%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU787^1 : 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 : n031.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 09:37:12 AM UTC 2026
% Result : Theorem 3.00s 1.88s
% Output : Refutation 3.00s
% Verified :
% SZS Type : Refutation
% Derivation depth : 14
% Number of leaves : 11
% Syntax : Number of formulae : 39 ( 7 unt; 0 typ; 3 def)
% Number of atoms : 5249 (1400 equ; 0 cnn)
% Maximal formula atoms : 313 ( 134 avg)
% Number of connectives : 2875 ( 72 ~; 61 |; 991 &; 488 @)
% ( 6 <=>;1257 =>; 0 <=; 0 <~>)
% Maximal formula depth : 323 ( 61 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 535 ( 532 usr; 317 con; 0-6 aty)
% Number of variables : 98 ( 0 ^; 51 !; 47 ?; 98 :)
% 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_365,type,
sP0: $i > $i > $i > $o ).
thf(func_def_368,type,
sP3: $i > $i > $i > $o ).
thf(func_def_371,type,
sP6: $i > $i > $o ).
thf(func_def_372,type,
sP7: $i > $o ).
thf(func_def_373,type,
sP8: $i > $i > $o ).
thf(func_def_374,type,
sP9: $i > $i > $o ).
thf(func_def_378,type,
sP13: $i > $i > $i > $i > $o ).
thf(func_def_380,type,
sP15: $o > $i > $i > $i > $o ).
thf(func_def_389,type,
sK24: $i > $i > $i > $i ).
thf(func_def_393,type,
sK28: $i > $i > $i > $i ).
thf(func_def_399,type,
sK34: ( $i > $o ) > $i ).
thf(func_def_400,type,
sK35: $i > $o ).
thf(func_def_401,type,
sK36: $i > $i ).
thf(func_def_407,type,
sK42: $i > $o ).
thf(func_def_409,type,
sK44: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_410,type,
sK45: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_413,type,
sK48: $i > $i > $o ).
thf(func_def_424,type,
sK59: $i > $i ).
thf(func_def_425,type,
sK60: ( $i > $i ) > $i > $i > $i ).
thf(func_def_439,type,
sK74: $i > $o ).
thf(func_def_442,type,
sK77: $i > ( $i > $o ) > $i ).
thf(func_def_443,type,
sK78: $i > ( $i > $o ) > $i ).
thf(func_def_444,type,
sK79: $i > $o ).
thf(func_def_445,type,
sK80: ( $i > $o ) > $i ).
thf(func_def_456,type,
sK91: $i > $i > $o ).
thf(func_def_480,type,
sK115: $i > $i > $i ).
thf(func_def_496,type,
sK131: $i > $o ).
thf(func_def_498,type,
sK133: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_499,type,
sK134: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_502,type,
sK137: $i > $i ).
thf(func_def_529,type,
sK164: $i > $i > $i > $i > $i ).
thf(func_def_539,type,
sK174: $i > $i > $o ).
thf(func_def_554,type,
sK189: $i > ( $i > $o ) > $i ).
thf(func_def_555,type,
sK190: $i > $o ).
thf(func_def_558,type,
sK193: $i > $o ).
thf(func_def_561,type,
sK196: $i > ( $i > $o ) > $i ).
thf(func_def_566,type,
sK201: ( $i > $o ) > $i > $i > $i ).
thf(func_def_567,type,
sK202: ( $i > $o ) > $i > $i > $i ).
thf(func_def_570,type,
sK205: $i > $o ).
thf(func_def_583,type,
sK218: $i > $o ).
thf(func_def_585,type,
sK220: ( $i > $o ) > $i > $i > $i ).
thf(func_def_586,type,
sK221: ( $i > $o ) > $i > $i > $i ).
thf(func_def_590,type,
sK225: $i > $i > $i > $i ).
thf(func_def_591,type,
sK226: $i > $i > $i > $i ).
thf(func_def_592,type,
sK227: $i > $i > $i > $i ).
thf(func_def_593,type,
sK228: $i > $i > $i > $i ).
thf(func_def_601,type,
sK236: $o > $i > $i > $i ).
thf(func_def_610,type,
sK245: $i > $i > $i ).
thf(func_def_612,type,
sK247: $i > $o ).
thf(func_def_614,type,
sK249: ( $i > $o ) > $i > $i ).
thf(func_def_615,type,
sK250: ( $i > $o ) > $i > $i ).
thf(func_def_616,type,
sK251: $i > $i > ( $i > $i ) > $i ).
thf(func_def_617,type,
sK252: $i > $i ).
thf(func_def_646,type,
sK281: $i > $o ).
thf(func_def_648,type,
sK283: $i > $i ).
thf(func_def_650,type,
sK285: $i > $o ).
thf(func_def_652,type,
sK287: $i > ( $i > $o ) > $i ).
thf(func_def_661,type,
sK296: $i > $o ).
thf(func_def_664,type,
sK299: $i > $o ).
thf(func_def_673,type,
sK308: $i > $i > $i > $i ).
thf(func_def_674,type,
sK309: $i > $i > $i > $i ).
thf(func_def_675,type,
sK310: $i > $i > $i ).
thf(func_def_676,type,
sK311: $i > $i > $i ).
thf(func_def_677,type,
sK312: $i > $i > $i ).
thf(func_def_678,type,
sK313: $i > $i > $i ).
thf(func_def_679,type,
sK314: $i > $i > $i ).
thf(func_def_680,type,
sK315: $i > $i > $i > $i ).
thf(func_def_681,type,
sK316: $i > $i ).
thf(func_def_682,type,
sK317: $i > $i ).
thf(func_def_683,type,
sK318: $i > $i ).
thf(func_def_684,type,
sK319: $i > $i ).
thf(func_def_685,type,
sK320: $i > $i > $i > $i ).
thf(func_def_686,type,
sK321: $i > $i > $i > $i ).
thf(func_def_687,type,
sK322: $i > $i > $i ).
thf(func_def_688,type,
sK323: $i > $i > $i ).
thf(func_def_689,type,
sK324: $i > $i > $i ).
thf(func_def_690,type,
sK325: $i > $i ).
thf(func_def_692,type,
sK327: $i > $i ).
thf(func_def_693,type,
sK328: $i > $i ).
thf(func_def_717,type,
sK352: $i > $i > $i > $i > $i ).
thf(func_def_737,type,
sK372: $i > $i ).
thf(func_def_739,type,
sK374: $i > ( $i > $i ) > $i > $i ).
thf(func_def_747,type,
sK382: $i > $i > $o ).
thf(func_def_762,type,
sK397: $i > $o ).
thf(func_def_776,type,
sK411: $i > $o ).
thf(func_def_791,type,
sK426: $i > $i > $i ).
thf(func_def_794,type,
sK429: $i > $o ).
thf(func_def_816,type,
sK451: $i > $o ).
thf(func_def_820,type,
sK455: $i > $o ).
thf(func_def_822,type,
sK457: ( $i > $o ) > $i > $i ).
thf(func_def_823,type,
sK458: $i > $o ).
thf(func_def_826,type,
sK461: $i > ( $i > $o ) > $i ).
thf(func_def_850,type,
sK485: $i > $i > $i ).
thf(func_def_851,type,
sK486: $i > $i > $i ).
thf(func_def_853,type,
sK488: $i > $o ).
thf(func_def_857,type,
sK492: $i > $i > $i ).
thf(func_def_858,type,
sK493: $i > $i > $i ).
thf(func_def_861,type,
sK496: ( $i > $i ) > $i > $i > $i ).
thf(func_def_864,type,
sK499: $i > $i ).
thf(func_def_885,type,
sK520: $i > $i > ( $i > $i ) > $i ).
thf(func_def_886,type,
sK521: $i > $i ).
thf(func_def_927,type,
sK562: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_928,type,
sK563: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_929,type,
sK564: $i > $o ).
thf(func_def_930,type,
sK565: $i > $o ).
thf(func_def_947,type,
sK582: $i > $i > $i > $i > $i ).
thf(func_def_948,type,
sK583: $i > $i > $i > $i > $i ).
thf(func_def_956,type,
sK591: $i > $i > $i > $i ).
thf(func_def_957,type,
sK592: $i > $i > $i > $i ).
thf(func_def_970,type,
sK605: $i > $i > $o ).
thf(func_def_973,type,
sK608: $i > ( $i > $o ) > $i > $i ).
thf(func_def_975,type,
sK610: $i > $o ).
thf(func_def_1007,type,
sK642: $i > $i > $o ).
thf(func_def_1008,type,
sK643: $i > $i ).
thf(func_def_1010,type,
sK645: $i > $i ).
thf(func_def_1021,type,
sK656: $i > $o ).
thf(func_def_1037,type,
sK672: $i > $i > $i > $i ).
thf(func_def_1041,type,
sK676: $i > $o ).
thf(func_def_1061,type,
sK696: $i > $i > $i ).
thf(func_def_1074,type,
sK709: $i > $i > $i > $i > $i > $i ).
thf(func_def_1075,type,
sK710: $i > $o ).
thf(func_def_1076,type,
sK711: $i > $o ).
thf(func_def_1077,type,
sK712: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_1078,type,
sK713: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_1087,type,
sK722: $i > $i ).
thf(func_def_1095,type,
sK730: $i > $i ).
thf(func_def_1098,type,
sK733: ( $i > $o ) > $i > $i ).
thf(func_def_1100,type,
sK735: $i > $o ).
thf(func_def_1104,type,
sK739: $i > $i > $o ).
thf(func_def_1118,type,
sK753: $i > $i > $o ).
thf(func_def_1120,type,
sK755: $i > $i ).
thf(func_def_1121,type,
sK756: $i > $i ).
thf(func_def_1122,type,
sK757: $i > ( $i > $i > $o ) > $i ).
thf(func_def_1123,type,
sK758: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_1124,type,
sK759: $i > ( $i > $i > $o ) > $i ).
thf(func_def_1130,type,
sK765: $i > $i > $i ).
thf(func_def_1136,type,
sK771: $i > $i > $o ).
thf(func_def_1155,type,
sK790: $i > $i > $o ).
thf(func_def_1157,type,
sK792: $i > $i ).
thf(func_def_1160,type,
sK795: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_1167,type,
sK802: $o > $i > $i > $i > $i > $i > $i ).
thf(func_def_1178,type,
sK813: $i > $i > $i ).
thf(func_def_1204,type,
sK839: $i > $i > $i > $i > $i ).
thf(func_def_1205,type,
sK840: $i > $i > $i > $i > $i ).
thf(func_def_1206,type,
sK841: $i > $i > $i > $i > $i ).
thf(func_def_1207,type,
sK842: $i > $i > $i > $i > $i ).
thf(func_def_1218,type,
sK853: $i > $i > $i > $i > $i ).
thf(func_def_1225,type,
sK860: $i > $o ).
thf(func_def_1235,type,
sK870: $i > $o ).
thf(func_def_1258,type,
sK893: $i > $i > $i > $i ).
thf(func_def_1271,type,
sK906: $i > $o ).
thf(func_def_1272,type,
sK907: $i > $i ).
thf(func_def_1273,type,
sK908: ( $i > $o ) > $i ).
thf(func_def_1295,type,
sK930: $i > $i ).
thf(func_def_1308,type,
sK943: $i > $o ).
thf(func_def_1309,type,
sK944: $i > $o ).
thf(func_def_1310,type,
sK945: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_1311,type,
sK946: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_1340,type,
sK975: $i > $o ).
thf(func_def_1342,type,
sK977: ( $i > $o ) > $i > $i ).
thf(func_def_1343,type,
sK978: $i > $o ).
thf(func_def_1345,type,
sK980: $i > ( $i > $o ) > $i ).
thf(func_def_1350,type,
sK985: $i > $i ).
thf(func_def_1352,type,
sK987: ( $i > $i ) > $i > $i > $i ).
thf(func_def_1365,type,
sK1000: $i > $o ).
thf(func_def_1367,type,
sK1002: ( $i > $o ) > $i ).
thf(func_def_1368,type,
sK1003: ( $i > $o ) > $i ).
thf(func_def_1369,type,
sK1004: $i > $o ).
thf(func_def_1375,type,
sK1010: $i > ( $i > $o ) > $i ).
thf(func_def_1376,type,
sK1011: $i > $o ).
thf(func_def_1389,type,
ph1024:
!>[X0: $tType] : X0 ).
thf(f4908,plain,
$false,
inference(avatar_sat_refutation,[status(thm)],[f4786,f4791,f4792,f4896]) ).
thf(f4896,plain,
( spl1023_1
| spl1023_3 ),
inference(avatar_split_clause,[status(thm)],[f4895,f4788,f4779]) ).
thf(f4779,definition,
( spl1023_1
<=> ( $true
= ( in @ ( kpair @ sK821 @ sK822 ) @ sK819 ) ) ),
introduced(definition,[new_symbols(naming,[spl1023_1])],[avatar_definition]) ).
thf(f4788,definition,
( spl1023_3
<=> ( ( in @ ( kpair @ sK821 @ sK822 ) @ sK820 )
= $true ) ),
introduced(definition,[new_symbols(naming,[spl1023_3])],[avatar_definition]) ).
thf(f4895,plain,
( ( ( in @ ( kpair @ sK821 @ sK822 ) @ sK820 )
= $true )
| ( $true
= ( in @ ( kpair @ sK821 @ sK822 ) @ sK819 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f4894]) ).
thf(f4894,plain,
( ( $true != $true )
| ( $true
= ( in @ ( kpair @ sK821 @ sK822 ) @ sK819 ) )
| ( ( in @ ( kpair @ sK821 @ sK822 ) @ sK820 )
= $true ) ),
inference(forward_demodulation,[status(thm)],[f4863,f4024]) ).
thf(f4024,plain,
binunionE = $true,
inference(cnf_transformation,[status(thm)],[f2612]) ).
thf(f2612,plain,
( ( quantDeMorgan3 = $true )
& ( setminusSubset2 = $true )
& ( setadjoinAx = $true )
& ( woz1_5 = $true )
& ( emptyE1 = $true )
& ( quantDeMorgan2 = $true )
& ( nonemptyI = $true )
& ( intersectInPowersetIntersectUnions = $true )
& ( ex1I = $true )
& ( setadjoinSub2 = $true )
& ( setoftrueEq = $true )
& ( contraSubsetComplement = $true )
& ( ( breln1 @ sK818 @ sK819 )
= $true )
& ( ( breln1 @ sK818 @ sK820 )
= $true )
& ( $true
= ( in @ sK821 @ sK818 ) )
& ( ( in @ ( kpair @ sK821 @ sK822 ) @ ( binunion @ sK819 @ sK820 ) )
= $true )
& ( ( in @ sK822 @ sK818 )
= $true )
& ( ( $true
!= ( in @ ( kpair @ sK821 @ sK822 ) @ sK819 ) )
| ( sK823 = $true ) )
& ( sK823 != $true )
& ( ( sK823 = $true )
| ( ( in @ ( kpair @ sK821 @ sK822 ) @ sK820 )
!= $true ) )
& ( notdexE = $true )
& ( prop2setI = $true )
& ( binunionIR = $true )
& ( breln1unionI = $true )
& ( funcGraphProp2 = $true )
& ( subsetTI = $true )
& ( kfstsingleton = $true )
& ( funcImageSingleton = $true )
& ( setminusSubset1 = $true )
& ( ksndpairEq = $true )
& ( setOfPairsIsBReln = $true )
& ( exuE3e = $true )
& ( upairsetE = $true )
& ( cartprodsndpairEq = $true )
& ( symdiffE = $true )
& ( funcGraphProp3 = $true )
& ( omega0Ax = $true )
& ( binintersectSubset1 = $true )
& ( iftrueProp2 = $true )
& ( bs114d = $true )
& ( setunionI = $true )
& ( binintersectTELcontra = $true )
& ( breln1unionprop = $true )
& ( iffalseProp2 = $true )
& ( dpsetconstrEL2 = $true )
& ( emptysetimpfalse = $true )
& ( app = $true )
& ( binintersectSubset2 = $true )
& ( binunionTIRcontra = $true )
& ( powersetTE1 = $true )
& ( ap2p = $true )
& ( exuE2 = $true )
& ( setextsub = $true )
& ( breln1invI = $true )
& ( breln1unionIR = $true )
& ( setadjoinIL = $true )
& ( setOfPairsIsBReln1 = $true )
& ( foundationAx = $true )
& ( setukpairinjL1 = $true )
& ( setukpairinjR = $true )
& ( exuI1 = $true )
& ( setunionsingleton2 = $true )
& ( breln1compI = $true )
& ( breln1compE = $true )
& ( binintersectRsub = $true )
& ( subsetemptysetimpeq = $true )
& ( cartprodmempair = $true )
& ( eqinunit = $true )
& ( upairsubunion = $true )
& ( nonemptyImpWitness = $true )
& ( setminusI = $true )
& ( secondinupair = $true )
& ( ex1E2 = $true )
& ( symdiffI1 = $true )
& ( complementTI1 = $true )
& ( notequalI2 = $true )
& ( cartprodmempaircEq = $true )
& ( nonemptyI1 = $true )
& ( eqbreln = $true )
& ( prop2set2propI = $true )
& ( demorgan1b = $true )
& ( complementTnotintersectT = $true )
& ( setminusELneg = $true )
& ( omegaIndAx = $true )
& ( eta2 = $true )
& ( funcGraphProp1 = $true )
& ( subsetE2 = $true )
& ( kpairiskpair = $true )
& ( woz1_4 = $true )
& ( setminusT_lem = $true )
& ( funcext2 = $true )
& ( kfstpairEq = $true )
& ( complementImpComplementIntersect = $true )
& ( cartprodpairmemEL = $true )
& ( prop2setE = $true )
& ( iftrueorfalse = $true )
& ( powersetI = $true )
& ( lamp = $true )
& ( setminusER = $true )
& ( powersetT_lem = $true )
& ( contrasubsetT1 = $true )
& ( subsetRefl = $true )
& ( woz13rule0 = $true )
& ( cartprodpairin = $true )
& ( lam2p = $true )
& ( descr__Cong = $true )
& ( dpsetconstrI = $true )
& ( doubleComplementI1 = $true )
& ( powersetE1 = $true )
& ( subbreln1 = $true )
& ( emptyInPowerset = $true )
& ( brelnall1 = $true )
& ( disjointsetsI1 = $true )
& ( binintersectSubset5 = $true )
& ( powersetAx = $true )
& ( setukpairIL = $true )
& ( binunionRsub = $true )
& ( upairsetIL = $true )
& ( setextAx = $true )
& ( demorgan1a = $true )
& ( ubforcartprodlem1 = $true )
& ( kpairp = $true )
& ( demorgan2a = $true )
& ( replAx = $true )
& ( notdallE = $true )
& ( woz1_3 = $true )
& ( complementTcontraSubset = $true )
& ( emptyI = $true )
& ( subsetI2 = $true )
& ( setadjoin__Cong = $true )
& ( singletonprop = $true )
& ( exuE1 = $true )
& ( woz1_2 = $true )
& ( binintersectI = $true )
& ( breln1invprop = $true )
& ( theprop = $true )
& ( setukpairIR = $true )
& ( contrasubsetT = $true )
& ( setunionsingleton1 = $true )
& ( subsetI1 = $true )
& ( upairset2IR = $true )
& ( upairequniteq = $true )
& ( ex1I2 = $true )
& ( ifSingleton = $true )
& ( inIntersectImpInIntersectUnions = $true )
& ( lam2lamEq = $true )
& ( setunionsingleton = $true )
& ( demorgan2 = $true )
& ( setextT = $true )
& ( dsetconstrI = $true )
& ( brelnall2 = $true )
& ( setukpairinjR2 = $true )
& ( singletonsubset = $true )
& ( exu__Cong = $true )
& ( setminusIRneg = $true )
& ( setminusEL = $true )
& ( singletoninpowunion = $true )
& ( ksndsingleton = $true )
& ( eta1 = $true )
& ( demorgan2a1 = $true )
& ( setunion__Cong = $true )
& ( iffalse = $true )
& ( woz13rule3 = $true )
& ( subsetTrans = $true )
& ( breln1all1 = $true )
& ( binunionTILcontra = $true )
& ( subbreln = $true )
& ( powersetsubset = $true )
& ( ex1E1 = $true )
& ( binunionTE = $true )
& ( funcGraphProp4 = $true )
& ( exuI2 = $true )
& ( breln1compprop = $true )
& ( woz1_1 = $true )
& ( complementInPowersetComplementIntersect = $true )
& ( binunionLsub = $true )
& ( doubleComplementSub2 = $true )
& ( powersetTI1 = $true )
& ( setminusERneg = $true )
& ( exuI3 = $true )
& ( eqimpsubset1 = $true )
& ( symdiffIneg1 = $true )
& ( doubleComplementE1 = $true )
& ( woz13rule2 = $true )
& ( quantDeMorgan1 = $true )
& ( binunionEcases = $true )
& ( emptyset__Cong = $true )
& ( setadjoinOr = $true )
& ( descrp = $true )
& ( iftrue = $true )
& ( symdiffI2 = $true )
& ( singletoninpowerset = $true )
& ( inComplementUnionImpNotIn1 = $true )
& ( noeltsimpempty = $true )
& ( subset2powerset = $true )
& ( dsetconstr__Cong = $true )
& ( dpsetconstrSub = $true )
& ( binunionT_lem = $true )
& ( doubleComplementEq = $true )
& ( omega__Cong = $true )
& ( notequalI1 = $true )
& ( setunionE = $true )
& ( setminusILneg = $true )
& ( apProp = $true )
& ( cartprodfstpairEq = $true )
& ( funcext = $true )
& ( binintersectT_lem = $true )
& ( setukpairinjL = $true )
& ( demorgan2a2 = $true )
& ( setukpairinjR1 = $true )
& ( binintersectEL = $true )
& ( setadjoinE = $true )
& ( vacuousDall = $true )
& ( breln1SetBreln1 = $true )
& ( upairinpowunion = $true )
& ( binintersectER = $true )
& ( powerset__Cong = $true )
& ( setunionAx = $true )
& ( omegaSAx = $true )
& ( inPowerset = $true )
& ( cartprodsndin = $true )
& ( woz13rule4 = $true )
& ( powersetE = $true )
& ( demorgan2b = $true )
& ( setadjoinIR = $true )
& ( binintersectLsub = $true )
& ( lamProp = $true )
& ( funcinfuncset = $true )
& ( inIntersectImpInUnion2 = $true )
& ( singletonsswitch = $true )
& ( quantDeMorgan4 = $true )
& ( inComplementUnionImpInComplement1 = $true )
& ( in__Cong = $true )
& ( sepInPowerset = $true )
& ( binintersectSubset3 = $true )
& ( breln1all2 = $true )
& ( binunionIL = $true )
& ( setbeta = $true )
& ( woz13rule1 = $true )
& ( inCongP = $true )
& ( ubforcartprodlem3 = $true )
& ( setukpairinjR11 = $true )
& ( breln1invE = $true )
& ( notinsingleton = $true )
& ( kpairsurjEq = $true )
& ( ubforcartprodlem2 = $true )
& ( emptysetE = $true )
& ( subsetE = $true )
& ( breln1unionE = $true )
& ( breln1unionIL = $true )
& ( upairset2E = $true )
& ( dpsetconstrERa = $true )
& ( setminusLsub = $true )
& ( sepSubset = $true )
& ( dsetconstrER = $true )
& ( ap2apEq2 = $true )
& ( setext = $true )
& ( binintersectSubset4 = $true )
& ( powersetI1 = $true )
& ( uniqinunit = $true )
& ( cartprodpairmemER = $true )
& ( exuE3u = $true )
& ( demorgan1 = $true )
& ( nonemptyE1 = $true )
& ( eqimpsubset2 = $true )
& ( subPowSU = $true )
& ( iffalseProp1 = $true )
& ( exuEu = $true )
& ( complementUnionInPowersetComplement = $true )
& ( cartprodpairsurjEq = $true )
& ( setukpairinjR12 = $true )
& ( beta2 = $true )
& ( emptysetAx = $true )
& ( infuncsetfunc = $true )
& ( wellorderingAx = $true )
& ( theeq = $true )
& ( dsetconstrEL = $true )
& ( breln1compEex = $true )
& ( setunionE2 = $true )
& ( iftrueProp1 = $true )
& ( contrasubsetT3 = $true )
& ( symdiffIneg2 = $true )
& ( cartprodfstin = $true )
& ( complementSubsetComplementIntersect = $true )
& ( cartprodmempair1 = $true )
& ( upairsetIR = $true )
& ( funcextLem = $true )
& ( ifp = $true )
& ( notinemptyset = $true )
& ( contrasubsetT2 = $true )
& ( binunionTEcontra = $true )
& ( notsubsetI = $true )
& ( singletonsuniq = $true )
& ( beta1 = $true )
& ( emptysetsubset = $true )
& ( choice2fnsingleton = $true )
& ( complementT_lem = $true )
& ( setadjoinSub = $true )
& ( binunionE = $true )
& ( emptyinPowerset = $true )
& ( ap2apEq1 = $true )
& ( setukpairinjL2 = $true )
& ( doubleComplementSub1 = $true )
& ( binintersectTERcontra = $true )
& ( dpsetconstrER = $true )
& ( demorgan2b2 = $true )
& ( inIntersectImpInUnion = $true )
& ( dpsetconstrEL1 = $true )
& ( eqbreln1 = $true )
& ( complementTE1 = $true )
& ( emptyinunitempty = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK818,sK819,sK820,sK821,sK822,sK823])],[f1067,f2611,f2610,f2609,f2608,f2607]) ).
thf(f2607,plain,
( ? [X0: $i,X1: $i] :
( ( ( breln1 @ X0 @ X1 )
= $true )
& ? [X2: $i] :
( ( ( breln1 @ X0 @ X2 )
= $true )
& ? [X3: $i] :
( ( $true
= ( in @ X3 @ X0 ) )
& ? [X4: $i] :
( ( $true
= ( in @ ( kpair @ X3 @ X4 ) @ ( binunion @ X1 @ X2 ) ) )
& ( $true
= ( in @ X4 @ X0 ) )
& ? [X5: $o] :
( ( ( ( in @ ( kpair @ X3 @ X4 ) @ X1 )
!= $true )
| ( $true = X5 ) )
& ( $true != X5 )
& ( ( $true = X5 )
| ( $true
!= ( in @ ( kpair @ X3 @ X4 ) @ X2 ) ) ) ) ) ) ) )
=> ( ( ( breln1 @ sK818 @ sK819 )
= $true )
& ? [X2: $i] :
( ( ( breln1 @ sK818 @ X2 )
= $true )
& ? [X3: $i] :
( ( ( in @ X3 @ sK818 )
= $true )
& ? [X4: $i] :
( ( ( in @ ( kpair @ X3 @ X4 ) @ ( binunion @ sK819 @ X2 ) )
= $true )
& ( ( in @ X4 @ sK818 )
= $true )
& ? [X5: $o] :
( ( ( ( in @ ( kpair @ X3 @ X4 ) @ sK819 )
!= $true )
| ( $true = X5 ) )
& ( $true != X5 )
& ( ( $true = X5 )
| ( $true
!= ( in @ ( kpair @ X3 @ X4 ) @ X2 ) ) ) ) ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f2608,plain,
( ? [X2: $i] :
( ( ( breln1 @ sK818 @ X2 )
= $true )
& ? [X3: $i] :
( ( ( in @ X3 @ sK818 )
= $true )
& ? [X4: $i] :
( ( ( in @ ( kpair @ X3 @ X4 ) @ ( binunion @ sK819 @ X2 ) )
= $true )
& ( ( in @ X4 @ sK818 )
= $true )
& ? [X5: $o] :
( ( ( ( in @ ( kpair @ X3 @ X4 ) @ sK819 )
!= $true )
| ( $true = X5 ) )
& ( $true != X5 )
& ( ( $true = X5 )
| ( $true
!= ( in @ ( kpair @ X3 @ X4 ) @ X2 ) ) ) ) ) ) )
=> ( ( ( breln1 @ sK818 @ sK820 )
= $true )
& ? [X3: $i] :
( ( ( in @ X3 @ sK818 )
= $true )
& ? [X4: $i] :
( ( $true
= ( in @ ( kpair @ X3 @ X4 ) @ ( binunion @ sK819 @ sK820 ) ) )
& ( ( in @ X4 @ sK818 )
= $true )
& ? [X5: $o] :
( ( ( ( in @ ( kpair @ X3 @ X4 ) @ sK819 )
!= $true )
| ( $true = X5 ) )
& ( $true != X5 )
& ( ( $true = X5 )
| ( ( in @ ( kpair @ X3 @ X4 ) @ sK820 )
!= $true ) ) ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f2609,plain,
( ? [X3: $i] :
( ( ( in @ X3 @ sK818 )
= $true )
& ? [X4: $i] :
( ( $true
= ( in @ ( kpair @ X3 @ X4 ) @ ( binunion @ sK819 @ sK820 ) ) )
& ( ( in @ X4 @ sK818 )
= $true )
& ? [X5: $o] :
( ( ( ( in @ ( kpair @ X3 @ X4 ) @ sK819 )
!= $true )
| ( $true = X5 ) )
& ( $true != X5 )
& ( ( $true = X5 )
| ( ( in @ ( kpair @ X3 @ X4 ) @ sK820 )
!= $true ) ) ) ) )
=> ( ( $true
= ( in @ sK821 @ sK818 ) )
& ? [X4: $i] :
( ( $true
= ( in @ ( kpair @ sK821 @ X4 ) @ ( binunion @ sK819 @ sK820 ) ) )
& ( ( in @ X4 @ sK818 )
= $true )
& ? [X5: $o] :
( ( ( $true
!= ( in @ ( kpair @ sK821 @ X4 ) @ sK819 ) )
| ( $true = X5 ) )
& ( $true != X5 )
& ( ( $true = X5 )
| ( $true
!= ( in @ ( kpair @ sK821 @ X4 ) @ sK820 ) ) ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f2610,plain,
( ? [X4: $i] :
( ( $true
= ( in @ ( kpair @ sK821 @ X4 ) @ ( binunion @ sK819 @ sK820 ) ) )
& ( ( in @ X4 @ sK818 )
= $true )
& ? [X5: $o] :
( ( ( $true
!= ( in @ ( kpair @ sK821 @ X4 ) @ sK819 ) )
| ( $true = X5 ) )
& ( $true != X5 )
& ( ( $true = X5 )
| ( $true
!= ( in @ ( kpair @ sK821 @ X4 ) @ sK820 ) ) ) ) )
=> ( ( ( in @ ( kpair @ sK821 @ sK822 ) @ ( binunion @ sK819 @ sK820 ) )
= $true )
& ( ( in @ sK822 @ sK818 )
= $true )
& ? [X5: $o] :
( ( ( $true
!= ( in @ ( kpair @ sK821 @ sK822 ) @ sK819 ) )
| ( $true = X5 ) )
& ( $true != X5 )
& ( ( $true = X5 )
| ( ( in @ ( kpair @ sK821 @ sK822 ) @ sK820 )
!= $true ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f2611,plain,
( ? [X5: $o] :
( ( ( $true
!= ( in @ ( kpair @ sK821 @ sK822 ) @ sK819 ) )
| ( $true = X5 ) )
& ( $true != X5 )
& ( ( $true = X5 )
| ( ( in @ ( kpair @ sK821 @ sK822 ) @ sK820 )
!= $true ) ) )
=> ( ( ( $true
!= ( in @ ( kpair @ sK821 @ sK822 ) @ sK819 ) )
| ( sK823 = $true ) )
& ( sK823 != $true )
& ( ( sK823 = $true )
| ( ( in @ ( kpair @ sK821 @ sK822 ) @ sK820 )
!= $true ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1067,plain,
( ( quantDeMorgan3 = $true )
& ( setminusSubset2 = $true )
& ( setadjoinAx = $true )
& ( woz1_5 = $true )
& ( emptyE1 = $true )
& ( quantDeMorgan2 = $true )
& ( nonemptyI = $true )
& ( intersectInPowersetIntersectUnions = $true )
& ( ex1I = $true )
& ( setadjoinSub2 = $true )
& ( setoftrueEq = $true )
& ( contraSubsetComplement = $true )
& ? [X0: $i,X1: $i] :
( ( ( breln1 @ X0 @ X1 )
= $true )
& ? [X2: $i] :
( ( ( breln1 @ X0 @ X2 )
= $true )
& ? [X3: $i] :
( ( $true
= ( in @ X3 @ X0 ) )
& ? [X4: $i] :
( ( $true
= ( in @ ( kpair @ X3 @ X4 ) @ ( binunion @ X1 @ X2 ) ) )
& ( $true
= ( in @ X4 @ X0 ) )
& ? [X5: $o] :
( ( ( ( in @ ( kpair @ X3 @ X4 ) @ X1 )
!= $true )
| ( $true = X5 ) )
& ( $true != X5 )
& ( ( $true = X5 )
| ( $true
!= ( in @ ( kpair @ X3 @ X4 ) @ X2 ) ) ) ) ) ) ) )
& ( notdexE = $true )
& ( prop2setI = $true )
& ( binunionIR = $true )
& ( breln1unionI = $true )
& ( funcGraphProp2 = $true )
& ( subsetTI = $true )
& ( kfstsingleton = $true )
& ( funcImageSingleton = $true )
& ( setminusSubset1 = $true )
& ( ksndpairEq = $true )
& ( setOfPairsIsBReln = $true )
& ( exuE3e = $true )
& ( upairsetE = $true )
& ( cartprodsndpairEq = $true )
& ( symdiffE = $true )
& ( funcGraphProp3 = $true )
& ( omega0Ax = $true )
& ( binintersectSubset1 = $true )
& ( iftrueProp2 = $true )
& ( bs114d = $true )
& ( setunionI = $true )
& ( binintersectTELcontra = $true )
& ( breln1unionprop = $true )
& ( iffalseProp2 = $true )
& ( dpsetconstrEL2 = $true )
& ( emptysetimpfalse = $true )
& ( app = $true )
& ( binintersectSubset2 = $true )
& ( binunionTIRcontra = $true )
& ( powersetTE1 = $true )
& ( ap2p = $true )
& ( exuE2 = $true )
& ( setextsub = $true )
& ( breln1invI = $true )
& ( breln1unionIR = $true )
& ( setadjoinIL = $true )
& ( setOfPairsIsBReln1 = $true )
& ( foundationAx = $true )
& ( setukpairinjL1 = $true )
& ( setukpairinjR = $true )
& ( exuI1 = $true )
& ( setunionsingleton2 = $true )
& ( breln1compI = $true )
& ( breln1compE = $true )
& ( binintersectRsub = $true )
& ( subsetemptysetimpeq = $true )
& ( cartprodmempair = $true )
& ( eqinunit = $true )
& ( upairsubunion = $true )
& ( nonemptyImpWitness = $true )
& ( setminusI = $true )
& ( secondinupair = $true )
& ( ex1E2 = $true )
& ( symdiffI1 = $true )
& ( complementTI1 = $true )
& ( notequalI2 = $true )
& ( cartprodmempaircEq = $true )
& ( nonemptyI1 = $true )
& ( eqbreln = $true )
& ( prop2set2propI = $true )
& ( demorgan1b = $true )
& ( complementTnotintersectT = $true )
& ( setminusELneg = $true )
& ( omegaIndAx = $true )
& ( eta2 = $true )
& ( funcGraphProp1 = $true )
& ( subsetE2 = $true )
& ( kpairiskpair = $true )
& ( woz1_4 = $true )
& ( setminusT_lem = $true )
& ( funcext2 = $true )
& ( kfstpairEq = $true )
& ( complementImpComplementIntersect = $true )
& ( cartprodpairmemEL = $true )
& ( prop2setE = $true )
& ( iftrueorfalse = $true )
& ( powersetI = $true )
& ( lamp = $true )
& ( setminusER = $true )
& ( powersetT_lem = $true )
& ( contrasubsetT1 = $true )
& ( subsetRefl = $true )
& ( woz13rule0 = $true )
& ( cartprodpairin = $true )
& ( lam2p = $true )
& ( descr__Cong = $true )
& ( dpsetconstrI = $true )
& ( doubleComplementI1 = $true )
& ( powersetE1 = $true )
& ( subbreln1 = $true )
& ( emptyInPowerset = $true )
& ( brelnall1 = $true )
& ( disjointsetsI1 = $true )
& ( binintersectSubset5 = $true )
& ( powersetAx = $true )
& ( setukpairIL = $true )
& ( binunionRsub = $true )
& ( upairsetIL = $true )
& ( setextAx = $true )
& ( demorgan1a = $true )
& ( ubforcartprodlem1 = $true )
& ( kpairp = $true )
& ( demorgan2a = $true )
& ( replAx = $true )
& ( notdallE = $true )
& ( woz1_3 = $true )
& ( complementTcontraSubset = $true )
& ( emptyI = $true )
& ( subsetI2 = $true )
& ( setadjoin__Cong = $true )
& ( singletonprop = $true )
& ( exuE1 = $true )
& ( woz1_2 = $true )
& ( binintersectI = $true )
& ( breln1invprop = $true )
& ( theprop = $true )
& ( setukpairIR = $true )
& ( contrasubsetT = $true )
& ( setunionsingleton1 = $true )
& ( subsetI1 = $true )
& ( upairset2IR = $true )
& ( upairequniteq = $true )
& ( ex1I2 = $true )
& ( ifSingleton = $true )
& ( inIntersectImpInIntersectUnions = $true )
& ( lam2lamEq = $true )
& ( setunionsingleton = $true )
& ( demorgan2 = $true )
& ( setextT = $true )
& ( dsetconstrI = $true )
& ( brelnall2 = $true )
& ( setukpairinjR2 = $true )
& ( singletonsubset = $true )
& ( exu__Cong = $true )
& ( setminusIRneg = $true )
& ( setminusEL = $true )
& ( singletoninpowunion = $true )
& ( ksndsingleton = $true )
& ( eta1 = $true )
& ( demorgan2a1 = $true )
& ( setunion__Cong = $true )
& ( iffalse = $true )
& ( woz13rule3 = $true )
& ( subsetTrans = $true )
& ( breln1all1 = $true )
& ( binunionTILcontra = $true )
& ( subbreln = $true )
& ( powersetsubset = $true )
& ( ex1E1 = $true )
& ( binunionTE = $true )
& ( funcGraphProp4 = $true )
& ( exuI2 = $true )
& ( breln1compprop = $true )
& ( woz1_1 = $true )
& ( complementInPowersetComplementIntersect = $true )
& ( binunionLsub = $true )
& ( doubleComplementSub2 = $true )
& ( powersetTI1 = $true )
& ( setminusERneg = $true )
& ( exuI3 = $true )
& ( eqimpsubset1 = $true )
& ( symdiffIneg1 = $true )
& ( doubleComplementE1 = $true )
& ( woz13rule2 = $true )
& ( quantDeMorgan1 = $true )
& ( binunionEcases = $true )
& ( emptyset__Cong = $true )
& ( setadjoinOr = $true )
& ( descrp = $true )
& ( iftrue = $true )
& ( symdiffI2 = $true )
& ( singletoninpowerset = $true )
& ( inComplementUnionImpNotIn1 = $true )
& ( noeltsimpempty = $true )
& ( subset2powerset = $true )
& ( dsetconstr__Cong = $true )
& ( dpsetconstrSub = $true )
& ( binunionT_lem = $true )
& ( doubleComplementEq = $true )
& ( omega__Cong = $true )
& ( notequalI1 = $true )
& ( setunionE = $true )
& ( setminusILneg = $true )
& ( apProp = $true )
& ( cartprodfstpairEq = $true )
& ( funcext = $true )
& ( binintersectT_lem = $true )
& ( setukpairinjL = $true )
& ( demorgan2a2 = $true )
& ( setukpairinjR1 = $true )
& ( binintersectEL = $true )
& ( setadjoinE = $true )
& ( vacuousDall = $true )
& ( breln1SetBreln1 = $true )
& ( upairinpowunion = $true )
& ( binintersectER = $true )
& ( powerset__Cong = $true )
& ( setunionAx = $true )
& ( omegaSAx = $true )
& ( inPowerset = $true )
& ( cartprodsndin = $true )
& ( woz13rule4 = $true )
& ( powersetE = $true )
& ( demorgan2b = $true )
& ( setadjoinIR = $true )
& ( binintersectLsub = $true )
& ( lamProp = $true )
& ( funcinfuncset = $true )
& ( inIntersectImpInUnion2 = $true )
& ( singletonsswitch = $true )
& ( quantDeMorgan4 = $true )
& ( inComplementUnionImpInComplement1 = $true )
& ( in__Cong = $true )
& ( sepInPowerset = $true )
& ( binintersectSubset3 = $true )
& ( breln1all2 = $true )
& ( binunionIL = $true )
& ( setbeta = $true )
& ( woz13rule1 = $true )
& ( inCongP = $true )
& ( ubforcartprodlem3 = $true )
& ( setukpairinjR11 = $true )
& ( breln1invE = $true )
& ( notinsingleton = $true )
& ( kpairsurjEq = $true )
& ( ubforcartprodlem2 = $true )
& ( emptysetE = $true )
& ( subsetE = $true )
& ( breln1unionE = $true )
& ( breln1unionIL = $true )
& ( upairset2E = $true )
& ( dpsetconstrERa = $true )
& ( setminusLsub = $true )
& ( sepSubset = $true )
& ( dsetconstrER = $true )
& ( ap2apEq2 = $true )
& ( setext = $true )
& ( binintersectSubset4 = $true )
& ( powersetI1 = $true )
& ( uniqinunit = $true )
& ( cartprodpairmemER = $true )
& ( exuE3u = $true )
& ( demorgan1 = $true )
& ( nonemptyE1 = $true )
& ( eqimpsubset2 = $true )
& ( subPowSU = $true )
& ( iffalseProp1 = $true )
& ( exuEu = $true )
& ( complementUnionInPowersetComplement = $true )
& ( cartprodpairsurjEq = $true )
& ( setukpairinjR12 = $true )
& ( beta2 = $true )
& ( emptysetAx = $true )
& ( infuncsetfunc = $true )
& ( wellorderingAx = $true )
& ( theeq = $true )
& ( dsetconstrEL = $true )
& ( breln1compEex = $true )
& ( setunionE2 = $true )
& ( iftrueProp1 = $true )
& ( contrasubsetT3 = $true )
& ( symdiffIneg2 = $true )
& ( cartprodfstin = $true )
& ( complementSubsetComplementIntersect = $true )
& ( cartprodmempair1 = $true )
& ( upairsetIR = $true )
& ( funcextLem = $true )
& ( ifp = $true )
& ( notinemptyset = $true )
& ( contrasubsetT2 = $true )
& ( binunionTEcontra = $true )
& ( notsubsetI = $true )
& ( singletonsuniq = $true )
& ( beta1 = $true )
& ( emptysetsubset = $true )
& ( choice2fnsingleton = $true )
& ( complementT_lem = $true )
& ( setadjoinSub = $true )
& ( binunionE = $true )
& ( emptyinPowerset = $true )
& ( ap2apEq1 = $true )
& ( setukpairinjL2 = $true )
& ( doubleComplementSub1 = $true )
& ( binintersectTERcontra = $true )
& ( dpsetconstrER = $true )
& ( demorgan2b2 = $true )
& ( inIntersectImpInUnion = $true )
& ( dpsetconstrEL1 = $true )
& ( eqbreln1 = $true )
& ( complementTE1 = $true )
& ( emptyinunitempty = $true ) ),
inference(flattening,[status(thm)],[f1066]) ).
thf(f1066,plain,
( ? [X0: $i,X1: $i] :
( ? [X2: $i] :
( ? [X3: $i] :
( ? [X4: $i] :
( ? [X5: $o] :
( ( $true != X5 )
& ( ( $true = X5 )
| ( $true
!= ( in @ ( kpair @ X3 @ X4 ) @ X2 ) ) )
& ( ( ( in @ ( kpair @ X3 @ X4 ) @ X1 )
!= $true )
| ( $true = X5 ) ) )
& ( $true
= ( in @ ( kpair @ X3 @ X4 ) @ ( binunion @ X1 @ X2 ) ) )
& ( $true
= ( in @ X4 @ X0 ) ) )
& ( $true
= ( in @ X3 @ X0 ) ) )
& ( ( breln1 @ X0 @ X2 )
= $true ) )
& ( ( breln1 @ X0 @ X1 )
= $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)],[f425]) ).
thf(f425,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 )
=> ! [X0: $i,X1: $i] :
( ( ( breln1 @ X0 @ X1 )
= $true )
=> ! [X2: $i] :
( ( ( breln1 @ X0 @ X2 )
= $true )
=> ! [X3: $i] :
( ( $true
= ( in @ X3 @ X0 ) )
=> ! [X4: $i] :
( ( $true
= ( in @ X4 @ X0 ) )
=> ( ( $true
= ( in @ ( kpair @ X3 @ X4 ) @ ( binunion @ X1 @ X2 ) ) )
=> ! [X5: $o] :
( ( ( ( in @ ( kpair @ X3 @ X4 ) @ X1 )
= $true )
=> ( $true = X5 ) )
=> ( ( ( $true
= ( in @ ( kpair @ X3 @ X4 ) @ X2 ) )
=> ( $true = X5 ) )
=> ( $true = X5 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f424]) ).
thf(f424,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
=> ! [X0: $i,X1: $i] :
( ( breln1 @ X0 @ X1 )
=> ! [X2: $i] :
( ( breln1 @ X0 @ X2 )
=> ! [X3: $i] :
( ( in @ X3 @ X0 )
=> ! [X4: $i] :
( ( in @ X4 @ X0 )
=> ( ( in @ ( kpair @ X3 @ X4 ) @ ( binunion @ X1 @ X2 ) )
=> ! [X5: $o] :
( ( ( in @ ( kpair @ X3 @ X4 ) @ X1 )
=> X5 )
=> ( ( ( in @ ( kpair @ X3 @ X4 ) @ X2 )
=> X5 )
=> X5 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f307]) ).
thf(f307,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
=> ! [X3: $i,X13: $i] :
( ( breln1 @ X3 @ X13 )
=> ! [X15: $i] :
( ( breln1 @ X3 @ X15 )
=> ! [X1: $i] :
( ( in @ X1 @ X3 )
=> ! [X2: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ ( kpair @ X1 @ X2 ) @ ( binunion @ X13 @ X15 ) )
=> ! [X0: $o] :
( ( ( in @ ( kpair @ X1 @ X2 ) @ X13 )
=> X0 )
=> ( ( ( in @ ( kpair @ X1 @ X2 ) @ X15 )
=> X0 )
=> X0 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f306]) ).
thf(f306,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
=> ! [X3: $i,X13: $i] :
( ( breln1 @ X3 @ X13 )
=> ! [X15: $i] :
( ( breln1 @ X3 @ X15 )
=> ! [X1: $i] :
( ( in @ X1 @ X3 )
=> ! [X2: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ ( kpair @ X1 @ X2 ) @ ( binunion @ X13 @ X15 ) )
=> ! [X0: $o] :
( ( ( in @ ( kpair @ X1 @ X2 ) @ X13 )
=> X0 )
=> ( ( ( in @ ( kpair @ X1 @ X2 ) @ X15 )
=> X0 )
=> X0 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.U76IOM5xNk/DTF2THF_4862.p',breln1unionEcases) ).
thf(f4863,plain,
( ( binunionE != $true )
| ( ( in @ ( kpair @ sK821 @ sK822 ) @ sK820 )
= $true )
| ( $true
= ( in @ ( kpair @ sK821 @ sK822 ) @ sK819 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f4858]) ).
thf(f4858,plain,
( ( $true != $true )
| ( binunionE != $true )
| ( $true
= ( in @ ( kpair @ sK821 @ sK822 ) @ sK819 ) )
| ( ( in @ ( kpair @ sK821 @ sK822 ) @ sK820 )
= $true ) ),
inference(superposition,[status(thm)],[f3613,f4307]) ).
thf(f4307,plain,
( ( in @ ( kpair @ sK821 @ sK822 ) @ ( binunion @ sK819 @ sK820 ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f2612]) ).
thf(f3613,plain,
! [X2: $i,X0: $i,X1: $i] :
( ( ( in @ X2 @ ( binunion @ X1 @ X0 ) )
!= $true )
| ( ( in @ X2 @ X1 )
= $true )
| ( binunionE != $true )
| ( ( in @ X2 @ X0 )
= $true ) ),
inference(cnf_transformation,[status(thm)],[f2151]) ).
thf(f2151,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X2 @ X0 )
= $true )
| ( ( in @ X2 @ ( binunion @ X1 @ X0 ) )
!= $true )
| ( ( in @ X2 @ X1 )
= $true ) )
| ( binunionE != $true ) )
& ( ( binunionE = $true )
| ( ( ( in @ sK519 @ sK517 )
!= $true )
& ( $true
= ( in @ sK519 @ ( binunion @ sK518 @ sK517 ) ) )
& ( $true
!= ( in @ sK519 @ sK518 ) ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK517,sK518,sK519])],[f2149,f2150]) ).
thf(f2150,plain,
( ? [X3: $i,X4: $i,X5: $i] :
( ( ( in @ X5 @ X3 )
!= $true )
& ( ( in @ X5 @ ( binunion @ X4 @ X3 ) )
= $true )
& ( ( in @ X5 @ X4 )
!= $true ) )
=> ( ( ( in @ sK519 @ sK517 )
!= $true )
& ( $true
= ( in @ sK519 @ ( binunion @ sK518 @ sK517 ) ) )
& ( $true
!= ( in @ sK519 @ sK518 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f2149,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X2 @ X0 )
= $true )
| ( ( in @ X2 @ ( binunion @ X1 @ X0 ) )
!= $true )
| ( ( in @ X2 @ X1 )
= $true ) )
| ( binunionE != $true ) )
& ( ( binunionE = $true )
| ? [X3: $i,X4: $i,X5: $i] :
( ( ( in @ X5 @ X3 )
!= $true )
& ( ( in @ X5 @ ( binunion @ X4 @ X3 ) )
= $true )
& ( ( in @ X5 @ X4 )
!= $true ) ) ) ),
inference(rectify,[status(thm)],[f2148]) ).
thf(f2148,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X2 @ X0 )
= $true )
| ( ( in @ X2 @ ( binunion @ X1 @ X0 ) )
!= $true )
| ( ( in @ X2 @ X1 )
= $true ) )
| ( binunionE != $true ) )
& ( ( binunionE = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X2 @ X0 )
!= $true )
& ( ( in @ X2 @ ( binunion @ X1 @ X0 ) )
= $true )
& ( ( in @ X2 @ X1 )
!= $true ) ) ) ),
inference(nnf_transformation,[status(thm)],[f1053]) ).
thf(f1053,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X2 @ X0 )
= $true )
| ( ( in @ X2 @ ( binunion @ X1 @ X0 ) )
!= $true )
| ( ( in @ X2 @ X1 )
= $true ) )
<=> ( binunionE = $true ) ),
inference(flattening,[status(thm)],[f1052]) ).
thf(f1052,plain,
( ( binunionE = $true )
<=> ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X2 @ X0 )
= $true )
| ( ( in @ X2 @ X1 )
= $true )
| ( ( in @ X2 @ ( binunion @ X1 @ X0 ) )
!= $true ) ) ),
inference(ennf_transformation,[status(thm)],[f822]) ).
thf(f822,plain,
( ( binunionE = $true )
<=> ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X2 @ ( binunion @ X1 @ X0 ) )
= $true )
=> ( ( ( in @ X2 @ X0 )
= $true )
| ( ( in @ X2 @ X1 )
= $true ) ) ) ),
inference(fool_elimination,[status(thm)],[f821]) ).
thf(f821,plain,
! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X2 @ ( binunion @ X1 @ X0 ) )
=> ( ( in @ X2 @ X1 )
| ( in @ X2 @ X0 ) ) )
= binunionE ),
inference(rectify,[status(thm)],[f105]) ).
thf(f105,axiom,
! [X4: $i,X3: $i,X1: $i] :
( ( ( in @ X1 @ ( binunion @ X3 @ X4 ) )
=> ( ( in @ X1 @ X3 )
| ( in @ X1 @ X4 ) ) )
= binunionE ),
file('/export/starexec/sandbox2/tmp/tmp.U76IOM5xNk/DTF2THF_4862.p',binunionE) ).
thf(f4792,plain,
~ spl1023_2,
inference(avatar_split_clause,[status(thm)],[f4304,f4783]) ).
thf(f4783,definition,
( spl1023_2
<=> ( sK823 = $true ) ),
introduced(definition,[new_symbols(naming,[spl1023_2])],[avatar_definition]) ).
thf(f4304,plain,
sK823 != $true,
inference(cnf_transformation,[status(thm)],[f2612]) ).
thf(f4791,plain,
( ~ spl1023_3
| spl1023_2 ),
inference(avatar_split_clause,[status(thm)],[f4303,f4783,f4788]) ).
thf(f4303,plain,
( ( sK823 = $true )
| ( ( in @ ( kpair @ sK821 @ sK822 ) @ sK820 )
!= $true ) ),
inference(cnf_transformation,[status(thm)],[f2612]) ).
thf(f4786,plain,
( ~ spl1023_1
| spl1023_2 ),
inference(avatar_split_clause,[status(thm)],[f4305,f4783,f4779]) ).
thf(f4305,plain,
( ( sK823 = $true )
| ( $true
!= ( in @ ( kpair @ sK821 @ sK822 ) @ sK819 ) ) ),
inference(cnf_transformation,[status(thm)],[f2612]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SEU787^1 : TPTP v9.2.1. Released v3.7.0.
% 0.00/0.13 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.13/0.33 % Computer : n031.cluster.edu
% 0.13/0.33 % Model : x86_64 x86_64
% 0.13/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.33 % Memory : 8042.1875MB
% 0.13/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.33 % CPULimit : 300
% 0.13/0.33 % WCLimit : 300
% 0.13/0.33 % DateTime : Tue Feb 24 12:35:03 EST 2026
% 0.13/0.34 % CPUTime :
% 0.13/0.34 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.16/0.43 ---- Original DTF file ---
% 0.16/0.43 thf(spec,logic,$$dhol).
% 0.16/0.43 %------------------------------------------------------------------------------
% 0.16/0.43 % File : SEU787^1 : TPTP v9.2.1. Released v3.7.0.
% 0.16/0.43 % Domain : Set Theory
% 0.16/0.43 % Problem : Binary Relations on a Set
% 0.16/0.43 % Version : Especial.
% 0.16/0.43 % English : (! A:i.! R:i.breln1 A R -> (! S:i.breln1 A S -> (! x:i.in x A ->
% 0.16/0.43 % (! y:i.in y A -> in (kpair x y) (binunion R S) ->
% 0.16/0.43 % (! phi:o.(in (kpair x y) R -> phi) ->
% 0.16/0.43 % (in (kpair x y) S -> phi) -> phi)))))
% 0.16/0.43
% 0.16/0.43 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.16/0.43 % Source : [Bro08]
% 0.16/0.43 % Names : ZFC289g [Bro08]
% 0.16/0.43
% 0.16/0.43 % Status : Theorem
% 0.16/0.43 % 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, 0.22 v7.2.0, 0.12 v7.1.0, 0.25 v7.0.0, 0.14 v6.4.0, 0.17 v6.3.0, 0.20 v6.2.0, 0.14 v6.1.0, 0.29 v5.5.0, 0.33 v5.4.0, 0.40 v5.2.0, 0.60 v5.1.0, 0.80 v5.0.0, 0.60 v4.1.0, 0.67 v3.7.0
% 0.16/0.43 % Syntax : Number of formulae : 657 ( 305 unt; 351 typ; 305 def)
% 0.16/0.43 % Number of atoms : 1886 ( 435 equ; 0 cnn)
% 0.16/0.43 % Maximal formula atoms : 310 ( 6 avg)
% 0.16/0.43 % Number of connectives : 4140 ( 73 ~; 14 |; 48 &;3003 @)
% 0.16/0.43 % ( 14 <=>; 988 =>; 0 <=; 0 <~>)
% 0.16/0.43 % Maximal formula depth : 322 ( 2 avg)
% 0.16/0.43 % Number of types : 2 ( 0 usr)
% 0.16/0.43 % Number of type conns : 157 ( 157 >; 0 *; 0 +; 0 <<)
% 0.16/0.43 % Number of symbols : 354 ( 351 usr; 307 con; 0-4 aty)
% 0.16/0.43 % Number of variables : 1077 ( 71 ^; 969 !; 37 ?;1077 :)
% 0.16/0.43 % SPC : TH0_THM_EQU_NAR
% 0.16/0.43
% 0.16/0.43 % Comments : http://mathgate.info/detsetitem.php?id=352
% 0.16/0.43 %------------------------------------------------------------------------------
% 0.16/0.43 thf(in_type,type,
% 0.16/0.43 in: $i > $i > $o ).
% 0.16/0.43
% 0.16/0.43 thf(exu_type,type,
% 0.16/0.43 exu: ( $i > $o ) > $o ).
% 0.16/0.43
% 0.16/0.43 thf(exu,definition,
% 0.16/0.43 ( exu
% 0.16/0.43 = ( ^ [Xphi: $i > $o] :
% 0.16/0.43 ? [Xx: $i] :
% 0.16/0.43 ( ( Xphi @ Xx )
% 0.16/0.43 & ! [Xy: $i] :
% 0.16/0.43 ( ( Xphi @ Xy )
% 0.16/0.43 => ( Xx = Xy ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setextAx_type,type,
% 0.16/0.43 setextAx: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setextAx,definition,
% 0.16/0.43 ( setextAx
% 0.16/0.43 = ( ! [A: $i,B: $i] :
% 0.16/0.43 ( ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 <=> ( in @ Xx @ B ) )
% 0.16/0.43 => ( A = B ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(emptyset_type,type,
% 0.16/0.43 emptyset: $i ).
% 0.16/0.43
% 0.16/0.43 thf(emptysetAx_type,type,
% 0.16/0.43 emptysetAx: $o ).
% 0.16/0.43
% 0.16/0.43 thf(emptysetAx,definition,
% 0.16/0.43 ( emptysetAx
% 0.16/0.43 = ( ! [Xx: $i] :
% 0.16/0.43 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setadjoin_type,type,
% 0.16/0.43 setadjoin: $i > $i > $i ).
% 0.16/0.43
% 0.16/0.43 thf(setadjoinAx_type,type,
% 0.16/0.43 setadjoinAx: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setadjoinAx,definition,
% 0.16/0.43 ( setadjoinAx
% 0.16/0.43 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.16/0.43 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.16/0.43 <=> ( ( Xy = Xx )
% 0.16/0.43 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(powerset_type,type,
% 0.16/0.43 powerset: $i > $i ).
% 0.16/0.43
% 0.16/0.43 thf(powersetAx_type,type,
% 0.16/0.43 powersetAx: $o ).
% 0.16/0.43
% 0.16/0.43 thf(powersetAx,definition,
% 0.16/0.43 ( powersetAx
% 0.16/0.43 = ( ! [A: $i,B: $i] :
% 0.16/0.43 ( ( in @ B @ ( powerset @ A ) )
% 0.16/0.43 <=> ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ B )
% 0.16/0.43 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setunion_type,type,
% 0.16/0.43 setunion: $i > $i ).
% 0.16/0.43
% 0.16/0.43 thf(setunionAx_type,type,
% 0.16/0.43 setunionAx: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setunionAx,definition,
% 0.16/0.43 ( setunionAx
% 0.16/0.43 = ( ! [A: $i,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ ( setunion @ A ) )
% 0.16/0.43 <=> ? [B: $i] :
% 0.16/0.43 ( ( in @ Xx @ B )
% 0.16/0.43 & ( in @ B @ A ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(omega_type,type,
% 0.16/0.43 omega: $i ).
% 0.16/0.43
% 0.16/0.43 thf(omega0Ax_type,type,
% 0.16/0.43 omega0Ax: $o ).
% 0.16/0.43
% 0.16/0.43 thf(omega0Ax,definition,
% 0.16/0.43 ( omega0Ax
% 0.16/0.43 = ( in @ emptyset @ omega ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(omegaSAx_type,type,
% 0.16/0.43 omegaSAx: $o ).
% 0.16/0.43
% 0.16/0.43 thf(omegaSAx,definition,
% 0.16/0.43 ( omegaSAx
% 0.16/0.43 = ( ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ omega )
% 0.16/0.43 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(omegaIndAx_type,type,
% 0.16/0.43 omegaIndAx: $o ).
% 0.16/0.43
% 0.16/0.43 thf(omegaIndAx,definition,
% 0.16/0.43 ( omegaIndAx
% 0.16/0.43 = ( ! [A: $i] :
% 0.16/0.43 ( ( ( in @ emptyset @ A )
% 0.16/0.43 & ! [Xx: $i] :
% 0.16/0.43 ( ( ( in @ Xx @ omega )
% 0.16/0.43 & ( in @ Xx @ A ) )
% 0.16/0.43 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.16/0.43 => ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ omega )
% 0.16/0.43 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(replAx_type,type,
% 0.16/0.43 replAx: $o ).
% 0.16/0.43
% 0.16/0.43 thf(replAx,definition,
% 0.16/0.43 ( replAx
% 0.16/0.43 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.16/0.43 ( ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ( exu
% 0.16/0.43 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.16/0.43 => ? [B: $i] :
% 0.16/0.43 ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ B )
% 0.16/0.43 <=> ? [Xy: $i] :
% 0.16/0.43 ( ( in @ Xy @ A )
% 0.16/0.43 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(foundationAx_type,type,
% 0.16/0.43 foundationAx: $o ).
% 0.16/0.43
% 0.16/0.43 thf(foundationAx,definition,
% 0.16/0.43 ( foundationAx
% 0.16/0.43 = ( ! [A: $i] :
% 0.16/0.43 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.16/0.43 => ? [B: $i] :
% 0.16/0.43 ( ( in @ B @ A )
% 0.16/0.43 & ~ ? [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ B )
% 0.16/0.43 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(wellorderingAx_type,type,
% 0.16/0.43 wellorderingAx: $o ).
% 0.16/0.43
% 0.16/0.43 thf(wellorderingAx,definition,
% 0.16/0.43 ( wellorderingAx
% 0.16/0.43 = ( ! [A: $i] :
% 0.16/0.43 ? [B: $i] :
% 0.16/0.43 ( ! [C: $i] :
% 0.16/0.43 ( ( in @ C @ B )
% 0.16/0.43 => ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ C )
% 0.16/0.43 => ( in @ Xx @ A ) ) )
% 0.16/0.43 & ! [Xx: $i,Xy: $i] :
% 0.16/0.43 ( ( ( in @ Xx @ A )
% 0.16/0.43 & ( in @ Xy @ A ) )
% 0.16/0.43 => ( ! [C: $i] :
% 0.16/0.43 ( ( in @ C @ B )
% 0.16/0.43 => ( ( in @ Xx @ C )
% 0.16/0.43 <=> ( in @ Xy @ C ) ) )
% 0.16/0.43 => ( Xx = Xy ) ) )
% 0.16/0.43 & ! [C: $i,D: $i] :
% 0.16/0.43 ( ( ( in @ C @ B )
% 0.16/0.43 & ( in @ D @ B ) )
% 0.16/0.43 => ( ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ C )
% 0.16/0.43 => ( in @ Xx @ D ) )
% 0.16/0.43 | ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ D )
% 0.16/0.43 => ( in @ Xx @ C ) ) ) )
% 0.16/0.43 & ! [C: $i] :
% 0.16/0.43 ( ( ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ C )
% 0.16/0.43 => ( in @ Xx @ A ) )
% 0.16/0.43 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.16/0.43 => ? [D: $i,Xx: $i] :
% 0.16/0.43 ( ( in @ D @ B )
% 0.16/0.43 & ( in @ Xx @ C )
% 0.16/0.43 & ~ ? [Xy: $i] :
% 0.16/0.43 ( ( in @ Xy @ D )
% 0.16/0.43 & ( in @ Xy @ C ) )
% 0.16/0.43 & ! [E: $i] :
% 0.16/0.43 ( ( in @ E @ B )
% 0.16/0.43 => ( ! [Xy: $i] :
% 0.16/0.43 ( ( in @ Xy @ E )
% 0.16/0.43 => ( in @ Xy @ D ) )
% 0.16/0.43 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(descr_type,type,
% 0.16/0.43 descr: ( $i > $o ) > $i ).
% 0.16/0.43
% 0.16/0.43 thf(descrp_type,type,
% 0.16/0.43 descrp: $o ).
% 0.16/0.43
% 0.16/0.43 thf(descrp,definition,
% 0.16/0.43 ( descrp
% 0.16/0.43 = ( ! [Xphi: $i > $o] :
% 0.16/0.43 ( ( exu
% 0.16/0.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.43 => ( Xphi
% 0.16/0.43 @ ( descr
% 0.16/0.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(dsetconstr_type,type,
% 0.16/0.43 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.16/0.43
% 0.16/0.43 thf(dsetconstrI_type,type,
% 0.16/0.43 dsetconstrI: $o ).
% 0.16/0.43
% 0.16/0.43 thf(dsetconstrI,definition,
% 0.16/0.43 ( dsetconstrI
% 0.16/0.43 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ( ( Xphi @ Xx )
% 0.16/0.43 => ( in @ Xx
% 0.16/0.43 @ ( dsetconstr @ A
% 0.16/0.43 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(dsetconstrEL_type,type,
% 0.16/0.43 dsetconstrEL: $o ).
% 0.16/0.43
% 0.16/0.43 thf(dsetconstrEL,definition,
% 0.16/0.43 ( dsetconstrEL
% 0.16/0.43 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx
% 0.16/0.43 @ ( dsetconstr @ A
% 0.16/0.43 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.16/0.43 => ( in @ Xx @ A ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(dsetconstrER_type,type,
% 0.16/0.43 dsetconstrER: $o ).
% 0.16/0.43
% 0.16/0.43 thf(dsetconstrER,definition,
% 0.16/0.43 ( dsetconstrER
% 0.16/0.43 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx
% 0.16/0.43 @ ( dsetconstr @ A
% 0.16/0.43 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.16/0.43 => ( Xphi @ Xx ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(exuE1_type,type,
% 0.16/0.43 exuE1: $o ).
% 0.16/0.43
% 0.16/0.43 thf(exuE1,definition,
% 0.16/0.43 ( exuE1
% 0.16/0.43 = ( ! [Xphi: $i > $o] :
% 0.16/0.43 ( ( exu
% 0.16/0.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.43 => ? [Xx: $i] :
% 0.16/0.43 ( ( Xphi @ Xx )
% 0.16/0.43 & ! [Xy: $i] :
% 0.16/0.43 ( ( Xphi @ Xy )
% 0.16/0.43 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(prop2set_type,type,
% 0.16/0.43 prop2set: $o > $i ).
% 0.16/0.43
% 0.16/0.43 thf(prop2setE_type,type,
% 0.16/0.43 prop2setE: $o ).
% 0.16/0.43
% 0.16/0.43 thf(prop2setE,definition,
% 0.16/0.43 ( prop2setE
% 0.16/0.43 = ( ! [Xphi: $o,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.16/0.43 => Xphi ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(emptysetE_type,type,
% 0.16/0.43 emptysetE: $o ).
% 0.16/0.43
% 0.16/0.43 thf(emptysetE,definition,
% 0.16/0.43 ( emptysetE
% 0.16/0.43 = ( ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ emptyset )
% 0.16/0.43 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(emptysetimpfalse_type,type,
% 0.16/0.43 emptysetimpfalse: $o ).
% 0.16/0.43
% 0.16/0.43 thf(emptysetimpfalse,definition,
% 0.16/0.43 ( emptysetimpfalse
% 0.16/0.43 = ( ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ emptyset )
% 0.16/0.43 => $false ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(notinemptyset_type,type,
% 0.16/0.43 notinemptyset: $o ).
% 0.16/0.43
% 0.16/0.43 thf(notinemptyset,definition,
% 0.16/0.43 ( notinemptyset
% 0.16/0.43 = ( ! [Xx: $i] :
% 0.16/0.43 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(exuE3e_type,type,
% 0.16/0.43 exuE3e: $o ).
% 0.16/0.43
% 0.16/0.43 thf(exuE3e,definition,
% 0.16/0.43 ( exuE3e
% 0.16/0.43 = ( ! [Xphi: $i > $o] :
% 0.16/0.43 ( ( exu
% 0.16/0.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.43 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setext_type,type,
% 0.16/0.43 setext: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setext,definition,
% 0.16/0.43 ( setext
% 0.16/0.43 = ( ! [A: $i,B: $i] :
% 0.16/0.43 ( ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ( in @ Xx @ B ) )
% 0.16/0.43 => ( ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ B )
% 0.16/0.43 => ( in @ Xx @ A ) )
% 0.16/0.43 => ( A = B ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(emptyI_type,type,
% 0.16/0.43 emptyI: $o ).
% 0.16/0.43
% 0.16/0.43 thf(emptyI,definition,
% 0.16/0.43 ( emptyI
% 0.16/0.43 = ( ! [A: $i] :
% 0.16/0.43 ( ! [Xx: $i] :
% 0.16/0.43 ~ ( in @ Xx @ A )
% 0.16/0.43 => ( A = emptyset ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(noeltsimpempty_type,type,
% 0.16/0.43 noeltsimpempty: $o ).
% 0.16/0.43
% 0.16/0.43 thf(noeltsimpempty,definition,
% 0.16/0.43 ( noeltsimpempty
% 0.16/0.43 = ( ! [A: $i] :
% 0.16/0.43 ( ! [Xx: $i] :
% 0.16/0.43 ~ ( in @ Xx @ A )
% 0.16/0.43 => ( A = emptyset ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setbeta_type,type,
% 0.16/0.43 setbeta: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setbeta,definition,
% 0.16/0.43 ( setbeta
% 0.16/0.43 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ( ( in @ Xx
% 0.16/0.43 @ ( dsetconstr @ A
% 0.16/0.43 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.16/0.43 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(nonempty_type,type,
% 0.16/0.43 nonempty: $i > $o ).
% 0.16/0.43
% 0.16/0.43 thf(nonempty,definition,
% 0.16/0.43 ( nonempty
% 0.16/0.43 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(nonemptyE1_type,type,
% 0.16/0.43 nonemptyE1: $o ).
% 0.16/0.43
% 0.16/0.43 thf(nonemptyE1,definition,
% 0.16/0.43 ( nonemptyE1
% 0.16/0.43 = ( ! [A: $i] :
% 0.16/0.43 ( ( nonempty @ A )
% 0.16/0.43 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(nonemptyI_type,type,
% 0.16/0.43 nonemptyI: $o ).
% 0.16/0.43
% 0.16/0.43 thf(nonemptyI,definition,
% 0.16/0.43 ( nonemptyI
% 0.16/0.43 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ( ( Xphi @ Xx )
% 0.16/0.43 => ( nonempty
% 0.16/0.43 @ ( dsetconstr @ A
% 0.16/0.43 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(nonemptyI1_type,type,
% 0.16/0.43 nonemptyI1: $o ).
% 0.16/0.43
% 0.16/0.43 thf(nonemptyI1,definition,
% 0.16/0.43 ( nonemptyI1
% 0.16/0.43 = ( ! [A: $i] :
% 0.16/0.43 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.16/0.43 => ( nonempty @ A ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setadjoinIL_type,type,
% 0.16/0.43 setadjoinIL: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setadjoinIL,definition,
% 0.16/0.43 ( setadjoinIL
% 0.16/0.43 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(emptyinunitempty_type,type,
% 0.16/0.43 emptyinunitempty: $o ).
% 0.16/0.43
% 0.16/0.43 thf(emptyinunitempty,definition,
% 0.16/0.43 ( emptyinunitempty
% 0.16/0.43 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setadjoinIR_type,type,
% 0.16/0.43 setadjoinIR: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setadjoinIR,definition,
% 0.16/0.43 ( setadjoinIR
% 0.16/0.43 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.16/0.43 ( ( in @ Xy @ A )
% 0.16/0.43 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setadjoinE_type,type,
% 0.16/0.43 setadjoinE: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setadjoinE,definition,
% 0.16/0.43 ( setadjoinE
% 0.16/0.43 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.16/0.43 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.16/0.43 => ! [Xphi: $o] :
% 0.16/0.43 ( ( ( Xy = Xx )
% 0.16/0.43 => Xphi )
% 0.16/0.43 => ( ( ( in @ Xy @ A )
% 0.16/0.43 => Xphi )
% 0.16/0.43 => Xphi ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setadjoinOr_type,type,
% 0.16/0.43 setadjoinOr: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setadjoinOr,definition,
% 0.16/0.43 ( setadjoinOr
% 0.16/0.43 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.16/0.43 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.16/0.43 => ( ( Xy = Xx )
% 0.16/0.43 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setoftrueEq_type,type,
% 0.16/0.43 setoftrueEq: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setoftrueEq,definition,
% 0.16/0.43 ( setoftrueEq
% 0.16/0.43 = ( ! [A: $i] :
% 0.16/0.43 ( ( dsetconstr @ A
% 0.16/0.43 @ ^ [Xx: $i] : $true )
% 0.16/0.43 = A ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(powersetI_type,type,
% 0.16/0.43 powersetI: $o ).
% 0.16/0.43
% 0.16/0.43 thf(powersetI,definition,
% 0.16/0.43 ( powersetI
% 0.16/0.43 = ( ! [A: $i,B: $i] :
% 0.16/0.43 ( ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ B )
% 0.16/0.43 => ( in @ Xx @ A ) )
% 0.16/0.43 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(emptyinPowerset_type,type,
% 0.16/0.43 emptyinPowerset: $o ).
% 0.16/0.43
% 0.16/0.43 thf(emptyinPowerset,definition,
% 0.16/0.43 ( emptyinPowerset
% 0.16/0.43 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(emptyInPowerset_type,type,
% 0.16/0.43 emptyInPowerset: $o ).
% 0.16/0.43
% 0.16/0.43 thf(emptyInPowerset,definition,
% 0.16/0.43 ( emptyInPowerset
% 0.16/0.43 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(powersetE_type,type,
% 0.16/0.43 powersetE: $o ).
% 0.16/0.43
% 0.16/0.43 thf(powersetE,definition,
% 0.16/0.43 ( powersetE
% 0.16/0.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.43 ( ( in @ B @ ( powerset @ A ) )
% 0.16/0.43 => ( ( in @ Xx @ B )
% 0.16/0.43 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setunionI_type,type,
% 0.16/0.43 setunionI: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setunionI,definition,
% 0.16/0.43 ( setunionI
% 0.16/0.43 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.16/0.43 ( ( in @ Xx @ B )
% 0.16/0.43 => ( ( in @ B @ A )
% 0.16/0.43 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setunionE_type,type,
% 0.16/0.43 setunionE: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setunionE,definition,
% 0.16/0.43 ( setunionE
% 0.16/0.43 = ( ! [A: $i,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ ( setunion @ A ) )
% 0.16/0.43 => ! [Xphi: $o] :
% 0.16/0.43 ( ! [B: $i] :
% 0.16/0.43 ( ( in @ Xx @ B )
% 0.16/0.43 => ( ( in @ B @ A )
% 0.16/0.43 => Xphi ) )
% 0.16/0.43 => Xphi ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(subPowSU_type,type,
% 0.16/0.43 subPowSU: $o ).
% 0.16/0.43
% 0.16/0.43 thf(subPowSU,definition,
% 0.16/0.43 ( subPowSU
% 0.16/0.43 = ( ! [A: $i,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(exuE2_type,type,
% 0.16/0.43 exuE2: $o ).
% 0.16/0.43
% 0.16/0.43 thf(exuE2,definition,
% 0.16/0.43 ( exuE2
% 0.16/0.43 = ( ! [Xphi: $i > $o] :
% 0.16/0.43 ( ( exu
% 0.16/0.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.43 => ? [Xx: $i] :
% 0.16/0.43 ! [Xy: $i] :
% 0.16/0.43 ( ( Xphi @ Xy )
% 0.16/0.43 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(nonemptyImpWitness_type,type,
% 0.16/0.43 nonemptyImpWitness: $o ).
% 0.16/0.43
% 0.16/0.43 thf(nonemptyImpWitness,definition,
% 0.16/0.43 ( nonemptyImpWitness
% 0.16/0.43 = ( ! [A: $i] :
% 0.16/0.43 ( ( nonempty @ A )
% 0.16/0.43 => ? [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 & $true ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(uniqinunit_type,type,
% 0.16/0.43 uniqinunit: $o ).
% 0.16/0.43
% 0.16/0.43 thf(uniqinunit,definition,
% 0.16/0.43 ( uniqinunit
% 0.16/0.43 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.43 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.16/0.43 => ( Xx = Xy ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(notinsingleton_type,type,
% 0.16/0.43 notinsingleton: $o ).
% 0.16/0.43
% 0.16/0.43 thf(notinsingleton,definition,
% 0.16/0.43 ( notinsingleton
% 0.16/0.43 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.43 ( ( Xx != Xy )
% 0.16/0.43 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(eqinunit_type,type,
% 0.16/0.43 eqinunit: $o ).
% 0.16/0.43
% 0.16/0.43 thf(eqinunit,definition,
% 0.16/0.43 ( eqinunit
% 0.16/0.43 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.43 ( ( Xx = Xy )
% 0.16/0.43 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(singletonsswitch_type,type,
% 0.16/0.43 singletonsswitch: $o ).
% 0.16/0.43
% 0.16/0.43 thf(singletonsswitch,definition,
% 0.16/0.43 ( singletonsswitch
% 0.16/0.43 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.43 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.16/0.43 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(upairsetE_type,type,
% 0.16/0.43 upairsetE: $o ).
% 0.16/0.43
% 0.16/0.43 thf(upairsetE,definition,
% 0.16/0.43 ( upairsetE
% 0.16/0.43 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.16/0.43 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.16/0.43 => ( ( Xz = Xx )
% 0.16/0.43 | ( Xz = Xy ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(upairsetIL_type,type,
% 0.16/0.43 upairsetIL: $o ).
% 0.16/0.43
% 0.16/0.43 thf(upairsetIL,definition,
% 0.16/0.43 ( upairsetIL
% 0.16/0.43 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(upairsetIR_type,type,
% 0.16/0.43 upairsetIR: $o ).
% 0.16/0.43
% 0.16/0.43 thf(upairsetIR,definition,
% 0.16/0.43 ( upairsetIR
% 0.16/0.43 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(emptyE1_type,type,
% 0.16/0.43 emptyE1: $o ).
% 0.16/0.43
% 0.16/0.43 thf(emptyE1,definition,
% 0.16/0.43 ( emptyE1
% 0.16/0.43 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.43 ( ? [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 & ( Xphi @ Xx ) )
% 0.16/0.43 => ( ( ( dsetconstr @ A
% 0.16/0.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.44 = emptyset )
% 0.16/0.44 => $false ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(vacuousDall_type,type,
% 0.16/0.44 vacuousDall: $o ).
% 0.16/0.44
% 0.16/0.44 thf(vacuousDall,definition,
% 0.16/0.44 ( vacuousDall
% 0.16/0.44 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ emptyset )
% 0.16/0.44 => ( Xphi @ Xx ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(quantDeMorgan1_type,type,
% 0.16/0.44 quantDeMorgan1: $o ).
% 0.16/0.44
% 0.16/0.44 thf(quantDeMorgan1,definition,
% 0.16/0.44 ( quantDeMorgan1
% 0.16/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.44 ( ~ ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( Xphi @ Xx ) )
% 0.16/0.44 => ? [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(quantDeMorgan2_type,type,
% 0.16/0.44 quantDeMorgan2: $o ).
% 0.16/0.44
% 0.16/0.44 thf(quantDeMorgan2,definition,
% 0.16/0.44 ( quantDeMorgan2
% 0.16/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.44 ( ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ~ ( Xphi @ Xx ) )
% 0.16/0.44 => ~ ? [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(quantDeMorgan3_type,type,
% 0.16/0.44 quantDeMorgan3: $o ).
% 0.16/0.44
% 0.16/0.44 thf(quantDeMorgan3,definition,
% 0.16/0.44 ( quantDeMorgan3
% 0.16/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.44 ( ~ ? [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 & ( Xphi @ Xx ) )
% 0.16/0.44 => ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(quantDeMorgan4_type,type,
% 0.16/0.44 quantDeMorgan4: $o ).
% 0.16/0.44
% 0.16/0.44 thf(quantDeMorgan4,definition,
% 0.16/0.44 ( quantDeMorgan4
% 0.16/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.44 ( ? [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 & ~ ( Xphi @ Xx ) )
% 0.16/0.44 => ~ ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(prop2setI_type,type,
% 0.16/0.44 prop2setI: $o ).
% 0.16/0.44
% 0.16/0.44 thf(prop2setI,definition,
% 0.16/0.44 ( prop2setI
% 0.16/0.44 = ( ! [Xphi: $o] :
% 0.16/0.44 ( Xphi
% 0.16/0.44 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(set2prop_type,type,
% 0.16/0.44 set2prop: $i > $o ).
% 0.16/0.44
% 0.16/0.44 thf(prop2set2propI_type,type,
% 0.16/0.44 prop2set2propI: $o ).
% 0.16/0.44
% 0.16/0.44 thf(prop2set2propI,definition,
% 0.16/0.44 ( prop2set2propI
% 0.16/0.44 = ( ! [Xphi: $o] :
% 0.16/0.44 ( Xphi
% 0.16/0.44 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(notdexE_type,type,
% 0.16/0.44 notdexE: $o ).
% 0.16/0.44
% 0.16/0.44 thf(notdexE,definition,
% 0.16/0.44 ( notdexE
% 0.16/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.44 ( ~ ? [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 & ( Xphi @ Xx ) )
% 0.16/0.44 => ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(notdallE_type,type,
% 0.16/0.44 notdallE: $o ).
% 0.16/0.44
% 0.16/0.44 thf(notdallE,definition,
% 0.16/0.44 ( notdallE
% 0.16/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.44 ( ~ ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( Xphi @ Xx ) )
% 0.16/0.44 => ? [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(exuI1_type,type,
% 0.16/0.44 exuI1: $o ).
% 0.16/0.44
% 0.16/0.44 thf(exuI1,definition,
% 0.16/0.44 ( exuI1
% 0.16/0.44 = ( ! [Xphi: $i > $o] :
% 0.16/0.44 ( ? [Xx: $i] :
% 0.16/0.44 ( ( Xphi @ Xx )
% 0.16/0.44 & ! [Xy: $i] :
% 0.16/0.44 ( ( Xphi @ Xy )
% 0.16/0.44 => ( Xx = Xy ) ) )
% 0.16/0.44 => ( exu
% 0.16/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(exuI3_type,type,
% 0.16/0.44 exuI3: $o ).
% 0.16/0.44
% 0.16/0.44 thf(exuI3,definition,
% 0.16/0.44 ( exuI3
% 0.16/0.44 = ( ! [Xphi: $i > $o] :
% 0.16/0.44 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.16/0.44 => ( ! [Xx: $i,Xy: $i] :
% 0.16/0.44 ( ( Xphi @ Xx )
% 0.16/0.44 => ( ( Xphi @ Xy )
% 0.16/0.44 => ( Xx = Xy ) ) )
% 0.16/0.44 => ( exu
% 0.16/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(exuI2_type,type,
% 0.16/0.44 exuI2: $o ).
% 0.16/0.44
% 0.16/0.44 thf(exuI2,definition,
% 0.16/0.44 ( exuI2
% 0.16/0.44 = ( ! [Xphi: $i > $o] :
% 0.16/0.44 ( ? [Xx: $i] :
% 0.16/0.44 ! [Xy: $i] :
% 0.16/0.44 ( ( Xphi @ Xy )
% 0.16/0.44 <=> ( Xy = Xx ) )
% 0.16/0.44 => ( exu
% 0.16/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(inCongP_type,type,
% 0.16/0.44 inCongP: $o ).
% 0.16/0.44
% 0.16/0.44 thf(inCongP,definition,
% 0.16/0.44 ( inCongP
% 0.16/0.44 = ( ! [A: $i,B: $i] :
% 0.16/0.44 ( ( A = B )
% 0.16/0.44 => ! [Xx: $i,Xy: $i] :
% 0.16/0.44 ( ( Xx = Xy )
% 0.16/0.44 => ( ( in @ Xx @ A )
% 0.16/0.44 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(in__Cong_type,type,
% 0.16/0.44 in__Cong: $o ).
% 0.16/0.44
% 0.16/0.44 thf(in__Cong,definition,
% 0.16/0.44 ( in__Cong
% 0.16/0.44 = ( ! [A: $i,B: $i] :
% 0.16/0.44 ( ( A = B )
% 0.16/0.44 => ! [Xx: $i,Xy: $i] :
% 0.16/0.44 ( ( Xx = Xy )
% 0.16/0.44 => ( ( in @ Xx @ A )
% 0.16/0.44 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(exuE3u_type,type,
% 0.16/0.44 exuE3u: $o ).
% 0.16/0.44
% 0.16/0.44 thf(exuE3u,definition,
% 0.16/0.44 ( exuE3u
% 0.16/0.44 = ( ! [Xphi: $i > $o] :
% 0.16/0.44 ( ( exu
% 0.16/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.44 => ! [Xx: $i,Xy: $i] :
% 0.16/0.44 ( ( Xphi @ Xx )
% 0.16/0.44 => ( ( Xphi @ Xy )
% 0.16/0.44 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(exu__Cong_type,type,
% 0.16/0.44 exu__Cong: $o ).
% 0.16/0.44
% 0.16/0.44 thf(exu__Cong,definition,
% 0.16/0.44 ( exu__Cong
% 0.16/0.44 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.16/0.44 ( ! [Xx: $i,Xy: $i] :
% 0.16/0.44 ( ( Xx = Xy )
% 0.16/0.44 => ( ( Xphi @ Xx )
% 0.16/0.44 <=> ( Xpsi @ Xy ) ) )
% 0.16/0.44 => ( ( exu
% 0.16/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.44 <=> ( exu
% 0.16/0.44 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(emptyset__Cong_type,type,
% 0.16/0.44 emptyset__Cong: $o ).
% 0.16/0.44
% 0.16/0.44 thf(emptyset__Cong,definition,
% 0.16/0.44 ( emptyset__Cong
% 0.16/0.44 = ( emptyset = emptyset ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(setadjoin__Cong_type,type,
% 0.16/0.44 setadjoin__Cong: $o ).
% 0.16/0.44
% 0.16/0.44 thf(setadjoin__Cong,definition,
% 0.16/0.44 ( setadjoin__Cong
% 0.16/0.44 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.44 ( ( Xx = Xy )
% 0.16/0.44 => ! [Xz: $i,Xu: $i] :
% 0.16/0.44 ( ( Xz = Xu )
% 0.16/0.44 => ( ( setadjoin @ Xx @ Xz )
% 0.16/0.44 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(powerset__Cong_type,type,
% 0.16/0.44 powerset__Cong: $o ).
% 0.16/0.44
% 0.16/0.44 thf(powerset__Cong,definition,
% 0.16/0.44 ( powerset__Cong
% 0.16/0.44 = ( ! [A: $i,B: $i] :
% 0.16/0.44 ( ( A = B )
% 0.16/0.44 => ( ( powerset @ A )
% 0.16/0.44 = ( powerset @ B ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(setunion__Cong_type,type,
% 0.16/0.44 setunion__Cong: $o ).
% 0.16/0.44
% 0.16/0.44 thf(setunion__Cong,definition,
% 0.16/0.44 ( setunion__Cong
% 0.16/0.44 = ( ! [A: $i,B: $i] :
% 0.16/0.44 ( ( A = B )
% 0.16/0.44 => ( ( setunion @ A )
% 0.16/0.44 = ( setunion @ B ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(omega__Cong_type,type,
% 0.16/0.44 omega__Cong: $o ).
% 0.16/0.44
% 0.16/0.44 thf(omega__Cong,definition,
% 0.16/0.44 ( omega__Cong
% 0.16/0.44 = ( omega = omega ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(exuEu_type,type,
% 0.16/0.44 exuEu: $o ).
% 0.16/0.44
% 0.16/0.44 thf(exuEu,definition,
% 0.16/0.44 ( exuEu
% 0.16/0.44 = ( ! [Xphi: $i > $o] :
% 0.16/0.44 ( ( exu
% 0.16/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.44 => ! [Xx: $i,Xy: $i] :
% 0.16/0.44 ( ( Xphi @ Xx )
% 0.16/0.44 => ( ( Xphi @ Xy )
% 0.16/0.44 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(descr__Cong_type,type,
% 0.16/0.44 descr__Cong: $o ).
% 0.16/0.44
% 0.16/0.44 thf(descr__Cong,definition,
% 0.16/0.44 ( descr__Cong
% 0.16/0.44 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.16/0.44 ( ! [Xx: $i,Xy: $i] :
% 0.16/0.44 ( ( Xx = Xy )
% 0.16/0.44 => ( ( Xphi @ Xx )
% 0.16/0.44 <=> ( Xpsi @ Xy ) ) )
% 0.16/0.44 => ( ( exu
% 0.16/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.44 => ( ( exu
% 0.16/0.44 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.16/0.44 => ( ( descr
% 0.16/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.44 = ( descr
% 0.16/0.44 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(dsetconstr__Cong_type,type,
% 0.16/0.44 dsetconstr__Cong: $o ).
% 0.16/0.44
% 0.16/0.44 thf(dsetconstr__Cong,definition,
% 0.16/0.44 ( dsetconstr__Cong
% 0.16/0.44 = ( ! [A: $i,B: $i] :
% 0.16/0.44 ( ( A = B )
% 0.16/0.44 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.16/0.44 ( ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ! [Xy: $i] :
% 0.16/0.44 ( ( in @ Xy @ B )
% 0.16/0.44 => ( ( Xx = Xy )
% 0.16/0.44 => ( ( Xphi @ Xx )
% 0.16/0.44 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.16/0.44 => ( ( dsetconstr @ A
% 0.16/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.44 = ( dsetconstr @ B
% 0.16/0.44 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(subset_type,type,
% 0.16/0.44 subset: $i > $i > $o ).
% 0.16/0.44
% 0.16/0.44 thf(disjoint_type,type,
% 0.16/0.44 disjoint: $i > $i > $o ).
% 0.16/0.44
% 0.16/0.44 thf(setsmeet_type,type,
% 0.16/0.44 setsmeet: $i > $i > $o ).
% 0.16/0.44
% 0.16/0.44 thf(subsetI1_type,type,
% 0.16/0.44 subsetI1: $o ).
% 0.16/0.44
% 0.16/0.44 thf(subsetI1,definition,
% 0.16/0.44 ( subsetI1
% 0.16/0.44 = ( ! [A: $i,B: $i] :
% 0.16/0.44 ( ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( in @ Xx @ B ) )
% 0.16/0.44 => ( subset @ A @ B ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(eqimpsubset2_type,type,
% 0.16/0.44 eqimpsubset2: $o ).
% 0.16/0.44
% 0.16/0.44 thf(eqimpsubset2,definition,
% 0.16/0.44 ( eqimpsubset2
% 0.16/0.44 = ( ! [A: $i,B: $i] :
% 0.16/0.44 ( ( A = B )
% 0.16/0.44 => ( subset @ B @ A ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(eqimpsubset1_type,type,
% 0.16/0.44 eqimpsubset1: $o ).
% 0.16/0.44
% 0.16/0.44 thf(eqimpsubset1,definition,
% 0.16/0.44 ( eqimpsubset1
% 0.16/0.44 = ( ! [A: $i,B: $i] :
% 0.16/0.44 ( ( A = B )
% 0.16/0.44 => ( subset @ A @ B ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(subsetI2_type,type,
% 0.16/0.44 subsetI2: $o ).
% 0.16/0.44
% 0.16/0.44 thf(subsetI2,definition,
% 0.16/0.44 ( subsetI2
% 0.16/0.44 = ( ! [A: $i,B: $i] :
% 0.16/0.44 ( ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( in @ Xx @ B ) )
% 0.16/0.44 => ( subset @ A @ B ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(emptysetsubset_type,type,
% 0.16/0.44 emptysetsubset: $o ).
% 0.16/0.44
% 0.16/0.44 thf(emptysetsubset,definition,
% 0.16/0.44 ( emptysetsubset
% 0.16/0.44 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(subsetE_type,type,
% 0.16/0.44 subsetE: $o ).
% 0.16/0.44
% 0.16/0.44 thf(subsetE,definition,
% 0.16/0.44 ( subsetE
% 0.16/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.44 ( ( subset @ A @ B )
% 0.16/0.44 => ( ( in @ Xx @ A )
% 0.16/0.44 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(subsetE2_type,type,
% 0.16/0.44 subsetE2: $o ).
% 0.16/0.44
% 0.16/0.44 thf(subsetE2,definition,
% 0.16/0.44 ( subsetE2
% 0.16/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.44 ( ( subset @ A @ B )
% 0.16/0.44 => ( ~ ( in @ Xx @ B )
% 0.16/0.44 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(notsubsetI_type,type,
% 0.16/0.44 notsubsetI: $o ).
% 0.16/0.44
% 0.16/0.44 thf(notsubsetI,definition,
% 0.16/0.44 ( notsubsetI
% 0.16/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( ~ ( in @ Xx @ B )
% 0.16/0.44 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(notequalI1_type,type,
% 0.16/0.44 notequalI1: $o ).
% 0.16/0.44
% 0.16/0.44 thf(notequalI1,definition,
% 0.16/0.44 ( notequalI1
% 0.16/0.44 = ( ! [A: $i,B: $i] :
% 0.16/0.44 ( ~ ( subset @ A @ B )
% 0.16/0.44 => ( A != B ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(notequalI2_type,type,
% 0.16/0.44 notequalI2: $o ).
% 0.16/0.44
% 0.16/0.44 thf(notequalI2,definition,
% 0.16/0.44 ( notequalI2
% 0.16/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( ~ ( in @ Xx @ B )
% 0.16/0.44 => ( A != B ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(subsetRefl_type,type,
% 0.16/0.44 subsetRefl: $o ).
% 0.16/0.44
% 0.16/0.44 thf(subsetRefl,definition,
% 0.16/0.44 ( subsetRefl
% 0.16/0.44 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(subsetTrans_type,type,
% 0.16/0.44 subsetTrans: $o ).
% 0.16/0.44
% 0.16/0.44 thf(subsetTrans,definition,
% 0.16/0.44 ( subsetTrans
% 0.16/0.44 = ( ! [A: $i,B: $i,C: $i] :
% 0.16/0.44 ( ( subset @ A @ B )
% 0.16/0.44 => ( ( subset @ B @ C )
% 0.16/0.44 => ( subset @ A @ C ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(setadjoinSub_type,type,
% 0.16/0.44 setadjoinSub: $o ).
% 0.16/0.44
% 0.16/0.44 thf(setadjoinSub,definition,
% 0.16/0.44 ( setadjoinSub
% 0.16/0.44 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(setadjoinSub2_type,type,
% 0.16/0.44 setadjoinSub2: $o ).
% 0.16/0.44
% 0.16/0.44 thf(setadjoinSub2,definition,
% 0.16/0.44 ( setadjoinSub2
% 0.16/0.44 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.16/0.44 ( ( subset @ A @ B )
% 0.16/0.44 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(subset2powerset_type,type,
% 0.16/0.44 subset2powerset: $o ).
% 0.16/0.44
% 0.16/0.44 thf(subset2powerset,definition,
% 0.16/0.44 ( subset2powerset
% 0.16/0.44 = ( ! [A: $i,B: $i] :
% 0.16/0.44 ( ( subset @ A @ B )
% 0.16/0.44 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(setextsub_type,type,
% 0.16/0.44 setextsub: $o ).
% 0.16/0.44
% 0.16/0.44 thf(setextsub,definition,
% 0.16/0.44 ( setextsub
% 0.16/0.44 = ( ! [A: $i,B: $i] :
% 0.16/0.44 ( ( subset @ A @ B )
% 0.16/0.44 => ( ( subset @ B @ A )
% 0.16/0.44 => ( A = B ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(subsetemptysetimpeq_type,type,
% 0.16/0.44 subsetemptysetimpeq: $o ).
% 0.16/0.44
% 0.16/0.44 thf(subsetemptysetimpeq,definition,
% 0.16/0.44 ( subsetemptysetimpeq
% 0.16/0.44 = ( ! [A: $i] :
% 0.16/0.44 ( ( subset @ A @ emptyset )
% 0.16/0.44 => ( A = emptyset ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(powersetI1_type,type,
% 0.16/0.44 powersetI1: $o ).
% 0.16/0.44
% 0.16/0.44 thf(powersetI1,definition,
% 0.16/0.44 ( powersetI1
% 0.16/0.44 = ( ! [A: $i,B: $i] :
% 0.16/0.44 ( ( subset @ B @ A )
% 0.16/0.44 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(powersetE1_type,type,
% 0.16/0.44 powersetE1: $o ).
% 0.16/0.44
% 0.16/0.44 thf(powersetE1,definition,
% 0.16/0.44 ( powersetE1
% 0.16/0.44 = ( ! [A: $i,B: $i] :
% 0.16/0.44 ( ( in @ B @ ( powerset @ A ) )
% 0.16/0.44 => ( subset @ B @ A ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(inPowerset_type,type,
% 0.16/0.44 inPowerset: $o ).
% 0.16/0.44
% 0.16/0.44 thf(inPowerset,definition,
% 0.16/0.44 ( inPowerset
% 0.16/0.44 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(powersetsubset_type,type,
% 0.16/0.44 powersetsubset: $o ).
% 0.16/0.44
% 0.16/0.44 thf(powersetsubset,definition,
% 0.16/0.44 ( powersetsubset
% 0.16/0.44 = ( ! [A: $i,B: $i] :
% 0.16/0.44 ( ( subset @ A @ B )
% 0.16/0.44 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(sepInPowerset_type,type,
% 0.16/0.44 sepInPowerset: $o ).
% 0.16/0.44
% 0.16/0.44 thf(sepInPowerset,definition,
% 0.16/0.44 ( sepInPowerset
% 0.16/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.44 ( in
% 0.16/0.44 @ ( dsetconstr @ A
% 0.16/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.44 @ ( powerset @ A ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(sepSubset_type,type,
% 0.16/0.44 sepSubset: $o ).
% 0.16/0.44
% 0.16/0.44 thf(sepSubset,definition,
% 0.16/0.44 ( sepSubset
% 0.16/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.44 ( subset
% 0.16/0.44 @ ( dsetconstr @ A
% 0.16/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.44 @ A ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(binunion_type,type,
% 0.16/0.44 binunion: $i > $i > $i ).
% 0.16/0.44
% 0.16/0.44 thf(binunionIL_type,type,
% 0.16/0.44 binunionIL: $o ).
% 0.16/0.44
% 0.16/0.44 thf(binunionIL,definition,
% 0.16/0.44 ( binunionIL
% 0.16/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(upairset2IR_type,type,
% 0.16/0.44 upairset2IR: $o ).
% 0.16/0.44
% 0.16/0.44 thf(upairset2IR,definition,
% 0.16/0.44 ( upairset2IR
% 0.16/0.44 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(binunionIR_type,type,
% 0.16/0.44 binunionIR: $o ).
% 0.16/0.44
% 0.16/0.44 thf(binunionIR,definition,
% 0.16/0.44 ( binunionIR
% 0.16/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ B )
% 0.16/0.44 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(binunionEcases_type,type,
% 0.16/0.44 binunionEcases: $o ).
% 0.16/0.44
% 0.16/0.44 thf(binunionEcases,definition,
% 0.16/0.44 ( binunionEcases
% 0.16/0.44 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.16/0.44 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.16/0.44 => ( ( ( in @ Xx @ A )
% 0.16/0.44 => Xphi )
% 0.16/0.44 => ( ( ( in @ Xx @ B )
% 0.16/0.44 => Xphi )
% 0.16/0.44 => Xphi ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(binunionE_type,type,
% 0.16/0.44 binunionE: $o ).
% 0.16/0.44
% 0.16/0.44 thf(binunionE,definition,
% 0.16/0.44 ( binunionE
% 0.16/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.16/0.44 => ( ( in @ Xx @ A )
% 0.16/0.44 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(binunionLsub_type,type,
% 0.16/0.44 binunionLsub: $o ).
% 0.16/0.44
% 0.16/0.44 thf(binunionLsub,definition,
% 0.16/0.44 ( binunionLsub
% 0.16/0.44 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(binunionRsub_type,type,
% 0.16/0.44 binunionRsub: $o ).
% 0.16/0.44
% 0.16/0.44 thf(binunionRsub,definition,
% 0.16/0.44 ( binunionRsub
% 0.16/0.44 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(binintersect_type,type,
% 0.16/0.44 binintersect: $i > $i > $i ).
% 0.16/0.44
% 0.16/0.44 thf(binintersectI_type,type,
% 0.16/0.44 binintersectI: $o ).
% 0.16/0.44
% 0.16/0.44 thf(binintersectI,definition,
% 0.16/0.44 ( binintersectI
% 0.16/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( ( in @ Xx @ B )
% 0.16/0.44 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(binintersectSubset5_type,type,
% 0.16/0.44 binintersectSubset5: $o ).
% 0.16/0.44
% 0.16/0.44 thf(binintersectSubset5,definition,
% 0.16/0.44 ( binintersectSubset5
% 0.16/0.44 = ( ! [A: $i,B: $i,C: $i] :
% 0.16/0.44 ( ( subset @ C @ A )
% 0.16/0.44 => ( ( subset @ C @ B )
% 0.16/0.44 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(binintersectEL_type,type,
% 0.16/0.44 binintersectEL: $o ).
% 0.16/0.44
% 0.16/0.44 thf(binintersectEL,definition,
% 0.16/0.44 ( binintersectEL
% 0.16/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.16/0.44 => ( in @ Xx @ A ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(binintersectLsub_type,type,
% 0.16/0.44 binintersectLsub: $o ).
% 0.16/0.44
% 0.16/0.44 thf(binintersectLsub,definition,
% 0.16/0.44 ( binintersectLsub
% 0.16/0.44 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(binintersectSubset2_type,type,
% 0.16/0.44 binintersectSubset2: $o ).
% 0.16/0.44
% 0.16/0.44 thf(binintersectSubset2,definition,
% 0.16/0.44 ( binintersectSubset2
% 0.16/0.44 = ( ! [A: $i,B: $i] :
% 0.16/0.44 ( ( subset @ A @ B )
% 0.16/0.44 => ( ( binintersect @ A @ B )
% 0.16/0.44 = A ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(binintersectSubset3_type,type,
% 0.16/0.44 binintersectSubset3: $o ).
% 0.16/0.44
% 0.16/0.44 thf(binintersectSubset3,definition,
% 0.16/0.44 ( binintersectSubset3
% 0.16/0.44 = ( ! [A: $i,B: $i] :
% 0.16/0.44 ( ( ( binintersect @ A @ B )
% 0.16/0.44 = B )
% 0.16/0.44 => ( subset @ B @ A ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(binintersectER_type,type,
% 0.16/0.44 binintersectER: $o ).
% 0.16/0.44
% 0.16/0.44 thf(binintersectER,definition,
% 0.16/0.44 ( binintersectER
% 0.16/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.16/0.44 => ( in @ Xx @ B ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(disjointsetsI1_type,type,
% 0.16/0.44 disjointsetsI1: $o ).
% 0.16/0.44
% 0.16/0.44 thf(disjointsetsI1,definition,
% 0.16/0.44 ( disjointsetsI1
% 0.16/0.44 = ( ! [A: $i,B: $i] :
% 0.16/0.44 ( ~ ? [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 & ( in @ Xx @ B ) )
% 0.16/0.44 => ( ( binintersect @ A @ B )
% 0.16/0.44 = emptyset ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(binintersectRsub_type,type,
% 0.16/0.44 binintersectRsub: $o ).
% 0.16/0.44
% 0.16/0.44 thf(binintersectRsub,definition,
% 0.16/0.44 ( binintersectRsub
% 0.16/0.44 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(binintersectSubset4_type,type,
% 0.16/0.44 binintersectSubset4: $o ).
% 0.16/0.44
% 0.16/0.44 thf(binintersectSubset4,definition,
% 0.16/0.44 ( binintersectSubset4
% 0.16/0.44 = ( ! [A: $i,B: $i] :
% 0.16/0.44 ( ( subset @ B @ A )
% 0.16/0.44 => ( ( binintersect @ A @ B )
% 0.16/0.44 = B ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(binintersectSubset1_type,type,
% 0.16/0.44 binintersectSubset1: $o ).
% 0.16/0.44
% 0.16/0.44 thf(binintersectSubset1,definition,
% 0.16/0.44 ( binintersectSubset1
% 0.16/0.44 = ( ! [A: $i,B: $i] :
% 0.16/0.44 ( ( ( binintersect @ A @ B )
% 0.16/0.44 = A )
% 0.16/0.44 => ( subset @ A @ B ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(bs114d_type,type,
% 0.16/0.44 bs114d: $o ).
% 0.16/0.44
% 0.16/0.44 thf(bs114d,definition,
% 0.16/0.44 ( bs114d
% 0.16/0.44 = ( ! [A: $i,B: $i,C: $i] :
% 0.16/0.44 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 0.16/0.44 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(regular_type,type,
% 0.16/0.44 regular: $i > $o ).
% 0.16/0.44
% 0.16/0.44 thf(setminus_type,type,
% 0.16/0.44 setminus: $i > $i > $i ).
% 0.16/0.44
% 0.16/0.44 thf(setminusI_type,type,
% 0.16/0.44 setminusI: $o ).
% 0.16/0.44
% 0.16/0.44 thf(setminusI,definition,
% 0.16/0.44 ( setminusI
% 0.16/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( ~ ( in @ Xx @ B )
% 0.16/0.44 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(setminusEL_type,type,
% 0.16/0.44 setminusEL: $o ).
% 0.16/0.44
% 0.16/0.44 thf(setminusEL,definition,
% 0.16/0.44 ( setminusEL
% 0.16/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.16/0.44 => ( in @ Xx @ A ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(setminusER_type,type,
% 0.16/0.44 setminusER: $o ).
% 0.16/0.44
% 0.16/0.44 thf(setminusER,definition,
% 0.16/0.44 ( setminusER
% 0.16/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.16/0.44 => ~ ( in @ Xx @ B ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(setminusSubset2_type,type,
% 0.16/0.44 setminusSubset2: $o ).
% 0.16/0.44
% 0.16/0.44 thf(setminusSubset2,definition,
% 0.16/0.44 ( setminusSubset2
% 0.16/0.44 = ( ! [A: $i,B: $i] :
% 0.16/0.44 ( ( subset @ A @ B )
% 0.16/0.44 => ( ( setminus @ A @ B )
% 0.16/0.44 = emptyset ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(setminusERneg_type,type,
% 0.16/0.44 setminusERneg: $o ).
% 0.16/0.44
% 0.16/0.44 thf(setminusERneg,definition,
% 0.16/0.44 ( setminusERneg
% 0.16/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.44 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.16/0.44 => ( ( in @ Xx @ A )
% 0.16/0.44 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(setminusELneg_type,type,
% 0.16/0.44 setminusELneg: $o ).
% 0.16/0.44
% 0.16/0.44 thf(setminusELneg,definition,
% 0.16/0.44 ( setminusELneg
% 0.16/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.44 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.16/0.44 => ( ~ ( in @ Xx @ B )
% 0.16/0.44 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(setminusILneg_type,type,
% 0.16/0.44 setminusILneg: $o ).
% 0.16/0.44
% 0.16/0.44 thf(setminusILneg,definition,
% 0.16/0.44 ( setminusILneg
% 0.16/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.44 ( ~ ( in @ Xx @ A )
% 0.16/0.44 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(setminusIRneg_type,type,
% 0.16/0.44 setminusIRneg: $o ).
% 0.16/0.44
% 0.16/0.44 thf(setminusIRneg,definition,
% 0.16/0.44 ( setminusIRneg
% 0.16/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ B )
% 0.16/0.44 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(setminusLsub_type,type,
% 0.16/0.44 setminusLsub: $o ).
% 0.16/0.44
% 0.16/0.44 thf(setminusLsub,definition,
% 0.16/0.44 ( setminusLsub
% 0.16/0.44 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(setminusSubset1_type,type,
% 0.16/0.44 setminusSubset1: $o ).
% 0.16/0.44
% 0.16/0.44 thf(setminusSubset1,definition,
% 0.16/0.44 ( setminusSubset1
% 0.16/0.44 = ( ! [A: $i,B: $i] :
% 0.16/0.44 ( ( ( setminus @ A @ B )
% 0.16/0.44 = emptyset )
% 0.16/0.44 => ( subset @ A @ B ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(symdiff_type,type,
% 0.16/0.44 symdiff: $i > $i > $i ).
% 0.16/0.44
% 0.16/0.44 thf(symdiffE_type,type,
% 0.16/0.44 symdiffE: $o ).
% 0.16/0.44
% 0.16/0.44 thf(symdiffE,definition,
% 0.16/0.44 ( symdiffE
% 0.16/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 0.16/0.44 => ! [Xphi: $o] :
% 0.16/0.44 ( ( ( in @ Xx @ A )
% 0.16/0.44 => ( ~ ( in @ Xx @ B )
% 0.16/0.44 => Xphi ) )
% 0.16/0.44 => ( ( ~ ( in @ Xx @ A )
% 0.16/0.44 => ( ( in @ Xx @ B )
% 0.16/0.44 => Xphi ) )
% 0.16/0.44 => Xphi ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(symdiffI1_type,type,
% 0.16/0.44 symdiffI1: $o ).
% 0.16/0.44
% 0.16/0.44 thf(symdiffI1,definition,
% 0.16/0.44 ( symdiffI1
% 0.16/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( ~ ( in @ Xx @ B )
% 0.16/0.44 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(symdiffI2_type,type,
% 0.16/0.44 symdiffI2: $o ).
% 0.16/0.44
% 0.16/0.44 thf(symdiffI2,definition,
% 0.16/0.44 ( symdiffI2
% 0.16/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.44 ( ~ ( in @ Xx @ A )
% 0.16/0.44 => ( ( in @ Xx @ B )
% 0.16/0.44 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(symdiffIneg1_type,type,
% 0.16/0.44 symdiffIneg1: $o ).
% 0.16/0.44
% 0.16/0.44 thf(symdiffIneg1,definition,
% 0.16/0.44 ( symdiffIneg1
% 0.16/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( ( in @ Xx @ B )
% 0.16/0.44 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(symdiffIneg2_type,type,
% 0.16/0.44 symdiffIneg2: $o ).
% 0.16/0.44
% 0.16/0.44 thf(symdiffIneg2,definition,
% 0.16/0.44 ( symdiffIneg2
% 0.16/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.44 ( ~ ( in @ Xx @ A )
% 0.16/0.44 => ( ~ ( in @ Xx @ B )
% 0.16/0.44 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(iskpair_type,type,
% 0.16/0.44 iskpair: $i > $o ).
% 0.16/0.44
% 0.16/0.44 thf(secondinupair_type,type,
% 0.16/0.44 secondinupair: $o ).
% 0.16/0.44
% 0.16/0.44 thf(secondinupair,definition,
% 0.16/0.44 ( secondinupair
% 0.16/0.44 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(setukpairIL_type,type,
% 0.16/0.44 setukpairIL: $o ).
% 0.16/0.44
% 0.16/0.44 thf(setukpairIL,definition,
% 0.16/0.44 ( setukpairIL
% 0.16/0.44 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(setukpairIR_type,type,
% 0.16/0.44 setukpairIR: $o ).
% 0.16/0.44
% 0.16/0.44 thf(setukpairIR,definition,
% 0.16/0.44 ( setukpairIR
% 0.16/0.44 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(kpairiskpair_type,type,
% 0.16/0.44 kpairiskpair: $o ).
% 0.16/0.44
% 0.16/0.44 thf(kpairiskpair,definition,
% 0.16/0.44 ( kpairiskpair
% 0.16/0.44 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(kpair_type,type,
% 0.16/0.44 kpair: $i > $i > $i ).
% 0.16/0.44
% 0.16/0.44 thf(kpairp_type,type,
% 0.16/0.44 kpairp: $o ).
% 0.16/0.44
% 0.16/0.44 thf(kpairp,definition,
% 0.16/0.44 ( kpairp
% 0.16/0.44 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(cartprod_type,type,
% 0.16/0.44 cartprod: $i > $i > $i ).
% 0.16/0.44
% 0.16/0.44 thf(singletonsubset_type,type,
% 0.16/0.44 singletonsubset: $o ).
% 0.16/0.44
% 0.16/0.44 thf(singletonsubset,definition,
% 0.16/0.44 ( singletonsubset
% 0.16/0.44 = ( ! [A: $i,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(singletoninpowerset_type,type,
% 0.16/0.44 singletoninpowerset: $o ).
% 0.16/0.44
% 0.16/0.44 thf(singletoninpowerset,definition,
% 0.16/0.44 ( singletoninpowerset
% 0.16/0.44 = ( ! [A: $i,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(singletoninpowunion_type,type,
% 0.16/0.44 singletoninpowunion: $o ).
% 0.16/0.44
% 0.16/0.44 thf(singletoninpowunion,definition,
% 0.16/0.44 ( singletoninpowunion
% 0.16/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(upairset2E_type,type,
% 0.16/0.44 upairset2E: $o ).
% 0.16/0.44
% 0.16/0.44 thf(upairset2E,definition,
% 0.16/0.44 ( upairset2E
% 0.16/0.44 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.16/0.44 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.16/0.44 => ( ( Xz = Xx )
% 0.16/0.44 | ( Xz = Xy ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(upairsubunion_type,type,
% 0.16/0.44 upairsubunion: $o ).
% 0.16/0.44
% 0.16/0.44 thf(upairsubunion,definition,
% 0.16/0.44 ( upairsubunion
% 0.16/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ! [Xy: $i] :
% 0.16/0.44 ( ( in @ Xy @ B )
% 0.16/0.44 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(upairinpowunion_type,type,
% 0.16/0.44 upairinpowunion: $o ).
% 0.16/0.44
% 0.16/0.44 thf(upairinpowunion,definition,
% 0.16/0.44 ( upairinpowunion
% 0.16/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ! [Xy: $i] :
% 0.16/0.44 ( ( in @ Xy @ B )
% 0.16/0.44 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(ubforcartprodlem1_type,type,
% 0.16/0.44 ubforcartprodlem1: $o ).
% 0.16/0.44
% 0.16/0.44 thf(ubforcartprodlem1,definition,
% 0.16/0.44 ( ubforcartprodlem1
% 0.16/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ! [Xy: $i] :
% 0.16/0.44 ( ( in @ Xy @ B )
% 0.16/0.44 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(ubforcartprodlem2_type,type,
% 0.16/0.44 ubforcartprodlem2: $o ).
% 0.16/0.44
% 0.16/0.44 thf(ubforcartprodlem2,definition,
% 0.16/0.44 ( ubforcartprodlem2
% 0.16/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ! [Xy: $i] :
% 0.16/0.44 ( ( in @ Xy @ B )
% 0.16/0.44 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(ubforcartprodlem3_type,type,
% 0.16/0.44 ubforcartprodlem3: $o ).
% 0.16/0.44
% 0.16/0.44 thf(ubforcartprodlem3,definition,
% 0.16/0.44 ( ubforcartprodlem3
% 0.16/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ! [Xy: $i] :
% 0.16/0.44 ( ( in @ Xy @ B )
% 0.16/0.44 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(cartprodpairin_type,type,
% 0.16/0.44 cartprodpairin: $o ).
% 0.16/0.44
% 0.16/0.44 thf(cartprodpairin,definition,
% 0.16/0.44 ( cartprodpairin
% 0.16/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ! [Xy: $i] :
% 0.16/0.44 ( ( in @ Xy @ B )
% 0.16/0.44 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(cartprodmempair1_type,type,
% 0.16/0.44 cartprodmempair1: $o ).
% 0.16/0.44
% 0.16/0.44 thf(cartprodmempair1,definition,
% 0.16/0.44 ( cartprodmempair1
% 0.16/0.44 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.16/0.44 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.16/0.44 => ? [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 & ? [Xy: $i] :
% 0.16/0.44 ( ( in @ Xy @ B )
% 0.16/0.44 & ( Xu
% 0.16/0.44 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(cartprodmempair_type,type,
% 0.16/0.44 cartprodmempair: $o ).
% 0.16/0.44
% 0.16/0.44 thf(cartprodmempair,definition,
% 0.16/0.44 ( cartprodmempair
% 0.16/0.44 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.16/0.44 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.16/0.44 => ( iskpair @ Xu ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(setunionE2_type,type,
% 0.16/0.44 setunionE2: $o ).
% 0.16/0.44
% 0.16/0.44 thf(setunionE2,definition,
% 0.16/0.44 ( setunionE2
% 0.16/0.44 = ( ! [A: $i,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ ( setunion @ A ) )
% 0.16/0.44 => ? [X: $i] :
% 0.16/0.44 ( ( in @ X @ A )
% 0.16/0.44 & ( in @ Xx @ X ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(setunionsingleton1_type,type,
% 0.16/0.44 setunionsingleton1: $o ).
% 0.16/0.44
% 0.16/0.44 thf(setunionsingleton1,definition,
% 0.16/0.44 ( setunionsingleton1
% 0.16/0.44 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(setunionsingleton2_type,type,
% 0.16/0.44 setunionsingleton2: $o ).
% 0.16/0.44
% 0.16/0.44 thf(setunionsingleton2,definition,
% 0.16/0.44 ( setunionsingleton2
% 0.16/0.44 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(setunionsingleton_type,type,
% 0.16/0.44 setunionsingleton: $o ).
% 0.16/0.44
% 0.16/0.44 thf(setunionsingleton,definition,
% 0.16/0.44 ( setunionsingleton
% 0.16/0.44 = ( ! [Xx: $i] :
% 0.16/0.44 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 0.16/0.44 = Xx ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(singleton_type,type,
% 0.16/0.44 singleton: $i > $o ).
% 0.16/0.44
% 0.16/0.44 thf(singletonprop_type,type,
% 0.16/0.44 singletonprop: $o ).
% 0.16/0.44
% 0.16/0.44 thf(singletonprop,definition,
% 0.16/0.44 ( singletonprop
% 0.16/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.44 ( ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ! [Xy: $i] :
% 0.16/0.44 ( ( in @ Xy @ A )
% 0.16/0.44 => ( ( Xphi @ Xx )
% 0.16/0.44 => ( ( Xphi @ Xy )
% 0.16/0.44 => ( Xx = Xy ) ) ) ) )
% 0.16/0.44 => ( ? [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 & ( Xphi @ Xx ) )
% 0.16/0.44 => ( singleton
% 0.16/0.44 @ ( dsetconstr @ A
% 0.16/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(ex1_type,type,
% 0.16/0.44 ex1: $i > ( $i > $o ) > $o ).
% 0.16/0.44
% 0.16/0.44 thf(ex1E1_type,type,
% 0.16/0.44 ex1E1: $o ).
% 0.16/0.44
% 0.16/0.44 thf(ex1E1,definition,
% 0.16/0.44 ( ex1E1
% 0.16/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.44 ( ( ex1 @ A
% 0.16/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.44 => ? [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(ex1I_type,type,
% 0.16/0.44 ex1I: $o ).
% 0.16/0.44
% 0.16/0.44 thf(ex1I,definition,
% 0.16/0.44 ( ex1I
% 0.16/0.44 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( ( Xphi @ Xx )
% 0.16/0.44 => ( ! [Xy: $i] :
% 0.16/0.44 ( ( in @ Xy @ A )
% 0.16/0.44 => ( ( Xphi @ Xy )
% 0.16/0.44 => ( Xy = Xx ) ) )
% 0.16/0.44 => ( ex1 @ A
% 0.16/0.44 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(ex1I2_type,type,
% 0.16/0.44 ex1I2: $o ).
% 0.16/0.44
% 0.16/0.44 thf(ex1I2,definition,
% 0.16/0.44 ( ex1I2
% 0.16/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.44 ( ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ! [Xy: $i] :
% 0.16/0.44 ( ( in @ Xy @ A )
% 0.16/0.44 => ( ( Xphi @ Xx )
% 0.16/0.44 => ( ( Xphi @ Xy )
% 0.16/0.44 => ( Xx = Xy ) ) ) ) )
% 0.16/0.44 => ( ? [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 & ( Xphi @ Xx ) )
% 0.16/0.44 => ( ex1 @ A
% 0.16/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(singletonsuniq_type,type,
% 0.16/0.44 singletonsuniq: $o ).
% 0.16/0.44
% 0.16/0.44 thf(singletonsuniq,definition,
% 0.16/0.44 ( singletonsuniq
% 0.16/0.44 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.44 ( ( ( setadjoin @ Xx @ emptyset )
% 0.16/0.44 = ( setadjoin @ Xy @ emptyset ) )
% 0.16/0.44 => ( Xx = Xy ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(atmost1p_type,type,
% 0.16/0.44 atmost1p: $i > $o ).
% 0.16/0.44
% 0.16/0.44 thf(atleast2p_type,type,
% 0.16/0.44 atleast2p: $i > $o ).
% 0.16/0.44
% 0.16/0.44 thf(atmost2p_type,type,
% 0.16/0.44 atmost2p: $i > $o ).
% 0.16/0.44
% 0.16/0.44 thf(upairsetp_type,type,
% 0.16/0.44 upairsetp: $i > $o ).
% 0.16/0.44
% 0.16/0.44 thf(setukpairinjL1_type,type,
% 0.16/0.44 setukpairinjL1: $o ).
% 0.16/0.44
% 0.16/0.44 thf(setukpairinjL1,definition,
% 0.16/0.44 ( setukpairinjL1
% 0.16/0.44 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.16/0.44 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 0.16/0.44 => ( Xx = Xz ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(kfstsingleton_type,type,
% 0.16/0.44 kfstsingleton: $o ).
% 0.16/0.44
% 0.16/0.44 thf(kfstsingleton,definition,
% 0.16/0.44 ( kfstsingleton
% 0.16/0.44 = ( ! [Xu: $i] :
% 0.16/0.44 ( ( iskpair @ Xu )
% 0.16/0.44 => ( singleton
% 0.16/0.44 @ ( dsetconstr @ ( setunion @ Xu )
% 0.16/0.44 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(theprop_type,type,
% 0.16/0.44 theprop: $o ).
% 0.16/0.44
% 0.16/0.44 thf(theprop,definition,
% 0.16/0.44 ( theprop
% 0.16/0.44 = ( ! [X: $i] :
% 0.16/0.44 ( ( singleton @ X )
% 0.16/0.44 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(kfst_type,type,
% 0.16/0.44 kfst: $i > $i ).
% 0.16/0.44
% 0.16/0.44 thf(kfstpairEq_type,type,
% 0.16/0.44 kfstpairEq: $o ).
% 0.16/0.44
% 0.16/0.44 thf(kfstpairEq,definition,
% 0.16/0.44 ( kfstpairEq
% 0.16/0.44 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.44 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.16/0.44 = Xx ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(cartprodfstin_type,type,
% 0.16/0.44 cartprodfstin: $o ).
% 0.16/0.44
% 0.16/0.44 thf(cartprodfstin,definition,
% 0.16/0.44 ( cartprodfstin
% 0.16/0.44 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.16/0.44 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.16/0.44 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(setukpairinjL2_type,type,
% 0.16/0.44 setukpairinjL2: $o ).
% 0.16/0.44
% 0.16/0.44 thf(setukpairinjL2,definition,
% 0.16/0.44 ( setukpairinjL2
% 0.16/0.44 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.16/0.44 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.16/0.44 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.16/0.44 => ( Xx = Xz ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(setukpairinjL_type,type,
% 0.16/0.44 setukpairinjL: $o ).
% 0.16/0.44
% 0.16/0.44 thf(setukpairinjL,definition,
% 0.16/0.44 ( setukpairinjL
% 0.16/0.44 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.16/0.44 ( ( ( kpair @ Xx @ Xy )
% 0.16/0.44 = ( kpair @ Xz @ Xu ) )
% 0.16/0.44 => ( Xx = Xz ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(setukpairinjR11_type,type,
% 0.16/0.44 setukpairinjR11: $o ).
% 0.16/0.44
% 0.16/0.44 thf(setukpairinjR11,definition,
% 0.16/0.44 ( setukpairinjR11
% 0.16/0.44 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.44 ( ( Xx = Xy )
% 0.16/0.44 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.16/0.44 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(setukpairinjR12_type,type,
% 0.16/0.44 setukpairinjR12: $o ).
% 0.16/0.44
% 0.16/0.44 thf(setukpairinjR12,definition,
% 0.16/0.44 ( setukpairinjR12
% 0.16/0.44 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.44 ( ( Xx = Xy )
% 0.16/0.44 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.16/0.44 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(setukpairinjR1_type,type,
% 0.16/0.44 setukpairinjR1: $o ).
% 0.16/0.44
% 0.16/0.44 thf(setukpairinjR1,definition,
% 0.16/0.44 ( setukpairinjR1
% 0.16/0.44 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.16/0.44 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.16/0.44 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.16/0.44 => ( ( Xz = Xu )
% 0.16/0.44 => ( Xy = Xu ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(upairequniteq_type,type,
% 0.16/0.44 upairequniteq: $o ).
% 0.16/0.44
% 0.16/0.44 thf(upairequniteq,definition,
% 0.16/0.44 ( upairequniteq
% 0.16/0.44 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.16/0.44 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.16/0.44 = ( setadjoin @ Xz @ emptyset ) )
% 0.16/0.44 => ( Xx = Xy ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(setukpairinjR2_type,type,
% 0.16/0.44 setukpairinjR2: $o ).
% 0.16/0.44
% 0.16/0.44 thf(setukpairinjR2,definition,
% 0.16/0.44 ( setukpairinjR2
% 0.16/0.44 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.16/0.44 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.16/0.44 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.16/0.44 => ( Xy = Xu ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(setukpairinjR_type,type,
% 0.16/0.44 setukpairinjR: $o ).
% 0.16/0.44
% 0.16/0.44 thf(setukpairinjR,definition,
% 0.16/0.44 ( setukpairinjR
% 0.16/0.44 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.16/0.44 ( ( ( kpair @ Xx @ Xy )
% 0.16/0.44 = ( kpair @ Xz @ Xu ) )
% 0.16/0.44 => ( Xy = Xu ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(ksndsingleton_type,type,
% 0.16/0.44 ksndsingleton: $o ).
% 0.16/0.44
% 0.16/0.44 thf(ksndsingleton,definition,
% 0.16/0.44 ( ksndsingleton
% 0.16/0.44 = ( ! [Xu: $i] :
% 0.16/0.44 ( ( iskpair @ Xu )
% 0.16/0.44 => ( singleton
% 0.16/0.44 @ ( dsetconstr @ ( setunion @ Xu )
% 0.16/0.44 @ ^ [Xx: $i] :
% 0.16/0.44 ( Xu
% 0.16/0.44 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(ksnd_type,type,
% 0.16/0.44 ksnd: $i > $i ).
% 0.16/0.44
% 0.16/0.44 thf(ksndpairEq_type,type,
% 0.16/0.44 ksndpairEq: $o ).
% 0.16/0.44
% 0.16/0.44 thf(ksndpairEq,definition,
% 0.16/0.44 ( ksndpairEq
% 0.16/0.44 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.44 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.16/0.44 = Xy ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(kpairsurjEq_type,type,
% 0.16/0.44 kpairsurjEq: $o ).
% 0.16/0.44
% 0.16/0.44 thf(kpairsurjEq,definition,
% 0.16/0.44 ( kpairsurjEq
% 0.16/0.44 = ( ! [Xu: $i] :
% 0.16/0.44 ( ( iskpair @ Xu )
% 0.16/0.44 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.16/0.44 = Xu ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(cartprodsndin_type,type,
% 0.16/0.44 cartprodsndin: $o ).
% 0.16/0.44
% 0.16/0.44 thf(cartprodsndin,definition,
% 0.16/0.44 ( cartprodsndin
% 0.16/0.44 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.16/0.44 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.16/0.44 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(cartprodpairmemEL_type,type,
% 0.16/0.44 cartprodpairmemEL: $o ).
% 0.16/0.44
% 0.16/0.44 thf(cartprodpairmemEL,definition,
% 0.16/0.44 ( cartprodpairmemEL
% 0.16/0.44 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.16/0.44 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.16/0.44 => ( in @ Xx @ A ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(cartprodpairmemER_type,type,
% 0.16/0.44 cartprodpairmemER: $o ).
% 0.16/0.44
% 0.16/0.44 thf(cartprodpairmemER,definition,
% 0.16/0.44 ( cartprodpairmemER
% 0.16/0.44 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.16/0.44 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.16/0.44 => ( in @ Xy @ B ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(cartprodmempaircEq_type,type,
% 0.16/0.44 cartprodmempaircEq: $o ).
% 0.16/0.44
% 0.16/0.44 thf(cartprodmempaircEq,definition,
% 0.16/0.44 ( cartprodmempaircEq
% 0.16/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ! [Xy: $i] :
% 0.16/0.44 ( ( in @ Xy @ B )
% 0.16/0.44 => ( ( kpair @ Xx @ Xy )
% 0.16/0.44 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(cartprodfstpairEq_type,type,
% 0.16/0.44 cartprodfstpairEq: $o ).
% 0.16/0.44
% 0.16/0.44 thf(cartprodfstpairEq,definition,
% 0.16/0.44 ( cartprodfstpairEq
% 0.16/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ! [Xy: $i] :
% 0.16/0.44 ( ( in @ Xy @ B )
% 0.16/0.44 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.16/0.44 = Xx ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(cartprodsndpairEq_type,type,
% 0.16/0.44 cartprodsndpairEq: $o ).
% 0.16/0.44
% 0.16/0.44 thf(cartprodsndpairEq,definition,
% 0.16/0.44 ( cartprodsndpairEq
% 0.16/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ! [Xy: $i] :
% 0.16/0.44 ( ( in @ Xy @ B )
% 0.16/0.44 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.16/0.44 = Xy ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(cartprodpairsurjEq_type,type,
% 0.16/0.44 cartprodpairsurjEq: $o ).
% 0.16/0.44
% 0.16/0.44 thf(cartprodpairsurjEq,definition,
% 0.16/0.44 ( cartprodpairsurjEq
% 0.16/0.44 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.16/0.44 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.16/0.44 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.16/0.44 = Xu ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(breln_type,type,
% 0.16/0.44 breln: $i > $i > $i > $o ).
% 0.16/0.44
% 0.16/0.44 thf(dpsetconstr_type,type,
% 0.16/0.44 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 0.16/0.44
% 0.16/0.44 thf(dpsetconstrI_type,type,
% 0.16/0.44 dpsetconstrI: $o ).
% 0.16/0.44
% 0.16/0.44 thf(dpsetconstrI,definition,
% 0.16/0.44 ( dpsetconstrI
% 0.16/0.44 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ! [Xy: $i] :
% 0.16/0.44 ( ( in @ Xy @ B )
% 0.16/0.44 => ( ( Xphi @ Xx @ Xy )
% 0.16/0.44 => ( in @ ( kpair @ Xx @ Xy )
% 0.16/0.44 @ ( dpsetconstr @ A @ B
% 0.16/0.44 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(dpsetconstrSub_type,type,
% 0.16/0.44 dpsetconstrSub: $o ).
% 0.16/0.44
% 0.16/0.44 thf(dpsetconstrSub,definition,
% 0.16/0.44 ( dpsetconstrSub
% 0.16/0.44 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.16/0.44 ( subset
% 0.16/0.44 @ ( dpsetconstr @ A @ B
% 0.16/0.44 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 0.16/0.44 @ ( cartprod @ A @ B ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(setOfPairsIsBReln_type,type,
% 0.16/0.44 setOfPairsIsBReln: $o ).
% 0.16/0.44
% 0.16/0.44 thf(setOfPairsIsBReln,definition,
% 0.16/0.44 ( setOfPairsIsBReln
% 0.16/0.44 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.16/0.44 ( breln @ A @ B
% 0.16/0.44 @ ( dpsetconstr @ A @ B
% 0.16/0.44 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(dpsetconstrERa_type,type,
% 0.16/0.44 dpsetconstrERa: $o ).
% 0.16/0.44
% 0.16/0.44 thf(dpsetconstrERa,definition,
% 0.16/0.44 ( dpsetconstrERa
% 0.16/0.44 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ! [Xy: $i] :
% 0.16/0.44 ( ( in @ Xy @ B )
% 0.16/0.44 => ( ( in @ ( kpair @ Xx @ Xy )
% 0.16/0.44 @ ( dpsetconstr @ A @ B
% 0.16/0.44 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.16/0.44 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(dpsetconstrEL1_type,type,
% 0.16/0.44 dpsetconstrEL1: $o ).
% 0.16/0.44
% 0.16/0.44 thf(dpsetconstrEL1,definition,
% 0.16/0.44 ( dpsetconstrEL1
% 0.16/0.44 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.16/0.44 ( ( in @ ( kpair @ Xx @ Xy )
% 0.16/0.44 @ ( dpsetconstr @ A @ B
% 0.16/0.44 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.16/0.44 => ( in @ Xx @ A ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(dpsetconstrEL2_type,type,
% 0.16/0.44 dpsetconstrEL2: $o ).
% 0.16/0.44
% 0.16/0.44 thf(dpsetconstrEL2,definition,
% 0.16/0.44 ( dpsetconstrEL2
% 0.16/0.44 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.16/0.44 ( ( in @ ( kpair @ Xx @ Xy )
% 0.16/0.44 @ ( dpsetconstr @ A @ B
% 0.16/0.44 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.16/0.44 => ( in @ Xy @ B ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(dpsetconstrER_type,type,
% 0.16/0.44 dpsetconstrER: $o ).
% 0.16/0.44
% 0.16/0.44 thf(dpsetconstrER,definition,
% 0.16/0.44 ( dpsetconstrER
% 0.16/0.44 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.16/0.44 ( ( in @ ( kpair @ Xx @ Xy )
% 0.16/0.44 @ ( dpsetconstr @ A @ B
% 0.16/0.44 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.16/0.44 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(func_type,type,
% 0.16/0.44 func: $i > $i > $i > $o ).
% 0.16/0.44
% 0.16/0.44 thf(funcSet_type,type,
% 0.16/0.44 funcSet: $i > $i > $i ).
% 0.16/0.44
% 0.16/0.44 thf(funcImageSingleton_type,type,
% 0.16/0.44 funcImageSingleton: $o ).
% 0.16/0.44
% 0.16/0.44 thf(funcImageSingleton,definition,
% 0.16/0.44 ( funcImageSingleton
% 0.16/0.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.44 ( ( func @ A @ B @ Xf )
% 0.16/0.44 => ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( singleton
% 0.16/0.44 @ ( dsetconstr @ B
% 0.16/0.44 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(apProp_type,type,
% 0.16/0.44 apProp: $o ).
% 0.16/0.44
% 0.16/0.44 thf(apProp,definition,
% 0.16/0.44 ( apProp
% 0.16/0.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.44 ( ( func @ A @ B @ Xf )
% 0.16/0.44 => ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( in
% 0.16/0.44 @ ( setunion
% 0.16/0.44 @ ( dsetconstr @ B
% 0.16/0.44 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 0.16/0.44 @ B ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(ap_type,type,
% 0.16/0.44 ap: $i > $i > $i > $i > $i ).
% 0.16/0.44
% 0.16/0.44 thf(app_type,type,
% 0.16/0.44 app: $o ).
% 0.16/0.44
% 0.16/0.44 thf(app,definition,
% 0.16/0.44 ( app
% 0.16/0.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.44 ( ( func @ A @ B @ Xf )
% 0.16/0.44 => ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(infuncsetfunc_type,type,
% 0.16/0.44 infuncsetfunc: $o ).
% 0.16/0.44
% 0.16/0.44 thf(infuncsetfunc,definition,
% 0.16/0.44 ( infuncsetfunc
% 0.16/0.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.44 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.16/0.44 => ( func @ A @ B @ Xf ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(ap2p_type,type,
% 0.16/0.44 ap2p: $o ).
% 0.16/0.44
% 0.16/0.44 thf(ap2p,definition,
% 0.16/0.44 ( ap2p
% 0.16/0.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.44 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.16/0.44 => ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(funcinfuncset_type,type,
% 0.16/0.44 funcinfuncset: $o ).
% 0.16/0.44
% 0.16/0.44 thf(funcinfuncset,definition,
% 0.16/0.44 ( funcinfuncset
% 0.16/0.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.44 ( ( func @ A @ B @ Xf )
% 0.16/0.44 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(lamProp_type,type,
% 0.16/0.44 lamProp: $o ).
% 0.16/0.44
% 0.16/0.44 thf(lamProp,definition,
% 0.16/0.44 ( lamProp
% 0.16/0.44 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.16/0.44 ( ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.16/0.44 => ( func @ A @ B
% 0.16/0.44 @ ( dpsetconstr @ A @ B
% 0.16/0.44 @ ^ [Xx: $i,Xy: $i] :
% 0.16/0.44 ( ( Xf @ Xx )
% 0.16/0.44 = Xy ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(lam_type,type,
% 0.16/0.44 lam: $i > $i > ( $i > $i ) > $i ).
% 0.16/0.44
% 0.16/0.44 thf(lamp_type,type,
% 0.16/0.44 lamp: $o ).
% 0.16/0.44
% 0.16/0.44 thf(lamp,definition,
% 0.16/0.44 ( lamp
% 0.16/0.44 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.16/0.44 ( ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.16/0.44 => ( func @ A @ B
% 0.16/0.44 @ ( lam @ A @ B
% 0.16/0.44 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(lam2p_type,type,
% 0.16/0.44 lam2p: $o ).
% 0.16/0.44
% 0.16/0.44 thf(lam2p,definition,
% 0.16/0.44 ( lam2p
% 0.16/0.44 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.16/0.44 ( ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.16/0.44 => ( in
% 0.16/0.44 @ ( lam @ A @ B
% 0.16/0.44 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.16/0.44 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(brelnall1_type,type,
% 0.16/0.44 brelnall1: $o ).
% 0.16/0.44
% 0.16/0.44 thf(brelnall1,definition,
% 0.16/0.44 ( brelnall1
% 0.16/0.44 = ( ! [A: $i,B: $i,R: $i] :
% 0.16/0.44 ( ( breln @ A @ B @ R )
% 0.16/0.44 => ! [Xphi: $i > $o] :
% 0.16/0.44 ( ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ! [Xy: $i] :
% 0.16/0.44 ( ( in @ Xy @ B )
% 0.16/0.44 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.16/0.44 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.16/0.44 => ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ R )
% 0.16/0.44 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(brelnall2_type,type,
% 0.16/0.44 brelnall2: $o ).
% 0.16/0.44
% 0.16/0.44 thf(brelnall2,definition,
% 0.16/0.44 ( brelnall2
% 0.16/0.44 = ( ! [A: $i,B: $i,R: $i] :
% 0.16/0.44 ( ( breln @ A @ B @ R )
% 0.16/0.44 => ! [Xphi: $i > $o] :
% 0.16/0.44 ( ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ! [Xy: $i] :
% 0.16/0.44 ( ( in @ Xy @ B )
% 0.16/0.44 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.16/0.44 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.16/0.44 => ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ R )
% 0.16/0.44 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(ex1E2_type,type,
% 0.16/0.44 ex1E2: $o ).
% 0.16/0.44
% 0.16/0.44 thf(ex1E2,definition,
% 0.16/0.44 ( ex1E2
% 0.16/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.44 ( ( ex1 @ A
% 0.16/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.44 => ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ! [Xy: $i] :
% 0.16/0.44 ( ( in @ Xy @ A )
% 0.16/0.44 => ( ( Xphi @ Xx )
% 0.16/0.44 => ( ( Xphi @ Xy )
% 0.16/0.44 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(funcGraphProp1_type,type,
% 0.16/0.44 funcGraphProp1: $o ).
% 0.16/0.44
% 0.16/0.44 thf(funcGraphProp1,definition,
% 0.16/0.44 ( funcGraphProp1
% 0.16/0.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.44 ( ( func @ A @ B @ Xf )
% 0.16/0.44 => ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(funcGraphProp3_type,type,
% 0.16/0.44 funcGraphProp3: $o ).
% 0.16/0.44
% 0.16/0.44 thf(funcGraphProp3,definition,
% 0.16/0.44 ( funcGraphProp3
% 0.16/0.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.44 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.16/0.44 => ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(funcGraphProp2_type,type,
% 0.16/0.44 funcGraphProp2: $o ).
% 0.16/0.44
% 0.16/0.44 thf(funcGraphProp2,definition,
% 0.16/0.44 ( funcGraphProp2
% 0.16/0.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.44 ( ( func @ A @ B @ Xf )
% 0.16/0.44 => ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ! [Xy: $i] :
% 0.16/0.44 ( ( in @ Xy @ B )
% 0.16/0.44 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.16/0.44 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.16/0.44 = Xy ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(funcextLem_type,type,
% 0.16/0.44 funcextLem: $o ).
% 0.16/0.44
% 0.16/0.44 thf(funcextLem,definition,
% 0.16/0.44 ( funcextLem
% 0.16/0.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.44 ( ( func @ A @ B @ Xf )
% 0.16/0.44 => ! [Xg: $i] :
% 0.16/0.44 ( ( func @ A @ B @ Xg )
% 0.16/0.44 => ( ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.16/0.44 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.16/0.44 => ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ! [Xy: $i] :
% 0.16/0.44 ( ( in @ Xy @ B )
% 0.16/0.44 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 0.16/0.44 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(funcGraphProp4_type,type,
% 0.16/0.44 funcGraphProp4: $o ).
% 0.16/0.44
% 0.16/0.44 thf(funcGraphProp4,definition,
% 0.16/0.44 ( funcGraphProp4
% 0.16/0.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.44 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.16/0.44 => ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ! [Xy: $i] :
% 0.16/0.44 ( ( in @ Xy @ B )
% 0.16/0.44 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.16/0.44 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.16/0.44 = Xy ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(subbreln_type,type,
% 0.16/0.44 subbreln: $o ).
% 0.16/0.44
% 0.16/0.44 thf(subbreln,definition,
% 0.16/0.44 ( subbreln
% 0.16/0.44 = ( ! [A: $i,B: $i,R: $i] :
% 0.16/0.44 ( ( breln @ A @ B @ R )
% 0.16/0.44 => ! [S: $i] :
% 0.16/0.44 ( ( breln @ A @ B @ S )
% 0.16/0.44 => ( ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ! [Xy: $i] :
% 0.16/0.44 ( ( in @ Xy @ B )
% 0.16/0.44 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.16/0.44 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.16/0.44 => ( subset @ R @ S ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(eqbreln_type,type,
% 0.16/0.44 eqbreln: $o ).
% 0.16/0.44
% 0.16/0.44 thf(eqbreln,definition,
% 0.16/0.44 ( eqbreln
% 0.16/0.44 = ( ! [A: $i,B: $i,R: $i] :
% 0.16/0.44 ( ( breln @ A @ B @ R )
% 0.16/0.44 => ! [S: $i] :
% 0.16/0.44 ( ( breln @ A @ B @ S )
% 0.16/0.44 => ( ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ! [Xy: $i] :
% 0.16/0.44 ( ( in @ Xy @ B )
% 0.16/0.44 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.16/0.44 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.16/0.44 => ( ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ! [Xy: $i] :
% 0.16/0.44 ( ( in @ Xy @ B )
% 0.16/0.44 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.16/0.44 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 0.16/0.44 => ( R = S ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(funcext_type,type,
% 0.16/0.44 funcext: $o ).
% 0.16/0.44
% 0.16/0.44 thf(funcext,definition,
% 0.16/0.44 ( funcext
% 0.16/0.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.44 ( ( func @ A @ B @ Xf )
% 0.16/0.44 => ! [Xg: $i] :
% 0.16/0.44 ( ( func @ A @ B @ Xg )
% 0.16/0.44 => ( ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.16/0.44 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.16/0.44 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(funcext2_type,type,
% 0.16/0.44 funcext2: $o ).
% 0.16/0.44
% 0.16/0.44 thf(funcext2,definition,
% 0.16/0.44 ( funcext2
% 0.16/0.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.44 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.16/0.44 => ! [Xg: $i] :
% 0.16/0.44 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 0.16/0.44 => ( ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.16/0.44 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.16/0.44 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(ap2apEq1_type,type,
% 0.16/0.44 ap2apEq1: $o ).
% 0.16/0.44
% 0.16/0.44 thf(ap2apEq1,definition,
% 0.16/0.44 ( ap2apEq1
% 0.16/0.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.44 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.16/0.44 => ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.16/0.44 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(ap2apEq2_type,type,
% 0.16/0.44 ap2apEq2: $o ).
% 0.16/0.44
% 0.16/0.44 thf(ap2apEq2,definition,
% 0.16/0.44 ( ap2apEq2
% 0.16/0.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.44 ( ( func @ A @ B @ Xf )
% 0.16/0.44 => ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.16/0.44 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(beta1_type,type,
% 0.16/0.44 beta1: $o ).
% 0.16/0.44
% 0.16/0.44 thf(beta1,definition,
% 0.16/0.44 ( beta1
% 0.16/0.44 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.16/0.44 ( ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.16/0.44 => ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( ( ap @ A @ B
% 0.16/0.44 @ ( lam @ A @ B
% 0.16/0.44 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.16/0.44 @ Xx )
% 0.16/0.44 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(eta1_type,type,
% 0.16/0.44 eta1: $o ).
% 0.16/0.44
% 0.16/0.44 thf(eta1,definition,
% 0.16/0.44 ( eta1
% 0.16/0.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.44 ( ( func @ A @ B @ Xf )
% 0.16/0.44 => ( ( lam @ A @ B
% 0.16/0.44 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.16/0.44 = Xf ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(lam2lamEq_type,type,
% 0.16/0.44 lam2lamEq: $o ).
% 0.16/0.44
% 0.16/0.44 thf(lam2lamEq,definition,
% 0.16/0.44 ( lam2lamEq
% 0.16/0.44 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.16/0.44 ( ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.16/0.44 => ( ( lam @ A @ B
% 0.16/0.44 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.16/0.44 = ( lam @ A @ B
% 0.16/0.44 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(beta2_type,type,
% 0.16/0.44 beta2: $o ).
% 0.16/0.44
% 0.16/0.44 thf(beta2,definition,
% 0.16/0.44 ( beta2
% 0.16/0.44 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.16/0.44 ( ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.16/0.44 => ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( ( ap @ A @ B
% 0.16/0.44 @ ( lam @ A @ B
% 0.16/0.44 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.16/0.44 @ Xx )
% 0.16/0.44 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(eta2_type,type,
% 0.16/0.44 eta2: $o ).
% 0.16/0.44
% 0.16/0.44 thf(eta2,definition,
% 0.16/0.44 ( eta2
% 0.16/0.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.44 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.16/0.44 => ( ( lam @ A @ B
% 0.16/0.44 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.16/0.44 = Xf ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(iffalseProp1_type,type,
% 0.16/0.44 iffalseProp1: $o ).
% 0.16/0.44
% 0.16/0.44 thf(iffalseProp1,definition,
% 0.16/0.44 ( iffalseProp1
% 0.16/0.44 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ! [Xy: $i] :
% 0.16/0.44 ( ( in @ Xy @ A )
% 0.16/0.44 => ( ~ Xphi
% 0.16/0.44 => ( in @ Xy
% 0.16/0.44 @ ( dsetconstr @ A
% 0.16/0.44 @ ^ [Xz: $i] :
% 0.16/0.44 ( ( Xphi
% 0.16/0.44 & ( Xz = Xx ) )
% 0.16/0.44 | ( ~ Xphi
% 0.16/0.44 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(iffalseProp2_type,type,
% 0.16/0.44 iffalseProp2: $o ).
% 0.16/0.44
% 0.16/0.44 thf(iffalseProp2,definition,
% 0.16/0.44 ( iffalseProp2
% 0.16/0.44 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ! [Xy: $i] :
% 0.16/0.44 ( ( in @ Xy @ A )
% 0.16/0.44 => ( ~ Xphi
% 0.16/0.44 => ( ( dsetconstr @ A
% 0.16/0.44 @ ^ [Xz: $i] :
% 0.16/0.44 ( ( Xphi
% 0.16/0.44 & ( Xz = Xx ) )
% 0.16/0.44 | ( ~ Xphi
% 0.16/0.44 & ( Xz = Xy ) ) ) )
% 0.16/0.44 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(iftrueProp1_type,type,
% 0.16/0.44 iftrueProp1: $o ).
% 0.16/0.44
% 0.16/0.44 thf(iftrueProp1,definition,
% 0.16/0.44 ( iftrueProp1
% 0.16/0.44 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ! [Xy: $i] :
% 0.16/0.44 ( ( in @ Xy @ A )
% 0.16/0.44 => ( Xphi
% 0.16/0.44 => ( in @ Xx
% 0.16/0.44 @ ( dsetconstr @ A
% 0.16/0.44 @ ^ [Xz: $i] :
% 0.16/0.44 ( ( Xphi
% 0.16/0.44 & ( Xz = Xx ) )
% 0.16/0.44 | ( ~ Xphi
% 0.16/0.44 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(iftrueProp2_type,type,
% 0.16/0.44 iftrueProp2: $o ).
% 0.16/0.44
% 0.16/0.44 thf(iftrueProp2,definition,
% 0.16/0.44 ( iftrueProp2
% 0.16/0.44 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ! [Xy: $i] :
% 0.16/0.44 ( ( in @ Xy @ A )
% 0.16/0.44 => ( Xphi
% 0.16/0.44 => ( ( dsetconstr @ A
% 0.16/0.44 @ ^ [Xz: $i] :
% 0.16/0.44 ( ( Xphi
% 0.16/0.44 & ( Xz = Xx ) )
% 0.16/0.44 | ( ~ Xphi
% 0.16/0.44 & ( Xz = Xy ) ) ) )
% 0.16/0.44 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(ifSingleton_type,type,
% 0.16/0.44 ifSingleton: $o ).
% 0.16/0.44
% 0.16/0.44 thf(ifSingleton,definition,
% 0.16/0.44 ( ifSingleton
% 0.16/0.44 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ! [Xy: $i] :
% 0.16/0.44 ( ( in @ Xy @ A )
% 0.16/0.44 => ( singleton
% 0.16/0.44 @ ( dsetconstr @ A
% 0.16/0.44 @ ^ [Xz: $i] :
% 0.16/0.44 ( ( Xphi
% 0.16/0.44 & ( Xz = Xx ) )
% 0.16/0.44 | ( ~ Xphi
% 0.16/0.44 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(if_type,type,
% 0.16/0.44 if: $i > $o > $i > $i > $i ).
% 0.16/0.44
% 0.16/0.44 thf(ifp_type,type,
% 0.16/0.44 ifp: $o ).
% 0.16/0.44
% 0.16/0.44 thf(ifp,definition,
% 0.16/0.44 ( ifp
% 0.16/0.44 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ! [Xy: $i] :
% 0.16/0.44 ( ( in @ Xy @ A )
% 0.16/0.44 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(theeq_type,type,
% 0.16/0.44 theeq: $o ).
% 0.16/0.44
% 0.16/0.44 thf(theeq,definition,
% 0.16/0.44 ( theeq
% 0.16/0.44 = ( ! [X: $i] :
% 0.16/0.44 ( ( singleton @ X )
% 0.16/0.44 => ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ X )
% 0.16/0.44 => ( ( setunion @ X )
% 0.16/0.44 = Xx ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(iftrue_type,type,
% 0.16/0.44 iftrue: $o ).
% 0.16/0.44
% 0.16/0.44 thf(iftrue,definition,
% 0.16/0.44 ( iftrue
% 0.16/0.44 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ! [Xy: $i] :
% 0.16/0.44 ( ( in @ Xy @ A )
% 0.16/0.44 => ( Xphi
% 0.16/0.44 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.16/0.44 = Xx ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(iffalse_type,type,
% 0.16/0.44 iffalse: $o ).
% 0.16/0.44
% 0.16/0.44 thf(iffalse,definition,
% 0.16/0.44 ( iffalse
% 0.16/0.44 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ! [Xy: $i] :
% 0.16/0.44 ( ( in @ Xy @ A )
% 0.16/0.44 => ( ~ Xphi
% 0.16/0.44 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.16/0.44 = Xy ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(iftrueorfalse_type,type,
% 0.16/0.44 iftrueorfalse: $o ).
% 0.16/0.44
% 0.16/0.44 thf(iftrueorfalse,definition,
% 0.16/0.44 ( iftrueorfalse
% 0.16/0.44 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ! [Xy: $i] :
% 0.16/0.44 ( ( in @ Xy @ A )
% 0.16/0.44 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(binintersectT_lem_type,type,
% 0.16/0.44 binintersectT_lem: $o ).
% 0.16/0.44
% 0.16/0.44 thf(binintersectT_lem,definition,
% 0.16/0.44 ( binintersectT_lem
% 0.16/0.44 = ( ! [A: $i,X: $i] :
% 0.16/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.44 => ! [Y: $i] :
% 0.16/0.44 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.44 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(binunionT_lem_type,type,
% 0.16/0.44 binunionT_lem: $o ).
% 0.16/0.44
% 0.16/0.44 thf(binunionT_lem,definition,
% 0.16/0.44 ( binunionT_lem
% 0.16/0.44 = ( ! [A: $i,X: $i] :
% 0.16/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.44 => ! [Y: $i] :
% 0.16/0.44 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.44 => ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(powersetT_lem_type,type,
% 0.16/0.44 powersetT_lem: $o ).
% 0.16/0.44
% 0.16/0.44 thf(powersetT_lem,definition,
% 0.16/0.44 ( powersetT_lem
% 0.16/0.44 = ( ! [A: $i,X: $i] :
% 0.16/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.44 => ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(setminusT_lem_type,type,
% 0.16/0.44 setminusT_lem: $o ).
% 0.16/0.44
% 0.16/0.44 thf(setminusT_lem,definition,
% 0.16/0.44 ( setminusT_lem
% 0.16/0.44 = ( ! [A: $i,X: $i] :
% 0.16/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.44 => ! [Y: $i] :
% 0.16/0.44 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.44 => ( in @ ( setminus @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(complementT_lem_type,type,
% 0.16/0.44 complementT_lem: $o ).
% 0.16/0.44
% 0.16/0.44 thf(complementT_lem,definition,
% 0.16/0.44 ( complementT_lem
% 0.16/0.44 = ( ! [A: $i,X: $i] :
% 0.16/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.44 => ( in @ ( setminus @ A @ X ) @ ( powerset @ A ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(setextT_type,type,
% 0.16/0.44 setextT: $o ).
% 0.16/0.44
% 0.16/0.44 thf(setextT,definition,
% 0.16/0.44 ( setextT
% 0.16/0.44 = ( ! [A: $i,X: $i] :
% 0.16/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.44 => ! [Y: $i] :
% 0.16/0.44 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.44 => ( ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( ( in @ Xx @ X )
% 0.16/0.44 => ( in @ Xx @ Y ) ) )
% 0.16/0.44 => ( ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( ( in @ Xx @ Y )
% 0.16/0.44 => ( in @ Xx @ X ) ) )
% 0.16/0.44 => ( X = Y ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(subsetTI_type,type,
% 0.16/0.44 subsetTI: $o ).
% 0.16/0.44
% 0.16/0.44 thf(subsetTI,definition,
% 0.16/0.44 ( subsetTI
% 0.16/0.44 = ( ! [A: $i,X: $i] :
% 0.16/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.44 => ! [Y: $i] :
% 0.16/0.44 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.44 => ( ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( ( in @ Xx @ X )
% 0.16/0.44 => ( in @ Xx @ Y ) ) )
% 0.16/0.44 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(powersetTI1_type,type,
% 0.16/0.44 powersetTI1: $o ).
% 0.16/0.44
% 0.16/0.44 thf(powersetTI1,definition,
% 0.16/0.44 ( powersetTI1
% 0.16/0.44 = ( ! [A: $i,X: $i] :
% 0.16/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.44 => ! [Y: $i] :
% 0.16/0.44 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.44 => ( ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( ( in @ Xx @ X )
% 0.16/0.44 => ( in @ Xx @ Y ) ) )
% 0.16/0.44 => ( in @ X @ ( powerset @ Y ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(powersetTE1_type,type,
% 0.16/0.44 powersetTE1: $o ).
% 0.16/0.44
% 0.16/0.44 thf(powersetTE1,definition,
% 0.16/0.44 ( powersetTE1
% 0.16/0.44 = ( ! [A: $i,X: $i] :
% 0.16/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.44 => ! [Y: $i] :
% 0.16/0.44 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.44 => ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( ( in @ X @ ( powerset @ Y ) )
% 0.16/0.44 => ( ( in @ Xx @ X )
% 0.16/0.44 => ( in @ Xx @ Y ) ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(complementTI1_type,type,
% 0.16/0.44 complementTI1: $o ).
% 0.16/0.44
% 0.16/0.44 thf(complementTI1,definition,
% 0.16/0.44 ( complementTI1
% 0.16/0.44 = ( ! [A: $i,X: $i] :
% 0.16/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.44 => ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( ( in @ Xx @ X )
% 0.16/0.44 => ~ ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(complementTE1_type,type,
% 0.16/0.44 complementTE1: $o ).
% 0.16/0.44
% 0.16/0.44 thf(complementTE1,definition,
% 0.16/0.44 ( complementTE1
% 0.16/0.44 = ( ! [A: $i,X: $i] :
% 0.16/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.44 => ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( ~ ( in @ Xx @ ( setminus @ A @ X ) )
% 0.16/0.44 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(binintersectTELcontra_type,type,
% 0.16/0.44 binintersectTELcontra: $o ).
% 0.16/0.44
% 0.16/0.44 thf(binintersectTELcontra,definition,
% 0.16/0.44 ( binintersectTELcontra
% 0.16/0.44 = ( ! [A: $i,X: $i] :
% 0.16/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.44 => ! [Y: $i] :
% 0.16/0.44 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.44 => ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( ~ ( in @ Xx @ X )
% 0.16/0.44 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(binintersectTERcontra_type,type,
% 0.16/0.44 binintersectTERcontra: $o ).
% 0.16/0.44
% 0.16/0.44 thf(binintersectTERcontra,definition,
% 0.16/0.44 ( binintersectTERcontra
% 0.16/0.44 = ( ! [A: $i,X: $i] :
% 0.16/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.44 => ! [Y: $i] :
% 0.16/0.44 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.44 => ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( ~ ( in @ Xx @ Y )
% 0.16/0.44 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(contrasubsetT_type,type,
% 0.16/0.44 contrasubsetT: $o ).
% 0.16/0.44
% 0.16/0.44 thf(contrasubsetT,definition,
% 0.16/0.44 ( contrasubsetT
% 0.16/0.44 = ( ! [A: $i,X: $i] :
% 0.16/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.44 => ! [Y: $i] :
% 0.16/0.44 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.44 => ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.16/0.44 => ( ( in @ Xx @ Y )
% 0.16/0.44 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(contrasubsetT1_type,type,
% 0.16/0.44 contrasubsetT1: $o ).
% 0.16/0.44
% 0.16/0.44 thf(contrasubsetT1,definition,
% 0.16/0.44 ( contrasubsetT1
% 0.16/0.44 = ( ! [A: $i,X: $i] :
% 0.16/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.44 => ! [Y: $i] :
% 0.16/0.44 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.44 => ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( ( subset @ X @ Y )
% 0.16/0.44 => ( ~ ( in @ Xx @ Y )
% 0.16/0.44 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(contrasubsetT2_type,type,
% 0.16/0.44 contrasubsetT2: $o ).
% 0.16/0.44
% 0.16/0.44 thf(contrasubsetT2,definition,
% 0.16/0.44 ( contrasubsetT2
% 0.16/0.44 = ( ! [A: $i,X: $i] :
% 0.16/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.44 => ! [Y: $i] :
% 0.16/0.44 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.44 => ( ( subset @ X @ Y )
% 0.16/0.44 => ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(contrasubsetT3_type,type,
% 0.16/0.44 contrasubsetT3: $o ).
% 0.16/0.44
% 0.16/0.44 thf(contrasubsetT3,definition,
% 0.16/0.44 ( contrasubsetT3
% 0.16/0.44 = ( ! [A: $i,X: $i] :
% 0.16/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.44 => ! [Y: $i] :
% 0.16/0.44 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.44 => ( ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) )
% 0.16/0.44 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(doubleComplementI1_type,type,
% 0.16/0.44 doubleComplementI1: $o ).
% 0.16/0.44
% 0.16/0.44 thf(doubleComplementI1,definition,
% 0.16/0.44 ( doubleComplementI1
% 0.16/0.44 = ( ! [A: $i,X: $i] :
% 0.16/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.44 => ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( ( in @ Xx @ X )
% 0.16/0.44 => ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(doubleComplementE1_type,type,
% 0.16/0.44 doubleComplementE1: $o ).
% 0.16/0.44
% 0.16/0.44 thf(doubleComplementE1,definition,
% 0.16/0.44 ( doubleComplementE1
% 0.16/0.44 = ( ! [A: $i,X: $i] :
% 0.16/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.44 => ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) )
% 0.16/0.44 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(doubleComplementSub1_type,type,
% 0.16/0.44 doubleComplementSub1: $o ).
% 0.16/0.44
% 0.16/0.44 thf(doubleComplementSub1,definition,
% 0.16/0.44 ( doubleComplementSub1
% 0.16/0.44 = ( ! [A: $i,X: $i] :
% 0.16/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.44 => ( subset @ X @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(doubleComplementSub2_type,type,
% 0.16/0.44 doubleComplementSub2: $o ).
% 0.16/0.44
% 0.16/0.44 thf(doubleComplementSub2,definition,
% 0.16/0.44 ( doubleComplementSub2
% 0.16/0.44 = ( ! [A: $i,X: $i] :
% 0.16/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.44 => ( subset @ ( setminus @ A @ ( setminus @ A @ X ) ) @ X ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(doubleComplementEq_type,type,
% 0.16/0.44 doubleComplementEq: $o ).
% 0.16/0.44
% 0.16/0.44 thf(doubleComplementEq,definition,
% 0.16/0.44 ( doubleComplementEq
% 0.16/0.44 = ( ! [A: $i,X: $i] :
% 0.16/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.44 => ( X
% 0.16/0.44 = ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(complementTnotintersectT_type,type,
% 0.16/0.44 complementTnotintersectT: $o ).
% 0.16/0.44
% 0.16/0.44 thf(complementTnotintersectT,definition,
% 0.16/0.44 ( complementTnotintersectT
% 0.16/0.44 = ( ! [A: $i,X: $i] :
% 0.16/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.44 => ! [Y: $i] :
% 0.16/0.44 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.44 => ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.16/0.44 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(complementImpComplementIntersect_type,type,
% 0.16/0.44 complementImpComplementIntersect: $o ).
% 0.16/0.44
% 0.16/0.44 thf(complementImpComplementIntersect,definition,
% 0.16/0.44 ( complementImpComplementIntersect
% 0.16/0.44 = ( ! [A: $i,X: $i] :
% 0.16/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.44 => ! [Y: $i] :
% 0.16/0.44 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.44 => ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.16/0.44 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(complementSubsetComplementIntersect_type,type,
% 0.16/0.44 complementSubsetComplementIntersect: $o ).
% 0.16/0.44
% 0.16/0.44 thf(complementSubsetComplementIntersect,definition,
% 0.16/0.44 ( complementSubsetComplementIntersect
% 0.16/0.44 = ( ! [A: $i,X: $i] :
% 0.16/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.44 => ! [Y: $i] :
% 0.16/0.44 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.44 => ( subset @ ( setminus @ A @ X ) @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(complementInPowersetComplementIntersect_type,type,
% 0.16/0.44 complementInPowersetComplementIntersect: $o ).
% 0.16/0.44
% 0.16/0.44 thf(complementInPowersetComplementIntersect,definition,
% 0.16/0.44 ( complementInPowersetComplementIntersect
% 0.16/0.44 = ( ! [A: $i,X: $i] :
% 0.16/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.44 => ! [Y: $i] :
% 0.16/0.44 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.44 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(contraSubsetComplement_type,type,
% 0.16/0.44 contraSubsetComplement: $o ).
% 0.16/0.44
% 0.16/0.44 thf(contraSubsetComplement,definition,
% 0.16/0.44 ( contraSubsetComplement
% 0.16/0.44 = ( ! [A: $i,X: $i] :
% 0.16/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.44 => ! [Y: $i] :
% 0.16/0.44 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.44 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.16/0.44 => ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( ( in @ Xx @ Y )
% 0.16/0.44 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(complementTcontraSubset_type,type,
% 0.16/0.44 complementTcontraSubset: $o ).
% 0.16/0.44
% 0.16/0.44 thf(complementTcontraSubset,definition,
% 0.16/0.44 ( complementTcontraSubset
% 0.16/0.44 = ( ! [A: $i,X: $i] :
% 0.16/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.44 => ! [Y: $i] :
% 0.16/0.44 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.44 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.16/0.44 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(binunionTILcontra_type,type,
% 0.16/0.44 binunionTILcontra: $o ).
% 0.16/0.44
% 0.16/0.44 thf(binunionTILcontra,definition,
% 0.16/0.44 ( binunionTILcontra
% 0.16/0.44 = ( ! [A: $i,X: $i] :
% 0.16/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.44 => ! [Y: $i] :
% 0.16/0.44 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.44 => ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.16/0.44 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(binunionTIRcontra_type,type,
% 0.16/0.44 binunionTIRcontra: $o ).
% 0.16/0.44
% 0.16/0.44 thf(binunionTIRcontra,definition,
% 0.16/0.44 ( binunionTIRcontra
% 0.16/0.44 = ( ! [A: $i,X: $i] :
% 0.16/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.44 => ! [Y: $i] :
% 0.16/0.44 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.44 => ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.16/0.44 => ~ ( in @ Xx @ Y ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(inIntersectImpInUnion_type,type,
% 0.16/0.44 inIntersectImpInUnion: $o ).
% 0.16/0.44
% 0.16/0.44 thf(inIntersectImpInUnion,definition,
% 0.16/0.44 ( inIntersectImpInUnion
% 0.16/0.44 = ( ! [A: $i,X: $i] :
% 0.16/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.44 => ! [Y: $i] :
% 0.16/0.44 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.44 => ! [Z: $i] :
% 0.16/0.44 ( ( in @ Z @ ( powerset @ A ) )
% 0.16/0.44 => ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.16/0.44 => ( in @ Xx @ ( binunion @ X @ Z ) ) ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(inIntersectImpInUnion2_type,type,
% 0.16/0.44 inIntersectImpInUnion2: $o ).
% 0.16/0.44
% 0.16/0.44 thf(inIntersectImpInUnion2,definition,
% 0.16/0.44 ( inIntersectImpInUnion2
% 0.16/0.44 = ( ! [A: $i,X: $i] :
% 0.16/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.44 => ! [Y: $i] :
% 0.16/0.44 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.44 => ! [Z: $i] :
% 0.16/0.44 ( ( in @ Z @ ( powerset @ A ) )
% 0.16/0.44 => ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.16/0.44 => ( in @ Xx @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(inIntersectImpInIntersectUnions_type,type,
% 0.16/0.44 inIntersectImpInIntersectUnions: $o ).
% 0.16/0.44
% 0.16/0.44 thf(inIntersectImpInIntersectUnions,definition,
% 0.16/0.44 ( inIntersectImpInIntersectUnions
% 0.16/0.44 = ( ! [A: $i,X: $i] :
% 0.16/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.44 => ! [Y: $i] :
% 0.16/0.44 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.44 => ! [Z: $i] :
% 0.16/0.44 ( ( in @ Z @ ( powerset @ A ) )
% 0.16/0.44 => ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.16/0.44 => ( in @ Xx @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(intersectInPowersetIntersectUnions_type,type,
% 0.16/0.44 intersectInPowersetIntersectUnions: $o ).
% 0.16/0.44
% 0.16/0.44 thf(intersectInPowersetIntersectUnions,definition,
% 0.16/0.44 ( intersectInPowersetIntersectUnions
% 0.16/0.44 = ( ! [A: $i,X: $i] :
% 0.16/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.44 => ! [Y: $i] :
% 0.16/0.44 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.44 => ! [Z: $i] :
% 0.16/0.44 ( ( in @ Z @ ( powerset @ A ) )
% 0.16/0.44 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(inComplementUnionImpNotIn1_type,type,
% 0.16/0.44 inComplementUnionImpNotIn1: $o ).
% 0.16/0.44
% 0.16/0.44 thf(inComplementUnionImpNotIn1,definition,
% 0.16/0.44 ( inComplementUnionImpNotIn1
% 0.16/0.44 = ( ! [A: $i,X: $i] :
% 0.16/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.44 => ! [Y: $i] :
% 0.16/0.44 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.44 => ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.16/0.44 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(inComplementUnionImpInComplement1_type,type,
% 0.16/0.44 inComplementUnionImpInComplement1: $o ).
% 0.16/0.44
% 0.16/0.44 thf(inComplementUnionImpInComplement1,definition,
% 0.16/0.44 ( inComplementUnionImpInComplement1
% 0.16/0.44 = ( ! [A: $i,X: $i] :
% 0.16/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.44 => ! [Y: $i] :
% 0.16/0.44 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.44 => ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.16/0.44 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(binunionTE_type,type,
% 0.16/0.44 binunionTE: $o ).
% 0.16/0.44
% 0.16/0.44 thf(binunionTE,definition,
% 0.16/0.44 ( binunionTE
% 0.16/0.44 = ( ! [A: $i,X: $i] :
% 0.16/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.44 => ! [Y: $i] :
% 0.16/0.44 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.44 => ! [Xphi: $o,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.16/0.44 => ( ( ( in @ Xx @ X )
% 0.16/0.44 => Xphi )
% 0.16/0.44 => ( ( ( in @ Xx @ Y )
% 0.16/0.44 => Xphi )
% 0.16/0.44 => Xphi ) ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(binunionTEcontra_type,type,
% 0.16/0.44 binunionTEcontra: $o ).
% 0.16/0.44
% 0.16/0.44 thf(binunionTEcontra,definition,
% 0.16/0.44 ( binunionTEcontra
% 0.16/0.44 = ( ! [A: $i,X: $i] :
% 0.16/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.44 => ! [Y: $i] :
% 0.16/0.44 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.44 => ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( ~ ( in @ Xx @ X )
% 0.16/0.44 => ( ~ ( in @ Xx @ Y )
% 0.16/0.44 => ~ ( in @ Xx @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(demorgan2a1_type,type,
% 0.16/0.44 demorgan2a1: $o ).
% 0.16/0.44
% 0.16/0.44 thf(demorgan2a1,definition,
% 0.16/0.44 ( demorgan2a1
% 0.16/0.44 = ( ! [A: $i,X: $i] :
% 0.16/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.44 => ! [Y: $i] :
% 0.16/0.44 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.44 => ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.16/0.44 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(complementUnionInPowersetComplement_type,type,
% 0.16/0.44 complementUnionInPowersetComplement: $o ).
% 0.16/0.44
% 0.16/0.44 thf(complementUnionInPowersetComplement,definition,
% 0.16/0.44 ( complementUnionInPowersetComplement
% 0.16/0.44 = ( ! [A: $i,X: $i] :
% 0.16/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.44 => ! [Y: $i] :
% 0.16/0.44 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.44 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(demorgan2a2_type,type,
% 0.16/0.44 demorgan2a2: $o ).
% 0.16/0.44
% 0.16/0.44 thf(demorgan2a2,definition,
% 0.16/0.44 ( demorgan2a2
% 0.16/0.44 = ( ! [A: $i,X: $i] :
% 0.16/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.44 => ! [Y: $i] :
% 0.16/0.44 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.44 => ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.16/0.44 => ( in @ Xx @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(demorgan1a_type,type,
% 0.16/0.44 demorgan1a: $o ).
% 0.16/0.44
% 0.16/0.44 thf(demorgan1a,definition,
% 0.16/0.44 ( demorgan1a
% 0.16/0.44 = ( ! [A: $i,X: $i] :
% 0.16/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.44 => ! [Y: $i] :
% 0.16/0.44 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.44 => ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) )
% 0.16/0.44 => ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(demorgan1b_type,type,
% 0.16/0.44 demorgan1b: $o ).
% 0.16/0.44
% 0.16/0.44 thf(demorgan1b,definition,
% 0.16/0.44 ( demorgan1b
% 0.16/0.44 = ( ! [A: $i,X: $i] :
% 0.16/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.44 => ! [Y: $i] :
% 0.16/0.44 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.44 => ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 0.16/0.44 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(demorgan1_type,type,
% 0.16/0.44 demorgan1: $o ).
% 0.16/0.44
% 0.16/0.44 thf(demorgan1,definition,
% 0.16/0.44 ( demorgan1
% 0.16/0.44 = ( ! [A: $i,X: $i] :
% 0.16/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.44 => ! [Y: $i] :
% 0.16/0.44 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.44 => ( ( setminus @ A @ ( binintersect @ X @ Y ) )
% 0.16/0.44 = ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(demorgan2a_type,type,
% 0.16/0.44 demorgan2a: $o ).
% 0.16/0.44
% 0.16/0.44 thf(demorgan2a,definition,
% 0.16/0.44 ( demorgan2a
% 0.16/0.44 = ( ! [A: $i,X: $i] :
% 0.16/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.44 => ! [Y: $i] :
% 0.16/0.44 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.44 => ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.16/0.44 => ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(demorgan2b2_type,type,
% 0.16/0.44 demorgan2b2: $o ).
% 0.16/0.44
% 0.16/0.44 thf(demorgan2b2,definition,
% 0.16/0.44 ( demorgan2b2
% 0.16/0.44 = ( ! [A: $i,X: $i] :
% 0.16/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.44 => ! [Y: $i] :
% 0.16/0.44 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.44 => ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.16/0.44 => ( ( in @ Xx @ ( setminus @ A @ Y ) )
% 0.16/0.44 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(demorgan2b_type,type,
% 0.16/0.44 demorgan2b: $o ).
% 0.16/0.44
% 0.16/0.44 thf(demorgan2b,definition,
% 0.16/0.44 ( demorgan2b
% 0.16/0.44 = ( ! [A: $i,X: $i] :
% 0.16/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.44 => ! [Y: $i] :
% 0.16/0.44 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.44 => ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 0.16/0.44 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(demorgan2_type,type,
% 0.16/0.44 demorgan2: $o ).
% 0.16/0.44
% 0.16/0.44 thf(demorgan2,definition,
% 0.16/0.44 ( demorgan2
% 0.16/0.44 = ( ! [A: $i,X: $i] :
% 0.16/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.44 => ! [Y: $i] :
% 0.16/0.44 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.44 => ( ( setminus @ A @ ( binunion @ X @ Y ) )
% 0.16/0.44 = ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(woz13rule0_type,type,
% 0.16/0.44 woz13rule0: $o ).
% 0.16/0.44
% 0.16/0.44 thf(woz13rule0,definition,
% 0.16/0.44 ( woz13rule0
% 0.16/0.44 = ( ! [A: $i,X: $i] :
% 0.16/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.44 => ! [Y: $i] :
% 0.16/0.44 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.44 => ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.16/0.44 => ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(woz13rule1_type,type,
% 0.16/0.44 woz13rule1: $o ).
% 0.16/0.44
% 0.16/0.44 thf(woz13rule1,definition,
% 0.16/0.44 ( woz13rule1
% 0.16/0.44 = ( ! [A: $i,X: $i] :
% 0.16/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.44 => ! [Y: $i] :
% 0.16/0.44 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.44 => ! [Z: $i] :
% 0.16/0.44 ( ( in @ Z @ ( powerset @ A ) )
% 0.16/0.44 => ( ( subset @ X @ Z )
% 0.16/0.44 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(woz13rule2_type,type,
% 0.16/0.44 woz13rule2: $o ).
% 0.16/0.44
% 0.16/0.44 thf(woz13rule2,definition,
% 0.16/0.44 ( woz13rule2
% 0.16/0.44 = ( ! [A: $i,X: $i] :
% 0.16/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.44 => ! [Y: $i] :
% 0.16/0.44 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.44 => ! [Z: $i] :
% 0.16/0.44 ( ( in @ Z @ ( powerset @ A ) )
% 0.16/0.44 => ( ( subset @ Y @ Z )
% 0.16/0.44 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(woz13rule3_type,type,
% 0.16/0.44 woz13rule3: $o ).
% 0.16/0.44
% 0.16/0.44 thf(woz13rule3,definition,
% 0.16/0.44 ( woz13rule3
% 0.16/0.44 = ( ! [A: $i,X: $i] :
% 0.16/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.44 => ! [Y: $i] :
% 0.16/0.44 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.44 => ! [Z: $i] :
% 0.16/0.44 ( ( in @ Z @ ( powerset @ A ) )
% 0.16/0.44 => ( ( subset @ X @ Y )
% 0.16/0.44 => ( ( subset @ X @ Z )
% 0.16/0.44 => ( subset @ X @ ( binintersect @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(woz13rule4_type,type,
% 0.16/0.44 woz13rule4: $o ).
% 0.16/0.44
% 0.16/0.44 thf(woz13rule4,definition,
% 0.16/0.44 ( woz13rule4
% 0.16/0.44 = ( ! [A: $i,X: $i] :
% 0.16/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.44 => ! [Y: $i] :
% 0.16/0.44 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.44 => ! [Z: $i] :
% 0.16/0.44 ( ( in @ Z @ ( powerset @ A ) )
% 0.16/0.44 => ! [W: $i] :
% 0.16/0.44 ( ( in @ W @ ( powerset @ A ) )
% 0.16/0.44 => ( ( subset @ X @ Z )
% 0.16/0.44 => ( ( subset @ Y @ W )
% 0.16/0.44 => ( subset @ ( binintersect @ X @ Y ) @ ( binintersect @ Z @ W ) ) ) ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(woz1_1_type,type,
% 0.16/0.44 woz1_1: $o ).
% 0.16/0.44
% 0.16/0.44 thf(woz1_1,definition,
% 0.16/0.44 ( woz1_1
% 0.16/0.44 = ( ! [A: $i,X: $i] :
% 0.16/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.44 => ! [Y: $i] :
% 0.16/0.44 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.44 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(woz1_2_type,type,
% 0.16/0.44 woz1_2: $o ).
% 0.16/0.44
% 0.16/0.44 thf(woz1_2,definition,
% 0.16/0.44 ( woz1_2
% 0.16/0.44 = ( ! [A: $i,X: $i] :
% 0.16/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.44 => ! [Y: $i] :
% 0.16/0.44 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.44 => ! [Z: $i] :
% 0.16/0.44 ( ( in @ Z @ ( powerset @ A ) )
% 0.16/0.44 => ! [W: $i] :
% 0.16/0.44 ( ( in @ W @ ( powerset @ A ) )
% 0.16/0.44 => ( ( setminus @ A @ ( binintersect @ ( binunion @ X @ Y ) @ ( binunion @ Z @ W ) ) )
% 0.16/0.44 = ( binunion @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) @ ( binintersect @ ( setminus @ A @ Z ) @ ( setminus @ A @ W ) ) ) ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(woz1_3_type,type,
% 0.16/0.44 woz1_3: $o ).
% 0.16/0.44
% 0.16/0.44 thf(woz1_3,definition,
% 0.16/0.44 ( woz1_3
% 0.16/0.44 = ( ! [A: $i,X: $i] :
% 0.16/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.44 => ! [Y: $i] :
% 0.16/0.44 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.44 => ! [Z: $i] :
% 0.16/0.44 ( ( in @ Z @ ( powerset @ A ) )
% 0.16/0.44 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(woz1_4_type,type,
% 0.16/0.44 woz1_4: $o ).
% 0.16/0.44
% 0.16/0.44 thf(woz1_4,definition,
% 0.16/0.44 ( woz1_4
% 0.16/0.44 = ( ! [A: $i,X: $i] :
% 0.16/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.44 => ! [Y: $i] :
% 0.16/0.44 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.44 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.16/0.44 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(woz1_5_type,type,
% 0.16/0.44 woz1_5: $o ).
% 0.16/0.44
% 0.16/0.44 thf(woz1_5,definition,
% 0.16/0.44 ( woz1_5
% 0.16/0.44 = ( ! [A: $i,X: $i] :
% 0.16/0.44 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.44 => ! [Y: $i] :
% 0.16/0.44 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.44 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(breln1_type,type,
% 0.16/0.44 breln1: $i > $i > $o ).
% 0.16/0.44
% 0.16/0.44 thf(breln1all2_type,type,
% 0.16/0.44 breln1all2: $o ).
% 0.16/0.44
% 0.16/0.44 thf(breln1all2,definition,
% 0.16/0.44 ( breln1all2
% 0.16/0.44 = ( ! [A: $i,R: $i] :
% 0.16/0.44 ( ( breln1 @ A @ R )
% 0.16/0.44 => ! [Xphi: $i > $o] :
% 0.16/0.44 ( ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ! [Xy: $i] :
% 0.16/0.44 ( ( in @ Xy @ A )
% 0.16/0.44 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.16/0.44 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.16/0.44 => ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ R )
% 0.16/0.44 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(breln1Set_type,type,
% 0.16/0.44 breln1Set: $i > $i ).
% 0.16/0.44
% 0.16/0.44 thf(breln1SetBreln1_type,type,
% 0.16/0.44 breln1SetBreln1: $o ).
% 0.16/0.44
% 0.16/0.44 thf(breln1SetBreln1,definition,
% 0.16/0.44 ( breln1SetBreln1
% 0.16/0.44 = ( ! [A: $i,R: $i] :
% 0.16/0.44 ( ( in @ R @ ( breln1Set @ A ) )
% 0.16/0.44 => ( breln1 @ A @ R ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(transitive_type,type,
% 0.16/0.44 transitive: $i > $i > $o ).
% 0.16/0.44
% 0.16/0.44 thf(antisymmetric_type,type,
% 0.16/0.44 antisymmetric: $i > $i > $o ).
% 0.16/0.44
% 0.16/0.44 thf(reflexive_type,type,
% 0.16/0.44 reflexive: $i > $i > $o ).
% 0.16/0.44
% 0.16/0.44 thf(refltransitive_type,type,
% 0.16/0.44 refltransitive: $i > $i > $o ).
% 0.16/0.44
% 0.16/0.44 thf(refllinearorder_type,type,
% 0.16/0.44 refllinearorder: $i > $i > $o ).
% 0.16/0.44
% 0.16/0.44 thf(reflwellordering_type,type,
% 0.16/0.44 reflwellordering: $i > $i > $o ).
% 0.16/0.44
% 0.16/0.44 thf(choice2fnsingleton_type,type,
% 0.16/0.44 choice2fnsingleton: $o ).
% 0.16/0.44
% 0.16/0.44 thf(choice2fnsingleton,definition,
% 0.16/0.44 ( choice2fnsingleton
% 0.16/0.44 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.16/0.44 ( ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ? [Xy: $i] :
% 0.16/0.44 ( ( in @ Xy @ B )
% 0.16/0.44 & ( Xphi @ Xx @ Xy ) ) )
% 0.16/0.44 => ! [R: $i] :
% 0.16/0.44 ( ( in @ R @ ( breln1Set @ B ) )
% 0.16/0.44 => ( ( reflwellordering @ B @ R )
% 0.16/0.44 => ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( singleton
% 0.16/0.44 @ ( dsetconstr @ B
% 0.16/0.44 @ ^ [Xy: $i] :
% 0.16/0.44 ( ( Xphi @ Xx @ Xy )
% 0.16/0.44 & ! [Xz: $i] :
% 0.16/0.44 ( ( in @ Xz @ B )
% 0.16/0.44 => ( ( Xphi @ Xx @ Xz )
% 0.16/0.44 => ( in @ ( kpair @ Xy @ Xz ) @ R ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(setOfPairsIsBReln1_type,type,
% 0.16/0.44 setOfPairsIsBReln1: $o ).
% 0.16/0.44
% 0.16/0.44 thf(setOfPairsIsBReln1,definition,
% 0.16/0.44 ( setOfPairsIsBReln1
% 0.16/0.44 = ( ! [A: $i,Xphi: $i > $i > $o] :
% 0.16/0.44 ( breln1 @ A
% 0.16/0.44 @ ( dpsetconstr @ A @ A
% 0.16/0.44 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(breln1all1_type,type,
% 0.16/0.44 breln1all1: $o ).
% 0.16/0.44
% 0.16/0.44 thf(breln1all1,definition,
% 0.16/0.44 ( breln1all1
% 0.16/0.44 = ( ! [A: $i,R: $i] :
% 0.16/0.44 ( ( breln1 @ A @ R )
% 0.16/0.44 => ! [Xphi: $i > $o] :
% 0.16/0.44 ( ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ! [Xy: $i] :
% 0.16/0.44 ( ( in @ Xy @ A )
% 0.16/0.44 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.16/0.44 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.16/0.44 => ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ R )
% 0.16/0.44 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(subbreln1_type,type,
% 0.16/0.44 subbreln1: $o ).
% 0.16/0.44
% 0.16/0.44 thf(subbreln1,definition,
% 0.16/0.44 ( subbreln1
% 0.16/0.44 = ( ! [A: $i,R: $i] :
% 0.16/0.44 ( ( breln1 @ A @ R )
% 0.16/0.44 => ! [S: $i] :
% 0.16/0.44 ( ( breln1 @ A @ S )
% 0.16/0.44 => ( ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ! [Xy: $i] :
% 0.16/0.44 ( ( in @ Xy @ A )
% 0.16/0.44 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.16/0.44 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.16/0.44 => ( subset @ R @ S ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(eqbreln1_type,type,
% 0.16/0.44 eqbreln1: $o ).
% 0.16/0.44
% 0.16/0.44 thf(eqbreln1,definition,
% 0.16/0.44 ( eqbreln1
% 0.16/0.44 = ( ! [A: $i,R: $i] :
% 0.16/0.44 ( ( breln1 @ A @ R )
% 0.16/0.44 => ! [S: $i] :
% 0.16/0.44 ( ( breln1 @ A @ S )
% 0.16/0.44 => ( ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ! [Xy: $i] :
% 0.16/0.44 ( ( in @ Xy @ A )
% 0.16/0.44 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.16/0.44 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.16/0.44 => ( ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ! [Xy: $i] :
% 0.16/0.44 ( ( in @ Xy @ A )
% 0.16/0.44 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.16/0.44 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 0.16/0.44 => ( R = S ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(breln1invset_type,type,
% 0.16/0.44 breln1invset: $i > $i > $i ).
% 0.16/0.44
% 0.16/0.44 thf(breln1invprop_type,type,
% 0.16/0.44 breln1invprop: $o ).
% 0.16/0.44
% 0.16/0.44 thf(breln1invprop,definition,
% 0.16/0.44 ( breln1invprop
% 0.16/0.44 = ( ! [A: $i,R: $i] :
% 0.16/0.44 ( ( breln1 @ A @ R )
% 0.16/0.44 => ( breln1 @ A @ ( breln1invset @ A @ R ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(breln1invI_type,type,
% 0.16/0.44 breln1invI: $o ).
% 0.16/0.44
% 0.16/0.44 thf(breln1invI,definition,
% 0.16/0.44 ( breln1invI
% 0.16/0.44 = ( ! [A: $i,R: $i] :
% 0.16/0.44 ( ( breln1 @ A @ R )
% 0.16/0.44 => ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ! [Xy: $i] :
% 0.16/0.44 ( ( in @ Xy @ A )
% 0.16/0.44 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.16/0.44 => ( in @ ( kpair @ Xy @ Xx ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(breln1invE_type,type,
% 0.16/0.44 breln1invE: $o ).
% 0.16/0.44
% 0.16/0.44 thf(breln1invE,definition,
% 0.16/0.44 ( breln1invE
% 0.16/0.44 = ( ! [A: $i,R: $i] :
% 0.16/0.44 ( ( breln1 @ A @ R )
% 0.16/0.44 => ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ! [Xy: $i] :
% 0.16/0.44 ( ( in @ Xy @ A )
% 0.16/0.44 => ( ( in @ ( kpair @ Xy @ Xx ) @ ( breln1invset @ A @ R ) )
% 0.16/0.44 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(breln1compset_type,type,
% 0.16/0.44 breln1compset: $i > $i > $i > $i ).
% 0.16/0.44
% 0.16/0.44 thf(breln1compprop_type,type,
% 0.16/0.44 breln1compprop: $o ).
% 0.16/0.44
% 0.16/0.44 thf(breln1compprop,definition,
% 0.16/0.44 ( breln1compprop
% 0.16/0.44 = ( ! [A: $i,R: $i] :
% 0.16/0.44 ( ( breln1 @ A @ R )
% 0.16/0.44 => ! [S: $i] :
% 0.16/0.44 ( ( breln1 @ A @ S )
% 0.16/0.44 => ( breln1 @ A @ ( breln1compset @ A @ R @ S ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(breln1compI_type,type,
% 0.16/0.44 breln1compI: $o ).
% 0.16/0.44
% 0.16/0.44 thf(breln1compI,definition,
% 0.16/0.44 ( breln1compI
% 0.16/0.44 = ( ! [A: $i,R: $i] :
% 0.16/0.44 ( ( breln1 @ A @ R )
% 0.16/0.44 => ! [S: $i] :
% 0.16/0.44 ( ( breln1 @ A @ S )
% 0.16/0.44 => ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ! [Xy: $i] :
% 0.16/0.44 ( ( in @ Xy @ A )
% 0.16/0.44 => ! [Xz: $i] :
% 0.16/0.44 ( ( in @ Xz @ A )
% 0.16/0.44 => ( ( in @ ( kpair @ Xx @ Xz ) @ R )
% 0.16/0.44 => ( ( in @ ( kpair @ Xz @ Xy ) @ S )
% 0.16/0.44 => ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) ) ) ) ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(breln1compE_type,type,
% 0.16/0.44 breln1compE: $o ).
% 0.16/0.44
% 0.16/0.44 thf(breln1compE,definition,
% 0.16/0.44 ( breln1compE
% 0.16/0.44 = ( ! [A: $i,R: $i] :
% 0.16/0.44 ( ( breln1 @ A @ R )
% 0.16/0.44 => ! [S: $i] :
% 0.16/0.44 ( ( breln1 @ A @ S )
% 0.16/0.44 => ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ! [Xy: $i] :
% 0.16/0.44 ( ( in @ Xy @ A )
% 0.16/0.44 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) )
% 0.16/0.44 => ? [Xz: $i] :
% 0.16/0.44 ( ( in @ Xz @ A )
% 0.16/0.44 & ( in @ ( kpair @ Xx @ Xz ) @ R )
% 0.16/0.44 & ( in @ ( kpair @ Xz @ Xy ) @ S ) ) ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(breln1compEex_type,type,
% 0.16/0.44 breln1compEex: $o ).
% 0.16/0.44
% 0.16/0.44 thf(breln1compEex,definition,
% 0.16/0.44 ( breln1compEex
% 0.16/0.44 = ( ! [A: $i,R: $i] :
% 0.16/0.44 ( ( breln1 @ A @ R )
% 0.16/0.44 => ! [S: $i] :
% 0.16/0.44 ( ( breln1 @ A @ S )
% 0.16/0.44 => ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ! [Xy: $i] :
% 0.16/0.44 ( ( in @ Xy @ A )
% 0.16/0.44 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) )
% 0.16/0.44 => ! [Xphi: $o] :
% 0.16/0.44 ( ! [Xz: $i] :
% 0.16/0.44 ( ( in @ Xz @ A )
% 0.16/0.44 => ( ( in @ ( kpair @ Xx @ Xz ) @ R )
% 0.16/0.44 => ( ( in @ ( kpair @ Xz @ Xy ) @ S )
% 0.16/0.44 => Xphi ) ) )
% 0.16/0.44 => Xphi ) ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(breln1unionprop_type,type,
% 0.16/0.44 breln1unionprop: $o ).
% 0.16/0.44
% 0.16/0.44 thf(breln1unionprop,definition,
% 0.16/0.44 ( breln1unionprop
% 0.16/0.44 = ( ! [A: $i,R: $i] :
% 0.16/0.44 ( ( breln1 @ A @ R )
% 0.16/0.44 => ! [S: $i] :
% 0.16/0.44 ( ( breln1 @ A @ S )
% 0.16/0.44 => ( breln1 @ A @ ( binunion @ R @ S ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(breln1unionIL_type,type,
% 0.16/0.44 breln1unionIL: $o ).
% 0.16/0.44
% 0.16/0.44 thf(breln1unionIL,definition,
% 0.16/0.44 ( breln1unionIL
% 0.16/0.44 = ( ! [A: $i,R: $i] :
% 0.16/0.44 ( ( breln1 @ A @ R )
% 0.16/0.44 => ! [S: $i] :
% 0.16/0.44 ( ( breln1 @ A @ S )
% 0.16/0.44 => ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ! [Xy: $i] :
% 0.16/0.44 ( ( in @ Xy @ A )
% 0.16/0.44 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.16/0.44 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(breln1unionIR_type,type,
% 0.16/0.44 breln1unionIR: $o ).
% 0.16/0.44
% 0.16/0.44 thf(breln1unionIR,definition,
% 0.16/0.44 ( breln1unionIR
% 0.16/0.44 = ( ! [A: $i,R: $i] :
% 0.16/0.44 ( ( breln1 @ A @ R )
% 0.16/0.44 => ! [S: $i] :
% 0.16/0.44 ( ( breln1 @ A @ S )
% 0.16/0.44 => ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ! [Xy: $i] :
% 0.16/0.44 ( ( in @ Xy @ A )
% 0.16/0.44 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.16/0.44 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(breln1unionI_type,type,
% 0.16/0.44 breln1unionI: $o ).
% 0.16/0.44
% 0.16/0.44 thf(breln1unionI,definition,
% 0.16/0.44 ( breln1unionI
% 0.16/0.44 = ( ! [A: $i,R: $i] :
% 0.16/0.44 ( ( breln1 @ A @ R )
% 0.16/0.44 => ! [S: $i] :
% 0.16/0.44 ( ( breln1 @ A @ S )
% 0.16/0.44 => ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ! [Xy: $i] :
% 0.16/0.44 ( ( in @ Xy @ A )
% 0.16/0.44 => ( ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.16/0.44 | ( in @ ( kpair @ Xx @ Xy ) @ S ) )
% 0.16/0.44 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(breln1unionE_type,type,
% 0.16/0.44 breln1unionE: $o ).
% 0.16/0.44
% 0.16/0.44 thf(breln1unionE,definition,
% 0.16/0.44 ( breln1unionE
% 0.16/0.44 = ( ! [A: $i,R: $i] :
% 0.16/0.44 ( ( breln1 @ A @ R )
% 0.16/0.44 => ! [S: $i] :
% 0.16/0.44 ( ( breln1 @ A @ S )
% 0.16/0.44 => ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ! [Xy: $i] :
% 0.16/0.44 ( ( in @ Xy @ A )
% 0.16/0.44 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) )
% 0.16/0.44 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.16/0.44 | ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(breln1unionEcases,conjecture,
% 0.16/0.44 ( setextAx
% 0.16/0.44 => ( emptysetAx
% 0.16/0.44 => ( setadjoinAx
% 0.16/0.44 => ( powersetAx
% 0.16/0.44 => ( setunionAx
% 0.16/0.44 => ( omega0Ax
% 0.16/0.44 => ( omegaSAx
% 0.16/0.44 => ( omegaIndAx
% 0.16/0.44 => ( replAx
% 0.16/0.44 => ( foundationAx
% 0.16/0.44 => ( wellorderingAx
% 0.16/0.44 => ( descrp
% 0.16/0.44 => ( dsetconstrI
% 0.16/0.44 => ( dsetconstrEL
% 0.16/0.44 => ( dsetconstrER
% 0.16/0.44 => ( exuE1
% 0.16/0.44 => ( prop2setE
% 0.16/0.44 => ( emptysetE
% 0.16/0.44 => ( emptysetimpfalse
% 0.16/0.44 => ( notinemptyset
% 0.16/0.44 => ( exuE3e
% 0.16/0.44 => ( setext
% 0.16/0.44 => ( emptyI
% 0.16/0.44 => ( noeltsimpempty
% 0.16/0.44 => ( setbeta
% 0.16/0.44 => ( nonemptyE1
% 0.16/0.44 => ( nonemptyI
% 0.16/0.44 => ( nonemptyI1
% 0.16/0.44 => ( setadjoinIL
% 0.16/0.44 => ( emptyinunitempty
% 0.16/0.44 => ( setadjoinIR
% 0.16/0.44 => ( setadjoinE
% 0.16/0.44 => ( setadjoinOr
% 0.16/0.44 => ( setoftrueEq
% 0.16/0.44 => ( powersetI
% 0.16/0.44 => ( emptyinPowerset
% 0.16/0.44 => ( emptyInPowerset
% 0.16/0.44 => ( powersetE
% 0.16/0.44 => ( setunionI
% 0.16/0.44 => ( setunionE
% 0.16/0.44 => ( subPowSU
% 0.16/0.44 => ( exuE2
% 0.16/0.44 => ( nonemptyImpWitness
% 0.16/0.44 => ( uniqinunit
% 0.16/0.44 => ( notinsingleton
% 0.16/0.44 => ( eqinunit
% 0.16/0.44 => ( singletonsswitch
% 0.16/0.44 => ( upairsetE
% 0.16/0.44 => ( upairsetIL
% 0.16/0.44 => ( upairsetIR
% 0.16/0.44 => ( emptyE1
% 0.16/0.44 => ( vacuousDall
% 0.16/0.44 => ( quantDeMorgan1
% 0.16/0.44 => ( quantDeMorgan2
% 0.16/0.44 => ( quantDeMorgan3
% 0.16/0.44 => ( quantDeMorgan4
% 0.16/0.44 => ( prop2setI
% 0.16/0.44 => ( prop2set2propI
% 0.16/0.44 => ( notdexE
% 0.16/0.44 => ( notdallE
% 0.16/0.44 => ( exuI1
% 0.16/0.44 => ( exuI3
% 0.16/0.44 => ( exuI2
% 0.16/0.44 => ( inCongP
% 0.16/0.44 => ( in__Cong
% 0.16/0.44 => ( exuE3u
% 0.16/0.44 => ( exu__Cong
% 0.16/0.44 => ( emptyset__Cong
% 0.16/0.44 => ( setadjoin__Cong
% 0.16/0.44 => ( powerset__Cong
% 0.16/0.44 => ( setunion__Cong
% 0.16/0.44 => ( omega__Cong
% 0.16/0.44 => ( exuEu
% 0.16/0.44 => ( descr__Cong
% 0.16/0.44 => ( dsetconstr__Cong
% 0.16/0.44 => ( subsetI1
% 0.16/0.44 => ( eqimpsubset2
% 0.16/0.44 => ( eqimpsubset1
% 0.16/0.44 => ( subsetI2
% 0.16/0.44 => ( emptysetsubset
% 0.16/0.44 => ( subsetE
% 0.16/0.44 => ( subsetE2
% 0.16/0.44 => ( notsubsetI
% 0.16/0.44 => ( notequalI1
% 0.16/0.44 => ( notequalI2
% 0.16/0.44 => ( subsetRefl
% 0.16/0.44 => ( subsetTrans
% 0.16/0.44 => ( setadjoinSub
% 0.16/0.44 => ( setadjoinSub2
% 0.16/0.44 => ( subset2powerset
% 0.16/0.44 => ( setextsub
% 0.16/0.44 => ( subsetemptysetimpeq
% 0.16/0.44 => ( powersetI1
% 0.16/0.44 => ( powersetE1
% 0.16/0.44 => ( inPowerset
% 0.16/0.44 => ( powersetsubset
% 0.16/0.44 => ( sepInPowerset
% 0.16/0.44 => ( sepSubset
% 0.16/0.44 => ( binunionIL
% 0.16/0.44 => ( upairset2IR
% 0.16/0.44 => ( binunionIR
% 0.16/0.44 => ( binunionEcases
% 0.16/0.44 => ( binunionE
% 0.16/0.44 => ( binunionLsub
% 0.16/0.44 => ( binunionRsub
% 0.16/0.44 => ( binintersectI
% 0.16/0.44 => ( binintersectSubset5
% 0.16/0.44 => ( binintersectEL
% 0.16/0.44 => ( binintersectLsub
% 0.16/0.44 => ( binintersectSubset2
% 0.16/0.44 => ( binintersectSubset3
% 0.16/0.44 => ( binintersectER
% 0.16/0.44 => ( disjointsetsI1
% 0.16/0.44 => ( binintersectRsub
% 0.16/0.44 => ( binintersectSubset4
% 0.16/0.44 => ( binintersectSubset1
% 0.16/0.44 => ( bs114d
% 0.16/0.44 => ( setminusI
% 0.16/0.44 => ( setminusEL
% 0.16/0.44 => ( setminusER
% 0.16/0.44 => ( setminusSubset2
% 0.16/0.44 => ( setminusERneg
% 0.16/0.44 => ( setminusELneg
% 0.16/0.44 => ( setminusILneg
% 0.16/0.44 => ( setminusIRneg
% 0.16/0.44 => ( setminusLsub
% 0.16/0.44 => ( setminusSubset1
% 0.16/0.44 => ( symdiffE
% 0.16/0.44 => ( symdiffI1
% 0.16/0.44 => ( symdiffI2
% 0.16/0.44 => ( symdiffIneg1
% 0.16/0.44 => ( symdiffIneg2
% 0.16/0.44 => ( secondinupair
% 0.16/0.44 => ( setukpairIL
% 0.16/0.44 => ( setukpairIR
% 0.16/0.44 => ( kpairiskpair
% 0.16/0.44 => ( kpairp
% 0.16/0.44 => ( singletonsubset
% 0.16/0.44 => ( singletoninpowerset
% 0.16/0.44 => ( singletoninpowunion
% 0.16/0.44 => ( upairset2E
% 0.16/0.44 => ( upairsubunion
% 0.16/0.44 => ( upairinpowunion
% 0.16/0.44 => ( ubforcartprodlem1
% 0.16/0.44 => ( ubforcartprodlem2
% 0.16/0.44 => ( ubforcartprodlem3
% 0.16/0.44 => ( cartprodpairin
% 0.16/0.44 => ( cartprodmempair1
% 0.16/0.44 => ( cartprodmempair
% 0.16/0.44 => ( setunionE2
% 0.16/0.44 => ( setunionsingleton1
% 0.16/0.44 => ( setunionsingleton2
% 0.16/0.44 => ( setunionsingleton
% 0.16/0.44 => ( singletonprop
% 0.16/0.44 => ( ex1E1
% 0.16/0.44 => ( ex1I
% 0.16/0.44 => ( ex1I2
% 0.16/0.44 => ( singletonsuniq
% 0.16/0.44 => ( setukpairinjL1
% 0.16/0.44 => ( kfstsingleton
% 0.16/0.44 => ( theprop
% 0.16/0.44 => ( kfstpairEq
% 0.16/0.44 => ( cartprodfstin
% 0.16/0.44 => ( setukpairinjL2
% 0.16/0.44 => ( setukpairinjL
% 0.16/0.44 => ( setukpairinjR11
% 0.16/0.44 => ( setukpairinjR12
% 0.16/0.44 => ( setukpairinjR1
% 0.16/0.44 => ( upairequniteq
% 0.16/0.44 => ( setukpairinjR2
% 0.16/0.44 => ( setukpairinjR
% 0.16/0.44 => ( ksndsingleton
% 0.16/0.44 => ( ksndpairEq
% 0.16/0.44 => ( kpairsurjEq
% 0.16/0.44 => ( cartprodsndin
% 0.16/0.44 => ( cartprodpairmemEL
% 0.16/0.44 => ( cartprodpairmemER
% 0.16/0.44 => ( cartprodmempaircEq
% 0.16/0.44 => ( cartprodfstpairEq
% 0.16/0.44 => ( cartprodsndpairEq
% 0.16/0.44 => ( cartprodpairsurjEq
% 0.16/0.44 => ( dpsetconstrI
% 0.16/0.44 => ( dpsetconstrSub
% 0.16/0.44 => ( setOfPairsIsBReln
% 0.16/0.44 => ( dpsetconstrERa
% 0.16/0.44 => ( dpsetconstrEL1
% 0.16/0.44 => ( dpsetconstrEL2
% 0.16/0.44 => ( dpsetconstrER
% 0.16/0.44 => ( funcImageSingleton
% 0.16/0.44 => ( apProp
% 0.16/0.44 => ( app
% 0.16/0.44 => ( infuncsetfunc
% 0.16/0.44 => ( ap2p
% 0.16/0.44 => ( funcinfuncset
% 0.16/0.44 => ( lamProp
% 0.16/0.44 => ( lamp
% 0.16/0.44 => ( lam2p
% 0.16/0.44 => ( brelnall1
% 0.16/0.44 => ( brelnall2
% 0.16/0.44 => ( ex1E2
% 0.16/0.44 => ( funcGraphProp1
% 0.16/0.44 => ( funcGraphProp3
% 0.16/0.44 => ( funcGraphProp2
% 0.16/0.44 => ( funcextLem
% 0.16/0.44 => ( funcGraphProp4
% 0.16/0.44 => ( subbreln
% 0.16/0.44 => ( eqbreln
% 0.16/0.44 => ( funcext
% 0.16/0.44 => ( funcext2
% 0.16/0.44 => ( ap2apEq1
% 0.16/0.44 => ( ap2apEq2
% 0.16/0.44 => ( beta1
% 0.16/0.44 => ( eta1
% 0.16/0.44 => ( lam2lamEq
% 0.16/0.44 => ( beta2
% 0.16/0.44 => ( eta2
% 0.16/0.44 => ( iffalseProp1
% 0.16/0.44 => ( iffalseProp2
% 0.16/0.44 => ( iftrueProp1
% 0.16/0.44 => ( iftrueProp2
% 0.16/0.44 => ( ifSingleton
% 0.16/0.44 => ( ifp
% 0.16/0.44 => ( theeq
% 0.16/0.44 => ( iftrue
% 0.16/0.44 => ( iffalse
% 0.16/0.44 => ( iftrueorfalse
% 0.16/0.44 => ( binintersectT_lem
% 0.16/0.44 => ( binunionT_lem
% 0.16/0.44 => ( powersetT_lem
% 0.16/0.44 => ( setminusT_lem
% 0.16/0.44 => ( complementT_lem
% 0.16/0.44 => ( setextT
% 0.16/0.44 => ( subsetTI
% 0.16/0.44 => ( powersetTI1
% 0.16/0.44 => ( powersetTE1
% 0.16/0.44 => ( complementTI1
% 0.16/0.44 => ( complementTE1
% 0.16/0.44 => ( binintersectTELcontra
% 0.16/0.44 => ( binintersectTERcontra
% 0.16/0.44 => ( contrasubsetT
% 0.16/0.44 => ( contrasubsetT1
% 0.16/0.44 => ( contrasubsetT2
% 0.16/0.44 => ( contrasubsetT3
% 0.16/0.44 => ( doubleComplementI1
% 0.16/0.44 => ( doubleComplementE1
% 0.16/0.44 => ( doubleComplementSub1
% 0.16/0.44 => ( doubleComplementSub2
% 0.16/0.44 => ( doubleComplementEq
% 0.16/0.44 => ( complementTnotintersectT
% 0.16/0.44 => ( complementImpComplementIntersect
% 0.16/0.44 => ( complementSubsetComplementIntersect
% 0.16/0.44 => ( complementInPowersetComplementIntersect
% 0.16/0.44 => ( contraSubsetComplement
% 0.16/0.44 => ( complementTcontraSubset
% 0.16/0.44 => ( binunionTILcontra
% 0.16/0.44 => ( binunionTIRcontra
% 0.16/0.44 => ( inIntersectImpInUnion
% 0.16/0.44 => ( inIntersectImpInUnion2
% 0.16/0.44 => ( inIntersectImpInIntersectUnions
% 0.16/0.44 => ( intersectInPowersetIntersectUnions
% 0.16/0.44 => ( inComplementUnionImpNotIn1
% 0.16/0.44 => ( inComplementUnionImpInComplement1
% 0.16/0.44 => ( binunionTE
% 0.16/0.44 => ( binunionTEcontra
% 0.16/0.44 => ( demorgan2a1
% 0.16/0.44 => ( complementUnionInPowersetComplement
% 0.16/0.44 => ( demorgan2a2
% 0.16/0.44 => ( demorgan1a
% 0.16/0.44 => ( demorgan1b
% 0.16/0.44 => ( demorgan1
% 0.16/0.44 => ( demorgan2a
% 0.16/0.44 => ( demorgan2b2
% 0.16/0.44 => ( demorgan2b
% 0.16/0.44 => ( demorgan2
% 0.16/0.44 => ( woz13rule0
% 0.16/0.44 => ( woz13rule1
% 0.16/0.44 => ( woz13rule2
% 0.16/0.44 => ( woz13rule3
% 0.16/0.44 => ( woz13rule4
% 0.16/0.44 => ( woz1_1
% 0.16/0.44 => ( woz1_2
% 0.16/0.44 => ( woz1_3
% 0.16/0.44 => ( woz1_4
% 0.16/0.44 => ( woz1_5
% 0.16/0.44 => ( breln1all2
% 0.16/0.44 => ( breln1SetBreln1
% 0.16/0.44 => ( choice2fnsingleton
% 0.16/0.44 => ( setOfPairsIsBReln1
% 0.16/0.44 => ( breln1all1
% 0.16/0.44 => ( subbreln1
% 0.16/0.44 => ( eqbreln1
% 0.16/0.44 => ( breln1invprop
% 0.16/0.44 => ( breln1invI
% 0.16/0.44 => ( breln1invE
% 0.16/0.44 => ( breln1compprop
% 0.16/0.44 => ( breln1compI
% 0.16/0.44 => ( breln1compE
% 0.16/0.44 => ( breln1compEex
% 0.16/0.44 => ( breln1unionprop
% 0.16/0.44 => ( breln1unionIL
% 0.16/0.44 => ( breln1unionIR
% 0.16/0.44 => ( breln1unionI
% 0.16/0.44 => ( breln1unionE
% 0.16/0.44 => ! [A: $i,R: $i] :
% 0.16/0.44 ( ( breln1 @ A @ R )
% 0.16/0.44 => ! [S: $i] :
% 0.16/0.44 ( ( breln1 @ A @ S )
% 0.16/0.44 => ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ! [Xy: $i] :
% 0.16/0.44 ( ( in @ Xy @ A )
% 0.16/0.44 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) )
% 0.16/0.44 => ! [Xphi: $o] :
% 0.16/0.44 ( ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.16/0.44 => Xphi )
% 0.16/0.44 => ( ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.16/0.44 => Xphi )
% 0.16/0.44 => Xphi ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 %------------------------------------------------------------------------------
% 0.16/0.44 ------------------------
% 2.71/1.47 ---- Embedded in THF ---
% 2.71/1.47 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.71/1.47 %%% Generated on Tue Feb 24 12:35:04 EST 2026
% 2.71/1.47 %%% using '$$dhol' embedding, version 1.3.0.
% 2.71/1.47 %%% Logic specification used:
% 2.71/1.47 %%% thf(spec, logic, $$dhol).
% 2.71/1.47
% 2.71/1.47 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.U76IOM5xNk/DTF2DTF4862.p
% See solution above
% 2.71/1.47 ------------------------
% 2.93/1.54 ---- Cleaned THF ---
% 2.93/1.54 thf(in_type,type,
% 2.93/1.54 in: $i > $i > $o ).
% 2.93/1.54
% 2.93/1.54 thf(exu_type,type,
% 2.93/1.54 exu: ( $i > $o ) > $o ).
% 2.93/1.54
% 2.93/1.54 thf(exu,definition,
% 2.93/1.54 ( exu
% 2.93/1.54 = ( ^ [Xphi: $i > $o] :
% 2.93/1.54 ? [Xx: $i] :
% 2.93/1.54 ( ( Xphi @ Xx )
% 2.93/1.54 & ! [Xy: $i] :
% 2.93/1.54 ( ( Xphi @ Xy )
% 2.93/1.54 => ( Xx = Xy ) ) ) ) ) ).
% 2.93/1.54
% 2.93/1.54 thf(setextAx_type,type,
% 2.93/1.54 setextAx: $o ).
% 2.93/1.54
% 2.93/1.54 thf(setextAx,definition,
% 2.93/1.54 ( setextAx
% 2.93/1.54 = ( ! [A: $i,B: $i] :
% 2.93/1.54 ( ! [Xx: $i] :
% 2.93/1.54 ( ( in @ Xx @ A )
% 2.93/1.54 <=> ( in @ Xx @ B ) )
% 2.93/1.54 => ( A = B ) ) ) ) ).
% 2.93/1.54
% 2.93/1.54 thf(emptyset_type,type,
% 2.93/1.54 emptyset: $i ).
% 2.93/1.54
% 2.93/1.54 thf(emptysetAx_type,type,
% 2.93/1.54 emptysetAx: $o ).
% 2.93/1.54
% 2.93/1.54 thf(emptysetAx,definition,
% 2.93/1.54 ( emptysetAx
% 2.93/1.54 = ( ! [Xx: $i] :
% 2.93/1.54 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.93/1.54
% 2.93/1.54 thf(setadjoin_type,type,
% 2.93/1.54 setadjoin: $i > $i > $i ).
% 2.93/1.54
% 2.93/1.54 thf(setadjoinAx_type,type,
% 2.93/1.54 setadjoinAx: $o ).
% 2.93/1.54
% 2.93/1.54 thf(setadjoinAx,definition,
% 2.93/1.54 ( setadjoinAx
% 2.93/1.54 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.93/1.54 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.93/1.54 <=> ( ( Xy = Xx )
% 2.93/1.54 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.93/1.54
% 2.93/1.54 thf(powerset_type,type,
% 2.93/1.54 powerset: $i > $i ).
% 2.93/1.54
% 2.93/1.54 thf(powersetAx_type,type,
% 2.93/1.54 powersetAx: $o ).
% 2.93/1.54
% 2.93/1.54 thf(powersetAx,definition,
% 2.93/1.54 ( powersetAx
% 2.93/1.54 = ( ! [A: $i,B: $i] :
% 2.93/1.54 ( ( in @ B @ ( powerset @ A ) )
% 2.93/1.54 <=> ! [Xx: $i] :
% 2.93/1.54 ( ( in @ Xx @ B )
% 2.93/1.54 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.93/1.54
% 2.93/1.54 thf(setunion_type,type,
% 2.93/1.54 setunion: $i > $i ).
% 2.93/1.54
% 2.93/1.54 thf(setunionAx_type,type,
% 2.93/1.54 setunionAx: $o ).
% 2.93/1.54
% 2.93/1.54 thf(setunionAx,definition,
% 2.93/1.54 ( setunionAx
% 2.93/1.54 = ( ! [A: $i,Xx: $i] :
% 2.93/1.54 ( ( in @ Xx @ ( setunion @ A ) )
% 2.93/1.54 <=> ? [B: $i] :
% 2.93/1.54 ( ( in @ Xx @ B )
% 2.93/1.54 & ( in @ B @ A ) ) ) ) ) ).
% 2.93/1.54
% 2.93/1.54 thf(omega_type,type,
% 2.93/1.54 omega: $i ).
% 2.93/1.54
% 2.93/1.54 thf(omega0Ax_type,type,
% 2.93/1.54 omega0Ax: $o ).
% 2.93/1.54
% 2.93/1.54 thf(omega0Ax,definition,
% 2.93/1.54 ( omega0Ax
% 2.93/1.54 = ( in @ emptyset @ omega ) ) ).
% 2.93/1.54
% 2.93/1.54 thf(omegaSAx_type,type,
% 2.93/1.54 omegaSAx: $o ).
% 2.93/1.54
% 2.93/1.54 thf(omegaSAx,definition,
% 2.93/1.54 ( omegaSAx
% 2.93/1.54 = ( ! [Xx: $i] :
% 2.93/1.54 ( ( in @ Xx @ omega )
% 2.93/1.54 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 2.93/1.54
% 2.93/1.54 thf(omegaIndAx_type,type,
% 2.93/1.54 omegaIndAx: $o ).
% 2.93/1.54
% 2.93/1.54 thf(omegaIndAx,definition,
% 2.93/1.54 ( omegaIndAx
% 2.93/1.54 = ( ! [A: $i] :
% 2.93/1.54 ( ( ( in @ emptyset @ A )
% 2.93/1.54 & ! [Xx: $i] :
% 2.93/1.54 ( ( ( in @ Xx @ omega )
% 2.93/1.54 & ( in @ Xx @ A ) )
% 2.93/1.54 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 2.93/1.54 => ! [Xx: $i] :
% 2.93/1.54 ( ( in @ Xx @ omega )
% 2.93/1.54 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.93/1.54
% 2.93/1.54 thf(replAx_type,type,
% 2.93/1.54 replAx: $o ).
% 2.93/1.54
% 2.93/1.54 thf(replAx,definition,
% 2.93/1.54 ( replAx
% 2.93/1.54 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 2.93/1.54 ( ! [Xx: $i] :
% 2.93/1.54 ( ( in @ Xx @ A )
% 2.93/1.54 => ( exu
% 2.93/1.54 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 2.93/1.54 => ? [B: $i] :
% 2.93/1.54 ! [Xx: $i] :
% 2.93/1.54 ( ( in @ Xx @ B )
% 2.93/1.54 <=> ? [Xy: $i] :
% 2.93/1.54 ( ( in @ Xy @ A )
% 2.93/1.54 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 2.93/1.54
% 2.93/1.54 thf(foundationAx_type,type,
% 2.93/1.54 foundationAx: $o ).
% 2.93/1.54
% 2.93/1.54 thf(foundationAx,definition,
% 2.93/1.54 ( foundationAx
% 2.93/1.54 = ( ! [A: $i] :
% 2.93/1.54 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.93/1.54 => ? [B: $i] :
% 2.93/1.54 ( ( in @ B @ A )
% 2.93/1.54 & ~ ? [Xx: $i] :
% 2.93/1.54 ( ( in @ Xx @ B )
% 2.93/1.54 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 2.93/1.54
% 2.93/1.54 thf(wellorderingAx_type,type,
% 2.93/1.54 wellorderingAx: $o ).
% 2.93/1.54
% 2.93/1.54 thf(wellorderingAx,definition,
% 2.93/1.54 ( wellorderingAx
% 2.93/1.54 = ( ! [A: $i] :
% 2.93/1.54 ? [B: $i] :
% 2.93/1.54 ( ! [C: $i] :
% 2.93/1.54 ( ( in @ C @ B )
% 2.93/1.54 => ! [Xx: $i] :
% 2.93/1.54 ( ( in @ Xx @ C )
% 2.93/1.54 => ( in @ Xx @ A ) ) )
% 2.93/1.54 & ! [Xx: $i,Xy: $i] :
% 2.93/1.54 ( ( ( in @ Xx @ A )
% 2.93/1.54 & ( in @ Xy @ A ) )
% 2.93/1.54 => ( ! [C: $i] :
% 2.93/1.54 ( ( in @ C @ B )
% 2.93/1.54 => ( ( in @ Xx @ C )
% 2.93/1.54 <=> ( in @ Xy @ C ) ) )
% 2.93/1.54 => ( Xx = Xy ) ) )
% 2.93/1.54 & ! [C: $i,D: $i] :
% 2.93/1.54 ( ( ( in @ C @ B )
% 2.93/1.54 & ( in @ D @ B ) )
% 2.93/1.54 => ( ! [Xx: $i] :
% 2.93/1.54 ( ( in @ Xx @ C )
% 2.93/1.54 => ( in @ Xx @ D ) )
% 2.93/1.54 | ! [Xx: $i] :
% 2.93/1.54 ( ( in @ Xx @ D )
% 2.93/1.54 => ( in @ Xx @ C ) ) ) )
% 2.93/1.54 & ! [C: $i] :
% 2.93/1.54 ( ( ! [Xx: $i] :
% 2.93/1.54 ( ( in @ Xx @ C )
% 2.93/1.54 => ( in @ Xx @ A ) )
% 2.93/1.54 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 2.93/1.54 => ? [D: $i,Xx: $i] :
% 2.93/1.54 ( ( in @ D @ B )
% 2.93/1.54 & ( in @ Xx @ C )
% 2.93/1.54 & ~ ? [Xy: $i] :
% 2.93/1.54 ( ( in @ Xy @ D )
% 2.93/1.54 & ( in @ Xy @ C ) )
% 2.93/1.54 & ! [E: $i] :
% 2.93/1.54 ( ( in @ E @ B )
% 2.93/1.54 => ( ! [Xy: $i] :
% 2.93/1.54 ( ( in @ Xy @ E )
% 2.93/1.54 => ( in @ Xy @ D ) )
% 2.93/1.54 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 2.93/1.54
% 2.93/1.54 thf(descr_type,type,
% 2.93/1.54 descr: ( $i > $o ) > $i ).
% 2.93/1.54
% 2.93/1.54 thf(descrp_type,type,
% 2.93/1.54 descrp: $o ).
% 2.93/1.54
% 2.93/1.54 thf(descrp,definition,
% 2.93/1.54 ( descrp
% 2.93/1.54 = ( ! [Xphi: $i > $o] :
% 2.93/1.54 ( ( exu
% 2.93/1.54 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.93/1.54 => ( Xphi
% 2.93/1.54 @ ( descr
% 2.93/1.54 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.93/1.54
% 2.93/1.54 thf(dsetconstr_type,type,
% 2.93/1.54 dsetconstr: $i > ( $i > $o ) > $i ).
% 2.93/1.54
% 2.93/1.54 thf(dsetconstrI_type,type,
% 2.93/1.54 dsetconstrI: $o ).
% 2.93/1.54
% 2.93/1.54 thf(dsetconstrI,definition,
% 2.93/1.54 ( dsetconstrI
% 2.93/1.54 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.93/1.54 ( ( in @ Xx @ A )
% 2.93/1.54 => ( ( Xphi @ Xx )
% 2.93/1.54 => ( in @ Xx
% 2.93/1.54 @ ( dsetconstr @ A
% 2.93/1.54 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.93/1.54
% 2.93/1.54 thf(dsetconstrEL_type,type,
% 2.93/1.54 dsetconstrEL: $o ).
% 2.93/1.54
% 2.93/1.54 thf(dsetconstrEL,definition,
% 2.93/1.54 ( dsetconstrEL
% 2.93/1.54 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.93/1.54 ( ( in @ Xx
% 2.93/1.54 @ ( dsetconstr @ A
% 2.93/1.54 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.93/1.54 => ( in @ Xx @ A ) ) ) ) ).
% 2.93/1.54
% 2.93/1.54 thf(dsetconstrER_type,type,
% 2.93/1.54 dsetconstrER: $o ).
% 2.93/1.54
% 2.93/1.54 thf(dsetconstrER,definition,
% 2.93/1.54 ( dsetconstrER
% 2.93/1.54 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.93/1.54 ( ( in @ Xx
% 2.93/1.54 @ ( dsetconstr @ A
% 2.93/1.54 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.93/1.54 => ( Xphi @ Xx ) ) ) ) ).
% 2.93/1.54
% 2.93/1.54 thf(exuE1_type,type,
% 2.93/1.54 exuE1: $o ).
% 2.93/1.54
% 2.93/1.54 thf(exuE1,definition,
% 2.93/1.54 ( exuE1
% 2.93/1.54 = ( ! [Xphi: $i > $o] :
% 2.93/1.54 ( ( exu
% 2.93/1.54 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.93/1.54 => ? [Xx: $i] :
% 2.93/1.54 ( ( Xphi @ Xx )
% 2.93/1.54 & ! [Xy: $i] :
% 2.93/1.54 ( ( Xphi @ Xy )
% 2.93/1.54 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.93/1.54
% 2.93/1.54 thf(prop2set_type,type,
% 2.93/1.54 prop2set: $o > $i ).
% 2.93/1.54
% 2.93/1.54 thf(prop2setE_type,type,
% 2.93/1.54 prop2setE: $o ).
% 2.93/1.54
% 2.93/1.54 thf(prop2setE,definition,
% 2.93/1.54 ( prop2setE
% 2.93/1.54 = ( ! [Xphi: $o,Xx: $i] :
% 2.93/1.54 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 2.93/1.54 => Xphi ) ) ) ).
% 2.93/1.54
% 2.93/1.54 thf(emptysetE_type,type,
% 2.93/1.54 emptysetE: $o ).
% 2.93/1.54
% 2.93/1.54 thf(emptysetE,definition,
% 2.93/1.54 ( emptysetE
% 2.93/1.54 = ( ! [Xx: $i] :
% 2.93/1.54 ( ( in @ Xx @ emptyset )
% 2.93/1.54 => ! [Xphi: $o] : Xphi ) ) ) ).
% 2.93/1.54
% 2.93/1.54 thf(emptysetimpfalse_type,type,
% 2.93/1.54 emptysetimpfalse: $o ).
% 2.93/1.54
% 2.93/1.54 thf(emptysetimpfalse,definition,
% 2.93/1.54 ( emptysetimpfalse
% 2.93/1.54 = ( ! [Xx: $i] :
% 2.93/1.54 ( ( in @ Xx @ emptyset )
% 2.93/1.54 => $false ) ) ) ).
% 2.93/1.54
% 2.93/1.54 thf(notinemptyset_type,type,
% 2.93/1.54 notinemptyset: $o ).
% 2.93/1.54
% 2.93/1.54 thf(notinemptyset,definition,
% 2.93/1.54 ( notinemptyset
% 2.93/1.54 = ( ! [Xx: $i] :
% 2.93/1.54 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.93/1.54
% 2.93/1.54 thf(exuE3e_type,type,
% 2.93/1.54 exuE3e: $o ).
% 2.93/1.54
% 2.93/1.54 thf(exuE3e,definition,
% 2.93/1.54 ( exuE3e
% 2.93/1.54 = ( ! [Xphi: $i > $o] :
% 2.93/1.54 ( ( exu
% 2.93/1.54 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.93/1.54 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 2.93/1.54
% 2.93/1.54 thf(setext_type,type,
% 2.93/1.54 setext: $o ).
% 2.93/1.54
% 2.93/1.54 thf(setext,definition,
% 2.93/1.54 ( setext
% 2.93/1.54 = ( ! [A: $i,B: $i] :
% 2.93/1.54 ( ! [Xx: $i] :
% 2.93/1.54 ( ( in @ Xx @ A )
% 2.93/1.54 => ( in @ Xx @ B ) )
% 2.93/1.54 => ( ! [Xx: $i] :
% 2.93/1.54 ( ( in @ Xx @ B )
% 2.93/1.54 => ( in @ Xx @ A ) )
% 2.93/1.54 => ( A = B ) ) ) ) ) ).
% 2.93/1.54
% 2.93/1.54 thf(emptyI_type,type,
% 2.93/1.54 emptyI: $o ).
% 2.93/1.54
% 2.93/1.54 thf(emptyI,definition,
% 2.93/1.54 ( emptyI
% 2.93/1.54 = ( ! [A: $i] :
% 2.93/1.54 ( ! [Xx: $i] :
% 2.93/1.54 ~ ( in @ Xx @ A )
% 2.93/1.54 => ( A = emptyset ) ) ) ) ).
% 2.93/1.54
% 2.93/1.54 thf(noeltsimpempty_type,type,
% 2.93/1.54 noeltsimpempty: $o ).
% 2.93/1.54
% 2.93/1.54 thf(noeltsimpempty,definition,
% 2.93/1.54 ( noeltsimpempty
% 2.93/1.54 = ( ! [A: $i] :
% 2.93/1.54 ( ! [Xx: $i] :
% 2.93/1.54 ~ ( in @ Xx @ A )
% 2.93/1.54 => ( A = emptyset ) ) ) ) ).
% 2.93/1.54
% 2.93/1.54 thf(setbeta_type,type,
% 2.93/1.54 setbeta: $o ).
% 2.93/1.54
% 2.93/1.54 thf(setbeta,definition,
% 2.93/1.54 ( setbeta
% 2.93/1.54 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.93/1.54 ( ( in @ Xx @ A )
% 2.93/1.54 => ( ( in @ Xx
% 2.93/1.54 @ ( dsetconstr @ A
% 2.93/1.54 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.93/1.54 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 2.93/1.54
% 2.93/1.54 thf(nonempty_type,type,
% 2.93/1.54 nonempty: $i > $o ).
% 2.93/1.54
% 2.93/1.54 thf(nonempty,definition,
% 2.93/1.54 ( nonempty
% 2.93/1.54 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 2.93/1.54
% 2.93/1.54 thf(nonemptyE1_type,type,
% 2.93/1.54 nonemptyE1: $o ).
% 2.93/1.54
% 2.93/1.54 thf(nonemptyE1,definition,
% 2.93/1.54 ( nonemptyE1
% 2.93/1.54 = ( ! [A: $i] :
% 2.93/1.54 ( ( nonempty @ A )
% 2.93/1.54 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 2.93/1.54
% 2.93/1.54 thf(nonemptyI_type,type,
% 2.93/1.54 nonemptyI: $o ).
% 2.93/1.54
% 2.93/1.54 thf(nonemptyI,definition,
% 2.93/1.54 ( nonemptyI
% 2.93/1.54 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.93/1.54 ( ( in @ Xx @ A )
% 2.93/1.54 => ( ( Xphi @ Xx )
% 2.93/1.54 => ( nonempty
% 2.93/1.54 @ ( dsetconstr @ A
% 2.93/1.54 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.93/1.54
% 2.93/1.54 thf(nonemptyI1_type,type,
% 2.93/1.54 nonemptyI1: $o ).
% 2.93/1.54
% 2.93/1.54 thf(nonemptyI1,definition,
% 2.93/1.54 ( nonemptyI1
% 2.93/1.54 = ( ! [A: $i] :
% 2.93/1.54 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.93/1.54 => ( nonempty @ A ) ) ) ) ).
% 2.93/1.54
% 2.93/1.54 thf(setadjoinIL_type,type,
% 2.93/1.54 setadjoinIL: $o ).
% 2.93/1.54
% 2.93/1.54 thf(setadjoinIL,definition,
% 2.93/1.54 ( setadjoinIL
% 2.93/1.54 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 2.93/1.54
% 2.93/1.54 thf(emptyinunitempty_type,type,
% 2.93/1.54 emptyinunitempty: $o ).
% 2.93/1.54
% 2.93/1.54 thf(emptyinunitempty,definition,
% 2.93/1.54 ( emptyinunitempty
% 2.93/1.54 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 2.93/1.54
% 2.93/1.54 thf(setadjoinIR_type,type,
% 2.93/1.54 setadjoinIR: $o ).
% 2.93/1.54
% 2.93/1.54 thf(setadjoinIR,definition,
% 2.93/1.54 ( setadjoinIR
% 2.93/1.54 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.93/1.54 ( ( in @ Xy @ A )
% 2.93/1.54 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 2.93/1.54
% 2.93/1.54 thf(setadjoinE_type,type,
% 2.93/1.54 setadjoinE: $o ).
% 2.93/1.54
% 2.93/1.54 thf(setadjoinE,definition,
% 2.93/1.54 ( setadjoinE
% 2.93/1.54 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.93/1.54 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.93/1.54 => ! [Xphi: $o] :
% 2.93/1.54 ( ( ( Xy = Xx )
% 2.93/1.54 => Xphi )
% 2.93/1.54 => ( ( ( in @ Xy @ A )
% 2.93/1.54 => Xphi )
% 2.93/1.54 => Xphi ) ) ) ) ) ).
% 2.93/1.54
% 2.93/1.54 thf(setadjoinOr_type,type,
% 2.93/1.54 setadjoinOr: $o ).
% 2.93/1.54
% 2.93/1.54 thf(setadjoinOr,definition,
% 2.93/1.54 ( setadjoinOr
% 2.93/1.54 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.93/1.54 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.93/1.54 => ( ( Xy = Xx )
% 2.93/1.54 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.93/1.54
% 2.93/1.54 thf(setoftrueEq_type,type,
% 2.93/1.54 setoftrueEq: $o ).
% 2.93/1.54
% 2.93/1.54 thf(setoftrueEq,definition,
% 2.93/1.54 ( setoftrueEq
% 2.93/1.54 = ( ! [A: $i] :
% 2.93/1.54 ( ( dsetconstr @ A
% 2.93/1.54 @ ^ [Xx: $i] : $true )
% 2.93/1.54 = A ) ) ) ).
% 2.93/1.54
% 2.93/1.54 thf(powersetI_type,type,
% 2.93/1.54 powersetI: $o ).
% 2.93/1.54
% 2.93/1.54 thf(powersetI,definition,
% 2.93/1.54 ( powersetI
% 2.93/1.54 = ( ! [A: $i,B: $i] :
% 2.93/1.54 ( ! [Xx: $i] :
% 2.93/1.54 ( ( in @ Xx @ B )
% 2.93/1.54 => ( in @ Xx @ A ) )
% 2.93/1.54 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.93/1.54
% 2.93/1.54 thf(emptyinPowerset_type,type,
% 2.93/1.54 emptyinPowerset: $o ).
% 2.93/1.54
% 2.93/1.54 thf(emptyinPowerset,definition,
% 2.93/1.54 ( emptyinPowerset
% 2.93/1.54 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.93/1.54
% 2.93/1.54 thf(emptyInPowerset_type,type,
% 2.93/1.54 emptyInPowerset: $o ).
% 2.93/1.54
% 2.93/1.54 thf(emptyInPowerset,definition,
% 2.93/1.54 ( emptyInPowerset
% 2.93/1.54 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.93/1.54
% 2.93/1.54 thf(powersetE_type,type,
% 2.93/1.54 powersetE: $o ).
% 2.93/1.54
% 2.93/1.54 thf(powersetE,definition,
% 2.93/1.54 ( powersetE
% 2.93/1.54 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.93/1.54 ( ( in @ B @ ( powerset @ A ) )
% 2.93/1.54 => ( ( in @ Xx @ B )
% 2.93/1.54 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.93/1.54
% 2.93/1.54 thf(setunionI_type,type,
% 2.93/1.54 setunionI: $o ).
% 2.93/1.54
% 2.93/1.54 thf(setunionI,definition,
% 2.93/1.54 ( setunionI
% 2.93/1.54 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.93/1.54 ( ( in @ Xx @ B )
% 2.93/1.54 => ( ( in @ B @ A )
% 2.93/1.54 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 2.93/1.54
% 2.93/1.54 thf(setunionE_type,type,
% 2.93/1.54 setunionE: $o ).
% 2.93/1.54
% 2.93/1.54 thf(setunionE,definition,
% 2.93/1.54 ( setunionE
% 2.93/1.54 = ( ! [A: $i,Xx: $i] :
% 2.93/1.54 ( ( in @ Xx @ ( setunion @ A ) )
% 2.93/1.54 => ! [Xphi: $o] :
% 2.93/1.54 ( ! [B: $i] :
% 2.93/1.54 ( ( in @ Xx @ B )
% 2.93/1.54 => ( ( in @ B @ A )
% 2.93/1.54 => Xphi ) )
% 2.93/1.54 => Xphi ) ) ) ) ).
% 2.93/1.54
% 2.93/1.54 thf(subPowSU_type,type,
% 2.93/1.54 subPowSU: $o ).
% 2.93/1.54
% 2.93/1.54 thf(subPowSU,definition,
% 2.93/1.54 ( subPowSU
% 2.93/1.54 = ( ! [A: $i,Xx: $i] :
% 2.93/1.54 ( ( in @ Xx @ A )
% 2.93/1.54 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 2.93/1.54
% 2.93/1.54 thf(exuE2_type,type,
% 2.93/1.54 exuE2: $o ).
% 2.93/1.54
% 2.93/1.54 thf(exuE2,definition,
% 2.93/1.54 ( exuE2
% 2.93/1.54 = ( ! [Xphi: $i > $o] :
% 2.93/1.54 ( ( exu
% 2.93/1.54 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.93/1.54 => ? [Xx: $i] :
% 2.93/1.54 ! [Xy: $i] :
% 2.93/1.54 ( ( Xphi @ Xy )
% 2.93/1.54 <=> ( Xy = Xx ) ) ) ) ) ).
% 2.93/1.54
% 2.93/1.54 thf(nonemptyImpWitness_type,type,
% 2.93/1.54 nonemptyImpWitness: $o ).
% 2.93/1.54
% 2.93/1.54 thf(nonemptyImpWitness,definition,
% 2.93/1.54 ( nonemptyImpWitness
% 2.93/1.54 = ( ! [A: $i] :
% 2.93/1.54 ( ( nonempty @ A )
% 2.93/1.54 => ? [Xx: $i] :
% 2.93/1.54 ( ( in @ Xx @ A )
% 2.93/1.54 & $true ) ) ) ) ).
% 2.93/1.54
% 2.93/1.54 thf(uniqinunit_type,type,
% 2.93/1.54 uniqinunit: $o ).
% 2.93/1.54
% 2.93/1.54 thf(uniqinunit,definition,
% 2.93/1.54 ( uniqinunit
% 2.93/1.54 = ( ! [Xx: $i,Xy: $i] :
% 2.93/1.54 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.93/1.54 => ( Xx = Xy ) ) ) ) ).
% 2.93/1.54
% 2.93/1.54 thf(notinsingleton_type,type,
% 2.93/1.54 notinsingleton: $o ).
% 2.93/1.54
% 2.93/1.54 thf(notinsingleton,definition,
% 2.93/1.54 ( notinsingleton
% 2.93/1.54 = ( ! [Xx: $i,Xy: $i] :
% 2.93/1.54 ( ( Xx != Xy )
% 2.93/1.54 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.93/1.54
% 2.93/1.54 thf(eqinunit_type,type,
% 2.93/1.54 eqinunit: $o ).
% 2.93/1.54
% 2.93/1.54 thf(eqinunit,definition,
% 2.93/1.54 ( eqinunit
% 2.93/1.54 = ( ! [Xx: $i,Xy: $i] :
% 2.93/1.54 ( ( Xx = Xy )
% 2.93/1.54 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.93/1.54
% 2.93/1.54 thf(singletonsswitch_type,type,
% 2.93/1.54 singletonsswitch: $o ).
% 2.93/1.54
% 2.93/1.54 thf(singletonsswitch,definition,
% 2.93/1.54 ( singletonsswitch
% 2.93/1.54 = ( ! [Xx: $i,Xy: $i] :
% 2.93/1.54 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.93/1.54 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.93/1.54
% 2.93/1.54 thf(upairsetE_type,type,
% 2.93/1.54 upairsetE: $o ).
% 2.93/1.54
% 2.93/1.54 thf(upairsetE,definition,
% 2.93/1.54 ( upairsetE
% 2.93/1.54 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.93/1.54 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.93/1.54 => ( ( Xz = Xx )
% 2.93/1.54 | ( Xz = Xy ) ) ) ) ) ).
% 2.93/1.54
% 2.93/1.54 thf(upairsetIL_type,type,
% 2.93/1.54 upairsetIL: $o ).
% 2.93/1.54
% 2.93/1.54 thf(upairsetIL,definition,
% 2.93/1.54 ( upairsetIL
% 2.93/1.54 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.93/1.54
% 2.93/1.54 thf(upairsetIR_type,type,
% 2.93/1.54 upairsetIR: $o ).
% 2.93/1.54
% 2.93/1.54 thf(upairsetIR,definition,
% 2.93/1.54 ( upairsetIR
% 2.93/1.54 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.93/1.54
% 2.93/1.54 thf(emptyE1_type,type,
% 2.93/1.54 emptyE1: $o ).
% 2.93/1.54
% 2.93/1.54 thf(emptyE1,definition,
% 2.93/1.54 ( emptyE1
% 2.93/1.54 = ( ! [A: $i,Xphi: $i > $o] :
% 2.93/1.54 ( ? [Xx: $i] :
% 2.93/1.54 ( ( in @ Xx @ A )
% 2.93/1.54 & ( Xphi @ Xx ) )
% 2.93/1.54 => ( ( ( dsetconstr @ A
% 2.93/1.54 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.93/1.54 = emptyset )
% 2.93/1.54 => $false ) ) ) ) ).
% 2.93/1.54
% 2.93/1.54 thf(vacuousDall_type,type,
% 2.93/1.54 vacuousDall: $o ).
% 2.93/1.54
% 2.93/1.54 thf(vacuousDall,definition,
% 2.93/1.54 ( vacuousDall
% 2.93/1.54 = ( ! [Xphi: $i > $o,Xx: $i] :
% 2.93/1.54 ( ( in @ Xx @ emptyset )
% 2.93/1.54 => ( Xphi @ Xx ) ) ) ) ).
% 2.93/1.54
% 2.93/1.54 thf(quantDeMorgan1_type,type,
% 2.93/1.54 quantDeMorgan1: $o ).
% 2.93/1.54
% 2.93/1.54 thf(quantDeMorgan1,definition,
% 2.93/1.54 ( quantDeMorgan1
% 2.93/1.54 = ( ! [A: $i,Xphi: $i > $o] :
% 2.93/1.54 ( ~ ! [Xx: $i] :
% 2.93/1.54 ( ( in @ Xx @ A )
% 2.93/1.54 => ( Xphi @ Xx ) )
% 2.93/1.54 => ? [Xx: $i] :
% 2.93/1.54 ( ( in @ Xx @ A )
% 2.93/1.54 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.93/1.54
% 2.93/1.54 thf(quantDeMorgan2_type,type,
% 2.93/1.54 quantDeMorgan2: $o ).
% 2.93/1.54
% 2.93/1.54 thf(quantDeMorgan2,definition,
% 2.93/1.54 ( quantDeMorgan2
% 2.93/1.54 = ( ! [A: $i,Xphi: $i > $o] :
% 2.93/1.54 ( ! [Xx: $i] :
% 2.93/1.54 ( ( in @ Xx @ A )
% 2.93/1.54 => ~ ( Xphi @ Xx ) )
% 2.93/1.54 => ~ ? [Xx: $i] :
% 2.93/1.54 ( ( in @ Xx @ A )
% 2.93/1.54 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.93/1.54
% 2.93/1.54 thf(quantDeMorgan3_type,type,
% 2.93/1.54 quantDeMorgan3: $o ).
% 2.93/1.54
% 2.93/1.54 thf(quantDeMorgan3,definition,
% 2.93/1.54 ( quantDeMorgan3
% 2.93/1.54 = ( ! [A: $i,Xphi: $i > $o] :
% 2.93/1.54 ( ~ ? [Xx: $i] :
% 2.93/1.54 ( ( in @ Xx @ A )
% 2.93/1.54 & ( Xphi @ Xx ) )
% 2.93/1.54 => ! [Xx: $i] :
% 2.93/1.54 ( ( in @ Xx @ A )
% 2.93/1.54 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.93/1.54
% 2.93/1.54 thf(quantDeMorgan4_type,type,
% 2.93/1.54 quantDeMorgan4: $o ).
% 2.93/1.54
% 2.93/1.54 thf(quantDeMorgan4,definition,
% 2.93/1.54 ( quantDeMorgan4
% 2.93/1.54 = ( ! [A: $i,Xphi: $i > $o] :
% 2.93/1.54 ( ? [Xx: $i] :
% 2.93/1.54 ( ( in @ Xx @ A )
% 2.93/1.54 & ~ ( Xphi @ Xx ) )
% 2.93/1.54 => ~ ! [Xx: $i] :
% 2.93/1.54 ( ( in @ Xx @ A )
% 2.93/1.54 => ( Xphi @ Xx ) ) ) ) ) ).
% 2.93/1.54
% 2.93/1.54 thf(prop2setI_type,type,
% 2.93/1.54 prop2setI: $o ).
% 2.93/1.54
% 2.93/1.54 thf(prop2setI,definition,
% 2.93/1.54 ( prop2setI
% 2.93/1.54 = ( ! [Xphi: $o] :
% 2.93/1.54 ( Xphi
% 2.93/1.54 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.93/1.54
% 2.93/1.54 thf(set2prop_type,type,
% 2.93/1.54 set2prop: $i > $o ).
% 2.93/1.54
% 2.93/1.54 thf(prop2set2propI_type,type,
% 2.93/1.54 prop2set2propI: $o ).
% 2.93/1.54
% 2.93/1.54 thf(prop2set2propI,definition,
% 2.93/1.54 ( prop2set2propI
% 2.93/1.54 = ( ! [Xphi: $o] :
% 2.93/1.54 ( Xphi
% 2.93/1.54 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.93/1.54
% 2.93/1.54 thf(notdexE_type,type,
% 2.93/1.54 notdexE: $o ).
% 2.93/1.54
% 2.93/1.54 thf(notdexE,definition,
% 2.93/1.54 ( notdexE
% 2.93/1.54 = ( ! [A: $i,Xphi: $i > $o] :
% 2.93/1.54 ( ~ ? [Xx: $i] :
% 2.93/1.54 ( ( in @ Xx @ A )
% 2.93/1.54 & ( Xphi @ Xx ) )
% 2.93/1.54 => ! [Xx: $i] :
% 2.93/1.54 ( ( in @ Xx @ A )
% 2.93/1.54 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.93/1.54
% 2.93/1.54 thf(notdallE_type,type,
% 2.93/1.54 notdallE: $o ).
% 2.93/1.54
% 2.93/1.54 thf(notdallE,definition,
% 2.93/1.54 ( notdallE
% 2.93/1.54 = ( ! [A: $i,Xphi: $i > $o] :
% 2.93/1.54 ( ~ ! [Xx: $i] :
% 2.93/1.54 ( ( in @ Xx @ A )
% 2.93/1.54 => ( Xphi @ Xx ) )
% 2.93/1.54 => ? [Xx: $i] :
% 2.93/1.54 ( ( in @ Xx @ A )
% 2.93/1.54 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.93/1.54
% 2.93/1.54 thf(exuI1_type,type,
% 2.93/1.54 exuI1: $o ).
% 2.93/1.54
% 2.93/1.54 thf(exuI1,definition,
% 2.93/1.54 ( exuI1
% 2.93/1.54 = ( ! [Xphi: $i > $o] :
% 2.93/1.54 ( ? [Xx: $i] :
% 2.93/1.54 ( ( Xphi @ Xx )
% 2.93/1.54 & ! [Xy: $i] :
% 2.93/1.54 ( ( Xphi @ Xy )
% 2.93/1.54 => ( Xx = Xy ) ) )
% 2.93/1.54 => ( exu
% 2.93/1.54 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.93/1.54
% 2.93/1.54 thf(exuI3_type,type,
% 2.93/1.54 exuI3: $o ).
% 2.93/1.54
% 2.93/1.54 thf(exuI3,definition,
% 2.93/1.54 ( exuI3
% 2.93/1.54 = ( ! [Xphi: $i > $o] :
% 2.93/1.54 ( ? [Xx: $i] : ( Xphi @ Xx )
% 2.93/1.54 => ( ! [Xx: $i,Xy: $i] :
% 2.93/1.54 ( ( Xphi @ Xx )
% 2.93/1.54 => ( ( Xphi @ Xy )
% 2.93/1.54 => ( Xx = Xy ) ) )
% 2.93/1.54 => ( exu
% 2.93/1.54 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.93/1.54
% 2.93/1.54 thf(exuI2_type,type,
% 2.93/1.54 exuI2: $o ).
% 2.93/1.54
% 2.93/1.54 thf(exuI2,definition,
% 2.93/1.54 ( exuI2
% 2.93/1.54 = ( ! [Xphi: $i > $o] :
% 2.93/1.54 ( ? [Xx: $i] :
% 2.93/1.54 ! [Xy: $i] :
% 2.93/1.54 ( ( Xphi @ Xy )
% 2.93/1.54 <=> ( Xy = Xx ) )
% 2.93/1.54 => ( exu
% 2.93/1.54 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.93/1.54
% 2.93/1.54 thf(inCongP_type,type,
% 2.93/1.54 inCongP: $o ).
% 2.93/1.54
% 2.93/1.54 thf(inCongP,definition,
% 2.93/1.54 ( inCongP
% 2.93/1.54 = ( ! [A: $i,B: $i] :
% 2.93/1.54 ( ( A = B )
% 2.93/1.54 => ! [Xx: $i,Xy: $i] :
% 2.93/1.54 ( ( Xx = Xy )
% 2.93/1.54 => ( ( in @ Xx @ A )
% 2.93/1.54 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.93/1.54
% 2.93/1.54 thf(in__Cong_type,type,
% 2.93/1.54 in__Cong: $o ).
% 2.93/1.54
% 2.93/1.54 thf(in__Cong,definition,
% 2.93/1.54 ( in__Cong
% 2.93/1.54 = ( ! [A: $i,B: $i] :
% 2.93/1.54 ( ( A = B )
% 2.93/1.54 => ! [Xx: $i,Xy: $i] :
% 2.93/1.54 ( ( Xx = Xy )
% 2.93/1.54 => ( ( in @ Xx @ A )
% 2.93/1.54 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.93/1.54
% 2.93/1.54 thf(exuE3u_type,type,
% 2.93/1.54 exuE3u: $o ).
% 2.93/1.54
% 2.93/1.54 thf(exuE3u,definition,
% 2.93/1.54 ( exuE3u
% 2.93/1.54 = ( ! [Xphi: $i > $o] :
% 2.93/1.54 ( ( exu
% 2.93/1.54 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.93/1.54 => ! [Xx: $i,Xy: $i] :
% 2.93/1.54 ( ( Xphi @ Xx )
% 2.93/1.54 => ( ( Xphi @ Xy )
% 2.93/1.54 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.93/1.54
% 2.93/1.54 thf(exu__Cong_type,type,
% 2.93/1.54 exu__Cong: $o ).
% 2.93/1.54
% 2.93/1.54 thf(exu__Cong,definition,
% 2.93/1.54 ( exu__Cong
% 2.93/1.54 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.93/1.54 ( ! [Xx: $i,Xy: $i] :
% 2.93/1.54 ( ( Xx = Xy )
% 2.93/1.54 => ( ( Xphi @ Xx )
% 2.93/1.54 <=> ( Xpsi @ Xy ) ) )
% 2.93/1.54 => ( ( exu
% 2.93/1.54 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.93/1.54 <=> ( exu
% 2.93/1.54 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 2.93/1.54
% 2.93/1.54 thf(emptyset__Cong_type,type,
% 2.93/1.54 emptyset__Cong: $o ).
% 2.93/1.54
% 2.93/1.54 thf(emptyset__Cong,definition,
% 2.93/1.54 ( emptyset__Cong
% 2.93/1.54 = ( emptyset = emptyset ) ) ).
% 2.93/1.54
% 2.93/1.54 thf(setadjoin__Cong_type,type,
% 2.93/1.54 setadjoin__Cong: $o ).
% 2.93/1.54
% 2.93/1.54 thf(setadjoin__Cong,definition,
% 2.93/1.54 ( setadjoin__Cong
% 2.93/1.54 = ( ! [Xx: $i,Xy: $i] :
% 2.93/1.54 ( ( Xx = Xy )
% 2.93/1.54 => ! [Xz: $i,Xu: $i] :
% 2.93/1.54 ( ( Xz = Xu )
% 2.93/1.54 => ( ( setadjoin @ Xx @ Xz )
% 2.93/1.54 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 2.93/1.54
% 2.93/1.54 thf(powerset__Cong_type,type,
% 2.93/1.54 powerset__Cong: $o ).
% 2.93/1.54
% 2.93/1.54 thf(powerset__Cong,definition,
% 2.93/1.54 ( powerset__Cong
% 2.93/1.54 = ( ! [A: $i,B: $i] :
% 2.93/1.54 ( ( A = B )
% 2.93/1.54 => ( ( powerset @ A )
% 2.93/1.54 = ( powerset @ B ) ) ) ) ) ).
% 2.93/1.54
% 2.93/1.54 thf(setunion__Cong_type,type,
% 2.93/1.54 setunion__Cong: $o ).
% 2.93/1.54
% 2.93/1.54 thf(setunion__Cong,definition,
% 2.93/1.54 ( setunion__Cong
% 2.93/1.54 = ( ! [A: $i,B: $i] :
% 2.93/1.54 ( ( A = B )
% 2.93/1.54 => ( ( setunion @ A )
% 2.93/1.54 = ( setunion @ B ) ) ) ) ) ).
% 2.93/1.54
% 2.93/1.54 thf(omega__Cong_type,type,
% 2.93/1.54 omega__Cong: $o ).
% 2.93/1.54
% 2.93/1.54 thf(omega__Cong,definition,
% 2.93/1.54 ( omega__Cong
% 2.93/1.54 = ( omega = omega ) ) ).
% 2.93/1.54
% 2.93/1.54 thf(exuEu_type,type,
% 2.93/1.54 exuEu: $o ).
% 2.93/1.54
% 2.93/1.54 thf(exuEu,definition,
% 2.93/1.54 ( exuEu
% 2.93/1.54 = ( ! [Xphi: $i > $o] :
% 2.93/1.54 ( ( exu
% 2.93/1.54 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.93/1.54 => ! [Xx: $i,Xy: $i] :
% 2.93/1.54 ( ( Xphi @ Xx )
% 2.93/1.54 => ( ( Xphi @ Xy )
% 2.93/1.54 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.93/1.54
% 2.93/1.54 thf(descr__Cong_type,type,
% 2.93/1.54 descr__Cong: $o ).
% 2.93/1.54
% 2.93/1.54 thf(descr__Cong,definition,
% 2.93/1.54 ( descr__Cong
% 2.93/1.54 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.93/1.54 ( ! [Xx: $i,Xy: $i] :
% 2.93/1.54 ( ( Xx = Xy )
% 2.93/1.54 => ( ( Xphi @ Xx )
% 2.93/1.54 <=> ( Xpsi @ Xy ) ) )
% 2.93/1.54 => ( ( exu
% 2.93/1.54 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.93/1.54 => ( ( exu
% 2.93/1.54 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 2.93/1.54 => ( ( descr
% 2.93/1.54 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.93/1.54 = ( descr
% 2.93/1.54 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 2.93/1.54
% 2.93/1.54 thf(dsetconstr__Cong_type,type,
% 2.93/1.54 dsetconstr__Cong: $o ).
% 2.93/1.54
% 2.93/1.54 thf(dsetconstr__Cong,definition,
% 2.93/1.54 ( dsetconstr__Cong
% 2.93/1.54 = ( ! [A: $i,B: $i] :
% 2.93/1.54 ( ( A = B )
% 2.93/1.54 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.93/1.54 ( ! [Xx: $i] :
% 2.93/1.54 ( ( in @ Xx @ A )
% 2.93/1.54 => ! [Xy: $i] :
% 2.93/1.54 ( ( in @ Xy @ B )
% 2.93/1.54 => ( ( Xx = Xy )
% 2.93/1.54 => ( ( Xphi @ Xx )
% 2.93/1.54 <=> ( Xpsi @ Xy ) ) ) ) )
% 2.93/1.54 => ( ( dsetconstr @ A
% 2.93/1.54 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.93/1.54 = ( dsetconstr @ B
% 2.93/1.54 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 2.93/1.54
% 2.93/1.54 thf(subset_type,type,
% 2.93/1.54 subset: $i > $i > $o ).
% 2.93/1.54
% 2.93/1.54 thf(disjoint_type,type,
% 2.93/1.54 disjoint: $i > $i > $o ).
% 2.93/1.54
% 2.93/1.54 thf(setsmeet_type,type,
% 2.93/1.54 setsmeet: $i > $i > $o ).
% 2.93/1.54
% 2.93/1.54 thf(subsetI1_type,type,
% 2.93/1.54 subsetI1: $o ).
% 2.93/1.54
% 2.93/1.54 thf(subsetI1,definition,
% 2.93/1.54 ( subsetI1
% 2.93/1.54 = ( ! [A: $i,B: $i] :
% 2.93/1.54 ( ! [Xx: $i] :
% 2.93/1.54 ( ( in @ Xx @ A )
% 2.93/1.54 => ( in @ Xx @ B ) )
% 2.93/1.55 => ( subset @ A @ B ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(eqimpsubset2_type,type,
% 2.93/1.55 eqimpsubset2: $o ).
% 2.93/1.55
% 2.93/1.55 thf(eqimpsubset2,definition,
% 2.93/1.55 ( eqimpsubset2
% 2.93/1.55 = ( ! [A: $i,B: $i] :
% 2.93/1.55 ( ( A = B )
% 2.93/1.55 => ( subset @ B @ A ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(eqimpsubset1_type,type,
% 2.93/1.55 eqimpsubset1: $o ).
% 2.93/1.55
% 2.93/1.55 thf(eqimpsubset1,definition,
% 2.93/1.55 ( eqimpsubset1
% 2.93/1.55 = ( ! [A: $i,B: $i] :
% 2.93/1.55 ( ( A = B )
% 2.93/1.55 => ( subset @ A @ B ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(subsetI2_type,type,
% 2.93/1.55 subsetI2: $o ).
% 2.93/1.55
% 2.93/1.55 thf(subsetI2,definition,
% 2.93/1.55 ( subsetI2
% 2.93/1.55 = ( ! [A: $i,B: $i] :
% 2.93/1.55 ( ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ( in @ Xx @ B ) )
% 2.93/1.55 => ( subset @ A @ B ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(emptysetsubset_type,type,
% 2.93/1.55 emptysetsubset: $o ).
% 2.93/1.55
% 2.93/1.55 thf(emptysetsubset,definition,
% 2.93/1.55 ( emptysetsubset
% 2.93/1.55 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(subsetE_type,type,
% 2.93/1.55 subsetE: $o ).
% 2.93/1.55
% 2.93/1.55 thf(subsetE,definition,
% 2.93/1.55 ( subsetE
% 2.93/1.55 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.93/1.55 ( ( subset @ A @ B )
% 2.93/1.55 => ( ( in @ Xx @ A )
% 2.93/1.55 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(subsetE2_type,type,
% 2.93/1.55 subsetE2: $o ).
% 2.93/1.55
% 2.93/1.55 thf(subsetE2,definition,
% 2.93/1.55 ( subsetE2
% 2.93/1.55 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.93/1.55 ( ( subset @ A @ B )
% 2.93/1.55 => ( ~ ( in @ Xx @ B )
% 2.93/1.55 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(notsubsetI_type,type,
% 2.93/1.55 notsubsetI: $o ).
% 2.93/1.55
% 2.93/1.55 thf(notsubsetI,definition,
% 2.93/1.55 ( notsubsetI
% 2.93/1.55 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ( ~ ( in @ Xx @ B )
% 2.93/1.55 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(notequalI1_type,type,
% 2.93/1.55 notequalI1: $o ).
% 2.93/1.55
% 2.93/1.55 thf(notequalI1,definition,
% 2.93/1.55 ( notequalI1
% 2.93/1.55 = ( ! [A: $i,B: $i] :
% 2.93/1.55 ( ~ ( subset @ A @ B )
% 2.93/1.55 => ( A != B ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(notequalI2_type,type,
% 2.93/1.55 notequalI2: $o ).
% 2.93/1.55
% 2.93/1.55 thf(notequalI2,definition,
% 2.93/1.55 ( notequalI2
% 2.93/1.55 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ( ~ ( in @ Xx @ B )
% 2.93/1.55 => ( A != B ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(subsetRefl_type,type,
% 2.93/1.55 subsetRefl: $o ).
% 2.93/1.55
% 2.93/1.55 thf(subsetRefl,definition,
% 2.93/1.55 ( subsetRefl
% 2.93/1.55 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(subsetTrans_type,type,
% 2.93/1.55 subsetTrans: $o ).
% 2.93/1.55
% 2.93/1.55 thf(subsetTrans,definition,
% 2.93/1.55 ( subsetTrans
% 2.93/1.55 = ( ! [A: $i,B: $i,C: $i] :
% 2.93/1.55 ( ( subset @ A @ B )
% 2.93/1.55 => ( ( subset @ B @ C )
% 2.93/1.55 => ( subset @ A @ C ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(setadjoinSub_type,type,
% 2.93/1.55 setadjoinSub: $o ).
% 2.93/1.55
% 2.93/1.55 thf(setadjoinSub,definition,
% 2.93/1.55 ( setadjoinSub
% 2.93/1.55 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(setadjoinSub2_type,type,
% 2.93/1.55 setadjoinSub2: $o ).
% 2.93/1.55
% 2.93/1.55 thf(setadjoinSub2,definition,
% 2.93/1.55 ( setadjoinSub2
% 2.93/1.55 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.93/1.55 ( ( subset @ A @ B )
% 2.93/1.55 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(subset2powerset_type,type,
% 2.93/1.55 subset2powerset: $o ).
% 2.93/1.55
% 2.93/1.55 thf(subset2powerset,definition,
% 2.93/1.55 ( subset2powerset
% 2.93/1.55 = ( ! [A: $i,B: $i] :
% 2.93/1.55 ( ( subset @ A @ B )
% 2.93/1.55 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(setextsub_type,type,
% 2.93/1.55 setextsub: $o ).
% 2.93/1.55
% 2.93/1.55 thf(setextsub,definition,
% 2.93/1.55 ( setextsub
% 2.93/1.55 = ( ! [A: $i,B: $i] :
% 2.93/1.55 ( ( subset @ A @ B )
% 2.93/1.55 => ( ( subset @ B @ A )
% 2.93/1.55 => ( A = B ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(subsetemptysetimpeq_type,type,
% 2.93/1.55 subsetemptysetimpeq: $o ).
% 2.93/1.55
% 2.93/1.55 thf(subsetemptysetimpeq,definition,
% 2.93/1.55 ( subsetemptysetimpeq
% 2.93/1.55 = ( ! [A: $i] :
% 2.93/1.55 ( ( subset @ A @ emptyset )
% 2.93/1.55 => ( A = emptyset ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(powersetI1_type,type,
% 2.93/1.55 powersetI1: $o ).
% 2.93/1.55
% 2.93/1.55 thf(powersetI1,definition,
% 2.93/1.55 ( powersetI1
% 2.93/1.55 = ( ! [A: $i,B: $i] :
% 2.93/1.55 ( ( subset @ B @ A )
% 2.93/1.55 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(powersetE1_type,type,
% 2.93/1.55 powersetE1: $o ).
% 2.93/1.55
% 2.93/1.55 thf(powersetE1,definition,
% 2.93/1.55 ( powersetE1
% 2.93/1.55 = ( ! [A: $i,B: $i] :
% 2.93/1.55 ( ( in @ B @ ( powerset @ A ) )
% 2.93/1.55 => ( subset @ B @ A ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(inPowerset_type,type,
% 2.93/1.55 inPowerset: $o ).
% 2.93/1.55
% 2.93/1.55 thf(inPowerset,definition,
% 2.93/1.55 ( inPowerset
% 2.93/1.55 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(powersetsubset_type,type,
% 2.93/1.55 powersetsubset: $o ).
% 2.93/1.55
% 2.93/1.55 thf(powersetsubset,definition,
% 2.93/1.55 ( powersetsubset
% 2.93/1.55 = ( ! [A: $i,B: $i] :
% 2.93/1.55 ( ( subset @ A @ B )
% 2.93/1.55 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(sepInPowerset_type,type,
% 2.93/1.55 sepInPowerset: $o ).
% 2.93/1.55
% 2.93/1.55 thf(sepInPowerset,definition,
% 2.93/1.55 ( sepInPowerset
% 2.93/1.55 = ( ! [A: $i,Xphi: $i > $o] :
% 2.93/1.55 ( in
% 2.93/1.55 @ ( dsetconstr @ A
% 2.93/1.55 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.93/1.55 @ ( powerset @ A ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(sepSubset_type,type,
% 2.93/1.55 sepSubset: $o ).
% 2.93/1.55
% 2.93/1.55 thf(sepSubset,definition,
% 2.93/1.55 ( sepSubset
% 2.93/1.55 = ( ! [A: $i,Xphi: $i > $o] :
% 2.93/1.55 ( subset
% 2.93/1.55 @ ( dsetconstr @ A
% 2.93/1.55 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.93/1.55 @ A ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(binunion_type,type,
% 2.93/1.55 binunion: $i > $i > $i ).
% 2.93/1.55
% 2.93/1.55 thf(binunionIL_type,type,
% 2.93/1.55 binunionIL: $o ).
% 2.93/1.55
% 2.93/1.55 thf(binunionIL,definition,
% 2.93/1.55 ( binunionIL
% 2.93/1.55 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(upairset2IR_type,type,
% 2.93/1.55 upairset2IR: $o ).
% 2.93/1.55
% 2.93/1.55 thf(upairset2IR,definition,
% 2.93/1.55 ( upairset2IR
% 2.93/1.55 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(binunionIR_type,type,
% 2.93/1.55 binunionIR: $o ).
% 2.93/1.55
% 2.93/1.55 thf(binunionIR,definition,
% 2.93/1.55 ( binunionIR
% 2.93/1.55 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ B )
% 2.93/1.55 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(binunionEcases_type,type,
% 2.93/1.55 binunionEcases: $o ).
% 2.93/1.55
% 2.93/1.55 thf(binunionEcases,definition,
% 2.93/1.55 ( binunionEcases
% 2.93/1.55 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 2.93/1.55 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.93/1.55 => ( ( ( in @ Xx @ A )
% 2.93/1.55 => Xphi )
% 2.93/1.55 => ( ( ( in @ Xx @ B )
% 2.93/1.55 => Xphi )
% 2.93/1.55 => Xphi ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(binunionE_type,type,
% 2.93/1.55 binunionE: $o ).
% 2.93/1.55
% 2.93/1.55 thf(binunionE,definition,
% 2.93/1.55 ( binunionE
% 2.93/1.55 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.93/1.55 => ( ( in @ Xx @ A )
% 2.93/1.55 | ( in @ Xx @ B ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(binunionLsub_type,type,
% 2.93/1.55 binunionLsub: $o ).
% 2.93/1.55
% 2.93/1.55 thf(binunionLsub,definition,
% 2.93/1.55 ( binunionLsub
% 2.93/1.55 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(binunionRsub_type,type,
% 2.93/1.55 binunionRsub: $o ).
% 2.93/1.55
% 2.93/1.55 thf(binunionRsub,definition,
% 2.93/1.55 ( binunionRsub
% 2.93/1.55 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(binintersect_type,type,
% 2.93/1.55 binintersect: $i > $i > $i ).
% 2.93/1.55
% 2.93/1.55 thf(binintersectI_type,type,
% 2.93/1.55 binintersectI: $o ).
% 2.93/1.55
% 2.93/1.55 thf(binintersectI,definition,
% 2.93/1.55 ( binintersectI
% 2.93/1.55 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ( ( in @ Xx @ B )
% 2.93/1.55 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(binintersectSubset5_type,type,
% 2.93/1.55 binintersectSubset5: $o ).
% 2.93/1.55
% 2.93/1.55 thf(binintersectSubset5,definition,
% 2.93/1.55 ( binintersectSubset5
% 2.93/1.55 = ( ! [A: $i,B: $i,C: $i] :
% 2.93/1.55 ( ( subset @ C @ A )
% 2.93/1.55 => ( ( subset @ C @ B )
% 2.93/1.55 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(binintersectEL_type,type,
% 2.93/1.55 binintersectEL: $o ).
% 2.93/1.55
% 2.93/1.55 thf(binintersectEL,definition,
% 2.93/1.55 ( binintersectEL
% 2.93/1.55 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.93/1.55 => ( in @ Xx @ A ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(binintersectLsub_type,type,
% 2.93/1.55 binintersectLsub: $o ).
% 2.93/1.55
% 2.93/1.55 thf(binintersectLsub,definition,
% 2.93/1.55 ( binintersectLsub
% 2.93/1.55 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(binintersectSubset2_type,type,
% 2.93/1.55 binintersectSubset2: $o ).
% 2.93/1.55
% 2.93/1.55 thf(binintersectSubset2,definition,
% 2.93/1.55 ( binintersectSubset2
% 2.93/1.55 = ( ! [A: $i,B: $i] :
% 2.93/1.55 ( ( subset @ A @ B )
% 2.93/1.55 => ( ( binintersect @ A @ B )
% 2.93/1.55 = A ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(binintersectSubset3_type,type,
% 2.93/1.55 binintersectSubset3: $o ).
% 2.93/1.55
% 2.93/1.55 thf(binintersectSubset3,definition,
% 2.93/1.55 ( binintersectSubset3
% 2.93/1.55 = ( ! [A: $i,B: $i] :
% 2.93/1.55 ( ( ( binintersect @ A @ B )
% 2.93/1.55 = B )
% 2.93/1.55 => ( subset @ B @ A ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(binintersectER_type,type,
% 2.93/1.55 binintersectER: $o ).
% 2.93/1.55
% 2.93/1.55 thf(binintersectER,definition,
% 2.93/1.55 ( binintersectER
% 2.93/1.55 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.93/1.55 => ( in @ Xx @ B ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(disjointsetsI1_type,type,
% 2.93/1.55 disjointsetsI1: $o ).
% 2.93/1.55
% 2.93/1.55 thf(disjointsetsI1,definition,
% 2.93/1.55 ( disjointsetsI1
% 2.93/1.55 = ( ! [A: $i,B: $i] :
% 2.93/1.55 ( ~ ? [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 & ( in @ Xx @ B ) )
% 2.93/1.55 => ( ( binintersect @ A @ B )
% 2.93/1.55 = emptyset ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(binintersectRsub_type,type,
% 2.93/1.55 binintersectRsub: $o ).
% 2.93/1.55
% 2.93/1.55 thf(binintersectRsub,definition,
% 2.93/1.55 ( binintersectRsub
% 2.93/1.55 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(binintersectSubset4_type,type,
% 2.93/1.55 binintersectSubset4: $o ).
% 2.93/1.55
% 2.93/1.55 thf(binintersectSubset4,definition,
% 2.93/1.55 ( binintersectSubset4
% 2.93/1.55 = ( ! [A: $i,B: $i] :
% 2.93/1.55 ( ( subset @ B @ A )
% 2.93/1.55 => ( ( binintersect @ A @ B )
% 2.93/1.55 = B ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(binintersectSubset1_type,type,
% 2.93/1.55 binintersectSubset1: $o ).
% 2.93/1.55
% 2.93/1.55 thf(binintersectSubset1,definition,
% 2.93/1.55 ( binintersectSubset1
% 2.93/1.55 = ( ! [A: $i,B: $i] :
% 2.93/1.55 ( ( ( binintersect @ A @ B )
% 2.93/1.55 = A )
% 2.93/1.55 => ( subset @ A @ B ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(bs114d_type,type,
% 2.93/1.55 bs114d: $o ).
% 2.93/1.55
% 2.93/1.55 thf(bs114d,definition,
% 2.93/1.55 ( bs114d
% 2.93/1.55 = ( ! [A: $i,B: $i,C: $i] :
% 2.93/1.55 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 2.93/1.55 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(regular_type,type,
% 2.93/1.55 regular: $i > $o ).
% 2.93/1.55
% 2.93/1.55 thf(setminus_type,type,
% 2.93/1.55 setminus: $i > $i > $i ).
% 2.93/1.55
% 2.93/1.55 thf(setminusI_type,type,
% 2.93/1.55 setminusI: $o ).
% 2.93/1.55
% 2.93/1.55 thf(setminusI,definition,
% 2.93/1.55 ( setminusI
% 2.93/1.55 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ( ~ ( in @ Xx @ B )
% 2.93/1.55 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(setminusEL_type,type,
% 2.93/1.55 setminusEL: $o ).
% 2.93/1.55
% 2.93/1.55 thf(setminusEL,definition,
% 2.93/1.55 ( setminusEL
% 2.93/1.55 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.93/1.55 => ( in @ Xx @ A ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(setminusER_type,type,
% 2.93/1.55 setminusER: $o ).
% 2.93/1.55
% 2.93/1.55 thf(setminusER,definition,
% 2.93/1.55 ( setminusER
% 2.93/1.55 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.93/1.55 => ~ ( in @ Xx @ B ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(setminusSubset2_type,type,
% 2.93/1.55 setminusSubset2: $o ).
% 2.93/1.55
% 2.93/1.55 thf(setminusSubset2,definition,
% 2.93/1.55 ( setminusSubset2
% 2.93/1.55 = ( ! [A: $i,B: $i] :
% 2.93/1.55 ( ( subset @ A @ B )
% 2.93/1.55 => ( ( setminus @ A @ B )
% 2.93/1.55 = emptyset ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(setminusERneg_type,type,
% 2.93/1.55 setminusERneg: $o ).
% 2.93/1.55
% 2.93/1.55 thf(setminusERneg,definition,
% 2.93/1.55 ( setminusERneg
% 2.93/1.55 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.93/1.55 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.93/1.55 => ( ( in @ Xx @ A )
% 2.93/1.55 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(setminusELneg_type,type,
% 2.93/1.55 setminusELneg: $o ).
% 2.93/1.55
% 2.93/1.55 thf(setminusELneg,definition,
% 2.93/1.55 ( setminusELneg
% 2.93/1.55 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.93/1.55 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.93/1.55 => ( ~ ( in @ Xx @ B )
% 2.93/1.55 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(setminusILneg_type,type,
% 2.93/1.55 setminusILneg: $o ).
% 2.93/1.55
% 2.93/1.55 thf(setminusILneg,definition,
% 2.93/1.55 ( setminusILneg
% 2.93/1.55 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.93/1.55 ( ~ ( in @ Xx @ A )
% 2.93/1.55 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(setminusIRneg_type,type,
% 2.93/1.55 setminusIRneg: $o ).
% 2.93/1.55
% 2.93/1.55 thf(setminusIRneg,definition,
% 2.93/1.55 ( setminusIRneg
% 2.93/1.55 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ B )
% 2.93/1.55 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(setminusLsub_type,type,
% 2.93/1.55 setminusLsub: $o ).
% 2.93/1.55
% 2.93/1.55 thf(setminusLsub,definition,
% 2.93/1.55 ( setminusLsub
% 2.93/1.55 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(setminusSubset1_type,type,
% 2.93/1.55 setminusSubset1: $o ).
% 2.93/1.55
% 2.93/1.55 thf(setminusSubset1,definition,
% 2.93/1.55 ( setminusSubset1
% 2.93/1.55 = ( ! [A: $i,B: $i] :
% 2.93/1.55 ( ( ( setminus @ A @ B )
% 2.93/1.55 = emptyset )
% 2.93/1.55 => ( subset @ A @ B ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(symdiff_type,type,
% 2.93/1.55 symdiff: $i > $i > $i ).
% 2.93/1.55
% 2.93/1.55 thf(symdiffE_type,type,
% 2.93/1.55 symdiffE: $o ).
% 2.93/1.55
% 2.93/1.55 thf(symdiffE,definition,
% 2.93/1.55 ( symdiffE
% 2.93/1.55 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 2.93/1.55 => ! [Xphi: $o] :
% 2.93/1.55 ( ( ( in @ Xx @ A )
% 2.93/1.55 => ( ~ ( in @ Xx @ B )
% 2.93/1.55 => Xphi ) )
% 2.93/1.55 => ( ( ~ ( in @ Xx @ A )
% 2.93/1.55 => ( ( in @ Xx @ B )
% 2.93/1.55 => Xphi ) )
% 2.93/1.55 => Xphi ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(symdiffI1_type,type,
% 2.93/1.55 symdiffI1: $o ).
% 2.93/1.55
% 2.93/1.55 thf(symdiffI1,definition,
% 2.93/1.55 ( symdiffI1
% 2.93/1.55 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ( ~ ( in @ Xx @ B )
% 2.93/1.55 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(symdiffI2_type,type,
% 2.93/1.55 symdiffI2: $o ).
% 2.93/1.55
% 2.93/1.55 thf(symdiffI2,definition,
% 2.93/1.55 ( symdiffI2
% 2.93/1.55 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.93/1.55 ( ~ ( in @ Xx @ A )
% 2.93/1.55 => ( ( in @ Xx @ B )
% 2.93/1.55 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(symdiffIneg1_type,type,
% 2.93/1.55 symdiffIneg1: $o ).
% 2.93/1.55
% 2.93/1.55 thf(symdiffIneg1,definition,
% 2.93/1.55 ( symdiffIneg1
% 2.93/1.55 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ( ( in @ Xx @ B )
% 2.93/1.55 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(symdiffIneg2_type,type,
% 2.93/1.55 symdiffIneg2: $o ).
% 2.93/1.55
% 2.93/1.55 thf(symdiffIneg2,definition,
% 2.93/1.55 ( symdiffIneg2
% 2.93/1.55 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.93/1.55 ( ~ ( in @ Xx @ A )
% 2.93/1.55 => ( ~ ( in @ Xx @ B )
% 2.93/1.55 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(iskpair_type,type,
% 2.93/1.55 iskpair: $i > $o ).
% 2.93/1.55
% 2.93/1.55 thf(secondinupair_type,type,
% 2.93/1.55 secondinupair: $o ).
% 2.93/1.55
% 2.93/1.55 thf(secondinupair,definition,
% 2.93/1.55 ( secondinupair
% 2.93/1.55 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(setukpairIL_type,type,
% 2.93/1.55 setukpairIL: $o ).
% 2.93/1.55
% 2.93/1.55 thf(setukpairIL,definition,
% 2.93/1.55 ( setukpairIL
% 2.93/1.55 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(setukpairIR_type,type,
% 2.93/1.55 setukpairIR: $o ).
% 2.93/1.55
% 2.93/1.55 thf(setukpairIR,definition,
% 2.93/1.55 ( setukpairIR
% 2.93/1.55 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(kpairiskpair_type,type,
% 2.93/1.55 kpairiskpair: $o ).
% 2.93/1.55
% 2.93/1.55 thf(kpairiskpair,definition,
% 2.93/1.55 ( kpairiskpair
% 2.93/1.55 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(kpair_type,type,
% 2.93/1.55 kpair: $i > $i > $i ).
% 2.93/1.55
% 2.93/1.55 thf(kpairp_type,type,
% 2.93/1.55 kpairp: $o ).
% 2.93/1.55
% 2.93/1.55 thf(kpairp,definition,
% 2.93/1.55 ( kpairp
% 2.93/1.55 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(cartprod_type,type,
% 2.93/1.55 cartprod: $i > $i > $i ).
% 2.93/1.55
% 2.93/1.55 thf(singletonsubset_type,type,
% 2.93/1.55 singletonsubset: $o ).
% 2.93/1.55
% 2.93/1.55 thf(singletonsubset,definition,
% 2.93/1.55 ( singletonsubset
% 2.93/1.55 = ( ! [A: $i,Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(singletoninpowerset_type,type,
% 2.93/1.55 singletoninpowerset: $o ).
% 2.93/1.55
% 2.93/1.55 thf(singletoninpowerset,definition,
% 2.93/1.55 ( singletoninpowerset
% 2.93/1.55 = ( ! [A: $i,Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(singletoninpowunion_type,type,
% 2.93/1.55 singletoninpowunion: $o ).
% 2.93/1.55
% 2.93/1.55 thf(singletoninpowunion,definition,
% 2.93/1.55 ( singletoninpowunion
% 2.93/1.55 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(upairset2E_type,type,
% 2.93/1.55 upairset2E: $o ).
% 2.93/1.55
% 2.93/1.55 thf(upairset2E,definition,
% 2.93/1.55 ( upairset2E
% 2.93/1.55 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.93/1.55 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.93/1.55 => ( ( Xz = Xx )
% 2.93/1.55 | ( Xz = Xy ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(upairsubunion_type,type,
% 2.93/1.55 upairsubunion: $o ).
% 2.93/1.55
% 2.93/1.55 thf(upairsubunion,definition,
% 2.93/1.55 ( upairsubunion
% 2.93/1.55 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ! [Xy: $i] :
% 2.93/1.55 ( ( in @ Xy @ B )
% 2.93/1.55 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(upairinpowunion_type,type,
% 2.93/1.55 upairinpowunion: $o ).
% 2.93/1.55
% 2.93/1.55 thf(upairinpowunion,definition,
% 2.93/1.55 ( upairinpowunion
% 2.93/1.55 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ! [Xy: $i] :
% 2.93/1.55 ( ( in @ Xy @ B )
% 2.93/1.55 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(ubforcartprodlem1_type,type,
% 2.93/1.55 ubforcartprodlem1: $o ).
% 2.93/1.55
% 2.93/1.55 thf(ubforcartprodlem1,definition,
% 2.93/1.55 ( ubforcartprodlem1
% 2.93/1.55 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ! [Xy: $i] :
% 2.93/1.55 ( ( in @ Xy @ B )
% 2.93/1.55 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(ubforcartprodlem2_type,type,
% 2.93/1.55 ubforcartprodlem2: $o ).
% 2.93/1.55
% 2.93/1.55 thf(ubforcartprodlem2,definition,
% 2.93/1.55 ( ubforcartprodlem2
% 2.93/1.55 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ! [Xy: $i] :
% 2.93/1.55 ( ( in @ Xy @ B )
% 2.93/1.55 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(ubforcartprodlem3_type,type,
% 2.93/1.55 ubforcartprodlem3: $o ).
% 2.93/1.55
% 2.93/1.55 thf(ubforcartprodlem3,definition,
% 2.93/1.55 ( ubforcartprodlem3
% 2.93/1.55 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ! [Xy: $i] :
% 2.93/1.55 ( ( in @ Xy @ B )
% 2.93/1.55 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(cartprodpairin_type,type,
% 2.93/1.55 cartprodpairin: $o ).
% 2.93/1.55
% 2.93/1.55 thf(cartprodpairin,definition,
% 2.93/1.55 ( cartprodpairin
% 2.93/1.55 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ! [Xy: $i] :
% 2.93/1.55 ( ( in @ Xy @ B )
% 2.93/1.55 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(cartprodmempair1_type,type,
% 2.93/1.55 cartprodmempair1: $o ).
% 2.93/1.55
% 2.93/1.55 thf(cartprodmempair1,definition,
% 2.93/1.55 ( cartprodmempair1
% 2.93/1.55 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.93/1.55 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.93/1.55 => ? [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 & ? [Xy: $i] :
% 2.93/1.55 ( ( in @ Xy @ B )
% 2.93/1.55 & ( Xu
% 2.93/1.55 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(cartprodmempair_type,type,
% 2.93/1.55 cartprodmempair: $o ).
% 2.93/1.55
% 2.93/1.55 thf(cartprodmempair,definition,
% 2.93/1.55 ( cartprodmempair
% 2.93/1.55 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.93/1.55 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.93/1.55 => ( iskpair @ Xu ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(setunionE2_type,type,
% 2.93/1.55 setunionE2: $o ).
% 2.93/1.55
% 2.93/1.55 thf(setunionE2,definition,
% 2.93/1.55 ( setunionE2
% 2.93/1.55 = ( ! [A: $i,Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ ( setunion @ A ) )
% 2.93/1.55 => ? [X: $i] :
% 2.93/1.55 ( ( in @ X @ A )
% 2.93/1.55 & ( in @ Xx @ X ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(setunionsingleton1_type,type,
% 2.93/1.55 setunionsingleton1: $o ).
% 2.93/1.55
% 2.93/1.55 thf(setunionsingleton1,definition,
% 2.93/1.55 ( setunionsingleton1
% 2.93/1.55 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(setunionsingleton2_type,type,
% 2.93/1.55 setunionsingleton2: $o ).
% 2.93/1.55
% 2.93/1.55 thf(setunionsingleton2,definition,
% 2.93/1.55 ( setunionsingleton2
% 2.93/1.55 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(setunionsingleton_type,type,
% 2.93/1.55 setunionsingleton: $o ).
% 2.93/1.55
% 2.93/1.55 thf(setunionsingleton,definition,
% 2.93/1.55 ( setunionsingleton
% 2.93/1.55 = ( ! [Xx: $i] :
% 2.93/1.55 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 2.93/1.55 = Xx ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(singleton_type,type,
% 2.93/1.55 singleton: $i > $o ).
% 2.93/1.55
% 2.93/1.55 thf(singletonprop_type,type,
% 2.93/1.55 singletonprop: $o ).
% 2.93/1.55
% 2.93/1.55 thf(singletonprop,definition,
% 2.93/1.55 ( singletonprop
% 2.93/1.55 = ( ! [A: $i,Xphi: $i > $o] :
% 2.93/1.55 ( ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ! [Xy: $i] :
% 2.93/1.55 ( ( in @ Xy @ A )
% 2.93/1.55 => ( ( Xphi @ Xx )
% 2.93/1.55 => ( ( Xphi @ Xy )
% 2.93/1.55 => ( Xx = Xy ) ) ) ) )
% 2.93/1.55 => ( ? [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 & ( Xphi @ Xx ) )
% 2.93/1.55 => ( singleton
% 2.93/1.55 @ ( dsetconstr @ A
% 2.93/1.55 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(ex1_type,type,
% 2.93/1.55 ex1: $i > ( $i > $o ) > $o ).
% 2.93/1.55
% 2.93/1.55 thf(ex1E1_type,type,
% 2.93/1.55 ex1E1: $o ).
% 2.93/1.55
% 2.93/1.55 thf(ex1E1,definition,
% 2.93/1.55 ( ex1E1
% 2.93/1.55 = ( ! [A: $i,Xphi: $i > $o] :
% 2.93/1.55 ( ( ex1 @ A
% 2.93/1.55 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.93/1.55 => ? [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(ex1I_type,type,
% 2.93/1.55 ex1I: $o ).
% 2.93/1.55
% 2.93/1.55 thf(ex1I,definition,
% 2.93/1.55 ( ex1I
% 2.93/1.55 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ( ( Xphi @ Xx )
% 2.93/1.55 => ( ! [Xy: $i] :
% 2.93/1.55 ( ( in @ Xy @ A )
% 2.93/1.55 => ( ( Xphi @ Xy )
% 2.93/1.55 => ( Xy = Xx ) ) )
% 2.93/1.55 => ( ex1 @ A
% 2.93/1.55 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(ex1I2_type,type,
% 2.93/1.55 ex1I2: $o ).
% 2.93/1.55
% 2.93/1.55 thf(ex1I2,definition,
% 2.93/1.55 ( ex1I2
% 2.93/1.55 = ( ! [A: $i,Xphi: $i > $o] :
% 2.93/1.55 ( ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ! [Xy: $i] :
% 2.93/1.55 ( ( in @ Xy @ A )
% 2.93/1.55 => ( ( Xphi @ Xx )
% 2.93/1.55 => ( ( Xphi @ Xy )
% 2.93/1.55 => ( Xx = Xy ) ) ) ) )
% 2.93/1.55 => ( ? [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 & ( Xphi @ Xx ) )
% 2.93/1.55 => ( ex1 @ A
% 2.93/1.55 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(singletonsuniq_type,type,
% 2.93/1.55 singletonsuniq: $o ).
% 2.93/1.55
% 2.93/1.55 thf(singletonsuniq,definition,
% 2.93/1.55 ( singletonsuniq
% 2.93/1.55 = ( ! [Xx: $i,Xy: $i] :
% 2.93/1.55 ( ( ( setadjoin @ Xx @ emptyset )
% 2.93/1.55 = ( setadjoin @ Xy @ emptyset ) )
% 2.93/1.55 => ( Xx = Xy ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(atmost1p_type,type,
% 2.93/1.55 atmost1p: $i > $o ).
% 2.93/1.55
% 2.93/1.55 thf(atleast2p_type,type,
% 2.93/1.55 atleast2p: $i > $o ).
% 2.93/1.55
% 2.93/1.55 thf(atmost2p_type,type,
% 2.93/1.55 atmost2p: $i > $o ).
% 2.93/1.55
% 2.93/1.55 thf(upairsetp_type,type,
% 2.93/1.55 upairsetp: $i > $o ).
% 2.93/1.55
% 2.93/1.55 thf(setukpairinjL1_type,type,
% 2.93/1.55 setukpairinjL1: $o ).
% 2.93/1.55
% 2.93/1.55 thf(setukpairinjL1,definition,
% 2.93/1.55 ( setukpairinjL1
% 2.93/1.55 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.93/1.55 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 2.93/1.55 => ( Xx = Xz ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(kfstsingleton_type,type,
% 2.93/1.55 kfstsingleton: $o ).
% 2.93/1.55
% 2.93/1.55 thf(kfstsingleton,definition,
% 2.93/1.55 ( kfstsingleton
% 2.93/1.55 = ( ! [Xu: $i] :
% 2.93/1.55 ( ( iskpair @ Xu )
% 2.93/1.55 => ( singleton
% 2.93/1.55 @ ( dsetconstr @ ( setunion @ Xu )
% 2.93/1.55 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(theprop_type,type,
% 2.93/1.55 theprop: $o ).
% 2.93/1.55
% 2.93/1.55 thf(theprop,definition,
% 2.93/1.55 ( theprop
% 2.93/1.55 = ( ! [X: $i] :
% 2.93/1.55 ( ( singleton @ X )
% 2.93/1.55 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(kfst_type,type,
% 2.93/1.55 kfst: $i > $i ).
% 2.93/1.55
% 2.93/1.55 thf(kfstpairEq_type,type,
% 2.93/1.55 kfstpairEq: $o ).
% 2.93/1.55
% 2.93/1.55 thf(kfstpairEq,definition,
% 2.93/1.55 ( kfstpairEq
% 2.93/1.55 = ( ! [Xx: $i,Xy: $i] :
% 2.93/1.55 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 2.93/1.55 = Xx ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(cartprodfstin_type,type,
% 2.93/1.55 cartprodfstin: $o ).
% 2.93/1.55
% 2.93/1.55 thf(cartprodfstin,definition,
% 2.93/1.55 ( cartprodfstin
% 2.93/1.55 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.93/1.55 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.93/1.55 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(setukpairinjL2_type,type,
% 2.93/1.55 setukpairinjL2: $o ).
% 2.93/1.55
% 2.93/1.55 thf(setukpairinjL2,definition,
% 2.93/1.55 ( setukpairinjL2
% 2.93/1.55 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.93/1.55 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.93/1.55 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.93/1.55 => ( Xx = Xz ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(setukpairinjL_type,type,
% 2.93/1.55 setukpairinjL: $o ).
% 2.93/1.55
% 2.93/1.55 thf(setukpairinjL,definition,
% 2.93/1.55 ( setukpairinjL
% 2.93/1.55 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.93/1.55 ( ( ( kpair @ Xx @ Xy )
% 2.93/1.55 = ( kpair @ Xz @ Xu ) )
% 2.93/1.55 => ( Xx = Xz ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(setukpairinjR11_type,type,
% 2.93/1.55 setukpairinjR11: $o ).
% 2.93/1.55
% 2.93/1.55 thf(setukpairinjR11,definition,
% 2.93/1.55 ( setukpairinjR11
% 2.93/1.55 = ( ! [Xx: $i,Xy: $i] :
% 2.93/1.55 ( ( Xx = Xy )
% 2.93/1.55 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.93/1.55 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(setukpairinjR12_type,type,
% 2.93/1.55 setukpairinjR12: $o ).
% 2.93/1.55
% 2.93/1.55 thf(setukpairinjR12,definition,
% 2.93/1.55 ( setukpairinjR12
% 2.93/1.55 = ( ! [Xx: $i,Xy: $i] :
% 2.93/1.55 ( ( Xx = Xy )
% 2.93/1.55 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.93/1.55 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(setukpairinjR1_type,type,
% 2.93/1.55 setukpairinjR1: $o ).
% 2.93/1.55
% 2.93/1.55 thf(setukpairinjR1,definition,
% 2.93/1.55 ( setukpairinjR1
% 2.93/1.55 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.93/1.55 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.93/1.55 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.93/1.55 => ( ( Xz = Xu )
% 2.93/1.55 => ( Xy = Xu ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(upairequniteq_type,type,
% 2.93/1.55 upairequniteq: $o ).
% 2.93/1.55
% 2.93/1.55 thf(upairequniteq,definition,
% 2.93/1.55 ( upairequniteq
% 2.93/1.55 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.93/1.55 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.93/1.55 = ( setadjoin @ Xz @ emptyset ) )
% 2.93/1.55 => ( Xx = Xy ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(setukpairinjR2_type,type,
% 2.93/1.55 setukpairinjR2: $o ).
% 2.93/1.55
% 2.93/1.55 thf(setukpairinjR2,definition,
% 2.93/1.55 ( setukpairinjR2
% 2.93/1.55 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.93/1.55 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.93/1.55 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.93/1.55 => ( Xy = Xu ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(setukpairinjR_type,type,
% 2.93/1.55 setukpairinjR: $o ).
% 2.93/1.55
% 2.93/1.55 thf(setukpairinjR,definition,
% 2.93/1.55 ( setukpairinjR
% 2.93/1.55 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.93/1.55 ( ( ( kpair @ Xx @ Xy )
% 2.93/1.55 = ( kpair @ Xz @ Xu ) )
% 2.93/1.55 => ( Xy = Xu ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(ksndsingleton_type,type,
% 2.93/1.55 ksndsingleton: $o ).
% 2.93/1.55
% 2.93/1.55 thf(ksndsingleton,definition,
% 2.93/1.55 ( ksndsingleton
% 2.93/1.55 = ( ! [Xu: $i] :
% 2.93/1.55 ( ( iskpair @ Xu )
% 2.93/1.55 => ( singleton
% 2.93/1.55 @ ( dsetconstr @ ( setunion @ Xu )
% 2.93/1.55 @ ^ [Xx: $i] :
% 2.93/1.55 ( Xu
% 2.93/1.55 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(ksnd_type,type,
% 2.93/1.55 ksnd: $i > $i ).
% 2.93/1.55
% 2.93/1.55 thf(ksndpairEq_type,type,
% 2.93/1.55 ksndpairEq: $o ).
% 2.93/1.55
% 2.93/1.55 thf(ksndpairEq,definition,
% 2.93/1.55 ( ksndpairEq
% 2.93/1.55 = ( ! [Xx: $i,Xy: $i] :
% 2.93/1.55 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 2.93/1.55 = Xy ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(kpairsurjEq_type,type,
% 2.93/1.55 kpairsurjEq: $o ).
% 2.93/1.55
% 2.93/1.55 thf(kpairsurjEq,definition,
% 2.93/1.55 ( kpairsurjEq
% 2.93/1.55 = ( ! [Xu: $i] :
% 2.93/1.55 ( ( iskpair @ Xu )
% 2.93/1.55 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 2.93/1.55 = Xu ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(cartprodsndin_type,type,
% 2.93/1.55 cartprodsndin: $o ).
% 2.93/1.55
% 2.93/1.55 thf(cartprodsndin,definition,
% 2.93/1.55 ( cartprodsndin
% 2.93/1.55 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.93/1.55 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.93/1.55 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(cartprodpairmemEL_type,type,
% 2.93/1.55 cartprodpairmemEL: $o ).
% 2.93/1.55
% 2.93/1.55 thf(cartprodpairmemEL,definition,
% 2.93/1.55 ( cartprodpairmemEL
% 2.93/1.55 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 2.93/1.55 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 2.93/1.55 => ( in @ Xx @ A ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(cartprodpairmemER_type,type,
% 2.93/1.55 cartprodpairmemER: $o ).
% 2.93/1.55
% 2.93/1.55 thf(cartprodpairmemER,definition,
% 2.93/1.55 ( cartprodpairmemER
% 2.93/1.55 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 2.93/1.55 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 2.93/1.55 => ( in @ Xy @ B ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(cartprodmempaircEq_type,type,
% 2.93/1.55 cartprodmempaircEq: $o ).
% 2.93/1.55
% 2.93/1.55 thf(cartprodmempaircEq,definition,
% 2.93/1.55 ( cartprodmempaircEq
% 2.93/1.55 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ! [Xy: $i] :
% 2.93/1.55 ( ( in @ Xy @ B )
% 2.93/1.55 => ( ( kpair @ Xx @ Xy )
% 2.93/1.55 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(cartprodfstpairEq_type,type,
% 2.93/1.55 cartprodfstpairEq: $o ).
% 2.93/1.55
% 2.93/1.55 thf(cartprodfstpairEq,definition,
% 2.93/1.55 ( cartprodfstpairEq
% 2.93/1.55 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ! [Xy: $i] :
% 2.93/1.55 ( ( in @ Xy @ B )
% 2.93/1.55 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 2.93/1.55 = Xx ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(cartprodsndpairEq_type,type,
% 2.93/1.55 cartprodsndpairEq: $o ).
% 2.93/1.55
% 2.93/1.55 thf(cartprodsndpairEq,definition,
% 2.93/1.55 ( cartprodsndpairEq
% 2.93/1.55 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ! [Xy: $i] :
% 2.93/1.55 ( ( in @ Xy @ B )
% 2.93/1.55 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 2.93/1.55 = Xy ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(cartprodpairsurjEq_type,type,
% 2.93/1.55 cartprodpairsurjEq: $o ).
% 2.93/1.55
% 2.93/1.55 thf(cartprodpairsurjEq,definition,
% 2.93/1.55 ( cartprodpairsurjEq
% 2.93/1.55 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.93/1.55 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.93/1.55 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 2.93/1.55 = Xu ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(breln_type,type,
% 2.93/1.55 breln: $i > $i > $i > $o ).
% 2.93/1.55
% 2.93/1.55 thf(dpsetconstr_type,type,
% 2.93/1.55 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 2.93/1.55
% 2.93/1.55 thf(dpsetconstrI_type,type,
% 2.93/1.55 dpsetconstrI: $o ).
% 2.93/1.55
% 2.93/1.55 thf(dpsetconstrI,definition,
% 2.93/1.55 ( dpsetconstrI
% 2.93/1.55 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ! [Xy: $i] :
% 2.93/1.55 ( ( in @ Xy @ B )
% 2.93/1.55 => ( ( Xphi @ Xx @ Xy )
% 2.93/1.55 => ( in @ ( kpair @ Xx @ Xy )
% 2.93/1.55 @ ( dpsetconstr @ A @ B
% 2.93/1.55 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(dpsetconstrSub_type,type,
% 2.93/1.55 dpsetconstrSub: $o ).
% 2.93/1.55
% 2.93/1.55 thf(dpsetconstrSub,definition,
% 2.93/1.55 ( dpsetconstrSub
% 2.93/1.55 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 2.93/1.55 ( subset
% 2.93/1.55 @ ( dpsetconstr @ A @ B
% 2.93/1.55 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 2.93/1.55 @ ( cartprod @ A @ B ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(setOfPairsIsBReln_type,type,
% 2.93/1.55 setOfPairsIsBReln: $o ).
% 2.93/1.55
% 2.93/1.55 thf(setOfPairsIsBReln,definition,
% 2.93/1.55 ( setOfPairsIsBReln
% 2.93/1.55 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 2.93/1.55 ( breln @ A @ B
% 2.93/1.55 @ ( dpsetconstr @ A @ B
% 2.93/1.55 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(dpsetconstrERa_type,type,
% 2.93/1.55 dpsetconstrERa: $o ).
% 2.93/1.55
% 2.93/1.55 thf(dpsetconstrERa,definition,
% 2.93/1.55 ( dpsetconstrERa
% 2.93/1.55 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ! [Xy: $i] :
% 2.93/1.55 ( ( in @ Xy @ B )
% 2.93/1.55 => ( ( in @ ( kpair @ Xx @ Xy )
% 2.93/1.55 @ ( dpsetconstr @ A @ B
% 2.93/1.55 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.93/1.55 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(dpsetconstrEL1_type,type,
% 2.93/1.55 dpsetconstrEL1: $o ).
% 2.93/1.55
% 2.93/1.55 thf(dpsetconstrEL1,definition,
% 2.93/1.55 ( dpsetconstrEL1
% 2.93/1.55 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.93/1.55 ( ( in @ ( kpair @ Xx @ Xy )
% 2.93/1.55 @ ( dpsetconstr @ A @ B
% 2.93/1.55 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.93/1.55 => ( in @ Xx @ A ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(dpsetconstrEL2_type,type,
% 2.93/1.55 dpsetconstrEL2: $o ).
% 2.93/1.55
% 2.93/1.55 thf(dpsetconstrEL2,definition,
% 2.93/1.55 ( dpsetconstrEL2
% 2.93/1.55 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.93/1.55 ( ( in @ ( kpair @ Xx @ Xy )
% 2.93/1.55 @ ( dpsetconstr @ A @ B
% 2.93/1.55 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.93/1.55 => ( in @ Xy @ B ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(dpsetconstrER_type,type,
% 2.93/1.55 dpsetconstrER: $o ).
% 2.93/1.55
% 2.93/1.55 thf(dpsetconstrER,definition,
% 2.93/1.55 ( dpsetconstrER
% 2.93/1.55 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.93/1.55 ( ( in @ ( kpair @ Xx @ Xy )
% 2.93/1.55 @ ( dpsetconstr @ A @ B
% 2.93/1.55 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.93/1.55 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(func_type,type,
% 2.93/1.55 func: $i > $i > $i > $o ).
% 2.93/1.55
% 2.93/1.55 thf(funcSet_type,type,
% 2.93/1.55 funcSet: $i > $i > $i ).
% 2.93/1.55
% 2.93/1.55 thf(funcImageSingleton_type,type,
% 2.93/1.55 funcImageSingleton: $o ).
% 2.93/1.55
% 2.93/1.55 thf(funcImageSingleton,definition,
% 2.93/1.55 ( funcImageSingleton
% 2.93/1.55 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.93/1.55 ( ( func @ A @ B @ Xf )
% 2.93/1.55 => ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ( singleton
% 2.93/1.55 @ ( dsetconstr @ B
% 2.93/1.55 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(apProp_type,type,
% 2.93/1.55 apProp: $o ).
% 2.93/1.55
% 2.93/1.55 thf(apProp,definition,
% 2.93/1.55 ( apProp
% 2.93/1.55 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.93/1.55 ( ( func @ A @ B @ Xf )
% 2.93/1.55 => ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ( in
% 2.93/1.55 @ ( setunion
% 2.93/1.55 @ ( dsetconstr @ B
% 2.93/1.55 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 2.93/1.55 @ B ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(ap_type,type,
% 2.93/1.55 ap: $i > $i > $i > $i > $i ).
% 2.93/1.55
% 2.93/1.55 thf(app_type,type,
% 2.93/1.55 app: $o ).
% 2.93/1.55
% 2.93/1.55 thf(app,definition,
% 2.93/1.55 ( app
% 2.93/1.55 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.93/1.55 ( ( func @ A @ B @ Xf )
% 2.93/1.55 => ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(infuncsetfunc_type,type,
% 2.93/1.55 infuncsetfunc: $o ).
% 2.93/1.55
% 2.93/1.55 thf(infuncsetfunc,definition,
% 2.93/1.55 ( infuncsetfunc
% 2.93/1.55 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.93/1.55 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.93/1.55 => ( func @ A @ B @ Xf ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(ap2p_type,type,
% 2.93/1.55 ap2p: $o ).
% 2.93/1.55
% 2.93/1.55 thf(ap2p,definition,
% 2.93/1.55 ( ap2p
% 2.93/1.55 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.93/1.55 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.93/1.55 => ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(funcinfuncset_type,type,
% 2.93/1.55 funcinfuncset: $o ).
% 2.93/1.55
% 2.93/1.55 thf(funcinfuncset,definition,
% 2.93/1.55 ( funcinfuncset
% 2.93/1.55 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.93/1.55 ( ( func @ A @ B @ Xf )
% 2.93/1.55 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(lamProp_type,type,
% 2.93/1.55 lamProp: $o ).
% 2.93/1.55
% 2.93/1.55 thf(lamProp,definition,
% 2.93/1.55 ( lamProp
% 2.93/1.55 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.93/1.55 ( ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.93/1.55 => ( func @ A @ B
% 2.93/1.55 @ ( dpsetconstr @ A @ B
% 2.93/1.55 @ ^ [Xx: $i,Xy: $i] :
% 2.93/1.55 ( ( Xf @ Xx )
% 2.93/1.55 = Xy ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(lam_type,type,
% 2.93/1.55 lam: $i > $i > ( $i > $i ) > $i ).
% 2.93/1.55
% 2.93/1.55 thf(lamp_type,type,
% 2.93/1.55 lamp: $o ).
% 2.93/1.55
% 2.93/1.55 thf(lamp,definition,
% 2.93/1.55 ( lamp
% 2.93/1.55 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.93/1.55 ( ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.93/1.55 => ( func @ A @ B
% 2.93/1.55 @ ( lam @ A @ B
% 2.93/1.55 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(lam2p_type,type,
% 2.93/1.55 lam2p: $o ).
% 2.93/1.55
% 2.93/1.55 thf(lam2p,definition,
% 2.93/1.55 ( lam2p
% 2.93/1.55 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.93/1.55 ( ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.93/1.55 => ( in
% 2.93/1.55 @ ( lam @ A @ B
% 2.93/1.55 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 2.93/1.55 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(brelnall1_type,type,
% 2.93/1.55 brelnall1: $o ).
% 2.93/1.55
% 2.93/1.55 thf(brelnall1,definition,
% 2.93/1.55 ( brelnall1
% 2.93/1.55 = ( ! [A: $i,B: $i,R: $i] :
% 2.93/1.55 ( ( breln @ A @ B @ R )
% 2.93/1.55 => ! [Xphi: $i > $o] :
% 2.93/1.55 ( ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ! [Xy: $i] :
% 2.93/1.55 ( ( in @ Xy @ B )
% 2.93/1.55 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.93/1.55 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 2.93/1.55 => ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ R )
% 2.93/1.55 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(brelnall2_type,type,
% 2.93/1.55 brelnall2: $o ).
% 2.93/1.55
% 2.93/1.55 thf(brelnall2,definition,
% 2.93/1.55 ( brelnall2
% 2.93/1.55 = ( ! [A: $i,B: $i,R: $i] :
% 2.93/1.55 ( ( breln @ A @ B @ R )
% 2.93/1.55 => ! [Xphi: $i > $o] :
% 2.93/1.55 ( ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ! [Xy: $i] :
% 2.93/1.55 ( ( in @ Xy @ B )
% 2.93/1.55 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.93/1.55 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 2.93/1.55 => ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ R )
% 2.93/1.55 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(ex1E2_type,type,
% 2.93/1.55 ex1E2: $o ).
% 2.93/1.55
% 2.93/1.55 thf(ex1E2,definition,
% 2.93/1.55 ( ex1E2
% 2.93/1.55 = ( ! [A: $i,Xphi: $i > $o] :
% 2.93/1.55 ( ( ex1 @ A
% 2.93/1.55 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.93/1.55 => ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ! [Xy: $i] :
% 2.93/1.55 ( ( in @ Xy @ A )
% 2.93/1.55 => ( ( Xphi @ Xx )
% 2.93/1.55 => ( ( Xphi @ Xy )
% 2.93/1.55 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(funcGraphProp1_type,type,
% 2.93/1.55 funcGraphProp1: $o ).
% 2.93/1.55
% 2.93/1.55 thf(funcGraphProp1,definition,
% 2.93/1.55 ( funcGraphProp1
% 2.93/1.55 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.93/1.55 ( ( func @ A @ B @ Xf )
% 2.93/1.55 => ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(funcGraphProp3_type,type,
% 2.93/1.55 funcGraphProp3: $o ).
% 2.93/1.55
% 2.93/1.55 thf(funcGraphProp3,definition,
% 2.93/1.55 ( funcGraphProp3
% 2.93/1.55 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.93/1.55 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.93/1.55 => ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(funcGraphProp2_type,type,
% 2.93/1.55 funcGraphProp2: $o ).
% 2.93/1.55
% 2.93/1.55 thf(funcGraphProp2,definition,
% 2.93/1.55 ( funcGraphProp2
% 2.93/1.55 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.93/1.55 ( ( func @ A @ B @ Xf )
% 2.93/1.55 => ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ! [Xy: $i] :
% 2.93/1.55 ( ( in @ Xy @ B )
% 2.93/1.55 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 2.93/1.55 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.93/1.55 = Xy ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(funcextLem_type,type,
% 2.93/1.55 funcextLem: $o ).
% 2.93/1.55
% 2.93/1.55 thf(funcextLem,definition,
% 2.93/1.55 ( funcextLem
% 2.93/1.55 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.93/1.55 ( ( func @ A @ B @ Xf )
% 2.93/1.55 => ! [Xg: $i] :
% 2.93/1.55 ( ( func @ A @ B @ Xg )
% 2.93/1.55 => ( ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.93/1.55 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.93/1.55 => ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ! [Xy: $i] :
% 2.93/1.55 ( ( in @ Xy @ B )
% 2.93/1.55 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 2.93/1.55 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(funcGraphProp4_type,type,
% 2.93/1.55 funcGraphProp4: $o ).
% 2.93/1.55
% 2.93/1.55 thf(funcGraphProp4,definition,
% 2.93/1.55 ( funcGraphProp4
% 2.93/1.55 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.93/1.55 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.93/1.55 => ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ! [Xy: $i] :
% 2.93/1.55 ( ( in @ Xy @ B )
% 2.93/1.55 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 2.93/1.55 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.93/1.55 = Xy ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(subbreln_type,type,
% 2.93/1.55 subbreln: $o ).
% 2.93/1.55
% 2.93/1.55 thf(subbreln,definition,
% 2.93/1.55 ( subbreln
% 2.93/1.55 = ( ! [A: $i,B: $i,R: $i] :
% 2.93/1.55 ( ( breln @ A @ B @ R )
% 2.93/1.55 => ! [S: $i] :
% 2.93/1.55 ( ( breln @ A @ B @ S )
% 2.93/1.55 => ( ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ! [Xy: $i] :
% 2.93/1.55 ( ( in @ Xy @ B )
% 2.93/1.55 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.93/1.55 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 2.93/1.55 => ( subset @ R @ S ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(eqbreln_type,type,
% 2.93/1.55 eqbreln: $o ).
% 2.93/1.55
% 2.93/1.55 thf(eqbreln,definition,
% 2.93/1.55 ( eqbreln
% 2.93/1.55 = ( ! [A: $i,B: $i,R: $i] :
% 2.93/1.55 ( ( breln @ A @ B @ R )
% 2.93/1.55 => ! [S: $i] :
% 2.93/1.55 ( ( breln @ A @ B @ S )
% 2.93/1.55 => ( ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ! [Xy: $i] :
% 2.93/1.55 ( ( in @ Xy @ B )
% 2.93/1.55 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.93/1.55 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 2.93/1.55 => ( ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ! [Xy: $i] :
% 2.93/1.55 ( ( in @ Xy @ B )
% 2.93/1.55 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 2.93/1.55 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 2.93/1.55 => ( R = S ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(funcext_type,type,
% 2.93/1.55 funcext: $o ).
% 2.93/1.55
% 2.93/1.55 thf(funcext,definition,
% 2.93/1.55 ( funcext
% 2.93/1.55 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.93/1.55 ( ( func @ A @ B @ Xf )
% 2.93/1.55 => ! [Xg: $i] :
% 2.93/1.55 ( ( func @ A @ B @ Xg )
% 2.93/1.55 => ( ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.93/1.55 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.93/1.55 => ( Xf = Xg ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(funcext2_type,type,
% 2.93/1.55 funcext2: $o ).
% 2.93/1.55
% 2.93/1.55 thf(funcext2,definition,
% 2.93/1.55 ( funcext2
% 2.93/1.55 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.93/1.55 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.93/1.55 => ! [Xg: $i] :
% 2.93/1.55 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 2.93/1.55 => ( ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.93/1.55 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.93/1.55 => ( Xf = Xg ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(ap2apEq1_type,type,
% 2.93/1.55 ap2apEq1: $o ).
% 2.93/1.55
% 2.93/1.55 thf(ap2apEq1,definition,
% 2.93/1.55 ( ap2apEq1
% 2.93/1.55 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.93/1.55 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.93/1.55 => ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.93/1.55 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(ap2apEq2_type,type,
% 2.93/1.55 ap2apEq2: $o ).
% 2.93/1.55
% 2.93/1.55 thf(ap2apEq2,definition,
% 2.93/1.55 ( ap2apEq2
% 2.93/1.55 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.93/1.55 ( ( func @ A @ B @ Xf )
% 2.93/1.55 => ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.93/1.55 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(beta1_type,type,
% 2.93/1.55 beta1: $o ).
% 2.93/1.55
% 2.93/1.55 thf(beta1,definition,
% 2.93/1.55 ( beta1
% 2.93/1.55 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.93/1.55 ( ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.93/1.55 => ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ( ( ap @ A @ B
% 2.93/1.55 @ ( lam @ A @ B
% 2.93/1.55 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 2.93/1.55 @ Xx )
% 2.93/1.55 = ( Xf @ Xx ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(eta1_type,type,
% 2.93/1.55 eta1: $o ).
% 2.93/1.55
% 2.93/1.55 thf(eta1,definition,
% 2.93/1.55 ( eta1
% 2.93/1.55 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.93/1.55 ( ( func @ A @ B @ Xf )
% 2.93/1.55 => ( ( lam @ A @ B
% 2.93/1.55 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 2.93/1.55 = Xf ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(lam2lamEq_type,type,
% 2.93/1.55 lam2lamEq: $o ).
% 2.93/1.55
% 2.93/1.55 thf(lam2lamEq,definition,
% 2.93/1.55 ( lam2lamEq
% 2.93/1.55 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.93/1.55 ( ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.93/1.55 => ( ( lam @ A @ B
% 2.93/1.55 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 2.93/1.55 = ( lam @ A @ B
% 2.93/1.55 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(beta2_type,type,
% 2.93/1.55 beta2: $o ).
% 2.93/1.55
% 2.93/1.55 thf(beta2,definition,
% 2.93/1.55 ( beta2
% 2.93/1.55 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.93/1.55 ( ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.93/1.55 => ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ( ( ap @ A @ B
% 2.93/1.55 @ ( lam @ A @ B
% 2.93/1.55 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 2.93/1.55 @ Xx )
% 2.93/1.55 = ( Xf @ Xx ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(eta2_type,type,
% 2.93/1.55 eta2: $o ).
% 2.93/1.55
% 2.93/1.55 thf(eta2,definition,
% 2.93/1.55 ( eta2
% 2.93/1.55 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.93/1.55 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.93/1.55 => ( ( lam @ A @ B
% 2.93/1.55 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 2.93/1.55 = Xf ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(iffalseProp1_type,type,
% 2.93/1.55 iffalseProp1: $o ).
% 2.93/1.55
% 2.93/1.55 thf(iffalseProp1,definition,
% 2.93/1.55 ( iffalseProp1
% 2.93/1.55 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ! [Xy: $i] :
% 2.93/1.55 ( ( in @ Xy @ A )
% 2.93/1.55 => ( ~ Xphi
% 2.93/1.55 => ( in @ Xy
% 2.93/1.55 @ ( dsetconstr @ A
% 2.93/1.55 @ ^ [Xz: $i] :
% 2.93/1.55 ( ( Xphi
% 2.93/1.55 & ( Xz = Xx ) )
% 2.93/1.55 | ( ~ Xphi
% 2.93/1.55 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(iffalseProp2_type,type,
% 2.93/1.55 iffalseProp2: $o ).
% 2.93/1.55
% 2.93/1.55 thf(iffalseProp2,definition,
% 2.93/1.55 ( iffalseProp2
% 2.93/1.55 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ! [Xy: $i] :
% 2.93/1.55 ( ( in @ Xy @ A )
% 2.93/1.55 => ( ~ Xphi
% 2.93/1.55 => ( ( dsetconstr @ A
% 2.93/1.55 @ ^ [Xz: $i] :
% 2.93/1.55 ( ( Xphi
% 2.93/1.55 & ( Xz = Xx ) )
% 2.93/1.55 | ( ~ Xphi
% 2.93/1.55 & ( Xz = Xy ) ) ) )
% 2.93/1.55 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(iftrueProp1_type,type,
% 2.93/1.55 iftrueProp1: $o ).
% 2.93/1.55
% 2.93/1.55 thf(iftrueProp1,definition,
% 2.93/1.55 ( iftrueProp1
% 2.93/1.55 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ! [Xy: $i] :
% 2.93/1.55 ( ( in @ Xy @ A )
% 2.93/1.55 => ( Xphi
% 2.93/1.55 => ( in @ Xx
% 2.93/1.55 @ ( dsetconstr @ A
% 2.93/1.55 @ ^ [Xz: $i] :
% 2.93/1.55 ( ( Xphi
% 2.93/1.55 & ( Xz = Xx ) )
% 2.93/1.55 | ( ~ Xphi
% 2.93/1.55 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(iftrueProp2_type,type,
% 2.93/1.55 iftrueProp2: $o ).
% 2.93/1.55
% 2.93/1.55 thf(iftrueProp2,definition,
% 2.93/1.55 ( iftrueProp2
% 2.93/1.55 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ! [Xy: $i] :
% 2.93/1.55 ( ( in @ Xy @ A )
% 2.93/1.55 => ( Xphi
% 2.93/1.55 => ( ( dsetconstr @ A
% 2.93/1.55 @ ^ [Xz: $i] :
% 2.93/1.55 ( ( Xphi
% 2.93/1.55 & ( Xz = Xx ) )
% 2.93/1.55 | ( ~ Xphi
% 2.93/1.55 & ( Xz = Xy ) ) ) )
% 2.93/1.55 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(ifSingleton_type,type,
% 2.93/1.55 ifSingleton: $o ).
% 2.93/1.55
% 2.93/1.55 thf(ifSingleton,definition,
% 2.93/1.55 ( ifSingleton
% 2.93/1.55 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ! [Xy: $i] :
% 2.93/1.55 ( ( in @ Xy @ A )
% 2.93/1.55 => ( singleton
% 2.93/1.55 @ ( dsetconstr @ A
% 2.93/1.55 @ ^ [Xz: $i] :
% 2.93/1.55 ( ( Xphi
% 2.93/1.55 & ( Xz = Xx ) )
% 2.93/1.55 | ( ~ Xphi
% 2.93/1.55 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(if_type,type,
% 2.93/1.55 if: $i > $o > $i > $i > $i ).
% 2.93/1.55
% 2.93/1.55 thf(ifp_type,type,
% 2.93/1.55 ifp: $o ).
% 2.93/1.55
% 2.93/1.55 thf(ifp,definition,
% 2.93/1.55 ( ifp
% 2.93/1.55 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ! [Xy: $i] :
% 2.93/1.55 ( ( in @ Xy @ A )
% 2.93/1.55 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(theeq_type,type,
% 2.93/1.55 theeq: $o ).
% 2.93/1.55
% 2.93/1.55 thf(theeq,definition,
% 2.93/1.55 ( theeq
% 2.93/1.55 = ( ! [X: $i] :
% 2.93/1.55 ( ( singleton @ X )
% 2.93/1.55 => ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ X )
% 2.93/1.55 => ( ( setunion @ X )
% 2.93/1.55 = Xx ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(iftrue_type,type,
% 2.93/1.55 iftrue: $o ).
% 2.93/1.55
% 2.93/1.55 thf(iftrue,definition,
% 2.93/1.55 ( iftrue
% 2.93/1.55 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ! [Xy: $i] :
% 2.93/1.55 ( ( in @ Xy @ A )
% 2.93/1.55 => ( Xphi
% 2.93/1.55 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 2.93/1.55 = Xx ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(iffalse_type,type,
% 2.93/1.55 iffalse: $o ).
% 2.93/1.55
% 2.93/1.55 thf(iffalse,definition,
% 2.93/1.55 ( iffalse
% 2.93/1.55 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ! [Xy: $i] :
% 2.93/1.55 ( ( in @ Xy @ A )
% 2.93/1.55 => ( ~ Xphi
% 2.93/1.55 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 2.93/1.55 = Xy ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(iftrueorfalse_type,type,
% 2.93/1.55 iftrueorfalse: $o ).
% 2.93/1.55
% 2.93/1.55 thf(iftrueorfalse,definition,
% 2.93/1.55 ( iftrueorfalse
% 2.93/1.55 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ! [Xy: $i] :
% 2.93/1.55 ( ( in @ Xy @ A )
% 2.93/1.55 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(binintersectT_lem_type,type,
% 2.93/1.55 binintersectT_lem: $o ).
% 2.93/1.55
% 2.93/1.55 thf(binintersectT_lem,definition,
% 2.93/1.55 ( binintersectT_lem
% 2.93/1.55 = ( ! [A: $i,X: $i] :
% 2.93/1.55 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.55 => ! [Y: $i] :
% 2.93/1.55 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.55 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(binunionT_lem_type,type,
% 2.93/1.55 binunionT_lem: $o ).
% 2.93/1.55
% 2.93/1.55 thf(binunionT_lem,definition,
% 2.93/1.55 ( binunionT_lem
% 2.93/1.55 = ( ! [A: $i,X: $i] :
% 2.93/1.55 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.55 => ! [Y: $i] :
% 2.93/1.55 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.55 => ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(powersetT_lem_type,type,
% 2.93/1.55 powersetT_lem: $o ).
% 2.93/1.55
% 2.93/1.55 thf(powersetT_lem,definition,
% 2.93/1.55 ( powersetT_lem
% 2.93/1.55 = ( ! [A: $i,X: $i] :
% 2.93/1.55 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.55 => ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(setminusT_lem_type,type,
% 2.93/1.55 setminusT_lem: $o ).
% 2.93/1.55
% 2.93/1.55 thf(setminusT_lem,definition,
% 2.93/1.55 ( setminusT_lem
% 2.93/1.55 = ( ! [A: $i,X: $i] :
% 2.93/1.55 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.55 => ! [Y: $i] :
% 2.93/1.55 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.55 => ( in @ ( setminus @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(complementT_lem_type,type,
% 2.93/1.55 complementT_lem: $o ).
% 2.93/1.55
% 2.93/1.55 thf(complementT_lem,definition,
% 2.93/1.55 ( complementT_lem
% 2.93/1.55 = ( ! [A: $i,X: $i] :
% 2.93/1.55 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.55 => ( in @ ( setminus @ A @ X ) @ ( powerset @ A ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(setextT_type,type,
% 2.93/1.55 setextT: $o ).
% 2.93/1.55
% 2.93/1.55 thf(setextT,definition,
% 2.93/1.55 ( setextT
% 2.93/1.55 = ( ! [A: $i,X: $i] :
% 2.93/1.55 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.55 => ! [Y: $i] :
% 2.93/1.55 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.55 => ( ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ( ( in @ Xx @ X )
% 2.93/1.55 => ( in @ Xx @ Y ) ) )
% 2.93/1.55 => ( ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ( ( in @ Xx @ Y )
% 2.93/1.55 => ( in @ Xx @ X ) ) )
% 2.93/1.55 => ( X = Y ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(subsetTI_type,type,
% 2.93/1.55 subsetTI: $o ).
% 2.93/1.55
% 2.93/1.55 thf(subsetTI,definition,
% 2.93/1.55 ( subsetTI
% 2.93/1.55 = ( ! [A: $i,X: $i] :
% 2.93/1.55 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.55 => ! [Y: $i] :
% 2.93/1.55 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.55 => ( ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ( ( in @ Xx @ X )
% 2.93/1.55 => ( in @ Xx @ Y ) ) )
% 2.93/1.55 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(powersetTI1_type,type,
% 2.93/1.55 powersetTI1: $o ).
% 2.93/1.55
% 2.93/1.55 thf(powersetTI1,definition,
% 2.93/1.55 ( powersetTI1
% 2.93/1.55 = ( ! [A: $i,X: $i] :
% 2.93/1.55 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.55 => ! [Y: $i] :
% 2.93/1.55 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.55 => ( ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ( ( in @ Xx @ X )
% 2.93/1.55 => ( in @ Xx @ Y ) ) )
% 2.93/1.55 => ( in @ X @ ( powerset @ Y ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(powersetTE1_type,type,
% 2.93/1.55 powersetTE1: $o ).
% 2.93/1.55
% 2.93/1.55 thf(powersetTE1,definition,
% 2.93/1.55 ( powersetTE1
% 2.93/1.55 = ( ! [A: $i,X: $i] :
% 2.93/1.55 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.55 => ! [Y: $i] :
% 2.93/1.55 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.55 => ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ( ( in @ X @ ( powerset @ Y ) )
% 2.93/1.55 => ( ( in @ Xx @ X )
% 2.93/1.55 => ( in @ Xx @ Y ) ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(complementTI1_type,type,
% 2.93/1.55 complementTI1: $o ).
% 2.93/1.55
% 2.93/1.55 thf(complementTI1,definition,
% 2.93/1.55 ( complementTI1
% 2.93/1.55 = ( ! [A: $i,X: $i] :
% 2.93/1.55 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.55 => ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ( ( in @ Xx @ X )
% 2.93/1.55 => ~ ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(complementTE1_type,type,
% 2.93/1.55 complementTE1: $o ).
% 2.93/1.55
% 2.93/1.55 thf(complementTE1,definition,
% 2.93/1.55 ( complementTE1
% 2.93/1.55 = ( ! [A: $i,X: $i] :
% 2.93/1.55 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.55 => ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ( ~ ( in @ Xx @ ( setminus @ A @ X ) )
% 2.93/1.55 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(binintersectTELcontra_type,type,
% 2.93/1.55 binintersectTELcontra: $o ).
% 2.93/1.55
% 2.93/1.55 thf(binintersectTELcontra,definition,
% 2.93/1.55 ( binintersectTELcontra
% 2.93/1.55 = ( ! [A: $i,X: $i] :
% 2.93/1.55 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.55 => ! [Y: $i] :
% 2.93/1.55 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.55 => ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ( ~ ( in @ Xx @ X )
% 2.93/1.55 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(binintersectTERcontra_type,type,
% 2.93/1.55 binintersectTERcontra: $o ).
% 2.93/1.55
% 2.93/1.55 thf(binintersectTERcontra,definition,
% 2.93/1.55 ( binintersectTERcontra
% 2.93/1.55 = ( ! [A: $i,X: $i] :
% 2.93/1.55 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.55 => ! [Y: $i] :
% 2.93/1.55 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.55 => ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ( ~ ( in @ Xx @ Y )
% 2.93/1.55 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(contrasubsetT_type,type,
% 2.93/1.55 contrasubsetT: $o ).
% 2.93/1.55
% 2.93/1.55 thf(contrasubsetT,definition,
% 2.93/1.55 ( contrasubsetT
% 2.93/1.55 = ( ! [A: $i,X: $i] :
% 2.93/1.55 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.55 => ! [Y: $i] :
% 2.93/1.55 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.55 => ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 2.93/1.55 => ( ( in @ Xx @ Y )
% 2.93/1.55 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(contrasubsetT1_type,type,
% 2.93/1.55 contrasubsetT1: $o ).
% 2.93/1.55
% 2.93/1.55 thf(contrasubsetT1,definition,
% 2.93/1.55 ( contrasubsetT1
% 2.93/1.55 = ( ! [A: $i,X: $i] :
% 2.93/1.55 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.55 => ! [Y: $i] :
% 2.93/1.55 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.55 => ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ( ( subset @ X @ Y )
% 2.93/1.55 => ( ~ ( in @ Xx @ Y )
% 2.93/1.55 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(contrasubsetT2_type,type,
% 2.93/1.55 contrasubsetT2: $o ).
% 2.93/1.55
% 2.93/1.55 thf(contrasubsetT2,definition,
% 2.93/1.55 ( contrasubsetT2
% 2.93/1.55 = ( ! [A: $i,X: $i] :
% 2.93/1.55 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.55 => ! [Y: $i] :
% 2.93/1.55 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.55 => ( ( subset @ X @ Y )
% 2.93/1.55 => ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(contrasubsetT3_type,type,
% 2.93/1.55 contrasubsetT3: $o ).
% 2.93/1.55
% 2.93/1.55 thf(contrasubsetT3,definition,
% 2.93/1.55 ( contrasubsetT3
% 2.93/1.55 = ( ! [A: $i,X: $i] :
% 2.93/1.55 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.55 => ! [Y: $i] :
% 2.93/1.55 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.55 => ( ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) )
% 2.93/1.55 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(doubleComplementI1_type,type,
% 2.93/1.55 doubleComplementI1: $o ).
% 2.93/1.55
% 2.93/1.55 thf(doubleComplementI1,definition,
% 2.93/1.55 ( doubleComplementI1
% 2.93/1.55 = ( ! [A: $i,X: $i] :
% 2.93/1.55 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.55 => ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ( ( in @ Xx @ X )
% 2.93/1.55 => ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(doubleComplementE1_type,type,
% 2.93/1.55 doubleComplementE1: $o ).
% 2.93/1.55
% 2.93/1.55 thf(doubleComplementE1,definition,
% 2.93/1.55 ( doubleComplementE1
% 2.93/1.55 = ( ! [A: $i,X: $i] :
% 2.93/1.55 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.55 => ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ( ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) )
% 2.93/1.55 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(doubleComplementSub1_type,type,
% 2.93/1.55 doubleComplementSub1: $o ).
% 2.93/1.55
% 2.93/1.55 thf(doubleComplementSub1,definition,
% 2.93/1.55 ( doubleComplementSub1
% 2.93/1.55 = ( ! [A: $i,X: $i] :
% 2.93/1.55 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.55 => ( subset @ X @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(doubleComplementSub2_type,type,
% 2.93/1.55 doubleComplementSub2: $o ).
% 2.93/1.55
% 2.93/1.55 thf(doubleComplementSub2,definition,
% 2.93/1.55 ( doubleComplementSub2
% 2.93/1.55 = ( ! [A: $i,X: $i] :
% 2.93/1.55 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.55 => ( subset @ ( setminus @ A @ ( setminus @ A @ X ) ) @ X ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(doubleComplementEq_type,type,
% 2.93/1.55 doubleComplementEq: $o ).
% 2.93/1.55
% 2.93/1.55 thf(doubleComplementEq,definition,
% 2.93/1.55 ( doubleComplementEq
% 2.93/1.55 = ( ! [A: $i,X: $i] :
% 2.93/1.55 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.55 => ( X
% 2.93/1.55 = ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(complementTnotintersectT_type,type,
% 2.93/1.55 complementTnotintersectT: $o ).
% 2.93/1.55
% 2.93/1.55 thf(complementTnotintersectT,definition,
% 2.93/1.55 ( complementTnotintersectT
% 2.93/1.55 = ( ! [A: $i,X: $i] :
% 2.93/1.55 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.55 => ! [Y: $i] :
% 2.93/1.55 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.55 => ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 2.93/1.55 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(complementImpComplementIntersect_type,type,
% 2.93/1.55 complementImpComplementIntersect: $o ).
% 2.93/1.55
% 2.93/1.55 thf(complementImpComplementIntersect,definition,
% 2.93/1.55 ( complementImpComplementIntersect
% 2.93/1.55 = ( ! [A: $i,X: $i] :
% 2.93/1.55 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.55 => ! [Y: $i] :
% 2.93/1.55 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.55 => ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 2.93/1.55 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(complementSubsetComplementIntersect_type,type,
% 2.93/1.55 complementSubsetComplementIntersect: $o ).
% 2.93/1.55
% 2.93/1.55 thf(complementSubsetComplementIntersect,definition,
% 2.93/1.55 ( complementSubsetComplementIntersect
% 2.93/1.55 = ( ! [A: $i,X: $i] :
% 2.93/1.55 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.55 => ! [Y: $i] :
% 2.93/1.55 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.55 => ( subset @ ( setminus @ A @ X ) @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(complementInPowersetComplementIntersect_type,type,
% 2.93/1.55 complementInPowersetComplementIntersect: $o ).
% 2.93/1.55
% 2.93/1.55 thf(complementInPowersetComplementIntersect,definition,
% 2.93/1.55 ( complementInPowersetComplementIntersect
% 2.93/1.55 = ( ! [A: $i,X: $i] :
% 2.93/1.55 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.55 => ! [Y: $i] :
% 2.93/1.55 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.55 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(contraSubsetComplement_type,type,
% 2.93/1.55 contraSubsetComplement: $o ).
% 2.93/1.55
% 2.93/1.55 thf(contraSubsetComplement,definition,
% 2.93/1.55 ( contraSubsetComplement
% 2.93/1.55 = ( ! [A: $i,X: $i] :
% 2.93/1.55 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.55 => ! [Y: $i] :
% 2.93/1.55 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.55 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 2.93/1.55 => ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ( ( in @ Xx @ Y )
% 2.93/1.55 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(complementTcontraSubset_type,type,
% 2.93/1.55 complementTcontraSubset: $o ).
% 2.93/1.55
% 2.93/1.55 thf(complementTcontraSubset,definition,
% 2.93/1.55 ( complementTcontraSubset
% 2.93/1.55 = ( ! [A: $i,X: $i] :
% 2.93/1.55 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.55 => ! [Y: $i] :
% 2.93/1.55 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.55 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 2.93/1.55 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(binunionTILcontra_type,type,
% 2.93/1.55 binunionTILcontra: $o ).
% 2.93/1.55
% 2.93/1.55 thf(binunionTILcontra,definition,
% 2.93/1.55 ( binunionTILcontra
% 2.93/1.55 = ( ! [A: $i,X: $i] :
% 2.93/1.55 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.55 => ! [Y: $i] :
% 2.93/1.55 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.55 => ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 2.93/1.55 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(binunionTIRcontra_type,type,
% 2.93/1.55 binunionTIRcontra: $o ).
% 2.93/1.55
% 2.93/1.55 thf(binunionTIRcontra,definition,
% 2.93/1.55 ( binunionTIRcontra
% 2.93/1.55 = ( ! [A: $i,X: $i] :
% 2.93/1.55 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.55 => ! [Y: $i] :
% 2.93/1.55 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.55 => ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 2.93/1.55 => ~ ( in @ Xx @ Y ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(inIntersectImpInUnion_type,type,
% 2.93/1.55 inIntersectImpInUnion: $o ).
% 2.93/1.55
% 2.93/1.55 thf(inIntersectImpInUnion,definition,
% 2.93/1.55 ( inIntersectImpInUnion
% 2.93/1.55 = ( ! [A: $i,X: $i] :
% 2.93/1.55 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.55 => ! [Y: $i] :
% 2.93/1.55 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.55 => ! [Z: $i] :
% 2.93/1.55 ( ( in @ Z @ ( powerset @ A ) )
% 2.93/1.55 => ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 2.93/1.55 => ( in @ Xx @ ( binunion @ X @ Z ) ) ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(inIntersectImpInUnion2_type,type,
% 2.93/1.55 inIntersectImpInUnion2: $o ).
% 2.93/1.55
% 2.93/1.55 thf(inIntersectImpInUnion2,definition,
% 2.93/1.55 ( inIntersectImpInUnion2
% 2.93/1.55 = ( ! [A: $i,X: $i] :
% 2.93/1.55 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.55 => ! [Y: $i] :
% 2.93/1.55 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.55 => ! [Z: $i] :
% 2.93/1.55 ( ( in @ Z @ ( powerset @ A ) )
% 2.93/1.55 => ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 2.93/1.55 => ( in @ Xx @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(inIntersectImpInIntersectUnions_type,type,
% 2.93/1.55 inIntersectImpInIntersectUnions: $o ).
% 2.93/1.55
% 2.93/1.55 thf(inIntersectImpInIntersectUnions,definition,
% 2.93/1.55 ( inIntersectImpInIntersectUnions
% 2.93/1.55 = ( ! [A: $i,X: $i] :
% 2.93/1.55 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.55 => ! [Y: $i] :
% 2.93/1.55 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.55 => ! [Z: $i] :
% 2.93/1.55 ( ( in @ Z @ ( powerset @ A ) )
% 2.93/1.55 => ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 2.93/1.55 => ( in @ Xx @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(intersectInPowersetIntersectUnions_type,type,
% 2.93/1.55 intersectInPowersetIntersectUnions: $o ).
% 2.93/1.55
% 2.93/1.55 thf(intersectInPowersetIntersectUnions,definition,
% 2.93/1.55 ( intersectInPowersetIntersectUnions
% 2.93/1.55 = ( ! [A: $i,X: $i] :
% 2.93/1.55 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.55 => ! [Y: $i] :
% 2.93/1.55 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.55 => ! [Z: $i] :
% 2.93/1.55 ( ( in @ Z @ ( powerset @ A ) )
% 2.93/1.55 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(inComplementUnionImpNotIn1_type,type,
% 2.93/1.55 inComplementUnionImpNotIn1: $o ).
% 2.93/1.55
% 2.93/1.55 thf(inComplementUnionImpNotIn1,definition,
% 2.93/1.55 ( inComplementUnionImpNotIn1
% 2.93/1.55 = ( ! [A: $i,X: $i] :
% 2.93/1.55 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.55 => ! [Y: $i] :
% 2.93/1.55 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.55 => ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 2.93/1.55 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(inComplementUnionImpInComplement1_type,type,
% 2.93/1.55 inComplementUnionImpInComplement1: $o ).
% 2.93/1.55
% 2.93/1.55 thf(inComplementUnionImpInComplement1,definition,
% 2.93/1.55 ( inComplementUnionImpInComplement1
% 2.93/1.55 = ( ! [A: $i,X: $i] :
% 2.93/1.55 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.55 => ! [Y: $i] :
% 2.93/1.55 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.55 => ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 2.93/1.55 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(binunionTE_type,type,
% 2.93/1.55 binunionTE: $o ).
% 2.93/1.55
% 2.93/1.55 thf(binunionTE,definition,
% 2.93/1.55 ( binunionTE
% 2.93/1.55 = ( ! [A: $i,X: $i] :
% 2.93/1.55 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.55 => ! [Y: $i] :
% 2.93/1.55 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.55 => ! [Xphi: $o,Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ( ( in @ Xx @ ( binunion @ X @ Y ) )
% 2.93/1.55 => ( ( ( in @ Xx @ X )
% 2.93/1.55 => Xphi )
% 2.93/1.55 => ( ( ( in @ Xx @ Y )
% 2.93/1.55 => Xphi )
% 2.93/1.55 => Xphi ) ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(binunionTEcontra_type,type,
% 2.93/1.55 binunionTEcontra: $o ).
% 2.93/1.55
% 2.93/1.55 thf(binunionTEcontra,definition,
% 2.93/1.55 ( binunionTEcontra
% 2.93/1.55 = ( ! [A: $i,X: $i] :
% 2.93/1.55 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.55 => ! [Y: $i] :
% 2.93/1.55 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.55 => ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ( ~ ( in @ Xx @ X )
% 2.93/1.55 => ( ~ ( in @ Xx @ Y )
% 2.93/1.55 => ~ ( in @ Xx @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(demorgan2a1_type,type,
% 2.93/1.55 demorgan2a1: $o ).
% 2.93/1.55
% 2.93/1.55 thf(demorgan2a1,definition,
% 2.93/1.55 ( demorgan2a1
% 2.93/1.55 = ( ! [A: $i,X: $i] :
% 2.93/1.55 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.55 => ! [Y: $i] :
% 2.93/1.55 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.55 => ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 2.93/1.55 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(complementUnionInPowersetComplement_type,type,
% 2.93/1.55 complementUnionInPowersetComplement: $o ).
% 2.93/1.55
% 2.93/1.55 thf(complementUnionInPowersetComplement,definition,
% 2.93/1.55 ( complementUnionInPowersetComplement
% 2.93/1.55 = ( ! [A: $i,X: $i] :
% 2.93/1.55 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.55 => ! [Y: $i] :
% 2.93/1.55 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.55 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(demorgan2a2_type,type,
% 2.93/1.55 demorgan2a2: $o ).
% 2.93/1.55
% 2.93/1.55 thf(demorgan2a2,definition,
% 2.93/1.55 ( demorgan2a2
% 2.93/1.55 = ( ! [A: $i,X: $i] :
% 2.93/1.55 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.55 => ! [Y: $i] :
% 2.93/1.55 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.55 => ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 2.93/1.55 => ( in @ Xx @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(demorgan1a_type,type,
% 2.93/1.55 demorgan1a: $o ).
% 2.93/1.55
% 2.93/1.55 thf(demorgan1a,definition,
% 2.93/1.55 ( demorgan1a
% 2.93/1.55 = ( ! [A: $i,X: $i] :
% 2.93/1.55 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.55 => ! [Y: $i] :
% 2.93/1.55 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.55 => ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ( ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) )
% 2.93/1.55 => ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(demorgan1b_type,type,
% 2.93/1.55 demorgan1b: $o ).
% 2.93/1.55
% 2.93/1.55 thf(demorgan1b,definition,
% 2.93/1.55 ( demorgan1b
% 2.93/1.55 = ( ! [A: $i,X: $i] :
% 2.93/1.55 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.55 => ! [Y: $i] :
% 2.93/1.55 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.55 => ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ( ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 2.93/1.55 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(demorgan1_type,type,
% 2.93/1.55 demorgan1: $o ).
% 2.93/1.55
% 2.93/1.55 thf(demorgan1,definition,
% 2.93/1.55 ( demorgan1
% 2.93/1.55 = ( ! [A: $i,X: $i] :
% 2.93/1.55 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.55 => ! [Y: $i] :
% 2.93/1.55 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.55 => ( ( setminus @ A @ ( binintersect @ X @ Y ) )
% 2.93/1.55 = ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(demorgan2a_type,type,
% 2.93/1.55 demorgan2a: $o ).
% 2.93/1.55
% 2.93/1.55 thf(demorgan2a,definition,
% 2.93/1.55 ( demorgan2a
% 2.93/1.55 = ( ! [A: $i,X: $i] :
% 2.93/1.55 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.55 => ! [Y: $i] :
% 2.93/1.55 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.55 => ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 2.93/1.55 => ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(demorgan2b2_type,type,
% 2.93/1.55 demorgan2b2: $o ).
% 2.93/1.55
% 2.93/1.55 thf(demorgan2b2,definition,
% 2.93/1.55 ( demorgan2b2
% 2.93/1.55 = ( ! [A: $i,X: $i] :
% 2.93/1.55 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.55 => ! [Y: $i] :
% 2.93/1.55 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.55 => ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 2.93/1.55 => ( ( in @ Xx @ ( setminus @ A @ Y ) )
% 2.93/1.55 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(demorgan2b_type,type,
% 2.93/1.55 demorgan2b: $o ).
% 2.93/1.55
% 2.93/1.55 thf(demorgan2b,definition,
% 2.93/1.55 ( demorgan2b
% 2.93/1.55 = ( ! [A: $i,X: $i] :
% 2.93/1.55 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.55 => ! [Y: $i] :
% 2.93/1.55 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.55 => ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ( ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 2.93/1.55 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(demorgan2_type,type,
% 2.93/1.55 demorgan2: $o ).
% 2.93/1.55
% 2.93/1.55 thf(demorgan2,definition,
% 2.93/1.55 ( demorgan2
% 2.93/1.55 = ( ! [A: $i,X: $i] :
% 2.93/1.55 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.55 => ! [Y: $i] :
% 2.93/1.55 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.55 => ( ( setminus @ A @ ( binunion @ X @ Y ) )
% 2.93/1.55 = ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(woz13rule0_type,type,
% 2.93/1.55 woz13rule0: $o ).
% 2.93/1.55
% 2.93/1.55 thf(woz13rule0,definition,
% 2.93/1.55 ( woz13rule0
% 2.93/1.55 = ( ! [A: $i,X: $i] :
% 2.93/1.55 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.55 => ! [Y: $i] :
% 2.93/1.55 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.55 => ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 2.93/1.55 => ( in @ Xx @ A ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(woz13rule1_type,type,
% 2.93/1.55 woz13rule1: $o ).
% 2.93/1.55
% 2.93/1.55 thf(woz13rule1,definition,
% 2.93/1.55 ( woz13rule1
% 2.93/1.55 = ( ! [A: $i,X: $i] :
% 2.93/1.55 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.55 => ! [Y: $i] :
% 2.93/1.55 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.55 => ! [Z: $i] :
% 2.93/1.55 ( ( in @ Z @ ( powerset @ A ) )
% 2.93/1.55 => ( ( subset @ X @ Z )
% 2.93/1.55 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(woz13rule2_type,type,
% 2.93/1.55 woz13rule2: $o ).
% 2.93/1.55
% 2.93/1.55 thf(woz13rule2,definition,
% 2.93/1.55 ( woz13rule2
% 2.93/1.55 = ( ! [A: $i,X: $i] :
% 2.93/1.55 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.55 => ! [Y: $i] :
% 2.93/1.55 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.55 => ! [Z: $i] :
% 2.93/1.55 ( ( in @ Z @ ( powerset @ A ) )
% 2.93/1.55 => ( ( subset @ Y @ Z )
% 2.93/1.55 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(woz13rule3_type,type,
% 2.93/1.55 woz13rule3: $o ).
% 2.93/1.55
% 2.93/1.55 thf(woz13rule3,definition,
% 2.93/1.55 ( woz13rule3
% 2.93/1.55 = ( ! [A: $i,X: $i] :
% 2.93/1.55 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.55 => ! [Y: $i] :
% 2.93/1.55 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.55 => ! [Z: $i] :
% 2.93/1.55 ( ( in @ Z @ ( powerset @ A ) )
% 2.93/1.55 => ( ( subset @ X @ Y )
% 2.93/1.55 => ( ( subset @ X @ Z )
% 2.93/1.55 => ( subset @ X @ ( binintersect @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(woz13rule4_type,type,
% 2.93/1.55 woz13rule4: $o ).
% 2.93/1.55
% 2.93/1.55 thf(woz13rule4,definition,
% 2.93/1.55 ( woz13rule4
% 2.93/1.55 = ( ! [A: $i,X: $i] :
% 2.93/1.55 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.55 => ! [Y: $i] :
% 2.93/1.55 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.55 => ! [Z: $i] :
% 2.93/1.55 ( ( in @ Z @ ( powerset @ A ) )
% 2.93/1.55 => ! [W: $i] :
% 2.93/1.55 ( ( in @ W @ ( powerset @ A ) )
% 2.93/1.55 => ( ( subset @ X @ Z )
% 2.93/1.55 => ( ( subset @ Y @ W )
% 2.93/1.55 => ( subset @ ( binintersect @ X @ Y ) @ ( binintersect @ Z @ W ) ) ) ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(woz1_1_type,type,
% 2.93/1.55 woz1_1: $o ).
% 2.93/1.55
% 2.93/1.55 thf(woz1_1,definition,
% 2.93/1.55 ( woz1_1
% 2.93/1.55 = ( ! [A: $i,X: $i] :
% 2.93/1.55 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.55 => ! [Y: $i] :
% 2.93/1.55 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.55 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(woz1_2_type,type,
% 2.93/1.55 woz1_2: $o ).
% 2.93/1.55
% 2.93/1.55 thf(woz1_2,definition,
% 2.93/1.55 ( woz1_2
% 2.93/1.55 = ( ! [A: $i,X: $i] :
% 2.93/1.55 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.55 => ! [Y: $i] :
% 2.93/1.55 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.55 => ! [Z: $i] :
% 2.93/1.55 ( ( in @ Z @ ( powerset @ A ) )
% 2.93/1.55 => ! [W: $i] :
% 2.93/1.55 ( ( in @ W @ ( powerset @ A ) )
% 2.93/1.55 => ( ( setminus @ A @ ( binintersect @ ( binunion @ X @ Y ) @ ( binunion @ Z @ W ) ) )
% 2.93/1.55 = ( binunion @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) @ ( binintersect @ ( setminus @ A @ Z ) @ ( setminus @ A @ W ) ) ) ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(woz1_3_type,type,
% 2.93/1.55 woz1_3: $o ).
% 2.93/1.55
% 2.93/1.55 thf(woz1_3,definition,
% 2.93/1.55 ( woz1_3
% 2.93/1.55 = ( ! [A: $i,X: $i] :
% 2.93/1.55 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.55 => ! [Y: $i] :
% 2.93/1.55 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.55 => ! [Z: $i] :
% 2.93/1.55 ( ( in @ Z @ ( powerset @ A ) )
% 2.93/1.55 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(woz1_4_type,type,
% 2.93/1.55 woz1_4: $o ).
% 2.93/1.55
% 2.93/1.55 thf(woz1_4,definition,
% 2.93/1.55 ( woz1_4
% 2.93/1.55 = ( ! [A: $i,X: $i] :
% 2.93/1.55 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.55 => ! [Y: $i] :
% 2.93/1.55 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.55 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 2.93/1.55 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(woz1_5_type,type,
% 2.93/1.55 woz1_5: $o ).
% 2.93/1.55
% 2.93/1.55 thf(woz1_5,definition,
% 2.93/1.55 ( woz1_5
% 2.93/1.55 = ( ! [A: $i,X: $i] :
% 2.93/1.55 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.55 => ! [Y: $i] :
% 2.93/1.55 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.55 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(breln1_type,type,
% 2.93/1.55 breln1: $i > $i > $o ).
% 2.93/1.55
% 2.93/1.55 thf(breln1all2_type,type,
% 2.93/1.55 breln1all2: $o ).
% 2.93/1.55
% 2.93/1.55 thf(breln1all2,definition,
% 2.93/1.55 ( breln1all2
% 2.93/1.55 = ( ! [A: $i,R: $i] :
% 2.93/1.55 ( ( breln1 @ A @ R )
% 2.93/1.55 => ! [Xphi: $i > $o] :
% 2.93/1.55 ( ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ! [Xy: $i] :
% 2.93/1.55 ( ( in @ Xy @ A )
% 2.93/1.55 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.93/1.55 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 2.93/1.55 => ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ R )
% 2.93/1.55 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(breln1Set_type,type,
% 2.93/1.55 breln1Set: $i > $i ).
% 2.93/1.55
% 2.93/1.55 thf(breln1SetBreln1_type,type,
% 2.93/1.55 breln1SetBreln1: $o ).
% 2.93/1.55
% 2.93/1.55 thf(breln1SetBreln1,definition,
% 2.93/1.55 ( breln1SetBreln1
% 2.93/1.55 = ( ! [A: $i,R: $i] :
% 2.93/1.55 ( ( in @ R @ ( breln1Set @ A ) )
% 2.93/1.55 => ( breln1 @ A @ R ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(transitive_type,type,
% 2.93/1.55 transitive: $i > $i > $o ).
% 2.93/1.55
% 2.93/1.55 thf(antisymmetric_type,type,
% 2.93/1.55 antisymmetric: $i > $i > $o ).
% 2.93/1.55
% 2.93/1.55 thf(reflexive_type,type,
% 2.93/1.55 reflexive: $i > $i > $o ).
% 2.93/1.55
% 2.93/1.55 thf(refltransitive_type,type,
% 2.93/1.55 refltransitive: $i > $i > $o ).
% 2.93/1.55
% 2.93/1.55 thf(refllinearorder_type,type,
% 2.93/1.55 refllinearorder: $i > $i > $o ).
% 2.93/1.55
% 2.93/1.55 thf(reflwellordering_type,type,
% 2.93/1.55 reflwellordering: $i > $i > $o ).
% 2.93/1.55
% 2.93/1.55 thf(choice2fnsingleton_type,type,
% 2.93/1.55 choice2fnsingleton: $o ).
% 2.93/1.55
% 2.93/1.55 thf(choice2fnsingleton,definition,
% 2.93/1.55 ( choice2fnsingleton
% 2.93/1.55 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 2.93/1.55 ( ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ? [Xy: $i] :
% 2.93/1.55 ( ( in @ Xy @ B )
% 2.93/1.55 & ( Xphi @ Xx @ Xy ) ) )
% 2.93/1.55 => ! [R: $i] :
% 2.93/1.55 ( ( in @ R @ ( breln1Set @ B ) )
% 2.93/1.55 => ( ( reflwellordering @ B @ R )
% 2.93/1.55 => ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ( singleton
% 2.93/1.55 @ ( dsetconstr @ B
% 2.93/1.55 @ ^ [Xy: $i] :
% 2.93/1.55 ( ( Xphi @ Xx @ Xy )
% 2.93/1.55 & ! [Xz: $i] :
% 2.93/1.55 ( ( in @ Xz @ B )
% 2.93/1.55 => ( ( Xphi @ Xx @ Xz )
% 2.93/1.55 => ( in @ ( kpair @ Xy @ Xz ) @ R ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(setOfPairsIsBReln1_type,type,
% 2.93/1.55 setOfPairsIsBReln1: $o ).
% 2.93/1.55
% 2.93/1.55 thf(setOfPairsIsBReln1,definition,
% 2.93/1.55 ( setOfPairsIsBReln1
% 2.93/1.55 = ( ! [A: $i,Xphi: $i > $i > $o] :
% 2.93/1.55 ( breln1 @ A
% 2.93/1.55 @ ( dpsetconstr @ A @ A
% 2.93/1.55 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(breln1all1_type,type,
% 2.93/1.55 breln1all1: $o ).
% 2.93/1.55
% 2.93/1.55 thf(breln1all1,definition,
% 2.93/1.55 ( breln1all1
% 2.93/1.55 = ( ! [A: $i,R: $i] :
% 2.93/1.55 ( ( breln1 @ A @ R )
% 2.93/1.55 => ! [Xphi: $i > $o] :
% 2.93/1.55 ( ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ! [Xy: $i] :
% 2.93/1.55 ( ( in @ Xy @ A )
% 2.93/1.55 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.93/1.55 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 2.93/1.55 => ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ R )
% 2.93/1.55 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(subbreln1_type,type,
% 2.93/1.55 subbreln1: $o ).
% 2.93/1.55
% 2.93/1.55 thf(subbreln1,definition,
% 2.93/1.55 ( subbreln1
% 2.93/1.55 = ( ! [A: $i,R: $i] :
% 2.93/1.55 ( ( breln1 @ A @ R )
% 2.93/1.55 => ! [S: $i] :
% 2.93/1.55 ( ( breln1 @ A @ S )
% 2.93/1.55 => ( ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ! [Xy: $i] :
% 2.93/1.55 ( ( in @ Xy @ A )
% 2.93/1.55 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.93/1.55 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 2.93/1.55 => ( subset @ R @ S ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(eqbreln1_type,type,
% 2.93/1.55 eqbreln1: $o ).
% 2.93/1.55
% 2.93/1.55 thf(eqbreln1,definition,
% 2.93/1.55 ( eqbreln1
% 2.93/1.55 = ( ! [A: $i,R: $i] :
% 2.93/1.55 ( ( breln1 @ A @ R )
% 2.93/1.55 => ! [S: $i] :
% 2.93/1.55 ( ( breln1 @ A @ S )
% 2.93/1.55 => ( ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ! [Xy: $i] :
% 2.93/1.55 ( ( in @ Xy @ A )
% 2.93/1.55 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.93/1.55 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 2.93/1.55 => ( ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ! [Xy: $i] :
% 2.93/1.55 ( ( in @ Xy @ A )
% 2.93/1.55 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 2.93/1.55 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 2.93/1.55 => ( R = S ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(breln1invset_type,type,
% 2.93/1.55 breln1invset: $i > $i > $i ).
% 2.93/1.55
% 2.93/1.55 thf(breln1invprop_type,type,
% 2.93/1.55 breln1invprop: $o ).
% 2.93/1.55
% 2.93/1.55 thf(breln1invprop,definition,
% 2.93/1.55 ( breln1invprop
% 2.93/1.55 = ( ! [A: $i,R: $i] :
% 2.93/1.55 ( ( breln1 @ A @ R )
% 2.93/1.55 => ( breln1 @ A @ ( breln1invset @ A @ R ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(breln1invI_type,type,
% 2.93/1.55 breln1invI: $o ).
% 2.93/1.55
% 2.93/1.55 thf(breln1invI,definition,
% 2.93/1.55 ( breln1invI
% 2.93/1.55 = ( ! [A: $i,R: $i] :
% 2.93/1.55 ( ( breln1 @ A @ R )
% 2.93/1.55 => ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ! [Xy: $i] :
% 2.93/1.55 ( ( in @ Xy @ A )
% 2.93/1.55 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.93/1.55 => ( in @ ( kpair @ Xy @ Xx ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(breln1invE_type,type,
% 2.93/1.55 breln1invE: $o ).
% 2.93/1.55
% 2.93/1.55 thf(breln1invE,definition,
% 2.93/1.55 ( breln1invE
% 2.93/1.55 = ( ! [A: $i,R: $i] :
% 2.93/1.55 ( ( breln1 @ A @ R )
% 2.93/1.55 => ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ! [Xy: $i] :
% 2.93/1.55 ( ( in @ Xy @ A )
% 2.93/1.55 => ( ( in @ ( kpair @ Xy @ Xx ) @ ( breln1invset @ A @ R ) )
% 2.93/1.55 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(breln1compset_type,type,
% 2.93/1.55 breln1compset: $i > $i > $i > $i ).
% 2.93/1.55
% 2.93/1.55 thf(breln1compprop_type,type,
% 2.93/1.55 breln1compprop: $o ).
% 2.93/1.55
% 2.93/1.55 thf(breln1compprop,definition,
% 2.93/1.55 ( breln1compprop
% 2.93/1.55 = ( ! [A: $i,R: $i] :
% 2.93/1.55 ( ( breln1 @ A @ R )
% 2.93/1.55 => ! [S: $i] :
% 2.93/1.55 ( ( breln1 @ A @ S )
% 2.93/1.55 => ( breln1 @ A @ ( breln1compset @ A @ R @ S ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(breln1compI_type,type,
% 2.93/1.55 breln1compI: $o ).
% 2.93/1.55
% 2.93/1.55 thf(breln1compI,definition,
% 2.93/1.55 ( breln1compI
% 2.93/1.55 = ( ! [A: $i,R: $i] :
% 2.93/1.55 ( ( breln1 @ A @ R )
% 2.93/1.55 => ! [S: $i] :
% 2.93/1.55 ( ( breln1 @ A @ S )
% 2.93/1.55 => ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ! [Xy: $i] :
% 2.93/1.55 ( ( in @ Xy @ A )
% 2.93/1.55 => ! [Xz: $i] :
% 2.93/1.55 ( ( in @ Xz @ A )
% 2.93/1.55 => ( ( in @ ( kpair @ Xx @ Xz ) @ R )
% 2.93/1.55 => ( ( in @ ( kpair @ Xz @ Xy ) @ S )
% 2.93/1.55 => ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) ) ) ) ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(breln1compE_type,type,
% 2.93/1.55 breln1compE: $o ).
% 2.93/1.55
% 2.93/1.55 thf(breln1compE,definition,
% 2.93/1.55 ( breln1compE
% 2.93/1.55 = ( ! [A: $i,R: $i] :
% 2.93/1.55 ( ( breln1 @ A @ R )
% 2.93/1.55 => ! [S: $i] :
% 2.93/1.55 ( ( breln1 @ A @ S )
% 2.93/1.55 => ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ! [Xy: $i] :
% 2.93/1.55 ( ( in @ Xy @ A )
% 2.93/1.55 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) )
% 2.93/1.55 => ? [Xz: $i] :
% 2.93/1.55 ( ( in @ Xz @ A )
% 2.93/1.55 & ( in @ ( kpair @ Xx @ Xz ) @ R )
% 2.93/1.55 & ( in @ ( kpair @ Xz @ Xy ) @ S ) ) ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(breln1compEex_type,type,
% 2.93/1.55 breln1compEex: $o ).
% 2.93/1.55
% 2.93/1.55 thf(breln1compEex,definition,
% 2.93/1.55 ( breln1compEex
% 2.93/1.55 = ( ! [A: $i,R: $i] :
% 2.93/1.55 ( ( breln1 @ A @ R )
% 2.93/1.55 => ! [S: $i] :
% 2.93/1.55 ( ( breln1 @ A @ S )
% 2.93/1.55 => ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ! [Xy: $i] :
% 2.93/1.55 ( ( in @ Xy @ A )
% 2.93/1.55 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) )
% 2.93/1.55 => ! [Xphi: $o] :
% 2.93/1.55 ( ! [Xz: $i] :
% 2.93/1.55 ( ( in @ Xz @ A )
% 2.93/1.55 => ( ( in @ ( kpair @ Xx @ Xz ) @ R )
% 2.93/1.55 => ( ( in @ ( kpair @ Xz @ Xy ) @ S )
% 2.93/1.55 => Xphi ) ) )
% 2.93/1.55 => Xphi ) ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(breln1unionprop_type,type,
% 2.93/1.55 breln1unionprop: $o ).
% 2.93/1.55
% 2.93/1.55 thf(breln1unionprop,definition,
% 2.93/1.55 ( breln1unionprop
% 2.93/1.55 = ( ! [A: $i,R: $i] :
% 2.93/1.55 ( ( breln1 @ A @ R )
% 2.93/1.55 => ! [S: $i] :
% 2.93/1.55 ( ( breln1 @ A @ S )
% 2.93/1.55 => ( breln1 @ A @ ( binunion @ R @ S ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(breln1unionIL_type,type,
% 2.93/1.55 breln1unionIL: $o ).
% 2.93/1.55
% 2.93/1.55 thf(breln1unionIL,definition,
% 2.93/1.55 ( breln1unionIL
% 2.93/1.55 = ( ! [A: $i,R: $i] :
% 2.93/1.55 ( ( breln1 @ A @ R )
% 2.93/1.55 => ! [S: $i] :
% 2.93/1.55 ( ( breln1 @ A @ S )
% 2.93/1.55 => ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ! [Xy: $i] :
% 2.93/1.55 ( ( in @ Xy @ A )
% 2.93/1.55 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.93/1.55 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(breln1unionIR_type,type,
% 2.93/1.55 breln1unionIR: $o ).
% 2.93/1.55
% 2.93/1.55 thf(breln1unionIR,definition,
% 2.93/1.55 ( breln1unionIR
% 2.93/1.55 = ( ! [A: $i,R: $i] :
% 2.93/1.55 ( ( breln1 @ A @ R )
% 2.93/1.55 => ! [S: $i] :
% 2.93/1.55 ( ( breln1 @ A @ S )
% 2.93/1.55 => ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ! [Xy: $i] :
% 2.93/1.55 ( ( in @ Xy @ A )
% 2.93/1.55 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 2.93/1.55 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(breln1unionI_type,type,
% 2.93/1.55 breln1unionI: $o ).
% 2.93/1.55
% 2.93/1.55 thf(breln1unionI,definition,
% 2.93/1.55 ( breln1unionI
% 2.93/1.55 = ( ! [A: $i,R: $i] :
% 2.93/1.55 ( ( breln1 @ A @ R )
% 2.93/1.55 => ! [S: $i] :
% 2.93/1.55 ( ( breln1 @ A @ S )
% 2.93/1.55 => ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ! [Xy: $i] :
% 2.93/1.55 ( ( in @ Xy @ A )
% 2.93/1.55 => ( ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.93/1.55 | ( in @ ( kpair @ Xx @ Xy ) @ S ) )
% 2.93/1.55 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(breln1unionE_type,type,
% 2.93/1.55 breln1unionE: $o ).
% 2.93/1.55
% 2.93/1.55 thf(breln1unionE,definition,
% 2.93/1.55 ( breln1unionE
% 2.93/1.55 = ( ! [A: $i,R: $i] :
% 2.93/1.55 ( ( breln1 @ A @ R )
% 2.93/1.55 => ! [S: $i] :
% 2.93/1.55 ( ( breln1 @ A @ S )
% 2.93/1.55 => ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ! [Xy: $i] :
% 2.93/1.55 ( ( in @ Xy @ A )
% 2.93/1.55 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) )
% 2.93/1.55 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.93/1.55 | ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) ) ) ) ) ) ) ).
% 2.93/1.55
% 2.93/1.55 thf(breln1unionEcases,conjecture,
% 2.93/1.55 ( setextAx
% 2.93/1.55 => ( emptysetAx
% 2.93/1.55 => ( setadjoinAx
% 2.93/1.55 => ( powersetAx
% 2.93/1.55 => ( setunionAx
% 2.93/1.55 => ( omega0Ax
% 2.93/1.55 => ( omegaSAx
% 2.93/1.55 => ( omegaIndAx
% 2.93/1.55 => ( replAx
% 2.93/1.55 => ( foundationAx
% 2.93/1.55 => ( wellorderingAx
% 2.93/1.55 => ( descrp
% 2.93/1.55 => ( dsetconstrI
% 2.93/1.55 => ( dsetconstrEL
% 2.93/1.55 => ( dsetconstrER
% 2.93/1.55 => ( exuE1
% 2.93/1.55 => ( prop2setE
% 2.93/1.55 => ( emptysetE
% 2.93/1.55 => ( emptysetimpfalse
% 2.93/1.55 => ( notinemptyset
% 2.93/1.55 => ( exuE3e
% 2.93/1.55 => ( setext
% 2.93/1.55 => ( emptyI
% 2.93/1.55 => ( noeltsimpempty
% 2.93/1.55 => ( setbeta
% 2.93/1.55 => ( nonemptyE1
% 2.93/1.55 => ( nonemptyI
% 2.93/1.55 => ( nonemptyI1
% 2.93/1.55 => ( setadjoinIL
% 2.93/1.55 => ( emptyinunitempty
% 2.93/1.55 => ( setadjoinIR
% 2.93/1.55 => ( setadjoinE
% 2.93/1.55 => ( setadjoinOr
% 2.93/1.55 => ( setoftrueEq
% 2.93/1.55 => ( powersetI
% 2.93/1.55 => ( emptyinPowerset
% 2.93/1.55 => ( emptyInPowerset
% 2.93/1.55 => ( powersetE
% 2.93/1.55 => ( setunionI
% 2.93/1.55 => ( setunionE
% 2.93/1.55 => ( subPowSU
% 2.93/1.55 => ( exuE2
% 2.93/1.55 => ( nonemptyImpWitness
% 2.93/1.55 => ( uniqinunit
% 2.93/1.55 => ( notinsingleton
% 2.93/1.55 => ( eqinunit
% 2.93/1.55 => ( singletonsswitch
% 2.93/1.55 => ( upairsetE
% 2.93/1.55 => ( upairsetIL
% 2.93/1.55 => ( upairsetIR
% 2.93/1.55 => ( emptyE1
% 2.93/1.55 => ( vacuousDall
% 2.93/1.55 => ( quantDeMorgan1
% 2.93/1.55 => ( quantDeMorgan2
% 2.93/1.55 => ( quantDeMorgan3
% 2.93/1.55 => ( quantDeMorgan4
% 2.93/1.55 => ( prop2setI
% 2.93/1.55 => ( prop2set2propI
% 2.93/1.55 => ( notdexE
% 2.93/1.55 => ( notdallE
% 2.93/1.55 => ( exuI1
% 2.93/1.55 => ( exuI3
% 2.93/1.55 => ( exuI2
% 2.93/1.55 => ( inCongP
% 2.93/1.55 => ( in__Cong
% 2.93/1.55 => ( exuE3u
% 2.93/1.55 => ( exu__Cong
% 2.93/1.55 => ( emptyset__Cong
% 2.93/1.55 => ( setadjoin__Cong
% 2.93/1.55 => ( powerset__Cong
% 2.93/1.55 => ( setunion__Cong
% 2.93/1.55 => ( omega__Cong
% 2.93/1.55 => ( exuEu
% 2.93/1.55 => ( descr__Cong
% 2.93/1.55 => ( dsetconstr__Cong
% 2.93/1.55 => ( subsetI1
% 2.93/1.55 => ( eqimpsubset2
% 2.93/1.55 => ( eqimpsubset1
% 2.93/1.55 => ( subsetI2
% 2.93/1.55 => ( emptysetsubset
% 2.93/1.55 => ( subsetE
% 2.93/1.55 => ( subsetE2
% 2.93/1.55 => ( notsubsetI
% 2.93/1.55 => ( notequalI1
% 2.93/1.55 => ( notequalI2
% 2.93/1.55 => ( subsetRefl
% 2.93/1.55 => ( subsetTrans
% 2.93/1.55 => ( setadjoinSub
% 2.93/1.55 => ( setadjoinSub2
% 2.93/1.55 => ( subset2powerset
% 2.93/1.55 => ( setextsub
% 2.93/1.55 => ( subsetemptysetimpeq
% 2.93/1.55 => ( powersetI1
% 2.93/1.55 => ( powersetE1
% 2.93/1.55 => ( inPowerset
% 2.93/1.55 => ( powersetsubset
% 2.93/1.55 => ( sepInPowerset
% 2.93/1.55 => ( sepSubset
% 2.93/1.55 => ( binunionIL
% 2.93/1.55 => ( upairset2IR
% 2.93/1.55 => ( binunionIR
% 2.93/1.55 => ( binunionEcases
% 2.93/1.55 => ( binunionE
% 2.93/1.55 => ( binunionLsub
% 2.93/1.55 => ( binunionRsub
% 2.93/1.55 => ( binintersectI
% 2.93/1.55 => ( binintersectSubset5
% 2.93/1.55 => ( binintersectEL
% 2.93/1.55 => ( binintersectLsub
% 2.93/1.55 => ( binintersectSubset2
% 2.93/1.55 => ( binintersectSubset3
% 2.93/1.55 => ( binintersectER
% 2.93/1.55 => ( disjointsetsI1
% 2.93/1.55 => ( binintersectRsub
% 2.93/1.55 => ( binintersectSubset4
% 2.93/1.55 => ( binintersectSubset1
% 2.93/1.55 => ( bs114d
% 2.93/1.55 => ( setminusI
% 2.93/1.55 => ( setminusEL
% 2.93/1.55 => ( setminusER
% 2.93/1.55 => ( setminusSubset2
% 2.93/1.55 => ( setminusERneg
% 2.93/1.55 => ( setminusELneg
% 2.93/1.55 => ( setminusILneg
% 2.93/1.55 => ( setminusIRneg
% 2.93/1.55 => ( setminusLsub
% 2.93/1.55 => ( setminusSubset1
% 2.93/1.55 => ( symdiffE
% 2.93/1.55 => ( symdiffI1
% 2.93/1.55 => ( symdiffI2
% 2.93/1.55 => ( symdiffIneg1
% 2.93/1.55 => ( symdiffIneg2
% 2.93/1.55 => ( secondinupair
% 2.93/1.55 => ( setukpairIL
% 2.93/1.55 => ( setukpairIR
% 2.93/1.55 => ( kpairiskpair
% 2.93/1.55 => ( kpairp
% 2.93/1.55 => ( singletonsubset
% 2.93/1.55 => ( singletoninpowerset
% 2.93/1.55 => ( singletoninpowunion
% 2.93/1.55 => ( upairset2E
% 2.93/1.55 => ( upairsubunion
% 2.93/1.55 => ( upairinpowunion
% 2.93/1.55 => ( ubforcartprodlem1
% 2.93/1.55 => ( ubforcartprodlem2
% 2.93/1.55 => ( ubforcartprodlem3
% 2.93/1.55 => ( cartprodpairin
% 2.93/1.55 => ( cartprodmempair1
% 2.93/1.55 => ( cartprodmempair
% 2.93/1.55 => ( setunionE2
% 2.93/1.55 => ( setunionsingleton1
% 2.93/1.55 => ( setunionsingleton2
% 2.93/1.55 => ( setunionsingleton
% 2.93/1.55 => ( singletonprop
% 2.93/1.55 => ( ex1E1
% 2.93/1.55 => ( ex1I
% 2.93/1.55 => ( ex1I2
% 2.93/1.55 => ( singletonsuniq
% 2.93/1.55 => ( setukpairinjL1
% 2.93/1.55 => ( kfstsingleton
% 2.93/1.55 => ( theprop
% 2.93/1.55 => ( kfstpairEq
% 2.93/1.55 => ( cartprodfstin
% 2.93/1.55 => ( setukpairinjL2
% 2.93/1.55 => ( setukpairinjL
% 2.93/1.55 => ( setukpairinjR11
% 2.93/1.55 => ( setukpairinjR12
% 2.93/1.55 => ( setukpairinjR1
% 2.93/1.55 => ( upairequniteq
% 2.93/1.55 => ( setukpairinjR2
% 2.93/1.55 => ( setukpairinjR
% 2.93/1.55 => ( ksndsingleton
% 2.93/1.55 => ( ksndpairEq
% 2.93/1.55 => ( kpairsurjEq
% 2.93/1.55 => ( cartprodsndin
% 2.93/1.55 => ( cartprodpairmemEL
% 2.93/1.55 => ( cartprodpairmemER
% 2.93/1.55 => ( cartprodmempaircEq
% 2.93/1.55 => ( cartprodfstpairEq
% 2.93/1.55 => ( cartprodsndpairEq
% 2.93/1.55 => ( cartprodpairsurjEq
% 2.93/1.55 => ( dpsetconstrI
% 2.93/1.55 => ( dpsetconstrSub
% 2.93/1.55 => ( setOfPairsIsBReln
% 2.93/1.55 => ( dpsetconstrERa
% 2.93/1.55 => ( dpsetconstrEL1
% 2.93/1.55 => ( dpsetconstrEL2
% 2.93/1.55 => ( dpsetconstrER
% 2.93/1.55 => ( funcImageSingleton
% 2.93/1.55 => ( apProp
% 2.93/1.55 => ( app
% 2.93/1.55 => ( infuncsetfunc
% 2.93/1.55 => ( ap2p
% 2.93/1.55 => ( funcinfuncset
% 2.93/1.55 => ( lamProp
% 2.93/1.55 => ( lamp
% 2.93/1.55 => ( lam2p
% 2.93/1.55 => ( brelnall1
% 2.93/1.55 => ( brelnall2
% 2.93/1.55 => ( ex1E2
% 2.93/1.55 => ( funcGraphProp1
% 2.93/1.55 => ( funcGraphProp3
% 2.93/1.55 => ( funcGraphProp2
% 2.93/1.55 => ( funcextLem
% 2.93/1.55 => ( funcGraphProp4
% 2.93/1.55 => ( subbreln
% 2.93/1.55 => ( eqbreln
% 2.93/1.55 => ( funcext
% 2.93/1.55 => ( funcext2
% 2.93/1.55 => ( ap2apEq1
% 2.93/1.55 => ( ap2apEq2
% 2.93/1.55 => ( beta1
% 2.93/1.55 => ( eta1
% 2.93/1.55 => ( lam2lamEq
% 2.93/1.55 => ( beta2
% 2.93/1.55 => ( eta2
% 2.93/1.55 => ( iffalseProp1
% 2.93/1.55 => ( iffalseProp2
% 2.93/1.55 => ( iftrueProp1
% 2.93/1.55 => ( iftrueProp2
% 2.93/1.55 => ( ifSingleton
% 2.93/1.55 => ( ifp
% 2.93/1.55 => ( theeq
% 2.93/1.55 => ( iftrue
% 2.93/1.55 => ( iffalse
% 2.93/1.55 => ( iftrueorfalse
% 2.93/1.55 => ( binintersectT_lem
% 2.93/1.55 => ( binunionT_lem
% 2.93/1.55 => ( powersetT_lem
% 2.93/1.55 => ( setminusT_lem
% 2.93/1.55 => ( complementT_lem
% 2.93/1.55 => ( setextT
% 2.93/1.55 => ( subsetTI
% 2.93/1.55 => ( powersetTI1
% 2.93/1.55 => ( powersetTE1
% 2.93/1.55 => ( complementTI1
% 2.93/1.55 => ( complementTE1
% 2.93/1.55 => ( binintersectTELcontra
% 2.93/1.55 => ( binintersectTERcontra
% 2.93/1.55 => ( contrasubsetT
% 2.93/1.55 => ( contrasubsetT1
% 2.93/1.55 => ( contrasubsetT2
% 2.93/1.55 => ( contrasubsetT3
% 2.93/1.55 => ( doubleComplementI1
% 2.93/1.55 => ( doubleComplementE1
% 2.93/1.55 => ( doubleComplementSub1
% 2.93/1.55 => ( doubleComplementSub2
% 2.93/1.55 => ( doubleComplementEq
% 2.93/1.55 => ( complementTnotintersectT
% 2.93/1.55 => ( complementImpComplementIntersect
% 2.93/1.55 => ( complementSubsetComplementIntersect
% 2.93/1.55 => ( complementInPowersetComplementIntersect
% 2.93/1.55 => ( contraSubsetComplement
% 2.93/1.55 => ( complementTcontraSubset
% 2.93/1.55 => ( binunionTILcontra
% 2.93/1.55 => ( binunionTIRcontra
% 2.93/1.55 => ( inIntersectImpInUnion
% 2.93/1.55 => ( inIntersectImpInUnion2
% 2.93/1.55 => ( inIntersectImpInIntersectUnions
% 2.93/1.55 => ( intersectInPowersetIntersectUnions
% 2.93/1.55 => ( inComplementUnionImpNotIn1
% 2.93/1.55 => ( inComplementUnionImpInComplement1
% 2.93/1.55 => ( binunionTE
% 2.93/1.55 => ( binunionTEcontra
% 2.93/1.55 => ( demorgan2a1
% 2.93/1.55 => ( complementUnionInPowersetComplement
% 2.93/1.55 => ( demorgan2a2
% 2.93/1.55 => ( demorgan1a
% 2.93/1.55 => ( demorgan1b
% 2.93/1.55 => ( demorgan1
% 2.93/1.55 => ( demorgan2a
% 2.93/1.55 => ( demorgan2b2
% 2.93/1.55 => ( demorgan2b
% 2.93/1.55 => ( demorgan2
% 2.93/1.55 => ( woz13rule0
% 2.93/1.55 => ( woz13rule1
% 2.93/1.55 => ( woz13rule2
% 2.93/1.55 => ( woz13rule3
% 2.93/1.55 => ( woz13rule4
% 2.93/1.55 => ( woz1_1
% 2.93/1.55 => ( woz1_2
% 2.93/1.55 => ( woz1_3
% 2.93/1.55 => ( woz1_4
% 2.93/1.55 => ( woz1_5
% 2.93/1.55 => ( breln1all2
% 2.93/1.55 => ( breln1SetBreln1
% 2.93/1.55 => ( choice2fnsingleton
% 2.93/1.55 => ( setOfPairsIsBReln1
% 2.93/1.55 => ( breln1all1
% 2.93/1.55 => ( subbreln1
% 2.93/1.55 => ( eqbreln1
% 2.93/1.55 => ( breln1invprop
% 2.93/1.55 => ( breln1invI
% 2.93/1.55 => ( breln1invE
% 2.93/1.55 => ( breln1compprop
% 2.93/1.55 => ( breln1compI
% 2.93/1.55 => ( breln1compE
% 2.93/1.55 => ( breln1compEex
% 2.93/1.55 => ( breln1unionprop
% 2.93/1.55 => ( breln1unionIL
% 2.93/1.55 => ( breln1unionIR
% 2.93/1.55 => ( breln1unionI
% 2.93/1.55 => ( breln1unionE
% 2.93/1.55 => ! [A: $i,R: $i] :
% 2.93/1.55 ( ( breln1 @ A @ R )
% 2.93/1.55 => ! [S: $i] :
% 2.93/1.55 ( ( breln1 @ A @ S )
% 2.93/1.55 => ! [Xx: $i] :
% 2.93/1.55 ( ( in @ Xx @ A )
% 2.93/1.55 => ! [Xy: $i] :
% 2.93/1.55 ( ( in @ Xy @ A )
% 2.93/1.55 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) )
% 2.93/1.55 => ! [Xphi: $o] :
% 2.93/1.55 ( ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.93/1.55 => Xphi )
% 2.93/1.55 => ( ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 2.93/1.55 => Xphi )
% 2.93/1.55 => Xphi ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.93/1.55 ------------------------
% 2.93/1.58 % (4970)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_4862 for (2999ds/183Mi)
% 2.93/1.59 % (4971)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_4862 for (2999ds/4Mi)
% 2.93/1.59 % (4974)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_4862 for (2999ds/2Mi)
% 2.93/1.59 % (4972)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_4862 for (2999ds/27Mi)
% 2.93/1.59 % (4973)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_4862 for (2999ds/2Mi)
% 2.93/1.59 % (4975)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_4862 for (2999ds/275Mi)
% 2.93/1.59 % (4976)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_4862 for (2999ds/18Mi)
% 2.93/1.59 % (4973)Instruction limit reached!
% 2.93/1.59 % (4973)------------------------------
% 2.93/1.59 % (4973)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.59 % (4973)Termination reason: Unknown
% 2.93/1.59 % (4974)Instruction limit reached!
% 2.93/1.59 % (4974)------------------------------
% 2.93/1.59 % (4974)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.59 % (4973)Termination phase: shuffling
% 2.93/1.59
% 2.93/1.59 % (4973)Memory used [KB]: 1535
% 2.93/1.59 % (4973)Time elapsed: 0.003 s
% 2.93/1.59 % (4973)Instructions burned: 2 (million)
% 2.93/1.59 % (4973)------------------------------
% 2.93/1.59 % (4973)------------------------------
% 2.93/1.59 % (4974)Termination reason: Unknown
% 2.93/1.59 % (4974)Termination phase: shuffling
% 2.93/1.59
% 2.93/1.59 % (4974)Memory used [KB]: 1535
% 2.93/1.59 % (4974)Time elapsed: 0.003 s
% 2.93/1.59 % (4974)Instructions burned: 3 (million)
% 2.93/1.59 % (4974)------------------------------
% 2.93/1.59 % (4974)------------------------------
% 2.93/1.59 % (4971)Instruction limit reached!
% 2.93/1.59 % (4971)------------------------------
% 2.93/1.59 % (4971)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.59 % (4971)Termination reason: Unknown
% 2.93/1.59 % (4971)Termination phase: shuffling
% 2.93/1.59
% 2.93/1.59 % (4971)Memory used [KB]: 1663
% 2.93/1.59 % (4971)Time elapsed: 0.004 s
% 2.93/1.59 % (4971)Instructions burned: 5 (million)
% 2.93/1.59 % (4971)------------------------------
% 2.93/1.59 % (4971)------------------------------
% 3.00/1.60 % (4976)Instruction limit reached!
% 3.00/1.60 % (4976)------------------------------
% 3.00/1.60 % (4976)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.60 % (4976)Termination reason: Unknown
% 3.00/1.60 % (4976)Termination phase: shuffling
% 3.00/1.60
% 3.00/1.60 % (4976)Memory used [KB]: 1791
% 3.00/1.60 % (4976)Time elapsed: 0.011 s
% 3.00/1.60 % (4976)Instructions burned: 19 (million)
% 3.00/1.60 % (4976)------------------------------
% 3.00/1.60 % (4976)------------------------------
% 3.00/1.60 % (4972)Instruction limit reached!
% 3.00/1.60 % (4972)------------------------------
% 3.00/1.60 % (4972)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.60 % (4972)Termination reason: Unknown
% 3.00/1.60 % (4972)Termination phase: shuffling
% 3.00/1.60
% 3.00/1.60 % (4972)Memory used [KB]: 1918
% 3.00/1.60 % (4972)Time elapsed: 0.014 s
% 3.00/1.60 % (4972)Instructions burned: 27 (million)
% 3.00/1.60 % (4972)------------------------------
% 3.00/1.60 % (4972)------------------------------
% 3.00/1.60 % (4977)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_4862 for (2999ds/3Mi)
% 3.00/1.60 % (4978)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_4862 for (2999ds/37Mi)
% 3.00/1.61 % (4977)Instruction limit reached!
% 3.00/1.61 % (4977)------------------------------
% 3.00/1.61 % (4977)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.61 % (4977)Termination reason: Unknown
% 3.00/1.61 % (4977)Termination phase: shuffling
% 3.00/1.61
% 3.00/1.61 % (4977)Memory used [KB]: 1535
% 3.00/1.61 % (4977)Time elapsed: 0.003 s
% 3.00/1.61 % (4977)Instructions burned: 3 (million)
% 3.00/1.61 % (4977)------------------------------
% 3.00/1.61 % (4977)------------------------------
% 3.00/1.61 % (4979)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_4862 for (2999ds/15Mi)
% 3.00/1.61 % (4980)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_4862 for (2999ds/3Mi)
% 3.00/1.62 % (4981)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_4862 for (2999ds/1041Mi)
% 3.00/1.62 % (4979)Instruction limit reached!
% 3.00/1.62 % (4979)------------------------------
% 3.00/1.62 % (4979)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.62 % (4979)Termination reason: Unknown
% 3.00/1.62 % (4979)Termination phase: shuffling
% 3.00/1.62
% 3.00/1.62 % (4979)Memory used [KB]: 1791
% 3.00/1.62 % (4979)Time elapsed: 0.009 s
% 3.00/1.62 % (4979)Instructions burned: 15 (million)
% 3.00/1.62 % (4979)------------------------------
% 3.00/1.62 % (4979)------------------------------
% 3.00/1.62 % (4980)Instruction limit reached!
% 3.00/1.62 % (4980)------------------------------
% 3.00/1.62 % (4980)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.62 % (4980)Termination reason: Unknown
% 3.00/1.62 % (4980)Termination phase: shuffling
% 3.00/1.62
% 3.00/1.62 % (4980)Memory used [KB]: 1535
% 3.00/1.62 % (4980)Time elapsed: 0.003 s
% 3.00/1.62 % (4980)Instructions burned: 3 (million)
% 3.00/1.62 % (4980)------------------------------
% 3.00/1.62 % (4980)------------------------------
% 3.00/1.62 % (4978)Instruction limit reached!
% 3.00/1.62 % (4978)------------------------------
% 3.00/1.62 % (4978)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.62 % (4978)Termination reason: Unknown
% 3.00/1.62 % (4978)Termination phase: shuffling
% 3.00/1.62
% 3.00/1.62 % (4978)Memory used [KB]: 2302
% 3.00/1.62 % (4978)Time elapsed: 0.018 s
% 3.00/1.62 % (4978)Instructions burned: 37 (million)
% 3.00/1.62 % (4978)------------------------------
% 3.00/1.62 % (4978)------------------------------
% 3.00/1.62 % (4982)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_4862 for (2999ds/7Mi)
% 3.00/1.63 % (4982)Instruction limit reached!
% 3.00/1.63 % (4982)------------------------------
% 3.00/1.63 % (4982)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.63 % (4982)Termination reason: Unknown
% 3.00/1.63 % (4982)Termination phase: shuffling
% 3.00/1.63
% 3.00/1.63 % (4982)Memory used [KB]: 1663
% 3.00/1.63 % (4982)Time elapsed: 0.005 s
% 3.00/1.63 % (4982)Instructions burned: 7 (million)
% 3.00/1.63 % (4982)------------------------------
% 3.00/1.63 % (4982)------------------------------
% 3.00/1.63 % (4970)Instruction limit reached!
% 3.00/1.63 % (4970)------------------------------
% 3.00/1.63 % (4970)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.63 % (4970)Termination reason: Unknown
% 3.00/1.63 % (4970)Termination phase: Saturation
% 3.00/1.63
% 3.00/1.63 % (4970)Memory used [KB]: 2942
% 3.00/1.63 % (4970)Time elapsed: 0.049 s
% 3.00/1.63 % (4970)Instructions burned: 185 (million)
% 3.00/1.63 % (4970)------------------------------
% 3.00/1.63 % (4970)------------------------------
% 3.00/1.63 % (4983)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_4862 for (2999ds/16Mi)
% 3.00/1.63 % (4984)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_4862 for (2999ds/3Mi)
% 3.00/1.63 % (4984)Instruction limit reached!
% 3.00/1.63 % (4984)------------------------------
% 3.00/1.63 % (4984)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.63 % (4984)Termination reason: Unknown
% 3.00/1.63 % (4984)Termination phase: shuffling
% 3.00/1.63
% 3.00/1.63 % (4984)Memory used [KB]: 1535
% 3.00/1.63 % (4984)Time elapsed: 0.003 s
% 3.00/1.63 % (4984)Instructions burned: 3 (million)
% 3.00/1.63 % (4984)------------------------------
% 3.00/1.63 % (4984)------------------------------
% 3.00/1.64 % (4985)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_4862 for (2999ds/3Mi)
% 3.00/1.64 % (4987)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_4862 for (2999ds/3Mi)
% 3.00/1.64 % (4985)Instruction limit reached!
% 3.00/1.64 % (4985)------------------------------
% 3.00/1.64 % (4985)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.64 % (4985)Termination reason: Unknown
% 3.00/1.64 % (4985)Termination phase: shuffling
% 3.00/1.64
% 3.00/1.64 % (4985)Memory used [KB]: 1535
% 3.00/1.64 % (4985)Time elapsed: 0.003 s
% 3.00/1.64 % (4985)Instructions burned: 3 (million)
% 3.00/1.64 % (4985)------------------------------
% 3.00/1.64 % (4985)------------------------------
% 3.00/1.64 % (4987)Instruction limit reached!
% 3.00/1.64 % (4987)------------------------------
% 3.00/1.64 % (4987)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.64 % (4987)Termination reason: Unknown
% 3.00/1.64 % (4987)Termination phase: shuffling
% 3.00/1.64
% 3.00/1.64 % (4987)Memory used [KB]: 1535
% 3.00/1.64 % (4987)Time elapsed: 0.002 s
% 3.00/1.64 % (4987)Instructions burned: 3 (million)
% 3.00/1.64 % (4987)------------------------------
% 3.00/1.64 % (4987)------------------------------
% 3.00/1.64 % (4983)Instruction limit reached!
% 3.00/1.64 % (4983)------------------------------
% 3.00/1.64 % (4983)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.64 % (4983)Termination reason: Unknown
% 3.00/1.64 % (4983)Termination phase: shuffling
% 3.00/1.64
% 3.00/1.64 % (4983)Memory used [KB]: 1918
% 3.00/1.64 % (4983)Time elapsed: 0.009 s
% 3.00/1.64 % (4983)Instructions burned: 17 (million)
% 3.00/1.64 % (4983)------------------------------
% 3.00/1.64 % (4983)------------------------------
% 3.00/1.64 % (4986)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_4862 for (2999ds/7Mi)
% 3.00/1.65 % (4986)Instruction limit reached!
% 3.00/1.65 % (4986)------------------------------
% 3.00/1.65 % (4986)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.65 % (4986)Termination reason: Unknown
% 3.00/1.65 % (4986)Termination phase: shuffling
% 3.00/1.65
% 3.00/1.65 % (4986)Memory used [KB]: 1663
% 3.00/1.65 % (4986)Time elapsed: 0.005 s
% 3.00/1.65 % (4986)Instructions burned: 8 (million)
% 3.00/1.65 % (4986)------------------------------
% 3.00/1.65 % (4986)------------------------------
% 3.00/1.65 % (4989)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_4862 for (2999ds/18Mi)
% 3.00/1.65 % (4988)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_4862 for (2999ds/4Mi)
% 3.00/1.65 % (4988)Instruction limit reached!
% 3.00/1.65 % (4988)------------------------------
% 3.00/1.65 % (4988)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.65 % (4988)Termination reason: Unknown
% 3.00/1.65 % (4988)Termination phase: shuffling
% 3.00/1.65
% 3.00/1.65 % (4988)Memory used [KB]: 1535
% 3.00/1.65 % (4988)Time elapsed: 0.004 s
% 3.00/1.65 % (4988)Instructions burned: 6 (million)
% 3.00/1.65 % (4988)------------------------------
% 3.00/1.65 % (4988)------------------------------
% 3.00/1.65 % (4989)Instruction limit reached!
% 3.00/1.65 % (4989)------------------------------
% 3.00/1.65 % (4989)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.65 % (4989)Termination reason: Unknown
% 3.00/1.65 % (4989)Termination phase: shuffling
% 3.00/1.65
% 3.00/1.65 % (4989)Memory used [KB]: 1918
% 3.00/1.65 % (4989)Time elapsed: 0.007 s
% 3.00/1.65 % (4989)Instructions burned: 21 (million)
% 3.00/1.65 % (4989)------------------------------
% 3.00/1.65 % (4989)------------------------------
% 3.00/1.66 % (4990)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_4862 for (2999ds/710Mi)
% 3.00/1.66 % (4991)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_4862 for (2999ds/6Mi)
% 3.00/1.66 % (4991)Instruction limit reached!
% 3.00/1.66 % (4991)------------------------------
% 3.00/1.66 % (4991)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.66 % (4991)Termination reason: Unknown
% 3.00/1.66 % (4991)Termination phase: shuffling
% 3.00/1.66
% 3.00/1.66 % (4991)Memory used [KB]: 1535
% 3.00/1.66 % (4991)Time elapsed: 0.004 s
% 3.00/1.66 % (4991)Instructions burned: 6 (million)
% 3.00/1.66 % (4991)------------------------------
% 3.00/1.66 % (4991)------------------------------
% 3.00/1.66 % (4992)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_4862 for (2999ds/902Mi)
% 3.00/1.66 % (4994)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_4862 for (2999ds/5Mi)
% 3.00/1.67 % (4994)Instruction limit reached!
% 3.00/1.67 % (4994)------------------------------
% 3.00/1.67 % (4994)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.67 % (4994)Termination reason: Unknown
% 3.00/1.67 % (4994)Termination phase: shuffling
% 3.00/1.67
% 3.00/1.67 % (4994)Memory used [KB]: 1663
% 3.00/1.67 % (4994)Time elapsed: 0.003 s
% 3.00/1.67 % (4994)Instructions burned: 7 (million)
% 3.00/1.67 % (4994)------------------------------
% 3.00/1.67 % (4994)------------------------------
% 3.00/1.67 % (4993)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_4862 for (2999ds/21Mi)
% 3.00/1.67 % (4995)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_4862 for (2999ds/6Mi)
% 3.00/1.68 % (4996)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_4862 for (2998ds/377Mi)
% 3.00/1.68 % (4995)Instruction limit reached!
% 3.00/1.68 % (4995)------------------------------
% 3.00/1.68 % (4995)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.68 % (4995)Termination reason: Unknown
% 3.00/1.68 % (4995)Termination phase: shuffling
% 3.00/1.68
% 3.00/1.68 % (4995)Memory used [KB]: 1663
% 3.00/1.68 % (4995)Time elapsed: 0.005 s
% 3.00/1.68 % (4995)Instructions burned: 8 (million)
% 3.00/1.68 % (4995)------------------------------
% 3.00/1.68 % (4995)------------------------------
% 3.00/1.68 % (4993)Instruction limit reached!
% 3.00/1.68 % (4993)------------------------------
% 3.00/1.68 % (4993)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.68 % (4993)Termination reason: Unknown
% 3.00/1.68 % (4993)Termination phase: shuffling
% 3.00/1.68
% 3.00/1.68 % (4993)Memory used [KB]: 1918
% 3.00/1.68 % (4993)Time elapsed: 0.011 s
% 3.00/1.68 % (4993)Instructions burned: 21 (million)
% 3.00/1.68 % (4993)------------------------------
% 3.00/1.68 % (4993)------------------------------
% 3.00/1.69 % (4997)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_4862 for (2998ds/779Mi)
% 3.00/1.69 % (4998)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_4862 for (2998ds/19Mi)
% 3.00/1.70 % (4998)Instruction limit reached!
% 3.00/1.70 % (4998)------------------------------
% 3.00/1.70 % (4998)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.70 % (4998)Termination reason: Unknown
% 3.00/1.70 % (4998)Termination phase: shuffling
% 3.00/1.70
% 3.00/1.70 % (4998)Memory used [KB]: 1918
% 3.00/1.70 % (4998)Time elapsed: 0.010 s
% 3.00/1.70 % (4998)Instructions burned: 19 (million)
% 3.00/1.70 % (4998)------------------------------
% 3.00/1.70 % (4998)------------------------------
% 3.00/1.71 % (4975)Instruction limit reached!
% 3.00/1.71 % (4975)------------------------------
% 3.00/1.71 % (4975)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.71 % (4975)Termination reason: Unknown
% 3.00/1.71 % (4975)Termination phase: Saturation
% 3.00/1.71
% 3.00/1.71 % (4975)Memory used [KB]: 10362
% 3.00/1.71 % (4975)Time elapsed: 0.128 s
% 3.00/1.71 % (4975)Instructions burned: 275 (million)
% 3.00/1.71 % (4975)------------------------------
% 3.00/1.71 % (4975)------------------------------
% 3.00/1.72 % (4999)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_4862 for (2998ds/879Mi)
% 3.00/1.73 % (5000)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_4862 for (2998ds/17Mi)
% 3.00/1.74 % (5000)Instruction limit reached!
% 3.00/1.74 % (5000)------------------------------
% 3.00/1.74 % (5000)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.74 % (5000)Termination reason: Unknown
% 3.00/1.74 % (5000)Termination phase: shuffling
% 3.00/1.74
% 3.00/1.74 % (5000)Memory used [KB]: 1791
% 3.00/1.74 % (5000)Time elapsed: 0.010 s
% 3.00/1.74 % (5000)Instructions burned: 18 (million)
% 3.00/1.74 % (5000)------------------------------
% 3.00/1.74 % (5000)------------------------------
% 3.00/1.76 % (5001)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_4862 for (2998ds/3Mi)
% 3.00/1.76 % (5001)Instruction limit reached!
% 3.00/1.76 % (5001)------------------------------
% 3.00/1.76 % (5001)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.76 % (5001)Termination reason: Unknown
% 3.00/1.76 % (5001)Termination phase: shuffling
% 3.00/1.76
% 3.00/1.76 % (5001)Memory used [KB]: 1535
% 3.00/1.76 % (5001)Time elapsed: 0.004 s
% 3.00/1.76 % (5001)Instructions burned: 5 (million)
% 3.00/1.76 % (5001)------------------------------
% 3.00/1.76 % (5001)------------------------------
% 3.00/1.77 % (5002)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_4862 for (2998ds/30Mi)
% 3.00/1.79 % (5002)Instruction limit reached!
% 3.00/1.79 % (5002)------------------------------
% 3.00/1.79 % (5002)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.79 % (5002)Termination reason: Unknown
% 3.00/1.79 % (5002)Termination phase: shuffling
% 3.00/1.79
% 3.00/1.79 % (5002)Memory used [KB]: 2046
% 3.00/1.79 % (5002)Time elapsed: 0.016 s
% 3.00/1.79 % (5002)Instructions burned: 30 (million)
% 3.00/1.79 % (5002)------------------------------
% 3.00/1.79 % (5002)------------------------------
% 3.00/1.81 % (5004)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_4862 for (2997ds/127Mi)
% 3.00/1.81 % (4996)Instruction limit reached!
% 3.00/1.81 % (4996)------------------------------
% 3.00/1.81 % (4996)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.81 % (4996)Termination reason: Unknown
% 3.00/1.81 % (4996)Termination phase: Saturation
% 3.00/1.81
% 3.00/1.81 % (4996)Memory used [KB]: 9466
% 3.00/1.81 % (4996)Time elapsed: 0.134 s
% 3.00/1.81 % (4996)Instructions burned: 377 (million)
% 3.00/1.81 % (4996)------------------------------
% 3.00/1.81 % (4996)------------------------------
% 3.00/1.83 % (5005)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_4862 for (2997ds/100Mi)
% 3.00/1.86 % (5004)Instruction limit reached!
% 3.00/1.86 % (5004)------------------------------
% 3.00/1.86 % (5004)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.86 % (5004)Termination reason: Unknown
% 3.00/1.86 % (5004)Termination phase: Property scanning
% 3.00/1.86
% 3.00/1.86 % (5004)Memory used [KB]: 2814
% 3.00/1.86 % (5004)Time elapsed: 0.057 s
% 3.00/1.86 % (5004)Instructions burned: 128 (million)
% 3.00/1.86 % (5004)------------------------------
% 3.00/1.86 % (5004)------------------------------
% 3.00/1.87 % (5005)Instruction limit reached!
% 3.00/1.87 % (5005)------------------------------
% 3.00/1.87 % (5005)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.87 % (5005)Termination reason: Unknown
% 3.00/1.87 % (5005)Termination phase: Property scanning
% 3.00/1.87
% 3.00/1.87 % (5005)Memory used [KB]: 2814
% 3.00/1.87 % (5005)Time elapsed: 0.047 s
% 3.00/1.87 % (5005)Instructions burned: 100 (million)
% 3.00/1.87 % (5005)------------------------------
% 3.00/1.87 % (5005)------------------------------
% 3.00/1.87 % (4992)First to succeed.
% 3.00/1.88 % (5006)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_4862 for (2997ds/3Mi)
% 3.00/1.88 % (4992)Refutation found. Thanks to Tanya!
% 3.00/1.88 % SZS status Theorem for DTF2THF_4862
% 3.00/1.88 % SZS output start Proof for DTF2THF_4862
% 3.00/1.88 thf(func_def_0, type, in: $i > $i > $o).
% 3.00/1.88 thf(func_def_1, type, exu: ($i > $o) > $o).
% 3.00/1.88 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 3.00/1.88 thf(func_def_8, type, powerset: $i > $i).
% 3.00/1.88 thf(func_def_10, type, setunion: $i > $i).
% 3.00/1.88 thf(func_def_19, type, descr: ($i > $o) > $i).
% 3.00/1.88 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 3.00/1.88 thf(func_def_26, type, prop2set: $o > $i).
% 3.00/1.88 thf(func_def_36, type, nonempty: $i > $o).
% 3.00/1.88 thf(func_def_69, type, set2prop: $i > $o).
% 3.00/1.88 thf(func_def_88, type, subset: $i > $i > $o).
% 3.00/1.88 thf(func_def_89, type, disjoint: $i > $i > $o).
% 3.00/1.88 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 3.00/1.88 thf(func_def_114, type, binunion: $i > $i > $i).
% 3.00/1.88 thf(func_def_122, type, binintersect: $i > $i > $i).
% 3.00/1.88 thf(func_def_135, type, regular: $i > $o).
% 3.00/1.88 thf(func_def_136, type, setminus: $i > $i > $i).
% 3.00/1.88 thf(func_def_147, type, symdiff: $i > $i > $i).
% 3.00/1.88 thf(func_def_153, type, iskpair: $i > $o).
% 3.00/1.88 thf(func_def_158, type, kpair: $i > $i > $i).
% 3.00/1.88 thf(func_def_160, type, cartprod: $i > $i > $i).
% 3.00/1.88 thf(func_def_177, type, singleton: $i > $o).
% 3.00/1.88 thf(func_def_179, type, ex1: $i > ($i > $o) > $o).
% 3.00/1.88 thf(func_def_184, type, atmost1p: $i > $o).
% 3.00/1.88 thf(func_def_185, type, atleast2p: $i > $o).
% 3.00/1.88 thf(func_def_186, type, atmost2p: $i > $o).
% 3.00/1.88 thf(func_def_187, type, upairsetp: $i > $o).
% 3.00/1.88 thf(func_def_191, type, kfst: $i > $i).
% 3.00/1.88 thf(func_def_203, type, ksnd: $i > $i).
% 3.00/1.88 thf(func_def_213, type, breln: $i > $i > $i > $o).
% 3.00/1.88 thf(func_def_214, type, dpsetconstr: $i > $i > ($i > $i > $o) > $i).
% 3.00/1.88 thf(func_def_222, type, func: $i > $i > $i > $o).
% 3.00/1.88 thf(func_def_223, type, funcSet: $i > $i > $i).
% 3.00/1.88 thf(func_def_226, type, ap: $i > $i > $i > $i > $i).
% 3.00/1.88 thf(func_def_232, type, lam: $i > $i > ($i > $i) > $i).
% 3.00/1.88 thf(func_def_259, type, if: $i > $o > $i > $i > $i).
% 3.00/1.88 thf(func_def_323, type, breln1: $i > $i > $o).
% 3.00/1.88 thf(func_def_325, type, breln1Set: $i > $i).
% 3.00/1.88 thf(func_def_327, type, transitive: $i > $i > $o).
% 3.00/1.88 thf(func_def_328, type, antisymmetric: $i > $i > $o).
% 3.00/1.88 thf(func_def_329, type, reflexive: $i > $i > $o).
% 3.00/1.88 thf(func_def_330, type, refltransitive: $i > $i > $o).
% 3.00/1.88 thf(func_def_331, type, refllinearorder: $i > $i > $o).
% 3.00/1.88 thf(func_def_332, type, reflwellordering: $i > $i > $o).
% 3.00/1.88 thf(func_def_338, type, breln1invset: $i > $i > $i).
% 3.00/1.88 thf(func_def_342, type, breln1compset: $i > $i > $i > $i).
% 3.00/1.88 thf(func_def_365, type, sP0: $i > $i > $i > $o).
% 3.00/1.88 thf(func_def_368, type, sP3: $i > $i > $i > $o).
% 3.00/1.88 thf(func_def_371, type, sP6: $i > $i > $o).
% 3.00/1.88 thf(func_def_372, type, sP7: $i > $o).
% 3.00/1.88 thf(func_def_373, type, sP8: $i > $i > $o).
% 3.00/1.88 thf(func_def_374, type, sP9: $i > $i > $o).
% 3.00/1.88 thf(func_def_378, type, sP13: $i > $i > $i > $i > $o).
% 3.00/1.88 thf(func_def_380, type, sP15: $o > $i > $i > $i > $o).
% 3.00/1.88 thf(func_def_389, type, sK24: $i > $i > $i > $i).
% 3.00/1.88 thf(func_def_393, type, sK28: $i > $i > $i > $i).
% 3.00/1.88 thf(func_def_399, type, sK34: ($i > $o) > $i).
% 3.00/1.88 thf(func_def_400, type, sK35: $i > $o).
% 3.00/1.88 thf(func_def_401, type, sK36: $i > $i).
% 3.00/1.88 thf(func_def_407, type, sK42: $i > $o).
% 3.00/1.88 thf(func_def_409, type, sK44: ($i > $o) > $i > $i > $i > $i).
% 3.00/1.88 thf(func_def_410, type, sK45: ($i > $o) > $i > $i > $i > $i).
% 3.00/1.88 thf(func_def_413, type, sK48: $i > $i > $o).
% 3.00/1.88 thf(func_def_424, type, sK59: $i > $i).
% 3.00/1.88 thf(func_def_425, type, sK60: ($i > $i) > $i > $i > $i).
% 3.00/1.88 thf(func_def_439, type, sK74: $i > $o).
% 3.00/1.88 thf(func_def_442, type, sK77: $i > ($i > $o) > $i).
% 3.00/1.88 thf(func_def_443, type, sK78: $i > ($i > $o) > $i).
% 3.00/1.88 thf(func_def_444, type, sK79: $i > $o).
% 3.00/1.88 thf(func_def_445, type, sK80: ($i > $o) > $i).
% 3.00/1.88 thf(func_def_456, type, sK91: $i > $i > $o).
% 3.00/1.88 thf(func_def_480, type, sK115: $i > $i > $i).
% 3.00/1.88 thf(func_def_496, type, sK131: $i > $o).
% 3.00/1.88 thf(func_def_498, type, sK133: ($i > $o) > $i > $i > $i > $i).
% 3.00/1.88 thf(func_def_499, type, sK134: ($i > $o) > $i > $i > $i > $i).
% 3.00/1.88 thf(func_def_502, type, sK137: $i > $i).
% 3.00/1.88 thf(func_def_529, type, sK164: $i > $i > $i > $i > $i).
% 3.00/1.88 thf(func_def_539, type, sK174: $i > $i > $o).
% 3.00/1.88 thf(func_def_554, type, sK189: $i > ($i > $o) > $i).
% 3.00/1.88 thf(func_def_555, type, sK190: $i > $o).
% 3.00/1.88 thf(func_def_558, type, sK193: $i > $o).
% 3.00/1.88 thf(func_def_561, type, sK196: $i > ($i > $o) > $i).
% 3.00/1.88 thf(func_def_566, type, sK201: ($i > $o) > $i > $i > $i).
% 3.00/1.88 thf(func_def_567, type, sK202: ($i > $o) > $i > $i > $i).
% 3.00/1.88 thf(func_def_570, type, sK205: $i > $o).
% 3.00/1.88 thf(func_def_583, type, sK218: $i > $o).
% 3.00/1.88 thf(func_def_585, type, sK220: ($i > $o) > $i > $i > $i).
% 3.00/1.88 thf(func_def_586, type, sK221: ($i > $o) > $i > $i > $i).
% 3.00/1.88 thf(func_def_590, type, sK225: $i > $i > $i > $i).
% 3.00/1.88 thf(func_def_591, type, sK226: $i > $i > $i > $i).
% 3.00/1.88 thf(func_def_592, type, sK227: $i > $i > $i > $i).
% 3.00/1.88 thf(func_def_593, type, sK228: $i > $i > $i > $i).
% 3.00/1.88 thf(func_def_601, type, sK236: $o > $i > $i > $i).
% 3.00/1.88 thf(func_def_610, type, sK245: $i > $i > $i).
% 3.00/1.88 thf(func_def_612, type, sK247: $i > $o).
% 3.00/1.88 thf(func_def_614, type, sK249: ($i > $o) > $i > $i).
% 3.00/1.88 thf(func_def_615, type, sK250: ($i > $o) > $i > $i).
% 3.00/1.88 thf(func_def_616, type, sK251: $i > $i > ($i > $i) > $i).
% 3.00/1.88 thf(func_def_617, type, sK252: $i > $i).
% 3.00/1.88 thf(func_def_646, type, sK281: $i > $o).
% 3.00/1.88 thf(func_def_648, type, sK283: $i > $i).
% 3.00/1.88 thf(func_def_650, type, sK285: $i > $o).
% 3.00/1.88 thf(func_def_652, type, sK287: $i > ($i > $o) > $i).
% 3.00/1.88 thf(func_def_661, type, sK296: $i > $o).
% 3.00/1.88 thf(func_def_664, type, sK299: $i > $o).
% 3.00/1.88 thf(func_def_673, type, sK308: $i > $i > $i > $i).
% 3.00/1.88 thf(func_def_674, type, sK309: $i > $i > $i > $i).
% 3.00/1.88 thf(func_def_675, type, sK310: $i > $i > $i).
% 3.00/1.88 thf(func_def_676, type, sK311: $i > $i > $i).
% 3.00/1.88 thf(func_def_677, type, sK312: $i > $i > $i).
% 3.00/1.88 thf(func_def_678, type, sK313: $i > $i > $i).
% 3.00/1.88 thf(func_def_679, type, sK314: $i > $i > $i).
% 3.00/1.88 thf(func_def_680, type, sK315: $i > $i > $i > $i).
% 3.00/1.88 thf(func_def_681, type, sK316: $i > $i).
% 3.00/1.88 thf(func_def_682, type, sK317: $i > $i).
% 3.00/1.88 thf(func_def_683, type, sK318: $i > $i).
% 3.00/1.88 thf(func_def_684, type, sK319: $i > $i).
% 3.00/1.88 thf(func_def_685, type, sK320: $i > $i > $i > $i).
% 3.00/1.88 thf(func_def_686, type, sK321: $i > $i > $i > $i).
% 3.00/1.88 thf(func_def_687, type, sK322: $i > $i > $i).
% 3.00/1.88 thf(func_def_688, type, sK323: $i > $i > $i).
% 3.00/1.88 thf(func_def_689, type, sK324: $i > $i > $i).
% 3.00/1.88 thf(func_def_690, type, sK325: $i > $i).
% 3.00/1.88 thf(func_def_692, type, sK327: $i > $i).
% 3.00/1.88 thf(func_def_693, type, sK328: $i > $i).
% 3.00/1.88 thf(func_def_717, type, sK352: $i > $i > $i > $i > $i).
% 3.00/1.88 thf(func_def_737, type, sK372: $i > $i).
% 3.00/1.88 thf(func_def_739, type, sK374: $i > ($i > $i) > $i > $i).
% 3.00/1.88 thf(func_def_747, type, sK382: $i > $i > $o).
% 3.00/1.88 thf(func_def_762, type, sK397: $i > $o).
% 3.00/1.88 thf(func_def_776, type, sK411: $i > $o).
% 3.00/1.88 thf(func_def_791, type, sK426: $i > $i > $i).
% 3.00/1.88 thf(func_def_794, type, sK429: $i > $o).
% 3.00/1.88 thf(func_def_816, type, sK451: $i > $o).
% 3.00/1.88 thf(func_def_820, type, sK455: $i > $o).
% 3.00/1.88 thf(func_def_822, type, sK457: ($i > $o) > $i > $i).
% 3.00/1.88 thf(func_def_823, type, sK458: $i > $o).
% 3.00/1.88 thf(func_def_826, type, sK461: $i > ($i > $o) > $i).
% 3.00/1.88 thf(func_def_850, type, sK485: $i > $i > $i).
% 3.00/1.88 thf(func_def_851, type, sK486: $i > $i > $i).
% 3.00/1.88 thf(func_def_853, type, sK488: $i > $o).
% 3.00/1.88 thf(func_def_857, type, sK492: $i > $i > $i).
% 3.00/1.88 thf(func_def_858, type, sK493: $i > $i > $i).
% 3.00/1.88 thf(func_def_861, type, sK496: ($i > $i) > $i > $i > $i).
% 3.00/1.88 thf(func_def_864, type, sK499: $i > $i).
% 3.00/1.88 thf(func_def_885, type, sK520: $i > $i > ($i > $i) > $i).
% 3.00/1.88 thf(func_def_886, type, sK521: $i > $i).
% 3.00/1.88 thf(func_def_927, type, sK562: ($i > $o) > ($i > $o) > $i).
% 3.00/1.88 thf(func_def_928, type, sK563: ($i > $o) > ($i > $o) > $i).
% 3.00/1.88 thf(func_def_929, type, sK564: $i > $o).
% 3.00/1.88 thf(func_def_930, type, sK565: $i > $o).
% 3.00/1.88 thf(func_def_947, type, sK582: $i > $i > $i > $i > $i).
% 3.00/1.88 thf(func_def_948, type, sK583: $i > $i > $i > $i > $i).
% 3.00/1.88 thf(func_def_956, type, sK591: $i > $i > $i > $i).
% 3.00/1.88 thf(func_def_957, type, sK592: $i > $i > $i > $i).
% 3.00/1.88 thf(func_def_970, type, sK605: $i > $i > $o).
% 3.00/1.88 thf(func_def_973, type, sK608: $i > ($i > $o) > $i > $i).
% 3.00/1.88 thf(func_def_975, type, sK610: $i > $o).
% 3.00/1.88 thf(func_def_1007, type, sK642: $i > $i > $o).
% 3.00/1.88 thf(func_def_1008, type, sK643: $i > $i).
% 3.00/1.88 thf(func_def_1010, type, sK645: $i > $i).
% 3.00/1.88 thf(func_def_1021, type, sK656: $i > $o).
% 3.00/1.88 thf(func_def_1037, type, sK672: $i > $i > $i > $i).
% 3.00/1.88 thf(func_def_1041, type, sK676: $i > $o).
% 3.00/1.88 thf(func_def_1061, type, sK696: $i > $i > $i).
% 3.00/1.88 thf(func_def_1074, type, sK709: $i > $i > $i > $i > $i > $i).
% 3.00/1.88 thf(func_def_1075, type, sK710: $i > $o).
% 3.00/1.88 thf(func_def_1076, type, sK711: $i > $o).
% 3.00/1.88 thf(func_def_1077, type, sK712: ($i > $o) > ($i > $o) > $i).
% 3.00/1.88 thf(func_def_1078, type, sK713: ($i > $o) > ($i > $o) > $i).
% 3.00/1.88 thf(func_def_1087, type, sK722: $i > $i).
% 3.00/1.88 thf(func_def_1095, type, sK730: $i > $i).
% 3.00/1.88 thf(func_def_1098, type, sK733: ($i > $o) > $i > $i).
% 3.00/1.88 thf(func_def_1100, type, sK735: $i > $o).
% 3.00/1.88 thf(func_def_1104, type, sK739: $i > $i > $o).
% 3.00/1.88 thf(func_def_1118, type, sK753: $i > $i > $o).
% 3.00/1.88 thf(func_def_1120, type, sK755: $i > $i).
% 3.00/1.88 thf(func_def_1121, type, sK756: $i > $i).
% 3.00/1.88 thf(func_def_1122, type, sK757: $i > ($i > $i > $o) > $i).
% 3.00/1.88 thf(func_def_1123, type, sK758: $i > $i > ($i > $i > $o) > $i).
% 3.00/1.88 thf(func_def_1124, type, sK759: $i > ($i > $i > $o) > $i).
% 3.00/1.88 thf(func_def_1130, type, sK765: $i > $i > $i).
% 3.00/1.88 thf(func_def_1136, type, sK771: $i > $i > $o).
% 3.00/1.88 thf(func_def_1155, type, sK790: $i > $i > $o).
% 3.00/1.88 thf(func_def_1157, type, sK792: $i > $i).
% 3.00/1.88 thf(func_def_1160, type, sK795: $i > ($i > $i > $o) > $i > $i).
% 3.00/1.88 thf(func_def_1167, type, sK802: $o > $i > $i > $i > $i > $i > $i).
% 3.00/1.88 thf(func_def_1178, type, sK813: $i > $i > $i).
% 3.00/1.88 thf(func_def_1204, type, sK839: $i > $i > $i > $i > $i).
% 3.00/1.88 thf(func_def_1205, type, sK840: $i > $i > $i > $i > $i).
% 3.00/1.88 thf(func_def_1206, type, sK841: $i > $i > $i > $i > $i).
% 3.00/1.88 thf(func_def_1207, type, sK842: $i > $i > $i > $i > $i).
% 3.00/1.88 thf(func_def_1218, type, sK853: $i > $i > $i > $i > $i).
% 3.00/1.88 thf(func_def_1225, type, sK860: $i > $o).
% 3.00/1.88 thf(func_def_1235, type, sK870: $i > $o).
% 3.00/1.88 thf(func_def_1258, type, sK893: $i > $i > $i > $i).
% 3.00/1.88 thf(func_def_1271, type, sK906: $i > $o).
% 3.00/1.88 thf(func_def_1272, type, sK907: $i > $i).
% 3.00/1.88 thf(func_def_1273, type, sK908: ($i > $o) > $i).
% 3.00/1.88 thf(func_def_1295, type, sK930: $i > $i).
% 3.00/1.88 thf(func_def_1308, type, sK943: $i > $o).
% 3.00/1.88 thf(func_def_1309, type, sK944: $i > $o).
% 3.00/1.88 thf(func_def_1310, type, sK945: ($i > $o) > ($i > $o) > $i > $i > $i).
% 3.00/1.88 thf(func_def_1311, type, sK946: ($i > $o) > ($i > $o) > $i > $i > $i).
% 3.00/1.88 thf(func_def_1340, type, sK975: $i > $o).
% 3.00/1.88 thf(func_def_1342, type, sK977: ($i > $o) > $i > $i).
% 3.00/1.88 thf(func_def_1343, type, sK978: $i > $o).
% 3.00/1.88 thf(func_def_1345, type, sK980: $i > ($i > $o) > $i).
% 3.00/1.88 thf(func_def_1350, type, sK985: $i > $i).
% 3.00/1.88 thf(func_def_1352, type, sK987: ($i > $i) > $i > $i > $i).
% 3.00/1.88 thf(func_def_1365, type, sK1000: $i > $o).
% 3.00/1.88 thf(func_def_1367, type, sK1002: ($i > $o) > $i).
% 3.00/1.88 thf(func_def_1368, type, sK1003: ($i > $o) > $i).
% 3.00/1.88 thf(func_def_1369, type, sK1004: $i > $o).
% 3.00/1.88 thf(func_def_1375, type, sK1010: $i > ($i > $o) > $i).
% 3.00/1.88 thf(func_def_1376, type, sK1011: $i > $o).
% 3.00/1.88 thf(func_def_1389, type, ph1024: !>[X0: $tType]:(X0)).
% 3.00/1.88 thf(f4908,plain,(
% 3.00/1.88 $false),
% 3.00/1.88 inference(avatar_sat_refutation,[status(thm)],[f4786,f4791,f4792,f4896])).
% 3.00/1.88 thf(f4896,plain,(
% 3.00/1.88 spl1023_1 | spl1023_3),
% 3.00/1.88 inference(avatar_split_clause,[status(thm)],[f4895,f4788,f4779])).
% 3.00/1.88 thf(f4779,definition,(
% 3.00/1.88 spl1023_1 <=> ($true = (in @ (kpair @ sK821 @ sK822) @ sK819))),
% 3.00/1.88 introduced(definition,[new_symbols(naming,[spl1023_1])],[avatar_definition])).
% 3.00/1.88 thf(f4788,definition,(
% 3.00/1.88 spl1023_3 <=> ((in @ (kpair @ sK821 @ sK822) @ sK820) = $true)),
% 3.00/1.88 introduced(definition,[new_symbols(naming,[spl1023_3])],[avatar_definition])).
% 3.00/1.88 thf(f4895,plain,(
% 3.00/1.88 ((in @ (kpair @ sK821 @ sK822) @ sK820) = $true) | ($true = (in @ (kpair @ sK821 @ sK822) @ sK819))),
% 3.00/1.88 inference(trivial_inequality_removal,[status(thm)],[f4894])).
% 3.00/1.88 thf(f4894,plain,(
% 3.00/1.88 ($true != $true) | ($true = (in @ (kpair @ sK821 @ sK822) @ sK819)) | ((in @ (kpair @ sK821 @ sK822) @ sK820) = $true)),
% 3.00/1.88 inference(forward_demodulation,[status(thm)],[f4863,f4024])).
% 3.00/1.88 thf(f4024,plain,(
% 3.00/1.88 (binunionE = $true)),
% 3.00/1.88 inference(cnf_transformation,[status(thm)],[f2612])).
% 3.00/1.88 thf(f2612,plain,(
% 3.00/1.88 (quantDeMorgan3 = $true) & (setminusSubset2 = $true) & (setadjoinAx = $true) & (woz1_5 = $true) & (emptyE1 = $true) & (quantDeMorgan2 = $true) & (nonemptyI = $true) & (intersectInPowersetIntersectUnions = $true) & (ex1I = $true) & (setadjoinSub2 = $true) & (setoftrueEq = $true) & (contraSubsetComplement = $true) & (((breln1 @ sK818 @ sK819) = $true) & (((breln1 @ sK818 @ sK820) = $true) & (($true = (in @ sK821 @ sK818)) & (((in @ (kpair @ sK821 @ sK822) @ (binunion @ sK819 @ sK820)) = $true) & ((in @ sK822 @ sK818) = $true) & ((($true != (in @ (kpair @ sK821 @ sK822) @ sK819)) | (sK823 = $true)) & (sK823 != $true) & ((sK823 = $true) | ((in @ (kpair @ sK821 @ sK822) @ sK820) != $true))))))) & (notdexE = $true) & (prop2setI = $true) & (binunionIR = $true) & (breln1unionI = $true) & (funcGraphProp2 = $true) & (subsetTI = $true) & (kfstsingleton = $true) & (funcImageSingleton = $true) & (setminusSubset1 = $true) & (ksndpairEq = $true) & (setOfPairsIsBReln = $true) & (exuE3e = $true) & (upairsetE = $true) & (cartprodsndpairEq = $true) & (symdiffE = $true) & (funcGraphProp3 = $true) & (omega0Ax = $true) & (binintersectSubset1 = $true) & (iftrueProp2 = $true) & (bs114d = $true) & (setunionI = $true) & (binintersectTELcontra = $true) & (breln1unionprop = $true) & (iffalseProp2 = $true) & (dpsetconstrEL2 = $true) & (emptysetimpfalse = $true) & (app = $true) & (binintersectSubset2 = $true) & (binunionTIRcontra = $true) & (powersetTE1 = $true) & (ap2p = $true) & (exuE2 = $true) & (setextsub = $true) & (breln1invI = $true) & (breln1unionIR = $true) & (setadjoinIL = $true) & (setOfPairsIsBReln1 = $true) & (foundationAx = $true) & (setukpairinjL1 = $true) & (setukpairinjR = $true) & (exuI1 = $true) & (setunionsingleton2 = $true) & (breln1compI = $true) & (breln1compE = $true) & (binintersectRsub = $true) & (subsetemptysetimpeq = $true) & (cartprodmempair = $true) & (eqinunit = $true) & (upairsubunion = $true) & (nonemptyImpWitness = $true) & (setminusI = $true) & (secondinupair = $true) & (ex1E2 = $true) & (symdiffI1 = $true) & (complementTI1 = $true) & (notequalI2 = $true) & (cartprodmempaircEq = $true) & (nonemptyI1 = $true) & (eqbreln = $true) & (prop2set2propI = $true) & (demorgan1b = $true) & (complementTnotintersectT = $true) & (setminusELneg = $true) & (omegaIndAx = $true) & (eta2 = $true) & (funcGraphProp1 = $true) & (subsetE2 = $true) & (kpairiskpair = $true) & (woz1_4 = $true) & (setminusT_lem = $true) & (funcext2 = $true) & (kfstpairEq = $true) & (complementImpComplementIntersect = $true) & (cartprodpairmemEL = $true) & (prop2setE = $true) & (iftrueorfalse = $true) & (powersetI = $true) & (lamp = $true) & (setminusER = $true) & (powersetT_lem = $true) & (contrasubsetT1 = $true) & (subsetRefl = $true) & (woz13rule0 = $true) & (cartprodpairin = $true) & (lam2p = $true) & (descr__Cong = $true) & (dpsetconstrI = $true) & (doubleComplementI1 = $true) & (powersetE1 = $true) & (subbreln1 = $true) & (emptyInPowerset = $true) & (brelnall1 = $true) & (disjointsetsI1 = $true) & (binintersectSubset5 = $true) & (powersetAx = $true) & (setukpairIL = $true) & (binunionRsub = $true) & (upairsetIL = $true) & (setextAx = $true) & (demorgan1a = $true) & (ubforcartprodlem1 = $true) & (kpairp = $true) & (demorgan2a = $true) & (replAx = $true) & (notdallE = $true) & (woz1_3 = $true) & (complementTcontraSubset = $true) & (emptyI = $true) & (subsetI2 = $true) & (setadjoin__Cong = $true) & (singletonprop = $true) & (exuE1 = $true) & (woz1_2 = $true) & (binintersectI = $true) & (breln1invprop = $true) & (theprop = $true) & (setukpairIR = $true) & (contrasubsetT = $true) & (setunionsingleton1 = $true) & (subsetI1 = $true) & (upairset2IR = $true) & (upairequniteq = $true) & (ex1I2 = $true) & (ifSingleton = $true) & (inIntersectImpInIntersectUnions = $true) & (lam2lamEq = $true) & (setunionsingleton = $true) & (demorgan2 = $true) & (setextT = $true) & (dsetconstrI = $true) & (brelnall2 = $true) & (setukpairinjR2 = $true) & (singletonsubset = $true) & (exu__Cong = $true) & (setminusIRneg = $true) & (setminusEL = $true) & (singletoninpowunion = $true) & (ksndsingleton = $true) & (eta1 = $true) & (demorgan2a1 = $true) & (setunion__Cong = $true) & (iffalse = $true) & (woz13rule3 = $true) & (subsetTrans = $true) & (breln1all1 = $true) & (binunionTILcontra = $true) & (subbreln = $true) & (powersetsubset = $true) & (ex1E1 = $true) & (binunionTE = $true) & (funcGraphProp4 = $true) & (exuI2 = $true) & (breln1compprop = $true) & (woz1_1 = $true) & (complementInPowersetComplementIntersect = $true) & (binunionLsub = $true) & (doubleComplementSub2 = $true) & (powersetTI1 = $true) & (setminusERneg = $true) & (exuI3 = $true) & (eqimpsubset1 = $true) & (symdiffIneg1 = $true) & (doubleComplementE1 = $true) & (woz13rule2 = $true) & (quantDeMorgan1 = $true) & (binunionEcases = $true) & (emptyset__Cong = $true) & (setadjoinOr = $true) & (descrp = $true) & (iftrue = $true) & (symdiffI2 = $true) & (singletoninpowerset = $true) & (inComplementUnionImpNotIn1 = $true) & (noeltsimpempty = $true) & (subset2powerset = $true) & (dsetconstr__Cong = $true) & (dpsetconstrSub = $true) & (binunionT_lem = $true) & (doubleComplementEq = $true) & (omega__Cong = $true) & (notequalI1 = $true) & (setunionE = $true) & (setminusILneg = $true) & (apProp = $true) & (cartprodfstpairEq = $true) & (funcext = $true) & (binintersectT_lem = $true) & (setukpairinjL = $true) & (demorgan2a2 = $true) & (setukpairinjR1 = $true) & (binintersectEL = $true) & (setadjoinE = $true) & (vacuousDall = $true) & (breln1SetBreln1 = $true) & (upairinpowunion = $true) & (binintersectER = $true) & (powerset__Cong = $true) & (setunionAx = $true) & (omegaSAx = $true) & (inPowerset = $true) & (cartprodsndin = $true) & (woz13rule4 = $true) & (powersetE = $true) & (demorgan2b = $true) & (setadjoinIR = $true) & (binintersectLsub = $true) & (lamProp = $true) & (funcinfuncset = $true) & (inIntersectImpInUnion2 = $true) & (singletonsswitch = $true) & (quantDeMorgan4 = $true) & (inComplementUnionImpInComplement1 = $true) & (in__Cong = $true) & (sepInPowerset = $true) & (binintersectSubset3 = $true) & (breln1all2 = $true) & (binunionIL = $true) & (setbeta = $true) & (woz13rule1 = $true) & (inCongP = $true) & (ubforcartprodlem3 = $true) & (setukpairinjR11 = $true) & (breln1invE = $true) & (notinsingleton = $true) & (kpairsurjEq = $true) & (ubforcartprodlem2 = $true) & (emptysetE = $true) & (subsetE = $true) & (breln1unionE = $true) & (breln1unionIL = $true) & (upairset2E = $true) & (dpsetconstrERa = $true) & (setminusLsub = $true) & (sepSubset = $true) & (dsetconstrER = $true) & (ap2apEq2 = $true) & (setext = $true) & (binintersectSubset4 = $true) & (powersetI1 = $true) & (uniqinunit = $true) & (cartprodpairmemER = $true) & (exuE3u = $true) & (demorgan1 = $true) & (nonemptyE1 = $true) & (eqimpsubset2 = $true) & (subPowSU = $true) & (iffalseProp1 = $true) & (exuEu = $true) & (complementUnionInPowersetComplement = $true) & (cartprodpairsurjEq = $true) & (setukpairinjR12 = $true) & (beta2 = $true) & (emptysetAx = $true) & (infuncsetfunc = $true) & (wellorderingAx = $true) & (theeq = $true) & (dsetconstrEL = $true) & (breln1compEex = $true) & (setunionE2 = $true) & (iftrueProp1 = $true) & (contrasubsetT3 = $true) & (symdiffIneg2 = $true) & (cartprodfstin = $true) & (complementSubsetComplementIntersect = $true) & (cartprodmempair1 = $true) & (upairsetIR = $true) & (funcextLem = $true) & (ifp = $true) & (notinemptyset = $true) & (contrasubsetT2 = $true) & (binunionTEcontra = $true) & (notsubsetI = $true) & (singletonsuniq = $true) & (beta1 = $true) & (emptysetsubset = $true) & (choice2fnsingleton = $true) & (complementT_lem = $true) & (setadjoinSub = $true) & (binunionE = $true) & (emptyinPowerset = $true) & (ap2apEq1 = $true) & (setukpairinjL2 = $true) & (doubleComplementSub1 = $true) & (binintersectTERcontra = $true) & (dpsetconstrER = $true) & (demorgan2b2 = $true) & (inIntersectImpInUnion = $true) & (dpsetconstrEL1 = $true) & (eqbreln1 = $true) & (complementTE1 = $true) & (emptyinunitempty = $true)),
% 3.00/1.88 inference(skolemisation,[status(esa),new_symbols(skolem,[sK818,sK819,sK820,sK821,sK822,sK823])],[f1067,f2611,f2610,f2609,f2608,f2607])).
% 3.00/1.88 thf(f2607,plain,(
% 3.00/1.88 ? [X0 : $i,X1 : $i] : (((breln1 @ X0 @ X1) = $true) & ? [X2 : $i] : (((breln1 @ X0 @ X2) = $true) & ? [X3 : $i] : (($true = (in @ X3 @ X0)) & ? [X4 : $i] : (($true = (in @ (kpair @ X3 @ X4) @ (binunion @ X1 @ X2))) & ($true = (in @ X4 @ X0)) & ? [X5 : $o] : ((((in @ (kpair @ X3 @ X4) @ X1) != $true) | ($true = X5)) & ($true != X5) & (($true = X5) | ($true != (in @ (kpair @ X3 @ X4) @ X2)))))))) => (((breln1 @ sK818 @ sK819) = $true) & ? [X2 : $i] : (((breln1 @ sK818 @ X2) = $true) & ? [X3 : $i] : (((in @ X3 @ sK818) = $true) & ? [X4 : $i] : (((in @ (kpair @ X3 @ X4) @ (binunion @ sK819 @ X2)) = $true) & ((in @ X4 @ sK818) = $true) & ? [X5 : $o] : ((((in @ (kpair @ X3 @ X4) @ sK819) != $true) | ($true = X5)) & ($true != X5) & (($true = X5) | ($true != (in @ (kpair @ X3 @ X4) @ X2))))))))),
% 3.00/1.88 introduced(definition,[],[choice_axiom])).
% 3.00/1.88 thf(f2608,plain,(
% 3.00/1.88 ? [X2 : $i] : (((breln1 @ sK818 @ X2) = $true) & ? [X3 : $i] : (((in @ X3 @ sK818) = $true) & ? [X4 : $i] : (((in @ (kpair @ X3 @ X4) @ (binunion @ sK819 @ X2)) = $true) & ((in @ X4 @ sK818) = $true) & ? [X5 : $o] : ((((in @ (kpair @ X3 @ X4) @ sK819) != $true) | ($true = X5)) & ($true != X5) & (($true = X5) | ($true != (in @ (kpair @ X3 @ X4) @ X2))))))) => (((breln1 @ sK818 @ sK820) = $true) & ? [X3 : $i] : (((in @ X3 @ sK818) = $true) & ? [X4 : $i] : (($true = (in @ (kpair @ X3 @ X4) @ (binunion @ sK819 @ sK820))) & ((in @ X4 @ sK818) = $true) & ? [X5 : $o] : ((((in @ (kpair @ X3 @ X4) @ sK819) != $true) | ($true = X5)) & ($true != X5) & (($true = X5) | ((in @ (kpair @ X3 @ X4) @ sK820) != $true))))))),
% 3.00/1.88 introduced(definition,[],[choice_axiom])).
% 3.00/1.88 thf(f2609,plain,(
% 3.00/1.88 ? [X3 : $i] : (((in @ X3 @ sK818) = $true) & ? [X4 : $i] : (($true = (in @ (kpair @ X3 @ X4) @ (binunion @ sK819 @ sK820))) & ((in @ X4 @ sK818) = $true) & ? [X5 : $o] : ((((in @ (kpair @ X3 @ X4) @ sK819) != $true) | ($true = X5)) & ($true != X5) & (($true = X5) | ((in @ (kpair @ X3 @ X4) @ sK820) != $true))))) => (($true = (in @ sK821 @ sK818)) & ? [X4 : $i] : (($true = (in @ (kpair @ sK821 @ X4) @ (binunion @ sK819 @ sK820))) & ((in @ X4 @ sK818) = $true) & ? [X5 : $o] : ((($true != (in @ (kpair @ sK821 @ X4) @ sK819)) | ($true = X5)) & ($true != X5) & (($true = X5) | ($true != (in @ (kpair @ sK821 @ X4) @ sK820))))))),
% 3.00/1.88 introduced(definition,[],[choice_axiom])).
% 3.00/1.88 thf(f2610,plain,(
% 3.00/1.88 ? [X4 : $i] : (($true = (in @ (kpair @ sK821 @ X4) @ (binunion @ sK819 @ sK820))) & ((in @ X4 @ sK818) = $true) & ? [X5 : $o] : ((($true != (in @ (kpair @ sK821 @ X4) @ sK819)) | ($true = X5)) & ($true != X5) & (($true = X5) | ($true != (in @ (kpair @ sK821 @ X4) @ sK820))))) => (((in @ (kpair @ sK821 @ sK822) @ (binunion @ sK819 @ sK820)) = $true) & ((in @ sK822 @ sK818) = $true) & ? [X5 : $o] : ((($true != (in @ (kpair @ sK821 @ sK822) @ sK819)) | ($true = X5)) & ($true != X5) & (($true = X5) | ((in @ (kpair @ sK821 @ sK822) @ sK820) != $true))))),
% 3.00/1.88 introduced(definition,[],[choice_axiom])).
% 3.00/1.88 thf(f2611,plain,(
% 3.00/1.88 ? [X5 : $o] : ((($true != (in @ (kpair @ sK821 @ sK822) @ sK819)) | ($true = X5)) & ($true != X5) & (($true = X5) | ((in @ (kpair @ sK821 @ sK822) @ sK820) != $true))) => ((($true != (in @ (kpair @ sK821 @ sK822) @ sK819)) | (sK823 = $true)) & (sK823 != $true) & ((sK823 = $true) | ((in @ (kpair @ sK821 @ sK822) @ sK820) != $true)))),
% 3.00/1.88 introduced(definition,[],[choice_axiom])).
% 3.00/1.88 thf(f1067,plain,(
% 3.00/1.88 (quantDeMorgan3 = $true) & (setminusSubset2 = $true) & (setadjoinAx = $true) & (woz1_5 = $true) & (emptyE1 = $true) & (quantDeMorgan2 = $true) & (nonemptyI = $true) & (intersectInPowersetIntersectUnions = $true) & (ex1I = $true) & (setadjoinSub2 = $true) & (setoftrueEq = $true) & (contraSubsetComplement = $true) & ? [X0 : $i,X1 : $i] : (((breln1 @ X0 @ X1) = $true) & ? [X2 : $i] : (((breln1 @ X0 @ X2) = $true) & ? [X3 : $i] : (($true = (in @ X3 @ X0)) & ? [X4 : $i] : (($true = (in @ (kpair @ X3 @ X4) @ (binunion @ X1 @ X2))) & ($true = (in @ X4 @ X0)) & ? [X5 : $o] : ((((in @ (kpair @ X3 @ X4) @ X1) != $true) | ($true = X5)) & ($true != X5) & (($true = X5) | ($true != (in @ (kpair @ X3 @ X4) @ X2)))))))) & (notdexE = $true) & (prop2setI = $true) & (binunionIR = $true) & (breln1unionI = $true) & (funcGraphProp2 = $true) & (subsetTI = $true) & (kfstsingleton = $true) & (funcImageSingleton = $true) & (setminusSubset1 = $true) & (ksndpairEq = $true) & (setOfPairsIsBReln = $true) & (exuE3e = $true) & (upairsetE = $true) & (cartprodsndpairEq = $true) & (symdiffE = $true) & (funcGraphProp3 = $true) & (omega0Ax = $true) & (binintersectSubset1 = $true) & (iftrueProp2 = $true) & (bs114d = $true) & (setunionI = $true) & (binintersectTELcontra = $true) & (breln1unionprop = $true) & (iffalseProp2 = $true) & (dpsetconstrEL2 = $true) & (emptysetimpfalse = $true) & (app = $true) & (binintersectSubset2 = $true) & (binunionTIRcontra = $true) & (powersetTE1 = $true) & (ap2p = $true) & (exuE2 = $true) & (setextsub = $true) & (breln1invI = $true) & (breln1unionIR = $true) & (setadjoinIL = $true) & (setOfPairsIsBReln1 = $true) & (foundationAx = $true) & (setukpairinjL1 = $true) & (setukpairinjR = $true) & (exuI1 = $true) & (setunionsingleton2 = $true) & (breln1compI = $true) & (breln1compE = $true) & (binintersectRsub = $true) & (subsetemptysetimpeq = $true) & (cartprodmempair = $true) & (eqinunit = $true) & (upairsubunion = $true) & (nonemptyImpWitness = $true) & (setminusI = $true) & (secondinupair = $true) & (ex1E2 = $true) & (symdiffI1 = $true) & (complementTI1 = $true) & (notequalI2 = $true) & (cartprodmempaircEq = $true) & (nonemptyI1 = $true) & (eqbreln = $true) & (prop2set2propI = $true) & (demorgan1b = $true) & (complementTnotintersectT = $true) & (setminusELneg = $true) & (omegaIndAx = $true) & (eta2 = $true) & (funcGraphProp1 = $true) & (subsetE2 = $true) & (kpairiskpair = $true) & (woz1_4 = $true) & (setminusT_lem = $true) & (funcext2 = $true) & (kfstpairEq = $true) & (complementImpComplementIntersect = $true) & (cartprodpairmemEL = $true) & (prop2setE = $true) & (iftrueorfalse = $true) & (powersetI = $true) & (lamp = $true) & (setminusER = $true) & (powersetT_lem = $true) & (contrasubsetT1 = $true) & (subsetRefl = $true) & (woz13rule0 = $true) & (cartprodpairin = $true) & (lam2p = $true) & (descr__Cong = $true) & (dpsetconstrI = $true) & (doubleComplementI1 = $true) & (powersetE1 = $true) & (subbreln1 = $true) & (emptyInPowerset = $true) & (brelnall1 = $true) & (disjointsetsI1 = $true) & (binintersectSubset5 = $true) & (powersetAx = $true) & (setukpairIL = $true) & (binunionRsub = $true) & (upairsetIL = $true) & (setextAx = $true) & (demorgan1a = $true) & (ubforcartprodlem1 = $true) & (kpairp = $true) & (demorgan2a = $true) & (replAx = $true) & (notdallE = $true) & (woz1_3 = $true) & (complementTcontraSubset = $true) & (emptyI = $true) & (subsetI2 = $true) & (setadjoin__Cong = $true) & (singletonprop = $true) & (exuE1 = $true) & (woz1_2 = $true) & (binintersectI = $true) & (breln1invprop = $true) & (theprop = $true) & (setukpairIR = $true) & (contrasubsetT = $true) & (setunionsingleton1 = $true) & (subsetI1 = $true) & (upairset2IR = $true) & (upairequniteq = $true) & (ex1I2 = $true) & (ifSingleton = $true) & (inIntersectImpInIntersectUnions = $true) & (lam2lamEq = $true) & (setunionsingleton = $true) & (demorgan2 = $true) & (setextT = $true) & (dsetconstrI = $true) & (brelnall2 = $true) & (setukpairinjR2 = $true) & (singletonsubset = $true) & (exu__Cong = $true) & (setminusIRneg = $true) & (setminusEL = $true) & (singletoninpowunion = $true) & (ksndsingleton = $true) & (eta1 = $true) & (demorgan2a1 = $true) & (setunion__Cong = $true) & (iffalse = $true) & (woz13rule3 = $true) & (subsetTrans = $true) & (breln1all1 = $true) & (binunionTILcontra = $true) & (subbreln = $true) & (powersetsubset = $true) & (ex1E1 = $true) & (binunionTE = $true) & (funcGraphProp4 = $true) & (exuI2 = $true) & (breln1compprop = $true) & (woz1_1 = $true) & (complementInPowersetComplementIntersect = $true) & (binunionLsub = $true) & (doubleComplementSub2 = $true) & (powersetTI1 = $true) & (setminusERneg = $true) & (exuI3 = $true) & (eqimpsubset1 = $true) & (symdiffIneg1 = $true) & (doubleComplementE1 = $true) & (woz13rule2 = $true) & (quantDeMorgan1 = $true) & (binunionEcases = $true) & (emptyset__Cong = $true) & (setadjoinOr = $true) & (descrp = $true) & (iftrue = $true) & (symdiffI2 = $true) & (singletoninpowerset = $true) & (inComplementUnionImpNotIn1 = $true) & (noeltsimpempty = $true) & (subset2powerset = $true) & (dsetconstr__Cong = $true) & (dpsetconstrSub = $true) & (binunionT_lem = $true) & (doubleComplementEq = $true) & (omega__Cong = $true) & (notequalI1 = $true) & (setunionE = $true) & (setminusILneg = $true) & (apProp = $true) & (cartprodfstpairEq = $true) & (funcext = $true) & (binintersectT_lem = $true) & (setukpairinjL = $true) & (demorgan2a2 = $true) & (setukpairinjR1 = $true) & (binintersectEL = $true) & (setadjoinE = $true) & (vacuousDall = $true) & (breln1SetBreln1 = $true) & (upairinpowunion = $true) & (binintersectER = $true) & (powerset__Cong = $true) & (setunionAx = $true) & (omegaSAx = $true) & (inPowerset = $true) & (cartprodsndin = $true) & (woz13rule4 = $true) & (powersetE = $true) & (demorgan2b = $true) & (setadjoinIR = $true) & (binintersectLsub = $true) & (lamProp = $true) & (funcinfuncset = $true) & (inIntersectImpInUnion2 = $true) & (singletonsswitch = $true) & (quantDeMorgan4 = $true) & (inComplementUnionImpInComplement1 = $true) & (in__Cong = $true) & (sepInPowerset = $true) & (binintersectSubset3 = $true) & (breln1all2 = $true) & (binunionIL = $true) & (setbeta = $true) & (woz13rule1 = $true) & (inCongP = $true) & (ubforcartprodlem3 = $true) & (setukpairinjR11 = $true) & (breln1invE = $true) & (notinsingleton = $true) & (kpairsurjEq = $true) & (ubforcartprodlem2 = $true) & (emptysetE = $true) & (subsetE = $true) & (breln1unionE = $true) & (breln1unionIL = $true) & (upairset2E = $true) & (dpsetconstrERa = $true) & (setminusLsub = $true) & (sepSubset = $true) & (dsetconstrER = $true) & (ap2apEq2 = $true) & (setext = $true) & (binintersectSubset4 = $true) & (powersetI1 = $true) & (uniqinunit = $true) & (cartprodpairmemER = $true) & (exuE3u = $true) & (demorgan1 = $true) & (nonemptyE1 = $true) & (eqimpsubset2 = $true) & (subPowSU = $true) & (iffalseProp1 = $true) & (exuEu = $true) & (complementUnionInPowersetComplement = $true) & (cartprodpairsurjEq = $true) & (setukpairinjR12 = $true) & (beta2 = $true) & (emptysetAx = $true) & (infuncsetfunc = $true) & (wellorderingAx = $true) & (theeq = $true) & (dsetconstrEL = $true) & (breln1compEex = $true) & (setunionE2 = $true) & (iftrueProp1 = $true) & (contrasubsetT3 = $true) & (symdiffIneg2 = $true) & (cartprodfstin = $true) & (complementSubsetComplementIntersect = $true) & (cartprodmempair1 = $true) & (upairsetIR = $true) & (funcextLem = $true) & (ifp = $true) & (notinemptyset = $true) & (contrasubsetT2 = $true) & (binunionTEcontra = $true) & (notsubsetI = $true) & (singletonsuniq = $true) & (beta1 = $true) & (emptysetsubset = $true) & (choice2fnsingleton = $true) & (complementT_lem = $true) & (setadjoinSub = $true) & (binunionE = $true) & (emptyinPowerset = $true) & (ap2apEq1 = $true) & (setukpairinjL2 = $true) & (doubleComplementSub1 = $true) & (binintersectTERcontra = $true) & (dpsetconstrER = $true) & (demorgan2b2 = $true) & (inIntersectImpInUnion = $true) & (dpsetconstrEL1 = $true) & (eqbreln1 = $true) & (complementTE1 = $true) & (emptyinunitempty = $true)),
% 3.00/1.88 inference(flattening,[status(thm)],[f1066])).
% 3.00/1.88 thf(f1066,plain,(
% 3.00/1.88 ((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i,X1 : $i] : (? [X2 : $i] : (? [X3 : $i] : (? [X4 : $i] : ((? [X5 : $o] : ((($true != X5) & (($true = X5) | ($true != (in @ (kpair @ X3 @ X4) @ X2)))) & (((in @ (kpair @ X3 @ X4) @ X1) != $true) | ($true = X5))) & ($true = (in @ (kpair @ X3 @ X4) @ (binunion @ X1 @ X2)))) & ($true = (in @ X4 @ X0))) & ($true = (in @ X3 @ X0))) & ((breln1 @ X0 @ X2) = $true)) & ((breln1 @ X0 @ X1) = $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)),
% 3.00/1.88 inference(ennf_transformation,[status(thm)],[f425])).
% 3.00/1.88 thf(f425,plain,(
% 3.00/1.88 ~((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) => ! [X0 : $i,X1 : $i] : (((breln1 @ X0 @ X1) = $true) => ! [X2 : $i] : (((breln1 @ X0 @ X2) = $true) => ! [X3 : $i] : (($true = (in @ X3 @ X0)) => ! [X4 : $i] : (($true = (in @ X4 @ X0)) => (($true = (in @ (kpair @ X3 @ X4) @ (binunion @ X1 @ X2))) => ! [X5 : $o] : ((((in @ (kpair @ X3 @ X4) @ X1) = $true) => ($true = X5)) => ((($true = (in @ (kpair @ X3 @ X4) @ X2)) => ($true = X5)) => ($true = X5)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 3.00/1.88 inference(fool_elimination,[status(thm)],[f424])).
% 3.00/1.88 thf(f424,plain,(
% 3.00/1.88 ~(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 => ! [X0 : $i,X1 : $i] : ((breln1 @ X0 @ X1) => ! [X2 : $i] : ((breln1 @ X0 @ X2) => ! [X3 : $i] : ((in @ X3 @ X0) => ! [X4 : $i] : ((in @ X4 @ X0) => ((in @ (kpair @ X3 @ X4) @ (binunion @ X1 @ X2)) => ! [X5 : $o] : (((in @ (kpair @ X3 @ X4) @ X1) => X5) => (((in @ (kpair @ X3 @ X4) @ X2) => X5) => X5))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 3.00/1.88 inference(rectify,[status(thm)],[f307])).
% 3.00/1.88 thf(f307,negated_conjecture,(
% 3.00/1.88 ~(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 => ! [X3 : $i,X13 : $i] : ((breln1 @ X3 @ X13) => ! [X15 : $i] : ((breln1 @ X3 @ X15) => ! [X1 : $i] : ((in @ X1 @ X3) => ! [X2 : $i] : ((in @ X2 @ X3) => ((in @ (kpair @ X1 @ X2) @ (binunion @ X13 @ X15)) => ! [X0 : $o] : (((in @ (kpair @ X1 @ X2) @ X13) => X0) => (((in @ (kpair @ X1 @ X2) @ X15) => X0) => X0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 3.00/1.88 inference(negated_conjecture,[status(cth)],[f306])).
% 3.00/1.88 thf(f306,conjecture,(
% 3.00/1.88 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 => ! [X3 : $i,X13 : $i] : ((breln1 @ X3 @ X13) => ! [X15 : $i] : ((breln1 @ X3 @ X15) => ! [X1 : $i] : ((in @ X1 @ X3) => ! [X2 : $i] : ((in @ X2 @ X3) => ((in @ (kpair @ X1 @ X2) @ (binunion @ X13 @ X15)) => ! [X0 : $o] : (((in @ (kpair @ X1 @ X2) @ X13) => X0) => (((in @ (kpair @ X1 @ X2) @ X15) => X0) => X0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 3.00/1.88 file('/export/starexec/sandbox2/tmp/tmp.U76IOM5xNk/DTF2THF_4862.p',breln1unionEcases)).
% 3.00/1.88 thf(f4863,plain,(
% 3.00/1.88 (binunionE != $true) | ((in @ (kpair @ sK821 @ sK822) @ sK820) = $true) | ($true = (in @ (kpair @ sK821 @ sK822) @ sK819))),
% 3.00/1.88 inference(trivial_inequality_removal,[status(thm)],[f4858])).
% 3.00/1.88 thf(f4858,plain,(
% 3.00/1.88 ($true != $true) | (binunionE != $true) | ($true = (in @ (kpair @ sK821 @ sK822) @ sK819)) | ((in @ (kpair @ sK821 @ sK822) @ sK820) = $true)),
% 3.00/1.88 inference(superposition,[status(thm)],[f3613,f4307])).
% 3.00/1.88 thf(f4307,plain,(
% 3.00/1.88 ((in @ (kpair @ sK821 @ sK822) @ (binunion @ sK819 @ sK820)) = $true)),
% 3.00/1.88 inference(cnf_transformation,[status(thm)],[f2612])).
% 3.00/1.88 thf(f3613,plain,(
% 3.00/1.88 ( ! [X2 : $i,X0 : $i,X1 : $i] : (((in @ X2 @ (binunion @ X1 @ X0)) != $true) | ((in @ X2 @ X1) = $true) | (binunionE != $true) | ((in @ X2 @ X0) = $true)) )),
% 3.00/1.88 inference(cnf_transformation,[status(thm)],[f2151])).
% 3.00/1.88 thf(f2151,plain,(
% 3.00/1.88 (! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X2 @ X0) = $true) | ((in @ X2 @ (binunion @ X1 @ X0)) != $true) | ((in @ X2 @ X1) = $true)) | (binunionE != $true)) & ((binunionE = $true) | (((in @ sK519 @ sK517) != $true) & ($true = (in @ sK519 @ (binunion @ sK518 @ sK517))) & ($true != (in @ sK519 @ sK518))))),
% 3.00/1.88 inference(skolemisation,[status(esa),new_symbols(skolem,[sK517,sK518,sK519])],[f2149,f2150])).
% 3.00/1.88 thf(f2150,plain,(
% 3.00/1.88 ? [X3 : $i,X4 : $i,X5 : $i] : (((in @ X5 @ X3) != $true) & ((in @ X5 @ (binunion @ X4 @ X3)) = $true) & ((in @ X5 @ X4) != $true)) => (((in @ sK519 @ sK517) != $true) & ($true = (in @ sK519 @ (binunion @ sK518 @ sK517))) & ($true != (in @ sK519 @ sK518)))),
% 3.00/1.88 introduced(definition,[],[choice_axiom])).
% 3.00/1.88 thf(f2149,plain,(
% 3.00/1.88 (! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X2 @ X0) = $true) | ((in @ X2 @ (binunion @ X1 @ X0)) != $true) | ((in @ X2 @ X1) = $true)) | (binunionE != $true)) & ((binunionE = $true) | ? [X3 : $i,X4 : $i,X5 : $i] : (((in @ X5 @ X3) != $true) & ((in @ X5 @ (binunion @ X4 @ X3)) = $true) & ((in @ X5 @ X4) != $true)))),
% 3.00/1.88 inference(rectify,[status(thm)],[f2148])).
% 3.00/1.88 thf(f2148,plain,(
% 3.00/1.88 (! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X2 @ X0) = $true) | ((in @ X2 @ (binunion @ X1 @ X0)) != $true) | ((in @ X2 @ X1) = $true)) | (binunionE != $true)) & ((binunionE = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (((in @ X2 @ X0) != $true) & ((in @ X2 @ (binunion @ X1 @ X0)) = $true) & ((in @ X2 @ X1) != $true)))),
% 3.00/1.88 inference(nnf_transformation,[status(thm)],[f1053])).
% 3.00/1.88 thf(f1053,plain,(
% 3.00/1.88 ! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X2 @ X0) = $true) | ((in @ X2 @ (binunion @ X1 @ X0)) != $true) | ((in @ X2 @ X1) = $true)) <=> (binunionE = $true)),
% 3.00/1.88 inference(flattening,[status(thm)],[f1052])).
% 3.00/1.88 thf(f1052,plain,(
% 3.00/1.88 (binunionE = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i] : ((((in @ X2 @ X0) = $true) | ((in @ X2 @ X1) = $true)) | ((in @ X2 @ (binunion @ X1 @ X0)) != $true))),
% 3.00/1.88 inference(ennf_transformation,[status(thm)],[f822])).
% 3.00/1.88 thf(f822,plain,(
% 3.00/1.88 (binunionE = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X2 @ (binunion @ X1 @ X0)) = $true) => (((in @ X2 @ X0) = $true) | ((in @ X2 @ X1) = $true)))),
% 3.00/1.88 inference(fool_elimination,[status(thm)],[f821])).
% 3.00/1.88 thf(f821,plain,(
% 3.00/1.88 (! [X0 : $i,X1 : $i,X2 : $i] : ((in @ X2 @ (binunion @ X1 @ X0)) => ((in @ X2 @ X1) | (in @ X2 @ X0))) = binunionE)),
% 3.00/1.88 inference(rectify,[status(thm)],[f105])).
% 3.00/1.88 thf(f105,axiom,(
% 3.00/1.88 (! [X4 : $i,X3 : $i,X1 : $i] : ((in @ X1 @ (binunion @ X3 @ X4)) => ((in @ X1 @ X3) | (in @ X1 @ X4))) = binunionE)),
% 3.00/1.88 file('/export/starexec/sandbox2/tmp/tmp.U76IOM5xNk/DTF2THF_4862.p',binunionE)).
% 3.00/1.88 thf(f4792,plain,(
% 3.00/1.88 ~spl1023_2),
% 3.00/1.88 inference(avatar_split_clause,[status(thm)],[f4304,f4783])).
% 3.00/1.88 thf(f4783,definition,(
% 3.00/1.88 spl1023_2 <=> (sK823 = $true)),
% 3.00/1.88 introduced(definition,[new_symbols(naming,[spl1023_2])],[avatar_definition])).
% 3.00/1.88 thf(f4304,plain,(
% 3.00/1.88 (sK823 != $true)),
% 3.00/1.88 inference(cnf_transformation,[status(thm)],[f2612])).
% 3.00/1.88 thf(f4791,plain,(
% 3.00/1.88 ~spl1023_3 | spl1023_2),
% 3.00/1.88 inference(avatar_split_clause,[status(thm)],[f4303,f4783,f4788])).
% 3.00/1.88 thf(f4303,plain,(
% 3.00/1.88 (sK823 = $true) | ((in @ (kpair @ sK821 @ sK822) @ sK820) != $true)),
% 3.00/1.88 inference(cnf_transformation,[status(thm)],[f2612])).
% 3.00/1.88 thf(f4786,plain,(
% 3.00/1.88 ~spl1023_1 | spl1023_2),
% 3.00/1.88 inference(avatar_split_clause,[status(thm)],[f4305,f4783,f4779])).
% 3.00/1.88 thf(f4305,plain,(
% 3.00/1.88 (sK823 = $true) | ($true != (in @ (kpair @ sK821 @ sK822) @ sK819))),
% 3.00/1.88 inference(cnf_transformation,[status(thm)],[f2612])).
% 3.00/1.88 % SZS output end Proof for DTF2THF_4862
% 3.00/1.88 % (4992)------------------------------
% 3.00/1.88 % (4992)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.88 % (4992)Termination reason: Refutation
% 3.00/1.88
% 3.00/1.88 % (4992)Memory used [KB]: 11385
% 3.00/1.88 % (4992)Time elapsed: 0.216 s
% 3.00/1.88 % (4992)Instructions burned: 453 (million)
% 3.00/1.88 % (4992)------------------------------
% 3.00/1.88 % (4992)------------------------------
% 3.00/1.88 % (4969)Success in time 0.307 s
%------------------------------------------------------------------------------