
& ( ex1I2 = $true )
& ( noeltsimpempty = $true )
& ( ubforcartprodlem1 = $true )
& ( symdiffIneg1 = $true )
& ( kpairp = $true )
& ( cartprodmempair = $true )
& ( eqimpsubset1 = $true )
& ( setminusI = $true )
& ( binintersectI = $true )
& ( omega__Cong = $true )
& ( binunionIR = $true )
& ( cartprodpairmemER = $true )
& ( exuE1 = $true )
& ( emptyI = $true )
& ( powersetI = $true )
& ( singletonsuniq = $true )
& ( subsetI2 = $true )
& ( setminusSubset1 = $true )
& ( emptysetsubset = $true )
& ( emptyInPowerset = $true )
& ( nonemptyE1 = $true )
& ( upairsubunion = $true )
& ( setunionE = $true )
& ( foundationAx = $true )
& ( quantDeMorgan4 = $true )
& ( binintersectSubset4 = $true )
& ( powersetE1 = $true )
& ( prop2setE = $true )
& ( subsetI1 = $true )
& ( setukpairIL = $true ) ),
inference(flattening,[status(thm)],[f667]) ).
thf(f667,plain,
( ? [X0: $i,X1: $i,X2: $i > $i > $o,X3: $i,X4: $i] :
( ( ( in @ ( kpair @ X4 @ X1 )
@ ( dpsetconstr @ X3 @ X0
@ ^ [Y0: $i,Y1: $i] : ( X2 @ Y0 @ Y1 ) ) )
= $true )
& ( ( X2 @ X4 @ X1 )
!= $true ) )
& ( dpsetconstrEL2 = $true )
& ( dpsetconstrEL1 = $true )
& ( dpsetconstrERa = $true )
& ( setOfPairsIsBReln = $true )
& ( dpsetconstrSub = $true )
& ( dpsetconstrI = $true )
& ( cartprodpairsurjEq = $true )
& ( cartprodsndpairEq = $true )
& ( cartprodfstpairEq = $true )
& ( cartprodmempaircEq = $true )
& ( cartprodpairmemER = $true )
& ( cartprodpairmemEL = $true )
& ( cartprodsndin = $true )
& ( kpairsurjEq = $true )
& ( ksndpairEq = $true )
& ( ksndsingleton = $true )
& ( setukpairinjR = $true )
& ( setukpairinjR2 = $true )
& ( upairequniteq = $true )
& ( setukpairinjR1 = $true )
& ( setukpairinjR12 = $true )
& ( 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)],[f371]) ).
thf(f371,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 )
=> ( ( setukpairinjR12 = $true )
=> ( ( setukpairinjR1 = $true )
=> ( ( upairequniteq = $true )
=> ( ( setukpairinjR2 = $true )
=> ( ( setukpairinjR = $true )
=> ( ( ksndsingleton = $true )
=> ( ( ksndpairEq = $true )
=> ( ( kpairsurjEq = $true )
=> ( ( cartprodsndin = $true )
=> ( ( cartprodpairmemEL = $true )
=> ( ( cartprodpairmemER = $true )
=> ( ( cartprodmempaircEq = $true )
=> ( ( cartprodfstpairEq = $true )
=> ( ( cartprodsndpairEq = $true )
=> ( ( cartprodpairsurjEq = $true )
=> ( ( dpsetconstrI = $true )
=> ( ( dpsetconstrSub = $true )
=> ( ( setOfPairsIsBReln = $true )
=> ( ( dpsetconstrERa = $true )
=> ( ( dpsetconstrEL1 = $true )
=> ( ( dpsetconstrEL2 = $true )
=> ! [X0: $i,X1: $i,X2: $i > $i > $o,X3: $i,X4: $i] :
( ( ( in @ ( kpair @ X4 @ X1 )
@ ( dpsetconstr @ X3 @ X0
@ ^ [Y0: $i,Y1: $i] : ( X2 @ Y0 @ Y1 ) ) )
= $true )
=> ( ( X2 @ X4 @ X1 )
= $true ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f370]) ).
thf(f370,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
=> ( setukpairinjR12
=> ( setukpairinjR1
=> ( upairequniteq
=> ( setukpairinjR2
=> ( setukpairinjR
=> ( ksndsingleton
=> ( ksndpairEq
=> ( kpairsurjEq
=> ( cartprodsndin
=> ( cartprodpairmemEL
=> ( cartprodpairmemER
=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ( cartprodpairsurjEq
=> ( dpsetconstrI
=> ( dpsetconstrSub
=> ( setOfPairsIsBReln
=> ( dpsetconstrERa
=> ( dpsetconstrEL1
=> ( dpsetconstrEL2
=> ! [X0: $i,X1: $i,X2: $i > $i > $o,X3: $i,X4: $i] :
( ( in @ ( kpair @ X4 @ X1 )
@ ( dpsetconstr @ X3 @ X0
@ ^ [X5: $i,X6: $i] : ( X2 @ X5 @ X6 ) ) )
=> ( X2 @ X4 @ X1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f195]) ).
thf(f195,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
=> ( setukpairinjR12
=> ( setukpairinjR1
=> ( upairequniteq
=> ( setukpairinjR2
=> ( setukpairinjR
=> ( ksndsingleton
=> ( ksndpairEq
=> ( kpairsurjEq
=> ( cartprodsndin
=> ( cartprodpairmemEL
=> ( cartprodpairmemER
=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ( cartprodpairsurjEq
=> ( dpsetconstrI
=> ( dpsetconstrSub
=> ( setOfPairsIsBReln
=> ( dpsetconstrERa
=> ( dpsetconstrEL1
=> ( dpsetconstrEL2
=> ! [X4: $i,X2: $i,X0: $i > $i > $o,X3: $i,X1: $i] :
( ( in @ ( kpair @ X1 @ X2 )
@ ( dpsetconstr @ X3 @ X4
@ ^ [X8: $i,X10: $i] : ( X0 @ X8 @ X10 ) ) )
=> ( X0 @ X1 @ X2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f194]) ).
thf(f194,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
=> ( setukpairinjR12
=> ( setukpairinjR1
=> ( upairequniteq
=> ( setukpairinjR2
=> ( setukpairinjR
=> ( ksndsingleton
=> ( ksndpairEq
=> ( kpairsurjEq
=> ( cartprodsndin
=> ( cartprodpairmemEL
=> ( cartprodpairmemER
=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ( cartprodpairsurjEq
=> ( dpsetconstrI
=> ( dpsetconstrSub
=> ( setOfPairsIsBReln
=> ( dpsetconstrERa
=> ( dpsetconstrEL1
=> ( dpsetconstrEL2
=> ! [X4: $i,X2: $i,X0: $i > $i > $o,X3: $i,X1: $i] :
( ( in @ ( kpair @ X1 @ X2 )
@ ( dpsetconstr @ X3 @ X4
@ ^ [X8: $i,X10: $i] : ( X0 @ X8 @ X10 ) ) )
=> ( X0 @ X1 @ X2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.0Ah447e165/DTF2THF_7726.p',dpsetconstrER) ).
thf(f2743,plain,
spl540_2,
inference(avatar_split_clause,[status(thm)],[f2724,f2734]) ).
thf(f2734,definition,
( spl540_2
<=> ( ( in @ sK115 @ sK114 )
= $true ) ),
introduced(definition,[new_symbols(naming,[spl540_2])],[avatar_definition]) ).
thf(f2724,plain,
( ( in @ sK115 @ sK114 )
= $true ),
inference(trivial_inequality_removal,[status(thm)],[f2723]) ).
thf(f2723,plain,
( ( ( in @ sK115 @ sK114 )
= $true )
| ( $true != $true ) ),
inference(forward_demodulation,[status(thm)],[f2719,f1913]) ).
thf(f1913,plain,
dpsetconstrEL2 = $true,
inference(cnf_transformation,[status(thm)],[f988]) ).
thf(f2719,plain,
( ( dpsetconstrEL2 != $true )
| ( ( in @ sK115 @ sK114 )
= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f2713]) ).
thf(f2713,plain,
( ( $true != $true )
| ( dpsetconstrEL2 != $true )
| ( ( in @ sK115 @ sK114 )
= $true ) ),
inference(superposition,[status(thm)],[f2693,f2712]) ).
thf(f2712,plain,
( ( in @ ( kpair @ sK118 @ sK115 ) @ ( dpsetconstr @ sK117 @ sK114 @ sK116 ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f1954]) ).
thf(f1954,plain,
( ( in @ ( kpair @ sK118 @ sK115 )
@ ( dpsetconstr @ sK117 @ sK114
@ ^ [Y0: $i,Y1: $i] : ( sK116 @ Y0 @ Y1 ) ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f988]) ).
thf(f2693,plain,
! [X8: $i > $i > $o,X6: $i,X9: $i,X7: $i,X5: $i] :
( ( $true
!= ( in @ ( kpair @ X6 @ X9 ) @ ( dpsetconstr @ X7 @ X5 @ X8 ) ) )
| ( dpsetconstrEL2 != $true )
| ( $true
= ( in @ X9 @ X5 ) ) ),
inference(beta_eta_normalization,[status(thm)],[f1779]) ).
thf(f1779,plain,
! [X8: $i > $i > $o,X6: $i,X9: $i,X7: $i,X5: $i] :
( ( ( in @ ( kpair @ X6 @ X9 )
@ ( dpsetconstr @ X7 @ X5
@ ^ [Y0: $i,Y1: $i] : ( X8 @ Y0 @ Y1 ) ) )
!= $true )
| ( dpsetconstrEL2 != $true )
| ( $true
= ( in @ X9 @ X5 ) ) ),
inference(cnf_transformation,[status(thm)],[f944]) ).
thf(f944,plain,
( ( ( dpsetconstrEL2 = $true )
| ( ( ( in @ sK83 @ sK79 )
!= $true )
& ( ( in @ ( kpair @ sK80 @ sK83 )
@ ( dpsetconstr @ sK81 @ sK79
@ ^ [Y0: $i,Y1: $i] : ( sK82 @ Y0 @ Y1 ) ) )
= $true ) ) )
& ( ! [X5: $i,X6: $i,X7: $i,X8: $i > $i > $o,X9: $i] :
( ( $true
= ( in @ X9 @ X5 ) )
| ( ( in @ ( kpair @ X6 @ X9 )
@ ( dpsetconstr @ X7 @ X5
@ ^ [Y0: $i,Y1: $i] : ( X8 @ Y0 @ Y1 ) ) )
!= $true ) )
| ( dpsetconstrEL2 != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK79,sK80,sK81,sK82,sK83])],[f942,f943]) ).
thf(f943,plain,
( ? [X0: $i,X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( ( in @ X4 @ X0 )
!= $true )
& ( $true
= ( in @ ( kpair @ X1 @ X4 )
@ ( dpsetconstr @ X2 @ X0
@ ^ [Y0: $i,Y1: $i] : ( X3 @ Y0 @ Y1 ) ) ) ) )
=> ( ( ( in @ sK83 @ sK79 )
!= $true )
& ( ( in @ ( kpair @ sK80 @ sK83 )
@ ( dpsetconstr @ sK81 @ sK79
@ ^ [Y0: $i,Y1: $i] : ( sK82 @ Y0 @ Y1 ) ) )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f942,plain,
( ( ( dpsetconstrEL2 = $true )
| ? [X0: $i,X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( ( in @ X4 @ X0 )
!= $true )
& ( $true
= ( in @ ( kpair @ X1 @ X4 )
@ ( dpsetconstr @ X2 @ X0
@ ^ [Y0: $i,Y1: $i] : ( X3 @ Y0 @ Y1 ) ) ) ) ) )
& ( ! [X5: $i,X6: $i,X7: $i,X8: $i > $i > $o,X9: $i] :
( ( $true
= ( in @ X9 @ X5 ) )
| ( ( in @ ( kpair @ X6 @ X9 )
@ ( dpsetconstr @ X7 @ X5
@ ^ [Y0: $i,Y1: $i] : ( X8 @ Y0 @ Y1 ) ) )
!= $true ) )
| ( dpsetconstrEL2 != $true ) ) ),
inference(rectify,[status(thm)],[f941]) ).
thf(f941,plain,
( ( ( dpsetconstrEL2 = $true )
| ? [X0: $i,X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( ( in @ X4 @ X0 )
!= $true )
& ( $true
= ( in @ ( kpair @ X1 @ X4 )
@ ( dpsetconstr @ X2 @ X0
@ ^ [Y0: $i,Y1: $i] : ( X3 @ Y0 @ Y1 ) ) ) ) ) )
& ( ! [X0: $i,X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( ( in @ X4 @ X0 )
= $true )
| ( $true
!= ( in @ ( kpair @ X1 @ X4 )
@ ( dpsetconstr @ X2 @ X0
@ ^ [Y0: $i,Y1: $i] : ( X3 @ Y0 @ Y1 ) ) ) ) )
| ( dpsetconstrEL2 != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f730]) ).
thf(f730,plain,
( ( dpsetconstrEL2 = $true )
<=> ! [X0: $i,X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( ( in @ X4 @ X0 )
= $true )
| ( $true
!= ( in @ ( kpair @ X1 @ X4 )
@ ( dpsetconstr @ X2 @ X0
@ ^ [Y0: $i,Y1: $i] : ( X3 @ Y0 @ Y1 ) ) ) ) ) ),
inference(ennf_transformation,[status(thm)],[f248]) ).
thf(f248,plain,
( ( dpsetconstrEL2 = $true )
<=> ! [X0: $i,X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( $true
= ( in @ ( kpair @ X1 @ X4 )
@ ( dpsetconstr @ X2 @ X0
@ ^ [Y0: $i,Y1: $i] : ( X3 @ Y0 @ Y1 ) ) ) )
=> ( ( in @ X4 @ X0 )
= $true ) ) ),
inference(fool_elimination,[status(thm)],[f247]) ).
thf(f247,plain,
( dpsetconstrEL2
= ( ! [X0: $i,X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ ( kpair @ X1 @ X4 )
@ ( dpsetconstr @ X2 @ X0
@ ^ [X5: $i,X6: $i] : ( X3 @ X5 @ X6 ) ) )
=> ( in @ X4 @ X0 ) ) ) ),
inference(rectify,[status(thm)],[f193]) ).
thf(f193,axiom,
( dpsetconstrEL2
= ( ! [X4: $i,X1: $i,X3: $i,X0: $i > $i > $o,X2: $i] :
( ( in @ ( kpair @ X1 @ X2 )
@ ( dpsetconstr @ X3 @ X4
@ ^ [X8: $i,X10: $i] : ( X0 @ X8 @ X10 ) ) )
=> ( in @ X2 @ X4 ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.0Ah447e165/DTF2THF_7726.p',dpsetconstrEL2) ).
thf(f2742,plain,
spl540_3,
inference(avatar_split_clause,[status(thm)],[f2722,f2738]) ).
thf(f2738,definition,
( spl540_3
<=> ( ( in @ sK118 @ sK117 )
= $true ) ),
introduced(definition,[new_symbols(naming,[spl540_3])],[avatar_definition]) ).
thf(f2722,plain,
( ( in @ sK118 @ sK117 )
= $true ),
inference(trivial_inequality_removal,[status(thm)],[f2721]) ).
thf(f2721,plain,
( ( $true != $true )
| ( ( in @ sK118 @ sK117 )
= $true ) ),
inference(forward_demodulation,[status(thm)],[f2720,f1925]) ).
thf(f1925,plain,
dpsetconstrEL1 = $true,
inference(cnf_transformation,[status(thm)],[f988]) ).
thf(f2720,plain,
( ( ( in @ sK118 @ sK117 )
= $true )
| ( dpsetconstrEL1 != $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f2714]) ).
thf(f2714,plain,
( ( dpsetconstrEL1 != $true )
| ( ( in @ sK118 @ sK117 )
= $true )
| ( $true != $true ) ),
inference(superposition,[status(thm)],[f2656,f2712]) ).
thf(f2656,plain,
! [X8: $i,X6: $i,X9: $i,X7: $i > $i > $o,X5: $i] :
( ( ( in @ ( kpair @ X6 @ X8 ) @ ( dpsetconstr @ X5 @ X9 @ X7 ) )
!= $true )
| ( dpsetconstrEL1 != $true )
| ( ( in @ X6 @ X5 )
= $true ) ),
inference(beta_eta_normalization,[status(thm)],[f2427]) ).
thf(f2427,plain,
! [X8: $i,X6: $i,X9: $i,X7: $i > $i > $o,X5: $i] :
( ( ( in @ X6 @ X5 )
= $true )
| ( dpsetconstrEL1 != $true )
| ( ( in @ ( kpair @ X6 @ X8 )
@ ( dpsetconstr @ X5 @ X9
@ ^ [Y0: $i,Y1: $i] : ( X7 @ Y0 @ Y1 ) ) )
!= $true ) ),
inference(cnf_transformation,[status(thm)],[f1476]) ).
thf(f1476,plain,
( ( ( dpsetconstrEL1 = $true )
| ( ( ( in @ ( kpair @ sK415 @ sK417 )
@ ( dpsetconstr @ sK414 @ sK418
@ ^ [Y0: $i,Y1: $i] : ( sK416 @ Y0 @ Y1 ) ) )
= $true )
& ( $true
!= ( in @ sK415 @ sK414 ) ) ) )
& ( ! [X5: $i,X6: $i,X7: $i > $i > $o,X8: $i,X9: $i] :
( ( ( in @ ( kpair @ X6 @ X8 )
@ ( dpsetconstr @ X5 @ X9
@ ^ [Y0: $i,Y1: $i] : ( X7 @ Y0 @ Y1 ) ) )
!= $true )
| ( ( in @ X6 @ X5 )
= $true ) )
| ( dpsetconstrEL1 != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK414,sK415,sK416,sK417,sK418])],[f1474,f1475]) ).
thf(f1475,plain,
( ? [X0: $i,X1: $i,X2: $i > $i > $o,X3: $i,X4: $i] :
( ( ( in @ ( kpair @ X1 @ X3 )
@ ( dpsetconstr @ X0 @ X4
@ ^ [Y0: $i,Y1: $i] : ( X2 @ Y0 @ Y1 ) ) )
= $true )
& ( ( in @ X1 @ X0 )
!= $true ) )
=> ( ( ( in @ ( kpair @ sK415 @ sK417 )
@ ( dpsetconstr @ sK414 @ sK418
@ ^ [Y0: $i,Y1: $i] : ( sK416 @ Y0 @ Y1 ) ) )
= $true )
& ( $true
!= ( in @ sK415 @ sK414 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1474,plain,
( ( ( dpsetconstrEL1 = $true )
| ? [X0: $i,X1: $i,X2: $i > $i > $o,X3: $i,X4: $i] :
( ( ( in @ ( kpair @ X1 @ X3 )
@ ( dpsetconstr @ X0 @ X4
@ ^ [Y0: $i,Y1: $i] : ( X2 @ Y0 @ Y1 ) ) )
= $true )
& ( ( in @ X1 @ X0 )
!= $true ) ) )
& ( ! [X5: $i,X6: $i,X7: $i > $i > $o,X8: $i,X9: $i] :
( ( ( in @ ( kpair @ X6 @ X8 )
@ ( dpsetconstr @ X5 @ X9
@ ^ [Y0: $i,Y1: $i] : ( X7 @ Y0 @ Y1 ) ) )
!= $true )
| ( ( in @ X6 @ X5 )
= $true ) )
| ( dpsetconstrEL1 != $true ) ) ),
inference(rectify,[status(thm)],[f1473]) ).
thf(f1473,plain,
( ( ( dpsetconstrEL1 = $true )
| ? [X0: $i,X1: $i,X2: $i > $i > $o,X3: $i,X4: $i] :
( ( ( in @ ( kpair @ X1 @ X3 )
@ ( dpsetconstr @ X0 @ X4
@ ^ [Y0: $i,Y1: $i] : ( X2 @ Y0 @ Y1 ) ) )
= $true )
& ( ( in @ X1 @ X0 )
!= $true ) ) )
& ( ! [X0: $i,X1: $i,X2: $i > $i > $o,X3: $i,X4: $i] :
( ( ( in @ ( kpair @ X1 @ X3 )
@ ( dpsetconstr @ X0 @ X4
@ ^ [Y0: $i,Y1: $i] : ( X2 @ Y0 @ Y1 ) ) )
!= $true )
| ( ( in @ X1 @ X0 )
= $true ) )
| ( dpsetconstrEL1 != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f708]) ).
thf(f708,plain,
( ( dpsetconstrEL1 = $true )
<=> ! [X0: $i,X1: $i,X2: $i > $i > $o,X3: $i,X4: $i] :
( ( ( in @ ( kpair @ X1 @ X3 )
@ ( dpsetconstr @ X0 @ X4
@ ^ [Y0: $i,Y1: $i] : ( X2 @ Y0 @ Y1 ) ) )
!= $true )
| ( ( in @ X1 @ X0 )
= $true ) ) ),
inference(ennf_transformation,[status(thm)],[f324]) ).
thf(f324,plain,
( ! [X0: $i,X1: $i,X2: $i > $i > $o,X3: $i,X4: $i] :
( ( ( in @ ( kpair @ X1 @ X3 )
@ ( dpsetconstr @ X0 @ X4
@ ^ [Y0: $i,Y1: $i] : ( X2 @ Y0 @ Y1 ) ) )
= $true )
=> ( ( in @ X1 @ X0 )
= $true ) )
<=> ( dpsetconstrEL1 = $true ) ),
inference(fool_elimination,[status(thm)],[f323]) ).
thf(f323,plain,
! [X0: $i,X1: $i,X2: $i > $i > $o,X3: $i,X4: $i] :
( ( ( in @ ( kpair @ X1 @ X3 )
@ ( dpsetconstr @ X0 @ X4
@ ^ [X5: $i,X6: $i] : ( X2 @ X5 @ X6 ) ) )
=> ( in @ X1 @ X0 ) )
= dpsetconstrEL1 ),
inference(rectify,[status(thm)],[f192]) ).
thf(f192,axiom,
! [X3: $i,X1: $i,X0: $i > $i > $o,X2: $i,X4: $i] :
( ( ( in @ ( kpair @ X1 @ X2 )
@ ( dpsetconstr @ X3 @ X4
@ ^ [X8: $i,X10: $i] : ( X0 @ X8 @ X10 ) ) )
=> ( in @ X1 @ X3 ) )
= dpsetconstrEL1 ),
file('/export/starexec/sandbox/tmp/tmp.0Ah447e165/DTF2THF_7726.p',dpsetconstrEL1) ).
thf(f2741,plain,
( spl540_1
| ~ spl540_2
| ~ spl540_3 ),
inference(avatar_split_clause,[status(thm)],[f2728,f2738,f2734,f2730]) ).
thf(f2728,plain,
( ( ( sK116 @ sK118 @ sK115 )
= $true )
| ( ( in @ sK115 @ sK114 )
!= $true )
| ( ( in @ sK118 @ sK117 )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f2727]) ).
thf(f2727,plain,
( ( $true != $true )
| ( ( sK116 @ sK118 @ sK115 )
= $true )
| ( ( in @ sK118 @ sK117 )
!= $true )
| ( ( in @ sK115 @ sK114 )
!= $true ) ),
inference(forward_demodulation,[status(thm)],[f2717,f1959]) ).
thf(f1959,plain,
dpsetconstrERa = $true,
inference(cnf_transformation,[status(thm)],[f988]) ).
thf(f2717,plain,
( ( ( in @ sK118 @ sK117 )
!= $true )
| ( dpsetconstrERa != $true )
| ( ( in @ sK115 @ sK114 )
!= $true )
| ( ( sK116 @ sK118 @ sK115 )
= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f2715]) ).
thf(f2715,plain,
( ( ( in @ sK115 @ sK114 )
!= $true )
| ( ( in @ sK118 @ sK117 )
!= $true )
| ( dpsetconstrERa != $true )
| ( $true != $true )
| ( ( sK116 @ sK118 @ sK115 )
= $true ) ),
inference(superposition,[status(thm)],[f2640,f2712]) ).
thf(f2640,plain,
! [X8: $i > $i > $o,X6: $i,X9: $i,X7: $i,X5: $i] :
( ( ( in @ ( kpair @ X6 @ X9 ) @ ( dpsetconstr @ X5 @ X7 @ X8 ) )
!= $true )
| ( dpsetconstrERa != $true )
| ( ( X8 @ X6 @ X9 )
= $true )
| ( $true
!= ( in @ X9 @ X7 ) )
| ( ( in @ X6 @ X5 )
!= $true ) ),
inference(beta_eta_normalization,[status(thm)],[f1792]) ).
thf(f1792,plain,
! [X8: $i > $i > $o,X6: $i,X9: $i,X7: $i,X5: $i] :
( ( dpsetconstrERa != $true )
| ( $true
!= ( in @ ( kpair @ X6 @ X9 )
@ ( dpsetconstr @ X5 @ X7
@ ^ [Y0: $i,Y1: $i] : ( X8 @ Y0 @ Y1 ) ) ) )
| ( ( X8 @ X6 @ X9 )
= $true )
| ( $true
!= ( in @ X9 @ X7 ) )
| ( ( in @ X6 @ X5 )
!= $true ) ),
inference(cnf_transformation,[status(thm)],[f961]) ).
thf(f961,plain,
( ( ( dpsetconstrERa = $true )
| ( ( ( sK94 @ sK92 @ sK95 )
!= $true )
& ( ( in @ ( kpair @ sK92 @ sK95 )
@ ( dpsetconstr @ sK91 @ sK93
@ ^ [Y0: $i,Y1: $i] : ( sK94 @ Y0 @ Y1 ) ) )
= $true )
& ( ( in @ sK95 @ sK93 )
= $true )
& ( ( in @ sK92 @ sK91 )
= $true ) ) )
& ( ! [X5: $i,X6: $i,X7: $i,X8: $i > $i > $o] :
( ! [X9: $i] :
( ( ( X8 @ X6 @ X9 )
= $true )
| ( $true
!= ( in @ ( kpair @ X6 @ X9 )
@ ( dpsetconstr @ X5 @ X7
@ ^ [Y0: $i,Y1: $i] : ( X8 @ Y0 @ Y1 ) ) ) )
| ( $true
!= ( in @ X9 @ X7 ) ) )
| ( ( in @ X6 @ X5 )
!= $true ) )
| ( dpsetconstrERa != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK91,sK92,sK93,sK94,sK95])],[f958,f960,f959]) ).
thf(f959,plain,
( ? [X0: $i,X1: $i,X2: $i,X3: $i > $i > $o] :
( ? [X4: $i] :
( ( ( X3 @ X1 @ X4 )
!= $true )
& ( ( in @ ( kpair @ X1 @ X4 )
@ ( dpsetconstr @ X0 @ X2
@ ^ [Y0: $i,Y1: $i] : ( X3 @ Y0 @ Y1 ) ) )
= $true )
& ( $true
= ( in @ X4 @ X2 ) ) )
& ( ( in @ X1 @ X0 )
= $true ) )
=> ( ? [X4: $i] :
( ( ( sK94 @ sK92 @ X4 )
!= $true )
& ( ( in @ ( kpair @ sK92 @ X4 )
@ ( dpsetconstr @ sK91 @ sK93
@ ^ [Y0: $i,Y1: $i] : ( sK94 @ Y0 @ Y1 ) ) )
= $true )
& ( ( in @ X4 @ sK93 )
= $true ) )
& ( ( in @ sK92 @ sK91 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f960,plain,
( ? [X4: $i] :
( ( ( sK94 @ sK92 @ X4 )
!= $true )
& ( ( in @ ( kpair @ sK92 @ X4 )
@ ( dpsetconstr @ sK91 @ sK93
@ ^ [Y0: $i,Y1: $i] : ( sK94 @ Y0 @ Y1 ) ) )
= $true )
& ( ( in @ X4 @ sK93 )
= $true ) )
=> ( ( ( sK94 @ sK92 @ sK95 )
!= $true )
& ( ( in @ ( kpair @ sK92 @ sK95 )
@ ( dpsetconstr @ sK91 @ sK93
@ ^ [Y0: $i,Y1: $i] : ( sK94 @ Y0 @ Y1 ) ) )
= $true )
& ( ( in @ sK95 @ sK93 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f958,plain,
( ( ( dpsetconstrERa = $true )
| ? [X0: $i,X1: $i,X2: $i,X3: $i > $i > $o] :
( ? [X4: $i] :
( ( ( X3 @ X1 @ X4 )
!= $true )
& ( ( in @ ( kpair @ X1 @ X4 )
@ ( dpsetconstr @ X0 @ X2
@ ^ [Y0: $i,Y1: $i] : ( X3 @ Y0 @ Y1 ) ) )
= $true )
& ( $true
= ( in @ X4 @ X2 ) ) )
& ( ( in @ X1 @ X0 )
= $true ) ) )
& ( ! [X5: $i,X6: $i,X7: $i,X8: $i > $i > $o] :
( ! [X9: $i] :
( ( ( X8 @ X6 @ X9 )
= $true )
| ( $true
!= ( in @ ( kpair @ X6 @ X9 )
@ ( dpsetconstr @ X5 @ X7
@ ^ [Y0: $i,Y1: $i] : ( X8 @ Y0 @ Y1 ) ) ) )
| ( $true
!= ( in @ X9 @ X7 ) ) )
| ( ( in @ X6 @ X5 )
!= $true ) )
| ( dpsetconstrERa != $true ) ) ),
inference(rectify,[status(thm)],[f957]) ).
thf(f957,plain,
( ( ( dpsetconstrERa = $true )
| ? [X0: $i,X1: $i,X2: $i,X3: $i > $i > $o] :
( ? [X4: $i] :
( ( ( X3 @ X1 @ X4 )
!= $true )
& ( ( in @ ( kpair @ X1 @ X4 )
@ ( dpsetconstr @ X0 @ X2
@ ^ [Y0: $i,Y1: $i] : ( X3 @ Y0 @ Y1 ) ) )
= $true )
& ( $true
= ( in @ X4 @ X2 ) ) )
& ( ( in @ X1 @ X0 )
= $true ) ) )
& ( ! [X0: $i,X1: $i,X2: $i,X3: $i > $i > $o] :
( ! [X4: $i] :
( ( ( X3 @ X1 @ X4 )
= $true )
| ( ( in @ ( kpair @ X1 @ X4 )
@ ( dpsetconstr @ X0 @ X2
@ ^ [Y0: $i,Y1: $i] : ( X3 @ Y0 @ Y1 ) ) )
!= $true )
| ( $true
!= ( in @ X4 @ X2 ) ) )
| ( ( in @ X1 @ X0 )
!= $true ) )
| ( dpsetconstrERa != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f757]) ).
thf(f757,plain,
( ( dpsetconstrERa = $true )
<=> ! [X0: $i,X1: $i,X2: $i,X3: $i > $i > $o] :
( ! [X4: $i] :
( ( ( X3 @ X1 @ X4 )
= $true )
| ( ( in @ ( kpair @ X1 @ X4 )
@ ( dpsetconstr @ X0 @ X2
@ ^ [Y0: $i,Y1: $i] : ( X3 @ Y0 @ Y1 ) ) )
!= $true )
| ( $true
!= ( in @ X4 @ X2 ) ) )
| ( ( in @ X1 @ X0 )
!= $true ) ) ),
inference(flattening,[status(thm)],[f756]) ).
thf(f756,plain,
( ! [X0: $i,X1: $i,X2: $i,X3: $i > $i > $o] :
( ! [X4: $i] :
( ( ( X3 @ X1 @ X4 )
= $true )
| ( ( in @ ( kpair @ X1 @ X4 )
@ ( dpsetconstr @ X0 @ X2
@ ^ [Y0: $i,Y1: $i] : ( X3 @ Y0 @ Y1 ) ) )
!= $true )
| ( $true
!= ( in @ X4 @ X2 ) ) )
| ( ( in @ X1 @ X0 )
!= $true ) )
<=> ( dpsetconstrERa = $true ) ),
inference(ennf_transformation,[status(thm)],[f464]) ).
thf(f464,plain,
( ! [X0: $i,X1: $i,X2: $i,X3: $i > $i > $o] :
( ( ( in @ X1 @ X0 )
= $true )
=> ! [X4: $i] :
( ( $true
= ( in @ X4 @ X2 ) )
=> ( ( ( in @ ( kpair @ X1 @ X4 )
@ ( dpsetconstr @ X0 @ X2
@ ^ [Y0: $i,Y1: $i] : ( X3 @ Y0 @ Y1 ) ) )
= $true )
=> ( ( X3 @ X1 @ X4 )
= $true ) ) ) )
<=> ( dpsetconstrERa = $true ) ),
inference(fool_elimination,[status(thm)],[f463]) ).
thf(f463,plain,
( dpsetconstrERa
= ( ! [X0: $i,X1: $i,X2: $i,X3: $i > $i > $o] :
( ( in @ X1 @ X0 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( in @ ( kpair @ X1 @ X4 )
@ ( dpsetconstr @ X0 @ X2
@ ^ [X5: $i,X6: $i] : ( X3 @ X5 @ X6 ) ) )
=> ( X3 @ X1 @ X4 ) ) ) ) ) ),
inference(rectify,[status(thm)],[f191]) ).
thf(f191,axiom,
( dpsetconstrERa
= ( ! [X3: $i,X1: $i,X4: $i,X0: $i > $i > $o] :
( ( in @ X1 @ X3 )
=> ! [X2: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ ( kpair @ X1 @ X2 )
@ ( dpsetconstr @ X3 @ X4
@ ^ [X8: $i,X10: $i] : ( X0 @ X8 @ X10 ) ) )
=> ( X0 @ X1 @ X2 ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.0Ah447e165/DTF2THF_7726.p',dpsetconstrERa) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : SEU671^1 : 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
% 0.16/0.33 % Computer : n015.cluster.edu
% 0.16/0.33 % Model : x86_64 x86_64
% 0.16/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.33 % Memory : 8042.1875MB
% 0.16/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.33 % CPULimit : 300
% 0.16/0.33 % WCLimit : 300
% 0.16/0.33 % DateTime : Tue Feb 24 12:30:46 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
% 0.31/0.48 ---- Original DTF file ---
% 0.31/0.48 thf(spec,logic,$$dhol).
% 0.31/0.48 %------------------------------------------------------------------------------
% 0.31/0.48 % File : SEU671^1 : TPTP v9.2.1. Released v3.7.0.
% 0.31/0.48 % Domain : Set Theory
% 0.31/0.48 % Problem : Ordered Pairs - Sets of Pairs
% 0.31/0.48 % Version : Especial.
% 0.31/0.48 % English : (! A:i.! B:i.! phi:i>(i>o).! x:i.! y:i.in (kpair x y)
% 0.31/0.48 % (dpsetconstr A B (^ z,u:i.phi z u)) -> phi x y)
% 0.31/0.48
% 0.31/0.48 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.31/0.48 % Source : [Bro08]
% 0.31/0.48 % Names : ZFC173g [Bro08]
% 0.31/0.48
% 0.31/0.48 % Status : Theorem
% 0.31/0.48 % Rating : 0.11 v9.1.0, 0.12 v9.0.0, 0.30 v8.2.0, 0.38 v8.1.0, 0.27 v7.5.0, 0.14 v7.4.0, 0.33 v7.2.0, 0.25 v7.1.0, 0.38 v7.0.0, 0.29 v6.4.0, 0.33 v6.3.0, 0.40 v6.2.0, 0.29 v6.1.0, 0.43 v5.5.0, 0.50 v5.4.0, 0.60 v5.2.0, 0.80 v4.1.0, 1.00 v3.7.0
% 0.31/0.48 % Syntax : Number of formulae : 414 ( 193 unt; 220 typ; 193 def)
% 0.31/0.48 % Number of atoms : 1020 ( 287 equ; 0 cnn)
% 0.31/0.48 % Maximal formula atoms : 188 ( 5 avg)
% 0.31/0.48 % Number of connectives : 1833 ( 47 ~; 7 |; 38 &;1278 @)
% 0.31/0.48 % ( 14 <=>; 449 =>; 0 <=; 0 <~>)
% 0.31/0.48 % Maximal formula depth : 200 ( 2 avg)
% 0.31/0.48 % Number of types : 2 ( 0 usr)
% 0.31/0.48 % Number of type conns : 108 ( 108 >; 0 *; 0 +; 0 <<)
% 0.31/0.48 % Number of symbols : 223 ( 220 usr; 191 con; 0-3 aty)
% 0.31/0.48 % Number of variables : 595 ( 61 ^; 496 !; 38 ?; 595 :)
% 0.31/0.48 % SPC : TH0_THM_EQU_NAR
% 0.31/0.48
% 0.31/0.48 % Comments : http://mathgate.info/detsetitem.php?id=341
% 0.31/0.48 %------------------------------------------------------------------------------
% 0.31/0.48 thf(in_type,type,
% 0.31/0.48 in: $i > $i > $o ).
% 0.31/0.48
% 0.31/0.48 thf(exu_type,type,
% 0.31/0.48 exu: ( $i > $o ) > $o ).
% 0.31/0.48
% 0.31/0.48 thf(exu,definition,
% 0.31/0.48 ( exu
% 0.31/0.48 = ( ^ [Xphi: $i > $o] :
% 0.31/0.48 ? [Xx: $i] :
% 0.31/0.48 ( ( Xphi @ Xx )
% 0.31/0.48 & ! [Xy: $i] :
% 0.31/0.48 ( ( Xphi @ Xy )
% 0.31/0.48 => ( Xx = Xy ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(setextAx_type,type,
% 0.31/0.48 setextAx: $o ).
% 0.31/0.48
% 0.31/0.48 thf(setextAx,definition,
% 0.31/0.48 ( setextAx
% 0.31/0.48 = ( ! [A: $i,B: $i] :
% 0.31/0.48 ( ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 <=> ( in @ Xx @ B ) )
% 0.31/0.48 => ( A = B ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(emptyset_type,type,
% 0.31/0.48 emptyset: $i ).
% 0.31/0.48
% 0.31/0.48 thf(emptysetAx_type,type,
% 0.31/0.48 emptysetAx: $o ).
% 0.31/0.48
% 0.31/0.48 thf(emptysetAx,definition,
% 0.31/0.48 ( emptysetAx
% 0.31/0.48 = ( ! [Xx: $i] :
% 0.31/0.48 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(setadjoin_type,type,
% 0.31/0.48 setadjoin: $i > $i > $i ).
% 0.31/0.48
% 0.31/0.48 thf(setadjoinAx_type,type,
% 0.31/0.48 setadjoinAx: $o ).
% 0.31/0.48
% 0.31/0.48 thf(setadjoinAx,definition,
% 0.31/0.48 ( setadjoinAx
% 0.31/0.48 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.31/0.48 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.31/0.48 <=> ( ( Xy = Xx )
% 0.31/0.48 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(powerset_type,type,
% 0.31/0.48 powerset: $i > $i ).
% 0.31/0.48
% 0.31/0.48 thf(powersetAx_type,type,
% 0.31/0.48 powersetAx: $o ).
% 0.31/0.48
% 0.31/0.48 thf(powersetAx,definition,
% 0.31/0.48 ( powersetAx
% 0.31/0.48 = ( ! [A: $i,B: $i] :
% 0.31/0.48 ( ( in @ B @ ( powerset @ A ) )
% 0.31/0.48 <=> ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ B )
% 0.31/0.48 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(setunion_type,type,
% 0.31/0.48 setunion: $i > $i ).
% 0.31/0.48
% 0.31/0.48 thf(setunionAx_type,type,
% 0.31/0.48 setunionAx: $o ).
% 0.31/0.48
% 0.31/0.48 thf(setunionAx,definition,
% 0.31/0.48 ( setunionAx
% 0.31/0.48 = ( ! [A: $i,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ ( setunion @ A ) )
% 0.31/0.48 <=> ? [B: $i] :
% 0.31/0.48 ( ( in @ Xx @ B )
% 0.31/0.48 & ( in @ B @ A ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(omega_type,type,
% 0.31/0.48 omega: $i ).
% 0.31/0.48
% 0.31/0.48 thf(omega0Ax_type,type,
% 0.31/0.48 omega0Ax: $o ).
% 0.31/0.48
% 0.31/0.48 thf(omega0Ax,definition,
% 0.31/0.48 ( omega0Ax
% 0.31/0.48 = ( in @ emptyset @ omega ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(omegaSAx_type,type,
% 0.31/0.48 omegaSAx: $o ).
% 0.31/0.48
% 0.31/0.48 thf(omegaSAx,definition,
% 0.31/0.48 ( omegaSAx
% 0.31/0.48 = ( ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ omega )
% 0.31/0.48 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(omegaIndAx_type,type,
% 0.31/0.48 omegaIndAx: $o ).
% 0.31/0.48
% 0.31/0.48 thf(omegaIndAx,definition,
% 0.31/0.48 ( omegaIndAx
% 0.31/0.48 = ( ! [A: $i] :
% 0.31/0.48 ( ( ( in @ emptyset @ A )
% 0.31/0.48 & ! [Xx: $i] :
% 0.31/0.48 ( ( ( in @ Xx @ omega )
% 0.31/0.48 & ( in @ Xx @ A ) )
% 0.31/0.48 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.31/0.48 => ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ omega )
% 0.31/0.48 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(replAx_type,type,
% 0.31/0.48 replAx: $o ).
% 0.31/0.48
% 0.31/0.48 thf(replAx,definition,
% 0.31/0.48 ( replAx
% 0.31/0.48 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.31/0.48 ( ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ( exu
% 0.31/0.48 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.31/0.48 => ? [B: $i] :
% 0.31/0.48 ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ B )
% 0.31/0.48 <=> ? [Xy: $i] :
% 0.31/0.48 ( ( in @ Xy @ A )
% 0.31/0.48 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(foundationAx_type,type,
% 0.31/0.48 foundationAx: $o ).
% 0.31/0.48
% 0.31/0.48 thf(foundationAx,definition,
% 0.31/0.48 ( foundationAx
% 0.31/0.48 = ( ! [A: $i] :
% 0.31/0.48 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.31/0.48 => ? [B: $i] :
% 0.31/0.48 ( ( in @ B @ A )
% 0.31/0.48 & ~ ? [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ B )
% 0.31/0.48 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(wellorderingAx_type,type,
% 0.31/0.48 wellorderingAx: $o ).
% 0.31/0.48
% 0.31/0.48 thf(wellorderingAx,definition,
% 0.31/0.48 ( wellorderingAx
% 0.31/0.48 = ( ! [A: $i] :
% 0.31/0.48 ? [B: $i] :
% 0.31/0.48 ( ! [C: $i] :
% 0.31/0.48 ( ( in @ C @ B )
% 0.31/0.48 => ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ C )
% 0.31/0.48 => ( in @ Xx @ A ) ) )
% 0.31/0.48 & ! [Xx: $i,Xy: $i] :
% 0.31/0.48 ( ( ( in @ Xx @ A )
% 0.31/0.48 & ( in @ Xy @ A ) )
% 0.31/0.48 => ( ! [C: $i] :
% 0.31/0.48 ( ( in @ C @ B )
% 0.31/0.48 => ( ( in @ Xx @ C )
% 0.31/0.48 <=> ( in @ Xy @ C ) ) )
% 0.31/0.48 => ( Xx = Xy ) ) )
% 0.31/0.48 & ! [C: $i,D: $i] :
% 0.31/0.48 ( ( ( in @ C @ B )
% 0.31/0.48 & ( in @ D @ B ) )
% 0.31/0.48 => ( ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ C )
% 0.31/0.48 => ( in @ Xx @ D ) )
% 0.31/0.48 | ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ D )
% 0.31/0.48 => ( in @ Xx @ C ) ) ) )
% 0.31/0.48 & ! [C: $i] :
% 0.31/0.48 ( ( ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ C )
% 0.31/0.48 => ( in @ Xx @ A ) )
% 0.31/0.48 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.31/0.48 => ? [D: $i,Xx: $i] :
% 0.31/0.48 ( ( in @ D @ B )
% 0.31/0.48 & ( in @ Xx @ C )
% 0.31/0.48 & ~ ? [Xy: $i] :
% 0.31/0.48 ( ( in @ Xy @ D )
% 0.31/0.48 & ( in @ Xy @ C ) )
% 0.31/0.48 & ! [E: $i] :
% 0.31/0.48 ( ( in @ E @ B )
% 0.31/0.48 => ( ! [Xy: $i] :
% 0.31/0.48 ( ( in @ Xy @ E )
% 0.31/0.48 => ( in @ Xy @ D ) )
% 0.31/0.48 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(descr_type,type,
% 0.31/0.48 descr: ( $i > $o ) > $i ).
% 0.31/0.48
% 0.31/0.48 thf(descrp_type,type,
% 0.31/0.48 descrp: $o ).
% 0.31/0.48
% 0.31/0.48 thf(descrp,definition,
% 0.31/0.48 ( descrp
% 0.31/0.48 = ( ! [Xphi: $i > $o] :
% 0.31/0.48 ( ( exu
% 0.31/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.48 => ( Xphi
% 0.31/0.48 @ ( descr
% 0.31/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(dsetconstr_type,type,
% 0.31/0.48 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.31/0.48
% 0.31/0.48 thf(dsetconstrI_type,type,
% 0.31/0.48 dsetconstrI: $o ).
% 0.31/0.48
% 0.31/0.48 thf(dsetconstrI,definition,
% 0.31/0.48 ( dsetconstrI
% 0.31/0.48 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ( ( Xphi @ Xx )
% 0.31/0.48 => ( in @ Xx
% 0.31/0.48 @ ( dsetconstr @ A
% 0.31/0.48 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(dsetconstrEL_type,type,
% 0.31/0.48 dsetconstrEL: $o ).
% 0.31/0.48
% 0.31/0.48 thf(dsetconstrEL,definition,
% 0.31/0.48 ( dsetconstrEL
% 0.31/0.48 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx
% 0.31/0.48 @ ( dsetconstr @ A
% 0.31/0.48 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.31/0.48 => ( in @ Xx @ A ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(dsetconstrER_type,type,
% 0.31/0.48 dsetconstrER: $o ).
% 0.31/0.48
% 0.31/0.48 thf(dsetconstrER,definition,
% 0.31/0.48 ( dsetconstrER
% 0.31/0.48 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx
% 0.31/0.48 @ ( dsetconstr @ A
% 0.31/0.48 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.31/0.48 => ( Xphi @ Xx ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(exuE1_type,type,
% 0.31/0.48 exuE1: $o ).
% 0.31/0.48
% 0.31/0.48 thf(exuE1,definition,
% 0.31/0.48 ( exuE1
% 0.31/0.48 = ( ! [Xphi: $i > $o] :
% 0.31/0.48 ( ( exu
% 0.31/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.48 => ? [Xx: $i] :
% 0.31/0.48 ( ( Xphi @ Xx )
% 0.31/0.48 & ! [Xy: $i] :
% 0.31/0.48 ( ( Xphi @ Xy )
% 0.31/0.48 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(prop2set_type,type,
% 0.31/0.48 prop2set: $o > $i ).
% 0.31/0.48
% 0.31/0.48 thf(prop2setE_type,type,
% 0.31/0.48 prop2setE: $o ).
% 0.31/0.48
% 0.31/0.48 thf(prop2setE,definition,
% 0.31/0.48 ( prop2setE
% 0.31/0.48 = ( ! [Xphi: $o,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.31/0.48 => Xphi ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(emptysetE_type,type,
% 0.31/0.48 emptysetE: $o ).
% 0.31/0.48
% 0.31/0.48 thf(emptysetE,definition,
% 0.31/0.48 ( emptysetE
% 0.31/0.48 = ( ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ emptyset )
% 0.31/0.48 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(emptysetimpfalse_type,type,
% 0.31/0.48 emptysetimpfalse: $o ).
% 0.31/0.48
% 0.31/0.48 thf(emptysetimpfalse,definition,
% 0.31/0.48 ( emptysetimpfalse
% 0.31/0.48 = ( ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ emptyset )
% 0.31/0.48 => $false ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(notinemptyset_type,type,
% 0.31/0.48 notinemptyset: $o ).
% 0.31/0.48
% 0.31/0.48 thf(notinemptyset,definition,
% 0.31/0.48 ( notinemptyset
% 0.31/0.48 = ( ! [Xx: $i] :
% 0.31/0.48 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(exuE3e_type,type,
% 0.31/0.48 exuE3e: $o ).
% 0.31/0.48
% 0.31/0.48 thf(exuE3e,definition,
% 0.31/0.48 ( exuE3e
% 0.31/0.48 = ( ! [Xphi: $i > $o] :
% 0.31/0.48 ( ( exu
% 0.31/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.48 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(setext_type,type,
% 0.31/0.48 setext: $o ).
% 0.31/0.48
% 0.31/0.48 thf(setext,definition,
% 0.31/0.48 ( setext
% 0.31/0.48 = ( ! [A: $i,B: $i] :
% 0.31/0.48 ( ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ( in @ Xx @ B ) )
% 0.31/0.48 => ( ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ B )
% 0.31/0.48 => ( in @ Xx @ A ) )
% 0.31/0.48 => ( A = B ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(emptyI_type,type,
% 0.31/0.48 emptyI: $o ).
% 0.31/0.48
% 0.31/0.48 thf(emptyI,definition,
% 0.31/0.48 ( emptyI
% 0.31/0.48 = ( ! [A: $i] :
% 0.31/0.48 ( ! [Xx: $i] :
% 0.31/0.48 ~ ( in @ Xx @ A )
% 0.31/0.48 => ( A = emptyset ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(noeltsimpempty_type,type,
% 0.31/0.48 noeltsimpempty: $o ).
% 0.31/0.48
% 0.31/0.48 thf(noeltsimpempty,definition,
% 0.31/0.48 ( noeltsimpempty
% 0.31/0.48 = ( ! [A: $i] :
% 0.31/0.48 ( ! [Xx: $i] :
% 0.31/0.48 ~ ( in @ Xx @ A )
% 0.31/0.48 => ( A = emptyset ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(setbeta_type,type,
% 0.31/0.48 setbeta: $o ).
% 0.31/0.48
% 0.31/0.48 thf(setbeta,definition,
% 0.31/0.48 ( setbeta
% 0.31/0.48 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ( ( in @ Xx
% 0.31/0.48 @ ( dsetconstr @ A
% 0.31/0.48 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.31/0.48 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(nonempty_type,type,
% 0.31/0.48 nonempty: $i > $o ).
% 0.31/0.48
% 0.31/0.48 thf(nonempty,definition,
% 0.31/0.48 ( nonempty
% 0.31/0.48 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(nonemptyE1_type,type,
% 0.31/0.48 nonemptyE1: $o ).
% 0.31/0.48
% 0.31/0.48 thf(nonemptyE1,definition,
% 0.31/0.48 ( nonemptyE1
% 0.31/0.48 = ( ! [A: $i] :
% 0.31/0.48 ( ( nonempty @ A )
% 0.31/0.48 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(nonemptyI_type,type,
% 0.31/0.48 nonemptyI: $o ).
% 0.31/0.48
% 0.31/0.48 thf(nonemptyI,definition,
% 0.31/0.48 ( nonemptyI
% 0.31/0.48 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ( ( Xphi @ Xx )
% 0.31/0.48 => ( nonempty
% 0.31/0.48 @ ( dsetconstr @ A
% 0.31/0.48 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(nonemptyI1_type,type,
% 0.31/0.48 nonemptyI1: $o ).
% 0.31/0.48
% 0.31/0.48 thf(nonemptyI1,definition,
% 0.31/0.48 ( nonemptyI1
% 0.31/0.48 = ( ! [A: $i] :
% 0.31/0.48 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.31/0.48 => ( nonempty @ A ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(setadjoinIL_type,type,
% 0.31/0.48 setadjoinIL: $o ).
% 0.31/0.48
% 0.31/0.48 thf(setadjoinIL,definition,
% 0.31/0.48 ( setadjoinIL
% 0.31/0.48 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(emptyinunitempty_type,type,
% 0.31/0.48 emptyinunitempty: $o ).
% 0.31/0.48
% 0.31/0.48 thf(emptyinunitempty,definition,
% 0.31/0.48 ( emptyinunitempty
% 0.31/0.48 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(setadjoinIR_type,type,
% 0.31/0.48 setadjoinIR: $o ).
% 0.31/0.48
% 0.31/0.48 thf(setadjoinIR,definition,
% 0.31/0.48 ( setadjoinIR
% 0.31/0.48 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.31/0.48 ( ( in @ Xy @ A )
% 0.31/0.48 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(setadjoinE_type,type,
% 0.31/0.48 setadjoinE: $o ).
% 0.31/0.48
% 0.31/0.48 thf(setadjoinE,definition,
% 0.31/0.48 ( setadjoinE
% 0.31/0.48 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.31/0.48 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.31/0.48 => ! [Xphi: $o] :
% 0.31/0.48 ( ( ( Xy = Xx )
% 0.31/0.48 => Xphi )
% 0.31/0.48 => ( ( ( in @ Xy @ A )
% 0.31/0.48 => Xphi )
% 0.31/0.48 => Xphi ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(setadjoinOr_type,type,
% 0.31/0.48 setadjoinOr: $o ).
% 0.31/0.48
% 0.31/0.48 thf(setadjoinOr,definition,
% 0.31/0.48 ( setadjoinOr
% 0.31/0.48 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.31/0.48 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.31/0.48 => ( ( Xy = Xx )
% 0.31/0.48 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(setoftrueEq_type,type,
% 0.31/0.48 setoftrueEq: $o ).
% 0.31/0.48
% 0.31/0.48 thf(setoftrueEq,definition,
% 0.31/0.48 ( setoftrueEq
% 0.31/0.48 = ( ! [A: $i] :
% 0.31/0.48 ( ( dsetconstr @ A
% 0.31/0.48 @ ^ [Xx: $i] : $true )
% 0.31/0.48 = A ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(powersetI_type,type,
% 0.31/0.48 powersetI: $o ).
% 0.31/0.48
% 0.31/0.48 thf(powersetI,definition,
% 0.31/0.48 ( powersetI
% 0.31/0.48 = ( ! [A: $i,B: $i] :
% 0.31/0.48 ( ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ B )
% 0.31/0.48 => ( in @ Xx @ A ) )
% 0.31/0.48 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(emptyinPowerset_type,type,
% 0.31/0.48 emptyinPowerset: $o ).
% 0.31/0.48
% 0.31/0.48 thf(emptyinPowerset,definition,
% 0.31/0.48 ( emptyinPowerset
% 0.31/0.48 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(emptyInPowerset_type,type,
% 0.31/0.48 emptyInPowerset: $o ).
% 0.31/0.48
% 0.31/0.48 thf(emptyInPowerset,definition,
% 0.31/0.48 ( emptyInPowerset
% 0.31/0.48 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(powersetE_type,type,
% 0.31/0.48 powersetE: $o ).
% 0.31/0.48
% 0.31/0.48 thf(powersetE,definition,
% 0.31/0.48 ( powersetE
% 0.31/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.48 ( ( in @ B @ ( powerset @ A ) )
% 0.31/0.48 => ( ( in @ Xx @ B )
% 0.31/0.48 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(setunionI_type,type,
% 0.31/0.48 setunionI: $o ).
% 0.31/0.48
% 0.31/0.48 thf(setunionI,definition,
% 0.31/0.48 ( setunionI
% 0.31/0.48 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.31/0.48 ( ( in @ Xx @ B )
% 0.31/0.48 => ( ( in @ B @ A )
% 0.31/0.48 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(setunionE_type,type,
% 0.31/0.48 setunionE: $o ).
% 0.31/0.48
% 0.31/0.48 thf(setunionE,definition,
% 0.31/0.48 ( setunionE
% 0.31/0.48 = ( ! [A: $i,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ ( setunion @ A ) )
% 0.31/0.48 => ! [Xphi: $o] :
% 0.31/0.48 ( ! [B: $i] :
% 0.31/0.48 ( ( in @ Xx @ B )
% 0.31/0.48 => ( ( in @ B @ A )
% 0.31/0.48 => Xphi ) )
% 0.31/0.48 => Xphi ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(subPowSU_type,type,
% 0.31/0.48 subPowSU: $o ).
% 0.31/0.48
% 0.31/0.48 thf(subPowSU,definition,
% 0.31/0.48 ( subPowSU
% 0.31/0.48 = ( ! [A: $i,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(exuE2_type,type,
% 0.31/0.48 exuE2: $o ).
% 0.31/0.48
% 0.31/0.48 thf(exuE2,definition,
% 0.31/0.48 ( exuE2
% 0.31/0.48 = ( ! [Xphi: $i > $o] :
% 0.31/0.48 ( ( exu
% 0.31/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.48 => ? [Xx: $i] :
% 0.31/0.48 ! [Xy: $i] :
% 0.31/0.48 ( ( Xphi @ Xy )
% 0.31/0.48 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(nonemptyImpWitness_type,type,
% 0.31/0.48 nonemptyImpWitness: $o ).
% 0.31/0.48
% 0.31/0.48 thf(nonemptyImpWitness,definition,
% 0.31/0.48 ( nonemptyImpWitness
% 0.31/0.48 = ( ! [A: $i] :
% 0.31/0.48 ( ( nonempty @ A )
% 0.31/0.48 => ? [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 & $true ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(uniqinunit_type,type,
% 0.31/0.48 uniqinunit: $o ).
% 0.31/0.48
% 0.31/0.48 thf(uniqinunit,definition,
% 0.31/0.48 ( uniqinunit
% 0.31/0.48 = ( ! [Xx: $i,Xy: $i] :
% 0.31/0.48 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.31/0.48 => ( Xx = Xy ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(notinsingleton_type,type,
% 0.31/0.48 notinsingleton: $o ).
% 0.31/0.48
% 0.31/0.48 thf(notinsingleton,definition,
% 0.31/0.48 ( notinsingleton
% 0.31/0.48 = ( ! [Xx: $i,Xy: $i] :
% 0.31/0.48 ( ( Xx != Xy )
% 0.31/0.48 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(eqinunit_type,type,
% 0.31/0.48 eqinunit: $o ).
% 0.31/0.48
% 0.31/0.48 thf(eqinunit,definition,
% 0.31/0.48 ( eqinunit
% 0.31/0.48 = ( ! [Xx: $i,Xy: $i] :
% 0.31/0.48 ( ( Xx = Xy )
% 0.31/0.48 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(singletonsswitch_type,type,
% 0.31/0.48 singletonsswitch: $o ).
% 0.31/0.48
% 0.31/0.48 thf(singletonsswitch,definition,
% 0.31/0.48 ( singletonsswitch
% 0.31/0.48 = ( ! [Xx: $i,Xy: $i] :
% 0.31/0.48 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.31/0.48 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(upairsetE_type,type,
% 0.31/0.48 upairsetE: $o ).
% 0.31/0.48
% 0.31/0.48 thf(upairsetE,definition,
% 0.31/0.48 ( upairsetE
% 0.31/0.48 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.31/0.48 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.31/0.48 => ( ( Xz = Xx )
% 0.31/0.48 | ( Xz = Xy ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(upairsetIL_type,type,
% 0.31/0.48 upairsetIL: $o ).
% 0.31/0.48
% 0.31/0.48 thf(upairsetIL,definition,
% 0.31/0.48 ( upairsetIL
% 0.31/0.48 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(upairsetIR_type,type,
% 0.31/0.48 upairsetIR: $o ).
% 0.31/0.48
% 0.31/0.48 thf(upairsetIR,definition,
% 0.31/0.48 ( upairsetIR
% 0.31/0.48 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(emptyE1_type,type,
% 0.31/0.48 emptyE1: $o ).
% 0.31/0.48
% 0.31/0.48 thf(emptyE1,definition,
% 0.31/0.48 ( emptyE1
% 0.31/0.48 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/0.48 ( ? [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 & ( Xphi @ Xx ) )
% 0.31/0.48 => ( ( ( dsetconstr @ A
% 0.31/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.48 = emptyset )
% 0.31/0.48 => $false ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(vacuousDall_type,type,
% 0.31/0.48 vacuousDall: $o ).
% 0.31/0.48
% 0.31/0.48 thf(vacuousDall,definition,
% 0.31/0.48 ( vacuousDall
% 0.31/0.48 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ emptyset )
% 0.31/0.48 => ( Xphi @ Xx ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(quantDeMorgan1_type,type,
% 0.31/0.48 quantDeMorgan1: $o ).
% 0.31/0.48
% 0.31/0.48 thf(quantDeMorgan1,definition,
% 0.31/0.48 ( quantDeMorgan1
% 0.31/0.48 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/0.48 ( ~ ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ( Xphi @ Xx ) )
% 0.31/0.48 => ? [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(quantDeMorgan2_type,type,
% 0.31/0.48 quantDeMorgan2: $o ).
% 0.31/0.48
% 0.31/0.48 thf(quantDeMorgan2,definition,
% 0.31/0.48 ( quantDeMorgan2
% 0.31/0.48 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/0.48 ( ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ~ ( Xphi @ Xx ) )
% 0.31/0.48 => ~ ? [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(quantDeMorgan3_type,type,
% 0.31/0.48 quantDeMorgan3: $o ).
% 0.31/0.48
% 0.31/0.48 thf(quantDeMorgan3,definition,
% 0.31/0.48 ( quantDeMorgan3
% 0.31/0.48 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/0.48 ( ~ ? [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 & ( Xphi @ Xx ) )
% 0.31/0.48 => ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(quantDeMorgan4_type,type,
% 0.31/0.48 quantDeMorgan4: $o ).
% 0.31/0.48
% 0.31/0.48 thf(quantDeMorgan4,definition,
% 0.31/0.48 ( quantDeMorgan4
% 0.31/0.48 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/0.48 ( ? [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 & ~ ( Xphi @ Xx ) )
% 0.31/0.48 => ~ ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(prop2setI_type,type,
% 0.31/0.48 prop2setI: $o ).
% 0.31/0.48
% 0.31/0.48 thf(prop2setI,definition,
% 0.31/0.48 ( prop2setI
% 0.31/0.48 = ( ! [Xphi: $o] :
% 0.31/0.48 ( Xphi
% 0.31/0.48 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(set2prop_type,type,
% 0.31/0.48 set2prop: $i > $o ).
% 0.31/0.48
% 0.31/0.48 thf(prop2set2propI_type,type,
% 0.31/0.48 prop2set2propI: $o ).
% 0.31/0.48
% 0.31/0.48 thf(prop2set2propI,definition,
% 0.31/0.48 ( prop2set2propI
% 0.31/0.48 = ( ! [Xphi: $o] :
% 0.31/0.48 ( Xphi
% 0.31/0.48 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(notdexE_type,type,
% 0.31/0.48 notdexE: $o ).
% 0.31/0.48
% 0.31/0.48 thf(notdexE,definition,
% 0.31/0.48 ( notdexE
% 0.31/0.48 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/0.48 ( ~ ? [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 & ( Xphi @ Xx ) )
% 0.31/0.48 => ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(notdallE_type,type,
% 0.31/0.48 notdallE: $o ).
% 0.31/0.48
% 0.31/0.48 thf(notdallE,definition,
% 0.31/0.48 ( notdallE
% 0.31/0.48 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/0.48 ( ~ ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ( Xphi @ Xx ) )
% 0.31/0.48 => ? [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(exuI1_type,type,
% 0.31/0.48 exuI1: $o ).
% 0.31/0.48
% 0.31/0.48 thf(exuI1,definition,
% 0.31/0.48 ( exuI1
% 0.31/0.48 = ( ! [Xphi: $i > $o] :
% 0.31/0.48 ( ? [Xx: $i] :
% 0.31/0.48 ( ( Xphi @ Xx )
% 0.31/0.48 & ! [Xy: $i] :
% 0.31/0.48 ( ( Xphi @ Xy )
% 0.31/0.48 => ( Xx = Xy ) ) )
% 0.31/0.48 => ( exu
% 0.31/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(exuI3_type,type,
% 0.31/0.48 exuI3: $o ).
% 0.31/0.48
% 0.31/0.48 thf(exuI3,definition,
% 0.31/0.48 ( exuI3
% 0.31/0.48 = ( ! [Xphi: $i > $o] :
% 0.31/0.48 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.31/0.48 => ( ! [Xx: $i,Xy: $i] :
% 0.31/0.48 ( ( Xphi @ Xx )
% 0.31/0.48 => ( ( Xphi @ Xy )
% 0.31/0.48 => ( Xx = Xy ) ) )
% 0.31/0.48 => ( exu
% 0.31/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(exuI2_type,type,
% 0.31/0.48 exuI2: $o ).
% 0.31/0.48
% 0.31/0.48 thf(exuI2,definition,
% 0.31/0.48 ( exuI2
% 0.31/0.48 = ( ! [Xphi: $i > $o] :
% 0.31/0.48 ( ? [Xx: $i] :
% 0.31/0.48 ! [Xy: $i] :
% 0.31/0.48 ( ( Xphi @ Xy )
% 0.31/0.48 <=> ( Xy = Xx ) )
% 0.31/0.48 => ( exu
% 0.31/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(inCongP_type,type,
% 0.31/0.48 inCongP: $o ).
% 0.31/0.48
% 0.31/0.48 thf(inCongP,definition,
% 0.31/0.48 ( inCongP
% 0.31/0.48 = ( ! [A: $i,B: $i] :
% 0.31/0.48 ( ( A = B )
% 0.31/0.48 => ! [Xx: $i,Xy: $i] :
% 0.31/0.48 ( ( Xx = Xy )
% 0.31/0.48 => ( ( in @ Xx @ A )
% 0.31/0.48 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(in__Cong_type,type,
% 0.31/0.48 in__Cong: $o ).
% 0.31/0.48
% 0.31/0.48 thf(in__Cong,definition,
% 0.31/0.48 ( in__Cong
% 0.31/0.48 = ( ! [A: $i,B: $i] :
% 0.31/0.48 ( ( A = B )
% 0.31/0.48 => ! [Xx: $i,Xy: $i] :
% 0.31/0.48 ( ( Xx = Xy )
% 0.31/0.48 => ( ( in @ Xx @ A )
% 0.31/0.48 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(exuE3u_type,type,
% 0.31/0.48 exuE3u: $o ).
% 0.31/0.48
% 0.31/0.48 thf(exuE3u,definition,
% 0.31/0.48 ( exuE3u
% 0.31/0.48 = ( ! [Xphi: $i > $o] :
% 0.31/0.48 ( ( exu
% 0.31/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.48 => ! [Xx: $i,Xy: $i] :
% 0.31/0.48 ( ( Xphi @ Xx )
% 0.31/0.48 => ( ( Xphi @ Xy )
% 0.31/0.48 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(exu__Cong_type,type,
% 0.31/0.48 exu__Cong: $o ).
% 0.31/0.48
% 0.31/0.48 thf(exu__Cong,definition,
% 0.31/0.48 ( exu__Cong
% 0.31/0.48 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.31/0.48 ( ! [Xx: $i,Xy: $i] :
% 0.31/0.48 ( ( Xx = Xy )
% 0.31/0.48 => ( ( Xphi @ Xx )
% 0.31/0.48 <=> ( Xpsi @ Xy ) ) )
% 0.31/0.48 => ( ( exu
% 0.31/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.48 <=> ( exu
% 0.31/0.48 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(emptyset__Cong_type,type,
% 0.31/0.48 emptyset__Cong: $o ).
% 0.31/0.48
% 0.31/0.48 thf(emptyset__Cong,definition,
% 0.31/0.48 ( emptyset__Cong
% 0.31/0.48 = ( emptyset = emptyset ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(setadjoin__Cong_type,type,
% 0.31/0.48 setadjoin__Cong: $o ).
% 0.31/0.48
% 0.31/0.48 thf(setadjoin__Cong,definition,
% 0.31/0.48 ( setadjoin__Cong
% 0.31/0.48 = ( ! [Xx: $i,Xy: $i] :
% 0.31/0.48 ( ( Xx = Xy )
% 0.31/0.48 => ! [Xz: $i,Xu: $i] :
% 0.31/0.48 ( ( Xz = Xu )
% 0.31/0.48 => ( ( setadjoin @ Xx @ Xz )
% 0.31/0.48 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(powerset__Cong_type,type,
% 0.31/0.48 powerset__Cong: $o ).
% 0.31/0.48
% 0.31/0.48 thf(powerset__Cong,definition,
% 0.31/0.48 ( powerset__Cong
% 0.31/0.48 = ( ! [A: $i,B: $i] :
% 0.31/0.48 ( ( A = B )
% 0.31/0.48 => ( ( powerset @ A )
% 0.31/0.48 = ( powerset @ B ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(setunion__Cong_type,type,
% 0.31/0.48 setunion__Cong: $o ).
% 0.31/0.48
% 0.31/0.48 thf(setunion__Cong,definition,
% 0.31/0.48 ( setunion__Cong
% 0.31/0.48 = ( ! [A: $i,B: $i] :
% 0.31/0.48 ( ( A = B )
% 0.31/0.48 => ( ( setunion @ A )
% 0.31/0.48 = ( setunion @ B ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(omega__Cong_type,type,
% 0.31/0.48 omega__Cong: $o ).
% 0.31/0.48
% 0.31/0.48 thf(omega__Cong,definition,
% 0.31/0.48 ( omega__Cong
% 0.31/0.48 = ( omega = omega ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(exuEu_type,type,
% 0.31/0.48 exuEu: $o ).
% 0.31/0.48
% 0.31/0.48 thf(exuEu,definition,
% 0.31/0.48 ( exuEu
% 0.31/0.48 = ( ! [Xphi: $i > $o] :
% 0.31/0.48 ( ( exu
% 0.31/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.48 => ! [Xx: $i,Xy: $i] :
% 0.31/0.48 ( ( Xphi @ Xx )
% 0.31/0.48 => ( ( Xphi @ Xy )
% 0.31/0.48 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(descr__Cong_type,type,
% 0.31/0.48 descr__Cong: $o ).
% 0.31/0.48
% 0.31/0.48 thf(descr__Cong,definition,
% 0.31/0.48 ( descr__Cong
% 0.31/0.48 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.31/0.48 ( ! [Xx: $i,Xy: $i] :
% 0.31/0.48 ( ( Xx = Xy )
% 0.31/0.48 => ( ( Xphi @ Xx )
% 0.31/0.48 <=> ( Xpsi @ Xy ) ) )
% 0.31/0.48 => ( ( exu
% 0.31/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.48 => ( ( exu
% 0.31/0.48 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.31/0.48 => ( ( descr
% 0.31/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.48 = ( descr
% 0.31/0.48 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(dsetconstr__Cong_type,type,
% 0.31/0.48 dsetconstr__Cong: $o ).
% 0.31/0.48
% 0.31/0.48 thf(dsetconstr__Cong,definition,
% 0.31/0.48 ( dsetconstr__Cong
% 0.31/0.48 = ( ! [A: $i,B: $i] :
% 0.31/0.48 ( ( A = B )
% 0.31/0.48 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.31/0.48 ( ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ! [Xy: $i] :
% 0.31/0.48 ( ( in @ Xy @ B )
% 0.31/0.48 => ( ( Xx = Xy )
% 0.31/0.48 => ( ( Xphi @ Xx )
% 0.31/0.48 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.31/0.48 => ( ( dsetconstr @ A
% 0.31/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.48 = ( dsetconstr @ B
% 0.31/0.48 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(subset_type,type,
% 0.31/0.48 subset: $i > $i > $o ).
% 0.31/0.48
% 0.31/0.48 thf(disjoint_type,type,
% 0.31/0.48 disjoint: $i > $i > $o ).
% 0.31/0.48
% 0.31/0.48 thf(setsmeet_type,type,
% 0.31/0.48 setsmeet: $i > $i > $o ).
% 0.31/0.48
% 0.31/0.48 thf(subsetI1_type,type,
% 0.31/0.48 subsetI1: $o ).
% 0.31/0.48
% 0.31/0.48 thf(subsetI1,definition,
% 0.31/0.48 ( subsetI1
% 0.31/0.48 = ( ! [A: $i,B: $i] :
% 0.31/0.48 ( ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ( in @ Xx @ B ) )
% 0.31/0.48 => ( subset @ A @ B ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(eqimpsubset2_type,type,
% 0.31/0.48 eqimpsubset2: $o ).
% 0.31/0.48
% 0.31/0.48 thf(eqimpsubset2,definition,
% 0.31/0.48 ( eqimpsubset2
% 0.31/0.48 = ( ! [A: $i,B: $i] :
% 0.31/0.48 ( ( A = B )
% 0.31/0.48 => ( subset @ B @ A ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(eqimpsubset1_type,type,
% 0.31/0.48 eqimpsubset1: $o ).
% 0.31/0.48
% 0.31/0.48 thf(eqimpsubset1,definition,
% 0.31/0.48 ( eqimpsubset1
% 0.31/0.48 = ( ! [A: $i,B: $i] :
% 0.31/0.48 ( ( A = B )
% 0.31/0.48 => ( subset @ A @ B ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(subsetI2_type,type,
% 0.31/0.48 subsetI2: $o ).
% 0.31/0.48
% 0.31/0.48 thf(subsetI2,definition,
% 0.31/0.48 ( subsetI2
% 0.31/0.48 = ( ! [A: $i,B: $i] :
% 0.31/0.48 ( ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ( in @ Xx @ B ) )
% 0.31/0.48 => ( subset @ A @ B ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(emptysetsubset_type,type,
% 0.31/0.48 emptysetsubset: $o ).
% 0.31/0.48
% 0.31/0.48 thf(emptysetsubset,definition,
% 0.31/0.48 ( emptysetsubset
% 0.31/0.48 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(subsetE_type,type,
% 0.31/0.48 subsetE: $o ).
% 0.31/0.48
% 0.31/0.48 thf(subsetE,definition,
% 0.31/0.48 ( subsetE
% 0.31/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.48 ( ( subset @ A @ B )
% 0.31/0.48 => ( ( in @ Xx @ A )
% 0.31/0.48 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(subsetE2_type,type,
% 0.31/0.48 subsetE2: $o ).
% 0.31/0.48
% 0.31/0.48 thf(subsetE2,definition,
% 0.31/0.48 ( subsetE2
% 0.31/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.48 ( ( subset @ A @ B )
% 0.31/0.48 => ( ~ ( in @ Xx @ B )
% 0.31/0.48 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(notsubsetI_type,type,
% 0.31/0.48 notsubsetI: $o ).
% 0.31/0.48
% 0.31/0.48 thf(notsubsetI,definition,
% 0.31/0.48 ( notsubsetI
% 0.31/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ( ~ ( in @ Xx @ B )
% 0.31/0.48 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(notequalI1_type,type,
% 0.31/0.48 notequalI1: $o ).
% 0.31/0.48
% 0.31/0.48 thf(notequalI1,definition,
% 0.31/0.48 ( notequalI1
% 0.31/0.48 = ( ! [A: $i,B: $i] :
% 0.31/0.48 ( ~ ( subset @ A @ B )
% 0.31/0.48 => ( A != B ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(notequalI2_type,type,
% 0.31/0.48 notequalI2: $o ).
% 0.31/0.48
% 0.31/0.48 thf(notequalI2,definition,
% 0.31/0.48 ( notequalI2
% 0.31/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ( ~ ( in @ Xx @ B )
% 0.31/0.48 => ( A != B ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(subsetRefl_type,type,
% 0.31/0.48 subsetRefl: $o ).
% 0.31/0.48
% 0.31/0.48 thf(subsetRefl,definition,
% 0.31/0.48 ( subsetRefl
% 0.31/0.48 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(subsetTrans_type,type,
% 0.31/0.48 subsetTrans: $o ).
% 0.31/0.48
% 0.31/0.48 thf(subsetTrans,definition,
% 0.31/0.48 ( subsetTrans
% 0.31/0.48 = ( ! [A: $i,B: $i,C: $i] :
% 0.31/0.48 ( ( subset @ A @ B )
% 0.31/0.48 => ( ( subset @ B @ C )
% 0.31/0.48 => ( subset @ A @ C ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(setadjoinSub_type,type,
% 0.31/0.48 setadjoinSub: $o ).
% 0.31/0.48
% 0.31/0.48 thf(setadjoinSub,definition,
% 0.31/0.48 ( setadjoinSub
% 0.31/0.48 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(setadjoinSub2_type,type,
% 0.31/0.48 setadjoinSub2: $o ).
% 0.31/0.48
% 0.31/0.48 thf(setadjoinSub2,definition,
% 0.31/0.48 ( setadjoinSub2
% 0.31/0.48 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.31/0.48 ( ( subset @ A @ B )
% 0.31/0.48 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(subset2powerset_type,type,
% 0.31/0.48 subset2powerset: $o ).
% 0.31/0.48
% 0.31/0.48 thf(subset2powerset,definition,
% 0.31/0.48 ( subset2powerset
% 0.31/0.48 = ( ! [A: $i,B: $i] :
% 0.31/0.48 ( ( subset @ A @ B )
% 0.31/0.48 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(setextsub_type,type,
% 0.31/0.48 setextsub: $o ).
% 0.31/0.48
% 0.31/0.48 thf(setextsub,definition,
% 0.31/0.48 ( setextsub
% 0.31/0.48 = ( ! [A: $i,B: $i] :
% 0.31/0.48 ( ( subset @ A @ B )
% 0.31/0.48 => ( ( subset @ B @ A )
% 0.31/0.48 => ( A = B ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(subsetemptysetimpeq_type,type,
% 0.31/0.48 subsetemptysetimpeq: $o ).
% 0.31/0.48
% 0.31/0.48 thf(subsetemptysetimpeq,definition,
% 0.31/0.48 ( subsetemptysetimpeq
% 0.31/0.48 = ( ! [A: $i] :
% 0.31/0.48 ( ( subset @ A @ emptyset )
% 0.31/0.48 => ( A = emptyset ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(powersetI1_type,type,
% 0.31/0.48 powersetI1: $o ).
% 0.31/0.48
% 0.31/0.48 thf(powersetI1,definition,
% 0.31/0.48 ( powersetI1
% 0.31/0.48 = ( ! [A: $i,B: $i] :
% 0.31/0.48 ( ( subset @ B @ A )
% 0.31/0.48 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(powersetE1_type,type,
% 0.31/0.48 powersetE1: $o ).
% 0.31/0.48
% 0.31/0.48 thf(powersetE1,definition,
% 0.31/0.48 ( powersetE1
% 0.31/0.48 = ( ! [A: $i,B: $i] :
% 0.31/0.48 ( ( in @ B @ ( powerset @ A ) )
% 0.31/0.48 => ( subset @ B @ A ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(inPowerset_type,type,
% 0.31/0.48 inPowerset: $o ).
% 0.31/0.48
% 0.31/0.48 thf(inPowerset,definition,
% 0.31/0.48 ( inPowerset
% 0.31/0.48 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(powersetsubset_type,type,
% 0.31/0.48 powersetsubset: $o ).
% 0.31/0.48
% 0.31/0.48 thf(powersetsubset,definition,
% 0.31/0.48 ( powersetsubset
% 0.31/0.48 = ( ! [A: $i,B: $i] :
% 0.31/0.48 ( ( subset @ A @ B )
% 0.31/0.48 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(sepInPowerset_type,type,
% 0.31/0.48 sepInPowerset: $o ).
% 0.31/0.48
% 0.31/0.48 thf(sepInPowerset,definition,
% 0.31/0.48 ( sepInPowerset
% 0.31/0.48 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/0.48 ( in
% 0.31/0.48 @ ( dsetconstr @ A
% 0.31/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.48 @ ( powerset @ A ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(sepSubset_type,type,
% 0.31/0.48 sepSubset: $o ).
% 0.31/0.48
% 0.31/0.48 thf(sepSubset,definition,
% 0.31/0.48 ( sepSubset
% 0.31/0.48 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/0.48 ( subset
% 0.31/0.48 @ ( dsetconstr @ A
% 0.31/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.48 @ A ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(binunion_type,type,
% 0.31/0.48 binunion: $i > $i > $i ).
% 0.31/0.48
% 0.31/0.48 thf(binunionIL_type,type,
% 0.31/0.48 binunionIL: $o ).
% 0.31/0.48
% 0.31/0.48 thf(binunionIL,definition,
% 0.31/0.48 ( binunionIL
% 0.31/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(upairset2IR_type,type,
% 0.31/0.48 upairset2IR: $o ).
% 0.31/0.48
% 0.31/0.48 thf(upairset2IR,definition,
% 0.31/0.48 ( upairset2IR
% 0.31/0.48 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(binunionIR_type,type,
% 0.31/0.48 binunionIR: $o ).
% 0.31/0.48
% 0.31/0.48 thf(binunionIR,definition,
% 0.31/0.48 ( binunionIR
% 0.31/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ B )
% 0.31/0.48 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(binunionEcases_type,type,
% 0.31/0.48 binunionEcases: $o ).
% 0.31/0.48
% 0.31/0.48 thf(binunionEcases,definition,
% 0.31/0.48 ( binunionEcases
% 0.31/0.48 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.31/0.48 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.31/0.48 => ( ( ( in @ Xx @ A )
% 0.31/0.48 => Xphi )
% 0.31/0.48 => ( ( ( in @ Xx @ B )
% 0.31/0.48 => Xphi )
% 0.31/0.48 => Xphi ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(binunionE_type,type,
% 0.31/0.48 binunionE: $o ).
% 0.31/0.48
% 0.31/0.48 thf(binunionE,definition,
% 0.31/0.48 ( binunionE
% 0.31/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.31/0.48 => ( ( in @ Xx @ A )
% 0.31/0.48 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(binunionLsub_type,type,
% 0.31/0.48 binunionLsub: $o ).
% 0.31/0.48
% 0.31/0.48 thf(binunionLsub,definition,
% 0.31/0.48 ( binunionLsub
% 0.31/0.48 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(binunionRsub_type,type,
% 0.31/0.48 binunionRsub: $o ).
% 0.31/0.48
% 0.31/0.48 thf(binunionRsub,definition,
% 0.31/0.48 ( binunionRsub
% 0.31/0.48 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(binintersect_type,type,
% 0.31/0.48 binintersect: $i > $i > $i ).
% 0.31/0.48
% 0.31/0.48 thf(binintersectI_type,type,
% 0.31/0.48 binintersectI: $o ).
% 0.31/0.48
% 0.31/0.48 thf(binintersectI,definition,
% 0.31/0.48 ( binintersectI
% 0.31/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ( ( in @ Xx @ B )
% 0.31/0.48 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(binintersectSubset5_type,type,
% 0.31/0.48 binintersectSubset5: $o ).
% 0.31/0.48
% 0.31/0.48 thf(binintersectSubset5,definition,
% 0.31/0.48 ( binintersectSubset5
% 0.31/0.48 = ( ! [A: $i,B: $i,C: $i] :
% 0.31/0.48 ( ( subset @ C @ A )
% 0.31/0.48 => ( ( subset @ C @ B )
% 0.31/0.48 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(binintersectEL_type,type,
% 0.31/0.48 binintersectEL: $o ).
% 0.31/0.48
% 0.31/0.48 thf(binintersectEL,definition,
% 0.31/0.48 ( binintersectEL
% 0.31/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.31/0.48 => ( in @ Xx @ A ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(binintersectLsub_type,type,
% 0.31/0.48 binintersectLsub: $o ).
% 0.31/0.48
% 0.31/0.48 thf(binintersectLsub,definition,
% 0.31/0.48 ( binintersectLsub
% 0.31/0.48 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(binintersectSubset2_type,type,
% 0.31/0.48 binintersectSubset2: $o ).
% 0.31/0.48
% 0.31/0.48 thf(binintersectSubset2,definition,
% 0.31/0.48 ( binintersectSubset2
% 0.31/0.48 = ( ! [A: $i,B: $i] :
% 0.31/0.48 ( ( subset @ A @ B )
% 0.31/0.48 => ( ( binintersect @ A @ B )
% 0.31/0.48 = A ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(binintersectSubset3_type,type,
% 0.31/0.48 binintersectSubset3: $o ).
% 0.31/0.48
% 0.31/0.48 thf(binintersectSubset3,definition,
% 0.31/0.48 ( binintersectSubset3
% 0.31/0.48 = ( ! [A: $i,B: $i] :
% 0.31/0.48 ( ( ( binintersect @ A @ B )
% 0.31/0.48 = B )
% 0.31/0.48 => ( subset @ B @ A ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(binintersectER_type,type,
% 0.31/0.48 binintersectER: $o ).
% 0.31/0.48
% 0.31/0.48 thf(binintersectER,definition,
% 0.31/0.48 ( binintersectER
% 0.31/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.31/0.48 => ( in @ Xx @ B ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(disjointsetsI1_type,type,
% 0.31/0.48 disjointsetsI1: $o ).
% 0.31/0.48
% 0.31/0.48 thf(disjointsetsI1,definition,
% 0.31/0.48 ( disjointsetsI1
% 0.31/0.48 = ( ! [A: $i,B: $i] :
% 0.31/0.48 ( ~ ? [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 & ( in @ Xx @ B ) )
% 0.31/0.48 => ( ( binintersect @ A @ B )
% 0.31/0.48 = emptyset ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(binintersectRsub_type,type,
% 0.31/0.48 binintersectRsub: $o ).
% 0.31/0.48
% 0.31/0.48 thf(binintersectRsub,definition,
% 0.31/0.48 ( binintersectRsub
% 0.31/0.48 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(binintersectSubset4_type,type,
% 0.31/0.48 binintersectSubset4: $o ).
% 0.31/0.48
% 0.31/0.48 thf(binintersectSubset4,definition,
% 0.31/0.48 ( binintersectSubset4
% 0.31/0.48 = ( ! [A: $i,B: $i] :
% 0.31/0.48 ( ( subset @ B @ A )
% 0.31/0.48 => ( ( binintersect @ A @ B )
% 0.31/0.48 = B ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(binintersectSubset1_type,type,
% 0.31/0.48 binintersectSubset1: $o ).
% 0.31/0.48
% 0.31/0.48 thf(binintersectSubset1,definition,
% 0.31/0.48 ( binintersectSubset1
% 0.31/0.48 = ( ! [A: $i,B: $i] :
% 0.31/0.48 ( ( ( binintersect @ A @ B )
% 0.31/0.48 = A )
% 0.31/0.48 => ( subset @ A @ B ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(bs114d_type,type,
% 0.31/0.48 bs114d: $o ).
% 0.31/0.48
% 0.31/0.48 thf(bs114d,definition,
% 0.31/0.48 ( bs114d
% 0.31/0.48 = ( ! [A: $i,B: $i,C: $i] :
% 0.31/0.48 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 0.31/0.48 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(regular_type,type,
% 0.31/0.48 regular: $i > $o ).
% 0.31/0.48
% 0.31/0.48 thf(setminus_type,type,
% 0.31/0.48 setminus: $i > $i > $i ).
% 0.31/0.48
% 0.31/0.48 thf(setminusI_type,type,
% 0.31/0.48 setminusI: $o ).
% 0.31/0.48
% 0.31/0.48 thf(setminusI,definition,
% 0.31/0.48 ( setminusI
% 0.31/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ( ~ ( in @ Xx @ B )
% 0.31/0.48 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(setminusEL_type,type,
% 0.31/0.48 setminusEL: $o ).
% 0.31/0.48
% 0.31/0.48 thf(setminusEL,definition,
% 0.31/0.48 ( setminusEL
% 0.31/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.31/0.48 => ( in @ Xx @ A ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(setminusER_type,type,
% 0.31/0.48 setminusER: $o ).
% 0.31/0.48
% 0.31/0.48 thf(setminusER,definition,
% 0.31/0.48 ( setminusER
% 0.31/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.31/0.48 => ~ ( in @ Xx @ B ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(setminusSubset2_type,type,
% 0.31/0.48 setminusSubset2: $o ).
% 0.31/0.48
% 0.31/0.48 thf(setminusSubset2,definition,
% 0.31/0.48 ( setminusSubset2
% 0.31/0.48 = ( ! [A: $i,B: $i] :
% 0.31/0.48 ( ( subset @ A @ B )
% 0.31/0.48 => ( ( setminus @ A @ B )
% 0.31/0.48 = emptyset ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(setminusERneg_type,type,
% 0.31/0.48 setminusERneg: $o ).
% 0.31/0.48
% 0.31/0.48 thf(setminusERneg,definition,
% 0.31/0.48 ( setminusERneg
% 0.31/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.48 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.31/0.48 => ( ( in @ Xx @ A )
% 0.31/0.48 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(setminusELneg_type,type,
% 0.31/0.48 setminusELneg: $o ).
% 0.31/0.48
% 0.31/0.48 thf(setminusELneg,definition,
% 0.31/0.48 ( setminusELneg
% 0.31/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.48 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.31/0.48 => ( ~ ( in @ Xx @ B )
% 0.31/0.48 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(setminusILneg_type,type,
% 0.31/0.48 setminusILneg: $o ).
% 0.31/0.48
% 0.31/0.48 thf(setminusILneg,definition,
% 0.31/0.48 ( setminusILneg
% 0.31/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.48 ( ~ ( in @ Xx @ A )
% 0.31/0.48 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(setminusIRneg_type,type,
% 0.31/0.48 setminusIRneg: $o ).
% 0.31/0.48
% 0.31/0.48 thf(setminusIRneg,definition,
% 0.31/0.48 ( setminusIRneg
% 0.31/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ B )
% 0.31/0.48 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(setminusLsub_type,type,
% 0.31/0.48 setminusLsub: $o ).
% 0.31/0.48
% 0.31/0.48 thf(setminusLsub,definition,
% 0.31/0.48 ( setminusLsub
% 0.31/0.48 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(setminusSubset1_type,type,
% 0.31/0.48 setminusSubset1: $o ).
% 0.31/0.48
% 0.31/0.48 thf(setminusSubset1,definition,
% 0.31/0.48 ( setminusSubset1
% 0.31/0.48 = ( ! [A: $i,B: $i] :
% 0.31/0.48 ( ( ( setminus @ A @ B )
% 0.31/0.48 = emptyset )
% 0.31/0.48 => ( subset @ A @ B ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(symdiff_type,type,
% 0.31/0.48 symdiff: $i > $i > $i ).
% 0.31/0.48
% 0.31/0.48 thf(symdiffE_type,type,
% 0.31/0.48 symdiffE: $o ).
% 0.31/0.48
% 0.31/0.48 thf(symdiffE,definition,
% 0.31/0.48 ( symdiffE
% 0.31/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 0.31/0.48 => ! [Xphi: $o] :
% 0.31/0.48 ( ( ( in @ Xx @ A )
% 0.31/0.48 => ( ~ ( in @ Xx @ B )
% 0.31/0.48 => Xphi ) )
% 0.31/0.48 => ( ( ~ ( in @ Xx @ A )
% 0.31/0.48 => ( ( in @ Xx @ B )
% 0.31/0.48 => Xphi ) )
% 0.31/0.48 => Xphi ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(symdiffI1_type,type,
% 0.31/0.48 symdiffI1: $o ).
% 0.31/0.48
% 0.31/0.48 thf(symdiffI1,definition,
% 0.31/0.48 ( symdiffI1
% 0.31/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ( ~ ( in @ Xx @ B )
% 0.31/0.48 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(symdiffI2_type,type,
% 0.31/0.48 symdiffI2: $o ).
% 0.31/0.48
% 0.31/0.48 thf(symdiffI2,definition,
% 0.31/0.48 ( symdiffI2
% 0.31/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.48 ( ~ ( in @ Xx @ A )
% 0.31/0.48 => ( ( in @ Xx @ B )
% 0.31/0.48 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(symdiffIneg1_type,type,
% 0.31/0.48 symdiffIneg1: $o ).
% 0.31/0.48
% 0.31/0.48 thf(symdiffIneg1,definition,
% 0.31/0.48 ( symdiffIneg1
% 0.31/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ( ( in @ Xx @ B )
% 0.31/0.48 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(symdiffIneg2_type,type,
% 0.31/0.48 symdiffIneg2: $o ).
% 0.31/0.48
% 0.31/0.48 thf(symdiffIneg2,definition,
% 0.31/0.48 ( symdiffIneg2
% 0.31/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.48 ( ~ ( in @ Xx @ A )
% 0.31/0.48 => ( ~ ( in @ Xx @ B )
% 0.31/0.48 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(iskpair_type,type,
% 0.31/0.48 iskpair: $i > $o ).
% 0.31/0.48
% 0.31/0.48 thf(secondinupair_type,type,
% 0.31/0.48 secondinupair: $o ).
% 0.31/0.48
% 0.31/0.48 thf(secondinupair,definition,
% 0.31/0.48 ( secondinupair
% 0.31/0.48 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(setukpairIL_type,type,
% 0.31/0.48 setukpairIL: $o ).
% 0.31/0.48
% 0.31/0.48 thf(setukpairIL,definition,
% 0.31/0.48 ( setukpairIL
% 0.31/0.48 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(setukpairIR_type,type,
% 0.31/0.48 setukpairIR: $o ).
% 0.31/0.48
% 0.31/0.48 thf(setukpairIR,definition,
% 0.31/0.48 ( setukpairIR
% 0.31/0.48 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(kpairiskpair_type,type,
% 0.31/0.48 kpairiskpair: $o ).
% 0.31/0.48
% 0.31/0.48 thf(kpairiskpair,definition,
% 0.31/0.48 ( kpairiskpair
% 0.31/0.48 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(kpair_type,type,
% 0.31/0.48 kpair: $i > $i > $i ).
% 0.31/0.48
% 0.31/0.48 thf(kpairp_type,type,
% 0.31/0.48 kpairp: $o ).
% 0.31/0.48
% 0.31/0.48 thf(kpairp,definition,
% 0.31/0.48 ( kpairp
% 0.31/0.48 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(cartprod_type,type,
% 0.31/0.48 cartprod: $i > $i > $i ).
% 0.31/0.48
% 0.31/0.48 thf(singletonsubset_type,type,
% 0.31/0.48 singletonsubset: $o ).
% 0.31/0.48
% 0.31/0.48 thf(singletonsubset,definition,
% 0.31/0.48 ( singletonsubset
% 0.31/0.48 = ( ! [A: $i,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(singletoninpowerset_type,type,
% 0.31/0.48 singletoninpowerset: $o ).
% 0.31/0.48
% 0.31/0.48 thf(singletoninpowerset,definition,
% 0.31/0.48 ( singletoninpowerset
% 0.31/0.48 = ( ! [A: $i,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(singletoninpowunion_type,type,
% 0.31/0.48 singletoninpowunion: $o ).
% 0.31/0.48
% 0.31/0.48 thf(singletoninpowunion,definition,
% 0.31/0.48 ( singletoninpowunion
% 0.31/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(upairset2E_type,type,
% 0.31/0.48 upairset2E: $o ).
% 0.31/0.48
% 0.31/0.48 thf(upairset2E,definition,
% 0.31/0.48 ( upairset2E
% 0.31/0.48 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.31/0.48 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.31/0.48 => ( ( Xz = Xx )
% 0.31/0.48 | ( Xz = Xy ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(upairsubunion_type,type,
% 0.31/0.48 upairsubunion: $o ).
% 0.31/0.48
% 0.31/0.48 thf(upairsubunion,definition,
% 0.31/0.48 ( upairsubunion
% 0.31/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ! [Xy: $i] :
% 0.31/0.48 ( ( in @ Xy @ B )
% 0.31/0.48 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(upairinpowunion_type,type,
% 0.31/0.48 upairinpowunion: $o ).
% 0.31/0.48
% 0.31/0.48 thf(upairinpowunion,definition,
% 0.31/0.48 ( upairinpowunion
% 0.31/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ! [Xy: $i] :
% 0.31/0.48 ( ( in @ Xy @ B )
% 0.31/0.48 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(ubforcartprodlem1_type,type,
% 0.31/0.48 ubforcartprodlem1: $o ).
% 0.31/0.48
% 0.31/0.48 thf(ubforcartprodlem1,definition,
% 0.31/0.48 ( ubforcartprodlem1
% 0.31/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ! [Xy: $i] :
% 0.31/0.48 ( ( in @ Xy @ B )
% 0.31/0.48 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(ubforcartprodlem2_type,type,
% 0.31/0.48 ubforcartprodlem2: $o ).
% 0.31/0.48
% 0.31/0.48 thf(ubforcartprodlem2,definition,
% 0.31/0.48 ( ubforcartprodlem2
% 0.31/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ! [Xy: $i] :
% 0.31/0.48 ( ( in @ Xy @ B )
% 0.31/0.48 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(ubforcartprodlem3_type,type,
% 0.31/0.48 ubforcartprodlem3: $o ).
% 0.31/0.48
% 0.31/0.48 thf(ubforcartprodlem3,definition,
% 0.31/0.48 ( ubforcartprodlem3
% 0.31/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ! [Xy: $i] :
% 0.31/0.48 ( ( in @ Xy @ B )
% 0.31/0.48 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(cartprodpairin_type,type,
% 0.31/0.48 cartprodpairin: $o ).
% 0.31/0.48
% 0.31/0.48 thf(cartprodpairin,definition,
% 0.31/0.48 ( cartprodpairin
% 0.31/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ! [Xy: $i] :
% 0.31/0.48 ( ( in @ Xy @ B )
% 0.31/0.48 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(cartprodmempair1_type,type,
% 0.31/0.48 cartprodmempair1: $o ).
% 0.31/0.48
% 0.31/0.48 thf(cartprodmempair1,definition,
% 0.31/0.48 ( cartprodmempair1
% 0.31/0.48 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.31/0.48 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.31/0.48 => ? [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 & ? [Xy: $i] :
% 0.31/0.48 ( ( in @ Xy @ B )
% 0.31/0.48 & ( Xu
% 0.31/0.48 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(cartprodmempair_type,type,
% 0.31/0.48 cartprodmempair: $o ).
% 0.31/0.48
% 0.31/0.48 thf(cartprodmempair,definition,
% 0.31/0.48 ( cartprodmempair
% 0.31/0.48 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.31/0.48 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.31/0.48 => ( iskpair @ Xu ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(setunionE2_type,type,
% 0.31/0.48 setunionE2: $o ).
% 0.31/0.48
% 0.31/0.48 thf(setunionE2,definition,
% 0.31/0.48 ( setunionE2
% 0.31/0.48 = ( ! [A: $i,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ ( setunion @ A ) )
% 0.31/0.48 => ? [X: $i] :
% 0.31/0.48 ( ( in @ X @ A )
% 0.31/0.48 & ( in @ Xx @ X ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(setunionsingleton1_type,type,
% 0.31/0.48 setunionsingleton1: $o ).
% 0.31/0.48
% 0.31/0.48 thf(setunionsingleton1,definition,
% 0.31/0.48 ( setunionsingleton1
% 0.31/0.48 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(setunionsingleton2_type,type,
% 0.31/0.48 setunionsingleton2: $o ).
% 0.31/0.48
% 0.31/0.48 thf(setunionsingleton2,definition,
% 0.31/0.48 ( setunionsingleton2
% 0.31/0.48 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(setunionsingleton_type,type,
% 0.31/0.48 setunionsingleton: $o ).
% 0.31/0.48
% 0.31/0.48 thf(setunionsingleton,definition,
% 0.31/0.48 ( setunionsingleton
% 0.31/0.48 = ( ! [Xx: $i] :
% 0.31/0.48 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 0.31/0.48 = Xx ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(singleton_type,type,
% 0.31/0.48 singleton: $i > $o ).
% 0.31/0.48
% 0.31/0.48 thf(singleton,definition,
% 0.31/0.48 ( singleton
% 0.31/0.48 = ( ^ [A: $i] :
% 0.31/0.48 ? [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 & ( A
% 0.31/0.48 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(singletonprop_type,type,
% 0.31/0.48 singletonprop: $o ).
% 0.31/0.48
% 0.31/0.48 thf(singletonprop,definition,
% 0.31/0.48 ( singletonprop
% 0.31/0.48 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/0.48 ( ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ! [Xy: $i] :
% 0.31/0.48 ( ( in @ Xy @ A )
% 0.31/0.48 => ( ( Xphi @ Xx )
% 0.31/0.48 => ( ( Xphi @ Xy )
% 0.31/0.48 => ( Xx = Xy ) ) ) ) )
% 0.31/0.48 => ( ? [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 & ( Xphi @ Xx ) )
% 0.31/0.48 => ( singleton
% 0.31/0.48 @ ( dsetconstr @ A
% 0.31/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(ex1_type,type,
% 0.31/0.48 ex1: $i > ( $i > $o ) > $o ).
% 0.31/0.48
% 0.31/0.48 thf(ex1,definition,
% 0.31/0.48 ( ex1
% 0.31/0.48 = ( ^ [A: $i,Xphi: $i > $o] :
% 0.31/0.48 ( singleton
% 0.31/0.48 @ ( dsetconstr @ A
% 0.31/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(ex1E1_type,type,
% 0.31/0.48 ex1E1: $o ).
% 0.31/0.48
% 0.31/0.48 thf(ex1E1,definition,
% 0.31/0.48 ( ex1E1
% 0.31/0.48 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/0.48 ( ( ex1 @ A
% 0.31/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.48 => ? [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(ex1I_type,type,
% 0.31/0.48 ex1I: $o ).
% 0.31/0.48
% 0.31/0.48 thf(ex1I,definition,
% 0.31/0.48 ( ex1I
% 0.31/0.48 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ( ( Xphi @ Xx )
% 0.31/0.48 => ( ! [Xy: $i] :
% 0.31/0.48 ( ( in @ Xy @ A )
% 0.31/0.48 => ( ( Xphi @ Xy )
% 0.31/0.48 => ( Xy = Xx ) ) )
% 0.31/0.48 => ( ex1 @ A
% 0.31/0.48 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(ex1I2_type,type,
% 0.31/0.48 ex1I2: $o ).
% 0.31/0.48
% 0.31/0.48 thf(ex1I2,definition,
% 0.31/0.48 ( ex1I2
% 0.31/0.48 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/0.48 ( ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ! [Xy: $i] :
% 0.31/0.48 ( ( in @ Xy @ A )
% 0.31/0.48 => ( ( Xphi @ Xx )
% 0.31/0.48 => ( ( Xphi @ Xy )
% 0.31/0.48 => ( Xx = Xy ) ) ) ) )
% 0.31/0.48 => ( ? [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 & ( Xphi @ Xx ) )
% 0.31/0.48 => ( ex1 @ A
% 0.31/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(singletonsuniq_type,type,
% 0.31/0.48 singletonsuniq: $o ).
% 0.31/0.48
% 0.31/0.48 thf(singletonsuniq,definition,
% 0.31/0.48 ( singletonsuniq
% 0.31/0.48 = ( ! [Xx: $i,Xy: $i] :
% 0.31/0.48 ( ( ( setadjoin @ Xx @ emptyset )
% 0.31/0.48 = ( setadjoin @ Xy @ emptyset ) )
% 0.31/0.48 => ( Xx = Xy ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(atmost1p_type,type,
% 0.31/0.48 atmost1p: $i > $o ).
% 0.31/0.48
% 0.31/0.48 thf(atleast2p_type,type,
% 0.31/0.48 atleast2p: $i > $o ).
% 0.31/0.48
% 0.31/0.48 thf(atmost2p_type,type,
% 0.31/0.48 atmost2p: $i > $o ).
% 0.31/0.48
% 0.31/0.48 thf(upairsetp_type,type,
% 0.31/0.48 upairsetp: $i > $o ).
% 0.31/0.48
% 0.31/0.48 thf(setukpairinjL1_type,type,
% 0.31/0.48 setukpairinjL1: $o ).
% 0.31/0.48
% 0.31/0.48 thf(setukpairinjL1,definition,
% 0.31/0.48 ( setukpairinjL1
% 0.31/0.48 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.31/0.48 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 0.31/0.48 => ( Xx = Xz ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(kfstsingleton_type,type,
% 0.31/0.48 kfstsingleton: $o ).
% 0.31/0.48
% 0.31/0.48 thf(kfstsingleton,definition,
% 0.31/0.48 ( kfstsingleton
% 0.31/0.48 = ( ! [Xu: $i] :
% 0.31/0.48 ( ( iskpair @ Xu )
% 0.31/0.48 => ( singleton
% 0.31/0.48 @ ( dsetconstr @ ( setunion @ Xu )
% 0.31/0.48 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(theprop_type,type,
% 0.31/0.48 theprop: $o ).
% 0.31/0.48
% 0.31/0.48 thf(theprop,definition,
% 0.31/0.48 ( theprop
% 0.31/0.48 = ( ! [X: $i] :
% 0.31/0.48 ( ( singleton @ X )
% 0.31/0.48 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(kfst_type,type,
% 0.31/0.48 kfst: $i > $i ).
% 0.31/0.48
% 0.31/0.48 thf(kfstpairEq_type,type,
% 0.31/0.48 kfstpairEq: $o ).
% 0.31/0.48
% 0.31/0.48 thf(kfstpairEq,definition,
% 0.31/0.48 ( kfstpairEq
% 0.31/0.48 = ( ! [Xx: $i,Xy: $i] :
% 0.31/0.48 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.31/0.48 = Xx ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(cartprodfstin_type,type,
% 0.31/0.48 cartprodfstin: $o ).
% 0.31/0.48
% 0.31/0.48 thf(cartprodfstin,definition,
% 0.31/0.48 ( cartprodfstin
% 0.31/0.48 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.31/0.48 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.31/0.48 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(setukpairinjL2_type,type,
% 0.31/0.48 setukpairinjL2: $o ).
% 0.31/0.48
% 0.31/0.48 thf(setukpairinjL2,definition,
% 0.31/0.48 ( setukpairinjL2
% 0.31/0.48 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.31/0.48 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.31/0.48 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.31/0.48 => ( Xx = Xz ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(setukpairinjL_type,type,
% 0.31/0.48 setukpairinjL: $o ).
% 0.31/0.48
% 0.31/0.48 thf(setukpairinjL,definition,
% 0.31/0.48 ( setukpairinjL
% 0.31/0.48 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.31/0.48 ( ( ( kpair @ Xx @ Xy )
% 0.31/0.48 = ( kpair @ Xz @ Xu ) )
% 0.31/0.48 => ( Xx = Xz ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(setukpairinjR11_type,type,
% 0.31/0.48 setukpairinjR11: $o ).
% 0.31/0.48
% 0.31/0.48 thf(setukpairinjR11,definition,
% 0.31/0.48 ( setukpairinjR11
% 0.31/0.48 = ( ! [Xx: $i,Xy: $i] :
% 0.31/0.48 ( ( Xx = Xy )
% 0.31/0.48 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.31/0.48 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(setukpairinjR12_type,type,
% 0.31/0.48 setukpairinjR12: $o ).
% 0.31/0.48
% 0.31/0.48 thf(setukpairinjR12,definition,
% 0.31/0.48 ( setukpairinjR12
% 0.31/0.48 = ( ! [Xx: $i,Xy: $i] :
% 0.31/0.48 ( ( Xx = Xy )
% 0.31/0.48 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.31/0.48 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(setukpairinjR1_type,type,
% 0.31/0.48 setukpairinjR1: $o ).
% 0.31/0.48
% 0.31/0.48 thf(setukpairinjR1,definition,
% 0.31/0.48 ( setukpairinjR1
% 0.31/0.48 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.31/0.48 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.31/0.48 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.31/0.48 => ( ( Xz = Xu )
% 0.31/0.48 => ( Xy = Xu ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(upairequniteq_type,type,
% 0.31/0.48 upairequniteq: $o ).
% 0.31/0.48
% 0.31/0.48 thf(upairequniteq,definition,
% 0.31/0.48 ( upairequniteq
% 0.31/0.48 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.31/0.48 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.31/0.48 = ( setadjoin @ Xz @ emptyset ) )
% 0.31/0.48 => ( Xx = Xy ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(setukpairinjR2_type,type,
% 0.31/0.48 setukpairinjR2: $o ).
% 0.31/0.48
% 0.31/0.48 thf(setukpairinjR2,definition,
% 0.31/0.48 ( setukpairinjR2
% 0.31/0.48 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.31/0.48 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.31/0.48 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.31/0.48 => ( Xy = Xu ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(setukpairinjR_type,type,
% 0.31/0.48 setukpairinjR: $o ).
% 0.31/0.48
% 0.31/0.48 thf(setukpairinjR,definition,
% 0.31/0.48 ( setukpairinjR
% 0.31/0.48 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.31/0.48 ( ( ( kpair @ Xx @ Xy )
% 0.31/0.48 = ( kpair @ Xz @ Xu ) )
% 0.31/0.48 => ( Xy = Xu ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(ksndsingleton_type,type,
% 0.31/0.48 ksndsingleton: $o ).
% 0.31/0.48
% 0.31/0.48 thf(ksndsingleton,definition,
% 0.31/0.48 ( ksndsingleton
% 0.31/0.48 = ( ! [Xu: $i] :
% 0.31/0.48 ( ( iskpair @ Xu )
% 0.31/0.48 => ( singleton
% 0.31/0.48 @ ( dsetconstr @ ( setunion @ Xu )
% 0.31/0.48 @ ^ [Xx: $i] :
% 0.31/0.48 ( Xu
% 0.31/0.48 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(ksnd_type,type,
% 0.31/0.48 ksnd: $i > $i ).
% 0.31/0.48
% 0.31/0.48 thf(ksndpairEq_type,type,
% 0.31/0.48 ksndpairEq: $o ).
% 0.31/0.48
% 0.31/0.48 thf(ksndpairEq,definition,
% 0.31/0.48 ( ksndpairEq
% 0.31/0.48 = ( ! [Xx: $i,Xy: $i] :
% 0.31/0.48 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.31/0.48 = Xy ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(kpairsurjEq_type,type,
% 0.31/0.48 kpairsurjEq: $o ).
% 0.31/0.48
% 0.31/0.48 thf(kpairsurjEq,definition,
% 0.31/0.48 ( kpairsurjEq
% 0.31/0.48 = ( ! [Xu: $i] :
% 0.31/0.48 ( ( iskpair @ Xu )
% 0.31/0.48 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.31/0.48 = Xu ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(cartprodsndin_type,type,
% 0.31/0.48 cartprodsndin: $o ).
% 0.31/0.48
% 0.31/0.48 thf(cartprodsndin,definition,
% 0.31/0.48 ( cartprodsndin
% 0.31/0.48 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.31/0.48 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.31/0.48 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(cartprodpairmemEL_type,type,
% 0.31/0.48 cartprodpairmemEL: $o ).
% 0.31/0.48
% 0.31/0.48 thf(cartprodpairmemEL,definition,
% 0.31/0.48 ( cartprodpairmemEL
% 0.31/0.48 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.31/0.48 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.31/0.48 => ( in @ Xx @ A ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(cartprodpairmemER_type,type,
% 0.31/0.48 cartprodpairmemER: $o ).
% 0.31/0.48
% 0.31/0.48 thf(cartprodpairmemER,definition,
% 0.31/0.48 ( cartprodpairmemER
% 0.31/0.48 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.31/0.48 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.31/0.48 => ( in @ Xy @ B ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(cartprodmempaircEq_type,type,
% 0.31/0.48 cartprodmempaircEq: $o ).
% 0.31/0.48
% 0.31/0.48 thf(cartprodmempaircEq,definition,
% 0.31/0.48 ( cartprodmempaircEq
% 0.31/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ! [Xy: $i] :
% 0.31/0.48 ( ( in @ Xy @ B )
% 0.31/0.48 => ( ( kpair @ Xx @ Xy )
% 0.31/0.48 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(cartprodfstpairEq_type,type,
% 0.31/0.48 cartprodfstpairEq: $o ).
% 0.31/0.48
% 0.31/0.48 thf(cartprodfstpairEq,definition,
% 0.31/0.48 ( cartprodfstpairEq
% 0.31/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ! [Xy: $i] :
% 0.31/0.48 ( ( in @ Xy @ B )
% 0.31/0.48 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.31/0.48 = Xx ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(cartprodsndpairEq_type,type,
% 0.31/0.48 cartprodsndpairEq: $o ).
% 0.31/0.48
% 0.31/0.48 thf(cartprodsndpairEq,definition,
% 0.31/0.48 ( cartprodsndpairEq
% 0.31/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ! [Xy: $i] :
% 0.31/0.48 ( ( in @ Xy @ B )
% 0.31/0.48 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.31/0.48 = Xy ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(cartprodpairsurjEq_type,type,
% 0.31/0.48 cartprodpairsurjEq: $o ).
% 0.31/0.48
% 0.31/0.48 thf(cartprodpairsurjEq,definition,
% 0.31/0.48 ( cartprodpairsurjEq
% 0.31/0.48 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.31/0.48 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.31/0.48 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.31/0.48 = Xu ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(breln_type,type,
% 0.31/0.48 breln: $i > $i > $i > $o ).
% 0.31/0.48
% 0.31/0.48 thf(breln,definition,
% 0.31/0.48 ( breln
% 0.31/0.48 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(dpsetconstr_type,type,
% 0.31/0.48 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 0.31/0.48
% 0.31/0.48 thf(dpsetconstr,definition,
% 0.31/0.48 ( dpsetconstr
% 0.31/0.48 = ( ^ [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.31/0.48 ( dsetconstr @ ( cartprod @ A @ B )
% 0.31/0.48 @ ^ [Xu: $i] :
% 0.31/0.48 ? [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 & ? [Xy: $i] :
% 0.31/0.48 ( ( in @ Xy @ B )
% 0.31/0.48 & ( Xphi @ Xx @ Xy )
% 0.31/0.48 & ( Xu
% 0.31/0.48 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(dpsetconstrI_type,type,
% 0.31/0.48 dpsetconstrI: $o ).
% 0.31/0.48
% 0.31/0.48 thf(dpsetconstrI,definition,
% 0.31/0.48 ( dpsetconstrI
% 0.31/0.48 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ! [Xy: $i] :
% 0.31/0.48 ( ( in @ Xy @ B )
% 0.31/0.48 => ( ( Xphi @ Xx @ Xy )
% 0.31/0.48 => ( in @ ( kpair @ Xx @ Xy )
% 0.31/0.48 @ ( dpsetconstr @ A @ B
% 0.31/0.48 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(dpsetconstrSub_type,type,
% 0.31/0.48 dpsetconstrSub: $o ).
% 0.31/0.48
% 0.31/0.48 thf(dpsetconstrSub,definition,
% 0.31/0.48 ( dpsetconstrSub
% 0.31/0.48 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.31/0.48 ( subset
% 0.31/0.48 @ ( dpsetconstr @ A @ B
% 0.31/0.48 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 0.31/0.48 @ ( cartprod @ A @ B ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(setOfPairsIsBReln_type,type,
% 0.31/0.48 setOfPairsIsBReln: $o ).
% 0.31/0.48
% 0.31/0.48 thf(setOfPairsIsBReln,definition,
% 0.31/0.48 ( setOfPairsIsBReln
% 0.31/0.48 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.31/0.48 ( breln @ A @ B
% 0.31/0.48 @ ( dpsetconstr @ A @ B
% 0.31/0.48 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(dpsetconstrERa_type,type,
% 0.31/0.48 dpsetconstrERa: $o ).
% 0.31/0.48
% 0.31/0.48 thf(dpsetconstrERa,definition,
% 0.31/0.48 ( dpsetconstrERa
% 0.31/0.48 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ! [Xy: $i] :
% 0.31/0.48 ( ( in @ Xy @ B )
% 0.31/0.48 => ( ( in @ ( kpair @ Xx @ Xy )
% 0.31/0.48 @ ( dpsetconstr @ A @ B
% 0.31/0.48 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.31/0.48 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(dpsetconstrEL1_type,type,
% 0.31/0.48 dpsetconstrEL1: $o ).
% 0.31/0.48
% 0.31/0.48 thf(dpsetconstrEL1,definition,
% 0.31/0.48 ( dpsetconstrEL1
% 0.31/0.48 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.31/0.48 ( ( in @ ( kpair @ Xx @ Xy )
% 0.31/0.48 @ ( dpsetconstr @ A @ B
% 0.31/0.48 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.31/0.48 => ( in @ Xx @ A ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(dpsetconstrEL2_type,type,
% 0.31/0.48 dpsetconstrEL2: $o ).
% 0.31/0.48
% 0.31/0.48 thf(dpsetconstrEL2,definition,
% 0.31/0.48 ( dpsetconstrEL2
% 0.31/0.48 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.31/0.48 ( ( in @ ( kpair @ Xx @ Xy )
% 0.31/0.48 @ ( dpsetconstr @ A @ B
% 0.31/0.48 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.31/0.48 => ( in @ Xy @ B ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(dpsetconstrER,conjecture,
% 0.31/0.48 ( setextAx
% 0.31/0.48 => ( emptysetAx
% 0.31/0.48 => ( setadjoinAx
% 0.31/0.48 => ( powersetAx
% 0.31/0.48 => ( setunionAx
% 0.31/0.48 => ( omega0Ax
% 0.31/0.48 => ( omegaSAx
% 0.31/0.48 => ( omegaIndAx
% 0.31/0.48 => ( replAx
% 0.31/0.48 => ( foundationAx
% 0.31/0.48 => ( wellorderingAx
% 0.31/0.48 => ( descrp
% 0.31/0.48 => ( dsetconstrI
% 0.31/0.48 => ( dsetconstrEL
% 0.31/0.48 => ( dsetconstrER
% 0.31/0.48 => ( exuE1
% 0.31/0.48 => ( prop2setE
% 0.31/0.48 => ( emptysetE
% 0.31/0.48 => ( emptysetimpfalse
% 0.31/0.48 => ( notinemptyset
% 0.31/0.48 => ( exuE3e
% 0.31/0.48 => ( setext
% 0.31/0.48 => ( emptyI
% 0.31/0.48 => ( noeltsimpempty
% 0.31/0.48 => ( setbeta
% 0.31/0.48 => ( nonemptyE1
% 0.31/0.48 => ( nonemptyI
% 0.31/0.48 => ( nonemptyI1
% 0.31/0.48 => ( setadjoinIL
% 0.31/0.48 => ( emptyinunitempty
% 0.31/0.48 => ( setadjoinIR
% 0.31/0.48 => ( setadjoinE
% 0.31/0.48 => ( setadjoinOr
% 0.31/0.48 => ( setoftrueEq
% 0.31/0.48 => ( powersetI
% 0.31/0.48 => ( emptyinPowerset
% 0.31/0.48 => ( emptyInPowerset
% 0.31/0.48 => ( powersetE
% 0.31/0.48 => ( setunionI
% 0.31/0.48 => ( setunionE
% 0.31/0.48 => ( subPowSU
% 0.31/0.48 => ( exuE2
% 0.31/0.48 => ( nonemptyImpWitness
% 0.31/0.48 => ( uniqinunit
% 0.31/0.48 => ( notinsingleton
% 0.31/0.48 => ( eqinunit
% 0.31/0.48 => ( singletonsswitch
% 0.31/0.48 => ( upairsetE
% 0.31/0.48 => ( upairsetIL
% 0.31/0.48 => ( upairsetIR
% 0.31/0.48 => ( emptyE1
% 0.31/0.48 => ( vacuousDall
% 0.31/0.48 => ( quantDeMorgan1
% 0.31/0.48 => ( quantDeMorgan2
% 0.31/0.48 => ( quantDeMorgan3
% 0.31/0.48 => ( quantDeMorgan4
% 0.31/0.48 => ( prop2setI
% 0.31/0.48 => ( prop2set2propI
% 0.31/0.48 => ( notdexE
% 0.31/0.48 => ( notdallE
% 0.31/0.48 => ( exuI1
% 0.31/0.48 => ( exuI3
% 0.31/0.48 => ( exuI2
% 0.31/0.48 => ( inCongP
% 0.31/0.48 => ( in__Cong
% 0.31/0.48 => ( exuE3u
% 0.31/0.48 => ( exu__Cong
% 0.31/0.48 => ( emptyset__Cong
% 0.31/0.48 => ( setadjoin__Cong
% 0.31/0.48 => ( powerset__Cong
% 0.31/0.48 => ( setunion__Cong
% 0.31/0.48 => ( omega__Cong
% 0.31/0.48 => ( exuEu
% 0.31/0.48 => ( descr__Cong
% 0.31/0.48 => ( dsetconstr__Cong
% 0.31/0.48 => ( subsetI1
% 0.31/0.48 => ( eqimpsubset2
% 0.31/0.48 => ( eqimpsubset1
% 0.31/0.48 => ( subsetI2
% 0.31/0.48 => ( emptysetsubset
% 0.31/0.48 => ( subsetE
% 0.31/0.48 => ( subsetE2
% 0.31/0.48 => ( notsubsetI
% 0.31/0.48 => ( notequalI1
% 0.31/0.48 => ( notequalI2
% 0.31/0.48 => ( subsetRefl
% 0.31/0.48 => ( subsetTrans
% 0.31/0.48 => ( setadjoinSub
% 0.31/0.48 => ( setadjoinSub2
% 0.31/0.48 => ( subset2powerset
% 0.31/0.48 => ( setextsub
% 0.31/0.48 => ( subsetemptysetimpeq
% 0.31/0.48 => ( powersetI1
% 0.31/0.48 => ( powersetE1
% 0.31/0.48 => ( inPowerset
% 0.31/0.48 => ( powersetsubset
% 0.31/0.48 => ( sepInPowerset
% 0.31/0.48 => ( sepSubset
% 0.31/0.48 => ( binunionIL
% 0.31/0.48 => ( upairset2IR
% 0.31/0.48 => ( binunionIR
% 0.31/0.48 => ( binunionEcases
% 0.31/0.48 => ( binunionE
% 0.31/0.48 => ( binunionLsub
% 0.31/0.48 => ( binunionRsub
% 0.31/0.48 => ( binintersectI
% 0.31/0.48 => ( binintersectSubset5
% 0.31/0.48 => ( binintersectEL
% 0.31/0.48 => ( binintersectLsub
% 0.31/0.48 => ( binintersectSubset2
% 0.31/0.48 => ( binintersectSubset3
% 0.31/0.48 => ( binintersectER
% 0.31/0.48 => ( disjointsetsI1
% 0.31/0.48 => ( binintersectRsub
% 0.31/0.48 => ( binintersectSubset4
% 0.31/0.48 => ( binintersectSubset1
% 0.31/0.48 => ( bs114d
% 0.31/0.48 => ( setminusI
% 0.31/0.48 => ( setminusEL
% 0.31/0.48 => ( setminusER
% 0.31/0.48 => ( setminusSubset2
% 0.31/0.48 => ( setminusERneg
% 0.31/0.48 => ( setminusELneg
% 0.31/0.48 => ( setminusILneg
% 0.31/0.48 => ( setminusIRneg
% 0.31/0.48 => ( setminusLsub
% 0.31/0.48 => ( setminusSubset1
% 0.31/0.48 => ( symdiffE
% 0.31/0.48 => ( symdiffI1
% 0.31/0.48 => ( symdiffI2
% 0.31/0.48 => ( symdiffIneg1
% 0.31/0.48 => ( symdiffIneg2
% 0.31/0.48 => ( secondinupair
% 0.31/0.48 => ( setukpairIL
% 0.31/0.48 => ( setukpairIR
% 0.31/0.48 => ( kpairiskpair
% 0.31/0.48 => ( kpairp
% 0.31/0.48 => ( singletonsubset
% 0.31/0.48 => ( singletoninpowerset
% 0.31/0.48 => ( singletoninpowunion
% 0.31/0.48 => ( upairset2E
% 0.31/0.48 => ( upairsubunion
% 0.31/0.48 => ( upairinpowunion
% 0.31/0.48 => ( ubforcartprodlem1
% 0.31/0.48 => ( ubforcartprodlem2
% 0.31/0.48 => ( ubforcartprodlem3
% 0.31/0.48 => ( cartprodpairin
% 0.31/0.48 => ( cartprodmempair1
% 0.31/0.48 => ( cartprodmempair
% 0.31/0.48 => ( setunionE2
% 0.31/0.48 => ( setunionsingleton1
% 0.31/0.48 => ( setunionsingleton2
% 0.31/0.48 => ( setunionsingleton
% 0.31/0.48 => ( singletonprop
% 0.31/0.48 => ( ex1E1
% 0.31/0.48 => ( ex1I
% 0.31/0.48 => ( ex1I2
% 0.31/0.48 => ( singletonsuniq
% 0.31/0.48 => ( setukpairinjL1
% 0.31/0.48 => ( kfstsingleton
% 0.31/0.48 => ( theprop
% 0.31/0.48 => ( kfstpairEq
% 0.31/0.48 => ( cartprodfstin
% 0.31/0.48 => ( setukpairinjL2
% 0.31/0.48 => ( setukpairinjL
% 0.31/0.48 => ( setukpairinjR11
% 0.31/0.48 => ( setukpairinjR12
% 0.31/0.48 => ( setukpairinjR1
% 0.31/0.48 => ( upairequniteq
% 0.31/0.48 => ( setukpairinjR2
% 0.31/0.48 => ( setukpairinjR
% 0.31/0.48 => ( ksndsingleton
% 0.31/0.48 => ( ksndpairEq
% 0.31/0.48 => ( kpairsurjEq
% 0.31/0.48 => ( cartprodsndin
% 0.31/0.48 => ( cartprodpairmemEL
% 0.31/0.48 => ( cartprodpairmemER
% 0.31/0.48 => ( cartprodmempaircEq
% 0.31/0.48 => ( cartprodfstpairEq
% 0.31/0.48 => ( cartprodsndpairEq
% 0.31/0.48 => ( cartprodpairsurjEq
% 0.31/0.48 => ( dpsetconstrI
% 0.31/0.48 => ( dpsetconstrSub
% 0.31/0.48 => ( setOfPairsIsBReln
% 0.31/0.48 => ( dpsetconstrERa
% 0.31/0.48 => ( dpsetconstrEL1
% 0.31/0.48 => ( dpsetconstrEL2
% 0.31/0.48 => ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.31/0.48 ( ( in @ ( kpair @ Xx @ Xy )
% 0.31/0.48 @ ( dpsetconstr @ A @ B
% 0.31/0.48 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.31/0.48 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 %------------------------------------------------------------------------------
% 0.31/0.48 ------------------------
% 2.91/1.72 ---- Embedded in THF ---
% 2.91/1.72 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.91/1.72 %%% Generated on Tue Feb 24 12:30:47 EST 2026
% 2.91/1.72 %%% using '$$dhol' embedding, version 1.3.0.
% 2.91/1.72 %%% Logic specification used:
% 2.91/1.72 %%% thf(spec, logic, $$dhol).
% 2.91/1.72
% 2.91/1.72 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.0Ah447e165/DTF2DTF7726.p
% See solution above
% 2.91/1.72 ------------------------
% 2.91/1.76 ---- Cleaned THF ---
% 2.91/1.76 thf(in_type,type,
% 2.91/1.76 in: $i > $i > $o ).
% 2.91/1.76
% 2.91/1.76 thf(exu_type,type,
% 2.91/1.76 exu: ( $i > $o ) > $o ).
% 2.91/1.76
% 2.91/1.76 thf(exu,definition,
% 2.91/1.76 ( exu
% 2.91/1.76 = ( ^ [Xphi: $i > $o] :
% 2.91/1.76 ? [Xx: $i] :
% 2.91/1.76 ( ( Xphi @ Xx )
% 2.91/1.76 & ! [Xy: $i] :
% 2.91/1.76 ( ( Xphi @ Xy )
% 2.91/1.76 => ( Xx = Xy ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(setextAx_type,type,
% 2.91/1.76 setextAx: $o ).
% 2.91/1.76
% 2.91/1.76 thf(setextAx,definition,
% 2.91/1.76 ( setextAx
% 2.91/1.76 = ( ! [A: $i,B: $i] :
% 2.91/1.76 ( ! [Xx: $i] :
% 2.91/1.76 ( ( in @ Xx @ A )
% 2.91/1.76 <=> ( in @ Xx @ B ) )
% 2.91/1.76 => ( A = B ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(emptyset_type,type,
% 2.91/1.76 emptyset: $i ).
% 2.91/1.76
% 2.91/1.76 thf(emptysetAx_type,type,
% 2.91/1.76 emptysetAx: $o ).
% 2.91/1.76
% 2.91/1.76 thf(emptysetAx,definition,
% 2.91/1.76 ( emptysetAx
% 2.91/1.76 = ( ! [Xx: $i] :
% 2.91/1.76 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(setadjoin_type,type,
% 2.91/1.76 setadjoin: $i > $i > $i ).
% 2.91/1.76
% 2.91/1.76 thf(setadjoinAx_type,type,
% 2.91/1.76 setadjoinAx: $o ).
% 2.91/1.76
% 2.91/1.76 thf(setadjoinAx,definition,
% 2.91/1.76 ( setadjoinAx
% 2.91/1.76 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.91/1.76 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.91/1.76 <=> ( ( Xy = Xx )
% 2.91/1.76 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(powerset_type,type,
% 2.91/1.76 powerset: $i > $i ).
% 2.91/1.76
% 2.91/1.76 thf(powersetAx_type,type,
% 2.91/1.76 powersetAx: $o ).
% 2.91/1.76
% 2.91/1.76 thf(powersetAx,definition,
% 2.91/1.76 ( powersetAx
% 2.91/1.76 = ( ! [A: $i,B: $i] :
% 2.91/1.76 ( ( in @ B @ ( powerset @ A ) )
% 2.91/1.76 <=> ! [Xx: $i] :
% 2.91/1.76 ( ( in @ Xx @ B )
% 2.91/1.76 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(setunion_type,type,
% 2.91/1.76 setunion: $i > $i ).
% 2.91/1.76
% 2.91/1.76 thf(setunionAx_type,type,
% 2.91/1.76 setunionAx: $o ).
% 2.91/1.76
% 2.91/1.76 thf(setunionAx,definition,
% 2.91/1.76 ( setunionAx
% 2.91/1.76 = ( ! [A: $i,Xx: $i] :
% 2.91/1.76 ( ( in @ Xx @ ( setunion @ A ) )
% 2.91/1.76 <=> ? [B: $i] :
% 2.91/1.76 ( ( in @ Xx @ B )
% 2.91/1.76 & ( in @ B @ A ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(omega_type,type,
% 2.91/1.76 omega: $i ).
% 2.91/1.76
% 2.91/1.76 thf(omega0Ax_type,type,
% 2.91/1.76 omega0Ax: $o ).
% 2.91/1.76
% 2.91/1.76 thf(omega0Ax,definition,
% 2.91/1.76 ( omega0Ax
% 2.91/1.76 = ( in @ emptyset @ omega ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(omegaSAx_type,type,
% 2.91/1.76 omegaSAx: $o ).
% 2.91/1.76
% 2.91/1.76 thf(omegaSAx,definition,
% 2.91/1.76 ( omegaSAx
% 2.91/1.76 = ( ! [Xx: $i] :
% 2.91/1.76 ( ( in @ Xx @ omega )
% 2.91/1.76 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(omegaIndAx_type,type,
% 2.91/1.76 omegaIndAx: $o ).
% 2.91/1.76
% 2.91/1.76 thf(omegaIndAx,definition,
% 2.91/1.76 ( omegaIndAx
% 2.91/1.76 = ( ! [A: $i] :
% 2.91/1.76 ( ( ( in @ emptyset @ A )
% 2.91/1.76 & ! [Xx: $i] :
% 2.91/1.76 ( ( ( in @ Xx @ omega )
% 2.91/1.76 & ( in @ Xx @ A ) )
% 2.91/1.76 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 2.91/1.76 => ! [Xx: $i] :
% 2.91/1.76 ( ( in @ Xx @ omega )
% 2.91/1.76 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(replAx_type,type,
% 2.91/1.76 replAx: $o ).
% 2.91/1.76
% 2.91/1.76 thf(replAx,definition,
% 2.91/1.76 ( replAx
% 2.91/1.76 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 2.91/1.76 ( ! [Xx: $i] :
% 2.91/1.76 ( ( in @ Xx @ A )
% 2.91/1.76 => ( exu
% 2.91/1.76 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 2.91/1.76 => ? [B: $i] :
% 2.91/1.76 ! [Xx: $i] :
% 2.91/1.76 ( ( in @ Xx @ B )
% 2.91/1.76 <=> ? [Xy: $i] :
% 2.91/1.76 ( ( in @ Xy @ A )
% 2.91/1.76 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(foundationAx_type,type,
% 2.91/1.76 foundationAx: $o ).
% 2.91/1.76
% 2.91/1.76 thf(foundationAx,definition,
% 2.91/1.76 ( foundationAx
% 2.91/1.76 = ( ! [A: $i] :
% 2.91/1.76 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.91/1.76 => ? [B: $i] :
% 2.91/1.76 ( ( in @ B @ A )
% 2.91/1.76 & ~ ? [Xx: $i] :
% 2.91/1.76 ( ( in @ Xx @ B )
% 2.91/1.76 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(wellorderingAx_type,type,
% 2.91/1.76 wellorderingAx: $o ).
% 2.91/1.76
% 2.91/1.76 thf(wellorderingAx,definition,
% 2.91/1.76 ( wellorderingAx
% 2.91/1.76 = ( ! [A: $i] :
% 2.91/1.76 ? [B: $i] :
% 2.91/1.76 ( ! [C: $i] :
% 2.91/1.76 ( ( in @ C @ B )
% 2.91/1.76 => ! [Xx: $i] :
% 2.91/1.76 ( ( in @ Xx @ C )
% 2.91/1.76 => ( in @ Xx @ A ) ) )
% 2.91/1.76 & ! [Xx: $i,Xy: $i] :
% 2.91/1.76 ( ( ( in @ Xx @ A )
% 2.91/1.76 & ( in @ Xy @ A ) )
% 2.91/1.76 => ( ! [C: $i] :
% 2.91/1.76 ( ( in @ C @ B )
% 2.91/1.76 => ( ( in @ Xx @ C )
% 2.91/1.76 <=> ( in @ Xy @ C ) ) )
% 2.91/1.76 => ( Xx = Xy ) ) )
% 2.91/1.76 & ! [C: $i,D: $i] :
% 2.91/1.76 ( ( ( in @ C @ B )
% 2.91/1.76 & ( in @ D @ B ) )
% 2.91/1.76 => ( ! [Xx: $i] :
% 2.91/1.76 ( ( in @ Xx @ C )
% 2.91/1.76 => ( in @ Xx @ D ) )
% 2.91/1.76 | ! [Xx: $i] :
% 2.91/1.76 ( ( in @ Xx @ D )
% 2.91/1.76 => ( in @ Xx @ C ) ) ) )
% 2.91/1.76 & ! [C: $i] :
% 2.91/1.76 ( ( ! [Xx: $i] :
% 2.91/1.76 ( ( in @ Xx @ C )
% 2.91/1.76 => ( in @ Xx @ A ) )
% 2.91/1.76 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 2.91/1.76 => ? [D: $i,Xx: $i] :
% 2.91/1.76 ( ( in @ D @ B )
% 2.91/1.76 & ( in @ Xx @ C )
% 2.91/1.76 & ~ ? [Xy: $i] :
% 2.91/1.76 ( ( in @ Xy @ D )
% 2.91/1.76 & ( in @ Xy @ C ) )
% 2.91/1.76 & ! [E: $i] :
% 2.91/1.76 ( ( in @ E @ B )
% 2.91/1.76 => ( ! [Xy: $i] :
% 2.91/1.76 ( ( in @ Xy @ E )
% 2.91/1.76 => ( in @ Xy @ D ) )
% 2.91/1.76 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(descr_type,type,
% 2.91/1.76 descr: ( $i > $o ) > $i ).
% 2.91/1.76
% 2.91/1.76 thf(descrp_type,type,
% 2.91/1.76 descrp: $o ).
% 2.91/1.76
% 2.91/1.76 thf(descrp,definition,
% 2.91/1.76 ( descrp
% 2.91/1.76 = ( ! [Xphi: $i > $o] :
% 2.91/1.76 ( ( exu
% 2.91/1.76 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.91/1.76 => ( Xphi
% 2.91/1.76 @ ( descr
% 2.91/1.76 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(dsetconstr_type,type,
% 2.91/1.76 dsetconstr: $i > ( $i > $o ) > $i ).
% 2.91/1.76
% 2.91/1.76 thf(dsetconstrI_type,type,
% 2.91/1.76 dsetconstrI: $o ).
% 2.91/1.76
% 2.91/1.76 thf(dsetconstrI,definition,
% 2.91/1.76 ( dsetconstrI
% 2.91/1.76 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.91/1.76 ( ( in @ Xx @ A )
% 2.91/1.76 => ( ( Xphi @ Xx )
% 2.91/1.76 => ( in @ Xx
% 2.91/1.76 @ ( dsetconstr @ A
% 2.91/1.76 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(dsetconstrEL_type,type,
% 2.91/1.76 dsetconstrEL: $o ).
% 2.91/1.76
% 2.91/1.76 thf(dsetconstrEL,definition,
% 2.91/1.76 ( dsetconstrEL
% 2.91/1.76 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.91/1.76 ( ( in @ Xx
% 2.91/1.76 @ ( dsetconstr @ A
% 2.91/1.76 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.91/1.76 => ( in @ Xx @ A ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(dsetconstrER_type,type,
% 2.91/1.76 dsetconstrER: $o ).
% 2.91/1.76
% 2.91/1.76 thf(dsetconstrER,definition,
% 2.91/1.76 ( dsetconstrER
% 2.91/1.76 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.91/1.76 ( ( in @ Xx
% 2.91/1.76 @ ( dsetconstr @ A
% 2.91/1.76 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.91/1.76 => ( Xphi @ Xx ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(exuE1_type,type,
% 2.91/1.76 exuE1: $o ).
% 2.91/1.76
% 2.91/1.76 thf(exuE1,definition,
% 2.91/1.76 ( exuE1
% 2.91/1.76 = ( ! [Xphi: $i > $o] :
% 2.91/1.76 ( ( exu
% 2.91/1.76 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.91/1.76 => ? [Xx: $i] :
% 2.91/1.76 ( ( Xphi @ Xx )
% 2.91/1.76 & ! [Xy: $i] :
% 2.91/1.76 ( ( Xphi @ Xy )
% 2.91/1.76 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(prop2set_type,type,
% 2.91/1.76 prop2set: $o > $i ).
% 2.91/1.76
% 2.91/1.76 thf(prop2setE_type,type,
% 2.91/1.76 prop2setE: $o ).
% 2.91/1.76
% 2.91/1.76 thf(prop2setE,definition,
% 2.91/1.76 ( prop2setE
% 2.91/1.76 = ( ! [Xphi: $o,Xx: $i] :
% 2.91/1.76 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 2.91/1.76 => Xphi ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(emptysetE_type,type,
% 2.91/1.76 emptysetE: $o ).
% 2.91/1.76
% 2.91/1.76 thf(emptysetE,definition,
% 2.91/1.76 ( emptysetE
% 2.91/1.76 = ( ! [Xx: $i] :
% 2.91/1.76 ( ( in @ Xx @ emptyset )
% 2.91/1.76 => ! [Xphi: $o] : Xphi ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(emptysetimpfalse_type,type,
% 2.91/1.76 emptysetimpfalse: $o ).
% 2.91/1.76
% 2.91/1.76 thf(emptysetimpfalse,definition,
% 2.91/1.76 ( emptysetimpfalse
% 2.91/1.76 = ( ! [Xx: $i] :
% 2.91/1.76 ( ( in @ Xx @ emptyset )
% 2.91/1.76 => $false ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(notinemptyset_type,type,
% 2.91/1.76 notinemptyset: $o ).
% 2.91/1.76
% 2.91/1.76 thf(notinemptyset,definition,
% 2.91/1.76 ( notinemptyset
% 2.91/1.76 = ( ! [Xx: $i] :
% 2.91/1.76 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(exuE3e_type,type,
% 2.91/1.76 exuE3e: $o ).
% 2.91/1.76
% 2.91/1.76 thf(exuE3e,definition,
% 2.91/1.76 ( exuE3e
% 2.91/1.76 = ( ! [Xphi: $i > $o] :
% 2.91/1.76 ( ( exu
% 2.91/1.76 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.91/1.76 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(setext_type,type,
% 2.91/1.76 setext: $o ).
% 2.91/1.76
% 2.91/1.76 thf(setext,definition,
% 2.91/1.76 ( setext
% 2.91/1.76 = ( ! [A: $i,B: $i] :
% 2.91/1.76 ( ! [Xx: $i] :
% 2.91/1.76 ( ( in @ Xx @ A )
% 2.91/1.76 => ( in @ Xx @ B ) )
% 2.91/1.76 => ( ! [Xx: $i] :
% 2.91/1.76 ( ( in @ Xx @ B )
% 2.91/1.76 => ( in @ Xx @ A ) )
% 2.91/1.76 => ( A = B ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(emptyI_type,type,
% 2.91/1.76 emptyI: $o ).
% 2.91/1.76
% 2.91/1.76 thf(emptyI,definition,
% 2.91/1.76 ( emptyI
% 2.91/1.76 = ( ! [A: $i] :
% 2.91/1.76 ( ! [Xx: $i] :
% 2.91/1.76 ~ ( in @ Xx @ A )
% 2.91/1.76 => ( A = emptyset ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(noeltsimpempty_type,type,
% 2.91/1.76 noeltsimpempty: $o ).
% 2.91/1.76
% 2.91/1.76 thf(noeltsimpempty,definition,
% 2.91/1.76 ( noeltsimpempty
% 2.91/1.76 = ( ! [A: $i] :
% 2.91/1.76 ( ! [Xx: $i] :
% 2.91/1.76 ~ ( in @ Xx @ A )
% 2.91/1.76 => ( A = emptyset ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(setbeta_type,type,
% 2.91/1.76 setbeta: $o ).
% 2.91/1.76
% 2.91/1.76 thf(setbeta,definition,
% 2.91/1.76 ( setbeta
% 2.91/1.76 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.91/1.76 ( ( in @ Xx @ A )
% 2.91/1.76 => ( ( in @ Xx
% 2.91/1.76 @ ( dsetconstr @ A
% 2.91/1.76 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.91/1.76 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(nonempty_type,type,
% 2.91/1.76 nonempty: $i > $o ).
% 2.91/1.76
% 2.91/1.76 thf(nonempty,definition,
% 2.91/1.76 ( nonempty
% 2.91/1.76 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(nonemptyE1_type,type,
% 2.91/1.76 nonemptyE1: $o ).
% 2.91/1.76
% 2.91/1.76 thf(nonemptyE1,definition,
% 2.91/1.76 ( nonemptyE1
% 2.91/1.76 = ( ! [A: $i] :
% 2.91/1.76 ( ( nonempty @ A )
% 2.91/1.76 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(nonemptyI_type,type,
% 2.91/1.76 nonemptyI: $o ).
% 2.91/1.76
% 2.91/1.76 thf(nonemptyI,definition,
% 2.91/1.76 ( nonemptyI
% 2.91/1.76 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.91/1.76 ( ( in @ Xx @ A )
% 2.91/1.76 => ( ( Xphi @ Xx )
% 2.91/1.76 => ( nonempty
% 2.91/1.76 @ ( dsetconstr @ A
% 2.91/1.76 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(nonemptyI1_type,type,
% 2.91/1.76 nonemptyI1: $o ).
% 2.91/1.76
% 2.91/1.76 thf(nonemptyI1,definition,
% 2.91/1.76 ( nonemptyI1
% 2.91/1.76 = ( ! [A: $i] :
% 2.91/1.76 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.91/1.76 => ( nonempty @ A ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(setadjoinIL_type,type,
% 2.91/1.76 setadjoinIL: $o ).
% 2.91/1.76
% 2.91/1.76 thf(setadjoinIL,definition,
% 2.91/1.76 ( setadjoinIL
% 2.91/1.76 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(emptyinunitempty_type,type,
% 2.91/1.76 emptyinunitempty: $o ).
% 2.91/1.76
% 2.91/1.76 thf(emptyinunitempty,definition,
% 2.91/1.76 ( emptyinunitempty
% 2.91/1.76 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(setadjoinIR_type,type,
% 2.91/1.76 setadjoinIR: $o ).
% 2.91/1.76
% 2.91/1.76 thf(setadjoinIR,definition,
% 2.91/1.76 ( setadjoinIR
% 2.91/1.76 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.91/1.76 ( ( in @ Xy @ A )
% 2.91/1.76 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(setadjoinE_type,type,
% 2.91/1.76 setadjoinE: $o ).
% 2.91/1.76
% 2.91/1.76 thf(setadjoinE,definition,
% 2.91/1.76 ( setadjoinE
% 2.91/1.76 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.91/1.76 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.91/1.76 => ! [Xphi: $o] :
% 2.91/1.76 ( ( ( Xy = Xx )
% 2.91/1.76 => Xphi )
% 2.91/1.76 => ( ( ( in @ Xy @ A )
% 2.91/1.76 => Xphi )
% 2.91/1.76 => Xphi ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(setadjoinOr_type,type,
% 2.91/1.76 setadjoinOr: $o ).
% 2.91/1.76
% 2.91/1.76 thf(setadjoinOr,definition,
% 2.91/1.76 ( setadjoinOr
% 2.91/1.76 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.91/1.76 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.91/1.76 => ( ( Xy = Xx )
% 2.91/1.76 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(setoftrueEq_type,type,
% 2.91/1.76 setoftrueEq: $o ).
% 2.91/1.76
% 2.91/1.76 thf(setoftrueEq,definition,
% 2.91/1.76 ( setoftrueEq
% 2.91/1.76 = ( ! [A: $i] :
% 2.91/1.76 ( ( dsetconstr @ A
% 2.91/1.76 @ ^ [Xx: $i] : $true )
% 2.91/1.76 = A ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(powersetI_type,type,
% 2.91/1.76 powersetI: $o ).
% 2.91/1.76
% 2.91/1.76 thf(powersetI,definition,
% 2.91/1.76 ( powersetI
% 2.91/1.76 = ( ! [A: $i,B: $i] :
% 2.91/1.76 ( ! [Xx: $i] :
% 2.91/1.76 ( ( in @ Xx @ B )
% 2.91/1.76 => ( in @ Xx @ A ) )
% 2.91/1.76 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(emptyinPowerset_type,type,
% 2.91/1.76 emptyinPowerset: $o ).
% 2.91/1.76
% 2.91/1.76 thf(emptyinPowerset,definition,
% 2.91/1.76 ( emptyinPowerset
% 2.91/1.76 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(emptyInPowerset_type,type,
% 2.91/1.76 emptyInPowerset: $o ).
% 2.91/1.76
% 2.91/1.76 thf(emptyInPowerset,definition,
% 2.91/1.76 ( emptyInPowerset
% 2.91/1.76 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(powersetE_type,type,
% 2.91/1.76 powersetE: $o ).
% 2.91/1.76
% 2.91/1.76 thf(powersetE,definition,
% 2.91/1.76 ( powersetE
% 2.91/1.76 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.91/1.76 ( ( in @ B @ ( powerset @ A ) )
% 2.91/1.76 => ( ( in @ Xx @ B )
% 2.91/1.76 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(setunionI_type,type,
% 2.91/1.76 setunionI: $o ).
% 2.91/1.76
% 2.91/1.76 thf(setunionI,definition,
% 2.91/1.76 ( setunionI
% 2.91/1.76 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.91/1.76 ( ( in @ Xx @ B )
% 2.91/1.76 => ( ( in @ B @ A )
% 2.91/1.76 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(setunionE_type,type,
% 2.91/1.76 setunionE: $o ).
% 2.91/1.76
% 2.91/1.76 thf(setunionE,definition,
% 2.91/1.76 ( setunionE
% 2.91/1.76 = ( ! [A: $i,Xx: $i] :
% 2.91/1.76 ( ( in @ Xx @ ( setunion @ A ) )
% 2.91/1.76 => ! [Xphi: $o] :
% 2.91/1.76 ( ! [B: $i] :
% 2.91/1.76 ( ( in @ Xx @ B )
% 2.91/1.76 => ( ( in @ B @ A )
% 2.91/1.76 => Xphi ) )
% 2.91/1.76 => Xphi ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(subPowSU_type,type,
% 2.91/1.76 subPowSU: $o ).
% 2.91/1.76
% 2.91/1.76 thf(subPowSU,definition,
% 2.91/1.76 ( subPowSU
% 2.91/1.76 = ( ! [A: $i,Xx: $i] :
% 2.91/1.76 ( ( in @ Xx @ A )
% 2.91/1.76 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(exuE2_type,type,
% 2.91/1.76 exuE2: $o ).
% 2.91/1.76
% 2.91/1.76 thf(exuE2,definition,
% 2.91/1.76 ( exuE2
% 2.91/1.76 = ( ! [Xphi: $i > $o] :
% 2.91/1.76 ( ( exu
% 2.91/1.76 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.91/1.76 => ? [Xx: $i] :
% 2.91/1.76 ! [Xy: $i] :
% 2.91/1.76 ( ( Xphi @ Xy )
% 2.91/1.76 <=> ( Xy = Xx ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(nonemptyImpWitness_type,type,
% 2.91/1.76 nonemptyImpWitness: $o ).
% 2.91/1.76
% 2.91/1.76 thf(nonemptyImpWitness,definition,
% 2.91/1.76 ( nonemptyImpWitness
% 2.91/1.76 = ( ! [A: $i] :
% 2.91/1.76 ( ( nonempty @ A )
% 2.91/1.76 => ? [Xx: $i] :
% 2.91/1.76 ( ( in @ Xx @ A )
% 2.91/1.76 & $true ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(uniqinunit_type,type,
% 2.91/1.76 uniqinunit: $o ).
% 2.91/1.76
% 2.91/1.76 thf(uniqinunit,definition,
% 2.91/1.76 ( uniqinunit
% 2.91/1.76 = ( ! [Xx: $i,Xy: $i] :
% 2.91/1.76 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.91/1.76 => ( Xx = Xy ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(notinsingleton_type,type,
% 2.91/1.76 notinsingleton: $o ).
% 2.91/1.76
% 2.91/1.76 thf(notinsingleton,definition,
% 2.91/1.76 ( notinsingleton
% 2.91/1.76 = ( ! [Xx: $i,Xy: $i] :
% 2.91/1.76 ( ( Xx != Xy )
% 2.91/1.76 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(eqinunit_type,type,
% 2.91/1.76 eqinunit: $o ).
% 2.91/1.76
% 2.91/1.76 thf(eqinunit,definition,
% 2.91/1.76 ( eqinunit
% 2.91/1.76 = ( ! [Xx: $i,Xy: $i] :
% 2.91/1.76 ( ( Xx = Xy )
% 2.91/1.76 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(singletonsswitch_type,type,
% 2.91/1.76 singletonsswitch: $o ).
% 2.91/1.76
% 2.91/1.76 thf(singletonsswitch,definition,
% 2.91/1.76 ( singletonsswitch
% 2.91/1.76 = ( ! [Xx: $i,Xy: $i] :
% 2.91/1.76 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.91/1.76 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(upairsetE_type,type,
% 2.91/1.76 upairsetE: $o ).
% 2.91/1.76
% 2.91/1.76 thf(upairsetE,definition,
% 2.91/1.76 ( upairsetE
% 2.91/1.76 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.91/1.76 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.91/1.76 => ( ( Xz = Xx )
% 2.91/1.76 | ( Xz = Xy ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(upairsetIL_type,type,
% 2.91/1.76 upairsetIL: $o ).
% 2.91/1.76
% 2.91/1.76 thf(upairsetIL,definition,
% 2.91/1.76 ( upairsetIL
% 2.91/1.76 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(upairsetIR_type,type,
% 2.91/1.76 upairsetIR: $o ).
% 2.91/1.76
% 2.91/1.76 thf(upairsetIR,definition,
% 2.91/1.76 ( upairsetIR
% 2.91/1.76 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(emptyE1_type,type,
% 2.91/1.76 emptyE1: $o ).
% 2.91/1.76
% 2.91/1.76 thf(emptyE1,definition,
% 2.91/1.76 ( emptyE1
% 2.91/1.76 = ( ! [A: $i,Xphi: $i > $o] :
% 2.91/1.76 ( ? [Xx: $i] :
% 2.91/1.76 ( ( in @ Xx @ A )
% 2.91/1.76 & ( Xphi @ Xx ) )
% 2.91/1.76 => ( ( ( dsetconstr @ A
% 2.91/1.76 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.91/1.76 = emptyset )
% 2.91/1.76 => $false ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(vacuousDall_type,type,
% 2.91/1.76 vacuousDall: $o ).
% 2.91/1.76
% 2.91/1.76 thf(vacuousDall,definition,
% 2.91/1.76 ( vacuousDall
% 2.91/1.76 = ( ! [Xphi: $i > $o,Xx: $i] :
% 2.91/1.76 ( ( in @ Xx @ emptyset )
% 2.91/1.76 => ( Xphi @ Xx ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(quantDeMorgan1_type,type,
% 2.91/1.76 quantDeMorgan1: $o ).
% 2.91/1.76
% 2.91/1.76 thf(quantDeMorgan1,definition,
% 2.91/1.76 ( quantDeMorgan1
% 2.91/1.76 = ( ! [A: $i,Xphi: $i > $o] :
% 2.91/1.76 ( ~ ! [Xx: $i] :
% 2.91/1.76 ( ( in @ Xx @ A )
% 2.91/1.76 => ( Xphi @ Xx ) )
% 2.91/1.76 => ? [Xx: $i] :
% 2.91/1.76 ( ( in @ Xx @ A )
% 2.91/1.76 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(quantDeMorgan2_type,type,
% 2.91/1.76 quantDeMorgan2: $o ).
% 2.91/1.76
% 2.91/1.76 thf(quantDeMorgan2,definition,
% 2.91/1.76 ( quantDeMorgan2
% 2.91/1.76 = ( ! [A: $i,Xphi: $i > $o] :
% 2.91/1.76 ( ! [Xx: $i] :
% 2.91/1.76 ( ( in @ Xx @ A )
% 2.91/1.76 => ~ ( Xphi @ Xx ) )
% 2.91/1.76 => ~ ? [Xx: $i] :
% 2.91/1.76 ( ( in @ Xx @ A )
% 2.91/1.76 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(quantDeMorgan3_type,type,
% 2.91/1.76 quantDeMorgan3: $o ).
% 2.91/1.76
% 2.91/1.76 thf(quantDeMorgan3,definition,
% 2.91/1.76 ( quantDeMorgan3
% 2.91/1.76 = ( ! [A: $i,Xphi: $i > $o] :
% 2.91/1.76 ( ~ ? [Xx: $i] :
% 2.91/1.76 ( ( in @ Xx @ A )
% 2.91/1.76 & ( Xphi @ Xx ) )
% 2.91/1.76 => ! [Xx: $i] :
% 2.91/1.76 ( ( in @ Xx @ A )
% 2.91/1.76 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(quantDeMorgan4_type,type,
% 2.91/1.76 quantDeMorgan4: $o ).
% 2.91/1.76
% 2.91/1.76 thf(quantDeMorgan4,definition,
% 2.91/1.76 ( quantDeMorgan4
% 2.91/1.76 = ( ! [A: $i,Xphi: $i > $o] :
% 2.91/1.76 ( ? [Xx: $i] :
% 2.91/1.76 ( ( in @ Xx @ A )
% 2.91/1.76 & ~ ( Xphi @ Xx ) )
% 2.91/1.76 => ~ ! [Xx: $i] :
% 2.91/1.76 ( ( in @ Xx @ A )
% 2.91/1.76 => ( Xphi @ Xx ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(prop2setI_type,type,
% 2.91/1.76 prop2setI: $o ).
% 2.91/1.76
% 2.91/1.76 thf(prop2setI,definition,
% 2.91/1.76 ( prop2setI
% 2.91/1.76 = ( ! [Xphi: $o] :
% 2.91/1.76 ( Xphi
% 2.91/1.76 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(set2prop_type,type,
% 2.91/1.76 set2prop: $i > $o ).
% 2.91/1.76
% 2.91/1.76 thf(prop2set2propI_type,type,
% 2.91/1.76 prop2set2propI: $o ).
% 2.91/1.76
% 2.91/1.76 thf(prop2set2propI,definition,
% 2.91/1.76 ( prop2set2propI
% 2.91/1.76 = ( ! [Xphi: $o] :
% 2.91/1.76 ( Xphi
% 2.91/1.76 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(notdexE_type,type,
% 2.91/1.76 notdexE: $o ).
% 2.91/1.76
% 2.91/1.76 thf(notdexE,definition,
% 2.91/1.76 ( notdexE
% 2.91/1.76 = ( ! [A: $i,Xphi: $i > $o] :
% 2.91/1.76 ( ~ ? [Xx: $i] :
% 2.91/1.76 ( ( in @ Xx @ A )
% 2.91/1.76 & ( Xphi @ Xx ) )
% 2.91/1.76 => ! [Xx: $i] :
% 2.91/1.76 ( ( in @ Xx @ A )
% 2.91/1.76 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(notdallE_type,type,
% 2.91/1.76 notdallE: $o ).
% 2.91/1.76
% 2.91/1.76 thf(notdallE,definition,
% 2.91/1.76 ( notdallE
% 2.91/1.76 = ( ! [A: $i,Xphi: $i > $o] :
% 2.91/1.76 ( ~ ! [Xx: $i] :
% 2.91/1.76 ( ( in @ Xx @ A )
% 2.91/1.76 => ( Xphi @ Xx ) )
% 2.91/1.76 => ? [Xx: $i] :
% 2.91/1.76 ( ( in @ Xx @ A )
% 2.91/1.76 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(exuI1_type,type,
% 2.91/1.76 exuI1: $o ).
% 2.91/1.76
% 2.91/1.76 thf(exuI1,definition,
% 2.91/1.76 ( exuI1
% 2.91/1.76 = ( ! [Xphi: $i > $o] :
% 2.91/1.76 ( ? [Xx: $i] :
% 2.91/1.76 ( ( Xphi @ Xx )
% 2.91/1.76 & ! [Xy: $i] :
% 2.91/1.76 ( ( Xphi @ Xy )
% 2.91/1.76 => ( Xx = Xy ) ) )
% 2.91/1.76 => ( exu
% 2.91/1.76 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(exuI3_type,type,
% 2.91/1.76 exuI3: $o ).
% 2.91/1.76
% 2.91/1.76 thf(exuI3,definition,
% 2.91/1.76 ( exuI3
% 2.91/1.76 = ( ! [Xphi: $i > $o] :
% 2.91/1.76 ( ? [Xx: $i] : ( Xphi @ Xx )
% 2.91/1.76 => ( ! [Xx: $i,Xy: $i] :
% 2.91/1.76 ( ( Xphi @ Xx )
% 2.91/1.76 => ( ( Xphi @ Xy )
% 2.91/1.76 => ( Xx = Xy ) ) )
% 2.91/1.76 => ( exu
% 2.91/1.76 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(exuI2_type,type,
% 2.91/1.76 exuI2: $o ).
% 2.91/1.76
% 2.91/1.76 thf(exuI2,definition,
% 2.91/1.76 ( exuI2
% 2.91/1.76 = ( ! [Xphi: $i > $o] :
% 2.91/1.76 ( ? [Xx: $i] :
% 2.91/1.76 ! [Xy: $i] :
% 2.91/1.76 ( ( Xphi @ Xy )
% 2.91/1.76 <=> ( Xy = Xx ) )
% 2.91/1.76 => ( exu
% 2.91/1.76 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(inCongP_type,type,
% 2.91/1.76 inCongP: $o ).
% 2.91/1.76
% 2.91/1.76 thf(inCongP,definition,
% 2.91/1.76 ( inCongP
% 2.91/1.76 = ( ! [A: $i,B: $i] :
% 2.91/1.76 ( ( A = B )
% 2.91/1.76 => ! [Xx: $i,Xy: $i] :
% 2.91/1.76 ( ( Xx = Xy )
% 2.91/1.76 => ( ( in @ Xx @ A )
% 2.91/1.76 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(in__Cong_type,type,
% 2.91/1.76 in__Cong: $o ).
% 2.91/1.76
% 2.91/1.76 thf(in__Cong,definition,
% 2.91/1.76 ( in__Cong
% 2.91/1.76 = ( ! [A: $i,B: $i] :
% 2.91/1.76 ( ( A = B )
% 2.91/1.76 => ! [Xx: $i,Xy: $i] :
% 2.91/1.76 ( ( Xx = Xy )
% 2.91/1.76 => ( ( in @ Xx @ A )
% 2.91/1.76 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(exuE3u_type,type,
% 2.91/1.76 exuE3u: $o ).
% 2.91/1.76
% 2.91/1.76 thf(exuE3u,definition,
% 2.91/1.76 ( exuE3u
% 2.91/1.76 = ( ! [Xphi: $i > $o] :
% 2.91/1.76 ( ( exu
% 2.91/1.76 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.91/1.76 => ! [Xx: $i,Xy: $i] :
% 2.91/1.76 ( ( Xphi @ Xx )
% 2.91/1.76 => ( ( Xphi @ Xy )
% 2.91/1.76 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(exu__Cong_type,type,
% 2.91/1.76 exu__Cong: $o ).
% 2.91/1.76
% 2.91/1.76 thf(exu__Cong,definition,
% 2.91/1.76 ( exu__Cong
% 2.91/1.76 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.91/1.76 ( ! [Xx: $i,Xy: $i] :
% 2.91/1.76 ( ( Xx = Xy )
% 2.91/1.76 => ( ( Xphi @ Xx )
% 2.91/1.76 <=> ( Xpsi @ Xy ) ) )
% 2.91/1.76 => ( ( exu
% 2.91/1.76 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.91/1.76 <=> ( exu
% 2.91/1.76 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(emptyset__Cong_type,type,
% 2.91/1.76 emptyset__Cong: $o ).
% 2.91/1.76
% 2.91/1.76 thf(emptyset__Cong,definition,
% 2.91/1.76 ( emptyset__Cong
% 2.91/1.76 = ( emptyset = emptyset ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(setadjoin__Cong_type,type,
% 2.91/1.76 setadjoin__Cong: $o ).
% 2.91/1.76
% 2.91/1.76 thf(setadjoin__Cong,definition,
% 2.91/1.76 ( setadjoin__Cong
% 2.91/1.76 = ( ! [Xx: $i,Xy: $i] :
% 2.91/1.76 ( ( Xx = Xy )
% 2.91/1.76 => ! [Xz: $i,Xu: $i] :
% 2.91/1.76 ( ( Xz = Xu )
% 2.91/1.76 => ( ( setadjoin @ Xx @ Xz )
% 2.91/1.76 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(powerset__Cong_type,type,
% 2.91/1.76 powerset__Cong: $o ).
% 2.91/1.76
% 2.91/1.76 thf(powerset__Cong,definition,
% 2.91/1.76 ( powerset__Cong
% 2.91/1.76 = ( ! [A: $i,B: $i] :
% 2.91/1.76 ( ( A = B )
% 2.91/1.76 => ( ( powerset @ A )
% 2.91/1.76 = ( powerset @ B ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(setunion__Cong_type,type,
% 2.91/1.76 setunion__Cong: $o ).
% 2.91/1.76
% 2.91/1.76 thf(setunion__Cong,definition,
% 2.91/1.76 ( setunion__Cong
% 2.91/1.76 = ( ! [A: $i,B: $i] :
% 2.91/1.76 ( ( A = B )
% 2.91/1.76 => ( ( setunion @ A )
% 2.91/1.76 = ( setunion @ B ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(omega__Cong_type,type,
% 2.91/1.76 omega__Cong: $o ).
% 2.91/1.76
% 2.91/1.76 thf(omega__Cong,definition,
% 2.91/1.76 ( omega__Cong
% 2.91/1.76 = ( omega = omega ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(exuEu_type,type,
% 2.91/1.76 exuEu: $o ).
% 2.91/1.76
% 2.91/1.76 thf(exuEu,definition,
% 2.91/1.76 ( exuEu
% 2.91/1.76 = ( ! [Xphi: $i > $o] :
% 2.91/1.76 ( ( exu
% 2.91/1.76 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.91/1.76 => ! [Xx: $i,Xy: $i] :
% 2.91/1.76 ( ( Xphi @ Xx )
% 2.91/1.76 => ( ( Xphi @ Xy )
% 2.91/1.76 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(descr__Cong_type,type,
% 2.91/1.76 descr__Cong: $o ).
% 2.91/1.76
% 2.91/1.76 thf(descr__Cong,definition,
% 2.91/1.76 ( descr__Cong
% 2.91/1.76 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.91/1.76 ( ! [Xx: $i,Xy: $i] :
% 2.91/1.76 ( ( Xx = Xy )
% 2.91/1.76 => ( ( Xphi @ Xx )
% 2.91/1.76 <=> ( Xpsi @ Xy ) ) )
% 2.91/1.76 => ( ( exu
% 2.91/1.76 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.91/1.76 => ( ( exu
% 2.91/1.76 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 2.91/1.76 => ( ( descr
% 2.91/1.76 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.91/1.76 = ( descr
% 2.91/1.76 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(dsetconstr__Cong_type,type,
% 2.91/1.76 dsetconstr__Cong: $o ).
% 2.91/1.76
% 2.91/1.76 thf(dsetconstr__Cong,definition,
% 2.91/1.76 ( dsetconstr__Cong
% 2.91/1.76 = ( ! [A: $i,B: $i] :
% 2.91/1.76 ( ( A = B )
% 2.91/1.76 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.91/1.76 ( ! [Xx: $i] :
% 2.91/1.76 ( ( in @ Xx @ A )
% 2.91/1.76 => ! [Xy: $i] :
% 2.91/1.76 ( ( in @ Xy @ B )
% 2.91/1.76 => ( ( Xx = Xy )
% 2.91/1.76 => ( ( Xphi @ Xx )
% 2.91/1.76 <=> ( Xpsi @ Xy ) ) ) ) )
% 2.91/1.76 => ( ( dsetconstr @ A
% 2.91/1.76 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.91/1.76 = ( dsetconstr @ B
% 2.91/1.76 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(subset_type,type,
% 2.91/1.76 subset: $i > $i > $o ).
% 2.91/1.76
% 2.91/1.76 thf(disjoint_type,type,
% 2.91/1.76 disjoint: $i > $i > $o ).
% 2.91/1.76
% 2.91/1.76 thf(setsmeet_type,type,
% 2.91/1.76 setsmeet: $i > $i > $o ).
% 2.91/1.76
% 2.91/1.76 thf(subsetI1_type,type,
% 2.91/1.76 subsetI1: $o ).
% 2.91/1.76
% 2.91/1.76 thf(subsetI1,definition,
% 2.91/1.76 ( subsetI1
% 2.91/1.76 = ( ! [A: $i,B: $i] :
% 2.91/1.76 ( ! [Xx: $i] :
% 2.91/1.76 ( ( in @ Xx @ A )
% 2.91/1.76 => ( in @ Xx @ B ) )
% 2.91/1.76 => ( subset @ A @ B ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(eqimpsubset2_type,type,
% 2.91/1.76 eqimpsubset2: $o ).
% 2.91/1.76
% 2.91/1.76 thf(eqimpsubset2,definition,
% 2.91/1.76 ( eqimpsubset2
% 2.91/1.76 = ( ! [A: $i,B: $i] :
% 2.91/1.76 ( ( A = B )
% 2.91/1.76 => ( subset @ B @ A ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(eqimpsubset1_type,type,
% 2.91/1.76 eqimpsubset1: $o ).
% 2.91/1.76
% 2.91/1.76 thf(eqimpsubset1,definition,
% 2.91/1.76 ( eqimpsubset1
% 2.91/1.76 = ( ! [A: $i,B: $i] :
% 2.91/1.76 ( ( A = B )
% 2.91/1.76 => ( subset @ A @ B ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(subsetI2_type,type,
% 2.91/1.76 subsetI2: $o ).
% 2.91/1.76
% 2.91/1.76 thf(subsetI2,definition,
% 2.91/1.76 ( subsetI2
% 2.91/1.76 = ( ! [A: $i,B: $i] :
% 2.91/1.76 ( ! [Xx: $i] :
% 2.91/1.76 ( ( in @ Xx @ A )
% 2.91/1.76 => ( in @ Xx @ B ) )
% 2.91/1.76 => ( subset @ A @ B ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(emptysetsubset_type,type,
% 2.91/1.76 emptysetsubset: $o ).
% 2.91/1.76
% 2.91/1.76 thf(emptysetsubset,definition,
% 2.91/1.76 ( emptysetsubset
% 2.91/1.76 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(subsetE_type,type,
% 2.91/1.76 subsetE: $o ).
% 2.91/1.76
% 2.91/1.76 thf(subsetE,definition,
% 2.91/1.76 ( subsetE
% 2.91/1.76 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.91/1.76 ( ( subset @ A @ B )
% 2.91/1.76 => ( ( in @ Xx @ A )
% 2.91/1.76 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(subsetE2_type,type,
% 2.91/1.76 subsetE2: $o ).
% 2.91/1.76
% 2.91/1.76 thf(subsetE2,definition,
% 2.91/1.76 ( subsetE2
% 2.91/1.76 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.91/1.76 ( ( subset @ A @ B )
% 2.91/1.76 => ( ~ ( in @ Xx @ B )
% 2.91/1.76 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(notsubsetI_type,type,
% 2.91/1.76 notsubsetI: $o ).
% 2.91/1.76
% 2.91/1.76 thf(notsubsetI,definition,
% 2.91/1.76 ( notsubsetI
% 2.91/1.76 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.91/1.76 ( ( in @ Xx @ A )
% 2.91/1.76 => ( ~ ( in @ Xx @ B )
% 2.91/1.76 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(notequalI1_type,type,
% 2.91/1.76 notequalI1: $o ).
% 2.91/1.76
% 2.91/1.76 thf(notequalI1,definition,
% 2.91/1.76 ( notequalI1
% 2.91/1.76 = ( ! [A: $i,B: $i] :
% 2.91/1.76 ( ~ ( subset @ A @ B )
% 2.91/1.76 => ( A != B ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(notequalI2_type,type,
% 2.91/1.76 notequalI2: $o ).
% 2.91/1.76
% 2.91/1.76 thf(notequalI2,definition,
% 2.91/1.76 ( notequalI2
% 2.91/1.76 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.91/1.76 ( ( in @ Xx @ A )
% 2.91/1.76 => ( ~ ( in @ Xx @ B )
% 2.91/1.76 => ( A != B ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(subsetRefl_type,type,
% 2.91/1.76 subsetRefl: $o ).
% 2.91/1.76
% 2.91/1.76 thf(subsetRefl,definition,
% 2.91/1.76 ( subsetRefl
% 2.91/1.76 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(subsetTrans_type,type,
% 2.91/1.76 subsetTrans: $o ).
% 2.91/1.76
% 2.91/1.76 thf(subsetTrans,definition,
% 2.91/1.76 ( subsetTrans
% 2.91/1.76 = ( ! [A: $i,B: $i,C: $i] :
% 2.91/1.76 ( ( subset @ A @ B )
% 2.91/1.76 => ( ( subset @ B @ C )
% 2.91/1.76 => ( subset @ A @ C ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(setadjoinSub_type,type,
% 2.91/1.76 setadjoinSub: $o ).
% 2.91/1.76
% 2.91/1.76 thf(setadjoinSub,definition,
% 2.91/1.76 ( setadjoinSub
% 2.91/1.76 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(setadjoinSub2_type,type,
% 2.91/1.76 setadjoinSub2: $o ).
% 2.91/1.76
% 2.91/1.76 thf(setadjoinSub2,definition,
% 2.91/1.76 ( setadjoinSub2
% 2.91/1.76 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.91/1.76 ( ( subset @ A @ B )
% 2.91/1.76 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(subset2powerset_type,type,
% 2.91/1.76 subset2powerset: $o ).
% 2.91/1.76
% 2.91/1.76 thf(subset2powerset,definition,
% 2.91/1.76 ( subset2powerset
% 2.91/1.76 = ( ! [A: $i,B: $i] :
% 2.91/1.76 ( ( subset @ A @ B )
% 2.91/1.76 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(setextsub_type,type,
% 2.91/1.76 setextsub: $o ).
% 2.91/1.76
% 2.91/1.76 thf(setextsub,definition,
% 2.91/1.76 ( setextsub
% 2.91/1.76 = ( ! [A: $i,B: $i] :
% 2.91/1.76 ( ( subset @ A @ B )
% 2.91/1.76 => ( ( subset @ B @ A )
% 2.91/1.76 => ( A = B ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(subsetemptysetimpeq_type,type,
% 2.91/1.76 subsetemptysetimpeq: $o ).
% 2.91/1.76
% 2.91/1.76 thf(subsetemptysetimpeq,definition,
% 2.91/1.76 ( subsetemptysetimpeq
% 2.91/1.76 = ( ! [A: $i] :
% 2.91/1.76 ( ( subset @ A @ emptyset )
% 2.91/1.76 => ( A = emptyset ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(powersetI1_type,type,
% 2.91/1.76 powersetI1: $o ).
% 2.91/1.76
% 2.91/1.76 thf(powersetI1,definition,
% 2.91/1.76 ( powersetI1
% 2.91/1.76 = ( ! [A: $i,B: $i] :
% 2.91/1.76 ( ( subset @ B @ A )
% 2.91/1.76 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(powersetE1_type,type,
% 2.91/1.76 powersetE1: $o ).
% 2.91/1.76
% 2.91/1.76 thf(powersetE1,definition,
% 2.91/1.76 ( powersetE1
% 2.91/1.76 = ( ! [A: $i,B: $i] :
% 2.91/1.76 ( ( in @ B @ ( powerset @ A ) )
% 2.91/1.76 => ( subset @ B @ A ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(inPowerset_type,type,
% 2.91/1.76 inPowerset: $o ).
% 2.91/1.76
% 2.91/1.76 thf(inPowerset,definition,
% 2.91/1.76 ( inPowerset
% 2.91/1.76 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(powersetsubset_type,type,
% 2.91/1.76 powersetsubset: $o ).
% 2.91/1.76
% 2.91/1.76 thf(powersetsubset,definition,
% 2.91/1.76 ( powersetsubset
% 2.91/1.76 = ( ! [A: $i,B: $i] :
% 2.91/1.76 ( ( subset @ A @ B )
% 2.91/1.76 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(sepInPowerset_type,type,
% 2.91/1.76 sepInPowerset: $o ).
% 2.91/1.76
% 2.91/1.76 thf(sepInPowerset,definition,
% 2.91/1.76 ( sepInPowerset
% 2.91/1.76 = ( ! [A: $i,Xphi: $i > $o] :
% 2.91/1.76 ( in
% 2.91/1.76 @ ( dsetconstr @ A
% 2.91/1.76 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.91/1.76 @ ( powerset @ A ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(sepSubset_type,type,
% 2.91/1.76 sepSubset: $o ).
% 2.91/1.76
% 2.91/1.76 thf(sepSubset,definition,
% 2.91/1.76 ( sepSubset
% 2.91/1.76 = ( ! [A: $i,Xphi: $i > $o] :
% 2.91/1.76 ( subset
% 2.91/1.76 @ ( dsetconstr @ A
% 2.91/1.76 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.91/1.76 @ A ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(binunion_type,type,
% 2.91/1.76 binunion: $i > $i > $i ).
% 2.91/1.76
% 2.91/1.76 thf(binunionIL_type,type,
% 2.91/1.76 binunionIL: $o ).
% 2.91/1.76
% 2.91/1.76 thf(binunionIL,definition,
% 2.91/1.76 ( binunionIL
% 2.91/1.76 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.91/1.76 ( ( in @ Xx @ A )
% 2.91/1.76 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(upairset2IR_type,type,
% 2.91/1.76 upairset2IR: $o ).
% 2.91/1.76
% 2.91/1.76 thf(upairset2IR,definition,
% 2.91/1.76 ( upairset2IR
% 2.91/1.76 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(binunionIR_type,type,
% 2.91/1.76 binunionIR: $o ).
% 2.91/1.76
% 2.91/1.76 thf(binunionIR,definition,
% 2.91/1.76 ( binunionIR
% 2.91/1.76 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.91/1.76 ( ( in @ Xx @ B )
% 2.91/1.76 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(binunionEcases_type,type,
% 2.91/1.76 binunionEcases: $o ).
% 2.91/1.76
% 2.91/1.76 thf(binunionEcases,definition,
% 2.91/1.76 ( binunionEcases
% 2.91/1.76 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 2.91/1.76 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.91/1.76 => ( ( ( in @ Xx @ A )
% 2.91/1.76 => Xphi )
% 2.91/1.76 => ( ( ( in @ Xx @ B )
% 2.91/1.76 => Xphi )
% 2.91/1.76 => Xphi ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(binunionE_type,type,
% 2.91/1.76 binunionE: $o ).
% 2.91/1.76
% 2.91/1.76 thf(binunionE,definition,
% 2.91/1.76 ( binunionE
% 2.91/1.76 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.91/1.76 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.91/1.76 => ( ( in @ Xx @ A )
% 2.91/1.76 | ( in @ Xx @ B ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(binunionLsub_type,type,
% 2.91/1.76 binunionLsub: $o ).
% 2.91/1.76
% 2.91/1.76 thf(binunionLsub,definition,
% 2.91/1.76 ( binunionLsub
% 2.91/1.76 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(binunionRsub_type,type,
% 2.91/1.76 binunionRsub: $o ).
% 2.91/1.76
% 2.91/1.76 thf(binunionRsub,definition,
% 2.91/1.76 ( binunionRsub
% 2.91/1.76 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(binintersect_type,type,
% 2.91/1.76 binintersect: $i > $i > $i ).
% 2.91/1.76
% 2.91/1.76 thf(binintersectI_type,type,
% 2.91/1.76 binintersectI: $o ).
% 2.91/1.76
% 2.91/1.76 thf(binintersectI,definition,
% 2.91/1.76 ( binintersectI
% 2.91/1.76 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.91/1.76 ( ( in @ Xx @ A )
% 2.91/1.76 => ( ( in @ Xx @ B )
% 2.91/1.76 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(binintersectSubset5_type,type,
% 2.91/1.76 binintersectSubset5: $o ).
% 2.91/1.76
% 2.91/1.76 thf(binintersectSubset5,definition,
% 2.91/1.76 ( binintersectSubset5
% 2.91/1.76 = ( ! [A: $i,B: $i,C: $i] :
% 2.91/1.76 ( ( subset @ C @ A )
% 2.91/1.76 => ( ( subset @ C @ B )
% 2.91/1.76 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(binintersectEL_type,type,
% 2.91/1.76 binintersectEL: $o ).
% 2.91/1.76
% 2.91/1.76 thf(binintersectEL,definition,
% 2.91/1.76 ( binintersectEL
% 2.91/1.76 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.91/1.76 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.91/1.76 => ( in @ Xx @ A ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(binintersectLsub_type,type,
% 2.91/1.76 binintersectLsub: $o ).
% 2.91/1.76
% 2.91/1.76 thf(binintersectLsub,definition,
% 2.91/1.76 ( binintersectLsub
% 2.91/1.76 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(binintersectSubset2_type,type,
% 2.91/1.76 binintersectSubset2: $o ).
% 2.91/1.76
% 2.91/1.76 thf(binintersectSubset2,definition,
% 2.91/1.76 ( binintersectSubset2
% 2.91/1.76 = ( ! [A: $i,B: $i] :
% 2.91/1.76 ( ( subset @ A @ B )
% 2.91/1.76 => ( ( binintersect @ A @ B )
% 2.91/1.76 = A ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(binintersectSubset3_type,type,
% 2.91/1.76 binintersectSubset3: $o ).
% 2.91/1.76
% 2.91/1.76 thf(binintersectSubset3,definition,
% 2.91/1.76 ( binintersectSubset3
% 2.91/1.76 = ( ! [A: $i,B: $i] :
% 2.91/1.76 ( ( ( binintersect @ A @ B )
% 2.91/1.76 = B )
% 2.91/1.76 => ( subset @ B @ A ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(binintersectER_type,type,
% 2.91/1.76 binintersectER: $o ).
% 2.91/1.76
% 2.91/1.76 thf(binintersectER,definition,
% 2.91/1.76 ( binintersectER
% 2.91/1.76 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.91/1.76 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.91/1.76 => ( in @ Xx @ B ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(disjointsetsI1_type,type,
% 2.91/1.76 disjointsetsI1: $o ).
% 2.91/1.76
% 2.91/1.76 thf(disjointsetsI1,definition,
% 2.91/1.76 ( disjointsetsI1
% 2.91/1.76 = ( ! [A: $i,B: $i] :
% 2.91/1.76 ( ~ ? [Xx: $i] :
% 2.91/1.76 ( ( in @ Xx @ A )
% 2.91/1.76 & ( in @ Xx @ B ) )
% 2.91/1.76 => ( ( binintersect @ A @ B )
% 2.91/1.76 = emptyset ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(binintersectRsub_type,type,
% 2.91/1.76 binintersectRsub: $o ).
% 2.91/1.76
% 2.91/1.76 thf(binintersectRsub,definition,
% 2.91/1.76 ( binintersectRsub
% 2.91/1.76 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(binintersectSubset4_type,type,
% 2.91/1.76 binintersectSubset4: $o ).
% 2.91/1.76
% 2.91/1.76 thf(binintersectSubset4,definition,
% 2.91/1.76 ( binintersectSubset4
% 2.91/1.76 = ( ! [A: $i,B: $i] :
% 2.91/1.76 ( ( subset @ B @ A )
% 2.91/1.76 => ( ( binintersect @ A @ B )
% 2.91/1.76 = B ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(binintersectSubset1_type,type,
% 2.91/1.76 binintersectSubset1: $o ).
% 2.91/1.76
% 2.91/1.76 thf(binintersectSubset1,definition,
% 2.91/1.76 ( binintersectSubset1
% 2.91/1.76 = ( ! [A: $i,B: $i] :
% 2.91/1.76 ( ( ( binintersect @ A @ B )
% 2.91/1.76 = A )
% 2.91/1.76 => ( subset @ A @ B ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(bs114d_type,type,
% 2.91/1.76 bs114d: $o ).
% 2.91/1.76
% 2.91/1.76 thf(bs114d,definition,
% 2.91/1.76 ( bs114d
% 2.91/1.76 = ( ! [A: $i,B: $i,C: $i] :
% 2.91/1.76 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 2.91/1.76 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(regular_type,type,
% 2.91/1.76 regular: $i > $o ).
% 2.91/1.76
% 2.91/1.76 thf(setminus_type,type,
% 2.91/1.76 setminus: $i > $i > $i ).
% 2.91/1.76
% 2.91/1.76 thf(setminusI_type,type,
% 2.91/1.76 setminusI: $o ).
% 2.91/1.76
% 2.91/1.76 thf(setminusI,definition,
% 2.91/1.76 ( setminusI
% 2.91/1.76 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.91/1.76 ( ( in @ Xx @ A )
% 2.91/1.76 => ( ~ ( in @ Xx @ B )
% 2.91/1.76 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(setminusEL_type,type,
% 2.91/1.76 setminusEL: $o ).
% 2.91/1.76
% 2.91/1.76 thf(setminusEL,definition,
% 2.91/1.76 ( setminusEL
% 2.91/1.76 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.91/1.76 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.91/1.76 => ( in @ Xx @ A ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(setminusER_type,type,
% 2.91/1.76 setminusER: $o ).
% 2.91/1.76
% 2.91/1.76 thf(setminusER,definition,
% 2.91/1.76 ( setminusER
% 2.91/1.76 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.91/1.76 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.91/1.76 => ~ ( in @ Xx @ B ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(setminusSubset2_type,type,
% 2.91/1.76 setminusSubset2: $o ).
% 2.91/1.76
% 2.91/1.76 thf(setminusSubset2,definition,
% 2.91/1.76 ( setminusSubset2
% 2.91/1.76 = ( ! [A: $i,B: $i] :
% 2.91/1.76 ( ( subset @ A @ B )
% 2.91/1.76 => ( ( setminus @ A @ B )
% 2.91/1.76 = emptyset ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(setminusERneg_type,type,
% 2.91/1.76 setminusERneg: $o ).
% 2.91/1.76
% 2.91/1.76 thf(setminusERneg,definition,
% 2.91/1.76 ( setminusERneg
% 2.91/1.76 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.91/1.76 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.91/1.76 => ( ( in @ Xx @ A )
% 2.91/1.76 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(setminusELneg_type,type,
% 2.91/1.76 setminusELneg: $o ).
% 2.91/1.76
% 2.91/1.76 thf(setminusELneg,definition,
% 2.91/1.76 ( setminusELneg
% 2.91/1.76 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.91/1.76 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.91/1.76 => ( ~ ( in @ Xx @ B )
% 2.91/1.76 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(setminusILneg_type,type,
% 2.91/1.76 setminusILneg: $o ).
% 2.91/1.76
% 2.91/1.76 thf(setminusILneg,definition,
% 2.91/1.76 ( setminusILneg
% 2.91/1.76 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.91/1.76 ( ~ ( in @ Xx @ A )
% 2.91/1.76 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(setminusIRneg_type,type,
% 2.91/1.76 setminusIRneg: $o ).
% 2.91/1.76
% 2.91/1.76 thf(setminusIRneg,definition,
% 2.91/1.76 ( setminusIRneg
% 2.91/1.76 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.91/1.76 ( ( in @ Xx @ B )
% 2.91/1.76 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(setminusLsub_type,type,
% 2.91/1.76 setminusLsub: $o ).
% 2.91/1.76
% 2.91/1.76 thf(setminusLsub,definition,
% 2.91/1.76 ( setminusLsub
% 2.91/1.76 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(setminusSubset1_type,type,
% 2.91/1.76 setminusSubset1: $o ).
% 2.91/1.76
% 2.91/1.76 thf(setminusSubset1,definition,
% 2.91/1.76 ( setminusSubset1
% 2.91/1.76 = ( ! [A: $i,B: $i] :
% 2.91/1.76 ( ( ( setminus @ A @ B )
% 2.91/1.76 = emptyset )
% 2.91/1.76 => ( subset @ A @ B ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(symdiff_type,type,
% 2.91/1.76 symdiff: $i > $i > $i ).
% 2.91/1.76
% 2.91/1.76 thf(symdiffE_type,type,
% 2.91/1.76 symdiffE: $o ).
% 2.91/1.76
% 2.91/1.76 thf(symdiffE,definition,
% 2.91/1.76 ( symdiffE
% 2.91/1.76 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.91/1.76 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 2.91/1.76 => ! [Xphi: $o] :
% 2.91/1.76 ( ( ( in @ Xx @ A )
% 2.91/1.76 => ( ~ ( in @ Xx @ B )
% 2.91/1.76 => Xphi ) )
% 2.91/1.76 => ( ( ~ ( in @ Xx @ A )
% 2.91/1.76 => ( ( in @ Xx @ B )
% 2.91/1.76 => Xphi ) )
% 2.91/1.76 => Xphi ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(symdiffI1_type,type,
% 2.91/1.76 symdiffI1: $o ).
% 2.91/1.76
% 2.91/1.76 thf(symdiffI1,definition,
% 2.91/1.76 ( symdiffI1
% 2.91/1.76 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.91/1.76 ( ( in @ Xx @ A )
% 2.91/1.76 => ( ~ ( in @ Xx @ B )
% 2.91/1.76 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(symdiffI2_type,type,
% 2.91/1.76 symdiffI2: $o ).
% 2.91/1.76
% 2.91/1.76 thf(symdiffI2,definition,
% 2.91/1.76 ( symdiffI2
% 2.91/1.76 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.91/1.76 ( ~ ( in @ Xx @ A )
% 2.91/1.76 => ( ( in @ Xx @ B )
% 2.91/1.76 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(symdiffIneg1_type,type,
% 2.91/1.76 symdiffIneg1: $o ).
% 2.91/1.76
% 2.91/1.76 thf(symdiffIneg1,definition,
% 2.91/1.76 ( symdiffIneg1
% 2.91/1.76 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.91/1.76 ( ( in @ Xx @ A )
% 2.91/1.76 => ( ( in @ Xx @ B )
% 2.91/1.76 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(symdiffIneg2_type,type,
% 2.91/1.76 symdiffIneg2: $o ).
% 2.91/1.76
% 2.91/1.76 thf(symdiffIneg2,definition,
% 2.91/1.76 ( symdiffIneg2
% 2.91/1.76 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.91/1.76 ( ~ ( in @ Xx @ A )
% 2.91/1.76 => ( ~ ( in @ Xx @ B )
% 2.91/1.76 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(iskpair_type,type,
% 2.91/1.76 iskpair: $i > $o ).
% 2.91/1.76
% 2.91/1.76 thf(secondinupair_type,type,
% 2.91/1.76 secondinupair: $o ).
% 2.91/1.76
% 2.91/1.76 thf(secondinupair,definition,
% 2.91/1.76 ( secondinupair
% 2.91/1.76 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(setukpairIL_type,type,
% 2.91/1.76 setukpairIL: $o ).
% 2.91/1.76
% 2.91/1.76 thf(setukpairIL,definition,
% 2.91/1.76 ( setukpairIL
% 2.91/1.76 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(setukpairIR_type,type,
% 2.91/1.76 setukpairIR: $o ).
% 2.91/1.76
% 2.91/1.76 thf(setukpairIR,definition,
% 2.91/1.76 ( setukpairIR
% 2.91/1.76 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(kpairiskpair_type,type,
% 2.91/1.76 kpairiskpair: $o ).
% 2.91/1.76
% 2.91/1.76 thf(kpairiskpair,definition,
% 2.91/1.76 ( kpairiskpair
% 2.91/1.76 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(kpair_type,type,
% 2.91/1.76 kpair: $i > $i > $i ).
% 2.91/1.76
% 2.91/1.76 thf(kpairp_type,type,
% 2.91/1.76 kpairp: $o ).
% 2.91/1.76
% 2.91/1.76 thf(kpairp,definition,
% 2.91/1.76 ( kpairp
% 2.91/1.76 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(cartprod_type,type,
% 2.91/1.76 cartprod: $i > $i > $i ).
% 2.91/1.76
% 2.91/1.76 thf(singletonsubset_type,type,
% 2.91/1.76 singletonsubset: $o ).
% 2.91/1.76
% 2.91/1.76 thf(singletonsubset,definition,
% 2.91/1.76 ( singletonsubset
% 2.91/1.76 = ( ! [A: $i,Xx: $i] :
% 2.91/1.76 ( ( in @ Xx @ A )
% 2.91/1.76 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(singletoninpowerset_type,type,
% 2.91/1.76 singletoninpowerset: $o ).
% 2.91/1.76
% 2.91/1.76 thf(singletoninpowerset,definition,
% 2.91/1.76 ( singletoninpowerset
% 2.91/1.76 = ( ! [A: $i,Xx: $i] :
% 2.91/1.76 ( ( in @ Xx @ A )
% 2.91/1.76 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(singletoninpowunion_type,type,
% 2.91/1.76 singletoninpowunion: $o ).
% 2.91/1.76
% 2.91/1.76 thf(singletoninpowunion,definition,
% 2.91/1.76 ( singletoninpowunion
% 2.91/1.76 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.91/1.76 ( ( in @ Xx @ A )
% 2.91/1.76 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.91/1.76
% 2.91/1.76 thf(upairset2E_type,type,
% 2.91/1.77 upairset2E: $o ).
% 2.91/1.77
% 2.91/1.77 thf(upairset2E,definition,
% 2.91/1.77 ( upairset2E
% 2.91/1.77 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.91/1.77 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.91/1.77 => ( ( Xz = Xx )
% 2.91/1.77 | ( Xz = Xy ) ) ) ) ) ).
% 2.91/1.77
% 2.91/1.77 thf(upairsubunion_type,type,
% 2.91/1.77 upairsubunion: $o ).
% 2.91/1.77
% 2.91/1.77 thf(upairsubunion,definition,
% 2.91/1.77 ( upairsubunion
% 2.91/1.77 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.91/1.77 ( ( in @ Xx @ A )
% 2.91/1.77 => ! [Xy: $i] :
% 2.91/1.77 ( ( in @ Xy @ B )
% 2.91/1.77 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.91/1.77
% 2.91/1.77 thf(upairinpowunion_type,type,
% 2.91/1.77 upairinpowunion: $o ).
% 2.91/1.77
% 2.91/1.77 thf(upairinpowunion,definition,
% 2.91/1.77 ( upairinpowunion
% 2.91/1.77 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.91/1.77 ( ( in @ Xx @ A )
% 2.91/1.77 => ! [Xy: $i] :
% 2.91/1.77 ( ( in @ Xy @ B )
% 2.91/1.77 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.91/1.77
% 2.91/1.77 thf(ubforcartprodlem1_type,type,
% 2.91/1.77 ubforcartprodlem1: $o ).
% 2.91/1.77
% 2.91/1.77 thf(ubforcartprodlem1,definition,
% 2.91/1.77 ( ubforcartprodlem1
% 2.91/1.77 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.91/1.77 ( ( in @ Xx @ A )
% 2.91/1.77 => ! [Xy: $i] :
% 2.91/1.77 ( ( in @ Xy @ B )
% 2.91/1.77 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.91/1.77
% 2.91/1.77 thf(ubforcartprodlem2_type,type,
% 2.91/1.77 ubforcartprodlem2: $o ).
% 2.91/1.77
% 2.91/1.77 thf(ubforcartprodlem2,definition,
% 2.91/1.77 ( ubforcartprodlem2
% 2.91/1.77 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.91/1.77 ( ( in @ Xx @ A )
% 2.91/1.77 => ! [Xy: $i] :
% 2.91/1.77 ( ( in @ Xy @ B )
% 2.91/1.77 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.91/1.77
% 2.91/1.77 thf(ubforcartprodlem3_type,type,
% 2.91/1.77 ubforcartprodlem3: $o ).
% 2.91/1.77
% 2.91/1.77 thf(ubforcartprodlem3,definition,
% 2.91/1.77 ( ubforcartprodlem3
% 2.91/1.77 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.91/1.77 ( ( in @ Xx @ A )
% 2.91/1.77 => ! [Xy: $i] :
% 2.91/1.77 ( ( in @ Xy @ B )
% 2.91/1.77 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.91/1.77
% 2.91/1.77 thf(cartprodpairin_type,type,
% 2.91/1.77 cartprodpairin: $o ).
% 2.91/1.77
% 2.91/1.77 thf(cartprodpairin,definition,
% 2.91/1.77 ( cartprodpairin
% 2.91/1.77 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.91/1.77 ( ( in @ Xx @ A )
% 2.91/1.77 => ! [Xy: $i] :
% 2.91/1.77 ( ( in @ Xy @ B )
% 2.91/1.77 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 2.91/1.77
% 2.91/1.77 thf(cartprodmempair1_type,type,
% 2.91/1.77 cartprodmempair1: $o ).
% 2.91/1.77
% 2.91/1.77 thf(cartprodmempair1,definition,
% 2.91/1.77 ( cartprodmempair1
% 2.91/1.77 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.91/1.77 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.91/1.77 => ? [Xx: $i] :
% 2.91/1.77 ( ( in @ Xx @ A )
% 2.91/1.77 & ? [Xy: $i] :
% 2.91/1.77 ( ( in @ Xy @ B )
% 2.91/1.77 & ( Xu
% 2.91/1.77 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 2.91/1.77
% 2.91/1.77 thf(cartprodmempair_type,type,
% 2.91/1.77 cartprodmempair: $o ).
% 2.91/1.77
% 2.91/1.77 thf(cartprodmempair,definition,
% 2.91/1.77 ( cartprodmempair
% 2.91/1.77 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.91/1.77 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.91/1.77 => ( iskpair @ Xu ) ) ) ) ).
% 2.91/1.77
% 2.91/1.77 thf(setunionE2_type,type,
% 2.91/1.77 setunionE2: $o ).
% 2.91/1.77
% 2.91/1.77 thf(setunionE2,definition,
% 2.91/1.77 ( setunionE2
% 2.91/1.77 = ( ! [A: $i,Xx: $i] :
% 2.91/1.77 ( ( in @ Xx @ ( setunion @ A ) )
% 2.91/1.77 => ? [X: $i] :
% 2.91/1.77 ( ( in @ X @ A )
% 2.91/1.77 & ( in @ Xx @ X ) ) ) ) ) ).
% 2.91/1.77
% 2.91/1.77 thf(setunionsingleton1_type,type,
% 2.91/1.77 setunionsingleton1: $o ).
% 2.91/1.77
% 2.91/1.77 thf(setunionsingleton1,definition,
% 2.91/1.77 ( setunionsingleton1
% 2.91/1.77 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 2.91/1.77
% 2.91/1.77 thf(setunionsingleton2_type,type,
% 2.91/1.77 setunionsingleton2: $o ).
% 2.91/1.77
% 2.91/1.77 thf(setunionsingleton2,definition,
% 2.91/1.77 ( setunionsingleton2
% 2.91/1.77 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 2.91/1.77
% 2.91/1.77 thf(setunionsingleton_type,type,
% 2.91/1.77 setunionsingleton: $o ).
% 2.91/1.77
% 2.91/1.77 thf(setunionsingleton,definition,
% 2.91/1.77 ( setunionsingleton
% 2.91/1.77 = ( ! [Xx: $i] :
% 2.91/1.77 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 2.91/1.77 = Xx ) ) ) ).
% 2.91/1.77
% 2.91/1.77 thf(singleton_type,type,
% 2.91/1.77 singleton: $i > $o ).
% 2.91/1.77
% 2.91/1.77 thf(singleton,definition,
% 2.91/1.77 ( singleton
% 2.91/1.77 = ( ^ [A: $i] :
% 2.91/1.77 ? [Xx: $i] :
% 2.91/1.77 ( ( in @ Xx @ A )
% 2.91/1.77 & ( A
% 2.91/1.77 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.91/1.77
% 2.91/1.77 thf(singletonprop_type,type,
% 2.91/1.77 singletonprop: $o ).
% 2.91/1.77
% 2.91/1.77 thf(singletonprop,definition,
% 2.91/1.77 ( singletonprop
% 2.91/1.77 = ( ! [A: $i,Xphi: $i > $o] :
% 2.91/1.77 ( ! [Xx: $i] :
% 2.91/1.77 ( ( in @ Xx @ A )
% 2.91/1.77 => ! [Xy: $i] :
% 2.91/1.77 ( ( in @ Xy @ A )
% 2.91/1.77 => ( ( Xphi @ Xx )
% 2.91/1.77 => ( ( Xphi @ Xy )
% 2.91/1.77 => ( Xx = Xy ) ) ) ) )
% 2.91/1.77 => ( ? [Xx: $i] :
% 2.91/1.77 ( ( in @ Xx @ A )
% 2.91/1.77 & ( Xphi @ Xx ) )
% 2.91/1.77 => ( singleton
% 2.91/1.77 @ ( dsetconstr @ A
% 2.91/1.77 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 2.91/1.77
% 2.91/1.77 thf(ex1_type,type,
% 2.91/1.77 ex1: $i > ( $i > $o ) > $o ).
% 2.91/1.77
% 2.91/1.77 thf(ex1,definition,
% 2.91/1.77 ( ex1
% 2.91/1.77 = ( ^ [A: $i,Xphi: $i > $o] :
% 2.91/1.77 ( singleton
% 2.91/1.77 @ ( dsetconstr @ A
% 2.91/1.77 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.91/1.77
% 2.91/1.77 thf(ex1E1_type,type,
% 2.91/1.77 ex1E1: $o ).
% 2.91/1.77
% 2.91/1.77 thf(ex1E1,definition,
% 2.91/1.77 ( ex1E1
% 2.91/1.77 = ( ! [A: $i,Xphi: $i > $o] :
% 2.91/1.77 ( ( ex1 @ A
% 2.91/1.77 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.91/1.77 => ? [Xx: $i] :
% 2.91/1.77 ( ( in @ Xx @ A )
% 2.91/1.77 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.91/1.77
% 2.91/1.77 thf(ex1I_type,type,
% 2.91/1.77 ex1I: $o ).
% 2.91/1.77
% 2.91/1.77 thf(ex1I,definition,
% 2.91/1.77 ( ex1I
% 2.91/1.77 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.91/1.77 ( ( in @ Xx @ A )
% 2.91/1.77 => ( ( Xphi @ Xx )
% 2.91/1.77 => ( ! [Xy: $i] :
% 2.91/1.77 ( ( in @ Xy @ A )
% 2.91/1.77 => ( ( Xphi @ Xy )
% 2.91/1.77 => ( Xy = Xx ) ) )
% 2.91/1.77 => ( ex1 @ A
% 2.91/1.77 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.91/1.77
% 2.91/1.77 thf(ex1I2_type,type,
% 2.91/1.77 ex1I2: $o ).
% 2.91/1.77
% 2.91/1.77 thf(ex1I2,definition,
% 2.91/1.77 ( ex1I2
% 2.91/1.77 = ( ! [A: $i,Xphi: $i > $o] :
% 2.91/1.77 ( ! [Xx: $i] :
% 2.91/1.77 ( ( in @ Xx @ A )
% 2.91/1.77 => ! [Xy: $i] :
% 2.91/1.77 ( ( in @ Xy @ A )
% 2.91/1.77 => ( ( Xphi @ Xx )
% 2.91/1.77 => ( ( Xphi @ Xy )
% 2.91/1.77 => ( Xx = Xy ) ) ) ) )
% 2.91/1.77 => ( ? [Xx: $i] :
% 2.91/1.77 ( ( in @ Xx @ A )
% 2.91/1.77 & ( Xphi @ Xx ) )
% 2.91/1.77 => ( ex1 @ A
% 2.91/1.77 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.91/1.77
% 2.91/1.77 thf(singletonsuniq_type,type,
% 2.91/1.77 singletonsuniq: $o ).
% 2.91/1.77
% 2.91/1.77 thf(singletonsuniq,definition,
% 2.91/1.77 ( singletonsuniq
% 2.91/1.77 = ( ! [Xx: $i,Xy: $i] :
% 2.91/1.77 ( ( ( setadjoin @ Xx @ emptyset )
% 2.91/1.77 = ( setadjoin @ Xy @ emptyset ) )
% 2.91/1.77 => ( Xx = Xy ) ) ) ) ).
% 2.91/1.77
% 2.91/1.77 thf(atmost1p_type,type,
% 2.91/1.77 atmost1p: $i > $o ).
% 2.91/1.77
% 2.91/1.77 thf(atleast2p_type,type,
% 2.91/1.77 atleast2p: $i > $o ).
% 2.91/1.77
% 2.91/1.77 thf(atmost2p_type,type,
% 2.91/1.77 atmost2p: $i > $o ).
% 2.91/1.77
% 2.91/1.77 thf(upairsetp_type,type,
% 2.91/1.77 upairsetp: $i > $o ).
% 2.91/1.77
% 2.91/1.77 thf(setukpairinjL1_type,type,
% 2.91/1.77 setukpairinjL1: $o ).
% 2.91/1.77
% 2.91/1.77 thf(setukpairinjL1,definition,
% 2.91/1.77 ( setukpairinjL1
% 2.91/1.77 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.91/1.77 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 2.91/1.77 => ( Xx = Xz ) ) ) ) ).
% 2.91/1.77
% 2.91/1.77 thf(kfstsingleton_type,type,
% 2.91/1.77 kfstsingleton: $o ).
% 2.91/1.77
% 2.91/1.77 thf(kfstsingleton,definition,
% 2.91/1.77 ( kfstsingleton
% 2.91/1.77 = ( ! [Xu: $i] :
% 2.91/1.77 ( ( iskpair @ Xu )
% 2.91/1.77 => ( singleton
% 2.91/1.77 @ ( dsetconstr @ ( setunion @ Xu )
% 2.91/1.77 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 2.91/1.77
% 2.91/1.77 thf(theprop_type,type,
% 2.91/1.77 theprop: $o ).
% 2.91/1.77
% 2.91/1.77 thf(theprop,definition,
% 2.91/1.77 ( theprop
% 2.91/1.77 = ( ! [X: $i] :
% 2.91/1.77 ( ( singleton @ X )
% 2.91/1.77 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 2.91/1.77
% 2.91/1.77 thf(kfst_type,type,
% 2.91/1.77 kfst: $i > $i ).
% 2.91/1.77
% 2.91/1.77 thf(kfstpairEq_type,type,
% 2.91/1.77 kfstpairEq: $o ).
% 2.91/1.77
% 2.91/1.77 thf(kfstpairEq,definition,
% 2.91/1.77 ( kfstpairEq
% 2.91/1.77 = ( ! [Xx: $i,Xy: $i] :
% 2.91/1.77 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 2.91/1.77 = Xx ) ) ) ).
% 2.91/1.77
% 2.91/1.77 thf(cartprodfstin_type,type,
% 2.91/1.77 cartprodfstin: $o ).
% 2.91/1.77
% 2.91/1.77 thf(cartprodfstin,definition,
% 2.91/1.77 ( cartprodfstin
% 2.91/1.77 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.91/1.77 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.91/1.77 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 2.91/1.77
% 2.91/1.77 thf(setukpairinjL2_type,type,
% 2.91/1.77 setukpairinjL2: $o ).
% 2.91/1.77
% 2.91/1.77 thf(setukpairinjL2,definition,
% 2.91/1.77 ( setukpairinjL2
% 2.91/1.77 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.91/1.77 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.91/1.77 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.91/1.77 => ( Xx = Xz ) ) ) ) ).
% 2.91/1.77
% 2.91/1.77 thf(setukpairinjL_type,type,
% 2.91/1.77 setukpairinjL: $o ).
% 2.91/1.77
% 2.91/1.77 thf(setukpairinjL,definition,
% 2.91/1.77 ( setukpairinjL
% 2.91/1.77 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.91/1.77 ( ( ( kpair @ Xx @ Xy )
% 2.91/1.77 = ( kpair @ Xz @ Xu ) )
% 2.91/1.77 => ( Xx = Xz ) ) ) ) ).
% 2.91/1.77
% 2.91/1.77 thf(setukpairinjR11_type,type,
% 2.91/1.77 setukpairinjR11: $o ).
% 2.91/1.77
% 2.91/1.77 thf(setukpairinjR11,definition,
% 2.91/1.77 ( setukpairinjR11
% 2.91/1.77 = ( ! [Xx: $i,Xy: $i] :
% 2.91/1.77 ( ( Xx = Xy )
% 2.91/1.77 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.91/1.77 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.91/1.77
% 2.91/1.77 thf(setukpairinjR12_type,type,
% 2.91/1.77 setukpairinjR12: $o ).
% 2.91/1.77
% 2.91/1.77 thf(setukpairinjR12,definition,
% 2.91/1.77 ( setukpairinjR12
% 2.91/1.77 = ( ! [Xx: $i,Xy: $i] :
% 2.91/1.77 ( ( Xx = Xy )
% 2.91/1.77 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.91/1.77 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 2.91/1.77
% 2.91/1.77 thf(setukpairinjR1_type,type,
% 2.91/1.77 setukpairinjR1: $o ).
% 2.91/1.77
% 2.91/1.77 thf(setukpairinjR1,definition,
% 2.91/1.77 ( setukpairinjR1
% 2.91/1.77 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.91/1.77 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.91/1.77 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.91/1.77 => ( ( Xz = Xu )
% 2.91/1.77 => ( Xy = Xu ) ) ) ) ) ).
% 2.91/1.77
% 2.91/1.77 thf(upairequniteq_type,type,
% 2.91/1.77 upairequniteq: $o ).
% 2.91/1.77
% 2.91/1.77 thf(upairequniteq,definition,
% 2.91/1.77 ( upairequniteq
% 2.91/1.77 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.91/1.77 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.91/1.77 = ( setadjoin @ Xz @ emptyset ) )
% 2.91/1.77 => ( Xx = Xy ) ) ) ) ).
% 2.91/1.77
% 2.91/1.77 thf(setukpairinjR2_type,type,
% 2.91/1.77 setukpairinjR2: $o ).
% 2.91/1.77
% 2.91/1.77 thf(setukpairinjR2,definition,
% 2.91/1.77 ( setukpairinjR2
% 2.91/1.77 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.91/1.77 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.91/1.77 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.91/1.77 => ( Xy = Xu ) ) ) ) ).
% 2.91/1.77
% 2.91/1.77 thf(setukpairinjR_type,type,
% 2.91/1.77 setukpairinjR: $o ).
% 2.91/1.77
% 2.91/1.77 thf(setukpairinjR,definition,
% 2.91/1.77 ( setukpairinjR
% 2.91/1.77 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.91/1.77 ( ( ( kpair @ Xx @ Xy )
% 2.91/1.77 = ( kpair @ Xz @ Xu ) )
% 2.91/1.77 => ( Xy = Xu ) ) ) ) ).
% 2.91/1.77
% 2.91/1.77 thf(ksndsingleton_type,type,
% 2.91/1.77 ksndsingleton: $o ).
% 2.91/1.77
% 2.91/1.77 thf(ksndsingleton,definition,
% 2.91/1.77 ( ksndsingleton
% 2.91/1.77 = ( ! [Xu: $i] :
% 2.91/1.77 ( ( iskpair @ Xu )
% 2.91/1.77 => ( singleton
% 2.91/1.77 @ ( dsetconstr @ ( setunion @ Xu )
% 2.91/1.77 @ ^ [Xx: $i] :
% 2.91/1.77 ( Xu
% 2.91/1.77 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 2.91/1.77
% 2.91/1.77 thf(ksnd_type,type,
% 2.91/1.77 ksnd: $i > $i ).
% 2.91/1.77
% 2.91/1.77 thf(ksndpairEq_type,type,
% 2.91/1.77 ksndpairEq: $o ).
% 2.91/1.77
% 2.91/1.77 thf(ksndpairEq,definition,
% 2.91/1.77 ( ksndpairEq
% 2.91/1.77 = ( ! [Xx: $i,Xy: $i] :
% 2.91/1.77 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 2.91/1.77 = Xy ) ) ) ).
% 2.91/1.77
% 2.91/1.77 thf(kpairsurjEq_type,type,
% 2.91/1.77 kpairsurjEq: $o ).
% 2.91/1.77
% 2.91/1.77 thf(kpairsurjEq,definition,
% 2.91/1.77 ( kpairsurjEq
% 2.91/1.77 = ( ! [Xu: $i] :
% 2.91/1.77 ( ( iskpair @ Xu )
% 2.91/1.77 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 2.91/1.77 = Xu ) ) ) ) ).
% 2.91/1.77
% 2.91/1.77 thf(cartprodsndin_type,type,
% 2.91/1.77 cartprodsndin: $o ).
% 2.91/1.77
% 2.91/1.77 thf(cartprodsndin,definition,
% 2.91/1.77 ( cartprodsndin
% 2.91/1.77 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.91/1.77 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.91/1.77 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 2.91/1.77
% 2.91/1.77 thf(cartprodpairmemEL_type,type,
% 2.91/1.77 cartprodpairmemEL: $o ).
% 2.91/1.77
% 2.91/1.77 thf(cartprodpairmemEL,definition,
% 2.91/1.77 ( cartprodpairmemEL
% 2.91/1.77 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 2.91/1.77 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 2.91/1.77 => ( in @ Xx @ A ) ) ) ) ).
% 2.91/1.77
% 2.91/1.77 thf(cartprodpairmemER_type,type,
% 2.91/1.77 cartprodpairmemER: $o ).
% 2.91/1.77
% 2.91/1.77 thf(cartprodpairmemER,definition,
% 2.91/1.77 ( cartprodpairmemER
% 2.91/1.77 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 2.91/1.77 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 2.91/1.77 => ( in @ Xy @ B ) ) ) ) ).
% 2.91/1.77
% 2.91/1.77 thf(cartprodmempaircEq_type,type,
% 2.91/1.77 cartprodmempaircEq: $o ).
% 2.91/1.77
% 2.91/1.77 thf(cartprodmempaircEq,definition,
% 2.91/1.77 ( cartprodmempaircEq
% 2.91/1.77 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.91/1.77 ( ( in @ Xx @ A )
% 2.91/1.77 => ! [Xy: $i] :
% 2.91/1.77 ( ( in @ Xy @ B )
% 2.91/1.77 => ( ( kpair @ Xx @ Xy )
% 2.91/1.77 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 2.91/1.77
% 2.91/1.77 thf(cartprodfstpairEq_type,type,
% 2.91/1.77 cartprodfstpairEq: $o ).
% 2.91/1.77
% 2.91/1.77 thf(cartprodfstpairEq,definition,
% 2.91/1.77 ( cartprodfstpairEq
% 2.91/1.77 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.91/1.77 ( ( in @ Xx @ A )
% 2.91/1.77 => ! [Xy: $i] :
% 2.91/1.77 ( ( in @ Xy @ B )
% 2.91/1.77 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 2.91/1.77 = Xx ) ) ) ) ) ).
% 2.91/1.77
% 2.91/1.77 thf(cartprodsndpairEq_type,type,
% 2.91/1.77 cartprodsndpairEq: $o ).
% 2.91/1.77
% 2.91/1.77 thf(cartprodsndpairEq,definition,
% 2.91/1.77 ( cartprodsndpairEq
% 2.91/1.77 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.91/1.77 ( ( in @ Xx @ A )
% 2.91/1.77 => ! [Xy: $i] :
% 2.91/1.77 ( ( in @ Xy @ B )
% 2.91/1.77 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 2.91/1.77 = Xy ) ) ) ) ) ).
% 2.91/1.77
% 2.91/1.77 thf(cartprodpairsurjEq_type,type,
% 2.91/1.77 cartprodpairsurjEq: $o ).
% 2.91/1.77
% 2.91/1.77 thf(cartprodpairsurjEq,definition,
% 2.91/1.77 ( cartprodpairsurjEq
% 2.91/1.77 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.91/1.77 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.91/1.77 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 2.91/1.77 = Xu ) ) ) ) ).
% 2.91/1.77
% 2.91/1.77 thf(breln_type,type,
% 2.91/1.77 breln: $i > $i > $i > $o ).
% 2.91/1.77
% 2.91/1.77 thf(breln,definition,
% 2.91/1.77 ( breln
% 2.91/1.77 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 2.91/1.77
% 2.91/1.77 thf(dpsetconstr_type,type,
% 2.91/1.77 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 2.91/1.77
% 2.91/1.77 thf(dpsetconstr,definition,
% 2.91/1.77 ( dpsetconstr
% 2.91/1.77 = ( ^ [A: $i,B: $i,Xphi: $i > $i > $o] :
% 2.91/1.77 ( dsetconstr @ ( cartprod @ A @ B )
% 2.91/1.77 @ ^ [Xu: $i] :
% 2.91/1.77 ? [Xx: $i] :
% 2.91/1.77 ( ( in @ Xx @ A )
% 2.91/1.77 & ? [Xy: $i] :
% 2.91/1.77 ( ( in @ Xy @ B )
% 2.91/1.77 & ( Xphi @ Xx @ Xy )
% 2.91/1.77 & ( Xu
% 2.91/1.77 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 2.91/1.77
% 2.91/1.77 thf(dpsetconstrI_type,type,
% 2.91/1.77 dpsetconstrI: $o ).
% 2.91/1.77
% 2.91/1.77 thf(dpsetconstrI,definition,
% 2.91/1.77 ( dpsetconstrI
% 2.91/1.77 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 2.91/1.77 ( ( in @ Xx @ A )
% 2.91/1.77 => ! [Xy: $i] :
% 2.91/1.77 ( ( in @ Xy @ B )
% 2.91/1.77 => ( ( Xphi @ Xx @ Xy )
% 2.91/1.77 => ( in @ ( kpair @ Xx @ Xy )
% 2.91/1.77 @ ( dpsetconstr @ A @ B
% 2.91/1.77 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 2.91/1.77
% 2.91/1.77 thf(dpsetconstrSub_type,type,
% 2.91/1.77 dpsetconstrSub: $o ).
% 2.91/1.77
% 2.91/1.77 thf(dpsetconstrSub,definition,
% 2.91/1.77 ( dpsetconstrSub
% 2.91/1.77 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 2.91/1.77 ( subset
% 2.91/1.77 @ ( dpsetconstr @ A @ B
% 2.91/1.77 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 2.91/1.77 @ ( cartprod @ A @ B ) ) ) ) ).
% 2.91/1.77
% 2.91/1.77 thf(setOfPairsIsBReln_type,type,
% 2.91/1.77 setOfPairsIsBReln: $o ).
% 2.91/1.77
% 2.91/1.77 thf(setOfPairsIsBReln,definition,
% 2.91/1.77 ( setOfPairsIsBReln
% 2.91/1.77 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 2.91/1.77 ( breln @ A @ B
% 2.91/1.77 @ ( dpsetconstr @ A @ B
% 2.91/1.77 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 2.91/1.77
% 2.91/1.77 thf(dpsetconstrERa_type,type,
% 2.91/1.77 dpsetconstrERa: $o ).
% 2.91/1.77
% 2.91/1.77 thf(dpsetconstrERa,definition,
% 2.91/1.77 ( dpsetconstrERa
% 2.91/1.77 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 2.91/1.77 ( ( in @ Xx @ A )
% 2.91/1.77 => ! [Xy: $i] :
% 2.91/1.77 ( ( in @ Xy @ B )
% 2.91/1.77 => ( ( in @ ( kpair @ Xx @ Xy )
% 2.91/1.77 @ ( dpsetconstr @ A @ B
% 2.91/1.77 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.91/1.77 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 2.91/1.77
% 2.91/1.77 thf(dpsetconstrEL1_type,type,
% 2.91/1.77 dpsetconstrEL1: $o ).
% 2.91/1.77
% 2.91/1.77 thf(dpsetconstrEL1,definition,
% 2.91/1.77 ( dpsetconstrEL1
% 2.91/1.77 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.91/1.77 ( ( in @ ( kpair @ Xx @ Xy )
% 2.91/1.77 @ ( dpsetconstr @ A @ B
% 2.91/1.77 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.91/1.77 => ( in @ Xx @ A ) ) ) ) ).
% 2.91/1.77
% 2.91/1.77 thf(dpsetconstrEL2_type,type,
% 2.91/1.77 dpsetconstrEL2: $o ).
% 2.91/1.77
% 2.91/1.77 thf(dpsetconstrEL2,definition,
% 2.91/1.77 ( dpsetconstrEL2
% 2.91/1.77 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.91/1.77 ( ( in @ ( kpair @ Xx @ Xy )
% 2.91/1.77 @ ( dpsetconstr @ A @ B
% 2.91/1.77 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.91/1.77 => ( in @ Xy @ B ) ) ) ) ).
% 2.91/1.77
% 2.91/1.77 thf(dpsetconstrER,conjecture,
% 2.91/1.77 ( setextAx
% 2.91/1.77 => ( emptysetAx
% 2.91/1.77 => ( setadjoinAx
% 2.91/1.77 => ( powersetAx
% 2.91/1.77 => ( setunionAx
% 2.91/1.77 => ( omega0Ax
% 2.91/1.77 => ( omegaSAx
% 2.91/1.77 => ( omegaIndAx
% 2.91/1.77 => ( replAx
% 2.91/1.77 => ( foundationAx
% 2.91/1.77 => ( wellorderingAx
% 2.91/1.77 => ( descrp
% 2.91/1.77 => ( dsetconstrI
% 2.91/1.77 => ( dsetconstrEL
% 2.91/1.77 => ( dsetconstrER
% 2.91/1.77 => ( exuE1
% 2.91/1.77 => ( prop2setE
% 2.91/1.77 => ( emptysetE
% 2.91/1.77 => ( emptysetimpfalse
% 2.91/1.77 => ( notinemptyset
% 2.91/1.77 => ( exuE3e
% 2.91/1.77 => ( setext
% 2.91/1.77 => ( emptyI
% 2.91/1.77 => ( noeltsimpempty
% 2.91/1.77 => ( setbeta
% 2.91/1.77 => ( nonemptyE1
% 2.91/1.77 => ( nonemptyI
% 2.91/1.77 => ( nonemptyI1
% 2.91/1.77 => ( setadjoinIL
% 2.91/1.77 => ( emptyinunitempty
% 2.91/1.77 => ( setadjoinIR
% 2.91/1.77 => ( setadjoinE
% 2.91/1.77 => ( setadjoinOr
% 2.91/1.77 => ( setoftrueEq
% 2.91/1.77 => ( powersetI
% 2.91/1.77 => ( emptyinPowerset
% 2.91/1.77 => ( emptyInPowerset
% 2.91/1.77 => ( powersetE
% 2.91/1.77 => ( setunionI
% 2.91/1.77 => ( setunionE
% 2.91/1.77 => ( subPowSU
% 2.91/1.77 => ( exuE2
% 2.91/1.77 => ( nonemptyImpWitness
% 2.91/1.77 => ( uniqinunit
% 2.91/1.77 => ( notinsingleton
% 2.91/1.77 => ( eqinunit
% 2.91/1.77 => ( singletonsswitch
% 2.91/1.77 => ( upairsetE
% 2.91/1.77 => ( upairsetIL
% 2.91/1.77 => ( upairsetIR
% 2.91/1.77 => ( emptyE1
% 2.91/1.77 => ( vacuousDall
% 2.91/1.77 => ( quantDeMorgan1
% 2.91/1.77 => ( quantDeMorgan2
% 2.91/1.77 => ( quantDeMorgan3
% 2.91/1.77 => ( quantDeMorgan4
% 2.91/1.77 => ( prop2setI
% 2.91/1.77 => ( prop2set2propI
% 2.91/1.77 => ( notdexE
% 2.91/1.77 => ( notdallE
% 2.91/1.77 => ( exuI1
% 2.91/1.77 => ( exuI3
% 2.91/1.77 => ( exuI2
% 2.91/1.77 => ( inCongP
% 2.91/1.77 => ( in__Cong
% 2.91/1.77 => ( exuE3u
% 2.91/1.77 => ( exu__Cong
% 2.91/1.77 => ( emptyset__Cong
% 2.91/1.77 => ( setadjoin__Cong
% 2.91/1.77 => ( powerset__Cong
% 2.91/1.77 => ( setunion__Cong
% 2.91/1.77 => ( omega__Cong
% 2.91/1.77 => ( exuEu
% 2.91/1.77 => ( descr__Cong
% 2.91/1.77 => ( dsetconstr__Cong
% 2.91/1.77 => ( subsetI1
% 2.91/1.77 => ( eqimpsubset2
% 2.91/1.77 => ( eqimpsubset1
% 2.91/1.77 => ( subsetI2
% 2.91/1.77 => ( emptysetsubset
% 2.91/1.77 => ( subsetE
% 2.91/1.77 => ( subsetE2
% 2.91/1.77 => ( notsubsetI
% 2.91/1.77 => ( notequalI1
% 2.91/1.77 => ( notequalI2
% 2.91/1.77 => ( subsetRefl
% 2.91/1.77 => ( subsetTrans
% 2.91/1.77 => ( setadjoinSub
% 2.91/1.77 => ( setadjoinSub2
% 2.91/1.77 => ( subset2powerset
% 2.91/1.77 => ( setextsub
% 2.91/1.77 => ( subsetemptysetimpeq
% 2.91/1.77 => ( powersetI1
% 2.91/1.77 => ( powersetE1
% 2.91/1.77 => ( inPowerset
% 2.91/1.77 => ( powersetsubset
% 2.91/1.77 => ( sepInPowerset
% 2.91/1.77 => ( sepSubset
% 2.91/1.77 => ( binunionIL
% 2.91/1.77 => ( upairset2IR
% 2.91/1.77 => ( binunionIR
% 2.91/1.77 => ( binunionEcases
% 2.91/1.77 => ( binunionE
% 2.91/1.77 => ( binunionLsub
% 2.91/1.77 => ( binunionRsub
% 2.91/1.77 => ( binintersectI
% 2.91/1.77 => ( binintersectSubset5
% 2.91/1.77 => ( binintersectEL
% 2.91/1.77 => ( binintersectLsub
% 2.91/1.77 => ( binintersectSubset2
% 2.91/1.77 => ( binintersectSubset3
% 2.91/1.77 => ( binintersectER
% 2.91/1.77 => ( disjointsetsI1
% 2.91/1.77 => ( binintersectRsub
% 2.91/1.77 => ( binintersectSubset4
% 2.91/1.77 => ( binintersectSubset1
% 2.91/1.77 => ( bs114d
% 2.91/1.77 => ( setminusI
% 2.91/1.77 => ( setminusEL
% 2.91/1.77 => ( setminusER
% 2.91/1.77 => ( setminusSubset2
% 2.91/1.77 => ( setminusERneg
% 2.91/1.77 => ( setminusELneg
% 2.91/1.77 => ( setminusILneg
% 2.91/1.77 => ( setminusIRneg
% 2.91/1.77 => ( setminusLsub
% 2.91/1.77 => ( setminusSubset1
% 2.91/1.77 => ( symdiffE
% 2.91/1.77 => ( symdiffI1
% 2.91/1.77 => ( symdiffI2
% 2.91/1.77 => ( symdiffIneg1
% 2.91/1.77 => ( symdiffIneg2
% 2.91/1.77 => ( secondinupair
% 2.91/1.77 => ( setukpairIL
% 2.91/1.77 => ( setukpairIR
% 2.91/1.77 => ( kpairiskpair
% 2.91/1.77 => ( kpairp
% 2.91/1.77 => ( singletonsubset
% 2.91/1.77 => ( singletoninpowerset
% 2.91/1.77 => ( singletoninpowunion
% 2.91/1.77 => ( upairset2E
% 2.91/1.77 => ( upairsubunion
% 2.91/1.77 => ( upairinpowunion
% 2.91/1.77 => ( ubforcartprodlem1
% 2.91/1.77 => ( ubforcartprodlem2
% 2.91/1.77 => ( ubforcartprodlem3
% 2.91/1.77 => ( cartprodpairin
% 2.91/1.77 => ( cartprodmempair1
% 2.91/1.77 => ( cartprodmempair
% 2.91/1.77 => ( setunionE2
% 2.91/1.77 => ( setunionsingleton1
% 2.91/1.77 => ( setunionsingleton2
% 2.91/1.77 => ( setunionsingleton
% 2.91/1.77 => ( singletonprop
% 2.91/1.77 => ( ex1E1
% 2.91/1.77 => ( ex1I
% 2.91/1.77 => ( ex1I2
% 2.91/1.77 => ( singletonsuniq
% 2.91/1.77 => ( setukpairinjL1
% 2.91/1.77 => ( kfstsingleton
% 2.91/1.77 => ( theprop
% 2.91/1.77 => ( kfstpairEq
% 2.91/1.77 => ( cartprodfstin
% 2.91/1.77 => ( setukpairinjL2
% 2.91/1.77 => ( setukpairinjL
% 2.91/1.77 => ( setukpairinjR11
% 2.91/1.77 => ( setukpairinjR12
% 2.91/1.77 => ( setukpairinjR1
% 2.91/1.77 => ( upairequniteq
% 2.91/1.77 => ( setukpairinjR2
% 2.91/1.77 => ( setukpairinjR
% 2.91/1.77 => ( ksndsingleton
% 2.91/1.77 => ( ksndpairEq
% 2.91/1.77 => ( kpairsurjEq
% 2.91/1.77 => ( cartprodsndin
% 2.91/1.77 => ( cartprodpairmemEL
% 2.91/1.77 => ( cartprodpairmemER
% 2.91/1.77 => ( cartprodmempaircEq
% 2.91/1.77 => ( cartprodfstpairEq
% 2.91/1.77 => ( cartprodsndpairEq
% 2.91/1.77 => ( cartprodpairsurjEq
% 2.91/1.77 => ( dpsetconstrI
% 2.91/1.77 => ( dpsetconstrSub
% 2.91/1.77 => ( setOfPairsIsBReln
% 2.91/1.77 => ( dpsetconstrERa
% 2.91/1.77 => ( dpsetconstrEL1
% 2.91/1.77 => ( dpsetconstrEL2
% 2.91/1.77 => ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.91/1.77 ( ( in @ ( kpair @ Xx @ Xy )
% 2.91/1.77 @ ( dpsetconstr @ A @ B
% 2.91/1.77 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.91/1.77 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.91/1.77 ------------------------
% 2.99/1.80 % (7842)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_7726 for (2999ds/27Mi)
% 2.99/1.80 % (7841)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_7726 for (2999ds/4Mi)
% 2.99/1.80 % (7840)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_7726 for (2999ds/183Mi)
% 2.99/1.80 % (7844)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_7726 for (2999ds/2Mi)
% 2.99/1.80 % (7843)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_7726 for (2999ds/2Mi)
% 2.99/1.80 % (7846)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_7726 for (2999ds/18Mi)
% 2.99/1.80 % (7845)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_7726 for (2999ds/275Mi)
% 2.99/1.80 % (7843)Instruction limit reached!
% 2.99/1.80 % (7843)------------------------------
% 2.99/1.80 % (7843)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.99/1.80 % (7843)Termination reason: Unknown
% 2.99/1.80 % (7844)Instruction limit reached!
% 2.99/1.80 % (7844)------------------------------
% 2.99/1.80 % (7844)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.99/1.80 % (7843)Termination phase: shuffling
% 2.99/1.80
% 2.99/1.80 % (7843)Memory used [KB]: 1279
% 2.99/1.80 % (7843)Time elapsed: 0.003 s
% 2.99/1.80 % (7843)Instructions burned: 2 (million)
% 2.99/1.80 % (7843)------------------------------
% 2.99/1.80 % (7843)------------------------------
% 2.99/1.80 % (7844)Termination reason: Unknown
% 2.99/1.80 % (7844)Termination phase: shuffling
% 2.99/1.80
% 2.99/1.80 % (7844)Memory used [KB]: 1279
% 2.99/1.80 % (7844)Time elapsed: 0.003 s
% 2.99/1.80 % (7844)Instructions burned: 2 (million)
% 2.99/1.80 % (7844)------------------------------
% 2.99/1.80 % (7844)------------------------------
% 2.99/1.80 % (7841)Instruction limit reached!
% 2.99/1.80 % (7841)------------------------------
% 2.99/1.80 % (7841)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.99/1.80 % (7841)Termination reason: Unknown
% 2.99/1.80 % (7841)Termination phase: shuffling
% 2.99/1.80
% 2.99/1.80 % (7841)Memory used [KB]: 1407
% 2.99/1.80 % (7841)Time elapsed: 0.004 s
% 2.99/1.80 % (7841)Instructions burned: 4 (million)
% 2.99/1.80 % (7841)------------------------------
% 2.99/1.80 % (7841)------------------------------
% 2.99/1.81 % (7846)Instruction limit reached!
% 2.99/1.81 % (7846)------------------------------
% 2.99/1.81 % (7846)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.99/1.81 % (7846)Termination reason: Unknown
% 2.99/1.81 % (7846)Termination phase: Property scanning
% 2.99/1.81
% 2.99/1.81 % (7846)Memory used [KB]: 1663
% 2.99/1.81 % (7846)Time elapsed: 0.011 s
% 2.99/1.81 % (7846)Instructions burned: 18 (million)
% 2.99/1.81 % (7846)------------------------------
% 2.99/1.81 % (7846)------------------------------
% 2.99/1.81 % (7842)Instruction limit reached!
% 2.99/1.81 % (7842)------------------------------
% 2.99/1.81 % (7842)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.99/1.81 % (7842)Termination reason: Unknown
% 2.99/1.81 % (7842)Termination phase: shuffling
% 2.99/1.81
% 2.99/1.81 % (7842)Memory used [KB]: 1918
% 2.99/1.81 % (7842)Time elapsed: 0.015 s
% 2.99/1.81 % (7842)Instructions burned: 28 (million)
% 2.99/1.81 % (7842)------------------------------
% 2.99/1.81 % (7842)------------------------------
% 2.99/1.81 % (7847)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_7726 for (2999ds/3Mi)
% 2.99/1.81 % (7848)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_7726 for (2999ds/37Mi)
% 2.99/1.82 % (7849)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_7726 for (2999ds/15Mi)
% 2.99/1.82 % (7847)Instruction limit reached!
% 2.99/1.82 % (7847)------------------------------
% 2.99/1.82 % (7847)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.99/1.82 % (7847)Termination reason: Unknown
% 2.99/1.82 % (7847)Termination phase: shuffling
% 2.99/1.82
% 2.99/1.82 % (7847)Memory used [KB]: 1279
% 2.99/1.82 % (7847)Time elapsed: 0.003 s
% 2.99/1.82 % (7847)Instructions burned: 3 (million)
% 2.99/1.82 % (7847)------------------------------
% 2.99/1.82 % (7847)------------------------------
% 2.99/1.82 % (7849)Instruction limit reached!
% 2.99/1.82 % (7849)------------------------------
% 2.99/1.82 % (7849)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.99/1.82 % (7849)Termination reason: Unknown
% 2.99/1.82 % (7849)Termination phase: shuffling
% 2.99/1.82
% 2.99/1.82 % (7849)Memory used [KB]: 1663
% 2.99/1.82 % (7849)Time elapsed: 0.009 s
% 2.99/1.82 % (7849)Instructions burned: 15 (million)
% 2.99/1.82 % (7850)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_7726 for (2999ds/3Mi)
% 2.99/1.82 % (7849)------------------------------
% 2.99/1.82 % (7849)------------------------------
% 2.99/1.82 % (7850)Instruction limit reached!
% 2.99/1.82 % (7850)------------------------------
% 2.99/1.82 % (7850)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.99/1.82 % (7850)Termination reason: Unknown
% 2.99/1.82 % (7850)Termination phase: shuffling
% 2.99/1.82
% 2.99/1.82 % (7850)Memory used [KB]: 1279
% 2.99/1.82 % (7850)Time elapsed: 0.003 s
% 2.99/1.82 % (7850)Instructions burned: 3 (million)
% 2.99/1.82 % (7850)------------------------------
% 2.99/1.82 % (7850)------------------------------
% 2.99/1.83 % (7851)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_7726 for (2999ds/1041Mi)
% 2.99/1.83 % (7848)Instruction limit reached!
% 2.99/1.83 % (7848)------------------------------
% 2.99/1.83 % (7848)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.99/1.83 % (7848)Termination reason: Unknown
% 2.99/1.83 % (7848)Termination phase: Property scanning
% 2.99/1.83
% 2.99/1.83 % (7848)Memory used [KB]: 1918
% 2.99/1.83 % (7848)Time elapsed: 0.018 s
% 2.99/1.83 % (7848)Instructions burned: 37 (million)
% 2.99/1.83 % (7848)------------------------------
% 2.99/1.83 % (7848)------------------------------
% 2.99/1.83 % (7852)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_7726 for (2999ds/7Mi)
% 2.99/1.84 % (7852)Instruction limit reached!
% 2.99/1.84 % (7852)------------------------------
% 2.99/1.84 % (7852)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.99/1.84 % (7852)Termination reason: Unknown
% 2.99/1.84 % (7852)Termination phase: shuffling
% 2.99/1.84
% 2.99/1.84 % (7852)Memory used [KB]: 1407
% 2.99/1.84 % (7852)Time elapsed: 0.005 s
% 2.99/1.84 % (7852)Instructions burned: 7 (million)
% 2.99/1.84 % (7852)------------------------------
% 2.99/1.84 % (7852)------------------------------
% 2.99/1.84 % (7853)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_7726 for (2999ds/16Mi)
% 2.99/1.84 % (7854)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_7726 for (2999ds/3Mi)
% 2.99/1.84 % (7854)Instruction limit reached!
% 2.99/1.84 % (7854)------------------------------
% 2.99/1.84 % (7854)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.99/1.84 % (7854)Termination reason: Unknown
% 2.99/1.84 % (7854)Termination phase: shuffling
% 2.99/1.84
% 2.99/1.84 % (7854)Memory used [KB]: 1279
% 2.99/1.84 % (7854)Time elapsed: 0.003 s
% 2.99/1.84 % (7854)Instructions burned: 3 (million)
% 2.99/1.84 % (7854)------------------------------
% 2.99/1.84 % (7854)------------------------------
% 2.99/1.85 % (7853)Instruction limit reached!
% 2.99/1.85 % (7853)------------------------------
% 2.99/1.85 % (7853)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.99/1.85 % (7853)Termination reason: Unknown
% 2.99/1.85 % (7853)Termination phase: shuffling
% 2.99/1.85
% 2.99/1.85 % (7853)Memory used [KB]: 1663
% 2.99/1.85 % (7853)Time elapsed: 0.009 s
% 2.99/1.85 % (7853)Instructions burned: 16 (million)
% 2.99/1.85 % (7853)------------------------------
% 2.99/1.85 % (7853)------------------------------
% 2.99/1.85 % (7855)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_7726 for (2999ds/3Mi)
% 2.99/1.85 % (7855)Instruction limit reached!
% 2.99/1.85 % (7855)------------------------------
% 2.99/1.85 % (7855)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.99/1.85 % (7855)Termination reason: Unknown
% 2.99/1.85 % (7855)Termination phase: shuffling
% 2.99/1.85
% 2.99/1.85 % (7855)Memory used [KB]: 1279
% 2.99/1.85 % (7855)Time elapsed: 0.003 s
% 2.99/1.85 % (7855)Instructions burned: 3 (million)
% 2.99/1.85 % (7855)------------------------------
% 2.99/1.85 % (7855)------------------------------
% 2.99/1.85 % (7856)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_7726 for (2999ds/7Mi)
% 2.99/1.86 % (7856)Instruction limit reached!
% 2.99/1.86 % (7856)------------------------------
% 2.99/1.86 % (7856)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.99/1.86 % (7856)Termination reason: Unknown
% 2.99/1.86 % (7856)Termination phase: shuffling
% 2.99/1.86
% 2.99/1.86 % (7856)Memory used [KB]: 1407
% 2.99/1.86 % (7856)Time elapsed: 0.005 s
% 2.99/1.86 % (7856)Instructions burned: 8 (million)
% 2.99/1.86 % (7856)------------------------------
% 2.99/1.86 % (7856)------------------------------
% 2.99/1.86 % (7857)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_7726 for (2999ds/3Mi)
% 2.99/1.86 % (7857)Instruction limit reached!
% 2.99/1.86 % (7857)------------------------------
% 2.99/1.86 % (7857)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.99/1.86 % (7857)Termination reason: Unknown
% 2.99/1.86 % (7857)Termination phase: shuffling
% 2.99/1.86
% 2.99/1.86 % (7857)Memory used [KB]: 1279
% 2.99/1.86 % (7857)Time elapsed: 0.003 s
% 2.99/1.86 % (7857)Instructions burned: 3 (million)
% 2.99/1.86 % (7857)------------------------------
% 2.99/1.86 % (7857)------------------------------
% 2.99/1.86 % (7858)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_7726 for (2999ds/4Mi)
% 2.99/1.86 % (7858)Instruction limit reached!
% 2.99/1.86 % (7858)------------------------------
% 2.99/1.86 % (7858)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.99/1.87 % (7859)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_7726 for (2999ds/18Mi)
% 2.99/1.87 % (7858)Termination reason: Unknown
% 2.99/1.87 % (7858)Termination phase: shuffling
% 2.99/1.87
% 2.99/1.87 % (7858)Memory used [KB]: 1407
% 2.99/1.87 % (7858)Time elapsed: 0.004 s
% 2.99/1.87 % (7858)Instructions burned: 5 (million)
% 2.99/1.87 % (7858)------------------------------
% 2.99/1.87 % (7858)------------------------------
% 2.99/1.87 % (7860)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_7726 for (2999ds/710Mi)
% 2.99/1.87 % (7859)Instruction limit reached!
% 2.99/1.87 % (7859)------------------------------
% 2.99/1.87 % (7859)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.99/1.87 % (7859)Termination reason: Unknown
% 2.99/1.87 % (7859)Termination phase: shuffling
% 2.99/1.87
% 2.99/1.87 % (7859)Memory used [KB]: 1663
% 2.99/1.87 % (7859)Time elapsed: 0.010 s
% 2.99/1.87 % (7859)Instructions burned: 18 (million)
% 2.99/1.87 % (7859)------------------------------
% 2.99/1.87 % (7859)------------------------------
% 2.99/1.88 % (7861)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_7726 for (2999ds/6Mi)
% 2.99/1.88 % (7861)Instruction limit reached!
% 2.99/1.88 % (7861)------------------------------
% 2.99/1.88 % (7861)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.99/1.88 % (7861)Termination reason: Unknown
% 2.99/1.88 % (7861)Termination phase: shuffling
% 2.99/1.88
% 2.99/1.88 % (7861)Memory used [KB]: 1407
% 2.99/1.88 % (7861)Time elapsed: 0.005 s
% 2.99/1.88 % (7861)Instructions burned: 8 (million)
% 2.99/1.88 % (7861)------------------------------
% 2.99/1.88 % (7861)------------------------------
% 2.99/1.88 % (7840)Instruction limit reached!
% 2.99/1.88 % (7840)------------------------------
% 2.99/1.88 % (7840)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.99/1.88 % (7840)Termination reason: Unknown
% 2.99/1.88 % (7840)Termination phase: Saturation
% 2.99/1.88
% 2.99/1.88 % (7840)Memory used [KB]: 7419
% 2.99/1.88 % (7840)Time elapsed: 0.084 s
% 2.99/1.88 % (7840)Instructions burned: 183 (million)
% 2.99/1.88 % (7840)------------------------------
% 2.99/1.88 % (7840)------------------------------
% 2.99/1.88 % (7862)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_7726 for (2999ds/902Mi)
% 2.99/1.89 % (7863)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_7726 for (2999ds/21Mi)
% 2.99/1.90 % (7864)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_7726 for (2998ds/5Mi)
% 2.99/1.90 % (7865)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_7726 for (2998ds/6Mi)
% 2.99/1.90 % (7864)Instruction limit reached!
% 2.99/1.90 % (7864)------------------------------
% 2.99/1.90 % (7864)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.99/1.90 % (7864)Termination reason: Unknown
% 2.99/1.90 % (7864)Termination phase: shuffling
% 2.99/1.90
% 2.99/1.90 % (7864)Memory used [KB]: 1407
% 2.99/1.90 % (7864)Time elapsed: 0.004 s
% 2.99/1.90 % (7864)Instructions burned: 5 (million)
% 2.99/1.90 % (7864)------------------------------
% 2.99/1.90 % (7864)------------------------------
% 2.99/1.90 % (7865)Instruction limit reached!
% 2.99/1.90 % (7865)------------------------------
% 2.99/1.90 % (7865)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.99/1.90 % (7865)Termination reason: Unknown
% 2.99/1.90 % (7865)Termination phase: shuffling
% 2.99/1.90
% 2.99/1.90 % (7865)Memory used [KB]: 1407
% 2.99/1.90 % (7865)Time elapsed: 0.005 s
% 2.99/1.90 % (7865)Instructions burned: 7 (million)
% 2.99/1.90 % (7865)------------------------------
% 2.99/1.90 % (7865)------------------------------
% 2.99/1.90 % (7863)Instruction limit reached!
% 2.99/1.90 % (7863)------------------------------
% 2.99/1.90 % (7863)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.99/1.90 % (7863)Termination reason: Unknown
% 2.99/1.90 % (7863)Termination phase: shuffling
% 2.99/1.90
% 2.99/1.90 % (7863)Memory used [KB]: 1791
% 2.99/1.90 % (7863)Time elapsed: 0.012 s
% 2.99/1.90 % (7863)Instructions burned: 22 (million)
% 2.99/1.90 % (7863)------------------------------
% 2.99/1.90 % (7863)------------------------------
% 2.99/1.91 % (7866)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_7726 for (2998ds/377Mi)
% 2.99/1.92 % (7867)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_7726 for (2998ds/779Mi)
% 2.99/1.92 % (7868)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_7726 for (2998ds/19Mi)
% 2.99/1.92 % (7868)Instruction limit reached!
% 2.99/1.92 % (7868)------------------------------
% 2.99/1.92 % (7868)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.99/1.92 % (7868)Termination reason: Unknown
% 2.99/1.92 % (7868)Termination phase: shuffling
% 2.99/1.92
% 2.99/1.92 % (7868)Memory used [KB]: 1663
% 2.99/1.92 % (7868)Time elapsed: 0.010 s
% 2.99/1.92 % (7868)Instructions burned: 19 (million)
% 2.99/1.92 % (7868)------------------------------
% 2.99/1.92 % (7868)------------------------------
% 2.99/1.93 % (7845)Instruction limit reached!
% 2.99/1.93 % (7845)------------------------------
% 2.99/1.93 % (7845)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.99/1.93 % (7845)Termination reason: Unknown
% 2.99/1.93 % (7845)Termination phase: Saturation
% 2.99/1.93
% 2.99/1.93 % (7845)Memory used [KB]: 8443
% 2.99/1.93 % (7845)Time elapsed: 0.136 s
% 2.99/1.93 % (7845)Instructions burned: 275 (million)
% 2.99/1.93 % (7845)------------------------------
% 2.99/1.93 % (7845)------------------------------
% 2.99/1.94 % (7869)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_7726 for (2998ds/879Mi)
% 2.99/1.95 % (7870)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_7726 for (2998ds/17Mi)
% 2.99/1.96 % (7870)Instruction limit reached!
% 2.99/1.96 % (7870)------------------------------
% 2.99/1.96 % (7870)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.99/1.96 % (7870)Termination reason: Unknown
% 2.99/1.96 % (7870)Termination phase: shuffling
% 2.99/1.96
% 2.99/1.96 % (7870)Memory used [KB]: 1663
% 2.99/1.96 % (7870)Time elapsed: 0.010 s
% 2.99/1.96 % (7870)Instructions burned: 18 (million)
% 2.99/1.96 % (7870)------------------------------
% 2.99/1.96 % (7870)------------------------------
% 2.99/1.97 % (7871)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_7726 for (2998ds/3Mi)
% 2.99/1.98 % (7871)Instruction limit reached!
% 2.99/1.98 % (7871)------------------------------
% 2.99/1.98 % (7871)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.99/1.98 % (7871)Termination reason: Unknown
% 2.99/1.98 % (7871)Termination phase: shuffling
% 2.99/1.98
% 2.99/1.98 % (7871)Memory used [KB]: 1279
% 2.99/1.98 % (7871)Time elapsed: 0.003 s
% 2.99/1.98 % (7871)Instructions burned: 3 (million)
% 2.99/1.98 % (7871)------------------------------
% 2.99/1.98 % (7871)------------------------------
% 2.99/1.99 % (7862)First to succeed.
% 2.99/1.99 % (7872)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_7726 for (2998ds/30Mi)
% 2.99/1.99 % (7862)Refutation found. Thanks to Tanya!
% 2.99/1.99 % SZS status Theorem for DTF2THF_7726
% 2.99/1.99 % SZS output start Proof for DTF2THF_7726
% 2.99/1.99 thf(func_def_0, type, in: $i > $i > $o).
% 2.99/1.99 thf(func_def_1, type, exu: ($i > $o) > $o).
% 2.99/1.99 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 2.99/1.99 thf(func_def_8, type, powerset: $i > $i).
% 2.99/1.99 thf(func_def_10, type, setunion: $i > $i).
% 2.99/1.99 thf(func_def_19, type, descr: ($i > $o) > $i).
% 2.99/1.99 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 2.99/1.99 thf(func_def_26, type, prop2set: $o > $i).
% 2.99/1.99 thf(func_def_36, type, nonempty: $i > $o).
% 2.99/1.99 thf(func_def_69, type, set2prop: $i > $o).
% 2.99/1.99 thf(func_def_88, type, subset: $i > $i > $o).
% 2.99/1.99 thf(func_def_89, type, disjoint: $i > $i > $o).
% 2.99/1.99 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 2.99/1.99 thf(func_def_114, type, binunion: $i > $i > $i).
% 2.99/1.99 thf(func_def_122, type, binintersect: $i > $i > $i).
% 2.99/1.99 thf(func_def_135, type, regular: $i > $o).
% 2.99/1.99 thf(func_def_136, type, setminus: $i > $i > $i).
% 2.99/1.99 thf(func_def_147, type, symdiff: $i > $i > $i).
% 2.99/1.99 thf(func_def_153, type, iskpair: $i > $o).
% 2.99/1.99 thf(func_def_158, type, kpair: $i > $i > $i).
% 2.99/1.99 thf(func_def_160, type, cartprod: $i > $i > $i).
% 2.99/1.99 thf(func_def_177, type, singleton: $i > $o).
% 2.99/1.99 thf(func_def_179, type, ex1: $i > ($i > $o) > $o).
% 2.99/1.99 thf(func_def_184, type, atmost1p: $i > $o).
% 2.99/1.99 thf(func_def_185, type, atleast2p: $i > $o).
% 2.99/1.99 thf(func_def_186, type, atmost2p: $i > $o).
% 2.99/1.99 thf(func_def_187, type, upairsetp: $i > $o).
% 2.99/1.99 thf(func_def_191, type, kfst: $i > $i).
% 2.99/1.99 thf(func_def_203, type, ksnd: $i > $i).
% 2.99/1.99 thf(func_def_213, type, breln: $i > $i > $i > $o).
% 2.99/1.99 thf(func_def_214, type, dpsetconstr: $i > $i > ($i > $i > $o) > $i).
% 2.99/1.99 thf(func_def_235, type, sP0: $o > $i > $i > $i > $o).
% 2.99/1.99 thf(func_def_238, type, sP3: $i > $i > $o).
% 2.99/1.99 thf(func_def_239, type, sP4: $i > $o).
% 2.99/1.99 thf(func_def_240, type, sP5: $i > $i > $o).
% 2.99/1.99 thf(func_def_241, type, sP6: $i > $i > $o).
% 2.99/1.99 thf(func_def_255, type, sK20: $i > $o).
% 2.99/1.99 thf(func_def_282, type, sK47: $i > $o).
% 2.99/1.99 thf(func_def_297, type, sK62: $i > $i > $i).
% 2.99/1.99 thf(func_def_298, type, sK63: $i > $i > $i).
% 2.99/1.99 thf(func_def_301, type, sK66: $i > $o).
% 2.99/1.99 thf(func_def_308, type, sK73: $i > $i > $i).
% 2.99/1.99 thf(func_def_317, type, sK82: $i > $i > $o).
% 2.99/1.99 thf(func_def_329, type, sK94: $i > $i > $o).
% 2.99/1.99 thf(func_def_331, type, sK96: $i > $o).
% 2.99/1.99 thf(func_def_334, type, sK99: $i > $o).
% 2.99/1.99 thf(func_def_337, type, sK102: $i > ($i > $o) > $i).
% 2.99/1.99 thf(func_def_338, type, sK103: $i > ($i > $o) > $i > $i).
% 2.99/1.99 thf(func_def_340, type, sK105: $i > $o).
% 2.99/1.99 thf(func_def_347, type, sK112: $i > $i > $i > $i).
% 2.99/1.99 thf(func_def_348, type, sK113: $i > $i > $i > $i).
% 2.99/1.99 thf(func_def_351, type, sK116: $i > $i > $o).
% 2.99/1.99 thf(func_def_365, type, sK130: $i > $o).
% 2.99/1.99 thf(func_def_367, type, sK132: ($i > $o) > $i > $i).
% 2.99/1.99 thf(func_def_368, type, sK133: ($i > $o) > $i > $i).
% 2.99/1.99 thf(func_def_385, type, sK150: $i > $o).
% 2.99/1.99 thf(func_def_387, type, sK152: ($i > $o) > $i > $i).
% 2.99/1.99 thf(func_def_389, type, sK154: $i > $i).
% 2.99/1.99 thf(func_def_391, type, sK156: ($i > $o) > $i).
% 2.99/1.99 thf(func_def_392, type, sK157: ($i > $o) > $i).
% 2.99/1.99 thf(func_def_393, type, sK158: $i > $o).
% 2.99/1.99 thf(func_def_398, type, sK163: $i > $i).
% 2.99/1.99 thf(func_def_406, type, sK171: ($i > $o) > $i).
% 2.99/1.99 thf(func_def_407, type, sK172: $i > $o).
% 2.99/1.99 thf(func_def_408, type, sK173: $i > $i).
% 2.99/1.99 thf(func_def_424, type, sK189: $i > $i > $o).
% 2.99/1.99 thf(func_def_425, type, sK190: $i > $i).
% 2.99/1.99 thf(func_def_426, type, sK191: $i > $i).
% 2.99/1.99 thf(func_def_427, type, sK192: ($i > $i > $o) > $i > $i).
% 2.99/1.99 thf(func_def_428, type, sK193: $i > ($i > $i > $o) > $i > $i).
% 2.99/1.99 thf(func_def_429, type, sK194: ($i > $i > $o) > $i > $i).
% 2.99/1.99 thf(func_def_436, type, sK201: $i > ($i > $o) > $i).
% 2.99/1.99 thf(func_def_437, type, sK202: $i > $o).
% 2.99/1.99 thf(func_def_439, type, sK204: ($i > $o) > $i > $i).
% 2.99/1.99 thf(func_def_441, type, sK206: $i > $o).
% 2.99/1.99 thf(func_def_451, type, sK216: $i > $i > $i).
% 2.99/1.99 thf(func_def_459, type, sK224: $i > $i).
% 2.99/1.99 thf(func_def_462, type, sK227: $i > $i > $i).
% 2.99/1.99 thf(func_def_472, type, sK237: $i > $i > $o).
% 2.99/1.99 thf(func_def_490, type, sK255: $i > $o).
% 2.99/1.99 thf(func_def_492, type, sK257: ($i > $o) > $i > $i).
% 2.99/1.99 thf(func_def_494, type, sK259: $i > $o).
% 2.99/1.99 thf(func_def_501, type, sK266: $i > $i > $i).
% 2.99/1.99 thf(func_def_505, type, sK270: $i > $o).
% 2.99/1.99 thf(func_def_507, type, sK272: ($i > $o) > $i > $i).
% 2.99/1.99 thf(func_def_510, type, sK275: $i > $o).
% 2.99/1.99 thf(func_def_525, type, sK290: ($i > $o) > ($i > $o) > $i).
% 2.99/1.99 thf(func_def_526, type, sK291: ($i > $o) > ($i > $o) > $i).
% 2.99/1.99 thf(func_def_527, type, sK292: $i > $o).
% 2.99/1.99 thf(func_def_528, type, sK293: $i > $o).
% 2.99/1.99 thf(func_def_532, type, sK297: $i > $i > $i).
% 2.99/1.99 thf(func_def_543, type, sK308: $i > $o).
% 2.99/1.99 thf(func_def_544, type, sK309: $i > $o).
% 2.99/1.99 thf(func_def_552, type, sK317: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.99/1.99 thf(func_def_553, type, sK318: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.99/1.99 thf(func_def_556, type, sK321: $i > $o).
% 2.99/1.99 thf(func_def_557, type, sK322: $i > $o).
% 2.99/1.99 thf(func_def_558, type, sK323: $i > $o).
% 2.99/1.99 thf(func_def_562, type, sK327: $i > $i).
% 2.99/1.99 thf(func_def_570, type, sK335: $i > ($i > $o) > $i).
% 2.99/1.99 thf(func_def_571, type, sK336: $i > $o).
% 2.99/1.99 thf(func_def_575, type, sK340: $i > $i > $i).
% 2.99/1.99 thf(func_def_578, type, sK343: $i > $i).
% 2.99/1.99 thf(func_def_579, type, sK344: $i > $i).
% 2.99/1.99 thf(func_def_593, type, sK358: $i > $i > $o).
% 2.99/1.99 thf(func_def_596, type, sK361: $i > $i > $i).
% 2.99/1.99 thf(func_def_597, type, sK362: $i > $i > $i).
% 2.99/1.99 thf(func_def_598, type, sK363: $i > $i > $i).
% 2.99/1.99 thf(func_def_599, type, sK364: $i > $i > $i).
% 2.99/1.99 thf(func_def_600, type, sK365: $i > $i > $i > $i).
% 2.99/1.99 thf(func_def_601, type, sK366: $i > $i).
% 2.99/1.99 thf(func_def_602, type, sK367: $i > $i).
% 2.99/1.99 thf(func_def_603, type, sK368: $i > $i).
% 2.99/1.99 thf(func_def_604, type, sK369: $i > $i).
% 2.99/1.99 thf(func_def_605, type, sK370: $i > $i > $i > $i).
% 2.99/1.99 thf(func_def_606, type, sK371: $i > $i > $i > $i).
% 2.99/1.99 thf(func_def_607, type, sK372: $i > $i > $i).
% 2.99/1.99 thf(func_def_608, type, sK373: $i > $i > $i).
% 2.99/1.99 thf(func_def_609, type, sK374: $i > $i > $i).
% 2.99/1.99 thf(func_def_611, type, sK376: $i > $i).
% 2.99/1.99 thf(func_def_612, type, sK377: $i > $i).
% 2.99/1.99 thf(func_def_613, type, sK378: $i > $i).
% 2.99/1.99 thf(func_def_614, type, sK379: $i > $i > $i).
% 2.99/1.99 thf(func_def_615, type, sK380: $i > $o).
% 2.99/1.99 thf(func_def_626, type, sK391: $i > $o).
% 2.99/1.99 thf(func_def_629, type, sK394: $i > ($i > $o) > $i).
% 2.99/1.99 thf(func_def_630, type, sK395: $i > ($i > $o) > $i).
% 2.99/1.99 thf(func_def_643, type, sK408: ($i > $o) > $i).
% 2.99/1.99 thf(func_def_644, type, sK409: $i > $o).
% 2.99/1.99 thf(func_def_645, type, sK410: $i > $i).
% 2.99/1.99 thf(func_def_651, type, sK416: $i > $i > $o).
% 2.99/1.99 thf(func_def_664, type, sK429: $i > ($i > $o) > $i).
% 2.99/1.99 thf(func_def_665, type, sK430: $i > $o).
% 2.99/1.99 thf(func_def_667, type, sK432: $i > $o).
% 2.99/1.99 thf(func_def_668, type, sK433: $i > $o).
% 2.99/1.99 thf(func_def_669, type, sK434: ($i > $o) > ($i > $o) > $i).
% 2.99/1.99 thf(func_def_670, type, sK435: ($i > $o) > ($i > $o) > $i).
% 2.99/1.99 thf(func_def_673, type, sK438: $i > $i > $i).
% 2.99/1.99 thf(func_def_676, type, sK441: $i > $i).
% 2.99/1.99 thf(func_def_687, type, sK452: $i > $i > $i).
% 2.99/1.99 thf(func_def_700, type, sK465: $o > $i > $i > $i).
% 2.99/1.99 thf(func_def_704, type, sK469: ($i > $o) > $i).
% 2.99/1.99 thf(func_def_705, type, sK470: $i > $o).
% 2.99/1.99 thf(func_def_731, type, sK496: $i > $o).
% 2.99/1.99 thf(func_def_743, type, sK508: $i > $i > $o).
% 2.99/1.99 thf(func_def_754, type, sK519: $i > $o).
% 2.99/1.99 thf(func_def_758, type, sK523: $i > $o).
% 2.99/1.99 thf(func_def_761, type, sK526: $i > ($i > $o) > $i).
% 2.99/1.99 thf(func_def_776, type, ph541: !>[X0: $tType]:(X0)).
% 2.99/1.99 thf(f2747,plain,(
% 2.99/1.99 $false),
% 2.99/1.99 inference(avatar_sat_refutation,[status(thm)],[f2741,f2742,f2743,f2746])).
% 2.99/1.99 thf(f2746,plain,(
% 2.99/1.99 ~spl540_1),
% 2.99/1.99 inference(avatar_contradiction_clause,[status(thm)],[f2745])).
% 2.99/1.99 thf(f2745,plain,(
% 2.99/1.99 $false | ~spl540_1),
% 2.99/1.99 inference(trivial_inequality_removal,[status(thm)],[f2744])).
% 2.99/1.99 thf(f2744,plain,(
% 2.99/1.99 ($true != $true) | ~spl540_1),
% 2.99/1.99 inference(superposition,[status(thm)],[f1953,f2732])).
% 2.99/1.99 thf(f2732,plain,(
% 2.99/1.99 ((sK116 @ sK118 @ sK115) = $true) | ~spl540_1),
% 2.99/1.99 inference(avatar_component_clause,[status(thm)],[f2730])).
% 2.99/1.99 thf(f2730,definition,(
% 2.99/1.99 spl540_1 <=> ((sK116 @ sK118 @ sK115) = $true)),
% 2.99/1.99 introduced(definition,[new_symbols(naming,[spl540_1])],[avatar_definition])).
% 2.99/1.99 thf(f1953,plain,(
% 2.99/1.99 ((sK116 @ sK118 @ sK115) != $true)),
% 2.99/1.99 inference(cnf_transformation,[status(thm)],[f988])).
% 2.99/1.99 thf(f988,plain,(
% 2.99/1.99 (prop2setI = $true) & (ubforcartprodlem3 = $true) & (emptyE1 = $true) & (emptyinunitempty = $true) & (inCongP = $true) & (exu__Cong = $true) & (setadjoinAx = $true) & (subset2powerset = $true) & (setunionI = $true) & (binintersectSubset5 = $true) & (setukpairinjR12 = $true) & (binintersectEL = $true) & (setadjoinIL = $true) & (setukpairinjR1 = $true) & (binintersectSubset2 = $true) & (emptysetAx = $true) & (upairequniteq = $true) & (prop2set2propI = $true) & (singletoninpowunion = $true) & (setadjoin__Cong = $true) & (setukpairinjL2 = $true) & (in__Cong = $true) & (upairsetIR = $true) & (setextAx = $true) & (cartprodsndpairEq = $true) & (quantDeMorgan3 = $true) & (singletonprop = $true) & (setminusELneg = $true) & (binintersectER = $true) & (emptysetimpfalse = $true) & (quantDeMorgan1 = $true) & (kpairiskpair = $true) & (binintersectSubset3 = $true) & (setminusSubset2 = $true) & (symdiffI1 = $true) & (eqimpsubset2 = $true) & (nonemptyI1 = $true) & (vacuousDall = $true) & (singletonsubset = $true) & (powerset__Cong = $true) & (setukpairinjR11 = $true) & (setext = $true) & (symdiffE = $true) & (setadjoinSub2 = $true) & (subsetemptysetimpeq = $true) & (binunionLsub = $true) & (emptyset__Cong = $true) & (subsetE = $true) & (setukpairinjR = $true) & (dpsetconstrERa = $true) & (dsetconstrER = $true) & (setukpairinjL1 = $true) & (setunionsingleton2 = $true) & (setadjoinIR = $true) & (((in @ (kpair @ sK118 @ sK115) @ (dpsetconstr @ sK117 @ sK114 @ (^[Y0 : $i]: ((^[Y1 : $i]: (sK116 @ Y0 @ Y1)))))) = $true) & ((sK116 @ sK118 @ sK115) != $true)) & (sepInPowerset = $true) & (setukpairIR = $true) & (disjointsetsI1 = $true) & (setminusEL = $true) & (cartprodpairin = $true) & (notequalI2 = $true) & (setextsub = $true) & (setminusERneg = $true) & (binintersectLsub = $true) & (subsetTrans = $true) & (exuI1 = $true) & (setOfPairsIsBReln = $true) & (setadjoinOr = $true) & (quantDeMorgan2 = $true) & (notsubsetI = $true) & (notequalI1 = $true) & (subPowSU = $true) & (binintersectRsub = $true) & (kfstpairEq = $true) & (setminusILneg = $true) & (cartprodfstin = $true) & (nonemptyImpWitness = $true) & (binunionRsub = $true) & (dsetconstrEL = $true) & (notinemptyset = $true) & (powersetI1 = $true) & (secondinupair = $true) & (dpsetconstrEL1 = $true) & (wellorderingAx = $true) & (setunion__Cong = $true) & (upairsetE = $true) & (binunionIL = $true) & (setukpairinjR2 = $true) & (powersetE = $true) & (exuE3e = $true) & (nonemptyI = $true) & (setoftrueEq = $true) & (replAx = $true) & (ubforcartprodlem2 = $true) & (dpsetconstrEL2 = $true) & (ksndsingleton = $true) & (upairset2IR = $true) & (ex1E1 = $true) & (notdallE = $true) & (upairsetIL = $true) & (cartprodmempair1 = $true) & (setadjoinSub = $true) & (upairinpowunion = $true) & (cartprodsndin = $true) & (setadjoinE = $true) & (eqinunit = $true) & (dpsetconstrSub = $true) & (omega0Ax = $true) & (binunionEcases = $true) & (upairset2E = $true) & (notdexE = $true) & (dpsetconstrI = $true) & (ksndpairEq = $true) & (cartprodpairsurjEq = $true) & (omegaSAx = $true) & (omegaIndAx = $true) & (powersetAx = $true) & (exuE3u = $true) & (inPowerset = $true) & (emptysetE = $true) & (exuE2 = $true) & (dsetconstrI = $true) & (descrp = $true) & (descr__Cong = $true) & (cartprodmempaircEq = $true) & (emptyinPowerset = $true) & (binintersectSubset1 = $true) & (dsetconstr__Cong = $true) & (setunionsingleton = $true) & (binunionE = $true) & (ex1I = $true) & (setminusIRneg = $true) & (symdiffI2 = $true) & (kfstsingleton = $true) & (bs114d = $true) & (subsetRefl = $true) & (cartprodpairmemEL = $true) & (setminusLsub = $true) & (subsetE2 = $true) & (setunionsingleton1 = $true) & (sepSubset = $true) & (notinsingleton = $true) & (powersetsubset = $true) & (setbeta = $true) & (exuI2 = $true) & (setminusER = $true) & (kpairsurjEq = $true) & (singletoninpowerset = $true) & (setukpairinjL = $true) & (exuI3 = $true) & (symdiffIneg2 = $true) & (cartprodfstpairEq = $true) & (setunionAx = $true) & (singletonsswitch = $true) & (exuEu = $true) & (uniqinunit = $true) & (theprop = $true) & (setunionE2 = $true) & (ex1I2 = $true) & (noeltsimpempty = $true) & (ubforcartprodlem1 = $true) & (symdiffIneg1 = $true) & (kpairp = $true) & (cartprodmempair = $true) & (eqimpsubset1 = $true) & (setminusI = $true) & (binintersectI = $true) & (omega__Cong = $true) & (binunionIR = $true) & (cartprodpairmemER = $true) & (exuE1 = $true) & (emptyI = $true) & (powersetI = $true) & (singletonsuniq = $true) & (subsetI2 = $true) & (setminusSubset1 = $true) & (emptysetsubset = $true) & (emptyInPowerset = $true) & (nonemptyE1 = $true) & (upairsubunion = $true) & (setunionE = $true) & (foundationAx = $true) & (quantDeMorgan4 = $true) & (binintersectSubset4 = $true) & (powersetE1 = $true) & (prop2setE = $true) & (subsetI1 = $true) & (setukpairIL = $true)),
% 2.99/1.99 inference(skolemisation,[status(esa),new_symbols(skolem,[sK114,sK115,sK116,sK117,sK118])],[f668,f987])).
% 2.99/1.99 thf(f987,plain,(
% 2.99/1.99 ? [X0 : $i,X1 : $i,X2 : $i > $i > $o,X3 : $i,X4 : $i] : (((in @ (kpair @ X4 @ X1) @ (dpsetconstr @ X3 @ X0 @ (^[Y0 : $i]: ((^[Y1 : $i]: (X2 @ Y0 @ Y1)))))) = $true) & ((X2 @ X4 @ X1) != $true)) => (((in @ (kpair @ sK118 @ sK115) @ (dpsetconstr @ sK117 @ sK114 @ (^[Y0 : $i]: ((^[Y1 : $i]: (sK116 @ Y0 @ Y1)))))) = $true) & ((sK116 @ sK118 @ sK115) != $true))),
% 2.99/1.99 introduced(definition,[],[choice_axiom])).
% 2.99/1.99 thf(f668,plain,(
% 2.99/1.99 (prop2setI = $true) & (ubforcartprodlem3 = $true) & (emptyE1 = $true) & (emptyinunitempty = $true) & (inCongP = $true) & (exu__Cong = $true) & (setadjoinAx = $true) & (subset2powerset = $true) & (setunionI = $true) & (binintersectSubset5 = $true) & (setukpairinjR12 = $true) & (binintersectEL = $true) & (setadjoinIL = $true) & (setukpairinjR1 = $true) & (binintersectSubset2 = $true) & (emptysetAx = $true) & (upairequniteq = $true) & (prop2set2propI = $true) & (singletoninpowunion = $true) & (setadjoin__Cong = $true) & (setukpairinjL2 = $true) & (in__Cong = $true) & (upairsetIR = $true) & (setextAx = $true) & (cartprodsndpairEq = $true) & (quantDeMorgan3 = $true) & (singletonprop = $true) & (setminusELneg = $true) & (binintersectER = $true) & (emptysetimpfalse = $true) & (quantDeMorgan1 = $true) & (kpairiskpair = $true) & (binintersectSubset3 = $true) & (setminusSubset2 = $true) & (symdiffI1 = $true) & (eqimpsubset2 = $true) & (nonemptyI1 = $true) & (vacuousDall = $true) & (singletonsubset = $true) & (powerset__Cong = $true) & (setukpairinjR11 = $true) & (setext = $true) & (symdiffE = $true) & (setadjoinSub2 = $true) & (subsetemptysetimpeq = $true) & (binunionLsub = $true) & (emptyset__Cong = $true) & (subsetE = $true) & (setukpairinjR = $true) & (dpsetconstrERa = $true) & (dsetconstrER = $true) & (setukpairinjL1 = $true) & (setunionsingleton2 = $true) & (setadjoinIR = $true) & ? [X0 : $i,X1 : $i,X2 : $i > $i > $o,X3 : $i,X4 : $i] : (((in @ (kpair @ X4 @ X1) @ (dpsetconstr @ X3 @ X0 @ (^[Y0 : $i]: ((^[Y1 : $i]: (X2 @ Y0 @ Y1)))))) = $true) & ((X2 @ X4 @ X1) != $true)) & (sepInPowerset = $true) & (setukpairIR = $true) & (disjointsetsI1 = $true) & (setminusEL = $true) & (cartprodpairin = $true) & (notequalI2 = $true) & (setextsub = $true) & (setminusERneg = $true) & (binintersectLsub = $true) & (subsetTrans = $true) & (exuI1 = $true) & (setOfPairsIsBReln = $true) & (setadjoinOr = $true) & (quantDeMorgan2 = $true) & (notsubsetI = $true) & (notequalI1 = $true) & (subPowSU = $true) & (binintersectRsub = $true) & (kfstpairEq = $true) & (setminusILneg = $true) & (cartprodfstin = $true) & (nonemptyImpWitness = $true) & (binunionRsub = $true) & (dsetconstrEL = $true) & (notinemptyset = $true) & (powersetI1 = $true) & (secondinupair = $true) & (dpsetconstrEL1 = $true) & (wellorderingAx = $true) & (setunion__Cong = $true) & (upairsetE = $true) & (binunionIL = $true) & (setukpairinjR2 = $true) & (powersetE = $true) & (exuE3e = $true) & (nonemptyI = $true) & (setoftrueEq = $true) & (replAx = $true) & (ubforcartprodlem2 = $true) & (dpsetconstrEL2 = $true) & (ksndsingleton = $true) & (upairset2IR = $true) & (ex1E1 = $true) & (notdallE = $true) & (upairsetIL = $true) & (cartprodmempair1 = $true) & (setadjoinSub = $true) & (upairinpowunion = $true) & (cartprodsndin = $true) & (setadjoinE = $true) & (eqinunit = $true) & (dpsetconstrSub = $true) & (omega0Ax = $true) & (binunionEcases = $true) & (upairset2E = $true) & (notdexE = $true) & (dpsetconstrI = $true) & (ksndpairEq = $true) & (cartprodpairsurjEq = $true) & (omegaSAx = $true) & (omegaIndAx = $true) & (powersetAx = $true) & (exuE3u = $true) & (inPowerset = $true) & (emptysetE = $true) & (exuE2 = $true) & (dsetconstrI = $true) & (descrp = $true) & (descr__Cong = $true) & (cartprodmempaircEq = $true) & (emptyinPowerset = $true) & (binintersectSubset1 = $true) & (dsetconstr__Cong = $true) & (setunionsingleton = $true) & (binunionE = $true) & (ex1I = $true) & (setminusIRneg = $true) & (symdiffI2 = $true) & (kfstsingleton = $true) & (bs114d = $true) & (subsetRefl = $true) & (cartprodpairmemEL = $true) & (setminusLsub = $true) & (subsetE2 = $true) & (setunionsingleton1 = $true) & (sepSubset = $true) & (notinsingleton = $true) & (powersetsubset = $true) & (setbeta = $true) & (exuI2 = $true) & (setminusER = $true) & (kpairsurjEq = $true) & (singletoninpowerset = $true) & (setukpairinjL = $true) & (exuI3 = $true) & (symdiffIneg2 = $true) & (cartprodfstpairEq = $true) & (setunionAx = $true) & (singletonsswitch = $true) & (exuEu = $true) & (uniqinunit = $true) & (theprop = $true) & (setunionE2 = $true) & (ex1I2 = $true) & (noeltsimpempty = $true) & (ubforcartprodlem1 = $true) & (symdiffIneg1 = $true) & (kpairp = $true) & (cartprodmempair = $true) & (eqimpsubset1 = $true) & (setminusI = $true) & (binintersectI = $true) & (omega__Cong = $true) & (binunionIR = $true) & (cartprodpairmemER = $true) & (exuE1 = $true) & (emptyI = $true) & (powersetI = $true) & (singletonsuniq = $true) & (subsetI2 = $true) & (setminusSubset1 = $true) & (emptysetsubset = $true) & (emptyInPowerset = $true) & (nonemptyE1 = $true) & (upairsubunion = $true) & (setunionE = $true) & (foundationAx = $true) & (quantDeMorgan4 = $true) & (binintersectSubset4 = $true) & (powersetE1 = $true) & (prop2setE = $true) & (subsetI1 = $true) & (setukpairIL = $true)),
% 2.99/1.99 inference(flattening,[status(thm)],[f667])).
% 2.99/1.99 thf(f667,plain,(
% 2.99/1.99 ((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i,X1 : $i,X2 : $i > $i > $o,X3 : $i,X4 : $i] : (((in @ (kpair @ X4 @ X1) @ (dpsetconstr @ X3 @ X0 @ (^[Y0 : $i]: ((^[Y1 : $i]: (X2 @ Y0 @ Y1)))))) = $true) & ((X2 @ X4 @ X1) != $true)) & (dpsetconstrEL2 = $true)) & (dpsetconstrEL1 = $true)) & (dpsetconstrERa = $true)) & (setOfPairsIsBReln = $true)) & (dpsetconstrSub = $true)) & (dpsetconstrI = $true)) & (cartprodpairsurjEq = $true)) & (cartprodsndpairEq = $true)) & (cartprodfstpairEq = $true)) & (cartprodmempaircEq = $true)) & (cartprodpairmemER = $true)) & (cartprodpairmemEL = $true)) & (cartprodsndin = $true)) & (kpairsurjEq = $true)) & (ksndpairEq = $true)) & (ksndsingleton = $true)) & (setukpairinjR = $true)) & (setukpairinjR2 = $true)) & (upairequniteq = $true)) & (setukpairinjR1 = $true)) & (setukpairinjR12 = $true)) & (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)),
% 2.99/1.99 inference(ennf_transformation,[status(thm)],[f371])).
% 2.99/1.99 thf(f371,plain,(
% 2.99/1.99 ~((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) => ((setukpairinjR12 = $true) => ((setukpairinjR1 = $true) => ((upairequniteq = $true) => ((setukpairinjR2 = $true) => ((setukpairinjR = $true) => ((ksndsingleton = $true) => ((ksndpairEq = $true) => ((kpairsurjEq = $true) => ((cartprodsndin = $true) => ((cartprodpairmemEL = $true) => ((cartprodpairmemER = $true) => ((cartprodmempaircEq = $true) => ((cartprodfstpairEq = $true) => ((cartprodsndpairEq = $true) => ((cartprodpairsurjEq = $true) => ((dpsetconstrI = $true) => ((dpsetconstrSub = $true) => ((setOfPairsIsBReln = $true) => ((dpsetconstrERa = $true) => ((dpsetconstrEL1 = $true) => ((dpsetconstrEL2 = $true) => ! [X0 : $i,X1 : $i,X2 : $i > $i > $o,X3 : $i,X4 : $i] : (((in @ (kpair @ X4 @ X1) @ (dpsetconstr @ X3 @ X0 @ (^[Y0 : $i]: ((^[Y1 : $i]: (X2 @ Y0 @ Y1)))))) = $true) => ((X2 @ X4 @ X1) = $true)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.99/1.99 inference(fool_elimination,[status(thm)],[f370])).
% 2.99/1.99 thf(f370,plain,(
% 2.99/1.99 ~(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 => (setukpairinjR12 => (setukpairinjR1 => (upairequniteq => (setukpairinjR2 => (setukpairinjR => (ksndsingleton => (ksndpairEq => (kpairsurjEq => (cartprodsndin => (cartprodpairmemEL => (cartprodpairmemER => (cartprodmempaircEq => (cartprodfstpairEq => (cartprodsndpairEq => (cartprodpairsurjEq => (dpsetconstrI => (dpsetconstrSub => (setOfPairsIsBReln => (dpsetconstrERa => (dpsetconstrEL1 => (dpsetconstrEL2 => ! [X0 : $i,X1 : $i,X2 : $i > $i > $o,X3 : $i,X4 : $i] : ((in @ (kpair @ X4 @ X1) @ (dpsetconstr @ X3 @ X0 @ (^[X5 : $i, X6 : $i] : (X2 @ X5 @ X6)))) => (X2 @ X4 @ X1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.99/1.99 inference(rectify,[status(thm)],[f195])).
% 2.99/1.99 thf(f195,negated_conjecture,(
% 2.99/1.99 ~(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 => (setukpairinjR12 => (setukpairinjR1 => (upairequniteq => (setukpairinjR2 => (setukpairinjR => (ksndsingleton => (ksndpairEq => (kpairsurjEq => (cartprodsndin => (cartprodpairmemEL => (cartprodpairmemER => (cartprodmempaircEq => (cartprodfstpairEq => (cartprodsndpairEq => (cartprodpairsurjEq => (dpsetconstrI => (dpsetconstrSub => (setOfPairsIsBReln => (dpsetconstrERa => (dpsetconstrEL1 => (dpsetconstrEL2 => ! [X4 : $i,X2 : $i,X0 : $i > $i > $o,X3 : $i,X1 : $i] : ((in @ (kpair @ X1 @ X2) @ (dpsetconstr @ X3 @ X4 @ (^[X8 : $i, X10 : $i] : (X0 @ X8 @ X10)))) => (X0 @ X1 @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.99/1.99 inference(negated_conjecture,[status(cth)],[f194])).
% 2.99/1.99 thf(f194,conjecture,(
% 2.99/1.99 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 => (setukpairinjR12 => (setukpairinjR1 => (upairequniteq => (setukpairinjR2 => (setukpairinjR => (ksndsingleton => (ksndpairEq => (kpairsurjEq => (cartprodsndin => (cartprodpairmemEL => (cartprodpairmemER => (cartprodmempaircEq => (cartprodfstpairEq => (cartprodsndpairEq => (cartprodpairsurjEq => (dpsetconstrI => (dpsetconstrSub => (setOfPairsIsBReln => (dpsetconstrERa => (dpsetconstrEL1 => (dpsetconstrEL2 => ! [X4 : $i,X2 : $i,X0 : $i > $i > $o,X3 : $i,X1 : $i] : ((in @ (kpair @ X1 @ X2) @ (dpsetconstr @ X3 @ X4 @ (^[X8 : $i, X10 : $i] : (X0 @ X8 @ X10)))) => (X0 @ X1 @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.99/1.99 file('/export/starexec/sandbox/tmp/tmp.0Ah447e165/DTF2THF_7726.p',dpsetconstrER)).
% 2.99/1.99 thf(f2743,plain,(
% 2.99/1.99 spl540_2),
% 2.99/1.99 inference(avatar_split_clause,[status(thm)],[f2724,f2734])).
% 2.99/1.99 thf(f2734,definition,(
% 2.99/1.99 spl540_2 <=> ((in @ sK115 @ sK114) = $true)),
% 2.99/1.99 introduced(definition,[new_symbols(naming,[spl540_2])],[avatar_definition])).
% 2.99/1.99 thf(f2724,plain,(
% 2.99/1.99 ((in @ sK115 @ sK114) = $true)),
% 2.99/1.99 inference(trivial_inequality_removal,[status(thm)],[f2723])).
% 2.99/1.99 thf(f2723,plain,(
% 2.99/1.99 ((in @ sK115 @ sK114) = $true) | ($true != $true)),
% 2.99/1.99 inference(forward_demodulation,[status(thm)],[f2719,f1913])).
% 2.99/1.99 thf(f1913,plain,(
% 2.99/1.99 (dpsetconstrEL2 = $true)),
% 2.99/1.99 inference(cnf_transformation,[status(thm)],[f988])).
% 2.99/1.99 thf(f2719,plain,(
% 2.99/1.99 (dpsetconstrEL2 != $true) | ((in @ sK115 @ sK114) = $true)),
% 2.99/1.99 inference(trivial_inequality_removal,[status(thm)],[f2713])).
% 2.99/1.99 thf(f2713,plain,(
% 2.99/1.99 ($true != $true) | (dpsetconstrEL2 != $true) | ((in @ sK115 @ sK114) = $true)),
% 2.99/1.99 inference(superposition,[status(thm)],[f2693,f2712])).
% 2.99/1.99 thf(f2712,plain,(
% 2.99/1.99 ((in @ (kpair @ sK118 @ sK115) @ (dpsetconstr @ sK117 @ sK114 @ sK116)) = $true)),
% 2.99/1.99 inference(beta_eta_normalization,[status(thm)],[f1954])).
% 2.99/1.99 thf(f1954,plain,(
% 2.99/1.99 ((in @ (kpair @ sK118 @ sK115) @ (dpsetconstr @ sK117 @ sK114 @ (^[Y0 : $i]: ((^[Y1 : $i]: (sK116 @ Y0 @ Y1)))))) = $true)),
% 2.99/1.99 inference(cnf_transformation,[status(thm)],[f988])).
% 2.99/1.99 thf(f2693,plain,(
% 2.99/1.99 ( ! [X8 : $i > $i > $o,X6 : $i,X9 : $i,X7 : $i,X5 : $i] : (($true != (in @ (kpair @ X6 @ X9) @ (dpsetconstr @ X7 @ X5 @ X8))) | (dpsetconstrEL2 != $true) | ($true = (in @ X9 @ X5))) )),
% 2.99/1.99 inference(beta_eta_normalization,[status(thm)],[f1779])).
% 2.99/1.99 thf(f1779,plain,(
% 2.99/1.99 ( ! [X8 : $i > $i > $o,X6 : $i,X9 : $i,X7 : $i,X5 : $i] : (((in @ (kpair @ X6 @ X9) @ (dpsetconstr @ X7 @ X5 @ (^[Y0 : $i]: ((^[Y1 : $i]: (X8 @ Y0 @ Y1)))))) != $true) | (dpsetconstrEL2 != $true) | ($true = (in @ X9 @ X5))) )),
% 2.99/1.99 inference(cnf_transformation,[status(thm)],[f944])).
% 2.99/1.99 thf(f944,plain,(
% 2.99/1.99 ((dpsetconstrEL2 = $true) | (((in @ sK83 @ sK79) != $true) & ((in @ (kpair @ sK80 @ sK83) @ (dpsetconstr @ sK81 @ sK79 @ (^[Y0 : $i]: ((^[Y1 : $i]: (sK82 @ Y0 @ Y1)))))) = $true))) & (! [X5 : $i,X6 : $i,X7 : $i,X8 : $i > $i > $o,X9 : $i] : (($true = (in @ X9 @ X5)) | ((in @ (kpair @ X6 @ X9) @ (dpsetconstr @ X7 @ X5 @ (^[Y0 : $i]: ((^[Y1 : $i]: (X8 @ Y0 @ Y1)))))) != $true)) | (dpsetconstrEL2 != $true))),
% 2.99/1.99 inference(skolemisation,[status(esa),new_symbols(skolem,[sK79,sK80,sK81,sK82,sK83])],[f942,f943])).
% 2.99/1.99 thf(f943,plain,(
% 2.99/1.99 ? [X0 : $i,X1 : $i,X2 : $i,X3 : $i > $i > $o,X4 : $i] : (((in @ X4 @ X0) != $true) & ($true = (in @ (kpair @ X1 @ X4) @ (dpsetconstr @ X2 @ X0 @ (^[Y0 : $i]: ((^[Y1 : $i]: (X3 @ Y0 @ Y1)))))))) => (((in @ sK83 @ sK79) != $true) & ((in @ (kpair @ sK80 @ sK83) @ (dpsetconstr @ sK81 @ sK79 @ (^[Y0 : $i]: ((^[Y1 : $i]: (sK82 @ Y0 @ Y1)))))) = $true))),
% 2.99/1.99 introduced(definition,[],[choice_axiom])).
% 2.99/1.99 thf(f942,plain,(
% 2.99/1.99 ((dpsetconstrEL2 = $true) | ? [X0 : $i,X1 : $i,X2 : $i,X3 : $i > $i > $o,X4 : $i] : (((in @ X4 @ X0) != $true) & ($true = (in @ (kpair @ X1 @ X4) @ (dpsetconstr @ X2 @ X0 @ (^[Y0 : $i]: ((^[Y1 : $i]: (X3 @ Y0 @ Y1))))))))) & (! [X5 : $i,X6 : $i,X7 : $i,X8 : $i > $i > $o,X9 : $i] : (($true = (in @ X9 @ X5)) | ((in @ (kpair @ X6 @ X9) @ (dpsetconstr @ X7 @ X5 @ (^[Y0 : $i]: ((^[Y1 : $i]: (X8 @ Y0 @ Y1)))))) != $true)) | (dpsetconstrEL2 != $true))),
% 2.99/1.99 inference(rectify,[status(thm)],[f941])).
% 2.99/1.99 thf(f941,plain,(
% 2.99/1.99 ((dpsetconstrEL2 = $true) | ? [X0 : $i,X1 : $i,X2 : $i,X3 : $i > $i > $o,X4 : $i] : (((in @ X4 @ X0) != $true) & ($true = (in @ (kpair @ X1 @ X4) @ (dpsetconstr @ X2 @ X0 @ (^[Y0 : $i]: ((^[Y1 : $i]: (X3 @ Y0 @ Y1))))))))) & (! [X0 : $i,X1 : $i,X2 : $i,X3 : $i > $i > $o,X4 : $i] : (((in @ X4 @ X0) = $true) | ($true != (in @ (kpair @ X1 @ X4) @ (dpsetconstr @ X2 @ X0 @ (^[Y0 : $i]: ((^[Y1 : $i]: (X3 @ Y0 @ Y1)))))))) | (dpsetconstrEL2 != $true))),
% 2.99/1.99 inference(nnf_transformation,[status(thm)],[f730])).
% 2.99/1.99 thf(f730,plain,(
% 2.99/1.99 (dpsetconstrEL2 = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i,X3 : $i > $i > $o,X4 : $i] : (((in @ X4 @ X0) = $true) | ($true != (in @ (kpair @ X1 @ X4) @ (dpsetconstr @ X2 @ X0 @ (^[Y0 : $i]: ((^[Y1 : $i]: (X3 @ Y0 @ Y1))))))))),
% 2.99/1.99 inference(ennf_transformation,[status(thm)],[f248])).
% 2.99/1.99 thf(f248,plain,(
% 2.99/1.99 (dpsetconstrEL2 = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i,X3 : $i > $i > $o,X4 : $i] : (($true = (in @ (kpair @ X1 @ X4) @ (dpsetconstr @ X2 @ X0 @ (^[Y0 : $i]: ((^[Y1 : $i]: (X3 @ Y0 @ Y1))))))) => ((in @ X4 @ X0) = $true))),
% 2.99/1.99 inference(fool_elimination,[status(thm)],[f247])).
% 2.99/1.99 thf(f247,plain,(
% 2.99/1.99 (dpsetconstrEL2 = ! [X0 : $i,X1 : $i,X2 : $i,X3 : $i > $i > $o,X4 : $i] : ((in @ (kpair @ X1 @ X4) @ (dpsetconstr @ X2 @ X0 @ (^[X5 : $i, X6 : $i] : (X3 @ X5 @ X6)))) => (in @ X4 @ X0)))),
% 2.99/1.99 inference(rectify,[status(thm)],[f193])).
% 2.99/1.99 thf(f193,axiom,(
% 2.99/1.99 (dpsetconstrEL2 = ! [X4 : $i,X1 : $i,X3 : $i,X0 : $i > $i > $o,X2 : $i] : ((in @ (kpair @ X1 @ X2) @ (dpsetconstr @ X3 @ X4 @ (^[X8 : $i, X10 : $i] : (X0 @ X8 @ X10)))) => (in @ X2 @ X4)))),
% 2.99/1.99 file('/export/starexec/sandbox/tmp/tmp.0Ah447e165/DTF2THF_7726.p',dpsetconstrEL2)).
% 2.99/1.99 thf(f2742,plain,(
% 2.99/1.99 spl540_3),
% 2.99/1.99 inference(avatar_split_clause,[status(thm)],[f2722,f2738])).
% 2.99/1.99 thf(f2738,definition,(
% 2.99/1.99 spl540_3 <=> ((in @ sK118 @ sK117) = $true)),
% 2.99/1.99 introduced(definition,[new_symbols(naming,[spl540_3])],[avatar_definition])).
% 2.99/1.99 thf(f2722,plain,(
% 2.99/1.99 ((in @ sK118 @ sK117) = $true)),
% 2.99/1.99 inference(trivial_inequality_removal,[status(thm)],[f2721])).
% 2.99/1.99 thf(f2721,plain,(
% 2.99/1.99 ($true != $true) | ((in @ sK118 @ sK117) = $true)),
% 2.99/1.99 inference(forward_demodulation,[status(thm)],[f2720,f1925])).
% 2.99/1.99 thf(f1925,plain,(
% 2.99/1.99 (dpsetconstrEL1 = $true)),
% 2.99/1.99 inference(cnf_transformation,[status(thm)],[f988])).
% 2.99/1.99 thf(f2720,plain,(
% 2.99/1.99 ((in @ sK118 @ sK117) = $true) | (dpsetconstrEL1 != $true)),
% 2.99/1.99 inference(trivial_inequality_removal,[status(thm)],[f2714])).
% 2.99/1.99 thf(f2714,plain,(
% 2.99/1.99 (dpsetconstrEL1 != $true) | ((in @ sK118 @ sK117) = $true) | ($true != $true)),
% 2.99/1.99 inference(superposition,[status(thm)],[f2656,f2712])).
% 2.99/1.99 thf(f2656,plain,(
% 2.99/1.99 ( ! [X8 : $i,X6 : $i,X9 : $i,X7 : $i > $i > $o,X5 : $i] : (((in @ (kpair @ X6 @ X8) @ (dpsetconstr @ X5 @ X9 @ X7)) != $true) | (dpsetconstrEL1 != $true) | ((in @ X6 @ X5) = $true)) )),
% 2.99/1.99 inference(beta_eta_normalization,[status(thm)],[f2427])).
% 2.99/1.99 thf(f2427,plain,(
% 2.99/1.99 ( ! [X8 : $i,X6 : $i,X9 : $i,X7 : $i > $i > $o,X5 : $i] : (((in @ X6 @ X5) = $true) | (dpsetconstrEL1 != $true) | ((in @ (kpair @ X6 @ X8) @ (dpsetconstr @ X5 @ X9 @ (^[Y0 : $i]: ((^[Y1 : $i]: (X7 @ Y0 @ Y1)))))) != $true)) )),
% 2.99/1.99 inference(cnf_transformation,[status(thm)],[f1476])).
% 2.99/1.99 thf(f1476,plain,(
% 2.99/1.99 ((dpsetconstrEL1 = $true) | (((in @ (kpair @ sK415 @ sK417) @ (dpsetconstr @ sK414 @ sK418 @ (^[Y0 : $i]: ((^[Y1 : $i]: (sK416 @ Y0 @ Y1)))))) = $true) & ($true != (in @ sK415 @ sK414)))) & (! [X5 : $i,X6 : $i,X7 : $i > $i > $o,X8 : $i,X9 : $i] : (((in @ (kpair @ X6 @ X8) @ (dpsetconstr @ X5 @ X9 @ (^[Y0 : $i]: ((^[Y1 : $i]: (X7 @ Y0 @ Y1)))))) != $true) | ((in @ X6 @ X5) = $true)) | (dpsetconstrEL1 != $true))),
% 2.99/1.99 inference(skolemisation,[status(esa),new_symbols(skolem,[sK414,sK415,sK416,sK417,sK418])],[f1474,f1475])).
% 2.99/1.99 thf(f1475,plain,(
% 2.99/1.99 ? [X0 : $i,X1 : $i,X2 : $i > $i > $o,X3 : $i,X4 : $i] : (((in @ (kpair @ X1 @ X3) @ (dpsetconstr @ X0 @ X4 @ (^[Y0 : $i]: ((^[Y1 : $i]: (X2 @ Y0 @ Y1)))))) = $true) & ((in @ X1 @ X0) != $true)) => (((in @ (kpair @ sK415 @ sK417) @ (dpsetconstr @ sK414 @ sK418 @ (^[Y0 : $i]: ((^[Y1 : $i]: (sK416 @ Y0 @ Y1)))))) = $true) & ($true != (in @ sK415 @ sK414)))),
% 2.99/1.99 introduced(definition,[],[choice_axiom])).
% 2.99/1.99 thf(f1474,plain,(
% 2.99/1.99 ((dpsetconstrEL1 = $true) | ? [X0 : $i,X1 : $i,X2 : $i > $i > $o,X3 : $i,X4 : $i] : (((in @ (kpair @ X1 @ X3) @ (dpsetconstr @ X0 @ X4 @ (^[Y0 : $i]: ((^[Y1 : $i]: (X2 @ Y0 @ Y1)))))) = $true) & ((in @ X1 @ X0) != $true))) & (! [X5 : $i,X6 : $i,X7 : $i > $i > $o,X8 : $i,X9 : $i] : (((in @ (kpair @ X6 @ X8) @ (dpsetconstr @ X5 @ X9 @ (^[Y0 : $i]: ((^[Y1 : $i]: (X7 @ Y0 @ Y1)))))) != $true) | ((in @ X6 @ X5) = $true)) | (dpsetconstrEL1 != $true))),
% 2.99/1.99 inference(rectify,[status(thm)],[f1473])).
% 2.99/1.99 thf(f1473,plain,(
% 2.99/1.99 ((dpsetconstrEL1 = $true) | ? [X0 : $i,X1 : $i,X2 : $i > $i > $o,X3 : $i,X4 : $i] : (((in @ (kpair @ X1 @ X3) @ (dpsetconstr @ X0 @ X4 @ (^[Y0 : $i]: ((^[Y1 : $i]: (X2 @ Y0 @ Y1)))))) = $true) & ((in @ X1 @ X0) != $true))) & (! [X0 : $i,X1 : $i,X2 : $i > $i > $o,X3 : $i,X4 : $i] : (((in @ (kpair @ X1 @ X3) @ (dpsetconstr @ X0 @ X4 @ (^[Y0 : $i]: ((^[Y1 : $i]: (X2 @ Y0 @ Y1)))))) != $true) | ((in @ X1 @ X0) = $true)) | (dpsetconstrEL1 != $true))),
% 2.99/1.99 inference(nnf_transformation,[status(thm)],[f708])).
% 2.99/1.99 thf(f708,plain,(
% 2.99/1.99 (dpsetconstrEL1 = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i > $i > $o,X3 : $i,X4 : $i] : (((in @ (kpair @ X1 @ X3) @ (dpsetconstr @ X0 @ X4 @ (^[Y0 : $i]: ((^[Y1 : $i]: (X2 @ Y0 @ Y1)))))) != $true) | ((in @ X1 @ X0) = $true))),
% 2.99/1.99 inference(ennf_transformation,[status(thm)],[f324])).
% 2.99/1.99 thf(f324,plain,(
% 2.99/1.99 ! [X0 : $i,X1 : $i,X2 : $i > $i > $o,X3 : $i,X4 : $i] : (((in @ (kpair @ X1 @ X3) @ (dpsetconstr @ X0 @ X4 @ (^[Y0 : $i]: ((^[Y1 : $i]: (X2 @ Y0 @ Y1)))))) = $true) => ((in @ X1 @ X0) = $true)) <=> (dpsetconstrEL1 = $true)),
% 2.99/1.99 inference(fool_elimination,[status(thm)],[f323])).
% 2.99/1.99 thf(f323,plain,(
% 2.99/1.99 (! [X0 : $i,X1 : $i,X2 : $i > $i > $o,X3 : $i,X4 : $i] : ((in @ (kpair @ X1 @ X3) @ (dpsetconstr @ X0 @ X4 @ (^[X5 : $i, X6 : $i] : (X2 @ X5 @ X6)))) => (in @ X1 @ X0)) = dpsetconstrEL1)),
% 2.99/1.99 inference(rectify,[status(thm)],[f192])).
% 2.99/1.99 thf(f192,axiom,(
% 2.99/1.99 (! [X3 : $i,X1 : $i,X0 : $i > $i > $o,X2 : $i,X4 : $i] : ((in @ (kpair @ X1 @ X2) @ (dpsetconstr @ X3 @ X4 @ (^[X8 : $i, X10 : $i] : (X0 @ X8 @ X10)))) => (in @ X1 @ X3)) = dpsetconstrEL1)),
% 2.99/1.99 file('/export/starexec/sandbox/tmp/tmp.0Ah447e165/DTF2THF_7726.p',dpsetconstrEL1)).
% 2.99/1.99 thf(f2741,plain,(
% 2.99/1.99 spl540_1 | ~spl540_2 | ~spl540_3),
% 2.99/1.99 inference(avatar_split_clause,[status(thm)],[f2728,f2738,f2734,f2730])).
% 2.99/1.99 thf(f2728,plain,(
% 2.99/1.99 ((sK116 @ sK118 @ sK115) = $true) | ((in @ sK115 @ sK114) != $true) | ((in @ sK118 @ sK117) != $true)),
% 2.99/1.99 inference(trivial_inequality_removal,[status(thm)],[f2727])).
% 2.99/1.99 thf(f2727,plain,(
% 2.99/1.99 ($true != $true) | ((sK116 @ sK118 @ sK115) = $true) | ((in @ sK118 @ sK117) != $true) | ((in @ sK115 @ sK114) != $true)),
% 2.99/1.99 inference(forward_demodulation,[status(thm)],[f2717,f1959])).
% 2.99/1.99 thf(f1959,plain,(
% 2.99/1.99 (dpsetconstrERa = $true)),
% 2.99/1.99 inference(cnf_transformation,[status(thm)],[f988])).
% 2.99/1.99 thf(f2717,plain,(
% 2.99/1.99 ((in @ sK118 @ sK117) != $true) | (dpsetconstrERa != $true) | ((in @ sK115 @ sK114) != $true) | ((sK116 @ sK118 @ sK115) = $true)),
% 2.99/1.99 inference(trivial_inequality_removal,[status(thm)],[f2715])).
% 2.99/1.99 thf(f2715,plain,(
% 2.99/1.99 ((in @ sK115 @ sK114) != $true) | ((in @ sK118 @ sK117) != $true) | (dpsetconstrERa != $true) | ($true != $true) | ((sK116 @ sK118 @ sK115) = $true)),
% 2.99/1.99 inference(superposition,[status(thm)],[f2640,f2712])).
% 2.99/1.99 thf(f2640,plain,(
% 2.99/1.99 ( ! [X8 : $i > $i > $o,X6 : $i,X9 : $i,X7 : $i,X5 : $i] : (((in @ (kpair @ X6 @ X9) @ (dpsetconstr @ X5 @ X7 @ X8)) != $true) | (dpsetconstrERa != $true) | ((X8 @ X6 @ X9) = $true) | ($true != (in @ X9 @ X7)) | ((in @ X6 @ X5) != $true)) )),
% 2.99/1.99 inference(beta_eta_normalization,[status(thm)],[f1792])).
% 2.99/1.99 thf(f1792,plain,(
% 2.99/1.99 ( ! [X8 : $i > $i > $o,X6 : $i,X9 : $i,X7 : $i,X5 : $i] : ((dpsetconstrERa != $true) | ($true != (in @ (kpair @ X6 @ X9) @ (dpsetconstr @ X5 @ X7 @ (^[Y0 : $i]: ((^[Y1 : $i]: (X8 @ Y0 @ Y1))))))) | ((X8 @ X6 @ X9) = $true) | ($true != (in @ X9 @ X7)) | ((in @ X6 @ X5) != $true)) )),
% 2.99/1.99 inference(cnf_transformation,[status(thm)],[f961])).
% 2.99/1.99 thf(f961,plain,(
% 2.99/1.99 ((dpsetconstrERa = $true) | ((((sK94 @ sK92 @ sK95) != $true) & ((in @ (kpair @ sK92 @ sK95) @ (dpsetconstr @ sK91 @ sK93 @ (^[Y0 : $i]: ((^[Y1 : $i]: (sK94 @ Y0 @ Y1)))))) = $true) & ((in @ sK95 @ sK93) = $true)) & ((in @ sK92 @ sK91) = $true))) & (! [X5 : $i,X6 : $i,X7 : $i,X8 : $i > $i > $o] : (! [X9 : $i] : (((X8 @ X6 @ X9) = $true) | ($true != (in @ (kpair @ X6 @ X9) @ (dpsetconstr @ X5 @ X7 @ (^[Y0 : $i]: ((^[Y1 : $i]: (X8 @ Y0 @ Y1))))))) | ($true != (in @ X9 @ X7))) | ((in @ X6 @ X5) != $true)) | (dpsetconstrERa != $true))),
% 2.99/1.99 inference(skolemisation,[status(esa),new_symbols(skolem,[sK91,sK92,sK93,sK94,sK95])],[f958,f960,f959])).
% 2.99/1.99 thf(f959,plain,(
% 2.99/1.99 ? [X0 : $i,X1 : $i,X2 : $i,X3 : $i > $i > $o] : (? [X4 : $i] : (((X3 @ X1 @ X4) != $true) & ((in @ (kpair @ X1 @ X4) @ (dpsetconstr @ X0 @ X2 @ (^[Y0 : $i]: ((^[Y1 : $i]: (X3 @ Y0 @ Y1)))))) = $true) & ($true = (in @ X4 @ X2))) & ((in @ X1 @ X0) = $true)) => (? [X4 : $i] : (((sK94 @ sK92 @ X4) != $true) & ((in @ (kpair @ sK92 @ X4) @ (dpsetconstr @ sK91 @ sK93 @ (^[Y0 : $i]: ((^[Y1 : $i]: (sK94 @ Y0 @ Y1)))))) = $true) & ((in @ X4 @ sK93) = $true)) & ((in @ sK92 @ sK91) = $true))),
% 2.99/1.99 introduced(definition,[],[choice_axiom])).
% 2.99/1.99 thf(f960,plain,(
% 2.99/1.99 ? [X4 : $i] : (((sK94 @ sK92 @ X4) != $true) & ((in @ (kpair @ sK92 @ X4) @ (dpsetconstr @ sK91 @ sK93 @ (^[Y0 : $i]: ((^[Y1 : $i]: (sK94 @ Y0 @ Y1)))))) = $true) & ((in @ X4 @ sK93) = $true)) => (((sK94 @ sK92 @ sK95) != $true) & ((in @ (kpair @ sK92 @ sK95) @ (dpsetconstr @ sK91 @ sK93 @ (^[Y0 : $i]: ((^[Y1 : $i]: (sK94 @ Y0 @ Y1)))))) = $true) & ((in @ sK95 @ sK93) = $true))),
% 2.99/1.99 introduced(definition,[],[choice_axiom])).
% 2.99/1.99 thf(f958,plain,(
% 2.99/1.99 ((dpsetconstrERa = $true) | ? [X0 : $i,X1 : $i,X2 : $i,X3 : $i > $i > $o] : (? [X4 : $i] : (((X3 @ X1 @ X4) != $true) & ((in @ (kpair @ X1 @ X4) @ (dpsetconstr @ X0 @ X2 @ (^[Y0 : $i]: ((^[Y1 : $i]: (X3 @ Y0 @ Y1)))))) = $true) & ($true = (in @ X4 @ X2))) & ((in @ X1 @ X0) = $true))) & (! [X5 : $i,X6 : $i,X7 : $i,X8 : $i > $i > $o] : (! [X9 : $i] : (((X8 @ X6 @ X9) = $true) | ($true != (in @ (kpair @ X6 @ X9) @ (dpsetconstr @ X5 @ X7 @ (^[Y0 : $i]: ((^[Y1 : $i]: (X8 @ Y0 @ Y1))))))) | ($true != (in @ X9 @ X7))) | ((in @ X6 @ X5) != $true)) | (dpsetconstrERa != $true))),
% 2.99/1.99 inference(rectify,[status(thm)],[f957])).
% 2.99/1.99 thf(f957,plain,(
% 2.99/1.99 ((dpsetconstrERa = $true) | ? [X0 : $i,X1 : $i,X2 : $i,X3 : $i > $i > $o] : (? [X4 : $i] : (((X3 @ X1 @ X4) != $true) & ((in @ (kpair @ X1 @ X4) @ (dpsetconstr @ X0 @ X2 @ (^[Y0 : $i]: ((^[Y1 : $i]: (X3 @ Y0 @ Y1)))))) = $true) & ($true = (in @ X4 @ X2))) & ((in @ X1 @ X0) = $true))) & (! [X0 : $i,X1 : $i,X2 : $i,X3 : $i > $i > $o] : (! [X4 : $i] : (((X3 @ X1 @ X4) = $true) | ((in @ (kpair @ X1 @ X4) @ (dpsetconstr @ X0 @ X2 @ (^[Y0 : $i]: ((^[Y1 : $i]: (X3 @ Y0 @ Y1)))))) != $true) | ($true != (in @ X4 @ X2))) | ((in @ X1 @ X0) != $true)) | (dpsetconstrERa != $true))),
% 2.99/1.99 inference(nnf_transformation,[status(thm)],[f757])).
% 2.99/1.99 thf(f757,plain,(
% 2.99/1.99 (dpsetconstrERa = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i,X3 : $i > $i > $o] : (! [X4 : $i] : (((X3 @ X1 @ X4) = $true) | ((in @ (kpair @ X1 @ X4) @ (dpsetconstr @ X0 @ X2 @ (^[Y0 : $i]: ((^[Y1 : $i]: (X3 @ Y0 @ Y1)))))) != $true) | ($true != (in @ X4 @ X2))) | ((in @ X1 @ X0) != $true))),
% 2.99/1.99 inference(flattening,[status(thm)],[f756])).
% 2.99/1.99 thf(f756,plain,(
% 2.99/1.99 ! [X0 : $i,X1 : $i,X2 : $i,X3 : $i > $i > $o] : (! [X4 : $i] : ((((X3 @ X1 @ X4) = $true) | ((in @ (kpair @ X1 @ X4) @ (dpsetconstr @ X0 @ X2 @ (^[Y0 : $i]: ((^[Y1 : $i]: (X3 @ Y0 @ Y1)))))) != $true)) | ($true != (in @ X4 @ X2))) | ((in @ X1 @ X0) != $true)) <=> (dpsetconstrERa = $true)),
% 2.99/1.99 inference(ennf_transformation,[status(thm)],[f464])).
% 2.99/1.99 thf(f464,plain,(
% 2.99/1.99 ! [X0 : $i,X1 : $i,X2 : $i,X3 : $i > $i > $o] : (((in @ X1 @ X0) = $true) => ! [X4 : $i] : (($true = (in @ X4 @ X2)) => (((in @ (kpair @ X1 @ X4) @ (dpsetconstr @ X0 @ X2 @ (^[Y0 : $i]: ((^[Y1 : $i]: (X3 @ Y0 @ Y1)))))) = $true) => ((X3 @ X1 @ X4) = $true)))) <=> (dpsetconstrERa = $true)),
% 2.99/1.99 inference(fool_elimination,[status(thm)],[f463])).
% 2.99/1.99 thf(f463,plain,(
% 2.99/1.99 (dpsetconstrERa = ! [X0 : $i,X1 : $i,X2 : $i,X3 : $i > $i > $o] : ((in @ X1 @ X0) => ! [X4 : $i] : ((in @ X4 @ X2) => ((in @ (kpair @ X1 @ X4) @ (dpsetconstr @ X0 @ X2 @ (^[X5 : $i, X6 : $i] : (X3 @ X5 @ X6)))) => (X3 @ X1 @ X4)))))),
% 2.99/1.99 inference(rectify,[status(thm)],[f191])).
% 2.99/1.99 thf(f191,axiom,(
% 2.99/1.99 (dpsetconstrERa = ! [X3 : $i,X1 : $i,X4 : $i,X0 : $i > $i > $o] : ((in @ X1 @ X3) => ! [X2 : $i] : ((in @ X2 @ X4) => ((in @ (kpair @ X1 @ X2) @ (dpsetconstr @ X3 @ X4 @ (^[X8 : $i, X10 : $i] : (X0 @ X8 @ X10)))) => (X0 @ X1 @ X2)))))),
% 2.99/1.99 file('/export/starexec/sandbox/tmp/tmp.0Ah447e165/DTF2THF_7726.p',dpsetconstrERa)).
% 2.99/1.99 % SZS output end Proof for DTF2THF_7726
% 2.99/1.99 % (7862)------------------------------
% 2.99/1.99 % (7862)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.99/1.99 % (7862)Termination reason: Refutation
% 2.99/1.99
% 2.99/1.99 % (7862)Memory used [KB]: 8571
% 2.99/1.99 % (7862)Time elapsed: 0.111 s
% 2.99/1.99 % (7862)Instructions burned: 211 (million)
% 2.99/1.99 % (7862)------------------------------
% 2.99/1.99 % (7862)------------------------------
% 2.99/1.99 % (7839)Success in time 0.209 s
%------------------------------------------------------------------------------