%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU809^1 : TPTP v9.2.1. Released v3.7.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n026.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 09:37:23 AM UTC 2026
% Result : Theorem 5.26s 2.08s
% Output : Refutation 5.26s
% Verified :
% SZS Type : Refutation
% Derivation depth : 13
% Number of leaves : 4
% Syntax : Number of formulae : 26 ( 8 unt; 0 typ; 0 def)
% Number of atoms : 5124 (1357 equ; 0 cnn)
% Maximal formula atoms : 329 ( 197 avg)
% Number of connectives : 2493 ( 32 ~; 16 |; 988 &; 146 @)
% ( 2 <=>;1309 =>; 0 <=; 0 <~>)
% Maximal formula depth : 333 ( 91 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 569 ( 566 usr; 330 con; 0-6 aty)
% Number of variables : 18 ( 0 ^; 12 !; 6 ?; 18 :)
% 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_360,type,
image1: $i > ( $i > $i ) > $i ).
thf(func_def_364,type,
injective: $i > $i > $i > $o ).
thf(func_def_365,type,
injFuncSet: $i > $i > $i ).
thf(func_def_369,type,
surjective: $i > $i > $i > $o ).
thf(func_def_370,type,
surjFuncSet: $i > $i > $i ).
thf(func_def_378,type,
omegaS: $i > $i ).
thf(func_def_394,type,
sP0: $i > $i > $i > $o ).
thf(func_def_397,type,
sP3: $i > $i > $o ).
thf(func_def_398,type,
sP4: $i > $o ).
thf(func_def_399,type,
sP5: $i > $i > $o ).
thf(func_def_400,type,
sP6: $i > $i > $o ).
thf(func_def_404,type,
sP10: $i > $i > $i > $o ).
thf(func_def_406,type,
sP12: $i > $i > $i > $i > $o ).
thf(func_def_410,type,
sP16: $i > $o > $i > $i > $o ).
thf(func_def_412,type,
sK18: $i > $o ).
thf(func_def_415,type,
sK21: $i > ( $i > $o ) > $i ).
thf(func_def_416,type,
sK22: $i > ( $i > $o ) > $i ).
thf(func_def_417,type,
sK23: $i > $o ).
thf(func_def_425,type,
sK31: $i > $i > $i ).
thf(func_def_432,type,
sK38: $i > $i ).
thf(func_def_436,type,
sK42: $i > $o ).
thf(func_def_443,type,
sK49: $i > $i > $o ).
thf(func_def_444,type,
sK50: $i > ( $i > $o ) > $i ).
thf(func_def_445,type,
sK51: $i > $o ).
thf(func_def_455,type,
sK61: $i > $i ).
thf(func_def_458,type,
sK64: $i > ( $i > $i ) > $i > $i ).
thf(func_def_459,type,
sK65: $i > $i > ( $i > $i ) > $i > $i ).
thf(func_def_467,type,
sK73: $i > $i > $o ).
thf(func_def_481,type,
sK87: $i > $i > ( $i > $o ) > $i ).
thf(func_def_482,type,
sK88: $i > $o ).
thf(func_def_497,type,
sK103: $i > $i ).
thf(func_def_509,type,
sK115: $i > $i > $i > $i ).
thf(func_def_510,type,
sK116: $i > $i > $i > $i ).
thf(func_def_516,type,
sK122: $i > $i > $i > $i ).
thf(func_def_520,type,
sK126: $i > $i > $i > $i ).
thf(func_def_533,type,
sK139: $i > $o ).
thf(func_def_534,type,
sK140: $i > $i > $o ).
thf(func_def_547,type,
sK153: $i > $i > $o ).
thf(func_def_557,type,
sK163: $i > $i > ( $i > $i ) > $i ).
thf(func_def_558,type,
sK164: $i > $i ).
thf(func_def_566,type,
sK172: $i > $i > $i ).
thf(func_def_599,type,
sK205: $i > $i > $i ).
thf(func_def_604,type,
sK210: $i > $i > $i ).
thf(func_def_605,type,
sK211: $i > $i > $i ).
thf(func_def_607,type,
sK213: $i > $o ).
thf(func_def_624,type,
sK230: $i > $o ).
thf(func_def_642,type,
sK248: $i > $i > $i > $i ).
thf(func_def_646,type,
sK252: $i > $i > $i ).
thf(func_def_669,type,
sK275: $i > $i > $i > $i > $i > $i ).
thf(func_def_677,type,
sK283: $i > $i ).
thf(func_def_694,type,
sK300: $i > $i ).
thf(func_def_695,type,
sK301: ( $i > $i ) > $i > $i > $i ).
thf(func_def_703,type,
sK309: $i > $i > $i ).
thf(func_def_704,type,
sK310: $i > $i > $i ).
thf(func_def_705,type,
sK311: $i > $i > $i ).
thf(func_def_706,type,
sK312: $i > $i > $i ).
thf(func_def_707,type,
sK313: $i > $i > $i > $i ).
thf(func_def_708,type,
sK314: $i > $i ).
thf(func_def_709,type,
sK315: $i > $i ).
thf(func_def_710,type,
sK316: $i > $i ).
thf(func_def_711,type,
sK317: $i > $i ).
thf(func_def_712,type,
sK318: $i > $i > $i > $i ).
thf(func_def_713,type,
sK319: $i > $i > $i > $i ).
thf(func_def_714,type,
sK320: $i > $i > $i ).
thf(func_def_715,type,
sK321: $i > $i > $i ).
thf(func_def_716,type,
sK322: $i > $i > $i ).
thf(func_def_717,type,
sK323: $i > $i ).
thf(func_def_718,type,
sK324: $i > $i > $i ).
thf(func_def_720,type,
sK326: $i > $i ).
thf(func_def_721,type,
sK327: $i > $i ).
thf(func_def_742,type,
sK348: $i > $i ).
thf(func_def_750,type,
sK356: $i > $o ).
thf(func_def_751,type,
sK357: ( $i > $o ) > $i ).
thf(func_def_758,type,
sK364: $i > $o ).
thf(func_def_766,type,
sK372: $i > $i ).
thf(func_def_769,type,
sK375: $i > $i > ( $i > $i ) > $i ).
thf(func_def_771,type,
sK377: $i > ( $i > $i ) > $i > $i ).
thf(func_def_773,type,
sK379: $i > $i ).
thf(func_def_778,type,
sK384: $i > $i > $o ).
thf(func_def_784,type,
sK390: $i > $i > $i ).
thf(func_def_807,type,
sK413: $i > $i > $i > $i > $i ).
thf(func_def_829,type,
sK435: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_830,type,
sK436: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_831,type,
sK437: $i > $o ).
thf(func_def_832,type,
sK438: $i > $o ).
thf(func_def_837,type,
sK443: $i > $i ).
thf(func_def_838,type,
sK444: $i > $i ).
thf(func_def_843,type,
sK449: $i > $i > $i > $i > $i ).
thf(func_def_884,type,
sK490: $i > $o ).
thf(func_def_893,type,
sK499: $i > $o ).
thf(func_def_894,type,
sK500: $i > $o ).
thf(func_def_895,type,
sK501: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_896,type,
sK502: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_908,type,
sK514: $o > $i > $i > $i > $i > $i > $i ).
thf(func_def_935,type,
sK541: $i > $o ).
thf(func_def_938,type,
sK544: $i > ( $i > $o ) > $i ).
thf(func_def_939,type,
sK545: $i > ( $i > $o ) > $i ).
thf(func_def_941,type,
sK547: $i > $o ).
thf(func_def_944,type,
sK550: $i > ( $i > $o ) > $i ).
thf(func_def_945,type,
sK551: $i > $o ).
thf(func_def_948,type,
sK554: $i > $o ).
thf(func_def_951,type,
sK557: $i > $o ).
thf(func_def_952,type,
sK558: $i > $i ).
thf(func_def_953,type,
sK559: ( $i > $o ) > $i ).
thf(func_def_969,type,
sK575: $i > $i ).
thf(func_def_971,type,
sK577: $i > ( $i > $o ) > $i ).
thf(func_def_972,type,
sK578: $i > $o ).
thf(func_def_976,type,
sK582: $i > $i > $o ).
thf(func_def_988,type,
sK594: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_989,type,
sK595: $i > $i > $o ).
thf(func_def_992,type,
sK598: $i > $i ).
thf(func_def_996,type,
sK602: $i > $i > $o ).
thf(func_def_1001,type,
sK607: $i > $i ).
thf(func_def_1002,type,
sK608: $i > $i ).
thf(func_def_1003,type,
sK609: $i > $i ).
thf(func_def_1004,type,
sK610: ( $i > $i ) > $i > $i ).
thf(func_def_1005,type,
sK611: $i > ( $i > $i ) > $i > $i ).
thf(func_def_1017,type,
sK623: $i > $o ).
thf(func_def_1019,type,
sK625: $i > ( $i > $o ) > $i ).
thf(func_def_1024,type,
sK630: $i > $i ).
thf(func_def_1033,type,
sK639: $i > $i ).
thf(func_def_1036,type,
sK642: $i > ( $i > $i ) > $i > $i ).
thf(func_def_1037,type,
sK643: $i > $o ).
thf(func_def_1039,type,
sK645: ( $i > $o ) > $i ).
thf(func_def_1040,type,
sK646: ( $i > $o ) > $i ).
thf(func_def_1043,type,
sK649: $i > ( $i > $o ) > $i ).
thf(func_def_1044,type,
sK650: $i > $o ).
thf(func_def_1061,type,
sK667: $i > $i ).
thf(func_def_1065,type,
sK671: $i > $i > ( $i > $i ) > $i ).
thf(func_def_1069,type,
sK675: $i > $i ).
thf(func_def_1072,type,
sK678: $i > $i > ( $i > $i ) > $i ).
thf(func_def_1076,type,
sK682: $i > $i > $i > $i ).
thf(func_def_1077,type,
sK683: $i > $i > $i > $i ).
thf(func_def_1078,type,
sK684: $i > $i > $i > $i ).
thf(func_def_1079,type,
sK685: $i > $i > $i > $i ).
thf(func_def_1089,type,
sK695: $i > $i > $i > $i ).
thf(func_def_1097,type,
sK703: $i > $i > $i > $i > $i ).
thf(func_def_1098,type,
sK704: $i > $i > $i > $i > $i ).
thf(func_def_1099,type,
sK705: $i > $i > $i > $i > $i ).
thf(func_def_1100,type,
sK706: $i > $i > $i > $i > $i ).
thf(func_def_1103,type,
sK709: $i > $i > $i ).
thf(func_def_1111,type,
sK717: $i > $i ).
thf(func_def_1114,type,
sK720: $i > $i > $i ).
thf(func_def_1133,type,
sK739: $i > $i ).
thf(func_def_1135,type,
sK741: $i > ( $i > $i ) > $i > $i ).
thf(func_def_1150,type,
sK756: $o > $i > $i > $i ).
thf(func_def_1206,type,
sK812: $i > $i > $i ).
thf(func_def_1209,type,
sK815: $i > $o ).
thf(func_def_1217,type,
sK823: $i > $o ).
thf(func_def_1220,type,
sK826: $i > ( $i > $o ) > $i ).
thf(func_def_1221,type,
sK827: $i > ( $i > $o ) > $i ).
thf(func_def_1232,type,
sK838: $i > $o ).
thf(func_def_1234,type,
sK840: ( $i > $o ) > $i > $i > $i ).
thf(func_def_1235,type,
sK841: ( $i > $o ) > $i > $i > $i ).
thf(func_def_1236,type,
sK842: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_1237,type,
sK843: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_1241,type,
sK847: $i > $o ).
thf(func_def_1256,type,
sK862: $i > $o ).
thf(func_def_1258,type,
sK864: ( $i > $o ) > $i > $i > $i ).
thf(func_def_1259,type,
sK865: ( $i > $o ) > $i > $i > $i ).
thf(func_def_1277,type,
sK883: $i > $i ).
thf(func_def_1281,type,
sK887: $i > $o ).
thf(func_def_1284,type,
sK890: $i > ( $i > $o ) > $i ).
thf(func_def_1296,type,
sK902: $i > ( $i > $o ) > $i ).
thf(func_def_1297,type,
sK903: $i > $o ).
thf(func_def_1307,type,
sK913: $i > $i > $i > $i > $i ).
thf(func_def_1308,type,
sK914: $i > $i > $i > $i > $i ).
thf(func_def_1319,type,
sK925: $i > $o ).
thf(func_def_1321,type,
sK927: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_1322,type,
sK928: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_1323,type,
sK929: $i > $i > $o ).
thf(func_def_1325,type,
sK931: $i > $i ).
thf(func_def_1326,type,
sK932: $i > $i ).
thf(func_def_1327,type,
sK933: $i > ( $i > $i > $o ) > $i ).
thf(func_def_1328,type,
sK934: $i > ( $i > $i > $o ) > $i ).
thf(func_def_1329,type,
sK935: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_1330,type,
sK936: $i > $o ).
thf(func_def_1341,type,
sK947: $i > $i > $i > $i > $i ).
thf(func_def_1357,type,
sK963: $i > $o ).
thf(func_def_1361,type,
sK967: $i > $i > $o ).
thf(func_def_1366,type,
sK972: $i > $i > $i > $i ).
thf(func_def_1367,type,
sK973: $i > $i > $i > $i ).
thf(func_def_1374,type,
sK980: $i > $o ).
thf(func_def_1390,type,
sK996: $i > $o ).
thf(func_def_1439,type,
sK1045: $i > $o ).
thf(func_def_1440,type,
sK1046: $i > $i ).
thf(func_def_1441,type,
sK1047: ( $i > $o ) > $i ).
thf(func_def_1466,type,
sK1072: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_1467,type,
sK1073: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_1470,type,
sK1076: $i > $o ).
thf(func_def_1471,type,
sK1077: $i > $o ).
thf(func_def_1486,type,
ph1092:
!>[X0: $tType] : X0 ).
thf(f5176,plain,
$false,
inference(trivial_inequality_removal,[status(thm)],[f5169]) ).
thf(f5169,plain,
$true != $true,
inference(superposition,[status(thm)],[f5152,f4125]) ).
thf(f4125,plain,
( $true
= ( in @ sK655 @ omega ) ),
inference(cnf_transformation,[status(thm)],[f2443]) ).
thf(f2443,plain,
( ( setminusER = $true )
& ( wellorderingAx = $true )
& ( setukpairinjL1 = $true )
& ( image1E = $true )
& ( inComplementUnionImpNotIn1 = $true )
& ( notinself = $true )
& ( setminusI = $true )
& ( upairequniteq = $true )
& ( surjFuncSetFuncIn = $true )
& ( setunionI = $true )
& ( setadjoinSub2 = $true )
& ( quantDeMorgan4 = $true )
& ( ksndsingleton = $true )
& ( binunionTIRcontra = $true )
& ( emptyset__Cong = $true )
& ( iftrueProp1 = $true )
& ( funcImageSingleton = $true )
& ( setOfPairsIsBReln1 = $true )
& ( powersetTI1 = $true )
& ( quantDeMorgan1 = $true )
& ( demorgan2a1 = $true )
& ( setunionsingleton2 = $true )
& ( ap2apEq2 = $true )
& ( surjCantorThm = $true )
& ( inIntersectImpInUnion = $true )
& ( upairset2IR = $true )
& ( setextT = $true )
& ( iftrueProp2 = $true )
& ( theprop = $true )
& ( woz1_5 = $true )
& ( subsetTrans = $true )
& ( setextAx = $true )
& ( complementTcontraSubset = $true )
& ( omegaIndAx = $true )
& ( exuE3e = $true )
& ( emptyInPowerset = $true )
& ( kpairiskpair = $true )
& ( binintersectEL = $true )
& ( lam2lamEq = $true )
& ( notinemptyset = $true )
& ( setadjoinOr = $true )
& ( singletonsswitch = $true )
& ( complementTE1 = $true )
& ( setadjoinIR = $true )
& ( notsubsetI = $true )
& ( setunionsingleton = $true )
& ( setextsub = $true )
& ( funcGraphProp3 = $true )
& ( setOfPairsIsBReln = $true )
& ( ubforcartprodlem1 = $true )
& ( setbeta = $true )
& ( emptyinPowerset = $true )
& ( cartprodpairmemEL = $true )
& ( binintersectSubset1 = $true )
& ( contrasubsetT1 = $true )
& ( emptyE1 = $true )
& ( binintersectT_lem = $true )
& ( subsetRefl = $true )
& ( symdiffI2 = $true )
& ( ex1I = $true )
& ( omegaSAx = $true )
& ( ap2apEq1 = $true )
& ( eqimpsubset1 = $true )
& ( powerset__Cong = $true )
& ( singletonprop = $true )
& ( funcext = $true )
& ( funcext2 = $true )
& ( binunionTE = $true )
& ( subbreln1 = $true )
& ( injFuncSetFuncIn = $true )
& ( demorgan2 = $true )
& ( woz1_1 = $true )
& ( image1Ex = $true )
& ( ex1I2 = $true )
& ( emptyI = $true )
& ( cartprodfstpairEq = $true )
& ( woz1_2 = $true )
& ( breln1unionEcases = $true )
& ( breln1unionE = $true )
& ( complementTI1 = $true )
& ( binintersectSubset2 = $true )
& ( exuI1 = $true )
& ( nonemptyE1 = $true )
& ( powersetAx = $true )
& ( upairinpowunion = $true )
& ( omegaSp = $true )
& ( sepInPowerset = $true )
& ( notdallE = $true )
& ( powersetsubset = $true )
& ( woz1_3 = $true )
& ( cartprodsndpairEq = $true )
& ( setukpairinjR = $true )
& ( kpairp = $true )
& ( iffalseProp1 = $true )
& ( complementUnionInPowersetComplement = $true )
& ( contrasubsetT = $true )
& ( powersetI = $true )
& ( complementImpComplementIntersect = $true )
& ( setminusSubset1 = $true )
& ( setunionAx = $true )
& ( sepSubset = $true )
& ( setadjoinE = $true )
& ( breln1compI = $true )
& ( complementT_lem = $true )
& ( setukpairIL = $true )
& ( eqbreln = $true )
& ( exuE2 = $true )
& ( singletoninpowunion = $true )
& ( symdiffIneg2 = $true )
& ( doubleComplementI1 = $true )
& ( inIntersectImpInUnion2 = $true )
& ( subsetE = $true )
& ( eta2 = $true )
& ( setadjoinAx = $true )
& ( subPowSU = $true )
& ( doubleComplementSub2 = $true )
& ( lam2p = $true )
& ( intersectInPowersetIntersectUnions = $true )
& ( iffalseProp2 = $true )
& ( secondinupair = $true )
& ( setukpairinjR12 = $true )
& ( upairsubunion = $true )
& ( demorgan1 = $true )
& ( emptysetimpfalse = $true )
& ( cartprodpairin = $true )
& ( kfstpairEq = $true )
& ( eta1 = $true )
& ( powersetE = $true )
& ( demorgan2a2 = $true )
& ( breln1invE = $true )
& ( injFuncSetFuncInj = $true )
& ( setunion__Cong = $true )
& ( cartprodfstin = $true )
& ( contrasubsetT3 = $true )
& ( setunionsingleton1 = $true )
& ( emptysetAx = $true )
& ( quantDeMorgan2 = $true )
& ( dsetconstrEL = $true )
& ( setminusERneg = $true )
& ( dpsetconstrERa = $true )
& ( eqimpsubset2 = $true )
& ( theeq = $true )
& ( symdiffE = $true )
& ( exu__Cong = $true )
& ( choice2fnsingleton = $true )
& ( demorgan2b = $true )
& ( powersetT_lem = $true )
& ( replAx = $true )
& ( iftrueorfalse = $true )
& ( inIntersectImpInIntersectUnions = $true )
& ( exuEu = $true )
& ( noeltsimpempty = $true )
& ( binunionEcases = $true )
& ( binunionIL = $true )
& ( dsetconstrI = $true )
& ( exuE3u = $true )
& ( ex1E2 = $true )
& ( subbreln = $true )
& ( breln1unionI = $true )
& ( funcinfuncset = $true )
& ( emptyinunitempty = $true )
& ( breln1unionIR = $true )
& ( notequalI1 = $true )
& ( setunionE2 = $true )
& ( omega0Ax = $true )
& ( powersetI1 = $true )
& ( setext = $true )
& ( doubleComplementSub1 = $true )
& ( woz2W = $true )
& ( binintersectSubset5 = $true )
& ( subsetI1 = $true )
& ( setukpairinjR2 = $true )
& ( ex1E1 = $true )
& ( cartprodmempair = $true )
& ( iftrue = $true )
& ( foundationAx = $true )
& ( notequalI2 = $true )
& ( nonemptyImpWitness = $true )
& ( binunionIR = $true )
& ( ksndpairEq = $true )
& ( subsetTI = $true )
& ( funcextLem = $true )
& ( dpsetconstrI = $true )
& ( binintersectI = $true )
& ( descrp = $true )
& ( image1Ex1 = $true )
& ( ifp = $true )
& ( binintersectTELcontra = $true )
& ( $true
= ( in @ sK655 @ omega ) )
& ( $true
!= ( in @ ( setadjoin @ sK655 @ sK655 ) @ omega ) )
& ( woz13rule2 = $true )
& ( upairsetE = $true )
& ( breln1unionprop = $true )
& ( funcGraphProp1 = $true )
& ( cartprodmempair1 = $true )
& ( lamProp = $true )
& ( woz13rule4 = $true )
& ( leftInvIsSurj = $true )
& ( demorgan1a = $true )
& ( image1I = $true )
& ( upairsetIR = $true )
& ( cartprodpairsurjEq = $true )
& ( breln1all2 = $true )
& ( ubforcartprodlem3 = $true )
& ( woz1_4 = $true )
& ( demorgan2a = $true )
& ( setunionE = $true )
& ( disjointsetsI1 = $true )
& ( doubleComplementE1 = $true )
& ( dpsetconstrER = $true )
& ( subset2powerset = $true )
& ( subsetE2 = $true )
& ( woz2Ex = $true )
& ( brelnall1 = $true )
& ( setukpairinjR1 = $true )
& ( symdiffIneg1 = $true )
& ( iffalse = $true )
& ( notinself2 = $true )
& ( beta2 = $true )
& ( exuI2 = $true )
& ( powersetE1 = $true )
& ( demorgan2b2 = $true )
& ( woz2B = $true )
& ( kpairsurjEq = $true )
& ( binunionTEcontra = $true )
& ( infuncsetfunc = $true )
& ( setminusELneg = $true )
& ( nonemptyI1 = $true )
& ( setoftrueEq = $true )
& ( breln1compprop = $true )
& ( binunionE = $true )
& ( cartprodsndin = $true )
& ( cartprodpairmemER = $true )
& ( singletonsuniq = $true )
& ( inCongP = $true )
& ( vacuousDall = $true )
& ( binintersectSubset4 = $true )
& ( emptysetE = $true )
& ( complementTnotintersectT = $true )
& ( in__Cong = $true )
& ( prop2setE = $true )
& ( dsetconstrER = $true )
& ( woz13rule0 = $true )
& ( subsetemptysetimpeq = $true )
& ( breln1invprop = $true )
& ( dpsetconstrEL1 = $true )
& ( app = $true )
& ( setminusIRneg = $true )
& ( eqinunit = $true )
& ( cartprodmempaircEq = $true )
& ( singletonsubset = $true )
& ( setukpairinjR11 = $true )
& ( setadjoinIL = $true )
& ( powersetTE1 = $true )
& ( prop2setI = $true )
& ( prop2set2propI = $true )
& ( complementSubsetComplementIntersect = $true )
& ( binintersectER = $true )
& ( uniqinunit = $true )
& ( binunionLsub = $true )
& ( setukpairinjL = $true )
& ( surjFuncSetFuncSurj = $true )
& ( beta1 = $true )
& ( notinsingleton = $true )
& ( lamp = $true )
& ( setukpairIR = $true )
& ( nonemptyI = $true )
& ( exuE1 = $true )
& ( setukpairinjL2 = $true )
& ( descr__Cong = $true )
& ( bs114d = $true )
& ( dpsetconstrEL2 = $true )
& ( doubleComplementEq = $true )
& ( dpsetconstrSub = $true )
& ( binunionRsub = $true )
& ( binintersectRsub = $true )
& ( breln1SetBreln1 = $true )
& ( demorgan1b = $true )
& ( setminusT_lem = $true )
& ( complementInPowersetComplementIntersect = $true )
& ( binunionTILcontra = $true )
& ( setminusILneg = $true )
& ( exuI3 = $true )
& ( ifSingleton = $true )
& ( setminusEL = $true )
& ( binintersectTERcontra = $true )
& ( breln1all1 = $true )
& ( singletoninpowerset = $true )
& ( inComplementUnionImpInComplement1 = $true )
& ( eqbreln1 = $true )
& ( inPowerset = $true )
& ( breln1unionCommutes = $true )
& ( brelnall2 = $true )
& ( setminusLsub = $true )
& ( contrasubsetT2 = $true )
& ( binintersectLsub = $true )
& ( breln1compEex = $true )
& ( contraSubsetComplement = $true )
& ( injFuncInInjFuncSet = $true )
& ( setadjoinSub = $true )
& ( breln1invI = $true )
& ( dsetconstr__Cong = $true )
& ( binunionT_lem = $true )
& ( breln1unionIL = $true )
& ( funcGraphProp4 = $true )
& ( ap2p = $true )
& ( omega__Cong = $true )
& ( foundation2 = $true )
& ( ubforcartprodlem2 = $true )
& ( upairsetIL = $true )
& ( setminusSubset2 = $true )
& ( breln1compE = $true )
& ( subsetI2 = $true )
& ( image1Equiv = $true )
& ( woz2A = $true )
& ( funcGraphProp2 = $true )
& ( symdiffI1 = $true )
& ( kfstsingleton = $true )
& ( quantDeMorgan3 = $true )
& ( emptysetsubset = $true )
& ( notdexE = $true )
& ( upairset2E = $true )
& ( apProp = $true )
& ( woz13rule3 = $true )
& ( setadjoin__Cong = $true )
& ( woz13rule1 = $true )
& ( binintersectSubset3 = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK655])],[f1072,f2442]) ).
thf(f2442,plain,
( ? [X0: $i] :
( ( ( in @ X0 @ omega )
= $true )
& ( $true
!= ( in @ ( setadjoin @ X0 @ X0 ) @ omega ) ) )
=> ( ( $true
= ( in @ sK655 @ omega ) )
& ( $true
!= ( in @ ( setadjoin @ sK655 @ sK655 ) @ omega ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1072,plain,
( ( setminusER = $true )
& ( wellorderingAx = $true )
& ( setukpairinjL1 = $true )
& ( image1E = $true )
& ( inComplementUnionImpNotIn1 = $true )
& ( notinself = $true )
& ( setminusI = $true )
& ( upairequniteq = $true )
& ( surjFuncSetFuncIn = $true )
& ( setunionI = $true )
& ( setadjoinSub2 = $true )
& ( quantDeMorgan4 = $true )
& ( ksndsingleton = $true )
& ( binunionTIRcontra = $true )
& ( emptyset__Cong = $true )
& ( iftrueProp1 = $true )
& ( funcImageSingleton = $true )
& ( setOfPairsIsBReln1 = $true )
& ( powersetTI1 = $true )
& ( quantDeMorgan1 = $true )
& ( demorgan2a1 = $true )
& ( setunionsingleton2 = $true )
& ( ap2apEq2 = $true )
& ( surjCantorThm = $true )
& ( inIntersectImpInUnion = $true )
& ( upairset2IR = $true )
& ( setextT = $true )
& ( iftrueProp2 = $true )
& ( theprop = $true )
& ( woz1_5 = $true )
& ( subsetTrans = $true )
& ( setextAx = $true )
& ( complementTcontraSubset = $true )
& ( omegaIndAx = $true )
& ( exuE3e = $true )
& ( emptyInPowerset = $true )
& ( kpairiskpair = $true )
& ( binintersectEL = $true )
& ( lam2lamEq = $true )
& ( notinemptyset = $true )
& ( setadjoinOr = $true )
& ( singletonsswitch = $true )
& ( complementTE1 = $true )
& ( setadjoinIR = $true )
& ( notsubsetI = $true )
& ( setunionsingleton = $true )
& ( setextsub = $true )
& ( funcGraphProp3 = $true )
& ( setOfPairsIsBReln = $true )
& ( ubforcartprodlem1 = $true )
& ( setbeta = $true )
& ( emptyinPowerset = $true )
& ( cartprodpairmemEL = $true )
& ( binintersectSubset1 = $true )
& ( contrasubsetT1 = $true )
& ( emptyE1 = $true )
& ( binintersectT_lem = $true )
& ( subsetRefl = $true )
& ( symdiffI2 = $true )
& ( ex1I = $true )
& ( omegaSAx = $true )
& ( ap2apEq1 = $true )
& ( eqimpsubset1 = $true )
& ( powerset__Cong = $true )
& ( singletonprop = $true )
& ( funcext = $true )
& ( funcext2 = $true )
& ( binunionTE = $true )
& ( subbreln1 = $true )
& ( injFuncSetFuncIn = $true )
& ( demorgan2 = $true )
& ( woz1_1 = $true )
& ( image1Ex = $true )
& ( ex1I2 = $true )
& ( emptyI = $true )
& ( cartprodfstpairEq = $true )
& ( woz1_2 = $true )
& ( breln1unionEcases = $true )
& ( breln1unionE = $true )
& ( complementTI1 = $true )
& ( binintersectSubset2 = $true )
& ( exuI1 = $true )
& ( nonemptyE1 = $true )
& ( powersetAx = $true )
& ( upairinpowunion = $true )
& ( omegaSp = $true )
& ( sepInPowerset = $true )
& ( notdallE = $true )
& ( powersetsubset = $true )
& ( woz1_3 = $true )
& ( cartprodsndpairEq = $true )
& ( setukpairinjR = $true )
& ( kpairp = $true )
& ( iffalseProp1 = $true )
& ( complementUnionInPowersetComplement = $true )
& ( contrasubsetT = $true )
& ( powersetI = $true )
& ( complementImpComplementIntersect = $true )
& ( setminusSubset1 = $true )
& ( setunionAx = $true )
& ( sepSubset = $true )
& ( setadjoinE = $true )
& ( breln1compI = $true )
& ( complementT_lem = $true )
& ( setukpairIL = $true )
& ( eqbreln = $true )
& ( exuE2 = $true )
& ( singletoninpowunion = $true )
& ( symdiffIneg2 = $true )
& ( doubleComplementI1 = $true )
& ( inIntersectImpInUnion2 = $true )
& ( subsetE = $true )
& ( eta2 = $true )
& ( setadjoinAx = $true )
& ( subPowSU = $true )
& ( doubleComplementSub2 = $true )
& ( lam2p = $true )
& ( intersectInPowersetIntersectUnions = $true )
& ( iffalseProp2 = $true )
& ( secondinupair = $true )
& ( setukpairinjR12 = $true )
& ( upairsubunion = $true )
& ( demorgan1 = $true )
& ( emptysetimpfalse = $true )
& ( cartprodpairin = $true )
& ( kfstpairEq = $true )
& ( eta1 = $true )
& ( powersetE = $true )
& ( demorgan2a2 = $true )
& ( breln1invE = $true )
& ( injFuncSetFuncInj = $true )
& ( setunion__Cong = $true )
& ( cartprodfstin = $true )
& ( contrasubsetT3 = $true )
& ( setunionsingleton1 = $true )
& ( emptysetAx = $true )
& ( quantDeMorgan2 = $true )
& ( dsetconstrEL = $true )
& ( setminusERneg = $true )
& ( dpsetconstrERa = $true )
& ( eqimpsubset2 = $true )
& ( theeq = $true )
& ( symdiffE = $true )
& ( exu__Cong = $true )
& ( choice2fnsingleton = $true )
& ( demorgan2b = $true )
& ( powersetT_lem = $true )
& ( replAx = $true )
& ( iftrueorfalse = $true )
& ( inIntersectImpInIntersectUnions = $true )
& ( exuEu = $true )
& ( noeltsimpempty = $true )
& ( binunionEcases = $true )
& ( binunionIL = $true )
& ( dsetconstrI = $true )
& ( exuE3u = $true )
& ( ex1E2 = $true )
& ( subbreln = $true )
& ( breln1unionI = $true )
& ( funcinfuncset = $true )
& ( emptyinunitempty = $true )
& ( breln1unionIR = $true )
& ( notequalI1 = $true )
& ( setunionE2 = $true )
& ( omega0Ax = $true )
& ( powersetI1 = $true )
& ( setext = $true )
& ( doubleComplementSub1 = $true )
& ( woz2W = $true )
& ( binintersectSubset5 = $true )
& ( subsetI1 = $true )
& ( setukpairinjR2 = $true )
& ( ex1E1 = $true )
& ( cartprodmempair = $true )
& ( iftrue = $true )
& ( foundationAx = $true )
& ( notequalI2 = $true )
& ( nonemptyImpWitness = $true )
& ( binunionIR = $true )
& ( ksndpairEq = $true )
& ( subsetTI = $true )
& ( funcextLem = $true )
& ( dpsetconstrI = $true )
& ( binintersectI = $true )
& ( descrp = $true )
& ( image1Ex1 = $true )
& ( ifp = $true )
& ( binintersectTELcontra = $true )
& ? [X0: $i] :
( ( ( in @ X0 @ omega )
= $true )
& ( $true
!= ( in @ ( setadjoin @ X0 @ X0 ) @ omega ) ) )
& ( woz13rule2 = $true )
& ( upairsetE = $true )
& ( breln1unionprop = $true )
& ( funcGraphProp1 = $true )
& ( cartprodmempair1 = $true )
& ( lamProp = $true )
& ( woz13rule4 = $true )
& ( leftInvIsSurj = $true )
& ( demorgan1a = $true )
& ( image1I = $true )
& ( upairsetIR = $true )
& ( cartprodpairsurjEq = $true )
& ( breln1all2 = $true )
& ( ubforcartprodlem3 = $true )
& ( woz1_4 = $true )
& ( demorgan2a = $true )
& ( setunionE = $true )
& ( disjointsetsI1 = $true )
& ( doubleComplementE1 = $true )
& ( dpsetconstrER = $true )
& ( subset2powerset = $true )
& ( subsetE2 = $true )
& ( woz2Ex = $true )
& ( brelnall1 = $true )
& ( setukpairinjR1 = $true )
& ( symdiffIneg1 = $true )
& ( iffalse = $true )
& ( notinself2 = $true )
& ( beta2 = $true )
& ( exuI2 = $true )
& ( powersetE1 = $true )
& ( demorgan2b2 = $true )
& ( woz2B = $true )
& ( kpairsurjEq = $true )
& ( binunionTEcontra = $true )
& ( infuncsetfunc = $true )
& ( setminusELneg = $true )
& ( nonemptyI1 = $true )
& ( setoftrueEq = $true )
& ( breln1compprop = $true )
& ( binunionE = $true )
& ( cartprodsndin = $true )
& ( cartprodpairmemER = $true )
& ( singletonsuniq = $true )
& ( inCongP = $true )
& ( vacuousDall = $true )
& ( binintersectSubset4 = $true )
& ( emptysetE = $true )
& ( complementTnotintersectT = $true )
& ( in__Cong = $true )
& ( prop2setE = $true )
& ( dsetconstrER = $true )
& ( woz13rule0 = $true )
& ( subsetemptysetimpeq = $true )
& ( breln1invprop = $true )
& ( dpsetconstrEL1 = $true )
& ( app = $true )
& ( setminusIRneg = $true )
& ( eqinunit = $true )
& ( cartprodmempaircEq = $true )
& ( singletonsubset = $true )
& ( setukpairinjR11 = $true )
& ( setadjoinIL = $true )
& ( powersetTE1 = $true )
& ( prop2setI = $true )
& ( prop2set2propI = $true )
& ( complementSubsetComplementIntersect = $true )
& ( binintersectER = $true )
& ( uniqinunit = $true )
& ( binunionLsub = $true )
& ( setukpairinjL = $true )
& ( surjFuncSetFuncSurj = $true )
& ( beta1 = $true )
& ( notinsingleton = $true )
& ( lamp = $true )
& ( setukpairIR = $true )
& ( nonemptyI = $true )
& ( exuE1 = $true )
& ( setukpairinjL2 = $true )
& ( descr__Cong = $true )
& ( bs114d = $true )
& ( dpsetconstrEL2 = $true )
& ( doubleComplementEq = $true )
& ( dpsetconstrSub = $true )
& ( binunionRsub = $true )
& ( binintersectRsub = $true )
& ( breln1SetBreln1 = $true )
& ( demorgan1b = $true )
& ( setminusT_lem = $true )
& ( complementInPowersetComplementIntersect = $true )
& ( binunionTILcontra = $true )
& ( setminusILneg = $true )
& ( exuI3 = $true )
& ( ifSingleton = $true )
& ( setminusEL = $true )
& ( binintersectTERcontra = $true )
& ( breln1all1 = $true )
& ( singletoninpowerset = $true )
& ( inComplementUnionImpInComplement1 = $true )
& ( eqbreln1 = $true )
& ( inPowerset = $true )
& ( breln1unionCommutes = $true )
& ( brelnall2 = $true )
& ( setminusLsub = $true )
& ( contrasubsetT2 = $true )
& ( binintersectLsub = $true )
& ( breln1compEex = $true )
& ( contraSubsetComplement = $true )
& ( injFuncInInjFuncSet = $true )
& ( setadjoinSub = $true )
& ( breln1invI = $true )
& ( dsetconstr__Cong = $true )
& ( binunionT_lem = $true )
& ( breln1unionIL = $true )
& ( funcGraphProp4 = $true )
& ( ap2p = $true )
& ( omega__Cong = $true )
& ( foundation2 = $true )
& ( ubforcartprodlem2 = $true )
& ( upairsetIL = $true )
& ( setminusSubset2 = $true )
& ( breln1compE = $true )
& ( subsetI2 = $true )
& ( image1Equiv = $true )
& ( woz2A = $true )
& ( funcGraphProp2 = $true )
& ( symdiffI1 = $true )
& ( kfstsingleton = $true )
& ( quantDeMorgan3 = $true )
& ( emptysetsubset = $true )
& ( notdexE = $true )
& ( upairset2E = $true )
& ( apProp = $true )
& ( woz13rule3 = $true )
& ( setadjoin__Cong = $true )
& ( woz13rule1 = $true )
& ( binintersectSubset3 = $true ) ),
inference(flattening,[status(thm)],[f1071]) ).
thf(f1071,plain,
( ? [X0: $i] :
( ( ( in @ X0 @ omega )
= $true )
& ( $true
!= ( in @ ( setadjoin @ X0 @ X0 ) @ omega ) ) )
& ( omegaSp = $true )
& ( notinself2 = $true )
& ( notinself = $true )
& ( foundation2 = $true )
& ( surjCantorThm = $true )
& ( leftInvIsSurj = $true )
& ( surjFuncSetFuncSurj = $true )
& ( surjFuncSetFuncIn = $true )
& ( injFuncSetFuncInj = $true )
& ( injFuncSetFuncIn = $true )
& ( injFuncInInjFuncSet = $true )
& ( image1I = $true )
& ( image1E = $true )
& ( image1Equiv = $true )
& ( image1Ex1 = $true )
& ( image1Ex = $true )
& ( woz2B = $true )
& ( woz2A = $true )
& ( woz2W = $true )
& ( woz2Ex = $true )
& ( breln1unionCommutes = $true )
& ( breln1unionEcases = $true )
& ( breln1unionE = $true )
& ( breln1unionI = $true )
& ( breln1unionIR = $true )
& ( breln1unionIL = $true )
& ( breln1unionprop = $true )
& ( breln1compEex = $true )
& ( breln1compE = $true )
& ( breln1compI = $true )
& ( breln1compprop = $true )
& ( breln1invE = $true )
& ( breln1invI = $true )
& ( breln1invprop = $true )
& ( eqbreln1 = $true )
& ( subbreln1 = $true )
& ( breln1all1 = $true )
& ( setOfPairsIsBReln1 = $true )
& ( choice2fnsingleton = $true )
& ( breln1SetBreln1 = $true )
& ( breln1all2 = $true )
& ( woz1_5 = $true )
& ( woz1_4 = $true )
& ( woz1_3 = $true )
& ( woz1_2 = $true )
& ( woz1_1 = $true )
& ( woz13rule4 = $true )
& ( woz13rule3 = $true )
& ( woz13rule2 = $true )
& ( woz13rule1 = $true )
& ( woz13rule0 = $true )
& ( demorgan2 = $true )
& ( demorgan2b = $true )
& ( demorgan2b2 = $true )
& ( demorgan2a = $true )
& ( demorgan1 = $true )
& ( demorgan1b = $true )
& ( demorgan1a = $true )
& ( demorgan2a2 = $true )
& ( complementUnionInPowersetComplement = $true )
& ( demorgan2a1 = $true )
& ( binunionTEcontra = $true )
& ( binunionTE = $true )
& ( inComplementUnionImpInComplement1 = $true )
& ( inComplementUnionImpNotIn1 = $true )
& ( intersectInPowersetIntersectUnions = $true )
& ( inIntersectImpInIntersectUnions = $true )
& ( inIntersectImpInUnion2 = $true )
& ( inIntersectImpInUnion = $true )
& ( binunionTIRcontra = $true )
& ( binunionTILcontra = $true )
& ( complementTcontraSubset = $true )
& ( contraSubsetComplement = $true )
& ( complementInPowersetComplementIntersect = $true )
& ( complementSubsetComplementIntersect = $true )
& ( complementImpComplementIntersect = $true )
& ( complementTnotintersectT = $true )
& ( doubleComplementEq = $true )
& ( doubleComplementSub2 = $true )
& ( doubleComplementSub1 = $true )
& ( doubleComplementE1 = $true )
& ( doubleComplementI1 = $true )
& ( contrasubsetT3 = $true )
& ( contrasubsetT2 = $true )
& ( contrasubsetT1 = $true )
& ( contrasubsetT = $true )
& ( binintersectTERcontra = $true )
& ( binintersectTELcontra = $true )
& ( complementTE1 = $true )
& ( complementTI1 = $true )
& ( powersetTE1 = $true )
& ( powersetTI1 = $true )
& ( subsetTI = $true )
& ( setextT = $true )
& ( complementT_lem = $true )
& ( setminusT_lem = $true )
& ( powersetT_lem = $true )
& ( binunionT_lem = $true )
& ( binintersectT_lem = $true )
& ( iftrueorfalse = $true )
& ( iffalse = $true )
& ( iftrue = $true )
& ( theeq = $true )
& ( ifp = $true )
& ( ifSingleton = $true )
& ( iftrueProp2 = $true )
& ( iftrueProp1 = $true )
& ( iffalseProp2 = $true )
& ( iffalseProp1 = $true )
& ( eta2 = $true )
& ( beta2 = $true )
& ( lam2lamEq = $true )
& ( eta1 = $true )
& ( beta1 = $true )
& ( ap2apEq2 = $true )
& ( ap2apEq1 = $true )
& ( funcext2 = $true )
& ( funcext = $true )
& ( eqbreln = $true )
& ( subbreln = $true )
& ( funcGraphProp4 = $true )
& ( funcextLem = $true )
& ( funcGraphProp2 = $true )
& ( funcGraphProp3 = $true )
& ( funcGraphProp1 = $true )
& ( ex1E2 = $true )
& ( brelnall2 = $true )
& ( brelnall1 = $true )
& ( lam2p = $true )
& ( lamp = $true )
& ( lamProp = $true )
& ( funcinfuncset = $true )
& ( ap2p = $true )
& ( infuncsetfunc = $true )
& ( app = $true )
& ( apProp = $true )
& ( funcImageSingleton = $true )
& ( dpsetconstrER = $true )
& ( dpsetconstrEL2 = $true )
& ( dpsetconstrEL1 = $true )
& ( dpsetconstrERa = $true )
& ( setOfPairsIsBReln = $true )
& ( dpsetconstrSub = $true )
& ( dpsetconstrI = $true )
& ( cartprodpairsurjEq = $true )
& ( cartprodsndpairEq = $true )
& ( cartprodfstpairEq = $true )
& ( cartprodmempaircEq = $true )
& ( cartprodpairmemER = $true )
& ( cartprodpairmemEL = $true )
& ( cartprodsndin = $true )
& ( kpairsurjEq = $true )
& ( ksndpairEq = $true )
& ( ksndsingleton = $true )
& ( setukpairinjR = $true )
& ( setukpairinjR2 = $true )
& ( upairequniteq = $true )
& ( setukpairinjR1 = $true )
& ( setukpairinjR12 = $true )
& ( setukpairinjR11 = $true )
& ( setukpairinjL = $true )
& ( setukpairinjL2 = $true )
& ( cartprodfstin = $true )
& ( kfstpairEq = $true )
& ( theprop = $true )
& ( kfstsingleton = $true )
& ( setukpairinjL1 = $true )
& ( singletonsuniq = $true )
& ( ex1I2 = $true )
& ( ex1I = $true )
& ( ex1E1 = $true )
& ( singletonprop = $true )
& ( setunionsingleton = $true )
& ( setunionsingleton2 = $true )
& ( setunionsingleton1 = $true )
& ( setunionE2 = $true )
& ( cartprodmempair = $true )
& ( cartprodmempair1 = $true )
& ( cartprodpairin = $true )
& ( ubforcartprodlem3 = $true )
& ( ubforcartprodlem2 = $true )
& ( ubforcartprodlem1 = $true )
& ( upairinpowunion = $true )
& ( upairsubunion = $true )
& ( upairset2E = $true )
& ( singletoninpowunion = $true )
& ( singletoninpowerset = $true )
& ( singletonsubset = $true )
& ( kpairp = $true )
& ( kpairiskpair = $true )
& ( setukpairIR = $true )
& ( setukpairIL = $true )
& ( secondinupair = $true )
& ( symdiffIneg2 = $true )
& ( symdiffIneg1 = $true )
& ( symdiffI2 = $true )
& ( symdiffI1 = $true )
& ( symdiffE = $true )
& ( setminusSubset1 = $true )
& ( setminusLsub = $true )
& ( setminusIRneg = $true )
& ( setminusILneg = $true )
& ( setminusELneg = $true )
& ( setminusERneg = $true )
& ( setminusSubset2 = $true )
& ( setminusER = $true )
& ( setminusEL = $true )
& ( setminusI = $true )
& ( bs114d = $true )
& ( binintersectSubset1 = $true )
& ( binintersectSubset4 = $true )
& ( binintersectRsub = $true )
& ( disjointsetsI1 = $true )
& ( binintersectER = $true )
& ( binintersectSubset3 = $true )
& ( binintersectSubset2 = $true )
& ( binintersectLsub = $true )
& ( binintersectEL = $true )
& ( binintersectSubset5 = $true )
& ( binintersectI = $true )
& ( binunionRsub = $true )
& ( binunionLsub = $true )
& ( binunionE = $true )
& ( binunionEcases = $true )
& ( binunionIR = $true )
& ( upairset2IR = $true )
& ( binunionIL = $true )
& ( sepSubset = $true )
& ( sepInPowerset = $true )
& ( powersetsubset = $true )
& ( inPowerset = $true )
& ( powersetE1 = $true )
& ( powersetI1 = $true )
& ( subsetemptysetimpeq = $true )
& ( setextsub = $true )
& ( subset2powerset = $true )
& ( setadjoinSub2 = $true )
& ( setadjoinSub = $true )
& ( subsetTrans = $true )
& ( subsetRefl = $true )
& ( notequalI2 = $true )
& ( notequalI1 = $true )
& ( notsubsetI = $true )
& ( subsetE2 = $true )
& ( subsetE = $true )
& ( emptysetsubset = $true )
& ( subsetI2 = $true )
& ( eqimpsubset1 = $true )
& ( eqimpsubset2 = $true )
& ( subsetI1 = $true )
& ( dsetconstr__Cong = $true )
& ( descr__Cong = $true )
& ( exuEu = $true )
& ( omega__Cong = $true )
& ( setunion__Cong = $true )
& ( powerset__Cong = $true )
& ( setadjoin__Cong = $true )
& ( emptyset__Cong = $true )
& ( exu__Cong = $true )
& ( exuE3u = $true )
& ( in__Cong = $true )
& ( inCongP = $true )
& ( exuI2 = $true )
& ( exuI3 = $true )
& ( exuI1 = $true )
& ( notdallE = $true )
& ( notdexE = $true )
& ( prop2set2propI = $true )
& ( prop2setI = $true )
& ( quantDeMorgan4 = $true )
& ( quantDeMorgan3 = $true )
& ( quantDeMorgan2 = $true )
& ( quantDeMorgan1 = $true )
& ( vacuousDall = $true )
& ( emptyE1 = $true )
& ( upairsetIR = $true )
& ( upairsetIL = $true )
& ( upairsetE = $true )
& ( singletonsswitch = $true )
& ( eqinunit = $true )
& ( notinsingleton = $true )
& ( uniqinunit = $true )
& ( nonemptyImpWitness = $true )
& ( exuE2 = $true )
& ( subPowSU = $true )
& ( setunionE = $true )
& ( setunionI = $true )
& ( powersetE = $true )
& ( emptyInPowerset = $true )
& ( emptyinPowerset = $true )
& ( powersetI = $true )
& ( setoftrueEq = $true )
& ( setadjoinOr = $true )
& ( setadjoinE = $true )
& ( setadjoinIR = $true )
& ( emptyinunitempty = $true )
& ( setadjoinIL = $true )
& ( nonemptyI1 = $true )
& ( nonemptyI = $true )
& ( nonemptyE1 = $true )
& ( setbeta = $true )
& ( noeltsimpempty = $true )
& ( emptyI = $true )
& ( setext = $true )
& ( exuE3e = $true )
& ( notinemptyset = $true )
& ( emptysetimpfalse = $true )
& ( emptysetE = $true )
& ( prop2setE = $true )
& ( exuE1 = $true )
& ( dsetconstrER = $true )
& ( dsetconstrEL = $true )
& ( dsetconstrI = $true )
& ( descrp = $true )
& ( wellorderingAx = $true )
& ( foundationAx = $true )
& ( replAx = $true )
& ( omegaIndAx = $true )
& ( omegaSAx = $true )
& ( omega0Ax = $true )
& ( setunionAx = $true )
& ( powersetAx = $true )
& ( setadjoinAx = $true )
& ( emptysetAx = $true )
& ( setextAx = $true ) ),
inference(ennf_transformation,[status(thm)],[f401]) ).
thf(f401,plain,
~ ( ( setextAx = $true )
=> ( ( emptysetAx = $true )
=> ( ( setadjoinAx = $true )
=> ( ( powersetAx = $true )
=> ( ( setunionAx = $true )
=> ( ( omega0Ax = $true )
=> ( ( omegaSAx = $true )
=> ( ( omegaIndAx = $true )
=> ( ( replAx = $true )
=> ( ( foundationAx = $true )
=> ( ( wellorderingAx = $true )
=> ( ( descrp = $true )
=> ( ( dsetconstrI = $true )
=> ( ( dsetconstrEL = $true )
=> ( ( dsetconstrER = $true )
=> ( ( exuE1 = $true )
=> ( ( prop2setE = $true )
=> ( ( emptysetE = $true )
=> ( ( emptysetimpfalse = $true )
=> ( ( notinemptyset = $true )
=> ( ( exuE3e = $true )
=> ( ( setext = $true )
=> ( ( emptyI = $true )
=> ( ( noeltsimpempty = $true )
=> ( ( setbeta = $true )
=> ( ( nonemptyE1 = $true )
=> ( ( nonemptyI = $true )
=> ( ( nonemptyI1 = $true )
=> ( ( setadjoinIL = $true )
=> ( ( emptyinunitempty = $true )
=> ( ( setadjoinIR = $true )
=> ( ( setadjoinE = $true )
=> ( ( setadjoinOr = $true )
=> ( ( setoftrueEq = $true )
=> ( ( powersetI = $true )
=> ( ( emptyinPowerset = $true )
=> ( ( emptyInPowerset = $true )
=> ( ( powersetE = $true )
=> ( ( setunionI = $true )
=> ( ( setunionE = $true )
=> ( ( subPowSU = $true )
=> ( ( exuE2 = $true )
=> ( ( nonemptyImpWitness = $true )
=> ( ( uniqinunit = $true )
=> ( ( notinsingleton = $true )
=> ( ( eqinunit = $true )
=> ( ( singletonsswitch = $true )
=> ( ( upairsetE = $true )
=> ( ( upairsetIL = $true )
=> ( ( upairsetIR = $true )
=> ( ( emptyE1 = $true )
=> ( ( vacuousDall = $true )
=> ( ( quantDeMorgan1 = $true )
=> ( ( quantDeMorgan2 = $true )
=> ( ( quantDeMorgan3 = $true )
=> ( ( quantDeMorgan4 = $true )
=> ( ( prop2setI = $true )
=> ( ( prop2set2propI = $true )
=> ( ( notdexE = $true )
=> ( ( notdallE = $true )
=> ( ( exuI1 = $true )
=> ( ( exuI3 = $true )
=> ( ( exuI2 = $true )
=> ( ( inCongP = $true )
=> ( ( in__Cong = $true )
=> ( ( exuE3u = $true )
=> ( ( exu__Cong = $true )
=> ( ( emptyset__Cong = $true )
=> ( ( setadjoin__Cong = $true )
=> ( ( powerset__Cong = $true )
=> ( ( setunion__Cong = $true )
=> ( ( omega__Cong = $true )
=> ( ( exuEu = $true )
=> ( ( descr__Cong = $true )
=> ( ( dsetconstr__Cong = $true )
=> ( ( subsetI1 = $true )
=> ( ( eqimpsubset2 = $true )
=> ( ( eqimpsubset1 = $true )
=> ( ( subsetI2 = $true )
=> ( ( emptysetsubset = $true )
=> ( ( subsetE = $true )
=> ( ( subsetE2 = $true )
=> ( ( notsubsetI = $true )
=> ( ( notequalI1 = $true )
=> ( ( notequalI2 = $true )
=> ( ( subsetRefl = $true )
=> ( ( subsetTrans = $true )
=> ( ( setadjoinSub = $true )
=> ( ( setadjoinSub2 = $true )
=> ( ( subset2powerset = $true )
=> ( ( setextsub = $true )
=> ( ( subsetemptysetimpeq = $true )
=> ( ( powersetI1 = $true )
=> ( ( powersetE1 = $true )
=> ( ( inPowerset = $true )
=> ( ( powersetsubset = $true )
=> ( ( sepInPowerset = $true )
=> ( ( sepSubset = $true )
=> ( ( binunionIL = $true )
=> ( ( upairset2IR = $true )
=> ( ( binunionIR = $true )
=> ( ( binunionEcases = $true )
=> ( ( binunionE = $true )
=> ( ( binunionLsub = $true )
=> ( ( binunionRsub = $true )
=> ( ( binintersectI = $true )
=> ( ( binintersectSubset5 = $true )
=> ( ( binintersectEL = $true )
=> ( ( binintersectLsub = $true )
=> ( ( binintersectSubset2 = $true )
=> ( ( binintersectSubset3 = $true )
=> ( ( binintersectER = $true )
=> ( ( disjointsetsI1 = $true )
=> ( ( binintersectRsub = $true )
=> ( ( binintersectSubset4 = $true )
=> ( ( binintersectSubset1 = $true )
=> ( ( bs114d = $true )
=> ( ( setminusI = $true )
=> ( ( setminusEL = $true )
=> ( ( setminusER = $true )
=> ( ( setminusSubset2 = $true )
=> ( ( setminusERneg = $true )
=> ( ( setminusELneg = $true )
=> ( ( setminusILneg = $true )
=> ( ( setminusIRneg = $true )
=> ( ( setminusLsub = $true )
=> ( ( setminusSubset1 = $true )
=> ( ( symdiffE = $true )
=> ( ( symdiffI1 = $true )
=> ( ( symdiffI2 = $true )
=> ( ( symdiffIneg1 = $true )
=> ( ( symdiffIneg2 = $true )
=> ( ( secondinupair = $true )
=> ( ( setukpairIL = $true )
=> ( ( setukpairIR = $true )
=> ( ( kpairiskpair = $true )
=> ( ( kpairp = $true )
=> ( ( singletonsubset = $true )
=> ( ( singletoninpowerset = $true )
=> ( ( singletoninpowunion = $true )
=> ( ( upairset2E = $true )
=> ( ( upairsubunion = $true )
=> ( ( upairinpowunion = $true )
=> ( ( ubforcartprodlem1 = $true )
=> ( ( ubforcartprodlem2 = $true )
=> ( ( ubforcartprodlem3 = $true )
=> ( ( cartprodpairin = $true )
=> ( ( cartprodmempair1 = $true )
=> ( ( cartprodmempair = $true )
=> ( ( setunionE2 = $true )
=> ( ( setunionsingleton1 = $true )
=> ( ( setunionsingleton2 = $true )
=> ( ( setunionsingleton = $true )
=> ( ( singletonprop = $true )
=> ( ( ex1E1 = $true )
=> ( ( ex1I = $true )
=> ( ( ex1I2 = $true )
=> ( ( singletonsuniq = $true )
=> ( ( setukpairinjL1 = $true )
=> ( ( kfstsingleton = $true )
=> ( ( theprop = $true )
=> ( ( kfstpairEq = $true )
=> ( ( cartprodfstin = $true )
=> ( ( setukpairinjL2 = $true )
=> ( ( setukpairinjL = $true )
=> ( ( setukpairinjR11 = $true )
=> ( ( setukpairinjR12 = $true )
=> ( ( setukpairinjR1 = $true )
=> ( ( upairequniteq = $true )
=> ( ( setukpairinjR2 = $true )
=> ( ( setukpairinjR = $true )
=> ( ( ksndsingleton = $true )
=> ( ( ksndpairEq = $true )
=> ( ( kpairsurjEq = $true )
=> ( ( cartprodsndin = $true )
=> ( ( cartprodpairmemEL = $true )
=> ( ( cartprodpairmemER = $true )
=> ( ( cartprodmempaircEq = $true )
=> ( ( cartprodfstpairEq = $true )
=> ( ( cartprodsndpairEq = $true )
=> ( ( cartprodpairsurjEq = $true )
=> ( ( dpsetconstrI = $true )
=> ( ( dpsetconstrSub = $true )
=> ( ( setOfPairsIsBReln = $true )
=> ( ( dpsetconstrERa = $true )
=> ( ( dpsetconstrEL1 = $true )
=> ( ( dpsetconstrEL2 = $true )
=> ( ( dpsetconstrER = $true )
=> ( ( funcImageSingleton = $true )
=> ( ( apProp = $true )
=> ( ( app = $true )
=> ( ( infuncsetfunc = $true )
=> ( ( ap2p = $true )
=> ( ( funcinfuncset = $true )
=> ( ( lamProp = $true )
=> ( ( lamp = $true )
=> ( ( lam2p = $true )
=> ( ( brelnall1 = $true )
=> ( ( brelnall2 = $true )
=> ( ( ex1E2 = $true )
=> ( ( funcGraphProp1 = $true )
=> ( ( funcGraphProp3 = $true )
=> ( ( funcGraphProp2 = $true )
=> ( ( funcextLem = $true )
=> ( ( funcGraphProp4 = $true )
=> ( ( subbreln = $true )
=> ( ( eqbreln = $true )
=> ( ( funcext = $true )
=> ( ( funcext2 = $true )
=> ( ( ap2apEq1 = $true )
=> ( ( ap2apEq2 = $true )
=> ( ( beta1 = $true )
=> ( ( eta1 = $true )
=> ( ( lam2lamEq = $true )
=> ( ( beta2 = $true )
=> ( ( eta2 = $true )
=> ( ( iffalseProp1 = $true )
=> ( ( iffalseProp2 = $true )
=> ( ( iftrueProp1 = $true )
=> ( ( iftrueProp2 = $true )
=> ( ( ifSingleton = $true )
=> ( ( ifp = $true )
=> ( ( theeq = $true )
=> ( ( iftrue = $true )
=> ( ( iffalse = $true )
=> ( ( iftrueorfalse = $true )
=> ( ( binintersectT_lem = $true )
=> ( ( binunionT_lem = $true )
=> ( ( powersetT_lem = $true )
=> ( ( setminusT_lem = $true )
=> ( ( complementT_lem = $true )
=> ( ( setextT = $true )
=> ( ( subsetTI = $true )
=> ( ( powersetTI1 = $true )
=> ( ( powersetTE1 = $true )
=> ( ( complementTI1 = $true )
=> ( ( complementTE1 = $true )
=> ( ( binintersectTELcontra = $true )
=> ( ( binintersectTERcontra = $true )
=> ( ( contrasubsetT = $true )
=> ( ( contrasubsetT1 = $true )
=> ( ( contrasubsetT2 = $true )
=> ( ( contrasubsetT3 = $true )
=> ( ( doubleComplementI1 = $true )
=> ( ( doubleComplementE1 = $true )
=> ( ( doubleComplementSub1 = $true )
=> ( ( doubleComplementSub2 = $true )
=> ( ( doubleComplementEq = $true )
=> ( ( complementTnotintersectT = $true )
=> ( ( complementImpComplementIntersect = $true )
=> ( ( complementSubsetComplementIntersect = $true )
=> ( ( complementInPowersetComplementIntersect = $true )
=> ( ( contraSubsetComplement = $true )
=> ( ( complementTcontraSubset = $true )
=> ( ( binunionTILcontra = $true )
=> ( ( binunionTIRcontra = $true )
=> ( ( inIntersectImpInUnion = $true )
=> ( ( inIntersectImpInUnion2 = $true )
=> ( ( inIntersectImpInIntersectUnions = $true )
=> ( ( intersectInPowersetIntersectUnions = $true )
=> ( ( inComplementUnionImpNotIn1 = $true )
=> ( ( inComplementUnionImpInComplement1 = $true )
=> ( ( binunionTE = $true )
=> ( ( binunionTEcontra = $true )
=> ( ( demorgan2a1 = $true )
=> ( ( complementUnionInPowersetComplement = $true )
=> ( ( demorgan2a2 = $true )
=> ( ( demorgan1a = $true )
=> ( ( demorgan1b = $true )
=> ( ( demorgan1 = $true )
=> ( ( demorgan2a = $true )
=> ( ( demorgan2b2 = $true )
=> ( ( demorgan2b = $true )
=> ( ( demorgan2 = $true )
=> ( ( woz13rule0 = $true )
=> ( ( woz13rule1 = $true )
=> ( ( woz13rule2 = $true )
=> ( ( woz13rule3 = $true )
=> ( ( woz13rule4 = $true )
=> ( ( woz1_1 = $true )
=> ( ( woz1_2 = $true )
=> ( ( woz1_3 = $true )
=> ( ( woz1_4 = $true )
=> ( ( woz1_5 = $true )
=> ( ( breln1all2 = $true )
=> ( ( breln1SetBreln1 = $true )
=> ( ( choice2fnsingleton = $true )
=> ( ( setOfPairsIsBReln1 = $true )
=> ( ( breln1all1 = $true )
=> ( ( subbreln1 = $true )
=> ( ( eqbreln1 = $true )
=> ( ( breln1invprop = $true )
=> ( ( breln1invI = $true )
=> ( ( breln1invE = $true )
=> ( ( breln1compprop = $true )
=> ( ( breln1compI = $true )
=> ( ( breln1compE = $true )
=> ( ( breln1compEex = $true )
=> ( ( breln1unionprop = $true )
=> ( ( breln1unionIL = $true )
=> ( ( breln1unionIR = $true )
=> ( ( breln1unionI = $true )
=> ( ( breln1unionE = $true )
=> ( ( breln1unionEcases = $true )
=> ( ( breln1unionCommutes = $true )
=> ( ( woz2Ex = $true )
=> ( ( woz2W = $true )
=> ( ( woz2A = $true )
=> ( ( woz2B = $true )
=> ( ( image1Ex = $true )
=> ( ( image1Ex1 = $true )
=> ( ( image1Equiv = $true )
=> ( ( image1E = $true )
=> ( ( image1I = $true )
=> ( ( injFuncInInjFuncSet = $true )
=> ( ( injFuncSetFuncIn = $true )
=> ( ( injFuncSetFuncInj = $true )
=> ( ( surjFuncSetFuncIn = $true )
=> ( ( surjFuncSetFuncSurj = $true )
=> ( ( leftInvIsSurj = $true )
=> ( ( surjCantorThm = $true )
=> ( ( foundation2 = $true )
=> ( ( notinself = $true )
=> ( ( notinself2 = $true )
=> ( ( omegaSp = $true )
=> ! [X0: $i] :
( ( ( in @ X0 @ omega )
= $true )
=> ( $true
= ( in @ ( setadjoin @ X0 @ X0 ) @ omega ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f400]) ).
thf(f400,plain,
~ ( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> ( setunionE2
=> ( setunionsingleton1
=> ( setunionsingleton2
=> ( setunionsingleton
=> ( singletonprop
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> ( setukpairinjL1
=> ( kfstsingleton
=> ( theprop
=> ( kfstpairEq
=> ( cartprodfstin
=> ( setukpairinjL2
=> ( setukpairinjL
=> ( setukpairinjR11
=> ( setukpairinjR12
=> ( setukpairinjR1
=> ( upairequniteq
=> ( setukpairinjR2
=> ( setukpairinjR
=> ( ksndsingleton
=> ( ksndpairEq
=> ( kpairsurjEq
=> ( cartprodsndin
=> ( cartprodpairmemEL
=> ( cartprodpairmemER
=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ( cartprodpairsurjEq
=> ( dpsetconstrI
=> ( dpsetconstrSub
=> ( setOfPairsIsBReln
=> ( dpsetconstrERa
=> ( dpsetconstrEL1
=> ( dpsetconstrEL2
=> ( dpsetconstrER
=> ( funcImageSingleton
=> ( apProp
=> ( app
=> ( infuncsetfunc
=> ( ap2p
=> ( funcinfuncset
=> ( lamProp
=> ( lamp
=> ( lam2p
=> ( brelnall1
=> ( brelnall2
=> ( ex1E2
=> ( funcGraphProp1
=> ( funcGraphProp3
=> ( funcGraphProp2
=> ( funcextLem
=> ( funcGraphProp4
=> ( subbreln
=> ( eqbreln
=> ( funcext
=> ( funcext2
=> ( ap2apEq1
=> ( ap2apEq2
=> ( beta1
=> ( eta1
=> ( lam2lamEq
=> ( beta2
=> ( eta2
=> ( iffalseProp1
=> ( iffalseProp2
=> ( iftrueProp1
=> ( iftrueProp2
=> ( ifSingleton
=> ( ifp
=> ( theeq
=> ( iftrue
=> ( iffalse
=> ( iftrueorfalse
=> ( binintersectT_lem
=> ( binunionT_lem
=> ( powersetT_lem
=> ( setminusT_lem
=> ( complementT_lem
=> ( setextT
=> ( subsetTI
=> ( powersetTI1
=> ( powersetTE1
=> ( complementTI1
=> ( complementTE1
=> ( binintersectTELcontra
=> ( binintersectTERcontra
=> ( contrasubsetT
=> ( contrasubsetT1
=> ( contrasubsetT2
=> ( contrasubsetT3
=> ( doubleComplementI1
=> ( doubleComplementE1
=> ( doubleComplementSub1
=> ( doubleComplementSub2
=> ( doubleComplementEq
=> ( complementTnotintersectT
=> ( complementImpComplementIntersect
=> ( complementSubsetComplementIntersect
=> ( complementInPowersetComplementIntersect
=> ( contraSubsetComplement
=> ( complementTcontraSubset
=> ( binunionTILcontra
=> ( binunionTIRcontra
=> ( inIntersectImpInUnion
=> ( inIntersectImpInUnion2
=> ( inIntersectImpInIntersectUnions
=> ( intersectInPowersetIntersectUnions
=> ( inComplementUnionImpNotIn1
=> ( inComplementUnionImpInComplement1
=> ( binunionTE
=> ( binunionTEcontra
=> ( demorgan2a1
=> ( complementUnionInPowersetComplement
=> ( demorgan2a2
=> ( demorgan1a
=> ( demorgan1b
=> ( demorgan1
=> ( demorgan2a
=> ( demorgan2b2
=> ( demorgan2b
=> ( demorgan2
=> ( woz13rule0
=> ( woz13rule1
=> ( woz13rule2
=> ( woz13rule3
=> ( woz13rule4
=> ( woz1_1
=> ( woz1_2
=> ( woz1_3
=> ( woz1_4
=> ( woz1_5
=> ( breln1all2
=> ( breln1SetBreln1
=> ( choice2fnsingleton
=> ( setOfPairsIsBReln1
=> ( breln1all1
=> ( subbreln1
=> ( eqbreln1
=> ( breln1invprop
=> ( breln1invI
=> ( breln1invE
=> ( breln1compprop
=> ( breln1compI
=> ( breln1compE
=> ( breln1compEex
=> ( breln1unionprop
=> ( breln1unionIL
=> ( breln1unionIR
=> ( breln1unionI
=> ( breln1unionE
=> ( breln1unionEcases
=> ( breln1unionCommutes
=> ( woz2Ex
=> ( woz2W
=> ( woz2A
=> ( woz2B
=> ( image1Ex
=> ( image1Ex1
=> ( image1Equiv
=> ( image1E
=> ( image1I
=> ( injFuncInInjFuncSet
=> ( injFuncSetFuncIn
=> ( injFuncSetFuncInj
=> ( surjFuncSetFuncIn
=> ( surjFuncSetFuncSurj
=> ( leftInvIsSurj
=> ( surjCantorThm
=> ( foundation2
=> ( notinself
=> ( notinself2
=> ( omegaSp
=> ! [X0: $i] :
( ( in @ X0 @ omega )
=> ( in @ ( setadjoin @ X0 @ X0 ) @ omega ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f330]) ).
thf(f330,negated_conjecture,
~ ( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> ( setunionE2
=> ( setunionsingleton1
=> ( setunionsingleton2
=> ( setunionsingleton
=> ( singletonprop
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> ( setukpairinjL1
=> ( kfstsingleton
=> ( theprop
=> ( kfstpairEq
=> ( cartprodfstin
=> ( setukpairinjL2
=> ( setukpairinjL
=> ( setukpairinjR11
=> ( setukpairinjR12
=> ( setukpairinjR1
=> ( upairequniteq
=> ( setukpairinjR2
=> ( setukpairinjR
=> ( ksndsingleton
=> ( ksndpairEq
=> ( kpairsurjEq
=> ( cartprodsndin
=> ( cartprodpairmemEL
=> ( cartprodpairmemER
=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ( cartprodpairsurjEq
=> ( dpsetconstrI
=> ( dpsetconstrSub
=> ( setOfPairsIsBReln
=> ( dpsetconstrERa
=> ( dpsetconstrEL1
=> ( dpsetconstrEL2
=> ( dpsetconstrER
=> ( funcImageSingleton
=> ( apProp
=> ( app
=> ( infuncsetfunc
=> ( ap2p
=> ( funcinfuncset
=> ( lamProp
=> ( lamp
=> ( lam2p
=> ( brelnall1
=> ( brelnall2
=> ( ex1E2
=> ( funcGraphProp1
=> ( funcGraphProp3
=> ( funcGraphProp2
=> ( funcextLem
=> ( funcGraphProp4
=> ( subbreln
=> ( eqbreln
=> ( funcext
=> ( funcext2
=> ( ap2apEq1
=> ( ap2apEq2
=> ( beta1
=> ( eta1
=> ( lam2lamEq
=> ( beta2
=> ( eta2
=> ( iffalseProp1
=> ( iffalseProp2
=> ( iftrueProp1
=> ( iftrueProp2
=> ( ifSingleton
=> ( ifp
=> ( theeq
=> ( iftrue
=> ( iffalse
=> ( iftrueorfalse
=> ( binintersectT_lem
=> ( binunionT_lem
=> ( powersetT_lem
=> ( setminusT_lem
=> ( complementT_lem
=> ( setextT
=> ( subsetTI
=> ( powersetTI1
=> ( powersetTE1
=> ( complementTI1
=> ( complementTE1
=> ( binintersectTELcontra
=> ( binintersectTERcontra
=> ( contrasubsetT
=> ( contrasubsetT1
=> ( contrasubsetT2
=> ( contrasubsetT3
=> ( doubleComplementI1
=> ( doubleComplementE1
=> ( doubleComplementSub1
=> ( doubleComplementSub2
=> ( doubleComplementEq
=> ( complementTnotintersectT
=> ( complementImpComplementIntersect
=> ( complementSubsetComplementIntersect
=> ( complementInPowersetComplementIntersect
=> ( contraSubsetComplement
=> ( complementTcontraSubset
=> ( binunionTILcontra
=> ( binunionTIRcontra
=> ( inIntersectImpInUnion
=> ( inIntersectImpInUnion2
=> ( inIntersectImpInIntersectUnions
=> ( intersectInPowersetIntersectUnions
=> ( inComplementUnionImpNotIn1
=> ( inComplementUnionImpInComplement1
=> ( binunionTE
=> ( binunionTEcontra
=> ( demorgan2a1
=> ( complementUnionInPowersetComplement
=> ( demorgan2a2
=> ( demorgan1a
=> ( demorgan1b
=> ( demorgan1
=> ( demorgan2a
=> ( demorgan2b2
=> ( demorgan2b
=> ( demorgan2
=> ( woz13rule0
=> ( woz13rule1
=> ( woz13rule2
=> ( woz13rule3
=> ( woz13rule4
=> ( woz1_1
=> ( woz1_2
=> ( woz1_3
=> ( woz1_4
=> ( woz1_5
=> ( breln1all2
=> ( breln1SetBreln1
=> ( choice2fnsingleton
=> ( setOfPairsIsBReln1
=> ( breln1all1
=> ( subbreln1
=> ( eqbreln1
=> ( breln1invprop
=> ( breln1invI
=> ( breln1invE
=> ( breln1compprop
=> ( breln1compI
=> ( breln1compE
=> ( breln1compEex
=> ( breln1unionprop
=> ( breln1unionIL
=> ( breln1unionIR
=> ( breln1unionI
=> ( breln1unionE
=> ( breln1unionEcases
=> ( breln1unionCommutes
=> ( woz2Ex
=> ( woz2W
=> ( woz2A
=> ( woz2B
=> ( image1Ex
=> ( image1Ex1
=> ( image1Equiv
=> ( image1E
=> ( image1I
=> ( injFuncInInjFuncSet
=> ( injFuncSetFuncIn
=> ( injFuncSetFuncInj
=> ( surjFuncSetFuncIn
=> ( surjFuncSetFuncSurj
=> ( leftInvIsSurj
=> ( surjCantorThm
=> ( foundation2
=> ( notinself
=> ( notinself2
=> ( omegaSp
=> ! [X1: $i] :
( ( in @ X1 @ omega )
=> ( in @ ( setadjoin @ X1 @ X1 ) @ omega ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f329]) ).
thf(f329,conjecture,
( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> ( setunionE2
=> ( setunionsingleton1
=> ( setunionsingleton2
=> ( setunionsingleton
=> ( singletonprop
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> ( setukpairinjL1
=> ( kfstsingleton
=> ( theprop
=> ( kfstpairEq
=> ( cartprodfstin
=> ( setukpairinjL2
=> ( setukpairinjL
=> ( setukpairinjR11
=> ( setukpairinjR12
=> ( setukpairinjR1
=> ( upairequniteq
=> ( setukpairinjR2
=> ( setukpairinjR
=> ( ksndsingleton
=> ( ksndpairEq
=> ( kpairsurjEq
=> ( cartprodsndin
=> ( cartprodpairmemEL
=> ( cartprodpairmemER
=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ( cartprodpairsurjEq
=> ( dpsetconstrI
=> ( dpsetconstrSub
=> ( setOfPairsIsBReln
=> ( dpsetconstrERa
=> ( dpsetconstrEL1
=> ( dpsetconstrEL2
=> ( dpsetconstrER
=> ( funcImageSingleton
=> ( apProp
=> ( app
=> ( infuncsetfunc
=> ( ap2p
=> ( funcinfuncset
=> ( lamProp
=> ( lamp
=> ( lam2p
=> ( brelnall1
=> ( brelnall2
=> ( ex1E2
=> ( funcGraphProp1
=> ( funcGraphProp3
=> ( funcGraphProp2
=> ( funcextLem
=> ( funcGraphProp4
=> ( subbreln
=> ( eqbreln
=> ( funcext
=> ( funcext2
=> ( ap2apEq1
=> ( ap2apEq2
=> ( beta1
=> ( eta1
=> ( lam2lamEq
=> ( beta2
=> ( eta2
=> ( iffalseProp1
=> ( iffalseProp2
=> ( iftrueProp1
=> ( iftrueProp2
=> ( ifSingleton
=> ( ifp
=> ( theeq
=> ( iftrue
=> ( iffalse
=> ( iftrueorfalse
=> ( binintersectT_lem
=> ( binunionT_lem
=> ( powersetT_lem
=> ( setminusT_lem
=> ( complementT_lem
=> ( setextT
=> ( subsetTI
=> ( powersetTI1
=> ( powersetTE1
=> ( complementTI1
=> ( complementTE1
=> ( binintersectTELcontra
=> ( binintersectTERcontra
=> ( contrasubsetT
=> ( contrasubsetT1
=> ( contrasubsetT2
=> ( contrasubsetT3
=> ( doubleComplementI1
=> ( doubleComplementE1
=> ( doubleComplementSub1
=> ( doubleComplementSub2
=> ( doubleComplementEq
=> ( complementTnotintersectT
=> ( complementImpComplementIntersect
=> ( complementSubsetComplementIntersect
=> ( complementInPowersetComplementIntersect
=> ( contraSubsetComplement
=> ( complementTcontraSubset
=> ( binunionTILcontra
=> ( binunionTIRcontra
=> ( inIntersectImpInUnion
=> ( inIntersectImpInUnion2
=> ( inIntersectImpInIntersectUnions
=> ( intersectInPowersetIntersectUnions
=> ( inComplementUnionImpNotIn1
=> ( inComplementUnionImpInComplement1
=> ( binunionTE
=> ( binunionTEcontra
=> ( demorgan2a1
=> ( complementUnionInPowersetComplement
=> ( demorgan2a2
=> ( demorgan1a
=> ( demorgan1b
=> ( demorgan1
=> ( demorgan2a
=> ( demorgan2b2
=> ( demorgan2b
=> ( demorgan2
=> ( woz13rule0
=> ( woz13rule1
=> ( woz13rule2
=> ( woz13rule3
=> ( woz13rule4
=> ( woz1_1
=> ( woz1_2
=> ( woz1_3
=> ( woz1_4
=> ( woz1_5
=> ( breln1all2
=> ( breln1SetBreln1
=> ( choice2fnsingleton
=> ( setOfPairsIsBReln1
=> ( breln1all1
=> ( subbreln1
=> ( eqbreln1
=> ( breln1invprop
=> ( breln1invI
=> ( breln1invE
=> ( breln1compprop
=> ( breln1compI
=> ( breln1compE
=> ( breln1compEex
=> ( breln1unionprop
=> ( breln1unionIL
=> ( breln1unionIR
=> ( breln1unionI
=> ( breln1unionE
=> ( breln1unionEcases
=> ( breln1unionCommutes
=> ( woz2Ex
=> ( woz2W
=> ( woz2A
=> ( woz2B
=> ( image1Ex
=> ( image1Ex1
=> ( image1Equiv
=> ( image1E
=> ( image1I
=> ( injFuncInInjFuncSet
=> ( injFuncSetFuncIn
=> ( injFuncSetFuncInj
=> ( surjFuncSetFuncIn
=> ( surjFuncSetFuncSurj
=> ( leftInvIsSurj
=> ( surjCantorThm
=> ( foundation2
=> ( notinself
=> ( notinself2
=> ( omegaSp
=> ! [X1: $i] :
( ( in @ X1 @ omega )
=> ( in @ ( setadjoin @ X1 @ X1 ) @ omega ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.N0zPVEAfqW/DTF2THF_28502.p',omegaSclos) ).
thf(f5152,plain,
( $true
!= ( in @ sK655 @ omega ) ),
inference(trivial_inequality_removal,[status(thm)],[f5151]) ).
thf(f5151,plain,
( ( $true != $true )
| ( $true
!= ( in @ sK655 @ omega ) ) ),
inference(forward_demodulation,[status(thm)],[f5140,f4253]) ).
thf(f4253,plain,
omegaSAx = $true,
inference(cnf_transformation,[status(thm)],[f2443]) ).
thf(f5140,plain,
( ( omegaSAx != $true )
| ( $true
!= ( in @ sK655 @ omega ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f5133]) ).
thf(f5133,plain,
( ( omegaSAx != $true )
| ( $true
!= ( in @ sK655 @ omega ) )
| ( $true != $true ) ),
inference(superposition,[status(thm)],[f4124,f4402]) ).
thf(f4402,plain,
! [X0: $i] :
( ( $true
= ( in @ ( setadjoin @ X0 @ X0 ) @ omega ) )
| ( omegaSAx != $true )
| ( ( in @ X0 @ omega )
!= $true ) ),
inference(cnf_transformation,[status(thm)],[f2535]) ).
thf(f2535,plain,
( ( ! [X0: $i] :
( ( ( in @ X0 @ omega )
!= $true )
| ( $true
= ( in @ ( setadjoin @ X0 @ X0 ) @ omega ) ) )
| ( omegaSAx != $true ) )
& ( ( omegaSAx = $true )
| ( ( $true
= ( in @ sK716 @ omega ) )
& ( ( in @ ( setadjoin @ sK716 @ sK716 ) @ omega )
!= $true ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK716])],[f2533,f2534]) ).
thf(f2534,plain,
( ? [X1: $i] :
( ( ( in @ X1 @ omega )
= $true )
& ( ( in @ ( setadjoin @ X1 @ X1 ) @ omega )
!= $true ) )
=> ( ( $true
= ( in @ sK716 @ omega ) )
& ( ( in @ ( setadjoin @ sK716 @ sK716 ) @ omega )
!= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f2533,plain,
( ( ! [X0: $i] :
( ( ( in @ X0 @ omega )
!= $true )
| ( $true
= ( in @ ( setadjoin @ X0 @ X0 ) @ omega ) ) )
| ( omegaSAx != $true ) )
& ( ( omegaSAx = $true )
| ? [X1: $i] :
( ( ( in @ X1 @ omega )
= $true )
& ( ( in @ ( setadjoin @ X1 @ X1 ) @ omega )
!= $true ) ) ) ),
inference(rectify,[status(thm)],[f2532]) ).
thf(f2532,plain,
( ( ! [X0: $i] :
( ( ( in @ X0 @ omega )
!= $true )
| ( $true
= ( in @ ( setadjoin @ X0 @ X0 ) @ omega ) ) )
| ( omegaSAx != $true ) )
& ( ( omegaSAx = $true )
| ? [X0: $i] :
( ( ( in @ X0 @ omega )
= $true )
& ( $true
!= ( in @ ( setadjoin @ X0 @ X0 ) @ omega ) ) ) ) ),
inference(nnf_transformation,[status(thm)],[f1204]) ).
thf(f1204,plain,
( ! [X0: $i] :
( ( ( in @ X0 @ omega )
!= $true )
| ( $true
= ( in @ ( setadjoin @ X0 @ X0 ) @ omega ) ) )
<=> ( omegaSAx = $true ) ),
inference(ennf_transformation,[status(thm)],[f699]) ).
thf(f699,plain,
( ( omegaSAx = $true )
<=> ! [X0: $i] :
( ( ( in @ X0 @ omega )
= $true )
=> ( $true
= ( in @ ( setadjoin @ X0 @ X0 ) @ omega ) ) ) ),
inference(fool_elimination,[status(thm)],[f698]) ).
thf(f698,plain,
! [X0: $i] :
( ( ( in @ X0 @ omega )
=> ( in @ ( setadjoin @ X0 @ X0 ) @ omega ) )
= omegaSAx ),
inference(rectify,[status(thm)],[f8]) ).
thf(f8,axiom,
! [X1: $i] :
( ( ( in @ X1 @ omega )
=> ( in @ ( setadjoin @ X1 @ X1 ) @ omega ) )
= omegaSAx ),
file('/export/starexec/sandbox2/tmp/tmp.N0zPVEAfqW/DTF2THF_28502.p',omegaSAx) ).
thf(f4124,plain,
( $true
!= ( in @ ( setadjoin @ sK655 @ sK655 ) @ omega ) ),
inference(cnf_transformation,[status(thm)],[f2443]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SEU809^1 : TPTP v9.2.1. Released v3.7.0.
% 0.12/0.13 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.16/0.34 % Computer : n026.cluster.edu
% 0.16/0.34 % Model : x86_64 x86_64
% 0.16/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.34 % Memory : 8042.1875MB
% 0.16/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.34 % CPULimit : 300
% 0.16/0.34 % WCLimit : 300
% 0.16/0.34 % DateTime : Tue Feb 24 12:34:55 EST 2026
% 0.16/0.34 % CPUTime :
% 0.16/0.34 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.19/0.46 ---- Original DTF file ---
% 0.30/0.47 thf(spec,logic,$$dhol).
% 0.30/0.47 %------------------------------------------------------------------------------
% 0.30/0.47 % File : SEU809^1 : TPTP v9.2.1. Released v3.7.0.
% 0.30/0.47 % Domain : Set Theory
% 0.30/0.47 % Problem : Omega and Peano
% 0.30/0.47 % Version : Especial.
% 0.30/0.47 % English : (! x:i.in x omega -> in (setadjoin x x) omega)
% 0.30/0.47
% 0.30/0.47 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.30/0.47 % Source : [Bro08]
% 0.30/0.47 % Names : ZFC311g [Bro08]
% 0.30/0.47
% 0.30/0.47 % Status : Theorem
% 0.30/0.47 % Rating : 0.11 v9.1.0, 0.12 v9.0.0, 0.20 v8.2.0, 0.23 v8.1.0, 0.00 v7.4.0, 0.11 v7.2.0, 0.00 v7.1.0, 0.25 v7.0.0, 0.14 v6.4.0, 0.17 v6.3.0, 0.20 v6.2.0, 0.14 v6.1.0, 0.29 v5.5.0, 0.33 v5.4.0, 0.40 v5.2.0, 0.60 v5.1.0, 0.80 v5.0.0, 0.60 v4.1.0, 0.67 v3.7.0
% 0.30/0.47 % Syntax : Number of formulae : 708 ( 328 unt; 379 typ; 328 def)
% 0.30/0.47 % Number of atoms : 2014 ( 470 equ; 0 cnn)
% 0.30/0.47 % Maximal formula atoms : 327 ( 6 avg)
% 0.30/0.47 % Number of connectives : 4411 ( 76 ~; 14 |; 54 &;3211 @)
% 0.30/0.47 % ( 17 <=>;1039 =>; 0 <=; 0 <~>)
% 0.30/0.47 % Maximal formula depth : 332 ( 2 avg)
% 0.30/0.47 % Number of types : 2 ( 0 usr)
% 0.30/0.47 % Number of type conns : 177 ( 177 >; 0 *; 0 +; 0 <<)
% 0.30/0.47 % Number of symbols : 382 ( 379 usr; 329 con; 0-4 aty)
% 0.30/0.47 % Number of variables : 1149 ( 76 ^;1029 !; 44 ?;1149 :)
% 0.30/0.47 % SPC : TH0_THM_EQU_NAR
% 0.30/0.47
% 0.30/0.47 % Comments : http://mathgate.info/detsetitem.php?id=442
% 0.30/0.47 %------------------------------------------------------------------------------
% 0.30/0.47 thf(in_type,type,
% 0.30/0.47 in: $i > $i > $o ).
% 0.30/0.47
% 0.30/0.47 thf(exu_type,type,
% 0.30/0.47 exu: ( $i > $o ) > $o ).
% 0.30/0.47
% 0.30/0.47 thf(exu,definition,
% 0.30/0.47 ( exu
% 0.30/0.47 = ( ^ [Xphi: $i > $o] :
% 0.30/0.47 ? [Xx: $i] :
% 0.30/0.47 ( ( Xphi @ Xx )
% 0.30/0.47 & ! [Xy: $i] :
% 0.30/0.47 ( ( Xphi @ Xy )
% 0.30/0.47 => ( Xx = Xy ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(setextAx_type,type,
% 0.30/0.47 setextAx: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setextAx,definition,
% 0.30/0.47 ( setextAx
% 0.30/0.47 = ( ! [A: $i,B: $i] :
% 0.30/0.47 ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 <=> ( in @ Xx @ B ) )
% 0.30/0.47 => ( A = B ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(emptyset_type,type,
% 0.30/0.47 emptyset: $i ).
% 0.30/0.47
% 0.30/0.47 thf(emptysetAx_type,type,
% 0.30/0.47 emptysetAx: $o ).
% 0.30/0.47
% 0.30/0.47 thf(emptysetAx,definition,
% 0.30/0.47 ( emptysetAx
% 0.30/0.47 = ( ! [Xx: $i] :
% 0.30/0.47 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(setadjoin_type,type,
% 0.30/0.47 setadjoin: $i > $i > $i ).
% 0.30/0.47
% 0.30/0.47 thf(setadjoinAx_type,type,
% 0.30/0.47 setadjoinAx: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setadjoinAx,definition,
% 0.30/0.47 ( setadjoinAx
% 0.30/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.30/0.47 <=> ( ( Xy = Xx )
% 0.30/0.47 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(powerset_type,type,
% 0.30/0.47 powerset: $i > $i ).
% 0.30/0.47
% 0.30/0.47 thf(powersetAx_type,type,
% 0.30/0.47 powersetAx: $o ).
% 0.30/0.47
% 0.30/0.47 thf(powersetAx,definition,
% 0.30/0.47 ( powersetAx
% 0.30/0.47 = ( ! [A: $i,B: $i] :
% 0.30/0.47 ( ( in @ B @ ( powerset @ A ) )
% 0.30/0.47 <=> ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ B )
% 0.30/0.47 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(setunion_type,type,
% 0.30/0.47 setunion: $i > $i ).
% 0.30/0.47
% 0.30/0.47 thf(setunionAx_type,type,
% 0.30/0.47 setunionAx: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setunionAx,definition,
% 0.30/0.47 ( setunionAx
% 0.30/0.47 = ( ! [A: $i,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ ( setunion @ A ) )
% 0.30/0.47 <=> ? [B: $i] :
% 0.30/0.47 ( ( in @ Xx @ B )
% 0.30/0.47 & ( in @ B @ A ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(omega_type,type,
% 0.30/0.47 omega: $i ).
% 0.30/0.47
% 0.30/0.47 thf(omega0Ax_type,type,
% 0.30/0.47 omega0Ax: $o ).
% 0.30/0.47
% 0.30/0.47 thf(omega0Ax,definition,
% 0.30/0.47 ( omega0Ax
% 0.30/0.47 = ( in @ emptyset @ omega ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(omegaSAx_type,type,
% 0.30/0.47 omegaSAx: $o ).
% 0.30/0.47
% 0.30/0.47 thf(omegaSAx,definition,
% 0.30/0.47 ( omegaSAx
% 0.30/0.47 = ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ omega )
% 0.30/0.47 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(omegaIndAx_type,type,
% 0.30/0.47 omegaIndAx: $o ).
% 0.30/0.47
% 0.30/0.47 thf(omegaIndAx,definition,
% 0.30/0.47 ( omegaIndAx
% 0.30/0.47 = ( ! [A: $i] :
% 0.30/0.47 ( ( ( in @ emptyset @ A )
% 0.30/0.47 & ! [Xx: $i] :
% 0.30/0.47 ( ( ( in @ Xx @ omega )
% 0.30/0.47 & ( in @ Xx @ A ) )
% 0.30/0.47 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ omega )
% 0.30/0.47 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(replAx_type,type,
% 0.30/0.47 replAx: $o ).
% 0.30/0.47
% 0.30/0.47 thf(replAx,definition,
% 0.30/0.47 ( replAx
% 0.30/0.47 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.30/0.47 ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( exu
% 0.30/0.47 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.30/0.47 => ? [B: $i] :
% 0.30/0.47 ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ B )
% 0.30/0.47 <=> ? [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ A )
% 0.30/0.47 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(foundationAx_type,type,
% 0.30/0.47 foundationAx: $o ).
% 0.30/0.47
% 0.30/0.47 thf(foundationAx,definition,
% 0.30/0.47 ( foundationAx
% 0.30/0.47 = ( ! [A: $i] :
% 0.30/0.47 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.30/0.47 => ? [B: $i] :
% 0.30/0.47 ( ( in @ B @ A )
% 0.30/0.47 & ~ ? [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ B )
% 0.30/0.47 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(wellorderingAx_type,type,
% 0.30/0.47 wellorderingAx: $o ).
% 0.30/0.47
% 0.30/0.47 thf(wellorderingAx,definition,
% 0.30/0.47 ( wellorderingAx
% 0.30/0.47 = ( ! [A: $i] :
% 0.30/0.47 ? [B: $i] :
% 0.30/0.47 ( ! [C: $i] :
% 0.30/0.47 ( ( in @ C @ B )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ C )
% 0.30/0.47 => ( in @ Xx @ A ) ) )
% 0.30/0.47 & ! [Xx: $i,Xy: $i] :
% 0.30/0.47 ( ( ( in @ Xx @ A )
% 0.30/0.47 & ( in @ Xy @ A ) )
% 0.30/0.47 => ( ! [C: $i] :
% 0.30/0.47 ( ( in @ C @ B )
% 0.30/0.47 => ( ( in @ Xx @ C )
% 0.30/0.47 <=> ( in @ Xy @ C ) ) )
% 0.30/0.47 => ( Xx = Xy ) ) )
% 0.30/0.47 & ! [C: $i,D: $i] :
% 0.30/0.47 ( ( ( in @ C @ B )
% 0.30/0.47 & ( in @ D @ B ) )
% 0.30/0.47 => ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ C )
% 0.30/0.47 => ( in @ Xx @ D ) )
% 0.30/0.47 | ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ D )
% 0.30/0.47 => ( in @ Xx @ C ) ) ) )
% 0.30/0.47 & ! [C: $i] :
% 0.30/0.47 ( ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ C )
% 0.30/0.47 => ( in @ Xx @ A ) )
% 0.30/0.47 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.30/0.47 => ? [D: $i,Xx: $i] :
% 0.30/0.47 ( ( in @ D @ B )
% 0.30/0.47 & ( in @ Xx @ C )
% 0.30/0.47 & ~ ? [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ D )
% 0.30/0.47 & ( in @ Xy @ C ) )
% 0.30/0.47 & ! [E: $i] :
% 0.30/0.47 ( ( in @ E @ B )
% 0.30/0.47 => ( ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ E )
% 0.30/0.47 => ( in @ Xy @ D ) )
% 0.30/0.47 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(descr_type,type,
% 0.30/0.47 descr: ( $i > $o ) > $i ).
% 0.30/0.47
% 0.30/0.47 thf(descrp_type,type,
% 0.30/0.47 descrp: $o ).
% 0.30/0.47
% 0.30/0.47 thf(descrp,definition,
% 0.30/0.47 ( descrp
% 0.30/0.47 = ( ! [Xphi: $i > $o] :
% 0.30/0.47 ( ( exu
% 0.30/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.47 => ( Xphi
% 0.30/0.47 @ ( descr
% 0.30/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(dsetconstr_type,type,
% 0.30/0.47 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.30/0.47
% 0.30/0.47 thf(dsetconstrI_type,type,
% 0.30/0.47 dsetconstrI: $o ).
% 0.30/0.47
% 0.30/0.47 thf(dsetconstrI,definition,
% 0.30/0.47 ( dsetconstrI
% 0.30/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( Xphi @ Xx )
% 0.30/0.47 => ( in @ Xx
% 0.30/0.47 @ ( dsetconstr @ A
% 0.30/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(dsetconstrEL_type,type,
% 0.30/0.47 dsetconstrEL: $o ).
% 0.30/0.47
% 0.30/0.47 thf(dsetconstrEL,definition,
% 0.30/0.47 ( dsetconstrEL
% 0.30/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx
% 0.30/0.47 @ ( dsetconstr @ A
% 0.30/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.30/0.47 => ( in @ Xx @ A ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(dsetconstrER_type,type,
% 0.30/0.47 dsetconstrER: $o ).
% 0.30/0.47
% 0.30/0.47 thf(dsetconstrER,definition,
% 0.30/0.47 ( dsetconstrER
% 0.30/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx
% 0.30/0.47 @ ( dsetconstr @ A
% 0.30/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.30/0.47 => ( Xphi @ Xx ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(exuE1_type,type,
% 0.30/0.47 exuE1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(exuE1,definition,
% 0.30/0.47 ( exuE1
% 0.30/0.47 = ( ! [Xphi: $i > $o] :
% 0.30/0.47 ( ( exu
% 0.30/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.47 => ? [Xx: $i] :
% 0.30/0.47 ( ( Xphi @ Xx )
% 0.30/0.47 & ! [Xy: $i] :
% 0.30/0.47 ( ( Xphi @ Xy )
% 0.30/0.47 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(prop2set_type,type,
% 0.30/0.47 prop2set: $o > $i ).
% 0.30/0.47
% 0.30/0.47 thf(prop2setE_type,type,
% 0.30/0.47 prop2setE: $o ).
% 0.30/0.47
% 0.30/0.47 thf(prop2setE,definition,
% 0.30/0.47 ( prop2setE
% 0.30/0.47 = ( ! [Xphi: $o,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.30/0.47 => Xphi ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(emptysetE_type,type,
% 0.30/0.47 emptysetE: $o ).
% 0.30/0.47
% 0.30/0.47 thf(emptysetE,definition,
% 0.30/0.47 ( emptysetE
% 0.30/0.47 = ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ emptyset )
% 0.30/0.47 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(emptysetimpfalse_type,type,
% 0.30/0.47 emptysetimpfalse: $o ).
% 0.30/0.47
% 0.30/0.47 thf(emptysetimpfalse,definition,
% 0.30/0.47 ( emptysetimpfalse
% 0.30/0.47 = ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ emptyset )
% 0.30/0.47 => $false ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(notinemptyset_type,type,
% 0.30/0.47 notinemptyset: $o ).
% 0.30/0.47
% 0.30/0.47 thf(notinemptyset,definition,
% 0.30/0.47 ( notinemptyset
% 0.30/0.47 = ( ! [Xx: $i] :
% 0.30/0.47 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(exuE3e_type,type,
% 0.30/0.47 exuE3e: $o ).
% 0.30/0.47
% 0.30/0.47 thf(exuE3e,definition,
% 0.30/0.47 ( exuE3e
% 0.30/0.47 = ( ! [Xphi: $i > $o] :
% 0.30/0.47 ( ( exu
% 0.30/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.47 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(setext_type,type,
% 0.30/0.47 setext: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setext,definition,
% 0.30/0.47 ( setext
% 0.30/0.47 = ( ! [A: $i,B: $i] :
% 0.30/0.47 ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( in @ Xx @ B ) )
% 0.30/0.47 => ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ B )
% 0.30/0.47 => ( in @ Xx @ A ) )
% 0.30/0.47 => ( A = B ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(emptyI_type,type,
% 0.30/0.47 emptyI: $o ).
% 0.30/0.47
% 0.30/0.47 thf(emptyI,definition,
% 0.30/0.47 ( emptyI
% 0.30/0.47 = ( ! [A: $i] :
% 0.30/0.47 ( ! [Xx: $i] :
% 0.30/0.47 ~ ( in @ Xx @ A )
% 0.30/0.47 => ( A = emptyset ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(noeltsimpempty_type,type,
% 0.30/0.47 noeltsimpempty: $o ).
% 0.30/0.47
% 0.30/0.47 thf(noeltsimpempty,definition,
% 0.30/0.47 ( noeltsimpempty
% 0.30/0.47 = ( ! [A: $i] :
% 0.30/0.47 ( ! [Xx: $i] :
% 0.30/0.47 ~ ( in @ Xx @ A )
% 0.30/0.47 => ( A = emptyset ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(setbeta_type,type,
% 0.30/0.47 setbeta: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setbeta,definition,
% 0.30/0.47 ( setbeta
% 0.30/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( in @ Xx
% 0.30/0.47 @ ( dsetconstr @ A
% 0.30/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.30/0.47 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(nonempty_type,type,
% 0.30/0.47 nonempty: $i > $o ).
% 0.30/0.47
% 0.30/0.47 thf(nonempty,definition,
% 0.30/0.47 ( nonempty
% 0.30/0.47 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(nonemptyE1_type,type,
% 0.30/0.47 nonemptyE1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(nonemptyE1,definition,
% 0.30/0.47 ( nonemptyE1
% 0.30/0.47 = ( ! [A: $i] :
% 0.30/0.47 ( ( nonempty @ A )
% 0.30/0.47 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(nonemptyI_type,type,
% 0.30/0.47 nonemptyI: $o ).
% 0.30/0.47
% 0.30/0.47 thf(nonemptyI,definition,
% 0.30/0.47 ( nonemptyI
% 0.30/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( Xphi @ Xx )
% 0.30/0.47 => ( nonempty
% 0.30/0.47 @ ( dsetconstr @ A
% 0.30/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(nonemptyI1_type,type,
% 0.30/0.47 nonemptyI1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(nonemptyI1,definition,
% 0.30/0.47 ( nonemptyI1
% 0.30/0.47 = ( ! [A: $i] :
% 0.30/0.47 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.30/0.47 => ( nonempty @ A ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(setadjoinIL_type,type,
% 0.30/0.47 setadjoinIL: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setadjoinIL,definition,
% 0.30/0.47 ( setadjoinIL
% 0.30/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(emptyinunitempty_type,type,
% 0.30/0.47 emptyinunitempty: $o ).
% 0.30/0.47
% 0.30/0.47 thf(emptyinunitempty,definition,
% 0.30/0.47 ( emptyinunitempty
% 0.30/0.47 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(setadjoinIR_type,type,
% 0.30/0.47 setadjoinIR: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setadjoinIR,definition,
% 0.30/0.47 ( setadjoinIR
% 0.30/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ A )
% 0.30/0.47 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(setadjoinE_type,type,
% 0.30/0.47 setadjoinE: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setadjoinE,definition,
% 0.30/0.47 ( setadjoinE
% 0.30/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.30/0.47 => ! [Xphi: $o] :
% 0.30/0.47 ( ( ( Xy = Xx )
% 0.30/0.47 => Xphi )
% 0.30/0.47 => ( ( ( in @ Xy @ A )
% 0.30/0.47 => Xphi )
% 0.30/0.47 => Xphi ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(setadjoinOr_type,type,
% 0.30/0.47 setadjoinOr: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setadjoinOr,definition,
% 0.30/0.47 ( setadjoinOr
% 0.30/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.30/0.47 => ( ( Xy = Xx )
% 0.30/0.47 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(setoftrueEq_type,type,
% 0.30/0.47 setoftrueEq: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setoftrueEq,definition,
% 0.30/0.47 ( setoftrueEq
% 0.30/0.47 = ( ! [A: $i] :
% 0.30/0.47 ( ( dsetconstr @ A
% 0.30/0.47 @ ^ [Xx: $i] : $true )
% 0.30/0.47 = A ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(powersetI_type,type,
% 0.30/0.47 powersetI: $o ).
% 0.30/0.47
% 0.30/0.47 thf(powersetI,definition,
% 0.30/0.47 ( powersetI
% 0.30/0.47 = ( ! [A: $i,B: $i] :
% 0.30/0.47 ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ B )
% 0.30/0.47 => ( in @ Xx @ A ) )
% 0.30/0.47 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(emptyinPowerset_type,type,
% 0.30/0.47 emptyinPowerset: $o ).
% 0.30/0.47
% 0.30/0.47 thf(emptyinPowerset,definition,
% 0.30/0.47 ( emptyinPowerset
% 0.30/0.47 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(emptyInPowerset_type,type,
% 0.30/0.47 emptyInPowerset: $o ).
% 0.30/0.47
% 0.30/0.47 thf(emptyInPowerset,definition,
% 0.30/0.47 ( emptyInPowerset
% 0.30/0.47 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(powersetE_type,type,
% 0.30/0.47 powersetE: $o ).
% 0.30/0.47
% 0.30/0.47 thf(powersetE,definition,
% 0.30/0.47 ( powersetE
% 0.30/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.47 ( ( in @ B @ ( powerset @ A ) )
% 0.30/0.47 => ( ( in @ Xx @ B )
% 0.30/0.47 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(setunionI_type,type,
% 0.30/0.47 setunionI: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setunionI,definition,
% 0.30/0.47 ( setunionI
% 0.30/0.47 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.30/0.47 ( ( in @ Xx @ B )
% 0.30/0.47 => ( ( in @ B @ A )
% 0.30/0.47 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(setunionE_type,type,
% 0.30/0.47 setunionE: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setunionE,definition,
% 0.30/0.47 ( setunionE
% 0.30/0.47 = ( ! [A: $i,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ ( setunion @ A ) )
% 0.30/0.47 => ! [Xphi: $o] :
% 0.30/0.47 ( ! [B: $i] :
% 0.30/0.47 ( ( in @ Xx @ B )
% 0.30/0.47 => ( ( in @ B @ A )
% 0.30/0.47 => Xphi ) )
% 0.30/0.47 => Xphi ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(subPowSU_type,type,
% 0.30/0.47 subPowSU: $o ).
% 0.30/0.47
% 0.30/0.47 thf(subPowSU,definition,
% 0.30/0.47 ( subPowSU
% 0.30/0.47 = ( ! [A: $i,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(exuE2_type,type,
% 0.30/0.47 exuE2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(exuE2,definition,
% 0.30/0.47 ( exuE2
% 0.30/0.47 = ( ! [Xphi: $i > $o] :
% 0.30/0.47 ( ( exu
% 0.30/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.47 => ? [Xx: $i] :
% 0.30/0.47 ! [Xy: $i] :
% 0.30/0.47 ( ( Xphi @ Xy )
% 0.30/0.47 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(nonemptyImpWitness_type,type,
% 0.30/0.47 nonemptyImpWitness: $o ).
% 0.30/0.47
% 0.30/0.47 thf(nonemptyImpWitness,definition,
% 0.30/0.47 ( nonemptyImpWitness
% 0.30/0.47 = ( ! [A: $i] :
% 0.30/0.47 ( ( nonempty @ A )
% 0.30/0.47 => ? [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 & $true ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(uniqinunit_type,type,
% 0.30/0.47 uniqinunit: $o ).
% 0.30/0.47
% 0.30/0.47 thf(uniqinunit,definition,
% 0.30/0.47 ( uniqinunit
% 0.30/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.47 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.30/0.47 => ( Xx = Xy ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(notinsingleton_type,type,
% 0.30/0.47 notinsingleton: $o ).
% 0.30/0.47
% 0.30/0.47 thf(notinsingleton,definition,
% 0.30/0.47 ( notinsingleton
% 0.30/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.47 ( ( Xx != Xy )
% 0.30/0.47 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(eqinunit_type,type,
% 0.30/0.47 eqinunit: $o ).
% 0.30/0.47
% 0.30/0.47 thf(eqinunit,definition,
% 0.30/0.47 ( eqinunit
% 0.30/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.47 ( ( Xx = Xy )
% 0.30/0.47 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(singletonsswitch_type,type,
% 0.30/0.47 singletonsswitch: $o ).
% 0.30/0.47
% 0.30/0.47 thf(singletonsswitch,definition,
% 0.30/0.47 ( singletonsswitch
% 0.30/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.47 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.30/0.47 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(upairsetE_type,type,
% 0.30/0.47 upairsetE: $o ).
% 0.30/0.47
% 0.30/0.47 thf(upairsetE,definition,
% 0.30/0.47 ( upairsetE
% 0.30/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.30/0.47 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.30/0.47 => ( ( Xz = Xx )
% 0.30/0.47 | ( Xz = Xy ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(upairsetIL_type,type,
% 0.30/0.47 upairsetIL: $o ).
% 0.30/0.47
% 0.30/0.47 thf(upairsetIL,definition,
% 0.30/0.47 ( upairsetIL
% 0.30/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(upairsetIR_type,type,
% 0.30/0.47 upairsetIR: $o ).
% 0.30/0.47
% 0.30/0.47 thf(upairsetIR,definition,
% 0.30/0.47 ( upairsetIR
% 0.30/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(emptyE1_type,type,
% 0.30/0.47 emptyE1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(emptyE1,definition,
% 0.30/0.47 ( emptyE1
% 0.30/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.47 ( ? [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 & ( Xphi @ Xx ) )
% 0.30/0.47 => ( ( ( dsetconstr @ A
% 0.30/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.47 = emptyset )
% 0.30/0.47 => $false ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(vacuousDall_type,type,
% 0.30/0.47 vacuousDall: $o ).
% 0.30/0.47
% 0.30/0.47 thf(vacuousDall,definition,
% 0.30/0.47 ( vacuousDall
% 0.30/0.47 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ emptyset )
% 0.30/0.47 => ( Xphi @ Xx ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(quantDeMorgan1_type,type,
% 0.30/0.47 quantDeMorgan1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(quantDeMorgan1,definition,
% 0.30/0.47 ( quantDeMorgan1
% 0.30/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.47 ( ~ ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( Xphi @ Xx ) )
% 0.30/0.47 => ? [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(quantDeMorgan2_type,type,
% 0.30/0.47 quantDeMorgan2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(quantDeMorgan2,definition,
% 0.30/0.47 ( quantDeMorgan2
% 0.30/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.47 ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ~ ( Xphi @ Xx ) )
% 0.30/0.47 => ~ ? [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(quantDeMorgan3_type,type,
% 0.30/0.47 quantDeMorgan3: $o ).
% 0.30/0.47
% 0.30/0.47 thf(quantDeMorgan3,definition,
% 0.30/0.47 ( quantDeMorgan3
% 0.30/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.47 ( ~ ? [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 & ( Xphi @ Xx ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(quantDeMorgan4_type,type,
% 0.30/0.47 quantDeMorgan4: $o ).
% 0.30/0.47
% 0.30/0.47 thf(quantDeMorgan4,definition,
% 0.30/0.47 ( quantDeMorgan4
% 0.30/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.47 ( ? [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 & ~ ( Xphi @ Xx ) )
% 0.30/0.47 => ~ ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(prop2setI_type,type,
% 0.30/0.47 prop2setI: $o ).
% 0.30/0.47
% 0.30/0.47 thf(prop2setI,definition,
% 0.30/0.47 ( prop2setI
% 0.30/0.47 = ( ! [Xphi: $o] :
% 0.30/0.47 ( Xphi
% 0.30/0.47 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(set2prop_type,type,
% 0.30/0.47 set2prop: $i > $o ).
% 0.30/0.47
% 0.30/0.47 thf(prop2set2propI_type,type,
% 0.30/0.47 prop2set2propI: $o ).
% 0.30/0.47
% 0.30/0.47 thf(prop2set2propI,definition,
% 0.30/0.47 ( prop2set2propI
% 0.30/0.47 = ( ! [Xphi: $o] :
% 0.30/0.47 ( Xphi
% 0.30/0.47 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(notdexE_type,type,
% 0.30/0.47 notdexE: $o ).
% 0.30/0.47
% 0.30/0.47 thf(notdexE,definition,
% 0.30/0.47 ( notdexE
% 0.30/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.47 ( ~ ? [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 & ( Xphi @ Xx ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(notdallE_type,type,
% 0.30/0.47 notdallE: $o ).
% 0.30/0.47
% 0.30/0.47 thf(notdallE,definition,
% 0.30/0.47 ( notdallE
% 0.30/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.47 ( ~ ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( Xphi @ Xx ) )
% 0.30/0.47 => ? [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(exuI1_type,type,
% 0.30/0.47 exuI1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(exuI1,definition,
% 0.30/0.47 ( exuI1
% 0.30/0.47 = ( ! [Xphi: $i > $o] :
% 0.30/0.47 ( ? [Xx: $i] :
% 0.30/0.47 ( ( Xphi @ Xx )
% 0.30/0.47 & ! [Xy: $i] :
% 0.30/0.47 ( ( Xphi @ Xy )
% 0.30/0.47 => ( Xx = Xy ) ) )
% 0.30/0.47 => ( exu
% 0.30/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(exuI3_type,type,
% 0.30/0.47 exuI3: $o ).
% 0.30/0.47
% 0.30/0.47 thf(exuI3,definition,
% 0.30/0.47 ( exuI3
% 0.30/0.47 = ( ! [Xphi: $i > $o] :
% 0.30/0.47 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.30/0.47 => ( ! [Xx: $i,Xy: $i] :
% 0.30/0.47 ( ( Xphi @ Xx )
% 0.30/0.47 => ( ( Xphi @ Xy )
% 0.30/0.47 => ( Xx = Xy ) ) )
% 0.30/0.47 => ( exu
% 0.30/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(exuI2_type,type,
% 0.30/0.47 exuI2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(exuI2,definition,
% 0.30/0.47 ( exuI2
% 0.30/0.47 = ( ! [Xphi: $i > $o] :
% 0.30/0.47 ( ? [Xx: $i] :
% 0.30/0.47 ! [Xy: $i] :
% 0.30/0.47 ( ( Xphi @ Xy )
% 0.30/0.47 <=> ( Xy = Xx ) )
% 0.30/0.47 => ( exu
% 0.30/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(inCongP_type,type,
% 0.30/0.47 inCongP: $o ).
% 0.30/0.47
% 0.30/0.47 thf(inCongP,definition,
% 0.30/0.47 ( inCongP
% 0.30/0.47 = ( ! [A: $i,B: $i] :
% 0.30/0.47 ( ( A = B )
% 0.30/0.47 => ! [Xx: $i,Xy: $i] :
% 0.30/0.47 ( ( Xx = Xy )
% 0.30/0.47 => ( ( in @ Xx @ A )
% 0.30/0.47 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(in__Cong_type,type,
% 0.30/0.47 in__Cong: $o ).
% 0.30/0.47
% 0.30/0.47 thf(in__Cong,definition,
% 0.30/0.47 ( in__Cong
% 0.30/0.47 = ( ! [A: $i,B: $i] :
% 0.30/0.47 ( ( A = B )
% 0.30/0.47 => ! [Xx: $i,Xy: $i] :
% 0.30/0.47 ( ( Xx = Xy )
% 0.30/0.47 => ( ( in @ Xx @ A )
% 0.30/0.47 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(exuE3u_type,type,
% 0.30/0.47 exuE3u: $o ).
% 0.30/0.47
% 0.30/0.47 thf(exuE3u,definition,
% 0.30/0.47 ( exuE3u
% 0.30/0.47 = ( ! [Xphi: $i > $o] :
% 0.30/0.47 ( ( exu
% 0.30/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.47 => ! [Xx: $i,Xy: $i] :
% 0.30/0.47 ( ( Xphi @ Xx )
% 0.30/0.47 => ( ( Xphi @ Xy )
% 0.30/0.47 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(exu__Cong_type,type,
% 0.30/0.47 exu__Cong: $o ).
% 0.30/0.47
% 0.30/0.47 thf(exu__Cong,definition,
% 0.30/0.47 ( exu__Cong
% 0.30/0.47 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.30/0.47 ( ! [Xx: $i,Xy: $i] :
% 0.30/0.47 ( ( Xx = Xy )
% 0.30/0.47 => ( ( Xphi @ Xx )
% 0.30/0.47 <=> ( Xpsi @ Xy ) ) )
% 0.30/0.47 => ( ( exu
% 0.30/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.47 <=> ( exu
% 0.30/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(emptyset__Cong_type,type,
% 0.30/0.47 emptyset__Cong: $o ).
% 0.30/0.47
% 0.30/0.47 thf(emptyset__Cong,definition,
% 0.30/0.47 ( emptyset__Cong
% 0.30/0.47 = ( emptyset = emptyset ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(setadjoin__Cong_type,type,
% 0.30/0.47 setadjoin__Cong: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setadjoin__Cong,definition,
% 0.30/0.47 ( setadjoin__Cong
% 0.30/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.47 ( ( Xx = Xy )
% 0.30/0.47 => ! [Xz: $i,Xu: $i] :
% 0.30/0.47 ( ( Xz = Xu )
% 0.30/0.47 => ( ( setadjoin @ Xx @ Xz )
% 0.30/0.47 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(powerset__Cong_type,type,
% 0.30/0.47 powerset__Cong: $o ).
% 0.30/0.47
% 0.30/0.47 thf(powerset__Cong,definition,
% 0.30/0.47 ( powerset__Cong
% 0.30/0.47 = ( ! [A: $i,B: $i] :
% 0.30/0.47 ( ( A = B )
% 0.30/0.47 => ( ( powerset @ A )
% 0.30/0.47 = ( powerset @ B ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(setunion__Cong_type,type,
% 0.30/0.47 setunion__Cong: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setunion__Cong,definition,
% 0.30/0.47 ( setunion__Cong
% 0.30/0.47 = ( ! [A: $i,B: $i] :
% 0.30/0.47 ( ( A = B )
% 0.30/0.47 => ( ( setunion @ A )
% 0.30/0.47 = ( setunion @ B ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(omega__Cong_type,type,
% 0.30/0.47 omega__Cong: $o ).
% 0.30/0.47
% 0.30/0.47 thf(omega__Cong,definition,
% 0.30/0.47 ( omega__Cong
% 0.30/0.47 = ( omega = omega ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(exuEu_type,type,
% 0.30/0.47 exuEu: $o ).
% 0.30/0.47
% 0.30/0.47 thf(exuEu,definition,
% 0.30/0.47 ( exuEu
% 0.30/0.47 = ( ! [Xphi: $i > $o] :
% 0.30/0.47 ( ( exu
% 0.30/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.47 => ! [Xx: $i,Xy: $i] :
% 0.30/0.47 ( ( Xphi @ Xx )
% 0.30/0.47 => ( ( Xphi @ Xy )
% 0.30/0.47 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(descr__Cong_type,type,
% 0.30/0.47 descr__Cong: $o ).
% 0.30/0.47
% 0.30/0.47 thf(descr__Cong,definition,
% 0.30/0.47 ( descr__Cong
% 0.30/0.47 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.30/0.47 ( ! [Xx: $i,Xy: $i] :
% 0.30/0.47 ( ( Xx = Xy )
% 0.30/0.47 => ( ( Xphi @ Xx )
% 0.30/0.47 <=> ( Xpsi @ Xy ) ) )
% 0.30/0.47 => ( ( exu
% 0.30/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.47 => ( ( exu
% 0.30/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.30/0.47 => ( ( descr
% 0.30/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.47 = ( descr
% 0.30/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(dsetconstr__Cong_type,type,
% 0.30/0.47 dsetconstr__Cong: $o ).
% 0.30/0.47
% 0.30/0.47 thf(dsetconstr__Cong,definition,
% 0.30/0.47 ( dsetconstr__Cong
% 0.30/0.47 = ( ! [A: $i,B: $i] :
% 0.30/0.47 ( ( A = B )
% 0.30/0.47 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 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 @ B )
% 0.30/0.47 => ( ( Xx = Xy )
% 0.30/0.47 => ( ( Xphi @ Xx )
% 0.30/0.47 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.30/0.47 => ( ( dsetconstr @ A
% 0.30/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.47 = ( dsetconstr @ B
% 0.30/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(subset_type,type,
% 0.30/0.47 subset: $i > $i > $o ).
% 0.30/0.47
% 0.30/0.47 thf(disjoint_type,type,
% 0.30/0.47 disjoint: $i > $i > $o ).
% 0.30/0.47
% 0.30/0.47 thf(setsmeet_type,type,
% 0.30/0.47 setsmeet: $i > $i > $o ).
% 0.30/0.47
% 0.30/0.47 thf(subsetI1_type,type,
% 0.30/0.47 subsetI1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(subsetI1,definition,
% 0.30/0.47 ( subsetI1
% 0.30/0.47 = ( ! [A: $i,B: $i] :
% 0.30/0.47 ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( in @ Xx @ B ) )
% 0.30/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(eqimpsubset2_type,type,
% 0.30/0.47 eqimpsubset2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(eqimpsubset2,definition,
% 0.30/0.47 ( eqimpsubset2
% 0.30/0.47 = ( ! [A: $i,B: $i] :
% 0.30/0.47 ( ( A = B )
% 0.30/0.47 => ( subset @ B @ A ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(eqimpsubset1_type,type,
% 0.30/0.47 eqimpsubset1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(eqimpsubset1,definition,
% 0.30/0.47 ( eqimpsubset1
% 0.30/0.47 = ( ! [A: $i,B: $i] :
% 0.30/0.47 ( ( A = B )
% 0.30/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(subsetI2_type,type,
% 0.30/0.47 subsetI2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(subsetI2,definition,
% 0.30/0.47 ( subsetI2
% 0.30/0.47 = ( ! [A: $i,B: $i] :
% 0.30/0.47 ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( in @ Xx @ B ) )
% 0.30/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(emptysetsubset_type,type,
% 0.30/0.47 emptysetsubset: $o ).
% 0.30/0.47
% 0.30/0.47 thf(emptysetsubset,definition,
% 0.30/0.47 ( emptysetsubset
% 0.30/0.47 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(subsetE_type,type,
% 0.30/0.47 subsetE: $o ).
% 0.30/0.47
% 0.30/0.47 thf(subsetE,definition,
% 0.30/0.47 ( subsetE
% 0.30/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.47 ( ( subset @ A @ B )
% 0.30/0.47 => ( ( in @ Xx @ A )
% 0.30/0.47 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(subsetE2_type,type,
% 0.30/0.47 subsetE2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(subsetE2,definition,
% 0.30/0.47 ( subsetE2
% 0.30/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.47 ( ( subset @ A @ B )
% 0.30/0.47 => ( ~ ( in @ Xx @ B )
% 0.30/0.47 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(notsubsetI_type,type,
% 0.30/0.47 notsubsetI: $o ).
% 0.30/0.47
% 0.30/0.47 thf(notsubsetI,definition,
% 0.30/0.47 ( notsubsetI
% 0.30/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ~ ( in @ Xx @ B )
% 0.30/0.47 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(notequalI1_type,type,
% 0.30/0.47 notequalI1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(notequalI1,definition,
% 0.30/0.47 ( notequalI1
% 0.30/0.47 = ( ! [A: $i,B: $i] :
% 0.30/0.47 ( ~ ( subset @ A @ B )
% 0.30/0.47 => ( A != B ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(notequalI2_type,type,
% 0.30/0.47 notequalI2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(notequalI2,definition,
% 0.30/0.47 ( notequalI2
% 0.30/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ~ ( in @ Xx @ B )
% 0.30/0.47 => ( A != B ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(subsetRefl_type,type,
% 0.30/0.47 subsetRefl: $o ).
% 0.30/0.47
% 0.30/0.47 thf(subsetRefl,definition,
% 0.30/0.47 ( subsetRefl
% 0.30/0.47 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(subsetTrans_type,type,
% 0.30/0.47 subsetTrans: $o ).
% 0.30/0.47
% 0.30/0.47 thf(subsetTrans,definition,
% 0.30/0.47 ( subsetTrans
% 0.30/0.47 = ( ! [A: $i,B: $i,C: $i] :
% 0.30/0.47 ( ( subset @ A @ B )
% 0.30/0.47 => ( ( subset @ B @ C )
% 0.30/0.47 => ( subset @ A @ C ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(setadjoinSub_type,type,
% 0.30/0.47 setadjoinSub: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setadjoinSub,definition,
% 0.30/0.47 ( setadjoinSub
% 0.30/0.47 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(setadjoinSub2_type,type,
% 0.30/0.47 setadjoinSub2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setadjoinSub2,definition,
% 0.30/0.47 ( setadjoinSub2
% 0.30/0.47 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.30/0.47 ( ( subset @ A @ B )
% 0.30/0.47 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(subset2powerset_type,type,
% 0.30/0.47 subset2powerset: $o ).
% 0.30/0.47
% 0.30/0.47 thf(subset2powerset,definition,
% 0.30/0.47 ( subset2powerset
% 0.30/0.47 = ( ! [A: $i,B: $i] :
% 0.30/0.47 ( ( subset @ A @ B )
% 0.30/0.47 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(setextsub_type,type,
% 0.30/0.47 setextsub: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setextsub,definition,
% 0.30/0.47 ( setextsub
% 0.30/0.47 = ( ! [A: $i,B: $i] :
% 0.30/0.47 ( ( subset @ A @ B )
% 0.30/0.47 => ( ( subset @ B @ A )
% 0.30/0.47 => ( A = B ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(subsetemptysetimpeq_type,type,
% 0.30/0.47 subsetemptysetimpeq: $o ).
% 0.30/0.47
% 0.30/0.47 thf(subsetemptysetimpeq,definition,
% 0.30/0.47 ( subsetemptysetimpeq
% 0.30/0.47 = ( ! [A: $i] :
% 0.30/0.47 ( ( subset @ A @ emptyset )
% 0.30/0.47 => ( A = emptyset ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(powersetI1_type,type,
% 0.30/0.47 powersetI1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(powersetI1,definition,
% 0.30/0.47 ( powersetI1
% 0.30/0.47 = ( ! [A: $i,B: $i] :
% 0.30/0.47 ( ( subset @ B @ A )
% 0.30/0.47 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(powersetE1_type,type,
% 0.30/0.47 powersetE1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(powersetE1,definition,
% 0.30/0.47 ( powersetE1
% 0.30/0.47 = ( ! [A: $i,B: $i] :
% 0.30/0.47 ( ( in @ B @ ( powerset @ A ) )
% 0.30/0.47 => ( subset @ B @ A ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(inPowerset_type,type,
% 0.30/0.47 inPowerset: $o ).
% 0.30/0.47
% 0.30/0.47 thf(inPowerset,definition,
% 0.30/0.47 ( inPowerset
% 0.30/0.47 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(powersetsubset_type,type,
% 0.30/0.47 powersetsubset: $o ).
% 0.30/0.47
% 0.30/0.47 thf(powersetsubset,definition,
% 0.30/0.47 ( powersetsubset
% 0.30/0.47 = ( ! [A: $i,B: $i] :
% 0.30/0.47 ( ( subset @ A @ B )
% 0.30/0.47 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(sepInPowerset_type,type,
% 0.30/0.47 sepInPowerset: $o ).
% 0.30/0.47
% 0.30/0.47 thf(sepInPowerset,definition,
% 0.30/0.47 ( sepInPowerset
% 0.30/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.47 ( in
% 0.30/0.47 @ ( dsetconstr @ A
% 0.30/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.47 @ ( powerset @ A ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(sepSubset_type,type,
% 0.30/0.47 sepSubset: $o ).
% 0.30/0.47
% 0.30/0.47 thf(sepSubset,definition,
% 0.30/0.47 ( sepSubset
% 0.30/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.47 ( subset
% 0.30/0.47 @ ( dsetconstr @ A
% 0.30/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.47 @ A ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(binunion_type,type,
% 0.30/0.47 binunion: $i > $i > $i ).
% 0.30/0.47
% 0.30/0.47 thf(binunionIL_type,type,
% 0.30/0.47 binunionIL: $o ).
% 0.30/0.47
% 0.30/0.47 thf(binunionIL,definition,
% 0.30/0.47 ( binunionIL
% 0.30/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(upairset2IR_type,type,
% 0.30/0.47 upairset2IR: $o ).
% 0.30/0.47
% 0.30/0.47 thf(upairset2IR,definition,
% 0.30/0.47 ( upairset2IR
% 0.30/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(binunionIR_type,type,
% 0.30/0.47 binunionIR: $o ).
% 0.30/0.47
% 0.30/0.47 thf(binunionIR,definition,
% 0.30/0.47 ( binunionIR
% 0.30/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ B )
% 0.30/0.47 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(binunionEcases_type,type,
% 0.30/0.47 binunionEcases: $o ).
% 0.30/0.47
% 0.30/0.47 thf(binunionEcases,definition,
% 0.30/0.47 ( binunionEcases
% 0.30/0.47 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.30/0.47 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.30/0.47 => ( ( ( in @ Xx @ A )
% 0.30/0.47 => Xphi )
% 0.30/0.47 => ( ( ( in @ Xx @ B )
% 0.30/0.47 => Xphi )
% 0.30/0.47 => Xphi ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(binunionE_type,type,
% 0.30/0.47 binunionE: $o ).
% 0.30/0.47
% 0.30/0.47 thf(binunionE,definition,
% 0.30/0.47 ( binunionE
% 0.30/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.30/0.47 => ( ( in @ Xx @ A )
% 0.30/0.47 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(binunionLsub_type,type,
% 0.30/0.47 binunionLsub: $o ).
% 0.30/0.47
% 0.30/0.47 thf(binunionLsub,definition,
% 0.30/0.47 ( binunionLsub
% 0.30/0.47 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(binunionRsub_type,type,
% 0.30/0.47 binunionRsub: $o ).
% 0.30/0.47
% 0.30/0.47 thf(binunionRsub,definition,
% 0.30/0.47 ( binunionRsub
% 0.30/0.47 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(binintersect_type,type,
% 0.30/0.47 binintersect: $i > $i > $i ).
% 0.30/0.47
% 0.30/0.47 thf(binintersectI_type,type,
% 0.30/0.47 binintersectI: $o ).
% 0.30/0.47
% 0.30/0.47 thf(binintersectI,definition,
% 0.30/0.47 ( binintersectI
% 0.30/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( in @ Xx @ B )
% 0.30/0.47 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(binintersectSubset5_type,type,
% 0.30/0.47 binintersectSubset5: $o ).
% 0.30/0.47
% 0.30/0.47 thf(binintersectSubset5,definition,
% 0.30/0.47 ( binintersectSubset5
% 0.30/0.47 = ( ! [A: $i,B: $i,C: $i] :
% 0.30/0.47 ( ( subset @ C @ A )
% 0.30/0.47 => ( ( subset @ C @ B )
% 0.30/0.47 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(binintersectEL_type,type,
% 0.30/0.47 binintersectEL: $o ).
% 0.30/0.47
% 0.30/0.47 thf(binintersectEL,definition,
% 0.30/0.47 ( binintersectEL
% 0.30/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.30/0.47 => ( in @ Xx @ A ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(binintersectLsub_type,type,
% 0.30/0.47 binintersectLsub: $o ).
% 0.30/0.47
% 0.30/0.47 thf(binintersectLsub,definition,
% 0.30/0.47 ( binintersectLsub
% 0.30/0.47 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(binintersectSubset2_type,type,
% 0.30/0.47 binintersectSubset2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(binintersectSubset2,definition,
% 0.30/0.47 ( binintersectSubset2
% 0.30/0.47 = ( ! [A: $i,B: $i] :
% 0.30/0.47 ( ( subset @ A @ B )
% 0.30/0.47 => ( ( binintersect @ A @ B )
% 0.30/0.47 = A ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(binintersectSubset3_type,type,
% 0.30/0.47 binintersectSubset3: $o ).
% 0.30/0.47
% 0.30/0.47 thf(binintersectSubset3,definition,
% 0.30/0.47 ( binintersectSubset3
% 0.30/0.47 = ( ! [A: $i,B: $i] :
% 0.30/0.47 ( ( ( binintersect @ A @ B )
% 0.30/0.47 = B )
% 0.30/0.47 => ( subset @ B @ A ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(binintersectER_type,type,
% 0.30/0.47 binintersectER: $o ).
% 0.30/0.47
% 0.30/0.47 thf(binintersectER,definition,
% 0.30/0.47 ( binintersectER
% 0.30/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.30/0.47 => ( in @ Xx @ B ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(disjointsetsI1_type,type,
% 0.30/0.47 disjointsetsI1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(disjointsetsI1,definition,
% 0.30/0.47 ( disjointsetsI1
% 0.30/0.47 = ( ! [A: $i,B: $i] :
% 0.30/0.47 ( ~ ? [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 & ( in @ Xx @ B ) )
% 0.30/0.47 => ( ( binintersect @ A @ B )
% 0.30/0.47 = emptyset ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(binintersectRsub_type,type,
% 0.30/0.47 binintersectRsub: $o ).
% 0.30/0.47
% 0.30/0.47 thf(binintersectRsub,definition,
% 0.30/0.47 ( binintersectRsub
% 0.30/0.47 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(binintersectSubset4_type,type,
% 0.30/0.47 binintersectSubset4: $o ).
% 0.30/0.47
% 0.30/0.47 thf(binintersectSubset4,definition,
% 0.30/0.47 ( binintersectSubset4
% 0.30/0.47 = ( ! [A: $i,B: $i] :
% 0.30/0.47 ( ( subset @ B @ A )
% 0.30/0.47 => ( ( binintersect @ A @ B )
% 0.30/0.47 = B ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(binintersectSubset1_type,type,
% 0.30/0.47 binintersectSubset1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(binintersectSubset1,definition,
% 0.30/0.47 ( binintersectSubset1
% 0.30/0.47 = ( ! [A: $i,B: $i] :
% 0.30/0.47 ( ( ( binintersect @ A @ B )
% 0.30/0.47 = A )
% 0.30/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(bs114d_type,type,
% 0.30/0.47 bs114d: $o ).
% 0.30/0.47
% 0.30/0.47 thf(bs114d,definition,
% 0.30/0.47 ( bs114d
% 0.30/0.47 = ( ! [A: $i,B: $i,C: $i] :
% 0.30/0.47 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 0.30/0.47 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(regular_type,type,
% 0.30/0.47 regular: $i > $o ).
% 0.30/0.47
% 0.30/0.47 thf(setminus_type,type,
% 0.30/0.47 setminus: $i > $i > $i ).
% 0.30/0.47
% 0.30/0.47 thf(setminusI_type,type,
% 0.30/0.47 setminusI: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setminusI,definition,
% 0.30/0.47 ( setminusI
% 0.30/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ~ ( in @ Xx @ B )
% 0.30/0.47 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(setminusEL_type,type,
% 0.30/0.47 setminusEL: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setminusEL,definition,
% 0.30/0.47 ( setminusEL
% 0.30/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.30/0.47 => ( in @ Xx @ A ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(setminusER_type,type,
% 0.30/0.47 setminusER: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setminusER,definition,
% 0.30/0.47 ( setminusER
% 0.30/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.30/0.47 => ~ ( in @ Xx @ B ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(setminusSubset2_type,type,
% 0.30/0.47 setminusSubset2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setminusSubset2,definition,
% 0.30/0.47 ( setminusSubset2
% 0.30/0.47 = ( ! [A: $i,B: $i] :
% 0.30/0.47 ( ( subset @ A @ B )
% 0.30/0.47 => ( ( setminus @ A @ B )
% 0.30/0.47 = emptyset ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(setminusERneg_type,type,
% 0.30/0.47 setminusERneg: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setminusERneg,definition,
% 0.30/0.47 ( setminusERneg
% 0.30/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.47 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.30/0.47 => ( ( in @ Xx @ A )
% 0.30/0.47 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(setminusELneg_type,type,
% 0.30/0.47 setminusELneg: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setminusELneg,definition,
% 0.30/0.47 ( setminusELneg
% 0.30/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.47 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.30/0.47 => ( ~ ( in @ Xx @ B )
% 0.30/0.47 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(setminusILneg_type,type,
% 0.30/0.47 setminusILneg: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setminusILneg,definition,
% 0.30/0.47 ( setminusILneg
% 0.30/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.47 ( ~ ( in @ Xx @ A )
% 0.30/0.47 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(setminusIRneg_type,type,
% 0.30/0.47 setminusIRneg: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setminusIRneg,definition,
% 0.30/0.47 ( setminusIRneg
% 0.30/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ B )
% 0.30/0.47 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(setminusLsub_type,type,
% 0.30/0.47 setminusLsub: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setminusLsub,definition,
% 0.30/0.47 ( setminusLsub
% 0.30/0.47 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(setminusSubset1_type,type,
% 0.30/0.47 setminusSubset1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setminusSubset1,definition,
% 0.30/0.47 ( setminusSubset1
% 0.30/0.47 = ( ! [A: $i,B: $i] :
% 0.30/0.47 ( ( ( setminus @ A @ B )
% 0.30/0.47 = emptyset )
% 0.30/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(symdiff_type,type,
% 0.30/0.47 symdiff: $i > $i > $i ).
% 0.30/0.47
% 0.30/0.47 thf(symdiffE_type,type,
% 0.30/0.47 symdiffE: $o ).
% 0.30/0.47
% 0.30/0.47 thf(symdiffE,definition,
% 0.30/0.47 ( symdiffE
% 0.30/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 0.30/0.47 => ! [Xphi: $o] :
% 0.30/0.47 ( ( ( in @ Xx @ A )
% 0.30/0.47 => ( ~ ( in @ Xx @ B )
% 0.30/0.47 => Xphi ) )
% 0.30/0.47 => ( ( ~ ( in @ Xx @ A )
% 0.30/0.47 => ( ( in @ Xx @ B )
% 0.30/0.47 => Xphi ) )
% 0.30/0.47 => Xphi ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(symdiffI1_type,type,
% 0.30/0.47 symdiffI1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(symdiffI1,definition,
% 0.30/0.47 ( symdiffI1
% 0.30/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ~ ( in @ Xx @ B )
% 0.30/0.47 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(symdiffI2_type,type,
% 0.30/0.47 symdiffI2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(symdiffI2,definition,
% 0.30/0.47 ( symdiffI2
% 0.30/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.47 ( ~ ( in @ Xx @ A )
% 0.30/0.47 => ( ( in @ Xx @ B )
% 0.30/0.47 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(symdiffIneg1_type,type,
% 0.30/0.47 symdiffIneg1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(symdiffIneg1,definition,
% 0.30/0.47 ( symdiffIneg1
% 0.30/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( in @ Xx @ B )
% 0.30/0.47 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(symdiffIneg2_type,type,
% 0.30/0.47 symdiffIneg2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(symdiffIneg2,definition,
% 0.30/0.47 ( symdiffIneg2
% 0.30/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.47 ( ~ ( in @ Xx @ A )
% 0.30/0.47 => ( ~ ( in @ Xx @ B )
% 0.30/0.47 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(iskpair_type,type,
% 0.30/0.47 iskpair: $i > $o ).
% 0.30/0.47
% 0.30/0.47 thf(secondinupair_type,type,
% 0.30/0.47 secondinupair: $o ).
% 0.30/0.47
% 0.30/0.47 thf(secondinupair,definition,
% 0.30/0.47 ( secondinupair
% 0.30/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(setukpairIL_type,type,
% 0.30/0.47 setukpairIL: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setukpairIL,definition,
% 0.30/0.47 ( setukpairIL
% 0.30/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(setukpairIR_type,type,
% 0.30/0.47 setukpairIR: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setukpairIR,definition,
% 0.30/0.47 ( setukpairIR
% 0.30/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(kpairiskpair_type,type,
% 0.30/0.47 kpairiskpair: $o ).
% 0.30/0.47
% 0.30/0.47 thf(kpairiskpair,definition,
% 0.30/0.47 ( kpairiskpair
% 0.30/0.47 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(kpair_type,type,
% 0.30/0.47 kpair: $i > $i > $i ).
% 0.30/0.47
% 0.30/0.47 thf(kpairp_type,type,
% 0.30/0.47 kpairp: $o ).
% 0.30/0.47
% 0.30/0.47 thf(kpairp,definition,
% 0.30/0.47 ( kpairp
% 0.30/0.47 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(cartprod_type,type,
% 0.30/0.47 cartprod: $i > $i > $i ).
% 0.30/0.47
% 0.30/0.47 thf(singletonsubset_type,type,
% 0.30/0.47 singletonsubset: $o ).
% 0.30/0.47
% 0.30/0.47 thf(singletonsubset,definition,
% 0.30/0.47 ( singletonsubset
% 0.30/0.47 = ( ! [A: $i,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(singletoninpowerset_type,type,
% 0.30/0.47 singletoninpowerset: $o ).
% 0.30/0.47
% 0.30/0.47 thf(singletoninpowerset,definition,
% 0.30/0.47 ( singletoninpowerset
% 0.30/0.47 = ( ! [A: $i,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(singletoninpowunion_type,type,
% 0.30/0.47 singletoninpowunion: $o ).
% 0.30/0.47
% 0.30/0.47 thf(singletoninpowunion,definition,
% 0.30/0.47 ( singletoninpowunion
% 0.30/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(upairset2E_type,type,
% 0.30/0.47 upairset2E: $o ).
% 0.30/0.47
% 0.30/0.47 thf(upairset2E,definition,
% 0.30/0.47 ( upairset2E
% 0.30/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.30/0.47 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.30/0.47 => ( ( Xz = Xx )
% 0.30/0.47 | ( Xz = Xy ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(upairsubunion_type,type,
% 0.30/0.47 upairsubunion: $o ).
% 0.30/0.47
% 0.30/0.47 thf(upairsubunion,definition,
% 0.30/0.47 ( upairsubunion
% 0.30/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ B )
% 0.30/0.47 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(upairinpowunion_type,type,
% 0.30/0.47 upairinpowunion: $o ).
% 0.30/0.47
% 0.30/0.47 thf(upairinpowunion,definition,
% 0.30/0.47 ( upairinpowunion
% 0.30/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ B )
% 0.30/0.47 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(ubforcartprodlem1_type,type,
% 0.30/0.47 ubforcartprodlem1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(ubforcartprodlem1,definition,
% 0.30/0.47 ( ubforcartprodlem1
% 0.30/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ B )
% 0.30/0.47 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(ubforcartprodlem2_type,type,
% 0.30/0.47 ubforcartprodlem2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(ubforcartprodlem2,definition,
% 0.30/0.47 ( ubforcartprodlem2
% 0.30/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ B )
% 0.30/0.47 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(ubforcartprodlem3_type,type,
% 0.30/0.47 ubforcartprodlem3: $o ).
% 0.30/0.47
% 0.30/0.47 thf(ubforcartprodlem3,definition,
% 0.30/0.47 ( ubforcartprodlem3
% 0.30/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ B )
% 0.30/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(cartprodpairin_type,type,
% 0.30/0.47 cartprodpairin: $o ).
% 0.30/0.47
% 0.30/0.47 thf(cartprodpairin,definition,
% 0.30/0.47 ( cartprodpairin
% 0.30/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ B )
% 0.30/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(cartprodmempair1_type,type,
% 0.30/0.47 cartprodmempair1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(cartprodmempair1,definition,
% 0.30/0.47 ( cartprodmempair1
% 0.30/0.47 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.30/0.47 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 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 @ B )
% 0.30/0.47 & ( Xu
% 0.30/0.47 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(cartprodmempair_type,type,
% 0.30/0.47 cartprodmempair: $o ).
% 0.30/0.47
% 0.30/0.47 thf(cartprodmempair,definition,
% 0.30/0.47 ( cartprodmempair
% 0.30/0.47 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.30/0.47 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.30/0.47 => ( iskpair @ Xu ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(setunionE2_type,type,
% 0.30/0.47 setunionE2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setunionE2,definition,
% 0.30/0.47 ( setunionE2
% 0.30/0.47 = ( ! [A: $i,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ ( setunion @ A ) )
% 0.30/0.47 => ? [X: $i] :
% 0.30/0.47 ( ( in @ X @ A )
% 0.30/0.47 & ( in @ Xx @ X ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(setunionsingleton1_type,type,
% 0.30/0.47 setunionsingleton1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setunionsingleton1,definition,
% 0.30/0.47 ( setunionsingleton1
% 0.30/0.47 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(setunionsingleton2_type,type,
% 0.30/0.47 setunionsingleton2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setunionsingleton2,definition,
% 0.30/0.47 ( setunionsingleton2
% 0.30/0.47 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(setunionsingleton_type,type,
% 0.30/0.47 setunionsingleton: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setunionsingleton,definition,
% 0.30/0.47 ( setunionsingleton
% 0.30/0.47 = ( ! [Xx: $i] :
% 0.30/0.47 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 0.30/0.47 = Xx ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(singleton_type,type,
% 0.30/0.47 singleton: $i > $o ).
% 0.30/0.47
% 0.30/0.47 thf(singletonprop_type,type,
% 0.30/0.47 singletonprop: $o ).
% 0.30/0.47
% 0.30/0.47 thf(singletonprop,definition,
% 0.30/0.47 ( singletonprop
% 0.30/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 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 => ( ( Xphi @ Xx )
% 0.30/0.47 => ( ( Xphi @ Xy )
% 0.30/0.47 => ( Xx = Xy ) ) ) ) )
% 0.30/0.47 => ( ? [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 & ( Xphi @ Xx ) )
% 0.30/0.47 => ( singleton
% 0.30/0.47 @ ( dsetconstr @ A
% 0.30/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(ex1_type,type,
% 0.30/0.47 ex1: $i > ( $i > $o ) > $o ).
% 0.30/0.47
% 0.30/0.47 thf(ex1E1_type,type,
% 0.30/0.47 ex1E1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(ex1E1,definition,
% 0.30/0.47 ( ex1E1
% 0.30/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.47 ( ( ex1 @ A
% 0.30/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.47 => ? [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(ex1I_type,type,
% 0.30/0.47 ex1I: $o ).
% 0.30/0.47
% 0.30/0.47 thf(ex1I,definition,
% 0.30/0.47 ( ex1I
% 0.30/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( Xphi @ Xx )
% 0.30/0.47 => ( ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ A )
% 0.30/0.47 => ( ( Xphi @ Xy )
% 0.30/0.47 => ( Xy = Xx ) ) )
% 0.30/0.47 => ( ex1 @ A
% 0.30/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(ex1I2_type,type,
% 0.30/0.47 ex1I2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(ex1I2,definition,
% 0.30/0.47 ( ex1I2
% 0.30/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 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 => ( ( Xphi @ Xx )
% 0.30/0.47 => ( ( Xphi @ Xy )
% 0.30/0.47 => ( Xx = Xy ) ) ) ) )
% 0.30/0.47 => ( ? [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 & ( Xphi @ Xx ) )
% 0.30/0.47 => ( ex1 @ A
% 0.30/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(singletonsuniq_type,type,
% 0.30/0.47 singletonsuniq: $o ).
% 0.30/0.47
% 0.30/0.47 thf(singletonsuniq,definition,
% 0.30/0.47 ( singletonsuniq
% 0.30/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.47 ( ( ( setadjoin @ Xx @ emptyset )
% 0.30/0.47 = ( setadjoin @ Xy @ emptyset ) )
% 0.30/0.47 => ( Xx = Xy ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(atmost1p_type,type,
% 0.30/0.47 atmost1p: $i > $o ).
% 0.30/0.47
% 0.30/0.47 thf(atleast2p_type,type,
% 0.30/0.47 atleast2p: $i > $o ).
% 0.30/0.47
% 0.30/0.47 thf(atmost2p_type,type,
% 0.30/0.47 atmost2p: $i > $o ).
% 0.30/0.47
% 0.30/0.47 thf(upairsetp_type,type,
% 0.30/0.47 upairsetp: $i > $o ).
% 0.30/0.47
% 0.30/0.47 thf(setukpairinjL1_type,type,
% 0.30/0.47 setukpairinjL1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setukpairinjL1,definition,
% 0.30/0.47 ( setukpairinjL1
% 0.30/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.30/0.47 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 0.30/0.47 => ( Xx = Xz ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(kfstsingleton_type,type,
% 0.30/0.47 kfstsingleton: $o ).
% 0.30/0.47
% 0.30/0.47 thf(kfstsingleton,definition,
% 0.30/0.47 ( kfstsingleton
% 0.30/0.47 = ( ! [Xu: $i] :
% 0.30/0.47 ( ( iskpair @ Xu )
% 0.30/0.47 => ( singleton
% 0.30/0.47 @ ( dsetconstr @ ( setunion @ Xu )
% 0.30/0.47 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(theprop_type,type,
% 0.30/0.47 theprop: $o ).
% 0.30/0.47
% 0.30/0.47 thf(theprop,definition,
% 0.30/0.47 ( theprop
% 0.30/0.47 = ( ! [X: $i] :
% 0.30/0.47 ( ( singleton @ X )
% 0.30/0.47 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(kfst_type,type,
% 0.30/0.47 kfst: $i > $i ).
% 0.30/0.47
% 0.30/0.47 thf(kfstpairEq_type,type,
% 0.30/0.47 kfstpairEq: $o ).
% 0.30/0.47
% 0.30/0.47 thf(kfstpairEq,definition,
% 0.30/0.47 ( kfstpairEq
% 0.30/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.47 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.30/0.47 = Xx ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(cartprodfstin_type,type,
% 0.30/0.47 cartprodfstin: $o ).
% 0.30/0.47
% 0.30/0.47 thf(cartprodfstin,definition,
% 0.30/0.47 ( cartprodfstin
% 0.30/0.47 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.30/0.47 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.30/0.47 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(setukpairinjL2_type,type,
% 0.30/0.47 setukpairinjL2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setukpairinjL2,definition,
% 0.30/0.47 ( setukpairinjL2
% 0.30/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.30/0.47 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.30/0.47 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.30/0.47 => ( Xx = Xz ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(setukpairinjL_type,type,
% 0.30/0.47 setukpairinjL: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setukpairinjL,definition,
% 0.30/0.47 ( setukpairinjL
% 0.30/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.30/0.47 ( ( ( kpair @ Xx @ Xy )
% 0.30/0.47 = ( kpair @ Xz @ Xu ) )
% 0.30/0.47 => ( Xx = Xz ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(setukpairinjR11_type,type,
% 0.30/0.47 setukpairinjR11: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setukpairinjR11,definition,
% 0.30/0.47 ( setukpairinjR11
% 0.30/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.47 ( ( Xx = Xy )
% 0.30/0.47 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.30/0.47 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(setukpairinjR12_type,type,
% 0.30/0.47 setukpairinjR12: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setukpairinjR12,definition,
% 0.30/0.47 ( setukpairinjR12
% 0.30/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.47 ( ( Xx = Xy )
% 0.30/0.47 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.30/0.47 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(setukpairinjR1_type,type,
% 0.30/0.47 setukpairinjR1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setukpairinjR1,definition,
% 0.30/0.47 ( setukpairinjR1
% 0.30/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.30/0.47 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.30/0.47 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.30/0.47 => ( ( Xz = Xu )
% 0.30/0.47 => ( Xy = Xu ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(upairequniteq_type,type,
% 0.30/0.47 upairequniteq: $o ).
% 0.30/0.47
% 0.30/0.47 thf(upairequniteq,definition,
% 0.30/0.47 ( upairequniteq
% 0.30/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.30/0.47 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.30/0.47 = ( setadjoin @ Xz @ emptyset ) )
% 0.30/0.47 => ( Xx = Xy ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(setukpairinjR2_type,type,
% 0.30/0.47 setukpairinjR2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setukpairinjR2,definition,
% 0.30/0.47 ( setukpairinjR2
% 0.30/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.30/0.47 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.30/0.47 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.30/0.47 => ( Xy = Xu ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(setukpairinjR_type,type,
% 0.30/0.47 setukpairinjR: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setukpairinjR,definition,
% 0.30/0.47 ( setukpairinjR
% 0.30/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.30/0.47 ( ( ( kpair @ Xx @ Xy )
% 0.30/0.47 = ( kpair @ Xz @ Xu ) )
% 0.30/0.47 => ( Xy = Xu ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(ksndsingleton_type,type,
% 0.30/0.47 ksndsingleton: $o ).
% 0.30/0.47
% 0.30/0.47 thf(ksndsingleton,definition,
% 0.30/0.47 ( ksndsingleton
% 0.30/0.47 = ( ! [Xu: $i] :
% 0.30/0.47 ( ( iskpair @ Xu )
% 0.30/0.47 => ( singleton
% 0.30/0.47 @ ( dsetconstr @ ( setunion @ Xu )
% 0.30/0.47 @ ^ [Xx: $i] :
% 0.30/0.47 ( Xu
% 0.30/0.47 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(ksnd_type,type,
% 0.30/0.47 ksnd: $i > $i ).
% 0.30/0.47
% 0.30/0.47 thf(ksndpairEq_type,type,
% 0.30/0.47 ksndpairEq: $o ).
% 0.30/0.47
% 0.30/0.47 thf(ksndpairEq,definition,
% 0.30/0.47 ( ksndpairEq
% 0.30/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.47 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.30/0.47 = Xy ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(kpairsurjEq_type,type,
% 0.30/0.47 kpairsurjEq: $o ).
% 0.30/0.47
% 0.30/0.47 thf(kpairsurjEq,definition,
% 0.30/0.47 ( kpairsurjEq
% 0.30/0.47 = ( ! [Xu: $i] :
% 0.30/0.47 ( ( iskpair @ Xu )
% 0.30/0.47 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.30/0.47 = Xu ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(cartprodsndin_type,type,
% 0.30/0.47 cartprodsndin: $o ).
% 0.30/0.47
% 0.30/0.47 thf(cartprodsndin,definition,
% 0.30/0.47 ( cartprodsndin
% 0.30/0.47 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.30/0.47 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.30/0.47 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(cartprodpairmemEL_type,type,
% 0.30/0.47 cartprodpairmemEL: $o ).
% 0.30/0.47
% 0.30/0.47 thf(cartprodpairmemEL,definition,
% 0.30/0.47 ( cartprodpairmemEL
% 0.30/0.47 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.30/0.47 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.30/0.47 => ( in @ Xx @ A ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(cartprodpairmemER_type,type,
% 0.30/0.47 cartprodpairmemER: $o ).
% 0.30/0.47
% 0.30/0.47 thf(cartprodpairmemER,definition,
% 0.30/0.47 ( cartprodpairmemER
% 0.30/0.47 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.30/0.47 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.30/0.47 => ( in @ Xy @ B ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(cartprodmempaircEq_type,type,
% 0.30/0.47 cartprodmempaircEq: $o ).
% 0.30/0.47
% 0.30/0.47 thf(cartprodmempaircEq,definition,
% 0.30/0.47 ( cartprodmempaircEq
% 0.30/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ B )
% 0.30/0.47 => ( ( kpair @ Xx @ Xy )
% 0.30/0.47 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(cartprodfstpairEq_type,type,
% 0.30/0.47 cartprodfstpairEq: $o ).
% 0.30/0.47
% 0.30/0.47 thf(cartprodfstpairEq,definition,
% 0.30/0.47 ( cartprodfstpairEq
% 0.30/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ B )
% 0.30/0.47 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.30/0.47 = Xx ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(cartprodsndpairEq_type,type,
% 0.30/0.47 cartprodsndpairEq: $o ).
% 0.30/0.47
% 0.30/0.47 thf(cartprodsndpairEq,definition,
% 0.30/0.47 ( cartprodsndpairEq
% 0.30/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ B )
% 0.30/0.47 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.30/0.47 = Xy ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(cartprodpairsurjEq_type,type,
% 0.30/0.47 cartprodpairsurjEq: $o ).
% 0.30/0.47
% 0.30/0.47 thf(cartprodpairsurjEq,definition,
% 0.30/0.47 ( cartprodpairsurjEq
% 0.30/0.47 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.30/0.47 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.30/0.47 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.30/0.47 = Xu ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(breln_type,type,
% 0.30/0.47 breln: $i > $i > $i > $o ).
% 0.30/0.47
% 0.30/0.47 thf(dpsetconstr_type,type,
% 0.30/0.47 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 0.30/0.47
% 0.30/0.47 thf(dpsetconstrI_type,type,
% 0.30/0.47 dpsetconstrI: $o ).
% 0.30/0.47
% 0.30/0.47 thf(dpsetconstrI,definition,
% 0.30/0.47 ( dpsetconstrI
% 0.30/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ B )
% 0.30/0.47 => ( ( Xphi @ Xx @ Xy )
% 0.30/0.47 => ( in @ ( kpair @ Xx @ Xy )
% 0.30/0.47 @ ( dpsetconstr @ A @ B
% 0.30/0.47 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(dpsetconstrSub_type,type,
% 0.30/0.47 dpsetconstrSub: $o ).
% 0.30/0.47
% 0.30/0.47 thf(dpsetconstrSub,definition,
% 0.30/0.47 ( dpsetconstrSub
% 0.30/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.30/0.47 ( subset
% 0.30/0.47 @ ( dpsetconstr @ A @ B
% 0.30/0.47 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 0.30/0.47 @ ( cartprod @ A @ B ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(setOfPairsIsBReln_type,type,
% 0.30/0.47 setOfPairsIsBReln: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setOfPairsIsBReln,definition,
% 0.30/0.47 ( setOfPairsIsBReln
% 0.30/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.30/0.47 ( breln @ A @ B
% 0.30/0.47 @ ( dpsetconstr @ A @ B
% 0.30/0.47 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(dpsetconstrERa_type,type,
% 0.30/0.47 dpsetconstrERa: $o ).
% 0.30/0.47
% 0.30/0.47 thf(dpsetconstrERa,definition,
% 0.30/0.47 ( dpsetconstrERa
% 0.30/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ B )
% 0.30/0.47 => ( ( in @ ( kpair @ Xx @ Xy )
% 0.30/0.47 @ ( dpsetconstr @ A @ B
% 0.30/0.47 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.30/0.47 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(dpsetconstrEL1_type,type,
% 0.30/0.47 dpsetconstrEL1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(dpsetconstrEL1,definition,
% 0.30/0.47 ( dpsetconstrEL1
% 0.30/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.30/0.47 ( ( in @ ( kpair @ Xx @ Xy )
% 0.30/0.47 @ ( dpsetconstr @ A @ B
% 0.30/0.47 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.30/0.47 => ( in @ Xx @ A ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(dpsetconstrEL2_type,type,
% 0.30/0.47 dpsetconstrEL2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(dpsetconstrEL2,definition,
% 0.30/0.47 ( dpsetconstrEL2
% 0.30/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.30/0.47 ( ( in @ ( kpair @ Xx @ Xy )
% 0.30/0.47 @ ( dpsetconstr @ A @ B
% 0.30/0.47 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.30/0.47 => ( in @ Xy @ B ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(dpsetconstrER_type,type,
% 0.30/0.47 dpsetconstrER: $o ).
% 0.30/0.47
% 0.30/0.47 thf(dpsetconstrER,definition,
% 0.30/0.47 ( dpsetconstrER
% 0.30/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.30/0.47 ( ( in @ ( kpair @ Xx @ Xy )
% 0.30/0.47 @ ( dpsetconstr @ A @ B
% 0.30/0.47 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.30/0.47 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(func_type,type,
% 0.30/0.47 func: $i > $i > $i > $o ).
% 0.30/0.47
% 0.30/0.47 thf(funcSet_type,type,
% 0.30/0.47 funcSet: $i > $i > $i ).
% 0.30/0.47
% 0.30/0.47 thf(funcImageSingleton_type,type,
% 0.30/0.47 funcImageSingleton: $o ).
% 0.30/0.47
% 0.30/0.47 thf(funcImageSingleton,definition,
% 0.30/0.47 ( funcImageSingleton
% 0.30/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.47 ( ( func @ A @ B @ Xf )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( singleton
% 0.30/0.47 @ ( dsetconstr @ B
% 0.30/0.47 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(apProp_type,type,
% 0.30/0.47 apProp: $o ).
% 0.30/0.47
% 0.30/0.47 thf(apProp,definition,
% 0.30/0.47 ( apProp
% 0.30/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.47 ( ( func @ A @ B @ Xf )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( in
% 0.30/0.47 @ ( setunion
% 0.30/0.47 @ ( dsetconstr @ B
% 0.30/0.47 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 0.30/0.47 @ B ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(ap_type,type,
% 0.30/0.47 ap: $i > $i > $i > $i > $i ).
% 0.30/0.47
% 0.30/0.47 thf(app_type,type,
% 0.30/0.47 app: $o ).
% 0.30/0.47
% 0.30/0.47 thf(app,definition,
% 0.30/0.47 ( app
% 0.30/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.47 ( ( func @ A @ B @ Xf )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(infuncsetfunc_type,type,
% 0.30/0.47 infuncsetfunc: $o ).
% 0.30/0.47
% 0.30/0.47 thf(infuncsetfunc,definition,
% 0.30/0.47 ( infuncsetfunc
% 0.30/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.30/0.47 => ( func @ A @ B @ Xf ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(ap2p_type,type,
% 0.30/0.47 ap2p: $o ).
% 0.30/0.47
% 0.30/0.47 thf(ap2p,definition,
% 0.30/0.47 ( ap2p
% 0.30/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(funcinfuncset_type,type,
% 0.30/0.47 funcinfuncset: $o ).
% 0.30/0.47
% 0.30/0.47 thf(funcinfuncset,definition,
% 0.30/0.47 ( funcinfuncset
% 0.30/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.47 ( ( func @ A @ B @ Xf )
% 0.30/0.47 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(lamProp_type,type,
% 0.30/0.47 lamProp: $o ).
% 0.30/0.47
% 0.30/0.47 thf(lamProp,definition,
% 0.30/0.47 ( lamProp
% 0.30/0.47 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.30/0.47 ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.30/0.47 => ( func @ A @ B
% 0.30/0.47 @ ( dpsetconstr @ A @ B
% 0.30/0.47 @ ^ [Xx: $i,Xy: $i] :
% 0.30/0.47 ( ( Xf @ Xx )
% 0.30/0.47 = Xy ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(lam_type,type,
% 0.30/0.47 lam: $i > $i > ( $i > $i ) > $i ).
% 0.30/0.47
% 0.30/0.47 thf(lamp_type,type,
% 0.30/0.47 lamp: $o ).
% 0.30/0.47
% 0.30/0.47 thf(lamp,definition,
% 0.30/0.47 ( lamp
% 0.30/0.47 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.30/0.47 ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.30/0.47 => ( func @ A @ B
% 0.30/0.47 @ ( lam @ A @ B
% 0.30/0.47 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(lam2p_type,type,
% 0.30/0.47 lam2p: $o ).
% 0.30/0.47
% 0.30/0.47 thf(lam2p,definition,
% 0.30/0.47 ( lam2p
% 0.30/0.47 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.30/0.47 ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.30/0.47 => ( in
% 0.30/0.47 @ ( lam @ A @ B
% 0.30/0.47 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.30/0.47 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(brelnall1_type,type,
% 0.30/0.47 brelnall1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(brelnall1,definition,
% 0.30/0.47 ( brelnall1
% 0.30/0.47 = ( ! [A: $i,B: $i,R: $i] :
% 0.30/0.47 ( ( breln @ A @ B @ R )
% 0.30/0.47 => ! [Xphi: $i > $o] :
% 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 @ B )
% 0.30/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.30/0.47 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ R )
% 0.30/0.47 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(brelnall2_type,type,
% 0.30/0.47 brelnall2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(brelnall2,definition,
% 0.30/0.47 ( brelnall2
% 0.30/0.47 = ( ! [A: $i,B: $i,R: $i] :
% 0.30/0.47 ( ( breln @ A @ B @ R )
% 0.30/0.47 => ! [Xphi: $i > $o] :
% 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 @ B )
% 0.30/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.30/0.47 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ R )
% 0.30/0.47 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(ex1E2_type,type,
% 0.30/0.47 ex1E2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(ex1E2,definition,
% 0.30/0.47 ( ex1E2
% 0.30/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.47 ( ( ex1 @ A
% 0.30/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 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 => ( ( Xphi @ Xx )
% 0.30/0.47 => ( ( Xphi @ Xy )
% 0.30/0.47 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(funcGraphProp1_type,type,
% 0.30/0.47 funcGraphProp1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(funcGraphProp1,definition,
% 0.30/0.47 ( funcGraphProp1
% 0.30/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.47 ( ( func @ A @ B @ Xf )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(funcGraphProp3_type,type,
% 0.30/0.47 funcGraphProp3: $o ).
% 0.30/0.47
% 0.30/0.47 thf(funcGraphProp3,definition,
% 0.30/0.47 ( funcGraphProp3
% 0.30/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(funcGraphProp2_type,type,
% 0.30/0.47 funcGraphProp2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(funcGraphProp2,definition,
% 0.30/0.47 ( funcGraphProp2
% 0.30/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.47 ( ( func @ A @ B @ Xf )
% 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 @ B )
% 0.30/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.30/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.30/0.47 = Xy ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(funcextLem_type,type,
% 0.30/0.47 funcextLem: $o ).
% 0.30/0.47
% 0.30/0.47 thf(funcextLem,definition,
% 0.30/0.47 ( funcextLem
% 0.30/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.47 ( ( func @ A @ B @ Xf )
% 0.30/0.47 => ! [Xg: $i] :
% 0.30/0.47 ( ( func @ A @ B @ Xg )
% 0.30/0.47 => ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.30/0.47 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 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 @ B )
% 0.30/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 0.30/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(funcGraphProp4_type,type,
% 0.30/0.47 funcGraphProp4: $o ).
% 0.30/0.47
% 0.30/0.47 thf(funcGraphProp4,definition,
% 0.30/0.47 ( funcGraphProp4
% 0.30/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 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 @ B )
% 0.30/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.30/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.30/0.47 = Xy ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(subbreln_type,type,
% 0.30/0.47 subbreln: $o ).
% 0.30/0.47
% 0.30/0.47 thf(subbreln,definition,
% 0.30/0.47 ( subbreln
% 0.30/0.47 = ( ! [A: $i,B: $i,R: $i] :
% 0.30/0.47 ( ( breln @ A @ B @ R )
% 0.30/0.47 => ! [S: $i] :
% 0.30/0.47 ( ( breln @ A @ B @ 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 @ B )
% 0.30/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.30/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.30/0.47 => ( subset @ R @ S ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(eqbreln_type,type,
% 0.30/0.47 eqbreln: $o ).
% 0.30/0.47
% 0.30/0.47 thf(eqbreln,definition,
% 0.30/0.47 ( eqbreln
% 0.30/0.47 = ( ! [A: $i,B: $i,R: $i] :
% 0.30/0.47 ( ( breln @ A @ B @ R )
% 0.30/0.47 => ! [S: $i] :
% 0.30/0.47 ( ( breln @ A @ B @ 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 @ B )
% 0.30/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.30/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ 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 @ B )
% 0.30/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.30/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 0.30/0.47 => ( R = S ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(funcext_type,type,
% 0.30/0.47 funcext: $o ).
% 0.30/0.47
% 0.30/0.47 thf(funcext,definition,
% 0.30/0.47 ( funcext
% 0.30/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.47 ( ( func @ A @ B @ Xf )
% 0.30/0.47 => ! [Xg: $i] :
% 0.30/0.47 ( ( func @ A @ B @ Xg )
% 0.30/0.47 => ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.30/0.47 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.30/0.47 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(funcext2_type,type,
% 0.30/0.47 funcext2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(funcext2,definition,
% 0.30/0.47 ( funcext2
% 0.30/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.30/0.47 => ! [Xg: $i] :
% 0.30/0.47 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 0.30/0.47 => ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.30/0.47 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.30/0.47 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(ap2apEq1_type,type,
% 0.30/0.47 ap2apEq1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(ap2apEq1,definition,
% 0.30/0.47 ( ap2apEq1
% 0.30/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.30/0.47 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(ap2apEq2_type,type,
% 0.30/0.47 ap2apEq2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(ap2apEq2,definition,
% 0.30/0.47 ( ap2apEq2
% 0.30/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.47 ( ( func @ A @ B @ Xf )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.30/0.47 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(beta1_type,type,
% 0.30/0.47 beta1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(beta1,definition,
% 0.30/0.47 ( beta1
% 0.30/0.47 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.30/0.47 ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( ap @ A @ B
% 0.30/0.47 @ ( lam @ A @ B
% 0.30/0.47 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.30/0.47 @ Xx )
% 0.30/0.47 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(eta1_type,type,
% 0.30/0.47 eta1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(eta1,definition,
% 0.30/0.47 ( eta1
% 0.30/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.47 ( ( func @ A @ B @ Xf )
% 0.30/0.47 => ( ( lam @ A @ B
% 0.30/0.47 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.30/0.47 = Xf ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(lam2lamEq_type,type,
% 0.30/0.47 lam2lamEq: $o ).
% 0.30/0.47
% 0.30/0.47 thf(lam2lamEq,definition,
% 0.30/0.47 ( lam2lamEq
% 0.30/0.47 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.30/0.47 ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.30/0.47 => ( ( lam @ A @ B
% 0.30/0.47 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.30/0.47 = ( lam @ A @ B
% 0.30/0.47 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(beta2_type,type,
% 0.30/0.47 beta2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(beta2,definition,
% 0.30/0.47 ( beta2
% 0.30/0.47 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.30/0.47 ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( ap @ A @ B
% 0.30/0.47 @ ( lam @ A @ B
% 0.30/0.47 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.30/0.47 @ Xx )
% 0.30/0.47 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(eta2_type,type,
% 0.30/0.47 eta2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(eta2,definition,
% 0.30/0.47 ( eta2
% 0.30/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.30/0.47 => ( ( lam @ A @ B
% 0.30/0.47 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.30/0.47 = Xf ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(iffalseProp1_type,type,
% 0.30/0.47 iffalseProp1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(iffalseProp1,definition,
% 0.30/0.47 ( iffalseProp1
% 0.30/0.47 = ( ! [A: $i,Xphi: $o,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 => ( ~ Xphi
% 0.30/0.47 => ( in @ Xy
% 0.30/0.47 @ ( dsetconstr @ A
% 0.30/0.47 @ ^ [Xz: $i] :
% 0.30/0.47 ( ( Xphi
% 0.30/0.47 & ( Xz = Xx ) )
% 0.30/0.47 | ( ~ Xphi
% 0.30/0.47 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(iffalseProp2_type,type,
% 0.30/0.47 iffalseProp2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(iffalseProp2,definition,
% 0.30/0.47 ( iffalseProp2
% 0.30/0.47 = ( ! [A: $i,Xphi: $o,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 => ( ~ Xphi
% 0.30/0.47 => ( ( dsetconstr @ A
% 0.30/0.47 @ ^ [Xz: $i] :
% 0.30/0.47 ( ( Xphi
% 0.30/0.47 & ( Xz = Xx ) )
% 0.30/0.47 | ( ~ Xphi
% 0.30/0.47 & ( Xz = Xy ) ) ) )
% 0.30/0.47 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(iftrueProp1_type,type,
% 0.30/0.47 iftrueProp1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(iftrueProp1,definition,
% 0.30/0.47 ( iftrueProp1
% 0.30/0.47 = ( ! [A: $i,Xphi: $o,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 => ( Xphi
% 0.30/0.47 => ( in @ Xx
% 0.30/0.47 @ ( dsetconstr @ A
% 0.30/0.47 @ ^ [Xz: $i] :
% 0.30/0.47 ( ( Xphi
% 0.30/0.47 & ( Xz = Xx ) )
% 0.30/0.47 | ( ~ Xphi
% 0.30/0.47 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(iftrueProp2_type,type,
% 0.30/0.47 iftrueProp2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(iftrueProp2,definition,
% 0.30/0.47 ( iftrueProp2
% 0.30/0.47 = ( ! [A: $i,Xphi: $o,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 => ( Xphi
% 0.30/0.47 => ( ( dsetconstr @ A
% 0.30/0.47 @ ^ [Xz: $i] :
% 0.30/0.47 ( ( Xphi
% 0.30/0.47 & ( Xz = Xx ) )
% 0.30/0.47 | ( ~ Xphi
% 0.30/0.47 & ( Xz = Xy ) ) ) )
% 0.30/0.47 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(ifSingleton_type,type,
% 0.30/0.47 ifSingleton: $o ).
% 0.30/0.47
% 0.30/0.47 thf(ifSingleton,definition,
% 0.30/0.47 ( ifSingleton
% 0.30/0.47 = ( ! [A: $i,Xphi: $o,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 => ( singleton
% 0.30/0.47 @ ( dsetconstr @ A
% 0.30/0.47 @ ^ [Xz: $i] :
% 0.30/0.47 ( ( Xphi
% 0.30/0.47 & ( Xz = Xx ) )
% 0.30/0.47 | ( ~ Xphi
% 0.30/0.47 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(if_type,type,
% 0.30/0.47 if: $i > $o > $i > $i > $i ).
% 0.30/0.47
% 0.30/0.47 thf(ifp_type,type,
% 0.30/0.47 ifp: $o ).
% 0.30/0.47
% 0.30/0.47 thf(ifp,definition,
% 0.30/0.47 ( ifp
% 0.30/0.47 = ( ! [A: $i,Xphi: $o,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 @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(theeq_type,type,
% 0.30/0.47 theeq: $o ).
% 0.30/0.47
% 0.30/0.47 thf(theeq,definition,
% 0.30/0.47 ( theeq
% 0.30/0.47 = ( ! [X: $i] :
% 0.30/0.47 ( ( singleton @ X )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ X )
% 0.30/0.47 => ( ( setunion @ X )
% 0.30/0.47 = Xx ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(iftrue_type,type,
% 0.30/0.47 iftrue: $o ).
% 0.30/0.47
% 0.30/0.47 thf(iftrue,definition,
% 0.30/0.47 ( iftrue
% 0.30/0.47 = ( ! [A: $i,Xphi: $o,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 => ( Xphi
% 0.30/0.47 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.30/0.47 = Xx ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(iffalse_type,type,
% 0.30/0.47 iffalse: $o ).
% 0.30/0.47
% 0.30/0.47 thf(iffalse,definition,
% 0.30/0.47 ( iffalse
% 0.30/0.47 = ( ! [A: $i,Xphi: $o,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 => ( ~ Xphi
% 0.30/0.47 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.30/0.47 = Xy ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(iftrueorfalse_type,type,
% 0.30/0.47 iftrueorfalse: $o ).
% 0.30/0.47
% 0.30/0.47 thf(iftrueorfalse,definition,
% 0.30/0.47 ( iftrueorfalse
% 0.30/0.47 = ( ! [A: $i,Xphi: $o,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 @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(binintersectT_lem_type,type,
% 0.30/0.47 binintersectT_lem: $o ).
% 0.30/0.47
% 0.30/0.47 thf(binintersectT_lem,definition,
% 0.30/0.47 ( binintersectT_lem
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(binunionT_lem_type,type,
% 0.30/0.47 binunionT_lem: $o ).
% 0.30/0.47
% 0.30/0.47 thf(binunionT_lem,definition,
% 0.30/0.47 ( binunionT_lem
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(powersetT_lem_type,type,
% 0.30/0.47 powersetT_lem: $o ).
% 0.30/0.47
% 0.30/0.47 thf(powersetT_lem,definition,
% 0.30/0.47 ( powersetT_lem
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(setminusT_lem_type,type,
% 0.30/0.47 setminusT_lem: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setminusT_lem,definition,
% 0.30/0.47 ( setminusT_lem
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ( in @ ( setminus @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(complementT_lem_type,type,
% 0.30/0.47 complementT_lem: $o ).
% 0.30/0.47
% 0.30/0.47 thf(complementT_lem,definition,
% 0.30/0.47 ( complementT_lem
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ( in @ ( setminus @ A @ X ) @ ( powerset @ A ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(setextT_type,type,
% 0.30/0.47 setextT: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setextT,definition,
% 0.30/0.47 ( setextT
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( in @ Xx @ X )
% 0.30/0.47 => ( in @ Xx @ Y ) ) )
% 0.30/0.47 => ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( in @ Xx @ Y )
% 0.30/0.47 => ( in @ Xx @ X ) ) )
% 0.30/0.47 => ( X = Y ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(subsetTI_type,type,
% 0.30/0.47 subsetTI: $o ).
% 0.30/0.47
% 0.30/0.47 thf(subsetTI,definition,
% 0.30/0.47 ( subsetTI
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( in @ Xx @ X )
% 0.30/0.47 => ( in @ Xx @ Y ) ) )
% 0.30/0.47 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(powersetTI1_type,type,
% 0.30/0.47 powersetTI1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(powersetTI1,definition,
% 0.30/0.47 ( powersetTI1
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( in @ Xx @ X )
% 0.30/0.47 => ( in @ Xx @ Y ) ) )
% 0.30/0.47 => ( in @ X @ ( powerset @ Y ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(powersetTE1_type,type,
% 0.30/0.47 powersetTE1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(powersetTE1,definition,
% 0.30/0.47 ( powersetTE1
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( in @ X @ ( powerset @ Y ) )
% 0.30/0.47 => ( ( in @ Xx @ X )
% 0.30/0.47 => ( in @ Xx @ Y ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(complementTI1_type,type,
% 0.30/0.47 complementTI1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(complementTI1,definition,
% 0.30/0.47 ( complementTI1
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( in @ Xx @ X )
% 0.30/0.47 => ~ ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(complementTE1_type,type,
% 0.30/0.47 complementTE1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(complementTE1,definition,
% 0.30/0.47 ( complementTE1
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ~ ( in @ Xx @ ( setminus @ A @ X ) )
% 0.30/0.47 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(binintersectTELcontra_type,type,
% 0.30/0.47 binintersectTELcontra: $o ).
% 0.30/0.47
% 0.30/0.47 thf(binintersectTELcontra,definition,
% 0.30/0.47 ( binintersectTELcontra
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ~ ( in @ Xx @ X )
% 0.30/0.47 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(binintersectTERcontra_type,type,
% 0.30/0.47 binintersectTERcontra: $o ).
% 0.30/0.47
% 0.30/0.47 thf(binintersectTERcontra,definition,
% 0.30/0.47 ( binintersectTERcontra
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ~ ( in @ Xx @ Y )
% 0.30/0.47 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(contrasubsetT_type,type,
% 0.30/0.47 contrasubsetT: $o ).
% 0.30/0.47
% 0.30/0.47 thf(contrasubsetT,definition,
% 0.30/0.47 ( contrasubsetT
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.30/0.47 => ( ( in @ Xx @ Y )
% 0.30/0.47 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(contrasubsetT1_type,type,
% 0.30/0.47 contrasubsetT1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(contrasubsetT1,definition,
% 0.30/0.47 ( contrasubsetT1
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( subset @ X @ Y )
% 0.30/0.47 => ( ~ ( in @ Xx @ Y )
% 0.30/0.47 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(contrasubsetT2_type,type,
% 0.30/0.47 contrasubsetT2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(contrasubsetT2,definition,
% 0.30/0.47 ( contrasubsetT2
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ( ( subset @ X @ Y )
% 0.30/0.47 => ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(contrasubsetT3_type,type,
% 0.30/0.47 contrasubsetT3: $o ).
% 0.30/0.47
% 0.30/0.47 thf(contrasubsetT3,definition,
% 0.30/0.47 ( contrasubsetT3
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ( ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) )
% 0.30/0.47 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(doubleComplementI1_type,type,
% 0.30/0.47 doubleComplementI1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(doubleComplementI1,definition,
% 0.30/0.47 ( doubleComplementI1
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( in @ Xx @ X )
% 0.30/0.47 => ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(doubleComplementE1_type,type,
% 0.30/0.47 doubleComplementE1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(doubleComplementE1,definition,
% 0.30/0.47 ( doubleComplementE1
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) )
% 0.30/0.47 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(doubleComplementSub1_type,type,
% 0.30/0.47 doubleComplementSub1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(doubleComplementSub1,definition,
% 0.30/0.47 ( doubleComplementSub1
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ( subset @ X @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(doubleComplementSub2_type,type,
% 0.30/0.47 doubleComplementSub2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(doubleComplementSub2,definition,
% 0.30/0.47 ( doubleComplementSub2
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ( subset @ ( setminus @ A @ ( setminus @ A @ X ) ) @ X ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(doubleComplementEq_type,type,
% 0.30/0.47 doubleComplementEq: $o ).
% 0.30/0.47
% 0.30/0.47 thf(doubleComplementEq,definition,
% 0.30/0.47 ( doubleComplementEq
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ( X
% 0.30/0.47 = ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(complementTnotintersectT_type,type,
% 0.30/0.47 complementTnotintersectT: $o ).
% 0.30/0.47
% 0.30/0.47 thf(complementTnotintersectT,definition,
% 0.30/0.47 ( complementTnotintersectT
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.30/0.47 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(complementImpComplementIntersect_type,type,
% 0.30/0.47 complementImpComplementIntersect: $o ).
% 0.30/0.47
% 0.30/0.47 thf(complementImpComplementIntersect,definition,
% 0.30/0.47 ( complementImpComplementIntersect
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.30/0.47 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(complementSubsetComplementIntersect_type,type,
% 0.30/0.47 complementSubsetComplementIntersect: $o ).
% 0.30/0.47
% 0.30/0.47 thf(complementSubsetComplementIntersect,definition,
% 0.30/0.47 ( complementSubsetComplementIntersect
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ( subset @ ( setminus @ A @ X ) @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(complementInPowersetComplementIntersect_type,type,
% 0.30/0.47 complementInPowersetComplementIntersect: $o ).
% 0.30/0.47
% 0.30/0.47 thf(complementInPowersetComplementIntersect,definition,
% 0.30/0.47 ( complementInPowersetComplementIntersect
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(contraSubsetComplement_type,type,
% 0.30/0.47 contraSubsetComplement: $o ).
% 0.30/0.47
% 0.30/0.47 thf(contraSubsetComplement,definition,
% 0.30/0.47 ( contraSubsetComplement
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( in @ Xx @ Y )
% 0.30/0.47 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(complementTcontraSubset_type,type,
% 0.30/0.47 complementTcontraSubset: $o ).
% 0.30/0.47
% 0.30/0.47 thf(complementTcontraSubset,definition,
% 0.30/0.47 ( complementTcontraSubset
% 0.30/0.48 = ( ! [A: $i,X: $i] :
% 0.30/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.48 => ! [Y: $i] :
% 0.30/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.48 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.30/0.48 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(binunionTILcontra_type,type,
% 0.30/0.48 binunionTILcontra: $o ).
% 0.30/0.48
% 0.30/0.48 thf(binunionTILcontra,definition,
% 0.30/0.48 ( binunionTILcontra
% 0.30/0.48 = ( ! [A: $i,X: $i] :
% 0.30/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.48 => ! [Y: $i] :
% 0.30/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.48 => ! [Xx: $i] :
% 0.30/0.48 ( ( in @ Xx @ A )
% 0.30/0.48 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.30/0.48 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(binunionTIRcontra_type,type,
% 0.30/0.48 binunionTIRcontra: $o ).
% 0.30/0.48
% 0.30/0.48 thf(binunionTIRcontra,definition,
% 0.30/0.48 ( binunionTIRcontra
% 0.30/0.48 = ( ! [A: $i,X: $i] :
% 0.30/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.48 => ! [Y: $i] :
% 0.30/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.48 => ! [Xx: $i] :
% 0.30/0.48 ( ( in @ Xx @ A )
% 0.30/0.48 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.30/0.48 => ~ ( in @ Xx @ Y ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(inIntersectImpInUnion_type,type,
% 0.30/0.48 inIntersectImpInUnion: $o ).
% 0.30/0.48
% 0.30/0.48 thf(inIntersectImpInUnion,definition,
% 0.30/0.48 ( inIntersectImpInUnion
% 0.30/0.48 = ( ! [A: $i,X: $i] :
% 0.30/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.48 => ! [Y: $i] :
% 0.30/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.48 => ! [Z: $i] :
% 0.30/0.48 ( ( in @ Z @ ( powerset @ A ) )
% 0.30/0.48 => ! [Xx: $i] :
% 0.30/0.48 ( ( in @ Xx @ A )
% 0.30/0.48 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.30/0.48 => ( in @ Xx @ ( binunion @ X @ Z ) ) ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(inIntersectImpInUnion2_type,type,
% 0.30/0.48 inIntersectImpInUnion2: $o ).
% 0.30/0.48
% 0.30/0.48 thf(inIntersectImpInUnion2,definition,
% 0.30/0.48 ( inIntersectImpInUnion2
% 0.30/0.48 = ( ! [A: $i,X: $i] :
% 0.30/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.48 => ! [Y: $i] :
% 0.30/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.48 => ! [Z: $i] :
% 0.30/0.48 ( ( in @ Z @ ( powerset @ A ) )
% 0.30/0.48 => ! [Xx: $i] :
% 0.30/0.48 ( ( in @ Xx @ A )
% 0.30/0.48 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.30/0.48 => ( in @ Xx @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(inIntersectImpInIntersectUnions_type,type,
% 0.30/0.48 inIntersectImpInIntersectUnions: $o ).
% 0.30/0.48
% 0.30/0.48 thf(inIntersectImpInIntersectUnions,definition,
% 0.30/0.48 ( inIntersectImpInIntersectUnions
% 0.30/0.48 = ( ! [A: $i,X: $i] :
% 0.30/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.48 => ! [Y: $i] :
% 0.30/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.48 => ! [Z: $i] :
% 0.30/0.48 ( ( in @ Z @ ( powerset @ A ) )
% 0.30/0.48 => ! [Xx: $i] :
% 0.30/0.48 ( ( in @ Xx @ A )
% 0.30/0.48 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.30/0.48 => ( in @ Xx @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(intersectInPowersetIntersectUnions_type,type,
% 0.30/0.48 intersectInPowersetIntersectUnions: $o ).
% 0.30/0.48
% 0.30/0.48 thf(intersectInPowersetIntersectUnions,definition,
% 0.30/0.48 ( intersectInPowersetIntersectUnions
% 0.30/0.48 = ( ! [A: $i,X: $i] :
% 0.30/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.48 => ! [Y: $i] :
% 0.30/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.48 => ! [Z: $i] :
% 0.30/0.48 ( ( in @ Z @ ( powerset @ A ) )
% 0.30/0.48 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(inComplementUnionImpNotIn1_type,type,
% 0.30/0.48 inComplementUnionImpNotIn1: $o ).
% 0.30/0.48
% 0.30/0.48 thf(inComplementUnionImpNotIn1,definition,
% 0.30/0.48 ( inComplementUnionImpNotIn1
% 0.30/0.48 = ( ! [A: $i,X: $i] :
% 0.30/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.48 => ! [Y: $i] :
% 0.30/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.48 => ! [Xx: $i] :
% 0.30/0.48 ( ( in @ Xx @ A )
% 0.30/0.48 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.30/0.48 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(inComplementUnionImpInComplement1_type,type,
% 0.30/0.48 inComplementUnionImpInComplement1: $o ).
% 0.30/0.48
% 0.30/0.48 thf(inComplementUnionImpInComplement1,definition,
% 0.30/0.48 ( inComplementUnionImpInComplement1
% 0.30/0.48 = ( ! [A: $i,X: $i] :
% 0.30/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.48 => ! [Y: $i] :
% 0.30/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.48 => ! [Xx: $i] :
% 0.30/0.48 ( ( in @ Xx @ A )
% 0.30/0.48 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.30/0.48 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(binunionTE_type,type,
% 0.30/0.48 binunionTE: $o ).
% 0.30/0.48
% 0.30/0.48 thf(binunionTE,definition,
% 0.30/0.48 ( binunionTE
% 0.30/0.48 = ( ! [A: $i,X: $i] :
% 0.30/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.48 => ! [Y: $i] :
% 0.30/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.48 => ! [Xphi: $o,Xx: $i] :
% 0.30/0.48 ( ( in @ Xx @ A )
% 0.30/0.48 => ( ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.30/0.48 => ( ( ( in @ Xx @ X )
% 0.30/0.48 => Xphi )
% 0.30/0.48 => ( ( ( in @ Xx @ Y )
% 0.30/0.48 => Xphi )
% 0.30/0.48 => Xphi ) ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(binunionTEcontra_type,type,
% 0.30/0.48 binunionTEcontra: $o ).
% 0.30/0.48
% 0.30/0.48 thf(binunionTEcontra,definition,
% 0.30/0.48 ( binunionTEcontra
% 0.30/0.48 = ( ! [A: $i,X: $i] :
% 0.30/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.48 => ! [Y: $i] :
% 0.30/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.48 => ! [Xx: $i] :
% 0.30/0.48 ( ( in @ Xx @ A )
% 0.30/0.48 => ( ~ ( in @ Xx @ X )
% 0.30/0.48 => ( ~ ( in @ Xx @ Y )
% 0.30/0.48 => ~ ( in @ Xx @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(demorgan2a1_type,type,
% 0.30/0.48 demorgan2a1: $o ).
% 0.30/0.48
% 0.30/0.48 thf(demorgan2a1,definition,
% 0.30/0.48 ( demorgan2a1
% 0.30/0.48 = ( ! [A: $i,X: $i] :
% 0.30/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.48 => ! [Y: $i] :
% 0.30/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.48 => ! [Xx: $i] :
% 0.30/0.48 ( ( in @ Xx @ A )
% 0.30/0.48 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.30/0.48 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(complementUnionInPowersetComplement_type,type,
% 0.30/0.48 complementUnionInPowersetComplement: $o ).
% 0.30/0.48
% 0.30/0.48 thf(complementUnionInPowersetComplement,definition,
% 0.30/0.48 ( complementUnionInPowersetComplement
% 0.30/0.48 = ( ! [A: $i,X: $i] :
% 0.30/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.48 => ! [Y: $i] :
% 0.30/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.48 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(demorgan2a2_type,type,
% 0.30/0.48 demorgan2a2: $o ).
% 0.30/0.48
% 0.30/0.48 thf(demorgan2a2,definition,
% 0.30/0.48 ( demorgan2a2
% 0.30/0.48 = ( ! [A: $i,X: $i] :
% 0.30/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.48 => ! [Y: $i] :
% 0.30/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.48 => ! [Xx: $i] :
% 0.30/0.48 ( ( in @ Xx @ A )
% 0.30/0.48 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.30/0.48 => ( in @ Xx @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(demorgan1a_type,type,
% 0.30/0.48 demorgan1a: $o ).
% 0.30/0.48
% 0.30/0.48 thf(demorgan1a,definition,
% 0.30/0.48 ( demorgan1a
% 0.30/0.48 = ( ! [A: $i,X: $i] :
% 0.30/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.48 => ! [Y: $i] :
% 0.30/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.48 => ! [Xx: $i] :
% 0.30/0.48 ( ( in @ Xx @ A )
% 0.30/0.48 => ( ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) )
% 0.30/0.48 => ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(demorgan1b_type,type,
% 0.30/0.48 demorgan1b: $o ).
% 0.30/0.48
% 0.30/0.48 thf(demorgan1b,definition,
% 0.30/0.48 ( demorgan1b
% 0.30/0.48 = ( ! [A: $i,X: $i] :
% 0.30/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.48 => ! [Y: $i] :
% 0.30/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.48 => ! [Xx: $i] :
% 0.30/0.48 ( ( in @ Xx @ A )
% 0.30/0.48 => ( ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 0.30/0.48 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(demorgan1_type,type,
% 0.30/0.48 demorgan1: $o ).
% 0.30/0.48
% 0.30/0.48 thf(demorgan1,definition,
% 0.30/0.48 ( demorgan1
% 0.30/0.48 = ( ! [A: $i,X: $i] :
% 0.30/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.48 => ! [Y: $i] :
% 0.30/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.48 => ( ( setminus @ A @ ( binintersect @ X @ Y ) )
% 0.30/0.48 = ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(demorgan2a_type,type,
% 0.30/0.48 demorgan2a: $o ).
% 0.30/0.48
% 0.30/0.48 thf(demorgan2a,definition,
% 0.30/0.48 ( demorgan2a
% 0.30/0.48 = ( ! [A: $i,X: $i] :
% 0.30/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.48 => ! [Y: $i] :
% 0.30/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.48 => ! [Xx: $i] :
% 0.30/0.48 ( ( in @ Xx @ A )
% 0.30/0.48 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.30/0.48 => ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(demorgan2b2_type,type,
% 0.30/0.48 demorgan2b2: $o ).
% 0.30/0.48
% 0.30/0.48 thf(demorgan2b2,definition,
% 0.30/0.48 ( demorgan2b2
% 0.30/0.48 = ( ! [A: $i,X: $i] :
% 0.30/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.48 => ! [Y: $i] :
% 0.30/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.48 => ! [Xx: $i] :
% 0.30/0.48 ( ( in @ Xx @ A )
% 0.30/0.48 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.30/0.48 => ( ( in @ Xx @ ( setminus @ A @ Y ) )
% 0.30/0.48 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(demorgan2b_type,type,
% 0.30/0.48 demorgan2b: $o ).
% 0.30/0.48
% 0.30/0.48 thf(demorgan2b,definition,
% 0.30/0.48 ( demorgan2b
% 0.30/0.48 = ( ! [A: $i,X: $i] :
% 0.30/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.48 => ! [Y: $i] :
% 0.30/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.48 => ! [Xx: $i] :
% 0.30/0.48 ( ( in @ Xx @ A )
% 0.30/0.48 => ( ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 0.30/0.48 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(demorgan2_type,type,
% 0.30/0.48 demorgan2: $o ).
% 0.30/0.48
% 0.30/0.48 thf(demorgan2,definition,
% 0.30/0.48 ( demorgan2
% 0.30/0.48 = ( ! [A: $i,X: $i] :
% 0.30/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.48 => ! [Y: $i] :
% 0.30/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.48 => ( ( setminus @ A @ ( binunion @ X @ Y ) )
% 0.30/0.48 = ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(woz13rule0_type,type,
% 0.30/0.48 woz13rule0: $o ).
% 0.30/0.48
% 0.30/0.48 thf(woz13rule0,definition,
% 0.30/0.48 ( woz13rule0
% 0.30/0.48 = ( ! [A: $i,X: $i] :
% 0.30/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.48 => ! [Y: $i] :
% 0.30/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.48 => ! [Xx: $i] :
% 0.30/0.48 ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.30/0.48 => ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(woz13rule1_type,type,
% 0.30/0.48 woz13rule1: $o ).
% 0.30/0.48
% 0.30/0.48 thf(woz13rule1,definition,
% 0.30/0.48 ( woz13rule1
% 0.30/0.48 = ( ! [A: $i,X: $i] :
% 0.30/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.48 => ! [Y: $i] :
% 0.30/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.48 => ! [Z: $i] :
% 0.30/0.48 ( ( in @ Z @ ( powerset @ A ) )
% 0.30/0.48 => ( ( subset @ X @ Z )
% 0.30/0.48 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(woz13rule2_type,type,
% 0.30/0.48 woz13rule2: $o ).
% 0.30/0.48
% 0.30/0.48 thf(woz13rule2,definition,
% 0.30/0.48 ( woz13rule2
% 0.30/0.48 = ( ! [A: $i,X: $i] :
% 0.30/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.48 => ! [Y: $i] :
% 0.30/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.48 => ! [Z: $i] :
% 0.30/0.48 ( ( in @ Z @ ( powerset @ A ) )
% 0.30/0.48 => ( ( subset @ Y @ Z )
% 0.30/0.48 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(woz13rule3_type,type,
% 0.30/0.48 woz13rule3: $o ).
% 0.30/0.48
% 0.30/0.48 thf(woz13rule3,definition,
% 0.30/0.48 ( woz13rule3
% 0.30/0.48 = ( ! [A: $i,X: $i] :
% 0.30/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.48 => ! [Y: $i] :
% 0.30/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.48 => ! [Z: $i] :
% 0.30/0.48 ( ( in @ Z @ ( powerset @ A ) )
% 0.30/0.48 => ( ( subset @ X @ Y )
% 0.30/0.48 => ( ( subset @ X @ Z )
% 0.30/0.48 => ( subset @ X @ ( binintersect @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(woz13rule4_type,type,
% 0.30/0.48 woz13rule4: $o ).
% 0.30/0.48
% 0.30/0.48 thf(woz13rule4,definition,
% 0.30/0.48 ( woz13rule4
% 0.30/0.48 = ( ! [A: $i,X: $i] :
% 0.30/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.48 => ! [Y: $i] :
% 0.30/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.48 => ! [Z: $i] :
% 0.30/0.48 ( ( in @ Z @ ( powerset @ A ) )
% 0.30/0.48 => ! [W: $i] :
% 0.30/0.48 ( ( in @ W @ ( powerset @ A ) )
% 0.30/0.48 => ( ( subset @ X @ Z )
% 0.30/0.48 => ( ( subset @ Y @ W )
% 0.30/0.48 => ( subset @ ( binintersect @ X @ Y ) @ ( binintersect @ Z @ W ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(woz1_1_type,type,
% 0.30/0.48 woz1_1: $o ).
% 0.30/0.48
% 0.30/0.48 thf(woz1_1,definition,
% 0.30/0.48 ( woz1_1
% 0.30/0.48 = ( ! [A: $i,X: $i] :
% 0.30/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.48 => ! [Y: $i] :
% 0.30/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.48 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(woz1_2_type,type,
% 0.30/0.48 woz1_2: $o ).
% 0.30/0.48
% 0.30/0.48 thf(woz1_2,definition,
% 0.30/0.48 ( woz1_2
% 0.30/0.48 = ( ! [A: $i,X: $i] :
% 0.30/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.48 => ! [Y: $i] :
% 0.30/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.48 => ! [Z: $i] :
% 0.30/0.48 ( ( in @ Z @ ( powerset @ A ) )
% 0.30/0.48 => ! [W: $i] :
% 0.30/0.48 ( ( in @ W @ ( powerset @ A ) )
% 0.30/0.48 => ( ( setminus @ A @ ( binintersect @ ( binunion @ X @ Y ) @ ( binunion @ Z @ W ) ) )
% 0.30/0.48 = ( binunion @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) @ ( binintersect @ ( setminus @ A @ Z ) @ ( setminus @ A @ W ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(woz1_3_type,type,
% 0.30/0.48 woz1_3: $o ).
% 0.30/0.48
% 0.30/0.48 thf(woz1_3,definition,
% 0.30/0.48 ( woz1_3
% 0.30/0.48 = ( ! [A: $i,X: $i] :
% 0.30/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.48 => ! [Y: $i] :
% 0.30/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.48 => ! [Z: $i] :
% 0.30/0.48 ( ( in @ Z @ ( powerset @ A ) )
% 0.30/0.48 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(woz1_4_type,type,
% 0.30/0.48 woz1_4: $o ).
% 0.30/0.48
% 0.30/0.48 thf(woz1_4,definition,
% 0.30/0.48 ( woz1_4
% 0.30/0.48 = ( ! [A: $i,X: $i] :
% 0.30/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.48 => ! [Y: $i] :
% 0.30/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.48 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.30/0.48 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(woz1_5_type,type,
% 0.30/0.48 woz1_5: $o ).
% 0.30/0.48
% 0.30/0.48 thf(woz1_5,definition,
% 0.30/0.48 ( woz1_5
% 0.30/0.48 = ( ! [A: $i,X: $i] :
% 0.30/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.48 => ! [Y: $i] :
% 0.30/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.48 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(breln1_type,type,
% 0.30/0.48 breln1: $i > $i > $o ).
% 0.30/0.48
% 0.30/0.48 thf(breln1all2_type,type,
% 0.30/0.48 breln1all2: $o ).
% 0.30/0.48
% 0.30/0.48 thf(breln1all2,definition,
% 0.30/0.48 ( breln1all2
% 0.30/0.48 = ( ! [A: $i,R: $i] :
% 0.30/0.48 ( ( breln1 @ A @ R )
% 0.30/0.48 => ! [Xphi: $i > $o] :
% 0.30/0.48 ( ! [Xx: $i] :
% 0.30/0.48 ( ( in @ Xx @ A )
% 0.30/0.48 => ! [Xy: $i] :
% 0.30/0.48 ( ( in @ Xy @ A )
% 0.30/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.30/0.48 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.30/0.48 => ! [Xx: $i] :
% 0.30/0.48 ( ( in @ Xx @ R )
% 0.30/0.48 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(breln1Set_type,type,
% 0.30/0.48 breln1Set: $i > $i ).
% 0.30/0.48
% 0.30/0.48 thf(breln1SetBreln1_type,type,
% 0.30/0.48 breln1SetBreln1: $o ).
% 0.30/0.48
% 0.30/0.48 thf(breln1SetBreln1,definition,
% 0.30/0.48 ( breln1SetBreln1
% 0.30/0.48 = ( ! [A: $i,R: $i] :
% 0.30/0.48 ( ( in @ R @ ( breln1Set @ A ) )
% 0.30/0.48 => ( breln1 @ A @ R ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(transitive_type,type,
% 0.30/0.48 transitive: $i > $i > $o ).
% 0.30/0.48
% 0.30/0.48 thf(antisymmetric_type,type,
% 0.30/0.48 antisymmetric: $i > $i > $o ).
% 0.30/0.48
% 0.30/0.48 thf(reflexive_type,type,
% 0.30/0.48 reflexive: $i > $i > $o ).
% 0.30/0.48
% 0.30/0.48 thf(refltransitive_type,type,
% 0.30/0.48 refltransitive: $i > $i > $o ).
% 0.30/0.48
% 0.30/0.48 thf(refllinearorder_type,type,
% 0.30/0.48 refllinearorder: $i > $i > $o ).
% 0.30/0.48
% 0.30/0.48 thf(reflwellordering_type,type,
% 0.30/0.48 reflwellordering: $i > $i > $o ).
% 0.30/0.48
% 0.30/0.48 thf(choice2fnsingleton_type,type,
% 0.30/0.48 choice2fnsingleton: $o ).
% 0.30/0.48
% 0.30/0.48 thf(choice2fnsingleton,definition,
% 0.30/0.48 ( choice2fnsingleton
% 0.30/0.48 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.30/0.48 ( ! [Xx: $i] :
% 0.30/0.48 ( ( in @ Xx @ A )
% 0.30/0.48 => ? [Xy: $i] :
% 0.30/0.48 ( ( in @ Xy @ B )
% 0.30/0.48 & ( Xphi @ Xx @ Xy ) ) )
% 0.30/0.48 => ! [R: $i] :
% 0.30/0.48 ( ( in @ R @ ( breln1Set @ B ) )
% 0.30/0.48 => ( ( reflwellordering @ B @ R )
% 0.30/0.48 => ! [Xx: $i] :
% 0.30/0.48 ( ( in @ Xx @ A )
% 0.30/0.48 => ( singleton
% 0.30/0.48 @ ( dsetconstr @ B
% 0.30/0.48 @ ^ [Xy: $i] :
% 0.30/0.48 ( ( Xphi @ Xx @ Xy )
% 0.30/0.48 & ! [Xz: $i] :
% 0.30/0.48 ( ( in @ Xz @ B )
% 0.30/0.48 => ( ( Xphi @ Xx @ Xz )
% 0.30/0.48 => ( in @ ( kpair @ Xy @ Xz ) @ R ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(setOfPairsIsBReln1_type,type,
% 0.30/0.48 setOfPairsIsBReln1: $o ).
% 0.30/0.48
% 0.30/0.48 thf(setOfPairsIsBReln1,definition,
% 0.30/0.48 ( setOfPairsIsBReln1
% 0.30/0.48 = ( ! [A: $i,Xphi: $i > $i > $o] :
% 0.30/0.48 ( breln1 @ A
% 0.30/0.48 @ ( dpsetconstr @ A @ A
% 0.30/0.48 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(breln1all1_type,type,
% 0.30/0.48 breln1all1: $o ).
% 0.30/0.48
% 0.30/0.48 thf(breln1all1,definition,
% 0.30/0.48 ( breln1all1
% 0.30/0.48 = ( ! [A: $i,R: $i] :
% 0.30/0.48 ( ( breln1 @ A @ R )
% 0.30/0.48 => ! [Xphi: $i > $o] :
% 0.30/0.48 ( ! [Xx: $i] :
% 0.30/0.48 ( ( in @ Xx @ A )
% 0.30/0.48 => ! [Xy: $i] :
% 0.30/0.48 ( ( in @ Xy @ A )
% 0.30/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.30/0.48 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.30/0.48 => ! [Xx: $i] :
% 0.30/0.48 ( ( in @ Xx @ R )
% 0.30/0.48 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(subbreln1_type,type,
% 0.30/0.48 subbreln1: $o ).
% 0.30/0.48
% 0.30/0.48 thf(subbreln1,definition,
% 0.30/0.48 ( subbreln1
% 0.30/0.48 = ( ! [A: $i,R: $i] :
% 0.30/0.48 ( ( breln1 @ A @ R )
% 0.30/0.48 => ! [S: $i] :
% 0.30/0.48 ( ( breln1 @ A @ S )
% 0.30/0.48 => ( ! [Xx: $i] :
% 0.30/0.48 ( ( in @ Xx @ A )
% 0.30/0.48 => ! [Xy: $i] :
% 0.30/0.48 ( ( in @ Xy @ A )
% 0.30/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.30/0.48 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.30/0.48 => ( subset @ R @ S ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(eqbreln1_type,type,
% 0.30/0.48 eqbreln1: $o ).
% 0.30/0.48
% 0.30/0.48 thf(eqbreln1,definition,
% 0.30/0.48 ( eqbreln1
% 0.30/0.48 = ( ! [A: $i,R: $i] :
% 0.30/0.48 ( ( breln1 @ A @ R )
% 0.30/0.48 => ! [S: $i] :
% 0.30/0.48 ( ( breln1 @ A @ S )
% 0.30/0.48 => ( ! [Xx: $i] :
% 0.30/0.48 ( ( in @ Xx @ A )
% 0.30/0.48 => ! [Xy: $i] :
% 0.30/0.48 ( ( in @ Xy @ A )
% 0.30/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.30/0.48 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.30/0.48 => ( ! [Xx: $i] :
% 0.30/0.48 ( ( in @ Xx @ A )
% 0.30/0.48 => ! [Xy: $i] :
% 0.30/0.48 ( ( in @ Xy @ A )
% 0.30/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.30/0.48 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 0.30/0.48 => ( R = S ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(breln1invset_type,type,
% 0.30/0.48 breln1invset: $i > $i > $i ).
% 0.30/0.48
% 0.30/0.48 thf(breln1invprop_type,type,
% 0.30/0.48 breln1invprop: $o ).
% 0.30/0.48
% 0.30/0.48 thf(breln1invprop,definition,
% 0.30/0.48 ( breln1invprop
% 0.30/0.48 = ( ! [A: $i,R: $i] :
% 0.30/0.48 ( ( breln1 @ A @ R )
% 0.30/0.48 => ( breln1 @ A @ ( breln1invset @ A @ R ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(breln1invI_type,type,
% 0.30/0.48 breln1invI: $o ).
% 0.30/0.48
% 0.30/0.48 thf(breln1invI,definition,
% 0.30/0.48 ( breln1invI
% 0.30/0.48 = ( ! [A: $i,R: $i] :
% 0.30/0.48 ( ( breln1 @ A @ R )
% 0.30/0.48 => ! [Xx: $i] :
% 0.30/0.48 ( ( in @ Xx @ A )
% 0.30/0.48 => ! [Xy: $i] :
% 0.30/0.48 ( ( in @ Xy @ A )
% 0.30/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.30/0.48 => ( in @ ( kpair @ Xy @ Xx ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(breln1invE_type,type,
% 0.30/0.48 breln1invE: $o ).
% 0.30/0.48
% 0.30/0.48 thf(breln1invE,definition,
% 0.30/0.48 ( breln1invE
% 0.30/0.48 = ( ! [A: $i,R: $i] :
% 0.30/0.48 ( ( breln1 @ A @ R )
% 0.30/0.48 => ! [Xx: $i] :
% 0.30/0.48 ( ( in @ Xx @ A )
% 0.30/0.48 => ! [Xy: $i] :
% 0.30/0.48 ( ( in @ Xy @ A )
% 0.30/0.48 => ( ( in @ ( kpair @ Xy @ Xx ) @ ( breln1invset @ A @ R ) )
% 0.30/0.48 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(breln1compset_type,type,
% 0.30/0.48 breln1compset: $i > $i > $i > $i ).
% 0.30/0.48
% 0.30/0.48 thf(breln1compprop_type,type,
% 0.30/0.48 breln1compprop: $o ).
% 0.30/0.48
% 0.30/0.48 thf(breln1compprop,definition,
% 0.30/0.48 ( breln1compprop
% 0.30/0.48 = ( ! [A: $i,R: $i] :
% 0.30/0.48 ( ( breln1 @ A @ R )
% 0.30/0.48 => ! [S: $i] :
% 0.30/0.48 ( ( breln1 @ A @ S )
% 0.30/0.48 => ( breln1 @ A @ ( breln1compset @ A @ R @ S ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(breln1compI_type,type,
% 0.30/0.48 breln1compI: $o ).
% 0.30/0.48
% 0.30/0.48 thf(breln1compI,definition,
% 0.30/0.48 ( breln1compI
% 0.30/0.48 = ( ! [A: $i,R: $i] :
% 0.30/0.48 ( ( breln1 @ A @ R )
% 0.30/0.48 => ! [S: $i] :
% 0.30/0.48 ( ( breln1 @ A @ S )
% 0.30/0.48 => ! [Xx: $i] :
% 0.30/0.48 ( ( in @ Xx @ A )
% 0.30/0.48 => ! [Xy: $i] :
% 0.30/0.48 ( ( in @ Xy @ A )
% 0.30/0.48 => ! [Xz: $i] :
% 0.30/0.48 ( ( in @ Xz @ A )
% 0.30/0.48 => ( ( in @ ( kpair @ Xx @ Xz ) @ R )
% 0.30/0.48 => ( ( in @ ( kpair @ Xz @ Xy ) @ S )
% 0.30/0.48 => ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(breln1compE_type,type,
% 0.30/0.48 breln1compE: $o ).
% 0.30/0.48
% 0.30/0.48 thf(breln1compE,definition,
% 0.30/0.48 ( breln1compE
% 0.30/0.48 = ( ! [A: $i,R: $i] :
% 0.30/0.48 ( ( breln1 @ A @ R )
% 0.30/0.48 => ! [S: $i] :
% 0.30/0.48 ( ( breln1 @ A @ S )
% 0.30/0.48 => ! [Xx: $i] :
% 0.30/0.48 ( ( in @ Xx @ A )
% 0.30/0.48 => ! [Xy: $i] :
% 0.30/0.48 ( ( in @ Xy @ A )
% 0.30/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) )
% 0.30/0.48 => ? [Xz: $i] :
% 0.30/0.48 ( ( in @ Xz @ A )
% 0.30/0.48 & ( in @ ( kpair @ Xx @ Xz ) @ R )
% 0.30/0.48 & ( in @ ( kpair @ Xz @ Xy ) @ S ) ) ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(breln1compEex_type,type,
% 0.30/0.48 breln1compEex: $o ).
% 0.30/0.48
% 0.30/0.48 thf(breln1compEex,definition,
% 0.30/0.48 ( breln1compEex
% 0.30/0.48 = ( ! [A: $i,R: $i] :
% 0.30/0.48 ( ( breln1 @ A @ R )
% 0.30/0.48 => ! [S: $i] :
% 0.30/0.48 ( ( breln1 @ A @ S )
% 0.30/0.48 => ! [Xx: $i] :
% 0.30/0.48 ( ( in @ Xx @ A )
% 0.30/0.48 => ! [Xy: $i] :
% 0.30/0.48 ( ( in @ Xy @ A )
% 0.30/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) )
% 0.30/0.48 => ! [Xphi: $o] :
% 0.30/0.48 ( ! [Xz: $i] :
% 0.30/0.48 ( ( in @ Xz @ A )
% 0.30/0.48 => ( ( in @ ( kpair @ Xx @ Xz ) @ R )
% 0.30/0.48 => ( ( in @ ( kpair @ Xz @ Xy ) @ S )
% 0.30/0.48 => Xphi ) ) )
% 0.30/0.48 => Xphi ) ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(breln1unionprop_type,type,
% 0.30/0.48 breln1unionprop: $o ).
% 0.30/0.48
% 0.30/0.48 thf(breln1unionprop,definition,
% 0.30/0.48 ( breln1unionprop
% 0.30/0.48 = ( ! [A: $i,R: $i] :
% 0.30/0.48 ( ( breln1 @ A @ R )
% 0.30/0.48 => ! [S: $i] :
% 0.30/0.48 ( ( breln1 @ A @ S )
% 0.30/0.48 => ( breln1 @ A @ ( binunion @ R @ S ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(breln1unionIL_type,type,
% 0.30/0.48 breln1unionIL: $o ).
% 0.30/0.48
% 0.30/0.48 thf(breln1unionIL,definition,
% 0.30/0.48 ( breln1unionIL
% 0.30/0.48 = ( ! [A: $i,R: $i] :
% 0.30/0.48 ( ( breln1 @ A @ R )
% 0.30/0.48 => ! [S: $i] :
% 0.30/0.48 ( ( breln1 @ A @ S )
% 0.30/0.48 => ! [Xx: $i] :
% 0.30/0.48 ( ( in @ Xx @ A )
% 0.30/0.48 => ! [Xy: $i] :
% 0.30/0.48 ( ( in @ Xy @ A )
% 0.30/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.30/0.48 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(breln1unionIR_type,type,
% 0.30/0.48 breln1unionIR: $o ).
% 0.30/0.48
% 0.30/0.48 thf(breln1unionIR,definition,
% 0.30/0.48 ( breln1unionIR
% 0.30/0.48 = ( ! [A: $i,R: $i] :
% 0.30/0.48 ( ( breln1 @ A @ R )
% 0.30/0.48 => ! [S: $i] :
% 0.30/0.48 ( ( breln1 @ A @ S )
% 0.30/0.48 => ! [Xx: $i] :
% 0.30/0.48 ( ( in @ Xx @ A )
% 0.30/0.48 => ! [Xy: $i] :
% 0.30/0.48 ( ( in @ Xy @ A )
% 0.30/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.30/0.48 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(breln1unionI_type,type,
% 0.30/0.48 breln1unionI: $o ).
% 0.30/0.48
% 0.30/0.48 thf(breln1unionI,definition,
% 0.30/0.48 ( breln1unionI
% 0.30/0.48 = ( ! [A: $i,R: $i] :
% 0.30/0.48 ( ( breln1 @ A @ R )
% 0.30/0.48 => ! [S: $i] :
% 0.30/0.48 ( ( breln1 @ A @ S )
% 0.30/0.48 => ! [Xx: $i] :
% 0.30/0.48 ( ( in @ Xx @ A )
% 0.30/0.48 => ! [Xy: $i] :
% 0.30/0.48 ( ( in @ Xy @ A )
% 0.30/0.48 => ( ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.30/0.48 | ( in @ ( kpair @ Xx @ Xy ) @ S ) )
% 0.30/0.48 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(breln1unionE_type,type,
% 0.30/0.48 breln1unionE: $o ).
% 0.30/0.48
% 0.30/0.48 thf(breln1unionE,definition,
% 0.30/0.48 ( breln1unionE
% 0.30/0.48 = ( ! [A: $i,R: $i] :
% 0.30/0.48 ( ( breln1 @ A @ R )
% 0.30/0.48 => ! [S: $i] :
% 0.30/0.48 ( ( breln1 @ A @ S )
% 0.30/0.48 => ! [Xx: $i] :
% 0.30/0.48 ( ( in @ Xx @ A )
% 0.30/0.48 => ! [Xy: $i] :
% 0.30/0.48 ( ( in @ Xy @ A )
% 0.30/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) )
% 0.30/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.30/0.48 | ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(breln1unionEcases_type,type,
% 0.30/0.48 breln1unionEcases: $o ).
% 0.30/0.48
% 0.30/0.48 thf(breln1unionEcases,definition,
% 0.30/0.48 ( breln1unionEcases
% 0.30/0.48 = ( ! [A: $i,R: $i] :
% 0.30/0.48 ( ( breln1 @ A @ R )
% 0.30/0.48 => ! [S: $i] :
% 0.30/0.48 ( ( breln1 @ A @ S )
% 0.30/0.48 => ! [Xx: $i] :
% 0.30/0.48 ( ( in @ Xx @ A )
% 0.30/0.48 => ! [Xy: $i] :
% 0.30/0.48 ( ( in @ Xy @ A )
% 0.30/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) )
% 0.30/0.48 => ! [Xphi: $o] :
% 0.30/0.48 ( ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.30/0.48 => Xphi )
% 0.30/0.48 => ( ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.30/0.48 => Xphi )
% 0.30/0.48 => Xphi ) ) ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(breln1unionCommutes_type,type,
% 0.30/0.48 breln1unionCommutes: $o ).
% 0.30/0.48
% 0.30/0.48 thf(breln1unionCommutes,definition,
% 0.30/0.48 ( breln1unionCommutes
% 0.30/0.48 = ( ! [A: $i,R: $i] :
% 0.30/0.48 ( ( breln1 @ A @ R )
% 0.30/0.48 => ! [S: $i] :
% 0.30/0.48 ( ( breln1 @ A @ S )
% 0.30/0.48 => ( ( binunion @ R @ S )
% 0.30/0.48 = ( binunion @ S @ R ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(woz2Ex_type,type,
% 0.30/0.48 woz2Ex: $o ).
% 0.30/0.48
% 0.30/0.48 thf(woz2Ex,definition,
% 0.30/0.48 ( woz2Ex
% 0.30/0.48 = ( ! [A: $i,R: $i] :
% 0.30/0.48 ( ( breln1 @ A @ R )
% 0.30/0.48 => ( R
% 0.30/0.48 = ( breln1invset @ A @ ( breln1invset @ A @ R ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(woz2W_type,type,
% 0.30/0.48 woz2W: $o ).
% 0.30/0.48
% 0.30/0.48 thf(woz2W,definition,
% 0.30/0.48 ( woz2W
% 0.30/0.48 = ( ! [A: $i,R: $i] :
% 0.30/0.48 ( ( breln1 @ A @ R )
% 0.30/0.48 => ! [S: $i] :
% 0.30/0.48 ( ( breln1 @ A @ S )
% 0.30/0.48 => ( ( breln1invset @ A @ ( breln1compset @ A @ R @ S ) )
% 0.30/0.48 = ( breln1compset @ A @ ( breln1invset @ A @ S ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(woz2A_type,type,
% 0.30/0.48 woz2A: $o ).
% 0.30/0.48
% 0.30/0.48 thf(woz2A,definition,
% 0.30/0.48 ( woz2A
% 0.30/0.48 = ( ! [A: $i,R: $i] :
% 0.30/0.48 ( ( breln1 @ A @ R )
% 0.30/0.48 => ! [S: $i] :
% 0.30/0.48 ( ( breln1 @ A @ S )
% 0.30/0.48 => ! [T: $i] :
% 0.30/0.48 ( ( breln1 @ A @ T )
% 0.30/0.48 => ( ( breln1compset @ A @ ( binunion @ R @ S ) @ T )
% 0.30/0.48 = ( binunion @ ( breln1compset @ A @ R @ T ) @ ( breln1compset @ A @ S @ T ) ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(woz2B_type,type,
% 0.30/0.48 woz2B: $o ).
% 0.30/0.48
% 0.30/0.48 thf(woz2B,definition,
% 0.30/0.48 ( woz2B
% 0.30/0.48 = ( ! [A: $i,R: $i] :
% 0.30/0.48 ( ( breln1 @ A @ R )
% 0.30/0.48 => ! [S: $i] :
% 0.30/0.48 ( ( breln1 @ A @ S )
% 0.30/0.48 => ! [T: $i] :
% 0.30/0.48 ( ( breln1 @ A @ T )
% 0.30/0.48 => ( ( breln1compset @ A @ ( binunion @ R @ S ) @ T )
% 0.30/0.48 = ( binunion @ ( breln1invset @ A @ ( breln1compset @ A @ ( breln1invset @ A @ T ) @ ( breln1invset @ A @ S ) ) ) @ ( breln1invset @ A @ ( breln1compset @ A @ ( breln1invset @ A @ T ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(image1Ex_type,type,
% 0.30/0.48 image1Ex: $o ).
% 0.30/0.48
% 0.30/0.48 thf(image1Ex,definition,
% 0.30/0.48 ( image1Ex
% 0.30/0.48 = ( ! [A: $i,Xf: $i > $i] :
% 0.30/0.48 ? [B: $i] :
% 0.30/0.48 ! [Xx: $i] :
% 0.30/0.48 ( ( in @ Xx @ B )
% 0.30/0.48 <=> ? [Xy: $i] :
% 0.30/0.48 ( ( in @ Xy @ A )
% 0.30/0.48 & ( Xx
% 0.30/0.48 = ( Xf @ Xy ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(image1Ex1_type,type,
% 0.30/0.48 image1Ex1: $o ).
% 0.30/0.48
% 0.30/0.48 thf(image1Ex1,definition,
% 0.30/0.48 ( image1Ex1
% 0.30/0.48 = ( ! [A: $i,Xf: $i > $i] :
% 0.30/0.48 ( exu
% 0.30/0.48 @ ^ [B: $i] :
% 0.30/0.48 ! [Xx: $i] :
% 0.30/0.48 ( ( in @ Xx @ B )
% 0.30/0.48 <=> ? [Xy: $i] :
% 0.30/0.48 ( ( in @ Xy @ A )
% 0.30/0.48 & ( Xx
% 0.30/0.48 = ( Xf @ Xy ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(image1_type,type,
% 0.30/0.48 image1: $i > ( $i > $i ) > $i ).
% 0.30/0.48
% 0.30/0.48 thf(image1Equiv_type,type,
% 0.30/0.48 image1Equiv: $o ).
% 0.30/0.48
% 0.30/0.48 thf(image1Equiv,definition,
% 0.30/0.48 ( image1Equiv
% 0.30/0.48 = ( ! [A: $i,Xf: $i > $i,Xx: $i] :
% 0.30/0.48 ( ( in @ Xx
% 0.30/0.48 @ ( image1 @ A
% 0.30/0.48 @ ^ [Xy: $i] : ( Xf @ Xy ) ) )
% 0.30/0.48 <=> ? [Xy: $i] :
% 0.30/0.48 ( ( in @ Xy @ A )
% 0.30/0.48 & ( Xx
% 0.30/0.48 = ( Xf @ Xy ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(image1E_type,type,
% 0.30/0.48 image1E: $o ).
% 0.30/0.48
% 0.30/0.48 thf(image1E,definition,
% 0.30/0.48 ( image1E
% 0.30/0.48 = ( ! [A: $i,Xf: $i > $i,Xx: $i] :
% 0.30/0.48 ( ( in @ Xx
% 0.30/0.48 @ ( image1 @ A
% 0.30/0.48 @ ^ [Xy: $i] : ( Xf @ Xy ) ) )
% 0.30/0.48 => ? [Xy: $i] :
% 0.30/0.48 ( ( in @ Xy @ A )
% 0.30/0.48 & ( Xx
% 0.30/0.48 = ( Xf @ Xy ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(image1I_type,type,
% 0.30/0.48 image1I: $o ).
% 0.30/0.48
% 0.30/0.48 thf(image1I,definition,
% 0.30/0.48 ( image1I
% 0.30/0.48 = ( ! [A: $i,Xf: $i > $i,Xx: $i] :
% 0.30/0.48 ( ? [Xy: $i] :
% 0.30/0.48 ( ( in @ Xy @ A )
% 0.30/0.48 & ( Xx
% 0.30/0.48 = ( Xf @ Xy ) ) )
% 0.30/0.48 => ( in @ Xx
% 0.30/0.48 @ ( image1 @ A
% 0.30/0.48 @ ^ [Xy: $i] : ( Xf @ Xy ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(injective_type,type,
% 0.30/0.48 injective: $i > $i > $i > $o ).
% 0.30/0.48
% 0.30/0.48 thf(injFuncSet_type,type,
% 0.30/0.48 injFuncSet: $i > $i > $i ).
% 0.30/0.48
% 0.30/0.48 thf(injFuncInInjFuncSet_type,type,
% 0.30/0.48 injFuncInInjFuncSet: $o ).
% 0.30/0.48
% 0.30/0.48 thf(injFuncInInjFuncSet,definition,
% 0.30/0.48 ( injFuncInInjFuncSet
% 0.30/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.48 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.30/0.48 => ( ( injective @ A @ B @ Xf )
% 0.30/0.48 => ( in @ Xf @ ( injFuncSet @ A @ B ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(injFuncSetFuncIn_type,type,
% 0.30/0.48 injFuncSetFuncIn: $o ).
% 0.30/0.48
% 0.30/0.48 thf(injFuncSetFuncIn,definition,
% 0.30/0.48 ( injFuncSetFuncIn
% 0.30/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.48 ( ( in @ Xf @ ( injFuncSet @ A @ B ) )
% 0.30/0.48 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(injFuncSetFuncInj_type,type,
% 0.30/0.48 injFuncSetFuncInj: $o ).
% 0.30/0.48
% 0.30/0.48 thf(injFuncSetFuncInj,definition,
% 0.30/0.48 ( injFuncSetFuncInj
% 0.30/0.48 = ( ! [Xx: $i,Xy: $i,Xf: $i] :
% 0.30/0.48 ( ( in @ Xf @ ( injFuncSet @ Xx @ Xy ) )
% 0.30/0.48 => ( injective @ Xx @ Xy @ Xf ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(surjective_type,type,
% 0.30/0.48 surjective: $i > $i > $i > $o ).
% 0.30/0.48
% 0.30/0.48 thf(surjFuncSet_type,type,
% 0.30/0.48 surjFuncSet: $i > $i > $i ).
% 0.30/0.48
% 0.30/0.48 thf(surjFuncSetFuncIn_type,type,
% 0.30/0.48 surjFuncSetFuncIn: $o ).
% 0.30/0.48
% 0.30/0.48 thf(surjFuncSetFuncIn,definition,
% 0.30/0.48 ( surjFuncSetFuncIn
% 0.30/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.48 ( ( in @ Xf @ ( surjFuncSet @ A @ B ) )
% 0.30/0.48 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(surjFuncSetFuncSurj_type,type,
% 0.30/0.48 surjFuncSetFuncSurj: $o ).
% 0.30/0.48
% 0.30/0.48 thf(surjFuncSetFuncSurj,definition,
% 0.30/0.48 ( surjFuncSetFuncSurj
% 0.30/0.48 = ( ! [Xx: $i,Xy: $i,Xf: $i] :
% 0.30/0.48 ( ( in @ Xf @ ( surjFuncSet @ Xx @ Xy ) )
% 0.30/0.48 => ( surjective @ Xx @ Xy @ Xf ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(leftInvIsSurj_type,type,
% 0.30/0.48 leftInvIsSurj: $o ).
% 0.30/0.48
% 0.30/0.48 thf(leftInvIsSurj,definition,
% 0.30/0.48 ( leftInvIsSurj
% 0.30/0.48 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.30/0.48 ( ! [Xx: $i] :
% 0.30/0.48 ( ( in @ Xx @ A )
% 0.30/0.48 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.30/0.48 => ! [Xg: $i] :
% 0.30/0.48 ( ( in @ Xg @ ( funcSet @ B @ A ) )
% 0.30/0.48 => ( ! [Xx: $i] :
% 0.30/0.48 ( ( in @ Xx @ A )
% 0.30/0.48 => ( ( ap @ B @ A @ Xg @ ( Xf @ Xx ) )
% 0.30/0.48 = Xx ) )
% 0.30/0.48 => ( surjective @ B @ A @ Xg ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(surjCantorThm_type,type,
% 0.30/0.48 surjCantorThm: $o ).
% 0.30/0.48
% 0.30/0.48 thf(surjCantorThm,definition,
% 0.30/0.48 ( surjCantorThm
% 0.30/0.48 = ( ! [A: $i,Xf: $i] :
% 0.30/0.48 ( ( in @ Xf @ ( funcSet @ A @ ( powerset @ A ) ) )
% 0.30/0.48 => ~ ( surjective @ A @ ( powerset @ A ) @ Xf ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(foundation2_type,type,
% 0.30/0.48 foundation2: $o ).
% 0.30/0.48
% 0.30/0.48 thf(foundation2,definition,
% 0.30/0.48 ( foundation2
% 0.30/0.48 = ( ! [A: $i] :
% 0.30/0.48 ( ( nonempty @ A )
% 0.30/0.48 => ? [X: $i] :
% 0.30/0.48 ( ( in @ X @ A )
% 0.30/0.48 & ( ( binintersect @ X @ A )
% 0.30/0.48 = emptyset ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(notinself_type,type,
% 0.30/0.48 notinself: $o ).
% 0.30/0.48
% 0.30/0.48 thf(notinself,definition,
% 0.30/0.48 ( notinself
% 0.30/0.48 = ( ! [A: $i] :
% 0.30/0.48 ~ ( in @ A @ A ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(notinself2_type,type,
% 0.30/0.48 notinself2: $o ).
% 0.30/0.48
% 0.30/0.48 thf(notinself2,definition,
% 0.30/0.48 ( notinself2
% 0.30/0.48 = ( ! [A: $i,B: $i] :
% 0.30/0.48 ( ( in @ A @ B )
% 0.30/0.48 => ~ ( in @ B @ A ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(omegaS_type,type,
% 0.30/0.48 omegaS: $i > $i ).
% 0.30/0.48
% 0.30/0.48 thf(omegaS,definition,
% 0.30/0.48 ( omegaS
% 0.30/0.48 = ( ^ [Xx: $i] : ( setadjoin @ Xx @ Xx ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(omegaSp_type,type,
% 0.30/0.48 omegaSp: $o ).
% 0.30/0.48
% 0.30/0.48 thf(omegaSp,definition,
% 0.30/0.48 ( omegaSp
% 0.30/0.48 = ( ! [Xx: $i] :
% 0.30/0.48 ( ( in @ Xx @ omega )
% 0.30/0.48 => ( in @ ( omegaS @ Xx ) @ omega ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 thf(omegaSclos,conjecture,
% 0.30/0.48 ( setextAx
% 0.30/0.48 => ( emptysetAx
% 0.30/0.48 => ( setadjoinAx
% 0.30/0.48 => ( powersetAx
% 0.30/0.48 => ( setunionAx
% 0.30/0.48 => ( omega0Ax
% 0.30/0.48 => ( omegaSAx
% 0.30/0.48 => ( omegaIndAx
% 0.30/0.48 => ( replAx
% 0.30/0.48 => ( foundationAx
% 0.30/0.48 => ( wellorderingAx
% 0.30/0.48 => ( descrp
% 0.30/0.48 => ( dsetconstrI
% 0.30/0.48 => ( dsetconstrEL
% 0.30/0.48 => ( dsetconstrER
% 0.30/0.48 => ( exuE1
% 0.30/0.48 => ( prop2setE
% 0.30/0.48 => ( emptysetE
% 0.30/0.48 => ( emptysetimpfalse
% 0.30/0.48 => ( notinemptyset
% 0.30/0.48 => ( exuE3e
% 0.30/0.48 => ( setext
% 0.30/0.48 => ( emptyI
% 0.30/0.48 => ( noeltsimpempty
% 0.30/0.48 => ( setbeta
% 0.30/0.48 => ( nonemptyE1
% 0.30/0.48 => ( nonemptyI
% 0.30/0.48 => ( nonemptyI1
% 0.30/0.48 => ( setadjoinIL
% 0.30/0.48 => ( emptyinunitempty
% 0.30/0.48 => ( setadjoinIR
% 0.30/0.48 => ( setadjoinE
% 0.30/0.48 => ( setadjoinOr
% 0.30/0.48 => ( setoftrueEq
% 0.30/0.48 => ( powersetI
% 0.30/0.48 => ( emptyinPowerset
% 0.30/0.48 => ( emptyInPowerset
% 0.30/0.48 => ( powersetE
% 0.30/0.48 => ( setunionI
% 0.30/0.48 => ( setunionE
% 0.30/0.48 => ( subPowSU
% 0.30/0.48 => ( exuE2
% 0.30/0.48 => ( nonemptyImpWitness
% 0.30/0.48 => ( uniqinunit
% 0.30/0.48 => ( notinsingleton
% 0.30/0.48 => ( eqinunit
% 0.30/0.48 => ( singletonsswitch
% 0.30/0.48 => ( upairsetE
% 0.30/0.48 => ( upairsetIL
% 0.30/0.48 => ( upairsetIR
% 0.30/0.48 => ( emptyE1
% 0.30/0.48 => ( vacuousDall
% 0.30/0.48 => ( quantDeMorgan1
% 0.30/0.48 => ( quantDeMorgan2
% 0.30/0.48 => ( quantDeMorgan3
% 0.30/0.48 => ( quantDeMorgan4
% 0.30/0.48 => ( prop2setI
% 0.30/0.48 => ( prop2set2propI
% 0.30/0.48 => ( notdexE
% 0.30/0.48 => ( notdallE
% 0.30/0.48 => ( exuI1
% 0.30/0.48 => ( exuI3
% 0.30/0.48 => ( exuI2
% 0.30/0.48 => ( inCongP
% 0.30/0.48 => ( in__Cong
% 0.30/0.48 => ( exuE3u
% 0.30/0.48 => ( exu__Cong
% 0.30/0.48 => ( emptyset__Cong
% 0.30/0.48 => ( setadjoin__Cong
% 0.30/0.48 => ( powerset__Cong
% 0.30/0.48 => ( setunion__Cong
% 0.30/0.48 => ( omega__Cong
% 0.30/0.48 => ( exuEu
% 0.30/0.48 => ( descr__Cong
% 0.30/0.48 => ( dsetconstr__Cong
% 0.30/0.48 => ( subsetI1
% 0.30/0.48 => ( eqimpsubset2
% 0.30/0.48 => ( eqimpsubset1
% 0.30/0.48 => ( subsetI2
% 0.30/0.48 => ( emptysetsubset
% 0.30/0.48 => ( subsetE
% 0.30/0.48 => ( subsetE2
% 0.30/0.48 => ( notsubsetI
% 0.30/0.48 => ( notequalI1
% 0.30/0.48 => ( notequalI2
% 0.30/0.48 => ( subsetRefl
% 0.30/0.48 => ( subsetTrans
% 0.30/0.48 => ( setadjoinSub
% 0.30/0.48 => ( setadjoinSub2
% 0.30/0.48 => ( subset2powerset
% 0.30/0.48 => ( setextsub
% 0.30/0.48 => ( subsetemptysetimpeq
% 0.30/0.48 => ( powersetI1
% 0.30/0.48 => ( powersetE1
% 0.30/0.48 => ( inPowerset
% 0.30/0.48 => ( powersetsubset
% 0.30/0.48 => ( sepInPowerset
% 0.30/0.48 => ( sepSubset
% 0.30/0.48 => ( binunionIL
% 0.30/0.48 => ( upairset2IR
% 0.30/0.48 => ( binunionIR
% 0.30/0.48 => ( binunionEcases
% 0.30/0.48 => ( binunionE
% 0.30/0.48 => ( binunionLsub
% 0.30/0.48 => ( binunionRsub
% 0.30/0.48 => ( binintersectI
% 0.30/0.48 => ( binintersectSubset5
% 0.30/0.48 => ( binintersectEL
% 0.30/0.48 => ( binintersectLsub
% 0.30/0.48 => ( binintersectSubset2
% 0.30/0.48 => ( binintersectSubset3
% 0.30/0.48 => ( binintersectER
% 0.30/0.48 => ( disjointsetsI1
% 0.30/0.48 => ( binintersectRsub
% 0.30/0.48 => ( binintersectSubset4
% 0.30/0.48 => ( binintersectSubset1
% 0.30/0.48 => ( bs114d
% 0.30/0.48 => ( setminusI
% 0.30/0.48 => ( setminusEL
% 0.30/0.48 => ( setminusER
% 0.30/0.48 => ( setminusSubset2
% 0.30/0.48 => ( setminusERneg
% 0.30/0.48 => ( setminusELneg
% 0.30/0.48 => ( setminusILneg
% 0.30/0.48 => ( setminusIRneg
% 0.30/0.48 => ( setminusLsub
% 0.30/0.48 => ( setminusSubset1
% 0.30/0.48 => ( symdiffE
% 0.30/0.48 => ( symdiffI1
% 0.30/0.48 => ( symdiffI2
% 0.30/0.48 => ( symdiffIneg1
% 0.30/0.48 => ( symdiffIneg2
% 0.30/0.48 => ( secondinupair
% 0.30/0.48 => ( setukpairIL
% 0.30/0.48 => ( setukpairIR
% 0.30/0.48 => ( kpairiskpair
% 0.30/0.48 => ( kpairp
% 0.30/0.48 => ( singletonsubset
% 0.30/0.48 => ( singletoninpowerset
% 0.30/0.48 => ( singletoninpowunion
% 0.30/0.48 => ( upairset2E
% 0.30/0.48 => ( upairsubunion
% 0.30/0.48 => ( upairinpowunion
% 0.30/0.48 => ( ubforcartprodlem1
% 0.30/0.48 => ( ubforcartprodlem2
% 0.30/0.48 => ( ubforcartprodlem3
% 0.30/0.48 => ( cartprodpairin
% 0.30/0.48 => ( cartprodmempair1
% 0.30/0.48 => ( cartprodmempair
% 0.30/0.48 => ( setunionE2
% 0.30/0.48 => ( setunionsingleton1
% 0.30/0.48 => ( setunionsingleton2
% 0.30/0.48 => ( setunionsingleton
% 0.30/0.48 => ( singletonprop
% 0.30/0.48 => ( ex1E1
% 0.30/0.48 => ( ex1I
% 0.30/0.48 => ( ex1I2
% 0.30/0.48 => ( singletonsuniq
% 0.30/0.48 => ( setukpairinjL1
% 0.30/0.48 => ( kfstsingleton
% 0.30/0.48 => ( theprop
% 0.30/0.48 => ( kfstpairEq
% 0.30/0.48 => ( cartprodfstin
% 0.30/0.48 => ( setukpairinjL2
% 0.30/0.48 => ( setukpairinjL
% 0.30/0.48 => ( setukpairinjR11
% 0.30/0.48 => ( setukpairinjR12
% 0.30/0.48 => ( setukpairinjR1
% 0.30/0.48 => ( upairequniteq
% 0.30/0.48 => ( setukpairinjR2
% 0.30/0.48 => ( setukpairinjR
% 0.30/0.48 => ( ksndsingleton
% 0.30/0.48 => ( ksndpairEq
% 0.30/0.48 => ( kpairsurjEq
% 0.30/0.48 => ( cartprodsndin
% 0.30/0.48 => ( cartprodpairmemEL
% 0.30/0.48 => ( cartprodpairmemER
% 0.30/0.48 => ( cartprodmempaircEq
% 0.30/0.48 => ( cartprodfstpairEq
% 0.30/0.48 => ( cartprodsndpairEq
% 0.30/0.48 => ( cartprodpairsurjEq
% 0.30/0.48 => ( dpsetconstrI
% 0.30/0.48 => ( dpsetconstrSub
% 0.30/0.48 => ( setOfPairsIsBReln
% 0.30/0.48 => ( dpsetconstrERa
% 0.30/0.48 => ( dpsetconstrEL1
% 0.30/0.48 => ( dpsetconstrEL2
% 0.30/0.48 => ( dpsetconstrER
% 0.30/0.48 => ( funcImageSingleton
% 0.30/0.48 => ( apProp
% 0.30/0.48 => ( app
% 0.30/0.48 => ( infuncsetfunc
% 0.30/0.48 => ( ap2p
% 0.30/0.48 => ( funcinfuncset
% 0.30/0.48 => ( lamProp
% 0.30/0.48 => ( lamp
% 0.30/0.48 => ( lam2p
% 0.30/0.48 => ( brelnall1
% 0.30/0.48 => ( brelnall2
% 0.30/0.48 => ( ex1E2
% 0.30/0.48 => ( funcGraphProp1
% 0.30/0.48 => ( funcGraphProp3
% 0.30/0.48 => ( funcGraphProp2
% 0.30/0.48 => ( funcextLem
% 0.30/0.48 => ( funcGraphProp4
% 0.30/0.48 => ( subbreln
% 0.30/0.48 => ( eqbreln
% 0.30/0.48 => ( funcext
% 0.30/0.48 => ( funcext2
% 0.30/0.48 => ( ap2apEq1
% 0.30/0.48 => ( ap2apEq2
% 0.30/0.48 => ( beta1
% 0.30/0.48 => ( eta1
% 0.30/0.48 => ( lam2lamEq
% 0.30/0.48 => ( beta2
% 0.30/0.48 => ( eta2
% 0.30/0.48 => ( iffalseProp1
% 0.30/0.48 => ( iffalseProp2
% 0.30/0.48 => ( iftrueProp1
% 0.30/0.48 => ( iftrueProp2
% 0.30/0.48 => ( ifSingleton
% 0.30/0.48 => ( ifp
% 0.30/0.48 => ( theeq
% 0.30/0.48 => ( iftrue
% 0.30/0.48 => ( iffalse
% 0.30/0.48 => ( iftrueorfalse
% 0.30/0.48 => ( binintersectT_lem
% 0.30/0.48 => ( binunionT_lem
% 0.30/0.48 => ( powersetT_lem
% 0.30/0.48 => ( setminusT_lem
% 0.30/0.48 => ( complementT_lem
% 0.30/0.48 => ( setextT
% 0.30/0.48 => ( subsetTI
% 0.30/0.48 => ( powersetTI1
% 0.30/0.48 => ( powersetTE1
% 0.30/0.48 => ( complementTI1
% 0.30/0.48 => ( complementTE1
% 0.30/0.48 => ( binintersectTELcontra
% 0.30/0.48 => ( binintersectTERcontra
% 0.30/0.48 => ( contrasubsetT
% 0.30/0.48 => ( contrasubsetT1
% 0.30/0.48 => ( contrasubsetT2
% 0.30/0.48 => ( contrasubsetT3
% 0.30/0.48 => ( doubleComplementI1
% 0.30/0.48 => ( doubleComplementE1
% 0.30/0.48 => ( doubleComplementSub1
% 0.30/0.48 => ( doubleComplementSub2
% 0.30/0.48 => ( doubleComplementEq
% 0.30/0.48 => ( complementTnotintersectT
% 0.30/0.48 => ( complementImpComplementIntersect
% 0.30/0.48 => ( complementSubsetComplementIntersect
% 0.30/0.48 => ( complementInPowersetComplementIntersect
% 0.30/0.48 => ( contraSubsetComplement
% 0.30/0.48 => ( complementTcontraSubset
% 0.30/0.48 => ( binunionTILcontra
% 0.30/0.48 => ( binunionTIRcontra
% 0.30/0.48 => ( inIntersectImpInUnion
% 0.30/0.48 => ( inIntersectImpInUnion2
% 0.30/0.48 => ( inIntersectImpInIntersectUnions
% 0.30/0.48 => ( intersectInPowersetIntersectUnions
% 0.30/0.48 => ( inComplementUnionImpNotIn1
% 0.30/0.48 => ( inComplementUnionImpInComplement1
% 0.30/0.48 => ( binunionTE
% 0.30/0.48 => ( binunionTEcontra
% 0.30/0.48 => ( demorgan2a1
% 0.30/0.48 => ( complementUnionInPowersetComplement
% 0.30/0.48 => ( demorgan2a2
% 0.30/0.48 => ( demorgan1a
% 0.30/0.48 => ( demorgan1b
% 0.30/0.48 => ( demorgan1
% 0.30/0.48 => ( demorgan2a
% 0.30/0.48 => ( demorgan2b2
% 0.30/0.48 => ( demorgan2b
% 0.30/0.48 => ( demorgan2
% 0.30/0.48 => ( woz13rule0
% 0.30/0.48 => ( woz13rule1
% 0.30/0.48 => ( woz13rule2
% 0.30/0.48 => ( woz13rule3
% 0.30/0.48 => ( woz13rule4
% 0.30/0.48 => ( woz1_1
% 0.30/0.48 => ( woz1_2
% 0.30/0.48 => ( woz1_3
% 0.30/0.48 => ( woz1_4
% 0.30/0.48 => ( woz1_5
% 0.30/0.48 => ( breln1all2
% 0.30/0.48 => ( breln1SetBreln1
% 0.30/0.48 => ( choice2fnsingleton
% 0.30/0.48 => ( setOfPairsIsBReln1
% 0.30/0.48 => ( breln1all1
% 0.30/0.48 => ( subbreln1
% 0.30/0.48 => ( eqbreln1
% 0.30/0.48 => ( breln1invprop
% 0.30/0.48 => ( breln1invI
% 0.30/0.48 => ( breln1invE
% 0.30/0.48 => ( breln1compprop
% 0.30/0.48 => ( breln1compI
% 0.30/0.48 => ( breln1compE
% 0.30/0.48 => ( breln1compEex
% 0.30/0.48 => ( breln1unionprop
% 0.30/0.48 => ( breln1unionIL
% 0.30/0.48 => ( breln1unionIR
% 0.30/0.48 => ( breln1unionI
% 0.30/0.48 => ( breln1unionE
% 0.30/0.48 => ( breln1unionEcases
% 0.30/0.48 => ( breln1unionCommutes
% 0.30/0.48 => ( woz2Ex
% 0.30/0.48 => ( woz2W
% 0.30/0.48 => ( woz2A
% 0.30/0.48 => ( woz2B
% 0.30/0.48 => ( image1Ex
% 0.30/0.48 => ( image1Ex1
% 0.30/0.48 => ( image1Equiv
% 0.30/0.48 => ( image1E
% 0.30/0.48 => ( image1I
% 0.30/0.48 => ( injFuncInInjFuncSet
% 0.30/0.48 => ( injFuncSetFuncIn
% 0.30/0.48 => ( injFuncSetFuncInj
% 0.30/0.48 => ( surjFuncSetFuncIn
% 0.30/0.48 => ( surjFuncSetFuncSurj
% 0.30/0.48 => ( leftInvIsSurj
% 0.30/0.48 => ( surjCantorThm
% 0.30/0.48 => ( foundation2
% 0.30/0.48 => ( notinself
% 0.30/0.48 => ( notinself2
% 0.30/0.48 => ( omegaSp
% 0.30/0.48 => ! [Xx: $i] :
% 0.30/0.48 ( ( in @ Xx @ omega )
% 0.30/0.48 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 %------------------------------------------------------------------------------
% 0.30/0.48 ------------------------
% 3.22/1.65 ---- Embedded in THF ---
% 3.22/1.65 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 3.22/1.65 %%% Generated on Tue Feb 24 12:34:56 EST 2026
% 3.22/1.65 %%% using '$$dhol' embedding, version 1.3.0.
% 3.22/1.65 %%% Logic specification used:
% 3.22/1.65 %%% thf(spec, logic, $$dhol).
% 3.22/1.65
% 3.22/1.65 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.N0zPVEAfqW/DTF2DTF28502.p
% See solution above
% 3.22/1.65 ------------------------
% 3.34/1.71 ---- Cleaned THF ---
% 3.34/1.72 thf(in_type,type,
% 3.34/1.72 in: $i > $i > $o ).
% 3.34/1.72
% 3.34/1.72 thf(exu_type,type,
% 3.34/1.72 exu: ( $i > $o ) > $o ).
% 3.34/1.72
% 3.34/1.72 thf(exu,definition,
% 3.34/1.72 ( exu
% 3.34/1.72 = ( ^ [Xphi: $i > $o] :
% 3.34/1.72 ? [Xx: $i] :
% 3.34/1.72 ( ( Xphi @ Xx )
% 3.34/1.72 & ! [Xy: $i] :
% 3.34/1.72 ( ( Xphi @ Xy )
% 3.34/1.72 => ( Xx = Xy ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(setextAx_type,type,
% 3.34/1.72 setextAx: $o ).
% 3.34/1.72
% 3.34/1.72 thf(setextAx,definition,
% 3.34/1.72 ( setextAx
% 3.34/1.72 = ( ! [A: $i,B: $i] :
% 3.34/1.72 ( ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 <=> ( in @ Xx @ B ) )
% 3.34/1.72 => ( A = B ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(emptyset_type,type,
% 3.34/1.72 emptyset: $i ).
% 3.34/1.72
% 3.34/1.72 thf(emptysetAx_type,type,
% 3.34/1.72 emptysetAx: $o ).
% 3.34/1.72
% 3.34/1.72 thf(emptysetAx,definition,
% 3.34/1.72 ( emptysetAx
% 3.34/1.72 = ( ! [Xx: $i] :
% 3.34/1.72 ~ ( in @ Xx @ emptyset ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(setadjoin_type,type,
% 3.34/1.72 setadjoin: $i > $i > $i ).
% 3.34/1.72
% 3.34/1.72 thf(setadjoinAx_type,type,
% 3.34/1.72 setadjoinAx: $o ).
% 3.34/1.72
% 3.34/1.72 thf(setadjoinAx,definition,
% 3.34/1.72 ( setadjoinAx
% 3.34/1.72 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 3.34/1.72 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 3.34/1.72 <=> ( ( Xy = Xx )
% 3.34/1.72 | ( in @ Xy @ A ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(powerset_type,type,
% 3.34/1.72 powerset: $i > $i ).
% 3.34/1.72
% 3.34/1.72 thf(powersetAx_type,type,
% 3.34/1.72 powersetAx: $o ).
% 3.34/1.72
% 3.34/1.72 thf(powersetAx,definition,
% 3.34/1.72 ( powersetAx
% 3.34/1.72 = ( ! [A: $i,B: $i] :
% 3.34/1.72 ( ( in @ B @ ( powerset @ A ) )
% 3.34/1.72 <=> ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ B )
% 3.34/1.72 => ( in @ Xx @ A ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(setunion_type,type,
% 3.34/1.72 setunion: $i > $i ).
% 3.34/1.72
% 3.34/1.72 thf(setunionAx_type,type,
% 3.34/1.72 setunionAx: $o ).
% 3.34/1.72
% 3.34/1.72 thf(setunionAx,definition,
% 3.34/1.72 ( setunionAx
% 3.34/1.72 = ( ! [A: $i,Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ ( setunion @ A ) )
% 3.34/1.72 <=> ? [B: $i] :
% 3.34/1.72 ( ( in @ Xx @ B )
% 3.34/1.72 & ( in @ B @ A ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(omega_type,type,
% 3.34/1.72 omega: $i ).
% 3.34/1.72
% 3.34/1.72 thf(omega0Ax_type,type,
% 3.34/1.72 omega0Ax: $o ).
% 3.34/1.72
% 3.34/1.72 thf(omega0Ax,definition,
% 3.34/1.72 ( omega0Ax
% 3.34/1.72 = ( in @ emptyset @ omega ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(omegaSAx_type,type,
% 3.34/1.72 omegaSAx: $o ).
% 3.34/1.72
% 3.34/1.72 thf(omegaSAx,definition,
% 3.34/1.72 ( omegaSAx
% 3.34/1.72 = ( ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ omega )
% 3.34/1.72 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(omegaIndAx_type,type,
% 3.34/1.72 omegaIndAx: $o ).
% 3.34/1.72
% 3.34/1.72 thf(omegaIndAx,definition,
% 3.34/1.72 ( omegaIndAx
% 3.34/1.72 = ( ! [A: $i] :
% 3.34/1.72 ( ( ( in @ emptyset @ A )
% 3.34/1.72 & ! [Xx: $i] :
% 3.34/1.72 ( ( ( in @ Xx @ omega )
% 3.34/1.72 & ( in @ Xx @ A ) )
% 3.34/1.72 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 3.34/1.72 => ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ omega )
% 3.34/1.72 => ( in @ Xx @ A ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(replAx_type,type,
% 3.34/1.72 replAx: $o ).
% 3.34/1.72
% 3.34/1.72 thf(replAx,definition,
% 3.34/1.72 ( replAx
% 3.34/1.72 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 3.34/1.72 ( ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ( exu
% 3.34/1.72 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 3.34/1.72 => ? [B: $i] :
% 3.34/1.72 ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ B )
% 3.34/1.72 <=> ? [Xy: $i] :
% 3.34/1.72 ( ( in @ Xy @ A )
% 3.34/1.72 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(foundationAx_type,type,
% 3.34/1.72 foundationAx: $o ).
% 3.34/1.72
% 3.34/1.72 thf(foundationAx,definition,
% 3.34/1.72 ( foundationAx
% 3.34/1.72 = ( ! [A: $i] :
% 3.34/1.72 ( ? [Xx: $i] : ( in @ Xx @ A )
% 3.34/1.72 => ? [B: $i] :
% 3.34/1.72 ( ( in @ B @ A )
% 3.34/1.72 & ~ ? [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ B )
% 3.34/1.72 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(wellorderingAx_type,type,
% 3.34/1.72 wellorderingAx: $o ).
% 3.34/1.72
% 3.34/1.72 thf(wellorderingAx,definition,
% 3.34/1.72 ( wellorderingAx
% 3.34/1.72 = ( ! [A: $i] :
% 3.34/1.72 ? [B: $i] :
% 3.34/1.72 ( ! [C: $i] :
% 3.34/1.72 ( ( in @ C @ B )
% 3.34/1.72 => ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ C )
% 3.34/1.72 => ( in @ Xx @ A ) ) )
% 3.34/1.72 & ! [Xx: $i,Xy: $i] :
% 3.34/1.72 ( ( ( in @ Xx @ A )
% 3.34/1.72 & ( in @ Xy @ A ) )
% 3.34/1.72 => ( ! [C: $i] :
% 3.34/1.72 ( ( in @ C @ B )
% 3.34/1.72 => ( ( in @ Xx @ C )
% 3.34/1.72 <=> ( in @ Xy @ C ) ) )
% 3.34/1.72 => ( Xx = Xy ) ) )
% 3.34/1.72 & ! [C: $i,D: $i] :
% 3.34/1.72 ( ( ( in @ C @ B )
% 3.34/1.72 & ( in @ D @ B ) )
% 3.34/1.72 => ( ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ C )
% 3.34/1.72 => ( in @ Xx @ D ) )
% 3.34/1.72 | ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ D )
% 3.34/1.72 => ( in @ Xx @ C ) ) ) )
% 3.34/1.72 & ! [C: $i] :
% 3.34/1.72 ( ( ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ C )
% 3.34/1.72 => ( in @ Xx @ A ) )
% 3.34/1.72 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 3.34/1.72 => ? [D: $i,Xx: $i] :
% 3.34/1.72 ( ( in @ D @ B )
% 3.34/1.72 & ( in @ Xx @ C )
% 3.34/1.72 & ~ ? [Xy: $i] :
% 3.34/1.72 ( ( in @ Xy @ D )
% 3.34/1.72 & ( in @ Xy @ C ) )
% 3.34/1.72 & ! [E: $i] :
% 3.34/1.72 ( ( in @ E @ B )
% 3.34/1.72 => ( ! [Xy: $i] :
% 3.34/1.72 ( ( in @ Xy @ E )
% 3.34/1.72 => ( in @ Xy @ D ) )
% 3.34/1.72 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(descr_type,type,
% 3.34/1.72 descr: ( $i > $o ) > $i ).
% 3.34/1.72
% 3.34/1.72 thf(descrp_type,type,
% 3.34/1.72 descrp: $o ).
% 3.34/1.72
% 3.34/1.72 thf(descrp,definition,
% 3.34/1.72 ( descrp
% 3.34/1.72 = ( ! [Xphi: $i > $o] :
% 3.34/1.72 ( ( exu
% 3.34/1.72 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.34/1.72 => ( Xphi
% 3.34/1.72 @ ( descr
% 3.34/1.72 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(dsetconstr_type,type,
% 3.34/1.72 dsetconstr: $i > ( $i > $o ) > $i ).
% 3.34/1.72
% 3.34/1.72 thf(dsetconstrI_type,type,
% 3.34/1.72 dsetconstrI: $o ).
% 3.34/1.72
% 3.34/1.72 thf(dsetconstrI,definition,
% 3.34/1.72 ( dsetconstrI
% 3.34/1.72 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ( ( Xphi @ Xx )
% 3.34/1.72 => ( in @ Xx
% 3.34/1.72 @ ( dsetconstr @ A
% 3.34/1.72 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(dsetconstrEL_type,type,
% 3.34/1.72 dsetconstrEL: $o ).
% 3.34/1.72
% 3.34/1.72 thf(dsetconstrEL,definition,
% 3.34/1.72 ( dsetconstrEL
% 3.34/1.72 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.34/1.72 ( ( in @ Xx
% 3.34/1.72 @ ( dsetconstr @ A
% 3.34/1.72 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 3.34/1.72 => ( in @ Xx @ A ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(dsetconstrER_type,type,
% 3.34/1.72 dsetconstrER: $o ).
% 3.34/1.72
% 3.34/1.72 thf(dsetconstrER,definition,
% 3.34/1.72 ( dsetconstrER
% 3.34/1.72 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.34/1.72 ( ( in @ Xx
% 3.34/1.72 @ ( dsetconstr @ A
% 3.34/1.72 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 3.34/1.72 => ( Xphi @ Xx ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(exuE1_type,type,
% 3.34/1.72 exuE1: $o ).
% 3.34/1.72
% 3.34/1.72 thf(exuE1,definition,
% 3.34/1.72 ( exuE1
% 3.34/1.72 = ( ! [Xphi: $i > $o] :
% 3.34/1.72 ( ( exu
% 3.34/1.72 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.34/1.72 => ? [Xx: $i] :
% 3.34/1.72 ( ( Xphi @ Xx )
% 3.34/1.72 & ! [Xy: $i] :
% 3.34/1.72 ( ( Xphi @ Xy )
% 3.34/1.72 => ( Xx = Xy ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(prop2set_type,type,
% 3.34/1.72 prop2set: $o > $i ).
% 3.34/1.72
% 3.34/1.72 thf(prop2setE_type,type,
% 3.34/1.72 prop2setE: $o ).
% 3.34/1.72
% 3.34/1.72 thf(prop2setE,definition,
% 3.34/1.72 ( prop2setE
% 3.34/1.72 = ( ! [Xphi: $o,Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 3.34/1.72 => Xphi ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(emptysetE_type,type,
% 3.34/1.72 emptysetE: $o ).
% 3.34/1.72
% 3.34/1.72 thf(emptysetE,definition,
% 3.34/1.72 ( emptysetE
% 3.34/1.72 = ( ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ emptyset )
% 3.34/1.72 => ! [Xphi: $o] : Xphi ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(emptysetimpfalse_type,type,
% 3.34/1.72 emptysetimpfalse: $o ).
% 3.34/1.72
% 3.34/1.72 thf(emptysetimpfalse,definition,
% 3.34/1.72 ( emptysetimpfalse
% 3.34/1.72 = ( ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ emptyset )
% 3.34/1.72 => $false ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(notinemptyset_type,type,
% 3.34/1.72 notinemptyset: $o ).
% 3.34/1.72
% 3.34/1.72 thf(notinemptyset,definition,
% 3.34/1.72 ( notinemptyset
% 3.34/1.72 = ( ! [Xx: $i] :
% 3.34/1.72 ~ ( in @ Xx @ emptyset ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(exuE3e_type,type,
% 3.34/1.72 exuE3e: $o ).
% 3.34/1.72
% 3.34/1.72 thf(exuE3e,definition,
% 3.34/1.72 ( exuE3e
% 3.34/1.72 = ( ! [Xphi: $i > $o] :
% 3.34/1.72 ( ( exu
% 3.34/1.72 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.34/1.72 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(setext_type,type,
% 3.34/1.72 setext: $o ).
% 3.34/1.72
% 3.34/1.72 thf(setext,definition,
% 3.34/1.72 ( setext
% 3.34/1.72 = ( ! [A: $i,B: $i] :
% 3.34/1.72 ( ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ( in @ Xx @ B ) )
% 3.34/1.72 => ( ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ B )
% 3.34/1.72 => ( in @ Xx @ A ) )
% 3.34/1.72 => ( A = B ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(emptyI_type,type,
% 3.34/1.72 emptyI: $o ).
% 3.34/1.72
% 3.34/1.72 thf(emptyI,definition,
% 3.34/1.72 ( emptyI
% 3.34/1.72 = ( ! [A: $i] :
% 3.34/1.72 ( ! [Xx: $i] :
% 3.34/1.72 ~ ( in @ Xx @ A )
% 3.34/1.72 => ( A = emptyset ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(noeltsimpempty_type,type,
% 3.34/1.72 noeltsimpempty: $o ).
% 3.34/1.72
% 3.34/1.72 thf(noeltsimpempty,definition,
% 3.34/1.72 ( noeltsimpempty
% 3.34/1.72 = ( ! [A: $i] :
% 3.34/1.72 ( ! [Xx: $i] :
% 3.34/1.72 ~ ( in @ Xx @ A )
% 3.34/1.72 => ( A = emptyset ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(setbeta_type,type,
% 3.34/1.72 setbeta: $o ).
% 3.34/1.72
% 3.34/1.72 thf(setbeta,definition,
% 3.34/1.72 ( setbeta
% 3.34/1.72 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ( ( in @ Xx
% 3.34/1.72 @ ( dsetconstr @ A
% 3.34/1.72 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 3.34/1.72 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(nonempty_type,type,
% 3.34/1.72 nonempty: $i > $o ).
% 3.34/1.72
% 3.34/1.72 thf(nonempty,definition,
% 3.34/1.72 ( nonempty
% 3.34/1.72 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(nonemptyE1_type,type,
% 3.34/1.72 nonemptyE1: $o ).
% 3.34/1.72
% 3.34/1.72 thf(nonemptyE1,definition,
% 3.34/1.72 ( nonemptyE1
% 3.34/1.72 = ( ! [A: $i] :
% 3.34/1.72 ( ( nonempty @ A )
% 3.34/1.72 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(nonemptyI_type,type,
% 3.34/1.72 nonemptyI: $o ).
% 3.34/1.72
% 3.34/1.72 thf(nonemptyI,definition,
% 3.34/1.72 ( nonemptyI
% 3.34/1.72 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ( ( Xphi @ Xx )
% 3.34/1.72 => ( nonempty
% 3.34/1.72 @ ( dsetconstr @ A
% 3.34/1.72 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(nonemptyI1_type,type,
% 3.34/1.72 nonemptyI1: $o ).
% 3.34/1.72
% 3.34/1.72 thf(nonemptyI1,definition,
% 3.34/1.72 ( nonemptyI1
% 3.34/1.72 = ( ! [A: $i] :
% 3.34/1.72 ( ? [Xx: $i] : ( in @ Xx @ A )
% 3.34/1.72 => ( nonempty @ A ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(setadjoinIL_type,type,
% 3.34/1.72 setadjoinIL: $o ).
% 3.34/1.72
% 3.34/1.72 thf(setadjoinIL,definition,
% 3.34/1.72 ( setadjoinIL
% 3.34/1.72 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(emptyinunitempty_type,type,
% 3.34/1.72 emptyinunitempty: $o ).
% 3.34/1.72
% 3.34/1.72 thf(emptyinunitempty,definition,
% 3.34/1.72 ( emptyinunitempty
% 3.34/1.72 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(setadjoinIR_type,type,
% 3.34/1.72 setadjoinIR: $o ).
% 3.34/1.72
% 3.34/1.72 thf(setadjoinIR,definition,
% 3.34/1.72 ( setadjoinIR
% 3.34/1.72 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 3.34/1.72 ( ( in @ Xy @ A )
% 3.34/1.72 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(setadjoinE_type,type,
% 3.34/1.72 setadjoinE: $o ).
% 3.34/1.72
% 3.34/1.72 thf(setadjoinE,definition,
% 3.34/1.72 ( setadjoinE
% 3.34/1.72 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 3.34/1.72 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 3.34/1.72 => ! [Xphi: $o] :
% 3.34/1.72 ( ( ( Xy = Xx )
% 3.34/1.72 => Xphi )
% 3.34/1.72 => ( ( ( in @ Xy @ A )
% 3.34/1.72 => Xphi )
% 3.34/1.72 => Xphi ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(setadjoinOr_type,type,
% 3.34/1.72 setadjoinOr: $o ).
% 3.34/1.72
% 3.34/1.72 thf(setadjoinOr,definition,
% 3.34/1.72 ( setadjoinOr
% 3.34/1.72 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 3.34/1.72 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 3.34/1.72 => ( ( Xy = Xx )
% 3.34/1.72 | ( in @ Xy @ A ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(setoftrueEq_type,type,
% 3.34/1.72 setoftrueEq: $o ).
% 3.34/1.72
% 3.34/1.72 thf(setoftrueEq,definition,
% 3.34/1.72 ( setoftrueEq
% 3.34/1.72 = ( ! [A: $i] :
% 3.34/1.72 ( ( dsetconstr @ A
% 3.34/1.72 @ ^ [Xx: $i] : $true )
% 3.34/1.72 = A ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(powersetI_type,type,
% 3.34/1.72 powersetI: $o ).
% 3.34/1.72
% 3.34/1.72 thf(powersetI,definition,
% 3.34/1.72 ( powersetI
% 3.34/1.72 = ( ! [A: $i,B: $i] :
% 3.34/1.72 ( ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ B )
% 3.34/1.72 => ( in @ Xx @ A ) )
% 3.34/1.72 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(emptyinPowerset_type,type,
% 3.34/1.72 emptyinPowerset: $o ).
% 3.34/1.72
% 3.34/1.72 thf(emptyinPowerset,definition,
% 3.34/1.72 ( emptyinPowerset
% 3.34/1.72 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(emptyInPowerset_type,type,
% 3.34/1.72 emptyInPowerset: $o ).
% 3.34/1.72
% 3.34/1.72 thf(emptyInPowerset,definition,
% 3.34/1.72 ( emptyInPowerset
% 3.34/1.72 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(powersetE_type,type,
% 3.34/1.72 powersetE: $o ).
% 3.34/1.72
% 3.34/1.72 thf(powersetE,definition,
% 3.34/1.72 ( powersetE
% 3.34/1.72 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.34/1.72 ( ( in @ B @ ( powerset @ A ) )
% 3.34/1.72 => ( ( in @ Xx @ B )
% 3.34/1.72 => ( in @ Xx @ A ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(setunionI_type,type,
% 3.34/1.72 setunionI: $o ).
% 3.34/1.72
% 3.34/1.72 thf(setunionI,definition,
% 3.34/1.72 ( setunionI
% 3.34/1.72 = ( ! [A: $i,Xx: $i,B: $i] :
% 3.34/1.72 ( ( in @ Xx @ B )
% 3.34/1.72 => ( ( in @ B @ A )
% 3.34/1.72 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(setunionE_type,type,
% 3.34/1.72 setunionE: $o ).
% 3.34/1.72
% 3.34/1.72 thf(setunionE,definition,
% 3.34/1.72 ( setunionE
% 3.34/1.72 = ( ! [A: $i,Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ ( setunion @ A ) )
% 3.34/1.72 => ! [Xphi: $o] :
% 3.34/1.72 ( ! [B: $i] :
% 3.34/1.72 ( ( in @ Xx @ B )
% 3.34/1.72 => ( ( in @ B @ A )
% 3.34/1.72 => Xphi ) )
% 3.34/1.72 => Xphi ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(subPowSU_type,type,
% 3.34/1.72 subPowSU: $o ).
% 3.34/1.72
% 3.34/1.72 thf(subPowSU,definition,
% 3.34/1.72 ( subPowSU
% 3.34/1.72 = ( ! [A: $i,Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(exuE2_type,type,
% 3.34/1.72 exuE2: $o ).
% 3.34/1.72
% 3.34/1.72 thf(exuE2,definition,
% 3.34/1.72 ( exuE2
% 3.34/1.72 = ( ! [Xphi: $i > $o] :
% 3.34/1.72 ( ( exu
% 3.34/1.72 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.34/1.72 => ? [Xx: $i] :
% 3.34/1.72 ! [Xy: $i] :
% 3.34/1.72 ( ( Xphi @ Xy )
% 3.34/1.72 <=> ( Xy = Xx ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(nonemptyImpWitness_type,type,
% 3.34/1.72 nonemptyImpWitness: $o ).
% 3.34/1.72
% 3.34/1.72 thf(nonemptyImpWitness,definition,
% 3.34/1.72 ( nonemptyImpWitness
% 3.34/1.72 = ( ! [A: $i] :
% 3.34/1.72 ( ( nonempty @ A )
% 3.34/1.72 => ? [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 & $true ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(uniqinunit_type,type,
% 3.34/1.72 uniqinunit: $o ).
% 3.34/1.72
% 3.34/1.72 thf(uniqinunit,definition,
% 3.34/1.72 ( uniqinunit
% 3.34/1.72 = ( ! [Xx: $i,Xy: $i] :
% 3.34/1.72 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 3.34/1.72 => ( Xx = Xy ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(notinsingleton_type,type,
% 3.34/1.72 notinsingleton: $o ).
% 3.34/1.72
% 3.34/1.72 thf(notinsingleton,definition,
% 3.34/1.72 ( notinsingleton
% 3.34/1.72 = ( ! [Xx: $i,Xy: $i] :
% 3.34/1.72 ( ( Xx != Xy )
% 3.34/1.72 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(eqinunit_type,type,
% 3.34/1.72 eqinunit: $o ).
% 3.34/1.72
% 3.34/1.72 thf(eqinunit,definition,
% 3.34/1.72 ( eqinunit
% 3.34/1.72 = ( ! [Xx: $i,Xy: $i] :
% 3.34/1.72 ( ( Xx = Xy )
% 3.34/1.72 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(singletonsswitch_type,type,
% 3.34/1.72 singletonsswitch: $o ).
% 3.34/1.72
% 3.34/1.72 thf(singletonsswitch,definition,
% 3.34/1.72 ( singletonsswitch
% 3.34/1.72 = ( ! [Xx: $i,Xy: $i] :
% 3.34/1.72 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 3.34/1.72 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(upairsetE_type,type,
% 3.34/1.72 upairsetE: $o ).
% 3.34/1.72
% 3.34/1.72 thf(upairsetE,definition,
% 3.34/1.72 ( upairsetE
% 3.34/1.72 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 3.34/1.72 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 3.34/1.72 => ( ( Xz = Xx )
% 3.34/1.72 | ( Xz = Xy ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(upairsetIL_type,type,
% 3.34/1.72 upairsetIL: $o ).
% 3.34/1.72
% 3.34/1.72 thf(upairsetIL,definition,
% 3.34/1.72 ( upairsetIL
% 3.34/1.72 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(upairsetIR_type,type,
% 3.34/1.72 upairsetIR: $o ).
% 3.34/1.72
% 3.34/1.72 thf(upairsetIR,definition,
% 3.34/1.72 ( upairsetIR
% 3.34/1.72 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(emptyE1_type,type,
% 3.34/1.72 emptyE1: $o ).
% 3.34/1.72
% 3.34/1.72 thf(emptyE1,definition,
% 3.34/1.72 ( emptyE1
% 3.34/1.72 = ( ! [A: $i,Xphi: $i > $o] :
% 3.34/1.72 ( ? [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 & ( Xphi @ Xx ) )
% 3.34/1.72 => ( ( ( dsetconstr @ A
% 3.34/1.72 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.34/1.72 = emptyset )
% 3.34/1.72 => $false ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(vacuousDall_type,type,
% 3.34/1.72 vacuousDall: $o ).
% 3.34/1.72
% 3.34/1.72 thf(vacuousDall,definition,
% 3.34/1.72 ( vacuousDall
% 3.34/1.72 = ( ! [Xphi: $i > $o,Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ emptyset )
% 3.34/1.72 => ( Xphi @ Xx ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(quantDeMorgan1_type,type,
% 3.34/1.72 quantDeMorgan1: $o ).
% 3.34/1.72
% 3.34/1.72 thf(quantDeMorgan1,definition,
% 3.34/1.72 ( quantDeMorgan1
% 3.34/1.72 = ( ! [A: $i,Xphi: $i > $o] :
% 3.34/1.72 ( ~ ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ( Xphi @ Xx ) )
% 3.34/1.72 => ? [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(quantDeMorgan2_type,type,
% 3.34/1.72 quantDeMorgan2: $o ).
% 3.34/1.72
% 3.34/1.72 thf(quantDeMorgan2,definition,
% 3.34/1.72 ( quantDeMorgan2
% 3.34/1.72 = ( ! [A: $i,Xphi: $i > $o] :
% 3.34/1.72 ( ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ~ ( Xphi @ Xx ) )
% 3.34/1.72 => ~ ? [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 & ( Xphi @ Xx ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(quantDeMorgan3_type,type,
% 3.34/1.72 quantDeMorgan3: $o ).
% 3.34/1.72
% 3.34/1.72 thf(quantDeMorgan3,definition,
% 3.34/1.72 ( quantDeMorgan3
% 3.34/1.72 = ( ! [A: $i,Xphi: $i > $o] :
% 3.34/1.72 ( ~ ? [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 & ( Xphi @ Xx ) )
% 3.34/1.72 => ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(quantDeMorgan4_type,type,
% 3.34/1.72 quantDeMorgan4: $o ).
% 3.34/1.72
% 3.34/1.72 thf(quantDeMorgan4,definition,
% 3.34/1.72 ( quantDeMorgan4
% 3.34/1.72 = ( ! [A: $i,Xphi: $i > $o] :
% 3.34/1.72 ( ? [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 & ~ ( Xphi @ Xx ) )
% 3.34/1.72 => ~ ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ( Xphi @ Xx ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(prop2setI_type,type,
% 3.34/1.72 prop2setI: $o ).
% 3.34/1.72
% 3.34/1.72 thf(prop2setI,definition,
% 3.34/1.72 ( prop2setI
% 3.34/1.72 = ( ! [Xphi: $o] :
% 3.34/1.72 ( Xphi
% 3.34/1.72 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(set2prop_type,type,
% 3.34/1.72 set2prop: $i > $o ).
% 3.34/1.72
% 3.34/1.72 thf(prop2set2propI_type,type,
% 3.34/1.72 prop2set2propI: $o ).
% 3.34/1.72
% 3.34/1.72 thf(prop2set2propI,definition,
% 3.34/1.72 ( prop2set2propI
% 3.34/1.72 = ( ! [Xphi: $o] :
% 3.34/1.72 ( Xphi
% 3.34/1.72 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(notdexE_type,type,
% 3.34/1.72 notdexE: $o ).
% 3.34/1.72
% 3.34/1.72 thf(notdexE,definition,
% 3.34/1.72 ( notdexE
% 3.34/1.72 = ( ! [A: $i,Xphi: $i > $o] :
% 3.34/1.72 ( ~ ? [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 & ( Xphi @ Xx ) )
% 3.34/1.72 => ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(notdallE_type,type,
% 3.34/1.72 notdallE: $o ).
% 3.34/1.72
% 3.34/1.72 thf(notdallE,definition,
% 3.34/1.72 ( notdallE
% 3.34/1.72 = ( ! [A: $i,Xphi: $i > $o] :
% 3.34/1.72 ( ~ ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ( Xphi @ Xx ) )
% 3.34/1.72 => ? [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(exuI1_type,type,
% 3.34/1.72 exuI1: $o ).
% 3.34/1.72
% 3.34/1.72 thf(exuI1,definition,
% 3.34/1.72 ( exuI1
% 3.34/1.72 = ( ! [Xphi: $i > $o] :
% 3.34/1.72 ( ? [Xx: $i] :
% 3.34/1.72 ( ( Xphi @ Xx )
% 3.34/1.72 & ! [Xy: $i] :
% 3.34/1.72 ( ( Xphi @ Xy )
% 3.34/1.72 => ( Xx = Xy ) ) )
% 3.34/1.72 => ( exu
% 3.34/1.72 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(exuI3_type,type,
% 3.34/1.72 exuI3: $o ).
% 3.34/1.72
% 3.34/1.72 thf(exuI3,definition,
% 3.34/1.72 ( exuI3
% 3.34/1.72 = ( ! [Xphi: $i > $o] :
% 3.34/1.72 ( ? [Xx: $i] : ( Xphi @ Xx )
% 3.34/1.72 => ( ! [Xx: $i,Xy: $i] :
% 3.34/1.72 ( ( Xphi @ Xx )
% 3.34/1.72 => ( ( Xphi @ Xy )
% 3.34/1.72 => ( Xx = Xy ) ) )
% 3.34/1.72 => ( exu
% 3.34/1.72 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(exuI2_type,type,
% 3.34/1.72 exuI2: $o ).
% 3.34/1.72
% 3.34/1.72 thf(exuI2,definition,
% 3.34/1.72 ( exuI2
% 3.34/1.72 = ( ! [Xphi: $i > $o] :
% 3.34/1.72 ( ? [Xx: $i] :
% 3.34/1.72 ! [Xy: $i] :
% 3.34/1.72 ( ( Xphi @ Xy )
% 3.34/1.72 <=> ( Xy = Xx ) )
% 3.34/1.72 => ( exu
% 3.34/1.72 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(inCongP_type,type,
% 3.34/1.72 inCongP: $o ).
% 3.34/1.72
% 3.34/1.72 thf(inCongP,definition,
% 3.34/1.72 ( inCongP
% 3.34/1.72 = ( ! [A: $i,B: $i] :
% 3.34/1.72 ( ( A = B )
% 3.34/1.72 => ! [Xx: $i,Xy: $i] :
% 3.34/1.72 ( ( Xx = Xy )
% 3.34/1.72 => ( ( in @ Xx @ A )
% 3.34/1.72 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(in__Cong_type,type,
% 3.34/1.72 in__Cong: $o ).
% 3.34/1.72
% 3.34/1.72 thf(in__Cong,definition,
% 3.34/1.72 ( in__Cong
% 3.34/1.72 = ( ! [A: $i,B: $i] :
% 3.34/1.72 ( ( A = B )
% 3.34/1.72 => ! [Xx: $i,Xy: $i] :
% 3.34/1.72 ( ( Xx = Xy )
% 3.34/1.72 => ( ( in @ Xx @ A )
% 3.34/1.72 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(exuE3u_type,type,
% 3.34/1.72 exuE3u: $o ).
% 3.34/1.72
% 3.34/1.72 thf(exuE3u,definition,
% 3.34/1.72 ( exuE3u
% 3.34/1.72 = ( ! [Xphi: $i > $o] :
% 3.34/1.72 ( ( exu
% 3.34/1.72 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.34/1.72 => ! [Xx: $i,Xy: $i] :
% 3.34/1.72 ( ( Xphi @ Xx )
% 3.34/1.72 => ( ( Xphi @ Xy )
% 3.34/1.72 => ( Xx = Xy ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(exu__Cong_type,type,
% 3.34/1.72 exu__Cong: $o ).
% 3.34/1.72
% 3.34/1.72 thf(exu__Cong,definition,
% 3.34/1.72 ( exu__Cong
% 3.34/1.72 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 3.34/1.72 ( ! [Xx: $i,Xy: $i] :
% 3.34/1.72 ( ( Xx = Xy )
% 3.34/1.72 => ( ( Xphi @ Xx )
% 3.34/1.72 <=> ( Xpsi @ Xy ) ) )
% 3.34/1.72 => ( ( exu
% 3.34/1.72 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.34/1.72 <=> ( exu
% 3.34/1.72 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(emptyset__Cong_type,type,
% 3.34/1.72 emptyset__Cong: $o ).
% 3.34/1.72
% 3.34/1.72 thf(emptyset__Cong,definition,
% 3.34/1.72 ( emptyset__Cong
% 3.34/1.72 = ( emptyset = emptyset ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(setadjoin__Cong_type,type,
% 3.34/1.72 setadjoin__Cong: $o ).
% 3.34/1.72
% 3.34/1.72 thf(setadjoin__Cong,definition,
% 3.34/1.72 ( setadjoin__Cong
% 3.34/1.72 = ( ! [Xx: $i,Xy: $i] :
% 3.34/1.72 ( ( Xx = Xy )
% 3.34/1.72 => ! [Xz: $i,Xu: $i] :
% 3.34/1.72 ( ( Xz = Xu )
% 3.34/1.72 => ( ( setadjoin @ Xx @ Xz )
% 3.34/1.72 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(powerset__Cong_type,type,
% 3.34/1.72 powerset__Cong: $o ).
% 3.34/1.72
% 3.34/1.72 thf(powerset__Cong,definition,
% 3.34/1.72 ( powerset__Cong
% 3.34/1.72 = ( ! [A: $i,B: $i] :
% 3.34/1.72 ( ( A = B )
% 3.34/1.72 => ( ( powerset @ A )
% 3.34/1.72 = ( powerset @ B ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(setunion__Cong_type,type,
% 3.34/1.72 setunion__Cong: $o ).
% 3.34/1.72
% 3.34/1.72 thf(setunion__Cong,definition,
% 3.34/1.72 ( setunion__Cong
% 3.34/1.72 = ( ! [A: $i,B: $i] :
% 3.34/1.72 ( ( A = B )
% 3.34/1.72 => ( ( setunion @ A )
% 3.34/1.72 = ( setunion @ B ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(omega__Cong_type,type,
% 3.34/1.72 omega__Cong: $o ).
% 3.34/1.72
% 3.34/1.72 thf(omega__Cong,definition,
% 3.34/1.72 ( omega__Cong
% 3.34/1.72 = ( omega = omega ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(exuEu_type,type,
% 3.34/1.72 exuEu: $o ).
% 3.34/1.72
% 3.34/1.72 thf(exuEu,definition,
% 3.34/1.72 ( exuEu
% 3.34/1.72 = ( ! [Xphi: $i > $o] :
% 3.34/1.72 ( ( exu
% 3.34/1.72 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.34/1.72 => ! [Xx: $i,Xy: $i] :
% 3.34/1.72 ( ( Xphi @ Xx )
% 3.34/1.72 => ( ( Xphi @ Xy )
% 3.34/1.72 => ( Xx = Xy ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(descr__Cong_type,type,
% 3.34/1.72 descr__Cong: $o ).
% 3.34/1.72
% 3.34/1.72 thf(descr__Cong,definition,
% 3.34/1.72 ( descr__Cong
% 3.34/1.72 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 3.34/1.72 ( ! [Xx: $i,Xy: $i] :
% 3.34/1.72 ( ( Xx = Xy )
% 3.34/1.72 => ( ( Xphi @ Xx )
% 3.34/1.72 <=> ( Xpsi @ Xy ) ) )
% 3.34/1.72 => ( ( exu
% 3.34/1.72 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.34/1.72 => ( ( exu
% 3.34/1.72 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 3.34/1.72 => ( ( descr
% 3.34/1.72 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.34/1.72 = ( descr
% 3.34/1.72 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(dsetconstr__Cong_type,type,
% 3.34/1.72 dsetconstr__Cong: $o ).
% 3.34/1.72
% 3.34/1.72 thf(dsetconstr__Cong,definition,
% 3.34/1.72 ( dsetconstr__Cong
% 3.34/1.72 = ( ! [A: $i,B: $i] :
% 3.34/1.72 ( ( A = B )
% 3.34/1.72 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 3.34/1.72 ( ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ! [Xy: $i] :
% 3.34/1.72 ( ( in @ Xy @ B )
% 3.34/1.72 => ( ( Xx = Xy )
% 3.34/1.72 => ( ( Xphi @ Xx )
% 3.34/1.72 <=> ( Xpsi @ Xy ) ) ) ) )
% 3.34/1.72 => ( ( dsetconstr @ A
% 3.34/1.72 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.34/1.72 = ( dsetconstr @ B
% 3.34/1.72 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(subset_type,type,
% 3.34/1.72 subset: $i > $i > $o ).
% 3.34/1.72
% 3.34/1.72 thf(disjoint_type,type,
% 3.34/1.72 disjoint: $i > $i > $o ).
% 3.34/1.72
% 3.34/1.72 thf(setsmeet_type,type,
% 3.34/1.72 setsmeet: $i > $i > $o ).
% 3.34/1.72
% 3.34/1.72 thf(subsetI1_type,type,
% 3.34/1.72 subsetI1: $o ).
% 3.34/1.72
% 3.34/1.72 thf(subsetI1,definition,
% 3.34/1.72 ( subsetI1
% 3.34/1.72 = ( ! [A: $i,B: $i] :
% 3.34/1.72 ( ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ( in @ Xx @ B ) )
% 3.34/1.72 => ( subset @ A @ B ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(eqimpsubset2_type,type,
% 3.34/1.72 eqimpsubset2: $o ).
% 3.34/1.72
% 3.34/1.72 thf(eqimpsubset2,definition,
% 3.34/1.72 ( eqimpsubset2
% 3.34/1.72 = ( ! [A: $i,B: $i] :
% 3.34/1.72 ( ( A = B )
% 3.34/1.72 => ( subset @ B @ A ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(eqimpsubset1_type,type,
% 3.34/1.72 eqimpsubset1: $o ).
% 3.34/1.72
% 3.34/1.72 thf(eqimpsubset1,definition,
% 3.34/1.72 ( eqimpsubset1
% 3.34/1.72 = ( ! [A: $i,B: $i] :
% 3.34/1.72 ( ( A = B )
% 3.34/1.72 => ( subset @ A @ B ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(subsetI2_type,type,
% 3.34/1.72 subsetI2: $o ).
% 3.34/1.72
% 3.34/1.72 thf(subsetI2,definition,
% 3.34/1.72 ( subsetI2
% 3.34/1.72 = ( ! [A: $i,B: $i] :
% 3.34/1.72 ( ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ( in @ Xx @ B ) )
% 3.34/1.72 => ( subset @ A @ B ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(emptysetsubset_type,type,
% 3.34/1.72 emptysetsubset: $o ).
% 3.34/1.72
% 3.34/1.72 thf(emptysetsubset,definition,
% 3.34/1.72 ( emptysetsubset
% 3.34/1.72 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(subsetE_type,type,
% 3.34/1.72 subsetE: $o ).
% 3.34/1.72
% 3.34/1.72 thf(subsetE,definition,
% 3.34/1.72 ( subsetE
% 3.34/1.72 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.34/1.72 ( ( subset @ A @ B )
% 3.34/1.72 => ( ( in @ Xx @ A )
% 3.34/1.72 => ( in @ Xx @ B ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(subsetE2_type,type,
% 3.34/1.72 subsetE2: $o ).
% 3.34/1.72
% 3.34/1.72 thf(subsetE2,definition,
% 3.34/1.72 ( subsetE2
% 3.34/1.72 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.34/1.72 ( ( subset @ A @ B )
% 3.34/1.72 => ( ~ ( in @ Xx @ B )
% 3.34/1.72 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(notsubsetI_type,type,
% 3.34/1.72 notsubsetI: $o ).
% 3.34/1.72
% 3.34/1.72 thf(notsubsetI,definition,
% 3.34/1.72 ( notsubsetI
% 3.34/1.72 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ( ~ ( in @ Xx @ B )
% 3.34/1.72 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(notequalI1_type,type,
% 3.34/1.72 notequalI1: $o ).
% 3.34/1.72
% 3.34/1.72 thf(notequalI1,definition,
% 3.34/1.72 ( notequalI1
% 3.34/1.72 = ( ! [A: $i,B: $i] :
% 3.34/1.72 ( ~ ( subset @ A @ B )
% 3.34/1.72 => ( A != B ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(notequalI2_type,type,
% 3.34/1.72 notequalI2: $o ).
% 3.34/1.72
% 3.34/1.72 thf(notequalI2,definition,
% 3.34/1.72 ( notequalI2
% 3.34/1.72 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ( ~ ( in @ Xx @ B )
% 3.34/1.72 => ( A != B ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(subsetRefl_type,type,
% 3.34/1.72 subsetRefl: $o ).
% 3.34/1.72
% 3.34/1.72 thf(subsetRefl,definition,
% 3.34/1.72 ( subsetRefl
% 3.34/1.72 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(subsetTrans_type,type,
% 3.34/1.72 subsetTrans: $o ).
% 3.34/1.72
% 3.34/1.72 thf(subsetTrans,definition,
% 3.34/1.72 ( subsetTrans
% 3.34/1.72 = ( ! [A: $i,B: $i,C: $i] :
% 3.34/1.72 ( ( subset @ A @ B )
% 3.34/1.72 => ( ( subset @ B @ C )
% 3.34/1.72 => ( subset @ A @ C ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(setadjoinSub_type,type,
% 3.34/1.72 setadjoinSub: $o ).
% 3.34/1.72
% 3.34/1.72 thf(setadjoinSub,definition,
% 3.34/1.72 ( setadjoinSub
% 3.34/1.72 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(setadjoinSub2_type,type,
% 3.34/1.72 setadjoinSub2: $o ).
% 3.34/1.72
% 3.34/1.72 thf(setadjoinSub2,definition,
% 3.34/1.72 ( setadjoinSub2
% 3.34/1.72 = ( ! [A: $i,Xx: $i,B: $i] :
% 3.34/1.72 ( ( subset @ A @ B )
% 3.34/1.72 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(subset2powerset_type,type,
% 3.34/1.72 subset2powerset: $o ).
% 3.34/1.72
% 3.34/1.72 thf(subset2powerset,definition,
% 3.34/1.72 ( subset2powerset
% 3.34/1.72 = ( ! [A: $i,B: $i] :
% 3.34/1.72 ( ( subset @ A @ B )
% 3.34/1.72 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(setextsub_type,type,
% 3.34/1.72 setextsub: $o ).
% 3.34/1.72
% 3.34/1.72 thf(setextsub,definition,
% 3.34/1.72 ( setextsub
% 3.34/1.72 = ( ! [A: $i,B: $i] :
% 3.34/1.72 ( ( subset @ A @ B )
% 3.34/1.72 => ( ( subset @ B @ A )
% 3.34/1.72 => ( A = B ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(subsetemptysetimpeq_type,type,
% 3.34/1.72 subsetemptysetimpeq: $o ).
% 3.34/1.72
% 3.34/1.72 thf(subsetemptysetimpeq,definition,
% 3.34/1.72 ( subsetemptysetimpeq
% 3.34/1.72 = ( ! [A: $i] :
% 3.34/1.72 ( ( subset @ A @ emptyset )
% 3.34/1.72 => ( A = emptyset ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(powersetI1_type,type,
% 3.34/1.72 powersetI1: $o ).
% 3.34/1.72
% 3.34/1.72 thf(powersetI1,definition,
% 3.34/1.72 ( powersetI1
% 3.34/1.72 = ( ! [A: $i,B: $i] :
% 3.34/1.72 ( ( subset @ B @ A )
% 3.34/1.72 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(powersetE1_type,type,
% 3.34/1.72 powersetE1: $o ).
% 3.34/1.72
% 3.34/1.72 thf(powersetE1,definition,
% 3.34/1.72 ( powersetE1
% 3.34/1.72 = ( ! [A: $i,B: $i] :
% 3.34/1.72 ( ( in @ B @ ( powerset @ A ) )
% 3.34/1.72 => ( subset @ B @ A ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(inPowerset_type,type,
% 3.34/1.72 inPowerset: $o ).
% 3.34/1.72
% 3.34/1.72 thf(inPowerset,definition,
% 3.34/1.72 ( inPowerset
% 3.34/1.72 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(powersetsubset_type,type,
% 3.34/1.72 powersetsubset: $o ).
% 3.34/1.72
% 3.34/1.72 thf(powersetsubset,definition,
% 3.34/1.72 ( powersetsubset
% 3.34/1.72 = ( ! [A: $i,B: $i] :
% 3.34/1.72 ( ( subset @ A @ B )
% 3.34/1.72 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(sepInPowerset_type,type,
% 3.34/1.72 sepInPowerset: $o ).
% 3.34/1.72
% 3.34/1.72 thf(sepInPowerset,definition,
% 3.34/1.72 ( sepInPowerset
% 3.34/1.72 = ( ! [A: $i,Xphi: $i > $o] :
% 3.34/1.72 ( in
% 3.34/1.72 @ ( dsetconstr @ A
% 3.34/1.72 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.34/1.72 @ ( powerset @ A ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(sepSubset_type,type,
% 3.34/1.72 sepSubset: $o ).
% 3.34/1.72
% 3.34/1.72 thf(sepSubset,definition,
% 3.34/1.72 ( sepSubset
% 3.34/1.72 = ( ! [A: $i,Xphi: $i > $o] :
% 3.34/1.72 ( subset
% 3.34/1.72 @ ( dsetconstr @ A
% 3.34/1.72 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.34/1.72 @ A ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(binunion_type,type,
% 3.34/1.72 binunion: $i > $i > $i ).
% 3.34/1.72
% 3.34/1.72 thf(binunionIL_type,type,
% 3.34/1.72 binunionIL: $o ).
% 3.34/1.72
% 3.34/1.72 thf(binunionIL,definition,
% 3.34/1.72 ( binunionIL
% 3.34/1.72 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(upairset2IR_type,type,
% 3.34/1.72 upairset2IR: $o ).
% 3.34/1.72
% 3.34/1.72 thf(upairset2IR,definition,
% 3.34/1.72 ( upairset2IR
% 3.34/1.72 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(binunionIR_type,type,
% 3.34/1.72 binunionIR: $o ).
% 3.34/1.72
% 3.34/1.72 thf(binunionIR,definition,
% 3.34/1.72 ( binunionIR
% 3.34/1.72 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ B )
% 3.34/1.72 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(binunionEcases_type,type,
% 3.34/1.72 binunionEcases: $o ).
% 3.34/1.72
% 3.34/1.72 thf(binunionEcases,definition,
% 3.34/1.72 ( binunionEcases
% 3.34/1.72 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 3.34/1.72 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 3.34/1.72 => ( ( ( in @ Xx @ A )
% 3.34/1.72 => Xphi )
% 3.34/1.72 => ( ( ( in @ Xx @ B )
% 3.34/1.72 => Xphi )
% 3.34/1.72 => Xphi ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(binunionE_type,type,
% 3.34/1.72 binunionE: $o ).
% 3.34/1.72
% 3.34/1.72 thf(binunionE,definition,
% 3.34/1.72 ( binunionE
% 3.34/1.72 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 3.34/1.72 => ( ( in @ Xx @ A )
% 3.34/1.72 | ( in @ Xx @ B ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(binunionLsub_type,type,
% 3.34/1.72 binunionLsub: $o ).
% 3.34/1.72
% 3.34/1.72 thf(binunionLsub,definition,
% 3.34/1.72 ( binunionLsub
% 3.34/1.72 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(binunionRsub_type,type,
% 3.34/1.72 binunionRsub: $o ).
% 3.34/1.72
% 3.34/1.72 thf(binunionRsub,definition,
% 3.34/1.72 ( binunionRsub
% 3.34/1.72 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(binintersect_type,type,
% 3.34/1.72 binintersect: $i > $i > $i ).
% 3.34/1.72
% 3.34/1.72 thf(binintersectI_type,type,
% 3.34/1.72 binintersectI: $o ).
% 3.34/1.72
% 3.34/1.72 thf(binintersectI,definition,
% 3.34/1.72 ( binintersectI
% 3.34/1.72 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ( ( in @ Xx @ B )
% 3.34/1.72 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(binintersectSubset5_type,type,
% 3.34/1.72 binintersectSubset5: $o ).
% 3.34/1.72
% 3.34/1.72 thf(binintersectSubset5,definition,
% 3.34/1.72 ( binintersectSubset5
% 3.34/1.72 = ( ! [A: $i,B: $i,C: $i] :
% 3.34/1.72 ( ( subset @ C @ A )
% 3.34/1.72 => ( ( subset @ C @ B )
% 3.34/1.72 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(binintersectEL_type,type,
% 3.34/1.72 binintersectEL: $o ).
% 3.34/1.72
% 3.34/1.72 thf(binintersectEL,definition,
% 3.34/1.72 ( binintersectEL
% 3.34/1.72 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 3.34/1.72 => ( in @ Xx @ A ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(binintersectLsub_type,type,
% 3.34/1.72 binintersectLsub: $o ).
% 3.34/1.72
% 3.34/1.72 thf(binintersectLsub,definition,
% 3.34/1.72 ( binintersectLsub
% 3.34/1.72 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(binintersectSubset2_type,type,
% 3.34/1.72 binintersectSubset2: $o ).
% 3.34/1.72
% 3.34/1.72 thf(binintersectSubset2,definition,
% 3.34/1.72 ( binintersectSubset2
% 3.34/1.72 = ( ! [A: $i,B: $i] :
% 3.34/1.72 ( ( subset @ A @ B )
% 3.34/1.72 => ( ( binintersect @ A @ B )
% 3.34/1.72 = A ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(binintersectSubset3_type,type,
% 3.34/1.72 binintersectSubset3: $o ).
% 3.34/1.72
% 3.34/1.72 thf(binintersectSubset3,definition,
% 3.34/1.72 ( binintersectSubset3
% 3.34/1.72 = ( ! [A: $i,B: $i] :
% 3.34/1.72 ( ( ( binintersect @ A @ B )
% 3.34/1.72 = B )
% 3.34/1.72 => ( subset @ B @ A ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(binintersectER_type,type,
% 3.34/1.72 binintersectER: $o ).
% 3.34/1.72
% 3.34/1.72 thf(binintersectER,definition,
% 3.34/1.72 ( binintersectER
% 3.34/1.72 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 3.34/1.72 => ( in @ Xx @ B ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(disjointsetsI1_type,type,
% 3.34/1.72 disjointsetsI1: $o ).
% 3.34/1.72
% 3.34/1.72 thf(disjointsetsI1,definition,
% 3.34/1.72 ( disjointsetsI1
% 3.34/1.72 = ( ! [A: $i,B: $i] :
% 3.34/1.72 ( ~ ? [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 & ( in @ Xx @ B ) )
% 3.34/1.72 => ( ( binintersect @ A @ B )
% 3.34/1.72 = emptyset ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(binintersectRsub_type,type,
% 3.34/1.72 binintersectRsub: $o ).
% 3.34/1.72
% 3.34/1.72 thf(binintersectRsub,definition,
% 3.34/1.72 ( binintersectRsub
% 3.34/1.72 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(binintersectSubset4_type,type,
% 3.34/1.72 binintersectSubset4: $o ).
% 3.34/1.72
% 3.34/1.72 thf(binintersectSubset4,definition,
% 3.34/1.72 ( binintersectSubset4
% 3.34/1.72 = ( ! [A: $i,B: $i] :
% 3.34/1.72 ( ( subset @ B @ A )
% 3.34/1.72 => ( ( binintersect @ A @ B )
% 3.34/1.72 = B ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(binintersectSubset1_type,type,
% 3.34/1.72 binintersectSubset1: $o ).
% 3.34/1.72
% 3.34/1.72 thf(binintersectSubset1,definition,
% 3.34/1.72 ( binintersectSubset1
% 3.34/1.72 = ( ! [A: $i,B: $i] :
% 3.34/1.72 ( ( ( binintersect @ A @ B )
% 3.34/1.72 = A )
% 3.34/1.72 => ( subset @ A @ B ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(bs114d_type,type,
% 3.34/1.72 bs114d: $o ).
% 3.34/1.72
% 3.34/1.72 thf(bs114d,definition,
% 3.34/1.72 ( bs114d
% 3.34/1.72 = ( ! [A: $i,B: $i,C: $i] :
% 3.34/1.72 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 3.34/1.72 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(regular_type,type,
% 3.34/1.72 regular: $i > $o ).
% 3.34/1.72
% 3.34/1.72 thf(setminus_type,type,
% 3.34/1.72 setminus: $i > $i > $i ).
% 3.34/1.72
% 3.34/1.72 thf(setminusI_type,type,
% 3.34/1.72 setminusI: $o ).
% 3.34/1.72
% 3.34/1.72 thf(setminusI,definition,
% 3.34/1.72 ( setminusI
% 3.34/1.72 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ( ~ ( in @ Xx @ B )
% 3.34/1.72 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(setminusEL_type,type,
% 3.34/1.72 setminusEL: $o ).
% 3.34/1.72
% 3.34/1.72 thf(setminusEL,definition,
% 3.34/1.72 ( setminusEL
% 3.34/1.72 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 3.34/1.72 => ( in @ Xx @ A ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(setminusER_type,type,
% 3.34/1.72 setminusER: $o ).
% 3.34/1.72
% 3.34/1.72 thf(setminusER,definition,
% 3.34/1.72 ( setminusER
% 3.34/1.72 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 3.34/1.72 => ~ ( in @ Xx @ B ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(setminusSubset2_type,type,
% 3.34/1.72 setminusSubset2: $o ).
% 3.34/1.72
% 3.34/1.72 thf(setminusSubset2,definition,
% 3.34/1.72 ( setminusSubset2
% 3.34/1.72 = ( ! [A: $i,B: $i] :
% 3.34/1.72 ( ( subset @ A @ B )
% 3.34/1.72 => ( ( setminus @ A @ B )
% 3.34/1.72 = emptyset ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(setminusERneg_type,type,
% 3.34/1.72 setminusERneg: $o ).
% 3.34/1.72
% 3.34/1.72 thf(setminusERneg,definition,
% 3.34/1.72 ( setminusERneg
% 3.34/1.72 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.34/1.72 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 3.34/1.72 => ( ( in @ Xx @ A )
% 3.34/1.72 => ( in @ Xx @ B ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(setminusELneg_type,type,
% 3.34/1.72 setminusELneg: $o ).
% 3.34/1.72
% 3.34/1.72 thf(setminusELneg,definition,
% 3.34/1.72 ( setminusELneg
% 3.34/1.72 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.34/1.72 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 3.34/1.72 => ( ~ ( in @ Xx @ B )
% 3.34/1.72 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(setminusILneg_type,type,
% 3.34/1.72 setminusILneg: $o ).
% 3.34/1.72
% 3.34/1.72 thf(setminusILneg,definition,
% 3.34/1.72 ( setminusILneg
% 3.34/1.72 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.34/1.72 ( ~ ( in @ Xx @ A )
% 3.34/1.72 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(setminusIRneg_type,type,
% 3.34/1.72 setminusIRneg: $o ).
% 3.34/1.72
% 3.34/1.72 thf(setminusIRneg,definition,
% 3.34/1.72 ( setminusIRneg
% 3.34/1.72 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ B )
% 3.34/1.72 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(setminusLsub_type,type,
% 3.34/1.72 setminusLsub: $o ).
% 3.34/1.72
% 3.34/1.72 thf(setminusLsub,definition,
% 3.34/1.72 ( setminusLsub
% 3.34/1.72 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(setminusSubset1_type,type,
% 3.34/1.72 setminusSubset1: $o ).
% 3.34/1.72
% 3.34/1.72 thf(setminusSubset1,definition,
% 3.34/1.72 ( setminusSubset1
% 3.34/1.72 = ( ! [A: $i,B: $i] :
% 3.34/1.72 ( ( ( setminus @ A @ B )
% 3.34/1.72 = emptyset )
% 3.34/1.72 => ( subset @ A @ B ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(symdiff_type,type,
% 3.34/1.72 symdiff: $i > $i > $i ).
% 3.34/1.72
% 3.34/1.72 thf(symdiffE_type,type,
% 3.34/1.72 symdiffE: $o ).
% 3.34/1.72
% 3.34/1.72 thf(symdiffE,definition,
% 3.34/1.72 ( symdiffE
% 3.34/1.72 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 3.34/1.72 => ! [Xphi: $o] :
% 3.34/1.72 ( ( ( in @ Xx @ A )
% 3.34/1.72 => ( ~ ( in @ Xx @ B )
% 3.34/1.72 => Xphi ) )
% 3.34/1.72 => ( ( ~ ( in @ Xx @ A )
% 3.34/1.72 => ( ( in @ Xx @ B )
% 3.34/1.72 => Xphi ) )
% 3.34/1.72 => Xphi ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(symdiffI1_type,type,
% 3.34/1.72 symdiffI1: $o ).
% 3.34/1.72
% 3.34/1.72 thf(symdiffI1,definition,
% 3.34/1.72 ( symdiffI1
% 3.34/1.72 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ( ~ ( in @ Xx @ B )
% 3.34/1.72 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(symdiffI2_type,type,
% 3.34/1.72 symdiffI2: $o ).
% 3.34/1.72
% 3.34/1.72 thf(symdiffI2,definition,
% 3.34/1.72 ( symdiffI2
% 3.34/1.72 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.34/1.72 ( ~ ( in @ Xx @ A )
% 3.34/1.72 => ( ( in @ Xx @ B )
% 3.34/1.72 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(symdiffIneg1_type,type,
% 3.34/1.72 symdiffIneg1: $o ).
% 3.34/1.72
% 3.34/1.72 thf(symdiffIneg1,definition,
% 3.34/1.72 ( symdiffIneg1
% 3.34/1.72 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ( ( in @ Xx @ B )
% 3.34/1.72 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(symdiffIneg2_type,type,
% 3.34/1.72 symdiffIneg2: $o ).
% 3.34/1.72
% 3.34/1.72 thf(symdiffIneg2,definition,
% 3.34/1.72 ( symdiffIneg2
% 3.34/1.72 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.34/1.72 ( ~ ( in @ Xx @ A )
% 3.34/1.72 => ( ~ ( in @ Xx @ B )
% 3.34/1.72 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(iskpair_type,type,
% 3.34/1.72 iskpair: $i > $o ).
% 3.34/1.72
% 3.34/1.72 thf(secondinupair_type,type,
% 3.34/1.72 secondinupair: $o ).
% 3.34/1.72
% 3.34/1.72 thf(secondinupair,definition,
% 3.34/1.72 ( secondinupair
% 3.34/1.72 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(setukpairIL_type,type,
% 3.34/1.72 setukpairIL: $o ).
% 3.34/1.72
% 3.34/1.72 thf(setukpairIL,definition,
% 3.34/1.72 ( setukpairIL
% 3.34/1.72 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(setukpairIR_type,type,
% 3.34/1.72 setukpairIR: $o ).
% 3.34/1.72
% 3.34/1.72 thf(setukpairIR,definition,
% 3.34/1.72 ( setukpairIR
% 3.34/1.72 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(kpairiskpair_type,type,
% 3.34/1.72 kpairiskpair: $o ).
% 3.34/1.72
% 3.34/1.72 thf(kpairiskpair,definition,
% 3.34/1.72 ( kpairiskpair
% 3.34/1.72 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(kpair_type,type,
% 3.34/1.72 kpair: $i > $i > $i ).
% 3.34/1.72
% 3.34/1.72 thf(kpairp_type,type,
% 3.34/1.72 kpairp: $o ).
% 3.34/1.72
% 3.34/1.72 thf(kpairp,definition,
% 3.34/1.72 ( kpairp
% 3.34/1.72 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(cartprod_type,type,
% 3.34/1.72 cartprod: $i > $i > $i ).
% 3.34/1.72
% 3.34/1.72 thf(singletonsubset_type,type,
% 3.34/1.72 singletonsubset: $o ).
% 3.34/1.72
% 3.34/1.72 thf(singletonsubset,definition,
% 3.34/1.72 ( singletonsubset
% 3.34/1.72 = ( ! [A: $i,Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(singletoninpowerset_type,type,
% 3.34/1.72 singletoninpowerset: $o ).
% 3.34/1.72
% 3.34/1.72 thf(singletoninpowerset,definition,
% 3.34/1.72 ( singletoninpowerset
% 3.34/1.72 = ( ! [A: $i,Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(singletoninpowunion_type,type,
% 3.34/1.72 singletoninpowunion: $o ).
% 3.34/1.72
% 3.34/1.72 thf(singletoninpowunion,definition,
% 3.34/1.72 ( singletoninpowunion
% 3.34/1.72 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(upairset2E_type,type,
% 3.34/1.72 upairset2E: $o ).
% 3.34/1.72
% 3.34/1.72 thf(upairset2E,definition,
% 3.34/1.72 ( upairset2E
% 3.34/1.72 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 3.34/1.72 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 3.34/1.72 => ( ( Xz = Xx )
% 3.34/1.72 | ( Xz = Xy ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(upairsubunion_type,type,
% 3.34/1.72 upairsubunion: $o ).
% 3.34/1.72
% 3.34/1.72 thf(upairsubunion,definition,
% 3.34/1.72 ( upairsubunion
% 3.34/1.72 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ! [Xy: $i] :
% 3.34/1.72 ( ( in @ Xy @ B )
% 3.34/1.72 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(upairinpowunion_type,type,
% 3.34/1.72 upairinpowunion: $o ).
% 3.34/1.72
% 3.34/1.72 thf(upairinpowunion,definition,
% 3.34/1.72 ( upairinpowunion
% 3.34/1.72 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ! [Xy: $i] :
% 3.34/1.72 ( ( in @ Xy @ B )
% 3.34/1.72 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(ubforcartprodlem1_type,type,
% 3.34/1.72 ubforcartprodlem1: $o ).
% 3.34/1.72
% 3.34/1.72 thf(ubforcartprodlem1,definition,
% 3.34/1.72 ( ubforcartprodlem1
% 3.34/1.72 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ! [Xy: $i] :
% 3.34/1.72 ( ( in @ Xy @ B )
% 3.34/1.72 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(ubforcartprodlem2_type,type,
% 3.34/1.72 ubforcartprodlem2: $o ).
% 3.34/1.72
% 3.34/1.72 thf(ubforcartprodlem2,definition,
% 3.34/1.72 ( ubforcartprodlem2
% 3.34/1.72 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ! [Xy: $i] :
% 3.34/1.72 ( ( in @ Xy @ B )
% 3.34/1.72 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(ubforcartprodlem3_type,type,
% 3.34/1.72 ubforcartprodlem3: $o ).
% 3.34/1.72
% 3.34/1.72 thf(ubforcartprodlem3,definition,
% 3.34/1.72 ( ubforcartprodlem3
% 3.34/1.72 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ! [Xy: $i] :
% 3.34/1.72 ( ( in @ Xy @ B )
% 3.34/1.72 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(cartprodpairin_type,type,
% 3.34/1.72 cartprodpairin: $o ).
% 3.34/1.72
% 3.34/1.72 thf(cartprodpairin,definition,
% 3.34/1.72 ( cartprodpairin
% 3.34/1.72 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ! [Xy: $i] :
% 3.34/1.72 ( ( in @ Xy @ B )
% 3.34/1.72 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(cartprodmempair1_type,type,
% 3.34/1.72 cartprodmempair1: $o ).
% 3.34/1.72
% 3.34/1.72 thf(cartprodmempair1,definition,
% 3.34/1.72 ( cartprodmempair1
% 3.34/1.72 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.34/1.72 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.34/1.72 => ? [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 & ? [Xy: $i] :
% 3.34/1.72 ( ( in @ Xy @ B )
% 3.34/1.72 & ( Xu
% 3.34/1.72 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(cartprodmempair_type,type,
% 3.34/1.72 cartprodmempair: $o ).
% 3.34/1.72
% 3.34/1.72 thf(cartprodmempair,definition,
% 3.34/1.72 ( cartprodmempair
% 3.34/1.72 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.34/1.72 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.34/1.72 => ( iskpair @ Xu ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(setunionE2_type,type,
% 3.34/1.72 setunionE2: $o ).
% 3.34/1.72
% 3.34/1.72 thf(setunionE2,definition,
% 3.34/1.72 ( setunionE2
% 3.34/1.72 = ( ! [A: $i,Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ ( setunion @ A ) )
% 3.34/1.72 => ? [X: $i] :
% 3.34/1.72 ( ( in @ X @ A )
% 3.34/1.72 & ( in @ Xx @ X ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(setunionsingleton1_type,type,
% 3.34/1.72 setunionsingleton1: $o ).
% 3.34/1.72
% 3.34/1.72 thf(setunionsingleton1,definition,
% 3.34/1.72 ( setunionsingleton1
% 3.34/1.72 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(setunionsingleton2_type,type,
% 3.34/1.72 setunionsingleton2: $o ).
% 3.34/1.72
% 3.34/1.72 thf(setunionsingleton2,definition,
% 3.34/1.72 ( setunionsingleton2
% 3.34/1.72 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(setunionsingleton_type,type,
% 3.34/1.72 setunionsingleton: $o ).
% 3.34/1.72
% 3.34/1.72 thf(setunionsingleton,definition,
% 3.34/1.72 ( setunionsingleton
% 3.34/1.72 = ( ! [Xx: $i] :
% 3.34/1.72 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 3.34/1.72 = Xx ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(singleton_type,type,
% 3.34/1.72 singleton: $i > $o ).
% 3.34/1.72
% 3.34/1.72 thf(singletonprop_type,type,
% 3.34/1.72 singletonprop: $o ).
% 3.34/1.72
% 3.34/1.72 thf(singletonprop,definition,
% 3.34/1.72 ( singletonprop
% 3.34/1.72 = ( ! [A: $i,Xphi: $i > $o] :
% 3.34/1.72 ( ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ! [Xy: $i] :
% 3.34/1.72 ( ( in @ Xy @ A )
% 3.34/1.72 => ( ( Xphi @ Xx )
% 3.34/1.72 => ( ( Xphi @ Xy )
% 3.34/1.72 => ( Xx = Xy ) ) ) ) )
% 3.34/1.72 => ( ? [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 & ( Xphi @ Xx ) )
% 3.34/1.72 => ( singleton
% 3.34/1.72 @ ( dsetconstr @ A
% 3.34/1.72 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(ex1_type,type,
% 3.34/1.72 ex1: $i > ( $i > $o ) > $o ).
% 3.34/1.72
% 3.34/1.72 thf(ex1E1_type,type,
% 3.34/1.72 ex1E1: $o ).
% 3.34/1.72
% 3.34/1.72 thf(ex1E1,definition,
% 3.34/1.72 ( ex1E1
% 3.34/1.72 = ( ! [A: $i,Xphi: $i > $o] :
% 3.34/1.72 ( ( ex1 @ A
% 3.34/1.72 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.34/1.72 => ? [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 & ( Xphi @ Xx ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(ex1I_type,type,
% 3.34/1.72 ex1I: $o ).
% 3.34/1.72
% 3.34/1.72 thf(ex1I,definition,
% 3.34/1.72 ( ex1I
% 3.34/1.72 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ( ( Xphi @ Xx )
% 3.34/1.72 => ( ! [Xy: $i] :
% 3.34/1.72 ( ( in @ Xy @ A )
% 3.34/1.72 => ( ( Xphi @ Xy )
% 3.34/1.72 => ( Xy = Xx ) ) )
% 3.34/1.72 => ( ex1 @ A
% 3.34/1.72 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(ex1I2_type,type,
% 3.34/1.72 ex1I2: $o ).
% 3.34/1.72
% 3.34/1.72 thf(ex1I2,definition,
% 3.34/1.72 ( ex1I2
% 3.34/1.72 = ( ! [A: $i,Xphi: $i > $o] :
% 3.34/1.72 ( ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ! [Xy: $i] :
% 3.34/1.72 ( ( in @ Xy @ A )
% 3.34/1.72 => ( ( Xphi @ Xx )
% 3.34/1.72 => ( ( Xphi @ Xy )
% 3.34/1.72 => ( Xx = Xy ) ) ) ) )
% 3.34/1.72 => ( ? [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 & ( Xphi @ Xx ) )
% 3.34/1.72 => ( ex1 @ A
% 3.34/1.72 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(singletonsuniq_type,type,
% 3.34/1.72 singletonsuniq: $o ).
% 3.34/1.72
% 3.34/1.72 thf(singletonsuniq,definition,
% 3.34/1.72 ( singletonsuniq
% 3.34/1.72 = ( ! [Xx: $i,Xy: $i] :
% 3.34/1.72 ( ( ( setadjoin @ Xx @ emptyset )
% 3.34/1.72 = ( setadjoin @ Xy @ emptyset ) )
% 3.34/1.72 => ( Xx = Xy ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(atmost1p_type,type,
% 3.34/1.72 atmost1p: $i > $o ).
% 3.34/1.72
% 3.34/1.72 thf(atleast2p_type,type,
% 3.34/1.72 atleast2p: $i > $o ).
% 3.34/1.72
% 3.34/1.72 thf(atmost2p_type,type,
% 3.34/1.72 atmost2p: $i > $o ).
% 3.34/1.72
% 3.34/1.72 thf(upairsetp_type,type,
% 3.34/1.72 upairsetp: $i > $o ).
% 3.34/1.72
% 3.34/1.72 thf(setukpairinjL1_type,type,
% 3.34/1.72 setukpairinjL1: $o ).
% 3.34/1.72
% 3.34/1.72 thf(setukpairinjL1,definition,
% 3.34/1.72 ( setukpairinjL1
% 3.34/1.72 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 3.34/1.72 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 3.34/1.72 => ( Xx = Xz ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(kfstsingleton_type,type,
% 3.34/1.72 kfstsingleton: $o ).
% 3.34/1.72
% 3.34/1.72 thf(kfstsingleton,definition,
% 3.34/1.72 ( kfstsingleton
% 3.34/1.72 = ( ! [Xu: $i] :
% 3.34/1.72 ( ( iskpair @ Xu )
% 3.34/1.72 => ( singleton
% 3.34/1.72 @ ( dsetconstr @ ( setunion @ Xu )
% 3.34/1.72 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(theprop_type,type,
% 3.34/1.72 theprop: $o ).
% 3.34/1.72
% 3.34/1.72 thf(theprop,definition,
% 3.34/1.72 ( theprop
% 3.34/1.72 = ( ! [X: $i] :
% 3.34/1.72 ( ( singleton @ X )
% 3.34/1.72 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(kfst_type,type,
% 3.34/1.72 kfst: $i > $i ).
% 3.34/1.72
% 3.34/1.72 thf(kfstpairEq_type,type,
% 3.34/1.72 kfstpairEq: $o ).
% 3.34/1.72
% 3.34/1.72 thf(kfstpairEq,definition,
% 3.34/1.72 ( kfstpairEq
% 3.34/1.72 = ( ! [Xx: $i,Xy: $i] :
% 3.34/1.72 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 3.34/1.72 = Xx ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(cartprodfstin_type,type,
% 3.34/1.72 cartprodfstin: $o ).
% 3.34/1.72
% 3.34/1.72 thf(cartprodfstin,definition,
% 3.34/1.72 ( cartprodfstin
% 3.34/1.72 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.34/1.72 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.34/1.72 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(setukpairinjL2_type,type,
% 3.34/1.72 setukpairinjL2: $o ).
% 3.34/1.72
% 3.34/1.72 thf(setukpairinjL2,definition,
% 3.34/1.72 ( setukpairinjL2
% 3.34/1.72 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.34/1.72 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 3.34/1.72 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 3.34/1.72 => ( Xx = Xz ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(setukpairinjL_type,type,
% 3.34/1.72 setukpairinjL: $o ).
% 3.34/1.72
% 3.34/1.72 thf(setukpairinjL,definition,
% 3.34/1.72 ( setukpairinjL
% 3.34/1.72 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.34/1.72 ( ( ( kpair @ Xx @ Xy )
% 3.34/1.72 = ( kpair @ Xz @ Xu ) )
% 3.34/1.72 => ( Xx = Xz ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(setukpairinjR11_type,type,
% 3.34/1.72 setukpairinjR11: $o ).
% 3.34/1.72
% 3.34/1.72 thf(setukpairinjR11,definition,
% 3.34/1.72 ( setukpairinjR11
% 3.34/1.72 = ( ! [Xx: $i,Xy: $i] :
% 3.34/1.72 ( ( Xx = Xy )
% 3.34/1.72 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 3.34/1.72 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(setukpairinjR12_type,type,
% 3.34/1.72 setukpairinjR12: $o ).
% 3.34/1.72
% 3.34/1.72 thf(setukpairinjR12,definition,
% 3.34/1.72 ( setukpairinjR12
% 3.34/1.72 = ( ! [Xx: $i,Xy: $i] :
% 3.34/1.72 ( ( Xx = Xy )
% 3.34/1.72 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 3.34/1.72 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(setukpairinjR1_type,type,
% 3.34/1.72 setukpairinjR1: $o ).
% 3.34/1.72
% 3.34/1.72 thf(setukpairinjR1,definition,
% 3.34/1.72 ( setukpairinjR1
% 3.34/1.72 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.34/1.72 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 3.34/1.72 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 3.34/1.72 => ( ( Xz = Xu )
% 3.34/1.72 => ( Xy = Xu ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(upairequniteq_type,type,
% 3.34/1.72 upairequniteq: $o ).
% 3.34/1.72
% 3.34/1.72 thf(upairequniteq,definition,
% 3.34/1.72 ( upairequniteq
% 3.34/1.72 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 3.34/1.72 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 3.34/1.72 = ( setadjoin @ Xz @ emptyset ) )
% 3.34/1.72 => ( Xx = Xy ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(setukpairinjR2_type,type,
% 3.34/1.72 setukpairinjR2: $o ).
% 3.34/1.72
% 3.34/1.72 thf(setukpairinjR2,definition,
% 3.34/1.72 ( setukpairinjR2
% 3.34/1.72 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.34/1.72 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 3.34/1.72 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 3.34/1.72 => ( Xy = Xu ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(setukpairinjR_type,type,
% 3.34/1.72 setukpairinjR: $o ).
% 3.34/1.72
% 3.34/1.72 thf(setukpairinjR,definition,
% 3.34/1.72 ( setukpairinjR
% 3.34/1.72 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.34/1.72 ( ( ( kpair @ Xx @ Xy )
% 3.34/1.72 = ( kpair @ Xz @ Xu ) )
% 3.34/1.72 => ( Xy = Xu ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(ksndsingleton_type,type,
% 3.34/1.72 ksndsingleton: $o ).
% 3.34/1.72
% 3.34/1.72 thf(ksndsingleton,definition,
% 3.34/1.72 ( ksndsingleton
% 3.34/1.72 = ( ! [Xu: $i] :
% 3.34/1.72 ( ( iskpair @ Xu )
% 3.34/1.72 => ( singleton
% 3.34/1.72 @ ( dsetconstr @ ( setunion @ Xu )
% 3.34/1.72 @ ^ [Xx: $i] :
% 3.34/1.72 ( Xu
% 3.34/1.72 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(ksnd_type,type,
% 3.34/1.72 ksnd: $i > $i ).
% 3.34/1.72
% 3.34/1.72 thf(ksndpairEq_type,type,
% 3.34/1.72 ksndpairEq: $o ).
% 3.34/1.72
% 3.34/1.72 thf(ksndpairEq,definition,
% 3.34/1.72 ( ksndpairEq
% 3.34/1.72 = ( ! [Xx: $i,Xy: $i] :
% 3.34/1.72 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 3.34/1.72 = Xy ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(kpairsurjEq_type,type,
% 3.34/1.72 kpairsurjEq: $o ).
% 3.34/1.72
% 3.34/1.72 thf(kpairsurjEq,definition,
% 3.34/1.72 ( kpairsurjEq
% 3.34/1.72 = ( ! [Xu: $i] :
% 3.34/1.72 ( ( iskpair @ Xu )
% 3.34/1.72 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 3.34/1.72 = Xu ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(cartprodsndin_type,type,
% 3.34/1.72 cartprodsndin: $o ).
% 3.34/1.72
% 3.34/1.72 thf(cartprodsndin,definition,
% 3.34/1.72 ( cartprodsndin
% 3.34/1.72 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.34/1.72 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.34/1.72 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(cartprodpairmemEL_type,type,
% 3.34/1.72 cartprodpairmemEL: $o ).
% 3.34/1.72
% 3.34/1.72 thf(cartprodpairmemEL,definition,
% 3.34/1.72 ( cartprodpairmemEL
% 3.34/1.72 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 3.34/1.72 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 3.34/1.72 => ( in @ Xx @ A ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(cartprodpairmemER_type,type,
% 3.34/1.72 cartprodpairmemER: $o ).
% 3.34/1.72
% 3.34/1.72 thf(cartprodpairmemER,definition,
% 3.34/1.72 ( cartprodpairmemER
% 3.34/1.72 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 3.34/1.72 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 3.34/1.72 => ( in @ Xy @ B ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(cartprodmempaircEq_type,type,
% 3.34/1.72 cartprodmempaircEq: $o ).
% 3.34/1.72
% 3.34/1.72 thf(cartprodmempaircEq,definition,
% 3.34/1.72 ( cartprodmempaircEq
% 3.34/1.72 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ! [Xy: $i] :
% 3.34/1.72 ( ( in @ Xy @ B )
% 3.34/1.72 => ( ( kpair @ Xx @ Xy )
% 3.34/1.72 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(cartprodfstpairEq_type,type,
% 3.34/1.72 cartprodfstpairEq: $o ).
% 3.34/1.72
% 3.34/1.72 thf(cartprodfstpairEq,definition,
% 3.34/1.72 ( cartprodfstpairEq
% 3.34/1.72 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ! [Xy: $i] :
% 3.34/1.72 ( ( in @ Xy @ B )
% 3.34/1.72 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 3.34/1.72 = Xx ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(cartprodsndpairEq_type,type,
% 3.34/1.72 cartprodsndpairEq: $o ).
% 3.34/1.72
% 3.34/1.72 thf(cartprodsndpairEq,definition,
% 3.34/1.72 ( cartprodsndpairEq
% 3.34/1.72 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ! [Xy: $i] :
% 3.34/1.72 ( ( in @ Xy @ B )
% 3.34/1.72 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 3.34/1.72 = Xy ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(cartprodpairsurjEq_type,type,
% 3.34/1.72 cartprodpairsurjEq: $o ).
% 3.34/1.72
% 3.34/1.72 thf(cartprodpairsurjEq,definition,
% 3.34/1.72 ( cartprodpairsurjEq
% 3.34/1.72 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.34/1.72 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.34/1.72 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 3.34/1.72 = Xu ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(breln_type,type,
% 3.34/1.72 breln: $i > $i > $i > $o ).
% 3.34/1.72
% 3.34/1.72 thf(dpsetconstr_type,type,
% 3.34/1.72 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 3.34/1.72
% 3.34/1.72 thf(dpsetconstrI_type,type,
% 3.34/1.72 dpsetconstrI: $o ).
% 3.34/1.72
% 3.34/1.72 thf(dpsetconstrI,definition,
% 3.34/1.72 ( dpsetconstrI
% 3.34/1.72 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ! [Xy: $i] :
% 3.34/1.72 ( ( in @ Xy @ B )
% 3.34/1.72 => ( ( Xphi @ Xx @ Xy )
% 3.34/1.72 => ( in @ ( kpair @ Xx @ Xy )
% 3.34/1.72 @ ( dpsetconstr @ A @ B
% 3.34/1.72 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(dpsetconstrSub_type,type,
% 3.34/1.72 dpsetconstrSub: $o ).
% 3.34/1.72
% 3.34/1.72 thf(dpsetconstrSub,definition,
% 3.34/1.72 ( dpsetconstrSub
% 3.34/1.72 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 3.34/1.72 ( subset
% 3.34/1.72 @ ( dpsetconstr @ A @ B
% 3.34/1.72 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 3.34/1.72 @ ( cartprod @ A @ B ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(setOfPairsIsBReln_type,type,
% 3.34/1.72 setOfPairsIsBReln: $o ).
% 3.34/1.72
% 3.34/1.72 thf(setOfPairsIsBReln,definition,
% 3.34/1.72 ( setOfPairsIsBReln
% 3.34/1.72 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 3.34/1.72 ( breln @ A @ B
% 3.34/1.72 @ ( dpsetconstr @ A @ B
% 3.34/1.72 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(dpsetconstrERa_type,type,
% 3.34/1.72 dpsetconstrERa: $o ).
% 3.34/1.72
% 3.34/1.72 thf(dpsetconstrERa,definition,
% 3.34/1.72 ( dpsetconstrERa
% 3.34/1.72 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ! [Xy: $i] :
% 3.34/1.72 ( ( in @ Xy @ B )
% 3.34/1.72 => ( ( in @ ( kpair @ Xx @ Xy )
% 3.34/1.72 @ ( dpsetconstr @ A @ B
% 3.34/1.72 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 3.34/1.72 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(dpsetconstrEL1_type,type,
% 3.34/1.72 dpsetconstrEL1: $o ).
% 3.34/1.72
% 3.34/1.72 thf(dpsetconstrEL1,definition,
% 3.34/1.72 ( dpsetconstrEL1
% 3.34/1.72 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 3.34/1.72 ( ( in @ ( kpair @ Xx @ Xy )
% 3.34/1.72 @ ( dpsetconstr @ A @ B
% 3.34/1.72 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 3.34/1.72 => ( in @ Xx @ A ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(dpsetconstrEL2_type,type,
% 3.34/1.72 dpsetconstrEL2: $o ).
% 3.34/1.72
% 3.34/1.72 thf(dpsetconstrEL2,definition,
% 3.34/1.72 ( dpsetconstrEL2
% 3.34/1.72 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 3.34/1.72 ( ( in @ ( kpair @ Xx @ Xy )
% 3.34/1.72 @ ( dpsetconstr @ A @ B
% 3.34/1.72 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 3.34/1.72 => ( in @ Xy @ B ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(dpsetconstrER_type,type,
% 3.34/1.72 dpsetconstrER: $o ).
% 3.34/1.72
% 3.34/1.72 thf(dpsetconstrER,definition,
% 3.34/1.72 ( dpsetconstrER
% 3.34/1.72 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 3.34/1.72 ( ( in @ ( kpair @ Xx @ Xy )
% 3.34/1.72 @ ( dpsetconstr @ A @ B
% 3.34/1.72 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 3.34/1.72 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(func_type,type,
% 3.34/1.72 func: $i > $i > $i > $o ).
% 3.34/1.72
% 3.34/1.72 thf(funcSet_type,type,
% 3.34/1.72 funcSet: $i > $i > $i ).
% 3.34/1.72
% 3.34/1.72 thf(funcImageSingleton_type,type,
% 3.34/1.72 funcImageSingleton: $o ).
% 3.34/1.72
% 3.34/1.72 thf(funcImageSingleton,definition,
% 3.34/1.72 ( funcImageSingleton
% 3.34/1.72 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.34/1.72 ( ( func @ A @ B @ Xf )
% 3.34/1.72 => ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ( singleton
% 3.34/1.72 @ ( dsetconstr @ B
% 3.34/1.72 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(apProp_type,type,
% 3.34/1.72 apProp: $o ).
% 3.34/1.72
% 3.34/1.72 thf(apProp,definition,
% 3.34/1.72 ( apProp
% 3.34/1.72 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.34/1.72 ( ( func @ A @ B @ Xf )
% 3.34/1.72 => ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ( in
% 3.34/1.72 @ ( setunion
% 3.34/1.72 @ ( dsetconstr @ B
% 3.34/1.72 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 3.34/1.72 @ B ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(ap_type,type,
% 3.34/1.72 ap: $i > $i > $i > $i > $i ).
% 3.34/1.72
% 3.34/1.72 thf(app_type,type,
% 3.34/1.72 app: $o ).
% 3.34/1.72
% 3.34/1.72 thf(app,definition,
% 3.34/1.72 ( app
% 3.34/1.72 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.34/1.72 ( ( func @ A @ B @ Xf )
% 3.34/1.72 => ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(infuncsetfunc_type,type,
% 3.34/1.72 infuncsetfunc: $o ).
% 3.34/1.72
% 3.34/1.72 thf(infuncsetfunc,definition,
% 3.34/1.72 ( infuncsetfunc
% 3.34/1.72 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.34/1.72 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.34/1.72 => ( func @ A @ B @ Xf ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(ap2p_type,type,
% 3.34/1.72 ap2p: $o ).
% 3.34/1.72
% 3.34/1.72 thf(ap2p,definition,
% 3.34/1.72 ( ap2p
% 3.34/1.72 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.34/1.72 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.34/1.72 => ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(funcinfuncset_type,type,
% 3.34/1.72 funcinfuncset: $o ).
% 3.34/1.72
% 3.34/1.72 thf(funcinfuncset,definition,
% 3.34/1.72 ( funcinfuncset
% 3.34/1.72 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.34/1.72 ( ( func @ A @ B @ Xf )
% 3.34/1.72 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(lamProp_type,type,
% 3.34/1.72 lamProp: $o ).
% 3.34/1.72
% 3.34/1.72 thf(lamProp,definition,
% 3.34/1.72 ( lamProp
% 3.34/1.72 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.34/1.72 ( ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.34/1.72 => ( func @ A @ B
% 3.34/1.72 @ ( dpsetconstr @ A @ B
% 3.34/1.72 @ ^ [Xx: $i,Xy: $i] :
% 3.34/1.72 ( ( Xf @ Xx )
% 3.34/1.72 = Xy ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(lam_type,type,
% 3.34/1.72 lam: $i > $i > ( $i > $i ) > $i ).
% 3.34/1.72
% 3.34/1.72 thf(lamp_type,type,
% 3.34/1.72 lamp: $o ).
% 3.34/1.72
% 3.34/1.72 thf(lamp,definition,
% 3.34/1.72 ( lamp
% 3.34/1.72 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.34/1.72 ( ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.34/1.72 => ( func @ A @ B
% 3.34/1.72 @ ( lam @ A @ B
% 3.34/1.72 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(lam2p_type,type,
% 3.34/1.72 lam2p: $o ).
% 3.34/1.72
% 3.34/1.72 thf(lam2p,definition,
% 3.34/1.72 ( lam2p
% 3.34/1.72 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.34/1.72 ( ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.34/1.72 => ( in
% 3.34/1.72 @ ( lam @ A @ B
% 3.34/1.72 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 3.34/1.72 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(brelnall1_type,type,
% 3.34/1.72 brelnall1: $o ).
% 3.34/1.72
% 3.34/1.72 thf(brelnall1,definition,
% 3.34/1.72 ( brelnall1
% 3.34/1.72 = ( ! [A: $i,B: $i,R: $i] :
% 3.34/1.72 ( ( breln @ A @ B @ R )
% 3.34/1.72 => ! [Xphi: $i > $o] :
% 3.34/1.72 ( ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ! [Xy: $i] :
% 3.34/1.72 ( ( in @ Xy @ B )
% 3.34/1.72 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.34/1.72 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 3.34/1.72 => ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ R )
% 3.34/1.72 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(brelnall2_type,type,
% 3.34/1.72 brelnall2: $o ).
% 3.34/1.72
% 3.34/1.72 thf(brelnall2,definition,
% 3.34/1.72 ( brelnall2
% 3.34/1.72 = ( ! [A: $i,B: $i,R: $i] :
% 3.34/1.72 ( ( breln @ A @ B @ R )
% 3.34/1.72 => ! [Xphi: $i > $o] :
% 3.34/1.72 ( ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ! [Xy: $i] :
% 3.34/1.72 ( ( in @ Xy @ B )
% 3.34/1.72 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.34/1.72 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 3.34/1.72 => ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ R )
% 3.34/1.72 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(ex1E2_type,type,
% 3.34/1.72 ex1E2: $o ).
% 3.34/1.72
% 3.34/1.72 thf(ex1E2,definition,
% 3.34/1.72 ( ex1E2
% 3.34/1.72 = ( ! [A: $i,Xphi: $i > $o] :
% 3.34/1.72 ( ( ex1 @ A
% 3.34/1.72 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.34/1.72 => ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ! [Xy: $i] :
% 3.34/1.72 ( ( in @ Xy @ A )
% 3.34/1.72 => ( ( Xphi @ Xx )
% 3.34/1.72 => ( ( Xphi @ Xy )
% 3.34/1.72 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(funcGraphProp1_type,type,
% 3.34/1.72 funcGraphProp1: $o ).
% 3.34/1.72
% 3.34/1.72 thf(funcGraphProp1,definition,
% 3.34/1.72 ( funcGraphProp1
% 3.34/1.72 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.34/1.72 ( ( func @ A @ B @ Xf )
% 3.34/1.72 => ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(funcGraphProp3_type,type,
% 3.34/1.72 funcGraphProp3: $o ).
% 3.34/1.72
% 3.34/1.72 thf(funcGraphProp3,definition,
% 3.34/1.72 ( funcGraphProp3
% 3.34/1.72 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.34/1.72 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.34/1.72 => ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(funcGraphProp2_type,type,
% 3.34/1.72 funcGraphProp2: $o ).
% 3.34/1.72
% 3.34/1.72 thf(funcGraphProp2,definition,
% 3.34/1.72 ( funcGraphProp2
% 3.34/1.72 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.34/1.72 ( ( func @ A @ B @ Xf )
% 3.34/1.72 => ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ! [Xy: $i] :
% 3.34/1.72 ( ( in @ Xy @ B )
% 3.34/1.72 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 3.34/1.72 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.34/1.72 = Xy ) ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(funcextLem_type,type,
% 3.34/1.72 funcextLem: $o ).
% 3.34/1.72
% 3.34/1.72 thf(funcextLem,definition,
% 3.34/1.72 ( funcextLem
% 3.34/1.72 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.34/1.72 ( ( func @ A @ B @ Xf )
% 3.34/1.72 => ! [Xg: $i] :
% 3.34/1.72 ( ( func @ A @ B @ Xg )
% 3.34/1.72 => ( ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.34/1.72 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 3.34/1.72 => ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ! [Xy: $i] :
% 3.34/1.72 ( ( in @ Xy @ B )
% 3.34/1.72 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 3.34/1.72 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(funcGraphProp4_type,type,
% 3.34/1.72 funcGraphProp4: $o ).
% 3.34/1.72
% 3.34/1.72 thf(funcGraphProp4,definition,
% 3.34/1.72 ( funcGraphProp4
% 3.34/1.72 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.34/1.72 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.34/1.72 => ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ! [Xy: $i] :
% 3.34/1.72 ( ( in @ Xy @ B )
% 3.34/1.72 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 3.34/1.72 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.34/1.72 = Xy ) ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(subbreln_type,type,
% 3.34/1.72 subbreln: $o ).
% 3.34/1.72
% 3.34/1.72 thf(subbreln,definition,
% 3.34/1.72 ( subbreln
% 3.34/1.72 = ( ! [A: $i,B: $i,R: $i] :
% 3.34/1.72 ( ( breln @ A @ B @ R )
% 3.34/1.72 => ! [S: $i] :
% 3.34/1.72 ( ( breln @ A @ B @ S )
% 3.34/1.72 => ( ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ! [Xy: $i] :
% 3.34/1.72 ( ( in @ Xy @ B )
% 3.34/1.72 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.34/1.72 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 3.34/1.72 => ( subset @ R @ S ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(eqbreln_type,type,
% 3.34/1.72 eqbreln: $o ).
% 3.34/1.72
% 3.34/1.72 thf(eqbreln,definition,
% 3.34/1.72 ( eqbreln
% 3.34/1.72 = ( ! [A: $i,B: $i,R: $i] :
% 3.34/1.72 ( ( breln @ A @ B @ R )
% 3.34/1.72 => ! [S: $i] :
% 3.34/1.72 ( ( breln @ A @ B @ S )
% 3.34/1.72 => ( ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ! [Xy: $i] :
% 3.34/1.72 ( ( in @ Xy @ B )
% 3.34/1.72 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.34/1.72 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 3.34/1.72 => ( ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ! [Xy: $i] :
% 3.34/1.72 ( ( in @ Xy @ B )
% 3.34/1.72 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 3.34/1.72 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 3.34/1.72 => ( R = S ) ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(funcext_type,type,
% 3.34/1.72 funcext: $o ).
% 3.34/1.72
% 3.34/1.72 thf(funcext,definition,
% 3.34/1.72 ( funcext
% 3.34/1.72 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.34/1.72 ( ( func @ A @ B @ Xf )
% 3.34/1.72 => ! [Xg: $i] :
% 3.34/1.72 ( ( func @ A @ B @ Xg )
% 3.34/1.72 => ( ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.34/1.72 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 3.34/1.72 => ( Xf = Xg ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(funcext2_type,type,
% 3.34/1.72 funcext2: $o ).
% 3.34/1.72
% 3.34/1.72 thf(funcext2,definition,
% 3.34/1.72 ( funcext2
% 3.34/1.72 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.34/1.72 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.34/1.72 => ! [Xg: $i] :
% 3.34/1.72 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 3.34/1.72 => ( ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.34/1.72 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 3.34/1.72 => ( Xf = Xg ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(ap2apEq1_type,type,
% 3.34/1.72 ap2apEq1: $o ).
% 3.34/1.72
% 3.34/1.72 thf(ap2apEq1,definition,
% 3.34/1.72 ( ap2apEq1
% 3.34/1.72 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.34/1.72 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.34/1.72 => ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.34/1.72 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(ap2apEq2_type,type,
% 3.34/1.72 ap2apEq2: $o ).
% 3.34/1.72
% 3.34/1.72 thf(ap2apEq2,definition,
% 3.34/1.72 ( ap2apEq2
% 3.34/1.72 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.34/1.72 ( ( func @ A @ B @ Xf )
% 3.34/1.72 => ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.34/1.72 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(beta1_type,type,
% 3.34/1.72 beta1: $o ).
% 3.34/1.72
% 3.34/1.72 thf(beta1,definition,
% 3.34/1.72 ( beta1
% 3.34/1.72 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.34/1.72 ( ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.34/1.72 => ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ( ( ap @ A @ B
% 3.34/1.72 @ ( lam @ A @ B
% 3.34/1.72 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 3.34/1.72 @ Xx )
% 3.34/1.72 = ( Xf @ Xx ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(eta1_type,type,
% 3.34/1.72 eta1: $o ).
% 3.34/1.72
% 3.34/1.72 thf(eta1,definition,
% 3.34/1.72 ( eta1
% 3.34/1.72 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.34/1.72 ( ( func @ A @ B @ Xf )
% 3.34/1.72 => ( ( lam @ A @ B
% 3.34/1.72 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 3.34/1.72 = Xf ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(lam2lamEq_type,type,
% 3.34/1.72 lam2lamEq: $o ).
% 3.34/1.72
% 3.34/1.72 thf(lam2lamEq,definition,
% 3.34/1.72 ( lam2lamEq
% 3.34/1.72 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.34/1.72 ( ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.34/1.72 => ( ( lam @ A @ B
% 3.34/1.72 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 3.34/1.72 = ( lam @ A @ B
% 3.34/1.72 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(beta2_type,type,
% 3.34/1.72 beta2: $o ).
% 3.34/1.72
% 3.34/1.72 thf(beta2,definition,
% 3.34/1.72 ( beta2
% 3.34/1.72 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.34/1.72 ( ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.34/1.72 => ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ( ( ap @ A @ B
% 3.34/1.72 @ ( lam @ A @ B
% 3.34/1.72 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 3.34/1.72 @ Xx )
% 3.34/1.72 = ( Xf @ Xx ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(eta2_type,type,
% 3.34/1.72 eta2: $o ).
% 3.34/1.72
% 3.34/1.72 thf(eta2,definition,
% 3.34/1.72 ( eta2
% 3.34/1.72 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.34/1.72 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.34/1.72 => ( ( lam @ A @ B
% 3.34/1.72 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 3.34/1.72 = Xf ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(iffalseProp1_type,type,
% 3.34/1.72 iffalseProp1: $o ).
% 3.34/1.72
% 3.34/1.72 thf(iffalseProp1,definition,
% 3.34/1.72 ( iffalseProp1
% 3.34/1.72 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ! [Xy: $i] :
% 3.34/1.72 ( ( in @ Xy @ A )
% 3.34/1.72 => ( ~ Xphi
% 3.34/1.72 => ( in @ Xy
% 3.34/1.72 @ ( dsetconstr @ A
% 3.34/1.72 @ ^ [Xz: $i] :
% 3.34/1.72 ( ( Xphi
% 3.34/1.72 & ( Xz = Xx ) )
% 3.34/1.72 | ( ~ Xphi
% 3.34/1.72 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(iffalseProp2_type,type,
% 3.34/1.72 iffalseProp2: $o ).
% 3.34/1.72
% 3.34/1.72 thf(iffalseProp2,definition,
% 3.34/1.72 ( iffalseProp2
% 3.34/1.72 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ! [Xy: $i] :
% 3.34/1.72 ( ( in @ Xy @ A )
% 3.34/1.72 => ( ~ Xphi
% 3.34/1.72 => ( ( dsetconstr @ A
% 3.34/1.72 @ ^ [Xz: $i] :
% 3.34/1.72 ( ( Xphi
% 3.34/1.72 & ( Xz = Xx ) )
% 3.34/1.72 | ( ~ Xphi
% 3.34/1.72 & ( Xz = Xy ) ) ) )
% 3.34/1.72 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(iftrueProp1_type,type,
% 3.34/1.72 iftrueProp1: $o ).
% 3.34/1.72
% 3.34/1.72 thf(iftrueProp1,definition,
% 3.34/1.72 ( iftrueProp1
% 3.34/1.72 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ! [Xy: $i] :
% 3.34/1.72 ( ( in @ Xy @ A )
% 3.34/1.72 => ( Xphi
% 3.34/1.72 => ( in @ Xx
% 3.34/1.72 @ ( dsetconstr @ A
% 3.34/1.72 @ ^ [Xz: $i] :
% 3.34/1.72 ( ( Xphi
% 3.34/1.72 & ( Xz = Xx ) )
% 3.34/1.72 | ( ~ Xphi
% 3.34/1.72 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(iftrueProp2_type,type,
% 3.34/1.72 iftrueProp2: $o ).
% 3.34/1.72
% 3.34/1.72 thf(iftrueProp2,definition,
% 3.34/1.72 ( iftrueProp2
% 3.34/1.72 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ! [Xy: $i] :
% 3.34/1.72 ( ( in @ Xy @ A )
% 3.34/1.72 => ( Xphi
% 3.34/1.72 => ( ( dsetconstr @ A
% 3.34/1.72 @ ^ [Xz: $i] :
% 3.34/1.72 ( ( Xphi
% 3.34/1.72 & ( Xz = Xx ) )
% 3.34/1.72 | ( ~ Xphi
% 3.34/1.72 & ( Xz = Xy ) ) ) )
% 3.34/1.72 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(ifSingleton_type,type,
% 3.34/1.72 ifSingleton: $o ).
% 3.34/1.72
% 3.34/1.72 thf(ifSingleton,definition,
% 3.34/1.72 ( ifSingleton
% 3.34/1.72 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ! [Xy: $i] :
% 3.34/1.72 ( ( in @ Xy @ A )
% 3.34/1.72 => ( singleton
% 3.34/1.72 @ ( dsetconstr @ A
% 3.34/1.72 @ ^ [Xz: $i] :
% 3.34/1.72 ( ( Xphi
% 3.34/1.72 & ( Xz = Xx ) )
% 3.34/1.72 | ( ~ Xphi
% 3.34/1.72 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(if_type,type,
% 3.34/1.72 if: $i > $o > $i > $i > $i ).
% 3.34/1.72
% 3.34/1.72 thf(ifp_type,type,
% 3.34/1.72 ifp: $o ).
% 3.34/1.72
% 3.34/1.72 thf(ifp,definition,
% 3.34/1.72 ( ifp
% 3.34/1.72 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ! [Xy: $i] :
% 3.34/1.72 ( ( in @ Xy @ A )
% 3.34/1.72 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(theeq_type,type,
% 3.34/1.72 theeq: $o ).
% 3.34/1.72
% 3.34/1.72 thf(theeq,definition,
% 3.34/1.72 ( theeq
% 3.34/1.72 = ( ! [X: $i] :
% 3.34/1.72 ( ( singleton @ X )
% 3.34/1.72 => ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ X )
% 3.34/1.72 => ( ( setunion @ X )
% 3.34/1.72 = Xx ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(iftrue_type,type,
% 3.34/1.72 iftrue: $o ).
% 3.34/1.72
% 3.34/1.72 thf(iftrue,definition,
% 3.34/1.72 ( iftrue
% 3.34/1.72 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ! [Xy: $i] :
% 3.34/1.72 ( ( in @ Xy @ A )
% 3.34/1.72 => ( Xphi
% 3.34/1.72 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 3.34/1.72 = Xx ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(iffalse_type,type,
% 3.34/1.72 iffalse: $o ).
% 3.34/1.72
% 3.34/1.72 thf(iffalse,definition,
% 3.34/1.72 ( iffalse
% 3.34/1.72 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ! [Xy: $i] :
% 3.34/1.72 ( ( in @ Xy @ A )
% 3.34/1.72 => ( ~ Xphi
% 3.34/1.72 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 3.34/1.72 = Xy ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(iftrueorfalse_type,type,
% 3.34/1.72 iftrueorfalse: $o ).
% 3.34/1.72
% 3.34/1.72 thf(iftrueorfalse,definition,
% 3.34/1.72 ( iftrueorfalse
% 3.34/1.72 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ! [Xy: $i] :
% 3.34/1.72 ( ( in @ Xy @ A )
% 3.34/1.72 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(binintersectT_lem_type,type,
% 3.34/1.72 binintersectT_lem: $o ).
% 3.34/1.72
% 3.34/1.72 thf(binintersectT_lem,definition,
% 3.34/1.72 ( binintersectT_lem
% 3.34/1.72 = ( ! [A: $i,X: $i] :
% 3.34/1.72 ( ( in @ X @ ( powerset @ A ) )
% 3.34/1.72 => ! [Y: $i] :
% 3.34/1.72 ( ( in @ Y @ ( powerset @ A ) )
% 3.34/1.72 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(binunionT_lem_type,type,
% 3.34/1.72 binunionT_lem: $o ).
% 3.34/1.72
% 3.34/1.72 thf(binunionT_lem,definition,
% 3.34/1.72 ( binunionT_lem
% 3.34/1.72 = ( ! [A: $i,X: $i] :
% 3.34/1.72 ( ( in @ X @ ( powerset @ A ) )
% 3.34/1.72 => ! [Y: $i] :
% 3.34/1.72 ( ( in @ Y @ ( powerset @ A ) )
% 3.34/1.72 => ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(powersetT_lem_type,type,
% 3.34/1.72 powersetT_lem: $o ).
% 3.34/1.72
% 3.34/1.72 thf(powersetT_lem,definition,
% 3.34/1.72 ( powersetT_lem
% 3.34/1.72 = ( ! [A: $i,X: $i] :
% 3.34/1.72 ( ( in @ X @ ( powerset @ A ) )
% 3.34/1.72 => ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(setminusT_lem_type,type,
% 3.34/1.72 setminusT_lem: $o ).
% 3.34/1.72
% 3.34/1.72 thf(setminusT_lem,definition,
% 3.34/1.72 ( setminusT_lem
% 3.34/1.72 = ( ! [A: $i,X: $i] :
% 3.34/1.72 ( ( in @ X @ ( powerset @ A ) )
% 3.34/1.72 => ! [Y: $i] :
% 3.34/1.72 ( ( in @ Y @ ( powerset @ A ) )
% 3.34/1.72 => ( in @ ( setminus @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(complementT_lem_type,type,
% 3.34/1.72 complementT_lem: $o ).
% 3.34/1.72
% 3.34/1.72 thf(complementT_lem,definition,
% 3.34/1.72 ( complementT_lem
% 3.34/1.72 = ( ! [A: $i,X: $i] :
% 3.34/1.72 ( ( in @ X @ ( powerset @ A ) )
% 3.34/1.72 => ( in @ ( setminus @ A @ X ) @ ( powerset @ A ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(setextT_type,type,
% 3.34/1.72 setextT: $o ).
% 3.34/1.72
% 3.34/1.72 thf(setextT,definition,
% 3.34/1.72 ( setextT
% 3.34/1.72 = ( ! [A: $i,X: $i] :
% 3.34/1.72 ( ( in @ X @ ( powerset @ A ) )
% 3.34/1.72 => ! [Y: $i] :
% 3.34/1.72 ( ( in @ Y @ ( powerset @ A ) )
% 3.34/1.72 => ( ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ( ( in @ Xx @ X )
% 3.34/1.72 => ( in @ Xx @ Y ) ) )
% 3.34/1.72 => ( ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ( ( in @ Xx @ Y )
% 3.34/1.72 => ( in @ Xx @ X ) ) )
% 3.34/1.72 => ( X = Y ) ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(subsetTI_type,type,
% 3.34/1.72 subsetTI: $o ).
% 3.34/1.72
% 3.34/1.72 thf(subsetTI,definition,
% 3.34/1.72 ( subsetTI
% 3.34/1.72 = ( ! [A: $i,X: $i] :
% 3.34/1.72 ( ( in @ X @ ( powerset @ A ) )
% 3.34/1.72 => ! [Y: $i] :
% 3.34/1.72 ( ( in @ Y @ ( powerset @ A ) )
% 3.34/1.72 => ( ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ( ( in @ Xx @ X )
% 3.34/1.72 => ( in @ Xx @ Y ) ) )
% 3.34/1.72 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(powersetTI1_type,type,
% 3.34/1.72 powersetTI1: $o ).
% 3.34/1.72
% 3.34/1.72 thf(powersetTI1,definition,
% 3.34/1.72 ( powersetTI1
% 3.34/1.72 = ( ! [A: $i,X: $i] :
% 3.34/1.72 ( ( in @ X @ ( powerset @ A ) )
% 3.34/1.72 => ! [Y: $i] :
% 3.34/1.72 ( ( in @ Y @ ( powerset @ A ) )
% 3.34/1.72 => ( ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ( ( in @ Xx @ X )
% 3.34/1.72 => ( in @ Xx @ Y ) ) )
% 3.34/1.72 => ( in @ X @ ( powerset @ Y ) ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(powersetTE1_type,type,
% 3.34/1.72 powersetTE1: $o ).
% 3.34/1.72
% 3.34/1.72 thf(powersetTE1,definition,
% 3.34/1.72 ( powersetTE1
% 3.34/1.72 = ( ! [A: $i,X: $i] :
% 3.34/1.72 ( ( in @ X @ ( powerset @ A ) )
% 3.34/1.72 => ! [Y: $i] :
% 3.34/1.72 ( ( in @ Y @ ( powerset @ A ) )
% 3.34/1.72 => ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ( ( in @ X @ ( powerset @ Y ) )
% 3.34/1.72 => ( ( in @ Xx @ X )
% 3.34/1.72 => ( in @ Xx @ Y ) ) ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(complementTI1_type,type,
% 3.34/1.72 complementTI1: $o ).
% 3.34/1.72
% 3.34/1.72 thf(complementTI1,definition,
% 3.34/1.72 ( complementTI1
% 3.34/1.72 = ( ! [A: $i,X: $i] :
% 3.34/1.72 ( ( in @ X @ ( powerset @ A ) )
% 3.34/1.72 => ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ( ( in @ Xx @ X )
% 3.34/1.72 => ~ ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(complementTE1_type,type,
% 3.34/1.72 complementTE1: $o ).
% 3.34/1.72
% 3.34/1.72 thf(complementTE1,definition,
% 3.34/1.72 ( complementTE1
% 3.34/1.72 = ( ! [A: $i,X: $i] :
% 3.34/1.72 ( ( in @ X @ ( powerset @ A ) )
% 3.34/1.72 => ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ( ~ ( in @ Xx @ ( setminus @ A @ X ) )
% 3.34/1.72 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(binintersectTELcontra_type,type,
% 3.34/1.72 binintersectTELcontra: $o ).
% 3.34/1.72
% 3.34/1.72 thf(binintersectTELcontra,definition,
% 3.34/1.72 ( binintersectTELcontra
% 3.34/1.72 = ( ! [A: $i,X: $i] :
% 3.34/1.72 ( ( in @ X @ ( powerset @ A ) )
% 3.34/1.72 => ! [Y: $i] :
% 3.34/1.72 ( ( in @ Y @ ( powerset @ A ) )
% 3.34/1.72 => ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ( ~ ( in @ Xx @ X )
% 3.34/1.72 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(binintersectTERcontra_type,type,
% 3.34/1.72 binintersectTERcontra: $o ).
% 3.34/1.72
% 3.34/1.72 thf(binintersectTERcontra,definition,
% 3.34/1.72 ( binintersectTERcontra
% 3.34/1.72 = ( ! [A: $i,X: $i] :
% 3.34/1.72 ( ( in @ X @ ( powerset @ A ) )
% 3.34/1.72 => ! [Y: $i] :
% 3.34/1.72 ( ( in @ Y @ ( powerset @ A ) )
% 3.34/1.72 => ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ( ~ ( in @ Xx @ Y )
% 3.34/1.72 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(contrasubsetT_type,type,
% 3.34/1.72 contrasubsetT: $o ).
% 3.34/1.72
% 3.34/1.72 thf(contrasubsetT,definition,
% 3.34/1.72 ( contrasubsetT
% 3.34/1.72 = ( ! [A: $i,X: $i] :
% 3.34/1.72 ( ( in @ X @ ( powerset @ A ) )
% 3.34/1.72 => ! [Y: $i] :
% 3.34/1.72 ( ( in @ Y @ ( powerset @ A ) )
% 3.34/1.72 => ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 3.34/1.72 => ( ( in @ Xx @ Y )
% 3.34/1.72 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(contrasubsetT1_type,type,
% 3.34/1.72 contrasubsetT1: $o ).
% 3.34/1.72
% 3.34/1.72 thf(contrasubsetT1,definition,
% 3.34/1.72 ( contrasubsetT1
% 3.34/1.72 = ( ! [A: $i,X: $i] :
% 3.34/1.72 ( ( in @ X @ ( powerset @ A ) )
% 3.34/1.72 => ! [Y: $i] :
% 3.34/1.72 ( ( in @ Y @ ( powerset @ A ) )
% 3.34/1.72 => ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ( ( subset @ X @ Y )
% 3.34/1.72 => ( ~ ( in @ Xx @ Y )
% 3.34/1.72 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(contrasubsetT2_type,type,
% 3.34/1.72 contrasubsetT2: $o ).
% 3.34/1.72
% 3.34/1.72 thf(contrasubsetT2,definition,
% 3.34/1.72 ( contrasubsetT2
% 3.34/1.72 = ( ! [A: $i,X: $i] :
% 3.34/1.72 ( ( in @ X @ ( powerset @ A ) )
% 3.34/1.72 => ! [Y: $i] :
% 3.34/1.72 ( ( in @ Y @ ( powerset @ A ) )
% 3.34/1.72 => ( ( subset @ X @ Y )
% 3.34/1.72 => ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(contrasubsetT3_type,type,
% 3.34/1.72 contrasubsetT3: $o ).
% 3.34/1.72
% 3.34/1.72 thf(contrasubsetT3,definition,
% 3.34/1.72 ( contrasubsetT3
% 3.34/1.72 = ( ! [A: $i,X: $i] :
% 3.34/1.72 ( ( in @ X @ ( powerset @ A ) )
% 3.34/1.72 => ! [Y: $i] :
% 3.34/1.72 ( ( in @ Y @ ( powerset @ A ) )
% 3.34/1.72 => ( ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) )
% 3.34/1.72 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(doubleComplementI1_type,type,
% 3.34/1.72 doubleComplementI1: $o ).
% 3.34/1.72
% 3.34/1.72 thf(doubleComplementI1,definition,
% 3.34/1.72 ( doubleComplementI1
% 3.34/1.72 = ( ! [A: $i,X: $i] :
% 3.34/1.72 ( ( in @ X @ ( powerset @ A ) )
% 3.34/1.72 => ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ( ( in @ Xx @ X )
% 3.34/1.72 => ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(doubleComplementE1_type,type,
% 3.34/1.72 doubleComplementE1: $o ).
% 3.34/1.72
% 3.34/1.72 thf(doubleComplementE1,definition,
% 3.34/1.72 ( doubleComplementE1
% 3.34/1.72 = ( ! [A: $i,X: $i] :
% 3.34/1.72 ( ( in @ X @ ( powerset @ A ) )
% 3.34/1.72 => ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ( ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) )
% 3.34/1.72 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(doubleComplementSub1_type,type,
% 3.34/1.72 doubleComplementSub1: $o ).
% 3.34/1.72
% 3.34/1.72 thf(doubleComplementSub1,definition,
% 3.34/1.72 ( doubleComplementSub1
% 3.34/1.72 = ( ! [A: $i,X: $i] :
% 3.34/1.72 ( ( in @ X @ ( powerset @ A ) )
% 3.34/1.72 => ( subset @ X @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(doubleComplementSub2_type,type,
% 3.34/1.72 doubleComplementSub2: $o ).
% 3.34/1.72
% 3.34/1.72 thf(doubleComplementSub2,definition,
% 3.34/1.72 ( doubleComplementSub2
% 3.34/1.72 = ( ! [A: $i,X: $i] :
% 3.34/1.72 ( ( in @ X @ ( powerset @ A ) )
% 3.34/1.72 => ( subset @ ( setminus @ A @ ( setminus @ A @ X ) ) @ X ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(doubleComplementEq_type,type,
% 3.34/1.72 doubleComplementEq: $o ).
% 3.34/1.72
% 3.34/1.72 thf(doubleComplementEq,definition,
% 3.34/1.72 ( doubleComplementEq
% 3.34/1.72 = ( ! [A: $i,X: $i] :
% 3.34/1.72 ( ( in @ X @ ( powerset @ A ) )
% 3.34/1.72 => ( X
% 3.34/1.72 = ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(complementTnotintersectT_type,type,
% 3.34/1.72 complementTnotintersectT: $o ).
% 3.34/1.72
% 3.34/1.72 thf(complementTnotintersectT,definition,
% 3.34/1.72 ( complementTnotintersectT
% 3.34/1.72 = ( ! [A: $i,X: $i] :
% 3.34/1.72 ( ( in @ X @ ( powerset @ A ) )
% 3.34/1.72 => ! [Y: $i] :
% 3.34/1.72 ( ( in @ Y @ ( powerset @ A ) )
% 3.34/1.72 => ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 3.34/1.72 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(complementImpComplementIntersect_type,type,
% 3.34/1.72 complementImpComplementIntersect: $o ).
% 3.34/1.72
% 3.34/1.72 thf(complementImpComplementIntersect,definition,
% 3.34/1.72 ( complementImpComplementIntersect
% 3.34/1.72 = ( ! [A: $i,X: $i] :
% 3.34/1.72 ( ( in @ X @ ( powerset @ A ) )
% 3.34/1.72 => ! [Y: $i] :
% 3.34/1.72 ( ( in @ Y @ ( powerset @ A ) )
% 3.34/1.72 => ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 3.34/1.72 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(complementSubsetComplementIntersect_type,type,
% 3.34/1.72 complementSubsetComplementIntersect: $o ).
% 3.34/1.72
% 3.34/1.72 thf(complementSubsetComplementIntersect,definition,
% 3.34/1.72 ( complementSubsetComplementIntersect
% 3.34/1.72 = ( ! [A: $i,X: $i] :
% 3.34/1.72 ( ( in @ X @ ( powerset @ A ) )
% 3.34/1.72 => ! [Y: $i] :
% 3.34/1.72 ( ( in @ Y @ ( powerset @ A ) )
% 3.34/1.72 => ( subset @ ( setminus @ A @ X ) @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(complementInPowersetComplementIntersect_type,type,
% 3.34/1.72 complementInPowersetComplementIntersect: $o ).
% 3.34/1.72
% 3.34/1.72 thf(complementInPowersetComplementIntersect,definition,
% 3.34/1.72 ( complementInPowersetComplementIntersect
% 3.34/1.72 = ( ! [A: $i,X: $i] :
% 3.34/1.72 ( ( in @ X @ ( powerset @ A ) )
% 3.34/1.72 => ! [Y: $i] :
% 3.34/1.72 ( ( in @ Y @ ( powerset @ A ) )
% 3.34/1.72 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(contraSubsetComplement_type,type,
% 3.34/1.72 contraSubsetComplement: $o ).
% 3.34/1.72
% 3.34/1.72 thf(contraSubsetComplement,definition,
% 3.34/1.72 ( contraSubsetComplement
% 3.34/1.72 = ( ! [A: $i,X: $i] :
% 3.34/1.72 ( ( in @ X @ ( powerset @ A ) )
% 3.34/1.72 => ! [Y: $i] :
% 3.34/1.72 ( ( in @ Y @ ( powerset @ A ) )
% 3.34/1.72 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 3.34/1.72 => ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ( ( in @ Xx @ Y )
% 3.34/1.72 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(complementTcontraSubset_type,type,
% 3.34/1.72 complementTcontraSubset: $o ).
% 3.34/1.72
% 3.34/1.72 thf(complementTcontraSubset,definition,
% 3.34/1.72 ( complementTcontraSubset
% 3.34/1.72 = ( ! [A: $i,X: $i] :
% 3.34/1.72 ( ( in @ X @ ( powerset @ A ) )
% 3.34/1.72 => ! [Y: $i] :
% 3.34/1.72 ( ( in @ Y @ ( powerset @ A ) )
% 3.34/1.72 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 3.34/1.72 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(binunionTILcontra_type,type,
% 3.34/1.72 binunionTILcontra: $o ).
% 3.34/1.72
% 3.34/1.72 thf(binunionTILcontra,definition,
% 3.34/1.72 ( binunionTILcontra
% 3.34/1.72 = ( ! [A: $i,X: $i] :
% 3.34/1.72 ( ( in @ X @ ( powerset @ A ) )
% 3.34/1.72 => ! [Y: $i] :
% 3.34/1.72 ( ( in @ Y @ ( powerset @ A ) )
% 3.34/1.72 => ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 3.34/1.72 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(binunionTIRcontra_type,type,
% 3.34/1.72 binunionTIRcontra: $o ).
% 3.34/1.72
% 3.34/1.72 thf(binunionTIRcontra,definition,
% 3.34/1.72 ( binunionTIRcontra
% 3.34/1.72 = ( ! [A: $i,X: $i] :
% 3.34/1.72 ( ( in @ X @ ( powerset @ A ) )
% 3.34/1.72 => ! [Y: $i] :
% 3.34/1.72 ( ( in @ Y @ ( powerset @ A ) )
% 3.34/1.72 => ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 3.34/1.72 => ~ ( in @ Xx @ Y ) ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(inIntersectImpInUnion_type,type,
% 3.34/1.72 inIntersectImpInUnion: $o ).
% 3.34/1.72
% 3.34/1.72 thf(inIntersectImpInUnion,definition,
% 3.34/1.72 ( inIntersectImpInUnion
% 3.34/1.72 = ( ! [A: $i,X: $i] :
% 3.34/1.72 ( ( in @ X @ ( powerset @ A ) )
% 3.34/1.72 => ! [Y: $i] :
% 3.34/1.72 ( ( in @ Y @ ( powerset @ A ) )
% 3.34/1.72 => ! [Z: $i] :
% 3.34/1.72 ( ( in @ Z @ ( powerset @ A ) )
% 3.34/1.72 => ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 3.34/1.72 => ( in @ Xx @ ( binunion @ X @ Z ) ) ) ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(inIntersectImpInUnion2_type,type,
% 3.34/1.72 inIntersectImpInUnion2: $o ).
% 3.34/1.72
% 3.34/1.72 thf(inIntersectImpInUnion2,definition,
% 3.34/1.72 ( inIntersectImpInUnion2
% 3.34/1.72 = ( ! [A: $i,X: $i] :
% 3.34/1.72 ( ( in @ X @ ( powerset @ A ) )
% 3.34/1.72 => ! [Y: $i] :
% 3.34/1.72 ( ( in @ Y @ ( powerset @ A ) )
% 3.34/1.72 => ! [Z: $i] :
% 3.34/1.72 ( ( in @ Z @ ( powerset @ A ) )
% 3.34/1.72 => ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 3.34/1.72 => ( in @ Xx @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(inIntersectImpInIntersectUnions_type,type,
% 3.34/1.72 inIntersectImpInIntersectUnions: $o ).
% 3.34/1.72
% 3.34/1.72 thf(inIntersectImpInIntersectUnions,definition,
% 3.34/1.72 ( inIntersectImpInIntersectUnions
% 3.34/1.72 = ( ! [A: $i,X: $i] :
% 3.34/1.72 ( ( in @ X @ ( powerset @ A ) )
% 3.34/1.72 => ! [Y: $i] :
% 3.34/1.72 ( ( in @ Y @ ( powerset @ A ) )
% 3.34/1.72 => ! [Z: $i] :
% 3.34/1.72 ( ( in @ Z @ ( powerset @ A ) )
% 3.34/1.72 => ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 3.34/1.72 => ( in @ Xx @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(intersectInPowersetIntersectUnions_type,type,
% 3.34/1.72 intersectInPowersetIntersectUnions: $o ).
% 3.34/1.72
% 3.34/1.72 thf(intersectInPowersetIntersectUnions,definition,
% 3.34/1.72 ( intersectInPowersetIntersectUnions
% 3.34/1.72 = ( ! [A: $i,X: $i] :
% 3.34/1.72 ( ( in @ X @ ( powerset @ A ) )
% 3.34/1.72 => ! [Y: $i] :
% 3.34/1.72 ( ( in @ Y @ ( powerset @ A ) )
% 3.34/1.72 => ! [Z: $i] :
% 3.34/1.72 ( ( in @ Z @ ( powerset @ A ) )
% 3.34/1.72 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(inComplementUnionImpNotIn1_type,type,
% 3.34/1.72 inComplementUnionImpNotIn1: $o ).
% 3.34/1.72
% 3.34/1.72 thf(inComplementUnionImpNotIn1,definition,
% 3.34/1.72 ( inComplementUnionImpNotIn1
% 3.34/1.72 = ( ! [A: $i,X: $i] :
% 3.34/1.72 ( ( in @ X @ ( powerset @ A ) )
% 3.34/1.72 => ! [Y: $i] :
% 3.34/1.72 ( ( in @ Y @ ( powerset @ A ) )
% 3.34/1.72 => ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 3.34/1.72 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(inComplementUnionImpInComplement1_type,type,
% 3.34/1.72 inComplementUnionImpInComplement1: $o ).
% 3.34/1.72
% 3.34/1.72 thf(inComplementUnionImpInComplement1,definition,
% 3.34/1.72 ( inComplementUnionImpInComplement1
% 3.34/1.72 = ( ! [A: $i,X: $i] :
% 3.34/1.72 ( ( in @ X @ ( powerset @ A ) )
% 3.34/1.72 => ! [Y: $i] :
% 3.34/1.72 ( ( in @ Y @ ( powerset @ A ) )
% 3.34/1.72 => ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 3.34/1.72 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(binunionTE_type,type,
% 3.34/1.72 binunionTE: $o ).
% 3.34/1.72
% 3.34/1.72 thf(binunionTE,definition,
% 3.34/1.72 ( binunionTE
% 3.34/1.72 = ( ! [A: $i,X: $i] :
% 3.34/1.72 ( ( in @ X @ ( powerset @ A ) )
% 3.34/1.72 => ! [Y: $i] :
% 3.34/1.72 ( ( in @ Y @ ( powerset @ A ) )
% 3.34/1.72 => ! [Xphi: $o,Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ( ( in @ Xx @ ( binunion @ X @ Y ) )
% 3.34/1.72 => ( ( ( in @ Xx @ X )
% 3.34/1.72 => Xphi )
% 3.34/1.72 => ( ( ( in @ Xx @ Y )
% 3.34/1.72 => Xphi )
% 3.34/1.72 => Xphi ) ) ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(binunionTEcontra_type,type,
% 3.34/1.72 binunionTEcontra: $o ).
% 3.34/1.72
% 3.34/1.72 thf(binunionTEcontra,definition,
% 3.34/1.72 ( binunionTEcontra
% 3.34/1.72 = ( ! [A: $i,X: $i] :
% 3.34/1.72 ( ( in @ X @ ( powerset @ A ) )
% 3.34/1.72 => ! [Y: $i] :
% 3.34/1.72 ( ( in @ Y @ ( powerset @ A ) )
% 3.34/1.72 => ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ( ~ ( in @ Xx @ X )
% 3.34/1.72 => ( ~ ( in @ Xx @ Y )
% 3.34/1.72 => ~ ( in @ Xx @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(demorgan2a1_type,type,
% 3.34/1.72 demorgan2a1: $o ).
% 3.34/1.72
% 3.34/1.72 thf(demorgan2a1,definition,
% 3.34/1.72 ( demorgan2a1
% 3.34/1.72 = ( ! [A: $i,X: $i] :
% 3.34/1.72 ( ( in @ X @ ( powerset @ A ) )
% 3.34/1.72 => ! [Y: $i] :
% 3.34/1.72 ( ( in @ Y @ ( powerset @ A ) )
% 3.34/1.72 => ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 3.34/1.72 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(complementUnionInPowersetComplement_type,type,
% 3.34/1.72 complementUnionInPowersetComplement: $o ).
% 3.34/1.72
% 3.34/1.72 thf(complementUnionInPowersetComplement,definition,
% 3.34/1.72 ( complementUnionInPowersetComplement
% 3.34/1.72 = ( ! [A: $i,X: $i] :
% 3.34/1.72 ( ( in @ X @ ( powerset @ A ) )
% 3.34/1.72 => ! [Y: $i] :
% 3.34/1.72 ( ( in @ Y @ ( powerset @ A ) )
% 3.34/1.72 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(demorgan2a2_type,type,
% 3.34/1.72 demorgan2a2: $o ).
% 3.34/1.72
% 3.34/1.72 thf(demorgan2a2,definition,
% 3.34/1.72 ( demorgan2a2
% 3.34/1.72 = ( ! [A: $i,X: $i] :
% 3.34/1.72 ( ( in @ X @ ( powerset @ A ) )
% 3.34/1.72 => ! [Y: $i] :
% 3.34/1.72 ( ( in @ Y @ ( powerset @ A ) )
% 3.34/1.72 => ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 3.34/1.72 => ( in @ Xx @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(demorgan1a_type,type,
% 3.34/1.72 demorgan1a: $o ).
% 3.34/1.72
% 3.34/1.72 thf(demorgan1a,definition,
% 3.34/1.72 ( demorgan1a
% 3.34/1.72 = ( ! [A: $i,X: $i] :
% 3.34/1.72 ( ( in @ X @ ( powerset @ A ) )
% 3.34/1.72 => ! [Y: $i] :
% 3.34/1.72 ( ( in @ Y @ ( powerset @ A ) )
% 3.34/1.72 => ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ( ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) )
% 3.34/1.72 => ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(demorgan1b_type,type,
% 3.34/1.72 demorgan1b: $o ).
% 3.34/1.72
% 3.34/1.72 thf(demorgan1b,definition,
% 3.34/1.72 ( demorgan1b
% 3.34/1.72 = ( ! [A: $i,X: $i] :
% 3.34/1.72 ( ( in @ X @ ( powerset @ A ) )
% 3.34/1.72 => ! [Y: $i] :
% 3.34/1.72 ( ( in @ Y @ ( powerset @ A ) )
% 3.34/1.72 => ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ( ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 3.34/1.72 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(demorgan1_type,type,
% 3.34/1.72 demorgan1: $o ).
% 3.34/1.72
% 3.34/1.72 thf(demorgan1,definition,
% 3.34/1.72 ( demorgan1
% 3.34/1.72 = ( ! [A: $i,X: $i] :
% 3.34/1.72 ( ( in @ X @ ( powerset @ A ) )
% 3.34/1.72 => ! [Y: $i] :
% 3.34/1.72 ( ( in @ Y @ ( powerset @ A ) )
% 3.34/1.72 => ( ( setminus @ A @ ( binintersect @ X @ Y ) )
% 3.34/1.72 = ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(demorgan2a_type,type,
% 3.34/1.72 demorgan2a: $o ).
% 3.34/1.72
% 3.34/1.72 thf(demorgan2a,definition,
% 3.34/1.72 ( demorgan2a
% 3.34/1.72 = ( ! [A: $i,X: $i] :
% 3.34/1.72 ( ( in @ X @ ( powerset @ A ) )
% 3.34/1.72 => ! [Y: $i] :
% 3.34/1.72 ( ( in @ Y @ ( powerset @ A ) )
% 3.34/1.72 => ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 3.34/1.72 => ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(demorgan2b2_type,type,
% 3.34/1.72 demorgan2b2: $o ).
% 3.34/1.72
% 3.34/1.72 thf(demorgan2b2,definition,
% 3.34/1.72 ( demorgan2b2
% 3.34/1.72 = ( ! [A: $i,X: $i] :
% 3.34/1.72 ( ( in @ X @ ( powerset @ A ) )
% 3.34/1.72 => ! [Y: $i] :
% 3.34/1.72 ( ( in @ Y @ ( powerset @ A ) )
% 3.34/1.72 => ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 3.34/1.72 => ( ( in @ Xx @ ( setminus @ A @ Y ) )
% 3.34/1.72 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(demorgan2b_type,type,
% 3.34/1.72 demorgan2b: $o ).
% 3.34/1.72
% 3.34/1.72 thf(demorgan2b,definition,
% 3.34/1.72 ( demorgan2b
% 3.34/1.72 = ( ! [A: $i,X: $i] :
% 3.34/1.72 ( ( in @ X @ ( powerset @ A ) )
% 3.34/1.72 => ! [Y: $i] :
% 3.34/1.72 ( ( in @ Y @ ( powerset @ A ) )
% 3.34/1.72 => ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ( ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 3.34/1.72 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(demorgan2_type,type,
% 3.34/1.72 demorgan2: $o ).
% 3.34/1.72
% 3.34/1.72 thf(demorgan2,definition,
% 3.34/1.72 ( demorgan2
% 3.34/1.72 = ( ! [A: $i,X: $i] :
% 3.34/1.72 ( ( in @ X @ ( powerset @ A ) )
% 3.34/1.72 => ! [Y: $i] :
% 3.34/1.72 ( ( in @ Y @ ( powerset @ A ) )
% 3.34/1.72 => ( ( setminus @ A @ ( binunion @ X @ Y ) )
% 3.34/1.72 = ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(woz13rule0_type,type,
% 3.34/1.72 woz13rule0: $o ).
% 3.34/1.72
% 3.34/1.72 thf(woz13rule0,definition,
% 3.34/1.72 ( woz13rule0
% 3.34/1.72 = ( ! [A: $i,X: $i] :
% 3.34/1.72 ( ( in @ X @ ( powerset @ A ) )
% 3.34/1.72 => ! [Y: $i] :
% 3.34/1.72 ( ( in @ Y @ ( powerset @ A ) )
% 3.34/1.72 => ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 3.34/1.72 => ( in @ Xx @ A ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(woz13rule1_type,type,
% 3.34/1.72 woz13rule1: $o ).
% 3.34/1.72
% 3.34/1.72 thf(woz13rule1,definition,
% 3.34/1.72 ( woz13rule1
% 3.34/1.72 = ( ! [A: $i,X: $i] :
% 3.34/1.72 ( ( in @ X @ ( powerset @ A ) )
% 3.34/1.72 => ! [Y: $i] :
% 3.34/1.72 ( ( in @ Y @ ( powerset @ A ) )
% 3.34/1.72 => ! [Z: $i] :
% 3.34/1.72 ( ( in @ Z @ ( powerset @ A ) )
% 3.34/1.72 => ( ( subset @ X @ Z )
% 3.34/1.72 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(woz13rule2_type,type,
% 3.34/1.72 woz13rule2: $o ).
% 3.34/1.72
% 3.34/1.72 thf(woz13rule2,definition,
% 3.34/1.72 ( woz13rule2
% 3.34/1.72 = ( ! [A: $i,X: $i] :
% 3.34/1.72 ( ( in @ X @ ( powerset @ A ) )
% 3.34/1.72 => ! [Y: $i] :
% 3.34/1.72 ( ( in @ Y @ ( powerset @ A ) )
% 3.34/1.72 => ! [Z: $i] :
% 3.34/1.72 ( ( in @ Z @ ( powerset @ A ) )
% 3.34/1.72 => ( ( subset @ Y @ Z )
% 3.34/1.72 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(woz13rule3_type,type,
% 3.34/1.72 woz13rule3: $o ).
% 3.34/1.72
% 3.34/1.72 thf(woz13rule3,definition,
% 3.34/1.72 ( woz13rule3
% 3.34/1.72 = ( ! [A: $i,X: $i] :
% 3.34/1.72 ( ( in @ X @ ( powerset @ A ) )
% 3.34/1.72 => ! [Y: $i] :
% 3.34/1.72 ( ( in @ Y @ ( powerset @ A ) )
% 3.34/1.72 => ! [Z: $i] :
% 3.34/1.72 ( ( in @ Z @ ( powerset @ A ) )
% 3.34/1.72 => ( ( subset @ X @ Y )
% 3.34/1.72 => ( ( subset @ X @ Z )
% 3.34/1.72 => ( subset @ X @ ( binintersect @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(woz13rule4_type,type,
% 3.34/1.72 woz13rule4: $o ).
% 3.34/1.72
% 3.34/1.72 thf(woz13rule4,definition,
% 3.34/1.72 ( woz13rule4
% 3.34/1.72 = ( ! [A: $i,X: $i] :
% 3.34/1.72 ( ( in @ X @ ( powerset @ A ) )
% 3.34/1.72 => ! [Y: $i] :
% 3.34/1.72 ( ( in @ Y @ ( powerset @ A ) )
% 3.34/1.72 => ! [Z: $i] :
% 3.34/1.72 ( ( in @ Z @ ( powerset @ A ) )
% 3.34/1.72 => ! [W: $i] :
% 3.34/1.72 ( ( in @ W @ ( powerset @ A ) )
% 3.34/1.72 => ( ( subset @ X @ Z )
% 3.34/1.72 => ( ( subset @ Y @ W )
% 3.34/1.72 => ( subset @ ( binintersect @ X @ Y ) @ ( binintersect @ Z @ W ) ) ) ) ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(woz1_1_type,type,
% 3.34/1.72 woz1_1: $o ).
% 3.34/1.72
% 3.34/1.72 thf(woz1_1,definition,
% 3.34/1.72 ( woz1_1
% 3.34/1.72 = ( ! [A: $i,X: $i] :
% 3.34/1.72 ( ( in @ X @ ( powerset @ A ) )
% 3.34/1.72 => ! [Y: $i] :
% 3.34/1.72 ( ( in @ Y @ ( powerset @ A ) )
% 3.34/1.72 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(woz1_2_type,type,
% 3.34/1.72 woz1_2: $o ).
% 3.34/1.72
% 3.34/1.72 thf(woz1_2,definition,
% 3.34/1.72 ( woz1_2
% 3.34/1.72 = ( ! [A: $i,X: $i] :
% 3.34/1.72 ( ( in @ X @ ( powerset @ A ) )
% 3.34/1.72 => ! [Y: $i] :
% 3.34/1.72 ( ( in @ Y @ ( powerset @ A ) )
% 3.34/1.72 => ! [Z: $i] :
% 3.34/1.72 ( ( in @ Z @ ( powerset @ A ) )
% 3.34/1.72 => ! [W: $i] :
% 3.34/1.72 ( ( in @ W @ ( powerset @ A ) )
% 3.34/1.72 => ( ( setminus @ A @ ( binintersect @ ( binunion @ X @ Y ) @ ( binunion @ Z @ W ) ) )
% 3.34/1.72 = ( binunion @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) @ ( binintersect @ ( setminus @ A @ Z ) @ ( setminus @ A @ W ) ) ) ) ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(woz1_3_type,type,
% 3.34/1.72 woz1_3: $o ).
% 3.34/1.72
% 3.34/1.72 thf(woz1_3,definition,
% 3.34/1.72 ( woz1_3
% 3.34/1.72 = ( ! [A: $i,X: $i] :
% 3.34/1.72 ( ( in @ X @ ( powerset @ A ) )
% 3.34/1.72 => ! [Y: $i] :
% 3.34/1.72 ( ( in @ Y @ ( powerset @ A ) )
% 3.34/1.72 => ! [Z: $i] :
% 3.34/1.72 ( ( in @ Z @ ( powerset @ A ) )
% 3.34/1.72 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(woz1_4_type,type,
% 3.34/1.72 woz1_4: $o ).
% 3.34/1.72
% 3.34/1.72 thf(woz1_4,definition,
% 3.34/1.72 ( woz1_4
% 3.34/1.72 = ( ! [A: $i,X: $i] :
% 3.34/1.72 ( ( in @ X @ ( powerset @ A ) )
% 3.34/1.72 => ! [Y: $i] :
% 3.34/1.72 ( ( in @ Y @ ( powerset @ A ) )
% 3.34/1.72 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 3.34/1.72 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(woz1_5_type,type,
% 3.34/1.72 woz1_5: $o ).
% 3.34/1.72
% 3.34/1.72 thf(woz1_5,definition,
% 3.34/1.72 ( woz1_5
% 3.34/1.72 = ( ! [A: $i,X: $i] :
% 3.34/1.72 ( ( in @ X @ ( powerset @ A ) )
% 3.34/1.72 => ! [Y: $i] :
% 3.34/1.72 ( ( in @ Y @ ( powerset @ A ) )
% 3.34/1.72 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(breln1_type,type,
% 3.34/1.72 breln1: $i > $i > $o ).
% 3.34/1.72
% 3.34/1.72 thf(breln1all2_type,type,
% 3.34/1.72 breln1all2: $o ).
% 3.34/1.72
% 3.34/1.72 thf(breln1all2,definition,
% 3.34/1.72 ( breln1all2
% 3.34/1.72 = ( ! [A: $i,R: $i] :
% 3.34/1.72 ( ( breln1 @ A @ R )
% 3.34/1.72 => ! [Xphi: $i > $o] :
% 3.34/1.72 ( ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ A )
% 3.34/1.72 => ! [Xy: $i] :
% 3.34/1.72 ( ( in @ Xy @ A )
% 3.34/1.72 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.34/1.72 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 3.34/1.72 => ! [Xx: $i] :
% 3.34/1.72 ( ( in @ Xx @ R )
% 3.34/1.72 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(breln1Set_type,type,
% 3.34/1.72 breln1Set: $i > $i ).
% 3.34/1.72
% 3.34/1.72 thf(breln1SetBreln1_type,type,
% 3.34/1.72 breln1SetBreln1: $o ).
% 3.34/1.72
% 3.34/1.72 thf(breln1SetBreln1,definition,
% 3.34/1.72 ( breln1SetBreln1
% 3.34/1.72 = ( ! [A: $i,R: $i] :
% 3.34/1.72 ( ( in @ R @ ( breln1Set @ A ) )
% 3.34/1.72 => ( breln1 @ A @ R ) ) ) ) ).
% 3.34/1.72
% 3.34/1.72 thf(transitive_type,type,
% 3.34/1.72 transitive: $i > $i > $o ).
% 3.34/1.72
% 3.34/1.72 thf(antisymmetric_type,type,
% 3.34/1.72 antisymmetric: $i > $i > $o ).
% 3.34/1.72
% 3.34/1.72 thf(reflexive_type,type,
% 3.34/1.72 reflexive: $i > $i > $o ).
% 3.34/1.72
% 3.34/1.72 thf(refltransitive_type,type,
% 3.34/1.72 refltransitive: $i > $i > $o ).
% 3.34/1.72
% 3.34/1.72 thf(refllinearorder_type,type,
% 3.34/1.72 refllinearorder: $i > $i > $o ).
% 3.34/1.72
% 3.34/1.72 thf(reflwellordering_type,type,
% 3.34/1.72 reflwellordering: $i > $i > $o ).
% 3.34/1.72
% 3.34/1.72 thf(choice2fnsingleton_type,type,
% 3.34/1.72 choice2fnsingleton: $o ).
% 3.34/1.73
% 3.34/1.73 thf(choice2fnsingleton,definition,
% 3.34/1.73 ( choice2fnsingleton
% 3.34/1.73 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 3.34/1.73 ( ! [Xx: $i] :
% 3.34/1.73 ( ( in @ Xx @ A )
% 3.34/1.73 => ? [Xy: $i] :
% 3.34/1.73 ( ( in @ Xy @ B )
% 3.34/1.73 & ( Xphi @ Xx @ Xy ) ) )
% 3.34/1.73 => ! [R: $i] :
% 3.34/1.73 ( ( in @ R @ ( breln1Set @ B ) )
% 3.34/1.73 => ( ( reflwellordering @ B @ R )
% 3.34/1.73 => ! [Xx: $i] :
% 3.34/1.73 ( ( in @ Xx @ A )
% 3.34/1.73 => ( singleton
% 3.34/1.73 @ ( dsetconstr @ B
% 3.34/1.73 @ ^ [Xy: $i] :
% 3.34/1.73 ( ( Xphi @ Xx @ Xy )
% 3.34/1.73 & ! [Xz: $i] :
% 3.34/1.73 ( ( in @ Xz @ B )
% 3.34/1.73 => ( ( Xphi @ Xx @ Xz )
% 3.34/1.73 => ( in @ ( kpair @ Xy @ Xz ) @ R ) ) ) ) ) ) ) ) ) ) ) ) ).
% 3.34/1.73
% 3.34/1.73 thf(setOfPairsIsBReln1_type,type,
% 3.34/1.73 setOfPairsIsBReln1: $o ).
% 3.34/1.73
% 3.34/1.73 thf(setOfPairsIsBReln1,definition,
% 3.34/1.73 ( setOfPairsIsBReln1
% 3.34/1.73 = ( ! [A: $i,Xphi: $i > $i > $o] :
% 3.34/1.73 ( breln1 @ A
% 3.34/1.73 @ ( dpsetconstr @ A @ A
% 3.34/1.73 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 3.34/1.73
% 3.34/1.73 thf(breln1all1_type,type,
% 3.34/1.73 breln1all1: $o ).
% 3.34/1.73
% 3.34/1.73 thf(breln1all1,definition,
% 3.34/1.73 ( breln1all1
% 3.34/1.73 = ( ! [A: $i,R: $i] :
% 3.34/1.73 ( ( breln1 @ A @ R )
% 3.34/1.73 => ! [Xphi: $i > $o] :
% 3.34/1.73 ( ! [Xx: $i] :
% 3.34/1.73 ( ( in @ Xx @ A )
% 3.34/1.73 => ! [Xy: $i] :
% 3.34/1.73 ( ( in @ Xy @ A )
% 3.34/1.73 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.34/1.73 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 3.34/1.73 => ! [Xx: $i] :
% 3.34/1.73 ( ( in @ Xx @ R )
% 3.34/1.73 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.34/1.73
% 3.34/1.73 thf(subbreln1_type,type,
% 3.34/1.73 subbreln1: $o ).
% 3.34/1.73
% 3.34/1.73 thf(subbreln1,definition,
% 3.34/1.73 ( subbreln1
% 3.34/1.73 = ( ! [A: $i,R: $i] :
% 3.34/1.73 ( ( breln1 @ A @ R )
% 3.34/1.73 => ! [S: $i] :
% 3.34/1.73 ( ( breln1 @ A @ S )
% 3.34/1.73 => ( ! [Xx: $i] :
% 3.34/1.73 ( ( in @ Xx @ A )
% 3.34/1.73 => ! [Xy: $i] :
% 3.34/1.73 ( ( in @ Xy @ A )
% 3.34/1.73 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.34/1.73 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 3.34/1.73 => ( subset @ R @ S ) ) ) ) ) ) ).
% 3.34/1.73
% 3.34/1.73 thf(eqbreln1_type,type,
% 3.34/1.73 eqbreln1: $o ).
% 3.34/1.73
% 3.34/1.73 thf(eqbreln1,definition,
% 3.34/1.73 ( eqbreln1
% 3.34/1.73 = ( ! [A: $i,R: $i] :
% 3.34/1.73 ( ( breln1 @ A @ R )
% 3.34/1.73 => ! [S: $i] :
% 3.34/1.73 ( ( breln1 @ A @ S )
% 3.34/1.73 => ( ! [Xx: $i] :
% 3.34/1.73 ( ( in @ Xx @ A )
% 3.34/1.73 => ! [Xy: $i] :
% 3.34/1.73 ( ( in @ Xy @ A )
% 3.34/1.73 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.34/1.73 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 3.34/1.73 => ( ! [Xx: $i] :
% 3.34/1.73 ( ( in @ Xx @ A )
% 3.34/1.73 => ! [Xy: $i] :
% 3.34/1.73 ( ( in @ Xy @ A )
% 3.34/1.73 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 3.34/1.73 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 3.34/1.73 => ( R = S ) ) ) ) ) ) ) ).
% 3.34/1.73
% 3.34/1.73 thf(breln1invset_type,type,
% 3.34/1.73 breln1invset: $i > $i > $i ).
% 3.34/1.73
% 3.34/1.73 thf(breln1invprop_type,type,
% 3.34/1.73 breln1invprop: $o ).
% 3.34/1.73
% 3.34/1.73 thf(breln1invprop,definition,
% 3.34/1.73 ( breln1invprop
% 3.34/1.73 = ( ! [A: $i,R: $i] :
% 3.34/1.73 ( ( breln1 @ A @ R )
% 3.34/1.73 => ( breln1 @ A @ ( breln1invset @ A @ R ) ) ) ) ) ).
% 3.34/1.73
% 3.34/1.73 thf(breln1invI_type,type,
% 3.34/1.73 breln1invI: $o ).
% 3.34/1.73
% 3.34/1.73 thf(breln1invI,definition,
% 3.34/1.73 ( breln1invI
% 3.34/1.73 = ( ! [A: $i,R: $i] :
% 3.34/1.73 ( ( breln1 @ A @ R )
% 3.34/1.73 => ! [Xx: $i] :
% 3.34/1.73 ( ( in @ Xx @ A )
% 3.34/1.73 => ! [Xy: $i] :
% 3.34/1.73 ( ( in @ Xy @ A )
% 3.34/1.73 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.34/1.73 => ( in @ ( kpair @ Xy @ Xx ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ) ).
% 3.34/1.73
% 3.34/1.73 thf(breln1invE_type,type,
% 3.34/1.73 breln1invE: $o ).
% 3.34/1.73
% 3.34/1.73 thf(breln1invE,definition,
% 3.34/1.73 ( breln1invE
% 3.34/1.73 = ( ! [A: $i,R: $i] :
% 3.34/1.73 ( ( breln1 @ A @ R )
% 3.34/1.73 => ! [Xx: $i] :
% 3.34/1.73 ( ( in @ Xx @ A )
% 3.34/1.73 => ! [Xy: $i] :
% 3.34/1.73 ( ( in @ Xy @ A )
% 3.34/1.73 => ( ( in @ ( kpair @ Xy @ Xx ) @ ( breln1invset @ A @ R ) )
% 3.34/1.73 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) ) ) ) ) ).
% 3.34/1.73
% 3.34/1.73 thf(breln1compset_type,type,
% 3.34/1.73 breln1compset: $i > $i > $i > $i ).
% 3.34/1.73
% 3.34/1.73 thf(breln1compprop_type,type,
% 3.34/1.73 breln1compprop: $o ).
% 3.34/1.73
% 3.34/1.73 thf(breln1compprop,definition,
% 3.34/1.73 ( breln1compprop
% 3.34/1.73 = ( ! [A: $i,R: $i] :
% 3.34/1.73 ( ( breln1 @ A @ R )
% 3.34/1.73 => ! [S: $i] :
% 3.34/1.73 ( ( breln1 @ A @ S )
% 3.34/1.73 => ( breln1 @ A @ ( breln1compset @ A @ R @ S ) ) ) ) ) ) ).
% 3.34/1.73
% 3.34/1.73 thf(breln1compI_type,type,
% 3.34/1.73 breln1compI: $o ).
% 3.34/1.73
% 3.34/1.73 thf(breln1compI,definition,
% 3.34/1.73 ( breln1compI
% 3.34/1.73 = ( ! [A: $i,R: $i] :
% 3.34/1.73 ( ( breln1 @ A @ R )
% 3.34/1.73 => ! [S: $i] :
% 3.34/1.73 ( ( breln1 @ A @ S )
% 3.34/1.73 => ! [Xx: $i] :
% 3.34/1.73 ( ( in @ Xx @ A )
% 3.34/1.73 => ! [Xy: $i] :
% 3.34/1.73 ( ( in @ Xy @ A )
% 3.34/1.73 => ! [Xz: $i] :
% 3.34/1.73 ( ( in @ Xz @ A )
% 3.34/1.73 => ( ( in @ ( kpair @ Xx @ Xz ) @ R )
% 3.34/1.73 => ( ( in @ ( kpair @ Xz @ Xy ) @ S )
% 3.34/1.73 => ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) ) ) ) ) ) ) ) ) ) ) ).
% 3.34/1.73
% 3.34/1.73 thf(breln1compE_type,type,
% 3.34/1.73 breln1compE: $o ).
% 3.34/1.73
% 3.34/1.73 thf(breln1compE,definition,
% 3.34/1.73 ( breln1compE
% 3.34/1.73 = ( ! [A: $i,R: $i] :
% 3.34/1.73 ( ( breln1 @ A @ R )
% 3.34/1.73 => ! [S: $i] :
% 3.34/1.73 ( ( breln1 @ A @ S )
% 3.34/1.73 => ! [Xx: $i] :
% 3.34/1.73 ( ( in @ Xx @ A )
% 3.34/1.73 => ! [Xy: $i] :
% 3.34/1.73 ( ( in @ Xy @ A )
% 3.34/1.73 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) )
% 3.34/1.73 => ? [Xz: $i] :
% 3.34/1.73 ( ( in @ Xz @ A )
% 3.34/1.73 & ( in @ ( kpair @ Xx @ Xz ) @ R )
% 3.34/1.73 & ( in @ ( kpair @ Xz @ Xy ) @ S ) ) ) ) ) ) ) ) ) ).
% 3.34/1.73
% 3.34/1.73 thf(breln1compEex_type,type,
% 3.34/1.73 breln1compEex: $o ).
% 3.34/1.73
% 3.34/1.73 thf(breln1compEex,definition,
% 3.34/1.73 ( breln1compEex
% 3.34/1.73 = ( ! [A: $i,R: $i] :
% 3.34/1.73 ( ( breln1 @ A @ R )
% 3.34/1.73 => ! [S: $i] :
% 3.34/1.73 ( ( breln1 @ A @ S )
% 3.34/1.73 => ! [Xx: $i] :
% 3.34/1.73 ( ( in @ Xx @ A )
% 3.34/1.73 => ! [Xy: $i] :
% 3.34/1.73 ( ( in @ Xy @ A )
% 3.34/1.73 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) )
% 3.34/1.73 => ! [Xphi: $o] :
% 3.34/1.73 ( ! [Xz: $i] :
% 3.34/1.73 ( ( in @ Xz @ A )
% 3.34/1.73 => ( ( in @ ( kpair @ Xx @ Xz ) @ R )
% 3.34/1.73 => ( ( in @ ( kpair @ Xz @ Xy ) @ S )
% 3.34/1.73 => Xphi ) ) )
% 3.34/1.73 => Xphi ) ) ) ) ) ) ) ) ).
% 3.34/1.73
% 3.34/1.73 thf(breln1unionprop_type,type,
% 3.34/1.73 breln1unionprop: $o ).
% 3.34/1.73
% 3.34/1.73 thf(breln1unionprop,definition,
% 3.34/1.73 ( breln1unionprop
% 3.34/1.73 = ( ! [A: $i,R: $i] :
% 3.34/1.73 ( ( breln1 @ A @ R )
% 3.34/1.73 => ! [S: $i] :
% 3.34/1.73 ( ( breln1 @ A @ S )
% 3.34/1.73 => ( breln1 @ A @ ( binunion @ R @ S ) ) ) ) ) ) ).
% 3.34/1.73
% 3.34/1.73 thf(breln1unionIL_type,type,
% 3.34/1.73 breln1unionIL: $o ).
% 3.34/1.73
% 3.34/1.73 thf(breln1unionIL,definition,
% 3.34/1.73 ( breln1unionIL
% 3.34/1.73 = ( ! [A: $i,R: $i] :
% 3.34/1.73 ( ( breln1 @ A @ R )
% 3.34/1.73 => ! [S: $i] :
% 3.34/1.73 ( ( breln1 @ A @ S )
% 3.34/1.73 => ! [Xx: $i] :
% 3.34/1.73 ( ( in @ Xx @ A )
% 3.34/1.73 => ! [Xy: $i] :
% 3.34/1.73 ( ( in @ Xy @ A )
% 3.34/1.73 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.34/1.73 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 3.34/1.73
% 3.34/1.73 thf(breln1unionIR_type,type,
% 3.34/1.73 breln1unionIR: $o ).
% 3.34/1.73
% 3.34/1.73 thf(breln1unionIR,definition,
% 3.34/1.73 ( breln1unionIR
% 3.34/1.73 = ( ! [A: $i,R: $i] :
% 3.34/1.73 ( ( breln1 @ A @ R )
% 3.34/1.73 => ! [S: $i] :
% 3.34/1.73 ( ( breln1 @ A @ S )
% 3.34/1.73 => ! [Xx: $i] :
% 3.34/1.73 ( ( in @ Xx @ A )
% 3.34/1.73 => ! [Xy: $i] :
% 3.34/1.73 ( ( in @ Xy @ A )
% 3.34/1.73 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 3.34/1.73 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 3.34/1.73
% 3.34/1.73 thf(breln1unionI_type,type,
% 3.34/1.73 breln1unionI: $o ).
% 3.34/1.73
% 3.34/1.73 thf(breln1unionI,definition,
% 3.34/1.73 ( breln1unionI
% 3.34/1.73 = ( ! [A: $i,R: $i] :
% 3.34/1.73 ( ( breln1 @ A @ R )
% 3.34/1.73 => ! [S: $i] :
% 3.34/1.73 ( ( breln1 @ A @ S )
% 3.34/1.73 => ! [Xx: $i] :
% 3.34/1.73 ( ( in @ Xx @ A )
% 3.34/1.73 => ! [Xy: $i] :
% 3.34/1.73 ( ( in @ Xy @ A )
% 3.34/1.73 => ( ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.34/1.73 | ( in @ ( kpair @ Xx @ Xy ) @ S ) )
% 3.34/1.73 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 3.34/1.73
% 3.34/1.73 thf(breln1unionE_type,type,
% 3.34/1.73 breln1unionE: $o ).
% 3.34/1.73
% 3.34/1.73 thf(breln1unionE,definition,
% 3.34/1.73 ( breln1unionE
% 3.34/1.73 = ( ! [A: $i,R: $i] :
% 3.34/1.73 ( ( breln1 @ A @ R )
% 3.34/1.73 => ! [S: $i] :
% 3.34/1.73 ( ( breln1 @ A @ S )
% 3.34/1.73 => ! [Xx: $i] :
% 3.34/1.73 ( ( in @ Xx @ A )
% 3.34/1.73 => ! [Xy: $i] :
% 3.34/1.73 ( ( in @ Xy @ A )
% 3.34/1.73 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) )
% 3.34/1.73 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.34/1.73 | ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) ) ) ) ) ) ) ).
% 3.34/1.73
% 3.34/1.73 thf(breln1unionEcases_type,type,
% 3.34/1.73 breln1unionEcases: $o ).
% 3.34/1.73
% 3.34/1.73 thf(breln1unionEcases,definition,
% 3.34/1.73 ( breln1unionEcases
% 3.34/1.73 = ( ! [A: $i,R: $i] :
% 3.34/1.73 ( ( breln1 @ A @ R )
% 3.34/1.73 => ! [S: $i] :
% 3.34/1.73 ( ( breln1 @ A @ S )
% 3.34/1.73 => ! [Xx: $i] :
% 3.34/1.73 ( ( in @ Xx @ A )
% 3.34/1.73 => ! [Xy: $i] :
% 3.34/1.73 ( ( in @ Xy @ A )
% 3.34/1.73 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) )
% 3.34/1.73 => ! [Xphi: $o] :
% 3.34/1.73 ( ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.34/1.73 => Xphi )
% 3.34/1.73 => ( ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 3.34/1.73 => Xphi )
% 3.34/1.73 => Xphi ) ) ) ) ) ) ) ) ) ).
% 3.34/1.73
% 3.34/1.73 thf(breln1unionCommutes_type,type,
% 3.34/1.73 breln1unionCommutes: $o ).
% 3.34/1.73
% 3.34/1.73 thf(breln1unionCommutes,definition,
% 3.34/1.73 ( breln1unionCommutes
% 3.34/1.73 = ( ! [A: $i,R: $i] :
% 3.34/1.73 ( ( breln1 @ A @ R )
% 3.34/1.73 => ! [S: $i] :
% 3.34/1.73 ( ( breln1 @ A @ S )
% 3.34/1.73 => ( ( binunion @ R @ S )
% 3.34/1.73 = ( binunion @ S @ R ) ) ) ) ) ) ).
% 3.34/1.73
% 3.34/1.73 thf(woz2Ex_type,type,
% 3.34/1.73 woz2Ex: $o ).
% 3.34/1.73
% 3.34/1.73 thf(woz2Ex,definition,
% 3.34/1.73 ( woz2Ex
% 3.34/1.73 = ( ! [A: $i,R: $i] :
% 3.34/1.73 ( ( breln1 @ A @ R )
% 3.34/1.73 => ( R
% 3.34/1.73 = ( breln1invset @ A @ ( breln1invset @ A @ R ) ) ) ) ) ) ).
% 3.34/1.73
% 3.34/1.73 thf(woz2W_type,type,
% 3.34/1.73 woz2W: $o ).
% 3.34/1.73
% 3.34/1.73 thf(woz2W,definition,
% 3.34/1.73 ( woz2W
% 3.34/1.73 = ( ! [A: $i,R: $i] :
% 3.34/1.73 ( ( breln1 @ A @ R )
% 3.34/1.73 => ! [S: $i] :
% 3.34/1.73 ( ( breln1 @ A @ S )
% 3.34/1.73 => ( ( breln1invset @ A @ ( breln1compset @ A @ R @ S ) )
% 3.34/1.73 = ( breln1compset @ A @ ( breln1invset @ A @ S ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ).
% 3.34/1.73
% 3.34/1.73 thf(woz2A_type,type,
% 3.34/1.73 woz2A: $o ).
% 3.34/1.73
% 3.34/1.73 thf(woz2A,definition,
% 3.34/1.73 ( woz2A
% 3.34/1.73 = ( ! [A: $i,R: $i] :
% 3.34/1.73 ( ( breln1 @ A @ R )
% 3.34/1.73 => ! [S: $i] :
% 3.34/1.73 ( ( breln1 @ A @ S )
% 3.34/1.73 => ! [T: $i] :
% 3.34/1.73 ( ( breln1 @ A @ T )
% 3.34/1.73 => ( ( breln1compset @ A @ ( binunion @ R @ S ) @ T )
% 3.34/1.73 = ( binunion @ ( breln1compset @ A @ R @ T ) @ ( breln1compset @ A @ S @ T ) ) ) ) ) ) ) ) ).
% 3.34/1.73
% 3.34/1.73 thf(woz2B_type,type,
% 3.34/1.73 woz2B: $o ).
% 3.34/1.73
% 3.34/1.73 thf(woz2B,definition,
% 3.34/1.73 ( woz2B
% 3.34/1.73 = ( ! [A: $i,R: $i] :
% 3.34/1.73 ( ( breln1 @ A @ R )
% 3.34/1.73 => ! [S: $i] :
% 3.34/1.73 ( ( breln1 @ A @ S )
% 3.34/1.73 => ! [T: $i] :
% 3.34/1.73 ( ( breln1 @ A @ T )
% 3.34/1.73 => ( ( breln1compset @ A @ ( binunion @ R @ S ) @ T )
% 3.34/1.73 = ( binunion @ ( breln1invset @ A @ ( breln1compset @ A @ ( breln1invset @ A @ T ) @ ( breln1invset @ A @ S ) ) ) @ ( breln1invset @ A @ ( breln1compset @ A @ ( breln1invset @ A @ T ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ) ) ) ).
% 3.34/1.73
% 3.34/1.73 thf(image1Ex_type,type,
% 3.34/1.73 image1Ex: $o ).
% 3.34/1.73
% 3.34/1.73 thf(image1Ex,definition,
% 3.34/1.73 ( image1Ex
% 3.34/1.73 = ( ! [A: $i,Xf: $i > $i] :
% 3.34/1.73 ? [B: $i] :
% 3.34/1.73 ! [Xx: $i] :
% 3.34/1.73 ( ( in @ Xx @ B )
% 3.34/1.73 <=> ? [Xy: $i] :
% 3.34/1.73 ( ( in @ Xy @ A )
% 3.34/1.73 & ( Xx
% 3.34/1.73 = ( Xf @ Xy ) ) ) ) ) ) ).
% 3.34/1.73
% 3.34/1.73 thf(image1Ex1_type,type,
% 3.34/1.73 image1Ex1: $o ).
% 3.34/1.73
% 3.34/1.73 thf(image1Ex1,definition,
% 3.34/1.73 ( image1Ex1
% 3.34/1.73 = ( ! [A: $i,Xf: $i > $i] :
% 3.34/1.73 ( exu
% 3.34/1.73 @ ^ [B: $i] :
% 3.34/1.73 ! [Xx: $i] :
% 3.34/1.73 ( ( in @ Xx @ B )
% 3.34/1.73 <=> ? [Xy: $i] :
% 3.34/1.73 ( ( in @ Xy @ A )
% 3.34/1.73 & ( Xx
% 3.34/1.73 = ( Xf @ Xy ) ) ) ) ) ) ) ).
% 3.34/1.73
% 3.34/1.73 thf(image1_type,type,
% 3.34/1.73 image1: $i > ( $i > $i ) > $i ).
% 3.34/1.73
% 3.34/1.73 thf(image1Equiv_type,type,
% 3.34/1.73 image1Equiv: $o ).
% 3.34/1.73
% 3.34/1.73 thf(image1Equiv,definition,
% 3.34/1.73 ( image1Equiv
% 3.34/1.73 = ( ! [A: $i,Xf: $i > $i,Xx: $i] :
% 3.34/1.73 ( ( in @ Xx
% 3.34/1.73 @ ( image1 @ A
% 3.34/1.73 @ ^ [Xy: $i] : ( Xf @ Xy ) ) )
% 3.34/1.73 <=> ? [Xy: $i] :
% 3.34/1.73 ( ( in @ Xy @ A )
% 3.34/1.73 & ( Xx
% 3.34/1.73 = ( Xf @ Xy ) ) ) ) ) ) ).
% 3.34/1.73
% 3.34/1.73 thf(image1E_type,type,
% 3.34/1.73 image1E: $o ).
% 3.34/1.73
% 3.34/1.73 thf(image1E,definition,
% 3.34/1.73 ( image1E
% 3.34/1.73 = ( ! [A: $i,Xf: $i > $i,Xx: $i] :
% 3.34/1.73 ( ( in @ Xx
% 3.34/1.73 @ ( image1 @ A
% 3.34/1.73 @ ^ [Xy: $i] : ( Xf @ Xy ) ) )
% 3.34/1.73 => ? [Xy: $i] :
% 3.34/1.73 ( ( in @ Xy @ A )
% 3.34/1.73 & ( Xx
% 3.34/1.73 = ( Xf @ Xy ) ) ) ) ) ) ).
% 3.34/1.73
% 3.34/1.73 thf(image1I_type,type,
% 3.34/1.73 image1I: $o ).
% 3.34/1.73
% 3.34/1.73 thf(image1I,definition,
% 3.34/1.73 ( image1I
% 3.34/1.73 = ( ! [A: $i,Xf: $i > $i,Xx: $i] :
% 3.34/1.73 ( ? [Xy: $i] :
% 3.34/1.73 ( ( in @ Xy @ A )
% 3.34/1.73 & ( Xx
% 3.34/1.73 = ( Xf @ Xy ) ) )
% 3.34/1.73 => ( in @ Xx
% 3.34/1.73 @ ( image1 @ A
% 3.34/1.73 @ ^ [Xy: $i] : ( Xf @ Xy ) ) ) ) ) ) ).
% 3.34/1.73
% 3.34/1.73 thf(injective_type,type,
% 3.34/1.73 injective: $i > $i > $i > $o ).
% 3.34/1.73
% 3.34/1.73 thf(injFuncSet_type,type,
% 3.34/1.73 injFuncSet: $i > $i > $i ).
% 3.34/1.73
% 3.34/1.73 thf(injFuncInInjFuncSet_type,type,
% 3.34/1.73 injFuncInInjFuncSet: $o ).
% 3.34/1.73
% 3.34/1.73 thf(injFuncInInjFuncSet,definition,
% 3.34/1.73 ( injFuncInInjFuncSet
% 3.34/1.73 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.34/1.73 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.34/1.73 => ( ( injective @ A @ B @ Xf )
% 3.34/1.73 => ( in @ Xf @ ( injFuncSet @ A @ B ) ) ) ) ) ) ).
% 3.34/1.73
% 3.34/1.73 thf(injFuncSetFuncIn_type,type,
% 3.34/1.73 injFuncSetFuncIn: $o ).
% 3.34/1.73
% 3.34/1.73 thf(injFuncSetFuncIn,definition,
% 3.34/1.73 ( injFuncSetFuncIn
% 3.34/1.73 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.34/1.73 ( ( in @ Xf @ ( injFuncSet @ A @ B ) )
% 3.34/1.73 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 3.34/1.73
% 3.34/1.73 thf(injFuncSetFuncInj_type,type,
% 3.34/1.73 injFuncSetFuncInj: $o ).
% 3.34/1.73
% 3.34/1.73 thf(injFuncSetFuncInj,definition,
% 3.34/1.73 ( injFuncSetFuncInj
% 3.34/1.73 = ( ! [Xx: $i,Xy: $i,Xf: $i] :
% 3.34/1.73 ( ( in @ Xf @ ( injFuncSet @ Xx @ Xy ) )
% 3.34/1.73 => ( injective @ Xx @ Xy @ Xf ) ) ) ) ).
% 3.34/1.73
% 3.34/1.73 thf(surjective_type,type,
% 3.34/1.73 surjective: $i > $i > $i > $o ).
% 3.34/1.73
% 3.34/1.73 thf(surjFuncSet_type,type,
% 3.34/1.73 surjFuncSet: $i > $i > $i ).
% 3.34/1.73
% 3.34/1.73 thf(surjFuncSetFuncIn_type,type,
% 3.34/1.73 surjFuncSetFuncIn: $o ).
% 3.34/1.73
% 3.34/1.73 thf(surjFuncSetFuncIn,definition,
% 3.34/1.73 ( surjFuncSetFuncIn
% 3.34/1.73 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.34/1.73 ( ( in @ Xf @ ( surjFuncSet @ A @ B ) )
% 3.34/1.73 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 3.34/1.73
% 3.34/1.73 thf(surjFuncSetFuncSurj_type,type,
% 3.34/1.73 surjFuncSetFuncSurj: $o ).
% 3.34/1.73
% 3.34/1.73 thf(surjFuncSetFuncSurj,definition,
% 3.34/1.73 ( surjFuncSetFuncSurj
% 3.34/1.73 = ( ! [Xx: $i,Xy: $i,Xf: $i] :
% 3.34/1.73 ( ( in @ Xf @ ( surjFuncSet @ Xx @ Xy ) )
% 3.34/1.73 => ( surjective @ Xx @ Xy @ Xf ) ) ) ) ).
% 3.34/1.73
% 3.34/1.73 thf(leftInvIsSurj_type,type,
% 3.34/1.73 leftInvIsSurj: $o ).
% 3.34/1.73
% 3.34/1.73 thf(leftInvIsSurj,definition,
% 3.34/1.73 ( leftInvIsSurj
% 3.34/1.73 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.34/1.73 ( ! [Xx: $i] :
% 3.34/1.73 ( ( in @ Xx @ A )
% 3.34/1.73 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.34/1.73 => ! [Xg: $i] :
% 3.34/1.73 ( ( in @ Xg @ ( funcSet @ B @ A ) )
% 3.34/1.73 => ( ! [Xx: $i] :
% 3.34/1.73 ( ( in @ Xx @ A )
% 3.34/1.73 => ( ( ap @ B @ A @ Xg @ ( Xf @ Xx ) )
% 3.34/1.73 = Xx ) )
% 3.34/1.73 => ( surjective @ B @ A @ Xg ) ) ) ) ) ) ).
% 3.34/1.73
% 3.34/1.73 thf(surjCantorThm_type,type,
% 3.34/1.73 surjCantorThm: $o ).
% 3.34/1.73
% 3.34/1.73 thf(surjCantorThm,definition,
% 3.34/1.73 ( surjCantorThm
% 3.34/1.73 = ( ! [A: $i,Xf: $i] :
% 3.34/1.73 ( ( in @ Xf @ ( funcSet @ A @ ( powerset @ A ) ) )
% 3.34/1.73 => ~ ( surjective @ A @ ( powerset @ A ) @ Xf ) ) ) ) ).
% 3.34/1.73
% 3.34/1.73 thf(foundation2_type,type,
% 3.34/1.73 foundation2: $o ).
% 3.34/1.73
% 3.34/1.73 thf(foundation2,definition,
% 3.34/1.73 ( foundation2
% 3.34/1.73 = ( ! [A: $i] :
% 3.34/1.73 ( ( nonempty @ A )
% 3.34/1.73 => ? [X: $i] :
% 3.34/1.73 ( ( in @ X @ A )
% 3.34/1.73 & ( ( binintersect @ X @ A )
% 3.34/1.73 = emptyset ) ) ) ) ) ).
% 3.34/1.73
% 3.34/1.73 thf(notinself_type,type,
% 3.34/1.73 notinself: $o ).
% 3.34/1.73
% 3.34/1.73 thf(notinself,definition,
% 3.34/1.73 ( notinself
% 3.34/1.73 = ( ! [A: $i] :
% 3.34/1.73 ~ ( in @ A @ A ) ) ) ).
% 3.34/1.73
% 3.34/1.73 thf(notinself2_type,type,
% 3.34/1.73 notinself2: $o ).
% 3.34/1.73
% 3.34/1.73 thf(notinself2,definition,
% 3.34/1.73 ( notinself2
% 3.34/1.73 = ( ! [A: $i,B: $i] :
% 3.34/1.73 ( ( in @ A @ B )
% 3.34/1.73 => ~ ( in @ B @ A ) ) ) ) ).
% 3.34/1.73
% 3.34/1.73 thf(omegaS_type,type,
% 3.34/1.73 omegaS: $i > $i ).
% 3.34/1.73
% 3.34/1.73 thf(omegaS,definition,
% 3.34/1.73 ( omegaS
% 3.34/1.73 = ( ^ [Xx: $i] : ( setadjoin @ Xx @ Xx ) ) ) ).
% 3.34/1.73
% 3.34/1.73 thf(omegaSp_type,type,
% 3.34/1.73 omegaSp: $o ).
% 3.34/1.73
% 3.34/1.73 thf(omegaSp,definition,
% 3.34/1.73 ( omegaSp
% 3.34/1.73 = ( ! [Xx: $i] :
% 3.34/1.73 ( ( in @ Xx @ omega )
% 3.34/1.73 => ( in @ ( omegaS @ Xx ) @ omega ) ) ) ) ).
% 3.34/1.73
% 3.34/1.73 thf(omegaSclos,conjecture,
% 3.34/1.73 ( setextAx
% 3.34/1.73 => ( emptysetAx
% 3.34/1.73 => ( setadjoinAx
% 3.34/1.73 => ( powersetAx
% 3.34/1.73 => ( setunionAx
% 3.34/1.73 => ( omega0Ax
% 3.34/1.73 => ( omegaSAx
% 3.34/1.73 => ( omegaIndAx
% 3.34/1.73 => ( replAx
% 3.34/1.73 => ( foundationAx
% 3.34/1.73 => ( wellorderingAx
% 3.34/1.73 => ( descrp
% 3.34/1.73 => ( dsetconstrI
% 3.34/1.73 => ( dsetconstrEL
% 3.34/1.73 => ( dsetconstrER
% 3.34/1.73 => ( exuE1
% 3.34/1.73 => ( prop2setE
% 3.34/1.73 => ( emptysetE
% 3.34/1.73 => ( emptysetimpfalse
% 3.34/1.73 => ( notinemptyset
% 3.34/1.73 => ( exuE3e
% 3.34/1.73 => ( setext
% 3.34/1.73 => ( emptyI
% 3.34/1.73 => ( noeltsimpempty
% 3.34/1.73 => ( setbeta
% 3.34/1.73 => ( nonemptyE1
% 3.34/1.73 => ( nonemptyI
% 3.34/1.73 => ( nonemptyI1
% 3.34/1.73 => ( setadjoinIL
% 3.34/1.73 => ( emptyinunitempty
% 3.34/1.73 => ( setadjoinIR
% 3.34/1.73 => ( setadjoinE
% 3.34/1.73 => ( setadjoinOr
% 3.34/1.73 => ( setoftrueEq
% 3.34/1.73 => ( powersetI
% 3.34/1.73 => ( emptyinPowerset
% 3.34/1.73 => ( emptyInPowerset
% 3.34/1.73 => ( powersetE
% 3.34/1.73 => ( setunionI
% 3.34/1.73 => ( setunionE
% 3.34/1.73 => ( subPowSU
% 3.34/1.73 => ( exuE2
% 3.34/1.73 => ( nonemptyImpWitness
% 3.34/1.73 => ( uniqinunit
% 3.34/1.73 => ( notinsingleton
% 3.34/1.73 => ( eqinunit
% 3.34/1.73 => ( singletonsswitch
% 3.34/1.73 => ( upairsetE
% 3.34/1.73 => ( upairsetIL
% 3.34/1.73 => ( upairsetIR
% 3.34/1.73 => ( emptyE1
% 3.34/1.73 => ( vacuousDall
% 3.34/1.73 => ( quantDeMorgan1
% 3.34/1.73 => ( quantDeMorgan2
% 3.34/1.73 => ( quantDeMorgan3
% 3.34/1.73 => ( quantDeMorgan4
% 3.34/1.73 => ( prop2setI
% 3.34/1.73 => ( prop2set2propI
% 3.34/1.73 => ( notdexE
% 3.34/1.73 => ( notdallE
% 3.34/1.73 => ( exuI1
% 3.34/1.73 => ( exuI3
% 3.34/1.73 => ( exuI2
% 3.34/1.73 => ( inCongP
% 3.34/1.73 => ( in__Cong
% 3.34/1.73 => ( exuE3u
% 3.34/1.73 => ( exu__Cong
% 3.34/1.73 => ( emptyset__Cong
% 3.34/1.73 => ( setadjoin__Cong
% 3.34/1.73 => ( powerset__Cong
% 3.34/1.73 => ( setunion__Cong
% 3.34/1.73 => ( omega__Cong
% 3.34/1.73 => ( exuEu
% 3.34/1.73 => ( descr__Cong
% 3.34/1.73 => ( dsetconstr__Cong
% 3.34/1.73 => ( subsetI1
% 3.34/1.73 => ( eqimpsubset2
% 3.34/1.73 => ( eqimpsubset1
% 3.34/1.73 => ( subsetI2
% 3.34/1.73 => ( emptysetsubset
% 3.34/1.73 => ( subsetE
% 3.34/1.73 => ( subsetE2
% 3.34/1.73 => ( notsubsetI
% 3.34/1.73 => ( notequalI1
% 3.34/1.73 => ( notequalI2
% 3.34/1.73 => ( subsetRefl
% 3.34/1.73 => ( subsetTrans
% 3.34/1.73 => ( setadjoinSub
% 3.34/1.73 => ( setadjoinSub2
% 3.34/1.73 => ( subset2powerset
% 3.34/1.73 => ( setextsub
% 3.34/1.73 => ( subsetemptysetimpeq
% 3.34/1.73 => ( powersetI1
% 3.34/1.73 => ( powersetE1
% 3.34/1.73 => ( inPowerset
% 3.34/1.73 => ( powersetsubset
% 3.34/1.73 => ( sepInPowerset
% 3.34/1.73 => ( sepSubset
% 3.34/1.73 => ( binunionIL
% 3.34/1.73 => ( upairset2IR
% 3.34/1.73 => ( binunionIR
% 3.34/1.73 => ( binunionEcases
% 3.34/1.73 => ( binunionE
% 3.34/1.73 => ( binunionLsub
% 3.34/1.73 => ( binunionRsub
% 3.34/1.73 => ( binintersectI
% 3.34/1.73 => ( binintersectSubset5
% 3.34/1.73 => ( binintersectEL
% 3.34/1.73 => ( binintersectLsub
% 3.34/1.73 => ( binintersectSubset2
% 3.34/1.73 => ( binintersectSubset3
% 3.34/1.73 => ( binintersectER
% 3.34/1.73 => ( disjointsetsI1
% 3.34/1.73 => ( binintersectRsub
% 3.34/1.73 => ( binintersectSubset4
% 3.34/1.73 => ( binintersectSubset1
% 3.34/1.73 => ( bs114d
% 3.34/1.73 => ( setminusI
% 3.34/1.73 => ( setminusEL
% 3.34/1.73 => ( setminusER
% 3.34/1.73 => ( setminusSubset2
% 3.34/1.73 => ( setminusERneg
% 3.34/1.73 => ( setminusELneg
% 3.34/1.73 => ( setminusILneg
% 3.34/1.73 => ( setminusIRneg
% 3.34/1.73 => ( setminusLsub
% 3.34/1.73 => ( setminusSubset1
% 3.34/1.73 => ( symdiffE
% 3.34/1.73 => ( symdiffI1
% 3.34/1.73 => ( symdiffI2
% 3.34/1.73 => ( symdiffIneg1
% 3.34/1.73 => ( symdiffIneg2
% 3.34/1.73 => ( secondinupair
% 3.34/1.73 => ( setukpairIL
% 3.34/1.73 => ( setukpairIR
% 3.34/1.73 => ( kpairiskpair
% 3.34/1.73 => ( kpairp
% 3.34/1.73 => ( singletonsubset
% 3.34/1.73 => ( singletoninpowerset
% 3.34/1.73 => ( singletoninpowunion
% 3.34/1.73 => ( upairset2E
% 3.34/1.73 => ( upairsubunion
% 3.34/1.73 => ( upairinpowunion
% 3.34/1.73 => ( ubforcartprodlem1
% 3.34/1.73 => ( ubforcartprodlem2
% 3.34/1.73 => ( ubforcartprodlem3
% 3.34/1.73 => ( cartprodpairin
% 3.34/1.73 => ( cartprodmempair1
% 3.34/1.73 => ( cartprodmempair
% 3.34/1.73 => ( setunionE2
% 3.34/1.73 => ( setunionsingleton1
% 3.34/1.73 => ( setunionsingleton2
% 3.34/1.73 => ( setunionsingleton
% 3.34/1.73 => ( singletonprop
% 3.34/1.73 => ( ex1E1
% 3.34/1.73 => ( ex1I
% 3.34/1.73 => ( ex1I2
% 3.34/1.73 => ( singletonsuniq
% 3.34/1.73 => ( setukpairinjL1
% 3.34/1.73 => ( kfstsingleton
% 3.34/1.73 => ( theprop
% 3.34/1.73 => ( kfstpairEq
% 3.34/1.73 => ( cartprodfstin
% 3.34/1.73 => ( setukpairinjL2
% 3.34/1.73 => ( setukpairinjL
% 3.34/1.73 => ( setukpairinjR11
% 3.34/1.73 => ( setukpairinjR12
% 3.34/1.73 => ( setukpairinjR1
% 3.34/1.73 => ( upairequniteq
% 3.34/1.73 => ( setukpairinjR2
% 3.34/1.73 => ( setukpairinjR
% 3.34/1.73 => ( ksndsingleton
% 3.34/1.73 => ( ksndpairEq
% 3.34/1.73 => ( kpairsurjEq
% 3.34/1.73 => ( cartprodsndin
% 3.34/1.73 => ( cartprodpairmemEL
% 3.34/1.73 => ( cartprodpairmemER
% 3.34/1.73 => ( cartprodmempaircEq
% 3.34/1.73 => ( cartprodfstpairEq
% 3.34/1.73 => ( cartprodsndpairEq
% 3.34/1.73 => ( cartprodpairsurjEq
% 3.34/1.73 => ( dpsetconstrI
% 3.34/1.73 => ( dpsetconstrSub
% 3.34/1.73 => ( setOfPairsIsBReln
% 3.34/1.73 => ( dpsetconstrERa
% 3.34/1.73 => ( dpsetconstrEL1
% 3.34/1.73 => ( dpsetconstrEL2
% 3.34/1.73 => ( dpsetconstrER
% 3.34/1.73 => ( funcImageSingleton
% 3.34/1.73 => ( apProp
% 3.34/1.73 => ( app
% 3.34/1.73 => ( infuncsetfunc
% 3.34/1.73 => ( ap2p
% 3.34/1.73 => ( funcinfuncset
% 3.34/1.73 => ( lamProp
% 3.34/1.73 => ( lamp
% 3.34/1.73 => ( lam2p
% 3.34/1.73 => ( brelnall1
% 3.34/1.73 => ( brelnall2
% 3.34/1.73 => ( ex1E2
% 3.34/1.73 => ( funcGraphProp1
% 3.34/1.73 => ( funcGraphProp3
% 3.34/1.73 => ( funcGraphProp2
% 3.34/1.73 => ( funcextLem
% 3.34/1.73 => ( funcGraphProp4
% 3.34/1.73 => ( subbreln
% 3.34/1.73 => ( eqbreln
% 3.34/1.73 => ( funcext
% 3.34/1.73 => ( funcext2
% 3.34/1.73 => ( ap2apEq1
% 3.34/1.73 => ( ap2apEq2
% 3.34/1.73 => ( beta1
% 3.34/1.73 => ( eta1
% 3.34/1.73 => ( lam2lamEq
% 3.34/1.73 => ( beta2
% 3.34/1.73 => ( eta2
% 3.34/1.73 => ( iffalseProp1
% 3.34/1.73 => ( iffalseProp2
% 3.34/1.73 => ( iftrueProp1
% 3.34/1.73 => ( iftrueProp2
% 3.34/1.73 => ( ifSingleton
% 3.34/1.73 => ( ifp
% 3.34/1.73 => ( theeq
% 3.34/1.73 => ( iftrue
% 3.34/1.73 => ( iffalse
% 3.34/1.73 => ( iftrueorfalse
% 3.34/1.73 => ( binintersectT_lem
% 3.34/1.73 => ( binunionT_lem
% 3.34/1.73 => ( powersetT_lem
% 3.34/1.73 => ( setminusT_lem
% 3.34/1.73 => ( complementT_lem
% 3.34/1.73 => ( setextT
% 3.34/1.73 => ( subsetTI
% 3.34/1.73 => ( powersetTI1
% 3.34/1.73 => ( powersetTE1
% 3.34/1.73 => ( complementTI1
% 3.34/1.73 => ( complementTE1
% 3.34/1.73 => ( binintersectTELcontra
% 3.34/1.73 => ( binintersectTERcontra
% 3.34/1.73 => ( contrasubsetT
% 3.34/1.73 => ( contrasubsetT1
% 3.34/1.73 => ( contrasubsetT2
% 3.34/1.73 => ( contrasubsetT3
% 3.34/1.73 => ( doubleComplementI1
% 3.34/1.73 => ( doubleComplementE1
% 3.34/1.73 => ( doubleComplementSub1
% 3.34/1.73 => ( doubleComplementSub2
% 3.34/1.73 => ( doubleComplementEq
% 3.34/1.73 => ( complementTnotintersectT
% 3.34/1.73 => ( complementImpComplementIntersect
% 3.34/1.73 => ( complementSubsetComplementIntersect
% 3.34/1.73 => ( complementInPowersetComplementIntersect
% 3.34/1.73 => ( contraSubsetComplement
% 3.34/1.73 => ( complementTcontraSubset
% 3.34/1.73 => ( binunionTILcontra
% 3.34/1.73 => ( binunionTIRcontra
% 3.34/1.73 => ( inIntersectImpInUnion
% 3.34/1.73 => ( inIntersectImpInUnion2
% 3.34/1.73 => ( inIntersectImpInIntersectUnions
% 3.34/1.73 => ( intersectInPowersetIntersectUnions
% 3.34/1.73 => ( inComplementUnionImpNotIn1
% 3.34/1.73 => ( inComplementUnionImpInComplement1
% 3.34/1.73 => ( binunionTE
% 3.34/1.73 => ( binunionTEcontra
% 3.34/1.73 => ( demorgan2a1
% 3.34/1.73 => ( complementUnionInPowersetComplement
% 3.34/1.73 => ( demorgan2a2
% 3.34/1.73 => ( demorgan1a
% 3.34/1.73 => ( demorgan1b
% 3.34/1.73 => ( demorgan1
% 3.34/1.73 => ( demorgan2a
% 3.34/1.73 => ( demorgan2b2
% 3.34/1.73 => ( demorgan2b
% 3.34/1.73 => ( demorgan2
% 3.34/1.73 => ( woz13rule0
% 3.34/1.73 => ( woz13rule1
% 3.34/1.73 => ( woz13rule2
% 3.34/1.73 => ( woz13rule3
% 3.34/1.73 => ( woz13rule4
% 3.34/1.73 => ( woz1_1
% 3.34/1.73 => ( woz1_2
% 3.34/1.73 => ( woz1_3
% 3.34/1.73 => ( woz1_4
% 3.34/1.73 => ( woz1_5
% 3.34/1.73 => ( breln1all2
% 3.34/1.73 => ( breln1SetBreln1
% 3.34/1.73 => ( choice2fnsingleton
% 3.34/1.73 => ( setOfPairsIsBReln1
% 3.34/1.73 => ( breln1all1
% 3.34/1.73 => ( subbreln1
% 3.34/1.73 => ( eqbreln1
% 3.34/1.73 => ( breln1invprop
% 3.34/1.73 => ( breln1invI
% 3.34/1.73 => ( breln1invE
% 3.34/1.73 => ( breln1compprop
% 3.34/1.73 => ( breln1compI
% 3.34/1.73 => ( breln1compE
% 3.34/1.73 => ( breln1compEex
% 3.34/1.73 => ( breln1unionprop
% 3.34/1.73 => ( breln1unionIL
% 3.34/1.73 => ( breln1unionIR
% 3.34/1.73 => ( breln1unionI
% 3.34/1.73 => ( breln1unionE
% 3.34/1.73 => ( breln1unionEcases
% 3.34/1.73 => ( breln1unionCommutes
% 3.34/1.73 => ( woz2Ex
% 3.34/1.73 => ( woz2W
% 3.34/1.73 => ( woz2A
% 3.34/1.73 => ( woz2B
% 3.34/1.73 => ( image1Ex
% 3.34/1.73 => ( image1Ex1
% 3.34/1.73 => ( image1Equiv
% 3.34/1.73 => ( image1E
% 3.34/1.73 => ( image1I
% 3.34/1.73 => ( injFuncInInjFuncSet
% 3.34/1.73 => ( injFuncSetFuncIn
% 3.34/1.73 => ( injFuncSetFuncInj
% 3.34/1.73 => ( surjFuncSetFuncIn
% 3.34/1.73 => ( surjFuncSetFuncSurj
% 3.34/1.73 => ( leftInvIsSurj
% 3.34/1.73 => ( surjCantorThm
% 3.34/1.73 => ( foundation2
% 3.34/1.73 => ( notinself
% 3.34/1.73 => ( notinself2
% 3.34/1.73 => ( omegaSp
% 3.34/1.73 => ! [Xx: $i] :
% 3.34/1.73 ( ( in @ Xx @ omega )
% 3.34/1.73 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 3.34/1.73 ------------------------
% 3.34/1.76 % (28694)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_28502 for (2999ds/183Mi)
% 3.34/1.76 % (28695)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_28502 for (2999ds/4Mi)
% 3.34/1.76 % (28700)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_28502 for (2999ds/18Mi)
% 3.34/1.76 % (28696)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_28502 for (2999ds/27Mi)
% 3.34/1.76 % (28697)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_28502 for (2999ds/2Mi)
% 3.34/1.76 % (28699)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_28502 for (2999ds/275Mi)
% 3.34/1.76 % (28698)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_28502 for (2999ds/2Mi)
% 3.34/1.76 % (28697)Instruction limit reached!
% 3.34/1.76 % (28697)------------------------------
% 3.34/1.76 % (28697)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.34/1.76 % (28697)Termination reason: Unknown
% 3.34/1.76 % (28697)Termination phase: shuffling
% 3.34/1.76
% 3.34/1.76 % (28697)Memory used [KB]: 1663
% 3.34/1.76 % (28697)Time elapsed: 0.003 s
% 3.34/1.76 % (28697)Instructions burned: 2 (million)
% 3.34/1.76 % (28697)------------------------------
% 3.34/1.76 % (28697)------------------------------
% 3.34/1.76 % (28695)Instruction limit reached!
% 3.34/1.76 % (28695)------------------------------
% 3.34/1.76 % (28695)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.34/1.76 % (28695)Termination reason: Unknown
% 3.34/1.76 % (28695)Termination phase: shuffling
% 3.34/1.76
% 3.34/1.76 % (28695)Memory used [KB]: 1663
% 3.34/1.76 % (28695)Time elapsed: 0.004 s
% 3.34/1.76 % (28695)Instructions burned: 5 (million)
% 3.34/1.76 % (28695)------------------------------
% 3.34/1.76 % (28695)------------------------------
% 3.34/1.76 % (28698)Instruction limit reached!
% 3.34/1.76 % (28698)------------------------------
% 3.34/1.76 % (28698)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.34/1.76 % (28698)Termination reason: Unknown
% 3.34/1.76 % (28698)Termination phase: shuffling
% 3.34/1.76
% 3.34/1.76 % (28698)Memory used [KB]: 1663
% 3.34/1.76 % (28698)Time elapsed: 0.003 s
% 3.34/1.76 % (28698)Instructions burned: 3 (million)
% 3.34/1.76 % (28698)------------------------------
% 3.34/1.76 % (28698)------------------------------
% 3.34/1.77 % (28700)Instruction limit reached!
% 3.34/1.77 % (28700)------------------------------
% 3.34/1.77 % (28700)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.34/1.77 % (28700)Termination reason: Unknown
% 3.34/1.77 % (28700)Termination phase: shuffling
% 3.34/1.77
% 3.34/1.77 % (28700)Memory used [KB]: 1918
% 3.34/1.77 % (28700)Time elapsed: 0.011 s
% 3.34/1.77 % (28700)Instructions burned: 19 (million)
% 3.34/1.77 % (28700)------------------------------
% 3.34/1.77 % (28700)------------------------------
% 3.34/1.78 % (28696)Instruction limit reached!
% 3.34/1.78 % (28696)------------------------------
% 3.34/1.78 % (28696)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.34/1.78 % (28696)Termination reason: Unknown
% 3.34/1.78 % (28696)Termination phase: shuffling
% 3.34/1.78
% 3.34/1.78 % (28696)Memory used [KB]: 2174
% 3.34/1.78 % (28696)Time elapsed: 0.015 s
% 3.34/1.78 % (28696)Instructions burned: 28 (million)
% 3.34/1.78 % (28696)------------------------------
% 3.34/1.78 % (28696)------------------------------
% 3.34/1.78 % (28702)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_28502 for (2999ds/3Mi)
% 3.34/1.78 % (28703)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_28502 for (2999ds/37Mi)
% 3.34/1.78 % (28704)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_28502 for (2999ds/15Mi)
% 3.34/1.78 % (28702)Instruction limit reached!
% 3.34/1.78 % (28702)------------------------------
% 3.34/1.78 % (28702)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.34/1.78 % (28702)Termination reason: Unknown
% 3.34/1.78 % (28702)Termination phase: shuffling
% 3.34/1.78
% 3.34/1.78 % (28702)Memory used [KB]: 1663
% 3.34/1.78 % (28702)Time elapsed: 0.003 s
% 3.34/1.78 % (28702)Instructions burned: 3 (million)
% 3.34/1.78 % (28702)------------------------------
% 3.34/1.78 % (28702)------------------------------
% 3.34/1.79 % (28706)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_28502 for (2999ds/3Mi)
% 3.34/1.79 % (28704)Instruction limit reached!
% 3.34/1.79 % (28704)------------------------------
% 3.34/1.79 % (28704)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.34/1.79 % (28704)Termination reason: Unknown
% 3.34/1.79 % (28704)Termination phase: shuffling
% 3.34/1.79
% 3.34/1.79 % (28704)Memory used [KB]: 1918
% 3.34/1.79 % (28704)Time elapsed: 0.010 s
% 3.34/1.79 % (28704)Instructions burned: 16 (million)
% 3.34/1.79 % (28704)------------------------------
% 3.34/1.79 % (28704)------------------------------
% 3.34/1.79 % (28706)Instruction limit reached!
% 3.34/1.79 % (28706)------------------------------
% 3.34/1.79 % (28706)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.34/1.79 % (28706)Termination reason: Unknown
% 3.34/1.79 % (28706)Termination phase: shuffling
% 3.34/1.79
% 3.34/1.79 % (28706)Memory used [KB]: 1663
% 3.34/1.79 % (28706)Time elapsed: 0.003 s
% 3.34/1.79 % (28706)Instructions burned: 3 (million)
% 3.34/1.79 % (28706)------------------------------
% 3.34/1.79 % (28706)------------------------------
% 3.34/1.79 % (28707)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_28502 for (2999ds/1041Mi)
% 3.40/1.80 % (28703)Instruction limit reached!
% 3.40/1.80 % (28703)------------------------------
% 3.40/1.80 % (28703)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.40/1.80 % (28703)Termination reason: Unknown
% 3.40/1.80 % (28703)Termination phase: shuffling
% 3.40/1.80
% 3.40/1.80 % (28703)Memory used [KB]: 2302
% 3.40/1.80 % (28703)Time elapsed: 0.019 s
% 3.40/1.80 % (28703)Instructions burned: 38 (million)
% 3.40/1.80 % (28703)------------------------------
% 3.40/1.80 % (28703)------------------------------
% 3.40/1.80 % (28708)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_28502 for (2999ds/7Mi)
% 3.40/1.80 % (28708)Instruction limit reached!
% 3.40/1.80 % (28708)------------------------------
% 3.40/1.80 % (28708)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.40/1.80 % (28708)Termination reason: Unknown
% 3.40/1.80 % (28708)Termination phase: shuffling
% 3.40/1.80
% 3.40/1.80 % (28708)Memory used [KB]: 1663
% 3.40/1.80 % (28708)Time elapsed: 0.005 s
% 3.40/1.80 % (28708)Instructions burned: 7 (million)
% 3.40/1.80 % (28708)------------------------------
% 3.40/1.80 % (28708)------------------------------
% 3.40/1.80 % (28694)Instruction limit reached!
% 3.40/1.80 % (28694)------------------------------
% 3.40/1.80 % (28694)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.40/1.80 % (28694)Termination reason: Unknown
% 3.40/1.80 % (28694)Termination phase: Property scanning
% 3.40/1.80
% 3.40/1.80 % (28694)Memory used [KB]: 3198
% 3.40/1.80 % (28694)Time elapsed: 0.049 s
% 3.40/1.80 % (28694)Instructions burned: 183 (million)
% 3.40/1.80 % (28694)------------------------------
% 3.40/1.80 % (28694)------------------------------
% 3.40/1.81 % (28711)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_28502 for (2999ds/3Mi)
% 3.40/1.81 % (28711)Instruction limit reached!
% 3.40/1.81 % (28711)------------------------------
% 3.40/1.81 % (28711)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.40/1.81 % (28711)Termination reason: Unknown
% 3.40/1.81 % (28711)Termination phase: shuffling
% 3.40/1.81
% 3.40/1.81 % (28711)Memory used [KB]: 1663
% 3.40/1.81 % (28711)Time elapsed: 0.003 s
% 3.40/1.81 % (28711)Instructions burned: 3 (million)
% 3.40/1.81 % (28711)------------------------------
% 3.40/1.81 % (28711)------------------------------
% 3.40/1.81 % (28710)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_28502 for (2999ds/16Mi)
% 3.40/1.81 % (28714)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_28502 for (2999ds/3Mi)
% 3.40/1.81 % (28712)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_28502 for (2999ds/3Mi)
% 3.40/1.81 % (28710)Instruction limit reached!
% 3.40/1.81 % (28710)------------------------------
% 3.40/1.81 % (28710)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.40/1.81 % (28710)Termination reason: Unknown
% 3.40/1.81 % (28710)Termination phase: shuffling
% 3.40/1.81
% 3.40/1.81 % (28710)Memory used [KB]: 1918
% 3.40/1.81 % (28710)Time elapsed: 0.009 s
% 3.40/1.81 % (28710)Instructions burned: 17 (million)
% 3.40/1.81 % (28710)------------------------------
% 3.40/1.81 % (28710)------------------------------
% 3.40/1.82 % (28714)Instruction limit reached!
% 3.40/1.82 % (28714)------------------------------
% 3.40/1.82 % (28714)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.40/1.82 % (28714)Termination reason: Unknown
% 3.40/1.82 % (28714)Termination phase: shuffling
% 3.40/1.82
% 3.40/1.82 % (28714)Memory used [KB]: 1663
% 3.40/1.82 % (28714)Time elapsed: 0.003 s
% 3.40/1.82 % (28714)Instructions burned: 6 (million)
% 3.40/1.82 % (28714)------------------------------
% 3.40/1.82 % (28714)------------------------------
% 3.40/1.82 % (28712)Instruction limit reached!
% 3.40/1.82 % (28712)------------------------------
% 3.40/1.82 % (28712)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.40/1.82 % (28712)Termination reason: Unknown
% 3.40/1.82 % (28712)Termination phase: shuffling
% 3.40/1.82
% 3.40/1.82 % (28712)Memory used [KB]: 1663
% 3.40/1.82 % (28712)Time elapsed: 0.003 s
% 3.40/1.82 % (28712)Instructions burned: 3 (million)
% 3.40/1.82 % (28712)------------------------------
% 3.40/1.82 % (28712)------------------------------
% 3.40/1.82 % (28713)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_28502 for (2999ds/7Mi)
% 3.40/1.82 % (28713)Instruction limit reached!
% 3.40/1.82 % (28713)------------------------------
% 3.40/1.82 % (28713)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.40/1.82 % (28713)Termination reason: Unknown
% 3.40/1.82 % (28713)Termination phase: shuffling
% 3.40/1.82
% 3.40/1.82 % (28713)Memory used [KB]: 1663
% 3.40/1.82 % (28713)Time elapsed: 0.005 s
% 3.40/1.82 % (28713)Instructions burned: 7 (million)
% 3.40/1.82 % (28713)------------------------------
% 3.40/1.82 % (28713)------------------------------
% 3.40/1.82 % (28715)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_28502 for (2999ds/4Mi)
% 3.40/1.82 % (28718)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_28502 for (2999ds/710Mi)
% 3.40/1.83 % (28715)Instruction limit reached!
% 3.40/1.83 % (28715)------------------------------
% 3.40/1.83 % (28715)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.40/1.83 % (28715)Termination reason: Unknown
% 3.40/1.83 % (28715)Termination phase: shuffling
% 3.40/1.83
% 3.40/1.83 % (28715)Memory used [KB]: 1663
% 3.40/1.83 % (28715)Time elapsed: 0.004 s
% 3.40/1.83 % (28715)Instructions burned: 5 (million)
% 3.40/1.83 % (28715)------------------------------
% 3.40/1.83 % (28715)------------------------------
% 3.40/1.83 % (28717)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_28502 for (2999ds/18Mi)
% 3.40/1.83 % (28719)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_28502 for (2999ds/6Mi)
% 3.40/1.84 % (28719)Instruction limit reached!
% 3.40/1.84 % (28719)------------------------------
% 3.40/1.84 % (28719)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.40/1.84 % (28719)Termination reason: Unknown
% 3.40/1.84 % (28719)Termination phase: shuffling
% 3.40/1.84
% 3.40/1.84 % (28719)Memory used [KB]: 1663
% 3.40/1.84 % (28719)Time elapsed: 0.005 s
% 3.40/1.84 % (28719)Instructions burned: 7 (million)
% 3.40/1.84 % (28719)------------------------------
% 3.40/1.84 % (28719)------------------------------
% 3.40/1.84 % (28720)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_28502 for (2999ds/902Mi)
% 3.40/1.84 % (28717)Instruction limit reached!
% 3.40/1.84 % (28717)------------------------------
% 3.40/1.84 % (28717)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.40/1.84 % (28717)Termination reason: Unknown
% 3.40/1.84 % (28717)Termination phase: shuffling
% 3.40/1.84
% 3.40/1.84 % (28717)Memory used [KB]: 1918
% 3.40/1.84 % (28717)Time elapsed: 0.010 s
% 3.40/1.84 % (28717)Instructions burned: 19 (million)
% 3.40/1.84 % (28717)------------------------------
% 3.40/1.84 % (28717)------------------------------
% 3.40/1.84 % (28721)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_28502 for (2999ds/21Mi)
% 3.40/1.85 % (28723)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_28502 for (2999ds/5Mi)
% 3.40/1.85 % (28721)Instruction limit reached!
% 3.40/1.85 % (28721)------------------------------
% 3.40/1.85 % (28721)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.40/1.85 % (28721)Termination reason: Unknown
% 3.40/1.85 % (28721)Termination phase: shuffling
% 3.40/1.85
% 3.40/1.85 % (28721)Memory used [KB]: 1918
% 3.40/1.85 % (28721)Time elapsed: 0.011 s
% 3.40/1.85 % (28721)Instructions burned: 21 (million)
% 3.40/1.85 % (28721)------------------------------
% 3.40/1.85 % (28721)------------------------------
% 3.40/1.85 % (28723)Instruction limit reached!
% 3.40/1.85 % (28723)------------------------------
% 3.40/1.85 % (28723)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.40/1.85 % (28723)Termination reason: Unknown
% 3.40/1.85 % (28723)Termination phase: shuffling
% 3.40/1.85
% 3.40/1.85 % (28723)Memory used [KB]: 1663
% 3.40/1.85 % (28723)Time elapsed: 0.004 s
% 3.40/1.85 % (28723)Instructions burned: 6 (million)
% 3.40/1.85 % (28723)------------------------------
% 3.40/1.85 % (28723)------------------------------
% 3.40/1.86 % (28724)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_28502 for (2999ds/6Mi)
% 3.40/1.86 % (28724)Instruction limit reached!
% 3.40/1.86 % (28724)------------------------------
% 3.40/1.86 % (28724)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.40/1.86 % (28724)Termination reason: Unknown
% 3.40/1.86 % (28724)Termination phase: shuffling
% 3.40/1.86
% 3.40/1.86 % (28724)Memory used [KB]: 1663
% 3.40/1.86 % (28724)Time elapsed: 0.004 s
% 3.40/1.86 % (28724)Instructions burned: 6 (million)
% 3.40/1.86 % (28724)------------------------------
% 3.40/1.86 % (28724)------------------------------
% 3.40/1.87 % (28727)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_28502 for (2998ds/377Mi)
% 3.40/1.87 % (28728)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_28502 for (2998ds/779Mi)
% 3.40/1.88 % (28729)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_28502 for (2998ds/19Mi)
% 3.40/1.88 % (28729)Instruction limit reached!
% 3.40/1.88 % (28729)------------------------------
% 3.40/1.88 % (28729)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.40/1.88 % (28729)Termination reason: Unknown
% 3.40/1.88 % (28729)Termination phase: shuffling
% 3.40/1.88
% 3.40/1.88 % (28729)Memory used [KB]: 1918
% 3.40/1.88 % (28729)Time elapsed: 0.011 s
% 3.40/1.88 % (28729)Instructions burned: 19 (million)
% 3.40/1.88 % (28729)------------------------------
% 3.40/1.88 % (28729)------------------------------
% 3.40/1.89 % (28699)Instruction limit reached!
% 3.40/1.89 % (28699)------------------------------
% 3.40/1.89 % (28699)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.40/1.89 % (28699)Termination reason: Unknown
% 3.40/1.89 % (28699)Termination phase: Property scanning
% 3.40/1.89
% 3.40/1.89 % (28699)Memory used [KB]: 5884
% 3.40/1.89 % (28699)Time elapsed: 0.131 s
% 3.40/1.89 % (28699)Instructions burned: 276 (million)
% 3.40/1.89 % (28699)------------------------------
% 3.40/1.89 % (28699)------------------------------
% 3.40/1.90 % (28731)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_28502 for (2998ds/879Mi)
% 3.40/1.91 % (28734)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_28502 for (2998ds/17Mi)
% 3.40/1.92 % (28734)Instruction limit reached!
% 3.40/1.92 % (28734)------------------------------
% 3.40/1.92 % (28734)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.40/1.92 % (28734)Termination reason: Unknown
% 3.40/1.92 % (28734)Termination phase: shuffling
% 3.40/1.92
% 3.40/1.92 % (28734)Memory used [KB]: 1918
% 3.40/1.92 % (28734)Time elapsed: 0.009 s
% 3.40/1.92 % (28734)Instructions burned: 19 (million)
% 3.40/1.92 % (28734)------------------------------
% 3.40/1.92 % (28734)------------------------------
% 3.40/1.93 % (28737)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_28502 for (2998ds/3Mi)
% 3.40/1.93 % (28737)Instruction limit reached!
% 3.40/1.93 % (28737)------------------------------
% 3.40/1.93 % (28737)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.40/1.93 % (28737)Termination reason: Unknown
% 3.40/1.93 % (28737)Termination phase: shuffling
% 3.40/1.93
% 3.40/1.93 % (28737)Memory used [KB]: 1663
% 3.40/1.93 % (28737)Time elapsed: 0.004 s
% 3.40/1.93 % (28737)Instructions burned: 4 (million)
% 3.40/1.93 % (28737)------------------------------
% 3.40/1.93 % (28737)------------------------------
% 3.40/1.95 % (28741)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_28502 for (2998ds/30Mi)
% 3.40/1.96 % (28741)Instruction limit reached!
% 3.40/1.96 % (28741)------------------------------
% 3.40/1.96 % (28741)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.40/1.96 % (28741)Termination reason: Unknown
% 3.40/1.96 % (28741)Termination phase: shuffling
% 3.40/1.96
% 3.40/1.96 % (28741)Memory used [KB]: 2302
% 3.40/1.96 % (28741)Time elapsed: 0.016 s
% 3.40/1.96 % (28741)Instructions burned: 31 (million)
% 3.40/1.96 % (28741)------------------------------
% 3.40/1.96 % (28741)------------------------------
% 3.40/1.98 % (28743)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_28502 for (2997ds/127Mi)
% 5.26/2.04 % (28743)Instruction limit reached!
% 5.26/2.04 % (28743)------------------------------
% 5.26/2.04 % (28743)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.26/2.04 % (28743)Termination reason: Unknown
% 5.26/2.04 % (28743)Termination phase: Property scanning
% 5.26/2.04
% 5.26/2.04 % (28743)Memory used [KB]: 2942
% 5.26/2.04 % (28743)Time elapsed: 0.057 s
% 5.26/2.04 % (28743)Instructions burned: 127 (million)
% 5.26/2.04 % (28743)------------------------------
% 5.26/2.04 % (28743)------------------------------
% 5.26/2.04 % (28727)Instruction limit reached!
% 5.26/2.04 % (28727)------------------------------
% 5.26/2.04 % (28727)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.26/2.04 % (28727)Termination reason: Unknown
% 5.26/2.04 % (28727)Termination phase: Saturation
% 5.26/2.04
% 5.26/2.04 % (28727)Memory used [KB]: 9594
% 5.26/2.04 % (28727)Time elapsed: 0.170 s
% 5.26/2.04 % (28727)Instructions burned: 377 (million)
% 5.26/2.04 % (28727)------------------------------
% 5.26/2.04 % (28727)------------------------------
% 5.26/2.05 % (28750)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_28502 for (2996ds/100Mi)
% 5.26/2.05 % (28751)dis+10_1:1_anc=none:cnfonf=lazy_gen:fd=preordered:fe=off:hud=10:ins=3:ixr=off:nwc=5.0:plsq=on:plsqc=1:plsqr=32,1:sp=const_frequency:uhcvi=on:i=3:si=on:rtra=on_0 on DTF2THF_28502 for (2996ds/3Mi)
% 5.26/2.06 % (28751)Instruction limit reached!
% 5.26/2.06 % (28751)------------------------------
% 5.26/2.06 % (28751)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.26/2.06 % (28751)Termination reason: Unknown
% 5.26/2.06 % (28751)Termination phase: shuffling
% 5.26/2.06
% 5.26/2.06 % (28751)Memory used [KB]: 1663
% 5.26/2.06 % (28751)Time elapsed: 0.004 s
% 5.26/2.06 % (28751)Instructions burned: 5 (million)
% 5.26/2.06 % (28751)------------------------------
% 5.26/2.06 % (28751)------------------------------
% 5.26/2.07 % (28753)lrs+10_8:1_au=on:avsq=on:e2e=on:ins=3:s2a=on:s2at=3.0:ss=axioms:i=20:si=on:rtra=on_0 on DTF2THF_28502 for (2996ds/20Mi)
% 5.26/2.08 % (28720)First to succeed.
% 5.26/2.08 % (28720)Refutation found. Thanks to Tanya!
% 5.26/2.08 % SZS status Theorem for DTF2THF_28502
% 5.26/2.08 % SZS output start Proof for DTF2THF_28502
% 5.26/2.08 thf(func_def_0, type, in: $i > $i > $o).
% 5.26/2.08 thf(func_def_1, type, exu: ($i > $o) > $o).
% 5.26/2.08 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 5.26/2.08 thf(func_def_8, type, powerset: $i > $i).
% 5.26/2.08 thf(func_def_10, type, setunion: $i > $i).
% 5.26/2.08 thf(func_def_19, type, descr: ($i > $o) > $i).
% 5.26/2.08 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 5.26/2.08 thf(func_def_26, type, prop2set: $o > $i).
% 5.26/2.08 thf(func_def_36, type, nonempty: $i > $o).
% 5.26/2.08 thf(func_def_69, type, set2prop: $i > $o).
% 5.26/2.08 thf(func_def_88, type, subset: $i > $i > $o).
% 5.26/2.08 thf(func_def_89, type, disjoint: $i > $i > $o).
% 5.26/2.08 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 5.26/2.08 thf(func_def_114, type, binunion: $i > $i > $i).
% 5.26/2.08 thf(func_def_122, type, binintersect: $i > $i > $i).
% 5.26/2.08 thf(func_def_135, type, regular: $i > $o).
% 5.26/2.08 thf(func_def_136, type, setminus: $i > $i > $i).
% 5.26/2.08 thf(func_def_147, type, symdiff: $i > $i > $i).
% 5.26/2.08 thf(func_def_153, type, iskpair: $i > $o).
% 5.26/2.08 thf(func_def_158, type, kpair: $i > $i > $i).
% 5.26/2.08 thf(func_def_160, type, cartprod: $i > $i > $i).
% 5.26/2.08 thf(func_def_177, type, singleton: $i > $o).
% 5.26/2.08 thf(func_def_179, type, ex1: $i > ($i > $o) > $o).
% 5.26/2.08 thf(func_def_184, type, atmost1p: $i > $o).
% 5.26/2.08 thf(func_def_185, type, atleast2p: $i > $o).
% 5.26/2.08 thf(func_def_186, type, atmost2p: $i > $o).
% 5.26/2.08 thf(func_def_187, type, upairsetp: $i > $o).
% 5.26/2.08 thf(func_def_191, type, kfst: $i > $i).
% 5.26/2.08 thf(func_def_203, type, ksnd: $i > $i).
% 5.26/2.08 thf(func_def_213, type, breln: $i > $i > $i > $o).
% 5.26/2.08 thf(func_def_214, type, dpsetconstr: $i > $i > ($i > $i > $o) > $i).
% 5.26/2.08 thf(func_def_222, type, func: $i > $i > $i > $o).
% 5.26/2.08 thf(func_def_223, type, funcSet: $i > $i > $i).
% 5.26/2.08 thf(func_def_226, type, ap: $i > $i > $i > $i > $i).
% 5.26/2.08 thf(func_def_232, type, lam: $i > $i > ($i > $i) > $i).
% 5.26/2.08 thf(func_def_259, type, if: $i > $o > $i > $i > $i).
% 5.26/2.08 thf(func_def_323, type, breln1: $i > $i > $o).
% 5.26/2.08 thf(func_def_325, type, breln1Set: $i > $i).
% 5.26/2.08 thf(func_def_327, type, transitive: $i > $i > $o).
% 5.26/2.08 thf(func_def_328, type, antisymmetric: $i > $i > $o).
% 5.26/2.08 thf(func_def_329, type, reflexive: $i > $i > $o).
% 5.26/2.08 thf(func_def_330, type, refltransitive: $i > $i > $o).
% 5.26/2.08 thf(func_def_331, type, refllinearorder: $i > $i > $o).
% 5.26/2.08 thf(func_def_332, type, reflwellordering: $i > $i > $o).
% 5.26/2.08 thf(func_def_338, type, breln1invset: $i > $i > $i).
% 5.26/2.08 thf(func_def_342, type, breln1compset: $i > $i > $i > $i).
% 5.26/2.08 thf(func_def_360, type, image1: $i > ($i > $i) > $i).
% 5.26/2.08 thf(func_def_364, type, injective: $i > $i > $i > $o).
% 5.26/2.08 thf(func_def_365, type, injFuncSet: $i > $i > $i).
% 5.26/2.08 thf(func_def_369, type, surjective: $i > $i > $i > $o).
% 5.26/2.08 thf(func_def_370, type, surjFuncSet: $i > $i > $i).
% 5.26/2.08 thf(func_def_378, type, omegaS: $i > $i).
% 5.26/2.08 thf(func_def_394, type, sP0: $i > $i > $i > $o).
% 5.26/2.08 thf(func_def_397, type, sP3: $i > $i > $o).
% 5.26/2.08 thf(func_def_398, type, sP4: $i > $o).
% 5.26/2.08 thf(func_def_399, type, sP5: $i > $i > $o).
% 5.26/2.08 thf(func_def_400, type, sP6: $i > $i > $o).
% 5.26/2.08 thf(func_def_404, type, sP10: $i > $i > $i > $o).
% 5.26/2.08 thf(func_def_406, type, sP12: $i > $i > $i > $i > $o).
% 5.26/2.08 thf(func_def_410, type, sP16: $i > $o > $i > $i > $o).
% 5.26/2.08 thf(func_def_412, type, sK18: $i > $o).
% 5.26/2.08 thf(func_def_415, type, sK21: $i > ($i > $o) > $i).
% 5.26/2.08 thf(func_def_416, type, sK22: $i > ($i > $o) > $i).
% 5.26/2.08 thf(func_def_417, type, sK23: $i > $o).
% 5.26/2.08 thf(func_def_425, type, sK31: $i > $i > $i).
% 5.26/2.08 thf(func_def_432, type, sK38: $i > $i).
% 5.26/2.08 thf(func_def_436, type, sK42: $i > $o).
% 5.26/2.08 thf(func_def_443, type, sK49: $i > $i > $o).
% 5.26/2.08 thf(func_def_444, type, sK50: $i > ($i > $o) > $i).
% 5.26/2.08 thf(func_def_445, type, sK51: $i > $o).
% 5.26/2.08 thf(func_def_455, type, sK61: $i > $i).
% 5.26/2.08 thf(func_def_458, type, sK64: $i > ($i > $i) > $i > $i).
% 5.26/2.08 thf(func_def_459, type, sK65: $i > $i > ($i > $i) > $i > $i).
% 5.26/2.08 thf(func_def_467, type, sK73: $i > $i > $o).
% 5.26/2.08 thf(func_def_481, type, sK87: $i > $i > ($i > $o) > $i).
% 5.26/2.08 thf(func_def_482, type, sK88: $i > $o).
% 5.26/2.08 thf(func_def_497, type, sK103: $i > $i).
% 5.26/2.08 thf(func_def_509, type, sK115: $i > $i > $i > $i).
% 5.26/2.08 thf(func_def_510, type, sK116: $i > $i > $i > $i).
% 5.26/2.08 thf(func_def_516, type, sK122: $i > $i > $i > $i).
% 5.26/2.08 thf(func_def_520, type, sK126: $i > $i > $i > $i).
% 5.26/2.08 thf(func_def_533, type, sK139: $i > $o).
% 5.26/2.08 thf(func_def_534, type, sK140: $i > $i > $o).
% 5.26/2.08 thf(func_def_547, type, sK153: $i > $i > $o).
% 5.26/2.08 thf(func_def_557, type, sK163: $i > $i > ($i > $i) > $i).
% 5.26/2.08 thf(func_def_558, type, sK164: $i > $i).
% 5.26/2.08 thf(func_def_566, type, sK172: $i > $i > $i).
% 5.26/2.08 thf(func_def_599, type, sK205: $i > $i > $i).
% 5.26/2.08 thf(func_def_604, type, sK210: $i > $i > $i).
% 5.26/2.08 thf(func_def_605, type, sK211: $i > $i > $i).
% 5.26/2.08 thf(func_def_607, type, sK213: $i > $o).
% 5.26/2.08 thf(func_def_624, type, sK230: $i > $o).
% 5.26/2.08 thf(func_def_642, type, sK248: $i > $i > $i > $i).
% 5.26/2.08 thf(func_def_646, type, sK252: $i > $i > $i).
% 5.26/2.08 thf(func_def_669, type, sK275: $i > $i > $i > $i > $i > $i).
% 5.26/2.08 thf(func_def_677, type, sK283: $i > $i).
% 5.26/2.08 thf(func_def_694, type, sK300: $i > $i).
% 5.26/2.08 thf(func_def_695, type, sK301: ($i > $i) > $i > $i > $i).
% 5.26/2.08 thf(func_def_703, type, sK309: $i > $i > $i).
% 5.26/2.08 thf(func_def_704, type, sK310: $i > $i > $i).
% 5.26/2.08 thf(func_def_705, type, sK311: $i > $i > $i).
% 5.26/2.08 thf(func_def_706, type, sK312: $i > $i > $i).
% 5.26/2.08 thf(func_def_707, type, sK313: $i > $i > $i > $i).
% 5.26/2.08 thf(func_def_708, type, sK314: $i > $i).
% 5.26/2.08 thf(func_def_709, type, sK315: $i > $i).
% 5.26/2.08 thf(func_def_710, type, sK316: $i > $i).
% 5.26/2.08 thf(func_def_711, type, sK317: $i > $i).
% 5.26/2.08 thf(func_def_712, type, sK318: $i > $i > $i > $i).
% 5.26/2.08 thf(func_def_713, type, sK319: $i > $i > $i > $i).
% 5.26/2.08 thf(func_def_714, type, sK320: $i > $i > $i).
% 5.26/2.08 thf(func_def_715, type, sK321: $i > $i > $i).
% 5.26/2.08 thf(func_def_716, type, sK322: $i > $i > $i).
% 5.26/2.08 thf(func_def_717, type, sK323: $i > $i).
% 5.26/2.08 thf(func_def_718, type, sK324: $i > $i > $i).
% 5.26/2.08 thf(func_def_720, type, sK326: $i > $i).
% 5.26/2.08 thf(func_def_721, type, sK327: $i > $i).
% 5.26/2.08 thf(func_def_742, type, sK348: $i > $i).
% 5.26/2.08 thf(func_def_750, type, sK356: $i > $o).
% 5.26/2.08 thf(func_def_751, type, sK357: ($i > $o) > $i).
% 5.26/2.08 thf(func_def_758, type, sK364: $i > $o).
% 5.26/2.08 thf(func_def_766, type, sK372: $i > $i).
% 5.26/2.08 thf(func_def_769, type, sK375: $i > $i > ($i > $i) > $i).
% 5.26/2.08 thf(func_def_771, type, sK377: $i > ($i > $i) > $i > $i).
% 5.26/2.08 thf(func_def_773, type, sK379: $i > $i).
% 5.26/2.08 thf(func_def_778, type, sK384: $i > $i > $o).
% 5.26/2.08 thf(func_def_784, type, sK390: $i > $i > $i).
% 5.26/2.08 thf(func_def_807, type, sK413: $i > $i > $i > $i > $i).
% 5.26/2.08 thf(func_def_829, type, sK435: ($i > $o) > ($i > $o) > $i).
% 5.26/2.08 thf(func_def_830, type, sK436: ($i > $o) > ($i > $o) > $i).
% 5.26/2.08 thf(func_def_831, type, sK437: $i > $o).
% 5.26/2.08 thf(func_def_832, type, sK438: $i > $o).
% 5.26/2.08 thf(func_def_837, type, sK443: $i > $i).
% 5.26/2.08 thf(func_def_838, type, sK444: $i > $i).
% 5.26/2.08 thf(func_def_843, type, sK449: $i > $i > $i > $i > $i).
% 5.26/2.08 thf(func_def_884, type, sK490: $i > $o).
% 5.26/2.08 thf(func_def_893, type, sK499: $i > $o).
% 5.26/2.08 thf(func_def_894, type, sK500: $i > $o).
% 5.26/2.08 thf(func_def_895, type, sK501: ($i > $o) > ($i > $o) > $i).
% 5.26/2.08 thf(func_def_896, type, sK502: ($i > $o) > ($i > $o) > $i).
% 5.26/2.08 thf(func_def_908, type, sK514: $o > $i > $i > $i > $i > $i > $i).
% 5.26/2.08 thf(func_def_935, type, sK541: $i > $o).
% 5.26/2.08 thf(func_def_938, type, sK544: $i > ($i > $o) > $i).
% 5.26/2.08 thf(func_def_939, type, sK545: $i > ($i > $o) > $i).
% 5.26/2.08 thf(func_def_941, type, sK547: $i > $o).
% 5.26/2.08 thf(func_def_944, type, sK550: $i > ($i > $o) > $i).
% 5.26/2.08 thf(func_def_945, type, sK551: $i > $o).
% 5.26/2.08 thf(func_def_948, type, sK554: $i > $o).
% 5.26/2.08 thf(func_def_951, type, sK557: $i > $o).
% 5.26/2.08 thf(func_def_952, type, sK558: $i > $i).
% 5.26/2.08 thf(func_def_953, type, sK559: ($i > $o) > $i).
% 5.26/2.08 thf(func_def_969, type, sK575: $i > $i).
% 5.26/2.08 thf(func_def_971, type, sK577: $i > ($i > $o) > $i).
% 5.26/2.08 thf(func_def_972, type, sK578: $i > $o).
% 5.26/2.08 thf(func_def_976, type, sK582: $i > $i > $o).
% 5.26/2.08 thf(func_def_988, type, sK594: $i > $i > ($i > $i > $o) > $i).
% 5.26/2.08 thf(func_def_989, type, sK595: $i > $i > $o).
% 5.26/2.08 thf(func_def_992, type, sK598: $i > $i).
% 5.26/2.08 thf(func_def_996, type, sK602: $i > $i > $o).
% 5.26/2.08 thf(func_def_1001, type, sK607: $i > $i).
% 5.26/2.08 thf(func_def_1002, type, sK608: $i > $i).
% 5.26/2.08 thf(func_def_1003, type, sK609: $i > $i).
% 5.26/2.08 thf(func_def_1004, type, sK610: ($i > $i) > $i > $i).
% 5.26/2.08 thf(func_def_1005, type, sK611: $i > ($i > $i) > $i > $i).
% 5.26/2.08 thf(func_def_1017, type, sK623: $i > $o).
% 5.26/2.08 thf(func_def_1019, type, sK625: $i > ($i > $o) > $i).
% 5.26/2.08 thf(func_def_1024, type, sK630: $i > $i).
% 5.26/2.08 thf(func_def_1033, type, sK639: $i > $i).
% 5.26/2.08 thf(func_def_1036, type, sK642: $i > ($i > $i) > $i > $i).
% 5.26/2.08 thf(func_def_1037, type, sK643: $i > $o).
% 5.26/2.08 thf(func_def_1039, type, sK645: ($i > $o) > $i).
% 5.26/2.08 thf(func_def_1040, type, sK646: ($i > $o) > $i).
% 5.26/2.08 thf(func_def_1043, type, sK649: $i > ($i > $o) > $i).
% 5.26/2.08 thf(func_def_1044, type, sK650: $i > $o).
% 5.26/2.08 thf(func_def_1061, type, sK667: $i > $i).
% 5.26/2.08 thf(func_def_1065, type, sK671: $i > $i > ($i > $i) > $i).
% 5.26/2.08 thf(func_def_1069, type, sK675: $i > $i).
% 5.26/2.08 thf(func_def_1072, type, sK678: $i > $i > ($i > $i) > $i).
% 5.26/2.08 thf(func_def_1076, type, sK682: $i > $i > $i > $i).
% 5.26/2.08 thf(func_def_1077, type, sK683: $i > $i > $i > $i).
% 5.26/2.08 thf(func_def_1078, type, sK684: $i > $i > $i > $i).
% 5.26/2.08 thf(func_def_1079, type, sK685: $i > $i > $i > $i).
% 5.26/2.08 thf(func_def_1089, type, sK695: $i > $i > $i > $i).
% 5.26/2.08 thf(func_def_1097, type, sK703: $i > $i > $i > $i > $i).
% 5.26/2.08 thf(func_def_1098, type, sK704: $i > $i > $i > $i > $i).
% 5.26/2.08 thf(func_def_1099, type, sK705: $i > $i > $i > $i > $i).
% 5.26/2.08 thf(func_def_1100, type, sK706: $i > $i > $i > $i > $i).
% 5.26/2.08 thf(func_def_1103, type, sK709: $i > $i > $i).
% 5.26/2.08 thf(func_def_1111, type, sK717: $i > $i).
% 5.26/2.08 thf(func_def_1114, type, sK720: $i > $i > $i).
% 5.26/2.08 thf(func_def_1133, type, sK739: $i > $i).
% 5.26/2.08 thf(func_def_1135, type, sK741: $i > ($i > $i) > $i > $i).
% 5.26/2.08 thf(func_def_1150, type, sK756: $o > $i > $i > $i).
% 5.26/2.08 thf(func_def_1206, type, sK812: $i > $i > $i).
% 5.26/2.08 thf(func_def_1209, type, sK815: $i > $o).
% 5.26/2.08 thf(func_def_1217, type, sK823: $i > $o).
% 5.26/2.08 thf(func_def_1220, type, sK826: $i > ($i > $o) > $i).
% 5.26/2.08 thf(func_def_1221, type, sK827: $i > ($i > $o) > $i).
% 5.26/2.08 thf(func_def_1232, type, sK838: $i > $o).
% 5.26/2.08 thf(func_def_1234, type, sK840: ($i > $o) > $i > $i > $i).
% 5.26/2.08 thf(func_def_1235, type, sK841: ($i > $o) > $i > $i > $i).
% 5.26/2.08 thf(func_def_1236, type, sK842: ($i > $o) > $i > $i > $i > $i).
% 5.26/2.08 thf(func_def_1237, type, sK843: ($i > $o) > $i > $i > $i > $i).
% 5.26/2.08 thf(func_def_1241, type, sK847: $i > $o).
% 5.26/2.08 thf(func_def_1256, type, sK862: $i > $o).
% 5.26/2.08 thf(func_def_1258, type, sK864: ($i > $o) > $i > $i > $i).
% 5.26/2.08 thf(func_def_1259, type, sK865: ($i > $o) > $i > $i > $i).
% 5.26/2.08 thf(func_def_1277, type, sK883: $i > $i).
% 5.26/2.08 thf(func_def_1281, type, sK887: $i > $o).
% 5.26/2.08 thf(func_def_1284, type, sK890: $i > ($i > $o) > $i).
% 5.26/2.08 thf(func_def_1296, type, sK902: $i > ($i > $o) > $i).
% 5.26/2.08 thf(func_def_1297, type, sK903: $i > $o).
% 5.26/2.08 thf(func_def_1307, type, sK913: $i > $i > $i > $i > $i).
% 5.26/2.08 thf(func_def_1308, type, sK914: $i > $i > $i > $i > $i).
% 5.26/2.08 thf(func_def_1319, type, sK925: $i > $o).
% 5.26/2.08 thf(func_def_1321, type, sK927: ($i > $o) > $i > $i > $i > $i).
% 5.26/2.08 thf(func_def_1322, type, sK928: ($i > $o) > $i > $i > $i > $i).
% 5.26/2.08 thf(func_def_1323, type, sK929: $i > $i > $o).
% 5.26/2.08 thf(func_def_1325, type, sK931: $i > $i).
% 5.26/2.08 thf(func_def_1326, type, sK932: $i > $i).
% 5.26/2.08 thf(func_def_1327, type, sK933: $i > ($i > $i > $o) > $i).
% 5.26/2.08 thf(func_def_1328, type, sK934: $i > ($i > $i > $o) > $i).
% 5.26/2.08 thf(func_def_1329, type, sK935: $i > $i > ($i > $i > $o) > $i).
% 5.26/2.08 thf(func_def_1330, type, sK936: $i > $o).
% 5.26/2.08 thf(func_def_1341, type, sK947: $i > $i > $i > $i > $i).
% 5.26/2.08 thf(func_def_1357, type, sK963: $i > $o).
% 5.26/2.08 thf(func_def_1361, type, sK967: $i > $i > $o).
% 5.26/2.08 thf(func_def_1366, type, sK972: $i > $i > $i > $i).
% 5.26/2.08 thf(func_def_1367, type, sK973: $i > $i > $i > $i).
% 5.26/2.08 thf(func_def_1374, type, sK980: $i > $o).
% 5.26/2.08 thf(func_def_1390, type, sK996: $i > $o).
% 5.26/2.08 thf(func_def_1439, type, sK1045: $i > $o).
% 5.26/2.08 thf(func_def_1440, type, sK1046: $i > $i).
% 5.26/2.08 thf(func_def_1441, type, sK1047: ($i > $o) > $i).
% 5.26/2.08 thf(func_def_1466, type, sK1072: ($i > $o) > ($i > $o) > $i > $i > $i).
% 5.26/2.08 thf(func_def_1467, type, sK1073: ($i > $o) > ($i > $o) > $i > $i > $i).
% 5.26/2.08 thf(func_def_1470, type, sK1076: $i > $o).
% 5.26/2.08 thf(func_def_1471, type, sK1077: $i > $o).
% 5.26/2.08 thf(func_def_1486, type, ph1092: !>[X0: $tType]:(X0)).
% 5.26/2.08 thf(f5176,plain,(
% 5.26/2.08 $false),
% 5.26/2.08 inference(trivial_inequality_removal,[status(thm)],[f5169])).
% 5.26/2.08 thf(f5169,plain,(
% 5.26/2.08 ($true != $true)),
% 5.26/2.08 inference(superposition,[status(thm)],[f5152,f4125])).
% 5.26/2.08 thf(f4125,plain,(
% 5.26/2.08 ($true = (in @ sK655 @ omega))),
% 5.26/2.08 inference(cnf_transformation,[status(thm)],[f2443])).
% 5.26/2.08 thf(f2443,plain,(
% 5.26/2.08 (setminusER = $true) & (wellorderingAx = $true) & (setukpairinjL1 = $true) & (image1E = $true) & (inComplementUnionImpNotIn1 = $true) & (notinself = $true) & (setminusI = $true) & (upairequniteq = $true) & (surjFuncSetFuncIn = $true) & (setunionI = $true) & (setadjoinSub2 = $true) & (quantDeMorgan4 = $true) & (ksndsingleton = $true) & (binunionTIRcontra = $true) & (emptyset__Cong = $true) & (iftrueProp1 = $true) & (funcImageSingleton = $true) & (setOfPairsIsBReln1 = $true) & (powersetTI1 = $true) & (quantDeMorgan1 = $true) & (demorgan2a1 = $true) & (setunionsingleton2 = $true) & (ap2apEq2 = $true) & (surjCantorThm = $true) & (inIntersectImpInUnion = $true) & (upairset2IR = $true) & (setextT = $true) & (iftrueProp2 = $true) & (theprop = $true) & (woz1_5 = $true) & (subsetTrans = $true) & (setextAx = $true) & (complementTcontraSubset = $true) & (omegaIndAx = $true) & (exuE3e = $true) & (emptyInPowerset = $true) & (kpairiskpair = $true) & (binintersectEL = $true) & (lam2lamEq = $true) & (notinemptyset = $true) & (setadjoinOr = $true) & (singletonsswitch = $true) & (complementTE1 = $true) & (setadjoinIR = $true) & (notsubsetI = $true) & (setunionsingleton = $true) & (setextsub = $true) & (funcGraphProp3 = $true) & (setOfPairsIsBReln = $true) & (ubforcartprodlem1 = $true) & (setbeta = $true) & (emptyinPowerset = $true) & (cartprodpairmemEL = $true) & (binintersectSubset1 = $true) & (contrasubsetT1 = $true) & (emptyE1 = $true) & (binintersectT_lem = $true) & (subsetRefl = $true) & (symdiffI2 = $true) & (ex1I = $true) & (omegaSAx = $true) & (ap2apEq1 = $true) & (eqimpsubset1 = $true) & (powerset__Cong = $true) & (singletonprop = $true) & (funcext = $true) & (funcext2 = $true) & (binunionTE = $true) & (subbreln1 = $true) & (injFuncSetFuncIn = $true) & (demorgan2 = $true) & (woz1_1 = $true) & (image1Ex = $true) & (ex1I2 = $true) & (emptyI = $true) & (cartprodfstpairEq = $true) & (woz1_2 = $true) & (breln1unionEcases = $true) & (breln1unionE = $true) & (complementTI1 = $true) & (binintersectSubset2 = $true) & (exuI1 = $true) & (nonemptyE1 = $true) & (powersetAx = $true) & (upairinpowunion = $true) & (omegaSp = $true) & (sepInPowerset = $true) & (notdallE = $true) & (powersetsubset = $true) & (woz1_3 = $true) & (cartprodsndpairEq = $true) & (setukpairinjR = $true) & (kpairp = $true) & (iffalseProp1 = $true) & (complementUnionInPowersetComplement = $true) & (contrasubsetT = $true) & (powersetI = $true) & (complementImpComplementIntersect = $true) & (setminusSubset1 = $true) & (setunionAx = $true) & (sepSubset = $true) & (setadjoinE = $true) & (breln1compI = $true) & (complementT_lem = $true) & (setukpairIL = $true) & (eqbreln = $true) & (exuE2 = $true) & (singletoninpowunion = $true) & (symdiffIneg2 = $true) & (doubleComplementI1 = $true) & (inIntersectImpInUnion2 = $true) & (subsetE = $true) & (eta2 = $true) & (setadjoinAx = $true) & (subPowSU = $true) & (doubleComplementSub2 = $true) & (lam2p = $true) & (intersectInPowersetIntersectUnions = $true) & (iffalseProp2 = $true) & (secondinupair = $true) & (setukpairinjR12 = $true) & (upairsubunion = $true) & (demorgan1 = $true) & (emptysetimpfalse = $true) & (cartprodpairin = $true) & (kfstpairEq = $true) & (eta1 = $true) & (powersetE = $true) & (demorgan2a2 = $true) & (breln1invE = $true) & (injFuncSetFuncInj = $true) & (setunion__Cong = $true) & (cartprodfstin = $true) & (contrasubsetT3 = $true) & (setunionsingleton1 = $true) & (emptysetAx = $true) & (quantDeMorgan2 = $true) & (dsetconstrEL = $true) & (setminusERneg = $true) & (dpsetconstrERa = $true) & (eqimpsubset2 = $true) & (theeq = $true) & (symdiffE = $true) & (exu__Cong = $true) & (choice2fnsingleton = $true) & (demorgan2b = $true) & (powersetT_lem = $true) & (replAx = $true) & (iftrueorfalse = $true) & (inIntersectImpInIntersectUnions = $true) & (exuEu = $true) & (noeltsimpempty = $true) & (binunionEcases = $true) & (binunionIL = $true) & (dsetconstrI = $true) & (exuE3u = $true) & (ex1E2 = $true) & (subbreln = $true) & (breln1unionI = $true) & (funcinfuncset = $true) & (emptyinunitempty = $true) & (breln1unionIR = $true) & (notequalI1 = $true) & (setunionE2 = $true) & (omega0Ax = $true) & (powersetI1 = $true) & (setext = $true) & (doubleComplementSub1 = $true) & (woz2W = $true) & (binintersectSubset5 = $true) & (subsetI1 = $true) & (setukpairinjR2 = $true) & (ex1E1 = $true) & (cartprodmempair = $true) & (iftrue = $true) & (foundationAx = $true) & (notequalI2 = $true) & (nonemptyImpWitness = $true) & (binunionIR = $true) & (ksndpairEq = $true) & (subsetTI = $true) & (funcextLem = $true) & (dpsetconstrI = $true) & (binintersectI = $true) & (descrp = $true) & (image1Ex1 = $true) & (ifp = $true) & (binintersectTELcontra = $true) & (($true = (in @ sK655 @ omega)) & ($true != (in @ (setadjoin @ sK655 @ sK655) @ omega))) & (woz13rule2 = $true) & (upairsetE = $true) & (breln1unionprop = $true) & (funcGraphProp1 = $true) & (cartprodmempair1 = $true) & (lamProp = $true) & (woz13rule4 = $true) & (leftInvIsSurj = $true) & (demorgan1a = $true) & (image1I = $true) & (upairsetIR = $true) & (cartprodpairsurjEq = $true) & (breln1all2 = $true) & (ubforcartprodlem3 = $true) & (woz1_4 = $true) & (demorgan2a = $true) & (setunionE = $true) & (disjointsetsI1 = $true) & (doubleComplementE1 = $true) & (dpsetconstrER = $true) & (subset2powerset = $true) & (subsetE2 = $true) & (woz2Ex = $true) & (brelnall1 = $true) & (setukpairinjR1 = $true) & (symdiffIneg1 = $true) & (iffalse = $true) & (notinself2 = $true) & (beta2 = $true) & (exuI2 = $true) & (powersetE1 = $true) & (demorgan2b2 = $true) & (woz2B = $true) & (kpairsurjEq = $true) & (binunionTEcontra = $true) & (infuncsetfunc = $true) & (setminusELneg = $true) & (nonemptyI1 = $true) & (setoftrueEq = $true) & (breln1compprop = $true) & (binunionE = $true) & (cartprodsndin = $true) & (cartprodpairmemER = $true) & (singletonsuniq = $true) & (inCongP = $true) & (vacuousDall = $true) & (binintersectSubset4 = $true) & (emptysetE = $true) & (complementTnotintersectT = $true) & (in__Cong = $true) & (prop2setE = $true) & (dsetconstrER = $true) & (woz13rule0 = $true) & (subsetemptysetimpeq = $true) & (breln1invprop = $true) & (dpsetconstrEL1 = $true) & (app = $true) & (setminusIRneg = $true) & (eqinunit = $true) & (cartprodmempaircEq = $true) & (singletonsubset = $true) & (setukpairinjR11 = $true) & (setadjoinIL = $true) & (powersetTE1 = $true) & (prop2setI = $true) & (prop2set2propI = $true) & (complementSubsetComplementIntersect = $true) & (binintersectER = $true) & (uniqinunit = $true) & (binunionLsub = $true) & (setukpairinjL = $true) & (surjFuncSetFuncSurj = $true) & (beta1 = $true) & (notinsingleton = $true) & (lamp = $true) & (setukpairIR = $true) & (nonemptyI = $true) & (exuE1 = $true) & (setukpairinjL2 = $true) & (descr__Cong = $true) & (bs114d = $true) & (dpsetconstrEL2 = $true) & (doubleComplementEq = $true) & (dpsetconstrSub = $true) & (binunionRsub = $true) & (binintersectRsub = $true) & (breln1SetBreln1 = $true) & (demorgan1b = $true) & (setminusT_lem = $true) & (complementInPowersetComplementIntersect = $true) & (binunionTILcontra = $true) & (setminusILneg = $true) & (exuI3 = $true) & (ifSingleton = $true) & (setminusEL = $true) & (binintersectTERcontra = $true) & (breln1all1 = $true) & (singletoninpowerset = $true) & (inComplementUnionImpInComplement1 = $true) & (eqbreln1 = $true) & (inPowerset = $true) & (breln1unionCommutes = $true) & (brelnall2 = $true) & (setminusLsub = $true) & (contrasubsetT2 = $true) & (binintersectLsub = $true) & (breln1compEex = $true) & (contraSubsetComplement = $true) & (injFuncInInjFuncSet = $true) & (setadjoinSub = $true) & (breln1invI = $true) & (dsetconstr__Cong = $true) & (binunionT_lem = $true) & (breln1unionIL = $true) & (funcGraphProp4 = $true) & (ap2p = $true) & (omega__Cong = $true) & (foundation2 = $true) & (ubforcartprodlem2 = $true) & (upairsetIL = $true) & (setminusSubset2 = $true) & (breln1compE = $true) & (subsetI2 = $true) & (image1Equiv = $true) & (woz2A = $true) & (funcGraphProp2 = $true) & (symdiffI1 = $true) & (kfstsingleton = $true) & (quantDeMorgan3 = $true) & (emptysetsubset = $true) & (notdexE = $true) & (upairset2E = $true) & (apProp = $true) & (woz13rule3 = $true) & (setadjoin__Cong = $true) & (woz13rule1 = $true) & (binintersectSubset3 = $true)),
% 5.26/2.08 inference(skolemisation,[status(esa),new_symbols(skolem,[sK655])],[f1072,f2442])).
% 5.26/2.08 thf(f2442,plain,(
% 5.26/2.08 ? [X0 : $i] : (((in @ X0 @ omega) = $true) & ($true != (in @ (setadjoin @ X0 @ X0) @ omega))) => (($true = (in @ sK655 @ omega)) & ($true != (in @ (setadjoin @ sK655 @ sK655) @ omega)))),
% 5.26/2.08 introduced(definition,[],[choice_axiom])).
% 5.26/2.08 thf(f1072,plain,(
% 5.26/2.08 (setminusER = $true) & (wellorderingAx = $true) & (setukpairinjL1 = $true) & (image1E = $true) & (inComplementUnionImpNotIn1 = $true) & (notinself = $true) & (setminusI = $true) & (upairequniteq = $true) & (surjFuncSetFuncIn = $true) & (setunionI = $true) & (setadjoinSub2 = $true) & (quantDeMorgan4 = $true) & (ksndsingleton = $true) & (binunionTIRcontra = $true) & (emptyset__Cong = $true) & (iftrueProp1 = $true) & (funcImageSingleton = $true) & (setOfPairsIsBReln1 = $true) & (powersetTI1 = $true) & (quantDeMorgan1 = $true) & (demorgan2a1 = $true) & (setunionsingleton2 = $true) & (ap2apEq2 = $true) & (surjCantorThm = $true) & (inIntersectImpInUnion = $true) & (upairset2IR = $true) & (setextT = $true) & (iftrueProp2 = $true) & (theprop = $true) & (woz1_5 = $true) & (subsetTrans = $true) & (setextAx = $true) & (complementTcontraSubset = $true) & (omegaIndAx = $true) & (exuE3e = $true) & (emptyInPowerset = $true) & (kpairiskpair = $true) & (binintersectEL = $true) & (lam2lamEq = $true) & (notinemptyset = $true) & (setadjoinOr = $true) & (singletonsswitch = $true) & (complementTE1 = $true) & (setadjoinIR = $true) & (notsubsetI = $true) & (setunionsingleton = $true) & (setextsub = $true) & (funcGraphProp3 = $true) & (setOfPairsIsBReln = $true) & (ubforcartprodlem1 = $true) & (setbeta = $true) & (emptyinPowerset = $true) & (cartprodpairmemEL = $true) & (binintersectSubset1 = $true) & (contrasubsetT1 = $true) & (emptyE1 = $true) & (binintersectT_lem = $true) & (subsetRefl = $true) & (symdiffI2 = $true) & (ex1I = $true) & (omegaSAx = $true) & (ap2apEq1 = $true) & (eqimpsubset1 = $true) & (powerset__Cong = $true) & (singletonprop = $true) & (funcext = $true) & (funcext2 = $true) & (binunionTE = $true) & (subbreln1 = $true) & (injFuncSetFuncIn = $true) & (demorgan2 = $true) & (woz1_1 = $true) & (image1Ex = $true) & (ex1I2 = $true) & (emptyI = $true) & (cartprodfstpairEq = $true) & (woz1_2 = $true) & (breln1unionEcases = $true) & (breln1unionE = $true) & (complementTI1 = $true) & (binintersectSubset2 = $true) & (exuI1 = $true) & (nonemptyE1 = $true) & (powersetAx = $true) & (upairinpowunion = $true) & (omegaSp = $true) & (sepInPowerset = $true) & (notdallE = $true) & (powersetsubset = $true) & (woz1_3 = $true) & (cartprodsndpairEq = $true) & (setukpairinjR = $true) & (kpairp = $true) & (iffalseProp1 = $true) & (complementUnionInPowersetComplement = $true) & (contrasubsetT = $true) & (powersetI = $true) & (complementImpComplementIntersect = $true) & (setminusSubset1 = $true) & (setunionAx = $true) & (sepSubset = $true) & (setadjoinE = $true) & (breln1compI = $true) & (complementT_lem = $true) & (setukpairIL = $true) & (eqbreln = $true) & (exuE2 = $true) & (singletoninpowunion = $true) & (symdiffIneg2 = $true) & (doubleComplementI1 = $true) & (inIntersectImpInUnion2 = $true) & (subsetE = $true) & (eta2 = $true) & (setadjoinAx = $true) & (subPowSU = $true) & (doubleComplementSub2 = $true) & (lam2p = $true) & (intersectInPowersetIntersectUnions = $true) & (iffalseProp2 = $true) & (secondinupair = $true) & (setukpairinjR12 = $true) & (upairsubunion = $true) & (demorgan1 = $true) & (emptysetimpfalse = $true) & (cartprodpairin = $true) & (kfstpairEq = $true) & (eta1 = $true) & (powersetE = $true) & (demorgan2a2 = $true) & (breln1invE = $true) & (injFuncSetFuncInj = $true) & (setunion__Cong = $true) & (cartprodfstin = $true) & (contrasubsetT3 = $true) & (setunionsingleton1 = $true) & (emptysetAx = $true) & (quantDeMorgan2 = $true) & (dsetconstrEL = $true) & (setminusERneg = $true) & (dpsetconstrERa = $true) & (eqimpsubset2 = $true) & (theeq = $true) & (symdiffE = $true) & (exu__Cong = $true) & (choice2fnsingleton = $true) & (demorgan2b = $true) & (powersetT_lem = $true) & (replAx = $true) & (iftrueorfalse = $true) & (inIntersectImpInIntersectUnions = $true) & (exuEu = $true) & (noeltsimpempty = $true) & (binunionEcases = $true) & (binunionIL = $true) & (dsetconstrI = $true) & (exuE3u = $true) & (ex1E2 = $true) & (subbreln = $true) & (breln1unionI = $true) & (funcinfuncset = $true) & (emptyinunitempty = $true) & (breln1unionIR = $true) & (notequalI1 = $true) & (setunionE2 = $true) & (omega0Ax = $true) & (powersetI1 = $true) & (setext = $true) & (doubleComplementSub1 = $true) & (woz2W = $true) & (binintersectSubset5 = $true) & (subsetI1 = $true) & (setukpairinjR2 = $true) & (ex1E1 = $true) & (cartprodmempair = $true) & (iftrue = $true) & (foundationAx = $true) & (notequalI2 = $true) & (nonemptyImpWitness = $true) & (binunionIR = $true) & (ksndpairEq = $true) & (subsetTI = $true) & (funcextLem = $true) & (dpsetconstrI = $true) & (binintersectI = $true) & (descrp = $true) & (image1Ex1 = $true) & (ifp = $true) & (binintersectTELcontra = $true) & ? [X0 : $i] : (((in @ X0 @ omega) = $true) & ($true != (in @ (setadjoin @ X0 @ X0) @ omega))) & (woz13rule2 = $true) & (upairsetE = $true) & (breln1unionprop = $true) & (funcGraphProp1 = $true) & (cartprodmempair1 = $true) & (lamProp = $true) & (woz13rule4 = $true) & (leftInvIsSurj = $true) & (demorgan1a = $true) & (image1I = $true) & (upairsetIR = $true) & (cartprodpairsurjEq = $true) & (breln1all2 = $true) & (ubforcartprodlem3 = $true) & (woz1_4 = $true) & (demorgan2a = $true) & (setunionE = $true) & (disjointsetsI1 = $true) & (doubleComplementE1 = $true) & (dpsetconstrER = $true) & (subset2powerset = $true) & (subsetE2 = $true) & (woz2Ex = $true) & (brelnall1 = $true) & (setukpairinjR1 = $true) & (symdiffIneg1 = $true) & (iffalse = $true) & (notinself2 = $true) & (beta2 = $true) & (exuI2 = $true) & (powersetE1 = $true) & (demorgan2b2 = $true) & (woz2B = $true) & (kpairsurjEq = $true) & (binunionTEcontra = $true) & (infuncsetfunc = $true) & (setminusELneg = $true) & (nonemptyI1 = $true) & (setoftrueEq = $true) & (breln1compprop = $true) & (binunionE = $true) & (cartprodsndin = $true) & (cartprodpairmemER = $true) & (singletonsuniq = $true) & (inCongP = $true) & (vacuousDall = $true) & (binintersectSubset4 = $true) & (emptysetE = $true) & (complementTnotintersectT = $true) & (in__Cong = $true) & (prop2setE = $true) & (dsetconstrER = $true) & (woz13rule0 = $true) & (subsetemptysetimpeq = $true) & (breln1invprop = $true) & (dpsetconstrEL1 = $true) & (app = $true) & (setminusIRneg = $true) & (eqinunit = $true) & (cartprodmempaircEq = $true) & (singletonsubset = $true) & (setukpairinjR11 = $true) & (setadjoinIL = $true) & (powersetTE1 = $true) & (prop2setI = $true) & (prop2set2propI = $true) & (complementSubsetComplementIntersect = $true) & (binintersectER = $true) & (uniqinunit = $true) & (binunionLsub = $true) & (setukpairinjL = $true) & (surjFuncSetFuncSurj = $true) & (beta1 = $true) & (notinsingleton = $true) & (lamp = $true) & (setukpairIR = $true) & (nonemptyI = $true) & (exuE1 = $true) & (setukpairinjL2 = $true) & (descr__Cong = $true) & (bs114d = $true) & (dpsetconstrEL2 = $true) & (doubleComplementEq = $true) & (dpsetconstrSub = $true) & (binunionRsub = $true) & (binintersectRsub = $true) & (breln1SetBreln1 = $true) & (demorgan1b = $true) & (setminusT_lem = $true) & (complementInPowersetComplementIntersect = $true) & (binunionTILcontra = $true) & (setminusILneg = $true) & (exuI3 = $true) & (ifSingleton = $true) & (setminusEL = $true) & (binintersectTERcontra = $true) & (breln1all1 = $true) & (singletoninpowerset = $true) & (inComplementUnionImpInComplement1 = $true) & (eqbreln1 = $true) & (inPowerset = $true) & (breln1unionCommutes = $true) & (brelnall2 = $true) & (setminusLsub = $true) & (contrasubsetT2 = $true) & (binintersectLsub = $true) & (breln1compEex = $true) & (contraSubsetComplement = $true) & (injFuncInInjFuncSet = $true) & (setadjoinSub = $true) & (breln1invI = $true) & (dsetconstr__Cong = $true) & (binunionT_lem = $true) & (breln1unionIL = $true) & (funcGraphProp4 = $true) & (ap2p = $true) & (omega__Cong = $true) & (foundation2 = $true) & (ubforcartprodlem2 = $true) & (upairsetIL = $true) & (setminusSubset2 = $true) & (breln1compE = $true) & (subsetI2 = $true) & (image1Equiv = $true) & (woz2A = $true) & (funcGraphProp2 = $true) & (symdiffI1 = $true) & (kfstsingleton = $true) & (quantDeMorgan3 = $true) & (emptysetsubset = $true) & (notdexE = $true) & (upairset2E = $true) & (apProp = $true) & (woz13rule3 = $true) & (setadjoin__Cong = $true) & (woz13rule1 = $true) & (binintersectSubset3 = $true)),
% 5.26/2.08 inference(flattening,[status(thm)],[f1071])).
% 5.26/2.08 thf(f1071,plain,(
% 5.26/2.08 ((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i] : (((in @ X0 @ omega) = $true) & ($true != (in @ (setadjoin @ X0 @ X0) @ omega))) & (omegaSp = $true)) & (notinself2 = $true)) & (notinself = $true)) & (foundation2 = $true)) & (surjCantorThm = $true)) & (leftInvIsSurj = $true)) & (surjFuncSetFuncSurj = $true)) & (surjFuncSetFuncIn = $true)) & (injFuncSetFuncInj = $true)) & (injFuncSetFuncIn = $true)) & (injFuncInInjFuncSet = $true)) & (image1I = $true)) & (image1E = $true)) & (image1Equiv = $true)) & (image1Ex1 = $true)) & (image1Ex = $true)) & (woz2B = $true)) & (woz2A = $true)) & (woz2W = $true)) & (woz2Ex = $true)) & (breln1unionCommutes = $true)) & (breln1unionEcases = $true)) & (breln1unionE = $true)) & (breln1unionI = $true)) & (breln1unionIR = $true)) & (breln1unionIL = $true)) & (breln1unionprop = $true)) & (breln1compEex = $true)) & (breln1compE = $true)) & (breln1compI = $true)) & (breln1compprop = $true)) & (breln1invE = $true)) & (breln1invI = $true)) & (breln1invprop = $true)) & (eqbreln1 = $true)) & (subbreln1 = $true)) & (breln1all1 = $true)) & (setOfPairsIsBReln1 = $true)) & (choice2fnsingleton = $true)) & (breln1SetBreln1 = $true)) & (breln1all2 = $true)) & (woz1_5 = $true)) & (woz1_4 = $true)) & (woz1_3 = $true)) & (woz1_2 = $true)) & (woz1_1 = $true)) & (woz13rule4 = $true)) & (woz13rule3 = $true)) & (woz13rule2 = $true)) & (woz13rule1 = $true)) & (woz13rule0 = $true)) & (demorgan2 = $true)) & (demorgan2b = $true)) & (demorgan2b2 = $true)) & (demorgan2a = $true)) & (demorgan1 = $true)) & (demorgan1b = $true)) & (demorgan1a = $true)) & (demorgan2a2 = $true)) & (complementUnionInPowersetComplement = $true)) & (demorgan2a1 = $true)) & (binunionTEcontra = $true)) & (binunionTE = $true)) & (inComplementUnionImpInComplement1 = $true)) & (inComplementUnionImpNotIn1 = $true)) & (intersectInPowersetIntersectUnions = $true)) & (inIntersectImpInIntersectUnions = $true)) & (inIntersectImpInUnion2 = $true)) & (inIntersectImpInUnion = $true)) & (binunionTIRcontra = $true)) & (binunionTILcontra = $true)) & (complementTcontraSubset = $true)) & (contraSubsetComplement = $true)) & (complementInPowersetComplementIntersect = $true)) & (complementSubsetComplementIntersect = $true)) & (complementImpComplementIntersect = $true)) & (complementTnotintersectT = $true)) & (doubleComplementEq = $true)) & (doubleComplementSub2 = $true)) & (doubleComplementSub1 = $true)) & (doubleComplementE1 = $true)) & (doubleComplementI1 = $true)) & (contrasubsetT3 = $true)) & (contrasubsetT2 = $true)) & (contrasubsetT1 = $true)) & (contrasubsetT = $true)) & (binintersectTERcontra = $true)) & (binintersectTELcontra = $true)) & (complementTE1 = $true)) & (complementTI1 = $true)) & (powersetTE1 = $true)) & (powersetTI1 = $true)) & (subsetTI = $true)) & (setextT = $true)) & (complementT_lem = $true)) & (setminusT_lem = $true)) & (powersetT_lem = $true)) & (binunionT_lem = $true)) & (binintersectT_lem = $true)) & (iftrueorfalse = $true)) & (iffalse = $true)) & (iftrue = $true)) & (theeq = $true)) & (ifp = $true)) & (ifSingleton = $true)) & (iftrueProp2 = $true)) & (iftrueProp1 = $true)) & (iffalseProp2 = $true)) & (iffalseProp1 = $true)) & (eta2 = $true)) & (beta2 = $true)) & (lam2lamEq = $true)) & (eta1 = $true)) & (beta1 = $true)) & (ap2apEq2 = $true)) & (ap2apEq1 = $true)) & (funcext2 = $true)) & (funcext = $true)) & (eqbreln = $true)) & (subbreln = $true)) & (funcGraphProp4 = $true)) & (funcextLem = $true)) & (funcGraphProp2 = $true)) & (funcGraphProp3 = $true)) & (funcGraphProp1 = $true)) & (ex1E2 = $true)) & (brelnall2 = $true)) & (brelnall1 = $true)) & (lam2p = $true)) & (lamp = $true)) & (lamProp = $true)) & (funcinfuncset = $true)) & (ap2p = $true)) & (infuncsetfunc = $true)) & (app = $true)) & (apProp = $true)) & (funcImageSingleton = $true)) & (dpsetconstrER = $true)) & (dpsetconstrEL2 = $true)) & (dpsetconstrEL1 = $true)) & (dpsetconstrERa = $true)) & (setOfPairsIsBReln = $true)) & (dpsetconstrSub = $true)) & (dpsetconstrI = $true)) & (cartprodpairsurjEq = $true)) & (cartprodsndpairEq = $true)) & (cartprodfstpairEq = $true)) & (cartprodmempaircEq = $true)) & (cartprodpairmemER = $true)) & (cartprodpairmemEL = $true)) & (cartprodsndin = $true)) & (kpairsurjEq = $true)) & (ksndpairEq = $true)) & (ksndsingleton = $true)) & (setukpairinjR = $true)) & (setukpairinjR2 = $true)) & (upairequniteq = $true)) & (setukpairinjR1 = $true)) & (setukpairinjR12 = $true)) & (setukpairinjR11 = $true)) & (setukpairinjL = $true)) & (setukpairinjL2 = $true)) & (cartprodfstin = $true)) & (kfstpairEq = $true)) & (theprop = $true)) & (kfstsingleton = $true)) & (setukpairinjL1 = $true)) & (singletonsuniq = $true)) & (ex1I2 = $true)) & (ex1I = $true)) & (ex1E1 = $true)) & (singletonprop = $true)) & (setunionsingleton = $true)) & (setunionsingleton2 = $true)) & (setunionsingleton1 = $true)) & (setunionE2 = $true)) & (cartprodmempair = $true)) & (cartprodmempair1 = $true)) & (cartprodpairin = $true)) & (ubforcartprodlem3 = $true)) & (ubforcartprodlem2 = $true)) & (ubforcartprodlem1 = $true)) & (upairinpowunion = $true)) & (upairsubunion = $true)) & (upairset2E = $true)) & (singletoninpowunion = $true)) & (singletoninpowerset = $true)) & (singletonsubset = $true)) & (kpairp = $true)) & (kpairiskpair = $true)) & (setukpairIR = $true)) & (setukpairIL = $true)) & (secondinupair = $true)) & (symdiffIneg2 = $true)) & (symdiffIneg1 = $true)) & (symdiffI2 = $true)) & (symdiffI1 = $true)) & (symdiffE = $true)) & (setminusSubset1 = $true)) & (setminusLsub = $true)) & (setminusIRneg = $true)) & (setminusILneg = $true)) & (setminusELneg = $true)) & (setminusERneg = $true)) & (setminusSubset2 = $true)) & (setminusER = $true)) & (setminusEL = $true)) & (setminusI = $true)) & (bs114d = $true)) & (binintersectSubset1 = $true)) & (binintersectSubset4 = $true)) & (binintersectRsub = $true)) & (disjointsetsI1 = $true)) & (binintersectER = $true)) & (binintersectSubset3 = $true)) & (binintersectSubset2 = $true)) & (binintersectLsub = $true)) & (binintersectEL = $true)) & (binintersectSubset5 = $true)) & (binintersectI = $true)) & (binunionRsub = $true)) & (binunionLsub = $true)) & (binunionE = $true)) & (binunionEcases = $true)) & (binunionIR = $true)) & (upairset2IR = $true)) & (binunionIL = $true)) & (sepSubset = $true)) & (sepInPowerset = $true)) & (powersetsubset = $true)) & (inPowerset = $true)) & (powersetE1 = $true)) & (powersetI1 = $true)) & (subsetemptysetimpeq = $true)) & (setextsub = $true)) & (subset2powerset = $true)) & (setadjoinSub2 = $true)) & (setadjoinSub = $true)) & (subsetTrans = $true)) & (subsetRefl = $true)) & (notequalI2 = $true)) & (notequalI1 = $true)) & (notsubsetI = $true)) & (subsetE2 = $true)) & (subsetE = $true)) & (emptysetsubset = $true)) & (subsetI2 = $true)) & (eqimpsubset1 = $true)) & (eqimpsubset2 = $true)) & (subsetI1 = $true)) & (dsetconstr__Cong = $true)) & (descr__Cong = $true)) & (exuEu = $true)) & (omega__Cong = $true)) & (setunion__Cong = $true)) & (powerset__Cong = $true)) & (setadjoin__Cong = $true)) & (emptyset__Cong = $true)) & (exu__Cong = $true)) & (exuE3u = $true)) & (in__Cong = $true)) & (inCongP = $true)) & (exuI2 = $true)) & (exuI3 = $true)) & (exuI1 = $true)) & (notdallE = $true)) & (notdexE = $true)) & (prop2set2propI = $true)) & (prop2setI = $true)) & (quantDeMorgan4 = $true)) & (quantDeMorgan3 = $true)) & (quantDeMorgan2 = $true)) & (quantDeMorgan1 = $true)) & (vacuousDall = $true)) & (emptyE1 = $true)) & (upairsetIR = $true)) & (upairsetIL = $true)) & (upairsetE = $true)) & (singletonsswitch = $true)) & (eqinunit = $true)) & (notinsingleton = $true)) & (uniqinunit = $true)) & (nonemptyImpWitness = $true)) & (exuE2 = $true)) & (subPowSU = $true)) & (setunionE = $true)) & (setunionI = $true)) & (powersetE = $true)) & (emptyInPowerset = $true)) & (emptyinPowerset = $true)) & (powersetI = $true)) & (setoftrueEq = $true)) & (setadjoinOr = $true)) & (setadjoinE = $true)) & (setadjoinIR = $true)) & (emptyinunitempty = $true)) & (setadjoinIL = $true)) & (nonemptyI1 = $true)) & (nonemptyI = $true)) & (nonemptyE1 = $true)) & (setbeta = $true)) & (noeltsimpempty = $true)) & (emptyI = $true)) & (setext = $true)) & (exuE3e = $true)) & (notinemptyset = $true)) & (emptysetimpfalse = $true)) & (emptysetE = $true)) & (prop2setE = $true)) & (exuE1 = $true)) & (dsetconstrER = $true)) & (dsetconstrEL = $true)) & (dsetconstrI = $true)) & (descrp = $true)) & (wellorderingAx = $true)) & (foundationAx = $true)) & (replAx = $true)) & (omegaIndAx = $true)) & (omegaSAx = $true)) & (omega0Ax = $true)) & (setunionAx = $true)) & (powersetAx = $true)) & (setadjoinAx = $true)) & (emptysetAx = $true)) & (setextAx = $true)),
% 5.26/2.08 inference(ennf_transformation,[status(thm)],[f401])).
% 5.26/2.08 thf(f401,plain,(
% 5.26/2.08 ~((setextAx = $true) => ((emptysetAx = $true) => ((setadjoinAx = $true) => ((powersetAx = $true) => ((setunionAx = $true) => ((omega0Ax = $true) => ((omegaSAx = $true) => ((omegaIndAx = $true) => ((replAx = $true) => ((foundationAx = $true) => ((wellorderingAx = $true) => ((descrp = $true) => ((dsetconstrI = $true) => ((dsetconstrEL = $true) => ((dsetconstrER = $true) => ((exuE1 = $true) => ((prop2setE = $true) => ((emptysetE = $true) => ((emptysetimpfalse = $true) => ((notinemptyset = $true) => ((exuE3e = $true) => ((setext = $true) => ((emptyI = $true) => ((noeltsimpempty = $true) => ((setbeta = $true) => ((nonemptyE1 = $true) => ((nonemptyI = $true) => ((nonemptyI1 = $true) => ((setadjoinIL = $true) => ((emptyinunitempty = $true) => ((setadjoinIR = $true) => ((setadjoinE = $true) => ((setadjoinOr = $true) => ((setoftrueEq = $true) => ((powersetI = $true) => ((emptyinPowerset = $true) => ((emptyInPowerset = $true) => ((powersetE = $true) => ((setunionI = $true) => ((setunionE = $true) => ((subPowSU = $true) => ((exuE2 = $true) => ((nonemptyImpWitness = $true) => ((uniqinunit = $true) => ((notinsingleton = $true) => ((eqinunit = $true) => ((singletonsswitch = $true) => ((upairsetE = $true) => ((upairsetIL = $true) => ((upairsetIR = $true) => ((emptyE1 = $true) => ((vacuousDall = $true) => ((quantDeMorgan1 = $true) => ((quantDeMorgan2 = $true) => ((quantDeMorgan3 = $true) => ((quantDeMorgan4 = $true) => ((prop2setI = $true) => ((prop2set2propI = $true) => ((notdexE = $true) => ((notdallE = $true) => ((exuI1 = $true) => ((exuI3 = $true) => ((exuI2 = $true) => ((inCongP = $true) => ((in__Cong = $true) => ((exuE3u = $true) => ((exu__Cong = $true) => ((emptyset__Cong = $true) => ((setadjoin__Cong = $true) => ((powerset__Cong = $true) => ((setunion__Cong = $true) => ((omega__Cong = $true) => ((exuEu = $true) => ((descr__Cong = $true) => ((dsetconstr__Cong = $true) => ((subsetI1 = $true) => ((eqimpsubset2 = $true) => ((eqimpsubset1 = $true) => ((subsetI2 = $true) => ((emptysetsubset = $true) => ((subsetE = $true) => ((subsetE2 = $true) => ((notsubsetI = $true) => ((notequalI1 = $true) => ((notequalI2 = $true) => ((subsetRefl = $true) => ((subsetTrans = $true) => ((setadjoinSub = $true) => ((setadjoinSub2 = $true) => ((subset2powerset = $true) => ((setextsub = $true) => ((subsetemptysetimpeq = $true) => ((powersetI1 = $true) => ((powersetE1 = $true) => ((inPowerset = $true) => ((powersetsubset = $true) => ((sepInPowerset = $true) => ((sepSubset = $true) => ((binunionIL = $true) => ((upairset2IR = $true) => ((binunionIR = $true) => ((binunionEcases = $true) => ((binunionE = $true) => ((binunionLsub = $true) => ((binunionRsub = $true) => ((binintersectI = $true) => ((binintersectSubset5 = $true) => ((binintersectEL = $true) => ((binintersectLsub = $true) => ((binintersectSubset2 = $true) => ((binintersectSubset3 = $true) => ((binintersectER = $true) => ((disjointsetsI1 = $true) => ((binintersectRsub = $true) => ((binintersectSubset4 = $true) => ((binintersectSubset1 = $true) => ((bs114d = $true) => ((setminusI = $true) => ((setminusEL = $true) => ((setminusER = $true) => ((setminusSubset2 = $true) => ((setminusERneg = $true) => ((setminusELneg = $true) => ((setminusILneg = $true) => ((setminusIRneg = $true) => ((setminusLsub = $true) => ((setminusSubset1 = $true) => ((symdiffE = $true) => ((symdiffI1 = $true) => ((symdiffI2 = $true) => ((symdiffIneg1 = $true) => ((symdiffIneg2 = $true) => ((secondinupair = $true) => ((setukpairIL = $true) => ((setukpairIR = $true) => ((kpairiskpair = $true) => ((kpairp = $true) => ((singletonsubset = $true) => ((singletoninpowerset = $true) => ((singletoninpowunion = $true) => ((upairset2E = $true) => ((upairsubunion = $true) => ((upairinpowunion = $true) => ((ubforcartprodlem1 = $true) => ((ubforcartprodlem2 = $true) => ((ubforcartprodlem3 = $true) => ((cartprodpairin = $true) => ((cartprodmempair1 = $true) => ((cartprodmempair = $true) => ((setunionE2 = $true) => ((setunionsingleton1 = $true) => ((setunionsingleton2 = $true) => ((setunionsingleton = $true) => ((singletonprop = $true) => ((ex1E1 = $true) => ((ex1I = $true) => ((ex1I2 = $true) => ((singletonsuniq = $true) => ((setukpairinjL1 = $true) => ((kfstsingleton = $true) => ((theprop = $true) => ((kfstpairEq = $true) => ((cartprodfstin = $true) => ((setukpairinjL2 = $true) => ((setukpairinjL = $true) => ((setukpairinjR11 = $true) => ((setukpairinjR12 = $true) => ((setukpairinjR1 = $true) => ((upairequniteq = $true) => ((setukpairinjR2 = $true) => ((setukpairinjR = $true) => ((ksndsingleton = $true) => ((ksndpairEq = $true) => ((kpairsurjEq = $true) => ((cartprodsndin = $true) => ((cartprodpairmemEL = $true) => ((cartprodpairmemER = $true) => ((cartprodmempaircEq = $true) => ((cartprodfstpairEq = $true) => ((cartprodsndpairEq = $true) => ((cartprodpairsurjEq = $true) => ((dpsetconstrI = $true) => ((dpsetconstrSub = $true) => ((setOfPairsIsBReln = $true) => ((dpsetconstrERa = $true) => ((dpsetconstrEL1 = $true) => ((dpsetconstrEL2 = $true) => ((dpsetconstrER = $true) => ((funcImageSingleton = $true) => ((apProp = $true) => ((app = $true) => ((infuncsetfunc = $true) => ((ap2p = $true) => ((funcinfuncset = $true) => ((lamProp = $true) => ((lamp = $true) => ((lam2p = $true) => ((brelnall1 = $true) => ((brelnall2 = $true) => ((ex1E2 = $true) => ((funcGraphProp1 = $true) => ((funcGraphProp3 = $true) => ((funcGraphProp2 = $true) => ((funcextLem = $true) => ((funcGraphProp4 = $true) => ((subbreln = $true) => ((eqbreln = $true) => ((funcext = $true) => ((funcext2 = $true) => ((ap2apEq1 = $true) => ((ap2apEq2 = $true) => ((beta1 = $true) => ((eta1 = $true) => ((lam2lamEq = $true) => ((beta2 = $true) => ((eta2 = $true) => ((iffalseProp1 = $true) => ((iffalseProp2 = $true) => ((iftrueProp1 = $true) => ((iftrueProp2 = $true) => ((ifSingleton = $true) => ((ifp = $true) => ((theeq = $true) => ((iftrue = $true) => ((iffalse = $true) => ((iftrueorfalse = $true) => ((binintersectT_lem = $true) => ((binunionT_lem = $true) => ((powersetT_lem = $true) => ((setminusT_lem = $true) => ((complementT_lem = $true) => ((setextT = $true) => ((subsetTI = $true) => ((powersetTI1 = $true) => ((powersetTE1 = $true) => ((complementTI1 = $true) => ((complementTE1 = $true) => ((binintersectTELcontra = $true) => ((binintersectTERcontra = $true) => ((contrasubsetT = $true) => ((contrasubsetT1 = $true) => ((contrasubsetT2 = $true) => ((contrasubsetT3 = $true) => ((doubleComplementI1 = $true) => ((doubleComplementE1 = $true) => ((doubleComplementSub1 = $true) => ((doubleComplementSub2 = $true) => ((doubleComplementEq = $true) => ((complementTnotintersectT = $true) => ((complementImpComplementIntersect = $true) => ((complementSubsetComplementIntersect = $true) => ((complementInPowersetComplementIntersect = $true) => ((contraSubsetComplement = $true) => ((complementTcontraSubset = $true) => ((binunionTILcontra = $true) => ((binunionTIRcontra = $true) => ((inIntersectImpInUnion = $true) => ((inIntersectImpInUnion2 = $true) => ((inIntersectImpInIntersectUnions = $true) => ((intersectInPowersetIntersectUnions = $true) => ((inComplementUnionImpNotIn1 = $true) => ((inComplementUnionImpInComplement1 = $true) => ((binunionTE = $true) => ((binunionTEcontra = $true) => ((demorgan2a1 = $true) => ((complementUnionInPowersetComplement = $true) => ((demorgan2a2 = $true) => ((demorgan1a = $true) => ((demorgan1b = $true) => ((demorgan1 = $true) => ((demorgan2a = $true) => ((demorgan2b2 = $true) => ((demorgan2b = $true) => ((demorgan2 = $true) => ((woz13rule0 = $true) => ((woz13rule1 = $true) => ((woz13rule2 = $true) => ((woz13rule3 = $true) => ((woz13rule4 = $true) => ((woz1_1 = $true) => ((woz1_2 = $true) => ((woz1_3 = $true) => ((woz1_4 = $true) => ((woz1_5 = $true) => ((breln1all2 = $true) => ((breln1SetBreln1 = $true) => ((choice2fnsingleton = $true) => ((setOfPairsIsBReln1 = $true) => ((breln1all1 = $true) => ((subbreln1 = $true) => ((eqbreln1 = $true) => ((breln1invprop = $true) => ((breln1invI = $true) => ((breln1invE = $true) => ((breln1compprop = $true) => ((breln1compI = $true) => ((breln1compE = $true) => ((breln1compEex = $true) => ((breln1unionprop = $true) => ((breln1unionIL = $true) => ((breln1unionIR = $true) => ((breln1unionI = $true) => ((breln1unionE = $true) => ((breln1unionEcases = $true) => ((breln1unionCommutes = $true) => ((woz2Ex = $true) => ((woz2W = $true) => ((woz2A = $true) => ((woz2B = $true) => ((image1Ex = $true) => ((image1Ex1 = $true) => ((image1Equiv = $true) => ((image1E = $true) => ((image1I = $true) => ((injFuncInInjFuncSet = $true) => ((injFuncSetFuncIn = $true) => ((injFuncSetFuncInj = $true) => ((surjFuncSetFuncIn = $true) => ((surjFuncSetFuncSurj = $true) => ((leftInvIsSurj = $true) => ((surjCantorThm = $true) => ((foundation2 = $true) => ((notinself = $true) => ((notinself2 = $true) => ((omegaSp = $true) => ! [X0 : $i] : (((in @ X0 @ omega) = $true) => ($true = (in @ (setadjoin @ X0 @ X0) @ omega))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 5.26/2.08 inference(fool_elimination,[status(thm)],[f400])).
% 5.26/2.08 thf(f400,plain,(
% 5.26/2.08 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => (ubforcartprodlem2 => (ubforcartprodlem3 => (cartprodpairin => (cartprodmempair1 => (cartprodmempair => (setunionE2 => (setunionsingleton1 => (setunionsingleton2 => (setunionsingleton => (singletonprop => (ex1E1 => (ex1I => (ex1I2 => (singletonsuniq => (setukpairinjL1 => (kfstsingleton => (theprop => (kfstpairEq => (cartprodfstin => (setukpairinjL2 => (setukpairinjL => (setukpairinjR11 => (setukpairinjR12 => (setukpairinjR1 => (upairequniteq => (setukpairinjR2 => (setukpairinjR => (ksndsingleton => (ksndpairEq => (kpairsurjEq => (cartprodsndin => (cartprodpairmemEL => (cartprodpairmemER => (cartprodmempaircEq => (cartprodfstpairEq => (cartprodsndpairEq => (cartprodpairsurjEq => (dpsetconstrI => (dpsetconstrSub => (setOfPairsIsBReln => (dpsetconstrERa => (dpsetconstrEL1 => (dpsetconstrEL2 => (dpsetconstrER => (funcImageSingleton => (apProp => (app => (infuncsetfunc => (ap2p => (funcinfuncset => (lamProp => (lamp => (lam2p => (brelnall1 => (brelnall2 => (ex1E2 => (funcGraphProp1 => (funcGraphProp3 => (funcGraphProp2 => (funcextLem => (funcGraphProp4 => (subbreln => (eqbreln => (funcext => (funcext2 => (ap2apEq1 => (ap2apEq2 => (beta1 => (eta1 => (lam2lamEq => (beta2 => (eta2 => (iffalseProp1 => (iffalseProp2 => (iftrueProp1 => (iftrueProp2 => (ifSingleton => (ifp => (theeq => (iftrue => (iffalse => (iftrueorfalse => (binintersectT_lem => (binunionT_lem => (powersetT_lem => (setminusT_lem => (complementT_lem => (setextT => (subsetTI => (powersetTI1 => (powersetTE1 => (complementTI1 => (complementTE1 => (binintersectTELcontra => (binintersectTERcontra => (contrasubsetT => (contrasubsetT1 => (contrasubsetT2 => (contrasubsetT3 => (doubleComplementI1 => (doubleComplementE1 => (doubleComplementSub1 => (doubleComplementSub2 => (doubleComplementEq => (complementTnotintersectT => (complementImpComplementIntersect => (complementSubsetComplementIntersect => (complementInPowersetComplementIntersect => (contraSubsetComplement => (complementTcontraSubset => (binunionTILcontra => (binunionTIRcontra => (inIntersectImpInUnion => (inIntersectImpInUnion2 => (inIntersectImpInIntersectUnions => (intersectInPowersetIntersectUnions => (inComplementUnionImpNotIn1 => (inComplementUnionImpInComplement1 => (binunionTE => (binunionTEcontra => (demorgan2a1 => (complementUnionInPowersetComplement => (demorgan2a2 => (demorgan1a => (demorgan1b => (demorgan1 => (demorgan2a => (demorgan2b2 => (demorgan2b => (demorgan2 => (woz13rule0 => (woz13rule1 => (woz13rule2 => (woz13rule3 => (woz13rule4 => (woz1_1 => (woz1_2 => (woz1_3 => (woz1_4 => (woz1_5 => (breln1all2 => (breln1SetBreln1 => (choice2fnsingleton => (setOfPairsIsBReln1 => (breln1all1 => (subbreln1 => (eqbreln1 => (breln1invprop => (breln1invI => (breln1invE => (breln1compprop => (breln1compI => (breln1compE => (breln1compEex => (breln1unionprop => (breln1unionIL => (breln1unionIR => (breln1unionI => (breln1unionE => (breln1unionEcases => (breln1unionCommutes => (woz2Ex => (woz2W => (woz2A => (woz2B => (image1Ex => (image1Ex1 => (image1Equiv => (image1E => (image1I => (injFuncInInjFuncSet => (injFuncSetFuncIn => (injFuncSetFuncInj => (surjFuncSetFuncIn => (surjFuncSetFuncSurj => (leftInvIsSurj => (surjCantorThm => (foundation2 => (notinself => (notinself2 => (omegaSp => ! [X0 : $i] : ((in @ X0 @ omega) => (in @ (setadjoin @ X0 @ X0) @ omega)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 5.26/2.08 inference(rectify,[status(thm)],[f330])).
% 5.26/2.08 thf(f330,negated_conjecture,(
% 5.26/2.08 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => (ubforcartprodlem2 => (ubforcartprodlem3 => (cartprodpairin => (cartprodmempair1 => (cartprodmempair => (setunionE2 => (setunionsingleton1 => (setunionsingleton2 => (setunionsingleton => (singletonprop => (ex1E1 => (ex1I => (ex1I2 => (singletonsuniq => (setukpairinjL1 => (kfstsingleton => (theprop => (kfstpairEq => (cartprodfstin => (setukpairinjL2 => (setukpairinjL => (setukpairinjR11 => (setukpairinjR12 => (setukpairinjR1 => (upairequniteq => (setukpairinjR2 => (setukpairinjR => (ksndsingleton => (ksndpairEq => (kpairsurjEq => (cartprodsndin => (cartprodpairmemEL => (cartprodpairmemER => (cartprodmempaircEq => (cartprodfstpairEq => (cartprodsndpairEq => (cartprodpairsurjEq => (dpsetconstrI => (dpsetconstrSub => (setOfPairsIsBReln => (dpsetconstrERa => (dpsetconstrEL1 => (dpsetconstrEL2 => (dpsetconstrER => (funcImageSingleton => (apProp => (app => (infuncsetfunc => (ap2p => (funcinfuncset => (lamProp => (lamp => (lam2p => (brelnall1 => (brelnall2 => (ex1E2 => (funcGraphProp1 => (funcGraphProp3 => (funcGraphProp2 => (funcextLem => (funcGraphProp4 => (subbreln => (eqbreln => (funcext => (funcext2 => (ap2apEq1 => (ap2apEq2 => (beta1 => (eta1 => (lam2lamEq => (beta2 => (eta2 => (iffalseProp1 => (iffalseProp2 => (iftrueProp1 => (iftrueProp2 => (ifSingleton => (ifp => (theeq => (iftrue => (iffalse => (iftrueorfalse => (binintersectT_lem => (binunionT_lem => (powersetT_lem => (setminusT_lem => (complementT_lem => (setextT => (subsetTI => (powersetTI1 => (powersetTE1 => (complementTI1 => (complementTE1 => (binintersectTELcontra => (binintersectTERcontra => (contrasubsetT => (contrasubsetT1 => (contrasubsetT2 => (contrasubsetT3 => (doubleComplementI1 => (doubleComplementE1 => (doubleComplementSub1 => (doubleComplementSub2 => (doubleComplementEq => (complementTnotintersectT => (complementImpComplementIntersect => (complementSubsetComplementIntersect => (complementInPowersetComplementIntersect => (contraSubsetComplement => (complementTcontraSubset => (binunionTILcontra => (binunionTIRcontra => (inIntersectImpInUnion => (inIntersectImpInUnion2 => (inIntersectImpInIntersectUnions => (intersectInPowersetIntersectUnions => (inComplementUnionImpNotIn1 => (inComplementUnionImpInComplement1 => (binunionTE => (binunionTEcontra => (demorgan2a1 => (complementUnionInPowersetComplement => (demorgan2a2 => (demorgan1a => (demorgan1b => (demorgan1 => (demorgan2a => (demorgan2b2 => (demorgan2b => (demorgan2 => (woz13rule0 => (woz13rule1 => (woz13rule2 => (woz13rule3 => (woz13rule4 => (woz1_1 => (woz1_2 => (woz1_3 => (woz1_4 => (woz1_5 => (breln1all2 => (breln1SetBreln1 => (choice2fnsingleton => (setOfPairsIsBReln1 => (breln1all1 => (subbreln1 => (eqbreln1 => (breln1invprop => (breln1invI => (breln1invE => (breln1compprop => (breln1compI => (breln1compE => (breln1compEex => (breln1unionprop => (breln1unionIL => (breln1unionIR => (breln1unionI => (breln1unionE => (breln1unionEcases => (breln1unionCommutes => (woz2Ex => (woz2W => (woz2A => (woz2B => (image1Ex => (image1Ex1 => (image1Equiv => (image1E => (image1I => (injFuncInInjFuncSet => (injFuncSetFuncIn => (injFuncSetFuncInj => (surjFuncSetFuncIn => (surjFuncSetFuncSurj => (leftInvIsSurj => (surjCantorThm => (foundation2 => (notinself => (notinself2 => (omegaSp => ! [X1 : $i] : ((in @ X1 @ omega) => (in @ (setadjoin @ X1 @ X1) @ omega)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 5.26/2.08 inference(negated_conjecture,[status(cth)],[f329])).
% 5.26/2.08 thf(f329,conjecture,(
% 5.26/2.08 setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => (ubforcartprodlem2 => (ubforcartprodlem3 => (cartprodpairin => (cartprodmempair1 => (cartprodmempair => (setunionE2 => (setunionsingleton1 => (setunionsingleton2 => (setunionsingleton => (singletonprop => (ex1E1 => (ex1I => (ex1I2 => (singletonsuniq => (setukpairinjL1 => (kfstsingleton => (theprop => (kfstpairEq => (cartprodfstin => (setukpairinjL2 => (setukpairinjL => (setukpairinjR11 => (setukpairinjR12 => (setukpairinjR1 => (upairequniteq => (setukpairinjR2 => (setukpairinjR => (ksndsingleton => (ksndpairEq => (kpairsurjEq => (cartprodsndin => (cartprodpairmemEL => (cartprodpairmemER => (cartprodmempaircEq => (cartprodfstpairEq => (cartprodsndpairEq => (cartprodpairsurjEq => (dpsetconstrI => (dpsetconstrSub => (setOfPairsIsBReln => (dpsetconstrERa => (dpsetconstrEL1 => (dpsetconstrEL2 => (dpsetconstrER => (funcImageSingleton => (apProp => (app => (infuncsetfunc => (ap2p => (funcinfuncset => (lamProp => (lamp => (lam2p => (brelnall1 => (brelnall2 => (ex1E2 => (funcGraphProp1 => (funcGraphProp3 => (funcGraphProp2 => (funcextLem => (funcGraphProp4 => (subbreln => (eqbreln => (funcext => (funcext2 => (ap2apEq1 => (ap2apEq2 => (beta1 => (eta1 => (lam2lamEq => (beta2 => (eta2 => (iffalseProp1 => (iffalseProp2 => (iftrueProp1 => (iftrueProp2 => (ifSingleton => (ifp => (theeq => (iftrue => (iffalse => (iftrueorfalse => (binintersectT_lem => (binunionT_lem => (powersetT_lem => (setminusT_lem => (complementT_lem => (setextT => (subsetTI => (powersetTI1 => (powersetTE1 => (complementTI1 => (complementTE1 => (binintersectTELcontra => (binintersectTERcontra => (contrasubsetT => (contrasubsetT1 => (contrasubsetT2 => (contrasubsetT3 => (doubleComplementI1 => (doubleComplementE1 => (doubleComplementSub1 => (doubleComplementSub2 => (doubleComplementEq => (complementTnotintersectT => (complementImpComplementIntersect => (complementSubsetComplementIntersect => (complementInPowersetComplementIntersect => (contraSubsetComplement => (complementTcontraSubset => (binunionTILcontra => (binunionTIRcontra => (inIntersectImpInUnion => (inIntersectImpInUnion2 => (inIntersectImpInIntersectUnions => (intersectInPowersetIntersectUnions => (inComplementUnionImpNotIn1 => (inComplementUnionImpInComplement1 => (binunionTE => (binunionTEcontra => (demorgan2a1 => (complementUnionInPowersetComplement => (demorgan2a2 => (demorgan1a => (demorgan1b => (demorgan1 => (demorgan2a => (demorgan2b2 => (demorgan2b => (demorgan2 => (woz13rule0 => (woz13rule1 => (woz13rule2 => (woz13rule3 => (woz13rule4 => (woz1_1 => (woz1_2 => (woz1_3 => (woz1_4 => (woz1_5 => (breln1all2 => (breln1SetBreln1 => (choice2fnsingleton => (setOfPairsIsBReln1 => (breln1all1 => (subbreln1 => (eqbreln1 => (breln1invprop => (breln1invI => (breln1invE => (breln1compprop => (breln1compI => (breln1compE => (breln1compEex => (breln1unionprop => (breln1unionIL => (breln1unionIR => (breln1unionI => (breln1unionE => (breln1unionEcases => (breln1unionCommutes => (woz2Ex => (woz2W => (woz2A => (woz2B => (image1Ex => (image1Ex1 => (image1Equiv => (image1E => (image1I => (injFuncInInjFuncSet => (injFuncSetFuncIn => (injFuncSetFuncInj => (surjFuncSetFuncIn => (surjFuncSetFuncSurj => (leftInvIsSurj => (surjCantorThm => (foundation2 => (notinself => (notinself2 => (omegaSp => ! [X1 : $i] : ((in @ X1 @ omega) => (in @ (setadjoin @ X1 @ X1) @ omega))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 5.26/2.08 file('/export/starexec/sandbox2/tmp/tmp.N0zPVEAfqW/DTF2THF_28502.p',omegaSclos)).
% 5.26/2.08 thf(f5152,plain,(
% 5.26/2.08 ($true != (in @ sK655 @ omega))),
% 5.26/2.08 inference(trivial_inequality_removal,[status(thm)],[f5151])).
% 5.26/2.08 thf(f5151,plain,(
% 5.26/2.08 ($true != $true) | ($true != (in @ sK655 @ omega))),
% 5.26/2.08 inference(forward_demodulation,[status(thm)],[f5140,f4253])).
% 5.26/2.08 thf(f4253,plain,(
% 5.26/2.08 (omegaSAx = $true)),
% 5.26/2.08 inference(cnf_transformation,[status(thm)],[f2443])).
% 5.26/2.08 thf(f5140,plain,(
% 5.26/2.08 (omegaSAx != $true) | ($true != (in @ sK655 @ omega))),
% 5.26/2.08 inference(trivial_inequality_removal,[status(thm)],[f5133])).
% 5.26/2.08 thf(f5133,plain,(
% 5.26/2.08 (omegaSAx != $true) | ($true != (in @ sK655 @ omega)) | ($true != $true)),
% 5.26/2.08 inference(superposition,[status(thm)],[f4124,f4402])).
% 5.26/2.08 thf(f4402,plain,(
% 5.26/2.08 ( ! [X0 : $i] : (($true = (in @ (setadjoin @ X0 @ X0) @ omega)) | (omegaSAx != $true) | ((in @ X0 @ omega) != $true)) )),
% 5.26/2.08 inference(cnf_transformation,[status(thm)],[f2535])).
% 5.26/2.08 thf(f2535,plain,(
% 5.26/2.08 (! [X0 : $i] : (((in @ X0 @ omega) != $true) | ($true = (in @ (setadjoin @ X0 @ X0) @ omega))) | (omegaSAx != $true)) & ((omegaSAx = $true) | (($true = (in @ sK716 @ omega)) & ((in @ (setadjoin @ sK716 @ sK716) @ omega) != $true)))),
% 5.26/2.08 inference(skolemisation,[status(esa),new_symbols(skolem,[sK716])],[f2533,f2534])).
% 5.26/2.08 thf(f2534,plain,(
% 5.26/2.08 ? [X1 : $i] : (((in @ X1 @ omega) = $true) & ((in @ (setadjoin @ X1 @ X1) @ omega) != $true)) => (($true = (in @ sK716 @ omega)) & ((in @ (setadjoin @ sK716 @ sK716) @ omega) != $true))),
% 5.26/2.08 introduced(definition,[],[choice_axiom])).
% 5.26/2.08 thf(f2533,plain,(
% 5.26/2.08 (! [X0 : $i] : (((in @ X0 @ omega) != $true) | ($true = (in @ (setadjoin @ X0 @ X0) @ omega))) | (omegaSAx != $true)) & ((omegaSAx = $true) | ? [X1 : $i] : (((in @ X1 @ omega) = $true) & ((in @ (setadjoin @ X1 @ X1) @ omega) != $true)))),
% 5.26/2.08 inference(rectify,[status(thm)],[f2532])).
% 5.26/2.08 thf(f2532,plain,(
% 5.26/2.08 (! [X0 : $i] : (((in @ X0 @ omega) != $true) | ($true = (in @ (setadjoin @ X0 @ X0) @ omega))) | (omegaSAx != $true)) & ((omegaSAx = $true) | ? [X0 : $i] : (((in @ X0 @ omega) = $true) & ($true != (in @ (setadjoin @ X0 @ X0) @ omega))))),
% 5.26/2.08 inference(nnf_transformation,[status(thm)],[f1204])).
% 5.26/2.08 thf(f1204,plain,(
% 5.26/2.08 ! [X0 : $i] : (((in @ X0 @ omega) != $true) | ($true = (in @ (setadjoin @ X0 @ X0) @ omega))) <=> (omegaSAx = $true)),
% 5.26/2.08 inference(ennf_transformation,[status(thm)],[f699])).
% 5.26/2.08 thf(f699,plain,(
% 5.26/2.08 (omegaSAx = $true) <=> ! [X0 : $i] : (((in @ X0 @ omega) = $true) => ($true = (in @ (setadjoin @ X0 @ X0) @ omega)))),
% 5.26/2.08 inference(fool_elimination,[status(thm)],[f698])).
% 5.26/2.08 thf(f698,plain,(
% 5.26/2.08 (! [X0 : $i] : ((in @ X0 @ omega) => (in @ (setadjoin @ X0 @ X0) @ omega)) = omegaSAx)),
% 5.26/2.08 inference(rectify,[status(thm)],[f8])).
% 5.26/2.08 thf(f8,axiom,(
% 5.26/2.08 (! [X1 : $i] : ((in @ X1 @ omega) => (in @ (setadjoin @ X1 @ X1) @ omega)) = omegaSAx)),
% 5.26/2.08 file('/export/starexec/sandbox2/tmp/tmp.N0zPVEAfqW/DTF2THF_28502.p',omegaSAx)).
% 5.26/2.08 thf(f4124,plain,(
% 5.26/2.08 ($true != (in @ (setadjoin @ sK655 @ sK655) @ omega))),
% 5.26/2.08 inference(cnf_transformation,[status(thm)],[f2443])).
% 5.26/2.08 % SZS output end Proof for DTF2THF_28502
% 5.26/2.08 % (28720)------------------------------
% 5.26/2.08 % (28720)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.26/2.08 % (28720)Termination reason: Refutation
% 5.26/2.08
% 5.26/2.08 % (28720)Memory used [KB]: 11897
% 5.26/2.08 % (28720)Time elapsed: 0.244 s
% 5.26/2.08 % (28720)Instructions burned: 487 (million)
% 5.26/2.08 % (28720)------------------------------
% 5.26/2.08 % (28720)------------------------------
% 5.26/2.08 % (28691)Success in time 0.355 s
%------------------------------------------------------------------------------