%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU688^1 : 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 : n014.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 09:36:24 AM UTC 2026
% Result : Theorem 2.72s 1.73s
% Output : Refutation 2.72s
% Verified :
% SZS Type : Refutation
% Derivation depth : 15
% Number of leaves : 12
% Syntax : Number of formulae : 58 ( 15 unt; 0 typ; 2 def)
% Number of atoms : 3881 (1037 equ; 0 cnn)
% Maximal formula atoms : 209 ( 66 avg)
% Number of connectives : 2363 ( 112 ~; 80 |; 683 &; 627 @)
% ( 7 <=>; 854 =>; 0 <=; 0 <~>)
% Maximal formula depth : 219 ( 31 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 389 ( 386 usr; 221 con; 0-4 aty)
% Number of variables : 147 ( 0 ^; 96 !; 51 ?; 147 :)
% 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_222,type,
func: $i > $i > $i > $o ).
thf(func_def_223,type,
funcSet: $i > $i > $i ).
thf(func_def_226,type,
ap: $i > $i > $i > $i > $i ).
thf(func_def_232,type,
lam: $i > $i > ( $i > $i ) > $i ).
thf(func_def_255,type,
sP0: $o > $i > $i > $i > $o ).
thf(func_def_260,type,
sP5: $i > $i > $o ).
thf(func_def_261,type,
sP6: $i > $o ).
thf(func_def_262,type,
sP7: $i > $i > $o ).
thf(func_def_263,type,
sP8: $i > $i > $o ).
thf(func_def_279,type,
sK24: $i > $o ).
thf(func_def_280,type,
sK25: $i > $o ).
thf(func_def_283,type,
sK28: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_284,type,
sK29: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_285,type,
sK30: $i > $o ).
thf(func_def_286,type,
sK31: $i > $o ).
thf(func_def_287,type,
sK32: $i > $i > $i ).
thf(func_def_310,type,
sK55: $i > $i > $i ).
thf(func_def_324,type,
sK69: $i > ( $i > $o ) > $i ).
thf(func_def_325,type,
sK70: $i > $o ).
thf(func_def_334,type,
sK79: ( $i > $o ) > $i ).
thf(func_def_335,type,
sK80: $i > $o ).
thf(func_def_336,type,
sK81: $i > $i ).
thf(func_def_340,type,
sK85: $i > $i ).
thf(func_def_342,type,
sK87: $i > ( $i > $i ) > $i > $i ).
thf(func_def_354,type,
sK99: $i > $i > $i ).
thf(func_def_355,type,
sK100: ( $i > $o ) > $i ).
thf(func_def_356,type,
sK101: $i > $o ).
thf(func_def_357,type,
sK102: $i > $i ).
thf(func_def_360,type,
sK105: $i > $o ).
thf(func_def_363,type,
sK108: $i > $i ).
thf(func_def_370,type,
sK115: $i > $o ).
thf(func_def_385,type,
sK130: $i > $i > $i ).
thf(func_def_386,type,
sK131: $i > $i > $i ).
thf(func_def_389,type,
sK134: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_390,type,
sK135: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_393,type,
sK138: $i > $o ).
thf(func_def_394,type,
sK139: $i > $o ).
thf(func_def_413,type,
sK158: ( $i > $o ) > $i > $i ).
thf(func_def_415,type,
sK160: $i > $o ).
thf(func_def_420,type,
sK165: $i > $o ).
thf(func_def_427,type,
sK172: $i > $i > $i > $i > $i ).
thf(func_def_430,type,
sK175: $i > $i > $i ).
thf(func_def_431,type,
sK176: $i > $i > $i ).
thf(func_def_442,type,
sK187: $i > $i > $o ).
thf(func_def_460,type,
sK205: $i > $o ).
thf(func_def_463,type,
sK208: $i > ( $i > $o ) > $i ).
thf(func_def_464,type,
sK209: $i > ( $i > $o ) > $i ).
thf(func_def_488,type,
sK233: $i > $i > $i ).
thf(func_def_502,type,
sK247: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_503,type,
sK248: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_507,type,
sK252: $i > $o ).
thf(func_def_509,type,
sK254: $i > $o ).
thf(func_def_513,type,
sK258: $i > $o ).
thf(func_def_521,type,
sK266: $i > $i ).
thf(func_def_523,type,
sK268: $i > $i ).
thf(func_def_524,type,
sK269: $i > ( $i > $i ) > $i > $i ).
thf(func_def_526,type,
sK271: $i > $i ).
thf(func_def_528,type,
sK273: $i > $i > $o ).
thf(func_def_533,type,
sK278: ( $i > $o ) > $i ).
thf(func_def_534,type,
sK279: $i > $o ).
thf(func_def_543,type,
sK288: $i > $i > $o ).
thf(func_def_545,type,
sK290: $i > $i ).
thf(func_def_546,type,
sK291: $i > $i ).
thf(func_def_547,type,
sK292: $i > ( $i > $i > $o ) > $i ).
thf(func_def_548,type,
sK293: $i > ( $i > $i > $o ) > $i ).
thf(func_def_549,type,
sK294: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_550,type,
sK295: $i > ( $i > $o ) > $i ).
thf(func_def_551,type,
sK296: $i > $o ).
thf(func_def_556,type,
sK301: $i > $o ).
thf(func_def_558,type,
sK303: ( $i > $o ) > $i ).
thf(func_def_559,type,
sK304: ( $i > $o ) > $i ).
thf(func_def_567,type,
sK312: $i > $i > $o ).
thf(func_def_586,type,
sK331: $i > $o ).
thf(func_def_589,type,
sK334: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_590,type,
sK335: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_594,type,
sK339: $i > $o ).
thf(func_def_596,type,
sK341: $i > $o ).
thf(func_def_623,type,
sK368: $i > $i ).
thf(func_def_625,type,
sK370: $i > $i > $i ).
thf(func_def_629,type,
sK374: $i > $o ).
thf(func_def_632,type,
sK377: $i > ( $i > $o ) > $i ).
thf(func_def_633,type,
sK378: $i > ( $i > $o ) > $i ).
thf(func_def_652,type,
sK397: $i > $i > $o ).
thf(func_def_654,type,
sK399: $i > $o ).
thf(func_def_657,type,
sK402: $i > ( $i > $o ) > $i ).
thf(func_def_661,type,
sK406: $i > $i > $i > $i ).
thf(func_def_662,type,
sK407: $i > $i > $i > $i ).
thf(func_def_664,type,
sK409: ( $i > $o ) > $i > $i > $i ).
thf(func_def_667,type,
sK412: $i > $o ).
thf(func_def_668,type,
sK413: $i > $i ).
thf(func_def_671,type,
sK416: $i > $i > ( $i > $i ) > $i ).
thf(func_def_673,type,
sK418: $i > $o ).
thf(func_def_675,type,
sK420: ( $i > $o ) > $i > $i ).
thf(func_def_698,type,
sK443: $o > $i > $i > $i ).
thf(func_def_703,type,
sK448: $i > $o ).
thf(func_def_706,type,
sK451: $i > $o ).
thf(func_def_714,type,
sK459: $i > $i > $o ).
thf(func_def_730,type,
sK475: $i > $o ).
thf(func_def_732,type,
sK477: $i > ( $i > $o ) > $i ).
thf(func_def_737,type,
sK482: $i > $i > $o ).
thf(func_def_739,type,
sK484: $i > $o ).
thf(func_def_740,type,
sK485: $i > $o ).
thf(func_def_741,type,
sK486: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_742,type,
sK487: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_752,type,
sK497: $i > $o ).
thf(func_def_757,type,
sK502: $i > $i ).
thf(func_def_763,type,
sK508: ( $i > $o ) > $i > $i ).
thf(func_def_765,type,
sK510: $i > $o ).
thf(func_def_777,type,
sK522: $i > $i > $i ).
thf(func_def_778,type,
sK523: $i > $i > $i ).
thf(func_def_779,type,
sK524: $i > $i > $i ).
thf(func_def_780,type,
sK525: $i > $i > $i ).
thf(func_def_781,type,
sK526: $i > $i > $i > $i ).
thf(func_def_782,type,
sK527: $i > $i ).
thf(func_def_783,type,
sK528: $i > $i ).
thf(func_def_784,type,
sK529: $i > $i ).
thf(func_def_785,type,
sK530: $i > $i ).
thf(func_def_786,type,
sK531: $i > $i > $i ).
thf(func_def_787,type,
sK532: $i > $i > $i > $i ).
thf(func_def_788,type,
sK533: $i > $i > $i > $i ).
thf(func_def_789,type,
sK534: $i > $i > $i ).
thf(func_def_790,type,
sK535: $i > $i > $i ).
thf(func_def_791,type,
sK536: $i > $i ).
thf(func_def_792,type,
sK537: $i > $i > $i ).
thf(func_def_794,type,
sK539: $i > $i ).
thf(func_def_795,type,
sK540: $i > $i ).
thf(func_def_800,type,
sK545: $i > $i ).
thf(func_def_809,type,
sK554: $i > $i > $o ).
thf(func_def_814,type,
sK559: ( $i > $o ) > $i > $i ).
thf(func_def_816,type,
sK561: $i > $o ).
thf(func_def_820,type,
sK565: $i > $i ).
thf(func_def_833,type,
sK578: $i > $o ).
thf(func_def_838,type,
sK583: $i > $i > $i ).
thf(func_def_860,type,
sK605: ( $i > $o ) > $i > $i ).
thf(func_def_862,type,
sK607: $i > $o ).
thf(func_def_874,type,
ph619:
!>[X0: $tType] : X0 ).
thf(f3086,plain,
$false,
inference(avatar_sat_refutation,[status(thm)],[f3039,f3040,f3076]) ).
thf(f3076,plain,
~ spl618_1,
inference(avatar_contradiction_clause,[status(thm)],[f3075]) ).
thf(f3075,plain,
( $false
| ~ spl618_1 ),
inference(trivial_inequality_removal,[status(thm)],[f3067]) ).
thf(f3067,plain,
( ( $true != $true )
| ~ spl618_1 ),
inference(superposition,[status(thm)],[f3051,f2668]) ).
thf(f2668,plain,
( ( in @ sK496 @ sK493 )
= $true ),
inference(cnf_transformation,[status(thm)],[f1654]) ).
thf(f1654,plain,
( ( binintersectEL = $true )
& ( foundationAx = $true )
& ( setminusERneg = $true )
& ( dpsetconstrEL2 = $true )
& ( upairsetIR = $true )
& ( setminusEL = $true )
& ( quantDeMorgan4 = $true )
& ( setukpairinjL2 = $true )
& ( setminusER = $true )
& ( kfstsingleton = $true )
& ( subsetI1 = $true )
& ( exuI3 = $true )
& ( setunion__Cong = $true )
& ( cartprodmempair1 = $true )
& ( setminusSubset1 = $true )
& ( subsetE2 = $true )
& ( infuncsetfunc = $true )
& ( $true
= ( in @ sK495 @ sK494 ) )
& ( sK496
!= ( ap @ sK494 @ sK493 @ sK492 @ sK495 ) )
& ( $true
= ( in @ ( kpair @ sK495 @ sK496 ) @ sK492 ) )
& ( ( in @ sK496 @ sK493 )
= $true )
& ( ( in @ sK492 @ ( funcSet @ sK494 @ sK493 ) )
= $true )
& ( exu__Cong = $true )
& ( upairsetIL = $true )
& ( binunionRsub = $true )
& ( binunionEcases = $true )
& ( singletonsubset = $true )
& ( binunionIR = $true )
& ( emptysetsubset = $true )
& ( setadjoinSub2 = $true )
& ( setminusLsub = $true )
& ( kpairp = $true )
& ( singletonsuniq = $true )
& ( ex1E1 = $true )
& ( setbeta = $true )
& ( omegaSAx = $true )
& ( kpairiskpair = $true )
& ( quantDeMorgan2 = $true )
& ( omegaIndAx = $true )
& ( exuEu = $true )
& ( binintersectI = $true )
& ( cartprodsndin = $true )
& ( inCongP = $true )
& ( uniqinunit = $true )
& ( singletonsswitch = $true )
& ( upairsetE = $true )
& ( binunionLsub = $true )
& ( upairset2IR = $true )
& ( ksndpairEq = $true )
& ( setadjoinAx = $true )
& ( setukpairinjR = $true )
& ( nonemptyI = $true )
& ( cartprodsndpairEq = $true )
& ( emptysetimpfalse = $true )
& ( ex1E2 = $true )
& ( cartprodpairmemEL = $true )
& ( lam2p = $true )
& ( setoftrueEq = $true )
& ( exuE1 = $true )
& ( notinemptyset = $true )
& ( subsetemptysetimpeq = $true )
& ( ksndsingleton = $true )
& ( sepInPowerset = $true )
& ( setukpairIL = $true )
& ( binintersectSubset4 = $true )
& ( brelnall1 = $true )
& ( setunionsingleton = $true )
& ( setextAx = $true )
& ( cartprodfstpairEq = $true )
& ( prop2setE = $true )
& ( powersetI1 = $true )
& ( vacuousDall = $true )
& ( setadjoinIR = $true )
& ( cartprodpairin = $true )
& ( ubforcartprodlem1 = $true )
& ( binintersectSubset5 = $true )
& ( dsetconstrI = $true )
& ( binunionIL = $true )
& ( exuE3u = $true )
& ( dpsetconstrSub = $true )
& ( binintersectSubset3 = $true )
& ( dpsetconstrEL1 = $true )
& ( subsetI2 = $true )
& ( powersetsubset = $true )
& ( setminusILneg = $true )
& ( setminusI = $true )
& ( notinsingleton = $true )
& ( symdiffI1 = $true )
& ( funcextLem = $true )
& ( binintersectRsub = $true )
& ( prop2setI = $true )
& ( powersetI = $true )
& ( sepSubset = $true )
& ( powerset__Cong = $true )
& ( dsetconstr__Cong = $true )
& ( eqimpsubset2 = $true )
& ( kpairsurjEq = $true )
& ( prop2set2propI = $true )
& ( ex1I2 = $true )
& ( setunionsingleton1 = $true )
& ( powersetE1 = $true )
& ( cartprodmempaircEq = $true )
& ( upairequniteq = $true )
& ( subPowSU = $true )
& ( exuE2 = $true )
& ( setunionsingleton2 = $true )
& ( singletonprop = $true )
& ( emptysetAx = $true )
& ( subsetRefl = $true )
& ( subsetTrans = $true )
& ( cartprodfstin = $true )
& ( emptyinunitempty = $true )
& ( kfstpairEq = $true )
& ( powersetAx = $true )
& ( descrp = $true )
& ( quantDeMorgan1 = $true )
& ( setadjoinIL = $true )
& ( dsetconstrER = $true )
& ( setukpairinjL1 = $true )
& ( setOfPairsIsBReln = $true )
& ( emptyset__Cong = $true )
& ( ap2p = $true )
& ( emptyInPowerset = $true )
& ( dpsetconstrER = $true )
& ( setunionI = $true )
& ( dsetconstrEL = $true )
& ( notdallE = $true )
& ( upairsubunion = $true )
& ( inPowerset = $true )
& ( cartprodpairmemER = $true )
& ( cartprodpairsurjEq = $true )
& ( setext = $true )
& ( nonemptyE1 = $true )
& ( setminusIRneg = $true )
& ( exuI1 = $true )
& ( dpsetconstrI = $true )
& ( disjointsetsI1 = $true )
& ( singletoninpowerset = $true )
& ( exuI2 = $true )
& ( binunionE = $true )
& ( quantDeMorgan3 = $true )
& ( ubforcartprodlem2 = $true )
& ( setukpairinjR11 = $true )
& ( symdiffI2 = $true )
& ( setukpairinjR12 = $true )
& ( symdiffIneg1 = $true )
& ( emptyinPowerset = $true )
& ( setadjoinE = $true )
& ( setadjoinOr = $true )
& ( omega0Ax = $true )
& ( emptyE1 = $true )
& ( lamp = $true )
& ( funcGraphProp3 = $true )
& ( binintersectSubset2 = $true )
& ( wellorderingAx = $true )
& ( ex1I = $true )
& ( symdiffE = $true )
& ( funcGraphProp2 = $true )
& ( nonemptyImpWitness = $true )
& ( setunionE = $true )
& ( app = $true )
& ( setunionE2 = $true )
& ( setukpairinjR1 = $true )
& ( lamProp = $true )
& ( setminusELneg = $true )
& ( brelnall2 = $true )
& ( setadjoin__Cong = $true )
& ( subset2powerset = $true )
& ( notsubsetI = $true )
& ( descr__Cong = $true )
& ( replAx = $true )
& ( omega__Cong = $true )
& ( theprop = $true )
& ( noeltsimpempty = $true )
& ( nonemptyI1 = $true )
& ( setminusSubset2 = $true )
& ( singletoninpowunion = $true )
& ( subsetE = $true )
& ( setukpairIR = $true )
& ( in__Cong = $true )
& ( setadjoinSub = $true )
& ( funcImageSingleton = $true )
& ( exuE3e = $true )
& ( upairinpowunion = $true )
& ( funcGraphProp1 = $true )
& ( binintersectSubset1 = $true )
& ( dpsetconstrERa = $true )
& ( powersetE = $true )
& ( upairset2E = $true )
& ( binintersectER = $true )
& ( emptyI = $true )
& ( setukpairinjR2 = $true )
& ( bs114d = $true )
& ( eqimpsubset1 = $true )
& ( binintersectLsub = $true )
& ( funcinfuncset = $true )
& ( ubforcartprodlem3 = $true )
& ( notequalI1 = $true )
& ( setukpairinjL = $true )
& ( setextsub = $true )
& ( emptysetE = $true )
& ( cartprodmempair = $true )
& ( setunionAx = $true )
& ( notdexE = $true )
& ( notequalI2 = $true )
& ( eqinunit = $true )
& ( secondinupair = $true )
& ( symdiffIneg2 = $true )
& ( apProp = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK492,sK493,sK494,sK495,sK496])],[f733,f1653,f1652,f1651]) ).
thf(f1651,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ? [X3: $i] :
( ( ( in @ X3 @ X2 )
= $true )
& ? [X4: $i] :
( ( ( ap @ X2 @ X1 @ X0 @ X3 )
!= X4 )
& ( ( in @ ( kpair @ X3 @ X4 ) @ X0 )
= $true )
& ( ( in @ X4 @ X1 )
= $true ) ) )
& ( ( in @ X0 @ ( funcSet @ X2 @ X1 ) )
= $true ) )
=> ( ? [X3: $i] :
( ( $true
= ( in @ X3 @ sK494 ) )
& ? [X4: $i] :
( ( ( ap @ sK494 @ sK493 @ sK492 @ X3 )
!= X4 )
& ( ( in @ ( kpair @ X3 @ X4 ) @ sK492 )
= $true )
& ( $true
= ( in @ X4 @ sK493 ) ) ) )
& ( ( in @ sK492 @ ( funcSet @ sK494 @ sK493 ) )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1652,plain,
( ? [X3: $i] :
( ( $true
= ( in @ X3 @ sK494 ) )
& ? [X4: $i] :
( ( ( ap @ sK494 @ sK493 @ sK492 @ X3 )
!= X4 )
& ( ( in @ ( kpair @ X3 @ X4 ) @ sK492 )
= $true )
& ( $true
= ( in @ X4 @ sK493 ) ) ) )
=> ( ( $true
= ( in @ sK495 @ sK494 ) )
& ? [X4: $i] :
( ( ( ap @ sK494 @ sK493 @ sK492 @ sK495 )
!= X4 )
& ( $true
= ( in @ ( kpair @ sK495 @ X4 ) @ sK492 ) )
& ( $true
= ( in @ X4 @ sK493 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1653,plain,
( ? [X4: $i] :
( ( ( ap @ sK494 @ sK493 @ sK492 @ sK495 )
!= X4 )
& ( $true
= ( in @ ( kpair @ sK495 @ X4 ) @ sK492 ) )
& ( $true
= ( in @ X4 @ sK493 ) ) )
=> ( ( sK496
!= ( ap @ sK494 @ sK493 @ sK492 @ sK495 ) )
& ( $true
= ( in @ ( kpair @ sK495 @ sK496 ) @ sK492 ) )
& ( ( in @ sK496 @ sK493 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f733,plain,
( ( binintersectEL = $true )
& ( foundationAx = $true )
& ( setminusERneg = $true )
& ( dpsetconstrEL2 = $true )
& ( upairsetIR = $true )
& ( setminusEL = $true )
& ( quantDeMorgan4 = $true )
& ( setukpairinjL2 = $true )
& ( setminusER = $true )
& ( kfstsingleton = $true )
& ( subsetI1 = $true )
& ( exuI3 = $true )
& ( setunion__Cong = $true )
& ( cartprodmempair1 = $true )
& ( setminusSubset1 = $true )
& ( subsetE2 = $true )
& ( infuncsetfunc = $true )
& ? [X0: $i,X1: $i,X2: $i] :
( ? [X3: $i] :
( ( ( in @ X3 @ X2 )
= $true )
& ? [X4: $i] :
( ( ( ap @ X2 @ X1 @ X0 @ X3 )
!= X4 )
& ( ( in @ ( kpair @ X3 @ X4 ) @ X0 )
= $true )
& ( ( in @ X4 @ X1 )
= $true ) ) )
& ( ( in @ X0 @ ( funcSet @ X2 @ X1 ) )
= $true ) )
& ( exu__Cong = $true )
& ( upairsetIL = $true )
& ( binunionRsub = $true )
& ( binunionEcases = $true )
& ( singletonsubset = $true )
& ( binunionIR = $true )
& ( emptysetsubset = $true )
& ( setadjoinSub2 = $true )
& ( setminusLsub = $true )
& ( kpairp = $true )
& ( singletonsuniq = $true )
& ( ex1E1 = $true )
& ( setbeta = $true )
& ( omegaSAx = $true )
& ( kpairiskpair = $true )
& ( quantDeMorgan2 = $true )
& ( omegaIndAx = $true )
& ( exuEu = $true )
& ( binintersectI = $true )
& ( cartprodsndin = $true )
& ( inCongP = $true )
& ( uniqinunit = $true )
& ( singletonsswitch = $true )
& ( upairsetE = $true )
& ( binunionLsub = $true )
& ( upairset2IR = $true )
& ( ksndpairEq = $true )
& ( setadjoinAx = $true )
& ( setukpairinjR = $true )
& ( nonemptyI = $true )
& ( cartprodsndpairEq = $true )
& ( emptysetimpfalse = $true )
& ( ex1E2 = $true )
& ( cartprodpairmemEL = $true )
& ( lam2p = $true )
& ( setoftrueEq = $true )
& ( exuE1 = $true )
& ( notinemptyset = $true )
& ( subsetemptysetimpeq = $true )
& ( ksndsingleton = $true )
& ( sepInPowerset = $true )
& ( setukpairIL = $true )
& ( binintersectSubset4 = $true )
& ( brelnall1 = $true )
& ( setunionsingleton = $true )
& ( setextAx = $true )
& ( cartprodfstpairEq = $true )
& ( prop2setE = $true )
& ( powersetI1 = $true )
& ( vacuousDall = $true )
& ( setadjoinIR = $true )
& ( cartprodpairin = $true )
& ( ubforcartprodlem1 = $true )
& ( binintersectSubset5 = $true )
& ( dsetconstrI = $true )
& ( binunionIL = $true )
& ( exuE3u = $true )
& ( dpsetconstrSub = $true )
& ( binintersectSubset3 = $true )
& ( dpsetconstrEL1 = $true )
& ( subsetI2 = $true )
& ( powersetsubset = $true )
& ( setminusILneg = $true )
& ( setminusI = $true )
& ( notinsingleton = $true )
& ( symdiffI1 = $true )
& ( funcextLem = $true )
& ( binintersectRsub = $true )
& ( prop2setI = $true )
& ( powersetI = $true )
& ( sepSubset = $true )
& ( powerset__Cong = $true )
& ( dsetconstr__Cong = $true )
& ( eqimpsubset2 = $true )
& ( kpairsurjEq = $true )
& ( prop2set2propI = $true )
& ( ex1I2 = $true )
& ( setunionsingleton1 = $true )
& ( powersetE1 = $true )
& ( cartprodmempaircEq = $true )
& ( upairequniteq = $true )
& ( subPowSU = $true )
& ( exuE2 = $true )
& ( setunionsingleton2 = $true )
& ( singletonprop = $true )
& ( emptysetAx = $true )
& ( subsetRefl = $true )
& ( subsetTrans = $true )
& ( cartprodfstin = $true )
& ( emptyinunitempty = $true )
& ( kfstpairEq = $true )
& ( powersetAx = $true )
& ( descrp = $true )
& ( quantDeMorgan1 = $true )
& ( setadjoinIL = $true )
& ( dsetconstrER = $true )
& ( setukpairinjL1 = $true )
& ( setOfPairsIsBReln = $true )
& ( emptyset__Cong = $true )
& ( ap2p = $true )
& ( emptyInPowerset = $true )
& ( dpsetconstrER = $true )
& ( setunionI = $true )
& ( dsetconstrEL = $true )
& ( notdallE = $true )
& ( upairsubunion = $true )
& ( inPowerset = $true )
& ( cartprodpairmemER = $true )
& ( cartprodpairsurjEq = $true )
& ( setext = $true )
& ( nonemptyE1 = $true )
& ( setminusIRneg = $true )
& ( exuI1 = $true )
& ( dpsetconstrI = $true )
& ( disjointsetsI1 = $true )
& ( singletoninpowerset = $true )
& ( exuI2 = $true )
& ( binunionE = $true )
& ( quantDeMorgan3 = $true )
& ( ubforcartprodlem2 = $true )
& ( setukpairinjR11 = $true )
& ( symdiffI2 = $true )
& ( setukpairinjR12 = $true )
& ( symdiffIneg1 = $true )
& ( emptyinPowerset = $true )
& ( setadjoinE = $true )
& ( setadjoinOr = $true )
& ( omega0Ax = $true )
& ( emptyE1 = $true )
& ( lamp = $true )
& ( funcGraphProp3 = $true )
& ( binintersectSubset2 = $true )
& ( wellorderingAx = $true )
& ( ex1I = $true )
& ( symdiffE = $true )
& ( funcGraphProp2 = $true )
& ( nonemptyImpWitness = $true )
& ( setunionE = $true )
& ( app = $true )
& ( setunionE2 = $true )
& ( setukpairinjR1 = $true )
& ( lamProp = $true )
& ( setminusELneg = $true )
& ( brelnall2 = $true )
& ( setadjoin__Cong = $true )
& ( subset2powerset = $true )
& ( notsubsetI = $true )
& ( descr__Cong = $true )
& ( replAx = $true )
& ( omega__Cong = $true )
& ( theprop = $true )
& ( noeltsimpempty = $true )
& ( nonemptyI1 = $true )
& ( setminusSubset2 = $true )
& ( singletoninpowunion = $true )
& ( subsetE = $true )
& ( setukpairIR = $true )
& ( in__Cong = $true )
& ( setadjoinSub = $true )
& ( funcImageSingleton = $true )
& ( exuE3e = $true )
& ( upairinpowunion = $true )
& ( funcGraphProp1 = $true )
& ( binintersectSubset1 = $true )
& ( dpsetconstrERa = $true )
& ( powersetE = $true )
& ( upairset2E = $true )
& ( binintersectER = $true )
& ( emptyI = $true )
& ( setukpairinjR2 = $true )
& ( bs114d = $true )
& ( eqimpsubset1 = $true )
& ( binintersectLsub = $true )
& ( funcinfuncset = $true )
& ( ubforcartprodlem3 = $true )
& ( notequalI1 = $true )
& ( setukpairinjL = $true )
& ( setextsub = $true )
& ( emptysetE = $true )
& ( cartprodmempair = $true )
& ( setunionAx = $true )
& ( notdexE = $true )
& ( notequalI2 = $true )
& ( eqinunit = $true )
& ( secondinupair = $true )
& ( symdiffIneg2 = $true )
& ( apProp = $true ) ),
inference(flattening,[status(thm)],[f732]) ).
thf(f732,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ? [X3: $i] :
( ? [X4: $i] :
( ( ( ap @ X2 @ X1 @ X0 @ X3 )
!= X4 )
& ( ( in @ ( kpair @ X3 @ X4 ) @ X0 )
= $true )
& ( ( in @ X4 @ X1 )
= $true ) )
& ( ( in @ X3 @ X2 )
= $true ) )
& ( ( in @ X0 @ ( funcSet @ X2 @ X1 ) )
= $true ) )
& ( funcextLem = $true )
& ( funcGraphProp2 = $true )
& ( funcGraphProp3 = $true )
& ( funcGraphProp1 = $true )
& ( ex1E2 = $true )
& ( brelnall2 = $true )
& ( brelnall1 = $true )
& ( lam2p = $true )
& ( lamp = $true )
& ( lamProp = $true )
& ( funcinfuncset = $true )
& ( ap2p = $true )
& ( infuncsetfunc = $true )
& ( app = $true )
& ( apProp = $true )
& ( funcImageSingleton = $true )
& ( dpsetconstrER = $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)],[f568]) ).
thf(f568,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 )
=> ( ( dpsetconstrER = $true )
=> ( ( funcImageSingleton = $true )
=> ( ( apProp = $true )
=> ( ( app = $true )
=> ( ( infuncsetfunc = $true )
=> ( ( ap2p = $true )
=> ( ( funcinfuncset = $true )
=> ( ( lamProp = $true )
=> ( ( lamp = $true )
=> ( ( lam2p = $true )
=> ( ( brelnall1 = $true )
=> ( ( brelnall2 = $true )
=> ( ( ex1E2 = $true )
=> ( ( funcGraphProp1 = $true )
=> ( ( funcGraphProp3 = $true )
=> ( ( funcGraphProp2 = $true )
=> ( ( funcextLem = $true )
=> ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X0 @ ( funcSet @ X2 @ X1 ) )
= $true )
=> ! [X3: $i] :
( ( ( in @ X3 @ X2 )
= $true )
=> ! [X4: $i] :
( ( ( in @ X4 @ X1 )
= $true )
=> ( ( ( in @ ( kpair @ X3 @ X4 ) @ X0 )
= $true )
=> ( ( ap @ X2 @ X1 @ X0 @ X3 )
= X4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f567]) ).
thf(f567,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
=> ( dpsetconstrER
=> ( funcImageSingleton
=> ( apProp
=> ( app
=> ( infuncsetfunc
=> ( ap2p
=> ( funcinfuncset
=> ( lamProp
=> ( lamp
=> ( lam2p
=> ( brelnall1
=> ( brelnall2
=> ( ex1E2
=> ( funcGraphProp1
=> ( funcGraphProp3
=> ( funcGraphProp2
=> ( funcextLem
=> ! [X0: $i,X1: $i,X2: $i] :
( ( in @ X0 @ ( funcSet @ X2 @ X1 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( in @ ( kpair @ X3 @ X4 ) @ X0 )
=> ( ( ap @ X2 @ X1 @ X0 @ X3 )
= X4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f213]) ).
thf(f213,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
=> ( dpsetconstrER
=> ( funcImageSingleton
=> ( apProp
=> ( app
=> ( infuncsetfunc
=> ( ap2p
=> ( funcinfuncset
=> ( lamProp
=> ( lamp
=> ( lam2p
=> ( brelnall1
=> ( brelnall2
=> ( ex1E2
=> ( funcGraphProp1
=> ( funcGraphProp3
=> ( funcGraphProp2
=> ( funcextLem
=> ! [X13: $i,X4: $i,X3: $i] :
( ( in @ X13 @ ( funcSet @ X3 @ X4 ) )
=> ! [X1: $i] :
( ( in @ X1 @ X3 )
=> ! [X2: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ ( kpair @ X1 @ X2 ) @ X13 )
=> ( ( ap @ X3 @ X4 @ X13 @ X1 )
= X2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f212]) ).
thf(f212,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
=> ( dpsetconstrER
=> ( funcImageSingleton
=> ( apProp
=> ( app
=> ( infuncsetfunc
=> ( ap2p
=> ( funcinfuncset
=> ( lamProp
=> ( lamp
=> ( lam2p
=> ( brelnall1
=> ( brelnall2
=> ( ex1E2
=> ( funcGraphProp1
=> ( funcGraphProp3
=> ( funcGraphProp2
=> ( funcextLem
=> ! [X13: $i,X4: $i,X3: $i] :
( ( in @ X13 @ ( funcSet @ X3 @ X4 ) )
=> ! [X1: $i] :
( ( in @ X1 @ X3 )
=> ! [X2: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ ( kpair @ X1 @ X2 ) @ X13 )
=> ( ( ap @ X3 @ X4 @ X13 @ X1 )
= X2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.CZb2EvKrRp/DTF2THF_1488.p',funcGraphProp4) ).
thf(f3051,plain,
( ( ( in @ sK496 @ sK493 )
!= $true )
| ~ spl618_1 ),
inference(trivial_inequality_removal,[status(thm)],[f3041]) ).
thf(f3041,plain,
( ( ( in @ sK496 @ sK493 )
!= $true )
| ( $true != $true )
| ( sK496 != sK496 )
| ~ spl618_1 ),
inference(superposition,[status(thm)],[f3034,f2669]) ).
thf(f2669,plain,
( $true
= ( in @ ( kpair @ sK495 @ sK496 ) @ sK492 ) ),
inference(cnf_transformation,[status(thm)],[f1654]) ).
thf(f3034,plain,
( ! [X0: $i] :
( ( ( in @ ( kpair @ sK495 @ X0 ) @ sK492 )
!= $true )
| ( $true
!= ( in @ X0 @ sK493 ) )
| ( sK496 != X0 ) )
| ~ spl618_1 ),
inference(avatar_component_clause,[status(thm)],[f3033]) ).
thf(f3033,definition,
( spl618_1
<=> ! [X0: $i] :
( ( $true
!= ( in @ X0 @ sK493 ) )
| ( sK496 != X0 )
| ( ( in @ ( kpair @ sK495 @ X0 ) @ sK492 )
!= $true ) ) ),
introduced(definition,[new_symbols(naming,[spl618_1])],[avatar_definition]) ).
thf(f3040,plain,
spl618_2,
inference(avatar_split_clause,[status(thm)],[f3020,f3036]) ).
thf(f3036,definition,
( spl618_2
<=> ( ( func @ sK494 @ sK493 @ sK492 )
= $true ) ),
introduced(definition,[new_symbols(naming,[spl618_2])],[avatar_definition]) ).
thf(f3020,plain,
( ( func @ sK494 @ sK493 @ sK492 )
= $true ),
inference(trivial_inequality_removal,[status(thm)],[f3019]) ).
thf(f3019,plain,
( ( ( func @ sK494 @ sK493 @ sK492 )
= $true )
| ( $true != $true ) ),
inference(forward_demodulation,[status(thm)],[f3018,f2672]) ).
thf(f2672,plain,
infuncsetfunc = $true,
inference(cnf_transformation,[status(thm)],[f1654]) ).
thf(f3018,plain,
( ( ( func @ sK494 @ sK493 @ sK492 )
= $true )
| ( infuncsetfunc != $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f3015]) ).
thf(f3015,plain,
( ( ( func @ sK494 @ sK493 @ sK492 )
= $true )
| ( infuncsetfunc != $true )
| ( $true != $true ) ),
inference(superposition,[status(thm)],[f2084,f2667]) ).
thf(f2667,plain,
( ( in @ sK492 @ ( funcSet @ sK494 @ sK493 ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f1654]) ).
thf(f2084,plain,
! [X3: $i,X4: $i,X5: $i] :
( ( ( in @ X5 @ ( funcSet @ X4 @ X3 ) )
!= $true )
| ( infuncsetfunc != $true )
| ( $true
= ( func @ X4 @ X3 @ X5 ) ) ),
inference(cnf_transformation,[status(thm)],[f1182]) ).
thf(f1182,plain,
( ( ( infuncsetfunc = $true )
| ( ( $true
!= ( func @ sK185 @ sK184 @ sK186 ) )
& ( $true
= ( in @ sK186 @ ( funcSet @ sK185 @ sK184 ) ) ) ) )
& ( ! [X3: $i,X4: $i,X5: $i] :
( ( $true
= ( func @ X4 @ X3 @ X5 ) )
| ( ( in @ X5 @ ( funcSet @ X4 @ X3 ) )
!= $true ) )
| ( infuncsetfunc != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK184,sK185,sK186])],[f1180,f1181]) ).
thf(f1181,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ( ( func @ X1 @ X0 @ X2 )
!= $true )
& ( ( in @ X2 @ ( funcSet @ X1 @ X0 ) )
= $true ) )
=> ( ( $true
!= ( func @ sK185 @ sK184 @ sK186 ) )
& ( $true
= ( in @ sK186 @ ( funcSet @ sK185 @ sK184 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1180,plain,
( ( ( infuncsetfunc = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( ( func @ X1 @ X0 @ X2 )
!= $true )
& ( ( in @ X2 @ ( funcSet @ X1 @ X0 ) )
= $true ) ) )
& ( ! [X3: $i,X4: $i,X5: $i] :
( ( $true
= ( func @ X4 @ X3 @ X5 ) )
| ( ( in @ X5 @ ( funcSet @ X4 @ X3 ) )
!= $true ) )
| ( infuncsetfunc != $true ) ) ),
inference(rectify,[status(thm)],[f1179]) ).
thf(f1179,plain,
( ( ( infuncsetfunc = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( ( func @ X1 @ X0 @ X2 )
!= $true )
& ( ( in @ X2 @ ( funcSet @ X1 @ X0 ) )
= $true ) ) )
& ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( func @ X1 @ X0 @ X2 )
= $true )
| ( ( in @ X2 @ ( funcSet @ X1 @ X0 ) )
!= $true ) )
| ( infuncsetfunc != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f753]) ).
thf(f753,plain,
( ( infuncsetfunc = $true )
<=> ! [X0: $i,X1: $i,X2: $i] :
( ( ( func @ X1 @ X0 @ X2 )
= $true )
| ( ( in @ X2 @ ( funcSet @ X1 @ X0 ) )
!= $true ) ) ),
inference(ennf_transformation,[status(thm)],[f254]) ).
thf(f254,plain,
( ( infuncsetfunc = $true )
<=> ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X2 @ ( funcSet @ X1 @ X0 ) )
= $true )
=> ( ( func @ X1 @ X0 @ X2 )
= $true ) ) ),
inference(fool_elimination,[status(thm)],[f253]) ).
thf(f253,plain,
( infuncsetfunc
= ( ! [X0: $i,X1: $i,X2: $i] :
( ( in @ X2 @ ( funcSet @ X1 @ X0 ) )
=> ( func @ X1 @ X0 @ X2 ) ) ) ),
inference(rectify,[status(thm)],[f198]) ).
thf(f198,axiom,
( infuncsetfunc
= ( ! [X4: $i,X3: $i,X13: $i] :
( ( in @ X13 @ ( funcSet @ X3 @ X4 ) )
=> ( func @ X3 @ X4 @ X13 ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.CZb2EvKrRp/DTF2THF_1488.p',infuncsetfunc) ).
thf(f3039,plain,
( spl618_1
| ~ spl618_2 ),
inference(avatar_split_clause,[status(thm)],[f3031,f3036,f3033]) ).
thf(f3031,plain,
! [X0: $i] :
( ( $true
!= ( in @ X0 @ sK493 ) )
| ( ( func @ sK494 @ sK493 @ sK492 )
!= $true )
| ( ( in @ ( kpair @ sK495 @ X0 ) @ sK492 )
!= $true )
| ( sK496 != X0 ) ),
inference(trivial_inequality_removal,[status(thm)],[f3030]) ).
thf(f3030,plain,
! [X0: $i] :
( ( ( in @ ( kpair @ sK495 @ X0 ) @ sK492 )
!= $true )
| ( sK496 != X0 )
| ( $true != $true )
| ( ( func @ sK494 @ sK493 @ sK492 )
!= $true )
| ( $true
!= ( in @ X0 @ sK493 ) ) ),
inference(forward_demodulation,[status(thm)],[f3029,f2531]) ).
thf(f2531,plain,
funcGraphProp2 = $true,
inference(cnf_transformation,[status(thm)],[f1654]) ).
thf(f3029,plain,
! [X0: $i] :
( ( $true
!= ( in @ X0 @ sK493 ) )
| ( sK496 != X0 )
| ( funcGraphProp2 != $true )
| ( ( in @ ( kpair @ sK495 @ X0 ) @ sK492 )
!= $true )
| ( ( func @ sK494 @ sK493 @ sK492 )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f3028]) ).
thf(f3028,plain,
! [X0: $i] :
( ( ( in @ ( kpair @ sK495 @ X0 ) @ sK492 )
!= $true )
| ( funcGraphProp2 != $true )
| ( $true != $true )
| ( sK496 != X0 )
| ( ( func @ sK494 @ sK493 @ sK492 )
!= $true )
| ( $true
!= ( in @ X0 @ sK493 ) ) ),
inference(forward_demodulation,[status(thm)],[f3027,f2671]) ).
thf(f2671,plain,
( $true
= ( in @ sK495 @ sK494 ) ),
inference(cnf_transformation,[status(thm)],[f1654]) ).
thf(f3027,plain,
! [X0: $i] :
( ( ( func @ sK494 @ sK493 @ sK492 )
!= $true )
| ( $true
!= ( in @ X0 @ sK493 ) )
| ( ( in @ ( kpair @ sK495 @ X0 ) @ sK492 )
!= $true )
| ( $true
!= ( in @ sK495 @ sK494 ) )
| ( funcGraphProp2 != $true )
| ( sK496 != X0 ) ),
inference(superposition,[status(thm)],[f2670,f2299]) ).
thf(f2299,plain,
! [X8: $i,X6: $i,X9: $i,X7: $i,X5: $i] :
( ( ( ap @ X5 @ X7 @ X6 @ X8 )
= X9 )
| ( $true
!= ( in @ X9 @ X7 ) )
| ( funcGraphProp2 != $true )
| ( ( func @ X5 @ X7 @ X6 )
!= $true )
| ( $true
!= ( in @ ( kpair @ X8 @ X9 ) @ X6 ) )
| ( $true
!= ( in @ X8 @ X5 ) ) ),
inference(cnf_transformation,[status(thm)],[f1435]) ).
thf(f1435,plain,
( ( ( funcGraphProp2 = $true )
| ( ( ( func @ sK344 @ sK346 @ sK345 )
= $true )
& ( ( in @ ( kpair @ sK347 @ sK348 ) @ sK345 )
= $true )
& ( $true
= ( in @ sK348 @ sK346 ) )
& ( ( ap @ sK344 @ sK346 @ sK345 @ sK347 )
!= sK348 )
& ( $true
= ( in @ sK347 @ sK344 ) ) ) )
& ( ! [X5: $i,X6: $i,X7: $i] :
( ( ( func @ X5 @ X7 @ X6 )
!= $true )
| ! [X8: $i] :
( ! [X9: $i] :
( ( $true
!= ( in @ ( kpair @ X8 @ X9 ) @ X6 ) )
| ( $true
!= ( in @ X9 @ X7 ) )
| ( ( ap @ X5 @ X7 @ X6 @ X8 )
= X9 ) )
| ( $true
!= ( in @ X8 @ X5 ) ) ) )
| ( funcGraphProp2 != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK344,sK345,sK346,sK347,sK348])],[f1431,f1434,f1433,f1432]) ).
thf(f1432,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( func @ X0 @ X2 @ X1 ) )
& ? [X3: $i] :
( ? [X4: $i] :
( ( $true
= ( in @ ( kpair @ X3 @ X4 ) @ X1 ) )
& ( $true
= ( in @ X4 @ X2 ) )
& ( ( ap @ X0 @ X2 @ X1 @ X3 )
!= X4 ) )
& ( ( in @ X3 @ X0 )
= $true ) ) )
=> ( ( ( func @ sK344 @ sK346 @ sK345 )
= $true )
& ? [X3: $i] :
( ? [X4: $i] :
( ( ( in @ ( kpair @ X3 @ X4 ) @ sK345 )
= $true )
& ( $true
= ( in @ X4 @ sK346 ) )
& ( ( ap @ sK344 @ sK346 @ sK345 @ X3 )
!= X4 ) )
& ( ( in @ X3 @ sK344 )
= $true ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1433,plain,
( ? [X3: $i] :
( ? [X4: $i] :
( ( ( in @ ( kpair @ X3 @ X4 ) @ sK345 )
= $true )
& ( $true
= ( in @ X4 @ sK346 ) )
& ( ( ap @ sK344 @ sK346 @ sK345 @ X3 )
!= X4 ) )
& ( ( in @ X3 @ sK344 )
= $true ) )
=> ( ? [X4: $i] :
( ( ( in @ ( kpair @ sK347 @ X4 ) @ sK345 )
= $true )
& ( $true
= ( in @ X4 @ sK346 ) )
& ( ( ap @ sK344 @ sK346 @ sK345 @ sK347 )
!= X4 ) )
& ( $true
= ( in @ sK347 @ sK344 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1434,plain,
( ? [X4: $i] :
( ( ( in @ ( kpair @ sK347 @ X4 ) @ sK345 )
= $true )
& ( $true
= ( in @ X4 @ sK346 ) )
& ( ( ap @ sK344 @ sK346 @ sK345 @ sK347 )
!= X4 ) )
=> ( ( ( in @ ( kpair @ sK347 @ sK348 ) @ sK345 )
= $true )
& ( $true
= ( in @ sK348 @ sK346 ) )
& ( ( ap @ sK344 @ sK346 @ sK345 @ sK347 )
!= sK348 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1431,plain,
( ( ( funcGraphProp2 = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( func @ X0 @ X2 @ X1 ) )
& ? [X3: $i] :
( ? [X4: $i] :
( ( $true
= ( in @ ( kpair @ X3 @ X4 ) @ X1 ) )
& ( $true
= ( in @ X4 @ X2 ) )
& ( ( ap @ X0 @ X2 @ X1 @ X3 )
!= X4 ) )
& ( ( in @ X3 @ X0 )
= $true ) ) ) )
& ( ! [X5: $i,X6: $i,X7: $i] :
( ( ( func @ X5 @ X7 @ X6 )
!= $true )
| ! [X8: $i] :
( ! [X9: $i] :
( ( $true
!= ( in @ ( kpair @ X8 @ X9 ) @ X6 ) )
| ( $true
!= ( in @ X9 @ X7 ) )
| ( ( ap @ X5 @ X7 @ X6 @ X8 )
= X9 ) )
| ( $true
!= ( in @ X8 @ X5 ) ) ) )
| ( funcGraphProp2 != $true ) ) ),
inference(rectify,[status(thm)],[f1430]) ).
thf(f1430,plain,
( ( ( funcGraphProp2 = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( func @ X0 @ X2 @ X1 ) )
& ? [X3: $i] :
( ? [X4: $i] :
( ( $true
= ( in @ ( kpair @ X3 @ X4 ) @ X1 ) )
& ( $true
= ( in @ X4 @ X2 ) )
& ( ( ap @ X0 @ X2 @ X1 @ X3 )
!= X4 ) )
& ( ( in @ X3 @ X0 )
= $true ) ) ) )
& ( ! [X0: $i,X1: $i,X2: $i] :
( ( $true
!= ( func @ X0 @ X2 @ X1 ) )
| ! [X3: $i] :
( ! [X4: $i] :
( ( $true
!= ( in @ ( kpair @ X3 @ X4 ) @ X1 ) )
| ( $true
!= ( in @ X4 @ X2 ) )
| ( ( ap @ X0 @ X2 @ X1 @ X3 )
= X4 ) )
| ( ( in @ X3 @ X0 )
!= $true ) ) )
| ( funcGraphProp2 != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f748]) ).
thf(f748,plain,
( ( funcGraphProp2 = $true )
<=> ! [X0: $i,X1: $i,X2: $i] :
( ( $true
!= ( func @ X0 @ X2 @ X1 ) )
| ! [X3: $i] :
( ! [X4: $i] :
( ( $true
!= ( in @ ( kpair @ X3 @ X4 ) @ X1 ) )
| ( $true
!= ( in @ X4 @ X2 ) )
| ( ( ap @ X0 @ X2 @ X1 @ X3 )
= X4 ) )
| ( ( in @ X3 @ X0 )
!= $true ) ) ) ),
inference(flattening,[status(thm)],[f747]) ).
thf(f747,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ! [X3: $i] :
( ! [X4: $i] :
( ( ( ap @ X0 @ X2 @ X1 @ X3 )
= X4 )
| ( $true
!= ( in @ ( kpair @ X3 @ X4 ) @ X1 ) )
| ( $true
!= ( in @ X4 @ X2 ) ) )
| ( ( in @ X3 @ X0 )
!= $true ) )
| ( $true
!= ( func @ X0 @ X2 @ X1 ) ) )
<=> ( funcGraphProp2 = $true ) ),
inference(ennf_transformation,[status(thm)],[f242]) ).
thf(f242,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( func @ X0 @ X2 @ X1 ) )
=> ! [X3: $i] :
( ( ( in @ X3 @ X0 )
= $true )
=> ! [X4: $i] :
( ( $true
= ( in @ X4 @ X2 ) )
=> ( ( $true
= ( in @ ( kpair @ X3 @ X4 ) @ X1 ) )
=> ( ( ap @ X0 @ X2 @ X1 @ X3 )
= X4 ) ) ) ) )
<=> ( funcGraphProp2 = $true ) ),
inference(fool_elimination,[status(thm)],[f241]) ).
thf(f241,plain,
( funcGraphProp2
= ( ! [X0: $i,X1: $i,X2: $i] :
( ( func @ X0 @ X2 @ X1 )
=> ! [X3: $i] :
( ( in @ X3 @ X0 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( in @ ( kpair @ X3 @ X4 ) @ X1 )
=> ( ( ap @ X0 @ X2 @ X1 @ X3 )
= X4 ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f210]) ).
thf(f210,axiom,
( funcGraphProp2
= ( ! [X3: $i,X13: $i,X4: $i] :
( ( func @ X3 @ X4 @ X13 )
=> ! [X1: $i] :
( ( in @ X1 @ X3 )
=> ! [X2: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ ( kpair @ X1 @ X2 ) @ X13 )
=> ( ( ap @ X3 @ X4 @ X13 @ X1 )
= X2 ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.CZb2EvKrRp/DTF2THF_1488.p',funcGraphProp2) ).
thf(f2670,plain,
( sK496
!= ( ap @ sK494 @ sK493 @ sK492 @ sK495 ) ),
inference(cnf_transformation,[status(thm)],[f1654]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SEU688^1 : TPTP v9.2.1. Released v3.7.0.
% 0.12/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.16/0.34 % Computer : n014.cluster.edu
% 0.16/0.34 % Model : x86_64 x86_64
% 0.16/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.34 % Memory : 8042.1875MB
% 0.16/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.34 % CPULimit : 300
% 0.16/0.34 % WCLimit : 300
% 0.16/0.34 % DateTime : Tue Feb 24 12:31:12 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
% 0.27/0.46 ---- Original DTF file ---
% 0.27/0.46 thf(spec,logic,$$dhol).
% 0.27/0.46 %------------------------------------------------------------------------------
% 0.27/0.46 % File : SEU688^1 : TPTP v9.2.1. Released v3.7.0.
% 0.27/0.46 % Domain : Set Theory
% 0.27/0.46 % Problem : Functions - Extensionality and Beta Reduction
% 0.27/0.46 % Version : Especial.
% 0.27/0.46 % English : (! A:i.! B:i.! f:i.in f (funcSet A B) -> (! x:i.in x A ->
% 0.27/0.46 % (! y:i.in y B -> in (kpair x y) f -> ap A B f x = y)))
% 0.27/0.46
% 0.27/0.46 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.27/0.46 % Source : [Bro08]
% 0.27/0.46 % Names : ZFC190g [Bro08]
% 0.27/0.46
% 0.27/0.46 % Status : Theorem
% 0.27/0.46 % Rating : 0.33 v9.1.0, 0.38 v9.0.0, 0.50 v8.2.0, 0.77 v8.1.0, 0.64 v7.5.0, 0.57 v7.4.0, 0.44 v7.2.0, 0.38 v7.0.0, 0.29 v6.4.0, 0.33 v6.3.0, 0.40 v6.2.0, 0.57 v6.1.0, 0.71 v5.5.0, 0.83 v5.4.0, 0.80 v5.2.0, 1.00 v3.7.0
% 0.27/0.46 % Syntax : Number of formulae : 453 ( 211 unt; 241 typ; 211 def)
% 0.27/0.46 % Number of atoms : 1141 ( 310 equ; 0 cnn)
% 0.27/0.46 % Maximal formula atoms : 209 ( 5 avg)
% 0.27/0.46 % Number of connectives : 2133 ( 47 ~; 7 |; 36 &;1510 @)
% 0.27/0.46 % ( 14 <=>; 519 =>; 0 <=; 0 <~>)
% 0.27/0.46 % Maximal formula depth : 218 ( 2 avg)
% 0.27/0.46 % Number of types : 2 ( 0 usr)
% 0.27/0.46 % Number of type conns : 126 ( 126 >; 0 *; 0 +; 0 <<)
% 0.27/0.46 % Number of symbols : 244 ( 241 usr; 208 con; 0-4 aty)
% 0.27/0.46 % Number of variables : 683 ( 73 ^; 574 !; 36 ?; 683 :)
% 0.27/0.46 % SPC : TH0_THM_EQU_NAR
% 0.27/0.46
% 0.27/0.46 % Comments : http://mathgate.info/detsetitem.php?id=413
% 0.27/0.46 %------------------------------------------------------------------------------
% 0.27/0.46 thf(in_type,type,
% 0.27/0.46 in: $i > $i > $o ).
% 0.27/0.46
% 0.27/0.46 thf(exu_type,type,
% 0.27/0.46 exu: ( $i > $o ) > $o ).
% 0.27/0.46
% 0.27/0.46 thf(exu,definition,
% 0.27/0.46 ( exu
% 0.27/0.46 = ( ^ [Xphi: $i > $o] :
% 0.27/0.46 ? [Xx: $i] :
% 0.27/0.46 ( ( Xphi @ Xx )
% 0.27/0.46 & ! [Xy: $i] :
% 0.27/0.46 ( ( Xphi @ Xy )
% 0.27/0.46 => ( Xx = Xy ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(setextAx_type,type,
% 0.27/0.46 setextAx: $o ).
% 0.27/0.46
% 0.27/0.46 thf(setextAx,definition,
% 0.27/0.46 ( setextAx
% 0.27/0.46 = ( ! [A: $i,B: $i] :
% 0.27/0.46 ( ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 <=> ( in @ Xx @ B ) )
% 0.27/0.46 => ( A = B ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(emptyset_type,type,
% 0.27/0.46 emptyset: $i ).
% 0.27/0.46
% 0.27/0.46 thf(emptysetAx_type,type,
% 0.27/0.46 emptysetAx: $o ).
% 0.27/0.46
% 0.27/0.46 thf(emptysetAx,definition,
% 0.27/0.46 ( emptysetAx
% 0.27/0.46 = ( ! [Xx: $i] :
% 0.27/0.46 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(setadjoin_type,type,
% 0.27/0.46 setadjoin: $i > $i > $i ).
% 0.27/0.46
% 0.27/0.46 thf(setadjoinAx_type,type,
% 0.27/0.46 setadjoinAx: $o ).
% 0.27/0.46
% 0.27/0.46 thf(setadjoinAx,definition,
% 0.27/0.46 ( setadjoinAx
% 0.27/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.27/0.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.27/0.46 <=> ( ( Xy = Xx )
% 0.27/0.46 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(powerset_type,type,
% 0.27/0.46 powerset: $i > $i ).
% 0.27/0.46
% 0.27/0.46 thf(powersetAx_type,type,
% 0.27/0.46 powersetAx: $o ).
% 0.27/0.46
% 0.27/0.46 thf(powersetAx,definition,
% 0.27/0.46 ( powersetAx
% 0.27/0.46 = ( ! [A: $i,B: $i] :
% 0.27/0.46 ( ( in @ B @ ( powerset @ A ) )
% 0.27/0.46 <=> ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ B )
% 0.27/0.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(setunion_type,type,
% 0.27/0.46 setunion: $i > $i ).
% 0.27/0.46
% 0.27/0.46 thf(setunionAx_type,type,
% 0.27/0.46 setunionAx: $o ).
% 0.27/0.46
% 0.27/0.46 thf(setunionAx,definition,
% 0.27/0.46 ( setunionAx
% 0.27/0.46 = ( ! [A: $i,Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ ( setunion @ A ) )
% 0.27/0.46 <=> ? [B: $i] :
% 0.27/0.46 ( ( in @ Xx @ B )
% 0.27/0.46 & ( in @ B @ A ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(omega_type,type,
% 0.27/0.46 omega: $i ).
% 0.27/0.46
% 0.27/0.46 thf(omega0Ax_type,type,
% 0.27/0.46 omega0Ax: $o ).
% 0.27/0.46
% 0.27/0.46 thf(omega0Ax,definition,
% 0.27/0.46 ( omega0Ax
% 0.27/0.46 = ( in @ emptyset @ omega ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(omegaSAx_type,type,
% 0.27/0.46 omegaSAx: $o ).
% 0.27/0.46
% 0.27/0.46 thf(omegaSAx,definition,
% 0.27/0.46 ( omegaSAx
% 0.27/0.46 = ( ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ omega )
% 0.27/0.46 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(omegaIndAx_type,type,
% 0.27/0.46 omegaIndAx: $o ).
% 0.27/0.46
% 0.27/0.46 thf(omegaIndAx,definition,
% 0.27/0.46 ( omegaIndAx
% 0.27/0.46 = ( ! [A: $i] :
% 0.27/0.46 ( ( ( in @ emptyset @ A )
% 0.27/0.46 & ! [Xx: $i] :
% 0.27/0.46 ( ( ( in @ Xx @ omega )
% 0.27/0.46 & ( in @ Xx @ A ) )
% 0.27/0.46 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.27/0.46 => ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ omega )
% 0.27/0.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(replAx_type,type,
% 0.27/0.46 replAx: $o ).
% 0.27/0.46
% 0.27/0.46 thf(replAx,definition,
% 0.27/0.46 ( replAx
% 0.27/0.46 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.27/0.46 ( ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( exu
% 0.27/0.46 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.27/0.46 => ? [B: $i] :
% 0.27/0.46 ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ B )
% 0.27/0.46 <=> ? [Xy: $i] :
% 0.27/0.46 ( ( in @ Xy @ A )
% 0.27/0.46 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(foundationAx_type,type,
% 0.27/0.46 foundationAx: $o ).
% 0.27/0.46
% 0.27/0.46 thf(foundationAx,definition,
% 0.27/0.46 ( foundationAx
% 0.27/0.46 = ( ! [A: $i] :
% 0.27/0.46 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.27/0.46 => ? [B: $i] :
% 0.27/0.46 ( ( in @ B @ A )
% 0.27/0.46 & ~ ? [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ B )
% 0.27/0.46 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(wellorderingAx_type,type,
% 0.27/0.46 wellorderingAx: $o ).
% 0.27/0.46
% 0.27/0.46 thf(wellorderingAx,definition,
% 0.27/0.46 ( wellorderingAx
% 0.27/0.46 = ( ! [A: $i] :
% 0.27/0.46 ? [B: $i] :
% 0.27/0.46 ( ! [C: $i] :
% 0.27/0.46 ( ( in @ C @ B )
% 0.27/0.46 => ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ C )
% 0.27/0.46 => ( in @ Xx @ A ) ) )
% 0.27/0.46 & ! [Xx: $i,Xy: $i] :
% 0.27/0.46 ( ( ( in @ Xx @ A )
% 0.27/0.46 & ( in @ Xy @ A ) )
% 0.27/0.46 => ( ! [C: $i] :
% 0.27/0.46 ( ( in @ C @ B )
% 0.27/0.46 => ( ( in @ Xx @ C )
% 0.27/0.46 <=> ( in @ Xy @ C ) ) )
% 0.27/0.46 => ( Xx = Xy ) ) )
% 0.27/0.46 & ! [C: $i,D: $i] :
% 0.27/0.46 ( ( ( in @ C @ B )
% 0.27/0.46 & ( in @ D @ B ) )
% 0.27/0.46 => ( ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ C )
% 0.27/0.46 => ( in @ Xx @ D ) )
% 0.27/0.46 | ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ D )
% 0.27/0.46 => ( in @ Xx @ C ) ) ) )
% 0.27/0.46 & ! [C: $i] :
% 0.27/0.46 ( ( ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ C )
% 0.27/0.46 => ( in @ Xx @ A ) )
% 0.27/0.46 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.27/0.46 => ? [D: $i,Xx: $i] :
% 0.27/0.46 ( ( in @ D @ B )
% 0.27/0.46 & ( in @ Xx @ C )
% 0.27/0.46 & ~ ? [Xy: $i] :
% 0.27/0.46 ( ( in @ Xy @ D )
% 0.27/0.46 & ( in @ Xy @ C ) )
% 0.27/0.46 & ! [E: $i] :
% 0.27/0.46 ( ( in @ E @ B )
% 0.27/0.46 => ( ! [Xy: $i] :
% 0.27/0.46 ( ( in @ Xy @ E )
% 0.27/0.46 => ( in @ Xy @ D ) )
% 0.27/0.46 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(descr_type,type,
% 0.27/0.46 descr: ( $i > $o ) > $i ).
% 0.27/0.46
% 0.27/0.46 thf(descrp_type,type,
% 0.27/0.46 descrp: $o ).
% 0.27/0.46
% 0.27/0.46 thf(descrp,definition,
% 0.27/0.46 ( descrp
% 0.27/0.46 = ( ! [Xphi: $i > $o] :
% 0.27/0.46 ( ( exu
% 0.27/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.27/0.46 => ( Xphi
% 0.27/0.46 @ ( descr
% 0.27/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(dsetconstr_type,type,
% 0.27/0.46 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.27/0.46
% 0.27/0.46 thf(dsetconstrI_type,type,
% 0.27/0.46 dsetconstrI: $o ).
% 0.27/0.46
% 0.27/0.46 thf(dsetconstrI,definition,
% 0.27/0.46 ( dsetconstrI
% 0.27/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( ( Xphi @ Xx )
% 0.27/0.46 => ( in @ Xx
% 0.27/0.46 @ ( dsetconstr @ A
% 0.27/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(dsetconstrEL_type,type,
% 0.27/0.46 dsetconstrEL: $o ).
% 0.27/0.46
% 0.27/0.46 thf(dsetconstrEL,definition,
% 0.27/0.46 ( dsetconstrEL
% 0.27/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.27/0.46 ( ( in @ Xx
% 0.27/0.46 @ ( dsetconstr @ A
% 0.27/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.27/0.46 => ( in @ Xx @ A ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(dsetconstrER_type,type,
% 0.27/0.46 dsetconstrER: $o ).
% 0.27/0.46
% 0.27/0.46 thf(dsetconstrER,definition,
% 0.27/0.46 ( dsetconstrER
% 0.27/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.27/0.46 ( ( in @ Xx
% 0.27/0.46 @ ( dsetconstr @ A
% 0.27/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.27/0.46 => ( Xphi @ Xx ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(exuE1_type,type,
% 0.27/0.46 exuE1: $o ).
% 0.27/0.46
% 0.27/0.46 thf(exuE1,definition,
% 0.27/0.46 ( exuE1
% 0.27/0.46 = ( ! [Xphi: $i > $o] :
% 0.27/0.46 ( ( exu
% 0.27/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.27/0.46 => ? [Xx: $i] :
% 0.27/0.46 ( ( Xphi @ Xx )
% 0.27/0.46 & ! [Xy: $i] :
% 0.27/0.46 ( ( Xphi @ Xy )
% 0.27/0.46 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(prop2set_type,type,
% 0.27/0.46 prop2set: $o > $i ).
% 0.27/0.46
% 0.27/0.46 thf(prop2setE_type,type,
% 0.27/0.46 prop2setE: $o ).
% 0.27/0.46
% 0.27/0.46 thf(prop2setE,definition,
% 0.27/0.46 ( prop2setE
% 0.27/0.46 = ( ! [Xphi: $o,Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.27/0.46 => Xphi ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(emptysetE_type,type,
% 0.27/0.46 emptysetE: $o ).
% 0.27/0.46
% 0.27/0.46 thf(emptysetE,definition,
% 0.27/0.46 ( emptysetE
% 0.27/0.46 = ( ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ emptyset )
% 0.27/0.46 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(emptysetimpfalse_type,type,
% 0.27/0.46 emptysetimpfalse: $o ).
% 0.27/0.46
% 0.27/0.46 thf(emptysetimpfalse,definition,
% 0.27/0.46 ( emptysetimpfalse
% 0.27/0.46 = ( ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ emptyset )
% 0.27/0.46 => $false ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(notinemptyset_type,type,
% 0.27/0.46 notinemptyset: $o ).
% 0.27/0.46
% 0.27/0.46 thf(notinemptyset,definition,
% 0.27/0.46 ( notinemptyset
% 0.27/0.46 = ( ! [Xx: $i] :
% 0.27/0.46 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(exuE3e_type,type,
% 0.27/0.46 exuE3e: $o ).
% 0.27/0.46
% 0.27/0.46 thf(exuE3e,definition,
% 0.27/0.46 ( exuE3e
% 0.27/0.46 = ( ! [Xphi: $i > $o] :
% 0.27/0.46 ( ( exu
% 0.27/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.27/0.46 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(setext_type,type,
% 0.27/0.46 setext: $o ).
% 0.27/0.46
% 0.27/0.46 thf(setext,definition,
% 0.27/0.46 ( setext
% 0.27/0.46 = ( ! [A: $i,B: $i] :
% 0.27/0.46 ( ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( in @ Xx @ B ) )
% 0.27/0.46 => ( ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ B )
% 0.27/0.46 => ( in @ Xx @ A ) )
% 0.27/0.46 => ( A = B ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(emptyI_type,type,
% 0.27/0.46 emptyI: $o ).
% 0.27/0.46
% 0.27/0.46 thf(emptyI,definition,
% 0.27/0.46 ( emptyI
% 0.27/0.46 = ( ! [A: $i] :
% 0.27/0.46 ( ! [Xx: $i] :
% 0.27/0.46 ~ ( in @ Xx @ A )
% 0.27/0.46 => ( A = emptyset ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(noeltsimpempty_type,type,
% 0.27/0.46 noeltsimpempty: $o ).
% 0.27/0.46
% 0.27/0.46 thf(noeltsimpempty,definition,
% 0.27/0.46 ( noeltsimpempty
% 0.27/0.46 = ( ! [A: $i] :
% 0.27/0.46 ( ! [Xx: $i] :
% 0.27/0.46 ~ ( in @ Xx @ A )
% 0.27/0.46 => ( A = emptyset ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(setbeta_type,type,
% 0.27/0.46 setbeta: $o ).
% 0.27/0.46
% 0.27/0.46 thf(setbeta,definition,
% 0.27/0.46 ( setbeta
% 0.27/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( ( in @ Xx
% 0.27/0.46 @ ( dsetconstr @ A
% 0.27/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.27/0.46 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(nonempty_type,type,
% 0.27/0.46 nonempty: $i > $o ).
% 0.27/0.46
% 0.27/0.46 thf(nonempty,definition,
% 0.27/0.46 ( nonempty
% 0.27/0.46 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(nonemptyE1_type,type,
% 0.27/0.46 nonemptyE1: $o ).
% 0.27/0.46
% 0.27/0.46 thf(nonemptyE1,definition,
% 0.27/0.46 ( nonemptyE1
% 0.27/0.46 = ( ! [A: $i] :
% 0.27/0.46 ( ( nonempty @ A )
% 0.27/0.46 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(nonemptyI_type,type,
% 0.27/0.46 nonemptyI: $o ).
% 0.27/0.46
% 0.27/0.46 thf(nonemptyI,definition,
% 0.27/0.46 ( nonemptyI
% 0.27/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( ( Xphi @ Xx )
% 0.27/0.46 => ( nonempty
% 0.27/0.46 @ ( dsetconstr @ A
% 0.27/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(nonemptyI1_type,type,
% 0.27/0.46 nonemptyI1: $o ).
% 0.27/0.46
% 0.27/0.46 thf(nonemptyI1,definition,
% 0.27/0.46 ( nonemptyI1
% 0.27/0.46 = ( ! [A: $i] :
% 0.27/0.46 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.27/0.46 => ( nonempty @ A ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(setadjoinIL_type,type,
% 0.27/0.46 setadjoinIL: $o ).
% 0.27/0.46
% 0.27/0.46 thf(setadjoinIL,definition,
% 0.27/0.46 ( setadjoinIL
% 0.27/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(emptyinunitempty_type,type,
% 0.27/0.46 emptyinunitempty: $o ).
% 0.27/0.46
% 0.27/0.46 thf(emptyinunitempty,definition,
% 0.27/0.46 ( emptyinunitempty
% 0.27/0.46 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(setadjoinIR_type,type,
% 0.27/0.46 setadjoinIR: $o ).
% 0.27/0.46
% 0.27/0.46 thf(setadjoinIR,definition,
% 0.27/0.46 ( setadjoinIR
% 0.27/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.27/0.46 ( ( in @ Xy @ A )
% 0.27/0.46 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(setadjoinE_type,type,
% 0.27/0.46 setadjoinE: $o ).
% 0.27/0.46
% 0.27/0.46 thf(setadjoinE,definition,
% 0.27/0.46 ( setadjoinE
% 0.27/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.27/0.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.27/0.46 => ! [Xphi: $o] :
% 0.27/0.46 ( ( ( Xy = Xx )
% 0.27/0.46 => Xphi )
% 0.27/0.46 => ( ( ( in @ Xy @ A )
% 0.27/0.46 => Xphi )
% 0.27/0.46 => Xphi ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(setadjoinOr_type,type,
% 0.27/0.46 setadjoinOr: $o ).
% 0.27/0.46
% 0.27/0.46 thf(setadjoinOr,definition,
% 0.27/0.46 ( setadjoinOr
% 0.27/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.27/0.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.27/0.46 => ( ( Xy = Xx )
% 0.27/0.46 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(setoftrueEq_type,type,
% 0.27/0.46 setoftrueEq: $o ).
% 0.27/0.46
% 0.27/0.46 thf(setoftrueEq,definition,
% 0.27/0.46 ( setoftrueEq
% 0.27/0.46 = ( ! [A: $i] :
% 0.27/0.46 ( ( dsetconstr @ A
% 0.27/0.46 @ ^ [Xx: $i] : $true )
% 0.27/0.46 = A ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(powersetI_type,type,
% 0.27/0.46 powersetI: $o ).
% 0.27/0.46
% 0.27/0.46 thf(powersetI,definition,
% 0.27/0.46 ( powersetI
% 0.27/0.46 = ( ! [A: $i,B: $i] :
% 0.27/0.46 ( ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ B )
% 0.27/0.46 => ( in @ Xx @ A ) )
% 0.27/0.46 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(emptyinPowerset_type,type,
% 0.27/0.46 emptyinPowerset: $o ).
% 0.27/0.46
% 0.27/0.46 thf(emptyinPowerset,definition,
% 0.27/0.46 ( emptyinPowerset
% 0.27/0.46 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(emptyInPowerset_type,type,
% 0.27/0.46 emptyInPowerset: $o ).
% 0.27/0.46
% 0.27/0.46 thf(emptyInPowerset,definition,
% 0.27/0.46 ( emptyInPowerset
% 0.27/0.46 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(powersetE_type,type,
% 0.27/0.46 powersetE: $o ).
% 0.27/0.46
% 0.27/0.46 thf(powersetE,definition,
% 0.27/0.46 ( powersetE
% 0.27/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.46 ( ( in @ B @ ( powerset @ A ) )
% 0.27/0.46 => ( ( in @ Xx @ B )
% 0.27/0.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(setunionI_type,type,
% 0.27/0.46 setunionI: $o ).
% 0.27/0.46
% 0.27/0.46 thf(setunionI,definition,
% 0.27/0.46 ( setunionI
% 0.27/0.46 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.27/0.46 ( ( in @ Xx @ B )
% 0.27/0.46 => ( ( in @ B @ A )
% 0.27/0.46 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(setunionE_type,type,
% 0.27/0.46 setunionE: $o ).
% 0.27/0.46
% 0.27/0.46 thf(setunionE,definition,
% 0.27/0.46 ( setunionE
% 0.27/0.46 = ( ! [A: $i,Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ ( setunion @ A ) )
% 0.27/0.46 => ! [Xphi: $o] :
% 0.27/0.46 ( ! [B: $i] :
% 0.27/0.46 ( ( in @ Xx @ B )
% 0.27/0.46 => ( ( in @ B @ A )
% 0.27/0.46 => Xphi ) )
% 0.27/0.46 => Xphi ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(subPowSU_type,type,
% 0.27/0.46 subPowSU: $o ).
% 0.27/0.46
% 0.27/0.46 thf(subPowSU,definition,
% 0.27/0.46 ( subPowSU
% 0.27/0.46 = ( ! [A: $i,Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(exuE2_type,type,
% 0.27/0.46 exuE2: $o ).
% 0.27/0.46
% 0.27/0.46 thf(exuE2,definition,
% 0.27/0.46 ( exuE2
% 0.27/0.46 = ( ! [Xphi: $i > $o] :
% 0.27/0.46 ( ( exu
% 0.27/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.27/0.46 => ? [Xx: $i] :
% 0.27/0.46 ! [Xy: $i] :
% 0.27/0.46 ( ( Xphi @ Xy )
% 0.27/0.46 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(nonemptyImpWitness_type,type,
% 0.27/0.46 nonemptyImpWitness: $o ).
% 0.27/0.46
% 0.27/0.46 thf(nonemptyImpWitness,definition,
% 0.27/0.46 ( nonemptyImpWitness
% 0.27/0.46 = ( ! [A: $i] :
% 0.27/0.46 ( ( nonempty @ A )
% 0.27/0.46 => ? [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 & $true ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(uniqinunit_type,type,
% 0.27/0.46 uniqinunit: $o ).
% 0.27/0.46
% 0.27/0.46 thf(uniqinunit,definition,
% 0.27/0.46 ( uniqinunit
% 0.27/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.27/0.46 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.27/0.46 => ( Xx = Xy ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(notinsingleton_type,type,
% 0.27/0.46 notinsingleton: $o ).
% 0.27/0.46
% 0.27/0.46 thf(notinsingleton,definition,
% 0.27/0.46 ( notinsingleton
% 0.27/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.27/0.46 ( ( Xx != Xy )
% 0.27/0.46 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(eqinunit_type,type,
% 0.27/0.46 eqinunit: $o ).
% 0.27/0.46
% 0.27/0.46 thf(eqinunit,definition,
% 0.27/0.46 ( eqinunit
% 0.27/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.27/0.46 ( ( Xx = Xy )
% 0.27/0.46 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(singletonsswitch_type,type,
% 0.27/0.46 singletonsswitch: $o ).
% 0.27/0.46
% 0.27/0.46 thf(singletonsswitch,definition,
% 0.27/0.46 ( singletonsswitch
% 0.27/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.27/0.46 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.27/0.46 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(upairsetE_type,type,
% 0.27/0.46 upairsetE: $o ).
% 0.27/0.46
% 0.27/0.46 thf(upairsetE,definition,
% 0.27/0.46 ( upairsetE
% 0.27/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.27/0.46 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.27/0.46 => ( ( Xz = Xx )
% 0.27/0.46 | ( Xz = Xy ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(upairsetIL_type,type,
% 0.27/0.46 upairsetIL: $o ).
% 0.27/0.46
% 0.27/0.46 thf(upairsetIL,definition,
% 0.27/0.46 ( upairsetIL
% 0.27/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(upairsetIR_type,type,
% 0.27/0.46 upairsetIR: $o ).
% 0.27/0.46
% 0.27/0.46 thf(upairsetIR,definition,
% 0.27/0.46 ( upairsetIR
% 0.27/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(emptyE1_type,type,
% 0.27/0.46 emptyE1: $o ).
% 0.27/0.46
% 0.27/0.46 thf(emptyE1,definition,
% 0.27/0.46 ( emptyE1
% 0.27/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.27/0.46 ( ? [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 & ( Xphi @ Xx ) )
% 0.27/0.46 => ( ( ( dsetconstr @ A
% 0.27/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.27/0.46 = emptyset )
% 0.27/0.46 => $false ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(vacuousDall_type,type,
% 0.27/0.46 vacuousDall: $o ).
% 0.27/0.46
% 0.27/0.46 thf(vacuousDall,definition,
% 0.27/0.46 ( vacuousDall
% 0.27/0.46 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ emptyset )
% 0.27/0.46 => ( Xphi @ Xx ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(quantDeMorgan1_type,type,
% 0.27/0.46 quantDeMorgan1: $o ).
% 0.27/0.46
% 0.27/0.46 thf(quantDeMorgan1,definition,
% 0.27/0.46 ( quantDeMorgan1
% 0.27/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.27/0.46 ( ~ ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( Xphi @ Xx ) )
% 0.27/0.46 => ? [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(quantDeMorgan2_type,type,
% 0.27/0.46 quantDeMorgan2: $o ).
% 0.27/0.46
% 0.27/0.46 thf(quantDeMorgan2,definition,
% 0.27/0.46 ( quantDeMorgan2
% 0.27/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.27/0.46 ( ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ~ ( Xphi @ Xx ) )
% 0.27/0.46 => ~ ? [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(quantDeMorgan3_type,type,
% 0.27/0.46 quantDeMorgan3: $o ).
% 0.27/0.46
% 0.27/0.46 thf(quantDeMorgan3,definition,
% 0.27/0.46 ( quantDeMorgan3
% 0.27/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.27/0.46 ( ~ ? [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 & ( Xphi @ Xx ) )
% 0.27/0.46 => ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(quantDeMorgan4_type,type,
% 0.27/0.46 quantDeMorgan4: $o ).
% 0.27/0.46
% 0.27/0.46 thf(quantDeMorgan4,definition,
% 0.27/0.46 ( quantDeMorgan4
% 0.27/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.27/0.46 ( ? [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 & ~ ( Xphi @ Xx ) )
% 0.27/0.46 => ~ ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(prop2setI_type,type,
% 0.27/0.46 prop2setI: $o ).
% 0.27/0.46
% 0.27/0.46 thf(prop2setI,definition,
% 0.27/0.46 ( prop2setI
% 0.27/0.46 = ( ! [Xphi: $o] :
% 0.27/0.46 ( Xphi
% 0.27/0.46 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(set2prop_type,type,
% 0.27/0.46 set2prop: $i > $o ).
% 0.27/0.46
% 0.27/0.46 thf(prop2set2propI_type,type,
% 0.27/0.46 prop2set2propI: $o ).
% 0.27/0.46
% 0.27/0.46 thf(prop2set2propI,definition,
% 0.27/0.46 ( prop2set2propI
% 0.27/0.46 = ( ! [Xphi: $o] :
% 0.27/0.46 ( Xphi
% 0.27/0.46 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(notdexE_type,type,
% 0.27/0.46 notdexE: $o ).
% 0.27/0.46
% 0.27/0.46 thf(notdexE,definition,
% 0.27/0.46 ( notdexE
% 0.27/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.27/0.46 ( ~ ? [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 & ( Xphi @ Xx ) )
% 0.27/0.46 => ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(notdallE_type,type,
% 0.27/0.46 notdallE: $o ).
% 0.27/0.46
% 0.27/0.46 thf(notdallE,definition,
% 0.27/0.46 ( notdallE
% 0.27/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.27/0.46 ( ~ ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( Xphi @ Xx ) )
% 0.27/0.46 => ? [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(exuI1_type,type,
% 0.27/0.46 exuI1: $o ).
% 0.27/0.46
% 0.27/0.46 thf(exuI1,definition,
% 0.27/0.46 ( exuI1
% 0.27/0.46 = ( ! [Xphi: $i > $o] :
% 0.27/0.46 ( ? [Xx: $i] :
% 0.27/0.46 ( ( Xphi @ Xx )
% 0.27/0.46 & ! [Xy: $i] :
% 0.27/0.46 ( ( Xphi @ Xy )
% 0.27/0.46 => ( Xx = Xy ) ) )
% 0.27/0.46 => ( exu
% 0.27/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(exuI3_type,type,
% 0.27/0.46 exuI3: $o ).
% 0.27/0.46
% 0.27/0.46 thf(exuI3,definition,
% 0.27/0.46 ( exuI3
% 0.27/0.46 = ( ! [Xphi: $i > $o] :
% 0.27/0.46 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.27/0.46 => ( ! [Xx: $i,Xy: $i] :
% 0.27/0.46 ( ( Xphi @ Xx )
% 0.27/0.46 => ( ( Xphi @ Xy )
% 0.27/0.46 => ( Xx = Xy ) ) )
% 0.27/0.46 => ( exu
% 0.27/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(exuI2_type,type,
% 0.27/0.46 exuI2: $o ).
% 0.27/0.46
% 0.27/0.46 thf(exuI2,definition,
% 0.27/0.46 ( exuI2
% 0.27/0.46 = ( ! [Xphi: $i > $o] :
% 0.27/0.46 ( ? [Xx: $i] :
% 0.27/0.46 ! [Xy: $i] :
% 0.27/0.46 ( ( Xphi @ Xy )
% 0.27/0.46 <=> ( Xy = Xx ) )
% 0.27/0.46 => ( exu
% 0.27/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(inCongP_type,type,
% 0.27/0.46 inCongP: $o ).
% 0.27/0.46
% 0.27/0.46 thf(inCongP,definition,
% 0.27/0.46 ( inCongP
% 0.27/0.46 = ( ! [A: $i,B: $i] :
% 0.27/0.46 ( ( A = B )
% 0.27/0.46 => ! [Xx: $i,Xy: $i] :
% 0.27/0.46 ( ( Xx = Xy )
% 0.27/0.46 => ( ( in @ Xx @ A )
% 0.27/0.46 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(in__Cong_type,type,
% 0.27/0.46 in__Cong: $o ).
% 0.27/0.46
% 0.27/0.46 thf(in__Cong,definition,
% 0.27/0.46 ( in__Cong
% 0.27/0.46 = ( ! [A: $i,B: $i] :
% 0.27/0.46 ( ( A = B )
% 0.27/0.46 => ! [Xx: $i,Xy: $i] :
% 0.27/0.46 ( ( Xx = Xy )
% 0.27/0.46 => ( ( in @ Xx @ A )
% 0.27/0.46 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(exuE3u_type,type,
% 0.27/0.46 exuE3u: $o ).
% 0.27/0.46
% 0.27/0.46 thf(exuE3u,definition,
% 0.27/0.46 ( exuE3u
% 0.27/0.46 = ( ! [Xphi: $i > $o] :
% 0.27/0.46 ( ( exu
% 0.27/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.27/0.46 => ! [Xx: $i,Xy: $i] :
% 0.27/0.46 ( ( Xphi @ Xx )
% 0.27/0.46 => ( ( Xphi @ Xy )
% 0.27/0.46 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(exu__Cong_type,type,
% 0.27/0.46 exu__Cong: $o ).
% 0.27/0.46
% 0.27/0.46 thf(exu__Cong,definition,
% 0.27/0.46 ( exu__Cong
% 0.27/0.46 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.27/0.46 ( ! [Xx: $i,Xy: $i] :
% 0.27/0.46 ( ( Xx = Xy )
% 0.27/0.46 => ( ( Xphi @ Xx )
% 0.27/0.46 <=> ( Xpsi @ Xy ) ) )
% 0.27/0.46 => ( ( exu
% 0.27/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.27/0.46 <=> ( exu
% 0.27/0.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(emptyset__Cong_type,type,
% 0.27/0.46 emptyset__Cong: $o ).
% 0.27/0.46
% 0.27/0.46 thf(emptyset__Cong,definition,
% 0.27/0.46 ( emptyset__Cong
% 0.27/0.46 = ( emptyset = emptyset ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(setadjoin__Cong_type,type,
% 0.27/0.46 setadjoin__Cong: $o ).
% 0.27/0.46
% 0.27/0.46 thf(setadjoin__Cong,definition,
% 0.27/0.46 ( setadjoin__Cong
% 0.27/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.27/0.46 ( ( Xx = Xy )
% 0.27/0.46 => ! [Xz: $i,Xu: $i] :
% 0.27/0.46 ( ( Xz = Xu )
% 0.27/0.46 => ( ( setadjoin @ Xx @ Xz )
% 0.27/0.46 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(powerset__Cong_type,type,
% 0.27/0.46 powerset__Cong: $o ).
% 0.27/0.46
% 0.27/0.46 thf(powerset__Cong,definition,
% 0.27/0.46 ( powerset__Cong
% 0.27/0.46 = ( ! [A: $i,B: $i] :
% 0.27/0.46 ( ( A = B )
% 0.27/0.46 => ( ( powerset @ A )
% 0.27/0.46 = ( powerset @ B ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(setunion__Cong_type,type,
% 0.27/0.46 setunion__Cong: $o ).
% 0.27/0.46
% 0.27/0.46 thf(setunion__Cong,definition,
% 0.27/0.46 ( setunion__Cong
% 0.27/0.46 = ( ! [A: $i,B: $i] :
% 0.27/0.46 ( ( A = B )
% 0.27/0.46 => ( ( setunion @ A )
% 0.27/0.46 = ( setunion @ B ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(omega__Cong_type,type,
% 0.27/0.46 omega__Cong: $o ).
% 0.27/0.46
% 0.27/0.46 thf(omega__Cong,definition,
% 0.27/0.46 ( omega__Cong
% 0.27/0.46 = ( omega = omega ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(exuEu_type,type,
% 0.27/0.46 exuEu: $o ).
% 0.27/0.46
% 0.27/0.46 thf(exuEu,definition,
% 0.27/0.46 ( exuEu
% 0.27/0.46 = ( ! [Xphi: $i > $o] :
% 0.27/0.46 ( ( exu
% 0.27/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.27/0.46 => ! [Xx: $i,Xy: $i] :
% 0.27/0.46 ( ( Xphi @ Xx )
% 0.27/0.46 => ( ( Xphi @ Xy )
% 0.27/0.46 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(descr__Cong_type,type,
% 0.27/0.46 descr__Cong: $o ).
% 0.27/0.46
% 0.27/0.46 thf(descr__Cong,definition,
% 0.27/0.46 ( descr__Cong
% 0.27/0.46 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.27/0.46 ( ! [Xx: $i,Xy: $i] :
% 0.27/0.46 ( ( Xx = Xy )
% 0.27/0.46 => ( ( Xphi @ Xx )
% 0.27/0.46 <=> ( Xpsi @ Xy ) ) )
% 0.27/0.46 => ( ( exu
% 0.27/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.27/0.46 => ( ( exu
% 0.27/0.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.27/0.46 => ( ( descr
% 0.27/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.27/0.46 = ( descr
% 0.27/0.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(dsetconstr__Cong_type,type,
% 0.27/0.46 dsetconstr__Cong: $o ).
% 0.27/0.46
% 0.27/0.46 thf(dsetconstr__Cong,definition,
% 0.27/0.46 ( dsetconstr__Cong
% 0.27/0.46 = ( ! [A: $i,B: $i] :
% 0.27/0.46 ( ( A = B )
% 0.27/0.46 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.27/0.46 ( ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ! [Xy: $i] :
% 0.27/0.46 ( ( in @ Xy @ B )
% 0.27/0.46 => ( ( Xx = Xy )
% 0.27/0.46 => ( ( Xphi @ Xx )
% 0.27/0.47 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.27/0.47 => ( ( dsetconstr @ A
% 0.27/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.27/0.47 = ( dsetconstr @ B
% 0.27/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(subset_type,type,
% 0.27/0.47 subset: $i > $i > $o ).
% 0.27/0.47
% 0.27/0.47 thf(disjoint_type,type,
% 0.27/0.47 disjoint: $i > $i > $o ).
% 0.27/0.47
% 0.27/0.47 thf(setsmeet_type,type,
% 0.27/0.47 setsmeet: $i > $i > $o ).
% 0.27/0.47
% 0.27/0.47 thf(subsetI1_type,type,
% 0.27/0.47 subsetI1: $o ).
% 0.27/0.47
% 0.27/0.47 thf(subsetI1,definition,
% 0.27/0.47 ( subsetI1
% 0.27/0.47 = ( ! [A: $i,B: $i] :
% 0.27/0.47 ( ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( in @ Xx @ B ) )
% 0.27/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(eqimpsubset2_type,type,
% 0.27/0.47 eqimpsubset2: $o ).
% 0.27/0.47
% 0.27/0.47 thf(eqimpsubset2,definition,
% 0.27/0.47 ( eqimpsubset2
% 0.27/0.47 = ( ! [A: $i,B: $i] :
% 0.27/0.47 ( ( A = B )
% 0.27/0.47 => ( subset @ B @ A ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(eqimpsubset1_type,type,
% 0.27/0.47 eqimpsubset1: $o ).
% 0.27/0.47
% 0.27/0.47 thf(eqimpsubset1,definition,
% 0.27/0.47 ( eqimpsubset1
% 0.27/0.47 = ( ! [A: $i,B: $i] :
% 0.27/0.47 ( ( A = B )
% 0.27/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(subsetI2_type,type,
% 0.27/0.47 subsetI2: $o ).
% 0.27/0.47
% 0.27/0.47 thf(subsetI2,definition,
% 0.27/0.47 ( subsetI2
% 0.27/0.47 = ( ! [A: $i,B: $i] :
% 0.27/0.47 ( ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( in @ Xx @ B ) )
% 0.27/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(emptysetsubset_type,type,
% 0.27/0.47 emptysetsubset: $o ).
% 0.27/0.47
% 0.27/0.47 thf(emptysetsubset,definition,
% 0.27/0.47 ( emptysetsubset
% 0.27/0.47 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(subsetE_type,type,
% 0.27/0.47 subsetE: $o ).
% 0.27/0.47
% 0.27/0.47 thf(subsetE,definition,
% 0.27/0.47 ( subsetE
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ( subset @ A @ B )
% 0.27/0.47 => ( ( in @ Xx @ A )
% 0.27/0.47 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(subsetE2_type,type,
% 0.27/0.47 subsetE2: $o ).
% 0.27/0.47
% 0.27/0.47 thf(subsetE2,definition,
% 0.27/0.47 ( subsetE2
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ( subset @ A @ B )
% 0.27/0.47 => ( ~ ( in @ Xx @ B )
% 0.27/0.47 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(notsubsetI_type,type,
% 0.27/0.47 notsubsetI: $o ).
% 0.27/0.47
% 0.27/0.47 thf(notsubsetI,definition,
% 0.27/0.47 ( notsubsetI
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( ~ ( in @ Xx @ B )
% 0.27/0.47 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(notequalI1_type,type,
% 0.27/0.47 notequalI1: $o ).
% 0.27/0.47
% 0.27/0.47 thf(notequalI1,definition,
% 0.27/0.47 ( notequalI1
% 0.27/0.47 = ( ! [A: $i,B: $i] :
% 0.27/0.47 ( ~ ( subset @ A @ B )
% 0.27/0.47 => ( A != B ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(notequalI2_type,type,
% 0.27/0.47 notequalI2: $o ).
% 0.27/0.47
% 0.27/0.47 thf(notequalI2,definition,
% 0.27/0.47 ( notequalI2
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( ~ ( in @ Xx @ B )
% 0.27/0.47 => ( A != B ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(subsetRefl_type,type,
% 0.27/0.47 subsetRefl: $o ).
% 0.27/0.47
% 0.27/0.47 thf(subsetRefl,definition,
% 0.27/0.47 ( subsetRefl
% 0.27/0.47 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(subsetTrans_type,type,
% 0.27/0.47 subsetTrans: $o ).
% 0.27/0.47
% 0.27/0.47 thf(subsetTrans,definition,
% 0.27/0.47 ( subsetTrans
% 0.27/0.47 = ( ! [A: $i,B: $i,C: $i] :
% 0.27/0.47 ( ( subset @ A @ B )
% 0.27/0.47 => ( ( subset @ B @ C )
% 0.27/0.47 => ( subset @ A @ C ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setadjoinSub_type,type,
% 0.27/0.47 setadjoinSub: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setadjoinSub,definition,
% 0.27/0.47 ( setadjoinSub
% 0.27/0.47 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setadjoinSub2_type,type,
% 0.27/0.47 setadjoinSub2: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setadjoinSub2,definition,
% 0.27/0.47 ( setadjoinSub2
% 0.27/0.47 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.27/0.47 ( ( subset @ A @ B )
% 0.27/0.47 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(subset2powerset_type,type,
% 0.27/0.47 subset2powerset: $o ).
% 0.27/0.47
% 0.27/0.47 thf(subset2powerset,definition,
% 0.27/0.47 ( subset2powerset
% 0.27/0.47 = ( ! [A: $i,B: $i] :
% 0.27/0.47 ( ( subset @ A @ B )
% 0.27/0.47 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setextsub_type,type,
% 0.27/0.47 setextsub: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setextsub,definition,
% 0.27/0.47 ( setextsub
% 0.27/0.47 = ( ! [A: $i,B: $i] :
% 0.27/0.47 ( ( subset @ A @ B )
% 0.27/0.47 => ( ( subset @ B @ A )
% 0.27/0.47 => ( A = B ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(subsetemptysetimpeq_type,type,
% 0.27/0.47 subsetemptysetimpeq: $o ).
% 0.27/0.47
% 0.27/0.47 thf(subsetemptysetimpeq,definition,
% 0.27/0.47 ( subsetemptysetimpeq
% 0.27/0.47 = ( ! [A: $i] :
% 0.27/0.47 ( ( subset @ A @ emptyset )
% 0.27/0.47 => ( A = emptyset ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(powersetI1_type,type,
% 0.27/0.47 powersetI1: $o ).
% 0.27/0.47
% 0.27/0.47 thf(powersetI1,definition,
% 0.27/0.47 ( powersetI1
% 0.27/0.47 = ( ! [A: $i,B: $i] :
% 0.27/0.47 ( ( subset @ B @ A )
% 0.27/0.47 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(powersetE1_type,type,
% 0.27/0.47 powersetE1: $o ).
% 0.27/0.47
% 0.27/0.47 thf(powersetE1,definition,
% 0.27/0.47 ( powersetE1
% 0.27/0.47 = ( ! [A: $i,B: $i] :
% 0.27/0.47 ( ( in @ B @ ( powerset @ A ) )
% 0.27/0.47 => ( subset @ B @ A ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(inPowerset_type,type,
% 0.27/0.47 inPowerset: $o ).
% 0.27/0.47
% 0.27/0.47 thf(inPowerset,definition,
% 0.27/0.47 ( inPowerset
% 0.27/0.47 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(powersetsubset_type,type,
% 0.27/0.47 powersetsubset: $o ).
% 0.27/0.47
% 0.27/0.47 thf(powersetsubset,definition,
% 0.27/0.47 ( powersetsubset
% 0.27/0.47 = ( ! [A: $i,B: $i] :
% 0.27/0.47 ( ( subset @ A @ B )
% 0.27/0.47 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(sepInPowerset_type,type,
% 0.27/0.47 sepInPowerset: $o ).
% 0.27/0.47
% 0.27/0.47 thf(sepInPowerset,definition,
% 0.27/0.47 ( sepInPowerset
% 0.27/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.27/0.47 ( in
% 0.27/0.47 @ ( dsetconstr @ A
% 0.27/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.27/0.47 @ ( powerset @ A ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(sepSubset_type,type,
% 0.27/0.47 sepSubset: $o ).
% 0.27/0.47
% 0.27/0.47 thf(sepSubset,definition,
% 0.27/0.47 ( sepSubset
% 0.27/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.27/0.47 ( subset
% 0.27/0.47 @ ( dsetconstr @ A
% 0.27/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.27/0.47 @ A ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(binunion_type,type,
% 0.27/0.47 binunion: $i > $i > $i ).
% 0.27/0.47
% 0.27/0.47 thf(binunionIL_type,type,
% 0.27/0.47 binunionIL: $o ).
% 0.27/0.47
% 0.27/0.47 thf(binunionIL,definition,
% 0.27/0.47 ( binunionIL
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(upairset2IR_type,type,
% 0.27/0.47 upairset2IR: $o ).
% 0.27/0.47
% 0.27/0.47 thf(upairset2IR,definition,
% 0.27/0.47 ( upairset2IR
% 0.27/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(binunionIR_type,type,
% 0.27/0.47 binunionIR: $o ).
% 0.27/0.47
% 0.27/0.47 thf(binunionIR,definition,
% 0.27/0.47 ( binunionIR
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ B )
% 0.27/0.47 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(binunionEcases_type,type,
% 0.27/0.47 binunionEcases: $o ).
% 0.27/0.47
% 0.27/0.47 thf(binunionEcases,definition,
% 0.27/0.47 ( binunionEcases
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.27/0.47 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.27/0.47 => ( ( ( in @ Xx @ A )
% 0.27/0.47 => Xphi )
% 0.27/0.47 => ( ( ( in @ Xx @ B )
% 0.27/0.47 => Xphi )
% 0.27/0.47 => Xphi ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(binunionE_type,type,
% 0.27/0.47 binunionE: $o ).
% 0.27/0.47
% 0.27/0.47 thf(binunionE,definition,
% 0.27/0.47 ( binunionE
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.27/0.47 => ( ( in @ Xx @ A )
% 0.27/0.47 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(binunionLsub_type,type,
% 0.27/0.47 binunionLsub: $o ).
% 0.27/0.47
% 0.27/0.47 thf(binunionLsub,definition,
% 0.27/0.47 ( binunionLsub
% 0.27/0.47 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(binunionRsub_type,type,
% 0.27/0.47 binunionRsub: $o ).
% 0.27/0.47
% 0.27/0.47 thf(binunionRsub,definition,
% 0.27/0.47 ( binunionRsub
% 0.27/0.47 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(binintersect_type,type,
% 0.27/0.47 binintersect: $i > $i > $i ).
% 0.27/0.47
% 0.27/0.47 thf(binintersectI_type,type,
% 0.27/0.47 binintersectI: $o ).
% 0.27/0.47
% 0.27/0.47 thf(binintersectI,definition,
% 0.27/0.47 ( binintersectI
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( ( in @ Xx @ B )
% 0.27/0.47 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(binintersectSubset5_type,type,
% 0.27/0.47 binintersectSubset5: $o ).
% 0.27/0.47
% 0.27/0.47 thf(binintersectSubset5,definition,
% 0.27/0.47 ( binintersectSubset5
% 0.27/0.47 = ( ! [A: $i,B: $i,C: $i] :
% 0.27/0.47 ( ( subset @ C @ A )
% 0.27/0.47 => ( ( subset @ C @ B )
% 0.27/0.47 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(binintersectEL_type,type,
% 0.27/0.47 binintersectEL: $o ).
% 0.27/0.47
% 0.27/0.47 thf(binintersectEL,definition,
% 0.27/0.47 ( binintersectEL
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.27/0.47 => ( in @ Xx @ A ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(binintersectLsub_type,type,
% 0.27/0.47 binintersectLsub: $o ).
% 0.27/0.47
% 0.27/0.47 thf(binintersectLsub,definition,
% 0.27/0.47 ( binintersectLsub
% 0.27/0.47 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(binintersectSubset2_type,type,
% 0.27/0.47 binintersectSubset2: $o ).
% 0.27/0.47
% 0.27/0.47 thf(binintersectSubset2,definition,
% 0.27/0.47 ( binintersectSubset2
% 0.27/0.47 = ( ! [A: $i,B: $i] :
% 0.27/0.47 ( ( subset @ A @ B )
% 0.27/0.47 => ( ( binintersect @ A @ B )
% 0.27/0.47 = A ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(binintersectSubset3_type,type,
% 0.27/0.47 binintersectSubset3: $o ).
% 0.27/0.47
% 0.27/0.47 thf(binintersectSubset3,definition,
% 0.27/0.47 ( binintersectSubset3
% 0.27/0.47 = ( ! [A: $i,B: $i] :
% 0.27/0.47 ( ( ( binintersect @ A @ B )
% 0.27/0.47 = B )
% 0.27/0.47 => ( subset @ B @ A ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(binintersectER_type,type,
% 0.27/0.47 binintersectER: $o ).
% 0.27/0.47
% 0.27/0.47 thf(binintersectER,definition,
% 0.27/0.47 ( binintersectER
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.27/0.47 => ( in @ Xx @ B ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(disjointsetsI1_type,type,
% 0.27/0.47 disjointsetsI1: $o ).
% 0.27/0.47
% 0.27/0.47 thf(disjointsetsI1,definition,
% 0.27/0.47 ( disjointsetsI1
% 0.27/0.47 = ( ! [A: $i,B: $i] :
% 0.27/0.47 ( ~ ? [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 & ( in @ Xx @ B ) )
% 0.27/0.47 => ( ( binintersect @ A @ B )
% 0.27/0.47 = emptyset ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(binintersectRsub_type,type,
% 0.27/0.47 binintersectRsub: $o ).
% 0.27/0.47
% 0.27/0.47 thf(binintersectRsub,definition,
% 0.27/0.47 ( binintersectRsub
% 0.27/0.47 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(binintersectSubset4_type,type,
% 0.27/0.47 binintersectSubset4: $o ).
% 0.27/0.47
% 0.27/0.47 thf(binintersectSubset4,definition,
% 0.27/0.47 ( binintersectSubset4
% 0.27/0.47 = ( ! [A: $i,B: $i] :
% 0.27/0.47 ( ( subset @ B @ A )
% 0.27/0.47 => ( ( binintersect @ A @ B )
% 0.27/0.47 = B ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(binintersectSubset1_type,type,
% 0.27/0.47 binintersectSubset1: $o ).
% 0.27/0.47
% 0.27/0.47 thf(binintersectSubset1,definition,
% 0.27/0.47 ( binintersectSubset1
% 0.27/0.47 = ( ! [A: $i,B: $i] :
% 0.27/0.47 ( ( ( binintersect @ A @ B )
% 0.27/0.47 = A )
% 0.27/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(bs114d_type,type,
% 0.27/0.47 bs114d: $o ).
% 0.27/0.47
% 0.27/0.47 thf(bs114d,definition,
% 0.27/0.47 ( bs114d
% 0.27/0.47 = ( ! [A: $i,B: $i,C: $i] :
% 0.27/0.47 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 0.27/0.47 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(regular_type,type,
% 0.27/0.47 regular: $i > $o ).
% 0.27/0.47
% 0.27/0.47 thf(setminus_type,type,
% 0.27/0.47 setminus: $i > $i > $i ).
% 0.27/0.47
% 0.27/0.47 thf(setminusI_type,type,
% 0.27/0.47 setminusI: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setminusI,definition,
% 0.27/0.47 ( setminusI
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( ~ ( in @ Xx @ B )
% 0.27/0.47 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setminusEL_type,type,
% 0.27/0.47 setminusEL: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setminusEL,definition,
% 0.27/0.47 ( setminusEL
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.27/0.47 => ( in @ Xx @ A ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setminusER_type,type,
% 0.27/0.47 setminusER: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setminusER,definition,
% 0.27/0.47 ( setminusER
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.27/0.47 => ~ ( in @ Xx @ B ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setminusSubset2_type,type,
% 0.27/0.47 setminusSubset2: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setminusSubset2,definition,
% 0.27/0.47 ( setminusSubset2
% 0.27/0.47 = ( ! [A: $i,B: $i] :
% 0.27/0.47 ( ( subset @ A @ B )
% 0.27/0.47 => ( ( setminus @ A @ B )
% 0.27/0.47 = emptyset ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setminusERneg_type,type,
% 0.27/0.47 setminusERneg: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setminusERneg,definition,
% 0.27/0.47 ( setminusERneg
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.27/0.47 => ( ( in @ Xx @ A )
% 0.27/0.47 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setminusELneg_type,type,
% 0.27/0.47 setminusELneg: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setminusELneg,definition,
% 0.27/0.47 ( setminusELneg
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.27/0.47 => ( ~ ( in @ Xx @ B )
% 0.27/0.47 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setminusILneg_type,type,
% 0.27/0.47 setminusILneg: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setminusILneg,definition,
% 0.27/0.47 ( setminusILneg
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ~ ( in @ Xx @ A )
% 0.27/0.47 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setminusIRneg_type,type,
% 0.27/0.47 setminusIRneg: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setminusIRneg,definition,
% 0.27/0.47 ( setminusIRneg
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ B )
% 0.27/0.47 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setminusLsub_type,type,
% 0.27/0.47 setminusLsub: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setminusLsub,definition,
% 0.27/0.47 ( setminusLsub
% 0.27/0.47 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setminusSubset1_type,type,
% 0.27/0.47 setminusSubset1: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setminusSubset1,definition,
% 0.27/0.47 ( setminusSubset1
% 0.27/0.47 = ( ! [A: $i,B: $i] :
% 0.27/0.47 ( ( ( setminus @ A @ B )
% 0.27/0.47 = emptyset )
% 0.27/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(symdiff_type,type,
% 0.27/0.47 symdiff: $i > $i > $i ).
% 0.27/0.47
% 0.27/0.47 thf(symdiffE_type,type,
% 0.27/0.47 symdiffE: $o ).
% 0.27/0.47
% 0.27/0.47 thf(symdiffE,definition,
% 0.27/0.47 ( symdiffE
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 0.27/0.47 => ! [Xphi: $o] :
% 0.27/0.47 ( ( ( in @ Xx @ A )
% 0.27/0.47 => ( ~ ( in @ Xx @ B )
% 0.27/0.47 => Xphi ) )
% 0.27/0.47 => ( ( ~ ( in @ Xx @ A )
% 0.27/0.47 => ( ( in @ Xx @ B )
% 0.27/0.47 => Xphi ) )
% 0.27/0.47 => Xphi ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(symdiffI1_type,type,
% 0.27/0.47 symdiffI1: $o ).
% 0.27/0.47
% 0.27/0.47 thf(symdiffI1,definition,
% 0.27/0.47 ( symdiffI1
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( ~ ( in @ Xx @ B )
% 0.27/0.47 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(symdiffI2_type,type,
% 0.27/0.47 symdiffI2: $o ).
% 0.27/0.47
% 0.27/0.47 thf(symdiffI2,definition,
% 0.27/0.47 ( symdiffI2
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ~ ( in @ Xx @ A )
% 0.27/0.47 => ( ( in @ Xx @ B )
% 0.27/0.47 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(symdiffIneg1_type,type,
% 0.27/0.47 symdiffIneg1: $o ).
% 0.27/0.47
% 0.27/0.47 thf(symdiffIneg1,definition,
% 0.27/0.47 ( symdiffIneg1
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( ( in @ Xx @ B )
% 0.27/0.47 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(symdiffIneg2_type,type,
% 0.27/0.47 symdiffIneg2: $o ).
% 0.27/0.47
% 0.27/0.47 thf(symdiffIneg2,definition,
% 0.27/0.47 ( symdiffIneg2
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ~ ( in @ Xx @ A )
% 0.27/0.47 => ( ~ ( in @ Xx @ B )
% 0.27/0.47 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(iskpair_type,type,
% 0.27/0.47 iskpair: $i > $o ).
% 0.27/0.47
% 0.27/0.47 thf(secondinupair_type,type,
% 0.27/0.47 secondinupair: $o ).
% 0.27/0.47
% 0.27/0.47 thf(secondinupair,definition,
% 0.27/0.47 ( secondinupair
% 0.27/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setukpairIL_type,type,
% 0.27/0.47 setukpairIL: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setukpairIL,definition,
% 0.27/0.47 ( setukpairIL
% 0.27/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setukpairIR_type,type,
% 0.27/0.47 setukpairIR: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setukpairIR,definition,
% 0.27/0.47 ( setukpairIR
% 0.27/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(kpairiskpair_type,type,
% 0.27/0.47 kpairiskpair: $o ).
% 0.27/0.47
% 0.27/0.47 thf(kpairiskpair,definition,
% 0.27/0.47 ( kpairiskpair
% 0.27/0.47 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(kpair_type,type,
% 0.27/0.47 kpair: $i > $i > $i ).
% 0.27/0.47
% 0.27/0.47 thf(kpairp_type,type,
% 0.27/0.47 kpairp: $o ).
% 0.27/0.47
% 0.27/0.47 thf(kpairp,definition,
% 0.27/0.47 ( kpairp
% 0.27/0.47 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(cartprod_type,type,
% 0.27/0.47 cartprod: $i > $i > $i ).
% 0.27/0.47
% 0.27/0.47 thf(singletonsubset_type,type,
% 0.27/0.47 singletonsubset: $o ).
% 0.27/0.47
% 0.27/0.47 thf(singletonsubset,definition,
% 0.27/0.47 ( singletonsubset
% 0.27/0.47 = ( ! [A: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(singletoninpowerset_type,type,
% 0.27/0.47 singletoninpowerset: $o ).
% 0.27/0.47
% 0.27/0.47 thf(singletoninpowerset,definition,
% 0.27/0.47 ( singletoninpowerset
% 0.27/0.47 = ( ! [A: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(singletoninpowunion_type,type,
% 0.27/0.47 singletoninpowunion: $o ).
% 0.27/0.47
% 0.27/0.47 thf(singletoninpowunion,definition,
% 0.27/0.47 ( singletoninpowunion
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(upairset2E_type,type,
% 0.27/0.47 upairset2E: $o ).
% 0.27/0.47
% 0.27/0.47 thf(upairset2E,definition,
% 0.27/0.47 ( upairset2E
% 0.27/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.27/0.47 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.27/0.47 => ( ( Xz = Xx )
% 0.27/0.47 | ( Xz = Xy ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(upairsubunion_type,type,
% 0.27/0.47 upairsubunion: $o ).
% 0.27/0.47
% 0.27/0.47 thf(upairsubunion,definition,
% 0.27/0.47 ( upairsubunion
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ! [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ B )
% 0.27/0.47 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(upairinpowunion_type,type,
% 0.27/0.47 upairinpowunion: $o ).
% 0.27/0.47
% 0.27/0.47 thf(upairinpowunion,definition,
% 0.27/0.47 ( upairinpowunion
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ! [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ B )
% 0.27/0.47 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(ubforcartprodlem1_type,type,
% 0.27/0.47 ubforcartprodlem1: $o ).
% 0.27/0.47
% 0.27/0.47 thf(ubforcartprodlem1,definition,
% 0.27/0.47 ( ubforcartprodlem1
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ! [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ B )
% 0.27/0.47 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(ubforcartprodlem2_type,type,
% 0.27/0.47 ubforcartprodlem2: $o ).
% 0.27/0.47
% 0.27/0.47 thf(ubforcartprodlem2,definition,
% 0.27/0.47 ( ubforcartprodlem2
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ! [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ B )
% 0.27/0.47 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(ubforcartprodlem3_type,type,
% 0.27/0.47 ubforcartprodlem3: $o ).
% 0.27/0.47
% 0.27/0.47 thf(ubforcartprodlem3,definition,
% 0.27/0.47 ( ubforcartprodlem3
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ! [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ B )
% 0.27/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(cartprodpairin_type,type,
% 0.27/0.47 cartprodpairin: $o ).
% 0.27/0.47
% 0.27/0.47 thf(cartprodpairin,definition,
% 0.27/0.47 ( cartprodpairin
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ! [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ B )
% 0.27/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(cartprodmempair1_type,type,
% 0.27/0.47 cartprodmempair1: $o ).
% 0.27/0.47
% 0.27/0.47 thf(cartprodmempair1,definition,
% 0.27/0.47 ( cartprodmempair1
% 0.27/0.47 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.27/0.47 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.27/0.47 => ? [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 & ? [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ B )
% 0.27/0.47 & ( Xu
% 0.27/0.47 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(cartprodmempair_type,type,
% 0.27/0.47 cartprodmempair: $o ).
% 0.27/0.47
% 0.27/0.47 thf(cartprodmempair,definition,
% 0.27/0.47 ( cartprodmempair
% 0.27/0.47 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.27/0.47 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.27/0.47 => ( iskpair @ Xu ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setunionE2_type,type,
% 0.27/0.47 setunionE2: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setunionE2,definition,
% 0.27/0.47 ( setunionE2
% 0.27/0.47 = ( ! [A: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ ( setunion @ A ) )
% 0.27/0.47 => ? [X: $i] :
% 0.27/0.47 ( ( in @ X @ A )
% 0.27/0.47 & ( in @ Xx @ X ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setunionsingleton1_type,type,
% 0.27/0.47 setunionsingleton1: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setunionsingleton1,definition,
% 0.27/0.47 ( setunionsingleton1
% 0.27/0.47 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setunionsingleton2_type,type,
% 0.27/0.47 setunionsingleton2: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setunionsingleton2,definition,
% 0.27/0.47 ( setunionsingleton2
% 0.27/0.47 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setunionsingleton_type,type,
% 0.27/0.47 setunionsingleton: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setunionsingleton,definition,
% 0.27/0.47 ( setunionsingleton
% 0.27/0.47 = ( ! [Xx: $i] :
% 0.27/0.47 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 0.27/0.47 = Xx ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(singleton_type,type,
% 0.27/0.47 singleton: $i > $o ).
% 0.27/0.47
% 0.27/0.47 thf(singleton,definition,
% 0.27/0.47 ( singleton
% 0.27/0.47 = ( ^ [A: $i] :
% 0.27/0.47 ? [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 & ( A
% 0.27/0.47 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(singletonprop_type,type,
% 0.27/0.47 singletonprop: $o ).
% 0.27/0.47
% 0.27/0.47 thf(singletonprop,definition,
% 0.27/0.47 ( singletonprop
% 0.27/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.27/0.47 ( ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ! [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ A )
% 0.27/0.47 => ( ( Xphi @ Xx )
% 0.27/0.47 => ( ( Xphi @ Xy )
% 0.27/0.47 => ( Xx = Xy ) ) ) ) )
% 0.27/0.47 => ( ? [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 & ( Xphi @ Xx ) )
% 0.27/0.47 => ( singleton
% 0.27/0.47 @ ( dsetconstr @ A
% 0.27/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(ex1_type,type,
% 0.27/0.47 ex1: $i > ( $i > $o ) > $o ).
% 0.27/0.47
% 0.27/0.47 thf(ex1,definition,
% 0.27/0.47 ( ex1
% 0.27/0.47 = ( ^ [A: $i,Xphi: $i > $o] :
% 0.27/0.47 ( singleton
% 0.27/0.47 @ ( dsetconstr @ A
% 0.27/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(ex1E1_type,type,
% 0.27/0.47 ex1E1: $o ).
% 0.27/0.47
% 0.27/0.47 thf(ex1E1,definition,
% 0.27/0.47 ( ex1E1
% 0.27/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.27/0.47 ( ( ex1 @ A
% 0.27/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.27/0.47 => ? [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(ex1I_type,type,
% 0.27/0.47 ex1I: $o ).
% 0.27/0.47
% 0.27/0.47 thf(ex1I,definition,
% 0.27/0.47 ( ex1I
% 0.27/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( ( Xphi @ Xx )
% 0.27/0.47 => ( ! [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ A )
% 0.27/0.47 => ( ( Xphi @ Xy )
% 0.27/0.47 => ( Xy = Xx ) ) )
% 0.27/0.47 => ( ex1 @ A
% 0.27/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(ex1I2_type,type,
% 0.27/0.47 ex1I2: $o ).
% 0.27/0.47
% 0.27/0.47 thf(ex1I2,definition,
% 0.27/0.47 ( ex1I2
% 0.27/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.27/0.47 ( ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ! [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ A )
% 0.27/0.47 => ( ( Xphi @ Xx )
% 0.27/0.47 => ( ( Xphi @ Xy )
% 0.27/0.47 => ( Xx = Xy ) ) ) ) )
% 0.27/0.47 => ( ? [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 & ( Xphi @ Xx ) )
% 0.27/0.47 => ( ex1 @ A
% 0.27/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(singletonsuniq_type,type,
% 0.27/0.47 singletonsuniq: $o ).
% 0.27/0.47
% 0.27/0.47 thf(singletonsuniq,definition,
% 0.27/0.47 ( singletonsuniq
% 0.27/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.27/0.47 ( ( ( setadjoin @ Xx @ emptyset )
% 0.27/0.47 = ( setadjoin @ Xy @ emptyset ) )
% 0.27/0.47 => ( Xx = Xy ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(atmost1p_type,type,
% 0.27/0.47 atmost1p: $i > $o ).
% 0.27/0.47
% 0.27/0.47 thf(atleast2p_type,type,
% 0.27/0.47 atleast2p: $i > $o ).
% 0.27/0.47
% 0.27/0.47 thf(atmost2p_type,type,
% 0.27/0.47 atmost2p: $i > $o ).
% 0.27/0.47
% 0.27/0.47 thf(upairsetp_type,type,
% 0.27/0.47 upairsetp: $i > $o ).
% 0.27/0.47
% 0.27/0.47 thf(setukpairinjL1_type,type,
% 0.27/0.47 setukpairinjL1: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setukpairinjL1,definition,
% 0.27/0.47 ( setukpairinjL1
% 0.27/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.27/0.47 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 0.27/0.47 => ( Xx = Xz ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(kfstsingleton_type,type,
% 0.27/0.47 kfstsingleton: $o ).
% 0.27/0.47
% 0.27/0.47 thf(kfstsingleton,definition,
% 0.27/0.47 ( kfstsingleton
% 0.27/0.47 = ( ! [Xu: $i] :
% 0.27/0.47 ( ( iskpair @ Xu )
% 0.27/0.47 => ( singleton
% 0.27/0.47 @ ( dsetconstr @ ( setunion @ Xu )
% 0.27/0.47 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(theprop_type,type,
% 0.27/0.47 theprop: $o ).
% 0.27/0.47
% 0.27/0.47 thf(theprop,definition,
% 0.27/0.47 ( theprop
% 0.27/0.47 = ( ! [X: $i] :
% 0.27/0.47 ( ( singleton @ X )
% 0.27/0.47 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(kfst_type,type,
% 0.27/0.47 kfst: $i > $i ).
% 0.27/0.47
% 0.27/0.47 thf(kfstpairEq_type,type,
% 0.27/0.47 kfstpairEq: $o ).
% 0.27/0.47
% 0.27/0.47 thf(kfstpairEq,definition,
% 0.27/0.47 ( kfstpairEq
% 0.27/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.27/0.47 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.27/0.47 = Xx ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(cartprodfstin_type,type,
% 0.27/0.47 cartprodfstin: $o ).
% 0.27/0.47
% 0.27/0.47 thf(cartprodfstin,definition,
% 0.27/0.47 ( cartprodfstin
% 0.27/0.47 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.27/0.47 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.27/0.47 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setukpairinjL2_type,type,
% 0.27/0.47 setukpairinjL2: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setukpairinjL2,definition,
% 0.27/0.47 ( setukpairinjL2
% 0.27/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.27/0.47 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.27/0.47 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.27/0.47 => ( Xx = Xz ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setukpairinjL_type,type,
% 0.27/0.47 setukpairinjL: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setukpairinjL,definition,
% 0.27/0.47 ( setukpairinjL
% 0.27/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.27/0.47 ( ( ( kpair @ Xx @ Xy )
% 0.27/0.47 = ( kpair @ Xz @ Xu ) )
% 0.27/0.47 => ( Xx = Xz ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setukpairinjR11_type,type,
% 0.27/0.47 setukpairinjR11: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setukpairinjR11,definition,
% 0.27/0.47 ( setukpairinjR11
% 0.27/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.27/0.47 ( ( Xx = Xy )
% 0.27/0.47 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.27/0.47 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setukpairinjR12_type,type,
% 0.27/0.47 setukpairinjR12: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setukpairinjR12,definition,
% 0.27/0.47 ( setukpairinjR12
% 0.27/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.27/0.47 ( ( Xx = Xy )
% 0.27/0.47 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.27/0.47 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setukpairinjR1_type,type,
% 0.27/0.47 setukpairinjR1: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setukpairinjR1,definition,
% 0.27/0.47 ( setukpairinjR1
% 0.27/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.27/0.47 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.27/0.47 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.27/0.47 => ( ( Xz = Xu )
% 0.27/0.47 => ( Xy = Xu ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(upairequniteq_type,type,
% 0.27/0.47 upairequniteq: $o ).
% 0.27/0.47
% 0.27/0.47 thf(upairequniteq,definition,
% 0.27/0.47 ( upairequniteq
% 0.27/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.27/0.47 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.27/0.47 = ( setadjoin @ Xz @ emptyset ) )
% 0.27/0.47 => ( Xx = Xy ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setukpairinjR2_type,type,
% 0.27/0.47 setukpairinjR2: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setukpairinjR2,definition,
% 0.27/0.47 ( setukpairinjR2
% 0.27/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.27/0.47 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.27/0.47 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.27/0.47 => ( Xy = Xu ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setukpairinjR_type,type,
% 0.27/0.47 setukpairinjR: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setukpairinjR,definition,
% 0.27/0.47 ( setukpairinjR
% 0.27/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.27/0.47 ( ( ( kpair @ Xx @ Xy )
% 0.27/0.47 = ( kpair @ Xz @ Xu ) )
% 0.27/0.47 => ( Xy = Xu ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(ksndsingleton_type,type,
% 0.27/0.47 ksndsingleton: $o ).
% 0.27/0.47
% 0.27/0.47 thf(ksndsingleton,definition,
% 0.27/0.47 ( ksndsingleton
% 0.27/0.47 = ( ! [Xu: $i] :
% 0.27/0.47 ( ( iskpair @ Xu )
% 0.27/0.47 => ( singleton
% 0.27/0.47 @ ( dsetconstr @ ( setunion @ Xu )
% 0.27/0.47 @ ^ [Xx: $i] :
% 0.27/0.47 ( Xu
% 0.27/0.47 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(ksnd_type,type,
% 0.27/0.47 ksnd: $i > $i ).
% 0.27/0.47
% 0.27/0.47 thf(ksndpairEq_type,type,
% 0.27/0.47 ksndpairEq: $o ).
% 0.27/0.47
% 0.27/0.47 thf(ksndpairEq,definition,
% 0.27/0.47 ( ksndpairEq
% 0.27/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.27/0.47 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.27/0.47 = Xy ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(kpairsurjEq_type,type,
% 0.27/0.47 kpairsurjEq: $o ).
% 0.27/0.47
% 0.27/0.47 thf(kpairsurjEq,definition,
% 0.27/0.47 ( kpairsurjEq
% 0.27/0.47 = ( ! [Xu: $i] :
% 0.27/0.47 ( ( iskpair @ Xu )
% 0.27/0.47 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.27/0.47 = Xu ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(cartprodsndin_type,type,
% 0.27/0.47 cartprodsndin: $o ).
% 0.27/0.47
% 0.27/0.47 thf(cartprodsndin,definition,
% 0.27/0.47 ( cartprodsndin
% 0.27/0.47 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.27/0.47 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.27/0.47 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(cartprodpairmemEL_type,type,
% 0.27/0.47 cartprodpairmemEL: $o ).
% 0.27/0.47
% 0.27/0.47 thf(cartprodpairmemEL,definition,
% 0.27/0.47 ( cartprodpairmemEL
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.27/0.47 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.27/0.47 => ( in @ Xx @ A ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(cartprodpairmemER_type,type,
% 0.27/0.47 cartprodpairmemER: $o ).
% 0.27/0.47
% 0.27/0.47 thf(cartprodpairmemER,definition,
% 0.27/0.47 ( cartprodpairmemER
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.27/0.47 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.27/0.47 => ( in @ Xy @ B ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(cartprodmempaircEq_type,type,
% 0.27/0.47 cartprodmempaircEq: $o ).
% 0.27/0.47
% 0.27/0.47 thf(cartprodmempaircEq,definition,
% 0.27/0.47 ( cartprodmempaircEq
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ! [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ B )
% 0.27/0.47 => ( ( kpair @ Xx @ Xy )
% 0.27/0.47 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(cartprodfstpairEq_type,type,
% 0.27/0.47 cartprodfstpairEq: $o ).
% 0.27/0.47
% 0.27/0.47 thf(cartprodfstpairEq,definition,
% 0.27/0.47 ( cartprodfstpairEq
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ! [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ B )
% 0.27/0.47 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.27/0.47 = Xx ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(cartprodsndpairEq_type,type,
% 0.27/0.47 cartprodsndpairEq: $o ).
% 0.27/0.47
% 0.27/0.47 thf(cartprodsndpairEq,definition,
% 0.27/0.47 ( cartprodsndpairEq
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ! [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ B )
% 0.27/0.47 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.27/0.47 = Xy ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(cartprodpairsurjEq_type,type,
% 0.27/0.47 cartprodpairsurjEq: $o ).
% 0.27/0.47
% 0.27/0.47 thf(cartprodpairsurjEq,definition,
% 0.27/0.47 ( cartprodpairsurjEq
% 0.27/0.47 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.27/0.47 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.27/0.47 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.27/0.47 = Xu ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(breln_type,type,
% 0.27/0.47 breln: $i > $i > $i > $o ).
% 0.27/0.47
% 0.27/0.47 thf(breln,definition,
% 0.27/0.47 ( breln
% 0.27/0.47 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(dpsetconstr_type,type,
% 0.27/0.47 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 0.27/0.47
% 0.27/0.47 thf(dpsetconstrI_type,type,
% 0.27/0.47 dpsetconstrI: $o ).
% 0.27/0.47
% 0.27/0.47 thf(dpsetconstrI,definition,
% 0.27/0.47 ( dpsetconstrI
% 0.27/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ! [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ B )
% 0.27/0.47 => ( ( Xphi @ Xx @ Xy )
% 0.27/0.47 => ( in @ ( kpair @ Xx @ Xy )
% 0.27/0.47 @ ( dpsetconstr @ A @ B
% 0.27/0.47 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(dpsetconstrSub_type,type,
% 0.27/0.47 dpsetconstrSub: $o ).
% 0.27/0.47
% 0.27/0.47 thf(dpsetconstrSub,definition,
% 0.27/0.47 ( dpsetconstrSub
% 0.27/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.27/0.47 ( subset
% 0.27/0.47 @ ( dpsetconstr @ A @ B
% 0.27/0.47 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 0.27/0.47 @ ( cartprod @ A @ B ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setOfPairsIsBReln_type,type,
% 0.27/0.47 setOfPairsIsBReln: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setOfPairsIsBReln,definition,
% 0.27/0.47 ( setOfPairsIsBReln
% 0.27/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.27/0.47 ( breln @ A @ B
% 0.27/0.47 @ ( dpsetconstr @ A @ B
% 0.27/0.47 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(dpsetconstrERa_type,type,
% 0.27/0.47 dpsetconstrERa: $o ).
% 0.27/0.47
% 0.27/0.47 thf(dpsetconstrERa,definition,
% 0.27/0.47 ( dpsetconstrERa
% 0.27/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ! [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ B )
% 0.27/0.47 => ( ( in @ ( kpair @ Xx @ Xy )
% 0.27/0.47 @ ( dpsetconstr @ A @ B
% 0.27/0.47 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.27/0.47 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(dpsetconstrEL1_type,type,
% 0.27/0.47 dpsetconstrEL1: $o ).
% 0.27/0.47
% 0.27/0.47 thf(dpsetconstrEL1,definition,
% 0.27/0.47 ( dpsetconstrEL1
% 0.27/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.27/0.47 ( ( in @ ( kpair @ Xx @ Xy )
% 0.27/0.47 @ ( dpsetconstr @ A @ B
% 0.27/0.47 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.27/0.47 => ( in @ Xx @ A ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(dpsetconstrEL2_type,type,
% 0.27/0.47 dpsetconstrEL2: $o ).
% 0.27/0.47
% 0.27/0.47 thf(dpsetconstrEL2,definition,
% 0.27/0.47 ( dpsetconstrEL2
% 0.27/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.27/0.47 ( ( in @ ( kpair @ Xx @ Xy )
% 0.27/0.47 @ ( dpsetconstr @ A @ B
% 0.27/0.47 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.27/0.47 => ( in @ Xy @ B ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(dpsetconstrER_type,type,
% 0.27/0.47 dpsetconstrER: $o ).
% 0.27/0.47
% 0.27/0.47 thf(dpsetconstrER,definition,
% 0.27/0.47 ( dpsetconstrER
% 0.27/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.27/0.47 ( ( in @ ( kpair @ Xx @ Xy )
% 0.27/0.47 @ ( dpsetconstr @ A @ B
% 0.27/0.47 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.27/0.47 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(func_type,type,
% 0.27/0.47 func: $i > $i > $i > $o ).
% 0.27/0.47
% 0.27/0.47 thf(func,definition,
% 0.27/0.47 ( func
% 0.27/0.47 = ( ^ [A: $i,B: $i,R: $i] :
% 0.27/0.47 ( ( breln @ A @ B @ R )
% 0.27/0.47 & ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( ex1 @ B
% 0.27/0.47 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(funcSet_type,type,
% 0.27/0.47 funcSet: $i > $i > $i ).
% 0.27/0.47
% 0.27/0.47 thf(funcImageSingleton_type,type,
% 0.27/0.47 funcImageSingleton: $o ).
% 0.27/0.47
% 0.27/0.47 thf(funcImageSingleton,definition,
% 0.27/0.47 ( funcImageSingleton
% 0.27/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.27/0.47 ( ( func @ A @ B @ Xf )
% 0.27/0.47 => ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( singleton
% 0.27/0.47 @ ( dsetconstr @ B
% 0.27/0.47 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(apProp_type,type,
% 0.27/0.47 apProp: $o ).
% 0.27/0.47
% 0.27/0.47 thf(apProp,definition,
% 0.27/0.47 ( apProp
% 0.27/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.27/0.47 ( ( func @ A @ B @ Xf )
% 0.27/0.47 => ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( in
% 0.27/0.47 @ ( setunion
% 0.27/0.47 @ ( dsetconstr @ B
% 0.27/0.47 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 0.27/0.47 @ B ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(ap_type,type,
% 0.27/0.47 ap: $i > $i > $i > $i > $i ).
% 0.27/0.47
% 0.27/0.47 thf(app_type,type,
% 0.27/0.47 app: $o ).
% 0.27/0.47
% 0.27/0.47 thf(app,definition,
% 0.27/0.47 ( app
% 0.27/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.27/0.47 ( ( func @ A @ B @ Xf )
% 0.27/0.47 => ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(infuncsetfunc_type,type,
% 0.27/0.47 infuncsetfunc: $o ).
% 0.27/0.47
% 0.27/0.47 thf(infuncsetfunc,definition,
% 0.27/0.47 ( infuncsetfunc
% 0.27/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.27/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.27/0.47 => ( func @ A @ B @ Xf ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(ap2p_type,type,
% 0.27/0.47 ap2p: $o ).
% 0.27/0.47
% 0.27/0.47 thf(ap2p,definition,
% 0.27/0.47 ( ap2p
% 0.27/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.27/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.27/0.47 => ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(funcinfuncset_type,type,
% 0.27/0.47 funcinfuncset: $o ).
% 0.27/0.47
% 0.27/0.47 thf(funcinfuncset,definition,
% 0.27/0.47 ( funcinfuncset
% 0.27/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.27/0.47 ( ( func @ A @ B @ Xf )
% 0.27/0.47 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(lamProp_type,type,
% 0.27/0.47 lamProp: $o ).
% 0.27/0.47
% 0.27/0.47 thf(lamProp,definition,
% 0.27/0.47 ( lamProp
% 0.27/0.47 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.27/0.47 ( ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.27/0.47 => ( func @ A @ B
% 0.27/0.47 @ ( dpsetconstr @ A @ B
% 0.27/0.47 @ ^ [Xx: $i,Xy: $i] :
% 0.27/0.47 ( ( Xf @ Xx )
% 0.27/0.47 = Xy ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(lam_type,type,
% 0.27/0.47 lam: $i > $i > ( $i > $i ) > $i ).
% 0.27/0.47
% 0.27/0.47 thf(lam,definition,
% 0.27/0.47 ( lam
% 0.27/0.47 = ( ^ [A: $i,B: $i,Xf: $i > $i] :
% 0.27/0.47 ( dpsetconstr @ A @ B
% 0.27/0.47 @ ^ [Xx: $i,Xy: $i] :
% 0.27/0.47 ( ( Xf @ Xx )
% 0.27/0.47 = Xy ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(lamp_type,type,
% 0.27/0.47 lamp: $o ).
% 0.27/0.47
% 0.27/0.47 thf(lamp,definition,
% 0.27/0.47 ( lamp
% 0.27/0.47 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.27/0.47 ( ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.27/0.47 => ( func @ A @ B
% 0.27/0.47 @ ( lam @ A @ B
% 0.27/0.47 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(lam2p_type,type,
% 0.27/0.47 lam2p: $o ).
% 0.27/0.47
% 0.27/0.47 thf(lam2p,definition,
% 0.27/0.47 ( lam2p
% 0.27/0.47 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.27/0.47 ( ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.27/0.47 => ( in
% 0.27/0.47 @ ( lam @ A @ B
% 0.27/0.47 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.27/0.47 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(brelnall1_type,type,
% 0.27/0.47 brelnall1: $o ).
% 0.27/0.47
% 0.27/0.47 thf(brelnall1,definition,
% 0.27/0.47 ( brelnall1
% 0.27/0.47 = ( ! [A: $i,B: $i,R: $i] :
% 0.27/0.47 ( ( breln @ A @ B @ R )
% 0.27/0.47 => ! [Xphi: $i > $o] :
% 0.27/0.47 ( ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ! [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ B )
% 0.27/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.27/0.47 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.27/0.47 => ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ R )
% 0.27/0.47 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(brelnall2_type,type,
% 0.27/0.47 brelnall2: $o ).
% 0.27/0.47
% 0.27/0.47 thf(brelnall2,definition,
% 0.27/0.47 ( brelnall2
% 0.27/0.47 = ( ! [A: $i,B: $i,R: $i] :
% 0.27/0.47 ( ( breln @ A @ B @ R )
% 0.27/0.47 => ! [Xphi: $i > $o] :
% 0.27/0.47 ( ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ! [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ B )
% 0.27/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.27/0.47 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.27/0.47 => ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ R )
% 0.27/0.47 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(ex1E2_type,type,
% 0.27/0.47 ex1E2: $o ).
% 0.27/0.47
% 0.27/0.47 thf(ex1E2,definition,
% 0.27/0.47 ( ex1E2
% 0.27/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.27/0.47 ( ( ex1 @ A
% 0.27/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.27/0.47 => ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ! [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ A )
% 0.27/0.47 => ( ( Xphi @ Xx )
% 0.27/0.47 => ( ( Xphi @ Xy )
% 0.27/0.47 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(funcGraphProp1_type,type,
% 0.27/0.47 funcGraphProp1: $o ).
% 0.27/0.47
% 0.27/0.47 thf(funcGraphProp1,definition,
% 0.27/0.47 ( funcGraphProp1
% 0.27/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.27/0.47 ( ( func @ A @ B @ Xf )
% 0.27/0.47 => ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(funcGraphProp3_type,type,
% 0.27/0.47 funcGraphProp3: $o ).
% 0.27/0.47
% 0.27/0.47 thf(funcGraphProp3,definition,
% 0.27/0.47 ( funcGraphProp3
% 0.27/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.27/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.27/0.47 => ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(funcGraphProp2_type,type,
% 0.27/0.47 funcGraphProp2: $o ).
% 0.27/0.47
% 0.27/0.47 thf(funcGraphProp2,definition,
% 0.27/0.47 ( funcGraphProp2
% 0.27/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.27/0.47 ( ( func @ A @ B @ Xf )
% 0.27/0.47 => ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ! [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ B )
% 0.27/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.27/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.27/0.47 = Xy ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(funcextLem_type,type,
% 0.27/0.47 funcextLem: $o ).
% 0.27/0.47
% 0.27/0.47 thf(funcextLem,definition,
% 0.27/0.47 ( funcextLem
% 0.27/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.27/0.47 ( ( func @ A @ B @ Xf )
% 0.27/0.47 => ! [Xg: $i] :
% 0.27/0.47 ( ( func @ A @ B @ Xg )
% 0.27/0.47 => ( ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.27/0.47 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.27/0.47 => ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ! [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ B )
% 0.27/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 0.27/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(funcGraphProp4,conjecture,
% 0.27/0.47 ( setextAx
% 0.27/0.47 => ( emptysetAx
% 0.27/0.47 => ( setadjoinAx
% 0.27/0.47 => ( powersetAx
% 0.27/0.47 => ( setunionAx
% 0.27/0.47 => ( omega0Ax
% 0.27/0.47 => ( omegaSAx
% 0.27/0.47 => ( omegaIndAx
% 0.27/0.47 => ( replAx
% 0.27/0.47 => ( foundationAx
% 0.27/0.47 => ( wellorderingAx
% 0.27/0.47 => ( descrp
% 0.27/0.47 => ( dsetconstrI
% 0.27/0.47 => ( dsetconstrEL
% 0.27/0.47 => ( dsetconstrER
% 0.27/0.47 => ( exuE1
% 0.27/0.47 => ( prop2setE
% 0.27/0.47 => ( emptysetE
% 0.27/0.47 => ( emptysetimpfalse
% 0.27/0.47 => ( notinemptyset
% 0.27/0.47 => ( exuE3e
% 0.27/0.47 => ( setext
% 0.27/0.47 => ( emptyI
% 0.27/0.47 => ( noeltsimpempty
% 0.27/0.47 => ( setbeta
% 0.27/0.47 => ( nonemptyE1
% 0.27/0.47 => ( nonemptyI
% 0.27/0.47 => ( nonemptyI1
% 0.27/0.47 => ( setadjoinIL
% 0.27/0.47 => ( emptyinunitempty
% 0.27/0.47 => ( setadjoinIR
% 0.27/0.47 => ( setadjoinE
% 0.27/0.47 => ( setadjoinOr
% 0.27/0.47 => ( setoftrueEq
% 0.27/0.47 => ( powersetI
% 0.27/0.47 => ( emptyinPowerset
% 0.27/0.47 => ( emptyInPowerset
% 0.27/0.47 => ( powersetE
% 0.27/0.47 => ( setunionI
% 0.27/0.47 => ( setunionE
% 0.27/0.47 => ( subPowSU
% 0.27/0.47 => ( exuE2
% 0.27/0.47 => ( nonemptyImpWitness
% 0.27/0.47 => ( uniqinunit
% 0.27/0.47 => ( notinsingleton
% 0.27/0.47 => ( eqinunit
% 0.27/0.47 => ( singletonsswitch
% 0.27/0.47 => ( upairsetE
% 0.27/0.47 => ( upairsetIL
% 0.27/0.47 => ( upairsetIR
% 0.27/0.47 => ( emptyE1
% 0.27/0.47 => ( vacuousDall
% 0.27/0.47 => ( quantDeMorgan1
% 0.27/0.47 => ( quantDeMorgan2
% 0.27/0.47 => ( quantDeMorgan3
% 0.27/0.47 => ( quantDeMorgan4
% 0.27/0.47 => ( prop2setI
% 0.27/0.47 => ( prop2set2propI
% 0.27/0.47 => ( notdexE
% 0.27/0.47 => ( notdallE
% 0.27/0.47 => ( exuI1
% 0.27/0.47 => ( exuI3
% 0.27/0.47 => ( exuI2
% 0.27/0.47 => ( inCongP
% 0.27/0.47 => ( in__Cong
% 0.27/0.47 => ( exuE3u
% 0.27/0.47 => ( exu__Cong
% 0.27/0.47 => ( emptyset__Cong
% 0.27/0.47 => ( setadjoin__Cong
% 0.27/0.47 => ( powerset__Cong
% 0.27/0.47 => ( setunion__Cong
% 0.27/0.47 => ( omega__Cong
% 0.27/0.47 => ( exuEu
% 0.27/0.47 => ( descr__Cong
% 0.27/0.47 => ( dsetconstr__Cong
% 0.27/0.47 => ( subsetI1
% 0.27/0.47 => ( eqimpsubset2
% 0.27/0.47 => ( eqimpsubset1
% 0.27/0.47 => ( subsetI2
% 0.27/0.47 => ( emptysetsubset
% 0.27/0.47 => ( subsetE
% 0.27/0.47 => ( subsetE2
% 0.27/0.47 => ( notsubsetI
% 0.27/0.47 => ( notequalI1
% 0.27/0.47 => ( notequalI2
% 0.27/0.47 => ( subsetRefl
% 0.27/0.47 => ( subsetTrans
% 0.27/0.47 => ( setadjoinSub
% 0.27/0.47 => ( setadjoinSub2
% 0.27/0.47 => ( subset2powerset
% 0.27/0.47 => ( setextsub
% 0.27/0.47 => ( subsetemptysetimpeq
% 0.27/0.47 => ( powersetI1
% 0.27/0.47 => ( powersetE1
% 0.27/0.47 => ( inPowerset
% 0.27/0.47 => ( powersetsubset
% 0.27/0.47 => ( sepInPowerset
% 0.27/0.47 => ( sepSubset
% 0.27/0.47 => ( binunionIL
% 0.27/0.47 => ( upairset2IR
% 0.27/0.47 => ( binunionIR
% 0.27/0.47 => ( binunionEcases
% 0.27/0.47 => ( binunionE
% 0.27/0.47 => ( binunionLsub
% 0.27/0.47 => ( binunionRsub
% 0.27/0.47 => ( binintersectI
% 0.27/0.47 => ( binintersectSubset5
% 0.27/0.47 => ( binintersectEL
% 0.27/0.47 => ( binintersectLsub
% 0.27/0.47 => ( binintersectSubset2
% 0.27/0.47 => ( binintersectSubset3
% 0.27/0.47 => ( binintersectER
% 0.27/0.47 => ( disjointsetsI1
% 0.27/0.47 => ( binintersectRsub
% 0.27/0.47 => ( binintersectSubset4
% 0.27/0.47 => ( binintersectSubset1
% 0.27/0.47 => ( bs114d
% 0.27/0.47 => ( setminusI
% 0.27/0.47 => ( setminusEL
% 0.27/0.47 => ( setminusER
% 0.27/0.47 => ( setminusSubset2
% 0.27/0.47 => ( setminusERneg
% 0.27/0.47 => ( setminusELneg
% 0.27/0.47 => ( setminusILneg
% 0.27/0.47 => ( setminusIRneg
% 0.27/0.47 => ( setminusLsub
% 0.27/0.47 => ( setminusSubset1
% 0.27/0.47 => ( symdiffE
% 0.27/0.47 => ( symdiffI1
% 0.27/0.47 => ( symdiffI2
% 0.27/0.47 => ( symdiffIneg1
% 0.27/0.47 => ( symdiffIneg2
% 0.27/0.47 => ( secondinupair
% 0.27/0.47 => ( setukpairIL
% 0.27/0.47 => ( setukpairIR
% 0.27/0.47 => ( kpairiskpair
% 0.27/0.47 => ( kpairp
% 0.27/0.47 => ( singletonsubset
% 0.27/0.47 => ( singletoninpowerset
% 0.27/0.47 => ( singletoninpowunion
% 0.27/0.47 => ( upairset2E
% 0.27/0.47 => ( upairsubunion
% 0.27/0.47 => ( upairinpowunion
% 0.27/0.47 => ( ubforcartprodlem1
% 0.27/0.47 => ( ubforcartprodlem2
% 0.27/0.47 => ( ubforcartprodlem3
% 0.27/0.47 => ( cartprodpairin
% 0.27/0.47 => ( cartprodmempair1
% 0.27/0.47 => ( cartprodmempair
% 0.27/0.47 => ( setunionE2
% 0.27/0.47 => ( setunionsingleton1
% 0.27/0.47 => ( setunionsingleton2
% 0.27/0.47 => ( setunionsingleton
% 0.27/0.47 => ( singletonprop
% 0.27/0.47 => ( ex1E1
% 0.27/0.47 => ( ex1I
% 0.27/0.47 => ( ex1I2
% 0.27/0.47 => ( singletonsuniq
% 0.27/0.47 => ( setukpairinjL1
% 0.27/0.47 => ( kfstsingleton
% 0.27/0.47 => ( theprop
% 0.27/0.47 => ( kfstpairEq
% 0.27/0.47 => ( cartprodfstin
% 0.27/0.47 => ( setukpairinjL2
% 0.27/0.47 => ( setukpairinjL
% 0.27/0.47 => ( setukpairinjR11
% 0.27/0.47 => ( setukpairinjR12
% 0.27/0.47 => ( setukpairinjR1
% 0.27/0.47 => ( upairequniteq
% 0.27/0.47 => ( setukpairinjR2
% 0.27/0.47 => ( setukpairinjR
% 0.27/0.47 => ( ksndsingleton
% 0.27/0.47 => ( ksndpairEq
% 0.27/0.47 => ( kpairsurjEq
% 0.27/0.47 => ( cartprodsndin
% 0.27/0.47 => ( cartprodpairmemEL
% 0.27/0.47 => ( cartprodpairmemER
% 0.27/0.47 => ( cartprodmempaircEq
% 0.27/0.47 => ( cartprodfstpairEq
% 0.27/0.47 => ( cartprodsndpairEq
% 0.27/0.47 => ( cartprodpairsurjEq
% 0.27/0.47 => ( dpsetconstrI
% 0.27/0.47 => ( dpsetconstrSub
% 0.27/0.47 => ( setOfPairsIsBReln
% 0.27/0.47 => ( dpsetconstrERa
% 0.27/0.47 => ( dpsetconstrEL1
% 0.27/0.47 => ( dpsetconstrEL2
% 0.27/0.47 => ( dpsetconstrER
% 0.27/0.47 => ( funcImageSingleton
% 0.27/0.47 => ( apProp
% 0.27/0.47 => ( app
% 0.27/0.47 => ( infuncsetfunc
% 0.27/0.47 => ( ap2p
% 0.27/0.47 => ( funcinfuncset
% 0.27/0.47 => ( lamProp
% 0.27/0.47 => ( lamp
% 0.27/0.47 => ( lam2p
% 0.27/0.47 => ( brelnall1
% 0.27/0.47 => ( brelnall2
% 0.27/0.47 => ( ex1E2
% 0.27/0.47 => ( funcGraphProp1
% 0.27/0.47 => ( funcGraphProp3
% 0.27/0.47 => ( funcGraphProp2
% 0.27/0.47 => ( funcextLem
% 0.27/0.47 => ! [A: $i,B: $i,Xf: $i] :
% 0.27/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.27/0.47 => ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ! [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ B )
% 0.27/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.27/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.27/0.47 = Xy ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 %------------------------------------------------------------------------------
% 0.27/0.47 ------------------------
% 2.40/1.49 ---- Embedded in THF ---
% 2.40/1.50 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.40/1.50 %%% Generated on Tue Feb 24 12:31:13 EST 2026
% 2.40/1.50 %%% using '$$dhol' embedding, version 1.3.0.
% 2.40/1.50 %%% Logic specification used:
% 2.40/1.50 %%% thf(spec, logic, $$dhol).
% 2.40/1.50
% 2.40/1.50 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.CZb2EvKrRp/DTF2DTF1488.p
% See solution above
% 2.68/1.50 ------------------------
% 2.68/1.54 ---- Cleaned THF ---
% 2.68/1.54 thf(in_type,type,
% 2.68/1.54 in: $i > $i > $o ).
% 2.68/1.54
% 2.68/1.54 thf(exu_type,type,
% 2.68/1.54 exu: ( $i > $o ) > $o ).
% 2.68/1.54
% 2.68/1.54 thf(exu,definition,
% 2.68/1.54 ( exu
% 2.68/1.54 = ( ^ [Xphi: $i > $o] :
% 2.68/1.54 ? [Xx: $i] :
% 2.68/1.54 ( ( Xphi @ Xx )
% 2.68/1.54 & ! [Xy: $i] :
% 2.68/1.54 ( ( Xphi @ Xy )
% 2.68/1.54 => ( Xx = Xy ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(setextAx_type,type,
% 2.68/1.54 setextAx: $o ).
% 2.68/1.54
% 2.68/1.54 thf(setextAx,definition,
% 2.68/1.54 ( setextAx
% 2.68/1.54 = ( ! [A: $i,B: $i] :
% 2.68/1.54 ( ! [Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ A )
% 2.68/1.54 <=> ( in @ Xx @ B ) )
% 2.68/1.54 => ( A = B ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(emptyset_type,type,
% 2.68/1.54 emptyset: $i ).
% 2.68/1.54
% 2.68/1.54 thf(emptysetAx_type,type,
% 2.68/1.54 emptysetAx: $o ).
% 2.68/1.54
% 2.68/1.54 thf(emptysetAx,definition,
% 2.68/1.54 ( emptysetAx
% 2.68/1.54 = ( ! [Xx: $i] :
% 2.68/1.54 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(setadjoin_type,type,
% 2.68/1.54 setadjoin: $i > $i > $i ).
% 2.68/1.54
% 2.68/1.54 thf(setadjoinAx_type,type,
% 2.68/1.54 setadjoinAx: $o ).
% 2.68/1.54
% 2.68/1.54 thf(setadjoinAx,definition,
% 2.68/1.54 ( setadjoinAx
% 2.68/1.54 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.68/1.54 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.68/1.54 <=> ( ( Xy = Xx )
% 2.68/1.54 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(powerset_type,type,
% 2.68/1.54 powerset: $i > $i ).
% 2.68/1.54
% 2.68/1.54 thf(powersetAx_type,type,
% 2.68/1.54 powersetAx: $o ).
% 2.68/1.54
% 2.68/1.54 thf(powersetAx,definition,
% 2.68/1.54 ( powersetAx
% 2.68/1.54 = ( ! [A: $i,B: $i] :
% 2.68/1.54 ( ( in @ B @ ( powerset @ A ) )
% 2.68/1.54 <=> ! [Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ B )
% 2.68/1.54 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(setunion_type,type,
% 2.68/1.54 setunion: $i > $i ).
% 2.68/1.54
% 2.68/1.54 thf(setunionAx_type,type,
% 2.68/1.54 setunionAx: $o ).
% 2.68/1.54
% 2.68/1.54 thf(setunionAx,definition,
% 2.68/1.54 ( setunionAx
% 2.68/1.54 = ( ! [A: $i,Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ ( setunion @ A ) )
% 2.68/1.54 <=> ? [B: $i] :
% 2.68/1.54 ( ( in @ Xx @ B )
% 2.68/1.54 & ( in @ B @ A ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(omega_type,type,
% 2.68/1.54 omega: $i ).
% 2.68/1.54
% 2.68/1.54 thf(omega0Ax_type,type,
% 2.68/1.54 omega0Ax: $o ).
% 2.68/1.54
% 2.68/1.54 thf(omega0Ax,definition,
% 2.68/1.54 ( omega0Ax
% 2.68/1.54 = ( in @ emptyset @ omega ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(omegaSAx_type,type,
% 2.68/1.54 omegaSAx: $o ).
% 2.68/1.54
% 2.68/1.54 thf(omegaSAx,definition,
% 2.68/1.54 ( omegaSAx
% 2.68/1.54 = ( ! [Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ omega )
% 2.68/1.54 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(omegaIndAx_type,type,
% 2.68/1.54 omegaIndAx: $o ).
% 2.68/1.54
% 2.68/1.54 thf(omegaIndAx,definition,
% 2.68/1.54 ( omegaIndAx
% 2.68/1.54 = ( ! [A: $i] :
% 2.68/1.54 ( ( ( in @ emptyset @ A )
% 2.68/1.54 & ! [Xx: $i] :
% 2.68/1.54 ( ( ( in @ Xx @ omega )
% 2.68/1.54 & ( in @ Xx @ A ) )
% 2.68/1.54 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 2.68/1.54 => ! [Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ omega )
% 2.68/1.54 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(replAx_type,type,
% 2.68/1.54 replAx: $o ).
% 2.68/1.54
% 2.68/1.54 thf(replAx,definition,
% 2.68/1.54 ( replAx
% 2.68/1.54 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 2.68/1.54 ( ! [Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ A )
% 2.68/1.54 => ( exu
% 2.68/1.54 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 2.68/1.54 => ? [B: $i] :
% 2.68/1.54 ! [Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ B )
% 2.68/1.54 <=> ? [Xy: $i] :
% 2.68/1.54 ( ( in @ Xy @ A )
% 2.68/1.54 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(foundationAx_type,type,
% 2.68/1.54 foundationAx: $o ).
% 2.68/1.54
% 2.68/1.54 thf(foundationAx,definition,
% 2.68/1.54 ( foundationAx
% 2.68/1.54 = ( ! [A: $i] :
% 2.68/1.54 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.68/1.54 => ? [B: $i] :
% 2.68/1.54 ( ( in @ B @ A )
% 2.68/1.54 & ~ ? [Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ B )
% 2.68/1.54 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(wellorderingAx_type,type,
% 2.68/1.54 wellorderingAx: $o ).
% 2.68/1.54
% 2.68/1.54 thf(wellorderingAx,definition,
% 2.68/1.54 ( wellorderingAx
% 2.68/1.54 = ( ! [A: $i] :
% 2.68/1.54 ? [B: $i] :
% 2.68/1.54 ( ! [C: $i] :
% 2.68/1.54 ( ( in @ C @ B )
% 2.68/1.54 => ! [Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ C )
% 2.68/1.54 => ( in @ Xx @ A ) ) )
% 2.68/1.54 & ! [Xx: $i,Xy: $i] :
% 2.68/1.54 ( ( ( in @ Xx @ A )
% 2.68/1.54 & ( in @ Xy @ A ) )
% 2.68/1.54 => ( ! [C: $i] :
% 2.68/1.54 ( ( in @ C @ B )
% 2.68/1.54 => ( ( in @ Xx @ C )
% 2.68/1.54 <=> ( in @ Xy @ C ) ) )
% 2.68/1.54 => ( Xx = Xy ) ) )
% 2.68/1.54 & ! [C: $i,D: $i] :
% 2.68/1.54 ( ( ( in @ C @ B )
% 2.68/1.54 & ( in @ D @ B ) )
% 2.68/1.54 => ( ! [Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ C )
% 2.68/1.54 => ( in @ Xx @ D ) )
% 2.68/1.54 | ! [Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ D )
% 2.68/1.54 => ( in @ Xx @ C ) ) ) )
% 2.68/1.54 & ! [C: $i] :
% 2.68/1.54 ( ( ! [Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ C )
% 2.68/1.54 => ( in @ Xx @ A ) )
% 2.68/1.54 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 2.68/1.54 => ? [D: $i,Xx: $i] :
% 2.68/1.54 ( ( in @ D @ B )
% 2.68/1.54 & ( in @ Xx @ C )
% 2.68/1.54 & ~ ? [Xy: $i] :
% 2.68/1.54 ( ( in @ Xy @ D )
% 2.68/1.54 & ( in @ Xy @ C ) )
% 2.68/1.54 & ! [E: $i] :
% 2.68/1.54 ( ( in @ E @ B )
% 2.68/1.54 => ( ! [Xy: $i] :
% 2.68/1.54 ( ( in @ Xy @ E )
% 2.68/1.54 => ( in @ Xy @ D ) )
% 2.68/1.54 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(descr_type,type,
% 2.68/1.54 descr: ( $i > $o ) > $i ).
% 2.68/1.54
% 2.68/1.54 thf(descrp_type,type,
% 2.68/1.54 descrp: $o ).
% 2.68/1.54
% 2.68/1.54 thf(descrp,definition,
% 2.68/1.54 ( descrp
% 2.68/1.54 = ( ! [Xphi: $i > $o] :
% 2.68/1.54 ( ( exu
% 2.68/1.54 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.68/1.54 => ( Xphi
% 2.68/1.54 @ ( descr
% 2.68/1.54 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(dsetconstr_type,type,
% 2.68/1.54 dsetconstr: $i > ( $i > $o ) > $i ).
% 2.68/1.54
% 2.68/1.54 thf(dsetconstrI_type,type,
% 2.68/1.54 dsetconstrI: $o ).
% 2.68/1.54
% 2.68/1.54 thf(dsetconstrI,definition,
% 2.68/1.54 ( dsetconstrI
% 2.68/1.54 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ A )
% 2.68/1.54 => ( ( Xphi @ Xx )
% 2.68/1.54 => ( in @ Xx
% 2.68/1.54 @ ( dsetconstr @ A
% 2.68/1.54 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(dsetconstrEL_type,type,
% 2.68/1.54 dsetconstrEL: $o ).
% 2.68/1.54
% 2.68/1.54 thf(dsetconstrEL,definition,
% 2.68/1.54 ( dsetconstrEL
% 2.68/1.54 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.68/1.54 ( ( in @ Xx
% 2.68/1.54 @ ( dsetconstr @ A
% 2.68/1.54 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.68/1.54 => ( in @ Xx @ A ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(dsetconstrER_type,type,
% 2.68/1.54 dsetconstrER: $o ).
% 2.68/1.54
% 2.68/1.54 thf(dsetconstrER,definition,
% 2.68/1.54 ( dsetconstrER
% 2.68/1.54 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.68/1.54 ( ( in @ Xx
% 2.68/1.54 @ ( dsetconstr @ A
% 2.68/1.54 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.68/1.54 => ( Xphi @ Xx ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(exuE1_type,type,
% 2.68/1.54 exuE1: $o ).
% 2.68/1.54
% 2.68/1.54 thf(exuE1,definition,
% 2.68/1.54 ( exuE1
% 2.68/1.54 = ( ! [Xphi: $i > $o] :
% 2.68/1.54 ( ( exu
% 2.68/1.54 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.68/1.54 => ? [Xx: $i] :
% 2.68/1.54 ( ( Xphi @ Xx )
% 2.68/1.54 & ! [Xy: $i] :
% 2.68/1.54 ( ( Xphi @ Xy )
% 2.68/1.54 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(prop2set_type,type,
% 2.68/1.54 prop2set: $o > $i ).
% 2.68/1.54
% 2.68/1.54 thf(prop2setE_type,type,
% 2.68/1.54 prop2setE: $o ).
% 2.68/1.54
% 2.68/1.54 thf(prop2setE,definition,
% 2.68/1.54 ( prop2setE
% 2.68/1.54 = ( ! [Xphi: $o,Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 2.68/1.54 => Xphi ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(emptysetE_type,type,
% 2.68/1.54 emptysetE: $o ).
% 2.68/1.54
% 2.68/1.54 thf(emptysetE,definition,
% 2.68/1.54 ( emptysetE
% 2.68/1.54 = ( ! [Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ emptyset )
% 2.68/1.54 => ! [Xphi: $o] : Xphi ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(emptysetimpfalse_type,type,
% 2.68/1.54 emptysetimpfalse: $o ).
% 2.68/1.54
% 2.68/1.54 thf(emptysetimpfalse,definition,
% 2.68/1.54 ( emptysetimpfalse
% 2.68/1.54 = ( ! [Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ emptyset )
% 2.68/1.54 => $false ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(notinemptyset_type,type,
% 2.68/1.54 notinemptyset: $o ).
% 2.68/1.54
% 2.68/1.54 thf(notinemptyset,definition,
% 2.68/1.54 ( notinemptyset
% 2.68/1.54 = ( ! [Xx: $i] :
% 2.68/1.54 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(exuE3e_type,type,
% 2.68/1.54 exuE3e: $o ).
% 2.68/1.54
% 2.68/1.54 thf(exuE3e,definition,
% 2.68/1.54 ( exuE3e
% 2.68/1.54 = ( ! [Xphi: $i > $o] :
% 2.68/1.54 ( ( exu
% 2.68/1.54 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.68/1.54 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(setext_type,type,
% 2.68/1.54 setext: $o ).
% 2.68/1.54
% 2.68/1.54 thf(setext,definition,
% 2.68/1.54 ( setext
% 2.68/1.54 = ( ! [A: $i,B: $i] :
% 2.68/1.54 ( ! [Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ A )
% 2.68/1.54 => ( in @ Xx @ B ) )
% 2.68/1.54 => ( ! [Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ B )
% 2.68/1.54 => ( in @ Xx @ A ) )
% 2.68/1.54 => ( A = B ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(emptyI_type,type,
% 2.68/1.54 emptyI: $o ).
% 2.68/1.54
% 2.68/1.54 thf(emptyI,definition,
% 2.68/1.54 ( emptyI
% 2.68/1.54 = ( ! [A: $i] :
% 2.68/1.54 ( ! [Xx: $i] :
% 2.68/1.54 ~ ( in @ Xx @ A )
% 2.68/1.54 => ( A = emptyset ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(noeltsimpempty_type,type,
% 2.68/1.54 noeltsimpempty: $o ).
% 2.68/1.54
% 2.68/1.54 thf(noeltsimpempty,definition,
% 2.68/1.54 ( noeltsimpempty
% 2.68/1.54 = ( ! [A: $i] :
% 2.68/1.54 ( ! [Xx: $i] :
% 2.68/1.54 ~ ( in @ Xx @ A )
% 2.68/1.54 => ( A = emptyset ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(setbeta_type,type,
% 2.68/1.54 setbeta: $o ).
% 2.68/1.54
% 2.68/1.54 thf(setbeta,definition,
% 2.68/1.54 ( setbeta
% 2.68/1.54 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ A )
% 2.68/1.54 => ( ( in @ Xx
% 2.68/1.54 @ ( dsetconstr @ A
% 2.68/1.54 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.68/1.54 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(nonempty_type,type,
% 2.68/1.54 nonempty: $i > $o ).
% 2.68/1.54
% 2.68/1.54 thf(nonempty,definition,
% 2.68/1.54 ( nonempty
% 2.68/1.54 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(nonemptyE1_type,type,
% 2.68/1.54 nonemptyE1: $o ).
% 2.68/1.54
% 2.68/1.54 thf(nonemptyE1,definition,
% 2.68/1.54 ( nonemptyE1
% 2.68/1.54 = ( ! [A: $i] :
% 2.68/1.54 ( ( nonempty @ A )
% 2.68/1.54 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(nonemptyI_type,type,
% 2.68/1.54 nonemptyI: $o ).
% 2.68/1.54
% 2.68/1.54 thf(nonemptyI,definition,
% 2.68/1.54 ( nonemptyI
% 2.68/1.54 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ A )
% 2.68/1.54 => ( ( Xphi @ Xx )
% 2.68/1.54 => ( nonempty
% 2.68/1.54 @ ( dsetconstr @ A
% 2.68/1.54 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(nonemptyI1_type,type,
% 2.68/1.54 nonemptyI1: $o ).
% 2.68/1.54
% 2.68/1.54 thf(nonemptyI1,definition,
% 2.68/1.54 ( nonemptyI1
% 2.68/1.54 = ( ! [A: $i] :
% 2.68/1.54 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.68/1.54 => ( nonempty @ A ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(setadjoinIL_type,type,
% 2.68/1.54 setadjoinIL: $o ).
% 2.68/1.54
% 2.68/1.54 thf(setadjoinIL,definition,
% 2.68/1.54 ( setadjoinIL
% 2.68/1.54 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(emptyinunitempty_type,type,
% 2.68/1.54 emptyinunitempty: $o ).
% 2.68/1.54
% 2.68/1.54 thf(emptyinunitempty,definition,
% 2.68/1.54 ( emptyinunitempty
% 2.68/1.54 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(setadjoinIR_type,type,
% 2.68/1.54 setadjoinIR: $o ).
% 2.68/1.54
% 2.68/1.54 thf(setadjoinIR,definition,
% 2.68/1.54 ( setadjoinIR
% 2.68/1.54 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.68/1.54 ( ( in @ Xy @ A )
% 2.68/1.54 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(setadjoinE_type,type,
% 2.68/1.54 setadjoinE: $o ).
% 2.68/1.54
% 2.68/1.54 thf(setadjoinE,definition,
% 2.68/1.54 ( setadjoinE
% 2.68/1.54 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.68/1.54 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.68/1.54 => ! [Xphi: $o] :
% 2.68/1.54 ( ( ( Xy = Xx )
% 2.68/1.54 => Xphi )
% 2.68/1.54 => ( ( ( in @ Xy @ A )
% 2.68/1.54 => Xphi )
% 2.68/1.54 => Xphi ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(setadjoinOr_type,type,
% 2.68/1.54 setadjoinOr: $o ).
% 2.68/1.54
% 2.68/1.54 thf(setadjoinOr,definition,
% 2.68/1.54 ( setadjoinOr
% 2.68/1.54 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.68/1.54 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.68/1.54 => ( ( Xy = Xx )
% 2.68/1.54 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(setoftrueEq_type,type,
% 2.68/1.54 setoftrueEq: $o ).
% 2.68/1.54
% 2.68/1.54 thf(setoftrueEq,definition,
% 2.68/1.54 ( setoftrueEq
% 2.68/1.54 = ( ! [A: $i] :
% 2.68/1.54 ( ( dsetconstr @ A
% 2.68/1.54 @ ^ [Xx: $i] : $true )
% 2.68/1.54 = A ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(powersetI_type,type,
% 2.68/1.54 powersetI: $o ).
% 2.68/1.54
% 2.68/1.54 thf(powersetI,definition,
% 2.68/1.54 ( powersetI
% 2.68/1.54 = ( ! [A: $i,B: $i] :
% 2.68/1.54 ( ! [Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ B )
% 2.68/1.54 => ( in @ Xx @ A ) )
% 2.68/1.54 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(emptyinPowerset_type,type,
% 2.68/1.54 emptyinPowerset: $o ).
% 2.68/1.54
% 2.68/1.54 thf(emptyinPowerset,definition,
% 2.68/1.54 ( emptyinPowerset
% 2.68/1.54 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(emptyInPowerset_type,type,
% 2.68/1.54 emptyInPowerset: $o ).
% 2.68/1.54
% 2.68/1.54 thf(emptyInPowerset,definition,
% 2.68/1.54 ( emptyInPowerset
% 2.68/1.54 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(powersetE_type,type,
% 2.68/1.54 powersetE: $o ).
% 2.68/1.54
% 2.68/1.54 thf(powersetE,definition,
% 2.68/1.54 ( powersetE
% 2.68/1.54 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.68/1.54 ( ( in @ B @ ( powerset @ A ) )
% 2.68/1.54 => ( ( in @ Xx @ B )
% 2.68/1.54 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(setunionI_type,type,
% 2.68/1.54 setunionI: $o ).
% 2.68/1.54
% 2.68/1.54 thf(setunionI,definition,
% 2.68/1.54 ( setunionI
% 2.68/1.54 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.68/1.54 ( ( in @ Xx @ B )
% 2.68/1.54 => ( ( in @ B @ A )
% 2.68/1.54 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(setunionE_type,type,
% 2.68/1.54 setunionE: $o ).
% 2.68/1.54
% 2.68/1.54 thf(setunionE,definition,
% 2.68/1.54 ( setunionE
% 2.68/1.54 = ( ! [A: $i,Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ ( setunion @ A ) )
% 2.68/1.54 => ! [Xphi: $o] :
% 2.68/1.54 ( ! [B: $i] :
% 2.68/1.54 ( ( in @ Xx @ B )
% 2.68/1.54 => ( ( in @ B @ A )
% 2.68/1.54 => Xphi ) )
% 2.68/1.54 => Xphi ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(subPowSU_type,type,
% 2.68/1.54 subPowSU: $o ).
% 2.68/1.54
% 2.68/1.54 thf(subPowSU,definition,
% 2.68/1.54 ( subPowSU
% 2.68/1.54 = ( ! [A: $i,Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ A )
% 2.68/1.54 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(exuE2_type,type,
% 2.68/1.54 exuE2: $o ).
% 2.68/1.54
% 2.68/1.54 thf(exuE2,definition,
% 2.68/1.54 ( exuE2
% 2.68/1.54 = ( ! [Xphi: $i > $o] :
% 2.68/1.54 ( ( exu
% 2.68/1.54 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.68/1.54 => ? [Xx: $i] :
% 2.68/1.54 ! [Xy: $i] :
% 2.68/1.54 ( ( Xphi @ Xy )
% 2.68/1.54 <=> ( Xy = Xx ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(nonemptyImpWitness_type,type,
% 2.68/1.54 nonemptyImpWitness: $o ).
% 2.68/1.54
% 2.68/1.54 thf(nonemptyImpWitness,definition,
% 2.68/1.54 ( nonemptyImpWitness
% 2.68/1.54 = ( ! [A: $i] :
% 2.68/1.54 ( ( nonempty @ A )
% 2.68/1.54 => ? [Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ A )
% 2.68/1.54 & $true ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(uniqinunit_type,type,
% 2.68/1.54 uniqinunit: $o ).
% 2.68/1.54
% 2.68/1.54 thf(uniqinunit,definition,
% 2.68/1.54 ( uniqinunit
% 2.68/1.54 = ( ! [Xx: $i,Xy: $i] :
% 2.68/1.54 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.68/1.54 => ( Xx = Xy ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(notinsingleton_type,type,
% 2.68/1.54 notinsingleton: $o ).
% 2.68/1.54
% 2.68/1.54 thf(notinsingleton,definition,
% 2.68/1.54 ( notinsingleton
% 2.68/1.54 = ( ! [Xx: $i,Xy: $i] :
% 2.68/1.54 ( ( Xx != Xy )
% 2.68/1.54 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(eqinunit_type,type,
% 2.68/1.54 eqinunit: $o ).
% 2.68/1.54
% 2.68/1.54 thf(eqinunit,definition,
% 2.68/1.54 ( eqinunit
% 2.68/1.54 = ( ! [Xx: $i,Xy: $i] :
% 2.68/1.54 ( ( Xx = Xy )
% 2.68/1.54 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(singletonsswitch_type,type,
% 2.68/1.54 singletonsswitch: $o ).
% 2.68/1.54
% 2.68/1.54 thf(singletonsswitch,definition,
% 2.68/1.54 ( singletonsswitch
% 2.68/1.54 = ( ! [Xx: $i,Xy: $i] :
% 2.68/1.54 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.68/1.54 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(upairsetE_type,type,
% 2.68/1.54 upairsetE: $o ).
% 2.68/1.54
% 2.68/1.54 thf(upairsetE,definition,
% 2.68/1.54 ( upairsetE
% 2.68/1.54 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.68/1.54 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.68/1.54 => ( ( Xz = Xx )
% 2.68/1.54 | ( Xz = Xy ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(upairsetIL_type,type,
% 2.68/1.54 upairsetIL: $o ).
% 2.68/1.54
% 2.68/1.54 thf(upairsetIL,definition,
% 2.68/1.54 ( upairsetIL
% 2.68/1.54 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(upairsetIR_type,type,
% 2.68/1.54 upairsetIR: $o ).
% 2.68/1.54
% 2.68/1.54 thf(upairsetIR,definition,
% 2.68/1.54 ( upairsetIR
% 2.68/1.54 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(emptyE1_type,type,
% 2.68/1.54 emptyE1: $o ).
% 2.68/1.54
% 2.68/1.54 thf(emptyE1,definition,
% 2.68/1.54 ( emptyE1
% 2.68/1.54 = ( ! [A: $i,Xphi: $i > $o] :
% 2.68/1.54 ( ? [Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ A )
% 2.68/1.54 & ( Xphi @ Xx ) )
% 2.68/1.54 => ( ( ( dsetconstr @ A
% 2.68/1.54 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.68/1.54 = emptyset )
% 2.68/1.54 => $false ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(vacuousDall_type,type,
% 2.68/1.54 vacuousDall: $o ).
% 2.68/1.54
% 2.68/1.54 thf(vacuousDall,definition,
% 2.68/1.54 ( vacuousDall
% 2.68/1.54 = ( ! [Xphi: $i > $o,Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ emptyset )
% 2.68/1.54 => ( Xphi @ Xx ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(quantDeMorgan1_type,type,
% 2.68/1.54 quantDeMorgan1: $o ).
% 2.68/1.54
% 2.68/1.54 thf(quantDeMorgan1,definition,
% 2.68/1.54 ( quantDeMorgan1
% 2.68/1.54 = ( ! [A: $i,Xphi: $i > $o] :
% 2.68/1.54 ( ~ ! [Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ A )
% 2.68/1.54 => ( Xphi @ Xx ) )
% 2.68/1.54 => ? [Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ A )
% 2.68/1.54 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(quantDeMorgan2_type,type,
% 2.68/1.54 quantDeMorgan2: $o ).
% 2.68/1.54
% 2.68/1.54 thf(quantDeMorgan2,definition,
% 2.68/1.54 ( quantDeMorgan2
% 2.68/1.54 = ( ! [A: $i,Xphi: $i > $o] :
% 2.68/1.54 ( ! [Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ A )
% 2.68/1.54 => ~ ( Xphi @ Xx ) )
% 2.68/1.54 => ~ ? [Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ A )
% 2.68/1.54 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(quantDeMorgan3_type,type,
% 2.68/1.54 quantDeMorgan3: $o ).
% 2.68/1.54
% 2.68/1.54 thf(quantDeMorgan3,definition,
% 2.68/1.54 ( quantDeMorgan3
% 2.68/1.54 = ( ! [A: $i,Xphi: $i > $o] :
% 2.68/1.54 ( ~ ? [Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ A )
% 2.68/1.54 & ( Xphi @ Xx ) )
% 2.68/1.54 => ! [Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ A )
% 2.68/1.54 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(quantDeMorgan4_type,type,
% 2.68/1.54 quantDeMorgan4: $o ).
% 2.68/1.54
% 2.68/1.54 thf(quantDeMorgan4,definition,
% 2.68/1.54 ( quantDeMorgan4
% 2.68/1.54 = ( ! [A: $i,Xphi: $i > $o] :
% 2.68/1.54 ( ? [Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ A )
% 2.68/1.54 & ~ ( Xphi @ Xx ) )
% 2.68/1.54 => ~ ! [Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ A )
% 2.68/1.54 => ( Xphi @ Xx ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(prop2setI_type,type,
% 2.68/1.54 prop2setI: $o ).
% 2.68/1.54
% 2.68/1.54 thf(prop2setI,definition,
% 2.68/1.54 ( prop2setI
% 2.68/1.54 = ( ! [Xphi: $o] :
% 2.68/1.54 ( Xphi
% 2.68/1.54 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(set2prop_type,type,
% 2.68/1.54 set2prop: $i > $o ).
% 2.68/1.54
% 2.68/1.54 thf(prop2set2propI_type,type,
% 2.68/1.54 prop2set2propI: $o ).
% 2.68/1.54
% 2.68/1.54 thf(prop2set2propI,definition,
% 2.68/1.54 ( prop2set2propI
% 2.68/1.54 = ( ! [Xphi: $o] :
% 2.68/1.54 ( Xphi
% 2.68/1.54 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(notdexE_type,type,
% 2.68/1.54 notdexE: $o ).
% 2.68/1.54
% 2.68/1.54 thf(notdexE,definition,
% 2.68/1.54 ( notdexE
% 2.68/1.54 = ( ! [A: $i,Xphi: $i > $o] :
% 2.68/1.54 ( ~ ? [Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ A )
% 2.68/1.54 & ( Xphi @ Xx ) )
% 2.68/1.54 => ! [Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ A )
% 2.68/1.54 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(notdallE_type,type,
% 2.68/1.54 notdallE: $o ).
% 2.68/1.54
% 2.68/1.54 thf(notdallE,definition,
% 2.68/1.54 ( notdallE
% 2.68/1.54 = ( ! [A: $i,Xphi: $i > $o] :
% 2.68/1.54 ( ~ ! [Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ A )
% 2.68/1.54 => ( Xphi @ Xx ) )
% 2.68/1.54 => ? [Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ A )
% 2.68/1.54 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(exuI1_type,type,
% 2.68/1.54 exuI1: $o ).
% 2.68/1.54
% 2.68/1.54 thf(exuI1,definition,
% 2.68/1.54 ( exuI1
% 2.68/1.54 = ( ! [Xphi: $i > $o] :
% 2.68/1.54 ( ? [Xx: $i] :
% 2.68/1.54 ( ( Xphi @ Xx )
% 2.68/1.54 & ! [Xy: $i] :
% 2.68/1.54 ( ( Xphi @ Xy )
% 2.68/1.54 => ( Xx = Xy ) ) )
% 2.68/1.54 => ( exu
% 2.68/1.54 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(exuI3_type,type,
% 2.68/1.54 exuI3: $o ).
% 2.68/1.54
% 2.68/1.54 thf(exuI3,definition,
% 2.68/1.54 ( exuI3
% 2.68/1.54 = ( ! [Xphi: $i > $o] :
% 2.68/1.54 ( ? [Xx: $i] : ( Xphi @ Xx )
% 2.68/1.54 => ( ! [Xx: $i,Xy: $i] :
% 2.68/1.54 ( ( Xphi @ Xx )
% 2.68/1.54 => ( ( Xphi @ Xy )
% 2.68/1.54 => ( Xx = Xy ) ) )
% 2.68/1.54 => ( exu
% 2.68/1.54 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(exuI2_type,type,
% 2.68/1.54 exuI2: $o ).
% 2.68/1.54
% 2.68/1.54 thf(exuI2,definition,
% 2.68/1.54 ( exuI2
% 2.68/1.54 = ( ! [Xphi: $i > $o] :
% 2.68/1.54 ( ? [Xx: $i] :
% 2.68/1.54 ! [Xy: $i] :
% 2.68/1.54 ( ( Xphi @ Xy )
% 2.68/1.54 <=> ( Xy = Xx ) )
% 2.68/1.54 => ( exu
% 2.68/1.54 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(inCongP_type,type,
% 2.68/1.54 inCongP: $o ).
% 2.68/1.54
% 2.68/1.54 thf(inCongP,definition,
% 2.68/1.54 ( inCongP
% 2.68/1.54 = ( ! [A: $i,B: $i] :
% 2.68/1.54 ( ( A = B )
% 2.68/1.54 => ! [Xx: $i,Xy: $i] :
% 2.68/1.54 ( ( Xx = Xy )
% 2.68/1.54 => ( ( in @ Xx @ A )
% 2.68/1.54 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(in__Cong_type,type,
% 2.68/1.54 in__Cong: $o ).
% 2.68/1.54
% 2.68/1.54 thf(in__Cong,definition,
% 2.68/1.54 ( in__Cong
% 2.68/1.54 = ( ! [A: $i,B: $i] :
% 2.68/1.54 ( ( A = B )
% 2.68/1.54 => ! [Xx: $i,Xy: $i] :
% 2.68/1.54 ( ( Xx = Xy )
% 2.68/1.54 => ( ( in @ Xx @ A )
% 2.68/1.54 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(exuE3u_type,type,
% 2.68/1.54 exuE3u: $o ).
% 2.68/1.54
% 2.68/1.54 thf(exuE3u,definition,
% 2.68/1.54 ( exuE3u
% 2.68/1.54 = ( ! [Xphi: $i > $o] :
% 2.68/1.54 ( ( exu
% 2.68/1.54 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.68/1.54 => ! [Xx: $i,Xy: $i] :
% 2.68/1.54 ( ( Xphi @ Xx )
% 2.68/1.54 => ( ( Xphi @ Xy )
% 2.68/1.54 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(exu__Cong_type,type,
% 2.68/1.54 exu__Cong: $o ).
% 2.68/1.54
% 2.68/1.54 thf(exu__Cong,definition,
% 2.68/1.54 ( exu__Cong
% 2.68/1.54 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.68/1.54 ( ! [Xx: $i,Xy: $i] :
% 2.68/1.54 ( ( Xx = Xy )
% 2.68/1.54 => ( ( Xphi @ Xx )
% 2.68/1.54 <=> ( Xpsi @ Xy ) ) )
% 2.68/1.54 => ( ( exu
% 2.68/1.54 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.68/1.54 <=> ( exu
% 2.68/1.54 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(emptyset__Cong_type,type,
% 2.68/1.54 emptyset__Cong: $o ).
% 2.68/1.54
% 2.68/1.54 thf(emptyset__Cong,definition,
% 2.68/1.54 ( emptyset__Cong
% 2.68/1.54 = ( emptyset = emptyset ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(setadjoin__Cong_type,type,
% 2.68/1.54 setadjoin__Cong: $o ).
% 2.68/1.54
% 2.68/1.54 thf(setadjoin__Cong,definition,
% 2.68/1.54 ( setadjoin__Cong
% 2.68/1.54 = ( ! [Xx: $i,Xy: $i] :
% 2.68/1.54 ( ( Xx = Xy )
% 2.68/1.54 => ! [Xz: $i,Xu: $i] :
% 2.68/1.54 ( ( Xz = Xu )
% 2.68/1.54 => ( ( setadjoin @ Xx @ Xz )
% 2.68/1.54 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(powerset__Cong_type,type,
% 2.68/1.54 powerset__Cong: $o ).
% 2.68/1.54
% 2.68/1.54 thf(powerset__Cong,definition,
% 2.68/1.54 ( powerset__Cong
% 2.68/1.54 = ( ! [A: $i,B: $i] :
% 2.68/1.54 ( ( A = B )
% 2.68/1.54 => ( ( powerset @ A )
% 2.68/1.54 = ( powerset @ B ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(setunion__Cong_type,type,
% 2.68/1.54 setunion__Cong: $o ).
% 2.68/1.54
% 2.68/1.54 thf(setunion__Cong,definition,
% 2.68/1.54 ( setunion__Cong
% 2.68/1.54 = ( ! [A: $i,B: $i] :
% 2.68/1.54 ( ( A = B )
% 2.68/1.54 => ( ( setunion @ A )
% 2.68/1.54 = ( setunion @ B ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(omega__Cong_type,type,
% 2.68/1.54 omega__Cong: $o ).
% 2.68/1.54
% 2.68/1.54 thf(omega__Cong,definition,
% 2.68/1.54 ( omega__Cong
% 2.68/1.54 = ( omega = omega ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(exuEu_type,type,
% 2.68/1.54 exuEu: $o ).
% 2.68/1.54
% 2.68/1.54 thf(exuEu,definition,
% 2.68/1.54 ( exuEu
% 2.68/1.54 = ( ! [Xphi: $i > $o] :
% 2.68/1.54 ( ( exu
% 2.68/1.54 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.68/1.54 => ! [Xx: $i,Xy: $i] :
% 2.68/1.54 ( ( Xphi @ Xx )
% 2.68/1.54 => ( ( Xphi @ Xy )
% 2.68/1.54 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(descr__Cong_type,type,
% 2.68/1.54 descr__Cong: $o ).
% 2.68/1.54
% 2.68/1.54 thf(descr__Cong,definition,
% 2.68/1.54 ( descr__Cong
% 2.68/1.54 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.68/1.54 ( ! [Xx: $i,Xy: $i] :
% 2.68/1.54 ( ( Xx = Xy )
% 2.68/1.54 => ( ( Xphi @ Xx )
% 2.68/1.54 <=> ( Xpsi @ Xy ) ) )
% 2.68/1.54 => ( ( exu
% 2.68/1.54 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.68/1.54 => ( ( exu
% 2.68/1.54 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 2.68/1.54 => ( ( descr
% 2.68/1.54 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.68/1.54 = ( descr
% 2.68/1.54 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(dsetconstr__Cong_type,type,
% 2.68/1.54 dsetconstr__Cong: $o ).
% 2.68/1.54
% 2.68/1.54 thf(dsetconstr__Cong,definition,
% 2.68/1.54 ( dsetconstr__Cong
% 2.68/1.54 = ( ! [A: $i,B: $i] :
% 2.68/1.54 ( ( A = B )
% 2.68/1.54 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.68/1.54 ( ! [Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ A )
% 2.68/1.54 => ! [Xy: $i] :
% 2.68/1.54 ( ( in @ Xy @ B )
% 2.68/1.54 => ( ( Xx = Xy )
% 2.68/1.54 => ( ( Xphi @ Xx )
% 2.68/1.54 <=> ( Xpsi @ Xy ) ) ) ) )
% 2.68/1.54 => ( ( dsetconstr @ A
% 2.68/1.54 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.68/1.54 = ( dsetconstr @ B
% 2.68/1.54 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(subset_type,type,
% 2.68/1.54 subset: $i > $i > $o ).
% 2.68/1.54
% 2.68/1.54 thf(disjoint_type,type,
% 2.68/1.54 disjoint: $i > $i > $o ).
% 2.68/1.54
% 2.68/1.54 thf(setsmeet_type,type,
% 2.68/1.54 setsmeet: $i > $i > $o ).
% 2.68/1.54
% 2.68/1.54 thf(subsetI1_type,type,
% 2.68/1.54 subsetI1: $o ).
% 2.68/1.54
% 2.68/1.54 thf(subsetI1,definition,
% 2.68/1.54 ( subsetI1
% 2.68/1.54 = ( ! [A: $i,B: $i] :
% 2.68/1.54 ( ! [Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ A )
% 2.68/1.54 => ( in @ Xx @ B ) )
% 2.68/1.54 => ( subset @ A @ B ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(eqimpsubset2_type,type,
% 2.68/1.54 eqimpsubset2: $o ).
% 2.68/1.54
% 2.68/1.54 thf(eqimpsubset2,definition,
% 2.68/1.54 ( eqimpsubset2
% 2.68/1.54 = ( ! [A: $i,B: $i] :
% 2.68/1.54 ( ( A = B )
% 2.68/1.54 => ( subset @ B @ A ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(eqimpsubset1_type,type,
% 2.68/1.54 eqimpsubset1: $o ).
% 2.68/1.54
% 2.68/1.54 thf(eqimpsubset1,definition,
% 2.68/1.54 ( eqimpsubset1
% 2.68/1.54 = ( ! [A: $i,B: $i] :
% 2.68/1.54 ( ( A = B )
% 2.68/1.54 => ( subset @ A @ B ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(subsetI2_type,type,
% 2.68/1.54 subsetI2: $o ).
% 2.68/1.54
% 2.68/1.54 thf(subsetI2,definition,
% 2.68/1.54 ( subsetI2
% 2.68/1.54 = ( ! [A: $i,B: $i] :
% 2.68/1.54 ( ! [Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ A )
% 2.68/1.54 => ( in @ Xx @ B ) )
% 2.68/1.54 => ( subset @ A @ B ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(emptysetsubset_type,type,
% 2.68/1.54 emptysetsubset: $o ).
% 2.68/1.54
% 2.68/1.54 thf(emptysetsubset,definition,
% 2.68/1.54 ( emptysetsubset
% 2.68/1.54 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(subsetE_type,type,
% 2.68/1.54 subsetE: $o ).
% 2.68/1.54
% 2.68/1.54 thf(subsetE,definition,
% 2.68/1.54 ( subsetE
% 2.68/1.54 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.68/1.54 ( ( subset @ A @ B )
% 2.68/1.54 => ( ( in @ Xx @ A )
% 2.68/1.54 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(subsetE2_type,type,
% 2.68/1.54 subsetE2: $o ).
% 2.68/1.54
% 2.68/1.54 thf(subsetE2,definition,
% 2.68/1.54 ( subsetE2
% 2.68/1.54 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.68/1.54 ( ( subset @ A @ B )
% 2.68/1.54 => ( ~ ( in @ Xx @ B )
% 2.68/1.54 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(notsubsetI_type,type,
% 2.68/1.54 notsubsetI: $o ).
% 2.68/1.54
% 2.68/1.54 thf(notsubsetI,definition,
% 2.68/1.54 ( notsubsetI
% 2.68/1.54 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ A )
% 2.68/1.54 => ( ~ ( in @ Xx @ B )
% 2.68/1.54 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(notequalI1_type,type,
% 2.68/1.54 notequalI1: $o ).
% 2.68/1.54
% 2.68/1.54 thf(notequalI1,definition,
% 2.68/1.54 ( notequalI1
% 2.68/1.54 = ( ! [A: $i,B: $i] :
% 2.68/1.54 ( ~ ( subset @ A @ B )
% 2.68/1.54 => ( A != B ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(notequalI2_type,type,
% 2.68/1.54 notequalI2: $o ).
% 2.68/1.54
% 2.68/1.54 thf(notequalI2,definition,
% 2.68/1.54 ( notequalI2
% 2.68/1.54 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ A )
% 2.68/1.54 => ( ~ ( in @ Xx @ B )
% 2.68/1.54 => ( A != B ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(subsetRefl_type,type,
% 2.68/1.54 subsetRefl: $o ).
% 2.68/1.54
% 2.68/1.54 thf(subsetRefl,definition,
% 2.68/1.54 ( subsetRefl
% 2.68/1.54 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(subsetTrans_type,type,
% 2.68/1.54 subsetTrans: $o ).
% 2.68/1.54
% 2.68/1.54 thf(subsetTrans,definition,
% 2.68/1.54 ( subsetTrans
% 2.68/1.54 = ( ! [A: $i,B: $i,C: $i] :
% 2.68/1.54 ( ( subset @ A @ B )
% 2.68/1.54 => ( ( subset @ B @ C )
% 2.68/1.54 => ( subset @ A @ C ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(setadjoinSub_type,type,
% 2.68/1.54 setadjoinSub: $o ).
% 2.68/1.54
% 2.68/1.54 thf(setadjoinSub,definition,
% 2.68/1.54 ( setadjoinSub
% 2.68/1.54 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(setadjoinSub2_type,type,
% 2.68/1.54 setadjoinSub2: $o ).
% 2.68/1.54
% 2.68/1.54 thf(setadjoinSub2,definition,
% 2.68/1.54 ( setadjoinSub2
% 2.68/1.54 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.68/1.54 ( ( subset @ A @ B )
% 2.68/1.54 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(subset2powerset_type,type,
% 2.68/1.54 subset2powerset: $o ).
% 2.68/1.54
% 2.68/1.54 thf(subset2powerset,definition,
% 2.68/1.54 ( subset2powerset
% 2.68/1.54 = ( ! [A: $i,B: $i] :
% 2.68/1.54 ( ( subset @ A @ B )
% 2.68/1.54 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(setextsub_type,type,
% 2.68/1.54 setextsub: $o ).
% 2.68/1.54
% 2.68/1.54 thf(setextsub,definition,
% 2.68/1.54 ( setextsub
% 2.68/1.54 = ( ! [A: $i,B: $i] :
% 2.68/1.54 ( ( subset @ A @ B )
% 2.68/1.54 => ( ( subset @ B @ A )
% 2.68/1.54 => ( A = B ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(subsetemptysetimpeq_type,type,
% 2.68/1.54 subsetemptysetimpeq: $o ).
% 2.68/1.54
% 2.68/1.54 thf(subsetemptysetimpeq,definition,
% 2.68/1.54 ( subsetemptysetimpeq
% 2.68/1.54 = ( ! [A: $i] :
% 2.68/1.54 ( ( subset @ A @ emptyset )
% 2.68/1.54 => ( A = emptyset ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(powersetI1_type,type,
% 2.68/1.54 powersetI1: $o ).
% 2.68/1.54
% 2.68/1.54 thf(powersetI1,definition,
% 2.68/1.54 ( powersetI1
% 2.68/1.54 = ( ! [A: $i,B: $i] :
% 2.68/1.54 ( ( subset @ B @ A )
% 2.68/1.54 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(powersetE1_type,type,
% 2.68/1.54 powersetE1: $o ).
% 2.68/1.54
% 2.68/1.54 thf(powersetE1,definition,
% 2.68/1.54 ( powersetE1
% 2.68/1.54 = ( ! [A: $i,B: $i] :
% 2.68/1.54 ( ( in @ B @ ( powerset @ A ) )
% 2.68/1.54 => ( subset @ B @ A ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(inPowerset_type,type,
% 2.68/1.54 inPowerset: $o ).
% 2.68/1.54
% 2.68/1.54 thf(inPowerset,definition,
% 2.68/1.54 ( inPowerset
% 2.68/1.54 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(powersetsubset_type,type,
% 2.68/1.54 powersetsubset: $o ).
% 2.68/1.54
% 2.68/1.54 thf(powersetsubset,definition,
% 2.68/1.54 ( powersetsubset
% 2.68/1.54 = ( ! [A: $i,B: $i] :
% 2.68/1.54 ( ( subset @ A @ B )
% 2.68/1.54 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(sepInPowerset_type,type,
% 2.68/1.54 sepInPowerset: $o ).
% 2.68/1.54
% 2.68/1.54 thf(sepInPowerset,definition,
% 2.68/1.54 ( sepInPowerset
% 2.68/1.54 = ( ! [A: $i,Xphi: $i > $o] :
% 2.68/1.54 ( in
% 2.68/1.54 @ ( dsetconstr @ A
% 2.68/1.54 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.68/1.54 @ ( powerset @ A ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(sepSubset_type,type,
% 2.68/1.54 sepSubset: $o ).
% 2.68/1.54
% 2.68/1.54 thf(sepSubset,definition,
% 2.68/1.54 ( sepSubset
% 2.68/1.54 = ( ! [A: $i,Xphi: $i > $o] :
% 2.68/1.54 ( subset
% 2.68/1.54 @ ( dsetconstr @ A
% 2.68/1.54 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.68/1.54 @ A ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(binunion_type,type,
% 2.68/1.54 binunion: $i > $i > $i ).
% 2.68/1.54
% 2.68/1.54 thf(binunionIL_type,type,
% 2.68/1.54 binunionIL: $o ).
% 2.68/1.54
% 2.68/1.54 thf(binunionIL,definition,
% 2.68/1.54 ( binunionIL
% 2.68/1.54 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ A )
% 2.68/1.54 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(upairset2IR_type,type,
% 2.68/1.54 upairset2IR: $o ).
% 2.68/1.54
% 2.68/1.54 thf(upairset2IR,definition,
% 2.68/1.54 ( upairset2IR
% 2.68/1.54 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(binunionIR_type,type,
% 2.68/1.54 binunionIR: $o ).
% 2.68/1.54
% 2.68/1.54 thf(binunionIR,definition,
% 2.68/1.54 ( binunionIR
% 2.68/1.54 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ B )
% 2.68/1.54 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(binunionEcases_type,type,
% 2.68/1.54 binunionEcases: $o ).
% 2.68/1.54
% 2.68/1.54 thf(binunionEcases,definition,
% 2.68/1.54 ( binunionEcases
% 2.68/1.54 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 2.68/1.54 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.68/1.54 => ( ( ( in @ Xx @ A )
% 2.68/1.54 => Xphi )
% 2.68/1.54 => ( ( ( in @ Xx @ B )
% 2.68/1.54 => Xphi )
% 2.68/1.54 => Xphi ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(binunionE_type,type,
% 2.68/1.54 binunionE: $o ).
% 2.68/1.54
% 2.68/1.54 thf(binunionE,definition,
% 2.68/1.54 ( binunionE
% 2.68/1.54 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.68/1.54 => ( ( in @ Xx @ A )
% 2.68/1.54 | ( in @ Xx @ B ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(binunionLsub_type,type,
% 2.68/1.54 binunionLsub: $o ).
% 2.68/1.54
% 2.68/1.54 thf(binunionLsub,definition,
% 2.68/1.54 ( binunionLsub
% 2.68/1.54 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(binunionRsub_type,type,
% 2.68/1.54 binunionRsub: $o ).
% 2.68/1.54
% 2.68/1.54 thf(binunionRsub,definition,
% 2.68/1.54 ( binunionRsub
% 2.68/1.54 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(binintersect_type,type,
% 2.68/1.54 binintersect: $i > $i > $i ).
% 2.68/1.54
% 2.68/1.54 thf(binintersectI_type,type,
% 2.68/1.54 binintersectI: $o ).
% 2.68/1.54
% 2.68/1.54 thf(binintersectI,definition,
% 2.68/1.54 ( binintersectI
% 2.68/1.54 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ A )
% 2.68/1.54 => ( ( in @ Xx @ B )
% 2.68/1.54 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(binintersectSubset5_type,type,
% 2.68/1.54 binintersectSubset5: $o ).
% 2.68/1.54
% 2.68/1.54 thf(binintersectSubset5,definition,
% 2.68/1.54 ( binintersectSubset5
% 2.68/1.54 = ( ! [A: $i,B: $i,C: $i] :
% 2.68/1.54 ( ( subset @ C @ A )
% 2.68/1.54 => ( ( subset @ C @ B )
% 2.68/1.54 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(binintersectEL_type,type,
% 2.68/1.54 binintersectEL: $o ).
% 2.68/1.54
% 2.68/1.54 thf(binintersectEL,definition,
% 2.68/1.54 ( binintersectEL
% 2.68/1.54 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.68/1.54 => ( in @ Xx @ A ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(binintersectLsub_type,type,
% 2.68/1.54 binintersectLsub: $o ).
% 2.68/1.54
% 2.68/1.54 thf(binintersectLsub,definition,
% 2.68/1.54 ( binintersectLsub
% 2.68/1.54 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(binintersectSubset2_type,type,
% 2.68/1.54 binintersectSubset2: $o ).
% 2.68/1.54
% 2.68/1.54 thf(binintersectSubset2,definition,
% 2.68/1.54 ( binintersectSubset2
% 2.68/1.54 = ( ! [A: $i,B: $i] :
% 2.68/1.54 ( ( subset @ A @ B )
% 2.68/1.54 => ( ( binintersect @ A @ B )
% 2.68/1.54 = A ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(binintersectSubset3_type,type,
% 2.68/1.54 binintersectSubset3: $o ).
% 2.68/1.54
% 2.68/1.54 thf(binintersectSubset3,definition,
% 2.68/1.54 ( binintersectSubset3
% 2.68/1.54 = ( ! [A: $i,B: $i] :
% 2.68/1.54 ( ( ( binintersect @ A @ B )
% 2.68/1.54 = B )
% 2.68/1.54 => ( subset @ B @ A ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(binintersectER_type,type,
% 2.68/1.54 binintersectER: $o ).
% 2.68/1.54
% 2.68/1.54 thf(binintersectER,definition,
% 2.68/1.54 ( binintersectER
% 2.68/1.54 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.68/1.54 => ( in @ Xx @ B ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(disjointsetsI1_type,type,
% 2.68/1.54 disjointsetsI1: $o ).
% 2.68/1.54
% 2.68/1.54 thf(disjointsetsI1,definition,
% 2.68/1.54 ( disjointsetsI1
% 2.68/1.54 = ( ! [A: $i,B: $i] :
% 2.68/1.54 ( ~ ? [Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ A )
% 2.68/1.54 & ( in @ Xx @ B ) )
% 2.68/1.54 => ( ( binintersect @ A @ B )
% 2.68/1.54 = emptyset ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(binintersectRsub_type,type,
% 2.68/1.54 binintersectRsub: $o ).
% 2.68/1.54
% 2.68/1.54 thf(binintersectRsub,definition,
% 2.68/1.54 ( binintersectRsub
% 2.68/1.54 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(binintersectSubset4_type,type,
% 2.68/1.54 binintersectSubset4: $o ).
% 2.68/1.54
% 2.68/1.54 thf(binintersectSubset4,definition,
% 2.68/1.54 ( binintersectSubset4
% 2.68/1.54 = ( ! [A: $i,B: $i] :
% 2.68/1.54 ( ( subset @ B @ A )
% 2.68/1.54 => ( ( binintersect @ A @ B )
% 2.68/1.54 = B ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(binintersectSubset1_type,type,
% 2.68/1.54 binintersectSubset1: $o ).
% 2.68/1.54
% 2.68/1.54 thf(binintersectSubset1,definition,
% 2.68/1.54 ( binintersectSubset1
% 2.68/1.54 = ( ! [A: $i,B: $i] :
% 2.68/1.54 ( ( ( binintersect @ A @ B )
% 2.68/1.54 = A )
% 2.68/1.54 => ( subset @ A @ B ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(bs114d_type,type,
% 2.68/1.54 bs114d: $o ).
% 2.68/1.54
% 2.68/1.54 thf(bs114d,definition,
% 2.68/1.54 ( bs114d
% 2.68/1.54 = ( ! [A: $i,B: $i,C: $i] :
% 2.68/1.54 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 2.68/1.54 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(regular_type,type,
% 2.68/1.54 regular: $i > $o ).
% 2.68/1.54
% 2.68/1.54 thf(setminus_type,type,
% 2.68/1.54 setminus: $i > $i > $i ).
% 2.68/1.54
% 2.68/1.54 thf(setminusI_type,type,
% 2.68/1.54 setminusI: $o ).
% 2.68/1.54
% 2.68/1.54 thf(setminusI,definition,
% 2.68/1.54 ( setminusI
% 2.68/1.54 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ A )
% 2.68/1.54 => ( ~ ( in @ Xx @ B )
% 2.68/1.54 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(setminusEL_type,type,
% 2.68/1.54 setminusEL: $o ).
% 2.68/1.54
% 2.68/1.54 thf(setminusEL,definition,
% 2.68/1.54 ( setminusEL
% 2.68/1.54 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.68/1.54 => ( in @ Xx @ A ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(setminusER_type,type,
% 2.68/1.54 setminusER: $o ).
% 2.68/1.54
% 2.68/1.54 thf(setminusER,definition,
% 2.68/1.54 ( setminusER
% 2.68/1.54 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.68/1.54 => ~ ( in @ Xx @ B ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(setminusSubset2_type,type,
% 2.68/1.54 setminusSubset2: $o ).
% 2.68/1.54
% 2.68/1.54 thf(setminusSubset2,definition,
% 2.68/1.54 ( setminusSubset2
% 2.68/1.54 = ( ! [A: $i,B: $i] :
% 2.68/1.54 ( ( subset @ A @ B )
% 2.68/1.54 => ( ( setminus @ A @ B )
% 2.68/1.54 = emptyset ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(setminusERneg_type,type,
% 2.68/1.54 setminusERneg: $o ).
% 2.68/1.54
% 2.68/1.54 thf(setminusERneg,definition,
% 2.68/1.54 ( setminusERneg
% 2.68/1.54 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.68/1.54 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.68/1.54 => ( ( in @ Xx @ A )
% 2.68/1.54 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(setminusELneg_type,type,
% 2.68/1.54 setminusELneg: $o ).
% 2.68/1.54
% 2.68/1.54 thf(setminusELneg,definition,
% 2.68/1.54 ( setminusELneg
% 2.68/1.54 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.68/1.54 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.68/1.54 => ( ~ ( in @ Xx @ B )
% 2.68/1.54 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(setminusILneg_type,type,
% 2.68/1.54 setminusILneg: $o ).
% 2.68/1.54
% 2.68/1.54 thf(setminusILneg,definition,
% 2.68/1.54 ( setminusILneg
% 2.68/1.54 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.68/1.54 ( ~ ( in @ Xx @ A )
% 2.68/1.54 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(setminusIRneg_type,type,
% 2.68/1.54 setminusIRneg: $o ).
% 2.68/1.54
% 2.68/1.54 thf(setminusIRneg,definition,
% 2.68/1.54 ( setminusIRneg
% 2.68/1.54 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ B )
% 2.68/1.54 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(setminusLsub_type,type,
% 2.68/1.54 setminusLsub: $o ).
% 2.68/1.54
% 2.68/1.54 thf(setminusLsub,definition,
% 2.68/1.54 ( setminusLsub
% 2.68/1.54 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(setminusSubset1_type,type,
% 2.68/1.54 setminusSubset1: $o ).
% 2.68/1.54
% 2.68/1.54 thf(setminusSubset1,definition,
% 2.68/1.54 ( setminusSubset1
% 2.68/1.54 = ( ! [A: $i,B: $i] :
% 2.68/1.54 ( ( ( setminus @ A @ B )
% 2.68/1.54 = emptyset )
% 2.68/1.54 => ( subset @ A @ B ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(symdiff_type,type,
% 2.68/1.54 symdiff: $i > $i > $i ).
% 2.68/1.54
% 2.68/1.54 thf(symdiffE_type,type,
% 2.68/1.54 symdiffE: $o ).
% 2.68/1.54
% 2.68/1.54 thf(symdiffE,definition,
% 2.68/1.54 ( symdiffE
% 2.68/1.54 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 2.68/1.54 => ! [Xphi: $o] :
% 2.68/1.54 ( ( ( in @ Xx @ A )
% 2.68/1.54 => ( ~ ( in @ Xx @ B )
% 2.68/1.54 => Xphi ) )
% 2.68/1.54 => ( ( ~ ( in @ Xx @ A )
% 2.68/1.54 => ( ( in @ Xx @ B )
% 2.68/1.54 => Xphi ) )
% 2.68/1.54 => Xphi ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(symdiffI1_type,type,
% 2.68/1.54 symdiffI1: $o ).
% 2.68/1.54
% 2.68/1.54 thf(symdiffI1,definition,
% 2.68/1.54 ( symdiffI1
% 2.68/1.54 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ A )
% 2.68/1.54 => ( ~ ( in @ Xx @ B )
% 2.68/1.54 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(symdiffI2_type,type,
% 2.68/1.54 symdiffI2: $o ).
% 2.68/1.54
% 2.68/1.54 thf(symdiffI2,definition,
% 2.68/1.54 ( symdiffI2
% 2.68/1.54 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.68/1.54 ( ~ ( in @ Xx @ A )
% 2.68/1.54 => ( ( in @ Xx @ B )
% 2.68/1.54 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(symdiffIneg1_type,type,
% 2.68/1.54 symdiffIneg1: $o ).
% 2.68/1.54
% 2.68/1.54 thf(symdiffIneg1,definition,
% 2.68/1.54 ( symdiffIneg1
% 2.68/1.54 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ A )
% 2.68/1.54 => ( ( in @ Xx @ B )
% 2.68/1.54 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(symdiffIneg2_type,type,
% 2.68/1.54 symdiffIneg2: $o ).
% 2.68/1.54
% 2.68/1.54 thf(symdiffIneg2,definition,
% 2.68/1.54 ( symdiffIneg2
% 2.68/1.54 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.68/1.54 ( ~ ( in @ Xx @ A )
% 2.68/1.54 => ( ~ ( in @ Xx @ B )
% 2.68/1.54 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(iskpair_type,type,
% 2.68/1.54 iskpair: $i > $o ).
% 2.68/1.54
% 2.68/1.54 thf(secondinupair_type,type,
% 2.68/1.54 secondinupair: $o ).
% 2.68/1.54
% 2.68/1.54 thf(secondinupair,definition,
% 2.68/1.54 ( secondinupair
% 2.68/1.54 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(setukpairIL_type,type,
% 2.68/1.54 setukpairIL: $o ).
% 2.68/1.54
% 2.68/1.54 thf(setukpairIL,definition,
% 2.68/1.54 ( setukpairIL
% 2.68/1.54 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(setukpairIR_type,type,
% 2.68/1.54 setukpairIR: $o ).
% 2.68/1.54
% 2.68/1.54 thf(setukpairIR,definition,
% 2.68/1.54 ( setukpairIR
% 2.68/1.54 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(kpairiskpair_type,type,
% 2.68/1.54 kpairiskpair: $o ).
% 2.68/1.54
% 2.68/1.54 thf(kpairiskpair,definition,
% 2.68/1.54 ( kpairiskpair
% 2.68/1.54 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(kpair_type,type,
% 2.68/1.54 kpair: $i > $i > $i ).
% 2.68/1.54
% 2.68/1.54 thf(kpairp_type,type,
% 2.68/1.54 kpairp: $o ).
% 2.68/1.54
% 2.68/1.54 thf(kpairp,definition,
% 2.68/1.54 ( kpairp
% 2.68/1.54 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(cartprod_type,type,
% 2.68/1.54 cartprod: $i > $i > $i ).
% 2.68/1.54
% 2.68/1.54 thf(singletonsubset_type,type,
% 2.68/1.54 singletonsubset: $o ).
% 2.68/1.54
% 2.68/1.54 thf(singletonsubset,definition,
% 2.68/1.54 ( singletonsubset
% 2.68/1.54 = ( ! [A: $i,Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ A )
% 2.68/1.54 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(singletoninpowerset_type,type,
% 2.68/1.54 singletoninpowerset: $o ).
% 2.68/1.54
% 2.68/1.54 thf(singletoninpowerset,definition,
% 2.68/1.54 ( singletoninpowerset
% 2.68/1.54 = ( ! [A: $i,Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ A )
% 2.68/1.54 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(singletoninpowunion_type,type,
% 2.68/1.54 singletoninpowunion: $o ).
% 2.68/1.54
% 2.68/1.54 thf(singletoninpowunion,definition,
% 2.68/1.54 ( singletoninpowunion
% 2.68/1.54 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ A )
% 2.68/1.54 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(upairset2E_type,type,
% 2.68/1.54 upairset2E: $o ).
% 2.68/1.54
% 2.68/1.54 thf(upairset2E,definition,
% 2.68/1.54 ( upairset2E
% 2.68/1.54 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.68/1.54 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.68/1.54 => ( ( Xz = Xx )
% 2.68/1.54 | ( Xz = Xy ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(upairsubunion_type,type,
% 2.68/1.54 upairsubunion: $o ).
% 2.68/1.54
% 2.68/1.54 thf(upairsubunion,definition,
% 2.68/1.54 ( upairsubunion
% 2.68/1.54 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ A )
% 2.68/1.54 => ! [Xy: $i] :
% 2.68/1.54 ( ( in @ Xy @ B )
% 2.68/1.54 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(upairinpowunion_type,type,
% 2.68/1.54 upairinpowunion: $o ).
% 2.68/1.54
% 2.68/1.54 thf(upairinpowunion,definition,
% 2.68/1.54 ( upairinpowunion
% 2.68/1.54 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ A )
% 2.68/1.54 => ! [Xy: $i] :
% 2.68/1.54 ( ( in @ Xy @ B )
% 2.68/1.54 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(ubforcartprodlem1_type,type,
% 2.68/1.54 ubforcartprodlem1: $o ).
% 2.68/1.54
% 2.68/1.54 thf(ubforcartprodlem1,definition,
% 2.68/1.54 ( ubforcartprodlem1
% 2.68/1.54 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ A )
% 2.68/1.54 => ! [Xy: $i] :
% 2.68/1.54 ( ( in @ Xy @ B )
% 2.68/1.54 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(ubforcartprodlem2_type,type,
% 2.68/1.54 ubforcartprodlem2: $o ).
% 2.68/1.54
% 2.68/1.54 thf(ubforcartprodlem2,definition,
% 2.68/1.54 ( ubforcartprodlem2
% 2.68/1.54 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ A )
% 2.68/1.54 => ! [Xy: $i] :
% 2.68/1.54 ( ( in @ Xy @ B )
% 2.68/1.54 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(ubforcartprodlem3_type,type,
% 2.68/1.54 ubforcartprodlem3: $o ).
% 2.68/1.54
% 2.68/1.54 thf(ubforcartprodlem3,definition,
% 2.68/1.54 ( ubforcartprodlem3
% 2.68/1.54 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ A )
% 2.68/1.54 => ! [Xy: $i] :
% 2.68/1.54 ( ( in @ Xy @ B )
% 2.68/1.54 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(cartprodpairin_type,type,
% 2.68/1.54 cartprodpairin: $o ).
% 2.68/1.54
% 2.68/1.54 thf(cartprodpairin,definition,
% 2.68/1.54 ( cartprodpairin
% 2.68/1.54 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ A )
% 2.68/1.54 => ! [Xy: $i] :
% 2.68/1.54 ( ( in @ Xy @ B )
% 2.68/1.54 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(cartprodmempair1_type,type,
% 2.68/1.54 cartprodmempair1: $o ).
% 2.68/1.54
% 2.68/1.54 thf(cartprodmempair1,definition,
% 2.68/1.54 ( cartprodmempair1
% 2.68/1.54 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.68/1.54 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.68/1.54 => ? [Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ A )
% 2.68/1.54 & ? [Xy: $i] :
% 2.68/1.54 ( ( in @ Xy @ B )
% 2.68/1.54 & ( Xu
% 2.68/1.54 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(cartprodmempair_type,type,
% 2.68/1.54 cartprodmempair: $o ).
% 2.68/1.54
% 2.68/1.54 thf(cartprodmempair,definition,
% 2.68/1.54 ( cartprodmempair
% 2.68/1.54 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.68/1.54 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.68/1.54 => ( iskpair @ Xu ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(setunionE2_type,type,
% 2.68/1.54 setunionE2: $o ).
% 2.68/1.54
% 2.68/1.54 thf(setunionE2,definition,
% 2.68/1.54 ( setunionE2
% 2.68/1.54 = ( ! [A: $i,Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ ( setunion @ A ) )
% 2.68/1.54 => ? [X: $i] :
% 2.68/1.54 ( ( in @ X @ A )
% 2.68/1.54 & ( in @ Xx @ X ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(setunionsingleton1_type,type,
% 2.68/1.54 setunionsingleton1: $o ).
% 2.68/1.54
% 2.68/1.54 thf(setunionsingleton1,definition,
% 2.68/1.54 ( setunionsingleton1
% 2.68/1.54 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(setunionsingleton2_type,type,
% 2.68/1.54 setunionsingleton2: $o ).
% 2.68/1.54
% 2.68/1.54 thf(setunionsingleton2,definition,
% 2.68/1.54 ( setunionsingleton2
% 2.68/1.54 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(setunionsingleton_type,type,
% 2.68/1.54 setunionsingleton: $o ).
% 2.68/1.54
% 2.68/1.54 thf(setunionsingleton,definition,
% 2.68/1.54 ( setunionsingleton
% 2.68/1.54 = ( ! [Xx: $i] :
% 2.68/1.54 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 2.68/1.54 = Xx ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(singleton_type,type,
% 2.68/1.54 singleton: $i > $o ).
% 2.68/1.54
% 2.68/1.54 thf(singleton,definition,
% 2.68/1.54 ( singleton
% 2.68/1.54 = ( ^ [A: $i] :
% 2.68/1.54 ? [Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ A )
% 2.68/1.54 & ( A
% 2.68/1.54 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(singletonprop_type,type,
% 2.68/1.54 singletonprop: $o ).
% 2.68/1.54
% 2.68/1.54 thf(singletonprop,definition,
% 2.68/1.54 ( singletonprop
% 2.68/1.54 = ( ! [A: $i,Xphi: $i > $o] :
% 2.68/1.54 ( ! [Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ A )
% 2.68/1.54 => ! [Xy: $i] :
% 2.68/1.54 ( ( in @ Xy @ A )
% 2.68/1.54 => ( ( Xphi @ Xx )
% 2.68/1.54 => ( ( Xphi @ Xy )
% 2.68/1.54 => ( Xx = Xy ) ) ) ) )
% 2.68/1.54 => ( ? [Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ A )
% 2.68/1.54 & ( Xphi @ Xx ) )
% 2.68/1.54 => ( singleton
% 2.68/1.54 @ ( dsetconstr @ A
% 2.68/1.54 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(ex1_type,type,
% 2.68/1.54 ex1: $i > ( $i > $o ) > $o ).
% 2.68/1.54
% 2.68/1.54 thf(ex1,definition,
% 2.68/1.54 ( ex1
% 2.68/1.54 = ( ^ [A: $i,Xphi: $i > $o] :
% 2.68/1.54 ( singleton
% 2.68/1.54 @ ( dsetconstr @ A
% 2.68/1.54 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(ex1E1_type,type,
% 2.68/1.54 ex1E1: $o ).
% 2.68/1.54
% 2.68/1.54 thf(ex1E1,definition,
% 2.68/1.54 ( ex1E1
% 2.68/1.54 = ( ! [A: $i,Xphi: $i > $o] :
% 2.68/1.54 ( ( ex1 @ A
% 2.68/1.54 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.68/1.54 => ? [Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ A )
% 2.68/1.54 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(ex1I_type,type,
% 2.68/1.54 ex1I: $o ).
% 2.68/1.54
% 2.68/1.54 thf(ex1I,definition,
% 2.68/1.54 ( ex1I
% 2.68/1.54 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ A )
% 2.68/1.54 => ( ( Xphi @ Xx )
% 2.68/1.54 => ( ! [Xy: $i] :
% 2.68/1.54 ( ( in @ Xy @ A )
% 2.68/1.54 => ( ( Xphi @ Xy )
% 2.68/1.54 => ( Xy = Xx ) ) )
% 2.68/1.54 => ( ex1 @ A
% 2.68/1.54 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(ex1I2_type,type,
% 2.68/1.54 ex1I2: $o ).
% 2.68/1.54
% 2.68/1.54 thf(ex1I2,definition,
% 2.68/1.54 ( ex1I2
% 2.68/1.54 = ( ! [A: $i,Xphi: $i > $o] :
% 2.68/1.54 ( ! [Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ A )
% 2.68/1.54 => ! [Xy: $i] :
% 2.68/1.54 ( ( in @ Xy @ A )
% 2.68/1.54 => ( ( Xphi @ Xx )
% 2.68/1.54 => ( ( Xphi @ Xy )
% 2.68/1.54 => ( Xx = Xy ) ) ) ) )
% 2.68/1.54 => ( ? [Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ A )
% 2.68/1.54 & ( Xphi @ Xx ) )
% 2.68/1.54 => ( ex1 @ A
% 2.68/1.54 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(singletonsuniq_type,type,
% 2.68/1.54 singletonsuniq: $o ).
% 2.68/1.54
% 2.68/1.54 thf(singletonsuniq,definition,
% 2.68/1.54 ( singletonsuniq
% 2.68/1.54 = ( ! [Xx: $i,Xy: $i] :
% 2.68/1.54 ( ( ( setadjoin @ Xx @ emptyset )
% 2.68/1.54 = ( setadjoin @ Xy @ emptyset ) )
% 2.68/1.54 => ( Xx = Xy ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(atmost1p_type,type,
% 2.68/1.54 atmost1p: $i > $o ).
% 2.68/1.54
% 2.68/1.54 thf(atleast2p_type,type,
% 2.68/1.54 atleast2p: $i > $o ).
% 2.68/1.54
% 2.68/1.54 thf(atmost2p_type,type,
% 2.68/1.54 atmost2p: $i > $o ).
% 2.68/1.54
% 2.68/1.54 thf(upairsetp_type,type,
% 2.68/1.54 upairsetp: $i > $o ).
% 2.68/1.54
% 2.68/1.54 thf(setukpairinjL1_type,type,
% 2.68/1.54 setukpairinjL1: $o ).
% 2.68/1.54
% 2.68/1.54 thf(setukpairinjL1,definition,
% 2.68/1.54 ( setukpairinjL1
% 2.68/1.54 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.68/1.54 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 2.68/1.54 => ( Xx = Xz ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(kfstsingleton_type,type,
% 2.68/1.54 kfstsingleton: $o ).
% 2.68/1.54
% 2.68/1.54 thf(kfstsingleton,definition,
% 2.68/1.54 ( kfstsingleton
% 2.68/1.54 = ( ! [Xu: $i] :
% 2.68/1.54 ( ( iskpair @ Xu )
% 2.68/1.54 => ( singleton
% 2.68/1.54 @ ( dsetconstr @ ( setunion @ Xu )
% 2.68/1.54 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(theprop_type,type,
% 2.68/1.54 theprop: $o ).
% 2.68/1.54
% 2.68/1.54 thf(theprop,definition,
% 2.68/1.54 ( theprop
% 2.68/1.54 = ( ! [X: $i] :
% 2.68/1.54 ( ( singleton @ X )
% 2.68/1.54 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(kfst_type,type,
% 2.68/1.54 kfst: $i > $i ).
% 2.68/1.54
% 2.68/1.54 thf(kfstpairEq_type,type,
% 2.68/1.54 kfstpairEq: $o ).
% 2.68/1.54
% 2.68/1.54 thf(kfstpairEq,definition,
% 2.68/1.54 ( kfstpairEq
% 2.68/1.54 = ( ! [Xx: $i,Xy: $i] :
% 2.68/1.54 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 2.68/1.54 = Xx ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(cartprodfstin_type,type,
% 2.68/1.54 cartprodfstin: $o ).
% 2.68/1.54
% 2.68/1.54 thf(cartprodfstin,definition,
% 2.68/1.54 ( cartprodfstin
% 2.68/1.54 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.68/1.54 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.68/1.54 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(setukpairinjL2_type,type,
% 2.68/1.54 setukpairinjL2: $o ).
% 2.68/1.54
% 2.68/1.54 thf(setukpairinjL2,definition,
% 2.68/1.54 ( setukpairinjL2
% 2.68/1.54 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.68/1.54 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.68/1.54 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.68/1.54 => ( Xx = Xz ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(setukpairinjL_type,type,
% 2.68/1.54 setukpairinjL: $o ).
% 2.68/1.54
% 2.68/1.54 thf(setukpairinjL,definition,
% 2.68/1.54 ( setukpairinjL
% 2.68/1.54 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.68/1.54 ( ( ( kpair @ Xx @ Xy )
% 2.68/1.54 = ( kpair @ Xz @ Xu ) )
% 2.68/1.54 => ( Xx = Xz ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(setukpairinjR11_type,type,
% 2.68/1.54 setukpairinjR11: $o ).
% 2.68/1.54
% 2.68/1.54 thf(setukpairinjR11,definition,
% 2.68/1.54 ( setukpairinjR11
% 2.68/1.54 = ( ! [Xx: $i,Xy: $i] :
% 2.68/1.54 ( ( Xx = Xy )
% 2.68/1.54 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.68/1.54 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(setukpairinjR12_type,type,
% 2.68/1.54 setukpairinjR12: $o ).
% 2.68/1.54
% 2.68/1.54 thf(setukpairinjR12,definition,
% 2.68/1.54 ( setukpairinjR12
% 2.68/1.54 = ( ! [Xx: $i,Xy: $i] :
% 2.68/1.54 ( ( Xx = Xy )
% 2.68/1.54 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.68/1.54 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(setukpairinjR1_type,type,
% 2.68/1.54 setukpairinjR1: $o ).
% 2.68/1.54
% 2.68/1.54 thf(setukpairinjR1,definition,
% 2.68/1.54 ( setukpairinjR1
% 2.68/1.54 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.68/1.54 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.68/1.54 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.68/1.54 => ( ( Xz = Xu )
% 2.68/1.54 => ( Xy = Xu ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(upairequniteq_type,type,
% 2.68/1.54 upairequniteq: $o ).
% 2.68/1.54
% 2.68/1.54 thf(upairequniteq,definition,
% 2.68/1.54 ( upairequniteq
% 2.68/1.54 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.68/1.54 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.68/1.54 = ( setadjoin @ Xz @ emptyset ) )
% 2.68/1.54 => ( Xx = Xy ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(setukpairinjR2_type,type,
% 2.68/1.54 setukpairinjR2: $o ).
% 2.68/1.54
% 2.68/1.54 thf(setukpairinjR2,definition,
% 2.68/1.54 ( setukpairinjR2
% 2.68/1.54 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.68/1.54 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.68/1.54 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.68/1.54 => ( Xy = Xu ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(setukpairinjR_type,type,
% 2.68/1.54 setukpairinjR: $o ).
% 2.68/1.54
% 2.68/1.54 thf(setukpairinjR,definition,
% 2.68/1.54 ( setukpairinjR
% 2.68/1.54 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.68/1.54 ( ( ( kpair @ Xx @ Xy )
% 2.68/1.54 = ( kpair @ Xz @ Xu ) )
% 2.68/1.54 => ( Xy = Xu ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(ksndsingleton_type,type,
% 2.68/1.54 ksndsingleton: $o ).
% 2.68/1.54
% 2.68/1.54 thf(ksndsingleton,definition,
% 2.68/1.54 ( ksndsingleton
% 2.68/1.54 = ( ! [Xu: $i] :
% 2.68/1.54 ( ( iskpair @ Xu )
% 2.68/1.54 => ( singleton
% 2.68/1.54 @ ( dsetconstr @ ( setunion @ Xu )
% 2.68/1.54 @ ^ [Xx: $i] :
% 2.68/1.54 ( Xu
% 2.68/1.54 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(ksnd_type,type,
% 2.68/1.54 ksnd: $i > $i ).
% 2.68/1.54
% 2.68/1.54 thf(ksndpairEq_type,type,
% 2.68/1.54 ksndpairEq: $o ).
% 2.68/1.54
% 2.68/1.54 thf(ksndpairEq,definition,
% 2.68/1.54 ( ksndpairEq
% 2.68/1.54 = ( ! [Xx: $i,Xy: $i] :
% 2.68/1.54 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 2.68/1.54 = Xy ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(kpairsurjEq_type,type,
% 2.68/1.54 kpairsurjEq: $o ).
% 2.68/1.54
% 2.68/1.54 thf(kpairsurjEq,definition,
% 2.68/1.54 ( kpairsurjEq
% 2.68/1.54 = ( ! [Xu: $i] :
% 2.68/1.54 ( ( iskpair @ Xu )
% 2.68/1.54 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 2.68/1.54 = Xu ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(cartprodsndin_type,type,
% 2.68/1.54 cartprodsndin: $o ).
% 2.68/1.54
% 2.68/1.54 thf(cartprodsndin,definition,
% 2.68/1.54 ( cartprodsndin
% 2.68/1.54 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.68/1.54 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.68/1.54 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(cartprodpairmemEL_type,type,
% 2.68/1.54 cartprodpairmemEL: $o ).
% 2.68/1.54
% 2.68/1.54 thf(cartprodpairmemEL,definition,
% 2.68/1.54 ( cartprodpairmemEL
% 2.68/1.54 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 2.68/1.54 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 2.68/1.54 => ( in @ Xx @ A ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(cartprodpairmemER_type,type,
% 2.68/1.54 cartprodpairmemER: $o ).
% 2.68/1.54
% 2.68/1.54 thf(cartprodpairmemER,definition,
% 2.68/1.54 ( cartprodpairmemER
% 2.68/1.54 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 2.68/1.54 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 2.68/1.54 => ( in @ Xy @ B ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(cartprodmempaircEq_type,type,
% 2.68/1.54 cartprodmempaircEq: $o ).
% 2.68/1.54
% 2.68/1.54 thf(cartprodmempaircEq,definition,
% 2.68/1.54 ( cartprodmempaircEq
% 2.68/1.54 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ A )
% 2.68/1.54 => ! [Xy: $i] :
% 2.68/1.54 ( ( in @ Xy @ B )
% 2.68/1.54 => ( ( kpair @ Xx @ Xy )
% 2.68/1.54 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(cartprodfstpairEq_type,type,
% 2.68/1.54 cartprodfstpairEq: $o ).
% 2.68/1.54
% 2.68/1.54 thf(cartprodfstpairEq,definition,
% 2.68/1.54 ( cartprodfstpairEq
% 2.68/1.54 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ A )
% 2.68/1.54 => ! [Xy: $i] :
% 2.68/1.54 ( ( in @ Xy @ B )
% 2.68/1.54 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 2.68/1.54 = Xx ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(cartprodsndpairEq_type,type,
% 2.68/1.54 cartprodsndpairEq: $o ).
% 2.68/1.54
% 2.68/1.54 thf(cartprodsndpairEq,definition,
% 2.68/1.54 ( cartprodsndpairEq
% 2.68/1.54 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ A )
% 2.68/1.54 => ! [Xy: $i] :
% 2.68/1.54 ( ( in @ Xy @ B )
% 2.68/1.54 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 2.68/1.54 = Xy ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(cartprodpairsurjEq_type,type,
% 2.68/1.54 cartprodpairsurjEq: $o ).
% 2.68/1.54
% 2.68/1.54 thf(cartprodpairsurjEq,definition,
% 2.68/1.54 ( cartprodpairsurjEq
% 2.68/1.54 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.68/1.54 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.68/1.54 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 2.68/1.54 = Xu ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(breln_type,type,
% 2.68/1.54 breln: $i > $i > $i > $o ).
% 2.68/1.54
% 2.68/1.54 thf(breln,definition,
% 2.68/1.54 ( breln
% 2.68/1.54 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(dpsetconstr_type,type,
% 2.68/1.54 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 2.68/1.54
% 2.68/1.54 thf(dpsetconstrI_type,type,
% 2.68/1.54 dpsetconstrI: $o ).
% 2.68/1.54
% 2.68/1.54 thf(dpsetconstrI,definition,
% 2.68/1.54 ( dpsetconstrI
% 2.68/1.54 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ A )
% 2.68/1.54 => ! [Xy: $i] :
% 2.68/1.54 ( ( in @ Xy @ B )
% 2.68/1.54 => ( ( Xphi @ Xx @ Xy )
% 2.68/1.54 => ( in @ ( kpair @ Xx @ Xy )
% 2.68/1.54 @ ( dpsetconstr @ A @ B
% 2.68/1.54 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(dpsetconstrSub_type,type,
% 2.68/1.54 dpsetconstrSub: $o ).
% 2.68/1.54
% 2.68/1.54 thf(dpsetconstrSub,definition,
% 2.68/1.54 ( dpsetconstrSub
% 2.68/1.54 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 2.68/1.54 ( subset
% 2.68/1.54 @ ( dpsetconstr @ A @ B
% 2.68/1.54 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 2.68/1.54 @ ( cartprod @ A @ B ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(setOfPairsIsBReln_type,type,
% 2.68/1.54 setOfPairsIsBReln: $o ).
% 2.68/1.54
% 2.68/1.54 thf(setOfPairsIsBReln,definition,
% 2.68/1.54 ( setOfPairsIsBReln
% 2.68/1.54 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 2.68/1.54 ( breln @ A @ B
% 2.68/1.54 @ ( dpsetconstr @ A @ B
% 2.68/1.54 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(dpsetconstrERa_type,type,
% 2.68/1.54 dpsetconstrERa: $o ).
% 2.68/1.54
% 2.68/1.54 thf(dpsetconstrERa,definition,
% 2.68/1.54 ( dpsetconstrERa
% 2.68/1.54 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ A )
% 2.68/1.54 => ! [Xy: $i] :
% 2.68/1.54 ( ( in @ Xy @ B )
% 2.68/1.54 => ( ( in @ ( kpair @ Xx @ Xy )
% 2.68/1.54 @ ( dpsetconstr @ A @ B
% 2.68/1.54 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.68/1.54 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(dpsetconstrEL1_type,type,
% 2.68/1.54 dpsetconstrEL1: $o ).
% 2.68/1.54
% 2.68/1.54 thf(dpsetconstrEL1,definition,
% 2.68/1.54 ( dpsetconstrEL1
% 2.68/1.54 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.68/1.54 ( ( in @ ( kpair @ Xx @ Xy )
% 2.68/1.54 @ ( dpsetconstr @ A @ B
% 2.68/1.54 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.68/1.54 => ( in @ Xx @ A ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(dpsetconstrEL2_type,type,
% 2.68/1.54 dpsetconstrEL2: $o ).
% 2.68/1.54
% 2.68/1.54 thf(dpsetconstrEL2,definition,
% 2.68/1.54 ( dpsetconstrEL2
% 2.68/1.54 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.68/1.54 ( ( in @ ( kpair @ Xx @ Xy )
% 2.68/1.54 @ ( dpsetconstr @ A @ B
% 2.68/1.54 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.68/1.54 => ( in @ Xy @ B ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(dpsetconstrER_type,type,
% 2.68/1.54 dpsetconstrER: $o ).
% 2.68/1.54
% 2.68/1.54 thf(dpsetconstrER,definition,
% 2.68/1.54 ( dpsetconstrER
% 2.68/1.54 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.68/1.54 ( ( in @ ( kpair @ Xx @ Xy )
% 2.68/1.54 @ ( dpsetconstr @ A @ B
% 2.68/1.54 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.68/1.54 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(func_type,type,
% 2.68/1.54 func: $i > $i > $i > $o ).
% 2.68/1.54
% 2.68/1.54 thf(func,definition,
% 2.68/1.54 ( func
% 2.68/1.54 = ( ^ [A: $i,B: $i,R: $i] :
% 2.68/1.54 ( ( breln @ A @ B @ R )
% 2.68/1.54 & ! [Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ A )
% 2.68/1.54 => ( ex1 @ B
% 2.68/1.54 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(funcSet_type,type,
% 2.68/1.54 funcSet: $i > $i > $i ).
% 2.68/1.54
% 2.68/1.54 thf(funcImageSingleton_type,type,
% 2.68/1.54 funcImageSingleton: $o ).
% 2.68/1.54
% 2.68/1.54 thf(funcImageSingleton,definition,
% 2.68/1.54 ( funcImageSingleton
% 2.68/1.54 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.68/1.54 ( ( func @ A @ B @ Xf )
% 2.68/1.54 => ! [Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ A )
% 2.68/1.54 => ( singleton
% 2.68/1.54 @ ( dsetconstr @ B
% 2.68/1.54 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(apProp_type,type,
% 2.68/1.54 apProp: $o ).
% 2.68/1.54
% 2.68/1.54 thf(apProp,definition,
% 2.68/1.54 ( apProp
% 2.68/1.54 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.68/1.54 ( ( func @ A @ B @ Xf )
% 2.68/1.54 => ! [Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ A )
% 2.68/1.54 => ( in
% 2.68/1.54 @ ( setunion
% 2.68/1.54 @ ( dsetconstr @ B
% 2.68/1.54 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 2.68/1.54 @ B ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(ap_type,type,
% 2.68/1.54 ap: $i > $i > $i > $i > $i ).
% 2.68/1.54
% 2.68/1.54 thf(app_type,type,
% 2.68/1.54 app: $o ).
% 2.68/1.54
% 2.68/1.54 thf(app,definition,
% 2.68/1.54 ( app
% 2.68/1.54 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.68/1.54 ( ( func @ A @ B @ Xf )
% 2.68/1.54 => ! [Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ A )
% 2.68/1.54 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(infuncsetfunc_type,type,
% 2.68/1.54 infuncsetfunc: $o ).
% 2.68/1.54
% 2.68/1.54 thf(infuncsetfunc,definition,
% 2.68/1.54 ( infuncsetfunc
% 2.68/1.54 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.68/1.54 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.68/1.54 => ( func @ A @ B @ Xf ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(ap2p_type,type,
% 2.68/1.54 ap2p: $o ).
% 2.68/1.54
% 2.68/1.54 thf(ap2p,definition,
% 2.68/1.54 ( ap2p
% 2.68/1.54 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.68/1.54 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.68/1.54 => ! [Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ A )
% 2.68/1.54 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(funcinfuncset_type,type,
% 2.68/1.54 funcinfuncset: $o ).
% 2.68/1.54
% 2.68/1.54 thf(funcinfuncset,definition,
% 2.68/1.54 ( funcinfuncset
% 2.68/1.54 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.68/1.54 ( ( func @ A @ B @ Xf )
% 2.68/1.54 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(lamProp_type,type,
% 2.68/1.54 lamProp: $o ).
% 2.68/1.54
% 2.68/1.54 thf(lamProp,definition,
% 2.68/1.54 ( lamProp
% 2.68/1.54 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.68/1.54 ( ! [Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ A )
% 2.68/1.54 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.68/1.54 => ( func @ A @ B
% 2.68/1.54 @ ( dpsetconstr @ A @ B
% 2.68/1.54 @ ^ [Xx: $i,Xy: $i] :
% 2.68/1.54 ( ( Xf @ Xx )
% 2.68/1.54 = Xy ) ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(lam_type,type,
% 2.68/1.54 lam: $i > $i > ( $i > $i ) > $i ).
% 2.68/1.54
% 2.68/1.54 thf(lam,definition,
% 2.68/1.54 ( lam
% 2.68/1.54 = ( ^ [A: $i,B: $i,Xf: $i > $i] :
% 2.68/1.54 ( dpsetconstr @ A @ B
% 2.68/1.54 @ ^ [Xx: $i,Xy: $i] :
% 2.68/1.54 ( ( Xf @ Xx )
% 2.68/1.54 = Xy ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(lamp_type,type,
% 2.68/1.54 lamp: $o ).
% 2.68/1.54
% 2.68/1.54 thf(lamp,definition,
% 2.68/1.54 ( lamp
% 2.68/1.54 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.68/1.54 ( ! [Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ A )
% 2.68/1.54 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.68/1.54 => ( func @ A @ B
% 2.68/1.54 @ ( lam @ A @ B
% 2.68/1.54 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(lam2p_type,type,
% 2.68/1.54 lam2p: $o ).
% 2.68/1.54
% 2.68/1.54 thf(lam2p,definition,
% 2.68/1.54 ( lam2p
% 2.68/1.54 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.68/1.54 ( ! [Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ A )
% 2.68/1.54 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.68/1.54 => ( in
% 2.68/1.54 @ ( lam @ A @ B
% 2.68/1.54 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 2.68/1.54 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(brelnall1_type,type,
% 2.68/1.54 brelnall1: $o ).
% 2.68/1.54
% 2.68/1.54 thf(brelnall1,definition,
% 2.68/1.54 ( brelnall1
% 2.68/1.54 = ( ! [A: $i,B: $i,R: $i] :
% 2.68/1.54 ( ( breln @ A @ B @ R )
% 2.68/1.54 => ! [Xphi: $i > $o] :
% 2.68/1.54 ( ! [Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ A )
% 2.68/1.54 => ! [Xy: $i] :
% 2.68/1.54 ( ( in @ Xy @ B )
% 2.68/1.54 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.68/1.54 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 2.68/1.54 => ! [Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ R )
% 2.68/1.54 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(brelnall2_type,type,
% 2.68/1.54 brelnall2: $o ).
% 2.68/1.54
% 2.68/1.54 thf(brelnall2,definition,
% 2.68/1.54 ( brelnall2
% 2.68/1.54 = ( ! [A: $i,B: $i,R: $i] :
% 2.68/1.54 ( ( breln @ A @ B @ R )
% 2.68/1.54 => ! [Xphi: $i > $o] :
% 2.68/1.54 ( ! [Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ A )
% 2.68/1.54 => ! [Xy: $i] :
% 2.68/1.54 ( ( in @ Xy @ B )
% 2.68/1.54 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.68/1.54 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 2.68/1.54 => ! [Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ R )
% 2.68/1.54 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(ex1E2_type,type,
% 2.68/1.54 ex1E2: $o ).
% 2.68/1.54
% 2.68/1.54 thf(ex1E2,definition,
% 2.68/1.54 ( ex1E2
% 2.68/1.54 = ( ! [A: $i,Xphi: $i > $o] :
% 2.68/1.54 ( ( ex1 @ A
% 2.68/1.54 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.68/1.54 => ! [Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ A )
% 2.68/1.54 => ! [Xy: $i] :
% 2.68/1.54 ( ( in @ Xy @ A )
% 2.68/1.54 => ( ( Xphi @ Xx )
% 2.68/1.54 => ( ( Xphi @ Xy )
% 2.68/1.54 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(funcGraphProp1_type,type,
% 2.68/1.54 funcGraphProp1: $o ).
% 2.68/1.54
% 2.68/1.54 thf(funcGraphProp1,definition,
% 2.68/1.54 ( funcGraphProp1
% 2.68/1.54 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.68/1.54 ( ( func @ A @ B @ Xf )
% 2.68/1.54 => ! [Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ A )
% 2.68/1.54 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(funcGraphProp3_type,type,
% 2.68/1.54 funcGraphProp3: $o ).
% 2.68/1.54
% 2.68/1.54 thf(funcGraphProp3,definition,
% 2.68/1.54 ( funcGraphProp3
% 2.68/1.54 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.68/1.54 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.68/1.54 => ! [Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ A )
% 2.68/1.54 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(funcGraphProp2_type,type,
% 2.68/1.54 funcGraphProp2: $o ).
% 2.68/1.54
% 2.68/1.54 thf(funcGraphProp2,definition,
% 2.68/1.54 ( funcGraphProp2
% 2.68/1.54 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.68/1.54 ( ( func @ A @ B @ Xf )
% 2.68/1.54 => ! [Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ A )
% 2.68/1.54 => ! [Xy: $i] :
% 2.68/1.54 ( ( in @ Xy @ B )
% 2.68/1.54 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 2.68/1.54 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.68/1.54 = Xy ) ) ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(funcextLem_type,type,
% 2.68/1.54 funcextLem: $o ).
% 2.68/1.54
% 2.68/1.54 thf(funcextLem,definition,
% 2.68/1.54 ( funcextLem
% 2.68/1.54 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.68/1.54 ( ( func @ A @ B @ Xf )
% 2.68/1.54 => ! [Xg: $i] :
% 2.68/1.54 ( ( func @ A @ B @ Xg )
% 2.68/1.54 => ( ! [Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ A )
% 2.68/1.54 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.68/1.54 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.68/1.54 => ! [Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ A )
% 2.68/1.54 => ! [Xy: $i] :
% 2.68/1.54 ( ( in @ Xy @ B )
% 2.68/1.54 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 2.68/1.54 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 2.68/1.54
% 2.68/1.54 thf(funcGraphProp4,conjecture,
% 2.68/1.54 ( setextAx
% 2.68/1.54 => ( emptysetAx
% 2.68/1.54 => ( setadjoinAx
% 2.68/1.54 => ( powersetAx
% 2.68/1.54 => ( setunionAx
% 2.68/1.54 => ( omega0Ax
% 2.68/1.54 => ( omegaSAx
% 2.68/1.54 => ( omegaIndAx
% 2.68/1.54 => ( replAx
% 2.68/1.54 => ( foundationAx
% 2.68/1.54 => ( wellorderingAx
% 2.68/1.54 => ( descrp
% 2.68/1.54 => ( dsetconstrI
% 2.68/1.54 => ( dsetconstrEL
% 2.68/1.54 => ( dsetconstrER
% 2.68/1.54 => ( exuE1
% 2.68/1.54 => ( prop2setE
% 2.68/1.54 => ( emptysetE
% 2.68/1.54 => ( emptysetimpfalse
% 2.68/1.54 => ( notinemptyset
% 2.68/1.54 => ( exuE3e
% 2.68/1.54 => ( setext
% 2.68/1.54 => ( emptyI
% 2.68/1.54 => ( noeltsimpempty
% 2.68/1.54 => ( setbeta
% 2.68/1.54 => ( nonemptyE1
% 2.68/1.54 => ( nonemptyI
% 2.68/1.54 => ( nonemptyI1
% 2.68/1.54 => ( setadjoinIL
% 2.68/1.54 => ( emptyinunitempty
% 2.68/1.54 => ( setadjoinIR
% 2.68/1.54 => ( setadjoinE
% 2.68/1.54 => ( setadjoinOr
% 2.68/1.54 => ( setoftrueEq
% 2.68/1.54 => ( powersetI
% 2.68/1.54 => ( emptyinPowerset
% 2.68/1.54 => ( emptyInPowerset
% 2.68/1.54 => ( powersetE
% 2.68/1.54 => ( setunionI
% 2.68/1.54 => ( setunionE
% 2.68/1.54 => ( subPowSU
% 2.68/1.54 => ( exuE2
% 2.68/1.54 => ( nonemptyImpWitness
% 2.68/1.54 => ( uniqinunit
% 2.68/1.54 => ( notinsingleton
% 2.68/1.54 => ( eqinunit
% 2.68/1.54 => ( singletonsswitch
% 2.68/1.54 => ( upairsetE
% 2.68/1.54 => ( upairsetIL
% 2.68/1.54 => ( upairsetIR
% 2.68/1.54 => ( emptyE1
% 2.68/1.54 => ( vacuousDall
% 2.68/1.54 => ( quantDeMorgan1
% 2.68/1.54 => ( quantDeMorgan2
% 2.68/1.54 => ( quantDeMorgan3
% 2.68/1.54 => ( quantDeMorgan4
% 2.68/1.54 => ( prop2setI
% 2.68/1.54 => ( prop2set2propI
% 2.68/1.54 => ( notdexE
% 2.68/1.54 => ( notdallE
% 2.68/1.54 => ( exuI1
% 2.68/1.54 => ( exuI3
% 2.68/1.54 => ( exuI2
% 2.68/1.54 => ( inCongP
% 2.68/1.54 => ( in__Cong
% 2.68/1.54 => ( exuE3u
% 2.68/1.54 => ( exu__Cong
% 2.68/1.54 => ( emptyset__Cong
% 2.68/1.54 => ( setadjoin__Cong
% 2.68/1.54 => ( powerset__Cong
% 2.68/1.54 => ( setunion__Cong
% 2.68/1.54 => ( omega__Cong
% 2.68/1.54 => ( exuEu
% 2.68/1.54 => ( descr__Cong
% 2.68/1.54 => ( dsetconstr__Cong
% 2.68/1.54 => ( subsetI1
% 2.68/1.54 => ( eqimpsubset2
% 2.68/1.54 => ( eqimpsubset1
% 2.68/1.54 => ( subsetI2
% 2.68/1.54 => ( emptysetsubset
% 2.68/1.54 => ( subsetE
% 2.68/1.54 => ( subsetE2
% 2.68/1.54 => ( notsubsetI
% 2.68/1.54 => ( notequalI1
% 2.68/1.54 => ( notequalI2
% 2.68/1.54 => ( subsetRefl
% 2.68/1.54 => ( subsetTrans
% 2.68/1.54 => ( setadjoinSub
% 2.68/1.54 => ( setadjoinSub2
% 2.68/1.54 => ( subset2powerset
% 2.68/1.54 => ( setextsub
% 2.68/1.54 => ( subsetemptysetimpeq
% 2.68/1.54 => ( powersetI1
% 2.68/1.54 => ( powersetE1
% 2.68/1.54 => ( inPowerset
% 2.68/1.54 => ( powersetsubset
% 2.68/1.54 => ( sepInPowerset
% 2.68/1.54 => ( sepSubset
% 2.68/1.54 => ( binunionIL
% 2.68/1.54 => ( upairset2IR
% 2.68/1.54 => ( binunionIR
% 2.68/1.54 => ( binunionEcases
% 2.68/1.54 => ( binunionE
% 2.68/1.54 => ( binunionLsub
% 2.68/1.54 => ( binunionRsub
% 2.68/1.54 => ( binintersectI
% 2.68/1.54 => ( binintersectSubset5
% 2.68/1.54 => ( binintersectEL
% 2.68/1.54 => ( binintersectLsub
% 2.68/1.54 => ( binintersectSubset2
% 2.68/1.54 => ( binintersectSubset3
% 2.68/1.54 => ( binintersectER
% 2.68/1.54 => ( disjointsetsI1
% 2.68/1.54 => ( binintersectRsub
% 2.68/1.54 => ( binintersectSubset4
% 2.68/1.54 => ( binintersectSubset1
% 2.68/1.54 => ( bs114d
% 2.68/1.54 => ( setminusI
% 2.68/1.54 => ( setminusEL
% 2.68/1.54 => ( setminusER
% 2.68/1.54 => ( setminusSubset2
% 2.68/1.54 => ( setminusERneg
% 2.68/1.54 => ( setminusELneg
% 2.68/1.54 => ( setminusILneg
% 2.68/1.54 => ( setminusIRneg
% 2.68/1.54 => ( setminusLsub
% 2.68/1.54 => ( setminusSubset1
% 2.68/1.54 => ( symdiffE
% 2.68/1.54 => ( symdiffI1
% 2.68/1.54 => ( symdiffI2
% 2.68/1.54 => ( symdiffIneg1
% 2.68/1.54 => ( symdiffIneg2
% 2.68/1.54 => ( secondinupair
% 2.68/1.54 => ( setukpairIL
% 2.68/1.54 => ( setukpairIR
% 2.68/1.54 => ( kpairiskpair
% 2.68/1.54 => ( kpairp
% 2.68/1.54 => ( singletonsubset
% 2.68/1.54 => ( singletoninpowerset
% 2.68/1.54 => ( singletoninpowunion
% 2.68/1.54 => ( upairset2E
% 2.68/1.54 => ( upairsubunion
% 2.68/1.54 => ( upairinpowunion
% 2.68/1.54 => ( ubforcartprodlem1
% 2.68/1.54 => ( ubforcartprodlem2
% 2.68/1.54 => ( ubforcartprodlem3
% 2.68/1.54 => ( cartprodpairin
% 2.68/1.54 => ( cartprodmempair1
% 2.68/1.54 => ( cartprodmempair
% 2.68/1.54 => ( setunionE2
% 2.68/1.54 => ( setunionsingleton1
% 2.68/1.54 => ( setunionsingleton2
% 2.68/1.54 => ( setunionsingleton
% 2.68/1.54 => ( singletonprop
% 2.68/1.54 => ( ex1E1
% 2.68/1.54 => ( ex1I
% 2.68/1.54 => ( ex1I2
% 2.68/1.54 => ( singletonsuniq
% 2.68/1.54 => ( setukpairinjL1
% 2.68/1.54 => ( kfstsingleton
% 2.68/1.54 => ( theprop
% 2.68/1.54 => ( kfstpairEq
% 2.68/1.54 => ( cartprodfstin
% 2.68/1.54 => ( setukpairinjL2
% 2.68/1.54 => ( setukpairinjL
% 2.68/1.54 => ( setukpairinjR11
% 2.68/1.54 => ( setukpairinjR12
% 2.68/1.54 => ( setukpairinjR1
% 2.68/1.54 => ( upairequniteq
% 2.68/1.54 => ( setukpairinjR2
% 2.68/1.54 => ( setukpairinjR
% 2.68/1.54 => ( ksndsingleton
% 2.68/1.54 => ( ksndpairEq
% 2.68/1.54 => ( kpairsurjEq
% 2.68/1.54 => ( cartprodsndin
% 2.68/1.54 => ( cartprodpairmemEL
% 2.68/1.54 => ( cartprodpairmemER
% 2.68/1.54 => ( cartprodmempaircEq
% 2.68/1.54 => ( cartprodfstpairEq
% 2.68/1.54 => ( cartprodsndpairEq
% 2.68/1.54 => ( cartprodpairsurjEq
% 2.68/1.54 => ( dpsetconstrI
% 2.68/1.54 => ( dpsetconstrSub
% 2.68/1.54 => ( setOfPairsIsBReln
% 2.68/1.54 => ( dpsetconstrERa
% 2.68/1.54 => ( dpsetconstrEL1
% 2.68/1.54 => ( dpsetconstrEL2
% 2.68/1.54 => ( dpsetconstrER
% 2.68/1.54 => ( funcImageSingleton
% 2.68/1.54 => ( apProp
% 2.68/1.54 => ( app
% 2.68/1.54 => ( infuncsetfunc
% 2.68/1.54 => ( ap2p
% 2.68/1.54 => ( funcinfuncset
% 2.68/1.54 => ( lamProp
% 2.68/1.54 => ( lamp
% 2.68/1.54 => ( lam2p
% 2.68/1.54 => ( brelnall1
% 2.68/1.54 => ( brelnall2
% 2.68/1.54 => ( ex1E2
% 2.68/1.54 => ( funcGraphProp1
% 2.68/1.54 => ( funcGraphProp3
% 2.68/1.54 => ( funcGraphProp2
% 2.68/1.54 => ( funcextLem
% 2.68/1.54 => ! [A: $i,B: $i,Xf: $i] :
% 2.68/1.54 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.68/1.54 => ! [Xx: $i] :
% 2.68/1.54 ( ( in @ Xx @ A )
% 2.68/1.54 => ! [Xy: $i] :
% 2.68/1.54 ( ( in @ Xy @ B )
% 2.68/1.54 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 2.68/1.54 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.68/1.54 = Xy ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.68/1.54 ------------------------
% 2.68/1.57 % (1775)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_1488 for (2999ds/27Mi)
% 2.68/1.57 % (1773)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_1488 for (2999ds/183Mi)
% 2.68/1.57 % (1777)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_1488 for (2999ds/2Mi)
% 2.68/1.58 % (1777)Instruction limit reached!
% 2.68/1.58 % (1777)------------------------------
% 2.68/1.58 % (1777)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.68/1.58 % (1777)Termination reason: Unknown
% 2.68/1.58 % (1777)Termination phase: shuffling
% 2.68/1.58
% 2.68/1.58 % (1777)Memory used [KB]: 1407
% 2.68/1.58 % (1777)Time elapsed: 0.003 s
% 2.68/1.58 % (1777)Instructions burned: 3 (million)
% 2.68/1.58 % (1777)------------------------------
% 2.68/1.58 % (1777)------------------------------
% 2.68/1.58 % (1774)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_1488 for (2999ds/4Mi)
% 2.68/1.58 % (1776)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_1488 for (2999ds/2Mi)
% 2.68/1.58 % (1775)Instruction limit reached!
% 2.68/1.58 % (1775)------------------------------
% 2.68/1.58 % (1775)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.68/1.58 % (1775)Termination reason: Unknown
% 2.68/1.58 % (1775)Termination phase: shuffling
% 2.68/1.58
% 2.68/1.58 % (1775)Memory used [KB]: 1918
% 2.68/1.58 % (1775)Time elapsed: 0.010 s
% 2.68/1.58 % (1775)Instructions burned: 30 (million)
% 2.68/1.58 % (1775)------------------------------
% 2.68/1.58 % (1775)------------------------------
% 2.68/1.58 % (1778)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_1488 for (2999ds/275Mi)
% 2.68/1.58 % (1779)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_1488 for (2999ds/18Mi)
% 2.68/1.58 % (1776)Instruction limit reached!
% 2.68/1.58 % (1776)------------------------------
% 2.68/1.58 % (1776)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.68/1.58 % (1776)Termination reason: Unknown
% 2.68/1.58 % (1776)Termination phase: shuffling
% 2.68/1.58
% 2.68/1.58 % (1776)Memory used [KB]: 1407
% 2.68/1.58 % (1776)Time elapsed: 0.003 s
% 2.68/1.58 % (1776)Instructions burned: 3 (million)
% 2.68/1.58 % (1776)------------------------------
% 2.68/1.58 % (1776)------------------------------
% 2.68/1.58 % (1774)Instruction limit reached!
% 2.68/1.58 % (1774)------------------------------
% 2.68/1.58 % (1774)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.68/1.58 % (1774)Termination reason: Unknown
% 2.68/1.58 % (1774)Termination phase: shuffling
% 2.68/1.58
% 2.68/1.58 % (1774)Memory used [KB]: 1407
% 2.68/1.58 % (1774)Time elapsed: 0.004 s
% 2.68/1.58 % (1774)Instructions burned: 4 (million)
% 2.68/1.58 % (1774)------------------------------
% 2.68/1.58 % (1774)------------------------------
% 2.68/1.59 % (1780)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_1488 for (2999ds/3Mi)
% 2.68/1.59 % (1780)Instruction limit reached!
% 2.68/1.59 % (1780)------------------------------
% 2.68/1.59 % (1780)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.68/1.59 % (1780)Termination reason: Unknown
% 2.68/1.59 % (1780)Termination phase: shuffling
% 2.68/1.59
% 2.68/1.59 % (1780)Memory used [KB]: 1407
% 2.68/1.59 % (1780)Time elapsed: 0.002 s
% 2.68/1.59 % (1780)Instructions burned: 3 (million)
% 2.68/1.59 % (1780)------------------------------
% 2.68/1.59 % (1780)------------------------------
% 2.68/1.59 % (1781)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_1488 for (2999ds/37Mi)
% 2.68/1.59 % (1779)Instruction limit reached!
% 2.68/1.59 % (1779)------------------------------
% 2.68/1.59 % (1779)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.68/1.59 % (1779)Termination reason: Unknown
% 2.68/1.59 % (1779)Termination phase: shuffling
% 2.68/1.59
% 2.68/1.59 % (1779)Memory used [KB]: 1791
% 2.68/1.59 % (1779)Time elapsed: 0.013 s
% 2.68/1.59 % (1779)Instructions burned: 19 (million)
% 2.68/1.59 % (1779)------------------------------
% 2.68/1.59 % (1779)------------------------------
% 2.72/1.60 % (1784)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_1488 for (2999ds/1041Mi)
% 2.72/1.60 % (1782)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_1488 for (2999ds/15Mi)
% 2.72/1.60 % (1783)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_1488 for (2999ds/3Mi)
% 2.72/1.60 % (1781)Instruction limit reached!
% 2.72/1.60 % (1781)------------------------------
% 2.72/1.60 % (1781)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.60 % (1781)Termination reason: Unknown
% 2.72/1.60 % (1781)Termination phase: Property scanning
% 2.72/1.60
% 2.72/1.60 % (1781)Memory used [KB]: 2174
% 2.72/1.60 % (1781)Time elapsed: 0.012 s
% 2.72/1.60 % (1781)Instructions burned: 41 (million)
% 2.72/1.60 % (1781)------------------------------
% 2.72/1.60 % (1781)------------------------------
% 2.72/1.60 % (1783)Instruction limit reached!
% 2.72/1.60 % (1783)------------------------------
% 2.72/1.60 % (1783)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.60 % (1783)Termination reason: Unknown
% 2.72/1.60 % (1783)Termination phase: shuffling
% 2.72/1.60
% 2.72/1.60 % (1783)Memory used [KB]: 1407
% 2.72/1.60 % (1783)Time elapsed: 0.004 s
% 2.72/1.60 % (1783)Instructions burned: 3 (million)
% 2.72/1.60 % (1783)------------------------------
% 2.72/1.60 % (1783)------------------------------
% 2.72/1.61 % (1785)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_1488 for (2999ds/7Mi)
% 2.72/1.61 % (1782)Instruction limit reached!
% 2.72/1.61 % (1782)------------------------------
% 2.72/1.61 % (1782)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.61 % (1782)Termination reason: Unknown
% 2.72/1.61 % (1782)Termination phase: shuffling
% 2.72/1.61
% 2.72/1.61 % (1782)Memory used [KB]: 1663
% 2.72/1.61 % (1782)Time elapsed: 0.012 s
% 2.72/1.61 % (1782)Instructions burned: 16 (million)
% 2.72/1.61 % (1782)------------------------------
% 2.72/1.61 % (1782)------------------------------
% 2.72/1.61 % (1786)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_1488 for (2999ds/16Mi)
% 2.72/1.61 % (1785)Instruction limit reached!
% 2.72/1.61 % (1785)------------------------------
% 2.72/1.61 % (1785)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.61 % (1785)Termination reason: Unknown
% 2.72/1.61 % (1785)Termination phase: shuffling
% 2.72/1.61
% 2.72/1.61 % (1785)Memory used [KB]: 1535
% 2.72/1.61 % (1785)Time elapsed: 0.005 s
% 2.72/1.61 % (1785)Instructions burned: 8 (million)
% 2.72/1.61 % (1785)------------------------------
% 2.72/1.61 % (1785)------------------------------
% 2.72/1.62 % (1786)Instruction limit reached!
% 2.72/1.62 % (1786)------------------------------
% 2.72/1.62 % (1786)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.62 % (1786)Termination reason: Unknown
% 2.72/1.62 % (1786)Termination phase: shuffling
% 2.72/1.62
% 2.72/1.62 % (1786)Memory used [KB]: 1791
% 2.72/1.62 % (1786)Time elapsed: 0.006 s
% 2.72/1.62 % (1786)Instructions burned: 17 (million)
% 2.72/1.62 % (1786)------------------------------
% 2.72/1.62 % (1786)------------------------------
% 2.72/1.62 % (1787)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_1488 for (2999ds/3Mi)
% 2.72/1.62 % (1787)Instruction limit reached!
% 2.72/1.62 % (1787)------------------------------
% 2.72/1.62 % (1787)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.62 % (1787)Termination reason: Unknown
% 2.72/1.62 % (1787)Termination phase: shuffling
% 2.72/1.62
% 2.72/1.62 % (1787)Memory used [KB]: 1407
% 2.72/1.62 % (1787)Time elapsed: 0.004 s
% 2.72/1.62 % (1787)Instructions burned: 3 (million)
% 2.72/1.62 % (1787)------------------------------
% 2.72/1.62 % (1787)------------------------------
% 2.72/1.63 % (1789)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_1488 for (2999ds/7Mi)
% 2.72/1.63 % (1790)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_1488 for (2999ds/3Mi)
% 2.72/1.63 % (1773)Instruction limit reached!
% 2.72/1.63 % (1773)------------------------------
% 2.72/1.63 % (1773)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.63 % (1773)Termination reason: Unknown
% 2.72/1.63 % (1773)Termination phase: Saturation
% 2.72/1.63
% 2.72/1.63 % (1773)Memory used [KB]: 7419
% 2.72/1.63 % (1773)Time elapsed: 0.054 s
% 2.72/1.63 % (1773)Instructions burned: 184 (million)
% 2.72/1.63 % (1773)------------------------------
% 2.72/1.63 % (1773)------------------------------
% 2.72/1.63 % (1790)Instruction limit reached!
% 2.72/1.63 % (1790)------------------------------
% 2.72/1.63 % (1790)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.63 % (1790)Termination reason: Unknown
% 2.72/1.63 % (1790)Termination phase: shuffling
% 2.72/1.63
% 2.72/1.63 % (1790)Memory used [KB]: 1407
% 2.72/1.63 % (1790)Time elapsed: 0.002 s
% 2.72/1.63 % (1790)Instructions burned: 3 (million)
% 2.72/1.63 % (1790)------------------------------
% 2.72/1.63 % (1790)------------------------------
% 2.72/1.63 % (1789)Instruction limit reached!
% 2.72/1.63 % (1789)------------------------------
% 2.72/1.63 % (1789)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.63 % (1789)Termination reason: Unknown
% 2.72/1.63 % (1789)Termination phase: shuffling
% 2.72/1.63
% 2.72/1.63 % (1789)Memory used [KB]: 1535
% 2.72/1.63 % (1789)Time elapsed: 0.004 s
% 2.72/1.63 % (1789)Instructions burned: 9 (million)
% 2.72/1.63 % (1789)------------------------------
% 2.72/1.63 % (1789)------------------------------
% 2.72/1.63 % (1788)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_1488 for (2999ds/3Mi)
% 2.72/1.63 % (1788)Instruction limit reached!
% 2.72/1.63 % (1788)------------------------------
% 2.72/1.63 % (1788)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.63 % (1788)Termination reason: Unknown
% 2.72/1.63 % (1788)Termination phase: shuffling
% 2.72/1.63
% 2.72/1.63 % (1788)Memory used [KB]: 1407
% 2.72/1.63 % (1788)Time elapsed: 0.004 s
% 2.72/1.63 % (1788)Instructions burned: 4 (million)
% 2.72/1.63 % (1788)------------------------------
% 2.72/1.63 % (1788)------------------------------
% 2.72/1.64 % (1793)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_1488 for (2999ds/710Mi)
% 2.72/1.64 % (1792)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_1488 for (2999ds/18Mi)
% 2.72/1.64 % (1794)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_1488 for (2999ds/6Mi)
% 2.72/1.64 % (1791)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_1488 for (2999ds/4Mi)
% 2.72/1.64 % (1794)Instruction limit reached!
% 2.72/1.64 % (1794)------------------------------
% 2.72/1.64 % (1794)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.64 % (1794)Termination reason: Unknown
% 2.72/1.64 % (1794)Termination phase: shuffling
% 2.72/1.64
% 2.72/1.64 % (1794)Memory used [KB]: 1535
% 2.72/1.64 % (1794)Time elapsed: 0.003 s
% 2.72/1.64 % (1794)Instructions burned: 7 (million)
% 2.72/1.64 % (1794)------------------------------
% 2.72/1.64 % (1794)------------------------------
% 2.72/1.64 % (1792)Instruction limit reached!
% 2.72/1.64 % (1792)------------------------------
% 2.72/1.64 % (1792)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.64 % (1792)Termination reason: Unknown
% 2.72/1.64 % (1792)Termination phase: shuffling
% 2.72/1.64
% 2.72/1.64 % (1792)Memory used [KB]: 1791
% 2.72/1.64 % (1792)Time elapsed: 0.007 s
% 2.72/1.64 % (1792)Instructions burned: 21 (million)
% 2.72/1.64 % (1792)------------------------------
% 2.72/1.64 % (1792)------------------------------
% 2.72/1.64 % (1791)Instruction limit reached!
% 2.72/1.64 % (1791)------------------------------
% 2.72/1.64 % (1791)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.64 % (1791)Termination reason: Unknown
% 2.72/1.64 % (1791)Termination phase: shuffling
% 2.72/1.64
% 2.72/1.64 % (1791)Memory used [KB]: 1535
% 2.72/1.64 % (1791)Time elapsed: 0.005 s
% 2.72/1.64 % (1791)Instructions burned: 5 (million)
% 2.72/1.64 % (1791)------------------------------
% 2.72/1.64 % (1791)------------------------------
% 2.72/1.65 % (1795)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_1488 for (2999ds/902Mi)
% 2.72/1.65 % (1796)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_1488 for (2999ds/21Mi)
% 2.72/1.65 % (1797)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_1488 for (2999ds/5Mi)
% 2.72/1.66 % (1797)Instruction limit reached!
% 2.72/1.66 % (1797)------------------------------
% 2.72/1.66 % (1797)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.66 % (1797)Termination reason: Unknown
% 2.72/1.66 % (1797)Termination phase: shuffling
% 2.72/1.66
% 2.72/1.66 % (1797)Memory used [KB]: 1407
% 2.72/1.66 % (1797)Time elapsed: 0.004 s
% 2.72/1.66 % (1797)Instructions burned: 7 (million)
% 2.72/1.66 % (1797)------------------------------
% 2.72/1.66 % (1797)------------------------------
% 2.72/1.66 % (1796)Instruction limit reached!
% 2.72/1.66 % (1796)------------------------------
% 2.72/1.66 % (1796)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.66 % (1796)Termination reason: Unknown
% 2.72/1.66 % (1796)Termination phase: shuffling
% 2.72/1.66
% 2.72/1.66 % (1796)Memory used [KB]: 1791
% 2.72/1.66 % (1796)Time elapsed: 0.008 s
% 2.72/1.66 % (1796)Instructions burned: 24 (million)
% 2.72/1.66 % (1796)------------------------------
% 2.72/1.66 % (1796)------------------------------
% 2.72/1.66 % (1798)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_1488 for (2999ds/6Mi)
% 2.72/1.66 % (1798)Instruction limit reached!
% 2.72/1.66 % (1798)------------------------------
% 2.72/1.66 % (1798)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.66 % (1798)Termination reason: Unknown
% 2.72/1.66 % (1798)Termination phase: shuffling
% 2.72/1.66
% 2.72/1.66 % (1798)Memory used [KB]: 1407
% 2.72/1.66 % (1798)Time elapsed: 0.005 s
% 2.72/1.66 % (1798)Instructions burned: 7 (million)
% 2.72/1.66 % (1798)------------------------------
% 2.72/1.66 % (1798)------------------------------
% 2.72/1.67 % (1799)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_1488 for (2999ds/377Mi)
% 2.72/1.67 % (1800)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_1488 for (2998ds/779Mi)
% 2.72/1.67 % (1801)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_1488 for (2998ds/19Mi)
% 2.72/1.68 % (1801)Instruction limit reached!
% 2.72/1.68 % (1801)------------------------------
% 2.72/1.68 % (1801)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.68 % (1801)Termination reason: Unknown
% 2.72/1.68 % (1801)Termination phase: shuffling
% 2.72/1.68
% 2.72/1.68 % (1801)Memory used [KB]: 1791
% 2.72/1.68 % (1801)Time elapsed: 0.007 s
% 2.72/1.68 % (1801)Instructions burned: 19 (million)
% 2.72/1.68 % (1801)------------------------------
% 2.72/1.68 % (1801)------------------------------
% 2.72/1.69 % (1802)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_1488 for (2998ds/879Mi)
% 2.72/1.72 % (1778)Instruction limit reached!
% 2.72/1.72 % (1778)------------------------------
% 2.72/1.72 % (1778)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.72 % (1778)Termination reason: Unknown
% 2.72/1.72 % (1778)Termination phase: Saturation
% 2.72/1.72
% 2.72/1.72 % (1778)Memory used [KB]: 8699
% 2.72/1.72 % (1778)Time elapsed: 0.143 s
% 2.72/1.72 % (1778)Instructions burned: 276 (million)
% 2.72/1.72 % (1778)------------------------------
% 2.72/1.72 % (1778)------------------------------
% 2.72/1.73 % (1795)First to succeed.
% 2.72/1.73 % (1795)Refutation found. Thanks to Tanya!
% 2.72/1.73 % SZS status Theorem for DTF2THF_1488
% 2.72/1.73 % SZS output start Proof for DTF2THF_1488
% 2.72/1.73 thf(func_def_0, type, in: $i > $i > $o).
% 2.72/1.73 thf(func_def_1, type, exu: ($i > $o) > $o).
% 2.72/1.73 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 2.72/1.73 thf(func_def_8, type, powerset: $i > $i).
% 2.72/1.73 thf(func_def_10, type, setunion: $i > $i).
% 2.72/1.73 thf(func_def_19, type, descr: ($i > $o) > $i).
% 2.72/1.73 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 2.72/1.73 thf(func_def_26, type, prop2set: $o > $i).
% 2.72/1.73 thf(func_def_36, type, nonempty: $i > $o).
% 2.72/1.73 thf(func_def_69, type, set2prop: $i > $o).
% 2.72/1.73 thf(func_def_88, type, subset: $i > $i > $o).
% 2.72/1.73 thf(func_def_89, type, disjoint: $i > $i > $o).
% 2.72/1.73 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 2.72/1.73 thf(func_def_114, type, binunion: $i > $i > $i).
% 2.72/1.73 thf(func_def_122, type, binintersect: $i > $i > $i).
% 2.72/1.73 thf(func_def_135, type, regular: $i > $o).
% 2.72/1.73 thf(func_def_136, type, setminus: $i > $i > $i).
% 2.72/1.73 thf(func_def_147, type, symdiff: $i > $i > $i).
% 2.72/1.73 thf(func_def_153, type, iskpair: $i > $o).
% 2.72/1.73 thf(func_def_158, type, kpair: $i > $i > $i).
% 2.72/1.73 thf(func_def_160, type, cartprod: $i > $i > $i).
% 2.72/1.73 thf(func_def_177, type, singleton: $i > $o).
% 2.72/1.73 thf(func_def_179, type, ex1: $i > ($i > $o) > $o).
% 2.72/1.73 thf(func_def_184, type, atmost1p: $i > $o).
% 2.72/1.73 thf(func_def_185, type, atleast2p: $i > $o).
% 2.72/1.73 thf(func_def_186, type, atmost2p: $i > $o).
% 2.72/1.73 thf(func_def_187, type, upairsetp: $i > $o).
% 2.72/1.73 thf(func_def_191, type, kfst: $i > $i).
% 2.72/1.73 thf(func_def_203, type, ksnd: $i > $i).
% 2.72/1.73 thf(func_def_213, type, breln: $i > $i > $i > $o).
% 2.72/1.73 thf(func_def_214, type, dpsetconstr: $i > $i > ($i > $i > $o) > $i).
% 2.72/1.73 thf(func_def_222, type, func: $i > $i > $i > $o).
% 2.72/1.73 thf(func_def_223, type, funcSet: $i > $i > $i).
% 2.72/1.73 thf(func_def_226, type, ap: $i > $i > $i > $i > $i).
% 2.72/1.73 thf(func_def_232, type, lam: $i > $i > ($i > $i) > $i).
% 2.72/1.73 thf(func_def_255, type, sP0: $o > $i > $i > $i > $o).
% 2.72/1.73 thf(func_def_260, type, sP5: $i > $i > $o).
% 2.72/1.73 thf(func_def_261, type, sP6: $i > $o).
% 2.72/1.73 thf(func_def_262, type, sP7: $i > $i > $o).
% 2.72/1.73 thf(func_def_263, type, sP8: $i > $i > $o).
% 2.72/1.73 thf(func_def_279, type, sK24: $i > $o).
% 2.72/1.73 thf(func_def_280, type, sK25: $i > $o).
% 2.72/1.73 thf(func_def_283, type, sK28: ($i > $o) > ($i > $o) > $i).
% 2.72/1.73 thf(func_def_284, type, sK29: ($i > $o) > ($i > $o) > $i).
% 2.72/1.73 thf(func_def_285, type, sK30: $i > $o).
% 2.72/1.73 thf(func_def_286, type, sK31: $i > $o).
% 2.72/1.73 thf(func_def_287, type, sK32: $i > $i > $i).
% 2.72/1.73 thf(func_def_310, type, sK55: $i > $i > $i).
% 2.72/1.73 thf(func_def_324, type, sK69: $i > ($i > $o) > $i).
% 2.72/1.73 thf(func_def_325, type, sK70: $i > $o).
% 2.72/1.73 thf(func_def_334, type, sK79: ($i > $o) > $i).
% 2.72/1.73 thf(func_def_335, type, sK80: $i > $o).
% 2.72/1.73 thf(func_def_336, type, sK81: $i > $i).
% 2.72/1.73 thf(func_def_340, type, sK85: $i > $i).
% 2.72/1.73 thf(func_def_342, type, sK87: $i > ($i > $i) > $i > $i).
% 2.72/1.73 thf(func_def_354, type, sK99: $i > $i > $i).
% 2.72/1.73 thf(func_def_355, type, sK100: ($i > $o) > $i).
% 2.72/1.73 thf(func_def_356, type, sK101: $i > $o).
% 2.72/1.73 thf(func_def_357, type, sK102: $i > $i).
% 2.72/1.73 thf(func_def_360, type, sK105: $i > $o).
% 2.72/1.73 thf(func_def_363, type, sK108: $i > $i).
% 2.72/1.73 thf(func_def_370, type, sK115: $i > $o).
% 2.72/1.73 thf(func_def_385, type, sK130: $i > $i > $i).
% 2.72/1.73 thf(func_def_386, type, sK131: $i > $i > $i).
% 2.72/1.73 thf(func_def_389, type, sK134: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.72/1.73 thf(func_def_390, type, sK135: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.72/1.73 thf(func_def_393, type, sK138: $i > $o).
% 2.72/1.73 thf(func_def_394, type, sK139: $i > $o).
% 2.72/1.73 thf(func_def_413, type, sK158: ($i > $o) > $i > $i).
% 2.72/1.73 thf(func_def_415, type, sK160: $i > $o).
% 2.72/1.73 thf(func_def_420, type, sK165: $i > $o).
% 2.72/1.73 thf(func_def_427, type, sK172: $i > $i > $i > $i > $i).
% 2.72/1.73 thf(func_def_430, type, sK175: $i > $i > $i).
% 2.72/1.73 thf(func_def_431, type, sK176: $i > $i > $i).
% 2.72/1.73 thf(func_def_442, type, sK187: $i > $i > $o).
% 2.72/1.73 thf(func_def_460, type, sK205: $i > $o).
% 2.72/1.73 thf(func_def_463, type, sK208: $i > ($i > $o) > $i).
% 2.72/1.73 thf(func_def_464, type, sK209: $i > ($i > $o) > $i).
% 2.72/1.73 thf(func_def_488, type, sK233: $i > $i > $i).
% 2.72/1.73 thf(func_def_502, type, sK247: ($i > $o) > $i > $i > $i > $i).
% 2.72/1.73 thf(func_def_503, type, sK248: ($i > $o) > $i > $i > $i > $i).
% 2.72/1.73 thf(func_def_507, type, sK252: $i > $o).
% 2.72/1.73 thf(func_def_509, type, sK254: $i > $o).
% 2.72/1.73 thf(func_def_513, type, sK258: $i > $o).
% 2.72/1.73 thf(func_def_521, type, sK266: $i > $i).
% 2.72/1.73 thf(func_def_523, type, sK268: $i > $i).
% 2.72/1.73 thf(func_def_524, type, sK269: $i > ($i > $i) > $i > $i).
% 2.72/1.73 thf(func_def_526, type, sK271: $i > $i).
% 2.72/1.73 thf(func_def_528, type, sK273: $i > $i > $o).
% 2.72/1.73 thf(func_def_533, type, sK278: ($i > $o) > $i).
% 2.72/1.73 thf(func_def_534, type, sK279: $i > $o).
% 2.72/1.73 thf(func_def_543, type, sK288: $i > $i > $o).
% 2.72/1.73 thf(func_def_545, type, sK290: $i > $i).
% 2.72/1.73 thf(func_def_546, type, sK291: $i > $i).
% 2.72/1.73 thf(func_def_547, type, sK292: $i > ($i > $i > $o) > $i).
% 2.72/1.73 thf(func_def_548, type, sK293: $i > ($i > $i > $o) > $i).
% 2.72/1.73 thf(func_def_549, type, sK294: $i > $i > ($i > $i > $o) > $i).
% 2.72/1.73 thf(func_def_550, type, sK295: $i > ($i > $o) > $i).
% 2.72/1.73 thf(func_def_551, type, sK296: $i > $o).
% 2.72/1.73 thf(func_def_556, type, sK301: $i > $o).
% 2.72/1.73 thf(func_def_558, type, sK303: ($i > $o) > $i).
% 2.72/1.73 thf(func_def_559, type, sK304: ($i > $o) > $i).
% 2.72/1.73 thf(func_def_567, type, sK312: $i > $i > $o).
% 2.72/1.73 thf(func_def_586, type, sK331: $i > $o).
% 2.72/1.73 thf(func_def_589, type, sK334: ($i > $o) > $i > $i > $i > $i).
% 2.72/1.73 thf(func_def_590, type, sK335: ($i > $o) > $i > $i > $i > $i).
% 2.72/1.73 thf(func_def_594, type, sK339: $i > $o).
% 2.72/1.73 thf(func_def_596, type, sK341: $i > $o).
% 2.72/1.73 thf(func_def_623, type, sK368: $i > $i).
% 2.72/1.73 thf(func_def_625, type, sK370: $i > $i > $i).
% 2.72/1.73 thf(func_def_629, type, sK374: $i > $o).
% 2.72/1.73 thf(func_def_632, type, sK377: $i > ($i > $o) > $i).
% 2.72/1.73 thf(func_def_633, type, sK378: $i > ($i > $o) > $i).
% 2.72/1.73 thf(func_def_652, type, sK397: $i > $i > $o).
% 2.72/1.73 thf(func_def_654, type, sK399: $i > $o).
% 2.72/1.73 thf(func_def_657, type, sK402: $i > ($i > $o) > $i).
% 2.72/1.73 thf(func_def_661, type, sK406: $i > $i > $i > $i).
% 2.72/1.73 thf(func_def_662, type, sK407: $i > $i > $i > $i).
% 2.72/1.73 thf(func_def_664, type, sK409: ($i > $o) > $i > $i > $i).
% 2.72/1.73 thf(func_def_667, type, sK412: $i > $o).
% 2.72/1.73 thf(func_def_668, type, sK413: $i > $i).
% 2.72/1.73 thf(func_def_671, type, sK416: $i > $i > ($i > $i) > $i).
% 2.72/1.73 thf(func_def_673, type, sK418: $i > $o).
% 2.72/1.73 thf(func_def_675, type, sK420: ($i > $o) > $i > $i).
% 2.72/1.73 thf(func_def_698, type, sK443: $o > $i > $i > $i).
% 2.72/1.73 thf(func_def_703, type, sK448: $i > $o).
% 2.72/1.73 thf(func_def_706, type, sK451: $i > $o).
% 2.72/1.73 thf(func_def_714, type, sK459: $i > $i > $o).
% 2.72/1.73 thf(func_def_730, type, sK475: $i > $o).
% 2.72/1.73 thf(func_def_732, type, sK477: $i > ($i > $o) > $i).
% 2.72/1.73 thf(func_def_737, type, sK482: $i > $i > $o).
% 2.72/1.73 thf(func_def_739, type, sK484: $i > $o).
% 2.72/1.73 thf(func_def_740, type, sK485: $i > $o).
% 2.72/1.73 thf(func_def_741, type, sK486: ($i > $o) > ($i > $o) > $i).
% 2.72/1.73 thf(func_def_742, type, sK487: ($i > $o) > ($i > $o) > $i).
% 2.72/1.73 thf(func_def_752, type, sK497: $i > $o).
% 2.72/1.73 thf(func_def_757, type, sK502: $i > $i).
% 2.72/1.73 thf(func_def_763, type, sK508: ($i > $o) > $i > $i).
% 2.72/1.73 thf(func_def_765, type, sK510: $i > $o).
% 2.72/1.73 thf(func_def_777, type, sK522: $i > $i > $i).
% 2.72/1.73 thf(func_def_778, type, sK523: $i > $i > $i).
% 2.72/1.73 thf(func_def_779, type, sK524: $i > $i > $i).
% 2.72/1.73 thf(func_def_780, type, sK525: $i > $i > $i).
% 2.72/1.73 thf(func_def_781, type, sK526: $i > $i > $i > $i).
% 2.72/1.73 thf(func_def_782, type, sK527: $i > $i).
% 2.72/1.73 thf(func_def_783, type, sK528: $i > $i).
% 2.72/1.73 thf(func_def_784, type, sK529: $i > $i).
% 2.72/1.73 thf(func_def_785, type, sK530: $i > $i).
% 2.72/1.73 thf(func_def_786, type, sK531: $i > $i > $i).
% 2.72/1.73 thf(func_def_787, type, sK532: $i > $i > $i > $i).
% 2.72/1.73 thf(func_def_788, type, sK533: $i > $i > $i > $i).
% 2.72/1.73 thf(func_def_789, type, sK534: $i > $i > $i).
% 2.72/1.73 thf(func_def_790, type, sK535: $i > $i > $i).
% 2.72/1.73 thf(func_def_791, type, sK536: $i > $i).
% 2.72/1.73 thf(func_def_792, type, sK537: $i > $i > $i).
% 2.72/1.73 thf(func_def_794, type, sK539: $i > $i).
% 2.72/1.73 thf(func_def_795, type, sK540: $i > $i).
% 2.72/1.73 thf(func_def_800, type, sK545: $i > $i).
% 2.72/1.73 thf(func_def_809, type, sK554: $i > $i > $o).
% 2.72/1.73 thf(func_def_814, type, sK559: ($i > $o) > $i > $i).
% 2.72/1.73 thf(func_def_816, type, sK561: $i > $o).
% 2.72/1.73 thf(func_def_820, type, sK565: $i > $i).
% 2.72/1.73 thf(func_def_833, type, sK578: $i > $o).
% 2.72/1.73 thf(func_def_838, type, sK583: $i > $i > $i).
% 2.72/1.73 thf(func_def_860, type, sK605: ($i > $o) > $i > $i).
% 2.72/1.73 thf(func_def_862, type, sK607: $i > $o).
% 2.72/1.73 thf(func_def_874, type, ph619: !>[X0: $tType]:(X0)).
% 2.72/1.73 thf(f3086,plain,(
% 2.72/1.73 $false),
% 2.72/1.73 inference(avatar_sat_refutation,[status(thm)],[f3039,f3040,f3076])).
% 2.72/1.73 thf(f3076,plain,(
% 2.72/1.73 ~spl618_1),
% 2.72/1.73 inference(avatar_contradiction_clause,[status(thm)],[f3075])).
% 2.72/1.73 thf(f3075,plain,(
% 2.72/1.73 $false | ~spl618_1),
% 2.72/1.73 inference(trivial_inequality_removal,[status(thm)],[f3067])).
% 2.72/1.73 thf(f3067,plain,(
% 2.72/1.73 ($true != $true) | ~spl618_1),
% 2.72/1.73 inference(superposition,[status(thm)],[f3051,f2668])).
% 2.72/1.73 thf(f2668,plain,(
% 2.72/1.73 ((in @ sK496 @ sK493) = $true)),
% 2.72/1.73 inference(cnf_transformation,[status(thm)],[f1654])).
% 2.72/1.73 thf(f1654,plain,(
% 2.72/1.73 (binintersectEL = $true) & (foundationAx = $true) & (setminusERneg = $true) & (dpsetconstrEL2 = $true) & (upairsetIR = $true) & (setminusEL = $true) & (quantDeMorgan4 = $true) & (setukpairinjL2 = $true) & (setminusER = $true) & (kfstsingleton = $true) & (subsetI1 = $true) & (exuI3 = $true) & (setunion__Cong = $true) & (cartprodmempair1 = $true) & (setminusSubset1 = $true) & (subsetE2 = $true) & (infuncsetfunc = $true) & ((($true = (in @ sK495 @ sK494)) & ((sK496 != (ap @ sK494 @ sK493 @ sK492 @ sK495)) & ($true = (in @ (kpair @ sK495 @ sK496) @ sK492)) & ((in @ sK496 @ sK493) = $true))) & ((in @ sK492 @ (funcSet @ sK494 @ sK493)) = $true)) & (exu__Cong = $true) & (upairsetIL = $true) & (binunionRsub = $true) & (binunionEcases = $true) & (singletonsubset = $true) & (binunionIR = $true) & (emptysetsubset = $true) & (setadjoinSub2 = $true) & (setminusLsub = $true) & (kpairp = $true) & (singletonsuniq = $true) & (ex1E1 = $true) & (setbeta = $true) & (omegaSAx = $true) & (kpairiskpair = $true) & (quantDeMorgan2 = $true) & (omegaIndAx = $true) & (exuEu = $true) & (binintersectI = $true) & (cartprodsndin = $true) & (inCongP = $true) & (uniqinunit = $true) & (singletonsswitch = $true) & (upairsetE = $true) & (binunionLsub = $true) & (upairset2IR = $true) & (ksndpairEq = $true) & (setadjoinAx = $true) & (setukpairinjR = $true) & (nonemptyI = $true) & (cartprodsndpairEq = $true) & (emptysetimpfalse = $true) & (ex1E2 = $true) & (cartprodpairmemEL = $true) & (lam2p = $true) & (setoftrueEq = $true) & (exuE1 = $true) & (notinemptyset = $true) & (subsetemptysetimpeq = $true) & (ksndsingleton = $true) & (sepInPowerset = $true) & (setukpairIL = $true) & (binintersectSubset4 = $true) & (brelnall1 = $true) & (setunionsingleton = $true) & (setextAx = $true) & (cartprodfstpairEq = $true) & (prop2setE = $true) & (powersetI1 = $true) & (vacuousDall = $true) & (setadjoinIR = $true) & (cartprodpairin = $true) & (ubforcartprodlem1 = $true) & (binintersectSubset5 = $true) & (dsetconstrI = $true) & (binunionIL = $true) & (exuE3u = $true) & (dpsetconstrSub = $true) & (binintersectSubset3 = $true) & (dpsetconstrEL1 = $true) & (subsetI2 = $true) & (powersetsubset = $true) & (setminusILneg = $true) & (setminusI = $true) & (notinsingleton = $true) & (symdiffI1 = $true) & (funcextLem = $true) & (binintersectRsub = $true) & (prop2setI = $true) & (powersetI = $true) & (sepSubset = $true) & (powerset__Cong = $true) & (dsetconstr__Cong = $true) & (eqimpsubset2 = $true) & (kpairsurjEq = $true) & (prop2set2propI = $true) & (ex1I2 = $true) & (setunionsingleton1 = $true) & (powersetE1 = $true) & (cartprodmempaircEq = $true) & (upairequniteq = $true) & (subPowSU = $true) & (exuE2 = $true) & (setunionsingleton2 = $true) & (singletonprop = $true) & (emptysetAx = $true) & (subsetRefl = $true) & (subsetTrans = $true) & (cartprodfstin = $true) & (emptyinunitempty = $true) & (kfstpairEq = $true) & (powersetAx = $true) & (descrp = $true) & (quantDeMorgan1 = $true) & (setadjoinIL = $true) & (dsetconstrER = $true) & (setukpairinjL1 = $true) & (setOfPairsIsBReln = $true) & (emptyset__Cong = $true) & (ap2p = $true) & (emptyInPowerset = $true) & (dpsetconstrER = $true) & (setunionI = $true) & (dsetconstrEL = $true) & (notdallE = $true) & (upairsubunion = $true) & (inPowerset = $true) & (cartprodpairmemER = $true) & (cartprodpairsurjEq = $true) & (setext = $true) & (nonemptyE1 = $true) & (setminusIRneg = $true) & (exuI1 = $true) & (dpsetconstrI = $true) & (disjointsetsI1 = $true) & (singletoninpowerset = $true) & (exuI2 = $true) & (binunionE = $true) & (quantDeMorgan3 = $true) & (ubforcartprodlem2 = $true) & (setukpairinjR11 = $true) & (symdiffI2 = $true) & (setukpairinjR12 = $true) & (symdiffIneg1 = $true) & (emptyinPowerset = $true) & (setadjoinE = $true) & (setadjoinOr = $true) & (omega0Ax = $true) & (emptyE1 = $true) & (lamp = $true) & (funcGraphProp3 = $true) & (binintersectSubset2 = $true) & (wellorderingAx = $true) & (ex1I = $true) & (symdiffE = $true) & (funcGraphProp2 = $true) & (nonemptyImpWitness = $true) & (setunionE = $true) & (app = $true) & (setunionE2 = $true) & (setukpairinjR1 = $true) & (lamProp = $true) & (setminusELneg = $true) & (brelnall2 = $true) & (setadjoin__Cong = $true) & (subset2powerset = $true) & (notsubsetI = $true) & (descr__Cong = $true) & (replAx = $true) & (omega__Cong = $true) & (theprop = $true) & (noeltsimpempty = $true) & (nonemptyI1 = $true) & (setminusSubset2 = $true) & (singletoninpowunion = $true) & (subsetE = $true) & (setukpairIR = $true) & (in__Cong = $true) & (setadjoinSub = $true) & (funcImageSingleton = $true) & (exuE3e = $true) & (upairinpowunion = $true) & (funcGraphProp1 = $true) & (binintersectSubset1 = $true) & (dpsetconstrERa = $true) & (powersetE = $true) & (upairset2E = $true) & (binintersectER = $true) & (emptyI = $true) & (setukpairinjR2 = $true) & (bs114d = $true) & (eqimpsubset1 = $true) & (binintersectLsub = $true) & (funcinfuncset = $true) & (ubforcartprodlem3 = $true) & (notequalI1 = $true) & (setukpairinjL = $true) & (setextsub = $true) & (emptysetE = $true) & (cartprodmempair = $true) & (setunionAx = $true) & (notdexE = $true) & (notequalI2 = $true) & (eqinunit = $true) & (secondinupair = $true) & (symdiffIneg2 = $true) & (apProp = $true)),
% 2.72/1.73 inference(skolemisation,[status(esa),new_symbols(skolem,[sK492,sK493,sK494,sK495,sK496])],[f733,f1653,f1652,f1651])).
% 2.72/1.73 thf(f1651,plain,(
% 2.72/1.73 ? [X0 : $i,X1 : $i,X2 : $i] : (? [X3 : $i] : (((in @ X3 @ X2) = $true) & ? [X4 : $i] : (((ap @ X2 @ X1 @ X0 @ X3) != X4) & ((in @ (kpair @ X3 @ X4) @ X0) = $true) & ((in @ X4 @ X1) = $true))) & ((in @ X0 @ (funcSet @ X2 @ X1)) = $true)) => (? [X3 : $i] : (($true = (in @ X3 @ sK494)) & ? [X4 : $i] : (((ap @ sK494 @ sK493 @ sK492 @ X3) != X4) & ((in @ (kpair @ X3 @ X4) @ sK492) = $true) & ($true = (in @ X4 @ sK493)))) & ((in @ sK492 @ (funcSet @ sK494 @ sK493)) = $true))),
% 2.72/1.73 introduced(definition,[],[choice_axiom])).
% 2.72/1.73 thf(f1652,plain,(
% 2.72/1.73 ? [X3 : $i] : (($true = (in @ X3 @ sK494)) & ? [X4 : $i] : (((ap @ sK494 @ sK493 @ sK492 @ X3) != X4) & ((in @ (kpair @ X3 @ X4) @ sK492) = $true) & ($true = (in @ X4 @ sK493)))) => (($true = (in @ sK495 @ sK494)) & ? [X4 : $i] : (((ap @ sK494 @ sK493 @ sK492 @ sK495) != X4) & ($true = (in @ (kpair @ sK495 @ X4) @ sK492)) & ($true = (in @ X4 @ sK493))))),
% 2.72/1.73 introduced(definition,[],[choice_axiom])).
% 2.72/1.73 thf(f1653,plain,(
% 2.72/1.73 ? [X4 : $i] : (((ap @ sK494 @ sK493 @ sK492 @ sK495) != X4) & ($true = (in @ (kpair @ sK495 @ X4) @ sK492)) & ($true = (in @ X4 @ sK493))) => ((sK496 != (ap @ sK494 @ sK493 @ sK492 @ sK495)) & ($true = (in @ (kpair @ sK495 @ sK496) @ sK492)) & ((in @ sK496 @ sK493) = $true))),
% 2.72/1.73 introduced(definition,[],[choice_axiom])).
% 2.72/1.73 thf(f733,plain,(
% 2.72/1.73 (binintersectEL = $true) & (foundationAx = $true) & (setminusERneg = $true) & (dpsetconstrEL2 = $true) & (upairsetIR = $true) & (setminusEL = $true) & (quantDeMorgan4 = $true) & (setukpairinjL2 = $true) & (setminusER = $true) & (kfstsingleton = $true) & (subsetI1 = $true) & (exuI3 = $true) & (setunion__Cong = $true) & (cartprodmempair1 = $true) & (setminusSubset1 = $true) & (subsetE2 = $true) & (infuncsetfunc = $true) & ? [X0 : $i,X1 : $i,X2 : $i] : (? [X3 : $i] : (((in @ X3 @ X2) = $true) & ? [X4 : $i] : (((ap @ X2 @ X1 @ X0 @ X3) != X4) & ((in @ (kpair @ X3 @ X4) @ X0) = $true) & ((in @ X4 @ X1) = $true))) & ((in @ X0 @ (funcSet @ X2 @ X1)) = $true)) & (exu__Cong = $true) & (upairsetIL = $true) & (binunionRsub = $true) & (binunionEcases = $true) & (singletonsubset = $true) & (binunionIR = $true) & (emptysetsubset = $true) & (setadjoinSub2 = $true) & (setminusLsub = $true) & (kpairp = $true) & (singletonsuniq = $true) & (ex1E1 = $true) & (setbeta = $true) & (omegaSAx = $true) & (kpairiskpair = $true) & (quantDeMorgan2 = $true) & (omegaIndAx = $true) & (exuEu = $true) & (binintersectI = $true) & (cartprodsndin = $true) & (inCongP = $true) & (uniqinunit = $true) & (singletonsswitch = $true) & (upairsetE = $true) & (binunionLsub = $true) & (upairset2IR = $true) & (ksndpairEq = $true) & (setadjoinAx = $true) & (setukpairinjR = $true) & (nonemptyI = $true) & (cartprodsndpairEq = $true) & (emptysetimpfalse = $true) & (ex1E2 = $true) & (cartprodpairmemEL = $true) & (lam2p = $true) & (setoftrueEq = $true) & (exuE1 = $true) & (notinemptyset = $true) & (subsetemptysetimpeq = $true) & (ksndsingleton = $true) & (sepInPowerset = $true) & (setukpairIL = $true) & (binintersectSubset4 = $true) & (brelnall1 = $true) & (setunionsingleton = $true) & (setextAx = $true) & (cartprodfstpairEq = $true) & (prop2setE = $true) & (powersetI1 = $true) & (vacuousDall = $true) & (setadjoinIR = $true) & (cartprodpairin = $true) & (ubforcartprodlem1 = $true) & (binintersectSubset5 = $true) & (dsetconstrI = $true) & (binunionIL = $true) & (exuE3u = $true) & (dpsetconstrSub = $true) & (binintersectSubset3 = $true) & (dpsetconstrEL1 = $true) & (subsetI2 = $true) & (powersetsubset = $true) & (setminusILneg = $true) & (setminusI = $true) & (notinsingleton = $true) & (symdiffI1 = $true) & (funcextLem = $true) & (binintersectRsub = $true) & (prop2setI = $true) & (powersetI = $true) & (sepSubset = $true) & (powerset__Cong = $true) & (dsetconstr__Cong = $true) & (eqimpsubset2 = $true) & (kpairsurjEq = $true) & (prop2set2propI = $true) & (ex1I2 = $true) & (setunionsingleton1 = $true) & (powersetE1 = $true) & (cartprodmempaircEq = $true) & (upairequniteq = $true) & (subPowSU = $true) & (exuE2 = $true) & (setunionsingleton2 = $true) & (singletonprop = $true) & (emptysetAx = $true) & (subsetRefl = $true) & (subsetTrans = $true) & (cartprodfstin = $true) & (emptyinunitempty = $true) & (kfstpairEq = $true) & (powersetAx = $true) & (descrp = $true) & (quantDeMorgan1 = $true) & (setadjoinIL = $true) & (dsetconstrER = $true) & (setukpairinjL1 = $true) & (setOfPairsIsBReln = $true) & (emptyset__Cong = $true) & (ap2p = $true) & (emptyInPowerset = $true) & (dpsetconstrER = $true) & (setunionI = $true) & (dsetconstrEL = $true) & (notdallE = $true) & (upairsubunion = $true) & (inPowerset = $true) & (cartprodpairmemER = $true) & (cartprodpairsurjEq = $true) & (setext = $true) & (nonemptyE1 = $true) & (setminusIRneg = $true) & (exuI1 = $true) & (dpsetconstrI = $true) & (disjointsetsI1 = $true) & (singletoninpowerset = $true) & (exuI2 = $true) & (binunionE = $true) & (quantDeMorgan3 = $true) & (ubforcartprodlem2 = $true) & (setukpairinjR11 = $true) & (symdiffI2 = $true) & (setukpairinjR12 = $true) & (symdiffIneg1 = $true) & (emptyinPowerset = $true) & (setadjoinE = $true) & (setadjoinOr = $true) & (omega0Ax = $true) & (emptyE1 = $true) & (lamp = $true) & (funcGraphProp3 = $true) & (binintersectSubset2 = $true) & (wellorderingAx = $true) & (ex1I = $true) & (symdiffE = $true) & (funcGraphProp2 = $true) & (nonemptyImpWitness = $true) & (setunionE = $true) & (app = $true) & (setunionE2 = $true) & (setukpairinjR1 = $true) & (lamProp = $true) & (setminusELneg = $true) & (brelnall2 = $true) & (setadjoin__Cong = $true) & (subset2powerset = $true) & (notsubsetI = $true) & (descr__Cong = $true) & (replAx = $true) & (omega__Cong = $true) & (theprop = $true) & (noeltsimpempty = $true) & (nonemptyI1 = $true) & (setminusSubset2 = $true) & (singletoninpowunion = $true) & (subsetE = $true) & (setukpairIR = $true) & (in__Cong = $true) & (setadjoinSub = $true) & (funcImageSingleton = $true) & (exuE3e = $true) & (upairinpowunion = $true) & (funcGraphProp1 = $true) & (binintersectSubset1 = $true) & (dpsetconstrERa = $true) & (powersetE = $true) & (upairset2E = $true) & (binintersectER = $true) & (emptyI = $true) & (setukpairinjR2 = $true) & (bs114d = $true) & (eqimpsubset1 = $true) & (binintersectLsub = $true) & (funcinfuncset = $true) & (ubforcartprodlem3 = $true) & (notequalI1 = $true) & (setukpairinjL = $true) & (setextsub = $true) & (emptysetE = $true) & (cartprodmempair = $true) & (setunionAx = $true) & (notdexE = $true) & (notequalI2 = $true) & (eqinunit = $true) & (secondinupair = $true) & (symdiffIneg2 = $true) & (apProp = $true)),
% 2.72/1.73 inference(flattening,[status(thm)],[f732])).
% 2.72/1.73 thf(f732,plain,(
% 2.72/1.73 (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i,X1 : $i,X2 : $i] : (? [X3 : $i] : (? [X4 : $i] : ((((ap @ X2 @ X1 @ X0 @ X3) != X4) & ((in @ (kpair @ X3 @ X4) @ X0) = $true)) & ((in @ X4 @ X1) = $true)) & ((in @ X3 @ X2) = $true)) & ((in @ X0 @ (funcSet @ X2 @ X1)) = $true)) & (funcextLem = $true)) & (funcGraphProp2 = $true)) & (funcGraphProp3 = $true)) & (funcGraphProp1 = $true)) & (ex1E2 = $true)) & (brelnall2 = $true)) & (brelnall1 = $true)) & (lam2p = $true)) & (lamp = $true)) & (lamProp = $true)) & (funcinfuncset = $true)) & (ap2p = $true)) & (infuncsetfunc = $true)) & (app = $true)) & (apProp = $true)) & (funcImageSingleton = $true)) & (dpsetconstrER = $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)),
% 2.72/1.73 inference(ennf_transformation,[status(thm)],[f568])).
% 2.72/1.73 thf(f568,plain,(
% 2.72/1.73 ~((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) => ((dpsetconstrER = $true) => ((funcImageSingleton = $true) => ((apProp = $true) => ((app = $true) => ((infuncsetfunc = $true) => ((ap2p = $true) => ((funcinfuncset = $true) => ((lamProp = $true) => ((lamp = $true) => ((lam2p = $true) => ((brelnall1 = $true) => ((brelnall2 = $true) => ((ex1E2 = $true) => ((funcGraphProp1 = $true) => ((funcGraphProp3 = $true) => ((funcGraphProp2 = $true) => ((funcextLem = $true) => ! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X0 @ (funcSet @ X2 @ X1)) = $true) => ! [X3 : $i] : (((in @ X3 @ X2) = $true) => ! [X4 : $i] : (((in @ X4 @ X1) = $true) => (((in @ (kpair @ X3 @ X4) @ X0) = $true) => ((ap @ X2 @ X1 @ X0 @ X3) = X4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.72/1.73 inference(fool_elimination,[status(thm)],[f567])).
% 2.72/1.73 thf(f567,plain,(
% 2.72/1.73 ~(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 => (dpsetconstrER => (funcImageSingleton => (apProp => (app => (infuncsetfunc => (ap2p => (funcinfuncset => (lamProp => (lamp => (lam2p => (brelnall1 => (brelnall2 => (ex1E2 => (funcGraphProp1 => (funcGraphProp3 => (funcGraphProp2 => (funcextLem => ! [X0 : $i,X1 : $i,X2 : $i] : ((in @ X0 @ (funcSet @ X2 @ X1)) => ! [X3 : $i] : ((in @ X3 @ X2) => ! [X4 : $i] : ((in @ X4 @ X1) => ((in @ (kpair @ X3 @ X4) @ X0) => ((ap @ X2 @ X1 @ X0 @ X3) = X4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.72/1.73 inference(rectify,[status(thm)],[f213])).
% 2.72/1.73 thf(f213,negated_conjecture,(
% 2.72/1.73 ~(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 => (dpsetconstrER => (funcImageSingleton => (apProp => (app => (infuncsetfunc => (ap2p => (funcinfuncset => (lamProp => (lamp => (lam2p => (brelnall1 => (brelnall2 => (ex1E2 => (funcGraphProp1 => (funcGraphProp3 => (funcGraphProp2 => (funcextLem => ! [X13 : $i,X4 : $i,X3 : $i] : ((in @ X13 @ (funcSet @ X3 @ X4)) => ! [X1 : $i] : ((in @ X1 @ X3) => ! [X2 : $i] : ((in @ X2 @ X4) => ((in @ (kpair @ X1 @ X2) @ X13) => ((ap @ X3 @ X4 @ X13 @ X1) = X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.72/1.73 inference(negated_conjecture,[status(cth)],[f212])).
% 2.72/1.73 thf(f212,conjecture,(
% 2.72/1.73 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 => (dpsetconstrER => (funcImageSingleton => (apProp => (app => (infuncsetfunc => (ap2p => (funcinfuncset => (lamProp => (lamp => (lam2p => (brelnall1 => (brelnall2 => (ex1E2 => (funcGraphProp1 => (funcGraphProp3 => (funcGraphProp2 => (funcextLem => ! [X13 : $i,X4 : $i,X3 : $i] : ((in @ X13 @ (funcSet @ X3 @ X4)) => ! [X1 : $i] : ((in @ X1 @ X3) => ! [X2 : $i] : ((in @ X2 @ X4) => ((in @ (kpair @ X1 @ X2) @ X13) => ((ap @ X3 @ X4 @ X13 @ X1) = X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.72/1.73 file('/export/starexec/sandbox/tmp/tmp.CZb2EvKrRp/DTF2THF_1488.p',funcGraphProp4)).
% 2.72/1.73 thf(f3051,plain,(
% 2.72/1.73 ((in @ sK496 @ sK493) != $true) | ~spl618_1),
% 2.72/1.73 inference(trivial_inequality_removal,[status(thm)],[f3041])).
% 2.72/1.73 thf(f3041,plain,(
% 2.72/1.73 ((in @ sK496 @ sK493) != $true) | ($true != $true) | (sK496 != sK496) | ~spl618_1),
% 2.72/1.73 inference(superposition,[status(thm)],[f3034,f2669])).
% 2.72/1.73 thf(f2669,plain,(
% 2.72/1.73 ($true = (in @ (kpair @ sK495 @ sK496) @ sK492))),
% 2.72/1.73 inference(cnf_transformation,[status(thm)],[f1654])).
% 2.72/1.73 thf(f3034,plain,(
% 2.72/1.73 ( ! [X0 : $i] : (((in @ (kpair @ sK495 @ X0) @ sK492) != $true) | ($true != (in @ X0 @ sK493)) | (sK496 != X0)) ) | ~spl618_1),
% 2.72/1.73 inference(avatar_component_clause,[status(thm)],[f3033])).
% 2.72/1.73 thf(f3033,definition,(
% 2.72/1.73 spl618_1 <=> ! [X0 : $i] : (($true != (in @ X0 @ sK493)) | (sK496 != X0) | ((in @ (kpair @ sK495 @ X0) @ sK492) != $true))),
% 2.72/1.73 introduced(definition,[new_symbols(naming,[spl618_1])],[avatar_definition])).
% 2.72/1.73 thf(f3040,plain,(
% 2.72/1.73 spl618_2),
% 2.72/1.73 inference(avatar_split_clause,[status(thm)],[f3020,f3036])).
% 2.72/1.73 thf(f3036,definition,(
% 2.72/1.73 spl618_2 <=> ((func @ sK494 @ sK493 @ sK492) = $true)),
% 2.72/1.73 introduced(definition,[new_symbols(naming,[spl618_2])],[avatar_definition])).
% 2.72/1.73 thf(f3020,plain,(
% 2.72/1.73 ((func @ sK494 @ sK493 @ sK492) = $true)),
% 2.72/1.73 inference(trivial_inequality_removal,[status(thm)],[f3019])).
% 2.72/1.73 thf(f3019,plain,(
% 2.72/1.73 ((func @ sK494 @ sK493 @ sK492) = $true) | ($true != $true)),
% 2.72/1.73 inference(forward_demodulation,[status(thm)],[f3018,f2672])).
% 2.72/1.73 thf(f2672,plain,(
% 2.72/1.73 (infuncsetfunc = $true)),
% 2.72/1.73 inference(cnf_transformation,[status(thm)],[f1654])).
% 2.72/1.73 thf(f3018,plain,(
% 2.72/1.73 ((func @ sK494 @ sK493 @ sK492) = $true) | (infuncsetfunc != $true)),
% 2.72/1.73 inference(trivial_inequality_removal,[status(thm)],[f3015])).
% 2.72/1.73 thf(f3015,plain,(
% 2.72/1.73 ((func @ sK494 @ sK493 @ sK492) = $true) | (infuncsetfunc != $true) | ($true != $true)),
% 2.72/1.73 inference(superposition,[status(thm)],[f2084,f2667])).
% 2.72/1.73 thf(f2667,plain,(
% 2.72/1.73 ((in @ sK492 @ (funcSet @ sK494 @ sK493)) = $true)),
% 2.72/1.73 inference(cnf_transformation,[status(thm)],[f1654])).
% 2.72/1.73 thf(f2084,plain,(
% 2.72/1.73 ( ! [X3 : $i,X4 : $i,X5 : $i] : (((in @ X5 @ (funcSet @ X4 @ X3)) != $true) | (infuncsetfunc != $true) | ($true = (func @ X4 @ X3 @ X5))) )),
% 2.72/1.73 inference(cnf_transformation,[status(thm)],[f1182])).
% 2.72/1.73 thf(f1182,plain,(
% 2.72/1.73 ((infuncsetfunc = $true) | (($true != (func @ sK185 @ sK184 @ sK186)) & ($true = (in @ sK186 @ (funcSet @ sK185 @ sK184))))) & (! [X3 : $i,X4 : $i,X5 : $i] : (($true = (func @ X4 @ X3 @ X5)) | ((in @ X5 @ (funcSet @ X4 @ X3)) != $true)) | (infuncsetfunc != $true))),
% 2.72/1.73 inference(skolemisation,[status(esa),new_symbols(skolem,[sK184,sK185,sK186])],[f1180,f1181])).
% 2.72/1.73 thf(f1181,plain,(
% 2.72/1.73 ? [X0 : $i,X1 : $i,X2 : $i] : (((func @ X1 @ X0 @ X2) != $true) & ((in @ X2 @ (funcSet @ X1 @ X0)) = $true)) => (($true != (func @ sK185 @ sK184 @ sK186)) & ($true = (in @ sK186 @ (funcSet @ sK185 @ sK184))))),
% 2.72/1.73 introduced(definition,[],[choice_axiom])).
% 2.72/1.73 thf(f1180,plain,(
% 2.72/1.73 ((infuncsetfunc = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (((func @ X1 @ X0 @ X2) != $true) & ((in @ X2 @ (funcSet @ X1 @ X0)) = $true))) & (! [X3 : $i,X4 : $i,X5 : $i] : (($true = (func @ X4 @ X3 @ X5)) | ((in @ X5 @ (funcSet @ X4 @ X3)) != $true)) | (infuncsetfunc != $true))),
% 2.72/1.73 inference(rectify,[status(thm)],[f1179])).
% 2.72/1.73 thf(f1179,plain,(
% 2.72/1.73 ((infuncsetfunc = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (((func @ X1 @ X0 @ X2) != $true) & ((in @ X2 @ (funcSet @ X1 @ X0)) = $true))) & (! [X0 : $i,X1 : $i,X2 : $i] : (((func @ X1 @ X0 @ X2) = $true) | ((in @ X2 @ (funcSet @ X1 @ X0)) != $true)) | (infuncsetfunc != $true))),
% 2.72/1.73 inference(nnf_transformation,[status(thm)],[f753])).
% 2.72/1.73 thf(f753,plain,(
% 2.72/1.73 (infuncsetfunc = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i] : (((func @ X1 @ X0 @ X2) = $true) | ((in @ X2 @ (funcSet @ X1 @ X0)) != $true))),
% 2.72/1.73 inference(ennf_transformation,[status(thm)],[f254])).
% 2.72/1.73 thf(f254,plain,(
% 2.72/1.73 (infuncsetfunc = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X2 @ (funcSet @ X1 @ X0)) = $true) => ((func @ X1 @ X0 @ X2) = $true))),
% 2.72/1.73 inference(fool_elimination,[status(thm)],[f253])).
% 2.72/1.73 thf(f253,plain,(
% 2.72/1.73 (infuncsetfunc = ! [X0 : $i,X1 : $i,X2 : $i] : ((in @ X2 @ (funcSet @ X1 @ X0)) => (func @ X1 @ X0 @ X2)))),
% 2.72/1.73 inference(rectify,[status(thm)],[f198])).
% 2.72/1.73 thf(f198,axiom,(
% 2.72/1.73 (infuncsetfunc = ! [X4 : $i,X3 : $i,X13 : $i] : ((in @ X13 @ (funcSet @ X3 @ X4)) => (func @ X3 @ X4 @ X13)))),
% 2.72/1.73 file('/export/starexec/sandbox/tmp/tmp.CZb2EvKrRp/DTF2THF_1488.p',infuncsetfunc)).
% 2.72/1.73 thf(f3039,plain,(
% 2.72/1.73 spl618_1 | ~spl618_2),
% 2.72/1.73 inference(avatar_split_clause,[status(thm)],[f3031,f3036,f3033])).
% 2.72/1.73 thf(f3031,plain,(
% 2.72/1.73 ( ! [X0 : $i] : (($true != (in @ X0 @ sK493)) | ((func @ sK494 @ sK493 @ sK492) != $true) | ((in @ (kpair @ sK495 @ X0) @ sK492) != $true) | (sK496 != X0)) )),
% 2.72/1.73 inference(trivial_inequality_removal,[status(thm)],[f3030])).
% 2.72/1.73 thf(f3030,plain,(
% 2.72/1.73 ( ! [X0 : $i] : (((in @ (kpair @ sK495 @ X0) @ sK492) != $true) | (sK496 != X0) | ($true != $true) | ((func @ sK494 @ sK493 @ sK492) != $true) | ($true != (in @ X0 @ sK493))) )),
% 2.72/1.73 inference(forward_demodulation,[status(thm)],[f3029,f2531])).
% 2.72/1.73 thf(f2531,plain,(
% 2.72/1.73 (funcGraphProp2 = $true)),
% 2.72/1.73 inference(cnf_transformation,[status(thm)],[f1654])).
% 2.72/1.73 thf(f3029,plain,(
% 2.72/1.73 ( ! [X0 : $i] : (($true != (in @ X0 @ sK493)) | (sK496 != X0) | (funcGraphProp2 != $true) | ((in @ (kpair @ sK495 @ X0) @ sK492) != $true) | ((func @ sK494 @ sK493 @ sK492) != $true)) )),
% 2.72/1.73 inference(trivial_inequality_removal,[status(thm)],[f3028])).
% 2.72/1.73 thf(f3028,plain,(
% 2.72/1.73 ( ! [X0 : $i] : (((in @ (kpair @ sK495 @ X0) @ sK492) != $true) | (funcGraphProp2 != $true) | ($true != $true) | (sK496 != X0) | ((func @ sK494 @ sK493 @ sK492) != $true) | ($true != (in @ X0 @ sK493))) )),
% 2.72/1.73 inference(forward_demodulation,[status(thm)],[f3027,f2671])).
% 2.72/1.73 thf(f2671,plain,(
% 2.72/1.73 ($true = (in @ sK495 @ sK494))),
% 2.72/1.73 inference(cnf_transformation,[status(thm)],[f1654])).
% 2.72/1.73 thf(f3027,plain,(
% 2.72/1.73 ( ! [X0 : $i] : (((func @ sK494 @ sK493 @ sK492) != $true) | ($true != (in @ X0 @ sK493)) | ((in @ (kpair @ sK495 @ X0) @ sK492) != $true) | ($true != (in @ sK495 @ sK494)) | (funcGraphProp2 != $true) | (sK496 != X0)) )),
% 2.72/1.73 inference(superposition,[status(thm)],[f2670,f2299])).
% 2.72/1.73 thf(f2299,plain,(
% 2.72/1.73 ( ! [X8 : $i,X6 : $i,X9 : $i,X7 : $i,X5 : $i] : (((ap @ X5 @ X7 @ X6 @ X8) = X9) | ($true != (in @ X9 @ X7)) | (funcGraphProp2 != $true) | ((func @ X5 @ X7 @ X6) != $true) | ($true != (in @ (kpair @ X8 @ X9) @ X6)) | ($true != (in @ X8 @ X5))) )),
% 2.72/1.73 inference(cnf_transformation,[status(thm)],[f1435])).
% 2.72/1.73 thf(f1435,plain,(
% 2.72/1.73 ((funcGraphProp2 = $true) | (((func @ sK344 @ sK346 @ sK345) = $true) & ((((in @ (kpair @ sK347 @ sK348) @ sK345) = $true) & ($true = (in @ sK348 @ sK346)) & ((ap @ sK344 @ sK346 @ sK345 @ sK347) != sK348)) & ($true = (in @ sK347 @ sK344))))) & (! [X5 : $i,X6 : $i,X7 : $i] : (((func @ X5 @ X7 @ X6) != $true) | ! [X8 : $i] : (! [X9 : $i] : (($true != (in @ (kpair @ X8 @ X9) @ X6)) | ($true != (in @ X9 @ X7)) | ((ap @ X5 @ X7 @ X6 @ X8) = X9)) | ($true != (in @ X8 @ X5)))) | (funcGraphProp2 != $true))),
% 2.72/1.73 inference(skolemisation,[status(esa),new_symbols(skolem,[sK344,sK345,sK346,sK347,sK348])],[f1431,f1434,f1433,f1432])).
% 2.72/1.73 thf(f1432,plain,(
% 2.72/1.73 ? [X0 : $i,X1 : $i,X2 : $i] : (($true = (func @ X0 @ X2 @ X1)) & ? [X3 : $i] : (? [X4 : $i] : (($true = (in @ (kpair @ X3 @ X4) @ X1)) & ($true = (in @ X4 @ X2)) & ((ap @ X0 @ X2 @ X1 @ X3) != X4)) & ((in @ X3 @ X0) = $true))) => (((func @ sK344 @ sK346 @ sK345) = $true) & ? [X3 : $i] : (? [X4 : $i] : (((in @ (kpair @ X3 @ X4) @ sK345) = $true) & ($true = (in @ X4 @ sK346)) & ((ap @ sK344 @ sK346 @ sK345 @ X3) != X4)) & ((in @ X3 @ sK344) = $true)))),
% 2.72/1.73 introduced(definition,[],[choice_axiom])).
% 2.72/1.73 thf(f1433,plain,(
% 2.72/1.73 ? [X3 : $i] : (? [X4 : $i] : (((in @ (kpair @ X3 @ X4) @ sK345) = $true) & ($true = (in @ X4 @ sK346)) & ((ap @ sK344 @ sK346 @ sK345 @ X3) != X4)) & ((in @ X3 @ sK344) = $true)) => (? [X4 : $i] : (((in @ (kpair @ sK347 @ X4) @ sK345) = $true) & ($true = (in @ X4 @ sK346)) & ((ap @ sK344 @ sK346 @ sK345 @ sK347) != X4)) & ($true = (in @ sK347 @ sK344)))),
% 2.72/1.73 introduced(definition,[],[choice_axiom])).
% 2.72/1.73 thf(f1434,plain,(
% 2.72/1.73 ? [X4 : $i] : (((in @ (kpair @ sK347 @ X4) @ sK345) = $true) & ($true = (in @ X4 @ sK346)) & ((ap @ sK344 @ sK346 @ sK345 @ sK347) != X4)) => (((in @ (kpair @ sK347 @ sK348) @ sK345) = $true) & ($true = (in @ sK348 @ sK346)) & ((ap @ sK344 @ sK346 @ sK345 @ sK347) != sK348))),
% 2.72/1.73 introduced(definition,[],[choice_axiom])).
% 2.72/1.73 thf(f1431,plain,(
% 2.72/1.73 ((funcGraphProp2 = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (($true = (func @ X0 @ X2 @ X1)) & ? [X3 : $i] : (? [X4 : $i] : (($true = (in @ (kpair @ X3 @ X4) @ X1)) & ($true = (in @ X4 @ X2)) & ((ap @ X0 @ X2 @ X1 @ X3) != X4)) & ((in @ X3 @ X0) = $true)))) & (! [X5 : $i,X6 : $i,X7 : $i] : (((func @ X5 @ X7 @ X6) != $true) | ! [X8 : $i] : (! [X9 : $i] : (($true != (in @ (kpair @ X8 @ X9) @ X6)) | ($true != (in @ X9 @ X7)) | ((ap @ X5 @ X7 @ X6 @ X8) = X9)) | ($true != (in @ X8 @ X5)))) | (funcGraphProp2 != $true))),
% 2.72/1.73 inference(rectify,[status(thm)],[f1430])).
% 2.72/1.73 thf(f1430,plain,(
% 2.72/1.73 ((funcGraphProp2 = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (($true = (func @ X0 @ X2 @ X1)) & ? [X3 : $i] : (? [X4 : $i] : (($true = (in @ (kpair @ X3 @ X4) @ X1)) & ($true = (in @ X4 @ X2)) & ((ap @ X0 @ X2 @ X1 @ X3) != X4)) & ((in @ X3 @ X0) = $true)))) & (! [X0 : $i,X1 : $i,X2 : $i] : (($true != (func @ X0 @ X2 @ X1)) | ! [X3 : $i] : (! [X4 : $i] : (($true != (in @ (kpair @ X3 @ X4) @ X1)) | ($true != (in @ X4 @ X2)) | ((ap @ X0 @ X2 @ X1 @ X3) = X4)) | ((in @ X3 @ X0) != $true))) | (funcGraphProp2 != $true))),
% 2.72/1.73 inference(nnf_transformation,[status(thm)],[f748])).
% 2.72/1.73 thf(f748,plain,(
% 2.72/1.73 (funcGraphProp2 = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i] : (($true != (func @ X0 @ X2 @ X1)) | ! [X3 : $i] : (! [X4 : $i] : (($true != (in @ (kpair @ X3 @ X4) @ X1)) | ($true != (in @ X4 @ X2)) | ((ap @ X0 @ X2 @ X1 @ X3) = X4)) | ((in @ X3 @ X0) != $true)))),
% 2.72/1.73 inference(flattening,[status(thm)],[f747])).
% 2.72/1.73 thf(f747,plain,(
% 2.72/1.73 ! [X0 : $i,X1 : $i,X2 : $i] : (! [X3 : $i] : (! [X4 : $i] : ((((ap @ X0 @ X2 @ X1 @ X3) = X4) | ($true != (in @ (kpair @ X3 @ X4) @ X1))) | ($true != (in @ X4 @ X2))) | ((in @ X3 @ X0) != $true)) | ($true != (func @ X0 @ X2 @ X1))) <=> (funcGraphProp2 = $true)),
% 2.72/1.73 inference(ennf_transformation,[status(thm)],[f242])).
% 2.72/1.73 thf(f242,plain,(
% 2.72/1.73 ! [X0 : $i,X1 : $i,X2 : $i] : (($true = (func @ X0 @ X2 @ X1)) => ! [X3 : $i] : (((in @ X3 @ X0) = $true) => ! [X4 : $i] : (($true = (in @ X4 @ X2)) => (($true = (in @ (kpair @ X3 @ X4) @ X1)) => ((ap @ X0 @ X2 @ X1 @ X3) = X4))))) <=> (funcGraphProp2 = $true)),
% 2.72/1.73 inference(fool_elimination,[status(thm)],[f241])).
% 2.72/1.73 thf(f241,plain,(
% 2.72/1.73 (funcGraphProp2 = ! [X0 : $i,X1 : $i,X2 : $i] : ((func @ X0 @ X2 @ X1) => ! [X3 : $i] : ((in @ X3 @ X0) => ! [X4 : $i] : ((in @ X4 @ X2) => ((in @ (kpair @ X3 @ X4) @ X1) => ((ap @ X0 @ X2 @ X1 @ X3) = X4))))))),
% 2.72/1.73 inference(rectify,[status(thm)],[f210])).
% 2.72/1.73 thf(f210,axiom,(
% 2.72/1.73 (funcGraphProp2 = ! [X3 : $i,X13 : $i,X4 : $i] : ((func @ X3 @ X4 @ X13) => ! [X1 : $i] : ((in @ X1 @ X3) => ! [X2 : $i] : ((in @ X2 @ X4) => ((in @ (kpair @ X1 @ X2) @ X13) => ((ap @ X3 @ X4 @ X13 @ X1) = X2))))))),
% 2.72/1.73 file('/export/starexec/sandbox/tmp/tmp.CZb2EvKrRp/DTF2THF_1488.p',funcGraphProp2)).
% 2.72/1.73 thf(f2670,plain,(
% 2.72/1.73 (sK496 != (ap @ sK494 @ sK493 @ sK492 @ sK495))),
% 2.72/1.73 inference(cnf_transformation,[status(thm)],[f1654])).
% 2.72/1.73 % SZS output end Proof for DTF2THF_1488
% 2.72/1.73 % (1795)------------------------------
% 2.72/1.73 % (1795)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.73 % (1795)Termination reason: Refutation
% 2.72/1.73
% 2.72/1.73 % (1795)Memory used [KB]: 8955
% 2.72/1.73 % (1795)Time elapsed: 0.086 s
% 2.72/1.73 % (1795)Instructions burned: 256 (million)
% 2.72/1.73 % (1795)------------------------------
% 2.72/1.73 % (1795)------------------------------
% 2.72/1.73 % (1772)Success in time 0.172 s
%------------------------------------------------------------------------------