%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU718^1 : TPTP v9.2.1. Released v3.7.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n015.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 09:36:38 AM UTC 2026
% Result : Theorem 3.02s 2.00s
% Output : Refutation 3.02s
% Verified :
% SZS Type : Refutation
% Derivation depth : 16
% Number of leaves : 8
% Syntax : Number of formulae : 46 ( 13 unt; 0 typ; 0 def)
% Number of atoms : 4115 (1100 equ; 0 cnn)
% Maximal formula atoms : 240 ( 89 avg)
% Number of connectives : 2210 ( 68 ~; 44 |; 762 &; 361 @)
% ( 5 <=>; 970 =>; 0 <=; 0 <~>)
% Maximal formula depth : 247 ( 41 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 434 ( 431 usr; 245 con; 0-4 aty)
% Number of variables : 96 ( 0 ^; 63 !; 33 ?; 96 :)
% 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_259,type,
if: $i > $o > $i > $i > $i ).
thf(func_def_287,type,
sP1: $i > $i > $i > $i > $o ).
thf(func_def_292,type,
sP6: $i > $o > $i > $i > $o ).
thf(func_def_293,type,
sP7: $i > $i > $o ).
thf(func_def_294,type,
sP8: $i > $o ).
thf(func_def_295,type,
sP9: $i > $i > $o ).
thf(func_def_296,type,
sP10: $i > $i > $o ).
thf(func_def_297,type,
sP11: $i > $i > $i > $o ).
thf(func_def_300,type,
sK14: $i > $i > $i ).
thf(func_def_312,type,
sK26: $i > $i ).
thf(func_def_313,type,
sK27: $i > $i ).
thf(func_def_320,type,
sK34: ( $i > $o ) > $i > $i ).
thf(func_def_322,type,
sK36: $i > $o ).
thf(func_def_325,type,
sK39: $i > $o ).
thf(func_def_333,type,
sK47: $i > $i > $i > $i > $i ).
thf(func_def_343,type,
sK57: $i > $i > $i > $i > $i ).
thf(func_def_346,type,
sK60: $i > $i ).
thf(func_def_348,type,
sK62: ( $i > $i ) > $i > $i > $i ).
thf(func_def_354,type,
sK68: $i > $i > $i > $i > $i ).
thf(func_def_355,type,
sK69: $i > $i > $i > $i > $i ).
thf(func_def_375,type,
sK89: $i > $i > $i > $i > $i ).
thf(func_def_376,type,
sK90: $i > $i > $i > $i > $i ).
thf(func_def_377,type,
sK91: $i > $i > $i > $i > $i ).
thf(func_def_378,type,
sK92: $i > $i > $i > $i > $i ).
thf(func_def_387,type,
sK101: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_388,type,
sK102: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_392,type,
sK106: $i > $o ).
thf(func_def_402,type,
sK116: $i > $o ).
thf(func_def_417,type,
sK131: $i > $i > $i > $i ).
thf(func_def_427,type,
sK141: $i > $i > $i ).
thf(func_def_428,type,
sK142: $i > $i > $i ).
thf(func_def_431,type,
sK145: $i > $o ).
thf(func_def_432,type,
sK146: $i > $i > $o ).
thf(func_def_439,type,
sK153: $i > $o ).
thf(func_def_444,type,
sK158: $i > $o ).
thf(func_def_446,type,
sK160: $i > ( $i > $o ) > $i ).
thf(func_def_449,type,
sK163: $i > $i > $i ).
thf(func_def_461,type,
sK175: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_462,type,
sK176: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_463,type,
sK177: $i > $o ).
thf(func_def_464,type,
sK178: $i > $o ).
thf(func_def_467,type,
sK181: $i > $o ).
thf(func_def_499,type,
sK213: $i > $o ).
thf(func_def_502,type,
sK216: $i > ( $i > $o ) > $i ).
thf(func_def_504,type,
sK218: $i > $o ).
thf(func_def_517,type,
sK231: $i > $o ).
thf(func_def_523,type,
sK237: ( $i > $o ) > $i ).
thf(func_def_524,type,
sK238: $i > $o ).
thf(func_def_525,type,
sK239: $i > $i ).
thf(func_def_529,type,
sK243: $i > $i > $o ).
thf(func_def_545,type,
sK259: $i > $i > $i > $i ).
thf(func_def_552,type,
sK266: $i > $i ).
thf(func_def_554,type,
sK268: $i > ( $i > $i ) > $i > $i ).
thf(func_def_555,type,
sK269: ( $i > $i ) > $i > $i > $i ).
thf(func_def_558,type,
sK272: $i > $i ).
thf(func_def_561,type,
sK275: $i > $o ).
thf(func_def_583,type,
sK297: $i > $i > $i ).
thf(func_def_589,type,
sK303: $i > $i > $o ).
thf(func_def_619,type,
sK333: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_620,type,
sK334: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_621,type,
sK335: $i > $o ).
thf(func_def_622,type,
sK336: $i > $o ).
thf(func_def_623,type,
sK337: $i > $i > $i ).
thf(func_def_637,type,
sK351: ( $i > $o ) > $i ).
thf(func_def_638,type,
sK352: $i > $o ).
thf(func_def_639,type,
sK353: $i > $i ).
thf(func_def_645,type,
sK359: $i > $o ).
thf(func_def_647,type,
sK361: ( $i > $o ) > $i > $i ).
thf(func_def_648,type,
sK362: ( $i > $o ) > $i > $i ).
thf(func_def_678,type,
sK392: $i > $o ).
thf(func_def_686,type,
sK400: $i > $i ).
thf(func_def_688,type,
sK402: $i > $i ).
thf(func_def_705,type,
sK419: $i > $i > $o ).
thf(func_def_706,type,
sK420: $i > $i ).
thf(func_def_707,type,
sK421: $i > $i ).
thf(func_def_708,type,
sK422: ( $i > $i > $o ) > $i > $i ).
thf(func_def_709,type,
sK423: ( $i > $i > $o ) > $i > $i ).
thf(func_def_710,type,
sK424: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_718,type,
sK432: $i > $i > $o ).
thf(func_def_722,type,
sK436: $i > $o ).
thf(func_def_723,type,
sK437: ( $i > $o ) > $i ).
thf(func_def_726,type,
sK440: $i > $o ).
thf(func_def_731,type,
sK445: $i > ( $i > $i ) > $i > $i ).
thf(func_def_733,type,
sK447: $i > $i ).
thf(func_def_736,type,
sK450: $i > $i ).
thf(func_def_743,type,
sK457: $i > $i > $i ).
thf(func_def_746,type,
sK460: $i > $o ).
thf(func_def_749,type,
sK463: $i > ( $i > $o ) > $i ).
thf(func_def_763,type,
sK477: $i > $o ).
thf(func_def_764,type,
sK478: $i > $o ).
thf(func_def_765,type,
sK479: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_766,type,
sK480: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_777,type,
sK491: $i > $i ).
thf(func_def_780,type,
sK494: ( $i > $o ) > $i ).
thf(func_def_781,type,
sK495: ( $i > $o ) > $i ).
thf(func_def_782,type,
sK496: $i > $o ).
thf(func_def_796,type,
sK510: $i > $o ).
thf(func_def_799,type,
sK513: $i > $i > ( $i > $o ) > $i ).
thf(func_def_801,type,
sK515: $i > $o ).
thf(func_def_808,type,
sK522: $i > $i > $i ).
thf(func_def_809,type,
sK523: ( $i > $o ) > $i > $i ).
thf(func_def_811,type,
sK525: $i > $o ).
thf(func_def_815,type,
sK529: $i > $i > $o ).
thf(func_def_821,type,
sK535: $o > $i > $i > $i ).
thf(func_def_822,type,
sK536: $i > $o ).
thf(func_def_825,type,
sK539: $i > ( $i > $o ) > $i ).
thf(func_def_826,type,
sK540: $i > ( $i > $o ) > $i ).
thf(func_def_836,type,
sK550: $i > $i > $i > $i > $i ).
thf(func_def_855,type,
sK569: $i > $i ).
thf(func_def_861,type,
sK575: $i > $i > $i > $i ).
thf(func_def_862,type,
sK576: $i > $i > $i > $i ).
thf(func_def_865,type,
sK579: $i > $i ).
thf(func_def_866,type,
sK580: ( $i > $i ) > $i > $i > $i ).
thf(func_def_869,type,
sK583: $i > $i ).
thf(func_def_872,type,
sK586: $i > $i > ( $i > $i ) > $i ).
thf(func_def_873,type,
sK587: $i > $o ).
thf(func_def_875,type,
sK589: $i > ( $i > $o ) > $i ).
thf(func_def_899,type,
sK613: $i > $i > $o ).
thf(func_def_907,type,
sK621: $i > $o ).
thf(func_def_910,type,
sK624: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_911,type,
sK625: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_915,type,
sK629: $i > $o ).
thf(func_def_926,type,
sK640: $i > $i > $i ).
thf(func_def_947,type,
sK661: $i > $o ).
thf(func_def_949,type,
sK663: ( $i > $o ) > $i > $i ).
thf(func_def_950,type,
sK664: $i > ( $i > $o ) > $i ).
thf(func_def_951,type,
sK665: $i > $o ).
thf(func_def_956,type,
sK670: $i > $i > $i ).
thf(func_def_957,type,
sK671: $i > $i > $i ).
thf(func_def_958,type,
sK672: $i > $i > $i ).
thf(func_def_959,type,
sK673: $i > $i > $i ).
thf(func_def_960,type,
sK674: $i > $i > $i ).
thf(func_def_961,type,
sK675: $i > $i > $i > $i ).
thf(func_def_962,type,
sK676: $i > $i ).
thf(func_def_963,type,
sK677: $i > $i ).
thf(func_def_964,type,
sK678: $i > $i ).
thf(func_def_965,type,
sK679: $i > $i ).
thf(func_def_966,type,
sK680: $i > $i > $i > $i ).
thf(func_def_967,type,
sK681: $i > $i > $i > $i ).
thf(func_def_968,type,
sK682: $i > $i > $i ).
thf(func_def_969,type,
sK683: $i > $i > $i ).
thf(func_def_970,type,
sK684: $i > $i > $i ).
thf(func_def_971,type,
sK685: $i > $i ).
thf(func_def_973,type,
sK687: $i > $i ).
thf(func_def_974,type,
sK688: $i > $i ).
thf(func_def_981,type,
sK695: $i > $i > $i ).
thf(func_def_992,type,
sK706: ( $i > $o ) > $i > $i ).
thf(func_def_994,type,
sK708: $i > $o ).
thf(func_def_1011,type,
sK725: $i > $i > $o ).
thf(func_def_1015,type,
sK729: $i > $i > $i > $i ).
thf(func_def_1019,type,
sK733: $i > $i > $i > $i ).
thf(func_def_1020,type,
sK734: $i > $o ).
thf(func_def_1028,type,
ph742:
!>[X0: $tType] : X0 ).
thf(f5979,plain,
$false,
inference(trivial_inequality_removal,[status(thm)],[f5968]) ).
thf(f5968,plain,
$true != $true,
inference(superposition,[status(thm)],[f3722,f2550]) ).
thf(f2550,plain,
( $true
= ( in @ sK167 @ sK165 ) ),
inference(cnf_transformation,[status(thm)],[f1270]) ).
thf(f1270,plain,
( ( setukpairinjR11 = $true )
& ( exuE1 = $true )
& ( singletonprop = $true )
& ( setminusERneg = $true )
& ( powerset__Cong = $true )
& ( binunionRsub = $true )
& ( upairsetE = $true )
& ( ex1I2 = $true )
& ( dsetconstrI = $true )
& ( setadjoinIR = $true )
& ( foundationAx = $true )
& ( setukpairIR = $true )
& ( setminusER = $true )
& ( exuI3 = $true )
& ( ubforcartprodlem1 = $true )
& ( eqinunit = $true )
& ( funcGraphProp2 = $true )
& ( binintersectRsub = $true )
& ( emptyinPowerset = $true )
& ( kpairp = $true )
& ( binintersectI = $true )
& ( binintersectSubset1 = $true )
& ( subsetRefl = $true )
& ( funcGraphProp3 = $true )
& ( sepSubset = $true )
& ( inPowerset = $true )
& ( singletoninpowunion = $true )
& ( cartprodpairsurjEq = $true )
& ( sepInPowerset = $true )
& ( setbeta = $true )
& ( singletonsubset = $true )
& ( funcinfuncset = $true )
& ( upairset2E = $true )
& ( emptysetimpfalse = $true )
& ( noeltsimpempty = $true )
& ( cartprodpairin = $true )
& ( binintersectSubset4 = $true )
& ( binunionE = $true )
& ( prop2setI = $true )
& ( ubforcartprodlem3 = $true )
& ( disjointsetsI1 = $true )
& ( ( in @ sK167 @ sK164 )
= $true )
& ( $true
= ( in @ sK165 @ ( powerset @ sK166 ) ) )
& ( ( in @ sK167 @ sK166 )
!= $true )
& ( $true
= ( in @ sK167 @ sK165 ) )
& ( ( in @ sK166 @ ( powerset @ sK164 ) )
= $true )
& ( ( in @ sK165 @ ( powerset @ sK164 ) )
= $true )
& ( funcext2 = $true )
& ( dpsetconstrEL2 = $true )
& ( setminusT_lem = $true )
& ( beta2 = $true )
& ( eqimpsubset2 = $true )
& ( iftrueProp1 = $true )
& ( cartprodpairmemEL = $true )
& ( binunionIR = $true )
& ( binintersectER = $true )
& ( setukpairinjR = $true )
& ( setminusI = $true )
& ( ap2apEq2 = $true )
& ( ex1I = $true )
& ( setunionsingleton1 = $true )
& ( lamProp = $true )
& ( powersetI1 = $true )
& ( emptyInPowerset = $true )
& ( uniqinunit = $true )
& ( binunionIL = $true )
& ( prop2setE = $true )
& ( subset2powerset = $true )
& ( cartprodfstin = $true )
& ( setunion__Cong = $true )
& ( exuEu = $true )
& ( notdallE = $true )
& ( subsetE = $true )
& ( symdiffIneg1 = $true )
& ( symdiffE = $true )
& ( setukpairinjR1 = $true )
& ( ksndpairEq = $true )
& ( complementT_lem = $true )
& ( setunionE = $true )
& ( upairequniteq = $true )
& ( ex1E2 = $true )
& ( setukpairinjL = $true )
& ( apProp = $true )
& ( vacuousDall = $true )
& ( descrp = $true )
& ( subsetI2 = $true )
& ( emptyI = $true )
& ( upairsubunion = $true )
& ( notsubsetI = $true )
& ( ifSingleton = $true )
& ( dpsetconstrEL1 = $true )
& ( nonemptyI = $true )
& ( exu__Cong = $true )
& ( setunionsingleton = $true )
& ( singletonsswitch = $true )
& ( beta1 = $true )
& ( subsetE2 = $true )
& ( iftrue = $true )
& ( symdiffIneg2 = $true )
& ( infuncsetfunc = $true )
& ( setadjoinE = $true )
& ( setminusILneg = $true )
& ( powersetI = $true )
& ( upairsetIL = $true )
& ( quantDeMorgan4 = $true )
& ( powersetTI1 = $true )
& ( binintersectLsub = $true )
& ( bs114d = $true )
& ( singletonsuniq = $true )
& ( nonemptyE1 = $true )
& ( in__Cong = $true )
& ( setunionAx = $true )
& ( binintersectEL = $true )
& ( binintersectSubset3 = $true )
& ( funcextLem = $true )
& ( replAx = $true )
& ( subbreln = $true )
& ( binintersectT_lem = $true )
& ( emptysetsubset = $true )
& ( descr__Cong = $true )
& ( setunionI = $true )
& ( setextAx = $true )
& ( singletoninpowerset = $true )
& ( setadjoinIL = $true )
& ( setminusSubset1 = $true )
& ( upairsetIR = $true )
& ( setadjoinOr = $true )
& ( dpsetconstrERa = $true )
& ( binintersectSubset5 = $true )
& ( exuI1 = $true )
& ( kfstpairEq = $true )
& ( iffalse = $true )
& ( setadjoinSub = $true )
& ( setunionsingleton2 = $true )
& ( binintersectSubset2 = $true )
& ( nonemptyI1 = $true )
& ( cartprodsndin = $true )
& ( dsetconstrEL = $true )
& ( omega0Ax = $true )
& ( emptyinunitempty = $true )
& ( setadjoin__Cong = $true )
& ( setukpairinjR2 = $true )
& ( subPowSU = $true )
& ( iftrueProp2 = $true )
& ( omegaIndAx = $true )
& ( setOfPairsIsBReln = $true )
& ( kpairsurjEq = $true )
& ( notequalI1 = $true )
& ( binunionT_lem = $true )
& ( powersetE1 = $true )
& ( dpsetconstrI = $true )
& ( exuE3u = $true )
& ( binunionLsub = $true )
& ( symdiffI1 = $true )
& ( funcext = $true )
& ( dsetconstr__Cong = $true )
& ( funcGraphProp4 = $true )
& ( iffalseProp1 = $true )
& ( setext = $true )
& ( setminusIRneg = $true )
& ( theeq = $true )
& ( quantDeMorgan3 = $true )
& ( powersetE = $true )
& ( theprop = $true )
& ( notequalI2 = $true )
& ( lamp = $true )
& ( emptyset__Cong = $true )
& ( nonemptyImpWitness = $true )
& ( iffalseProp2 = $true )
& ( quantDeMorgan2 = $true )
& ( kpairiskpair = $true )
& ( exuI2 = $true )
& ( dpsetconstrER = $true )
& ( inCongP = $true )
& ( brelnall1 = $true )
& ( app = $true )
& ( cartprodfstpairEq = $true )
& ( ksndsingleton = $true )
& ( upairset2IR = $true )
& ( exuE2 = $true )
& ( dpsetconstrSub = $true )
& ( powersetsubset = $true )
& ( brelnall2 = $true )
& ( eta1 = $true )
& ( setukpairinjL1 = $true )
& ( cartprodmempaircEq = $true )
& ( cartprodmempair1 = $true )
& ( lam2lamEq = $true )
& ( binunionEcases = $true )
& ( exuE3e = $true )
& ( dsetconstrER = $true )
& ( symdiffI2 = $true )
& ( quantDeMorgan1 = $true )
& ( funcGraphProp1 = $true )
& ( setadjoinAx = $true )
& ( ubforcartprodlem2 = $true )
& ( notinsingleton = $true )
& ( subsetI1 = $true )
& ( funcImageSingleton = $true )
& ( cartprodsndpairEq = $true )
& ( ifp = $true )
& ( cartprodmempair = $true )
& ( subsetTrans = $true )
& ( iftrueorfalse = $true )
& ( setminusELneg = $true )
& ( emptyE1 = $true )
& ( setukpairinjL2 = $true )
& ( setextsub = $true )
& ( emptysetAx = $true )
& ( setminusSubset2 = $true )
& ( setminusLsub = $true )
& ( ap2p = $true )
& ( omegaSAx = $true )
& ( setextT = $true )
& ( powersetT_lem = $true )
& ( setukpairinjR12 = $true )
& ( eqbreln = $true )
& ( subsetTI = $true )
& ( cartprodpairmemER = $true )
& ( emptysetE = $true )
& ( notdexE = $true )
& ( setadjoinSub2 = $true )
& ( lam2p = $true )
& ( kfstsingleton = $true )
& ( setoftrueEq = $true )
& ( omega__Cong = $true )
& ( ex1E1 = $true )
& ( upairinpowunion = $true )
& ( prop2set2propI = $true )
& ( wellorderingAx = $true )
& ( setminusEL = $true )
& ( powersetAx = $true )
& ( secondinupair = $true )
& ( eqimpsubset1 = $true )
& ( subsetemptysetimpeq = $true )
& ( ap2apEq1 = $true )
& ( eta2 = $true )
& ( setunionE2 = $true )
& ( setukpairIL = $true )
& ( notinemptyset = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK164,sK165,sK166,sK167])],[f885,f1269,f1268,f1267]) ).
thf(f1267,plain,
( ? [X0: $i,X1: $i] :
( ? [X2: $i] :
( ? [X3: $i] :
( ( $true
= ( in @ X3 @ X0 ) )
& ( $true
= ( in @ X1 @ ( powerset @ X2 ) ) )
& ( ( in @ X3 @ X2 )
!= $true )
& ( ( in @ X3 @ X1 )
= $true ) )
& ( ( in @ X2 @ ( powerset @ X0 ) )
= $true ) )
& ( $true
= ( in @ X1 @ ( powerset @ X0 ) ) ) )
=> ( ? [X2: $i] :
( ? [X3: $i] :
( ( ( in @ X3 @ sK164 )
= $true )
& ( $true
= ( in @ sK165 @ ( powerset @ X2 ) ) )
& ( ( in @ X3 @ X2 )
!= $true )
& ( ( in @ X3 @ sK165 )
= $true ) )
& ( ( in @ X2 @ ( powerset @ sK164 ) )
= $true ) )
& ( ( in @ sK165 @ ( powerset @ sK164 ) )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1268,plain,
( ? [X2: $i] :
( ? [X3: $i] :
( ( ( in @ X3 @ sK164 )
= $true )
& ( $true
= ( in @ sK165 @ ( powerset @ X2 ) ) )
& ( ( in @ X3 @ X2 )
!= $true )
& ( ( in @ X3 @ sK165 )
= $true ) )
& ( ( in @ X2 @ ( powerset @ sK164 ) )
= $true ) )
=> ( ? [X3: $i] :
( ( ( in @ X3 @ sK164 )
= $true )
& ( $true
= ( in @ sK165 @ ( powerset @ sK166 ) ) )
& ( ( in @ X3 @ sK166 )
!= $true )
& ( ( in @ X3 @ sK165 )
= $true ) )
& ( ( in @ sK166 @ ( powerset @ sK164 ) )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1269,plain,
( ? [X3: $i] :
( ( ( in @ X3 @ sK164 )
= $true )
& ( $true
= ( in @ sK165 @ ( powerset @ sK166 ) ) )
& ( ( in @ X3 @ sK166 )
!= $true )
& ( ( in @ X3 @ sK165 )
= $true ) )
=> ( ( ( in @ sK167 @ sK164 )
= $true )
& ( $true
= ( in @ sK165 @ ( powerset @ sK166 ) ) )
& ( ( in @ sK167 @ sK166 )
!= $true )
& ( $true
= ( in @ sK167 @ sK165 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f885,plain,
( ( setukpairinjR11 = $true )
& ( exuE1 = $true )
& ( singletonprop = $true )
& ( setminusERneg = $true )
& ( powerset__Cong = $true )
& ( binunionRsub = $true )
& ( upairsetE = $true )
& ( ex1I2 = $true )
& ( dsetconstrI = $true )
& ( setadjoinIR = $true )
& ( foundationAx = $true )
& ( setukpairIR = $true )
& ( setminusER = $true )
& ( exuI3 = $true )
& ( ubforcartprodlem1 = $true )
& ( eqinunit = $true )
& ( funcGraphProp2 = $true )
& ( binintersectRsub = $true )
& ( emptyinPowerset = $true )
& ( kpairp = $true )
& ( binintersectI = $true )
& ( binintersectSubset1 = $true )
& ( subsetRefl = $true )
& ( funcGraphProp3 = $true )
& ( sepSubset = $true )
& ( inPowerset = $true )
& ( singletoninpowunion = $true )
& ( cartprodpairsurjEq = $true )
& ( sepInPowerset = $true )
& ( setbeta = $true )
& ( singletonsubset = $true )
& ( funcinfuncset = $true )
& ( upairset2E = $true )
& ( emptysetimpfalse = $true )
& ( noeltsimpempty = $true )
& ( cartprodpairin = $true )
& ( binintersectSubset4 = $true )
& ( binunionE = $true )
& ( prop2setI = $true )
& ( ubforcartprodlem3 = $true )
& ( disjointsetsI1 = $true )
& ? [X0: $i,X1: $i] :
( ? [X2: $i] :
( ? [X3: $i] :
( ( $true
= ( in @ X3 @ X0 ) )
& ( $true
= ( in @ X1 @ ( powerset @ X2 ) ) )
& ( ( in @ X3 @ X2 )
!= $true )
& ( ( in @ X3 @ X1 )
= $true ) )
& ( ( in @ X2 @ ( powerset @ X0 ) )
= $true ) )
& ( $true
= ( in @ X1 @ ( powerset @ X0 ) ) ) )
& ( funcext2 = $true )
& ( dpsetconstrEL2 = $true )
& ( setminusT_lem = $true )
& ( beta2 = $true )
& ( eqimpsubset2 = $true )
& ( iftrueProp1 = $true )
& ( cartprodpairmemEL = $true )
& ( binunionIR = $true )
& ( binintersectER = $true )
& ( setukpairinjR = $true )
& ( setminusI = $true )
& ( ap2apEq2 = $true )
& ( ex1I = $true )
& ( setunionsingleton1 = $true )
& ( lamProp = $true )
& ( powersetI1 = $true )
& ( emptyInPowerset = $true )
& ( uniqinunit = $true )
& ( binunionIL = $true )
& ( prop2setE = $true )
& ( subset2powerset = $true )
& ( cartprodfstin = $true )
& ( setunion__Cong = $true )
& ( exuEu = $true )
& ( notdallE = $true )
& ( subsetE = $true )
& ( symdiffIneg1 = $true )
& ( symdiffE = $true )
& ( setukpairinjR1 = $true )
& ( ksndpairEq = $true )
& ( complementT_lem = $true )
& ( setunionE = $true )
& ( upairequniteq = $true )
& ( ex1E2 = $true )
& ( setukpairinjL = $true )
& ( apProp = $true )
& ( vacuousDall = $true )
& ( descrp = $true )
& ( subsetI2 = $true )
& ( emptyI = $true )
& ( upairsubunion = $true )
& ( notsubsetI = $true )
& ( ifSingleton = $true )
& ( dpsetconstrEL1 = $true )
& ( nonemptyI = $true )
& ( exu__Cong = $true )
& ( setunionsingleton = $true )
& ( singletonsswitch = $true )
& ( beta1 = $true )
& ( subsetE2 = $true )
& ( iftrue = $true )
& ( symdiffIneg2 = $true )
& ( infuncsetfunc = $true )
& ( setadjoinE = $true )
& ( setminusILneg = $true )
& ( powersetI = $true )
& ( upairsetIL = $true )
& ( quantDeMorgan4 = $true )
& ( powersetTI1 = $true )
& ( binintersectLsub = $true )
& ( bs114d = $true )
& ( singletonsuniq = $true )
& ( nonemptyE1 = $true )
& ( in__Cong = $true )
& ( setunionAx = $true )
& ( binintersectEL = $true )
& ( binintersectSubset3 = $true )
& ( funcextLem = $true )
& ( replAx = $true )
& ( subbreln = $true )
& ( binintersectT_lem = $true )
& ( emptysetsubset = $true )
& ( descr__Cong = $true )
& ( setunionI = $true )
& ( setextAx = $true )
& ( singletoninpowerset = $true )
& ( setadjoinIL = $true )
& ( setminusSubset1 = $true )
& ( upairsetIR = $true )
& ( setadjoinOr = $true )
& ( dpsetconstrERa = $true )
& ( binintersectSubset5 = $true )
& ( exuI1 = $true )
& ( kfstpairEq = $true )
& ( iffalse = $true )
& ( setadjoinSub = $true )
& ( setunionsingleton2 = $true )
& ( binintersectSubset2 = $true )
& ( nonemptyI1 = $true )
& ( cartprodsndin = $true )
& ( dsetconstrEL = $true )
& ( omega0Ax = $true )
& ( emptyinunitempty = $true )
& ( setadjoin__Cong = $true )
& ( setukpairinjR2 = $true )
& ( subPowSU = $true )
& ( iftrueProp2 = $true )
& ( omegaIndAx = $true )
& ( setOfPairsIsBReln = $true )
& ( kpairsurjEq = $true )
& ( notequalI1 = $true )
& ( binunionT_lem = $true )
& ( powersetE1 = $true )
& ( dpsetconstrI = $true )
& ( exuE3u = $true )
& ( binunionLsub = $true )
& ( symdiffI1 = $true )
& ( funcext = $true )
& ( dsetconstr__Cong = $true )
& ( funcGraphProp4 = $true )
& ( iffalseProp1 = $true )
& ( setext = $true )
& ( setminusIRneg = $true )
& ( theeq = $true )
& ( quantDeMorgan3 = $true )
& ( powersetE = $true )
& ( theprop = $true )
& ( notequalI2 = $true )
& ( lamp = $true )
& ( emptyset__Cong = $true )
& ( nonemptyImpWitness = $true )
& ( iffalseProp2 = $true )
& ( quantDeMorgan2 = $true )
& ( kpairiskpair = $true )
& ( exuI2 = $true )
& ( dpsetconstrER = $true )
& ( inCongP = $true )
& ( brelnall1 = $true )
& ( app = $true )
& ( cartprodfstpairEq = $true )
& ( ksndsingleton = $true )
& ( upairset2IR = $true )
& ( exuE2 = $true )
& ( dpsetconstrSub = $true )
& ( powersetsubset = $true )
& ( brelnall2 = $true )
& ( eta1 = $true )
& ( setukpairinjL1 = $true )
& ( cartprodmempaircEq = $true )
& ( cartprodmempair1 = $true )
& ( lam2lamEq = $true )
& ( binunionEcases = $true )
& ( exuE3e = $true )
& ( dsetconstrER = $true )
& ( symdiffI2 = $true )
& ( quantDeMorgan1 = $true )
& ( funcGraphProp1 = $true )
& ( setadjoinAx = $true )
& ( ubforcartprodlem2 = $true )
& ( notinsingleton = $true )
& ( subsetI1 = $true )
& ( funcImageSingleton = $true )
& ( cartprodsndpairEq = $true )
& ( ifp = $true )
& ( cartprodmempair = $true )
& ( subsetTrans = $true )
& ( iftrueorfalse = $true )
& ( setminusELneg = $true )
& ( emptyE1 = $true )
& ( setukpairinjL2 = $true )
& ( setextsub = $true )
& ( emptysetAx = $true )
& ( setminusSubset2 = $true )
& ( setminusLsub = $true )
& ( ap2p = $true )
& ( omegaSAx = $true )
& ( setextT = $true )
& ( powersetT_lem = $true )
& ( setukpairinjR12 = $true )
& ( eqbreln = $true )
& ( subsetTI = $true )
& ( cartprodpairmemER = $true )
& ( emptysetE = $true )
& ( notdexE = $true )
& ( setadjoinSub2 = $true )
& ( lam2p = $true )
& ( kfstsingleton = $true )
& ( setoftrueEq = $true )
& ( omega__Cong = $true )
& ( ex1E1 = $true )
& ( upairinpowunion = $true )
& ( prop2set2propI = $true )
& ( wellorderingAx = $true )
& ( setminusEL = $true )
& ( powersetAx = $true )
& ( secondinupair = $true )
& ( eqimpsubset1 = $true )
& ( subsetemptysetimpeq = $true )
& ( ap2apEq1 = $true )
& ( eta2 = $true )
& ( setunionE2 = $true )
& ( setukpairIL = $true )
& ( notinemptyset = $true ) ),
inference(flattening,[status(thm)],[f884]) ).
thf(f884,plain,
( ? [X0: $i,X1: $i] :
( ? [X2: $i] :
( ? [X3: $i] :
( ( ( in @ X3 @ X2 )
!= $true )
& ( ( in @ X3 @ X1 )
= $true )
& ( $true
= ( in @ X1 @ ( powerset @ X2 ) ) )
& ( $true
= ( in @ X3 @ X0 ) ) )
& ( ( in @ X2 @ ( powerset @ X0 ) )
= $true ) )
& ( $true
= ( in @ X1 @ ( powerset @ X0 ) ) ) )
& ( powersetTI1 = $true )
& ( subsetTI = $true )
& ( setextT = $true )
& ( complementT_lem = $true )
& ( setminusT_lem = $true )
& ( powersetT_lem = $true )
& ( binunionT_lem = $true )
& ( binintersectT_lem = $true )
& ( iftrueorfalse = $true )
& ( iffalse = $true )
& ( iftrue = $true )
& ( theeq = $true )
& ( ifp = $true )
& ( ifSingleton = $true )
& ( iftrueProp2 = $true )
& ( iftrueProp1 = $true )
& ( iffalseProp2 = $true )
& ( iffalseProp1 = $true )
& ( eta2 = $true )
& ( beta2 = $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)],[f711]) ).
thf(f711,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 )
=> ( ( beta2 = $true )
=> ( ( eta2 = $true )
=> ( ( iffalseProp1 = $true )
=> ( ( iffalseProp2 = $true )
=> ( ( iftrueProp1 = $true )
=> ( ( iftrueProp2 = $true )
=> ( ( ifSingleton = $true )
=> ( ( ifp = $true )
=> ( ( theeq = $true )
=> ( ( iftrue = $true )
=> ( ( iffalse = $true )
=> ( ( iftrueorfalse = $true )
=> ( ( binintersectT_lem = $true )
=> ( ( binunionT_lem = $true )
=> ( ( powersetT_lem = $true )
=> ( ( setminusT_lem = $true )
=> ( ( complementT_lem = $true )
=> ( ( setextT = $true )
=> ( ( subsetTI = $true )
=> ( ( powersetTI1 = $true )
=> ! [X0: $i,X1: $i] :
( ( $true
= ( in @ X1 @ ( powerset @ X0 ) ) )
=> ! [X2: $i] :
( ( ( in @ X2 @ ( powerset @ X0 ) )
= $true )
=> ! [X3: $i] :
( ( $true
= ( in @ X3 @ X0 ) )
=> ( ( $true
= ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ( ( in @ X3 @ X1 )
= $true )
=> ( ( in @ X3 @ X2 )
= $true ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f710]) ).
thf(f710,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
=> ( beta2
=> ( eta2
=> ( iffalseProp1
=> ( iffalseProp2
=> ( iftrueProp1
=> ( iftrueProp2
=> ( ifSingleton
=> ( ifp
=> ( theeq
=> ( iftrue
=> ( iffalse
=> ( iftrueorfalse
=> ( binintersectT_lem
=> ( binunionT_lem
=> ( powersetT_lem
=> ( setminusT_lem
=> ( complementT_lem
=> ( setextT
=> ( subsetTI
=> ( powersetTI1
=> ! [X0: $i,X1: $i] :
( ( in @ X1 @ ( powerset @ X0 ) )
=> ! [X2: $i] :
( ( in @ X2 @ ( powerset @ X0 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X0 )
=> ( ( in @ X1 @ ( powerset @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f240]) ).
thf(f240,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
=> ( beta2
=> ( eta2
=> ( iffalseProp1
=> ( iffalseProp2
=> ( iftrueProp1
=> ( iftrueProp2
=> ( ifSingleton
=> ( ifp
=> ( theeq
=> ( iftrue
=> ( iffalse
=> ( iftrueorfalse
=> ( binintersectT_lem
=> ( binunionT_lem
=> ( powersetT_lem
=> ( setminusT_lem
=> ( complementT_lem
=> ( setextT
=> ( subsetTI
=> ( powersetTI1
=> ! [X3: $i,X11: $i] :
( ( in @ X11 @ ( powerset @ X3 ) )
=> ! [X16: $i] :
( ( in @ X16 @ ( powerset @ X3 ) )
=> ! [X1: $i] :
( ( in @ X1 @ X3 )
=> ( ( in @ X11 @ ( powerset @ X16 ) )
=> ( ( in @ X1 @ X11 )
=> ( in @ X1 @ X16 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f239]) ).
thf(f239,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
=> ( beta2
=> ( eta2
=> ( iffalseProp1
=> ( iffalseProp2
=> ( iftrueProp1
=> ( iftrueProp2
=> ( ifSingleton
=> ( ifp
=> ( theeq
=> ( iftrue
=> ( iffalse
=> ( iftrueorfalse
=> ( binintersectT_lem
=> ( binunionT_lem
=> ( powersetT_lem
=> ( setminusT_lem
=> ( complementT_lem
=> ( setextT
=> ( subsetTI
=> ( powersetTI1
=> ! [X3: $i,X11: $i] :
( ( in @ X11 @ ( powerset @ X3 ) )
=> ! [X16: $i] :
( ( in @ X16 @ ( powerset @ X3 ) )
=> ! [X1: $i] :
( ( in @ X1 @ X3 )
=> ( ( in @ X11 @ ( powerset @ X16 ) )
=> ( ( in @ X1 @ X11 )
=> ( in @ X1 @ X16 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.OYw9T6ZJNR/DTF2THF_12262.p',powersetTE1) ).
thf(f3722,plain,
( $true
!= ( in @ sK167 @ sK165 ) ),
inference(trivial_inequality_removal,[status(thm)],[f3691]) ).
thf(f3691,plain,
( ( $true
!= ( in @ sK167 @ sK165 ) )
| ( $true != $true ) ),
inference(superposition,[status(thm)],[f3547,f3647]) ).
thf(f3647,plain,
( $true
= ( subset @ sK165 @ sK166 ) ),
inference(trivial_inequality_removal,[status(thm)],[f3646]) ).
thf(f3646,plain,
( ( $true != $true )
| ( $true
= ( subset @ sK165 @ sK166 ) ) ),
inference(forward_demodulation,[status(thm)],[f3630,f2445]) ).
thf(f2445,plain,
powersetE1 = $true,
inference(cnf_transformation,[status(thm)],[f1270]) ).
thf(f3630,plain,
( ( $true
= ( subset @ sK165 @ sK166 ) )
| ( powersetE1 != $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f3623]) ).
thf(f3623,plain,
( ( $true
= ( subset @ sK165 @ sK166 ) )
| ( $true != $true )
| ( powersetE1 != $true ) ),
inference(superposition,[status(thm)],[f3077,f2552]) ).
thf(f2552,plain,
( $true
= ( in @ sK165 @ ( powerset @ sK166 ) ) ),
inference(cnf_transformation,[status(thm)],[f1270]) ).
thf(f3077,plain,
! [X2: $i,X3: $i] :
( ( $true
!= ( in @ X2 @ ( powerset @ X3 ) ) )
| ( powersetE1 != $true )
| ( ( subset @ X2 @ X3 )
= $true ) ),
inference(cnf_transformation,[status(thm)],[f1849]) ).
thf(f1849,plain,
( ( ( powersetE1 = $true )
| ( ( $true
= ( in @ sK544 @ ( powerset @ sK545 ) ) )
& ( ( subset @ sK544 @ sK545 )
!= $true ) ) )
& ( ! [X2: $i,X3: $i] :
( ( $true
!= ( in @ X2 @ ( powerset @ X3 ) ) )
| ( ( subset @ X2 @ X3 )
= $true ) )
| ( powersetE1 != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK544,sK545])],[f1847,f1848]) ).
thf(f1848,plain,
( ? [X0: $i,X1: $i] :
( ( ( in @ X0 @ ( powerset @ X1 ) )
= $true )
& ( ( subset @ X0 @ X1 )
!= $true ) )
=> ( ( $true
= ( in @ sK544 @ ( powerset @ sK545 ) ) )
& ( ( subset @ sK544 @ sK545 )
!= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1847,plain,
( ( ( powersetE1 = $true )
| ? [X0: $i,X1: $i] :
( ( ( in @ X0 @ ( powerset @ X1 ) )
= $true )
& ( ( subset @ X0 @ X1 )
!= $true ) ) )
& ( ! [X2: $i,X3: $i] :
( ( $true
!= ( in @ X2 @ ( powerset @ X3 ) ) )
| ( ( subset @ X2 @ X3 )
= $true ) )
| ( powersetE1 != $true ) ) ),
inference(rectify,[status(thm)],[f1846]) ).
thf(f1846,plain,
( ( ( powersetE1 = $true )
| ? [X0: $i,X1: $i] :
( ( ( in @ X0 @ ( powerset @ X1 ) )
= $true )
& ( ( subset @ X0 @ X1 )
!= $true ) ) )
& ( ! [X0: $i,X1: $i] :
( ( ( in @ X0 @ ( powerset @ X1 ) )
!= $true )
| ( ( subset @ X0 @ X1 )
= $true ) )
| ( powersetE1 != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f819]) ).
thf(f819,plain,
( ( powersetE1 = $true )
<=> ! [X0: $i,X1: $i] :
( ( ( in @ X0 @ ( powerset @ X1 ) )
!= $true )
| ( ( subset @ X0 @ X1 )
= $true ) ) ),
inference(ennf_transformation,[status(thm)],[f361]) ).
thf(f361,plain,
( ! [X0: $i,X1: $i] :
( ( ( in @ X0 @ ( powerset @ X1 ) )
= $true )
=> ( ( subset @ X0 @ X1 )
= $true ) )
<=> ( powersetE1 = $true ) ),
inference(fool_elimination,[status(thm)],[f360]) ).
thf(f360,plain,
! [X0: $i,X1: $i] :
( ( ( in @ X0 @ ( powerset @ X1 ) )
=> ( subset @ X0 @ X1 ) )
= powersetE1 ),
inference(rectify,[status(thm)],[f96]) ).
thf(f96,axiom,
! [X4: $i,X3: $i] :
( ( ( in @ X4 @ ( powerset @ X3 ) )
=> ( subset @ X4 @ X3 ) )
= powersetE1 ),
file('/export/starexec/sandbox2/tmp/tmp.OYw9T6ZJNR/DTF2THF_12262.p',powersetE1) ).
thf(f3547,plain,
! [X0: $i] :
( ( $true
!= ( subset @ X0 @ sK166 ) )
| ( $true
!= ( in @ sK167 @ X0 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f3546]) ).
thf(f3546,plain,
! [X0: $i] :
( ( $true
!= ( in @ sK167 @ X0 ) )
| ( $true
!= ( subset @ X0 @ sK166 ) )
| ( $true != $true ) ),
inference(forward_demodulation,[status(thm)],[f3544,f2522]) ).
thf(f2522,plain,
subsetE = $true,
inference(cnf_transformation,[status(thm)],[f1270]) ).
thf(f3544,plain,
! [X0: $i] :
( ( $true
!= ( subset @ X0 @ sK166 ) )
| ( subsetE != $true )
| ( $true
!= ( in @ sK167 @ X0 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f3538]) ).
thf(f3538,plain,
! [X0: $i] :
( ( $true
!= ( subset @ X0 @ sK166 ) )
| ( subsetE != $true )
| ( $true != $true )
| ( $true
!= ( in @ sK167 @ X0 ) ) ),
inference(superposition,[status(thm)],[f2551,f2635]) ).
thf(f2635,plain,
! [X3: $i,X4: $i,X5: $i] :
( ( ( in @ X5 @ X4 )
= $true )
| ( subsetE != $true )
| ( $true
!= ( in @ X5 @ X3 ) )
| ( ( subset @ X3 @ X4 )
!= $true ) ),
inference(cnf_transformation,[status(thm)],[f1318]) ).
thf(f1318,plain,
( ( ( subsetE = $true )
| ( ( ( subset @ sK197 @ sK198 )
= $true )
& ( $true
!= ( in @ sK199 @ sK198 ) )
& ( ( in @ sK199 @ sK197 )
= $true ) ) )
& ( ! [X3: $i,X4: $i,X5: $i] :
( ( ( subset @ X3 @ X4 )
!= $true )
| ( ( in @ X5 @ X4 )
= $true )
| ( $true
!= ( in @ X5 @ X3 ) ) )
| ( subsetE != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK197,sK198,sK199])],[f1316,f1317]) ).
thf(f1317,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ( ( subset @ X0 @ X1 )
= $true )
& ( ( in @ X2 @ X1 )
!= $true )
& ( ( in @ X2 @ X0 )
= $true ) )
=> ( ( ( subset @ sK197 @ sK198 )
= $true )
& ( $true
!= ( in @ sK199 @ sK198 ) )
& ( ( in @ sK199 @ sK197 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1316,plain,
( ( ( subsetE = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( ( subset @ X0 @ X1 )
= $true )
& ( ( in @ X2 @ X1 )
!= $true )
& ( ( in @ X2 @ X0 )
= $true ) ) )
& ( ! [X3: $i,X4: $i,X5: $i] :
( ( ( subset @ X3 @ X4 )
!= $true )
| ( ( in @ X5 @ X4 )
= $true )
| ( $true
!= ( in @ X5 @ X3 ) ) )
| ( subsetE != $true ) ) ),
inference(rectify,[status(thm)],[f1315]) ).
thf(f1315,plain,
( ( ( subsetE = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( ( subset @ X0 @ X1 )
= $true )
& ( ( in @ X2 @ X1 )
!= $true )
& ( ( in @ X2 @ X0 )
= $true ) ) )
& ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( subset @ X0 @ X1 )
!= $true )
| ( ( in @ X2 @ X1 )
= $true )
| ( ( in @ X2 @ X0 )
!= $true ) )
| ( subsetE != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f955]) ).
thf(f955,plain,
( ( subsetE = $true )
<=> ! [X0: $i,X1: $i,X2: $i] :
( ( ( subset @ X0 @ X1 )
!= $true )
| ( ( in @ X2 @ X1 )
= $true )
| ( ( in @ X2 @ X0 )
!= $true ) ) ),
inference(flattening,[status(thm)],[f954]) ).
thf(f954,plain,
( ( subsetE = $true )
<=> ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X2 @ X1 )
= $true )
| ( ( in @ X2 @ X0 )
!= $true )
| ( ( subset @ X0 @ X1 )
!= $true ) ) ),
inference(ennf_transformation,[status(thm)],[f335]) ).
thf(f335,plain,
( ( subsetE = $true )
<=> ! [X0: $i,X1: $i,X2: $i] :
( ( ( subset @ X0 @ X1 )
= $true )
=> ( ( ( in @ X2 @ X0 )
= $true )
=> ( ( in @ X2 @ X1 )
= $true ) ) ) ),
inference(fool_elimination,[status(thm)],[f334]) ).
thf(f334,plain,
( subsetE
= ( ! [X0: $i,X1: $i,X2: $i] :
( ( subset @ X0 @ X1 )
=> ( ( in @ X2 @ X0 )
=> ( in @ X2 @ X1 ) ) ) ) ),
inference(rectify,[status(thm)],[f83]) ).
thf(f83,axiom,
( subsetE
= ( ! [X3: $i,X4: $i,X1: $i] :
( ( subset @ X3 @ X4 )
=> ( ( in @ X1 @ X3 )
=> ( in @ X1 @ X4 ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.OYw9T6ZJNR/DTF2THF_12262.p',subsetE) ).
thf(f2551,plain,
( ( in @ sK167 @ sK166 )
!= $true ),
inference(cnf_transformation,[status(thm)],[f1270]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : SEU718^1 : TPTP v9.2.1. Released v3.7.0.
% 0.11/0.12 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.16/0.33 % Computer : n015.cluster.edu
% 0.16/0.33 % Model : x86_64 x86_64
% 0.16/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.33 % Memory : 8042.1875MB
% 0.16/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.33 % CPULimit : 300
% 0.16/0.33 % WCLimit : 300
% 0.16/0.33 % DateTime : Tue Feb 24 12:32:01 EST 2026
% 0.16/0.34 % CPUTime :
% 0.16/0.34 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.23/0.46 ---- Original DTF file ---
% 0.23/0.46 thf(spec,logic,$$dhol).
% 0.23/0.46 %------------------------------------------------------------------------------
% 0.23/0.46 % File : SEU718^1 : TPTP v9.2.1. Released v3.7.0.
% 0.23/0.46 % Domain : Set Theory
% 0.23/0.46 % Problem : Typed Set Theory - Laws for Typed Sets
% 0.23/0.46 % Version : Especial.
% 0.23/0.46 % English : (! A:i.! X:i.in X (powerset A) -> (! Y:i.in Y (powerset A) ->
% 0.23/0.46 % (! x:i.in x A -> in X (powerset Y) -> in x X -> in x Y)))
% 0.23/0.46
% 0.23/0.46 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.23/0.46 % Source : [Bro08]
% 0.23/0.46 % Names : ZFC220g [Bro08]
% 0.23/0.46
% 0.23/0.46 % Status : Theorem
% 0.23/0.46 % Rating : 0.11 v9.1.0, 0.12 v9.0.0, 0.20 v8.2.0, 0.31 v8.1.0, 0.09 v7.5.0, 0.00 v7.4.0, 0.22 v7.2.0, 0.12 v7.1.0, 0.25 v7.0.0, 0.14 v6.4.0, 0.17 v6.3.0, 0.20 v6.2.0, 0.14 v6.1.0, 0.29 v6.0.0, 0.43 v5.5.0, 0.50 v5.4.0, 0.60 v5.2.0, 1.00 v3.7.0
% 0.23/0.46 % Syntax : Number of formulae : 511 ( 238 unt; 272 typ; 238 def)
% 0.23/0.46 % Number of atoms : 1348 ( 363 equ; 0 cnn)
% 0.23/0.46 % Maximal formula atoms : 240 ( 5 avg)
% 0.23/0.46 % Number of connectives : 2630 ( 55 ~; 12 |; 44 &;1862 @)
% 0.23/0.46 % ( 14 <=>; 643 =>; 0 <=; 0 <~>)
% 0.23/0.46 % Maximal formula depth : 246 ( 2 avg)
% 0.23/0.46 % Number of types : 2 ( 0 usr)
% 0.23/0.46 % Number of type conns : 132 ( 132 >; 0 *; 0 +; 0 <<)
% 0.23/0.46 % Number of symbols : 275 ( 272 usr; 238 con; 0-4 aty)
% 0.23/0.46 % Number of variables : 802 ( 74 ^; 693 !; 35 ?; 802 :)
% 0.23/0.46 % SPC : TH0_THM_EQU_NAR
% 0.23/0.46
% 0.23/0.46 % Comments : http://mathgate.info/detsetitem.php?id=276
% 0.23/0.46 %------------------------------------------------------------------------------
% 0.23/0.46 thf(in_type,type,
% 0.23/0.46 in: $i > $i > $o ).
% 0.23/0.46
% 0.23/0.46 thf(exu_type,type,
% 0.23/0.46 exu: ( $i > $o ) > $o ).
% 0.23/0.46
% 0.23/0.46 thf(exu,definition,
% 0.23/0.46 ( exu
% 0.23/0.46 = ( ^ [Xphi: $i > $o] :
% 0.23/0.46 ? [Xx: $i] :
% 0.23/0.46 ( ( Xphi @ Xx )
% 0.23/0.46 & ! [Xy: $i] :
% 0.23/0.46 ( ( Xphi @ Xy )
% 0.23/0.46 => ( Xx = Xy ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(setextAx_type,type,
% 0.23/0.46 setextAx: $o ).
% 0.23/0.46
% 0.23/0.46 thf(setextAx,definition,
% 0.23/0.46 ( setextAx
% 0.23/0.46 = ( ! [A: $i,B: $i] :
% 0.23/0.46 ( ! [Xx: $i] :
% 0.23/0.46 ( ( in @ Xx @ A )
% 0.23/0.46 <=> ( in @ Xx @ B ) )
% 0.23/0.46 => ( A = B ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(emptyset_type,type,
% 0.23/0.46 emptyset: $i ).
% 0.23/0.46
% 0.23/0.46 thf(emptysetAx_type,type,
% 0.23/0.46 emptysetAx: $o ).
% 0.23/0.46
% 0.23/0.46 thf(emptysetAx,definition,
% 0.23/0.46 ( emptysetAx
% 0.23/0.46 = ( ! [Xx: $i] :
% 0.23/0.46 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(setadjoin_type,type,
% 0.23/0.46 setadjoin: $i > $i > $i ).
% 0.23/0.46
% 0.23/0.46 thf(setadjoinAx_type,type,
% 0.23/0.46 setadjoinAx: $o ).
% 0.23/0.46
% 0.23/0.46 thf(setadjoinAx,definition,
% 0.23/0.46 ( setadjoinAx
% 0.23/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.23/0.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.23/0.46 <=> ( ( Xy = Xx )
% 0.23/0.46 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(powerset_type,type,
% 0.23/0.46 powerset: $i > $i ).
% 0.23/0.46
% 0.23/0.46 thf(powersetAx_type,type,
% 0.23/0.46 powersetAx: $o ).
% 0.23/0.46
% 0.23/0.46 thf(powersetAx,definition,
% 0.23/0.46 ( powersetAx
% 0.23/0.46 = ( ! [A: $i,B: $i] :
% 0.23/0.46 ( ( in @ B @ ( powerset @ A ) )
% 0.23/0.46 <=> ! [Xx: $i] :
% 0.23/0.46 ( ( in @ Xx @ B )
% 0.23/0.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(setunion_type,type,
% 0.23/0.46 setunion: $i > $i ).
% 0.23/0.46
% 0.23/0.46 thf(setunionAx_type,type,
% 0.23/0.46 setunionAx: $o ).
% 0.23/0.46
% 0.23/0.46 thf(setunionAx,definition,
% 0.23/0.46 ( setunionAx
% 0.23/0.46 = ( ! [A: $i,Xx: $i] :
% 0.23/0.46 ( ( in @ Xx @ ( setunion @ A ) )
% 0.23/0.46 <=> ? [B: $i] :
% 0.23/0.46 ( ( in @ Xx @ B )
% 0.23/0.46 & ( in @ B @ A ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(omega_type,type,
% 0.23/0.46 omega: $i ).
% 0.23/0.46
% 0.23/0.46 thf(omega0Ax_type,type,
% 0.23/0.46 omega0Ax: $o ).
% 0.23/0.46
% 0.23/0.46 thf(omega0Ax,definition,
% 0.23/0.46 ( omega0Ax
% 0.23/0.46 = ( in @ emptyset @ omega ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(omegaSAx_type,type,
% 0.23/0.46 omegaSAx: $o ).
% 0.23/0.46
% 0.23/0.46 thf(omegaSAx,definition,
% 0.23/0.46 ( omegaSAx
% 0.23/0.46 = ( ! [Xx: $i] :
% 0.23/0.46 ( ( in @ Xx @ omega )
% 0.23/0.46 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(omegaIndAx_type,type,
% 0.23/0.46 omegaIndAx: $o ).
% 0.23/0.46
% 0.23/0.46 thf(omegaIndAx,definition,
% 0.23/0.46 ( omegaIndAx
% 0.23/0.46 = ( ! [A: $i] :
% 0.23/0.46 ( ( ( in @ emptyset @ A )
% 0.23/0.46 & ! [Xx: $i] :
% 0.23/0.46 ( ( ( in @ Xx @ omega )
% 0.23/0.46 & ( in @ Xx @ A ) )
% 0.23/0.46 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.23/0.46 => ! [Xx: $i] :
% 0.23/0.46 ( ( in @ Xx @ omega )
% 0.23/0.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(replAx_type,type,
% 0.23/0.46 replAx: $o ).
% 0.23/0.46
% 0.23/0.46 thf(replAx,definition,
% 0.23/0.46 ( replAx
% 0.23/0.46 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.23/0.46 ( ! [Xx: $i] :
% 0.23/0.46 ( ( in @ Xx @ A )
% 0.23/0.46 => ( exu
% 0.23/0.46 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.23/0.46 => ? [B: $i] :
% 0.23/0.46 ! [Xx: $i] :
% 0.23/0.46 ( ( in @ Xx @ B )
% 0.23/0.46 <=> ? [Xy: $i] :
% 0.23/0.46 ( ( in @ Xy @ A )
% 0.23/0.46 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(foundationAx_type,type,
% 0.23/0.46 foundationAx: $o ).
% 0.23/0.46
% 0.23/0.46 thf(foundationAx,definition,
% 0.23/0.46 ( foundationAx
% 0.23/0.46 = ( ! [A: $i] :
% 0.23/0.46 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.23/0.46 => ? [B: $i] :
% 0.23/0.46 ( ( in @ B @ A )
% 0.23/0.46 & ~ ? [Xx: $i] :
% 0.23/0.46 ( ( in @ Xx @ B )
% 0.23/0.46 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(wellorderingAx_type,type,
% 0.23/0.46 wellorderingAx: $o ).
% 0.23/0.46
% 0.23/0.46 thf(wellorderingAx,definition,
% 0.23/0.46 ( wellorderingAx
% 0.23/0.46 = ( ! [A: $i] :
% 0.23/0.46 ? [B: $i] :
% 0.23/0.46 ( ! [C: $i] :
% 0.23/0.46 ( ( in @ C @ B )
% 0.23/0.46 => ! [Xx: $i] :
% 0.23/0.46 ( ( in @ Xx @ C )
% 0.23/0.46 => ( in @ Xx @ A ) ) )
% 0.23/0.46 & ! [Xx: $i,Xy: $i] :
% 0.23/0.46 ( ( ( in @ Xx @ A )
% 0.23/0.46 & ( in @ Xy @ A ) )
% 0.23/0.46 => ( ! [C: $i] :
% 0.23/0.46 ( ( in @ C @ B )
% 0.23/0.46 => ( ( in @ Xx @ C )
% 0.23/0.46 <=> ( in @ Xy @ C ) ) )
% 0.23/0.46 => ( Xx = Xy ) ) )
% 0.23/0.46 & ! [C: $i,D: $i] :
% 0.23/0.46 ( ( ( in @ C @ B )
% 0.23/0.46 & ( in @ D @ B ) )
% 0.23/0.46 => ( ! [Xx: $i] :
% 0.23/0.46 ( ( in @ Xx @ C )
% 0.23/0.46 => ( in @ Xx @ D ) )
% 0.23/0.46 | ! [Xx: $i] :
% 0.23/0.46 ( ( in @ Xx @ D )
% 0.23/0.46 => ( in @ Xx @ C ) ) ) )
% 0.23/0.46 & ! [C: $i] :
% 0.23/0.46 ( ( ! [Xx: $i] :
% 0.23/0.46 ( ( in @ Xx @ C )
% 0.23/0.46 => ( in @ Xx @ A ) )
% 0.23/0.46 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.23/0.46 => ? [D: $i,Xx: $i] :
% 0.23/0.46 ( ( in @ D @ B )
% 0.23/0.46 & ( in @ Xx @ C )
% 0.23/0.46 & ~ ? [Xy: $i] :
% 0.23/0.46 ( ( in @ Xy @ D )
% 0.23/0.46 & ( in @ Xy @ C ) )
% 0.23/0.46 & ! [E: $i] :
% 0.23/0.46 ( ( in @ E @ B )
% 0.23/0.46 => ( ! [Xy: $i] :
% 0.23/0.46 ( ( in @ Xy @ E )
% 0.23/0.46 => ( in @ Xy @ D ) )
% 0.23/0.46 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(descr_type,type,
% 0.23/0.46 descr: ( $i > $o ) > $i ).
% 0.23/0.46
% 0.23/0.46 thf(descrp_type,type,
% 0.23/0.46 descrp: $o ).
% 0.23/0.46
% 0.23/0.46 thf(descrp,definition,
% 0.23/0.46 ( descrp
% 0.23/0.46 = ( ! [Xphi: $i > $o] :
% 0.23/0.46 ( ( exu
% 0.23/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.23/0.46 => ( Xphi
% 0.23/0.46 @ ( descr
% 0.23/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(dsetconstr_type,type,
% 0.23/0.46 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.23/0.46
% 0.23/0.46 thf(dsetconstrI_type,type,
% 0.23/0.46 dsetconstrI: $o ).
% 0.23/0.46
% 0.23/0.46 thf(dsetconstrI,definition,
% 0.23/0.46 ( dsetconstrI
% 0.23/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.23/0.46 ( ( in @ Xx @ A )
% 0.23/0.46 => ( ( Xphi @ Xx )
% 0.23/0.46 => ( in @ Xx
% 0.23/0.46 @ ( dsetconstr @ A
% 0.23/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(dsetconstrEL_type,type,
% 0.23/0.46 dsetconstrEL: $o ).
% 0.23/0.46
% 0.23/0.46 thf(dsetconstrEL,definition,
% 0.23/0.46 ( dsetconstrEL
% 0.23/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.23/0.46 ( ( in @ Xx
% 0.23/0.46 @ ( dsetconstr @ A
% 0.23/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.23/0.46 => ( in @ Xx @ A ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(dsetconstrER_type,type,
% 0.23/0.46 dsetconstrER: $o ).
% 0.23/0.46
% 0.23/0.46 thf(dsetconstrER,definition,
% 0.23/0.46 ( dsetconstrER
% 0.23/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.23/0.46 ( ( in @ Xx
% 0.23/0.46 @ ( dsetconstr @ A
% 0.23/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.23/0.46 => ( Xphi @ Xx ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(exuE1_type,type,
% 0.23/0.46 exuE1: $o ).
% 0.23/0.46
% 0.23/0.46 thf(exuE1,definition,
% 0.23/0.46 ( exuE1
% 0.23/0.46 = ( ! [Xphi: $i > $o] :
% 0.23/0.46 ( ( exu
% 0.23/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.23/0.46 => ? [Xx: $i] :
% 0.23/0.46 ( ( Xphi @ Xx )
% 0.23/0.46 & ! [Xy: $i] :
% 0.23/0.46 ( ( Xphi @ Xy )
% 0.23/0.46 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(prop2set_type,type,
% 0.23/0.46 prop2set: $o > $i ).
% 0.23/0.46
% 0.23/0.46 thf(prop2setE_type,type,
% 0.23/0.46 prop2setE: $o ).
% 0.23/0.46
% 0.23/0.46 thf(prop2setE,definition,
% 0.23/0.46 ( prop2setE
% 0.23/0.46 = ( ! [Xphi: $o,Xx: $i] :
% 0.23/0.46 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.23/0.46 => Xphi ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(emptysetE_type,type,
% 0.23/0.46 emptysetE: $o ).
% 0.23/0.46
% 0.23/0.46 thf(emptysetE,definition,
% 0.23/0.46 ( emptysetE
% 0.23/0.46 = ( ! [Xx: $i] :
% 0.23/0.46 ( ( in @ Xx @ emptyset )
% 0.23/0.46 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(emptysetimpfalse_type,type,
% 0.23/0.46 emptysetimpfalse: $o ).
% 0.23/0.46
% 0.23/0.46 thf(emptysetimpfalse,definition,
% 0.23/0.46 ( emptysetimpfalse
% 0.23/0.46 = ( ! [Xx: $i] :
% 0.23/0.46 ( ( in @ Xx @ emptyset )
% 0.23/0.46 => $false ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(notinemptyset_type,type,
% 0.23/0.46 notinemptyset: $o ).
% 0.23/0.46
% 0.23/0.46 thf(notinemptyset,definition,
% 0.23/0.46 ( notinemptyset
% 0.23/0.46 = ( ! [Xx: $i] :
% 0.23/0.46 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(exuE3e_type,type,
% 0.23/0.46 exuE3e: $o ).
% 0.23/0.46
% 0.23/0.46 thf(exuE3e,definition,
% 0.23/0.46 ( exuE3e
% 0.23/0.46 = ( ! [Xphi: $i > $o] :
% 0.23/0.46 ( ( exu
% 0.23/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.23/0.46 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(setext_type,type,
% 0.23/0.46 setext: $o ).
% 0.23/0.46
% 0.23/0.46 thf(setext,definition,
% 0.23/0.46 ( setext
% 0.23/0.46 = ( ! [A: $i,B: $i] :
% 0.23/0.46 ( ! [Xx: $i] :
% 0.23/0.46 ( ( in @ Xx @ A )
% 0.23/0.46 => ( in @ Xx @ B ) )
% 0.23/0.46 => ( ! [Xx: $i] :
% 0.23/0.46 ( ( in @ Xx @ B )
% 0.23/0.46 => ( in @ Xx @ A ) )
% 0.23/0.46 => ( A = B ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(emptyI_type,type,
% 0.23/0.46 emptyI: $o ).
% 0.23/0.46
% 0.23/0.46 thf(emptyI,definition,
% 0.23/0.46 ( emptyI
% 0.23/0.46 = ( ! [A: $i] :
% 0.23/0.46 ( ! [Xx: $i] :
% 0.23/0.46 ~ ( in @ Xx @ A )
% 0.23/0.46 => ( A = emptyset ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(noeltsimpempty_type,type,
% 0.23/0.46 noeltsimpempty: $o ).
% 0.23/0.46
% 0.23/0.46 thf(noeltsimpempty,definition,
% 0.23/0.46 ( noeltsimpempty
% 0.23/0.46 = ( ! [A: $i] :
% 0.23/0.46 ( ! [Xx: $i] :
% 0.23/0.46 ~ ( in @ Xx @ A )
% 0.23/0.46 => ( A = emptyset ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(setbeta_type,type,
% 0.23/0.46 setbeta: $o ).
% 0.23/0.46
% 0.23/0.46 thf(setbeta,definition,
% 0.23/0.46 ( setbeta
% 0.23/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.23/0.46 ( ( in @ Xx @ A )
% 0.23/0.46 => ( ( in @ Xx
% 0.23/0.46 @ ( dsetconstr @ A
% 0.23/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.23/0.46 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(nonempty_type,type,
% 0.23/0.46 nonempty: $i > $o ).
% 0.23/0.46
% 0.23/0.46 thf(nonempty,definition,
% 0.23/0.46 ( nonempty
% 0.23/0.46 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(nonemptyE1_type,type,
% 0.23/0.46 nonemptyE1: $o ).
% 0.23/0.46
% 0.23/0.46 thf(nonemptyE1,definition,
% 0.23/0.46 ( nonemptyE1
% 0.23/0.46 = ( ! [A: $i] :
% 0.23/0.46 ( ( nonempty @ A )
% 0.23/0.46 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(nonemptyI_type,type,
% 0.23/0.46 nonemptyI: $o ).
% 0.23/0.46
% 0.23/0.46 thf(nonemptyI,definition,
% 0.23/0.46 ( nonemptyI
% 0.23/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.23/0.46 ( ( in @ Xx @ A )
% 0.23/0.46 => ( ( Xphi @ Xx )
% 0.23/0.46 => ( nonempty
% 0.23/0.46 @ ( dsetconstr @ A
% 0.23/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(nonemptyI1_type,type,
% 0.23/0.46 nonemptyI1: $o ).
% 0.23/0.46
% 0.23/0.46 thf(nonemptyI1,definition,
% 0.23/0.46 ( nonemptyI1
% 0.23/0.46 = ( ! [A: $i] :
% 0.23/0.46 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.23/0.46 => ( nonempty @ A ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(setadjoinIL_type,type,
% 0.23/0.46 setadjoinIL: $o ).
% 0.23/0.46
% 0.23/0.46 thf(setadjoinIL,definition,
% 0.23/0.46 ( setadjoinIL
% 0.23/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(emptyinunitempty_type,type,
% 0.23/0.46 emptyinunitempty: $o ).
% 0.23/0.46
% 0.23/0.46 thf(emptyinunitempty,definition,
% 0.23/0.46 ( emptyinunitempty
% 0.23/0.46 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(setadjoinIR_type,type,
% 0.23/0.46 setadjoinIR: $o ).
% 0.23/0.46
% 0.23/0.46 thf(setadjoinIR,definition,
% 0.23/0.46 ( setadjoinIR
% 0.23/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.23/0.46 ( ( in @ Xy @ A )
% 0.23/0.46 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(setadjoinE_type,type,
% 0.23/0.46 setadjoinE: $o ).
% 0.23/0.46
% 0.23/0.46 thf(setadjoinE,definition,
% 0.23/0.46 ( setadjoinE
% 0.23/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.23/0.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.23/0.46 => ! [Xphi: $o] :
% 0.23/0.46 ( ( ( Xy = Xx )
% 0.23/0.46 => Xphi )
% 0.23/0.46 => ( ( ( in @ Xy @ A )
% 0.23/0.46 => Xphi )
% 0.23/0.46 => Xphi ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(setadjoinOr_type,type,
% 0.23/0.46 setadjoinOr: $o ).
% 0.23/0.46
% 0.23/0.46 thf(setadjoinOr,definition,
% 0.23/0.46 ( setadjoinOr
% 0.23/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.23/0.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.23/0.46 => ( ( Xy = Xx )
% 0.23/0.46 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(setoftrueEq_type,type,
% 0.23/0.46 setoftrueEq: $o ).
% 0.23/0.46
% 0.23/0.46 thf(setoftrueEq,definition,
% 0.23/0.46 ( setoftrueEq
% 0.23/0.46 = ( ! [A: $i] :
% 0.23/0.46 ( ( dsetconstr @ A
% 0.23/0.46 @ ^ [Xx: $i] : $true )
% 0.23/0.46 = A ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(powersetI_type,type,
% 0.23/0.46 powersetI: $o ).
% 0.23/0.46
% 0.23/0.46 thf(powersetI,definition,
% 0.23/0.46 ( powersetI
% 0.23/0.46 = ( ! [A: $i,B: $i] :
% 0.23/0.46 ( ! [Xx: $i] :
% 0.23/0.46 ( ( in @ Xx @ B )
% 0.23/0.46 => ( in @ Xx @ A ) )
% 0.23/0.46 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(emptyinPowerset_type,type,
% 0.23/0.46 emptyinPowerset: $o ).
% 0.23/0.46
% 0.23/0.46 thf(emptyinPowerset,definition,
% 0.23/0.46 ( emptyinPowerset
% 0.23/0.46 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(emptyInPowerset_type,type,
% 0.23/0.46 emptyInPowerset: $o ).
% 0.23/0.46
% 0.23/0.46 thf(emptyInPowerset,definition,
% 0.23/0.46 ( emptyInPowerset
% 0.23/0.46 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(powersetE_type,type,
% 0.23/0.46 powersetE: $o ).
% 0.23/0.46
% 0.23/0.46 thf(powersetE,definition,
% 0.23/0.46 ( powersetE
% 0.23/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.23/0.46 ( ( in @ B @ ( powerset @ A ) )
% 0.23/0.46 => ( ( in @ Xx @ B )
% 0.23/0.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(setunionI_type,type,
% 0.23/0.46 setunionI: $o ).
% 0.23/0.46
% 0.23/0.46 thf(setunionI,definition,
% 0.23/0.46 ( setunionI
% 0.23/0.46 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.23/0.46 ( ( in @ Xx @ B )
% 0.23/0.46 => ( ( in @ B @ A )
% 0.23/0.46 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(setunionE_type,type,
% 0.23/0.46 setunionE: $o ).
% 0.23/0.46
% 0.23/0.46 thf(setunionE,definition,
% 0.23/0.46 ( setunionE
% 0.23/0.46 = ( ! [A: $i,Xx: $i] :
% 0.23/0.46 ( ( in @ Xx @ ( setunion @ A ) )
% 0.23/0.46 => ! [Xphi: $o] :
% 0.23/0.46 ( ! [B: $i] :
% 0.23/0.46 ( ( in @ Xx @ B )
% 0.23/0.46 => ( ( in @ B @ A )
% 0.23/0.46 => Xphi ) )
% 0.23/0.46 => Xphi ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(subPowSU_type,type,
% 0.23/0.46 subPowSU: $o ).
% 0.23/0.46
% 0.23/0.46 thf(subPowSU,definition,
% 0.23/0.46 ( subPowSU
% 0.23/0.46 = ( ! [A: $i,Xx: $i] :
% 0.23/0.46 ( ( in @ Xx @ A )
% 0.23/0.46 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(exuE2_type,type,
% 0.23/0.46 exuE2: $o ).
% 0.23/0.46
% 0.23/0.46 thf(exuE2,definition,
% 0.23/0.46 ( exuE2
% 0.23/0.46 = ( ! [Xphi: $i > $o] :
% 0.23/0.46 ( ( exu
% 0.23/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.23/0.46 => ? [Xx: $i] :
% 0.23/0.46 ! [Xy: $i] :
% 0.23/0.46 ( ( Xphi @ Xy )
% 0.23/0.46 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(nonemptyImpWitness_type,type,
% 0.23/0.46 nonemptyImpWitness: $o ).
% 0.23/0.46
% 0.23/0.46 thf(nonemptyImpWitness,definition,
% 0.23/0.46 ( nonemptyImpWitness
% 0.23/0.46 = ( ! [A: $i] :
% 0.23/0.46 ( ( nonempty @ A )
% 0.23/0.46 => ? [Xx: $i] :
% 0.23/0.46 ( ( in @ Xx @ A )
% 0.23/0.46 & $true ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(uniqinunit_type,type,
% 0.23/0.46 uniqinunit: $o ).
% 0.23/0.46
% 0.23/0.46 thf(uniqinunit,definition,
% 0.23/0.46 ( uniqinunit
% 0.23/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.23/0.46 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.23/0.46 => ( Xx = Xy ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(notinsingleton_type,type,
% 0.23/0.46 notinsingleton: $o ).
% 0.23/0.46
% 0.23/0.46 thf(notinsingleton,definition,
% 0.23/0.46 ( notinsingleton
% 0.23/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.23/0.46 ( ( Xx != Xy )
% 0.23/0.46 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(eqinunit_type,type,
% 0.23/0.46 eqinunit: $o ).
% 0.23/0.46
% 0.23/0.46 thf(eqinunit,definition,
% 0.23/0.46 ( eqinunit
% 0.23/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.23/0.46 ( ( Xx = Xy )
% 0.23/0.46 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(singletonsswitch_type,type,
% 0.23/0.46 singletonsswitch: $o ).
% 0.23/0.46
% 0.23/0.46 thf(singletonsswitch,definition,
% 0.23/0.46 ( singletonsswitch
% 0.23/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.23/0.46 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.23/0.46 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(upairsetE_type,type,
% 0.23/0.46 upairsetE: $o ).
% 0.23/0.46
% 0.23/0.46 thf(upairsetE,definition,
% 0.23/0.46 ( upairsetE
% 0.23/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.23/0.46 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.23/0.46 => ( ( Xz = Xx )
% 0.23/0.46 | ( Xz = Xy ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(upairsetIL_type,type,
% 0.23/0.46 upairsetIL: $o ).
% 0.23/0.46
% 0.23/0.46 thf(upairsetIL,definition,
% 0.23/0.46 ( upairsetIL
% 0.23/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(upairsetIR_type,type,
% 0.23/0.46 upairsetIR: $o ).
% 0.23/0.46
% 0.23/0.46 thf(upairsetIR,definition,
% 0.23/0.46 ( upairsetIR
% 0.23/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(emptyE1_type,type,
% 0.23/0.46 emptyE1: $o ).
% 0.23/0.46
% 0.23/0.46 thf(emptyE1,definition,
% 0.23/0.46 ( emptyE1
% 0.23/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.23/0.46 ( ? [Xx: $i] :
% 0.23/0.46 ( ( in @ Xx @ A )
% 0.23/0.46 & ( Xphi @ Xx ) )
% 0.23/0.46 => ( ( ( dsetconstr @ A
% 0.23/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.23/0.46 = emptyset )
% 0.23/0.46 => $false ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(vacuousDall_type,type,
% 0.23/0.46 vacuousDall: $o ).
% 0.23/0.46
% 0.23/0.46 thf(vacuousDall,definition,
% 0.23/0.46 ( vacuousDall
% 0.23/0.46 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.23/0.46 ( ( in @ Xx @ emptyset )
% 0.23/0.46 => ( Xphi @ Xx ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(quantDeMorgan1_type,type,
% 0.23/0.46 quantDeMorgan1: $o ).
% 0.23/0.46
% 0.23/0.46 thf(quantDeMorgan1,definition,
% 0.23/0.46 ( quantDeMorgan1
% 0.23/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.23/0.46 ( ~ ! [Xx: $i] :
% 0.23/0.46 ( ( in @ Xx @ A )
% 0.23/0.46 => ( Xphi @ Xx ) )
% 0.23/0.46 => ? [Xx: $i] :
% 0.23/0.46 ( ( in @ Xx @ A )
% 0.23/0.46 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(quantDeMorgan2_type,type,
% 0.23/0.46 quantDeMorgan2: $o ).
% 0.23/0.46
% 0.23/0.46 thf(quantDeMorgan2,definition,
% 0.23/0.46 ( quantDeMorgan2
% 0.23/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.23/0.46 ( ! [Xx: $i] :
% 0.23/0.46 ( ( in @ Xx @ A )
% 0.23/0.46 => ~ ( Xphi @ Xx ) )
% 0.23/0.46 => ~ ? [Xx: $i] :
% 0.23/0.46 ( ( in @ Xx @ A )
% 0.23/0.46 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(quantDeMorgan3_type,type,
% 0.23/0.46 quantDeMorgan3: $o ).
% 0.23/0.46
% 0.23/0.46 thf(quantDeMorgan3,definition,
% 0.23/0.46 ( quantDeMorgan3
% 0.23/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.23/0.46 ( ~ ? [Xx: $i] :
% 0.23/0.46 ( ( in @ Xx @ A )
% 0.23/0.46 & ( Xphi @ Xx ) )
% 0.23/0.46 => ! [Xx: $i] :
% 0.23/0.46 ( ( in @ Xx @ A )
% 0.23/0.46 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(quantDeMorgan4_type,type,
% 0.23/0.46 quantDeMorgan4: $o ).
% 0.23/0.46
% 0.23/0.46 thf(quantDeMorgan4,definition,
% 0.23/0.46 ( quantDeMorgan4
% 0.23/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.23/0.46 ( ? [Xx: $i] :
% 0.23/0.46 ( ( in @ Xx @ A )
% 0.23/0.46 & ~ ( Xphi @ Xx ) )
% 0.23/0.46 => ~ ! [Xx: $i] :
% 0.23/0.46 ( ( in @ Xx @ A )
% 0.23/0.46 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(prop2setI_type,type,
% 0.23/0.46 prop2setI: $o ).
% 0.23/0.46
% 0.23/0.46 thf(prop2setI,definition,
% 0.23/0.46 ( prop2setI
% 0.23/0.46 = ( ! [Xphi: $o] :
% 0.23/0.46 ( Xphi
% 0.23/0.46 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(set2prop_type,type,
% 0.23/0.46 set2prop: $i > $o ).
% 0.23/0.46
% 0.23/0.46 thf(prop2set2propI_type,type,
% 0.23/0.46 prop2set2propI: $o ).
% 0.23/0.46
% 0.23/0.46 thf(prop2set2propI,definition,
% 0.23/0.46 ( prop2set2propI
% 0.23/0.46 = ( ! [Xphi: $o] :
% 0.23/0.46 ( Xphi
% 0.23/0.46 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(notdexE_type,type,
% 0.23/0.46 notdexE: $o ).
% 0.23/0.46
% 0.23/0.46 thf(notdexE,definition,
% 0.23/0.46 ( notdexE
% 0.23/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.23/0.46 ( ~ ? [Xx: $i] :
% 0.23/0.46 ( ( in @ Xx @ A )
% 0.23/0.46 & ( Xphi @ Xx ) )
% 0.23/0.46 => ! [Xx: $i] :
% 0.23/0.46 ( ( in @ Xx @ A )
% 0.23/0.46 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(notdallE_type,type,
% 0.23/0.46 notdallE: $o ).
% 0.23/0.46
% 0.23/0.46 thf(notdallE,definition,
% 0.23/0.46 ( notdallE
% 0.23/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.23/0.46 ( ~ ! [Xx: $i] :
% 0.23/0.46 ( ( in @ Xx @ A )
% 0.23/0.46 => ( Xphi @ Xx ) )
% 0.23/0.46 => ? [Xx: $i] :
% 0.23/0.46 ( ( in @ Xx @ A )
% 0.23/0.46 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(exuI1_type,type,
% 0.23/0.46 exuI1: $o ).
% 0.23/0.46
% 0.23/0.46 thf(exuI1,definition,
% 0.23/0.46 ( exuI1
% 0.23/0.46 = ( ! [Xphi: $i > $o] :
% 0.23/0.46 ( ? [Xx: $i] :
% 0.23/0.46 ( ( Xphi @ Xx )
% 0.23/0.46 & ! [Xy: $i] :
% 0.23/0.46 ( ( Xphi @ Xy )
% 0.23/0.46 => ( Xx = Xy ) ) )
% 0.23/0.46 => ( exu
% 0.23/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(exuI3_type,type,
% 0.23/0.46 exuI3: $o ).
% 0.23/0.46
% 0.23/0.46 thf(exuI3,definition,
% 0.23/0.46 ( exuI3
% 0.23/0.46 = ( ! [Xphi: $i > $o] :
% 0.23/0.46 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.23/0.46 => ( ! [Xx: $i,Xy: $i] :
% 0.23/0.46 ( ( Xphi @ Xx )
% 0.23/0.46 => ( ( Xphi @ Xy )
% 0.23/0.46 => ( Xx = Xy ) ) )
% 0.23/0.46 => ( exu
% 0.23/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(exuI2_type,type,
% 0.23/0.46 exuI2: $o ).
% 0.23/0.46
% 0.23/0.46 thf(exuI2,definition,
% 0.23/0.46 ( exuI2
% 0.23/0.46 = ( ! [Xphi: $i > $o] :
% 0.23/0.46 ( ? [Xx: $i] :
% 0.23/0.46 ! [Xy: $i] :
% 0.23/0.46 ( ( Xphi @ Xy )
% 0.23/0.46 <=> ( Xy = Xx ) )
% 0.23/0.46 => ( exu
% 0.23/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(inCongP_type,type,
% 0.23/0.46 inCongP: $o ).
% 0.23/0.46
% 0.23/0.46 thf(inCongP,definition,
% 0.23/0.46 ( inCongP
% 0.23/0.46 = ( ! [A: $i,B: $i] :
% 0.23/0.46 ( ( A = B )
% 0.23/0.46 => ! [Xx: $i,Xy: $i] :
% 0.23/0.46 ( ( Xx = Xy )
% 0.23/0.46 => ( ( in @ Xx @ A )
% 0.23/0.46 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(in__Cong_type,type,
% 0.23/0.46 in__Cong: $o ).
% 0.23/0.46
% 0.23/0.46 thf(in__Cong,definition,
% 0.23/0.46 ( in__Cong
% 0.23/0.46 = ( ! [A: $i,B: $i] :
% 0.23/0.46 ( ( A = B )
% 0.23/0.46 => ! [Xx: $i,Xy: $i] :
% 0.23/0.46 ( ( Xx = Xy )
% 0.23/0.46 => ( ( in @ Xx @ A )
% 0.23/0.46 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(exuE3u_type,type,
% 0.23/0.46 exuE3u: $o ).
% 0.23/0.46
% 0.23/0.46 thf(exuE3u,definition,
% 0.23/0.46 ( exuE3u
% 0.23/0.46 = ( ! [Xphi: $i > $o] :
% 0.23/0.46 ( ( exu
% 0.23/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.23/0.46 => ! [Xx: $i,Xy: $i] :
% 0.23/0.46 ( ( Xphi @ Xx )
% 0.23/0.46 => ( ( Xphi @ Xy )
% 0.23/0.46 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(exu__Cong_type,type,
% 0.23/0.46 exu__Cong: $o ).
% 0.23/0.46
% 0.23/0.46 thf(exu__Cong,definition,
% 0.23/0.46 ( exu__Cong
% 0.23/0.46 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.23/0.46 ( ! [Xx: $i,Xy: $i] :
% 0.23/0.46 ( ( Xx = Xy )
% 0.23/0.46 => ( ( Xphi @ Xx )
% 0.23/0.46 <=> ( Xpsi @ Xy ) ) )
% 0.23/0.46 => ( ( exu
% 0.23/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.23/0.46 <=> ( exu
% 0.23/0.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(emptyset__Cong_type,type,
% 0.23/0.46 emptyset__Cong: $o ).
% 0.23/0.46
% 0.23/0.46 thf(emptyset__Cong,definition,
% 0.23/0.46 ( emptyset__Cong
% 0.23/0.46 = ( emptyset = emptyset ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(setadjoin__Cong_type,type,
% 0.23/0.46 setadjoin__Cong: $o ).
% 0.23/0.46
% 0.23/0.46 thf(setadjoin__Cong,definition,
% 0.23/0.46 ( setadjoin__Cong
% 0.23/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.23/0.46 ( ( Xx = Xy )
% 0.23/0.46 => ! [Xz: $i,Xu: $i] :
% 0.23/0.46 ( ( Xz = Xu )
% 0.23/0.46 => ( ( setadjoin @ Xx @ Xz )
% 0.23/0.46 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(powerset__Cong_type,type,
% 0.23/0.46 powerset__Cong: $o ).
% 0.23/0.46
% 0.23/0.46 thf(powerset__Cong,definition,
% 0.23/0.46 ( powerset__Cong
% 0.23/0.46 = ( ! [A: $i,B: $i] :
% 0.23/0.46 ( ( A = B )
% 0.23/0.46 => ( ( powerset @ A )
% 0.23/0.46 = ( powerset @ B ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(setunion__Cong_type,type,
% 0.23/0.46 setunion__Cong: $o ).
% 0.23/0.46
% 0.23/0.46 thf(setunion__Cong,definition,
% 0.23/0.46 ( setunion__Cong
% 0.23/0.46 = ( ! [A: $i,B: $i] :
% 0.23/0.46 ( ( A = B )
% 0.23/0.46 => ( ( setunion @ A )
% 0.23/0.46 = ( setunion @ B ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(omega__Cong_type,type,
% 0.23/0.46 omega__Cong: $o ).
% 0.23/0.46
% 0.23/0.46 thf(omega__Cong,definition,
% 0.23/0.46 ( omega__Cong
% 0.23/0.46 = ( omega = omega ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(exuEu_type,type,
% 0.23/0.46 exuEu: $o ).
% 0.23/0.46
% 0.23/0.46 thf(exuEu,definition,
% 0.23/0.46 ( exuEu
% 0.23/0.46 = ( ! [Xphi: $i > $o] :
% 0.23/0.46 ( ( exu
% 0.23/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.23/0.46 => ! [Xx: $i,Xy: $i] :
% 0.23/0.46 ( ( Xphi @ Xx )
% 0.23/0.46 => ( ( Xphi @ Xy )
% 0.23/0.46 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(descr__Cong_type,type,
% 0.23/0.46 descr__Cong: $o ).
% 0.23/0.46
% 0.23/0.46 thf(descr__Cong,definition,
% 0.23/0.46 ( descr__Cong
% 0.23/0.46 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.23/0.46 ( ! [Xx: $i,Xy: $i] :
% 0.23/0.46 ( ( Xx = Xy )
% 0.23/0.46 => ( ( Xphi @ Xx )
% 0.23/0.46 <=> ( Xpsi @ Xy ) ) )
% 0.23/0.46 => ( ( exu
% 0.23/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.23/0.46 => ( ( exu
% 0.23/0.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.23/0.46 => ( ( descr
% 0.23/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.23/0.46 = ( descr
% 0.23/0.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(dsetconstr__Cong_type,type,
% 0.23/0.46 dsetconstr__Cong: $o ).
% 0.23/0.46
% 0.23/0.46 thf(dsetconstr__Cong,definition,
% 0.23/0.46 ( dsetconstr__Cong
% 0.23/0.46 = ( ! [A: $i,B: $i] :
% 0.23/0.46 ( ( A = B )
% 0.23/0.46 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.23/0.46 ( ! [Xx: $i] :
% 0.23/0.46 ( ( in @ Xx @ A )
% 0.23/0.46 => ! [Xy: $i] :
% 0.23/0.46 ( ( in @ Xy @ B )
% 0.23/0.46 => ( ( Xx = Xy )
% 0.23/0.46 => ( ( Xphi @ Xx )
% 0.23/0.46 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.23/0.46 => ( ( dsetconstr @ A
% 0.23/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.23/0.46 = ( dsetconstr @ B
% 0.23/0.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(subset_type,type,
% 0.23/0.46 subset: $i > $i > $o ).
% 0.23/0.46
% 0.23/0.46 thf(disjoint_type,type,
% 0.23/0.46 disjoint: $i > $i > $o ).
% 0.23/0.46
% 0.23/0.46 thf(setsmeet_type,type,
% 0.23/0.46 setsmeet: $i > $i > $o ).
% 0.23/0.46
% 0.23/0.46 thf(subsetI1_type,type,
% 0.23/0.46 subsetI1: $o ).
% 0.23/0.46
% 0.23/0.46 thf(subsetI1,definition,
% 0.23/0.46 ( subsetI1
% 0.23/0.46 = ( ! [A: $i,B: $i] :
% 0.23/0.46 ( ! [Xx: $i] :
% 0.23/0.46 ( ( in @ Xx @ A )
% 0.23/0.46 => ( in @ Xx @ B ) )
% 0.23/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(eqimpsubset2_type,type,
% 0.23/0.46 eqimpsubset2: $o ).
% 0.23/0.46
% 0.23/0.46 thf(eqimpsubset2,definition,
% 0.23/0.46 ( eqimpsubset2
% 0.23/0.46 = ( ! [A: $i,B: $i] :
% 0.23/0.46 ( ( A = B )
% 0.23/0.46 => ( subset @ B @ A ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(eqimpsubset1_type,type,
% 0.23/0.46 eqimpsubset1: $o ).
% 0.23/0.46
% 0.23/0.46 thf(eqimpsubset1,definition,
% 0.23/0.46 ( eqimpsubset1
% 0.23/0.46 = ( ! [A: $i,B: $i] :
% 0.23/0.46 ( ( A = B )
% 0.23/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(subsetI2_type,type,
% 0.23/0.46 subsetI2: $o ).
% 0.23/0.46
% 0.23/0.46 thf(subsetI2,definition,
% 0.23/0.46 ( subsetI2
% 0.23/0.46 = ( ! [A: $i,B: $i] :
% 0.23/0.46 ( ! [Xx: $i] :
% 0.23/0.46 ( ( in @ Xx @ A )
% 0.23/0.46 => ( in @ Xx @ B ) )
% 0.23/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(emptysetsubset_type,type,
% 0.23/0.46 emptysetsubset: $o ).
% 0.23/0.46
% 0.23/0.46 thf(emptysetsubset,definition,
% 0.23/0.46 ( emptysetsubset
% 0.23/0.46 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(subsetE_type,type,
% 0.23/0.46 subsetE: $o ).
% 0.23/0.46
% 0.23/0.46 thf(subsetE,definition,
% 0.23/0.46 ( subsetE
% 0.23/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.23/0.46 ( ( subset @ A @ B )
% 0.23/0.46 => ( ( in @ Xx @ A )
% 0.23/0.46 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(subsetE2_type,type,
% 0.23/0.46 subsetE2: $o ).
% 0.23/0.46
% 0.23/0.46 thf(subsetE2,definition,
% 0.23/0.46 ( subsetE2
% 0.23/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.23/0.46 ( ( subset @ A @ B )
% 0.23/0.46 => ( ~ ( in @ Xx @ B )
% 0.23/0.46 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(notsubsetI_type,type,
% 0.23/0.46 notsubsetI: $o ).
% 0.23/0.46
% 0.23/0.46 thf(notsubsetI,definition,
% 0.23/0.46 ( notsubsetI
% 0.23/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.23/0.46 ( ( in @ Xx @ A )
% 0.23/0.46 => ( ~ ( in @ Xx @ B )
% 0.23/0.46 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(notequalI1_type,type,
% 0.23/0.46 notequalI1: $o ).
% 0.23/0.46
% 0.23/0.46 thf(notequalI1,definition,
% 0.23/0.46 ( notequalI1
% 0.23/0.46 = ( ! [A: $i,B: $i] :
% 0.23/0.46 ( ~ ( subset @ A @ B )
% 0.23/0.46 => ( A != B ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(notequalI2_type,type,
% 0.23/0.46 notequalI2: $o ).
% 0.23/0.46
% 0.23/0.46 thf(notequalI2,definition,
% 0.23/0.46 ( notequalI2
% 0.23/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.23/0.46 ( ( in @ Xx @ A )
% 0.23/0.46 => ( ~ ( in @ Xx @ B )
% 0.23/0.46 => ( A != B ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(subsetRefl_type,type,
% 0.23/0.46 subsetRefl: $o ).
% 0.23/0.46
% 0.23/0.46 thf(subsetRefl,definition,
% 0.23/0.46 ( subsetRefl
% 0.23/0.46 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(subsetTrans_type,type,
% 0.23/0.46 subsetTrans: $o ).
% 0.23/0.46
% 0.23/0.46 thf(subsetTrans,definition,
% 0.23/0.46 ( subsetTrans
% 0.23/0.46 = ( ! [A: $i,B: $i,C: $i] :
% 0.23/0.46 ( ( subset @ A @ B )
% 0.23/0.46 => ( ( subset @ B @ C )
% 0.23/0.46 => ( subset @ A @ C ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(setadjoinSub_type,type,
% 0.23/0.46 setadjoinSub: $o ).
% 0.23/0.46
% 0.23/0.46 thf(setadjoinSub,definition,
% 0.23/0.46 ( setadjoinSub
% 0.23/0.46 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(setadjoinSub2_type,type,
% 0.23/0.46 setadjoinSub2: $o ).
% 0.23/0.46
% 0.23/0.46 thf(setadjoinSub2,definition,
% 0.23/0.46 ( setadjoinSub2
% 0.23/0.46 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.23/0.46 ( ( subset @ A @ B )
% 0.23/0.46 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(subset2powerset_type,type,
% 0.23/0.46 subset2powerset: $o ).
% 0.23/0.46
% 0.23/0.46 thf(subset2powerset,definition,
% 0.23/0.46 ( subset2powerset
% 0.23/0.46 = ( ! [A: $i,B: $i] :
% 0.23/0.46 ( ( subset @ A @ B )
% 0.23/0.46 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(setextsub_type,type,
% 0.23/0.46 setextsub: $o ).
% 0.23/0.46
% 0.23/0.46 thf(setextsub,definition,
% 0.23/0.46 ( setextsub
% 0.23/0.46 = ( ! [A: $i,B: $i] :
% 0.23/0.46 ( ( subset @ A @ B )
% 0.23/0.46 => ( ( subset @ B @ A )
% 0.23/0.46 => ( A = B ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(subsetemptysetimpeq_type,type,
% 0.23/0.46 subsetemptysetimpeq: $o ).
% 0.23/0.46
% 0.23/0.46 thf(subsetemptysetimpeq,definition,
% 0.23/0.46 ( subsetemptysetimpeq
% 0.23/0.46 = ( ! [A: $i] :
% 0.23/0.46 ( ( subset @ A @ emptyset )
% 0.23/0.46 => ( A = emptyset ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(powersetI1_type,type,
% 0.23/0.46 powersetI1: $o ).
% 0.23/0.46
% 0.23/0.46 thf(powersetI1,definition,
% 0.23/0.46 ( powersetI1
% 0.23/0.46 = ( ! [A: $i,B: $i] :
% 0.23/0.46 ( ( subset @ B @ A )
% 0.23/0.46 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(powersetE1_type,type,
% 0.23/0.46 powersetE1: $o ).
% 0.23/0.46
% 0.23/0.46 thf(powersetE1,definition,
% 0.23/0.46 ( powersetE1
% 0.23/0.46 = ( ! [A: $i,B: $i] :
% 0.23/0.46 ( ( in @ B @ ( powerset @ A ) )
% 0.23/0.46 => ( subset @ B @ A ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(inPowerset_type,type,
% 0.23/0.46 inPowerset: $o ).
% 0.23/0.46
% 0.23/0.46 thf(inPowerset,definition,
% 0.23/0.46 ( inPowerset
% 0.23/0.46 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(powersetsubset_type,type,
% 0.23/0.46 powersetsubset: $o ).
% 0.23/0.46
% 0.23/0.46 thf(powersetsubset,definition,
% 0.23/0.46 ( powersetsubset
% 0.23/0.46 = ( ! [A: $i,B: $i] :
% 0.23/0.46 ( ( subset @ A @ B )
% 0.23/0.46 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(sepInPowerset_type,type,
% 0.23/0.46 sepInPowerset: $o ).
% 0.23/0.46
% 0.23/0.46 thf(sepInPowerset,definition,
% 0.23/0.46 ( sepInPowerset
% 0.23/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.23/0.46 ( in
% 0.23/0.46 @ ( dsetconstr @ A
% 0.23/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.23/0.46 @ ( powerset @ A ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(sepSubset_type,type,
% 0.23/0.46 sepSubset: $o ).
% 0.23/0.46
% 0.23/0.46 thf(sepSubset,definition,
% 0.23/0.46 ( sepSubset
% 0.23/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.23/0.46 ( subset
% 0.23/0.46 @ ( dsetconstr @ A
% 0.23/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.23/0.46 @ A ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(binunion_type,type,
% 0.23/0.46 binunion: $i > $i > $i ).
% 0.23/0.46
% 0.23/0.46 thf(binunionIL_type,type,
% 0.23/0.46 binunionIL: $o ).
% 0.23/0.46
% 0.23/0.46 thf(binunionIL,definition,
% 0.23/0.46 ( binunionIL
% 0.23/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.23/0.46 ( ( in @ Xx @ A )
% 0.23/0.46 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(upairset2IR_type,type,
% 0.23/0.46 upairset2IR: $o ).
% 0.23/0.46
% 0.23/0.46 thf(upairset2IR,definition,
% 0.23/0.46 ( upairset2IR
% 0.23/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(binunionIR_type,type,
% 0.23/0.46 binunionIR: $o ).
% 0.23/0.46
% 0.23/0.46 thf(binunionIR,definition,
% 0.23/0.46 ( binunionIR
% 0.23/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.23/0.46 ( ( in @ Xx @ B )
% 0.23/0.46 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(binunionEcases_type,type,
% 0.23/0.46 binunionEcases: $o ).
% 0.23/0.46
% 0.23/0.46 thf(binunionEcases,definition,
% 0.23/0.46 ( binunionEcases
% 0.23/0.46 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.23/0.46 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.23/0.46 => ( ( ( in @ Xx @ A )
% 0.23/0.46 => Xphi )
% 0.23/0.46 => ( ( ( in @ Xx @ B )
% 0.23/0.46 => Xphi )
% 0.23/0.46 => Xphi ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(binunionE_type,type,
% 0.23/0.46 binunionE: $o ).
% 0.23/0.46
% 0.23/0.46 thf(binunionE,definition,
% 0.23/0.46 ( binunionE
% 0.23/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.23/0.46 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.23/0.46 => ( ( in @ Xx @ A )
% 0.23/0.46 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(binunionLsub_type,type,
% 0.23/0.46 binunionLsub: $o ).
% 0.23/0.46
% 0.23/0.46 thf(binunionLsub,definition,
% 0.23/0.46 ( binunionLsub
% 0.23/0.46 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(binunionRsub_type,type,
% 0.23/0.46 binunionRsub: $o ).
% 0.23/0.46
% 0.23/0.46 thf(binunionRsub,definition,
% 0.23/0.46 ( binunionRsub
% 0.23/0.46 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(binintersect_type,type,
% 0.23/0.46 binintersect: $i > $i > $i ).
% 0.23/0.46
% 0.23/0.46 thf(binintersectI_type,type,
% 0.23/0.46 binintersectI: $o ).
% 0.23/0.46
% 0.23/0.46 thf(binintersectI,definition,
% 0.23/0.46 ( binintersectI
% 0.23/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.23/0.46 ( ( in @ Xx @ A )
% 0.23/0.46 => ( ( in @ Xx @ B )
% 0.23/0.46 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(binintersectSubset5_type,type,
% 0.23/0.46 binintersectSubset5: $o ).
% 0.23/0.46
% 0.23/0.46 thf(binintersectSubset5,definition,
% 0.23/0.46 ( binintersectSubset5
% 0.23/0.46 = ( ! [A: $i,B: $i,C: $i] :
% 0.23/0.46 ( ( subset @ C @ A )
% 0.23/0.46 => ( ( subset @ C @ B )
% 0.23/0.46 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(binintersectEL_type,type,
% 0.23/0.46 binintersectEL: $o ).
% 0.23/0.46
% 0.23/0.46 thf(binintersectEL,definition,
% 0.23/0.46 ( binintersectEL
% 0.23/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.23/0.46 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.23/0.46 => ( in @ Xx @ A ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(binintersectLsub_type,type,
% 0.23/0.46 binintersectLsub: $o ).
% 0.23/0.46
% 0.23/0.46 thf(binintersectLsub,definition,
% 0.23/0.46 ( binintersectLsub
% 0.23/0.46 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(binintersectSubset2_type,type,
% 0.23/0.46 binintersectSubset2: $o ).
% 0.23/0.46
% 0.23/0.46 thf(binintersectSubset2,definition,
% 0.23/0.46 ( binintersectSubset2
% 0.23/0.46 = ( ! [A: $i,B: $i] :
% 0.23/0.46 ( ( subset @ A @ B )
% 0.23/0.46 => ( ( binintersect @ A @ B )
% 0.23/0.46 = A ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(binintersectSubset3_type,type,
% 0.23/0.46 binintersectSubset3: $o ).
% 0.23/0.46
% 0.23/0.46 thf(binintersectSubset3,definition,
% 0.23/0.46 ( binintersectSubset3
% 0.23/0.46 = ( ! [A: $i,B: $i] :
% 0.23/0.46 ( ( ( binintersect @ A @ B )
% 0.23/0.46 = B )
% 0.23/0.46 => ( subset @ B @ A ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(binintersectER_type,type,
% 0.23/0.46 binintersectER: $o ).
% 0.23/0.46
% 0.23/0.46 thf(binintersectER,definition,
% 0.23/0.46 ( binintersectER
% 0.23/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.23/0.46 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.23/0.46 => ( in @ Xx @ B ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(disjointsetsI1_type,type,
% 0.23/0.46 disjointsetsI1: $o ).
% 0.23/0.46
% 0.23/0.46 thf(disjointsetsI1,definition,
% 0.23/0.46 ( disjointsetsI1
% 0.23/0.46 = ( ! [A: $i,B: $i] :
% 0.23/0.46 ( ~ ? [Xx: $i] :
% 0.23/0.46 ( ( in @ Xx @ A )
% 0.23/0.46 & ( in @ Xx @ B ) )
% 0.23/0.46 => ( ( binintersect @ A @ B )
% 0.23/0.46 = emptyset ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(binintersectRsub_type,type,
% 0.23/0.46 binintersectRsub: $o ).
% 0.23/0.46
% 0.23/0.46 thf(binintersectRsub,definition,
% 0.23/0.46 ( binintersectRsub
% 0.23/0.46 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(binintersectSubset4_type,type,
% 0.23/0.46 binintersectSubset4: $o ).
% 0.23/0.46
% 0.23/0.46 thf(binintersectSubset4,definition,
% 0.23/0.46 ( binintersectSubset4
% 0.23/0.46 = ( ! [A: $i,B: $i] :
% 0.23/0.46 ( ( subset @ B @ A )
% 0.23/0.46 => ( ( binintersect @ A @ B )
% 0.23/0.46 = B ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(binintersectSubset1_type,type,
% 0.23/0.46 binintersectSubset1: $o ).
% 0.23/0.46
% 0.23/0.46 thf(binintersectSubset1,definition,
% 0.23/0.46 ( binintersectSubset1
% 0.23/0.46 = ( ! [A: $i,B: $i] :
% 0.23/0.46 ( ( ( binintersect @ A @ B )
% 0.23/0.46 = A )
% 0.23/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(bs114d_type,type,
% 0.23/0.46 bs114d: $o ).
% 0.23/0.46
% 0.23/0.46 thf(bs114d,definition,
% 0.23/0.46 ( bs114d
% 0.23/0.46 = ( ! [A: $i,B: $i,C: $i] :
% 0.23/0.46 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 0.23/0.46 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(regular_type,type,
% 0.23/0.46 regular: $i > $o ).
% 0.23/0.46
% 0.23/0.46 thf(setminus_type,type,
% 0.23/0.46 setminus: $i > $i > $i ).
% 0.23/0.46
% 0.23/0.46 thf(setminusI_type,type,
% 0.23/0.46 setminusI: $o ).
% 0.23/0.46
% 0.23/0.46 thf(setminusI,definition,
% 0.23/0.46 ( setminusI
% 0.23/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.23/0.46 ( ( in @ Xx @ A )
% 0.23/0.46 => ( ~ ( in @ Xx @ B )
% 0.23/0.46 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(setminusEL_type,type,
% 0.23/0.46 setminusEL: $o ).
% 0.23/0.46
% 0.23/0.46 thf(setminusEL,definition,
% 0.23/0.46 ( setminusEL
% 0.23/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.23/0.46 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.23/0.46 => ( in @ Xx @ A ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(setminusER_type,type,
% 0.23/0.46 setminusER: $o ).
% 0.23/0.46
% 0.23/0.46 thf(setminusER,definition,
% 0.23/0.46 ( setminusER
% 0.23/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.23/0.46 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.23/0.46 => ~ ( in @ Xx @ B ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(setminusSubset2_type,type,
% 0.23/0.46 setminusSubset2: $o ).
% 0.23/0.46
% 0.23/0.46 thf(setminusSubset2,definition,
% 0.23/0.46 ( setminusSubset2
% 0.23/0.46 = ( ! [A: $i,B: $i] :
% 0.23/0.46 ( ( subset @ A @ B )
% 0.23/0.46 => ( ( setminus @ A @ B )
% 0.23/0.46 = emptyset ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(setminusERneg_type,type,
% 0.23/0.46 setminusERneg: $o ).
% 0.23/0.46
% 0.23/0.46 thf(setminusERneg,definition,
% 0.23/0.46 ( setminusERneg
% 0.23/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.23/0.46 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.23/0.46 => ( ( in @ Xx @ A )
% 0.23/0.46 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(setminusELneg_type,type,
% 0.23/0.46 setminusELneg: $o ).
% 0.23/0.46
% 0.23/0.46 thf(setminusELneg,definition,
% 0.23/0.46 ( setminusELneg
% 0.23/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.23/0.46 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.23/0.46 => ( ~ ( in @ Xx @ B )
% 0.23/0.46 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(setminusILneg_type,type,
% 0.23/0.46 setminusILneg: $o ).
% 0.23/0.46
% 0.23/0.46 thf(setminusILneg,definition,
% 0.23/0.46 ( setminusILneg
% 0.23/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.23/0.46 ( ~ ( in @ Xx @ A )
% 0.23/0.46 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(setminusIRneg_type,type,
% 0.23/0.46 setminusIRneg: $o ).
% 0.23/0.46
% 0.23/0.46 thf(setminusIRneg,definition,
% 0.23/0.46 ( setminusIRneg
% 0.23/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.23/0.46 ( ( in @ Xx @ B )
% 0.23/0.46 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(setminusLsub_type,type,
% 0.23/0.46 setminusLsub: $o ).
% 0.23/0.46
% 0.23/0.46 thf(setminusLsub,definition,
% 0.23/0.46 ( setminusLsub
% 0.23/0.46 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 0.23/0.46
% 0.23/0.46 thf(setminusSubset1_type,type,
% 0.23/0.46 setminusSubset1: $o ).
% 0.23/0.46
% 0.23/0.46 thf(setminusSubset1,definition,
% 0.23/0.46 ( setminusSubset1
% 0.23/0.46 = ( ! [A: $i,B: $i] :
% 0.23/0.46 ( ( ( setminus @ A @ B )
% 0.23/0.46 = emptyset )
% 0.23/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(symdiff_type,type,
% 0.23/0.47 symdiff: $i > $i > $i ).
% 0.23/0.47
% 0.23/0.47 thf(symdiffE_type,type,
% 0.23/0.47 symdiffE: $o ).
% 0.23/0.47
% 0.23/0.47 thf(symdiffE,definition,
% 0.23/0.47 ( symdiffE
% 0.23/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.23/0.47 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 0.23/0.47 => ! [Xphi: $o] :
% 0.23/0.47 ( ( ( in @ Xx @ A )
% 0.23/0.47 => ( ~ ( in @ Xx @ B )
% 0.23/0.47 => Xphi ) )
% 0.23/0.47 => ( ( ~ ( in @ Xx @ A )
% 0.23/0.47 => ( ( in @ Xx @ B )
% 0.23/0.47 => Xphi ) )
% 0.23/0.47 => Xphi ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(symdiffI1_type,type,
% 0.23/0.47 symdiffI1: $o ).
% 0.23/0.47
% 0.23/0.47 thf(symdiffI1,definition,
% 0.23/0.47 ( symdiffI1
% 0.23/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.23/0.47 ( ( in @ Xx @ A )
% 0.23/0.47 => ( ~ ( in @ Xx @ B )
% 0.23/0.47 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(symdiffI2_type,type,
% 0.23/0.47 symdiffI2: $o ).
% 0.23/0.47
% 0.23/0.47 thf(symdiffI2,definition,
% 0.23/0.47 ( symdiffI2
% 0.23/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.23/0.47 ( ~ ( in @ Xx @ A )
% 0.23/0.47 => ( ( in @ Xx @ B )
% 0.23/0.47 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(symdiffIneg1_type,type,
% 0.23/0.47 symdiffIneg1: $o ).
% 0.23/0.47
% 0.23/0.47 thf(symdiffIneg1,definition,
% 0.23/0.47 ( symdiffIneg1
% 0.23/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.23/0.47 ( ( in @ Xx @ A )
% 0.23/0.47 => ( ( in @ Xx @ B )
% 0.23/0.47 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(symdiffIneg2_type,type,
% 0.23/0.47 symdiffIneg2: $o ).
% 0.23/0.47
% 0.23/0.47 thf(symdiffIneg2,definition,
% 0.23/0.47 ( symdiffIneg2
% 0.23/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.23/0.47 ( ~ ( in @ Xx @ A )
% 0.23/0.47 => ( ~ ( in @ Xx @ B )
% 0.23/0.47 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(iskpair_type,type,
% 0.23/0.47 iskpair: $i > $o ).
% 0.23/0.47
% 0.23/0.47 thf(secondinupair_type,type,
% 0.23/0.47 secondinupair: $o ).
% 0.23/0.47
% 0.23/0.47 thf(secondinupair,definition,
% 0.23/0.47 ( secondinupair
% 0.23/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(setukpairIL_type,type,
% 0.23/0.47 setukpairIL: $o ).
% 0.23/0.47
% 0.23/0.47 thf(setukpairIL,definition,
% 0.23/0.47 ( setukpairIL
% 0.23/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(setukpairIR_type,type,
% 0.23/0.47 setukpairIR: $o ).
% 0.23/0.47
% 0.23/0.47 thf(setukpairIR,definition,
% 0.23/0.47 ( setukpairIR
% 0.23/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(kpairiskpair_type,type,
% 0.23/0.47 kpairiskpair: $o ).
% 0.23/0.47
% 0.23/0.47 thf(kpairiskpair,definition,
% 0.23/0.47 ( kpairiskpair
% 0.23/0.47 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(kpair_type,type,
% 0.23/0.47 kpair: $i > $i > $i ).
% 0.23/0.47
% 0.23/0.47 thf(kpairp_type,type,
% 0.23/0.47 kpairp: $o ).
% 0.23/0.47
% 0.23/0.47 thf(kpairp,definition,
% 0.23/0.47 ( kpairp
% 0.23/0.47 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(cartprod_type,type,
% 0.23/0.47 cartprod: $i > $i > $i ).
% 0.23/0.47
% 0.23/0.47 thf(singletonsubset_type,type,
% 0.23/0.47 singletonsubset: $o ).
% 0.23/0.47
% 0.23/0.47 thf(singletonsubset,definition,
% 0.23/0.47 ( singletonsubset
% 0.23/0.47 = ( ! [A: $i,Xx: $i] :
% 0.23/0.47 ( ( in @ Xx @ A )
% 0.23/0.47 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(singletoninpowerset_type,type,
% 0.23/0.47 singletoninpowerset: $o ).
% 0.23/0.47
% 0.23/0.47 thf(singletoninpowerset,definition,
% 0.23/0.47 ( singletoninpowerset
% 0.23/0.47 = ( ! [A: $i,Xx: $i] :
% 0.23/0.47 ( ( in @ Xx @ A )
% 0.23/0.47 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(singletoninpowunion_type,type,
% 0.23/0.47 singletoninpowunion: $o ).
% 0.23/0.47
% 0.23/0.47 thf(singletoninpowunion,definition,
% 0.23/0.47 ( singletoninpowunion
% 0.23/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.23/0.47 ( ( in @ Xx @ A )
% 0.23/0.47 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(upairset2E_type,type,
% 0.23/0.47 upairset2E: $o ).
% 0.23/0.47
% 0.23/0.47 thf(upairset2E,definition,
% 0.23/0.47 ( upairset2E
% 0.23/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.23/0.47 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.23/0.47 => ( ( Xz = Xx )
% 0.23/0.47 | ( Xz = Xy ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(upairsubunion_type,type,
% 0.23/0.47 upairsubunion: $o ).
% 0.23/0.47
% 0.23/0.47 thf(upairsubunion,definition,
% 0.23/0.47 ( upairsubunion
% 0.23/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.23/0.47 ( ( in @ Xx @ A )
% 0.23/0.47 => ! [Xy: $i] :
% 0.23/0.47 ( ( in @ Xy @ B )
% 0.23/0.47 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(upairinpowunion_type,type,
% 0.23/0.47 upairinpowunion: $o ).
% 0.23/0.47
% 0.23/0.47 thf(upairinpowunion,definition,
% 0.23/0.47 ( upairinpowunion
% 0.23/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.23/0.47 ( ( in @ Xx @ A )
% 0.23/0.47 => ! [Xy: $i] :
% 0.23/0.47 ( ( in @ Xy @ B )
% 0.23/0.47 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(ubforcartprodlem1_type,type,
% 0.23/0.47 ubforcartprodlem1: $o ).
% 0.23/0.47
% 0.23/0.47 thf(ubforcartprodlem1,definition,
% 0.23/0.47 ( ubforcartprodlem1
% 0.23/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.23/0.47 ( ( in @ Xx @ A )
% 0.23/0.47 => ! [Xy: $i] :
% 0.23/0.47 ( ( in @ Xy @ B )
% 0.23/0.47 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(ubforcartprodlem2_type,type,
% 0.23/0.47 ubforcartprodlem2: $o ).
% 0.23/0.47
% 0.23/0.47 thf(ubforcartprodlem2,definition,
% 0.23/0.47 ( ubforcartprodlem2
% 0.23/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.23/0.47 ( ( in @ Xx @ A )
% 0.23/0.47 => ! [Xy: $i] :
% 0.23/0.47 ( ( in @ Xy @ B )
% 0.23/0.47 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(ubforcartprodlem3_type,type,
% 0.23/0.47 ubforcartprodlem3: $o ).
% 0.23/0.47
% 0.23/0.47 thf(ubforcartprodlem3,definition,
% 0.23/0.47 ( ubforcartprodlem3
% 0.23/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.23/0.47 ( ( in @ Xx @ A )
% 0.23/0.47 => ! [Xy: $i] :
% 0.23/0.47 ( ( in @ Xy @ B )
% 0.23/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(cartprodpairin_type,type,
% 0.23/0.47 cartprodpairin: $o ).
% 0.23/0.47
% 0.23/0.47 thf(cartprodpairin,definition,
% 0.23/0.47 ( cartprodpairin
% 0.23/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.23/0.47 ( ( in @ Xx @ A )
% 0.23/0.47 => ! [Xy: $i] :
% 0.23/0.47 ( ( in @ Xy @ B )
% 0.23/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(cartprodmempair1_type,type,
% 0.23/0.47 cartprodmempair1: $o ).
% 0.23/0.47
% 0.23/0.47 thf(cartprodmempair1,definition,
% 0.23/0.47 ( cartprodmempair1
% 0.23/0.47 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.23/0.47 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.23/0.47 => ? [Xx: $i] :
% 0.23/0.47 ( ( in @ Xx @ A )
% 0.23/0.47 & ? [Xy: $i] :
% 0.23/0.47 ( ( in @ Xy @ B )
% 0.23/0.47 & ( Xu
% 0.23/0.47 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(cartprodmempair_type,type,
% 0.23/0.47 cartprodmempair: $o ).
% 0.23/0.47
% 0.23/0.47 thf(cartprodmempair,definition,
% 0.23/0.47 ( cartprodmempair
% 0.23/0.47 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.23/0.47 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.23/0.47 => ( iskpair @ Xu ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(setunionE2_type,type,
% 0.23/0.47 setunionE2: $o ).
% 0.23/0.47
% 0.23/0.47 thf(setunionE2,definition,
% 0.23/0.47 ( setunionE2
% 0.23/0.47 = ( ! [A: $i,Xx: $i] :
% 0.23/0.47 ( ( in @ Xx @ ( setunion @ A ) )
% 0.23/0.47 => ? [X: $i] :
% 0.23/0.47 ( ( in @ X @ A )
% 0.23/0.47 & ( in @ Xx @ X ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(setunionsingleton1_type,type,
% 0.23/0.47 setunionsingleton1: $o ).
% 0.23/0.47
% 0.23/0.47 thf(setunionsingleton1,definition,
% 0.23/0.47 ( setunionsingleton1
% 0.23/0.47 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(setunionsingleton2_type,type,
% 0.23/0.47 setunionsingleton2: $o ).
% 0.23/0.47
% 0.23/0.47 thf(setunionsingleton2,definition,
% 0.23/0.47 ( setunionsingleton2
% 0.23/0.47 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(setunionsingleton_type,type,
% 0.23/0.47 setunionsingleton: $o ).
% 0.23/0.47
% 0.23/0.47 thf(setunionsingleton,definition,
% 0.23/0.47 ( setunionsingleton
% 0.23/0.47 = ( ! [Xx: $i] :
% 0.23/0.47 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 0.23/0.47 = Xx ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(singleton_type,type,
% 0.23/0.47 singleton: $i > $o ).
% 0.23/0.47
% 0.23/0.47 thf(singletonprop_type,type,
% 0.23/0.47 singletonprop: $o ).
% 0.23/0.47
% 0.23/0.47 thf(singletonprop,definition,
% 0.23/0.47 ( singletonprop
% 0.23/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.23/0.47 ( ! [Xx: $i] :
% 0.23/0.47 ( ( in @ Xx @ A )
% 0.23/0.47 => ! [Xy: $i] :
% 0.23/0.47 ( ( in @ Xy @ A )
% 0.23/0.47 => ( ( Xphi @ Xx )
% 0.23/0.47 => ( ( Xphi @ Xy )
% 0.23/0.47 => ( Xx = Xy ) ) ) ) )
% 0.23/0.47 => ( ? [Xx: $i] :
% 0.23/0.47 ( ( in @ Xx @ A )
% 0.23/0.47 & ( Xphi @ Xx ) )
% 0.23/0.47 => ( singleton
% 0.23/0.47 @ ( dsetconstr @ A
% 0.23/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(ex1_type,type,
% 0.23/0.47 ex1: $i > ( $i > $o ) > $o ).
% 0.23/0.47
% 0.23/0.47 thf(ex1,definition,
% 0.23/0.47 ( ex1
% 0.23/0.47 = ( ^ [A: $i,Xphi: $i > $o] :
% 0.23/0.47 ( singleton
% 0.23/0.47 @ ( dsetconstr @ A
% 0.23/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(ex1E1_type,type,
% 0.23/0.47 ex1E1: $o ).
% 0.23/0.47
% 0.23/0.47 thf(ex1E1,definition,
% 0.23/0.47 ( ex1E1
% 0.23/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.23/0.47 ( ( ex1 @ A
% 0.23/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.23/0.47 => ? [Xx: $i] :
% 0.23/0.47 ( ( in @ Xx @ A )
% 0.23/0.47 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(ex1I_type,type,
% 0.23/0.47 ex1I: $o ).
% 0.23/0.47
% 0.23/0.47 thf(ex1I,definition,
% 0.23/0.47 ( ex1I
% 0.23/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.23/0.47 ( ( in @ Xx @ A )
% 0.23/0.47 => ( ( Xphi @ Xx )
% 0.23/0.47 => ( ! [Xy: $i] :
% 0.23/0.47 ( ( in @ Xy @ A )
% 0.23/0.47 => ( ( Xphi @ Xy )
% 0.23/0.47 => ( Xy = Xx ) ) )
% 0.23/0.47 => ( ex1 @ A
% 0.23/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(ex1I2_type,type,
% 0.23/0.47 ex1I2: $o ).
% 0.23/0.47
% 0.23/0.47 thf(ex1I2,definition,
% 0.23/0.47 ( ex1I2
% 0.23/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.23/0.47 ( ! [Xx: $i] :
% 0.23/0.47 ( ( in @ Xx @ A )
% 0.23/0.47 => ! [Xy: $i] :
% 0.23/0.47 ( ( in @ Xy @ A )
% 0.23/0.47 => ( ( Xphi @ Xx )
% 0.23/0.47 => ( ( Xphi @ Xy )
% 0.23/0.47 => ( Xx = Xy ) ) ) ) )
% 0.23/0.47 => ( ? [Xx: $i] :
% 0.23/0.47 ( ( in @ Xx @ A )
% 0.23/0.47 & ( Xphi @ Xx ) )
% 0.23/0.47 => ( ex1 @ A
% 0.23/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(singletonsuniq_type,type,
% 0.23/0.47 singletonsuniq: $o ).
% 0.23/0.47
% 0.23/0.47 thf(singletonsuniq,definition,
% 0.23/0.47 ( singletonsuniq
% 0.23/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.23/0.47 ( ( ( setadjoin @ Xx @ emptyset )
% 0.23/0.47 = ( setadjoin @ Xy @ emptyset ) )
% 0.23/0.47 => ( Xx = Xy ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(atmost1p_type,type,
% 0.23/0.47 atmost1p: $i > $o ).
% 0.23/0.47
% 0.23/0.47 thf(atleast2p_type,type,
% 0.23/0.47 atleast2p: $i > $o ).
% 0.23/0.47
% 0.23/0.47 thf(atmost2p_type,type,
% 0.23/0.47 atmost2p: $i > $o ).
% 0.23/0.47
% 0.23/0.47 thf(upairsetp_type,type,
% 0.23/0.47 upairsetp: $i > $o ).
% 0.23/0.47
% 0.23/0.47 thf(setukpairinjL1_type,type,
% 0.23/0.47 setukpairinjL1: $o ).
% 0.23/0.47
% 0.23/0.47 thf(setukpairinjL1,definition,
% 0.23/0.47 ( setukpairinjL1
% 0.23/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.23/0.47 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 0.23/0.47 => ( Xx = Xz ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(kfstsingleton_type,type,
% 0.23/0.47 kfstsingleton: $o ).
% 0.23/0.47
% 0.23/0.47 thf(kfstsingleton,definition,
% 0.23/0.47 ( kfstsingleton
% 0.23/0.47 = ( ! [Xu: $i] :
% 0.23/0.47 ( ( iskpair @ Xu )
% 0.23/0.47 => ( singleton
% 0.23/0.47 @ ( dsetconstr @ ( setunion @ Xu )
% 0.23/0.47 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(theprop_type,type,
% 0.23/0.47 theprop: $o ).
% 0.23/0.47
% 0.23/0.47 thf(theprop,definition,
% 0.23/0.47 ( theprop
% 0.23/0.47 = ( ! [X: $i] :
% 0.23/0.47 ( ( singleton @ X )
% 0.23/0.47 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(kfst_type,type,
% 0.23/0.47 kfst: $i > $i ).
% 0.23/0.47
% 0.23/0.47 thf(kfstpairEq_type,type,
% 0.23/0.47 kfstpairEq: $o ).
% 0.23/0.47
% 0.23/0.47 thf(kfstpairEq,definition,
% 0.23/0.47 ( kfstpairEq
% 0.23/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.23/0.47 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.23/0.47 = Xx ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(cartprodfstin_type,type,
% 0.23/0.47 cartprodfstin: $o ).
% 0.23/0.47
% 0.23/0.47 thf(cartprodfstin,definition,
% 0.23/0.47 ( cartprodfstin
% 0.23/0.47 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.23/0.47 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.23/0.47 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(setukpairinjL2_type,type,
% 0.23/0.47 setukpairinjL2: $o ).
% 0.23/0.47
% 0.23/0.47 thf(setukpairinjL2,definition,
% 0.23/0.47 ( setukpairinjL2
% 0.23/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.23/0.47 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.23/0.47 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.23/0.47 => ( Xx = Xz ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(setukpairinjL_type,type,
% 0.23/0.47 setukpairinjL: $o ).
% 0.23/0.47
% 0.23/0.47 thf(setukpairinjL,definition,
% 0.23/0.47 ( setukpairinjL
% 0.23/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.23/0.47 ( ( ( kpair @ Xx @ Xy )
% 0.23/0.47 = ( kpair @ Xz @ Xu ) )
% 0.23/0.47 => ( Xx = Xz ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(setukpairinjR11_type,type,
% 0.23/0.47 setukpairinjR11: $o ).
% 0.23/0.47
% 0.23/0.47 thf(setukpairinjR11,definition,
% 0.23/0.47 ( setukpairinjR11
% 0.23/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.23/0.47 ( ( Xx = Xy )
% 0.23/0.47 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.23/0.47 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(setukpairinjR12_type,type,
% 0.23/0.47 setukpairinjR12: $o ).
% 0.23/0.47
% 0.23/0.47 thf(setukpairinjR12,definition,
% 0.23/0.47 ( setukpairinjR12
% 0.23/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.23/0.47 ( ( Xx = Xy )
% 0.23/0.47 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.23/0.47 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(setukpairinjR1_type,type,
% 0.23/0.47 setukpairinjR1: $o ).
% 0.23/0.47
% 0.23/0.47 thf(setukpairinjR1,definition,
% 0.23/0.47 ( setukpairinjR1
% 0.23/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.23/0.47 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.23/0.47 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.23/0.47 => ( ( Xz = Xu )
% 0.23/0.47 => ( Xy = Xu ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(upairequniteq_type,type,
% 0.23/0.47 upairequniteq: $o ).
% 0.23/0.47
% 0.23/0.47 thf(upairequniteq,definition,
% 0.23/0.47 ( upairequniteq
% 0.23/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.23/0.47 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.23/0.47 = ( setadjoin @ Xz @ emptyset ) )
% 0.23/0.47 => ( Xx = Xy ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(setukpairinjR2_type,type,
% 0.23/0.47 setukpairinjR2: $o ).
% 0.23/0.47
% 0.23/0.47 thf(setukpairinjR2,definition,
% 0.23/0.47 ( setukpairinjR2
% 0.23/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.23/0.47 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.23/0.47 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.23/0.47 => ( Xy = Xu ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(setukpairinjR_type,type,
% 0.23/0.47 setukpairinjR: $o ).
% 0.23/0.47
% 0.23/0.47 thf(setukpairinjR,definition,
% 0.23/0.47 ( setukpairinjR
% 0.23/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.23/0.47 ( ( ( kpair @ Xx @ Xy )
% 0.23/0.47 = ( kpair @ Xz @ Xu ) )
% 0.23/0.47 => ( Xy = Xu ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(ksndsingleton_type,type,
% 0.23/0.47 ksndsingleton: $o ).
% 0.23/0.47
% 0.23/0.47 thf(ksndsingleton,definition,
% 0.23/0.47 ( ksndsingleton
% 0.23/0.47 = ( ! [Xu: $i] :
% 0.23/0.47 ( ( iskpair @ Xu )
% 0.23/0.47 => ( singleton
% 0.23/0.47 @ ( dsetconstr @ ( setunion @ Xu )
% 0.23/0.47 @ ^ [Xx: $i] :
% 0.23/0.47 ( Xu
% 0.23/0.47 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(ksnd_type,type,
% 0.23/0.47 ksnd: $i > $i ).
% 0.23/0.47
% 0.23/0.47 thf(ksndpairEq_type,type,
% 0.23/0.47 ksndpairEq: $o ).
% 0.23/0.47
% 0.23/0.47 thf(ksndpairEq,definition,
% 0.23/0.47 ( ksndpairEq
% 0.23/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.23/0.47 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.23/0.47 = Xy ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(kpairsurjEq_type,type,
% 0.23/0.47 kpairsurjEq: $o ).
% 0.23/0.47
% 0.23/0.47 thf(kpairsurjEq,definition,
% 0.23/0.47 ( kpairsurjEq
% 0.23/0.47 = ( ! [Xu: $i] :
% 0.23/0.47 ( ( iskpair @ Xu )
% 0.23/0.47 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.23/0.47 = Xu ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(cartprodsndin_type,type,
% 0.23/0.47 cartprodsndin: $o ).
% 0.23/0.47
% 0.23/0.47 thf(cartprodsndin,definition,
% 0.23/0.47 ( cartprodsndin
% 0.23/0.47 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.23/0.47 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.23/0.47 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(cartprodpairmemEL_type,type,
% 0.23/0.47 cartprodpairmemEL: $o ).
% 0.23/0.47
% 0.23/0.47 thf(cartprodpairmemEL,definition,
% 0.23/0.47 ( cartprodpairmemEL
% 0.23/0.47 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.23/0.47 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.23/0.47 => ( in @ Xx @ A ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(cartprodpairmemER_type,type,
% 0.23/0.47 cartprodpairmemER: $o ).
% 0.23/0.47
% 0.23/0.47 thf(cartprodpairmemER,definition,
% 0.23/0.47 ( cartprodpairmemER
% 0.23/0.47 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.23/0.47 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.23/0.47 => ( in @ Xy @ B ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(cartprodmempaircEq_type,type,
% 0.23/0.47 cartprodmempaircEq: $o ).
% 0.23/0.47
% 0.23/0.47 thf(cartprodmempaircEq,definition,
% 0.23/0.47 ( cartprodmempaircEq
% 0.23/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.23/0.47 ( ( in @ Xx @ A )
% 0.23/0.47 => ! [Xy: $i] :
% 0.23/0.47 ( ( in @ Xy @ B )
% 0.23/0.47 => ( ( kpair @ Xx @ Xy )
% 0.23/0.47 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(cartprodfstpairEq_type,type,
% 0.23/0.47 cartprodfstpairEq: $o ).
% 0.23/0.47
% 0.23/0.47 thf(cartprodfstpairEq,definition,
% 0.23/0.47 ( cartprodfstpairEq
% 0.23/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.23/0.47 ( ( in @ Xx @ A )
% 0.23/0.47 => ! [Xy: $i] :
% 0.23/0.47 ( ( in @ Xy @ B )
% 0.23/0.47 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.23/0.47 = Xx ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(cartprodsndpairEq_type,type,
% 0.23/0.47 cartprodsndpairEq: $o ).
% 0.23/0.47
% 0.23/0.47 thf(cartprodsndpairEq,definition,
% 0.23/0.47 ( cartprodsndpairEq
% 0.23/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.23/0.47 ( ( in @ Xx @ A )
% 0.23/0.47 => ! [Xy: $i] :
% 0.23/0.47 ( ( in @ Xy @ B )
% 0.23/0.47 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.23/0.47 = Xy ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(cartprodpairsurjEq_type,type,
% 0.23/0.47 cartprodpairsurjEq: $o ).
% 0.23/0.47
% 0.23/0.47 thf(cartprodpairsurjEq,definition,
% 0.23/0.47 ( cartprodpairsurjEq
% 0.23/0.47 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.23/0.47 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.23/0.47 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.23/0.47 = Xu ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(breln_type,type,
% 0.23/0.47 breln: $i > $i > $i > $o ).
% 0.23/0.47
% 0.23/0.47 thf(breln,definition,
% 0.23/0.47 ( breln
% 0.23/0.47 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(dpsetconstr_type,type,
% 0.23/0.47 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 0.23/0.47
% 0.23/0.47 thf(dpsetconstrI_type,type,
% 0.23/0.47 dpsetconstrI: $o ).
% 0.23/0.47
% 0.23/0.47 thf(dpsetconstrI,definition,
% 0.23/0.47 ( dpsetconstrI
% 0.23/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.23/0.47 ( ( in @ Xx @ A )
% 0.23/0.47 => ! [Xy: $i] :
% 0.23/0.47 ( ( in @ Xy @ B )
% 0.23/0.47 => ( ( Xphi @ Xx @ Xy )
% 0.23/0.47 => ( in @ ( kpair @ Xx @ Xy )
% 0.23/0.47 @ ( dpsetconstr @ A @ B
% 0.23/0.47 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(dpsetconstrSub_type,type,
% 0.23/0.47 dpsetconstrSub: $o ).
% 0.23/0.47
% 0.23/0.47 thf(dpsetconstrSub,definition,
% 0.23/0.47 ( dpsetconstrSub
% 0.23/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.23/0.47 ( subset
% 0.23/0.47 @ ( dpsetconstr @ A @ B
% 0.23/0.47 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 0.23/0.47 @ ( cartprod @ A @ B ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(setOfPairsIsBReln_type,type,
% 0.23/0.47 setOfPairsIsBReln: $o ).
% 0.23/0.47
% 0.23/0.47 thf(setOfPairsIsBReln,definition,
% 0.23/0.47 ( setOfPairsIsBReln
% 0.23/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.23/0.47 ( breln @ A @ B
% 0.23/0.47 @ ( dpsetconstr @ A @ B
% 0.23/0.47 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(dpsetconstrERa_type,type,
% 0.23/0.47 dpsetconstrERa: $o ).
% 0.23/0.47
% 0.23/0.47 thf(dpsetconstrERa,definition,
% 0.23/0.47 ( dpsetconstrERa
% 0.23/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.23/0.47 ( ( in @ Xx @ A )
% 0.23/0.47 => ! [Xy: $i] :
% 0.23/0.47 ( ( in @ Xy @ B )
% 0.23/0.47 => ( ( in @ ( kpair @ Xx @ Xy )
% 0.23/0.47 @ ( dpsetconstr @ A @ B
% 0.23/0.47 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.23/0.47 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(dpsetconstrEL1_type,type,
% 0.23/0.47 dpsetconstrEL1: $o ).
% 0.23/0.47
% 0.23/0.47 thf(dpsetconstrEL1,definition,
% 0.23/0.47 ( dpsetconstrEL1
% 0.23/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.23/0.47 ( ( in @ ( kpair @ Xx @ Xy )
% 0.23/0.47 @ ( dpsetconstr @ A @ B
% 0.23/0.47 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.23/0.47 => ( in @ Xx @ A ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(dpsetconstrEL2_type,type,
% 0.23/0.47 dpsetconstrEL2: $o ).
% 0.23/0.47
% 0.23/0.47 thf(dpsetconstrEL2,definition,
% 0.23/0.47 ( dpsetconstrEL2
% 0.23/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.23/0.47 ( ( in @ ( kpair @ Xx @ Xy )
% 0.23/0.47 @ ( dpsetconstr @ A @ B
% 0.23/0.47 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.23/0.47 => ( in @ Xy @ B ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(dpsetconstrER_type,type,
% 0.23/0.47 dpsetconstrER: $o ).
% 0.23/0.47
% 0.23/0.47 thf(dpsetconstrER,definition,
% 0.23/0.47 ( dpsetconstrER
% 0.23/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.23/0.47 ( ( in @ ( kpair @ Xx @ Xy )
% 0.23/0.47 @ ( dpsetconstr @ A @ B
% 0.23/0.47 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.23/0.47 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(func_type,type,
% 0.23/0.47 func: $i > $i > $i > $o ).
% 0.23/0.47
% 0.23/0.47 thf(funcSet_type,type,
% 0.23/0.47 funcSet: $i > $i > $i ).
% 0.23/0.47
% 0.23/0.47 thf(funcImageSingleton_type,type,
% 0.23/0.47 funcImageSingleton: $o ).
% 0.23/0.47
% 0.23/0.47 thf(funcImageSingleton,definition,
% 0.23/0.47 ( funcImageSingleton
% 0.23/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.23/0.47 ( ( func @ A @ B @ Xf )
% 0.23/0.47 => ! [Xx: $i] :
% 0.23/0.47 ( ( in @ Xx @ A )
% 0.23/0.47 => ( singleton
% 0.23/0.47 @ ( dsetconstr @ B
% 0.23/0.47 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(apProp_type,type,
% 0.23/0.47 apProp: $o ).
% 0.23/0.47
% 0.23/0.47 thf(apProp,definition,
% 0.23/0.47 ( apProp
% 0.23/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.23/0.47 ( ( func @ A @ B @ Xf )
% 0.23/0.47 => ! [Xx: $i] :
% 0.23/0.47 ( ( in @ Xx @ A )
% 0.23/0.47 => ( in
% 0.23/0.47 @ ( setunion
% 0.23/0.47 @ ( dsetconstr @ B
% 0.23/0.47 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 0.23/0.47 @ B ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(ap_type,type,
% 0.23/0.47 ap: $i > $i > $i > $i > $i ).
% 0.23/0.47
% 0.23/0.47 thf(app_type,type,
% 0.23/0.47 app: $o ).
% 0.23/0.47
% 0.23/0.47 thf(app,definition,
% 0.23/0.47 ( app
% 0.23/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.23/0.47 ( ( func @ A @ B @ Xf )
% 0.23/0.47 => ! [Xx: $i] :
% 0.23/0.47 ( ( in @ Xx @ A )
% 0.23/0.47 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(infuncsetfunc_type,type,
% 0.23/0.47 infuncsetfunc: $o ).
% 0.23/0.47
% 0.23/0.47 thf(infuncsetfunc,definition,
% 0.23/0.47 ( infuncsetfunc
% 0.23/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.23/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.23/0.47 => ( func @ A @ B @ Xf ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(ap2p_type,type,
% 0.23/0.47 ap2p: $o ).
% 0.23/0.47
% 0.23/0.47 thf(ap2p,definition,
% 0.23/0.47 ( ap2p
% 0.23/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.23/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.23/0.47 => ! [Xx: $i] :
% 0.23/0.47 ( ( in @ Xx @ A )
% 0.23/0.47 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(funcinfuncset_type,type,
% 0.23/0.47 funcinfuncset: $o ).
% 0.23/0.47
% 0.23/0.47 thf(funcinfuncset,definition,
% 0.23/0.47 ( funcinfuncset
% 0.23/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.23/0.47 ( ( func @ A @ B @ Xf )
% 0.23/0.47 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(lamProp_type,type,
% 0.23/0.47 lamProp: $o ).
% 0.23/0.47
% 0.23/0.47 thf(lamProp,definition,
% 0.23/0.47 ( lamProp
% 0.23/0.47 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.23/0.47 ( ! [Xx: $i] :
% 0.23/0.47 ( ( in @ Xx @ A )
% 0.23/0.47 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.23/0.47 => ( func @ A @ B
% 0.23/0.47 @ ( dpsetconstr @ A @ B
% 0.23/0.47 @ ^ [Xx: $i,Xy: $i] :
% 0.23/0.47 ( ( Xf @ Xx )
% 0.23/0.47 = Xy ) ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(lam_type,type,
% 0.23/0.47 lam: $i > $i > ( $i > $i ) > $i ).
% 0.23/0.47
% 0.23/0.47 thf(lamp_type,type,
% 0.23/0.47 lamp: $o ).
% 0.23/0.47
% 0.23/0.47 thf(lamp,definition,
% 0.23/0.47 ( lamp
% 0.23/0.47 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.23/0.47 ( ! [Xx: $i] :
% 0.23/0.47 ( ( in @ Xx @ A )
% 0.23/0.47 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.23/0.47 => ( func @ A @ B
% 0.23/0.47 @ ( lam @ A @ B
% 0.23/0.47 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(lam2p_type,type,
% 0.23/0.47 lam2p: $o ).
% 0.23/0.47
% 0.23/0.47 thf(lam2p,definition,
% 0.23/0.47 ( lam2p
% 0.23/0.47 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.23/0.47 ( ! [Xx: $i] :
% 0.23/0.47 ( ( in @ Xx @ A )
% 0.23/0.47 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.23/0.47 => ( in
% 0.23/0.47 @ ( lam @ A @ B
% 0.23/0.47 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.23/0.47 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(brelnall1_type,type,
% 0.23/0.47 brelnall1: $o ).
% 0.23/0.47
% 0.23/0.47 thf(brelnall1,definition,
% 0.23/0.47 ( brelnall1
% 0.23/0.47 = ( ! [A: $i,B: $i,R: $i] :
% 0.23/0.47 ( ( breln @ A @ B @ R )
% 0.23/0.47 => ! [Xphi: $i > $o] :
% 0.23/0.47 ( ! [Xx: $i] :
% 0.23/0.47 ( ( in @ Xx @ A )
% 0.23/0.47 => ! [Xy: $i] :
% 0.23/0.47 ( ( in @ Xy @ B )
% 0.23/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.23/0.47 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.23/0.47 => ! [Xx: $i] :
% 0.23/0.47 ( ( in @ Xx @ R )
% 0.23/0.47 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(brelnall2_type,type,
% 0.23/0.47 brelnall2: $o ).
% 0.23/0.47
% 0.23/0.47 thf(brelnall2,definition,
% 0.23/0.47 ( brelnall2
% 0.23/0.47 = ( ! [A: $i,B: $i,R: $i] :
% 0.23/0.47 ( ( breln @ A @ B @ R )
% 0.23/0.47 => ! [Xphi: $i > $o] :
% 0.23/0.47 ( ! [Xx: $i] :
% 0.23/0.47 ( ( in @ Xx @ A )
% 0.23/0.47 => ! [Xy: $i] :
% 0.23/0.47 ( ( in @ Xy @ B )
% 0.23/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.23/0.47 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.23/0.47 => ! [Xx: $i] :
% 0.23/0.47 ( ( in @ Xx @ R )
% 0.23/0.47 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(ex1E2_type,type,
% 0.23/0.47 ex1E2: $o ).
% 0.23/0.47
% 0.23/0.47 thf(ex1E2,definition,
% 0.23/0.47 ( ex1E2
% 0.23/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.23/0.47 ( ( ex1 @ A
% 0.23/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.23/0.47 => ! [Xx: $i] :
% 0.23/0.47 ( ( in @ Xx @ A )
% 0.23/0.47 => ! [Xy: $i] :
% 0.23/0.47 ( ( in @ Xy @ A )
% 0.23/0.47 => ( ( Xphi @ Xx )
% 0.23/0.47 => ( ( Xphi @ Xy )
% 0.23/0.47 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(funcGraphProp1_type,type,
% 0.23/0.47 funcGraphProp1: $o ).
% 0.23/0.47
% 0.23/0.47 thf(funcGraphProp1,definition,
% 0.23/0.47 ( funcGraphProp1
% 0.23/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.23/0.47 ( ( func @ A @ B @ Xf )
% 0.23/0.47 => ! [Xx: $i] :
% 0.23/0.47 ( ( in @ Xx @ A )
% 0.23/0.47 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(funcGraphProp3_type,type,
% 0.23/0.47 funcGraphProp3: $o ).
% 0.23/0.47
% 0.23/0.47 thf(funcGraphProp3,definition,
% 0.23/0.47 ( funcGraphProp3
% 0.23/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.23/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.23/0.47 => ! [Xx: $i] :
% 0.23/0.47 ( ( in @ Xx @ A )
% 0.23/0.47 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(funcGraphProp2_type,type,
% 0.23/0.47 funcGraphProp2: $o ).
% 0.23/0.47
% 0.23/0.47 thf(funcGraphProp2,definition,
% 0.23/0.47 ( funcGraphProp2
% 0.23/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.23/0.47 ( ( func @ A @ B @ Xf )
% 0.23/0.47 => ! [Xx: $i] :
% 0.23/0.47 ( ( in @ Xx @ A )
% 0.23/0.47 => ! [Xy: $i] :
% 0.23/0.47 ( ( in @ Xy @ B )
% 0.23/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.23/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.23/0.47 = Xy ) ) ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(funcextLem_type,type,
% 0.23/0.47 funcextLem: $o ).
% 0.23/0.47
% 0.23/0.47 thf(funcextLem,definition,
% 0.23/0.47 ( funcextLem
% 0.23/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.23/0.47 ( ( func @ A @ B @ Xf )
% 0.23/0.47 => ! [Xg: $i] :
% 0.23/0.47 ( ( func @ A @ B @ Xg )
% 0.23/0.47 => ( ! [Xx: $i] :
% 0.23/0.47 ( ( in @ Xx @ A )
% 0.23/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.23/0.47 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.23/0.47 => ! [Xx: $i] :
% 0.23/0.47 ( ( in @ Xx @ A )
% 0.23/0.47 => ! [Xy: $i] :
% 0.23/0.47 ( ( in @ Xy @ B )
% 0.23/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 0.23/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(funcGraphProp4_type,type,
% 0.23/0.47 funcGraphProp4: $o ).
% 0.23/0.47
% 0.23/0.47 thf(funcGraphProp4,definition,
% 0.23/0.47 ( funcGraphProp4
% 0.23/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.23/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.23/0.47 => ! [Xx: $i] :
% 0.23/0.47 ( ( in @ Xx @ A )
% 0.23/0.47 => ! [Xy: $i] :
% 0.23/0.47 ( ( in @ Xy @ B )
% 0.23/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.23/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.23/0.47 = Xy ) ) ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(subbreln_type,type,
% 0.23/0.47 subbreln: $o ).
% 0.23/0.47
% 0.23/0.47 thf(subbreln,definition,
% 0.23/0.47 ( subbreln
% 0.23/0.47 = ( ! [A: $i,B: $i,R: $i] :
% 0.23/0.47 ( ( breln @ A @ B @ R )
% 0.23/0.47 => ! [S: $i] :
% 0.23/0.47 ( ( breln @ A @ B @ S )
% 0.23/0.47 => ( ! [Xx: $i] :
% 0.23/0.47 ( ( in @ Xx @ A )
% 0.23/0.47 => ! [Xy: $i] :
% 0.23/0.47 ( ( in @ Xy @ B )
% 0.23/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.23/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.23/0.47 => ( subset @ R @ S ) ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(eqbreln_type,type,
% 0.23/0.47 eqbreln: $o ).
% 0.23/0.47
% 0.23/0.47 thf(eqbreln,definition,
% 0.23/0.47 ( eqbreln
% 0.23/0.47 = ( ! [A: $i,B: $i,R: $i] :
% 0.23/0.47 ( ( breln @ A @ B @ R )
% 0.23/0.47 => ! [S: $i] :
% 0.23/0.47 ( ( breln @ A @ B @ S )
% 0.23/0.47 => ( ! [Xx: $i] :
% 0.23/0.47 ( ( in @ Xx @ A )
% 0.23/0.47 => ! [Xy: $i] :
% 0.23/0.47 ( ( in @ Xy @ B )
% 0.23/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.23/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.23/0.47 => ( ! [Xx: $i] :
% 0.23/0.47 ( ( in @ Xx @ A )
% 0.23/0.47 => ! [Xy: $i] :
% 0.23/0.47 ( ( in @ Xy @ B )
% 0.23/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.23/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 0.23/0.47 => ( R = S ) ) ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(funcext_type,type,
% 0.23/0.47 funcext: $o ).
% 0.23/0.47
% 0.23/0.47 thf(funcext,definition,
% 0.23/0.47 ( funcext
% 0.23/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.23/0.47 ( ( func @ A @ B @ Xf )
% 0.23/0.47 => ! [Xg: $i] :
% 0.23/0.47 ( ( func @ A @ B @ Xg )
% 0.23/0.47 => ( ! [Xx: $i] :
% 0.23/0.47 ( ( in @ Xx @ A )
% 0.23/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.23/0.47 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.23/0.47 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(funcext2_type,type,
% 0.23/0.47 funcext2: $o ).
% 0.23/0.47
% 0.23/0.47 thf(funcext2,definition,
% 0.23/0.47 ( funcext2
% 0.23/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.23/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.23/0.47 => ! [Xg: $i] :
% 0.23/0.47 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 0.23/0.47 => ( ! [Xx: $i] :
% 0.23/0.47 ( ( in @ Xx @ A )
% 0.23/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.23/0.47 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.23/0.47 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(ap2apEq1_type,type,
% 0.23/0.47 ap2apEq1: $o ).
% 0.23/0.47
% 0.23/0.47 thf(ap2apEq1,definition,
% 0.23/0.47 ( ap2apEq1
% 0.23/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.23/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.23/0.47 => ! [Xx: $i] :
% 0.23/0.47 ( ( in @ Xx @ A )
% 0.23/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.23/0.47 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(ap2apEq2_type,type,
% 0.23/0.47 ap2apEq2: $o ).
% 0.23/0.47
% 0.23/0.47 thf(ap2apEq2,definition,
% 0.23/0.47 ( ap2apEq2
% 0.23/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.23/0.47 ( ( func @ A @ B @ Xf )
% 0.23/0.47 => ! [Xx: $i] :
% 0.23/0.47 ( ( in @ Xx @ A )
% 0.23/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.23/0.47 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(beta1_type,type,
% 0.23/0.47 beta1: $o ).
% 0.23/0.47
% 0.23/0.47 thf(beta1,definition,
% 0.23/0.47 ( beta1
% 0.23/0.47 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.23/0.47 ( ! [Xx: $i] :
% 0.23/0.47 ( ( in @ Xx @ A )
% 0.23/0.47 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.23/0.47 => ! [Xx: $i] :
% 0.23/0.47 ( ( in @ Xx @ A )
% 0.23/0.47 => ( ( ap @ A @ B
% 0.23/0.47 @ ( lam @ A @ B
% 0.23/0.47 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.23/0.47 @ Xx )
% 0.23/0.47 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(eta1_type,type,
% 0.23/0.47 eta1: $o ).
% 0.23/0.47
% 0.23/0.47 thf(eta1,definition,
% 0.23/0.47 ( eta1
% 0.23/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.23/0.47 ( ( func @ A @ B @ Xf )
% 0.23/0.47 => ( ( lam @ A @ B
% 0.23/0.47 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.23/0.47 = Xf ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(lam2lamEq_type,type,
% 0.23/0.47 lam2lamEq: $o ).
% 0.23/0.47
% 0.23/0.47 thf(lam2lamEq,definition,
% 0.23/0.47 ( lam2lamEq
% 0.23/0.47 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.23/0.47 ( ! [Xx: $i] :
% 0.23/0.47 ( ( in @ Xx @ A )
% 0.23/0.47 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.23/0.47 => ( ( lam @ A @ B
% 0.23/0.47 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.23/0.47 = ( lam @ A @ B
% 0.23/0.47 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(beta2_type,type,
% 0.23/0.47 beta2: $o ).
% 0.23/0.47
% 0.23/0.47 thf(beta2,definition,
% 0.23/0.47 ( beta2
% 0.23/0.47 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.23/0.47 ( ! [Xx: $i] :
% 0.23/0.47 ( ( in @ Xx @ A )
% 0.23/0.47 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.23/0.47 => ! [Xx: $i] :
% 0.23/0.47 ( ( in @ Xx @ A )
% 0.23/0.47 => ( ( ap @ A @ B
% 0.23/0.47 @ ( lam @ A @ B
% 0.23/0.47 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.23/0.47 @ Xx )
% 0.23/0.47 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(eta2_type,type,
% 0.23/0.47 eta2: $o ).
% 0.23/0.47
% 0.23/0.47 thf(eta2,definition,
% 0.23/0.47 ( eta2
% 0.23/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.23/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.23/0.47 => ( ( lam @ A @ B
% 0.23/0.47 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.23/0.47 = Xf ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(iffalseProp1_type,type,
% 0.23/0.47 iffalseProp1: $o ).
% 0.23/0.47
% 0.23/0.47 thf(iffalseProp1,definition,
% 0.23/0.47 ( iffalseProp1
% 0.23/0.47 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.23/0.47 ( ( in @ Xx @ A )
% 0.23/0.47 => ! [Xy: $i] :
% 0.23/0.47 ( ( in @ Xy @ A )
% 0.23/0.47 => ( ~ Xphi
% 0.23/0.47 => ( in @ Xy
% 0.23/0.47 @ ( dsetconstr @ A
% 0.23/0.47 @ ^ [Xz: $i] :
% 0.23/0.47 ( ( Xphi
% 0.23/0.47 & ( Xz = Xx ) )
% 0.23/0.47 | ( ~ Xphi
% 0.23/0.47 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(iffalseProp2_type,type,
% 0.23/0.47 iffalseProp2: $o ).
% 0.23/0.47
% 0.23/0.47 thf(iffalseProp2,definition,
% 0.23/0.47 ( iffalseProp2
% 0.23/0.47 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.23/0.47 ( ( in @ Xx @ A )
% 0.23/0.47 => ! [Xy: $i] :
% 0.23/0.47 ( ( in @ Xy @ A )
% 0.23/0.47 => ( ~ Xphi
% 0.23/0.47 => ( ( dsetconstr @ A
% 0.23/0.47 @ ^ [Xz: $i] :
% 0.23/0.47 ( ( Xphi
% 0.23/0.47 & ( Xz = Xx ) )
% 0.23/0.47 | ( ~ Xphi
% 0.23/0.47 & ( Xz = Xy ) ) ) )
% 0.23/0.47 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(iftrueProp1_type,type,
% 0.23/0.47 iftrueProp1: $o ).
% 0.23/0.47
% 0.23/0.47 thf(iftrueProp1,definition,
% 0.23/0.47 ( iftrueProp1
% 0.23/0.47 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.23/0.47 ( ( in @ Xx @ A )
% 0.23/0.47 => ! [Xy: $i] :
% 0.23/0.47 ( ( in @ Xy @ A )
% 0.23/0.47 => ( Xphi
% 0.23/0.47 => ( in @ Xx
% 0.23/0.47 @ ( dsetconstr @ A
% 0.23/0.47 @ ^ [Xz: $i] :
% 0.23/0.47 ( ( Xphi
% 0.23/0.47 & ( Xz = Xx ) )
% 0.23/0.47 | ( ~ Xphi
% 0.23/0.47 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(iftrueProp2_type,type,
% 0.23/0.47 iftrueProp2: $o ).
% 0.23/0.47
% 0.23/0.47 thf(iftrueProp2,definition,
% 0.23/0.47 ( iftrueProp2
% 0.23/0.47 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.23/0.47 ( ( in @ Xx @ A )
% 0.23/0.47 => ! [Xy: $i] :
% 0.23/0.47 ( ( in @ Xy @ A )
% 0.23/0.47 => ( Xphi
% 0.23/0.47 => ( ( dsetconstr @ A
% 0.23/0.47 @ ^ [Xz: $i] :
% 0.23/0.47 ( ( Xphi
% 0.23/0.47 & ( Xz = Xx ) )
% 0.23/0.47 | ( ~ Xphi
% 0.23/0.47 & ( Xz = Xy ) ) ) )
% 0.23/0.47 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(ifSingleton_type,type,
% 0.23/0.47 ifSingleton: $o ).
% 0.23/0.47
% 0.23/0.47 thf(ifSingleton,definition,
% 0.23/0.47 ( ifSingleton
% 0.23/0.47 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.23/0.47 ( ( in @ Xx @ A )
% 0.23/0.47 => ! [Xy: $i] :
% 0.23/0.47 ( ( in @ Xy @ A )
% 0.23/0.47 => ( singleton
% 0.23/0.47 @ ( dsetconstr @ A
% 0.23/0.47 @ ^ [Xz: $i] :
% 0.23/0.47 ( ( Xphi
% 0.23/0.47 & ( Xz = Xx ) )
% 0.23/0.47 | ( ~ Xphi
% 0.23/0.47 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(if_type,type,
% 0.23/0.47 if: $i > $o > $i > $i > $i ).
% 0.23/0.47
% 0.23/0.47 thf(ifp_type,type,
% 0.23/0.47 ifp: $o ).
% 0.23/0.47
% 0.23/0.47 thf(ifp,definition,
% 0.23/0.47 ( ifp
% 0.23/0.47 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.23/0.47 ( ( in @ Xx @ A )
% 0.23/0.47 => ! [Xy: $i] :
% 0.23/0.47 ( ( in @ Xy @ A )
% 0.23/0.47 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(theeq_type,type,
% 0.23/0.47 theeq: $o ).
% 0.23/0.47
% 0.23/0.47 thf(theeq,definition,
% 0.23/0.47 ( theeq
% 0.23/0.47 = ( ! [X: $i] :
% 0.23/0.47 ( ( singleton @ X )
% 0.23/0.47 => ! [Xx: $i] :
% 0.23/0.47 ( ( in @ Xx @ X )
% 0.23/0.47 => ( ( setunion @ X )
% 0.23/0.47 = Xx ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(iftrue_type,type,
% 0.23/0.47 iftrue: $o ).
% 0.23/0.47
% 0.23/0.47 thf(iftrue,definition,
% 0.23/0.47 ( iftrue
% 0.23/0.47 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.23/0.47 ( ( in @ Xx @ A )
% 0.23/0.47 => ! [Xy: $i] :
% 0.23/0.47 ( ( in @ Xy @ A )
% 0.23/0.47 => ( Xphi
% 0.23/0.47 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.23/0.47 = Xx ) ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(iffalse_type,type,
% 0.23/0.47 iffalse: $o ).
% 0.23/0.47
% 0.23/0.47 thf(iffalse,definition,
% 0.23/0.47 ( iffalse
% 0.23/0.47 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.23/0.47 ( ( in @ Xx @ A )
% 0.23/0.47 => ! [Xy: $i] :
% 0.23/0.47 ( ( in @ Xy @ A )
% 0.23/0.47 => ( ~ Xphi
% 0.23/0.47 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.23/0.47 = Xy ) ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(iftrueorfalse_type,type,
% 0.23/0.47 iftrueorfalse: $o ).
% 0.23/0.47
% 0.23/0.47 thf(iftrueorfalse,definition,
% 0.23/0.47 ( iftrueorfalse
% 0.23/0.47 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.23/0.47 ( ( in @ Xx @ A )
% 0.23/0.47 => ! [Xy: $i] :
% 0.23/0.47 ( ( in @ Xy @ A )
% 0.23/0.47 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(binintersectT_lem_type,type,
% 0.23/0.47 binintersectT_lem: $o ).
% 0.23/0.47
% 0.23/0.47 thf(binintersectT_lem,definition,
% 0.23/0.47 ( binintersectT_lem
% 0.23/0.47 = ( ! [A: $i,X: $i] :
% 0.23/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.23/0.47 => ! [Y: $i] :
% 0.23/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.23/0.47 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(binunionT_lem_type,type,
% 0.23/0.47 binunionT_lem: $o ).
% 0.23/0.47
% 0.23/0.47 thf(binunionT_lem,definition,
% 0.23/0.47 ( binunionT_lem
% 0.23/0.47 = ( ! [A: $i,X: $i] :
% 0.23/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.23/0.47 => ! [Y: $i] :
% 0.23/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.23/0.47 => ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(powersetT_lem_type,type,
% 0.23/0.47 powersetT_lem: $o ).
% 0.23/0.47
% 0.23/0.47 thf(powersetT_lem,definition,
% 0.23/0.47 ( powersetT_lem
% 0.23/0.47 = ( ! [A: $i,X: $i] :
% 0.23/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.23/0.47 => ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(setminusT_lem_type,type,
% 0.23/0.47 setminusT_lem: $o ).
% 0.23/0.47
% 0.23/0.47 thf(setminusT_lem,definition,
% 0.23/0.47 ( setminusT_lem
% 0.23/0.47 = ( ! [A: $i,X: $i] :
% 0.23/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.23/0.47 => ! [Y: $i] :
% 0.23/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.23/0.47 => ( in @ ( setminus @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(complementT_lem_type,type,
% 0.23/0.47 complementT_lem: $o ).
% 0.23/0.47
% 0.23/0.47 thf(complementT_lem,definition,
% 0.23/0.47 ( complementT_lem
% 0.23/0.47 = ( ! [A: $i,X: $i] :
% 0.23/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.23/0.47 => ( in @ ( setminus @ A @ X ) @ ( powerset @ A ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(setextT_type,type,
% 0.23/0.47 setextT: $o ).
% 0.23/0.47
% 0.23/0.47 thf(setextT,definition,
% 0.23/0.47 ( setextT
% 0.23/0.47 = ( ! [A: $i,X: $i] :
% 0.23/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.23/0.47 => ! [Y: $i] :
% 0.23/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.23/0.47 => ( ! [Xx: $i] :
% 0.23/0.47 ( ( in @ Xx @ A )
% 0.23/0.47 => ( ( in @ Xx @ X )
% 0.23/0.47 => ( in @ Xx @ Y ) ) )
% 0.23/0.47 => ( ! [Xx: $i] :
% 0.23/0.47 ( ( in @ Xx @ A )
% 0.23/0.47 => ( ( in @ Xx @ Y )
% 0.23/0.47 => ( in @ Xx @ X ) ) )
% 0.23/0.47 => ( X = Y ) ) ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(subsetTI_type,type,
% 0.23/0.47 subsetTI: $o ).
% 0.23/0.47
% 0.23/0.47 thf(subsetTI,definition,
% 0.23/0.47 ( subsetTI
% 0.23/0.47 = ( ! [A: $i,X: $i] :
% 0.23/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.23/0.47 => ! [Y: $i] :
% 0.23/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.23/0.47 => ( ! [Xx: $i] :
% 0.23/0.47 ( ( in @ Xx @ A )
% 0.23/0.47 => ( ( in @ Xx @ X )
% 0.23/0.47 => ( in @ Xx @ Y ) ) )
% 0.23/0.47 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(powersetTI1_type,type,
% 0.23/0.47 powersetTI1: $o ).
% 0.23/0.47
% 0.23/0.47 thf(powersetTI1,definition,
% 0.23/0.47 ( powersetTI1
% 0.23/0.47 = ( ! [A: $i,X: $i] :
% 0.23/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.23/0.47 => ! [Y: $i] :
% 0.23/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.23/0.47 => ( ! [Xx: $i] :
% 0.23/0.47 ( ( in @ Xx @ A )
% 0.23/0.47 => ( ( in @ Xx @ X )
% 0.23/0.47 => ( in @ Xx @ Y ) ) )
% 0.23/0.47 => ( in @ X @ ( powerset @ Y ) ) ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 thf(powersetTE1,conjecture,
% 0.23/0.47 ( setextAx
% 0.23/0.47 => ( emptysetAx
% 0.23/0.47 => ( setadjoinAx
% 0.23/0.47 => ( powersetAx
% 0.23/0.47 => ( setunionAx
% 0.23/0.47 => ( omega0Ax
% 0.23/0.47 => ( omegaSAx
% 0.23/0.47 => ( omegaIndAx
% 0.23/0.47 => ( replAx
% 0.23/0.47 => ( foundationAx
% 0.23/0.47 => ( wellorderingAx
% 0.23/0.47 => ( descrp
% 0.23/0.47 => ( dsetconstrI
% 0.23/0.47 => ( dsetconstrEL
% 0.23/0.47 => ( dsetconstrER
% 0.23/0.47 => ( exuE1
% 0.23/0.47 => ( prop2setE
% 0.23/0.47 => ( emptysetE
% 0.23/0.47 => ( emptysetimpfalse
% 0.23/0.47 => ( notinemptyset
% 0.23/0.47 => ( exuE3e
% 0.23/0.47 => ( setext
% 0.23/0.47 => ( emptyI
% 0.23/0.47 => ( noeltsimpempty
% 0.23/0.47 => ( setbeta
% 0.23/0.47 => ( nonemptyE1
% 0.23/0.47 => ( nonemptyI
% 0.23/0.47 => ( nonemptyI1
% 0.23/0.47 => ( setadjoinIL
% 0.23/0.47 => ( emptyinunitempty
% 0.23/0.47 => ( setadjoinIR
% 0.23/0.47 => ( setadjoinE
% 0.23/0.47 => ( setadjoinOr
% 0.23/0.47 => ( setoftrueEq
% 0.23/0.47 => ( powersetI
% 0.23/0.47 => ( emptyinPowerset
% 0.23/0.47 => ( emptyInPowerset
% 0.23/0.47 => ( powersetE
% 0.23/0.47 => ( setunionI
% 0.23/0.47 => ( setunionE
% 0.23/0.47 => ( subPowSU
% 0.23/0.47 => ( exuE2
% 0.23/0.47 => ( nonemptyImpWitness
% 0.23/0.47 => ( uniqinunit
% 0.23/0.47 => ( notinsingleton
% 0.23/0.47 => ( eqinunit
% 0.23/0.47 => ( singletonsswitch
% 0.23/0.47 => ( upairsetE
% 0.23/0.47 => ( upairsetIL
% 0.23/0.47 => ( upairsetIR
% 0.23/0.47 => ( emptyE1
% 0.23/0.47 => ( vacuousDall
% 0.23/0.47 => ( quantDeMorgan1
% 0.23/0.47 => ( quantDeMorgan2
% 0.23/0.47 => ( quantDeMorgan3
% 0.23/0.47 => ( quantDeMorgan4
% 0.23/0.47 => ( prop2setI
% 0.23/0.47 => ( prop2set2propI
% 0.23/0.47 => ( notdexE
% 0.23/0.47 => ( notdallE
% 0.23/0.47 => ( exuI1
% 0.23/0.47 => ( exuI3
% 0.23/0.47 => ( exuI2
% 0.23/0.47 => ( inCongP
% 0.23/0.47 => ( in__Cong
% 0.23/0.47 => ( exuE3u
% 0.23/0.47 => ( exu__Cong
% 0.23/0.47 => ( emptyset__Cong
% 0.23/0.47 => ( setadjoin__Cong
% 0.23/0.47 => ( powerset__Cong
% 0.23/0.47 => ( setunion__Cong
% 0.23/0.47 => ( omega__Cong
% 0.23/0.47 => ( exuEu
% 0.23/0.47 => ( descr__Cong
% 0.23/0.47 => ( dsetconstr__Cong
% 0.23/0.47 => ( subsetI1
% 0.23/0.47 => ( eqimpsubset2
% 0.23/0.47 => ( eqimpsubset1
% 0.23/0.47 => ( subsetI2
% 0.23/0.47 => ( emptysetsubset
% 0.23/0.47 => ( subsetE
% 0.23/0.47 => ( subsetE2
% 0.23/0.47 => ( notsubsetI
% 0.23/0.47 => ( notequalI1
% 0.23/0.47 => ( notequalI2
% 0.23/0.47 => ( subsetRefl
% 0.23/0.47 => ( subsetTrans
% 0.23/0.47 => ( setadjoinSub
% 0.23/0.47 => ( setadjoinSub2
% 0.23/0.47 => ( subset2powerset
% 0.23/0.47 => ( setextsub
% 0.23/0.47 => ( subsetemptysetimpeq
% 0.23/0.47 => ( powersetI1
% 0.23/0.47 => ( powersetE1
% 0.23/0.47 => ( inPowerset
% 0.23/0.47 => ( powersetsubset
% 0.23/0.47 => ( sepInPowerset
% 0.23/0.47 => ( sepSubset
% 0.23/0.47 => ( binunionIL
% 0.23/0.47 => ( upairset2IR
% 0.23/0.47 => ( binunionIR
% 0.23/0.47 => ( binunionEcases
% 0.23/0.47 => ( binunionE
% 0.23/0.47 => ( binunionLsub
% 0.23/0.47 => ( binunionRsub
% 0.23/0.47 => ( binintersectI
% 0.23/0.47 => ( binintersectSubset5
% 0.23/0.47 => ( binintersectEL
% 0.23/0.47 => ( binintersectLsub
% 0.23/0.47 => ( binintersectSubset2
% 0.23/0.47 => ( binintersectSubset3
% 0.23/0.47 => ( binintersectER
% 0.23/0.47 => ( disjointsetsI1
% 0.23/0.47 => ( binintersectRsub
% 0.23/0.47 => ( binintersectSubset4
% 0.23/0.47 => ( binintersectSubset1
% 0.23/0.47 => ( bs114d
% 0.23/0.47 => ( setminusI
% 0.23/0.47 => ( setminusEL
% 0.23/0.47 => ( setminusER
% 0.23/0.47 => ( setminusSubset2
% 0.23/0.47 => ( setminusERneg
% 0.23/0.47 => ( setminusELneg
% 0.23/0.47 => ( setminusILneg
% 0.23/0.47 => ( setminusIRneg
% 0.23/0.47 => ( setminusLsub
% 0.23/0.47 => ( setminusSubset1
% 0.23/0.47 => ( symdiffE
% 0.23/0.47 => ( symdiffI1
% 0.23/0.47 => ( symdiffI2
% 0.23/0.47 => ( symdiffIneg1
% 0.23/0.47 => ( symdiffIneg2
% 0.23/0.47 => ( secondinupair
% 0.23/0.47 => ( setukpairIL
% 0.23/0.47 => ( setukpairIR
% 0.23/0.47 => ( kpairiskpair
% 0.23/0.47 => ( kpairp
% 0.23/0.47 => ( singletonsubset
% 0.23/0.47 => ( singletoninpowerset
% 0.23/0.47 => ( singletoninpowunion
% 0.23/0.47 => ( upairset2E
% 0.23/0.47 => ( upairsubunion
% 0.23/0.47 => ( upairinpowunion
% 0.23/0.47 => ( ubforcartprodlem1
% 0.23/0.47 => ( ubforcartprodlem2
% 0.23/0.47 => ( ubforcartprodlem3
% 0.23/0.47 => ( cartprodpairin
% 0.23/0.47 => ( cartprodmempair1
% 0.23/0.47 => ( cartprodmempair
% 0.23/0.47 => ( setunionE2
% 0.23/0.47 => ( setunionsingleton1
% 0.23/0.47 => ( setunionsingleton2
% 0.23/0.47 => ( setunionsingleton
% 0.23/0.47 => ( singletonprop
% 0.23/0.47 => ( ex1E1
% 0.23/0.47 => ( ex1I
% 0.23/0.47 => ( ex1I2
% 0.23/0.47 => ( singletonsuniq
% 0.23/0.47 => ( setukpairinjL1
% 0.23/0.47 => ( kfstsingleton
% 0.23/0.47 => ( theprop
% 0.23/0.47 => ( kfstpairEq
% 0.23/0.47 => ( cartprodfstin
% 0.23/0.47 => ( setukpairinjL2
% 0.23/0.47 => ( setukpairinjL
% 0.23/0.47 => ( setukpairinjR11
% 0.23/0.47 => ( setukpairinjR12
% 0.23/0.47 => ( setukpairinjR1
% 0.23/0.47 => ( upairequniteq
% 0.23/0.47 => ( setukpairinjR2
% 0.23/0.47 => ( setukpairinjR
% 0.23/0.47 => ( ksndsingleton
% 0.23/0.47 => ( ksndpairEq
% 0.23/0.47 => ( kpairsurjEq
% 0.23/0.47 => ( cartprodsndin
% 0.23/0.47 => ( cartprodpairmemEL
% 0.23/0.47 => ( cartprodpairmemER
% 0.23/0.47 => ( cartprodmempaircEq
% 0.23/0.47 => ( cartprodfstpairEq
% 0.23/0.47 => ( cartprodsndpairEq
% 0.23/0.47 => ( cartprodpairsurjEq
% 0.23/0.47 => ( dpsetconstrI
% 0.23/0.47 => ( dpsetconstrSub
% 0.23/0.47 => ( setOfPairsIsBReln
% 0.23/0.47 => ( dpsetconstrERa
% 0.23/0.47 => ( dpsetconstrEL1
% 0.23/0.47 => ( dpsetconstrEL2
% 0.23/0.47 => ( dpsetconstrER
% 0.23/0.47 => ( funcImageSingleton
% 0.23/0.47 => ( apProp
% 0.23/0.47 => ( app
% 0.23/0.47 => ( infuncsetfunc
% 0.23/0.47 => ( ap2p
% 0.23/0.47 => ( funcinfuncset
% 0.23/0.47 => ( lamProp
% 0.23/0.47 => ( lamp
% 0.23/0.47 => ( lam2p
% 0.23/0.47 => ( brelnall1
% 0.23/0.47 => ( brelnall2
% 0.23/0.47 => ( ex1E2
% 0.23/0.47 => ( funcGraphProp1
% 0.23/0.47 => ( funcGraphProp3
% 0.23/0.47 => ( funcGraphProp2
% 0.23/0.47 => ( funcextLem
% 0.23/0.47 => ( funcGraphProp4
% 0.23/0.47 => ( subbreln
% 0.23/0.47 => ( eqbreln
% 0.23/0.47 => ( funcext
% 0.23/0.47 => ( funcext2
% 0.23/0.47 => ( ap2apEq1
% 0.23/0.47 => ( ap2apEq2
% 0.23/0.47 => ( beta1
% 0.23/0.47 => ( eta1
% 0.23/0.47 => ( lam2lamEq
% 0.23/0.47 => ( beta2
% 0.23/0.47 => ( eta2
% 0.23/0.47 => ( iffalseProp1
% 0.23/0.47 => ( iffalseProp2
% 0.23/0.47 => ( iftrueProp1
% 0.23/0.47 => ( iftrueProp2
% 0.23/0.47 => ( ifSingleton
% 0.23/0.47 => ( ifp
% 0.23/0.47 => ( theeq
% 0.23/0.47 => ( iftrue
% 0.23/0.47 => ( iffalse
% 0.23/0.47 => ( iftrueorfalse
% 0.23/0.47 => ( binintersectT_lem
% 0.23/0.47 => ( binunionT_lem
% 0.23/0.47 => ( powersetT_lem
% 0.23/0.47 => ( setminusT_lem
% 0.23/0.47 => ( complementT_lem
% 0.23/0.47 => ( setextT
% 0.23/0.47 => ( subsetTI
% 0.23/0.47 => ( powersetTI1
% 0.23/0.47 => ! [A: $i,X: $i] :
% 0.23/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.23/0.47 => ! [Y: $i] :
% 0.23/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.23/0.47 => ! [Xx: $i] :
% 0.23/0.47 ( ( in @ Xx @ A )
% 0.23/0.47 => ( ( in @ X @ ( powerset @ Y ) )
% 0.23/0.47 => ( ( in @ Xx @ X )
% 0.23/0.47 => ( in @ Xx @ Y ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.23/0.47
% 0.23/0.47 %------------------------------------------------------------------------------
% 0.23/0.47 ------------------------
% 2.76/1.58 ---- Embedded in THF ---
% 2.76/1.58 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.76/1.58 %%% Generated on Tue Feb 24 12:32:02 EST 2026
% 2.76/1.58 %%% using '$$dhol' embedding, version 1.3.0.
% 2.76/1.58 %%% Logic specification used:
% 2.76/1.58 %%% thf(spec, logic, $$dhol).
% 2.76/1.58
% 2.76/1.58 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.OYw9T6ZJNR/DTF2DTF12262.p
% See solution above
% 2.76/1.58 ------------------------
% 3.00/1.61 ---- Cleaned THF ---
% 3.00/1.61 thf(in_type,type,
% 3.00/1.61 in: $i > $i > $o ).
% 3.00/1.61
% 3.00/1.61 thf(exu_type,type,
% 3.00/1.61 exu: ( $i > $o ) > $o ).
% 3.00/1.61
% 3.00/1.61 thf(exu,definition,
% 3.00/1.61 ( exu
% 3.00/1.61 = ( ^ [Xphi: $i > $o] :
% 3.00/1.61 ? [Xx: $i] :
% 3.00/1.61 ( ( Xphi @ Xx )
% 3.00/1.61 & ! [Xy: $i] :
% 3.00/1.61 ( ( Xphi @ Xy )
% 3.00/1.61 => ( Xx = Xy ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(setextAx_type,type,
% 3.00/1.61 setextAx: $o ).
% 3.00/1.61
% 3.00/1.61 thf(setextAx,definition,
% 3.00/1.61 ( setextAx
% 3.00/1.61 = ( ! [A: $i,B: $i] :
% 3.00/1.61 ( ! [Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 <=> ( in @ Xx @ B ) )
% 3.00/1.61 => ( A = B ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(emptyset_type,type,
% 3.00/1.61 emptyset: $i ).
% 3.00/1.61
% 3.00/1.61 thf(emptysetAx_type,type,
% 3.00/1.61 emptysetAx: $o ).
% 3.00/1.61
% 3.00/1.61 thf(emptysetAx,definition,
% 3.00/1.61 ( emptysetAx
% 3.00/1.61 = ( ! [Xx: $i] :
% 3.00/1.61 ~ ( in @ Xx @ emptyset ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(setadjoin_type,type,
% 3.00/1.61 setadjoin: $i > $i > $i ).
% 3.00/1.61
% 3.00/1.61 thf(setadjoinAx_type,type,
% 3.00/1.61 setadjoinAx: $o ).
% 3.00/1.61
% 3.00/1.61 thf(setadjoinAx,definition,
% 3.00/1.61 ( setadjoinAx
% 3.00/1.61 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 3.00/1.61 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 3.00/1.61 <=> ( ( Xy = Xx )
% 3.00/1.61 | ( in @ Xy @ A ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(powerset_type,type,
% 3.00/1.61 powerset: $i > $i ).
% 3.00/1.61
% 3.00/1.61 thf(powersetAx_type,type,
% 3.00/1.61 powersetAx: $o ).
% 3.00/1.61
% 3.00/1.61 thf(powersetAx,definition,
% 3.00/1.61 ( powersetAx
% 3.00/1.61 = ( ! [A: $i,B: $i] :
% 3.00/1.61 ( ( in @ B @ ( powerset @ A ) )
% 3.00/1.61 <=> ! [Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ B )
% 3.00/1.61 => ( in @ Xx @ A ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(setunion_type,type,
% 3.00/1.61 setunion: $i > $i ).
% 3.00/1.61
% 3.00/1.61 thf(setunionAx_type,type,
% 3.00/1.61 setunionAx: $o ).
% 3.00/1.61
% 3.00/1.61 thf(setunionAx,definition,
% 3.00/1.61 ( setunionAx
% 3.00/1.61 = ( ! [A: $i,Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ ( setunion @ A ) )
% 3.00/1.61 <=> ? [B: $i] :
% 3.00/1.61 ( ( in @ Xx @ B )
% 3.00/1.61 & ( in @ B @ A ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(omega_type,type,
% 3.00/1.61 omega: $i ).
% 3.00/1.61
% 3.00/1.61 thf(omega0Ax_type,type,
% 3.00/1.61 omega0Ax: $o ).
% 3.00/1.61
% 3.00/1.61 thf(omega0Ax,definition,
% 3.00/1.61 ( omega0Ax
% 3.00/1.61 = ( in @ emptyset @ omega ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(omegaSAx_type,type,
% 3.00/1.61 omegaSAx: $o ).
% 3.00/1.61
% 3.00/1.61 thf(omegaSAx,definition,
% 3.00/1.61 ( omegaSAx
% 3.00/1.61 = ( ! [Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ omega )
% 3.00/1.61 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(omegaIndAx_type,type,
% 3.00/1.61 omegaIndAx: $o ).
% 3.00/1.61
% 3.00/1.61 thf(omegaIndAx,definition,
% 3.00/1.61 ( omegaIndAx
% 3.00/1.61 = ( ! [A: $i] :
% 3.00/1.61 ( ( ( in @ emptyset @ A )
% 3.00/1.61 & ! [Xx: $i] :
% 3.00/1.61 ( ( ( in @ Xx @ omega )
% 3.00/1.61 & ( in @ Xx @ A ) )
% 3.00/1.61 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 3.00/1.61 => ! [Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ omega )
% 3.00/1.61 => ( in @ Xx @ A ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(replAx_type,type,
% 3.00/1.61 replAx: $o ).
% 3.00/1.61
% 3.00/1.61 thf(replAx,definition,
% 3.00/1.61 ( replAx
% 3.00/1.61 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 3.00/1.61 ( ! [Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 => ( exu
% 3.00/1.61 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 3.00/1.61 => ? [B: $i] :
% 3.00/1.61 ! [Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ B )
% 3.00/1.61 <=> ? [Xy: $i] :
% 3.00/1.61 ( ( in @ Xy @ A )
% 3.00/1.61 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(foundationAx_type,type,
% 3.00/1.61 foundationAx: $o ).
% 3.00/1.61
% 3.00/1.61 thf(foundationAx,definition,
% 3.00/1.61 ( foundationAx
% 3.00/1.61 = ( ! [A: $i] :
% 3.00/1.61 ( ? [Xx: $i] : ( in @ Xx @ A )
% 3.00/1.61 => ? [B: $i] :
% 3.00/1.61 ( ( in @ B @ A )
% 3.00/1.61 & ~ ? [Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ B )
% 3.00/1.61 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(wellorderingAx_type,type,
% 3.00/1.61 wellorderingAx: $o ).
% 3.00/1.61
% 3.00/1.61 thf(wellorderingAx,definition,
% 3.00/1.61 ( wellorderingAx
% 3.00/1.61 = ( ! [A: $i] :
% 3.00/1.61 ? [B: $i] :
% 3.00/1.61 ( ! [C: $i] :
% 3.00/1.61 ( ( in @ C @ B )
% 3.00/1.61 => ! [Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ C )
% 3.00/1.61 => ( in @ Xx @ A ) ) )
% 3.00/1.61 & ! [Xx: $i,Xy: $i] :
% 3.00/1.61 ( ( ( in @ Xx @ A )
% 3.00/1.61 & ( in @ Xy @ A ) )
% 3.00/1.61 => ( ! [C: $i] :
% 3.00/1.61 ( ( in @ C @ B )
% 3.00/1.61 => ( ( in @ Xx @ C )
% 3.00/1.61 <=> ( in @ Xy @ C ) ) )
% 3.00/1.61 => ( Xx = Xy ) ) )
% 3.00/1.61 & ! [C: $i,D: $i] :
% 3.00/1.61 ( ( ( in @ C @ B )
% 3.00/1.61 & ( in @ D @ B ) )
% 3.00/1.61 => ( ! [Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ C )
% 3.00/1.61 => ( in @ Xx @ D ) )
% 3.00/1.61 | ! [Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ D )
% 3.00/1.61 => ( in @ Xx @ C ) ) ) )
% 3.00/1.61 & ! [C: $i] :
% 3.00/1.61 ( ( ! [Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ C )
% 3.00/1.61 => ( in @ Xx @ A ) )
% 3.00/1.61 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 3.00/1.61 => ? [D: $i,Xx: $i] :
% 3.00/1.61 ( ( in @ D @ B )
% 3.00/1.61 & ( in @ Xx @ C )
% 3.00/1.61 & ~ ? [Xy: $i] :
% 3.00/1.61 ( ( in @ Xy @ D )
% 3.00/1.61 & ( in @ Xy @ C ) )
% 3.00/1.61 & ! [E: $i] :
% 3.00/1.61 ( ( in @ E @ B )
% 3.00/1.61 => ( ! [Xy: $i] :
% 3.00/1.61 ( ( in @ Xy @ E )
% 3.00/1.61 => ( in @ Xy @ D ) )
% 3.00/1.61 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(descr_type,type,
% 3.00/1.61 descr: ( $i > $o ) > $i ).
% 3.00/1.61
% 3.00/1.61 thf(descrp_type,type,
% 3.00/1.61 descrp: $o ).
% 3.00/1.61
% 3.00/1.61 thf(descrp,definition,
% 3.00/1.61 ( descrp
% 3.00/1.61 = ( ! [Xphi: $i > $o] :
% 3.00/1.61 ( ( exu
% 3.00/1.61 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.00/1.61 => ( Xphi
% 3.00/1.61 @ ( descr
% 3.00/1.61 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(dsetconstr_type,type,
% 3.00/1.61 dsetconstr: $i > ( $i > $o ) > $i ).
% 3.00/1.61
% 3.00/1.61 thf(dsetconstrI_type,type,
% 3.00/1.61 dsetconstrI: $o ).
% 3.00/1.61
% 3.00/1.61 thf(dsetconstrI,definition,
% 3.00/1.61 ( dsetconstrI
% 3.00/1.61 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 => ( ( Xphi @ Xx )
% 3.00/1.61 => ( in @ Xx
% 3.00/1.61 @ ( dsetconstr @ A
% 3.00/1.61 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(dsetconstrEL_type,type,
% 3.00/1.61 dsetconstrEL: $o ).
% 3.00/1.61
% 3.00/1.61 thf(dsetconstrEL,definition,
% 3.00/1.61 ( dsetconstrEL
% 3.00/1.61 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.00/1.61 ( ( in @ Xx
% 3.00/1.61 @ ( dsetconstr @ A
% 3.00/1.61 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 3.00/1.61 => ( in @ Xx @ A ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(dsetconstrER_type,type,
% 3.00/1.61 dsetconstrER: $o ).
% 3.00/1.61
% 3.00/1.61 thf(dsetconstrER,definition,
% 3.00/1.61 ( dsetconstrER
% 3.00/1.61 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.00/1.61 ( ( in @ Xx
% 3.00/1.61 @ ( dsetconstr @ A
% 3.00/1.61 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 3.00/1.61 => ( Xphi @ Xx ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(exuE1_type,type,
% 3.00/1.61 exuE1: $o ).
% 3.00/1.61
% 3.00/1.61 thf(exuE1,definition,
% 3.00/1.61 ( exuE1
% 3.00/1.61 = ( ! [Xphi: $i > $o] :
% 3.00/1.61 ( ( exu
% 3.00/1.61 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.00/1.61 => ? [Xx: $i] :
% 3.00/1.61 ( ( Xphi @ Xx )
% 3.00/1.61 & ! [Xy: $i] :
% 3.00/1.61 ( ( Xphi @ Xy )
% 3.00/1.61 => ( Xx = Xy ) ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(prop2set_type,type,
% 3.00/1.61 prop2set: $o > $i ).
% 3.00/1.61
% 3.00/1.61 thf(prop2setE_type,type,
% 3.00/1.61 prop2setE: $o ).
% 3.00/1.61
% 3.00/1.61 thf(prop2setE,definition,
% 3.00/1.61 ( prop2setE
% 3.00/1.61 = ( ! [Xphi: $o,Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 3.00/1.61 => Xphi ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(emptysetE_type,type,
% 3.00/1.61 emptysetE: $o ).
% 3.00/1.61
% 3.00/1.61 thf(emptysetE,definition,
% 3.00/1.61 ( emptysetE
% 3.00/1.61 = ( ! [Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ emptyset )
% 3.00/1.61 => ! [Xphi: $o] : Xphi ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(emptysetimpfalse_type,type,
% 3.00/1.61 emptysetimpfalse: $o ).
% 3.00/1.61
% 3.00/1.61 thf(emptysetimpfalse,definition,
% 3.00/1.61 ( emptysetimpfalse
% 3.00/1.61 = ( ! [Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ emptyset )
% 3.00/1.61 => $false ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(notinemptyset_type,type,
% 3.00/1.61 notinemptyset: $o ).
% 3.00/1.61
% 3.00/1.61 thf(notinemptyset,definition,
% 3.00/1.61 ( notinemptyset
% 3.00/1.61 = ( ! [Xx: $i] :
% 3.00/1.61 ~ ( in @ Xx @ emptyset ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(exuE3e_type,type,
% 3.00/1.61 exuE3e: $o ).
% 3.00/1.61
% 3.00/1.61 thf(exuE3e,definition,
% 3.00/1.61 ( exuE3e
% 3.00/1.61 = ( ! [Xphi: $i > $o] :
% 3.00/1.61 ( ( exu
% 3.00/1.61 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.00/1.61 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(setext_type,type,
% 3.00/1.61 setext: $o ).
% 3.00/1.61
% 3.00/1.61 thf(setext,definition,
% 3.00/1.61 ( setext
% 3.00/1.61 = ( ! [A: $i,B: $i] :
% 3.00/1.61 ( ! [Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 => ( in @ Xx @ B ) )
% 3.00/1.61 => ( ! [Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ B )
% 3.00/1.61 => ( in @ Xx @ A ) )
% 3.00/1.61 => ( A = B ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(emptyI_type,type,
% 3.00/1.61 emptyI: $o ).
% 3.00/1.61
% 3.00/1.61 thf(emptyI,definition,
% 3.00/1.61 ( emptyI
% 3.00/1.61 = ( ! [A: $i] :
% 3.00/1.61 ( ! [Xx: $i] :
% 3.00/1.61 ~ ( in @ Xx @ A )
% 3.00/1.61 => ( A = emptyset ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(noeltsimpempty_type,type,
% 3.00/1.61 noeltsimpempty: $o ).
% 3.00/1.61
% 3.00/1.61 thf(noeltsimpempty,definition,
% 3.00/1.61 ( noeltsimpempty
% 3.00/1.61 = ( ! [A: $i] :
% 3.00/1.61 ( ! [Xx: $i] :
% 3.00/1.61 ~ ( in @ Xx @ A )
% 3.00/1.61 => ( A = emptyset ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(setbeta_type,type,
% 3.00/1.61 setbeta: $o ).
% 3.00/1.61
% 3.00/1.61 thf(setbeta,definition,
% 3.00/1.61 ( setbeta
% 3.00/1.61 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 => ( ( in @ Xx
% 3.00/1.61 @ ( dsetconstr @ A
% 3.00/1.61 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 3.00/1.61 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(nonempty_type,type,
% 3.00/1.61 nonempty: $i > $o ).
% 3.00/1.61
% 3.00/1.61 thf(nonempty,definition,
% 3.00/1.61 ( nonempty
% 3.00/1.61 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(nonemptyE1_type,type,
% 3.00/1.61 nonemptyE1: $o ).
% 3.00/1.61
% 3.00/1.61 thf(nonemptyE1,definition,
% 3.00/1.61 ( nonemptyE1
% 3.00/1.61 = ( ! [A: $i] :
% 3.00/1.61 ( ( nonempty @ A )
% 3.00/1.61 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(nonemptyI_type,type,
% 3.00/1.61 nonemptyI: $o ).
% 3.00/1.61
% 3.00/1.61 thf(nonemptyI,definition,
% 3.00/1.61 ( nonemptyI
% 3.00/1.61 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 => ( ( Xphi @ Xx )
% 3.00/1.61 => ( nonempty
% 3.00/1.61 @ ( dsetconstr @ A
% 3.00/1.61 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(nonemptyI1_type,type,
% 3.00/1.61 nonemptyI1: $o ).
% 3.00/1.61
% 3.00/1.61 thf(nonemptyI1,definition,
% 3.00/1.61 ( nonemptyI1
% 3.00/1.61 = ( ! [A: $i] :
% 3.00/1.61 ( ? [Xx: $i] : ( in @ Xx @ A )
% 3.00/1.61 => ( nonempty @ A ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(setadjoinIL_type,type,
% 3.00/1.61 setadjoinIL: $o ).
% 3.00/1.61
% 3.00/1.61 thf(setadjoinIL,definition,
% 3.00/1.61 ( setadjoinIL
% 3.00/1.61 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(emptyinunitempty_type,type,
% 3.00/1.61 emptyinunitempty: $o ).
% 3.00/1.61
% 3.00/1.61 thf(emptyinunitempty,definition,
% 3.00/1.61 ( emptyinunitempty
% 3.00/1.61 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(setadjoinIR_type,type,
% 3.00/1.61 setadjoinIR: $o ).
% 3.00/1.61
% 3.00/1.61 thf(setadjoinIR,definition,
% 3.00/1.61 ( setadjoinIR
% 3.00/1.61 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 3.00/1.61 ( ( in @ Xy @ A )
% 3.00/1.61 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(setadjoinE_type,type,
% 3.00/1.61 setadjoinE: $o ).
% 3.00/1.61
% 3.00/1.61 thf(setadjoinE,definition,
% 3.00/1.61 ( setadjoinE
% 3.00/1.61 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 3.00/1.61 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 3.00/1.61 => ! [Xphi: $o] :
% 3.00/1.61 ( ( ( Xy = Xx )
% 3.00/1.61 => Xphi )
% 3.00/1.61 => ( ( ( in @ Xy @ A )
% 3.00/1.61 => Xphi )
% 3.00/1.61 => Xphi ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(setadjoinOr_type,type,
% 3.00/1.61 setadjoinOr: $o ).
% 3.00/1.61
% 3.00/1.61 thf(setadjoinOr,definition,
% 3.00/1.61 ( setadjoinOr
% 3.00/1.61 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 3.00/1.61 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 3.00/1.61 => ( ( Xy = Xx )
% 3.00/1.61 | ( in @ Xy @ A ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(setoftrueEq_type,type,
% 3.00/1.61 setoftrueEq: $o ).
% 3.00/1.61
% 3.00/1.61 thf(setoftrueEq,definition,
% 3.00/1.61 ( setoftrueEq
% 3.00/1.61 = ( ! [A: $i] :
% 3.00/1.61 ( ( dsetconstr @ A
% 3.00/1.61 @ ^ [Xx: $i] : $true )
% 3.00/1.61 = A ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(powersetI_type,type,
% 3.00/1.61 powersetI: $o ).
% 3.00/1.61
% 3.00/1.61 thf(powersetI,definition,
% 3.00/1.61 ( powersetI
% 3.00/1.61 = ( ! [A: $i,B: $i] :
% 3.00/1.61 ( ! [Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ B )
% 3.00/1.61 => ( in @ Xx @ A ) )
% 3.00/1.61 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(emptyinPowerset_type,type,
% 3.00/1.61 emptyinPowerset: $o ).
% 3.00/1.61
% 3.00/1.61 thf(emptyinPowerset,definition,
% 3.00/1.61 ( emptyinPowerset
% 3.00/1.61 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(emptyInPowerset_type,type,
% 3.00/1.61 emptyInPowerset: $o ).
% 3.00/1.61
% 3.00/1.61 thf(emptyInPowerset,definition,
% 3.00/1.61 ( emptyInPowerset
% 3.00/1.61 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(powersetE_type,type,
% 3.00/1.61 powersetE: $o ).
% 3.00/1.61
% 3.00/1.61 thf(powersetE,definition,
% 3.00/1.61 ( powersetE
% 3.00/1.61 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.00/1.61 ( ( in @ B @ ( powerset @ A ) )
% 3.00/1.61 => ( ( in @ Xx @ B )
% 3.00/1.61 => ( in @ Xx @ A ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(setunionI_type,type,
% 3.00/1.61 setunionI: $o ).
% 3.00/1.61
% 3.00/1.61 thf(setunionI,definition,
% 3.00/1.61 ( setunionI
% 3.00/1.61 = ( ! [A: $i,Xx: $i,B: $i] :
% 3.00/1.61 ( ( in @ Xx @ B )
% 3.00/1.61 => ( ( in @ B @ A )
% 3.00/1.61 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(setunionE_type,type,
% 3.00/1.61 setunionE: $o ).
% 3.00/1.61
% 3.00/1.61 thf(setunionE,definition,
% 3.00/1.61 ( setunionE
% 3.00/1.61 = ( ! [A: $i,Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ ( setunion @ A ) )
% 3.00/1.61 => ! [Xphi: $o] :
% 3.00/1.61 ( ! [B: $i] :
% 3.00/1.61 ( ( in @ Xx @ B )
% 3.00/1.61 => ( ( in @ B @ A )
% 3.00/1.61 => Xphi ) )
% 3.00/1.61 => Xphi ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(subPowSU_type,type,
% 3.00/1.61 subPowSU: $o ).
% 3.00/1.61
% 3.00/1.61 thf(subPowSU,definition,
% 3.00/1.61 ( subPowSU
% 3.00/1.61 = ( ! [A: $i,Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(exuE2_type,type,
% 3.00/1.61 exuE2: $o ).
% 3.00/1.61
% 3.00/1.61 thf(exuE2,definition,
% 3.00/1.61 ( exuE2
% 3.00/1.61 = ( ! [Xphi: $i > $o] :
% 3.00/1.61 ( ( exu
% 3.00/1.61 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.00/1.61 => ? [Xx: $i] :
% 3.00/1.61 ! [Xy: $i] :
% 3.00/1.61 ( ( Xphi @ Xy )
% 3.00/1.61 <=> ( Xy = Xx ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(nonemptyImpWitness_type,type,
% 3.00/1.61 nonemptyImpWitness: $o ).
% 3.00/1.61
% 3.00/1.61 thf(nonemptyImpWitness,definition,
% 3.00/1.61 ( nonemptyImpWitness
% 3.00/1.61 = ( ! [A: $i] :
% 3.00/1.61 ( ( nonempty @ A )
% 3.00/1.61 => ? [Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 & $true ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(uniqinunit_type,type,
% 3.00/1.61 uniqinunit: $o ).
% 3.00/1.61
% 3.00/1.61 thf(uniqinunit,definition,
% 3.00/1.61 ( uniqinunit
% 3.00/1.61 = ( ! [Xx: $i,Xy: $i] :
% 3.00/1.61 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 3.00/1.61 => ( Xx = Xy ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(notinsingleton_type,type,
% 3.00/1.61 notinsingleton: $o ).
% 3.00/1.61
% 3.00/1.61 thf(notinsingleton,definition,
% 3.00/1.61 ( notinsingleton
% 3.00/1.61 = ( ! [Xx: $i,Xy: $i] :
% 3.00/1.61 ( ( Xx != Xy )
% 3.00/1.61 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(eqinunit_type,type,
% 3.00/1.61 eqinunit: $o ).
% 3.00/1.61
% 3.00/1.61 thf(eqinunit,definition,
% 3.00/1.61 ( eqinunit
% 3.00/1.61 = ( ! [Xx: $i,Xy: $i] :
% 3.00/1.61 ( ( Xx = Xy )
% 3.00/1.61 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(singletonsswitch_type,type,
% 3.00/1.61 singletonsswitch: $o ).
% 3.00/1.61
% 3.00/1.61 thf(singletonsswitch,definition,
% 3.00/1.61 ( singletonsswitch
% 3.00/1.61 = ( ! [Xx: $i,Xy: $i] :
% 3.00/1.61 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 3.00/1.61 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(upairsetE_type,type,
% 3.00/1.61 upairsetE: $o ).
% 3.00/1.61
% 3.00/1.61 thf(upairsetE,definition,
% 3.00/1.61 ( upairsetE
% 3.00/1.61 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 3.00/1.61 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 3.00/1.61 => ( ( Xz = Xx )
% 3.00/1.61 | ( Xz = Xy ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(upairsetIL_type,type,
% 3.00/1.61 upairsetIL: $o ).
% 3.00/1.61
% 3.00/1.61 thf(upairsetIL,definition,
% 3.00/1.61 ( upairsetIL
% 3.00/1.61 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(upairsetIR_type,type,
% 3.00/1.61 upairsetIR: $o ).
% 3.00/1.61
% 3.00/1.61 thf(upairsetIR,definition,
% 3.00/1.61 ( upairsetIR
% 3.00/1.61 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(emptyE1_type,type,
% 3.00/1.61 emptyE1: $o ).
% 3.00/1.61
% 3.00/1.61 thf(emptyE1,definition,
% 3.00/1.61 ( emptyE1
% 3.00/1.61 = ( ! [A: $i,Xphi: $i > $o] :
% 3.00/1.61 ( ? [Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 & ( Xphi @ Xx ) )
% 3.00/1.61 => ( ( ( dsetconstr @ A
% 3.00/1.61 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.00/1.61 = emptyset )
% 3.00/1.61 => $false ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(vacuousDall_type,type,
% 3.00/1.61 vacuousDall: $o ).
% 3.00/1.61
% 3.00/1.61 thf(vacuousDall,definition,
% 3.00/1.61 ( vacuousDall
% 3.00/1.61 = ( ! [Xphi: $i > $o,Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ emptyset )
% 3.00/1.61 => ( Xphi @ Xx ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(quantDeMorgan1_type,type,
% 3.00/1.61 quantDeMorgan1: $o ).
% 3.00/1.61
% 3.00/1.61 thf(quantDeMorgan1,definition,
% 3.00/1.61 ( quantDeMorgan1
% 3.00/1.61 = ( ! [A: $i,Xphi: $i > $o] :
% 3.00/1.61 ( ~ ! [Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 => ( Xphi @ Xx ) )
% 3.00/1.61 => ? [Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(quantDeMorgan2_type,type,
% 3.00/1.61 quantDeMorgan2: $o ).
% 3.00/1.61
% 3.00/1.61 thf(quantDeMorgan2,definition,
% 3.00/1.61 ( quantDeMorgan2
% 3.00/1.61 = ( ! [A: $i,Xphi: $i > $o] :
% 3.00/1.61 ( ! [Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 => ~ ( Xphi @ Xx ) )
% 3.00/1.61 => ~ ? [Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 & ( Xphi @ Xx ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(quantDeMorgan3_type,type,
% 3.00/1.61 quantDeMorgan3: $o ).
% 3.00/1.61
% 3.00/1.61 thf(quantDeMorgan3,definition,
% 3.00/1.61 ( quantDeMorgan3
% 3.00/1.61 = ( ! [A: $i,Xphi: $i > $o] :
% 3.00/1.61 ( ~ ? [Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 & ( Xphi @ Xx ) )
% 3.00/1.61 => ! [Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(quantDeMorgan4_type,type,
% 3.00/1.61 quantDeMorgan4: $o ).
% 3.00/1.61
% 3.00/1.61 thf(quantDeMorgan4,definition,
% 3.00/1.61 ( quantDeMorgan4
% 3.00/1.61 = ( ! [A: $i,Xphi: $i > $o] :
% 3.00/1.61 ( ? [Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 & ~ ( Xphi @ Xx ) )
% 3.00/1.61 => ~ ! [Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 => ( Xphi @ Xx ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(prop2setI_type,type,
% 3.00/1.61 prop2setI: $o ).
% 3.00/1.61
% 3.00/1.61 thf(prop2setI,definition,
% 3.00/1.61 ( prop2setI
% 3.00/1.61 = ( ! [Xphi: $o] :
% 3.00/1.61 ( Xphi
% 3.00/1.61 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(set2prop_type,type,
% 3.00/1.61 set2prop: $i > $o ).
% 3.00/1.61
% 3.00/1.61 thf(prop2set2propI_type,type,
% 3.00/1.61 prop2set2propI: $o ).
% 3.00/1.61
% 3.00/1.61 thf(prop2set2propI,definition,
% 3.00/1.61 ( prop2set2propI
% 3.00/1.61 = ( ! [Xphi: $o] :
% 3.00/1.61 ( Xphi
% 3.00/1.61 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(notdexE_type,type,
% 3.00/1.61 notdexE: $o ).
% 3.00/1.61
% 3.00/1.61 thf(notdexE,definition,
% 3.00/1.61 ( notdexE
% 3.00/1.61 = ( ! [A: $i,Xphi: $i > $o] :
% 3.00/1.61 ( ~ ? [Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 & ( Xphi @ Xx ) )
% 3.00/1.61 => ! [Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(notdallE_type,type,
% 3.00/1.61 notdallE: $o ).
% 3.00/1.61
% 3.00/1.61 thf(notdallE,definition,
% 3.00/1.61 ( notdallE
% 3.00/1.61 = ( ! [A: $i,Xphi: $i > $o] :
% 3.00/1.61 ( ~ ! [Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 => ( Xphi @ Xx ) )
% 3.00/1.61 => ? [Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(exuI1_type,type,
% 3.00/1.61 exuI1: $o ).
% 3.00/1.61
% 3.00/1.61 thf(exuI1,definition,
% 3.00/1.61 ( exuI1
% 3.00/1.61 = ( ! [Xphi: $i > $o] :
% 3.00/1.61 ( ? [Xx: $i] :
% 3.00/1.61 ( ( Xphi @ Xx )
% 3.00/1.61 & ! [Xy: $i] :
% 3.00/1.61 ( ( Xphi @ Xy )
% 3.00/1.61 => ( Xx = Xy ) ) )
% 3.00/1.61 => ( exu
% 3.00/1.61 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(exuI3_type,type,
% 3.00/1.61 exuI3: $o ).
% 3.00/1.61
% 3.00/1.61 thf(exuI3,definition,
% 3.00/1.61 ( exuI3
% 3.00/1.61 = ( ! [Xphi: $i > $o] :
% 3.00/1.61 ( ? [Xx: $i] : ( Xphi @ Xx )
% 3.00/1.61 => ( ! [Xx: $i,Xy: $i] :
% 3.00/1.61 ( ( Xphi @ Xx )
% 3.00/1.61 => ( ( Xphi @ Xy )
% 3.00/1.61 => ( Xx = Xy ) ) )
% 3.00/1.61 => ( exu
% 3.00/1.61 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(exuI2_type,type,
% 3.00/1.61 exuI2: $o ).
% 3.00/1.61
% 3.00/1.61 thf(exuI2,definition,
% 3.00/1.61 ( exuI2
% 3.00/1.61 = ( ! [Xphi: $i > $o] :
% 3.00/1.61 ( ? [Xx: $i] :
% 3.00/1.61 ! [Xy: $i] :
% 3.00/1.61 ( ( Xphi @ Xy )
% 3.00/1.61 <=> ( Xy = Xx ) )
% 3.00/1.61 => ( exu
% 3.00/1.61 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(inCongP_type,type,
% 3.00/1.61 inCongP: $o ).
% 3.00/1.61
% 3.00/1.61 thf(inCongP,definition,
% 3.00/1.61 ( inCongP
% 3.00/1.61 = ( ! [A: $i,B: $i] :
% 3.00/1.61 ( ( A = B )
% 3.00/1.61 => ! [Xx: $i,Xy: $i] :
% 3.00/1.61 ( ( Xx = Xy )
% 3.00/1.61 => ( ( in @ Xx @ A )
% 3.00/1.61 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(in__Cong_type,type,
% 3.00/1.61 in__Cong: $o ).
% 3.00/1.61
% 3.00/1.61 thf(in__Cong,definition,
% 3.00/1.61 ( in__Cong
% 3.00/1.61 = ( ! [A: $i,B: $i] :
% 3.00/1.61 ( ( A = B )
% 3.00/1.61 => ! [Xx: $i,Xy: $i] :
% 3.00/1.61 ( ( Xx = Xy )
% 3.00/1.61 => ( ( in @ Xx @ A )
% 3.00/1.61 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(exuE3u_type,type,
% 3.00/1.61 exuE3u: $o ).
% 3.00/1.61
% 3.00/1.61 thf(exuE3u,definition,
% 3.00/1.61 ( exuE3u
% 3.00/1.61 = ( ! [Xphi: $i > $o] :
% 3.00/1.61 ( ( exu
% 3.00/1.61 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.00/1.61 => ! [Xx: $i,Xy: $i] :
% 3.00/1.61 ( ( Xphi @ Xx )
% 3.00/1.61 => ( ( Xphi @ Xy )
% 3.00/1.61 => ( Xx = Xy ) ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(exu__Cong_type,type,
% 3.00/1.61 exu__Cong: $o ).
% 3.00/1.61
% 3.00/1.61 thf(exu__Cong,definition,
% 3.00/1.61 ( exu__Cong
% 3.00/1.61 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 3.00/1.61 ( ! [Xx: $i,Xy: $i] :
% 3.00/1.61 ( ( Xx = Xy )
% 3.00/1.61 => ( ( Xphi @ Xx )
% 3.00/1.61 <=> ( Xpsi @ Xy ) ) )
% 3.00/1.61 => ( ( exu
% 3.00/1.61 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.00/1.61 <=> ( exu
% 3.00/1.61 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(emptyset__Cong_type,type,
% 3.00/1.61 emptyset__Cong: $o ).
% 3.00/1.61
% 3.00/1.61 thf(emptyset__Cong,definition,
% 3.00/1.61 ( emptyset__Cong
% 3.00/1.61 = ( emptyset = emptyset ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(setadjoin__Cong_type,type,
% 3.00/1.61 setadjoin__Cong: $o ).
% 3.00/1.61
% 3.00/1.61 thf(setadjoin__Cong,definition,
% 3.00/1.61 ( setadjoin__Cong
% 3.00/1.61 = ( ! [Xx: $i,Xy: $i] :
% 3.00/1.61 ( ( Xx = Xy )
% 3.00/1.61 => ! [Xz: $i,Xu: $i] :
% 3.00/1.61 ( ( Xz = Xu )
% 3.00/1.61 => ( ( setadjoin @ Xx @ Xz )
% 3.00/1.61 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(powerset__Cong_type,type,
% 3.00/1.61 powerset__Cong: $o ).
% 3.00/1.61
% 3.00/1.61 thf(powerset__Cong,definition,
% 3.00/1.61 ( powerset__Cong
% 3.00/1.61 = ( ! [A: $i,B: $i] :
% 3.00/1.61 ( ( A = B )
% 3.00/1.61 => ( ( powerset @ A )
% 3.00/1.61 = ( powerset @ B ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(setunion__Cong_type,type,
% 3.00/1.61 setunion__Cong: $o ).
% 3.00/1.61
% 3.00/1.61 thf(setunion__Cong,definition,
% 3.00/1.61 ( setunion__Cong
% 3.00/1.61 = ( ! [A: $i,B: $i] :
% 3.00/1.61 ( ( A = B )
% 3.00/1.61 => ( ( setunion @ A )
% 3.00/1.61 = ( setunion @ B ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(omega__Cong_type,type,
% 3.00/1.61 omega__Cong: $o ).
% 3.00/1.61
% 3.00/1.61 thf(omega__Cong,definition,
% 3.00/1.61 ( omega__Cong
% 3.00/1.61 = ( omega = omega ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(exuEu_type,type,
% 3.00/1.61 exuEu: $o ).
% 3.00/1.61
% 3.00/1.61 thf(exuEu,definition,
% 3.00/1.61 ( exuEu
% 3.00/1.61 = ( ! [Xphi: $i > $o] :
% 3.00/1.61 ( ( exu
% 3.00/1.61 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.00/1.61 => ! [Xx: $i,Xy: $i] :
% 3.00/1.61 ( ( Xphi @ Xx )
% 3.00/1.61 => ( ( Xphi @ Xy )
% 3.00/1.61 => ( Xx = Xy ) ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(descr__Cong_type,type,
% 3.00/1.61 descr__Cong: $o ).
% 3.00/1.61
% 3.00/1.61 thf(descr__Cong,definition,
% 3.00/1.61 ( descr__Cong
% 3.00/1.61 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 3.00/1.61 ( ! [Xx: $i,Xy: $i] :
% 3.00/1.61 ( ( Xx = Xy )
% 3.00/1.61 => ( ( Xphi @ Xx )
% 3.00/1.61 <=> ( Xpsi @ Xy ) ) )
% 3.00/1.61 => ( ( exu
% 3.00/1.61 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.00/1.61 => ( ( exu
% 3.00/1.61 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 3.00/1.61 => ( ( descr
% 3.00/1.61 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.00/1.61 = ( descr
% 3.00/1.61 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(dsetconstr__Cong_type,type,
% 3.00/1.61 dsetconstr__Cong: $o ).
% 3.00/1.61
% 3.00/1.61 thf(dsetconstr__Cong,definition,
% 3.00/1.61 ( dsetconstr__Cong
% 3.00/1.61 = ( ! [A: $i,B: $i] :
% 3.00/1.61 ( ( A = B )
% 3.00/1.61 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 3.00/1.61 ( ! [Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 => ! [Xy: $i] :
% 3.00/1.61 ( ( in @ Xy @ B )
% 3.00/1.61 => ( ( Xx = Xy )
% 3.00/1.61 => ( ( Xphi @ Xx )
% 3.00/1.61 <=> ( Xpsi @ Xy ) ) ) ) )
% 3.00/1.61 => ( ( dsetconstr @ A
% 3.00/1.61 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.00/1.61 = ( dsetconstr @ B
% 3.00/1.61 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(subset_type,type,
% 3.00/1.61 subset: $i > $i > $o ).
% 3.00/1.61
% 3.00/1.61 thf(disjoint_type,type,
% 3.00/1.61 disjoint: $i > $i > $o ).
% 3.00/1.61
% 3.00/1.61 thf(setsmeet_type,type,
% 3.00/1.61 setsmeet: $i > $i > $o ).
% 3.00/1.61
% 3.00/1.61 thf(subsetI1_type,type,
% 3.00/1.61 subsetI1: $o ).
% 3.00/1.61
% 3.00/1.61 thf(subsetI1,definition,
% 3.00/1.61 ( subsetI1
% 3.00/1.61 = ( ! [A: $i,B: $i] :
% 3.00/1.61 ( ! [Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 => ( in @ Xx @ B ) )
% 3.00/1.61 => ( subset @ A @ B ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(eqimpsubset2_type,type,
% 3.00/1.61 eqimpsubset2: $o ).
% 3.00/1.61
% 3.00/1.61 thf(eqimpsubset2,definition,
% 3.00/1.61 ( eqimpsubset2
% 3.00/1.61 = ( ! [A: $i,B: $i] :
% 3.00/1.61 ( ( A = B )
% 3.00/1.61 => ( subset @ B @ A ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(eqimpsubset1_type,type,
% 3.00/1.61 eqimpsubset1: $o ).
% 3.00/1.61
% 3.00/1.61 thf(eqimpsubset1,definition,
% 3.00/1.61 ( eqimpsubset1
% 3.00/1.61 = ( ! [A: $i,B: $i] :
% 3.00/1.61 ( ( A = B )
% 3.00/1.61 => ( subset @ A @ B ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(subsetI2_type,type,
% 3.00/1.61 subsetI2: $o ).
% 3.00/1.61
% 3.00/1.61 thf(subsetI2,definition,
% 3.00/1.61 ( subsetI2
% 3.00/1.61 = ( ! [A: $i,B: $i] :
% 3.00/1.61 ( ! [Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 => ( in @ Xx @ B ) )
% 3.00/1.61 => ( subset @ A @ B ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(emptysetsubset_type,type,
% 3.00/1.61 emptysetsubset: $o ).
% 3.00/1.61
% 3.00/1.61 thf(emptysetsubset,definition,
% 3.00/1.61 ( emptysetsubset
% 3.00/1.61 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(subsetE_type,type,
% 3.00/1.61 subsetE: $o ).
% 3.00/1.61
% 3.00/1.61 thf(subsetE,definition,
% 3.00/1.61 ( subsetE
% 3.00/1.61 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.00/1.61 ( ( subset @ A @ B )
% 3.00/1.61 => ( ( in @ Xx @ A )
% 3.00/1.61 => ( in @ Xx @ B ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(subsetE2_type,type,
% 3.00/1.61 subsetE2: $o ).
% 3.00/1.61
% 3.00/1.61 thf(subsetE2,definition,
% 3.00/1.61 ( subsetE2
% 3.00/1.61 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.00/1.61 ( ( subset @ A @ B )
% 3.00/1.61 => ( ~ ( in @ Xx @ B )
% 3.00/1.61 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(notsubsetI_type,type,
% 3.00/1.61 notsubsetI: $o ).
% 3.00/1.61
% 3.00/1.61 thf(notsubsetI,definition,
% 3.00/1.61 ( notsubsetI
% 3.00/1.61 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 => ( ~ ( in @ Xx @ B )
% 3.00/1.61 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(notequalI1_type,type,
% 3.00/1.61 notequalI1: $o ).
% 3.00/1.61
% 3.00/1.61 thf(notequalI1,definition,
% 3.00/1.61 ( notequalI1
% 3.00/1.61 = ( ! [A: $i,B: $i] :
% 3.00/1.61 ( ~ ( subset @ A @ B )
% 3.00/1.61 => ( A != B ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(notequalI2_type,type,
% 3.00/1.61 notequalI2: $o ).
% 3.00/1.61
% 3.00/1.61 thf(notequalI2,definition,
% 3.00/1.61 ( notequalI2
% 3.00/1.61 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 => ( ~ ( in @ Xx @ B )
% 3.00/1.61 => ( A != B ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(subsetRefl_type,type,
% 3.00/1.61 subsetRefl: $o ).
% 3.00/1.61
% 3.00/1.61 thf(subsetRefl,definition,
% 3.00/1.61 ( subsetRefl
% 3.00/1.61 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(subsetTrans_type,type,
% 3.00/1.61 subsetTrans: $o ).
% 3.00/1.61
% 3.00/1.61 thf(subsetTrans,definition,
% 3.00/1.61 ( subsetTrans
% 3.00/1.61 = ( ! [A: $i,B: $i,C: $i] :
% 3.00/1.61 ( ( subset @ A @ B )
% 3.00/1.61 => ( ( subset @ B @ C )
% 3.00/1.61 => ( subset @ A @ C ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(setadjoinSub_type,type,
% 3.00/1.61 setadjoinSub: $o ).
% 3.00/1.61
% 3.00/1.61 thf(setadjoinSub,definition,
% 3.00/1.61 ( setadjoinSub
% 3.00/1.61 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(setadjoinSub2_type,type,
% 3.00/1.61 setadjoinSub2: $o ).
% 3.00/1.61
% 3.00/1.61 thf(setadjoinSub2,definition,
% 3.00/1.61 ( setadjoinSub2
% 3.00/1.61 = ( ! [A: $i,Xx: $i,B: $i] :
% 3.00/1.61 ( ( subset @ A @ B )
% 3.00/1.61 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(subset2powerset_type,type,
% 3.00/1.61 subset2powerset: $o ).
% 3.00/1.61
% 3.00/1.61 thf(subset2powerset,definition,
% 3.00/1.61 ( subset2powerset
% 3.00/1.61 = ( ! [A: $i,B: $i] :
% 3.00/1.61 ( ( subset @ A @ B )
% 3.00/1.61 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(setextsub_type,type,
% 3.00/1.61 setextsub: $o ).
% 3.00/1.61
% 3.00/1.61 thf(setextsub,definition,
% 3.00/1.61 ( setextsub
% 3.00/1.61 = ( ! [A: $i,B: $i] :
% 3.00/1.61 ( ( subset @ A @ B )
% 3.00/1.61 => ( ( subset @ B @ A )
% 3.00/1.61 => ( A = B ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(subsetemptysetimpeq_type,type,
% 3.00/1.61 subsetemptysetimpeq: $o ).
% 3.00/1.61
% 3.00/1.61 thf(subsetemptysetimpeq,definition,
% 3.00/1.61 ( subsetemptysetimpeq
% 3.00/1.61 = ( ! [A: $i] :
% 3.00/1.61 ( ( subset @ A @ emptyset )
% 3.00/1.61 => ( A = emptyset ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(powersetI1_type,type,
% 3.00/1.61 powersetI1: $o ).
% 3.00/1.61
% 3.00/1.61 thf(powersetI1,definition,
% 3.00/1.61 ( powersetI1
% 3.00/1.61 = ( ! [A: $i,B: $i] :
% 3.00/1.61 ( ( subset @ B @ A )
% 3.00/1.61 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(powersetE1_type,type,
% 3.00/1.61 powersetE1: $o ).
% 3.00/1.61
% 3.00/1.61 thf(powersetE1,definition,
% 3.00/1.61 ( powersetE1
% 3.00/1.61 = ( ! [A: $i,B: $i] :
% 3.00/1.61 ( ( in @ B @ ( powerset @ A ) )
% 3.00/1.61 => ( subset @ B @ A ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(inPowerset_type,type,
% 3.00/1.61 inPowerset: $o ).
% 3.00/1.61
% 3.00/1.61 thf(inPowerset,definition,
% 3.00/1.61 ( inPowerset
% 3.00/1.61 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(powersetsubset_type,type,
% 3.00/1.61 powersetsubset: $o ).
% 3.00/1.61
% 3.00/1.61 thf(powersetsubset,definition,
% 3.00/1.61 ( powersetsubset
% 3.00/1.61 = ( ! [A: $i,B: $i] :
% 3.00/1.61 ( ( subset @ A @ B )
% 3.00/1.61 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(sepInPowerset_type,type,
% 3.00/1.61 sepInPowerset: $o ).
% 3.00/1.61
% 3.00/1.61 thf(sepInPowerset,definition,
% 3.00/1.61 ( sepInPowerset
% 3.00/1.61 = ( ! [A: $i,Xphi: $i > $o] :
% 3.00/1.61 ( in
% 3.00/1.61 @ ( dsetconstr @ A
% 3.00/1.61 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.00/1.61 @ ( powerset @ A ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(sepSubset_type,type,
% 3.00/1.61 sepSubset: $o ).
% 3.00/1.61
% 3.00/1.61 thf(sepSubset,definition,
% 3.00/1.61 ( sepSubset
% 3.00/1.61 = ( ! [A: $i,Xphi: $i > $o] :
% 3.00/1.61 ( subset
% 3.00/1.61 @ ( dsetconstr @ A
% 3.00/1.61 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.00/1.61 @ A ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(binunion_type,type,
% 3.00/1.61 binunion: $i > $i > $i ).
% 3.00/1.61
% 3.00/1.61 thf(binunionIL_type,type,
% 3.00/1.61 binunionIL: $o ).
% 3.00/1.61
% 3.00/1.61 thf(binunionIL,definition,
% 3.00/1.61 ( binunionIL
% 3.00/1.61 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(upairset2IR_type,type,
% 3.00/1.61 upairset2IR: $o ).
% 3.00/1.61
% 3.00/1.61 thf(upairset2IR,definition,
% 3.00/1.61 ( upairset2IR
% 3.00/1.61 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(binunionIR_type,type,
% 3.00/1.61 binunionIR: $o ).
% 3.00/1.61
% 3.00/1.61 thf(binunionIR,definition,
% 3.00/1.61 ( binunionIR
% 3.00/1.61 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ B )
% 3.00/1.61 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(binunionEcases_type,type,
% 3.00/1.61 binunionEcases: $o ).
% 3.00/1.61
% 3.00/1.61 thf(binunionEcases,definition,
% 3.00/1.61 ( binunionEcases
% 3.00/1.61 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 3.00/1.61 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 3.00/1.61 => ( ( ( in @ Xx @ A )
% 3.00/1.61 => Xphi )
% 3.00/1.61 => ( ( ( in @ Xx @ B )
% 3.00/1.61 => Xphi )
% 3.00/1.61 => Xphi ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(binunionE_type,type,
% 3.00/1.61 binunionE: $o ).
% 3.00/1.61
% 3.00/1.61 thf(binunionE,definition,
% 3.00/1.61 ( binunionE
% 3.00/1.61 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 3.00/1.61 => ( ( in @ Xx @ A )
% 3.00/1.61 | ( in @ Xx @ B ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(binunionLsub_type,type,
% 3.00/1.61 binunionLsub: $o ).
% 3.00/1.61
% 3.00/1.61 thf(binunionLsub,definition,
% 3.00/1.61 ( binunionLsub
% 3.00/1.61 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(binunionRsub_type,type,
% 3.00/1.61 binunionRsub: $o ).
% 3.00/1.61
% 3.00/1.61 thf(binunionRsub,definition,
% 3.00/1.61 ( binunionRsub
% 3.00/1.61 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(binintersect_type,type,
% 3.00/1.61 binintersect: $i > $i > $i ).
% 3.00/1.61
% 3.00/1.61 thf(binintersectI_type,type,
% 3.00/1.61 binintersectI: $o ).
% 3.00/1.61
% 3.00/1.61 thf(binintersectI,definition,
% 3.00/1.61 ( binintersectI
% 3.00/1.61 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 => ( ( in @ Xx @ B )
% 3.00/1.61 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(binintersectSubset5_type,type,
% 3.00/1.61 binintersectSubset5: $o ).
% 3.00/1.61
% 3.00/1.61 thf(binintersectSubset5,definition,
% 3.00/1.61 ( binintersectSubset5
% 3.00/1.61 = ( ! [A: $i,B: $i,C: $i] :
% 3.00/1.61 ( ( subset @ C @ A )
% 3.00/1.61 => ( ( subset @ C @ B )
% 3.00/1.61 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(binintersectEL_type,type,
% 3.00/1.61 binintersectEL: $o ).
% 3.00/1.61
% 3.00/1.61 thf(binintersectEL,definition,
% 3.00/1.61 ( binintersectEL
% 3.00/1.61 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 3.00/1.61 => ( in @ Xx @ A ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(binintersectLsub_type,type,
% 3.00/1.61 binintersectLsub: $o ).
% 3.00/1.61
% 3.00/1.61 thf(binintersectLsub,definition,
% 3.00/1.61 ( binintersectLsub
% 3.00/1.61 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(binintersectSubset2_type,type,
% 3.00/1.61 binintersectSubset2: $o ).
% 3.00/1.61
% 3.00/1.61 thf(binintersectSubset2,definition,
% 3.00/1.61 ( binintersectSubset2
% 3.00/1.61 = ( ! [A: $i,B: $i] :
% 3.00/1.61 ( ( subset @ A @ B )
% 3.00/1.61 => ( ( binintersect @ A @ B )
% 3.00/1.61 = A ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(binintersectSubset3_type,type,
% 3.00/1.61 binintersectSubset3: $o ).
% 3.00/1.61
% 3.00/1.61 thf(binintersectSubset3,definition,
% 3.00/1.61 ( binintersectSubset3
% 3.00/1.61 = ( ! [A: $i,B: $i] :
% 3.00/1.61 ( ( ( binintersect @ A @ B )
% 3.00/1.61 = B )
% 3.00/1.61 => ( subset @ B @ A ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(binintersectER_type,type,
% 3.00/1.61 binintersectER: $o ).
% 3.00/1.61
% 3.00/1.61 thf(binintersectER,definition,
% 3.00/1.61 ( binintersectER
% 3.00/1.61 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 3.00/1.61 => ( in @ Xx @ B ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(disjointsetsI1_type,type,
% 3.00/1.61 disjointsetsI1: $o ).
% 3.00/1.61
% 3.00/1.61 thf(disjointsetsI1,definition,
% 3.00/1.61 ( disjointsetsI1
% 3.00/1.61 = ( ! [A: $i,B: $i] :
% 3.00/1.61 ( ~ ? [Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 & ( in @ Xx @ B ) )
% 3.00/1.61 => ( ( binintersect @ A @ B )
% 3.00/1.61 = emptyset ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(binintersectRsub_type,type,
% 3.00/1.61 binintersectRsub: $o ).
% 3.00/1.61
% 3.00/1.61 thf(binintersectRsub,definition,
% 3.00/1.61 ( binintersectRsub
% 3.00/1.61 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(binintersectSubset4_type,type,
% 3.00/1.61 binintersectSubset4: $o ).
% 3.00/1.61
% 3.00/1.61 thf(binintersectSubset4,definition,
% 3.00/1.61 ( binintersectSubset4
% 3.00/1.61 = ( ! [A: $i,B: $i] :
% 3.00/1.61 ( ( subset @ B @ A )
% 3.00/1.61 => ( ( binintersect @ A @ B )
% 3.00/1.61 = B ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(binintersectSubset1_type,type,
% 3.00/1.61 binintersectSubset1: $o ).
% 3.00/1.61
% 3.00/1.61 thf(binintersectSubset1,definition,
% 3.00/1.61 ( binintersectSubset1
% 3.00/1.61 = ( ! [A: $i,B: $i] :
% 3.00/1.61 ( ( ( binintersect @ A @ B )
% 3.00/1.61 = A )
% 3.00/1.61 => ( subset @ A @ B ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(bs114d_type,type,
% 3.00/1.61 bs114d: $o ).
% 3.00/1.61
% 3.00/1.61 thf(bs114d,definition,
% 3.00/1.61 ( bs114d
% 3.00/1.61 = ( ! [A: $i,B: $i,C: $i] :
% 3.00/1.61 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 3.00/1.61 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(regular_type,type,
% 3.00/1.61 regular: $i > $o ).
% 3.00/1.61
% 3.00/1.61 thf(setminus_type,type,
% 3.00/1.61 setminus: $i > $i > $i ).
% 3.00/1.61
% 3.00/1.61 thf(setminusI_type,type,
% 3.00/1.61 setminusI: $o ).
% 3.00/1.61
% 3.00/1.61 thf(setminusI,definition,
% 3.00/1.61 ( setminusI
% 3.00/1.61 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 => ( ~ ( in @ Xx @ B )
% 3.00/1.61 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(setminusEL_type,type,
% 3.00/1.61 setminusEL: $o ).
% 3.00/1.61
% 3.00/1.61 thf(setminusEL,definition,
% 3.00/1.61 ( setminusEL
% 3.00/1.61 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 3.00/1.61 => ( in @ Xx @ A ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(setminusER_type,type,
% 3.00/1.61 setminusER: $o ).
% 3.00/1.61
% 3.00/1.61 thf(setminusER,definition,
% 3.00/1.61 ( setminusER
% 3.00/1.61 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 3.00/1.61 => ~ ( in @ Xx @ B ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(setminusSubset2_type,type,
% 3.00/1.61 setminusSubset2: $o ).
% 3.00/1.61
% 3.00/1.61 thf(setminusSubset2,definition,
% 3.00/1.61 ( setminusSubset2
% 3.00/1.61 = ( ! [A: $i,B: $i] :
% 3.00/1.61 ( ( subset @ A @ B )
% 3.00/1.61 => ( ( setminus @ A @ B )
% 3.00/1.61 = emptyset ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(setminusERneg_type,type,
% 3.00/1.61 setminusERneg: $o ).
% 3.00/1.61
% 3.00/1.61 thf(setminusERneg,definition,
% 3.00/1.61 ( setminusERneg
% 3.00/1.61 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.00/1.61 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 3.00/1.61 => ( ( in @ Xx @ A )
% 3.00/1.61 => ( in @ Xx @ B ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(setminusELneg_type,type,
% 3.00/1.61 setminusELneg: $o ).
% 3.00/1.61
% 3.00/1.61 thf(setminusELneg,definition,
% 3.00/1.61 ( setminusELneg
% 3.00/1.61 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.00/1.61 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 3.00/1.61 => ( ~ ( in @ Xx @ B )
% 3.00/1.61 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(setminusILneg_type,type,
% 3.00/1.61 setminusILneg: $o ).
% 3.00/1.61
% 3.00/1.61 thf(setminusILneg,definition,
% 3.00/1.61 ( setminusILneg
% 3.00/1.61 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.00/1.61 ( ~ ( in @ Xx @ A )
% 3.00/1.61 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(setminusIRneg_type,type,
% 3.00/1.61 setminusIRneg: $o ).
% 3.00/1.61
% 3.00/1.61 thf(setminusIRneg,definition,
% 3.00/1.61 ( setminusIRneg
% 3.00/1.61 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ B )
% 3.00/1.61 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(setminusLsub_type,type,
% 3.00/1.61 setminusLsub: $o ).
% 3.00/1.61
% 3.00/1.61 thf(setminusLsub,definition,
% 3.00/1.61 ( setminusLsub
% 3.00/1.61 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(setminusSubset1_type,type,
% 3.00/1.61 setminusSubset1: $o ).
% 3.00/1.61
% 3.00/1.61 thf(setminusSubset1,definition,
% 3.00/1.61 ( setminusSubset1
% 3.00/1.61 = ( ! [A: $i,B: $i] :
% 3.00/1.61 ( ( ( setminus @ A @ B )
% 3.00/1.61 = emptyset )
% 3.00/1.61 => ( subset @ A @ B ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(symdiff_type,type,
% 3.00/1.61 symdiff: $i > $i > $i ).
% 3.00/1.61
% 3.00/1.61 thf(symdiffE_type,type,
% 3.00/1.61 symdiffE: $o ).
% 3.00/1.61
% 3.00/1.61 thf(symdiffE,definition,
% 3.00/1.61 ( symdiffE
% 3.00/1.61 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 3.00/1.61 => ! [Xphi: $o] :
% 3.00/1.61 ( ( ( in @ Xx @ A )
% 3.00/1.61 => ( ~ ( in @ Xx @ B )
% 3.00/1.61 => Xphi ) )
% 3.00/1.61 => ( ( ~ ( in @ Xx @ A )
% 3.00/1.61 => ( ( in @ Xx @ B )
% 3.00/1.61 => Xphi ) )
% 3.00/1.61 => Xphi ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(symdiffI1_type,type,
% 3.00/1.61 symdiffI1: $o ).
% 3.00/1.61
% 3.00/1.61 thf(symdiffI1,definition,
% 3.00/1.61 ( symdiffI1
% 3.00/1.61 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 => ( ~ ( in @ Xx @ B )
% 3.00/1.61 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(symdiffI2_type,type,
% 3.00/1.61 symdiffI2: $o ).
% 3.00/1.61
% 3.00/1.61 thf(symdiffI2,definition,
% 3.00/1.61 ( symdiffI2
% 3.00/1.61 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.00/1.61 ( ~ ( in @ Xx @ A )
% 3.00/1.61 => ( ( in @ Xx @ B )
% 3.00/1.61 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(symdiffIneg1_type,type,
% 3.00/1.61 symdiffIneg1: $o ).
% 3.00/1.61
% 3.00/1.61 thf(symdiffIneg1,definition,
% 3.00/1.61 ( symdiffIneg1
% 3.00/1.61 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 => ( ( in @ Xx @ B )
% 3.00/1.61 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(symdiffIneg2_type,type,
% 3.00/1.61 symdiffIneg2: $o ).
% 3.00/1.61
% 3.00/1.61 thf(symdiffIneg2,definition,
% 3.00/1.61 ( symdiffIneg2
% 3.00/1.61 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.00/1.61 ( ~ ( in @ Xx @ A )
% 3.00/1.61 => ( ~ ( in @ Xx @ B )
% 3.00/1.61 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(iskpair_type,type,
% 3.00/1.61 iskpair: $i > $o ).
% 3.00/1.61
% 3.00/1.61 thf(secondinupair_type,type,
% 3.00/1.61 secondinupair: $o ).
% 3.00/1.61
% 3.00/1.61 thf(secondinupair,definition,
% 3.00/1.61 ( secondinupair
% 3.00/1.61 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(setukpairIL_type,type,
% 3.00/1.61 setukpairIL: $o ).
% 3.00/1.61
% 3.00/1.61 thf(setukpairIL,definition,
% 3.00/1.61 ( setukpairIL
% 3.00/1.61 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(setukpairIR_type,type,
% 3.00/1.61 setukpairIR: $o ).
% 3.00/1.61
% 3.00/1.61 thf(setukpairIR,definition,
% 3.00/1.61 ( setukpairIR
% 3.00/1.61 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(kpairiskpair_type,type,
% 3.00/1.61 kpairiskpair: $o ).
% 3.00/1.61
% 3.00/1.61 thf(kpairiskpair,definition,
% 3.00/1.61 ( kpairiskpair
% 3.00/1.61 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(kpair_type,type,
% 3.00/1.61 kpair: $i > $i > $i ).
% 3.00/1.61
% 3.00/1.61 thf(kpairp_type,type,
% 3.00/1.61 kpairp: $o ).
% 3.00/1.61
% 3.00/1.61 thf(kpairp,definition,
% 3.00/1.61 ( kpairp
% 3.00/1.61 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(cartprod_type,type,
% 3.00/1.61 cartprod: $i > $i > $i ).
% 3.00/1.61
% 3.00/1.61 thf(singletonsubset_type,type,
% 3.00/1.61 singletonsubset: $o ).
% 3.00/1.61
% 3.00/1.61 thf(singletonsubset,definition,
% 3.00/1.61 ( singletonsubset
% 3.00/1.61 = ( ! [A: $i,Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(singletoninpowerset_type,type,
% 3.00/1.61 singletoninpowerset: $o ).
% 3.00/1.61
% 3.00/1.61 thf(singletoninpowerset,definition,
% 3.00/1.61 ( singletoninpowerset
% 3.00/1.61 = ( ! [A: $i,Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(singletoninpowunion_type,type,
% 3.00/1.61 singletoninpowunion: $o ).
% 3.00/1.61
% 3.00/1.61 thf(singletoninpowunion,definition,
% 3.00/1.61 ( singletoninpowunion
% 3.00/1.61 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(upairset2E_type,type,
% 3.00/1.61 upairset2E: $o ).
% 3.00/1.61
% 3.00/1.61 thf(upairset2E,definition,
% 3.00/1.61 ( upairset2E
% 3.00/1.61 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 3.00/1.61 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 3.00/1.61 => ( ( Xz = Xx )
% 3.00/1.61 | ( Xz = Xy ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(upairsubunion_type,type,
% 3.00/1.61 upairsubunion: $o ).
% 3.00/1.61
% 3.00/1.61 thf(upairsubunion,definition,
% 3.00/1.61 ( upairsubunion
% 3.00/1.61 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 => ! [Xy: $i] :
% 3.00/1.61 ( ( in @ Xy @ B )
% 3.00/1.61 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(upairinpowunion_type,type,
% 3.00/1.61 upairinpowunion: $o ).
% 3.00/1.61
% 3.00/1.61 thf(upairinpowunion,definition,
% 3.00/1.61 ( upairinpowunion
% 3.00/1.61 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 => ! [Xy: $i] :
% 3.00/1.61 ( ( in @ Xy @ B )
% 3.00/1.61 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(ubforcartprodlem1_type,type,
% 3.00/1.61 ubforcartprodlem1: $o ).
% 3.00/1.61
% 3.00/1.61 thf(ubforcartprodlem1,definition,
% 3.00/1.61 ( ubforcartprodlem1
% 3.00/1.61 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 => ! [Xy: $i] :
% 3.00/1.61 ( ( in @ Xy @ B )
% 3.00/1.61 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(ubforcartprodlem2_type,type,
% 3.00/1.61 ubforcartprodlem2: $o ).
% 3.00/1.61
% 3.00/1.61 thf(ubforcartprodlem2,definition,
% 3.00/1.61 ( ubforcartprodlem2
% 3.00/1.61 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 => ! [Xy: $i] :
% 3.00/1.61 ( ( in @ Xy @ B )
% 3.00/1.61 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(ubforcartprodlem3_type,type,
% 3.00/1.61 ubforcartprodlem3: $o ).
% 3.00/1.61
% 3.00/1.61 thf(ubforcartprodlem3,definition,
% 3.00/1.61 ( ubforcartprodlem3
% 3.00/1.61 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 => ! [Xy: $i] :
% 3.00/1.61 ( ( in @ Xy @ B )
% 3.00/1.61 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(cartprodpairin_type,type,
% 3.00/1.61 cartprodpairin: $o ).
% 3.00/1.61
% 3.00/1.61 thf(cartprodpairin,definition,
% 3.00/1.61 ( cartprodpairin
% 3.00/1.61 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 => ! [Xy: $i] :
% 3.00/1.61 ( ( in @ Xy @ B )
% 3.00/1.61 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(cartprodmempair1_type,type,
% 3.00/1.61 cartprodmempair1: $o ).
% 3.00/1.61
% 3.00/1.61 thf(cartprodmempair1,definition,
% 3.00/1.61 ( cartprodmempair1
% 3.00/1.61 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.00/1.61 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.00/1.61 => ? [Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 & ? [Xy: $i] :
% 3.00/1.61 ( ( in @ Xy @ B )
% 3.00/1.61 & ( Xu
% 3.00/1.61 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(cartprodmempair_type,type,
% 3.00/1.61 cartprodmempair: $o ).
% 3.00/1.61
% 3.00/1.61 thf(cartprodmempair,definition,
% 3.00/1.61 ( cartprodmempair
% 3.00/1.61 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.00/1.61 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.00/1.61 => ( iskpair @ Xu ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(setunionE2_type,type,
% 3.00/1.61 setunionE2: $o ).
% 3.00/1.61
% 3.00/1.61 thf(setunionE2,definition,
% 3.00/1.61 ( setunionE2
% 3.00/1.61 = ( ! [A: $i,Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ ( setunion @ A ) )
% 3.00/1.61 => ? [X: $i] :
% 3.00/1.61 ( ( in @ X @ A )
% 3.00/1.61 & ( in @ Xx @ X ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(setunionsingleton1_type,type,
% 3.00/1.61 setunionsingleton1: $o ).
% 3.00/1.61
% 3.00/1.61 thf(setunionsingleton1,definition,
% 3.00/1.61 ( setunionsingleton1
% 3.00/1.61 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(setunionsingleton2_type,type,
% 3.00/1.61 setunionsingleton2: $o ).
% 3.00/1.61
% 3.00/1.61 thf(setunionsingleton2,definition,
% 3.00/1.61 ( setunionsingleton2
% 3.00/1.61 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(setunionsingleton_type,type,
% 3.00/1.61 setunionsingleton: $o ).
% 3.00/1.61
% 3.00/1.61 thf(setunionsingleton,definition,
% 3.00/1.61 ( setunionsingleton
% 3.00/1.61 = ( ! [Xx: $i] :
% 3.00/1.61 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 3.00/1.61 = Xx ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(singleton_type,type,
% 3.00/1.61 singleton: $i > $o ).
% 3.00/1.61
% 3.00/1.61 thf(singletonprop_type,type,
% 3.00/1.61 singletonprop: $o ).
% 3.00/1.61
% 3.00/1.61 thf(singletonprop,definition,
% 3.00/1.61 ( singletonprop
% 3.00/1.61 = ( ! [A: $i,Xphi: $i > $o] :
% 3.00/1.61 ( ! [Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 => ! [Xy: $i] :
% 3.00/1.61 ( ( in @ Xy @ A )
% 3.00/1.61 => ( ( Xphi @ Xx )
% 3.00/1.61 => ( ( Xphi @ Xy )
% 3.00/1.61 => ( Xx = Xy ) ) ) ) )
% 3.00/1.61 => ( ? [Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 & ( Xphi @ Xx ) )
% 3.00/1.61 => ( singleton
% 3.00/1.61 @ ( dsetconstr @ A
% 3.00/1.61 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(ex1_type,type,
% 3.00/1.61 ex1: $i > ( $i > $o ) > $o ).
% 3.00/1.61
% 3.00/1.61 thf(ex1,definition,
% 3.00/1.61 ( ex1
% 3.00/1.61 = ( ^ [A: $i,Xphi: $i > $o] :
% 3.00/1.61 ( singleton
% 3.00/1.61 @ ( dsetconstr @ A
% 3.00/1.61 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(ex1E1_type,type,
% 3.00/1.61 ex1E1: $o ).
% 3.00/1.61
% 3.00/1.61 thf(ex1E1,definition,
% 3.00/1.61 ( ex1E1
% 3.00/1.61 = ( ! [A: $i,Xphi: $i > $o] :
% 3.00/1.61 ( ( ex1 @ A
% 3.00/1.61 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.00/1.61 => ? [Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 & ( Xphi @ Xx ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(ex1I_type,type,
% 3.00/1.61 ex1I: $o ).
% 3.00/1.61
% 3.00/1.61 thf(ex1I,definition,
% 3.00/1.61 ( ex1I
% 3.00/1.61 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 => ( ( Xphi @ Xx )
% 3.00/1.61 => ( ! [Xy: $i] :
% 3.00/1.61 ( ( in @ Xy @ A )
% 3.00/1.61 => ( ( Xphi @ Xy )
% 3.00/1.61 => ( Xy = Xx ) ) )
% 3.00/1.61 => ( ex1 @ A
% 3.00/1.61 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(ex1I2_type,type,
% 3.00/1.61 ex1I2: $o ).
% 3.00/1.61
% 3.00/1.61 thf(ex1I2,definition,
% 3.00/1.61 ( ex1I2
% 3.00/1.61 = ( ! [A: $i,Xphi: $i > $o] :
% 3.00/1.61 ( ! [Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 => ! [Xy: $i] :
% 3.00/1.61 ( ( in @ Xy @ A )
% 3.00/1.61 => ( ( Xphi @ Xx )
% 3.00/1.61 => ( ( Xphi @ Xy )
% 3.00/1.61 => ( Xx = Xy ) ) ) ) )
% 3.00/1.61 => ( ? [Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 & ( Xphi @ Xx ) )
% 3.00/1.61 => ( ex1 @ A
% 3.00/1.61 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(singletonsuniq_type,type,
% 3.00/1.61 singletonsuniq: $o ).
% 3.00/1.61
% 3.00/1.61 thf(singletonsuniq,definition,
% 3.00/1.61 ( singletonsuniq
% 3.00/1.61 = ( ! [Xx: $i,Xy: $i] :
% 3.00/1.61 ( ( ( setadjoin @ Xx @ emptyset )
% 3.00/1.61 = ( setadjoin @ Xy @ emptyset ) )
% 3.00/1.61 => ( Xx = Xy ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(atmost1p_type,type,
% 3.00/1.61 atmost1p: $i > $o ).
% 3.00/1.61
% 3.00/1.61 thf(atleast2p_type,type,
% 3.00/1.61 atleast2p: $i > $o ).
% 3.00/1.61
% 3.00/1.61 thf(atmost2p_type,type,
% 3.00/1.61 atmost2p: $i > $o ).
% 3.00/1.61
% 3.00/1.61 thf(upairsetp_type,type,
% 3.00/1.61 upairsetp: $i > $o ).
% 3.00/1.61
% 3.00/1.61 thf(setukpairinjL1_type,type,
% 3.00/1.61 setukpairinjL1: $o ).
% 3.00/1.61
% 3.00/1.61 thf(setukpairinjL1,definition,
% 3.00/1.61 ( setukpairinjL1
% 3.00/1.61 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 3.00/1.61 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 3.00/1.61 => ( Xx = Xz ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(kfstsingleton_type,type,
% 3.00/1.61 kfstsingleton: $o ).
% 3.00/1.61
% 3.00/1.61 thf(kfstsingleton,definition,
% 3.00/1.61 ( kfstsingleton
% 3.00/1.61 = ( ! [Xu: $i] :
% 3.00/1.61 ( ( iskpair @ Xu )
% 3.00/1.61 => ( singleton
% 3.00/1.61 @ ( dsetconstr @ ( setunion @ Xu )
% 3.00/1.61 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(theprop_type,type,
% 3.00/1.61 theprop: $o ).
% 3.00/1.61
% 3.00/1.61 thf(theprop,definition,
% 3.00/1.61 ( theprop
% 3.00/1.61 = ( ! [X: $i] :
% 3.00/1.61 ( ( singleton @ X )
% 3.00/1.61 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(kfst_type,type,
% 3.00/1.61 kfst: $i > $i ).
% 3.00/1.61
% 3.00/1.61 thf(kfstpairEq_type,type,
% 3.00/1.61 kfstpairEq: $o ).
% 3.00/1.61
% 3.00/1.61 thf(kfstpairEq,definition,
% 3.00/1.61 ( kfstpairEq
% 3.00/1.61 = ( ! [Xx: $i,Xy: $i] :
% 3.00/1.61 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 3.00/1.61 = Xx ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(cartprodfstin_type,type,
% 3.00/1.61 cartprodfstin: $o ).
% 3.00/1.61
% 3.00/1.61 thf(cartprodfstin,definition,
% 3.00/1.61 ( cartprodfstin
% 3.00/1.61 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.00/1.61 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.00/1.61 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(setukpairinjL2_type,type,
% 3.00/1.61 setukpairinjL2: $o ).
% 3.00/1.61
% 3.00/1.61 thf(setukpairinjL2,definition,
% 3.00/1.61 ( setukpairinjL2
% 3.00/1.61 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.00/1.61 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 3.00/1.61 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 3.00/1.61 => ( Xx = Xz ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(setukpairinjL_type,type,
% 3.00/1.61 setukpairinjL: $o ).
% 3.00/1.61
% 3.00/1.61 thf(setukpairinjL,definition,
% 3.00/1.61 ( setukpairinjL
% 3.00/1.61 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.00/1.61 ( ( ( kpair @ Xx @ Xy )
% 3.00/1.61 = ( kpair @ Xz @ Xu ) )
% 3.00/1.61 => ( Xx = Xz ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(setukpairinjR11_type,type,
% 3.00/1.61 setukpairinjR11: $o ).
% 3.00/1.61
% 3.00/1.61 thf(setukpairinjR11,definition,
% 3.00/1.61 ( setukpairinjR11
% 3.00/1.61 = ( ! [Xx: $i,Xy: $i] :
% 3.00/1.61 ( ( Xx = Xy )
% 3.00/1.61 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 3.00/1.61 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(setukpairinjR12_type,type,
% 3.00/1.61 setukpairinjR12: $o ).
% 3.00/1.61
% 3.00/1.61 thf(setukpairinjR12,definition,
% 3.00/1.61 ( setukpairinjR12
% 3.00/1.61 = ( ! [Xx: $i,Xy: $i] :
% 3.00/1.61 ( ( Xx = Xy )
% 3.00/1.61 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 3.00/1.61 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(setukpairinjR1_type,type,
% 3.00/1.61 setukpairinjR1: $o ).
% 3.00/1.61
% 3.00/1.61 thf(setukpairinjR1,definition,
% 3.00/1.61 ( setukpairinjR1
% 3.00/1.61 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.00/1.61 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 3.00/1.61 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 3.00/1.61 => ( ( Xz = Xu )
% 3.00/1.61 => ( Xy = Xu ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(upairequniteq_type,type,
% 3.00/1.61 upairequniteq: $o ).
% 3.00/1.61
% 3.00/1.61 thf(upairequniteq,definition,
% 3.00/1.61 ( upairequniteq
% 3.00/1.61 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 3.00/1.61 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 3.00/1.61 = ( setadjoin @ Xz @ emptyset ) )
% 3.00/1.61 => ( Xx = Xy ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(setukpairinjR2_type,type,
% 3.00/1.61 setukpairinjR2: $o ).
% 3.00/1.61
% 3.00/1.61 thf(setukpairinjR2,definition,
% 3.00/1.61 ( setukpairinjR2
% 3.00/1.61 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.00/1.61 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 3.00/1.61 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 3.00/1.61 => ( Xy = Xu ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(setukpairinjR_type,type,
% 3.00/1.61 setukpairinjR: $o ).
% 3.00/1.61
% 3.00/1.61 thf(setukpairinjR,definition,
% 3.00/1.61 ( setukpairinjR
% 3.00/1.61 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.00/1.61 ( ( ( kpair @ Xx @ Xy )
% 3.00/1.61 = ( kpair @ Xz @ Xu ) )
% 3.00/1.61 => ( Xy = Xu ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(ksndsingleton_type,type,
% 3.00/1.61 ksndsingleton: $o ).
% 3.00/1.61
% 3.00/1.61 thf(ksndsingleton,definition,
% 3.00/1.61 ( ksndsingleton
% 3.00/1.61 = ( ! [Xu: $i] :
% 3.00/1.61 ( ( iskpair @ Xu )
% 3.00/1.61 => ( singleton
% 3.00/1.61 @ ( dsetconstr @ ( setunion @ Xu )
% 3.00/1.61 @ ^ [Xx: $i] :
% 3.00/1.61 ( Xu
% 3.00/1.61 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(ksnd_type,type,
% 3.00/1.61 ksnd: $i > $i ).
% 3.00/1.61
% 3.00/1.61 thf(ksndpairEq_type,type,
% 3.00/1.61 ksndpairEq: $o ).
% 3.00/1.61
% 3.00/1.61 thf(ksndpairEq,definition,
% 3.00/1.61 ( ksndpairEq
% 3.00/1.61 = ( ! [Xx: $i,Xy: $i] :
% 3.00/1.61 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 3.00/1.61 = Xy ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(kpairsurjEq_type,type,
% 3.00/1.61 kpairsurjEq: $o ).
% 3.00/1.61
% 3.00/1.61 thf(kpairsurjEq,definition,
% 3.00/1.61 ( kpairsurjEq
% 3.00/1.61 = ( ! [Xu: $i] :
% 3.00/1.61 ( ( iskpair @ Xu )
% 3.00/1.61 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 3.00/1.61 = Xu ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(cartprodsndin_type,type,
% 3.00/1.61 cartprodsndin: $o ).
% 3.00/1.61
% 3.00/1.61 thf(cartprodsndin,definition,
% 3.00/1.61 ( cartprodsndin
% 3.00/1.61 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.00/1.61 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.00/1.61 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(cartprodpairmemEL_type,type,
% 3.00/1.61 cartprodpairmemEL: $o ).
% 3.00/1.61
% 3.00/1.61 thf(cartprodpairmemEL,definition,
% 3.00/1.61 ( cartprodpairmemEL
% 3.00/1.61 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 3.00/1.61 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 3.00/1.61 => ( in @ Xx @ A ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(cartprodpairmemER_type,type,
% 3.00/1.61 cartprodpairmemER: $o ).
% 3.00/1.61
% 3.00/1.61 thf(cartprodpairmemER,definition,
% 3.00/1.61 ( cartprodpairmemER
% 3.00/1.61 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 3.00/1.61 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 3.00/1.61 => ( in @ Xy @ B ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(cartprodmempaircEq_type,type,
% 3.00/1.61 cartprodmempaircEq: $o ).
% 3.00/1.61
% 3.00/1.61 thf(cartprodmempaircEq,definition,
% 3.00/1.61 ( cartprodmempaircEq
% 3.00/1.61 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 => ! [Xy: $i] :
% 3.00/1.61 ( ( in @ Xy @ B )
% 3.00/1.61 => ( ( kpair @ Xx @ Xy )
% 3.00/1.61 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(cartprodfstpairEq_type,type,
% 3.00/1.61 cartprodfstpairEq: $o ).
% 3.00/1.61
% 3.00/1.61 thf(cartprodfstpairEq,definition,
% 3.00/1.61 ( cartprodfstpairEq
% 3.00/1.61 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 => ! [Xy: $i] :
% 3.00/1.61 ( ( in @ Xy @ B )
% 3.00/1.61 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 3.00/1.61 = Xx ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(cartprodsndpairEq_type,type,
% 3.00/1.61 cartprodsndpairEq: $o ).
% 3.00/1.61
% 3.00/1.61 thf(cartprodsndpairEq,definition,
% 3.00/1.61 ( cartprodsndpairEq
% 3.00/1.61 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 => ! [Xy: $i] :
% 3.00/1.61 ( ( in @ Xy @ B )
% 3.00/1.61 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 3.00/1.61 = Xy ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(cartprodpairsurjEq_type,type,
% 3.00/1.61 cartprodpairsurjEq: $o ).
% 3.00/1.61
% 3.00/1.61 thf(cartprodpairsurjEq,definition,
% 3.00/1.61 ( cartprodpairsurjEq
% 3.00/1.61 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.00/1.61 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.00/1.61 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 3.00/1.61 = Xu ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(breln_type,type,
% 3.00/1.61 breln: $i > $i > $i > $o ).
% 3.00/1.61
% 3.00/1.61 thf(breln,definition,
% 3.00/1.61 ( breln
% 3.00/1.61 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(dpsetconstr_type,type,
% 3.00/1.61 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 3.00/1.61
% 3.00/1.61 thf(dpsetconstrI_type,type,
% 3.00/1.61 dpsetconstrI: $o ).
% 3.00/1.61
% 3.00/1.61 thf(dpsetconstrI,definition,
% 3.00/1.61 ( dpsetconstrI
% 3.00/1.61 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 => ! [Xy: $i] :
% 3.00/1.61 ( ( in @ Xy @ B )
% 3.00/1.61 => ( ( Xphi @ Xx @ Xy )
% 3.00/1.61 => ( in @ ( kpair @ Xx @ Xy )
% 3.00/1.61 @ ( dpsetconstr @ A @ B
% 3.00/1.61 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(dpsetconstrSub_type,type,
% 3.00/1.61 dpsetconstrSub: $o ).
% 3.00/1.61
% 3.00/1.61 thf(dpsetconstrSub,definition,
% 3.00/1.61 ( dpsetconstrSub
% 3.00/1.61 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 3.00/1.61 ( subset
% 3.00/1.61 @ ( dpsetconstr @ A @ B
% 3.00/1.61 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 3.00/1.61 @ ( cartprod @ A @ B ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(setOfPairsIsBReln_type,type,
% 3.00/1.61 setOfPairsIsBReln: $o ).
% 3.00/1.61
% 3.00/1.61 thf(setOfPairsIsBReln,definition,
% 3.00/1.61 ( setOfPairsIsBReln
% 3.00/1.61 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 3.00/1.61 ( breln @ A @ B
% 3.00/1.61 @ ( dpsetconstr @ A @ B
% 3.00/1.61 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(dpsetconstrERa_type,type,
% 3.00/1.61 dpsetconstrERa: $o ).
% 3.00/1.61
% 3.00/1.61 thf(dpsetconstrERa,definition,
% 3.00/1.61 ( dpsetconstrERa
% 3.00/1.61 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 => ! [Xy: $i] :
% 3.00/1.61 ( ( in @ Xy @ B )
% 3.00/1.61 => ( ( in @ ( kpair @ Xx @ Xy )
% 3.00/1.61 @ ( dpsetconstr @ A @ B
% 3.00/1.61 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 3.00/1.61 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(dpsetconstrEL1_type,type,
% 3.00/1.61 dpsetconstrEL1: $o ).
% 3.00/1.61
% 3.00/1.61 thf(dpsetconstrEL1,definition,
% 3.00/1.61 ( dpsetconstrEL1
% 3.00/1.61 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 3.00/1.61 ( ( in @ ( kpair @ Xx @ Xy )
% 3.00/1.61 @ ( dpsetconstr @ A @ B
% 3.00/1.61 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 3.00/1.61 => ( in @ Xx @ A ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(dpsetconstrEL2_type,type,
% 3.00/1.61 dpsetconstrEL2: $o ).
% 3.00/1.61
% 3.00/1.61 thf(dpsetconstrEL2,definition,
% 3.00/1.61 ( dpsetconstrEL2
% 3.00/1.61 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 3.00/1.61 ( ( in @ ( kpair @ Xx @ Xy )
% 3.00/1.61 @ ( dpsetconstr @ A @ B
% 3.00/1.61 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 3.00/1.61 => ( in @ Xy @ B ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(dpsetconstrER_type,type,
% 3.00/1.61 dpsetconstrER: $o ).
% 3.00/1.61
% 3.00/1.61 thf(dpsetconstrER,definition,
% 3.00/1.61 ( dpsetconstrER
% 3.00/1.61 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 3.00/1.61 ( ( in @ ( kpair @ Xx @ Xy )
% 3.00/1.61 @ ( dpsetconstr @ A @ B
% 3.00/1.61 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 3.00/1.61 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(func_type,type,
% 3.00/1.61 func: $i > $i > $i > $o ).
% 3.00/1.61
% 3.00/1.61 thf(funcSet_type,type,
% 3.00/1.61 funcSet: $i > $i > $i ).
% 3.00/1.61
% 3.00/1.61 thf(funcImageSingleton_type,type,
% 3.00/1.61 funcImageSingleton: $o ).
% 3.00/1.61
% 3.00/1.61 thf(funcImageSingleton,definition,
% 3.00/1.61 ( funcImageSingleton
% 3.00/1.61 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.00/1.61 ( ( func @ A @ B @ Xf )
% 3.00/1.61 => ! [Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 => ( singleton
% 3.00/1.61 @ ( dsetconstr @ B
% 3.00/1.61 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(apProp_type,type,
% 3.00/1.61 apProp: $o ).
% 3.00/1.61
% 3.00/1.61 thf(apProp,definition,
% 3.00/1.61 ( apProp
% 3.00/1.61 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.00/1.61 ( ( func @ A @ B @ Xf )
% 3.00/1.61 => ! [Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 => ( in
% 3.00/1.61 @ ( setunion
% 3.00/1.61 @ ( dsetconstr @ B
% 3.00/1.61 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 3.00/1.61 @ B ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(ap_type,type,
% 3.00/1.61 ap: $i > $i > $i > $i > $i ).
% 3.00/1.61
% 3.00/1.61 thf(app_type,type,
% 3.00/1.61 app: $o ).
% 3.00/1.61
% 3.00/1.61 thf(app,definition,
% 3.00/1.61 ( app
% 3.00/1.61 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.00/1.61 ( ( func @ A @ B @ Xf )
% 3.00/1.61 => ! [Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(infuncsetfunc_type,type,
% 3.00/1.61 infuncsetfunc: $o ).
% 3.00/1.61
% 3.00/1.61 thf(infuncsetfunc,definition,
% 3.00/1.61 ( infuncsetfunc
% 3.00/1.61 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.00/1.61 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.00/1.61 => ( func @ A @ B @ Xf ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(ap2p_type,type,
% 3.00/1.61 ap2p: $o ).
% 3.00/1.61
% 3.00/1.61 thf(ap2p,definition,
% 3.00/1.61 ( ap2p
% 3.00/1.61 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.00/1.61 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.00/1.61 => ! [Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(funcinfuncset_type,type,
% 3.00/1.61 funcinfuncset: $o ).
% 3.00/1.61
% 3.00/1.61 thf(funcinfuncset,definition,
% 3.00/1.61 ( funcinfuncset
% 3.00/1.61 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.00/1.61 ( ( func @ A @ B @ Xf )
% 3.00/1.61 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(lamProp_type,type,
% 3.00/1.61 lamProp: $o ).
% 3.00/1.61
% 3.00/1.61 thf(lamProp,definition,
% 3.00/1.61 ( lamProp
% 3.00/1.61 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.00/1.61 ( ! [Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.00/1.61 => ( func @ A @ B
% 3.00/1.61 @ ( dpsetconstr @ A @ B
% 3.00/1.61 @ ^ [Xx: $i,Xy: $i] :
% 3.00/1.61 ( ( Xf @ Xx )
% 3.00/1.61 = Xy ) ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(lam_type,type,
% 3.00/1.61 lam: $i > $i > ( $i > $i ) > $i ).
% 3.00/1.61
% 3.00/1.61 thf(lamp_type,type,
% 3.00/1.61 lamp: $o ).
% 3.00/1.61
% 3.00/1.61 thf(lamp,definition,
% 3.00/1.61 ( lamp
% 3.00/1.61 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.00/1.61 ( ! [Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.00/1.61 => ( func @ A @ B
% 3.00/1.61 @ ( lam @ A @ B
% 3.00/1.61 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(lam2p_type,type,
% 3.00/1.61 lam2p: $o ).
% 3.00/1.61
% 3.00/1.61 thf(lam2p,definition,
% 3.00/1.61 ( lam2p
% 3.00/1.61 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.00/1.61 ( ! [Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.00/1.61 => ( in
% 3.00/1.61 @ ( lam @ A @ B
% 3.00/1.61 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 3.00/1.61 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(brelnall1_type,type,
% 3.00/1.61 brelnall1: $o ).
% 3.00/1.61
% 3.00/1.61 thf(brelnall1,definition,
% 3.00/1.61 ( brelnall1
% 3.00/1.61 = ( ! [A: $i,B: $i,R: $i] :
% 3.00/1.61 ( ( breln @ A @ B @ R )
% 3.00/1.61 => ! [Xphi: $i > $o] :
% 3.00/1.61 ( ! [Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 => ! [Xy: $i] :
% 3.00/1.61 ( ( in @ Xy @ B )
% 3.00/1.61 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.00/1.61 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 3.00/1.61 => ! [Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ R )
% 3.00/1.61 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(brelnall2_type,type,
% 3.00/1.61 brelnall2: $o ).
% 3.00/1.61
% 3.00/1.61 thf(brelnall2,definition,
% 3.00/1.61 ( brelnall2
% 3.00/1.61 = ( ! [A: $i,B: $i,R: $i] :
% 3.00/1.61 ( ( breln @ A @ B @ R )
% 3.00/1.61 => ! [Xphi: $i > $o] :
% 3.00/1.61 ( ! [Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 => ! [Xy: $i] :
% 3.00/1.61 ( ( in @ Xy @ B )
% 3.00/1.61 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.00/1.61 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 3.00/1.61 => ! [Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ R )
% 3.00/1.61 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(ex1E2_type,type,
% 3.00/1.61 ex1E2: $o ).
% 3.00/1.61
% 3.00/1.61 thf(ex1E2,definition,
% 3.00/1.61 ( ex1E2
% 3.00/1.61 = ( ! [A: $i,Xphi: $i > $o] :
% 3.00/1.61 ( ( ex1 @ A
% 3.00/1.61 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.00/1.61 => ! [Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 => ! [Xy: $i] :
% 3.00/1.61 ( ( in @ Xy @ A )
% 3.00/1.61 => ( ( Xphi @ Xx )
% 3.00/1.61 => ( ( Xphi @ Xy )
% 3.00/1.61 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(funcGraphProp1_type,type,
% 3.00/1.61 funcGraphProp1: $o ).
% 3.00/1.61
% 3.00/1.61 thf(funcGraphProp1,definition,
% 3.00/1.61 ( funcGraphProp1
% 3.00/1.61 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.00/1.61 ( ( func @ A @ B @ Xf )
% 3.00/1.61 => ! [Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(funcGraphProp3_type,type,
% 3.00/1.61 funcGraphProp3: $o ).
% 3.00/1.61
% 3.00/1.61 thf(funcGraphProp3,definition,
% 3.00/1.61 ( funcGraphProp3
% 3.00/1.61 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.00/1.61 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.00/1.61 => ! [Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(funcGraphProp2_type,type,
% 3.00/1.61 funcGraphProp2: $o ).
% 3.00/1.61
% 3.00/1.61 thf(funcGraphProp2,definition,
% 3.00/1.61 ( funcGraphProp2
% 3.00/1.61 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.00/1.61 ( ( func @ A @ B @ Xf )
% 3.00/1.61 => ! [Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 => ! [Xy: $i] :
% 3.00/1.61 ( ( in @ Xy @ B )
% 3.00/1.61 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 3.00/1.61 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.00/1.61 = Xy ) ) ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(funcextLem_type,type,
% 3.00/1.61 funcextLem: $o ).
% 3.00/1.61
% 3.00/1.61 thf(funcextLem,definition,
% 3.00/1.61 ( funcextLem
% 3.00/1.61 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.00/1.61 ( ( func @ A @ B @ Xf )
% 3.00/1.61 => ! [Xg: $i] :
% 3.00/1.61 ( ( func @ A @ B @ Xg )
% 3.00/1.61 => ( ! [Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.00/1.61 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 3.00/1.61 => ! [Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 => ! [Xy: $i] :
% 3.00/1.61 ( ( in @ Xy @ B )
% 3.00/1.61 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 3.00/1.61 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(funcGraphProp4_type,type,
% 3.00/1.61 funcGraphProp4: $o ).
% 3.00/1.61
% 3.00/1.61 thf(funcGraphProp4,definition,
% 3.00/1.61 ( funcGraphProp4
% 3.00/1.61 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.00/1.61 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.00/1.61 => ! [Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 => ! [Xy: $i] :
% 3.00/1.61 ( ( in @ Xy @ B )
% 3.00/1.61 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 3.00/1.61 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.00/1.61 = Xy ) ) ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(subbreln_type,type,
% 3.00/1.61 subbreln: $o ).
% 3.00/1.61
% 3.00/1.61 thf(subbreln,definition,
% 3.00/1.61 ( subbreln
% 3.00/1.61 = ( ! [A: $i,B: $i,R: $i] :
% 3.00/1.61 ( ( breln @ A @ B @ R )
% 3.00/1.61 => ! [S: $i] :
% 3.00/1.61 ( ( breln @ A @ B @ S )
% 3.00/1.61 => ( ! [Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 => ! [Xy: $i] :
% 3.00/1.61 ( ( in @ Xy @ B )
% 3.00/1.61 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.00/1.61 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 3.00/1.61 => ( subset @ R @ S ) ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(eqbreln_type,type,
% 3.00/1.61 eqbreln: $o ).
% 3.00/1.61
% 3.00/1.61 thf(eqbreln,definition,
% 3.00/1.61 ( eqbreln
% 3.00/1.61 = ( ! [A: $i,B: $i,R: $i] :
% 3.00/1.61 ( ( breln @ A @ B @ R )
% 3.00/1.61 => ! [S: $i] :
% 3.00/1.61 ( ( breln @ A @ B @ S )
% 3.00/1.61 => ( ! [Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 => ! [Xy: $i] :
% 3.00/1.61 ( ( in @ Xy @ B )
% 3.00/1.61 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.00/1.61 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 3.00/1.61 => ( ! [Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 => ! [Xy: $i] :
% 3.00/1.61 ( ( in @ Xy @ B )
% 3.00/1.61 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 3.00/1.61 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 3.00/1.61 => ( R = S ) ) ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(funcext_type,type,
% 3.00/1.61 funcext: $o ).
% 3.00/1.61
% 3.00/1.61 thf(funcext,definition,
% 3.00/1.61 ( funcext
% 3.00/1.61 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.00/1.61 ( ( func @ A @ B @ Xf )
% 3.00/1.61 => ! [Xg: $i] :
% 3.00/1.61 ( ( func @ A @ B @ Xg )
% 3.00/1.61 => ( ! [Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.00/1.61 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 3.00/1.61 => ( Xf = Xg ) ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(funcext2_type,type,
% 3.00/1.61 funcext2: $o ).
% 3.00/1.61
% 3.00/1.61 thf(funcext2,definition,
% 3.00/1.61 ( funcext2
% 3.00/1.61 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.00/1.61 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.00/1.61 => ! [Xg: $i] :
% 3.00/1.61 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 3.00/1.61 => ( ! [Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.00/1.61 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 3.00/1.61 => ( Xf = Xg ) ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(ap2apEq1_type,type,
% 3.00/1.61 ap2apEq1: $o ).
% 3.00/1.61
% 3.00/1.61 thf(ap2apEq1,definition,
% 3.00/1.61 ( ap2apEq1
% 3.00/1.61 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.00/1.61 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.00/1.61 => ! [Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.00/1.61 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(ap2apEq2_type,type,
% 3.00/1.61 ap2apEq2: $o ).
% 3.00/1.61
% 3.00/1.61 thf(ap2apEq2,definition,
% 3.00/1.61 ( ap2apEq2
% 3.00/1.61 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.00/1.61 ( ( func @ A @ B @ Xf )
% 3.00/1.61 => ! [Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.00/1.61 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(beta1_type,type,
% 3.00/1.61 beta1: $o ).
% 3.00/1.61
% 3.00/1.61 thf(beta1,definition,
% 3.00/1.61 ( beta1
% 3.00/1.61 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.00/1.61 ( ! [Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.00/1.61 => ! [Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 => ( ( ap @ A @ B
% 3.00/1.61 @ ( lam @ A @ B
% 3.00/1.61 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 3.00/1.61 @ Xx )
% 3.00/1.61 = ( Xf @ Xx ) ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(eta1_type,type,
% 3.00/1.61 eta1: $o ).
% 3.00/1.61
% 3.00/1.61 thf(eta1,definition,
% 3.00/1.61 ( eta1
% 3.00/1.61 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.00/1.61 ( ( func @ A @ B @ Xf )
% 3.00/1.61 => ( ( lam @ A @ B
% 3.00/1.61 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 3.00/1.61 = Xf ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(lam2lamEq_type,type,
% 3.00/1.61 lam2lamEq: $o ).
% 3.00/1.61
% 3.00/1.61 thf(lam2lamEq,definition,
% 3.00/1.61 ( lam2lamEq
% 3.00/1.61 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.00/1.61 ( ! [Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.00/1.61 => ( ( lam @ A @ B
% 3.00/1.61 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 3.00/1.61 = ( lam @ A @ B
% 3.00/1.61 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(beta2_type,type,
% 3.00/1.61 beta2: $o ).
% 3.00/1.61
% 3.00/1.61 thf(beta2,definition,
% 3.00/1.61 ( beta2
% 3.00/1.61 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.00/1.61 ( ! [Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.00/1.61 => ! [Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 => ( ( ap @ A @ B
% 3.00/1.61 @ ( lam @ A @ B
% 3.00/1.61 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 3.00/1.61 @ Xx )
% 3.00/1.61 = ( Xf @ Xx ) ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(eta2_type,type,
% 3.00/1.61 eta2: $o ).
% 3.00/1.61
% 3.00/1.61 thf(eta2,definition,
% 3.00/1.61 ( eta2
% 3.00/1.61 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.00/1.61 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.00/1.61 => ( ( lam @ A @ B
% 3.00/1.61 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 3.00/1.61 = Xf ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(iffalseProp1_type,type,
% 3.00/1.61 iffalseProp1: $o ).
% 3.00/1.61
% 3.00/1.61 thf(iffalseProp1,definition,
% 3.00/1.61 ( iffalseProp1
% 3.00/1.61 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 => ! [Xy: $i] :
% 3.00/1.61 ( ( in @ Xy @ A )
% 3.00/1.61 => ( ~ Xphi
% 3.00/1.61 => ( in @ Xy
% 3.00/1.61 @ ( dsetconstr @ A
% 3.00/1.61 @ ^ [Xz: $i] :
% 3.00/1.61 ( ( Xphi
% 3.00/1.61 & ( Xz = Xx ) )
% 3.00/1.61 | ( ~ Xphi
% 3.00/1.61 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(iffalseProp2_type,type,
% 3.00/1.61 iffalseProp2: $o ).
% 3.00/1.61
% 3.00/1.61 thf(iffalseProp2,definition,
% 3.00/1.61 ( iffalseProp2
% 3.00/1.61 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 => ! [Xy: $i] :
% 3.00/1.61 ( ( in @ Xy @ A )
% 3.00/1.61 => ( ~ Xphi
% 3.00/1.61 => ( ( dsetconstr @ A
% 3.00/1.61 @ ^ [Xz: $i] :
% 3.00/1.61 ( ( Xphi
% 3.00/1.61 & ( Xz = Xx ) )
% 3.00/1.61 | ( ~ Xphi
% 3.00/1.61 & ( Xz = Xy ) ) ) )
% 3.00/1.61 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(iftrueProp1_type,type,
% 3.00/1.61 iftrueProp1: $o ).
% 3.00/1.61
% 3.00/1.61 thf(iftrueProp1,definition,
% 3.00/1.61 ( iftrueProp1
% 3.00/1.61 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 => ! [Xy: $i] :
% 3.00/1.61 ( ( in @ Xy @ A )
% 3.00/1.61 => ( Xphi
% 3.00/1.61 => ( in @ Xx
% 3.00/1.61 @ ( dsetconstr @ A
% 3.00/1.61 @ ^ [Xz: $i] :
% 3.00/1.61 ( ( Xphi
% 3.00/1.61 & ( Xz = Xx ) )
% 3.00/1.61 | ( ~ Xphi
% 3.00/1.61 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(iftrueProp2_type,type,
% 3.00/1.61 iftrueProp2: $o ).
% 3.00/1.61
% 3.00/1.61 thf(iftrueProp2,definition,
% 3.00/1.61 ( iftrueProp2
% 3.00/1.61 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 => ! [Xy: $i] :
% 3.00/1.61 ( ( in @ Xy @ A )
% 3.00/1.61 => ( Xphi
% 3.00/1.61 => ( ( dsetconstr @ A
% 3.00/1.61 @ ^ [Xz: $i] :
% 3.00/1.61 ( ( Xphi
% 3.00/1.61 & ( Xz = Xx ) )
% 3.00/1.61 | ( ~ Xphi
% 3.00/1.61 & ( Xz = Xy ) ) ) )
% 3.00/1.61 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(ifSingleton_type,type,
% 3.00/1.61 ifSingleton: $o ).
% 3.00/1.61
% 3.00/1.61 thf(ifSingleton,definition,
% 3.00/1.61 ( ifSingleton
% 3.00/1.61 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 => ! [Xy: $i] :
% 3.00/1.61 ( ( in @ Xy @ A )
% 3.00/1.61 => ( singleton
% 3.00/1.61 @ ( dsetconstr @ A
% 3.00/1.61 @ ^ [Xz: $i] :
% 3.00/1.61 ( ( Xphi
% 3.00/1.61 & ( Xz = Xx ) )
% 3.00/1.61 | ( ~ Xphi
% 3.00/1.61 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(if_type,type,
% 3.00/1.61 if: $i > $o > $i > $i > $i ).
% 3.00/1.61
% 3.00/1.61 thf(ifp_type,type,
% 3.00/1.61 ifp: $o ).
% 3.00/1.61
% 3.00/1.61 thf(ifp,definition,
% 3.00/1.61 ( ifp
% 3.00/1.61 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 => ! [Xy: $i] :
% 3.00/1.61 ( ( in @ Xy @ A )
% 3.00/1.61 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(theeq_type,type,
% 3.00/1.61 theeq: $o ).
% 3.00/1.61
% 3.00/1.61 thf(theeq,definition,
% 3.00/1.61 ( theeq
% 3.00/1.61 = ( ! [X: $i] :
% 3.00/1.61 ( ( singleton @ X )
% 3.00/1.61 => ! [Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ X )
% 3.00/1.61 => ( ( setunion @ X )
% 3.00/1.61 = Xx ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(iftrue_type,type,
% 3.00/1.61 iftrue: $o ).
% 3.00/1.61
% 3.00/1.61 thf(iftrue,definition,
% 3.00/1.61 ( iftrue
% 3.00/1.61 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 => ! [Xy: $i] :
% 3.00/1.61 ( ( in @ Xy @ A )
% 3.00/1.61 => ( Xphi
% 3.00/1.61 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 3.00/1.61 = Xx ) ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(iffalse_type,type,
% 3.00/1.61 iffalse: $o ).
% 3.00/1.61
% 3.00/1.61 thf(iffalse,definition,
% 3.00/1.61 ( iffalse
% 3.00/1.61 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 => ! [Xy: $i] :
% 3.00/1.61 ( ( in @ Xy @ A )
% 3.00/1.61 => ( ~ Xphi
% 3.00/1.61 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 3.00/1.61 = Xy ) ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(iftrueorfalse_type,type,
% 3.00/1.61 iftrueorfalse: $o ).
% 3.00/1.61
% 3.00/1.61 thf(iftrueorfalse,definition,
% 3.00/1.61 ( iftrueorfalse
% 3.00/1.61 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 => ! [Xy: $i] :
% 3.00/1.61 ( ( in @ Xy @ A )
% 3.00/1.61 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(binintersectT_lem_type,type,
% 3.00/1.61 binintersectT_lem: $o ).
% 3.00/1.61
% 3.00/1.61 thf(binintersectT_lem,definition,
% 3.00/1.61 ( binintersectT_lem
% 3.00/1.61 = ( ! [A: $i,X: $i] :
% 3.00/1.61 ( ( in @ X @ ( powerset @ A ) )
% 3.00/1.61 => ! [Y: $i] :
% 3.00/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 3.00/1.61 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(binunionT_lem_type,type,
% 3.00/1.61 binunionT_lem: $o ).
% 3.00/1.61
% 3.00/1.61 thf(binunionT_lem,definition,
% 3.00/1.61 ( binunionT_lem
% 3.00/1.61 = ( ! [A: $i,X: $i] :
% 3.00/1.61 ( ( in @ X @ ( powerset @ A ) )
% 3.00/1.61 => ! [Y: $i] :
% 3.00/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 3.00/1.61 => ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(powersetT_lem_type,type,
% 3.00/1.61 powersetT_lem: $o ).
% 3.00/1.61
% 3.00/1.61 thf(powersetT_lem,definition,
% 3.00/1.61 ( powersetT_lem
% 3.00/1.61 = ( ! [A: $i,X: $i] :
% 3.00/1.61 ( ( in @ X @ ( powerset @ A ) )
% 3.00/1.61 => ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(setminusT_lem_type,type,
% 3.00/1.61 setminusT_lem: $o ).
% 3.00/1.61
% 3.00/1.61 thf(setminusT_lem,definition,
% 3.00/1.61 ( setminusT_lem
% 3.00/1.61 = ( ! [A: $i,X: $i] :
% 3.00/1.61 ( ( in @ X @ ( powerset @ A ) )
% 3.00/1.61 => ! [Y: $i] :
% 3.00/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 3.00/1.61 => ( in @ ( setminus @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(complementT_lem_type,type,
% 3.00/1.61 complementT_lem: $o ).
% 3.00/1.61
% 3.00/1.61 thf(complementT_lem,definition,
% 3.00/1.61 ( complementT_lem
% 3.00/1.61 = ( ! [A: $i,X: $i] :
% 3.00/1.61 ( ( in @ X @ ( powerset @ A ) )
% 3.00/1.61 => ( in @ ( setminus @ A @ X ) @ ( powerset @ A ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(setextT_type,type,
% 3.00/1.61 setextT: $o ).
% 3.00/1.61
% 3.00/1.61 thf(setextT,definition,
% 3.00/1.61 ( setextT
% 3.00/1.61 = ( ! [A: $i,X: $i] :
% 3.00/1.61 ( ( in @ X @ ( powerset @ A ) )
% 3.00/1.61 => ! [Y: $i] :
% 3.00/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 3.00/1.61 => ( ! [Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 => ( ( in @ Xx @ X )
% 3.00/1.61 => ( in @ Xx @ Y ) ) )
% 3.00/1.61 => ( ! [Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 => ( ( in @ Xx @ Y )
% 3.00/1.61 => ( in @ Xx @ X ) ) )
% 3.00/1.61 => ( X = Y ) ) ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(subsetTI_type,type,
% 3.00/1.61 subsetTI: $o ).
% 3.00/1.61
% 3.00/1.61 thf(subsetTI,definition,
% 3.00/1.61 ( subsetTI
% 3.00/1.61 = ( ! [A: $i,X: $i] :
% 3.00/1.61 ( ( in @ X @ ( powerset @ A ) )
% 3.00/1.61 => ! [Y: $i] :
% 3.00/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 3.00/1.61 => ( ! [Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 => ( ( in @ Xx @ X )
% 3.00/1.61 => ( in @ Xx @ Y ) ) )
% 3.00/1.61 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(powersetTI1_type,type,
% 3.00/1.61 powersetTI1: $o ).
% 3.00/1.61
% 3.00/1.61 thf(powersetTI1,definition,
% 3.00/1.61 ( powersetTI1
% 3.00/1.61 = ( ! [A: $i,X: $i] :
% 3.00/1.61 ( ( in @ X @ ( powerset @ A ) )
% 3.00/1.61 => ! [Y: $i] :
% 3.00/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 3.00/1.61 => ( ! [Xx: $i] :
% 3.00/1.61 ( ( in @ Xx @ A )
% 3.00/1.61 => ( ( in @ Xx @ X )
% 3.00/1.61 => ( in @ Xx @ Y ) ) )
% 3.00/1.61 => ( in @ X @ ( powerset @ Y ) ) ) ) ) ) ) ).
% 3.00/1.61
% 3.00/1.61 thf(powersetTE1,conjecture,
% 3.00/1.61 ( setextAx
% 3.00/1.61 => ( emptysetAx
% 3.00/1.61 => ( setadjoinAx
% 3.00/1.61 => ( powersetAx
% 3.00/1.61 => ( setunionAx
% 3.00/1.61 => ( omega0Ax
% 3.00/1.61 => ( omegaSAx
% 3.00/1.61 => ( omegaIndAx
% 3.00/1.61 => ( replAx
% 3.00/1.61 => ( foundationAx
% 3.00/1.61 => ( wellorderingAx
% 3.00/1.61 => ( descrp
% 3.00/1.61 => ( dsetconstrI
% 3.00/1.61 => ( dsetconstrEL
% 3.00/1.61 => ( dsetconstrER
% 3.00/1.61 => ( exuE1
% 3.00/1.61 => ( prop2setE
% 3.00/1.61 => ( emptysetE
% 3.00/1.61 => ( emptysetimpfalse
% 3.00/1.61 => ( notinemptyset
% 3.00/1.61 => ( exuE3e
% 3.00/1.61 => ( setext
% 3.00/1.61 => ( emptyI
% 3.00/1.61 => ( noeltsimpempty
% 3.00/1.61 => ( setbeta
% 3.00/1.61 => ( nonemptyE1
% 3.00/1.61 => ( nonemptyI
% 3.00/1.61 => ( nonemptyI1
% 3.00/1.61 => ( setadjoinIL
% 3.00/1.61 => ( emptyinunitempty
% 3.00/1.61 => ( setadjoinIR
% 3.00/1.61 => ( setadjoinE
% 3.00/1.61 => ( setadjoinOr
% 3.00/1.61 => ( setoftrueEq
% 3.00/1.61 => ( powersetI
% 3.00/1.61 => ( emptyinPowerset
% 3.00/1.61 => ( emptyInPowerset
% 3.00/1.61 => ( powersetE
% 3.00/1.61 => ( setunionI
% 3.00/1.61 => ( setunionE
% 3.00/1.61 => ( subPowSU
% 3.00/1.61 => ( exuE2
% 3.00/1.61 => ( nonemptyImpWitness
% 3.00/1.61 => ( uniqinunit
% 3.00/1.61 => ( notinsingleton
% 3.00/1.61 => ( eqinunit
% 3.00/1.61 => ( singletonsswitch
% 3.00/1.62 => ( upairsetE
% 3.00/1.62 => ( upairsetIL
% 3.00/1.62 => ( upairsetIR
% 3.00/1.62 => ( emptyE1
% 3.00/1.62 => ( vacuousDall
% 3.00/1.62 => ( quantDeMorgan1
% 3.00/1.62 => ( quantDeMorgan2
% 3.00/1.62 => ( quantDeMorgan3
% 3.00/1.62 => ( quantDeMorgan4
% 3.00/1.62 => ( prop2setI
% 3.00/1.62 => ( prop2set2propI
% 3.00/1.62 => ( notdexE
% 3.00/1.62 => ( notdallE
% 3.00/1.62 => ( exuI1
% 3.00/1.62 => ( exuI3
% 3.00/1.62 => ( exuI2
% 3.00/1.62 => ( inCongP
% 3.00/1.62 => ( in__Cong
% 3.00/1.62 => ( exuE3u
% 3.00/1.62 => ( exu__Cong
% 3.00/1.62 => ( emptyset__Cong
% 3.00/1.62 => ( setadjoin__Cong
% 3.00/1.62 => ( powerset__Cong
% 3.00/1.62 => ( setunion__Cong
% 3.00/1.62 => ( omega__Cong
% 3.00/1.62 => ( exuEu
% 3.00/1.62 => ( descr__Cong
% 3.00/1.62 => ( dsetconstr__Cong
% 3.00/1.62 => ( subsetI1
% 3.00/1.62 => ( eqimpsubset2
% 3.00/1.62 => ( eqimpsubset1
% 3.00/1.62 => ( subsetI2
% 3.00/1.62 => ( emptysetsubset
% 3.00/1.62 => ( subsetE
% 3.00/1.62 => ( subsetE2
% 3.00/1.62 => ( notsubsetI
% 3.00/1.62 => ( notequalI1
% 3.00/1.62 => ( notequalI2
% 3.00/1.62 => ( subsetRefl
% 3.00/1.62 => ( subsetTrans
% 3.00/1.62 => ( setadjoinSub
% 3.00/1.62 => ( setadjoinSub2
% 3.00/1.62 => ( subset2powerset
% 3.00/1.62 => ( setextsub
% 3.00/1.62 => ( subsetemptysetimpeq
% 3.00/1.62 => ( powersetI1
% 3.00/1.62 => ( powersetE1
% 3.00/1.62 => ( inPowerset
% 3.00/1.62 => ( powersetsubset
% 3.00/1.62 => ( sepInPowerset
% 3.00/1.62 => ( sepSubset
% 3.00/1.62 => ( binunionIL
% 3.00/1.62 => ( upairset2IR
% 3.00/1.62 => ( binunionIR
% 3.00/1.62 => ( binunionEcases
% 3.00/1.62 => ( binunionE
% 3.00/1.62 => ( binunionLsub
% 3.00/1.62 => ( binunionRsub
% 3.00/1.62 => ( binintersectI
% 3.00/1.62 => ( binintersectSubset5
% 3.00/1.62 => ( binintersectEL
% 3.00/1.62 => ( binintersectLsub
% 3.00/1.62 => ( binintersectSubset2
% 3.00/1.62 => ( binintersectSubset3
% 3.00/1.62 => ( binintersectER
% 3.00/1.62 => ( disjointsetsI1
% 3.00/1.62 => ( binintersectRsub
% 3.00/1.62 => ( binintersectSubset4
% 3.00/1.62 => ( binintersectSubset1
% 3.00/1.62 => ( bs114d
% 3.00/1.62 => ( setminusI
% 3.00/1.62 => ( setminusEL
% 3.00/1.62 => ( setminusER
% 3.00/1.62 => ( setminusSubset2
% 3.00/1.62 => ( setminusERneg
% 3.00/1.62 => ( setminusELneg
% 3.00/1.62 => ( setminusILneg
% 3.00/1.62 => ( setminusIRneg
% 3.00/1.62 => ( setminusLsub
% 3.00/1.62 => ( setminusSubset1
% 3.00/1.62 => ( symdiffE
% 3.00/1.62 => ( symdiffI1
% 3.00/1.62 => ( symdiffI2
% 3.00/1.62 => ( symdiffIneg1
% 3.00/1.62 => ( symdiffIneg2
% 3.00/1.62 => ( secondinupair
% 3.00/1.62 => ( setukpairIL
% 3.00/1.62 => ( setukpairIR
% 3.00/1.62 => ( kpairiskpair
% 3.00/1.62 => ( kpairp
% 3.00/1.62 => ( singletonsubset
% 3.00/1.62 => ( singletoninpowerset
% 3.00/1.62 => ( singletoninpowunion
% 3.00/1.62 => ( upairset2E
% 3.00/1.62 => ( upairsubunion
% 3.00/1.62 => ( upairinpowunion
% 3.00/1.62 => ( ubforcartprodlem1
% 3.00/1.62 => ( ubforcartprodlem2
% 3.00/1.62 => ( ubforcartprodlem3
% 3.00/1.62 => ( cartprodpairin
% 3.00/1.62 => ( cartprodmempair1
% 3.00/1.62 => ( cartprodmempair
% 3.00/1.62 => ( setunionE2
% 3.00/1.62 => ( setunionsingleton1
% 3.00/1.62 => ( setunionsingleton2
% 3.00/1.62 => ( setunionsingleton
% 3.00/1.62 => ( singletonprop
% 3.00/1.62 => ( ex1E1
% 3.00/1.62 => ( ex1I
% 3.00/1.62 => ( ex1I2
% 3.00/1.62 => ( singletonsuniq
% 3.00/1.62 => ( setukpairinjL1
% 3.00/1.62 => ( kfstsingleton
% 3.00/1.62 => ( theprop
% 3.00/1.62 => ( kfstpairEq
% 3.00/1.62 => ( cartprodfstin
% 3.00/1.62 => ( setukpairinjL2
% 3.00/1.62 => ( setukpairinjL
% 3.00/1.62 => ( setukpairinjR11
% 3.00/1.62 => ( setukpairinjR12
% 3.00/1.62 => ( setukpairinjR1
% 3.00/1.62 => ( upairequniteq
% 3.00/1.62 => ( setukpairinjR2
% 3.00/1.62 => ( setukpairinjR
% 3.00/1.62 => ( ksndsingleton
% 3.00/1.62 => ( ksndpairEq
% 3.00/1.62 => ( kpairsurjEq
% 3.00/1.62 => ( cartprodsndin
% 3.00/1.62 => ( cartprodpairmemEL
% 3.00/1.62 => ( cartprodpairmemER
% 3.00/1.62 => ( cartprodmempaircEq
% 3.00/1.62 => ( cartprodfstpairEq
% 3.00/1.62 => ( cartprodsndpairEq
% 3.00/1.62 => ( cartprodpairsurjEq
% 3.00/1.62 => ( dpsetconstrI
% 3.00/1.62 => ( dpsetconstrSub
% 3.00/1.62 => ( setOfPairsIsBReln
% 3.00/1.62 => ( dpsetconstrERa
% 3.00/1.62 => ( dpsetconstrEL1
% 3.00/1.62 => ( dpsetconstrEL2
% 3.00/1.62 => ( dpsetconstrER
% 3.00/1.62 => ( funcImageSingleton
% 3.00/1.62 => ( apProp
% 3.00/1.62 => ( app
% 3.00/1.62 => ( infuncsetfunc
% 3.00/1.62 => ( ap2p
% 3.00/1.62 => ( funcinfuncset
% 3.00/1.62 => ( lamProp
% 3.00/1.62 => ( lamp
% 3.00/1.62 => ( lam2p
% 3.00/1.62 => ( brelnall1
% 3.00/1.62 => ( brelnall2
% 3.00/1.62 => ( ex1E2
% 3.00/1.62 => ( funcGraphProp1
% 3.00/1.62 => ( funcGraphProp3
% 3.00/1.62 => ( funcGraphProp2
% 3.00/1.62 => ( funcextLem
% 3.00/1.62 => ( funcGraphProp4
% 3.00/1.62 => ( subbreln
% 3.00/1.62 => ( eqbreln
% 3.00/1.62 => ( funcext
% 3.00/1.62 => ( funcext2
% 3.00/1.62 => ( ap2apEq1
% 3.00/1.62 => ( ap2apEq2
% 3.00/1.62 => ( beta1
% 3.00/1.62 => ( eta1
% 3.00/1.62 => ( lam2lamEq
% 3.00/1.62 => ( beta2
% 3.00/1.62 => ( eta2
% 3.00/1.62 => ( iffalseProp1
% 3.00/1.62 => ( iffalseProp2
% 3.00/1.62 => ( iftrueProp1
% 3.00/1.62 => ( iftrueProp2
% 3.00/1.62 => ( ifSingleton
% 3.00/1.62 => ( ifp
% 3.00/1.62 => ( theeq
% 3.00/1.62 => ( iftrue
% 3.00/1.62 => ( iffalse
% 3.00/1.62 => ( iftrueorfalse
% 3.00/1.62 => ( binintersectT_lem
% 3.00/1.62 => ( binunionT_lem
% 3.00/1.62 => ( powersetT_lem
% 3.00/1.62 => ( setminusT_lem
% 3.00/1.62 => ( complementT_lem
% 3.00/1.62 => ( setextT
% 3.00/1.62 => ( subsetTI
% 3.00/1.62 => ( powersetTI1
% 3.00/1.62 => ! [A: $i,X: $i] :
% 3.00/1.62 ( ( in @ X @ ( powerset @ A ) )
% 3.00/1.62 => ! [Y: $i] :
% 3.00/1.62 ( ( in @ Y @ ( powerset @ A ) )
% 3.00/1.62 => ! [Xx: $i] :
% 3.00/1.62 ( ( in @ Xx @ A )
% 3.00/1.62 => ( ( in @ X @ ( powerset @ Y ) )
% 3.00/1.62 => ( ( in @ Xx @ X )
% 3.00/1.62 => ( in @ Xx @ Y ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 3.00/1.62 ------------------------
% 3.00/1.64 % (12427)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_12262 for (2999ds/183Mi)
% 3.00/1.64 % (12430)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_12262 for (2999ds/2Mi)
% 3.00/1.64 % (12429)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_12262 for (2999ds/27Mi)
% 3.00/1.64 % (12428)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_12262 for (2999ds/4Mi)
% 3.00/1.64 % (12433)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_12262 for (2999ds/18Mi)
% 3.00/1.64 % (12430)Instruction limit reached!
% 3.00/1.64 % (12430)------------------------------
% 3.00/1.64 % (12430)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.64 % (12430)Termination reason: Unknown
% 3.00/1.64 % (12430)Termination phase: shuffling
% 3.00/1.64
% 3.00/1.64 % (12430)Memory used [KB]: 1407
% 3.00/1.64 % (12430)Time elapsed: 0.003 s
% 3.00/1.64 % (12430)Instructions burned: 3 (million)
% 3.00/1.64 % (12430)------------------------------
% 3.00/1.64 % (12430)------------------------------
% 3.00/1.65 % (12431)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_12262 for (2999ds/2Mi)
% 3.00/1.65 % (12432)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_12262 for (2999ds/275Mi)
% 3.00/1.65 % (12428)Instruction limit reached!
% 3.00/1.65 % (12428)------------------------------
% 3.00/1.65 % (12428)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.65 % (12428)Termination reason: Unknown
% 3.00/1.65 % (12428)Termination phase: shuffling
% 3.00/1.65
% 3.00/1.65 % (12428)Memory used [KB]: 1407
% 3.00/1.65 % (12428)Time elapsed: 0.004 s
% 3.00/1.65 % (12428)Instructions burned: 5 (million)
% 3.00/1.65 % (12428)------------------------------
% 3.00/1.65 % (12428)------------------------------
% 3.00/1.65 % (12431)Instruction limit reached!
% 3.00/1.65 % (12431)------------------------------
% 3.00/1.65 % (12431)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.65 % (12431)Termination reason: Unknown
% 3.00/1.65 % (12431)Termination phase: shuffling
% 3.00/1.65
% 3.00/1.65 % (12431)Memory used [KB]: 1407
% 3.00/1.65 % (12431)Time elapsed: 0.003 s
% 3.00/1.65 % (12431)Instructions burned: 2 (million)
% 3.00/1.65 % (12431)------------------------------
% 3.00/1.65 % (12431)------------------------------
% 3.00/1.65 % (12433)Instruction limit reached!
% 3.00/1.65 % (12433)------------------------------
% 3.00/1.65 % (12433)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.65 % (12433)Termination reason: Unknown
% 3.00/1.65 % (12433)Termination phase: shuffling
% 3.00/1.65
% 3.00/1.65 % (12433)Memory used [KB]: 1791
% 3.00/1.65 % (12433)Time elapsed: 0.011 s
% 3.00/1.65 % (12433)Instructions burned: 18 (million)
% 3.00/1.65 % (12433)------------------------------
% 3.00/1.65 % (12433)------------------------------
% 3.00/1.66 % (12429)Instruction limit reached!
% 3.00/1.66 % (12429)------------------------------
% 3.00/1.66 % (12429)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.66 % (12429)Termination reason: Unknown
% 3.00/1.66 % (12429)Termination phase: shuffling
% 3.00/1.66
% 3.00/1.66 % (12429)Memory used [KB]: 1918
% 3.00/1.66 % (12429)Time elapsed: 0.014 s
% 3.00/1.66 % (12429)Instructions burned: 27 (million)
% 3.00/1.66 % (12429)------------------------------
% 3.00/1.66 % (12429)------------------------------
% 3.00/1.66 % (12434)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_12262 for (2999ds/3Mi)
% 3.00/1.66 % (12436)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_12262 for (2999ds/15Mi)
% 3.00/1.66 % (12434)Instruction limit reached!
% 3.00/1.66 % (12434)------------------------------
% 3.00/1.66 % (12434)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.66 % (12434)Termination reason: Unknown
% 3.00/1.66 % (12434)Termination phase: shuffling
% 3.00/1.66
% 3.00/1.66 % (12434)Memory used [KB]: 1407
% 3.00/1.66 % (12434)Time elapsed: 0.003 s
% 3.00/1.66 % (12434)Instructions burned: 4 (million)
% 3.00/1.66 % (12434)------------------------------
% 3.00/1.66 % (12434)------------------------------
% 3.00/1.67 % (12435)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_12262 for (2999ds/37Mi)
% 3.00/1.67 % (12437)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_12262 for (2999ds/3Mi)
% 3.00/1.67 % (12436)Instruction limit reached!
% 3.00/1.67 % (12436)------------------------------
% 3.00/1.67 % (12436)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.67 % (12436)Termination reason: Unknown
% 3.00/1.67 % (12436)Termination phase: shuffling
% 3.00/1.67
% 3.00/1.67 % (12436)Memory used [KB]: 1663
% 3.00/1.67 % (12436)Time elapsed: 0.010 s
% 3.00/1.67 % (12436)Instructions burned: 15 (million)
% 3.00/1.67 % (12436)------------------------------
% 3.00/1.67 % (12436)------------------------------
% 3.00/1.67 % (12437)Instruction limit reached!
% 3.00/1.67 % (12437)------------------------------
% 3.00/1.67 % (12437)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.67 % (12437)Termination reason: Unknown
% 3.00/1.67 % (12437)Termination phase: shuffling
% 3.00/1.67
% 3.00/1.67 % (12437)Memory used [KB]: 1407
% 3.00/1.67 % (12437)Time elapsed: 0.003 s
% 3.00/1.67 % (12437)Instructions burned: 3 (million)
% 3.00/1.67 % (12437)------------------------------
% 3.00/1.67 % (12437)------------------------------
% 3.00/1.67 % (12438)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_12262 for (2999ds/1041Mi)
% 3.00/1.68 % (12435)Instruction limit reached!
% 3.00/1.68 % (12435)------------------------------
% 3.00/1.68 % (12435)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.68 % (12435)Termination reason: Unknown
% 3.00/1.68 % (12435)Termination phase: shuffling
% 3.00/1.68
% 3.00/1.68 % (12435)Memory used [KB]: 2174
% 3.00/1.68 % (12435)Time elapsed: 0.018 s
% 3.00/1.68 % (12435)Instructions burned: 37 (million)
% 3.00/1.68 % (12435)------------------------------
% 3.00/1.68 % (12435)------------------------------
% 3.00/1.69 % (12442)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_12262 for (2999ds/3Mi)
% 3.00/1.69 % (12442)Instruction limit reached!
% 3.00/1.69 % (12442)------------------------------
% 3.00/1.69 % (12442)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.69 % (12442)Termination reason: Unknown
% 3.00/1.69 % (12442)Termination phase: shuffling
% 3.00/1.69
% 3.00/1.69 % (12442)Memory used [KB]: 1407
% 3.00/1.69 % (12442)Time elapsed: 0.002 s
% 3.00/1.69 % (12442)Instructions burned: 4 (million)
% 3.00/1.69 % (12442)------------------------------
% 3.00/1.69 % (12442)------------------------------
% 3.00/1.69 % (12441)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_12262 for (2999ds/16Mi)
% 3.00/1.69 % (12427)Instruction limit reached!
% 3.00/1.69 % (12427)------------------------------
% 3.00/1.69 % (12427)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.69 % (12427)Termination reason: Unknown
% 3.00/1.69 % (12427)Termination phase: Saturation
% 3.00/1.69
% 3.00/1.69 % (12427)Memory used [KB]: 7547
% 3.00/1.69 % (12427)Time elapsed: 0.052 s
% 3.00/1.69 % (12427)Instructions burned: 183 (million)
% 3.00/1.69 % (12427)------------------------------
% 3.00/1.69 % (12427)------------------------------
% 3.00/1.69 % (12439)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_12262 for (2999ds/7Mi)
% 3.00/1.69 % (12439)Instruction limit reached!
% 3.00/1.69 % (12439)------------------------------
% 3.00/1.69 % (12439)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.69 % (12439)Termination reason: Unknown
% 3.00/1.69 % (12439)Termination phase: shuffling
% 3.00/1.69
% 3.00/1.69 % (12439)Memory used [KB]: 1535
% 3.00/1.69 % (12439)Time elapsed: 0.005 s
% 3.00/1.69 % (12439)Instructions burned: 8 (million)
% 3.00/1.69 % (12439)------------------------------
% 3.00/1.69 % (12439)------------------------------
% 3.00/1.70 % (12441)Instruction limit reached!
% 3.00/1.70 % (12441)------------------------------
% 3.00/1.70 % (12441)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.70 % (12441)Termination reason: Unknown
% 3.00/1.70 % (12441)Termination phase: shuffling
% 3.00/1.70
% 3.00/1.70 % (12441)Memory used [KB]: 1791
% 3.00/1.70 % (12441)Time elapsed: 0.009 s
% 3.00/1.70 % (12441)Instructions burned: 16 (million)
% 3.00/1.70 % (12441)------------------------------
% 3.00/1.70 % (12441)------------------------------
% 3.00/1.70 % (12445)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_12262 for (2999ds/7Mi)
% 3.02/1.70 % (12445)Instruction limit reached!
% 3.02/1.70 % (12445)------------------------------
% 3.02/1.70 % (12445)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.70 % (12445)Termination reason: Unknown
% 3.02/1.70 % (12445)Termination phase: shuffling
% 3.02/1.70
% 3.02/1.70 % (12445)Memory used [KB]: 1535
% 3.02/1.70 % (12445)Time elapsed: 0.003 s
% 3.02/1.70 % (12445)Instructions burned: 7 (million)
% 3.02/1.70 % (12445)------------------------------
% 3.02/1.70 % (12445)------------------------------
% 3.02/1.70 % (12444)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_12262 for (2999ds/3Mi)
% 3.02/1.70 % (12444)Instruction limit reached!
% 3.02/1.70 % (12444)------------------------------
% 3.02/1.70 % (12444)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.70 % (12444)Termination reason: Unknown
% 3.02/1.70 % (12444)Termination phase: shuffling
% 3.02/1.70
% 3.02/1.70 % (12444)Memory used [KB]: 1407
% 3.02/1.70 % (12444)Time elapsed: 0.003 s
% 3.02/1.70 % (12444)Instructions burned: 4 (million)
% 3.02/1.70 % (12444)------------------------------
% 3.02/1.70 % (12444)------------------------------
% 3.02/1.70 % (12446)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_12262 for (2999ds/3Mi)
% 3.02/1.71 % (12446)Instruction limit reached!
% 3.02/1.71 % (12446)------------------------------
% 3.02/1.71 % (12446)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.71 % (12446)Termination reason: Unknown
% 3.02/1.71 % (12446)Termination phase: shuffling
% 3.02/1.71
% 3.02/1.71 % (12446)Memory used [KB]: 1407
% 3.02/1.71 % (12446)Time elapsed: 0.003 s
% 3.02/1.71 % (12446)Instructions burned: 3 (million)
% 3.02/1.71 % (12446)------------------------------
% 3.02/1.71 % (12446)------------------------------
% 3.02/1.71 % (12450)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_12262 for (2999ds/710Mi)
% 3.02/1.71 % (12449)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_12262 for (2999ds/18Mi)
% 3.02/1.71 % (12447)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_12262 for (2999ds/4Mi)
% 3.02/1.72 % (12447)Instruction limit reached!
% 3.02/1.72 % (12447)------------------------------
% 3.02/1.72 % (12447)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.72 % (12447)Termination reason: Unknown
% 3.02/1.72 % (12447)Termination phase: shuffling
% 3.02/1.72
% 3.02/1.72 % (12447)Memory used [KB]: 1407
% 3.02/1.72 % (12447)Time elapsed: 0.004 s
% 3.02/1.72 % (12447)Instructions burned: 4 (million)
% 3.02/1.72 % (12447)------------------------------
% 3.02/1.72 % (12447)------------------------------
% 3.02/1.72 % (12449)Instruction limit reached!
% 3.02/1.72 % (12449)------------------------------
% 3.02/1.72 % (12449)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.72 % (12449)Termination reason: Unknown
% 3.02/1.72 % (12449)Termination phase: shuffling
% 3.02/1.72
% 3.02/1.72 % (12449)Memory used [KB]: 1791
% 3.02/1.72 % (12449)Time elapsed: 0.010 s
% 3.02/1.72 % (12449)Instructions burned: 19 (million)
% 3.02/1.72 % (12449)------------------------------
% 3.02/1.72 % (12449)------------------------------
% 3.02/1.72 % (12452)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_12262 for (2999ds/902Mi)
% 3.02/1.73 % (12451)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_12262 for (2999ds/6Mi)
% 3.02/1.73 % (12451)Instruction limit reached!
% 3.02/1.73 % (12451)------------------------------
% 3.02/1.73 % (12451)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.73 % (12451)Termination reason: Unknown
% 3.02/1.73 % (12451)Termination phase: shuffling
% 3.02/1.73
% 3.02/1.73 % (12451)Memory used [KB]: 1535
% 3.02/1.73 % (12451)Time elapsed: 0.005 s
% 3.02/1.73 % (12451)Instructions burned: 7 (million)
% 3.02/1.73 % (12451)------------------------------
% 3.02/1.73 % (12451)------------------------------
% 3.02/1.73 % (12454)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_12262 for (2999ds/21Mi)
% 3.02/1.74 % (12456)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_12262 for (2999ds/5Mi)
% 3.02/1.74 % (12456)Instruction limit reached!
% 3.02/1.74 % (12456)------------------------------
% 3.02/1.74 % (12456)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.74 % (12456)Termination reason: Unknown
% 3.02/1.74 % (12456)Termination phase: shuffling
% 3.02/1.74
% 3.02/1.74 % (12456)Memory used [KB]: 1535
% 3.02/1.74 % (12456)Time elapsed: 0.004 s
% 3.02/1.74 % (12456)Instructions burned: 5 (million)
% 3.02/1.74 % (12456)------------------------------
% 3.02/1.74 % (12456)------------------------------
% 3.02/1.75 % (12454)Instruction limit reached!
% 3.02/1.75 % (12454)------------------------------
% 3.02/1.75 % (12454)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.75 % (12454)Termination reason: Unknown
% 3.02/1.75 % (12454)Termination phase: shuffling
% 3.02/1.75
% 3.02/1.75 % (12454)Memory used [KB]: 1791
% 3.02/1.75 % (12454)Time elapsed: 0.013 s
% 3.02/1.75 % (12454)Instructions burned: 22 (million)
% 3.02/1.75 % (12454)------------------------------
% 3.02/1.75 % (12454)------------------------------
% 3.02/1.75 % (12457)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_12262 for (2998ds/6Mi)
% 3.02/1.75 % (12457)Instruction limit reached!
% 3.02/1.75 % (12457)------------------------------
% 3.02/1.75 % (12457)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.75 % (12457)Termination reason: Unknown
% 3.02/1.75 % (12457)Termination phase: shuffling
% 3.02/1.75
% 3.02/1.75 % (12457)Memory used [KB]: 1535
% 3.02/1.75 % (12457)Time elapsed: 0.005 s
% 3.02/1.75 % (12457)Instructions burned: 7 (million)
% 3.02/1.75 % (12457)------------------------------
% 3.02/1.75 % (12457)------------------------------
% 3.02/1.75 % (12458)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_12262 for (2998ds/377Mi)
% 3.02/1.76 % (12459)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_12262 for (2998ds/779Mi)
% 3.02/1.77 % (12460)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_12262 for (2998ds/19Mi)
% 3.02/1.77 % (12460)Instruction limit reached!
% 3.02/1.77 % (12460)------------------------------
% 3.02/1.77 % (12460)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.77 % (12460)Termination reason: Unknown
% 3.02/1.77 % (12460)Termination phase: shuffling
% 3.02/1.77
% 3.02/1.77 % (12460)Memory used [KB]: 1791
% 3.02/1.77 % (12460)Time elapsed: 0.010 s
% 3.02/1.77 % (12460)Instructions burned: 19 (million)
% 3.02/1.77 % (12460)------------------------------
% 3.02/1.77 % (12460)------------------------------
% 3.02/1.78 % (12432)Instruction limit reached!
% 3.02/1.78 % (12432)------------------------------
% 3.02/1.78 % (12432)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.78 % (12432)Termination reason: Unknown
% 3.02/1.78 % (12432)Termination phase: Saturation
% 3.02/1.78
% 3.02/1.78 % (12432)Memory used [KB]: 9466
% 3.02/1.78 % (12432)Time elapsed: 0.137 s
% 3.02/1.78 % (12432)Instructions burned: 276 (million)
% 3.02/1.78 % (12432)------------------------------
% 3.02/1.78 % (12432)------------------------------
% 3.02/1.79 % (12462)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_12262 for (2998ds/879Mi)
% 3.02/1.80 % (12463)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_12262 for (2998ds/17Mi)
% 3.02/1.81 % (12463)Instruction limit reached!
% 3.02/1.81 % (12463)------------------------------
% 3.02/1.81 % (12463)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.81 % (12463)Termination reason: Unknown
% 3.02/1.81 % (12463)Termination phase: shuffling
% 3.02/1.81
% 3.02/1.81 % (12463)Memory used [KB]: 1791
% 3.02/1.81 % (12463)Time elapsed: 0.010 s
% 3.02/1.81 % (12463)Instructions burned: 17 (million)
% 3.02/1.81 % (12463)------------------------------
% 3.02/1.81 % (12463)------------------------------
% 3.02/1.82 % (12466)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_12262 for (2998ds/3Mi)
% 3.02/1.82 % (12466)Instruction limit reached!
% 3.02/1.82 % (12466)------------------------------
% 3.02/1.82 % (12466)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.82 % (12466)Termination reason: Unknown
% 3.02/1.82 % (12466)Termination phase: shuffling
% 3.02/1.82
% 3.02/1.82 % (12466)Memory used [KB]: 1535
% 3.02/1.82 % (12466)Time elapsed: 0.003 s
% 3.02/1.82 % (12466)Instructions burned: 5 (million)
% 3.02/1.82 % (12466)------------------------------
% 3.02/1.82 % (12466)------------------------------
% 3.02/1.83 % (12469)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_12262 for (2998ds/30Mi)
% 3.02/1.84 % (12469)Instruction limit reached!
% 3.02/1.84 % (12469)------------------------------
% 3.02/1.84 % (12469)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.84 % (12469)Termination reason: Unknown
% 3.02/1.84 % (12469)Termination phase: shuffling
% 3.02/1.84
% 3.02/1.84 % (12469)Memory used [KB]: 2046
% 3.02/1.84 % (12469)Time elapsed: 0.010 s
% 3.02/1.84 % (12469)Instructions burned: 33 (million)
% 3.02/1.84 % (12469)------------------------------
% 3.02/1.84 % (12469)------------------------------
% 3.02/1.86 % (12470)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_12262 for (2997ds/127Mi)
% 3.02/1.91 % (12470)Instruction limit reached!
% 3.02/1.91 % (12470)------------------------------
% 3.02/1.91 % (12470)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.91 % (12470)Termination reason: Unknown
% 3.02/1.91 % (12470)Termination phase: Property scanning
% 3.02/1.91
% 3.02/1.91 % (12470)Memory used [KB]: 2558
% 3.02/1.91 % (12470)Time elapsed: 0.057 s
% 3.02/1.91 % (12470)Instructions burned: 128 (million)
% 3.02/1.91 % (12470)------------------------------
% 3.02/1.91 % (12470)------------------------------
% 3.02/1.92 % (12458)Instruction limit reached!
% 3.02/1.92 % (12458)------------------------------
% 3.02/1.92 % (12458)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.92 % (12458)Termination reason: Unknown
% 3.02/1.92 % (12458)Termination phase: Saturation
% 3.02/1.92
% 3.02/1.92 % (12458)Memory used [KB]: 8699
% 3.02/1.92 % (12458)Time elapsed: 0.168 s
% 3.02/1.92 % (12458)Instructions burned: 378 (million)
% 3.02/1.92 % (12458)------------------------------
% 3.02/1.92 % (12458)------------------------------
% 3.02/1.93 % (12477)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_12262 for (2997ds/100Mi)
% 3.02/1.94 % (12478)dis+10_1:1_anc=none:cnfonf=lazy_gen:fd=preordered:fe=off:hud=10:ins=3:ixr=off:nwc=5.0:plsq=on:plsqc=1:plsqr=32,1:sp=const_frequency:uhcvi=on:i=3:si=on:rtra=on_0 on DTF2THF_12262 for (2996ds/3Mi)
% 3.02/1.94 % (12478)Instruction limit reached!
% 3.02/1.94 % (12478)------------------------------
% 3.02/1.94 % (12478)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.94 % (12478)Termination reason: Unknown
% 3.02/1.94 % (12478)Termination phase: shuffling
% 3.02/1.94
% 3.02/1.94 % (12478)Memory used [KB]: 1407
% 3.02/1.94 % (12478)Time elapsed: 0.004 s
% 3.02/1.94 % (12478)Instructions burned: 5 (million)
% 3.02/1.94 % (12478)------------------------------
% 3.02/1.94 % (12478)------------------------------
% 3.02/1.96 % (12481)lrs+10_8:1_au=on:avsq=on:e2e=on:ins=3:s2a=on:s2at=3.0:ss=axioms:i=20:si=on:rtra=on_0 on DTF2THF_12262 for (2996ds/20Mi)
% 3.02/1.97 % (12481)Instruction limit reached!
% 3.02/1.97 % (12481)------------------------------
% 3.02/1.97 % (12481)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.97 % (12481)Termination reason: Unknown
% 3.02/1.97 % (12481)Termination phase: shuffling
% 3.02/1.97
% 3.02/1.97 % (12481)Memory used [KB]: 1791
% 3.02/1.97 % (12481)Time elapsed: 0.012 s
% 3.02/1.97 % (12481)Instructions burned: 20 (million)
% 3.02/1.97 % (12481)------------------------------
% 3.02/1.97 % (12481)------------------------------
% 3.02/1.97 % (12477)Instruction limit reached!
% 3.02/1.97 % (12477)------------------------------
% 3.02/1.97 % (12477)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.97 % (12477)Termination reason: Unknown
% 3.02/1.97 % (12477)Termination phase: Property scanning
% 3.02/1.97
% 3.02/1.97 % (12477)Memory used [KB]: 2558
% 3.02/1.97 % (12477)Time elapsed: 0.046 s
% 3.02/1.97 % (12477)Instructions burned: 101 (million)
% 3.02/1.97 % (12477)------------------------------
% 3.02/1.97 % (12477)------------------------------
% 3.02/1.99 % (12487)lrs+1010_1:1_au=on:cbe=off:nm=2:ntd=on:sd=2:ss=axioms:st=5.0:i=107:si=on:rtra=on_0 on DTF2THF_12262 for (2996ds/107Mi)
% 3.02/1.99 % (12486)dis+1002_1:1_cbe=off:hud=5:nm=4:plsq=on:plsqr=7,1:prag=on:sp=const_max:tnu=1:i=86:si=on:rtra=on_0 on DTF2THF_12262 for (2996ds/86Mi)
% 3.02/2.00 % (12452)First to succeed.
% 3.02/2.00 % (12452)Refutation found. Thanks to Tanya!
% 3.02/2.00 % SZS status Theorem for DTF2THF_12262
% 3.02/2.00 % SZS output start Proof for DTF2THF_12262
% 3.02/2.00 thf(func_def_0, type, in: $i > $i > $o).
% 3.02/2.00 thf(func_def_1, type, exu: ($i > $o) > $o).
% 3.02/2.00 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 3.02/2.00 thf(func_def_8, type, powerset: $i > $i).
% 3.02/2.00 thf(func_def_10, type, setunion: $i > $i).
% 3.02/2.00 thf(func_def_19, type, descr: ($i > $o) > $i).
% 3.02/2.00 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 3.02/2.00 thf(func_def_26, type, prop2set: $o > $i).
% 3.02/2.00 thf(func_def_36, type, nonempty: $i > $o).
% 3.02/2.00 thf(func_def_69, type, set2prop: $i > $o).
% 3.02/2.00 thf(func_def_88, type, subset: $i > $i > $o).
% 3.02/2.00 thf(func_def_89, type, disjoint: $i > $i > $o).
% 3.02/2.00 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 3.02/2.00 thf(func_def_114, type, binunion: $i > $i > $i).
% 3.02/2.00 thf(func_def_122, type, binintersect: $i > $i > $i).
% 3.02/2.00 thf(func_def_135, type, regular: $i > $o).
% 3.02/2.00 thf(func_def_136, type, setminus: $i > $i > $i).
% 3.02/2.00 thf(func_def_147, type, symdiff: $i > $i > $i).
% 3.02/2.00 thf(func_def_153, type, iskpair: $i > $o).
% 3.02/2.00 thf(func_def_158, type, kpair: $i > $i > $i).
% 3.02/2.00 thf(func_def_160, type, cartprod: $i > $i > $i).
% 3.02/2.00 thf(func_def_177, type, singleton: $i > $o).
% 3.02/2.00 thf(func_def_179, type, ex1: $i > ($i > $o) > $o).
% 3.02/2.00 thf(func_def_184, type, atmost1p: $i > $o).
% 3.02/2.00 thf(func_def_185, type, atleast2p: $i > $o).
% 3.02/2.00 thf(func_def_186, type, atmost2p: $i > $o).
% 3.02/2.00 thf(func_def_187, type, upairsetp: $i > $o).
% 3.02/2.00 thf(func_def_191, type, kfst: $i > $i).
% 3.02/2.00 thf(func_def_203, type, ksnd: $i > $i).
% 3.02/2.00 thf(func_def_213, type, breln: $i > $i > $i > $o).
% 3.02/2.00 thf(func_def_214, type, dpsetconstr: $i > $i > ($i > $i > $o) > $i).
% 3.02/2.00 thf(func_def_222, type, func: $i > $i > $i > $o).
% 3.02/2.00 thf(func_def_223, type, funcSet: $i > $i > $i).
% 3.02/2.00 thf(func_def_226, type, ap: $i > $i > $i > $i > $i).
% 3.02/2.00 thf(func_def_232, type, lam: $i > $i > ($i > $i) > $i).
% 3.02/2.00 thf(func_def_259, type, if: $i > $o > $i > $i > $i).
% 3.02/2.00 thf(func_def_287, type, sP1: $i > $i > $i > $i > $o).
% 3.02/2.00 thf(func_def_292, type, sP6: $i > $o > $i > $i > $o).
% 3.02/2.00 thf(func_def_293, type, sP7: $i > $i > $o).
% 3.02/2.00 thf(func_def_294, type, sP8: $i > $o).
% 3.02/2.00 thf(func_def_295, type, sP9: $i > $i > $o).
% 3.02/2.00 thf(func_def_296, type, sP10: $i > $i > $o).
% 3.02/2.00 thf(func_def_297, type, sP11: $i > $i > $i > $o).
% 3.02/2.00 thf(func_def_300, type, sK14: $i > $i > $i).
% 3.02/2.00 thf(func_def_312, type, sK26: $i > $i).
% 3.02/2.00 thf(func_def_313, type, sK27: $i > $i).
% 3.02/2.00 thf(func_def_320, type, sK34: ($i > $o) > $i > $i).
% 3.02/2.00 thf(func_def_322, type, sK36: $i > $o).
% 3.02/2.00 thf(func_def_325, type, sK39: $i > $o).
% 3.02/2.00 thf(func_def_333, type, sK47: $i > $i > $i > $i > $i).
% 3.02/2.00 thf(func_def_343, type, sK57: $i > $i > $i > $i > $i).
% 3.02/2.00 thf(func_def_346, type, sK60: $i > $i).
% 3.02/2.00 thf(func_def_348, type, sK62: ($i > $i) > $i > $i > $i).
% 3.02/2.00 thf(func_def_354, type, sK68: $i > $i > $i > $i > $i).
% 3.02/2.00 thf(func_def_355, type, sK69: $i > $i > $i > $i > $i).
% 3.02/2.00 thf(func_def_375, type, sK89: $i > $i > $i > $i > $i).
% 3.02/2.00 thf(func_def_376, type, sK90: $i > $i > $i > $i > $i).
% 3.02/2.00 thf(func_def_377, type, sK91: $i > $i > $i > $i > $i).
% 3.02/2.00 thf(func_def_378, type, sK92: $i > $i > $i > $i > $i).
% 3.02/2.00 thf(func_def_387, type, sK101: ($i > $o) > $i > $i > $i > $i).
% 3.02/2.00 thf(func_def_388, type, sK102: ($i > $o) > $i > $i > $i > $i).
% 3.02/2.00 thf(func_def_392, type, sK106: $i > $o).
% 3.02/2.00 thf(func_def_402, type, sK116: $i > $o).
% 3.02/2.00 thf(func_def_417, type, sK131: $i > $i > $i > $i).
% 3.02/2.00 thf(func_def_427, type, sK141: $i > $i > $i).
% 3.02/2.00 thf(func_def_428, type, sK142: $i > $i > $i).
% 3.02/2.00 thf(func_def_431, type, sK145: $i > $o).
% 3.02/2.00 thf(func_def_432, type, sK146: $i > $i > $o).
% 3.02/2.00 thf(func_def_439, type, sK153: $i > $o).
% 3.02/2.00 thf(func_def_444, type, sK158: $i > $o).
% 3.02/2.00 thf(func_def_446, type, sK160: $i > ($i > $o) > $i).
% 3.02/2.00 thf(func_def_449, type, sK163: $i > $i > $i).
% 3.02/2.00 thf(func_def_461, type, sK175: ($i > $o) > ($i > $o) > $i).
% 3.02/2.00 thf(func_def_462, type, sK176: ($i > $o) > ($i > $o) > $i).
% 3.02/2.00 thf(func_def_463, type, sK177: $i > $o).
% 3.02/2.00 thf(func_def_464, type, sK178: $i > $o).
% 3.02/2.00 thf(func_def_467, type, sK181: $i > $o).
% 3.02/2.00 thf(func_def_499, type, sK213: $i > $o).
% 3.02/2.00 thf(func_def_502, type, sK216: $i > ($i > $o) > $i).
% 3.02/2.00 thf(func_def_504, type, sK218: $i > $o).
% 3.02/2.00 thf(func_def_517, type, sK231: $i > $o).
% 3.02/2.00 thf(func_def_523, type, sK237: ($i > $o) > $i).
% 3.02/2.00 thf(func_def_524, type, sK238: $i > $o).
% 3.02/2.00 thf(func_def_525, type, sK239: $i > $i).
% 3.02/2.00 thf(func_def_529, type, sK243: $i > $i > $o).
% 3.02/2.00 thf(func_def_545, type, sK259: $i > $i > $i > $i).
% 3.02/2.00 thf(func_def_552, type, sK266: $i > $i).
% 3.02/2.00 thf(func_def_554, type, sK268: $i > ($i > $i) > $i > $i).
% 3.02/2.00 thf(func_def_555, type, sK269: ($i > $i) > $i > $i > $i).
% 3.02/2.00 thf(func_def_558, type, sK272: $i > $i).
% 3.02/2.00 thf(func_def_561, type, sK275: $i > $o).
% 3.02/2.00 thf(func_def_583, type, sK297: $i > $i > $i).
% 3.02/2.00 thf(func_def_589, type, sK303: $i > $i > $o).
% 3.02/2.00 thf(func_def_619, type, sK333: ($i > $o) > ($i > $o) > $i).
% 3.02/2.00 thf(func_def_620, type, sK334: ($i > $o) > ($i > $o) > $i).
% 3.02/2.00 thf(func_def_621, type, sK335: $i > $o).
% 3.02/2.00 thf(func_def_622, type, sK336: $i > $o).
% 3.02/2.00 thf(func_def_623, type, sK337: $i > $i > $i).
% 3.02/2.00 thf(func_def_637, type, sK351: ($i > $o) > $i).
% 3.02/2.00 thf(func_def_638, type, sK352: $i > $o).
% 3.02/2.00 thf(func_def_639, type, sK353: $i > $i).
% 3.02/2.00 thf(func_def_645, type, sK359: $i > $o).
% 3.02/2.00 thf(func_def_647, type, sK361: ($i > $o) > $i > $i).
% 3.02/2.00 thf(func_def_648, type, sK362: ($i > $o) > $i > $i).
% 3.02/2.00 thf(func_def_678, type, sK392: $i > $o).
% 3.02/2.00 thf(func_def_686, type, sK400: $i > $i).
% 3.02/2.00 thf(func_def_688, type, sK402: $i > $i).
% 3.02/2.00 thf(func_def_705, type, sK419: $i > $i > $o).
% 3.02/2.00 thf(func_def_706, type, sK420: $i > $i).
% 3.02/2.00 thf(func_def_707, type, sK421: $i > $i).
% 3.02/2.00 thf(func_def_708, type, sK422: ($i > $i > $o) > $i > $i).
% 3.02/2.00 thf(func_def_709, type, sK423: ($i > $i > $o) > $i > $i).
% 3.02/2.00 thf(func_def_710, type, sK424: $i > ($i > $i > $o) > $i > $i).
% 3.02/2.00 thf(func_def_718, type, sK432: $i > $i > $o).
% 3.02/2.00 thf(func_def_722, type, sK436: $i > $o).
% 3.02/2.00 thf(func_def_723, type, sK437: ($i > $o) > $i).
% 3.02/2.00 thf(func_def_726, type, sK440: $i > $o).
% 3.02/2.00 thf(func_def_731, type, sK445: $i > ($i > $i) > $i > $i).
% 3.02/2.00 thf(func_def_733, type, sK447: $i > $i).
% 3.02/2.00 thf(func_def_736, type, sK450: $i > $i).
% 3.02/2.00 thf(func_def_743, type, sK457: $i > $i > $i).
% 3.02/2.00 thf(func_def_746, type, sK460: $i > $o).
% 3.02/2.00 thf(func_def_749, type, sK463: $i > ($i > $o) > $i).
% 3.02/2.00 thf(func_def_763, type, sK477: $i > $o).
% 3.02/2.00 thf(func_def_764, type, sK478: $i > $o).
% 3.02/2.00 thf(func_def_765, type, sK479: ($i > $o) > ($i > $o) > $i > $i > $i).
% 3.02/2.00 thf(func_def_766, type, sK480: ($i > $o) > ($i > $o) > $i > $i > $i).
% 3.02/2.00 thf(func_def_777, type, sK491: $i > $i).
% 3.02/2.00 thf(func_def_780, type, sK494: ($i > $o) > $i).
% 3.02/2.00 thf(func_def_781, type, sK495: ($i > $o) > $i).
% 3.02/2.00 thf(func_def_782, type, sK496: $i > $o).
% 3.02/2.00 thf(func_def_796, type, sK510: $i > $o).
% 3.02/2.00 thf(func_def_799, type, sK513: $i > $i > ($i > $o) > $i).
% 3.02/2.00 thf(func_def_801, type, sK515: $i > $o).
% 3.02/2.00 thf(func_def_808, type, sK522: $i > $i > $i).
% 3.02/2.00 thf(func_def_809, type, sK523: ($i > $o) > $i > $i).
% 3.02/2.00 thf(func_def_811, type, sK525: $i > $o).
% 3.02/2.00 thf(func_def_815, type, sK529: $i > $i > $o).
% 3.02/2.00 thf(func_def_821, type, sK535: $o > $i > $i > $i).
% 3.02/2.00 thf(func_def_822, type, sK536: $i > $o).
% 3.02/2.00 thf(func_def_825, type, sK539: $i > ($i > $o) > $i).
% 3.02/2.00 thf(func_def_826, type, sK540: $i > ($i > $o) > $i).
% 3.02/2.00 thf(func_def_836, type, sK550: $i > $i > $i > $i > $i).
% 3.02/2.00 thf(func_def_855, type, sK569: $i > $i).
% 3.02/2.00 thf(func_def_861, type, sK575: $i > $i > $i > $i).
% 3.02/2.00 thf(func_def_862, type, sK576: $i > $i > $i > $i).
% 3.02/2.00 thf(func_def_865, type, sK579: $i > $i).
% 3.02/2.00 thf(func_def_866, type, sK580: ($i > $i) > $i > $i > $i).
% 3.02/2.00 thf(func_def_869, type, sK583: $i > $i).
% 3.02/2.00 thf(func_def_872, type, sK586: $i > $i > ($i > $i) > $i).
% 3.02/2.00 thf(func_def_873, type, sK587: $i > $o).
% 3.02/2.00 thf(func_def_875, type, sK589: $i > ($i > $o) > $i).
% 3.02/2.00 thf(func_def_899, type, sK613: $i > $i > $o).
% 3.02/2.00 thf(func_def_907, type, sK621: $i > $o).
% 3.02/2.00 thf(func_def_910, type, sK624: ($i > $o) > $i > $i > $i > $i).
% 3.02/2.00 thf(func_def_911, type, sK625: ($i > $o) > $i > $i > $i > $i).
% 3.02/2.00 thf(func_def_915, type, sK629: $i > $o).
% 3.02/2.00 thf(func_def_926, type, sK640: $i > $i > $i).
% 3.02/2.00 thf(func_def_947, type, sK661: $i > $o).
% 3.02/2.00 thf(func_def_949, type, sK663: ($i > $o) > $i > $i).
% 3.02/2.00 thf(func_def_950, type, sK664: $i > ($i > $o) > $i).
% 3.02/2.00 thf(func_def_951, type, sK665: $i > $o).
% 3.02/2.00 thf(func_def_956, type, sK670: $i > $i > $i).
% 3.02/2.00 thf(func_def_957, type, sK671: $i > $i > $i).
% 3.02/2.00 thf(func_def_958, type, sK672: $i > $i > $i).
% 3.02/2.00 thf(func_def_959, type, sK673: $i > $i > $i).
% 3.02/2.00 thf(func_def_960, type, sK674: $i > $i > $i).
% 3.02/2.00 thf(func_def_961, type, sK675: $i > $i > $i > $i).
% 3.02/2.00 thf(func_def_962, type, sK676: $i > $i).
% 3.02/2.00 thf(func_def_963, type, sK677: $i > $i).
% 3.02/2.00 thf(func_def_964, type, sK678: $i > $i).
% 3.02/2.00 thf(func_def_965, type, sK679: $i > $i).
% 3.02/2.00 thf(func_def_966, type, sK680: $i > $i > $i > $i).
% 3.02/2.00 thf(func_def_967, type, sK681: $i > $i > $i > $i).
% 3.02/2.00 thf(func_def_968, type, sK682: $i > $i > $i).
% 3.02/2.00 thf(func_def_969, type, sK683: $i > $i > $i).
% 3.02/2.00 thf(func_def_970, type, sK684: $i > $i > $i).
% 3.02/2.00 thf(func_def_971, type, sK685: $i > $i).
% 3.02/2.00 thf(func_def_973, type, sK687: $i > $i).
% 3.02/2.00 thf(func_def_974, type, sK688: $i > $i).
% 3.02/2.00 thf(func_def_981, type, sK695: $i > $i > $i).
% 3.02/2.00 thf(func_def_992, type, sK706: ($i > $o) > $i > $i).
% 3.02/2.00 thf(func_def_994, type, sK708: $i > $o).
% 3.02/2.00 thf(func_def_1011, type, sK725: $i > $i > $o).
% 3.02/2.00 thf(func_def_1015, type, sK729: $i > $i > $i > $i).
% 3.02/2.00 thf(func_def_1019, type, sK733: $i > $i > $i > $i).
% 3.02/2.00 thf(func_def_1020, type, sK734: $i > $o).
% 3.02/2.00 thf(func_def_1028, type, ph742: !>[X0: $tType]:(X0)).
% 3.02/2.00 thf(f5979,plain,(
% 3.02/2.00 $false),
% 3.02/2.00 inference(trivial_inequality_removal,[status(thm)],[f5968])).
% 3.02/2.00 thf(f5968,plain,(
% 3.02/2.00 ($true != $true)),
% 3.02/2.00 inference(superposition,[status(thm)],[f3722,f2550])).
% 3.02/2.00 thf(f2550,plain,(
% 3.02/2.00 ($true = (in @ sK167 @ sK165))),
% 3.02/2.00 inference(cnf_transformation,[status(thm)],[f1270])).
% 3.02/2.00 thf(f1270,plain,(
% 3.02/2.00 (setukpairinjR11 = $true) & (exuE1 = $true) & (singletonprop = $true) & (setminusERneg = $true) & (powerset__Cong = $true) & (binunionRsub = $true) & (upairsetE = $true) & (ex1I2 = $true) & (dsetconstrI = $true) & (setadjoinIR = $true) & (foundationAx = $true) & (setukpairIR = $true) & (setminusER = $true) & (exuI3 = $true) & (ubforcartprodlem1 = $true) & (eqinunit = $true) & (funcGraphProp2 = $true) & (binintersectRsub = $true) & (emptyinPowerset = $true) & (kpairp = $true) & (binintersectI = $true) & (binintersectSubset1 = $true) & (subsetRefl = $true) & (funcGraphProp3 = $true) & (sepSubset = $true) & (inPowerset = $true) & (singletoninpowunion = $true) & (cartprodpairsurjEq = $true) & (sepInPowerset = $true) & (setbeta = $true) & (singletonsubset = $true) & (funcinfuncset = $true) & (upairset2E = $true) & (emptysetimpfalse = $true) & (noeltsimpempty = $true) & (cartprodpairin = $true) & (binintersectSubset4 = $true) & (binunionE = $true) & (prop2setI = $true) & (ubforcartprodlem3 = $true) & (disjointsetsI1 = $true) & (((((in @ sK167 @ sK164) = $true) & ($true = (in @ sK165 @ (powerset @ sK166))) & ((in @ sK167 @ sK166) != $true) & ($true = (in @ sK167 @ sK165))) & ((in @ sK166 @ (powerset @ sK164)) = $true)) & ((in @ sK165 @ (powerset @ sK164)) = $true)) & (funcext2 = $true) & (dpsetconstrEL2 = $true) & (setminusT_lem = $true) & (beta2 = $true) & (eqimpsubset2 = $true) & (iftrueProp1 = $true) & (cartprodpairmemEL = $true) & (binunionIR = $true) & (binintersectER = $true) & (setukpairinjR = $true) & (setminusI = $true) & (ap2apEq2 = $true) & (ex1I = $true) & (setunionsingleton1 = $true) & (lamProp = $true) & (powersetI1 = $true) & (emptyInPowerset = $true) & (uniqinunit = $true) & (binunionIL = $true) & (prop2setE = $true) & (subset2powerset = $true) & (cartprodfstin = $true) & (setunion__Cong = $true) & (exuEu = $true) & (notdallE = $true) & (subsetE = $true) & (symdiffIneg1 = $true) & (symdiffE = $true) & (setukpairinjR1 = $true) & (ksndpairEq = $true) & (complementT_lem = $true) & (setunionE = $true) & (upairequniteq = $true) & (ex1E2 = $true) & (setukpairinjL = $true) & (apProp = $true) & (vacuousDall = $true) & (descrp = $true) & (subsetI2 = $true) & (emptyI = $true) & (upairsubunion = $true) & (notsubsetI = $true) & (ifSingleton = $true) & (dpsetconstrEL1 = $true) & (nonemptyI = $true) & (exu__Cong = $true) & (setunionsingleton = $true) & (singletonsswitch = $true) & (beta1 = $true) & (subsetE2 = $true) & (iftrue = $true) & (symdiffIneg2 = $true) & (infuncsetfunc = $true) & (setadjoinE = $true) & (setminusILneg = $true) & (powersetI = $true) & (upairsetIL = $true) & (quantDeMorgan4 = $true) & (powersetTI1 = $true) & (binintersectLsub = $true) & (bs114d = $true) & (singletonsuniq = $true) & (nonemptyE1 = $true) & (in__Cong = $true) & (setunionAx = $true) & (binintersectEL = $true) & (binintersectSubset3 = $true) & (funcextLem = $true) & (replAx = $true) & (subbreln = $true) & (binintersectT_lem = $true) & (emptysetsubset = $true) & (descr__Cong = $true) & (setunionI = $true) & (setextAx = $true) & (singletoninpowerset = $true) & (setadjoinIL = $true) & (setminusSubset1 = $true) & (upairsetIR = $true) & (setadjoinOr = $true) & (dpsetconstrERa = $true) & (binintersectSubset5 = $true) & (exuI1 = $true) & (kfstpairEq = $true) & (iffalse = $true) & (setadjoinSub = $true) & (setunionsingleton2 = $true) & (binintersectSubset2 = $true) & (nonemptyI1 = $true) & (cartprodsndin = $true) & (dsetconstrEL = $true) & (omega0Ax = $true) & (emptyinunitempty = $true) & (setadjoin__Cong = $true) & (setukpairinjR2 = $true) & (subPowSU = $true) & (iftrueProp2 = $true) & (omegaIndAx = $true) & (setOfPairsIsBReln = $true) & (kpairsurjEq = $true) & (notequalI1 = $true) & (binunionT_lem = $true) & (powersetE1 = $true) & (dpsetconstrI = $true) & (exuE3u = $true) & (binunionLsub = $true) & (symdiffI1 = $true) & (funcext = $true) & (dsetconstr__Cong = $true) & (funcGraphProp4 = $true) & (iffalseProp1 = $true) & (setext = $true) & (setminusIRneg = $true) & (theeq = $true) & (quantDeMorgan3 = $true) & (powersetE = $true) & (theprop = $true) & (notequalI2 = $true) & (lamp = $true) & (emptyset__Cong = $true) & (nonemptyImpWitness = $true) & (iffalseProp2 = $true) & (quantDeMorgan2 = $true) & (kpairiskpair = $true) & (exuI2 = $true) & (dpsetconstrER = $true) & (inCongP = $true) & (brelnall1 = $true) & (app = $true) & (cartprodfstpairEq = $true) & (ksndsingleton = $true) & (upairset2IR = $true) & (exuE2 = $true) & (dpsetconstrSub = $true) & (powersetsubset = $true) & (brelnall2 = $true) & (eta1 = $true) & (setukpairinjL1 = $true) & (cartprodmempaircEq = $true) & (cartprodmempair1 = $true) & (lam2lamEq = $true) & (binunionEcases = $true) & (exuE3e = $true) & (dsetconstrER = $true) & (symdiffI2 = $true) & (quantDeMorgan1 = $true) & (funcGraphProp1 = $true) & (setadjoinAx = $true) & (ubforcartprodlem2 = $true) & (notinsingleton = $true) & (subsetI1 = $true) & (funcImageSingleton = $true) & (cartprodsndpairEq = $true) & (ifp = $true) & (cartprodmempair = $true) & (subsetTrans = $true) & (iftrueorfalse = $true) & (setminusELneg = $true) & (emptyE1 = $true) & (setukpairinjL2 = $true) & (setextsub = $true) & (emptysetAx = $true) & (setminusSubset2 = $true) & (setminusLsub = $true) & (ap2p = $true) & (omegaSAx = $true) & (setextT = $true) & (powersetT_lem = $true) & (setukpairinjR12 = $true) & (eqbreln = $true) & (subsetTI = $true) & (cartprodpairmemER = $true) & (emptysetE = $true) & (notdexE = $true) & (setadjoinSub2 = $true) & (lam2p = $true) & (kfstsingleton = $true) & (setoftrueEq = $true) & (omega__Cong = $true) & (ex1E1 = $true) & (upairinpowunion = $true) & (prop2set2propI = $true) & (wellorderingAx = $true) & (setminusEL = $true) & (powersetAx = $true) & (secondinupair = $true) & (eqimpsubset1 = $true) & (subsetemptysetimpeq = $true) & (ap2apEq1 = $true) & (eta2 = $true) & (setunionE2 = $true) & (setukpairIL = $true) & (notinemptyset = $true)),
% 3.02/2.00 inference(skolemisation,[status(esa),new_symbols(skolem,[sK164,sK165,sK166,sK167])],[f885,f1269,f1268,f1267])).
% 3.02/2.00 thf(f1267,plain,(
% 3.02/2.00 ? [X0 : $i,X1 : $i] : (? [X2 : $i] : (? [X3 : $i] : (($true = (in @ X3 @ X0)) & ($true = (in @ X1 @ (powerset @ X2))) & ((in @ X3 @ X2) != $true) & ((in @ X3 @ X1) = $true)) & ((in @ X2 @ (powerset @ X0)) = $true)) & ($true = (in @ X1 @ (powerset @ X0)))) => (? [X2 : $i] : (? [X3 : $i] : (((in @ X3 @ sK164) = $true) & ($true = (in @ sK165 @ (powerset @ X2))) & ((in @ X3 @ X2) != $true) & ((in @ X3 @ sK165) = $true)) & ((in @ X2 @ (powerset @ sK164)) = $true)) & ((in @ sK165 @ (powerset @ sK164)) = $true))),
% 3.02/2.00 introduced(definition,[],[choice_axiom])).
% 3.02/2.00 thf(f1268,plain,(
% 3.02/2.00 ? [X2 : $i] : (? [X3 : $i] : (((in @ X3 @ sK164) = $true) & ($true = (in @ sK165 @ (powerset @ X2))) & ((in @ X3 @ X2) != $true) & ((in @ X3 @ sK165) = $true)) & ((in @ X2 @ (powerset @ sK164)) = $true)) => (? [X3 : $i] : (((in @ X3 @ sK164) = $true) & ($true = (in @ sK165 @ (powerset @ sK166))) & ((in @ X3 @ sK166) != $true) & ((in @ X3 @ sK165) = $true)) & ((in @ sK166 @ (powerset @ sK164)) = $true))),
% 3.02/2.00 introduced(definition,[],[choice_axiom])).
% 3.02/2.00 thf(f1269,plain,(
% 3.02/2.00 ? [X3 : $i] : (((in @ X3 @ sK164) = $true) & ($true = (in @ sK165 @ (powerset @ sK166))) & ((in @ X3 @ sK166) != $true) & ((in @ X3 @ sK165) = $true)) => (((in @ sK167 @ sK164) = $true) & ($true = (in @ sK165 @ (powerset @ sK166))) & ((in @ sK167 @ sK166) != $true) & ($true = (in @ sK167 @ sK165)))),
% 3.02/2.00 introduced(definition,[],[choice_axiom])).
% 3.02/2.00 thf(f885,plain,(
% 3.02/2.00 (setukpairinjR11 = $true) & (exuE1 = $true) & (singletonprop = $true) & (setminusERneg = $true) & (powerset__Cong = $true) & (binunionRsub = $true) & (upairsetE = $true) & (ex1I2 = $true) & (dsetconstrI = $true) & (setadjoinIR = $true) & (foundationAx = $true) & (setukpairIR = $true) & (setminusER = $true) & (exuI3 = $true) & (ubforcartprodlem1 = $true) & (eqinunit = $true) & (funcGraphProp2 = $true) & (binintersectRsub = $true) & (emptyinPowerset = $true) & (kpairp = $true) & (binintersectI = $true) & (binintersectSubset1 = $true) & (subsetRefl = $true) & (funcGraphProp3 = $true) & (sepSubset = $true) & (inPowerset = $true) & (singletoninpowunion = $true) & (cartprodpairsurjEq = $true) & (sepInPowerset = $true) & (setbeta = $true) & (singletonsubset = $true) & (funcinfuncset = $true) & (upairset2E = $true) & (emptysetimpfalse = $true) & (noeltsimpempty = $true) & (cartprodpairin = $true) & (binintersectSubset4 = $true) & (binunionE = $true) & (prop2setI = $true) & (ubforcartprodlem3 = $true) & (disjointsetsI1 = $true) & ? [X0 : $i,X1 : $i] : (? [X2 : $i] : (? [X3 : $i] : (($true = (in @ X3 @ X0)) & ($true = (in @ X1 @ (powerset @ X2))) & ((in @ X3 @ X2) != $true) & ((in @ X3 @ X1) = $true)) & ((in @ X2 @ (powerset @ X0)) = $true)) & ($true = (in @ X1 @ (powerset @ X0)))) & (funcext2 = $true) & (dpsetconstrEL2 = $true) & (setminusT_lem = $true) & (beta2 = $true) & (eqimpsubset2 = $true) & (iftrueProp1 = $true) & (cartprodpairmemEL = $true) & (binunionIR = $true) & (binintersectER = $true) & (setukpairinjR = $true) & (setminusI = $true) & (ap2apEq2 = $true) & (ex1I = $true) & (setunionsingleton1 = $true) & (lamProp = $true) & (powersetI1 = $true) & (emptyInPowerset = $true) & (uniqinunit = $true) & (binunionIL = $true) & (prop2setE = $true) & (subset2powerset = $true) & (cartprodfstin = $true) & (setunion__Cong = $true) & (exuEu = $true) & (notdallE = $true) & (subsetE = $true) & (symdiffIneg1 = $true) & (symdiffE = $true) & (setukpairinjR1 = $true) & (ksndpairEq = $true) & (complementT_lem = $true) & (setunionE = $true) & (upairequniteq = $true) & (ex1E2 = $true) & (setukpairinjL = $true) & (apProp = $true) & (vacuousDall = $true) & (descrp = $true) & (subsetI2 = $true) & (emptyI = $true) & (upairsubunion = $true) & (notsubsetI = $true) & (ifSingleton = $true) & (dpsetconstrEL1 = $true) & (nonemptyI = $true) & (exu__Cong = $true) & (setunionsingleton = $true) & (singletonsswitch = $true) & (beta1 = $true) & (subsetE2 = $true) & (iftrue = $true) & (symdiffIneg2 = $true) & (infuncsetfunc = $true) & (setadjoinE = $true) & (setminusILneg = $true) & (powersetI = $true) & (upairsetIL = $true) & (quantDeMorgan4 = $true) & (powersetTI1 = $true) & (binintersectLsub = $true) & (bs114d = $true) & (singletonsuniq = $true) & (nonemptyE1 = $true) & (in__Cong = $true) & (setunionAx = $true) & (binintersectEL = $true) & (binintersectSubset3 = $true) & (funcextLem = $true) & (replAx = $true) & (subbreln = $true) & (binintersectT_lem = $true) & (emptysetsubset = $true) & (descr__Cong = $true) & (setunionI = $true) & (setextAx = $true) & (singletoninpowerset = $true) & (setadjoinIL = $true) & (setminusSubset1 = $true) & (upairsetIR = $true) & (setadjoinOr = $true) & (dpsetconstrERa = $true) & (binintersectSubset5 = $true) & (exuI1 = $true) & (kfstpairEq = $true) & (iffalse = $true) & (setadjoinSub = $true) & (setunionsingleton2 = $true) & (binintersectSubset2 = $true) & (nonemptyI1 = $true) & (cartprodsndin = $true) & (dsetconstrEL = $true) & (omega0Ax = $true) & (emptyinunitempty = $true) & (setadjoin__Cong = $true) & (setukpairinjR2 = $true) & (subPowSU = $true) & (iftrueProp2 = $true) & (omegaIndAx = $true) & (setOfPairsIsBReln = $true) & (kpairsurjEq = $true) & (notequalI1 = $true) & (binunionT_lem = $true) & (powersetE1 = $true) & (dpsetconstrI = $true) & (exuE3u = $true) & (binunionLsub = $true) & (symdiffI1 = $true) & (funcext = $true) & (dsetconstr__Cong = $true) & (funcGraphProp4 = $true) & (iffalseProp1 = $true) & (setext = $true) & (setminusIRneg = $true) & (theeq = $true) & (quantDeMorgan3 = $true) & (powersetE = $true) & (theprop = $true) & (notequalI2 = $true) & (lamp = $true) & (emptyset__Cong = $true) & (nonemptyImpWitness = $true) & (iffalseProp2 = $true) & (quantDeMorgan2 = $true) & (kpairiskpair = $true) & (exuI2 = $true) & (dpsetconstrER = $true) & (inCongP = $true) & (brelnall1 = $true) & (app = $true) & (cartprodfstpairEq = $true) & (ksndsingleton = $true) & (upairset2IR = $true) & (exuE2 = $true) & (dpsetconstrSub = $true) & (powersetsubset = $true) & (brelnall2 = $true) & (eta1 = $true) & (setukpairinjL1 = $true) & (cartprodmempaircEq = $true) & (cartprodmempair1 = $true) & (lam2lamEq = $true) & (binunionEcases = $true) & (exuE3e = $true) & (dsetconstrER = $true) & (symdiffI2 = $true) & (quantDeMorgan1 = $true) & (funcGraphProp1 = $true) & (setadjoinAx = $true) & (ubforcartprodlem2 = $true) & (notinsingleton = $true) & (subsetI1 = $true) & (funcImageSingleton = $true) & (cartprodsndpairEq = $true) & (ifp = $true) & (cartprodmempair = $true) & (subsetTrans = $true) & (iftrueorfalse = $true) & (setminusELneg = $true) & (emptyE1 = $true) & (setukpairinjL2 = $true) & (setextsub = $true) & (emptysetAx = $true) & (setminusSubset2 = $true) & (setminusLsub = $true) & (ap2p = $true) & (omegaSAx = $true) & (setextT = $true) & (powersetT_lem = $true) & (setukpairinjR12 = $true) & (eqbreln = $true) & (subsetTI = $true) & (cartprodpairmemER = $true) & (emptysetE = $true) & (notdexE = $true) & (setadjoinSub2 = $true) & (lam2p = $true) & (kfstsingleton = $true) & (setoftrueEq = $true) & (omega__Cong = $true) & (ex1E1 = $true) & (upairinpowunion = $true) & (prop2set2propI = $true) & (wellorderingAx = $true) & (setminusEL = $true) & (powersetAx = $true) & (secondinupair = $true) & (eqimpsubset1 = $true) & (subsetemptysetimpeq = $true) & (ap2apEq1 = $true) & (eta2 = $true) & (setunionE2 = $true) & (setukpairIL = $true) & (notinemptyset = $true)),
% 3.02/2.00 inference(flattening,[status(thm)],[f884])).
% 3.02/2.00 thf(f884,plain,(
% 3.02/2.00 (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i,X1 : $i] : (? [X2 : $i] : (? [X3 : $i] : (((((in @ X3 @ X2) != $true) & ((in @ X3 @ X1) = $true)) & ($true = (in @ X1 @ (powerset @ X2)))) & ($true = (in @ X3 @ X0))) & ((in @ X2 @ (powerset @ X0)) = $true)) & ($true = (in @ X1 @ (powerset @ X0)))) & (powersetTI1 = $true)) & (subsetTI = $true)) & (setextT = $true)) & (complementT_lem = $true)) & (setminusT_lem = $true)) & (powersetT_lem = $true)) & (binunionT_lem = $true)) & (binintersectT_lem = $true)) & (iftrueorfalse = $true)) & (iffalse = $true)) & (iftrue = $true)) & (theeq = $true)) & (ifp = $true)) & (ifSingleton = $true)) & (iftrueProp2 = $true)) & (iftrueProp1 = $true)) & (iffalseProp2 = $true)) & (iffalseProp1 = $true)) & (eta2 = $true)) & (beta2 = $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)),
% 3.02/2.00 inference(ennf_transformation,[status(thm)],[f711])).
% 3.02/2.00 thf(f711,plain,(
% 3.02/2.00 ~((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) => ((beta2 = $true) => ((eta2 = $true) => ((iffalseProp1 = $true) => ((iffalseProp2 = $true) => ((iftrueProp1 = $true) => ((iftrueProp2 = $true) => ((ifSingleton = $true) => ((ifp = $true) => ((theeq = $true) => ((iftrue = $true) => ((iffalse = $true) => ((iftrueorfalse = $true) => ((binintersectT_lem = $true) => ((binunionT_lem = $true) => ((powersetT_lem = $true) => ((setminusT_lem = $true) => ((complementT_lem = $true) => ((setextT = $true) => ((subsetTI = $true) => ((powersetTI1 = $true) => ! [X0 : $i,X1 : $i] : (($true = (in @ X1 @ (powerset @ X0))) => ! [X2 : $i] : (((in @ X2 @ (powerset @ X0)) = $true) => ! [X3 : $i] : (($true = (in @ X3 @ X0)) => (($true = (in @ X1 @ (powerset @ X2))) => (((in @ X3 @ X1) = $true) => ((in @ X3 @ X2) = $true))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 3.02/2.00 inference(fool_elimination,[status(thm)],[f710])).
% 3.02/2.00 thf(f710,plain,(
% 3.02/2.00 ~(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 => (beta2 => (eta2 => (iffalseProp1 => (iffalseProp2 => (iftrueProp1 => (iftrueProp2 => (ifSingleton => (ifp => (theeq => (iftrue => (iffalse => (iftrueorfalse => (binintersectT_lem => (binunionT_lem => (powersetT_lem => (setminusT_lem => (complementT_lem => (setextT => (subsetTI => (powersetTI1 => ! [X0 : $i,X1 : $i] : ((in @ X1 @ (powerset @ X0)) => ! [X2 : $i] : ((in @ X2 @ (powerset @ X0)) => ! [X3 : $i] : ((in @ X3 @ X0) => ((in @ X1 @ (powerset @ X2)) => ((in @ X3 @ X1) => (in @ X3 @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 3.02/2.00 inference(rectify,[status(thm)],[f240])).
% 3.02/2.00 thf(f240,negated_conjecture,(
% 3.02/2.00 ~(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 => (beta2 => (eta2 => (iffalseProp1 => (iffalseProp2 => (iftrueProp1 => (iftrueProp2 => (ifSingleton => (ifp => (theeq => (iftrue => (iffalse => (iftrueorfalse => (binintersectT_lem => (binunionT_lem => (powersetT_lem => (setminusT_lem => (complementT_lem => (setextT => (subsetTI => (powersetTI1 => ! [X3 : $i,X11 : $i] : ((in @ X11 @ (powerset @ X3)) => ! [X16 : $i] : ((in @ X16 @ (powerset @ X3)) => ! [X1 : $i] : ((in @ X1 @ X3) => ((in @ X11 @ (powerset @ X16)) => ((in @ X1 @ X11) => (in @ X1 @ X16))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 3.02/2.00 inference(negated_conjecture,[status(cth)],[f239])).
% 3.02/2.00 thf(f239,conjecture,(
% 3.02/2.00 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 => (beta2 => (eta2 => (iffalseProp1 => (iffalseProp2 => (iftrueProp1 => (iftrueProp2 => (ifSingleton => (ifp => (theeq => (iftrue => (iffalse => (iftrueorfalse => (binintersectT_lem => (binunionT_lem => (powersetT_lem => (setminusT_lem => (complementT_lem => (setextT => (subsetTI => (powersetTI1 => ! [X3 : $i,X11 : $i] : ((in @ X11 @ (powerset @ X3)) => ! [X16 : $i] : ((in @ X16 @ (powerset @ X3)) => ! [X1 : $i] : ((in @ X1 @ X3) => ((in @ X11 @ (powerset @ X16)) => ((in @ X1 @ X11) => (in @ X1 @ X16)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 3.02/2.00 file('/export/starexec/sandbox2/tmp/tmp.OYw9T6ZJNR/DTF2THF_12262.p',powersetTE1)).
% 3.02/2.00 thf(f3722,plain,(
% 3.02/2.00 ($true != (in @ sK167 @ sK165))),
% 3.02/2.00 inference(trivial_inequality_removal,[status(thm)],[f3691])).
% 3.02/2.00 thf(f3691,plain,(
% 3.02/2.00 ($true != (in @ sK167 @ sK165)) | ($true != $true)),
% 3.02/2.00 inference(superposition,[status(thm)],[f3547,f3647])).
% 3.02/2.00 thf(f3647,plain,(
% 3.02/2.00 ($true = (subset @ sK165 @ sK166))),
% 3.02/2.00 inference(trivial_inequality_removal,[status(thm)],[f3646])).
% 3.02/2.00 thf(f3646,plain,(
% 3.02/2.00 ($true != $true) | ($true = (subset @ sK165 @ sK166))),
% 3.02/2.00 inference(forward_demodulation,[status(thm)],[f3630,f2445])).
% 3.02/2.00 thf(f2445,plain,(
% 3.02/2.00 (powersetE1 = $true)),
% 3.02/2.00 inference(cnf_transformation,[status(thm)],[f1270])).
% 3.02/2.00 thf(f3630,plain,(
% 3.02/2.00 ($true = (subset @ sK165 @ sK166)) | (powersetE1 != $true)),
% 3.02/2.00 inference(trivial_inequality_removal,[status(thm)],[f3623])).
% 3.02/2.00 thf(f3623,plain,(
% 3.02/2.00 ($true = (subset @ sK165 @ sK166)) | ($true != $true) | (powersetE1 != $true)),
% 3.02/2.00 inference(superposition,[status(thm)],[f3077,f2552])).
% 3.02/2.00 thf(f2552,plain,(
% 3.02/2.00 ($true = (in @ sK165 @ (powerset @ sK166)))),
% 3.02/2.00 inference(cnf_transformation,[status(thm)],[f1270])).
% 3.02/2.00 thf(f3077,plain,(
% 3.02/2.00 ( ! [X2 : $i,X3 : $i] : (($true != (in @ X2 @ (powerset @ X3))) | (powersetE1 != $true) | ((subset @ X2 @ X3) = $true)) )),
% 3.02/2.00 inference(cnf_transformation,[status(thm)],[f1849])).
% 3.02/2.00 thf(f1849,plain,(
% 3.02/2.00 ((powersetE1 = $true) | (($true = (in @ sK544 @ (powerset @ sK545))) & ((subset @ sK544 @ sK545) != $true))) & (! [X2 : $i,X3 : $i] : (($true != (in @ X2 @ (powerset @ X3))) | ((subset @ X2 @ X3) = $true)) | (powersetE1 != $true))),
% 3.02/2.00 inference(skolemisation,[status(esa),new_symbols(skolem,[sK544,sK545])],[f1847,f1848])).
% 3.02/2.00 thf(f1848,plain,(
% 3.02/2.00 ? [X0 : $i,X1 : $i] : (((in @ X0 @ (powerset @ X1)) = $true) & ((subset @ X0 @ X1) != $true)) => (($true = (in @ sK544 @ (powerset @ sK545))) & ((subset @ sK544 @ sK545) != $true))),
% 3.02/2.00 introduced(definition,[],[choice_axiom])).
% 3.02/2.00 thf(f1847,plain,(
% 3.02/2.00 ((powersetE1 = $true) | ? [X0 : $i,X1 : $i] : (((in @ X0 @ (powerset @ X1)) = $true) & ((subset @ X0 @ X1) != $true))) & (! [X2 : $i,X3 : $i] : (($true != (in @ X2 @ (powerset @ X3))) | ((subset @ X2 @ X3) = $true)) | (powersetE1 != $true))),
% 3.02/2.00 inference(rectify,[status(thm)],[f1846])).
% 3.02/2.00 thf(f1846,plain,(
% 3.02/2.00 ((powersetE1 = $true) | ? [X0 : $i,X1 : $i] : (((in @ X0 @ (powerset @ X1)) = $true) & ((subset @ X0 @ X1) != $true))) & (! [X0 : $i,X1 : $i] : (((in @ X0 @ (powerset @ X1)) != $true) | ((subset @ X0 @ X1) = $true)) | (powersetE1 != $true))),
% 3.02/2.00 inference(nnf_transformation,[status(thm)],[f819])).
% 3.02/2.00 thf(f819,plain,(
% 3.02/2.00 (powersetE1 = $true) <=> ! [X0 : $i,X1 : $i] : (((in @ X0 @ (powerset @ X1)) != $true) | ((subset @ X0 @ X1) = $true))),
% 3.02/2.00 inference(ennf_transformation,[status(thm)],[f361])).
% 3.02/2.00 thf(f361,plain,(
% 3.02/2.00 ! [X0 : $i,X1 : $i] : (((in @ X0 @ (powerset @ X1)) = $true) => ((subset @ X0 @ X1) = $true)) <=> (powersetE1 = $true)),
% 3.02/2.00 inference(fool_elimination,[status(thm)],[f360])).
% 3.02/2.00 thf(f360,plain,(
% 3.02/2.00 (! [X0 : $i,X1 : $i] : ((in @ X0 @ (powerset @ X1)) => (subset @ X0 @ X1)) = powersetE1)),
% 3.02/2.00 inference(rectify,[status(thm)],[f96])).
% 3.02/2.00 thf(f96,axiom,(
% 3.02/2.00 (! [X4 : $i,X3 : $i] : ((in @ X4 @ (powerset @ X3)) => (subset @ X4 @ X3)) = powersetE1)),
% 3.02/2.00 file('/export/starexec/sandbox2/tmp/tmp.OYw9T6ZJNR/DTF2THF_12262.p',powersetE1)).
% 3.02/2.00 thf(f3547,plain,(
% 3.02/2.00 ( ! [X0 : $i] : (($true != (subset @ X0 @ sK166)) | ($true != (in @ sK167 @ X0))) )),
% 3.02/2.00 inference(trivial_inequality_removal,[status(thm)],[f3546])).
% 3.02/2.00 thf(f3546,plain,(
% 3.02/2.00 ( ! [X0 : $i] : (($true != (in @ sK167 @ X0)) | ($true != (subset @ X0 @ sK166)) | ($true != $true)) )),
% 3.02/2.00 inference(forward_demodulation,[status(thm)],[f3544,f2522])).
% 3.02/2.00 thf(f2522,plain,(
% 3.02/2.00 (subsetE = $true)),
% 3.02/2.00 inference(cnf_transformation,[status(thm)],[f1270])).
% 3.02/2.00 thf(f3544,plain,(
% 3.02/2.00 ( ! [X0 : $i] : (($true != (subset @ X0 @ sK166)) | (subsetE != $true) | ($true != (in @ sK167 @ X0))) )),
% 3.02/2.00 inference(trivial_inequality_removal,[status(thm)],[f3538])).
% 3.02/2.00 thf(f3538,plain,(
% 3.02/2.00 ( ! [X0 : $i] : (($true != (subset @ X0 @ sK166)) | (subsetE != $true) | ($true != $true) | ($true != (in @ sK167 @ X0))) )),
% 3.02/2.00 inference(superposition,[status(thm)],[f2551,f2635])).
% 3.02/2.00 thf(f2635,plain,(
% 3.02/2.00 ( ! [X3 : $i,X4 : $i,X5 : $i] : (((in @ X5 @ X4) = $true) | (subsetE != $true) | ($true != (in @ X5 @ X3)) | ((subset @ X3 @ X4) != $true)) )),
% 3.02/2.00 inference(cnf_transformation,[status(thm)],[f1318])).
% 3.02/2.00 thf(f1318,plain,(
% 3.02/2.00 ((subsetE = $true) | (((subset @ sK197 @ sK198) = $true) & ($true != (in @ sK199 @ sK198)) & ((in @ sK199 @ sK197) = $true))) & (! [X3 : $i,X4 : $i,X5 : $i] : (((subset @ X3 @ X4) != $true) | ((in @ X5 @ X4) = $true) | ($true != (in @ X5 @ X3))) | (subsetE != $true))),
% 3.02/2.00 inference(skolemisation,[status(esa),new_symbols(skolem,[sK197,sK198,sK199])],[f1316,f1317])).
% 3.02/2.00 thf(f1317,plain,(
% 3.02/2.00 ? [X0 : $i,X1 : $i,X2 : $i] : (((subset @ X0 @ X1) = $true) & ((in @ X2 @ X1) != $true) & ((in @ X2 @ X0) = $true)) => (((subset @ sK197 @ sK198) = $true) & ($true != (in @ sK199 @ sK198)) & ((in @ sK199 @ sK197) = $true))),
% 3.02/2.00 introduced(definition,[],[choice_axiom])).
% 3.02/2.00 thf(f1316,plain,(
% 3.02/2.00 ((subsetE = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (((subset @ X0 @ X1) = $true) & ((in @ X2 @ X1) != $true) & ((in @ X2 @ X0) = $true))) & (! [X3 : $i,X4 : $i,X5 : $i] : (((subset @ X3 @ X4) != $true) | ((in @ X5 @ X4) = $true) | ($true != (in @ X5 @ X3))) | (subsetE != $true))),
% 3.02/2.00 inference(rectify,[status(thm)],[f1315])).
% 3.02/2.00 thf(f1315,plain,(
% 3.02/2.00 ((subsetE = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (((subset @ X0 @ X1) = $true) & ((in @ X2 @ X1) != $true) & ((in @ X2 @ X0) = $true))) & (! [X0 : $i,X1 : $i,X2 : $i] : (((subset @ X0 @ X1) != $true) | ((in @ X2 @ X1) = $true) | ((in @ X2 @ X0) != $true)) | (subsetE != $true))),
% 3.02/2.00 inference(nnf_transformation,[status(thm)],[f955])).
% 3.02/2.00 thf(f955,plain,(
% 3.02/2.00 (subsetE = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i] : (((subset @ X0 @ X1) != $true) | ((in @ X2 @ X1) = $true) | ((in @ X2 @ X0) != $true))),
% 3.02/2.00 inference(flattening,[status(thm)],[f954])).
% 3.02/2.00 thf(f954,plain,(
% 3.02/2.00 (subsetE = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i] : ((((in @ X2 @ X1) = $true) | ((in @ X2 @ X0) != $true)) | ((subset @ X0 @ X1) != $true))),
% 3.02/2.00 inference(ennf_transformation,[status(thm)],[f335])).
% 3.02/2.00 thf(f335,plain,(
% 3.02/2.00 (subsetE = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i] : (((subset @ X0 @ X1) = $true) => (((in @ X2 @ X0) = $true) => ((in @ X2 @ X1) = $true)))),
% 3.02/2.00 inference(fool_elimination,[status(thm)],[f334])).
% 3.02/2.00 thf(f334,plain,(
% 3.02/2.00 (subsetE = ! [X0 : $i,X1 : $i,X2 : $i] : ((subset @ X0 @ X1) => ((in @ X2 @ X0) => (in @ X2 @ X1))))),
% 3.02/2.00 inference(rectify,[status(thm)],[f83])).
% 3.02/2.00 thf(f83,axiom,(
% 3.02/2.00 (subsetE = ! [X3 : $i,X4 : $i,X1 : $i] : ((subset @ X3 @ X4) => ((in @ X1 @ X3) => (in @ X1 @ X4))))),
% 3.02/2.00 file('/export/starexec/sandbox2/tmp/tmp.OYw9T6ZJNR/DTF2THF_12262.p',subsetE)).
% 3.02/2.00 thf(f2551,plain,(
% 3.02/2.00 ((in @ sK167 @ sK166) != $true)),
% 3.02/2.00 inference(cnf_transformation,[status(thm)],[f1270])).
% 3.02/2.00 % SZS output end Proof for DTF2THF_12262
% 3.02/2.00 % (12452)------------------------------
% 3.02/2.00 % (12452)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/2.00 % (12452)Termination reason: Refutation
% 3.02/2.00
% 3.02/2.00 % (12452)Memory used [KB]: 10490
% 3.02/2.00 % (12452)Time elapsed: 0.279 s
% 3.02/2.00 % (12452)Instructions burned: 625 (million)
% 3.02/2.00 % (12452)------------------------------
% 3.02/2.00 % (12452)------------------------------
% 3.02/2.00 % (12426)Success in time 0.387 s
%------------------------------------------------------------------------------