%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU698^1 : 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 : n001.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 09:36:29 AM UTC 2026
% Result : Theorem 2.65s 1.72s
% Output : Refutation 2.65s
% Verified :
% SZS Type : Refutation
% Derivation depth : 18
% Number of leaves : 7
% Syntax : Number of formulae : 44 ( 11 unt; 0 typ; 0 def)
% Number of atoms : 3737 (1003 equ; 0 cnn)
% Maximal formula atoms : 218 ( 84 avg)
% Number of connectives : 2282 ( 81 ~; 56 |; 675 &; 586 @)
% ( 2 <=>; 882 =>; 0 <=; 0 <~>)
% Maximal formula depth : 225 ( 39 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 22 ( 22 >; 0 *; 0 +; 0 <<)
% Number of symbols : 404 ( 401 usr; 222 con; 0-4 aty)
% Number of variables : 141 ( 28 ^; 81 !; 32 ?; 141 :)
% 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_265,type,
sP1: $i > $i > $o ).
thf(func_def_266,type,
sP2: $i > $o ).
thf(func_def_267,type,
sP3: $i > $i > $o ).
thf(func_def_268,type,
sP4: $i > $i > $o ).
thf(func_def_269,type,
sP5: $i > $i > $i > $i > $o ).
thf(func_def_272,type,
sP8: $o > $i > $i > $i > $o ).
thf(func_def_278,type,
sK14: $o > $i > $i > $i ).
thf(func_def_282,type,
sK18: $i > $i > $i ).
thf(func_def_293,type,
sK29: ( $i > $o ) > $i > $i > $i ).
thf(func_def_296,type,
sK32: $i > $o ).
thf(func_def_297,type,
sK33: $i > $o ).
thf(func_def_299,type,
sK35: $i > ( $i > $o ) > $i ).
thf(func_def_322,type,
sK58: $i > $i ).
thf(func_def_327,type,
sK63: $i > $o ).
thf(func_def_333,type,
sK69: $i > $o ).
thf(func_def_335,type,
sK71: $i > $i ).
thf(func_def_342,type,
sK78: $i > $o ).
thf(func_def_344,type,
sK80: ( $i > $o ) > $i ).
thf(func_def_345,type,
sK81: ( $i > $o ) > $i ).
thf(func_def_346,type,
sK82: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_347,type,
sK83: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_351,type,
sK87: $i > $o ).
thf(func_def_361,type,
sK97: $i > $o ).
thf(func_def_364,type,
sK100: $i > ( $i > $o ) > $i ).
thf(func_def_365,type,
sK101: $i > ( $i > $o ) > $i ).
thf(func_def_368,type,
sK104: $i > $i > $i ).
thf(func_def_369,type,
sK105: $i > $i > $i ).
thf(func_def_374,type,
sK110: $i > $i > $o ).
thf(func_def_387,type,
sK123: $i > $i > $i ).
thf(func_def_394,type,
sK130: $i > $i > $i ).
thf(func_def_398,type,
sK134: $i > $o ).
thf(func_def_407,type,
sK143: $i > $i > $i > $i ).
thf(func_def_408,type,
sK144: $i > $i > $i > $i ).
thf(func_def_414,type,
sK150: $i > $o ).
thf(func_def_417,type,
sK153: $i > ( $i > $o ) > $i ).
thf(func_def_425,type,
sK161: $i > $i ).
thf(func_def_426,type,
sK162: $i > $o ).
thf(func_def_430,type,
sK166: $i > $o ).
thf(func_def_431,type,
sK167: $i > $o ).
thf(func_def_432,type,
sK168: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_433,type,
sK169: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_436,type,
sK172: $i > $i > $i ).
thf(func_def_440,type,
sK176: $i > $i > $i ).
thf(func_def_441,type,
sK177: $i > $i > $i ).
thf(func_def_442,type,
sK178: $i > $i > $i ).
thf(func_def_443,type,
sK179: $i > $i > $i ).
thf(func_def_444,type,
sK180: $i > $i > $i > $i ).
thf(func_def_445,type,
sK181: $i > $i ).
thf(func_def_446,type,
sK182: $i > $i ).
thf(func_def_447,type,
sK183: $i > $i ).
thf(func_def_448,type,
sK184: $i > $i ).
thf(func_def_449,type,
sK185: $i > $i > $i ).
thf(func_def_450,type,
sK186: $i > $i > $i > $i ).
thf(func_def_451,type,
sK187: $i > $i > $i > $i ).
thf(func_def_452,type,
sK188: $i > $i > $i ).
thf(func_def_453,type,
sK189: $i > $i > $i ).
thf(func_def_454,type,
sK190: $i > $i ).
thf(func_def_455,type,
sK191: $i > $i > $i ).
thf(func_def_457,type,
sK193: $i > $i ).
thf(func_def_458,type,
sK194: $i > $i ).
thf(func_def_483,type,
sK219: $i > $i > $i ).
thf(func_def_484,type,
sK220: $i > $i > ( $i > $i ) > $i ).
thf(func_def_485,type,
sK221: $i > $i ).
thf(func_def_499,type,
sK235: $i > $i > $o ).
thf(func_def_505,type,
sK241: $i > ( $i > $o ) > $i ).
thf(func_def_506,type,
sK242: $i > $o ).
thf(func_def_513,type,
sK249: $i > $o ).
thf(func_def_516,type,
sK252: $i > $i > $o ).
thf(func_def_520,type,
sK256: ( $i > $o ) > $i > $i ).
thf(func_def_522,type,
sK258: $i > $o ).
thf(func_def_538,type,
sK274: $i > $i ).
thf(func_def_540,type,
sK276: $i > ( $i > $i ) > $i > $i ).
thf(func_def_545,type,
sK281: $i > $o ).
thf(func_def_555,type,
sK291: $i > ( $i > $i ) > $i > $i ).
thf(func_def_557,type,
sK293: $i > $i ).
thf(func_def_567,type,
sK303: $i > $i > $i > $i > $i ).
thf(func_def_568,type,
sK304: $i > $i > $i > $i > $i ).
thf(func_def_569,type,
sK305: $i > $i > $i > $i > $i ).
thf(func_def_570,type,
sK306: $i > $i > $i > $i > $i ).
thf(func_def_575,type,
sK311: $i > $i ).
thf(func_def_579,type,
sK315: $i > $i > $i ).
thf(func_def_584,type,
sK320: ( $i > $o ) > $i ).
thf(func_def_585,type,
sK321: $i > $o ).
thf(func_def_586,type,
sK322: $i > $i > ( $i > $i ) > $i ).
thf(func_def_587,type,
sK323: $i > $i ).
thf(func_def_591,type,
sK327: $i > $o ).
thf(func_def_597,type,
sK333: $i > $i > $i ).
thf(func_def_605,type,
sK341: $i > $i > $i > $i > $i ).
thf(func_def_606,type,
sK342: $i > $i > $i > $i > $i ).
thf(func_def_627,type,
sK363: $i > $o ).
thf(func_def_643,type,
sK379: $i > $i > $o ).
thf(func_def_654,type,
sK390: $i > $o ).
thf(func_def_671,type,
sK407: $i > $i ).
thf(func_def_689,type,
sK425: $i > $o ).
thf(func_def_698,type,
sK434: $i > $i > $o ).
thf(func_def_701,type,
sK437: $i > $i > $o ).
thf(func_def_704,type,
sK440: $i > $o ).
thf(func_def_707,type,
sK443: $i > $o ).
thf(func_def_709,type,
sK445: ( $i > $o ) > $i > $i ).
thf(func_def_710,type,
sK446: ( $i > $o ) > $i > $i ).
thf(func_def_728,type,
sK464: $i > $o ).
thf(func_def_729,type,
sK465: $i > $o ).
thf(func_def_730,type,
sK466: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_731,type,
sK467: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_737,type,
sK473: $i > $o ).
thf(func_def_739,type,
sK475: ( $i > $o ) > $i > $i ).
thf(func_def_769,type,
sK505: $i > $i > $i > $i > $i ).
thf(func_def_772,type,
sK508: $i > $i ).
thf(func_def_775,type,
sK511: $i > $i ).
thf(func_def_788,type,
sK524: $i > ( $i > $i ) > $i > $i ).
thf(func_def_790,type,
sK526: $i > $i ).
thf(func_def_795,type,
sK531: $i > $o ).
thf(func_def_797,type,
sK533: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_798,type,
sK534: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_804,type,
sK540: $i > $o ).
thf(func_def_807,type,
sK543: $i > ( $i > $o ) > $i ).
thf(func_def_813,type,
sK549: $i > $i > $o ).
thf(func_def_816,type,
sK552: $i > $i > $i > $i > $i ).
thf(func_def_833,type,
sK569: $i > $o ).
thf(func_def_844,type,
sK580: $i > ( $i > $o ) > $i ).
thf(func_def_845,type,
sK581: $i > $o ).
thf(func_def_849,type,
sK585: $i > $o ).
thf(func_def_852,type,
sK588: $i > ( $i > $o ) > $i ).
thf(func_def_855,type,
sK591: $i > $o ).
thf(func_def_856,type,
sK592: $i > $i ).
thf(func_def_857,type,
sK593: ( $i > $o ) > $i ).
thf(func_def_860,type,
sK596: $i > $i ).
thf(func_def_865,type,
sK601: $i > $i > $o ).
thf(func_def_866,type,
sK602: $i > $i ).
thf(func_def_867,type,
sK603: $i > $i ).
thf(func_def_868,type,
sK604: ( $i > $i > $o ) > $i > $i ).
thf(func_def_869,type,
sK605: ( $i > $i > $o ) > $i > $i ).
thf(func_def_870,type,
sK606: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_877,type,
sK613: $i > $o ).
thf(func_def_878,type,
sK614: $i > $i ).
thf(func_def_879,type,
sK615: ( $i > $o ) > $i ).
thf(func_def_880,type,
sK616: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_881,type,
sK617: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_882,type,
sK618: $i > $o ).
thf(func_def_883,type,
sK619: $i > $o ).
thf(func_def_890,type,
sK626: $i > $i > $i > $i > $i ).
thf(func_def_897,type,
sK633: $i > $o ).
thf(func_def_899,type,
sK635: ( $i > $o ) > $i > $i ).
thf(func_def_913,type,
sK649: $i > $o ).
thf(func_def_919,type,
sK655: $i > $i > $i ).
thf(func_def_933,type,
ph669:
!>[X0: $tType] : X0 ).
thf(f3675,plain,
$false,
inference(trivial_inequality_removal,[status(thm)],[f3664]) ).
thf(f3664,plain,
$true != $true,
inference(superposition,[status(thm)],[f3620,f3238]) ).
thf(f3238,plain,
( ( in @ ( sK291 @ sK406 @ sK407 @ sK408 ) @ sK406 )
= $true ),
inference(trivial_inequality_removal,[status(thm)],[f3237]) ).
thf(f3237,plain,
( ( ( in @ ( sK291 @ sK406 @ sK407 @ sK408 ) @ sK406 )
= $true )
| ( $true != $true ) ),
inference(forward_demodulation,[status(thm)],[f3236,f2663]) ).
thf(f2663,plain,
( ( in @ sK409 @ sK406 )
= $true ),
inference(cnf_transformation,[status(thm)],[f1553]) ).
thf(f1553,plain,
( ( brelnall2 = $true )
& ( exuEu = $true )
& ( ksndpairEq = $true )
& ( setukpairinjR2 = $true )
& ( funcext2 = $true )
& ( funcext = $true )
& ( kpairsurjEq = $true )
& ( binintersectSubset4 = $true )
& ( brelnall1 = $true )
& ( quantDeMorgan3 = $true )
& ( upairset2IR = $true )
& ( binunionIR = $true )
& ( exuI3 = $true )
& ( notdallE = $true )
& ( nonemptyI = $true )
& ( singletonsubset = $true )
& ( subsetemptysetimpeq = $true )
& ( emptyinPowerset = $true )
& ( dpsetconstrERa = $true )
& ( notsubsetI = $true )
& ( kfstpairEq = $true )
& ( setunionsingleton1 = $true )
& ! [X3: $i] :
( ( ( in @ X3 @ sK406 )
!= $true )
| ( ( in @ ( sK407 @ X3 ) @ sK408 )
= $true ) )
& ( ( ap @ sK406 @ sK408
@ ( lam @ sK406 @ sK408
@ ^ [Y0: $i] : ( sK407 @ Y0 ) )
@ sK409 )
!= ( sK407 @ sK409 ) )
& ( ( in @ sK409 @ sK406 )
= $true )
& ( funcGraphProp3 = $true )
& ( setbeta = $true )
& ( symdiffIneg1 = $true )
& ( funcGraphProp4 = $true )
& ( omega0Ax = $true )
& ( sepInPowerset = $true )
& ( funcImageSingleton = $true )
& ( lamp = $true )
& ( notdexE = $true )
& ( setminusERneg = $true )
& ( binunionE = $true )
& ( dpsetconstrER = $true )
& ( ex1I2 = $true )
& ( dpsetconstrI = $true )
& ( lam2p = $true )
& ( emptyE1 = $true )
& ( upairequniteq = $true )
& ( ubforcartprodlem3 = $true )
& ( setunion__Cong = $true )
& ( binintersectSubset3 = $true )
& ( setadjoin__Cong = $true )
& ( singletoninpowunion = $true )
& ( emptyset__Cong = $true )
& ( wellorderingAx = $true )
& ( setadjoinSub = $true )
& ( replAx = $true )
& ( exuE3e = $true )
& ( emptyI = $true )
& ( exuI1 = $true )
& ( setminusILneg = $true )
& ( foundationAx = $true )
& ( setukpairinjL = $true )
& ( symdiffI2 = $true )
& ( upairset2E = $true )
& ( setukpairinjR12 = $true )
& ( dsetconstrEL = $true )
& ( setminusEL = $true )
& ( inPowerset = $true )
& ( funcGraphProp1 = $true )
& ( symdiffI1 = $true )
& ( upairsetIR = $true )
& ( eqinunit = $true )
& ( beta1 = $true )
& ( vacuousDall = $true )
& ( binunionLsub = $true )
& ( notinsingleton = $true )
& ( cartprodpairsurjEq = $true )
& ( exuE3u = $true )
& ( powersetE = $true )
& ( eta1 = $true )
& ( cartprodmempaircEq = $true )
& ( ubforcartprodlem1 = $true )
& ( powersetE1 = $true )
& ( binunionRsub = $true )
& ( emptyInPowerset = $true )
& ( singletonprop = $true )
& ( powersetI = $true )
& ( upairsetE = $true )
& ( binintersectLsub = $true )
& ( cartprodpairin = $true )
& ( nonemptyI1 = $true )
& ( binintersectSubset1 = $true )
& ( setadjoinAx = $true )
& ( cartprodpairmemER = $true )
& ( setext = $true )
& ( setminusI = $true )
& ( cartprodfstin = $true )
& ( setukpairIL = $true )
& ( setunionsingleton = $true )
& ( setextAx = $true )
& ( setminusELneg = $true )
& ( dsetconstrER = $true )
& ( setadjoinIR = $true )
& ( cartprodsndpairEq = $true )
& ( subbreln = $true )
& ( binunionEcases = $true )
& ( prop2setE = $true )
& ( secondinupair = $true )
& ( binintersectRsub = $true )
& ( setadjoinOr = $true )
& ( subsetI2 = $true )
& ( eqimpsubset1 = $true )
& ( sepSubset = $true )
& ( binunionIL = $true )
& ( setunionAx = $true )
& ( setoftrueEq = $true )
& ( ex1I = $true )
& ( subset2powerset = $true )
& ( lamProp = $true )
& ( dsetconstrI = $true )
& ( cartprodmempair = $true )
& ( emptysetimpfalse = $true )
& ( subsetTrans = $true )
& ( ap2p = $true )
& ( subsetRefl = $true )
& ( quantDeMorgan1 = $true )
& ( cartprodsndin = $true )
& ( notequalI1 = $true )
& ( nonemptyE1 = $true )
& ( emptysetAx = $true )
& ( apProp = $true )
& ( ubforcartprodlem2 = $true )
& ( cartprodmempair1 = $true )
& ( subsetE = $true )
& ( app = $true )
& ( subPowSU = $true )
& ( descr__Cong = $true )
& ( upairinpowunion = $true )
& ( setukpairIR = $true )
& ( setminusER = $true )
& ( dpsetconstrEL1 = $true )
& ( prop2setI = $true )
& ( kfstsingleton = $true )
& ( ap2apEq2 = $true )
& ( setukpairinjR = $true )
& ( noeltsimpempty = $true )
& ( omegaIndAx = $true )
& ( subsetI1 = $true )
& ( funcinfuncset = $true )
& ( funcGraphProp2 = $true )
& ( kpairiskpair = $true )
& ( binintersectER = $true )
& ( eqimpsubset2 = $true )
& ( setminusSubset1 = $true )
& ( funcextLem = $true )
& ( disjointsetsI1 = $true )
& ( omegaSAx = $true )
& ( omega__Cong = $true )
& ( cartprodfstpairEq = $true )
& ( subsetE2 = $true )
& ( notequalI2 = $true )
& ( singletoninpowerset = $true )
& ( setukpairinjR1 = $true )
& ( eqbreln = $true )
& ( theprop = $true )
& ( uniqinunit = $true )
& ( infuncsetfunc = $true )
& ( in__Cong = $true )
& ( setOfPairsIsBReln = $true )
& ( ksndsingleton = $true )
& ( ap2apEq1 = $true )
& ( dpsetconstrSub = $true )
& ( binintersectSubset2 = $true )
& ( emptysetsubset = $true )
& ( kpairp = $true )
& ( setminusLsub = $true )
& ( setminusIRneg = $true )
& ( notinemptyset = $true )
& ( exuI2 = $true )
& ( powersetAx = $true )
& ( setunionI = $true )
& ( symdiffE = $true )
& ( singletonsuniq = $true )
& ( powersetI1 = $true )
& ( prop2set2propI = $true )
& ( quantDeMorgan4 = $true )
& ( quantDeMorgan2 = $true )
& ( dsetconstr__Cong = $true )
& ( upairsubunion = $true )
& ( exuE1 = $true )
& ( cartprodpairmemEL = $true )
& ( dpsetconstrEL2 = $true )
& ( powerset__Cong = $true )
& ( setukpairinjR11 = $true )
& ( exu__Cong = $true )
& ( setminusSubset2 = $true )
& ( setunionE = $true )
& ( ex1E2 = $true )
& ( singletonsswitch = $true )
& ( powersetsubset = $true )
& ( binintersectSubset5 = $true )
& ( lam2lamEq = $true )
& ( bs114d = $true )
& ( upairsetIL = $true )
& ( binintersectI = $true )
& ( emptysetE = $true )
& ( setadjoinE = $true )
& ( inCongP = $true )
& ( ex1E1 = $true )
& ( setadjoinSub2 = $true )
& ( exuE2 = $true )
& ( symdiffIneg2 = $true )
& ( setunionE2 = $true )
& ( nonemptyImpWitness = $true )
& ( setunionsingleton2 = $true )
& ( setextsub = $true )
& ( setukpairinjL1 = $true )
& ( setadjoinIL = $true )
& ( setukpairinjL2 = $true )
& ( binintersectEL = $true )
& ( descrp = $true )
& ( emptyinunitempty = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK406,sK407,sK408,sK409])],[f864,f1552,f1551]) ).
thf(f1551,plain,
( ? [X0: $i,X1: $i > $i,X2: $i] :
( ! [X3: $i] :
( ( $true
!= ( in @ X3 @ X0 ) )
| ( ( in @ ( X1 @ X3 ) @ X2 )
= $true ) )
& ? [X4: $i] :
( ( ( X1 @ X4 )
!= ( ap @ X0 @ X2
@ ( lam @ X0 @ X2
@ ^ [Y0: $i] : ( X1 @ Y0 ) )
@ X4 ) )
& ( ( in @ X4 @ X0 )
= $true ) ) )
=> ( ! [X3: $i] :
( ( ( in @ X3 @ sK406 )
!= $true )
| ( ( in @ ( sK407 @ X3 ) @ sK408 )
= $true ) )
& ? [X4: $i] :
( ( ( ap @ sK406 @ sK408
@ ( lam @ sK406 @ sK408
@ ^ [Y0: $i] : ( sK407 @ Y0 ) )
@ X4 )
!= ( sK407 @ X4 ) )
& ( ( in @ X4 @ sK406 )
= $true ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1552,plain,
( ? [X4: $i] :
( ( ( ap @ sK406 @ sK408
@ ( lam @ sK406 @ sK408
@ ^ [Y0: $i] : ( sK407 @ Y0 ) )
@ X4 )
!= ( sK407 @ X4 ) )
& ( ( in @ X4 @ sK406 )
= $true ) )
=> ( ( ( ap @ sK406 @ sK408
@ ( lam @ sK406 @ sK408
@ ^ [Y0: $i] : ( sK407 @ Y0 ) )
@ sK409 )
!= ( sK407 @ sK409 ) )
& ( ( in @ sK409 @ sK406 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f864,plain,
( ( brelnall2 = $true )
& ( exuEu = $true )
& ( ksndpairEq = $true )
& ( setukpairinjR2 = $true )
& ( funcext2 = $true )
& ( funcext = $true )
& ( kpairsurjEq = $true )
& ( binintersectSubset4 = $true )
& ( brelnall1 = $true )
& ( quantDeMorgan3 = $true )
& ( upairset2IR = $true )
& ( binunionIR = $true )
& ( exuI3 = $true )
& ( notdallE = $true )
& ( nonemptyI = $true )
& ( singletonsubset = $true )
& ( subsetemptysetimpeq = $true )
& ( emptyinPowerset = $true )
& ( dpsetconstrERa = $true )
& ( notsubsetI = $true )
& ( kfstpairEq = $true )
& ( setunionsingleton1 = $true )
& ? [X0: $i,X1: $i > $i,X2: $i] :
( ! [X3: $i] :
( ( $true
!= ( in @ X3 @ X0 ) )
| ( ( in @ ( X1 @ X3 ) @ X2 )
= $true ) )
& ? [X4: $i] :
( ( ( X1 @ X4 )
!= ( ap @ X0 @ X2
@ ( lam @ X0 @ X2
@ ^ [Y0: $i] : ( X1 @ Y0 ) )
@ X4 ) )
& ( ( in @ X4 @ X0 )
= $true ) ) )
& ( funcGraphProp3 = $true )
& ( setbeta = $true )
& ( symdiffIneg1 = $true )
& ( funcGraphProp4 = $true )
& ( omega0Ax = $true )
& ( sepInPowerset = $true )
& ( funcImageSingleton = $true )
& ( lamp = $true )
& ( notdexE = $true )
& ( setminusERneg = $true )
& ( binunionE = $true )
& ( dpsetconstrER = $true )
& ( ex1I2 = $true )
& ( dpsetconstrI = $true )
& ( lam2p = $true )
& ( emptyE1 = $true )
& ( upairequniteq = $true )
& ( ubforcartprodlem3 = $true )
& ( setunion__Cong = $true )
& ( binintersectSubset3 = $true )
& ( setadjoin__Cong = $true )
& ( singletoninpowunion = $true )
& ( emptyset__Cong = $true )
& ( wellorderingAx = $true )
& ( setadjoinSub = $true )
& ( replAx = $true )
& ( exuE3e = $true )
& ( emptyI = $true )
& ( exuI1 = $true )
& ( setminusILneg = $true )
& ( foundationAx = $true )
& ( setukpairinjL = $true )
& ( symdiffI2 = $true )
& ( upairset2E = $true )
& ( setukpairinjR12 = $true )
& ( dsetconstrEL = $true )
& ( setminusEL = $true )
& ( inPowerset = $true )
& ( funcGraphProp1 = $true )
& ( symdiffI1 = $true )
& ( upairsetIR = $true )
& ( eqinunit = $true )
& ( beta1 = $true )
& ( vacuousDall = $true )
& ( binunionLsub = $true )
& ( notinsingleton = $true )
& ( cartprodpairsurjEq = $true )
& ( exuE3u = $true )
& ( powersetE = $true )
& ( eta1 = $true )
& ( cartprodmempaircEq = $true )
& ( ubforcartprodlem1 = $true )
& ( powersetE1 = $true )
& ( binunionRsub = $true )
& ( emptyInPowerset = $true )
& ( singletonprop = $true )
& ( powersetI = $true )
& ( upairsetE = $true )
& ( binintersectLsub = $true )
& ( cartprodpairin = $true )
& ( nonemptyI1 = $true )
& ( binintersectSubset1 = $true )
& ( setadjoinAx = $true )
& ( cartprodpairmemER = $true )
& ( setext = $true )
& ( setminusI = $true )
& ( cartprodfstin = $true )
& ( setukpairIL = $true )
& ( setunionsingleton = $true )
& ( setextAx = $true )
& ( setminusELneg = $true )
& ( dsetconstrER = $true )
& ( setadjoinIR = $true )
& ( cartprodsndpairEq = $true )
& ( subbreln = $true )
& ( binunionEcases = $true )
& ( prop2setE = $true )
& ( secondinupair = $true )
& ( binintersectRsub = $true )
& ( setadjoinOr = $true )
& ( subsetI2 = $true )
& ( eqimpsubset1 = $true )
& ( sepSubset = $true )
& ( binunionIL = $true )
& ( setunionAx = $true )
& ( setoftrueEq = $true )
& ( ex1I = $true )
& ( subset2powerset = $true )
& ( lamProp = $true )
& ( dsetconstrI = $true )
& ( cartprodmempair = $true )
& ( emptysetimpfalse = $true )
& ( subsetTrans = $true )
& ( ap2p = $true )
& ( subsetRefl = $true )
& ( quantDeMorgan1 = $true )
& ( cartprodsndin = $true )
& ( notequalI1 = $true )
& ( nonemptyE1 = $true )
& ( emptysetAx = $true )
& ( apProp = $true )
& ( ubforcartprodlem2 = $true )
& ( cartprodmempair1 = $true )
& ( subsetE = $true )
& ( app = $true )
& ( subPowSU = $true )
& ( descr__Cong = $true )
& ( upairinpowunion = $true )
& ( setukpairIR = $true )
& ( setminusER = $true )
& ( dpsetconstrEL1 = $true )
& ( prop2setI = $true )
& ( kfstsingleton = $true )
& ( ap2apEq2 = $true )
& ( setukpairinjR = $true )
& ( noeltsimpempty = $true )
& ( omegaIndAx = $true )
& ( subsetI1 = $true )
& ( funcinfuncset = $true )
& ( funcGraphProp2 = $true )
& ( kpairiskpair = $true )
& ( binintersectER = $true )
& ( eqimpsubset2 = $true )
& ( setminusSubset1 = $true )
& ( funcextLem = $true )
& ( disjointsetsI1 = $true )
& ( omegaSAx = $true )
& ( omega__Cong = $true )
& ( cartprodfstpairEq = $true )
& ( subsetE2 = $true )
& ( notequalI2 = $true )
& ( singletoninpowerset = $true )
& ( setukpairinjR1 = $true )
& ( eqbreln = $true )
& ( theprop = $true )
& ( uniqinunit = $true )
& ( infuncsetfunc = $true )
& ( in__Cong = $true )
& ( setOfPairsIsBReln = $true )
& ( ksndsingleton = $true )
& ( ap2apEq1 = $true )
& ( dpsetconstrSub = $true )
& ( binintersectSubset2 = $true )
& ( emptysetsubset = $true )
& ( kpairp = $true )
& ( setminusLsub = $true )
& ( setminusIRneg = $true )
& ( notinemptyset = $true )
& ( exuI2 = $true )
& ( powersetAx = $true )
& ( setunionI = $true )
& ( symdiffE = $true )
& ( singletonsuniq = $true )
& ( powersetI1 = $true )
& ( prop2set2propI = $true )
& ( quantDeMorgan4 = $true )
& ( quantDeMorgan2 = $true )
& ( dsetconstr__Cong = $true )
& ( upairsubunion = $true )
& ( exuE1 = $true )
& ( cartprodpairmemEL = $true )
& ( dpsetconstrEL2 = $true )
& ( powerset__Cong = $true )
& ( setukpairinjR11 = $true )
& ( exu__Cong = $true )
& ( setminusSubset2 = $true )
& ( setunionE = $true )
& ( ex1E2 = $true )
& ( singletonsswitch = $true )
& ( powersetsubset = $true )
& ( binintersectSubset5 = $true )
& ( lam2lamEq = $true )
& ( bs114d = $true )
& ( upairsetIL = $true )
& ( binintersectI = $true )
& ( emptysetE = $true )
& ( setadjoinE = $true )
& ( inCongP = $true )
& ( ex1E1 = $true )
& ( setadjoinSub2 = $true )
& ( exuE2 = $true )
& ( symdiffIneg2 = $true )
& ( setunionE2 = $true )
& ( nonemptyImpWitness = $true )
& ( setunionsingleton2 = $true )
& ( setextsub = $true )
& ( setukpairinjL1 = $true )
& ( setadjoinIL = $true )
& ( setukpairinjL2 = $true )
& ( binintersectEL = $true )
& ( descrp = $true )
& ( emptyinunitempty = $true ) ),
inference(flattening,[status(thm)],[f863]) ).
thf(f863,plain,
( ? [X0: $i,X1: $i > $i,X2: $i] :
( ! [X3: $i] :
( ( $true
!= ( in @ X3 @ X0 ) )
| ( ( in @ ( X1 @ X3 ) @ X2 )
= $true ) )
& ? [X4: $i] :
( ( ( X1 @ X4 )
!= ( ap @ X0 @ X2
@ ( lam @ X0 @ X2
@ ^ [Y0: $i] : ( X1 @ Y0 ) )
@ X4 ) )
& ( ( in @ X4 @ X0 )
= $true ) ) )
& ( lam2lamEq = $true )
& ( eta1 = $true )
& ( beta1 = $true )
& ( ap2apEq2 = $true )
& ( ap2apEq1 = $true )
& ( funcext2 = $true )
& ( funcext = $true )
& ( eqbreln = $true )
& ( subbreln = $true )
& ( funcGraphProp4 = $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)],[f650]) ).
thf(f650,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 )
=> ( ( funcGraphProp4 = $true )
=> ( ( subbreln = $true )
=> ( ( eqbreln = $true )
=> ( ( funcext = $true )
=> ( ( funcext2 = $true )
=> ( ( ap2apEq1 = $true )
=> ( ( ap2apEq2 = $true )
=> ( ( beta1 = $true )
=> ( ( eta1 = $true )
=> ( ( lam2lamEq = $true )
=> ! [X0: $i,X1: $i > $i,X2: $i] :
( ! [X3: $i] :
( ( $true
= ( in @ X3 @ X0 ) )
=> ( ( in @ ( X1 @ X3 ) @ X2 )
= $true ) )
=> ! [X4: $i] :
( ( ( in @ X4 @ X0 )
= $true )
=> ( ( X1 @ X4 )
= ( ap @ X0 @ X2
@ ( lam @ X0 @ X2
@ ^ [Y0: $i] : ( X1 @ Y0 ) )
@ X4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f649]) ).
thf(f649,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
=> ( funcGraphProp4
=> ( subbreln
=> ( eqbreln
=> ( funcext
=> ( funcext2
=> ( ap2apEq1
=> ( ap2apEq2
=> ( beta1
=> ( eta1
=> ( lam2lamEq
=> ! [X0: $i,X1: $i > $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X0 )
=> ( in @ ( X1 @ X3 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X0 )
=> ( ( ap @ X0 @ X2
@ ( lam @ X0 @ X2
@ ^ [X5: $i] : ( X1 @ X5 ) )
@ X4 )
= ( X1 @ X4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f222]) ).
thf(f222,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
=> ( funcGraphProp4
=> ( subbreln
=> ( eqbreln
=> ( funcext
=> ( funcext2
=> ( ap2apEq1
=> ( ap2apEq2
=> ( beta1
=> ( eta1
=> ( lam2lamEq
=> ! [X3: $i,X12: $i > $i,X4: $i] :
( ! [X1: $i] :
( ( in @ X1 @ X3 )
=> ( in @ ( X12 @ X1 ) @ X4 ) )
=> ! [X1: $i] :
( ( in @ X1 @ X3 )
=> ( ( ap @ X3 @ X4
@ ( lam @ X3 @ X4
@ ^ [X2: $i] : ( X12 @ X2 ) )
@ X1 )
= ( X12 @ X1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f221]) ).
thf(f221,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
=> ( funcGraphProp4
=> ( subbreln
=> ( eqbreln
=> ( funcext
=> ( funcext2
=> ( ap2apEq1
=> ( ap2apEq2
=> ( beta1
=> ( eta1
=> ( lam2lamEq
=> ! [X3: $i,X12: $i > $i,X4: $i] :
( ! [X1: $i] :
( ( in @ X1 @ X3 )
=> ( in @ ( X12 @ X1 ) @ X4 ) )
=> ! [X1: $i] :
( ( in @ X1 @ X3 )
=> ( ( ap @ X3 @ X4
@ ( lam @ X3 @ X4
@ ^ [X2: $i] : ( X12 @ X2 ) )
@ X1 )
= ( X12 @ X1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.9FuAMmeF6I/DTF2THF_3007.p',beta2) ).
thf(f3236,plain,
( ( ( in @ ( sK291 @ sK406 @ sK407 @ sK408 ) @ sK406 )
= $true )
| ( ( in @ sK409 @ sK406 )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f3235]) ).
thf(f3235,plain,
( ( $true != $true )
| ( ( in @ ( sK291 @ sK406 @ sK407 @ sK408 ) @ sK406 )
= $true )
| ( ( in @ sK409 @ sK406 )
!= $true ) ),
inference(forward_demodulation,[status(thm)],[f3213,f2620]) ).
thf(f2620,plain,
beta1 = $true,
inference(cnf_transformation,[status(thm)],[f1553]) ).
thf(f3213,plain,
( ( beta1 != $true )
| ( ( in @ ( sK291 @ sK406 @ sK407 @ sK408 ) @ sK406 )
= $true )
| ( ( in @ sK409 @ sK406 )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f3208]) ).
thf(f3208,plain,
( ( beta1 != $true )
| ( ( in @ ( sK291 @ sK406 @ sK407 @ sK408 ) @ sK406 )
= $true )
| ( ( in @ sK409 @ sK406 )
!= $true )
| ( ( sK407 @ sK409 )
!= ( sK407 @ sK409 ) ) ),
inference(superposition,[status(thm)],[f3191,f3118]) ).
thf(f3118,plain,
! [X2: $i,X3: $i,X0: $i,X1: $i > $i] :
( ( ( ap @ X2 @ X0 @ ( lam @ X2 @ X0 @ X1 ) @ X3 )
= ( X1 @ X3 ) )
| ( ( in @ X3 @ X2 )
!= $true )
| ( ( in @ ( sK291 @ X2 @ X1 @ X0 ) @ X2 )
= $true )
| ( beta1 != $true ) ),
inference(beta_eta_normalization,[status(thm)],[f2332]) ).
thf(f2332,plain,
! [X2: $i,X3: $i,X0: $i,X1: $i > $i] :
( ( beta1 != $true )
| ( ( in @ X3 @ X2 )
!= $true )
| ( ( in @ ( sK291 @ X2 @ X1 @ X0 ) @ X2 )
= $true )
| ( ( X1 @ X3 )
= ( ap @ X2 @ X0
@ ( lam @ X2 @ X0
@ ^ [Y0: $i] : ( X1 @ Y0 ) )
@ X3 ) ) ),
inference(cnf_transformation,[status(thm)],[f1381]) ).
thf(f1381,plain,
( ( ! [X0: $i,X1: $i > $i,X2: $i] :
( ! [X3: $i] :
( ( ( X1 @ X3 )
= ( ap @ X2 @ X0
@ ( lam @ X2 @ X0
@ ^ [Y0: $i] : ( X1 @ Y0 ) )
@ X3 ) )
| ( ( in @ X3 @ X2 )
!= $true ) )
| ( ( ( in @ ( X1 @ ( sK291 @ X2 @ X1 @ X0 ) ) @ X0 )
!= $true )
& ( ( in @ ( sK291 @ X2 @ X1 @ X0 ) @ X2 )
= $true ) ) )
| ( beta1 != $true ) )
& ( ( beta1 = $true )
| ( ( ( ap @ sK294 @ sK292
@ ( lam @ sK294 @ sK292
@ ^ [Y0: $i] : ( sK293 @ Y0 ) )
@ sK295 )
!= ( sK293 @ sK295 ) )
& ( ( in @ sK295 @ sK294 )
= $true )
& ! [X9: $i] :
( ( ( in @ ( sK293 @ X9 ) @ sK292 )
= $true )
| ( ( in @ X9 @ sK294 )
!= $true ) ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK291,sK292,sK293,sK294,sK295])],[f1377,f1380,f1379,f1378]) ).
thf(f1378,plain,
! [X0: $i,X1: $i > $i,X2: $i] :
( ? [X4: $i] :
( ( ( in @ ( X1 @ X4 ) @ X0 )
!= $true )
& ( ( in @ X4 @ X2 )
= $true ) )
=> ( ( ( in @ ( X1 @ ( sK291 @ X2 @ X1 @ X0 ) ) @ X0 )
!= $true )
& ( ( in @ ( sK291 @ X2 @ X1 @ X0 ) @ X2 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1379,plain,
( ? [X5: $i,X6: $i > $i,X7: $i] :
( ? [X8: $i] :
( ( ( X6 @ X8 )
!= ( ap @ X7 @ X5
@ ( lam @ X7 @ X5
@ ^ [Y0: $i] : ( X6 @ Y0 ) )
@ X8 ) )
& ( ( in @ X8 @ X7 )
= $true ) )
& ! [X9: $i] :
( ( ( in @ ( X6 @ X9 ) @ X5 )
= $true )
| ( $true
!= ( in @ X9 @ X7 ) ) ) )
=> ( ? [X8: $i] :
( ( ( ap @ sK294 @ sK292
@ ( lam @ sK294 @ sK292
@ ^ [Y0: $i] : ( sK293 @ Y0 ) )
@ X8 )
!= ( sK293 @ X8 ) )
& ( ( in @ X8 @ sK294 )
= $true ) )
& ! [X9: $i] :
( ( ( in @ ( sK293 @ X9 ) @ sK292 )
= $true )
| ( ( in @ X9 @ sK294 )
!= $true ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1380,plain,
( ? [X8: $i] :
( ( ( ap @ sK294 @ sK292
@ ( lam @ sK294 @ sK292
@ ^ [Y0: $i] : ( sK293 @ Y0 ) )
@ X8 )
!= ( sK293 @ X8 ) )
& ( ( in @ X8 @ sK294 )
= $true ) )
=> ( ( ( ap @ sK294 @ sK292
@ ( lam @ sK294 @ sK292
@ ^ [Y0: $i] : ( sK293 @ Y0 ) )
@ sK295 )
!= ( sK293 @ sK295 ) )
& ( ( in @ sK295 @ sK294 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1377,plain,
( ( ! [X0: $i,X1: $i > $i,X2: $i] :
( ! [X3: $i] :
( ( ( X1 @ X3 )
= ( ap @ X2 @ X0
@ ( lam @ X2 @ X0
@ ^ [Y0: $i] : ( X1 @ Y0 ) )
@ X3 ) )
| ( ( in @ X3 @ X2 )
!= $true ) )
| ? [X4: $i] :
( ( ( in @ ( X1 @ X4 ) @ X0 )
!= $true )
& ( ( in @ X4 @ X2 )
= $true ) ) )
| ( beta1 != $true ) )
& ( ( beta1 = $true )
| ? [X5: $i,X6: $i > $i,X7: $i] :
( ? [X8: $i] :
( ( ( X6 @ X8 )
!= ( ap @ X7 @ X5
@ ( lam @ X7 @ X5
@ ^ [Y0: $i] : ( X6 @ Y0 ) )
@ X8 ) )
& ( ( in @ X8 @ X7 )
= $true ) )
& ! [X9: $i] :
( ( ( in @ ( X6 @ X9 ) @ X5 )
= $true )
| ( $true
!= ( in @ X9 @ X7 ) ) ) ) ) ),
inference(rectify,[status(thm)],[f1376]) ).
thf(f1376,plain,
( ( ! [X0: $i,X1: $i > $i,X2: $i] :
( ! [X4: $i] :
( ( ( X1 @ X4 )
= ( ap @ X2 @ X0
@ ( lam @ X2 @ X0
@ ^ [Y0: $i] : ( X1 @ Y0 ) )
@ X4 ) )
| ( ( in @ X4 @ X2 )
!= $true ) )
| ? [X3: $i] :
( ( ( in @ ( X1 @ X3 ) @ X0 )
!= $true )
& ( ( in @ X3 @ X2 )
= $true ) ) )
| ( beta1 != $true ) )
& ( ( beta1 = $true )
| ? [X0: $i,X1: $i > $i,X2: $i] :
( ? [X4: $i] :
( ( ( X1 @ X4 )
!= ( ap @ X2 @ X0
@ ( lam @ X2 @ X0
@ ^ [Y0: $i] : ( X1 @ Y0 ) )
@ X4 ) )
& ( ( in @ X4 @ X2 )
= $true ) )
& ! [X3: $i] :
( ( ( in @ ( X1 @ X3 ) @ X0 )
= $true )
| ( ( in @ X3 @ X2 )
!= $true ) ) ) ) ),
inference(nnf_transformation,[status(thm)],[f843]) ).
thf(f843,plain,
( ! [X0: $i,X1: $i > $i,X2: $i] :
( ! [X4: $i] :
( ( ( X1 @ X4 )
= ( ap @ X2 @ X0
@ ( lam @ X2 @ X0
@ ^ [Y0: $i] : ( X1 @ Y0 ) )
@ X4 ) )
| ( ( in @ X4 @ X2 )
!= $true ) )
| ? [X3: $i] :
( ( ( in @ ( X1 @ X3 ) @ X0 )
!= $true )
& ( ( in @ X3 @ X2 )
= $true ) ) )
<=> ( beta1 = $true ) ),
inference(ennf_transformation,[status(thm)],[f468]) ).
thf(f468,plain,
( ! [X0: $i,X1: $i > $i,X2: $i] :
( ! [X3: $i] :
( ( ( in @ X3 @ X2 )
= $true )
=> ( ( in @ ( X1 @ X3 ) @ X0 )
= $true ) )
=> ! [X4: $i] :
( ( ( in @ X4 @ X2 )
= $true )
=> ( ( X1 @ X4 )
= ( ap @ X2 @ X0
@ ( lam @ X2 @ X0
@ ^ [Y0: $i] : ( X1 @ Y0 ) )
@ X4 ) ) ) )
<=> ( beta1 = $true ) ),
inference(fool_elimination,[status(thm)],[f467]) ).
thf(f467,plain,
( beta1
= ( ! [X0: $i,X1: $i > $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ ( X1 @ X3 ) @ X0 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ap @ X2 @ X0
@ ( lam @ X2 @ X0
@ ^ [X5: $i] : ( X1 @ X5 ) )
@ X4 )
= ( X1 @ X4 ) ) ) ) ) ),
inference(rectify,[status(thm)],[f218]) ).
thf(f218,axiom,
( beta1
= ( ! [X4: $i,X12: $i > $i,X3: $i] :
( ! [X1: $i] :
( ( in @ X1 @ X3 )
=> ( in @ ( X12 @ X1 ) @ X4 ) )
=> ! [X1: $i] :
( ( in @ X1 @ X3 )
=> ( ( ap @ X3 @ X4
@ ( lam @ X3 @ X4
@ ^ [X2: $i] : ( X12 @ X2 ) )
@ X1 )
= ( X12 @ X1 ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.9FuAMmeF6I/DTF2THF_3007.p',beta1) ).
thf(f3191,plain,
( ( ap @ sK406 @ sK408 @ ( lam @ sK406 @ sK408 @ sK407 ) @ sK409 )
!= ( sK407 @ sK409 ) ),
inference(beta_eta_normalization,[status(thm)],[f2664]) ).
thf(f2664,plain,
( ( ap @ sK406 @ sK408
@ ( lam @ sK406 @ sK408
@ ^ [Y0: $i] : ( sK407 @ Y0 ) )
@ sK409 )
!= ( sK407 @ sK409 ) ),
inference(cnf_transformation,[status(thm)],[f1553]) ).
thf(f3620,plain,
( ( in @ ( sK291 @ sK406 @ sK407 @ sK408 ) @ sK406 )
!= $true ),
inference(trivial_inequality_removal,[status(thm)],[f3609]) ).
thf(f3609,plain,
( ( $true != $true )
| ( ( in @ ( sK291 @ sK406 @ sK407 @ sK408 ) @ sK406 )
!= $true ) ),
inference(superposition,[status(thm)],[f3242,f2665]) ).
thf(f2665,plain,
! [X3: $i] :
( ( ( in @ ( sK407 @ X3 ) @ sK408 )
= $true )
| ( ( in @ X3 @ sK406 )
!= $true ) ),
inference(cnf_transformation,[status(thm)],[f1553]) ).
thf(f3242,plain,
( ( in @ ( sK407 @ ( sK291 @ sK406 @ sK407 @ sK408 ) ) @ sK408 )
!= $true ),
inference(trivial_inequality_removal,[status(thm)],[f3241]) ).
thf(f3241,plain,
( ( $true != $true )
| ( ( in @ ( sK407 @ ( sK291 @ sK406 @ sK407 @ sK408 ) ) @ sK408 )
!= $true ) ),
inference(forward_demodulation,[status(thm)],[f3240,f2663]) ).
thf(f3240,plain,
( ( ( in @ ( sK407 @ ( sK291 @ sK406 @ sK407 @ sK408 ) ) @ sK408 )
!= $true )
| ( ( in @ sK409 @ sK406 )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f3239]) ).
thf(f3239,plain,
( ( ( in @ ( sK407 @ ( sK291 @ sK406 @ sK407 @ sK408 ) ) @ sK408 )
!= $true )
| ( ( in @ sK409 @ sK406 )
!= $true )
| ( $true != $true ) ),
inference(forward_demodulation,[status(thm)],[f3212,f2620]) ).
thf(f3212,plain,
( ( beta1 != $true )
| ( ( in @ ( sK407 @ ( sK291 @ sK406 @ sK407 @ sK408 ) ) @ sK408 )
!= $true )
| ( ( in @ sK409 @ sK406 )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f3209]) ).
thf(f3209,plain,
( ( ( sK407 @ sK409 )
!= ( sK407 @ sK409 ) )
| ( ( in @ sK409 @ sK406 )
!= $true )
| ( ( in @ ( sK407 @ ( sK291 @ sK406 @ sK407 @ sK408 ) ) @ sK408 )
!= $true )
| ( beta1 != $true ) ),
inference(superposition,[status(thm)],[f3191,f3087]) ).
thf(f3087,plain,
! [X2: $i,X3: $i,X0: $i,X1: $i > $i] :
( ( ( ap @ X2 @ X0 @ ( lam @ X2 @ X0 @ X1 ) @ X3 )
= ( X1 @ X3 ) )
| ( ( in @ X3 @ X2 )
!= $true )
| ( ( in @ ( X1 @ ( sK291 @ X2 @ X1 @ X0 ) ) @ X0 )
!= $true )
| ( beta1 != $true ) ),
inference(beta_eta_normalization,[status(thm)],[f2333]) ).
thf(f2333,plain,
! [X2: $i,X3: $i,X0: $i,X1: $i > $i] :
( ( ( in @ ( X1 @ ( sK291 @ X2 @ X1 @ X0 ) ) @ X0 )
!= $true )
| ( ( in @ X3 @ X2 )
!= $true )
| ( ( X1 @ X3 )
= ( ap @ X2 @ X0
@ ( lam @ X2 @ X0
@ ^ [Y0: $i] : ( X1 @ Y0 ) )
@ X3 ) )
| ( beta1 != $true ) ),
inference(cnf_transformation,[status(thm)],[f1381]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : SEU698^1 : TPTP v9.2.1. Released v3.7.0.
% 0.00/0.12 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.14/0.32 % Computer : n001.cluster.edu
% 0.14/0.32 % Model : x86_64 x86_64
% 0.14/0.32 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.32 % Memory : 8042.1875MB
% 0.14/0.32 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.32 % CPULimit : 300
% 0.14/0.32 % WCLimit : 300
% 0.14/0.32 % DateTime : Tue Feb 24 12:32:05 EST 2026
% 0.14/0.32 % CPUTime :
% 0.14/0.32 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.17/0.43 ---- Original DTF file ---
% 0.17/0.43 thf(spec,logic,$$dhol).
% 0.17/0.43 %------------------------------------------------------------------------------
% 0.17/0.43 % File : SEU698^1 : TPTP v9.2.1. Released v3.7.0.
% 0.17/0.43 % Domain : Set Theory
% 0.17/0.43 % Problem : Functions - Extensionality and Beta Reduction
% 0.17/0.43 % Version : Especial.
% 0.17/0.43 % English : (! A:i.! B:i.! f:i>i.(! x:i.in x A -> in (f x) B) ->
% 0.17/0.43 % (! x:i.in x A -> ap A B (lam A B (^ y:i.f y)) x = f x))
% 0.17/0.43
% 0.17/0.43 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.17/0.43 % Source : [Bro08]
% 0.17/0.43 % Names : ZFC200g [Bro08]
% 0.17/0.43
% 0.17/0.43 % Status : Theorem
% 0.17/0.43 % Rating : 0.33 v9.1.0, 0.25 v9.0.0, 0.40 v8.2.0, 0.54 v8.1.0, 0.45 v7.5.0, 0.43 v7.4.0, 0.33 v7.2.0, 0.25 v7.1.0, 0.38 v7.0.0, 0.29 v6.4.0, 0.33 v6.3.0, 0.40 v6.2.0, 0.29 v6.1.0, 0.43 v5.5.0, 0.50 v5.4.0, 0.60 v5.2.0, 0.80 v5.0.0, 0.60 v4.1.0, 1.00 v4.0.0, 0.67 v3.7.0
% 0.17/0.43 % Syntax : Number of formulae : 472 ( 220 unt; 251 typ; 220 def)
% 0.17/0.43 % Number of atoms : 1212 ( 330 equ; 0 cnn)
% 0.17/0.43 % Maximal formula atoms : 218 ( 5 avg)
% 0.17/0.43 % Number of connectives : 2333 ( 47 ~; 7 |; 35 &;1665 @)
% 0.17/0.43 % ( 14 <=>; 565 =>; 0 <=; 0 <~>)
% 0.17/0.43 % Maximal formula depth : 224 ( 2 avg)
% 0.17/0.43 % Number of types : 2 ( 0 usr)
% 0.17/0.43 % Number of type conns : 129 ( 129 >; 0 *; 0 +; 0 <<)
% 0.17/0.43 % Number of symbols : 254 ( 251 usr; 218 con; 0-4 aty)
% 0.17/0.43 % Number of variables : 732 ( 74 ^; 622 !; 36 ?; 732 :)
% 0.17/0.43 % SPC : TH0_THM_EQU_NAR
% 0.17/0.43
% 0.17/0.43 % Comments : http://mathgate.info/detsetitem.php?id=250
% 0.17/0.43 %------------------------------------------------------------------------------
% 0.17/0.43 thf(in_type,type,
% 0.17/0.43 in: $i > $i > $o ).
% 0.17/0.43
% 0.17/0.43 thf(exu_type,type,
% 0.17/0.43 exu: ( $i > $o ) > $o ).
% 0.17/0.43
% 0.17/0.43 thf(exu,definition,
% 0.17/0.43 ( exu
% 0.17/0.43 = ( ^ [Xphi: $i > $o] :
% 0.17/0.43 ? [Xx: $i] :
% 0.17/0.43 ( ( Xphi @ Xx )
% 0.17/0.43 & ! [Xy: $i] :
% 0.17/0.43 ( ( Xphi @ Xy )
% 0.17/0.43 => ( Xx = Xy ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 thf(setextAx_type,type,
% 0.17/0.43 setextAx: $o ).
% 0.17/0.43
% 0.17/0.43 thf(setextAx,definition,
% 0.17/0.43 ( setextAx
% 0.17/0.43 = ( ! [A: $i,B: $i] :
% 0.17/0.43 ( ! [Xx: $i] :
% 0.17/0.43 ( ( in @ Xx @ A )
% 0.17/0.43 <=> ( in @ Xx @ B ) )
% 0.17/0.43 => ( A = B ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 thf(emptyset_type,type,
% 0.17/0.43 emptyset: $i ).
% 0.17/0.43
% 0.17/0.43 thf(emptysetAx_type,type,
% 0.17/0.43 emptysetAx: $o ).
% 0.17/0.43
% 0.17/0.43 thf(emptysetAx,definition,
% 0.17/0.43 ( emptysetAx
% 0.17/0.43 = ( ! [Xx: $i] :
% 0.17/0.43 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.17/0.43
% 0.17/0.43 thf(setadjoin_type,type,
% 0.17/0.43 setadjoin: $i > $i > $i ).
% 0.17/0.43
% 0.17/0.43 thf(setadjoinAx_type,type,
% 0.17/0.43 setadjoinAx: $o ).
% 0.17/0.43
% 0.17/0.43 thf(setadjoinAx,definition,
% 0.17/0.43 ( setadjoinAx
% 0.17/0.43 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.17/0.43 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.17/0.43 <=> ( ( Xy = Xx )
% 0.17/0.43 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 thf(powerset_type,type,
% 0.17/0.43 powerset: $i > $i ).
% 0.17/0.43
% 0.17/0.43 thf(powersetAx_type,type,
% 0.17/0.43 powersetAx: $o ).
% 0.17/0.43
% 0.17/0.43 thf(powersetAx,definition,
% 0.17/0.43 ( powersetAx
% 0.17/0.43 = ( ! [A: $i,B: $i] :
% 0.17/0.43 ( ( in @ B @ ( powerset @ A ) )
% 0.17/0.43 <=> ! [Xx: $i] :
% 0.17/0.43 ( ( in @ Xx @ B )
% 0.17/0.43 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 thf(setunion_type,type,
% 0.17/0.43 setunion: $i > $i ).
% 0.17/0.43
% 0.17/0.43 thf(setunionAx_type,type,
% 0.17/0.43 setunionAx: $o ).
% 0.17/0.43
% 0.17/0.43 thf(setunionAx,definition,
% 0.17/0.43 ( setunionAx
% 0.17/0.43 = ( ! [A: $i,Xx: $i] :
% 0.17/0.43 ( ( in @ Xx @ ( setunion @ A ) )
% 0.17/0.43 <=> ? [B: $i] :
% 0.17/0.43 ( ( in @ Xx @ B )
% 0.17/0.43 & ( in @ B @ A ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 thf(omega_type,type,
% 0.17/0.43 omega: $i ).
% 0.17/0.43
% 0.17/0.43 thf(omega0Ax_type,type,
% 0.17/0.43 omega0Ax: $o ).
% 0.17/0.43
% 0.17/0.43 thf(omega0Ax,definition,
% 0.17/0.43 ( omega0Ax
% 0.17/0.43 = ( in @ emptyset @ omega ) ) ).
% 0.17/0.43
% 0.17/0.43 thf(omegaSAx_type,type,
% 0.17/0.43 omegaSAx: $o ).
% 0.17/0.43
% 0.17/0.43 thf(omegaSAx,definition,
% 0.17/0.43 ( omegaSAx
% 0.17/0.43 = ( ! [Xx: $i] :
% 0.17/0.43 ( ( in @ Xx @ omega )
% 0.17/0.43 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 thf(omegaIndAx_type,type,
% 0.17/0.43 omegaIndAx: $o ).
% 0.17/0.43
% 0.17/0.43 thf(omegaIndAx,definition,
% 0.17/0.43 ( omegaIndAx
% 0.17/0.43 = ( ! [A: $i] :
% 0.17/0.43 ( ( ( in @ emptyset @ A )
% 0.17/0.43 & ! [Xx: $i] :
% 0.17/0.43 ( ( ( in @ Xx @ omega )
% 0.17/0.43 & ( in @ Xx @ A ) )
% 0.17/0.43 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.17/0.43 => ! [Xx: $i] :
% 0.17/0.43 ( ( in @ Xx @ omega )
% 0.17/0.43 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 thf(replAx_type,type,
% 0.17/0.43 replAx: $o ).
% 0.17/0.43
% 0.17/0.43 thf(replAx,definition,
% 0.17/0.43 ( replAx
% 0.17/0.43 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.17/0.43 ( ! [Xx: $i] :
% 0.17/0.43 ( ( in @ Xx @ A )
% 0.17/0.43 => ( exu
% 0.17/0.43 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.17/0.43 => ? [B: $i] :
% 0.17/0.43 ! [Xx: $i] :
% 0.17/0.43 ( ( in @ Xx @ B )
% 0.17/0.43 <=> ? [Xy: $i] :
% 0.17/0.43 ( ( in @ Xy @ A )
% 0.17/0.43 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 thf(foundationAx_type,type,
% 0.17/0.43 foundationAx: $o ).
% 0.17/0.43
% 0.17/0.43 thf(foundationAx,definition,
% 0.17/0.43 ( foundationAx
% 0.17/0.43 = ( ! [A: $i] :
% 0.17/0.43 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.17/0.43 => ? [B: $i] :
% 0.17/0.43 ( ( in @ B @ A )
% 0.17/0.43 & ~ ? [Xx: $i] :
% 0.17/0.43 ( ( in @ Xx @ B )
% 0.17/0.43 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 thf(wellorderingAx_type,type,
% 0.17/0.43 wellorderingAx: $o ).
% 0.17/0.43
% 0.17/0.43 thf(wellorderingAx,definition,
% 0.17/0.43 ( wellorderingAx
% 0.17/0.43 = ( ! [A: $i] :
% 0.17/0.43 ? [B: $i] :
% 0.17/0.43 ( ! [C: $i] :
% 0.17/0.43 ( ( in @ C @ B )
% 0.17/0.43 => ! [Xx: $i] :
% 0.17/0.43 ( ( in @ Xx @ C )
% 0.17/0.43 => ( in @ Xx @ A ) ) )
% 0.17/0.43 & ! [Xx: $i,Xy: $i] :
% 0.17/0.43 ( ( ( in @ Xx @ A )
% 0.17/0.43 & ( in @ Xy @ A ) )
% 0.17/0.43 => ( ! [C: $i] :
% 0.17/0.43 ( ( in @ C @ B )
% 0.17/0.43 => ( ( in @ Xx @ C )
% 0.17/0.43 <=> ( in @ Xy @ C ) ) )
% 0.17/0.43 => ( Xx = Xy ) ) )
% 0.17/0.43 & ! [C: $i,D: $i] :
% 0.17/0.43 ( ( ( in @ C @ B )
% 0.17/0.43 & ( in @ D @ B ) )
% 0.17/0.43 => ( ! [Xx: $i] :
% 0.17/0.43 ( ( in @ Xx @ C )
% 0.17/0.43 => ( in @ Xx @ D ) )
% 0.17/0.43 | ! [Xx: $i] :
% 0.17/0.43 ( ( in @ Xx @ D )
% 0.17/0.43 => ( in @ Xx @ C ) ) ) )
% 0.17/0.43 & ! [C: $i] :
% 0.17/0.43 ( ( ! [Xx: $i] :
% 0.17/0.43 ( ( in @ Xx @ C )
% 0.17/0.43 => ( in @ Xx @ A ) )
% 0.17/0.43 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.17/0.43 => ? [D: $i,Xx: $i] :
% 0.17/0.43 ( ( in @ D @ B )
% 0.17/0.43 & ( in @ Xx @ C )
% 0.17/0.43 & ~ ? [Xy: $i] :
% 0.17/0.43 ( ( in @ Xy @ D )
% 0.17/0.43 & ( in @ Xy @ C ) )
% 0.17/0.43 & ! [E: $i] :
% 0.17/0.43 ( ( in @ E @ B )
% 0.17/0.43 => ( ! [Xy: $i] :
% 0.17/0.43 ( ( in @ Xy @ E )
% 0.17/0.43 => ( in @ Xy @ D ) )
% 0.17/0.43 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 thf(descr_type,type,
% 0.17/0.43 descr: ( $i > $o ) > $i ).
% 0.17/0.43
% 0.17/0.43 thf(descrp_type,type,
% 0.17/0.43 descrp: $o ).
% 0.17/0.43
% 0.17/0.43 thf(descrp,definition,
% 0.17/0.43 ( descrp
% 0.17/0.43 = ( ! [Xphi: $i > $o] :
% 0.17/0.43 ( ( exu
% 0.17/0.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.43 => ( Xphi
% 0.17/0.43 @ ( descr
% 0.17/0.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 thf(dsetconstr_type,type,
% 0.17/0.43 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.17/0.43
% 0.17/0.43 thf(dsetconstrI_type,type,
% 0.17/0.43 dsetconstrI: $o ).
% 0.17/0.43
% 0.17/0.43 thf(dsetconstrI,definition,
% 0.17/0.43 ( dsetconstrI
% 0.17/0.43 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.17/0.43 ( ( in @ Xx @ A )
% 0.17/0.43 => ( ( Xphi @ Xx )
% 0.17/0.43 => ( in @ Xx
% 0.17/0.43 @ ( dsetconstr @ A
% 0.17/0.43 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.17/0.43
% 0.17/0.43 thf(dsetconstrEL_type,type,
% 0.17/0.44 dsetconstrEL: $o ).
% 0.17/0.44
% 0.17/0.44 thf(dsetconstrEL,definition,
% 0.17/0.44 ( dsetconstrEL
% 0.17/0.44 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx
% 0.17/0.44 @ ( dsetconstr @ A
% 0.17/0.44 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.17/0.44 => ( in @ Xx @ A ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(dsetconstrER_type,type,
% 0.17/0.44 dsetconstrER: $o ).
% 0.17/0.44
% 0.17/0.44 thf(dsetconstrER,definition,
% 0.17/0.44 ( dsetconstrER
% 0.17/0.44 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx
% 0.17/0.44 @ ( dsetconstr @ A
% 0.17/0.44 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.17/0.44 => ( Xphi @ Xx ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(exuE1_type,type,
% 0.17/0.44 exuE1: $o ).
% 0.17/0.44
% 0.17/0.44 thf(exuE1,definition,
% 0.17/0.44 ( exuE1
% 0.17/0.44 = ( ! [Xphi: $i > $o] :
% 0.17/0.44 ( ( exu
% 0.17/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.44 => ? [Xx: $i] :
% 0.17/0.44 ( ( Xphi @ Xx )
% 0.17/0.44 & ! [Xy: $i] :
% 0.17/0.44 ( ( Xphi @ Xy )
% 0.17/0.44 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(prop2set_type,type,
% 0.17/0.44 prop2set: $o > $i ).
% 0.17/0.44
% 0.17/0.44 thf(prop2setE_type,type,
% 0.17/0.44 prop2setE: $o ).
% 0.17/0.44
% 0.17/0.44 thf(prop2setE,definition,
% 0.17/0.44 ( prop2setE
% 0.17/0.44 = ( ! [Xphi: $o,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.17/0.44 => Xphi ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(emptysetE_type,type,
% 0.17/0.44 emptysetE: $o ).
% 0.17/0.44
% 0.17/0.44 thf(emptysetE,definition,
% 0.17/0.44 ( emptysetE
% 0.17/0.44 = ( ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ emptyset )
% 0.17/0.44 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(emptysetimpfalse_type,type,
% 0.17/0.44 emptysetimpfalse: $o ).
% 0.17/0.44
% 0.17/0.44 thf(emptysetimpfalse,definition,
% 0.17/0.44 ( emptysetimpfalse
% 0.17/0.44 = ( ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ emptyset )
% 0.17/0.44 => $false ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(notinemptyset_type,type,
% 0.17/0.44 notinemptyset: $o ).
% 0.17/0.44
% 0.17/0.44 thf(notinemptyset,definition,
% 0.17/0.44 ( notinemptyset
% 0.17/0.44 = ( ! [Xx: $i] :
% 0.17/0.44 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(exuE3e_type,type,
% 0.17/0.44 exuE3e: $o ).
% 0.17/0.44
% 0.17/0.44 thf(exuE3e,definition,
% 0.17/0.44 ( exuE3e
% 0.17/0.44 = ( ! [Xphi: $i > $o] :
% 0.17/0.44 ( ( exu
% 0.17/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.44 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setext_type,type,
% 0.17/0.44 setext: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setext,definition,
% 0.17/0.44 ( setext
% 0.17/0.44 = ( ! [A: $i,B: $i] :
% 0.17/0.44 ( ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ( in @ Xx @ B ) )
% 0.17/0.44 => ( ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ B )
% 0.17/0.44 => ( in @ Xx @ A ) )
% 0.17/0.44 => ( A = B ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(emptyI_type,type,
% 0.17/0.44 emptyI: $o ).
% 0.17/0.44
% 0.17/0.44 thf(emptyI,definition,
% 0.17/0.44 ( emptyI
% 0.17/0.44 = ( ! [A: $i] :
% 0.17/0.44 ( ! [Xx: $i] :
% 0.17/0.44 ~ ( in @ Xx @ A )
% 0.17/0.44 => ( A = emptyset ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(noeltsimpempty_type,type,
% 0.17/0.44 noeltsimpempty: $o ).
% 0.17/0.44
% 0.17/0.44 thf(noeltsimpempty,definition,
% 0.17/0.44 ( noeltsimpempty
% 0.17/0.44 = ( ! [A: $i] :
% 0.17/0.44 ( ! [Xx: $i] :
% 0.17/0.44 ~ ( in @ Xx @ A )
% 0.17/0.44 => ( A = emptyset ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setbeta_type,type,
% 0.17/0.44 setbeta: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setbeta,definition,
% 0.17/0.44 ( setbeta
% 0.17/0.44 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ( ( in @ Xx
% 0.17/0.44 @ ( dsetconstr @ A
% 0.17/0.44 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.17/0.44 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(nonempty_type,type,
% 0.17/0.44 nonempty: $i > $o ).
% 0.17/0.44
% 0.17/0.44 thf(nonempty,definition,
% 0.17/0.44 ( nonempty
% 0.17/0.44 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(nonemptyE1_type,type,
% 0.17/0.44 nonemptyE1: $o ).
% 0.17/0.44
% 0.17/0.44 thf(nonemptyE1,definition,
% 0.17/0.44 ( nonemptyE1
% 0.17/0.44 = ( ! [A: $i] :
% 0.17/0.44 ( ( nonempty @ A )
% 0.17/0.44 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(nonemptyI_type,type,
% 0.17/0.44 nonemptyI: $o ).
% 0.17/0.44
% 0.17/0.44 thf(nonemptyI,definition,
% 0.17/0.44 ( nonemptyI
% 0.17/0.44 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ( ( Xphi @ Xx )
% 0.17/0.44 => ( nonempty
% 0.17/0.44 @ ( dsetconstr @ A
% 0.17/0.44 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(nonemptyI1_type,type,
% 0.17/0.44 nonemptyI1: $o ).
% 0.17/0.44
% 0.17/0.44 thf(nonemptyI1,definition,
% 0.17/0.44 ( nonemptyI1
% 0.17/0.44 = ( ! [A: $i] :
% 0.17/0.44 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.17/0.44 => ( nonempty @ A ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setadjoinIL_type,type,
% 0.17/0.44 setadjoinIL: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setadjoinIL,definition,
% 0.17/0.44 ( setadjoinIL
% 0.17/0.44 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(emptyinunitempty_type,type,
% 0.17/0.44 emptyinunitempty: $o ).
% 0.17/0.44
% 0.17/0.44 thf(emptyinunitempty,definition,
% 0.17/0.44 ( emptyinunitempty
% 0.17/0.44 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setadjoinIR_type,type,
% 0.17/0.44 setadjoinIR: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setadjoinIR,definition,
% 0.17/0.44 ( setadjoinIR
% 0.17/0.44 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.17/0.44 ( ( in @ Xy @ A )
% 0.17/0.44 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setadjoinE_type,type,
% 0.17/0.44 setadjoinE: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setadjoinE,definition,
% 0.17/0.44 ( setadjoinE
% 0.17/0.44 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.17/0.44 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.17/0.44 => ! [Xphi: $o] :
% 0.17/0.44 ( ( ( Xy = Xx )
% 0.17/0.44 => Xphi )
% 0.17/0.44 => ( ( ( in @ Xy @ A )
% 0.17/0.44 => Xphi )
% 0.17/0.44 => Xphi ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setadjoinOr_type,type,
% 0.17/0.44 setadjoinOr: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setadjoinOr,definition,
% 0.17/0.44 ( setadjoinOr
% 0.17/0.44 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.17/0.44 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.17/0.44 => ( ( Xy = Xx )
% 0.17/0.44 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setoftrueEq_type,type,
% 0.17/0.44 setoftrueEq: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setoftrueEq,definition,
% 0.17/0.44 ( setoftrueEq
% 0.17/0.44 = ( ! [A: $i] :
% 0.17/0.44 ( ( dsetconstr @ A
% 0.17/0.44 @ ^ [Xx: $i] : $true )
% 0.17/0.44 = A ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(powersetI_type,type,
% 0.17/0.44 powersetI: $o ).
% 0.17/0.44
% 0.17/0.44 thf(powersetI,definition,
% 0.17/0.44 ( powersetI
% 0.17/0.44 = ( ! [A: $i,B: $i] :
% 0.17/0.44 ( ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ B )
% 0.17/0.44 => ( in @ Xx @ A ) )
% 0.17/0.44 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(emptyinPowerset_type,type,
% 0.17/0.44 emptyinPowerset: $o ).
% 0.17/0.44
% 0.17/0.44 thf(emptyinPowerset,definition,
% 0.17/0.44 ( emptyinPowerset
% 0.17/0.44 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(emptyInPowerset_type,type,
% 0.17/0.44 emptyInPowerset: $o ).
% 0.17/0.44
% 0.17/0.44 thf(emptyInPowerset,definition,
% 0.17/0.44 ( emptyInPowerset
% 0.17/0.44 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(powersetE_type,type,
% 0.17/0.44 powersetE: $o ).
% 0.17/0.44
% 0.17/0.44 thf(powersetE,definition,
% 0.17/0.44 ( powersetE
% 0.17/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.44 ( ( in @ B @ ( powerset @ A ) )
% 0.17/0.44 => ( ( in @ Xx @ B )
% 0.17/0.44 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setunionI_type,type,
% 0.17/0.44 setunionI: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setunionI,definition,
% 0.17/0.44 ( setunionI
% 0.17/0.44 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.17/0.44 ( ( in @ Xx @ B )
% 0.17/0.44 => ( ( in @ B @ A )
% 0.17/0.44 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setunionE_type,type,
% 0.17/0.44 setunionE: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setunionE,definition,
% 0.17/0.44 ( setunionE
% 0.17/0.44 = ( ! [A: $i,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ ( setunion @ A ) )
% 0.17/0.44 => ! [Xphi: $o] :
% 0.17/0.44 ( ! [B: $i] :
% 0.17/0.44 ( ( in @ Xx @ B )
% 0.17/0.44 => ( ( in @ B @ A )
% 0.17/0.44 => Xphi ) )
% 0.17/0.44 => Xphi ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(subPowSU_type,type,
% 0.17/0.44 subPowSU: $o ).
% 0.17/0.44
% 0.17/0.44 thf(subPowSU,definition,
% 0.17/0.44 ( subPowSU
% 0.17/0.44 = ( ! [A: $i,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(exuE2_type,type,
% 0.17/0.44 exuE2: $o ).
% 0.17/0.44
% 0.17/0.44 thf(exuE2,definition,
% 0.17/0.44 ( exuE2
% 0.17/0.44 = ( ! [Xphi: $i > $o] :
% 0.17/0.44 ( ( exu
% 0.17/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.44 => ? [Xx: $i] :
% 0.17/0.44 ! [Xy: $i] :
% 0.17/0.44 ( ( Xphi @ Xy )
% 0.17/0.44 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(nonemptyImpWitness_type,type,
% 0.17/0.44 nonemptyImpWitness: $o ).
% 0.17/0.44
% 0.17/0.44 thf(nonemptyImpWitness,definition,
% 0.17/0.44 ( nonemptyImpWitness
% 0.17/0.44 = ( ! [A: $i] :
% 0.17/0.44 ( ( nonempty @ A )
% 0.17/0.44 => ? [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 & $true ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(uniqinunit_type,type,
% 0.17/0.44 uniqinunit: $o ).
% 0.17/0.44
% 0.17/0.44 thf(uniqinunit,definition,
% 0.17/0.44 ( uniqinunit
% 0.17/0.44 = ( ! [Xx: $i,Xy: $i] :
% 0.17/0.44 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.17/0.44 => ( Xx = Xy ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(notinsingleton_type,type,
% 0.17/0.44 notinsingleton: $o ).
% 0.17/0.44
% 0.17/0.44 thf(notinsingleton,definition,
% 0.17/0.44 ( notinsingleton
% 0.17/0.44 = ( ! [Xx: $i,Xy: $i] :
% 0.17/0.44 ( ( Xx != Xy )
% 0.17/0.44 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(eqinunit_type,type,
% 0.17/0.44 eqinunit: $o ).
% 0.17/0.44
% 0.17/0.44 thf(eqinunit,definition,
% 0.17/0.44 ( eqinunit
% 0.17/0.44 = ( ! [Xx: $i,Xy: $i] :
% 0.17/0.44 ( ( Xx = Xy )
% 0.17/0.44 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(singletonsswitch_type,type,
% 0.17/0.44 singletonsswitch: $o ).
% 0.17/0.44
% 0.17/0.44 thf(singletonsswitch,definition,
% 0.17/0.44 ( singletonsswitch
% 0.17/0.44 = ( ! [Xx: $i,Xy: $i] :
% 0.17/0.44 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.17/0.44 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(upairsetE_type,type,
% 0.17/0.44 upairsetE: $o ).
% 0.17/0.44
% 0.17/0.44 thf(upairsetE,definition,
% 0.17/0.44 ( upairsetE
% 0.17/0.44 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.17/0.44 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.17/0.44 => ( ( Xz = Xx )
% 0.17/0.44 | ( Xz = Xy ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(upairsetIL_type,type,
% 0.17/0.44 upairsetIL: $o ).
% 0.17/0.44
% 0.17/0.44 thf(upairsetIL,definition,
% 0.17/0.44 ( upairsetIL
% 0.17/0.44 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(upairsetIR_type,type,
% 0.17/0.44 upairsetIR: $o ).
% 0.17/0.44
% 0.17/0.44 thf(upairsetIR,definition,
% 0.17/0.44 ( upairsetIR
% 0.17/0.44 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(emptyE1_type,type,
% 0.17/0.44 emptyE1: $o ).
% 0.17/0.44
% 0.17/0.44 thf(emptyE1,definition,
% 0.17/0.44 ( emptyE1
% 0.17/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.17/0.44 ( ? [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 & ( Xphi @ Xx ) )
% 0.17/0.44 => ( ( ( dsetconstr @ A
% 0.17/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.44 = emptyset )
% 0.17/0.44 => $false ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(vacuousDall_type,type,
% 0.17/0.44 vacuousDall: $o ).
% 0.17/0.44
% 0.17/0.44 thf(vacuousDall,definition,
% 0.17/0.44 ( vacuousDall
% 0.17/0.44 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ emptyset )
% 0.17/0.44 => ( Xphi @ Xx ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(quantDeMorgan1_type,type,
% 0.17/0.44 quantDeMorgan1: $o ).
% 0.17/0.44
% 0.17/0.44 thf(quantDeMorgan1,definition,
% 0.17/0.44 ( quantDeMorgan1
% 0.17/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.17/0.44 ( ~ ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ( Xphi @ Xx ) )
% 0.17/0.44 => ? [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(quantDeMorgan2_type,type,
% 0.17/0.44 quantDeMorgan2: $o ).
% 0.17/0.44
% 0.17/0.44 thf(quantDeMorgan2,definition,
% 0.17/0.44 ( quantDeMorgan2
% 0.17/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.17/0.44 ( ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ~ ( Xphi @ Xx ) )
% 0.17/0.44 => ~ ? [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(quantDeMorgan3_type,type,
% 0.17/0.44 quantDeMorgan3: $o ).
% 0.17/0.44
% 0.17/0.44 thf(quantDeMorgan3,definition,
% 0.17/0.44 ( quantDeMorgan3
% 0.17/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.17/0.44 ( ~ ? [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 & ( Xphi @ Xx ) )
% 0.17/0.44 => ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(quantDeMorgan4_type,type,
% 0.17/0.44 quantDeMorgan4: $o ).
% 0.17/0.44
% 0.17/0.44 thf(quantDeMorgan4,definition,
% 0.17/0.44 ( quantDeMorgan4
% 0.17/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.17/0.44 ( ? [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 & ~ ( Xphi @ Xx ) )
% 0.17/0.44 => ~ ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(prop2setI_type,type,
% 0.17/0.44 prop2setI: $o ).
% 0.17/0.44
% 0.17/0.44 thf(prop2setI,definition,
% 0.17/0.44 ( prop2setI
% 0.17/0.44 = ( ! [Xphi: $o] :
% 0.17/0.44 ( Xphi
% 0.17/0.44 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(set2prop_type,type,
% 0.17/0.44 set2prop: $i > $o ).
% 0.17/0.44
% 0.17/0.44 thf(prop2set2propI_type,type,
% 0.17/0.44 prop2set2propI: $o ).
% 0.17/0.44
% 0.17/0.44 thf(prop2set2propI,definition,
% 0.17/0.44 ( prop2set2propI
% 0.17/0.44 = ( ! [Xphi: $o] :
% 0.17/0.44 ( Xphi
% 0.17/0.44 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(notdexE_type,type,
% 0.17/0.44 notdexE: $o ).
% 0.17/0.44
% 0.17/0.44 thf(notdexE,definition,
% 0.17/0.44 ( notdexE
% 0.17/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.17/0.44 ( ~ ? [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 & ( Xphi @ Xx ) )
% 0.17/0.44 => ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(notdallE_type,type,
% 0.17/0.44 notdallE: $o ).
% 0.17/0.44
% 0.17/0.44 thf(notdallE,definition,
% 0.17/0.44 ( notdallE
% 0.17/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.17/0.44 ( ~ ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ( Xphi @ Xx ) )
% 0.17/0.44 => ? [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(exuI1_type,type,
% 0.17/0.44 exuI1: $o ).
% 0.17/0.44
% 0.17/0.44 thf(exuI1,definition,
% 0.17/0.44 ( exuI1
% 0.17/0.44 = ( ! [Xphi: $i > $o] :
% 0.17/0.44 ( ? [Xx: $i] :
% 0.17/0.44 ( ( Xphi @ Xx )
% 0.17/0.44 & ! [Xy: $i] :
% 0.17/0.44 ( ( Xphi @ Xy )
% 0.17/0.44 => ( Xx = Xy ) ) )
% 0.17/0.44 => ( exu
% 0.17/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(exuI3_type,type,
% 0.17/0.44 exuI3: $o ).
% 0.17/0.44
% 0.17/0.44 thf(exuI3,definition,
% 0.17/0.44 ( exuI3
% 0.17/0.44 = ( ! [Xphi: $i > $o] :
% 0.17/0.44 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.17/0.44 => ( ! [Xx: $i,Xy: $i] :
% 0.17/0.44 ( ( Xphi @ Xx )
% 0.17/0.44 => ( ( Xphi @ Xy )
% 0.17/0.44 => ( Xx = Xy ) ) )
% 0.17/0.44 => ( exu
% 0.17/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(exuI2_type,type,
% 0.17/0.44 exuI2: $o ).
% 0.17/0.44
% 0.17/0.44 thf(exuI2,definition,
% 0.17/0.44 ( exuI2
% 0.17/0.44 = ( ! [Xphi: $i > $o] :
% 0.17/0.44 ( ? [Xx: $i] :
% 0.17/0.44 ! [Xy: $i] :
% 0.17/0.44 ( ( Xphi @ Xy )
% 0.17/0.44 <=> ( Xy = Xx ) )
% 0.17/0.44 => ( exu
% 0.17/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(inCongP_type,type,
% 0.17/0.44 inCongP: $o ).
% 0.17/0.44
% 0.17/0.44 thf(inCongP,definition,
% 0.17/0.44 ( inCongP
% 0.17/0.44 = ( ! [A: $i,B: $i] :
% 0.17/0.44 ( ( A = B )
% 0.17/0.44 => ! [Xx: $i,Xy: $i] :
% 0.17/0.44 ( ( Xx = Xy )
% 0.17/0.44 => ( ( in @ Xx @ A )
% 0.17/0.44 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(in__Cong_type,type,
% 0.17/0.44 in__Cong: $o ).
% 0.17/0.44
% 0.17/0.44 thf(in__Cong,definition,
% 0.17/0.44 ( in__Cong
% 0.17/0.44 = ( ! [A: $i,B: $i] :
% 0.17/0.44 ( ( A = B )
% 0.17/0.44 => ! [Xx: $i,Xy: $i] :
% 0.17/0.44 ( ( Xx = Xy )
% 0.17/0.44 => ( ( in @ Xx @ A )
% 0.17/0.44 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(exuE3u_type,type,
% 0.17/0.44 exuE3u: $o ).
% 0.17/0.44
% 0.17/0.44 thf(exuE3u,definition,
% 0.17/0.44 ( exuE3u
% 0.17/0.44 = ( ! [Xphi: $i > $o] :
% 0.17/0.44 ( ( exu
% 0.17/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.44 => ! [Xx: $i,Xy: $i] :
% 0.17/0.44 ( ( Xphi @ Xx )
% 0.17/0.44 => ( ( Xphi @ Xy )
% 0.17/0.44 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(exu__Cong_type,type,
% 0.17/0.44 exu__Cong: $o ).
% 0.17/0.44
% 0.17/0.44 thf(exu__Cong,definition,
% 0.17/0.44 ( exu__Cong
% 0.17/0.44 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.17/0.44 ( ! [Xx: $i,Xy: $i] :
% 0.17/0.44 ( ( Xx = Xy )
% 0.17/0.44 => ( ( Xphi @ Xx )
% 0.17/0.44 <=> ( Xpsi @ Xy ) ) )
% 0.17/0.44 => ( ( exu
% 0.17/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.44 <=> ( exu
% 0.17/0.44 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(emptyset__Cong_type,type,
% 0.17/0.44 emptyset__Cong: $o ).
% 0.17/0.44
% 0.17/0.44 thf(emptyset__Cong,definition,
% 0.17/0.44 ( emptyset__Cong
% 0.17/0.44 = ( emptyset = emptyset ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setadjoin__Cong_type,type,
% 0.17/0.44 setadjoin__Cong: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setadjoin__Cong,definition,
% 0.17/0.44 ( setadjoin__Cong
% 0.17/0.44 = ( ! [Xx: $i,Xy: $i] :
% 0.17/0.44 ( ( Xx = Xy )
% 0.17/0.44 => ! [Xz: $i,Xu: $i] :
% 0.17/0.44 ( ( Xz = Xu )
% 0.17/0.44 => ( ( setadjoin @ Xx @ Xz )
% 0.17/0.44 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(powerset__Cong_type,type,
% 0.17/0.44 powerset__Cong: $o ).
% 0.17/0.44
% 0.17/0.44 thf(powerset__Cong,definition,
% 0.17/0.44 ( powerset__Cong
% 0.17/0.44 = ( ! [A: $i,B: $i] :
% 0.17/0.44 ( ( A = B )
% 0.17/0.44 => ( ( powerset @ A )
% 0.17/0.44 = ( powerset @ B ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setunion__Cong_type,type,
% 0.17/0.44 setunion__Cong: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setunion__Cong,definition,
% 0.17/0.44 ( setunion__Cong
% 0.17/0.44 = ( ! [A: $i,B: $i] :
% 0.17/0.44 ( ( A = B )
% 0.17/0.44 => ( ( setunion @ A )
% 0.17/0.44 = ( setunion @ B ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(omega__Cong_type,type,
% 0.17/0.44 omega__Cong: $o ).
% 0.17/0.44
% 0.17/0.44 thf(omega__Cong,definition,
% 0.17/0.44 ( omega__Cong
% 0.17/0.44 = ( omega = omega ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(exuEu_type,type,
% 0.17/0.44 exuEu: $o ).
% 0.17/0.44
% 0.17/0.44 thf(exuEu,definition,
% 0.17/0.44 ( exuEu
% 0.17/0.44 = ( ! [Xphi: $i > $o] :
% 0.17/0.44 ( ( exu
% 0.17/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.44 => ! [Xx: $i,Xy: $i] :
% 0.17/0.44 ( ( Xphi @ Xx )
% 0.17/0.44 => ( ( Xphi @ Xy )
% 0.17/0.44 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(descr__Cong_type,type,
% 0.17/0.44 descr__Cong: $o ).
% 0.17/0.44
% 0.17/0.44 thf(descr__Cong,definition,
% 0.17/0.44 ( descr__Cong
% 0.17/0.44 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.17/0.44 ( ! [Xx: $i,Xy: $i] :
% 0.17/0.44 ( ( Xx = Xy )
% 0.17/0.44 => ( ( Xphi @ Xx )
% 0.17/0.44 <=> ( Xpsi @ Xy ) ) )
% 0.17/0.44 => ( ( exu
% 0.17/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.44 => ( ( exu
% 0.17/0.44 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.17/0.44 => ( ( descr
% 0.17/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.44 = ( descr
% 0.17/0.44 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(dsetconstr__Cong_type,type,
% 0.17/0.44 dsetconstr__Cong: $o ).
% 0.17/0.44
% 0.17/0.44 thf(dsetconstr__Cong,definition,
% 0.17/0.44 ( dsetconstr__Cong
% 0.17/0.44 = ( ! [A: $i,B: $i] :
% 0.17/0.44 ( ( A = B )
% 0.17/0.44 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.17/0.44 ( ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ! [Xy: $i] :
% 0.17/0.44 ( ( in @ Xy @ B )
% 0.17/0.44 => ( ( Xx = Xy )
% 0.17/0.44 => ( ( Xphi @ Xx )
% 0.17/0.44 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.17/0.44 => ( ( dsetconstr @ A
% 0.17/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.44 = ( dsetconstr @ B
% 0.17/0.44 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(subset_type,type,
% 0.17/0.44 subset: $i > $i > $o ).
% 0.17/0.44
% 0.17/0.44 thf(disjoint_type,type,
% 0.17/0.44 disjoint: $i > $i > $o ).
% 0.17/0.44
% 0.17/0.44 thf(setsmeet_type,type,
% 0.17/0.44 setsmeet: $i > $i > $o ).
% 0.17/0.44
% 0.17/0.44 thf(subsetI1_type,type,
% 0.17/0.44 subsetI1: $o ).
% 0.17/0.44
% 0.17/0.44 thf(subsetI1,definition,
% 0.17/0.44 ( subsetI1
% 0.17/0.44 = ( ! [A: $i,B: $i] :
% 0.17/0.44 ( ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ( in @ Xx @ B ) )
% 0.17/0.44 => ( subset @ A @ B ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(eqimpsubset2_type,type,
% 0.17/0.44 eqimpsubset2: $o ).
% 0.17/0.44
% 0.17/0.44 thf(eqimpsubset2,definition,
% 0.17/0.44 ( eqimpsubset2
% 0.17/0.44 = ( ! [A: $i,B: $i] :
% 0.17/0.44 ( ( A = B )
% 0.17/0.44 => ( subset @ B @ A ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(eqimpsubset1_type,type,
% 0.17/0.44 eqimpsubset1: $o ).
% 0.17/0.44
% 0.17/0.44 thf(eqimpsubset1,definition,
% 0.17/0.44 ( eqimpsubset1
% 0.17/0.44 = ( ! [A: $i,B: $i] :
% 0.17/0.44 ( ( A = B )
% 0.17/0.44 => ( subset @ A @ B ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(subsetI2_type,type,
% 0.17/0.44 subsetI2: $o ).
% 0.17/0.44
% 0.17/0.44 thf(subsetI2,definition,
% 0.17/0.44 ( subsetI2
% 0.17/0.44 = ( ! [A: $i,B: $i] :
% 0.17/0.44 ( ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ( in @ Xx @ B ) )
% 0.17/0.44 => ( subset @ A @ B ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(emptysetsubset_type,type,
% 0.17/0.44 emptysetsubset: $o ).
% 0.17/0.44
% 0.17/0.44 thf(emptysetsubset,definition,
% 0.17/0.44 ( emptysetsubset
% 0.17/0.44 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(subsetE_type,type,
% 0.17/0.44 subsetE: $o ).
% 0.17/0.44
% 0.17/0.44 thf(subsetE,definition,
% 0.17/0.44 ( subsetE
% 0.17/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.44 ( ( subset @ A @ B )
% 0.17/0.44 => ( ( in @ Xx @ A )
% 0.17/0.44 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(subsetE2_type,type,
% 0.17/0.44 subsetE2: $o ).
% 0.17/0.44
% 0.17/0.44 thf(subsetE2,definition,
% 0.17/0.44 ( subsetE2
% 0.17/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.44 ( ( subset @ A @ B )
% 0.17/0.44 => ( ~ ( in @ Xx @ B )
% 0.17/0.44 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(notsubsetI_type,type,
% 0.17/0.44 notsubsetI: $o ).
% 0.17/0.44
% 0.17/0.44 thf(notsubsetI,definition,
% 0.17/0.44 ( notsubsetI
% 0.17/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ( ~ ( in @ Xx @ B )
% 0.17/0.44 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(notequalI1_type,type,
% 0.17/0.44 notequalI1: $o ).
% 0.17/0.44
% 0.17/0.44 thf(notequalI1,definition,
% 0.17/0.44 ( notequalI1
% 0.17/0.44 = ( ! [A: $i,B: $i] :
% 0.17/0.44 ( ~ ( subset @ A @ B )
% 0.17/0.44 => ( A != B ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(notequalI2_type,type,
% 0.17/0.44 notequalI2: $o ).
% 0.17/0.44
% 0.17/0.44 thf(notequalI2,definition,
% 0.17/0.44 ( notequalI2
% 0.17/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ( ~ ( in @ Xx @ B )
% 0.17/0.44 => ( A != B ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(subsetRefl_type,type,
% 0.17/0.44 subsetRefl: $o ).
% 0.17/0.44
% 0.17/0.44 thf(subsetRefl,definition,
% 0.17/0.44 ( subsetRefl
% 0.17/0.44 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(subsetTrans_type,type,
% 0.17/0.44 subsetTrans: $o ).
% 0.17/0.44
% 0.17/0.44 thf(subsetTrans,definition,
% 0.17/0.44 ( subsetTrans
% 0.17/0.44 = ( ! [A: $i,B: $i,C: $i] :
% 0.17/0.44 ( ( subset @ A @ B )
% 0.17/0.44 => ( ( subset @ B @ C )
% 0.17/0.44 => ( subset @ A @ C ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setadjoinSub_type,type,
% 0.17/0.44 setadjoinSub: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setadjoinSub,definition,
% 0.17/0.44 ( setadjoinSub
% 0.17/0.44 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setadjoinSub2_type,type,
% 0.17/0.44 setadjoinSub2: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setadjoinSub2,definition,
% 0.17/0.44 ( setadjoinSub2
% 0.17/0.44 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.17/0.44 ( ( subset @ A @ B )
% 0.17/0.44 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(subset2powerset_type,type,
% 0.17/0.44 subset2powerset: $o ).
% 0.17/0.44
% 0.17/0.44 thf(subset2powerset,definition,
% 0.17/0.44 ( subset2powerset
% 0.17/0.44 = ( ! [A: $i,B: $i] :
% 0.17/0.44 ( ( subset @ A @ B )
% 0.17/0.44 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setextsub_type,type,
% 0.17/0.44 setextsub: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setextsub,definition,
% 0.17/0.44 ( setextsub
% 0.17/0.44 = ( ! [A: $i,B: $i] :
% 0.17/0.44 ( ( subset @ A @ B )
% 0.17/0.44 => ( ( subset @ B @ A )
% 0.17/0.44 => ( A = B ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(subsetemptysetimpeq_type,type,
% 0.17/0.44 subsetemptysetimpeq: $o ).
% 0.17/0.44
% 0.17/0.44 thf(subsetemptysetimpeq,definition,
% 0.17/0.44 ( subsetemptysetimpeq
% 0.17/0.44 = ( ! [A: $i] :
% 0.17/0.44 ( ( subset @ A @ emptyset )
% 0.17/0.44 => ( A = emptyset ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(powersetI1_type,type,
% 0.17/0.44 powersetI1: $o ).
% 0.17/0.44
% 0.17/0.44 thf(powersetI1,definition,
% 0.17/0.44 ( powersetI1
% 0.17/0.44 = ( ! [A: $i,B: $i] :
% 0.17/0.44 ( ( subset @ B @ A )
% 0.17/0.44 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(powersetE1_type,type,
% 0.17/0.44 powersetE1: $o ).
% 0.17/0.44
% 0.17/0.44 thf(powersetE1,definition,
% 0.17/0.44 ( powersetE1
% 0.17/0.44 = ( ! [A: $i,B: $i] :
% 0.17/0.44 ( ( in @ B @ ( powerset @ A ) )
% 0.17/0.44 => ( subset @ B @ A ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(inPowerset_type,type,
% 0.17/0.44 inPowerset: $o ).
% 0.17/0.44
% 0.17/0.44 thf(inPowerset,definition,
% 0.17/0.44 ( inPowerset
% 0.17/0.44 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(powersetsubset_type,type,
% 0.17/0.44 powersetsubset: $o ).
% 0.17/0.44
% 0.17/0.44 thf(powersetsubset,definition,
% 0.17/0.44 ( powersetsubset
% 0.17/0.44 = ( ! [A: $i,B: $i] :
% 0.17/0.44 ( ( subset @ A @ B )
% 0.17/0.44 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(sepInPowerset_type,type,
% 0.17/0.44 sepInPowerset: $o ).
% 0.17/0.44
% 0.17/0.44 thf(sepInPowerset,definition,
% 0.17/0.44 ( sepInPowerset
% 0.17/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.17/0.44 ( in
% 0.17/0.44 @ ( dsetconstr @ A
% 0.17/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.44 @ ( powerset @ A ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(sepSubset_type,type,
% 0.17/0.44 sepSubset: $o ).
% 0.17/0.44
% 0.17/0.44 thf(sepSubset,definition,
% 0.17/0.44 ( sepSubset
% 0.17/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.17/0.44 ( subset
% 0.17/0.44 @ ( dsetconstr @ A
% 0.17/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.44 @ A ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(binunion_type,type,
% 0.17/0.44 binunion: $i > $i > $i ).
% 0.17/0.44
% 0.17/0.44 thf(binunionIL_type,type,
% 0.17/0.44 binunionIL: $o ).
% 0.17/0.44
% 0.17/0.44 thf(binunionIL,definition,
% 0.17/0.44 ( binunionIL
% 0.17/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(upairset2IR_type,type,
% 0.17/0.44 upairset2IR: $o ).
% 0.17/0.44
% 0.17/0.44 thf(upairset2IR,definition,
% 0.17/0.44 ( upairset2IR
% 0.17/0.44 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(binunionIR_type,type,
% 0.17/0.44 binunionIR: $o ).
% 0.17/0.44
% 0.17/0.44 thf(binunionIR,definition,
% 0.17/0.44 ( binunionIR
% 0.17/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ B )
% 0.17/0.44 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(binunionEcases_type,type,
% 0.17/0.44 binunionEcases: $o ).
% 0.17/0.44
% 0.17/0.44 thf(binunionEcases,definition,
% 0.17/0.44 ( binunionEcases
% 0.17/0.44 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.17/0.44 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.17/0.44 => ( ( ( in @ Xx @ A )
% 0.17/0.44 => Xphi )
% 0.17/0.44 => ( ( ( in @ Xx @ B )
% 0.17/0.44 => Xphi )
% 0.17/0.44 => Xphi ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(binunionE_type,type,
% 0.17/0.44 binunionE: $o ).
% 0.17/0.44
% 0.17/0.44 thf(binunionE,definition,
% 0.17/0.44 ( binunionE
% 0.17/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.17/0.44 => ( ( in @ Xx @ A )
% 0.17/0.44 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(binunionLsub_type,type,
% 0.17/0.44 binunionLsub: $o ).
% 0.17/0.44
% 0.17/0.44 thf(binunionLsub,definition,
% 0.17/0.44 ( binunionLsub
% 0.17/0.44 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(binunionRsub_type,type,
% 0.17/0.44 binunionRsub: $o ).
% 0.17/0.44
% 0.17/0.44 thf(binunionRsub,definition,
% 0.17/0.44 ( binunionRsub
% 0.17/0.44 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(binintersect_type,type,
% 0.17/0.44 binintersect: $i > $i > $i ).
% 0.17/0.44
% 0.17/0.44 thf(binintersectI_type,type,
% 0.17/0.44 binintersectI: $o ).
% 0.17/0.44
% 0.17/0.44 thf(binintersectI,definition,
% 0.17/0.44 ( binintersectI
% 0.17/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ( ( in @ Xx @ B )
% 0.17/0.44 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(binintersectSubset5_type,type,
% 0.17/0.44 binintersectSubset5: $o ).
% 0.17/0.44
% 0.17/0.44 thf(binintersectSubset5,definition,
% 0.17/0.44 ( binintersectSubset5
% 0.17/0.44 = ( ! [A: $i,B: $i,C: $i] :
% 0.17/0.44 ( ( subset @ C @ A )
% 0.17/0.44 => ( ( subset @ C @ B )
% 0.17/0.44 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(binintersectEL_type,type,
% 0.17/0.44 binintersectEL: $o ).
% 0.17/0.44
% 0.17/0.44 thf(binintersectEL,definition,
% 0.17/0.44 ( binintersectEL
% 0.17/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.17/0.44 => ( in @ Xx @ A ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(binintersectLsub_type,type,
% 0.17/0.44 binintersectLsub: $o ).
% 0.17/0.44
% 0.17/0.44 thf(binintersectLsub,definition,
% 0.17/0.44 ( binintersectLsub
% 0.17/0.44 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(binintersectSubset2_type,type,
% 0.17/0.44 binintersectSubset2: $o ).
% 0.17/0.44
% 0.17/0.44 thf(binintersectSubset2,definition,
% 0.17/0.44 ( binintersectSubset2
% 0.17/0.44 = ( ! [A: $i,B: $i] :
% 0.17/0.44 ( ( subset @ A @ B )
% 0.17/0.44 => ( ( binintersect @ A @ B )
% 0.17/0.44 = A ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(binintersectSubset3_type,type,
% 0.17/0.44 binintersectSubset3: $o ).
% 0.17/0.44
% 0.17/0.44 thf(binintersectSubset3,definition,
% 0.17/0.44 ( binintersectSubset3
% 0.17/0.44 = ( ! [A: $i,B: $i] :
% 0.17/0.44 ( ( ( binintersect @ A @ B )
% 0.17/0.44 = B )
% 0.17/0.44 => ( subset @ B @ A ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(binintersectER_type,type,
% 0.17/0.44 binintersectER: $o ).
% 0.17/0.44
% 0.17/0.44 thf(binintersectER,definition,
% 0.17/0.44 ( binintersectER
% 0.17/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.17/0.44 => ( in @ Xx @ B ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(disjointsetsI1_type,type,
% 0.17/0.44 disjointsetsI1: $o ).
% 0.17/0.44
% 0.17/0.44 thf(disjointsetsI1,definition,
% 0.17/0.44 ( disjointsetsI1
% 0.17/0.44 = ( ! [A: $i,B: $i] :
% 0.17/0.44 ( ~ ? [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 & ( in @ Xx @ B ) )
% 0.17/0.44 => ( ( binintersect @ A @ B )
% 0.17/0.44 = emptyset ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(binintersectRsub_type,type,
% 0.17/0.44 binintersectRsub: $o ).
% 0.17/0.44
% 0.17/0.44 thf(binintersectRsub,definition,
% 0.17/0.44 ( binintersectRsub
% 0.17/0.44 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(binintersectSubset4_type,type,
% 0.17/0.44 binintersectSubset4: $o ).
% 0.17/0.44
% 0.17/0.44 thf(binintersectSubset4,definition,
% 0.17/0.44 ( binintersectSubset4
% 0.17/0.44 = ( ! [A: $i,B: $i] :
% 0.17/0.44 ( ( subset @ B @ A )
% 0.17/0.44 => ( ( binintersect @ A @ B )
% 0.17/0.44 = B ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(binintersectSubset1_type,type,
% 0.17/0.44 binintersectSubset1: $o ).
% 0.17/0.44
% 0.17/0.44 thf(binintersectSubset1,definition,
% 0.17/0.44 ( binintersectSubset1
% 0.17/0.44 = ( ! [A: $i,B: $i] :
% 0.17/0.44 ( ( ( binintersect @ A @ B )
% 0.17/0.44 = A )
% 0.17/0.44 => ( subset @ A @ B ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(bs114d_type,type,
% 0.17/0.44 bs114d: $o ).
% 0.17/0.44
% 0.17/0.44 thf(bs114d,definition,
% 0.17/0.44 ( bs114d
% 0.17/0.44 = ( ! [A: $i,B: $i,C: $i] :
% 0.17/0.44 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 0.17/0.44 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(regular_type,type,
% 0.17/0.44 regular: $i > $o ).
% 0.17/0.44
% 0.17/0.44 thf(setminus_type,type,
% 0.17/0.44 setminus: $i > $i > $i ).
% 0.17/0.44
% 0.17/0.44 thf(setminusI_type,type,
% 0.17/0.44 setminusI: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setminusI,definition,
% 0.17/0.44 ( setminusI
% 0.17/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ( ~ ( in @ Xx @ B )
% 0.17/0.44 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setminusEL_type,type,
% 0.17/0.44 setminusEL: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setminusEL,definition,
% 0.17/0.44 ( setminusEL
% 0.17/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.17/0.44 => ( in @ Xx @ A ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setminusER_type,type,
% 0.17/0.44 setminusER: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setminusER,definition,
% 0.17/0.44 ( setminusER
% 0.17/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.17/0.44 => ~ ( in @ Xx @ B ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setminusSubset2_type,type,
% 0.17/0.44 setminusSubset2: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setminusSubset2,definition,
% 0.17/0.44 ( setminusSubset2
% 0.17/0.44 = ( ! [A: $i,B: $i] :
% 0.17/0.44 ( ( subset @ A @ B )
% 0.17/0.44 => ( ( setminus @ A @ B )
% 0.17/0.44 = emptyset ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setminusERneg_type,type,
% 0.17/0.44 setminusERneg: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setminusERneg,definition,
% 0.17/0.44 ( setminusERneg
% 0.17/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.44 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.17/0.44 => ( ( in @ Xx @ A )
% 0.17/0.44 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setminusELneg_type,type,
% 0.17/0.44 setminusELneg: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setminusELneg,definition,
% 0.17/0.44 ( setminusELneg
% 0.17/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.44 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.17/0.44 => ( ~ ( in @ Xx @ B )
% 0.17/0.44 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setminusILneg_type,type,
% 0.17/0.44 setminusILneg: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setminusILneg,definition,
% 0.17/0.44 ( setminusILneg
% 0.17/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.44 ( ~ ( in @ Xx @ A )
% 0.17/0.44 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setminusIRneg_type,type,
% 0.17/0.44 setminusIRneg: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setminusIRneg,definition,
% 0.17/0.44 ( setminusIRneg
% 0.17/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ B )
% 0.17/0.44 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setminusLsub_type,type,
% 0.17/0.44 setminusLsub: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setminusLsub,definition,
% 0.17/0.44 ( setminusLsub
% 0.17/0.44 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setminusSubset1_type,type,
% 0.17/0.44 setminusSubset1: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setminusSubset1,definition,
% 0.17/0.44 ( setminusSubset1
% 0.17/0.44 = ( ! [A: $i,B: $i] :
% 0.17/0.44 ( ( ( setminus @ A @ B )
% 0.17/0.44 = emptyset )
% 0.17/0.44 => ( subset @ A @ B ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(symdiff_type,type,
% 0.17/0.44 symdiff: $i > $i > $i ).
% 0.17/0.44
% 0.17/0.44 thf(symdiffE_type,type,
% 0.17/0.44 symdiffE: $o ).
% 0.17/0.44
% 0.17/0.44 thf(symdiffE,definition,
% 0.17/0.44 ( symdiffE
% 0.17/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 0.17/0.44 => ! [Xphi: $o] :
% 0.17/0.44 ( ( ( in @ Xx @ A )
% 0.17/0.44 => ( ~ ( in @ Xx @ B )
% 0.17/0.44 => Xphi ) )
% 0.17/0.44 => ( ( ~ ( in @ Xx @ A )
% 0.17/0.44 => ( ( in @ Xx @ B )
% 0.17/0.44 => Xphi ) )
% 0.17/0.44 => Xphi ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(symdiffI1_type,type,
% 0.17/0.44 symdiffI1: $o ).
% 0.17/0.44
% 0.17/0.44 thf(symdiffI1,definition,
% 0.17/0.44 ( symdiffI1
% 0.17/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ( ~ ( in @ Xx @ B )
% 0.17/0.44 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(symdiffI2_type,type,
% 0.17/0.44 symdiffI2: $o ).
% 0.17/0.44
% 0.17/0.44 thf(symdiffI2,definition,
% 0.17/0.44 ( symdiffI2
% 0.17/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.44 ( ~ ( in @ Xx @ A )
% 0.17/0.44 => ( ( in @ Xx @ B )
% 0.17/0.44 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(symdiffIneg1_type,type,
% 0.17/0.44 symdiffIneg1: $o ).
% 0.17/0.44
% 0.17/0.44 thf(symdiffIneg1,definition,
% 0.17/0.44 ( symdiffIneg1
% 0.17/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ( ( in @ Xx @ B )
% 0.17/0.44 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(symdiffIneg2_type,type,
% 0.17/0.44 symdiffIneg2: $o ).
% 0.17/0.44
% 0.17/0.44 thf(symdiffIneg2,definition,
% 0.17/0.44 ( symdiffIneg2
% 0.17/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.44 ( ~ ( in @ Xx @ A )
% 0.17/0.44 => ( ~ ( in @ Xx @ B )
% 0.17/0.44 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(iskpair_type,type,
% 0.17/0.44 iskpair: $i > $o ).
% 0.17/0.44
% 0.17/0.44 thf(secondinupair_type,type,
% 0.17/0.44 secondinupair: $o ).
% 0.17/0.44
% 0.17/0.44 thf(secondinupair,definition,
% 0.17/0.44 ( secondinupair
% 0.17/0.44 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setukpairIL_type,type,
% 0.17/0.44 setukpairIL: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setukpairIL,definition,
% 0.17/0.44 ( setukpairIL
% 0.17/0.44 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setukpairIR_type,type,
% 0.17/0.44 setukpairIR: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setukpairIR,definition,
% 0.17/0.44 ( setukpairIR
% 0.17/0.44 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(kpairiskpair_type,type,
% 0.17/0.44 kpairiskpair: $o ).
% 0.17/0.44
% 0.17/0.44 thf(kpairiskpair,definition,
% 0.17/0.44 ( kpairiskpair
% 0.17/0.44 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(kpair_type,type,
% 0.17/0.44 kpair: $i > $i > $i ).
% 0.17/0.44
% 0.17/0.44 thf(kpairp_type,type,
% 0.17/0.44 kpairp: $o ).
% 0.17/0.44
% 0.17/0.44 thf(kpairp,definition,
% 0.17/0.44 ( kpairp
% 0.17/0.44 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(cartprod_type,type,
% 0.17/0.44 cartprod: $i > $i > $i ).
% 0.17/0.44
% 0.17/0.44 thf(singletonsubset_type,type,
% 0.17/0.44 singletonsubset: $o ).
% 0.17/0.44
% 0.17/0.44 thf(singletonsubset,definition,
% 0.17/0.44 ( singletonsubset
% 0.17/0.44 = ( ! [A: $i,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(singletoninpowerset_type,type,
% 0.17/0.44 singletoninpowerset: $o ).
% 0.17/0.44
% 0.17/0.44 thf(singletoninpowerset,definition,
% 0.17/0.44 ( singletoninpowerset
% 0.17/0.44 = ( ! [A: $i,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(singletoninpowunion_type,type,
% 0.17/0.44 singletoninpowunion: $o ).
% 0.17/0.44
% 0.17/0.44 thf(singletoninpowunion,definition,
% 0.17/0.44 ( singletoninpowunion
% 0.17/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(upairset2E_type,type,
% 0.17/0.44 upairset2E: $o ).
% 0.17/0.44
% 0.17/0.44 thf(upairset2E,definition,
% 0.17/0.44 ( upairset2E
% 0.17/0.44 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.17/0.44 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.17/0.44 => ( ( Xz = Xx )
% 0.17/0.44 | ( Xz = Xy ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(upairsubunion_type,type,
% 0.17/0.44 upairsubunion: $o ).
% 0.17/0.44
% 0.17/0.44 thf(upairsubunion,definition,
% 0.17/0.44 ( upairsubunion
% 0.17/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ! [Xy: $i] :
% 0.17/0.44 ( ( in @ Xy @ B )
% 0.17/0.44 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(upairinpowunion_type,type,
% 0.17/0.44 upairinpowunion: $o ).
% 0.17/0.44
% 0.17/0.44 thf(upairinpowunion,definition,
% 0.17/0.44 ( upairinpowunion
% 0.17/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ! [Xy: $i] :
% 0.17/0.44 ( ( in @ Xy @ B )
% 0.17/0.44 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(ubforcartprodlem1_type,type,
% 0.17/0.44 ubforcartprodlem1: $o ).
% 0.17/0.44
% 0.17/0.44 thf(ubforcartprodlem1,definition,
% 0.17/0.44 ( ubforcartprodlem1
% 0.17/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ! [Xy: $i] :
% 0.17/0.44 ( ( in @ Xy @ B )
% 0.17/0.44 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(ubforcartprodlem2_type,type,
% 0.17/0.44 ubforcartprodlem2: $o ).
% 0.17/0.44
% 0.17/0.44 thf(ubforcartprodlem2,definition,
% 0.17/0.44 ( ubforcartprodlem2
% 0.17/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ! [Xy: $i] :
% 0.17/0.44 ( ( in @ Xy @ B )
% 0.17/0.44 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(ubforcartprodlem3_type,type,
% 0.17/0.44 ubforcartprodlem3: $o ).
% 0.17/0.44
% 0.17/0.44 thf(ubforcartprodlem3,definition,
% 0.17/0.44 ( ubforcartprodlem3
% 0.17/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ! [Xy: $i] :
% 0.17/0.44 ( ( in @ Xy @ B )
% 0.17/0.44 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(cartprodpairin_type,type,
% 0.17/0.44 cartprodpairin: $o ).
% 0.17/0.44
% 0.17/0.44 thf(cartprodpairin,definition,
% 0.17/0.44 ( cartprodpairin
% 0.17/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ! [Xy: $i] :
% 0.17/0.44 ( ( in @ Xy @ B )
% 0.17/0.44 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(cartprodmempair1_type,type,
% 0.17/0.44 cartprodmempair1: $o ).
% 0.17/0.44
% 0.17/0.44 thf(cartprodmempair1,definition,
% 0.17/0.44 ( cartprodmempair1
% 0.17/0.44 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.17/0.44 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.17/0.44 => ? [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 & ? [Xy: $i] :
% 0.17/0.44 ( ( in @ Xy @ B )
% 0.17/0.44 & ( Xu
% 0.17/0.44 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(cartprodmempair_type,type,
% 0.17/0.44 cartprodmempair: $o ).
% 0.17/0.44
% 0.17/0.44 thf(cartprodmempair,definition,
% 0.17/0.44 ( cartprodmempair
% 0.17/0.44 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.17/0.44 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.17/0.44 => ( iskpair @ Xu ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setunionE2_type,type,
% 0.17/0.44 setunionE2: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setunionE2,definition,
% 0.17/0.44 ( setunionE2
% 0.17/0.44 = ( ! [A: $i,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ ( setunion @ A ) )
% 0.17/0.44 => ? [X: $i] :
% 0.17/0.44 ( ( in @ X @ A )
% 0.17/0.44 & ( in @ Xx @ X ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setunionsingleton1_type,type,
% 0.17/0.44 setunionsingleton1: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setunionsingleton1,definition,
% 0.17/0.44 ( setunionsingleton1
% 0.17/0.44 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setunionsingleton2_type,type,
% 0.17/0.44 setunionsingleton2: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setunionsingleton2,definition,
% 0.17/0.44 ( setunionsingleton2
% 0.17/0.44 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setunionsingleton_type,type,
% 0.17/0.44 setunionsingleton: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setunionsingleton,definition,
% 0.17/0.44 ( setunionsingleton
% 0.17/0.44 = ( ! [Xx: $i] :
% 0.17/0.44 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 0.17/0.44 = Xx ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(singleton_type,type,
% 0.17/0.44 singleton: $i > $o ).
% 0.17/0.44
% 0.17/0.44 thf(singleton,definition,
% 0.17/0.44 ( singleton
% 0.17/0.44 = ( ^ [A: $i] :
% 0.17/0.44 ? [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 & ( A
% 0.17/0.44 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(singletonprop_type,type,
% 0.17/0.44 singletonprop: $o ).
% 0.17/0.44
% 0.17/0.44 thf(singletonprop,definition,
% 0.17/0.44 ( singletonprop
% 0.17/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.17/0.44 ( ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ! [Xy: $i] :
% 0.17/0.44 ( ( in @ Xy @ A )
% 0.17/0.44 => ( ( Xphi @ Xx )
% 0.17/0.44 => ( ( Xphi @ Xy )
% 0.17/0.44 => ( Xx = Xy ) ) ) ) )
% 0.17/0.44 => ( ? [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 & ( Xphi @ Xx ) )
% 0.17/0.44 => ( singleton
% 0.17/0.44 @ ( dsetconstr @ A
% 0.17/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(ex1_type,type,
% 0.17/0.44 ex1: $i > ( $i > $o ) > $o ).
% 0.17/0.44
% 0.17/0.44 thf(ex1,definition,
% 0.17/0.44 ( ex1
% 0.17/0.44 = ( ^ [A: $i,Xphi: $i > $o] :
% 0.17/0.44 ( singleton
% 0.17/0.44 @ ( dsetconstr @ A
% 0.17/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(ex1E1_type,type,
% 0.17/0.44 ex1E1: $o ).
% 0.17/0.44
% 0.17/0.44 thf(ex1E1,definition,
% 0.17/0.44 ( ex1E1
% 0.17/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.17/0.44 ( ( ex1 @ A
% 0.17/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.44 => ? [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(ex1I_type,type,
% 0.17/0.44 ex1I: $o ).
% 0.17/0.44
% 0.17/0.44 thf(ex1I,definition,
% 0.17/0.44 ( ex1I
% 0.17/0.44 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ( ( Xphi @ Xx )
% 0.17/0.44 => ( ! [Xy: $i] :
% 0.17/0.44 ( ( in @ Xy @ A )
% 0.17/0.44 => ( ( Xphi @ Xy )
% 0.17/0.44 => ( Xy = Xx ) ) )
% 0.17/0.44 => ( ex1 @ A
% 0.17/0.44 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(ex1I2_type,type,
% 0.17/0.44 ex1I2: $o ).
% 0.17/0.44
% 0.17/0.44 thf(ex1I2,definition,
% 0.17/0.44 ( ex1I2
% 0.17/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.17/0.44 ( ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ! [Xy: $i] :
% 0.17/0.44 ( ( in @ Xy @ A )
% 0.17/0.44 => ( ( Xphi @ Xx )
% 0.17/0.44 => ( ( Xphi @ Xy )
% 0.17/0.44 => ( Xx = Xy ) ) ) ) )
% 0.17/0.44 => ( ? [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 & ( Xphi @ Xx ) )
% 0.17/0.44 => ( ex1 @ A
% 0.17/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(singletonsuniq_type,type,
% 0.17/0.44 singletonsuniq: $o ).
% 0.17/0.44
% 0.17/0.44 thf(singletonsuniq,definition,
% 0.17/0.44 ( singletonsuniq
% 0.17/0.44 = ( ! [Xx: $i,Xy: $i] :
% 0.17/0.44 ( ( ( setadjoin @ Xx @ emptyset )
% 0.17/0.44 = ( setadjoin @ Xy @ emptyset ) )
% 0.17/0.44 => ( Xx = Xy ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(atmost1p_type,type,
% 0.17/0.44 atmost1p: $i > $o ).
% 0.17/0.44
% 0.17/0.44 thf(atleast2p_type,type,
% 0.17/0.44 atleast2p: $i > $o ).
% 0.17/0.44
% 0.17/0.44 thf(atmost2p_type,type,
% 0.17/0.44 atmost2p: $i > $o ).
% 0.17/0.44
% 0.17/0.44 thf(upairsetp_type,type,
% 0.17/0.44 upairsetp: $i > $o ).
% 0.17/0.44
% 0.17/0.44 thf(setukpairinjL1_type,type,
% 0.17/0.44 setukpairinjL1: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setukpairinjL1,definition,
% 0.17/0.44 ( setukpairinjL1
% 0.17/0.44 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.17/0.44 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 0.17/0.44 => ( Xx = Xz ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(kfstsingleton_type,type,
% 0.17/0.44 kfstsingleton: $o ).
% 0.17/0.44
% 0.17/0.44 thf(kfstsingleton,definition,
% 0.17/0.44 ( kfstsingleton
% 0.17/0.44 = ( ! [Xu: $i] :
% 0.17/0.44 ( ( iskpair @ Xu )
% 0.17/0.44 => ( singleton
% 0.17/0.44 @ ( dsetconstr @ ( setunion @ Xu )
% 0.17/0.44 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(theprop_type,type,
% 0.17/0.44 theprop: $o ).
% 0.17/0.44
% 0.17/0.44 thf(theprop,definition,
% 0.17/0.44 ( theprop
% 0.17/0.44 = ( ! [X: $i] :
% 0.17/0.44 ( ( singleton @ X )
% 0.17/0.44 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(kfst_type,type,
% 0.17/0.44 kfst: $i > $i ).
% 0.17/0.44
% 0.17/0.44 thf(kfstpairEq_type,type,
% 0.17/0.44 kfstpairEq: $o ).
% 0.17/0.44
% 0.17/0.44 thf(kfstpairEq,definition,
% 0.17/0.44 ( kfstpairEq
% 0.17/0.44 = ( ! [Xx: $i,Xy: $i] :
% 0.17/0.44 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.17/0.44 = Xx ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(cartprodfstin_type,type,
% 0.17/0.44 cartprodfstin: $o ).
% 0.17/0.44
% 0.17/0.44 thf(cartprodfstin,definition,
% 0.17/0.44 ( cartprodfstin
% 0.17/0.44 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.17/0.44 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.17/0.44 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setukpairinjL2_type,type,
% 0.17/0.44 setukpairinjL2: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setukpairinjL2,definition,
% 0.17/0.44 ( setukpairinjL2
% 0.17/0.44 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.17/0.44 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.17/0.44 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.17/0.44 => ( Xx = Xz ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setukpairinjL_type,type,
% 0.17/0.44 setukpairinjL: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setukpairinjL,definition,
% 0.17/0.44 ( setukpairinjL
% 0.17/0.44 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.17/0.44 ( ( ( kpair @ Xx @ Xy )
% 0.17/0.44 = ( kpair @ Xz @ Xu ) )
% 0.17/0.44 => ( Xx = Xz ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setukpairinjR11_type,type,
% 0.17/0.44 setukpairinjR11: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setukpairinjR11,definition,
% 0.17/0.44 ( setukpairinjR11
% 0.17/0.44 = ( ! [Xx: $i,Xy: $i] :
% 0.17/0.44 ( ( Xx = Xy )
% 0.17/0.44 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.17/0.44 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setukpairinjR12_type,type,
% 0.17/0.44 setukpairinjR12: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setukpairinjR12,definition,
% 0.17/0.44 ( setukpairinjR12
% 0.17/0.44 = ( ! [Xx: $i,Xy: $i] :
% 0.17/0.44 ( ( Xx = Xy )
% 0.17/0.44 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.17/0.44 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setukpairinjR1_type,type,
% 0.17/0.44 setukpairinjR1: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setukpairinjR1,definition,
% 0.17/0.44 ( setukpairinjR1
% 0.17/0.44 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.17/0.44 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.17/0.44 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.17/0.44 => ( ( Xz = Xu )
% 0.17/0.44 => ( Xy = Xu ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(upairequniteq_type,type,
% 0.17/0.44 upairequniteq: $o ).
% 0.17/0.44
% 0.17/0.44 thf(upairequniteq,definition,
% 0.17/0.44 ( upairequniteq
% 0.17/0.44 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.17/0.44 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.17/0.44 = ( setadjoin @ Xz @ emptyset ) )
% 0.17/0.44 => ( Xx = Xy ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setukpairinjR2_type,type,
% 0.17/0.44 setukpairinjR2: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setukpairinjR2,definition,
% 0.17/0.44 ( setukpairinjR2
% 0.17/0.44 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.17/0.44 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.17/0.44 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.17/0.44 => ( Xy = Xu ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setukpairinjR_type,type,
% 0.17/0.44 setukpairinjR: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setukpairinjR,definition,
% 0.17/0.44 ( setukpairinjR
% 0.17/0.44 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.17/0.44 ( ( ( kpair @ Xx @ Xy )
% 0.17/0.44 = ( kpair @ Xz @ Xu ) )
% 0.17/0.44 => ( Xy = Xu ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(ksndsingleton_type,type,
% 0.17/0.44 ksndsingleton: $o ).
% 0.17/0.44
% 0.17/0.44 thf(ksndsingleton,definition,
% 0.17/0.44 ( ksndsingleton
% 0.17/0.44 = ( ! [Xu: $i] :
% 0.17/0.44 ( ( iskpair @ Xu )
% 0.17/0.44 => ( singleton
% 0.17/0.44 @ ( dsetconstr @ ( setunion @ Xu )
% 0.17/0.44 @ ^ [Xx: $i] :
% 0.17/0.44 ( Xu
% 0.17/0.44 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(ksnd_type,type,
% 0.17/0.44 ksnd: $i > $i ).
% 0.17/0.44
% 0.17/0.44 thf(ksndpairEq_type,type,
% 0.17/0.44 ksndpairEq: $o ).
% 0.17/0.44
% 0.17/0.44 thf(ksndpairEq,definition,
% 0.17/0.44 ( ksndpairEq
% 0.17/0.44 = ( ! [Xx: $i,Xy: $i] :
% 0.17/0.44 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.17/0.44 = Xy ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(kpairsurjEq_type,type,
% 0.17/0.44 kpairsurjEq: $o ).
% 0.17/0.44
% 0.17/0.44 thf(kpairsurjEq,definition,
% 0.17/0.44 ( kpairsurjEq
% 0.17/0.44 = ( ! [Xu: $i] :
% 0.17/0.44 ( ( iskpair @ Xu )
% 0.17/0.44 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.17/0.44 = Xu ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(cartprodsndin_type,type,
% 0.17/0.44 cartprodsndin: $o ).
% 0.17/0.44
% 0.17/0.44 thf(cartprodsndin,definition,
% 0.17/0.44 ( cartprodsndin
% 0.17/0.44 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.17/0.44 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.17/0.44 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(cartprodpairmemEL_type,type,
% 0.17/0.44 cartprodpairmemEL: $o ).
% 0.17/0.44
% 0.17/0.44 thf(cartprodpairmemEL,definition,
% 0.17/0.44 ( cartprodpairmemEL
% 0.17/0.44 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.17/0.44 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.17/0.44 => ( in @ Xx @ A ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(cartprodpairmemER_type,type,
% 0.17/0.44 cartprodpairmemER: $o ).
% 0.17/0.44
% 0.17/0.44 thf(cartprodpairmemER,definition,
% 0.17/0.44 ( cartprodpairmemER
% 0.17/0.44 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.17/0.44 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.17/0.44 => ( in @ Xy @ B ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(cartprodmempaircEq_type,type,
% 0.17/0.44 cartprodmempaircEq: $o ).
% 0.17/0.44
% 0.17/0.44 thf(cartprodmempaircEq,definition,
% 0.17/0.44 ( cartprodmempaircEq
% 0.17/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ! [Xy: $i] :
% 0.17/0.44 ( ( in @ Xy @ B )
% 0.17/0.44 => ( ( kpair @ Xx @ Xy )
% 0.17/0.44 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(cartprodfstpairEq_type,type,
% 0.17/0.44 cartprodfstpairEq: $o ).
% 0.17/0.44
% 0.17/0.44 thf(cartprodfstpairEq,definition,
% 0.17/0.44 ( cartprodfstpairEq
% 0.17/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ! [Xy: $i] :
% 0.17/0.44 ( ( in @ Xy @ B )
% 0.17/0.44 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.17/0.44 = Xx ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(cartprodsndpairEq_type,type,
% 0.17/0.44 cartprodsndpairEq: $o ).
% 0.17/0.44
% 0.17/0.44 thf(cartprodsndpairEq,definition,
% 0.17/0.44 ( cartprodsndpairEq
% 0.17/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ! [Xy: $i] :
% 0.17/0.44 ( ( in @ Xy @ B )
% 0.17/0.44 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.17/0.44 = Xy ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(cartprodpairsurjEq_type,type,
% 0.17/0.44 cartprodpairsurjEq: $o ).
% 0.17/0.44
% 0.17/0.44 thf(cartprodpairsurjEq,definition,
% 0.17/0.44 ( cartprodpairsurjEq
% 0.17/0.44 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.17/0.44 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.17/0.44 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.17/0.44 = Xu ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(breln_type,type,
% 0.17/0.44 breln: $i > $i > $i > $o ).
% 0.17/0.44
% 0.17/0.44 thf(breln,definition,
% 0.17/0.44 ( breln
% 0.17/0.44 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(dpsetconstr_type,type,
% 0.17/0.44 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 0.17/0.44
% 0.17/0.44 thf(dpsetconstrI_type,type,
% 0.17/0.44 dpsetconstrI: $o ).
% 0.17/0.44
% 0.17/0.44 thf(dpsetconstrI,definition,
% 0.17/0.44 ( dpsetconstrI
% 0.17/0.44 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ! [Xy: $i] :
% 0.17/0.44 ( ( in @ Xy @ B )
% 0.17/0.44 => ( ( Xphi @ Xx @ Xy )
% 0.17/0.44 => ( in @ ( kpair @ Xx @ Xy )
% 0.17/0.44 @ ( dpsetconstr @ A @ B
% 0.17/0.44 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(dpsetconstrSub_type,type,
% 0.17/0.44 dpsetconstrSub: $o ).
% 0.17/0.44
% 0.17/0.44 thf(dpsetconstrSub,definition,
% 0.17/0.44 ( dpsetconstrSub
% 0.17/0.44 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.17/0.44 ( subset
% 0.17/0.44 @ ( dpsetconstr @ A @ B
% 0.17/0.44 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 0.17/0.44 @ ( cartprod @ A @ B ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setOfPairsIsBReln_type,type,
% 0.17/0.44 setOfPairsIsBReln: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setOfPairsIsBReln,definition,
% 0.17/0.44 ( setOfPairsIsBReln
% 0.17/0.44 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.17/0.44 ( breln @ A @ B
% 0.17/0.44 @ ( dpsetconstr @ A @ B
% 0.17/0.44 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(dpsetconstrERa_type,type,
% 0.17/0.44 dpsetconstrERa: $o ).
% 0.17/0.44
% 0.17/0.44 thf(dpsetconstrERa,definition,
% 0.17/0.44 ( dpsetconstrERa
% 0.17/0.44 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ! [Xy: $i] :
% 0.17/0.44 ( ( in @ Xy @ B )
% 0.17/0.44 => ( ( in @ ( kpair @ Xx @ Xy )
% 0.17/0.44 @ ( dpsetconstr @ A @ B
% 0.17/0.44 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.17/0.44 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(dpsetconstrEL1_type,type,
% 0.17/0.44 dpsetconstrEL1: $o ).
% 0.17/0.44
% 0.17/0.44 thf(dpsetconstrEL1,definition,
% 0.17/0.44 ( dpsetconstrEL1
% 0.17/0.44 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.17/0.44 ( ( in @ ( kpair @ Xx @ Xy )
% 0.17/0.44 @ ( dpsetconstr @ A @ B
% 0.17/0.44 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.17/0.44 => ( in @ Xx @ A ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(dpsetconstrEL2_type,type,
% 0.17/0.44 dpsetconstrEL2: $o ).
% 0.17/0.44
% 0.17/0.44 thf(dpsetconstrEL2,definition,
% 0.17/0.44 ( dpsetconstrEL2
% 0.17/0.44 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.17/0.44 ( ( in @ ( kpair @ Xx @ Xy )
% 0.17/0.44 @ ( dpsetconstr @ A @ B
% 0.17/0.44 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.17/0.44 => ( in @ Xy @ B ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(dpsetconstrER_type,type,
% 0.17/0.44 dpsetconstrER: $o ).
% 0.17/0.44
% 0.17/0.44 thf(dpsetconstrER,definition,
% 0.17/0.44 ( dpsetconstrER
% 0.17/0.44 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.17/0.44 ( ( in @ ( kpair @ Xx @ Xy )
% 0.17/0.44 @ ( dpsetconstr @ A @ B
% 0.17/0.44 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.17/0.44 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(func_type,type,
% 0.17/0.44 func: $i > $i > $i > $o ).
% 0.17/0.44
% 0.17/0.44 thf(funcSet_type,type,
% 0.17/0.44 funcSet: $i > $i > $i ).
% 0.17/0.44
% 0.17/0.44 thf(funcImageSingleton_type,type,
% 0.17/0.44 funcImageSingleton: $o ).
% 0.17/0.44
% 0.17/0.44 thf(funcImageSingleton,definition,
% 0.17/0.44 ( funcImageSingleton
% 0.17/0.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.17/0.44 ( ( func @ A @ B @ Xf )
% 0.17/0.44 => ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ( singleton
% 0.17/0.44 @ ( dsetconstr @ B
% 0.17/0.44 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(apProp_type,type,
% 0.17/0.44 apProp: $o ).
% 0.17/0.44
% 0.17/0.44 thf(apProp,definition,
% 0.17/0.44 ( apProp
% 0.17/0.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.17/0.44 ( ( func @ A @ B @ Xf )
% 0.17/0.44 => ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ( in
% 0.17/0.44 @ ( setunion
% 0.17/0.44 @ ( dsetconstr @ B
% 0.17/0.44 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 0.17/0.44 @ B ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(ap_type,type,
% 0.17/0.44 ap: $i > $i > $i > $i > $i ).
% 0.17/0.44
% 0.17/0.44 thf(app_type,type,
% 0.17/0.44 app: $o ).
% 0.17/0.44
% 0.17/0.44 thf(app,definition,
% 0.17/0.44 ( app
% 0.17/0.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.17/0.44 ( ( func @ A @ B @ Xf )
% 0.17/0.44 => ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(infuncsetfunc_type,type,
% 0.17/0.44 infuncsetfunc: $o ).
% 0.17/0.44
% 0.17/0.44 thf(infuncsetfunc,definition,
% 0.17/0.44 ( infuncsetfunc
% 0.17/0.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.17/0.44 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.17/0.44 => ( func @ A @ B @ Xf ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(ap2p_type,type,
% 0.17/0.44 ap2p: $o ).
% 0.17/0.44
% 0.17/0.44 thf(ap2p,definition,
% 0.17/0.44 ( ap2p
% 0.17/0.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.17/0.44 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.17/0.44 => ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(funcinfuncset_type,type,
% 0.17/0.44 funcinfuncset: $o ).
% 0.17/0.44
% 0.17/0.44 thf(funcinfuncset,definition,
% 0.17/0.44 ( funcinfuncset
% 0.17/0.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.17/0.44 ( ( func @ A @ B @ Xf )
% 0.17/0.44 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(lamProp_type,type,
% 0.17/0.44 lamProp: $o ).
% 0.17/0.44
% 0.17/0.44 thf(lamProp,definition,
% 0.17/0.44 ( lamProp
% 0.17/0.44 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.17/0.44 ( ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.17/0.44 => ( func @ A @ B
% 0.17/0.44 @ ( dpsetconstr @ A @ B
% 0.17/0.44 @ ^ [Xx: $i,Xy: $i] :
% 0.17/0.44 ( ( Xf @ Xx )
% 0.17/0.44 = Xy ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(lam_type,type,
% 0.17/0.44 lam: $i > $i > ( $i > $i ) > $i ).
% 0.17/0.44
% 0.17/0.44 thf(lam,definition,
% 0.17/0.44 ( lam
% 0.17/0.44 = ( ^ [A: $i,B: $i,Xf: $i > $i] :
% 0.17/0.44 ( dpsetconstr @ A @ B
% 0.17/0.44 @ ^ [Xx: $i,Xy: $i] :
% 0.17/0.44 ( ( Xf @ Xx )
% 0.17/0.44 = Xy ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(lamp_type,type,
% 0.17/0.44 lamp: $o ).
% 0.17/0.44
% 0.17/0.44 thf(lamp,definition,
% 0.17/0.44 ( lamp
% 0.17/0.44 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.17/0.44 ( ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.17/0.44 => ( func @ A @ B
% 0.17/0.44 @ ( lam @ A @ B
% 0.17/0.44 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(lam2p_type,type,
% 0.17/0.44 lam2p: $o ).
% 0.17/0.44
% 0.17/0.44 thf(lam2p,definition,
% 0.17/0.44 ( lam2p
% 0.17/0.44 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.17/0.44 ( ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.17/0.44 => ( in
% 0.17/0.44 @ ( lam @ A @ B
% 0.17/0.44 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.17/0.44 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(brelnall1_type,type,
% 0.17/0.44 brelnall1: $o ).
% 0.17/0.44
% 0.17/0.44 thf(brelnall1,definition,
% 0.17/0.44 ( brelnall1
% 0.17/0.44 = ( ! [A: $i,B: $i,R: $i] :
% 0.17/0.44 ( ( breln @ A @ B @ R )
% 0.17/0.44 => ! [Xphi: $i > $o] :
% 0.17/0.44 ( ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ! [Xy: $i] :
% 0.17/0.44 ( ( in @ Xy @ B )
% 0.17/0.44 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.17/0.44 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.17/0.44 => ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ R )
% 0.17/0.44 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(brelnall2_type,type,
% 0.17/0.44 brelnall2: $o ).
% 0.17/0.44
% 0.17/0.44 thf(brelnall2,definition,
% 0.17/0.44 ( brelnall2
% 0.17/0.44 = ( ! [A: $i,B: $i,R: $i] :
% 0.17/0.44 ( ( breln @ A @ B @ R )
% 0.17/0.44 => ! [Xphi: $i > $o] :
% 0.17/0.44 ( ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ! [Xy: $i] :
% 0.17/0.44 ( ( in @ Xy @ B )
% 0.17/0.44 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.17/0.44 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.17/0.44 => ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ R )
% 0.17/0.44 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(ex1E2_type,type,
% 0.17/0.44 ex1E2: $o ).
% 0.17/0.44
% 0.17/0.44 thf(ex1E2,definition,
% 0.17/0.44 ( ex1E2
% 0.17/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.17/0.44 ( ( ex1 @ A
% 0.17/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.44 => ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ! [Xy: $i] :
% 0.17/0.44 ( ( in @ Xy @ A )
% 0.17/0.44 => ( ( Xphi @ Xx )
% 0.17/0.44 => ( ( Xphi @ Xy )
% 0.17/0.44 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(funcGraphProp1_type,type,
% 0.17/0.44 funcGraphProp1: $o ).
% 0.17/0.44
% 0.17/0.44 thf(funcGraphProp1,definition,
% 0.17/0.44 ( funcGraphProp1
% 0.17/0.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.17/0.44 ( ( func @ A @ B @ Xf )
% 0.17/0.44 => ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(funcGraphProp3_type,type,
% 0.17/0.44 funcGraphProp3: $o ).
% 0.17/0.44
% 0.17/0.44 thf(funcGraphProp3,definition,
% 0.17/0.44 ( funcGraphProp3
% 0.17/0.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.17/0.44 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.17/0.44 => ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(funcGraphProp2_type,type,
% 0.17/0.44 funcGraphProp2: $o ).
% 0.17/0.44
% 0.17/0.44 thf(funcGraphProp2,definition,
% 0.17/0.44 ( funcGraphProp2
% 0.17/0.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.17/0.44 ( ( func @ A @ B @ Xf )
% 0.17/0.44 => ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ! [Xy: $i] :
% 0.17/0.44 ( ( in @ Xy @ B )
% 0.17/0.44 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.17/0.44 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.17/0.44 = Xy ) ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(funcextLem_type,type,
% 0.17/0.44 funcextLem: $o ).
% 0.17/0.44
% 0.17/0.44 thf(funcextLem,definition,
% 0.17/0.44 ( funcextLem
% 0.17/0.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.17/0.44 ( ( func @ A @ B @ Xf )
% 0.17/0.44 => ! [Xg: $i] :
% 0.17/0.44 ( ( func @ A @ B @ Xg )
% 0.17/0.44 => ( ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.17/0.44 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.17/0.44 => ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ! [Xy: $i] :
% 0.17/0.44 ( ( in @ Xy @ B )
% 0.17/0.44 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 0.17/0.44 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(funcGraphProp4_type,type,
% 0.17/0.44 funcGraphProp4: $o ).
% 0.17/0.44
% 0.17/0.44 thf(funcGraphProp4,definition,
% 0.17/0.44 ( funcGraphProp4
% 0.17/0.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.17/0.44 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.17/0.44 => ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ! [Xy: $i] :
% 0.17/0.44 ( ( in @ Xy @ B )
% 0.17/0.44 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.17/0.44 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.17/0.44 = Xy ) ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(subbreln_type,type,
% 0.17/0.44 subbreln: $o ).
% 0.17/0.44
% 0.17/0.44 thf(subbreln,definition,
% 0.17/0.44 ( subbreln
% 0.17/0.44 = ( ! [A: $i,B: $i,R: $i] :
% 0.17/0.44 ( ( breln @ A @ B @ R )
% 0.17/0.44 => ! [S: $i] :
% 0.17/0.44 ( ( breln @ A @ B @ S )
% 0.17/0.44 => ( ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ! [Xy: $i] :
% 0.17/0.44 ( ( in @ Xy @ B )
% 0.17/0.44 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.17/0.44 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.17/0.44 => ( subset @ R @ S ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(eqbreln_type,type,
% 0.17/0.44 eqbreln: $o ).
% 0.17/0.44
% 0.17/0.44 thf(eqbreln,definition,
% 0.17/0.44 ( eqbreln
% 0.17/0.44 = ( ! [A: $i,B: $i,R: $i] :
% 0.17/0.44 ( ( breln @ A @ B @ R )
% 0.17/0.44 => ! [S: $i] :
% 0.17/0.44 ( ( breln @ A @ B @ S )
% 0.17/0.44 => ( ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ! [Xy: $i] :
% 0.17/0.44 ( ( in @ Xy @ B )
% 0.17/0.44 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.17/0.44 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.17/0.44 => ( ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ! [Xy: $i] :
% 0.17/0.44 ( ( in @ Xy @ B )
% 0.17/0.44 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.17/0.44 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 0.17/0.44 => ( R = S ) ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(funcext_type,type,
% 0.17/0.44 funcext: $o ).
% 0.17/0.44
% 0.17/0.44 thf(funcext,definition,
% 0.17/0.44 ( funcext
% 0.17/0.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.17/0.44 ( ( func @ A @ B @ Xf )
% 0.17/0.44 => ! [Xg: $i] :
% 0.17/0.44 ( ( func @ A @ B @ Xg )
% 0.17/0.44 => ( ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.17/0.44 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.17/0.44 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(funcext2_type,type,
% 0.17/0.44 funcext2: $o ).
% 0.17/0.44
% 0.17/0.44 thf(funcext2,definition,
% 0.17/0.44 ( funcext2
% 0.17/0.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.17/0.44 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.17/0.44 => ! [Xg: $i] :
% 0.17/0.44 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 0.17/0.44 => ( ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.17/0.44 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.17/0.44 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(ap2apEq1_type,type,
% 0.17/0.44 ap2apEq1: $o ).
% 0.17/0.44
% 0.17/0.44 thf(ap2apEq1,definition,
% 0.17/0.44 ( ap2apEq1
% 0.17/0.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.17/0.44 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.17/0.44 => ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.17/0.44 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(ap2apEq2_type,type,
% 0.17/0.44 ap2apEq2: $o ).
% 0.17/0.44
% 0.17/0.44 thf(ap2apEq2,definition,
% 0.17/0.44 ( ap2apEq2
% 0.17/0.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.17/0.44 ( ( func @ A @ B @ Xf )
% 0.17/0.44 => ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.17/0.44 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(beta1_type,type,
% 0.17/0.44 beta1: $o ).
% 0.17/0.44
% 0.17/0.44 thf(beta1,definition,
% 0.17/0.44 ( beta1
% 0.17/0.44 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.17/0.44 ( ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.17/0.44 => ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ( ( ap @ A @ B
% 0.17/0.44 @ ( lam @ A @ B
% 0.17/0.44 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.17/0.44 @ Xx )
% 0.17/0.44 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(eta1_type,type,
% 0.17/0.44 eta1: $o ).
% 0.17/0.44
% 0.17/0.44 thf(eta1,definition,
% 0.17/0.44 ( eta1
% 0.17/0.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.17/0.44 ( ( func @ A @ B @ Xf )
% 0.17/0.44 => ( ( lam @ A @ B
% 0.17/0.44 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.17/0.44 = Xf ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(lam2lamEq_type,type,
% 0.17/0.44 lam2lamEq: $o ).
% 0.17/0.44
% 0.17/0.44 thf(lam2lamEq,definition,
% 0.17/0.44 ( lam2lamEq
% 0.17/0.44 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.17/0.44 ( ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.17/0.44 => ( ( lam @ A @ B
% 0.17/0.44 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.17/0.44 = ( lam @ A @ B
% 0.17/0.44 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(beta2,conjecture,
% 0.17/0.44 ( setextAx
% 0.17/0.44 => ( emptysetAx
% 0.17/0.44 => ( setadjoinAx
% 0.17/0.44 => ( powersetAx
% 0.17/0.44 => ( setunionAx
% 0.17/0.44 => ( omega0Ax
% 0.17/0.44 => ( omegaSAx
% 0.17/0.44 => ( omegaIndAx
% 0.17/0.44 => ( replAx
% 0.17/0.44 => ( foundationAx
% 0.17/0.44 => ( wellorderingAx
% 0.17/0.44 => ( descrp
% 0.17/0.44 => ( dsetconstrI
% 0.17/0.44 => ( dsetconstrEL
% 0.17/0.44 => ( dsetconstrER
% 0.17/0.44 => ( exuE1
% 0.17/0.44 => ( prop2setE
% 0.17/0.44 => ( emptysetE
% 0.17/0.44 => ( emptysetimpfalse
% 0.17/0.44 => ( notinemptyset
% 0.17/0.44 => ( exuE3e
% 0.17/0.44 => ( setext
% 0.17/0.44 => ( emptyI
% 0.17/0.44 => ( noeltsimpempty
% 0.17/0.44 => ( setbeta
% 0.17/0.44 => ( nonemptyE1
% 0.17/0.44 => ( nonemptyI
% 0.17/0.44 => ( nonemptyI1
% 0.17/0.44 => ( setadjoinIL
% 0.17/0.44 => ( emptyinunitempty
% 0.17/0.44 => ( setadjoinIR
% 0.17/0.44 => ( setadjoinE
% 0.17/0.44 => ( setadjoinOr
% 0.17/0.44 => ( setoftrueEq
% 0.17/0.44 => ( powersetI
% 0.17/0.44 => ( emptyinPowerset
% 0.17/0.44 => ( emptyInPowerset
% 0.17/0.44 => ( powersetE
% 0.17/0.44 => ( setunionI
% 0.17/0.44 => ( setunionE
% 0.17/0.44 => ( subPowSU
% 0.17/0.44 => ( exuE2
% 0.17/0.44 => ( nonemptyImpWitness
% 0.17/0.44 => ( uniqinunit
% 0.17/0.44 => ( notinsingleton
% 0.17/0.44 => ( eqinunit
% 0.17/0.44 => ( singletonsswitch
% 0.17/0.44 => ( upairsetE
% 0.17/0.44 => ( upairsetIL
% 0.17/0.44 => ( upairsetIR
% 0.17/0.44 => ( emptyE1
% 0.17/0.44 => ( vacuousDall
% 0.17/0.44 => ( quantDeMorgan1
% 0.17/0.44 => ( quantDeMorgan2
% 0.17/0.44 => ( quantDeMorgan3
% 0.17/0.44 => ( quantDeMorgan4
% 0.17/0.44 => ( prop2setI
% 0.17/0.44 => ( prop2set2propI
% 0.17/0.44 => ( notdexE
% 0.17/0.44 => ( notdallE
% 0.17/0.44 => ( exuI1
% 0.17/0.44 => ( exuI3
% 0.17/0.44 => ( exuI2
% 0.17/0.44 => ( inCongP
% 0.17/0.44 => ( in__Cong
% 0.17/0.44 => ( exuE3u
% 0.17/0.44 => ( exu__Cong
% 0.17/0.44 => ( emptyset__Cong
% 0.17/0.44 => ( setadjoin__Cong
% 0.17/0.44 => ( powerset__Cong
% 0.17/0.44 => ( setunion__Cong
% 0.17/0.44 => ( omega__Cong
% 0.17/0.44 => ( exuEu
% 0.17/0.44 => ( descr__Cong
% 0.17/0.44 => ( dsetconstr__Cong
% 0.17/0.44 => ( subsetI1
% 0.17/0.44 => ( eqimpsubset2
% 0.17/0.44 => ( eqimpsubset1
% 0.17/0.44 => ( subsetI2
% 0.17/0.44 => ( emptysetsubset
% 0.17/0.44 => ( subsetE
% 0.17/0.44 => ( subsetE2
% 0.17/0.44 => ( notsubsetI
% 0.17/0.44 => ( notequalI1
% 0.17/0.44 => ( notequalI2
% 0.17/0.44 => ( subsetRefl
% 0.17/0.44 => ( subsetTrans
% 0.17/0.44 => ( setadjoinSub
% 0.17/0.44 => ( setadjoinSub2
% 0.17/0.44 => ( subset2powerset
% 0.17/0.44 => ( setextsub
% 0.17/0.44 => ( subsetemptysetimpeq
% 0.17/0.44 => ( powersetI1
% 0.17/0.44 => ( powersetE1
% 0.17/0.44 => ( inPowerset
% 0.17/0.44 => ( powersetsubset
% 0.17/0.44 => ( sepInPowerset
% 0.17/0.44 => ( sepSubset
% 0.17/0.44 => ( binunionIL
% 0.17/0.44 => ( upairset2IR
% 0.17/0.44 => ( binunionIR
% 0.17/0.44 => ( binunionEcases
% 0.17/0.44 => ( binunionE
% 0.17/0.44 => ( binunionLsub
% 0.17/0.44 => ( binunionRsub
% 0.17/0.44 => ( binintersectI
% 0.17/0.44 => ( binintersectSubset5
% 0.17/0.44 => ( binintersectEL
% 0.17/0.44 => ( binintersectLsub
% 0.17/0.44 => ( binintersectSubset2
% 0.17/0.44 => ( binintersectSubset3
% 0.17/0.44 => ( binintersectER
% 0.17/0.44 => ( disjointsetsI1
% 0.17/0.44 => ( binintersectRsub
% 0.17/0.44 => ( binintersectSubset4
% 0.17/0.44 => ( binintersectSubset1
% 0.17/0.44 => ( bs114d
% 0.17/0.44 => ( setminusI
% 0.17/0.44 => ( setminusEL
% 0.17/0.44 => ( setminusER
% 0.17/0.44 => ( setminusSubset2
% 0.17/0.44 => ( setminusERneg
% 0.17/0.44 => ( setminusELneg
% 0.17/0.44 => ( setminusILneg
% 0.17/0.44 => ( setminusIRneg
% 0.17/0.44 => ( setminusLsub
% 0.17/0.44 => ( setminusSubset1
% 0.17/0.44 => ( symdiffE
% 0.17/0.44 => ( symdiffI1
% 0.17/0.44 => ( symdiffI2
% 0.17/0.44 => ( symdiffIneg1
% 0.17/0.44 => ( symdiffIneg2
% 0.17/0.44 => ( secondinupair
% 0.17/0.44 => ( setukpairIL
% 0.17/0.44 => ( setukpairIR
% 0.17/0.44 => ( kpairiskpair
% 0.17/0.44 => ( kpairp
% 0.17/0.44 => ( singletonsubset
% 0.17/0.44 => ( singletoninpowerset
% 0.17/0.44 => ( singletoninpowunion
% 0.17/0.44 => ( upairset2E
% 0.17/0.44 => ( upairsubunion
% 0.17/0.44 => ( upairinpowunion
% 0.17/0.44 => ( ubforcartprodlem1
% 0.17/0.44 => ( ubforcartprodlem2
% 0.17/0.44 => ( ubforcartprodlem3
% 0.17/0.44 => ( cartprodpairin
% 0.17/0.44 => ( cartprodmempair1
% 0.17/0.44 => ( cartprodmempair
% 0.17/0.44 => ( setunionE2
% 0.17/0.44 => ( setunionsingleton1
% 0.17/0.44 => ( setunionsingleton2
% 0.17/0.44 => ( setunionsingleton
% 0.17/0.44 => ( singletonprop
% 0.17/0.44 => ( ex1E1
% 0.17/0.44 => ( ex1I
% 0.17/0.44 => ( ex1I2
% 0.17/0.44 => ( singletonsuniq
% 0.17/0.44 => ( setukpairinjL1
% 0.17/0.44 => ( kfstsingleton
% 0.17/0.44 => ( theprop
% 0.17/0.44 => ( kfstpairEq
% 0.17/0.44 => ( cartprodfstin
% 0.17/0.44 => ( setukpairinjL2
% 0.17/0.44 => ( setukpairinjL
% 0.17/0.44 => ( setukpairinjR11
% 0.17/0.44 => ( setukpairinjR12
% 0.17/0.44 => ( setukpairinjR1
% 0.17/0.44 => ( upairequniteq
% 0.17/0.44 => ( setukpairinjR2
% 0.17/0.44 => ( setukpairinjR
% 0.17/0.44 => ( ksndsingleton
% 0.17/0.44 => ( ksndpairEq
% 0.17/0.44 => ( kpairsurjEq
% 0.17/0.44 => ( cartprodsndin
% 0.17/0.44 => ( cartprodpairmemEL
% 0.17/0.44 => ( cartprodpairmemER
% 0.17/0.44 => ( cartprodmempaircEq
% 0.17/0.44 => ( cartprodfstpairEq
% 0.17/0.44 => ( cartprodsndpairEq
% 0.17/0.44 => ( cartprodpairsurjEq
% 0.17/0.44 => ( dpsetconstrI
% 0.17/0.44 => ( dpsetconstrSub
% 0.17/0.44 => ( setOfPairsIsBReln
% 0.17/0.44 => ( dpsetconstrERa
% 0.17/0.44 => ( dpsetconstrEL1
% 0.17/0.44 => ( dpsetconstrEL2
% 0.17/0.44 => ( dpsetconstrER
% 0.17/0.44 => ( funcImageSingleton
% 0.17/0.44 => ( apProp
% 0.17/0.44 => ( app
% 0.17/0.44 => ( infuncsetfunc
% 0.17/0.44 => ( ap2p
% 0.17/0.44 => ( funcinfuncset
% 0.17/0.44 => ( lamProp
% 0.17/0.44 => ( lamp
% 0.17/0.44 => ( lam2p
% 0.17/0.44 => ( brelnall1
% 0.17/0.44 => ( brelnall2
% 0.17/0.44 => ( ex1E2
% 0.17/0.44 => ( funcGraphProp1
% 0.17/0.44 => ( funcGraphProp3
% 0.17/0.44 => ( funcGraphProp2
% 0.17/0.44 => ( funcextLem
% 0.17/0.44 => ( funcGraphProp4
% 0.17/0.44 => ( subbreln
% 0.17/0.44 => ( eqbreln
% 0.17/0.44 => ( funcext
% 0.17/0.44 => ( funcext2
% 0.17/0.44 => ( ap2apEq1
% 0.17/0.44 => ( ap2apEq2
% 0.17/0.44 => ( beta1
% 0.17/0.44 => ( eta1
% 0.17/0.44 => ( lam2lamEq
% 0.17/0.44 => ! [A: $i,B: $i,Xf: $i > $i] :
% 0.17/0.44 ( ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.17/0.44 => ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ( ( ap @ A @ B
% 0.17/0.44 @ ( lam @ A @ B
% 0.17/0.44 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.17/0.44 @ Xx )
% 0.17/0.44 = ( Xf @ Xx ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 %------------------------------------------------------------------------------
% 0.17/0.44 ------------------------
% 2.44/1.48 ---- Embedded in THF ---
% 2.44/1.48 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.44/1.48 %%% Generated on Tue Feb 24 12:32:06 EST 2026
% 2.44/1.48 %%% using '$$dhol' embedding, version 1.3.0.
% 2.44/1.48 %%% Logic specification used:
% 2.44/1.48 %%% thf(spec, logic, $$dhol).
% 2.44/1.48
% 2.44/1.48 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.9FuAMmeF6I/DTF2DTF3007.p
% See solution above
% 2.44/1.48 ------------------------
% 2.62/1.51 ---- Cleaned THF ---
% 2.62/1.51 thf(in_type,type,
% 2.62/1.51 in: $i > $i > $o ).
% 2.62/1.51
% 2.62/1.51 thf(exu_type,type,
% 2.62/1.51 exu: ( $i > $o ) > $o ).
% 2.62/1.51
% 2.62/1.51 thf(exu,definition,
% 2.62/1.51 ( exu
% 2.62/1.51 = ( ^ [Xphi: $i > $o] :
% 2.62/1.51 ? [Xx: $i] :
% 2.62/1.51 ( ( Xphi @ Xx )
% 2.62/1.51 & ! [Xy: $i] :
% 2.62/1.51 ( ( Xphi @ Xy )
% 2.62/1.51 => ( Xx = Xy ) ) ) ) ) ).
% 2.62/1.51
% 2.62/1.51 thf(setextAx_type,type,
% 2.62/1.51 setextAx: $o ).
% 2.62/1.51
% 2.62/1.51 thf(setextAx,definition,
% 2.62/1.51 ( setextAx
% 2.62/1.51 = ( ! [A: $i,B: $i] :
% 2.62/1.51 ( ! [Xx: $i] :
% 2.62/1.51 ( ( in @ Xx @ A )
% 2.62/1.51 <=> ( in @ Xx @ B ) )
% 2.62/1.51 => ( A = B ) ) ) ) ).
% 2.62/1.51
% 2.62/1.51 thf(emptyset_type,type,
% 2.62/1.51 emptyset: $i ).
% 2.62/1.51
% 2.62/1.51 thf(emptysetAx_type,type,
% 2.62/1.51 emptysetAx: $o ).
% 2.62/1.51
% 2.62/1.51 thf(emptysetAx,definition,
% 2.62/1.51 ( emptysetAx
% 2.62/1.51 = ( ! [Xx: $i] :
% 2.62/1.51 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.62/1.51
% 2.62/1.51 thf(setadjoin_type,type,
% 2.62/1.51 setadjoin: $i > $i > $i ).
% 2.62/1.51
% 2.62/1.51 thf(setadjoinAx_type,type,
% 2.62/1.51 setadjoinAx: $o ).
% 2.62/1.51
% 2.62/1.51 thf(setadjoinAx,definition,
% 2.62/1.51 ( setadjoinAx
% 2.62/1.51 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.62/1.51 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.62/1.51 <=> ( ( Xy = Xx )
% 2.62/1.51 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.62/1.51
% 2.62/1.51 thf(powerset_type,type,
% 2.62/1.51 powerset: $i > $i ).
% 2.62/1.51
% 2.62/1.51 thf(powersetAx_type,type,
% 2.62/1.51 powersetAx: $o ).
% 2.62/1.51
% 2.62/1.51 thf(powersetAx,definition,
% 2.62/1.51 ( powersetAx
% 2.62/1.51 = ( ! [A: $i,B: $i] :
% 2.62/1.51 ( ( in @ B @ ( powerset @ A ) )
% 2.62/1.51 <=> ! [Xx: $i] :
% 2.62/1.51 ( ( in @ Xx @ B )
% 2.62/1.51 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.62/1.51
% 2.62/1.51 thf(setunion_type,type,
% 2.62/1.51 setunion: $i > $i ).
% 2.62/1.51
% 2.62/1.51 thf(setunionAx_type,type,
% 2.62/1.51 setunionAx: $o ).
% 2.62/1.51
% 2.62/1.51 thf(setunionAx,definition,
% 2.62/1.51 ( setunionAx
% 2.62/1.51 = ( ! [A: $i,Xx: $i] :
% 2.62/1.51 ( ( in @ Xx @ ( setunion @ A ) )
% 2.62/1.51 <=> ? [B: $i] :
% 2.62/1.51 ( ( in @ Xx @ B )
% 2.62/1.51 & ( in @ B @ A ) ) ) ) ) ).
% 2.62/1.51
% 2.62/1.51 thf(omega_type,type,
% 2.62/1.51 omega: $i ).
% 2.62/1.51
% 2.62/1.51 thf(omega0Ax_type,type,
% 2.62/1.51 omega0Ax: $o ).
% 2.62/1.51
% 2.62/1.51 thf(omega0Ax,definition,
% 2.62/1.51 ( omega0Ax
% 2.62/1.51 = ( in @ emptyset @ omega ) ) ).
% 2.62/1.51
% 2.62/1.51 thf(omegaSAx_type,type,
% 2.62/1.51 omegaSAx: $o ).
% 2.62/1.51
% 2.62/1.51 thf(omegaSAx,definition,
% 2.62/1.51 ( omegaSAx
% 2.62/1.51 = ( ! [Xx: $i] :
% 2.62/1.51 ( ( in @ Xx @ omega )
% 2.62/1.51 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 2.62/1.51
% 2.62/1.51 thf(omegaIndAx_type,type,
% 2.62/1.51 omegaIndAx: $o ).
% 2.62/1.51
% 2.62/1.51 thf(omegaIndAx,definition,
% 2.62/1.51 ( omegaIndAx
% 2.62/1.51 = ( ! [A: $i] :
% 2.62/1.51 ( ( ( in @ emptyset @ A )
% 2.62/1.51 & ! [Xx: $i] :
% 2.62/1.51 ( ( ( in @ Xx @ omega )
% 2.62/1.51 & ( in @ Xx @ A ) )
% 2.62/1.51 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 2.62/1.51 => ! [Xx: $i] :
% 2.62/1.51 ( ( in @ Xx @ omega )
% 2.62/1.51 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.62/1.51
% 2.62/1.51 thf(replAx_type,type,
% 2.62/1.51 replAx: $o ).
% 2.62/1.51
% 2.62/1.51 thf(replAx,definition,
% 2.62/1.51 ( replAx
% 2.62/1.51 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 2.62/1.51 ( ! [Xx: $i] :
% 2.62/1.51 ( ( in @ Xx @ A )
% 2.62/1.51 => ( exu
% 2.62/1.51 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 2.62/1.51 => ? [B: $i] :
% 2.62/1.51 ! [Xx: $i] :
% 2.62/1.51 ( ( in @ Xx @ B )
% 2.62/1.51 <=> ? [Xy: $i] :
% 2.62/1.51 ( ( in @ Xy @ A )
% 2.62/1.51 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 2.62/1.51
% 2.62/1.51 thf(foundationAx_type,type,
% 2.62/1.51 foundationAx: $o ).
% 2.62/1.51
% 2.62/1.51 thf(foundationAx,definition,
% 2.62/1.51 ( foundationAx
% 2.62/1.51 = ( ! [A: $i] :
% 2.62/1.51 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.62/1.51 => ? [B: $i] :
% 2.62/1.51 ( ( in @ B @ A )
% 2.62/1.51 & ~ ? [Xx: $i] :
% 2.62/1.51 ( ( in @ Xx @ B )
% 2.62/1.51 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 2.62/1.51
% 2.62/1.51 thf(wellorderingAx_type,type,
% 2.62/1.51 wellorderingAx: $o ).
% 2.62/1.51
% 2.62/1.51 thf(wellorderingAx,definition,
% 2.62/1.51 ( wellorderingAx
% 2.62/1.51 = ( ! [A: $i] :
% 2.62/1.51 ? [B: $i] :
% 2.62/1.51 ( ! [C: $i] :
% 2.62/1.51 ( ( in @ C @ B )
% 2.62/1.51 => ! [Xx: $i] :
% 2.62/1.51 ( ( in @ Xx @ C )
% 2.62/1.51 => ( in @ Xx @ A ) ) )
% 2.62/1.51 & ! [Xx: $i,Xy: $i] :
% 2.62/1.51 ( ( ( in @ Xx @ A )
% 2.62/1.51 & ( in @ Xy @ A ) )
% 2.62/1.51 => ( ! [C: $i] :
% 2.62/1.51 ( ( in @ C @ B )
% 2.62/1.51 => ( ( in @ Xx @ C )
% 2.62/1.51 <=> ( in @ Xy @ C ) ) )
% 2.62/1.51 => ( Xx = Xy ) ) )
% 2.62/1.51 & ! [C: $i,D: $i] :
% 2.62/1.51 ( ( ( in @ C @ B )
% 2.62/1.51 & ( in @ D @ B ) )
% 2.62/1.51 => ( ! [Xx: $i] :
% 2.62/1.51 ( ( in @ Xx @ C )
% 2.62/1.51 => ( in @ Xx @ D ) )
% 2.62/1.51 | ! [Xx: $i] :
% 2.62/1.51 ( ( in @ Xx @ D )
% 2.62/1.51 => ( in @ Xx @ C ) ) ) )
% 2.62/1.51 & ! [C: $i] :
% 2.62/1.51 ( ( ! [Xx: $i] :
% 2.62/1.51 ( ( in @ Xx @ C )
% 2.62/1.51 => ( in @ Xx @ A ) )
% 2.62/1.51 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 2.62/1.51 => ? [D: $i,Xx: $i] :
% 2.62/1.51 ( ( in @ D @ B )
% 2.62/1.51 & ( in @ Xx @ C )
% 2.62/1.51 & ~ ? [Xy: $i] :
% 2.62/1.51 ( ( in @ Xy @ D )
% 2.62/1.51 & ( in @ Xy @ C ) )
% 2.62/1.51 & ! [E: $i] :
% 2.62/1.51 ( ( in @ E @ B )
% 2.62/1.51 => ( ! [Xy: $i] :
% 2.62/1.51 ( ( in @ Xy @ E )
% 2.62/1.51 => ( in @ Xy @ D ) )
% 2.62/1.51 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 2.62/1.51
% 2.62/1.51 thf(descr_type,type,
% 2.62/1.51 descr: ( $i > $o ) > $i ).
% 2.62/1.51
% 2.62/1.51 thf(descrp_type,type,
% 2.62/1.51 descrp: $o ).
% 2.62/1.51
% 2.62/1.51 thf(descrp,definition,
% 2.62/1.51 ( descrp
% 2.62/1.51 = ( ! [Xphi: $i > $o] :
% 2.62/1.51 ( ( exu
% 2.62/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.62/1.51 => ( Xphi
% 2.62/1.51 @ ( descr
% 2.62/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.62/1.51
% 2.62/1.51 thf(dsetconstr_type,type,
% 2.62/1.51 dsetconstr: $i > ( $i > $o ) > $i ).
% 2.62/1.51
% 2.62/1.51 thf(dsetconstrI_type,type,
% 2.62/1.51 dsetconstrI: $o ).
% 2.62/1.51
% 2.62/1.51 thf(dsetconstrI,definition,
% 2.62/1.51 ( dsetconstrI
% 2.62/1.51 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.62/1.51 ( ( in @ Xx @ A )
% 2.62/1.51 => ( ( Xphi @ Xx )
% 2.62/1.51 => ( in @ Xx
% 2.62/1.51 @ ( dsetconstr @ A
% 2.62/1.51 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.62/1.51
% 2.62/1.51 thf(dsetconstrEL_type,type,
% 2.62/1.51 dsetconstrEL: $o ).
% 2.62/1.51
% 2.62/1.51 thf(dsetconstrEL,definition,
% 2.62/1.51 ( dsetconstrEL
% 2.62/1.51 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.62/1.51 ( ( in @ Xx
% 2.62/1.51 @ ( dsetconstr @ A
% 2.62/1.51 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.62/1.51 => ( in @ Xx @ A ) ) ) ) ).
% 2.62/1.51
% 2.62/1.51 thf(dsetconstrER_type,type,
% 2.62/1.51 dsetconstrER: $o ).
% 2.62/1.51
% 2.62/1.51 thf(dsetconstrER,definition,
% 2.62/1.51 ( dsetconstrER
% 2.62/1.51 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.62/1.51 ( ( in @ Xx
% 2.62/1.51 @ ( dsetconstr @ A
% 2.62/1.51 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.62/1.51 => ( Xphi @ Xx ) ) ) ) ).
% 2.62/1.51
% 2.62/1.51 thf(exuE1_type,type,
% 2.62/1.51 exuE1: $o ).
% 2.62/1.51
% 2.62/1.51 thf(exuE1,definition,
% 2.62/1.51 ( exuE1
% 2.62/1.51 = ( ! [Xphi: $i > $o] :
% 2.62/1.51 ( ( exu
% 2.62/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.62/1.51 => ? [Xx: $i] :
% 2.62/1.51 ( ( Xphi @ Xx )
% 2.62/1.51 & ! [Xy: $i] :
% 2.62/1.51 ( ( Xphi @ Xy )
% 2.62/1.51 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.62/1.51
% 2.62/1.51 thf(prop2set_type,type,
% 2.62/1.51 prop2set: $o > $i ).
% 2.62/1.51
% 2.62/1.51 thf(prop2setE_type,type,
% 2.62/1.51 prop2setE: $o ).
% 2.62/1.51
% 2.62/1.51 thf(prop2setE,definition,
% 2.62/1.51 ( prop2setE
% 2.62/1.51 = ( ! [Xphi: $o,Xx: $i] :
% 2.62/1.51 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 2.62/1.51 => Xphi ) ) ) ).
% 2.62/1.51
% 2.62/1.51 thf(emptysetE_type,type,
% 2.62/1.51 emptysetE: $o ).
% 2.62/1.51
% 2.62/1.51 thf(emptysetE,definition,
% 2.62/1.51 ( emptysetE
% 2.62/1.51 = ( ! [Xx: $i] :
% 2.62/1.51 ( ( in @ Xx @ emptyset )
% 2.62/1.51 => ! [Xphi: $o] : Xphi ) ) ) ).
% 2.62/1.51
% 2.62/1.51 thf(emptysetimpfalse_type,type,
% 2.62/1.51 emptysetimpfalse: $o ).
% 2.62/1.51
% 2.62/1.51 thf(emptysetimpfalse,definition,
% 2.62/1.51 ( emptysetimpfalse
% 2.62/1.51 = ( ! [Xx: $i] :
% 2.62/1.51 ( ( in @ Xx @ emptyset )
% 2.62/1.51 => $false ) ) ) ).
% 2.62/1.51
% 2.62/1.51 thf(notinemptyset_type,type,
% 2.62/1.51 notinemptyset: $o ).
% 2.62/1.51
% 2.62/1.51 thf(notinemptyset,definition,
% 2.62/1.51 ( notinemptyset
% 2.62/1.51 = ( ! [Xx: $i] :
% 2.62/1.51 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.62/1.51
% 2.62/1.51 thf(exuE3e_type,type,
% 2.62/1.51 exuE3e: $o ).
% 2.62/1.51
% 2.62/1.51 thf(exuE3e,definition,
% 2.62/1.51 ( exuE3e
% 2.62/1.51 = ( ! [Xphi: $i > $o] :
% 2.62/1.51 ( ( exu
% 2.62/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.62/1.51 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 2.62/1.51
% 2.62/1.51 thf(setext_type,type,
% 2.62/1.51 setext: $o ).
% 2.62/1.51
% 2.62/1.51 thf(setext,definition,
% 2.62/1.51 ( setext
% 2.62/1.51 = ( ! [A: $i,B: $i] :
% 2.62/1.51 ( ! [Xx: $i] :
% 2.62/1.51 ( ( in @ Xx @ A )
% 2.62/1.51 => ( in @ Xx @ B ) )
% 2.62/1.51 => ( ! [Xx: $i] :
% 2.62/1.51 ( ( in @ Xx @ B )
% 2.62/1.51 => ( in @ Xx @ A ) )
% 2.62/1.51 => ( A = B ) ) ) ) ) ).
% 2.62/1.51
% 2.62/1.51 thf(emptyI_type,type,
% 2.62/1.51 emptyI: $o ).
% 2.62/1.51
% 2.62/1.51 thf(emptyI,definition,
% 2.62/1.51 ( emptyI
% 2.62/1.51 = ( ! [A: $i] :
% 2.62/1.51 ( ! [Xx: $i] :
% 2.62/1.51 ~ ( in @ Xx @ A )
% 2.62/1.51 => ( A = emptyset ) ) ) ) ).
% 2.62/1.51
% 2.62/1.51 thf(noeltsimpempty_type,type,
% 2.62/1.51 noeltsimpempty: $o ).
% 2.62/1.51
% 2.62/1.51 thf(noeltsimpempty,definition,
% 2.62/1.51 ( noeltsimpempty
% 2.62/1.51 = ( ! [A: $i] :
% 2.62/1.51 ( ! [Xx: $i] :
% 2.62/1.51 ~ ( in @ Xx @ A )
% 2.62/1.51 => ( A = emptyset ) ) ) ) ).
% 2.62/1.51
% 2.62/1.51 thf(setbeta_type,type,
% 2.62/1.51 setbeta: $o ).
% 2.62/1.51
% 2.62/1.51 thf(setbeta,definition,
% 2.62/1.51 ( setbeta
% 2.62/1.51 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.62/1.51 ( ( in @ Xx @ A )
% 2.62/1.51 => ( ( in @ Xx
% 2.62/1.51 @ ( dsetconstr @ A
% 2.62/1.51 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.62/1.51 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 2.62/1.51
% 2.62/1.51 thf(nonempty_type,type,
% 2.62/1.51 nonempty: $i > $o ).
% 2.62/1.51
% 2.62/1.51 thf(nonempty,definition,
% 2.62/1.51 ( nonempty
% 2.62/1.51 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 2.62/1.51
% 2.62/1.51 thf(nonemptyE1_type,type,
% 2.62/1.51 nonemptyE1: $o ).
% 2.62/1.51
% 2.62/1.51 thf(nonemptyE1,definition,
% 2.62/1.51 ( nonemptyE1
% 2.62/1.51 = ( ! [A: $i] :
% 2.62/1.51 ( ( nonempty @ A )
% 2.62/1.51 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 2.62/1.51
% 2.62/1.51 thf(nonemptyI_type,type,
% 2.62/1.51 nonemptyI: $o ).
% 2.62/1.51
% 2.62/1.51 thf(nonemptyI,definition,
% 2.62/1.51 ( nonemptyI
% 2.62/1.51 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.62/1.51 ( ( in @ Xx @ A )
% 2.62/1.51 => ( ( Xphi @ Xx )
% 2.62/1.51 => ( nonempty
% 2.62/1.51 @ ( dsetconstr @ A
% 2.62/1.51 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.62/1.51
% 2.62/1.51 thf(nonemptyI1_type,type,
% 2.62/1.51 nonemptyI1: $o ).
% 2.62/1.51
% 2.62/1.51 thf(nonemptyI1,definition,
% 2.62/1.51 ( nonemptyI1
% 2.62/1.51 = ( ! [A: $i] :
% 2.62/1.51 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.62/1.51 => ( nonempty @ A ) ) ) ) ).
% 2.62/1.51
% 2.62/1.51 thf(setadjoinIL_type,type,
% 2.62/1.51 setadjoinIL: $o ).
% 2.62/1.51
% 2.62/1.51 thf(setadjoinIL,definition,
% 2.62/1.51 ( setadjoinIL
% 2.62/1.51 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 2.62/1.51
% 2.62/1.51 thf(emptyinunitempty_type,type,
% 2.62/1.51 emptyinunitempty: $o ).
% 2.62/1.51
% 2.62/1.51 thf(emptyinunitempty,definition,
% 2.62/1.51 ( emptyinunitempty
% 2.62/1.51 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 2.62/1.51
% 2.62/1.51 thf(setadjoinIR_type,type,
% 2.62/1.51 setadjoinIR: $o ).
% 2.62/1.51
% 2.62/1.51 thf(setadjoinIR,definition,
% 2.62/1.51 ( setadjoinIR
% 2.62/1.51 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.62/1.51 ( ( in @ Xy @ A )
% 2.62/1.51 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 2.62/1.51
% 2.62/1.51 thf(setadjoinE_type,type,
% 2.62/1.51 setadjoinE: $o ).
% 2.62/1.51
% 2.62/1.51 thf(setadjoinE,definition,
% 2.62/1.51 ( setadjoinE
% 2.62/1.51 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.62/1.51 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.62/1.51 => ! [Xphi: $o] :
% 2.62/1.51 ( ( ( Xy = Xx )
% 2.62/1.51 => Xphi )
% 2.62/1.51 => ( ( ( in @ Xy @ A )
% 2.62/1.51 => Xphi )
% 2.62/1.51 => Xphi ) ) ) ) ) ).
% 2.62/1.51
% 2.62/1.51 thf(setadjoinOr_type,type,
% 2.62/1.51 setadjoinOr: $o ).
% 2.62/1.51
% 2.62/1.51 thf(setadjoinOr,definition,
% 2.62/1.51 ( setadjoinOr
% 2.62/1.51 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.62/1.51 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.62/1.51 => ( ( Xy = Xx )
% 2.62/1.51 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.62/1.51
% 2.62/1.51 thf(setoftrueEq_type,type,
% 2.62/1.51 setoftrueEq: $o ).
% 2.62/1.51
% 2.62/1.51 thf(setoftrueEq,definition,
% 2.62/1.51 ( setoftrueEq
% 2.62/1.51 = ( ! [A: $i] :
% 2.62/1.51 ( ( dsetconstr @ A
% 2.62/1.51 @ ^ [Xx: $i] : $true )
% 2.62/1.51 = A ) ) ) ).
% 2.62/1.51
% 2.62/1.51 thf(powersetI_type,type,
% 2.62/1.51 powersetI: $o ).
% 2.62/1.51
% 2.62/1.51 thf(powersetI,definition,
% 2.62/1.51 ( powersetI
% 2.62/1.51 = ( ! [A: $i,B: $i] :
% 2.62/1.51 ( ! [Xx: $i] :
% 2.62/1.51 ( ( in @ Xx @ B )
% 2.62/1.51 => ( in @ Xx @ A ) )
% 2.62/1.51 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.62/1.51
% 2.62/1.51 thf(emptyinPowerset_type,type,
% 2.62/1.51 emptyinPowerset: $o ).
% 2.62/1.51
% 2.62/1.51 thf(emptyinPowerset,definition,
% 2.62/1.51 ( emptyinPowerset
% 2.62/1.51 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.62/1.51
% 2.62/1.51 thf(emptyInPowerset_type,type,
% 2.62/1.51 emptyInPowerset: $o ).
% 2.62/1.51
% 2.62/1.51 thf(emptyInPowerset,definition,
% 2.62/1.51 ( emptyInPowerset
% 2.62/1.51 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.62/1.51
% 2.62/1.51 thf(powersetE_type,type,
% 2.62/1.51 powersetE: $o ).
% 2.62/1.51
% 2.62/1.51 thf(powersetE,definition,
% 2.62/1.51 ( powersetE
% 2.62/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.62/1.51 ( ( in @ B @ ( powerset @ A ) )
% 2.62/1.51 => ( ( in @ Xx @ B )
% 2.62/1.51 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.62/1.51
% 2.62/1.51 thf(setunionI_type,type,
% 2.62/1.51 setunionI: $o ).
% 2.62/1.51
% 2.62/1.51 thf(setunionI,definition,
% 2.62/1.51 ( setunionI
% 2.62/1.51 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.62/1.51 ( ( in @ Xx @ B )
% 2.62/1.51 => ( ( in @ B @ A )
% 2.62/1.51 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 2.62/1.51
% 2.62/1.51 thf(setunionE_type,type,
% 2.62/1.51 setunionE: $o ).
% 2.62/1.51
% 2.62/1.51 thf(setunionE,definition,
% 2.62/1.51 ( setunionE
% 2.62/1.51 = ( ! [A: $i,Xx: $i] :
% 2.62/1.51 ( ( in @ Xx @ ( setunion @ A ) )
% 2.62/1.51 => ! [Xphi: $o] :
% 2.62/1.51 ( ! [B: $i] :
% 2.62/1.51 ( ( in @ Xx @ B )
% 2.62/1.51 => ( ( in @ B @ A )
% 2.62/1.51 => Xphi ) )
% 2.62/1.51 => Xphi ) ) ) ) ).
% 2.62/1.51
% 2.62/1.51 thf(subPowSU_type,type,
% 2.62/1.51 subPowSU: $o ).
% 2.62/1.51
% 2.62/1.51 thf(subPowSU,definition,
% 2.62/1.51 ( subPowSU
% 2.62/1.51 = ( ! [A: $i,Xx: $i] :
% 2.62/1.51 ( ( in @ Xx @ A )
% 2.62/1.51 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 2.62/1.51
% 2.62/1.51 thf(exuE2_type,type,
% 2.62/1.51 exuE2: $o ).
% 2.62/1.51
% 2.62/1.51 thf(exuE2,definition,
% 2.62/1.51 ( exuE2
% 2.62/1.51 = ( ! [Xphi: $i > $o] :
% 2.62/1.51 ( ( exu
% 2.62/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.62/1.51 => ? [Xx: $i] :
% 2.62/1.51 ! [Xy: $i] :
% 2.62/1.51 ( ( Xphi @ Xy )
% 2.62/1.51 <=> ( Xy = Xx ) ) ) ) ) ).
% 2.62/1.51
% 2.62/1.51 thf(nonemptyImpWitness_type,type,
% 2.62/1.51 nonemptyImpWitness: $o ).
% 2.62/1.51
% 2.62/1.51 thf(nonemptyImpWitness,definition,
% 2.62/1.51 ( nonemptyImpWitness
% 2.62/1.51 = ( ! [A: $i] :
% 2.62/1.51 ( ( nonempty @ A )
% 2.62/1.51 => ? [Xx: $i] :
% 2.62/1.51 ( ( in @ Xx @ A )
% 2.62/1.51 & $true ) ) ) ) ).
% 2.62/1.51
% 2.62/1.51 thf(uniqinunit_type,type,
% 2.62/1.51 uniqinunit: $o ).
% 2.62/1.51
% 2.62/1.51 thf(uniqinunit,definition,
% 2.62/1.51 ( uniqinunit
% 2.62/1.51 = ( ! [Xx: $i,Xy: $i] :
% 2.62/1.51 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.62/1.51 => ( Xx = Xy ) ) ) ) ).
% 2.62/1.51
% 2.62/1.51 thf(notinsingleton_type,type,
% 2.62/1.51 notinsingleton: $o ).
% 2.62/1.51
% 2.62/1.51 thf(notinsingleton,definition,
% 2.62/1.51 ( notinsingleton
% 2.62/1.51 = ( ! [Xx: $i,Xy: $i] :
% 2.62/1.51 ( ( Xx != Xy )
% 2.62/1.51 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.62/1.51
% 2.62/1.51 thf(eqinunit_type,type,
% 2.62/1.51 eqinunit: $o ).
% 2.62/1.51
% 2.62/1.51 thf(eqinunit,definition,
% 2.62/1.51 ( eqinunit
% 2.62/1.51 = ( ! [Xx: $i,Xy: $i] :
% 2.62/1.51 ( ( Xx = Xy )
% 2.62/1.51 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.62/1.51
% 2.62/1.51 thf(singletonsswitch_type,type,
% 2.62/1.51 singletonsswitch: $o ).
% 2.62/1.51
% 2.62/1.51 thf(singletonsswitch,definition,
% 2.62/1.51 ( singletonsswitch
% 2.62/1.51 = ( ! [Xx: $i,Xy: $i] :
% 2.62/1.51 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.62/1.51 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.62/1.51
% 2.62/1.51 thf(upairsetE_type,type,
% 2.62/1.51 upairsetE: $o ).
% 2.62/1.51
% 2.62/1.51 thf(upairsetE,definition,
% 2.62/1.51 ( upairsetE
% 2.62/1.51 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.62/1.51 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.62/1.51 => ( ( Xz = Xx )
% 2.62/1.51 | ( Xz = Xy ) ) ) ) ) ).
% 2.62/1.51
% 2.62/1.51 thf(upairsetIL_type,type,
% 2.62/1.51 upairsetIL: $o ).
% 2.62/1.51
% 2.62/1.51 thf(upairsetIL,definition,
% 2.62/1.51 ( upairsetIL
% 2.62/1.51 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.62/1.51
% 2.62/1.51 thf(upairsetIR_type,type,
% 2.62/1.51 upairsetIR: $o ).
% 2.62/1.51
% 2.62/1.51 thf(upairsetIR,definition,
% 2.62/1.51 ( upairsetIR
% 2.62/1.51 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.62/1.51
% 2.62/1.51 thf(emptyE1_type,type,
% 2.62/1.51 emptyE1: $o ).
% 2.62/1.51
% 2.62/1.51 thf(emptyE1,definition,
% 2.62/1.51 ( emptyE1
% 2.62/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 2.62/1.51 ( ? [Xx: $i] :
% 2.62/1.51 ( ( in @ Xx @ A )
% 2.62/1.51 & ( Xphi @ Xx ) )
% 2.62/1.51 => ( ( ( dsetconstr @ A
% 2.62/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.62/1.51 = emptyset )
% 2.62/1.51 => $false ) ) ) ) ).
% 2.62/1.51
% 2.62/1.51 thf(vacuousDall_type,type,
% 2.62/1.51 vacuousDall: $o ).
% 2.62/1.51
% 2.62/1.51 thf(vacuousDall,definition,
% 2.62/1.51 ( vacuousDall
% 2.62/1.51 = ( ! [Xphi: $i > $o,Xx: $i] :
% 2.62/1.51 ( ( in @ Xx @ emptyset )
% 2.62/1.51 => ( Xphi @ Xx ) ) ) ) ).
% 2.62/1.51
% 2.62/1.51 thf(quantDeMorgan1_type,type,
% 2.62/1.51 quantDeMorgan1: $o ).
% 2.62/1.51
% 2.62/1.51 thf(quantDeMorgan1,definition,
% 2.62/1.51 ( quantDeMorgan1
% 2.62/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 2.62/1.51 ( ~ ! [Xx: $i] :
% 2.62/1.51 ( ( in @ Xx @ A )
% 2.62/1.51 => ( Xphi @ Xx ) )
% 2.62/1.51 => ? [Xx: $i] :
% 2.62/1.51 ( ( in @ Xx @ A )
% 2.62/1.51 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.62/1.51
% 2.62/1.51 thf(quantDeMorgan2_type,type,
% 2.62/1.51 quantDeMorgan2: $o ).
% 2.62/1.51
% 2.62/1.51 thf(quantDeMorgan2,definition,
% 2.62/1.51 ( quantDeMorgan2
% 2.62/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 2.62/1.51 ( ! [Xx: $i] :
% 2.62/1.51 ( ( in @ Xx @ A )
% 2.62/1.51 => ~ ( Xphi @ Xx ) )
% 2.62/1.51 => ~ ? [Xx: $i] :
% 2.62/1.51 ( ( in @ Xx @ A )
% 2.62/1.51 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.62/1.51
% 2.62/1.51 thf(quantDeMorgan3_type,type,
% 2.62/1.51 quantDeMorgan3: $o ).
% 2.62/1.51
% 2.62/1.51 thf(quantDeMorgan3,definition,
% 2.62/1.51 ( quantDeMorgan3
% 2.62/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 2.62/1.51 ( ~ ? [Xx: $i] :
% 2.62/1.51 ( ( in @ Xx @ A )
% 2.62/1.51 & ( Xphi @ Xx ) )
% 2.62/1.51 => ! [Xx: $i] :
% 2.62/1.51 ( ( in @ Xx @ A )
% 2.62/1.51 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.62/1.51
% 2.62/1.51 thf(quantDeMorgan4_type,type,
% 2.62/1.51 quantDeMorgan4: $o ).
% 2.62/1.51
% 2.62/1.51 thf(quantDeMorgan4,definition,
% 2.62/1.51 ( quantDeMorgan4
% 2.62/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 2.62/1.51 ( ? [Xx: $i] :
% 2.62/1.51 ( ( in @ Xx @ A )
% 2.62/1.51 & ~ ( Xphi @ Xx ) )
% 2.62/1.51 => ~ ! [Xx: $i] :
% 2.62/1.51 ( ( in @ Xx @ A )
% 2.62/1.51 => ( Xphi @ Xx ) ) ) ) ) ).
% 2.62/1.51
% 2.62/1.51 thf(prop2setI_type,type,
% 2.62/1.51 prop2setI: $o ).
% 2.62/1.51
% 2.62/1.51 thf(prop2setI,definition,
% 2.62/1.52 ( prop2setI
% 2.62/1.52 = ( ! [Xphi: $o] :
% 2.62/1.52 ( Xphi
% 2.62/1.52 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(set2prop_type,type,
% 2.62/1.52 set2prop: $i > $o ).
% 2.62/1.52
% 2.62/1.52 thf(prop2set2propI_type,type,
% 2.62/1.52 prop2set2propI: $o ).
% 2.62/1.52
% 2.62/1.52 thf(prop2set2propI,definition,
% 2.62/1.52 ( prop2set2propI
% 2.62/1.52 = ( ! [Xphi: $o] :
% 2.62/1.52 ( Xphi
% 2.62/1.52 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(notdexE_type,type,
% 2.62/1.52 notdexE: $o ).
% 2.62/1.52
% 2.62/1.52 thf(notdexE,definition,
% 2.62/1.52 ( notdexE
% 2.62/1.52 = ( ! [A: $i,Xphi: $i > $o] :
% 2.62/1.52 ( ~ ? [Xx: $i] :
% 2.62/1.52 ( ( in @ Xx @ A )
% 2.62/1.52 & ( Xphi @ Xx ) )
% 2.62/1.52 => ! [Xx: $i] :
% 2.62/1.52 ( ( in @ Xx @ A )
% 2.62/1.52 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(notdallE_type,type,
% 2.62/1.52 notdallE: $o ).
% 2.62/1.52
% 2.62/1.52 thf(notdallE,definition,
% 2.62/1.52 ( notdallE
% 2.62/1.52 = ( ! [A: $i,Xphi: $i > $o] :
% 2.62/1.52 ( ~ ! [Xx: $i] :
% 2.62/1.52 ( ( in @ Xx @ A )
% 2.62/1.52 => ( Xphi @ Xx ) )
% 2.62/1.52 => ? [Xx: $i] :
% 2.62/1.52 ( ( in @ Xx @ A )
% 2.62/1.52 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(exuI1_type,type,
% 2.62/1.52 exuI1: $o ).
% 2.62/1.52
% 2.62/1.52 thf(exuI1,definition,
% 2.62/1.52 ( exuI1
% 2.62/1.52 = ( ! [Xphi: $i > $o] :
% 2.62/1.52 ( ? [Xx: $i] :
% 2.62/1.52 ( ( Xphi @ Xx )
% 2.62/1.52 & ! [Xy: $i] :
% 2.62/1.52 ( ( Xphi @ Xy )
% 2.62/1.52 => ( Xx = Xy ) ) )
% 2.62/1.52 => ( exu
% 2.62/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(exuI3_type,type,
% 2.62/1.52 exuI3: $o ).
% 2.62/1.52
% 2.62/1.52 thf(exuI3,definition,
% 2.62/1.52 ( exuI3
% 2.62/1.52 = ( ! [Xphi: $i > $o] :
% 2.62/1.52 ( ? [Xx: $i] : ( Xphi @ Xx )
% 2.62/1.52 => ( ! [Xx: $i,Xy: $i] :
% 2.62/1.52 ( ( Xphi @ Xx )
% 2.62/1.52 => ( ( Xphi @ Xy )
% 2.62/1.52 => ( Xx = Xy ) ) )
% 2.62/1.52 => ( exu
% 2.62/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(exuI2_type,type,
% 2.62/1.52 exuI2: $o ).
% 2.62/1.52
% 2.62/1.52 thf(exuI2,definition,
% 2.62/1.52 ( exuI2
% 2.62/1.52 = ( ! [Xphi: $i > $o] :
% 2.62/1.52 ( ? [Xx: $i] :
% 2.62/1.52 ! [Xy: $i] :
% 2.62/1.52 ( ( Xphi @ Xy )
% 2.62/1.52 <=> ( Xy = Xx ) )
% 2.62/1.52 => ( exu
% 2.62/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(inCongP_type,type,
% 2.62/1.52 inCongP: $o ).
% 2.62/1.52
% 2.62/1.52 thf(inCongP,definition,
% 2.62/1.52 ( inCongP
% 2.62/1.52 = ( ! [A: $i,B: $i] :
% 2.62/1.52 ( ( A = B )
% 2.62/1.52 => ! [Xx: $i,Xy: $i] :
% 2.62/1.52 ( ( Xx = Xy )
% 2.62/1.52 => ( ( in @ Xx @ A )
% 2.62/1.52 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(in__Cong_type,type,
% 2.62/1.52 in__Cong: $o ).
% 2.62/1.52
% 2.62/1.52 thf(in__Cong,definition,
% 2.62/1.52 ( in__Cong
% 2.62/1.52 = ( ! [A: $i,B: $i] :
% 2.62/1.52 ( ( A = B )
% 2.62/1.52 => ! [Xx: $i,Xy: $i] :
% 2.62/1.52 ( ( Xx = Xy )
% 2.62/1.52 => ( ( in @ Xx @ A )
% 2.62/1.52 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(exuE3u_type,type,
% 2.62/1.52 exuE3u: $o ).
% 2.62/1.52
% 2.62/1.52 thf(exuE3u,definition,
% 2.62/1.52 ( exuE3u
% 2.62/1.52 = ( ! [Xphi: $i > $o] :
% 2.62/1.52 ( ( exu
% 2.62/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.62/1.52 => ! [Xx: $i,Xy: $i] :
% 2.62/1.52 ( ( Xphi @ Xx )
% 2.62/1.52 => ( ( Xphi @ Xy )
% 2.62/1.52 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(exu__Cong_type,type,
% 2.62/1.52 exu__Cong: $o ).
% 2.62/1.52
% 2.62/1.52 thf(exu__Cong,definition,
% 2.62/1.52 ( exu__Cong
% 2.62/1.52 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.62/1.52 ( ! [Xx: $i,Xy: $i] :
% 2.62/1.52 ( ( Xx = Xy )
% 2.62/1.52 => ( ( Xphi @ Xx )
% 2.62/1.52 <=> ( Xpsi @ Xy ) ) )
% 2.62/1.52 => ( ( exu
% 2.62/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.62/1.52 <=> ( exu
% 2.62/1.52 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(emptyset__Cong_type,type,
% 2.62/1.52 emptyset__Cong: $o ).
% 2.62/1.52
% 2.62/1.52 thf(emptyset__Cong,definition,
% 2.62/1.52 ( emptyset__Cong
% 2.62/1.52 = ( emptyset = emptyset ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(setadjoin__Cong_type,type,
% 2.62/1.52 setadjoin__Cong: $o ).
% 2.62/1.52
% 2.62/1.52 thf(setadjoin__Cong,definition,
% 2.62/1.52 ( setadjoin__Cong
% 2.62/1.52 = ( ! [Xx: $i,Xy: $i] :
% 2.62/1.52 ( ( Xx = Xy )
% 2.62/1.52 => ! [Xz: $i,Xu: $i] :
% 2.62/1.52 ( ( Xz = Xu )
% 2.62/1.52 => ( ( setadjoin @ Xx @ Xz )
% 2.62/1.52 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(powerset__Cong_type,type,
% 2.62/1.52 powerset__Cong: $o ).
% 2.62/1.52
% 2.62/1.52 thf(powerset__Cong,definition,
% 2.62/1.52 ( powerset__Cong
% 2.62/1.52 = ( ! [A: $i,B: $i] :
% 2.62/1.52 ( ( A = B )
% 2.62/1.52 => ( ( powerset @ A )
% 2.62/1.52 = ( powerset @ B ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(setunion__Cong_type,type,
% 2.62/1.52 setunion__Cong: $o ).
% 2.62/1.52
% 2.62/1.52 thf(setunion__Cong,definition,
% 2.62/1.52 ( setunion__Cong
% 2.62/1.52 = ( ! [A: $i,B: $i] :
% 2.62/1.52 ( ( A = B )
% 2.62/1.52 => ( ( setunion @ A )
% 2.62/1.52 = ( setunion @ B ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(omega__Cong_type,type,
% 2.62/1.52 omega__Cong: $o ).
% 2.62/1.52
% 2.62/1.52 thf(omega__Cong,definition,
% 2.62/1.52 ( omega__Cong
% 2.62/1.52 = ( omega = omega ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(exuEu_type,type,
% 2.62/1.52 exuEu: $o ).
% 2.62/1.52
% 2.62/1.52 thf(exuEu,definition,
% 2.62/1.52 ( exuEu
% 2.62/1.52 = ( ! [Xphi: $i > $o] :
% 2.62/1.52 ( ( exu
% 2.62/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.62/1.52 => ! [Xx: $i,Xy: $i] :
% 2.62/1.52 ( ( Xphi @ Xx )
% 2.62/1.52 => ( ( Xphi @ Xy )
% 2.62/1.52 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(descr__Cong_type,type,
% 2.62/1.52 descr__Cong: $o ).
% 2.62/1.52
% 2.62/1.52 thf(descr__Cong,definition,
% 2.62/1.52 ( descr__Cong
% 2.62/1.52 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.62/1.52 ( ! [Xx: $i,Xy: $i] :
% 2.62/1.52 ( ( Xx = Xy )
% 2.62/1.52 => ( ( Xphi @ Xx )
% 2.62/1.52 <=> ( Xpsi @ Xy ) ) )
% 2.62/1.52 => ( ( exu
% 2.62/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.62/1.52 => ( ( exu
% 2.62/1.52 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 2.62/1.52 => ( ( descr
% 2.62/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.62/1.52 = ( descr
% 2.62/1.52 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(dsetconstr__Cong_type,type,
% 2.62/1.52 dsetconstr__Cong: $o ).
% 2.62/1.52
% 2.62/1.52 thf(dsetconstr__Cong,definition,
% 2.62/1.52 ( dsetconstr__Cong
% 2.62/1.52 = ( ! [A: $i,B: $i] :
% 2.62/1.52 ( ( A = B )
% 2.62/1.52 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.62/1.52 ( ! [Xx: $i] :
% 2.62/1.52 ( ( in @ Xx @ A )
% 2.62/1.52 => ! [Xy: $i] :
% 2.62/1.52 ( ( in @ Xy @ B )
% 2.62/1.52 => ( ( Xx = Xy )
% 2.62/1.52 => ( ( Xphi @ Xx )
% 2.62/1.52 <=> ( Xpsi @ Xy ) ) ) ) )
% 2.62/1.52 => ( ( dsetconstr @ A
% 2.62/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.62/1.52 = ( dsetconstr @ B
% 2.62/1.52 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(subset_type,type,
% 2.62/1.52 subset: $i > $i > $o ).
% 2.62/1.52
% 2.62/1.52 thf(disjoint_type,type,
% 2.62/1.52 disjoint: $i > $i > $o ).
% 2.62/1.52
% 2.62/1.52 thf(setsmeet_type,type,
% 2.62/1.52 setsmeet: $i > $i > $o ).
% 2.62/1.52
% 2.62/1.52 thf(subsetI1_type,type,
% 2.62/1.52 subsetI1: $o ).
% 2.62/1.52
% 2.62/1.52 thf(subsetI1,definition,
% 2.62/1.52 ( subsetI1
% 2.62/1.52 = ( ! [A: $i,B: $i] :
% 2.62/1.52 ( ! [Xx: $i] :
% 2.62/1.52 ( ( in @ Xx @ A )
% 2.62/1.52 => ( in @ Xx @ B ) )
% 2.62/1.52 => ( subset @ A @ B ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(eqimpsubset2_type,type,
% 2.62/1.52 eqimpsubset2: $o ).
% 2.62/1.52
% 2.62/1.52 thf(eqimpsubset2,definition,
% 2.62/1.52 ( eqimpsubset2
% 2.62/1.52 = ( ! [A: $i,B: $i] :
% 2.62/1.52 ( ( A = B )
% 2.62/1.52 => ( subset @ B @ A ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(eqimpsubset1_type,type,
% 2.62/1.52 eqimpsubset1: $o ).
% 2.62/1.52
% 2.62/1.52 thf(eqimpsubset1,definition,
% 2.62/1.52 ( eqimpsubset1
% 2.62/1.52 = ( ! [A: $i,B: $i] :
% 2.62/1.52 ( ( A = B )
% 2.62/1.52 => ( subset @ A @ B ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(subsetI2_type,type,
% 2.62/1.52 subsetI2: $o ).
% 2.62/1.52
% 2.62/1.52 thf(subsetI2,definition,
% 2.62/1.52 ( subsetI2
% 2.62/1.52 = ( ! [A: $i,B: $i] :
% 2.62/1.52 ( ! [Xx: $i] :
% 2.62/1.52 ( ( in @ Xx @ A )
% 2.62/1.52 => ( in @ Xx @ B ) )
% 2.62/1.52 => ( subset @ A @ B ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(emptysetsubset_type,type,
% 2.62/1.52 emptysetsubset: $o ).
% 2.62/1.52
% 2.62/1.52 thf(emptysetsubset,definition,
% 2.62/1.52 ( emptysetsubset
% 2.62/1.52 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(subsetE_type,type,
% 2.62/1.52 subsetE: $o ).
% 2.62/1.52
% 2.62/1.52 thf(subsetE,definition,
% 2.62/1.52 ( subsetE
% 2.62/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.62/1.52 ( ( subset @ A @ B )
% 2.62/1.52 => ( ( in @ Xx @ A )
% 2.62/1.52 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(subsetE2_type,type,
% 2.62/1.52 subsetE2: $o ).
% 2.62/1.52
% 2.62/1.52 thf(subsetE2,definition,
% 2.62/1.52 ( subsetE2
% 2.62/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.62/1.52 ( ( subset @ A @ B )
% 2.62/1.52 => ( ~ ( in @ Xx @ B )
% 2.62/1.52 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(notsubsetI_type,type,
% 2.62/1.52 notsubsetI: $o ).
% 2.62/1.52
% 2.62/1.52 thf(notsubsetI,definition,
% 2.62/1.52 ( notsubsetI
% 2.62/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.62/1.52 ( ( in @ Xx @ A )
% 2.62/1.52 => ( ~ ( in @ Xx @ B )
% 2.62/1.52 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(notequalI1_type,type,
% 2.62/1.52 notequalI1: $o ).
% 2.62/1.52
% 2.62/1.52 thf(notequalI1,definition,
% 2.62/1.52 ( notequalI1
% 2.62/1.52 = ( ! [A: $i,B: $i] :
% 2.62/1.52 ( ~ ( subset @ A @ B )
% 2.62/1.52 => ( A != B ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(notequalI2_type,type,
% 2.62/1.52 notequalI2: $o ).
% 2.62/1.52
% 2.62/1.52 thf(notequalI2,definition,
% 2.62/1.52 ( notequalI2
% 2.62/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.62/1.52 ( ( in @ Xx @ A )
% 2.62/1.52 => ( ~ ( in @ Xx @ B )
% 2.62/1.52 => ( A != B ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(subsetRefl_type,type,
% 2.62/1.52 subsetRefl: $o ).
% 2.62/1.52
% 2.62/1.52 thf(subsetRefl,definition,
% 2.62/1.52 ( subsetRefl
% 2.62/1.52 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(subsetTrans_type,type,
% 2.62/1.52 subsetTrans: $o ).
% 2.62/1.52
% 2.62/1.52 thf(subsetTrans,definition,
% 2.62/1.52 ( subsetTrans
% 2.62/1.52 = ( ! [A: $i,B: $i,C: $i] :
% 2.62/1.52 ( ( subset @ A @ B )
% 2.62/1.52 => ( ( subset @ B @ C )
% 2.62/1.52 => ( subset @ A @ C ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(setadjoinSub_type,type,
% 2.62/1.52 setadjoinSub: $o ).
% 2.62/1.52
% 2.62/1.52 thf(setadjoinSub,definition,
% 2.62/1.52 ( setadjoinSub
% 2.62/1.52 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(setadjoinSub2_type,type,
% 2.62/1.52 setadjoinSub2: $o ).
% 2.62/1.52
% 2.62/1.52 thf(setadjoinSub2,definition,
% 2.62/1.52 ( setadjoinSub2
% 2.62/1.52 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.62/1.52 ( ( subset @ A @ B )
% 2.62/1.52 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(subset2powerset_type,type,
% 2.62/1.52 subset2powerset: $o ).
% 2.62/1.52
% 2.62/1.52 thf(subset2powerset,definition,
% 2.62/1.52 ( subset2powerset
% 2.62/1.52 = ( ! [A: $i,B: $i] :
% 2.62/1.52 ( ( subset @ A @ B )
% 2.62/1.52 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(setextsub_type,type,
% 2.62/1.52 setextsub: $o ).
% 2.62/1.52
% 2.62/1.52 thf(setextsub,definition,
% 2.62/1.52 ( setextsub
% 2.62/1.52 = ( ! [A: $i,B: $i] :
% 2.62/1.52 ( ( subset @ A @ B )
% 2.62/1.52 => ( ( subset @ B @ A )
% 2.62/1.52 => ( A = B ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(subsetemptysetimpeq_type,type,
% 2.62/1.52 subsetemptysetimpeq: $o ).
% 2.62/1.52
% 2.62/1.52 thf(subsetemptysetimpeq,definition,
% 2.62/1.52 ( subsetemptysetimpeq
% 2.62/1.52 = ( ! [A: $i] :
% 2.62/1.52 ( ( subset @ A @ emptyset )
% 2.62/1.52 => ( A = emptyset ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(powersetI1_type,type,
% 2.62/1.52 powersetI1: $o ).
% 2.62/1.52
% 2.62/1.52 thf(powersetI1,definition,
% 2.62/1.52 ( powersetI1
% 2.62/1.52 = ( ! [A: $i,B: $i] :
% 2.62/1.52 ( ( subset @ B @ A )
% 2.62/1.52 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(powersetE1_type,type,
% 2.62/1.52 powersetE1: $o ).
% 2.62/1.52
% 2.62/1.52 thf(powersetE1,definition,
% 2.62/1.52 ( powersetE1
% 2.62/1.52 = ( ! [A: $i,B: $i] :
% 2.62/1.52 ( ( in @ B @ ( powerset @ A ) )
% 2.62/1.52 => ( subset @ B @ A ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(inPowerset_type,type,
% 2.62/1.52 inPowerset: $o ).
% 2.62/1.52
% 2.62/1.52 thf(inPowerset,definition,
% 2.62/1.52 ( inPowerset
% 2.62/1.52 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(powersetsubset_type,type,
% 2.62/1.52 powersetsubset: $o ).
% 2.62/1.52
% 2.62/1.52 thf(powersetsubset,definition,
% 2.62/1.52 ( powersetsubset
% 2.62/1.52 = ( ! [A: $i,B: $i] :
% 2.62/1.52 ( ( subset @ A @ B )
% 2.62/1.52 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(sepInPowerset_type,type,
% 2.62/1.52 sepInPowerset: $o ).
% 2.62/1.52
% 2.62/1.52 thf(sepInPowerset,definition,
% 2.62/1.52 ( sepInPowerset
% 2.62/1.52 = ( ! [A: $i,Xphi: $i > $o] :
% 2.62/1.52 ( in
% 2.62/1.52 @ ( dsetconstr @ A
% 2.62/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.62/1.52 @ ( powerset @ A ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(sepSubset_type,type,
% 2.62/1.52 sepSubset: $o ).
% 2.62/1.52
% 2.62/1.52 thf(sepSubset,definition,
% 2.62/1.52 ( sepSubset
% 2.62/1.52 = ( ! [A: $i,Xphi: $i > $o] :
% 2.62/1.52 ( subset
% 2.62/1.52 @ ( dsetconstr @ A
% 2.62/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.62/1.52 @ A ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(binunion_type,type,
% 2.62/1.52 binunion: $i > $i > $i ).
% 2.62/1.52
% 2.62/1.52 thf(binunionIL_type,type,
% 2.62/1.52 binunionIL: $o ).
% 2.62/1.52
% 2.62/1.52 thf(binunionIL,definition,
% 2.62/1.52 ( binunionIL
% 2.62/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.62/1.52 ( ( in @ Xx @ A )
% 2.62/1.52 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(upairset2IR_type,type,
% 2.62/1.52 upairset2IR: $o ).
% 2.62/1.52
% 2.62/1.52 thf(upairset2IR,definition,
% 2.62/1.52 ( upairset2IR
% 2.62/1.52 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(binunionIR_type,type,
% 2.62/1.52 binunionIR: $o ).
% 2.62/1.52
% 2.62/1.52 thf(binunionIR,definition,
% 2.62/1.52 ( binunionIR
% 2.62/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.62/1.52 ( ( in @ Xx @ B )
% 2.62/1.52 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(binunionEcases_type,type,
% 2.62/1.52 binunionEcases: $o ).
% 2.62/1.52
% 2.62/1.52 thf(binunionEcases,definition,
% 2.62/1.52 ( binunionEcases
% 2.62/1.52 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 2.62/1.52 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.62/1.52 => ( ( ( in @ Xx @ A )
% 2.62/1.52 => Xphi )
% 2.62/1.52 => ( ( ( in @ Xx @ B )
% 2.62/1.52 => Xphi )
% 2.62/1.52 => Xphi ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(binunionE_type,type,
% 2.62/1.52 binunionE: $o ).
% 2.62/1.52
% 2.62/1.52 thf(binunionE,definition,
% 2.62/1.52 ( binunionE
% 2.62/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.62/1.52 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.62/1.52 => ( ( in @ Xx @ A )
% 2.62/1.52 | ( in @ Xx @ B ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(binunionLsub_type,type,
% 2.62/1.52 binunionLsub: $o ).
% 2.62/1.52
% 2.62/1.52 thf(binunionLsub,definition,
% 2.62/1.52 ( binunionLsub
% 2.62/1.52 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(binunionRsub_type,type,
% 2.62/1.52 binunionRsub: $o ).
% 2.62/1.52
% 2.62/1.52 thf(binunionRsub,definition,
% 2.62/1.52 ( binunionRsub
% 2.62/1.52 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(binintersect_type,type,
% 2.62/1.52 binintersect: $i > $i > $i ).
% 2.62/1.52
% 2.62/1.52 thf(binintersectI_type,type,
% 2.62/1.52 binintersectI: $o ).
% 2.62/1.52
% 2.62/1.52 thf(binintersectI,definition,
% 2.62/1.52 ( binintersectI
% 2.62/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.62/1.52 ( ( in @ Xx @ A )
% 2.62/1.52 => ( ( in @ Xx @ B )
% 2.62/1.52 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(binintersectSubset5_type,type,
% 2.62/1.52 binintersectSubset5: $o ).
% 2.62/1.52
% 2.62/1.52 thf(binintersectSubset5,definition,
% 2.62/1.52 ( binintersectSubset5
% 2.62/1.52 = ( ! [A: $i,B: $i,C: $i] :
% 2.62/1.52 ( ( subset @ C @ A )
% 2.62/1.52 => ( ( subset @ C @ B )
% 2.62/1.52 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(binintersectEL_type,type,
% 2.62/1.52 binintersectEL: $o ).
% 2.62/1.52
% 2.62/1.52 thf(binintersectEL,definition,
% 2.62/1.52 ( binintersectEL
% 2.62/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.62/1.52 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.62/1.52 => ( in @ Xx @ A ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(binintersectLsub_type,type,
% 2.62/1.52 binintersectLsub: $o ).
% 2.62/1.52
% 2.62/1.52 thf(binintersectLsub,definition,
% 2.62/1.52 ( binintersectLsub
% 2.62/1.52 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(binintersectSubset2_type,type,
% 2.62/1.52 binintersectSubset2: $o ).
% 2.62/1.52
% 2.62/1.52 thf(binintersectSubset2,definition,
% 2.62/1.52 ( binintersectSubset2
% 2.62/1.52 = ( ! [A: $i,B: $i] :
% 2.62/1.52 ( ( subset @ A @ B )
% 2.62/1.52 => ( ( binintersect @ A @ B )
% 2.62/1.52 = A ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(binintersectSubset3_type,type,
% 2.62/1.52 binintersectSubset3: $o ).
% 2.62/1.52
% 2.62/1.52 thf(binintersectSubset3,definition,
% 2.62/1.52 ( binintersectSubset3
% 2.62/1.52 = ( ! [A: $i,B: $i] :
% 2.62/1.52 ( ( ( binintersect @ A @ B )
% 2.62/1.52 = B )
% 2.62/1.52 => ( subset @ B @ A ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(binintersectER_type,type,
% 2.62/1.52 binintersectER: $o ).
% 2.62/1.52
% 2.62/1.52 thf(binintersectER,definition,
% 2.62/1.52 ( binintersectER
% 2.62/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.62/1.52 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.62/1.52 => ( in @ Xx @ B ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(disjointsetsI1_type,type,
% 2.62/1.52 disjointsetsI1: $o ).
% 2.62/1.52
% 2.62/1.52 thf(disjointsetsI1,definition,
% 2.62/1.52 ( disjointsetsI1
% 2.62/1.52 = ( ! [A: $i,B: $i] :
% 2.62/1.52 ( ~ ? [Xx: $i] :
% 2.62/1.52 ( ( in @ Xx @ A )
% 2.62/1.52 & ( in @ Xx @ B ) )
% 2.62/1.52 => ( ( binintersect @ A @ B )
% 2.62/1.52 = emptyset ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(binintersectRsub_type,type,
% 2.62/1.52 binintersectRsub: $o ).
% 2.62/1.52
% 2.62/1.52 thf(binintersectRsub,definition,
% 2.62/1.52 ( binintersectRsub
% 2.62/1.52 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(binintersectSubset4_type,type,
% 2.62/1.52 binintersectSubset4: $o ).
% 2.62/1.52
% 2.62/1.52 thf(binintersectSubset4,definition,
% 2.62/1.52 ( binintersectSubset4
% 2.62/1.52 = ( ! [A: $i,B: $i] :
% 2.62/1.52 ( ( subset @ B @ A )
% 2.62/1.52 => ( ( binintersect @ A @ B )
% 2.62/1.52 = B ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(binintersectSubset1_type,type,
% 2.62/1.52 binintersectSubset1: $o ).
% 2.62/1.52
% 2.62/1.52 thf(binintersectSubset1,definition,
% 2.62/1.52 ( binintersectSubset1
% 2.62/1.52 = ( ! [A: $i,B: $i] :
% 2.62/1.52 ( ( ( binintersect @ A @ B )
% 2.62/1.52 = A )
% 2.62/1.52 => ( subset @ A @ B ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(bs114d_type,type,
% 2.62/1.52 bs114d: $o ).
% 2.62/1.52
% 2.62/1.52 thf(bs114d,definition,
% 2.62/1.52 ( bs114d
% 2.62/1.52 = ( ! [A: $i,B: $i,C: $i] :
% 2.62/1.52 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 2.62/1.52 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(regular_type,type,
% 2.62/1.52 regular: $i > $o ).
% 2.62/1.52
% 2.62/1.52 thf(setminus_type,type,
% 2.62/1.52 setminus: $i > $i > $i ).
% 2.62/1.52
% 2.62/1.52 thf(setminusI_type,type,
% 2.62/1.52 setminusI: $o ).
% 2.62/1.52
% 2.62/1.52 thf(setminusI,definition,
% 2.62/1.52 ( setminusI
% 2.62/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.62/1.52 ( ( in @ Xx @ A )
% 2.62/1.52 => ( ~ ( in @ Xx @ B )
% 2.62/1.52 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(setminusEL_type,type,
% 2.62/1.52 setminusEL: $o ).
% 2.62/1.52
% 2.62/1.52 thf(setminusEL,definition,
% 2.62/1.52 ( setminusEL
% 2.62/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.62/1.52 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.62/1.52 => ( in @ Xx @ A ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(setminusER_type,type,
% 2.62/1.52 setminusER: $o ).
% 2.62/1.52
% 2.62/1.52 thf(setminusER,definition,
% 2.62/1.52 ( setminusER
% 2.62/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.62/1.52 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.62/1.52 => ~ ( in @ Xx @ B ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(setminusSubset2_type,type,
% 2.62/1.52 setminusSubset2: $o ).
% 2.62/1.52
% 2.62/1.52 thf(setminusSubset2,definition,
% 2.62/1.52 ( setminusSubset2
% 2.62/1.52 = ( ! [A: $i,B: $i] :
% 2.62/1.52 ( ( subset @ A @ B )
% 2.62/1.52 => ( ( setminus @ A @ B )
% 2.62/1.52 = emptyset ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(setminusERneg_type,type,
% 2.62/1.52 setminusERneg: $o ).
% 2.62/1.52
% 2.62/1.52 thf(setminusERneg,definition,
% 2.62/1.52 ( setminusERneg
% 2.62/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.62/1.52 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.62/1.52 => ( ( in @ Xx @ A )
% 2.62/1.52 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(setminusELneg_type,type,
% 2.62/1.52 setminusELneg: $o ).
% 2.62/1.52
% 2.62/1.52 thf(setminusELneg,definition,
% 2.62/1.52 ( setminusELneg
% 2.62/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.62/1.52 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.62/1.52 => ( ~ ( in @ Xx @ B )
% 2.62/1.52 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(setminusILneg_type,type,
% 2.62/1.52 setminusILneg: $o ).
% 2.62/1.52
% 2.62/1.52 thf(setminusILneg,definition,
% 2.62/1.52 ( setminusILneg
% 2.62/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.62/1.52 ( ~ ( in @ Xx @ A )
% 2.62/1.52 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(setminusIRneg_type,type,
% 2.62/1.52 setminusIRneg: $o ).
% 2.62/1.52
% 2.62/1.52 thf(setminusIRneg,definition,
% 2.62/1.52 ( setminusIRneg
% 2.62/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.62/1.52 ( ( in @ Xx @ B )
% 2.62/1.52 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(setminusLsub_type,type,
% 2.62/1.52 setminusLsub: $o ).
% 2.62/1.52
% 2.62/1.52 thf(setminusLsub,definition,
% 2.62/1.52 ( setminusLsub
% 2.62/1.52 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(setminusSubset1_type,type,
% 2.62/1.52 setminusSubset1: $o ).
% 2.62/1.52
% 2.62/1.52 thf(setminusSubset1,definition,
% 2.62/1.52 ( setminusSubset1
% 2.62/1.52 = ( ! [A: $i,B: $i] :
% 2.62/1.52 ( ( ( setminus @ A @ B )
% 2.62/1.52 = emptyset )
% 2.62/1.52 => ( subset @ A @ B ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(symdiff_type,type,
% 2.62/1.52 symdiff: $i > $i > $i ).
% 2.62/1.52
% 2.62/1.52 thf(symdiffE_type,type,
% 2.62/1.52 symdiffE: $o ).
% 2.62/1.52
% 2.62/1.52 thf(symdiffE,definition,
% 2.62/1.52 ( symdiffE
% 2.62/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.62/1.52 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 2.62/1.52 => ! [Xphi: $o] :
% 2.62/1.52 ( ( ( in @ Xx @ A )
% 2.62/1.52 => ( ~ ( in @ Xx @ B )
% 2.62/1.52 => Xphi ) )
% 2.62/1.52 => ( ( ~ ( in @ Xx @ A )
% 2.62/1.52 => ( ( in @ Xx @ B )
% 2.62/1.52 => Xphi ) )
% 2.62/1.52 => Xphi ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(symdiffI1_type,type,
% 2.62/1.52 symdiffI1: $o ).
% 2.62/1.52
% 2.62/1.52 thf(symdiffI1,definition,
% 2.62/1.52 ( symdiffI1
% 2.62/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.62/1.52 ( ( in @ Xx @ A )
% 2.62/1.52 => ( ~ ( in @ Xx @ B )
% 2.62/1.52 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(symdiffI2_type,type,
% 2.62/1.52 symdiffI2: $o ).
% 2.62/1.52
% 2.62/1.52 thf(symdiffI2,definition,
% 2.62/1.52 ( symdiffI2
% 2.62/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.62/1.52 ( ~ ( in @ Xx @ A )
% 2.62/1.52 => ( ( in @ Xx @ B )
% 2.62/1.52 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(symdiffIneg1_type,type,
% 2.62/1.52 symdiffIneg1: $o ).
% 2.62/1.52
% 2.62/1.52 thf(symdiffIneg1,definition,
% 2.62/1.52 ( symdiffIneg1
% 2.62/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.62/1.52 ( ( in @ Xx @ A )
% 2.62/1.52 => ( ( in @ Xx @ B )
% 2.62/1.52 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(symdiffIneg2_type,type,
% 2.62/1.52 symdiffIneg2: $o ).
% 2.62/1.52
% 2.62/1.52 thf(symdiffIneg2,definition,
% 2.62/1.52 ( symdiffIneg2
% 2.62/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.62/1.52 ( ~ ( in @ Xx @ A )
% 2.62/1.52 => ( ~ ( in @ Xx @ B )
% 2.62/1.52 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(iskpair_type,type,
% 2.62/1.52 iskpair: $i > $o ).
% 2.62/1.52
% 2.62/1.52 thf(secondinupair_type,type,
% 2.62/1.52 secondinupair: $o ).
% 2.62/1.52
% 2.62/1.52 thf(secondinupair,definition,
% 2.62/1.52 ( secondinupair
% 2.62/1.52 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(setukpairIL_type,type,
% 2.62/1.52 setukpairIL: $o ).
% 2.62/1.52
% 2.62/1.52 thf(setukpairIL,definition,
% 2.62/1.52 ( setukpairIL
% 2.62/1.52 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(setukpairIR_type,type,
% 2.62/1.52 setukpairIR: $o ).
% 2.62/1.52
% 2.62/1.52 thf(setukpairIR,definition,
% 2.62/1.52 ( setukpairIR
% 2.62/1.52 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(kpairiskpair_type,type,
% 2.62/1.52 kpairiskpair: $o ).
% 2.62/1.52
% 2.62/1.52 thf(kpairiskpair,definition,
% 2.62/1.52 ( kpairiskpair
% 2.62/1.52 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(kpair_type,type,
% 2.62/1.52 kpair: $i > $i > $i ).
% 2.62/1.52
% 2.62/1.52 thf(kpairp_type,type,
% 2.62/1.52 kpairp: $o ).
% 2.62/1.52
% 2.62/1.52 thf(kpairp,definition,
% 2.62/1.52 ( kpairp
% 2.62/1.52 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(cartprod_type,type,
% 2.62/1.52 cartprod: $i > $i > $i ).
% 2.62/1.52
% 2.62/1.52 thf(singletonsubset_type,type,
% 2.62/1.52 singletonsubset: $o ).
% 2.62/1.52
% 2.62/1.52 thf(singletonsubset,definition,
% 2.62/1.52 ( singletonsubset
% 2.62/1.52 = ( ! [A: $i,Xx: $i] :
% 2.62/1.52 ( ( in @ Xx @ A )
% 2.62/1.52 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(singletoninpowerset_type,type,
% 2.62/1.52 singletoninpowerset: $o ).
% 2.62/1.52
% 2.62/1.52 thf(singletoninpowerset,definition,
% 2.62/1.52 ( singletoninpowerset
% 2.62/1.52 = ( ! [A: $i,Xx: $i] :
% 2.62/1.52 ( ( in @ Xx @ A )
% 2.62/1.52 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(singletoninpowunion_type,type,
% 2.62/1.52 singletoninpowunion: $o ).
% 2.62/1.52
% 2.62/1.52 thf(singletoninpowunion,definition,
% 2.62/1.52 ( singletoninpowunion
% 2.62/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.62/1.52 ( ( in @ Xx @ A )
% 2.62/1.52 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(upairset2E_type,type,
% 2.62/1.52 upairset2E: $o ).
% 2.62/1.52
% 2.62/1.52 thf(upairset2E,definition,
% 2.62/1.52 ( upairset2E
% 2.62/1.52 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.62/1.52 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.62/1.52 => ( ( Xz = Xx )
% 2.62/1.52 | ( Xz = Xy ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(upairsubunion_type,type,
% 2.62/1.52 upairsubunion: $o ).
% 2.62/1.52
% 2.62/1.52 thf(upairsubunion,definition,
% 2.62/1.52 ( upairsubunion
% 2.62/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.62/1.52 ( ( in @ Xx @ A )
% 2.62/1.52 => ! [Xy: $i] :
% 2.62/1.52 ( ( in @ Xy @ B )
% 2.62/1.52 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(upairinpowunion_type,type,
% 2.62/1.52 upairinpowunion: $o ).
% 2.62/1.52
% 2.62/1.52 thf(upairinpowunion,definition,
% 2.62/1.52 ( upairinpowunion
% 2.62/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.62/1.52 ( ( in @ Xx @ A )
% 2.62/1.52 => ! [Xy: $i] :
% 2.62/1.52 ( ( in @ Xy @ B )
% 2.62/1.52 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(ubforcartprodlem1_type,type,
% 2.62/1.52 ubforcartprodlem1: $o ).
% 2.62/1.52
% 2.62/1.52 thf(ubforcartprodlem1,definition,
% 2.62/1.52 ( ubforcartprodlem1
% 2.62/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.62/1.52 ( ( in @ Xx @ A )
% 2.62/1.52 => ! [Xy: $i] :
% 2.62/1.52 ( ( in @ Xy @ B )
% 2.62/1.52 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(ubforcartprodlem2_type,type,
% 2.62/1.52 ubforcartprodlem2: $o ).
% 2.62/1.52
% 2.62/1.52 thf(ubforcartprodlem2,definition,
% 2.62/1.52 ( ubforcartprodlem2
% 2.62/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.62/1.52 ( ( in @ Xx @ A )
% 2.62/1.52 => ! [Xy: $i] :
% 2.62/1.52 ( ( in @ Xy @ B )
% 2.62/1.52 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(ubforcartprodlem3_type,type,
% 2.62/1.52 ubforcartprodlem3: $o ).
% 2.62/1.52
% 2.62/1.52 thf(ubforcartprodlem3,definition,
% 2.62/1.52 ( ubforcartprodlem3
% 2.62/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.62/1.52 ( ( in @ Xx @ A )
% 2.62/1.52 => ! [Xy: $i] :
% 2.62/1.52 ( ( in @ Xy @ B )
% 2.62/1.52 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(cartprodpairin_type,type,
% 2.62/1.52 cartprodpairin: $o ).
% 2.62/1.52
% 2.62/1.52 thf(cartprodpairin,definition,
% 2.62/1.52 ( cartprodpairin
% 2.62/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.62/1.52 ( ( in @ Xx @ A )
% 2.62/1.52 => ! [Xy: $i] :
% 2.62/1.52 ( ( in @ Xy @ B )
% 2.62/1.52 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(cartprodmempair1_type,type,
% 2.62/1.52 cartprodmempair1: $o ).
% 2.62/1.52
% 2.62/1.52 thf(cartprodmempair1,definition,
% 2.62/1.52 ( cartprodmempair1
% 2.62/1.52 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.62/1.52 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.62/1.52 => ? [Xx: $i] :
% 2.62/1.52 ( ( in @ Xx @ A )
% 2.62/1.52 & ? [Xy: $i] :
% 2.62/1.52 ( ( in @ Xy @ B )
% 2.62/1.52 & ( Xu
% 2.62/1.52 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(cartprodmempair_type,type,
% 2.62/1.52 cartprodmempair: $o ).
% 2.62/1.52
% 2.62/1.52 thf(cartprodmempair,definition,
% 2.62/1.52 ( cartprodmempair
% 2.62/1.52 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.62/1.52 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.62/1.52 => ( iskpair @ Xu ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(setunionE2_type,type,
% 2.62/1.52 setunionE2: $o ).
% 2.62/1.52
% 2.62/1.52 thf(setunionE2,definition,
% 2.62/1.52 ( setunionE2
% 2.62/1.52 = ( ! [A: $i,Xx: $i] :
% 2.62/1.52 ( ( in @ Xx @ ( setunion @ A ) )
% 2.62/1.52 => ? [X: $i] :
% 2.62/1.52 ( ( in @ X @ A )
% 2.62/1.52 & ( in @ Xx @ X ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(setunionsingleton1_type,type,
% 2.62/1.52 setunionsingleton1: $o ).
% 2.62/1.52
% 2.62/1.52 thf(setunionsingleton1,definition,
% 2.62/1.52 ( setunionsingleton1
% 2.62/1.52 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(setunionsingleton2_type,type,
% 2.62/1.52 setunionsingleton2: $o ).
% 2.62/1.52
% 2.62/1.52 thf(setunionsingleton2,definition,
% 2.62/1.52 ( setunionsingleton2
% 2.62/1.52 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(setunionsingleton_type,type,
% 2.62/1.52 setunionsingleton: $o ).
% 2.62/1.52
% 2.62/1.52 thf(setunionsingleton,definition,
% 2.62/1.52 ( setunionsingleton
% 2.62/1.52 = ( ! [Xx: $i] :
% 2.62/1.52 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 2.62/1.52 = Xx ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(singleton_type,type,
% 2.62/1.52 singleton: $i > $o ).
% 2.62/1.52
% 2.62/1.52 thf(singleton,definition,
% 2.62/1.52 ( singleton
% 2.62/1.52 = ( ^ [A: $i] :
% 2.62/1.52 ? [Xx: $i] :
% 2.62/1.52 ( ( in @ Xx @ A )
% 2.62/1.52 & ( A
% 2.62/1.52 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(singletonprop_type,type,
% 2.62/1.52 singletonprop: $o ).
% 2.62/1.52
% 2.62/1.52 thf(singletonprop,definition,
% 2.62/1.52 ( singletonprop
% 2.62/1.52 = ( ! [A: $i,Xphi: $i > $o] :
% 2.62/1.52 ( ! [Xx: $i] :
% 2.62/1.52 ( ( in @ Xx @ A )
% 2.62/1.52 => ! [Xy: $i] :
% 2.62/1.52 ( ( in @ Xy @ A )
% 2.62/1.52 => ( ( Xphi @ Xx )
% 2.62/1.52 => ( ( Xphi @ Xy )
% 2.62/1.52 => ( Xx = Xy ) ) ) ) )
% 2.62/1.52 => ( ? [Xx: $i] :
% 2.62/1.52 ( ( in @ Xx @ A )
% 2.62/1.52 & ( Xphi @ Xx ) )
% 2.62/1.52 => ( singleton
% 2.62/1.52 @ ( dsetconstr @ A
% 2.62/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(ex1_type,type,
% 2.62/1.52 ex1: $i > ( $i > $o ) > $o ).
% 2.62/1.52
% 2.62/1.52 thf(ex1,definition,
% 2.62/1.52 ( ex1
% 2.62/1.52 = ( ^ [A: $i,Xphi: $i > $o] :
% 2.62/1.52 ( singleton
% 2.62/1.52 @ ( dsetconstr @ A
% 2.62/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(ex1E1_type,type,
% 2.62/1.52 ex1E1: $o ).
% 2.62/1.52
% 2.62/1.52 thf(ex1E1,definition,
% 2.62/1.52 ( ex1E1
% 2.62/1.52 = ( ! [A: $i,Xphi: $i > $o] :
% 2.62/1.52 ( ( ex1 @ A
% 2.62/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.62/1.52 => ? [Xx: $i] :
% 2.62/1.52 ( ( in @ Xx @ A )
% 2.62/1.52 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(ex1I_type,type,
% 2.62/1.52 ex1I: $o ).
% 2.62/1.52
% 2.62/1.52 thf(ex1I,definition,
% 2.62/1.52 ( ex1I
% 2.62/1.52 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.62/1.52 ( ( in @ Xx @ A )
% 2.62/1.52 => ( ( Xphi @ Xx )
% 2.62/1.52 => ( ! [Xy: $i] :
% 2.62/1.52 ( ( in @ Xy @ A )
% 2.62/1.52 => ( ( Xphi @ Xy )
% 2.62/1.52 => ( Xy = Xx ) ) )
% 2.62/1.52 => ( ex1 @ A
% 2.62/1.52 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(ex1I2_type,type,
% 2.62/1.52 ex1I2: $o ).
% 2.62/1.52
% 2.62/1.52 thf(ex1I2,definition,
% 2.62/1.52 ( ex1I2
% 2.62/1.52 = ( ! [A: $i,Xphi: $i > $o] :
% 2.62/1.52 ( ! [Xx: $i] :
% 2.62/1.52 ( ( in @ Xx @ A )
% 2.62/1.52 => ! [Xy: $i] :
% 2.62/1.52 ( ( in @ Xy @ A )
% 2.62/1.52 => ( ( Xphi @ Xx )
% 2.62/1.52 => ( ( Xphi @ Xy )
% 2.62/1.52 => ( Xx = Xy ) ) ) ) )
% 2.62/1.52 => ( ? [Xx: $i] :
% 2.62/1.52 ( ( in @ Xx @ A )
% 2.62/1.52 & ( Xphi @ Xx ) )
% 2.62/1.52 => ( ex1 @ A
% 2.62/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(singletonsuniq_type,type,
% 2.62/1.52 singletonsuniq: $o ).
% 2.62/1.52
% 2.62/1.52 thf(singletonsuniq,definition,
% 2.62/1.52 ( singletonsuniq
% 2.62/1.52 = ( ! [Xx: $i,Xy: $i] :
% 2.62/1.52 ( ( ( setadjoin @ Xx @ emptyset )
% 2.62/1.52 = ( setadjoin @ Xy @ emptyset ) )
% 2.62/1.52 => ( Xx = Xy ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(atmost1p_type,type,
% 2.62/1.52 atmost1p: $i > $o ).
% 2.62/1.52
% 2.62/1.52 thf(atleast2p_type,type,
% 2.62/1.52 atleast2p: $i > $o ).
% 2.62/1.52
% 2.62/1.52 thf(atmost2p_type,type,
% 2.62/1.52 atmost2p: $i > $o ).
% 2.62/1.52
% 2.62/1.52 thf(upairsetp_type,type,
% 2.62/1.52 upairsetp: $i > $o ).
% 2.62/1.52
% 2.62/1.52 thf(setukpairinjL1_type,type,
% 2.62/1.52 setukpairinjL1: $o ).
% 2.62/1.52
% 2.62/1.52 thf(setukpairinjL1,definition,
% 2.62/1.52 ( setukpairinjL1
% 2.62/1.52 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.62/1.52 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 2.62/1.52 => ( Xx = Xz ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(kfstsingleton_type,type,
% 2.62/1.52 kfstsingleton: $o ).
% 2.62/1.52
% 2.62/1.52 thf(kfstsingleton,definition,
% 2.62/1.52 ( kfstsingleton
% 2.62/1.52 = ( ! [Xu: $i] :
% 2.62/1.52 ( ( iskpair @ Xu )
% 2.62/1.52 => ( singleton
% 2.62/1.52 @ ( dsetconstr @ ( setunion @ Xu )
% 2.62/1.52 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(theprop_type,type,
% 2.62/1.52 theprop: $o ).
% 2.62/1.52
% 2.62/1.52 thf(theprop,definition,
% 2.62/1.52 ( theprop
% 2.62/1.52 = ( ! [X: $i] :
% 2.62/1.52 ( ( singleton @ X )
% 2.62/1.52 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(kfst_type,type,
% 2.62/1.52 kfst: $i > $i ).
% 2.62/1.52
% 2.62/1.52 thf(kfstpairEq_type,type,
% 2.62/1.52 kfstpairEq: $o ).
% 2.62/1.52
% 2.62/1.52 thf(kfstpairEq,definition,
% 2.62/1.52 ( kfstpairEq
% 2.62/1.52 = ( ! [Xx: $i,Xy: $i] :
% 2.62/1.52 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 2.62/1.52 = Xx ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(cartprodfstin_type,type,
% 2.62/1.52 cartprodfstin: $o ).
% 2.62/1.52
% 2.62/1.52 thf(cartprodfstin,definition,
% 2.62/1.52 ( cartprodfstin
% 2.62/1.52 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.62/1.52 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.62/1.52 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(setukpairinjL2_type,type,
% 2.62/1.52 setukpairinjL2: $o ).
% 2.62/1.52
% 2.62/1.52 thf(setukpairinjL2,definition,
% 2.62/1.52 ( setukpairinjL2
% 2.62/1.52 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.62/1.52 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.62/1.52 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.62/1.52 => ( Xx = Xz ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(setukpairinjL_type,type,
% 2.62/1.52 setukpairinjL: $o ).
% 2.62/1.52
% 2.62/1.52 thf(setukpairinjL,definition,
% 2.62/1.52 ( setukpairinjL
% 2.62/1.52 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.62/1.52 ( ( ( kpair @ Xx @ Xy )
% 2.62/1.52 = ( kpair @ Xz @ Xu ) )
% 2.62/1.52 => ( Xx = Xz ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(setukpairinjR11_type,type,
% 2.62/1.52 setukpairinjR11: $o ).
% 2.62/1.52
% 2.62/1.52 thf(setukpairinjR11,definition,
% 2.62/1.52 ( setukpairinjR11
% 2.62/1.52 = ( ! [Xx: $i,Xy: $i] :
% 2.62/1.52 ( ( Xx = Xy )
% 2.62/1.52 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.62/1.52 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(setukpairinjR12_type,type,
% 2.62/1.52 setukpairinjR12: $o ).
% 2.62/1.52
% 2.62/1.52 thf(setukpairinjR12,definition,
% 2.62/1.52 ( setukpairinjR12
% 2.62/1.52 = ( ! [Xx: $i,Xy: $i] :
% 2.62/1.52 ( ( Xx = Xy )
% 2.62/1.52 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.62/1.52 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(setukpairinjR1_type,type,
% 2.62/1.52 setukpairinjR1: $o ).
% 2.62/1.52
% 2.62/1.52 thf(setukpairinjR1,definition,
% 2.62/1.52 ( setukpairinjR1
% 2.62/1.52 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.62/1.52 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.62/1.52 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.62/1.52 => ( ( Xz = Xu )
% 2.62/1.52 => ( Xy = Xu ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(upairequniteq_type,type,
% 2.62/1.52 upairequniteq: $o ).
% 2.62/1.52
% 2.62/1.52 thf(upairequniteq,definition,
% 2.62/1.52 ( upairequniteq
% 2.62/1.52 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.62/1.52 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.62/1.52 = ( setadjoin @ Xz @ emptyset ) )
% 2.62/1.52 => ( Xx = Xy ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(setukpairinjR2_type,type,
% 2.62/1.52 setukpairinjR2: $o ).
% 2.62/1.52
% 2.62/1.52 thf(setukpairinjR2,definition,
% 2.62/1.52 ( setukpairinjR2
% 2.62/1.52 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.62/1.52 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.62/1.52 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.62/1.52 => ( Xy = Xu ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(setukpairinjR_type,type,
% 2.62/1.52 setukpairinjR: $o ).
% 2.62/1.52
% 2.62/1.52 thf(setukpairinjR,definition,
% 2.62/1.52 ( setukpairinjR
% 2.62/1.52 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.62/1.52 ( ( ( kpair @ Xx @ Xy )
% 2.62/1.52 = ( kpair @ Xz @ Xu ) )
% 2.62/1.52 => ( Xy = Xu ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(ksndsingleton_type,type,
% 2.62/1.52 ksndsingleton: $o ).
% 2.62/1.52
% 2.62/1.52 thf(ksndsingleton,definition,
% 2.62/1.52 ( ksndsingleton
% 2.62/1.52 = ( ! [Xu: $i] :
% 2.62/1.52 ( ( iskpair @ Xu )
% 2.62/1.52 => ( singleton
% 2.62/1.52 @ ( dsetconstr @ ( setunion @ Xu )
% 2.62/1.52 @ ^ [Xx: $i] :
% 2.62/1.52 ( Xu
% 2.62/1.52 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(ksnd_type,type,
% 2.62/1.52 ksnd: $i > $i ).
% 2.62/1.52
% 2.62/1.52 thf(ksndpairEq_type,type,
% 2.62/1.52 ksndpairEq: $o ).
% 2.62/1.52
% 2.62/1.52 thf(ksndpairEq,definition,
% 2.62/1.52 ( ksndpairEq
% 2.62/1.52 = ( ! [Xx: $i,Xy: $i] :
% 2.62/1.52 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 2.62/1.52 = Xy ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(kpairsurjEq_type,type,
% 2.62/1.52 kpairsurjEq: $o ).
% 2.62/1.52
% 2.62/1.52 thf(kpairsurjEq,definition,
% 2.62/1.52 ( kpairsurjEq
% 2.62/1.52 = ( ! [Xu: $i] :
% 2.62/1.52 ( ( iskpair @ Xu )
% 2.62/1.52 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 2.62/1.52 = Xu ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(cartprodsndin_type,type,
% 2.62/1.52 cartprodsndin: $o ).
% 2.62/1.52
% 2.62/1.52 thf(cartprodsndin,definition,
% 2.62/1.52 ( cartprodsndin
% 2.62/1.52 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.62/1.52 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.62/1.52 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(cartprodpairmemEL_type,type,
% 2.62/1.52 cartprodpairmemEL: $o ).
% 2.62/1.52
% 2.62/1.52 thf(cartprodpairmemEL,definition,
% 2.62/1.52 ( cartprodpairmemEL
% 2.62/1.52 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 2.62/1.52 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 2.62/1.52 => ( in @ Xx @ A ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(cartprodpairmemER_type,type,
% 2.62/1.52 cartprodpairmemER: $o ).
% 2.62/1.52
% 2.62/1.52 thf(cartprodpairmemER,definition,
% 2.62/1.52 ( cartprodpairmemER
% 2.62/1.52 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 2.62/1.52 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 2.62/1.52 => ( in @ Xy @ B ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(cartprodmempaircEq_type,type,
% 2.62/1.52 cartprodmempaircEq: $o ).
% 2.62/1.52
% 2.62/1.52 thf(cartprodmempaircEq,definition,
% 2.62/1.52 ( cartprodmempaircEq
% 2.62/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.62/1.52 ( ( in @ Xx @ A )
% 2.62/1.52 => ! [Xy: $i] :
% 2.62/1.52 ( ( in @ Xy @ B )
% 2.62/1.52 => ( ( kpair @ Xx @ Xy )
% 2.62/1.52 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(cartprodfstpairEq_type,type,
% 2.62/1.52 cartprodfstpairEq: $o ).
% 2.62/1.52
% 2.62/1.52 thf(cartprodfstpairEq,definition,
% 2.62/1.52 ( cartprodfstpairEq
% 2.62/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.62/1.52 ( ( in @ Xx @ A )
% 2.62/1.52 => ! [Xy: $i] :
% 2.62/1.52 ( ( in @ Xy @ B )
% 2.62/1.52 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 2.62/1.52 = Xx ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(cartprodsndpairEq_type,type,
% 2.62/1.52 cartprodsndpairEq: $o ).
% 2.62/1.52
% 2.62/1.52 thf(cartprodsndpairEq,definition,
% 2.62/1.52 ( cartprodsndpairEq
% 2.62/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.62/1.52 ( ( in @ Xx @ A )
% 2.62/1.52 => ! [Xy: $i] :
% 2.62/1.52 ( ( in @ Xy @ B )
% 2.62/1.52 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 2.62/1.52 = Xy ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(cartprodpairsurjEq_type,type,
% 2.62/1.52 cartprodpairsurjEq: $o ).
% 2.62/1.52
% 2.62/1.52 thf(cartprodpairsurjEq,definition,
% 2.62/1.52 ( cartprodpairsurjEq
% 2.62/1.52 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.62/1.52 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.62/1.52 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 2.62/1.52 = Xu ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(breln_type,type,
% 2.62/1.52 breln: $i > $i > $i > $o ).
% 2.62/1.52
% 2.62/1.52 thf(breln,definition,
% 2.62/1.52 ( breln
% 2.62/1.52 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(dpsetconstr_type,type,
% 2.62/1.52 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 2.62/1.52
% 2.62/1.52 thf(dpsetconstrI_type,type,
% 2.62/1.52 dpsetconstrI: $o ).
% 2.62/1.52
% 2.62/1.52 thf(dpsetconstrI,definition,
% 2.62/1.52 ( dpsetconstrI
% 2.62/1.52 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 2.62/1.52 ( ( in @ Xx @ A )
% 2.62/1.52 => ! [Xy: $i] :
% 2.62/1.52 ( ( in @ Xy @ B )
% 2.62/1.52 => ( ( Xphi @ Xx @ Xy )
% 2.62/1.52 => ( in @ ( kpair @ Xx @ Xy )
% 2.62/1.52 @ ( dpsetconstr @ A @ B
% 2.62/1.52 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(dpsetconstrSub_type,type,
% 2.62/1.52 dpsetconstrSub: $o ).
% 2.62/1.52
% 2.62/1.52 thf(dpsetconstrSub,definition,
% 2.62/1.52 ( dpsetconstrSub
% 2.62/1.52 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 2.62/1.52 ( subset
% 2.62/1.52 @ ( dpsetconstr @ A @ B
% 2.62/1.52 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 2.62/1.52 @ ( cartprod @ A @ B ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(setOfPairsIsBReln_type,type,
% 2.62/1.52 setOfPairsIsBReln: $o ).
% 2.62/1.52
% 2.62/1.52 thf(setOfPairsIsBReln,definition,
% 2.62/1.52 ( setOfPairsIsBReln
% 2.62/1.52 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 2.62/1.52 ( breln @ A @ B
% 2.62/1.52 @ ( dpsetconstr @ A @ B
% 2.62/1.52 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(dpsetconstrERa_type,type,
% 2.62/1.52 dpsetconstrERa: $o ).
% 2.62/1.52
% 2.62/1.52 thf(dpsetconstrERa,definition,
% 2.62/1.52 ( dpsetconstrERa
% 2.62/1.52 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 2.62/1.52 ( ( in @ Xx @ A )
% 2.62/1.52 => ! [Xy: $i] :
% 2.62/1.52 ( ( in @ Xy @ B )
% 2.62/1.52 => ( ( in @ ( kpair @ Xx @ Xy )
% 2.62/1.52 @ ( dpsetconstr @ A @ B
% 2.62/1.52 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.62/1.52 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(dpsetconstrEL1_type,type,
% 2.62/1.52 dpsetconstrEL1: $o ).
% 2.62/1.52
% 2.62/1.52 thf(dpsetconstrEL1,definition,
% 2.62/1.52 ( dpsetconstrEL1
% 2.62/1.52 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.62/1.52 ( ( in @ ( kpair @ Xx @ Xy )
% 2.62/1.52 @ ( dpsetconstr @ A @ B
% 2.62/1.52 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.62/1.52 => ( in @ Xx @ A ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(dpsetconstrEL2_type,type,
% 2.62/1.52 dpsetconstrEL2: $o ).
% 2.62/1.52
% 2.62/1.52 thf(dpsetconstrEL2,definition,
% 2.62/1.52 ( dpsetconstrEL2
% 2.62/1.52 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.62/1.52 ( ( in @ ( kpair @ Xx @ Xy )
% 2.62/1.52 @ ( dpsetconstr @ A @ B
% 2.62/1.52 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.62/1.52 => ( in @ Xy @ B ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(dpsetconstrER_type,type,
% 2.62/1.52 dpsetconstrER: $o ).
% 2.62/1.52
% 2.62/1.52 thf(dpsetconstrER,definition,
% 2.62/1.52 ( dpsetconstrER
% 2.62/1.52 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.62/1.52 ( ( in @ ( kpair @ Xx @ Xy )
% 2.62/1.52 @ ( dpsetconstr @ A @ B
% 2.62/1.52 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.62/1.52 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(func_type,type,
% 2.62/1.52 func: $i > $i > $i > $o ).
% 2.62/1.52
% 2.62/1.52 thf(funcSet_type,type,
% 2.62/1.52 funcSet: $i > $i > $i ).
% 2.62/1.52
% 2.62/1.52 thf(funcImageSingleton_type,type,
% 2.62/1.52 funcImageSingleton: $o ).
% 2.62/1.52
% 2.62/1.52 thf(funcImageSingleton,definition,
% 2.62/1.52 ( funcImageSingleton
% 2.62/1.52 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.62/1.52 ( ( func @ A @ B @ Xf )
% 2.62/1.52 => ! [Xx: $i] :
% 2.62/1.52 ( ( in @ Xx @ A )
% 2.62/1.52 => ( singleton
% 2.62/1.52 @ ( dsetconstr @ B
% 2.62/1.52 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(apProp_type,type,
% 2.62/1.52 apProp: $o ).
% 2.62/1.52
% 2.62/1.52 thf(apProp,definition,
% 2.62/1.52 ( apProp
% 2.62/1.52 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.62/1.52 ( ( func @ A @ B @ Xf )
% 2.62/1.52 => ! [Xx: $i] :
% 2.62/1.52 ( ( in @ Xx @ A )
% 2.62/1.52 => ( in
% 2.62/1.52 @ ( setunion
% 2.62/1.52 @ ( dsetconstr @ B
% 2.62/1.52 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 2.62/1.52 @ B ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(ap_type,type,
% 2.62/1.52 ap: $i > $i > $i > $i > $i ).
% 2.62/1.52
% 2.62/1.52 thf(app_type,type,
% 2.62/1.52 app: $o ).
% 2.62/1.52
% 2.62/1.52 thf(app,definition,
% 2.62/1.52 ( app
% 2.62/1.52 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.62/1.52 ( ( func @ A @ B @ Xf )
% 2.62/1.52 => ! [Xx: $i] :
% 2.62/1.52 ( ( in @ Xx @ A )
% 2.62/1.52 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(infuncsetfunc_type,type,
% 2.62/1.52 infuncsetfunc: $o ).
% 2.62/1.52
% 2.62/1.52 thf(infuncsetfunc,definition,
% 2.62/1.52 ( infuncsetfunc
% 2.62/1.52 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.62/1.52 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.62/1.52 => ( func @ A @ B @ Xf ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(ap2p_type,type,
% 2.62/1.52 ap2p: $o ).
% 2.62/1.52
% 2.62/1.52 thf(ap2p,definition,
% 2.62/1.52 ( ap2p
% 2.62/1.52 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.62/1.52 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.62/1.52 => ! [Xx: $i] :
% 2.62/1.52 ( ( in @ Xx @ A )
% 2.62/1.52 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(funcinfuncset_type,type,
% 2.62/1.52 funcinfuncset: $o ).
% 2.62/1.52
% 2.62/1.52 thf(funcinfuncset,definition,
% 2.62/1.52 ( funcinfuncset
% 2.62/1.52 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.62/1.52 ( ( func @ A @ B @ Xf )
% 2.62/1.52 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(lamProp_type,type,
% 2.62/1.52 lamProp: $o ).
% 2.62/1.52
% 2.62/1.52 thf(lamProp,definition,
% 2.62/1.52 ( lamProp
% 2.62/1.52 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.62/1.52 ( ! [Xx: $i] :
% 2.62/1.52 ( ( in @ Xx @ A )
% 2.62/1.52 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.62/1.52 => ( func @ A @ B
% 2.62/1.52 @ ( dpsetconstr @ A @ B
% 2.62/1.52 @ ^ [Xx: $i,Xy: $i] :
% 2.62/1.52 ( ( Xf @ Xx )
% 2.62/1.52 = Xy ) ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(lam_type,type,
% 2.62/1.52 lam: $i > $i > ( $i > $i ) > $i ).
% 2.62/1.52
% 2.62/1.52 thf(lam,definition,
% 2.62/1.52 ( lam
% 2.62/1.52 = ( ^ [A: $i,B: $i,Xf: $i > $i] :
% 2.62/1.52 ( dpsetconstr @ A @ B
% 2.62/1.52 @ ^ [Xx: $i,Xy: $i] :
% 2.62/1.52 ( ( Xf @ Xx )
% 2.62/1.52 = Xy ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(lamp_type,type,
% 2.62/1.52 lamp: $o ).
% 2.62/1.52
% 2.62/1.52 thf(lamp,definition,
% 2.62/1.52 ( lamp
% 2.62/1.52 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.62/1.52 ( ! [Xx: $i] :
% 2.62/1.52 ( ( in @ Xx @ A )
% 2.62/1.52 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.62/1.52 => ( func @ A @ B
% 2.62/1.52 @ ( lam @ A @ B
% 2.62/1.52 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(lam2p_type,type,
% 2.62/1.52 lam2p: $o ).
% 2.62/1.52
% 2.62/1.52 thf(lam2p,definition,
% 2.62/1.52 ( lam2p
% 2.62/1.52 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.62/1.52 ( ! [Xx: $i] :
% 2.62/1.52 ( ( in @ Xx @ A )
% 2.62/1.52 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.62/1.52 => ( in
% 2.62/1.52 @ ( lam @ A @ B
% 2.62/1.52 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 2.62/1.52 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(brelnall1_type,type,
% 2.62/1.52 brelnall1: $o ).
% 2.62/1.52
% 2.62/1.52 thf(brelnall1,definition,
% 2.62/1.52 ( brelnall1
% 2.62/1.52 = ( ! [A: $i,B: $i,R: $i] :
% 2.62/1.52 ( ( breln @ A @ B @ R )
% 2.62/1.52 => ! [Xphi: $i > $o] :
% 2.62/1.52 ( ! [Xx: $i] :
% 2.62/1.52 ( ( in @ Xx @ A )
% 2.62/1.52 => ! [Xy: $i] :
% 2.62/1.52 ( ( in @ Xy @ B )
% 2.62/1.52 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.62/1.52 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 2.62/1.52 => ! [Xx: $i] :
% 2.62/1.52 ( ( in @ Xx @ R )
% 2.62/1.52 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(brelnall2_type,type,
% 2.62/1.52 brelnall2: $o ).
% 2.62/1.52
% 2.62/1.52 thf(brelnall2,definition,
% 2.62/1.52 ( brelnall2
% 2.62/1.52 = ( ! [A: $i,B: $i,R: $i] :
% 2.62/1.52 ( ( breln @ A @ B @ R )
% 2.62/1.52 => ! [Xphi: $i > $o] :
% 2.62/1.52 ( ! [Xx: $i] :
% 2.62/1.52 ( ( in @ Xx @ A )
% 2.62/1.52 => ! [Xy: $i] :
% 2.62/1.52 ( ( in @ Xy @ B )
% 2.62/1.52 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.62/1.52 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 2.62/1.52 => ! [Xx: $i] :
% 2.62/1.52 ( ( in @ Xx @ R )
% 2.62/1.52 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(ex1E2_type,type,
% 2.62/1.52 ex1E2: $o ).
% 2.62/1.52
% 2.62/1.52 thf(ex1E2,definition,
% 2.62/1.52 ( ex1E2
% 2.62/1.52 = ( ! [A: $i,Xphi: $i > $o] :
% 2.62/1.52 ( ( ex1 @ A
% 2.62/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.62/1.52 => ! [Xx: $i] :
% 2.62/1.52 ( ( in @ Xx @ A )
% 2.62/1.52 => ! [Xy: $i] :
% 2.62/1.52 ( ( in @ Xy @ A )
% 2.62/1.52 => ( ( Xphi @ Xx )
% 2.62/1.52 => ( ( Xphi @ Xy )
% 2.62/1.52 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(funcGraphProp1_type,type,
% 2.62/1.52 funcGraphProp1: $o ).
% 2.62/1.52
% 2.62/1.52 thf(funcGraphProp1,definition,
% 2.62/1.52 ( funcGraphProp1
% 2.62/1.52 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.62/1.52 ( ( func @ A @ B @ Xf )
% 2.62/1.52 => ! [Xx: $i] :
% 2.62/1.52 ( ( in @ Xx @ A )
% 2.62/1.52 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(funcGraphProp3_type,type,
% 2.62/1.52 funcGraphProp3: $o ).
% 2.62/1.52
% 2.62/1.52 thf(funcGraphProp3,definition,
% 2.62/1.52 ( funcGraphProp3
% 2.62/1.52 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.62/1.52 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.62/1.52 => ! [Xx: $i] :
% 2.62/1.52 ( ( in @ Xx @ A )
% 2.62/1.52 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(funcGraphProp2_type,type,
% 2.62/1.52 funcGraphProp2: $o ).
% 2.62/1.52
% 2.62/1.52 thf(funcGraphProp2,definition,
% 2.62/1.52 ( funcGraphProp2
% 2.62/1.52 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.62/1.52 ( ( func @ A @ B @ Xf )
% 2.62/1.52 => ! [Xx: $i] :
% 2.62/1.52 ( ( in @ Xx @ A )
% 2.62/1.52 => ! [Xy: $i] :
% 2.62/1.52 ( ( in @ Xy @ B )
% 2.62/1.52 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 2.62/1.52 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.62/1.52 = Xy ) ) ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(funcextLem_type,type,
% 2.62/1.52 funcextLem: $o ).
% 2.62/1.52
% 2.62/1.52 thf(funcextLem,definition,
% 2.62/1.52 ( funcextLem
% 2.62/1.52 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.62/1.52 ( ( func @ A @ B @ Xf )
% 2.62/1.52 => ! [Xg: $i] :
% 2.62/1.52 ( ( func @ A @ B @ Xg )
% 2.62/1.52 => ( ! [Xx: $i] :
% 2.62/1.52 ( ( in @ Xx @ A )
% 2.62/1.52 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.62/1.52 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.62/1.52 => ! [Xx: $i] :
% 2.62/1.52 ( ( in @ Xx @ A )
% 2.62/1.52 => ! [Xy: $i] :
% 2.62/1.52 ( ( in @ Xy @ B )
% 2.62/1.52 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 2.62/1.52 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(funcGraphProp4_type,type,
% 2.62/1.52 funcGraphProp4: $o ).
% 2.62/1.52
% 2.62/1.52 thf(funcGraphProp4,definition,
% 2.62/1.52 ( funcGraphProp4
% 2.62/1.52 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.62/1.52 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.62/1.52 => ! [Xx: $i] :
% 2.62/1.52 ( ( in @ Xx @ A )
% 2.62/1.52 => ! [Xy: $i] :
% 2.62/1.52 ( ( in @ Xy @ B )
% 2.62/1.52 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 2.62/1.52 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.62/1.52 = Xy ) ) ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(subbreln_type,type,
% 2.62/1.52 subbreln: $o ).
% 2.62/1.52
% 2.62/1.52 thf(subbreln,definition,
% 2.62/1.52 ( subbreln
% 2.62/1.52 = ( ! [A: $i,B: $i,R: $i] :
% 2.62/1.52 ( ( breln @ A @ B @ R )
% 2.62/1.52 => ! [S: $i] :
% 2.62/1.52 ( ( breln @ A @ B @ S )
% 2.62/1.52 => ( ! [Xx: $i] :
% 2.62/1.52 ( ( in @ Xx @ A )
% 2.62/1.52 => ! [Xy: $i] :
% 2.62/1.52 ( ( in @ Xy @ B )
% 2.62/1.52 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.62/1.52 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 2.62/1.52 => ( subset @ R @ S ) ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(eqbreln_type,type,
% 2.62/1.52 eqbreln: $o ).
% 2.62/1.52
% 2.62/1.52 thf(eqbreln,definition,
% 2.62/1.52 ( eqbreln
% 2.62/1.52 = ( ! [A: $i,B: $i,R: $i] :
% 2.62/1.52 ( ( breln @ A @ B @ R )
% 2.62/1.52 => ! [S: $i] :
% 2.62/1.52 ( ( breln @ A @ B @ S )
% 2.62/1.52 => ( ! [Xx: $i] :
% 2.62/1.52 ( ( in @ Xx @ A )
% 2.62/1.52 => ! [Xy: $i] :
% 2.62/1.52 ( ( in @ Xy @ B )
% 2.62/1.52 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.62/1.52 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 2.62/1.52 => ( ! [Xx: $i] :
% 2.62/1.52 ( ( in @ Xx @ A )
% 2.62/1.52 => ! [Xy: $i] :
% 2.62/1.52 ( ( in @ Xy @ B )
% 2.62/1.52 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 2.62/1.52 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 2.62/1.52 => ( R = S ) ) ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(funcext_type,type,
% 2.62/1.52 funcext: $o ).
% 2.62/1.52
% 2.62/1.52 thf(funcext,definition,
% 2.62/1.52 ( funcext
% 2.62/1.52 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.62/1.52 ( ( func @ A @ B @ Xf )
% 2.62/1.52 => ! [Xg: $i] :
% 2.62/1.52 ( ( func @ A @ B @ Xg )
% 2.62/1.52 => ( ! [Xx: $i] :
% 2.62/1.52 ( ( in @ Xx @ A )
% 2.62/1.52 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.62/1.52 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.62/1.52 => ( Xf = Xg ) ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(funcext2_type,type,
% 2.62/1.52 funcext2: $o ).
% 2.62/1.52
% 2.62/1.52 thf(funcext2,definition,
% 2.62/1.52 ( funcext2
% 2.62/1.52 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.62/1.52 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.62/1.52 => ! [Xg: $i] :
% 2.62/1.52 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 2.62/1.52 => ( ! [Xx: $i] :
% 2.62/1.52 ( ( in @ Xx @ A )
% 2.62/1.52 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.62/1.52 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.62/1.52 => ( Xf = Xg ) ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(ap2apEq1_type,type,
% 2.62/1.52 ap2apEq1: $o ).
% 2.62/1.52
% 2.62/1.52 thf(ap2apEq1,definition,
% 2.62/1.52 ( ap2apEq1
% 2.62/1.52 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.62/1.52 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.62/1.52 => ! [Xx: $i] :
% 2.62/1.52 ( ( in @ Xx @ A )
% 2.62/1.52 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.62/1.52 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(ap2apEq2_type,type,
% 2.62/1.52 ap2apEq2: $o ).
% 2.62/1.52
% 2.62/1.52 thf(ap2apEq2,definition,
% 2.62/1.52 ( ap2apEq2
% 2.62/1.52 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.62/1.52 ( ( func @ A @ B @ Xf )
% 2.62/1.52 => ! [Xx: $i] :
% 2.62/1.52 ( ( in @ Xx @ A )
% 2.62/1.52 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.62/1.52 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(beta1_type,type,
% 2.62/1.52 beta1: $o ).
% 2.62/1.52
% 2.62/1.52 thf(beta1,definition,
% 2.62/1.52 ( beta1
% 2.62/1.52 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.62/1.52 ( ! [Xx: $i] :
% 2.62/1.52 ( ( in @ Xx @ A )
% 2.62/1.52 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.62/1.52 => ! [Xx: $i] :
% 2.62/1.52 ( ( in @ Xx @ A )
% 2.62/1.52 => ( ( ap @ A @ B
% 2.62/1.52 @ ( lam @ A @ B
% 2.62/1.52 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 2.62/1.52 @ Xx )
% 2.62/1.52 = ( Xf @ Xx ) ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(eta1_type,type,
% 2.62/1.52 eta1: $o ).
% 2.62/1.52
% 2.62/1.52 thf(eta1,definition,
% 2.62/1.52 ( eta1
% 2.62/1.52 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.62/1.52 ( ( func @ A @ B @ Xf )
% 2.62/1.52 => ( ( lam @ A @ B
% 2.62/1.52 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 2.62/1.52 = Xf ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(lam2lamEq_type,type,
% 2.62/1.52 lam2lamEq: $o ).
% 2.62/1.52
% 2.62/1.52 thf(lam2lamEq,definition,
% 2.62/1.52 ( lam2lamEq
% 2.62/1.52 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.62/1.52 ( ! [Xx: $i] :
% 2.62/1.52 ( ( in @ Xx @ A )
% 2.62/1.52 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.62/1.52 => ( ( lam @ A @ B
% 2.62/1.52 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 2.62/1.52 = ( lam @ A @ B
% 2.62/1.52 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 2.62/1.52
% 2.62/1.52 thf(beta2,conjecture,
% 2.62/1.52 ( setextAx
% 2.62/1.52 => ( emptysetAx
% 2.62/1.52 => ( setadjoinAx
% 2.62/1.52 => ( powersetAx
% 2.62/1.52 => ( setunionAx
% 2.62/1.52 => ( omega0Ax
% 2.62/1.52 => ( omegaSAx
% 2.62/1.52 => ( omegaIndAx
% 2.62/1.52 => ( replAx
% 2.62/1.52 => ( foundationAx
% 2.62/1.52 => ( wellorderingAx
% 2.62/1.52 => ( descrp
% 2.62/1.52 => ( dsetconstrI
% 2.62/1.52 => ( dsetconstrEL
% 2.62/1.52 => ( dsetconstrER
% 2.62/1.52 => ( exuE1
% 2.62/1.52 => ( prop2setE
% 2.62/1.52 => ( emptysetE
% 2.62/1.52 => ( emptysetimpfalse
% 2.62/1.52 => ( notinemptyset
% 2.62/1.52 => ( exuE3e
% 2.62/1.52 => ( setext
% 2.62/1.52 => ( emptyI
% 2.62/1.52 => ( noeltsimpempty
% 2.62/1.52 => ( setbeta
% 2.62/1.52 => ( nonemptyE1
% 2.62/1.52 => ( nonemptyI
% 2.62/1.52 => ( nonemptyI1
% 2.62/1.52 => ( setadjoinIL
% 2.62/1.52 => ( emptyinunitempty
% 2.62/1.52 => ( setadjoinIR
% 2.62/1.52 => ( setadjoinE
% 2.62/1.52 => ( setadjoinOr
% 2.62/1.52 => ( setoftrueEq
% 2.62/1.52 => ( powersetI
% 2.62/1.52 => ( emptyinPowerset
% 2.62/1.52 => ( emptyInPowerset
% 2.62/1.52 => ( powersetE
% 2.62/1.52 => ( setunionI
% 2.62/1.52 => ( setunionE
% 2.62/1.52 => ( subPowSU
% 2.62/1.52 => ( exuE2
% 2.62/1.52 => ( nonemptyImpWitness
% 2.62/1.52 => ( uniqinunit
% 2.62/1.52 => ( notinsingleton
% 2.62/1.52 => ( eqinunit
% 2.62/1.52 => ( singletonsswitch
% 2.62/1.52 => ( upairsetE
% 2.62/1.52 => ( upairsetIL
% 2.62/1.52 => ( upairsetIR
% 2.62/1.52 => ( emptyE1
% 2.62/1.52 => ( vacuousDall
% 2.62/1.52 => ( quantDeMorgan1
% 2.62/1.52 => ( quantDeMorgan2
% 2.62/1.52 => ( quantDeMorgan3
% 2.62/1.52 => ( quantDeMorgan4
% 2.62/1.52 => ( prop2setI
% 2.62/1.52 => ( prop2set2propI
% 2.62/1.52 => ( notdexE
% 2.62/1.52 => ( notdallE
% 2.62/1.52 => ( exuI1
% 2.62/1.52 => ( exuI3
% 2.62/1.52 => ( exuI2
% 2.62/1.52 => ( inCongP
% 2.62/1.52 => ( in__Cong
% 2.62/1.52 => ( exuE3u
% 2.62/1.52 => ( exu__Cong
% 2.62/1.52 => ( emptyset__Cong
% 2.62/1.52 => ( setadjoin__Cong
% 2.62/1.52 => ( powerset__Cong
% 2.62/1.52 => ( setunion__Cong
% 2.62/1.52 => ( omega__Cong
% 2.62/1.52 => ( exuEu
% 2.62/1.52 => ( descr__Cong
% 2.62/1.52 => ( dsetconstr__Cong
% 2.62/1.52 => ( subsetI1
% 2.62/1.52 => ( eqimpsubset2
% 2.62/1.52 => ( eqimpsubset1
% 2.62/1.52 => ( subsetI2
% 2.62/1.52 => ( emptysetsubset
% 2.62/1.52 => ( subsetE
% 2.62/1.52 => ( subsetE2
% 2.62/1.52 => ( notsubsetI
% 2.62/1.52 => ( notequalI1
% 2.62/1.52 => ( notequalI2
% 2.62/1.52 => ( subsetRefl
% 2.62/1.52 => ( subsetTrans
% 2.62/1.52 => ( setadjoinSub
% 2.62/1.52 => ( setadjoinSub2
% 2.62/1.52 => ( subset2powerset
% 2.62/1.52 => ( setextsub
% 2.62/1.52 => ( subsetemptysetimpeq
% 2.62/1.52 => ( powersetI1
% 2.62/1.52 => ( powersetE1
% 2.62/1.52 => ( inPowerset
% 2.62/1.52 => ( powersetsubset
% 2.62/1.52 => ( sepInPowerset
% 2.62/1.52 => ( sepSubset
% 2.62/1.52 => ( binunionIL
% 2.62/1.52 => ( upairset2IR
% 2.62/1.52 => ( binunionIR
% 2.62/1.52 => ( binunionEcases
% 2.62/1.52 => ( binunionE
% 2.62/1.52 => ( binunionLsub
% 2.62/1.52 => ( binunionRsub
% 2.62/1.52 => ( binintersectI
% 2.62/1.52 => ( binintersectSubset5
% 2.62/1.52 => ( binintersectEL
% 2.62/1.52 => ( binintersectLsub
% 2.62/1.52 => ( binintersectSubset2
% 2.62/1.52 => ( binintersectSubset3
% 2.62/1.52 => ( binintersectER
% 2.62/1.52 => ( disjointsetsI1
% 2.62/1.52 => ( binintersectRsub
% 2.62/1.52 => ( binintersectSubset4
% 2.62/1.52 => ( binintersectSubset1
% 2.62/1.52 => ( bs114d
% 2.62/1.52 => ( setminusI
% 2.62/1.52 => ( setminusEL
% 2.62/1.52 => ( setminusER
% 2.62/1.52 => ( setminusSubset2
% 2.62/1.52 => ( setminusERneg
% 2.62/1.52 => ( setminusELneg
% 2.62/1.52 => ( setminusILneg
% 2.62/1.52 => ( setminusIRneg
% 2.62/1.52 => ( setminusLsub
% 2.62/1.52 => ( setminusSubset1
% 2.62/1.52 => ( symdiffE
% 2.62/1.52 => ( symdiffI1
% 2.62/1.52 => ( symdiffI2
% 2.62/1.52 => ( symdiffIneg1
% 2.62/1.52 => ( symdiffIneg2
% 2.62/1.52 => ( secondinupair
% 2.62/1.52 => ( setukpairIL
% 2.62/1.52 => ( setukpairIR
% 2.62/1.52 => ( kpairiskpair
% 2.62/1.52 => ( kpairp
% 2.62/1.52 => ( singletonsubset
% 2.62/1.52 => ( singletoninpowerset
% 2.62/1.52 => ( singletoninpowunion
% 2.62/1.52 => ( upairset2E
% 2.62/1.52 => ( upairsubunion
% 2.62/1.52 => ( upairinpowunion
% 2.62/1.52 => ( ubforcartprodlem1
% 2.62/1.52 => ( ubforcartprodlem2
% 2.62/1.52 => ( ubforcartprodlem3
% 2.62/1.52 => ( cartprodpairin
% 2.62/1.52 => ( cartprodmempair1
% 2.62/1.52 => ( cartprodmempair
% 2.62/1.52 => ( setunionE2
% 2.62/1.52 => ( setunionsingleton1
% 2.62/1.52 => ( setunionsingleton2
% 2.62/1.52 => ( setunionsingleton
% 2.62/1.52 => ( singletonprop
% 2.62/1.52 => ( ex1E1
% 2.62/1.52 => ( ex1I
% 2.62/1.52 => ( ex1I2
% 2.62/1.52 => ( singletonsuniq
% 2.62/1.52 => ( setukpairinjL1
% 2.62/1.52 => ( kfstsingleton
% 2.62/1.52 => ( theprop
% 2.62/1.52 => ( kfstpairEq
% 2.62/1.52 => ( cartprodfstin
% 2.62/1.52 => ( setukpairinjL2
% 2.62/1.52 => ( setukpairinjL
% 2.62/1.52 => ( setukpairinjR11
% 2.62/1.52 => ( setukpairinjR12
% 2.62/1.52 => ( setukpairinjR1
% 2.62/1.52 => ( upairequniteq
% 2.62/1.52 => ( setukpairinjR2
% 2.62/1.52 => ( setukpairinjR
% 2.62/1.52 => ( ksndsingleton
% 2.62/1.52 => ( ksndpairEq
% 2.62/1.52 => ( kpairsurjEq
% 2.62/1.52 => ( cartprodsndin
% 2.62/1.52 => ( cartprodpairmemEL
% 2.62/1.52 => ( cartprodpairmemER
% 2.62/1.52 => ( cartprodmempaircEq
% 2.62/1.52 => ( cartprodfstpairEq
% 2.62/1.52 => ( cartprodsndpairEq
% 2.62/1.52 => ( cartprodpairsurjEq
% 2.62/1.52 => ( dpsetconstrI
% 2.62/1.52 => ( dpsetconstrSub
% 2.62/1.52 => ( setOfPairsIsBReln
% 2.62/1.52 => ( dpsetconstrERa
% 2.62/1.52 => ( dpsetconstrEL1
% 2.62/1.52 => ( dpsetconstrEL2
% 2.62/1.52 => ( dpsetconstrER
% 2.62/1.52 => ( funcImageSingleton
% 2.62/1.52 => ( apProp
% 2.62/1.52 => ( app
% 2.62/1.52 => ( infuncsetfunc
% 2.62/1.52 => ( ap2p
% 2.62/1.52 => ( funcinfuncset
% 2.62/1.52 => ( lamProp
% 2.62/1.52 => ( lamp
% 2.62/1.52 => ( lam2p
% 2.62/1.52 => ( brelnall1
% 2.62/1.52 => ( brelnall2
% 2.62/1.52 => ( ex1E2
% 2.62/1.52 => ( funcGraphProp1
% 2.62/1.52 => ( funcGraphProp3
% 2.62/1.52 => ( funcGraphProp2
% 2.62/1.52 => ( funcextLem
% 2.62/1.52 => ( funcGraphProp4
% 2.62/1.52 => ( subbreln
% 2.62/1.52 => ( eqbreln
% 2.62/1.52 => ( funcext
% 2.62/1.52 => ( funcext2
% 2.62/1.52 => ( ap2apEq1
% 2.62/1.52 => ( ap2apEq2
% 2.62/1.52 => ( beta1
% 2.62/1.52 => ( eta1
% 2.62/1.52 => ( lam2lamEq
% 2.62/1.52 => ! [A: $i,B: $i,Xf: $i > $i] :
% 2.62/1.52 ( ! [Xx: $i] :
% 2.62/1.52 ( ( in @ Xx @ A )
% 2.62/1.52 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.62/1.52 => ! [Xx: $i] :
% 2.62/1.52 ( ( in @ Xx @ A )
% 2.62/1.52 => ( ( ap @ A @ B
% 2.62/1.52 @ ( lam @ A @ B
% 2.62/1.52 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 2.62/1.52 @ Xx )
% 2.62/1.52 = ( Xf @ Xx ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.62/1.52 ------------------------
% 2.62/1.55 % (3300)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_3007 for (2999ds/27Mi)
% 2.62/1.55 % (3301)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_3007 for (2999ds/2Mi)
% 2.62/1.55 % (3301)Instruction limit reached!
% 2.62/1.55 % (3301)------------------------------
% 2.62/1.55 % (3301)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.62/1.55 % (3301)Termination reason: Unknown
% 2.62/1.55 % (3301)Termination phase: shuffling
% 2.62/1.55
% 2.62/1.55 % (3301)Memory used [KB]: 1407
% 2.62/1.55 % (3301)Time elapsed: 0.003 s
% 2.62/1.55 % (3301)Instructions burned: 4 (million)
% 2.62/1.55 % (3301)------------------------------
% 2.62/1.55 % (3301)------------------------------
% 2.62/1.55 % (3298)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_3007 for (2999ds/183Mi)
% 2.62/1.56 % (3299)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_3007 for (2999ds/4Mi)
% 2.62/1.56 % (3302)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_3007 for (2999ds/2Mi)
% 2.62/1.56 % (3303)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_3007 for (2999ds/275Mi)
% 2.62/1.56 % (3304)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_3007 for (2999ds/18Mi)
% 2.62/1.56 % (3300)Instruction limit reached!
% 2.62/1.56 % (3300)------------------------------
% 2.62/1.56 % (3300)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.62/1.56 % (3300)Termination reason: Unknown
% 2.62/1.56 % (3300)Termination phase: shuffling
% 2.62/1.56
% 2.62/1.56 % (3300)Memory used [KB]: 1918
% 2.62/1.56 % (3300)Time elapsed: 0.010 s
% 2.62/1.56 % (3300)Instructions burned: 30 (million)
% 2.62/1.56 % (3300)------------------------------
% 2.62/1.56 % (3300)------------------------------
% 2.62/1.56 % (3302)Instruction limit reached!
% 2.62/1.56 % (3302)------------------------------
% 2.62/1.56 % (3302)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.62/1.56 % (3302)Termination reason: Unknown
% 2.62/1.56 % (3302)Termination phase: shuffling
% 2.62/1.56
% 2.62/1.56 % (3302)Memory used [KB]: 1407
% 2.62/1.56 % (3302)Time elapsed: 0.004 s
% 2.62/1.56 % (3302)Instructions burned: 3 (million)
% 2.62/1.56 % (3302)------------------------------
% 2.62/1.56 % (3302)------------------------------
% 2.62/1.56 % (3299)Instruction limit reached!
% 2.62/1.56 % (3299)------------------------------
% 2.62/1.56 % (3299)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.62/1.56 % (3299)Termination reason: Unknown
% 2.62/1.56 % (3299)Termination phase: shuffling
% 2.62/1.56
% 2.62/1.56 % (3299)Memory used [KB]: 1407
% 2.62/1.56 % (3299)Time elapsed: 0.005 s
% 2.62/1.56 % (3299)Instructions burned: 5 (million)
% 2.62/1.56 % (3299)------------------------------
% 2.62/1.56 % (3299)------------------------------
% 2.62/1.56 % (3305)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_3007 for (2999ds/3Mi)
% 2.62/1.56 % (3305)Instruction limit reached!
% 2.62/1.56 % (3305)------------------------------
% 2.62/1.56 % (3305)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.62/1.56 % (3305)Termination reason: Unknown
% 2.62/1.56 % (3305)Termination phase: shuffling
% 2.62/1.56
% 2.62/1.56 % (3305)Memory used [KB]: 1407
% 2.62/1.56 % (3305)Time elapsed: 0.002 s
% 2.62/1.56 % (3305)Instructions burned: 3 (million)
% 2.62/1.56 % (3305)------------------------------
% 2.62/1.56 % (3305)------------------------------
% 2.62/1.57 % (3306)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_3007 for (2999ds/37Mi)
% 2.62/1.57 % (3304)Instruction limit reached!
% 2.62/1.57 % (3304)------------------------------
% 2.62/1.57 % (3304)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.62/1.57 % (3304)Termination reason: Unknown
% 2.62/1.57 % (3304)Termination phase: shuffling
% 2.62/1.57
% 2.62/1.57 % (3304)Memory used [KB]: 1791
% 2.62/1.57 % (3304)Time elapsed: 0.014 s
% 2.62/1.57 % (3304)Instructions burned: 19 (million)
% 2.62/1.57 % (3304)------------------------------
% 2.62/1.57 % (3304)------------------------------
% 2.62/1.57 % (3309)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_3007 for (2999ds/1041Mi)
% 2.62/1.58 % (3307)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_3007 for (2999ds/15Mi)
% 2.62/1.58 % (3308)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_3007 for (2999ds/3Mi)
% 2.62/1.58 % (3306)Instruction limit reached!
% 2.62/1.58 % (3306)------------------------------
% 2.62/1.58 % (3306)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.62/1.58 % (3306)Termination reason: Unknown
% 2.62/1.58 % (3306)Termination phase: Property scanning
% 2.62/1.58
% 2.62/1.58 % (3306)Memory used [KB]: 2174
% 2.62/1.58 % (3306)Time elapsed: 0.012 s
% 2.62/1.58 % (3306)Instructions burned: 39 (million)
% 2.62/1.58 % (3306)------------------------------
% 2.62/1.58 % (3306)------------------------------
% 2.62/1.58 % (3308)Instruction limit reached!
% 2.62/1.58 % (3308)------------------------------
% 2.62/1.58 % (3308)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.62/1.58 % (3308)Termination reason: Unknown
% 2.62/1.58 % (3308)Termination phase: shuffling
% 2.62/1.58
% 2.62/1.58 % (3308)Memory used [KB]: 1407
% 2.62/1.58 % (3308)Time elapsed: 0.004 s
% 2.62/1.58 % (3308)Instructions burned: 3 (million)
% 2.62/1.58 % (3308)------------------------------
% 2.62/1.58 % (3308)------------------------------
% 2.62/1.59 % (3310)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_3007 for (2999ds/7Mi)
% 2.62/1.59 % (3307)Instruction limit reached!
% 2.62/1.59 % (3307)------------------------------
% 2.62/1.59 % (3307)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.62/1.59 % (3307)Termination reason: Unknown
% 2.62/1.59 % (3307)Termination phase: shuffling
% 2.62/1.59
% 2.62/1.59 % (3307)Memory used [KB]: 1663
% 2.62/1.59 % (3307)Time elapsed: 0.012 s
% 2.62/1.59 % (3307)Instructions burned: 16 (million)
% 2.62/1.59 % (3307)------------------------------
% 2.62/1.59 % (3307)------------------------------
% 2.62/1.59 % (3310)Instruction limit reached!
% 2.62/1.59 % (3310)------------------------------
% 2.62/1.59 % (3310)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.62/1.59 % (3310)Termination reason: Unknown
% 2.62/1.59 % (3310)Termination phase: shuffling
% 2.62/1.59
% 2.62/1.59 % (3310)Memory used [KB]: 1535
% 2.62/1.59 % (3310)Time elapsed: 0.004 s
% 2.62/1.59 % (3310)Instructions burned: 7 (million)
% 2.62/1.59 % (3310)------------------------------
% 2.62/1.59 % (3310)------------------------------
% 2.62/1.59 % (3311)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_3007 for (2999ds/16Mi)
% 2.62/1.59 % (3311)Instruction limit reached!
% 2.62/1.59 % (3311)------------------------------
% 2.62/1.59 % (3311)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.62/1.59 % (3311)Termination reason: Unknown
% 2.62/1.59 % (3311)Termination phase: shuffling
% 2.62/1.59
% 2.62/1.59 % (3311)Memory used [KB]: 1791
% 2.62/1.59 % (3311)Time elapsed: 0.006 s
% 2.62/1.59 % (3311)Instructions burned: 18 (million)
% 2.62/1.59 % (3311)------------------------------
% 2.62/1.59 % (3311)------------------------------
% 2.65/1.60 % (3312)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_3007 for (2999ds/3Mi)
% 2.65/1.60 % (3312)Instruction limit reached!
% 2.65/1.60 % (3312)------------------------------
% 2.65/1.60 % (3312)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.60 % (3312)Termination reason: Unknown
% 2.65/1.60 % (3312)Termination phase: shuffling
% 2.65/1.60
% 2.65/1.60 % (3312)Memory used [KB]: 1407
% 2.65/1.60 % (3312)Time elapsed: 0.004 s
% 2.65/1.60 % (3312)Instructions burned: 4 (million)
% 2.65/1.60 % (3312)------------------------------
% 2.65/1.60 % (3312)------------------------------
% 2.65/1.60 % (3314)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_3007 for (2999ds/7Mi)
% 2.65/1.60 % (3314)Instruction limit reached!
% 2.65/1.60 % (3314)------------------------------
% 2.65/1.60 % (3314)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.60 % (3314)Termination reason: Unknown
% 2.65/1.60 % (3314)Termination phase: shuffling
% 2.65/1.60
% 2.65/1.60 % (3314)Memory used [KB]: 1407
% 2.65/1.60 % (3314)Time elapsed: 0.003 s
% 2.65/1.60 % (3314)Instructions burned: 7 (million)
% 2.65/1.60 % (3314)------------------------------
% 2.65/1.60 % (3314)------------------------------
% 2.65/1.60 % (3315)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_3007 for (2999ds/3Mi)
% 2.65/1.61 % (3315)Instruction limit reached!
% 2.65/1.61 % (3315)------------------------------
% 2.65/1.61 % (3315)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.61 % (3315)Termination reason: Unknown
% 2.65/1.61 % (3315)Termination phase: shuffling
% 2.65/1.61
% 2.65/1.61 % (3315)Memory used [KB]: 1407
% 2.65/1.61 % (3315)Time elapsed: 0.002 s
% 2.65/1.61 % (3315)Instructions burned: 4 (million)
% 2.65/1.61 % (3315)------------------------------
% 2.65/1.61 % (3315)------------------------------
% 2.65/1.61 % (3313)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_3007 for (2999ds/3Mi)
% 2.65/1.61 % (3313)Instruction limit reached!
% 2.65/1.61 % (3313)------------------------------
% 2.65/1.61 % (3313)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.61 % (3313)Termination reason: Unknown
% 2.65/1.61 % (3313)Termination phase: shuffling
% 2.65/1.61
% 2.65/1.61 % (3313)Memory used [KB]: 1407
% 2.65/1.61 % (3313)Time elapsed: 0.004 s
% 2.65/1.61 % (3313)Instructions burned: 4 (million)
% 2.65/1.61 % (3313)------------------------------
% 2.65/1.61 % (3313)------------------------------
% 2.65/1.61 % (3298)Instruction limit reached!
% 2.65/1.61 % (3298)------------------------------
% 2.65/1.61 % (3298)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.61 % (3298)Termination reason: Unknown
% 2.65/1.61 % (3298)Termination phase: Saturation
% 2.65/1.61
% 2.65/1.61 % (3298)Memory used [KB]: 7547
% 2.65/1.61 % (3298)Time elapsed: 0.057 s
% 2.65/1.61 % (3298)Instructions burned: 183 (million)
% 2.65/1.61 % (3298)------------------------------
% 2.65/1.61 % (3298)------------------------------
% 2.65/1.61 % (3317)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_3007 for (2999ds/18Mi)
% 2.65/1.61 % (3318)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_3007 for (2999ds/710Mi)
% 2.65/1.62 % (3317)Instruction limit reached!
% 2.65/1.62 % (3317)------------------------------
% 2.65/1.62 % (3317)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.62 % (3317)Termination reason: Unknown
% 2.65/1.62 % (3317)Termination phase: shuffling
% 2.65/1.62
% 2.65/1.62 % (3317)Memory used [KB]: 1791
% 2.65/1.62 % (3317)Time elapsed: 0.006 s
% 2.65/1.62 % (3317)Instructions burned: 18 (million)
% 2.65/1.62 % (3317)------------------------------
% 2.65/1.62 % (3317)------------------------------
% 2.65/1.62 % (3316)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_3007 for (2999ds/4Mi)
% 2.65/1.62 % (3320)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_3007 for (2999ds/902Mi)
% 2.65/1.62 % (3316)Instruction limit reached!
% 2.65/1.62 % (3316)------------------------------
% 2.65/1.62 % (3316)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.62 % (3316)Termination reason: Unknown
% 2.65/1.62 % (3316)Termination phase: shuffling
% 2.65/1.62
% 2.65/1.62 % (3316)Memory used [KB]: 1407
% 2.65/1.62 % (3316)Time elapsed: 0.004 s
% 2.65/1.62 % (3316)Instructions burned: 4 (million)
% 2.65/1.62 % (3316)------------------------------
% 2.65/1.62 % (3316)------------------------------
% 2.65/1.62 % (3319)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_3007 for (2999ds/6Mi)
% 2.65/1.63 % (3319)Instruction limit reached!
% 2.65/1.63 % (3319)------------------------------
% 2.65/1.63 % (3319)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.63 % (3319)Termination reason: Unknown
% 2.65/1.63 % (3319)Termination phase: shuffling
% 2.65/1.63
% 2.65/1.63 % (3319)Memory used [KB]: 1535
% 2.65/1.63 % (3319)Time elapsed: 0.005 s
% 2.65/1.63 % (3319)Instructions burned: 7 (million)
% 2.65/1.63 % (3319)------------------------------
% 2.65/1.63 % (3319)------------------------------
% 2.65/1.63 % (3321)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_3007 for (2999ds/21Mi)
% 2.65/1.63 % (3321)Instruction limit reached!
% 2.65/1.63 % (3321)------------------------------
% 2.65/1.63 % (3321)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.63 % (3321)Termination reason: Unknown
% 2.65/1.63 % (3321)Termination phase: shuffling
% 2.65/1.63
% 2.65/1.63 % (3321)Memory used [KB]: 1791
% 2.65/1.63 % (3321)Time elapsed: 0.007 s
% 2.65/1.63 % (3321)Instructions burned: 21 (million)
% 2.65/1.63 % (3321)------------------------------
% 2.65/1.63 % (3321)------------------------------
% 2.65/1.64 % (3323)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_3007 for (2999ds/6Mi)
% 2.65/1.64 % (3322)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_3007 for (2999ds/5Mi)
% 2.65/1.64 % (3323)Instruction limit reached!
% 2.65/1.64 % (3323)------------------------------
% 2.65/1.64 % (3323)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.64 % (3323)Termination reason: Unknown
% 2.65/1.64 % (3323)Termination phase: shuffling
% 2.65/1.64
% 2.65/1.64 % (3323)Memory used [KB]: 1535
% 2.65/1.64 % (3323)Time elapsed: 0.004 s
% 2.65/1.64 % (3323)Instructions burned: 9 (million)
% 2.65/1.64 % (3323)------------------------------
% 2.65/1.64 % (3323)------------------------------
% 2.65/1.64 % (3322)Instruction limit reached!
% 2.65/1.64 % (3322)------------------------------
% 2.65/1.64 % (3322)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.64 % (3322)Termination reason: Unknown
% 2.65/1.64 % (3322)Termination phase: shuffling
% 2.65/1.64
% 2.65/1.64 % (3322)Memory used [KB]: 1407
% 2.65/1.64 % (3322)Time elapsed: 0.005 s
% 2.65/1.64 % (3322)Instructions burned: 6 (million)
% 2.65/1.64 % (3322)------------------------------
% 2.65/1.64 % (3322)------------------------------
% 2.65/1.64 % (3324)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_3007 for (2999ds/377Mi)
% 2.65/1.65 % (3325)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_3007 for (2998ds/779Mi)
% 2.65/1.66 % (3326)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_3007 for (2998ds/19Mi)
% 2.65/1.66 % (3326)Instruction limit reached!
% 2.65/1.66 % (3326)------------------------------
% 2.65/1.66 % (3326)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.66 % (3326)Termination reason: Unknown
% 2.65/1.66 % (3326)Termination phase: shuffling
% 2.65/1.66
% 2.65/1.66 % (3326)Memory used [KB]: 1791
% 2.65/1.66 % (3326)Time elapsed: 0.008 s
% 2.65/1.66 % (3326)Instructions burned: 20 (million)
% 2.65/1.66 % (3326)------------------------------
% 2.65/1.66 % (3326)------------------------------
% 2.65/1.67 % (3327)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_3007 for (2998ds/879Mi)
% 2.65/1.68 % (3303)Instruction limit reached!
% 2.65/1.68 % (3303)------------------------------
% 2.65/1.68 % (3303)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.68 % (3303)Termination reason: Unknown
% 2.65/1.68 % (3303)Termination phase: Saturation
% 2.65/1.68
% 2.65/1.68 % (3303)Memory used [KB]: 8955
% 2.65/1.68 % (3303)Time elapsed: 0.119 s
% 2.65/1.68 % (3303)Instructions burned: 276 (million)
% 2.65/1.68 % (3303)------------------------------
% 2.65/1.68 % (3303)------------------------------
% 2.65/1.69 % (3328)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_3007 for (2998ds/17Mi)
% 2.65/1.69 % (3328)Instruction limit reached!
% 2.65/1.69 % (3328)------------------------------
% 2.65/1.69 % (3328)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.69 % (3328)Termination reason: Unknown
% 2.65/1.69 % (3328)Termination phase: shuffling
% 2.65/1.69
% 2.65/1.69 % (3328)Memory used [KB]: 1791
% 2.65/1.69 % (3328)Time elapsed: 0.006 s
% 2.65/1.69 % (3328)Instructions burned: 17 (million)
% 2.65/1.69 % (3328)------------------------------
% 2.65/1.69 % (3328)------------------------------
% 2.65/1.70 % (3330)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_3007 for (2998ds/3Mi)
% 2.65/1.70 % (3330)Instruction limit reached!
% 2.65/1.70 % (3330)------------------------------
% 2.65/1.70 % (3330)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.70 % (3330)Termination reason: Unknown
% 2.65/1.70 % (3330)Termination phase: shuffling
% 2.65/1.70
% 2.65/1.70 % (3330)Memory used [KB]: 1407
% 2.65/1.70 % (3330)Time elapsed: 0.002 s
% 2.65/1.70 % (3330)Instructions burned: 4 (million)
% 2.65/1.70 % (3330)------------------------------
% 2.65/1.70 % (3330)------------------------------
% 2.65/1.71 % (3331)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_3007 for (2998ds/30Mi)
% 2.65/1.72 % (3320)First to succeed.
% 2.65/1.72 % (3320)Refutation found. Thanks to Tanya!
% 2.65/1.72 % SZS status Theorem for DTF2THF_3007
% 2.65/1.72 % SZS output start Proof for DTF2THF_3007
% 2.65/1.72 thf(func_def_0, type, in: $i > $i > $o).
% 2.65/1.72 thf(func_def_1, type, exu: ($i > $o) > $o).
% 2.65/1.72 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 2.65/1.72 thf(func_def_8, type, powerset: $i > $i).
% 2.65/1.72 thf(func_def_10, type, setunion: $i > $i).
% 2.65/1.72 thf(func_def_19, type, descr: ($i > $o) > $i).
% 2.65/1.72 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 2.65/1.72 thf(func_def_26, type, prop2set: $o > $i).
% 2.65/1.72 thf(func_def_36, type, nonempty: $i > $o).
% 2.65/1.72 thf(func_def_69, type, set2prop: $i > $o).
% 2.65/1.72 thf(func_def_88, type, subset: $i > $i > $o).
% 2.65/1.72 thf(func_def_89, type, disjoint: $i > $i > $o).
% 2.65/1.72 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 2.65/1.72 thf(func_def_114, type, binunion: $i > $i > $i).
% 2.65/1.72 thf(func_def_122, type, binintersect: $i > $i > $i).
% 2.65/1.72 thf(func_def_135, type, regular: $i > $o).
% 2.65/1.72 thf(func_def_136, type, setminus: $i > $i > $i).
% 2.65/1.72 thf(func_def_147, type, symdiff: $i > $i > $i).
% 2.65/1.72 thf(func_def_153, type, iskpair: $i > $o).
% 2.65/1.72 thf(func_def_158, type, kpair: $i > $i > $i).
% 2.65/1.72 thf(func_def_160, type, cartprod: $i > $i > $i).
% 2.65/1.72 thf(func_def_177, type, singleton: $i > $o).
% 2.65/1.72 thf(func_def_179, type, ex1: $i > ($i > $o) > $o).
% 2.65/1.72 thf(func_def_184, type, atmost1p: $i > $o).
% 2.65/1.72 thf(func_def_185, type, atleast2p: $i > $o).
% 2.65/1.72 thf(func_def_186, type, atmost2p: $i > $o).
% 2.65/1.72 thf(func_def_187, type, upairsetp: $i > $o).
% 2.65/1.72 thf(func_def_191, type, kfst: $i > $i).
% 2.65/1.72 thf(func_def_203, type, ksnd: $i > $i).
% 2.65/1.72 thf(func_def_213, type, breln: $i > $i > $i > $o).
% 2.65/1.72 thf(func_def_214, type, dpsetconstr: $i > $i > ($i > $i > $o) > $i).
% 2.65/1.72 thf(func_def_222, type, func: $i > $i > $i > $o).
% 2.65/1.72 thf(func_def_223, type, funcSet: $i > $i > $i).
% 2.65/1.72 thf(func_def_226, type, ap: $i > $i > $i > $i > $i).
% 2.65/1.72 thf(func_def_232, type, lam: $i > $i > ($i > $i) > $i).
% 2.65/1.72 thf(func_def_265, type, sP1: $i > $i > $o).
% 2.65/1.72 thf(func_def_266, type, sP2: $i > $o).
% 2.65/1.72 thf(func_def_267, type, sP3: $i > $i > $o).
% 2.65/1.72 thf(func_def_268, type, sP4: $i > $i > $o).
% 2.65/1.72 thf(func_def_269, type, sP5: $i > $i > $i > $i > $o).
% 2.65/1.72 thf(func_def_272, type, sP8: $o > $i > $i > $i > $o).
% 2.65/1.72 thf(func_def_278, type, sK14: $o > $i > $i > $i).
% 2.65/1.72 thf(func_def_282, type, sK18: $i > $i > $i).
% 2.65/1.72 thf(func_def_293, type, sK29: ($i > $o) > $i > $i > $i).
% 2.65/1.72 thf(func_def_296, type, sK32: $i > $o).
% 2.65/1.72 thf(func_def_297, type, sK33: $i > $o).
% 2.65/1.72 thf(func_def_299, type, sK35: $i > ($i > $o) > $i).
% 2.65/1.72 thf(func_def_322, type, sK58: $i > $i).
% 2.65/1.72 thf(func_def_327, type, sK63: $i > $o).
% 2.65/1.72 thf(func_def_333, type, sK69: $i > $o).
% 2.65/1.72 thf(func_def_335, type, sK71: $i > $i).
% 2.65/1.72 thf(func_def_342, type, sK78: $i > $o).
% 2.65/1.72 thf(func_def_344, type, sK80: ($i > $o) > $i).
% 2.65/1.72 thf(func_def_345, type, sK81: ($i > $o) > $i).
% 2.65/1.72 thf(func_def_346, type, sK82: ($i > $o) > $i > $i > $i > $i).
% 2.65/1.72 thf(func_def_347, type, sK83: ($i > $o) > $i > $i > $i > $i).
% 2.65/1.72 thf(func_def_351, type, sK87: $i > $o).
% 2.65/1.72 thf(func_def_361, type, sK97: $i > $o).
% 2.65/1.72 thf(func_def_364, type, sK100: $i > ($i > $o) > $i).
% 2.65/1.72 thf(func_def_365, type, sK101: $i > ($i > $o) > $i).
% 2.65/1.72 thf(func_def_368, type, sK104: $i > $i > $i).
% 2.65/1.72 thf(func_def_369, type, sK105: $i > $i > $i).
% 2.65/1.72 thf(func_def_374, type, sK110: $i > $i > $o).
% 2.65/1.72 thf(func_def_387, type, sK123: $i > $i > $i).
% 2.65/1.72 thf(func_def_394, type, sK130: $i > $i > $i).
% 2.65/1.72 thf(func_def_398, type, sK134: $i > $o).
% 2.65/1.72 thf(func_def_407, type, sK143: $i > $i > $i > $i).
% 2.65/1.72 thf(func_def_408, type, sK144: $i > $i > $i > $i).
% 2.65/1.72 thf(func_def_414, type, sK150: $i > $o).
% 2.65/1.72 thf(func_def_417, type, sK153: $i > ($i > $o) > $i).
% 2.65/1.72 thf(func_def_425, type, sK161: $i > $i).
% 2.65/1.72 thf(func_def_426, type, sK162: $i > $o).
% 2.65/1.72 thf(func_def_430, type, sK166: $i > $o).
% 2.65/1.72 thf(func_def_431, type, sK167: $i > $o).
% 2.65/1.72 thf(func_def_432, type, sK168: ($i > $o) > ($i > $o) > $i).
% 2.65/1.72 thf(func_def_433, type, sK169: ($i > $o) > ($i > $o) > $i).
% 2.65/1.72 thf(func_def_436, type, sK172: $i > $i > $i).
% 2.65/1.72 thf(func_def_440, type, sK176: $i > $i > $i).
% 2.65/1.72 thf(func_def_441, type, sK177: $i > $i > $i).
% 2.65/1.72 thf(func_def_442, type, sK178: $i > $i > $i).
% 2.65/1.72 thf(func_def_443, type, sK179: $i > $i > $i).
% 2.65/1.72 thf(func_def_444, type, sK180: $i > $i > $i > $i).
% 2.65/1.72 thf(func_def_445, type, sK181: $i > $i).
% 2.65/1.72 thf(func_def_446, type, sK182: $i > $i).
% 2.65/1.72 thf(func_def_447, type, sK183: $i > $i).
% 2.65/1.72 thf(func_def_448, type, sK184: $i > $i).
% 2.65/1.72 thf(func_def_449, type, sK185: $i > $i > $i).
% 2.65/1.72 thf(func_def_450, type, sK186: $i > $i > $i > $i).
% 2.65/1.72 thf(func_def_451, type, sK187: $i > $i > $i > $i).
% 2.65/1.72 thf(func_def_452, type, sK188: $i > $i > $i).
% 2.65/1.72 thf(func_def_453, type, sK189: $i > $i > $i).
% 2.65/1.72 thf(func_def_454, type, sK190: $i > $i).
% 2.65/1.72 thf(func_def_455, type, sK191: $i > $i > $i).
% 2.65/1.72 thf(func_def_457, type, sK193: $i > $i).
% 2.65/1.72 thf(func_def_458, type, sK194: $i > $i).
% 2.65/1.72 thf(func_def_483, type, sK219: $i > $i > $i).
% 2.65/1.72 thf(func_def_484, type, sK220: $i > $i > ($i > $i) > $i).
% 2.65/1.72 thf(func_def_485, type, sK221: $i > $i).
% 2.65/1.72 thf(func_def_499, type, sK235: $i > $i > $o).
% 2.65/1.72 thf(func_def_505, type, sK241: $i > ($i > $o) > $i).
% 2.65/1.72 thf(func_def_506, type, sK242: $i > $o).
% 2.65/1.72 thf(func_def_513, type, sK249: $i > $o).
% 2.65/1.72 thf(func_def_516, type, sK252: $i > $i > $o).
% 2.65/1.72 thf(func_def_520, type, sK256: ($i > $o) > $i > $i).
% 2.65/1.72 thf(func_def_522, type, sK258: $i > $o).
% 2.65/1.72 thf(func_def_538, type, sK274: $i > $i).
% 2.65/1.72 thf(func_def_540, type, sK276: $i > ($i > $i) > $i > $i).
% 2.65/1.72 thf(func_def_545, type, sK281: $i > $o).
% 2.65/1.72 thf(func_def_555, type, sK291: $i > ($i > $i) > $i > $i).
% 2.65/1.72 thf(func_def_557, type, sK293: $i > $i).
% 2.65/1.72 thf(func_def_567, type, sK303: $i > $i > $i > $i > $i).
% 2.65/1.72 thf(func_def_568, type, sK304: $i > $i > $i > $i > $i).
% 2.65/1.72 thf(func_def_569, type, sK305: $i > $i > $i > $i > $i).
% 2.65/1.72 thf(func_def_570, type, sK306: $i > $i > $i > $i > $i).
% 2.65/1.72 thf(func_def_575, type, sK311: $i > $i).
% 2.65/1.72 thf(func_def_579, type, sK315: $i > $i > $i).
% 2.65/1.72 thf(func_def_584, type, sK320: ($i > $o) > $i).
% 2.65/1.72 thf(func_def_585, type, sK321: $i > $o).
% 2.65/1.72 thf(func_def_586, type, sK322: $i > $i > ($i > $i) > $i).
% 2.65/1.72 thf(func_def_587, type, sK323: $i > $i).
% 2.65/1.72 thf(func_def_591, type, sK327: $i > $o).
% 2.65/1.72 thf(func_def_597, type, sK333: $i > $i > $i).
% 2.65/1.72 thf(func_def_605, type, sK341: $i > $i > $i > $i > $i).
% 2.65/1.72 thf(func_def_606, type, sK342: $i > $i > $i > $i > $i).
% 2.65/1.72 thf(func_def_627, type, sK363: $i > $o).
% 2.65/1.72 thf(func_def_643, type, sK379: $i > $i > $o).
% 2.65/1.72 thf(func_def_654, type, sK390: $i > $o).
% 2.65/1.72 thf(func_def_671, type, sK407: $i > $i).
% 2.65/1.72 thf(func_def_689, type, sK425: $i > $o).
% 2.65/1.72 thf(func_def_698, type, sK434: $i > $i > $o).
% 2.65/1.72 thf(func_def_701, type, sK437: $i > $i > $o).
% 2.65/1.72 thf(func_def_704, type, sK440: $i > $o).
% 2.65/1.72 thf(func_def_707, type, sK443: $i > $o).
% 2.65/1.72 thf(func_def_709, type, sK445: ($i > $o) > $i > $i).
% 2.65/1.72 thf(func_def_710, type, sK446: ($i > $o) > $i > $i).
% 2.65/1.72 thf(func_def_728, type, sK464: $i > $o).
% 2.65/1.72 thf(func_def_729, type, sK465: $i > $o).
% 2.65/1.72 thf(func_def_730, type, sK466: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.65/1.72 thf(func_def_731, type, sK467: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.65/1.72 thf(func_def_737, type, sK473: $i > $o).
% 2.65/1.72 thf(func_def_739, type, sK475: ($i > $o) > $i > $i).
% 2.65/1.72 thf(func_def_769, type, sK505: $i > $i > $i > $i > $i).
% 2.65/1.72 thf(func_def_772, type, sK508: $i > $i).
% 2.65/1.72 thf(func_def_775, type, sK511: $i > $i).
% 2.65/1.72 thf(func_def_788, type, sK524: $i > ($i > $i) > $i > $i).
% 2.65/1.72 thf(func_def_790, type, sK526: $i > $i).
% 2.65/1.72 thf(func_def_795, type, sK531: $i > $o).
% 2.65/1.72 thf(func_def_797, type, sK533: ($i > $o) > $i > $i > $i > $i).
% 2.65/1.72 thf(func_def_798, type, sK534: ($i > $o) > $i > $i > $i > $i).
% 2.65/1.72 thf(func_def_804, type, sK540: $i > $o).
% 2.65/1.72 thf(func_def_807, type, sK543: $i > ($i > $o) > $i).
% 2.65/1.72 thf(func_def_813, type, sK549: $i > $i > $o).
% 2.65/1.72 thf(func_def_816, type, sK552: $i > $i > $i > $i > $i).
% 2.65/1.72 thf(func_def_833, type, sK569: $i > $o).
% 2.65/1.72 thf(func_def_844, type, sK580: $i > ($i > $o) > $i).
% 2.65/1.72 thf(func_def_845, type, sK581: $i > $o).
% 2.65/1.72 thf(func_def_849, type, sK585: $i > $o).
% 2.65/1.72 thf(func_def_852, type, sK588: $i > ($i > $o) > $i).
% 2.65/1.72 thf(func_def_855, type, sK591: $i > $o).
% 2.65/1.72 thf(func_def_856, type, sK592: $i > $i).
% 2.65/1.72 thf(func_def_857, type, sK593: ($i > $o) > $i).
% 2.65/1.72 thf(func_def_860, type, sK596: $i > $i).
% 2.65/1.72 thf(func_def_865, type, sK601: $i > $i > $o).
% 2.65/1.72 thf(func_def_866, type, sK602: $i > $i).
% 2.65/1.72 thf(func_def_867, type, sK603: $i > $i).
% 2.65/1.72 thf(func_def_868, type, sK604: ($i > $i > $o) > $i > $i).
% 2.65/1.72 thf(func_def_869, type, sK605: ($i > $i > $o) > $i > $i).
% 2.65/1.72 thf(func_def_870, type, sK606: $i > ($i > $i > $o) > $i > $i).
% 2.65/1.72 thf(func_def_877, type, sK613: $i > $o).
% 2.65/1.72 thf(func_def_878, type, sK614: $i > $i).
% 2.65/1.72 thf(func_def_879, type, sK615: ($i > $o) > $i).
% 2.65/1.72 thf(func_def_880, type, sK616: ($i > $o) > ($i > $o) > $i).
% 2.65/1.72 thf(func_def_881, type, sK617: ($i > $o) > ($i > $o) > $i).
% 2.65/1.72 thf(func_def_882, type, sK618: $i > $o).
% 2.65/1.72 thf(func_def_883, type, sK619: $i > $o).
% 2.65/1.72 thf(func_def_890, type, sK626: $i > $i > $i > $i > $i).
% 2.65/1.72 thf(func_def_897, type, sK633: $i > $o).
% 2.65/1.72 thf(func_def_899, type, sK635: ($i > $o) > $i > $i).
% 2.65/1.72 thf(func_def_913, type, sK649: $i > $o).
% 2.65/1.72 thf(func_def_919, type, sK655: $i > $i > $i).
% 2.65/1.72 thf(func_def_933, type, ph669: !>[X0: $tType]:(X0)).
% 2.65/1.72 thf(f3675,plain,(
% 2.65/1.72 $false),
% 2.65/1.72 inference(trivial_inequality_removal,[status(thm)],[f3664])).
% 2.65/1.72 thf(f3664,plain,(
% 2.65/1.72 ($true != $true)),
% 2.65/1.72 inference(superposition,[status(thm)],[f3620,f3238])).
% 2.65/1.72 thf(f3238,plain,(
% 2.65/1.72 ((in @ (sK291 @ sK406 @ sK407 @ sK408) @ sK406) = $true)),
% 2.65/1.72 inference(trivial_inequality_removal,[status(thm)],[f3237])).
% 2.65/1.72 thf(f3237,plain,(
% 2.65/1.72 ((in @ (sK291 @ sK406 @ sK407 @ sK408) @ sK406) = $true) | ($true != $true)),
% 2.65/1.72 inference(forward_demodulation,[status(thm)],[f3236,f2663])).
% 2.65/1.72 thf(f2663,plain,(
% 2.65/1.72 ((in @ sK409 @ sK406) = $true)),
% 2.65/1.72 inference(cnf_transformation,[status(thm)],[f1553])).
% 2.65/1.72 thf(f1553,plain,(
% 2.65/1.72 (brelnall2 = $true) & (exuEu = $true) & (ksndpairEq = $true) & (setukpairinjR2 = $true) & (funcext2 = $true) & (funcext = $true) & (kpairsurjEq = $true) & (binintersectSubset4 = $true) & (brelnall1 = $true) & (quantDeMorgan3 = $true) & (upairset2IR = $true) & (binunionIR = $true) & (exuI3 = $true) & (notdallE = $true) & (nonemptyI = $true) & (singletonsubset = $true) & (subsetemptysetimpeq = $true) & (emptyinPowerset = $true) & (dpsetconstrERa = $true) & (notsubsetI = $true) & (kfstpairEq = $true) & (setunionsingleton1 = $true) & (! [X3 : $i] : (((in @ X3 @ sK406) != $true) | ((in @ (sK407 @ X3) @ sK408) = $true)) & (((ap @ sK406 @ sK408 @ (lam @ sK406 @ sK408 @ (^[Y0 : $i]: (sK407 @ Y0))) @ sK409) != (sK407 @ sK409)) & ((in @ sK409 @ sK406) = $true))) & (funcGraphProp3 = $true) & (setbeta = $true) & (symdiffIneg1 = $true) & (funcGraphProp4 = $true) & (omega0Ax = $true) & (sepInPowerset = $true) & (funcImageSingleton = $true) & (lamp = $true) & (notdexE = $true) & (setminusERneg = $true) & (binunionE = $true) & (dpsetconstrER = $true) & (ex1I2 = $true) & (dpsetconstrI = $true) & (lam2p = $true) & (emptyE1 = $true) & (upairequniteq = $true) & (ubforcartprodlem3 = $true) & (setunion__Cong = $true) & (binintersectSubset3 = $true) & (setadjoin__Cong = $true) & (singletoninpowunion = $true) & (emptyset__Cong = $true) & (wellorderingAx = $true) & (setadjoinSub = $true) & (replAx = $true) & (exuE3e = $true) & (emptyI = $true) & (exuI1 = $true) & (setminusILneg = $true) & (foundationAx = $true) & (setukpairinjL = $true) & (symdiffI2 = $true) & (upairset2E = $true) & (setukpairinjR12 = $true) & (dsetconstrEL = $true) & (setminusEL = $true) & (inPowerset = $true) & (funcGraphProp1 = $true) & (symdiffI1 = $true) & (upairsetIR = $true) & (eqinunit = $true) & (beta1 = $true) & (vacuousDall = $true) & (binunionLsub = $true) & (notinsingleton = $true) & (cartprodpairsurjEq = $true) & (exuE3u = $true) & (powersetE = $true) & (eta1 = $true) & (cartprodmempaircEq = $true) & (ubforcartprodlem1 = $true) & (powersetE1 = $true) & (binunionRsub = $true) & (emptyInPowerset = $true) & (singletonprop = $true) & (powersetI = $true) & (upairsetE = $true) & (binintersectLsub = $true) & (cartprodpairin = $true) & (nonemptyI1 = $true) & (binintersectSubset1 = $true) & (setadjoinAx = $true) & (cartprodpairmemER = $true) & (setext = $true) & (setminusI = $true) & (cartprodfstin = $true) & (setukpairIL = $true) & (setunionsingleton = $true) & (setextAx = $true) & (setminusELneg = $true) & (dsetconstrER = $true) & (setadjoinIR = $true) & (cartprodsndpairEq = $true) & (subbreln = $true) & (binunionEcases = $true) & (prop2setE = $true) & (secondinupair = $true) & (binintersectRsub = $true) & (setadjoinOr = $true) & (subsetI2 = $true) & (eqimpsubset1 = $true) & (sepSubset = $true) & (binunionIL = $true) & (setunionAx = $true) & (setoftrueEq = $true) & (ex1I = $true) & (subset2powerset = $true) & (lamProp = $true) & (dsetconstrI = $true) & (cartprodmempair = $true) & (emptysetimpfalse = $true) & (subsetTrans = $true) & (ap2p = $true) & (subsetRefl = $true) & (quantDeMorgan1 = $true) & (cartprodsndin = $true) & (notequalI1 = $true) & (nonemptyE1 = $true) & (emptysetAx = $true) & (apProp = $true) & (ubforcartprodlem2 = $true) & (cartprodmempair1 = $true) & (subsetE = $true) & (app = $true) & (subPowSU = $true) & (descr__Cong = $true) & (upairinpowunion = $true) & (setukpairIR = $true) & (setminusER = $true) & (dpsetconstrEL1 = $true) & (prop2setI = $true) & (kfstsingleton = $true) & (ap2apEq2 = $true) & (setukpairinjR = $true) & (noeltsimpempty = $true) & (omegaIndAx = $true) & (subsetI1 = $true) & (funcinfuncset = $true) & (funcGraphProp2 = $true) & (kpairiskpair = $true) & (binintersectER = $true) & (eqimpsubset2 = $true) & (setminusSubset1 = $true) & (funcextLem = $true) & (disjointsetsI1 = $true) & (omegaSAx = $true) & (omega__Cong = $true) & (cartprodfstpairEq = $true) & (subsetE2 = $true) & (notequalI2 = $true) & (singletoninpowerset = $true) & (setukpairinjR1 = $true) & (eqbreln = $true) & (theprop = $true) & (uniqinunit = $true) & (infuncsetfunc = $true) & (in__Cong = $true) & (setOfPairsIsBReln = $true) & (ksndsingleton = $true) & (ap2apEq1 = $true) & (dpsetconstrSub = $true) & (binintersectSubset2 = $true) & (emptysetsubset = $true) & (kpairp = $true) & (setminusLsub = $true) & (setminusIRneg = $true) & (notinemptyset = $true) & (exuI2 = $true) & (powersetAx = $true) & (setunionI = $true) & (symdiffE = $true) & (singletonsuniq = $true) & (powersetI1 = $true) & (prop2set2propI = $true) & (quantDeMorgan4 = $true) & (quantDeMorgan2 = $true) & (dsetconstr__Cong = $true) & (upairsubunion = $true) & (exuE1 = $true) & (cartprodpairmemEL = $true) & (dpsetconstrEL2 = $true) & (powerset__Cong = $true) & (setukpairinjR11 = $true) & (exu__Cong = $true) & (setminusSubset2 = $true) & (setunionE = $true) & (ex1E2 = $true) & (singletonsswitch = $true) & (powersetsubset = $true) & (binintersectSubset5 = $true) & (lam2lamEq = $true) & (bs114d = $true) & (upairsetIL = $true) & (binintersectI = $true) & (emptysetE = $true) & (setadjoinE = $true) & (inCongP = $true) & (ex1E1 = $true) & (setadjoinSub2 = $true) & (exuE2 = $true) & (symdiffIneg2 = $true) & (setunionE2 = $true) & (nonemptyImpWitness = $true) & (setunionsingleton2 = $true) & (setextsub = $true) & (setukpairinjL1 = $true) & (setadjoinIL = $true) & (setukpairinjL2 = $true) & (binintersectEL = $true) & (descrp = $true) & (emptyinunitempty = $true)),
% 2.65/1.72 inference(skolemisation,[status(esa),new_symbols(skolem,[sK406,sK407,sK408,sK409])],[f864,f1552,f1551])).
% 2.65/1.72 thf(f1551,plain,(
% 2.65/1.72 ? [X0 : $i,X1 : $i > $i,X2 : $i] : (! [X3 : $i] : (($true != (in @ X3 @ X0)) | ((in @ (X1 @ X3) @ X2) = $true)) & ? [X4 : $i] : (((X1 @ X4) != (ap @ X0 @ X2 @ (lam @ X0 @ X2 @ (^[Y0 : $i]: (X1 @ Y0))) @ X4)) & ((in @ X4 @ X0) = $true))) => (! [X3 : $i] : (((in @ X3 @ sK406) != $true) | ((in @ (sK407 @ X3) @ sK408) = $true)) & ? [X4 : $i] : (((ap @ sK406 @ sK408 @ (lam @ sK406 @ sK408 @ (^[Y0 : $i]: (sK407 @ Y0))) @ X4) != (sK407 @ X4)) & ((in @ X4 @ sK406) = $true)))),
% 2.65/1.72 introduced(definition,[],[choice_axiom])).
% 2.65/1.72 thf(f1552,plain,(
% 2.65/1.72 ? [X4 : $i] : (((ap @ sK406 @ sK408 @ (lam @ sK406 @ sK408 @ (^[Y0 : $i]: (sK407 @ Y0))) @ X4) != (sK407 @ X4)) & ((in @ X4 @ sK406) = $true)) => (((ap @ sK406 @ sK408 @ (lam @ sK406 @ sK408 @ (^[Y0 : $i]: (sK407 @ Y0))) @ sK409) != (sK407 @ sK409)) & ((in @ sK409 @ sK406) = $true))),
% 2.65/1.72 introduced(definition,[],[choice_axiom])).
% 2.65/1.72 thf(f864,plain,(
% 2.65/1.72 (brelnall2 = $true) & (exuEu = $true) & (ksndpairEq = $true) & (setukpairinjR2 = $true) & (funcext2 = $true) & (funcext = $true) & (kpairsurjEq = $true) & (binintersectSubset4 = $true) & (brelnall1 = $true) & (quantDeMorgan3 = $true) & (upairset2IR = $true) & (binunionIR = $true) & (exuI3 = $true) & (notdallE = $true) & (nonemptyI = $true) & (singletonsubset = $true) & (subsetemptysetimpeq = $true) & (emptyinPowerset = $true) & (dpsetconstrERa = $true) & (notsubsetI = $true) & (kfstpairEq = $true) & (setunionsingleton1 = $true) & ? [X0 : $i,X1 : $i > $i,X2 : $i] : (! [X3 : $i] : (($true != (in @ X3 @ X0)) | ((in @ (X1 @ X3) @ X2) = $true)) & ? [X4 : $i] : (((X1 @ X4) != (ap @ X0 @ X2 @ (lam @ X0 @ X2 @ (^[Y0 : $i]: (X1 @ Y0))) @ X4)) & ((in @ X4 @ X0) = $true))) & (funcGraphProp3 = $true) & (setbeta = $true) & (symdiffIneg1 = $true) & (funcGraphProp4 = $true) & (omega0Ax = $true) & (sepInPowerset = $true) & (funcImageSingleton = $true) & (lamp = $true) & (notdexE = $true) & (setminusERneg = $true) & (binunionE = $true) & (dpsetconstrER = $true) & (ex1I2 = $true) & (dpsetconstrI = $true) & (lam2p = $true) & (emptyE1 = $true) & (upairequniteq = $true) & (ubforcartprodlem3 = $true) & (setunion__Cong = $true) & (binintersectSubset3 = $true) & (setadjoin__Cong = $true) & (singletoninpowunion = $true) & (emptyset__Cong = $true) & (wellorderingAx = $true) & (setadjoinSub = $true) & (replAx = $true) & (exuE3e = $true) & (emptyI = $true) & (exuI1 = $true) & (setminusILneg = $true) & (foundationAx = $true) & (setukpairinjL = $true) & (symdiffI2 = $true) & (upairset2E = $true) & (setukpairinjR12 = $true) & (dsetconstrEL = $true) & (setminusEL = $true) & (inPowerset = $true) & (funcGraphProp1 = $true) & (symdiffI1 = $true) & (upairsetIR = $true) & (eqinunit = $true) & (beta1 = $true) & (vacuousDall = $true) & (binunionLsub = $true) & (notinsingleton = $true) & (cartprodpairsurjEq = $true) & (exuE3u = $true) & (powersetE = $true) & (eta1 = $true) & (cartprodmempaircEq = $true) & (ubforcartprodlem1 = $true) & (powersetE1 = $true) & (binunionRsub = $true) & (emptyInPowerset = $true) & (singletonprop = $true) & (powersetI = $true) & (upairsetE = $true) & (binintersectLsub = $true) & (cartprodpairin = $true) & (nonemptyI1 = $true) & (binintersectSubset1 = $true) & (setadjoinAx = $true) & (cartprodpairmemER = $true) & (setext = $true) & (setminusI = $true) & (cartprodfstin = $true) & (setukpairIL = $true) & (setunionsingleton = $true) & (setextAx = $true) & (setminusELneg = $true) & (dsetconstrER = $true) & (setadjoinIR = $true) & (cartprodsndpairEq = $true) & (subbreln = $true) & (binunionEcases = $true) & (prop2setE = $true) & (secondinupair = $true) & (binintersectRsub = $true) & (setadjoinOr = $true) & (subsetI2 = $true) & (eqimpsubset1 = $true) & (sepSubset = $true) & (binunionIL = $true) & (setunionAx = $true) & (setoftrueEq = $true) & (ex1I = $true) & (subset2powerset = $true) & (lamProp = $true) & (dsetconstrI = $true) & (cartprodmempair = $true) & (emptysetimpfalse = $true) & (subsetTrans = $true) & (ap2p = $true) & (subsetRefl = $true) & (quantDeMorgan1 = $true) & (cartprodsndin = $true) & (notequalI1 = $true) & (nonemptyE1 = $true) & (emptysetAx = $true) & (apProp = $true) & (ubforcartprodlem2 = $true) & (cartprodmempair1 = $true) & (subsetE = $true) & (app = $true) & (subPowSU = $true) & (descr__Cong = $true) & (upairinpowunion = $true) & (setukpairIR = $true) & (setminusER = $true) & (dpsetconstrEL1 = $true) & (prop2setI = $true) & (kfstsingleton = $true) & (ap2apEq2 = $true) & (setukpairinjR = $true) & (noeltsimpempty = $true) & (omegaIndAx = $true) & (subsetI1 = $true) & (funcinfuncset = $true) & (funcGraphProp2 = $true) & (kpairiskpair = $true) & (binintersectER = $true) & (eqimpsubset2 = $true) & (setminusSubset1 = $true) & (funcextLem = $true) & (disjointsetsI1 = $true) & (omegaSAx = $true) & (omega__Cong = $true) & (cartprodfstpairEq = $true) & (subsetE2 = $true) & (notequalI2 = $true) & (singletoninpowerset = $true) & (setukpairinjR1 = $true) & (eqbreln = $true) & (theprop = $true) & (uniqinunit = $true) & (infuncsetfunc = $true) & (in__Cong = $true) & (setOfPairsIsBReln = $true) & (ksndsingleton = $true) & (ap2apEq1 = $true) & (dpsetconstrSub = $true) & (binintersectSubset2 = $true) & (emptysetsubset = $true) & (kpairp = $true) & (setminusLsub = $true) & (setminusIRneg = $true) & (notinemptyset = $true) & (exuI2 = $true) & (powersetAx = $true) & (setunionI = $true) & (symdiffE = $true) & (singletonsuniq = $true) & (powersetI1 = $true) & (prop2set2propI = $true) & (quantDeMorgan4 = $true) & (quantDeMorgan2 = $true) & (dsetconstr__Cong = $true) & (upairsubunion = $true) & (exuE1 = $true) & (cartprodpairmemEL = $true) & (dpsetconstrEL2 = $true) & (powerset__Cong = $true) & (setukpairinjR11 = $true) & (exu__Cong = $true) & (setminusSubset2 = $true) & (setunionE = $true) & (ex1E2 = $true) & (singletonsswitch = $true) & (powersetsubset = $true) & (binintersectSubset5 = $true) & (lam2lamEq = $true) & (bs114d = $true) & (upairsetIL = $true) & (binintersectI = $true) & (emptysetE = $true) & (setadjoinE = $true) & (inCongP = $true) & (ex1E1 = $true) & (setadjoinSub2 = $true) & (exuE2 = $true) & (symdiffIneg2 = $true) & (setunionE2 = $true) & (nonemptyImpWitness = $true) & (setunionsingleton2 = $true) & (setextsub = $true) & (setukpairinjL1 = $true) & (setadjoinIL = $true) & (setukpairinjL2 = $true) & (binintersectEL = $true) & (descrp = $true) & (emptyinunitempty = $true)),
% 2.65/1.72 inference(flattening,[status(thm)],[f863])).
% 2.65/1.72 thf(f863,plain,(
% 2.65/1.72 (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i,X1 : $i > $i,X2 : $i] : (! [X3 : $i] : (($true != (in @ X3 @ X0)) | ((in @ (X1 @ X3) @ X2) = $true)) & ? [X4 : $i] : (((X1 @ X4) != (ap @ X0 @ X2 @ (lam @ X0 @ X2 @ (^[Y0 : $i]: (X1 @ Y0))) @ X4)) & ((in @ X4 @ X0) = $true))) & (lam2lamEq = $true)) & (eta1 = $true)) & (beta1 = $true)) & (ap2apEq2 = $true)) & (ap2apEq1 = $true)) & (funcext2 = $true)) & (funcext = $true)) & (eqbreln = $true)) & (subbreln = $true)) & (funcGraphProp4 = $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)),
% 2.65/1.72 inference(ennf_transformation,[status(thm)],[f650])).
% 2.65/1.72 thf(f650,plain,(
% 2.65/1.72 ~((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) => ((funcGraphProp4 = $true) => ((subbreln = $true) => ((eqbreln = $true) => ((funcext = $true) => ((funcext2 = $true) => ((ap2apEq1 = $true) => ((ap2apEq2 = $true) => ((beta1 = $true) => ((eta1 = $true) => ((lam2lamEq = $true) => ! [X0 : $i,X1 : $i > $i,X2 : $i] : (! [X3 : $i] : (($true = (in @ X3 @ X0)) => ((in @ (X1 @ X3) @ X2) = $true)) => ! [X4 : $i] : (((in @ X4 @ X0) = $true) => ((X1 @ X4) = (ap @ X0 @ X2 @ (lam @ X0 @ X2 @ (^[Y0 : $i]: (X1 @ Y0))) @ X4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.65/1.72 inference(fool_elimination,[status(thm)],[f649])).
% 2.65/1.72 thf(f649,plain,(
% 2.65/1.72 ~(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 => (funcGraphProp4 => (subbreln => (eqbreln => (funcext => (funcext2 => (ap2apEq1 => (ap2apEq2 => (beta1 => (eta1 => (lam2lamEq => ! [X0 : $i,X1 : $i > $i,X2 : $i] : (! [X3 : $i] : ((in @ X3 @ X0) => (in @ (X1 @ X3) @ X2)) => ! [X4 : $i] : ((in @ X4 @ X0) => ((ap @ X0 @ X2 @ (lam @ X0 @ X2 @ (^[X5 : $i] : (X1 @ X5))) @ X4) = (X1 @ X4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.65/1.72 inference(rectify,[status(thm)],[f222])).
% 2.65/1.72 thf(f222,negated_conjecture,(
% 2.65/1.72 ~(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 => (funcGraphProp4 => (subbreln => (eqbreln => (funcext => (funcext2 => (ap2apEq1 => (ap2apEq2 => (beta1 => (eta1 => (lam2lamEq => ! [X3 : $i,X12 : $i > $i,X4 : $i] : (! [X1 : $i] : ((in @ X1 @ X3) => (in @ (X12 @ X1) @ X4)) => ! [X1 : $i] : ((in @ X1 @ X3) => ((ap @ X3 @ X4 @ (lam @ X3 @ X4 @ (^[X2 : $i] : (X12 @ X2))) @ X1) = (X12 @ X1))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.65/1.72 inference(negated_conjecture,[status(cth)],[f221])).
% 2.65/1.72 thf(f221,conjecture,(
% 2.65/1.72 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 => (funcGraphProp4 => (subbreln => (eqbreln => (funcext => (funcext2 => (ap2apEq1 => (ap2apEq2 => (beta1 => (eta1 => (lam2lamEq => ! [X3 : $i,X12 : $i > $i,X4 : $i] : (! [X1 : $i] : ((in @ X1 @ X3) => (in @ (X12 @ X1) @ X4)) => ! [X1 : $i] : ((in @ X1 @ X3) => ((ap @ X3 @ X4 @ (lam @ X3 @ X4 @ (^[X2 : $i] : (X12 @ X2))) @ X1) = (X12 @ X1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.65/1.72 file('/export/starexec/sandbox/tmp/tmp.9FuAMmeF6I/DTF2THF_3007.p',beta2)).
% 2.65/1.72 thf(f3236,plain,(
% 2.65/1.72 ((in @ (sK291 @ sK406 @ sK407 @ sK408) @ sK406) = $true) | ((in @ sK409 @ sK406) != $true)),
% 2.65/1.72 inference(trivial_inequality_removal,[status(thm)],[f3235])).
% 2.65/1.72 thf(f3235,plain,(
% 2.65/1.72 ($true != $true) | ((in @ (sK291 @ sK406 @ sK407 @ sK408) @ sK406) = $true) | ((in @ sK409 @ sK406) != $true)),
% 2.65/1.72 inference(forward_demodulation,[status(thm)],[f3213,f2620])).
% 2.65/1.72 thf(f2620,plain,(
% 2.65/1.72 (beta1 = $true)),
% 2.65/1.72 inference(cnf_transformation,[status(thm)],[f1553])).
% 2.65/1.72 thf(f3213,plain,(
% 2.65/1.72 (beta1 != $true) | ((in @ (sK291 @ sK406 @ sK407 @ sK408) @ sK406) = $true) | ((in @ sK409 @ sK406) != $true)),
% 2.65/1.72 inference(trivial_inequality_removal,[status(thm)],[f3208])).
% 2.65/1.72 thf(f3208,plain,(
% 2.65/1.72 (beta1 != $true) | ((in @ (sK291 @ sK406 @ sK407 @ sK408) @ sK406) = $true) | ((in @ sK409 @ sK406) != $true) | ((sK407 @ sK409) != (sK407 @ sK409))),
% 2.65/1.72 inference(superposition,[status(thm)],[f3191,f3118])).
% 2.65/1.72 thf(f3118,plain,(
% 2.65/1.72 ( ! [X2 : $i,X3 : $i,X0 : $i,X1 : $i > $i] : (((ap @ X2 @ X0 @ (lam @ X2 @ X0 @ X1) @ X3) = (X1 @ X3)) | ((in @ X3 @ X2) != $true) | ((in @ (sK291 @ X2 @ X1 @ X0) @ X2) = $true) | (beta1 != $true)) )),
% 2.65/1.72 inference(beta_eta_normalization,[status(thm)],[f2332])).
% 2.65/1.72 thf(f2332,plain,(
% 2.65/1.72 ( ! [X2 : $i,X3 : $i,X0 : $i,X1 : $i > $i] : ((beta1 != $true) | ((in @ X3 @ X2) != $true) | ((in @ (sK291 @ X2 @ X1 @ X0) @ X2) = $true) | ((X1 @ X3) = (ap @ X2 @ X0 @ (lam @ X2 @ X0 @ (^[Y0 : $i]: (X1 @ Y0))) @ X3))) )),
% 2.65/1.72 inference(cnf_transformation,[status(thm)],[f1381])).
% 2.65/1.72 thf(f1381,plain,(
% 2.65/1.72 (! [X0 : $i,X1 : $i > $i,X2 : $i] : (! [X3 : $i] : (((X1 @ X3) = (ap @ X2 @ X0 @ (lam @ X2 @ X0 @ (^[Y0 : $i]: (X1 @ Y0))) @ X3)) | ((in @ X3 @ X2) != $true)) | (((in @ (X1 @ (sK291 @ X2 @ X1 @ X0)) @ X0) != $true) & ((in @ (sK291 @ X2 @ X1 @ X0) @ X2) = $true))) | (beta1 != $true)) & ((beta1 = $true) | ((((ap @ sK294 @ sK292 @ (lam @ sK294 @ sK292 @ (^[Y0 : $i]: (sK293 @ Y0))) @ sK295) != (sK293 @ sK295)) & ((in @ sK295 @ sK294) = $true)) & ! [X9 : $i] : (((in @ (sK293 @ X9) @ sK292) = $true) | ((in @ X9 @ sK294) != $true))))),
% 2.65/1.72 inference(skolemisation,[status(esa),new_symbols(skolem,[sK291,sK292,sK293,sK294,sK295])],[f1377,f1380,f1379,f1378])).
% 2.65/1.72 thf(f1378,plain,(
% 2.65/1.72 ! [X0 : $i,X1 : $i > $i,X2 : $i] : (? [X4 : $i] : (((in @ (X1 @ X4) @ X0) != $true) & ((in @ X4 @ X2) = $true)) => (((in @ (X1 @ (sK291 @ X2 @ X1 @ X0)) @ X0) != $true) & ((in @ (sK291 @ X2 @ X1 @ X0) @ X2) = $true)))),
% 2.65/1.72 introduced(definition,[],[choice_axiom])).
% 2.65/1.72 thf(f1379,plain,(
% 2.65/1.72 ? [X5 : $i,X6 : $i > $i,X7 : $i] : (? [X8 : $i] : (((X6 @ X8) != (ap @ X7 @ X5 @ (lam @ X7 @ X5 @ (^[Y0 : $i]: (X6 @ Y0))) @ X8)) & ((in @ X8 @ X7) = $true)) & ! [X9 : $i] : (((in @ (X6 @ X9) @ X5) = $true) | ($true != (in @ X9 @ X7)))) => (? [X8 : $i] : (((ap @ sK294 @ sK292 @ (lam @ sK294 @ sK292 @ (^[Y0 : $i]: (sK293 @ Y0))) @ X8) != (sK293 @ X8)) & ((in @ X8 @ sK294) = $true)) & ! [X9 : $i] : (((in @ (sK293 @ X9) @ sK292) = $true) | ((in @ X9 @ sK294) != $true)))),
% 2.65/1.72 introduced(definition,[],[choice_axiom])).
% 2.65/1.72 thf(f1380,plain,(
% 2.65/1.72 ? [X8 : $i] : (((ap @ sK294 @ sK292 @ (lam @ sK294 @ sK292 @ (^[Y0 : $i]: (sK293 @ Y0))) @ X8) != (sK293 @ X8)) & ((in @ X8 @ sK294) = $true)) => (((ap @ sK294 @ sK292 @ (lam @ sK294 @ sK292 @ (^[Y0 : $i]: (sK293 @ Y0))) @ sK295) != (sK293 @ sK295)) & ((in @ sK295 @ sK294) = $true))),
% 2.65/1.72 introduced(definition,[],[choice_axiom])).
% 2.65/1.72 thf(f1377,plain,(
% 2.65/1.72 (! [X0 : $i,X1 : $i > $i,X2 : $i] : (! [X3 : $i] : (((X1 @ X3) = (ap @ X2 @ X0 @ (lam @ X2 @ X0 @ (^[Y0 : $i]: (X1 @ Y0))) @ X3)) | ((in @ X3 @ X2) != $true)) | ? [X4 : $i] : (((in @ (X1 @ X4) @ X0) != $true) & ((in @ X4 @ X2) = $true))) | (beta1 != $true)) & ((beta1 = $true) | ? [X5 : $i,X6 : $i > $i,X7 : $i] : (? [X8 : $i] : (((X6 @ X8) != (ap @ X7 @ X5 @ (lam @ X7 @ X5 @ (^[Y0 : $i]: (X6 @ Y0))) @ X8)) & ((in @ X8 @ X7) = $true)) & ! [X9 : $i] : (((in @ (X6 @ X9) @ X5) = $true) | ($true != (in @ X9 @ X7)))))),
% 2.65/1.72 inference(rectify,[status(thm)],[f1376])).
% 2.65/1.72 thf(f1376,plain,(
% 2.65/1.72 (! [X0 : $i,X1 : $i > $i,X2 : $i] : (! [X4 : $i] : (((X1 @ X4) = (ap @ X2 @ X0 @ (lam @ X2 @ X0 @ (^[Y0 : $i]: (X1 @ Y0))) @ X4)) | ((in @ X4 @ X2) != $true)) | ? [X3 : $i] : (((in @ (X1 @ X3) @ X0) != $true) & ((in @ X3 @ X2) = $true))) | (beta1 != $true)) & ((beta1 = $true) | ? [X0 : $i,X1 : $i > $i,X2 : $i] : (? [X4 : $i] : (((X1 @ X4) != (ap @ X2 @ X0 @ (lam @ X2 @ X0 @ (^[Y0 : $i]: (X1 @ Y0))) @ X4)) & ((in @ X4 @ X2) = $true)) & ! [X3 : $i] : (((in @ (X1 @ X3) @ X0) = $true) | ((in @ X3 @ X2) != $true))))),
% 2.65/1.72 inference(nnf_transformation,[status(thm)],[f843])).
% 2.65/1.72 thf(f843,plain,(
% 2.65/1.72 ! [X0 : $i,X1 : $i > $i,X2 : $i] : (! [X4 : $i] : (((X1 @ X4) = (ap @ X2 @ X0 @ (lam @ X2 @ X0 @ (^[Y0 : $i]: (X1 @ Y0))) @ X4)) | ((in @ X4 @ X2) != $true)) | ? [X3 : $i] : (((in @ (X1 @ X3) @ X0) != $true) & ((in @ X3 @ X2) = $true))) <=> (beta1 = $true)),
% 2.65/1.72 inference(ennf_transformation,[status(thm)],[f468])).
% 2.65/1.72 thf(f468,plain,(
% 2.65/1.72 ! [X0 : $i,X1 : $i > $i,X2 : $i] : (! [X3 : $i] : (((in @ X3 @ X2) = $true) => ((in @ (X1 @ X3) @ X0) = $true)) => ! [X4 : $i] : (((in @ X4 @ X2) = $true) => ((X1 @ X4) = (ap @ X2 @ X0 @ (lam @ X2 @ X0 @ (^[Y0 : $i]: (X1 @ Y0))) @ X4)))) <=> (beta1 = $true)),
% 2.65/1.72 inference(fool_elimination,[status(thm)],[f467])).
% 2.65/1.72 thf(f467,plain,(
% 2.65/1.72 (beta1 = ! [X0 : $i,X1 : $i > $i,X2 : $i] : (! [X3 : $i] : ((in @ X3 @ X2) => (in @ (X1 @ X3) @ X0)) => ! [X4 : $i] : ((in @ X4 @ X2) => ((ap @ X2 @ X0 @ (lam @ X2 @ X0 @ (^[X5 : $i] : (X1 @ X5))) @ X4) = (X1 @ X4)))))),
% 2.65/1.72 inference(rectify,[status(thm)],[f218])).
% 2.65/1.72 thf(f218,axiom,(
% 2.65/1.72 (beta1 = ! [X4 : $i,X12 : $i > $i,X3 : $i] : (! [X1 : $i] : ((in @ X1 @ X3) => (in @ (X12 @ X1) @ X4)) => ! [X1 : $i] : ((in @ X1 @ X3) => ((ap @ X3 @ X4 @ (lam @ X3 @ X4 @ (^[X2 : $i] : (X12 @ X2))) @ X1) = (X12 @ X1)))))),
% 2.65/1.72 file('/export/starexec/sandbox/tmp/tmp.9FuAMmeF6I/DTF2THF_3007.p',beta1)).
% 2.65/1.72 thf(f3191,plain,(
% 2.65/1.72 ((ap @ sK406 @ sK408 @ (lam @ sK406 @ sK408 @ sK407) @ sK409) != (sK407 @ sK409))),
% 2.65/1.72 inference(beta_eta_normalization,[status(thm)],[f2664])).
% 2.65/1.72 thf(f2664,plain,(
% 2.65/1.72 ((ap @ sK406 @ sK408 @ (lam @ sK406 @ sK408 @ (^[Y0 : $i]: (sK407 @ Y0))) @ sK409) != (sK407 @ sK409))),
% 2.65/1.72 inference(cnf_transformation,[status(thm)],[f1553])).
% 2.65/1.72 thf(f3620,plain,(
% 2.65/1.72 ((in @ (sK291 @ sK406 @ sK407 @ sK408) @ sK406) != $true)),
% 2.65/1.72 inference(trivial_inequality_removal,[status(thm)],[f3609])).
% 2.65/1.72 thf(f3609,plain,(
% 2.65/1.72 ($true != $true) | ((in @ (sK291 @ sK406 @ sK407 @ sK408) @ sK406) != $true)),
% 2.65/1.72 inference(superposition,[status(thm)],[f3242,f2665])).
% 2.65/1.72 thf(f2665,plain,(
% 2.65/1.72 ( ! [X3 : $i] : (((in @ (sK407 @ X3) @ sK408) = $true) | ((in @ X3 @ sK406) != $true)) )),
% 2.65/1.72 inference(cnf_transformation,[status(thm)],[f1553])).
% 2.65/1.72 thf(f3242,plain,(
% 2.65/1.72 ((in @ (sK407 @ (sK291 @ sK406 @ sK407 @ sK408)) @ sK408) != $true)),
% 2.65/1.72 inference(trivial_inequality_removal,[status(thm)],[f3241])).
% 2.65/1.72 thf(f3241,plain,(
% 2.65/1.72 ($true != $true) | ((in @ (sK407 @ (sK291 @ sK406 @ sK407 @ sK408)) @ sK408) != $true)),
% 2.65/1.72 inference(forward_demodulation,[status(thm)],[f3240,f2663])).
% 2.65/1.72 thf(f3240,plain,(
% 2.65/1.72 ((in @ (sK407 @ (sK291 @ sK406 @ sK407 @ sK408)) @ sK408) != $true) | ((in @ sK409 @ sK406) != $true)),
% 2.65/1.72 inference(trivial_inequality_removal,[status(thm)],[f3239])).
% 2.65/1.72 thf(f3239,plain,(
% 2.65/1.72 ((in @ (sK407 @ (sK291 @ sK406 @ sK407 @ sK408)) @ sK408) != $true) | ((in @ sK409 @ sK406) != $true) | ($true != $true)),
% 2.65/1.72 inference(forward_demodulation,[status(thm)],[f3212,f2620])).
% 2.65/1.72 thf(f3212,plain,(
% 2.65/1.72 (beta1 != $true) | ((in @ (sK407 @ (sK291 @ sK406 @ sK407 @ sK408)) @ sK408) != $true) | ((in @ sK409 @ sK406) != $true)),
% 2.65/1.72 inference(trivial_inequality_removal,[status(thm)],[f3209])).
% 2.65/1.72 thf(f3209,plain,(
% 2.65/1.72 ((sK407 @ sK409) != (sK407 @ sK409)) | ((in @ sK409 @ sK406) != $true) | ((in @ (sK407 @ (sK291 @ sK406 @ sK407 @ sK408)) @ sK408) != $true) | (beta1 != $true)),
% 2.65/1.72 inference(superposition,[status(thm)],[f3191,f3087])).
% 2.65/1.72 thf(f3087,plain,(
% 2.65/1.72 ( ! [X2 : $i,X3 : $i,X0 : $i,X1 : $i > $i] : (((ap @ X2 @ X0 @ (lam @ X2 @ X0 @ X1) @ X3) = (X1 @ X3)) | ((in @ X3 @ X2) != $true) | ((in @ (X1 @ (sK291 @ X2 @ X1 @ X0)) @ X0) != $true) | (beta1 != $true)) )),
% 2.65/1.72 inference(beta_eta_normalization,[status(thm)],[f2333])).
% 2.65/1.72 thf(f2333,plain,(
% 2.65/1.72 ( ! [X2 : $i,X3 : $i,X0 : $i,X1 : $i > $i] : (((in @ (X1 @ (sK291 @ X2 @ X1 @ X0)) @ X0) != $true) | ((in @ X3 @ X2) != $true) | ((X1 @ X3) = (ap @ X2 @ X0 @ (lam @ X2 @ X0 @ (^[Y0 : $i]: (X1 @ Y0))) @ X3)) | (beta1 != $true)) )),
% 2.65/1.72 inference(cnf_transformation,[status(thm)],[f1381])).
% 2.65/1.72 % SZS output end Proof for DTF2THF_3007
% 2.65/1.72 % (3320)------------------------------
% 2.65/1.72 % (3320)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.72 % (3320)Termination reason: Refutation
% 2.65/1.72
% 2.65/1.72 % (3320)Memory used [KB]: 9594
% 2.65/1.72 % (3320)Time elapsed: 0.103 s
% 2.65/1.72 % (3320)Instructions burned: 304 (million)
% 2.65/1.72 % (3320)------------------------------
% 2.65/1.72 % (3320)------------------------------
% 2.65/1.72 % (3297)Success in time 0.186 s
%------------------------------------------------------------------------------