%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU786^1 : TPTP v9.2.1. Released v3.7.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n026.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 09:37:11 AM UTC 2026
% Result : Theorem 5.12s 2.11s
% Output : Refutation 5.12s
% Verified :
% SZS Type : Refutation
% Derivation depth : 14
% Number of leaves : 9
% Syntax : Number of formulae : 41 ( 9 unt; 0 typ; 2 def)
% Number of atoms : 5130 (1350 equ; 0 cnn)
% Maximal formula atoms : 309 ( 125 avg)
% Number of connectives : 2796 ( 61 ~; 41 |; 973 &; 480 @)
% ( 5 <=>;1236 =>; 0 <=; 0 <~>)
% Maximal formula depth : 319 ( 57 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 532 ( 529 usr; 314 con; 0-6 aty)
% Number of variables : 83 ( 0 ^; 47 !; 36 ?; 83 :)
% 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_323,type,
breln1: $i > $i > $o ).
thf(func_def_325,type,
breln1Set: $i > $i ).
thf(func_def_327,type,
transitive: $i > $i > $o ).
thf(func_def_328,type,
antisymmetric: $i > $i > $o ).
thf(func_def_329,type,
reflexive: $i > $i > $o ).
thf(func_def_330,type,
refltransitive: $i > $i > $o ).
thf(func_def_331,type,
refllinearorder: $i > $i > $o ).
thf(func_def_332,type,
reflwellordering: $i > $i > $o ).
thf(func_def_338,type,
breln1invset: $i > $i > $i ).
thf(func_def_342,type,
breln1compset: $i > $i > $i > $i ).
thf(func_def_368,type,
sP4: $i > $i > $i > $o ).
thf(func_def_369,type,
sP5: $i > $i > $o ).
thf(func_def_370,type,
sP6: $i > $o ).
thf(func_def_371,type,
sP7: $i > $i > $o ).
thf(func_def_372,type,
sP8: $i > $i > $o ).
thf(func_def_374,type,
sP10: $i > $i > $i > $o ).
thf(func_def_377,type,
sP13: $i > $i > $i > $i > $o ).
thf(func_def_379,type,
sP15: $o > $i > $i > $i > $o ).
thf(func_def_389,type,
sK25: $i > $i > $i ).
thf(func_def_394,type,
sK30: $i > $o ).
thf(func_def_396,type,
sK32: $i > ( $i > $o ) > $i ).
thf(func_def_405,type,
sK41: $i > $i > $i > $i > $i > $i ).
thf(func_def_417,type,
sK53: $o > $i > $i > $i > $i > $i > $i ).
thf(func_def_420,type,
sK56: $i > $i > $i ).
thf(func_def_421,type,
sK57: $i > $i > $i ).
thf(func_def_434,type,
sK70: $i > $i ).
thf(func_def_443,type,
sK79: $i > $i > $o ).
thf(func_def_446,type,
sK82: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_447,type,
sK83: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_448,type,
sK84: $i > $o ).
thf(func_def_449,type,
sK85: $i > $o ).
thf(func_def_469,type,
sK105: $i > $i > $i > $i > $i ).
thf(func_def_477,type,
sK113: $i > $o ).
thf(func_def_479,type,
sK115: ( $i > $o ) > $i > $i ).
thf(func_def_480,type,
sK116: ( $i > $o ) > $i > $i ).
thf(func_def_488,type,
sK124: $i > $i > $o ).
thf(func_def_495,type,
sK131: $o > $i > $i > $i ).
thf(func_def_500,type,
sK136: $i > $i > $o ).
thf(func_def_509,type,
sK145: $i > $i > $o ).
thf(func_def_513,type,
sK149: $i > $i ).
thf(func_def_514,type,
sK150: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_515,type,
sK151: ( $i > $o ) > $i ).
thf(func_def_516,type,
sK152: $i > $o ).
thf(func_def_517,type,
sK153: $i > $i ).
thf(func_def_529,type,
sK165: $i > ( $i > $i ) > $i > $i ).
thf(func_def_531,type,
sK167: $i > $i ).
thf(func_def_537,type,
sK173: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_538,type,
sK174: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_542,type,
sK178: $i > $o ).
thf(func_def_553,type,
sK189: $i > $i > $i > $i ).
thf(func_def_554,type,
sK190: $i > $i > $i > $i ).
thf(func_def_573,type,
sK209: $i > $i > $i ).
thf(func_def_574,type,
sK210: $i > $i > $i ).
thf(func_def_575,type,
sK211: $i > $i > $i ).
thf(func_def_576,type,
sK212: $i > $i > $i ).
thf(func_def_577,type,
sK213: $i > $i > $i ).
thf(func_def_578,type,
sK214: $i > $i > $i > $i ).
thf(func_def_579,type,
sK215: $i > $i ).
thf(func_def_580,type,
sK216: $i > $i ).
thf(func_def_581,type,
sK217: $i > $i ).
thf(func_def_582,type,
sK218: $i > $i ).
thf(func_def_583,type,
sK219: $i > $i > $i > $i ).
thf(func_def_584,type,
sK220: $i > $i > $i > $i ).
thf(func_def_585,type,
sK221: $i > $i > $i ).
thf(func_def_586,type,
sK222: $i > $i > $i ).
thf(func_def_587,type,
sK223: $i > $i > $i ).
thf(func_def_589,type,
sK225: $i > $i ).
thf(func_def_590,type,
sK226: $i > $i ).
thf(func_def_591,type,
sK227: $i > $i ).
thf(func_def_635,type,
sK271: $i > $i > $i > $i > $i ).
thf(func_def_638,type,
sK274: $i > $o ).
thf(func_def_647,type,
sK283: $i > $o ).
thf(func_def_649,type,
sK285: $i > ( $i > $o ) > $i ).
thf(func_def_672,type,
sK308: $i > $i > $i > $i > $i ).
thf(func_def_673,type,
sK309: $i > $i > $i > $i > $i ).
thf(func_def_679,type,
sK315: $i > $i ).
thf(func_def_681,type,
sK317: $i > ( $i > $i ) > $i > $i ).
thf(func_def_691,type,
sK327: $i > $o ).
thf(func_def_697,type,
sK333: $i > $i > $i > $i ).
thf(func_def_698,type,
sK334: $i > $i > $i > $i ).
thf(func_def_699,type,
sK335: $i > $i > $i > $i ).
thf(func_def_700,type,
sK336: $i > $i > $i > $i ).
thf(func_def_716,type,
sK352: $i > $i ).
thf(func_def_740,type,
sK376: $i > $i > $i > $i ).
thf(func_def_784,type,
sK420: $i > $o ).
thf(func_def_789,type,
sK425: $i > $i > $i ).
thf(func_def_796,type,
sK432: $i > $o ).
thf(func_def_802,type,
sK438: ( $i > $o ) > $i ).
thf(func_def_803,type,
sK439: $i > $o ).
thf(func_def_804,type,
sK440: ( $i > $i ) > $i > $i > $i ).
thf(func_def_807,type,
sK443: $i > $i ).
thf(func_def_816,type,
sK452: ( $i > $o ) > $i ).
thf(func_def_817,type,
sK453: $i > $o ).
thf(func_def_818,type,
sK454: $i > $i ).
thf(func_def_819,type,
sK455: $i > $i > $o ).
thf(func_def_821,type,
sK457: $i > $i ).
thf(func_def_822,type,
sK458: $i > $i ).
thf(func_def_823,type,
sK459: $i > ( $i > $i > $o ) > $i ).
thf(func_def_824,type,
sK460: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_825,type,
sK461: $i > ( $i > $i > $o ) > $i ).
thf(func_def_827,type,
sK463: $i > $o ).
thf(func_def_829,type,
sK465: ( $i > $o ) > $i > $i ).
thf(func_def_830,type,
sK466: $i > $i > $i ).
thf(func_def_842,type,
sK478: ( $i > $o ) > $i > $i > $i ).
thf(func_def_843,type,
sK479: ( $i > $o ) > $i > $i > $i ).
thf(func_def_846,type,
sK482: $i > $o ).
thf(func_def_854,type,
sK490: $i > $i > $i > $i ).
thf(func_def_855,type,
sK491: $i > $i > $i > $i ).
thf(func_def_856,type,
sK492: $i > $o ).
thf(func_def_863,type,
sK499: $i > $o ).
thf(func_def_866,type,
sK502: $i > ( $i > $o ) > $i ).
thf(func_def_868,type,
sK504: $i > $i ).
thf(func_def_872,type,
sK508: $i > $i > ( $i > $i ) > $i ).
thf(func_def_873,type,
sK509: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_874,type,
sK510: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_877,type,
sK513: $i > $o ).
thf(func_def_878,type,
sK514: $i > $o ).
thf(func_def_893,type,
sK529: $i > $i > $i ).
thf(func_def_896,type,
sK532: ( $i > $o ) > $i > $i ).
thf(func_def_898,type,
sK534: $i > $o ).
thf(func_def_913,type,
sK549: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_914,type,
sK550: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_918,type,
sK554: $i > $o ).
thf(func_def_921,type,
sK557: $i > $i ).
thf(func_def_924,type,
sK560: $i > ( $i > $i ) > $i > $i ).
thf(func_def_931,type,
sK567: $i > $i ).
thf(func_def_933,type,
sK569: $i > $i ).
thf(func_def_934,type,
sK570: $i > $i ).
thf(func_def_947,type,
sK583: $i > $o ).
thf(func_def_949,type,
sK585: $i > ( $i > $o ) > $i > $i ).
thf(func_def_957,type,
sK593: $i > $i > $o ).
thf(func_def_963,type,
sK599: $i > $i > $i > $i > $i ).
thf(func_def_964,type,
sK600: $i > $i > $i > $i > $i ).
thf(func_def_965,type,
sK601: $i > $i > $i > $i > $i ).
thf(func_def_966,type,
sK602: $i > $i > $i > $i > $i ).
thf(func_def_970,type,
sK606: $i > $o ).
thf(func_def_980,type,
sK616: $i > $i > $o ).
thf(func_def_987,type,
sK623: $i > $i > $i ).
thf(func_def_994,type,
sK630: $i > $o ).
thf(func_def_1015,type,
sK651: $i > $i > $i ).
thf(func_def_1022,type,
sK658: $i > $i > $i > $i > $i ).
thf(func_def_1040,type,
sK676: $i > $i > $i > $i ).
thf(func_def_1041,type,
sK677: ( $i > $o ) > $i > $i ).
thf(func_def_1043,type,
sK679: $i > $o ).
thf(func_def_1068,type,
sK704: $i > $o ).
thf(func_def_1078,type,
sK714: $i > $i > $i ).
thf(func_def_1106,type,
sK742: $i > $o ).
thf(func_def_1107,type,
sK743: $i > $o ).
thf(func_def_1108,type,
sK744: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_1109,type,
sK745: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_1119,type,
sK755: $i > $o ).
thf(func_def_1147,type,
sK783: $i > $o ).
thf(func_def_1151,type,
sK787: $i > $o ).
thf(func_def_1153,type,
sK789: $i > ( $i > $o ) > $i ).
thf(func_def_1164,type,
sK800: $i > $i ).
thf(func_def_1192,type,
sK828: $i > $i > $i > $i ).
thf(func_def_1193,type,
sK829: $i > $i > $i > $i ).
thf(func_def_1198,type,
sK834: $i > $o ).
thf(func_def_1200,type,
sK836: $i > $o ).
thf(func_def_1206,type,
sK842: $i > $i > $o ).
thf(func_def_1222,type,
sK858: $i > $i > $i ).
thf(func_def_1247,type,
sK883: ( $i > $o ) > $i > $i ).
thf(func_def_1249,type,
sK885: $i > $o ).
thf(func_def_1254,type,
sK890: $i > $i > $o ).
thf(func_def_1264,type,
sK900: $i > $i ).
thf(func_def_1267,type,
sK903: ( $i > $o ) > $i > $i > $i ).
thf(func_def_1268,type,
sK904: ( $i > $o ) > $i > $i > $i ).
thf(func_def_1271,type,
sK907: $i > $o ).
thf(func_def_1273,type,
sK909: $i > $o ).
thf(func_def_1276,type,
sK912: $i > ( $i > $o ) > $i ).
thf(func_def_1277,type,
sK913: $i > ( $i > $o ) > $i ).
thf(func_def_1299,type,
sK935: $i > $o ).
thf(func_def_1334,type,
sK970: $i > $o ).
thf(func_def_1336,type,
sK972: ( $i > $o ) > $i > $i ).
thf(func_def_1337,type,
sK973: $i > $i ).
thf(func_def_1340,type,
sK976: $i > $i > ( $i > $i ) > $i ).
thf(func_def_1364,type,
sK1000: $i > $i > $o ).
thf(func_def_1373,type,
sK1009: $i > $o ).
thf(func_def_1375,type,
sK1011: ( $i > $o ) > $i ).
thf(func_def_1376,type,
sK1012: ( $i > $o ) > $i ).
thf(func_def_1382,type,
ph1018:
!>[X0: $tType] : X0 ).
thf(f4926,plain,
$false,
inference(avatar_sat_refutation,[status(thm)],[f4847,f4876,f4912]) ).
thf(f4912,plain,
~ spl1017_6,
inference(avatar_contradiction_clause,[status(thm)],[f4911]) ).
thf(f4911,plain,
( $false
| ~ spl1017_6 ),
inference(trivial_inequality_removal,[status(thm)],[f4906]) ).
thf(f4906,plain,
( ( $true != $true )
| ~ spl1017_6 ),
inference(superposition,[status(thm)],[f3136,f4843]) ).
thf(f4843,plain,
( ( $true
= ( in @ ( kpair @ sK68 @ sK69 ) @ sK67 ) )
| ~ spl1017_6 ),
inference(avatar_component_clause,[status(thm)],[f4841]) ).
thf(f4841,definition,
( spl1017_6
<=> ( $true
= ( in @ ( kpair @ sK68 @ sK69 ) @ sK67 ) ) ),
introduced(definition,[new_symbols(naming,[spl1017_6])],[avatar_definition]) ).
thf(f3136,plain,
( $true
!= ( in @ ( kpair @ sK68 @ sK69 ) @ sK67 ) ),
inference(cnf_transformation,[status(thm)],[f1443]) ).
thf(f1443,plain,
( ( complementTcontraSubset = $true )
& ( cartprodsndpairEq = $true )
& ( setukpairinjR2 = $true )
& ( setminusELneg = $true )
& ( inIntersectImpInUnion2 = $true )
& ( setminusSubset1 = $true )
& ( ap2apEq1 = $true )
& ( demorgan2b = $true )
& ( vacuousDall = $true )
& ( iftrueorfalse = $true )
& ( breln1invprop = $true )
& ( subsetTI = $true )
& ( exuI1 = $true )
& ( setukpairinjL1 = $true )
& ( quantDeMorgan2 = $true )
& ( ksndpairEq = $true )
& ( symdiffI1 = $true )
& ( notinemptyset = $true )
& ( setadjoinSub2 = $true )
& ( demorgan2 = $true )
& ( exuE3u = $true )
& ( singletonsuniq = $true )
& ( upairsetIL = $true )
& ( setunionsingleton1 = $true )
& ( inIntersectImpInIntersectUnions = $true )
& ( woz1_2 = $true )
& ( contrasubsetT = $true )
& ( emptyset__Cong = $true )
& ( descr__Cong = $true )
& ( binintersectSubset3 = $true )
& ( beta2 = $true )
& ( ifSingleton = $true )
& ( complementUnionInPowersetComplement = $true )
& ( singletoninpowunion = $true )
& ( funcGraphProp1 = $true )
& ( doubleComplementEq = $true )
& ( setextT = $true )
& ( foundationAx = $true )
& ( replAx = $true )
& ( setminusI = $true )
& ( dsetconstrI = $true )
& ( inCongP = $true )
& ( setextAx = $true )
& ( binintersectI = $true )
& ( binunionTEcontra = $true )
& ( eqbreln = $true )
& ( funcext = $true )
& ( breln1compE = $true )
& ( powersetI1 = $true )
& ( setadjoinAx = $true )
& ( prop2setE = $true )
& ( cartprodpairsurjEq = $true )
& ( woz13rule2 = $true )
& ( setukpairinjR12 = $true )
& ( binintersectEL = $true )
& ( quantDeMorgan1 = $true )
& ( inComplementUnionImpInComplement1 = $true )
& ( setukpairinjR11 = $true )
& ( binunionRsub = $true )
& ( emptyinunitempty = $true )
& ( breln1unionprop = $true )
& ( subsetI2 = $true )
& ( doubleComplementI1 = $true )
& ( uniqinunit = $true )
& ( eqbreln1 = $true )
& ( sepInPowerset = $true )
& ( binunionTE = $true )
& ( upairsetE = $true )
& ( exuE2 = $true )
& ( complementInPowersetComplementIntersect = $true )
& ( omegaSAx = $true )
& ( breln1invE = $true )
& ( dpsetconstrEL2 = $true )
& ( prop2set2propI = $true )
& ( cartprodmempair = $true )
& ( omega__Cong = $true )
& ( doubleComplementSub1 = $true )
& ( setukpairinjR1 = $true )
& ( descrp = $true )
& ( powersetT_lem = $true )
& ( setminusILneg = $true )
& ( kpairsurjEq = $true )
& ( binunionTILcontra = $true )
& ( woz13rule4 = $true )
& ( setadjoinE = $true )
& ( setextsub = $true )
& ( dsetconstrER = $true )
& ( dpsetconstrEL1 = $true )
& ( complementSubsetComplementIntersect = $true )
& ( binintersectT_lem = $true )
& ( setukpairinjR = $true )
& ( ksndsingleton = $true )
& ( notequalI2 = $true )
& ( binintersectSubset1 = $true )
& ( inComplementUnionImpNotIn1 = $true )
& ( subbreln = $true )
& ( breln1all2 = $true )
& ( eta2 = $true )
& ( cartprodmempair1 = $true )
& ( setunionE = $true )
& ( exuI3 = $true )
& ( setadjoinIR = $true )
& ( infuncsetfunc = $true )
& ( setminusT_lem = $true )
& ( setminusEL = $true )
& ( symdiffI2 = $true )
& ( funcext2 = $true )
& ( upairsetIR = $true )
& ( complementTE1 = $true )
& ( setunionsingleton2 = $true )
& ( emptysetimpfalse = $true )
& ( singletonprop = $true )
& ( prop2setI = $true )
& ( cartprodpairin = $true )
& ( binunionIL = $true )
& ( contrasubsetT1 = $true )
& ( woz13rule0 = $true )
& ( binintersectSubset2 = $true )
& ( upairset2IR = $true )
& ( dpsetconstrI = $true )
& ( secondinupair = $true )
& ( upairinpowunion = $true )
& ( breln1unionIL = $true )
& ( notdexE = $true )
& ( symdiffIneg1 = $true )
& ( disjointsetsI1 = $true )
& ( demorgan1b = $true )
& ( doubleComplementSub2 = $true )
& ( binintersectTERcontra = $true )
& ( ap2p = $true )
& ( dpsetconstrSub = $true )
& ( subsetI1 = $true )
& ( exuEu = $true )
& ( setminusSubset2 = $true )
& ( funcinfuncset = $true )
& ( setukpairinjL = $true )
& ( ubforcartprodlem2 = $true )
& ( subsetemptysetimpeq = $true )
& ( complementImpComplementIntersect = $true )
& ( subbreln1 = $true )
& ( powerset__Cong = $true )
& ( ifp = $true )
& ( breln1unionIR = $true )
& ( subsetTrans = $true )
& ( symdiffE = $true )
& ( cartprodpairmemER = $true )
& ( setoftrueEq = $true )
& ( brelnall2 = $true )
& ( omega0Ax = $true )
& ( kpairp = $true )
& ( eqinunit = $true )
& ( upairequniteq = $true )
& ( $true
!= ( in @ ( kpair @ sK68 @ sK69 ) @ sK65 ) )
& ( $true
= ( in @ ( kpair @ sK68 @ sK69 ) @ ( binunion @ sK65 @ sK67 ) ) )
& ( $true
!= ( in @ ( kpair @ sK68 @ sK69 ) @ sK67 ) )
& ( $true
= ( in @ sK69 @ sK66 ) )
& ( ( in @ sK68 @ sK66 )
= $true )
& ( $true
= ( breln1 @ sK66 @ sK67 ) )
& ( ( breln1 @ sK66 @ sK65 )
= $true )
& ( contraSubsetComplement = $true )
& ( quantDeMorgan4 = $true )
& ( emptysetsubset = $true )
& ( binintersectSubset4 = $true )
& ( iftrueProp1 = $true )
& ( setukpairIR = $true )
& ( complementT_lem = $true )
& ( theprop = $true )
& ( woz1_4 = $true )
& ( kfstpairEq = $true )
& ( kfstsingleton = $true )
& ( powersetTE1 = $true )
& ( demorgan1a = $true )
& ( quantDeMorgan3 = $true )
& ( emptyE1 = $true )
& ( setukpairinjL2 = $true )
& ( funcGraphProp2 = $true )
& ( setbeta = $true )
& ( breln1compI = $true )
& ( breln1compEex = $true )
& ( setminusERneg = $true )
& ( singletonsswitch = $true )
& ( emptyI = $true )
& ( binintersectER = $true )
& ( breln1invI = $true )
& ( funcGraphProp3 = $true )
& ( eta1 = $true )
& ( binintersectSubset5 = $true )
& ( notdallE = $true )
& ( iffalse = $true )
& ( nonemptyI = $true )
& ( setunionAx = $true )
& ( cartprodpairmemEL = $true )
& ( in__Cong = $true )
& ( binunionEcases = $true )
& ( cartprodsndin = $true )
& ( setunionsingleton = $true )
& ( iffalseProp1 = $true )
& ( sepSubset = $true )
& ( doubleComplementE1 = $true )
& ( demorgan2a1 = $true )
& ( binunionTIRcontra = $true )
& ( exuE1 = $true )
& ( ubforcartprodlem3 = $true )
& ( demorgan2b2 = $true )
& ( powersetE1 = $true )
& ( setminusIRneg = $true )
& ( demorgan2a = $true )
& ( woz1_1 = $true )
& ( funcGraphProp4 = $true )
& ( singletoninpowerset = $true )
& ( notequalI1 = $true )
& ( dpsetconstrERa = $true )
& ( powersetsubset = $true )
& ( theeq = $true )
& ( inIntersectImpInUnion = $true )
& ( lam2lamEq = $true )
& ( setminusLsub = $true )
& ( emptyInPowerset = $true )
& ( nonemptyE1 = $true )
& ( contrasubsetT2 = $true )
& ( cartprodfstpairEq = $true )
& ( powersetTI1 = $true )
& ( inPowerset = $true )
& ( ex1E1 = $true )
& ( iftrueProp2 = $true )
& ( setadjoinOr = $true )
& ( wellorderingAx = $true )
& ( setminusER = $true )
& ( noeltsimpempty = $true )
& ( singletonsubset = $true )
& ( symdiffIneg2 = $true )
& ( eqimpsubset2 = $true )
& ( subsetE2 = $true )
& ( demorgan2a2 = $true )
& ( woz13rule3 = $true )
& ( cartprodmempaircEq = $true )
& ( omegaIndAx = $true )
& ( intersectInPowersetIntersectUnions = $true )
& ( iffalseProp2 = $true )
& ( breln1compprop = $true )
& ( ex1E2 = $true )
& ( powersetAx = $true )
& ( eqimpsubset1 = $true )
& ( emptysetE = $true )
& ( emptyinPowerset = $true )
& ( upairsubunion = $true )
& ( setukpairIL = $true )
& ( ex1I = $true )
& ( nonemptyI1 = $true )
& ( bs114d = $true )
& ( exuE3e = $true )
& ( subset2powerset = $true )
& ( powersetE = $true )
& ( funcImageSingleton = $true )
& ( setunionI = $true )
& ( binunionLsub = $true )
& ( ex1I2 = $true )
& ( emptysetAx = $true )
& ( complementTI1 = $true )
& ( kpairiskpair = $true )
& ( woz1_3 = $true )
& ( setadjoin__Cong = $true )
& ( demorgan1 = $true )
& ( dsetconstrEL = $true )
& ( subsetRefl = $true )
& ( exuI2 = $true )
& ( setunion__Cong = $true )
& ( breln1all1 = $true )
& ( notinsingleton = $true )
& ( woz13rule1 = $true )
& ( lamProp = $true )
& ( apProp = $true )
& ( lam2p = $true )
& ( beta1 = $true )
& ( dsetconstr__Cong = $true )
& ( notsubsetI = $true )
& ( setext = $true )
& ( funcextLem = $true )
& ( binunionIR = $true )
& ( subPowSU = $true )
& ( woz1_5 = $true )
& ( app = $true )
& ( powersetI = $true )
& ( choice2fnsingleton = $true )
& ( setadjoinIL = $true )
& ( contrasubsetT3 = $true )
& ( setunionE2 = $true )
& ( binunionT_lem = $true )
& ( brelnall1 = $true )
& ( binintersectTELcontra = $true )
& ( binintersectLsub = $true )
& ( setadjoinSub = $true )
& ( breln1unionI = $true )
& ( ap2apEq2 = $true )
& ( cartprodfstin = $true )
& ( binintersectRsub = $true )
& ( upairset2E = $true )
& ( binunionE = $true )
& ( subsetE = $true )
& ( ubforcartprodlem1 = $true )
& ( exu__Cong = $true )
& ( setOfPairsIsBReln = $true )
& ( setOfPairsIsBReln1 = $true )
& ( dpsetconstrER = $true )
& ( nonemptyImpWitness = $true )
& ( breln1SetBreln1 = $true )
& ( complementTnotintersectT = $true )
& ( lamp = $true )
& ( iftrue = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK65,sK66,sK67,sK68,sK69])],[f1154,f1442,f1441,f1440,f1439]) ).
thf(f1439,plain,
( ? [X0: $i,X1: $i] :
( ? [X2: $i] :
( ? [X3: $i] :
( ? [X4: $i] :
( ( ( in @ ( kpair @ X3 @ X4 ) @ X0 )
!= $true )
& ( $true
= ( in @ ( kpair @ X3 @ X4 ) @ ( binunion @ X0 @ X2 ) ) )
& ( ( in @ ( kpair @ X3 @ X4 ) @ X2 )
!= $true )
& ( $true
= ( in @ X4 @ X1 ) ) )
& ( $true
= ( in @ X3 @ X1 ) ) )
& ( ( breln1 @ X1 @ X2 )
= $true ) )
& ( ( breln1 @ X1 @ X0 )
= $true ) )
=> ( ? [X2: $i] :
( ? [X3: $i] :
( ? [X4: $i] :
( ( $true
!= ( in @ ( kpair @ X3 @ X4 ) @ sK65 ) )
& ( ( in @ ( kpair @ X3 @ X4 ) @ ( binunion @ sK65 @ X2 ) )
= $true )
& ( ( in @ ( kpair @ X3 @ X4 ) @ X2 )
!= $true )
& ( $true
= ( in @ X4 @ sK66 ) ) )
& ( $true
= ( in @ X3 @ sK66 ) ) )
& ( ( breln1 @ sK66 @ X2 )
= $true ) )
& ( ( breln1 @ sK66 @ sK65 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1440,plain,
( ? [X2: $i] :
( ? [X3: $i] :
( ? [X4: $i] :
( ( $true
!= ( in @ ( kpair @ X3 @ X4 ) @ sK65 ) )
& ( ( in @ ( kpair @ X3 @ X4 ) @ ( binunion @ sK65 @ X2 ) )
= $true )
& ( ( in @ ( kpair @ X3 @ X4 ) @ X2 )
!= $true )
& ( $true
= ( in @ X4 @ sK66 ) ) )
& ( $true
= ( in @ X3 @ sK66 ) ) )
& ( ( breln1 @ sK66 @ X2 )
= $true ) )
=> ( ? [X3: $i] :
( ? [X4: $i] :
( ( $true
!= ( in @ ( kpair @ X3 @ X4 ) @ sK65 ) )
& ( ( in @ ( kpair @ X3 @ X4 ) @ ( binunion @ sK65 @ sK67 ) )
= $true )
& ( $true
!= ( in @ ( kpair @ X3 @ X4 ) @ sK67 ) )
& ( $true
= ( in @ X4 @ sK66 ) ) )
& ( $true
= ( in @ X3 @ sK66 ) ) )
& ( $true
= ( breln1 @ sK66 @ sK67 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1441,plain,
( ? [X3: $i] :
( ? [X4: $i] :
( ( $true
!= ( in @ ( kpair @ X3 @ X4 ) @ sK65 ) )
& ( ( in @ ( kpair @ X3 @ X4 ) @ ( binunion @ sK65 @ sK67 ) )
= $true )
& ( $true
!= ( in @ ( kpair @ X3 @ X4 ) @ sK67 ) )
& ( $true
= ( in @ X4 @ sK66 ) ) )
& ( $true
= ( in @ X3 @ sK66 ) ) )
=> ( ? [X4: $i] :
( ( $true
!= ( in @ ( kpair @ sK68 @ X4 ) @ sK65 ) )
& ( ( in @ ( kpair @ sK68 @ X4 ) @ ( binunion @ sK65 @ sK67 ) )
= $true )
& ( ( in @ ( kpair @ sK68 @ X4 ) @ sK67 )
!= $true )
& ( $true
= ( in @ X4 @ sK66 ) ) )
& ( ( in @ sK68 @ sK66 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1442,plain,
( ? [X4: $i] :
( ( $true
!= ( in @ ( kpair @ sK68 @ X4 ) @ sK65 ) )
& ( ( in @ ( kpair @ sK68 @ X4 ) @ ( binunion @ sK65 @ sK67 ) )
= $true )
& ( ( in @ ( kpair @ sK68 @ X4 ) @ sK67 )
!= $true )
& ( $true
= ( in @ X4 @ sK66 ) ) )
=> ( ( $true
!= ( in @ ( kpair @ sK68 @ sK69 ) @ sK65 ) )
& ( $true
= ( in @ ( kpair @ sK68 @ sK69 ) @ ( binunion @ sK65 @ sK67 ) ) )
& ( $true
!= ( in @ ( kpair @ sK68 @ sK69 ) @ sK67 ) )
& ( $true
= ( in @ sK69 @ sK66 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1154,plain,
( ( complementTcontraSubset = $true )
& ( cartprodsndpairEq = $true )
& ( setukpairinjR2 = $true )
& ( setminusELneg = $true )
& ( inIntersectImpInUnion2 = $true )
& ( setminusSubset1 = $true )
& ( ap2apEq1 = $true )
& ( demorgan2b = $true )
& ( vacuousDall = $true )
& ( iftrueorfalse = $true )
& ( breln1invprop = $true )
& ( subsetTI = $true )
& ( exuI1 = $true )
& ( setukpairinjL1 = $true )
& ( quantDeMorgan2 = $true )
& ( ksndpairEq = $true )
& ( symdiffI1 = $true )
& ( notinemptyset = $true )
& ( setadjoinSub2 = $true )
& ( demorgan2 = $true )
& ( exuE3u = $true )
& ( singletonsuniq = $true )
& ( upairsetIL = $true )
& ( setunionsingleton1 = $true )
& ( inIntersectImpInIntersectUnions = $true )
& ( woz1_2 = $true )
& ( contrasubsetT = $true )
& ( emptyset__Cong = $true )
& ( descr__Cong = $true )
& ( binintersectSubset3 = $true )
& ( beta2 = $true )
& ( ifSingleton = $true )
& ( complementUnionInPowersetComplement = $true )
& ( singletoninpowunion = $true )
& ( funcGraphProp1 = $true )
& ( doubleComplementEq = $true )
& ( setextT = $true )
& ( foundationAx = $true )
& ( replAx = $true )
& ( setminusI = $true )
& ( dsetconstrI = $true )
& ( inCongP = $true )
& ( setextAx = $true )
& ( binintersectI = $true )
& ( binunionTEcontra = $true )
& ( eqbreln = $true )
& ( funcext = $true )
& ( breln1compE = $true )
& ( powersetI1 = $true )
& ( setadjoinAx = $true )
& ( prop2setE = $true )
& ( cartprodpairsurjEq = $true )
& ( woz13rule2 = $true )
& ( setukpairinjR12 = $true )
& ( binintersectEL = $true )
& ( quantDeMorgan1 = $true )
& ( inComplementUnionImpInComplement1 = $true )
& ( setukpairinjR11 = $true )
& ( binunionRsub = $true )
& ( emptyinunitempty = $true )
& ( breln1unionprop = $true )
& ( subsetI2 = $true )
& ( doubleComplementI1 = $true )
& ( uniqinunit = $true )
& ( eqbreln1 = $true )
& ( sepInPowerset = $true )
& ( binunionTE = $true )
& ( upairsetE = $true )
& ( exuE2 = $true )
& ( complementInPowersetComplementIntersect = $true )
& ( omegaSAx = $true )
& ( breln1invE = $true )
& ( dpsetconstrEL2 = $true )
& ( prop2set2propI = $true )
& ( cartprodmempair = $true )
& ( omega__Cong = $true )
& ( doubleComplementSub1 = $true )
& ( setukpairinjR1 = $true )
& ( descrp = $true )
& ( powersetT_lem = $true )
& ( setminusILneg = $true )
& ( kpairsurjEq = $true )
& ( binunionTILcontra = $true )
& ( woz13rule4 = $true )
& ( setadjoinE = $true )
& ( setextsub = $true )
& ( dsetconstrER = $true )
& ( dpsetconstrEL1 = $true )
& ( complementSubsetComplementIntersect = $true )
& ( binintersectT_lem = $true )
& ( setukpairinjR = $true )
& ( ksndsingleton = $true )
& ( notequalI2 = $true )
& ( binintersectSubset1 = $true )
& ( inComplementUnionImpNotIn1 = $true )
& ( subbreln = $true )
& ( breln1all2 = $true )
& ( eta2 = $true )
& ( cartprodmempair1 = $true )
& ( setunionE = $true )
& ( exuI3 = $true )
& ( setadjoinIR = $true )
& ( infuncsetfunc = $true )
& ( setminusT_lem = $true )
& ( setminusEL = $true )
& ( symdiffI2 = $true )
& ( funcext2 = $true )
& ( upairsetIR = $true )
& ( complementTE1 = $true )
& ( setunionsingleton2 = $true )
& ( emptysetimpfalse = $true )
& ( singletonprop = $true )
& ( prop2setI = $true )
& ( cartprodpairin = $true )
& ( binunionIL = $true )
& ( contrasubsetT1 = $true )
& ( woz13rule0 = $true )
& ( binintersectSubset2 = $true )
& ( upairset2IR = $true )
& ( dpsetconstrI = $true )
& ( secondinupair = $true )
& ( upairinpowunion = $true )
& ( breln1unionIL = $true )
& ( notdexE = $true )
& ( symdiffIneg1 = $true )
& ( disjointsetsI1 = $true )
& ( demorgan1b = $true )
& ( doubleComplementSub2 = $true )
& ( binintersectTERcontra = $true )
& ( ap2p = $true )
& ( dpsetconstrSub = $true )
& ( subsetI1 = $true )
& ( exuEu = $true )
& ( setminusSubset2 = $true )
& ( funcinfuncset = $true )
& ( setukpairinjL = $true )
& ( ubforcartprodlem2 = $true )
& ( subsetemptysetimpeq = $true )
& ( complementImpComplementIntersect = $true )
& ( subbreln1 = $true )
& ( powerset__Cong = $true )
& ( ifp = $true )
& ( breln1unionIR = $true )
& ( subsetTrans = $true )
& ( symdiffE = $true )
& ( cartprodpairmemER = $true )
& ( setoftrueEq = $true )
& ( brelnall2 = $true )
& ( omega0Ax = $true )
& ( kpairp = $true )
& ( eqinunit = $true )
& ( upairequniteq = $true )
& ? [X0: $i,X1: $i] :
( ? [X2: $i] :
( ? [X3: $i] :
( ? [X4: $i] :
( ( ( in @ ( kpair @ X3 @ X4 ) @ X0 )
!= $true )
& ( $true
= ( in @ ( kpair @ X3 @ X4 ) @ ( binunion @ X0 @ X2 ) ) )
& ( ( in @ ( kpair @ X3 @ X4 ) @ X2 )
!= $true )
& ( $true
= ( in @ X4 @ X1 ) ) )
& ( $true
= ( in @ X3 @ X1 ) ) )
& ( ( breln1 @ X1 @ X2 )
= $true ) )
& ( ( breln1 @ X1 @ X0 )
= $true ) )
& ( contraSubsetComplement = $true )
& ( quantDeMorgan4 = $true )
& ( emptysetsubset = $true )
& ( binintersectSubset4 = $true )
& ( iftrueProp1 = $true )
& ( setukpairIR = $true )
& ( complementT_lem = $true )
& ( theprop = $true )
& ( woz1_4 = $true )
& ( kfstpairEq = $true )
& ( kfstsingleton = $true )
& ( powersetTE1 = $true )
& ( demorgan1a = $true )
& ( quantDeMorgan3 = $true )
& ( emptyE1 = $true )
& ( setukpairinjL2 = $true )
& ( funcGraphProp2 = $true )
& ( setbeta = $true )
& ( breln1compI = $true )
& ( breln1compEex = $true )
& ( setminusERneg = $true )
& ( singletonsswitch = $true )
& ( emptyI = $true )
& ( binintersectER = $true )
& ( breln1invI = $true )
& ( funcGraphProp3 = $true )
& ( eta1 = $true )
& ( binintersectSubset5 = $true )
& ( notdallE = $true )
& ( iffalse = $true )
& ( nonemptyI = $true )
& ( setunionAx = $true )
& ( cartprodpairmemEL = $true )
& ( in__Cong = $true )
& ( binunionEcases = $true )
& ( cartprodsndin = $true )
& ( setunionsingleton = $true )
& ( iffalseProp1 = $true )
& ( sepSubset = $true )
& ( doubleComplementE1 = $true )
& ( demorgan2a1 = $true )
& ( binunionTIRcontra = $true )
& ( exuE1 = $true )
& ( ubforcartprodlem3 = $true )
& ( demorgan2b2 = $true )
& ( powersetE1 = $true )
& ( setminusIRneg = $true )
& ( demorgan2a = $true )
& ( woz1_1 = $true )
& ( funcGraphProp4 = $true )
& ( singletoninpowerset = $true )
& ( notequalI1 = $true )
& ( dpsetconstrERa = $true )
& ( powersetsubset = $true )
& ( theeq = $true )
& ( inIntersectImpInUnion = $true )
& ( lam2lamEq = $true )
& ( setminusLsub = $true )
& ( emptyInPowerset = $true )
& ( nonemptyE1 = $true )
& ( contrasubsetT2 = $true )
& ( cartprodfstpairEq = $true )
& ( powersetTI1 = $true )
& ( inPowerset = $true )
& ( ex1E1 = $true )
& ( iftrueProp2 = $true )
& ( setadjoinOr = $true )
& ( wellorderingAx = $true )
& ( setminusER = $true )
& ( noeltsimpempty = $true )
& ( singletonsubset = $true )
& ( symdiffIneg2 = $true )
& ( eqimpsubset2 = $true )
& ( subsetE2 = $true )
& ( demorgan2a2 = $true )
& ( woz13rule3 = $true )
& ( cartprodmempaircEq = $true )
& ( omegaIndAx = $true )
& ( intersectInPowersetIntersectUnions = $true )
& ( iffalseProp2 = $true )
& ( breln1compprop = $true )
& ( ex1E2 = $true )
& ( powersetAx = $true )
& ( eqimpsubset1 = $true )
& ( emptysetE = $true )
& ( emptyinPowerset = $true )
& ( upairsubunion = $true )
& ( setukpairIL = $true )
& ( ex1I = $true )
& ( nonemptyI1 = $true )
& ( bs114d = $true )
& ( exuE3e = $true )
& ( subset2powerset = $true )
& ( powersetE = $true )
& ( funcImageSingleton = $true )
& ( setunionI = $true )
& ( binunionLsub = $true )
& ( ex1I2 = $true )
& ( emptysetAx = $true )
& ( complementTI1 = $true )
& ( kpairiskpair = $true )
& ( woz1_3 = $true )
& ( setadjoin__Cong = $true )
& ( demorgan1 = $true )
& ( dsetconstrEL = $true )
& ( subsetRefl = $true )
& ( exuI2 = $true )
& ( setunion__Cong = $true )
& ( breln1all1 = $true )
& ( notinsingleton = $true )
& ( woz13rule1 = $true )
& ( lamProp = $true )
& ( apProp = $true )
& ( lam2p = $true )
& ( beta1 = $true )
& ( dsetconstr__Cong = $true )
& ( notsubsetI = $true )
& ( setext = $true )
& ( funcextLem = $true )
& ( binunionIR = $true )
& ( subPowSU = $true )
& ( woz1_5 = $true )
& ( app = $true )
& ( powersetI = $true )
& ( choice2fnsingleton = $true )
& ( setadjoinIL = $true )
& ( contrasubsetT3 = $true )
& ( setunionE2 = $true )
& ( binunionT_lem = $true )
& ( brelnall1 = $true )
& ( binintersectTELcontra = $true )
& ( binintersectLsub = $true )
& ( setadjoinSub = $true )
& ( breln1unionI = $true )
& ( ap2apEq2 = $true )
& ( cartprodfstin = $true )
& ( binintersectRsub = $true )
& ( upairset2E = $true )
& ( binunionE = $true )
& ( subsetE = $true )
& ( ubforcartprodlem1 = $true )
& ( exu__Cong = $true )
& ( setOfPairsIsBReln = $true )
& ( setOfPairsIsBReln1 = $true )
& ( dpsetconstrER = $true )
& ( nonemptyImpWitness = $true )
& ( breln1SetBreln1 = $true )
& ( complementTnotintersectT = $true )
& ( lamp = $true )
& ( iftrue = $true ) ),
inference(flattening,[status(thm)],[f1153]) ).
thf(f1153,plain,
( ? [X0: $i,X1: $i] :
( ? [X2: $i] :
( ? [X3: $i] :
( ? [X4: $i] :
( ( ( in @ ( kpair @ X3 @ X4 ) @ X2 )
!= $true )
& ( ( in @ ( kpair @ X3 @ X4 ) @ X0 )
!= $true )
& ( $true
= ( in @ ( kpair @ X3 @ X4 ) @ ( binunion @ X0 @ X2 ) ) )
& ( $true
= ( in @ X4 @ X1 ) ) )
& ( $true
= ( in @ X3 @ X1 ) ) )
& ( ( breln1 @ X1 @ X2 )
= $true ) )
& ( ( breln1 @ X1 @ X0 )
= $true ) )
& ( breln1unionI = $true )
& ( breln1unionIR = $true )
& ( breln1unionIL = $true )
& ( breln1unionprop = $true )
& ( breln1compEex = $true )
& ( breln1compE = $true )
& ( breln1compI = $true )
& ( breln1compprop = $true )
& ( breln1invE = $true )
& ( breln1invI = $true )
& ( breln1invprop = $true )
& ( eqbreln1 = $true )
& ( subbreln1 = $true )
& ( breln1all1 = $true )
& ( setOfPairsIsBReln1 = $true )
& ( choice2fnsingleton = $true )
& ( breln1SetBreln1 = $true )
& ( breln1all2 = $true )
& ( woz1_5 = $true )
& ( woz1_4 = $true )
& ( woz1_3 = $true )
& ( woz1_2 = $true )
& ( woz1_1 = $true )
& ( woz13rule4 = $true )
& ( woz13rule3 = $true )
& ( woz13rule2 = $true )
& ( woz13rule1 = $true )
& ( woz13rule0 = $true )
& ( demorgan2 = $true )
& ( demorgan2b = $true )
& ( demorgan2b2 = $true )
& ( demorgan2a = $true )
& ( demorgan1 = $true )
& ( demorgan1b = $true )
& ( demorgan1a = $true )
& ( demorgan2a2 = $true )
& ( complementUnionInPowersetComplement = $true )
& ( demorgan2a1 = $true )
& ( binunionTEcontra = $true )
& ( binunionTE = $true )
& ( inComplementUnionImpInComplement1 = $true )
& ( inComplementUnionImpNotIn1 = $true )
& ( intersectInPowersetIntersectUnions = $true )
& ( inIntersectImpInIntersectUnions = $true )
& ( inIntersectImpInUnion2 = $true )
& ( inIntersectImpInUnion = $true )
& ( binunionTIRcontra = $true )
& ( binunionTILcontra = $true )
& ( complementTcontraSubset = $true )
& ( contraSubsetComplement = $true )
& ( complementInPowersetComplementIntersect = $true )
& ( complementSubsetComplementIntersect = $true )
& ( complementImpComplementIntersect = $true )
& ( complementTnotintersectT = $true )
& ( doubleComplementEq = $true )
& ( doubleComplementSub2 = $true )
& ( doubleComplementSub1 = $true )
& ( doubleComplementE1 = $true )
& ( doubleComplementI1 = $true )
& ( contrasubsetT3 = $true )
& ( contrasubsetT2 = $true )
& ( contrasubsetT1 = $true )
& ( contrasubsetT = $true )
& ( binintersectTERcontra = $true )
& ( binintersectTELcontra = $true )
& ( complementTE1 = $true )
& ( complementTI1 = $true )
& ( powersetTE1 = $true )
& ( 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)],[f780]) ).
thf(f780,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 )
=> ( ( powersetTE1 = $true )
=> ( ( complementTI1 = $true )
=> ( ( complementTE1 = $true )
=> ( ( binintersectTELcontra = $true )
=> ( ( binintersectTERcontra = $true )
=> ( ( contrasubsetT = $true )
=> ( ( contrasubsetT1 = $true )
=> ( ( contrasubsetT2 = $true )
=> ( ( contrasubsetT3 = $true )
=> ( ( doubleComplementI1 = $true )
=> ( ( doubleComplementE1 = $true )
=> ( ( doubleComplementSub1 = $true )
=> ( ( doubleComplementSub2 = $true )
=> ( ( doubleComplementEq = $true )
=> ( ( complementTnotintersectT = $true )
=> ( ( complementImpComplementIntersect = $true )
=> ( ( complementSubsetComplementIntersect = $true )
=> ( ( complementInPowersetComplementIntersect = $true )
=> ( ( contraSubsetComplement = $true )
=> ( ( complementTcontraSubset = $true )
=> ( ( binunionTILcontra = $true )
=> ( ( binunionTIRcontra = $true )
=> ( ( inIntersectImpInUnion = $true )
=> ( ( inIntersectImpInUnion2 = $true )
=> ( ( inIntersectImpInIntersectUnions = $true )
=> ( ( intersectInPowersetIntersectUnions = $true )
=> ( ( inComplementUnionImpNotIn1 = $true )
=> ( ( inComplementUnionImpInComplement1 = $true )
=> ( ( binunionTE = $true )
=> ( ( binunionTEcontra = $true )
=> ( ( demorgan2a1 = $true )
=> ( ( complementUnionInPowersetComplement = $true )
=> ( ( demorgan2a2 = $true )
=> ( ( demorgan1a = $true )
=> ( ( demorgan1b = $true )
=> ( ( demorgan1 = $true )
=> ( ( demorgan2a = $true )
=> ( ( demorgan2b2 = $true )
=> ( ( demorgan2b = $true )
=> ( ( demorgan2 = $true )
=> ( ( woz13rule0 = $true )
=> ( ( woz13rule1 = $true )
=> ( ( woz13rule2 = $true )
=> ( ( woz13rule3 = $true )
=> ( ( woz13rule4 = $true )
=> ( ( woz1_1 = $true )
=> ( ( woz1_2 = $true )
=> ( ( woz1_3 = $true )
=> ( ( woz1_4 = $true )
=> ( ( woz1_5 = $true )
=> ( ( breln1all2 = $true )
=> ( ( breln1SetBreln1 = $true )
=> ( ( choice2fnsingleton = $true )
=> ( ( setOfPairsIsBReln1 = $true )
=> ( ( breln1all1 = $true )
=> ( ( subbreln1 = $true )
=> ( ( eqbreln1 = $true )
=> ( ( breln1invprop = $true )
=> ( ( breln1invI = $true )
=> ( ( breln1invE = $true )
=> ( ( breln1compprop = $true )
=> ( ( breln1compI = $true )
=> ( ( breln1compE = $true )
=> ( ( breln1compEex = $true )
=> ( ( breln1unionprop = $true )
=> ( ( breln1unionIL = $true )
=> ( ( breln1unionIR = $true )
=> ( ( breln1unionI = $true )
=> ! [X0: $i,X1: $i] :
( ( ( breln1 @ X1 @ X0 )
= $true )
=> ! [X2: $i] :
( ( ( breln1 @ X1 @ X2 )
= $true )
=> ! [X3: $i] :
( ( $true
= ( in @ X3 @ X1 ) )
=> ! [X4: $i] :
( ( $true
= ( in @ X4 @ X1 ) )
=> ( ( $true
= ( in @ ( kpair @ X3 @ X4 ) @ ( binunion @ X0 @ X2 ) ) )
=> ( ( ( in @ ( kpair @ X3 @ X4 ) @ X2 )
= $true )
| ( ( in @ ( kpair @ X3 @ X4 ) @ X0 )
= $true ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f779]) ).
thf(f779,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
=> ( powersetTE1
=> ( complementTI1
=> ( complementTE1
=> ( binintersectTELcontra
=> ( binintersectTERcontra
=> ( contrasubsetT
=> ( contrasubsetT1
=> ( contrasubsetT2
=> ( contrasubsetT3
=> ( doubleComplementI1
=> ( doubleComplementE1
=> ( doubleComplementSub1
=> ( doubleComplementSub2
=> ( doubleComplementEq
=> ( complementTnotintersectT
=> ( complementImpComplementIntersect
=> ( complementSubsetComplementIntersect
=> ( complementInPowersetComplementIntersect
=> ( contraSubsetComplement
=> ( complementTcontraSubset
=> ( binunionTILcontra
=> ( binunionTIRcontra
=> ( inIntersectImpInUnion
=> ( inIntersectImpInUnion2
=> ( inIntersectImpInIntersectUnions
=> ( intersectInPowersetIntersectUnions
=> ( inComplementUnionImpNotIn1
=> ( inComplementUnionImpInComplement1
=> ( binunionTE
=> ( binunionTEcontra
=> ( demorgan2a1
=> ( complementUnionInPowersetComplement
=> ( demorgan2a2
=> ( demorgan1a
=> ( demorgan1b
=> ( demorgan1
=> ( demorgan2a
=> ( demorgan2b2
=> ( demorgan2b
=> ( demorgan2
=> ( woz13rule0
=> ( woz13rule1
=> ( woz13rule2
=> ( woz13rule3
=> ( woz13rule4
=> ( woz1_1
=> ( woz1_2
=> ( woz1_3
=> ( woz1_4
=> ( woz1_5
=> ( breln1all2
=> ( breln1SetBreln1
=> ( choice2fnsingleton
=> ( setOfPairsIsBReln1
=> ( breln1all1
=> ( subbreln1
=> ( eqbreln1
=> ( breln1invprop
=> ( breln1invI
=> ( breln1invE
=> ( breln1compprop
=> ( breln1compI
=> ( breln1compE
=> ( breln1compEex
=> ( breln1unionprop
=> ( breln1unionIL
=> ( breln1unionIR
=> ( breln1unionI
=> ! [X0: $i,X1: $i] :
( ( breln1 @ X1 @ X0 )
=> ! [X2: $i] :
( ( breln1 @ X1 @ X2 )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( in @ ( kpair @ X3 @ X4 ) @ ( binunion @ X0 @ X2 ) )
=> ( ( in @ ( kpair @ X3 @ X4 ) @ X0 )
| ( in @ ( kpair @ X3 @ X4 ) @ X2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f306]) ).
thf(f306,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
=> ( powersetTE1
=> ( complementTI1
=> ( complementTE1
=> ( binintersectTELcontra
=> ( binintersectTERcontra
=> ( contrasubsetT
=> ( contrasubsetT1
=> ( contrasubsetT2
=> ( contrasubsetT3
=> ( doubleComplementI1
=> ( doubleComplementE1
=> ( doubleComplementSub1
=> ( doubleComplementSub2
=> ( doubleComplementEq
=> ( complementTnotintersectT
=> ( complementImpComplementIntersect
=> ( complementSubsetComplementIntersect
=> ( complementInPowersetComplementIntersect
=> ( contraSubsetComplement
=> ( complementTcontraSubset
=> ( binunionTILcontra
=> ( binunionTIRcontra
=> ( inIntersectImpInUnion
=> ( inIntersectImpInUnion2
=> ( inIntersectImpInIntersectUnions
=> ( intersectInPowersetIntersectUnions
=> ( inComplementUnionImpNotIn1
=> ( inComplementUnionImpInComplement1
=> ( binunionTE
=> ( binunionTEcontra
=> ( demorgan2a1
=> ( complementUnionInPowersetComplement
=> ( demorgan2a2
=> ( demorgan1a
=> ( demorgan1b
=> ( demorgan1
=> ( demorgan2a
=> ( demorgan2b2
=> ( demorgan2b
=> ( demorgan2
=> ( woz13rule0
=> ( woz13rule1
=> ( woz13rule2
=> ( woz13rule3
=> ( woz13rule4
=> ( woz1_1
=> ( woz1_2
=> ( woz1_3
=> ( woz1_4
=> ( woz1_5
=> ( breln1all2
=> ( breln1SetBreln1
=> ( choice2fnsingleton
=> ( setOfPairsIsBReln1
=> ( breln1all1
=> ( subbreln1
=> ( eqbreln1
=> ( breln1invprop
=> ( breln1invI
=> ( breln1invE
=> ( breln1compprop
=> ( breln1compI
=> ( breln1compE
=> ( breln1compEex
=> ( breln1unionprop
=> ( breln1unionIL
=> ( breln1unionIR
=> ( breln1unionI
=> ! [X13: $i,X3: $i] :
( ( breln1 @ X3 @ X13 )
=> ! [X15: $i] :
( ( breln1 @ X3 @ X15 )
=> ! [X1: $i] :
( ( in @ X1 @ X3 )
=> ! [X2: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ ( kpair @ X1 @ X2 ) @ ( binunion @ X13 @ X15 ) )
=> ( ( in @ ( kpair @ X1 @ X2 ) @ X13 )
| ( in @ ( kpair @ X1 @ X2 ) @ X15 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f305]) ).
thf(f305,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
=> ( powersetTE1
=> ( complementTI1
=> ( complementTE1
=> ( binintersectTELcontra
=> ( binintersectTERcontra
=> ( contrasubsetT
=> ( contrasubsetT1
=> ( contrasubsetT2
=> ( contrasubsetT3
=> ( doubleComplementI1
=> ( doubleComplementE1
=> ( doubleComplementSub1
=> ( doubleComplementSub2
=> ( doubleComplementEq
=> ( complementTnotintersectT
=> ( complementImpComplementIntersect
=> ( complementSubsetComplementIntersect
=> ( complementInPowersetComplementIntersect
=> ( contraSubsetComplement
=> ( complementTcontraSubset
=> ( binunionTILcontra
=> ( binunionTIRcontra
=> ( inIntersectImpInUnion
=> ( inIntersectImpInUnion2
=> ( inIntersectImpInIntersectUnions
=> ( intersectInPowersetIntersectUnions
=> ( inComplementUnionImpNotIn1
=> ( inComplementUnionImpInComplement1
=> ( binunionTE
=> ( binunionTEcontra
=> ( demorgan2a1
=> ( complementUnionInPowersetComplement
=> ( demorgan2a2
=> ( demorgan1a
=> ( demorgan1b
=> ( demorgan1
=> ( demorgan2a
=> ( demorgan2b2
=> ( demorgan2b
=> ( demorgan2
=> ( woz13rule0
=> ( woz13rule1
=> ( woz13rule2
=> ( woz13rule3
=> ( woz13rule4
=> ( woz1_1
=> ( woz1_2
=> ( woz1_3
=> ( woz1_4
=> ( woz1_5
=> ( breln1all2
=> ( breln1SetBreln1
=> ( choice2fnsingleton
=> ( setOfPairsIsBReln1
=> ( breln1all1
=> ( subbreln1
=> ( eqbreln1
=> ( breln1invprop
=> ( breln1invI
=> ( breln1invE
=> ( breln1compprop
=> ( breln1compI
=> ( breln1compE
=> ( breln1compEex
=> ( breln1unionprop
=> ( breln1unionIL
=> ( breln1unionIR
=> ( breln1unionI
=> ! [X13: $i,X3: $i] :
( ( breln1 @ X3 @ X13 )
=> ! [X15: $i] :
( ( breln1 @ X3 @ X15 )
=> ! [X1: $i] :
( ( in @ X1 @ X3 )
=> ! [X2: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ ( kpair @ X1 @ X2 ) @ ( binunion @ X13 @ X15 ) )
=> ( ( in @ ( kpair @ X1 @ X2 ) @ X13 )
| ( in @ ( kpair @ X1 @ X2 ) @ X15 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.onid0QyU2k/DTF2THF_25019.p',breln1unionE) ).
thf(f4876,plain,
~ spl1017_4,
inference(avatar_contradiction_clause,[status(thm)],[f4875]) ).
thf(f4875,plain,
( $false
| ~ spl1017_4 ),
inference(trivial_inequality_removal,[status(thm)],[f4870]) ).
thf(f4870,plain,
( ( $true != $true )
| ~ spl1017_4 ),
inference(superposition,[status(thm)],[f3138,f4836]) ).
thf(f4836,plain,
( ( $true
= ( in @ ( kpair @ sK68 @ sK69 ) @ sK65 ) )
| ~ spl1017_4 ),
inference(avatar_component_clause,[status(thm)],[f4834]) ).
thf(f4834,definition,
( spl1017_4
<=> ( $true
= ( in @ ( kpair @ sK68 @ sK69 ) @ sK65 ) ) ),
introduced(definition,[new_symbols(naming,[spl1017_4])],[avatar_definition]) ).
thf(f3138,plain,
( $true
!= ( in @ ( kpair @ sK68 @ sK69 ) @ sK65 ) ),
inference(cnf_transformation,[status(thm)],[f1443]) ).
thf(f4847,plain,
( spl1017_6
| spl1017_4 ),
inference(avatar_split_clause,[status(thm)],[f4846,f4834,f4841]) ).
thf(f4846,plain,
( ( $true
= ( in @ ( kpair @ sK68 @ sK69 ) @ sK65 ) )
| ( $true
= ( in @ ( kpair @ sK68 @ sK69 ) @ sK67 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f4845]) ).
thf(f4845,plain,
( ( $true
= ( in @ ( kpair @ sK68 @ sK69 ) @ sK65 ) )
| ( $true != $true )
| ( $true
= ( in @ ( kpair @ sK68 @ sK69 ) @ sK67 ) ) ),
inference(forward_demodulation,[status(thm)],[f4826,f2993]) ).
thf(f2993,plain,
binunionE = $true,
inference(cnf_transformation,[status(thm)],[f1443]) ).
thf(f4826,plain,
( ( binunionE != $true )
| ( $true
= ( in @ ( kpair @ sK68 @ sK69 ) @ sK67 ) )
| ( $true
= ( in @ ( kpair @ sK68 @ sK69 ) @ sK65 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f4820]) ).
thf(f4820,plain,
( ( $true != $true )
| ( binunionE != $true )
| ( $true
= ( in @ ( kpair @ sK68 @ sK69 ) @ sK67 ) )
| ( $true
= ( in @ ( kpair @ sK68 @ sK69 ) @ sK65 ) ) ),
inference(superposition,[status(thm)],[f4199,f3137]) ).
thf(f3137,plain,
( $true
= ( in @ ( kpair @ sK68 @ sK69 ) @ ( binunion @ sK65 @ sK67 ) ) ),
inference(cnf_transformation,[status(thm)],[f1443]) ).
thf(f4199,plain,
! [X2: $i,X0: $i,X1: $i] :
( ( ( in @ X2 @ ( binunion @ X1 @ X0 ) )
!= $true )
| ( $true
= ( in @ X2 @ X1 ) )
| ( $true
= ( in @ X2 @ X0 ) )
| ( binunionE != $true ) ),
inference(cnf_transformation,[status(thm)],[f2469]) ).
thf(f2469,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X2 @ X0 ) )
| ( $true
= ( in @ X2 @ X1 ) )
| ( ( in @ X2 @ ( binunion @ X1 @ X0 ) )
!= $true ) )
| ( binunionE != $true ) )
& ( ( binunionE = $true )
| ( ( $true
!= ( in @ sK741 @ sK739 ) )
& ( ( in @ sK741 @ sK740 )
!= $true )
& ( $true
= ( in @ sK741 @ ( binunion @ sK740 @ sK739 ) ) ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK739,sK740,sK741])],[f2467,f2468]) ).
thf(f2468,plain,
( ? [X3: $i,X4: $i,X5: $i] :
( ( $true
!= ( in @ X5 @ X3 ) )
& ( ( in @ X5 @ X4 )
!= $true )
& ( $true
= ( in @ X5 @ ( binunion @ X4 @ X3 ) ) ) )
=> ( ( $true
!= ( in @ sK741 @ sK739 ) )
& ( ( in @ sK741 @ sK740 )
!= $true )
& ( $true
= ( in @ sK741 @ ( binunion @ sK740 @ sK739 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f2467,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X2 @ X0 ) )
| ( $true
= ( in @ X2 @ X1 ) )
| ( ( in @ X2 @ ( binunion @ X1 @ X0 ) )
!= $true ) )
| ( binunionE != $true ) )
& ( ( binunionE = $true )
| ? [X3: $i,X4: $i,X5: $i] :
( ( $true
!= ( in @ X5 @ X3 ) )
& ( ( in @ X5 @ X4 )
!= $true )
& ( $true
= ( in @ X5 @ ( binunion @ X4 @ X3 ) ) ) ) ) ),
inference(rectify,[status(thm)],[f2466]) ).
thf(f2466,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X2 @ X0 ) )
| ( $true
= ( in @ X2 @ X1 ) )
| ( ( in @ X2 @ ( binunion @ X1 @ X0 ) )
!= $true ) )
| ( binunionE != $true ) )
& ( ( binunionE = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( $true
!= ( in @ X2 @ X0 ) )
& ( $true
!= ( in @ X2 @ X1 ) )
& ( ( in @ X2 @ ( binunion @ X1 @ X0 ) )
= $true ) ) ) ),
inference(nnf_transformation,[status(thm)],[f1287]) ).
thf(f1287,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X2 @ X0 ) )
| ( $true
= ( in @ X2 @ X1 ) )
| ( ( in @ X2 @ ( binunion @ X1 @ X0 ) )
!= $true ) )
<=> ( binunionE = $true ) ),
inference(flattening,[status(thm)],[f1286]) ).
thf(f1286,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X2 @ X0 ) )
| ( $true
= ( in @ X2 @ X1 ) )
| ( ( in @ X2 @ ( binunion @ X1 @ X0 ) )
!= $true ) )
<=> ( binunionE = $true ) ),
inference(ennf_transformation,[status(thm)],[f568]) ).
thf(f568,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X2 @ ( binunion @ X1 @ X0 ) )
= $true )
=> ( ( $true
= ( in @ X2 @ X0 ) )
| ( $true
= ( in @ X2 @ X1 ) ) ) )
<=> ( binunionE = $true ) ),
inference(fool_elimination,[status(thm)],[f567]) ).
thf(f567,plain,
( binunionE
= ( ! [X0: $i,X1: $i,X2: $i] :
( ( in @ X2 @ ( binunion @ X1 @ X0 ) )
=> ( ( in @ X2 @ X1 )
| ( in @ X2 @ X0 ) ) ) ) ),
inference(rectify,[status(thm)],[f105]) ).
thf(f105,axiom,
( binunionE
= ( ! [X4: $i,X3: $i,X1: $i] :
( ( in @ X1 @ ( binunion @ X3 @ X4 ) )
=> ( ( in @ X1 @ X3 )
| ( in @ X1 @ X4 ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.onid0QyU2k/DTF2THF_25019.p',binunionE) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.09 % Problem : SEU786^1 : TPTP v9.2.1. Released v3.7.0.
% 0.00/0.09 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.11/0.29 % Computer : n026.cluster.edu
% 0.11/0.29 % Model : x86_64 x86_64
% 0.11/0.29 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.29 % Memory : 8042.1875MB
% 0.11/0.29 % OS : Linux 3.10.0-693.el7.x86_64
% 0.11/0.29 % CPULimit : 300
% 0.11/0.29 % WCLimit : 300
% 0.11/0.29 % DateTime : Tue Feb 24 12:34:10 EST 2026
% 0.11/0.29 % CPUTime :
% 0.11/0.29 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.19/0.38 ---- Original DTF file ---
% 0.19/0.38 thf(spec,logic,$$dhol).
% 0.19/0.38 %------------------------------------------------------------------------------
% 0.19/0.38 % File : SEU786^1 : TPTP v9.2.1. Released v3.7.0.
% 0.19/0.38 % Domain : Set Theory
% 0.19/0.38 % Problem : Binary Relations on a Set
% 0.19/0.38 % Version : Especial.
% 0.19/0.38 % English : (! A:i.! R:i.breln1 A R -> (! S:i.breln1 A S -> (! x:i.in x A ->
% 0.19/0.38 % (! y:i.in y A -> in (kpair x y) (binunion R S) ->
% 0.19/0.38 % in (kpair x y) R | in (kpair x y) S))))
% 0.19/0.38
% 0.19/0.38 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.19/0.38 % Source : [Bro08]
% 0.19/0.38 % Names : ZFC288g [Bro08]
% 0.19/0.38
% 0.19/0.38 % Status : Theorem
% 0.19/0.38 % Rating : 0.33 v9.1.0, 0.25 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 v5.5.0, 0.33 v5.4.0, 0.40 v5.2.0, 0.80 v5.1.0, 1.00 v5.0.0, 0.80 v4.1.0, 0.67 v3.7.0
% 0.19/0.38 % Syntax : Number of formulae : 655 ( 304 unt; 350 typ; 304 def)
% 0.19/0.38 % Number of atoms : 1876 ( 434 equ; 0 cnn)
% 0.19/0.38 % Maximal formula atoms : 309 ( 6 avg)
% 0.19/0.38 % Number of connectives : 4108 ( 73 ~; 14 |; 48 &;2981 @)
% 0.19/0.38 % ( 14 <=>; 978 =>; 0 <=; 0 <~>)
% 0.19/0.38 % Maximal formula depth : 318 ( 2 avg)
% 0.19/0.38 % Number of types : 2 ( 0 usr)
% 0.19/0.38 % Number of type conns : 157 ( 157 >; 0 *; 0 +; 0 <<)
% 0.19/0.38 % Number of symbols : 353 ( 350 usr; 306 con; 0-4 aty)
% 0.19/0.38 % Number of variables : 1071 ( 71 ^; 963 !; 37 ?;1071 :)
% 0.19/0.38 % SPC : TH0_THM_EQU_NAR
% 0.19/0.38
% 0.19/0.38 % Comments : http://mathgate.info/detsetitem.php?id=351
% 0.19/0.38 %------------------------------------------------------------------------------
% 0.19/0.38 thf(in_type,type,
% 0.19/0.38 in: $i > $i > $o ).
% 0.19/0.38
% 0.19/0.38 thf(exu_type,type,
% 0.19/0.38 exu: ( $i > $o ) > $o ).
% 0.19/0.38
% 0.19/0.38 thf(exu,definition,
% 0.19/0.38 ( exu
% 0.19/0.38 = ( ^ [Xphi: $i > $o] :
% 0.19/0.38 ? [Xx: $i] :
% 0.19/0.38 ( ( Xphi @ Xx )
% 0.19/0.38 & ! [Xy: $i] :
% 0.19/0.38 ( ( Xphi @ Xy )
% 0.19/0.38 => ( Xx = Xy ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(setextAx_type,type,
% 0.19/0.38 setextAx: $o ).
% 0.19/0.38
% 0.19/0.38 thf(setextAx,definition,
% 0.19/0.38 ( setextAx
% 0.19/0.38 = ( ! [A: $i,B: $i] :
% 0.19/0.38 ( ! [Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ A )
% 0.19/0.38 <=> ( in @ Xx @ B ) )
% 0.19/0.38 => ( A = B ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(emptyset_type,type,
% 0.19/0.38 emptyset: $i ).
% 0.19/0.38
% 0.19/0.38 thf(emptysetAx_type,type,
% 0.19/0.38 emptysetAx: $o ).
% 0.19/0.38
% 0.19/0.38 thf(emptysetAx,definition,
% 0.19/0.38 ( emptysetAx
% 0.19/0.38 = ( ! [Xx: $i] :
% 0.19/0.38 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(setadjoin_type,type,
% 0.19/0.38 setadjoin: $i > $i > $i ).
% 0.19/0.38
% 0.19/0.38 thf(setadjoinAx_type,type,
% 0.19/0.38 setadjoinAx: $o ).
% 0.19/0.38
% 0.19/0.38 thf(setadjoinAx,definition,
% 0.19/0.38 ( setadjoinAx
% 0.19/0.38 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.19/0.38 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.19/0.38 <=> ( ( Xy = Xx )
% 0.19/0.38 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(powerset_type,type,
% 0.19/0.38 powerset: $i > $i ).
% 0.19/0.38
% 0.19/0.38 thf(powersetAx_type,type,
% 0.19/0.38 powersetAx: $o ).
% 0.19/0.38
% 0.19/0.38 thf(powersetAx,definition,
% 0.19/0.38 ( powersetAx
% 0.19/0.38 = ( ! [A: $i,B: $i] :
% 0.19/0.38 ( ( in @ B @ ( powerset @ A ) )
% 0.19/0.38 <=> ! [Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ B )
% 0.19/0.38 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(setunion_type,type,
% 0.19/0.38 setunion: $i > $i ).
% 0.19/0.38
% 0.19/0.38 thf(setunionAx_type,type,
% 0.19/0.38 setunionAx: $o ).
% 0.19/0.38
% 0.19/0.38 thf(setunionAx,definition,
% 0.19/0.38 ( setunionAx
% 0.19/0.38 = ( ! [A: $i,Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ ( setunion @ A ) )
% 0.19/0.38 <=> ? [B: $i] :
% 0.19/0.38 ( ( in @ Xx @ B )
% 0.19/0.38 & ( in @ B @ A ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(omega_type,type,
% 0.19/0.38 omega: $i ).
% 0.19/0.38
% 0.19/0.38 thf(omega0Ax_type,type,
% 0.19/0.38 omega0Ax: $o ).
% 0.19/0.38
% 0.19/0.38 thf(omega0Ax,definition,
% 0.19/0.38 ( omega0Ax
% 0.19/0.38 = ( in @ emptyset @ omega ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(omegaSAx_type,type,
% 0.19/0.38 omegaSAx: $o ).
% 0.19/0.38
% 0.19/0.38 thf(omegaSAx,definition,
% 0.19/0.38 ( omegaSAx
% 0.19/0.38 = ( ! [Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ omega )
% 0.19/0.38 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(omegaIndAx_type,type,
% 0.19/0.38 omegaIndAx: $o ).
% 0.19/0.38
% 0.19/0.38 thf(omegaIndAx,definition,
% 0.19/0.38 ( omegaIndAx
% 0.19/0.38 = ( ! [A: $i] :
% 0.19/0.38 ( ( ( in @ emptyset @ A )
% 0.19/0.38 & ! [Xx: $i] :
% 0.19/0.38 ( ( ( in @ Xx @ omega )
% 0.19/0.38 & ( in @ Xx @ A ) )
% 0.19/0.38 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.19/0.38 => ! [Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ omega )
% 0.19/0.38 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(replAx_type,type,
% 0.19/0.38 replAx: $o ).
% 0.19/0.38
% 0.19/0.38 thf(replAx,definition,
% 0.19/0.38 ( replAx
% 0.19/0.38 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.19/0.38 ( ! [Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ A )
% 0.19/0.38 => ( exu
% 0.19/0.38 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.19/0.38 => ? [B: $i] :
% 0.19/0.38 ! [Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ B )
% 0.19/0.38 <=> ? [Xy: $i] :
% 0.19/0.38 ( ( in @ Xy @ A )
% 0.19/0.38 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(foundationAx_type,type,
% 0.19/0.38 foundationAx: $o ).
% 0.19/0.38
% 0.19/0.38 thf(foundationAx,definition,
% 0.19/0.38 ( foundationAx
% 0.19/0.38 = ( ! [A: $i] :
% 0.19/0.38 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.19/0.38 => ? [B: $i] :
% 0.19/0.38 ( ( in @ B @ A )
% 0.19/0.38 & ~ ? [Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ B )
% 0.19/0.38 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(wellorderingAx_type,type,
% 0.19/0.38 wellorderingAx: $o ).
% 0.19/0.38
% 0.19/0.38 thf(wellorderingAx,definition,
% 0.19/0.38 ( wellorderingAx
% 0.19/0.38 = ( ! [A: $i] :
% 0.19/0.38 ? [B: $i] :
% 0.19/0.38 ( ! [C: $i] :
% 0.19/0.38 ( ( in @ C @ B )
% 0.19/0.38 => ! [Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ C )
% 0.19/0.38 => ( in @ Xx @ A ) ) )
% 0.19/0.38 & ! [Xx: $i,Xy: $i] :
% 0.19/0.38 ( ( ( in @ Xx @ A )
% 0.19/0.38 & ( in @ Xy @ A ) )
% 0.19/0.38 => ( ! [C: $i] :
% 0.19/0.38 ( ( in @ C @ B )
% 0.19/0.38 => ( ( in @ Xx @ C )
% 0.19/0.38 <=> ( in @ Xy @ C ) ) )
% 0.19/0.38 => ( Xx = Xy ) ) )
% 0.19/0.38 & ! [C: $i,D: $i] :
% 0.19/0.38 ( ( ( in @ C @ B )
% 0.19/0.38 & ( in @ D @ B ) )
% 0.19/0.38 => ( ! [Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ C )
% 0.19/0.38 => ( in @ Xx @ D ) )
% 0.19/0.38 | ! [Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ D )
% 0.19/0.38 => ( in @ Xx @ C ) ) ) )
% 0.19/0.38 & ! [C: $i] :
% 0.19/0.38 ( ( ! [Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ C )
% 0.19/0.38 => ( in @ Xx @ A ) )
% 0.19/0.38 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.19/0.38 => ? [D: $i,Xx: $i] :
% 0.19/0.38 ( ( in @ D @ B )
% 0.19/0.38 & ( in @ Xx @ C )
% 0.19/0.38 & ~ ? [Xy: $i] :
% 0.19/0.38 ( ( in @ Xy @ D )
% 0.19/0.38 & ( in @ Xy @ C ) )
% 0.19/0.38 & ! [E: $i] :
% 0.19/0.38 ( ( in @ E @ B )
% 0.19/0.38 => ( ! [Xy: $i] :
% 0.19/0.38 ( ( in @ Xy @ E )
% 0.19/0.38 => ( in @ Xy @ D ) )
% 0.19/0.38 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(descr_type,type,
% 0.19/0.38 descr: ( $i > $o ) > $i ).
% 0.19/0.38
% 0.19/0.38 thf(descrp_type,type,
% 0.19/0.38 descrp: $o ).
% 0.19/0.38
% 0.19/0.38 thf(descrp,definition,
% 0.19/0.38 ( descrp
% 0.19/0.38 = ( ! [Xphi: $i > $o] :
% 0.19/0.38 ( ( exu
% 0.19/0.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.38 => ( Xphi
% 0.19/0.38 @ ( descr
% 0.19/0.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(dsetconstr_type,type,
% 0.19/0.38 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.19/0.38
% 0.19/0.38 thf(dsetconstrI_type,type,
% 0.19/0.38 dsetconstrI: $o ).
% 0.19/0.38
% 0.19/0.38 thf(dsetconstrI,definition,
% 0.19/0.38 ( dsetconstrI
% 0.19/0.38 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ A )
% 0.19/0.38 => ( ( Xphi @ Xx )
% 0.19/0.38 => ( in @ Xx
% 0.19/0.38 @ ( dsetconstr @ A
% 0.19/0.38 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(dsetconstrEL_type,type,
% 0.19/0.38 dsetconstrEL: $o ).
% 0.19/0.38
% 0.19/0.38 thf(dsetconstrEL,definition,
% 0.19/0.38 ( dsetconstrEL
% 0.19/0.38 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.38 ( ( in @ Xx
% 0.19/0.38 @ ( dsetconstr @ A
% 0.19/0.38 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.19/0.38 => ( in @ Xx @ A ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(dsetconstrER_type,type,
% 0.19/0.38 dsetconstrER: $o ).
% 0.19/0.38
% 0.19/0.38 thf(dsetconstrER,definition,
% 0.19/0.38 ( dsetconstrER
% 0.19/0.38 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.38 ( ( in @ Xx
% 0.19/0.38 @ ( dsetconstr @ A
% 0.19/0.38 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.19/0.38 => ( Xphi @ Xx ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(exuE1_type,type,
% 0.19/0.38 exuE1: $o ).
% 0.19/0.38
% 0.19/0.38 thf(exuE1,definition,
% 0.19/0.38 ( exuE1
% 0.19/0.38 = ( ! [Xphi: $i > $o] :
% 0.19/0.38 ( ( exu
% 0.19/0.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.38 => ? [Xx: $i] :
% 0.19/0.38 ( ( Xphi @ Xx )
% 0.19/0.38 & ! [Xy: $i] :
% 0.19/0.38 ( ( Xphi @ Xy )
% 0.19/0.38 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(prop2set_type,type,
% 0.19/0.38 prop2set: $o > $i ).
% 0.19/0.38
% 0.19/0.38 thf(prop2setE_type,type,
% 0.19/0.38 prop2setE: $o ).
% 0.19/0.38
% 0.19/0.38 thf(prop2setE,definition,
% 0.19/0.38 ( prop2setE
% 0.19/0.38 = ( ! [Xphi: $o,Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.19/0.38 => Xphi ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(emptysetE_type,type,
% 0.19/0.38 emptysetE: $o ).
% 0.19/0.38
% 0.19/0.38 thf(emptysetE,definition,
% 0.19/0.38 ( emptysetE
% 0.19/0.38 = ( ! [Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ emptyset )
% 0.19/0.38 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(emptysetimpfalse_type,type,
% 0.19/0.38 emptysetimpfalse: $o ).
% 0.19/0.38
% 0.19/0.38 thf(emptysetimpfalse,definition,
% 0.19/0.38 ( emptysetimpfalse
% 0.19/0.38 = ( ! [Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ emptyset )
% 0.19/0.38 => $false ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(notinemptyset_type,type,
% 0.19/0.38 notinemptyset: $o ).
% 0.19/0.38
% 0.19/0.38 thf(notinemptyset,definition,
% 0.19/0.38 ( notinemptyset
% 0.19/0.38 = ( ! [Xx: $i] :
% 0.19/0.38 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(exuE3e_type,type,
% 0.19/0.38 exuE3e: $o ).
% 0.19/0.38
% 0.19/0.38 thf(exuE3e,definition,
% 0.19/0.38 ( exuE3e
% 0.19/0.38 = ( ! [Xphi: $i > $o] :
% 0.19/0.38 ( ( exu
% 0.19/0.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.38 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(setext_type,type,
% 0.19/0.38 setext: $o ).
% 0.19/0.38
% 0.19/0.38 thf(setext,definition,
% 0.19/0.38 ( setext
% 0.19/0.38 = ( ! [A: $i,B: $i] :
% 0.19/0.38 ( ! [Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ A )
% 0.19/0.38 => ( in @ Xx @ B ) )
% 0.19/0.38 => ( ! [Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ B )
% 0.19/0.38 => ( in @ Xx @ A ) )
% 0.19/0.38 => ( A = B ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(emptyI_type,type,
% 0.19/0.38 emptyI: $o ).
% 0.19/0.38
% 0.19/0.38 thf(emptyI,definition,
% 0.19/0.38 ( emptyI
% 0.19/0.38 = ( ! [A: $i] :
% 0.19/0.38 ( ! [Xx: $i] :
% 0.19/0.38 ~ ( in @ Xx @ A )
% 0.19/0.38 => ( A = emptyset ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(noeltsimpempty_type,type,
% 0.19/0.38 noeltsimpempty: $o ).
% 0.19/0.38
% 0.19/0.38 thf(noeltsimpempty,definition,
% 0.19/0.38 ( noeltsimpempty
% 0.19/0.38 = ( ! [A: $i] :
% 0.19/0.38 ( ! [Xx: $i] :
% 0.19/0.38 ~ ( in @ Xx @ A )
% 0.19/0.38 => ( A = emptyset ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(setbeta_type,type,
% 0.19/0.38 setbeta: $o ).
% 0.19/0.38
% 0.19/0.38 thf(setbeta,definition,
% 0.19/0.38 ( setbeta
% 0.19/0.38 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ A )
% 0.19/0.38 => ( ( in @ Xx
% 0.19/0.38 @ ( dsetconstr @ A
% 0.19/0.38 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.19/0.38 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(nonempty_type,type,
% 0.19/0.38 nonempty: $i > $o ).
% 0.19/0.38
% 0.19/0.38 thf(nonempty,definition,
% 0.19/0.38 ( nonempty
% 0.19/0.38 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(nonemptyE1_type,type,
% 0.19/0.38 nonemptyE1: $o ).
% 0.19/0.38
% 0.19/0.38 thf(nonemptyE1,definition,
% 0.19/0.38 ( nonemptyE1
% 0.19/0.38 = ( ! [A: $i] :
% 0.19/0.38 ( ( nonempty @ A )
% 0.19/0.38 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(nonemptyI_type,type,
% 0.19/0.38 nonemptyI: $o ).
% 0.19/0.38
% 0.19/0.38 thf(nonemptyI,definition,
% 0.19/0.38 ( nonemptyI
% 0.19/0.38 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ A )
% 0.19/0.38 => ( ( Xphi @ Xx )
% 0.19/0.38 => ( nonempty
% 0.19/0.38 @ ( dsetconstr @ A
% 0.19/0.38 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(nonemptyI1_type,type,
% 0.19/0.38 nonemptyI1: $o ).
% 0.19/0.38
% 0.19/0.38 thf(nonemptyI1,definition,
% 0.19/0.38 ( nonemptyI1
% 0.19/0.38 = ( ! [A: $i] :
% 0.19/0.38 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.19/0.38 => ( nonempty @ A ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(setadjoinIL_type,type,
% 0.19/0.38 setadjoinIL: $o ).
% 0.19/0.38
% 0.19/0.38 thf(setadjoinIL,definition,
% 0.19/0.38 ( setadjoinIL
% 0.19/0.38 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(emptyinunitempty_type,type,
% 0.19/0.38 emptyinunitempty: $o ).
% 0.19/0.38
% 0.19/0.38 thf(emptyinunitempty,definition,
% 0.19/0.38 ( emptyinunitempty
% 0.19/0.38 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(setadjoinIR_type,type,
% 0.19/0.38 setadjoinIR: $o ).
% 0.19/0.38
% 0.19/0.38 thf(setadjoinIR,definition,
% 0.19/0.38 ( setadjoinIR
% 0.19/0.38 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.19/0.38 ( ( in @ Xy @ A )
% 0.19/0.38 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(setadjoinE_type,type,
% 0.19/0.38 setadjoinE: $o ).
% 0.19/0.38
% 0.19/0.38 thf(setadjoinE,definition,
% 0.19/0.38 ( setadjoinE
% 0.19/0.38 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.19/0.38 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.19/0.38 => ! [Xphi: $o] :
% 0.19/0.38 ( ( ( Xy = Xx )
% 0.19/0.38 => Xphi )
% 0.19/0.38 => ( ( ( in @ Xy @ A )
% 0.19/0.38 => Xphi )
% 0.19/0.38 => Xphi ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(setadjoinOr_type,type,
% 0.19/0.38 setadjoinOr: $o ).
% 0.19/0.38
% 0.19/0.38 thf(setadjoinOr,definition,
% 0.19/0.38 ( setadjoinOr
% 0.19/0.38 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.19/0.38 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.19/0.38 => ( ( Xy = Xx )
% 0.19/0.38 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(setoftrueEq_type,type,
% 0.19/0.38 setoftrueEq: $o ).
% 0.19/0.38
% 0.19/0.38 thf(setoftrueEq,definition,
% 0.19/0.38 ( setoftrueEq
% 0.19/0.38 = ( ! [A: $i] :
% 0.19/0.38 ( ( dsetconstr @ A
% 0.19/0.38 @ ^ [Xx: $i] : $true )
% 0.19/0.38 = A ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(powersetI_type,type,
% 0.19/0.38 powersetI: $o ).
% 0.19/0.38
% 0.19/0.38 thf(powersetI,definition,
% 0.19/0.38 ( powersetI
% 0.19/0.38 = ( ! [A: $i,B: $i] :
% 0.19/0.38 ( ! [Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ B )
% 0.19/0.38 => ( in @ Xx @ A ) )
% 0.19/0.38 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(emptyinPowerset_type,type,
% 0.19/0.38 emptyinPowerset: $o ).
% 0.19/0.38
% 0.19/0.38 thf(emptyinPowerset,definition,
% 0.19/0.38 ( emptyinPowerset
% 0.19/0.38 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(emptyInPowerset_type,type,
% 0.19/0.38 emptyInPowerset: $o ).
% 0.19/0.38
% 0.19/0.38 thf(emptyInPowerset,definition,
% 0.19/0.38 ( emptyInPowerset
% 0.19/0.38 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(powersetE_type,type,
% 0.19/0.38 powersetE: $o ).
% 0.19/0.38
% 0.19/0.38 thf(powersetE,definition,
% 0.19/0.38 ( powersetE
% 0.19/0.38 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.38 ( ( in @ B @ ( powerset @ A ) )
% 0.19/0.38 => ( ( in @ Xx @ B )
% 0.19/0.38 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(setunionI_type,type,
% 0.19/0.38 setunionI: $o ).
% 0.19/0.38
% 0.19/0.38 thf(setunionI,definition,
% 0.19/0.38 ( setunionI
% 0.19/0.38 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.19/0.38 ( ( in @ Xx @ B )
% 0.19/0.38 => ( ( in @ B @ A )
% 0.19/0.38 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(setunionE_type,type,
% 0.19/0.38 setunionE: $o ).
% 0.19/0.38
% 0.19/0.38 thf(setunionE,definition,
% 0.19/0.38 ( setunionE
% 0.19/0.38 = ( ! [A: $i,Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ ( setunion @ A ) )
% 0.19/0.38 => ! [Xphi: $o] :
% 0.19/0.38 ( ! [B: $i] :
% 0.19/0.38 ( ( in @ Xx @ B )
% 0.19/0.38 => ( ( in @ B @ A )
% 0.19/0.38 => Xphi ) )
% 0.19/0.38 => Xphi ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(subPowSU_type,type,
% 0.19/0.38 subPowSU: $o ).
% 0.19/0.38
% 0.19/0.38 thf(subPowSU,definition,
% 0.19/0.38 ( subPowSU
% 0.19/0.38 = ( ! [A: $i,Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ A )
% 0.19/0.38 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(exuE2_type,type,
% 0.19/0.38 exuE2: $o ).
% 0.19/0.38
% 0.19/0.38 thf(exuE2,definition,
% 0.19/0.38 ( exuE2
% 0.19/0.38 = ( ! [Xphi: $i > $o] :
% 0.19/0.38 ( ( exu
% 0.19/0.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.38 => ? [Xx: $i] :
% 0.19/0.38 ! [Xy: $i] :
% 0.19/0.38 ( ( Xphi @ Xy )
% 0.19/0.38 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(nonemptyImpWitness_type,type,
% 0.19/0.38 nonemptyImpWitness: $o ).
% 0.19/0.38
% 0.19/0.38 thf(nonemptyImpWitness,definition,
% 0.19/0.38 ( nonemptyImpWitness
% 0.19/0.38 = ( ! [A: $i] :
% 0.19/0.38 ( ( nonempty @ A )
% 0.19/0.38 => ? [Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ A )
% 0.19/0.38 & $true ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(uniqinunit_type,type,
% 0.19/0.38 uniqinunit: $o ).
% 0.19/0.38
% 0.19/0.38 thf(uniqinunit,definition,
% 0.19/0.38 ( uniqinunit
% 0.19/0.38 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.38 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.19/0.38 => ( Xx = Xy ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(notinsingleton_type,type,
% 0.19/0.38 notinsingleton: $o ).
% 0.19/0.38
% 0.19/0.38 thf(notinsingleton,definition,
% 0.19/0.38 ( notinsingleton
% 0.19/0.38 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.38 ( ( Xx != Xy )
% 0.19/0.38 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(eqinunit_type,type,
% 0.19/0.38 eqinunit: $o ).
% 0.19/0.38
% 0.19/0.38 thf(eqinunit,definition,
% 0.19/0.38 ( eqinunit
% 0.19/0.38 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.38 ( ( Xx = Xy )
% 0.19/0.38 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(singletonsswitch_type,type,
% 0.19/0.38 singletonsswitch: $o ).
% 0.19/0.38
% 0.19/0.38 thf(singletonsswitch,definition,
% 0.19/0.38 ( singletonsswitch
% 0.19/0.38 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.38 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.19/0.38 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(upairsetE_type,type,
% 0.19/0.38 upairsetE: $o ).
% 0.19/0.38
% 0.19/0.38 thf(upairsetE,definition,
% 0.19/0.38 ( upairsetE
% 0.19/0.38 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.19/0.38 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.19/0.38 => ( ( Xz = Xx )
% 0.19/0.38 | ( Xz = Xy ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(upairsetIL_type,type,
% 0.19/0.38 upairsetIL: $o ).
% 0.19/0.38
% 0.19/0.38 thf(upairsetIL,definition,
% 0.19/0.38 ( upairsetIL
% 0.19/0.38 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(upairsetIR_type,type,
% 0.19/0.38 upairsetIR: $o ).
% 0.19/0.38
% 0.19/0.38 thf(upairsetIR,definition,
% 0.19/0.38 ( upairsetIR
% 0.19/0.38 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(emptyE1_type,type,
% 0.19/0.38 emptyE1: $o ).
% 0.19/0.38
% 0.19/0.38 thf(emptyE1,definition,
% 0.19/0.38 ( emptyE1
% 0.19/0.38 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.38 ( ? [Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ A )
% 0.19/0.38 & ( Xphi @ Xx ) )
% 0.19/0.38 => ( ( ( dsetconstr @ A
% 0.19/0.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.38 = emptyset )
% 0.19/0.38 => $false ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(vacuousDall_type,type,
% 0.19/0.38 vacuousDall: $o ).
% 0.19/0.38
% 0.19/0.38 thf(vacuousDall,definition,
% 0.19/0.38 ( vacuousDall
% 0.19/0.38 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ emptyset )
% 0.19/0.38 => ( Xphi @ Xx ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(quantDeMorgan1_type,type,
% 0.19/0.38 quantDeMorgan1: $o ).
% 0.19/0.38
% 0.19/0.38 thf(quantDeMorgan1,definition,
% 0.19/0.38 ( quantDeMorgan1
% 0.19/0.38 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.38 ( ~ ! [Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ A )
% 0.19/0.38 => ( Xphi @ Xx ) )
% 0.19/0.38 => ? [Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ A )
% 0.19/0.38 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(quantDeMorgan2_type,type,
% 0.19/0.38 quantDeMorgan2: $o ).
% 0.19/0.38
% 0.19/0.38 thf(quantDeMorgan2,definition,
% 0.19/0.38 ( quantDeMorgan2
% 0.19/0.38 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.38 ( ! [Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ A )
% 0.19/0.38 => ~ ( Xphi @ Xx ) )
% 0.19/0.38 => ~ ? [Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ A )
% 0.19/0.38 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(quantDeMorgan3_type,type,
% 0.19/0.38 quantDeMorgan3: $o ).
% 0.19/0.38
% 0.19/0.38 thf(quantDeMorgan3,definition,
% 0.19/0.38 ( quantDeMorgan3
% 0.19/0.38 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.38 ( ~ ? [Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ A )
% 0.19/0.38 & ( Xphi @ Xx ) )
% 0.19/0.38 => ! [Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ A )
% 0.19/0.38 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(quantDeMorgan4_type,type,
% 0.19/0.38 quantDeMorgan4: $o ).
% 0.19/0.38
% 0.19/0.38 thf(quantDeMorgan4,definition,
% 0.19/0.38 ( quantDeMorgan4
% 0.19/0.38 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.38 ( ? [Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ A )
% 0.19/0.38 & ~ ( Xphi @ Xx ) )
% 0.19/0.38 => ~ ! [Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ A )
% 0.19/0.38 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(prop2setI_type,type,
% 0.19/0.38 prop2setI: $o ).
% 0.19/0.38
% 0.19/0.38 thf(prop2setI,definition,
% 0.19/0.38 ( prop2setI
% 0.19/0.38 = ( ! [Xphi: $o] :
% 0.19/0.38 ( Xphi
% 0.19/0.38 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(set2prop_type,type,
% 0.19/0.38 set2prop: $i > $o ).
% 0.19/0.38
% 0.19/0.38 thf(prop2set2propI_type,type,
% 0.19/0.38 prop2set2propI: $o ).
% 0.19/0.38
% 0.19/0.38 thf(prop2set2propI,definition,
% 0.19/0.38 ( prop2set2propI
% 0.19/0.38 = ( ! [Xphi: $o] :
% 0.19/0.38 ( Xphi
% 0.19/0.38 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(notdexE_type,type,
% 0.19/0.38 notdexE: $o ).
% 0.19/0.38
% 0.19/0.38 thf(notdexE,definition,
% 0.19/0.38 ( notdexE
% 0.19/0.38 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.38 ( ~ ? [Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ A )
% 0.19/0.38 & ( Xphi @ Xx ) )
% 0.19/0.38 => ! [Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ A )
% 0.19/0.38 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(notdallE_type,type,
% 0.19/0.38 notdallE: $o ).
% 0.19/0.38
% 0.19/0.38 thf(notdallE,definition,
% 0.19/0.38 ( notdallE
% 0.19/0.38 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.38 ( ~ ! [Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ A )
% 0.19/0.38 => ( Xphi @ Xx ) )
% 0.19/0.38 => ? [Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ A )
% 0.19/0.38 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(exuI1_type,type,
% 0.19/0.38 exuI1: $o ).
% 0.19/0.38
% 0.19/0.38 thf(exuI1,definition,
% 0.19/0.38 ( exuI1
% 0.19/0.38 = ( ! [Xphi: $i > $o] :
% 0.19/0.38 ( ? [Xx: $i] :
% 0.19/0.38 ( ( Xphi @ Xx )
% 0.19/0.38 & ! [Xy: $i] :
% 0.19/0.38 ( ( Xphi @ Xy )
% 0.19/0.38 => ( Xx = Xy ) ) )
% 0.19/0.38 => ( exu
% 0.19/0.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(exuI3_type,type,
% 0.19/0.38 exuI3: $o ).
% 0.19/0.38
% 0.19/0.38 thf(exuI3,definition,
% 0.19/0.38 ( exuI3
% 0.19/0.38 = ( ! [Xphi: $i > $o] :
% 0.19/0.38 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.19/0.38 => ( ! [Xx: $i,Xy: $i] :
% 0.19/0.38 ( ( Xphi @ Xx )
% 0.19/0.38 => ( ( Xphi @ Xy )
% 0.19/0.38 => ( Xx = Xy ) ) )
% 0.19/0.38 => ( exu
% 0.19/0.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(exuI2_type,type,
% 0.19/0.38 exuI2: $o ).
% 0.19/0.38
% 0.19/0.38 thf(exuI2,definition,
% 0.19/0.38 ( exuI2
% 0.19/0.38 = ( ! [Xphi: $i > $o] :
% 0.19/0.38 ( ? [Xx: $i] :
% 0.19/0.38 ! [Xy: $i] :
% 0.19/0.38 ( ( Xphi @ Xy )
% 0.19/0.38 <=> ( Xy = Xx ) )
% 0.19/0.38 => ( exu
% 0.19/0.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(inCongP_type,type,
% 0.19/0.38 inCongP: $o ).
% 0.19/0.38
% 0.19/0.38 thf(inCongP,definition,
% 0.19/0.38 ( inCongP
% 0.19/0.38 = ( ! [A: $i,B: $i] :
% 0.19/0.38 ( ( A = B )
% 0.19/0.38 => ! [Xx: $i,Xy: $i] :
% 0.19/0.38 ( ( Xx = Xy )
% 0.19/0.38 => ( ( in @ Xx @ A )
% 0.19/0.38 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(in__Cong_type,type,
% 0.19/0.38 in__Cong: $o ).
% 0.19/0.38
% 0.19/0.38 thf(in__Cong,definition,
% 0.19/0.38 ( in__Cong
% 0.19/0.38 = ( ! [A: $i,B: $i] :
% 0.19/0.38 ( ( A = B )
% 0.19/0.38 => ! [Xx: $i,Xy: $i] :
% 0.19/0.38 ( ( Xx = Xy )
% 0.19/0.38 => ( ( in @ Xx @ A )
% 0.19/0.38 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(exuE3u_type,type,
% 0.19/0.38 exuE3u: $o ).
% 0.19/0.38
% 0.19/0.38 thf(exuE3u,definition,
% 0.19/0.38 ( exuE3u
% 0.19/0.38 = ( ! [Xphi: $i > $o] :
% 0.19/0.38 ( ( exu
% 0.19/0.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.38 => ! [Xx: $i,Xy: $i] :
% 0.19/0.38 ( ( Xphi @ Xx )
% 0.19/0.38 => ( ( Xphi @ Xy )
% 0.19/0.38 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(exu__Cong_type,type,
% 0.19/0.38 exu__Cong: $o ).
% 0.19/0.38
% 0.19/0.38 thf(exu__Cong,definition,
% 0.19/0.38 ( exu__Cong
% 0.19/0.38 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.19/0.38 ( ! [Xx: $i,Xy: $i] :
% 0.19/0.38 ( ( Xx = Xy )
% 0.19/0.38 => ( ( Xphi @ Xx )
% 0.19/0.38 <=> ( Xpsi @ Xy ) ) )
% 0.19/0.38 => ( ( exu
% 0.19/0.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.38 <=> ( exu
% 0.19/0.38 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(emptyset__Cong_type,type,
% 0.19/0.38 emptyset__Cong: $o ).
% 0.19/0.38
% 0.19/0.38 thf(emptyset__Cong,definition,
% 0.19/0.38 ( emptyset__Cong
% 0.19/0.38 = ( emptyset = emptyset ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(setadjoin__Cong_type,type,
% 0.19/0.38 setadjoin__Cong: $o ).
% 0.19/0.38
% 0.19/0.38 thf(setadjoin__Cong,definition,
% 0.19/0.38 ( setadjoin__Cong
% 0.19/0.38 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.38 ( ( Xx = Xy )
% 0.19/0.38 => ! [Xz: $i,Xu: $i] :
% 0.19/0.38 ( ( Xz = Xu )
% 0.19/0.38 => ( ( setadjoin @ Xx @ Xz )
% 0.19/0.38 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(powerset__Cong_type,type,
% 0.19/0.38 powerset__Cong: $o ).
% 0.19/0.38
% 0.19/0.38 thf(powerset__Cong,definition,
% 0.19/0.38 ( powerset__Cong
% 0.19/0.38 = ( ! [A: $i,B: $i] :
% 0.19/0.38 ( ( A = B )
% 0.19/0.38 => ( ( powerset @ A )
% 0.19/0.38 = ( powerset @ B ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(setunion__Cong_type,type,
% 0.19/0.38 setunion__Cong: $o ).
% 0.19/0.38
% 0.19/0.38 thf(setunion__Cong,definition,
% 0.19/0.38 ( setunion__Cong
% 0.19/0.38 = ( ! [A: $i,B: $i] :
% 0.19/0.38 ( ( A = B )
% 0.19/0.38 => ( ( setunion @ A )
% 0.19/0.38 = ( setunion @ B ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(omega__Cong_type,type,
% 0.19/0.38 omega__Cong: $o ).
% 0.19/0.38
% 0.19/0.38 thf(omega__Cong,definition,
% 0.19/0.38 ( omega__Cong
% 0.19/0.38 = ( omega = omega ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(exuEu_type,type,
% 0.19/0.38 exuEu: $o ).
% 0.19/0.38
% 0.19/0.38 thf(exuEu,definition,
% 0.19/0.38 ( exuEu
% 0.19/0.38 = ( ! [Xphi: $i > $o] :
% 0.19/0.38 ( ( exu
% 0.19/0.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.38 => ! [Xx: $i,Xy: $i] :
% 0.19/0.38 ( ( Xphi @ Xx )
% 0.19/0.38 => ( ( Xphi @ Xy )
% 0.19/0.38 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(descr__Cong_type,type,
% 0.19/0.38 descr__Cong: $o ).
% 0.19/0.38
% 0.19/0.38 thf(descr__Cong,definition,
% 0.19/0.38 ( descr__Cong
% 0.19/0.38 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.19/0.38 ( ! [Xx: $i,Xy: $i] :
% 0.19/0.38 ( ( Xx = Xy )
% 0.19/0.38 => ( ( Xphi @ Xx )
% 0.19/0.38 <=> ( Xpsi @ Xy ) ) )
% 0.19/0.38 => ( ( exu
% 0.19/0.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.38 => ( ( exu
% 0.19/0.38 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.19/0.38 => ( ( descr
% 0.19/0.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.38 = ( descr
% 0.19/0.38 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(dsetconstr__Cong_type,type,
% 0.19/0.38 dsetconstr__Cong: $o ).
% 0.19/0.38
% 0.19/0.38 thf(dsetconstr__Cong,definition,
% 0.19/0.38 ( dsetconstr__Cong
% 0.19/0.38 = ( ! [A: $i,B: $i] :
% 0.19/0.38 ( ( A = B )
% 0.19/0.38 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.19/0.38 ( ! [Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ A )
% 0.19/0.38 => ! [Xy: $i] :
% 0.19/0.38 ( ( in @ Xy @ B )
% 0.19/0.38 => ( ( Xx = Xy )
% 0.19/0.38 => ( ( Xphi @ Xx )
% 0.19/0.38 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.19/0.38 => ( ( dsetconstr @ A
% 0.19/0.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.38 = ( dsetconstr @ B
% 0.19/0.38 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(subset_type,type,
% 0.19/0.38 subset: $i > $i > $o ).
% 0.19/0.38
% 0.19/0.38 thf(disjoint_type,type,
% 0.19/0.38 disjoint: $i > $i > $o ).
% 0.19/0.38
% 0.19/0.38 thf(setsmeet_type,type,
% 0.19/0.38 setsmeet: $i > $i > $o ).
% 0.19/0.38
% 0.19/0.38 thf(subsetI1_type,type,
% 0.19/0.38 subsetI1: $o ).
% 0.19/0.38
% 0.19/0.38 thf(subsetI1,definition,
% 0.19/0.38 ( subsetI1
% 0.19/0.38 = ( ! [A: $i,B: $i] :
% 0.19/0.38 ( ! [Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ A )
% 0.19/0.38 => ( in @ Xx @ B ) )
% 0.19/0.38 => ( subset @ A @ B ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(eqimpsubset2_type,type,
% 0.19/0.38 eqimpsubset2: $o ).
% 0.19/0.38
% 0.19/0.38 thf(eqimpsubset2,definition,
% 0.19/0.38 ( eqimpsubset2
% 0.19/0.38 = ( ! [A: $i,B: $i] :
% 0.19/0.38 ( ( A = B )
% 0.19/0.38 => ( subset @ B @ A ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(eqimpsubset1_type,type,
% 0.19/0.38 eqimpsubset1: $o ).
% 0.19/0.38
% 0.19/0.38 thf(eqimpsubset1,definition,
% 0.19/0.38 ( eqimpsubset1
% 0.19/0.38 = ( ! [A: $i,B: $i] :
% 0.19/0.38 ( ( A = B )
% 0.19/0.38 => ( subset @ A @ B ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(subsetI2_type,type,
% 0.19/0.38 subsetI2: $o ).
% 0.19/0.38
% 0.19/0.38 thf(subsetI2,definition,
% 0.19/0.38 ( subsetI2
% 0.19/0.38 = ( ! [A: $i,B: $i] :
% 0.19/0.38 ( ! [Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ A )
% 0.19/0.38 => ( in @ Xx @ B ) )
% 0.19/0.38 => ( subset @ A @ B ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(emptysetsubset_type,type,
% 0.19/0.38 emptysetsubset: $o ).
% 0.19/0.38
% 0.19/0.38 thf(emptysetsubset,definition,
% 0.19/0.38 ( emptysetsubset
% 0.19/0.38 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(subsetE_type,type,
% 0.19/0.38 subsetE: $o ).
% 0.19/0.38
% 0.19/0.38 thf(subsetE,definition,
% 0.19/0.38 ( subsetE
% 0.19/0.38 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.38 ( ( subset @ A @ B )
% 0.19/0.38 => ( ( in @ Xx @ A )
% 0.19/0.38 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(subsetE2_type,type,
% 0.19/0.38 subsetE2: $o ).
% 0.19/0.38
% 0.19/0.38 thf(subsetE2,definition,
% 0.19/0.38 ( subsetE2
% 0.19/0.38 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.38 ( ( subset @ A @ B )
% 0.19/0.38 => ( ~ ( in @ Xx @ B )
% 0.19/0.38 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(notsubsetI_type,type,
% 0.19/0.38 notsubsetI: $o ).
% 0.19/0.38
% 0.19/0.38 thf(notsubsetI,definition,
% 0.19/0.38 ( notsubsetI
% 0.19/0.38 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ A )
% 0.19/0.38 => ( ~ ( in @ Xx @ B )
% 0.19/0.38 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(notequalI1_type,type,
% 0.19/0.38 notequalI1: $o ).
% 0.19/0.38
% 0.19/0.38 thf(notequalI1,definition,
% 0.19/0.38 ( notequalI1
% 0.19/0.38 = ( ! [A: $i,B: $i] :
% 0.19/0.38 ( ~ ( subset @ A @ B )
% 0.19/0.38 => ( A != B ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(notequalI2_type,type,
% 0.19/0.38 notequalI2: $o ).
% 0.19/0.38
% 0.19/0.38 thf(notequalI2,definition,
% 0.19/0.38 ( notequalI2
% 0.19/0.38 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ A )
% 0.19/0.38 => ( ~ ( in @ Xx @ B )
% 0.19/0.38 => ( A != B ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(subsetRefl_type,type,
% 0.19/0.38 subsetRefl: $o ).
% 0.19/0.38
% 0.19/0.38 thf(subsetRefl,definition,
% 0.19/0.38 ( subsetRefl
% 0.19/0.38 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(subsetTrans_type,type,
% 0.19/0.38 subsetTrans: $o ).
% 0.19/0.38
% 0.19/0.38 thf(subsetTrans,definition,
% 0.19/0.38 ( subsetTrans
% 0.19/0.38 = ( ! [A: $i,B: $i,C: $i] :
% 0.19/0.38 ( ( subset @ A @ B )
% 0.19/0.38 => ( ( subset @ B @ C )
% 0.19/0.38 => ( subset @ A @ C ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(setadjoinSub_type,type,
% 0.19/0.38 setadjoinSub: $o ).
% 0.19/0.38
% 0.19/0.38 thf(setadjoinSub,definition,
% 0.19/0.38 ( setadjoinSub
% 0.19/0.38 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(setadjoinSub2_type,type,
% 0.19/0.38 setadjoinSub2: $o ).
% 0.19/0.38
% 0.19/0.38 thf(setadjoinSub2,definition,
% 0.19/0.38 ( setadjoinSub2
% 0.19/0.38 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.19/0.38 ( ( subset @ A @ B )
% 0.19/0.38 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(subset2powerset_type,type,
% 0.19/0.38 subset2powerset: $o ).
% 0.19/0.38
% 0.19/0.38 thf(subset2powerset,definition,
% 0.19/0.38 ( subset2powerset
% 0.19/0.38 = ( ! [A: $i,B: $i] :
% 0.19/0.38 ( ( subset @ A @ B )
% 0.19/0.38 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(setextsub_type,type,
% 0.19/0.38 setextsub: $o ).
% 0.19/0.38
% 0.19/0.38 thf(setextsub,definition,
% 0.19/0.38 ( setextsub
% 0.19/0.38 = ( ! [A: $i,B: $i] :
% 0.19/0.38 ( ( subset @ A @ B )
% 0.19/0.38 => ( ( subset @ B @ A )
% 0.19/0.38 => ( A = B ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(subsetemptysetimpeq_type,type,
% 0.19/0.38 subsetemptysetimpeq: $o ).
% 0.19/0.38
% 0.19/0.38 thf(subsetemptysetimpeq,definition,
% 0.19/0.38 ( subsetemptysetimpeq
% 0.19/0.38 = ( ! [A: $i] :
% 0.19/0.38 ( ( subset @ A @ emptyset )
% 0.19/0.38 => ( A = emptyset ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(powersetI1_type,type,
% 0.19/0.38 powersetI1: $o ).
% 0.19/0.38
% 0.19/0.38 thf(powersetI1,definition,
% 0.19/0.38 ( powersetI1
% 0.19/0.38 = ( ! [A: $i,B: $i] :
% 0.19/0.38 ( ( subset @ B @ A )
% 0.19/0.38 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(powersetE1_type,type,
% 0.19/0.38 powersetE1: $o ).
% 0.19/0.38
% 0.19/0.38 thf(powersetE1,definition,
% 0.19/0.38 ( powersetE1
% 0.19/0.38 = ( ! [A: $i,B: $i] :
% 0.19/0.38 ( ( in @ B @ ( powerset @ A ) )
% 0.19/0.38 => ( subset @ B @ A ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(inPowerset_type,type,
% 0.19/0.38 inPowerset: $o ).
% 0.19/0.38
% 0.19/0.38 thf(inPowerset,definition,
% 0.19/0.38 ( inPowerset
% 0.19/0.38 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(powersetsubset_type,type,
% 0.19/0.38 powersetsubset: $o ).
% 0.19/0.38
% 0.19/0.38 thf(powersetsubset,definition,
% 0.19/0.38 ( powersetsubset
% 0.19/0.38 = ( ! [A: $i,B: $i] :
% 0.19/0.38 ( ( subset @ A @ B )
% 0.19/0.38 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(sepInPowerset_type,type,
% 0.19/0.38 sepInPowerset: $o ).
% 0.19/0.38
% 0.19/0.38 thf(sepInPowerset,definition,
% 0.19/0.38 ( sepInPowerset
% 0.19/0.38 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.38 ( in
% 0.19/0.38 @ ( dsetconstr @ A
% 0.19/0.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.38 @ ( powerset @ A ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(sepSubset_type,type,
% 0.19/0.38 sepSubset: $o ).
% 0.19/0.38
% 0.19/0.38 thf(sepSubset,definition,
% 0.19/0.38 ( sepSubset
% 0.19/0.38 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.38 ( subset
% 0.19/0.38 @ ( dsetconstr @ A
% 0.19/0.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.38 @ A ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(binunion_type,type,
% 0.19/0.38 binunion: $i > $i > $i ).
% 0.19/0.38
% 0.19/0.38 thf(binunionIL_type,type,
% 0.19/0.38 binunionIL: $o ).
% 0.19/0.38
% 0.19/0.38 thf(binunionIL,definition,
% 0.19/0.38 ( binunionIL
% 0.19/0.38 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ A )
% 0.19/0.38 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(upairset2IR_type,type,
% 0.19/0.38 upairset2IR: $o ).
% 0.19/0.38
% 0.19/0.38 thf(upairset2IR,definition,
% 0.19/0.38 ( upairset2IR
% 0.19/0.38 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(binunionIR_type,type,
% 0.19/0.38 binunionIR: $o ).
% 0.19/0.38
% 0.19/0.38 thf(binunionIR,definition,
% 0.19/0.38 ( binunionIR
% 0.19/0.38 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ B )
% 0.19/0.38 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(binunionEcases_type,type,
% 0.19/0.38 binunionEcases: $o ).
% 0.19/0.38
% 0.19/0.38 thf(binunionEcases,definition,
% 0.19/0.38 ( binunionEcases
% 0.19/0.38 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.19/0.38 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.19/0.38 => ( ( ( in @ Xx @ A )
% 0.19/0.38 => Xphi )
% 0.19/0.38 => ( ( ( in @ Xx @ B )
% 0.19/0.38 => Xphi )
% 0.19/0.38 => Xphi ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(binunionE_type,type,
% 0.19/0.38 binunionE: $o ).
% 0.19/0.38
% 0.19/0.38 thf(binunionE,definition,
% 0.19/0.38 ( binunionE
% 0.19/0.38 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.19/0.38 => ( ( in @ Xx @ A )
% 0.19/0.38 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(binunionLsub_type,type,
% 0.19/0.38 binunionLsub: $o ).
% 0.19/0.38
% 0.19/0.38 thf(binunionLsub,definition,
% 0.19/0.38 ( binunionLsub
% 0.19/0.38 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(binunionRsub_type,type,
% 0.19/0.38 binunionRsub: $o ).
% 0.19/0.38
% 0.19/0.38 thf(binunionRsub,definition,
% 0.19/0.38 ( binunionRsub
% 0.19/0.38 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(binintersect_type,type,
% 0.19/0.38 binintersect: $i > $i > $i ).
% 0.19/0.38
% 0.19/0.38 thf(binintersectI_type,type,
% 0.19/0.38 binintersectI: $o ).
% 0.19/0.38
% 0.19/0.38 thf(binintersectI,definition,
% 0.19/0.38 ( binintersectI
% 0.19/0.38 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ A )
% 0.19/0.38 => ( ( in @ Xx @ B )
% 0.19/0.38 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(binintersectSubset5_type,type,
% 0.19/0.38 binintersectSubset5: $o ).
% 0.19/0.38
% 0.19/0.38 thf(binintersectSubset5,definition,
% 0.19/0.38 ( binintersectSubset5
% 0.19/0.38 = ( ! [A: $i,B: $i,C: $i] :
% 0.19/0.38 ( ( subset @ C @ A )
% 0.19/0.38 => ( ( subset @ C @ B )
% 0.19/0.38 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(binintersectEL_type,type,
% 0.19/0.38 binintersectEL: $o ).
% 0.19/0.38
% 0.19/0.38 thf(binintersectEL,definition,
% 0.19/0.38 ( binintersectEL
% 0.19/0.38 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.19/0.38 => ( in @ Xx @ A ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(binintersectLsub_type,type,
% 0.19/0.38 binintersectLsub: $o ).
% 0.19/0.38
% 0.19/0.38 thf(binintersectLsub,definition,
% 0.19/0.38 ( binintersectLsub
% 0.19/0.38 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(binintersectSubset2_type,type,
% 0.19/0.38 binintersectSubset2: $o ).
% 0.19/0.38
% 0.19/0.38 thf(binintersectSubset2,definition,
% 0.19/0.38 ( binintersectSubset2
% 0.19/0.38 = ( ! [A: $i,B: $i] :
% 0.19/0.38 ( ( subset @ A @ B )
% 0.19/0.38 => ( ( binintersect @ A @ B )
% 0.19/0.38 = A ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(binintersectSubset3_type,type,
% 0.19/0.38 binintersectSubset3: $o ).
% 0.19/0.38
% 0.19/0.38 thf(binintersectSubset3,definition,
% 0.19/0.38 ( binintersectSubset3
% 0.19/0.38 = ( ! [A: $i,B: $i] :
% 0.19/0.38 ( ( ( binintersect @ A @ B )
% 0.19/0.38 = B )
% 0.19/0.38 => ( subset @ B @ A ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(binintersectER_type,type,
% 0.19/0.38 binintersectER: $o ).
% 0.19/0.38
% 0.19/0.38 thf(binintersectER,definition,
% 0.19/0.38 ( binintersectER
% 0.19/0.38 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.19/0.38 => ( in @ Xx @ B ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(disjointsetsI1_type,type,
% 0.19/0.38 disjointsetsI1: $o ).
% 0.19/0.38
% 0.19/0.38 thf(disjointsetsI1,definition,
% 0.19/0.38 ( disjointsetsI1
% 0.19/0.38 = ( ! [A: $i,B: $i] :
% 0.19/0.38 ( ~ ? [Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ A )
% 0.19/0.38 & ( in @ Xx @ B ) )
% 0.19/0.38 => ( ( binintersect @ A @ B )
% 0.19/0.38 = emptyset ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(binintersectRsub_type,type,
% 0.19/0.38 binintersectRsub: $o ).
% 0.19/0.38
% 0.19/0.38 thf(binintersectRsub,definition,
% 0.19/0.38 ( binintersectRsub
% 0.19/0.38 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(binintersectSubset4_type,type,
% 0.19/0.38 binintersectSubset4: $o ).
% 0.19/0.38
% 0.19/0.38 thf(binintersectSubset4,definition,
% 0.19/0.38 ( binintersectSubset4
% 0.19/0.38 = ( ! [A: $i,B: $i] :
% 0.19/0.38 ( ( subset @ B @ A )
% 0.19/0.38 => ( ( binintersect @ A @ B )
% 0.19/0.38 = B ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(binintersectSubset1_type,type,
% 0.19/0.38 binintersectSubset1: $o ).
% 0.19/0.38
% 0.19/0.38 thf(binintersectSubset1,definition,
% 0.19/0.38 ( binintersectSubset1
% 0.19/0.38 = ( ! [A: $i,B: $i] :
% 0.19/0.38 ( ( ( binintersect @ A @ B )
% 0.19/0.38 = A )
% 0.19/0.38 => ( subset @ A @ B ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(bs114d_type,type,
% 0.19/0.38 bs114d: $o ).
% 0.19/0.38
% 0.19/0.38 thf(bs114d,definition,
% 0.19/0.38 ( bs114d
% 0.19/0.38 = ( ! [A: $i,B: $i,C: $i] :
% 0.19/0.38 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 0.19/0.38 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(regular_type,type,
% 0.19/0.38 regular: $i > $o ).
% 0.19/0.38
% 0.19/0.38 thf(setminus_type,type,
% 0.19/0.38 setminus: $i > $i > $i ).
% 0.19/0.38
% 0.19/0.38 thf(setminusI_type,type,
% 0.19/0.38 setminusI: $o ).
% 0.19/0.38
% 0.19/0.38 thf(setminusI,definition,
% 0.19/0.38 ( setminusI
% 0.19/0.38 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ A )
% 0.19/0.38 => ( ~ ( in @ Xx @ B )
% 0.19/0.38 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(setminusEL_type,type,
% 0.19/0.38 setminusEL: $o ).
% 0.19/0.38
% 0.19/0.38 thf(setminusEL,definition,
% 0.19/0.38 ( setminusEL
% 0.19/0.38 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.19/0.38 => ( in @ Xx @ A ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(setminusER_type,type,
% 0.19/0.38 setminusER: $o ).
% 0.19/0.38
% 0.19/0.38 thf(setminusER,definition,
% 0.19/0.38 ( setminusER
% 0.19/0.38 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.19/0.38 => ~ ( in @ Xx @ B ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(setminusSubset2_type,type,
% 0.19/0.38 setminusSubset2: $o ).
% 0.19/0.38
% 0.19/0.38 thf(setminusSubset2,definition,
% 0.19/0.38 ( setminusSubset2
% 0.19/0.38 = ( ! [A: $i,B: $i] :
% 0.19/0.38 ( ( subset @ A @ B )
% 0.19/0.38 => ( ( setminus @ A @ B )
% 0.19/0.38 = emptyset ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(setminusERneg_type,type,
% 0.19/0.38 setminusERneg: $o ).
% 0.19/0.38
% 0.19/0.38 thf(setminusERneg,definition,
% 0.19/0.38 ( setminusERneg
% 0.19/0.38 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.38 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.19/0.38 => ( ( in @ Xx @ A )
% 0.19/0.38 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(setminusELneg_type,type,
% 0.19/0.38 setminusELneg: $o ).
% 0.19/0.38
% 0.19/0.38 thf(setminusELneg,definition,
% 0.19/0.38 ( setminusELneg
% 0.19/0.38 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.38 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.19/0.38 => ( ~ ( in @ Xx @ B )
% 0.19/0.38 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(setminusILneg_type,type,
% 0.19/0.38 setminusILneg: $o ).
% 0.19/0.38
% 0.19/0.38 thf(setminusILneg,definition,
% 0.19/0.38 ( setminusILneg
% 0.19/0.38 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.38 ( ~ ( in @ Xx @ A )
% 0.19/0.38 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(setminusIRneg_type,type,
% 0.19/0.38 setminusIRneg: $o ).
% 0.19/0.38
% 0.19/0.38 thf(setminusIRneg,definition,
% 0.19/0.38 ( setminusIRneg
% 0.19/0.38 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ B )
% 0.19/0.38 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(setminusLsub_type,type,
% 0.19/0.38 setminusLsub: $o ).
% 0.19/0.38
% 0.19/0.38 thf(setminusLsub,definition,
% 0.19/0.38 ( setminusLsub
% 0.19/0.38 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(setminusSubset1_type,type,
% 0.19/0.38 setminusSubset1: $o ).
% 0.19/0.38
% 0.19/0.38 thf(setminusSubset1,definition,
% 0.19/0.38 ( setminusSubset1
% 0.19/0.38 = ( ! [A: $i,B: $i] :
% 0.19/0.38 ( ( ( setminus @ A @ B )
% 0.19/0.38 = emptyset )
% 0.19/0.38 => ( subset @ A @ B ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(symdiff_type,type,
% 0.19/0.38 symdiff: $i > $i > $i ).
% 0.19/0.38
% 0.19/0.38 thf(symdiffE_type,type,
% 0.19/0.38 symdiffE: $o ).
% 0.19/0.38
% 0.19/0.38 thf(symdiffE,definition,
% 0.19/0.38 ( symdiffE
% 0.19/0.38 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 0.19/0.38 => ! [Xphi: $o] :
% 0.19/0.38 ( ( ( in @ Xx @ A )
% 0.19/0.38 => ( ~ ( in @ Xx @ B )
% 0.19/0.38 => Xphi ) )
% 0.19/0.38 => ( ( ~ ( in @ Xx @ A )
% 0.19/0.38 => ( ( in @ Xx @ B )
% 0.19/0.38 => Xphi ) )
% 0.19/0.38 => Xphi ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(symdiffI1_type,type,
% 0.19/0.38 symdiffI1: $o ).
% 0.19/0.38
% 0.19/0.38 thf(symdiffI1,definition,
% 0.19/0.38 ( symdiffI1
% 0.19/0.38 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ A )
% 0.19/0.38 => ( ~ ( in @ Xx @ B )
% 0.19/0.38 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(symdiffI2_type,type,
% 0.19/0.38 symdiffI2: $o ).
% 0.19/0.38
% 0.19/0.38 thf(symdiffI2,definition,
% 0.19/0.38 ( symdiffI2
% 0.19/0.38 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.38 ( ~ ( in @ Xx @ A )
% 0.19/0.38 => ( ( in @ Xx @ B )
% 0.19/0.38 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(symdiffIneg1_type,type,
% 0.19/0.38 symdiffIneg1: $o ).
% 0.19/0.38
% 0.19/0.38 thf(symdiffIneg1,definition,
% 0.19/0.38 ( symdiffIneg1
% 0.19/0.38 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ A )
% 0.19/0.38 => ( ( in @ Xx @ B )
% 0.19/0.38 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(symdiffIneg2_type,type,
% 0.19/0.38 symdiffIneg2: $o ).
% 0.19/0.38
% 0.19/0.38 thf(symdiffIneg2,definition,
% 0.19/0.38 ( symdiffIneg2
% 0.19/0.38 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.38 ( ~ ( in @ Xx @ A )
% 0.19/0.38 => ( ~ ( in @ Xx @ B )
% 0.19/0.38 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(iskpair_type,type,
% 0.19/0.38 iskpair: $i > $o ).
% 0.19/0.38
% 0.19/0.38 thf(secondinupair_type,type,
% 0.19/0.38 secondinupair: $o ).
% 0.19/0.38
% 0.19/0.38 thf(secondinupair,definition,
% 0.19/0.38 ( secondinupair
% 0.19/0.38 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(setukpairIL_type,type,
% 0.19/0.38 setukpairIL: $o ).
% 0.19/0.38
% 0.19/0.38 thf(setukpairIL,definition,
% 0.19/0.38 ( setukpairIL
% 0.19/0.38 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(setukpairIR_type,type,
% 0.19/0.38 setukpairIR: $o ).
% 0.19/0.38
% 0.19/0.38 thf(setukpairIR,definition,
% 0.19/0.38 ( setukpairIR
% 0.19/0.38 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(kpairiskpair_type,type,
% 0.19/0.38 kpairiskpair: $o ).
% 0.19/0.38
% 0.19/0.38 thf(kpairiskpair,definition,
% 0.19/0.38 ( kpairiskpair
% 0.19/0.38 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(kpair_type,type,
% 0.19/0.38 kpair: $i > $i > $i ).
% 0.19/0.38
% 0.19/0.38 thf(kpairp_type,type,
% 0.19/0.38 kpairp: $o ).
% 0.19/0.38
% 0.19/0.38 thf(kpairp,definition,
% 0.19/0.38 ( kpairp
% 0.19/0.38 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(cartprod_type,type,
% 0.19/0.38 cartprod: $i > $i > $i ).
% 0.19/0.38
% 0.19/0.38 thf(singletonsubset_type,type,
% 0.19/0.38 singletonsubset: $o ).
% 0.19/0.38
% 0.19/0.38 thf(singletonsubset,definition,
% 0.19/0.38 ( singletonsubset
% 0.19/0.38 = ( ! [A: $i,Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(singletoninpowerset_type,type,
% 0.19/0.39 singletoninpowerset: $o ).
% 0.19/0.39
% 0.19/0.39 thf(singletoninpowerset,definition,
% 0.19/0.39 ( singletoninpowerset
% 0.19/0.39 = ( ! [A: $i,Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(singletoninpowunion_type,type,
% 0.19/0.39 singletoninpowunion: $o ).
% 0.19/0.39
% 0.19/0.39 thf(singletoninpowunion,definition,
% 0.19/0.39 ( singletoninpowunion
% 0.19/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(upairset2E_type,type,
% 0.19/0.39 upairset2E: $o ).
% 0.19/0.39
% 0.19/0.39 thf(upairset2E,definition,
% 0.19/0.39 ( upairset2E
% 0.19/0.39 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.19/0.39 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.19/0.39 => ( ( Xz = Xx )
% 0.19/0.39 | ( Xz = Xy ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(upairsubunion_type,type,
% 0.19/0.39 upairsubunion: $o ).
% 0.19/0.39
% 0.19/0.39 thf(upairsubunion,definition,
% 0.19/0.39 ( upairsubunion
% 0.19/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ! [Xy: $i] :
% 0.19/0.39 ( ( in @ Xy @ B )
% 0.19/0.39 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(upairinpowunion_type,type,
% 0.19/0.39 upairinpowunion: $o ).
% 0.19/0.39
% 0.19/0.39 thf(upairinpowunion,definition,
% 0.19/0.39 ( upairinpowunion
% 0.19/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ! [Xy: $i] :
% 0.19/0.39 ( ( in @ Xy @ B )
% 0.19/0.39 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(ubforcartprodlem1_type,type,
% 0.19/0.39 ubforcartprodlem1: $o ).
% 0.19/0.39
% 0.19/0.39 thf(ubforcartprodlem1,definition,
% 0.19/0.39 ( ubforcartprodlem1
% 0.19/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ! [Xy: $i] :
% 0.19/0.39 ( ( in @ Xy @ B )
% 0.19/0.39 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(ubforcartprodlem2_type,type,
% 0.19/0.39 ubforcartprodlem2: $o ).
% 0.19/0.39
% 0.19/0.39 thf(ubforcartprodlem2,definition,
% 0.19/0.39 ( ubforcartprodlem2
% 0.19/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ! [Xy: $i] :
% 0.19/0.39 ( ( in @ Xy @ B )
% 0.19/0.39 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(ubforcartprodlem3_type,type,
% 0.19/0.39 ubforcartprodlem3: $o ).
% 0.19/0.39
% 0.19/0.39 thf(ubforcartprodlem3,definition,
% 0.19/0.39 ( ubforcartprodlem3
% 0.19/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ! [Xy: $i] :
% 0.19/0.39 ( ( in @ Xy @ B )
% 0.19/0.39 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(cartprodpairin_type,type,
% 0.19/0.39 cartprodpairin: $o ).
% 0.19/0.39
% 0.19/0.39 thf(cartprodpairin,definition,
% 0.19/0.39 ( cartprodpairin
% 0.19/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ! [Xy: $i] :
% 0.19/0.39 ( ( in @ Xy @ B )
% 0.19/0.39 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(cartprodmempair1_type,type,
% 0.19/0.39 cartprodmempair1: $o ).
% 0.19/0.39
% 0.19/0.39 thf(cartprodmempair1,definition,
% 0.19/0.39 ( cartprodmempair1
% 0.19/0.39 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.19/0.39 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.19/0.39 => ? [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 & ? [Xy: $i] :
% 0.19/0.39 ( ( in @ Xy @ B )
% 0.19/0.39 & ( Xu
% 0.19/0.39 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(cartprodmempair_type,type,
% 0.19/0.39 cartprodmempair: $o ).
% 0.19/0.39
% 0.19/0.39 thf(cartprodmempair,definition,
% 0.19/0.39 ( cartprodmempair
% 0.19/0.39 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.19/0.39 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.19/0.39 => ( iskpair @ Xu ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(setunionE2_type,type,
% 0.19/0.39 setunionE2: $o ).
% 0.19/0.39
% 0.19/0.39 thf(setunionE2,definition,
% 0.19/0.39 ( setunionE2
% 0.19/0.39 = ( ! [A: $i,Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ ( setunion @ A ) )
% 0.19/0.39 => ? [X: $i] :
% 0.19/0.39 ( ( in @ X @ A )
% 0.19/0.39 & ( in @ Xx @ X ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(setunionsingleton1_type,type,
% 0.19/0.39 setunionsingleton1: $o ).
% 0.19/0.39
% 0.19/0.39 thf(setunionsingleton1,definition,
% 0.19/0.39 ( setunionsingleton1
% 0.19/0.39 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(setunionsingleton2_type,type,
% 0.19/0.39 setunionsingleton2: $o ).
% 0.19/0.39
% 0.19/0.39 thf(setunionsingleton2,definition,
% 0.19/0.39 ( setunionsingleton2
% 0.19/0.39 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(setunionsingleton_type,type,
% 0.19/0.39 setunionsingleton: $o ).
% 0.19/0.39
% 0.19/0.39 thf(setunionsingleton,definition,
% 0.19/0.39 ( setunionsingleton
% 0.19/0.39 = ( ! [Xx: $i] :
% 0.19/0.39 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 0.19/0.39 = Xx ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(singleton_type,type,
% 0.19/0.39 singleton: $i > $o ).
% 0.19/0.39
% 0.19/0.39 thf(singletonprop_type,type,
% 0.19/0.39 singletonprop: $o ).
% 0.19/0.39
% 0.19/0.39 thf(singletonprop,definition,
% 0.19/0.39 ( singletonprop
% 0.19/0.39 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.39 ( ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ! [Xy: $i] :
% 0.19/0.39 ( ( in @ Xy @ A )
% 0.19/0.39 => ( ( Xphi @ Xx )
% 0.19/0.39 => ( ( Xphi @ Xy )
% 0.19/0.39 => ( Xx = Xy ) ) ) ) )
% 0.19/0.39 => ( ? [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 & ( Xphi @ Xx ) )
% 0.19/0.39 => ( singleton
% 0.19/0.39 @ ( dsetconstr @ A
% 0.19/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(ex1_type,type,
% 0.19/0.39 ex1: $i > ( $i > $o ) > $o ).
% 0.19/0.39
% 0.19/0.39 thf(ex1E1_type,type,
% 0.19/0.39 ex1E1: $o ).
% 0.19/0.39
% 0.19/0.39 thf(ex1E1,definition,
% 0.19/0.39 ( ex1E1
% 0.19/0.39 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.39 ( ( ex1 @ A
% 0.19/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.39 => ? [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(ex1I_type,type,
% 0.19/0.39 ex1I: $o ).
% 0.19/0.39
% 0.19/0.39 thf(ex1I,definition,
% 0.19/0.39 ( ex1I
% 0.19/0.39 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ( ( Xphi @ Xx )
% 0.19/0.39 => ( ! [Xy: $i] :
% 0.19/0.39 ( ( in @ Xy @ A )
% 0.19/0.39 => ( ( Xphi @ Xy )
% 0.19/0.39 => ( Xy = Xx ) ) )
% 0.19/0.39 => ( ex1 @ A
% 0.19/0.39 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(ex1I2_type,type,
% 0.19/0.39 ex1I2: $o ).
% 0.19/0.39
% 0.19/0.39 thf(ex1I2,definition,
% 0.19/0.39 ( ex1I2
% 0.19/0.39 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.39 ( ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ! [Xy: $i] :
% 0.19/0.39 ( ( in @ Xy @ A )
% 0.19/0.39 => ( ( Xphi @ Xx )
% 0.19/0.39 => ( ( Xphi @ Xy )
% 0.19/0.39 => ( Xx = Xy ) ) ) ) )
% 0.19/0.39 => ( ? [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 & ( Xphi @ Xx ) )
% 0.19/0.39 => ( ex1 @ A
% 0.19/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(singletonsuniq_type,type,
% 0.19/0.39 singletonsuniq: $o ).
% 0.19/0.39
% 0.19/0.39 thf(singletonsuniq,definition,
% 0.19/0.39 ( singletonsuniq
% 0.19/0.39 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.39 ( ( ( setadjoin @ Xx @ emptyset )
% 0.19/0.39 = ( setadjoin @ Xy @ emptyset ) )
% 0.19/0.39 => ( Xx = Xy ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(atmost1p_type,type,
% 0.19/0.39 atmost1p: $i > $o ).
% 0.19/0.39
% 0.19/0.39 thf(atleast2p_type,type,
% 0.19/0.39 atleast2p: $i > $o ).
% 0.19/0.39
% 0.19/0.39 thf(atmost2p_type,type,
% 0.19/0.39 atmost2p: $i > $o ).
% 0.19/0.39
% 0.19/0.39 thf(upairsetp_type,type,
% 0.19/0.39 upairsetp: $i > $o ).
% 0.19/0.39
% 0.19/0.39 thf(setukpairinjL1_type,type,
% 0.19/0.39 setukpairinjL1: $o ).
% 0.19/0.39
% 0.19/0.39 thf(setukpairinjL1,definition,
% 0.19/0.39 ( setukpairinjL1
% 0.19/0.39 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.19/0.39 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 0.19/0.39 => ( Xx = Xz ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(kfstsingleton_type,type,
% 0.19/0.39 kfstsingleton: $o ).
% 0.19/0.39
% 0.19/0.39 thf(kfstsingleton,definition,
% 0.19/0.39 ( kfstsingleton
% 0.19/0.39 = ( ! [Xu: $i] :
% 0.19/0.39 ( ( iskpair @ Xu )
% 0.19/0.39 => ( singleton
% 0.19/0.39 @ ( dsetconstr @ ( setunion @ Xu )
% 0.19/0.39 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(theprop_type,type,
% 0.19/0.39 theprop: $o ).
% 0.19/0.39
% 0.19/0.39 thf(theprop,definition,
% 0.19/0.39 ( theprop
% 0.19/0.39 = ( ! [X: $i] :
% 0.19/0.39 ( ( singleton @ X )
% 0.19/0.39 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(kfst_type,type,
% 0.19/0.39 kfst: $i > $i ).
% 0.19/0.39
% 0.19/0.39 thf(kfstpairEq_type,type,
% 0.19/0.39 kfstpairEq: $o ).
% 0.19/0.39
% 0.19/0.39 thf(kfstpairEq,definition,
% 0.19/0.39 ( kfstpairEq
% 0.19/0.39 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.39 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.19/0.39 = Xx ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(cartprodfstin_type,type,
% 0.19/0.39 cartprodfstin: $o ).
% 0.19/0.39
% 0.19/0.39 thf(cartprodfstin,definition,
% 0.19/0.39 ( cartprodfstin
% 0.19/0.39 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.19/0.39 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.19/0.39 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(setukpairinjL2_type,type,
% 0.19/0.39 setukpairinjL2: $o ).
% 0.19/0.39
% 0.19/0.39 thf(setukpairinjL2,definition,
% 0.19/0.39 ( setukpairinjL2
% 0.19/0.39 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.19/0.39 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.19/0.39 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.19/0.39 => ( Xx = Xz ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(setukpairinjL_type,type,
% 0.19/0.39 setukpairinjL: $o ).
% 0.19/0.39
% 0.19/0.39 thf(setukpairinjL,definition,
% 0.19/0.39 ( setukpairinjL
% 0.19/0.39 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.19/0.39 ( ( ( kpair @ Xx @ Xy )
% 0.19/0.39 = ( kpair @ Xz @ Xu ) )
% 0.19/0.39 => ( Xx = Xz ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(setukpairinjR11_type,type,
% 0.19/0.39 setukpairinjR11: $o ).
% 0.19/0.39
% 0.19/0.39 thf(setukpairinjR11,definition,
% 0.19/0.39 ( setukpairinjR11
% 0.19/0.39 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.39 ( ( Xx = Xy )
% 0.19/0.39 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.19/0.39 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(setukpairinjR12_type,type,
% 0.19/0.39 setukpairinjR12: $o ).
% 0.19/0.39
% 0.19/0.39 thf(setukpairinjR12,definition,
% 0.19/0.39 ( setukpairinjR12
% 0.19/0.39 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.39 ( ( Xx = Xy )
% 0.19/0.39 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.19/0.39 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(setukpairinjR1_type,type,
% 0.19/0.39 setukpairinjR1: $o ).
% 0.19/0.39
% 0.19/0.39 thf(setukpairinjR1,definition,
% 0.19/0.39 ( setukpairinjR1
% 0.19/0.39 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.19/0.39 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.19/0.39 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.19/0.39 => ( ( Xz = Xu )
% 0.19/0.39 => ( Xy = Xu ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(upairequniteq_type,type,
% 0.19/0.39 upairequniteq: $o ).
% 0.19/0.39
% 0.19/0.39 thf(upairequniteq,definition,
% 0.19/0.39 ( upairequniteq
% 0.19/0.39 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.19/0.39 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.19/0.39 = ( setadjoin @ Xz @ emptyset ) )
% 0.19/0.39 => ( Xx = Xy ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(setukpairinjR2_type,type,
% 0.19/0.39 setukpairinjR2: $o ).
% 0.19/0.39
% 0.19/0.39 thf(setukpairinjR2,definition,
% 0.19/0.39 ( setukpairinjR2
% 0.19/0.39 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.19/0.39 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.19/0.39 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.19/0.39 => ( Xy = Xu ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(setukpairinjR_type,type,
% 0.19/0.39 setukpairinjR: $o ).
% 0.19/0.39
% 0.19/0.39 thf(setukpairinjR,definition,
% 0.19/0.39 ( setukpairinjR
% 0.19/0.39 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.19/0.39 ( ( ( kpair @ Xx @ Xy )
% 0.19/0.39 = ( kpair @ Xz @ Xu ) )
% 0.19/0.39 => ( Xy = Xu ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(ksndsingleton_type,type,
% 0.19/0.39 ksndsingleton: $o ).
% 0.19/0.39
% 0.19/0.39 thf(ksndsingleton,definition,
% 0.19/0.39 ( ksndsingleton
% 0.19/0.39 = ( ! [Xu: $i] :
% 0.19/0.39 ( ( iskpair @ Xu )
% 0.19/0.39 => ( singleton
% 0.19/0.39 @ ( dsetconstr @ ( setunion @ Xu )
% 0.19/0.39 @ ^ [Xx: $i] :
% 0.19/0.39 ( Xu
% 0.19/0.39 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(ksnd_type,type,
% 0.19/0.39 ksnd: $i > $i ).
% 0.19/0.39
% 0.19/0.39 thf(ksndpairEq_type,type,
% 0.19/0.39 ksndpairEq: $o ).
% 0.19/0.39
% 0.19/0.39 thf(ksndpairEq,definition,
% 0.19/0.39 ( ksndpairEq
% 0.19/0.39 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.39 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.19/0.39 = Xy ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(kpairsurjEq_type,type,
% 0.19/0.39 kpairsurjEq: $o ).
% 0.19/0.39
% 0.19/0.39 thf(kpairsurjEq,definition,
% 0.19/0.39 ( kpairsurjEq
% 0.19/0.39 = ( ! [Xu: $i] :
% 0.19/0.39 ( ( iskpair @ Xu )
% 0.19/0.39 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.19/0.39 = Xu ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(cartprodsndin_type,type,
% 0.19/0.39 cartprodsndin: $o ).
% 0.19/0.39
% 0.19/0.39 thf(cartprodsndin,definition,
% 0.19/0.39 ( cartprodsndin
% 0.19/0.39 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.19/0.39 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.19/0.39 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(cartprodpairmemEL_type,type,
% 0.19/0.39 cartprodpairmemEL: $o ).
% 0.19/0.39
% 0.19/0.39 thf(cartprodpairmemEL,definition,
% 0.19/0.39 ( cartprodpairmemEL
% 0.19/0.39 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.19/0.39 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.19/0.39 => ( in @ Xx @ A ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(cartprodpairmemER_type,type,
% 0.19/0.39 cartprodpairmemER: $o ).
% 0.19/0.39
% 0.19/0.39 thf(cartprodpairmemER,definition,
% 0.19/0.39 ( cartprodpairmemER
% 0.19/0.39 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.19/0.39 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.19/0.39 => ( in @ Xy @ B ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(cartprodmempaircEq_type,type,
% 0.19/0.39 cartprodmempaircEq: $o ).
% 0.19/0.39
% 0.19/0.39 thf(cartprodmempaircEq,definition,
% 0.19/0.39 ( cartprodmempaircEq
% 0.19/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ! [Xy: $i] :
% 0.19/0.39 ( ( in @ Xy @ B )
% 0.19/0.39 => ( ( kpair @ Xx @ Xy )
% 0.19/0.39 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(cartprodfstpairEq_type,type,
% 0.19/0.39 cartprodfstpairEq: $o ).
% 0.19/0.39
% 0.19/0.39 thf(cartprodfstpairEq,definition,
% 0.19/0.39 ( cartprodfstpairEq
% 0.19/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ! [Xy: $i] :
% 0.19/0.39 ( ( in @ Xy @ B )
% 0.19/0.39 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.19/0.39 = Xx ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(cartprodsndpairEq_type,type,
% 0.19/0.39 cartprodsndpairEq: $o ).
% 0.19/0.39
% 0.19/0.39 thf(cartprodsndpairEq,definition,
% 0.19/0.39 ( cartprodsndpairEq
% 0.19/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ! [Xy: $i] :
% 0.19/0.39 ( ( in @ Xy @ B )
% 0.19/0.39 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.19/0.39 = Xy ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(cartprodpairsurjEq_type,type,
% 0.19/0.39 cartprodpairsurjEq: $o ).
% 0.19/0.39
% 0.19/0.39 thf(cartprodpairsurjEq,definition,
% 0.19/0.39 ( cartprodpairsurjEq
% 0.19/0.39 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.19/0.39 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.19/0.39 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.19/0.39 = Xu ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(breln_type,type,
% 0.19/0.39 breln: $i > $i > $i > $o ).
% 0.19/0.39
% 0.19/0.39 thf(dpsetconstr_type,type,
% 0.19/0.39 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 0.19/0.39
% 0.19/0.39 thf(dpsetconstrI_type,type,
% 0.19/0.39 dpsetconstrI: $o ).
% 0.19/0.39
% 0.19/0.39 thf(dpsetconstrI,definition,
% 0.19/0.39 ( dpsetconstrI
% 0.19/0.39 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ! [Xy: $i] :
% 0.19/0.39 ( ( in @ Xy @ B )
% 0.19/0.39 => ( ( Xphi @ Xx @ Xy )
% 0.19/0.39 => ( in @ ( kpair @ Xx @ Xy )
% 0.19/0.39 @ ( dpsetconstr @ A @ B
% 0.19/0.39 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(dpsetconstrSub_type,type,
% 0.19/0.39 dpsetconstrSub: $o ).
% 0.19/0.39
% 0.19/0.39 thf(dpsetconstrSub,definition,
% 0.19/0.39 ( dpsetconstrSub
% 0.19/0.39 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.19/0.39 ( subset
% 0.19/0.39 @ ( dpsetconstr @ A @ B
% 0.19/0.39 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 0.19/0.39 @ ( cartprod @ A @ B ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(setOfPairsIsBReln_type,type,
% 0.19/0.39 setOfPairsIsBReln: $o ).
% 0.19/0.39
% 0.19/0.39 thf(setOfPairsIsBReln,definition,
% 0.19/0.39 ( setOfPairsIsBReln
% 0.19/0.39 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.19/0.39 ( breln @ A @ B
% 0.19/0.39 @ ( dpsetconstr @ A @ B
% 0.19/0.39 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(dpsetconstrERa_type,type,
% 0.19/0.39 dpsetconstrERa: $o ).
% 0.19/0.39
% 0.19/0.39 thf(dpsetconstrERa,definition,
% 0.19/0.39 ( dpsetconstrERa
% 0.19/0.39 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ! [Xy: $i] :
% 0.19/0.39 ( ( in @ Xy @ B )
% 0.19/0.39 => ( ( in @ ( kpair @ Xx @ Xy )
% 0.19/0.39 @ ( dpsetconstr @ A @ B
% 0.19/0.39 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.19/0.39 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(dpsetconstrEL1_type,type,
% 0.19/0.39 dpsetconstrEL1: $o ).
% 0.19/0.39
% 0.19/0.39 thf(dpsetconstrEL1,definition,
% 0.19/0.39 ( dpsetconstrEL1
% 0.19/0.39 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.19/0.39 ( ( in @ ( kpair @ Xx @ Xy )
% 0.19/0.39 @ ( dpsetconstr @ A @ B
% 0.19/0.39 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.19/0.39 => ( in @ Xx @ A ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(dpsetconstrEL2_type,type,
% 0.19/0.39 dpsetconstrEL2: $o ).
% 0.19/0.39
% 0.19/0.39 thf(dpsetconstrEL2,definition,
% 0.19/0.39 ( dpsetconstrEL2
% 0.19/0.39 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.19/0.39 ( ( in @ ( kpair @ Xx @ Xy )
% 0.19/0.39 @ ( dpsetconstr @ A @ B
% 0.19/0.39 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.19/0.39 => ( in @ Xy @ B ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(dpsetconstrER_type,type,
% 0.19/0.39 dpsetconstrER: $o ).
% 0.19/0.39
% 0.19/0.39 thf(dpsetconstrER,definition,
% 0.19/0.39 ( dpsetconstrER
% 0.19/0.39 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.19/0.39 ( ( in @ ( kpair @ Xx @ Xy )
% 0.19/0.39 @ ( dpsetconstr @ A @ B
% 0.19/0.39 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.19/0.39 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(func_type,type,
% 0.19/0.39 func: $i > $i > $i > $o ).
% 0.19/0.39
% 0.19/0.39 thf(funcSet_type,type,
% 0.19/0.39 funcSet: $i > $i > $i ).
% 0.19/0.39
% 0.19/0.39 thf(funcImageSingleton_type,type,
% 0.19/0.39 funcImageSingleton: $o ).
% 0.19/0.39
% 0.19/0.39 thf(funcImageSingleton,definition,
% 0.19/0.39 ( funcImageSingleton
% 0.19/0.39 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.39 ( ( func @ A @ B @ Xf )
% 0.19/0.39 => ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ( singleton
% 0.19/0.39 @ ( dsetconstr @ B
% 0.19/0.39 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(apProp_type,type,
% 0.19/0.39 apProp: $o ).
% 0.19/0.39
% 0.19/0.39 thf(apProp,definition,
% 0.19/0.39 ( apProp
% 0.19/0.39 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.39 ( ( func @ A @ B @ Xf )
% 0.19/0.39 => ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ( in
% 0.19/0.39 @ ( setunion
% 0.19/0.39 @ ( dsetconstr @ B
% 0.19/0.39 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 0.19/0.39 @ B ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(ap_type,type,
% 0.19/0.39 ap: $i > $i > $i > $i > $i ).
% 0.19/0.39
% 0.19/0.39 thf(app_type,type,
% 0.19/0.39 app: $o ).
% 0.19/0.39
% 0.19/0.39 thf(app,definition,
% 0.19/0.39 ( app
% 0.19/0.39 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.39 ( ( func @ A @ B @ Xf )
% 0.19/0.39 => ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(infuncsetfunc_type,type,
% 0.19/0.39 infuncsetfunc: $o ).
% 0.19/0.39
% 0.19/0.39 thf(infuncsetfunc,definition,
% 0.19/0.39 ( infuncsetfunc
% 0.19/0.39 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.39 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.19/0.39 => ( func @ A @ B @ Xf ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(ap2p_type,type,
% 0.19/0.39 ap2p: $o ).
% 0.19/0.39
% 0.19/0.39 thf(ap2p,definition,
% 0.19/0.39 ( ap2p
% 0.19/0.39 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.39 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.19/0.39 => ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(funcinfuncset_type,type,
% 0.19/0.39 funcinfuncset: $o ).
% 0.19/0.39
% 0.19/0.39 thf(funcinfuncset,definition,
% 0.19/0.39 ( funcinfuncset
% 0.19/0.39 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.39 ( ( func @ A @ B @ Xf )
% 0.19/0.39 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(lamProp_type,type,
% 0.19/0.39 lamProp: $o ).
% 0.19/0.39
% 0.19/0.39 thf(lamProp,definition,
% 0.19/0.39 ( lamProp
% 0.19/0.39 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.19/0.39 ( ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.19/0.39 => ( func @ A @ B
% 0.19/0.39 @ ( dpsetconstr @ A @ B
% 0.19/0.39 @ ^ [Xx: $i,Xy: $i] :
% 0.19/0.39 ( ( Xf @ Xx )
% 0.19/0.39 = Xy ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(lam_type,type,
% 0.19/0.39 lam: $i > $i > ( $i > $i ) > $i ).
% 0.19/0.39
% 0.19/0.39 thf(lamp_type,type,
% 0.19/0.39 lamp: $o ).
% 0.19/0.39
% 0.19/0.39 thf(lamp,definition,
% 0.19/0.39 ( lamp
% 0.19/0.39 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.19/0.39 ( ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.19/0.39 => ( func @ A @ B
% 0.19/0.39 @ ( lam @ A @ B
% 0.19/0.39 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(lam2p_type,type,
% 0.19/0.39 lam2p: $o ).
% 0.19/0.39
% 0.19/0.39 thf(lam2p,definition,
% 0.19/0.39 ( lam2p
% 0.19/0.39 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.19/0.39 ( ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.19/0.39 => ( in
% 0.19/0.39 @ ( lam @ A @ B
% 0.19/0.39 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.19/0.39 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(brelnall1_type,type,
% 0.19/0.39 brelnall1: $o ).
% 0.19/0.39
% 0.19/0.39 thf(brelnall1,definition,
% 0.19/0.39 ( brelnall1
% 0.19/0.39 = ( ! [A: $i,B: $i,R: $i] :
% 0.19/0.39 ( ( breln @ A @ B @ R )
% 0.19/0.39 => ! [Xphi: $i > $o] :
% 0.19/0.39 ( ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ! [Xy: $i] :
% 0.19/0.39 ( ( in @ Xy @ B )
% 0.19/0.39 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.19/0.39 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.19/0.39 => ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ R )
% 0.19/0.39 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(brelnall2_type,type,
% 0.19/0.39 brelnall2: $o ).
% 0.19/0.39
% 0.19/0.39 thf(brelnall2,definition,
% 0.19/0.39 ( brelnall2
% 0.19/0.39 = ( ! [A: $i,B: $i,R: $i] :
% 0.19/0.39 ( ( breln @ A @ B @ R )
% 0.19/0.39 => ! [Xphi: $i > $o] :
% 0.19/0.39 ( ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ! [Xy: $i] :
% 0.19/0.39 ( ( in @ Xy @ B )
% 0.19/0.39 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.19/0.39 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.19/0.39 => ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ R )
% 0.19/0.39 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(ex1E2_type,type,
% 0.19/0.39 ex1E2: $o ).
% 0.19/0.39
% 0.19/0.39 thf(ex1E2,definition,
% 0.19/0.39 ( ex1E2
% 0.19/0.39 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.39 ( ( ex1 @ A
% 0.19/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.39 => ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ! [Xy: $i] :
% 0.19/0.39 ( ( in @ Xy @ A )
% 0.19/0.39 => ( ( Xphi @ Xx )
% 0.19/0.39 => ( ( Xphi @ Xy )
% 0.19/0.39 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(funcGraphProp1_type,type,
% 0.19/0.39 funcGraphProp1: $o ).
% 0.19/0.39
% 0.19/0.39 thf(funcGraphProp1,definition,
% 0.19/0.39 ( funcGraphProp1
% 0.19/0.39 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.39 ( ( func @ A @ B @ Xf )
% 0.19/0.39 => ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(funcGraphProp3_type,type,
% 0.19/0.39 funcGraphProp3: $o ).
% 0.19/0.39
% 0.19/0.39 thf(funcGraphProp3,definition,
% 0.19/0.39 ( funcGraphProp3
% 0.19/0.39 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.39 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.19/0.39 => ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(funcGraphProp2_type,type,
% 0.19/0.39 funcGraphProp2: $o ).
% 0.19/0.39
% 0.19/0.39 thf(funcGraphProp2,definition,
% 0.19/0.39 ( funcGraphProp2
% 0.19/0.39 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.39 ( ( func @ A @ B @ Xf )
% 0.19/0.39 => ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ! [Xy: $i] :
% 0.19/0.39 ( ( in @ Xy @ B )
% 0.19/0.39 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.19/0.39 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.19/0.39 = Xy ) ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(funcextLem_type,type,
% 0.19/0.39 funcextLem: $o ).
% 0.19/0.39
% 0.19/0.39 thf(funcextLem,definition,
% 0.19/0.39 ( funcextLem
% 0.19/0.39 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.39 ( ( func @ A @ B @ Xf )
% 0.19/0.39 => ! [Xg: $i] :
% 0.19/0.39 ( ( func @ A @ B @ Xg )
% 0.19/0.39 => ( ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.19/0.39 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.19/0.39 => ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ! [Xy: $i] :
% 0.19/0.39 ( ( in @ Xy @ B )
% 0.19/0.39 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 0.19/0.39 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(funcGraphProp4_type,type,
% 0.19/0.39 funcGraphProp4: $o ).
% 0.19/0.39
% 0.19/0.39 thf(funcGraphProp4,definition,
% 0.19/0.39 ( funcGraphProp4
% 0.19/0.39 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.39 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.19/0.39 => ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ! [Xy: $i] :
% 0.19/0.39 ( ( in @ Xy @ B )
% 0.19/0.39 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.19/0.39 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.19/0.39 = Xy ) ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(subbreln_type,type,
% 0.19/0.39 subbreln: $o ).
% 0.19/0.39
% 0.19/0.39 thf(subbreln,definition,
% 0.19/0.39 ( subbreln
% 0.19/0.39 = ( ! [A: $i,B: $i,R: $i] :
% 0.19/0.39 ( ( breln @ A @ B @ R )
% 0.19/0.39 => ! [S: $i] :
% 0.19/0.39 ( ( breln @ A @ B @ S )
% 0.19/0.39 => ( ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ! [Xy: $i] :
% 0.19/0.39 ( ( in @ Xy @ B )
% 0.19/0.39 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.19/0.39 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.19/0.39 => ( subset @ R @ S ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(eqbreln_type,type,
% 0.19/0.39 eqbreln: $o ).
% 0.19/0.39
% 0.19/0.39 thf(eqbreln,definition,
% 0.19/0.39 ( eqbreln
% 0.19/0.39 = ( ! [A: $i,B: $i,R: $i] :
% 0.19/0.39 ( ( breln @ A @ B @ R )
% 0.19/0.39 => ! [S: $i] :
% 0.19/0.39 ( ( breln @ A @ B @ S )
% 0.19/0.39 => ( ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ! [Xy: $i] :
% 0.19/0.39 ( ( in @ Xy @ B )
% 0.19/0.39 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.19/0.39 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.19/0.39 => ( ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ! [Xy: $i] :
% 0.19/0.39 ( ( in @ Xy @ B )
% 0.19/0.39 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.19/0.39 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 0.19/0.39 => ( R = S ) ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(funcext_type,type,
% 0.19/0.39 funcext: $o ).
% 0.19/0.39
% 0.19/0.39 thf(funcext,definition,
% 0.19/0.39 ( funcext
% 0.19/0.39 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.39 ( ( func @ A @ B @ Xf )
% 0.19/0.39 => ! [Xg: $i] :
% 0.19/0.39 ( ( func @ A @ B @ Xg )
% 0.19/0.39 => ( ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.19/0.39 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.19/0.39 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(funcext2_type,type,
% 0.19/0.39 funcext2: $o ).
% 0.19/0.39
% 0.19/0.39 thf(funcext2,definition,
% 0.19/0.39 ( funcext2
% 0.19/0.39 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.39 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.19/0.39 => ! [Xg: $i] :
% 0.19/0.39 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 0.19/0.39 => ( ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.19/0.39 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.19/0.39 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(ap2apEq1_type,type,
% 0.19/0.39 ap2apEq1: $o ).
% 0.19/0.39
% 0.19/0.39 thf(ap2apEq1,definition,
% 0.19/0.39 ( ap2apEq1
% 0.19/0.39 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.39 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.19/0.39 => ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.19/0.39 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(ap2apEq2_type,type,
% 0.19/0.39 ap2apEq2: $o ).
% 0.19/0.39
% 0.19/0.39 thf(ap2apEq2,definition,
% 0.19/0.39 ( ap2apEq2
% 0.19/0.39 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.39 ( ( func @ A @ B @ Xf )
% 0.19/0.39 => ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.19/0.39 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(beta1_type,type,
% 0.19/0.39 beta1: $o ).
% 0.19/0.39
% 0.19/0.39 thf(beta1,definition,
% 0.19/0.39 ( beta1
% 0.19/0.39 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.19/0.39 ( ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.19/0.39 => ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ( ( ap @ A @ B
% 0.19/0.39 @ ( lam @ A @ B
% 0.19/0.39 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.19/0.39 @ Xx )
% 0.19/0.39 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(eta1_type,type,
% 0.19/0.39 eta1: $o ).
% 0.19/0.39
% 0.19/0.39 thf(eta1,definition,
% 0.19/0.39 ( eta1
% 0.19/0.39 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.39 ( ( func @ A @ B @ Xf )
% 0.19/0.39 => ( ( lam @ A @ B
% 0.19/0.39 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.19/0.39 = Xf ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(lam2lamEq_type,type,
% 0.19/0.39 lam2lamEq: $o ).
% 0.19/0.39
% 0.19/0.39 thf(lam2lamEq,definition,
% 0.19/0.39 ( lam2lamEq
% 0.19/0.39 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.19/0.39 ( ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.19/0.39 => ( ( lam @ A @ B
% 0.19/0.39 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.19/0.39 = ( lam @ A @ B
% 0.19/0.39 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(beta2_type,type,
% 0.19/0.39 beta2: $o ).
% 0.19/0.39
% 0.19/0.39 thf(beta2,definition,
% 0.19/0.39 ( beta2
% 0.19/0.39 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.19/0.39 ( ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.19/0.39 => ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ( ( ap @ A @ B
% 0.19/0.39 @ ( lam @ A @ B
% 0.19/0.39 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.19/0.39 @ Xx )
% 0.19/0.39 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(eta2_type,type,
% 0.19/0.39 eta2: $o ).
% 0.19/0.39
% 0.19/0.39 thf(eta2,definition,
% 0.19/0.39 ( eta2
% 0.19/0.39 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.39 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.19/0.39 => ( ( lam @ A @ B
% 0.19/0.39 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.19/0.39 = Xf ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(iffalseProp1_type,type,
% 0.19/0.39 iffalseProp1: $o ).
% 0.19/0.39
% 0.19/0.39 thf(iffalseProp1,definition,
% 0.19/0.39 ( iffalseProp1
% 0.19/0.39 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ! [Xy: $i] :
% 0.19/0.39 ( ( in @ Xy @ A )
% 0.19/0.39 => ( ~ Xphi
% 0.19/0.39 => ( in @ Xy
% 0.19/0.39 @ ( dsetconstr @ A
% 0.19/0.39 @ ^ [Xz: $i] :
% 0.19/0.39 ( ( Xphi
% 0.19/0.39 & ( Xz = Xx ) )
% 0.19/0.39 | ( ~ Xphi
% 0.19/0.39 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(iffalseProp2_type,type,
% 0.19/0.39 iffalseProp2: $o ).
% 0.19/0.39
% 0.19/0.39 thf(iffalseProp2,definition,
% 0.19/0.39 ( iffalseProp2
% 0.19/0.39 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ! [Xy: $i] :
% 0.19/0.39 ( ( in @ Xy @ A )
% 0.19/0.39 => ( ~ Xphi
% 0.19/0.39 => ( ( dsetconstr @ A
% 0.19/0.39 @ ^ [Xz: $i] :
% 0.19/0.39 ( ( Xphi
% 0.19/0.39 & ( Xz = Xx ) )
% 0.19/0.39 | ( ~ Xphi
% 0.19/0.39 & ( Xz = Xy ) ) ) )
% 0.19/0.39 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(iftrueProp1_type,type,
% 0.19/0.39 iftrueProp1: $o ).
% 0.19/0.39
% 0.19/0.39 thf(iftrueProp1,definition,
% 0.19/0.39 ( iftrueProp1
% 0.19/0.39 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ! [Xy: $i] :
% 0.19/0.39 ( ( in @ Xy @ A )
% 0.19/0.39 => ( Xphi
% 0.19/0.39 => ( in @ Xx
% 0.19/0.39 @ ( dsetconstr @ A
% 0.19/0.39 @ ^ [Xz: $i] :
% 0.19/0.39 ( ( Xphi
% 0.19/0.39 & ( Xz = Xx ) )
% 0.19/0.39 | ( ~ Xphi
% 0.19/0.39 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(iftrueProp2_type,type,
% 0.19/0.39 iftrueProp2: $o ).
% 0.19/0.39
% 0.19/0.39 thf(iftrueProp2,definition,
% 0.19/0.39 ( iftrueProp2
% 0.19/0.39 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ! [Xy: $i] :
% 0.19/0.39 ( ( in @ Xy @ A )
% 0.19/0.39 => ( Xphi
% 0.19/0.39 => ( ( dsetconstr @ A
% 0.19/0.39 @ ^ [Xz: $i] :
% 0.19/0.39 ( ( Xphi
% 0.19/0.39 & ( Xz = Xx ) )
% 0.19/0.39 | ( ~ Xphi
% 0.19/0.39 & ( Xz = Xy ) ) ) )
% 0.19/0.39 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(ifSingleton_type,type,
% 0.19/0.39 ifSingleton: $o ).
% 0.19/0.39
% 0.19/0.39 thf(ifSingleton,definition,
% 0.19/0.39 ( ifSingleton
% 0.19/0.39 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ! [Xy: $i] :
% 0.19/0.39 ( ( in @ Xy @ A )
% 0.19/0.39 => ( singleton
% 0.19/0.39 @ ( dsetconstr @ A
% 0.19/0.39 @ ^ [Xz: $i] :
% 0.19/0.39 ( ( Xphi
% 0.19/0.39 & ( Xz = Xx ) )
% 0.19/0.39 | ( ~ Xphi
% 0.19/0.39 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(if_type,type,
% 0.19/0.39 if: $i > $o > $i > $i > $i ).
% 0.19/0.39
% 0.19/0.39 thf(ifp_type,type,
% 0.19/0.39 ifp: $o ).
% 0.19/0.39
% 0.19/0.39 thf(ifp,definition,
% 0.19/0.39 ( ifp
% 0.19/0.39 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ! [Xy: $i] :
% 0.19/0.39 ( ( in @ Xy @ A )
% 0.19/0.39 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(theeq_type,type,
% 0.19/0.39 theeq: $o ).
% 0.19/0.39
% 0.19/0.39 thf(theeq,definition,
% 0.19/0.39 ( theeq
% 0.19/0.39 = ( ! [X: $i] :
% 0.19/0.39 ( ( singleton @ X )
% 0.19/0.39 => ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ X )
% 0.19/0.39 => ( ( setunion @ X )
% 0.19/0.39 = Xx ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(iftrue_type,type,
% 0.19/0.39 iftrue: $o ).
% 0.19/0.39
% 0.19/0.39 thf(iftrue,definition,
% 0.19/0.39 ( iftrue
% 0.19/0.39 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ! [Xy: $i] :
% 0.19/0.39 ( ( in @ Xy @ A )
% 0.19/0.39 => ( Xphi
% 0.19/0.39 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.19/0.39 = Xx ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(iffalse_type,type,
% 0.19/0.39 iffalse: $o ).
% 0.19/0.39
% 0.19/0.39 thf(iffalse,definition,
% 0.19/0.39 ( iffalse
% 0.19/0.39 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ! [Xy: $i] :
% 0.19/0.39 ( ( in @ Xy @ A )
% 0.19/0.39 => ( ~ Xphi
% 0.19/0.39 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.19/0.39 = Xy ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(iftrueorfalse_type,type,
% 0.19/0.39 iftrueorfalse: $o ).
% 0.19/0.39
% 0.19/0.39 thf(iftrueorfalse,definition,
% 0.19/0.39 ( iftrueorfalse
% 0.19/0.39 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ! [Xy: $i] :
% 0.19/0.39 ( ( in @ Xy @ A )
% 0.19/0.39 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(binintersectT_lem_type,type,
% 0.19/0.39 binintersectT_lem: $o ).
% 0.19/0.39
% 0.19/0.39 thf(binintersectT_lem,definition,
% 0.19/0.39 ( binintersectT_lem
% 0.19/0.39 = ( ! [A: $i,X: $i] :
% 0.19/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.39 => ! [Y: $i] :
% 0.19/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.39 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(binunionT_lem_type,type,
% 0.19/0.39 binunionT_lem: $o ).
% 0.19/0.39
% 0.19/0.39 thf(binunionT_lem,definition,
% 0.19/0.39 ( binunionT_lem
% 0.19/0.39 = ( ! [A: $i,X: $i] :
% 0.19/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.39 => ! [Y: $i] :
% 0.19/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.39 => ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(powersetT_lem_type,type,
% 0.19/0.39 powersetT_lem: $o ).
% 0.19/0.39
% 0.19/0.39 thf(powersetT_lem,definition,
% 0.19/0.39 ( powersetT_lem
% 0.19/0.39 = ( ! [A: $i,X: $i] :
% 0.19/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.39 => ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(setminusT_lem_type,type,
% 0.19/0.39 setminusT_lem: $o ).
% 0.19/0.39
% 0.19/0.39 thf(setminusT_lem,definition,
% 0.19/0.39 ( setminusT_lem
% 0.19/0.39 = ( ! [A: $i,X: $i] :
% 0.19/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.39 => ! [Y: $i] :
% 0.19/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.39 => ( in @ ( setminus @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(complementT_lem_type,type,
% 0.19/0.39 complementT_lem: $o ).
% 0.19/0.39
% 0.19/0.39 thf(complementT_lem,definition,
% 0.19/0.39 ( complementT_lem
% 0.19/0.39 = ( ! [A: $i,X: $i] :
% 0.19/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.39 => ( in @ ( setminus @ A @ X ) @ ( powerset @ A ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(setextT_type,type,
% 0.19/0.39 setextT: $o ).
% 0.19/0.39
% 0.19/0.39 thf(setextT,definition,
% 0.19/0.39 ( setextT
% 0.19/0.39 = ( ! [A: $i,X: $i] :
% 0.19/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.39 => ! [Y: $i] :
% 0.19/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.39 => ( ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ( ( in @ Xx @ X )
% 0.19/0.39 => ( in @ Xx @ Y ) ) )
% 0.19/0.39 => ( ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ( ( in @ Xx @ Y )
% 0.19/0.39 => ( in @ Xx @ X ) ) )
% 0.19/0.39 => ( X = Y ) ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(subsetTI_type,type,
% 0.19/0.39 subsetTI: $o ).
% 0.19/0.39
% 0.19/0.39 thf(subsetTI,definition,
% 0.19/0.39 ( subsetTI
% 0.19/0.39 = ( ! [A: $i,X: $i] :
% 0.19/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.39 => ! [Y: $i] :
% 0.19/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.39 => ( ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ( ( in @ Xx @ X )
% 0.19/0.39 => ( in @ Xx @ Y ) ) )
% 0.19/0.39 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(powersetTI1_type,type,
% 0.19/0.39 powersetTI1: $o ).
% 0.19/0.39
% 0.19/0.39 thf(powersetTI1,definition,
% 0.19/0.39 ( powersetTI1
% 0.19/0.39 = ( ! [A: $i,X: $i] :
% 0.19/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.39 => ! [Y: $i] :
% 0.19/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.39 => ( ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ( ( in @ Xx @ X )
% 0.19/0.39 => ( in @ Xx @ Y ) ) )
% 0.19/0.39 => ( in @ X @ ( powerset @ Y ) ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(powersetTE1_type,type,
% 0.19/0.39 powersetTE1: $o ).
% 0.19/0.39
% 0.19/0.39 thf(powersetTE1,definition,
% 0.19/0.39 ( powersetTE1
% 0.19/0.39 = ( ! [A: $i,X: $i] :
% 0.19/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.39 => ! [Y: $i] :
% 0.19/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.39 => ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ( ( in @ X @ ( powerset @ Y ) )
% 0.19/0.39 => ( ( in @ Xx @ X )
% 0.19/0.39 => ( in @ Xx @ Y ) ) ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(complementTI1_type,type,
% 0.19/0.39 complementTI1: $o ).
% 0.19/0.39
% 0.19/0.39 thf(complementTI1,definition,
% 0.19/0.39 ( complementTI1
% 0.19/0.39 = ( ! [A: $i,X: $i] :
% 0.19/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.39 => ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ( ( in @ Xx @ X )
% 0.19/0.39 => ~ ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(complementTE1_type,type,
% 0.19/0.39 complementTE1: $o ).
% 0.19/0.39
% 0.19/0.39 thf(complementTE1,definition,
% 0.19/0.39 ( complementTE1
% 0.19/0.39 = ( ! [A: $i,X: $i] :
% 0.19/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.39 => ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ( ~ ( in @ Xx @ ( setminus @ A @ X ) )
% 0.19/0.39 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(binintersectTELcontra_type,type,
% 0.19/0.39 binintersectTELcontra: $o ).
% 0.19/0.39
% 0.19/0.39 thf(binintersectTELcontra,definition,
% 0.19/0.39 ( binintersectTELcontra
% 0.19/0.39 = ( ! [A: $i,X: $i] :
% 0.19/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.39 => ! [Y: $i] :
% 0.19/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.39 => ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ( ~ ( in @ Xx @ X )
% 0.19/0.39 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(binintersectTERcontra_type,type,
% 0.19/0.39 binintersectTERcontra: $o ).
% 0.19/0.39
% 0.19/0.39 thf(binintersectTERcontra,definition,
% 0.19/0.39 ( binintersectTERcontra
% 0.19/0.39 = ( ! [A: $i,X: $i] :
% 0.19/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.39 => ! [Y: $i] :
% 0.19/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.39 => ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ( ~ ( in @ Xx @ Y )
% 0.19/0.39 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(contrasubsetT_type,type,
% 0.19/0.39 contrasubsetT: $o ).
% 0.19/0.39
% 0.19/0.39 thf(contrasubsetT,definition,
% 0.19/0.39 ( contrasubsetT
% 0.19/0.39 = ( ! [A: $i,X: $i] :
% 0.19/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.39 => ! [Y: $i] :
% 0.19/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.39 => ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.19/0.39 => ( ( in @ Xx @ Y )
% 0.19/0.39 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(contrasubsetT1_type,type,
% 0.19/0.39 contrasubsetT1: $o ).
% 0.19/0.39
% 0.19/0.39 thf(contrasubsetT1,definition,
% 0.19/0.39 ( contrasubsetT1
% 0.19/0.39 = ( ! [A: $i,X: $i] :
% 0.19/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.39 => ! [Y: $i] :
% 0.19/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.39 => ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ( ( subset @ X @ Y )
% 0.19/0.39 => ( ~ ( in @ Xx @ Y )
% 0.19/0.39 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(contrasubsetT2_type,type,
% 0.19/0.39 contrasubsetT2: $o ).
% 0.19/0.39
% 0.19/0.39 thf(contrasubsetT2,definition,
% 0.19/0.39 ( contrasubsetT2
% 0.19/0.39 = ( ! [A: $i,X: $i] :
% 0.19/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.39 => ! [Y: $i] :
% 0.19/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.39 => ( ( subset @ X @ Y )
% 0.19/0.39 => ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(contrasubsetT3_type,type,
% 0.19/0.39 contrasubsetT3: $o ).
% 0.19/0.39
% 0.19/0.39 thf(contrasubsetT3,definition,
% 0.19/0.39 ( contrasubsetT3
% 0.19/0.39 = ( ! [A: $i,X: $i] :
% 0.19/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.39 => ! [Y: $i] :
% 0.19/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.39 => ( ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) )
% 0.19/0.39 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(doubleComplementI1_type,type,
% 0.19/0.39 doubleComplementI1: $o ).
% 0.19/0.39
% 0.19/0.39 thf(doubleComplementI1,definition,
% 0.19/0.39 ( doubleComplementI1
% 0.19/0.39 = ( ! [A: $i,X: $i] :
% 0.19/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.39 => ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ( ( in @ Xx @ X )
% 0.19/0.39 => ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(doubleComplementE1_type,type,
% 0.19/0.39 doubleComplementE1: $o ).
% 0.19/0.39
% 0.19/0.39 thf(doubleComplementE1,definition,
% 0.19/0.39 ( doubleComplementE1
% 0.19/0.39 = ( ! [A: $i,X: $i] :
% 0.19/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.39 => ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ( ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) )
% 0.19/0.39 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(doubleComplementSub1_type,type,
% 0.19/0.39 doubleComplementSub1: $o ).
% 0.19/0.39
% 0.19/0.39 thf(doubleComplementSub1,definition,
% 0.19/0.39 ( doubleComplementSub1
% 0.19/0.39 = ( ! [A: $i,X: $i] :
% 0.19/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.39 => ( subset @ X @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(doubleComplementSub2_type,type,
% 0.19/0.39 doubleComplementSub2: $o ).
% 0.19/0.39
% 0.19/0.39 thf(doubleComplementSub2,definition,
% 0.19/0.39 ( doubleComplementSub2
% 0.19/0.39 = ( ! [A: $i,X: $i] :
% 0.19/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.39 => ( subset @ ( setminus @ A @ ( setminus @ A @ X ) ) @ X ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(doubleComplementEq_type,type,
% 0.19/0.39 doubleComplementEq: $o ).
% 0.19/0.39
% 0.19/0.39 thf(doubleComplementEq,definition,
% 0.19/0.39 ( doubleComplementEq
% 0.19/0.39 = ( ! [A: $i,X: $i] :
% 0.19/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.39 => ( X
% 0.19/0.39 = ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(complementTnotintersectT_type,type,
% 0.19/0.39 complementTnotintersectT: $o ).
% 0.19/0.39
% 0.19/0.39 thf(complementTnotintersectT,definition,
% 0.19/0.39 ( complementTnotintersectT
% 0.19/0.39 = ( ! [A: $i,X: $i] :
% 0.19/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.39 => ! [Y: $i] :
% 0.19/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.39 => ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.19/0.39 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(complementImpComplementIntersect_type,type,
% 0.19/0.39 complementImpComplementIntersect: $o ).
% 0.19/0.39
% 0.19/0.39 thf(complementImpComplementIntersect,definition,
% 0.19/0.39 ( complementImpComplementIntersect
% 0.19/0.39 = ( ! [A: $i,X: $i] :
% 0.19/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.39 => ! [Y: $i] :
% 0.19/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.39 => ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.19/0.39 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(complementSubsetComplementIntersect_type,type,
% 0.19/0.39 complementSubsetComplementIntersect: $o ).
% 0.19/0.39
% 0.19/0.39 thf(complementSubsetComplementIntersect,definition,
% 0.19/0.39 ( complementSubsetComplementIntersect
% 0.19/0.39 = ( ! [A: $i,X: $i] :
% 0.19/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.39 => ! [Y: $i] :
% 0.19/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.39 => ( subset @ ( setminus @ A @ X ) @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(complementInPowersetComplementIntersect_type,type,
% 0.19/0.39 complementInPowersetComplementIntersect: $o ).
% 0.19/0.39
% 0.19/0.39 thf(complementInPowersetComplementIntersect,definition,
% 0.19/0.39 ( complementInPowersetComplementIntersect
% 0.19/0.39 = ( ! [A: $i,X: $i] :
% 0.19/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.39 => ! [Y: $i] :
% 0.19/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.39 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(contraSubsetComplement_type,type,
% 0.19/0.39 contraSubsetComplement: $o ).
% 0.19/0.39
% 0.19/0.39 thf(contraSubsetComplement,definition,
% 0.19/0.39 ( contraSubsetComplement
% 0.19/0.39 = ( ! [A: $i,X: $i] :
% 0.19/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.39 => ! [Y: $i] :
% 0.19/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.39 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.19/0.39 => ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ( ( in @ Xx @ Y )
% 0.19/0.39 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(complementTcontraSubset_type,type,
% 0.19/0.39 complementTcontraSubset: $o ).
% 0.19/0.39
% 0.19/0.39 thf(complementTcontraSubset,definition,
% 0.19/0.39 ( complementTcontraSubset
% 0.19/0.39 = ( ! [A: $i,X: $i] :
% 0.19/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.39 => ! [Y: $i] :
% 0.19/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.39 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.19/0.39 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(binunionTILcontra_type,type,
% 0.19/0.39 binunionTILcontra: $o ).
% 0.19/0.39
% 0.19/0.39 thf(binunionTILcontra,definition,
% 0.19/0.39 ( binunionTILcontra
% 0.19/0.39 = ( ! [A: $i,X: $i] :
% 0.19/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.39 => ! [Y: $i] :
% 0.19/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.39 => ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.19/0.39 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(binunionTIRcontra_type,type,
% 0.19/0.39 binunionTIRcontra: $o ).
% 0.19/0.39
% 0.19/0.39 thf(binunionTIRcontra,definition,
% 0.19/0.39 ( binunionTIRcontra
% 0.19/0.39 = ( ! [A: $i,X: $i] :
% 0.19/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.39 => ! [Y: $i] :
% 0.19/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.39 => ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.19/0.39 => ~ ( in @ Xx @ Y ) ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(inIntersectImpInUnion_type,type,
% 0.19/0.39 inIntersectImpInUnion: $o ).
% 0.19/0.39
% 0.19/0.39 thf(inIntersectImpInUnion,definition,
% 0.19/0.39 ( inIntersectImpInUnion
% 0.19/0.39 = ( ! [A: $i,X: $i] :
% 0.19/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.39 => ! [Y: $i] :
% 0.19/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.39 => ! [Z: $i] :
% 0.19/0.39 ( ( in @ Z @ ( powerset @ A ) )
% 0.19/0.39 => ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.19/0.39 => ( in @ Xx @ ( binunion @ X @ Z ) ) ) ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(inIntersectImpInUnion2_type,type,
% 0.19/0.39 inIntersectImpInUnion2: $o ).
% 0.19/0.39
% 0.19/0.39 thf(inIntersectImpInUnion2,definition,
% 0.19/0.39 ( inIntersectImpInUnion2
% 0.19/0.39 = ( ! [A: $i,X: $i] :
% 0.19/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.39 => ! [Y: $i] :
% 0.19/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.39 => ! [Z: $i] :
% 0.19/0.39 ( ( in @ Z @ ( powerset @ A ) )
% 0.19/0.39 => ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.19/0.39 => ( in @ Xx @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(inIntersectImpInIntersectUnions_type,type,
% 0.19/0.39 inIntersectImpInIntersectUnions: $o ).
% 0.19/0.39
% 0.19/0.39 thf(inIntersectImpInIntersectUnions,definition,
% 0.19/0.39 ( inIntersectImpInIntersectUnions
% 0.19/0.39 = ( ! [A: $i,X: $i] :
% 0.19/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.39 => ! [Y: $i] :
% 0.19/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.39 => ! [Z: $i] :
% 0.19/0.39 ( ( in @ Z @ ( powerset @ A ) )
% 0.19/0.39 => ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.19/0.39 => ( in @ Xx @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(intersectInPowersetIntersectUnions_type,type,
% 0.19/0.39 intersectInPowersetIntersectUnions: $o ).
% 0.19/0.39
% 0.19/0.39 thf(intersectInPowersetIntersectUnions,definition,
% 0.19/0.39 ( intersectInPowersetIntersectUnions
% 0.19/0.39 = ( ! [A: $i,X: $i] :
% 0.19/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.39 => ! [Y: $i] :
% 0.19/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.39 => ! [Z: $i] :
% 0.19/0.39 ( ( in @ Z @ ( powerset @ A ) )
% 0.19/0.39 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(inComplementUnionImpNotIn1_type,type,
% 0.19/0.39 inComplementUnionImpNotIn1: $o ).
% 0.19/0.39
% 0.19/0.39 thf(inComplementUnionImpNotIn1,definition,
% 0.19/0.39 ( inComplementUnionImpNotIn1
% 0.19/0.39 = ( ! [A: $i,X: $i] :
% 0.19/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.39 => ! [Y: $i] :
% 0.19/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.39 => ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.19/0.39 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(inComplementUnionImpInComplement1_type,type,
% 0.19/0.39 inComplementUnionImpInComplement1: $o ).
% 0.19/0.39
% 0.19/0.39 thf(inComplementUnionImpInComplement1,definition,
% 0.19/0.39 ( inComplementUnionImpInComplement1
% 0.19/0.39 = ( ! [A: $i,X: $i] :
% 0.19/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.39 => ! [Y: $i] :
% 0.19/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.39 => ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.19/0.39 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(binunionTE_type,type,
% 0.19/0.39 binunionTE: $o ).
% 0.19/0.39
% 0.19/0.39 thf(binunionTE,definition,
% 0.19/0.39 ( binunionTE
% 0.19/0.39 = ( ! [A: $i,X: $i] :
% 0.19/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.39 => ! [Y: $i] :
% 0.19/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.39 => ! [Xphi: $o,Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ( ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.19/0.39 => ( ( ( in @ Xx @ X )
% 0.19/0.39 => Xphi )
% 0.19/0.39 => ( ( ( in @ Xx @ Y )
% 0.19/0.39 => Xphi )
% 0.19/0.39 => Xphi ) ) ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(binunionTEcontra_type,type,
% 0.19/0.39 binunionTEcontra: $o ).
% 0.19/0.39
% 0.19/0.39 thf(binunionTEcontra,definition,
% 0.19/0.39 ( binunionTEcontra
% 0.19/0.39 = ( ! [A: $i,X: $i] :
% 0.19/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.39 => ! [Y: $i] :
% 0.19/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.39 => ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ( ~ ( in @ Xx @ X )
% 0.19/0.39 => ( ~ ( in @ Xx @ Y )
% 0.19/0.39 => ~ ( in @ Xx @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(demorgan2a1_type,type,
% 0.19/0.39 demorgan2a1: $o ).
% 0.19/0.39
% 0.19/0.39 thf(demorgan2a1,definition,
% 0.19/0.39 ( demorgan2a1
% 0.19/0.39 = ( ! [A: $i,X: $i] :
% 0.19/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.39 => ! [Y: $i] :
% 0.19/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.39 => ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.19/0.39 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(complementUnionInPowersetComplement_type,type,
% 0.19/0.39 complementUnionInPowersetComplement: $o ).
% 0.19/0.39
% 0.19/0.39 thf(complementUnionInPowersetComplement,definition,
% 0.19/0.39 ( complementUnionInPowersetComplement
% 0.19/0.39 = ( ! [A: $i,X: $i] :
% 0.19/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.39 => ! [Y: $i] :
% 0.19/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.39 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(demorgan2a2_type,type,
% 0.19/0.39 demorgan2a2: $o ).
% 0.19/0.39
% 0.19/0.39 thf(demorgan2a2,definition,
% 0.19/0.39 ( demorgan2a2
% 0.19/0.39 = ( ! [A: $i,X: $i] :
% 0.19/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.39 => ! [Y: $i] :
% 0.19/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.39 => ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.19/0.39 => ( in @ Xx @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(demorgan1a_type,type,
% 0.19/0.39 demorgan1a: $o ).
% 0.19/0.39
% 0.19/0.39 thf(demorgan1a,definition,
% 0.19/0.39 ( demorgan1a
% 0.19/0.39 = ( ! [A: $i,X: $i] :
% 0.19/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.39 => ! [Y: $i] :
% 0.19/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.39 => ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ( ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) )
% 0.19/0.39 => ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(demorgan1b_type,type,
% 0.19/0.39 demorgan1b: $o ).
% 0.19/0.39
% 0.19/0.39 thf(demorgan1b,definition,
% 0.19/0.39 ( demorgan1b
% 0.19/0.39 = ( ! [A: $i,X: $i] :
% 0.19/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.39 => ! [Y: $i] :
% 0.19/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.39 => ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ( ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 0.19/0.39 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(demorgan1_type,type,
% 0.19/0.39 demorgan1: $o ).
% 0.19/0.39
% 0.19/0.39 thf(demorgan1,definition,
% 0.19/0.39 ( demorgan1
% 0.19/0.39 = ( ! [A: $i,X: $i] :
% 0.19/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.39 => ! [Y: $i] :
% 0.19/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.39 => ( ( setminus @ A @ ( binintersect @ X @ Y ) )
% 0.19/0.39 = ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(demorgan2a_type,type,
% 0.19/0.39 demorgan2a: $o ).
% 0.19/0.39
% 0.19/0.39 thf(demorgan2a,definition,
% 0.19/0.39 ( demorgan2a
% 0.19/0.39 = ( ! [A: $i,X: $i] :
% 0.19/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.39 => ! [Y: $i] :
% 0.19/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.39 => ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.19/0.39 => ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(demorgan2b2_type,type,
% 0.19/0.39 demorgan2b2: $o ).
% 0.19/0.39
% 0.19/0.39 thf(demorgan2b2,definition,
% 0.19/0.39 ( demorgan2b2
% 0.19/0.39 = ( ! [A: $i,X: $i] :
% 0.19/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.39 => ! [Y: $i] :
% 0.19/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.39 => ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.19/0.39 => ( ( in @ Xx @ ( setminus @ A @ Y ) )
% 0.19/0.39 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(demorgan2b_type,type,
% 0.19/0.39 demorgan2b: $o ).
% 0.19/0.39
% 0.19/0.39 thf(demorgan2b,definition,
% 0.19/0.39 ( demorgan2b
% 0.19/0.39 = ( ! [A: $i,X: $i] :
% 0.19/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.39 => ! [Y: $i] :
% 0.19/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.39 => ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ( ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 0.19/0.39 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(demorgan2_type,type,
% 0.19/0.39 demorgan2: $o ).
% 0.19/0.39
% 0.19/0.39 thf(demorgan2,definition,
% 0.19/0.39 ( demorgan2
% 0.19/0.39 = ( ! [A: $i,X: $i] :
% 0.19/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.39 => ! [Y: $i] :
% 0.19/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.39 => ( ( setminus @ A @ ( binunion @ X @ Y ) )
% 0.19/0.39 = ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(woz13rule0_type,type,
% 0.19/0.39 woz13rule0: $o ).
% 0.19/0.39
% 0.19/0.39 thf(woz13rule0,definition,
% 0.19/0.39 ( woz13rule0
% 0.19/0.39 = ( ! [A: $i,X: $i] :
% 0.19/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.39 => ! [Y: $i] :
% 0.19/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.39 => ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.19/0.39 => ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(woz13rule1_type,type,
% 0.19/0.39 woz13rule1: $o ).
% 0.19/0.39
% 0.19/0.39 thf(woz13rule1,definition,
% 0.19/0.39 ( woz13rule1
% 0.19/0.39 = ( ! [A: $i,X: $i] :
% 0.19/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.39 => ! [Y: $i] :
% 0.19/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.39 => ! [Z: $i] :
% 0.19/0.39 ( ( in @ Z @ ( powerset @ A ) )
% 0.19/0.39 => ( ( subset @ X @ Z )
% 0.19/0.39 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(woz13rule2_type,type,
% 0.19/0.39 woz13rule2: $o ).
% 0.19/0.39
% 0.19/0.39 thf(woz13rule2,definition,
% 0.19/0.39 ( woz13rule2
% 0.19/0.39 = ( ! [A: $i,X: $i] :
% 0.19/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.39 => ! [Y: $i] :
% 0.19/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.39 => ! [Z: $i] :
% 0.19/0.39 ( ( in @ Z @ ( powerset @ A ) )
% 0.19/0.39 => ( ( subset @ Y @ Z )
% 0.19/0.39 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(woz13rule3_type,type,
% 0.19/0.39 woz13rule3: $o ).
% 0.19/0.39
% 0.19/0.39 thf(woz13rule3,definition,
% 0.19/0.39 ( woz13rule3
% 0.19/0.39 = ( ! [A: $i,X: $i] :
% 0.19/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.39 => ! [Y: $i] :
% 0.19/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.39 => ! [Z: $i] :
% 0.19/0.39 ( ( in @ Z @ ( powerset @ A ) )
% 0.19/0.39 => ( ( subset @ X @ Y )
% 0.19/0.39 => ( ( subset @ X @ Z )
% 0.19/0.39 => ( subset @ X @ ( binintersect @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(woz13rule4_type,type,
% 0.19/0.39 woz13rule4: $o ).
% 0.19/0.39
% 0.19/0.39 thf(woz13rule4,definition,
% 0.19/0.39 ( woz13rule4
% 0.19/0.39 = ( ! [A: $i,X: $i] :
% 0.19/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.39 => ! [Y: $i] :
% 0.19/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.39 => ! [Z: $i] :
% 0.19/0.39 ( ( in @ Z @ ( powerset @ A ) )
% 0.19/0.39 => ! [W: $i] :
% 0.19/0.39 ( ( in @ W @ ( powerset @ A ) )
% 0.19/0.39 => ( ( subset @ X @ Z )
% 0.19/0.39 => ( ( subset @ Y @ W )
% 0.19/0.39 => ( subset @ ( binintersect @ X @ Y ) @ ( binintersect @ Z @ W ) ) ) ) ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(woz1_1_type,type,
% 0.19/0.39 woz1_1: $o ).
% 0.19/0.39
% 0.19/0.39 thf(woz1_1,definition,
% 0.19/0.39 ( woz1_1
% 0.19/0.39 = ( ! [A: $i,X: $i] :
% 0.19/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.39 => ! [Y: $i] :
% 0.19/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.39 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(woz1_2_type,type,
% 0.19/0.39 woz1_2: $o ).
% 0.19/0.39
% 0.19/0.39 thf(woz1_2,definition,
% 0.19/0.39 ( woz1_2
% 0.19/0.39 = ( ! [A: $i,X: $i] :
% 0.19/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.39 => ! [Y: $i] :
% 0.19/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.39 => ! [Z: $i] :
% 0.19/0.39 ( ( in @ Z @ ( powerset @ A ) )
% 0.19/0.39 => ! [W: $i] :
% 0.19/0.39 ( ( in @ W @ ( powerset @ A ) )
% 0.19/0.39 => ( ( setminus @ A @ ( binintersect @ ( binunion @ X @ Y ) @ ( binunion @ Z @ W ) ) )
% 0.19/0.39 = ( binunion @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) @ ( binintersect @ ( setminus @ A @ Z ) @ ( setminus @ A @ W ) ) ) ) ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(woz1_3_type,type,
% 0.19/0.39 woz1_3: $o ).
% 0.19/0.39
% 0.19/0.39 thf(woz1_3,definition,
% 0.19/0.39 ( woz1_3
% 0.19/0.39 = ( ! [A: $i,X: $i] :
% 0.19/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.39 => ! [Y: $i] :
% 0.19/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.39 => ! [Z: $i] :
% 0.19/0.39 ( ( in @ Z @ ( powerset @ A ) )
% 0.19/0.39 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(woz1_4_type,type,
% 0.19/0.39 woz1_4: $o ).
% 0.19/0.39
% 0.19/0.39 thf(woz1_4,definition,
% 0.19/0.39 ( woz1_4
% 0.19/0.39 = ( ! [A: $i,X: $i] :
% 0.19/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.39 => ! [Y: $i] :
% 0.19/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.39 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.19/0.39 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(woz1_5_type,type,
% 0.19/0.39 woz1_5: $o ).
% 0.19/0.39
% 0.19/0.39 thf(woz1_5,definition,
% 0.19/0.39 ( woz1_5
% 0.19/0.39 = ( ! [A: $i,X: $i] :
% 0.19/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.39 => ! [Y: $i] :
% 0.19/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.39 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(breln1_type,type,
% 0.19/0.39 breln1: $i > $i > $o ).
% 0.19/0.39
% 0.19/0.39 thf(breln1all2_type,type,
% 0.19/0.39 breln1all2: $o ).
% 0.19/0.39
% 0.19/0.39 thf(breln1all2,definition,
% 0.19/0.39 ( breln1all2
% 0.19/0.39 = ( ! [A: $i,R: $i] :
% 0.19/0.39 ( ( breln1 @ A @ R )
% 0.19/0.39 => ! [Xphi: $i > $o] :
% 0.19/0.39 ( ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ! [Xy: $i] :
% 0.19/0.39 ( ( in @ Xy @ A )
% 0.19/0.39 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.19/0.39 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.19/0.39 => ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ R )
% 0.19/0.39 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(breln1Set_type,type,
% 0.19/0.39 breln1Set: $i > $i ).
% 0.19/0.39
% 0.19/0.39 thf(breln1SetBreln1_type,type,
% 0.19/0.39 breln1SetBreln1: $o ).
% 0.19/0.39
% 0.19/0.39 thf(breln1SetBreln1,definition,
% 0.19/0.39 ( breln1SetBreln1
% 0.19/0.39 = ( ! [A: $i,R: $i] :
% 0.19/0.39 ( ( in @ R @ ( breln1Set @ A ) )
% 0.19/0.39 => ( breln1 @ A @ R ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(transitive_type,type,
% 0.19/0.39 transitive: $i > $i > $o ).
% 0.19/0.39
% 0.19/0.39 thf(antisymmetric_type,type,
% 0.19/0.39 antisymmetric: $i > $i > $o ).
% 0.19/0.39
% 0.19/0.39 thf(reflexive_type,type,
% 0.19/0.39 reflexive: $i > $i > $o ).
% 0.19/0.39
% 0.19/0.39 thf(refltransitive_type,type,
% 0.19/0.39 refltransitive: $i > $i > $o ).
% 0.19/0.39
% 0.19/0.39 thf(refllinearorder_type,type,
% 0.19/0.39 refllinearorder: $i > $i > $o ).
% 0.19/0.39
% 0.19/0.39 thf(reflwellordering_type,type,
% 0.19/0.39 reflwellordering: $i > $i > $o ).
% 0.19/0.39
% 0.19/0.39 thf(choice2fnsingleton_type,type,
% 0.19/0.39 choice2fnsingleton: $o ).
% 0.19/0.39
% 0.19/0.39 thf(choice2fnsingleton,definition,
% 0.19/0.39 ( choice2fnsingleton
% 0.19/0.39 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.19/0.39 ( ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ? [Xy: $i] :
% 0.19/0.39 ( ( in @ Xy @ B )
% 0.19/0.39 & ( Xphi @ Xx @ Xy ) ) )
% 0.19/0.39 => ! [R: $i] :
% 0.19/0.39 ( ( in @ R @ ( breln1Set @ B ) )
% 0.19/0.39 => ( ( reflwellordering @ B @ R )
% 0.19/0.39 => ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ( singleton
% 0.19/0.39 @ ( dsetconstr @ B
% 0.19/0.39 @ ^ [Xy: $i] :
% 0.19/0.39 ( ( Xphi @ Xx @ Xy )
% 0.19/0.39 & ! [Xz: $i] :
% 0.19/0.39 ( ( in @ Xz @ B )
% 0.19/0.39 => ( ( Xphi @ Xx @ Xz )
% 0.19/0.39 => ( in @ ( kpair @ Xy @ Xz ) @ R ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(setOfPairsIsBReln1_type,type,
% 0.19/0.39 setOfPairsIsBReln1: $o ).
% 0.19/0.39
% 0.19/0.39 thf(setOfPairsIsBReln1,definition,
% 0.19/0.39 ( setOfPairsIsBReln1
% 0.19/0.39 = ( ! [A: $i,Xphi: $i > $i > $o] :
% 0.19/0.39 ( breln1 @ A
% 0.19/0.39 @ ( dpsetconstr @ A @ A
% 0.19/0.39 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(breln1all1_type,type,
% 0.19/0.39 breln1all1: $o ).
% 0.19/0.39
% 0.19/0.39 thf(breln1all1,definition,
% 0.19/0.39 ( breln1all1
% 0.19/0.39 = ( ! [A: $i,R: $i] :
% 0.19/0.39 ( ( breln1 @ A @ R )
% 0.19/0.39 => ! [Xphi: $i > $o] :
% 0.19/0.39 ( ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ! [Xy: $i] :
% 0.19/0.39 ( ( in @ Xy @ A )
% 0.19/0.39 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.19/0.39 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.19/0.39 => ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ R )
% 0.19/0.39 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(subbreln1_type,type,
% 0.19/0.39 subbreln1: $o ).
% 0.19/0.39
% 0.19/0.39 thf(subbreln1,definition,
% 0.19/0.39 ( subbreln1
% 0.19/0.39 = ( ! [A: $i,R: $i] :
% 0.19/0.39 ( ( breln1 @ A @ R )
% 0.19/0.39 => ! [S: $i] :
% 0.19/0.39 ( ( breln1 @ A @ S )
% 0.19/0.39 => ( ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ! [Xy: $i] :
% 0.19/0.39 ( ( in @ Xy @ A )
% 0.19/0.39 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.19/0.39 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.19/0.39 => ( subset @ R @ S ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(eqbreln1_type,type,
% 0.19/0.39 eqbreln1: $o ).
% 0.19/0.39
% 0.19/0.39 thf(eqbreln1,definition,
% 0.19/0.39 ( eqbreln1
% 0.19/0.39 = ( ! [A: $i,R: $i] :
% 0.19/0.39 ( ( breln1 @ A @ R )
% 0.19/0.39 => ! [S: $i] :
% 0.19/0.39 ( ( breln1 @ A @ S )
% 0.19/0.39 => ( ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ! [Xy: $i] :
% 0.19/0.39 ( ( in @ Xy @ A )
% 0.19/0.39 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.19/0.39 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.19/0.39 => ( ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ! [Xy: $i] :
% 0.19/0.39 ( ( in @ Xy @ A )
% 0.19/0.39 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.19/0.39 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 0.19/0.39 => ( R = S ) ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(breln1invset_type,type,
% 0.19/0.39 breln1invset: $i > $i > $i ).
% 0.19/0.39
% 0.19/0.39 thf(breln1invprop_type,type,
% 0.19/0.39 breln1invprop: $o ).
% 0.19/0.39
% 0.19/0.39 thf(breln1invprop,definition,
% 0.19/0.39 ( breln1invprop
% 0.19/0.39 = ( ! [A: $i,R: $i] :
% 0.19/0.39 ( ( breln1 @ A @ R )
% 0.19/0.39 => ( breln1 @ A @ ( breln1invset @ A @ R ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(breln1invI_type,type,
% 0.19/0.39 breln1invI: $o ).
% 0.19/0.39
% 0.19/0.39 thf(breln1invI,definition,
% 0.19/0.39 ( breln1invI
% 0.19/0.39 = ( ! [A: $i,R: $i] :
% 0.19/0.39 ( ( breln1 @ A @ R )
% 0.19/0.39 => ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ! [Xy: $i] :
% 0.19/0.39 ( ( in @ Xy @ A )
% 0.19/0.39 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.19/0.39 => ( in @ ( kpair @ Xy @ Xx ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(breln1invE_type,type,
% 0.19/0.39 breln1invE: $o ).
% 0.19/0.39
% 0.19/0.39 thf(breln1invE,definition,
% 0.19/0.39 ( breln1invE
% 0.19/0.39 = ( ! [A: $i,R: $i] :
% 0.19/0.39 ( ( breln1 @ A @ R )
% 0.19/0.39 => ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ! [Xy: $i] :
% 0.19/0.39 ( ( in @ Xy @ A )
% 0.19/0.39 => ( ( in @ ( kpair @ Xy @ Xx ) @ ( breln1invset @ A @ R ) )
% 0.19/0.39 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(breln1compset_type,type,
% 0.19/0.39 breln1compset: $i > $i > $i > $i ).
% 0.19/0.39
% 0.19/0.39 thf(breln1compprop_type,type,
% 0.19/0.39 breln1compprop: $o ).
% 0.19/0.39
% 0.19/0.39 thf(breln1compprop,definition,
% 0.19/0.39 ( breln1compprop
% 0.19/0.39 = ( ! [A: $i,R: $i] :
% 0.19/0.39 ( ( breln1 @ A @ R )
% 0.19/0.39 => ! [S: $i] :
% 0.19/0.39 ( ( breln1 @ A @ S )
% 0.19/0.39 => ( breln1 @ A @ ( breln1compset @ A @ R @ S ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(breln1compI_type,type,
% 0.19/0.39 breln1compI: $o ).
% 0.19/0.39
% 0.19/0.39 thf(breln1compI,definition,
% 0.19/0.39 ( breln1compI
% 0.19/0.39 = ( ! [A: $i,R: $i] :
% 0.19/0.39 ( ( breln1 @ A @ R )
% 0.19/0.39 => ! [S: $i] :
% 0.19/0.39 ( ( breln1 @ A @ S )
% 0.19/0.39 => ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ! [Xy: $i] :
% 0.19/0.39 ( ( in @ Xy @ A )
% 0.19/0.39 => ! [Xz: $i] :
% 0.19/0.39 ( ( in @ Xz @ A )
% 0.19/0.39 => ( ( in @ ( kpair @ Xx @ Xz ) @ R )
% 0.19/0.39 => ( ( in @ ( kpair @ Xz @ Xy ) @ S )
% 0.19/0.39 => ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) ) ) ) ) ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(breln1compE_type,type,
% 0.19/0.39 breln1compE: $o ).
% 0.19/0.39
% 0.19/0.39 thf(breln1compE,definition,
% 0.19/0.39 ( breln1compE
% 0.19/0.39 = ( ! [A: $i,R: $i] :
% 0.19/0.39 ( ( breln1 @ A @ R )
% 0.19/0.39 => ! [S: $i] :
% 0.19/0.39 ( ( breln1 @ A @ S )
% 0.19/0.39 => ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ! [Xy: $i] :
% 0.19/0.39 ( ( in @ Xy @ A )
% 0.19/0.39 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) )
% 0.19/0.39 => ? [Xz: $i] :
% 0.19/0.39 ( ( in @ Xz @ A )
% 0.19/0.39 & ( in @ ( kpair @ Xx @ Xz ) @ R )
% 0.19/0.39 & ( in @ ( kpair @ Xz @ Xy ) @ S ) ) ) ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(breln1compEex_type,type,
% 0.19/0.39 breln1compEex: $o ).
% 0.19/0.39
% 0.19/0.39 thf(breln1compEex,definition,
% 0.19/0.39 ( breln1compEex
% 0.19/0.39 = ( ! [A: $i,R: $i] :
% 0.19/0.39 ( ( breln1 @ A @ R )
% 0.19/0.39 => ! [S: $i] :
% 0.19/0.39 ( ( breln1 @ A @ S )
% 0.19/0.39 => ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ! [Xy: $i] :
% 0.19/0.39 ( ( in @ Xy @ A )
% 0.19/0.39 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) )
% 0.19/0.39 => ! [Xphi: $o] :
% 0.19/0.39 ( ! [Xz: $i] :
% 0.19/0.39 ( ( in @ Xz @ A )
% 0.19/0.39 => ( ( in @ ( kpair @ Xx @ Xz ) @ R )
% 0.19/0.39 => ( ( in @ ( kpair @ Xz @ Xy ) @ S )
% 0.19/0.39 => Xphi ) ) )
% 0.19/0.39 => Xphi ) ) ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(breln1unionprop_type,type,
% 0.19/0.39 breln1unionprop: $o ).
% 0.19/0.39
% 0.19/0.39 thf(breln1unionprop,definition,
% 0.19/0.39 ( breln1unionprop
% 0.19/0.39 = ( ! [A: $i,R: $i] :
% 0.19/0.39 ( ( breln1 @ A @ R )
% 0.19/0.39 => ! [S: $i] :
% 0.19/0.39 ( ( breln1 @ A @ S )
% 0.19/0.39 => ( breln1 @ A @ ( binunion @ R @ S ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(breln1unionIL_type,type,
% 0.19/0.39 breln1unionIL: $o ).
% 0.19/0.39
% 0.19/0.39 thf(breln1unionIL,definition,
% 0.19/0.39 ( breln1unionIL
% 0.19/0.39 = ( ! [A: $i,R: $i] :
% 0.19/0.39 ( ( breln1 @ A @ R )
% 0.19/0.39 => ! [S: $i] :
% 0.19/0.39 ( ( breln1 @ A @ S )
% 0.19/0.39 => ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ! [Xy: $i] :
% 0.19/0.39 ( ( in @ Xy @ A )
% 0.19/0.39 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.19/0.39 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(breln1unionIR_type,type,
% 0.19/0.39 breln1unionIR: $o ).
% 0.19/0.39
% 0.19/0.39 thf(breln1unionIR,definition,
% 0.19/0.39 ( breln1unionIR
% 0.19/0.39 = ( ! [A: $i,R: $i] :
% 0.19/0.39 ( ( breln1 @ A @ R )
% 0.19/0.39 => ! [S: $i] :
% 0.19/0.39 ( ( breln1 @ A @ S )
% 0.19/0.39 => ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ! [Xy: $i] :
% 0.19/0.39 ( ( in @ Xy @ A )
% 0.19/0.39 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.19/0.39 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(breln1unionI_type,type,
% 0.19/0.39 breln1unionI: $o ).
% 0.19/0.39
% 0.19/0.39 thf(breln1unionI,definition,
% 0.19/0.39 ( breln1unionI
% 0.19/0.39 = ( ! [A: $i,R: $i] :
% 0.19/0.39 ( ( breln1 @ A @ R )
% 0.19/0.39 => ! [S: $i] :
% 0.19/0.39 ( ( breln1 @ A @ S )
% 0.19/0.39 => ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ! [Xy: $i] :
% 0.19/0.39 ( ( in @ Xy @ A )
% 0.19/0.39 => ( ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.19/0.39 | ( in @ ( kpair @ Xx @ Xy ) @ S ) )
% 0.19/0.39 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(breln1unionE,conjecture,
% 0.19/0.39 ( setextAx
% 0.19/0.39 => ( emptysetAx
% 0.19/0.39 => ( setadjoinAx
% 0.19/0.39 => ( powersetAx
% 0.19/0.39 => ( setunionAx
% 0.19/0.39 => ( omega0Ax
% 0.19/0.39 => ( omegaSAx
% 0.19/0.39 => ( omegaIndAx
% 0.19/0.39 => ( replAx
% 0.19/0.39 => ( foundationAx
% 0.19/0.39 => ( wellorderingAx
% 0.19/0.39 => ( descrp
% 0.19/0.39 => ( dsetconstrI
% 0.19/0.39 => ( dsetconstrEL
% 0.19/0.39 => ( dsetconstrER
% 0.19/0.39 => ( exuE1
% 0.19/0.39 => ( prop2setE
% 0.19/0.39 => ( emptysetE
% 0.19/0.39 => ( emptysetimpfalse
% 0.19/0.39 => ( notinemptyset
% 0.19/0.39 => ( exuE3e
% 0.19/0.39 => ( setext
% 0.19/0.39 => ( emptyI
% 0.19/0.39 => ( noeltsimpempty
% 0.19/0.39 => ( setbeta
% 0.19/0.39 => ( nonemptyE1
% 0.19/0.39 => ( nonemptyI
% 0.19/0.39 => ( nonemptyI1
% 0.19/0.39 => ( setadjoinIL
% 0.19/0.39 => ( emptyinunitempty
% 0.19/0.39 => ( setadjoinIR
% 0.19/0.39 => ( setadjoinE
% 0.19/0.39 => ( setadjoinOr
% 0.19/0.39 => ( setoftrueEq
% 0.19/0.39 => ( powersetI
% 0.19/0.39 => ( emptyinPowerset
% 0.19/0.39 => ( emptyInPowerset
% 0.19/0.39 => ( powersetE
% 0.19/0.39 => ( setunionI
% 0.19/0.39 => ( setunionE
% 0.19/0.39 => ( subPowSU
% 0.19/0.39 => ( exuE2
% 0.19/0.39 => ( nonemptyImpWitness
% 0.19/0.39 => ( uniqinunit
% 0.19/0.39 => ( notinsingleton
% 0.19/0.39 => ( eqinunit
% 0.19/0.39 => ( singletonsswitch
% 0.19/0.39 => ( upairsetE
% 0.19/0.39 => ( upairsetIL
% 0.19/0.39 => ( upairsetIR
% 0.19/0.39 => ( emptyE1
% 0.19/0.39 => ( vacuousDall
% 0.19/0.39 => ( quantDeMorgan1
% 0.19/0.39 => ( quantDeMorgan2
% 0.19/0.39 => ( quantDeMorgan3
% 0.19/0.39 => ( quantDeMorgan4
% 0.19/0.39 => ( prop2setI
% 0.19/0.39 => ( prop2set2propI
% 0.19/0.39 => ( notdexE
% 0.19/0.39 => ( notdallE
% 0.19/0.39 => ( exuI1
% 0.19/0.39 => ( exuI3
% 0.19/0.39 => ( exuI2
% 0.19/0.39 => ( inCongP
% 0.19/0.39 => ( in__Cong
% 0.19/0.39 => ( exuE3u
% 0.19/0.39 => ( exu__Cong
% 0.19/0.39 => ( emptyset__Cong
% 0.19/0.39 => ( setadjoin__Cong
% 0.19/0.39 => ( powerset__Cong
% 0.19/0.39 => ( setunion__Cong
% 0.19/0.39 => ( omega__Cong
% 0.19/0.39 => ( exuEu
% 0.19/0.39 => ( descr__Cong
% 0.19/0.39 => ( dsetconstr__Cong
% 0.19/0.39 => ( subsetI1
% 0.19/0.39 => ( eqimpsubset2
% 0.19/0.39 => ( eqimpsubset1
% 0.19/0.39 => ( subsetI2
% 0.19/0.39 => ( emptysetsubset
% 0.19/0.39 => ( subsetE
% 0.19/0.39 => ( subsetE2
% 0.19/0.39 => ( notsubsetI
% 0.19/0.39 => ( notequalI1
% 0.19/0.39 => ( notequalI2
% 0.19/0.39 => ( subsetRefl
% 0.19/0.39 => ( subsetTrans
% 0.19/0.39 => ( setadjoinSub
% 0.19/0.39 => ( setadjoinSub2
% 0.19/0.39 => ( subset2powerset
% 0.19/0.39 => ( setextsub
% 0.19/0.39 => ( subsetemptysetimpeq
% 0.19/0.39 => ( powersetI1
% 0.19/0.39 => ( powersetE1
% 0.19/0.39 => ( inPowerset
% 0.19/0.39 => ( powersetsubset
% 0.19/0.39 => ( sepInPowerset
% 0.19/0.39 => ( sepSubset
% 0.19/0.39 => ( binunionIL
% 0.19/0.39 => ( upairset2IR
% 0.19/0.39 => ( binunionIR
% 0.19/0.39 => ( binunionEcases
% 0.19/0.39 => ( binunionE
% 0.19/0.39 => ( binunionLsub
% 0.19/0.39 => ( binunionRsub
% 0.19/0.39 => ( binintersectI
% 0.19/0.39 => ( binintersectSubset5
% 0.19/0.39 => ( binintersectEL
% 0.19/0.39 => ( binintersectLsub
% 0.19/0.39 => ( binintersectSubset2
% 0.19/0.39 => ( binintersectSubset3
% 0.19/0.39 => ( binintersectER
% 0.19/0.39 => ( disjointsetsI1
% 0.19/0.39 => ( binintersectRsub
% 0.19/0.39 => ( binintersectSubset4
% 0.19/0.39 => ( binintersectSubset1
% 0.19/0.39 => ( bs114d
% 0.19/0.39 => ( setminusI
% 0.19/0.39 => ( setminusEL
% 0.19/0.39 => ( setminusER
% 0.19/0.39 => ( setminusSubset2
% 0.19/0.39 => ( setminusERneg
% 0.19/0.39 => ( setminusELneg
% 0.19/0.39 => ( setminusILneg
% 0.19/0.39 => ( setminusIRneg
% 0.19/0.39 => ( setminusLsub
% 0.19/0.39 => ( setminusSubset1
% 0.19/0.39 => ( symdiffE
% 0.19/0.39 => ( symdiffI1
% 0.19/0.39 => ( symdiffI2
% 0.19/0.39 => ( symdiffIneg1
% 0.19/0.39 => ( symdiffIneg2
% 0.19/0.39 => ( secondinupair
% 0.19/0.39 => ( setukpairIL
% 0.19/0.39 => ( setukpairIR
% 0.19/0.39 => ( kpairiskpair
% 0.19/0.39 => ( kpairp
% 0.19/0.39 => ( singletonsubset
% 0.19/0.39 => ( singletoninpowerset
% 0.19/0.39 => ( singletoninpowunion
% 0.19/0.39 => ( upairset2E
% 0.19/0.39 => ( upairsubunion
% 0.19/0.39 => ( upairinpowunion
% 0.19/0.39 => ( ubforcartprodlem1
% 0.19/0.39 => ( ubforcartprodlem2
% 0.19/0.39 => ( ubforcartprodlem3
% 0.19/0.39 => ( cartprodpairin
% 0.19/0.39 => ( cartprodmempair1
% 0.19/0.39 => ( cartprodmempair
% 0.19/0.39 => ( setunionE2
% 0.19/0.39 => ( setunionsingleton1
% 0.19/0.39 => ( setunionsingleton2
% 0.19/0.39 => ( setunionsingleton
% 0.19/0.39 => ( singletonprop
% 0.19/0.39 => ( ex1E1
% 0.19/0.39 => ( ex1I
% 0.19/0.39 => ( ex1I2
% 0.19/0.39 => ( singletonsuniq
% 0.19/0.39 => ( setukpairinjL1
% 0.19/0.39 => ( kfstsingleton
% 0.19/0.39 => ( theprop
% 0.19/0.39 => ( kfstpairEq
% 0.19/0.39 => ( cartprodfstin
% 0.19/0.39 => ( setukpairinjL2
% 0.19/0.39 => ( setukpairinjL
% 0.19/0.39 => ( setukpairinjR11
% 0.19/0.39 => ( setukpairinjR12
% 0.19/0.39 => ( setukpairinjR1
% 0.19/0.39 => ( upairequniteq
% 0.19/0.39 => ( setukpairinjR2
% 0.19/0.39 => ( setukpairinjR
% 0.19/0.39 => ( ksndsingleton
% 0.19/0.39 => ( ksndpairEq
% 0.19/0.39 => ( kpairsurjEq
% 0.19/0.39 => ( cartprodsndin
% 0.19/0.39 => ( cartprodpairmemEL
% 0.19/0.39 => ( cartprodpairmemER
% 0.19/0.39 => ( cartprodmempaircEq
% 0.19/0.39 => ( cartprodfstpairEq
% 0.19/0.39 => ( cartprodsndpairEq
% 0.19/0.39 => ( cartprodpairsurjEq
% 0.19/0.39 => ( dpsetconstrI
% 0.19/0.39 => ( dpsetconstrSub
% 0.19/0.39 => ( setOfPairsIsBReln
% 0.19/0.39 => ( dpsetconstrERa
% 0.19/0.39 => ( dpsetconstrEL1
% 0.19/0.39 => ( dpsetconstrEL2
% 0.19/0.39 => ( dpsetconstrER
% 0.19/0.39 => ( funcImageSingleton
% 0.19/0.39 => ( apProp
% 0.19/0.39 => ( app
% 0.19/0.39 => ( infuncsetfunc
% 0.19/0.39 => ( ap2p
% 0.19/0.39 => ( funcinfuncset
% 0.19/0.39 => ( lamProp
% 0.19/0.39 => ( lamp
% 0.19/0.39 => ( lam2p
% 0.19/0.39 => ( brelnall1
% 0.19/0.39 => ( brelnall2
% 0.19/0.39 => ( ex1E2
% 0.19/0.39 => ( funcGraphProp1
% 0.19/0.39 => ( funcGraphProp3
% 0.19/0.39 => ( funcGraphProp2
% 0.19/0.39 => ( funcextLem
% 0.19/0.39 => ( funcGraphProp4
% 0.19/0.39 => ( subbreln
% 0.19/0.39 => ( eqbreln
% 0.19/0.39 => ( funcext
% 0.19/0.39 => ( funcext2
% 0.19/0.39 => ( ap2apEq1
% 0.19/0.39 => ( ap2apEq2
% 0.19/0.39 => ( beta1
% 0.19/0.39 => ( eta1
% 0.19/0.39 => ( lam2lamEq
% 0.19/0.39 => ( beta2
% 0.19/0.39 => ( eta2
% 0.19/0.39 => ( iffalseProp1
% 0.19/0.39 => ( iffalseProp2
% 0.19/0.39 => ( iftrueProp1
% 0.19/0.39 => ( iftrueProp2
% 0.19/0.39 => ( ifSingleton
% 0.19/0.39 => ( ifp
% 0.19/0.39 => ( theeq
% 0.19/0.39 => ( iftrue
% 0.19/0.39 => ( iffalse
% 0.19/0.39 => ( iftrueorfalse
% 0.19/0.39 => ( binintersectT_lem
% 0.19/0.39 => ( binunionT_lem
% 0.19/0.39 => ( powersetT_lem
% 0.19/0.39 => ( setminusT_lem
% 0.19/0.39 => ( complementT_lem
% 0.19/0.39 => ( setextT
% 0.19/0.39 => ( subsetTI
% 0.19/0.39 => ( powersetTI1
% 0.19/0.39 => ( powersetTE1
% 0.19/0.39 => ( complementTI1
% 0.19/0.39 => ( complementTE1
% 0.19/0.39 => ( binintersectTELcontra
% 0.19/0.39 => ( binintersectTERcontra
% 0.19/0.39 => ( contrasubsetT
% 0.19/0.39 => ( contrasubsetT1
% 0.19/0.39 => ( contrasubsetT2
% 0.19/0.39 => ( contrasubsetT3
% 0.19/0.39 => ( doubleComplementI1
% 0.19/0.39 => ( doubleComplementE1
% 0.19/0.39 => ( doubleComplementSub1
% 0.19/0.39 => ( doubleComplementSub2
% 0.19/0.39 => ( doubleComplementEq
% 0.19/0.39 => ( complementTnotintersectT
% 0.19/0.39 => ( complementImpComplementIntersect
% 0.19/0.39 => ( complementSubsetComplementIntersect
% 0.19/0.39 => ( complementInPowersetComplementIntersect
% 0.19/0.39 => ( contraSubsetComplement
% 0.19/0.39 => ( complementTcontraSubset
% 0.19/0.39 => ( binunionTILcontra
% 0.19/0.39 => ( binunionTIRcontra
% 0.19/0.39 => ( inIntersectImpInUnion
% 0.19/0.39 => ( inIntersectImpInUnion2
% 0.19/0.39 => ( inIntersectImpInIntersectUnions
% 0.19/0.39 => ( intersectInPowersetIntersectUnions
% 0.19/0.39 => ( inComplementUnionImpNotIn1
% 0.19/0.39 => ( inComplementUnionImpInComplement1
% 0.19/0.39 => ( binunionTE
% 0.19/0.39 => ( binunionTEcontra
% 0.19/0.39 => ( demorgan2a1
% 0.19/0.39 => ( complementUnionInPowersetComplement
% 0.19/0.39 => ( demorgan2a2
% 0.19/0.39 => ( demorgan1a
% 0.19/0.39 => ( demorgan1b
% 0.19/0.39 => ( demorgan1
% 0.19/0.39 => ( demorgan2a
% 0.19/0.39 => ( demorgan2b2
% 0.19/0.39 => ( demorgan2b
% 0.19/0.39 => ( demorgan2
% 0.19/0.39 => ( woz13rule0
% 0.19/0.39 => ( woz13rule1
% 0.19/0.39 => ( woz13rule2
% 0.19/0.39 => ( woz13rule3
% 0.19/0.39 => ( woz13rule4
% 0.19/0.39 => ( woz1_1
% 0.19/0.39 => ( woz1_2
% 0.19/0.39 => ( woz1_3
% 0.19/0.39 => ( woz1_4
% 0.19/0.39 => ( woz1_5
% 0.19/0.39 => ( breln1all2
% 0.19/0.39 => ( breln1SetBreln1
% 0.19/0.39 => ( choice2fnsingleton
% 0.19/0.39 => ( setOfPairsIsBReln1
% 0.19/0.39 => ( breln1all1
% 0.19/0.39 => ( subbreln1
% 0.19/0.39 => ( eqbreln1
% 0.19/0.39 => ( breln1invprop
% 0.19/0.39 => ( breln1invI
% 0.19/0.39 => ( breln1invE
% 0.19/0.39 => ( breln1compprop
% 0.19/0.39 => ( breln1compI
% 0.19/0.39 => ( breln1compE
% 0.19/0.39 => ( breln1compEex
% 0.19/0.39 => ( breln1unionprop
% 0.19/0.39 => ( breln1unionIL
% 0.19/0.39 => ( breln1unionIR
% 0.19/0.39 => ( breln1unionI
% 0.19/0.39 => ! [A: $i,R: $i] :
% 0.19/0.39 ( ( breln1 @ A @ R )
% 0.19/0.39 => ! [S: $i] :
% 0.19/0.39 ( ( breln1 @ A @ S )
% 0.19/0.39 => ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ! [Xy: $i] :
% 0.19/0.39 ( ( in @ Xy @ A )
% 0.19/0.39 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) )
% 0.19/0.39 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.19/0.39 | ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 %------------------------------------------------------------------------------
% 0.19/0.39 ------------------------
% 3.17/1.65 ---- Embedded in THF ---
% 3.17/1.65 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 3.17/1.65 %%% Generated on Tue Feb 24 12:34:11 EST 2026
% 3.17/1.65 %%% using '$$dhol' embedding, version 1.3.0.
% 3.17/1.65 %%% Logic specification used:
% 3.17/1.65 %%% thf(spec, logic, $$dhol).
% 3.17/1.65
% 3.17/1.65 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.onid0QyU2k/DTF2DTF25019.p
% See solution above
% 3.17/1.65 ------------------------
% 3.32/1.72 ---- Cleaned THF ---
% 3.32/1.73 thf(in_type,type,
% 3.32/1.73 in: $i > $i > $o ).
% 3.32/1.73
% 3.32/1.73 thf(exu_type,type,
% 3.32/1.73 exu: ( $i > $o ) > $o ).
% 3.32/1.73
% 3.32/1.73 thf(exu,definition,
% 3.32/1.73 ( exu
% 3.32/1.73 = ( ^ [Xphi: $i > $o] :
% 3.32/1.73 ? [Xx: $i] :
% 3.32/1.73 ( ( Xphi @ Xx )
% 3.32/1.73 & ! [Xy: $i] :
% 3.32/1.73 ( ( Xphi @ Xy )
% 3.32/1.73 => ( Xx = Xy ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(setextAx_type,type,
% 3.32/1.73 setextAx: $o ).
% 3.32/1.73
% 3.32/1.73 thf(setextAx,definition,
% 3.32/1.73 ( setextAx
% 3.32/1.73 = ( ! [A: $i,B: $i] :
% 3.32/1.73 ( ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 <=> ( in @ Xx @ B ) )
% 3.32/1.73 => ( A = B ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(emptyset_type,type,
% 3.32/1.73 emptyset: $i ).
% 3.32/1.73
% 3.32/1.73 thf(emptysetAx_type,type,
% 3.32/1.73 emptysetAx: $o ).
% 3.32/1.73
% 3.32/1.73 thf(emptysetAx,definition,
% 3.32/1.73 ( emptysetAx
% 3.32/1.73 = ( ! [Xx: $i] :
% 3.32/1.73 ~ ( in @ Xx @ emptyset ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(setadjoin_type,type,
% 3.32/1.73 setadjoin: $i > $i > $i ).
% 3.32/1.73
% 3.32/1.73 thf(setadjoinAx_type,type,
% 3.32/1.73 setadjoinAx: $o ).
% 3.32/1.73
% 3.32/1.73 thf(setadjoinAx,definition,
% 3.32/1.73 ( setadjoinAx
% 3.32/1.73 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 3.32/1.73 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 3.32/1.73 <=> ( ( Xy = Xx )
% 3.32/1.73 | ( in @ Xy @ A ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(powerset_type,type,
% 3.32/1.73 powerset: $i > $i ).
% 3.32/1.73
% 3.32/1.73 thf(powersetAx_type,type,
% 3.32/1.73 powersetAx: $o ).
% 3.32/1.73
% 3.32/1.73 thf(powersetAx,definition,
% 3.32/1.73 ( powersetAx
% 3.32/1.73 = ( ! [A: $i,B: $i] :
% 3.32/1.73 ( ( in @ B @ ( powerset @ A ) )
% 3.32/1.73 <=> ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ B )
% 3.32/1.73 => ( in @ Xx @ A ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(setunion_type,type,
% 3.32/1.73 setunion: $i > $i ).
% 3.32/1.73
% 3.32/1.73 thf(setunionAx_type,type,
% 3.32/1.73 setunionAx: $o ).
% 3.32/1.73
% 3.32/1.73 thf(setunionAx,definition,
% 3.32/1.73 ( setunionAx
% 3.32/1.73 = ( ! [A: $i,Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ ( setunion @ A ) )
% 3.32/1.73 <=> ? [B: $i] :
% 3.32/1.73 ( ( in @ Xx @ B )
% 3.32/1.73 & ( in @ B @ A ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(omega_type,type,
% 3.32/1.73 omega: $i ).
% 3.32/1.73
% 3.32/1.73 thf(omega0Ax_type,type,
% 3.32/1.73 omega0Ax: $o ).
% 3.32/1.73
% 3.32/1.73 thf(omega0Ax,definition,
% 3.32/1.73 ( omega0Ax
% 3.32/1.73 = ( in @ emptyset @ omega ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(omegaSAx_type,type,
% 3.32/1.73 omegaSAx: $o ).
% 3.32/1.73
% 3.32/1.73 thf(omegaSAx,definition,
% 3.32/1.73 ( omegaSAx
% 3.32/1.73 = ( ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ omega )
% 3.32/1.73 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(omegaIndAx_type,type,
% 3.32/1.73 omegaIndAx: $o ).
% 3.32/1.73
% 3.32/1.73 thf(omegaIndAx,definition,
% 3.32/1.73 ( omegaIndAx
% 3.32/1.73 = ( ! [A: $i] :
% 3.32/1.73 ( ( ( in @ emptyset @ A )
% 3.32/1.73 & ! [Xx: $i] :
% 3.32/1.73 ( ( ( in @ Xx @ omega )
% 3.32/1.73 & ( in @ Xx @ A ) )
% 3.32/1.73 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 3.32/1.73 => ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ omega )
% 3.32/1.73 => ( in @ Xx @ A ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(replAx_type,type,
% 3.32/1.73 replAx: $o ).
% 3.32/1.73
% 3.32/1.73 thf(replAx,definition,
% 3.32/1.73 ( replAx
% 3.32/1.73 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 3.32/1.73 ( ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ( exu
% 3.32/1.73 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 3.32/1.73 => ? [B: $i] :
% 3.32/1.73 ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ B )
% 3.32/1.73 <=> ? [Xy: $i] :
% 3.32/1.73 ( ( in @ Xy @ A )
% 3.32/1.73 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(foundationAx_type,type,
% 3.32/1.73 foundationAx: $o ).
% 3.32/1.73
% 3.32/1.73 thf(foundationAx,definition,
% 3.32/1.73 ( foundationAx
% 3.32/1.73 = ( ! [A: $i] :
% 3.32/1.73 ( ? [Xx: $i] : ( in @ Xx @ A )
% 3.32/1.73 => ? [B: $i] :
% 3.32/1.73 ( ( in @ B @ A )
% 3.32/1.73 & ~ ? [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ B )
% 3.32/1.73 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(wellorderingAx_type,type,
% 3.32/1.73 wellorderingAx: $o ).
% 3.32/1.73
% 3.32/1.73 thf(wellorderingAx,definition,
% 3.32/1.73 ( wellorderingAx
% 3.32/1.73 = ( ! [A: $i] :
% 3.32/1.73 ? [B: $i] :
% 3.32/1.73 ( ! [C: $i] :
% 3.32/1.73 ( ( in @ C @ B )
% 3.32/1.73 => ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ C )
% 3.32/1.73 => ( in @ Xx @ A ) ) )
% 3.32/1.73 & ! [Xx: $i,Xy: $i] :
% 3.32/1.73 ( ( ( in @ Xx @ A )
% 3.32/1.73 & ( in @ Xy @ A ) )
% 3.32/1.73 => ( ! [C: $i] :
% 3.32/1.73 ( ( in @ C @ B )
% 3.32/1.73 => ( ( in @ Xx @ C )
% 3.32/1.73 <=> ( in @ Xy @ C ) ) )
% 3.32/1.73 => ( Xx = Xy ) ) )
% 3.32/1.73 & ! [C: $i,D: $i] :
% 3.32/1.73 ( ( ( in @ C @ B )
% 3.32/1.73 & ( in @ D @ B ) )
% 3.32/1.73 => ( ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ C )
% 3.32/1.73 => ( in @ Xx @ D ) )
% 3.32/1.73 | ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ D )
% 3.32/1.73 => ( in @ Xx @ C ) ) ) )
% 3.32/1.73 & ! [C: $i] :
% 3.32/1.73 ( ( ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ C )
% 3.32/1.73 => ( in @ Xx @ A ) )
% 3.32/1.73 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 3.32/1.73 => ? [D: $i,Xx: $i] :
% 3.32/1.73 ( ( in @ D @ B )
% 3.32/1.73 & ( in @ Xx @ C )
% 3.32/1.73 & ~ ? [Xy: $i] :
% 3.32/1.73 ( ( in @ Xy @ D )
% 3.32/1.73 & ( in @ Xy @ C ) )
% 3.32/1.73 & ! [E: $i] :
% 3.32/1.73 ( ( in @ E @ B )
% 3.32/1.73 => ( ! [Xy: $i] :
% 3.32/1.73 ( ( in @ Xy @ E )
% 3.32/1.73 => ( in @ Xy @ D ) )
% 3.32/1.73 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(descr_type,type,
% 3.32/1.73 descr: ( $i > $o ) > $i ).
% 3.32/1.73
% 3.32/1.73 thf(descrp_type,type,
% 3.32/1.73 descrp: $o ).
% 3.32/1.73
% 3.32/1.73 thf(descrp,definition,
% 3.32/1.73 ( descrp
% 3.32/1.73 = ( ! [Xphi: $i > $o] :
% 3.32/1.73 ( ( exu
% 3.32/1.73 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.32/1.73 => ( Xphi
% 3.32/1.73 @ ( descr
% 3.32/1.73 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(dsetconstr_type,type,
% 3.32/1.73 dsetconstr: $i > ( $i > $o ) > $i ).
% 3.32/1.73
% 3.32/1.73 thf(dsetconstrI_type,type,
% 3.32/1.73 dsetconstrI: $o ).
% 3.32/1.73
% 3.32/1.73 thf(dsetconstrI,definition,
% 3.32/1.73 ( dsetconstrI
% 3.32/1.73 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ( ( Xphi @ Xx )
% 3.32/1.73 => ( in @ Xx
% 3.32/1.73 @ ( dsetconstr @ A
% 3.32/1.73 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(dsetconstrEL_type,type,
% 3.32/1.73 dsetconstrEL: $o ).
% 3.32/1.73
% 3.32/1.73 thf(dsetconstrEL,definition,
% 3.32/1.73 ( dsetconstrEL
% 3.32/1.73 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.32/1.73 ( ( in @ Xx
% 3.32/1.73 @ ( dsetconstr @ A
% 3.32/1.73 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 3.32/1.73 => ( in @ Xx @ A ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(dsetconstrER_type,type,
% 3.32/1.73 dsetconstrER: $o ).
% 3.32/1.73
% 3.32/1.73 thf(dsetconstrER,definition,
% 3.32/1.73 ( dsetconstrER
% 3.32/1.73 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.32/1.73 ( ( in @ Xx
% 3.32/1.73 @ ( dsetconstr @ A
% 3.32/1.73 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 3.32/1.73 => ( Xphi @ Xx ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(exuE1_type,type,
% 3.32/1.73 exuE1: $o ).
% 3.32/1.73
% 3.32/1.73 thf(exuE1,definition,
% 3.32/1.73 ( exuE1
% 3.32/1.73 = ( ! [Xphi: $i > $o] :
% 3.32/1.73 ( ( exu
% 3.32/1.73 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.32/1.73 => ? [Xx: $i] :
% 3.32/1.73 ( ( Xphi @ Xx )
% 3.32/1.73 & ! [Xy: $i] :
% 3.32/1.73 ( ( Xphi @ Xy )
% 3.32/1.73 => ( Xx = Xy ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(prop2set_type,type,
% 3.32/1.73 prop2set: $o > $i ).
% 3.32/1.73
% 3.32/1.73 thf(prop2setE_type,type,
% 3.32/1.73 prop2setE: $o ).
% 3.32/1.73
% 3.32/1.73 thf(prop2setE,definition,
% 3.32/1.73 ( prop2setE
% 3.32/1.73 = ( ! [Xphi: $o,Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 3.32/1.73 => Xphi ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(emptysetE_type,type,
% 3.32/1.73 emptysetE: $o ).
% 3.32/1.73
% 3.32/1.73 thf(emptysetE,definition,
% 3.32/1.73 ( emptysetE
% 3.32/1.73 = ( ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ emptyset )
% 3.32/1.73 => ! [Xphi: $o] : Xphi ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(emptysetimpfalse_type,type,
% 3.32/1.73 emptysetimpfalse: $o ).
% 3.32/1.73
% 3.32/1.73 thf(emptysetimpfalse,definition,
% 3.32/1.73 ( emptysetimpfalse
% 3.32/1.73 = ( ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ emptyset )
% 3.32/1.73 => $false ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(notinemptyset_type,type,
% 3.32/1.73 notinemptyset: $o ).
% 3.32/1.73
% 3.32/1.73 thf(notinemptyset,definition,
% 3.32/1.73 ( notinemptyset
% 3.32/1.73 = ( ! [Xx: $i] :
% 3.32/1.73 ~ ( in @ Xx @ emptyset ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(exuE3e_type,type,
% 3.32/1.73 exuE3e: $o ).
% 3.32/1.73
% 3.32/1.73 thf(exuE3e,definition,
% 3.32/1.73 ( exuE3e
% 3.32/1.73 = ( ! [Xphi: $i > $o] :
% 3.32/1.73 ( ( exu
% 3.32/1.73 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.32/1.73 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(setext_type,type,
% 3.32/1.73 setext: $o ).
% 3.32/1.73
% 3.32/1.73 thf(setext,definition,
% 3.32/1.73 ( setext
% 3.32/1.73 = ( ! [A: $i,B: $i] :
% 3.32/1.73 ( ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ( in @ Xx @ B ) )
% 3.32/1.73 => ( ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ B )
% 3.32/1.73 => ( in @ Xx @ A ) )
% 3.32/1.73 => ( A = B ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(emptyI_type,type,
% 3.32/1.73 emptyI: $o ).
% 3.32/1.73
% 3.32/1.73 thf(emptyI,definition,
% 3.32/1.73 ( emptyI
% 3.32/1.73 = ( ! [A: $i] :
% 3.32/1.73 ( ! [Xx: $i] :
% 3.32/1.73 ~ ( in @ Xx @ A )
% 3.32/1.73 => ( A = emptyset ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(noeltsimpempty_type,type,
% 3.32/1.73 noeltsimpempty: $o ).
% 3.32/1.73
% 3.32/1.73 thf(noeltsimpempty,definition,
% 3.32/1.73 ( noeltsimpempty
% 3.32/1.73 = ( ! [A: $i] :
% 3.32/1.73 ( ! [Xx: $i] :
% 3.32/1.73 ~ ( in @ Xx @ A )
% 3.32/1.73 => ( A = emptyset ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(setbeta_type,type,
% 3.32/1.73 setbeta: $o ).
% 3.32/1.73
% 3.32/1.73 thf(setbeta,definition,
% 3.32/1.73 ( setbeta
% 3.32/1.73 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ( ( in @ Xx
% 3.32/1.73 @ ( dsetconstr @ A
% 3.32/1.73 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 3.32/1.73 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(nonempty_type,type,
% 3.32/1.73 nonempty: $i > $o ).
% 3.32/1.73
% 3.32/1.73 thf(nonempty,definition,
% 3.32/1.73 ( nonempty
% 3.32/1.73 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(nonemptyE1_type,type,
% 3.32/1.73 nonemptyE1: $o ).
% 3.32/1.73
% 3.32/1.73 thf(nonemptyE1,definition,
% 3.32/1.73 ( nonemptyE1
% 3.32/1.73 = ( ! [A: $i] :
% 3.32/1.73 ( ( nonempty @ A )
% 3.32/1.73 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(nonemptyI_type,type,
% 3.32/1.73 nonemptyI: $o ).
% 3.32/1.73
% 3.32/1.73 thf(nonemptyI,definition,
% 3.32/1.73 ( nonemptyI
% 3.32/1.73 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ( ( Xphi @ Xx )
% 3.32/1.73 => ( nonempty
% 3.32/1.73 @ ( dsetconstr @ A
% 3.32/1.73 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(nonemptyI1_type,type,
% 3.32/1.73 nonemptyI1: $o ).
% 3.32/1.73
% 3.32/1.73 thf(nonemptyI1,definition,
% 3.32/1.73 ( nonemptyI1
% 3.32/1.73 = ( ! [A: $i] :
% 3.32/1.73 ( ? [Xx: $i] : ( in @ Xx @ A )
% 3.32/1.73 => ( nonempty @ A ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(setadjoinIL_type,type,
% 3.32/1.73 setadjoinIL: $o ).
% 3.32/1.73
% 3.32/1.73 thf(setadjoinIL,definition,
% 3.32/1.73 ( setadjoinIL
% 3.32/1.73 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(emptyinunitempty_type,type,
% 3.32/1.73 emptyinunitempty: $o ).
% 3.32/1.73
% 3.32/1.73 thf(emptyinunitempty,definition,
% 3.32/1.73 ( emptyinunitempty
% 3.32/1.73 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(setadjoinIR_type,type,
% 3.32/1.73 setadjoinIR: $o ).
% 3.32/1.73
% 3.32/1.73 thf(setadjoinIR,definition,
% 3.32/1.73 ( setadjoinIR
% 3.32/1.73 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 3.32/1.73 ( ( in @ Xy @ A )
% 3.32/1.73 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(setadjoinE_type,type,
% 3.32/1.73 setadjoinE: $o ).
% 3.32/1.73
% 3.32/1.73 thf(setadjoinE,definition,
% 3.32/1.73 ( setadjoinE
% 3.32/1.73 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 3.32/1.73 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 3.32/1.73 => ! [Xphi: $o] :
% 3.32/1.73 ( ( ( Xy = Xx )
% 3.32/1.73 => Xphi )
% 3.32/1.73 => ( ( ( in @ Xy @ A )
% 3.32/1.73 => Xphi )
% 3.32/1.73 => Xphi ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(setadjoinOr_type,type,
% 3.32/1.73 setadjoinOr: $o ).
% 3.32/1.73
% 3.32/1.73 thf(setadjoinOr,definition,
% 3.32/1.73 ( setadjoinOr
% 3.32/1.73 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 3.32/1.73 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 3.32/1.73 => ( ( Xy = Xx )
% 3.32/1.73 | ( in @ Xy @ A ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(setoftrueEq_type,type,
% 3.32/1.73 setoftrueEq: $o ).
% 3.32/1.73
% 3.32/1.73 thf(setoftrueEq,definition,
% 3.32/1.73 ( setoftrueEq
% 3.32/1.73 = ( ! [A: $i] :
% 3.32/1.73 ( ( dsetconstr @ A
% 3.32/1.73 @ ^ [Xx: $i] : $true )
% 3.32/1.73 = A ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(powersetI_type,type,
% 3.32/1.73 powersetI: $o ).
% 3.32/1.73
% 3.32/1.73 thf(powersetI,definition,
% 3.32/1.73 ( powersetI
% 3.32/1.73 = ( ! [A: $i,B: $i] :
% 3.32/1.73 ( ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ B )
% 3.32/1.73 => ( in @ Xx @ A ) )
% 3.32/1.73 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(emptyinPowerset_type,type,
% 3.32/1.73 emptyinPowerset: $o ).
% 3.32/1.73
% 3.32/1.73 thf(emptyinPowerset,definition,
% 3.32/1.73 ( emptyinPowerset
% 3.32/1.73 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(emptyInPowerset_type,type,
% 3.32/1.73 emptyInPowerset: $o ).
% 3.32/1.73
% 3.32/1.73 thf(emptyInPowerset,definition,
% 3.32/1.73 ( emptyInPowerset
% 3.32/1.73 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(powersetE_type,type,
% 3.32/1.73 powersetE: $o ).
% 3.32/1.73
% 3.32/1.73 thf(powersetE,definition,
% 3.32/1.73 ( powersetE
% 3.32/1.73 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.32/1.73 ( ( in @ B @ ( powerset @ A ) )
% 3.32/1.73 => ( ( in @ Xx @ B )
% 3.32/1.73 => ( in @ Xx @ A ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(setunionI_type,type,
% 3.32/1.73 setunionI: $o ).
% 3.32/1.73
% 3.32/1.73 thf(setunionI,definition,
% 3.32/1.73 ( setunionI
% 3.32/1.73 = ( ! [A: $i,Xx: $i,B: $i] :
% 3.32/1.73 ( ( in @ Xx @ B )
% 3.32/1.73 => ( ( in @ B @ A )
% 3.32/1.73 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(setunionE_type,type,
% 3.32/1.73 setunionE: $o ).
% 3.32/1.73
% 3.32/1.73 thf(setunionE,definition,
% 3.32/1.73 ( setunionE
% 3.32/1.73 = ( ! [A: $i,Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ ( setunion @ A ) )
% 3.32/1.73 => ! [Xphi: $o] :
% 3.32/1.73 ( ! [B: $i] :
% 3.32/1.73 ( ( in @ Xx @ B )
% 3.32/1.73 => ( ( in @ B @ A )
% 3.32/1.73 => Xphi ) )
% 3.32/1.73 => Xphi ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(subPowSU_type,type,
% 3.32/1.73 subPowSU: $o ).
% 3.32/1.73
% 3.32/1.73 thf(subPowSU,definition,
% 3.32/1.73 ( subPowSU
% 3.32/1.73 = ( ! [A: $i,Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(exuE2_type,type,
% 3.32/1.73 exuE2: $o ).
% 3.32/1.73
% 3.32/1.73 thf(exuE2,definition,
% 3.32/1.73 ( exuE2
% 3.32/1.73 = ( ! [Xphi: $i > $o] :
% 3.32/1.73 ( ( exu
% 3.32/1.73 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.32/1.73 => ? [Xx: $i] :
% 3.32/1.73 ! [Xy: $i] :
% 3.32/1.73 ( ( Xphi @ Xy )
% 3.32/1.73 <=> ( Xy = Xx ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(nonemptyImpWitness_type,type,
% 3.32/1.73 nonemptyImpWitness: $o ).
% 3.32/1.73
% 3.32/1.73 thf(nonemptyImpWitness,definition,
% 3.32/1.73 ( nonemptyImpWitness
% 3.32/1.73 = ( ! [A: $i] :
% 3.32/1.73 ( ( nonempty @ A )
% 3.32/1.73 => ? [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 & $true ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(uniqinunit_type,type,
% 3.32/1.73 uniqinunit: $o ).
% 3.32/1.73
% 3.32/1.73 thf(uniqinunit,definition,
% 3.32/1.73 ( uniqinunit
% 3.32/1.73 = ( ! [Xx: $i,Xy: $i] :
% 3.32/1.73 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 3.32/1.73 => ( Xx = Xy ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(notinsingleton_type,type,
% 3.32/1.73 notinsingleton: $o ).
% 3.32/1.73
% 3.32/1.73 thf(notinsingleton,definition,
% 3.32/1.73 ( notinsingleton
% 3.32/1.73 = ( ! [Xx: $i,Xy: $i] :
% 3.32/1.73 ( ( Xx != Xy )
% 3.32/1.73 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(eqinunit_type,type,
% 3.32/1.73 eqinunit: $o ).
% 3.32/1.73
% 3.32/1.73 thf(eqinunit,definition,
% 3.32/1.73 ( eqinunit
% 3.32/1.73 = ( ! [Xx: $i,Xy: $i] :
% 3.32/1.73 ( ( Xx = Xy )
% 3.32/1.73 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(singletonsswitch_type,type,
% 3.32/1.73 singletonsswitch: $o ).
% 3.32/1.73
% 3.32/1.73 thf(singletonsswitch,definition,
% 3.32/1.73 ( singletonsswitch
% 3.32/1.73 = ( ! [Xx: $i,Xy: $i] :
% 3.32/1.73 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 3.32/1.73 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(upairsetE_type,type,
% 3.32/1.73 upairsetE: $o ).
% 3.32/1.73
% 3.32/1.73 thf(upairsetE,definition,
% 3.32/1.73 ( upairsetE
% 3.32/1.73 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 3.32/1.73 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 3.32/1.73 => ( ( Xz = Xx )
% 3.32/1.73 | ( Xz = Xy ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(upairsetIL_type,type,
% 3.32/1.73 upairsetIL: $o ).
% 3.32/1.73
% 3.32/1.73 thf(upairsetIL,definition,
% 3.32/1.73 ( upairsetIL
% 3.32/1.73 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(upairsetIR_type,type,
% 3.32/1.73 upairsetIR: $o ).
% 3.32/1.73
% 3.32/1.73 thf(upairsetIR,definition,
% 3.32/1.73 ( upairsetIR
% 3.32/1.73 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(emptyE1_type,type,
% 3.32/1.73 emptyE1: $o ).
% 3.32/1.73
% 3.32/1.73 thf(emptyE1,definition,
% 3.32/1.73 ( emptyE1
% 3.32/1.73 = ( ! [A: $i,Xphi: $i > $o] :
% 3.32/1.73 ( ? [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 & ( Xphi @ Xx ) )
% 3.32/1.73 => ( ( ( dsetconstr @ A
% 3.32/1.73 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.32/1.73 = emptyset )
% 3.32/1.73 => $false ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(vacuousDall_type,type,
% 3.32/1.73 vacuousDall: $o ).
% 3.32/1.73
% 3.32/1.73 thf(vacuousDall,definition,
% 3.32/1.73 ( vacuousDall
% 3.32/1.73 = ( ! [Xphi: $i > $o,Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ emptyset )
% 3.32/1.73 => ( Xphi @ Xx ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(quantDeMorgan1_type,type,
% 3.32/1.73 quantDeMorgan1: $o ).
% 3.32/1.73
% 3.32/1.73 thf(quantDeMorgan1,definition,
% 3.32/1.73 ( quantDeMorgan1
% 3.32/1.73 = ( ! [A: $i,Xphi: $i > $o] :
% 3.32/1.73 ( ~ ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ( Xphi @ Xx ) )
% 3.32/1.73 => ? [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(quantDeMorgan2_type,type,
% 3.32/1.73 quantDeMorgan2: $o ).
% 3.32/1.73
% 3.32/1.73 thf(quantDeMorgan2,definition,
% 3.32/1.73 ( quantDeMorgan2
% 3.32/1.73 = ( ! [A: $i,Xphi: $i > $o] :
% 3.32/1.73 ( ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ~ ( Xphi @ Xx ) )
% 3.32/1.73 => ~ ? [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 & ( Xphi @ Xx ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(quantDeMorgan3_type,type,
% 3.32/1.73 quantDeMorgan3: $o ).
% 3.32/1.73
% 3.32/1.73 thf(quantDeMorgan3,definition,
% 3.32/1.73 ( quantDeMorgan3
% 3.32/1.73 = ( ! [A: $i,Xphi: $i > $o] :
% 3.32/1.73 ( ~ ? [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 & ( Xphi @ Xx ) )
% 3.32/1.73 => ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(quantDeMorgan4_type,type,
% 3.32/1.73 quantDeMorgan4: $o ).
% 3.32/1.73
% 3.32/1.73 thf(quantDeMorgan4,definition,
% 3.32/1.73 ( quantDeMorgan4
% 3.32/1.73 = ( ! [A: $i,Xphi: $i > $o] :
% 3.32/1.73 ( ? [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 & ~ ( Xphi @ Xx ) )
% 3.32/1.73 => ~ ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ( Xphi @ Xx ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(prop2setI_type,type,
% 3.32/1.73 prop2setI: $o ).
% 3.32/1.73
% 3.32/1.73 thf(prop2setI,definition,
% 3.32/1.73 ( prop2setI
% 3.32/1.73 = ( ! [Xphi: $o] :
% 3.32/1.73 ( Xphi
% 3.32/1.73 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(set2prop_type,type,
% 3.32/1.73 set2prop: $i > $o ).
% 3.32/1.73
% 3.32/1.73 thf(prop2set2propI_type,type,
% 3.32/1.73 prop2set2propI: $o ).
% 3.32/1.73
% 3.32/1.73 thf(prop2set2propI,definition,
% 3.32/1.73 ( prop2set2propI
% 3.32/1.73 = ( ! [Xphi: $o] :
% 3.32/1.73 ( Xphi
% 3.32/1.73 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(notdexE_type,type,
% 3.32/1.73 notdexE: $o ).
% 3.32/1.73
% 3.32/1.73 thf(notdexE,definition,
% 3.32/1.73 ( notdexE
% 3.32/1.73 = ( ! [A: $i,Xphi: $i > $o] :
% 3.32/1.73 ( ~ ? [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 & ( Xphi @ Xx ) )
% 3.32/1.73 => ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(notdallE_type,type,
% 3.32/1.73 notdallE: $o ).
% 3.32/1.73
% 3.32/1.73 thf(notdallE,definition,
% 3.32/1.73 ( notdallE
% 3.32/1.73 = ( ! [A: $i,Xphi: $i > $o] :
% 3.32/1.73 ( ~ ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ( Xphi @ Xx ) )
% 3.32/1.73 => ? [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(exuI1_type,type,
% 3.32/1.73 exuI1: $o ).
% 3.32/1.73
% 3.32/1.73 thf(exuI1,definition,
% 3.32/1.73 ( exuI1
% 3.32/1.73 = ( ! [Xphi: $i > $o] :
% 3.32/1.73 ( ? [Xx: $i] :
% 3.32/1.73 ( ( Xphi @ Xx )
% 3.32/1.73 & ! [Xy: $i] :
% 3.32/1.73 ( ( Xphi @ Xy )
% 3.32/1.73 => ( Xx = Xy ) ) )
% 3.32/1.73 => ( exu
% 3.32/1.73 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(exuI3_type,type,
% 3.32/1.73 exuI3: $o ).
% 3.32/1.73
% 3.32/1.73 thf(exuI3,definition,
% 3.32/1.73 ( exuI3
% 3.32/1.73 = ( ! [Xphi: $i > $o] :
% 3.32/1.73 ( ? [Xx: $i] : ( Xphi @ Xx )
% 3.32/1.73 => ( ! [Xx: $i,Xy: $i] :
% 3.32/1.73 ( ( Xphi @ Xx )
% 3.32/1.73 => ( ( Xphi @ Xy )
% 3.32/1.73 => ( Xx = Xy ) ) )
% 3.32/1.73 => ( exu
% 3.32/1.73 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(exuI2_type,type,
% 3.32/1.73 exuI2: $o ).
% 3.32/1.73
% 3.32/1.73 thf(exuI2,definition,
% 3.32/1.73 ( exuI2
% 3.32/1.73 = ( ! [Xphi: $i > $o] :
% 3.32/1.73 ( ? [Xx: $i] :
% 3.32/1.73 ! [Xy: $i] :
% 3.32/1.73 ( ( Xphi @ Xy )
% 3.32/1.73 <=> ( Xy = Xx ) )
% 3.32/1.73 => ( exu
% 3.32/1.73 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(inCongP_type,type,
% 3.32/1.73 inCongP: $o ).
% 3.32/1.73
% 3.32/1.73 thf(inCongP,definition,
% 3.32/1.73 ( inCongP
% 3.32/1.73 = ( ! [A: $i,B: $i] :
% 3.32/1.73 ( ( A = B )
% 3.32/1.73 => ! [Xx: $i,Xy: $i] :
% 3.32/1.73 ( ( Xx = Xy )
% 3.32/1.73 => ( ( in @ Xx @ A )
% 3.32/1.73 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(in__Cong_type,type,
% 3.32/1.73 in__Cong: $o ).
% 3.32/1.73
% 3.32/1.73 thf(in__Cong,definition,
% 3.32/1.73 ( in__Cong
% 3.32/1.73 = ( ! [A: $i,B: $i] :
% 3.32/1.73 ( ( A = B )
% 3.32/1.73 => ! [Xx: $i,Xy: $i] :
% 3.32/1.73 ( ( Xx = Xy )
% 3.32/1.73 => ( ( in @ Xx @ A )
% 3.32/1.73 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(exuE3u_type,type,
% 3.32/1.73 exuE3u: $o ).
% 3.32/1.73
% 3.32/1.73 thf(exuE3u,definition,
% 3.32/1.73 ( exuE3u
% 3.32/1.73 = ( ! [Xphi: $i > $o] :
% 3.32/1.73 ( ( exu
% 3.32/1.73 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.32/1.73 => ! [Xx: $i,Xy: $i] :
% 3.32/1.73 ( ( Xphi @ Xx )
% 3.32/1.73 => ( ( Xphi @ Xy )
% 3.32/1.73 => ( Xx = Xy ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(exu__Cong_type,type,
% 3.32/1.73 exu__Cong: $o ).
% 3.32/1.73
% 3.32/1.73 thf(exu__Cong,definition,
% 3.32/1.73 ( exu__Cong
% 3.32/1.73 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 3.32/1.73 ( ! [Xx: $i,Xy: $i] :
% 3.32/1.73 ( ( Xx = Xy )
% 3.32/1.73 => ( ( Xphi @ Xx )
% 3.32/1.73 <=> ( Xpsi @ Xy ) ) )
% 3.32/1.73 => ( ( exu
% 3.32/1.73 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.32/1.73 <=> ( exu
% 3.32/1.73 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(emptyset__Cong_type,type,
% 3.32/1.73 emptyset__Cong: $o ).
% 3.32/1.73
% 3.32/1.73 thf(emptyset__Cong,definition,
% 3.32/1.73 ( emptyset__Cong
% 3.32/1.73 = ( emptyset = emptyset ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(setadjoin__Cong_type,type,
% 3.32/1.73 setadjoin__Cong: $o ).
% 3.32/1.73
% 3.32/1.73 thf(setadjoin__Cong,definition,
% 3.32/1.73 ( setadjoin__Cong
% 3.32/1.73 = ( ! [Xx: $i,Xy: $i] :
% 3.32/1.73 ( ( Xx = Xy )
% 3.32/1.73 => ! [Xz: $i,Xu: $i] :
% 3.32/1.73 ( ( Xz = Xu )
% 3.32/1.73 => ( ( setadjoin @ Xx @ Xz )
% 3.32/1.73 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(powerset__Cong_type,type,
% 3.32/1.73 powerset__Cong: $o ).
% 3.32/1.73
% 3.32/1.73 thf(powerset__Cong,definition,
% 3.32/1.73 ( powerset__Cong
% 3.32/1.73 = ( ! [A: $i,B: $i] :
% 3.32/1.73 ( ( A = B )
% 3.32/1.73 => ( ( powerset @ A )
% 3.32/1.73 = ( powerset @ B ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(setunion__Cong_type,type,
% 3.32/1.73 setunion__Cong: $o ).
% 3.32/1.73
% 3.32/1.73 thf(setunion__Cong,definition,
% 3.32/1.73 ( setunion__Cong
% 3.32/1.73 = ( ! [A: $i,B: $i] :
% 3.32/1.73 ( ( A = B )
% 3.32/1.73 => ( ( setunion @ A )
% 3.32/1.73 = ( setunion @ B ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(omega__Cong_type,type,
% 3.32/1.73 omega__Cong: $o ).
% 3.32/1.73
% 3.32/1.73 thf(omega__Cong,definition,
% 3.32/1.73 ( omega__Cong
% 3.32/1.73 = ( omega = omega ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(exuEu_type,type,
% 3.32/1.73 exuEu: $o ).
% 3.32/1.73
% 3.32/1.73 thf(exuEu,definition,
% 3.32/1.73 ( exuEu
% 3.32/1.73 = ( ! [Xphi: $i > $o] :
% 3.32/1.73 ( ( exu
% 3.32/1.73 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.32/1.73 => ! [Xx: $i,Xy: $i] :
% 3.32/1.73 ( ( Xphi @ Xx )
% 3.32/1.73 => ( ( Xphi @ Xy )
% 3.32/1.73 => ( Xx = Xy ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(descr__Cong_type,type,
% 3.32/1.73 descr__Cong: $o ).
% 3.32/1.73
% 3.32/1.73 thf(descr__Cong,definition,
% 3.32/1.73 ( descr__Cong
% 3.32/1.73 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 3.32/1.73 ( ! [Xx: $i,Xy: $i] :
% 3.32/1.73 ( ( Xx = Xy )
% 3.32/1.73 => ( ( Xphi @ Xx )
% 3.32/1.73 <=> ( Xpsi @ Xy ) ) )
% 3.32/1.73 => ( ( exu
% 3.32/1.73 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.32/1.73 => ( ( exu
% 3.32/1.73 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 3.32/1.73 => ( ( descr
% 3.32/1.73 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.32/1.73 = ( descr
% 3.32/1.73 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(dsetconstr__Cong_type,type,
% 3.32/1.73 dsetconstr__Cong: $o ).
% 3.32/1.73
% 3.32/1.73 thf(dsetconstr__Cong,definition,
% 3.32/1.73 ( dsetconstr__Cong
% 3.32/1.73 = ( ! [A: $i,B: $i] :
% 3.32/1.73 ( ( A = B )
% 3.32/1.73 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 3.32/1.73 ( ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ! [Xy: $i] :
% 3.32/1.73 ( ( in @ Xy @ B )
% 3.32/1.73 => ( ( Xx = Xy )
% 3.32/1.73 => ( ( Xphi @ Xx )
% 3.32/1.73 <=> ( Xpsi @ Xy ) ) ) ) )
% 3.32/1.73 => ( ( dsetconstr @ A
% 3.32/1.73 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.32/1.73 = ( dsetconstr @ B
% 3.32/1.73 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(subset_type,type,
% 3.32/1.73 subset: $i > $i > $o ).
% 3.32/1.73
% 3.32/1.73 thf(disjoint_type,type,
% 3.32/1.73 disjoint: $i > $i > $o ).
% 3.32/1.73
% 3.32/1.73 thf(setsmeet_type,type,
% 3.32/1.73 setsmeet: $i > $i > $o ).
% 3.32/1.73
% 3.32/1.73 thf(subsetI1_type,type,
% 3.32/1.73 subsetI1: $o ).
% 3.32/1.73
% 3.32/1.73 thf(subsetI1,definition,
% 3.32/1.73 ( subsetI1
% 3.32/1.73 = ( ! [A: $i,B: $i] :
% 3.32/1.73 ( ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ( in @ Xx @ B ) )
% 3.32/1.73 => ( subset @ A @ B ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(eqimpsubset2_type,type,
% 3.32/1.73 eqimpsubset2: $o ).
% 3.32/1.73
% 3.32/1.73 thf(eqimpsubset2,definition,
% 3.32/1.73 ( eqimpsubset2
% 3.32/1.73 = ( ! [A: $i,B: $i] :
% 3.32/1.73 ( ( A = B )
% 3.32/1.73 => ( subset @ B @ A ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(eqimpsubset1_type,type,
% 3.32/1.73 eqimpsubset1: $o ).
% 3.32/1.73
% 3.32/1.73 thf(eqimpsubset1,definition,
% 3.32/1.73 ( eqimpsubset1
% 3.32/1.73 = ( ! [A: $i,B: $i] :
% 3.32/1.73 ( ( A = B )
% 3.32/1.73 => ( subset @ A @ B ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(subsetI2_type,type,
% 3.32/1.73 subsetI2: $o ).
% 3.32/1.73
% 3.32/1.73 thf(subsetI2,definition,
% 3.32/1.73 ( subsetI2
% 3.32/1.73 = ( ! [A: $i,B: $i] :
% 3.32/1.73 ( ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ( in @ Xx @ B ) )
% 3.32/1.73 => ( subset @ A @ B ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(emptysetsubset_type,type,
% 3.32/1.73 emptysetsubset: $o ).
% 3.32/1.73
% 3.32/1.73 thf(emptysetsubset,definition,
% 3.32/1.73 ( emptysetsubset
% 3.32/1.73 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(subsetE_type,type,
% 3.32/1.73 subsetE: $o ).
% 3.32/1.73
% 3.32/1.73 thf(subsetE,definition,
% 3.32/1.73 ( subsetE
% 3.32/1.73 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.32/1.73 ( ( subset @ A @ B )
% 3.32/1.73 => ( ( in @ Xx @ A )
% 3.32/1.73 => ( in @ Xx @ B ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(subsetE2_type,type,
% 3.32/1.73 subsetE2: $o ).
% 3.32/1.73
% 3.32/1.73 thf(subsetE2,definition,
% 3.32/1.73 ( subsetE2
% 3.32/1.73 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.32/1.73 ( ( subset @ A @ B )
% 3.32/1.73 => ( ~ ( in @ Xx @ B )
% 3.32/1.73 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(notsubsetI_type,type,
% 3.32/1.73 notsubsetI: $o ).
% 3.32/1.73
% 3.32/1.73 thf(notsubsetI,definition,
% 3.32/1.73 ( notsubsetI
% 3.32/1.73 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ( ~ ( in @ Xx @ B )
% 3.32/1.73 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(notequalI1_type,type,
% 3.32/1.73 notequalI1: $o ).
% 3.32/1.73
% 3.32/1.73 thf(notequalI1,definition,
% 3.32/1.73 ( notequalI1
% 3.32/1.73 = ( ! [A: $i,B: $i] :
% 3.32/1.73 ( ~ ( subset @ A @ B )
% 3.32/1.73 => ( A != B ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(notequalI2_type,type,
% 3.32/1.73 notequalI2: $o ).
% 3.32/1.73
% 3.32/1.73 thf(notequalI2,definition,
% 3.32/1.73 ( notequalI2
% 3.32/1.73 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ( ~ ( in @ Xx @ B )
% 3.32/1.73 => ( A != B ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(subsetRefl_type,type,
% 3.32/1.73 subsetRefl: $o ).
% 3.32/1.73
% 3.32/1.73 thf(subsetRefl,definition,
% 3.32/1.73 ( subsetRefl
% 3.32/1.73 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(subsetTrans_type,type,
% 3.32/1.73 subsetTrans: $o ).
% 3.32/1.73
% 3.32/1.73 thf(subsetTrans,definition,
% 3.32/1.73 ( subsetTrans
% 3.32/1.73 = ( ! [A: $i,B: $i,C: $i] :
% 3.32/1.73 ( ( subset @ A @ B )
% 3.32/1.73 => ( ( subset @ B @ C )
% 3.32/1.73 => ( subset @ A @ C ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(setadjoinSub_type,type,
% 3.32/1.73 setadjoinSub: $o ).
% 3.32/1.73
% 3.32/1.73 thf(setadjoinSub,definition,
% 3.32/1.73 ( setadjoinSub
% 3.32/1.73 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(setadjoinSub2_type,type,
% 3.32/1.73 setadjoinSub2: $o ).
% 3.32/1.73
% 3.32/1.73 thf(setadjoinSub2,definition,
% 3.32/1.73 ( setadjoinSub2
% 3.32/1.73 = ( ! [A: $i,Xx: $i,B: $i] :
% 3.32/1.73 ( ( subset @ A @ B )
% 3.32/1.73 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(subset2powerset_type,type,
% 3.32/1.73 subset2powerset: $o ).
% 3.32/1.73
% 3.32/1.73 thf(subset2powerset,definition,
% 3.32/1.73 ( subset2powerset
% 3.32/1.73 = ( ! [A: $i,B: $i] :
% 3.32/1.73 ( ( subset @ A @ B )
% 3.32/1.73 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(setextsub_type,type,
% 3.32/1.73 setextsub: $o ).
% 3.32/1.73
% 3.32/1.73 thf(setextsub,definition,
% 3.32/1.73 ( setextsub
% 3.32/1.73 = ( ! [A: $i,B: $i] :
% 3.32/1.73 ( ( subset @ A @ B )
% 3.32/1.73 => ( ( subset @ B @ A )
% 3.32/1.73 => ( A = B ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(subsetemptysetimpeq_type,type,
% 3.32/1.73 subsetemptysetimpeq: $o ).
% 3.32/1.73
% 3.32/1.73 thf(subsetemptysetimpeq,definition,
% 3.32/1.73 ( subsetemptysetimpeq
% 3.32/1.73 = ( ! [A: $i] :
% 3.32/1.73 ( ( subset @ A @ emptyset )
% 3.32/1.73 => ( A = emptyset ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(powersetI1_type,type,
% 3.32/1.73 powersetI1: $o ).
% 3.32/1.73
% 3.32/1.73 thf(powersetI1,definition,
% 3.32/1.73 ( powersetI1
% 3.32/1.73 = ( ! [A: $i,B: $i] :
% 3.32/1.73 ( ( subset @ B @ A )
% 3.32/1.73 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(powersetE1_type,type,
% 3.32/1.73 powersetE1: $o ).
% 3.32/1.73
% 3.32/1.73 thf(powersetE1,definition,
% 3.32/1.73 ( powersetE1
% 3.32/1.73 = ( ! [A: $i,B: $i] :
% 3.32/1.73 ( ( in @ B @ ( powerset @ A ) )
% 3.32/1.73 => ( subset @ B @ A ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(inPowerset_type,type,
% 3.32/1.73 inPowerset: $o ).
% 3.32/1.73
% 3.32/1.73 thf(inPowerset,definition,
% 3.32/1.73 ( inPowerset
% 3.32/1.73 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(powersetsubset_type,type,
% 3.32/1.73 powersetsubset: $o ).
% 3.32/1.73
% 3.32/1.73 thf(powersetsubset,definition,
% 3.32/1.73 ( powersetsubset
% 3.32/1.73 = ( ! [A: $i,B: $i] :
% 3.32/1.73 ( ( subset @ A @ B )
% 3.32/1.73 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(sepInPowerset_type,type,
% 3.32/1.73 sepInPowerset: $o ).
% 3.32/1.73
% 3.32/1.73 thf(sepInPowerset,definition,
% 3.32/1.73 ( sepInPowerset
% 3.32/1.73 = ( ! [A: $i,Xphi: $i > $o] :
% 3.32/1.73 ( in
% 3.32/1.73 @ ( dsetconstr @ A
% 3.32/1.73 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.32/1.73 @ ( powerset @ A ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(sepSubset_type,type,
% 3.32/1.73 sepSubset: $o ).
% 3.32/1.73
% 3.32/1.73 thf(sepSubset,definition,
% 3.32/1.73 ( sepSubset
% 3.32/1.73 = ( ! [A: $i,Xphi: $i > $o] :
% 3.32/1.73 ( subset
% 3.32/1.73 @ ( dsetconstr @ A
% 3.32/1.73 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.32/1.73 @ A ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(binunion_type,type,
% 3.32/1.73 binunion: $i > $i > $i ).
% 3.32/1.73
% 3.32/1.73 thf(binunionIL_type,type,
% 3.32/1.73 binunionIL: $o ).
% 3.32/1.73
% 3.32/1.73 thf(binunionIL,definition,
% 3.32/1.73 ( binunionIL
% 3.32/1.73 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(upairset2IR_type,type,
% 3.32/1.73 upairset2IR: $o ).
% 3.32/1.73
% 3.32/1.73 thf(upairset2IR,definition,
% 3.32/1.73 ( upairset2IR
% 3.32/1.73 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(binunionIR_type,type,
% 3.32/1.73 binunionIR: $o ).
% 3.32/1.73
% 3.32/1.73 thf(binunionIR,definition,
% 3.32/1.73 ( binunionIR
% 3.32/1.73 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ B )
% 3.32/1.73 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(binunionEcases_type,type,
% 3.32/1.73 binunionEcases: $o ).
% 3.32/1.73
% 3.32/1.73 thf(binunionEcases,definition,
% 3.32/1.73 ( binunionEcases
% 3.32/1.73 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 3.32/1.73 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 3.32/1.73 => ( ( ( in @ Xx @ A )
% 3.32/1.73 => Xphi )
% 3.32/1.73 => ( ( ( in @ Xx @ B )
% 3.32/1.73 => Xphi )
% 3.32/1.73 => Xphi ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(binunionE_type,type,
% 3.32/1.73 binunionE: $o ).
% 3.32/1.73
% 3.32/1.73 thf(binunionE,definition,
% 3.32/1.73 ( binunionE
% 3.32/1.73 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 3.32/1.73 => ( ( in @ Xx @ A )
% 3.32/1.73 | ( in @ Xx @ B ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(binunionLsub_type,type,
% 3.32/1.73 binunionLsub: $o ).
% 3.32/1.73
% 3.32/1.73 thf(binunionLsub,definition,
% 3.32/1.73 ( binunionLsub
% 3.32/1.73 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(binunionRsub_type,type,
% 3.32/1.73 binunionRsub: $o ).
% 3.32/1.73
% 3.32/1.73 thf(binunionRsub,definition,
% 3.32/1.73 ( binunionRsub
% 3.32/1.73 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(binintersect_type,type,
% 3.32/1.73 binintersect: $i > $i > $i ).
% 3.32/1.73
% 3.32/1.73 thf(binintersectI_type,type,
% 3.32/1.73 binintersectI: $o ).
% 3.32/1.73
% 3.32/1.73 thf(binintersectI,definition,
% 3.32/1.73 ( binintersectI
% 3.32/1.73 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ( ( in @ Xx @ B )
% 3.32/1.73 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(binintersectSubset5_type,type,
% 3.32/1.73 binintersectSubset5: $o ).
% 3.32/1.73
% 3.32/1.73 thf(binintersectSubset5,definition,
% 3.32/1.73 ( binintersectSubset5
% 3.32/1.73 = ( ! [A: $i,B: $i,C: $i] :
% 3.32/1.73 ( ( subset @ C @ A )
% 3.32/1.73 => ( ( subset @ C @ B )
% 3.32/1.73 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(binintersectEL_type,type,
% 3.32/1.73 binintersectEL: $o ).
% 3.32/1.73
% 3.32/1.73 thf(binintersectEL,definition,
% 3.32/1.73 ( binintersectEL
% 3.32/1.73 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 3.32/1.73 => ( in @ Xx @ A ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(binintersectLsub_type,type,
% 3.32/1.73 binintersectLsub: $o ).
% 3.32/1.73
% 3.32/1.73 thf(binintersectLsub,definition,
% 3.32/1.73 ( binintersectLsub
% 3.32/1.73 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(binintersectSubset2_type,type,
% 3.32/1.73 binintersectSubset2: $o ).
% 3.32/1.73
% 3.32/1.73 thf(binintersectSubset2,definition,
% 3.32/1.73 ( binintersectSubset2
% 3.32/1.73 = ( ! [A: $i,B: $i] :
% 3.32/1.73 ( ( subset @ A @ B )
% 3.32/1.73 => ( ( binintersect @ A @ B )
% 3.32/1.73 = A ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(binintersectSubset3_type,type,
% 3.32/1.73 binintersectSubset3: $o ).
% 3.32/1.73
% 3.32/1.73 thf(binintersectSubset3,definition,
% 3.32/1.73 ( binintersectSubset3
% 3.32/1.73 = ( ! [A: $i,B: $i] :
% 3.32/1.73 ( ( ( binintersect @ A @ B )
% 3.32/1.73 = B )
% 3.32/1.73 => ( subset @ B @ A ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(binintersectER_type,type,
% 3.32/1.73 binintersectER: $o ).
% 3.32/1.73
% 3.32/1.73 thf(binintersectER,definition,
% 3.32/1.73 ( binintersectER
% 3.32/1.73 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 3.32/1.73 => ( in @ Xx @ B ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(disjointsetsI1_type,type,
% 3.32/1.73 disjointsetsI1: $o ).
% 3.32/1.73
% 3.32/1.73 thf(disjointsetsI1,definition,
% 3.32/1.73 ( disjointsetsI1
% 3.32/1.73 = ( ! [A: $i,B: $i] :
% 3.32/1.73 ( ~ ? [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 & ( in @ Xx @ B ) )
% 3.32/1.73 => ( ( binintersect @ A @ B )
% 3.32/1.73 = emptyset ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(binintersectRsub_type,type,
% 3.32/1.73 binintersectRsub: $o ).
% 3.32/1.73
% 3.32/1.73 thf(binintersectRsub,definition,
% 3.32/1.73 ( binintersectRsub
% 3.32/1.73 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(binintersectSubset4_type,type,
% 3.32/1.73 binintersectSubset4: $o ).
% 3.32/1.73
% 3.32/1.73 thf(binintersectSubset4,definition,
% 3.32/1.73 ( binintersectSubset4
% 3.32/1.73 = ( ! [A: $i,B: $i] :
% 3.32/1.73 ( ( subset @ B @ A )
% 3.32/1.73 => ( ( binintersect @ A @ B )
% 3.32/1.73 = B ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(binintersectSubset1_type,type,
% 3.32/1.73 binintersectSubset1: $o ).
% 3.32/1.73
% 3.32/1.73 thf(binintersectSubset1,definition,
% 3.32/1.73 ( binintersectSubset1
% 3.32/1.73 = ( ! [A: $i,B: $i] :
% 3.32/1.73 ( ( ( binintersect @ A @ B )
% 3.32/1.73 = A )
% 3.32/1.73 => ( subset @ A @ B ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(bs114d_type,type,
% 3.32/1.73 bs114d: $o ).
% 3.32/1.73
% 3.32/1.73 thf(bs114d,definition,
% 3.32/1.73 ( bs114d
% 3.32/1.73 = ( ! [A: $i,B: $i,C: $i] :
% 3.32/1.73 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 3.32/1.73 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(regular_type,type,
% 3.32/1.73 regular: $i > $o ).
% 3.32/1.73
% 3.32/1.73 thf(setminus_type,type,
% 3.32/1.73 setminus: $i > $i > $i ).
% 3.32/1.73
% 3.32/1.73 thf(setminusI_type,type,
% 3.32/1.73 setminusI: $o ).
% 3.32/1.73
% 3.32/1.73 thf(setminusI,definition,
% 3.32/1.73 ( setminusI
% 3.32/1.73 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ( ~ ( in @ Xx @ B )
% 3.32/1.73 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(setminusEL_type,type,
% 3.32/1.73 setminusEL: $o ).
% 3.32/1.73
% 3.32/1.73 thf(setminusEL,definition,
% 3.32/1.73 ( setminusEL
% 3.32/1.73 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 3.32/1.73 => ( in @ Xx @ A ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(setminusER_type,type,
% 3.32/1.73 setminusER: $o ).
% 3.32/1.73
% 3.32/1.73 thf(setminusER,definition,
% 3.32/1.73 ( setminusER
% 3.32/1.73 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 3.32/1.73 => ~ ( in @ Xx @ B ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(setminusSubset2_type,type,
% 3.32/1.73 setminusSubset2: $o ).
% 3.32/1.73
% 3.32/1.73 thf(setminusSubset2,definition,
% 3.32/1.73 ( setminusSubset2
% 3.32/1.73 = ( ! [A: $i,B: $i] :
% 3.32/1.73 ( ( subset @ A @ B )
% 3.32/1.73 => ( ( setminus @ A @ B )
% 3.32/1.73 = emptyset ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(setminusERneg_type,type,
% 3.32/1.73 setminusERneg: $o ).
% 3.32/1.73
% 3.32/1.73 thf(setminusERneg,definition,
% 3.32/1.73 ( setminusERneg
% 3.32/1.73 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.32/1.73 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 3.32/1.73 => ( ( in @ Xx @ A )
% 3.32/1.73 => ( in @ Xx @ B ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(setminusELneg_type,type,
% 3.32/1.73 setminusELneg: $o ).
% 3.32/1.73
% 3.32/1.73 thf(setminusELneg,definition,
% 3.32/1.73 ( setminusELneg
% 3.32/1.73 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.32/1.73 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 3.32/1.73 => ( ~ ( in @ Xx @ B )
% 3.32/1.73 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(setminusILneg_type,type,
% 3.32/1.73 setminusILneg: $o ).
% 3.32/1.73
% 3.32/1.73 thf(setminusILneg,definition,
% 3.32/1.73 ( setminusILneg
% 3.32/1.73 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.32/1.73 ( ~ ( in @ Xx @ A )
% 3.32/1.73 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(setminusIRneg_type,type,
% 3.32/1.73 setminusIRneg: $o ).
% 3.32/1.73
% 3.32/1.73 thf(setminusIRneg,definition,
% 3.32/1.73 ( setminusIRneg
% 3.32/1.73 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ B )
% 3.32/1.73 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(setminusLsub_type,type,
% 3.32/1.73 setminusLsub: $o ).
% 3.32/1.73
% 3.32/1.73 thf(setminusLsub,definition,
% 3.32/1.73 ( setminusLsub
% 3.32/1.73 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(setminusSubset1_type,type,
% 3.32/1.73 setminusSubset1: $o ).
% 3.32/1.73
% 3.32/1.73 thf(setminusSubset1,definition,
% 3.32/1.73 ( setminusSubset1
% 3.32/1.73 = ( ! [A: $i,B: $i] :
% 3.32/1.73 ( ( ( setminus @ A @ B )
% 3.32/1.73 = emptyset )
% 3.32/1.73 => ( subset @ A @ B ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(symdiff_type,type,
% 3.32/1.73 symdiff: $i > $i > $i ).
% 3.32/1.73
% 3.32/1.73 thf(symdiffE_type,type,
% 3.32/1.73 symdiffE: $o ).
% 3.32/1.73
% 3.32/1.73 thf(symdiffE,definition,
% 3.32/1.73 ( symdiffE
% 3.32/1.73 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 3.32/1.73 => ! [Xphi: $o] :
% 3.32/1.73 ( ( ( in @ Xx @ A )
% 3.32/1.73 => ( ~ ( in @ Xx @ B )
% 3.32/1.73 => Xphi ) )
% 3.32/1.73 => ( ( ~ ( in @ Xx @ A )
% 3.32/1.73 => ( ( in @ Xx @ B )
% 3.32/1.73 => Xphi ) )
% 3.32/1.73 => Xphi ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(symdiffI1_type,type,
% 3.32/1.73 symdiffI1: $o ).
% 3.32/1.73
% 3.32/1.73 thf(symdiffI1,definition,
% 3.32/1.73 ( symdiffI1
% 3.32/1.73 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ( ~ ( in @ Xx @ B )
% 3.32/1.73 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(symdiffI2_type,type,
% 3.32/1.73 symdiffI2: $o ).
% 3.32/1.73
% 3.32/1.73 thf(symdiffI2,definition,
% 3.32/1.73 ( symdiffI2
% 3.32/1.73 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.32/1.73 ( ~ ( in @ Xx @ A )
% 3.32/1.73 => ( ( in @ Xx @ B )
% 3.32/1.73 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(symdiffIneg1_type,type,
% 3.32/1.73 symdiffIneg1: $o ).
% 3.32/1.73
% 3.32/1.73 thf(symdiffIneg1,definition,
% 3.32/1.73 ( symdiffIneg1
% 3.32/1.73 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ( ( in @ Xx @ B )
% 3.32/1.73 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(symdiffIneg2_type,type,
% 3.32/1.73 symdiffIneg2: $o ).
% 3.32/1.73
% 3.32/1.73 thf(symdiffIneg2,definition,
% 3.32/1.73 ( symdiffIneg2
% 3.32/1.73 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.32/1.73 ( ~ ( in @ Xx @ A )
% 3.32/1.73 => ( ~ ( in @ Xx @ B )
% 3.32/1.73 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(iskpair_type,type,
% 3.32/1.73 iskpair: $i > $o ).
% 3.32/1.73
% 3.32/1.73 thf(secondinupair_type,type,
% 3.32/1.73 secondinupair: $o ).
% 3.32/1.73
% 3.32/1.73 thf(secondinupair,definition,
% 3.32/1.73 ( secondinupair
% 3.32/1.73 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(setukpairIL_type,type,
% 3.32/1.73 setukpairIL: $o ).
% 3.32/1.73
% 3.32/1.73 thf(setukpairIL,definition,
% 3.32/1.73 ( setukpairIL
% 3.32/1.73 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(setukpairIR_type,type,
% 3.32/1.73 setukpairIR: $o ).
% 3.32/1.73
% 3.32/1.73 thf(setukpairIR,definition,
% 3.32/1.73 ( setukpairIR
% 3.32/1.73 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(kpairiskpair_type,type,
% 3.32/1.73 kpairiskpair: $o ).
% 3.32/1.73
% 3.32/1.73 thf(kpairiskpair,definition,
% 3.32/1.73 ( kpairiskpair
% 3.32/1.73 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(kpair_type,type,
% 3.32/1.73 kpair: $i > $i > $i ).
% 3.32/1.73
% 3.32/1.73 thf(kpairp_type,type,
% 3.32/1.73 kpairp: $o ).
% 3.32/1.73
% 3.32/1.73 thf(kpairp,definition,
% 3.32/1.73 ( kpairp
% 3.32/1.73 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(cartprod_type,type,
% 3.32/1.73 cartprod: $i > $i > $i ).
% 3.32/1.73
% 3.32/1.73 thf(singletonsubset_type,type,
% 3.32/1.73 singletonsubset: $o ).
% 3.32/1.73
% 3.32/1.73 thf(singletonsubset,definition,
% 3.32/1.73 ( singletonsubset
% 3.32/1.73 = ( ! [A: $i,Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(singletoninpowerset_type,type,
% 3.32/1.73 singletoninpowerset: $o ).
% 3.32/1.73
% 3.32/1.73 thf(singletoninpowerset,definition,
% 3.32/1.73 ( singletoninpowerset
% 3.32/1.73 = ( ! [A: $i,Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(singletoninpowunion_type,type,
% 3.32/1.73 singletoninpowunion: $o ).
% 3.32/1.73
% 3.32/1.73 thf(singletoninpowunion,definition,
% 3.32/1.73 ( singletoninpowunion
% 3.32/1.73 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(upairset2E_type,type,
% 3.32/1.73 upairset2E: $o ).
% 3.32/1.73
% 3.32/1.73 thf(upairset2E,definition,
% 3.32/1.73 ( upairset2E
% 3.32/1.73 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 3.32/1.73 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 3.32/1.73 => ( ( Xz = Xx )
% 3.32/1.73 | ( Xz = Xy ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(upairsubunion_type,type,
% 3.32/1.73 upairsubunion: $o ).
% 3.32/1.73
% 3.32/1.73 thf(upairsubunion,definition,
% 3.32/1.73 ( upairsubunion
% 3.32/1.73 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ! [Xy: $i] :
% 3.32/1.73 ( ( in @ Xy @ B )
% 3.32/1.73 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(upairinpowunion_type,type,
% 3.32/1.73 upairinpowunion: $o ).
% 3.32/1.73
% 3.32/1.73 thf(upairinpowunion,definition,
% 3.32/1.73 ( upairinpowunion
% 3.32/1.73 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ! [Xy: $i] :
% 3.32/1.73 ( ( in @ Xy @ B )
% 3.32/1.73 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(ubforcartprodlem1_type,type,
% 3.32/1.73 ubforcartprodlem1: $o ).
% 3.32/1.73
% 3.32/1.73 thf(ubforcartprodlem1,definition,
% 3.32/1.73 ( ubforcartprodlem1
% 3.32/1.73 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ! [Xy: $i] :
% 3.32/1.73 ( ( in @ Xy @ B )
% 3.32/1.73 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(ubforcartprodlem2_type,type,
% 3.32/1.73 ubforcartprodlem2: $o ).
% 3.32/1.73
% 3.32/1.73 thf(ubforcartprodlem2,definition,
% 3.32/1.73 ( ubforcartprodlem2
% 3.32/1.73 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ! [Xy: $i] :
% 3.32/1.73 ( ( in @ Xy @ B )
% 3.32/1.73 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(ubforcartprodlem3_type,type,
% 3.32/1.73 ubforcartprodlem3: $o ).
% 3.32/1.73
% 3.32/1.73 thf(ubforcartprodlem3,definition,
% 3.32/1.73 ( ubforcartprodlem3
% 3.32/1.73 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ! [Xy: $i] :
% 3.32/1.73 ( ( in @ Xy @ B )
% 3.32/1.73 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(cartprodpairin_type,type,
% 3.32/1.73 cartprodpairin: $o ).
% 3.32/1.73
% 3.32/1.73 thf(cartprodpairin,definition,
% 3.32/1.73 ( cartprodpairin
% 3.32/1.73 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ! [Xy: $i] :
% 3.32/1.73 ( ( in @ Xy @ B )
% 3.32/1.73 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(cartprodmempair1_type,type,
% 3.32/1.73 cartprodmempair1: $o ).
% 3.32/1.73
% 3.32/1.73 thf(cartprodmempair1,definition,
% 3.32/1.73 ( cartprodmempair1
% 3.32/1.73 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.32/1.73 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.32/1.73 => ? [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 & ? [Xy: $i] :
% 3.32/1.73 ( ( in @ Xy @ B )
% 3.32/1.73 & ( Xu
% 3.32/1.73 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(cartprodmempair_type,type,
% 3.32/1.73 cartprodmempair: $o ).
% 3.32/1.73
% 3.32/1.73 thf(cartprodmempair,definition,
% 3.32/1.73 ( cartprodmempair
% 3.32/1.73 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.32/1.73 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.32/1.73 => ( iskpair @ Xu ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(setunionE2_type,type,
% 3.32/1.73 setunionE2: $o ).
% 3.32/1.73
% 3.32/1.73 thf(setunionE2,definition,
% 3.32/1.73 ( setunionE2
% 3.32/1.73 = ( ! [A: $i,Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ ( setunion @ A ) )
% 3.32/1.73 => ? [X: $i] :
% 3.32/1.73 ( ( in @ X @ A )
% 3.32/1.73 & ( in @ Xx @ X ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(setunionsingleton1_type,type,
% 3.32/1.73 setunionsingleton1: $o ).
% 3.32/1.73
% 3.32/1.73 thf(setunionsingleton1,definition,
% 3.32/1.73 ( setunionsingleton1
% 3.32/1.73 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(setunionsingleton2_type,type,
% 3.32/1.73 setunionsingleton2: $o ).
% 3.32/1.73
% 3.32/1.73 thf(setunionsingleton2,definition,
% 3.32/1.73 ( setunionsingleton2
% 3.32/1.73 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(setunionsingleton_type,type,
% 3.32/1.73 setunionsingleton: $o ).
% 3.32/1.73
% 3.32/1.73 thf(setunionsingleton,definition,
% 3.32/1.73 ( setunionsingleton
% 3.32/1.73 = ( ! [Xx: $i] :
% 3.32/1.73 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 3.32/1.73 = Xx ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(singleton_type,type,
% 3.32/1.73 singleton: $i > $o ).
% 3.32/1.73
% 3.32/1.73 thf(singletonprop_type,type,
% 3.32/1.73 singletonprop: $o ).
% 3.32/1.73
% 3.32/1.73 thf(singletonprop,definition,
% 3.32/1.73 ( singletonprop
% 3.32/1.73 = ( ! [A: $i,Xphi: $i > $o] :
% 3.32/1.73 ( ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ! [Xy: $i] :
% 3.32/1.73 ( ( in @ Xy @ A )
% 3.32/1.73 => ( ( Xphi @ Xx )
% 3.32/1.73 => ( ( Xphi @ Xy )
% 3.32/1.73 => ( Xx = Xy ) ) ) ) )
% 3.32/1.73 => ( ? [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 & ( Xphi @ Xx ) )
% 3.32/1.73 => ( singleton
% 3.32/1.73 @ ( dsetconstr @ A
% 3.32/1.73 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(ex1_type,type,
% 3.32/1.73 ex1: $i > ( $i > $o ) > $o ).
% 3.32/1.73
% 3.32/1.73 thf(ex1E1_type,type,
% 3.32/1.73 ex1E1: $o ).
% 3.32/1.73
% 3.32/1.73 thf(ex1E1,definition,
% 3.32/1.73 ( ex1E1
% 3.32/1.73 = ( ! [A: $i,Xphi: $i > $o] :
% 3.32/1.73 ( ( ex1 @ A
% 3.32/1.73 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.32/1.73 => ? [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 & ( Xphi @ Xx ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(ex1I_type,type,
% 3.32/1.73 ex1I: $o ).
% 3.32/1.73
% 3.32/1.73 thf(ex1I,definition,
% 3.32/1.73 ( ex1I
% 3.32/1.73 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ( ( Xphi @ Xx )
% 3.32/1.73 => ( ! [Xy: $i] :
% 3.32/1.73 ( ( in @ Xy @ A )
% 3.32/1.73 => ( ( Xphi @ Xy )
% 3.32/1.73 => ( Xy = Xx ) ) )
% 3.32/1.73 => ( ex1 @ A
% 3.32/1.73 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(ex1I2_type,type,
% 3.32/1.73 ex1I2: $o ).
% 3.32/1.73
% 3.32/1.73 thf(ex1I2,definition,
% 3.32/1.73 ( ex1I2
% 3.32/1.73 = ( ! [A: $i,Xphi: $i > $o] :
% 3.32/1.73 ( ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ! [Xy: $i] :
% 3.32/1.73 ( ( in @ Xy @ A )
% 3.32/1.73 => ( ( Xphi @ Xx )
% 3.32/1.73 => ( ( Xphi @ Xy )
% 3.32/1.73 => ( Xx = Xy ) ) ) ) )
% 3.32/1.73 => ( ? [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 & ( Xphi @ Xx ) )
% 3.32/1.73 => ( ex1 @ A
% 3.32/1.73 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(singletonsuniq_type,type,
% 3.32/1.73 singletonsuniq: $o ).
% 3.32/1.73
% 3.32/1.73 thf(singletonsuniq,definition,
% 3.32/1.73 ( singletonsuniq
% 3.32/1.73 = ( ! [Xx: $i,Xy: $i] :
% 3.32/1.73 ( ( ( setadjoin @ Xx @ emptyset )
% 3.32/1.73 = ( setadjoin @ Xy @ emptyset ) )
% 3.32/1.73 => ( Xx = Xy ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(atmost1p_type,type,
% 3.32/1.73 atmost1p: $i > $o ).
% 3.32/1.73
% 3.32/1.73 thf(atleast2p_type,type,
% 3.32/1.73 atleast2p: $i > $o ).
% 3.32/1.73
% 3.32/1.73 thf(atmost2p_type,type,
% 3.32/1.73 atmost2p: $i > $o ).
% 3.32/1.73
% 3.32/1.73 thf(upairsetp_type,type,
% 3.32/1.73 upairsetp: $i > $o ).
% 3.32/1.73
% 3.32/1.73 thf(setukpairinjL1_type,type,
% 3.32/1.73 setukpairinjL1: $o ).
% 3.32/1.73
% 3.32/1.73 thf(setukpairinjL1,definition,
% 3.32/1.73 ( setukpairinjL1
% 3.32/1.73 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 3.32/1.73 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 3.32/1.73 => ( Xx = Xz ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(kfstsingleton_type,type,
% 3.32/1.73 kfstsingleton: $o ).
% 3.32/1.73
% 3.32/1.73 thf(kfstsingleton,definition,
% 3.32/1.73 ( kfstsingleton
% 3.32/1.73 = ( ! [Xu: $i] :
% 3.32/1.73 ( ( iskpair @ Xu )
% 3.32/1.73 => ( singleton
% 3.32/1.73 @ ( dsetconstr @ ( setunion @ Xu )
% 3.32/1.73 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(theprop_type,type,
% 3.32/1.73 theprop: $o ).
% 3.32/1.73
% 3.32/1.73 thf(theprop,definition,
% 3.32/1.73 ( theprop
% 3.32/1.73 = ( ! [X: $i] :
% 3.32/1.73 ( ( singleton @ X )
% 3.32/1.73 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(kfst_type,type,
% 3.32/1.73 kfst: $i > $i ).
% 3.32/1.73
% 3.32/1.73 thf(kfstpairEq_type,type,
% 3.32/1.73 kfstpairEq: $o ).
% 3.32/1.73
% 3.32/1.73 thf(kfstpairEq,definition,
% 3.32/1.73 ( kfstpairEq
% 3.32/1.73 = ( ! [Xx: $i,Xy: $i] :
% 3.32/1.73 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 3.32/1.73 = Xx ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(cartprodfstin_type,type,
% 3.32/1.73 cartprodfstin: $o ).
% 3.32/1.73
% 3.32/1.73 thf(cartprodfstin,definition,
% 3.32/1.73 ( cartprodfstin
% 3.32/1.73 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.32/1.73 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.32/1.73 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(setukpairinjL2_type,type,
% 3.32/1.73 setukpairinjL2: $o ).
% 3.32/1.73
% 3.32/1.73 thf(setukpairinjL2,definition,
% 3.32/1.73 ( setukpairinjL2
% 3.32/1.73 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.32/1.73 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 3.32/1.73 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 3.32/1.73 => ( Xx = Xz ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(setukpairinjL_type,type,
% 3.32/1.73 setukpairinjL: $o ).
% 3.32/1.73
% 3.32/1.73 thf(setukpairinjL,definition,
% 3.32/1.73 ( setukpairinjL
% 3.32/1.73 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.32/1.73 ( ( ( kpair @ Xx @ Xy )
% 3.32/1.73 = ( kpair @ Xz @ Xu ) )
% 3.32/1.73 => ( Xx = Xz ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(setukpairinjR11_type,type,
% 3.32/1.73 setukpairinjR11: $o ).
% 3.32/1.73
% 3.32/1.73 thf(setukpairinjR11,definition,
% 3.32/1.73 ( setukpairinjR11
% 3.32/1.73 = ( ! [Xx: $i,Xy: $i] :
% 3.32/1.73 ( ( Xx = Xy )
% 3.32/1.73 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 3.32/1.73 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(setukpairinjR12_type,type,
% 3.32/1.73 setukpairinjR12: $o ).
% 3.32/1.73
% 3.32/1.73 thf(setukpairinjR12,definition,
% 3.32/1.73 ( setukpairinjR12
% 3.32/1.73 = ( ! [Xx: $i,Xy: $i] :
% 3.32/1.73 ( ( Xx = Xy )
% 3.32/1.73 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 3.32/1.73 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(setukpairinjR1_type,type,
% 3.32/1.73 setukpairinjR1: $o ).
% 3.32/1.73
% 3.32/1.73 thf(setukpairinjR1,definition,
% 3.32/1.73 ( setukpairinjR1
% 3.32/1.73 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.32/1.73 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 3.32/1.73 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 3.32/1.73 => ( ( Xz = Xu )
% 3.32/1.73 => ( Xy = Xu ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(upairequniteq_type,type,
% 3.32/1.73 upairequniteq: $o ).
% 3.32/1.73
% 3.32/1.73 thf(upairequniteq,definition,
% 3.32/1.73 ( upairequniteq
% 3.32/1.73 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 3.32/1.73 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 3.32/1.73 = ( setadjoin @ Xz @ emptyset ) )
% 3.32/1.73 => ( Xx = Xy ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(setukpairinjR2_type,type,
% 3.32/1.73 setukpairinjR2: $o ).
% 3.32/1.73
% 3.32/1.73 thf(setukpairinjR2,definition,
% 3.32/1.73 ( setukpairinjR2
% 3.32/1.73 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.32/1.73 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 3.32/1.73 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 3.32/1.73 => ( Xy = Xu ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(setukpairinjR_type,type,
% 3.32/1.73 setukpairinjR: $o ).
% 3.32/1.73
% 3.32/1.73 thf(setukpairinjR,definition,
% 3.32/1.73 ( setukpairinjR
% 3.32/1.73 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.32/1.73 ( ( ( kpair @ Xx @ Xy )
% 3.32/1.73 = ( kpair @ Xz @ Xu ) )
% 3.32/1.73 => ( Xy = Xu ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(ksndsingleton_type,type,
% 3.32/1.73 ksndsingleton: $o ).
% 3.32/1.73
% 3.32/1.73 thf(ksndsingleton,definition,
% 3.32/1.73 ( ksndsingleton
% 3.32/1.73 = ( ! [Xu: $i] :
% 3.32/1.73 ( ( iskpair @ Xu )
% 3.32/1.73 => ( singleton
% 3.32/1.73 @ ( dsetconstr @ ( setunion @ Xu )
% 3.32/1.73 @ ^ [Xx: $i] :
% 3.32/1.73 ( Xu
% 3.32/1.73 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(ksnd_type,type,
% 3.32/1.73 ksnd: $i > $i ).
% 3.32/1.73
% 3.32/1.73 thf(ksndpairEq_type,type,
% 3.32/1.73 ksndpairEq: $o ).
% 3.32/1.73
% 3.32/1.73 thf(ksndpairEq,definition,
% 3.32/1.73 ( ksndpairEq
% 3.32/1.73 = ( ! [Xx: $i,Xy: $i] :
% 3.32/1.73 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 3.32/1.73 = Xy ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(kpairsurjEq_type,type,
% 3.32/1.73 kpairsurjEq: $o ).
% 3.32/1.73
% 3.32/1.73 thf(kpairsurjEq,definition,
% 3.32/1.73 ( kpairsurjEq
% 3.32/1.73 = ( ! [Xu: $i] :
% 3.32/1.73 ( ( iskpair @ Xu )
% 3.32/1.73 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 3.32/1.73 = Xu ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(cartprodsndin_type,type,
% 3.32/1.73 cartprodsndin: $o ).
% 3.32/1.73
% 3.32/1.73 thf(cartprodsndin,definition,
% 3.32/1.73 ( cartprodsndin
% 3.32/1.73 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.32/1.73 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.32/1.73 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(cartprodpairmemEL_type,type,
% 3.32/1.73 cartprodpairmemEL: $o ).
% 3.32/1.73
% 3.32/1.73 thf(cartprodpairmemEL,definition,
% 3.32/1.73 ( cartprodpairmemEL
% 3.32/1.73 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 3.32/1.73 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 3.32/1.73 => ( in @ Xx @ A ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(cartprodpairmemER_type,type,
% 3.32/1.73 cartprodpairmemER: $o ).
% 3.32/1.73
% 3.32/1.73 thf(cartprodpairmemER,definition,
% 3.32/1.73 ( cartprodpairmemER
% 3.32/1.73 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 3.32/1.73 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 3.32/1.73 => ( in @ Xy @ B ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(cartprodmempaircEq_type,type,
% 3.32/1.73 cartprodmempaircEq: $o ).
% 3.32/1.73
% 3.32/1.73 thf(cartprodmempaircEq,definition,
% 3.32/1.73 ( cartprodmempaircEq
% 3.32/1.73 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ! [Xy: $i] :
% 3.32/1.73 ( ( in @ Xy @ B )
% 3.32/1.73 => ( ( kpair @ Xx @ Xy )
% 3.32/1.73 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(cartprodfstpairEq_type,type,
% 3.32/1.73 cartprodfstpairEq: $o ).
% 3.32/1.73
% 3.32/1.73 thf(cartprodfstpairEq,definition,
% 3.32/1.73 ( cartprodfstpairEq
% 3.32/1.73 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ! [Xy: $i] :
% 3.32/1.73 ( ( in @ Xy @ B )
% 3.32/1.73 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 3.32/1.73 = Xx ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(cartprodsndpairEq_type,type,
% 3.32/1.73 cartprodsndpairEq: $o ).
% 3.32/1.73
% 3.32/1.73 thf(cartprodsndpairEq,definition,
% 3.32/1.73 ( cartprodsndpairEq
% 3.32/1.73 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ! [Xy: $i] :
% 3.32/1.73 ( ( in @ Xy @ B )
% 3.32/1.73 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 3.32/1.73 = Xy ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(cartprodpairsurjEq_type,type,
% 3.32/1.73 cartprodpairsurjEq: $o ).
% 3.32/1.73
% 3.32/1.73 thf(cartprodpairsurjEq,definition,
% 3.32/1.73 ( cartprodpairsurjEq
% 3.32/1.73 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.32/1.73 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.32/1.73 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 3.32/1.73 = Xu ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(breln_type,type,
% 3.32/1.73 breln: $i > $i > $i > $o ).
% 3.32/1.73
% 3.32/1.73 thf(dpsetconstr_type,type,
% 3.32/1.73 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 3.32/1.73
% 3.32/1.73 thf(dpsetconstrI_type,type,
% 3.32/1.73 dpsetconstrI: $o ).
% 3.32/1.73
% 3.32/1.73 thf(dpsetconstrI,definition,
% 3.32/1.73 ( dpsetconstrI
% 3.32/1.73 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ! [Xy: $i] :
% 3.32/1.73 ( ( in @ Xy @ B )
% 3.32/1.73 => ( ( Xphi @ Xx @ Xy )
% 3.32/1.73 => ( in @ ( kpair @ Xx @ Xy )
% 3.32/1.73 @ ( dpsetconstr @ A @ B
% 3.32/1.73 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(dpsetconstrSub_type,type,
% 3.32/1.73 dpsetconstrSub: $o ).
% 3.32/1.73
% 3.32/1.73 thf(dpsetconstrSub,definition,
% 3.32/1.73 ( dpsetconstrSub
% 3.32/1.73 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 3.32/1.73 ( subset
% 3.32/1.73 @ ( dpsetconstr @ A @ B
% 3.32/1.73 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 3.32/1.73 @ ( cartprod @ A @ B ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(setOfPairsIsBReln_type,type,
% 3.32/1.73 setOfPairsIsBReln: $o ).
% 3.32/1.73
% 3.32/1.73 thf(setOfPairsIsBReln,definition,
% 3.32/1.73 ( setOfPairsIsBReln
% 3.32/1.73 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 3.32/1.73 ( breln @ A @ B
% 3.32/1.73 @ ( dpsetconstr @ A @ B
% 3.32/1.73 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(dpsetconstrERa_type,type,
% 3.32/1.73 dpsetconstrERa: $o ).
% 3.32/1.73
% 3.32/1.73 thf(dpsetconstrERa,definition,
% 3.32/1.73 ( dpsetconstrERa
% 3.32/1.73 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ! [Xy: $i] :
% 3.32/1.73 ( ( in @ Xy @ B )
% 3.32/1.73 => ( ( in @ ( kpair @ Xx @ Xy )
% 3.32/1.73 @ ( dpsetconstr @ A @ B
% 3.32/1.73 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 3.32/1.73 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(dpsetconstrEL1_type,type,
% 3.32/1.73 dpsetconstrEL1: $o ).
% 3.32/1.73
% 3.32/1.73 thf(dpsetconstrEL1,definition,
% 3.32/1.73 ( dpsetconstrEL1
% 3.32/1.73 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 3.32/1.73 ( ( in @ ( kpair @ Xx @ Xy )
% 3.32/1.73 @ ( dpsetconstr @ A @ B
% 3.32/1.73 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 3.32/1.73 => ( in @ Xx @ A ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(dpsetconstrEL2_type,type,
% 3.32/1.73 dpsetconstrEL2: $o ).
% 3.32/1.73
% 3.32/1.73 thf(dpsetconstrEL2,definition,
% 3.32/1.73 ( dpsetconstrEL2
% 3.32/1.73 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 3.32/1.73 ( ( in @ ( kpair @ Xx @ Xy )
% 3.32/1.73 @ ( dpsetconstr @ A @ B
% 3.32/1.73 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 3.32/1.73 => ( in @ Xy @ B ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(dpsetconstrER_type,type,
% 3.32/1.73 dpsetconstrER: $o ).
% 3.32/1.73
% 3.32/1.73 thf(dpsetconstrER,definition,
% 3.32/1.73 ( dpsetconstrER
% 3.32/1.73 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 3.32/1.73 ( ( in @ ( kpair @ Xx @ Xy )
% 3.32/1.73 @ ( dpsetconstr @ A @ B
% 3.32/1.73 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 3.32/1.73 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(func_type,type,
% 3.32/1.73 func: $i > $i > $i > $o ).
% 3.32/1.73
% 3.32/1.73 thf(funcSet_type,type,
% 3.32/1.73 funcSet: $i > $i > $i ).
% 3.32/1.73
% 3.32/1.73 thf(funcImageSingleton_type,type,
% 3.32/1.73 funcImageSingleton: $o ).
% 3.32/1.73
% 3.32/1.73 thf(funcImageSingleton,definition,
% 3.32/1.73 ( funcImageSingleton
% 3.32/1.73 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.32/1.73 ( ( func @ A @ B @ Xf )
% 3.32/1.73 => ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ( singleton
% 3.32/1.73 @ ( dsetconstr @ B
% 3.32/1.73 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(apProp_type,type,
% 3.32/1.73 apProp: $o ).
% 3.32/1.73
% 3.32/1.73 thf(apProp,definition,
% 3.32/1.73 ( apProp
% 3.32/1.73 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.32/1.73 ( ( func @ A @ B @ Xf )
% 3.32/1.73 => ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ( in
% 3.32/1.73 @ ( setunion
% 3.32/1.73 @ ( dsetconstr @ B
% 3.32/1.73 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 3.32/1.73 @ B ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(ap_type,type,
% 3.32/1.73 ap: $i > $i > $i > $i > $i ).
% 3.32/1.73
% 3.32/1.73 thf(app_type,type,
% 3.32/1.73 app: $o ).
% 3.32/1.73
% 3.32/1.73 thf(app,definition,
% 3.32/1.73 ( app
% 3.32/1.73 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.32/1.73 ( ( func @ A @ B @ Xf )
% 3.32/1.73 => ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(infuncsetfunc_type,type,
% 3.32/1.73 infuncsetfunc: $o ).
% 3.32/1.73
% 3.32/1.73 thf(infuncsetfunc,definition,
% 3.32/1.73 ( infuncsetfunc
% 3.32/1.73 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.32/1.73 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.32/1.73 => ( func @ A @ B @ Xf ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(ap2p_type,type,
% 3.32/1.73 ap2p: $o ).
% 3.32/1.73
% 3.32/1.73 thf(ap2p,definition,
% 3.32/1.73 ( ap2p
% 3.32/1.73 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.32/1.73 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.32/1.73 => ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(funcinfuncset_type,type,
% 3.32/1.73 funcinfuncset: $o ).
% 3.32/1.73
% 3.32/1.73 thf(funcinfuncset,definition,
% 3.32/1.73 ( funcinfuncset
% 3.32/1.73 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.32/1.73 ( ( func @ A @ B @ Xf )
% 3.32/1.73 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(lamProp_type,type,
% 3.32/1.73 lamProp: $o ).
% 3.32/1.73
% 3.32/1.73 thf(lamProp,definition,
% 3.32/1.73 ( lamProp
% 3.32/1.73 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.32/1.73 ( ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.32/1.73 => ( func @ A @ B
% 3.32/1.73 @ ( dpsetconstr @ A @ B
% 3.32/1.73 @ ^ [Xx: $i,Xy: $i] :
% 3.32/1.73 ( ( Xf @ Xx )
% 3.32/1.73 = Xy ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(lam_type,type,
% 3.32/1.73 lam: $i > $i > ( $i > $i ) > $i ).
% 3.32/1.73
% 3.32/1.73 thf(lamp_type,type,
% 3.32/1.73 lamp: $o ).
% 3.32/1.73
% 3.32/1.73 thf(lamp,definition,
% 3.32/1.73 ( lamp
% 3.32/1.73 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.32/1.73 ( ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.32/1.73 => ( func @ A @ B
% 3.32/1.73 @ ( lam @ A @ B
% 3.32/1.73 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(lam2p_type,type,
% 3.32/1.73 lam2p: $o ).
% 3.32/1.73
% 3.32/1.73 thf(lam2p,definition,
% 3.32/1.73 ( lam2p
% 3.32/1.73 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.32/1.73 ( ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.32/1.73 => ( in
% 3.32/1.73 @ ( lam @ A @ B
% 3.32/1.73 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 3.32/1.73 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(brelnall1_type,type,
% 3.32/1.73 brelnall1: $o ).
% 3.32/1.73
% 3.32/1.73 thf(brelnall1,definition,
% 3.32/1.73 ( brelnall1
% 3.32/1.73 = ( ! [A: $i,B: $i,R: $i] :
% 3.32/1.73 ( ( breln @ A @ B @ R )
% 3.32/1.73 => ! [Xphi: $i > $o] :
% 3.32/1.73 ( ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ! [Xy: $i] :
% 3.32/1.73 ( ( in @ Xy @ B )
% 3.32/1.73 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.32/1.73 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 3.32/1.73 => ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ R )
% 3.32/1.73 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(brelnall2_type,type,
% 3.32/1.73 brelnall2: $o ).
% 3.32/1.73
% 3.32/1.73 thf(brelnall2,definition,
% 3.32/1.73 ( brelnall2
% 3.32/1.73 = ( ! [A: $i,B: $i,R: $i] :
% 3.32/1.73 ( ( breln @ A @ B @ R )
% 3.32/1.73 => ! [Xphi: $i > $o] :
% 3.32/1.73 ( ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ! [Xy: $i] :
% 3.32/1.73 ( ( in @ Xy @ B )
% 3.32/1.73 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.32/1.73 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 3.32/1.73 => ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ R )
% 3.32/1.73 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(ex1E2_type,type,
% 3.32/1.73 ex1E2: $o ).
% 3.32/1.73
% 3.32/1.73 thf(ex1E2,definition,
% 3.32/1.73 ( ex1E2
% 3.32/1.73 = ( ! [A: $i,Xphi: $i > $o] :
% 3.32/1.73 ( ( ex1 @ A
% 3.32/1.73 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.32/1.73 => ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ! [Xy: $i] :
% 3.32/1.73 ( ( in @ Xy @ A )
% 3.32/1.73 => ( ( Xphi @ Xx )
% 3.32/1.73 => ( ( Xphi @ Xy )
% 3.32/1.73 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(funcGraphProp1_type,type,
% 3.32/1.73 funcGraphProp1: $o ).
% 3.32/1.73
% 3.32/1.73 thf(funcGraphProp1,definition,
% 3.32/1.73 ( funcGraphProp1
% 3.32/1.73 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.32/1.73 ( ( func @ A @ B @ Xf )
% 3.32/1.73 => ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(funcGraphProp3_type,type,
% 3.32/1.73 funcGraphProp3: $o ).
% 3.32/1.73
% 3.32/1.73 thf(funcGraphProp3,definition,
% 3.32/1.73 ( funcGraphProp3
% 3.32/1.73 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.32/1.73 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.32/1.73 => ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(funcGraphProp2_type,type,
% 3.32/1.73 funcGraphProp2: $o ).
% 3.32/1.73
% 3.32/1.73 thf(funcGraphProp2,definition,
% 3.32/1.73 ( funcGraphProp2
% 3.32/1.73 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.32/1.73 ( ( func @ A @ B @ Xf )
% 3.32/1.73 => ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ! [Xy: $i] :
% 3.32/1.73 ( ( in @ Xy @ B )
% 3.32/1.73 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 3.32/1.73 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.32/1.73 = Xy ) ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(funcextLem_type,type,
% 3.32/1.73 funcextLem: $o ).
% 3.32/1.73
% 3.32/1.73 thf(funcextLem,definition,
% 3.32/1.73 ( funcextLem
% 3.32/1.73 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.32/1.73 ( ( func @ A @ B @ Xf )
% 3.32/1.73 => ! [Xg: $i] :
% 3.32/1.73 ( ( func @ A @ B @ Xg )
% 3.32/1.73 => ( ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.32/1.73 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 3.32/1.73 => ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ! [Xy: $i] :
% 3.32/1.73 ( ( in @ Xy @ B )
% 3.32/1.73 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 3.32/1.73 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(funcGraphProp4_type,type,
% 3.32/1.73 funcGraphProp4: $o ).
% 3.32/1.73
% 3.32/1.73 thf(funcGraphProp4,definition,
% 3.32/1.73 ( funcGraphProp4
% 3.32/1.73 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.32/1.73 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.32/1.73 => ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ! [Xy: $i] :
% 3.32/1.73 ( ( in @ Xy @ B )
% 3.32/1.73 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 3.32/1.73 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.32/1.73 = Xy ) ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(subbreln_type,type,
% 3.32/1.73 subbreln: $o ).
% 3.32/1.73
% 3.32/1.73 thf(subbreln,definition,
% 3.32/1.73 ( subbreln
% 3.32/1.73 = ( ! [A: $i,B: $i,R: $i] :
% 3.32/1.73 ( ( breln @ A @ B @ R )
% 3.32/1.73 => ! [S: $i] :
% 3.32/1.73 ( ( breln @ A @ B @ S )
% 3.32/1.73 => ( ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ! [Xy: $i] :
% 3.32/1.73 ( ( in @ Xy @ B )
% 3.32/1.73 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.32/1.73 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 3.32/1.73 => ( subset @ R @ S ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(eqbreln_type,type,
% 3.32/1.73 eqbreln: $o ).
% 3.32/1.73
% 3.32/1.73 thf(eqbreln,definition,
% 3.32/1.73 ( eqbreln
% 3.32/1.73 = ( ! [A: $i,B: $i,R: $i] :
% 3.32/1.73 ( ( breln @ A @ B @ R )
% 3.32/1.73 => ! [S: $i] :
% 3.32/1.73 ( ( breln @ A @ B @ S )
% 3.32/1.73 => ( ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ! [Xy: $i] :
% 3.32/1.73 ( ( in @ Xy @ B )
% 3.32/1.73 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.32/1.73 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 3.32/1.73 => ( ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ! [Xy: $i] :
% 3.32/1.73 ( ( in @ Xy @ B )
% 3.32/1.73 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 3.32/1.73 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 3.32/1.73 => ( R = S ) ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(funcext_type,type,
% 3.32/1.73 funcext: $o ).
% 3.32/1.73
% 3.32/1.73 thf(funcext,definition,
% 3.32/1.73 ( funcext
% 3.32/1.73 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.32/1.73 ( ( func @ A @ B @ Xf )
% 3.32/1.73 => ! [Xg: $i] :
% 3.32/1.73 ( ( func @ A @ B @ Xg )
% 3.32/1.73 => ( ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.32/1.73 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 3.32/1.73 => ( Xf = Xg ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(funcext2_type,type,
% 3.32/1.73 funcext2: $o ).
% 3.32/1.73
% 3.32/1.73 thf(funcext2,definition,
% 3.32/1.73 ( funcext2
% 3.32/1.73 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.32/1.73 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.32/1.73 => ! [Xg: $i] :
% 3.32/1.73 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 3.32/1.73 => ( ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.32/1.73 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 3.32/1.73 => ( Xf = Xg ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(ap2apEq1_type,type,
% 3.32/1.73 ap2apEq1: $o ).
% 3.32/1.73
% 3.32/1.73 thf(ap2apEq1,definition,
% 3.32/1.73 ( ap2apEq1
% 3.32/1.73 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.32/1.73 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.32/1.73 => ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.32/1.73 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(ap2apEq2_type,type,
% 3.32/1.73 ap2apEq2: $o ).
% 3.32/1.73
% 3.32/1.73 thf(ap2apEq2,definition,
% 3.32/1.73 ( ap2apEq2
% 3.32/1.73 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.32/1.73 ( ( func @ A @ B @ Xf )
% 3.32/1.73 => ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.32/1.73 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(beta1_type,type,
% 3.32/1.73 beta1: $o ).
% 3.32/1.73
% 3.32/1.73 thf(beta1,definition,
% 3.32/1.73 ( beta1
% 3.32/1.73 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.32/1.73 ( ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.32/1.73 => ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ( ( ap @ A @ B
% 3.32/1.73 @ ( lam @ A @ B
% 3.32/1.73 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 3.32/1.73 @ Xx )
% 3.32/1.73 = ( Xf @ Xx ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(eta1_type,type,
% 3.32/1.73 eta1: $o ).
% 3.32/1.73
% 3.32/1.73 thf(eta1,definition,
% 3.32/1.73 ( eta1
% 3.32/1.73 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.32/1.73 ( ( func @ A @ B @ Xf )
% 3.32/1.73 => ( ( lam @ A @ B
% 3.32/1.73 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 3.32/1.73 = Xf ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(lam2lamEq_type,type,
% 3.32/1.73 lam2lamEq: $o ).
% 3.32/1.73
% 3.32/1.73 thf(lam2lamEq,definition,
% 3.32/1.73 ( lam2lamEq
% 3.32/1.73 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.32/1.73 ( ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.32/1.73 => ( ( lam @ A @ B
% 3.32/1.73 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 3.32/1.73 = ( lam @ A @ B
% 3.32/1.73 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(beta2_type,type,
% 3.32/1.73 beta2: $o ).
% 3.32/1.73
% 3.32/1.73 thf(beta2,definition,
% 3.32/1.73 ( beta2
% 3.32/1.73 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.32/1.73 ( ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.32/1.73 => ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ( ( ap @ A @ B
% 3.32/1.73 @ ( lam @ A @ B
% 3.32/1.73 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 3.32/1.73 @ Xx )
% 3.32/1.73 = ( Xf @ Xx ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(eta2_type,type,
% 3.32/1.73 eta2: $o ).
% 3.32/1.73
% 3.32/1.73 thf(eta2,definition,
% 3.32/1.73 ( eta2
% 3.32/1.73 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.32/1.73 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.32/1.73 => ( ( lam @ A @ B
% 3.32/1.73 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 3.32/1.73 = Xf ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(iffalseProp1_type,type,
% 3.32/1.73 iffalseProp1: $o ).
% 3.32/1.73
% 3.32/1.73 thf(iffalseProp1,definition,
% 3.32/1.73 ( iffalseProp1
% 3.32/1.73 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ! [Xy: $i] :
% 3.32/1.73 ( ( in @ Xy @ A )
% 3.32/1.73 => ( ~ Xphi
% 3.32/1.73 => ( in @ Xy
% 3.32/1.73 @ ( dsetconstr @ A
% 3.32/1.73 @ ^ [Xz: $i] :
% 3.32/1.73 ( ( Xphi
% 3.32/1.73 & ( Xz = Xx ) )
% 3.32/1.73 | ( ~ Xphi
% 3.32/1.73 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(iffalseProp2_type,type,
% 3.32/1.73 iffalseProp2: $o ).
% 3.32/1.73
% 3.32/1.73 thf(iffalseProp2,definition,
% 3.32/1.73 ( iffalseProp2
% 3.32/1.73 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ! [Xy: $i] :
% 3.32/1.73 ( ( in @ Xy @ A )
% 3.32/1.73 => ( ~ Xphi
% 3.32/1.73 => ( ( dsetconstr @ A
% 3.32/1.73 @ ^ [Xz: $i] :
% 3.32/1.73 ( ( Xphi
% 3.32/1.73 & ( Xz = Xx ) )
% 3.32/1.73 | ( ~ Xphi
% 3.32/1.73 & ( Xz = Xy ) ) ) )
% 3.32/1.73 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(iftrueProp1_type,type,
% 3.32/1.73 iftrueProp1: $o ).
% 3.32/1.73
% 3.32/1.73 thf(iftrueProp1,definition,
% 3.32/1.73 ( iftrueProp1
% 3.32/1.73 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ! [Xy: $i] :
% 3.32/1.73 ( ( in @ Xy @ A )
% 3.32/1.73 => ( Xphi
% 3.32/1.73 => ( in @ Xx
% 3.32/1.73 @ ( dsetconstr @ A
% 3.32/1.73 @ ^ [Xz: $i] :
% 3.32/1.73 ( ( Xphi
% 3.32/1.73 & ( Xz = Xx ) )
% 3.32/1.73 | ( ~ Xphi
% 3.32/1.73 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(iftrueProp2_type,type,
% 3.32/1.73 iftrueProp2: $o ).
% 3.32/1.73
% 3.32/1.73 thf(iftrueProp2,definition,
% 3.32/1.73 ( iftrueProp2
% 3.32/1.73 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ! [Xy: $i] :
% 3.32/1.73 ( ( in @ Xy @ A )
% 3.32/1.73 => ( Xphi
% 3.32/1.73 => ( ( dsetconstr @ A
% 3.32/1.73 @ ^ [Xz: $i] :
% 3.32/1.73 ( ( Xphi
% 3.32/1.73 & ( Xz = Xx ) )
% 3.32/1.73 | ( ~ Xphi
% 3.32/1.73 & ( Xz = Xy ) ) ) )
% 3.32/1.73 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(ifSingleton_type,type,
% 3.32/1.73 ifSingleton: $o ).
% 3.32/1.73
% 3.32/1.73 thf(ifSingleton,definition,
% 3.32/1.73 ( ifSingleton
% 3.32/1.73 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ! [Xy: $i] :
% 3.32/1.73 ( ( in @ Xy @ A )
% 3.32/1.73 => ( singleton
% 3.32/1.73 @ ( dsetconstr @ A
% 3.32/1.73 @ ^ [Xz: $i] :
% 3.32/1.73 ( ( Xphi
% 3.32/1.73 & ( Xz = Xx ) )
% 3.32/1.73 | ( ~ Xphi
% 3.32/1.73 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(if_type,type,
% 3.32/1.73 if: $i > $o > $i > $i > $i ).
% 3.32/1.73
% 3.32/1.73 thf(ifp_type,type,
% 3.32/1.73 ifp: $o ).
% 3.32/1.73
% 3.32/1.73 thf(ifp,definition,
% 3.32/1.73 ( ifp
% 3.32/1.73 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ! [Xy: $i] :
% 3.32/1.73 ( ( in @ Xy @ A )
% 3.32/1.73 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(theeq_type,type,
% 3.32/1.73 theeq: $o ).
% 3.32/1.73
% 3.32/1.73 thf(theeq,definition,
% 3.32/1.73 ( theeq
% 3.32/1.73 = ( ! [X: $i] :
% 3.32/1.73 ( ( singleton @ X )
% 3.32/1.73 => ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ X )
% 3.32/1.73 => ( ( setunion @ X )
% 3.32/1.73 = Xx ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(iftrue_type,type,
% 3.32/1.73 iftrue: $o ).
% 3.32/1.73
% 3.32/1.73 thf(iftrue,definition,
% 3.32/1.73 ( iftrue
% 3.32/1.73 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ! [Xy: $i] :
% 3.32/1.73 ( ( in @ Xy @ A )
% 3.32/1.73 => ( Xphi
% 3.32/1.73 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 3.32/1.73 = Xx ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(iffalse_type,type,
% 3.32/1.73 iffalse: $o ).
% 3.32/1.73
% 3.32/1.73 thf(iffalse,definition,
% 3.32/1.73 ( iffalse
% 3.32/1.73 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ! [Xy: $i] :
% 3.32/1.73 ( ( in @ Xy @ A )
% 3.32/1.73 => ( ~ Xphi
% 3.32/1.73 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 3.32/1.73 = Xy ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(iftrueorfalse_type,type,
% 3.32/1.73 iftrueorfalse: $o ).
% 3.32/1.73
% 3.32/1.73 thf(iftrueorfalse,definition,
% 3.32/1.73 ( iftrueorfalse
% 3.32/1.73 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ! [Xy: $i] :
% 3.32/1.73 ( ( in @ Xy @ A )
% 3.32/1.73 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(binintersectT_lem_type,type,
% 3.32/1.73 binintersectT_lem: $o ).
% 3.32/1.73
% 3.32/1.73 thf(binintersectT_lem,definition,
% 3.32/1.73 ( binintersectT_lem
% 3.32/1.73 = ( ! [A: $i,X: $i] :
% 3.32/1.73 ( ( in @ X @ ( powerset @ A ) )
% 3.32/1.73 => ! [Y: $i] :
% 3.32/1.73 ( ( in @ Y @ ( powerset @ A ) )
% 3.32/1.73 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(binunionT_lem_type,type,
% 3.32/1.73 binunionT_lem: $o ).
% 3.32/1.73
% 3.32/1.73 thf(binunionT_lem,definition,
% 3.32/1.73 ( binunionT_lem
% 3.32/1.73 = ( ! [A: $i,X: $i] :
% 3.32/1.73 ( ( in @ X @ ( powerset @ A ) )
% 3.32/1.73 => ! [Y: $i] :
% 3.32/1.73 ( ( in @ Y @ ( powerset @ A ) )
% 3.32/1.73 => ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(powersetT_lem_type,type,
% 3.32/1.73 powersetT_lem: $o ).
% 3.32/1.73
% 3.32/1.73 thf(powersetT_lem,definition,
% 3.32/1.73 ( powersetT_lem
% 3.32/1.73 = ( ! [A: $i,X: $i] :
% 3.32/1.73 ( ( in @ X @ ( powerset @ A ) )
% 3.32/1.73 => ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(setminusT_lem_type,type,
% 3.32/1.73 setminusT_lem: $o ).
% 3.32/1.73
% 3.32/1.73 thf(setminusT_lem,definition,
% 3.32/1.73 ( setminusT_lem
% 3.32/1.73 = ( ! [A: $i,X: $i] :
% 3.32/1.73 ( ( in @ X @ ( powerset @ A ) )
% 3.32/1.73 => ! [Y: $i] :
% 3.32/1.73 ( ( in @ Y @ ( powerset @ A ) )
% 3.32/1.73 => ( in @ ( setminus @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(complementT_lem_type,type,
% 3.32/1.73 complementT_lem: $o ).
% 3.32/1.73
% 3.32/1.73 thf(complementT_lem,definition,
% 3.32/1.73 ( complementT_lem
% 3.32/1.73 = ( ! [A: $i,X: $i] :
% 3.32/1.73 ( ( in @ X @ ( powerset @ A ) )
% 3.32/1.73 => ( in @ ( setminus @ A @ X ) @ ( powerset @ A ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(setextT_type,type,
% 3.32/1.73 setextT: $o ).
% 3.32/1.73
% 3.32/1.73 thf(setextT,definition,
% 3.32/1.73 ( setextT
% 3.32/1.73 = ( ! [A: $i,X: $i] :
% 3.32/1.73 ( ( in @ X @ ( powerset @ A ) )
% 3.32/1.73 => ! [Y: $i] :
% 3.32/1.73 ( ( in @ Y @ ( powerset @ A ) )
% 3.32/1.73 => ( ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ( ( in @ Xx @ X )
% 3.32/1.73 => ( in @ Xx @ Y ) ) )
% 3.32/1.73 => ( ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ( ( in @ Xx @ Y )
% 3.32/1.73 => ( in @ Xx @ X ) ) )
% 3.32/1.73 => ( X = Y ) ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(subsetTI_type,type,
% 3.32/1.73 subsetTI: $o ).
% 3.32/1.73
% 3.32/1.73 thf(subsetTI,definition,
% 3.32/1.73 ( subsetTI
% 3.32/1.73 = ( ! [A: $i,X: $i] :
% 3.32/1.73 ( ( in @ X @ ( powerset @ A ) )
% 3.32/1.73 => ! [Y: $i] :
% 3.32/1.73 ( ( in @ Y @ ( powerset @ A ) )
% 3.32/1.73 => ( ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ( ( in @ Xx @ X )
% 3.32/1.73 => ( in @ Xx @ Y ) ) )
% 3.32/1.73 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(powersetTI1_type,type,
% 3.32/1.73 powersetTI1: $o ).
% 3.32/1.73
% 3.32/1.73 thf(powersetTI1,definition,
% 3.32/1.73 ( powersetTI1
% 3.32/1.73 = ( ! [A: $i,X: $i] :
% 3.32/1.73 ( ( in @ X @ ( powerset @ A ) )
% 3.32/1.73 => ! [Y: $i] :
% 3.32/1.73 ( ( in @ Y @ ( powerset @ A ) )
% 3.32/1.73 => ( ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ( ( in @ Xx @ X )
% 3.32/1.73 => ( in @ Xx @ Y ) ) )
% 3.32/1.73 => ( in @ X @ ( powerset @ Y ) ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(powersetTE1_type,type,
% 3.32/1.73 powersetTE1: $o ).
% 3.32/1.73
% 3.32/1.73 thf(powersetTE1,definition,
% 3.32/1.73 ( powersetTE1
% 3.32/1.73 = ( ! [A: $i,X: $i] :
% 3.32/1.73 ( ( in @ X @ ( powerset @ A ) )
% 3.32/1.73 => ! [Y: $i] :
% 3.32/1.73 ( ( in @ Y @ ( powerset @ A ) )
% 3.32/1.73 => ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ( ( in @ X @ ( powerset @ Y ) )
% 3.32/1.73 => ( ( in @ Xx @ X )
% 3.32/1.73 => ( in @ Xx @ Y ) ) ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(complementTI1_type,type,
% 3.32/1.73 complementTI1: $o ).
% 3.32/1.73
% 3.32/1.73 thf(complementTI1,definition,
% 3.32/1.73 ( complementTI1
% 3.32/1.73 = ( ! [A: $i,X: $i] :
% 3.32/1.73 ( ( in @ X @ ( powerset @ A ) )
% 3.32/1.73 => ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ( ( in @ Xx @ X )
% 3.32/1.73 => ~ ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(complementTE1_type,type,
% 3.32/1.73 complementTE1: $o ).
% 3.32/1.73
% 3.32/1.73 thf(complementTE1,definition,
% 3.32/1.73 ( complementTE1
% 3.32/1.73 = ( ! [A: $i,X: $i] :
% 3.32/1.73 ( ( in @ X @ ( powerset @ A ) )
% 3.32/1.73 => ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ( ~ ( in @ Xx @ ( setminus @ A @ X ) )
% 3.32/1.73 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(binintersectTELcontra_type,type,
% 3.32/1.73 binintersectTELcontra: $o ).
% 3.32/1.73
% 3.32/1.73 thf(binintersectTELcontra,definition,
% 3.32/1.73 ( binintersectTELcontra
% 3.32/1.73 = ( ! [A: $i,X: $i] :
% 3.32/1.73 ( ( in @ X @ ( powerset @ A ) )
% 3.32/1.73 => ! [Y: $i] :
% 3.32/1.73 ( ( in @ Y @ ( powerset @ A ) )
% 3.32/1.73 => ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ( ~ ( in @ Xx @ X )
% 3.32/1.73 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(binintersectTERcontra_type,type,
% 3.32/1.73 binintersectTERcontra: $o ).
% 3.32/1.73
% 3.32/1.73 thf(binintersectTERcontra,definition,
% 3.32/1.73 ( binintersectTERcontra
% 3.32/1.73 = ( ! [A: $i,X: $i] :
% 3.32/1.73 ( ( in @ X @ ( powerset @ A ) )
% 3.32/1.73 => ! [Y: $i] :
% 3.32/1.73 ( ( in @ Y @ ( powerset @ A ) )
% 3.32/1.73 => ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ( ~ ( in @ Xx @ Y )
% 3.32/1.73 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(contrasubsetT_type,type,
% 3.32/1.73 contrasubsetT: $o ).
% 3.32/1.73
% 3.32/1.73 thf(contrasubsetT,definition,
% 3.32/1.73 ( contrasubsetT
% 3.32/1.73 = ( ! [A: $i,X: $i] :
% 3.32/1.73 ( ( in @ X @ ( powerset @ A ) )
% 3.32/1.73 => ! [Y: $i] :
% 3.32/1.73 ( ( in @ Y @ ( powerset @ A ) )
% 3.32/1.73 => ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 3.32/1.73 => ( ( in @ Xx @ Y )
% 3.32/1.73 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(contrasubsetT1_type,type,
% 3.32/1.73 contrasubsetT1: $o ).
% 3.32/1.73
% 3.32/1.73 thf(contrasubsetT1,definition,
% 3.32/1.73 ( contrasubsetT1
% 3.32/1.73 = ( ! [A: $i,X: $i] :
% 3.32/1.73 ( ( in @ X @ ( powerset @ A ) )
% 3.32/1.73 => ! [Y: $i] :
% 3.32/1.73 ( ( in @ Y @ ( powerset @ A ) )
% 3.32/1.73 => ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ( ( subset @ X @ Y )
% 3.32/1.73 => ( ~ ( in @ Xx @ Y )
% 3.32/1.73 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(contrasubsetT2_type,type,
% 3.32/1.73 contrasubsetT2: $o ).
% 3.32/1.73
% 3.32/1.73 thf(contrasubsetT2,definition,
% 3.32/1.73 ( contrasubsetT2
% 3.32/1.73 = ( ! [A: $i,X: $i] :
% 3.32/1.73 ( ( in @ X @ ( powerset @ A ) )
% 3.32/1.73 => ! [Y: $i] :
% 3.32/1.73 ( ( in @ Y @ ( powerset @ A ) )
% 3.32/1.73 => ( ( subset @ X @ Y )
% 3.32/1.73 => ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(contrasubsetT3_type,type,
% 3.32/1.73 contrasubsetT3: $o ).
% 3.32/1.73
% 3.32/1.73 thf(contrasubsetT3,definition,
% 3.32/1.73 ( contrasubsetT3
% 3.32/1.73 = ( ! [A: $i,X: $i] :
% 3.32/1.73 ( ( in @ X @ ( powerset @ A ) )
% 3.32/1.73 => ! [Y: $i] :
% 3.32/1.73 ( ( in @ Y @ ( powerset @ A ) )
% 3.32/1.73 => ( ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) )
% 3.32/1.73 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(doubleComplementI1_type,type,
% 3.32/1.73 doubleComplementI1: $o ).
% 3.32/1.73
% 3.32/1.73 thf(doubleComplementI1,definition,
% 3.32/1.73 ( doubleComplementI1
% 3.32/1.73 = ( ! [A: $i,X: $i] :
% 3.32/1.73 ( ( in @ X @ ( powerset @ A ) )
% 3.32/1.73 => ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ( ( in @ Xx @ X )
% 3.32/1.73 => ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(doubleComplementE1_type,type,
% 3.32/1.73 doubleComplementE1: $o ).
% 3.32/1.73
% 3.32/1.73 thf(doubleComplementE1,definition,
% 3.32/1.73 ( doubleComplementE1
% 3.32/1.73 = ( ! [A: $i,X: $i] :
% 3.32/1.73 ( ( in @ X @ ( powerset @ A ) )
% 3.32/1.73 => ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ( ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) )
% 3.32/1.73 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(doubleComplementSub1_type,type,
% 3.32/1.73 doubleComplementSub1: $o ).
% 3.32/1.73
% 3.32/1.73 thf(doubleComplementSub1,definition,
% 3.32/1.73 ( doubleComplementSub1
% 3.32/1.73 = ( ! [A: $i,X: $i] :
% 3.32/1.73 ( ( in @ X @ ( powerset @ A ) )
% 3.32/1.73 => ( subset @ X @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(doubleComplementSub2_type,type,
% 3.32/1.73 doubleComplementSub2: $o ).
% 3.32/1.73
% 3.32/1.73 thf(doubleComplementSub2,definition,
% 3.32/1.73 ( doubleComplementSub2
% 3.32/1.73 = ( ! [A: $i,X: $i] :
% 3.32/1.73 ( ( in @ X @ ( powerset @ A ) )
% 3.32/1.73 => ( subset @ ( setminus @ A @ ( setminus @ A @ X ) ) @ X ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(doubleComplementEq_type,type,
% 3.32/1.73 doubleComplementEq: $o ).
% 3.32/1.73
% 3.32/1.73 thf(doubleComplementEq,definition,
% 3.32/1.73 ( doubleComplementEq
% 3.32/1.73 = ( ! [A: $i,X: $i] :
% 3.32/1.73 ( ( in @ X @ ( powerset @ A ) )
% 3.32/1.73 => ( X
% 3.32/1.73 = ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(complementTnotintersectT_type,type,
% 3.32/1.73 complementTnotintersectT: $o ).
% 3.32/1.73
% 3.32/1.73 thf(complementTnotintersectT,definition,
% 3.32/1.73 ( complementTnotintersectT
% 3.32/1.73 = ( ! [A: $i,X: $i] :
% 3.32/1.73 ( ( in @ X @ ( powerset @ A ) )
% 3.32/1.73 => ! [Y: $i] :
% 3.32/1.73 ( ( in @ Y @ ( powerset @ A ) )
% 3.32/1.73 => ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 3.32/1.73 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(complementImpComplementIntersect_type,type,
% 3.32/1.73 complementImpComplementIntersect: $o ).
% 3.32/1.73
% 3.32/1.73 thf(complementImpComplementIntersect,definition,
% 3.32/1.73 ( complementImpComplementIntersect
% 3.32/1.73 = ( ! [A: $i,X: $i] :
% 3.32/1.73 ( ( in @ X @ ( powerset @ A ) )
% 3.32/1.73 => ! [Y: $i] :
% 3.32/1.73 ( ( in @ Y @ ( powerset @ A ) )
% 3.32/1.73 => ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 3.32/1.73 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(complementSubsetComplementIntersect_type,type,
% 3.32/1.73 complementSubsetComplementIntersect: $o ).
% 3.32/1.73
% 3.32/1.73 thf(complementSubsetComplementIntersect,definition,
% 3.32/1.73 ( complementSubsetComplementIntersect
% 3.32/1.73 = ( ! [A: $i,X: $i] :
% 3.32/1.73 ( ( in @ X @ ( powerset @ A ) )
% 3.32/1.73 => ! [Y: $i] :
% 3.32/1.73 ( ( in @ Y @ ( powerset @ A ) )
% 3.32/1.73 => ( subset @ ( setminus @ A @ X ) @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(complementInPowersetComplementIntersect_type,type,
% 3.32/1.73 complementInPowersetComplementIntersect: $o ).
% 3.32/1.73
% 3.32/1.73 thf(complementInPowersetComplementIntersect,definition,
% 3.32/1.73 ( complementInPowersetComplementIntersect
% 3.32/1.73 = ( ! [A: $i,X: $i] :
% 3.32/1.73 ( ( in @ X @ ( powerset @ A ) )
% 3.32/1.73 => ! [Y: $i] :
% 3.32/1.73 ( ( in @ Y @ ( powerset @ A ) )
% 3.32/1.73 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(contraSubsetComplement_type,type,
% 3.32/1.73 contraSubsetComplement: $o ).
% 3.32/1.73
% 3.32/1.73 thf(contraSubsetComplement,definition,
% 3.32/1.73 ( contraSubsetComplement
% 3.32/1.73 = ( ! [A: $i,X: $i] :
% 3.32/1.73 ( ( in @ X @ ( powerset @ A ) )
% 3.32/1.73 => ! [Y: $i] :
% 3.32/1.73 ( ( in @ Y @ ( powerset @ A ) )
% 3.32/1.73 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 3.32/1.73 => ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ( ( in @ Xx @ Y )
% 3.32/1.73 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(complementTcontraSubset_type,type,
% 3.32/1.73 complementTcontraSubset: $o ).
% 3.32/1.73
% 3.32/1.73 thf(complementTcontraSubset,definition,
% 3.32/1.73 ( complementTcontraSubset
% 3.32/1.73 = ( ! [A: $i,X: $i] :
% 3.32/1.73 ( ( in @ X @ ( powerset @ A ) )
% 3.32/1.73 => ! [Y: $i] :
% 3.32/1.73 ( ( in @ Y @ ( powerset @ A ) )
% 3.32/1.73 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 3.32/1.73 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(binunionTILcontra_type,type,
% 3.32/1.73 binunionTILcontra: $o ).
% 3.32/1.73
% 3.32/1.73 thf(binunionTILcontra,definition,
% 3.32/1.73 ( binunionTILcontra
% 3.32/1.73 = ( ! [A: $i,X: $i] :
% 3.32/1.73 ( ( in @ X @ ( powerset @ A ) )
% 3.32/1.73 => ! [Y: $i] :
% 3.32/1.73 ( ( in @ Y @ ( powerset @ A ) )
% 3.32/1.73 => ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 3.32/1.73 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(binunionTIRcontra_type,type,
% 3.32/1.73 binunionTIRcontra: $o ).
% 3.32/1.73
% 3.32/1.73 thf(binunionTIRcontra,definition,
% 3.32/1.73 ( binunionTIRcontra
% 3.32/1.73 = ( ! [A: $i,X: $i] :
% 3.32/1.73 ( ( in @ X @ ( powerset @ A ) )
% 3.32/1.73 => ! [Y: $i] :
% 3.32/1.73 ( ( in @ Y @ ( powerset @ A ) )
% 3.32/1.73 => ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 3.32/1.73 => ~ ( in @ Xx @ Y ) ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(inIntersectImpInUnion_type,type,
% 3.32/1.73 inIntersectImpInUnion: $o ).
% 3.32/1.73
% 3.32/1.73 thf(inIntersectImpInUnion,definition,
% 3.32/1.73 ( inIntersectImpInUnion
% 3.32/1.73 = ( ! [A: $i,X: $i] :
% 3.32/1.73 ( ( in @ X @ ( powerset @ A ) )
% 3.32/1.73 => ! [Y: $i] :
% 3.32/1.73 ( ( in @ Y @ ( powerset @ A ) )
% 3.32/1.73 => ! [Z: $i] :
% 3.32/1.73 ( ( in @ Z @ ( powerset @ A ) )
% 3.32/1.73 => ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 3.32/1.73 => ( in @ Xx @ ( binunion @ X @ Z ) ) ) ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(inIntersectImpInUnion2_type,type,
% 3.32/1.73 inIntersectImpInUnion2: $o ).
% 3.32/1.73
% 3.32/1.73 thf(inIntersectImpInUnion2,definition,
% 3.32/1.73 ( inIntersectImpInUnion2
% 3.32/1.73 = ( ! [A: $i,X: $i] :
% 3.32/1.73 ( ( in @ X @ ( powerset @ A ) )
% 3.32/1.73 => ! [Y: $i] :
% 3.32/1.73 ( ( in @ Y @ ( powerset @ A ) )
% 3.32/1.73 => ! [Z: $i] :
% 3.32/1.73 ( ( in @ Z @ ( powerset @ A ) )
% 3.32/1.73 => ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 3.32/1.73 => ( in @ Xx @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(inIntersectImpInIntersectUnions_type,type,
% 3.32/1.73 inIntersectImpInIntersectUnions: $o ).
% 3.32/1.73
% 3.32/1.73 thf(inIntersectImpInIntersectUnions,definition,
% 3.32/1.73 ( inIntersectImpInIntersectUnions
% 3.32/1.73 = ( ! [A: $i,X: $i] :
% 3.32/1.73 ( ( in @ X @ ( powerset @ A ) )
% 3.32/1.73 => ! [Y: $i] :
% 3.32/1.73 ( ( in @ Y @ ( powerset @ A ) )
% 3.32/1.73 => ! [Z: $i] :
% 3.32/1.73 ( ( in @ Z @ ( powerset @ A ) )
% 3.32/1.73 => ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 3.32/1.73 => ( in @ Xx @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(intersectInPowersetIntersectUnions_type,type,
% 3.32/1.73 intersectInPowersetIntersectUnions: $o ).
% 3.32/1.73
% 3.32/1.73 thf(intersectInPowersetIntersectUnions,definition,
% 3.32/1.73 ( intersectInPowersetIntersectUnions
% 3.32/1.73 = ( ! [A: $i,X: $i] :
% 3.32/1.73 ( ( in @ X @ ( powerset @ A ) )
% 3.32/1.73 => ! [Y: $i] :
% 3.32/1.73 ( ( in @ Y @ ( powerset @ A ) )
% 3.32/1.73 => ! [Z: $i] :
% 3.32/1.73 ( ( in @ Z @ ( powerset @ A ) )
% 3.32/1.73 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(inComplementUnionImpNotIn1_type,type,
% 3.32/1.73 inComplementUnionImpNotIn1: $o ).
% 3.32/1.73
% 3.32/1.73 thf(inComplementUnionImpNotIn1,definition,
% 3.32/1.73 ( inComplementUnionImpNotIn1
% 3.32/1.73 = ( ! [A: $i,X: $i] :
% 3.32/1.73 ( ( in @ X @ ( powerset @ A ) )
% 3.32/1.73 => ! [Y: $i] :
% 3.32/1.73 ( ( in @ Y @ ( powerset @ A ) )
% 3.32/1.73 => ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 3.32/1.73 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(inComplementUnionImpInComplement1_type,type,
% 3.32/1.73 inComplementUnionImpInComplement1: $o ).
% 3.32/1.73
% 3.32/1.73 thf(inComplementUnionImpInComplement1,definition,
% 3.32/1.73 ( inComplementUnionImpInComplement1
% 3.32/1.73 = ( ! [A: $i,X: $i] :
% 3.32/1.73 ( ( in @ X @ ( powerset @ A ) )
% 3.32/1.73 => ! [Y: $i] :
% 3.32/1.73 ( ( in @ Y @ ( powerset @ A ) )
% 3.32/1.73 => ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 3.32/1.73 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(binunionTE_type,type,
% 3.32/1.73 binunionTE: $o ).
% 3.32/1.73
% 3.32/1.73 thf(binunionTE,definition,
% 3.32/1.73 ( binunionTE
% 3.32/1.73 = ( ! [A: $i,X: $i] :
% 3.32/1.73 ( ( in @ X @ ( powerset @ A ) )
% 3.32/1.73 => ! [Y: $i] :
% 3.32/1.73 ( ( in @ Y @ ( powerset @ A ) )
% 3.32/1.73 => ! [Xphi: $o,Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ( ( in @ Xx @ ( binunion @ X @ Y ) )
% 3.32/1.73 => ( ( ( in @ Xx @ X )
% 3.32/1.73 => Xphi )
% 3.32/1.73 => ( ( ( in @ Xx @ Y )
% 3.32/1.73 => Xphi )
% 3.32/1.73 => Xphi ) ) ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(binunionTEcontra_type,type,
% 3.32/1.73 binunionTEcontra: $o ).
% 3.32/1.73
% 3.32/1.73 thf(binunionTEcontra,definition,
% 3.32/1.73 ( binunionTEcontra
% 3.32/1.73 = ( ! [A: $i,X: $i] :
% 3.32/1.73 ( ( in @ X @ ( powerset @ A ) )
% 3.32/1.73 => ! [Y: $i] :
% 3.32/1.73 ( ( in @ Y @ ( powerset @ A ) )
% 3.32/1.73 => ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ( ~ ( in @ Xx @ X )
% 3.32/1.73 => ( ~ ( in @ Xx @ Y )
% 3.32/1.73 => ~ ( in @ Xx @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(demorgan2a1_type,type,
% 3.32/1.73 demorgan2a1: $o ).
% 3.32/1.73
% 3.32/1.73 thf(demorgan2a1,definition,
% 3.32/1.73 ( demorgan2a1
% 3.32/1.73 = ( ! [A: $i,X: $i] :
% 3.32/1.73 ( ( in @ X @ ( powerset @ A ) )
% 3.32/1.73 => ! [Y: $i] :
% 3.32/1.73 ( ( in @ Y @ ( powerset @ A ) )
% 3.32/1.73 => ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 3.32/1.73 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(complementUnionInPowersetComplement_type,type,
% 3.32/1.73 complementUnionInPowersetComplement: $o ).
% 3.32/1.73
% 3.32/1.73 thf(complementUnionInPowersetComplement,definition,
% 3.32/1.73 ( complementUnionInPowersetComplement
% 3.32/1.73 = ( ! [A: $i,X: $i] :
% 3.32/1.73 ( ( in @ X @ ( powerset @ A ) )
% 3.32/1.73 => ! [Y: $i] :
% 3.32/1.73 ( ( in @ Y @ ( powerset @ A ) )
% 3.32/1.73 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(demorgan2a2_type,type,
% 3.32/1.73 demorgan2a2: $o ).
% 3.32/1.73
% 3.32/1.73 thf(demorgan2a2,definition,
% 3.32/1.73 ( demorgan2a2
% 3.32/1.73 = ( ! [A: $i,X: $i] :
% 3.32/1.73 ( ( in @ X @ ( powerset @ A ) )
% 3.32/1.73 => ! [Y: $i] :
% 3.32/1.73 ( ( in @ Y @ ( powerset @ A ) )
% 3.32/1.73 => ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 3.32/1.73 => ( in @ Xx @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(demorgan1a_type,type,
% 3.32/1.73 demorgan1a: $o ).
% 3.32/1.73
% 3.32/1.73 thf(demorgan1a,definition,
% 3.32/1.73 ( demorgan1a
% 3.32/1.73 = ( ! [A: $i,X: $i] :
% 3.32/1.73 ( ( in @ X @ ( powerset @ A ) )
% 3.32/1.73 => ! [Y: $i] :
% 3.32/1.73 ( ( in @ Y @ ( powerset @ A ) )
% 3.32/1.73 => ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ( ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) )
% 3.32/1.73 => ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(demorgan1b_type,type,
% 3.32/1.73 demorgan1b: $o ).
% 3.32/1.73
% 3.32/1.73 thf(demorgan1b,definition,
% 3.32/1.73 ( demorgan1b
% 3.32/1.73 = ( ! [A: $i,X: $i] :
% 3.32/1.73 ( ( in @ X @ ( powerset @ A ) )
% 3.32/1.73 => ! [Y: $i] :
% 3.32/1.73 ( ( in @ Y @ ( powerset @ A ) )
% 3.32/1.73 => ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ( ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 3.32/1.73 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(demorgan1_type,type,
% 3.32/1.73 demorgan1: $o ).
% 3.32/1.73
% 3.32/1.73 thf(demorgan1,definition,
% 3.32/1.73 ( demorgan1
% 3.32/1.73 = ( ! [A: $i,X: $i] :
% 3.32/1.73 ( ( in @ X @ ( powerset @ A ) )
% 3.32/1.73 => ! [Y: $i] :
% 3.32/1.73 ( ( in @ Y @ ( powerset @ A ) )
% 3.32/1.73 => ( ( setminus @ A @ ( binintersect @ X @ Y ) )
% 3.32/1.73 = ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(demorgan2a_type,type,
% 3.32/1.73 demorgan2a: $o ).
% 3.32/1.73
% 3.32/1.73 thf(demorgan2a,definition,
% 3.32/1.73 ( demorgan2a
% 3.32/1.73 = ( ! [A: $i,X: $i] :
% 3.32/1.73 ( ( in @ X @ ( powerset @ A ) )
% 3.32/1.73 => ! [Y: $i] :
% 3.32/1.73 ( ( in @ Y @ ( powerset @ A ) )
% 3.32/1.73 => ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 3.32/1.73 => ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(demorgan2b2_type,type,
% 3.32/1.73 demorgan2b2: $o ).
% 3.32/1.73
% 3.32/1.73 thf(demorgan2b2,definition,
% 3.32/1.73 ( demorgan2b2
% 3.32/1.73 = ( ! [A: $i,X: $i] :
% 3.32/1.73 ( ( in @ X @ ( powerset @ A ) )
% 3.32/1.73 => ! [Y: $i] :
% 3.32/1.73 ( ( in @ Y @ ( powerset @ A ) )
% 3.32/1.73 => ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 3.32/1.73 => ( ( in @ Xx @ ( setminus @ A @ Y ) )
% 3.32/1.73 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(demorgan2b_type,type,
% 3.32/1.73 demorgan2b: $o ).
% 3.32/1.73
% 3.32/1.73 thf(demorgan2b,definition,
% 3.32/1.73 ( demorgan2b
% 3.32/1.73 = ( ! [A: $i,X: $i] :
% 3.32/1.73 ( ( in @ X @ ( powerset @ A ) )
% 3.32/1.73 => ! [Y: $i] :
% 3.32/1.73 ( ( in @ Y @ ( powerset @ A ) )
% 3.32/1.73 => ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ( ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 3.32/1.73 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(demorgan2_type,type,
% 3.32/1.73 demorgan2: $o ).
% 3.32/1.73
% 3.32/1.73 thf(demorgan2,definition,
% 3.32/1.73 ( demorgan2
% 3.32/1.73 = ( ! [A: $i,X: $i] :
% 3.32/1.73 ( ( in @ X @ ( powerset @ A ) )
% 3.32/1.73 => ! [Y: $i] :
% 3.32/1.73 ( ( in @ Y @ ( powerset @ A ) )
% 3.32/1.73 => ( ( setminus @ A @ ( binunion @ X @ Y ) )
% 3.32/1.73 = ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(woz13rule0_type,type,
% 3.32/1.73 woz13rule0: $o ).
% 3.32/1.73
% 3.32/1.73 thf(woz13rule0,definition,
% 3.32/1.73 ( woz13rule0
% 3.32/1.73 = ( ! [A: $i,X: $i] :
% 3.32/1.73 ( ( in @ X @ ( powerset @ A ) )
% 3.32/1.73 => ! [Y: $i] :
% 3.32/1.73 ( ( in @ Y @ ( powerset @ A ) )
% 3.32/1.73 => ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 3.32/1.73 => ( in @ Xx @ A ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(woz13rule1_type,type,
% 3.32/1.73 woz13rule1: $o ).
% 3.32/1.73
% 3.32/1.73 thf(woz13rule1,definition,
% 3.32/1.73 ( woz13rule1
% 3.32/1.73 = ( ! [A: $i,X: $i] :
% 3.32/1.73 ( ( in @ X @ ( powerset @ A ) )
% 3.32/1.73 => ! [Y: $i] :
% 3.32/1.73 ( ( in @ Y @ ( powerset @ A ) )
% 3.32/1.73 => ! [Z: $i] :
% 3.32/1.73 ( ( in @ Z @ ( powerset @ A ) )
% 3.32/1.73 => ( ( subset @ X @ Z )
% 3.32/1.73 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(woz13rule2_type,type,
% 3.32/1.73 woz13rule2: $o ).
% 3.32/1.73
% 3.32/1.73 thf(woz13rule2,definition,
% 3.32/1.73 ( woz13rule2
% 3.32/1.73 = ( ! [A: $i,X: $i] :
% 3.32/1.73 ( ( in @ X @ ( powerset @ A ) )
% 3.32/1.73 => ! [Y: $i] :
% 3.32/1.73 ( ( in @ Y @ ( powerset @ A ) )
% 3.32/1.73 => ! [Z: $i] :
% 3.32/1.73 ( ( in @ Z @ ( powerset @ A ) )
% 3.32/1.73 => ( ( subset @ Y @ Z )
% 3.32/1.73 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(woz13rule3_type,type,
% 3.32/1.73 woz13rule3: $o ).
% 3.32/1.73
% 3.32/1.73 thf(woz13rule3,definition,
% 3.32/1.73 ( woz13rule3
% 3.32/1.73 = ( ! [A: $i,X: $i] :
% 3.32/1.73 ( ( in @ X @ ( powerset @ A ) )
% 3.32/1.73 => ! [Y: $i] :
% 3.32/1.73 ( ( in @ Y @ ( powerset @ A ) )
% 3.32/1.73 => ! [Z: $i] :
% 3.32/1.73 ( ( in @ Z @ ( powerset @ A ) )
% 3.32/1.73 => ( ( subset @ X @ Y )
% 3.32/1.73 => ( ( subset @ X @ Z )
% 3.32/1.73 => ( subset @ X @ ( binintersect @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(woz13rule4_type,type,
% 3.32/1.73 woz13rule4: $o ).
% 3.32/1.73
% 3.32/1.73 thf(woz13rule4,definition,
% 3.32/1.73 ( woz13rule4
% 3.32/1.73 = ( ! [A: $i,X: $i] :
% 3.32/1.73 ( ( in @ X @ ( powerset @ A ) )
% 3.32/1.73 => ! [Y: $i] :
% 3.32/1.73 ( ( in @ Y @ ( powerset @ A ) )
% 3.32/1.73 => ! [Z: $i] :
% 3.32/1.73 ( ( in @ Z @ ( powerset @ A ) )
% 3.32/1.73 => ! [W: $i] :
% 3.32/1.73 ( ( in @ W @ ( powerset @ A ) )
% 3.32/1.73 => ( ( subset @ X @ Z )
% 3.32/1.73 => ( ( subset @ Y @ W )
% 3.32/1.73 => ( subset @ ( binintersect @ X @ Y ) @ ( binintersect @ Z @ W ) ) ) ) ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(woz1_1_type,type,
% 3.32/1.73 woz1_1: $o ).
% 3.32/1.73
% 3.32/1.73 thf(woz1_1,definition,
% 3.32/1.73 ( woz1_1
% 3.32/1.73 = ( ! [A: $i,X: $i] :
% 3.32/1.73 ( ( in @ X @ ( powerset @ A ) )
% 3.32/1.73 => ! [Y: $i] :
% 3.32/1.73 ( ( in @ Y @ ( powerset @ A ) )
% 3.32/1.73 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(woz1_2_type,type,
% 3.32/1.73 woz1_2: $o ).
% 3.32/1.73
% 3.32/1.73 thf(woz1_2,definition,
% 3.32/1.73 ( woz1_2
% 3.32/1.73 = ( ! [A: $i,X: $i] :
% 3.32/1.73 ( ( in @ X @ ( powerset @ A ) )
% 3.32/1.73 => ! [Y: $i] :
% 3.32/1.73 ( ( in @ Y @ ( powerset @ A ) )
% 3.32/1.73 => ! [Z: $i] :
% 3.32/1.73 ( ( in @ Z @ ( powerset @ A ) )
% 3.32/1.73 => ! [W: $i] :
% 3.32/1.73 ( ( in @ W @ ( powerset @ A ) )
% 3.32/1.73 => ( ( setminus @ A @ ( binintersect @ ( binunion @ X @ Y ) @ ( binunion @ Z @ W ) ) )
% 3.32/1.73 = ( binunion @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) @ ( binintersect @ ( setminus @ A @ Z ) @ ( setminus @ A @ W ) ) ) ) ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(woz1_3_type,type,
% 3.32/1.73 woz1_3: $o ).
% 3.32/1.73
% 3.32/1.73 thf(woz1_3,definition,
% 3.32/1.73 ( woz1_3
% 3.32/1.73 = ( ! [A: $i,X: $i] :
% 3.32/1.73 ( ( in @ X @ ( powerset @ A ) )
% 3.32/1.73 => ! [Y: $i] :
% 3.32/1.73 ( ( in @ Y @ ( powerset @ A ) )
% 3.32/1.73 => ! [Z: $i] :
% 3.32/1.73 ( ( in @ Z @ ( powerset @ A ) )
% 3.32/1.73 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(woz1_4_type,type,
% 3.32/1.73 woz1_4: $o ).
% 3.32/1.73
% 3.32/1.73 thf(woz1_4,definition,
% 3.32/1.73 ( woz1_4
% 3.32/1.73 = ( ! [A: $i,X: $i] :
% 3.32/1.73 ( ( in @ X @ ( powerset @ A ) )
% 3.32/1.73 => ! [Y: $i] :
% 3.32/1.73 ( ( in @ Y @ ( powerset @ A ) )
% 3.32/1.73 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 3.32/1.73 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(woz1_5_type,type,
% 3.32/1.73 woz1_5: $o ).
% 3.32/1.73
% 3.32/1.73 thf(woz1_5,definition,
% 3.32/1.73 ( woz1_5
% 3.32/1.73 = ( ! [A: $i,X: $i] :
% 3.32/1.73 ( ( in @ X @ ( powerset @ A ) )
% 3.32/1.73 => ! [Y: $i] :
% 3.32/1.73 ( ( in @ Y @ ( powerset @ A ) )
% 3.32/1.73 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(breln1_type,type,
% 3.32/1.73 breln1: $i > $i > $o ).
% 3.32/1.73
% 3.32/1.73 thf(breln1all2_type,type,
% 3.32/1.73 breln1all2: $o ).
% 3.32/1.73
% 3.32/1.73 thf(breln1all2,definition,
% 3.32/1.73 ( breln1all2
% 3.32/1.73 = ( ! [A: $i,R: $i] :
% 3.32/1.73 ( ( breln1 @ A @ R )
% 3.32/1.73 => ! [Xphi: $i > $o] :
% 3.32/1.73 ( ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ! [Xy: $i] :
% 3.32/1.73 ( ( in @ Xy @ A )
% 3.32/1.73 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.32/1.73 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 3.32/1.73 => ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ R )
% 3.32/1.73 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(breln1Set_type,type,
% 3.32/1.73 breln1Set: $i > $i ).
% 3.32/1.73
% 3.32/1.73 thf(breln1SetBreln1_type,type,
% 3.32/1.73 breln1SetBreln1: $o ).
% 3.32/1.73
% 3.32/1.73 thf(breln1SetBreln1,definition,
% 3.32/1.73 ( breln1SetBreln1
% 3.32/1.73 = ( ! [A: $i,R: $i] :
% 3.32/1.73 ( ( in @ R @ ( breln1Set @ A ) )
% 3.32/1.73 => ( breln1 @ A @ R ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(transitive_type,type,
% 3.32/1.73 transitive: $i > $i > $o ).
% 3.32/1.73
% 3.32/1.73 thf(antisymmetric_type,type,
% 3.32/1.73 antisymmetric: $i > $i > $o ).
% 3.32/1.73
% 3.32/1.73 thf(reflexive_type,type,
% 3.32/1.73 reflexive: $i > $i > $o ).
% 3.32/1.73
% 3.32/1.73 thf(refltransitive_type,type,
% 3.32/1.73 refltransitive: $i > $i > $o ).
% 3.32/1.73
% 3.32/1.73 thf(refllinearorder_type,type,
% 3.32/1.73 refllinearorder: $i > $i > $o ).
% 3.32/1.73
% 3.32/1.73 thf(reflwellordering_type,type,
% 3.32/1.73 reflwellordering: $i > $i > $o ).
% 3.32/1.73
% 3.32/1.73 thf(choice2fnsingleton_type,type,
% 3.32/1.73 choice2fnsingleton: $o ).
% 3.32/1.73
% 3.32/1.73 thf(choice2fnsingleton,definition,
% 3.32/1.73 ( choice2fnsingleton
% 3.32/1.73 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 3.32/1.73 ( ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ? [Xy: $i] :
% 3.32/1.73 ( ( in @ Xy @ B )
% 3.32/1.73 & ( Xphi @ Xx @ Xy ) ) )
% 3.32/1.73 => ! [R: $i] :
% 3.32/1.73 ( ( in @ R @ ( breln1Set @ B ) )
% 3.32/1.73 => ( ( reflwellordering @ B @ R )
% 3.32/1.73 => ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ( singleton
% 3.32/1.73 @ ( dsetconstr @ B
% 3.32/1.73 @ ^ [Xy: $i] :
% 3.32/1.73 ( ( Xphi @ Xx @ Xy )
% 3.32/1.73 & ! [Xz: $i] :
% 3.32/1.73 ( ( in @ Xz @ B )
% 3.32/1.73 => ( ( Xphi @ Xx @ Xz )
% 3.32/1.73 => ( in @ ( kpair @ Xy @ Xz ) @ R ) ) ) ) ) ) ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(setOfPairsIsBReln1_type,type,
% 3.32/1.73 setOfPairsIsBReln1: $o ).
% 3.32/1.73
% 3.32/1.73 thf(setOfPairsIsBReln1,definition,
% 3.32/1.73 ( setOfPairsIsBReln1
% 3.32/1.73 = ( ! [A: $i,Xphi: $i > $i > $o] :
% 3.32/1.73 ( breln1 @ A
% 3.32/1.73 @ ( dpsetconstr @ A @ A
% 3.32/1.73 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(breln1all1_type,type,
% 3.32/1.73 breln1all1: $o ).
% 3.32/1.73
% 3.32/1.73 thf(breln1all1,definition,
% 3.32/1.73 ( breln1all1
% 3.32/1.73 = ( ! [A: $i,R: $i] :
% 3.32/1.73 ( ( breln1 @ A @ R )
% 3.32/1.73 => ! [Xphi: $i > $o] :
% 3.32/1.73 ( ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ! [Xy: $i] :
% 3.32/1.73 ( ( in @ Xy @ A )
% 3.32/1.73 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.32/1.73 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 3.32/1.73 => ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ R )
% 3.32/1.73 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(subbreln1_type,type,
% 3.32/1.73 subbreln1: $o ).
% 3.32/1.73
% 3.32/1.73 thf(subbreln1,definition,
% 3.32/1.73 ( subbreln1
% 3.32/1.73 = ( ! [A: $i,R: $i] :
% 3.32/1.73 ( ( breln1 @ A @ R )
% 3.32/1.73 => ! [S: $i] :
% 3.32/1.73 ( ( breln1 @ A @ S )
% 3.32/1.73 => ( ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ! [Xy: $i] :
% 3.32/1.73 ( ( in @ Xy @ A )
% 3.32/1.73 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.32/1.73 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 3.32/1.73 => ( subset @ R @ S ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(eqbreln1_type,type,
% 3.32/1.73 eqbreln1: $o ).
% 3.32/1.73
% 3.32/1.73 thf(eqbreln1,definition,
% 3.32/1.73 ( eqbreln1
% 3.32/1.73 = ( ! [A: $i,R: $i] :
% 3.32/1.73 ( ( breln1 @ A @ R )
% 3.32/1.73 => ! [S: $i] :
% 3.32/1.73 ( ( breln1 @ A @ S )
% 3.32/1.73 => ( ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ! [Xy: $i] :
% 3.32/1.73 ( ( in @ Xy @ A )
% 3.32/1.73 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.32/1.73 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 3.32/1.73 => ( ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ! [Xy: $i] :
% 3.32/1.73 ( ( in @ Xy @ A )
% 3.32/1.73 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 3.32/1.73 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 3.32/1.73 => ( R = S ) ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(breln1invset_type,type,
% 3.32/1.73 breln1invset: $i > $i > $i ).
% 3.32/1.73
% 3.32/1.73 thf(breln1invprop_type,type,
% 3.32/1.73 breln1invprop: $o ).
% 3.32/1.73
% 3.32/1.73 thf(breln1invprop,definition,
% 3.32/1.73 ( breln1invprop
% 3.32/1.73 = ( ! [A: $i,R: $i] :
% 3.32/1.73 ( ( breln1 @ A @ R )
% 3.32/1.73 => ( breln1 @ A @ ( breln1invset @ A @ R ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(breln1invI_type,type,
% 3.32/1.73 breln1invI: $o ).
% 3.32/1.73
% 3.32/1.73 thf(breln1invI,definition,
% 3.32/1.73 ( breln1invI
% 3.32/1.73 = ( ! [A: $i,R: $i] :
% 3.32/1.73 ( ( breln1 @ A @ R )
% 3.32/1.73 => ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ! [Xy: $i] :
% 3.32/1.73 ( ( in @ Xy @ A )
% 3.32/1.73 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.32/1.73 => ( in @ ( kpair @ Xy @ Xx ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(breln1invE_type,type,
% 3.32/1.73 breln1invE: $o ).
% 3.32/1.73
% 3.32/1.73 thf(breln1invE,definition,
% 3.32/1.73 ( breln1invE
% 3.32/1.73 = ( ! [A: $i,R: $i] :
% 3.32/1.73 ( ( breln1 @ A @ R )
% 3.32/1.73 => ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ! [Xy: $i] :
% 3.32/1.73 ( ( in @ Xy @ A )
% 3.32/1.73 => ( ( in @ ( kpair @ Xy @ Xx ) @ ( breln1invset @ A @ R ) )
% 3.32/1.73 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(breln1compset_type,type,
% 3.32/1.73 breln1compset: $i > $i > $i > $i ).
% 3.32/1.73
% 3.32/1.73 thf(breln1compprop_type,type,
% 3.32/1.73 breln1compprop: $o ).
% 3.32/1.73
% 3.32/1.73 thf(breln1compprop,definition,
% 3.32/1.73 ( breln1compprop
% 3.32/1.73 = ( ! [A: $i,R: $i] :
% 3.32/1.73 ( ( breln1 @ A @ R )
% 3.32/1.73 => ! [S: $i] :
% 3.32/1.73 ( ( breln1 @ A @ S )
% 3.32/1.73 => ( breln1 @ A @ ( breln1compset @ A @ R @ S ) ) ) ) ) ) ).
% 3.32/1.73
% 3.32/1.73 thf(breln1compI_type,type,
% 3.32/1.73 breln1compI: $o ).
% 3.32/1.73
% 3.32/1.73 thf(breln1compI,definition,
% 3.32/1.73 ( breln1compI
% 3.32/1.73 = ( ! [A: $i,R: $i] :
% 3.32/1.73 ( ( breln1 @ A @ R )
% 3.32/1.73 => ! [S: $i] :
% 3.32/1.73 ( ( breln1 @ A @ S )
% 3.32/1.73 => ! [Xx: $i] :
% 3.32/1.73 ( ( in @ Xx @ A )
% 3.32/1.73 => ! [Xy: $i] :
% 3.32/1.73 ( ( in @ Xy @ A )
% 3.32/1.73 => ! [Xz: $i] :
% 3.32/1.73 ( ( in @ Xz @ A )
% 3.32/1.73 => ( ( in @ ( kpair @ Xx @ Xz ) @ R )
% 3.32/1.73 => ( ( in @ ( kpair @ Xz @ Xy ) @ S )
% 3.32/1.74 => ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) ) ) ) ) ) ) ) ) ) ) ).
% 3.32/1.74
% 3.32/1.74 thf(breln1compE_type,type,
% 3.32/1.74 breln1compE: $o ).
% 3.32/1.74
% 3.32/1.74 thf(breln1compE,definition,
% 3.32/1.74 ( breln1compE
% 3.32/1.74 = ( ! [A: $i,R: $i] :
% 3.32/1.74 ( ( breln1 @ A @ R )
% 3.32/1.74 => ! [S: $i] :
% 3.32/1.74 ( ( breln1 @ A @ S )
% 3.32/1.74 => ! [Xx: $i] :
% 3.32/1.74 ( ( in @ Xx @ A )
% 3.32/1.74 => ! [Xy: $i] :
% 3.32/1.74 ( ( in @ Xy @ A )
% 3.32/1.74 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) )
% 3.32/1.74 => ? [Xz: $i] :
% 3.32/1.74 ( ( in @ Xz @ A )
% 3.32/1.74 & ( in @ ( kpair @ Xx @ Xz ) @ R )
% 3.32/1.74 & ( in @ ( kpair @ Xz @ Xy ) @ S ) ) ) ) ) ) ) ) ) ).
% 3.32/1.74
% 3.32/1.74 thf(breln1compEex_type,type,
% 3.32/1.74 breln1compEex: $o ).
% 3.32/1.74
% 3.32/1.74 thf(breln1compEex,definition,
% 3.32/1.74 ( breln1compEex
% 3.32/1.74 = ( ! [A: $i,R: $i] :
% 3.32/1.74 ( ( breln1 @ A @ R )
% 3.32/1.74 => ! [S: $i] :
% 3.32/1.74 ( ( breln1 @ A @ S )
% 3.32/1.74 => ! [Xx: $i] :
% 3.32/1.74 ( ( in @ Xx @ A )
% 3.32/1.74 => ! [Xy: $i] :
% 3.32/1.74 ( ( in @ Xy @ A )
% 3.32/1.74 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) )
% 3.32/1.74 => ! [Xphi: $o] :
% 3.32/1.74 ( ! [Xz: $i] :
% 3.32/1.74 ( ( in @ Xz @ A )
% 3.32/1.74 => ( ( in @ ( kpair @ Xx @ Xz ) @ R )
% 3.32/1.74 => ( ( in @ ( kpair @ Xz @ Xy ) @ S )
% 3.32/1.74 => Xphi ) ) )
% 3.32/1.74 => Xphi ) ) ) ) ) ) ) ) ).
% 3.32/1.74
% 3.32/1.74 thf(breln1unionprop_type,type,
% 3.32/1.74 breln1unionprop: $o ).
% 3.32/1.74
% 3.32/1.74 thf(breln1unionprop,definition,
% 3.32/1.74 ( breln1unionprop
% 3.32/1.74 = ( ! [A: $i,R: $i] :
% 3.32/1.74 ( ( breln1 @ A @ R )
% 3.32/1.74 => ! [S: $i] :
% 3.32/1.74 ( ( breln1 @ A @ S )
% 3.32/1.74 => ( breln1 @ A @ ( binunion @ R @ S ) ) ) ) ) ) ).
% 3.32/1.74
% 3.32/1.74 thf(breln1unionIL_type,type,
% 3.32/1.74 breln1unionIL: $o ).
% 3.32/1.74
% 3.32/1.74 thf(breln1unionIL,definition,
% 3.32/1.74 ( breln1unionIL
% 3.32/1.74 = ( ! [A: $i,R: $i] :
% 3.32/1.74 ( ( breln1 @ A @ R )
% 3.32/1.74 => ! [S: $i] :
% 3.32/1.74 ( ( breln1 @ A @ S )
% 3.32/1.74 => ! [Xx: $i] :
% 3.32/1.74 ( ( in @ Xx @ A )
% 3.32/1.74 => ! [Xy: $i] :
% 3.32/1.74 ( ( in @ Xy @ A )
% 3.32/1.74 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.32/1.74 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 3.32/1.74
% 3.32/1.74 thf(breln1unionIR_type,type,
% 3.32/1.74 breln1unionIR: $o ).
% 3.32/1.74
% 3.32/1.74 thf(breln1unionIR,definition,
% 3.32/1.74 ( breln1unionIR
% 3.32/1.74 = ( ! [A: $i,R: $i] :
% 3.32/1.74 ( ( breln1 @ A @ R )
% 3.32/1.74 => ! [S: $i] :
% 3.32/1.74 ( ( breln1 @ A @ S )
% 3.32/1.74 => ! [Xx: $i] :
% 3.32/1.74 ( ( in @ Xx @ A )
% 3.32/1.74 => ! [Xy: $i] :
% 3.32/1.74 ( ( in @ Xy @ A )
% 3.32/1.74 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 3.32/1.74 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 3.32/1.74
% 3.32/1.74 thf(breln1unionI_type,type,
% 3.32/1.74 breln1unionI: $o ).
% 3.32/1.74
% 3.32/1.74 thf(breln1unionI,definition,
% 3.32/1.74 ( breln1unionI
% 3.32/1.74 = ( ! [A: $i,R: $i] :
% 3.32/1.74 ( ( breln1 @ A @ R )
% 3.32/1.74 => ! [S: $i] :
% 3.32/1.74 ( ( breln1 @ A @ S )
% 3.32/1.74 => ! [Xx: $i] :
% 3.32/1.74 ( ( in @ Xx @ A )
% 3.32/1.74 => ! [Xy: $i] :
% 3.32/1.74 ( ( in @ Xy @ A )
% 3.32/1.74 => ( ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.32/1.74 | ( in @ ( kpair @ Xx @ Xy ) @ S ) )
% 3.32/1.74 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 3.32/1.74
% 3.32/1.74 thf(breln1unionE,conjecture,
% 3.32/1.74 ( setextAx
% 3.32/1.74 => ( emptysetAx
% 3.32/1.74 => ( setadjoinAx
% 3.32/1.74 => ( powersetAx
% 3.32/1.74 => ( setunionAx
% 3.32/1.74 => ( omega0Ax
% 3.32/1.74 => ( omegaSAx
% 3.32/1.74 => ( omegaIndAx
% 3.32/1.74 => ( replAx
% 3.32/1.74 => ( foundationAx
% 3.32/1.74 => ( wellorderingAx
% 3.32/1.74 => ( descrp
% 3.32/1.74 => ( dsetconstrI
% 3.32/1.74 => ( dsetconstrEL
% 3.32/1.74 => ( dsetconstrER
% 3.32/1.74 => ( exuE1
% 3.32/1.74 => ( prop2setE
% 3.32/1.74 => ( emptysetE
% 3.32/1.74 => ( emptysetimpfalse
% 3.32/1.74 => ( notinemptyset
% 3.32/1.74 => ( exuE3e
% 3.32/1.74 => ( setext
% 3.32/1.74 => ( emptyI
% 3.32/1.74 => ( noeltsimpempty
% 3.32/1.74 => ( setbeta
% 3.32/1.74 => ( nonemptyE1
% 3.32/1.74 => ( nonemptyI
% 3.32/1.74 => ( nonemptyI1
% 3.32/1.74 => ( setadjoinIL
% 3.32/1.74 => ( emptyinunitempty
% 3.32/1.74 => ( setadjoinIR
% 3.32/1.74 => ( setadjoinE
% 3.32/1.74 => ( setadjoinOr
% 3.32/1.74 => ( setoftrueEq
% 3.32/1.74 => ( powersetI
% 3.32/1.74 => ( emptyinPowerset
% 3.32/1.74 => ( emptyInPowerset
% 3.32/1.74 => ( powersetE
% 3.32/1.74 => ( setunionI
% 3.32/1.74 => ( setunionE
% 3.32/1.74 => ( subPowSU
% 3.32/1.74 => ( exuE2
% 3.32/1.74 => ( nonemptyImpWitness
% 3.32/1.74 => ( uniqinunit
% 3.32/1.74 => ( notinsingleton
% 3.32/1.74 => ( eqinunit
% 3.32/1.74 => ( singletonsswitch
% 3.32/1.74 => ( upairsetE
% 3.32/1.74 => ( upairsetIL
% 3.32/1.74 => ( upairsetIR
% 3.32/1.74 => ( emptyE1
% 3.32/1.74 => ( vacuousDall
% 3.32/1.74 => ( quantDeMorgan1
% 3.32/1.74 => ( quantDeMorgan2
% 3.32/1.74 => ( quantDeMorgan3
% 3.32/1.74 => ( quantDeMorgan4
% 3.32/1.74 => ( prop2setI
% 3.32/1.74 => ( prop2set2propI
% 3.32/1.74 => ( notdexE
% 3.32/1.74 => ( notdallE
% 3.32/1.74 => ( exuI1
% 3.32/1.74 => ( exuI3
% 3.32/1.74 => ( exuI2
% 3.32/1.74 => ( inCongP
% 3.32/1.74 => ( in__Cong
% 3.32/1.74 => ( exuE3u
% 3.32/1.74 => ( exu__Cong
% 3.32/1.74 => ( emptyset__Cong
% 3.32/1.74 => ( setadjoin__Cong
% 3.32/1.74 => ( powerset__Cong
% 3.32/1.74 => ( setunion__Cong
% 3.32/1.74 => ( omega__Cong
% 3.32/1.74 => ( exuEu
% 3.32/1.74 => ( descr__Cong
% 3.32/1.74 => ( dsetconstr__Cong
% 3.32/1.74 => ( subsetI1
% 3.32/1.74 => ( eqimpsubset2
% 3.32/1.74 => ( eqimpsubset1
% 3.32/1.74 => ( subsetI2
% 3.32/1.74 => ( emptysetsubset
% 3.32/1.74 => ( subsetE
% 3.32/1.74 => ( subsetE2
% 3.32/1.74 => ( notsubsetI
% 3.32/1.74 => ( notequalI1
% 3.32/1.74 => ( notequalI2
% 3.32/1.74 => ( subsetRefl
% 3.32/1.74 => ( subsetTrans
% 3.32/1.74 => ( setadjoinSub
% 3.32/1.74 => ( setadjoinSub2
% 3.32/1.74 => ( subset2powerset
% 3.32/1.74 => ( setextsub
% 3.32/1.74 => ( subsetemptysetimpeq
% 3.32/1.74 => ( powersetI1
% 3.32/1.74 => ( powersetE1
% 3.32/1.74 => ( inPowerset
% 3.32/1.74 => ( powersetsubset
% 3.32/1.74 => ( sepInPowerset
% 3.32/1.74 => ( sepSubset
% 3.32/1.74 => ( binunionIL
% 3.32/1.74 => ( upairset2IR
% 3.32/1.74 => ( binunionIR
% 3.32/1.74 => ( binunionEcases
% 3.32/1.74 => ( binunionE
% 3.32/1.74 => ( binunionLsub
% 3.32/1.74 => ( binunionRsub
% 3.32/1.74 => ( binintersectI
% 3.32/1.74 => ( binintersectSubset5
% 3.32/1.74 => ( binintersectEL
% 3.32/1.74 => ( binintersectLsub
% 3.32/1.74 => ( binintersectSubset2
% 3.32/1.74 => ( binintersectSubset3
% 3.32/1.74 => ( binintersectER
% 3.32/1.74 => ( disjointsetsI1
% 3.32/1.74 => ( binintersectRsub
% 3.32/1.74 => ( binintersectSubset4
% 3.32/1.74 => ( binintersectSubset1
% 3.32/1.74 => ( bs114d
% 3.32/1.74 => ( setminusI
% 3.32/1.74 => ( setminusEL
% 3.32/1.74 => ( setminusER
% 3.32/1.74 => ( setminusSubset2
% 3.32/1.74 => ( setminusERneg
% 3.32/1.74 => ( setminusELneg
% 3.32/1.74 => ( setminusILneg
% 3.32/1.74 => ( setminusIRneg
% 3.32/1.74 => ( setminusLsub
% 3.32/1.74 => ( setminusSubset1
% 3.32/1.74 => ( symdiffE
% 3.32/1.74 => ( symdiffI1
% 3.32/1.74 => ( symdiffI2
% 3.32/1.74 => ( symdiffIneg1
% 3.32/1.74 => ( symdiffIneg2
% 3.32/1.74 => ( secondinupair
% 3.32/1.74 => ( setukpairIL
% 3.32/1.74 => ( setukpairIR
% 3.32/1.74 => ( kpairiskpair
% 3.32/1.74 => ( kpairp
% 3.32/1.74 => ( singletonsubset
% 3.32/1.74 => ( singletoninpowerset
% 3.32/1.74 => ( singletoninpowunion
% 3.32/1.74 => ( upairset2E
% 3.32/1.74 => ( upairsubunion
% 3.32/1.74 => ( upairinpowunion
% 3.32/1.74 => ( ubforcartprodlem1
% 3.32/1.74 => ( ubforcartprodlem2
% 3.32/1.74 => ( ubforcartprodlem3
% 3.32/1.74 => ( cartprodpairin
% 3.32/1.74 => ( cartprodmempair1
% 3.32/1.74 => ( cartprodmempair
% 3.32/1.74 => ( setunionE2
% 3.32/1.74 => ( setunionsingleton1
% 3.32/1.74 => ( setunionsingleton2
% 3.32/1.74 => ( setunionsingleton
% 3.32/1.74 => ( singletonprop
% 3.32/1.74 => ( ex1E1
% 3.32/1.74 => ( ex1I
% 3.32/1.74 => ( ex1I2
% 3.32/1.74 => ( singletonsuniq
% 3.32/1.74 => ( setukpairinjL1
% 3.32/1.74 => ( kfstsingleton
% 3.32/1.74 => ( theprop
% 3.32/1.74 => ( kfstpairEq
% 3.32/1.74 => ( cartprodfstin
% 3.32/1.74 => ( setukpairinjL2
% 3.32/1.74 => ( setukpairinjL
% 3.32/1.74 => ( setukpairinjR11
% 3.32/1.74 => ( setukpairinjR12
% 3.32/1.74 => ( setukpairinjR1
% 3.32/1.74 => ( upairequniteq
% 3.32/1.74 => ( setukpairinjR2
% 3.32/1.74 => ( setukpairinjR
% 3.32/1.74 => ( ksndsingleton
% 3.32/1.74 => ( ksndpairEq
% 3.32/1.74 => ( kpairsurjEq
% 3.32/1.74 => ( cartprodsndin
% 3.32/1.74 => ( cartprodpairmemEL
% 3.32/1.74 => ( cartprodpairmemER
% 3.32/1.74 => ( cartprodmempaircEq
% 3.32/1.74 => ( cartprodfstpairEq
% 3.32/1.74 => ( cartprodsndpairEq
% 3.32/1.74 => ( cartprodpairsurjEq
% 3.32/1.74 => ( dpsetconstrI
% 3.32/1.74 => ( dpsetconstrSub
% 3.32/1.74 => ( setOfPairsIsBReln
% 3.32/1.74 => ( dpsetconstrERa
% 3.32/1.74 => ( dpsetconstrEL1
% 3.32/1.74 => ( dpsetconstrEL2
% 3.32/1.74 => ( dpsetconstrER
% 3.32/1.74 => ( funcImageSingleton
% 3.32/1.74 => ( apProp
% 3.32/1.74 => ( app
% 3.32/1.74 => ( infuncsetfunc
% 3.32/1.74 => ( ap2p
% 3.32/1.74 => ( funcinfuncset
% 3.32/1.74 => ( lamProp
% 3.32/1.74 => ( lamp
% 3.32/1.74 => ( lam2p
% 3.32/1.74 => ( brelnall1
% 3.32/1.74 => ( brelnall2
% 3.32/1.74 => ( ex1E2
% 3.32/1.74 => ( funcGraphProp1
% 3.32/1.74 => ( funcGraphProp3
% 3.32/1.74 => ( funcGraphProp2
% 3.32/1.74 => ( funcextLem
% 3.32/1.74 => ( funcGraphProp4
% 3.32/1.74 => ( subbreln
% 3.32/1.74 => ( eqbreln
% 3.32/1.74 => ( funcext
% 3.32/1.74 => ( funcext2
% 3.32/1.74 => ( ap2apEq1
% 3.32/1.74 => ( ap2apEq2
% 3.32/1.74 => ( beta1
% 3.32/1.74 => ( eta1
% 3.32/1.74 => ( lam2lamEq
% 3.32/1.74 => ( beta2
% 3.32/1.74 => ( eta2
% 3.32/1.74 => ( iffalseProp1
% 3.32/1.74 => ( iffalseProp2
% 3.32/1.74 => ( iftrueProp1
% 3.32/1.74 => ( iftrueProp2
% 3.32/1.74 => ( ifSingleton
% 3.32/1.74 => ( ifp
% 3.32/1.74 => ( theeq
% 3.32/1.74 => ( iftrue
% 3.32/1.74 => ( iffalse
% 3.32/1.74 => ( iftrueorfalse
% 3.32/1.74 => ( binintersectT_lem
% 3.32/1.74 => ( binunionT_lem
% 3.32/1.74 => ( powersetT_lem
% 3.32/1.74 => ( setminusT_lem
% 3.32/1.74 => ( complementT_lem
% 3.32/1.74 => ( setextT
% 3.32/1.74 => ( subsetTI
% 3.32/1.74 => ( powersetTI1
% 3.32/1.74 => ( powersetTE1
% 3.32/1.74 => ( complementTI1
% 3.32/1.74 => ( complementTE1
% 3.32/1.74 => ( binintersectTELcontra
% 3.32/1.74 => ( binintersectTERcontra
% 3.32/1.74 => ( contrasubsetT
% 3.32/1.74 => ( contrasubsetT1
% 3.32/1.74 => ( contrasubsetT2
% 3.32/1.74 => ( contrasubsetT3
% 3.32/1.74 => ( doubleComplementI1
% 3.32/1.74 => ( doubleComplementE1
% 3.32/1.74 => ( doubleComplementSub1
% 3.32/1.74 => ( doubleComplementSub2
% 3.32/1.74 => ( doubleComplementEq
% 3.32/1.74 => ( complementTnotintersectT
% 3.32/1.74 => ( complementImpComplementIntersect
% 3.32/1.74 => ( complementSubsetComplementIntersect
% 3.32/1.74 => ( complementInPowersetComplementIntersect
% 3.32/1.74 => ( contraSubsetComplement
% 3.32/1.74 => ( complementTcontraSubset
% 3.32/1.74 => ( binunionTILcontra
% 3.32/1.74 => ( binunionTIRcontra
% 3.32/1.74 => ( inIntersectImpInUnion
% 3.32/1.74 => ( inIntersectImpInUnion2
% 3.32/1.74 => ( inIntersectImpInIntersectUnions
% 3.32/1.74 => ( intersectInPowersetIntersectUnions
% 3.32/1.74 => ( inComplementUnionImpNotIn1
% 3.32/1.74 => ( inComplementUnionImpInComplement1
% 3.32/1.74 => ( binunionTE
% 3.32/1.74 => ( binunionTEcontra
% 3.32/1.74 => ( demorgan2a1
% 3.32/1.74 => ( complementUnionInPowersetComplement
% 3.32/1.74 => ( demorgan2a2
% 3.32/1.74 => ( demorgan1a
% 3.32/1.74 => ( demorgan1b
% 3.32/1.74 => ( demorgan1
% 3.32/1.74 => ( demorgan2a
% 3.32/1.74 => ( demorgan2b2
% 3.32/1.74 => ( demorgan2b
% 3.32/1.74 => ( demorgan2
% 3.32/1.74 => ( woz13rule0
% 3.32/1.74 => ( woz13rule1
% 3.32/1.74 => ( woz13rule2
% 3.32/1.74 => ( woz13rule3
% 3.32/1.74 => ( woz13rule4
% 3.32/1.74 => ( woz1_1
% 3.32/1.74 => ( woz1_2
% 3.32/1.74 => ( woz1_3
% 3.32/1.74 => ( woz1_4
% 3.32/1.74 => ( woz1_5
% 3.32/1.74 => ( breln1all2
% 3.32/1.74 => ( breln1SetBreln1
% 3.32/1.74 => ( choice2fnsingleton
% 3.32/1.74 => ( setOfPairsIsBReln1
% 3.32/1.74 => ( breln1all1
% 3.32/1.74 => ( subbreln1
% 3.32/1.74 => ( eqbreln1
% 3.32/1.74 => ( breln1invprop
% 3.32/1.74 => ( breln1invI
% 3.32/1.74 => ( breln1invE
% 3.32/1.74 => ( breln1compprop
% 3.32/1.74 => ( breln1compI
% 3.32/1.74 => ( breln1compE
% 3.32/1.74 => ( breln1compEex
% 3.32/1.74 => ( breln1unionprop
% 3.32/1.74 => ( breln1unionIL
% 3.32/1.74 => ( breln1unionIR
% 3.32/1.74 => ( breln1unionI
% 3.32/1.74 => ! [A: $i,R: $i] :
% 3.32/1.74 ( ( breln1 @ A @ R )
% 3.32/1.74 => ! [S: $i] :
% 3.32/1.74 ( ( breln1 @ A @ S )
% 3.32/1.74 => ! [Xx: $i] :
% 3.32/1.74 ( ( in @ Xx @ A )
% 3.32/1.74 => ! [Xy: $i] :
% 3.32/1.74 ( ( in @ Xy @ A )
% 3.32/1.74 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) )
% 3.32/1.74 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.32/1.74 | ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 3.32/1.74 ------------------------
% 3.32/1.78 % (25120)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_25019 for (2999ds/4Mi)
% 3.32/1.78 % (25124)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_25019 for (2999ds/275Mi)
% 3.32/1.78 % (25121)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_25019 for (2999ds/27Mi)
% 3.32/1.78 % (25125)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_25019 for (2999ds/18Mi)
% 3.32/1.78 % (25123)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_25019 for (2999ds/2Mi)
% 3.32/1.78 % (25119)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_25019 for (2999ds/183Mi)
% 3.32/1.78 % (25122)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_25019 for (2999ds/2Mi)
% 3.32/1.78 % (25123)Instruction limit reached!
% 3.32/1.78 % (25123)------------------------------
% 3.32/1.78 % (25123)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.32/1.78 % (25123)Termination reason: Unknown
% 3.32/1.78 % (25123)Termination phase: shuffling
% 3.32/1.78
% 3.32/1.78 % (25123)Memory used [KB]: 1535
% 3.32/1.78 % (25123)Time elapsed: 0.003 s
% 3.32/1.78 % (25123)Instructions burned: 2 (million)
% 3.32/1.78 % (25123)------------------------------
% 3.32/1.78 % (25123)------------------------------
% 3.32/1.78 % (25122)Instruction limit reached!
% 3.32/1.78 % (25122)------------------------------
% 3.32/1.78 % (25122)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.32/1.78 % (25122)Termination reason: Unknown
% 3.32/1.78 % (25122)Termination phase: shuffling
% 3.32/1.78
% 3.32/1.78 % (25122)Memory used [KB]: 1663
% 3.32/1.78 % (25122)Time elapsed: 0.004 s
% 3.32/1.78 % (25122)Instructions burned: 3 (million)
% 3.32/1.78 % (25122)------------------------------
% 3.32/1.78 % (25122)------------------------------
% 3.32/1.79 % (25120)Instruction limit reached!
% 3.32/1.79 % (25120)------------------------------
% 3.32/1.79 % (25120)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.32/1.79 % (25120)Termination reason: Unknown
% 3.32/1.79 % (25120)Termination phase: shuffling
% 3.32/1.79
% 3.32/1.79 % (25120)Memory used [KB]: 1535
% 3.32/1.79 % (25120)Time elapsed: 0.004 s
% 3.32/1.79 % (25120)Instructions burned: 5 (million)
% 3.32/1.79 % (25120)------------------------------
% 3.32/1.79 % (25120)------------------------------
% 3.32/1.79 % (25125)Instruction limit reached!
% 3.32/1.79 % (25125)------------------------------
% 3.32/1.79 % (25125)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.32/1.79 % (25125)Termination reason: Unknown
% 3.32/1.79 % (25125)Termination phase: shuffling
% 3.32/1.79
% 3.32/1.79 % (25125)Memory used [KB]: 1918
% 3.32/1.79 % (25125)Time elapsed: 0.011 s
% 3.32/1.79 % (25125)Instructions burned: 18 (million)
% 3.32/1.79 % (25125)------------------------------
% 3.32/1.79 % (25125)------------------------------
% 3.32/1.80 % (25121)Instruction limit reached!
% 3.32/1.80 % (25121)------------------------------
% 3.32/1.80 % (25121)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.32/1.80 % (25121)Termination reason: Unknown
% 3.32/1.80 % (25121)Termination phase: shuffling
% 3.32/1.80
% 3.32/1.80 % (25121)Memory used [KB]: 2046
% 3.32/1.80 % (25121)Time elapsed: 0.015 s
% 3.32/1.80 % (25121)Instructions burned: 27 (million)
% 3.32/1.80 % (25121)------------------------------
% 3.32/1.80 % (25121)------------------------------
% 3.38/1.80 % (25126)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_25019 for (2999ds/3Mi)
% 3.38/1.80 % (25127)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_25019 for (2999ds/37Mi)
% 3.38/1.80 % (25128)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_25019 for (2999ds/15Mi)
% 3.38/1.80 % (25126)Instruction limit reached!
% 3.38/1.80 % (25126)------------------------------
% 3.38/1.80 % (25126)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.38/1.80 % (25126)Termination reason: Unknown
% 3.38/1.80 % (25126)Termination phase: shuffling
% 3.38/1.80
% 3.38/1.80 % (25126)Memory used [KB]: 1535
% 3.38/1.80 % (25126)Time elapsed: 0.004 s
% 3.38/1.80 % (25126)Instructions burned: 4 (million)
% 3.38/1.80 % (25126)------------------------------
% 3.38/1.80 % (25126)------------------------------
% 3.38/1.81 % (25129)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_25019 for (2999ds/3Mi)
% 3.38/1.81 % (25128)Instruction limit reached!
% 3.38/1.81 % (25128)------------------------------
% 3.38/1.81 % (25128)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.38/1.81 % (25128)Termination reason: Unknown
% 3.38/1.81 % (25128)Termination phase: shuffling
% 3.38/1.81
% 3.38/1.81 % (25128)Memory used [KB]: 1791
% 3.38/1.81 % (25128)Time elapsed: 0.009 s
% 3.38/1.81 % (25128)Instructions burned: 15 (million)
% 3.38/1.81 % (25128)------------------------------
% 3.38/1.81 % (25128)------------------------------
% 3.38/1.81 % (25129)Instruction limit reached!
% 3.38/1.81 % (25129)------------------------------
% 3.38/1.81 % (25129)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.38/1.81 % (25129)Termination reason: Unknown
% 3.38/1.81 % (25129)Termination phase: shuffling
% 3.38/1.81
% 3.38/1.81 % (25129)Memory used [KB]: 1535
% 3.38/1.81 % (25129)Time elapsed: 0.003 s
% 3.38/1.81 % (25129)Instructions burned: 3 (million)
% 3.38/1.81 % (25129)------------------------------
% 3.38/1.81 % (25129)------------------------------
% 3.38/1.81 % (25130)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_25019 for (2999ds/1041Mi)
% 3.38/1.82 % (25127)Instruction limit reached!
% 3.38/1.82 % (25127)------------------------------
% 3.38/1.82 % (25127)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.38/1.82 % (25127)Termination reason: Unknown
% 3.38/1.82 % (25127)Termination phase: shuffling
% 3.38/1.82
% 3.38/1.82 % (25127)Memory used [KB]: 2302
% 3.38/1.82 % (25127)Time elapsed: 0.019 s
% 3.38/1.82 % (25127)Instructions burned: 39 (million)
% 3.38/1.82 % (25127)------------------------------
% 3.38/1.82 % (25127)------------------------------
% 3.38/1.82 % (25131)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_25019 for (2999ds/7Mi)
% 3.38/1.82 % (25131)Instruction limit reached!
% 3.38/1.82 % (25131)------------------------------
% 3.38/1.82 % (25131)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.38/1.82 % (25131)Termination reason: Unknown
% 3.38/1.82 % (25131)Termination phase: shuffling
% 3.38/1.82
% 3.38/1.82 % (25131)Memory used [KB]: 1663
% 3.38/1.82 % (25131)Time elapsed: 0.005 s
% 3.38/1.82 % (25131)Instructions burned: 7 (million)
% 3.38/1.82 % (25131)------------------------------
% 3.38/1.82 % (25131)------------------------------
% 3.38/1.83 % (25132)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_25019 for (2999ds/16Mi)
% 3.38/1.83 % (25133)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_25019 for (2999ds/3Mi)
% 3.38/1.83 % (25133)Instruction limit reached!
% 3.38/1.83 % (25133)------------------------------
% 3.38/1.83 % (25133)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.38/1.83 % (25133)Termination reason: Unknown
% 3.38/1.83 % (25133)Termination phase: shuffling
% 3.38/1.83
% 3.38/1.83 % (25133)Memory used [KB]: 1535
% 3.38/1.83 % (25133)Time elapsed: 0.003 s
% 3.38/1.83 % (25133)Instructions burned: 3 (million)
% 3.38/1.83 % (25133)------------------------------
% 3.38/1.83 % (25133)------------------------------
% 3.38/1.83 % (25132)Instruction limit reached!
% 3.38/1.83 % (25132)------------------------------
% 3.38/1.83 % (25132)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.38/1.83 % (25132)Termination reason: Unknown
% 3.38/1.83 % (25132)Termination phase: shuffling
% 3.38/1.83
% 3.38/1.83 % (25132)Memory used [KB]: 1791
% 3.38/1.83 % (25132)Time elapsed: 0.009 s
% 3.38/1.83 % (25132)Instructions burned: 16 (million)
% 3.38/1.83 % (25132)------------------------------
% 3.38/1.83 % (25132)------------------------------
% 3.38/1.83 % (25134)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_25019 for (2999ds/3Mi)
% 3.38/1.84 % (25134)Instruction limit reached!
% 3.38/1.84 % (25134)------------------------------
% 3.38/1.84 % (25134)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.38/1.84 % (25134)Termination reason: Unknown
% 3.38/1.84 % (25134)Termination phase: shuffling
% 3.38/1.84
% 3.38/1.84 % (25134)Memory used [KB]: 1535
% 3.38/1.84 % (25134)Time elapsed: 0.003 s
% 3.38/1.84 % (25134)Instructions burned: 3 (million)
% 3.38/1.84 % (25134)------------------------------
% 3.38/1.84 % (25134)------------------------------
% 3.38/1.84 % (25135)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_25019 for (2999ds/7Mi)
% 3.38/1.84 % (25135)Instruction limit reached!
% 3.38/1.84 % (25135)------------------------------
% 3.38/1.84 % (25135)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.38/1.84 % (25135)Termination reason: Unknown
% 3.38/1.84 % (25135)Termination phase: shuffling
% 3.38/1.84
% 3.38/1.84 % (25135)Memory used [KB]: 1663
% 3.38/1.84 % (25135)Time elapsed: 0.005 s
% 3.38/1.84 % (25135)Instructions burned: 8 (million)
% 3.38/1.84 % (25135)------------------------------
% 3.38/1.84 % (25135)------------------------------
% 3.38/1.84 % (25136)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_25019 for (2999ds/3Mi)
% 3.38/1.84 % (25136)Instruction limit reached!
% 3.38/1.84 % (25136)------------------------------
% 3.38/1.84 % (25136)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.38/1.84 % (25136)Termination reason: Unknown
% 3.38/1.84 % (25136)Termination phase: shuffling
% 3.38/1.84
% 3.38/1.84 % (25136)Memory used [KB]: 1535
% 3.38/1.84 % (25136)Time elapsed: 0.003 s
% 3.38/1.84 % (25136)Instructions burned: 3 (million)
% 3.38/1.84 % (25136)------------------------------
% 3.38/1.84 % (25136)------------------------------
% 3.38/1.85 % (25137)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_25019 for (2999ds/4Mi)
% 3.38/1.85 % (25137)Instruction limit reached!
% 3.38/1.85 % (25137)------------------------------
% 3.38/1.85 % (25137)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.38/1.85 % (25137)Termination reason: Unknown
% 3.38/1.85 % (25137)Termination phase: shuffling
% 3.38/1.85
% 3.38/1.85 % (25137)Memory used [KB]: 1535
% 3.38/1.85 % (25137)Time elapsed: 0.004 s
% 3.38/1.85 % (25137)Instructions burned: 5 (million)
% 3.38/1.85 % (25137)------------------------------
% 3.38/1.85 % (25137)------------------------------
% 3.38/1.85 % (25138)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_25019 for (2999ds/18Mi)
% 3.38/1.86 % (25139)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_25019 for (2999ds/710Mi)
% 3.38/1.86 % (25138)Instruction limit reached!
% 3.38/1.86 % (25138)------------------------------
% 3.38/1.86 % (25138)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.38/1.86 % (25138)Termination reason: Unknown
% 3.38/1.86 % (25138)Termination phase: shuffling
% 3.38/1.86
% 3.38/1.86 % (25138)Memory used [KB]: 1791
% 3.38/1.86 % (25138)Time elapsed: 0.010 s
% 3.38/1.86 % (25138)Instructions burned: 18 (million)
% 3.38/1.86 % (25138)------------------------------
% 3.38/1.86 % (25138)------------------------------
% 3.38/1.86 % (25140)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_25019 for (2999ds/6Mi)
% 3.38/1.86 % (25119)Instruction limit reached!
% 3.38/1.86 % (25119)------------------------------
% 3.38/1.86 % (25119)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.38/1.86 % (25119)Termination reason: Unknown
% 3.38/1.86 % (25119)Termination phase: Property scanning
% 3.38/1.86
% 3.38/1.86 % (25119)Memory used [KB]: 2942
% 3.38/1.86 % (25119)Time elapsed: 0.081 s
% 3.38/1.86 % (25119)Instructions burned: 183 (million)
% 3.38/1.86 % (25119)------------------------------
% 3.38/1.86 % (25119)------------------------------
% 3.38/1.86 % (25140)Instruction limit reached!
% 3.38/1.86 % (25140)------------------------------
% 3.38/1.86 % (25140)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.38/1.86 % (25140)Termination reason: Unknown
% 3.38/1.86 % (25140)Termination phase: shuffling
% 3.38/1.86
% 3.38/1.86 % (25140)Memory used [KB]: 1663
% 3.38/1.86 % (25140)Time elapsed: 0.005 s
% 3.38/1.86 % (25140)Instructions burned: 7 (million)
% 3.38/1.86 % (25140)------------------------------
% 3.38/1.86 % (25140)------------------------------
% 3.38/1.87 % (25141)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_25019 for (2999ds/902Mi)
% 3.38/1.88 % (25142)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_25019 for (2998ds/21Mi)
% 3.38/1.88 % (25143)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_25019 for (2998ds/5Mi)
% 3.38/1.88 % (25144)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_25019 for (2998ds/6Mi)
% 3.38/1.88 % (25143)Instruction limit reached!
% 3.38/1.88 % (25143)------------------------------
% 3.38/1.88 % (25143)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.38/1.88 % (25143)Termination reason: Unknown
% 3.38/1.88 % (25143)Termination phase: shuffling
% 3.38/1.88
% 3.38/1.88 % (25143)Memory used [KB]: 1535
% 3.38/1.88 % (25143)Time elapsed: 0.004 s
% 3.38/1.88 % (25143)Instructions burned: 5 (million)
% 3.38/1.88 % (25143)------------------------------
% 3.38/1.88 % (25143)------------------------------
% 3.38/1.88 % (25144)Instruction limit reached!
% 3.38/1.88 % (25144)------------------------------
% 3.38/1.88 % (25144)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.38/1.88 % (25144)Termination reason: Unknown
% 3.38/1.88 % (25144)Termination phase: shuffling
% 3.38/1.88
% 3.38/1.88 % (25144)Memory used [KB]: 1663
% 3.38/1.88 % (25144)Time elapsed: 0.005 s
% 3.38/1.88 % (25144)Instructions burned: 7 (million)
% 3.38/1.88 % (25144)------------------------------
% 3.38/1.88 % (25144)------------------------------
% 3.38/1.89 % (25142)Instruction limit reached!
% 3.38/1.89 % (25142)------------------------------
% 3.38/1.89 % (25142)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.38/1.89 % (25142)Termination reason: Unknown
% 3.38/1.89 % (25142)Termination phase: shuffling
% 3.38/1.89
% 3.38/1.89 % (25142)Memory used [KB]: 1918
% 3.38/1.89 % (25142)Time elapsed: 0.012 s
% 3.38/1.89 % (25142)Instructions burned: 22 (million)
% 3.38/1.89 % (25142)------------------------------
% 3.38/1.89 % (25142)------------------------------
% 3.38/1.90 % (25145)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_25019 for (2998ds/377Mi)
% 3.38/1.90 % (25146)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_25019 for (2998ds/779Mi)
% 3.38/1.90 % (25147)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_25019 for (2998ds/19Mi)
% 3.38/1.91 % (25124)Instruction limit reached!
% 3.38/1.91 % (25124)------------------------------
% 3.38/1.91 % (25124)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.38/1.91 % (25124)Termination reason: Unknown
% 3.38/1.91 % (25124)Termination phase: Saturation
% 3.38/1.91
% 3.38/1.91 % (25124)Memory used [KB]: 10362
% 3.38/1.91 % (25124)Time elapsed: 0.130 s
% 3.38/1.91 % (25124)Instructions burned: 275 (million)
% 3.38/1.91 % (25124)------------------------------
% 3.38/1.91 % (25124)------------------------------
% 3.38/1.91 % (25147)Instruction limit reached!
% 3.38/1.91 % (25147)------------------------------
% 3.38/1.91 % (25147)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.38/1.91 % (25147)Termination reason: Unknown
% 3.38/1.91 % (25147)Termination phase: shuffling
% 3.38/1.91
% 3.38/1.91 % (25147)Memory used [KB]: 1918
% 3.38/1.91 % (25147)Time elapsed: 0.011 s
% 3.38/1.91 % (25147)Instructions burned: 21 (million)
% 3.38/1.91 % (25147)------------------------------
% 3.38/1.91 % (25147)------------------------------
% 3.38/1.93 % (25148)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_25019 for (2998ds/879Mi)
% 3.38/1.93 % (25149)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_25019 for (2998ds/17Mi)
% 3.38/1.94 % (25149)Instruction limit reached!
% 3.38/1.94 % (25149)------------------------------
% 3.38/1.94 % (25149)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.38/1.94 % (25149)Termination reason: Unknown
% 3.38/1.94 % (25149)Termination phase: shuffling
% 3.38/1.94
% 3.38/1.94 % (25149)Memory used [KB]: 1791
% 3.38/1.94 % (25149)Time elapsed: 0.010 s
% 3.38/1.94 % (25149)Instructions burned: 18 (million)
% 3.38/1.94 % (25149)------------------------------
% 3.38/1.94 % (25149)------------------------------
% 3.38/1.95 % (25150)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_25019 for (2998ds/3Mi)
% 3.38/1.95 % (25150)Instruction limit reached!
% 3.38/1.95 % (25150)------------------------------
% 3.38/1.95 % (25150)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.38/1.95 % (25150)Termination reason: Unknown
% 3.38/1.95 % (25150)Termination phase: shuffling
% 3.38/1.95
% 3.38/1.95 % (25150)Memory used [KB]: 1535
% 3.38/1.95 % (25150)Time elapsed: 0.003 s
% 3.38/1.95 % (25150)Instructions burned: 3 (million)
% 3.38/1.95 % (25150)------------------------------
% 3.38/1.95 % (25150)------------------------------
% 3.38/1.97 % (25151)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_25019 for (2998ds/30Mi)
% 3.38/1.99 % (25151)Instruction limit reached!
% 3.38/1.99 % (25151)------------------------------
% 3.38/1.99 % (25151)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.38/1.99 % (25151)Termination reason: Unknown
% 3.38/1.99 % (25151)Termination phase: shuffling
% 3.38/1.99
% 3.38/1.99 % (25151)Memory used [KB]: 2174
% 3.38/1.99 % (25151)Time elapsed: 0.016 s
% 3.38/1.99 % (25151)Instructions burned: 31 (million)
% 3.38/1.99 % (25151)------------------------------
% 3.38/1.99 % (25151)------------------------------
% 3.38/2.00 % (25152)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_25019 for (2997ds/127Mi)
% 5.12/2.06 % (25152)Instruction limit reached!
% 5.12/2.06 % (25152)------------------------------
% 5.12/2.06 % (25152)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.12/2.06 % (25152)Termination reason: Unknown
% 5.12/2.06 % (25152)Termination phase: Property scanning
% 5.12/2.06
% 5.12/2.06 % (25152)Memory used [KB]: 2814
% 5.12/2.06 % (25152)Time elapsed: 0.058 s
% 5.12/2.06 % (25152)Instructions burned: 127 (million)
% 5.12/2.06 % (25152)------------------------------
% 5.12/2.06 % (25152)------------------------------
% 5.12/2.07 % (25145)Instruction limit reached!
% 5.12/2.07 % (25145)------------------------------
% 5.12/2.07 % (25145)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.12/2.07 % (25145)Termination reason: Unknown
% 5.12/2.07 % (25145)Termination phase: Saturation
% 5.12/2.07
% 5.12/2.07 % (25145)Memory used [KB]: 9338
% 5.12/2.07 % (25145)Time elapsed: 0.170 s
% 5.12/2.07 % (25145)Instructions burned: 378 (million)
% 5.12/2.07 % (25145)------------------------------
% 5.12/2.07 % (25145)------------------------------
% 5.12/2.08 % (25153)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_25019 for (2996ds/100Mi)
% 5.12/2.08 % (25154)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_25019 for (2996ds/3Mi)
% 5.12/2.08 % (25154)Instruction limit reached!
% 5.12/2.08 % (25154)------------------------------
% 5.12/2.08 % (25154)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.12/2.08 % (25154)Termination reason: Unknown
% 5.12/2.08 % (25154)Termination phase: shuffling
% 5.12/2.08
% 5.12/2.08 % (25154)Memory used [KB]: 1535
% 5.12/2.08 % (25154)Time elapsed: 0.003 s
% 5.12/2.08 % (25154)Instructions burned: 3 (million)
% 5.12/2.08 % (25154)------------------------------
% 5.12/2.08 % (25154)------------------------------
% 5.12/2.10 % (25141)First to succeed.
% 5.12/2.10 % (25155)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_25019 for (2996ds/20Mi)
% 5.12/2.11 % (25141)Refutation found. Thanks to Tanya!
% 5.12/2.11 % SZS status Theorem for DTF2THF_25019
% 5.12/2.11 % SZS output start Proof for DTF2THF_25019
% 5.12/2.11 thf(func_def_0, type, in: $i > $i > $o).
% 5.12/2.11 thf(func_def_1, type, exu: ($i > $o) > $o).
% 5.12/2.11 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 5.12/2.11 thf(func_def_8, type, powerset: $i > $i).
% 5.12/2.11 thf(func_def_10, type, setunion: $i > $i).
% 5.12/2.11 thf(func_def_19, type, descr: ($i > $o) > $i).
% 5.12/2.11 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 5.12/2.11 thf(func_def_26, type, prop2set: $o > $i).
% 5.12/2.11 thf(func_def_36, type, nonempty: $i > $o).
% 5.12/2.11 thf(func_def_69, type, set2prop: $i > $o).
% 5.12/2.11 thf(func_def_88, type, subset: $i > $i > $o).
% 5.12/2.11 thf(func_def_89, type, disjoint: $i > $i > $o).
% 5.12/2.11 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 5.12/2.11 thf(func_def_114, type, binunion: $i > $i > $i).
% 5.12/2.11 thf(func_def_122, type, binintersect: $i > $i > $i).
% 5.12/2.11 thf(func_def_135, type, regular: $i > $o).
% 5.12/2.11 thf(func_def_136, type, setminus: $i > $i > $i).
% 5.12/2.11 thf(func_def_147, type, symdiff: $i > $i > $i).
% 5.12/2.11 thf(func_def_153, type, iskpair: $i > $o).
% 5.12/2.11 thf(func_def_158, type, kpair: $i > $i > $i).
% 5.12/2.11 thf(func_def_160, type, cartprod: $i > $i > $i).
% 5.12/2.11 thf(func_def_177, type, singleton: $i > $o).
% 5.12/2.11 thf(func_def_179, type, ex1: $i > ($i > $o) > $o).
% 5.12/2.11 thf(func_def_184, type, atmost1p: $i > $o).
% 5.12/2.11 thf(func_def_185, type, atleast2p: $i > $o).
% 5.12/2.11 thf(func_def_186, type, atmost2p: $i > $o).
% 5.12/2.11 thf(func_def_187, type, upairsetp: $i > $o).
% 5.12/2.11 thf(func_def_191, type, kfst: $i > $i).
% 5.12/2.11 thf(func_def_203, type, ksnd: $i > $i).
% 5.12/2.11 thf(func_def_213, type, breln: $i > $i > $i > $o).
% 5.12/2.11 thf(func_def_214, type, dpsetconstr: $i > $i > ($i > $i > $o) > $i).
% 5.12/2.11 thf(func_def_222, type, func: $i > $i > $i > $o).
% 5.12/2.11 thf(func_def_223, type, funcSet: $i > $i > $i).
% 5.12/2.11 thf(func_def_226, type, ap: $i > $i > $i > $i > $i).
% 5.12/2.11 thf(func_def_232, type, lam: $i > $i > ($i > $i) > $i).
% 5.12/2.11 thf(func_def_259, type, if: $i > $o > $i > $i > $i).
% 5.12/2.11 thf(func_def_323, type, breln1: $i > $i > $o).
% 5.12/2.11 thf(func_def_325, type, breln1Set: $i > $i).
% 5.12/2.11 thf(func_def_327, type, transitive: $i > $i > $o).
% 5.12/2.11 thf(func_def_328, type, antisymmetric: $i > $i > $o).
% 5.12/2.11 thf(func_def_329, type, reflexive: $i > $i > $o).
% 5.12/2.11 thf(func_def_330, type, refltransitive: $i > $i > $o).
% 5.12/2.11 thf(func_def_331, type, refllinearorder: $i > $i > $o).
% 5.12/2.11 thf(func_def_332, type, reflwellordering: $i > $i > $o).
% 5.12/2.11 thf(func_def_338, type, breln1invset: $i > $i > $i).
% 5.12/2.11 thf(func_def_342, type, breln1compset: $i > $i > $i > $i).
% 5.12/2.11 thf(func_def_368, type, sP4: $i > $i > $i > $o).
% 5.12/2.11 thf(func_def_369, type, sP5: $i > $i > $o).
% 5.12/2.11 thf(func_def_370, type, sP6: $i > $o).
% 5.12/2.11 thf(func_def_371, type, sP7: $i > $i > $o).
% 5.12/2.11 thf(func_def_372, type, sP8: $i > $i > $o).
% 5.12/2.11 thf(func_def_374, type, sP10: $i > $i > $i > $o).
% 5.12/2.11 thf(func_def_377, type, sP13: $i > $i > $i > $i > $o).
% 5.12/2.11 thf(func_def_379, type, sP15: $o > $i > $i > $i > $o).
% 5.12/2.11 thf(func_def_389, type, sK25: $i > $i > $i).
% 5.12/2.11 thf(func_def_394, type, sK30: $i > $o).
% 5.12/2.11 thf(func_def_396, type, sK32: $i > ($i > $o) > $i).
% 5.12/2.11 thf(func_def_405, type, sK41: $i > $i > $i > $i > $i > $i).
% 5.12/2.11 thf(func_def_417, type, sK53: $o > $i > $i > $i > $i > $i > $i).
% 5.12/2.11 thf(func_def_420, type, sK56: $i > $i > $i).
% 5.12/2.11 thf(func_def_421, type, sK57: $i > $i > $i).
% 5.12/2.11 thf(func_def_434, type, sK70: $i > $i).
% 5.12/2.11 thf(func_def_443, type, sK79: $i > $i > $o).
% 5.12/2.11 thf(func_def_446, type, sK82: ($i > $o) > ($i > $o) > $i).
% 5.12/2.11 thf(func_def_447, type, sK83: ($i > $o) > ($i > $o) > $i).
% 5.12/2.11 thf(func_def_448, type, sK84: $i > $o).
% 5.12/2.11 thf(func_def_449, type, sK85: $i > $o).
% 5.12/2.11 thf(func_def_469, type, sK105: $i > $i > $i > $i > $i).
% 5.12/2.11 thf(func_def_477, type, sK113: $i > $o).
% 5.12/2.11 thf(func_def_479, type, sK115: ($i > $o) > $i > $i).
% 5.12/2.11 thf(func_def_480, type, sK116: ($i > $o) > $i > $i).
% 5.12/2.11 thf(func_def_488, type, sK124: $i > $i > $o).
% 5.12/2.11 thf(func_def_495, type, sK131: $o > $i > $i > $i).
% 5.12/2.11 thf(func_def_500, type, sK136: $i > $i > $o).
% 5.12/2.11 thf(func_def_509, type, sK145: $i > $i > $o).
% 5.12/2.11 thf(func_def_513, type, sK149: $i > $i).
% 5.12/2.11 thf(func_def_514, type, sK150: $i > ($i > $i > $o) > $i > $i).
% 5.12/2.11 thf(func_def_515, type, sK151: ($i > $o) > $i).
% 5.12/2.11 thf(func_def_516, type, sK152: $i > $o).
% 5.12/2.11 thf(func_def_517, type, sK153: $i > $i).
% 5.12/2.11 thf(func_def_529, type, sK165: $i > ($i > $i) > $i > $i).
% 5.12/2.11 thf(func_def_531, type, sK167: $i > $i).
% 5.12/2.11 thf(func_def_537, type, sK173: ($i > $o) > $i > $i > $i > $i).
% 5.12/2.11 thf(func_def_538, type, sK174: ($i > $o) > $i > $i > $i > $i).
% 5.12/2.11 thf(func_def_542, type, sK178: $i > $o).
% 5.12/2.11 thf(func_def_553, type, sK189: $i > $i > $i > $i).
% 5.12/2.11 thf(func_def_554, type, sK190: $i > $i > $i > $i).
% 5.12/2.11 thf(func_def_573, type, sK209: $i > $i > $i).
% 5.12/2.11 thf(func_def_574, type, sK210: $i > $i > $i).
% 5.12/2.11 thf(func_def_575, type, sK211: $i > $i > $i).
% 5.12/2.11 thf(func_def_576, type, sK212: $i > $i > $i).
% 5.12/2.11 thf(func_def_577, type, sK213: $i > $i > $i).
% 5.12/2.11 thf(func_def_578, type, sK214: $i > $i > $i > $i).
% 5.12/2.11 thf(func_def_579, type, sK215: $i > $i).
% 5.12/2.11 thf(func_def_580, type, sK216: $i > $i).
% 5.12/2.11 thf(func_def_581, type, sK217: $i > $i).
% 5.12/2.11 thf(func_def_582, type, sK218: $i > $i).
% 5.12/2.11 thf(func_def_583, type, sK219: $i > $i > $i > $i).
% 5.12/2.11 thf(func_def_584, type, sK220: $i > $i > $i > $i).
% 5.12/2.11 thf(func_def_585, type, sK221: $i > $i > $i).
% 5.12/2.11 thf(func_def_586, type, sK222: $i > $i > $i).
% 5.12/2.11 thf(func_def_587, type, sK223: $i > $i > $i).
% 5.12/2.11 thf(func_def_589, type, sK225: $i > $i).
% 5.12/2.11 thf(func_def_590, type, sK226: $i > $i).
% 5.12/2.11 thf(func_def_591, type, sK227: $i > $i).
% 5.12/2.11 thf(func_def_635, type, sK271: $i > $i > $i > $i > $i).
% 5.12/2.11 thf(func_def_638, type, sK274: $i > $o).
% 5.12/2.11 thf(func_def_647, type, sK283: $i > $o).
% 5.12/2.11 thf(func_def_649, type, sK285: $i > ($i > $o) > $i).
% 5.12/2.11 thf(func_def_672, type, sK308: $i > $i > $i > $i > $i).
% 5.12/2.11 thf(func_def_673, type, sK309: $i > $i > $i > $i > $i).
% 5.12/2.11 thf(func_def_679, type, sK315: $i > $i).
% 5.12/2.11 thf(func_def_681, type, sK317: $i > ($i > $i) > $i > $i).
% 5.12/2.11 thf(func_def_691, type, sK327: $i > $o).
% 5.12/2.11 thf(func_def_697, type, sK333: $i > $i > $i > $i).
% 5.12/2.11 thf(func_def_698, type, sK334: $i > $i > $i > $i).
% 5.12/2.11 thf(func_def_699, type, sK335: $i > $i > $i > $i).
% 5.12/2.11 thf(func_def_700, type, sK336: $i > $i > $i > $i).
% 5.12/2.11 thf(func_def_716, type, sK352: $i > $i).
% 5.12/2.11 thf(func_def_740, type, sK376: $i > $i > $i > $i).
% 5.12/2.11 thf(func_def_784, type, sK420: $i > $o).
% 5.12/2.11 thf(func_def_789, type, sK425: $i > $i > $i).
% 5.12/2.11 thf(func_def_796, type, sK432: $i > $o).
% 5.12/2.11 thf(func_def_802, type, sK438: ($i > $o) > $i).
% 5.12/2.11 thf(func_def_803, type, sK439: $i > $o).
% 5.12/2.11 thf(func_def_804, type, sK440: ($i > $i) > $i > $i > $i).
% 5.12/2.11 thf(func_def_807, type, sK443: $i > $i).
% 5.12/2.11 thf(func_def_816, type, sK452: ($i > $o) > $i).
% 5.12/2.11 thf(func_def_817, type, sK453: $i > $o).
% 5.12/2.11 thf(func_def_818, type, sK454: $i > $i).
% 5.12/2.11 thf(func_def_819, type, sK455: $i > $i > $o).
% 5.12/2.11 thf(func_def_821, type, sK457: $i > $i).
% 5.12/2.11 thf(func_def_822, type, sK458: $i > $i).
% 5.12/2.11 thf(func_def_823, type, sK459: $i > ($i > $i > $o) > $i).
% 5.12/2.11 thf(func_def_824, type, sK460: $i > $i > ($i > $i > $o) > $i).
% 5.12/2.11 thf(func_def_825, type, sK461: $i > ($i > $i > $o) > $i).
% 5.12/2.11 thf(func_def_827, type, sK463: $i > $o).
% 5.12/2.11 thf(func_def_829, type, sK465: ($i > $o) > $i > $i).
% 5.12/2.11 thf(func_def_830, type, sK466: $i > $i > $i).
% 5.12/2.11 thf(func_def_842, type, sK478: ($i > $o) > $i > $i > $i).
% 5.12/2.11 thf(func_def_843, type, sK479: ($i > $o) > $i > $i > $i).
% 5.12/2.11 thf(func_def_846, type, sK482: $i > $o).
% 5.12/2.11 thf(func_def_854, type, sK490: $i > $i > $i > $i).
% 5.12/2.11 thf(func_def_855, type, sK491: $i > $i > $i > $i).
% 5.12/2.11 thf(func_def_856, type, sK492: $i > $o).
% 5.12/2.11 thf(func_def_863, type, sK499: $i > $o).
% 5.12/2.11 thf(func_def_866, type, sK502: $i > ($i > $o) > $i).
% 5.12/2.11 thf(func_def_868, type, sK504: $i > $i).
% 5.12/2.11 thf(func_def_872, type, sK508: $i > $i > ($i > $i) > $i).
% 5.12/2.11 thf(func_def_873, type, sK509: ($i > $o) > ($i > $o) > $i > $i > $i).
% 5.12/2.11 thf(func_def_874, type, sK510: ($i > $o) > ($i > $o) > $i > $i > $i).
% 5.12/2.11 thf(func_def_877, type, sK513: $i > $o).
% 5.12/2.11 thf(func_def_878, type, sK514: $i > $o).
% 5.12/2.11 thf(func_def_893, type, sK529: $i > $i > $i).
% 5.12/2.11 thf(func_def_896, type, sK532: ($i > $o) > $i > $i).
% 5.12/2.11 thf(func_def_898, type, sK534: $i > $o).
% 5.12/2.11 thf(func_def_913, type, sK549: ($i > $o) > $i > $i > $i > $i).
% 5.12/2.11 thf(func_def_914, type, sK550: ($i > $o) > $i > $i > $i > $i).
% 5.12/2.11 thf(func_def_918, type, sK554: $i > $o).
% 5.12/2.11 thf(func_def_921, type, sK557: $i > $i).
% 5.12/2.11 thf(func_def_924, type, sK560: $i > ($i > $i) > $i > $i).
% 5.12/2.11 thf(func_def_931, type, sK567: $i > $i).
% 5.12/2.11 thf(func_def_933, type, sK569: $i > $i).
% 5.12/2.11 thf(func_def_934, type, sK570: $i > $i).
% 5.12/2.11 thf(func_def_947, type, sK583: $i > $o).
% 5.12/2.11 thf(func_def_949, type, sK585: $i > ($i > $o) > $i > $i).
% 5.12/2.11 thf(func_def_957, type, sK593: $i > $i > $o).
% 5.12/2.11 thf(func_def_963, type, sK599: $i > $i > $i > $i > $i).
% 5.12/2.11 thf(func_def_964, type, sK600: $i > $i > $i > $i > $i).
% 5.12/2.11 thf(func_def_965, type, sK601: $i > $i > $i > $i > $i).
% 5.12/2.11 thf(func_def_966, type, sK602: $i > $i > $i > $i > $i).
% 5.12/2.11 thf(func_def_970, type, sK606: $i > $o).
% 5.12/2.11 thf(func_def_980, type, sK616: $i > $i > $o).
% 5.12/2.11 thf(func_def_987, type, sK623: $i > $i > $i).
% 5.12/2.11 thf(func_def_994, type, sK630: $i > $o).
% 5.12/2.11 thf(func_def_1015, type, sK651: $i > $i > $i).
% 5.12/2.11 thf(func_def_1022, type, sK658: $i > $i > $i > $i > $i).
% 5.12/2.11 thf(func_def_1040, type, sK676: $i > $i > $i > $i).
% 5.12/2.11 thf(func_def_1041, type, sK677: ($i > $o) > $i > $i).
% 5.12/2.11 thf(func_def_1043, type, sK679: $i > $o).
% 5.12/2.11 thf(func_def_1068, type, sK704: $i > $o).
% 5.12/2.11 thf(func_def_1078, type, sK714: $i > $i > $i).
% 5.12/2.11 thf(func_def_1106, type, sK742: $i > $o).
% 5.12/2.11 thf(func_def_1107, type, sK743: $i > $o).
% 5.12/2.11 thf(func_def_1108, type, sK744: ($i > $o) > ($i > $o) > $i).
% 5.12/2.11 thf(func_def_1109, type, sK745: ($i > $o) > ($i > $o) > $i).
% 5.12/2.11 thf(func_def_1119, type, sK755: $i > $o).
% 5.12/2.11 thf(func_def_1147, type, sK783: $i > $o).
% 5.12/2.11 thf(func_def_1151, type, sK787: $i > $o).
% 5.12/2.11 thf(func_def_1153, type, sK789: $i > ($i > $o) > $i).
% 5.12/2.11 thf(func_def_1164, type, sK800: $i > $i).
% 5.12/2.11 thf(func_def_1192, type, sK828: $i > $i > $i > $i).
% 5.12/2.11 thf(func_def_1193, type, sK829: $i > $i > $i > $i).
% 5.12/2.11 thf(func_def_1198, type, sK834: $i > $o).
% 5.12/2.11 thf(func_def_1200, type, sK836: $i > $o).
% 5.12/2.11 thf(func_def_1206, type, sK842: $i > $i > $o).
% 5.12/2.11 thf(func_def_1222, type, sK858: $i > $i > $i).
% 5.12/2.11 thf(func_def_1247, type, sK883: ($i > $o) > $i > $i).
% 5.12/2.11 thf(func_def_1249, type, sK885: $i > $o).
% 5.12/2.11 thf(func_def_1254, type, sK890: $i > $i > $o).
% 5.12/2.11 thf(func_def_1264, type, sK900: $i > $i).
% 5.12/2.11 thf(func_def_1267, type, sK903: ($i > $o) > $i > $i > $i).
% 5.12/2.11 thf(func_def_1268, type, sK904: ($i > $o) > $i > $i > $i).
% 5.12/2.11 thf(func_def_1271, type, sK907: $i > $o).
% 5.12/2.11 thf(func_def_1273, type, sK909: $i > $o).
% 5.12/2.11 thf(func_def_1276, type, sK912: $i > ($i > $o) > $i).
% 5.12/2.11 thf(func_def_1277, type, sK913: $i > ($i > $o) > $i).
% 5.12/2.11 thf(func_def_1299, type, sK935: $i > $o).
% 5.12/2.11 thf(func_def_1334, type, sK970: $i > $o).
% 5.12/2.11 thf(func_def_1336, type, sK972: ($i > $o) > $i > $i).
% 5.12/2.11 thf(func_def_1337, type, sK973: $i > $i).
% 5.12/2.11 thf(func_def_1340, type, sK976: $i > $i > ($i > $i) > $i).
% 5.12/2.11 thf(func_def_1364, type, sK1000: $i > $i > $o).
% 5.12/2.11 thf(func_def_1373, type, sK1009: $i > $o).
% 5.12/2.11 thf(func_def_1375, type, sK1011: ($i > $o) > $i).
% 5.12/2.11 thf(func_def_1376, type, sK1012: ($i > $o) > $i).
% 5.12/2.11 thf(func_def_1382, type, ph1018: !>[X0: $tType]:(X0)).
% 5.12/2.11 thf(f4926,plain,(
% 5.12/2.11 $false),
% 5.12/2.11 inference(avatar_sat_refutation,[status(thm)],[f4847,f4876,f4912])).
% 5.12/2.11 thf(f4912,plain,(
% 5.12/2.11 ~spl1017_6),
% 5.12/2.11 inference(avatar_contradiction_clause,[status(thm)],[f4911])).
% 5.12/2.11 thf(f4911,plain,(
% 5.12/2.11 $false | ~spl1017_6),
% 5.12/2.11 inference(trivial_inequality_removal,[status(thm)],[f4906])).
% 5.12/2.11 thf(f4906,plain,(
% 5.12/2.11 ($true != $true) | ~spl1017_6),
% 5.12/2.11 inference(superposition,[status(thm)],[f3136,f4843])).
% 5.12/2.11 thf(f4843,plain,(
% 5.12/2.11 ($true = (in @ (kpair @ sK68 @ sK69) @ sK67)) | ~spl1017_6),
% 5.12/2.11 inference(avatar_component_clause,[status(thm)],[f4841])).
% 5.12/2.11 thf(f4841,definition,(
% 5.12/2.11 spl1017_6 <=> ($true = (in @ (kpair @ sK68 @ sK69) @ sK67))),
% 5.12/2.11 introduced(definition,[new_symbols(naming,[spl1017_6])],[avatar_definition])).
% 5.12/2.11 thf(f3136,plain,(
% 5.12/2.11 ($true != (in @ (kpair @ sK68 @ sK69) @ sK67))),
% 5.12/2.11 inference(cnf_transformation,[status(thm)],[f1443])).
% 5.12/2.11 thf(f1443,plain,(
% 5.12/2.11 (complementTcontraSubset = $true) & (cartprodsndpairEq = $true) & (setukpairinjR2 = $true) & (setminusELneg = $true) & (inIntersectImpInUnion2 = $true) & (setminusSubset1 = $true) & (ap2apEq1 = $true) & (demorgan2b = $true) & (vacuousDall = $true) & (iftrueorfalse = $true) & (breln1invprop = $true) & (subsetTI = $true) & (exuI1 = $true) & (setukpairinjL1 = $true) & (quantDeMorgan2 = $true) & (ksndpairEq = $true) & (symdiffI1 = $true) & (notinemptyset = $true) & (setadjoinSub2 = $true) & (demorgan2 = $true) & (exuE3u = $true) & (singletonsuniq = $true) & (upairsetIL = $true) & (setunionsingleton1 = $true) & (inIntersectImpInIntersectUnions = $true) & (woz1_2 = $true) & (contrasubsetT = $true) & (emptyset__Cong = $true) & (descr__Cong = $true) & (binintersectSubset3 = $true) & (beta2 = $true) & (ifSingleton = $true) & (complementUnionInPowersetComplement = $true) & (singletoninpowunion = $true) & (funcGraphProp1 = $true) & (doubleComplementEq = $true) & (setextT = $true) & (foundationAx = $true) & (replAx = $true) & (setminusI = $true) & (dsetconstrI = $true) & (inCongP = $true) & (setextAx = $true) & (binintersectI = $true) & (binunionTEcontra = $true) & (eqbreln = $true) & (funcext = $true) & (breln1compE = $true) & (powersetI1 = $true) & (setadjoinAx = $true) & (prop2setE = $true) & (cartprodpairsurjEq = $true) & (woz13rule2 = $true) & (setukpairinjR12 = $true) & (binintersectEL = $true) & (quantDeMorgan1 = $true) & (inComplementUnionImpInComplement1 = $true) & (setukpairinjR11 = $true) & (binunionRsub = $true) & (emptyinunitempty = $true) & (breln1unionprop = $true) & (subsetI2 = $true) & (doubleComplementI1 = $true) & (uniqinunit = $true) & (eqbreln1 = $true) & (sepInPowerset = $true) & (binunionTE = $true) & (upairsetE = $true) & (exuE2 = $true) & (complementInPowersetComplementIntersect = $true) & (omegaSAx = $true) & (breln1invE = $true) & (dpsetconstrEL2 = $true) & (prop2set2propI = $true) & (cartprodmempair = $true) & (omega__Cong = $true) & (doubleComplementSub1 = $true) & (setukpairinjR1 = $true) & (descrp = $true) & (powersetT_lem = $true) & (setminusILneg = $true) & (kpairsurjEq = $true) & (binunionTILcontra = $true) & (woz13rule4 = $true) & (setadjoinE = $true) & (setextsub = $true) & (dsetconstrER = $true) & (dpsetconstrEL1 = $true) & (complementSubsetComplementIntersect = $true) & (binintersectT_lem = $true) & (setukpairinjR = $true) & (ksndsingleton = $true) & (notequalI2 = $true) & (binintersectSubset1 = $true) & (inComplementUnionImpNotIn1 = $true) & (subbreln = $true) & (breln1all2 = $true) & (eta2 = $true) & (cartprodmempair1 = $true) & (setunionE = $true) & (exuI3 = $true) & (setadjoinIR = $true) & (infuncsetfunc = $true) & (setminusT_lem = $true) & (setminusEL = $true) & (symdiffI2 = $true) & (funcext2 = $true) & (upairsetIR = $true) & (complementTE1 = $true) & (setunionsingleton2 = $true) & (emptysetimpfalse = $true) & (singletonprop = $true) & (prop2setI = $true) & (cartprodpairin = $true) & (binunionIL = $true) & (contrasubsetT1 = $true) & (woz13rule0 = $true) & (binintersectSubset2 = $true) & (upairset2IR = $true) & (dpsetconstrI = $true) & (secondinupair = $true) & (upairinpowunion = $true) & (breln1unionIL = $true) & (notdexE = $true) & (symdiffIneg1 = $true) & (disjointsetsI1 = $true) & (demorgan1b = $true) & (doubleComplementSub2 = $true) & (binintersectTERcontra = $true) & (ap2p = $true) & (dpsetconstrSub = $true) & (subsetI1 = $true) & (exuEu = $true) & (setminusSubset2 = $true) & (funcinfuncset = $true) & (setukpairinjL = $true) & (ubforcartprodlem2 = $true) & (subsetemptysetimpeq = $true) & (complementImpComplementIntersect = $true) & (subbreln1 = $true) & (powerset__Cong = $true) & (ifp = $true) & (breln1unionIR = $true) & (subsetTrans = $true) & (symdiffE = $true) & (cartprodpairmemER = $true) & (setoftrueEq = $true) & (brelnall2 = $true) & (omega0Ax = $true) & (kpairp = $true) & (eqinunit = $true) & (upairequniteq = $true) & ((((($true != (in @ (kpair @ sK68 @ sK69) @ sK65)) & ($true = (in @ (kpair @ sK68 @ sK69) @ (binunion @ sK65 @ sK67))) & ($true != (in @ (kpair @ sK68 @ sK69) @ sK67)) & ($true = (in @ sK69 @ sK66))) & ((in @ sK68 @ sK66) = $true)) & ($true = (breln1 @ sK66 @ sK67))) & ((breln1 @ sK66 @ sK65) = $true)) & (contraSubsetComplement = $true) & (quantDeMorgan4 = $true) & (emptysetsubset = $true) & (binintersectSubset4 = $true) & (iftrueProp1 = $true) & (setukpairIR = $true) & (complementT_lem = $true) & (theprop = $true) & (woz1_4 = $true) & (kfstpairEq = $true) & (kfstsingleton = $true) & (powersetTE1 = $true) & (demorgan1a = $true) & (quantDeMorgan3 = $true) & (emptyE1 = $true) & (setukpairinjL2 = $true) & (funcGraphProp2 = $true) & (setbeta = $true) & (breln1compI = $true) & (breln1compEex = $true) & (setminusERneg = $true) & (singletonsswitch = $true) & (emptyI = $true) & (binintersectER = $true) & (breln1invI = $true) & (funcGraphProp3 = $true) & (eta1 = $true) & (binintersectSubset5 = $true) & (notdallE = $true) & (iffalse = $true) & (nonemptyI = $true) & (setunionAx = $true) & (cartprodpairmemEL = $true) & (in__Cong = $true) & (binunionEcases = $true) & (cartprodsndin = $true) & (setunionsingleton = $true) & (iffalseProp1 = $true) & (sepSubset = $true) & (doubleComplementE1 = $true) & (demorgan2a1 = $true) & (binunionTIRcontra = $true) & (exuE1 = $true) & (ubforcartprodlem3 = $true) & (demorgan2b2 = $true) & (powersetE1 = $true) & (setminusIRneg = $true) & (demorgan2a = $true) & (woz1_1 = $true) & (funcGraphProp4 = $true) & (singletoninpowerset = $true) & (notequalI1 = $true) & (dpsetconstrERa = $true) & (powersetsubset = $true) & (theeq = $true) & (inIntersectImpInUnion = $true) & (lam2lamEq = $true) & (setminusLsub = $true) & (emptyInPowerset = $true) & (nonemptyE1 = $true) & (contrasubsetT2 = $true) & (cartprodfstpairEq = $true) & (powersetTI1 = $true) & (inPowerset = $true) & (ex1E1 = $true) & (iftrueProp2 = $true) & (setadjoinOr = $true) & (wellorderingAx = $true) & (setminusER = $true) & (noeltsimpempty = $true) & (singletonsubset = $true) & (symdiffIneg2 = $true) & (eqimpsubset2 = $true) & (subsetE2 = $true) & (demorgan2a2 = $true) & (woz13rule3 = $true) & (cartprodmempaircEq = $true) & (omegaIndAx = $true) & (intersectInPowersetIntersectUnions = $true) & (iffalseProp2 = $true) & (breln1compprop = $true) & (ex1E2 = $true) & (powersetAx = $true) & (eqimpsubset1 = $true) & (emptysetE = $true) & (emptyinPowerset = $true) & (upairsubunion = $true) & (setukpairIL = $true) & (ex1I = $true) & (nonemptyI1 = $true) & (bs114d = $true) & (exuE3e = $true) & (subset2powerset = $true) & (powersetE = $true) & (funcImageSingleton = $true) & (setunionI = $true) & (binunionLsub = $true) & (ex1I2 = $true) & (emptysetAx = $true) & (complementTI1 = $true) & (kpairiskpair = $true) & (woz1_3 = $true) & (setadjoin__Cong = $true) & (demorgan1 = $true) & (dsetconstrEL = $true) & (subsetRefl = $true) & (exuI2 = $true) & (setunion__Cong = $true) & (breln1all1 = $true) & (notinsingleton = $true) & (woz13rule1 = $true) & (lamProp = $true) & (apProp = $true) & (lam2p = $true) & (beta1 = $true) & (dsetconstr__Cong = $true) & (notsubsetI = $true) & (setext = $true) & (funcextLem = $true) & (binunionIR = $true) & (subPowSU = $true) & (woz1_5 = $true) & (app = $true) & (powersetI = $true) & (choice2fnsingleton = $true) & (setadjoinIL = $true) & (contrasubsetT3 = $true) & (setunionE2 = $true) & (binunionT_lem = $true) & (brelnall1 = $true) & (binintersectTELcontra = $true) & (binintersectLsub = $true) & (setadjoinSub = $true) & (breln1unionI = $true) & (ap2apEq2 = $true) & (cartprodfstin = $true) & (binintersectRsub = $true) & (upairset2E = $true) & (binunionE = $true) & (subsetE = $true) & (ubforcartprodlem1 = $true) & (exu__Cong = $true) & (setOfPairsIsBReln = $true) & (setOfPairsIsBReln1 = $true) & (dpsetconstrER = $true) & (nonemptyImpWitness = $true) & (breln1SetBreln1 = $true) & (complementTnotintersectT = $true) & (lamp = $true) & (iftrue = $true)),
% 5.12/2.11 inference(skolemisation,[status(esa),new_symbols(skolem,[sK65,sK66,sK67,sK68,sK69])],[f1154,f1442,f1441,f1440,f1439])).
% 5.12/2.11 thf(f1439,plain,(
% 5.12/2.11 ? [X0 : $i,X1 : $i] : (? [X2 : $i] : (? [X3 : $i] : (? [X4 : $i] : (((in @ (kpair @ X3 @ X4) @ X0) != $true) & ($true = (in @ (kpair @ X3 @ X4) @ (binunion @ X0 @ X2))) & ((in @ (kpair @ X3 @ X4) @ X2) != $true) & ($true = (in @ X4 @ X1))) & ($true = (in @ X3 @ X1))) & ((breln1 @ X1 @ X2) = $true)) & ((breln1 @ X1 @ X0) = $true)) => (? [X2 : $i] : (? [X3 : $i] : (? [X4 : $i] : (($true != (in @ (kpair @ X3 @ X4) @ sK65)) & ((in @ (kpair @ X3 @ X4) @ (binunion @ sK65 @ X2)) = $true) & ((in @ (kpair @ X3 @ X4) @ X2) != $true) & ($true = (in @ X4 @ sK66))) & ($true = (in @ X3 @ sK66))) & ((breln1 @ sK66 @ X2) = $true)) & ((breln1 @ sK66 @ sK65) = $true))),
% 5.12/2.11 introduced(definition,[],[choice_axiom])).
% 5.12/2.11 thf(f1440,plain,(
% 5.12/2.11 ? [X2 : $i] : (? [X3 : $i] : (? [X4 : $i] : (($true != (in @ (kpair @ X3 @ X4) @ sK65)) & ((in @ (kpair @ X3 @ X4) @ (binunion @ sK65 @ X2)) = $true) & ((in @ (kpair @ X3 @ X4) @ X2) != $true) & ($true = (in @ X4 @ sK66))) & ($true = (in @ X3 @ sK66))) & ((breln1 @ sK66 @ X2) = $true)) => (? [X3 : $i] : (? [X4 : $i] : (($true != (in @ (kpair @ X3 @ X4) @ sK65)) & ((in @ (kpair @ X3 @ X4) @ (binunion @ sK65 @ sK67)) = $true) & ($true != (in @ (kpair @ X3 @ X4) @ sK67)) & ($true = (in @ X4 @ sK66))) & ($true = (in @ X3 @ sK66))) & ($true = (breln1 @ sK66 @ sK67)))),
% 5.12/2.11 introduced(definition,[],[choice_axiom])).
% 5.12/2.11 thf(f1441,plain,(
% 5.12/2.11 ? [X3 : $i] : (? [X4 : $i] : (($true != (in @ (kpair @ X3 @ X4) @ sK65)) & ((in @ (kpair @ X3 @ X4) @ (binunion @ sK65 @ sK67)) = $true) & ($true != (in @ (kpair @ X3 @ X4) @ sK67)) & ($true = (in @ X4 @ sK66))) & ($true = (in @ X3 @ sK66))) => (? [X4 : $i] : (($true != (in @ (kpair @ sK68 @ X4) @ sK65)) & ((in @ (kpair @ sK68 @ X4) @ (binunion @ sK65 @ sK67)) = $true) & ((in @ (kpair @ sK68 @ X4) @ sK67) != $true) & ($true = (in @ X4 @ sK66))) & ((in @ sK68 @ sK66) = $true))),
% 5.12/2.11 introduced(definition,[],[choice_axiom])).
% 5.12/2.11 thf(f1442,plain,(
% 5.12/2.11 ? [X4 : $i] : (($true != (in @ (kpair @ sK68 @ X4) @ sK65)) & ((in @ (kpair @ sK68 @ X4) @ (binunion @ sK65 @ sK67)) = $true) & ((in @ (kpair @ sK68 @ X4) @ sK67) != $true) & ($true = (in @ X4 @ sK66))) => (($true != (in @ (kpair @ sK68 @ sK69) @ sK65)) & ($true = (in @ (kpair @ sK68 @ sK69) @ (binunion @ sK65 @ sK67))) & ($true != (in @ (kpair @ sK68 @ sK69) @ sK67)) & ($true = (in @ sK69 @ sK66)))),
% 5.12/2.11 introduced(definition,[],[choice_axiom])).
% 5.12/2.11 thf(f1154,plain,(
% 5.12/2.11 (complementTcontraSubset = $true) & (cartprodsndpairEq = $true) & (setukpairinjR2 = $true) & (setminusELneg = $true) & (inIntersectImpInUnion2 = $true) & (setminusSubset1 = $true) & (ap2apEq1 = $true) & (demorgan2b = $true) & (vacuousDall = $true) & (iftrueorfalse = $true) & (breln1invprop = $true) & (subsetTI = $true) & (exuI1 = $true) & (setukpairinjL1 = $true) & (quantDeMorgan2 = $true) & (ksndpairEq = $true) & (symdiffI1 = $true) & (notinemptyset = $true) & (setadjoinSub2 = $true) & (demorgan2 = $true) & (exuE3u = $true) & (singletonsuniq = $true) & (upairsetIL = $true) & (setunionsingleton1 = $true) & (inIntersectImpInIntersectUnions = $true) & (woz1_2 = $true) & (contrasubsetT = $true) & (emptyset__Cong = $true) & (descr__Cong = $true) & (binintersectSubset3 = $true) & (beta2 = $true) & (ifSingleton = $true) & (complementUnionInPowersetComplement = $true) & (singletoninpowunion = $true) & (funcGraphProp1 = $true) & (doubleComplementEq = $true) & (setextT = $true) & (foundationAx = $true) & (replAx = $true) & (setminusI = $true) & (dsetconstrI = $true) & (inCongP = $true) & (setextAx = $true) & (binintersectI = $true) & (binunionTEcontra = $true) & (eqbreln = $true) & (funcext = $true) & (breln1compE = $true) & (powersetI1 = $true) & (setadjoinAx = $true) & (prop2setE = $true) & (cartprodpairsurjEq = $true) & (woz13rule2 = $true) & (setukpairinjR12 = $true) & (binintersectEL = $true) & (quantDeMorgan1 = $true) & (inComplementUnionImpInComplement1 = $true) & (setukpairinjR11 = $true) & (binunionRsub = $true) & (emptyinunitempty = $true) & (breln1unionprop = $true) & (subsetI2 = $true) & (doubleComplementI1 = $true) & (uniqinunit = $true) & (eqbreln1 = $true) & (sepInPowerset = $true) & (binunionTE = $true) & (upairsetE = $true) & (exuE2 = $true) & (complementInPowersetComplementIntersect = $true) & (omegaSAx = $true) & (breln1invE = $true) & (dpsetconstrEL2 = $true) & (prop2set2propI = $true) & (cartprodmempair = $true) & (omega__Cong = $true) & (doubleComplementSub1 = $true) & (setukpairinjR1 = $true) & (descrp = $true) & (powersetT_lem = $true) & (setminusILneg = $true) & (kpairsurjEq = $true) & (binunionTILcontra = $true) & (woz13rule4 = $true) & (setadjoinE = $true) & (setextsub = $true) & (dsetconstrER = $true) & (dpsetconstrEL1 = $true) & (complementSubsetComplementIntersect = $true) & (binintersectT_lem = $true) & (setukpairinjR = $true) & (ksndsingleton = $true) & (notequalI2 = $true) & (binintersectSubset1 = $true) & (inComplementUnionImpNotIn1 = $true) & (subbreln = $true) & (breln1all2 = $true) & (eta2 = $true) & (cartprodmempair1 = $true) & (setunionE = $true) & (exuI3 = $true) & (setadjoinIR = $true) & (infuncsetfunc = $true) & (setminusT_lem = $true) & (setminusEL = $true) & (symdiffI2 = $true) & (funcext2 = $true) & (upairsetIR = $true) & (complementTE1 = $true) & (setunionsingleton2 = $true) & (emptysetimpfalse = $true) & (singletonprop = $true) & (prop2setI = $true) & (cartprodpairin = $true) & (binunionIL = $true) & (contrasubsetT1 = $true) & (woz13rule0 = $true) & (binintersectSubset2 = $true) & (upairset2IR = $true) & (dpsetconstrI = $true) & (secondinupair = $true) & (upairinpowunion = $true) & (breln1unionIL = $true) & (notdexE = $true) & (symdiffIneg1 = $true) & (disjointsetsI1 = $true) & (demorgan1b = $true) & (doubleComplementSub2 = $true) & (binintersectTERcontra = $true) & (ap2p = $true) & (dpsetconstrSub = $true) & (subsetI1 = $true) & (exuEu = $true) & (setminusSubset2 = $true) & (funcinfuncset = $true) & (setukpairinjL = $true) & (ubforcartprodlem2 = $true) & (subsetemptysetimpeq = $true) & (complementImpComplementIntersect = $true) & (subbreln1 = $true) & (powerset__Cong = $true) & (ifp = $true) & (breln1unionIR = $true) & (subsetTrans = $true) & (symdiffE = $true) & (cartprodpairmemER = $true) & (setoftrueEq = $true) & (brelnall2 = $true) & (omega0Ax = $true) & (kpairp = $true) & (eqinunit = $true) & (upairequniteq = $true) & ? [X0 : $i,X1 : $i] : (? [X2 : $i] : (? [X3 : $i] : (? [X4 : $i] : (((in @ (kpair @ X3 @ X4) @ X0) != $true) & ($true = (in @ (kpair @ X3 @ X4) @ (binunion @ X0 @ X2))) & ((in @ (kpair @ X3 @ X4) @ X2) != $true) & ($true = (in @ X4 @ X1))) & ($true = (in @ X3 @ X1))) & ((breln1 @ X1 @ X2) = $true)) & ((breln1 @ X1 @ X0) = $true)) & (contraSubsetComplement = $true) & (quantDeMorgan4 = $true) & (emptysetsubset = $true) & (binintersectSubset4 = $true) & (iftrueProp1 = $true) & (setukpairIR = $true) & (complementT_lem = $true) & (theprop = $true) & (woz1_4 = $true) & (kfstpairEq = $true) & (kfstsingleton = $true) & (powersetTE1 = $true) & (demorgan1a = $true) & (quantDeMorgan3 = $true) & (emptyE1 = $true) & (setukpairinjL2 = $true) & (funcGraphProp2 = $true) & (setbeta = $true) & (breln1compI = $true) & (breln1compEex = $true) & (setminusERneg = $true) & (singletonsswitch = $true) & (emptyI = $true) & (binintersectER = $true) & (breln1invI = $true) & (funcGraphProp3 = $true) & (eta1 = $true) & (binintersectSubset5 = $true) & (notdallE = $true) & (iffalse = $true) & (nonemptyI = $true) & (setunionAx = $true) & (cartprodpairmemEL = $true) & (in__Cong = $true) & (binunionEcases = $true) & (cartprodsndin = $true) & (setunionsingleton = $true) & (iffalseProp1 = $true) & (sepSubset = $true) & (doubleComplementE1 = $true) & (demorgan2a1 = $true) & (binunionTIRcontra = $true) & (exuE1 = $true) & (ubforcartprodlem3 = $true) & (demorgan2b2 = $true) & (powersetE1 = $true) & (setminusIRneg = $true) & (demorgan2a = $true) & (woz1_1 = $true) & (funcGraphProp4 = $true) & (singletoninpowerset = $true) & (notequalI1 = $true) & (dpsetconstrERa = $true) & (powersetsubset = $true) & (theeq = $true) & (inIntersectImpInUnion = $true) & (lam2lamEq = $true) & (setminusLsub = $true) & (emptyInPowerset = $true) & (nonemptyE1 = $true) & (contrasubsetT2 = $true) & (cartprodfstpairEq = $true) & (powersetTI1 = $true) & (inPowerset = $true) & (ex1E1 = $true) & (iftrueProp2 = $true) & (setadjoinOr = $true) & (wellorderingAx = $true) & (setminusER = $true) & (noeltsimpempty = $true) & (singletonsubset = $true) & (symdiffIneg2 = $true) & (eqimpsubset2 = $true) & (subsetE2 = $true) & (demorgan2a2 = $true) & (woz13rule3 = $true) & (cartprodmempaircEq = $true) & (omegaIndAx = $true) & (intersectInPowersetIntersectUnions = $true) & (iffalseProp2 = $true) & (breln1compprop = $true) & (ex1E2 = $true) & (powersetAx = $true) & (eqimpsubset1 = $true) & (emptysetE = $true) & (emptyinPowerset = $true) & (upairsubunion = $true) & (setukpairIL = $true) & (ex1I = $true) & (nonemptyI1 = $true) & (bs114d = $true) & (exuE3e = $true) & (subset2powerset = $true) & (powersetE = $true) & (funcImageSingleton = $true) & (setunionI = $true) & (binunionLsub = $true) & (ex1I2 = $true) & (emptysetAx = $true) & (complementTI1 = $true) & (kpairiskpair = $true) & (woz1_3 = $true) & (setadjoin__Cong = $true) & (demorgan1 = $true) & (dsetconstrEL = $true) & (subsetRefl = $true) & (exuI2 = $true) & (setunion__Cong = $true) & (breln1all1 = $true) & (notinsingleton = $true) & (woz13rule1 = $true) & (lamProp = $true) & (apProp = $true) & (lam2p = $true) & (beta1 = $true) & (dsetconstr__Cong = $true) & (notsubsetI = $true) & (setext = $true) & (funcextLem = $true) & (binunionIR = $true) & (subPowSU = $true) & (woz1_5 = $true) & (app = $true) & (powersetI = $true) & (choice2fnsingleton = $true) & (setadjoinIL = $true) & (contrasubsetT3 = $true) & (setunionE2 = $true) & (binunionT_lem = $true) & (brelnall1 = $true) & (binintersectTELcontra = $true) & (binintersectLsub = $true) & (setadjoinSub = $true) & (breln1unionI = $true) & (ap2apEq2 = $true) & (cartprodfstin = $true) & (binintersectRsub = $true) & (upairset2E = $true) & (binunionE = $true) & (subsetE = $true) & (ubforcartprodlem1 = $true) & (exu__Cong = $true) & (setOfPairsIsBReln = $true) & (setOfPairsIsBReln1 = $true) & (dpsetconstrER = $true) & (nonemptyImpWitness = $true) & (breln1SetBreln1 = $true) & (complementTnotintersectT = $true) & (lamp = $true) & (iftrue = $true)),
% 5.12/2.11 inference(flattening,[status(thm)],[f1153])).
% 5.12/2.11 thf(f1153,plain,(
% 5.12/2.11 (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i,X1 : $i] : (? [X2 : $i] : (? [X3 : $i] : (? [X4 : $i] : (((((in @ (kpair @ X3 @ X4) @ X2) != $true) & ((in @ (kpair @ X3 @ X4) @ X0) != $true)) & ($true = (in @ (kpair @ X3 @ X4) @ (binunion @ X0 @ X2)))) & ($true = (in @ X4 @ X1))) & ($true = (in @ X3 @ X1))) & ((breln1 @ X1 @ X2) = $true)) & ((breln1 @ X1 @ X0) = $true)) & (breln1unionI = $true)) & (breln1unionIR = $true)) & (breln1unionIL = $true)) & (breln1unionprop = $true)) & (breln1compEex = $true)) & (breln1compE = $true)) & (breln1compI = $true)) & (breln1compprop = $true)) & (breln1invE = $true)) & (breln1invI = $true)) & (breln1invprop = $true)) & (eqbreln1 = $true)) & (subbreln1 = $true)) & (breln1all1 = $true)) & (setOfPairsIsBReln1 = $true)) & (choice2fnsingleton = $true)) & (breln1SetBreln1 = $true)) & (breln1all2 = $true)) & (woz1_5 = $true)) & (woz1_4 = $true)) & (woz1_3 = $true)) & (woz1_2 = $true)) & (woz1_1 = $true)) & (woz13rule4 = $true)) & (woz13rule3 = $true)) & (woz13rule2 = $true)) & (woz13rule1 = $true)) & (woz13rule0 = $true)) & (demorgan2 = $true)) & (demorgan2b = $true)) & (demorgan2b2 = $true)) & (demorgan2a = $true)) & (demorgan1 = $true)) & (demorgan1b = $true)) & (demorgan1a = $true)) & (demorgan2a2 = $true)) & (complementUnionInPowersetComplement = $true)) & (demorgan2a1 = $true)) & (binunionTEcontra = $true)) & (binunionTE = $true)) & (inComplementUnionImpInComplement1 = $true)) & (inComplementUnionImpNotIn1 = $true)) & (intersectInPowersetIntersectUnions = $true)) & (inIntersectImpInIntersectUnions = $true)) & (inIntersectImpInUnion2 = $true)) & (inIntersectImpInUnion = $true)) & (binunionTIRcontra = $true)) & (binunionTILcontra = $true)) & (complementTcontraSubset = $true)) & (contraSubsetComplement = $true)) & (complementInPowersetComplementIntersect = $true)) & (complementSubsetComplementIntersect = $true)) & (complementImpComplementIntersect = $true)) & (complementTnotintersectT = $true)) & (doubleComplementEq = $true)) & (doubleComplementSub2 = $true)) & (doubleComplementSub1 = $true)) & (doubleComplementE1 = $true)) & (doubleComplementI1 = $true)) & (contrasubsetT3 = $true)) & (contrasubsetT2 = $true)) & (contrasubsetT1 = $true)) & (contrasubsetT = $true)) & (binintersectTERcontra = $true)) & (binintersectTELcontra = $true)) & (complementTE1 = $true)) & (complementTI1 = $true)) & (powersetTE1 = $true)) & (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)),
% 5.12/2.11 inference(ennf_transformation,[status(thm)],[f780])).
% 5.12/2.11 thf(f780,plain,(
% 5.12/2.11 ~((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) => ((powersetTE1 = $true) => ((complementTI1 = $true) => ((complementTE1 = $true) => ((binintersectTELcontra = $true) => ((binintersectTERcontra = $true) => ((contrasubsetT = $true) => ((contrasubsetT1 = $true) => ((contrasubsetT2 = $true) => ((contrasubsetT3 = $true) => ((doubleComplementI1 = $true) => ((doubleComplementE1 = $true) => ((doubleComplementSub1 = $true) => ((doubleComplementSub2 = $true) => ((doubleComplementEq = $true) => ((complementTnotintersectT = $true) => ((complementImpComplementIntersect = $true) => ((complementSubsetComplementIntersect = $true) => ((complementInPowersetComplementIntersect = $true) => ((contraSubsetComplement = $true) => ((complementTcontraSubset = $true) => ((binunionTILcontra = $true) => ((binunionTIRcontra = $true) => ((inIntersectImpInUnion = $true) => ((inIntersectImpInUnion2 = $true) => ((inIntersectImpInIntersectUnions = $true) => ((intersectInPowersetIntersectUnions = $true) => ((inComplementUnionImpNotIn1 = $true) => ((inComplementUnionImpInComplement1 = $true) => ((binunionTE = $true) => ((binunionTEcontra = $true) => ((demorgan2a1 = $true) => ((complementUnionInPowersetComplement = $true) => ((demorgan2a2 = $true) => ((demorgan1a = $true) => ((demorgan1b = $true) => ((demorgan1 = $true) => ((demorgan2a = $true) => ((demorgan2b2 = $true) => ((demorgan2b = $true) => ((demorgan2 = $true) => ((woz13rule0 = $true) => ((woz13rule1 = $true) => ((woz13rule2 = $true) => ((woz13rule3 = $true) => ((woz13rule4 = $true) => ((woz1_1 = $true) => ((woz1_2 = $true) => ((woz1_3 = $true) => ((woz1_4 = $true) => ((woz1_5 = $true) => ((breln1all2 = $true) => ((breln1SetBreln1 = $true) => ((choice2fnsingleton = $true) => ((setOfPairsIsBReln1 = $true) => ((breln1all1 = $true) => ((subbreln1 = $true) => ((eqbreln1 = $true) => ((breln1invprop = $true) => ((breln1invI = $true) => ((breln1invE = $true) => ((breln1compprop = $true) => ((breln1compI = $true) => ((breln1compE = $true) => ((breln1compEex = $true) => ((breln1unionprop = $true) => ((breln1unionIL = $true) => ((breln1unionIR = $true) => ((breln1unionI = $true) => ! [X0 : $i,X1 : $i] : (((breln1 @ X1 @ X0) = $true) => ! [X2 : $i] : (((breln1 @ X1 @ X2) = $true) => ! [X3 : $i] : (($true = (in @ X3 @ X1)) => ! [X4 : $i] : (($true = (in @ X4 @ X1)) => (($true = (in @ (kpair @ X3 @ X4) @ (binunion @ X0 @ X2))) => (((in @ (kpair @ X3 @ X4) @ X2) = $true) | ((in @ (kpair @ X3 @ X4) @ X0) = $true)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 5.12/2.11 inference(fool_elimination,[status(thm)],[f779])).
% 5.12/2.11 thf(f779,plain,(
% 5.12/2.11 ~(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 => (powersetTE1 => (complementTI1 => (complementTE1 => (binintersectTELcontra => (binintersectTERcontra => (contrasubsetT => (contrasubsetT1 => (contrasubsetT2 => (contrasubsetT3 => (doubleComplementI1 => (doubleComplementE1 => (doubleComplementSub1 => (doubleComplementSub2 => (doubleComplementEq => (complementTnotintersectT => (complementImpComplementIntersect => (complementSubsetComplementIntersect => (complementInPowersetComplementIntersect => (contraSubsetComplement => (complementTcontraSubset => (binunionTILcontra => (binunionTIRcontra => (inIntersectImpInUnion => (inIntersectImpInUnion2 => (inIntersectImpInIntersectUnions => (intersectInPowersetIntersectUnions => (inComplementUnionImpNotIn1 => (inComplementUnionImpInComplement1 => (binunionTE => (binunionTEcontra => (demorgan2a1 => (complementUnionInPowersetComplement => (demorgan2a2 => (demorgan1a => (demorgan1b => (demorgan1 => (demorgan2a => (demorgan2b2 => (demorgan2b => (demorgan2 => (woz13rule0 => (woz13rule1 => (woz13rule2 => (woz13rule3 => (woz13rule4 => (woz1_1 => (woz1_2 => (woz1_3 => (woz1_4 => (woz1_5 => (breln1all2 => (breln1SetBreln1 => (choice2fnsingleton => (setOfPairsIsBReln1 => (breln1all1 => (subbreln1 => (eqbreln1 => (breln1invprop => (breln1invI => (breln1invE => (breln1compprop => (breln1compI => (breln1compE => (breln1compEex => (breln1unionprop => (breln1unionIL => (breln1unionIR => (breln1unionI => ! [X0 : $i,X1 : $i] : ((breln1 @ X1 @ X0) => ! [X2 : $i] : ((breln1 @ X1 @ X2) => ! [X3 : $i] : ((in @ X3 @ X1) => ! [X4 : $i] : ((in @ X4 @ X1) => ((in @ (kpair @ X3 @ X4) @ (binunion @ X0 @ X2)) => ((in @ (kpair @ X3 @ X4) @ X0) | (in @ (kpair @ X3 @ X4) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 5.12/2.11 inference(rectify,[status(thm)],[f306])).
% 5.12/2.11 thf(f306,negated_conjecture,(
% 5.12/2.11 ~(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 => (powersetTE1 => (complementTI1 => (complementTE1 => (binintersectTELcontra => (binintersectTERcontra => (contrasubsetT => (contrasubsetT1 => (contrasubsetT2 => (contrasubsetT3 => (doubleComplementI1 => (doubleComplementE1 => (doubleComplementSub1 => (doubleComplementSub2 => (doubleComplementEq => (complementTnotintersectT => (complementImpComplementIntersect => (complementSubsetComplementIntersect => (complementInPowersetComplementIntersect => (contraSubsetComplement => (complementTcontraSubset => (binunionTILcontra => (binunionTIRcontra => (inIntersectImpInUnion => (inIntersectImpInUnion2 => (inIntersectImpInIntersectUnions => (intersectInPowersetIntersectUnions => (inComplementUnionImpNotIn1 => (inComplementUnionImpInComplement1 => (binunionTE => (binunionTEcontra => (demorgan2a1 => (complementUnionInPowersetComplement => (demorgan2a2 => (demorgan1a => (demorgan1b => (demorgan1 => (demorgan2a => (demorgan2b2 => (demorgan2b => (demorgan2 => (woz13rule0 => (woz13rule1 => (woz13rule2 => (woz13rule3 => (woz13rule4 => (woz1_1 => (woz1_2 => (woz1_3 => (woz1_4 => (woz1_5 => (breln1all2 => (breln1SetBreln1 => (choice2fnsingleton => (setOfPairsIsBReln1 => (breln1all1 => (subbreln1 => (eqbreln1 => (breln1invprop => (breln1invI => (breln1invE => (breln1compprop => (breln1compI => (breln1compE => (breln1compEex => (breln1unionprop => (breln1unionIL => (breln1unionIR => (breln1unionI => ! [X13 : $i,X3 : $i] : ((breln1 @ X3 @ X13) => ! [X15 : $i] : ((breln1 @ X3 @ X15) => ! [X1 : $i] : ((in @ X1 @ X3) => ! [X2 : $i] : ((in @ X2 @ X3) => ((in @ (kpair @ X1 @ X2) @ (binunion @ X13 @ X15)) => ((in @ (kpair @ X1 @ X2) @ X13) | (in @ (kpair @ X1 @ X2) @ X15)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 5.12/2.11 inference(negated_conjecture,[status(cth)],[f305])).
% 5.12/2.11 thf(f305,conjecture,(
% 5.12/2.11 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 => (powersetTE1 => (complementTI1 => (complementTE1 => (binintersectTELcontra => (binintersectTERcontra => (contrasubsetT => (contrasubsetT1 => (contrasubsetT2 => (contrasubsetT3 => (doubleComplementI1 => (doubleComplementE1 => (doubleComplementSub1 => (doubleComplementSub2 => (doubleComplementEq => (complementTnotintersectT => (complementImpComplementIntersect => (complementSubsetComplementIntersect => (complementInPowersetComplementIntersect => (contraSubsetComplement => (complementTcontraSubset => (binunionTILcontra => (binunionTIRcontra => (inIntersectImpInUnion => (inIntersectImpInUnion2 => (inIntersectImpInIntersectUnions => (intersectInPowersetIntersectUnions => (inComplementUnionImpNotIn1 => (inComplementUnionImpInComplement1 => (binunionTE => (binunionTEcontra => (demorgan2a1 => (complementUnionInPowersetComplement => (demorgan2a2 => (demorgan1a => (demorgan1b => (demorgan1 => (demorgan2a => (demorgan2b2 => (demorgan2b => (demorgan2 => (woz13rule0 => (woz13rule1 => (woz13rule2 => (woz13rule3 => (woz13rule4 => (woz1_1 => (woz1_2 => (woz1_3 => (woz1_4 => (woz1_5 => (breln1all2 => (breln1SetBreln1 => (choice2fnsingleton => (setOfPairsIsBReln1 => (breln1all1 => (subbreln1 => (eqbreln1 => (breln1invprop => (breln1invI => (breln1invE => (breln1compprop => (breln1compI => (breln1compE => (breln1compEex => (breln1unionprop => (breln1unionIL => (breln1unionIR => (breln1unionI => ! [X13 : $i,X3 : $i] : ((breln1 @ X3 @ X13) => ! [X15 : $i] : ((breln1 @ X3 @ X15) => ! [X1 : $i] : ((in @ X1 @ X3) => ! [X2 : $i] : ((in @ X2 @ X3) => ((in @ (kpair @ X1 @ X2) @ (binunion @ X13 @ X15)) => ((in @ (kpair @ X1 @ X2) @ X13) | (in @ (kpair @ X1 @ X2) @ X15))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 5.12/2.11 file('/export/starexec/sandbox/tmp/tmp.onid0QyU2k/DTF2THF_25019.p',breln1unionE)).
% 5.12/2.11 thf(f4876,plain,(
% 5.12/2.11 ~spl1017_4),
% 5.12/2.11 inference(avatar_contradiction_clause,[status(thm)],[f4875])).
% 5.12/2.11 thf(f4875,plain,(
% 5.12/2.11 $false | ~spl1017_4),
% 5.12/2.11 inference(trivial_inequality_removal,[status(thm)],[f4870])).
% 5.12/2.11 thf(f4870,plain,(
% 5.12/2.11 ($true != $true) | ~spl1017_4),
% 5.12/2.11 inference(superposition,[status(thm)],[f3138,f4836])).
% 5.12/2.11 thf(f4836,plain,(
% 5.12/2.11 ($true = (in @ (kpair @ sK68 @ sK69) @ sK65)) | ~spl1017_4),
% 5.12/2.11 inference(avatar_component_clause,[status(thm)],[f4834])).
% 5.12/2.11 thf(f4834,definition,(
% 5.12/2.11 spl1017_4 <=> ($true = (in @ (kpair @ sK68 @ sK69) @ sK65))),
% 5.12/2.11 introduced(definition,[new_symbols(naming,[spl1017_4])],[avatar_definition])).
% 5.12/2.11 thf(f3138,plain,(
% 5.12/2.11 ($true != (in @ (kpair @ sK68 @ sK69) @ sK65))),
% 5.12/2.11 inference(cnf_transformation,[status(thm)],[f1443])).
% 5.12/2.11 thf(f4847,plain,(
% 5.12/2.11 spl1017_6 | spl1017_4),
% 5.12/2.11 inference(avatar_split_clause,[status(thm)],[f4846,f4834,f4841])).
% 5.12/2.11 thf(f4846,plain,(
% 5.12/2.11 ($true = (in @ (kpair @ sK68 @ sK69) @ sK65)) | ($true = (in @ (kpair @ sK68 @ sK69) @ sK67))),
% 5.12/2.11 inference(trivial_inequality_removal,[status(thm)],[f4845])).
% 5.12/2.11 thf(f4845,plain,(
% 5.12/2.11 ($true = (in @ (kpair @ sK68 @ sK69) @ sK65)) | ($true != $true) | ($true = (in @ (kpair @ sK68 @ sK69) @ sK67))),
% 5.12/2.11 inference(forward_demodulation,[status(thm)],[f4826,f2993])).
% 5.12/2.11 thf(f2993,plain,(
% 5.12/2.11 (binunionE = $true)),
% 5.12/2.11 inference(cnf_transformation,[status(thm)],[f1443])).
% 5.12/2.11 thf(f4826,plain,(
% 5.12/2.11 (binunionE != $true) | ($true = (in @ (kpair @ sK68 @ sK69) @ sK67)) | ($true = (in @ (kpair @ sK68 @ sK69) @ sK65))),
% 5.12/2.11 inference(trivial_inequality_removal,[status(thm)],[f4820])).
% 5.12/2.11 thf(f4820,plain,(
% 5.12/2.11 ($true != $true) | (binunionE != $true) | ($true = (in @ (kpair @ sK68 @ sK69) @ sK67)) | ($true = (in @ (kpair @ sK68 @ sK69) @ sK65))),
% 5.12/2.11 inference(superposition,[status(thm)],[f4199,f3137])).
% 5.12/2.11 thf(f3137,plain,(
% 5.12/2.11 ($true = (in @ (kpair @ sK68 @ sK69) @ (binunion @ sK65 @ sK67)))),
% 5.12/2.11 inference(cnf_transformation,[status(thm)],[f1443])).
% 5.12/2.11 thf(f4199,plain,(
% 5.12/2.11 ( ! [X2 : $i,X0 : $i,X1 : $i] : (((in @ X2 @ (binunion @ X1 @ X0)) != $true) | ($true = (in @ X2 @ X1)) | ($true = (in @ X2 @ X0)) | (binunionE != $true)) )),
% 5.12/2.11 inference(cnf_transformation,[status(thm)],[f2469])).
% 5.12/2.11 thf(f2469,plain,(
% 5.12/2.11 (! [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X2 @ X0)) | ($true = (in @ X2 @ X1)) | ((in @ X2 @ (binunion @ X1 @ X0)) != $true)) | (binunionE != $true)) & ((binunionE = $true) | (($true != (in @ sK741 @ sK739)) & ((in @ sK741 @ sK740) != $true) & ($true = (in @ sK741 @ (binunion @ sK740 @ sK739)))))),
% 5.12/2.11 inference(skolemisation,[status(esa),new_symbols(skolem,[sK739,sK740,sK741])],[f2467,f2468])).
% 5.12/2.11 thf(f2468,plain,(
% 5.12/2.11 ? [X3 : $i,X4 : $i,X5 : $i] : (($true != (in @ X5 @ X3)) & ((in @ X5 @ X4) != $true) & ($true = (in @ X5 @ (binunion @ X4 @ X3)))) => (($true != (in @ sK741 @ sK739)) & ((in @ sK741 @ sK740) != $true) & ($true = (in @ sK741 @ (binunion @ sK740 @ sK739))))),
% 5.12/2.11 introduced(definition,[],[choice_axiom])).
% 5.12/2.11 thf(f2467,plain,(
% 5.12/2.11 (! [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X2 @ X0)) | ($true = (in @ X2 @ X1)) | ((in @ X2 @ (binunion @ X1 @ X0)) != $true)) | (binunionE != $true)) & ((binunionE = $true) | ? [X3 : $i,X4 : $i,X5 : $i] : (($true != (in @ X5 @ X3)) & ((in @ X5 @ X4) != $true) & ($true = (in @ X5 @ (binunion @ X4 @ X3)))))),
% 5.12/2.11 inference(rectify,[status(thm)],[f2466])).
% 5.12/2.11 thf(f2466,plain,(
% 5.12/2.11 (! [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X2 @ X0)) | ($true = (in @ X2 @ X1)) | ((in @ X2 @ (binunion @ X1 @ X0)) != $true)) | (binunionE != $true)) & ((binunionE = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (($true != (in @ X2 @ X0)) & ($true != (in @ X2 @ X1)) & ((in @ X2 @ (binunion @ X1 @ X0)) = $true)))),
% 5.12/2.11 inference(nnf_transformation,[status(thm)],[f1287])).
% 5.12/2.11 thf(f1287,plain,(
% 5.12/2.11 ! [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X2 @ X0)) | ($true = (in @ X2 @ X1)) | ((in @ X2 @ (binunion @ X1 @ X0)) != $true)) <=> (binunionE = $true)),
% 5.12/2.11 inference(flattening,[status(thm)],[f1286])).
% 5.12/2.11 thf(f1286,plain,(
% 5.12/2.11 ! [X0 : $i,X1 : $i,X2 : $i] : ((($true = (in @ X2 @ X0)) | ($true = (in @ X2 @ X1))) | ((in @ X2 @ (binunion @ X1 @ X0)) != $true)) <=> (binunionE = $true)),
% 5.12/2.11 inference(ennf_transformation,[status(thm)],[f568])).
% 5.12/2.11 thf(f568,plain,(
% 5.12/2.11 ! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X2 @ (binunion @ X1 @ X0)) = $true) => (($true = (in @ X2 @ X0)) | ($true = (in @ X2 @ X1)))) <=> (binunionE = $true)),
% 5.12/2.11 inference(fool_elimination,[status(thm)],[f567])).
% 5.12/2.11 thf(f567,plain,(
% 5.12/2.11 (binunionE = ! [X0 : $i,X1 : $i,X2 : $i] : ((in @ X2 @ (binunion @ X1 @ X0)) => ((in @ X2 @ X1) | (in @ X2 @ X0))))),
% 5.12/2.11 inference(rectify,[status(thm)],[f105])).
% 5.12/2.11 thf(f105,axiom,(
% 5.12/2.11 (binunionE = ! [X4 : $i,X3 : $i,X1 : $i] : ((in @ X1 @ (binunion @ X3 @ X4)) => ((in @ X1 @ X3) | (in @ X1 @ X4))))),
% 5.12/2.11 file('/export/starexec/sandbox/tmp/tmp.onid0QyU2k/DTF2THF_25019.p',binunionE)).
% 5.12/2.11 % SZS output end Proof for DTF2THF_25019
% 5.12/2.11 % (25141)------------------------------
% 5.12/2.11 % (25141)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.12/2.11 % (25141)Termination reason: Refutation
% 5.12/2.11
% 5.12/2.11 % (25141)Memory used [KB]: 11641
% 5.12/2.11 % (25141)Time elapsed: 0.238 s
% 5.12/2.11 % (25141)Instructions burned: 466 (million)
% 5.12/2.11 % (25141)------------------------------
% 5.12/2.11 % (25141)------------------------------
% 5.12/2.11 % (25118)Success in time 0.369 s
%------------------------------------------------------------------------------