%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU785^1 : 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 : n005.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 3.21s 1.98s
% Output : Refutation 3.21s
% Verified :
% SZS Type : Refutation
% Derivation depth : 13
% Number of leaves : 11
% Syntax : Number of formulae : 47 ( 12 unt; 0 typ; 2 def)
% Number of atoms : 5164 (1365 equ; 0 cnn)
% Maximal formula atoms : 308 ( 109 avg)
% Number of connectives : 2830 ( 61 ~; 49 |; 962 &; 516 @)
% ( 6 <=>;1236 =>; 0 <=; 0 <~>)
% Maximal formula depth : 318 ( 51 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 534 ( 531 usr; 316 con; 0-6 aty)
% Number of variables : 113 ( 0 ^; 68 !; 45 ?; 113 :)
% 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,
sP5: $o > $i > $i > $i > $o ).
thf(func_def_370,type,
sP7: $i > $i > $i > $i > $o ).
thf(func_def_372,type,
sP9: $i > $i > $i > $o ).
thf(func_def_374,type,
sP11: $i > $i > $o ).
thf(func_def_375,type,
sP12: $i > $o ).
thf(func_def_376,type,
sP13: $i > $i > $o ).
thf(func_def_377,type,
sP14: $i > $i > $o ).
thf(func_def_378,type,
sP15: $i > $i > $i > $o ).
thf(func_def_384,type,
sK21: $i > ( $i > $i ) > $i > $i ).
thf(func_def_386,type,
sK23: $i > $i ).
thf(func_def_399,type,
sK36: $i > ( $i > $o ) > $i ).
thf(func_def_400,type,
sK37: $i > $o ).
thf(func_def_434,type,
sK71: $i > $i > $i > $i > $i ).
thf(func_def_446,type,
sK83: $i > $i > $i > $i ).
thf(func_def_457,type,
sK94: $i > $o ).
thf(func_def_468,type,
sK105: $i > $o ).
thf(func_def_470,type,
sK107: ( $i > $o ) > $i > $i ).
thf(func_def_471,type,
sK108: ( $i > $o ) > $i > $i ).
thf(func_def_472,type,
sK109: $i > $i ).
thf(func_def_475,type,
sK112: $i > $i > ( $i > $i ) > $i ).
thf(func_def_476,type,
sK113: $i > $o ).
thf(func_def_481,type,
sK118: $i > $i > $o ).
thf(func_def_486,type,
sK123: $i > $i > $o ).
thf(func_def_491,type,
sK128: $i > $i > $o ).
thf(func_def_502,type,
sK139: $o > $i > $i > $i ).
thf(func_def_513,type,
sK150: ( $i > $o ) > $i ).
thf(func_def_514,type,
sK151: $i > $o ).
thf(func_def_515,type,
sK152: $i > $i ).
thf(func_def_527,type,
sK164: $i > $i > ( $i > $o ) > $i ).
thf(func_def_528,type,
sK165: $i > $o ).
thf(func_def_546,type,
sK183: $i > $i > $i ).
thf(func_def_550,type,
sK187: $i > $i ).
thf(func_def_553,type,
sK190: $i > ( $i > $i ) > $i > $i ).
thf(func_def_571,type,
sK208: $i > $i > $i > $i > $i > $i ).
thf(func_def_572,type,
sK209: $i > $i > $i > $i ).
thf(func_def_573,type,
sK210: $i > $i > $i > $i ).
thf(func_def_591,type,
sK228: $i > $i > $o ).
thf(func_def_602,type,
sK239: $i > $i > $i ).
thf(func_def_608,type,
sK245: $i > $i > $o ).
thf(func_def_613,type,
sK250: $i > $i ).
thf(func_def_614,type,
sK251: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_626,type,
sK263: $i > $i > $o ).
thf(func_def_628,type,
sK265: $i > $i ).
thf(func_def_629,type,
sK266: $i > $i ).
thf(func_def_630,type,
sK267: $i > ( $i > $i > $o ) > $i ).
thf(func_def_631,type,
sK268: $i > ( $i > $i > $o ) > $i ).
thf(func_def_632,type,
sK269: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_637,type,
sK274: $i > $i ).
thf(func_def_659,type,
sK296: $i > $i > $i ).
thf(func_def_662,type,
sK299: $i > ( $i > $o ) > $i ).
thf(func_def_663,type,
sK300: $i > $o ).
thf(func_def_669,type,
sK306: $i > $i > $i ).
thf(func_def_679,type,
sK316: $i > $o ).
thf(func_def_681,type,
sK318: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_682,type,
sK319: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_683,type,
sK320: ( $i > $o ) > $i ).
thf(func_def_684,type,
sK321: $i > $o ).
thf(func_def_685,type,
sK322: $i > $i ).
thf(func_def_697,type,
sK334: $i > $i > $i > $i > $i ).
thf(func_def_702,type,
sK339: ( $i > $o ) > $i > $i ).
thf(func_def_704,type,
sK341: $i > $o ).
thf(func_def_762,type,
sK399: $o > $i > $i > $i > $i > $i > $i ).
thf(func_def_768,type,
sK405: $i > $i ).
thf(func_def_769,type,
sK406: ( $i > $i ) > $i > $i > $i ).
thf(func_def_818,type,
sK455: $i > $o ).
thf(func_def_821,type,
sK458: $i > ( $i > $o ) > $i ).
thf(func_def_830,type,
sK467: $i > $i ).
thf(func_def_853,type,
sK490: $i > $o ).
thf(func_def_859,type,
sK496: $i > $o ).
thf(func_def_860,type,
sK497: $i > $i > $i ).
thf(func_def_876,type,
sK513: $i > $i > $i > $i > $i ).
thf(func_def_877,type,
sK514: $i > $i > $i > $i > $i ).
thf(func_def_884,type,
sK521: $i > $o ).
thf(func_def_895,type,
sK532: ( $i > $i ) > $i > $i > $i ).
thf(func_def_898,type,
sK535: $i > $i ).
thf(func_def_903,type,
sK540: $i > $o ).
thf(func_def_906,type,
sK543: $i > ( $i > $o ) > $i ).
thf(func_def_927,type,
sK564: $i > $i > $i > $i > $i ).
thf(func_def_928,type,
sK565: $i > $i > $i > $i > $i ).
thf(func_def_929,type,
sK566: $i > $i > $i > $i > $i ).
thf(func_def_930,type,
sK567: $i > $i > $i > $i > $i ).
thf(func_def_938,type,
sK575: $i > $o ).
thf(func_def_966,type,
sK603: $i > $i > $i > $i > $i ).
thf(func_def_967,type,
sK604: $i > ( $i > $o ) > $i ).
thf(func_def_968,type,
sK605: $i > $o ).
thf(func_def_973,type,
sK610: $i > $i > $i > $i ).
thf(func_def_974,type,
sK611: $i > $i > $i > $i ).
thf(func_def_975,type,
sK612: $i > $i > $i > $i ).
thf(func_def_976,type,
sK613: $i > $i > $i > $i ).
thf(func_def_980,type,
sK617: $i > $i > $i ).
thf(func_def_996,type,
sK633: ( $i > $o ) > $i ).
thf(func_def_997,type,
sK634: ( $i > $o ) > $i ).
thf(func_def_998,type,
sK635: $i > $o ).
thf(func_def_1026,type,
sK663: ( $i > $o ) > $i ).
thf(func_def_1027,type,
sK664: $i > $o ).
thf(func_def_1030,type,
sK667: $i > $o ).
thf(func_def_1032,type,
sK669: ( $i > $o ) > $i > $i ).
thf(func_def_1033,type,
sK670: $i > $i > $o ).
thf(func_def_1038,type,
sK675: $i > $i > $i ).
thf(func_def_1039,type,
sK676: $i > $i > $i ).
thf(func_def_1040,type,
sK677: $i > $i > $i ).
thf(func_def_1041,type,
sK678: $i > $i > $i ).
thf(func_def_1042,type,
sK679: $i > $i > $i ).
thf(func_def_1043,type,
sK680: $i > $i > $i > $i ).
thf(func_def_1044,type,
sK681: $i > $i ).
thf(func_def_1045,type,
sK682: $i > $i ).
thf(func_def_1046,type,
sK683: $i > $i ).
thf(func_def_1047,type,
sK684: $i > $i ).
thf(func_def_1048,type,
sK685: $i > $i > $i ).
thf(func_def_1049,type,
sK686: $i > $i > $i > $i ).
thf(func_def_1050,type,
sK687: $i > $i > $i > $i ).
thf(func_def_1051,type,
sK688: $i > $i > $i ).
thf(func_def_1052,type,
sK689: $i > $i > $i ).
thf(func_def_1054,type,
sK691: $i > $i ).
thf(func_def_1055,type,
sK692: $i > $i ).
thf(func_def_1056,type,
sK693: $i > $i ).
thf(func_def_1061,type,
sK698: $i > $o ).
thf(func_def_1068,type,
sK705: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_1069,type,
sK706: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_1073,type,
sK710: $i > $o ).
thf(func_def_1089,type,
sK726: $i > $o ).
thf(func_def_1102,type,
sK739: $i > $i ).
thf(func_def_1107,type,
sK744: $i > $i > $i ).
thf(func_def_1119,type,
sK756: $i > $i > $i ).
thf(func_def_1122,type,
sK759: $i > $i > $o ).
thf(func_def_1151,type,
sK788: $i > $i > $i > $i ).
thf(func_def_1152,type,
sK789: $i > $i > $i > $i ).
thf(func_def_1159,type,
sK796: $i > $i ).
thf(func_def_1162,type,
sK799: $i > $i ).
thf(func_def_1178,type,
sK815: $i > $o ).
thf(func_def_1181,type,
sK818: $i > $i > $o ).
thf(func_def_1188,type,
sK825: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_1189,type,
sK826: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_1190,type,
sK827: $i > $o ).
thf(func_def_1191,type,
sK828: $i > $o ).
thf(func_def_1195,type,
sK832: $i > $i > $i > $i ).
thf(func_def_1196,type,
sK833: $i > $i > $i > $i ).
thf(func_def_1208,type,
sK845: $i > $o ).
thf(func_def_1219,type,
sK856: $i > $o ).
thf(func_def_1242,type,
sK879: ( $i > $o ) > $i > $i ).
thf(func_def_1244,type,
sK881: $i > $o ).
thf(func_def_1254,type,
sK891: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_1255,type,
sK892: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_1256,type,
sK893: $i > $o ).
thf(func_def_1257,type,
sK894: $i > $o ).
thf(func_def_1272,type,
sK909: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_1273,type,
sK910: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_1276,type,
sK913: $i > $o ).
thf(func_def_1277,type,
sK914: $i > $o ).
thf(func_def_1278,type,
sK915: $i > $o ).
thf(func_def_1281,type,
sK918: $i > ( $i > $o ) > $i ).
thf(func_def_1282,type,
sK919: $i > ( $i > $o ) > $i ).
thf(func_def_1286,type,
sK923: $i > $i > $o ).
thf(func_def_1288,type,
sK925: $i > $i ).
thf(func_def_1294,type,
sK931: ( $i > $o ) > $i > $i > $i ).
thf(func_def_1295,type,
sK932: ( $i > $o ) > $i > $i > $i ).
thf(func_def_1298,type,
sK935: $i > $o ).
thf(func_def_1300,type,
sK937: ( $i > $o ) > $i > $i > $i ).
thf(func_def_1301,type,
sK938: ( $i > $o ) > $i > $i > $i ).
thf(func_def_1304,type,
sK941: $i > $o ).
thf(func_def_1308,type,
sK945: $i > $i ).
thf(func_def_1326,type,
sK963: $i > $o ).
thf(func_def_1339,type,
sK976: $i > $i > $i ).
thf(func_def_1340,type,
sK977: $i > $i > $i ).
thf(func_def_1346,type,
sK983: $i > $o ).
thf(func_def_1350,type,
sK987: $i > $i ).
thf(func_def_1352,type,
sK989: ( $i > $i ) > $i > $i > $i ).
thf(func_def_1354,type,
sK991: $i > ( $i > $o ) > $i ).
thf(func_def_1355,type,
sK992: $i > $o ).
thf(func_def_1368,type,
sK1005: $i > $i > $i > $i ).
thf(func_def_1376,type,
ph1013:
!>[X0: $tType] : X0 ).
thf(f4849,plain,
$false,
inference(avatar_sat_refutation,[status(thm)],[f4741,f4821,f4841]) ).
thf(f4841,plain,
~ spl1012_1,
inference(avatar_split_clause,[status(thm)],[f4840,f4734]) ).
thf(f4734,definition,
( spl1012_1
<=> ( ( in @ ( kpair @ sK259 @ sK260 ) @ sK258 )
= $true ) ),
introduced(definition,[new_symbols(naming,[spl1012_1])],[avatar_definition]) ).
thf(f4840,plain,
( ( in @ ( kpair @ sK259 @ sK260 ) @ sK258 )
!= $true ),
inference(trivial_inequality_removal,[status(thm)],[f4839]) ).
thf(f4839,plain,
( ( $true != $true )
| ( ( in @ ( kpair @ sK259 @ sK260 ) @ sK258 )
!= $true ) ),
inference(forward_demodulation,[status(thm)],[f4805,f3346]) ).
thf(f3346,plain,
binunionIR = $true,
inference(cnf_transformation,[status(thm)],[f1716]) ).
thf(f1716,plain,
( ( doubleComplementE1 = $true )
& ( cartprodpairmemER = $true )
& ( setextT = $true )
& ( setukpairinjR1 = $true )
& ( vacuousDall = $true )
& ( powersetI = $true )
& ( cartprodfstpairEq = $true )
& ( contrasubsetT1 = $true )
& ( powersetTI1 = $true )
& ( upairsubunion = $true )
& ( contraSubsetComplement = $true )
& ( setminusER = $true )
& ( emptyinunitempty = $true )
& ( dpsetconstrEL2 = $true )
& ( setukpairinjL2 = $true )
& ( emptysetsubset = $true )
& ( binunionLsub = $true )
& ( breln1invprop = $true )
& ( emptysetE = $true )
& ( ubforcartprodlem3 = $true )
& ( notequalI2 = $true )
& ( powerset__Cong = $true )
& ( emptyI = $true )
& ( funcImageSingleton = $true )
& ( contrasubsetT = $true )
& ( theeq = $true )
& ( ubforcartprodlem2 = $true )
& ( dpsetconstrER = $true )
& ( woz13rule4 = $true )
& ( setadjoinE = $true )
& ( binintersectRsub = $true )
& ( inIntersectImpInIntersectUnions = $true )
& ( doubleComplementEq = $true )
& ( symdiffIneg1 = $true )
& ( upairsetIL = $true )
& ( complementT_lem = $true )
& ( demorgan2a = $true )
& ( bs114d = $true )
& ( setadjoinAx = $true )
& ( ( in @ sK259 @ sK257 )
= $true )
& ( $true
!= ( in @ ( kpair @ sK259 @ sK260 ) @ ( binunion @ sK256 @ sK258 ) ) )
& ( ( in @ sK260 @ sK257 )
= $true )
& ( ( ( in @ ( kpair @ sK259 @ sK260 ) @ sK256 )
= $true )
| ( ( in @ ( kpair @ sK259 @ sK260 ) @ sK258 )
= $true ) )
& ( $true
= ( breln1 @ sK257 @ sK258 ) )
& ( ( breln1 @ sK257 @ sK256 )
= $true )
& ( setadjoinIR = $true )
& ( exu__Cong = $true )
& ( notinsingleton = $true )
& ( notdallE = $true )
& ( setextsub = $true )
& ( breln1compprop = $true )
& ( nonemptyImpWitness = $true )
& ( cartprodpairin = $true )
& ( binintersectTERcontra = $true )
& ( contrasubsetT2 = $true )
& ( dsetconstr__Cong = $true )
& ( lamProp = $true )
& ( subset2powerset = $true )
& ( secondinupair = $true )
& ( breln1compI = $true )
& ( iffalseProp2 = $true )
& ( binunionTEcontra = $true )
& ( breln1unionIL = $true )
& ( upairset2E = $true )
& ( emptyE1 = $true )
& ( inComplementUnionImpInComplement1 = $true )
& ( breln1invI = $true )
& ( setadjoin__Cong = $true )
& ( woz13rule2 = $true )
& ( setminusEL = $true )
& ( wellorderingAx = $true )
& ( setminusERneg = $true )
& ( notinemptyset = $true )
& ( ex1I = $true )
& ( powersetAx = $true )
& ( binintersectSubset1 = $true )
& ( breln1SetBreln1 = $true )
& ( binunionTIRcontra = $true )
& ( breln1unionIR = $true )
& ( emptyinPowerset = $true )
& ( inIntersectImpInUnion2 = $true )
& ( subsetE2 = $true )
& ( setunion__Cong = $true )
& ( setminusSubset1 = $true )
& ( woz13rule0 = $true )
& ( setadjoinIL = $true )
& ( funcext = $true )
& ( setOfPairsIsBReln1 = $true )
& ( dsetconstrI = $true )
& ( demorgan2a1 = $true )
& ( complementImpComplementIntersect = $true )
& ( beta1 = $true )
& ( iftrueProp2 = $true )
& ( setunionAx = $true )
& ( doubleComplementI1 = $true )
& ( powersetE1 = $true )
& ( cartprodmempair = $true )
& ( cartprodpairmemEL = $true )
& ( subsetemptysetimpeq = $true )
& ( binunionE = $true )
& ( subsetI2 = $true )
& ( prop2set2propI = $true )
& ( in__Cong = $true )
& ( demorgan2b2 = $true )
& ( setadjoinOr = $true )
& ( kpairsurjEq = $true )
& ( eqimpsubset2 = $true )
& ( singletoninpowerset = $true )
& ( kfstpairEq = $true )
& ( breln1all2 = $true )
& ( complementTnotintersectT = $true )
& ( breln1compEex = $true )
& ( quantDeMorgan3 = $true )
& ( exuI1 = $true )
& ( emptyInPowerset = $true )
& ( iftrueorfalse = $true )
& ( singletonprop = $true )
& ( demorgan1b = $true )
& ( funcGraphProp1 = $true )
& ( lamp = $true )
& ( powersetTE1 = $true )
& ( dpsetconstrEL1 = $true )
& ( omegaIndAx = $true )
& ( subsetE = $true )
& ( dpsetconstrI = $true )
& ( setminusELneg = $true )
& ( setukpairinjR12 = $true )
& ( subsetI1 = $true )
& ( binintersectSubset5 = $true )
& ( prop2setE = $true )
& ( ifp = $true )
& ( funcinfuncset = $true )
& ( setukpairIR = $true )
& ( binunionRsub = $true )
& ( setadjoinSub2 = $true )
& ( ksndpairEq = $true )
& ( upairequniteq = $true )
& ( symdiffIneg2 = $true )
& ( funcGraphProp3 = $true )
& ( demorgan1 = $true )
& ( binintersectER = $true )
& ( breln1unionprop = $true )
& ( exuE1 = $true )
& ( setminusILneg = $true )
& ( binunionIL = $true )
& ( ifSingleton = $true )
& ( quantDeMorgan1 = $true )
& ( powersetI1 = $true )
& ( setunionE = $true )
& ( woz1_3 = $true )
& ( binintersectI = $true )
& ( dsetconstrEL = $true )
& ( ap2apEq1 = $true )
& ( powersetT_lem = $true )
& ( prop2setI = $true )
& ( exuE3u = $true )
& ( apProp = $true )
& ( subbreln = $true )
& ( setbeta = $true )
& ( subbreln1 = $true )
& ( binunionIR = $true )
& ( binintersectSubset2 = $true )
& ( kpairp = $true )
& ( doubleComplementSub2 = $true )
& ( binintersectT_lem = $true )
& ( contrasubsetT3 = $true )
& ( cartprodpairsurjEq = $true )
& ( descr__Cong = $true )
& ( inCongP = $true )
& ( binunionTE = $true )
& ( ksndsingleton = $true )
& ( demorgan2 = $true )
& ( exuE3e = $true )
& ( foundationAx = $true )
& ( iftrueProp1 = $true )
& ( theprop = $true )
& ( eta2 = $true )
& ( omega__Cong = $true )
& ( woz1_1 = $true )
& ( binintersectLsub = $true )
& ( setextAx = $true )
& ( emptyset__Cong = $true )
& ( demorgan2a2 = $true )
& ( setunionsingleton = $true )
& ( setukpairinjR = $true )
& ( noeltsimpempty = $true )
& ( nonemptyI = $true )
& ( setminusLsub = $true )
& ( quantDeMorgan4 = $true )
& ( ex1E2 = $true )
& ( sepSubset = $true )
& ( setukpairinjL1 = $true )
& ( eqbreln = $true )
& ( setunionsingleton1 = $true )
& ( cartprodmempaircEq = $true )
& ( woz13rule3 = $true )
& ( powersetsubset = $true )
& ( inIntersectImpInUnion = $true )
& ( exuI2 = $true )
& ( subsetRefl = $true )
& ( doubleComplementSub1 = $true )
& ( subPowSU = $true )
& ( nonemptyI1 = $true )
& ( lam2lamEq = $true )
& ( complementInPowersetComplementIntersect = $true )
& ( infuncsetfunc = $true )
& ( dsetconstrER = $true )
& ( symdiffE = $true )
& ( complementTE1 = $true )
& ( lam2p = $true )
& ( binintersectSubset4 = $true )
& ( exuEu = $true )
& ( binunionTILcontra = $true )
& ( binunionEcases = $true )
& ( binintersectTELcontra = $true )
& ( complementUnionInPowersetComplement = $true )
& ( dpsetconstrSub = $true )
& ( emptysetimpfalse = $true )
& ( singletonsuniq = $true )
& ( iffalseProp1 = $true )
& ( brelnall1 = $true )
& ( complementTcontraSubset = $true )
& ( cartprodsndpairEq = $true )
& ( setadjoinSub = $true )
& ( setukpairinjR11 = $true )
& ( emptysetAx = $true )
& ( setunionI = $true )
& ( setminusI = $true )
& ( sepInPowerset = $true )
& ( exuE2 = $true )
& ( setoftrueEq = $true )
& ( descrp = $true )
& ( woz1_5 = $true )
& ( breln1compE = $true )
& ( setminusIRneg = $true )
& ( breln1invE = $true )
& ( setminusT_lem = $true )
& ( eqimpsubset1 = $true )
& ( choice2fnsingleton = $true )
& ( intersectInPowersetIntersectUnions = $true )
& ( upairset2IR = $true )
& ( symdiffI1 = $true )
& ( cartprodsndin = $true )
& ( funcGraphProp4 = $true )
& ( beta2 = $true )
& ( ap2p = $true )
& ( cartprodfstin = $true )
& ( brelnall2 = $true )
& ( subsetTI = $true )
& ( setukpairIL = $true )
& ( omegaSAx = $true )
& ( eqinunit = $true )
& ( funcextLem = $true )
& ( notsubsetI = $true )
& ( inPowerset = $true )
& ( dpsetconstrERa = $true )
& ( setunionsingleton2 = $true )
& ( powersetE = $true )
& ( complementSubsetComplementIntersect = $true )
& ( setunionE2 = $true )
& ( singletonsubset = $true )
& ( woz1_2 = $true )
& ( inComplementUnionImpNotIn1 = $true )
& ( cartprodmempair1 = $true )
& ( demorgan2b = $true )
& ( subsetTrans = $true )
& ( setext = $true )
& ( binintersectEL = $true )
& ( complementTI1 = $true )
& ( binunionT_lem = $true )
& ( upairinpowunion = $true )
& ( uniqinunit = $true )
& ( singletonsswitch = $true )
& ( upairsetIR = $true )
& ( setukpairinjL = $true )
& ( quantDeMorgan2 = $true )
& ( ex1I2 = $true )
& ( setukpairinjR2 = $true )
& ( disjointsetsI1 = $true )
& ( eta1 = $true )
& ( binintersectSubset3 = $true )
& ( ubforcartprodlem1 = $true )
& ( kfstsingleton = $true )
& ( setminusSubset2 = $true )
& ( woz13rule1 = $true )
& ( replAx = $true )
& ( exuI3 = $true )
& ( iffalse = $true )
& ( singletoninpowunion = $true )
& ( funcext2 = $true )
& ( ap2apEq2 = $true )
& ( woz1_4 = $true )
& ( breln1all1 = $true )
& ( demorgan1a = $true )
& ( nonemptyE1 = $true )
& ( app = $true )
& ( ex1E1 = $true )
& ( setOfPairsIsBReln = $true )
& ( notdexE = $true )
& ( iftrue = $true )
& ( funcGraphProp2 = $true )
& ( kpairiskpair = $true )
& ( omega0Ax = $true )
& ( notequalI1 = $true )
& ( symdiffI2 = $true )
& ( eqbreln1 = $true )
& ( upairsetE = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK256,sK257,sK258,sK259,sK260])],[f1037,f1715,f1714,f1713,f1712]) ).
thf(f1712,plain,
( ? [X0: $i,X1: $i] :
( ? [X2: $i] :
( ? [X3: $i] :
( ( ( in @ X3 @ X1 )
= $true )
& ? [X4: $i] :
( ( ( in @ ( kpair @ X3 @ X4 ) @ ( binunion @ X0 @ X2 ) )
!= $true )
& ( ( in @ X4 @ X1 )
= $true )
& ( ( ( in @ ( kpair @ X3 @ X4 ) @ X0 )
= $true )
| ( ( in @ ( kpair @ X3 @ X4 ) @ X2 )
= $true ) ) ) )
& ( ( breln1 @ X1 @ X2 )
= $true ) )
& ( ( breln1 @ X1 @ X0 )
= $true ) )
=> ( ? [X2: $i] :
( ? [X3: $i] :
( ( ( in @ X3 @ sK257 )
= $true )
& ? [X4: $i] :
( ( ( in @ ( kpair @ X3 @ X4 ) @ ( binunion @ sK256 @ X2 ) )
!= $true )
& ( $true
= ( in @ X4 @ sK257 ) )
& ( ( ( in @ ( kpair @ X3 @ X4 ) @ sK256 )
= $true )
| ( ( in @ ( kpair @ X3 @ X4 ) @ X2 )
= $true ) ) ) )
& ( ( breln1 @ sK257 @ X2 )
= $true ) )
& ( ( breln1 @ sK257 @ sK256 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1713,plain,
( ? [X2: $i] :
( ? [X3: $i] :
( ( ( in @ X3 @ sK257 )
= $true )
& ? [X4: $i] :
( ( ( in @ ( kpair @ X3 @ X4 ) @ ( binunion @ sK256 @ X2 ) )
!= $true )
& ( $true
= ( in @ X4 @ sK257 ) )
& ( ( ( in @ ( kpair @ X3 @ X4 ) @ sK256 )
= $true )
| ( ( in @ ( kpair @ X3 @ X4 ) @ X2 )
= $true ) ) ) )
& ( ( breln1 @ sK257 @ X2 )
= $true ) )
=> ( ? [X3: $i] :
( ( ( in @ X3 @ sK257 )
= $true )
& ? [X4: $i] :
( ( ( in @ ( kpair @ X3 @ X4 ) @ ( binunion @ sK256 @ sK258 ) )
!= $true )
& ( $true
= ( in @ X4 @ sK257 ) )
& ( ( ( in @ ( kpair @ X3 @ X4 ) @ sK256 )
= $true )
| ( $true
= ( in @ ( kpair @ X3 @ X4 ) @ sK258 ) ) ) ) )
& ( $true
= ( breln1 @ sK257 @ sK258 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1714,plain,
( ? [X3: $i] :
( ( ( in @ X3 @ sK257 )
= $true )
& ? [X4: $i] :
( ( ( in @ ( kpair @ X3 @ X4 ) @ ( binunion @ sK256 @ sK258 ) )
!= $true )
& ( $true
= ( in @ X4 @ sK257 ) )
& ( ( ( in @ ( kpair @ X3 @ X4 ) @ sK256 )
= $true )
| ( $true
= ( in @ ( kpair @ X3 @ X4 ) @ sK258 ) ) ) ) )
=> ( ( ( in @ sK259 @ sK257 )
= $true )
& ? [X4: $i] :
( ( $true
!= ( in @ ( kpair @ sK259 @ X4 ) @ ( binunion @ sK256 @ sK258 ) ) )
& ( $true
= ( in @ X4 @ sK257 ) )
& ( ( ( in @ ( kpair @ sK259 @ X4 ) @ sK256 )
= $true )
| ( ( in @ ( kpair @ sK259 @ X4 ) @ sK258 )
= $true ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1715,plain,
( ? [X4: $i] :
( ( $true
!= ( in @ ( kpair @ sK259 @ X4 ) @ ( binunion @ sK256 @ sK258 ) ) )
& ( $true
= ( in @ X4 @ sK257 ) )
& ( ( ( in @ ( kpair @ sK259 @ X4 ) @ sK256 )
= $true )
| ( ( in @ ( kpair @ sK259 @ X4 ) @ sK258 )
= $true ) ) )
=> ( ( $true
!= ( in @ ( kpair @ sK259 @ sK260 ) @ ( binunion @ sK256 @ sK258 ) ) )
& ( ( in @ sK260 @ sK257 )
= $true )
& ( ( ( in @ ( kpair @ sK259 @ sK260 ) @ sK256 )
= $true )
| ( ( in @ ( kpair @ sK259 @ sK260 ) @ sK258 )
= $true ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1037,plain,
( ( doubleComplementE1 = $true )
& ( cartprodpairmemER = $true )
& ( setextT = $true )
& ( setukpairinjR1 = $true )
& ( vacuousDall = $true )
& ( powersetI = $true )
& ( cartprodfstpairEq = $true )
& ( contrasubsetT1 = $true )
& ( powersetTI1 = $true )
& ( upairsubunion = $true )
& ( contraSubsetComplement = $true )
& ( setminusER = $true )
& ( emptyinunitempty = $true )
& ( dpsetconstrEL2 = $true )
& ( setukpairinjL2 = $true )
& ( emptysetsubset = $true )
& ( binunionLsub = $true )
& ( breln1invprop = $true )
& ( emptysetE = $true )
& ( ubforcartprodlem3 = $true )
& ( notequalI2 = $true )
& ( powerset__Cong = $true )
& ( emptyI = $true )
& ( funcImageSingleton = $true )
& ( contrasubsetT = $true )
& ( theeq = $true )
& ( ubforcartprodlem2 = $true )
& ( dpsetconstrER = $true )
& ( woz13rule4 = $true )
& ( setadjoinE = $true )
& ( binintersectRsub = $true )
& ( inIntersectImpInIntersectUnions = $true )
& ( doubleComplementEq = $true )
& ( symdiffIneg1 = $true )
& ( upairsetIL = $true )
& ( complementT_lem = $true )
& ( demorgan2a = $true )
& ( bs114d = $true )
& ( setadjoinAx = $true )
& ? [X0: $i,X1: $i] :
( ? [X2: $i] :
( ? [X3: $i] :
( ( ( in @ X3 @ X1 )
= $true )
& ? [X4: $i] :
( ( ( in @ ( kpair @ X3 @ X4 ) @ ( binunion @ X0 @ X2 ) )
!= $true )
& ( ( in @ X4 @ X1 )
= $true )
& ( ( ( in @ ( kpair @ X3 @ X4 ) @ X0 )
= $true )
| ( ( in @ ( kpair @ X3 @ X4 ) @ X2 )
= $true ) ) ) )
& ( ( breln1 @ X1 @ X2 )
= $true ) )
& ( ( breln1 @ X1 @ X0 )
= $true ) )
& ( setadjoinIR = $true )
& ( exu__Cong = $true )
& ( notinsingleton = $true )
& ( notdallE = $true )
& ( setextsub = $true )
& ( breln1compprop = $true )
& ( nonemptyImpWitness = $true )
& ( cartprodpairin = $true )
& ( binintersectTERcontra = $true )
& ( contrasubsetT2 = $true )
& ( dsetconstr__Cong = $true )
& ( lamProp = $true )
& ( subset2powerset = $true )
& ( secondinupair = $true )
& ( breln1compI = $true )
& ( iffalseProp2 = $true )
& ( binunionTEcontra = $true )
& ( breln1unionIL = $true )
& ( upairset2E = $true )
& ( emptyE1 = $true )
& ( inComplementUnionImpInComplement1 = $true )
& ( breln1invI = $true )
& ( setadjoin__Cong = $true )
& ( woz13rule2 = $true )
& ( setminusEL = $true )
& ( wellorderingAx = $true )
& ( setminusERneg = $true )
& ( notinemptyset = $true )
& ( ex1I = $true )
& ( powersetAx = $true )
& ( binintersectSubset1 = $true )
& ( breln1SetBreln1 = $true )
& ( binunionTIRcontra = $true )
& ( breln1unionIR = $true )
& ( emptyinPowerset = $true )
& ( inIntersectImpInUnion2 = $true )
& ( subsetE2 = $true )
& ( setunion__Cong = $true )
& ( setminusSubset1 = $true )
& ( woz13rule0 = $true )
& ( setadjoinIL = $true )
& ( funcext = $true )
& ( setOfPairsIsBReln1 = $true )
& ( dsetconstrI = $true )
& ( demorgan2a1 = $true )
& ( complementImpComplementIntersect = $true )
& ( beta1 = $true )
& ( iftrueProp2 = $true )
& ( setunionAx = $true )
& ( doubleComplementI1 = $true )
& ( powersetE1 = $true )
& ( cartprodmempair = $true )
& ( cartprodpairmemEL = $true )
& ( subsetemptysetimpeq = $true )
& ( binunionE = $true )
& ( subsetI2 = $true )
& ( prop2set2propI = $true )
& ( in__Cong = $true )
& ( demorgan2b2 = $true )
& ( setadjoinOr = $true )
& ( kpairsurjEq = $true )
& ( eqimpsubset2 = $true )
& ( singletoninpowerset = $true )
& ( kfstpairEq = $true )
& ( breln1all2 = $true )
& ( complementTnotintersectT = $true )
& ( breln1compEex = $true )
& ( quantDeMorgan3 = $true )
& ( exuI1 = $true )
& ( emptyInPowerset = $true )
& ( iftrueorfalse = $true )
& ( singletonprop = $true )
& ( demorgan1b = $true )
& ( funcGraphProp1 = $true )
& ( lamp = $true )
& ( powersetTE1 = $true )
& ( dpsetconstrEL1 = $true )
& ( omegaIndAx = $true )
& ( subsetE = $true )
& ( dpsetconstrI = $true )
& ( setminusELneg = $true )
& ( setukpairinjR12 = $true )
& ( subsetI1 = $true )
& ( binintersectSubset5 = $true )
& ( prop2setE = $true )
& ( ifp = $true )
& ( funcinfuncset = $true )
& ( setukpairIR = $true )
& ( binunionRsub = $true )
& ( setadjoinSub2 = $true )
& ( ksndpairEq = $true )
& ( upairequniteq = $true )
& ( symdiffIneg2 = $true )
& ( funcGraphProp3 = $true )
& ( demorgan1 = $true )
& ( binintersectER = $true )
& ( breln1unionprop = $true )
& ( exuE1 = $true )
& ( setminusILneg = $true )
& ( binunionIL = $true )
& ( ifSingleton = $true )
& ( quantDeMorgan1 = $true )
& ( powersetI1 = $true )
& ( setunionE = $true )
& ( woz1_3 = $true )
& ( binintersectI = $true )
& ( dsetconstrEL = $true )
& ( ap2apEq1 = $true )
& ( powersetT_lem = $true )
& ( prop2setI = $true )
& ( exuE3u = $true )
& ( apProp = $true )
& ( subbreln = $true )
& ( setbeta = $true )
& ( subbreln1 = $true )
& ( binunionIR = $true )
& ( binintersectSubset2 = $true )
& ( kpairp = $true )
& ( doubleComplementSub2 = $true )
& ( binintersectT_lem = $true )
& ( contrasubsetT3 = $true )
& ( cartprodpairsurjEq = $true )
& ( descr__Cong = $true )
& ( inCongP = $true )
& ( binunionTE = $true )
& ( ksndsingleton = $true )
& ( demorgan2 = $true )
& ( exuE3e = $true )
& ( foundationAx = $true )
& ( iftrueProp1 = $true )
& ( theprop = $true )
& ( eta2 = $true )
& ( omega__Cong = $true )
& ( woz1_1 = $true )
& ( binintersectLsub = $true )
& ( setextAx = $true )
& ( emptyset__Cong = $true )
& ( demorgan2a2 = $true )
& ( setunionsingleton = $true )
& ( setukpairinjR = $true )
& ( noeltsimpempty = $true )
& ( nonemptyI = $true )
& ( setminusLsub = $true )
& ( quantDeMorgan4 = $true )
& ( ex1E2 = $true )
& ( sepSubset = $true )
& ( setukpairinjL1 = $true )
& ( eqbreln = $true )
& ( setunionsingleton1 = $true )
& ( cartprodmempaircEq = $true )
& ( woz13rule3 = $true )
& ( powersetsubset = $true )
& ( inIntersectImpInUnion = $true )
& ( exuI2 = $true )
& ( subsetRefl = $true )
& ( doubleComplementSub1 = $true )
& ( subPowSU = $true )
& ( nonemptyI1 = $true )
& ( lam2lamEq = $true )
& ( complementInPowersetComplementIntersect = $true )
& ( infuncsetfunc = $true )
& ( dsetconstrER = $true )
& ( symdiffE = $true )
& ( complementTE1 = $true )
& ( lam2p = $true )
& ( binintersectSubset4 = $true )
& ( exuEu = $true )
& ( binunionTILcontra = $true )
& ( binunionEcases = $true )
& ( binintersectTELcontra = $true )
& ( complementUnionInPowersetComplement = $true )
& ( dpsetconstrSub = $true )
& ( emptysetimpfalse = $true )
& ( singletonsuniq = $true )
& ( iffalseProp1 = $true )
& ( brelnall1 = $true )
& ( complementTcontraSubset = $true )
& ( cartprodsndpairEq = $true )
& ( setadjoinSub = $true )
& ( setukpairinjR11 = $true )
& ( emptysetAx = $true )
& ( setunionI = $true )
& ( setminusI = $true )
& ( sepInPowerset = $true )
& ( exuE2 = $true )
& ( setoftrueEq = $true )
& ( descrp = $true )
& ( woz1_5 = $true )
& ( breln1compE = $true )
& ( setminusIRneg = $true )
& ( breln1invE = $true )
& ( setminusT_lem = $true )
& ( eqimpsubset1 = $true )
& ( choice2fnsingleton = $true )
& ( intersectInPowersetIntersectUnions = $true )
& ( upairset2IR = $true )
& ( symdiffI1 = $true )
& ( cartprodsndin = $true )
& ( funcGraphProp4 = $true )
& ( beta2 = $true )
& ( ap2p = $true )
& ( cartprodfstin = $true )
& ( brelnall2 = $true )
& ( subsetTI = $true )
& ( setukpairIL = $true )
& ( omegaSAx = $true )
& ( eqinunit = $true )
& ( funcextLem = $true )
& ( notsubsetI = $true )
& ( inPowerset = $true )
& ( dpsetconstrERa = $true )
& ( setunionsingleton2 = $true )
& ( powersetE = $true )
& ( complementSubsetComplementIntersect = $true )
& ( setunionE2 = $true )
& ( singletonsubset = $true )
& ( woz1_2 = $true )
& ( inComplementUnionImpNotIn1 = $true )
& ( cartprodmempair1 = $true )
& ( demorgan2b = $true )
& ( subsetTrans = $true )
& ( setext = $true )
& ( binintersectEL = $true )
& ( complementTI1 = $true )
& ( binunionT_lem = $true )
& ( upairinpowunion = $true )
& ( uniqinunit = $true )
& ( singletonsswitch = $true )
& ( upairsetIR = $true )
& ( setukpairinjL = $true )
& ( quantDeMorgan2 = $true )
& ( ex1I2 = $true )
& ( setukpairinjR2 = $true )
& ( disjointsetsI1 = $true )
& ( eta1 = $true )
& ( binintersectSubset3 = $true )
& ( ubforcartprodlem1 = $true )
& ( kfstsingleton = $true )
& ( setminusSubset2 = $true )
& ( woz13rule1 = $true )
& ( replAx = $true )
& ( exuI3 = $true )
& ( iffalse = $true )
& ( singletoninpowunion = $true )
& ( funcext2 = $true )
& ( ap2apEq2 = $true )
& ( woz1_4 = $true )
& ( breln1all1 = $true )
& ( demorgan1a = $true )
& ( nonemptyE1 = $true )
& ( app = $true )
& ( ex1E1 = $true )
& ( setOfPairsIsBReln = $true )
& ( notdexE = $true )
& ( iftrue = $true )
& ( funcGraphProp2 = $true )
& ( kpairiskpair = $true )
& ( omega0Ax = $true )
& ( notequalI1 = $true )
& ( symdiffI2 = $true )
& ( eqbreln1 = $true )
& ( upairsetE = $true ) ),
inference(flattening,[status(thm)],[f1036]) ).
thf(f1036,plain,
( ? [X0: $i,X1: $i] :
( ? [X2: $i] :
( ? [X3: $i] :
( ? [X4: $i] :
( ( ( in @ ( kpair @ X3 @ X4 ) @ ( binunion @ X0 @ X2 ) )
!= $true )
& ( ( ( in @ ( kpair @ X3 @ X4 ) @ X0 )
= $true )
| ( ( in @ ( kpair @ X3 @ X4 ) @ X2 )
= $true ) )
& ( ( in @ X4 @ X1 )
= $true ) )
& ( ( in @ X3 @ X1 )
= $true ) )
& ( ( breln1 @ X1 @ X2 )
= $true ) )
& ( ( breln1 @ X1 @ X0 )
= $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)],[f830]) ).
thf(f830,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 )
=> ! [X0: $i,X1: $i] :
( ( ( breln1 @ X1 @ X0 )
= $true )
=> ! [X2: $i] :
( ( ( breln1 @ X1 @ X2 )
= $true )
=> ! [X3: $i] :
( ( ( in @ X3 @ X1 )
= $true )
=> ! [X4: $i] :
( ( ( in @ X4 @ X1 )
= $true )
=> ( ( ( ( in @ ( kpair @ X3 @ X4 ) @ X0 )
= $true )
| ( ( in @ ( kpair @ X3 @ X4 ) @ X2 )
= $true ) )
=> ( ( in @ ( kpair @ X3 @ X4 ) @ ( binunion @ X0 @ X2 ) )
= $true ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f829]) ).
thf(f829,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
=> ! [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 ) @ X2 )
| ( in @ ( kpair @ X3 @ X4 ) @ X0 ) )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( binunion @ X0 @ X2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f305]) ).
thf(f305,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
=> ! [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 ) @ X15 )
| ( in @ ( kpair @ X1 @ X2 ) @ X13 ) )
=> ( in @ ( kpair @ X1 @ X2 ) @ ( binunion @ X13 @ X15 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f304]) ).
thf(f304,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
=> ! [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 ) @ X15 )
| ( in @ ( kpair @ X1 @ X2 ) @ X13 ) )
=> ( in @ ( kpair @ X1 @ X2 ) @ ( binunion @ X13 @ X15 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.uLFW0IB354/DTF2THF_24330.p',breln1unionI) ).
thf(f4805,plain,
( ( binunionIR != $true )
| ( ( in @ ( kpair @ sK259 @ sK260 ) @ sK258 )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f4795]) ).
thf(f4795,plain,
( ( $true != $true )
| ( binunionIR != $true )
| ( ( in @ ( kpair @ sK259 @ sK260 ) @ sK258 )
!= $true ) ),
inference(superposition,[status(thm)],[f3466,f3100]) ).
thf(f3100,plain,
! [X3: $i,X4: $i,X5: $i] :
( ( ( in @ X5 @ ( binunion @ X4 @ X3 ) )
= $true )
| ( ( in @ X5 @ X3 )
!= $true )
| ( binunionIR != $true ) ),
inference(cnf_transformation,[status(thm)],[f1600]) ).
thf(f1600,plain,
( ( ( binunionIR = $true )
| ( ( ( in @ sK182 @ sK180 )
= $true )
& ( ( in @ sK182 @ ( binunion @ sK181 @ sK180 ) )
!= $true ) ) )
& ( ! [X3: $i,X4: $i,X5: $i] :
( ( ( in @ X5 @ X3 )
!= $true )
| ( ( in @ X5 @ ( binunion @ X4 @ X3 ) )
= $true ) )
| ( binunionIR != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK180,sK181,sK182])],[f1598,f1599]) ).
thf(f1599,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X2 @ X0 )
= $true )
& ( ( in @ X2 @ ( binunion @ X1 @ X0 ) )
!= $true ) )
=> ( ( ( in @ sK182 @ sK180 )
= $true )
& ( ( in @ sK182 @ ( binunion @ sK181 @ sK180 ) )
!= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1598,plain,
( ( ( binunionIR = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X2 @ X0 )
= $true )
& ( ( in @ X2 @ ( binunion @ X1 @ X0 ) )
!= $true ) ) )
& ( ! [X3: $i,X4: $i,X5: $i] :
( ( ( in @ X5 @ X3 )
!= $true )
| ( ( in @ X5 @ ( binunion @ X4 @ X3 ) )
= $true ) )
| ( binunionIR != $true ) ) ),
inference(rectify,[status(thm)],[f1597]) ).
thf(f1597,plain,
( ( ( binunionIR = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X2 @ X0 )
= $true )
& ( ( in @ X2 @ ( binunion @ X1 @ X0 ) )
!= $true ) ) )
& ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X2 @ X0 )
!= $true )
| ( ( in @ X2 @ ( binunion @ X1 @ X0 ) )
= $true ) )
| ( binunionIR != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f1025]) ).
thf(f1025,plain,
( ( binunionIR = $true )
<=> ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X2 @ X0 )
!= $true )
| ( ( in @ X2 @ ( binunion @ X1 @ X0 ) )
= $true ) ) ),
inference(ennf_transformation,[status(thm)],[f436]) ).
thf(f436,plain,
( ( binunionIR = $true )
<=> ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X2 @ X0 )
= $true )
=> ( ( in @ X2 @ ( binunion @ X1 @ X0 ) )
= $true ) ) ),
inference(fool_elimination,[status(thm)],[f435]) ).
thf(f435,plain,
! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X2 @ X0 )
=> ( in @ X2 @ ( binunion @ X1 @ X0 ) ) )
= binunionIR ),
inference(rectify,[status(thm)],[f103]) ).
thf(f103,axiom,
! [X4: $i,X3: $i,X1: $i] :
( ( ( in @ X1 @ X4 )
=> ( in @ X1 @ ( binunion @ X3 @ X4 ) ) )
= binunionIR ),
file('/export/starexec/sandbox/tmp/tmp.uLFW0IB354/DTF2THF_24330.p',binunionIR) ).
thf(f3466,plain,
( $true
!= ( in @ ( kpair @ sK259 @ sK260 ) @ ( binunion @ sK256 @ sK258 ) ) ),
inference(cnf_transformation,[status(thm)],[f1716]) ).
thf(f4821,plain,
~ spl1012_2,
inference(avatar_split_clause,[status(thm)],[f4820,f4738]) ).
thf(f4738,definition,
( spl1012_2
<=> ( ( in @ ( kpair @ sK259 @ sK260 ) @ sK256 )
= $true ) ),
introduced(definition,[new_symbols(naming,[spl1012_2])],[avatar_definition]) ).
thf(f4820,plain,
( ( in @ ( kpair @ sK259 @ sK260 ) @ sK256 )
!= $true ),
inference(trivial_inequality_removal,[status(thm)],[f4819]) ).
thf(f4819,plain,
( ( ( in @ ( kpair @ sK259 @ sK260 ) @ sK256 )
!= $true )
| ( $true != $true ) ),
inference(forward_demodulation,[status(thm)],[f4810,f3362]) ).
thf(f3362,plain,
binunionIL = $true,
inference(cnf_transformation,[status(thm)],[f1716]) ).
thf(f4810,plain,
( ( ( in @ ( kpair @ sK259 @ sK260 ) @ sK256 )
!= $true )
| ( binunionIL != $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f4797]) ).
thf(f4797,plain,
( ( $true != $true )
| ( ( in @ ( kpair @ sK259 @ sK260 ) @ sK256 )
!= $true )
| ( binunionIL != $true ) ),
inference(superposition,[status(thm)],[f3466,f3669]) ).
thf(f3669,plain,
! [X2: $i,X0: $i,X1: $i] :
( ( ( in @ X0 @ ( binunion @ X2 @ X1 ) )
= $true )
| ( binunionIL != $true )
| ( ( in @ X0 @ X2 )
!= $true ) ),
inference(cnf_transformation,[status(thm)],[f1894]) ).
thf(f1894,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X0 @ ( binunion @ X2 @ X1 ) )
= $true )
| ( ( in @ X0 @ X2 )
!= $true ) )
| ( binunionIL != $true ) )
& ( ( binunionIL = $true )
| ( ( ( in @ sK372 @ ( binunion @ sK374 @ sK373 ) )
!= $true )
& ( ( in @ sK372 @ sK374 )
= $true ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK372,sK373,sK374])],[f1892,f1893]) ).
thf(f1893,plain,
( ? [X3: $i,X4: $i,X5: $i] :
( ( $true
!= ( in @ X3 @ ( binunion @ X5 @ X4 ) ) )
& ( ( in @ X3 @ X5 )
= $true ) )
=> ( ( ( in @ sK372 @ ( binunion @ sK374 @ sK373 ) )
!= $true )
& ( ( in @ sK372 @ sK374 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1892,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X0 @ ( binunion @ X2 @ X1 ) )
= $true )
| ( ( in @ X0 @ X2 )
!= $true ) )
| ( binunionIL != $true ) )
& ( ( binunionIL = $true )
| ? [X3: $i,X4: $i,X5: $i] :
( ( $true
!= ( in @ X3 @ ( binunion @ X5 @ X4 ) ) )
& ( ( in @ X3 @ X5 )
= $true ) ) ) ),
inference(rectify,[status(thm)],[f1891]) ).
thf(f1891,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X0 @ ( binunion @ X2 @ X1 ) )
= $true )
| ( ( in @ X0 @ X2 )
!= $true ) )
| ( binunionIL != $true ) )
& ( ( binunionIL = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X0 @ ( binunion @ X2 @ X1 ) )
!= $true )
& ( ( in @ X0 @ X2 )
= $true ) ) ) ),
inference(nnf_transformation,[status(thm)],[f1034]) ).
thf(f1034,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X0 @ ( binunion @ X2 @ X1 ) )
= $true )
| ( ( in @ X0 @ X2 )
!= $true ) )
<=> ( binunionIL = $true ) ),
inference(ennf_transformation,[status(thm)],[f364]) ).
thf(f364,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X0 @ X2 )
= $true )
=> ( ( in @ X0 @ ( binunion @ X2 @ X1 ) )
= $true ) )
<=> ( binunionIL = $true ) ),
inference(fool_elimination,[status(thm)],[f363]) ).
thf(f363,plain,
! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X0 @ X2 )
=> ( in @ X0 @ ( binunion @ X2 @ X1 ) ) )
= binunionIL ),
inference(rectify,[status(thm)],[f101]) ).
thf(f101,axiom,
! [X1: $i,X4: $i,X3: $i] :
( ( ( in @ X1 @ X3 )
=> ( in @ X1 @ ( binunion @ X3 @ X4 ) ) )
= binunionIL ),
file('/export/starexec/sandbox/tmp/tmp.uLFW0IB354/DTF2THF_24330.p',binunionIL) ).
thf(f4741,plain,
( spl1012_1
| spl1012_2 ),
inference(avatar_split_clause,[status(thm)],[f3464,f4738,f4734]) ).
thf(f3464,plain,
( ( ( in @ ( kpair @ sK259 @ sK260 ) @ sK256 )
= $true )
| ( ( in @ ( kpair @ sK259 @ sK260 ) @ sK258 )
= $true ) ),
inference(cnf_transformation,[status(thm)],[f1716]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : SEU785^1 : TPTP v9.2.1. Released v3.7.0.
% 0.00/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.18/0.34 % Computer : n005.cluster.edu
% 0.18/0.34 % Model : x86_64 x86_64
% 0.18/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.18/0.34 % Memory : 8042.1875MB
% 0.18/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.18/0.34 % CPULimit : 300
% 0.18/0.34 % WCLimit : 300
% 0.18/0.34 % DateTime : Tue Feb 24 12:33:44 EST 2026
% 0.18/0.34 % CPUTime :
% 0.18/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.30/0.46 ---- Original DTF file ---
% 0.30/0.46 thf(spec,logic,$$dhol).
% 0.30/0.46 %------------------------------------------------------------------------------
% 0.30/0.46 % File : SEU785^1 : TPTP v9.2.1. Released v3.7.0.
% 0.30/0.46 % Domain : Set Theory
% 0.30/0.46 % Problem : Binary Relations on a Set
% 0.30/0.46 % Version : Especial.
% 0.30/0.46 % English : (! A:i.! R:i.breln1 A R -> (! S:i.breln1 A S -> (! x:i.in x A ->
% 0.30/0.46 % (! y:i.in y A -> in (kpair x y) R | in (kpair x y) S ->
% 0.30/0.46 % in (kpair x y) (binunion R S)))))
% 0.30/0.46
% 0.30/0.46 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.30/0.46 % Source : [Bro08]
% 0.30/0.46 % Names : ZFC287g [Bro08]
% 0.30/0.46
% 0.30/0.46 % Status : Theorem
% 0.30/0.46 % Rating : 0.11 v9.1.0, 0.12 v9.0.0, 0.20 v8.2.0, 0.31 v8.1.0, 0.18 v7.5.0, 0.14 v7.4.0, 0.22 v7.2.0, 0.12 v7.1.0, 0.38 v7.0.0, 0.29 v6.4.0, 0.33 v6.3.0, 0.40 v6.2.0, 0.29 v6.1.0, 0.43 v5.5.0, 0.50 v5.4.0, 0.60 v5.2.0, 0.80 v4.1.0, 1.00 v3.7.0
% 0.30/0.46 % Syntax : Number of formulae : 653 ( 303 unt; 349 typ; 303 def)
% 0.30/0.46 % Number of atoms : 1866 ( 433 equ; 0 cnn)
% 0.30/0.46 % Maximal formula atoms : 308 ( 6 avg)
% 0.30/0.46 % Number of connectives : 4079 ( 73 ~; 13 |; 48 &;2959 @)
% 0.30/0.46 % ( 14 <=>; 972 =>; 0 <=; 0 <~>)
% 0.30/0.46 % Maximal formula depth : 317 ( 2 avg)
% 0.30/0.46 % Number of types : 2 ( 0 usr)
% 0.30/0.46 % Number of type conns : 157 ( 157 >; 0 *; 0 +; 0 <<)
% 0.30/0.46 % Number of symbols : 352 ( 349 usr; 305 con; 0-4 aty)
% 0.30/0.46 % Number of variables : 1066 ( 71 ^; 958 !; 37 ?;1066 :)
% 0.30/0.46 % SPC : TH0_THM_EQU_NAR
% 0.30/0.46
% 0.30/0.46 % Comments : http://mathgate.info/detsetitem.php?id=350
% 0.30/0.46 %------------------------------------------------------------------------------
% 0.30/0.46 thf(in_type,type,
% 0.30/0.46 in: $i > $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(exu_type,type,
% 0.30/0.46 exu: ( $i > $o ) > $o ).
% 0.30/0.46
% 0.30/0.46 thf(exu,definition,
% 0.30/0.46 ( exu
% 0.30/0.46 = ( ^ [Xphi: $i > $o] :
% 0.30/0.46 ? [Xx: $i] :
% 0.30/0.46 ( ( Xphi @ Xx )
% 0.30/0.46 & ! [Xy: $i] :
% 0.30/0.46 ( ( Xphi @ Xy )
% 0.30/0.46 => ( Xx = Xy ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setextAx_type,type,
% 0.30/0.46 setextAx: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setextAx,definition,
% 0.30/0.46 ( setextAx
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 <=> ( in @ Xx @ B ) )
% 0.30/0.46 => ( A = B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(emptyset_type,type,
% 0.30/0.46 emptyset: $i ).
% 0.30/0.46
% 0.30/0.46 thf(emptysetAx_type,type,
% 0.30/0.46 emptysetAx: $o ).
% 0.30/0.46
% 0.30/0.46 thf(emptysetAx,definition,
% 0.30/0.46 ( emptysetAx
% 0.30/0.46 = ( ! [Xx: $i] :
% 0.30/0.46 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoin_type,type,
% 0.30/0.46 setadjoin: $i > $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinAx_type,type,
% 0.30/0.46 setadjoinAx: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinAx,definition,
% 0.30/0.46 ( setadjoinAx
% 0.30/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.30/0.46 <=> ( ( Xy = Xx )
% 0.30/0.46 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(powerset_type,type,
% 0.30/0.46 powerset: $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(powersetAx_type,type,
% 0.30/0.46 powersetAx: $o ).
% 0.30/0.46
% 0.30/0.46 thf(powersetAx,definition,
% 0.30/0.46 ( powersetAx
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( in @ B @ ( powerset @ A ) )
% 0.30/0.46 <=> ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ B )
% 0.30/0.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setunion_type,type,
% 0.30/0.46 setunion: $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(setunionAx_type,type,
% 0.30/0.46 setunionAx: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setunionAx,definition,
% 0.30/0.46 ( setunionAx
% 0.30/0.46 = ( ! [A: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ ( setunion @ A ) )
% 0.30/0.46 <=> ? [B: $i] :
% 0.30/0.46 ( ( in @ Xx @ B )
% 0.30/0.46 & ( in @ B @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(omega_type,type,
% 0.30/0.46 omega: $i ).
% 0.30/0.46
% 0.30/0.46 thf(omega0Ax_type,type,
% 0.30/0.46 omega0Ax: $o ).
% 0.30/0.46
% 0.30/0.46 thf(omega0Ax,definition,
% 0.30/0.46 ( omega0Ax
% 0.30/0.46 = ( in @ emptyset @ omega ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(omegaSAx_type,type,
% 0.30/0.46 omegaSAx: $o ).
% 0.30/0.46
% 0.30/0.46 thf(omegaSAx,definition,
% 0.30/0.46 ( omegaSAx
% 0.30/0.46 = ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ omega )
% 0.30/0.46 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(omegaIndAx_type,type,
% 0.30/0.46 omegaIndAx: $o ).
% 0.30/0.46
% 0.30/0.46 thf(omegaIndAx,definition,
% 0.30/0.46 ( omegaIndAx
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ( ( ( in @ emptyset @ A )
% 0.30/0.46 & ! [Xx: $i] :
% 0.30/0.46 ( ( ( in @ Xx @ omega )
% 0.30/0.46 & ( in @ Xx @ A ) )
% 0.30/0.46 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ omega )
% 0.30/0.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(replAx_type,type,
% 0.30/0.46 replAx: $o ).
% 0.30/0.46
% 0.30/0.46 thf(replAx,definition,
% 0.30/0.46 ( replAx
% 0.30/0.46 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( exu
% 0.30/0.46 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.30/0.46 => ? [B: $i] :
% 0.30/0.46 ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ B )
% 0.30/0.46 <=> ? [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(foundationAx_type,type,
% 0.30/0.46 foundationAx: $o ).
% 0.30/0.46
% 0.30/0.46 thf(foundationAx,definition,
% 0.30/0.46 ( foundationAx
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.30/0.46 => ? [B: $i] :
% 0.30/0.46 ( ( in @ B @ A )
% 0.30/0.46 & ~ ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ B )
% 0.30/0.46 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(wellorderingAx_type,type,
% 0.30/0.46 wellorderingAx: $o ).
% 0.30/0.46
% 0.30/0.46 thf(wellorderingAx,definition,
% 0.30/0.46 ( wellorderingAx
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ? [B: $i] :
% 0.30/0.46 ( ! [C: $i] :
% 0.30/0.46 ( ( in @ C @ B )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ C )
% 0.30/0.46 => ( in @ Xx @ A ) ) )
% 0.30/0.46 & ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( ( in @ Xx @ A )
% 0.30/0.46 & ( in @ Xy @ A ) )
% 0.30/0.46 => ( ! [C: $i] :
% 0.30/0.46 ( ( in @ C @ B )
% 0.30/0.46 => ( ( in @ Xx @ C )
% 0.30/0.46 <=> ( in @ Xy @ C ) ) )
% 0.30/0.46 => ( Xx = Xy ) ) )
% 0.30/0.46 & ! [C: $i,D: $i] :
% 0.30/0.46 ( ( ( in @ C @ B )
% 0.30/0.46 & ( in @ D @ B ) )
% 0.30/0.46 => ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ C )
% 0.30/0.46 => ( in @ Xx @ D ) )
% 0.30/0.46 | ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ D )
% 0.30/0.46 => ( in @ Xx @ C ) ) ) )
% 0.30/0.46 & ! [C: $i] :
% 0.30/0.46 ( ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ C )
% 0.30/0.46 => ( in @ Xx @ A ) )
% 0.30/0.46 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.30/0.46 => ? [D: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ D @ B )
% 0.30/0.46 & ( in @ Xx @ C )
% 0.30/0.46 & ~ ? [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ D )
% 0.30/0.46 & ( in @ Xy @ C ) )
% 0.30/0.46 & ! [E: $i] :
% 0.30/0.46 ( ( in @ E @ B )
% 0.30/0.46 => ( ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ E )
% 0.30/0.46 => ( in @ Xy @ D ) )
% 0.30/0.46 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(descr_type,type,
% 0.30/0.46 descr: ( $i > $o ) > $i ).
% 0.30/0.46
% 0.30/0.46 thf(descrp_type,type,
% 0.30/0.46 descrp: $o ).
% 0.30/0.46
% 0.30/0.46 thf(descrp,definition,
% 0.30/0.46 ( descrp
% 0.30/0.46 = ( ! [Xphi: $i > $o] :
% 0.30/0.46 ( ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 => ( Xphi
% 0.30/0.46 @ ( descr
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(dsetconstr_type,type,
% 0.30/0.46 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.30/0.46
% 0.30/0.46 thf(dsetconstrI_type,type,
% 0.30/0.46 dsetconstrI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(dsetconstrI,definition,
% 0.30/0.46 ( dsetconstrI
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( Xphi @ Xx )
% 0.30/0.46 => ( in @ Xx
% 0.30/0.46 @ ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(dsetconstrEL_type,type,
% 0.30/0.46 dsetconstrEL: $o ).
% 0.30/0.46
% 0.30/0.46 thf(dsetconstrEL,definition,
% 0.30/0.46 ( dsetconstrEL
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx
% 0.30/0.46 @ ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.30/0.46 => ( in @ Xx @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(dsetconstrER_type,type,
% 0.30/0.46 dsetconstrER: $o ).
% 0.30/0.46
% 0.30/0.46 thf(dsetconstrER,definition,
% 0.30/0.46 ( dsetconstrER
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx
% 0.30/0.46 @ ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.30/0.46 => ( Xphi @ Xx ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(exuE1_type,type,
% 0.30/0.46 exuE1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(exuE1,definition,
% 0.30/0.46 ( exuE1
% 0.30/0.46 = ( ! [Xphi: $i > $o] :
% 0.30/0.46 ( ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 => ? [Xx: $i] :
% 0.30/0.46 ( ( Xphi @ Xx )
% 0.30/0.46 & ! [Xy: $i] :
% 0.30/0.46 ( ( Xphi @ Xy )
% 0.30/0.46 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(prop2set_type,type,
% 0.30/0.46 prop2set: $o > $i ).
% 0.30/0.46
% 0.30/0.46 thf(prop2setE_type,type,
% 0.30/0.46 prop2setE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(prop2setE,definition,
% 0.30/0.46 ( prop2setE
% 0.30/0.46 = ( ! [Xphi: $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.30/0.46 => Xphi ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(emptysetE_type,type,
% 0.30/0.46 emptysetE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(emptysetE,definition,
% 0.30/0.46 ( emptysetE
% 0.30/0.46 = ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ emptyset )
% 0.30/0.46 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(emptysetimpfalse_type,type,
% 0.30/0.46 emptysetimpfalse: $o ).
% 0.30/0.46
% 0.30/0.46 thf(emptysetimpfalse,definition,
% 0.30/0.46 ( emptysetimpfalse
% 0.30/0.46 = ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ emptyset )
% 0.30/0.46 => $false ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(notinemptyset_type,type,
% 0.30/0.46 notinemptyset: $o ).
% 0.30/0.46
% 0.30/0.46 thf(notinemptyset,definition,
% 0.30/0.46 ( notinemptyset
% 0.30/0.46 = ( ! [Xx: $i] :
% 0.30/0.46 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(exuE3e_type,type,
% 0.30/0.46 exuE3e: $o ).
% 0.30/0.46
% 0.30/0.46 thf(exuE3e,definition,
% 0.30/0.46 ( exuE3e
% 0.30/0.46 = ( ! [Xphi: $i > $o] :
% 0.30/0.46 ( ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setext_type,type,
% 0.30/0.46 setext: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setext,definition,
% 0.30/0.46 ( setext
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ Xx @ B ) )
% 0.30/0.46 => ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ B )
% 0.30/0.46 => ( in @ Xx @ A ) )
% 0.30/0.46 => ( A = B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(emptyI_type,type,
% 0.30/0.46 emptyI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(emptyI,definition,
% 0.30/0.46 ( emptyI
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ~ ( in @ Xx @ A )
% 0.30/0.46 => ( A = emptyset ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(noeltsimpempty_type,type,
% 0.30/0.46 noeltsimpempty: $o ).
% 0.30/0.46
% 0.30/0.46 thf(noeltsimpempty,definition,
% 0.30/0.46 ( noeltsimpempty
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ~ ( in @ Xx @ A )
% 0.30/0.46 => ( A = emptyset ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setbeta_type,type,
% 0.30/0.46 setbeta: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setbeta,definition,
% 0.30/0.46 ( setbeta
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx
% 0.30/0.46 @ ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.30/0.46 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(nonempty_type,type,
% 0.30/0.46 nonempty: $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(nonempty,definition,
% 0.30/0.46 ( nonempty
% 0.30/0.46 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(nonemptyE1_type,type,
% 0.30/0.46 nonemptyE1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(nonemptyE1,definition,
% 0.30/0.46 ( nonemptyE1
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ( ( nonempty @ A )
% 0.30/0.46 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(nonemptyI_type,type,
% 0.30/0.46 nonemptyI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(nonemptyI,definition,
% 0.30/0.46 ( nonemptyI
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( Xphi @ Xx )
% 0.30/0.46 => ( nonempty
% 0.30/0.46 @ ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(nonemptyI1_type,type,
% 0.30/0.46 nonemptyI1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(nonemptyI1,definition,
% 0.30/0.46 ( nonemptyI1
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.30/0.46 => ( nonempty @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinIL_type,type,
% 0.30/0.46 setadjoinIL: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinIL,definition,
% 0.30/0.46 ( setadjoinIL
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(emptyinunitempty_type,type,
% 0.30/0.46 emptyinunitempty: $o ).
% 0.30/0.46
% 0.30/0.46 thf(emptyinunitempty,definition,
% 0.30/0.46 ( emptyinunitempty
% 0.30/0.46 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinIR_type,type,
% 0.30/0.46 setadjoinIR: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinIR,definition,
% 0.30/0.46 ( setadjoinIR
% 0.30/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinE_type,type,
% 0.30/0.46 setadjoinE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinE,definition,
% 0.30/0.46 ( setadjoinE
% 0.30/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.30/0.46 => ! [Xphi: $o] :
% 0.30/0.46 ( ( ( Xy = Xx )
% 0.30/0.46 => Xphi )
% 0.30/0.46 => ( ( ( in @ Xy @ A )
% 0.30/0.46 => Xphi )
% 0.30/0.46 => Xphi ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinOr_type,type,
% 0.30/0.46 setadjoinOr: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinOr,definition,
% 0.30/0.46 ( setadjoinOr
% 0.30/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.30/0.46 => ( ( Xy = Xx )
% 0.30/0.46 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setoftrueEq_type,type,
% 0.30/0.46 setoftrueEq: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setoftrueEq,definition,
% 0.30/0.46 ( setoftrueEq
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ( ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xx: $i] : $true )
% 0.30/0.46 = A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(powersetI_type,type,
% 0.30/0.46 powersetI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(powersetI,definition,
% 0.30/0.46 ( powersetI
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ B )
% 0.30/0.46 => ( in @ Xx @ A ) )
% 0.30/0.46 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(emptyinPowerset_type,type,
% 0.30/0.46 emptyinPowerset: $o ).
% 0.30/0.46
% 0.30/0.46 thf(emptyinPowerset,definition,
% 0.30/0.46 ( emptyinPowerset
% 0.30/0.46 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(emptyInPowerset_type,type,
% 0.30/0.46 emptyInPowerset: $o ).
% 0.30/0.46
% 0.30/0.46 thf(emptyInPowerset,definition,
% 0.30/0.46 ( emptyInPowerset
% 0.30/0.46 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(powersetE_type,type,
% 0.30/0.46 powersetE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(powersetE,definition,
% 0.30/0.46 ( powersetE
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ B @ ( powerset @ A ) )
% 0.30/0.46 => ( ( in @ Xx @ B )
% 0.30/0.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setunionI_type,type,
% 0.30/0.46 setunionI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setunionI,definition,
% 0.30/0.46 ( setunionI
% 0.30/0.46 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.30/0.46 ( ( in @ Xx @ B )
% 0.30/0.46 => ( ( in @ B @ A )
% 0.30/0.46 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setunionE_type,type,
% 0.30/0.46 setunionE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setunionE,definition,
% 0.30/0.46 ( setunionE
% 0.30/0.46 = ( ! [A: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ ( setunion @ A ) )
% 0.30/0.46 => ! [Xphi: $o] :
% 0.30/0.46 ( ! [B: $i] :
% 0.30/0.46 ( ( in @ Xx @ B )
% 0.30/0.46 => ( ( in @ B @ A )
% 0.30/0.46 => Xphi ) )
% 0.30/0.46 => Xphi ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(subPowSU_type,type,
% 0.30/0.46 subPowSU: $o ).
% 0.30/0.46
% 0.30/0.46 thf(subPowSU,definition,
% 0.30/0.46 ( subPowSU
% 0.30/0.46 = ( ! [A: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(exuE2_type,type,
% 0.30/0.46 exuE2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(exuE2,definition,
% 0.30/0.46 ( exuE2
% 0.30/0.46 = ( ! [Xphi: $i > $o] :
% 0.30/0.46 ( ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 => ? [Xx: $i] :
% 0.30/0.46 ! [Xy: $i] :
% 0.30/0.46 ( ( Xphi @ Xy )
% 0.30/0.46 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(nonemptyImpWitness_type,type,
% 0.30/0.46 nonemptyImpWitness: $o ).
% 0.30/0.46
% 0.30/0.46 thf(nonemptyImpWitness,definition,
% 0.30/0.46 ( nonemptyImpWitness
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ( ( nonempty @ A )
% 0.30/0.46 => ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & $true ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(uniqinunit_type,type,
% 0.30/0.46 uniqinunit: $o ).
% 0.30/0.46
% 0.30/0.46 thf(uniqinunit,definition,
% 0.30/0.46 ( uniqinunit
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.30/0.46 => ( Xx = Xy ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(notinsingleton_type,type,
% 0.30/0.46 notinsingleton: $o ).
% 0.30/0.46
% 0.30/0.46 thf(notinsingleton,definition,
% 0.30/0.46 ( notinsingleton
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xx != Xy )
% 0.30/0.46 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(eqinunit_type,type,
% 0.30/0.46 eqinunit: $o ).
% 0.30/0.46
% 0.30/0.46 thf(eqinunit,definition,
% 0.30/0.46 ( eqinunit
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xx = Xy )
% 0.30/0.46 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(singletonsswitch_type,type,
% 0.30/0.46 singletonsswitch: $o ).
% 0.30/0.46
% 0.30/0.46 thf(singletonsswitch,definition,
% 0.30/0.46 ( singletonsswitch
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.30/0.46 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(upairsetE_type,type,
% 0.30/0.46 upairsetE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(upairsetE,definition,
% 0.30/0.46 ( upairsetE
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.30/0.46 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.30/0.46 => ( ( Xz = Xx )
% 0.30/0.46 | ( Xz = Xy ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(upairsetIL_type,type,
% 0.30/0.46 upairsetIL: $o ).
% 0.30/0.46
% 0.30/0.46 thf(upairsetIL,definition,
% 0.30/0.46 ( upairsetIL
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(upairsetIR_type,type,
% 0.30/0.46 upairsetIR: $o ).
% 0.30/0.46
% 0.30/0.46 thf(upairsetIR,definition,
% 0.30/0.46 ( upairsetIR
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(emptyE1_type,type,
% 0.30/0.46 emptyE1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(emptyE1,definition,
% 0.30/0.46 ( emptyE1
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & ( Xphi @ Xx ) )
% 0.30/0.46 => ( ( ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 = emptyset )
% 0.30/0.46 => $false ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(vacuousDall_type,type,
% 0.30/0.46 vacuousDall: $o ).
% 0.30/0.46
% 0.30/0.46 thf(vacuousDall,definition,
% 0.30/0.46 ( vacuousDall
% 0.30/0.46 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ emptyset )
% 0.30/0.46 => ( Xphi @ Xx ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(quantDeMorgan1_type,type,
% 0.30/0.46 quantDeMorgan1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(quantDeMorgan1,definition,
% 0.30/0.46 ( quantDeMorgan1
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( ~ ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( Xphi @ Xx ) )
% 0.30/0.46 => ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(quantDeMorgan2_type,type,
% 0.30/0.46 quantDeMorgan2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(quantDeMorgan2,definition,
% 0.30/0.46 ( quantDeMorgan2
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ~ ( Xphi @ Xx ) )
% 0.30/0.46 => ~ ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(quantDeMorgan3_type,type,
% 0.30/0.46 quantDeMorgan3: $o ).
% 0.30/0.46
% 0.30/0.46 thf(quantDeMorgan3,definition,
% 0.30/0.46 ( quantDeMorgan3
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( ~ ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & ( Xphi @ Xx ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(quantDeMorgan4_type,type,
% 0.30/0.46 quantDeMorgan4: $o ).
% 0.30/0.46
% 0.30/0.46 thf(quantDeMorgan4,definition,
% 0.30/0.46 ( quantDeMorgan4
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & ~ ( Xphi @ Xx ) )
% 0.30/0.46 => ~ ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(prop2setI_type,type,
% 0.30/0.46 prop2setI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(prop2setI,definition,
% 0.30/0.46 ( prop2setI
% 0.30/0.46 = ( ! [Xphi: $o] :
% 0.30/0.46 ( Xphi
% 0.30/0.46 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(set2prop_type,type,
% 0.30/0.46 set2prop: $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(prop2set2propI_type,type,
% 0.30/0.46 prop2set2propI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(prop2set2propI,definition,
% 0.30/0.46 ( prop2set2propI
% 0.30/0.46 = ( ! [Xphi: $o] :
% 0.30/0.46 ( Xphi
% 0.30/0.46 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(notdexE_type,type,
% 0.30/0.46 notdexE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(notdexE,definition,
% 0.30/0.46 ( notdexE
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( ~ ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & ( Xphi @ Xx ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(notdallE_type,type,
% 0.30/0.46 notdallE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(notdallE,definition,
% 0.30/0.46 ( notdallE
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( ~ ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( Xphi @ Xx ) )
% 0.30/0.46 => ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(exuI1_type,type,
% 0.30/0.46 exuI1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(exuI1,definition,
% 0.30/0.46 ( exuI1
% 0.30/0.46 = ( ! [Xphi: $i > $o] :
% 0.30/0.46 ( ? [Xx: $i] :
% 0.30/0.46 ( ( Xphi @ Xx )
% 0.30/0.46 & ! [Xy: $i] :
% 0.30/0.46 ( ( Xphi @ Xy )
% 0.30/0.46 => ( Xx = Xy ) ) )
% 0.30/0.46 => ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(exuI3_type,type,
% 0.30/0.46 exuI3: $o ).
% 0.30/0.46
% 0.30/0.46 thf(exuI3,definition,
% 0.30/0.46 ( exuI3
% 0.30/0.46 = ( ! [Xphi: $i > $o] :
% 0.30/0.46 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.30/0.46 => ( ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xphi @ Xx )
% 0.30/0.46 => ( ( Xphi @ Xy )
% 0.30/0.46 => ( Xx = Xy ) ) )
% 0.30/0.46 => ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(exuI2_type,type,
% 0.30/0.46 exuI2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(exuI2,definition,
% 0.30/0.46 ( exuI2
% 0.30/0.46 = ( ! [Xphi: $i > $o] :
% 0.30/0.46 ( ? [Xx: $i] :
% 0.30/0.46 ! [Xy: $i] :
% 0.30/0.46 ( ( Xphi @ Xy )
% 0.30/0.46 <=> ( Xy = Xx ) )
% 0.30/0.46 => ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(inCongP_type,type,
% 0.30/0.46 inCongP: $o ).
% 0.30/0.46
% 0.30/0.46 thf(inCongP,definition,
% 0.30/0.46 ( inCongP
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( A = B )
% 0.30/0.46 => ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xx = Xy )
% 0.30/0.46 => ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(in__Cong_type,type,
% 0.30/0.46 in__Cong: $o ).
% 0.30/0.46
% 0.30/0.46 thf(in__Cong,definition,
% 0.30/0.46 ( in__Cong
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( A = B )
% 0.30/0.46 => ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xx = Xy )
% 0.30/0.46 => ( ( in @ Xx @ A )
% 0.30/0.46 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(exuE3u_type,type,
% 0.30/0.46 exuE3u: $o ).
% 0.30/0.46
% 0.30/0.46 thf(exuE3u,definition,
% 0.30/0.46 ( exuE3u
% 0.30/0.46 = ( ! [Xphi: $i > $o] :
% 0.30/0.46 ( ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 => ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xphi @ Xx )
% 0.30/0.46 => ( ( Xphi @ Xy )
% 0.30/0.46 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(exu__Cong_type,type,
% 0.30/0.46 exu__Cong: $o ).
% 0.30/0.46
% 0.30/0.46 thf(exu__Cong,definition,
% 0.30/0.46 ( exu__Cong
% 0.30/0.46 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.30/0.46 ( ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xx = Xy )
% 0.30/0.46 => ( ( Xphi @ Xx )
% 0.30/0.46 <=> ( Xpsi @ Xy ) ) )
% 0.30/0.46 => ( ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 <=> ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(emptyset__Cong_type,type,
% 0.30/0.46 emptyset__Cong: $o ).
% 0.30/0.46
% 0.30/0.46 thf(emptyset__Cong,definition,
% 0.30/0.46 ( emptyset__Cong
% 0.30/0.46 = ( emptyset = emptyset ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoin__Cong_type,type,
% 0.30/0.46 setadjoin__Cong: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoin__Cong,definition,
% 0.30/0.46 ( setadjoin__Cong
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xx = Xy )
% 0.30/0.46 => ! [Xz: $i,Xu: $i] :
% 0.30/0.46 ( ( Xz = Xu )
% 0.30/0.46 => ( ( setadjoin @ Xx @ Xz )
% 0.30/0.46 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(powerset__Cong_type,type,
% 0.30/0.46 powerset__Cong: $o ).
% 0.30/0.46
% 0.30/0.46 thf(powerset__Cong,definition,
% 0.30/0.46 ( powerset__Cong
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( A = B )
% 0.30/0.46 => ( ( powerset @ A )
% 0.30/0.46 = ( powerset @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setunion__Cong_type,type,
% 0.30/0.46 setunion__Cong: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setunion__Cong,definition,
% 0.30/0.46 ( setunion__Cong
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( A = B )
% 0.30/0.46 => ( ( setunion @ A )
% 0.30/0.46 = ( setunion @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(omega__Cong_type,type,
% 0.30/0.46 omega__Cong: $o ).
% 0.30/0.46
% 0.30/0.46 thf(omega__Cong,definition,
% 0.30/0.46 ( omega__Cong
% 0.30/0.46 = ( omega = omega ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(exuEu_type,type,
% 0.30/0.46 exuEu: $o ).
% 0.30/0.46
% 0.30/0.46 thf(exuEu,definition,
% 0.30/0.46 ( exuEu
% 0.30/0.46 = ( ! [Xphi: $i > $o] :
% 0.30/0.46 ( ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 => ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xphi @ Xx )
% 0.30/0.46 => ( ( Xphi @ Xy )
% 0.30/0.46 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(descr__Cong_type,type,
% 0.30/0.46 descr__Cong: $o ).
% 0.30/0.46
% 0.30/0.46 thf(descr__Cong,definition,
% 0.30/0.46 ( descr__Cong
% 0.30/0.46 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.30/0.46 ( ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xx = Xy )
% 0.30/0.46 => ( ( Xphi @ Xx )
% 0.30/0.46 <=> ( Xpsi @ Xy ) ) )
% 0.30/0.46 => ( ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 => ( ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.30/0.46 => ( ( descr
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 = ( descr
% 0.30/0.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(dsetconstr__Cong_type,type,
% 0.30/0.46 dsetconstr__Cong: $o ).
% 0.30/0.46
% 0.30/0.46 thf(dsetconstr__Cong,definition,
% 0.30/0.46 ( dsetconstr__Cong
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( A = B )
% 0.30/0.46 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ B )
% 0.30/0.46 => ( ( Xx = Xy )
% 0.30/0.46 => ( ( Xphi @ Xx )
% 0.30/0.46 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.30/0.46 => ( ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 = ( dsetconstr @ B
% 0.30/0.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(subset_type,type,
% 0.30/0.46 subset: $i > $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(disjoint_type,type,
% 0.30/0.46 disjoint: $i > $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(setsmeet_type,type,
% 0.30/0.46 setsmeet: $i > $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(subsetI1_type,type,
% 0.30/0.46 subsetI1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(subsetI1,definition,
% 0.30/0.46 ( subsetI1
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ Xx @ B ) )
% 0.30/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(eqimpsubset2_type,type,
% 0.30/0.46 eqimpsubset2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(eqimpsubset2,definition,
% 0.30/0.46 ( eqimpsubset2
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( A = B )
% 0.30/0.46 => ( subset @ B @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(eqimpsubset1_type,type,
% 0.30/0.46 eqimpsubset1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(eqimpsubset1,definition,
% 0.30/0.46 ( eqimpsubset1
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( A = B )
% 0.30/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(subsetI2_type,type,
% 0.30/0.46 subsetI2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(subsetI2,definition,
% 0.30/0.46 ( subsetI2
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ Xx @ B ) )
% 0.30/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(emptysetsubset_type,type,
% 0.30/0.46 emptysetsubset: $o ).
% 0.30/0.46
% 0.30/0.46 thf(emptysetsubset,definition,
% 0.30/0.46 ( emptysetsubset
% 0.30/0.46 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(subsetE_type,type,
% 0.30/0.46 subsetE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(subsetE,definition,
% 0.30/0.46 ( subsetE
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( subset @ A @ B )
% 0.30/0.46 => ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(subsetE2_type,type,
% 0.30/0.46 subsetE2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(subsetE2,definition,
% 0.30/0.46 ( subsetE2
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( subset @ A @ B )
% 0.30/0.46 => ( ~ ( in @ Xx @ B )
% 0.30/0.46 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(notsubsetI_type,type,
% 0.30/0.46 notsubsetI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(notsubsetI,definition,
% 0.30/0.46 ( notsubsetI
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ~ ( in @ Xx @ B )
% 0.30/0.46 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(notequalI1_type,type,
% 0.30/0.46 notequalI1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(notequalI1,definition,
% 0.30/0.46 ( notequalI1
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ~ ( subset @ A @ B )
% 0.30/0.46 => ( A != B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(notequalI2_type,type,
% 0.30/0.46 notequalI2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(notequalI2,definition,
% 0.30/0.46 ( notequalI2
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ~ ( in @ Xx @ B )
% 0.30/0.46 => ( A != B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(subsetRefl_type,type,
% 0.30/0.46 subsetRefl: $o ).
% 0.30/0.46
% 0.30/0.46 thf(subsetRefl,definition,
% 0.30/0.46 ( subsetRefl
% 0.30/0.46 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(subsetTrans_type,type,
% 0.30/0.46 subsetTrans: $o ).
% 0.30/0.46
% 0.30/0.46 thf(subsetTrans,definition,
% 0.30/0.46 ( subsetTrans
% 0.30/0.46 = ( ! [A: $i,B: $i,C: $i] :
% 0.30/0.46 ( ( subset @ A @ B )
% 0.30/0.46 => ( ( subset @ B @ C )
% 0.30/0.46 => ( subset @ A @ C ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinSub_type,type,
% 0.30/0.46 setadjoinSub: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinSub,definition,
% 0.30/0.46 ( setadjoinSub
% 0.30/0.46 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinSub2_type,type,
% 0.30/0.46 setadjoinSub2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinSub2,definition,
% 0.30/0.46 ( setadjoinSub2
% 0.30/0.46 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.30/0.46 ( ( subset @ A @ B )
% 0.30/0.46 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(subset2powerset_type,type,
% 0.30/0.46 subset2powerset: $o ).
% 0.30/0.46
% 0.30/0.46 thf(subset2powerset,definition,
% 0.30/0.46 ( subset2powerset
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( subset @ A @ B )
% 0.30/0.46 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setextsub_type,type,
% 0.30/0.46 setextsub: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setextsub,definition,
% 0.30/0.46 ( setextsub
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( subset @ A @ B )
% 0.30/0.46 => ( ( subset @ B @ A )
% 0.30/0.46 => ( A = B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(subsetemptysetimpeq_type,type,
% 0.30/0.46 subsetemptysetimpeq: $o ).
% 0.30/0.46
% 0.30/0.46 thf(subsetemptysetimpeq,definition,
% 0.30/0.46 ( subsetemptysetimpeq
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ( ( subset @ A @ emptyset )
% 0.30/0.46 => ( A = emptyset ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(powersetI1_type,type,
% 0.30/0.46 powersetI1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(powersetI1,definition,
% 0.30/0.46 ( powersetI1
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( subset @ B @ A )
% 0.30/0.46 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(powersetE1_type,type,
% 0.30/0.46 powersetE1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(powersetE1,definition,
% 0.30/0.46 ( powersetE1
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( in @ B @ ( powerset @ A ) )
% 0.30/0.46 => ( subset @ B @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(inPowerset_type,type,
% 0.30/0.46 inPowerset: $o ).
% 0.30/0.46
% 0.30/0.46 thf(inPowerset,definition,
% 0.30/0.46 ( inPowerset
% 0.30/0.46 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(powersetsubset_type,type,
% 0.30/0.46 powersetsubset: $o ).
% 0.30/0.46
% 0.30/0.46 thf(powersetsubset,definition,
% 0.30/0.46 ( powersetsubset
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( subset @ A @ B )
% 0.30/0.46 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(sepInPowerset_type,type,
% 0.30/0.46 sepInPowerset: $o ).
% 0.30/0.46
% 0.30/0.46 thf(sepInPowerset,definition,
% 0.30/0.46 ( sepInPowerset
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( in
% 0.30/0.46 @ ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 @ ( powerset @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(sepSubset_type,type,
% 0.30/0.46 sepSubset: $o ).
% 0.30/0.46
% 0.30/0.46 thf(sepSubset,definition,
% 0.30/0.46 ( sepSubset
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( subset
% 0.30/0.46 @ ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binunion_type,type,
% 0.30/0.46 binunion: $i > $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(binunionIL_type,type,
% 0.30/0.46 binunionIL: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binunionIL,definition,
% 0.30/0.46 ( binunionIL
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(upairset2IR_type,type,
% 0.30/0.46 upairset2IR: $o ).
% 0.30/0.46
% 0.30/0.46 thf(upairset2IR,definition,
% 0.30/0.46 ( upairset2IR
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binunionIR_type,type,
% 0.30/0.46 binunionIR: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binunionIR,definition,
% 0.30/0.46 ( binunionIR
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ B )
% 0.30/0.46 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binunionEcases_type,type,
% 0.30/0.46 binunionEcases: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binunionEcases,definition,
% 0.30/0.46 ( binunionEcases
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.30/0.46 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.30/0.46 => ( ( ( in @ Xx @ A )
% 0.30/0.46 => Xphi )
% 0.30/0.46 => ( ( ( in @ Xx @ B )
% 0.30/0.46 => Xphi )
% 0.30/0.46 => Xphi ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binunionE_type,type,
% 0.30/0.46 binunionE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binunionE,definition,
% 0.30/0.46 ( binunionE
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.30/0.46 => ( ( in @ Xx @ A )
% 0.30/0.46 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binunionLsub_type,type,
% 0.30/0.46 binunionLsub: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binunionLsub,definition,
% 0.30/0.46 ( binunionLsub
% 0.30/0.46 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binunionRsub_type,type,
% 0.30/0.46 binunionRsub: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binunionRsub,definition,
% 0.30/0.46 ( binunionRsub
% 0.30/0.46 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binintersect_type,type,
% 0.30/0.46 binintersect: $i > $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectI_type,type,
% 0.30/0.46 binintersectI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectI,definition,
% 0.30/0.46 ( binintersectI
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ B )
% 0.30/0.46 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectSubset5_type,type,
% 0.30/0.46 binintersectSubset5: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectSubset5,definition,
% 0.30/0.46 ( binintersectSubset5
% 0.30/0.46 = ( ! [A: $i,B: $i,C: $i] :
% 0.30/0.46 ( ( subset @ C @ A )
% 0.30/0.46 => ( ( subset @ C @ B )
% 0.30/0.46 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectEL_type,type,
% 0.30/0.46 binintersectEL: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectEL,definition,
% 0.30/0.46 ( binintersectEL
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.30/0.46 => ( in @ Xx @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectLsub_type,type,
% 0.30/0.46 binintersectLsub: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectLsub,definition,
% 0.30/0.46 ( binintersectLsub
% 0.30/0.46 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectSubset2_type,type,
% 0.30/0.46 binintersectSubset2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectSubset2,definition,
% 0.30/0.46 ( binintersectSubset2
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( subset @ A @ B )
% 0.30/0.46 => ( ( binintersect @ A @ B )
% 0.30/0.46 = A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectSubset3_type,type,
% 0.30/0.46 binintersectSubset3: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectSubset3,definition,
% 0.30/0.46 ( binintersectSubset3
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( ( binintersect @ A @ B )
% 0.30/0.46 = B )
% 0.30/0.46 => ( subset @ B @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectER_type,type,
% 0.30/0.46 binintersectER: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectER,definition,
% 0.30/0.46 ( binintersectER
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.30/0.46 => ( in @ Xx @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(disjointsetsI1_type,type,
% 0.30/0.46 disjointsetsI1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(disjointsetsI1,definition,
% 0.30/0.46 ( disjointsetsI1
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ~ ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & ( in @ Xx @ B ) )
% 0.30/0.46 => ( ( binintersect @ A @ B )
% 0.30/0.46 = emptyset ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectRsub_type,type,
% 0.30/0.46 binintersectRsub: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectRsub,definition,
% 0.30/0.46 ( binintersectRsub
% 0.30/0.46 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectSubset4_type,type,
% 0.30/0.46 binintersectSubset4: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectSubset4,definition,
% 0.30/0.46 ( binintersectSubset4
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( subset @ B @ A )
% 0.30/0.46 => ( ( binintersect @ A @ B )
% 0.30/0.46 = B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectSubset1_type,type,
% 0.30/0.46 binintersectSubset1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectSubset1,definition,
% 0.30/0.46 ( binintersectSubset1
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( ( binintersect @ A @ B )
% 0.30/0.46 = A )
% 0.30/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(bs114d_type,type,
% 0.30/0.46 bs114d: $o ).
% 0.30/0.46
% 0.30/0.46 thf(bs114d,definition,
% 0.30/0.46 ( bs114d
% 0.30/0.46 = ( ! [A: $i,B: $i,C: $i] :
% 0.30/0.46 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 0.30/0.46 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(regular_type,type,
% 0.30/0.46 regular: $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(setminus_type,type,
% 0.30/0.46 setminus: $i > $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(setminusI_type,type,
% 0.30/0.46 setminusI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setminusI,definition,
% 0.30/0.46 ( setminusI
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ~ ( in @ Xx @ B )
% 0.30/0.46 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setminusEL_type,type,
% 0.30/0.46 setminusEL: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setminusEL,definition,
% 0.30/0.46 ( setminusEL
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.30/0.46 => ( in @ Xx @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setminusER_type,type,
% 0.30/0.46 setminusER: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setminusER,definition,
% 0.30/0.46 ( setminusER
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.30/0.46 => ~ ( in @ Xx @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setminusSubset2_type,type,
% 0.30/0.46 setminusSubset2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setminusSubset2,definition,
% 0.30/0.46 ( setminusSubset2
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( subset @ A @ B )
% 0.30/0.46 => ( ( setminus @ A @ B )
% 0.30/0.46 = emptyset ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setminusERneg_type,type,
% 0.30/0.46 setminusERneg: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setminusERneg,definition,
% 0.30/0.46 ( setminusERneg
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.30/0.46 => ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setminusELneg_type,type,
% 0.30/0.46 setminusELneg: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setminusELneg,definition,
% 0.30/0.46 ( setminusELneg
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.30/0.46 => ( ~ ( in @ Xx @ B )
% 0.30/0.46 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setminusILneg_type,type,
% 0.30/0.46 setminusILneg: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setminusILneg,definition,
% 0.30/0.46 ( setminusILneg
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ~ ( in @ Xx @ A )
% 0.30/0.46 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setminusIRneg_type,type,
% 0.30/0.46 setminusIRneg: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setminusIRneg,definition,
% 0.30/0.46 ( setminusIRneg
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ B )
% 0.30/0.46 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setminusLsub_type,type,
% 0.30/0.46 setminusLsub: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setminusLsub,definition,
% 0.30/0.46 ( setminusLsub
% 0.30/0.46 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setminusSubset1_type,type,
% 0.30/0.46 setminusSubset1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setminusSubset1,definition,
% 0.30/0.46 ( setminusSubset1
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( ( setminus @ A @ B )
% 0.30/0.46 = emptyset )
% 0.30/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(symdiff_type,type,
% 0.30/0.46 symdiff: $i > $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(symdiffE_type,type,
% 0.30/0.46 symdiffE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(symdiffE,definition,
% 0.30/0.46 ( symdiffE
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 0.30/0.46 => ! [Xphi: $o] :
% 0.30/0.46 ( ( ( in @ Xx @ A )
% 0.30/0.46 => ( ~ ( in @ Xx @ B )
% 0.30/0.46 => Xphi ) )
% 0.30/0.46 => ( ( ~ ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ B )
% 0.30/0.46 => Xphi ) )
% 0.30/0.46 => Xphi ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(symdiffI1_type,type,
% 0.30/0.46 symdiffI1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(symdiffI1,definition,
% 0.30/0.46 ( symdiffI1
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ~ ( in @ Xx @ B )
% 0.30/0.46 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(symdiffI2_type,type,
% 0.30/0.46 symdiffI2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(symdiffI2,definition,
% 0.30/0.46 ( symdiffI2
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ~ ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ B )
% 0.30/0.46 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(symdiffIneg1_type,type,
% 0.30/0.46 symdiffIneg1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(symdiffIneg1,definition,
% 0.30/0.46 ( symdiffIneg1
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ B )
% 0.30/0.46 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(symdiffIneg2_type,type,
% 0.30/0.46 symdiffIneg2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(symdiffIneg2,definition,
% 0.30/0.46 ( symdiffIneg2
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ~ ( in @ Xx @ A )
% 0.30/0.46 => ( ~ ( in @ Xx @ B )
% 0.30/0.46 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(iskpair_type,type,
% 0.30/0.46 iskpair: $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(secondinupair_type,type,
% 0.30/0.46 secondinupair: $o ).
% 0.30/0.46
% 0.30/0.46 thf(secondinupair,definition,
% 0.30/0.46 ( secondinupair
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setukpairIL_type,type,
% 0.30/0.46 setukpairIL: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setukpairIL,definition,
% 0.30/0.46 ( setukpairIL
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setukpairIR_type,type,
% 0.30/0.46 setukpairIR: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setukpairIR,definition,
% 0.30/0.46 ( setukpairIR
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(kpairiskpair_type,type,
% 0.30/0.46 kpairiskpair: $o ).
% 0.30/0.46
% 0.30/0.46 thf(kpairiskpair,definition,
% 0.30/0.46 ( kpairiskpair
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(kpair_type,type,
% 0.30/0.46 kpair: $i > $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(kpairp_type,type,
% 0.30/0.46 kpairp: $o ).
% 0.30/0.46
% 0.30/0.46 thf(kpairp,definition,
% 0.30/0.46 ( kpairp
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(cartprod_type,type,
% 0.30/0.46 cartprod: $i > $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(singletonsubset_type,type,
% 0.30/0.46 singletonsubset: $o ).
% 0.30/0.46
% 0.30/0.46 thf(singletonsubset,definition,
% 0.30/0.46 ( singletonsubset
% 0.30/0.46 = ( ! [A: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(singletoninpowerset_type,type,
% 0.30/0.46 singletoninpowerset: $o ).
% 0.30/0.46
% 0.30/0.46 thf(singletoninpowerset,definition,
% 0.30/0.46 ( singletoninpowerset
% 0.30/0.46 = ( ! [A: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(singletoninpowunion_type,type,
% 0.30/0.46 singletoninpowunion: $o ).
% 0.30/0.46
% 0.30/0.46 thf(singletoninpowunion,definition,
% 0.30/0.46 ( singletoninpowunion
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(upairset2E_type,type,
% 0.30/0.46 upairset2E: $o ).
% 0.30/0.46
% 0.30/0.46 thf(upairset2E,definition,
% 0.30/0.46 ( upairset2E
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.30/0.46 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.30/0.46 => ( ( Xz = Xx )
% 0.30/0.46 | ( Xz = Xy ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(upairsubunion_type,type,
% 0.30/0.46 upairsubunion: $o ).
% 0.30/0.46
% 0.30/0.46 thf(upairsubunion,definition,
% 0.30/0.46 ( upairsubunion
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ B )
% 0.30/0.46 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(upairinpowunion_type,type,
% 0.30/0.46 upairinpowunion: $o ).
% 0.30/0.46
% 0.30/0.46 thf(upairinpowunion,definition,
% 0.30/0.46 ( upairinpowunion
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ B )
% 0.30/0.46 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(ubforcartprodlem1_type,type,
% 0.30/0.46 ubforcartprodlem1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(ubforcartprodlem1,definition,
% 0.30/0.46 ( ubforcartprodlem1
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ B )
% 0.30/0.46 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(ubforcartprodlem2_type,type,
% 0.30/0.46 ubforcartprodlem2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(ubforcartprodlem2,definition,
% 0.30/0.46 ( ubforcartprodlem2
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ B )
% 0.30/0.46 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(ubforcartprodlem3_type,type,
% 0.30/0.46 ubforcartprodlem3: $o ).
% 0.30/0.46
% 0.30/0.46 thf(ubforcartprodlem3,definition,
% 0.30/0.46 ( ubforcartprodlem3
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ B )
% 0.30/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(cartprodpairin_type,type,
% 0.30/0.46 cartprodpairin: $o ).
% 0.30/0.46
% 0.30/0.46 thf(cartprodpairin,definition,
% 0.30/0.46 ( cartprodpairin
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ B )
% 0.30/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(cartprodmempair1_type,type,
% 0.30/0.46 cartprodmempair1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(cartprodmempair1,definition,
% 0.30/0.46 ( cartprodmempair1
% 0.30/0.46 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.30/0.46 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.30/0.46 => ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & ? [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ B )
% 0.30/0.46 & ( Xu
% 0.30/0.46 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(cartprodmempair_type,type,
% 0.30/0.46 cartprodmempair: $o ).
% 0.30/0.46
% 0.30/0.46 thf(cartprodmempair,definition,
% 0.30/0.46 ( cartprodmempair
% 0.30/0.46 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.30/0.46 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.30/0.46 => ( iskpair @ Xu ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setunionE2_type,type,
% 0.30/0.46 setunionE2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setunionE2,definition,
% 0.30/0.46 ( setunionE2
% 0.30/0.46 = ( ! [A: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ ( setunion @ A ) )
% 0.30/0.46 => ? [X: $i] :
% 0.30/0.46 ( ( in @ X @ A )
% 0.30/0.46 & ( in @ Xx @ X ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setunionsingleton1_type,type,
% 0.30/0.46 setunionsingleton1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setunionsingleton1,definition,
% 0.30/0.46 ( setunionsingleton1
% 0.30/0.46 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setunionsingleton2_type,type,
% 0.30/0.46 setunionsingleton2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setunionsingleton2,definition,
% 0.30/0.46 ( setunionsingleton2
% 0.30/0.46 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setunionsingleton_type,type,
% 0.30/0.46 setunionsingleton: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setunionsingleton,definition,
% 0.30/0.46 ( setunionsingleton
% 0.30/0.46 = ( ! [Xx: $i] :
% 0.30/0.46 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 0.30/0.46 = Xx ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(singleton_type,type,
% 0.30/0.46 singleton: $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(singletonprop_type,type,
% 0.30/0.46 singletonprop: $o ).
% 0.30/0.46
% 0.30/0.46 thf(singletonprop,definition,
% 0.30/0.46 ( singletonprop
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( ( Xphi @ Xx )
% 0.30/0.46 => ( ( Xphi @ Xy )
% 0.30/0.46 => ( Xx = Xy ) ) ) ) )
% 0.30/0.46 => ( ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & ( Xphi @ Xx ) )
% 0.30/0.46 => ( singleton
% 0.30/0.46 @ ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(ex1_type,type,
% 0.30/0.46 ex1: $i > ( $i > $o ) > $o ).
% 0.30/0.46
% 0.30/0.46 thf(ex1E1_type,type,
% 0.30/0.46 ex1E1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(ex1E1,definition,
% 0.30/0.46 ( ex1E1
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( ( ex1 @ A
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 => ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(ex1I_type,type,
% 0.30/0.46 ex1I: $o ).
% 0.30/0.46
% 0.30/0.46 thf(ex1I,definition,
% 0.30/0.46 ( ex1I
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( Xphi @ Xx )
% 0.30/0.46 => ( ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( ( Xphi @ Xy )
% 0.30/0.46 => ( Xy = Xx ) ) )
% 0.30/0.46 => ( ex1 @ A
% 0.30/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(ex1I2_type,type,
% 0.30/0.46 ex1I2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(ex1I2,definition,
% 0.30/0.46 ( ex1I2
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( ( Xphi @ Xx )
% 0.30/0.46 => ( ( Xphi @ Xy )
% 0.30/0.46 => ( Xx = Xy ) ) ) ) )
% 0.30/0.46 => ( ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & ( Xphi @ Xx ) )
% 0.30/0.46 => ( ex1 @ A
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(singletonsuniq_type,type,
% 0.30/0.46 singletonsuniq: $o ).
% 0.30/0.46
% 0.30/0.46 thf(singletonsuniq,definition,
% 0.30/0.46 ( singletonsuniq
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( ( setadjoin @ Xx @ emptyset )
% 0.30/0.46 = ( setadjoin @ Xy @ emptyset ) )
% 0.30/0.46 => ( Xx = Xy ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(atmost1p_type,type,
% 0.30/0.46 atmost1p: $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(atleast2p_type,type,
% 0.30/0.46 atleast2p: $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(atmost2p_type,type,
% 0.30/0.46 atmost2p: $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(upairsetp_type,type,
% 0.30/0.46 upairsetp: $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(setukpairinjL1_type,type,
% 0.30/0.46 setukpairinjL1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setukpairinjL1,definition,
% 0.30/0.46 ( setukpairinjL1
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.30/0.46 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 0.30/0.46 => ( Xx = Xz ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(kfstsingleton_type,type,
% 0.30/0.46 kfstsingleton: $o ).
% 0.30/0.46
% 0.30/0.46 thf(kfstsingleton,definition,
% 0.30/0.46 ( kfstsingleton
% 0.30/0.46 = ( ! [Xu: $i] :
% 0.30/0.46 ( ( iskpair @ Xu )
% 0.30/0.46 => ( singleton
% 0.30/0.46 @ ( dsetconstr @ ( setunion @ Xu )
% 0.30/0.46 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(theprop_type,type,
% 0.30/0.46 theprop: $o ).
% 0.30/0.46
% 0.30/0.46 thf(theprop,definition,
% 0.30/0.46 ( theprop
% 0.30/0.46 = ( ! [X: $i] :
% 0.30/0.46 ( ( singleton @ X )
% 0.30/0.46 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(kfst_type,type,
% 0.30/0.46 kfst: $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(kfstpairEq_type,type,
% 0.30/0.46 kfstpairEq: $o ).
% 0.30/0.46
% 0.30/0.46 thf(kfstpairEq,definition,
% 0.30/0.46 ( kfstpairEq
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.30/0.46 = Xx ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(cartprodfstin_type,type,
% 0.30/0.46 cartprodfstin: $o ).
% 0.30/0.46
% 0.30/0.46 thf(cartprodfstin,definition,
% 0.30/0.46 ( cartprodfstin
% 0.30/0.46 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.30/0.46 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.30/0.46 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setukpairinjL2_type,type,
% 0.30/0.46 setukpairinjL2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setukpairinjL2,definition,
% 0.30/0.46 ( setukpairinjL2
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.30/0.46 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.30/0.46 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.30/0.46 => ( Xx = Xz ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setukpairinjL_type,type,
% 0.30/0.46 setukpairinjL: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setukpairinjL,definition,
% 0.30/0.46 ( setukpairinjL
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.30/0.46 ( ( ( kpair @ Xx @ Xy )
% 0.30/0.46 = ( kpair @ Xz @ Xu ) )
% 0.30/0.46 => ( Xx = Xz ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setukpairinjR11_type,type,
% 0.30/0.46 setukpairinjR11: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setukpairinjR11,definition,
% 0.30/0.46 ( setukpairinjR11
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xx = Xy )
% 0.30/0.46 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.30/0.46 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setukpairinjR12_type,type,
% 0.30/0.46 setukpairinjR12: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setukpairinjR12,definition,
% 0.30/0.46 ( setukpairinjR12
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xx = Xy )
% 0.30/0.46 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.30/0.46 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setukpairinjR1_type,type,
% 0.30/0.46 setukpairinjR1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setukpairinjR1,definition,
% 0.30/0.46 ( setukpairinjR1
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.30/0.46 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.30/0.46 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.30/0.46 => ( ( Xz = Xu )
% 0.30/0.46 => ( Xy = Xu ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(upairequniteq_type,type,
% 0.30/0.46 upairequniteq: $o ).
% 0.30/0.46
% 0.30/0.46 thf(upairequniteq,definition,
% 0.30/0.46 ( upairequniteq
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.30/0.46 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.30/0.46 = ( setadjoin @ Xz @ emptyset ) )
% 0.30/0.46 => ( Xx = Xy ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setukpairinjR2_type,type,
% 0.30/0.46 setukpairinjR2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setukpairinjR2,definition,
% 0.30/0.46 ( setukpairinjR2
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.30/0.46 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.30/0.46 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.30/0.46 => ( Xy = Xu ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setukpairinjR_type,type,
% 0.30/0.46 setukpairinjR: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setukpairinjR,definition,
% 0.30/0.46 ( setukpairinjR
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.30/0.46 ( ( ( kpair @ Xx @ Xy )
% 0.30/0.46 = ( kpair @ Xz @ Xu ) )
% 0.30/0.46 => ( Xy = Xu ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(ksndsingleton_type,type,
% 0.30/0.46 ksndsingleton: $o ).
% 0.30/0.46
% 0.30/0.46 thf(ksndsingleton,definition,
% 0.30/0.46 ( ksndsingleton
% 0.30/0.46 = ( ! [Xu: $i] :
% 0.30/0.46 ( ( iskpair @ Xu )
% 0.30/0.46 => ( singleton
% 0.30/0.46 @ ( dsetconstr @ ( setunion @ Xu )
% 0.30/0.46 @ ^ [Xx: $i] :
% 0.30/0.46 ( Xu
% 0.30/0.46 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(ksnd_type,type,
% 0.30/0.46 ksnd: $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(ksndpairEq_type,type,
% 0.30/0.46 ksndpairEq: $o ).
% 0.30/0.46
% 0.30/0.46 thf(ksndpairEq,definition,
% 0.30/0.46 ( ksndpairEq
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.30/0.46 = Xy ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(kpairsurjEq_type,type,
% 0.30/0.46 kpairsurjEq: $o ).
% 0.30/0.46
% 0.30/0.46 thf(kpairsurjEq,definition,
% 0.30/0.46 ( kpairsurjEq
% 0.30/0.46 = ( ! [Xu: $i] :
% 0.30/0.46 ( ( iskpair @ Xu )
% 0.30/0.46 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.30/0.46 = Xu ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(cartprodsndin_type,type,
% 0.30/0.46 cartprodsndin: $o ).
% 0.30/0.46
% 0.30/0.46 thf(cartprodsndin,definition,
% 0.30/0.46 ( cartprodsndin
% 0.30/0.46 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.30/0.46 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.30/0.46 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(cartprodpairmemEL_type,type,
% 0.30/0.46 cartprodpairmemEL: $o ).
% 0.30/0.46
% 0.30/0.46 thf(cartprodpairmemEL,definition,
% 0.30/0.46 ( cartprodpairmemEL
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.30/0.46 => ( in @ Xx @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(cartprodpairmemER_type,type,
% 0.30/0.46 cartprodpairmemER: $o ).
% 0.30/0.46
% 0.30/0.46 thf(cartprodpairmemER,definition,
% 0.30/0.46 ( cartprodpairmemER
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.30/0.46 => ( in @ Xy @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(cartprodmempaircEq_type,type,
% 0.30/0.46 cartprodmempaircEq: $o ).
% 0.30/0.46
% 0.30/0.46 thf(cartprodmempaircEq,definition,
% 0.30/0.46 ( cartprodmempaircEq
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ B )
% 0.30/0.46 => ( ( kpair @ Xx @ Xy )
% 0.30/0.46 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(cartprodfstpairEq_type,type,
% 0.30/0.46 cartprodfstpairEq: $o ).
% 0.30/0.46
% 0.30/0.46 thf(cartprodfstpairEq,definition,
% 0.30/0.46 ( cartprodfstpairEq
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ B )
% 0.30/0.46 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.30/0.46 = Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(cartprodsndpairEq_type,type,
% 0.30/0.46 cartprodsndpairEq: $o ).
% 0.30/0.46
% 0.30/0.46 thf(cartprodsndpairEq,definition,
% 0.30/0.46 ( cartprodsndpairEq
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ B )
% 0.30/0.46 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.30/0.46 = Xy ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(cartprodpairsurjEq_type,type,
% 0.30/0.46 cartprodpairsurjEq: $o ).
% 0.30/0.46
% 0.30/0.46 thf(cartprodpairsurjEq,definition,
% 0.30/0.46 ( cartprodpairsurjEq
% 0.30/0.46 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.30/0.46 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.30/0.46 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.30/0.46 = Xu ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(breln_type,type,
% 0.30/0.46 breln: $i > $i > $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(dpsetconstr_type,type,
% 0.30/0.46 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 0.30/0.46
% 0.30/0.46 thf(dpsetconstrI_type,type,
% 0.30/0.46 dpsetconstrI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(dpsetconstrI,definition,
% 0.30/0.46 ( dpsetconstrI
% 0.30/0.46 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ B )
% 0.30/0.46 => ( ( Xphi @ Xx @ Xy )
% 0.30/0.46 => ( in @ ( kpair @ Xx @ Xy )
% 0.30/0.46 @ ( dpsetconstr @ A @ B
% 0.30/0.46 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(dpsetconstrSub_type,type,
% 0.30/0.46 dpsetconstrSub: $o ).
% 0.30/0.46
% 0.30/0.46 thf(dpsetconstrSub,definition,
% 0.30/0.46 ( dpsetconstrSub
% 0.30/0.46 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.30/0.46 ( subset
% 0.30/0.46 @ ( dpsetconstr @ A @ B
% 0.30/0.46 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 0.30/0.46 @ ( cartprod @ A @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setOfPairsIsBReln_type,type,
% 0.30/0.46 setOfPairsIsBReln: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setOfPairsIsBReln,definition,
% 0.30/0.46 ( setOfPairsIsBReln
% 0.30/0.46 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.30/0.46 ( breln @ A @ B
% 0.30/0.46 @ ( dpsetconstr @ A @ B
% 0.30/0.46 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(dpsetconstrERa_type,type,
% 0.30/0.46 dpsetconstrERa: $o ).
% 0.30/0.46
% 0.30/0.46 thf(dpsetconstrERa,definition,
% 0.30/0.46 ( dpsetconstrERa
% 0.30/0.46 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ B )
% 0.30/0.46 => ( ( in @ ( kpair @ Xx @ Xy )
% 0.30/0.46 @ ( dpsetconstr @ A @ B
% 0.30/0.46 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.30/0.46 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(dpsetconstrEL1_type,type,
% 0.30/0.46 dpsetconstrEL1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(dpsetconstrEL1,definition,
% 0.30/0.46 ( dpsetconstrEL1
% 0.30/0.46 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( in @ ( kpair @ Xx @ Xy )
% 0.30/0.46 @ ( dpsetconstr @ A @ B
% 0.30/0.46 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.30/0.46 => ( in @ Xx @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(dpsetconstrEL2_type,type,
% 0.30/0.46 dpsetconstrEL2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(dpsetconstrEL2,definition,
% 0.30/0.46 ( dpsetconstrEL2
% 0.30/0.46 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( in @ ( kpair @ Xx @ Xy )
% 0.30/0.46 @ ( dpsetconstr @ A @ B
% 0.30/0.46 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.30/0.46 => ( in @ Xy @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(dpsetconstrER_type,type,
% 0.30/0.46 dpsetconstrER: $o ).
% 0.30/0.46
% 0.30/0.46 thf(dpsetconstrER,definition,
% 0.30/0.46 ( dpsetconstrER
% 0.30/0.46 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( in @ ( kpair @ Xx @ Xy )
% 0.30/0.46 @ ( dpsetconstr @ A @ B
% 0.30/0.46 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.30/0.46 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(func_type,type,
% 0.30/0.46 func: $i > $i > $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(funcSet_type,type,
% 0.30/0.46 funcSet: $i > $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(funcImageSingleton_type,type,
% 0.30/0.46 funcImageSingleton: $o ).
% 0.30/0.46
% 0.30/0.46 thf(funcImageSingleton,definition,
% 0.30/0.46 ( funcImageSingleton
% 0.30/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.46 ( ( func @ A @ B @ Xf )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( singleton
% 0.30/0.46 @ ( dsetconstr @ B
% 0.30/0.46 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(apProp_type,type,
% 0.30/0.46 apProp: $o ).
% 0.30/0.46
% 0.30/0.46 thf(apProp,definition,
% 0.30/0.46 ( apProp
% 0.30/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.46 ( ( func @ A @ B @ Xf )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( in
% 0.30/0.46 @ ( setunion
% 0.30/0.46 @ ( dsetconstr @ B
% 0.30/0.46 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 0.30/0.46 @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(ap_type,type,
% 0.30/0.46 ap: $i > $i > $i > $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(app_type,type,
% 0.30/0.46 app: $o ).
% 0.30/0.46
% 0.30/0.46 thf(app,definition,
% 0.30/0.46 ( app
% 0.30/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.46 ( ( func @ A @ B @ Xf )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(infuncsetfunc_type,type,
% 0.30/0.46 infuncsetfunc: $o ).
% 0.30/0.46
% 0.30/0.46 thf(infuncsetfunc,definition,
% 0.30/0.46 ( infuncsetfunc
% 0.30/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.46 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.30/0.46 => ( func @ A @ B @ Xf ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(ap2p_type,type,
% 0.30/0.46 ap2p: $o ).
% 0.30/0.46
% 0.30/0.46 thf(ap2p,definition,
% 0.30/0.46 ( ap2p
% 0.30/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.46 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(funcinfuncset_type,type,
% 0.30/0.46 funcinfuncset: $o ).
% 0.30/0.46
% 0.30/0.46 thf(funcinfuncset,definition,
% 0.30/0.46 ( funcinfuncset
% 0.30/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.46 ( ( func @ A @ B @ Xf )
% 0.30/0.46 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(lamProp_type,type,
% 0.30/0.46 lamProp: $o ).
% 0.30/0.46
% 0.30/0.46 thf(lamProp,definition,
% 0.30/0.46 ( lamProp
% 0.30/0.46 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.30/0.46 => ( func @ A @ B
% 0.30/0.46 @ ( dpsetconstr @ A @ B
% 0.30/0.46 @ ^ [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xf @ Xx )
% 0.30/0.46 = Xy ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(lam_type,type,
% 0.30/0.46 lam: $i > $i > ( $i > $i ) > $i ).
% 0.30/0.46
% 0.30/0.46 thf(lamp_type,type,
% 0.30/0.46 lamp: $o ).
% 0.30/0.46
% 0.30/0.46 thf(lamp,definition,
% 0.30/0.46 ( lamp
% 0.30/0.46 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.30/0.46 => ( func @ A @ B
% 0.30/0.46 @ ( lam @ A @ B
% 0.30/0.46 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(lam2p_type,type,
% 0.30/0.46 lam2p: $o ).
% 0.30/0.46
% 0.30/0.46 thf(lam2p,definition,
% 0.30/0.46 ( lam2p
% 0.30/0.46 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.30/0.46 => ( in
% 0.30/0.46 @ ( lam @ A @ B
% 0.30/0.46 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.30/0.46 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(brelnall1_type,type,
% 0.30/0.46 brelnall1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(brelnall1,definition,
% 0.30/0.46 ( brelnall1
% 0.30/0.46 = ( ! [A: $i,B: $i,R: $i] :
% 0.30/0.46 ( ( breln @ A @ B @ R )
% 0.30/0.46 => ! [Xphi: $i > $o] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ B )
% 0.30/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.30/0.46 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ R )
% 0.30/0.46 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(brelnall2_type,type,
% 0.30/0.46 brelnall2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(brelnall2,definition,
% 0.30/0.46 ( brelnall2
% 0.30/0.46 = ( ! [A: $i,B: $i,R: $i] :
% 0.30/0.46 ( ( breln @ A @ B @ R )
% 0.30/0.46 => ! [Xphi: $i > $o] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ B )
% 0.30/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.30/0.46 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ R )
% 0.30/0.46 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(ex1E2_type,type,
% 0.30/0.46 ex1E2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(ex1E2,definition,
% 0.30/0.46 ( ex1E2
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( ( ex1 @ A
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( ( Xphi @ Xx )
% 0.30/0.46 => ( ( Xphi @ Xy )
% 0.30/0.46 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(funcGraphProp1_type,type,
% 0.30/0.46 funcGraphProp1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(funcGraphProp1,definition,
% 0.30/0.46 ( funcGraphProp1
% 0.30/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.46 ( ( func @ A @ B @ Xf )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(funcGraphProp3_type,type,
% 0.30/0.46 funcGraphProp3: $o ).
% 0.30/0.46
% 0.30/0.46 thf(funcGraphProp3,definition,
% 0.30/0.46 ( funcGraphProp3
% 0.30/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.46 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(funcGraphProp2_type,type,
% 0.30/0.46 funcGraphProp2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(funcGraphProp2,definition,
% 0.30/0.46 ( funcGraphProp2
% 0.30/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.46 ( ( func @ A @ B @ Xf )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ B )
% 0.30/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.30/0.46 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.30/0.46 = Xy ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(funcextLem_type,type,
% 0.30/0.46 funcextLem: $o ).
% 0.30/0.46
% 0.30/0.46 thf(funcextLem,definition,
% 0.30/0.46 ( funcextLem
% 0.30/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.46 ( ( func @ A @ B @ Xf )
% 0.30/0.46 => ! [Xg: $i] :
% 0.30/0.46 ( ( func @ A @ B @ Xg )
% 0.30/0.46 => ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.30/0.46 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ B )
% 0.30/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 0.30/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(funcGraphProp4_type,type,
% 0.30/0.46 funcGraphProp4: $o ).
% 0.30/0.46
% 0.30/0.46 thf(funcGraphProp4,definition,
% 0.30/0.46 ( funcGraphProp4
% 0.30/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.46 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ B )
% 0.30/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.30/0.46 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.30/0.46 = Xy ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(subbreln_type,type,
% 0.30/0.46 subbreln: $o ).
% 0.30/0.46
% 0.30/0.46 thf(subbreln,definition,
% 0.30/0.46 ( subbreln
% 0.30/0.46 = ( ! [A: $i,B: $i,R: $i] :
% 0.30/0.46 ( ( breln @ A @ B @ R )
% 0.30/0.46 => ! [S: $i] :
% 0.30/0.46 ( ( breln @ A @ B @ S )
% 0.30/0.46 => ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ B )
% 0.30/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.30/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.30/0.46 => ( subset @ R @ S ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(eqbreln_type,type,
% 0.30/0.46 eqbreln: $o ).
% 0.30/0.46
% 0.30/0.46 thf(eqbreln,definition,
% 0.30/0.46 ( eqbreln
% 0.30/0.46 = ( ! [A: $i,B: $i,R: $i] :
% 0.30/0.46 ( ( breln @ A @ B @ R )
% 0.30/0.46 => ! [S: $i] :
% 0.30/0.46 ( ( breln @ A @ B @ S )
% 0.30/0.46 => ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ B )
% 0.30/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.30/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.30/0.46 => ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ B )
% 0.30/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.30/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 0.30/0.46 => ( R = S ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(funcext_type,type,
% 0.30/0.46 funcext: $o ).
% 0.30/0.46
% 0.30/0.46 thf(funcext,definition,
% 0.30/0.46 ( funcext
% 0.30/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.46 ( ( func @ A @ B @ Xf )
% 0.30/0.46 => ! [Xg: $i] :
% 0.30/0.46 ( ( func @ A @ B @ Xg )
% 0.30/0.46 => ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.30/0.46 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.30/0.46 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(funcext2_type,type,
% 0.30/0.46 funcext2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(funcext2,definition,
% 0.30/0.46 ( funcext2
% 0.30/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.46 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.30/0.46 => ! [Xg: $i] :
% 0.30/0.46 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 0.30/0.46 => ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.30/0.46 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.30/0.46 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(ap2apEq1_type,type,
% 0.30/0.46 ap2apEq1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(ap2apEq1,definition,
% 0.30/0.46 ( ap2apEq1
% 0.30/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.46 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.30/0.46 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(ap2apEq2_type,type,
% 0.30/0.46 ap2apEq2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(ap2apEq2,definition,
% 0.30/0.46 ( ap2apEq2
% 0.30/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.46 ( ( func @ A @ B @ Xf )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.30/0.46 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(beta1_type,type,
% 0.30/0.46 beta1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(beta1,definition,
% 0.30/0.46 ( beta1
% 0.30/0.46 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( ap @ A @ B
% 0.30/0.46 @ ( lam @ A @ B
% 0.30/0.46 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.30/0.46 @ Xx )
% 0.30/0.46 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(eta1_type,type,
% 0.30/0.46 eta1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(eta1,definition,
% 0.30/0.46 ( eta1
% 0.30/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.46 ( ( func @ A @ B @ Xf )
% 0.30/0.46 => ( ( lam @ A @ B
% 0.30/0.46 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.30/0.46 = Xf ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(lam2lamEq_type,type,
% 0.30/0.46 lam2lamEq: $o ).
% 0.30/0.46
% 0.30/0.46 thf(lam2lamEq,definition,
% 0.30/0.46 ( lam2lamEq
% 0.30/0.46 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.30/0.46 => ( ( lam @ A @ B
% 0.30/0.46 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.30/0.46 = ( lam @ A @ B
% 0.30/0.46 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(beta2_type,type,
% 0.30/0.46 beta2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(beta2,definition,
% 0.30/0.46 ( beta2
% 0.30/0.46 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( ap @ A @ B
% 0.30/0.46 @ ( lam @ A @ B
% 0.30/0.46 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.30/0.46 @ Xx )
% 0.30/0.46 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(eta2_type,type,
% 0.30/0.46 eta2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(eta2,definition,
% 0.30/0.46 ( eta2
% 0.30/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.46 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.30/0.46 => ( ( lam @ A @ B
% 0.30/0.46 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.30/0.46 = Xf ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(iffalseProp1_type,type,
% 0.30/0.46 iffalseProp1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(iffalseProp1,definition,
% 0.30/0.46 ( iffalseProp1
% 0.30/0.46 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( ~ Xphi
% 0.30/0.46 => ( in @ Xy
% 0.30/0.46 @ ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xz: $i] :
% 0.30/0.46 ( ( Xphi
% 0.30/0.46 & ( Xz = Xx ) )
% 0.30/0.46 | ( ~ Xphi
% 0.30/0.46 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(iffalseProp2_type,type,
% 0.30/0.46 iffalseProp2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(iffalseProp2,definition,
% 0.30/0.46 ( iffalseProp2
% 0.30/0.46 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( ~ Xphi
% 0.30/0.46 => ( ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xz: $i] :
% 0.30/0.46 ( ( Xphi
% 0.30/0.46 & ( Xz = Xx ) )
% 0.30/0.46 | ( ~ Xphi
% 0.30/0.46 & ( Xz = Xy ) ) ) )
% 0.30/0.46 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(iftrueProp1_type,type,
% 0.30/0.46 iftrueProp1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(iftrueProp1,definition,
% 0.30/0.46 ( iftrueProp1
% 0.30/0.46 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( Xphi
% 0.30/0.46 => ( in @ Xx
% 0.30/0.46 @ ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xz: $i] :
% 0.30/0.46 ( ( Xphi
% 0.30/0.46 & ( Xz = Xx ) )
% 0.30/0.46 | ( ~ Xphi
% 0.30/0.46 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(iftrueProp2_type,type,
% 0.30/0.46 iftrueProp2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(iftrueProp2,definition,
% 0.30/0.46 ( iftrueProp2
% 0.30/0.46 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( Xphi
% 0.30/0.46 => ( ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xz: $i] :
% 0.30/0.46 ( ( Xphi
% 0.30/0.46 & ( Xz = Xx ) )
% 0.30/0.46 | ( ~ Xphi
% 0.30/0.46 & ( Xz = Xy ) ) ) )
% 0.30/0.46 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(ifSingleton_type,type,
% 0.30/0.46 ifSingleton: $o ).
% 0.30/0.46
% 0.30/0.46 thf(ifSingleton,definition,
% 0.30/0.46 ( ifSingleton
% 0.30/0.46 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( singleton
% 0.30/0.46 @ ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xz: $i] :
% 0.30/0.46 ( ( Xphi
% 0.30/0.46 & ( Xz = Xx ) )
% 0.30/0.46 | ( ~ Xphi
% 0.30/0.46 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(if_type,type,
% 0.30/0.46 if: $i > $o > $i > $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(ifp_type,type,
% 0.30/0.46 ifp: $o ).
% 0.30/0.46
% 0.30/0.46 thf(ifp,definition,
% 0.30/0.46 ( ifp
% 0.30/0.46 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(theeq_type,type,
% 0.30/0.46 theeq: $o ).
% 0.30/0.46
% 0.30/0.46 thf(theeq,definition,
% 0.30/0.46 ( theeq
% 0.30/0.46 = ( ! [X: $i] :
% 0.30/0.46 ( ( singleton @ X )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ X )
% 0.30/0.46 => ( ( setunion @ X )
% 0.30/0.46 = Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(iftrue_type,type,
% 0.30/0.46 iftrue: $o ).
% 0.30/0.46
% 0.30/0.46 thf(iftrue,definition,
% 0.30/0.46 ( iftrue
% 0.30/0.46 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( Xphi
% 0.30/0.46 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.30/0.46 = Xx ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(iffalse_type,type,
% 0.30/0.46 iffalse: $o ).
% 0.30/0.46
% 0.30/0.46 thf(iffalse,definition,
% 0.30/0.46 ( iffalse
% 0.30/0.46 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( ~ Xphi
% 0.30/0.46 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.30/0.46 = Xy ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(iftrueorfalse_type,type,
% 0.30/0.46 iftrueorfalse: $o ).
% 0.30/0.46
% 0.30/0.46 thf(iftrueorfalse,definition,
% 0.30/0.46 ( iftrueorfalse
% 0.30/0.46 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectT_lem_type,type,
% 0.30/0.46 binintersectT_lem: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectT_lem,definition,
% 0.30/0.46 ( binintersectT_lem
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binunionT_lem_type,type,
% 0.30/0.46 binunionT_lem: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binunionT_lem,definition,
% 0.30/0.46 ( binunionT_lem
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(powersetT_lem_type,type,
% 0.30/0.46 powersetT_lem: $o ).
% 0.30/0.46
% 0.30/0.46 thf(powersetT_lem,definition,
% 0.30/0.46 ( powersetT_lem
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setminusT_lem_type,type,
% 0.30/0.46 setminusT_lem: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setminusT_lem,definition,
% 0.30/0.46 ( setminusT_lem
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ( in @ ( setminus @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(complementT_lem_type,type,
% 0.30/0.46 complementT_lem: $o ).
% 0.30/0.46
% 0.30/0.46 thf(complementT_lem,definition,
% 0.30/0.46 ( complementT_lem
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ( in @ ( setminus @ A @ X ) @ ( powerset @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setextT_type,type,
% 0.30/0.46 setextT: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setextT,definition,
% 0.30/0.46 ( setextT
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ X )
% 0.30/0.46 => ( in @ Xx @ Y ) ) )
% 0.30/0.46 => ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ Y )
% 0.30/0.46 => ( in @ Xx @ X ) ) )
% 0.30/0.46 => ( X = Y ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(subsetTI_type,type,
% 0.30/0.46 subsetTI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(subsetTI,definition,
% 0.30/0.46 ( subsetTI
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ X )
% 0.30/0.46 => ( in @ Xx @ Y ) ) )
% 0.30/0.46 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(powersetTI1_type,type,
% 0.30/0.46 powersetTI1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(powersetTI1,definition,
% 0.30/0.46 ( powersetTI1
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ X )
% 0.30/0.46 => ( in @ Xx @ Y ) ) )
% 0.30/0.46 => ( in @ X @ ( powerset @ Y ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(powersetTE1_type,type,
% 0.30/0.46 powersetTE1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(powersetTE1,definition,
% 0.30/0.46 ( powersetTE1
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ X @ ( powerset @ Y ) )
% 0.30/0.46 => ( ( in @ Xx @ X )
% 0.30/0.46 => ( in @ Xx @ Y ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(complementTI1_type,type,
% 0.30/0.46 complementTI1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(complementTI1,definition,
% 0.30/0.46 ( complementTI1
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ X )
% 0.30/0.46 => ~ ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(complementTE1_type,type,
% 0.30/0.46 complementTE1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(complementTE1,definition,
% 0.30/0.46 ( complementTE1
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ~ ( in @ Xx @ ( setminus @ A @ X ) )
% 0.30/0.46 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectTELcontra_type,type,
% 0.30/0.46 binintersectTELcontra: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectTELcontra,definition,
% 0.30/0.46 ( binintersectTELcontra
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ~ ( in @ Xx @ X )
% 0.30/0.46 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectTERcontra_type,type,
% 0.30/0.46 binintersectTERcontra: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectTERcontra,definition,
% 0.30/0.46 ( binintersectTERcontra
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ~ ( in @ Xx @ Y )
% 0.30/0.46 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(contrasubsetT_type,type,
% 0.30/0.46 contrasubsetT: $o ).
% 0.30/0.46
% 0.30/0.46 thf(contrasubsetT,definition,
% 0.30/0.46 ( contrasubsetT
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.30/0.46 => ( ( in @ Xx @ Y )
% 0.30/0.46 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(contrasubsetT1_type,type,
% 0.30/0.46 contrasubsetT1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(contrasubsetT1,definition,
% 0.30/0.46 ( contrasubsetT1
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( subset @ X @ Y )
% 0.30/0.46 => ( ~ ( in @ Xx @ Y )
% 0.30/0.46 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(contrasubsetT2_type,type,
% 0.30/0.46 contrasubsetT2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(contrasubsetT2,definition,
% 0.30/0.46 ( contrasubsetT2
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ( ( subset @ X @ Y )
% 0.30/0.46 => ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(contrasubsetT3_type,type,
% 0.30/0.46 contrasubsetT3: $o ).
% 0.30/0.46
% 0.30/0.46 thf(contrasubsetT3,definition,
% 0.30/0.46 ( contrasubsetT3
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ( ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) )
% 0.30/0.46 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(doubleComplementI1_type,type,
% 0.30/0.46 doubleComplementI1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(doubleComplementI1,definition,
% 0.30/0.46 ( doubleComplementI1
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ X )
% 0.30/0.46 => ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(doubleComplementE1_type,type,
% 0.30/0.46 doubleComplementE1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(doubleComplementE1,definition,
% 0.30/0.46 ( doubleComplementE1
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) )
% 0.30/0.46 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(doubleComplementSub1_type,type,
% 0.30/0.46 doubleComplementSub1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(doubleComplementSub1,definition,
% 0.30/0.46 ( doubleComplementSub1
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ( subset @ X @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(doubleComplementSub2_type,type,
% 0.30/0.46 doubleComplementSub2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(doubleComplementSub2,definition,
% 0.30/0.46 ( doubleComplementSub2
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ( subset @ ( setminus @ A @ ( setminus @ A @ X ) ) @ X ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(doubleComplementEq_type,type,
% 0.30/0.46 doubleComplementEq: $o ).
% 0.30/0.46
% 0.30/0.46 thf(doubleComplementEq,definition,
% 0.30/0.46 ( doubleComplementEq
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ( X
% 0.30/0.46 = ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(complementTnotintersectT_type,type,
% 0.30/0.46 complementTnotintersectT: $o ).
% 0.30/0.46
% 0.30/0.46 thf(complementTnotintersectT,definition,
% 0.30/0.46 ( complementTnotintersectT
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.30/0.46 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(complementImpComplementIntersect_type,type,
% 0.30/0.46 complementImpComplementIntersect: $o ).
% 0.30/0.46
% 0.30/0.46 thf(complementImpComplementIntersect,definition,
% 0.30/0.46 ( complementImpComplementIntersect
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.30/0.46 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(complementSubsetComplementIntersect_type,type,
% 0.30/0.46 complementSubsetComplementIntersect: $o ).
% 0.30/0.46
% 0.30/0.46 thf(complementSubsetComplementIntersect,definition,
% 0.30/0.46 ( complementSubsetComplementIntersect
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ( subset @ ( setminus @ A @ X ) @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(complementInPowersetComplementIntersect_type,type,
% 0.30/0.46 complementInPowersetComplementIntersect: $o ).
% 0.30/0.46
% 0.30/0.46 thf(complementInPowersetComplementIntersect,definition,
% 0.30/0.46 ( complementInPowersetComplementIntersect
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(contraSubsetComplement_type,type,
% 0.30/0.46 contraSubsetComplement: $o ).
% 0.30/0.46
% 0.30/0.46 thf(contraSubsetComplement,definition,
% 0.30/0.46 ( contraSubsetComplement
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ Y )
% 0.30/0.46 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(complementTcontraSubset_type,type,
% 0.30/0.46 complementTcontraSubset: $o ).
% 0.30/0.46
% 0.30/0.46 thf(complementTcontraSubset,definition,
% 0.30/0.46 ( complementTcontraSubset
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.30/0.46 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binunionTILcontra_type,type,
% 0.30/0.46 binunionTILcontra: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binunionTILcontra,definition,
% 0.30/0.46 ( binunionTILcontra
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.30/0.46 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binunionTIRcontra_type,type,
% 0.30/0.46 binunionTIRcontra: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binunionTIRcontra,definition,
% 0.30/0.46 ( binunionTIRcontra
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.30/0.46 => ~ ( in @ Xx @ Y ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(inIntersectImpInUnion_type,type,
% 0.30/0.46 inIntersectImpInUnion: $o ).
% 0.30/0.46
% 0.30/0.46 thf(inIntersectImpInUnion,definition,
% 0.30/0.46 ( inIntersectImpInUnion
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Z: $i] :
% 0.30/0.46 ( ( in @ Z @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.30/0.46 => ( in @ Xx @ ( binunion @ X @ Z ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(inIntersectImpInUnion2_type,type,
% 0.30/0.46 inIntersectImpInUnion2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(inIntersectImpInUnion2,definition,
% 0.30/0.46 ( inIntersectImpInUnion2
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Z: $i] :
% 0.30/0.46 ( ( in @ Z @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.30/0.46 => ( in @ Xx @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(inIntersectImpInIntersectUnions_type,type,
% 0.30/0.46 inIntersectImpInIntersectUnions: $o ).
% 0.30/0.46
% 0.30/0.46 thf(inIntersectImpInIntersectUnions,definition,
% 0.30/0.46 ( inIntersectImpInIntersectUnions
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Z: $i] :
% 0.30/0.46 ( ( in @ Z @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.30/0.46 => ( in @ Xx @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(intersectInPowersetIntersectUnions_type,type,
% 0.30/0.46 intersectInPowersetIntersectUnions: $o ).
% 0.30/0.46
% 0.30/0.46 thf(intersectInPowersetIntersectUnions,definition,
% 0.30/0.46 ( intersectInPowersetIntersectUnions
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Z: $i] :
% 0.30/0.46 ( ( in @ Z @ ( powerset @ A ) )
% 0.30/0.46 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(inComplementUnionImpNotIn1_type,type,
% 0.30/0.46 inComplementUnionImpNotIn1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(inComplementUnionImpNotIn1,definition,
% 0.30/0.46 ( inComplementUnionImpNotIn1
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.30/0.46 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(inComplementUnionImpInComplement1_type,type,
% 0.30/0.46 inComplementUnionImpInComplement1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(inComplementUnionImpInComplement1,definition,
% 0.30/0.46 ( inComplementUnionImpInComplement1
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.30/0.46 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binunionTE_type,type,
% 0.30/0.46 binunionTE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binunionTE,definition,
% 0.30/0.46 ( binunionTE
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xphi: $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.30/0.46 => ( ( ( in @ Xx @ X )
% 0.30/0.46 => Xphi )
% 0.30/0.46 => ( ( ( in @ Xx @ Y )
% 0.30/0.46 => Xphi )
% 0.30/0.46 => Xphi ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binunionTEcontra_type,type,
% 0.30/0.46 binunionTEcontra: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binunionTEcontra,definition,
% 0.30/0.46 ( binunionTEcontra
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ~ ( in @ Xx @ X )
% 0.30/0.46 => ( ~ ( in @ Xx @ Y )
% 0.30/0.46 => ~ ( in @ Xx @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(demorgan2a1_type,type,
% 0.30/0.46 demorgan2a1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(demorgan2a1,definition,
% 0.30/0.46 ( demorgan2a1
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.30/0.46 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(complementUnionInPowersetComplement_type,type,
% 0.30/0.46 complementUnionInPowersetComplement: $o ).
% 0.30/0.46
% 0.30/0.46 thf(complementUnionInPowersetComplement,definition,
% 0.30/0.46 ( complementUnionInPowersetComplement
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(demorgan2a2_type,type,
% 0.30/0.46 demorgan2a2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(demorgan2a2,definition,
% 0.30/0.46 ( demorgan2a2
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.30/0.46 => ( in @ Xx @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(demorgan1a_type,type,
% 0.30/0.46 demorgan1a: $o ).
% 0.30/0.46
% 0.30/0.46 thf(demorgan1a,definition,
% 0.30/0.46 ( demorgan1a
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) )
% 0.30/0.46 => ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(demorgan1b_type,type,
% 0.30/0.46 demorgan1b: $o ).
% 0.30/0.46
% 0.30/0.46 thf(demorgan1b,definition,
% 0.30/0.46 ( demorgan1b
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 0.30/0.46 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(demorgan1_type,type,
% 0.30/0.46 demorgan1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(demorgan1,definition,
% 0.30/0.46 ( demorgan1
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ( ( setminus @ A @ ( binintersect @ X @ Y ) )
% 0.30/0.46 = ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(demorgan2a_type,type,
% 0.30/0.46 demorgan2a: $o ).
% 0.30/0.46
% 0.30/0.46 thf(demorgan2a,definition,
% 0.30/0.46 ( demorgan2a
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.30/0.46 => ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(demorgan2b2_type,type,
% 0.30/0.46 demorgan2b2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(demorgan2b2,definition,
% 0.30/0.46 ( demorgan2b2
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.30/0.46 => ( ( in @ Xx @ ( setminus @ A @ Y ) )
% 0.30/0.46 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(demorgan2b_type,type,
% 0.30/0.46 demorgan2b: $o ).
% 0.30/0.46
% 0.30/0.46 thf(demorgan2b,definition,
% 0.30/0.46 ( demorgan2b
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 0.30/0.46 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(demorgan2_type,type,
% 0.30/0.46 demorgan2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(demorgan2,definition,
% 0.30/0.46 ( demorgan2
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ( ( setminus @ A @ ( binunion @ X @ Y ) )
% 0.30/0.46 = ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(woz13rule0_type,type,
% 0.30/0.46 woz13rule0: $o ).
% 0.30/0.46
% 0.30/0.46 thf(woz13rule0,definition,
% 0.30/0.46 ( woz13rule0
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.30/0.46 => ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(woz13rule1_type,type,
% 0.30/0.46 woz13rule1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(woz13rule1,definition,
% 0.30/0.46 ( woz13rule1
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Z: $i] :
% 0.30/0.46 ( ( in @ Z @ ( powerset @ A ) )
% 0.30/0.46 => ( ( subset @ X @ Z )
% 0.30/0.46 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(woz13rule2_type,type,
% 0.30/0.46 woz13rule2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(woz13rule2,definition,
% 0.30/0.46 ( woz13rule2
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Z: $i] :
% 0.30/0.46 ( ( in @ Z @ ( powerset @ A ) )
% 0.30/0.46 => ( ( subset @ Y @ Z )
% 0.30/0.46 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(woz13rule3_type,type,
% 0.30/0.46 woz13rule3: $o ).
% 0.30/0.46
% 0.30/0.46 thf(woz13rule3,definition,
% 0.30/0.46 ( woz13rule3
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Z: $i] :
% 0.30/0.46 ( ( in @ Z @ ( powerset @ A ) )
% 0.30/0.46 => ( ( subset @ X @ Y )
% 0.30/0.46 => ( ( subset @ X @ Z )
% 0.30/0.46 => ( subset @ X @ ( binintersect @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(woz13rule4_type,type,
% 0.30/0.46 woz13rule4: $o ).
% 0.30/0.46
% 0.30/0.46 thf(woz13rule4,definition,
% 0.30/0.46 ( woz13rule4
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Z: $i] :
% 0.30/0.46 ( ( in @ Z @ ( powerset @ A ) )
% 0.30/0.46 => ! [W: $i] :
% 0.30/0.46 ( ( in @ W @ ( powerset @ A ) )
% 0.30/0.46 => ( ( subset @ X @ Z )
% 0.30/0.46 => ( ( subset @ Y @ W )
% 0.30/0.46 => ( subset @ ( binintersect @ X @ Y ) @ ( binintersect @ Z @ W ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(woz1_1_type,type,
% 0.30/0.46 woz1_1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(woz1_1,definition,
% 0.30/0.46 ( woz1_1
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(woz1_2_type,type,
% 0.30/0.46 woz1_2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(woz1_2,definition,
% 0.30/0.46 ( woz1_2
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Z: $i] :
% 0.30/0.46 ( ( in @ Z @ ( powerset @ A ) )
% 0.30/0.46 => ! [W: $i] :
% 0.30/0.46 ( ( in @ W @ ( powerset @ A ) )
% 0.30/0.46 => ( ( setminus @ A @ ( binintersect @ ( binunion @ X @ Y ) @ ( binunion @ Z @ W ) ) )
% 0.30/0.46 = ( binunion @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) @ ( binintersect @ ( setminus @ A @ Z ) @ ( setminus @ A @ W ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(woz1_3_type,type,
% 0.30/0.46 woz1_3: $o ).
% 0.30/0.46
% 0.30/0.46 thf(woz1_3,definition,
% 0.30/0.46 ( woz1_3
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Z: $i] :
% 0.30/0.46 ( ( in @ Z @ ( powerset @ A ) )
% 0.30/0.46 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(woz1_4_type,type,
% 0.30/0.46 woz1_4: $o ).
% 0.30/0.46
% 0.30/0.46 thf(woz1_4,definition,
% 0.30/0.46 ( woz1_4
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.30/0.46 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(woz1_5_type,type,
% 0.30/0.46 woz1_5: $o ).
% 0.30/0.46
% 0.30/0.46 thf(woz1_5,definition,
% 0.30/0.46 ( woz1_5
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(breln1_type,type,
% 0.30/0.46 breln1: $i > $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(breln1all2_type,type,
% 0.30/0.46 breln1all2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(breln1all2,definition,
% 0.30/0.46 ( breln1all2
% 0.30/0.46 = ( ! [A: $i,R: $i] :
% 0.30/0.46 ( ( breln1 @ A @ R )
% 0.30/0.46 => ! [Xphi: $i > $o] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.30/0.46 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ R )
% 0.30/0.46 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(breln1Set_type,type,
% 0.30/0.46 breln1Set: $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(breln1SetBreln1_type,type,
% 0.30/0.46 breln1SetBreln1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(breln1SetBreln1,definition,
% 0.30/0.46 ( breln1SetBreln1
% 0.30/0.46 = ( ! [A: $i,R: $i] :
% 0.30/0.46 ( ( in @ R @ ( breln1Set @ A ) )
% 0.30/0.46 => ( breln1 @ A @ R ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(transitive_type,type,
% 0.30/0.46 transitive: $i > $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(antisymmetric_type,type,
% 0.30/0.46 antisymmetric: $i > $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(reflexive_type,type,
% 0.30/0.46 reflexive: $i > $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(refltransitive_type,type,
% 0.30/0.46 refltransitive: $i > $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(refllinearorder_type,type,
% 0.30/0.46 refllinearorder: $i > $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(reflwellordering_type,type,
% 0.30/0.46 reflwellordering: $i > $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(choice2fnsingleton_type,type,
% 0.30/0.46 choice2fnsingleton: $o ).
% 0.30/0.46
% 0.30/0.46 thf(choice2fnsingleton,definition,
% 0.30/0.46 ( choice2fnsingleton
% 0.30/0.46 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ? [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ B )
% 0.30/0.46 & ( Xphi @ Xx @ Xy ) ) )
% 0.30/0.46 => ! [R: $i] :
% 0.30/0.46 ( ( in @ R @ ( breln1Set @ B ) )
% 0.30/0.46 => ( ( reflwellordering @ B @ R )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( singleton
% 0.30/0.46 @ ( dsetconstr @ B
% 0.30/0.46 @ ^ [Xy: $i] :
% 0.30/0.46 ( ( Xphi @ Xx @ Xy )
% 0.30/0.46 & ! [Xz: $i] :
% 0.30/0.46 ( ( in @ Xz @ B )
% 0.30/0.46 => ( ( Xphi @ Xx @ Xz )
% 0.30/0.46 => ( in @ ( kpair @ Xy @ Xz ) @ R ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setOfPairsIsBReln1_type,type,
% 0.30/0.46 setOfPairsIsBReln1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setOfPairsIsBReln1,definition,
% 0.30/0.46 ( setOfPairsIsBReln1
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $i > $o] :
% 0.30/0.46 ( breln1 @ A
% 0.30/0.46 @ ( dpsetconstr @ A @ A
% 0.30/0.46 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(breln1all1_type,type,
% 0.30/0.46 breln1all1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(breln1all1,definition,
% 0.30/0.46 ( breln1all1
% 0.30/0.46 = ( ! [A: $i,R: $i] :
% 0.30/0.46 ( ( breln1 @ A @ R )
% 0.30/0.46 => ! [Xphi: $i > $o] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.30/0.46 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ R )
% 0.30/0.46 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(subbreln1_type,type,
% 0.30/0.46 subbreln1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(subbreln1,definition,
% 0.30/0.46 ( subbreln1
% 0.30/0.46 = ( ! [A: $i,R: $i] :
% 0.30/0.46 ( ( breln1 @ A @ R )
% 0.30/0.46 => ! [S: $i] :
% 0.30/0.46 ( ( breln1 @ A @ S )
% 0.30/0.46 => ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.30/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.30/0.46 => ( subset @ R @ S ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(eqbreln1_type,type,
% 0.30/0.46 eqbreln1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(eqbreln1,definition,
% 0.30/0.46 ( eqbreln1
% 0.30/0.46 = ( ! [A: $i,R: $i] :
% 0.30/0.46 ( ( breln1 @ A @ R )
% 0.30/0.46 => ! [S: $i] :
% 0.30/0.46 ( ( breln1 @ A @ S )
% 0.30/0.46 => ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.30/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.30/0.46 => ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.30/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 0.30/0.46 => ( R = S ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(breln1invset_type,type,
% 0.30/0.46 breln1invset: $i > $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(breln1invprop_type,type,
% 0.30/0.46 breln1invprop: $o ).
% 0.30/0.46
% 0.30/0.46 thf(breln1invprop,definition,
% 0.30/0.46 ( breln1invprop
% 0.30/0.46 = ( ! [A: $i,R: $i] :
% 0.30/0.46 ( ( breln1 @ A @ R )
% 0.30/0.46 => ( breln1 @ A @ ( breln1invset @ A @ R ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(breln1invI_type,type,
% 0.30/0.46 breln1invI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(breln1invI,definition,
% 0.30/0.46 ( breln1invI
% 0.30/0.46 = ( ! [A: $i,R: $i] :
% 0.30/0.46 ( ( breln1 @ A @ R )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.30/0.46 => ( in @ ( kpair @ Xy @ Xx ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(breln1invE_type,type,
% 0.30/0.46 breln1invE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(breln1invE,definition,
% 0.30/0.46 ( breln1invE
% 0.30/0.46 = ( ! [A: $i,R: $i] :
% 0.30/0.46 ( ( breln1 @ A @ R )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( ( in @ ( kpair @ Xy @ Xx ) @ ( breln1invset @ A @ R ) )
% 0.30/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(breln1compset_type,type,
% 0.30/0.46 breln1compset: $i > $i > $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(breln1compprop_type,type,
% 0.30/0.46 breln1compprop: $o ).
% 0.30/0.46
% 0.30/0.46 thf(breln1compprop,definition,
% 0.30/0.46 ( breln1compprop
% 0.30/0.46 = ( ! [A: $i,R: $i] :
% 0.30/0.46 ( ( breln1 @ A @ R )
% 0.30/0.46 => ! [S: $i] :
% 0.30/0.46 ( ( breln1 @ A @ S )
% 0.30/0.46 => ( breln1 @ A @ ( breln1compset @ A @ R @ S ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(breln1compI_type,type,
% 0.30/0.46 breln1compI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(breln1compI,definition,
% 0.30/0.46 ( breln1compI
% 0.30/0.46 = ( ! [A: $i,R: $i] :
% 0.30/0.46 ( ( breln1 @ A @ R )
% 0.30/0.46 => ! [S: $i] :
% 0.30/0.46 ( ( breln1 @ A @ S )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ! [Xz: $i] :
% 0.30/0.46 ( ( in @ Xz @ A )
% 0.30/0.46 => ( ( in @ ( kpair @ Xx @ Xz ) @ R )
% 0.30/0.46 => ( ( in @ ( kpair @ Xz @ Xy ) @ S )
% 0.30/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(breln1compE_type,type,
% 0.30/0.46 breln1compE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(breln1compE,definition,
% 0.30/0.46 ( breln1compE
% 0.30/0.46 = ( ! [A: $i,R: $i] :
% 0.30/0.46 ( ( breln1 @ A @ R )
% 0.30/0.46 => ! [S: $i] :
% 0.30/0.46 ( ( breln1 @ A @ S )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) )
% 0.30/0.46 => ? [Xz: $i] :
% 0.30/0.46 ( ( in @ Xz @ A )
% 0.30/0.46 & ( in @ ( kpair @ Xx @ Xz ) @ R )
% 0.30/0.46 & ( in @ ( kpair @ Xz @ Xy ) @ S ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(breln1compEex_type,type,
% 0.30/0.46 breln1compEex: $o ).
% 0.30/0.46
% 0.30/0.46 thf(breln1compEex,definition,
% 0.30/0.46 ( breln1compEex
% 0.30/0.46 = ( ! [A: $i,R: $i] :
% 0.30/0.46 ( ( breln1 @ A @ R )
% 0.30/0.46 => ! [S: $i] :
% 0.30/0.46 ( ( breln1 @ A @ S )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) )
% 0.30/0.46 => ! [Xphi: $o] :
% 0.30/0.46 ( ! [Xz: $i] :
% 0.30/0.46 ( ( in @ Xz @ A )
% 0.30/0.46 => ( ( in @ ( kpair @ Xx @ Xz ) @ R )
% 0.30/0.46 => ( ( in @ ( kpair @ Xz @ Xy ) @ S )
% 0.30/0.46 => Xphi ) ) )
% 0.30/0.46 => Xphi ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(breln1unionprop_type,type,
% 0.30/0.46 breln1unionprop: $o ).
% 0.30/0.46
% 0.30/0.46 thf(breln1unionprop,definition,
% 0.30/0.46 ( breln1unionprop
% 0.30/0.46 = ( ! [A: $i,R: $i] :
% 0.30/0.46 ( ( breln1 @ A @ R )
% 0.30/0.46 => ! [S: $i] :
% 0.30/0.46 ( ( breln1 @ A @ S )
% 0.30/0.46 => ( breln1 @ A @ ( binunion @ R @ S ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(breln1unionIL_type,type,
% 0.30/0.46 breln1unionIL: $o ).
% 0.30/0.46
% 0.30/0.46 thf(breln1unionIL,definition,
% 0.30/0.46 ( breln1unionIL
% 0.30/0.46 = ( ! [A: $i,R: $i] :
% 0.30/0.46 ( ( breln1 @ A @ R )
% 0.30/0.46 => ! [S: $i] :
% 0.30/0.46 ( ( breln1 @ A @ S )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.30/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(breln1unionIR_type,type,
% 0.30/0.46 breln1unionIR: $o ).
% 0.30/0.46
% 0.30/0.46 thf(breln1unionIR,definition,
% 0.30/0.46 ( breln1unionIR
% 0.30/0.46 = ( ! [A: $i,R: $i] :
% 0.30/0.46 ( ( breln1 @ A @ R )
% 0.30/0.46 => ! [S: $i] :
% 0.30/0.46 ( ( breln1 @ A @ S )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.30/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(breln1unionI,conjecture,
% 0.30/0.46 ( setextAx
% 0.30/0.46 => ( emptysetAx
% 0.30/0.46 => ( setadjoinAx
% 0.30/0.46 => ( powersetAx
% 0.30/0.46 => ( setunionAx
% 0.30/0.46 => ( omega0Ax
% 0.30/0.46 => ( omegaSAx
% 0.30/0.46 => ( omegaIndAx
% 0.30/0.46 => ( replAx
% 0.30/0.46 => ( foundationAx
% 0.30/0.46 => ( wellorderingAx
% 0.30/0.46 => ( descrp
% 0.30/0.46 => ( dsetconstrI
% 0.30/0.46 => ( dsetconstrEL
% 0.30/0.46 => ( dsetconstrER
% 0.30/0.46 => ( exuE1
% 0.30/0.46 => ( prop2setE
% 0.30/0.46 => ( emptysetE
% 0.30/0.46 => ( emptysetimpfalse
% 0.30/0.46 => ( notinemptyset
% 0.30/0.46 => ( exuE3e
% 0.30/0.46 => ( setext
% 0.30/0.46 => ( emptyI
% 0.30/0.46 => ( noeltsimpempty
% 0.30/0.46 => ( setbeta
% 0.30/0.46 => ( nonemptyE1
% 0.30/0.46 => ( nonemptyI
% 0.30/0.46 => ( nonemptyI1
% 0.30/0.46 => ( setadjoinIL
% 0.30/0.46 => ( emptyinunitempty
% 0.30/0.46 => ( setadjoinIR
% 0.30/0.46 => ( setadjoinE
% 0.30/0.46 => ( setadjoinOr
% 0.30/0.46 => ( setoftrueEq
% 0.30/0.46 => ( powersetI
% 0.30/0.46 => ( emptyinPowerset
% 0.30/0.46 => ( emptyInPowerset
% 0.30/0.46 => ( powersetE
% 0.30/0.46 => ( setunionI
% 0.30/0.46 => ( setunionE
% 0.30/0.46 => ( subPowSU
% 0.30/0.46 => ( exuE2
% 0.30/0.46 => ( nonemptyImpWitness
% 0.30/0.46 => ( uniqinunit
% 0.30/0.46 => ( notinsingleton
% 0.30/0.46 => ( eqinunit
% 0.30/0.46 => ( singletonsswitch
% 0.30/0.46 => ( upairsetE
% 0.30/0.46 => ( upairsetIL
% 0.30/0.46 => ( upairsetIR
% 0.30/0.46 => ( emptyE1
% 0.30/0.46 => ( vacuousDall
% 0.30/0.46 => ( quantDeMorgan1
% 0.30/0.46 => ( quantDeMorgan2
% 0.30/0.46 => ( quantDeMorgan3
% 0.30/0.46 => ( quantDeMorgan4
% 0.30/0.46 => ( prop2setI
% 0.30/0.46 => ( prop2set2propI
% 0.30/0.46 => ( notdexE
% 0.30/0.46 => ( notdallE
% 0.30/0.46 => ( exuI1
% 0.30/0.46 => ( exuI3
% 0.30/0.46 => ( exuI2
% 0.30/0.46 => ( inCongP
% 0.30/0.46 => ( in__Cong
% 0.30/0.46 => ( exuE3u
% 0.30/0.46 => ( exu__Cong
% 0.30/0.46 => ( emptyset__Cong
% 0.30/0.46 => ( setadjoin__Cong
% 0.30/0.46 => ( powerset__Cong
% 0.30/0.46 => ( setunion__Cong
% 0.30/0.46 => ( omega__Cong
% 0.30/0.46 => ( exuEu
% 0.30/0.46 => ( descr__Cong
% 0.30/0.46 => ( dsetconstr__Cong
% 0.30/0.46 => ( subsetI1
% 0.30/0.46 => ( eqimpsubset2
% 0.30/0.46 => ( eqimpsubset1
% 0.30/0.46 => ( subsetI2
% 0.30/0.46 => ( emptysetsubset
% 0.30/0.46 => ( subsetE
% 0.30/0.46 => ( subsetE2
% 0.30/0.46 => ( notsubsetI
% 0.30/0.46 => ( notequalI1
% 0.30/0.46 => ( notequalI2
% 0.30/0.46 => ( subsetRefl
% 0.30/0.46 => ( subsetTrans
% 0.30/0.46 => ( setadjoinSub
% 0.30/0.46 => ( setadjoinSub2
% 0.30/0.46 => ( subset2powerset
% 0.30/0.46 => ( setextsub
% 0.30/0.46 => ( subsetemptysetimpeq
% 0.30/0.46 => ( powersetI1
% 0.30/0.46 => ( powersetE1
% 0.30/0.46 => ( inPowerset
% 0.30/0.46 => ( powersetsubset
% 0.30/0.46 => ( sepInPowerset
% 0.30/0.46 => ( sepSubset
% 0.30/0.46 => ( binunionIL
% 0.30/0.46 => ( upairset2IR
% 0.30/0.46 => ( binunionIR
% 0.30/0.46 => ( binunionEcases
% 0.30/0.46 => ( binunionE
% 0.30/0.46 => ( binunionLsub
% 0.30/0.46 => ( binunionRsub
% 0.30/0.46 => ( binintersectI
% 0.30/0.46 => ( binintersectSubset5
% 0.30/0.46 => ( binintersectEL
% 0.30/0.46 => ( binintersectLsub
% 0.30/0.46 => ( binintersectSubset2
% 0.30/0.46 => ( binintersectSubset3
% 0.30/0.46 => ( binintersectER
% 0.30/0.46 => ( disjointsetsI1
% 0.30/0.46 => ( binintersectRsub
% 0.30/0.46 => ( binintersectSubset4
% 0.30/0.46 => ( binintersectSubset1
% 0.30/0.46 => ( bs114d
% 0.30/0.46 => ( setminusI
% 0.30/0.46 => ( setminusEL
% 0.30/0.46 => ( setminusER
% 0.30/0.46 => ( setminusSubset2
% 0.30/0.46 => ( setminusERneg
% 0.30/0.46 => ( setminusELneg
% 0.30/0.46 => ( setminusILneg
% 0.30/0.46 => ( setminusIRneg
% 0.30/0.46 => ( setminusLsub
% 0.30/0.46 => ( setminusSubset1
% 0.30/0.46 => ( symdiffE
% 0.30/0.46 => ( symdiffI1
% 0.30/0.46 => ( symdiffI2
% 0.30/0.46 => ( symdiffIneg1
% 0.30/0.46 => ( symdiffIneg2
% 0.30/0.46 => ( secondinupair
% 0.30/0.46 => ( setukpairIL
% 0.30/0.46 => ( setukpairIR
% 0.30/0.46 => ( kpairiskpair
% 0.30/0.46 => ( kpairp
% 0.30/0.46 => ( singletonsubset
% 0.30/0.46 => ( singletoninpowerset
% 0.30/0.46 => ( singletoninpowunion
% 0.30/0.46 => ( upairset2E
% 0.30/0.46 => ( upairsubunion
% 0.30/0.46 => ( upairinpowunion
% 0.30/0.46 => ( ubforcartprodlem1
% 0.30/0.46 => ( ubforcartprodlem2
% 0.30/0.46 => ( ubforcartprodlem3
% 0.30/0.46 => ( cartprodpairin
% 0.30/0.46 => ( cartprodmempair1
% 0.30/0.46 => ( cartprodmempair
% 0.30/0.46 => ( setunionE2
% 0.30/0.46 => ( setunionsingleton1
% 0.30/0.46 => ( setunionsingleton2
% 0.30/0.46 => ( setunionsingleton
% 0.30/0.46 => ( singletonprop
% 0.30/0.46 => ( ex1E1
% 0.30/0.46 => ( ex1I
% 0.30/0.46 => ( ex1I2
% 0.30/0.46 => ( singletonsuniq
% 0.30/0.46 => ( setukpairinjL1
% 0.30/0.46 => ( kfstsingleton
% 0.30/0.46 => ( theprop
% 0.30/0.46 => ( kfstpairEq
% 0.30/0.46 => ( cartprodfstin
% 0.30/0.46 => ( setukpairinjL2
% 0.30/0.46 => ( setukpairinjL
% 0.30/0.46 => ( setukpairinjR11
% 0.30/0.46 => ( setukpairinjR12
% 0.30/0.46 => ( setukpairinjR1
% 0.30/0.46 => ( upairequniteq
% 0.30/0.46 => ( setukpairinjR2
% 0.30/0.46 => ( setukpairinjR
% 0.30/0.46 => ( ksndsingleton
% 0.30/0.46 => ( ksndpairEq
% 0.30/0.46 => ( kpairsurjEq
% 0.30/0.46 => ( cartprodsndin
% 0.30/0.46 => ( cartprodpairmemEL
% 0.30/0.46 => ( cartprodpairmemER
% 0.30/0.46 => ( cartprodmempaircEq
% 0.30/0.46 => ( cartprodfstpairEq
% 0.30/0.46 => ( cartprodsndpairEq
% 0.30/0.46 => ( cartprodpairsurjEq
% 0.30/0.46 => ( dpsetconstrI
% 0.30/0.46 => ( dpsetconstrSub
% 0.30/0.46 => ( setOfPairsIsBReln
% 0.30/0.46 => ( dpsetconstrERa
% 0.30/0.46 => ( dpsetconstrEL1
% 0.30/0.46 => ( dpsetconstrEL2
% 0.30/0.46 => ( dpsetconstrER
% 0.30/0.46 => ( funcImageSingleton
% 0.30/0.46 => ( apProp
% 0.30/0.46 => ( app
% 0.30/0.46 => ( infuncsetfunc
% 0.30/0.46 => ( ap2p
% 0.30/0.46 => ( funcinfuncset
% 0.30/0.46 => ( lamProp
% 0.30/0.46 => ( lamp
% 0.30/0.46 => ( lam2p
% 0.30/0.46 => ( brelnall1
% 0.30/0.46 => ( brelnall2
% 0.30/0.46 => ( ex1E2
% 0.30/0.46 => ( funcGraphProp1
% 0.30/0.46 => ( funcGraphProp3
% 0.30/0.46 => ( funcGraphProp2
% 0.30/0.46 => ( funcextLem
% 0.30/0.46 => ( funcGraphProp4
% 0.30/0.46 => ( subbreln
% 0.30/0.46 => ( eqbreln
% 0.30/0.46 => ( funcext
% 0.30/0.46 => ( funcext2
% 0.30/0.46 => ( ap2apEq1
% 0.30/0.46 => ( ap2apEq2
% 0.30/0.46 => ( beta1
% 0.30/0.46 => ( eta1
% 0.30/0.46 => ( lam2lamEq
% 0.30/0.46 => ( beta2
% 0.30/0.46 => ( eta2
% 0.30/0.46 => ( iffalseProp1
% 0.30/0.46 => ( iffalseProp2
% 0.30/0.46 => ( iftrueProp1
% 0.30/0.46 => ( iftrueProp2
% 0.30/0.46 => ( ifSingleton
% 0.30/0.46 => ( ifp
% 0.30/0.46 => ( theeq
% 0.30/0.46 => ( iftrue
% 0.30/0.46 => ( iffalse
% 0.30/0.46 => ( iftrueorfalse
% 0.30/0.46 => ( binintersectT_lem
% 0.30/0.46 => ( binunionT_lem
% 0.30/0.46 => ( powersetT_lem
% 0.30/0.46 => ( setminusT_lem
% 0.30/0.46 => ( complementT_lem
% 0.30/0.46 => ( setextT
% 0.30/0.46 => ( subsetTI
% 0.30/0.46 => ( powersetTI1
% 0.30/0.46 => ( powersetTE1
% 0.30/0.46 => ( complementTI1
% 0.30/0.46 => ( complementTE1
% 0.30/0.46 => ( binintersectTELcontra
% 0.30/0.46 => ( binintersectTERcontra
% 0.30/0.46 => ( contrasubsetT
% 0.30/0.46 => ( contrasubsetT1
% 0.30/0.46 => ( contrasubsetT2
% 0.30/0.46 => ( contrasubsetT3
% 0.30/0.46 => ( doubleComplementI1
% 0.30/0.46 => ( doubleComplementE1
% 0.30/0.46 => ( doubleComplementSub1
% 0.30/0.46 => ( doubleComplementSub2
% 0.30/0.46 => ( doubleComplementEq
% 0.30/0.46 => ( complementTnotintersectT
% 0.30/0.46 => ( complementImpComplementIntersect
% 0.30/0.46 => ( complementSubsetComplementIntersect
% 0.30/0.46 => ( complementInPowersetComplementIntersect
% 0.30/0.46 => ( contraSubsetComplement
% 0.30/0.46 => ( complementTcontraSubset
% 0.30/0.46 => ( binunionTILcontra
% 0.30/0.46 => ( binunionTIRcontra
% 0.30/0.46 => ( inIntersectImpInUnion
% 0.30/0.46 => ( inIntersectImpInUnion2
% 0.30/0.46 => ( inIntersectImpInIntersectUnions
% 0.30/0.46 => ( intersectInPowersetIntersectUnions
% 0.30/0.46 => ( inComplementUnionImpNotIn1
% 0.30/0.46 => ( inComplementUnionImpInComplement1
% 0.30/0.46 => ( binunionTE
% 0.30/0.46 => ( binunionTEcontra
% 0.30/0.46 => ( demorgan2a1
% 0.30/0.46 => ( complementUnionInPowersetComplement
% 0.30/0.46 => ( demorgan2a2
% 0.30/0.46 => ( demorgan1a
% 0.30/0.46 => ( demorgan1b
% 0.30/0.46 => ( demorgan1
% 0.30/0.46 => ( demorgan2a
% 0.30/0.46 => ( demorgan2b2
% 0.30/0.46 => ( demorgan2b
% 0.30/0.46 => ( demorgan2
% 0.30/0.46 => ( woz13rule0
% 0.30/0.46 => ( woz13rule1
% 0.30/0.46 => ( woz13rule2
% 0.30/0.46 => ( woz13rule3
% 0.30/0.46 => ( woz13rule4
% 0.30/0.46 => ( woz1_1
% 0.30/0.46 => ( woz1_2
% 0.30/0.46 => ( woz1_3
% 0.30/0.46 => ( woz1_4
% 0.30/0.46 => ( woz1_5
% 0.30/0.46 => ( breln1all2
% 0.30/0.46 => ( breln1SetBreln1
% 0.30/0.47 => ( choice2fnsingleton
% 0.30/0.47 => ( setOfPairsIsBReln1
% 0.30/0.47 => ( breln1all1
% 0.30/0.47 => ( subbreln1
% 0.30/0.47 => ( eqbreln1
% 0.30/0.47 => ( breln1invprop
% 0.30/0.47 => ( breln1invI
% 0.30/0.47 => ( breln1invE
% 0.30/0.47 => ( breln1compprop
% 0.30/0.47 => ( breln1compI
% 0.30/0.47 => ( breln1compE
% 0.30/0.47 => ( breln1compEex
% 0.30/0.47 => ( breln1unionprop
% 0.30/0.47 => ( breln1unionIL
% 0.30/0.47 => ( breln1unionIR
% 0.30/0.47 => ! [A: $i,R: $i] :
% 0.30/0.47 ( ( breln1 @ A @ R )
% 0.30/0.47 => ! [S: $i] :
% 0.30/0.47 ( ( breln1 @ A @ S )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ A )
% 0.30/0.47 => ( ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.30/0.47 | ( in @ ( kpair @ Xx @ Xy ) @ S ) )
% 0.30/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 %------------------------------------------------------------------------------
% 0.30/0.47 ------------------------
% 3.02/1.66 ---- Embedded in THF ---
% 3.02/1.66 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 3.02/1.66 %%% Generated on Tue Feb 24 12:33:46 EST 2026
% 3.02/1.66 %%% using '$$dhol' embedding, version 1.3.0.
% 3.02/1.66 %%% Logic specification used:
% 3.02/1.66 %%% thf(spec, logic, $$dhol).
% 3.02/1.66
% 3.02/1.66 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.uLFW0IB354/DTF2DTF24330.p
% See solution above
% 3.02/1.66 ------------------------
% 3.02/1.69 ---- Cleaned THF ---
% 3.02/1.70 thf(in_type,type,
% 3.02/1.70 in: $i > $i > $o ).
% 3.02/1.70
% 3.02/1.70 thf(exu_type,type,
% 3.02/1.70 exu: ( $i > $o ) > $o ).
% 3.02/1.70
% 3.02/1.70 thf(exu,definition,
% 3.02/1.70 ( exu
% 3.02/1.70 = ( ^ [Xphi: $i > $o] :
% 3.02/1.70 ? [Xx: $i] :
% 3.02/1.70 ( ( Xphi @ Xx )
% 3.02/1.70 & ! [Xy: $i] :
% 3.02/1.70 ( ( Xphi @ Xy )
% 3.02/1.70 => ( Xx = Xy ) ) ) ) ) ).
% 3.02/1.70
% 3.02/1.70 thf(setextAx_type,type,
% 3.02/1.70 setextAx: $o ).
% 3.02/1.70
% 3.02/1.70 thf(setextAx,definition,
% 3.02/1.70 ( setextAx
% 3.02/1.70 = ( ! [A: $i,B: $i] :
% 3.02/1.70 ( ! [Xx: $i] :
% 3.02/1.70 ( ( in @ Xx @ A )
% 3.02/1.70 <=> ( in @ Xx @ B ) )
% 3.02/1.70 => ( A = B ) ) ) ) ).
% 3.02/1.70
% 3.02/1.70 thf(emptyset_type,type,
% 3.02/1.70 emptyset: $i ).
% 3.02/1.70
% 3.02/1.70 thf(emptysetAx_type,type,
% 3.02/1.70 emptysetAx: $o ).
% 3.02/1.70
% 3.02/1.70 thf(emptysetAx,definition,
% 3.02/1.70 ( emptysetAx
% 3.02/1.70 = ( ! [Xx: $i] :
% 3.02/1.70 ~ ( in @ Xx @ emptyset ) ) ) ).
% 3.02/1.70
% 3.02/1.70 thf(setadjoin_type,type,
% 3.02/1.70 setadjoin: $i > $i > $i ).
% 3.02/1.70
% 3.02/1.70 thf(setadjoinAx_type,type,
% 3.02/1.70 setadjoinAx: $o ).
% 3.02/1.70
% 3.02/1.70 thf(setadjoinAx,definition,
% 3.02/1.70 ( setadjoinAx
% 3.02/1.70 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 3.02/1.70 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 3.02/1.70 <=> ( ( Xy = Xx )
% 3.02/1.70 | ( in @ Xy @ A ) ) ) ) ) ).
% 3.02/1.70
% 3.02/1.70 thf(powerset_type,type,
% 3.02/1.70 powerset: $i > $i ).
% 3.02/1.70
% 3.02/1.70 thf(powersetAx_type,type,
% 3.02/1.70 powersetAx: $o ).
% 3.02/1.70
% 3.02/1.70 thf(powersetAx,definition,
% 3.02/1.70 ( powersetAx
% 3.02/1.70 = ( ! [A: $i,B: $i] :
% 3.02/1.70 ( ( in @ B @ ( powerset @ A ) )
% 3.02/1.70 <=> ! [Xx: $i] :
% 3.02/1.70 ( ( in @ Xx @ B )
% 3.02/1.70 => ( in @ Xx @ A ) ) ) ) ) ).
% 3.02/1.70
% 3.02/1.70 thf(setunion_type,type,
% 3.02/1.70 setunion: $i > $i ).
% 3.02/1.70
% 3.02/1.70 thf(setunionAx_type,type,
% 3.02/1.70 setunionAx: $o ).
% 3.02/1.70
% 3.02/1.70 thf(setunionAx,definition,
% 3.02/1.70 ( setunionAx
% 3.02/1.70 = ( ! [A: $i,Xx: $i] :
% 3.02/1.70 ( ( in @ Xx @ ( setunion @ A ) )
% 3.02/1.70 <=> ? [B: $i] :
% 3.02/1.70 ( ( in @ Xx @ B )
% 3.02/1.70 & ( in @ B @ A ) ) ) ) ) ).
% 3.02/1.70
% 3.02/1.70 thf(omega_type,type,
% 3.02/1.70 omega: $i ).
% 3.02/1.70
% 3.02/1.70 thf(omega0Ax_type,type,
% 3.02/1.70 omega0Ax: $o ).
% 3.02/1.70
% 3.02/1.70 thf(omega0Ax,definition,
% 3.02/1.70 ( omega0Ax
% 3.02/1.70 = ( in @ emptyset @ omega ) ) ).
% 3.02/1.70
% 3.02/1.70 thf(omegaSAx_type,type,
% 3.02/1.70 omegaSAx: $o ).
% 3.02/1.70
% 3.02/1.70 thf(omegaSAx,definition,
% 3.02/1.70 ( omegaSAx
% 3.02/1.70 = ( ! [Xx: $i] :
% 3.02/1.70 ( ( in @ Xx @ omega )
% 3.02/1.70 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 3.02/1.70
% 3.02/1.70 thf(omegaIndAx_type,type,
% 3.02/1.70 omegaIndAx: $o ).
% 3.02/1.70
% 3.02/1.70 thf(omegaIndAx,definition,
% 3.02/1.70 ( omegaIndAx
% 3.02/1.70 = ( ! [A: $i] :
% 3.02/1.70 ( ( ( in @ emptyset @ A )
% 3.02/1.70 & ! [Xx: $i] :
% 3.02/1.70 ( ( ( in @ Xx @ omega )
% 3.02/1.70 & ( in @ Xx @ A ) )
% 3.02/1.70 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 3.02/1.70 => ! [Xx: $i] :
% 3.02/1.70 ( ( in @ Xx @ omega )
% 3.02/1.70 => ( in @ Xx @ A ) ) ) ) ) ).
% 3.02/1.70
% 3.02/1.70 thf(replAx_type,type,
% 3.02/1.70 replAx: $o ).
% 3.02/1.70
% 3.02/1.70 thf(replAx,definition,
% 3.02/1.70 ( replAx
% 3.02/1.70 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 3.02/1.70 ( ! [Xx: $i] :
% 3.02/1.70 ( ( in @ Xx @ A )
% 3.02/1.70 => ( exu
% 3.02/1.70 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 3.02/1.70 => ? [B: $i] :
% 3.02/1.70 ! [Xx: $i] :
% 3.02/1.70 ( ( in @ Xx @ B )
% 3.02/1.70 <=> ? [Xy: $i] :
% 3.02/1.70 ( ( in @ Xy @ A )
% 3.02/1.70 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 3.02/1.70
% 3.02/1.70 thf(foundationAx_type,type,
% 3.02/1.70 foundationAx: $o ).
% 3.02/1.70
% 3.02/1.70 thf(foundationAx,definition,
% 3.02/1.70 ( foundationAx
% 3.02/1.70 = ( ! [A: $i] :
% 3.02/1.70 ( ? [Xx: $i] : ( in @ Xx @ A )
% 3.02/1.70 => ? [B: $i] :
% 3.02/1.70 ( ( in @ B @ A )
% 3.02/1.70 & ~ ? [Xx: $i] :
% 3.02/1.70 ( ( in @ Xx @ B )
% 3.02/1.70 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 3.02/1.70
% 3.02/1.70 thf(wellorderingAx_type,type,
% 3.02/1.70 wellorderingAx: $o ).
% 3.02/1.70
% 3.02/1.70 thf(wellorderingAx,definition,
% 3.02/1.70 ( wellorderingAx
% 3.02/1.70 = ( ! [A: $i] :
% 3.02/1.70 ? [B: $i] :
% 3.02/1.70 ( ! [C: $i] :
% 3.02/1.70 ( ( in @ C @ B )
% 3.02/1.70 => ! [Xx: $i] :
% 3.02/1.70 ( ( in @ Xx @ C )
% 3.02/1.70 => ( in @ Xx @ A ) ) )
% 3.02/1.70 & ! [Xx: $i,Xy: $i] :
% 3.02/1.70 ( ( ( in @ Xx @ A )
% 3.02/1.70 & ( in @ Xy @ A ) )
% 3.02/1.70 => ( ! [C: $i] :
% 3.02/1.70 ( ( in @ C @ B )
% 3.02/1.70 => ( ( in @ Xx @ C )
% 3.02/1.70 <=> ( in @ Xy @ C ) ) )
% 3.02/1.70 => ( Xx = Xy ) ) )
% 3.02/1.70 & ! [C: $i,D: $i] :
% 3.02/1.70 ( ( ( in @ C @ B )
% 3.02/1.70 & ( in @ D @ B ) )
% 3.02/1.70 => ( ! [Xx: $i] :
% 3.02/1.70 ( ( in @ Xx @ C )
% 3.02/1.70 => ( in @ Xx @ D ) )
% 3.02/1.70 | ! [Xx: $i] :
% 3.02/1.70 ( ( in @ Xx @ D )
% 3.02/1.70 => ( in @ Xx @ C ) ) ) )
% 3.02/1.70 & ! [C: $i] :
% 3.02/1.70 ( ( ! [Xx: $i] :
% 3.02/1.70 ( ( in @ Xx @ C )
% 3.02/1.70 => ( in @ Xx @ A ) )
% 3.02/1.70 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 3.02/1.70 => ? [D: $i,Xx: $i] :
% 3.02/1.70 ( ( in @ D @ B )
% 3.02/1.70 & ( in @ Xx @ C )
% 3.02/1.70 & ~ ? [Xy: $i] :
% 3.02/1.70 ( ( in @ Xy @ D )
% 3.02/1.70 & ( in @ Xy @ C ) )
% 3.02/1.70 & ! [E: $i] :
% 3.02/1.70 ( ( in @ E @ B )
% 3.02/1.70 => ( ! [Xy: $i] :
% 3.02/1.70 ( ( in @ Xy @ E )
% 3.02/1.70 => ( in @ Xy @ D ) )
% 3.02/1.70 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 3.02/1.70
% 3.02/1.70 thf(descr_type,type,
% 3.02/1.70 descr: ( $i > $o ) > $i ).
% 3.02/1.70
% 3.02/1.70 thf(descrp_type,type,
% 3.02/1.70 descrp: $o ).
% 3.02/1.70
% 3.02/1.70 thf(descrp,definition,
% 3.02/1.70 ( descrp
% 3.02/1.70 = ( ! [Xphi: $i > $o] :
% 3.02/1.70 ( ( exu
% 3.02/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.02/1.70 => ( Xphi
% 3.02/1.70 @ ( descr
% 3.02/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.02/1.70
% 3.02/1.70 thf(dsetconstr_type,type,
% 3.02/1.70 dsetconstr: $i > ( $i > $o ) > $i ).
% 3.02/1.70
% 3.02/1.70 thf(dsetconstrI_type,type,
% 3.02/1.70 dsetconstrI: $o ).
% 3.02/1.70
% 3.02/1.70 thf(dsetconstrI,definition,
% 3.02/1.70 ( dsetconstrI
% 3.02/1.70 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.02/1.70 ( ( in @ Xx @ A )
% 3.02/1.70 => ( ( Xphi @ Xx )
% 3.02/1.70 => ( in @ Xx
% 3.02/1.70 @ ( dsetconstr @ A
% 3.02/1.70 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 3.02/1.70
% 3.02/1.70 thf(dsetconstrEL_type,type,
% 3.02/1.70 dsetconstrEL: $o ).
% 3.02/1.70
% 3.02/1.70 thf(dsetconstrEL,definition,
% 3.02/1.70 ( dsetconstrEL
% 3.02/1.70 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.02/1.70 ( ( in @ Xx
% 3.02/1.70 @ ( dsetconstr @ A
% 3.02/1.70 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 3.02/1.70 => ( in @ Xx @ A ) ) ) ) ).
% 3.02/1.70
% 3.02/1.70 thf(dsetconstrER_type,type,
% 3.02/1.70 dsetconstrER: $o ).
% 3.02/1.70
% 3.02/1.70 thf(dsetconstrER,definition,
% 3.02/1.70 ( dsetconstrER
% 3.02/1.70 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.02/1.70 ( ( in @ Xx
% 3.02/1.70 @ ( dsetconstr @ A
% 3.02/1.70 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 3.02/1.70 => ( Xphi @ Xx ) ) ) ) ).
% 3.02/1.70
% 3.02/1.70 thf(exuE1_type,type,
% 3.02/1.70 exuE1: $o ).
% 3.02/1.70
% 3.02/1.70 thf(exuE1,definition,
% 3.02/1.70 ( exuE1
% 3.02/1.70 = ( ! [Xphi: $i > $o] :
% 3.02/1.70 ( ( exu
% 3.02/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.02/1.70 => ? [Xx: $i] :
% 3.02/1.70 ( ( Xphi @ Xx )
% 3.02/1.70 & ! [Xy: $i] :
% 3.02/1.70 ( ( Xphi @ Xy )
% 3.02/1.70 => ( Xx = Xy ) ) ) ) ) ) ).
% 3.02/1.70
% 3.02/1.70 thf(prop2set_type,type,
% 3.02/1.70 prop2set: $o > $i ).
% 3.02/1.70
% 3.02/1.70 thf(prop2setE_type,type,
% 3.02/1.70 prop2setE: $o ).
% 3.02/1.70
% 3.02/1.70 thf(prop2setE,definition,
% 3.02/1.70 ( prop2setE
% 3.02/1.70 = ( ! [Xphi: $o,Xx: $i] :
% 3.02/1.70 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 3.02/1.70 => Xphi ) ) ) ).
% 3.02/1.70
% 3.02/1.70 thf(emptysetE_type,type,
% 3.02/1.70 emptysetE: $o ).
% 3.02/1.70
% 3.02/1.70 thf(emptysetE,definition,
% 3.02/1.70 ( emptysetE
% 3.02/1.70 = ( ! [Xx: $i] :
% 3.02/1.70 ( ( in @ Xx @ emptyset )
% 3.02/1.70 => ! [Xphi: $o] : Xphi ) ) ) ).
% 3.02/1.70
% 3.02/1.70 thf(emptysetimpfalse_type,type,
% 3.02/1.70 emptysetimpfalse: $o ).
% 3.02/1.70
% 3.02/1.70 thf(emptysetimpfalse,definition,
% 3.02/1.70 ( emptysetimpfalse
% 3.02/1.70 = ( ! [Xx: $i] :
% 3.02/1.70 ( ( in @ Xx @ emptyset )
% 3.02/1.70 => $false ) ) ) ).
% 3.02/1.70
% 3.02/1.70 thf(notinemptyset_type,type,
% 3.02/1.70 notinemptyset: $o ).
% 3.02/1.70
% 3.02/1.70 thf(notinemptyset,definition,
% 3.02/1.70 ( notinemptyset
% 3.02/1.70 = ( ! [Xx: $i] :
% 3.02/1.70 ~ ( in @ Xx @ emptyset ) ) ) ).
% 3.02/1.70
% 3.02/1.70 thf(exuE3e_type,type,
% 3.02/1.70 exuE3e: $o ).
% 3.02/1.70
% 3.02/1.70 thf(exuE3e,definition,
% 3.02/1.70 ( exuE3e
% 3.02/1.70 = ( ! [Xphi: $i > $o] :
% 3.02/1.70 ( ( exu
% 3.02/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.02/1.70 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 3.02/1.70
% 3.02/1.70 thf(setext_type,type,
% 3.02/1.70 setext: $o ).
% 3.02/1.70
% 3.02/1.70 thf(setext,definition,
% 3.02/1.70 ( setext
% 3.02/1.70 = ( ! [A: $i,B: $i] :
% 3.02/1.70 ( ! [Xx: $i] :
% 3.02/1.70 ( ( in @ Xx @ A )
% 3.02/1.70 => ( in @ Xx @ B ) )
% 3.02/1.70 => ( ! [Xx: $i] :
% 3.02/1.70 ( ( in @ Xx @ B )
% 3.02/1.70 => ( in @ Xx @ A ) )
% 3.02/1.70 => ( A = B ) ) ) ) ) ).
% 3.02/1.70
% 3.02/1.70 thf(emptyI_type,type,
% 3.02/1.70 emptyI: $o ).
% 3.02/1.70
% 3.02/1.70 thf(emptyI,definition,
% 3.02/1.70 ( emptyI
% 3.02/1.70 = ( ! [A: $i] :
% 3.02/1.70 ( ! [Xx: $i] :
% 3.02/1.70 ~ ( in @ Xx @ A )
% 3.02/1.70 => ( A = emptyset ) ) ) ) ).
% 3.02/1.70
% 3.02/1.70 thf(noeltsimpempty_type,type,
% 3.02/1.70 noeltsimpempty: $o ).
% 3.02/1.70
% 3.02/1.70 thf(noeltsimpempty,definition,
% 3.02/1.70 ( noeltsimpempty
% 3.02/1.70 = ( ! [A: $i] :
% 3.02/1.70 ( ! [Xx: $i] :
% 3.02/1.70 ~ ( in @ Xx @ A )
% 3.02/1.70 => ( A = emptyset ) ) ) ) ).
% 3.02/1.70
% 3.02/1.70 thf(setbeta_type,type,
% 3.02/1.70 setbeta: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setbeta,definition,
% 3.21/1.70 ( setbeta
% 3.21/1.70 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ( in @ Xx
% 3.21/1.70 @ ( dsetconstr @ A
% 3.21/1.70 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 3.21/1.70 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(nonempty_type,type,
% 3.21/1.70 nonempty: $i > $o ).
% 3.21/1.70
% 3.21/1.70 thf(nonempty,definition,
% 3.21/1.70 ( nonempty
% 3.21/1.70 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(nonemptyE1_type,type,
% 3.21/1.70 nonemptyE1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(nonemptyE1,definition,
% 3.21/1.70 ( nonemptyE1
% 3.21/1.70 = ( ! [A: $i] :
% 3.21/1.70 ( ( nonempty @ A )
% 3.21/1.70 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(nonemptyI_type,type,
% 3.21/1.70 nonemptyI: $o ).
% 3.21/1.70
% 3.21/1.70 thf(nonemptyI,definition,
% 3.21/1.70 ( nonemptyI
% 3.21/1.70 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ( Xphi @ Xx )
% 3.21/1.70 => ( nonempty
% 3.21/1.70 @ ( dsetconstr @ A
% 3.21/1.70 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(nonemptyI1_type,type,
% 3.21/1.70 nonemptyI1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(nonemptyI1,definition,
% 3.21/1.70 ( nonemptyI1
% 3.21/1.70 = ( ! [A: $i] :
% 3.21/1.70 ( ? [Xx: $i] : ( in @ Xx @ A )
% 3.21/1.70 => ( nonempty @ A ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(setadjoinIL_type,type,
% 3.21/1.70 setadjoinIL: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setadjoinIL,definition,
% 3.21/1.70 ( setadjoinIL
% 3.21/1.70 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(emptyinunitempty_type,type,
% 3.21/1.70 emptyinunitempty: $o ).
% 3.21/1.70
% 3.21/1.70 thf(emptyinunitempty,definition,
% 3.21/1.70 ( emptyinunitempty
% 3.21/1.70 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(setadjoinIR_type,type,
% 3.21/1.70 setadjoinIR: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setadjoinIR,definition,
% 3.21/1.70 ( setadjoinIR
% 3.21/1.70 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ A )
% 3.21/1.70 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(setadjoinE_type,type,
% 3.21/1.70 setadjoinE: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setadjoinE,definition,
% 3.21/1.70 ( setadjoinE
% 3.21/1.70 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 3.21/1.70 => ! [Xphi: $o] :
% 3.21/1.70 ( ( ( Xy = Xx )
% 3.21/1.70 => Xphi )
% 3.21/1.70 => ( ( ( in @ Xy @ A )
% 3.21/1.70 => Xphi )
% 3.21/1.70 => Xphi ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(setadjoinOr_type,type,
% 3.21/1.70 setadjoinOr: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setadjoinOr,definition,
% 3.21/1.70 ( setadjoinOr
% 3.21/1.70 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 3.21/1.70 => ( ( Xy = Xx )
% 3.21/1.70 | ( in @ Xy @ A ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(setoftrueEq_type,type,
% 3.21/1.70 setoftrueEq: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setoftrueEq,definition,
% 3.21/1.70 ( setoftrueEq
% 3.21/1.70 = ( ! [A: $i] :
% 3.21/1.70 ( ( dsetconstr @ A
% 3.21/1.70 @ ^ [Xx: $i] : $true )
% 3.21/1.70 = A ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(powersetI_type,type,
% 3.21/1.70 powersetI: $o ).
% 3.21/1.70
% 3.21/1.70 thf(powersetI,definition,
% 3.21/1.70 ( powersetI
% 3.21/1.70 = ( ! [A: $i,B: $i] :
% 3.21/1.70 ( ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ B )
% 3.21/1.70 => ( in @ Xx @ A ) )
% 3.21/1.70 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(emptyinPowerset_type,type,
% 3.21/1.70 emptyinPowerset: $o ).
% 3.21/1.70
% 3.21/1.70 thf(emptyinPowerset,definition,
% 3.21/1.70 ( emptyinPowerset
% 3.21/1.70 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(emptyInPowerset_type,type,
% 3.21/1.70 emptyInPowerset: $o ).
% 3.21/1.70
% 3.21/1.70 thf(emptyInPowerset,definition,
% 3.21/1.70 ( emptyInPowerset
% 3.21/1.70 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(powersetE_type,type,
% 3.21/1.70 powersetE: $o ).
% 3.21/1.70
% 3.21/1.70 thf(powersetE,definition,
% 3.21/1.70 ( powersetE
% 3.21/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.21/1.70 ( ( in @ B @ ( powerset @ A ) )
% 3.21/1.70 => ( ( in @ Xx @ B )
% 3.21/1.70 => ( in @ Xx @ A ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(setunionI_type,type,
% 3.21/1.70 setunionI: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setunionI,definition,
% 3.21/1.70 ( setunionI
% 3.21/1.70 = ( ! [A: $i,Xx: $i,B: $i] :
% 3.21/1.70 ( ( in @ Xx @ B )
% 3.21/1.70 => ( ( in @ B @ A )
% 3.21/1.70 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(setunionE_type,type,
% 3.21/1.70 setunionE: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setunionE,definition,
% 3.21/1.70 ( setunionE
% 3.21/1.70 = ( ! [A: $i,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ ( setunion @ A ) )
% 3.21/1.70 => ! [Xphi: $o] :
% 3.21/1.70 ( ! [B: $i] :
% 3.21/1.70 ( ( in @ Xx @ B )
% 3.21/1.70 => ( ( in @ B @ A )
% 3.21/1.70 => Xphi ) )
% 3.21/1.70 => Xphi ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(subPowSU_type,type,
% 3.21/1.70 subPowSU: $o ).
% 3.21/1.70
% 3.21/1.70 thf(subPowSU,definition,
% 3.21/1.70 ( subPowSU
% 3.21/1.70 = ( ! [A: $i,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(exuE2_type,type,
% 3.21/1.70 exuE2: $o ).
% 3.21/1.70
% 3.21/1.70 thf(exuE2,definition,
% 3.21/1.70 ( exuE2
% 3.21/1.70 = ( ! [Xphi: $i > $o] :
% 3.21/1.70 ( ( exu
% 3.21/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.21/1.70 => ? [Xx: $i] :
% 3.21/1.70 ! [Xy: $i] :
% 3.21/1.70 ( ( Xphi @ Xy )
% 3.21/1.70 <=> ( Xy = Xx ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(nonemptyImpWitness_type,type,
% 3.21/1.70 nonemptyImpWitness: $o ).
% 3.21/1.70
% 3.21/1.70 thf(nonemptyImpWitness,definition,
% 3.21/1.70 ( nonemptyImpWitness
% 3.21/1.70 = ( ! [A: $i] :
% 3.21/1.70 ( ( nonempty @ A )
% 3.21/1.70 => ? [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 & $true ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(uniqinunit_type,type,
% 3.21/1.70 uniqinunit: $o ).
% 3.21/1.70
% 3.21/1.70 thf(uniqinunit,definition,
% 3.21/1.70 ( uniqinunit
% 3.21/1.70 = ( ! [Xx: $i,Xy: $i] :
% 3.21/1.70 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 3.21/1.70 => ( Xx = Xy ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(notinsingleton_type,type,
% 3.21/1.70 notinsingleton: $o ).
% 3.21/1.70
% 3.21/1.70 thf(notinsingleton,definition,
% 3.21/1.70 ( notinsingleton
% 3.21/1.70 = ( ! [Xx: $i,Xy: $i] :
% 3.21/1.70 ( ( Xx != Xy )
% 3.21/1.70 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(eqinunit_type,type,
% 3.21/1.70 eqinunit: $o ).
% 3.21/1.70
% 3.21/1.70 thf(eqinunit,definition,
% 3.21/1.70 ( eqinunit
% 3.21/1.70 = ( ! [Xx: $i,Xy: $i] :
% 3.21/1.70 ( ( Xx = Xy )
% 3.21/1.70 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(singletonsswitch_type,type,
% 3.21/1.70 singletonsswitch: $o ).
% 3.21/1.70
% 3.21/1.70 thf(singletonsswitch,definition,
% 3.21/1.70 ( singletonsswitch
% 3.21/1.70 = ( ! [Xx: $i,Xy: $i] :
% 3.21/1.70 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 3.21/1.70 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(upairsetE_type,type,
% 3.21/1.70 upairsetE: $o ).
% 3.21/1.70
% 3.21/1.70 thf(upairsetE,definition,
% 3.21/1.70 ( upairsetE
% 3.21/1.70 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 3.21/1.70 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 3.21/1.70 => ( ( Xz = Xx )
% 3.21/1.70 | ( Xz = Xy ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(upairsetIL_type,type,
% 3.21/1.70 upairsetIL: $o ).
% 3.21/1.70
% 3.21/1.70 thf(upairsetIL,definition,
% 3.21/1.70 ( upairsetIL
% 3.21/1.70 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(upairsetIR_type,type,
% 3.21/1.70 upairsetIR: $o ).
% 3.21/1.70
% 3.21/1.70 thf(upairsetIR,definition,
% 3.21/1.70 ( upairsetIR
% 3.21/1.70 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(emptyE1_type,type,
% 3.21/1.70 emptyE1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(emptyE1,definition,
% 3.21/1.70 ( emptyE1
% 3.21/1.70 = ( ! [A: $i,Xphi: $i > $o] :
% 3.21/1.70 ( ? [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 & ( Xphi @ Xx ) )
% 3.21/1.70 => ( ( ( dsetconstr @ A
% 3.21/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.21/1.70 = emptyset )
% 3.21/1.70 => $false ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(vacuousDall_type,type,
% 3.21/1.70 vacuousDall: $o ).
% 3.21/1.70
% 3.21/1.70 thf(vacuousDall,definition,
% 3.21/1.70 ( vacuousDall
% 3.21/1.70 = ( ! [Xphi: $i > $o,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ emptyset )
% 3.21/1.70 => ( Xphi @ Xx ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(quantDeMorgan1_type,type,
% 3.21/1.70 quantDeMorgan1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(quantDeMorgan1,definition,
% 3.21/1.70 ( quantDeMorgan1
% 3.21/1.70 = ( ! [A: $i,Xphi: $i > $o] :
% 3.21/1.70 ( ~ ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( Xphi @ Xx ) )
% 3.21/1.70 => ? [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(quantDeMorgan2_type,type,
% 3.21/1.70 quantDeMorgan2: $o ).
% 3.21/1.70
% 3.21/1.70 thf(quantDeMorgan2,definition,
% 3.21/1.70 ( quantDeMorgan2
% 3.21/1.70 = ( ! [A: $i,Xphi: $i > $o] :
% 3.21/1.70 ( ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ~ ( Xphi @ Xx ) )
% 3.21/1.70 => ~ ? [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 & ( Xphi @ Xx ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(quantDeMorgan3_type,type,
% 3.21/1.70 quantDeMorgan3: $o ).
% 3.21/1.70
% 3.21/1.70 thf(quantDeMorgan3,definition,
% 3.21/1.70 ( quantDeMorgan3
% 3.21/1.70 = ( ! [A: $i,Xphi: $i > $o] :
% 3.21/1.70 ( ~ ? [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 & ( Xphi @ Xx ) )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(quantDeMorgan4_type,type,
% 3.21/1.70 quantDeMorgan4: $o ).
% 3.21/1.70
% 3.21/1.70 thf(quantDeMorgan4,definition,
% 3.21/1.70 ( quantDeMorgan4
% 3.21/1.70 = ( ! [A: $i,Xphi: $i > $o] :
% 3.21/1.70 ( ? [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 & ~ ( Xphi @ Xx ) )
% 3.21/1.70 => ~ ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( Xphi @ Xx ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(prop2setI_type,type,
% 3.21/1.70 prop2setI: $o ).
% 3.21/1.70
% 3.21/1.70 thf(prop2setI,definition,
% 3.21/1.70 ( prop2setI
% 3.21/1.70 = ( ! [Xphi: $o] :
% 3.21/1.70 ( Xphi
% 3.21/1.70 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(set2prop_type,type,
% 3.21/1.70 set2prop: $i > $o ).
% 3.21/1.70
% 3.21/1.70 thf(prop2set2propI_type,type,
% 3.21/1.70 prop2set2propI: $o ).
% 3.21/1.70
% 3.21/1.70 thf(prop2set2propI,definition,
% 3.21/1.70 ( prop2set2propI
% 3.21/1.70 = ( ! [Xphi: $o] :
% 3.21/1.70 ( Xphi
% 3.21/1.70 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(notdexE_type,type,
% 3.21/1.70 notdexE: $o ).
% 3.21/1.70
% 3.21/1.70 thf(notdexE,definition,
% 3.21/1.70 ( notdexE
% 3.21/1.70 = ( ! [A: $i,Xphi: $i > $o] :
% 3.21/1.70 ( ~ ? [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 & ( Xphi @ Xx ) )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(notdallE_type,type,
% 3.21/1.70 notdallE: $o ).
% 3.21/1.70
% 3.21/1.70 thf(notdallE,definition,
% 3.21/1.70 ( notdallE
% 3.21/1.70 = ( ! [A: $i,Xphi: $i > $o] :
% 3.21/1.70 ( ~ ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( Xphi @ Xx ) )
% 3.21/1.70 => ? [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(exuI1_type,type,
% 3.21/1.70 exuI1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(exuI1,definition,
% 3.21/1.70 ( exuI1
% 3.21/1.70 = ( ! [Xphi: $i > $o] :
% 3.21/1.70 ( ? [Xx: $i] :
% 3.21/1.70 ( ( Xphi @ Xx )
% 3.21/1.70 & ! [Xy: $i] :
% 3.21/1.70 ( ( Xphi @ Xy )
% 3.21/1.70 => ( Xx = Xy ) ) )
% 3.21/1.70 => ( exu
% 3.21/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(exuI3_type,type,
% 3.21/1.70 exuI3: $o ).
% 3.21/1.70
% 3.21/1.70 thf(exuI3,definition,
% 3.21/1.70 ( exuI3
% 3.21/1.70 = ( ! [Xphi: $i > $o] :
% 3.21/1.70 ( ? [Xx: $i] : ( Xphi @ Xx )
% 3.21/1.70 => ( ! [Xx: $i,Xy: $i] :
% 3.21/1.70 ( ( Xphi @ Xx )
% 3.21/1.70 => ( ( Xphi @ Xy )
% 3.21/1.70 => ( Xx = Xy ) ) )
% 3.21/1.70 => ( exu
% 3.21/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(exuI2_type,type,
% 3.21/1.70 exuI2: $o ).
% 3.21/1.70
% 3.21/1.70 thf(exuI2,definition,
% 3.21/1.70 ( exuI2
% 3.21/1.70 = ( ! [Xphi: $i > $o] :
% 3.21/1.70 ( ? [Xx: $i] :
% 3.21/1.70 ! [Xy: $i] :
% 3.21/1.70 ( ( Xphi @ Xy )
% 3.21/1.70 <=> ( Xy = Xx ) )
% 3.21/1.70 => ( exu
% 3.21/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(inCongP_type,type,
% 3.21/1.70 inCongP: $o ).
% 3.21/1.70
% 3.21/1.70 thf(inCongP,definition,
% 3.21/1.70 ( inCongP
% 3.21/1.70 = ( ! [A: $i,B: $i] :
% 3.21/1.70 ( ( A = B )
% 3.21/1.70 => ! [Xx: $i,Xy: $i] :
% 3.21/1.70 ( ( Xx = Xy )
% 3.21/1.70 => ( ( in @ Xx @ A )
% 3.21/1.70 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(in__Cong_type,type,
% 3.21/1.70 in__Cong: $o ).
% 3.21/1.70
% 3.21/1.70 thf(in__Cong,definition,
% 3.21/1.70 ( in__Cong
% 3.21/1.70 = ( ! [A: $i,B: $i] :
% 3.21/1.70 ( ( A = B )
% 3.21/1.70 => ! [Xx: $i,Xy: $i] :
% 3.21/1.70 ( ( Xx = Xy )
% 3.21/1.70 => ( ( in @ Xx @ A )
% 3.21/1.70 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(exuE3u_type,type,
% 3.21/1.70 exuE3u: $o ).
% 3.21/1.70
% 3.21/1.70 thf(exuE3u,definition,
% 3.21/1.70 ( exuE3u
% 3.21/1.70 = ( ! [Xphi: $i > $o] :
% 3.21/1.70 ( ( exu
% 3.21/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.21/1.70 => ! [Xx: $i,Xy: $i] :
% 3.21/1.70 ( ( Xphi @ Xx )
% 3.21/1.70 => ( ( Xphi @ Xy )
% 3.21/1.70 => ( Xx = Xy ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(exu__Cong_type,type,
% 3.21/1.70 exu__Cong: $o ).
% 3.21/1.70
% 3.21/1.70 thf(exu__Cong,definition,
% 3.21/1.70 ( exu__Cong
% 3.21/1.70 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 3.21/1.70 ( ! [Xx: $i,Xy: $i] :
% 3.21/1.70 ( ( Xx = Xy )
% 3.21/1.70 => ( ( Xphi @ Xx )
% 3.21/1.70 <=> ( Xpsi @ Xy ) ) )
% 3.21/1.70 => ( ( exu
% 3.21/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.21/1.70 <=> ( exu
% 3.21/1.70 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(emptyset__Cong_type,type,
% 3.21/1.70 emptyset__Cong: $o ).
% 3.21/1.70
% 3.21/1.70 thf(emptyset__Cong,definition,
% 3.21/1.70 ( emptyset__Cong
% 3.21/1.70 = ( emptyset = emptyset ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(setadjoin__Cong_type,type,
% 3.21/1.70 setadjoin__Cong: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setadjoin__Cong,definition,
% 3.21/1.70 ( setadjoin__Cong
% 3.21/1.70 = ( ! [Xx: $i,Xy: $i] :
% 3.21/1.70 ( ( Xx = Xy )
% 3.21/1.70 => ! [Xz: $i,Xu: $i] :
% 3.21/1.70 ( ( Xz = Xu )
% 3.21/1.70 => ( ( setadjoin @ Xx @ Xz )
% 3.21/1.70 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(powerset__Cong_type,type,
% 3.21/1.70 powerset__Cong: $o ).
% 3.21/1.70
% 3.21/1.70 thf(powerset__Cong,definition,
% 3.21/1.70 ( powerset__Cong
% 3.21/1.70 = ( ! [A: $i,B: $i] :
% 3.21/1.70 ( ( A = B )
% 3.21/1.70 => ( ( powerset @ A )
% 3.21/1.70 = ( powerset @ B ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(setunion__Cong_type,type,
% 3.21/1.70 setunion__Cong: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setunion__Cong,definition,
% 3.21/1.70 ( setunion__Cong
% 3.21/1.70 = ( ! [A: $i,B: $i] :
% 3.21/1.70 ( ( A = B )
% 3.21/1.70 => ( ( setunion @ A )
% 3.21/1.70 = ( setunion @ B ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(omega__Cong_type,type,
% 3.21/1.70 omega__Cong: $o ).
% 3.21/1.70
% 3.21/1.70 thf(omega__Cong,definition,
% 3.21/1.70 ( omega__Cong
% 3.21/1.70 = ( omega = omega ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(exuEu_type,type,
% 3.21/1.70 exuEu: $o ).
% 3.21/1.70
% 3.21/1.70 thf(exuEu,definition,
% 3.21/1.70 ( exuEu
% 3.21/1.70 = ( ! [Xphi: $i > $o] :
% 3.21/1.70 ( ( exu
% 3.21/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.21/1.70 => ! [Xx: $i,Xy: $i] :
% 3.21/1.70 ( ( Xphi @ Xx )
% 3.21/1.70 => ( ( Xphi @ Xy )
% 3.21/1.70 => ( Xx = Xy ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(descr__Cong_type,type,
% 3.21/1.70 descr__Cong: $o ).
% 3.21/1.70
% 3.21/1.70 thf(descr__Cong,definition,
% 3.21/1.70 ( descr__Cong
% 3.21/1.70 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 3.21/1.70 ( ! [Xx: $i,Xy: $i] :
% 3.21/1.70 ( ( Xx = Xy )
% 3.21/1.70 => ( ( Xphi @ Xx )
% 3.21/1.70 <=> ( Xpsi @ Xy ) ) )
% 3.21/1.70 => ( ( exu
% 3.21/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.21/1.70 => ( ( exu
% 3.21/1.70 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 3.21/1.70 => ( ( descr
% 3.21/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.21/1.70 = ( descr
% 3.21/1.70 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(dsetconstr__Cong_type,type,
% 3.21/1.70 dsetconstr__Cong: $o ).
% 3.21/1.70
% 3.21/1.70 thf(dsetconstr__Cong,definition,
% 3.21/1.70 ( dsetconstr__Cong
% 3.21/1.70 = ( ! [A: $i,B: $i] :
% 3.21/1.70 ( ( A = B )
% 3.21/1.70 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 3.21/1.70 ( ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ! [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ B )
% 3.21/1.70 => ( ( Xx = Xy )
% 3.21/1.70 => ( ( Xphi @ Xx )
% 3.21/1.70 <=> ( Xpsi @ Xy ) ) ) ) )
% 3.21/1.70 => ( ( dsetconstr @ A
% 3.21/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.21/1.70 = ( dsetconstr @ B
% 3.21/1.70 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(subset_type,type,
% 3.21/1.70 subset: $i > $i > $o ).
% 3.21/1.70
% 3.21/1.70 thf(disjoint_type,type,
% 3.21/1.70 disjoint: $i > $i > $o ).
% 3.21/1.70
% 3.21/1.70 thf(setsmeet_type,type,
% 3.21/1.70 setsmeet: $i > $i > $o ).
% 3.21/1.70
% 3.21/1.70 thf(subsetI1_type,type,
% 3.21/1.70 subsetI1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(subsetI1,definition,
% 3.21/1.70 ( subsetI1
% 3.21/1.70 = ( ! [A: $i,B: $i] :
% 3.21/1.70 ( ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( in @ Xx @ B ) )
% 3.21/1.70 => ( subset @ A @ B ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(eqimpsubset2_type,type,
% 3.21/1.70 eqimpsubset2: $o ).
% 3.21/1.70
% 3.21/1.70 thf(eqimpsubset2,definition,
% 3.21/1.70 ( eqimpsubset2
% 3.21/1.70 = ( ! [A: $i,B: $i] :
% 3.21/1.70 ( ( A = B )
% 3.21/1.70 => ( subset @ B @ A ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(eqimpsubset1_type,type,
% 3.21/1.70 eqimpsubset1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(eqimpsubset1,definition,
% 3.21/1.70 ( eqimpsubset1
% 3.21/1.70 = ( ! [A: $i,B: $i] :
% 3.21/1.70 ( ( A = B )
% 3.21/1.70 => ( subset @ A @ B ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(subsetI2_type,type,
% 3.21/1.70 subsetI2: $o ).
% 3.21/1.70
% 3.21/1.70 thf(subsetI2,definition,
% 3.21/1.70 ( subsetI2
% 3.21/1.70 = ( ! [A: $i,B: $i] :
% 3.21/1.70 ( ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( in @ Xx @ B ) )
% 3.21/1.70 => ( subset @ A @ B ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(emptysetsubset_type,type,
% 3.21/1.70 emptysetsubset: $o ).
% 3.21/1.70
% 3.21/1.70 thf(emptysetsubset,definition,
% 3.21/1.70 ( emptysetsubset
% 3.21/1.70 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(subsetE_type,type,
% 3.21/1.70 subsetE: $o ).
% 3.21/1.70
% 3.21/1.70 thf(subsetE,definition,
% 3.21/1.70 ( subsetE
% 3.21/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.21/1.70 ( ( subset @ A @ B )
% 3.21/1.70 => ( ( in @ Xx @ A )
% 3.21/1.70 => ( in @ Xx @ B ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(subsetE2_type,type,
% 3.21/1.70 subsetE2: $o ).
% 3.21/1.70
% 3.21/1.70 thf(subsetE2,definition,
% 3.21/1.70 ( subsetE2
% 3.21/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.21/1.70 ( ( subset @ A @ B )
% 3.21/1.70 => ( ~ ( in @ Xx @ B )
% 3.21/1.70 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(notsubsetI_type,type,
% 3.21/1.70 notsubsetI: $o ).
% 3.21/1.70
% 3.21/1.70 thf(notsubsetI,definition,
% 3.21/1.70 ( notsubsetI
% 3.21/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ~ ( in @ Xx @ B )
% 3.21/1.70 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(notequalI1_type,type,
% 3.21/1.70 notequalI1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(notequalI1,definition,
% 3.21/1.70 ( notequalI1
% 3.21/1.70 = ( ! [A: $i,B: $i] :
% 3.21/1.70 ( ~ ( subset @ A @ B )
% 3.21/1.70 => ( A != B ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(notequalI2_type,type,
% 3.21/1.70 notequalI2: $o ).
% 3.21/1.70
% 3.21/1.70 thf(notequalI2,definition,
% 3.21/1.70 ( notequalI2
% 3.21/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ~ ( in @ Xx @ B )
% 3.21/1.70 => ( A != B ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(subsetRefl_type,type,
% 3.21/1.70 subsetRefl: $o ).
% 3.21/1.70
% 3.21/1.70 thf(subsetRefl,definition,
% 3.21/1.70 ( subsetRefl
% 3.21/1.70 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(subsetTrans_type,type,
% 3.21/1.70 subsetTrans: $o ).
% 3.21/1.70
% 3.21/1.70 thf(subsetTrans,definition,
% 3.21/1.70 ( subsetTrans
% 3.21/1.70 = ( ! [A: $i,B: $i,C: $i] :
% 3.21/1.70 ( ( subset @ A @ B )
% 3.21/1.70 => ( ( subset @ B @ C )
% 3.21/1.70 => ( subset @ A @ C ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(setadjoinSub_type,type,
% 3.21/1.70 setadjoinSub: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setadjoinSub,definition,
% 3.21/1.70 ( setadjoinSub
% 3.21/1.70 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(setadjoinSub2_type,type,
% 3.21/1.70 setadjoinSub2: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setadjoinSub2,definition,
% 3.21/1.70 ( setadjoinSub2
% 3.21/1.70 = ( ! [A: $i,Xx: $i,B: $i] :
% 3.21/1.70 ( ( subset @ A @ B )
% 3.21/1.70 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(subset2powerset_type,type,
% 3.21/1.70 subset2powerset: $o ).
% 3.21/1.70
% 3.21/1.70 thf(subset2powerset,definition,
% 3.21/1.70 ( subset2powerset
% 3.21/1.70 = ( ! [A: $i,B: $i] :
% 3.21/1.70 ( ( subset @ A @ B )
% 3.21/1.70 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(setextsub_type,type,
% 3.21/1.70 setextsub: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setextsub,definition,
% 3.21/1.70 ( setextsub
% 3.21/1.70 = ( ! [A: $i,B: $i] :
% 3.21/1.70 ( ( subset @ A @ B )
% 3.21/1.70 => ( ( subset @ B @ A )
% 3.21/1.70 => ( A = B ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(subsetemptysetimpeq_type,type,
% 3.21/1.70 subsetemptysetimpeq: $o ).
% 3.21/1.70
% 3.21/1.70 thf(subsetemptysetimpeq,definition,
% 3.21/1.70 ( subsetemptysetimpeq
% 3.21/1.70 = ( ! [A: $i] :
% 3.21/1.70 ( ( subset @ A @ emptyset )
% 3.21/1.70 => ( A = emptyset ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(powersetI1_type,type,
% 3.21/1.70 powersetI1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(powersetI1,definition,
% 3.21/1.70 ( powersetI1
% 3.21/1.70 = ( ! [A: $i,B: $i] :
% 3.21/1.70 ( ( subset @ B @ A )
% 3.21/1.70 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(powersetE1_type,type,
% 3.21/1.70 powersetE1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(powersetE1,definition,
% 3.21/1.70 ( powersetE1
% 3.21/1.70 = ( ! [A: $i,B: $i] :
% 3.21/1.70 ( ( in @ B @ ( powerset @ A ) )
% 3.21/1.70 => ( subset @ B @ A ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(inPowerset_type,type,
% 3.21/1.70 inPowerset: $o ).
% 3.21/1.70
% 3.21/1.70 thf(inPowerset,definition,
% 3.21/1.70 ( inPowerset
% 3.21/1.70 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(powersetsubset_type,type,
% 3.21/1.70 powersetsubset: $o ).
% 3.21/1.70
% 3.21/1.70 thf(powersetsubset,definition,
% 3.21/1.70 ( powersetsubset
% 3.21/1.70 = ( ! [A: $i,B: $i] :
% 3.21/1.70 ( ( subset @ A @ B )
% 3.21/1.70 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(sepInPowerset_type,type,
% 3.21/1.70 sepInPowerset: $o ).
% 3.21/1.70
% 3.21/1.70 thf(sepInPowerset,definition,
% 3.21/1.70 ( sepInPowerset
% 3.21/1.70 = ( ! [A: $i,Xphi: $i > $o] :
% 3.21/1.70 ( in
% 3.21/1.70 @ ( dsetconstr @ A
% 3.21/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.21/1.70 @ ( powerset @ A ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(sepSubset_type,type,
% 3.21/1.70 sepSubset: $o ).
% 3.21/1.70
% 3.21/1.70 thf(sepSubset,definition,
% 3.21/1.70 ( sepSubset
% 3.21/1.70 = ( ! [A: $i,Xphi: $i > $o] :
% 3.21/1.70 ( subset
% 3.21/1.70 @ ( dsetconstr @ A
% 3.21/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.21/1.70 @ A ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(binunion_type,type,
% 3.21/1.70 binunion: $i > $i > $i ).
% 3.21/1.70
% 3.21/1.70 thf(binunionIL_type,type,
% 3.21/1.70 binunionIL: $o ).
% 3.21/1.70
% 3.21/1.70 thf(binunionIL,definition,
% 3.21/1.70 ( binunionIL
% 3.21/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(upairset2IR_type,type,
% 3.21/1.70 upairset2IR: $o ).
% 3.21/1.70
% 3.21/1.70 thf(upairset2IR,definition,
% 3.21/1.70 ( upairset2IR
% 3.21/1.70 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(binunionIR_type,type,
% 3.21/1.70 binunionIR: $o ).
% 3.21/1.70
% 3.21/1.70 thf(binunionIR,definition,
% 3.21/1.70 ( binunionIR
% 3.21/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ B )
% 3.21/1.70 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(binunionEcases_type,type,
% 3.21/1.70 binunionEcases: $o ).
% 3.21/1.70
% 3.21/1.70 thf(binunionEcases,definition,
% 3.21/1.70 ( binunionEcases
% 3.21/1.70 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 3.21/1.70 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 3.21/1.70 => ( ( ( in @ Xx @ A )
% 3.21/1.70 => Xphi )
% 3.21/1.70 => ( ( ( in @ Xx @ B )
% 3.21/1.70 => Xphi )
% 3.21/1.70 => Xphi ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(binunionE_type,type,
% 3.21/1.70 binunionE: $o ).
% 3.21/1.70
% 3.21/1.70 thf(binunionE,definition,
% 3.21/1.70 ( binunionE
% 3.21/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 3.21/1.70 => ( ( in @ Xx @ A )
% 3.21/1.70 | ( in @ Xx @ B ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(binunionLsub_type,type,
% 3.21/1.70 binunionLsub: $o ).
% 3.21/1.70
% 3.21/1.70 thf(binunionLsub,definition,
% 3.21/1.70 ( binunionLsub
% 3.21/1.70 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(binunionRsub_type,type,
% 3.21/1.70 binunionRsub: $o ).
% 3.21/1.70
% 3.21/1.70 thf(binunionRsub,definition,
% 3.21/1.70 ( binunionRsub
% 3.21/1.70 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(binintersect_type,type,
% 3.21/1.70 binintersect: $i > $i > $i ).
% 3.21/1.70
% 3.21/1.70 thf(binintersectI_type,type,
% 3.21/1.70 binintersectI: $o ).
% 3.21/1.70
% 3.21/1.70 thf(binintersectI,definition,
% 3.21/1.70 ( binintersectI
% 3.21/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ( in @ Xx @ B )
% 3.21/1.70 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(binintersectSubset5_type,type,
% 3.21/1.70 binintersectSubset5: $o ).
% 3.21/1.70
% 3.21/1.70 thf(binintersectSubset5,definition,
% 3.21/1.70 ( binintersectSubset5
% 3.21/1.70 = ( ! [A: $i,B: $i,C: $i] :
% 3.21/1.70 ( ( subset @ C @ A )
% 3.21/1.70 => ( ( subset @ C @ B )
% 3.21/1.70 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(binintersectEL_type,type,
% 3.21/1.70 binintersectEL: $o ).
% 3.21/1.70
% 3.21/1.70 thf(binintersectEL,definition,
% 3.21/1.70 ( binintersectEL
% 3.21/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 3.21/1.70 => ( in @ Xx @ A ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(binintersectLsub_type,type,
% 3.21/1.70 binintersectLsub: $o ).
% 3.21/1.70
% 3.21/1.70 thf(binintersectLsub,definition,
% 3.21/1.70 ( binintersectLsub
% 3.21/1.70 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(binintersectSubset2_type,type,
% 3.21/1.70 binintersectSubset2: $o ).
% 3.21/1.70
% 3.21/1.70 thf(binintersectSubset2,definition,
% 3.21/1.70 ( binintersectSubset2
% 3.21/1.70 = ( ! [A: $i,B: $i] :
% 3.21/1.70 ( ( subset @ A @ B )
% 3.21/1.70 => ( ( binintersect @ A @ B )
% 3.21/1.70 = A ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(binintersectSubset3_type,type,
% 3.21/1.70 binintersectSubset3: $o ).
% 3.21/1.70
% 3.21/1.70 thf(binintersectSubset3,definition,
% 3.21/1.70 ( binintersectSubset3
% 3.21/1.70 = ( ! [A: $i,B: $i] :
% 3.21/1.70 ( ( ( binintersect @ A @ B )
% 3.21/1.70 = B )
% 3.21/1.70 => ( subset @ B @ A ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(binintersectER_type,type,
% 3.21/1.70 binintersectER: $o ).
% 3.21/1.70
% 3.21/1.70 thf(binintersectER,definition,
% 3.21/1.70 ( binintersectER
% 3.21/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 3.21/1.70 => ( in @ Xx @ B ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(disjointsetsI1_type,type,
% 3.21/1.70 disjointsetsI1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(disjointsetsI1,definition,
% 3.21/1.70 ( disjointsetsI1
% 3.21/1.70 = ( ! [A: $i,B: $i] :
% 3.21/1.70 ( ~ ? [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 & ( in @ Xx @ B ) )
% 3.21/1.70 => ( ( binintersect @ A @ B )
% 3.21/1.70 = emptyset ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(binintersectRsub_type,type,
% 3.21/1.70 binintersectRsub: $o ).
% 3.21/1.70
% 3.21/1.70 thf(binintersectRsub,definition,
% 3.21/1.70 ( binintersectRsub
% 3.21/1.70 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(binintersectSubset4_type,type,
% 3.21/1.70 binintersectSubset4: $o ).
% 3.21/1.70
% 3.21/1.70 thf(binintersectSubset4,definition,
% 3.21/1.70 ( binintersectSubset4
% 3.21/1.70 = ( ! [A: $i,B: $i] :
% 3.21/1.70 ( ( subset @ B @ A )
% 3.21/1.70 => ( ( binintersect @ A @ B )
% 3.21/1.70 = B ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(binintersectSubset1_type,type,
% 3.21/1.70 binintersectSubset1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(binintersectSubset1,definition,
% 3.21/1.70 ( binintersectSubset1
% 3.21/1.70 = ( ! [A: $i,B: $i] :
% 3.21/1.70 ( ( ( binintersect @ A @ B )
% 3.21/1.70 = A )
% 3.21/1.70 => ( subset @ A @ B ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(bs114d_type,type,
% 3.21/1.70 bs114d: $o ).
% 3.21/1.70
% 3.21/1.70 thf(bs114d,definition,
% 3.21/1.70 ( bs114d
% 3.21/1.70 = ( ! [A: $i,B: $i,C: $i] :
% 3.21/1.70 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 3.21/1.70 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(regular_type,type,
% 3.21/1.70 regular: $i > $o ).
% 3.21/1.70
% 3.21/1.70 thf(setminus_type,type,
% 3.21/1.70 setminus: $i > $i > $i ).
% 3.21/1.70
% 3.21/1.70 thf(setminusI_type,type,
% 3.21/1.70 setminusI: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setminusI,definition,
% 3.21/1.70 ( setminusI
% 3.21/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ~ ( in @ Xx @ B )
% 3.21/1.70 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(setminusEL_type,type,
% 3.21/1.70 setminusEL: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setminusEL,definition,
% 3.21/1.70 ( setminusEL
% 3.21/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 3.21/1.70 => ( in @ Xx @ A ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(setminusER_type,type,
% 3.21/1.70 setminusER: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setminusER,definition,
% 3.21/1.70 ( setminusER
% 3.21/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 3.21/1.70 => ~ ( in @ Xx @ B ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(setminusSubset2_type,type,
% 3.21/1.70 setminusSubset2: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setminusSubset2,definition,
% 3.21/1.70 ( setminusSubset2
% 3.21/1.70 = ( ! [A: $i,B: $i] :
% 3.21/1.70 ( ( subset @ A @ B )
% 3.21/1.70 => ( ( setminus @ A @ B )
% 3.21/1.70 = emptyset ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(setminusERneg_type,type,
% 3.21/1.70 setminusERneg: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setminusERneg,definition,
% 3.21/1.70 ( setminusERneg
% 3.21/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.21/1.70 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 3.21/1.70 => ( ( in @ Xx @ A )
% 3.21/1.70 => ( in @ Xx @ B ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(setminusELneg_type,type,
% 3.21/1.70 setminusELneg: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setminusELneg,definition,
% 3.21/1.70 ( setminusELneg
% 3.21/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.21/1.70 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 3.21/1.70 => ( ~ ( in @ Xx @ B )
% 3.21/1.70 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(setminusILneg_type,type,
% 3.21/1.70 setminusILneg: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setminusILneg,definition,
% 3.21/1.70 ( setminusILneg
% 3.21/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.21/1.70 ( ~ ( in @ Xx @ A )
% 3.21/1.70 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(setminusIRneg_type,type,
% 3.21/1.70 setminusIRneg: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setminusIRneg,definition,
% 3.21/1.70 ( setminusIRneg
% 3.21/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ B )
% 3.21/1.70 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(setminusLsub_type,type,
% 3.21/1.70 setminusLsub: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setminusLsub,definition,
% 3.21/1.70 ( setminusLsub
% 3.21/1.70 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(setminusSubset1_type,type,
% 3.21/1.70 setminusSubset1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setminusSubset1,definition,
% 3.21/1.70 ( setminusSubset1
% 3.21/1.70 = ( ! [A: $i,B: $i] :
% 3.21/1.70 ( ( ( setminus @ A @ B )
% 3.21/1.70 = emptyset )
% 3.21/1.70 => ( subset @ A @ B ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(symdiff_type,type,
% 3.21/1.70 symdiff: $i > $i > $i ).
% 3.21/1.70
% 3.21/1.70 thf(symdiffE_type,type,
% 3.21/1.70 symdiffE: $o ).
% 3.21/1.70
% 3.21/1.70 thf(symdiffE,definition,
% 3.21/1.70 ( symdiffE
% 3.21/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 3.21/1.70 => ! [Xphi: $o] :
% 3.21/1.70 ( ( ( in @ Xx @ A )
% 3.21/1.70 => ( ~ ( in @ Xx @ B )
% 3.21/1.70 => Xphi ) )
% 3.21/1.70 => ( ( ~ ( in @ Xx @ A )
% 3.21/1.70 => ( ( in @ Xx @ B )
% 3.21/1.70 => Xphi ) )
% 3.21/1.70 => Xphi ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(symdiffI1_type,type,
% 3.21/1.70 symdiffI1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(symdiffI1,definition,
% 3.21/1.70 ( symdiffI1
% 3.21/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ~ ( in @ Xx @ B )
% 3.21/1.70 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(symdiffI2_type,type,
% 3.21/1.70 symdiffI2: $o ).
% 3.21/1.70
% 3.21/1.70 thf(symdiffI2,definition,
% 3.21/1.70 ( symdiffI2
% 3.21/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.21/1.70 ( ~ ( in @ Xx @ A )
% 3.21/1.70 => ( ( in @ Xx @ B )
% 3.21/1.70 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(symdiffIneg1_type,type,
% 3.21/1.70 symdiffIneg1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(symdiffIneg1,definition,
% 3.21/1.70 ( symdiffIneg1
% 3.21/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ( in @ Xx @ B )
% 3.21/1.70 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(symdiffIneg2_type,type,
% 3.21/1.70 symdiffIneg2: $o ).
% 3.21/1.70
% 3.21/1.70 thf(symdiffIneg2,definition,
% 3.21/1.70 ( symdiffIneg2
% 3.21/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.21/1.70 ( ~ ( in @ Xx @ A )
% 3.21/1.70 => ( ~ ( in @ Xx @ B )
% 3.21/1.70 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(iskpair_type,type,
% 3.21/1.70 iskpair: $i > $o ).
% 3.21/1.70
% 3.21/1.70 thf(secondinupair_type,type,
% 3.21/1.70 secondinupair: $o ).
% 3.21/1.70
% 3.21/1.70 thf(secondinupair,definition,
% 3.21/1.70 ( secondinupair
% 3.21/1.70 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(setukpairIL_type,type,
% 3.21/1.70 setukpairIL: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setukpairIL,definition,
% 3.21/1.70 ( setukpairIL
% 3.21/1.70 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(setukpairIR_type,type,
% 3.21/1.70 setukpairIR: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setukpairIR,definition,
% 3.21/1.70 ( setukpairIR
% 3.21/1.70 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(kpairiskpair_type,type,
% 3.21/1.70 kpairiskpair: $o ).
% 3.21/1.70
% 3.21/1.70 thf(kpairiskpair,definition,
% 3.21/1.70 ( kpairiskpair
% 3.21/1.70 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(kpair_type,type,
% 3.21/1.70 kpair: $i > $i > $i ).
% 3.21/1.70
% 3.21/1.70 thf(kpairp_type,type,
% 3.21/1.70 kpairp: $o ).
% 3.21/1.70
% 3.21/1.70 thf(kpairp,definition,
% 3.21/1.70 ( kpairp
% 3.21/1.70 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(cartprod_type,type,
% 3.21/1.70 cartprod: $i > $i > $i ).
% 3.21/1.70
% 3.21/1.70 thf(singletonsubset_type,type,
% 3.21/1.70 singletonsubset: $o ).
% 3.21/1.70
% 3.21/1.70 thf(singletonsubset,definition,
% 3.21/1.70 ( singletonsubset
% 3.21/1.70 = ( ! [A: $i,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(singletoninpowerset_type,type,
% 3.21/1.70 singletoninpowerset: $o ).
% 3.21/1.70
% 3.21/1.70 thf(singletoninpowerset,definition,
% 3.21/1.70 ( singletoninpowerset
% 3.21/1.70 = ( ! [A: $i,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(singletoninpowunion_type,type,
% 3.21/1.70 singletoninpowunion: $o ).
% 3.21/1.70
% 3.21/1.70 thf(singletoninpowunion,definition,
% 3.21/1.70 ( singletoninpowunion
% 3.21/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(upairset2E_type,type,
% 3.21/1.70 upairset2E: $o ).
% 3.21/1.70
% 3.21/1.70 thf(upairset2E,definition,
% 3.21/1.70 ( upairset2E
% 3.21/1.70 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 3.21/1.70 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 3.21/1.70 => ( ( Xz = Xx )
% 3.21/1.70 | ( Xz = Xy ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(upairsubunion_type,type,
% 3.21/1.70 upairsubunion: $o ).
% 3.21/1.70
% 3.21/1.70 thf(upairsubunion,definition,
% 3.21/1.70 ( upairsubunion
% 3.21/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ! [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ B )
% 3.21/1.70 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(upairinpowunion_type,type,
% 3.21/1.70 upairinpowunion: $o ).
% 3.21/1.70
% 3.21/1.70 thf(upairinpowunion,definition,
% 3.21/1.70 ( upairinpowunion
% 3.21/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ! [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ B )
% 3.21/1.70 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(ubforcartprodlem1_type,type,
% 3.21/1.70 ubforcartprodlem1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(ubforcartprodlem1,definition,
% 3.21/1.70 ( ubforcartprodlem1
% 3.21/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ! [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ B )
% 3.21/1.70 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(ubforcartprodlem2_type,type,
% 3.21/1.70 ubforcartprodlem2: $o ).
% 3.21/1.70
% 3.21/1.70 thf(ubforcartprodlem2,definition,
% 3.21/1.70 ( ubforcartprodlem2
% 3.21/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ! [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ B )
% 3.21/1.70 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(ubforcartprodlem3_type,type,
% 3.21/1.70 ubforcartprodlem3: $o ).
% 3.21/1.70
% 3.21/1.70 thf(ubforcartprodlem3,definition,
% 3.21/1.70 ( ubforcartprodlem3
% 3.21/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ! [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ B )
% 3.21/1.70 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(cartprodpairin_type,type,
% 3.21/1.70 cartprodpairin: $o ).
% 3.21/1.70
% 3.21/1.70 thf(cartprodpairin,definition,
% 3.21/1.70 ( cartprodpairin
% 3.21/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ! [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ B )
% 3.21/1.70 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(cartprodmempair1_type,type,
% 3.21/1.70 cartprodmempair1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(cartprodmempair1,definition,
% 3.21/1.70 ( cartprodmempair1
% 3.21/1.70 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.21/1.70 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.21/1.70 => ? [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 & ? [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ B )
% 3.21/1.70 & ( Xu
% 3.21/1.70 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(cartprodmempair_type,type,
% 3.21/1.70 cartprodmempair: $o ).
% 3.21/1.70
% 3.21/1.70 thf(cartprodmempair,definition,
% 3.21/1.70 ( cartprodmempair
% 3.21/1.70 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.21/1.70 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.21/1.70 => ( iskpair @ Xu ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(setunionE2_type,type,
% 3.21/1.70 setunionE2: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setunionE2,definition,
% 3.21/1.70 ( setunionE2
% 3.21/1.70 = ( ! [A: $i,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ ( setunion @ A ) )
% 3.21/1.70 => ? [X: $i] :
% 3.21/1.70 ( ( in @ X @ A )
% 3.21/1.70 & ( in @ Xx @ X ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(setunionsingleton1_type,type,
% 3.21/1.70 setunionsingleton1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setunionsingleton1,definition,
% 3.21/1.70 ( setunionsingleton1
% 3.21/1.70 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(setunionsingleton2_type,type,
% 3.21/1.70 setunionsingleton2: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setunionsingleton2,definition,
% 3.21/1.70 ( setunionsingleton2
% 3.21/1.70 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(setunionsingleton_type,type,
% 3.21/1.70 setunionsingleton: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setunionsingleton,definition,
% 3.21/1.70 ( setunionsingleton
% 3.21/1.70 = ( ! [Xx: $i] :
% 3.21/1.70 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 3.21/1.70 = Xx ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(singleton_type,type,
% 3.21/1.70 singleton: $i > $o ).
% 3.21/1.70
% 3.21/1.70 thf(singletonprop_type,type,
% 3.21/1.70 singletonprop: $o ).
% 3.21/1.70
% 3.21/1.70 thf(singletonprop,definition,
% 3.21/1.70 ( singletonprop
% 3.21/1.70 = ( ! [A: $i,Xphi: $i > $o] :
% 3.21/1.70 ( ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ! [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ A )
% 3.21/1.70 => ( ( Xphi @ Xx )
% 3.21/1.70 => ( ( Xphi @ Xy )
% 3.21/1.70 => ( Xx = Xy ) ) ) ) )
% 3.21/1.70 => ( ? [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 & ( Xphi @ Xx ) )
% 3.21/1.70 => ( singleton
% 3.21/1.70 @ ( dsetconstr @ A
% 3.21/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(ex1_type,type,
% 3.21/1.70 ex1: $i > ( $i > $o ) > $o ).
% 3.21/1.70
% 3.21/1.70 thf(ex1E1_type,type,
% 3.21/1.70 ex1E1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(ex1E1,definition,
% 3.21/1.70 ( ex1E1
% 3.21/1.70 = ( ! [A: $i,Xphi: $i > $o] :
% 3.21/1.70 ( ( ex1 @ A
% 3.21/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.21/1.70 => ? [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 & ( Xphi @ Xx ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(ex1I_type,type,
% 3.21/1.70 ex1I: $o ).
% 3.21/1.70
% 3.21/1.70 thf(ex1I,definition,
% 3.21/1.70 ( ex1I
% 3.21/1.70 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ( Xphi @ Xx )
% 3.21/1.70 => ( ! [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ A )
% 3.21/1.70 => ( ( Xphi @ Xy )
% 3.21/1.70 => ( Xy = Xx ) ) )
% 3.21/1.70 => ( ex1 @ A
% 3.21/1.70 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(ex1I2_type,type,
% 3.21/1.70 ex1I2: $o ).
% 3.21/1.70
% 3.21/1.70 thf(ex1I2,definition,
% 3.21/1.70 ( ex1I2
% 3.21/1.70 = ( ! [A: $i,Xphi: $i > $o] :
% 3.21/1.70 ( ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ! [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ A )
% 3.21/1.70 => ( ( Xphi @ Xx )
% 3.21/1.70 => ( ( Xphi @ Xy )
% 3.21/1.70 => ( Xx = Xy ) ) ) ) )
% 3.21/1.70 => ( ? [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 & ( Xphi @ Xx ) )
% 3.21/1.70 => ( ex1 @ A
% 3.21/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(singletonsuniq_type,type,
% 3.21/1.70 singletonsuniq: $o ).
% 3.21/1.70
% 3.21/1.70 thf(singletonsuniq,definition,
% 3.21/1.70 ( singletonsuniq
% 3.21/1.70 = ( ! [Xx: $i,Xy: $i] :
% 3.21/1.70 ( ( ( setadjoin @ Xx @ emptyset )
% 3.21/1.70 = ( setadjoin @ Xy @ emptyset ) )
% 3.21/1.70 => ( Xx = Xy ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(atmost1p_type,type,
% 3.21/1.70 atmost1p: $i > $o ).
% 3.21/1.70
% 3.21/1.70 thf(atleast2p_type,type,
% 3.21/1.70 atleast2p: $i > $o ).
% 3.21/1.70
% 3.21/1.70 thf(atmost2p_type,type,
% 3.21/1.70 atmost2p: $i > $o ).
% 3.21/1.70
% 3.21/1.70 thf(upairsetp_type,type,
% 3.21/1.70 upairsetp: $i > $o ).
% 3.21/1.70
% 3.21/1.70 thf(setukpairinjL1_type,type,
% 3.21/1.70 setukpairinjL1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setukpairinjL1,definition,
% 3.21/1.70 ( setukpairinjL1
% 3.21/1.70 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 3.21/1.70 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 3.21/1.70 => ( Xx = Xz ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(kfstsingleton_type,type,
% 3.21/1.70 kfstsingleton: $o ).
% 3.21/1.70
% 3.21/1.70 thf(kfstsingleton,definition,
% 3.21/1.70 ( kfstsingleton
% 3.21/1.70 = ( ! [Xu: $i] :
% 3.21/1.70 ( ( iskpair @ Xu )
% 3.21/1.70 => ( singleton
% 3.21/1.70 @ ( dsetconstr @ ( setunion @ Xu )
% 3.21/1.70 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(theprop_type,type,
% 3.21/1.70 theprop: $o ).
% 3.21/1.70
% 3.21/1.70 thf(theprop,definition,
% 3.21/1.70 ( theprop
% 3.21/1.70 = ( ! [X: $i] :
% 3.21/1.70 ( ( singleton @ X )
% 3.21/1.70 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(kfst_type,type,
% 3.21/1.70 kfst: $i > $i ).
% 3.21/1.70
% 3.21/1.70 thf(kfstpairEq_type,type,
% 3.21/1.70 kfstpairEq: $o ).
% 3.21/1.70
% 3.21/1.70 thf(kfstpairEq,definition,
% 3.21/1.70 ( kfstpairEq
% 3.21/1.70 = ( ! [Xx: $i,Xy: $i] :
% 3.21/1.70 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 3.21/1.70 = Xx ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(cartprodfstin_type,type,
% 3.21/1.70 cartprodfstin: $o ).
% 3.21/1.70
% 3.21/1.70 thf(cartprodfstin,definition,
% 3.21/1.70 ( cartprodfstin
% 3.21/1.70 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.21/1.70 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.21/1.70 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(setukpairinjL2_type,type,
% 3.21/1.70 setukpairinjL2: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setukpairinjL2,definition,
% 3.21/1.70 ( setukpairinjL2
% 3.21/1.70 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.21/1.70 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 3.21/1.70 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 3.21/1.70 => ( Xx = Xz ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(setukpairinjL_type,type,
% 3.21/1.70 setukpairinjL: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setukpairinjL,definition,
% 3.21/1.70 ( setukpairinjL
% 3.21/1.70 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.21/1.70 ( ( ( kpair @ Xx @ Xy )
% 3.21/1.70 = ( kpair @ Xz @ Xu ) )
% 3.21/1.70 => ( Xx = Xz ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(setukpairinjR11_type,type,
% 3.21/1.70 setukpairinjR11: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setukpairinjR11,definition,
% 3.21/1.70 ( setukpairinjR11
% 3.21/1.70 = ( ! [Xx: $i,Xy: $i] :
% 3.21/1.70 ( ( Xx = Xy )
% 3.21/1.70 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 3.21/1.70 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(setukpairinjR12_type,type,
% 3.21/1.70 setukpairinjR12: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setukpairinjR12,definition,
% 3.21/1.70 ( setukpairinjR12
% 3.21/1.70 = ( ! [Xx: $i,Xy: $i] :
% 3.21/1.70 ( ( Xx = Xy )
% 3.21/1.70 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 3.21/1.70 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(setukpairinjR1_type,type,
% 3.21/1.70 setukpairinjR1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setukpairinjR1,definition,
% 3.21/1.70 ( setukpairinjR1
% 3.21/1.70 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.21/1.70 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 3.21/1.70 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 3.21/1.70 => ( ( Xz = Xu )
% 3.21/1.70 => ( Xy = Xu ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(upairequniteq_type,type,
% 3.21/1.70 upairequniteq: $o ).
% 3.21/1.70
% 3.21/1.70 thf(upairequniteq,definition,
% 3.21/1.70 ( upairequniteq
% 3.21/1.70 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 3.21/1.70 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 3.21/1.70 = ( setadjoin @ Xz @ emptyset ) )
% 3.21/1.70 => ( Xx = Xy ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(setukpairinjR2_type,type,
% 3.21/1.70 setukpairinjR2: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setukpairinjR2,definition,
% 3.21/1.70 ( setukpairinjR2
% 3.21/1.70 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.21/1.70 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 3.21/1.70 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 3.21/1.70 => ( Xy = Xu ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(setukpairinjR_type,type,
% 3.21/1.70 setukpairinjR: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setukpairinjR,definition,
% 3.21/1.70 ( setukpairinjR
% 3.21/1.70 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.21/1.70 ( ( ( kpair @ Xx @ Xy )
% 3.21/1.70 = ( kpair @ Xz @ Xu ) )
% 3.21/1.70 => ( Xy = Xu ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(ksndsingleton_type,type,
% 3.21/1.70 ksndsingleton: $o ).
% 3.21/1.70
% 3.21/1.70 thf(ksndsingleton,definition,
% 3.21/1.70 ( ksndsingleton
% 3.21/1.70 = ( ! [Xu: $i] :
% 3.21/1.70 ( ( iskpair @ Xu )
% 3.21/1.70 => ( singleton
% 3.21/1.70 @ ( dsetconstr @ ( setunion @ Xu )
% 3.21/1.70 @ ^ [Xx: $i] :
% 3.21/1.70 ( Xu
% 3.21/1.70 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(ksnd_type,type,
% 3.21/1.70 ksnd: $i > $i ).
% 3.21/1.70
% 3.21/1.70 thf(ksndpairEq_type,type,
% 3.21/1.70 ksndpairEq: $o ).
% 3.21/1.70
% 3.21/1.70 thf(ksndpairEq,definition,
% 3.21/1.70 ( ksndpairEq
% 3.21/1.70 = ( ! [Xx: $i,Xy: $i] :
% 3.21/1.70 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 3.21/1.70 = Xy ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(kpairsurjEq_type,type,
% 3.21/1.70 kpairsurjEq: $o ).
% 3.21/1.70
% 3.21/1.70 thf(kpairsurjEq,definition,
% 3.21/1.70 ( kpairsurjEq
% 3.21/1.70 = ( ! [Xu: $i] :
% 3.21/1.70 ( ( iskpair @ Xu )
% 3.21/1.70 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 3.21/1.70 = Xu ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(cartprodsndin_type,type,
% 3.21/1.70 cartprodsndin: $o ).
% 3.21/1.70
% 3.21/1.70 thf(cartprodsndin,definition,
% 3.21/1.70 ( cartprodsndin
% 3.21/1.70 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.21/1.70 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.21/1.70 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(cartprodpairmemEL_type,type,
% 3.21/1.70 cartprodpairmemEL: $o ).
% 3.21/1.70
% 3.21/1.70 thf(cartprodpairmemEL,definition,
% 3.21/1.70 ( cartprodpairmemEL
% 3.21/1.70 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 3.21/1.70 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 3.21/1.70 => ( in @ Xx @ A ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(cartprodpairmemER_type,type,
% 3.21/1.70 cartprodpairmemER: $o ).
% 3.21/1.70
% 3.21/1.70 thf(cartprodpairmemER,definition,
% 3.21/1.70 ( cartprodpairmemER
% 3.21/1.70 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 3.21/1.70 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 3.21/1.70 => ( in @ Xy @ B ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(cartprodmempaircEq_type,type,
% 3.21/1.70 cartprodmempaircEq: $o ).
% 3.21/1.70
% 3.21/1.70 thf(cartprodmempaircEq,definition,
% 3.21/1.70 ( cartprodmempaircEq
% 3.21/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ! [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ B )
% 3.21/1.70 => ( ( kpair @ Xx @ Xy )
% 3.21/1.70 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(cartprodfstpairEq_type,type,
% 3.21/1.70 cartprodfstpairEq: $o ).
% 3.21/1.70
% 3.21/1.70 thf(cartprodfstpairEq,definition,
% 3.21/1.70 ( cartprodfstpairEq
% 3.21/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ! [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ B )
% 3.21/1.70 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 3.21/1.70 = Xx ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(cartprodsndpairEq_type,type,
% 3.21/1.70 cartprodsndpairEq: $o ).
% 3.21/1.70
% 3.21/1.70 thf(cartprodsndpairEq,definition,
% 3.21/1.70 ( cartprodsndpairEq
% 3.21/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ! [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ B )
% 3.21/1.70 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 3.21/1.70 = Xy ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(cartprodpairsurjEq_type,type,
% 3.21/1.70 cartprodpairsurjEq: $o ).
% 3.21/1.70
% 3.21/1.70 thf(cartprodpairsurjEq,definition,
% 3.21/1.70 ( cartprodpairsurjEq
% 3.21/1.70 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.21/1.70 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.21/1.70 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 3.21/1.70 = Xu ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(breln_type,type,
% 3.21/1.70 breln: $i > $i > $i > $o ).
% 3.21/1.70
% 3.21/1.70 thf(dpsetconstr_type,type,
% 3.21/1.70 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 3.21/1.70
% 3.21/1.70 thf(dpsetconstrI_type,type,
% 3.21/1.70 dpsetconstrI: $o ).
% 3.21/1.70
% 3.21/1.70 thf(dpsetconstrI,definition,
% 3.21/1.70 ( dpsetconstrI
% 3.21/1.70 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ! [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ B )
% 3.21/1.70 => ( ( Xphi @ Xx @ Xy )
% 3.21/1.70 => ( in @ ( kpair @ Xx @ Xy )
% 3.21/1.70 @ ( dpsetconstr @ A @ B
% 3.21/1.70 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(dpsetconstrSub_type,type,
% 3.21/1.70 dpsetconstrSub: $o ).
% 3.21/1.70
% 3.21/1.70 thf(dpsetconstrSub,definition,
% 3.21/1.70 ( dpsetconstrSub
% 3.21/1.70 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 3.21/1.70 ( subset
% 3.21/1.70 @ ( dpsetconstr @ A @ B
% 3.21/1.70 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 3.21/1.70 @ ( cartprod @ A @ B ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(setOfPairsIsBReln_type,type,
% 3.21/1.70 setOfPairsIsBReln: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setOfPairsIsBReln,definition,
% 3.21/1.70 ( setOfPairsIsBReln
% 3.21/1.70 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 3.21/1.70 ( breln @ A @ B
% 3.21/1.70 @ ( dpsetconstr @ A @ B
% 3.21/1.70 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(dpsetconstrERa_type,type,
% 3.21/1.70 dpsetconstrERa: $o ).
% 3.21/1.70
% 3.21/1.70 thf(dpsetconstrERa,definition,
% 3.21/1.70 ( dpsetconstrERa
% 3.21/1.70 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ! [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ B )
% 3.21/1.70 => ( ( in @ ( kpair @ Xx @ Xy )
% 3.21/1.70 @ ( dpsetconstr @ A @ B
% 3.21/1.70 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 3.21/1.70 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(dpsetconstrEL1_type,type,
% 3.21/1.70 dpsetconstrEL1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(dpsetconstrEL1,definition,
% 3.21/1.70 ( dpsetconstrEL1
% 3.21/1.70 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 3.21/1.70 ( ( in @ ( kpair @ Xx @ Xy )
% 3.21/1.70 @ ( dpsetconstr @ A @ B
% 3.21/1.70 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 3.21/1.70 => ( in @ Xx @ A ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(dpsetconstrEL2_type,type,
% 3.21/1.70 dpsetconstrEL2: $o ).
% 3.21/1.70
% 3.21/1.70 thf(dpsetconstrEL2,definition,
% 3.21/1.70 ( dpsetconstrEL2
% 3.21/1.70 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 3.21/1.70 ( ( in @ ( kpair @ Xx @ Xy )
% 3.21/1.70 @ ( dpsetconstr @ A @ B
% 3.21/1.70 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 3.21/1.70 => ( in @ Xy @ B ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(dpsetconstrER_type,type,
% 3.21/1.70 dpsetconstrER: $o ).
% 3.21/1.70
% 3.21/1.70 thf(dpsetconstrER,definition,
% 3.21/1.70 ( dpsetconstrER
% 3.21/1.70 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 3.21/1.70 ( ( in @ ( kpair @ Xx @ Xy )
% 3.21/1.70 @ ( dpsetconstr @ A @ B
% 3.21/1.70 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 3.21/1.70 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(func_type,type,
% 3.21/1.70 func: $i > $i > $i > $o ).
% 3.21/1.70
% 3.21/1.70 thf(funcSet_type,type,
% 3.21/1.70 funcSet: $i > $i > $i ).
% 3.21/1.70
% 3.21/1.70 thf(funcImageSingleton_type,type,
% 3.21/1.70 funcImageSingleton: $o ).
% 3.21/1.70
% 3.21/1.70 thf(funcImageSingleton,definition,
% 3.21/1.70 ( funcImageSingleton
% 3.21/1.70 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.21/1.70 ( ( func @ A @ B @ Xf )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( singleton
% 3.21/1.70 @ ( dsetconstr @ B
% 3.21/1.70 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(apProp_type,type,
% 3.21/1.70 apProp: $o ).
% 3.21/1.70
% 3.21/1.70 thf(apProp,definition,
% 3.21/1.70 ( apProp
% 3.21/1.70 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.21/1.70 ( ( func @ A @ B @ Xf )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( in
% 3.21/1.70 @ ( setunion
% 3.21/1.70 @ ( dsetconstr @ B
% 3.21/1.70 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 3.21/1.70 @ B ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(ap_type,type,
% 3.21/1.70 ap: $i > $i > $i > $i > $i ).
% 3.21/1.70
% 3.21/1.70 thf(app_type,type,
% 3.21/1.70 app: $o ).
% 3.21/1.70
% 3.21/1.70 thf(app,definition,
% 3.21/1.70 ( app
% 3.21/1.70 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.21/1.70 ( ( func @ A @ B @ Xf )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(infuncsetfunc_type,type,
% 3.21/1.70 infuncsetfunc: $o ).
% 3.21/1.70
% 3.21/1.70 thf(infuncsetfunc,definition,
% 3.21/1.70 ( infuncsetfunc
% 3.21/1.70 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.21/1.70 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.21/1.70 => ( func @ A @ B @ Xf ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(ap2p_type,type,
% 3.21/1.70 ap2p: $o ).
% 3.21/1.70
% 3.21/1.70 thf(ap2p,definition,
% 3.21/1.70 ( ap2p
% 3.21/1.70 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.21/1.70 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(funcinfuncset_type,type,
% 3.21/1.70 funcinfuncset: $o ).
% 3.21/1.70
% 3.21/1.70 thf(funcinfuncset,definition,
% 3.21/1.70 ( funcinfuncset
% 3.21/1.70 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.21/1.70 ( ( func @ A @ B @ Xf )
% 3.21/1.70 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(lamProp_type,type,
% 3.21/1.70 lamProp: $o ).
% 3.21/1.70
% 3.21/1.70 thf(lamProp,definition,
% 3.21/1.70 ( lamProp
% 3.21/1.70 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.21/1.70 ( ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.21/1.70 => ( func @ A @ B
% 3.21/1.70 @ ( dpsetconstr @ A @ B
% 3.21/1.70 @ ^ [Xx: $i,Xy: $i] :
% 3.21/1.70 ( ( Xf @ Xx )
% 3.21/1.70 = Xy ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(lam_type,type,
% 3.21/1.70 lam: $i > $i > ( $i > $i ) > $i ).
% 3.21/1.70
% 3.21/1.70 thf(lamp_type,type,
% 3.21/1.70 lamp: $o ).
% 3.21/1.70
% 3.21/1.70 thf(lamp,definition,
% 3.21/1.70 ( lamp
% 3.21/1.70 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.21/1.70 ( ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.21/1.70 => ( func @ A @ B
% 3.21/1.70 @ ( lam @ A @ B
% 3.21/1.70 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(lam2p_type,type,
% 3.21/1.70 lam2p: $o ).
% 3.21/1.70
% 3.21/1.70 thf(lam2p,definition,
% 3.21/1.70 ( lam2p
% 3.21/1.70 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.21/1.70 ( ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.21/1.70 => ( in
% 3.21/1.70 @ ( lam @ A @ B
% 3.21/1.70 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 3.21/1.70 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(brelnall1_type,type,
% 3.21/1.70 brelnall1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(brelnall1,definition,
% 3.21/1.70 ( brelnall1
% 3.21/1.70 = ( ! [A: $i,B: $i,R: $i] :
% 3.21/1.70 ( ( breln @ A @ B @ R )
% 3.21/1.70 => ! [Xphi: $i > $o] :
% 3.21/1.70 ( ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ! [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ B )
% 3.21/1.70 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.21/1.70 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ R )
% 3.21/1.70 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(brelnall2_type,type,
% 3.21/1.70 brelnall2: $o ).
% 3.21/1.70
% 3.21/1.70 thf(brelnall2,definition,
% 3.21/1.70 ( brelnall2
% 3.21/1.70 = ( ! [A: $i,B: $i,R: $i] :
% 3.21/1.70 ( ( breln @ A @ B @ R )
% 3.21/1.70 => ! [Xphi: $i > $o] :
% 3.21/1.70 ( ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ! [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ B )
% 3.21/1.70 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.21/1.70 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ R )
% 3.21/1.70 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(ex1E2_type,type,
% 3.21/1.70 ex1E2: $o ).
% 3.21/1.70
% 3.21/1.70 thf(ex1E2,definition,
% 3.21/1.70 ( ex1E2
% 3.21/1.70 = ( ! [A: $i,Xphi: $i > $o] :
% 3.21/1.70 ( ( ex1 @ A
% 3.21/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ! [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ A )
% 3.21/1.70 => ( ( Xphi @ Xx )
% 3.21/1.70 => ( ( Xphi @ Xy )
% 3.21/1.70 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(funcGraphProp1_type,type,
% 3.21/1.70 funcGraphProp1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(funcGraphProp1,definition,
% 3.21/1.70 ( funcGraphProp1
% 3.21/1.70 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.21/1.70 ( ( func @ A @ B @ Xf )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(funcGraphProp3_type,type,
% 3.21/1.70 funcGraphProp3: $o ).
% 3.21/1.70
% 3.21/1.70 thf(funcGraphProp3,definition,
% 3.21/1.70 ( funcGraphProp3
% 3.21/1.70 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.21/1.70 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(funcGraphProp2_type,type,
% 3.21/1.70 funcGraphProp2: $o ).
% 3.21/1.70
% 3.21/1.70 thf(funcGraphProp2,definition,
% 3.21/1.70 ( funcGraphProp2
% 3.21/1.70 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.21/1.70 ( ( func @ A @ B @ Xf )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ! [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ B )
% 3.21/1.70 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 3.21/1.70 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.21/1.70 = Xy ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(funcextLem_type,type,
% 3.21/1.70 funcextLem: $o ).
% 3.21/1.70
% 3.21/1.70 thf(funcextLem,definition,
% 3.21/1.70 ( funcextLem
% 3.21/1.70 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.21/1.70 ( ( func @ A @ B @ Xf )
% 3.21/1.70 => ! [Xg: $i] :
% 3.21/1.70 ( ( func @ A @ B @ Xg )
% 3.21/1.70 => ( ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.21/1.70 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ! [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ B )
% 3.21/1.70 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 3.21/1.70 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(funcGraphProp4_type,type,
% 3.21/1.70 funcGraphProp4: $o ).
% 3.21/1.70
% 3.21/1.70 thf(funcGraphProp4,definition,
% 3.21/1.70 ( funcGraphProp4
% 3.21/1.70 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.21/1.70 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ! [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ B )
% 3.21/1.70 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 3.21/1.70 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.21/1.70 = Xy ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(subbreln_type,type,
% 3.21/1.70 subbreln: $o ).
% 3.21/1.70
% 3.21/1.70 thf(subbreln,definition,
% 3.21/1.70 ( subbreln
% 3.21/1.70 = ( ! [A: $i,B: $i,R: $i] :
% 3.21/1.70 ( ( breln @ A @ B @ R )
% 3.21/1.70 => ! [S: $i] :
% 3.21/1.70 ( ( breln @ A @ B @ S )
% 3.21/1.70 => ( ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ! [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ B )
% 3.21/1.70 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.21/1.70 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 3.21/1.70 => ( subset @ R @ S ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(eqbreln_type,type,
% 3.21/1.70 eqbreln: $o ).
% 3.21/1.70
% 3.21/1.70 thf(eqbreln,definition,
% 3.21/1.70 ( eqbreln
% 3.21/1.70 = ( ! [A: $i,B: $i,R: $i] :
% 3.21/1.70 ( ( breln @ A @ B @ R )
% 3.21/1.70 => ! [S: $i] :
% 3.21/1.70 ( ( breln @ A @ B @ S )
% 3.21/1.70 => ( ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ! [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ B )
% 3.21/1.70 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.21/1.70 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 3.21/1.70 => ( ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ! [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ B )
% 3.21/1.70 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 3.21/1.70 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 3.21/1.70 => ( R = S ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(funcext_type,type,
% 3.21/1.70 funcext: $o ).
% 3.21/1.70
% 3.21/1.70 thf(funcext,definition,
% 3.21/1.70 ( funcext
% 3.21/1.70 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.21/1.70 ( ( func @ A @ B @ Xf )
% 3.21/1.70 => ! [Xg: $i] :
% 3.21/1.70 ( ( func @ A @ B @ Xg )
% 3.21/1.70 => ( ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.21/1.70 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 3.21/1.70 => ( Xf = Xg ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(funcext2_type,type,
% 3.21/1.70 funcext2: $o ).
% 3.21/1.70
% 3.21/1.70 thf(funcext2,definition,
% 3.21/1.70 ( funcext2
% 3.21/1.70 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.21/1.70 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.21/1.70 => ! [Xg: $i] :
% 3.21/1.70 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 3.21/1.70 => ( ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.21/1.70 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 3.21/1.70 => ( Xf = Xg ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(ap2apEq1_type,type,
% 3.21/1.70 ap2apEq1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(ap2apEq1,definition,
% 3.21/1.70 ( ap2apEq1
% 3.21/1.70 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.21/1.70 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.21/1.70 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(ap2apEq2_type,type,
% 3.21/1.70 ap2apEq2: $o ).
% 3.21/1.70
% 3.21/1.70 thf(ap2apEq2,definition,
% 3.21/1.70 ( ap2apEq2
% 3.21/1.70 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.21/1.70 ( ( func @ A @ B @ Xf )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.21/1.70 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(beta1_type,type,
% 3.21/1.70 beta1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(beta1,definition,
% 3.21/1.70 ( beta1
% 3.21/1.70 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.21/1.70 ( ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ( ap @ A @ B
% 3.21/1.70 @ ( lam @ A @ B
% 3.21/1.70 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 3.21/1.70 @ Xx )
% 3.21/1.70 = ( Xf @ Xx ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(eta1_type,type,
% 3.21/1.70 eta1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(eta1,definition,
% 3.21/1.70 ( eta1
% 3.21/1.70 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.21/1.70 ( ( func @ A @ B @ Xf )
% 3.21/1.70 => ( ( lam @ A @ B
% 3.21/1.70 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 3.21/1.70 = Xf ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(lam2lamEq_type,type,
% 3.21/1.70 lam2lamEq: $o ).
% 3.21/1.70
% 3.21/1.70 thf(lam2lamEq,definition,
% 3.21/1.70 ( lam2lamEq
% 3.21/1.70 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.21/1.70 ( ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.21/1.70 => ( ( lam @ A @ B
% 3.21/1.70 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 3.21/1.70 = ( lam @ A @ B
% 3.21/1.70 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(beta2_type,type,
% 3.21/1.70 beta2: $o ).
% 3.21/1.70
% 3.21/1.70 thf(beta2,definition,
% 3.21/1.70 ( beta2
% 3.21/1.70 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.21/1.70 ( ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ( ap @ A @ B
% 3.21/1.70 @ ( lam @ A @ B
% 3.21/1.70 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 3.21/1.70 @ Xx )
% 3.21/1.70 = ( Xf @ Xx ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(eta2_type,type,
% 3.21/1.70 eta2: $o ).
% 3.21/1.70
% 3.21/1.70 thf(eta2,definition,
% 3.21/1.70 ( eta2
% 3.21/1.70 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.21/1.70 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.21/1.70 => ( ( lam @ A @ B
% 3.21/1.70 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 3.21/1.70 = Xf ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(iffalseProp1_type,type,
% 3.21/1.70 iffalseProp1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(iffalseProp1,definition,
% 3.21/1.70 ( iffalseProp1
% 3.21/1.70 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ! [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ A )
% 3.21/1.70 => ( ~ Xphi
% 3.21/1.70 => ( in @ Xy
% 3.21/1.70 @ ( dsetconstr @ A
% 3.21/1.70 @ ^ [Xz: $i] :
% 3.21/1.70 ( ( Xphi
% 3.21/1.70 & ( Xz = Xx ) )
% 3.21/1.70 | ( ~ Xphi
% 3.21/1.70 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(iffalseProp2_type,type,
% 3.21/1.70 iffalseProp2: $o ).
% 3.21/1.70
% 3.21/1.70 thf(iffalseProp2,definition,
% 3.21/1.70 ( iffalseProp2
% 3.21/1.70 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ! [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ A )
% 3.21/1.70 => ( ~ Xphi
% 3.21/1.70 => ( ( dsetconstr @ A
% 3.21/1.70 @ ^ [Xz: $i] :
% 3.21/1.70 ( ( Xphi
% 3.21/1.70 & ( Xz = Xx ) )
% 3.21/1.70 | ( ~ Xphi
% 3.21/1.70 & ( Xz = Xy ) ) ) )
% 3.21/1.70 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(iftrueProp1_type,type,
% 3.21/1.70 iftrueProp1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(iftrueProp1,definition,
% 3.21/1.70 ( iftrueProp1
% 3.21/1.70 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ! [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ A )
% 3.21/1.70 => ( Xphi
% 3.21/1.70 => ( in @ Xx
% 3.21/1.70 @ ( dsetconstr @ A
% 3.21/1.70 @ ^ [Xz: $i] :
% 3.21/1.70 ( ( Xphi
% 3.21/1.70 & ( Xz = Xx ) )
% 3.21/1.70 | ( ~ Xphi
% 3.21/1.70 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(iftrueProp2_type,type,
% 3.21/1.70 iftrueProp2: $o ).
% 3.21/1.70
% 3.21/1.70 thf(iftrueProp2,definition,
% 3.21/1.70 ( iftrueProp2
% 3.21/1.70 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ! [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ A )
% 3.21/1.70 => ( Xphi
% 3.21/1.70 => ( ( dsetconstr @ A
% 3.21/1.70 @ ^ [Xz: $i] :
% 3.21/1.70 ( ( Xphi
% 3.21/1.70 & ( Xz = Xx ) )
% 3.21/1.70 | ( ~ Xphi
% 3.21/1.70 & ( Xz = Xy ) ) ) )
% 3.21/1.70 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(ifSingleton_type,type,
% 3.21/1.70 ifSingleton: $o ).
% 3.21/1.70
% 3.21/1.70 thf(ifSingleton,definition,
% 3.21/1.70 ( ifSingleton
% 3.21/1.70 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ! [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ A )
% 3.21/1.70 => ( singleton
% 3.21/1.70 @ ( dsetconstr @ A
% 3.21/1.70 @ ^ [Xz: $i] :
% 3.21/1.70 ( ( Xphi
% 3.21/1.70 & ( Xz = Xx ) )
% 3.21/1.70 | ( ~ Xphi
% 3.21/1.70 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(if_type,type,
% 3.21/1.70 if: $i > $o > $i > $i > $i ).
% 3.21/1.70
% 3.21/1.70 thf(ifp_type,type,
% 3.21/1.70 ifp: $o ).
% 3.21/1.70
% 3.21/1.70 thf(ifp,definition,
% 3.21/1.70 ( ifp
% 3.21/1.70 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ! [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ A )
% 3.21/1.70 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(theeq_type,type,
% 3.21/1.70 theeq: $o ).
% 3.21/1.70
% 3.21/1.70 thf(theeq,definition,
% 3.21/1.70 ( theeq
% 3.21/1.70 = ( ! [X: $i] :
% 3.21/1.70 ( ( singleton @ X )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ X )
% 3.21/1.70 => ( ( setunion @ X )
% 3.21/1.70 = Xx ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(iftrue_type,type,
% 3.21/1.70 iftrue: $o ).
% 3.21/1.70
% 3.21/1.70 thf(iftrue,definition,
% 3.21/1.70 ( iftrue
% 3.21/1.70 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ! [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ A )
% 3.21/1.70 => ( Xphi
% 3.21/1.70 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 3.21/1.70 = Xx ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(iffalse_type,type,
% 3.21/1.70 iffalse: $o ).
% 3.21/1.70
% 3.21/1.70 thf(iffalse,definition,
% 3.21/1.70 ( iffalse
% 3.21/1.70 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ! [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ A )
% 3.21/1.70 => ( ~ Xphi
% 3.21/1.70 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 3.21/1.70 = Xy ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(iftrueorfalse_type,type,
% 3.21/1.70 iftrueorfalse: $o ).
% 3.21/1.70
% 3.21/1.70 thf(iftrueorfalse,definition,
% 3.21/1.70 ( iftrueorfalse
% 3.21/1.70 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ! [Xy: $i] :
% 3.21/1.70 ( ( in @ Xy @ A )
% 3.21/1.70 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(binintersectT_lem_type,type,
% 3.21/1.70 binintersectT_lem: $o ).
% 3.21/1.70
% 3.21/1.70 thf(binintersectT_lem,definition,
% 3.21/1.70 ( binintersectT_lem
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ! [Y: $i] :
% 3.21/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.70 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(binunionT_lem_type,type,
% 3.21/1.70 binunionT_lem: $o ).
% 3.21/1.70
% 3.21/1.70 thf(binunionT_lem,definition,
% 3.21/1.70 ( binunionT_lem
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ! [Y: $i] :
% 3.21/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.70 => ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(powersetT_lem_type,type,
% 3.21/1.70 powersetT_lem: $o ).
% 3.21/1.70
% 3.21/1.70 thf(powersetT_lem,definition,
% 3.21/1.70 ( powersetT_lem
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(setminusT_lem_type,type,
% 3.21/1.70 setminusT_lem: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setminusT_lem,definition,
% 3.21/1.70 ( setminusT_lem
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ! [Y: $i] :
% 3.21/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.70 => ( in @ ( setminus @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(complementT_lem_type,type,
% 3.21/1.70 complementT_lem: $o ).
% 3.21/1.70
% 3.21/1.70 thf(complementT_lem,definition,
% 3.21/1.70 ( complementT_lem
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ( in @ ( setminus @ A @ X ) @ ( powerset @ A ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(setextT_type,type,
% 3.21/1.70 setextT: $o ).
% 3.21/1.70
% 3.21/1.70 thf(setextT,definition,
% 3.21/1.70 ( setextT
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ! [Y: $i] :
% 3.21/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.70 => ( ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ( in @ Xx @ X )
% 3.21/1.70 => ( in @ Xx @ Y ) ) )
% 3.21/1.70 => ( ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ( in @ Xx @ Y )
% 3.21/1.70 => ( in @ Xx @ X ) ) )
% 3.21/1.70 => ( X = Y ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(subsetTI_type,type,
% 3.21/1.70 subsetTI: $o ).
% 3.21/1.70
% 3.21/1.70 thf(subsetTI,definition,
% 3.21/1.70 ( subsetTI
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ! [Y: $i] :
% 3.21/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.70 => ( ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ( in @ Xx @ X )
% 3.21/1.70 => ( in @ Xx @ Y ) ) )
% 3.21/1.70 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(powersetTI1_type,type,
% 3.21/1.70 powersetTI1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(powersetTI1,definition,
% 3.21/1.70 ( powersetTI1
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ! [Y: $i] :
% 3.21/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.70 => ( ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ( in @ Xx @ X )
% 3.21/1.70 => ( in @ Xx @ Y ) ) )
% 3.21/1.70 => ( in @ X @ ( powerset @ Y ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(powersetTE1_type,type,
% 3.21/1.70 powersetTE1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(powersetTE1,definition,
% 3.21/1.70 ( powersetTE1
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ! [Y: $i] :
% 3.21/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ( in @ X @ ( powerset @ Y ) )
% 3.21/1.70 => ( ( in @ Xx @ X )
% 3.21/1.70 => ( in @ Xx @ Y ) ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(complementTI1_type,type,
% 3.21/1.70 complementTI1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(complementTI1,definition,
% 3.21/1.70 ( complementTI1
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ( in @ Xx @ X )
% 3.21/1.70 => ~ ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(complementTE1_type,type,
% 3.21/1.70 complementTE1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(complementTE1,definition,
% 3.21/1.70 ( complementTE1
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ~ ( in @ Xx @ ( setminus @ A @ X ) )
% 3.21/1.70 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(binintersectTELcontra_type,type,
% 3.21/1.70 binintersectTELcontra: $o ).
% 3.21/1.70
% 3.21/1.70 thf(binintersectTELcontra,definition,
% 3.21/1.70 ( binintersectTELcontra
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ! [Y: $i] :
% 3.21/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ~ ( in @ Xx @ X )
% 3.21/1.70 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(binintersectTERcontra_type,type,
% 3.21/1.70 binintersectTERcontra: $o ).
% 3.21/1.70
% 3.21/1.70 thf(binintersectTERcontra,definition,
% 3.21/1.70 ( binintersectTERcontra
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ! [Y: $i] :
% 3.21/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ~ ( in @ Xx @ Y )
% 3.21/1.70 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(contrasubsetT_type,type,
% 3.21/1.70 contrasubsetT: $o ).
% 3.21/1.70
% 3.21/1.70 thf(contrasubsetT,definition,
% 3.21/1.70 ( contrasubsetT
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ! [Y: $i] :
% 3.21/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 3.21/1.70 => ( ( in @ Xx @ Y )
% 3.21/1.70 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(contrasubsetT1_type,type,
% 3.21/1.70 contrasubsetT1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(contrasubsetT1,definition,
% 3.21/1.70 ( contrasubsetT1
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ! [Y: $i] :
% 3.21/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ( subset @ X @ Y )
% 3.21/1.70 => ( ~ ( in @ Xx @ Y )
% 3.21/1.70 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(contrasubsetT2_type,type,
% 3.21/1.70 contrasubsetT2: $o ).
% 3.21/1.70
% 3.21/1.70 thf(contrasubsetT2,definition,
% 3.21/1.70 ( contrasubsetT2
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ! [Y: $i] :
% 3.21/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.70 => ( ( subset @ X @ Y )
% 3.21/1.70 => ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(contrasubsetT3_type,type,
% 3.21/1.70 contrasubsetT3: $o ).
% 3.21/1.70
% 3.21/1.70 thf(contrasubsetT3,definition,
% 3.21/1.70 ( contrasubsetT3
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ! [Y: $i] :
% 3.21/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.70 => ( ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) )
% 3.21/1.70 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(doubleComplementI1_type,type,
% 3.21/1.70 doubleComplementI1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(doubleComplementI1,definition,
% 3.21/1.70 ( doubleComplementI1
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ( in @ Xx @ X )
% 3.21/1.70 => ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(doubleComplementE1_type,type,
% 3.21/1.70 doubleComplementE1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(doubleComplementE1,definition,
% 3.21/1.70 ( doubleComplementE1
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) )
% 3.21/1.70 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(doubleComplementSub1_type,type,
% 3.21/1.70 doubleComplementSub1: $o ).
% 3.21/1.70
% 3.21/1.70 thf(doubleComplementSub1,definition,
% 3.21/1.70 ( doubleComplementSub1
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ( subset @ X @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(doubleComplementSub2_type,type,
% 3.21/1.70 doubleComplementSub2: $o ).
% 3.21/1.70
% 3.21/1.70 thf(doubleComplementSub2,definition,
% 3.21/1.70 ( doubleComplementSub2
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ( subset @ ( setminus @ A @ ( setminus @ A @ X ) ) @ X ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(doubleComplementEq_type,type,
% 3.21/1.70 doubleComplementEq: $o ).
% 3.21/1.70
% 3.21/1.70 thf(doubleComplementEq,definition,
% 3.21/1.70 ( doubleComplementEq
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ( X
% 3.21/1.70 = ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(complementTnotintersectT_type,type,
% 3.21/1.70 complementTnotintersectT: $o ).
% 3.21/1.70
% 3.21/1.70 thf(complementTnotintersectT,definition,
% 3.21/1.70 ( complementTnotintersectT
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ! [Y: $i] :
% 3.21/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 3.21/1.70 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(complementImpComplementIntersect_type,type,
% 3.21/1.70 complementImpComplementIntersect: $o ).
% 3.21/1.70
% 3.21/1.70 thf(complementImpComplementIntersect,definition,
% 3.21/1.70 ( complementImpComplementIntersect
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ! [Y: $i] :
% 3.21/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 3.21/1.70 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(complementSubsetComplementIntersect_type,type,
% 3.21/1.70 complementSubsetComplementIntersect: $o ).
% 3.21/1.70
% 3.21/1.70 thf(complementSubsetComplementIntersect,definition,
% 3.21/1.70 ( complementSubsetComplementIntersect
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ! [Y: $i] :
% 3.21/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.70 => ( subset @ ( setminus @ A @ X ) @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(complementInPowersetComplementIntersect_type,type,
% 3.21/1.70 complementInPowersetComplementIntersect: $o ).
% 3.21/1.70
% 3.21/1.70 thf(complementInPowersetComplementIntersect,definition,
% 3.21/1.70 ( complementInPowersetComplementIntersect
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ! [Y: $i] :
% 3.21/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.70 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(contraSubsetComplement_type,type,
% 3.21/1.70 contraSubsetComplement: $o ).
% 3.21/1.70
% 3.21/1.70 thf(contraSubsetComplement,definition,
% 3.21/1.70 ( contraSubsetComplement
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ! [Y: $i] :
% 3.21/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.70 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ( in @ Xx @ Y )
% 3.21/1.70 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(complementTcontraSubset_type,type,
% 3.21/1.70 complementTcontraSubset: $o ).
% 3.21/1.70
% 3.21/1.70 thf(complementTcontraSubset,definition,
% 3.21/1.70 ( complementTcontraSubset
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ! [Y: $i] :
% 3.21/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.70 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 3.21/1.70 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(binunionTILcontra_type,type,
% 3.21/1.70 binunionTILcontra: $o ).
% 3.21/1.70
% 3.21/1.70 thf(binunionTILcontra,definition,
% 3.21/1.70 ( binunionTILcontra
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ! [Y: $i] :
% 3.21/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 3.21/1.70 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(binunionTIRcontra_type,type,
% 3.21/1.70 binunionTIRcontra: $o ).
% 3.21/1.70
% 3.21/1.70 thf(binunionTIRcontra,definition,
% 3.21/1.70 ( binunionTIRcontra
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ! [Y: $i] :
% 3.21/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.70 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 3.21/1.70 => ~ ( in @ Xx @ Y ) ) ) ) ) ) ) ).
% 3.21/1.70
% 3.21/1.70 thf(inIntersectImpInUnion_type,type,
% 3.21/1.70 inIntersectImpInUnion: $o ).
% 3.21/1.70
% 3.21/1.70 thf(inIntersectImpInUnion,definition,
% 3.21/1.70 ( inIntersectImpInUnion
% 3.21/1.70 = ( ! [A: $i,X: $i] :
% 3.21/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.70 => ! [Y: $i] :
% 3.21/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.70 => ! [Z: $i] :
% 3.21/1.70 ( ( in @ Z @ ( powerset @ A ) )
% 3.21/1.70 => ! [Xx: $i] :
% 3.21/1.70 ( ( in @ Xx @ A )
% 3.21/1.71 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 3.21/1.71 => ( in @ Xx @ ( binunion @ X @ Z ) ) ) ) ) ) ) ) ) ).
% 3.21/1.71
% 3.21/1.71 thf(inIntersectImpInUnion2_type,type,
% 3.21/1.71 inIntersectImpInUnion2: $o ).
% 3.21/1.71
% 3.21/1.71 thf(inIntersectImpInUnion2,definition,
% 3.21/1.71 ( inIntersectImpInUnion2
% 3.21/1.71 = ( ! [A: $i,X: $i] :
% 3.21/1.71 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.71 => ! [Y: $i] :
% 3.21/1.71 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.71 => ! [Z: $i] :
% 3.21/1.71 ( ( in @ Z @ ( powerset @ A ) )
% 3.21/1.71 => ! [Xx: $i] :
% 3.21/1.71 ( ( in @ Xx @ A )
% 3.21/1.71 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 3.21/1.71 => ( in @ Xx @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 3.21/1.71
% 3.21/1.71 thf(inIntersectImpInIntersectUnions_type,type,
% 3.21/1.71 inIntersectImpInIntersectUnions: $o ).
% 3.21/1.71
% 3.21/1.71 thf(inIntersectImpInIntersectUnions,definition,
% 3.21/1.71 ( inIntersectImpInIntersectUnions
% 3.21/1.71 = ( ! [A: $i,X: $i] :
% 3.21/1.71 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.71 => ! [Y: $i] :
% 3.21/1.71 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.71 => ! [Z: $i] :
% 3.21/1.71 ( ( in @ Z @ ( powerset @ A ) )
% 3.21/1.71 => ! [Xx: $i] :
% 3.21/1.71 ( ( in @ Xx @ A )
% 3.21/1.71 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 3.21/1.71 => ( in @ Xx @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ) ).
% 3.21/1.71
% 3.21/1.71 thf(intersectInPowersetIntersectUnions_type,type,
% 3.21/1.71 intersectInPowersetIntersectUnions: $o ).
% 3.21/1.71
% 3.21/1.71 thf(intersectInPowersetIntersectUnions,definition,
% 3.21/1.71 ( intersectInPowersetIntersectUnions
% 3.21/1.71 = ( ! [A: $i,X: $i] :
% 3.21/1.71 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.71 => ! [Y: $i] :
% 3.21/1.71 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.71 => ! [Z: $i] :
% 3.21/1.71 ( ( in @ Z @ ( powerset @ A ) )
% 3.21/1.71 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 3.21/1.71
% 3.21/1.71 thf(inComplementUnionImpNotIn1_type,type,
% 3.21/1.71 inComplementUnionImpNotIn1: $o ).
% 3.21/1.71
% 3.21/1.71 thf(inComplementUnionImpNotIn1,definition,
% 3.21/1.71 ( inComplementUnionImpNotIn1
% 3.21/1.71 = ( ! [A: $i,X: $i] :
% 3.21/1.71 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.71 => ! [Y: $i] :
% 3.21/1.71 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.71 => ! [Xx: $i] :
% 3.21/1.71 ( ( in @ Xx @ A )
% 3.21/1.71 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 3.21/1.71 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 3.21/1.71
% 3.21/1.71 thf(inComplementUnionImpInComplement1_type,type,
% 3.21/1.71 inComplementUnionImpInComplement1: $o ).
% 3.21/1.71
% 3.21/1.71 thf(inComplementUnionImpInComplement1,definition,
% 3.21/1.71 ( inComplementUnionImpInComplement1
% 3.21/1.71 = ( ! [A: $i,X: $i] :
% 3.21/1.71 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.71 => ! [Y: $i] :
% 3.21/1.71 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.71 => ! [Xx: $i] :
% 3.21/1.71 ( ( in @ Xx @ A )
% 3.21/1.71 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 3.21/1.71 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 3.21/1.71
% 3.21/1.71 thf(binunionTE_type,type,
% 3.21/1.71 binunionTE: $o ).
% 3.21/1.71
% 3.21/1.71 thf(binunionTE,definition,
% 3.21/1.71 ( binunionTE
% 3.21/1.71 = ( ! [A: $i,X: $i] :
% 3.21/1.71 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.71 => ! [Y: $i] :
% 3.21/1.71 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.71 => ! [Xphi: $o,Xx: $i] :
% 3.21/1.71 ( ( in @ Xx @ A )
% 3.21/1.71 => ( ( in @ Xx @ ( binunion @ X @ Y ) )
% 3.21/1.71 => ( ( ( in @ Xx @ X )
% 3.21/1.71 => Xphi )
% 3.21/1.71 => ( ( ( in @ Xx @ Y )
% 3.21/1.71 => Xphi )
% 3.21/1.71 => Xphi ) ) ) ) ) ) ) ) ).
% 3.21/1.71
% 3.21/1.71 thf(binunionTEcontra_type,type,
% 3.21/1.71 binunionTEcontra: $o ).
% 3.21/1.71
% 3.21/1.71 thf(binunionTEcontra,definition,
% 3.21/1.71 ( binunionTEcontra
% 3.21/1.71 = ( ! [A: $i,X: $i] :
% 3.21/1.71 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.71 => ! [Y: $i] :
% 3.21/1.71 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.71 => ! [Xx: $i] :
% 3.21/1.71 ( ( in @ Xx @ A )
% 3.21/1.71 => ( ~ ( in @ Xx @ X )
% 3.21/1.71 => ( ~ ( in @ Xx @ Y )
% 3.21/1.71 => ~ ( in @ Xx @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 3.21/1.71
% 3.21/1.71 thf(demorgan2a1_type,type,
% 3.21/1.71 demorgan2a1: $o ).
% 3.21/1.71
% 3.21/1.71 thf(demorgan2a1,definition,
% 3.21/1.71 ( demorgan2a1
% 3.21/1.71 = ( ! [A: $i,X: $i] :
% 3.21/1.71 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.71 => ! [Y: $i] :
% 3.21/1.71 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.71 => ! [Xx: $i] :
% 3.21/1.71 ( ( in @ Xx @ A )
% 3.21/1.71 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 3.21/1.71 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 3.21/1.71
% 3.21/1.71 thf(complementUnionInPowersetComplement_type,type,
% 3.21/1.71 complementUnionInPowersetComplement: $o ).
% 3.21/1.71
% 3.21/1.71 thf(complementUnionInPowersetComplement,definition,
% 3.21/1.71 ( complementUnionInPowersetComplement
% 3.21/1.71 = ( ! [A: $i,X: $i] :
% 3.21/1.71 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.71 => ! [Y: $i] :
% 3.21/1.71 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.71 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.21/1.71
% 3.21/1.71 thf(demorgan2a2_type,type,
% 3.21/1.71 demorgan2a2: $o ).
% 3.21/1.71
% 3.21/1.71 thf(demorgan2a2,definition,
% 3.21/1.71 ( demorgan2a2
% 3.21/1.71 = ( ! [A: $i,X: $i] :
% 3.21/1.71 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.71 => ! [Y: $i] :
% 3.21/1.71 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.71 => ! [Xx: $i] :
% 3.21/1.71 ( ( in @ Xx @ A )
% 3.21/1.71 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 3.21/1.71 => ( in @ Xx @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ).
% 3.21/1.71
% 3.21/1.71 thf(demorgan1a_type,type,
% 3.21/1.71 demorgan1a: $o ).
% 3.21/1.71
% 3.21/1.71 thf(demorgan1a,definition,
% 3.21/1.71 ( demorgan1a
% 3.21/1.71 = ( ! [A: $i,X: $i] :
% 3.21/1.71 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.71 => ! [Y: $i] :
% 3.21/1.71 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.71 => ! [Xx: $i] :
% 3.21/1.71 ( ( in @ Xx @ A )
% 3.21/1.71 => ( ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) )
% 3.21/1.71 => ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 3.21/1.71
% 3.21/1.71 thf(demorgan1b_type,type,
% 3.21/1.71 demorgan1b: $o ).
% 3.21/1.71
% 3.21/1.71 thf(demorgan1b,definition,
% 3.21/1.71 ( demorgan1b
% 3.21/1.71 = ( ! [A: $i,X: $i] :
% 3.21/1.71 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.71 => ! [Y: $i] :
% 3.21/1.71 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.71 => ! [Xx: $i] :
% 3.21/1.71 ( ( in @ Xx @ A )
% 3.21/1.71 => ( ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 3.21/1.71 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 3.21/1.71
% 3.21/1.71 thf(demorgan1_type,type,
% 3.21/1.71 demorgan1: $o ).
% 3.21/1.71
% 3.21/1.71 thf(demorgan1,definition,
% 3.21/1.71 ( demorgan1
% 3.21/1.71 = ( ! [A: $i,X: $i] :
% 3.21/1.71 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.71 => ! [Y: $i] :
% 3.21/1.71 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.71 => ( ( setminus @ A @ ( binintersect @ X @ Y ) )
% 3.21/1.71 = ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 3.21/1.71
% 3.21/1.71 thf(demorgan2a_type,type,
% 3.21/1.71 demorgan2a: $o ).
% 3.21/1.71
% 3.21/1.71 thf(demorgan2a,definition,
% 3.21/1.71 ( demorgan2a
% 3.21/1.71 = ( ! [A: $i,X: $i] :
% 3.21/1.71 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.71 => ! [Y: $i] :
% 3.21/1.71 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.71 => ! [Xx: $i] :
% 3.21/1.71 ( ( in @ Xx @ A )
% 3.21/1.71 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 3.21/1.71 => ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 3.21/1.71
% 3.21/1.71 thf(demorgan2b2_type,type,
% 3.21/1.71 demorgan2b2: $o ).
% 3.21/1.71
% 3.21/1.71 thf(demorgan2b2,definition,
% 3.21/1.71 ( demorgan2b2
% 3.21/1.71 = ( ! [A: $i,X: $i] :
% 3.21/1.71 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.71 => ! [Y: $i] :
% 3.21/1.71 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.71 => ! [Xx: $i] :
% 3.21/1.71 ( ( in @ Xx @ A )
% 3.21/1.71 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 3.21/1.71 => ( ( in @ Xx @ ( setminus @ A @ Y ) )
% 3.21/1.71 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ) ).
% 3.21/1.71
% 3.21/1.71 thf(demorgan2b_type,type,
% 3.21/1.71 demorgan2b: $o ).
% 3.21/1.71
% 3.21/1.71 thf(demorgan2b,definition,
% 3.21/1.71 ( demorgan2b
% 3.21/1.71 = ( ! [A: $i,X: $i] :
% 3.21/1.71 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.71 => ! [Y: $i] :
% 3.21/1.71 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.71 => ! [Xx: $i] :
% 3.21/1.71 ( ( in @ Xx @ A )
% 3.21/1.71 => ( ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 3.21/1.71 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 3.21/1.71
% 3.21/1.71 thf(demorgan2_type,type,
% 3.21/1.71 demorgan2: $o ).
% 3.21/1.71
% 3.21/1.71 thf(demorgan2,definition,
% 3.21/1.71 ( demorgan2
% 3.21/1.71 = ( ! [A: $i,X: $i] :
% 3.21/1.71 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.71 => ! [Y: $i] :
% 3.21/1.71 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.71 => ( ( setminus @ A @ ( binunion @ X @ Y ) )
% 3.21/1.71 = ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 3.21/1.71
% 3.21/1.71 thf(woz13rule0_type,type,
% 3.21/1.71 woz13rule0: $o ).
% 3.21/1.71
% 3.21/1.71 thf(woz13rule0,definition,
% 3.21/1.71 ( woz13rule0
% 3.21/1.71 = ( ! [A: $i,X: $i] :
% 3.21/1.71 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.71 => ! [Y: $i] :
% 3.21/1.71 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.71 => ! [Xx: $i] :
% 3.21/1.71 ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 3.21/1.71 => ( in @ Xx @ A ) ) ) ) ) ) ).
% 3.21/1.71
% 3.21/1.71 thf(woz13rule1_type,type,
% 3.21/1.71 woz13rule1: $o ).
% 3.21/1.71
% 3.21/1.71 thf(woz13rule1,definition,
% 3.21/1.71 ( woz13rule1
% 3.21/1.71 = ( ! [A: $i,X: $i] :
% 3.21/1.71 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.71 => ! [Y: $i] :
% 3.21/1.71 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.71 => ! [Z: $i] :
% 3.21/1.71 ( ( in @ Z @ ( powerset @ A ) )
% 3.21/1.71 => ( ( subset @ X @ Z )
% 3.21/1.71 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 3.21/1.71
% 3.21/1.71 thf(woz13rule2_type,type,
% 3.21/1.71 woz13rule2: $o ).
% 3.21/1.71
% 3.21/1.71 thf(woz13rule2,definition,
% 3.21/1.71 ( woz13rule2
% 3.21/1.71 = ( ! [A: $i,X: $i] :
% 3.21/1.71 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.71 => ! [Y: $i] :
% 3.21/1.71 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.71 => ! [Z: $i] :
% 3.21/1.71 ( ( in @ Z @ ( powerset @ A ) )
% 3.21/1.71 => ( ( subset @ Y @ Z )
% 3.21/1.71 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 3.21/1.71
% 3.21/1.71 thf(woz13rule3_type,type,
% 3.21/1.71 woz13rule3: $o ).
% 3.21/1.71
% 3.21/1.71 thf(woz13rule3,definition,
% 3.21/1.71 ( woz13rule3
% 3.21/1.71 = ( ! [A: $i,X: $i] :
% 3.21/1.71 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.71 => ! [Y: $i] :
% 3.21/1.71 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.71 => ! [Z: $i] :
% 3.21/1.71 ( ( in @ Z @ ( powerset @ A ) )
% 3.21/1.71 => ( ( subset @ X @ Y )
% 3.21/1.71 => ( ( subset @ X @ Z )
% 3.21/1.71 => ( subset @ X @ ( binintersect @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 3.21/1.71
% 3.21/1.71 thf(woz13rule4_type,type,
% 3.21/1.71 woz13rule4: $o ).
% 3.21/1.71
% 3.21/1.71 thf(woz13rule4,definition,
% 3.21/1.71 ( woz13rule4
% 3.21/1.71 = ( ! [A: $i,X: $i] :
% 3.21/1.71 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.71 => ! [Y: $i] :
% 3.21/1.71 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.71 => ! [Z: $i] :
% 3.21/1.71 ( ( in @ Z @ ( powerset @ A ) )
% 3.21/1.71 => ! [W: $i] :
% 3.21/1.71 ( ( in @ W @ ( powerset @ A ) )
% 3.21/1.71 => ( ( subset @ X @ Z )
% 3.21/1.71 => ( ( subset @ Y @ W )
% 3.21/1.71 => ( subset @ ( binintersect @ X @ Y ) @ ( binintersect @ Z @ W ) ) ) ) ) ) ) ) ) ) ).
% 3.21/1.71
% 3.21/1.71 thf(woz1_1_type,type,
% 3.21/1.71 woz1_1: $o ).
% 3.21/1.71
% 3.21/1.71 thf(woz1_1,definition,
% 3.21/1.71 ( woz1_1
% 3.21/1.71 = ( ! [A: $i,X: $i] :
% 3.21/1.71 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.71 => ! [Y: $i] :
% 3.21/1.71 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.71 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.21/1.71
% 3.21/1.71 thf(woz1_2_type,type,
% 3.21/1.71 woz1_2: $o ).
% 3.21/1.71
% 3.21/1.71 thf(woz1_2,definition,
% 3.21/1.71 ( woz1_2
% 3.21/1.71 = ( ! [A: $i,X: $i] :
% 3.21/1.71 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.71 => ! [Y: $i] :
% 3.21/1.71 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.71 => ! [Z: $i] :
% 3.21/1.71 ( ( in @ Z @ ( powerset @ A ) )
% 3.21/1.71 => ! [W: $i] :
% 3.21/1.71 ( ( in @ W @ ( powerset @ A ) )
% 3.21/1.71 => ( ( setminus @ A @ ( binintersect @ ( binunion @ X @ Y ) @ ( binunion @ Z @ W ) ) )
% 3.21/1.71 = ( binunion @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) @ ( binintersect @ ( setminus @ A @ Z ) @ ( setminus @ A @ W ) ) ) ) ) ) ) ) ) ) ).
% 3.21/1.71
% 3.21/1.71 thf(woz1_3_type,type,
% 3.21/1.71 woz1_3: $o ).
% 3.21/1.71
% 3.21/1.71 thf(woz1_3,definition,
% 3.21/1.71 ( woz1_3
% 3.21/1.71 = ( ! [A: $i,X: $i] :
% 3.21/1.71 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.71 => ! [Y: $i] :
% 3.21/1.71 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.71 => ! [Z: $i] :
% 3.21/1.71 ( ( in @ Z @ ( powerset @ A ) )
% 3.21/1.71 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 3.21/1.71
% 3.21/1.71 thf(woz1_4_type,type,
% 3.21/1.71 woz1_4: $o ).
% 3.21/1.71
% 3.21/1.71 thf(woz1_4,definition,
% 3.21/1.71 ( woz1_4
% 3.21/1.71 = ( ! [A: $i,X: $i] :
% 3.21/1.71 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.71 => ! [Y: $i] :
% 3.21/1.71 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.71 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 3.21/1.71 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.21/1.71
% 3.21/1.71 thf(woz1_5_type,type,
% 3.21/1.71 woz1_5: $o ).
% 3.21/1.71
% 3.21/1.71 thf(woz1_5,definition,
% 3.21/1.71 ( woz1_5
% 3.21/1.71 = ( ! [A: $i,X: $i] :
% 3.21/1.71 ( ( in @ X @ ( powerset @ A ) )
% 3.21/1.71 => ! [Y: $i] :
% 3.21/1.71 ( ( in @ Y @ ( powerset @ A ) )
% 3.21/1.71 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.21/1.71
% 3.21/1.71 thf(breln1_type,type,
% 3.21/1.71 breln1: $i > $i > $o ).
% 3.21/1.71
% 3.21/1.71 thf(breln1all2_type,type,
% 3.21/1.71 breln1all2: $o ).
% 3.21/1.71
% 3.21/1.71 thf(breln1all2,definition,
% 3.21/1.71 ( breln1all2
% 3.21/1.71 = ( ! [A: $i,R: $i] :
% 3.21/1.71 ( ( breln1 @ A @ R )
% 3.21/1.71 => ! [Xphi: $i > $o] :
% 3.21/1.71 ( ! [Xx: $i] :
% 3.21/1.71 ( ( in @ Xx @ A )
% 3.21/1.71 => ! [Xy: $i] :
% 3.21/1.71 ( ( in @ Xy @ A )
% 3.21/1.71 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.21/1.71 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 3.21/1.71 => ! [Xx: $i] :
% 3.21/1.71 ( ( in @ Xx @ R )
% 3.21/1.71 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.21/1.71
% 3.21/1.71 thf(breln1Set_type,type,
% 3.21/1.71 breln1Set: $i > $i ).
% 3.21/1.71
% 3.21/1.71 thf(breln1SetBreln1_type,type,
% 3.21/1.71 breln1SetBreln1: $o ).
% 3.21/1.71
% 3.21/1.71 thf(breln1SetBreln1,definition,
% 3.21/1.71 ( breln1SetBreln1
% 3.21/1.71 = ( ! [A: $i,R: $i] :
% 3.21/1.71 ( ( in @ R @ ( breln1Set @ A ) )
% 3.21/1.71 => ( breln1 @ A @ R ) ) ) ) ).
% 3.21/1.71
% 3.21/1.71 thf(transitive_type,type,
% 3.21/1.71 transitive: $i > $i > $o ).
% 3.21/1.71
% 3.21/1.71 thf(antisymmetric_type,type,
% 3.21/1.71 antisymmetric: $i > $i > $o ).
% 3.21/1.71
% 3.21/1.71 thf(reflexive_type,type,
% 3.21/1.71 reflexive: $i > $i > $o ).
% 3.21/1.71
% 3.21/1.71 thf(refltransitive_type,type,
% 3.21/1.71 refltransitive: $i > $i > $o ).
% 3.21/1.71
% 3.21/1.71 thf(refllinearorder_type,type,
% 3.21/1.71 refllinearorder: $i > $i > $o ).
% 3.21/1.71
% 3.21/1.71 thf(reflwellordering_type,type,
% 3.21/1.71 reflwellordering: $i > $i > $o ).
% 3.21/1.71
% 3.21/1.71 thf(choice2fnsingleton_type,type,
% 3.21/1.71 choice2fnsingleton: $o ).
% 3.21/1.71
% 3.21/1.71 thf(choice2fnsingleton,definition,
% 3.21/1.71 ( choice2fnsingleton
% 3.21/1.71 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 3.21/1.71 ( ! [Xx: $i] :
% 3.21/1.71 ( ( in @ Xx @ A )
% 3.21/1.71 => ? [Xy: $i] :
% 3.21/1.71 ( ( in @ Xy @ B )
% 3.21/1.71 & ( Xphi @ Xx @ Xy ) ) )
% 3.21/1.71 => ! [R: $i] :
% 3.21/1.71 ( ( in @ R @ ( breln1Set @ B ) )
% 3.21/1.71 => ( ( reflwellordering @ B @ R )
% 3.21/1.71 => ! [Xx: $i] :
% 3.21/1.71 ( ( in @ Xx @ A )
% 3.21/1.71 => ( singleton
% 3.21/1.71 @ ( dsetconstr @ B
% 3.21/1.71 @ ^ [Xy: $i] :
% 3.21/1.71 ( ( Xphi @ Xx @ Xy )
% 3.21/1.71 & ! [Xz: $i] :
% 3.21/1.71 ( ( in @ Xz @ B )
% 3.21/1.71 => ( ( Xphi @ Xx @ Xz )
% 3.21/1.71 => ( in @ ( kpair @ Xy @ Xz ) @ R ) ) ) ) ) ) ) ) ) ) ) ) ).
% 3.21/1.71
% 3.21/1.71 thf(setOfPairsIsBReln1_type,type,
% 3.21/1.71 setOfPairsIsBReln1: $o ).
% 3.21/1.71
% 3.21/1.71 thf(setOfPairsIsBReln1,definition,
% 3.21/1.71 ( setOfPairsIsBReln1
% 3.21/1.71 = ( ! [A: $i,Xphi: $i > $i > $o] :
% 3.21/1.71 ( breln1 @ A
% 3.21/1.71 @ ( dpsetconstr @ A @ A
% 3.21/1.71 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 3.21/1.71
% 3.21/1.71 thf(breln1all1_type,type,
% 3.21/1.71 breln1all1: $o ).
% 3.21/1.71
% 3.21/1.71 thf(breln1all1,definition,
% 3.21/1.71 ( breln1all1
% 3.21/1.71 = ( ! [A: $i,R: $i] :
% 3.21/1.71 ( ( breln1 @ A @ R )
% 3.21/1.71 => ! [Xphi: $i > $o] :
% 3.21/1.71 ( ! [Xx: $i] :
% 3.21/1.71 ( ( in @ Xx @ A )
% 3.21/1.71 => ! [Xy: $i] :
% 3.21/1.71 ( ( in @ Xy @ A )
% 3.21/1.71 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.21/1.71 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 3.21/1.71 => ! [Xx: $i] :
% 3.21/1.71 ( ( in @ Xx @ R )
% 3.21/1.71 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.21/1.71
% 3.21/1.71 thf(subbreln1_type,type,
% 3.21/1.71 subbreln1: $o ).
% 3.21/1.71
% 3.21/1.71 thf(subbreln1,definition,
% 3.21/1.71 ( subbreln1
% 3.21/1.71 = ( ! [A: $i,R: $i] :
% 3.21/1.71 ( ( breln1 @ A @ R )
% 3.21/1.71 => ! [S: $i] :
% 3.21/1.71 ( ( breln1 @ A @ S )
% 3.21/1.71 => ( ! [Xx: $i] :
% 3.21/1.71 ( ( in @ Xx @ A )
% 3.21/1.71 => ! [Xy: $i] :
% 3.21/1.71 ( ( in @ Xy @ A )
% 3.21/1.71 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.21/1.71 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 3.21/1.71 => ( subset @ R @ S ) ) ) ) ) ) ).
% 3.21/1.71
% 3.21/1.71 thf(eqbreln1_type,type,
% 3.21/1.71 eqbreln1: $o ).
% 3.21/1.71
% 3.21/1.71 thf(eqbreln1,definition,
% 3.21/1.71 ( eqbreln1
% 3.21/1.71 = ( ! [A: $i,R: $i] :
% 3.21/1.71 ( ( breln1 @ A @ R )
% 3.21/1.71 => ! [S: $i] :
% 3.21/1.71 ( ( breln1 @ A @ S )
% 3.21/1.71 => ( ! [Xx: $i] :
% 3.21/1.71 ( ( in @ Xx @ A )
% 3.21/1.71 => ! [Xy: $i] :
% 3.21/1.71 ( ( in @ Xy @ A )
% 3.21/1.71 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.21/1.71 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 3.21/1.71 => ( ! [Xx: $i] :
% 3.21/1.71 ( ( in @ Xx @ A )
% 3.21/1.71 => ! [Xy: $i] :
% 3.21/1.71 ( ( in @ Xy @ A )
% 3.21/1.71 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 3.21/1.71 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 3.21/1.71 => ( R = S ) ) ) ) ) ) ) ).
% 3.21/1.71
% 3.21/1.71 thf(breln1invset_type,type,
% 3.21/1.71 breln1invset: $i > $i > $i ).
% 3.21/1.71
% 3.21/1.71 thf(breln1invprop_type,type,
% 3.21/1.71 breln1invprop: $o ).
% 3.21/1.71
% 3.21/1.71 thf(breln1invprop,definition,
% 3.21/1.71 ( breln1invprop
% 3.21/1.71 = ( ! [A: $i,R: $i] :
% 3.21/1.71 ( ( breln1 @ A @ R )
% 3.21/1.71 => ( breln1 @ A @ ( breln1invset @ A @ R ) ) ) ) ) ).
% 3.21/1.71
% 3.21/1.71 thf(breln1invI_type,type,
% 3.21/1.71 breln1invI: $o ).
% 3.21/1.71
% 3.21/1.71 thf(breln1invI,definition,
% 3.21/1.71 ( breln1invI
% 3.21/1.71 = ( ! [A: $i,R: $i] :
% 3.21/1.71 ( ( breln1 @ A @ R )
% 3.21/1.71 => ! [Xx: $i] :
% 3.21/1.71 ( ( in @ Xx @ A )
% 3.21/1.71 => ! [Xy: $i] :
% 3.21/1.71 ( ( in @ Xy @ A )
% 3.21/1.71 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.21/1.71 => ( in @ ( kpair @ Xy @ Xx ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ) ).
% 3.21/1.71
% 3.21/1.71 thf(breln1invE_type,type,
% 3.21/1.71 breln1invE: $o ).
% 3.21/1.71
% 3.21/1.71 thf(breln1invE,definition,
% 3.21/1.71 ( breln1invE
% 3.21/1.71 = ( ! [A: $i,R: $i] :
% 3.21/1.71 ( ( breln1 @ A @ R )
% 3.21/1.71 => ! [Xx: $i] :
% 3.21/1.71 ( ( in @ Xx @ A )
% 3.21/1.71 => ! [Xy: $i] :
% 3.21/1.71 ( ( in @ Xy @ A )
% 3.21/1.71 => ( ( in @ ( kpair @ Xy @ Xx ) @ ( breln1invset @ A @ R ) )
% 3.21/1.71 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) ) ) ) ) ).
% 3.21/1.71
% 3.21/1.71 thf(breln1compset_type,type,
% 3.21/1.71 breln1compset: $i > $i > $i > $i ).
% 3.21/1.71
% 3.21/1.71 thf(breln1compprop_type,type,
% 3.21/1.71 breln1compprop: $o ).
% 3.21/1.71
% 3.21/1.71 thf(breln1compprop,definition,
% 3.21/1.71 ( breln1compprop
% 3.21/1.71 = ( ! [A: $i,R: $i] :
% 3.21/1.71 ( ( breln1 @ A @ R )
% 3.21/1.71 => ! [S: $i] :
% 3.21/1.71 ( ( breln1 @ A @ S )
% 3.21/1.71 => ( breln1 @ A @ ( breln1compset @ A @ R @ S ) ) ) ) ) ) ).
% 3.21/1.71
% 3.21/1.71 thf(breln1compI_type,type,
% 3.21/1.71 breln1compI: $o ).
% 3.21/1.71
% 3.21/1.71 thf(breln1compI,definition,
% 3.21/1.71 ( breln1compI
% 3.21/1.71 = ( ! [A: $i,R: $i] :
% 3.21/1.71 ( ( breln1 @ A @ R )
% 3.21/1.71 => ! [S: $i] :
% 3.21/1.71 ( ( breln1 @ A @ S )
% 3.21/1.71 => ! [Xx: $i] :
% 3.21/1.71 ( ( in @ Xx @ A )
% 3.21/1.71 => ! [Xy: $i] :
% 3.21/1.71 ( ( in @ Xy @ A )
% 3.21/1.71 => ! [Xz: $i] :
% 3.21/1.71 ( ( in @ Xz @ A )
% 3.21/1.71 => ( ( in @ ( kpair @ Xx @ Xz ) @ R )
% 3.21/1.71 => ( ( in @ ( kpair @ Xz @ Xy ) @ S )
% 3.21/1.71 => ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) ) ) ) ) ) ) ) ) ) ) ).
% 3.21/1.71
% 3.21/1.71 thf(breln1compE_type,type,
% 3.21/1.71 breln1compE: $o ).
% 3.21/1.71
% 3.21/1.71 thf(breln1compE,definition,
% 3.21/1.71 ( breln1compE
% 3.21/1.71 = ( ! [A: $i,R: $i] :
% 3.21/1.71 ( ( breln1 @ A @ R )
% 3.21/1.71 => ! [S: $i] :
% 3.21/1.71 ( ( breln1 @ A @ S )
% 3.21/1.71 => ! [Xx: $i] :
% 3.21/1.71 ( ( in @ Xx @ A )
% 3.21/1.71 => ! [Xy: $i] :
% 3.21/1.71 ( ( in @ Xy @ A )
% 3.21/1.71 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) )
% 3.21/1.71 => ? [Xz: $i] :
% 3.21/1.71 ( ( in @ Xz @ A )
% 3.21/1.71 & ( in @ ( kpair @ Xx @ Xz ) @ R )
% 3.21/1.71 & ( in @ ( kpair @ Xz @ Xy ) @ S ) ) ) ) ) ) ) ) ) ).
% 3.21/1.71
% 3.21/1.71 thf(breln1compEex_type,type,
% 3.21/1.71 breln1compEex: $o ).
% 3.21/1.71
% 3.21/1.71 thf(breln1compEex,definition,
% 3.21/1.71 ( breln1compEex
% 3.21/1.71 = ( ! [A: $i,R: $i] :
% 3.21/1.71 ( ( breln1 @ A @ R )
% 3.21/1.71 => ! [S: $i] :
% 3.21/1.71 ( ( breln1 @ A @ S )
% 3.21/1.71 => ! [Xx: $i] :
% 3.21/1.71 ( ( in @ Xx @ A )
% 3.21/1.71 => ! [Xy: $i] :
% 3.21/1.71 ( ( in @ Xy @ A )
% 3.21/1.71 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) )
% 3.21/1.71 => ! [Xphi: $o] :
% 3.21/1.71 ( ! [Xz: $i] :
% 3.21/1.71 ( ( in @ Xz @ A )
% 3.21/1.71 => ( ( in @ ( kpair @ Xx @ Xz ) @ R )
% 3.21/1.71 => ( ( in @ ( kpair @ Xz @ Xy ) @ S )
% 3.21/1.71 => Xphi ) ) )
% 3.21/1.71 => Xphi ) ) ) ) ) ) ) ) ).
% 3.21/1.71
% 3.21/1.71 thf(breln1unionprop_type,type,
% 3.21/1.71 breln1unionprop: $o ).
% 3.21/1.71
% 3.21/1.71 thf(breln1unionprop,definition,
% 3.21/1.71 ( breln1unionprop
% 3.21/1.71 = ( ! [A: $i,R: $i] :
% 3.21/1.71 ( ( breln1 @ A @ R )
% 3.21/1.71 => ! [S: $i] :
% 3.21/1.71 ( ( breln1 @ A @ S )
% 3.21/1.71 => ( breln1 @ A @ ( binunion @ R @ S ) ) ) ) ) ) ).
% 3.21/1.71
% 3.21/1.71 thf(breln1unionIL_type,type,
% 3.21/1.71 breln1unionIL: $o ).
% 3.21/1.71
% 3.21/1.71 thf(breln1unionIL,definition,
% 3.21/1.71 ( breln1unionIL
% 3.21/1.71 = ( ! [A: $i,R: $i] :
% 3.21/1.71 ( ( breln1 @ A @ R )
% 3.21/1.71 => ! [S: $i] :
% 3.21/1.71 ( ( breln1 @ A @ S )
% 3.21/1.71 => ! [Xx: $i] :
% 3.21/1.71 ( ( in @ Xx @ A )
% 3.21/1.71 => ! [Xy: $i] :
% 3.21/1.71 ( ( in @ Xy @ A )
% 3.21/1.71 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.21/1.71 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 3.21/1.71
% 3.21/1.71 thf(breln1unionIR_type,type,
% 3.21/1.71 breln1unionIR: $o ).
% 3.21/1.71
% 3.21/1.71 thf(breln1unionIR,definition,
% 3.21/1.71 ( breln1unionIR
% 3.21/1.71 = ( ! [A: $i,R: $i] :
% 3.21/1.71 ( ( breln1 @ A @ R )
% 3.21/1.71 => ! [S: $i] :
% 3.21/1.71 ( ( breln1 @ A @ S )
% 3.21/1.71 => ! [Xx: $i] :
% 3.21/1.71 ( ( in @ Xx @ A )
% 3.21/1.71 => ! [Xy: $i] :
% 3.21/1.71 ( ( in @ Xy @ A )
% 3.21/1.71 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 3.21/1.71 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 3.21/1.71
% 3.21/1.71 thf(breln1unionI,conjecture,
% 3.21/1.71 ( setextAx
% 3.21/1.71 => ( emptysetAx
% 3.21/1.71 => ( setadjoinAx
% 3.21/1.71 => ( powersetAx
% 3.21/1.71 => ( setunionAx
% 3.21/1.71 => ( omega0Ax
% 3.21/1.71 => ( omegaSAx
% 3.21/1.71 => ( omegaIndAx
% 3.21/1.71 => ( replAx
% 3.21/1.71 => ( foundationAx
% 3.21/1.71 => ( wellorderingAx
% 3.21/1.71 => ( descrp
% 3.21/1.71 => ( dsetconstrI
% 3.21/1.71 => ( dsetconstrEL
% 3.21/1.71 => ( dsetconstrER
% 3.21/1.71 => ( exuE1
% 3.21/1.71 => ( prop2setE
% 3.21/1.71 => ( emptysetE
% 3.21/1.71 => ( emptysetimpfalse
% 3.21/1.71 => ( notinemptyset
% 3.21/1.71 => ( exuE3e
% 3.21/1.71 => ( setext
% 3.21/1.71 => ( emptyI
% 3.21/1.71 => ( noeltsimpempty
% 3.21/1.71 => ( setbeta
% 3.21/1.71 => ( nonemptyE1
% 3.21/1.71 => ( nonemptyI
% 3.21/1.71 => ( nonemptyI1
% 3.21/1.71 => ( setadjoinIL
% 3.21/1.71 => ( emptyinunitempty
% 3.21/1.71 => ( setadjoinIR
% 3.21/1.71 => ( setadjoinE
% 3.21/1.71 => ( setadjoinOr
% 3.21/1.71 => ( setoftrueEq
% 3.21/1.71 => ( powersetI
% 3.21/1.71 => ( emptyinPowerset
% 3.21/1.71 => ( emptyInPowerset
% 3.21/1.71 => ( powersetE
% 3.21/1.71 => ( setunionI
% 3.21/1.71 => ( setunionE
% 3.21/1.71 => ( subPowSU
% 3.21/1.71 => ( exuE2
% 3.21/1.71 => ( nonemptyImpWitness
% 3.21/1.71 => ( uniqinunit
% 3.21/1.71 => ( notinsingleton
% 3.21/1.71 => ( eqinunit
% 3.21/1.71 => ( singletonsswitch
% 3.21/1.71 => ( upairsetE
% 3.21/1.71 => ( upairsetIL
% 3.21/1.71 => ( upairsetIR
% 3.21/1.71 => ( emptyE1
% 3.21/1.71 => ( vacuousDall
% 3.21/1.71 => ( quantDeMorgan1
% 3.21/1.71 => ( quantDeMorgan2
% 3.21/1.71 => ( quantDeMorgan3
% 3.21/1.71 => ( quantDeMorgan4
% 3.21/1.71 => ( prop2setI
% 3.21/1.71 => ( prop2set2propI
% 3.21/1.71 => ( notdexE
% 3.21/1.71 => ( notdallE
% 3.21/1.71 => ( exuI1
% 3.21/1.71 => ( exuI3
% 3.21/1.71 => ( exuI2
% 3.21/1.71 => ( inCongP
% 3.21/1.71 => ( in__Cong
% 3.21/1.71 => ( exuE3u
% 3.21/1.71 => ( exu__Cong
% 3.21/1.71 => ( emptyset__Cong
% 3.21/1.71 => ( setadjoin__Cong
% 3.21/1.71 => ( powerset__Cong
% 3.21/1.71 => ( setunion__Cong
% 3.21/1.71 => ( omega__Cong
% 3.21/1.71 => ( exuEu
% 3.21/1.71 => ( descr__Cong
% 3.21/1.71 => ( dsetconstr__Cong
% 3.21/1.71 => ( subsetI1
% 3.21/1.71 => ( eqimpsubset2
% 3.21/1.71 => ( eqimpsubset1
% 3.21/1.71 => ( subsetI2
% 3.21/1.71 => ( emptysetsubset
% 3.21/1.71 => ( subsetE
% 3.21/1.71 => ( subsetE2
% 3.21/1.71 => ( notsubsetI
% 3.21/1.71 => ( notequalI1
% 3.21/1.71 => ( notequalI2
% 3.21/1.71 => ( subsetRefl
% 3.21/1.71 => ( subsetTrans
% 3.21/1.71 => ( setadjoinSub
% 3.21/1.71 => ( setadjoinSub2
% 3.21/1.71 => ( subset2powerset
% 3.21/1.71 => ( setextsub
% 3.21/1.71 => ( subsetemptysetimpeq
% 3.21/1.71 => ( powersetI1
% 3.21/1.71 => ( powersetE1
% 3.21/1.71 => ( inPowerset
% 3.21/1.71 => ( powersetsubset
% 3.21/1.71 => ( sepInPowerset
% 3.21/1.71 => ( sepSubset
% 3.21/1.71 => ( binunionIL
% 3.21/1.71 => ( upairset2IR
% 3.21/1.71 => ( binunionIR
% 3.21/1.71 => ( binunionEcases
% 3.21/1.71 => ( binunionE
% 3.21/1.71 => ( binunionLsub
% 3.21/1.71 => ( binunionRsub
% 3.21/1.71 => ( binintersectI
% 3.21/1.71 => ( binintersectSubset5
% 3.21/1.71 => ( binintersectEL
% 3.21/1.71 => ( binintersectLsub
% 3.21/1.71 => ( binintersectSubset2
% 3.21/1.71 => ( binintersectSubset3
% 3.21/1.71 => ( binintersectER
% 3.21/1.71 => ( disjointsetsI1
% 3.21/1.71 => ( binintersectRsub
% 3.21/1.71 => ( binintersectSubset4
% 3.21/1.71 => ( binintersectSubset1
% 3.21/1.71 => ( bs114d
% 3.21/1.71 => ( setminusI
% 3.21/1.71 => ( setminusEL
% 3.21/1.71 => ( setminusER
% 3.21/1.71 => ( setminusSubset2
% 3.21/1.71 => ( setminusERneg
% 3.21/1.71 => ( setminusELneg
% 3.21/1.71 => ( setminusILneg
% 3.21/1.71 => ( setminusIRneg
% 3.21/1.71 => ( setminusLsub
% 3.21/1.71 => ( setminusSubset1
% 3.21/1.71 => ( symdiffE
% 3.21/1.71 => ( symdiffI1
% 3.21/1.71 => ( symdiffI2
% 3.21/1.71 => ( symdiffIneg1
% 3.21/1.71 => ( symdiffIneg2
% 3.21/1.71 => ( secondinupair
% 3.21/1.71 => ( setukpairIL
% 3.21/1.71 => ( setukpairIR
% 3.21/1.71 => ( kpairiskpair
% 3.21/1.71 => ( kpairp
% 3.21/1.71 => ( singletonsubset
% 3.21/1.71 => ( singletoninpowerset
% 3.21/1.71 => ( singletoninpowunion
% 3.21/1.71 => ( upairset2E
% 3.21/1.71 => ( upairsubunion
% 3.21/1.71 => ( upairinpowunion
% 3.21/1.71 => ( ubforcartprodlem1
% 3.21/1.71 => ( ubforcartprodlem2
% 3.21/1.71 => ( ubforcartprodlem3
% 3.21/1.71 => ( cartprodpairin
% 3.21/1.71 => ( cartprodmempair1
% 3.21/1.71 => ( cartprodmempair
% 3.21/1.71 => ( setunionE2
% 3.21/1.71 => ( setunionsingleton1
% 3.21/1.71 => ( setunionsingleton2
% 3.21/1.71 => ( setunionsingleton
% 3.21/1.71 => ( singletonprop
% 3.21/1.71 => ( ex1E1
% 3.21/1.71 => ( ex1I
% 3.21/1.71 => ( ex1I2
% 3.21/1.71 => ( singletonsuniq
% 3.21/1.71 => ( setukpairinjL1
% 3.21/1.71 => ( kfstsingleton
% 3.21/1.71 => ( theprop
% 3.21/1.71 => ( kfstpairEq
% 3.21/1.71 => ( cartprodfstin
% 3.21/1.71 => ( setukpairinjL2
% 3.21/1.71 => ( setukpairinjL
% 3.21/1.71 => ( setukpairinjR11
% 3.21/1.71 => ( setukpairinjR12
% 3.21/1.71 => ( setukpairinjR1
% 3.21/1.71 => ( upairequniteq
% 3.21/1.71 => ( setukpairinjR2
% 3.21/1.71 => ( setukpairinjR
% 3.21/1.71 => ( ksndsingleton
% 3.21/1.71 => ( ksndpairEq
% 3.21/1.71 => ( kpairsurjEq
% 3.21/1.71 => ( cartprodsndin
% 3.21/1.71 => ( cartprodpairmemEL
% 3.21/1.71 => ( cartprodpairmemER
% 3.21/1.71 => ( cartprodmempaircEq
% 3.21/1.71 => ( cartprodfstpairEq
% 3.21/1.71 => ( cartprodsndpairEq
% 3.21/1.71 => ( cartprodpairsurjEq
% 3.21/1.71 => ( dpsetconstrI
% 3.21/1.71 => ( dpsetconstrSub
% 3.21/1.71 => ( setOfPairsIsBReln
% 3.21/1.71 => ( dpsetconstrERa
% 3.21/1.71 => ( dpsetconstrEL1
% 3.21/1.71 => ( dpsetconstrEL2
% 3.21/1.71 => ( dpsetconstrER
% 3.21/1.71 => ( funcImageSingleton
% 3.21/1.71 => ( apProp
% 3.21/1.71 => ( app
% 3.21/1.71 => ( infuncsetfunc
% 3.21/1.71 => ( ap2p
% 3.21/1.71 => ( funcinfuncset
% 3.21/1.71 => ( lamProp
% 3.21/1.71 => ( lamp
% 3.21/1.71 => ( lam2p
% 3.21/1.71 => ( brelnall1
% 3.21/1.71 => ( brelnall2
% 3.21/1.71 => ( ex1E2
% 3.21/1.71 => ( funcGraphProp1
% 3.21/1.71 => ( funcGraphProp3
% 3.21/1.71 => ( funcGraphProp2
% 3.21/1.71 => ( funcextLem
% 3.21/1.71 => ( funcGraphProp4
% 3.21/1.71 => ( subbreln
% 3.21/1.71 => ( eqbreln
% 3.21/1.71 => ( funcext
% 3.21/1.71 => ( funcext2
% 3.21/1.71 => ( ap2apEq1
% 3.21/1.71 => ( ap2apEq2
% 3.21/1.71 => ( beta1
% 3.21/1.71 => ( eta1
% 3.21/1.71 => ( lam2lamEq
% 3.21/1.71 => ( beta2
% 3.21/1.71 => ( eta2
% 3.21/1.71 => ( iffalseProp1
% 3.21/1.71 => ( iffalseProp2
% 3.21/1.71 => ( iftrueProp1
% 3.21/1.71 => ( iftrueProp2
% 3.21/1.71 => ( ifSingleton
% 3.21/1.71 => ( ifp
% 3.21/1.71 => ( theeq
% 3.21/1.71 => ( iftrue
% 3.21/1.71 => ( iffalse
% 3.21/1.71 => ( iftrueorfalse
% 3.21/1.71 => ( binintersectT_lem
% 3.21/1.71 => ( binunionT_lem
% 3.21/1.71 => ( powersetT_lem
% 3.21/1.71 => ( setminusT_lem
% 3.21/1.71 => ( complementT_lem
% 3.21/1.71 => ( setextT
% 3.21/1.71 => ( subsetTI
% 3.21/1.71 => ( powersetTI1
% 3.21/1.71 => ( powersetTE1
% 3.21/1.71 => ( complementTI1
% 3.21/1.71 => ( complementTE1
% 3.21/1.71 => ( binintersectTELcontra
% 3.21/1.71 => ( binintersectTERcontra
% 3.21/1.71 => ( contrasubsetT
% 3.21/1.71 => ( contrasubsetT1
% 3.21/1.71 => ( contrasubsetT2
% 3.21/1.71 => ( contrasubsetT3
% 3.21/1.71 => ( doubleComplementI1
% 3.21/1.71 => ( doubleComplementE1
% 3.21/1.71 => ( doubleComplementSub1
% 3.21/1.71 => ( doubleComplementSub2
% 3.21/1.71 => ( doubleComplementEq
% 3.21/1.71 => ( complementTnotintersectT
% 3.21/1.71 => ( complementImpComplementIntersect
% 3.21/1.71 => ( complementSubsetComplementIntersect
% 3.21/1.71 => ( complementInPowersetComplementIntersect
% 3.21/1.71 => ( contraSubsetComplement
% 3.21/1.71 => ( complementTcontraSubset
% 3.21/1.71 => ( binunionTILcontra
% 3.21/1.71 => ( binunionTIRcontra
% 3.21/1.71 => ( inIntersectImpInUnion
% 3.21/1.71 => ( inIntersectImpInUnion2
% 3.21/1.71 => ( inIntersectImpInIntersectUnions
% 3.21/1.71 => ( intersectInPowersetIntersectUnions
% 3.21/1.71 => ( inComplementUnionImpNotIn1
% 3.21/1.71 => ( inComplementUnionImpInComplement1
% 3.21/1.71 => ( binunionTE
% 3.21/1.71 => ( binunionTEcontra
% 3.21/1.71 => ( demorgan2a1
% 3.21/1.71 => ( complementUnionInPowersetComplement
% 3.21/1.71 => ( demorgan2a2
% 3.21/1.71 => ( demorgan1a
% 3.21/1.71 => ( demorgan1b
% 3.21/1.71 => ( demorgan1
% 3.21/1.71 => ( demorgan2a
% 3.21/1.71 => ( demorgan2b2
% 3.21/1.71 => ( demorgan2b
% 3.21/1.71 => ( demorgan2
% 3.21/1.71 => ( woz13rule0
% 3.21/1.71 => ( woz13rule1
% 3.21/1.71 => ( woz13rule2
% 3.21/1.71 => ( woz13rule3
% 3.21/1.71 => ( woz13rule4
% 3.21/1.71 => ( woz1_1
% 3.21/1.71 => ( woz1_2
% 3.21/1.71 => ( woz1_3
% 3.21/1.71 => ( woz1_4
% 3.21/1.71 => ( woz1_5
% 3.21/1.71 => ( breln1all2
% 3.21/1.71 => ( breln1SetBreln1
% 3.21/1.71 => ( choice2fnsingleton
% 3.21/1.71 => ( setOfPairsIsBReln1
% 3.21/1.71 => ( breln1all1
% 3.21/1.71 => ( subbreln1
% 3.21/1.71 => ( eqbreln1
% 3.21/1.71 => ( breln1invprop
% 3.21/1.71 => ( breln1invI
% 3.21/1.71 => ( breln1invE
% 3.21/1.71 => ( breln1compprop
% 3.21/1.71 => ( breln1compI
% 3.21/1.71 => ( breln1compE
% 3.21/1.71 => ( breln1compEex
% 3.21/1.71 => ( breln1unionprop
% 3.21/1.71 => ( breln1unionIL
% 3.21/1.71 => ( breln1unionIR
% 3.21/1.71 => ! [A: $i,R: $i] :
% 3.21/1.71 ( ( breln1 @ A @ R )
% 3.21/1.71 => ! [S: $i] :
% 3.21/1.71 ( ( breln1 @ A @ S )
% 3.21/1.71 => ! [Xx: $i] :
% 3.21/1.71 ( ( in @ Xx @ A )
% 3.21/1.71 => ! [Xy: $i] :
% 3.21/1.71 ( ( in @ Xy @ A )
% 3.21/1.71 => ( ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.21/1.71 | ( in @ ( kpair @ Xx @ Xy ) @ S ) )
% 3.21/1.71 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 3.21/1.71 ------------------------
% 3.21/1.75 % (24600)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_24330 for (2999ds/27Mi)
% 3.21/1.75 % (24599)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_24330 for (2999ds/4Mi)
% 3.21/1.75 % (24598)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_24330 for (2999ds/183Mi)
% 3.21/1.75 % (24599)Instruction limit reached!
% 3.21/1.75 % (24599)------------------------------
% 3.21/1.75 % (24599)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.21/1.75 % (24599)Termination reason: Unknown
% 3.21/1.75 % (24599)Termination phase: shuffling
% 3.21/1.75
% 3.21/1.75 % (24599)Memory used [KB]: 1663
% 3.21/1.75 % (24599)Time elapsed: 0.003 s
% 3.21/1.75 % (24599)Instructions burned: 6 (million)
% 3.21/1.75 % (24599)------------------------------
% 3.21/1.75 % (24599)------------------------------
% 3.21/1.76 % (24602)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_24330 for (2999ds/2Mi)
% 3.21/1.76 % (24600)Instruction limit reached!
% 3.21/1.76 % (24600)------------------------------
% 3.21/1.76 % (24600)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.21/1.76 % (24600)Termination reason: Unknown
% 3.21/1.76 % (24600)Termination phase: shuffling
% 3.21/1.76
% 3.21/1.76 % (24600)Memory used [KB]: 2046
% 3.21/1.76 % (24600)Time elapsed: 0.009 s
% 3.21/1.76 % (24600)Instructions burned: 28 (million)
% 3.21/1.76 % (24600)------------------------------
% 3.21/1.76 % (24600)------------------------------
% 3.21/1.76 % (24603)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_24330 for (2999ds/2Mi)
% 3.21/1.76 % (24604)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_24330 for (2999ds/275Mi)
% 3.21/1.76 % (24605)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_24330 for (2999ds/18Mi)
% 3.21/1.76 % (24602)Instruction limit reached!
% 3.21/1.76 % (24602)------------------------------
% 3.21/1.76 % (24602)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.21/1.76 % (24602)Termination reason: Unknown
% 3.21/1.76 % (24602)Termination phase: shuffling
% 3.21/1.76
% 3.21/1.76 % (24602)Memory used [KB]: 1535
% 3.21/1.76 % (24602)Time elapsed: 0.003 s
% 3.21/1.76 % (24602)Instructions burned: 2 (million)
% 3.21/1.76 % (24602)------------------------------
% 3.21/1.76 % (24602)------------------------------
% 3.21/1.76 % (24603)Instruction limit reached!
% 3.21/1.76 % (24603)------------------------------
% 3.21/1.76 % (24603)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.21/1.76 % (24603)Termination reason: Unknown
% 3.21/1.76 % (24603)Termination phase: shuffling
% 3.21/1.76
% 3.21/1.76 % (24603)Memory used [KB]: 1535
% 3.21/1.76 % (24603)Time elapsed: 0.003 s
% 3.21/1.76 % (24603)Instructions burned: 2 (million)
% 3.21/1.76 % (24603)------------------------------
% 3.21/1.76 % (24603)------------------------------
% 3.21/1.76 % (24608)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_24330 for (2999ds/3Mi)
% 3.21/1.76 % (24608)Instruction limit reached!
% 3.21/1.76 % (24608)------------------------------
% 3.21/1.76 % (24608)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.21/1.76 % (24608)Termination reason: Unknown
% 3.21/1.76 % (24608)Termination phase: shuffling
% 3.21/1.76
% 3.21/1.76 % (24608)Memory used [KB]: 1535
% 3.21/1.76 % (24608)Time elapsed: 0.003 s
% 3.21/1.76 % (24608)Instructions burned: 6 (million)
% 3.21/1.76 % (24608)------------------------------
% 3.21/1.76 % (24608)------------------------------
% 3.21/1.77 % (24612)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_24330 for (2999ds/37Mi)
% 3.21/1.77 % (24605)Instruction limit reached!
% 3.21/1.77 % (24605)------------------------------
% 3.21/1.77 % (24605)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.21/1.77 % (24605)Termination reason: Unknown
% 3.21/1.77 % (24605)Termination phase: shuffling
% 3.21/1.77
% 3.21/1.77 % (24605)Memory used [KB]: 1791
% 3.21/1.77 % (24605)Time elapsed: 0.013 s
% 3.21/1.77 % (24605)Instructions burned: 19 (million)
% 3.21/1.77 % (24605)------------------------------
% 3.21/1.77 % (24605)------------------------------
% 3.21/1.77 % (24618)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_24330 for (2999ds/1041Mi)
% 3.21/1.78 % (24613)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_24330 for (2999ds/15Mi)
% 3.21/1.78 % (24614)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_24330 for (2999ds/3Mi)
% 3.21/1.78 % (24612)Instruction limit reached!
% 3.21/1.78 % (24612)------------------------------
% 3.21/1.78 % (24612)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.21/1.78 % (24612)Termination reason: Unknown
% 3.21/1.78 % (24612)Termination phase: shuffling
% 3.21/1.78
% 3.21/1.78 % (24612)Memory used [KB]: 2302
% 3.21/1.78 % (24612)Time elapsed: 0.011 s
% 3.21/1.78 % (24612)Instructions burned: 37 (million)
% 3.21/1.78 % (24612)------------------------------
% 3.21/1.78 % (24612)------------------------------
% 3.21/1.78 % (24614)Instruction limit reached!
% 3.21/1.78 % (24614)------------------------------
% 3.21/1.78 % (24614)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.21/1.78 % (24614)Termination reason: Unknown
% 3.21/1.78 % (24614)Termination phase: shuffling
% 3.21/1.78
% 3.21/1.78 % (24614)Memory used [KB]: 1535
% 3.21/1.78 % (24614)Time elapsed: 0.004 s
% 3.21/1.78 % (24614)Instructions burned: 3 (million)
% 3.21/1.78 % (24614)------------------------------
% 3.21/1.78 % (24614)------------------------------
% 3.21/1.79 % (24621)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_24330 for (2999ds/7Mi)
% 3.21/1.79 % (24613)Instruction limit reached!
% 3.21/1.79 % (24613)------------------------------
% 3.21/1.79 % (24613)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.21/1.79 % (24613)Termination reason: Unknown
% 3.21/1.79 % (24613)Termination phase: shuffling
% 3.21/1.79
% 3.21/1.79 % (24613)Memory used [KB]: 1791
% 3.21/1.79 % (24613)Time elapsed: 0.012 s
% 3.21/1.79 % (24613)Instructions burned: 16 (million)
% 3.21/1.79 % (24613)------------------------------
% 3.21/1.79 % (24613)------------------------------
% 3.21/1.79 % (24622)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_24330 for (2999ds/16Mi)
% 3.21/1.79 % (24621)Instruction limit reached!
% 3.21/1.79 % (24621)------------------------------
% 3.21/1.79 % (24621)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.21/1.79 % (24621)Termination reason: Unknown
% 3.21/1.79 % (24621)Termination phase: shuffling
% 3.21/1.79
% 3.21/1.79 % (24621)Memory used [KB]: 1663
% 3.21/1.79 % (24621)Time elapsed: 0.004 s
% 3.21/1.79 % (24621)Instructions burned: 9 (million)
% 3.21/1.79 % (24621)------------------------------
% 3.21/1.79 % (24621)------------------------------
% 3.21/1.79 % (24622)Instruction limit reached!
% 3.21/1.79 % (24622)------------------------------
% 3.21/1.79 % (24622)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.21/1.79 % (24622)Termination reason: Unknown
% 3.21/1.79 % (24622)Termination phase: shuffling
% 3.21/1.79
% 3.21/1.79 % (24622)Memory used [KB]: 1918
% 3.21/1.79 % (24622)Time elapsed: 0.006 s
% 3.21/1.79 % (24622)Instructions burned: 17 (million)
% 3.21/1.79 % (24622)------------------------------
% 3.21/1.79 % (24622)------------------------------
% 3.21/1.80 % (24623)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_24330 for (2999ds/3Mi)
% 3.21/1.80 % (24630)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_24330 for (2999ds/7Mi)
% 3.21/1.80 % (24623)Instruction limit reached!
% 3.21/1.80 % (24623)------------------------------
% 3.21/1.80 % (24623)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.21/1.80 % (24623)Termination reason: Unknown
% 3.21/1.80 % (24623)Termination phase: shuffling
% 3.21/1.80
% 3.21/1.80 % (24623)Memory used [KB]: 1535
% 3.21/1.80 % (24623)Time elapsed: 0.004 s
% 3.21/1.80 % (24623)Instructions burned: 4 (million)
% 3.21/1.80 % (24623)------------------------------
% 3.21/1.80 % (24623)------------------------------
% 3.21/1.80 % (24630)Instruction limit reached!
% 3.21/1.80 % (24630)------------------------------
% 3.21/1.80 % (24630)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.21/1.80 % (24630)Termination reason: Unknown
% 3.21/1.80 % (24630)Termination phase: shuffling
% 3.21/1.80
% 3.21/1.80 % (24630)Memory used [KB]: 1663
% 3.21/1.80 % (24630)Time elapsed: 0.004 s
% 3.21/1.80 % (24630)Instructions burned: 9 (million)
% 3.21/1.80 % (24630)------------------------------
% 3.21/1.80 % (24630)------------------------------
% 3.21/1.80 % (24598)Instruction limit reached!
% 3.21/1.80 % (24598)------------------------------
% 3.21/1.80 % (24598)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.21/1.80 % (24598)Termination reason: Unknown
% 3.21/1.80 % (24598)Termination phase: Saturation
% 3.21/1.80
% 3.21/1.80 % (24598)Memory used [KB]: 7419
% 3.21/1.80 % (24598)Time elapsed: 0.052 s
% 3.21/1.80 % (24598)Instructions burned: 184 (million)
% 3.21/1.80 % (24598)------------------------------
% 3.21/1.80 % (24598)------------------------------
% 3.21/1.80 % (24636)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_24330 for (2999ds/3Mi)
% 3.21/1.80 % (24636)Instruction limit reached!
% 3.21/1.80 % (24636)------------------------------
% 3.21/1.80 % (24636)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.21/1.80 % (24636)Termination reason: Unknown
% 3.21/1.80 % (24636)Termination phase: shuffling
% 3.21/1.80
% 3.21/1.80 % (24636)Memory used [KB]: 1535
% 3.21/1.80 % (24636)Time elapsed: 0.003 s
% 3.21/1.80 % (24636)Instructions burned: 6 (million)
% 3.21/1.80 % (24636)------------------------------
% 3.21/1.80 % (24636)------------------------------
% 3.21/1.81 % (24629)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_24330 for (2999ds/3Mi)
% 3.21/1.81 % (24629)Instruction limit reached!
% 3.21/1.81 % (24629)------------------------------
% 3.21/1.81 % (24629)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.21/1.81 % (24629)Termination reason: Unknown
% 3.21/1.81 % (24629)Termination phase: shuffling
% 3.21/1.81
% 3.21/1.81 % (24629)Memory used [KB]: 1535
% 3.21/1.81 % (24629)Time elapsed: 0.003 s
% 3.21/1.81 % (24629)Instructions burned: 3 (million)
% 3.21/1.81 % (24629)------------------------------
% 3.21/1.81 % (24629)------------------------------
% 3.21/1.81 % (24642)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_24330 for (2999ds/18Mi)
% 3.21/1.81 % (24639)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_24330 for (2999ds/4Mi)
% 3.21/1.81 % (24643)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_24330 for (2999ds/710Mi)
% 3.21/1.81 % (24639)Instruction limit reached!
% 3.21/1.81 % (24639)------------------------------
% 3.21/1.81 % (24639)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.21/1.81 % (24639)Termination reason: Unknown
% 3.21/1.81 % (24639)Termination phase: shuffling
% 3.21/1.81
% 3.21/1.81 % (24639)Memory used [KB]: 1535
% 3.21/1.81 % (24639)Time elapsed: 0.003 s
% 3.21/1.81 % (24639)Instructions burned: 4 (million)
% 3.21/1.81 % (24639)------------------------------
% 3.21/1.81 % (24639)------------------------------
% 3.21/1.81 % (24645)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_24330 for (2999ds/6Mi)
% 3.21/1.82 % (24645)Instruction limit reached!
% 3.21/1.82 % (24645)------------------------------
% 3.21/1.82 % (24645)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.21/1.82 % (24645)Termination reason: Unknown
% 3.21/1.82 % (24645)Termination phase: shuffling
% 3.21/1.82
% 3.21/1.82 % (24645)Memory used [KB]: 1663
% 3.21/1.82 % (24645)Time elapsed: 0.003 s
% 3.21/1.82 % (24645)Instructions burned: 7 (million)
% 3.21/1.82 % (24645)------------------------------
% 3.21/1.82 % (24645)------------------------------
% 3.21/1.82 % (24642)Instruction limit reached!
% 3.21/1.82 % (24642)------------------------------
% 3.21/1.82 % (24642)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.21/1.82 % (24642)Termination reason: Unknown
% 3.21/1.82 % (24642)Termination phase: shuffling
% 3.21/1.82
% 3.21/1.82 % (24642)Memory used [KB]: 1791
% 3.21/1.82 % (24642)Time elapsed: 0.006 s
% 3.21/1.82 % (24642)Instructions burned: 18 (million)
% 3.21/1.82 % (24642)------------------------------
% 3.21/1.82 % (24642)------------------------------
% 3.21/1.82 % (24649)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_24330 for (2999ds/902Mi)
% 3.21/1.82 % (24652)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_24330 for (2999ds/21Mi)
% 3.21/1.82 % (24653)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_24330 for (2999ds/5Mi)
% 3.21/1.83 % (24654)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_24330 for (2999ds/6Mi)
% 3.21/1.83 % (24653)Instruction limit reached!
% 3.21/1.83 % (24653)------------------------------
% 3.21/1.83 % (24653)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.21/1.83 % (24653)Termination reason: Unknown
% 3.21/1.83 % (24653)Termination phase: shuffling
% 3.21/1.83
% 3.21/1.83 % (24653)Memory used [KB]: 1663
% 3.21/1.83 % (24653)Time elapsed: 0.003 s
% 3.21/1.83 % (24653)Instructions burned: 7 (million)
% 3.21/1.83 % (24653)------------------------------
% 3.21/1.83 % (24653)------------------------------
% 3.21/1.83 % (24654)Instruction limit reached!
% 3.21/1.83 % (24654)------------------------------
% 3.21/1.83 % (24654)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.21/1.83 % (24654)Termination reason: Unknown
% 3.21/1.83 % (24654)Termination phase: shuffling
% 3.21/1.83
% 3.21/1.83 % (24654)Memory used [KB]: 1663
% 3.21/1.83 % (24654)Time elapsed: 0.003 s
% 3.21/1.83 % (24654)Instructions burned: 7 (million)
% 3.21/1.83 % (24654)------------------------------
% 3.21/1.83 % (24654)------------------------------
% 3.21/1.83 % (24652)Instruction limit reached!
% 3.21/1.83 % (24652)------------------------------
% 3.21/1.83 % (24652)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.21/1.83 % (24652)Termination reason: Unknown
% 3.21/1.83 % (24652)Termination phase: shuffling
% 3.21/1.83
% 3.21/1.83 % (24652)Memory used [KB]: 1918
% 3.21/1.83 % (24652)Time elapsed: 0.008 s
% 3.21/1.83 % (24652)Instructions burned: 24 (million)
% 3.21/1.83 % (24652)------------------------------
% 3.21/1.83 % (24652)------------------------------
% 3.21/1.84 % (24655)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_24330 for (2998ds/377Mi)
% 3.21/1.84 % (24656)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_24330 for (2998ds/779Mi)
% 3.21/1.84 % (24660)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_24330 for (2998ds/19Mi)
% 3.21/1.85 % (24660)Instruction limit reached!
% 3.21/1.85 % (24660)------------------------------
% 3.21/1.85 % (24660)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.21/1.85 % (24660)Termination reason: Unknown
% 3.21/1.85 % (24660)Termination phase: shuffling
% 3.21/1.85
% 3.21/1.85 % (24660)Memory used [KB]: 1918
% 3.21/1.85 % (24660)Time elapsed: 0.007 s
% 3.21/1.85 % (24660)Instructions burned: 21 (million)
% 3.21/1.85 % (24660)------------------------------
% 3.21/1.85 % (24660)------------------------------
% 3.21/1.86 % (24674)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_24330 for (2998ds/879Mi)
% 3.21/1.89 % (24604)Instruction limit reached!
% 3.21/1.89 % (24604)------------------------------
% 3.21/1.89 % (24604)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.21/1.89 % (24604)Termination reason: Unknown
% 3.21/1.89 % (24604)Termination phase: Saturation
% 3.21/1.89
% 3.21/1.89 % (24604)Memory used [KB]: 10362
% 3.21/1.89 % (24604)Time elapsed: 0.128 s
% 3.21/1.89 % (24604)Instructions burned: 276 (million)
% 3.21/1.89 % (24604)------------------------------
% 3.21/1.89 % (24604)------------------------------
% 3.21/1.90 % (24693)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_24330 for (2998ds/17Mi)
% 3.21/1.90 % (24693)Instruction limit reached!
% 3.21/1.90 % (24693)------------------------------
% 3.21/1.90 % (24693)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.21/1.90 % (24693)Termination reason: Unknown
% 3.21/1.90 % (24693)Termination phase: shuffling
% 3.21/1.90
% 3.21/1.90 % (24693)Memory used [KB]: 1918
% 3.21/1.90 % (24693)Time elapsed: 0.007 s
% 3.21/1.90 % (24693)Instructions burned: 19 (million)
% 3.21/1.90 % (24693)------------------------------
% 3.21/1.90 % (24693)------------------------------
% 3.21/1.91 % (24700)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_24330 for (2998ds/3Mi)
% 3.21/1.91 % (24700)Instruction limit reached!
% 3.21/1.91 % (24700)------------------------------
% 3.21/1.91 % (24700)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.21/1.91 % (24700)Termination reason: Unknown
% 3.21/1.91 % (24700)Termination phase: shuffling
% 3.21/1.91
% 3.21/1.91 % (24700)Memory used [KB]: 1535
% 3.21/1.91 % (24700)Time elapsed: 0.002 s
% 3.21/1.91 % (24700)Instructions burned: 3 (million)
% 3.21/1.91 % (24700)------------------------------
% 3.21/1.91 % (24700)------------------------------
% 3.21/1.92 % (24706)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_24330 for (2998ds/30Mi)
% 3.21/1.93 % (24706)Instruction limit reached!
% 3.21/1.93 % (24706)------------------------------
% 3.21/1.93 % (24706)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.21/1.93 % (24706)Termination reason: Unknown
% 3.21/1.93 % (24706)Termination phase: shuffling
% 3.21/1.93
% 3.21/1.93 % (24706)Memory used [KB]: 2046
% 3.21/1.93 % (24706)Time elapsed: 0.010 s
% 3.21/1.93 % (24706)Instructions burned: 31 (million)
% 3.21/1.93 % (24706)------------------------------
% 3.21/1.93 % (24706)------------------------------
% 3.21/1.94 % (24655)Instruction limit reached!
% 3.21/1.94 % (24655)------------------------------
% 3.21/1.94 % (24655)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.21/1.94 % (24655)Termination reason: Unknown
% 3.21/1.94 % (24655)Termination phase: Saturation
% 3.21/1.94
% 3.21/1.94 % (24655)Memory used [KB]: 9466
% 3.21/1.94 % (24655)Time elapsed: 0.103 s
% 3.21/1.94 % (24655)Instructions burned: 377 (million)
% 3.21/1.94 % (24655)------------------------------
% 3.21/1.94 % (24655)------------------------------
% 3.21/1.94 % (24716)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_24330 for (2997ds/127Mi)
% 3.21/1.95 % (24718)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_24330 for (2997ds/100Mi)
% 3.21/1.97 % (24649)First to succeed.
% 3.21/1.97 % (24718)Instruction limit reached!
% 3.21/1.97 % (24718)------------------------------
% 3.21/1.97 % (24718)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.21/1.98 % (24649)Refutation found. Thanks to Tanya!
% 3.21/1.98 % SZS status Theorem for DTF2THF_24330
% 3.21/1.98 % SZS output start Proof for DTF2THF_24330
% 3.21/1.98 thf(func_def_0, type, in: $i > $i > $o).
% 3.21/1.98 thf(func_def_1, type, exu: ($i > $o) > $o).
% 3.21/1.98 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 3.21/1.98 thf(func_def_8, type, powerset: $i > $i).
% 3.21/1.98 thf(func_def_10, type, setunion: $i > $i).
% 3.21/1.98 thf(func_def_19, type, descr: ($i > $o) > $i).
% 3.21/1.98 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 3.21/1.98 thf(func_def_26, type, prop2set: $o > $i).
% 3.21/1.98 thf(func_def_36, type, nonempty: $i > $o).
% 3.21/1.98 thf(func_def_69, type, set2prop: $i > $o).
% 3.21/1.98 thf(func_def_88, type, subset: $i > $i > $o).
% 3.21/1.98 thf(func_def_89, type, disjoint: $i > $i > $o).
% 3.21/1.98 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 3.21/1.98 thf(func_def_114, type, binunion: $i > $i > $i).
% 3.21/1.98 thf(func_def_122, type, binintersect: $i > $i > $i).
% 3.21/1.98 thf(func_def_135, type, regular: $i > $o).
% 3.21/1.98 thf(func_def_136, type, setminus: $i > $i > $i).
% 3.21/1.98 thf(func_def_147, type, symdiff: $i > $i > $i).
% 3.21/1.98 thf(func_def_153, type, iskpair: $i > $o).
% 3.21/1.98 thf(func_def_158, type, kpair: $i > $i > $i).
% 3.21/1.98 thf(func_def_160, type, cartprod: $i > $i > $i).
% 3.21/1.98 thf(func_def_177, type, singleton: $i > $o).
% 3.21/1.98 thf(func_def_179, type, ex1: $i > ($i > $o) > $o).
% 3.21/1.98 thf(func_def_184, type, atmost1p: $i > $o).
% 3.21/1.98 thf(func_def_185, type, atleast2p: $i > $o).
% 3.21/1.98 thf(func_def_186, type, atmost2p: $i > $o).
% 3.21/1.98 thf(func_def_187, type, upairsetp: $i > $o).
% 3.21/1.98 thf(func_def_191, type, kfst: $i > $i).
% 3.21/1.98 thf(func_def_203, type, ksnd: $i > $i).
% 3.21/1.98 thf(func_def_213, type, breln: $i > $i > $i > $o).
% 3.21/1.98 thf(func_def_214, type, dpsetconstr: $i > $i > ($i > $i > $o) > $i).
% 3.21/1.98 thf(func_def_222, type, func: $i > $i > $i > $o).
% 3.21/1.98 thf(func_def_223, type, funcSet: $i > $i > $i).
% 3.21/1.98 thf(func_def_226, type, ap: $i > $i > $i > $i > $i).
% 3.21/1.98 thf(func_def_232, type, lam: $i > $i > ($i > $i) > $i).
% 3.21/1.98 thf(func_def_259, type, if: $i > $o > $i > $i > $i).
% 3.21/1.98 thf(func_def_323, type, breln1: $i > $i > $o).
% 3.21/1.98 thf(func_def_325, type, breln1Set: $i > $i).
% 3.21/1.98 thf(func_def_327, type, transitive: $i > $i > $o).
% 3.21/1.98 thf(func_def_328, type, antisymmetric: $i > $i > $o).
% 3.21/1.98 thf(func_def_329, type, reflexive: $i > $i > $o).
% 3.21/1.98 thf(func_def_330, type, refltransitive: $i > $i > $o).
% 3.21/1.98 thf(func_def_331, type, refllinearorder: $i > $i > $o).
% 3.21/1.98 thf(func_def_332, type, reflwellordering: $i > $i > $o).
% 3.21/1.98 thf(func_def_338, type, breln1invset: $i > $i > $i).
% 3.21/1.98 thf(func_def_342, type, breln1compset: $i > $i > $i > $i).
% 3.21/1.98 thf(func_def_368, type, sP5: $o > $i > $i > $i > $o).
% 3.21/1.98 thf(func_def_370, type, sP7: $i > $i > $i > $i > $o).
% 3.21/1.98 thf(func_def_372, type, sP9: $i > $i > $i > $o).
% 3.21/1.98 thf(func_def_374, type, sP11: $i > $i > $o).
% 3.21/1.98 thf(func_def_375, type, sP12: $i > $o).
% 3.21/1.98 thf(func_def_376, type, sP13: $i > $i > $o).
% 3.21/1.98 thf(func_def_377, type, sP14: $i > $i > $o).
% 3.21/1.98 thf(func_def_378, type, sP15: $i > $i > $i > $o).
% 3.21/1.98 thf(func_def_384, type, sK21: $i > ($i > $i) > $i > $i).
% 3.21/1.98 thf(func_def_386, type, sK23: $i > $i).
% 3.21/1.98 thf(func_def_399, type, sK36: $i > ($i > $o) > $i).
% 3.21/1.98 thf(func_def_400, type, sK37: $i > $o).
% 3.21/1.98 thf(func_def_434, type, sK71: $i > $i > $i > $i > $i).
% 3.21/1.98 thf(func_def_446, type, sK83: $i > $i > $i > $i).
% 3.21/1.98 thf(func_def_457, type, sK94: $i > $o).
% 3.21/1.98 thf(func_def_468, type, sK105: $i > $o).
% 3.21/1.98 thf(func_def_470, type, sK107: ($i > $o) > $i > $i).
% 3.21/1.98 thf(func_def_471, type, sK108: ($i > $o) > $i > $i).
% 3.21/1.98 thf(func_def_472, type, sK109: $i > $i).
% 3.21/1.98 thf(func_def_475, type, sK112: $i > $i > ($i > $i) > $i).
% 3.21/1.98 thf(func_def_476, type, sK113: $i > $o).
% 3.21/1.98 thf(func_def_481, type, sK118: $i > $i > $o).
% 3.21/1.98 thf(func_def_486, type, sK123: $i > $i > $o).
% 3.21/1.98 thf(func_def_491, type, sK128: $i > $i > $o).
% 3.21/1.98 thf(func_def_502, type, sK139: $o > $i > $i > $i).
% 3.21/1.98 thf(func_def_513, type, sK150: ($i > $o) > $i).
% 3.21/1.98 thf(func_def_514, type, sK151: $i > $o).
% 3.21/1.98 thf(func_def_515, type, sK152: $i > $i).
% 3.21/1.98 thf(func_def_527, type, sK164: $i > $i > ($i > $o) > $i).
% 3.21/1.98 thf(func_def_528, type, sK165: $i > $o).
% 3.21/1.98 thf(func_def_546, type, sK183: $i > $i > $i).
% 3.21/1.98 thf(func_def_550, type, sK187: $i > $i).
% 3.21/1.98 thf(func_def_553, type, sK190: $i > ($i > $i) > $i > $i).
% 3.21/1.98 thf(func_def_571, type, sK208: $i > $i > $i > $i > $i > $i).
% 3.21/1.98 thf(func_def_572, type, sK209: $i > $i > $i > $i).
% 3.21/1.98 thf(func_def_573, type, sK210: $i > $i > $i > $i).
% 3.21/1.98 thf(func_def_591, type, sK228: $i > $i > $o).
% 3.21/1.98 thf(func_def_602, type, sK239: $i > $i > $i).
% 3.21/1.98 thf(func_def_608, type, sK245: $i > $i > $o).
% 3.21/1.98 thf(func_def_613, type, sK250: $i > $i).
% 3.21/1.98 thf(func_def_614, type, sK251: $i > $i > ($i > $i > $o) > $i).
% 3.21/1.98 thf(func_def_626, type, sK263: $i > $i > $o).
% 3.21/1.98 thf(func_def_628, type, sK265: $i > $i).
% 3.21/1.98 thf(func_def_629, type, sK266: $i > $i).
% 3.21/1.98 thf(func_def_630, type, sK267: $i > ($i > $i > $o) > $i).
% 3.21/1.98 thf(func_def_631, type, sK268: $i > ($i > $i > $o) > $i).
% 3.21/1.98 thf(func_def_632, type, sK269: $i > $i > ($i > $i > $o) > $i).
% 3.21/1.98 thf(func_def_637, type, sK274: $i > $i).
% 3.21/1.98 thf(func_def_659, type, sK296: $i > $i > $i).
% 3.21/1.98 thf(func_def_662, type, sK299: $i > ($i > $o) > $i).
% 3.21/1.98 thf(func_def_663, type, sK300: $i > $o).
% 3.21/1.98 thf(func_def_669, type, sK306: $i > $i > $i).
% 3.21/1.98 thf(func_def_679, type, sK316: $i > $o).
% 3.21/1.98 thf(func_def_681, type, sK318: ($i > $o) > $i > $i > $i > $i).
% 3.21/1.98 thf(func_def_682, type, sK319: ($i > $o) > $i > $i > $i > $i).
% 3.21/1.98 thf(func_def_683, type, sK320: ($i > $o) > $i).
% 3.21/1.98 thf(func_def_684, type, sK321: $i > $o).
% 3.21/1.98 thf(func_def_685, type, sK322: $i > $i).
% 3.21/1.98 thf(func_def_697, type, sK334: $i > $i > $i > $i > $i).
% 3.21/1.98 thf(func_def_702, type, sK339: ($i > $o) > $i > $i).
% 3.21/1.98 thf(func_def_704, type, sK341: $i > $o).
% 3.21/1.98 thf(func_def_762, type, sK399: $o > $i > $i > $i > $i > $i > $i).
% 3.21/1.98 thf(func_def_768, type, sK405: $i > $i).
% 3.21/1.98 thf(func_def_769, type, sK406: ($i > $i) > $i > $i > $i).
% 3.21/1.98 thf(func_def_818, type, sK455: $i > $o).
% 3.21/1.98 thf(func_def_821, type, sK458: $i > ($i > $o) > $i).
% 3.21/1.98 thf(func_def_830, type, sK467: $i > $i).
% 3.21/1.98 thf(func_def_853, type, sK490: $i > $o).
% 3.21/1.98 thf(func_def_859, type, sK496: $i > $o).
% 3.21/1.98 thf(func_def_860, type, sK497: $i > $i > $i).
% 3.21/1.98 thf(func_def_876, type, sK513: $i > $i > $i > $i > $i).
% 3.21/1.98 thf(func_def_877, type, sK514: $i > $i > $i > $i > $i).
% 3.21/1.98 thf(func_def_884, type, sK521: $i > $o).
% 3.21/1.98 thf(func_def_895, type, sK532: ($i > $i) > $i > $i > $i).
% 3.21/1.98 thf(func_def_898, type, sK535: $i > $i).
% 3.21/1.98 thf(func_def_903, type, sK540: $i > $o).
% 3.21/1.98 thf(func_def_906, type, sK543: $i > ($i > $o) > $i).
% 3.21/1.98 thf(func_def_927, type, sK564: $i > $i > $i > $i > $i).
% 3.21/1.98 thf(func_def_928, type, sK565: $i > $i > $i > $i > $i).
% 3.21/1.98 thf(func_def_929, type, sK566: $i > $i > $i > $i > $i).
% 3.21/1.98 thf(func_def_930, type, sK567: $i > $i > $i > $i > $i).
% 3.21/1.98 thf(func_def_938, type, sK575: $i > $o).
% 3.21/1.98 thf(func_def_966, type, sK603: $i > $i > $i > $i > $i).
% 3.21/1.98 thf(func_def_967, type, sK604: $i > ($i > $o) > $i).
% 3.21/1.98 thf(func_def_968, type, sK605: $i > $o).
% 3.21/1.98 thf(func_def_973, type, sK610: $i > $i > $i > $i).
% 3.21/1.98 thf(func_def_974, type, sK611: $i > $i > $i > $i).
% 3.21/1.98 thf(func_def_975, type, sK612: $i > $i > $i > $i).
% 3.21/1.98 thf(func_def_976, type, sK613: $i > $i > $i > $i).
% 3.21/1.98 thf(func_def_980, type, sK617: $i > $i > $i).
% 3.21/1.98 thf(func_def_996, type, sK633: ($i > $o) > $i).
% 3.21/1.98 thf(func_def_997, type, sK634: ($i > $o) > $i).
% 3.21/1.98 thf(func_def_998, type, sK635: $i > $o).
% 3.21/1.98 thf(func_def_1026, type, sK663: ($i > $o) > $i).
% 3.21/1.98 thf(func_def_1027, type, sK664: $i > $o).
% 3.21/1.98 thf(func_def_1030, type, sK667: $i > $o).
% 3.21/1.98 thf(func_def_1032, type, sK669: ($i > $o) > $i > $i).
% 3.21/1.98 thf(func_def_1033, type, sK670: $i > $i > $o).
% 3.21/1.98 thf(func_def_1038, type, sK675: $i > $i > $i).
% 3.21/1.98 thf(func_def_1039, type, sK676: $i > $i > $i).
% 3.21/1.98 thf(func_def_1040, type, sK677: $i > $i > $i).
% 3.21/1.98 thf(func_def_1041, type, sK678: $i > $i > $i).
% 3.21/1.98 thf(func_def_1042, type, sK679: $i > $i > $i).
% 3.21/1.98 thf(func_def_1043, type, sK680: $i > $i > $i > $i).
% 3.21/1.98 thf(func_def_1044, type, sK681: $i > $i).
% 3.21/1.98 thf(func_def_1045, type, sK682: $i > $i).
% 3.21/1.98 thf(func_def_1046, type, sK683: $i > $i).
% 3.21/1.98 thf(func_def_1047, type, sK684: $i > $i).
% 3.21/1.98 thf(func_def_1048, type, sK685: $i > $i > $i).
% 3.21/1.98 thf(func_def_1049, type, sK686: $i > $i > $i > $i).
% 3.21/1.98 thf(func_def_1050, type, sK687: $i > $i > $i > $i).
% 3.21/1.98 thf(func_def_1051, type, sK688: $i > $i > $i).
% 3.21/1.98 thf(func_def_1052, type, sK689: $i > $i > $i).
% 3.21/1.98 thf(func_def_1054, type, sK691: $i > $i).
% 3.21/1.98 thf(func_def_1055, type, sK692: $i > $i).
% 3.21/1.98 thf(func_def_1056, type, sK693: $i > $i).
% 3.21/1.98 thf(func_def_1061, type, sK698: $i > $o).
% 3.21/1.98 thf(func_def_1068, type, sK705: ($i > $o) > $i > $i > $i > $i).
% 3.21/1.98 thf(func_def_1069, type, sK706: ($i > $o) > $i > $i > $i > $i).
% 3.21/1.98 thf(func_def_1073, type, sK710: $i > $o).
% 3.21/1.98 thf(func_def_1089, type, sK726: $i > $o).
% 3.21/1.98 thf(func_def_1102, type, sK739: $i > $i).
% 3.21/1.98 thf(func_def_1107, type, sK744: $i > $i > $i).
% 3.21/1.98 thf(func_def_1119, type, sK756: $i > $i > $i).
% 3.21/1.98 thf(func_def_1122, type, sK759: $i > $i > $o).
% 3.21/1.98 thf(func_def_1151, type, sK788: $i > $i > $i > $i).
% 3.21/1.98 thf(func_def_1152, type, sK789: $i > $i > $i > $i).
% 3.21/1.98 thf(func_def_1159, type, sK796: $i > $i).
% 3.21/1.98 thf(func_def_1162, type, sK799: $i > $i).
% 3.21/1.98 thf(func_def_1178, type, sK815: $i > $o).
% 3.21/1.98 thf(func_def_1181, type, sK818: $i > $i > $o).
% 3.21/1.98 thf(func_def_1188, type, sK825: ($i > $o) > ($i > $o) > $i).
% 3.21/1.98 thf(func_def_1189, type, sK826: ($i > $o) > ($i > $o) > $i).
% 3.21/1.98 thf(func_def_1190, type, sK827: $i > $o).
% 3.21/1.98 thf(func_def_1191, type, sK828: $i > $o).
% 3.21/1.98 thf(func_def_1195, type, sK832: $i > $i > $i > $i).
% 3.21/1.98 thf(func_def_1196, type, sK833: $i > $i > $i > $i).
% 3.21/1.98 thf(func_def_1208, type, sK845: $i > $o).
% 3.21/1.98 thf(func_def_1219, type, sK856: $i > $o).
% 3.21/1.98 thf(func_def_1242, type, sK879: ($i > $o) > $i > $i).
% 3.21/1.98 thf(func_def_1244, type, sK881: $i > $o).
% 3.21/1.98 thf(func_def_1254, type, sK891: ($i > $o) > ($i > $o) > $i).
% 3.21/1.98 thf(func_def_1255, type, sK892: ($i > $o) > ($i > $o) > $i).
% 3.21/1.98 thf(func_def_1256, type, sK893: $i > $o).
% 3.21/1.98 thf(func_def_1257, type, sK894: $i > $o).
% 3.21/1.98 thf(func_def_1272, type, sK909: ($i > $o) > ($i > $o) > $i > $i > $i).
% 3.21/1.98 thf(func_def_1273, type, sK910: ($i > $o) > ($i > $o) > $i > $i > $i).
% 3.21/1.98 thf(func_def_1276, type, sK913: $i > $o).
% 3.21/1.98 thf(func_def_1277, type, sK914: $i > $o).
% 3.21/1.98 thf(func_def_1278, type, sK915: $i > $o).
% 3.21/1.98 thf(func_def_1281, type, sK918: $i > ($i > $o) > $i).
% 3.21/1.98 thf(func_def_1282, type, sK919: $i > ($i > $o) > $i).
% 3.21/1.98 thf(func_def_1286, type, sK923: $i > $i > $o).
% 3.21/1.98 thf(func_def_1288, type, sK925: $i > $i).
% 3.21/1.98 thf(func_def_1294, type, sK931: ($i > $o) > $i > $i > $i).
% 3.21/1.98 thf(func_def_1295, type, sK932: ($i > $o) > $i > $i > $i).
% 3.21/1.98 thf(func_def_1298, type, sK935: $i > $o).
% 3.21/1.98 thf(func_def_1300, type, sK937: ($i > $o) > $i > $i > $i).
% 3.21/1.98 thf(func_def_1301, type, sK938: ($i > $o) > $i > $i > $i).
% 3.21/1.98 thf(func_def_1304, type, sK941: $i > $o).
% 3.21/1.98 thf(func_def_1308, type, sK945: $i > $i).
% 3.21/1.98 thf(func_def_1326, type, sK963: $i > $o).
% 3.21/1.98 thf(func_def_1339, type, sK976: $i > $i > $i).
% 3.21/1.98 thf(func_def_1340, type, sK977: $i > $i > $i).
% 3.21/1.98 thf(func_def_1346, type, sK983: $i > $o).
% 3.21/1.98 thf(func_def_1350, type, sK987: $i > $i).
% 3.21/1.98 thf(func_def_1352, type, sK989: ($i > $i) > $i > $i > $i).
% 3.21/1.98 thf(func_def_1354, type, sK991: $i > ($i > $o) > $i).
% 3.21/1.98 thf(func_def_1355, type, sK992: $i > $o).
% 3.21/1.98 thf(func_def_1368, type, sK1005: $i > $i > $i > $i).
% 3.21/1.98 thf(func_def_1376, type, ph1013: !>[X0: $tType]:(X0)).
% 3.21/1.98 thf(f4849,plain,(
% 3.21/1.98 $false),
% 3.21/1.98 inference(avatar_sat_refutation,[status(thm)],[f4741,f4821,f4841])).
% 3.21/1.98 thf(f4841,plain,(
% 3.21/1.98 ~spl1012_1),
% 3.21/1.98 inference(avatar_split_clause,[status(thm)],[f4840,f4734])).
% 3.21/1.98 thf(f4734,definition,(
% 3.21/1.98 spl1012_1 <=> ((in @ (kpair @ sK259 @ sK260) @ sK258) = $true)),
% 3.21/1.98 introduced(definition,[new_symbols(naming,[spl1012_1])],[avatar_definition])).
% 3.21/1.98 thf(f4840,plain,(
% 3.21/1.98 ((in @ (kpair @ sK259 @ sK260) @ sK258) != $true)),
% 3.21/1.98 inference(trivial_inequality_removal,[status(thm)],[f4839])).
% 3.21/1.98 thf(f4839,plain,(
% 3.21/1.98 ($true != $true) | ((in @ (kpair @ sK259 @ sK260) @ sK258) != $true)),
% 3.21/1.98 inference(forward_demodulation,[status(thm)],[f4805,f3346])).
% 3.21/1.98 thf(f3346,plain,(
% 3.21/1.98 (binunionIR = $true)),
% 3.21/1.98 inference(cnf_transformation,[status(thm)],[f1716])).
% 3.21/1.98 thf(f1716,plain,(
% 3.21/1.98 (doubleComplementE1 = $true) & (cartprodpairmemER = $true) & (setextT = $true) & (setukpairinjR1 = $true) & (vacuousDall = $true) & (powersetI = $true) & (cartprodfstpairEq = $true) & (contrasubsetT1 = $true) & (powersetTI1 = $true) & (upairsubunion = $true) & (contraSubsetComplement = $true) & (setminusER = $true) & (emptyinunitempty = $true) & (dpsetconstrEL2 = $true) & (setukpairinjL2 = $true) & (emptysetsubset = $true) & (binunionLsub = $true) & (breln1invprop = $true) & (emptysetE = $true) & (ubforcartprodlem3 = $true) & (notequalI2 = $true) & (powerset__Cong = $true) & (emptyI = $true) & (funcImageSingleton = $true) & (contrasubsetT = $true) & (theeq = $true) & (ubforcartprodlem2 = $true) & (dpsetconstrER = $true) & (woz13rule4 = $true) & (setadjoinE = $true) & (binintersectRsub = $true) & (inIntersectImpInIntersectUnions = $true) & (doubleComplementEq = $true) & (symdiffIneg1 = $true) & (upairsetIL = $true) & (complementT_lem = $true) & (demorgan2a = $true) & (bs114d = $true) & (setadjoinAx = $true) & (((((in @ sK259 @ sK257) = $true) & (($true != (in @ (kpair @ sK259 @ sK260) @ (binunion @ sK256 @ sK258))) & ((in @ sK260 @ sK257) = $true) & (((in @ (kpair @ sK259 @ sK260) @ sK256) = $true) | ((in @ (kpair @ sK259 @ sK260) @ sK258) = $true)))) & ($true = (breln1 @ sK257 @ sK258))) & ((breln1 @ sK257 @ sK256) = $true)) & (setadjoinIR = $true) & (exu__Cong = $true) & (notinsingleton = $true) & (notdallE = $true) & (setextsub = $true) & (breln1compprop = $true) & (nonemptyImpWitness = $true) & (cartprodpairin = $true) & (binintersectTERcontra = $true) & (contrasubsetT2 = $true) & (dsetconstr__Cong = $true) & (lamProp = $true) & (subset2powerset = $true) & (secondinupair = $true) & (breln1compI = $true) & (iffalseProp2 = $true) & (binunionTEcontra = $true) & (breln1unionIL = $true) & (upairset2E = $true) & (emptyE1 = $true) & (inComplementUnionImpInComplement1 = $true) & (breln1invI = $true) & (setadjoin__Cong = $true) & (woz13rule2 = $true) & (setminusEL = $true) & (wellorderingAx = $true) & (setminusERneg = $true) & (notinemptyset = $true) & (ex1I = $true) & (powersetAx = $true) & (binintersectSubset1 = $true) & (breln1SetBreln1 = $true) & (binunionTIRcontra = $true) & (breln1unionIR = $true) & (emptyinPowerset = $true) & (inIntersectImpInUnion2 = $true) & (subsetE2 = $true) & (setunion__Cong = $true) & (setminusSubset1 = $true) & (woz13rule0 = $true) & (setadjoinIL = $true) & (funcext = $true) & (setOfPairsIsBReln1 = $true) & (dsetconstrI = $true) & (demorgan2a1 = $true) & (complementImpComplementIntersect = $true) & (beta1 = $true) & (iftrueProp2 = $true) & (setunionAx = $true) & (doubleComplementI1 = $true) & (powersetE1 = $true) & (cartprodmempair = $true) & (cartprodpairmemEL = $true) & (subsetemptysetimpeq = $true) & (binunionE = $true) & (subsetI2 = $true) & (prop2set2propI = $true) & (in__Cong = $true) & (demorgan2b2 = $true) & (setadjoinOr = $true) & (kpairsurjEq = $true) & (eqimpsubset2 = $true) & (singletoninpowerset = $true) & (kfstpairEq = $true) & (breln1all2 = $true) & (complementTnotintersectT = $true) & (breln1compEex = $true) & (quantDeMorgan3 = $true) & (exuI1 = $true) & (emptyInPowerset = $true) & (iftrueorfalse = $true) & (singletonprop = $true) & (demorgan1b = $true) & (funcGraphProp1 = $true) & (lamp = $true) & (powersetTE1 = $true) & (dpsetconstrEL1 = $true) & (omegaIndAx = $true) & (subsetE = $true) & (dpsetconstrI = $true) & (setminusELneg = $true) & (setukpairinjR12 = $true) & (subsetI1 = $true) & (binintersectSubset5 = $true) & (prop2setE = $true) & (ifp = $true) & (funcinfuncset = $true) & (setukpairIR = $true) & (binunionRsub = $true) & (setadjoinSub2 = $true) & (ksndpairEq = $true) & (upairequniteq = $true) & (symdiffIneg2 = $true) & (funcGraphProp3 = $true) & (demorgan1 = $true) & (binintersectER = $true) & (breln1unionprop = $true) & (exuE1 = $true) & (setminusILneg = $true) & (binunionIL = $true) & (ifSingleton = $true) & (quantDeMorgan1 = $true) & (powersetI1 = $true) & (setunionE = $true) & (woz1_3 = $true) & (binintersectI = $true) & (dsetconstrEL = $true) & (ap2apEq1 = $true) & (powersetT_lem = $true) & (prop2setI = $true) & (exuE3u = $true) & (apProp = $true) & (subbreln = $true) & (setbeta = $true) & (subbreln1 = $true) & (binunionIR = $true) & (binintersectSubset2 = $true) & (kpairp = $true) & (doubleComplementSub2 = $true) & (binintersectT_lem = $true) & (contrasubsetT3 = $true) & (cartprodpairsurjEq = $true) & (descr__Cong = $true) & (inCongP = $true) & (binunionTE = $true) & (ksndsingleton = $true) & (demorgan2 = $true) & (exuE3e = $true) & (foundationAx = $true) & (iftrueProp1 = $true) & (theprop = $true) & (eta2 = $true) & (omega__Cong = $true) & (woz1_1 = $true) & (binintersectLsub = $true) & (setextAx = $true) & (emptyset__Cong = $true) & (demorgan2a2 = $true) & (setunionsingleton = $true) & (setukpairinjR = $true) & (noeltsimpempty = $true) & (nonemptyI = $true) & (setminusLsub = $true) & (quantDeMorgan4 = $true) & (ex1E2 = $true) & (sepSubset = $true) & (setukpairinjL1 = $true) & (eqbreln = $true) & (setunionsingleton1 = $true) & (cartprodmempaircEq = $true) & (woz13rule3 = $true) & (powersetsubset = $true) & (inIntersectImpInUnion = $true) & (exuI2 = $true) & (subsetRefl = $true) & (doubleComplementSub1 = $true) & (subPowSU = $true) & (nonemptyI1 = $true) & (lam2lamEq = $true) & (complementInPowersetComplementIntersect = $true) & (infuncsetfunc = $true) & (dsetconstrER = $true) & (symdiffE = $true) & (complementTE1 = $true) & (lam2p = $true) & (binintersectSubset4 = $true) & (exuEu = $true) & (binunionTILcontra = $true) & (binunionEcases = $true) & (binintersectTELcontra = $true) & (complementUnionInPowersetComplement = $true) & (dpsetconstrSub = $true) & (emptysetimpfalse = $true) & (singletonsuniq = $true) & (iffalseProp1 = $true) & (brelnall1 = $true) & (complementTcontraSubset = $true) & (cartprodsndpairEq = $true) & (setadjoinSub = $true) & (setukpairinjR11 = $true) & (emptysetAx = $true) & (setunionI = $true) & (setminusI = $true) & (sepInPowerset = $true) & (exuE2 = $true) & (setoftrueEq = $true) & (descrp = $true) & (woz1_5 = $true) & (breln1compE = $true) & (setminusIRneg = $true) & (breln1invE = $true) & (setminusT_lem = $true) & (eqimpsubset1 = $true) & (choice2fnsingleton = $true) & (intersectInPowersetIntersectUnions = $true) & (upairset2IR = $true) & (symdiffI1 = $true) & (cartprodsndin = $true) & (funcGraphProp4 = $true) & (beta2 = $true) & (ap2p = $true) & (cartprodfstin = $true) & (brelnall2 = $true) & (subsetTI = $true) & (setukpairIL = $true) & (omegaSAx = $true) & (eqinunit = $true) & (funcextLem = $true) & (notsubsetI = $true) & (inPowerset = $true) & (dpsetconstrERa = $true) & (setunionsingleton2 = $true) & (powersetE = $true) & (complementSubsetComplementIntersect = $true) & (setunionE2 = $true) & (singletonsubset = $true) & (woz1_2 = $true) & (inComplementUnionImpNotIn1 = $true) & (cartprodmempair1 = $true) & (demorgan2b = $true) & (subsetTrans = $true) & (setext = $true) & (binintersectEL = $true) & (complementTI1 = $true) & (binunionT_lem = $true) & (upairinpowunion = $true) & (uniqinunit = $true) & (singletonsswitch = $true) & (upairsetIR = $true) & (setukpairinjL = $true) & (quantDeMorgan2 = $true) & (ex1I2 = $true) & (setukpairinjR2 = $true) & (disjointsetsI1 = $true) & (eta1 = $true) & (binintersectSubset3 = $true) & (ubforcartprodlem1 = $true) & (kfstsingleton = $true) & (setminusSubset2 = $true) & (woz13rule1 = $true) & (replAx = $true) & (exuI3 = $true) & (iffalse = $true) & (singletoninpowunion = $true) & (funcext2 = $true) & (ap2apEq2 = $true) & (woz1_4 = $true) & (breln1all1 = $true) & (demorgan1a = $true) & (nonemptyE1 = $true) & (app = $true) & (ex1E1 = $true) & (setOfPairsIsBReln = $true) & (notdexE = $true) & (iftrue = $true) & (funcGraphProp2 = $true) & (kpairiskpair = $true) & (omega0Ax = $true) & (notequalI1 = $true) & (symdiffI2 = $true) & (eqbreln1 = $true) & (upairsetE = $true)),
% 3.21/1.98 inference(skolemisation,[status(esa),new_symbols(skolem,[sK256,sK257,sK258,sK259,sK260])],[f1037,f1715,f1714,f1713,f1712])).
% 3.21/1.98 thf(f1712,plain,(
% 3.21/1.98 ? [X0 : $i,X1 : $i] : (? [X2 : $i] : (? [X3 : $i] : (((in @ X3 @ X1) = $true) & ? [X4 : $i] : (((in @ (kpair @ X3 @ X4) @ (binunion @ X0 @ X2)) != $true) & ((in @ X4 @ X1) = $true) & (((in @ (kpair @ X3 @ X4) @ X0) = $true) | ((in @ (kpair @ X3 @ X4) @ X2) = $true)))) & ((breln1 @ X1 @ X2) = $true)) & ((breln1 @ X1 @ X0) = $true)) => (? [X2 : $i] : (? [X3 : $i] : (((in @ X3 @ sK257) = $true) & ? [X4 : $i] : (((in @ (kpair @ X3 @ X4) @ (binunion @ sK256 @ X2)) != $true) & ($true = (in @ X4 @ sK257)) & (((in @ (kpair @ X3 @ X4) @ sK256) = $true) | ((in @ (kpair @ X3 @ X4) @ X2) = $true)))) & ((breln1 @ sK257 @ X2) = $true)) & ((breln1 @ sK257 @ sK256) = $true))),
% 3.21/1.98 introduced(definition,[],[choice_axiom])).
% 3.21/1.98 thf(f1713,plain,(
% 3.21/1.98 ? [X2 : $i] : (? [X3 : $i] : (((in @ X3 @ sK257) = $true) & ? [X4 : $i] : (((in @ (kpair @ X3 @ X4) @ (binunion @ sK256 @ X2)) != $true) & ($true = (in @ X4 @ sK257)) & (((in @ (kpair @ X3 @ X4) @ sK256) = $true) | ((in @ (kpair @ X3 @ X4) @ X2) = $true)))) & ((breln1 @ sK257 @ X2) = $true)) => (? [X3 : $i] : (((in @ X3 @ sK257) = $true) & ? [X4 : $i] : (((in @ (kpair @ X3 @ X4) @ (binunion @ sK256 @ sK258)) != $true) & ($true = (in @ X4 @ sK257)) & (((in @ (kpair @ X3 @ X4) @ sK256) = $true) | ($true = (in @ (kpair @ X3 @ X4) @ sK258))))) & ($true = (breln1 @ sK257 @ sK258)))),
% 3.21/1.98 introduced(definition,[],[choice_axiom])).
% 3.21/1.98 thf(f1714,plain,(
% 3.21/1.98 ? [X3 : $i] : (((in @ X3 @ sK257) = $true) & ? [X4 : $i] : (((in @ (kpair @ X3 @ X4) @ (binunion @ sK256 @ sK258)) != $true) & ($true = (in @ X4 @ sK257)) & (((in @ (kpair @ X3 @ X4) @ sK256) = $true) | ($true = (in @ (kpair @ X3 @ X4) @ sK258))))) => (((in @ sK259 @ sK257) = $true) & ? [X4 : $i] : (($true != (in @ (kpair @ sK259 @ X4) @ (binunion @ sK256 @ sK258))) & ($true = (in @ X4 @ sK257)) & (((in @ (kpair @ sK259 @ X4) @ sK256) = $true) | ((in @ (kpair @ sK259 @ X4) @ sK258) = $true))))),
% 3.21/1.98 introduced(definition,[],[choice_axiom])).
% 3.21/1.98 thf(f1715,plain,(
% 3.21/1.98 ? [X4 : $i] : (($true != (in @ (kpair @ sK259 @ X4) @ (binunion @ sK256 @ sK258))) & ($true = (in @ X4 @ sK257)) & (((in @ (kpair @ sK259 @ X4) @ sK256) = $true) | ((in @ (kpair @ sK259 @ X4) @ sK258) = $true))) => (($true != (in @ (kpair @ sK259 @ sK260) @ (binunion @ sK256 @ sK258))) & ((in @ sK260 @ sK257) = $true) & (((in @ (kpair @ sK259 @ sK260) @ sK256) = $true) | ((in @ (kpair @ sK259 @ sK260) @ sK258) = $true)))),
% 3.21/1.98 introduced(definition,[],[choice_axiom])).
% 3.21/1.98 thf(f1037,plain,(
% 3.21/1.98 (doubleComplementE1 = $true) & (cartprodpairmemER = $true) & (setextT = $true) & (setukpairinjR1 = $true) & (vacuousDall = $true) & (powersetI = $true) & (cartprodfstpairEq = $true) & (contrasubsetT1 = $true) & (powersetTI1 = $true) & (upairsubunion = $true) & (contraSubsetComplement = $true) & (setminusER = $true) & (emptyinunitempty = $true) & (dpsetconstrEL2 = $true) & (setukpairinjL2 = $true) & (emptysetsubset = $true) & (binunionLsub = $true) & (breln1invprop = $true) & (emptysetE = $true) & (ubforcartprodlem3 = $true) & (notequalI2 = $true) & (powerset__Cong = $true) & (emptyI = $true) & (funcImageSingleton = $true) & (contrasubsetT = $true) & (theeq = $true) & (ubforcartprodlem2 = $true) & (dpsetconstrER = $true) & (woz13rule4 = $true) & (setadjoinE = $true) & (binintersectRsub = $true) & (inIntersectImpInIntersectUnions = $true) & (doubleComplementEq = $true) & (symdiffIneg1 = $true) & (upairsetIL = $true) & (complementT_lem = $true) & (demorgan2a = $true) & (bs114d = $true) & (setadjoinAx = $true) & ? [X0 : $i,X1 : $i] : (? [X2 : $i] : (? [X3 : $i] : (((in @ X3 @ X1) = $true) & ? [X4 : $i] : (((in @ (kpair @ X3 @ X4) @ (binunion @ X0 @ X2)) != $true) & ((in @ X4 @ X1) = $true) & (((in @ (kpair @ X3 @ X4) @ X0) = $true) | ((in @ (kpair @ X3 @ X4) @ X2) = $true)))) & ((breln1 @ X1 @ X2) = $true)) & ((breln1 @ X1 @ X0) = $true)) & (setadjoinIR = $true) & (exu__Cong = $true) & (notinsingleton = $true) & (notdallE = $true) & (setextsub = $true) & (breln1compprop = $true) & (nonemptyImpWitness = $true) & (cartprodpairin = $true) & (binintersectTERcontra = $true) & (contrasubsetT2 = $true) & (dsetconstr__Cong = $true) & (lamProp = $true) & (subset2powerset = $true) & (secondinupair = $true) & (breln1compI = $true) & (iffalseProp2 = $true) & (binunionTEcontra = $true) & (breln1unionIL = $true) & (upairset2E = $true) & (emptyE1 = $true) & (inComplementUnionImpInComplement1 = $true) & (breln1invI = $true) & (setadjoin__Cong = $true) & (woz13rule2 = $true) & (setminusEL = $true) & (wellorderingAx = $true) & (setminusERneg = $true) & (notinemptyset = $true) & (ex1I = $true) & (powersetAx = $true) & (binintersectSubset1 = $true) & (breln1SetBreln1 = $true) & (binunionTIRcontra = $true) & (breln1unionIR = $true) & (emptyinPowerset = $true) & (inIntersectImpInUnion2 = $true) & (subsetE2 = $true) & (setunion__Cong = $true) & (setminusSubset1 = $true) & (woz13rule0 = $true) & (setadjoinIL = $true) & (funcext = $true) & (setOfPairsIsBReln1 = $true) & (dsetconstrI = $true) & (demorgan2a1 = $true) & (complementImpComplementIntersect = $true) & (beta1 = $true) & (iftrueProp2 = $true) & (setunionAx = $true) & (doubleComplementI1 = $true) & (powersetE1 = $true) & (cartprodmempair = $true) & (cartprodpairmemEL = $true) & (subsetemptysetimpeq = $true) & (binunionE = $true) & (subsetI2 = $true) & (prop2set2propI = $true) & (in__Cong = $true) & (demorgan2b2 = $true) & (setadjoinOr = $true) & (kpairsurjEq = $true) & (eqimpsubset2 = $true) & (singletoninpowerset = $true) & (kfstpairEq = $true) & (breln1all2 = $true) & (complementTnotintersectT = $true) & (breln1compEex = $true) & (quantDeMorgan3 = $true) & (exuI1 = $true) & (emptyInPowerset = $true) & (iftrueorfalse = $true) & (singletonprop = $true) & (demorgan1b = $true) & (funcGraphProp1 = $true) & (lamp = $true) & (powersetTE1 = $true) & (dpsetconstrEL1 = $true) & (omegaIndAx = $true) & (subsetE = $true) & (dpsetconstrI = $true) & (setminusELneg = $true) & (setukpairinjR12 = $true) & (subsetI1 = $true) & (binintersectSubset5 = $true) & (prop2setE = $true) & (ifp = $true) & (funcinfuncset = $true) & (setukpairIR = $true) & (binunionRsub = $true) & (setadjoinSub2 = $true) & (ksndpairEq = $true) & (upairequniteq = $true) & (symdiffIneg2 = $true) & (funcGraphProp3 = $true) & (demorgan1 = $true) & (binintersectER = $true) & (breln1unionprop = $true) & (exuE1 = $true) & (setminusILneg = $true) & (binunionIL = $true) & (ifSingleton = $true) & (quantDeMorgan1 = $true) & (powersetI1 = $true) & (setunionE = $true) & (woz1_3 = $true) & (binintersectI = $true) & (dsetconstrEL = $true) & (ap2apEq1 = $true) & (powersetT_lem = $true) & (prop2setI = $true) & (exuE3u = $true) & (apProp = $true) & (subbreln = $true) & (setbeta = $true) & (subbreln1 = $true) & (binunionIR = $true) & (binintersectSubset2 = $true) & (kpairp = $true) & (doubleComplementSub2 = $true) & (binintersectT_lem = $true) & (contrasubsetT3 = $true) & (cartprodpairsurjEq = $true) & (descr__Cong = $true) & (inCongP = $true) & (binunionTE = $true) & (ksndsingleton = $true) & (demorgan2 = $true) & (exuE3e = $true) & (foundationAx = $true) & (iftrueProp1 = $true) & (theprop = $true) & (eta2 = $true) & (omega__Cong = $true) & (woz1_1 = $true) & (binintersectLsub = $true) & (setextAx = $true) & (emptyset__Cong = $true) & (demorgan2a2 = $true) & (setunionsingleton = $true) & (setukpairinjR = $true) & (noeltsimpempty = $true) & (nonemptyI = $true) & (setminusLsub = $true) & (quantDeMorgan4 = $true) & (ex1E2 = $true) & (sepSubset = $true) & (setukpairinjL1 = $true) & (eqbreln = $true) & (setunionsingleton1 = $true) & (cartprodmempaircEq = $true) & (woz13rule3 = $true) & (powersetsubset = $true) & (inIntersectImpInUnion = $true) & (exuI2 = $true) & (subsetRefl = $true) & (doubleComplementSub1 = $true) & (subPowSU = $true) & (nonemptyI1 = $true) & (lam2lamEq = $true) & (complementInPowersetComplementIntersect = $true) & (infuncsetfunc = $true) & (dsetconstrER = $true) & (symdiffE = $true) & (complementTE1 = $true) & (lam2p = $true) & (binintersectSubset4 = $true) & (exuEu = $true) & (binunionTILcontra = $true) & (binunionEcases = $true) & (binintersectTELcontra = $true) & (complementUnionInPowersetComplement = $true) & (dpsetconstrSub = $true) & (emptysetimpfalse = $true) & (singletonsuniq = $true) & (iffalseProp1 = $true) & (brelnall1 = $true) & (complementTcontraSubset = $true) & (cartprodsndpairEq = $true) & (setadjoinSub = $true) & (setukpairinjR11 = $true) & (emptysetAx = $true) & (setunionI = $true) & (setminusI = $true) & (sepInPowerset = $true) & (exuE2 = $true) & (setoftrueEq = $true) & (descrp = $true) & (woz1_5 = $true) & (breln1compE = $true) & (setminusIRneg = $true) & (breln1invE = $true) & (setminusT_lem = $true) & (eqimpsubset1 = $true) & (choice2fnsingleton = $true) & (intersectInPowersetIntersectUnions = $true) & (upairset2IR = $true) & (symdiffI1 = $true) & (cartprodsndin = $true) & (funcGraphProp4 = $true) & (beta2 = $true) & (ap2p = $true) & (cartprodfstin = $true) & (brelnall2 = $true) & (subsetTI = $true) & (setukpairIL = $true) & (omegaSAx = $true) & (eqinunit = $true) & (funcextLem = $true) & (notsubsetI = $true) & (inPowerset = $true) & (dpsetconstrERa = $true) & (setunionsingleton2 = $true) & (powersetE = $true) & (complementSubsetComplementIntersect = $true) & (setunionE2 = $true) & (singletonsubset = $true) & (woz1_2 = $true) & (inComplementUnionImpNotIn1 = $true) & (cartprodmempair1 = $true) & (demorgan2b = $true) & (subsetTrans = $true) & (setext = $true) & (binintersectEL = $true) & (complementTI1 = $true) & (binunionT_lem = $true) & (upairinpowunion = $true) & (uniqinunit = $true) & (singletonsswitch = $true) & (upairsetIR = $true) & (setukpairinjL = $true) & (quantDeMorgan2 = $true) & (ex1I2 = $true) & (setukpairinjR2 = $true) & (disjointsetsI1 = $true) & (eta1 = $true) & (binintersectSubset3 = $true) & (ubforcartprodlem1 = $true) & (kfstsingleton = $true) & (setminusSubset2 = $true) & (woz13rule1 = $true) & (replAx = $true) & (exuI3 = $true) & (iffalse = $true) & (singletoninpowunion = $true) & (funcext2 = $true) & (ap2apEq2 = $true) & (woz1_4 = $true) & (breln1all1 = $true) & (demorgan1a = $true) & (nonemptyE1 = $true) & (app = $true) & (ex1E1 = $true) & (setOfPairsIsBReln = $true) & (notdexE = $true) & (iftrue = $true) & (funcGraphProp2 = $true) & (kpairiskpair = $true) & (omega0Ax = $true) & (notequalI1 = $true) & (symdiffI2 = $true) & (eqbreln1 = $true) & (upairsetE = $true)),
% 3.21/1.98 inference(flattening,[status(thm)],[f1036])).
% 3.21/1.98 thf(f1036,plain,(
% 3.21/1.98 ((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i,X1 : $i] : (? [X2 : $i] : (? [X3 : $i] : (? [X4 : $i] : ((((in @ (kpair @ X3 @ X4) @ (binunion @ X0 @ X2)) != $true) & (((in @ (kpair @ X3 @ X4) @ X0) = $true) | ((in @ (kpair @ X3 @ X4) @ X2) = $true))) & ((in @ X4 @ X1) = $true)) & ((in @ X3 @ X1) = $true)) & ((breln1 @ X1 @ X2) = $true)) & ((breln1 @ X1 @ X0) = $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)),
% 3.21/1.98 inference(ennf_transformation,[status(thm)],[f830])).
% 3.21/1.98 thf(f830,plain,(
% 3.21/1.98 ~((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) => ! [X0 : $i,X1 : $i] : (((breln1 @ X1 @ X0) = $true) => ! [X2 : $i] : (((breln1 @ X1 @ X2) = $true) => ! [X3 : $i] : (((in @ X3 @ X1) = $true) => ! [X4 : $i] : (((in @ X4 @ X1) = $true) => ((((in @ (kpair @ X3 @ X4) @ X0) = $true) | ((in @ (kpair @ X3 @ X4) @ X2) = $true)) => ((in @ (kpair @ X3 @ X4) @ (binunion @ X0 @ X2)) = $true)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 3.21/1.98 inference(fool_elimination,[status(thm)],[f829])).
% 3.21/1.98 thf(f829,plain,(
% 3.21/1.98 ~(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 => ! [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) @ X2) | (in @ (kpair @ X3 @ X4) @ X0)) => (in @ (kpair @ X3 @ X4) @ (binunion @ X0 @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 3.21/1.98 inference(rectify,[status(thm)],[f305])).
% 3.21/1.98 thf(f305,negated_conjecture,(
% 3.21/1.98 ~(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 => ! [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) @ X15) | (in @ (kpair @ X1 @ X2) @ X13)) => (in @ (kpair @ X1 @ X2) @ (binunion @ X13 @ X15))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 3.21/1.98 inference(negated_conjecture,[status(cth)],[f304])).
% 3.21/1.98 thf(f304,conjecture,(
% 3.21/1.98 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 => ! [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) @ X15) | (in @ (kpair @ X1 @ X2) @ X13)) => (in @ (kpair @ X1 @ X2) @ (binunion @ X13 @ X15)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 3.21/1.98 file('/export/starexec/sandbox/tmp/tmp.uLFW0IB354/DTF2THF_24330.p',breln1unionI)).
% 3.21/1.98 thf(f4805,plain,(
% 3.21/1.98 (binunionIR != $true) | ((in @ (kpair @ sK259 @ sK260) @ sK258) != $true)),
% 3.21/1.98 inference(trivial_inequality_removal,[status(thm)],[f4795])).
% 3.21/1.98 thf(f4795,plain,(
% 3.21/1.98 ($true != $true) | (binunionIR != $true) | ((in @ (kpair @ sK259 @ sK260) @ sK258) != $true)),
% 3.21/1.98 inference(superposition,[status(thm)],[f3466,f3100])).
% 3.21/1.98 thf(f3100,plain,(
% 3.21/1.98 ( ! [X3 : $i,X4 : $i,X5 : $i] : (((in @ X5 @ (binunion @ X4 @ X3)) = $true) | ((in @ X5 @ X3) != $true) | (binunionIR != $true)) )),
% 3.21/1.98 inference(cnf_transformation,[status(thm)],[f1600])).
% 3.21/1.98 thf(f1600,plain,(
% 3.21/1.98 ((binunionIR = $true) | (((in @ sK182 @ sK180) = $true) & ((in @ sK182 @ (binunion @ sK181 @ sK180)) != $true))) & (! [X3 : $i,X4 : $i,X5 : $i] : (((in @ X5 @ X3) != $true) | ((in @ X5 @ (binunion @ X4 @ X3)) = $true)) | (binunionIR != $true))),
% 3.21/1.98 inference(skolemisation,[status(esa),new_symbols(skolem,[sK180,sK181,sK182])],[f1598,f1599])).
% 3.21/1.98 thf(f1599,plain,(
% 3.21/1.98 ? [X0 : $i,X1 : $i,X2 : $i] : (((in @ X2 @ X0) = $true) & ((in @ X2 @ (binunion @ X1 @ X0)) != $true)) => (((in @ sK182 @ sK180) = $true) & ((in @ sK182 @ (binunion @ sK181 @ sK180)) != $true))),
% 3.21/1.98 introduced(definition,[],[choice_axiom])).
% 3.21/1.98 thf(f1598,plain,(
% 3.21/1.98 ((binunionIR = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (((in @ X2 @ X0) = $true) & ((in @ X2 @ (binunion @ X1 @ X0)) != $true))) & (! [X3 : $i,X4 : $i,X5 : $i] : (((in @ X5 @ X3) != $true) | ((in @ X5 @ (binunion @ X4 @ X3)) = $true)) | (binunionIR != $true))),
% 3.21/1.98 inference(rectify,[status(thm)],[f1597])).
% 3.21/1.98 thf(f1597,plain,(
% 3.21/1.98 ((binunionIR = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (((in @ X2 @ X0) = $true) & ((in @ X2 @ (binunion @ X1 @ X0)) != $true))) & (! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X2 @ X0) != $true) | ((in @ X2 @ (binunion @ X1 @ X0)) = $true)) | (binunionIR != $true))),
% 3.21/1.98 inference(nnf_transformation,[status(thm)],[f1025])).
% 3.21/1.98 thf(f1025,plain,(
% 3.21/1.98 (binunionIR = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X2 @ X0) != $true) | ((in @ X2 @ (binunion @ X1 @ X0)) = $true))),
% 3.21/1.98 inference(ennf_transformation,[status(thm)],[f436])).
% 3.21/1.98 thf(f436,plain,(
% 3.21/1.98 (binunionIR = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X2 @ X0) = $true) => ((in @ X2 @ (binunion @ X1 @ X0)) = $true))),
% 3.21/1.98 inference(fool_elimination,[status(thm)],[f435])).
% 3.21/1.98 thf(f435,plain,(
% 3.21/1.98 (! [X0 : $i,X1 : $i,X2 : $i] : ((in @ X2 @ X0) => (in @ X2 @ (binunion @ X1 @ X0))) = binunionIR)),
% 3.21/1.98 inference(rectify,[status(thm)],[f103])).
% 3.21/1.98 thf(f103,axiom,(
% 3.21/1.98 (! [X4 : $i,X3 : $i,X1 : $i] : ((in @ X1 @ X4) => (in @ X1 @ (binunion @ X3 @ X4))) = binunionIR)),
% 3.21/1.98 file('/export/starexec/sandbox/tmp/tmp.uLFW0IB354/DTF2THF_24330.p',binunionIR)).
% 3.21/1.98 thf(f3466,plain,(
% 3.21/1.98 ($true != (in @ (kpair @ sK259 @ sK260) @ (binunion @ sK256 @ sK258)))),
% 3.21/1.98 inference(cnf_transformation,[status(thm)],[f1716])).
% 3.21/1.98 thf(f4821,plain,(
% 3.21/1.98 ~spl1012_2),
% 3.21/1.98 inference(avatar_split_clause,[status(thm)],[f4820,f4738])).
% 3.21/1.98 thf(f4738,definition,(
% 3.21/1.98 spl1012_2 <=> ((in @ (kpair @ sK259 @ sK260) @ sK256) = $true)),
% 3.21/1.98 introduced(definition,[new_symbols(naming,[spl1012_2])],[avatar_definition])).
% 3.21/1.98 thf(f4820,plain,(
% 3.21/1.98 ((in @ (kpair @ sK259 @ sK260) @ sK256) != $true)),
% 3.21/1.98 inference(trivial_inequality_removal,[status(thm)],[f4819])).
% 3.21/1.98 thf(f4819,plain,(
% 3.21/1.98 ((in @ (kpair @ sK259 @ sK260) @ sK256) != $true) | ($true != $true)),
% 3.21/1.98 inference(forward_demodulation,[status(thm)],[f4810,f3362])).
% 3.21/1.98 thf(f3362,plain,(
% 3.21/1.98 (binunionIL = $true)),
% 3.21/1.98 inference(cnf_transformation,[status(thm)],[f1716])).
% 3.21/1.98 thf(f4810,plain,(
% 3.21/1.98 ((in @ (kpair @ sK259 @ sK260) @ sK256) != $true) | (binunionIL != $true)),
% 3.21/1.98 inference(trivial_inequality_removal,[status(thm)],[f4797])).
% 3.21/1.98 thf(f4797,plain,(
% 3.21/1.98 ($true != $true) | ((in @ (kpair @ sK259 @ sK260) @ sK256) != $true) | (binunionIL != $true)),
% 3.21/1.98 inference(superposition,[status(thm)],[f3466,f3669])).
% 3.21/1.98 thf(f3669,plain,(
% 3.21/1.98 ( ! [X2 : $i,X0 : $i,X1 : $i] : (((in @ X0 @ (binunion @ X2 @ X1)) = $true) | (binunionIL != $true) | ((in @ X0 @ X2) != $true)) )),
% 3.21/1.98 inference(cnf_transformation,[status(thm)],[f1894])).
% 3.21/1.98 thf(f1894,plain,(
% 3.21/1.98 (! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X0 @ (binunion @ X2 @ X1)) = $true) | ((in @ X0 @ X2) != $true)) | (binunionIL != $true)) & ((binunionIL = $true) | (((in @ sK372 @ (binunion @ sK374 @ sK373)) != $true) & ((in @ sK372 @ sK374) = $true)))),
% 3.21/1.98 inference(skolemisation,[status(esa),new_symbols(skolem,[sK372,sK373,sK374])],[f1892,f1893])).
% 3.21/1.98 thf(f1893,plain,(
% 3.21/1.98 ? [X3 : $i,X4 : $i,X5 : $i] : (($true != (in @ X3 @ (binunion @ X5 @ X4))) & ((in @ X3 @ X5) = $true)) => (((in @ sK372 @ (binunion @ sK374 @ sK373)) != $true) & ((in @ sK372 @ sK374) = $true))),
% 3.21/1.98 introduced(definition,[],[choice_axiom])).
% 3.21/1.98 thf(f1892,plain,(
% 3.21/1.98 (! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X0 @ (binunion @ X2 @ X1)) = $true) | ((in @ X0 @ X2) != $true)) | (binunionIL != $true)) & ((binunionIL = $true) | ? [X3 : $i,X4 : $i,X5 : $i] : (($true != (in @ X3 @ (binunion @ X5 @ X4))) & ((in @ X3 @ X5) = $true)))),
% 3.21/1.98 inference(rectify,[status(thm)],[f1891])).
% 3.21/1.98 thf(f1891,plain,(
% 3.21/1.98 (! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X0 @ (binunion @ X2 @ X1)) = $true) | ((in @ X0 @ X2) != $true)) | (binunionIL != $true)) & ((binunionIL = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (((in @ X0 @ (binunion @ X2 @ X1)) != $true) & ((in @ X0 @ X2) = $true)))),
% 3.21/1.98 inference(nnf_transformation,[status(thm)],[f1034])).
% 3.21/1.98 thf(f1034,plain,(
% 3.21/1.98 ! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X0 @ (binunion @ X2 @ X1)) = $true) | ((in @ X0 @ X2) != $true)) <=> (binunionIL = $true)),
% 3.21/1.98 inference(ennf_transformation,[status(thm)],[f364])).
% 3.21/1.98 thf(f364,plain,(
% 3.21/1.98 ! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X0 @ X2) = $true) => ((in @ X0 @ (binunion @ X2 @ X1)) = $true)) <=> (binunionIL = $true)),
% 3.21/1.98 inference(fool_elimination,[status(thm)],[f363])).
% 3.21/1.98 thf(f363,plain,(
% 3.21/1.98 (! [X0 : $i,X1 : $i,X2 : $i] : ((in @ X0 @ X2) => (in @ X0 @ (binunion @ X2 @ X1))) = binunionIL)),
% 3.21/1.98 inference(rectify,[status(thm)],[f101])).
% 3.21/1.98 thf(f101,axiom,(
% 3.21/1.98 (! [X1 : $i,X4 : $i,X3 : $i] : ((in @ X1 @ X3) => (in @ X1 @ (binunion @ X3 @ X4))) = binunionIL)),
% 3.21/1.98 file('/export/starexec/sandbox/tmp/tmp.uLFW0IB354/DTF2THF_24330.p',binunionIL)).
% 3.21/1.98 thf(f4741,plain,(
% 3.21/1.98 spl1012_1 | spl1012_2),
% 3.21/1.98 inference(avatar_split_clause,[status(thm)],[f3464,f4738,f4734])).
% 3.21/1.98 thf(f3464,plain,(
% 3.21/1.98 ((in @ (kpair @ sK259 @ sK260) @ sK256) = $true) | ((in @ (kpair @ sK259 @ sK260) @ sK258) = $true)),
% 3.21/1.98 inference(cnf_transformation,[status(thm)],[f1716])).
% 3.21/1.98 % SZS output end Proof for DTF2THF_24330
% 3.21/1.98 % (24649)------------------------------
% 3.21/1.98 % (24649)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.21/1.98 % (24649)Termination reason: Refutation
% 3.21/1.98
% 3.21/1.98 % (24649)Memory used [KB]: 11513
% 3.21/1.98 % (24649)Time elapsed: 0.153 s
% 3.21/1.98 % (24649)Instructions burned: 450 (million)
% 3.21/1.98 % (24649)------------------------------
% 3.21/1.98 % (24649)------------------------------
% 3.21/1.98 % (24585)Success in time 0.248 s
%------------------------------------------------------------------------------