%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU800^1 : 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 : n008.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 09:37:18 AM UTC 2026
% Result : Theorem 9.99s 2.89s
% Output : Refutation 9.99s
% Verified :
% SZS Type : Refutation
% Derivation depth : 13
% Number of leaves : 5
% Syntax : Number of formulae : 32 ( 13 unt; 0 typ; 0 def)
% Number of atoms : 4965 (1328 equ; 0 cnn)
% Maximal formula atoms : 318 ( 155 avg)
% Number of connectives : 2499 ( 28 ~; 17 |; 961 &; 218 @)
% ( 2 <=>;1273 =>; 0 <=; 0 <~>)
% Maximal formula depth : 325 ( 73 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 14 ( 14 >; 0 *; 0 +; 0 <<)
% Number of symbols : 661 ( 658 usr; 323 con; 0-6 aty)
% Number of variables : 93 ( 30 ^; 45 !; 18 ?; 93 :)
% Comments :
%------------------------------------------------------------------------------
thf(func_def_0,type,
in: $i > $i > $o ).
thf(func_def_1,type,
exu: ( $i > $o ) > $o ).
thf(func_def_6,type,
setadjoin: $i > $i > $i ).
thf(func_def_8,type,
powerset: $i > $i ).
thf(func_def_10,type,
setunion: $i > $i ).
thf(func_def_19,type,
descr: ( $i > $o ) > $i ).
thf(func_def_21,type,
dsetconstr: $i > ( $i > $o ) > $i ).
thf(func_def_26,type,
prop2set: $o > $i ).
thf(func_def_36,type,
nonempty: $i > $o ).
thf(func_def_69,type,
set2prop: $i > $o ).
thf(func_def_88,type,
subset: $i > $i > $o ).
thf(func_def_89,type,
disjoint: $i > $i > $o ).
thf(func_def_90,type,
setsmeet: $i > $i > $o ).
thf(func_def_114,type,
binunion: $i > $i > $i ).
thf(func_def_122,type,
binintersect: $i > $i > $i ).
thf(func_def_135,type,
regular: $i > $o ).
thf(func_def_136,type,
setminus: $i > $i > $i ).
thf(func_def_147,type,
symdiff: $i > $i > $i ).
thf(func_def_153,type,
iskpair: $i > $o ).
thf(func_def_158,type,
kpair: $i > $i > $i ).
thf(func_def_160,type,
cartprod: $i > $i > $i ).
thf(func_def_177,type,
singleton: $i > $o ).
thf(func_def_179,type,
ex1: $i > ( $i > $o ) > $o ).
thf(func_def_184,type,
atmost1p: $i > $o ).
thf(func_def_185,type,
atleast2p: $i > $o ).
thf(func_def_186,type,
atmost2p: $i > $o ).
thf(func_def_187,type,
upairsetp: $i > $o ).
thf(func_def_191,type,
kfst: $i > $i ).
thf(func_def_203,type,
ksnd: $i > $i ).
thf(func_def_213,type,
breln: $i > $i > $i > $o ).
thf(func_def_214,type,
dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_222,type,
func: $i > $i > $i > $o ).
thf(func_def_223,type,
funcSet: $i > $i > $i ).
thf(func_def_226,type,
ap: $i > $i > $i > $i > $i ).
thf(func_def_232,type,
lam: $i > $i > ( $i > $i ) > $i ).
thf(func_def_259,type,
if: $i > $o > $i > $i > $i ).
thf(func_def_323,type,
breln1: $i > $i > $o ).
thf(func_def_325,type,
breln1Set: $i > $i ).
thf(func_def_327,type,
transitive: $i > $i > $o ).
thf(func_def_328,type,
antisymmetric: $i > $i > $o ).
thf(func_def_329,type,
reflexive: $i > $i > $o ).
thf(func_def_330,type,
refltransitive: $i > $i > $o ).
thf(func_def_331,type,
refllinearorder: $i > $i > $o ).
thf(func_def_332,type,
reflwellordering: $i > $i > $o ).
thf(func_def_338,type,
breln1invset: $i > $i > $i ).
thf(func_def_342,type,
breln1compset: $i > $i > $i > $i ).
thf(func_def_360,type,
image1: $i > ( $i > $i ) > $i ).
thf(func_def_364,type,
injective: $i > $i > $i > $o ).
thf(func_def_365,type,
injFuncSet: $i > $i > $i ).
thf(func_def_382,type,
sP0: $i > $i > $i > $o ).
thf(func_def_383,type,
sP1: $i > $i > $o ).
thf(func_def_384,type,
sP2: $i > $i > ( $i > $o ) > $i > $o ).
thf(func_def_385,type,
sP3: $o > $i > $i > $i > $o ).
thf(func_def_386,type,
sP4: $i > $i > $i > $i > $o ).
thf(func_def_388,type,
sP6: $i > $i > $i > $o ).
thf(func_def_392,type,
sP10: ( $i > $i > $o ) > $i > $i > $o ).
thf(func_def_394,type,
sP12: $i > $i > $i > $o ).
thf(func_def_395,type,
sP13: $o > $i > $i > $i > $o ).
thf(func_def_396,type,
sP14: $i > $i > $i > $o ).
thf(func_def_397,type,
sP15: $i > ( $i > $o ) > $o ).
thf(func_def_398,type,
sP16: ( $i > $o ) > $i > $o ).
thf(func_def_400,type,
sP18: $o > $i > $i > $o ).
thf(func_def_401,type,
sP19: $i > $i > $i > $o ).
thf(func_def_402,type,
sP20: $i > $i > $i > $o ).
thf(func_def_403,type,
sP21: $i > $i > $o ).
thf(func_def_404,type,
sP22: $i > $o ).
thf(func_def_405,type,
sP23: $i > $i > $o ).
thf(func_def_406,type,
sP24: $i > $i > $o ).
thf(func_def_410,type,
sP28: $i > $i > $i > $i > $o ).
thf(func_def_412,type,
sP30: $o > $i > $i > $o ).
thf(func_def_416,type,
sP34: $i > $i > $i > $i > $o ).
thf(func_def_417,type,
sP35: $i > $i > $i > $i > $o ).
thf(func_def_418,type,
sP36: $i > $i > $i > $i > $o ).
thf(func_def_419,type,
sP37: $i > $i > $i > $o ).
thf(func_def_421,type,
sP39: $i > $i > ( $i > $o ) > $i > $o ).
thf(func_def_426,type,
sP44: $i > ( $i > $o ) > $o ).
thf(func_def_427,type,
sP45: $i > ( $i > $o ) > $o ).
thf(func_def_428,type,
sP46: $i > $i > $o ).
thf(func_def_433,type,
sP51: $o > $i > $i > $o ).
thf(func_def_434,type,
sP52: $i > $i > $i > $o ).
thf(func_def_435,type,
sP53: $i > $i > $o ).
thf(func_def_440,type,
sP58: $i > ( $i > $i > $o ) > $o ).
thf(func_def_445,type,
sP63: $i > $i > $i > $o ).
thf(func_def_446,type,
sP64: $i > $i > $i > $o ).
thf(func_def_447,type,
sP65: $i > $i > $o ).
thf(func_def_448,type,
sP66: ( $i > $o ) > $i > $i > $o ).
thf(func_def_452,type,
sP70: $i > $i > $i > $o ).
thf(func_def_456,type,
sP74: $i > $i > $i > $i > $o ).
thf(func_def_457,type,
sP75: $o > $i > $i > $i > $i > $i > $o ).
thf(func_def_459,type,
sP77: $i > ( $i > $o ) > ( $i > $o ) > $i > $o ).
thf(func_def_462,type,
sP80: $i > $i > $i > $i > $o ).
thf(func_def_463,type,
sP81: ( $i > $o ) > $i > $i > $o ).
thf(func_def_464,type,
sP82: $i > $o ).
thf(func_def_467,type,
sK85: $i > $i > $i ).
thf(func_def_468,type,
sK86: $i > $i > $i > $i ).
thf(func_def_469,type,
sK87: $i > $i > $i > $i ).
thf(func_def_476,type,
sK94: $i > $i > $o ).
thf(func_def_502,type,
sK120: $i > $i > ( $i > $o ) > $i > $i ).
thf(func_def_503,type,
sK121: $i > $i > ( $i > $o ) > $i > $i ).
thf(func_def_507,type,
sK125: $i > $o ).
thf(func_def_509,type,
sK127: $i > $i > $i > $i > $o ).
thf(func_def_522,type,
sK140: $i > $o ).
thf(func_def_525,type,
sK143: $i > $i > ( $i > $o ) > $i ).
thf(func_def_530,type,
sK148: $i > $i > $i > $i ).
thf(func_def_531,type,
sK149: $i > $i > $i > $i > $i ).
thf(func_def_549,type,
sK167: $i > $i > $i ).
thf(func_def_555,type,
sK173: $i > $o ).
thf(func_def_557,type,
sK175: ( $i > $o ) > $i > $i ).
thf(func_def_572,type,
sK190: $i > $o ).
thf(func_def_574,type,
sK192: ( $i > $o ) > $i ).
thf(func_def_575,type,
sK193: ( $i > $o ) > $i ).
thf(func_def_618,type,
sK236: $i > $i > $o ).
thf(func_def_638,type,
sK256: $i > $i > $o ).
thf(func_def_641,type,
sK259: $i > $i ).
thf(func_def_642,type,
sK260: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_643,type,
sK261: ( $i > $i > $o ) > $i > $i > $i ).
thf(func_def_644,type,
sK262: ( $i > $i > $o ) > $i > $i > $i ).
thf(func_def_665,type,
sK283: $i > $i > $i > $i ).
thf(func_def_666,type,
sK284: $i > $i > $i > $i > $i ).
thf(func_def_681,type,
sK299: $i > $i > $i > $o ).
thf(func_def_689,type,
sK307: ( $i > $o ) > $i > $i ).
thf(func_def_690,type,
sK308: ( $i > $o ) > $i > $i ).
thf(func_def_691,type,
sK309: $i > ( $i > $o ) > $i ).
thf(func_def_693,type,
sK311: $i > $o ).
thf(func_def_697,type,
sK315: $i > $i > $i ).
thf(func_def_699,type,
sK317: $i > $i ).
thf(func_def_706,type,
sK324: $i > $i > $o ).
thf(func_def_708,type,
sK326: $i > $i > $i > $i ).
thf(func_def_709,type,
sK327: $i > $i > $i > $o ).
thf(func_def_718,type,
sK336: ( $i > $o ) > $i ).
thf(func_def_719,type,
sK337: $i > $o ).
thf(func_def_720,type,
sK338: $i > $o ).
thf(func_def_735,type,
sK353: $i > $o ).
thf(func_def_748,type,
sK366: $i > $i ).
thf(func_def_751,type,
sK369: $i > $o ).
thf(func_def_772,type,
sK390: $i > $i ).
thf(func_def_785,type,
sK403: $i > $i ).
thf(func_def_786,type,
sK404: $i > $i ).
thf(func_def_787,type,
sK405: $i > $i ).
thf(func_def_788,type,
sK406: $i > $i > $i ).
thf(func_def_789,type,
sK407: $i > $i > $i ).
thf(func_def_790,type,
sK408: $i > $i > $i ).
thf(func_def_791,type,
sK409: $i > $i > $i ).
thf(func_def_792,type,
sK410: $i > $i > $i ).
thf(func_def_793,type,
sK411: $i > $i > $i > $i ).
thf(func_def_794,type,
sK412: $i > $i ).
thf(func_def_795,type,
sK413: $i > $i ).
thf(func_def_796,type,
sK414: $i > $i ).
thf(func_def_797,type,
sK415: $i > $i ).
thf(func_def_798,type,
sK416: $i > $i > $i ).
thf(func_def_799,type,
sK417: $i > $i > $i ).
thf(func_def_800,type,
sK418: $i > $i > $i ).
thf(func_def_801,type,
sK419: $i > $i > $i > $i ).
thf(func_def_802,type,
sK420: $i > $i > $i > $i ).
thf(func_def_811,type,
sK429: $i > $o ).
thf(func_def_824,type,
sK442: $i > $i ).
thf(func_def_827,type,
sK445: $i > ( $i > $i ) > $i > $i ).
thf(func_def_830,type,
sK448: $i > $i > $i ).
thf(func_def_835,type,
sK453: $i > $i > $i > $i > $i ).
thf(func_def_836,type,
sK454: $i > $i > $i > $i > $i > $i ).
thf(func_def_852,type,
sK470: $i > $i > $o ).
thf(func_def_862,type,
sK480: $i > $o ).
thf(func_def_869,type,
sK487: $i > $i ).
thf(func_def_871,type,
sK489: $i > ( $i > $i ) > $i > $i ).
thf(func_def_881,type,
sK499: $i > $i ).
thf(func_def_884,type,
sK502: $i > ( $i > $i ) > $i > $i ).
thf(func_def_903,type,
sK521: $i > $i > $i > $i > $i ).
thf(func_def_904,type,
sK522: $i > $i > $i > $i > $i ).
thf(func_def_905,type,
sK523: $i > $i > $i > $i > $i ).
thf(func_def_906,type,
sK524: $i > $i > $i > $i > $i ).
thf(func_def_928,type,
sK546: $i > $i > $i > $i ).
thf(func_def_929,type,
sK547: $i > $i > $i > $i > $i ).
thf(func_def_930,type,
sK548: $i > $i > $i > $i > $i ).
thf(func_def_931,type,
sK549: $i > $i > $i > $i > $i ).
thf(func_def_938,type,
sK556: $i > $i > $i > $i ).
thf(func_def_939,type,
sK557: $i > $i > $i > $i ).
thf(func_def_951,type,
sK569: $i > $o ).
thf(func_def_958,type,
sK576: $i > $i > $o ).
thf(func_def_968,type,
sK586: $i > $i > ( $i > $o ) > $i > $i ).
thf(func_def_969,type,
sK587: $i > $i > ( $i > $o ) > $i > $i ).
thf(func_def_973,type,
sK591: $i > $o ).
thf(func_def_976,type,
sK594: $i > $i ).
thf(func_def_978,type,
sK596: $i > $i ).
thf(func_def_981,type,
sK599: $i > $o ).
thf(func_def_994,type,
sK612: $i > $i > $i ).
thf(func_def_1002,type,
sK620: $i > $o ).
thf(func_def_1004,type,
sK622: $i > ( $i > $o ) > $i ).
thf(func_def_1017,type,
sK635: $i > $i ).
thf(func_def_1020,type,
sK638: $i > ( $i > $i ) > $i > $i ).
thf(func_def_1052,type,
sK670: $i > $o ).
thf(func_def_1067,type,
sK685: $i > ( $i > $o ) > $i ).
thf(func_def_1068,type,
sK686: $i > ( $i > $o ) > $i ).
thf(func_def_1069,type,
sK687: $i > ( $i > $o ) > $i ).
thf(func_def_1071,type,
sK689: $i > $o ).
thf(func_def_1072,type,
sK690: $i > $i ).
thf(func_def_1098,type,
sK716: $i > $i > $i ).
thf(func_def_1099,type,
sK717: $i > $i > $i > $i ).
thf(func_def_1115,type,
sK733: $o > $i > $i > $i ).
thf(func_def_1118,type,
sK736: $i > $i > $i ).
thf(func_def_1141,type,
sK759: $i > $o ).
thf(func_def_1142,type,
sK760: ( $i > $o ) > $i > $i ).
thf(func_def_1143,type,
sK761: $i > $i ).
thf(func_def_1146,type,
sK764: $i > $i > ( $i > $i ) > $i ).
thf(func_def_1147,type,
sK765: $i > $i > $i > $o ).
thf(func_def_1163,type,
sK781: $i > $i > $o ).
thf(func_def_1172,type,
sK790: $i > $i > $i ).
thf(func_def_1173,type,
sK791: $i > $i > $i ).
thf(func_def_1174,type,
sK792: $i > $o ).
thf(func_def_1177,type,
sK795: $i > ( $i > $o ) > $i ).
thf(func_def_1179,type,
sK797: $i > $i ).
thf(func_def_1180,type,
sK798: $i > $i ).
thf(func_def_1181,type,
sK799: $i > $i ).
thf(func_def_1182,type,
sK800: ( $i > $i ) > $i > $i ).
thf(func_def_1183,type,
sK801: $i > ( $i > $i ) > $i > $i ).
thf(func_def_1196,type,
sK814: $i > $o ).
thf(func_def_1198,type,
sK816: ( $i > $o ) > $i > $i ).
thf(func_def_1212,type,
sK830: $i > $i > $o ).
thf(func_def_1213,type,
sK831: ( $i > $i > $o ) > $i > $i ).
thf(func_def_1214,type,
sK832: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_1215,type,
sK833: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_1216,type,
sK834: $i > ( $i > $i > $o ) > $i ).
thf(func_def_1217,type,
sK835: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_1230,type,
sK848: $i > $i > $i ).
thf(func_def_1233,type,
sK851: $i > $i > $i ).
thf(func_def_1250,type,
sK868: $i > $o ).
thf(func_def_1251,type,
sK869: $i > $i ).
thf(func_def_1252,type,
sK870: ( $i > $o ) > $i ).
thf(func_def_1254,type,
sK872: $i > $o ).
thf(func_def_1267,type,
sK885: $i > $i > $i > $i ).
thf(func_def_1268,type,
sK886: $i > $i > $i > $i ).
thf(func_def_1269,type,
sK887: $i > $i > $i > $i ).
thf(func_def_1270,type,
sK888: $i > $i > $i > $i ).
thf(func_def_1278,type,
sK896: $i > $i ).
thf(func_def_1279,type,
sK897: $i > $o ).
thf(func_def_1282,type,
sK900: $i > $i > $i ).
thf(func_def_1283,type,
sK901: $i > $i > $i > $i ).
thf(func_def_1286,type,
sK904: $i > $o ).
thf(func_def_1310,type,
sK928: ( $i > $o ) > $i > $i > $i ).
thf(func_def_1311,type,
sK929: ( $i > $o ) > $i > $i > $i ).
thf(func_def_1314,type,
sK932: $i > $o ).
thf(func_def_1320,type,
sK938: $i > $o ).
thf(func_def_1323,type,
sK941: $i > ( $i > $o ) > $i ).
thf(func_def_1324,type,
sK942: $i > $o ).
thf(func_def_1325,type,
sK943: $i > $o ).
thf(func_def_1326,type,
sK944: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_1327,type,
sK945: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_1328,type,
sK946: $i > $i ).
thf(func_def_1331,type,
sK949: $i > $i > ( $i > $i ) > $i ).
thf(func_def_1350,type,
sK968: $i > $o ).
thf(func_def_1355,type,
sK973: $i > $i > $i > $i ).
thf(func_def_1356,type,
sK974: $i > $i > $i > $i ).
thf(func_def_1361,type,
sK979: $i > $i ).
thf(func_def_1370,type,
sK988: $i > $i > $i ).
thf(func_def_1371,type,
sK989: $i > $o ).
thf(func_def_1372,type,
sK990: $i > $i ).
thf(func_def_1373,type,
sK991: ( $i > $o ) > $i ).
thf(func_def_1380,type,
sK998: $i > $i ).
thf(func_def_1383,type,
sK1001: $i > $i > ( $i > $i ) > $i ).
thf(func_def_1398,type,
sK1016: $i > $o ).
thf(func_def_1399,type,
sK1017: $i > $o ).
thf(func_def_1400,type,
sK1018: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_1401,type,
sK1019: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_1405,type,
sK1023: $i > $i > $i > $i > $i ).
thf(func_def_1410,type,
sK1028: $o > $i > $i > $i > $i > $i > $i ).
thf(func_def_1418,type,
sK1036: $i > $o ).
thf(func_def_1420,type,
sK1038: ( $i > $o ) > $i > $i ).
thf(func_def_1427,type,
sK1045: $i > ( $i > $o ) > ( $i > $o ) > $i > $i ).
thf(func_def_1428,type,
sK1046: $i > ( $i > $o ) > ( $i > $o ) > $i > $i ).
thf(func_def_1431,type,
sK1049: $i > $o ).
thf(func_def_1432,type,
sK1050: $i > $o ).
thf(func_def_1447,type,
sK1065: $i > $o ).
thf(func_def_1450,type,
sK1068: $i > ( $i > $o ) > $i ).
thf(func_def_1460,type,
sK1078: $i > $o ).
thf(func_def_1462,type,
sK1080: $i > ( $i > $o ) > $i ).
thf(func_def_1463,type,
sK1081: $i > $i > $i > $i > $i ).
thf(func_def_1464,type,
sK1082: $i > $i > $i > $i > $i ).
thf(func_def_1476,type,
sK1094: ( $i > $o ) > $i > $i > $i ).
thf(func_def_1477,type,
sK1095: ( $i > $o ) > $i > $i > $i ).
thf(func_def_1480,type,
sK1098: $i > $o ).
thf(func_def_1482,type,
sK1100: $i > $o ).
thf(func_def_1485,type,
sK1103: $i > $i > $o ).
thf(func_def_1489,type,
sK1107: $i > $i ).
thf(func_def_1490,type,
sK1108: $i > $i ).
thf(func_def_1493,type,
sK1111: $i > $i ).
thf(func_def_1496,type,
sK1114: $i > $i > ( $i > $i ) > $i ).
thf(func_def_1514,type,
sK1132: $i > $i > $o ).
thf(func_def_1520,type,
sK1138: $i > $i ).
thf(func_def_1522,type,
sK1140: ( $i > $o ) > $i ).
thf(func_def_1523,type,
sK1141: ( $i > $o ) > $i > $i ).
thf(func_def_1524,type,
sK1142: ( $i > $o ) > $i > $i ).
thf(func_def_1525,type,
sK1143: ( $i > $o ) > $i ).
thf(func_def_1526,type,
sK1144: ( $i > $o ) > $i > $i ).
thf(func_def_1527,type,
sK1145: ( $i > $o ) > $i > $i ).
thf(func_def_1528,type,
sK1146: ( $i > $o ) > $i > $i ).
thf(func_def_1529,type,
sK1147: $i > ( $i > $o ) > $i ).
thf(func_def_1530,type,
sK1148: ( $i > $o ) > $i ).
thf(func_def_1531,type,
sK1149: $i > $i ).
thf(func_def_1532,type,
sK1150: ( $i > $o ) > $i > $i ).
thf(func_def_1533,type,
sK1151: ( $i > $o ) > $i > $i ).
thf(func_def_1534,type,
sK1152: ( $i > $o ) > $i ).
thf(func_def_1535,type,
sK1153: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_1537,type,
sK1155: ( $i > $o ) > $i > $i ).
thf(func_def_1538,type,
sK1156: $i > ( $i > $o ) > $i ).
thf(func_def_1539,type,
sK1157: $i > $i ).
thf(func_def_1540,type,
sK1158: $i > $i ).
thf(func_def_1543,type,
sK1161: ( $i > $o ) > $i ).
thf(func_def_1544,type,
sK1162: ( $i > $o ) > $i > $i ).
thf(func_def_1545,type,
sK1163: $i > ( $i > $o ) > $i ).
thf(func_def_1546,type,
sK1164: ( $i > $o ) > $i ).
thf(func_def_1547,type,
sK1165: ( $i > $o ) > $i > $i ).
thf(func_def_1548,type,
sK1166: $i > ( $i > $o ) > $i ).
thf(func_def_1549,type,
sK1167: ( $i > $o ) > $i ).
thf(func_def_1550,type,
sK1168: ( $i > $o ) > $i > $i ).
thf(func_def_1551,type,
sK1169: ( $i > $o ) > $i ).
thf(func_def_1552,type,
sK1170: ( $i > $o ) > $i > $i ).
thf(func_def_1553,type,
sK1171: ( $i > $o ) > $i > $i ).
thf(func_def_1554,type,
sK1172: $i > ( $i > $o ) > $i ).
thf(func_def_1555,type,
sK1173: $i > ( $i > $o ) > $i ).
thf(func_def_1556,type,
sK1174: ( $i > $i ) > $i > $i ).
thf(func_def_1557,type,
sK1175: ( $i > $i ) > $i > $i > $i ).
thf(func_def_1558,type,
sK1176: ( $i > $i ) > $i > $i > $i ).
thf(func_def_1559,type,
sK1177: ( $i > $i ) > $i > $i > $i ).
thf(func_def_1560,type,
sK1178: $i > $i ).
thf(func_def_1561,type,
sK1179: $i > ( $i > $o ) > $i ).
thf(func_def_1562,type,
sK1180: ( $i > $o ) > $i > $i ).
thf(func_def_1563,type,
sK1181: ( $i > $o ) > $i > $i ).
thf(func_def_1564,type,
sK1182: $i > ( $i > $o ) > $i ).
thf(func_def_1565,type,
sK1183: $i > ( $i > $o ) > $i ).
thf(func_def_1567,type,
sK1185: ( $i > $o ) > $i > $i ).
thf(func_def_1568,type,
sK1186: $i > $i ).
thf(func_def_1570,type,
sK1188: ( $i > $o ) > $i ).
thf(func_def_1571,type,
sK1189: ( $i > $o ) > $i ).
thf(func_def_1572,type,
sK1190: ( $i > $o ) > $i > $i ).
thf(func_def_1573,type,
sK1191: ( $i > $o ) > $i ).
thf(func_def_1574,type,
sK1192: ( $i > $o ) > $i ).
thf(func_def_1575,type,
sK1193: ( $i > $o ) > $i ).
thf(func_def_1576,type,
ph1194:
!>[X0: $tType] : X0 ).
thf(f9824,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f9822,f3736]) ).
thf(f3736,plain,
( $true
!= ( injective @ sK228 @ sK229 @ sK230 ) ),
inference(cnf_transformation,[status(thm)],[f1778]) ).
thf(f1778,plain,
( ( upairsetIR = $true )
& ( upairsetE = $true )
& ( funcextLem = $true )
& ( cartprodsndin = $true )
& ( dsetconstr__Cong = $true )
& ( setadjoin__Cong = $true )
& ( cartprodfstpairEq = $true )
& ( cartprodpairsurjEq = $true )
& ( secondinupair = $true )
& ( funcGraphProp2 = $true )
& ( nonemptyE1 = $true )
& ( ap2apEq1 = $true )
& ( demorgan1a = $true )
& ( contraSubsetComplement = $true )
& ( foundationAx = $true )
& ( funcinfuncset = $true )
& ( emptysetE = $true )
& ( binunionIR = $true )
& ( exuE3e = $true )
& ( dpsetconstrER = $true )
& ( subsetI1 = $true )
& ( subsetE = $true )
& ( setminusIRneg = $true )
& ( exuE2 = $true )
& ( funcImageSingleton = $true )
& ( contrasubsetT = $true )
& ( doubleComplementEq = $true )
& ( setminusEL = $true )
& ( subsetTI = $true )
& ( setadjoinOr = $true )
& ( ex1E1 = $true )
& ( sepSubset = $true )
& ( dpsetconstrERa = $true )
& ( dsetconstrER = $true )
& ( cartprodsndpairEq = $true )
& ( wellorderingAx = $true )
& ( cartprodmempair1 = $true )
& ( emptyset__Cong = $true )
& ( descrp = $true )
& ( demorgan2a2 = $true )
& ( powersetsubset = $true )
& ( subsetTrans = $true )
& ( setoftrueEq = $true )
& ( cartprodpairmemEL = $true )
& ( doubleComplementSub1 = $true )
& ( kpairsurjEq = $true )
& ( powersetI = $true )
& ( powersetTE1 = $true )
& ( dpsetconstrEL2 = $true )
& ( breln1all2 = $true )
& ( binintersectTERcontra = $true )
& ( singletonsswitch = $true )
& ( doubleComplementE1 = $true )
& ( cartprodfstin = $true )
& ( subsetE2 = $true )
& ( binunionTE = $true )
& ( binunionIL = $true )
& ( emptyI = $true )
& ( notequalI1 = $true )
& ( exuI3 = $true )
& ( powerset__Cong = $true )
& ( ubforcartprodlem3 = $true )
& ( in__Cong = $true )
& ( kpairp = $true )
& ( complementTE1 = $true )
& ( woz1_5 = $true )
& ( apProp = $true )
& ( cartprodpairmemER = $true )
& ( bs114d = $true )
& ( setukpairinjR1 = $true )
& ( quantDeMorgan2 = $true )
& ( setunionI = $true )
& ( binintersectRsub = $true )
& ( binintersectTELcontra = $true )
& ( dsetconstrI = $true )
& ( image1Equiv = $true )
& ( contrasubsetT2 = $true )
& ( ex1E2 = $true )
& ( inComplementUnionImpNotIn1 = $true )
& ( upairequniteq = $true )
& ( breln1unionprop = $true )
& ( ex1I = $true )
& ( emptyInPowerset = $true )
& ( breln1unionIL = $true )
& ( inComplementUnionImpInComplement1 = $true )
& ( symdiffIneg1 = $true )
& ( funcGraphProp3 = $true )
& ( woz1_4 = $true )
& ( binunionE = $true )
& ( ( in @ sK230 @ ( injFuncSet @ sK228 @ sK229 ) )
= $true )
& ( $true
!= ( injective @ sK228 @ sK229 @ sK230 ) )
& ( subsetRefl = $true )
& ( lam2p = $true )
& ( setukpairinjR = $true )
& ( image1Ex1 = $true )
& ( setukpairIR = $true )
& ( image1Ex = $true )
& ( exuEu = $true )
& ( binunionT_lem = $true )
& ( powersetE = $true )
& ( eqbreln = $true )
& ( kfstsingleton = $true )
& ( setOfPairsIsBReln1 = $true )
& ( setminusELneg = $true )
& ( breln1invI = $true )
& ( omegaSAx = $true )
& ( iftrueProp2 = $true )
& ( cartprodmempaircEq = $true )
& ( binintersectSubset5 = $true )
& ( prop2setI = $true )
& ( woz13rule1 = $true )
& ( setadjoinIL = $true )
& ( sepInPowerset = $true )
& ( contrasubsetT3 = $true )
& ( setadjoinAx = $true )
& ( setunionsingleton1 = $true )
& ( singletoninpowerset = $true )
& ( complementInPowersetComplementIntersect = $true )
& ( demorgan2b2 = $true )
& ( setminusLsub = $true )
& ( notsubsetI = $true )
& ( emptysetimpfalse = $true )
& ( binintersectI = $true )
& ( subbreln = $true )
& ( ap2apEq2 = $true )
& ( descr__Cong = $true )
& ( inIntersectImpInUnion = $true )
& ( vacuousDall = $true )
& ( cartprodmempair = $true )
& ( breln1unionIR = $true )
& ( complementTI1 = $true )
& ( powersetT_lem = $true )
& ( ifp = $true )
& ( brelnall2 = $true )
& ( setminusERneg = $true )
& ( exuI1 = $true )
& ( binunionEcases = $true )
& ( breln1unionE = $true )
& ( woz2A = $true )
& ( setminusER = $true )
& ( complementTnotintersectT = $true )
& ( complementTcontraSubset = $true )
& ( setext = $true )
& ( ubforcartprodlem2 = $true )
& ( noeltsimpempty = $true )
& ( beta2 = $true )
& ( symdiffI1 = $true )
& ( binintersectSubset3 = $true )
& ( ex1I2 = $true )
& ( setukpairinjL2 = $true )
& ( singletoninpowunion = $true )
& ( upairset2E = $true )
& ( brelnall1 = $true )
& ( theeq = $true )
& ( breln1all1 = $true )
& ( binunionTILcontra = $true )
& ( singletonsubset = $true )
& ( dpsetconstrSub = $true )
& ( ap2p = $true )
& ( setukpairinjL = $true )
& ( setextT = $true )
& ( upairinpowunion = $true )
& ( eqimpsubset1 = $true )
& ( notdexE = $true )
& ( complementSubsetComplementIntersect = $true )
& ( setadjoinE = $true )
& ( breln1compI = $true )
& ( setextsub = $true )
& ( dsetconstrEL = $true )
& ( setminusI = $true )
& ( theprop = $true )
& ( setminusILneg = $true )
& ( eta2 = $true )
& ( exuI2 = $true )
& ( woz13rule0 = $true )
& ( setminusT_lem = $true )
& ( iffalseProp1 = $true )
& ( setunionE = $true )
& ( notdallE = $true )
& ( setminusSubset2 = $true )
& ( notinemptyset = $true )
& ( emptyinunitempty = $true )
& ( ksndsingleton = $true )
& ( woz1_2 = $true )
& ( setadjoinIR = $true )
& ( beta1 = $true )
& ( subbreln1 = $true )
& ( powersetAx = $true )
& ( breln1compprop = $true )
& ( upairsetIL = $true )
& ( demorgan1 = $true )
& ( woz1_1 = $true )
& ( iftrue = $true )
& ( quantDeMorgan1 = $true )
& ( setunionsingleton2 = $true )
& ( breln1compEex = $true )
& ( nonemptyI1 = $true )
& ( singletonprop = $true )
& ( iffalse = $true )
& ( funcGraphProp1 = $true )
& ( app = $true )
& ( symdiffI2 = $true )
& ( eqbreln1 = $true )
& ( subPowSU = $true )
& ( inPowerset = $true )
& ( binunionRsub = $true )
& ( demorgan2b = $true )
& ( emptyinPowerset = $true )
& ( binintersectSubset2 = $true )
& ( setukpairIL = $true )
& ( funcext = $true )
& ( binintersectSubset1 = $true )
& ( iffalseProp2 = $true )
& ( quantDeMorgan3 = $true )
& ( setminusSubset1 = $true )
& ( binunionTIRcontra = $true )
& ( breln1invE = $true )
& ( binunionTEcontra = $true )
& ( quantDeMorgan4 = $true )
& ( binintersectLsub = $true )
& ( setunion__Cong = $true )
& ( complementT_lem = $true )
& ( funcGraphProp4 = $true )
& ( lamp = $true )
& ( subset2powerset = $true )
& ( exuE1 = $true )
& ( setukpairinjL1 = $true )
& ( woz1_3 = $true )
& ( setadjoinSub = $true )
& ( inIntersectImpInUnion2 = $true )
& ( breln1unionEcases = $true )
& ( powersetI1 = $true )
& ( injFuncSetFuncIn = $true )
& ( complementUnionInPowersetComplement = $true )
& ( demorgan2 = $true )
& ( ubforcartprodlem1 = $true )
& ( nonemptyI = $true )
& ( binintersectER = $true )
& ( ifSingleton = $true )
& ( setunionsingleton = $true )
& ( emptysetsubset = $true )
& ( omegaIndAx = $true )
& ( woz13rule2 = $true )
& ( binintersectT_lem = $true )
& ( setextAx = $true )
& ( demorgan2a = $true )
& ( upairset2IR = $true )
& ( eqinunit = $true )
& ( emptyE1 = $true )
& ( breln1compE = $true )
& ( setadjoinSub2 = $true )
& ( infuncsetfunc = $true )
& ( choice2fnsingleton = $true )
& ( symdiffE = $true )
& ( lamProp = $true )
& ( nonemptyImpWitness = $true )
& ( demorgan2a1 = $true )
& ( binunionLsub = $true )
& ( notequalI2 = $true )
& ( kfstpairEq = $true )
& ( singletonsuniq = $true )
& ( contrasubsetT1 = $true )
& ( disjointsetsI1 = $true )
& ( upairsubunion = $true )
& ( replAx = $true )
& ( inIntersectImpInIntersectUnions = $true )
& ( setukpairinjR11 = $true )
& ( dpsetconstrEL1 = $true )
& ( cartprodpairin = $true )
& ( ksndpairEq = $true )
& ( setunionAx = $true )
& ( image1I = $true )
& ( woz2B = $true )
& ( uniqinunit = $true )
& ( exuE3u = $true )
& ( injFuncInInjFuncSet = $true )
& ( setbeta = $true )
& ( woz13rule3 = $true )
& ( setOfPairsIsBReln = $true )
& ( breln1SetBreln1 = $true )
& ( eqimpsubset2 = $true )
& ( emptysetAx = $true )
& ( binintersectSubset4 = $true )
& ( woz2Ex = $true )
& ( binintersectEL = $true )
& ( intersectInPowersetIntersectUnions = $true )
& ( doubleComplementI1 = $true )
& ( setukpairinjR2 = $true )
& ( powersetTI1 = $true )
& ( powersetE1 = $true )
& ( iftrueorfalse = $true )
& ( subsetemptysetimpeq = $true )
& ( setunionE2 = $true )
& ( omega0Ax = $true )
& ( breln1unionI = $true )
& ( inCongP = $true )
& ( prop2set2propI = $true )
& ( complementImpComplementIntersect = $true )
& ( woz2W = $true )
& ( lam2lamEq = $true )
& ( setukpairinjR12 = $true )
& ( breln1unionCommutes = $true )
& ( eta1 = $true )
& ( doubleComplementSub2 = $true )
& ( demorgan1b = $true )
& ( dpsetconstrI = $true )
& ( subsetI2 = $true )
& ( omega__Cong = $true )
& ( notinsingleton = $true )
& ( iftrueProp1 = $true )
& ( breln1invprop = $true )
& ( symdiffIneg2 = $true )
& ( funcext2 = $true )
& ( exu__Cong = $true )
& ( kpairiskpair = $true )
& ( woz13rule4 = $true )
& ( prop2setE = $true )
& ( image1E = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK228,sK229,sK230])],[f1069,f1777]) ).
thf(f1777,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X2 @ ( injFuncSet @ X0 @ X1 ) ) )
& ( $true
!= ( injective @ X0 @ X1 @ X2 ) ) )
=> ( ( ( in @ sK230 @ ( injFuncSet @ sK228 @ sK229 ) )
= $true )
& ( $true
!= ( injective @ sK228 @ sK229 @ sK230 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1069,plain,
( ( upairsetIR = $true )
& ( upairsetE = $true )
& ( funcextLem = $true )
& ( cartprodsndin = $true )
& ( dsetconstr__Cong = $true )
& ( setadjoin__Cong = $true )
& ( cartprodfstpairEq = $true )
& ( cartprodpairsurjEq = $true )
& ( secondinupair = $true )
& ( funcGraphProp2 = $true )
& ( nonemptyE1 = $true )
& ( ap2apEq1 = $true )
& ( demorgan1a = $true )
& ( contraSubsetComplement = $true )
& ( foundationAx = $true )
& ( funcinfuncset = $true )
& ( emptysetE = $true )
& ( binunionIR = $true )
& ( exuE3e = $true )
& ( dpsetconstrER = $true )
& ( subsetI1 = $true )
& ( subsetE = $true )
& ( setminusIRneg = $true )
& ( exuE2 = $true )
& ( funcImageSingleton = $true )
& ( contrasubsetT = $true )
& ( doubleComplementEq = $true )
& ( setminusEL = $true )
& ( subsetTI = $true )
& ( setadjoinOr = $true )
& ( ex1E1 = $true )
& ( sepSubset = $true )
& ( dpsetconstrERa = $true )
& ( dsetconstrER = $true )
& ( cartprodsndpairEq = $true )
& ( wellorderingAx = $true )
& ( cartprodmempair1 = $true )
& ( emptyset__Cong = $true )
& ( descrp = $true )
& ( demorgan2a2 = $true )
& ( powersetsubset = $true )
& ( subsetTrans = $true )
& ( setoftrueEq = $true )
& ( cartprodpairmemEL = $true )
& ( doubleComplementSub1 = $true )
& ( kpairsurjEq = $true )
& ( powersetI = $true )
& ( powersetTE1 = $true )
& ( dpsetconstrEL2 = $true )
& ( breln1all2 = $true )
& ( binintersectTERcontra = $true )
& ( singletonsswitch = $true )
& ( doubleComplementE1 = $true )
& ( cartprodfstin = $true )
& ( subsetE2 = $true )
& ( binunionTE = $true )
& ( binunionIL = $true )
& ( emptyI = $true )
& ( notequalI1 = $true )
& ( exuI3 = $true )
& ( powerset__Cong = $true )
& ( ubforcartprodlem3 = $true )
& ( in__Cong = $true )
& ( kpairp = $true )
& ( complementTE1 = $true )
& ( woz1_5 = $true )
& ( apProp = $true )
& ( cartprodpairmemER = $true )
& ( bs114d = $true )
& ( setukpairinjR1 = $true )
& ( quantDeMorgan2 = $true )
& ( setunionI = $true )
& ( binintersectRsub = $true )
& ( binintersectTELcontra = $true )
& ( dsetconstrI = $true )
& ( image1Equiv = $true )
& ( contrasubsetT2 = $true )
& ( ex1E2 = $true )
& ( inComplementUnionImpNotIn1 = $true )
& ( upairequniteq = $true )
& ( breln1unionprop = $true )
& ( ex1I = $true )
& ( emptyInPowerset = $true )
& ( breln1unionIL = $true )
& ( inComplementUnionImpInComplement1 = $true )
& ( symdiffIneg1 = $true )
& ( funcGraphProp3 = $true )
& ( woz1_4 = $true )
& ( binunionE = $true )
& ? [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X2 @ ( injFuncSet @ X0 @ X1 ) ) )
& ( $true
!= ( injective @ X0 @ X1 @ X2 ) ) )
& ( subsetRefl = $true )
& ( lam2p = $true )
& ( setukpairinjR = $true )
& ( image1Ex1 = $true )
& ( setukpairIR = $true )
& ( image1Ex = $true )
& ( exuEu = $true )
& ( binunionT_lem = $true )
& ( powersetE = $true )
& ( eqbreln = $true )
& ( kfstsingleton = $true )
& ( setOfPairsIsBReln1 = $true )
& ( setminusELneg = $true )
& ( breln1invI = $true )
& ( omegaSAx = $true )
& ( iftrueProp2 = $true )
& ( cartprodmempaircEq = $true )
& ( binintersectSubset5 = $true )
& ( prop2setI = $true )
& ( woz13rule1 = $true )
& ( setadjoinIL = $true )
& ( sepInPowerset = $true )
& ( contrasubsetT3 = $true )
& ( setadjoinAx = $true )
& ( setunionsingleton1 = $true )
& ( singletoninpowerset = $true )
& ( complementInPowersetComplementIntersect = $true )
& ( demorgan2b2 = $true )
& ( setminusLsub = $true )
& ( notsubsetI = $true )
& ( emptysetimpfalse = $true )
& ( binintersectI = $true )
& ( subbreln = $true )
& ( ap2apEq2 = $true )
& ( descr__Cong = $true )
& ( inIntersectImpInUnion = $true )
& ( vacuousDall = $true )
& ( cartprodmempair = $true )
& ( breln1unionIR = $true )
& ( complementTI1 = $true )
& ( powersetT_lem = $true )
& ( ifp = $true )
& ( brelnall2 = $true )
& ( setminusERneg = $true )
& ( exuI1 = $true )
& ( binunionEcases = $true )
& ( breln1unionE = $true )
& ( woz2A = $true )
& ( setminusER = $true )
& ( complementTnotintersectT = $true )
& ( complementTcontraSubset = $true )
& ( setext = $true )
& ( ubforcartprodlem2 = $true )
& ( noeltsimpempty = $true )
& ( beta2 = $true )
& ( symdiffI1 = $true )
& ( binintersectSubset3 = $true )
& ( ex1I2 = $true )
& ( setukpairinjL2 = $true )
& ( singletoninpowunion = $true )
& ( upairset2E = $true )
& ( brelnall1 = $true )
& ( theeq = $true )
& ( breln1all1 = $true )
& ( binunionTILcontra = $true )
& ( singletonsubset = $true )
& ( dpsetconstrSub = $true )
& ( ap2p = $true )
& ( setukpairinjL = $true )
& ( setextT = $true )
& ( upairinpowunion = $true )
& ( eqimpsubset1 = $true )
& ( notdexE = $true )
& ( complementSubsetComplementIntersect = $true )
& ( setadjoinE = $true )
& ( breln1compI = $true )
& ( setextsub = $true )
& ( dsetconstrEL = $true )
& ( setminusI = $true )
& ( theprop = $true )
& ( setminusILneg = $true )
& ( eta2 = $true )
& ( exuI2 = $true )
& ( woz13rule0 = $true )
& ( setminusT_lem = $true )
& ( iffalseProp1 = $true )
& ( setunionE = $true )
& ( notdallE = $true )
& ( setminusSubset2 = $true )
& ( notinemptyset = $true )
& ( emptyinunitempty = $true )
& ( ksndsingleton = $true )
& ( woz1_2 = $true )
& ( setadjoinIR = $true )
& ( beta1 = $true )
& ( subbreln1 = $true )
& ( powersetAx = $true )
& ( breln1compprop = $true )
& ( upairsetIL = $true )
& ( demorgan1 = $true )
& ( woz1_1 = $true )
& ( iftrue = $true )
& ( quantDeMorgan1 = $true )
& ( setunionsingleton2 = $true )
& ( breln1compEex = $true )
& ( nonemptyI1 = $true )
& ( singletonprop = $true )
& ( iffalse = $true )
& ( funcGraphProp1 = $true )
& ( app = $true )
& ( symdiffI2 = $true )
& ( eqbreln1 = $true )
& ( subPowSU = $true )
& ( inPowerset = $true )
& ( binunionRsub = $true )
& ( demorgan2b = $true )
& ( emptyinPowerset = $true )
& ( binintersectSubset2 = $true )
& ( setukpairIL = $true )
& ( funcext = $true )
& ( binintersectSubset1 = $true )
& ( iffalseProp2 = $true )
& ( quantDeMorgan3 = $true )
& ( setminusSubset1 = $true )
& ( binunionTIRcontra = $true )
& ( breln1invE = $true )
& ( binunionTEcontra = $true )
& ( quantDeMorgan4 = $true )
& ( binintersectLsub = $true )
& ( setunion__Cong = $true )
& ( complementT_lem = $true )
& ( funcGraphProp4 = $true )
& ( lamp = $true )
& ( subset2powerset = $true )
& ( exuE1 = $true )
& ( setukpairinjL1 = $true )
& ( woz1_3 = $true )
& ( setadjoinSub = $true )
& ( inIntersectImpInUnion2 = $true )
& ( breln1unionEcases = $true )
& ( powersetI1 = $true )
& ( injFuncSetFuncIn = $true )
& ( complementUnionInPowersetComplement = $true )
& ( demorgan2 = $true )
& ( ubforcartprodlem1 = $true )
& ( nonemptyI = $true )
& ( binintersectER = $true )
& ( ifSingleton = $true )
& ( setunionsingleton = $true )
& ( emptysetsubset = $true )
& ( omegaIndAx = $true )
& ( woz13rule2 = $true )
& ( binintersectT_lem = $true )
& ( setextAx = $true )
& ( demorgan2a = $true )
& ( upairset2IR = $true )
& ( eqinunit = $true )
& ( emptyE1 = $true )
& ( breln1compE = $true )
& ( setadjoinSub2 = $true )
& ( infuncsetfunc = $true )
& ( choice2fnsingleton = $true )
& ( symdiffE = $true )
& ( lamProp = $true )
& ( nonemptyImpWitness = $true )
& ( demorgan2a1 = $true )
& ( binunionLsub = $true )
& ( notequalI2 = $true )
& ( kfstpairEq = $true )
& ( singletonsuniq = $true )
& ( contrasubsetT1 = $true )
& ( disjointsetsI1 = $true )
& ( upairsubunion = $true )
& ( replAx = $true )
& ( inIntersectImpInIntersectUnions = $true )
& ( setukpairinjR11 = $true )
& ( dpsetconstrEL1 = $true )
& ( cartprodpairin = $true )
& ( ksndpairEq = $true )
& ( setunionAx = $true )
& ( image1I = $true )
& ( woz2B = $true )
& ( uniqinunit = $true )
& ( exuE3u = $true )
& ( injFuncInInjFuncSet = $true )
& ( setbeta = $true )
& ( woz13rule3 = $true )
& ( setOfPairsIsBReln = $true )
& ( breln1SetBreln1 = $true )
& ( eqimpsubset2 = $true )
& ( emptysetAx = $true )
& ( binintersectSubset4 = $true )
& ( woz2Ex = $true )
& ( binintersectEL = $true )
& ( intersectInPowersetIntersectUnions = $true )
& ( doubleComplementI1 = $true )
& ( setukpairinjR2 = $true )
& ( powersetTI1 = $true )
& ( powersetE1 = $true )
& ( iftrueorfalse = $true )
& ( subsetemptysetimpeq = $true )
& ( setunionE2 = $true )
& ( omega0Ax = $true )
& ( breln1unionI = $true )
& ( inCongP = $true )
& ( prop2set2propI = $true )
& ( complementImpComplementIntersect = $true )
& ( woz2W = $true )
& ( lam2lamEq = $true )
& ( setukpairinjR12 = $true )
& ( breln1unionCommutes = $true )
& ( eta1 = $true )
& ( doubleComplementSub2 = $true )
& ( demorgan1b = $true )
& ( dpsetconstrI = $true )
& ( subsetI2 = $true )
& ( omega__Cong = $true )
& ( notinsingleton = $true )
& ( iftrueProp1 = $true )
& ( breln1invprop = $true )
& ( symdiffIneg2 = $true )
& ( funcext2 = $true )
& ( exu__Cong = $true )
& ( kpairiskpair = $true )
& ( woz13rule4 = $true )
& ( prop2setE = $true )
& ( image1E = $true ) ),
inference(flattening,[status(thm)],[f1068]) ).
thf(f1068,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X2 @ ( injFuncSet @ X0 @ X1 ) ) )
& ( $true
!= ( injective @ X0 @ X1 @ X2 ) ) )
& ( injFuncSetFuncIn = $true )
& ( injFuncInInjFuncSet = $true )
& ( image1I = $true )
& ( image1E = $true )
& ( image1Equiv = $true )
& ( image1Ex1 = $true )
& ( image1Ex = $true )
& ( woz2B = $true )
& ( woz2A = $true )
& ( woz2W = $true )
& ( woz2Ex = $true )
& ( breln1unionCommutes = $true )
& ( breln1unionEcases = $true )
& ( breln1unionE = $true )
& ( breln1unionI = $true )
& ( breln1unionIR = $true )
& ( breln1unionIL = $true )
& ( breln1unionprop = $true )
& ( breln1compEex = $true )
& ( breln1compE = $true )
& ( breln1compI = $true )
& ( breln1compprop = $true )
& ( breln1invE = $true )
& ( breln1invI = $true )
& ( breln1invprop = $true )
& ( eqbreln1 = $true )
& ( subbreln1 = $true )
& ( breln1all1 = $true )
& ( setOfPairsIsBReln1 = $true )
& ( choice2fnsingleton = $true )
& ( breln1SetBreln1 = $true )
& ( breln1all2 = $true )
& ( woz1_5 = $true )
& ( woz1_4 = $true )
& ( woz1_3 = $true )
& ( woz1_2 = $true )
& ( woz1_1 = $true )
& ( woz13rule4 = $true )
& ( woz13rule3 = $true )
& ( woz13rule2 = $true )
& ( woz13rule1 = $true )
& ( woz13rule0 = $true )
& ( demorgan2 = $true )
& ( demorgan2b = $true )
& ( demorgan2b2 = $true )
& ( demorgan2a = $true )
& ( demorgan1 = $true )
& ( demorgan1b = $true )
& ( demorgan1a = $true )
& ( demorgan2a2 = $true )
& ( complementUnionInPowersetComplement = $true )
& ( demorgan2a1 = $true )
& ( binunionTEcontra = $true )
& ( binunionTE = $true )
& ( inComplementUnionImpInComplement1 = $true )
& ( inComplementUnionImpNotIn1 = $true )
& ( intersectInPowersetIntersectUnions = $true )
& ( inIntersectImpInIntersectUnions = $true )
& ( inIntersectImpInUnion2 = $true )
& ( inIntersectImpInUnion = $true )
& ( binunionTIRcontra = $true )
& ( binunionTILcontra = $true )
& ( complementTcontraSubset = $true )
& ( contraSubsetComplement = $true )
& ( complementInPowersetComplementIntersect = $true )
& ( complementSubsetComplementIntersect = $true )
& ( complementImpComplementIntersect = $true )
& ( complementTnotintersectT = $true )
& ( doubleComplementEq = $true )
& ( doubleComplementSub2 = $true )
& ( doubleComplementSub1 = $true )
& ( doubleComplementE1 = $true )
& ( doubleComplementI1 = $true )
& ( contrasubsetT3 = $true )
& ( contrasubsetT2 = $true )
& ( contrasubsetT1 = $true )
& ( contrasubsetT = $true )
& ( binintersectTERcontra = $true )
& ( binintersectTELcontra = $true )
& ( complementTE1 = $true )
& ( complementTI1 = $true )
& ( powersetTE1 = $true )
& ( powersetTI1 = $true )
& ( subsetTI = $true )
& ( setextT = $true )
& ( complementT_lem = $true )
& ( setminusT_lem = $true )
& ( powersetT_lem = $true )
& ( binunionT_lem = $true )
& ( binintersectT_lem = $true )
& ( iftrueorfalse = $true )
& ( iffalse = $true )
& ( iftrue = $true )
& ( theeq = $true )
& ( ifp = $true )
& ( ifSingleton = $true )
& ( iftrueProp2 = $true )
& ( iftrueProp1 = $true )
& ( iffalseProp2 = $true )
& ( iffalseProp1 = $true )
& ( eta2 = $true )
& ( beta2 = $true )
& ( lam2lamEq = $true )
& ( eta1 = $true )
& ( beta1 = $true )
& ( ap2apEq2 = $true )
& ( ap2apEq1 = $true )
& ( funcext2 = $true )
& ( funcext = $true )
& ( eqbreln = $true )
& ( subbreln = $true )
& ( funcGraphProp4 = $true )
& ( funcextLem = $true )
& ( funcGraphProp2 = $true )
& ( funcGraphProp3 = $true )
& ( funcGraphProp1 = $true )
& ( ex1E2 = $true )
& ( brelnall2 = $true )
& ( brelnall1 = $true )
& ( lam2p = $true )
& ( lamp = $true )
& ( lamProp = $true )
& ( funcinfuncset = $true )
& ( ap2p = $true )
& ( infuncsetfunc = $true )
& ( app = $true )
& ( apProp = $true )
& ( funcImageSingleton = $true )
& ( dpsetconstrER = $true )
& ( dpsetconstrEL2 = $true )
& ( dpsetconstrEL1 = $true )
& ( dpsetconstrERa = $true )
& ( setOfPairsIsBReln = $true )
& ( dpsetconstrSub = $true )
& ( dpsetconstrI = $true )
& ( cartprodpairsurjEq = $true )
& ( cartprodsndpairEq = $true )
& ( cartprodfstpairEq = $true )
& ( cartprodmempaircEq = $true )
& ( cartprodpairmemER = $true )
& ( cartprodpairmemEL = $true )
& ( cartprodsndin = $true )
& ( kpairsurjEq = $true )
& ( ksndpairEq = $true )
& ( ksndsingleton = $true )
& ( setukpairinjR = $true )
& ( setukpairinjR2 = $true )
& ( upairequniteq = $true )
& ( setukpairinjR1 = $true )
& ( setukpairinjR12 = $true )
& ( setukpairinjR11 = $true )
& ( setukpairinjL = $true )
& ( setukpairinjL2 = $true )
& ( cartprodfstin = $true )
& ( kfstpairEq = $true )
& ( theprop = $true )
& ( kfstsingleton = $true )
& ( setukpairinjL1 = $true )
& ( singletonsuniq = $true )
& ( ex1I2 = $true )
& ( ex1I = $true )
& ( ex1E1 = $true )
& ( singletonprop = $true )
& ( setunionsingleton = $true )
& ( setunionsingleton2 = $true )
& ( setunionsingleton1 = $true )
& ( setunionE2 = $true )
& ( cartprodmempair = $true )
& ( cartprodmempair1 = $true )
& ( cartprodpairin = $true )
& ( ubforcartprodlem3 = $true )
& ( ubforcartprodlem2 = $true )
& ( ubforcartprodlem1 = $true )
& ( upairinpowunion = $true )
& ( upairsubunion = $true )
& ( upairset2E = $true )
& ( singletoninpowunion = $true )
& ( singletoninpowerset = $true )
& ( singletonsubset = $true )
& ( kpairp = $true )
& ( kpairiskpair = $true )
& ( setukpairIR = $true )
& ( setukpairIL = $true )
& ( secondinupair = $true )
& ( symdiffIneg2 = $true )
& ( symdiffIneg1 = $true )
& ( symdiffI2 = $true )
& ( symdiffI1 = $true )
& ( symdiffE = $true )
& ( setminusSubset1 = $true )
& ( setminusLsub = $true )
& ( setminusIRneg = $true )
& ( setminusILneg = $true )
& ( setminusELneg = $true )
& ( setminusERneg = $true )
& ( setminusSubset2 = $true )
& ( setminusER = $true )
& ( setminusEL = $true )
& ( setminusI = $true )
& ( bs114d = $true )
& ( binintersectSubset1 = $true )
& ( binintersectSubset4 = $true )
& ( binintersectRsub = $true )
& ( disjointsetsI1 = $true )
& ( binintersectER = $true )
& ( binintersectSubset3 = $true )
& ( binintersectSubset2 = $true )
& ( binintersectLsub = $true )
& ( binintersectEL = $true )
& ( binintersectSubset5 = $true )
& ( binintersectI = $true )
& ( binunionRsub = $true )
& ( binunionLsub = $true )
& ( binunionE = $true )
& ( binunionEcases = $true )
& ( binunionIR = $true )
& ( upairset2IR = $true )
& ( binunionIL = $true )
& ( sepSubset = $true )
& ( sepInPowerset = $true )
& ( powersetsubset = $true )
& ( inPowerset = $true )
& ( powersetE1 = $true )
& ( powersetI1 = $true )
& ( subsetemptysetimpeq = $true )
& ( setextsub = $true )
& ( subset2powerset = $true )
& ( setadjoinSub2 = $true )
& ( setadjoinSub = $true )
& ( subsetTrans = $true )
& ( subsetRefl = $true )
& ( notequalI2 = $true )
& ( notequalI1 = $true )
& ( notsubsetI = $true )
& ( subsetE2 = $true )
& ( subsetE = $true )
& ( emptysetsubset = $true )
& ( subsetI2 = $true )
& ( eqimpsubset1 = $true )
& ( eqimpsubset2 = $true )
& ( subsetI1 = $true )
& ( dsetconstr__Cong = $true )
& ( descr__Cong = $true )
& ( exuEu = $true )
& ( omega__Cong = $true )
& ( setunion__Cong = $true )
& ( powerset__Cong = $true )
& ( setadjoin__Cong = $true )
& ( emptyset__Cong = $true )
& ( exu__Cong = $true )
& ( exuE3u = $true )
& ( in__Cong = $true )
& ( inCongP = $true )
& ( exuI2 = $true )
& ( exuI3 = $true )
& ( exuI1 = $true )
& ( notdallE = $true )
& ( notdexE = $true )
& ( prop2set2propI = $true )
& ( prop2setI = $true )
& ( quantDeMorgan4 = $true )
& ( quantDeMorgan3 = $true )
& ( quantDeMorgan2 = $true )
& ( quantDeMorgan1 = $true )
& ( vacuousDall = $true )
& ( emptyE1 = $true )
& ( upairsetIR = $true )
& ( upairsetIL = $true )
& ( upairsetE = $true )
& ( singletonsswitch = $true )
& ( eqinunit = $true )
& ( notinsingleton = $true )
& ( uniqinunit = $true )
& ( nonemptyImpWitness = $true )
& ( exuE2 = $true )
& ( subPowSU = $true )
& ( setunionE = $true )
& ( setunionI = $true )
& ( powersetE = $true )
& ( emptyInPowerset = $true )
& ( emptyinPowerset = $true )
& ( powersetI = $true )
& ( setoftrueEq = $true )
& ( setadjoinOr = $true )
& ( setadjoinE = $true )
& ( setadjoinIR = $true )
& ( emptyinunitempty = $true )
& ( setadjoinIL = $true )
& ( nonemptyI1 = $true )
& ( nonemptyI = $true )
& ( nonemptyE1 = $true )
& ( setbeta = $true )
& ( noeltsimpempty = $true )
& ( emptyI = $true )
& ( setext = $true )
& ( exuE3e = $true )
& ( notinemptyset = $true )
& ( emptysetimpfalse = $true )
& ( emptysetE = $true )
& ( prop2setE = $true )
& ( exuE1 = $true )
& ( dsetconstrER = $true )
& ( dsetconstrEL = $true )
& ( dsetconstrI = $true )
& ( descrp = $true )
& ( wellorderingAx = $true )
& ( foundationAx = $true )
& ( replAx = $true )
& ( omegaIndAx = $true )
& ( omegaSAx = $true )
& ( omega0Ax = $true )
& ( setunionAx = $true )
& ( powersetAx = $true )
& ( setadjoinAx = $true )
& ( emptysetAx = $true )
& ( setextAx = $true ) ),
inference(ennf_transformation,[status(thm)],[f718]) ).
thf(f718,plain,
~ ( ( setextAx = $true )
=> ( ( emptysetAx = $true )
=> ( ( setadjoinAx = $true )
=> ( ( powersetAx = $true )
=> ( ( setunionAx = $true )
=> ( ( omega0Ax = $true )
=> ( ( omegaSAx = $true )
=> ( ( omegaIndAx = $true )
=> ( ( replAx = $true )
=> ( ( foundationAx = $true )
=> ( ( wellorderingAx = $true )
=> ( ( descrp = $true )
=> ( ( dsetconstrI = $true )
=> ( ( dsetconstrEL = $true )
=> ( ( dsetconstrER = $true )
=> ( ( exuE1 = $true )
=> ( ( prop2setE = $true )
=> ( ( emptysetE = $true )
=> ( ( emptysetimpfalse = $true )
=> ( ( notinemptyset = $true )
=> ( ( exuE3e = $true )
=> ( ( setext = $true )
=> ( ( emptyI = $true )
=> ( ( noeltsimpempty = $true )
=> ( ( setbeta = $true )
=> ( ( nonemptyE1 = $true )
=> ( ( nonemptyI = $true )
=> ( ( nonemptyI1 = $true )
=> ( ( setadjoinIL = $true )
=> ( ( emptyinunitempty = $true )
=> ( ( setadjoinIR = $true )
=> ( ( setadjoinE = $true )
=> ( ( setadjoinOr = $true )
=> ( ( setoftrueEq = $true )
=> ( ( powersetI = $true )
=> ( ( emptyinPowerset = $true )
=> ( ( emptyInPowerset = $true )
=> ( ( powersetE = $true )
=> ( ( setunionI = $true )
=> ( ( setunionE = $true )
=> ( ( subPowSU = $true )
=> ( ( exuE2 = $true )
=> ( ( nonemptyImpWitness = $true )
=> ( ( uniqinunit = $true )
=> ( ( notinsingleton = $true )
=> ( ( eqinunit = $true )
=> ( ( singletonsswitch = $true )
=> ( ( upairsetE = $true )
=> ( ( upairsetIL = $true )
=> ( ( upairsetIR = $true )
=> ( ( emptyE1 = $true )
=> ( ( vacuousDall = $true )
=> ( ( quantDeMorgan1 = $true )
=> ( ( quantDeMorgan2 = $true )
=> ( ( quantDeMorgan3 = $true )
=> ( ( quantDeMorgan4 = $true )
=> ( ( prop2setI = $true )
=> ( ( prop2set2propI = $true )
=> ( ( notdexE = $true )
=> ( ( notdallE = $true )
=> ( ( exuI1 = $true )
=> ( ( exuI3 = $true )
=> ( ( exuI2 = $true )
=> ( ( inCongP = $true )
=> ( ( in__Cong = $true )
=> ( ( exuE3u = $true )
=> ( ( exu__Cong = $true )
=> ( ( emptyset__Cong = $true )
=> ( ( setadjoin__Cong = $true )
=> ( ( powerset__Cong = $true )
=> ( ( setunion__Cong = $true )
=> ( ( omega__Cong = $true )
=> ( ( exuEu = $true )
=> ( ( descr__Cong = $true )
=> ( ( dsetconstr__Cong = $true )
=> ( ( subsetI1 = $true )
=> ( ( eqimpsubset2 = $true )
=> ( ( eqimpsubset1 = $true )
=> ( ( subsetI2 = $true )
=> ( ( emptysetsubset = $true )
=> ( ( subsetE = $true )
=> ( ( subsetE2 = $true )
=> ( ( notsubsetI = $true )
=> ( ( notequalI1 = $true )
=> ( ( notequalI2 = $true )
=> ( ( subsetRefl = $true )
=> ( ( subsetTrans = $true )
=> ( ( setadjoinSub = $true )
=> ( ( setadjoinSub2 = $true )
=> ( ( subset2powerset = $true )
=> ( ( setextsub = $true )
=> ( ( subsetemptysetimpeq = $true )
=> ( ( powersetI1 = $true )
=> ( ( powersetE1 = $true )
=> ( ( inPowerset = $true )
=> ( ( powersetsubset = $true )
=> ( ( sepInPowerset = $true )
=> ( ( sepSubset = $true )
=> ( ( binunionIL = $true )
=> ( ( upairset2IR = $true )
=> ( ( binunionIR = $true )
=> ( ( binunionEcases = $true )
=> ( ( binunionE = $true )
=> ( ( binunionLsub = $true )
=> ( ( binunionRsub = $true )
=> ( ( binintersectI = $true )
=> ( ( binintersectSubset5 = $true )
=> ( ( binintersectEL = $true )
=> ( ( binintersectLsub = $true )
=> ( ( binintersectSubset2 = $true )
=> ( ( binintersectSubset3 = $true )
=> ( ( binintersectER = $true )
=> ( ( disjointsetsI1 = $true )
=> ( ( binintersectRsub = $true )
=> ( ( binintersectSubset4 = $true )
=> ( ( binintersectSubset1 = $true )
=> ( ( bs114d = $true )
=> ( ( setminusI = $true )
=> ( ( setminusEL = $true )
=> ( ( setminusER = $true )
=> ( ( setminusSubset2 = $true )
=> ( ( setminusERneg = $true )
=> ( ( setminusELneg = $true )
=> ( ( setminusILneg = $true )
=> ( ( setminusIRneg = $true )
=> ( ( setminusLsub = $true )
=> ( ( setminusSubset1 = $true )
=> ( ( symdiffE = $true )
=> ( ( symdiffI1 = $true )
=> ( ( symdiffI2 = $true )
=> ( ( symdiffIneg1 = $true )
=> ( ( symdiffIneg2 = $true )
=> ( ( secondinupair = $true )
=> ( ( setukpairIL = $true )
=> ( ( setukpairIR = $true )
=> ( ( kpairiskpair = $true )
=> ( ( kpairp = $true )
=> ( ( singletonsubset = $true )
=> ( ( singletoninpowerset = $true )
=> ( ( singletoninpowunion = $true )
=> ( ( upairset2E = $true )
=> ( ( upairsubunion = $true )
=> ( ( upairinpowunion = $true )
=> ( ( ubforcartprodlem1 = $true )
=> ( ( ubforcartprodlem2 = $true )
=> ( ( ubforcartprodlem3 = $true )
=> ( ( cartprodpairin = $true )
=> ( ( cartprodmempair1 = $true )
=> ( ( cartprodmempair = $true )
=> ( ( setunionE2 = $true )
=> ( ( setunionsingleton1 = $true )
=> ( ( setunionsingleton2 = $true )
=> ( ( setunionsingleton = $true )
=> ( ( singletonprop = $true )
=> ( ( ex1E1 = $true )
=> ( ( ex1I = $true )
=> ( ( ex1I2 = $true )
=> ( ( singletonsuniq = $true )
=> ( ( setukpairinjL1 = $true )
=> ( ( kfstsingleton = $true )
=> ( ( theprop = $true )
=> ( ( kfstpairEq = $true )
=> ( ( cartprodfstin = $true )
=> ( ( setukpairinjL2 = $true )
=> ( ( setukpairinjL = $true )
=> ( ( setukpairinjR11 = $true )
=> ( ( setukpairinjR12 = $true )
=> ( ( setukpairinjR1 = $true )
=> ( ( upairequniteq = $true )
=> ( ( setukpairinjR2 = $true )
=> ( ( setukpairinjR = $true )
=> ( ( ksndsingleton = $true )
=> ( ( ksndpairEq = $true )
=> ( ( kpairsurjEq = $true )
=> ( ( cartprodsndin = $true )
=> ( ( cartprodpairmemEL = $true )
=> ( ( cartprodpairmemER = $true )
=> ( ( cartprodmempaircEq = $true )
=> ( ( cartprodfstpairEq = $true )
=> ( ( cartprodsndpairEq = $true )
=> ( ( cartprodpairsurjEq = $true )
=> ( ( dpsetconstrI = $true )
=> ( ( dpsetconstrSub = $true )
=> ( ( setOfPairsIsBReln = $true )
=> ( ( dpsetconstrERa = $true )
=> ( ( dpsetconstrEL1 = $true )
=> ( ( dpsetconstrEL2 = $true )
=> ( ( dpsetconstrER = $true )
=> ( ( funcImageSingleton = $true )
=> ( ( apProp = $true )
=> ( ( app = $true )
=> ( ( infuncsetfunc = $true )
=> ( ( ap2p = $true )
=> ( ( funcinfuncset = $true )
=> ( ( lamProp = $true )
=> ( ( lamp = $true )
=> ( ( lam2p = $true )
=> ( ( brelnall1 = $true )
=> ( ( brelnall2 = $true )
=> ( ( ex1E2 = $true )
=> ( ( funcGraphProp1 = $true )
=> ( ( funcGraphProp3 = $true )
=> ( ( funcGraphProp2 = $true )
=> ( ( funcextLem = $true )
=> ( ( funcGraphProp4 = $true )
=> ( ( subbreln = $true )
=> ( ( eqbreln = $true )
=> ( ( funcext = $true )
=> ( ( funcext2 = $true )
=> ( ( ap2apEq1 = $true )
=> ( ( ap2apEq2 = $true )
=> ( ( beta1 = $true )
=> ( ( eta1 = $true )
=> ( ( lam2lamEq = $true )
=> ( ( beta2 = $true )
=> ( ( eta2 = $true )
=> ( ( iffalseProp1 = $true )
=> ( ( iffalseProp2 = $true )
=> ( ( iftrueProp1 = $true )
=> ( ( iftrueProp2 = $true )
=> ( ( ifSingleton = $true )
=> ( ( ifp = $true )
=> ( ( theeq = $true )
=> ( ( iftrue = $true )
=> ( ( iffalse = $true )
=> ( ( iftrueorfalse = $true )
=> ( ( binintersectT_lem = $true )
=> ( ( binunionT_lem = $true )
=> ( ( powersetT_lem = $true )
=> ( ( setminusT_lem = $true )
=> ( ( complementT_lem = $true )
=> ( ( setextT = $true )
=> ( ( subsetTI = $true )
=> ( ( powersetTI1 = $true )
=> ( ( powersetTE1 = $true )
=> ( ( complementTI1 = $true )
=> ( ( complementTE1 = $true )
=> ( ( binintersectTELcontra = $true )
=> ( ( binintersectTERcontra = $true )
=> ( ( contrasubsetT = $true )
=> ( ( contrasubsetT1 = $true )
=> ( ( contrasubsetT2 = $true )
=> ( ( contrasubsetT3 = $true )
=> ( ( doubleComplementI1 = $true )
=> ( ( doubleComplementE1 = $true )
=> ( ( doubleComplementSub1 = $true )
=> ( ( doubleComplementSub2 = $true )
=> ( ( doubleComplementEq = $true )
=> ( ( complementTnotintersectT = $true )
=> ( ( complementImpComplementIntersect = $true )
=> ( ( complementSubsetComplementIntersect = $true )
=> ( ( complementInPowersetComplementIntersect = $true )
=> ( ( contraSubsetComplement = $true )
=> ( ( complementTcontraSubset = $true )
=> ( ( binunionTILcontra = $true )
=> ( ( binunionTIRcontra = $true )
=> ( ( inIntersectImpInUnion = $true )
=> ( ( inIntersectImpInUnion2 = $true )
=> ( ( inIntersectImpInIntersectUnions = $true )
=> ( ( intersectInPowersetIntersectUnions = $true )
=> ( ( inComplementUnionImpNotIn1 = $true )
=> ( ( inComplementUnionImpInComplement1 = $true )
=> ( ( binunionTE = $true )
=> ( ( binunionTEcontra = $true )
=> ( ( demorgan2a1 = $true )
=> ( ( complementUnionInPowersetComplement = $true )
=> ( ( demorgan2a2 = $true )
=> ( ( demorgan1a = $true )
=> ( ( demorgan1b = $true )
=> ( ( demorgan1 = $true )
=> ( ( demorgan2a = $true )
=> ( ( demorgan2b2 = $true )
=> ( ( demorgan2b = $true )
=> ( ( demorgan2 = $true )
=> ( ( woz13rule0 = $true )
=> ( ( woz13rule1 = $true )
=> ( ( woz13rule2 = $true )
=> ( ( woz13rule3 = $true )
=> ( ( woz13rule4 = $true )
=> ( ( woz1_1 = $true )
=> ( ( woz1_2 = $true )
=> ( ( woz1_3 = $true )
=> ( ( woz1_4 = $true )
=> ( ( woz1_5 = $true )
=> ( ( breln1all2 = $true )
=> ( ( breln1SetBreln1 = $true )
=> ( ( choice2fnsingleton = $true )
=> ( ( setOfPairsIsBReln1 = $true )
=> ( ( breln1all1 = $true )
=> ( ( subbreln1 = $true )
=> ( ( eqbreln1 = $true )
=> ( ( breln1invprop = $true )
=> ( ( breln1invI = $true )
=> ( ( breln1invE = $true )
=> ( ( breln1compprop = $true )
=> ( ( breln1compI = $true )
=> ( ( breln1compE = $true )
=> ( ( breln1compEex = $true )
=> ( ( breln1unionprop = $true )
=> ( ( breln1unionIL = $true )
=> ( ( breln1unionIR = $true )
=> ( ( breln1unionI = $true )
=> ( ( breln1unionE = $true )
=> ( ( breln1unionEcases = $true )
=> ( ( breln1unionCommutes = $true )
=> ( ( woz2Ex = $true )
=> ( ( woz2W = $true )
=> ( ( woz2A = $true )
=> ( ( woz2B = $true )
=> ( ( image1Ex = $true )
=> ( ( image1Ex1 = $true )
=> ( ( image1Equiv = $true )
=> ( ( image1E = $true )
=> ( ( image1I = $true )
=> ( ( injFuncInInjFuncSet = $true )
=> ( ( injFuncSetFuncIn = $true )
=> ! [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X2 @ ( injFuncSet @ X0 @ X1 ) ) )
=> ( $true
= ( injective @ X0 @ X1 @ X2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f717]) ).
thf(f717,plain,
~ ( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> ( setunionE2
=> ( setunionsingleton1
=> ( setunionsingleton2
=> ( setunionsingleton
=> ( singletonprop
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> ( setukpairinjL1
=> ( kfstsingleton
=> ( theprop
=> ( kfstpairEq
=> ( cartprodfstin
=> ( setukpairinjL2
=> ( setukpairinjL
=> ( setukpairinjR11
=> ( setukpairinjR12
=> ( setukpairinjR1
=> ( upairequniteq
=> ( setukpairinjR2
=> ( setukpairinjR
=> ( ksndsingleton
=> ( ksndpairEq
=> ( kpairsurjEq
=> ( cartprodsndin
=> ( cartprodpairmemEL
=> ( cartprodpairmemER
=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ( cartprodpairsurjEq
=> ( dpsetconstrI
=> ( dpsetconstrSub
=> ( setOfPairsIsBReln
=> ( dpsetconstrERa
=> ( dpsetconstrEL1
=> ( dpsetconstrEL2
=> ( dpsetconstrER
=> ( funcImageSingleton
=> ( apProp
=> ( app
=> ( infuncsetfunc
=> ( ap2p
=> ( funcinfuncset
=> ( lamProp
=> ( lamp
=> ( lam2p
=> ( brelnall1
=> ( brelnall2
=> ( ex1E2
=> ( funcGraphProp1
=> ( funcGraphProp3
=> ( funcGraphProp2
=> ( funcextLem
=> ( funcGraphProp4
=> ( subbreln
=> ( eqbreln
=> ( funcext
=> ( funcext2
=> ( ap2apEq1
=> ( ap2apEq2
=> ( beta1
=> ( eta1
=> ( lam2lamEq
=> ( beta2
=> ( eta2
=> ( iffalseProp1
=> ( iffalseProp2
=> ( iftrueProp1
=> ( iftrueProp2
=> ( ifSingleton
=> ( ifp
=> ( theeq
=> ( iftrue
=> ( iffalse
=> ( iftrueorfalse
=> ( binintersectT_lem
=> ( binunionT_lem
=> ( powersetT_lem
=> ( setminusT_lem
=> ( complementT_lem
=> ( setextT
=> ( subsetTI
=> ( powersetTI1
=> ( powersetTE1
=> ( complementTI1
=> ( complementTE1
=> ( binintersectTELcontra
=> ( binintersectTERcontra
=> ( contrasubsetT
=> ( contrasubsetT1
=> ( contrasubsetT2
=> ( contrasubsetT3
=> ( doubleComplementI1
=> ( doubleComplementE1
=> ( doubleComplementSub1
=> ( doubleComplementSub2
=> ( doubleComplementEq
=> ( complementTnotintersectT
=> ( complementImpComplementIntersect
=> ( complementSubsetComplementIntersect
=> ( complementInPowersetComplementIntersect
=> ( contraSubsetComplement
=> ( complementTcontraSubset
=> ( binunionTILcontra
=> ( binunionTIRcontra
=> ( inIntersectImpInUnion
=> ( inIntersectImpInUnion2
=> ( inIntersectImpInIntersectUnions
=> ( intersectInPowersetIntersectUnions
=> ( inComplementUnionImpNotIn1
=> ( inComplementUnionImpInComplement1
=> ( binunionTE
=> ( binunionTEcontra
=> ( demorgan2a1
=> ( complementUnionInPowersetComplement
=> ( demorgan2a2
=> ( demorgan1a
=> ( demorgan1b
=> ( demorgan1
=> ( demorgan2a
=> ( demorgan2b2
=> ( demorgan2b
=> ( demorgan2
=> ( woz13rule0
=> ( woz13rule1
=> ( woz13rule2
=> ( woz13rule3
=> ( woz13rule4
=> ( woz1_1
=> ( woz1_2
=> ( woz1_3
=> ( woz1_4
=> ( woz1_5
=> ( breln1all2
=> ( breln1SetBreln1
=> ( choice2fnsingleton
=> ( setOfPairsIsBReln1
=> ( breln1all1
=> ( subbreln1
=> ( eqbreln1
=> ( breln1invprop
=> ( breln1invI
=> ( breln1invE
=> ( breln1compprop
=> ( breln1compI
=> ( breln1compE
=> ( breln1compEex
=> ( breln1unionprop
=> ( breln1unionIL
=> ( breln1unionIR
=> ( breln1unionI
=> ( breln1unionE
=> ( breln1unionEcases
=> ( breln1unionCommutes
=> ( woz2Ex
=> ( woz2W
=> ( woz2A
=> ( woz2B
=> ( image1Ex
=> ( image1Ex1
=> ( image1Equiv
=> ( image1E
=> ( image1I
=> ( injFuncInInjFuncSet
=> ( injFuncSetFuncIn
=> ! [X0: $i,X1: $i,X2: $i] :
( ( in @ X2 @ ( injFuncSet @ X0 @ X1 ) )
=> ( injective @ X0 @ X1 @ X2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f321]) ).
thf(f321,negated_conjecture,
~ ( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> ( setunionE2
=> ( setunionsingleton1
=> ( setunionsingleton2
=> ( setunionsingleton
=> ( singletonprop
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> ( setukpairinjL1
=> ( kfstsingleton
=> ( theprop
=> ( kfstpairEq
=> ( cartprodfstin
=> ( setukpairinjL2
=> ( setukpairinjL
=> ( setukpairinjR11
=> ( setukpairinjR12
=> ( setukpairinjR1
=> ( upairequniteq
=> ( setukpairinjR2
=> ( setukpairinjR
=> ( ksndsingleton
=> ( ksndpairEq
=> ( kpairsurjEq
=> ( cartprodsndin
=> ( cartprodpairmemEL
=> ( cartprodpairmemER
=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ( cartprodpairsurjEq
=> ( dpsetconstrI
=> ( dpsetconstrSub
=> ( setOfPairsIsBReln
=> ( dpsetconstrERa
=> ( dpsetconstrEL1
=> ( dpsetconstrEL2
=> ( dpsetconstrER
=> ( funcImageSingleton
=> ( apProp
=> ( app
=> ( infuncsetfunc
=> ( ap2p
=> ( funcinfuncset
=> ( lamProp
=> ( lamp
=> ( lam2p
=> ( brelnall1
=> ( brelnall2
=> ( ex1E2
=> ( funcGraphProp1
=> ( funcGraphProp3
=> ( funcGraphProp2
=> ( funcextLem
=> ( funcGraphProp4
=> ( subbreln
=> ( eqbreln
=> ( funcext
=> ( funcext2
=> ( ap2apEq1
=> ( ap2apEq2
=> ( beta1
=> ( eta1
=> ( lam2lamEq
=> ( beta2
=> ( eta2
=> ( iffalseProp1
=> ( iffalseProp2
=> ( iftrueProp1
=> ( iftrueProp2
=> ( ifSingleton
=> ( ifp
=> ( theeq
=> ( iftrue
=> ( iffalse
=> ( iftrueorfalse
=> ( binintersectT_lem
=> ( binunionT_lem
=> ( powersetT_lem
=> ( setminusT_lem
=> ( complementT_lem
=> ( setextT
=> ( subsetTI
=> ( powersetTI1
=> ( powersetTE1
=> ( complementTI1
=> ( complementTE1
=> ( binintersectTELcontra
=> ( binintersectTERcontra
=> ( contrasubsetT
=> ( contrasubsetT1
=> ( contrasubsetT2
=> ( contrasubsetT3
=> ( doubleComplementI1
=> ( doubleComplementE1
=> ( doubleComplementSub1
=> ( doubleComplementSub2
=> ( doubleComplementEq
=> ( complementTnotintersectT
=> ( complementImpComplementIntersect
=> ( complementSubsetComplementIntersect
=> ( complementInPowersetComplementIntersect
=> ( contraSubsetComplement
=> ( complementTcontraSubset
=> ( binunionTILcontra
=> ( binunionTIRcontra
=> ( inIntersectImpInUnion
=> ( inIntersectImpInUnion2
=> ( inIntersectImpInIntersectUnions
=> ( intersectInPowersetIntersectUnions
=> ( inComplementUnionImpNotIn1
=> ( inComplementUnionImpInComplement1
=> ( binunionTE
=> ( binunionTEcontra
=> ( demorgan2a1
=> ( complementUnionInPowersetComplement
=> ( demorgan2a2
=> ( demorgan1a
=> ( demorgan1b
=> ( demorgan1
=> ( demorgan2a
=> ( demorgan2b2
=> ( demorgan2b
=> ( demorgan2
=> ( woz13rule0
=> ( woz13rule1
=> ( woz13rule2
=> ( woz13rule3
=> ( woz13rule4
=> ( woz1_1
=> ( woz1_2
=> ( woz1_3
=> ( woz1_4
=> ( woz1_5
=> ( breln1all2
=> ( breln1SetBreln1
=> ( choice2fnsingleton
=> ( setOfPairsIsBReln1
=> ( breln1all1
=> ( subbreln1
=> ( eqbreln1
=> ( breln1invprop
=> ( breln1invI
=> ( breln1invE
=> ( breln1compprop
=> ( breln1compI
=> ( breln1compE
=> ( breln1compEex
=> ( breln1unionprop
=> ( breln1unionIL
=> ( breln1unionIR
=> ( breln1unionI
=> ( breln1unionE
=> ( breln1unionEcases
=> ( breln1unionCommutes
=> ( woz2Ex
=> ( woz2W
=> ( woz2A
=> ( woz2B
=> ( image1Ex
=> ( image1Ex1
=> ( image1Equiv
=> ( image1E
=> ( image1I
=> ( injFuncInInjFuncSet
=> ( injFuncSetFuncIn
=> ! [X1: $i,X2: $i,X12: $i] :
( ( in @ X12 @ ( injFuncSet @ X1 @ X2 ) )
=> ( injective @ X1 @ X2 @ X12 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f320]) ).
thf(f320,conjecture,
( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> ( setunionE2
=> ( setunionsingleton1
=> ( setunionsingleton2
=> ( setunionsingleton
=> ( singletonprop
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> ( setukpairinjL1
=> ( kfstsingleton
=> ( theprop
=> ( kfstpairEq
=> ( cartprodfstin
=> ( setukpairinjL2
=> ( setukpairinjL
=> ( setukpairinjR11
=> ( setukpairinjR12
=> ( setukpairinjR1
=> ( upairequniteq
=> ( setukpairinjR2
=> ( setukpairinjR
=> ( ksndsingleton
=> ( ksndpairEq
=> ( kpairsurjEq
=> ( cartprodsndin
=> ( cartprodpairmemEL
=> ( cartprodpairmemER
=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ( cartprodpairsurjEq
=> ( dpsetconstrI
=> ( dpsetconstrSub
=> ( setOfPairsIsBReln
=> ( dpsetconstrERa
=> ( dpsetconstrEL1
=> ( dpsetconstrEL2
=> ( dpsetconstrER
=> ( funcImageSingleton
=> ( apProp
=> ( app
=> ( infuncsetfunc
=> ( ap2p
=> ( funcinfuncset
=> ( lamProp
=> ( lamp
=> ( lam2p
=> ( brelnall1
=> ( brelnall2
=> ( ex1E2
=> ( funcGraphProp1
=> ( funcGraphProp3
=> ( funcGraphProp2
=> ( funcextLem
=> ( funcGraphProp4
=> ( subbreln
=> ( eqbreln
=> ( funcext
=> ( funcext2
=> ( ap2apEq1
=> ( ap2apEq2
=> ( beta1
=> ( eta1
=> ( lam2lamEq
=> ( beta2
=> ( eta2
=> ( iffalseProp1
=> ( iffalseProp2
=> ( iftrueProp1
=> ( iftrueProp2
=> ( ifSingleton
=> ( ifp
=> ( theeq
=> ( iftrue
=> ( iffalse
=> ( iftrueorfalse
=> ( binintersectT_lem
=> ( binunionT_lem
=> ( powersetT_lem
=> ( setminusT_lem
=> ( complementT_lem
=> ( setextT
=> ( subsetTI
=> ( powersetTI1
=> ( powersetTE1
=> ( complementTI1
=> ( complementTE1
=> ( binintersectTELcontra
=> ( binintersectTERcontra
=> ( contrasubsetT
=> ( contrasubsetT1
=> ( contrasubsetT2
=> ( contrasubsetT3
=> ( doubleComplementI1
=> ( doubleComplementE1
=> ( doubleComplementSub1
=> ( doubleComplementSub2
=> ( doubleComplementEq
=> ( complementTnotintersectT
=> ( complementImpComplementIntersect
=> ( complementSubsetComplementIntersect
=> ( complementInPowersetComplementIntersect
=> ( contraSubsetComplement
=> ( complementTcontraSubset
=> ( binunionTILcontra
=> ( binunionTIRcontra
=> ( inIntersectImpInUnion
=> ( inIntersectImpInUnion2
=> ( inIntersectImpInIntersectUnions
=> ( intersectInPowersetIntersectUnions
=> ( inComplementUnionImpNotIn1
=> ( inComplementUnionImpInComplement1
=> ( binunionTE
=> ( binunionTEcontra
=> ( demorgan2a1
=> ( complementUnionInPowersetComplement
=> ( demorgan2a2
=> ( demorgan1a
=> ( demorgan1b
=> ( demorgan1
=> ( demorgan2a
=> ( demorgan2b2
=> ( demorgan2b
=> ( demorgan2
=> ( woz13rule0
=> ( woz13rule1
=> ( woz13rule2
=> ( woz13rule3
=> ( woz13rule4
=> ( woz1_1
=> ( woz1_2
=> ( woz1_3
=> ( woz1_4
=> ( woz1_5
=> ( breln1all2
=> ( breln1SetBreln1
=> ( choice2fnsingleton
=> ( setOfPairsIsBReln1
=> ( breln1all1
=> ( subbreln1
=> ( eqbreln1
=> ( breln1invprop
=> ( breln1invI
=> ( breln1invE
=> ( breln1compprop
=> ( breln1compI
=> ( breln1compE
=> ( breln1compEex
=> ( breln1unionprop
=> ( breln1unionIL
=> ( breln1unionIR
=> ( breln1unionI
=> ( breln1unionE
=> ( breln1unionEcases
=> ( breln1unionCommutes
=> ( woz2Ex
=> ( woz2W
=> ( woz2A
=> ( woz2B
=> ( image1Ex
=> ( image1Ex1
=> ( image1Equiv
=> ( image1E
=> ( image1I
=> ( injFuncInInjFuncSet
=> ( injFuncSetFuncIn
=> ! [X1: $i,X2: $i,X12: $i] :
( ( in @ X12 @ ( injFuncSet @ X1 @ X2 ) )
=> ( injective @ X1 @ X2 @ X12 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.2k8MKHSw6S/DTF2THF_18170.p',injFuncSetFuncInj) ).
thf(f9822,plain,
( $true
= ( injective @ sK228 @ sK229 @ sK230 ) ),
inference(trivial_inequality_removal,[status(thm)],[f9815]) ).
thf(f9815,plain,
( ( $true
= ( injective @ sK228 @ sK229 @ sK230 ) )
| ( $true != $true ) ),
inference(superposition,[status(thm)],[f6933,f7125]) ).
thf(f7125,plain,
( $true
= ( in @ sK230 @ ( dsetconstr @ ( funcSet @ sK228 @ sK229 ) @ ( injective @ sK228 @ sK229 ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f5349]) ).
thf(f5349,plain,
( $true
= ( in @ sK230
@ ( ^ [Y0: $i,Y1: $i] :
( dsetconstr @ ( funcSet @ Y0 @ Y1 )
@ ^ [Y2: $i] : ( injective @ Y0 @ Y1 @ Y2 ) )
@ sK228
@ sK229 ) ) ),
inference(definition_unfolding,[status(thm)],[f3737,f4974]) ).
thf(f4974,plain,
( injFuncSet
= ( ^ [Y0: $i,Y1: $i] :
( dsetconstr @ ( funcSet @ Y0 @ Y1 )
@ ^ [Y2: $i] : ( injective @ Y0 @ Y1 @ Y2 ) ) ) ),
inference(cnf_transformation,[status(thm)],[f454]) ).
thf(f454,plain,
( injFuncSet
= ( ^ [Y0: $i,Y1: $i] :
( dsetconstr @ ( funcSet @ Y0 @ Y1 )
@ ^ [Y2: $i] : ( injective @ Y0 @ Y1 @ Y2 ) ) ) ),
inference(fool_elimination,[status(thm)],[f453]) ).
thf(f453,plain,
( injFuncSet
= ( ^ [X0: $i,X1: $i] :
( dsetconstr @ ( funcSet @ X0 @ X1 )
@ ^ [X2: $i] : ( injective @ X0 @ X1 @ X2 ) ) ) ),
inference(rectify,[status(thm)],[f317]) ).
thf(f317,axiom,
( injFuncSet
= ( ^ [X3: $i,X4: $i] :
( dsetconstr @ ( funcSet @ X3 @ X4 )
@ ^ [X12: $i] : ( injective @ X3 @ X4 @ X12 ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.2k8MKHSw6S/DTF2THF_18170.p',injFuncSet) ).
thf(f3737,plain,
( ( in @ sK230 @ ( injFuncSet @ sK228 @ sK229 ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f1778]) ).
thf(f6933,plain,
! [X2: $i > $o,X0: $i,X1: $i] :
( ( ( in @ X1 @ ( dsetconstr @ X0 @ X2 ) )
!= $true )
| ( $true
= ( X2 @ X1 ) ) ),
inference(beta_eta_normalization,[status(thm)],[f6932]) ).
thf(f6932,plain,
! [X2: $i > $o,X0: $i,X1: $i] :
( ( $true
!= ( in @ X1
@ ( dsetconstr @ X0
@ ^ [Y0: $i] : ( X2 @ Y0 ) ) ) )
| ( $true
= ( X2 @ X1 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f5813]) ).
thf(f5813,plain,
! [X2: $i > $o,X0: $i,X1: $i] :
( ( $true != $true )
| ( $true
= ( X2 @ X1 ) )
| ( $true
!= ( in @ X1
@ ( dsetconstr @ X0
@ ^ [Y0: $i] : ( X2 @ Y0 ) ) ) ) ),
inference(definition_unfolding,[status(thm)],[f4466,f3793]) ).
thf(f3793,plain,
dsetconstrER = $true,
inference(cnf_transformation,[status(thm)],[f1778]) ).
thf(f4466,plain,
! [X2: $i > $o,X0: $i,X1: $i] :
( ( $true
!= ( in @ X1
@ ( dsetconstr @ X0
@ ^ [Y0: $i] : ( X2 @ Y0 ) ) ) )
| ( $true
= ( X2 @ X1 ) )
| ( dsetconstrER != $true ) ),
inference(cnf_transformation,[status(thm)],[f2495]) ).
thf(f2495,plain,
( ( ! [X0: $i,X1: $i,X2: $i > $o] :
( ( $true
!= ( in @ X1
@ ( dsetconstr @ X0
@ ^ [Y0: $i] : ( X2 @ Y0 ) ) ) )
| ( $true
= ( X2 @ X1 ) ) )
| ( dsetconstrER != $true ) )
& ( ( dsetconstrER = $true )
| ( ( $true
= ( in @ sK669
@ ( dsetconstr @ sK668
@ ^ [Y0: $i] : ( sK670 @ Y0 ) ) ) )
& ( $true
!= ( sK670 @ sK669 ) ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK668,sK669,sK670])],[f2493,f2494]) ).
thf(f2494,plain,
( ? [X3: $i,X4: $i,X5: $i > $o] :
( ( $true
= ( in @ X4
@ ( dsetconstr @ X3
@ ^ [Y0: $i] : ( X5 @ Y0 ) ) ) )
& ( $true
!= ( X5 @ X4 ) ) )
=> ( ( $true
= ( in @ sK669
@ ( dsetconstr @ sK668
@ ^ [Y0: $i] : ( sK670 @ Y0 ) ) ) )
& ( $true
!= ( sK670 @ sK669 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f2493,plain,
( ( ! [X0: $i,X1: $i,X2: $i > $o] :
( ( $true
!= ( in @ X1
@ ( dsetconstr @ X0
@ ^ [Y0: $i] : ( X2 @ Y0 ) ) ) )
| ( $true
= ( X2 @ X1 ) ) )
| ( dsetconstrER != $true ) )
& ( ( dsetconstrER = $true )
| ? [X3: $i,X4: $i,X5: $i > $o] :
( ( $true
= ( in @ X4
@ ( dsetconstr @ X3
@ ^ [Y0: $i] : ( X5 @ Y0 ) ) ) )
& ( $true
!= ( X5 @ X4 ) ) ) ) ),
inference(rectify,[status(thm)],[f2492]) ).
thf(f2492,plain,
( ( ! [X0: $i,X1: $i,X2: $i > $o] :
( ( $true
!= ( in @ X1
@ ( dsetconstr @ X0
@ ^ [Y0: $i] : ( X2 @ Y0 ) ) ) )
| ( $true
= ( X2 @ X1 ) ) )
| ( dsetconstrER != $true ) )
& ( ( dsetconstrER = $true )
| ? [X0: $i,X1: $i,X2: $i > $o] :
( ( $true
= ( in @ X1
@ ( dsetconstr @ X0
@ ^ [Y0: $i] : ( X2 @ Y0 ) ) ) )
& ( $true
!= ( X2 @ X1 ) ) ) ) ),
inference(nnf_transformation,[status(thm)],[f1029]) ).
thf(f1029,plain,
( ! [X0: $i,X1: $i,X2: $i > $o] :
( ( $true
!= ( in @ X1
@ ( dsetconstr @ X0
@ ^ [Y0: $i] : ( X2 @ Y0 ) ) ) )
| ( $true
= ( X2 @ X1 ) ) )
<=> ( dsetconstrER = $true ) ),
inference(ennf_transformation,[status(thm)],[f894]) ).
thf(f894,plain,
( ( dsetconstrER = $true )
<=> ! [X0: $i,X1: $i,X2: $i > $o] :
( ( $true
= ( in @ X1
@ ( dsetconstr @ X0
@ ^ [Y0: $i] : ( X2 @ Y0 ) ) ) )
=> ( $true
= ( X2 @ X1 ) ) ) ),
inference(fool_elimination,[status(thm)],[f893]) ).
thf(f893,plain,
( dsetconstrER
= ( ! [X0: $i,X1: $i,X2: $i > $o] :
( ( in @ X1
@ ( dsetconstr @ X0
@ ^ [X3: $i] : ( X2 @ X3 ) ) )
=> ( X2 @ X1 ) ) ) ),
inference(rectify,[status(thm)],[f16]) ).
thf(f16,axiom,
( dsetconstrER
= ( ! [X3: $i,X1: $i,X0: $i > $o] :
( ( in @ X1
@ ( dsetconstr @ X3
@ ^ [X2: $i] : ( X0 @ X2 ) ) )
=> ( X0 @ X1 ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.2k8MKHSw6S/DTF2THF_18170.p',dsetconstrER) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SEU800^1 : TPTP v9.2.1. Released v3.7.0.
% 0.12/0.13 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.15/0.33 % Computer : n008.cluster.edu
% 0.15/0.33 % Model : x86_64 x86_64
% 0.15/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.33 % Memory : 8042.1875MB
% 0.15/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.33 % CPULimit : 300
% 0.15/0.33 % WCLimit : 300
% 0.15/0.33 % DateTime : Tue Feb 24 12:34:24 EST 2026
% 0.15/0.33 % CPUTime :
% 0.15/0.33 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.19/0.45 ---- Original DTF file ---
% 0.19/0.45 thf(spec,logic,$$dhol).
% 0.19/0.45 %------------------------------------------------------------------------------
% 0.19/0.45 % File : SEU800^1 : TPTP v9.2.1. Released v3.7.0.
% 0.19/0.45 % Domain : Set Theory
% 0.19/0.45 % Problem : More about Functions - Injective Functions
% 0.19/0.45 % Version : Especial.
% 0.19/0.45 % English : (! x:i.! y:i.! f:i.in f (injFuncSet x y) -> injective x y f)
% 0.19/0.45
% 0.19/0.45 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.19/0.45 % Source : [Bro08]
% 0.19/0.45 % Names : ZFC302g [Bro08]
% 0.19/0.45
% 0.19/0.45 % Status : Theorem
% 0.19/0.45 % Rating : 0.11 v9.1.0, 0.12 v9.0.0, 0.30 v8.2.0, 0.54 v8.1.0, 0.45 v7.5.0, 0.14 v7.4.0, 0.22 v7.2.0, 0.12 v7.1.0, 0.38 v7.0.0, 0.29 v6.4.0, 0.33 v6.3.0, 0.40 v6.2.0, 0.43 v6.1.0, 0.57 v5.5.0, 0.67 v5.4.0, 0.60 v5.2.0, 1.00 v3.7.0
% 0.19/0.45 % Syntax : Number of formulae : 687 ( 319 unt; 367 typ; 319 def)
% 0.19/0.45 % Number of atoms : 1966 ( 459 equ; 0 cnn)
% 0.19/0.45 % Maximal formula atoms : 318 ( 6 avg)
% 0.19/0.45 % Number of connectives : 4326 ( 73 ~; 14 |; 53 &;3151 @)
% 0.19/0.45 % ( 17 <=>;1018 =>; 0 <=; 0 <~>)
% 0.19/0.45 % Maximal formula depth : 324 ( 2 avg)
% 0.19/0.45 % Number of types : 2 ( 0 usr)
% 0.19/0.45 % Number of type conns : 170 ( 170 >; 0 *; 0 +; 0 <<)
% 0.19/0.45 % Number of symbols : 370 ( 367 usr; 320 con; 0-4 aty)
% 0.19/0.45 % Number of variables : 1130 ( 78 ^;1009 !; 43 ?;1130 :)
% 0.19/0.45 % SPC : TH0_THM_EQU_NAR
% 0.19/0.45
% 0.19/0.45 % Comments : http://mathgate.info/detsetitem.php?id=425
% 0.19/0.45 %------------------------------------------------------------------------------
% 0.19/0.45 thf(in_type,type,
% 0.19/0.45 in: $i > $i > $o ).
% 0.19/0.45
% 0.19/0.45 thf(exu_type,type,
% 0.19/0.45 exu: ( $i > $o ) > $o ).
% 0.19/0.45
% 0.19/0.45 thf(exu,definition,
% 0.19/0.45 ( exu
% 0.19/0.45 = ( ^ [Xphi: $i > $o] :
% 0.19/0.45 ? [Xx: $i] :
% 0.19/0.45 ( ( Xphi @ Xx )
% 0.19/0.45 & ! [Xy: $i] :
% 0.19/0.45 ( ( Xphi @ Xy )
% 0.19/0.45 => ( Xx = Xy ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setextAx_type,type,
% 0.19/0.45 setextAx: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setextAx,definition,
% 0.19/0.45 ( setextAx
% 0.19/0.45 = ( ! [A: $i,B: $i] :
% 0.19/0.45 ( ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 <=> ( in @ Xx @ B ) )
% 0.19/0.45 => ( A = B ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(emptyset_type,type,
% 0.19/0.45 emptyset: $i ).
% 0.19/0.45
% 0.19/0.45 thf(emptysetAx_type,type,
% 0.19/0.45 emptysetAx: $o ).
% 0.19/0.45
% 0.19/0.45 thf(emptysetAx,definition,
% 0.19/0.45 ( emptysetAx
% 0.19/0.45 = ( ! [Xx: $i] :
% 0.19/0.45 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setadjoin_type,type,
% 0.19/0.45 setadjoin: $i > $i > $i ).
% 0.19/0.45
% 0.19/0.45 thf(setadjoinAx_type,type,
% 0.19/0.45 setadjoinAx: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setadjoinAx,definition,
% 0.19/0.45 ( setadjoinAx
% 0.19/0.45 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.19/0.45 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.19/0.45 <=> ( ( Xy = Xx )
% 0.19/0.45 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(powerset_type,type,
% 0.19/0.45 powerset: $i > $i ).
% 0.19/0.45
% 0.19/0.45 thf(powersetAx_type,type,
% 0.19/0.45 powersetAx: $o ).
% 0.19/0.45
% 0.19/0.45 thf(powersetAx,definition,
% 0.19/0.45 ( powersetAx
% 0.19/0.45 = ( ! [A: $i,B: $i] :
% 0.19/0.45 ( ( in @ B @ ( powerset @ A ) )
% 0.19/0.45 <=> ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ B )
% 0.19/0.45 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setunion_type,type,
% 0.19/0.45 setunion: $i > $i ).
% 0.19/0.45
% 0.19/0.45 thf(setunionAx_type,type,
% 0.19/0.45 setunionAx: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setunionAx,definition,
% 0.19/0.45 ( setunionAx
% 0.19/0.45 = ( ! [A: $i,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ ( setunion @ A ) )
% 0.19/0.45 <=> ? [B: $i] :
% 0.19/0.45 ( ( in @ Xx @ B )
% 0.19/0.45 & ( in @ B @ A ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(omega_type,type,
% 0.19/0.45 omega: $i ).
% 0.19/0.45
% 0.19/0.45 thf(omega0Ax_type,type,
% 0.19/0.45 omega0Ax: $o ).
% 0.19/0.45
% 0.19/0.45 thf(omega0Ax,definition,
% 0.19/0.45 ( omega0Ax
% 0.19/0.45 = ( in @ emptyset @ omega ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(omegaSAx_type,type,
% 0.19/0.45 omegaSAx: $o ).
% 0.19/0.45
% 0.19/0.45 thf(omegaSAx,definition,
% 0.19/0.45 ( omegaSAx
% 0.19/0.45 = ( ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ omega )
% 0.19/0.45 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(omegaIndAx_type,type,
% 0.19/0.45 omegaIndAx: $o ).
% 0.19/0.45
% 0.19/0.45 thf(omegaIndAx,definition,
% 0.19/0.45 ( omegaIndAx
% 0.19/0.45 = ( ! [A: $i] :
% 0.19/0.45 ( ( ( in @ emptyset @ A )
% 0.19/0.45 & ! [Xx: $i] :
% 0.19/0.45 ( ( ( in @ Xx @ omega )
% 0.19/0.45 & ( in @ Xx @ A ) )
% 0.19/0.45 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.19/0.45 => ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ omega )
% 0.19/0.45 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(replAx_type,type,
% 0.19/0.45 replAx: $o ).
% 0.19/0.45
% 0.19/0.45 thf(replAx,definition,
% 0.19/0.45 ( replAx
% 0.19/0.45 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.19/0.45 ( ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 => ( exu
% 0.19/0.45 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.19/0.45 => ? [B: $i] :
% 0.19/0.45 ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ B )
% 0.19/0.45 <=> ? [Xy: $i] :
% 0.19/0.45 ( ( in @ Xy @ A )
% 0.19/0.45 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(foundationAx_type,type,
% 0.19/0.45 foundationAx: $o ).
% 0.19/0.45
% 0.19/0.45 thf(foundationAx,definition,
% 0.19/0.45 ( foundationAx
% 0.19/0.45 = ( ! [A: $i] :
% 0.19/0.45 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.19/0.45 => ? [B: $i] :
% 0.19/0.45 ( ( in @ B @ A )
% 0.19/0.45 & ~ ? [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ B )
% 0.19/0.45 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(wellorderingAx_type,type,
% 0.19/0.45 wellorderingAx: $o ).
% 0.19/0.45
% 0.19/0.45 thf(wellorderingAx,definition,
% 0.19/0.45 ( wellorderingAx
% 0.19/0.45 = ( ! [A: $i] :
% 0.19/0.45 ? [B: $i] :
% 0.19/0.45 ( ! [C: $i] :
% 0.19/0.45 ( ( in @ C @ B )
% 0.19/0.45 => ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ C )
% 0.19/0.45 => ( in @ Xx @ A ) ) )
% 0.19/0.45 & ! [Xx: $i,Xy: $i] :
% 0.19/0.45 ( ( ( in @ Xx @ A )
% 0.19/0.45 & ( in @ Xy @ A ) )
% 0.19/0.45 => ( ! [C: $i] :
% 0.19/0.45 ( ( in @ C @ B )
% 0.19/0.45 => ( ( in @ Xx @ C )
% 0.19/0.45 <=> ( in @ Xy @ C ) ) )
% 0.19/0.45 => ( Xx = Xy ) ) )
% 0.19/0.45 & ! [C: $i,D: $i] :
% 0.19/0.45 ( ( ( in @ C @ B )
% 0.19/0.45 & ( in @ D @ B ) )
% 0.19/0.45 => ( ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ C )
% 0.19/0.45 => ( in @ Xx @ D ) )
% 0.19/0.45 | ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ D )
% 0.19/0.45 => ( in @ Xx @ C ) ) ) )
% 0.19/0.45 & ! [C: $i] :
% 0.19/0.45 ( ( ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ C )
% 0.19/0.45 => ( in @ Xx @ A ) )
% 0.19/0.45 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.19/0.45 => ? [D: $i,Xx: $i] :
% 0.19/0.45 ( ( in @ D @ B )
% 0.19/0.45 & ( in @ Xx @ C )
% 0.19/0.45 & ~ ? [Xy: $i] :
% 0.19/0.45 ( ( in @ Xy @ D )
% 0.19/0.45 & ( in @ Xy @ C ) )
% 0.19/0.45 & ! [E: $i] :
% 0.19/0.45 ( ( in @ E @ B )
% 0.19/0.45 => ( ! [Xy: $i] :
% 0.19/0.45 ( ( in @ Xy @ E )
% 0.19/0.45 => ( in @ Xy @ D ) )
% 0.19/0.45 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(descr_type,type,
% 0.19/0.45 descr: ( $i > $o ) > $i ).
% 0.19/0.45
% 0.19/0.45 thf(descrp_type,type,
% 0.19/0.45 descrp: $o ).
% 0.19/0.45
% 0.19/0.45 thf(descrp,definition,
% 0.19/0.45 ( descrp
% 0.19/0.45 = ( ! [Xphi: $i > $o] :
% 0.19/0.45 ( ( exu
% 0.19/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.45 => ( Xphi
% 0.19/0.45 @ ( descr
% 0.19/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(dsetconstr_type,type,
% 0.19/0.45 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.19/0.45
% 0.19/0.45 thf(dsetconstrI_type,type,
% 0.19/0.45 dsetconstrI: $o ).
% 0.19/0.45
% 0.19/0.45 thf(dsetconstrI,definition,
% 0.19/0.45 ( dsetconstrI
% 0.19/0.45 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 => ( ( Xphi @ Xx )
% 0.19/0.45 => ( in @ Xx
% 0.19/0.45 @ ( dsetconstr @ A
% 0.19/0.45 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(dsetconstrEL_type,type,
% 0.19/0.45 dsetconstrEL: $o ).
% 0.19/0.45
% 0.19/0.45 thf(dsetconstrEL,definition,
% 0.19/0.45 ( dsetconstrEL
% 0.19/0.45 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx
% 0.19/0.45 @ ( dsetconstr @ A
% 0.19/0.45 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.19/0.45 => ( in @ Xx @ A ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(dsetconstrER_type,type,
% 0.19/0.45 dsetconstrER: $o ).
% 0.19/0.45
% 0.19/0.45 thf(dsetconstrER,definition,
% 0.19/0.45 ( dsetconstrER
% 0.19/0.45 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx
% 0.19/0.45 @ ( dsetconstr @ A
% 0.19/0.45 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.19/0.45 => ( Xphi @ Xx ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(exuE1_type,type,
% 0.19/0.45 exuE1: $o ).
% 0.19/0.45
% 0.19/0.45 thf(exuE1,definition,
% 0.19/0.45 ( exuE1
% 0.19/0.45 = ( ! [Xphi: $i > $o] :
% 0.19/0.45 ( ( exu
% 0.19/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.45 => ? [Xx: $i] :
% 0.19/0.45 ( ( Xphi @ Xx )
% 0.19/0.45 & ! [Xy: $i] :
% 0.19/0.45 ( ( Xphi @ Xy )
% 0.19/0.45 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(prop2set_type,type,
% 0.19/0.45 prop2set: $o > $i ).
% 0.19/0.45
% 0.19/0.45 thf(prop2setE_type,type,
% 0.19/0.45 prop2setE: $o ).
% 0.19/0.45
% 0.19/0.45 thf(prop2setE,definition,
% 0.19/0.45 ( prop2setE
% 0.19/0.45 = ( ! [Xphi: $o,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.19/0.45 => Xphi ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(emptysetE_type,type,
% 0.19/0.45 emptysetE: $o ).
% 0.19/0.45
% 0.19/0.45 thf(emptysetE,definition,
% 0.19/0.45 ( emptysetE
% 0.19/0.45 = ( ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ emptyset )
% 0.19/0.45 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(emptysetimpfalse_type,type,
% 0.19/0.45 emptysetimpfalse: $o ).
% 0.19/0.45
% 0.19/0.45 thf(emptysetimpfalse,definition,
% 0.19/0.45 ( emptysetimpfalse
% 0.19/0.45 = ( ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ emptyset )
% 0.19/0.45 => $false ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(notinemptyset_type,type,
% 0.19/0.45 notinemptyset: $o ).
% 0.19/0.45
% 0.19/0.45 thf(notinemptyset,definition,
% 0.19/0.45 ( notinemptyset
% 0.19/0.45 = ( ! [Xx: $i] :
% 0.19/0.45 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(exuE3e_type,type,
% 0.19/0.45 exuE3e: $o ).
% 0.19/0.45
% 0.19/0.45 thf(exuE3e,definition,
% 0.19/0.45 ( exuE3e
% 0.19/0.45 = ( ! [Xphi: $i > $o] :
% 0.19/0.45 ( ( exu
% 0.19/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.45 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setext_type,type,
% 0.19/0.45 setext: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setext,definition,
% 0.19/0.45 ( setext
% 0.19/0.45 = ( ! [A: $i,B: $i] :
% 0.19/0.45 ( ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 => ( in @ Xx @ B ) )
% 0.19/0.45 => ( ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ B )
% 0.19/0.45 => ( in @ Xx @ A ) )
% 0.19/0.45 => ( A = B ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(emptyI_type,type,
% 0.19/0.45 emptyI: $o ).
% 0.19/0.45
% 0.19/0.45 thf(emptyI,definition,
% 0.19/0.45 ( emptyI
% 0.19/0.45 = ( ! [A: $i] :
% 0.19/0.45 ( ! [Xx: $i] :
% 0.19/0.45 ~ ( in @ Xx @ A )
% 0.19/0.45 => ( A = emptyset ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(noeltsimpempty_type,type,
% 0.19/0.45 noeltsimpempty: $o ).
% 0.19/0.45
% 0.19/0.45 thf(noeltsimpempty,definition,
% 0.19/0.45 ( noeltsimpempty
% 0.19/0.45 = ( ! [A: $i] :
% 0.19/0.45 ( ! [Xx: $i] :
% 0.19/0.45 ~ ( in @ Xx @ A )
% 0.19/0.45 => ( A = emptyset ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setbeta_type,type,
% 0.19/0.45 setbeta: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setbeta,definition,
% 0.19/0.45 ( setbeta
% 0.19/0.45 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 => ( ( in @ Xx
% 0.19/0.45 @ ( dsetconstr @ A
% 0.19/0.45 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.19/0.45 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(nonempty_type,type,
% 0.19/0.45 nonempty: $i > $o ).
% 0.19/0.45
% 0.19/0.45 thf(nonempty,definition,
% 0.19/0.45 ( nonempty
% 0.19/0.45 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(nonemptyE1_type,type,
% 0.19/0.45 nonemptyE1: $o ).
% 0.19/0.45
% 0.19/0.45 thf(nonemptyE1,definition,
% 0.19/0.45 ( nonemptyE1
% 0.19/0.45 = ( ! [A: $i] :
% 0.19/0.45 ( ( nonempty @ A )
% 0.19/0.45 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(nonemptyI_type,type,
% 0.19/0.45 nonemptyI: $o ).
% 0.19/0.45
% 0.19/0.45 thf(nonemptyI,definition,
% 0.19/0.45 ( nonemptyI
% 0.19/0.45 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 => ( ( Xphi @ Xx )
% 0.19/0.45 => ( nonempty
% 0.19/0.45 @ ( dsetconstr @ A
% 0.19/0.45 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(nonemptyI1_type,type,
% 0.19/0.45 nonemptyI1: $o ).
% 0.19/0.45
% 0.19/0.45 thf(nonemptyI1,definition,
% 0.19/0.45 ( nonemptyI1
% 0.19/0.45 = ( ! [A: $i] :
% 0.19/0.45 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.19/0.45 => ( nonempty @ A ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setadjoinIL_type,type,
% 0.19/0.45 setadjoinIL: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setadjoinIL,definition,
% 0.19/0.45 ( setadjoinIL
% 0.19/0.45 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(emptyinunitempty_type,type,
% 0.19/0.45 emptyinunitempty: $o ).
% 0.19/0.45
% 0.19/0.45 thf(emptyinunitempty,definition,
% 0.19/0.45 ( emptyinunitempty
% 0.19/0.45 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setadjoinIR_type,type,
% 0.19/0.45 setadjoinIR: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setadjoinIR,definition,
% 0.19/0.45 ( setadjoinIR
% 0.19/0.45 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.19/0.45 ( ( in @ Xy @ A )
% 0.19/0.45 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setadjoinE_type,type,
% 0.19/0.45 setadjoinE: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setadjoinE,definition,
% 0.19/0.45 ( setadjoinE
% 0.19/0.45 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.19/0.45 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.19/0.45 => ! [Xphi: $o] :
% 0.19/0.45 ( ( ( Xy = Xx )
% 0.19/0.45 => Xphi )
% 0.19/0.45 => ( ( ( in @ Xy @ A )
% 0.19/0.45 => Xphi )
% 0.19/0.45 => Xphi ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setadjoinOr_type,type,
% 0.19/0.45 setadjoinOr: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setadjoinOr,definition,
% 0.19/0.45 ( setadjoinOr
% 0.19/0.45 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.19/0.45 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.19/0.45 => ( ( Xy = Xx )
% 0.19/0.45 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setoftrueEq_type,type,
% 0.19/0.45 setoftrueEq: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setoftrueEq,definition,
% 0.19/0.45 ( setoftrueEq
% 0.19/0.45 = ( ! [A: $i] :
% 0.19/0.45 ( ( dsetconstr @ A
% 0.19/0.45 @ ^ [Xx: $i] : $true )
% 0.19/0.45 = A ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(powersetI_type,type,
% 0.19/0.45 powersetI: $o ).
% 0.19/0.45
% 0.19/0.45 thf(powersetI,definition,
% 0.19/0.45 ( powersetI
% 0.19/0.45 = ( ! [A: $i,B: $i] :
% 0.19/0.45 ( ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ B )
% 0.19/0.45 => ( in @ Xx @ A ) )
% 0.19/0.45 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(emptyinPowerset_type,type,
% 0.19/0.45 emptyinPowerset: $o ).
% 0.19/0.45
% 0.19/0.45 thf(emptyinPowerset,definition,
% 0.19/0.45 ( emptyinPowerset
% 0.19/0.45 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(emptyInPowerset_type,type,
% 0.19/0.45 emptyInPowerset: $o ).
% 0.19/0.45
% 0.19/0.45 thf(emptyInPowerset,definition,
% 0.19/0.45 ( emptyInPowerset
% 0.19/0.45 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(powersetE_type,type,
% 0.19/0.45 powersetE: $o ).
% 0.19/0.45
% 0.19/0.45 thf(powersetE,definition,
% 0.19/0.45 ( powersetE
% 0.19/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.45 ( ( in @ B @ ( powerset @ A ) )
% 0.19/0.45 => ( ( in @ Xx @ B )
% 0.19/0.45 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setunionI_type,type,
% 0.19/0.45 setunionI: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setunionI,definition,
% 0.19/0.45 ( setunionI
% 0.19/0.45 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.19/0.45 ( ( in @ Xx @ B )
% 0.19/0.45 => ( ( in @ B @ A )
% 0.19/0.45 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setunionE_type,type,
% 0.19/0.45 setunionE: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setunionE,definition,
% 0.19/0.45 ( setunionE
% 0.19/0.45 = ( ! [A: $i,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ ( setunion @ A ) )
% 0.19/0.45 => ! [Xphi: $o] :
% 0.19/0.45 ( ! [B: $i] :
% 0.19/0.45 ( ( in @ Xx @ B )
% 0.19/0.45 => ( ( in @ B @ A )
% 0.19/0.45 => Xphi ) )
% 0.19/0.45 => Xphi ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(subPowSU_type,type,
% 0.19/0.45 subPowSU: $o ).
% 0.19/0.45
% 0.19/0.45 thf(subPowSU,definition,
% 0.19/0.45 ( subPowSU
% 0.19/0.45 = ( ! [A: $i,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(exuE2_type,type,
% 0.19/0.45 exuE2: $o ).
% 0.19/0.45
% 0.19/0.45 thf(exuE2,definition,
% 0.19/0.45 ( exuE2
% 0.19/0.45 = ( ! [Xphi: $i > $o] :
% 0.19/0.45 ( ( exu
% 0.19/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.45 => ? [Xx: $i] :
% 0.19/0.45 ! [Xy: $i] :
% 0.19/0.45 ( ( Xphi @ Xy )
% 0.19/0.45 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(nonemptyImpWitness_type,type,
% 0.19/0.45 nonemptyImpWitness: $o ).
% 0.19/0.45
% 0.19/0.45 thf(nonemptyImpWitness,definition,
% 0.19/0.45 ( nonemptyImpWitness
% 0.19/0.45 = ( ! [A: $i] :
% 0.19/0.45 ( ( nonempty @ A )
% 0.19/0.45 => ? [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 & $true ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(uniqinunit_type,type,
% 0.19/0.45 uniqinunit: $o ).
% 0.19/0.45
% 0.19/0.45 thf(uniqinunit,definition,
% 0.19/0.45 ( uniqinunit
% 0.19/0.45 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.45 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.19/0.45 => ( Xx = Xy ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(notinsingleton_type,type,
% 0.19/0.45 notinsingleton: $o ).
% 0.19/0.45
% 0.19/0.45 thf(notinsingleton,definition,
% 0.19/0.45 ( notinsingleton
% 0.19/0.45 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.45 ( ( Xx != Xy )
% 0.19/0.45 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(eqinunit_type,type,
% 0.19/0.45 eqinunit: $o ).
% 0.19/0.45
% 0.19/0.45 thf(eqinunit,definition,
% 0.19/0.45 ( eqinunit
% 0.19/0.45 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.45 ( ( Xx = Xy )
% 0.19/0.45 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(singletonsswitch_type,type,
% 0.19/0.45 singletonsswitch: $o ).
% 0.19/0.45
% 0.19/0.45 thf(singletonsswitch,definition,
% 0.19/0.45 ( singletonsswitch
% 0.19/0.45 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.45 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.19/0.45 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(upairsetE_type,type,
% 0.19/0.45 upairsetE: $o ).
% 0.19/0.45
% 0.19/0.45 thf(upairsetE,definition,
% 0.19/0.45 ( upairsetE
% 0.19/0.45 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.19/0.45 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.19/0.45 => ( ( Xz = Xx )
% 0.19/0.45 | ( Xz = Xy ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(upairsetIL_type,type,
% 0.19/0.45 upairsetIL: $o ).
% 0.19/0.45
% 0.19/0.45 thf(upairsetIL,definition,
% 0.19/0.45 ( upairsetIL
% 0.19/0.45 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(upairsetIR_type,type,
% 0.19/0.45 upairsetIR: $o ).
% 0.19/0.45
% 0.19/0.45 thf(upairsetIR,definition,
% 0.19/0.45 ( upairsetIR
% 0.19/0.45 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(emptyE1_type,type,
% 0.19/0.45 emptyE1: $o ).
% 0.19/0.45
% 0.19/0.45 thf(emptyE1,definition,
% 0.19/0.45 ( emptyE1
% 0.19/0.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.45 ( ? [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 & ( Xphi @ Xx ) )
% 0.19/0.45 => ( ( ( dsetconstr @ A
% 0.19/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.45 = emptyset )
% 0.19/0.45 => $false ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(vacuousDall_type,type,
% 0.19/0.45 vacuousDall: $o ).
% 0.19/0.45
% 0.19/0.45 thf(vacuousDall,definition,
% 0.19/0.45 ( vacuousDall
% 0.19/0.45 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ emptyset )
% 0.19/0.45 => ( Xphi @ Xx ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(quantDeMorgan1_type,type,
% 0.19/0.45 quantDeMorgan1: $o ).
% 0.19/0.45
% 0.19/0.45 thf(quantDeMorgan1,definition,
% 0.19/0.45 ( quantDeMorgan1
% 0.19/0.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.45 ( ~ ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 => ( Xphi @ Xx ) )
% 0.19/0.45 => ? [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(quantDeMorgan2_type,type,
% 0.19/0.45 quantDeMorgan2: $o ).
% 0.19/0.45
% 0.19/0.45 thf(quantDeMorgan2,definition,
% 0.19/0.45 ( quantDeMorgan2
% 0.19/0.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.45 ( ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 => ~ ( Xphi @ Xx ) )
% 0.19/0.45 => ~ ? [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(quantDeMorgan3_type,type,
% 0.19/0.45 quantDeMorgan3: $o ).
% 0.19/0.45
% 0.19/0.45 thf(quantDeMorgan3,definition,
% 0.19/0.45 ( quantDeMorgan3
% 0.19/0.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.45 ( ~ ? [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 & ( Xphi @ Xx ) )
% 0.19/0.45 => ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(quantDeMorgan4_type,type,
% 0.19/0.45 quantDeMorgan4: $o ).
% 0.19/0.45
% 0.19/0.45 thf(quantDeMorgan4,definition,
% 0.19/0.45 ( quantDeMorgan4
% 0.19/0.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.45 ( ? [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 & ~ ( Xphi @ Xx ) )
% 0.19/0.45 => ~ ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(prop2setI_type,type,
% 0.19/0.45 prop2setI: $o ).
% 0.19/0.45
% 0.19/0.45 thf(prop2setI,definition,
% 0.19/0.45 ( prop2setI
% 0.19/0.45 = ( ! [Xphi: $o] :
% 0.19/0.45 ( Xphi
% 0.19/0.45 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(set2prop_type,type,
% 0.19/0.45 set2prop: $i > $o ).
% 0.19/0.45
% 0.19/0.45 thf(prop2set2propI_type,type,
% 0.19/0.45 prop2set2propI: $o ).
% 0.19/0.45
% 0.19/0.45 thf(prop2set2propI,definition,
% 0.19/0.45 ( prop2set2propI
% 0.19/0.45 = ( ! [Xphi: $o] :
% 0.19/0.45 ( Xphi
% 0.19/0.45 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(notdexE_type,type,
% 0.19/0.45 notdexE: $o ).
% 0.19/0.45
% 0.19/0.45 thf(notdexE,definition,
% 0.19/0.45 ( notdexE
% 0.19/0.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.45 ( ~ ? [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 & ( Xphi @ Xx ) )
% 0.19/0.45 => ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(notdallE_type,type,
% 0.19/0.45 notdallE: $o ).
% 0.19/0.45
% 0.19/0.45 thf(notdallE,definition,
% 0.19/0.45 ( notdallE
% 0.19/0.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.45 ( ~ ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 => ( Xphi @ Xx ) )
% 0.19/0.45 => ? [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(exuI1_type,type,
% 0.19/0.45 exuI1: $o ).
% 0.19/0.45
% 0.19/0.45 thf(exuI1,definition,
% 0.19/0.45 ( exuI1
% 0.19/0.45 = ( ! [Xphi: $i > $o] :
% 0.19/0.45 ( ? [Xx: $i] :
% 0.19/0.45 ( ( Xphi @ Xx )
% 0.19/0.45 & ! [Xy: $i] :
% 0.19/0.45 ( ( Xphi @ Xy )
% 0.19/0.45 => ( Xx = Xy ) ) )
% 0.19/0.45 => ( exu
% 0.19/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(exuI3_type,type,
% 0.19/0.45 exuI3: $o ).
% 0.19/0.45
% 0.19/0.45 thf(exuI3,definition,
% 0.19/0.45 ( exuI3
% 0.19/0.45 = ( ! [Xphi: $i > $o] :
% 0.19/0.45 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.19/0.45 => ( ! [Xx: $i,Xy: $i] :
% 0.19/0.45 ( ( Xphi @ Xx )
% 0.19/0.45 => ( ( Xphi @ Xy )
% 0.19/0.45 => ( Xx = Xy ) ) )
% 0.19/0.45 => ( exu
% 0.19/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(exuI2_type,type,
% 0.19/0.45 exuI2: $o ).
% 0.19/0.45
% 0.19/0.45 thf(exuI2,definition,
% 0.19/0.45 ( exuI2
% 0.19/0.45 = ( ! [Xphi: $i > $o] :
% 0.19/0.45 ( ? [Xx: $i] :
% 0.19/0.45 ! [Xy: $i] :
% 0.19/0.45 ( ( Xphi @ Xy )
% 0.19/0.45 <=> ( Xy = Xx ) )
% 0.19/0.45 => ( exu
% 0.19/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(inCongP_type,type,
% 0.19/0.45 inCongP: $o ).
% 0.19/0.45
% 0.19/0.45 thf(inCongP,definition,
% 0.19/0.45 ( inCongP
% 0.19/0.45 = ( ! [A: $i,B: $i] :
% 0.19/0.45 ( ( A = B )
% 0.19/0.45 => ! [Xx: $i,Xy: $i] :
% 0.19/0.45 ( ( Xx = Xy )
% 0.19/0.45 => ( ( in @ Xx @ A )
% 0.19/0.45 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(in__Cong_type,type,
% 0.19/0.45 in__Cong: $o ).
% 0.19/0.45
% 0.19/0.45 thf(in__Cong,definition,
% 0.19/0.45 ( in__Cong
% 0.19/0.45 = ( ! [A: $i,B: $i] :
% 0.19/0.45 ( ( A = B )
% 0.19/0.45 => ! [Xx: $i,Xy: $i] :
% 0.19/0.45 ( ( Xx = Xy )
% 0.19/0.45 => ( ( in @ Xx @ A )
% 0.19/0.45 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(exuE3u_type,type,
% 0.19/0.45 exuE3u: $o ).
% 0.19/0.46
% 0.19/0.46 thf(exuE3u,definition,
% 0.19/0.46 ( exuE3u
% 0.19/0.46 = ( ! [Xphi: $i > $o] :
% 0.19/0.46 ( ( exu
% 0.19/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.46 => ! [Xx: $i,Xy: $i] :
% 0.19/0.46 ( ( Xphi @ Xx )
% 0.19/0.46 => ( ( Xphi @ Xy )
% 0.19/0.46 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(exu__Cong_type,type,
% 0.19/0.46 exu__Cong: $o ).
% 0.19/0.46
% 0.19/0.46 thf(exu__Cong,definition,
% 0.19/0.46 ( exu__Cong
% 0.19/0.46 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.19/0.46 ( ! [Xx: $i,Xy: $i] :
% 0.19/0.46 ( ( Xx = Xy )
% 0.19/0.46 => ( ( Xphi @ Xx )
% 0.19/0.46 <=> ( Xpsi @ Xy ) ) )
% 0.19/0.46 => ( ( exu
% 0.19/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.46 <=> ( exu
% 0.19/0.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(emptyset__Cong_type,type,
% 0.19/0.46 emptyset__Cong: $o ).
% 0.19/0.46
% 0.19/0.46 thf(emptyset__Cong,definition,
% 0.19/0.46 ( emptyset__Cong
% 0.19/0.46 = ( emptyset = emptyset ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setadjoin__Cong_type,type,
% 0.19/0.46 setadjoin__Cong: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setadjoin__Cong,definition,
% 0.19/0.46 ( setadjoin__Cong
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.46 ( ( Xx = Xy )
% 0.19/0.46 => ! [Xz: $i,Xu: $i] :
% 0.19/0.46 ( ( Xz = Xu )
% 0.19/0.46 => ( ( setadjoin @ Xx @ Xz )
% 0.19/0.46 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(powerset__Cong_type,type,
% 0.19/0.46 powerset__Cong: $o ).
% 0.19/0.46
% 0.19/0.46 thf(powerset__Cong,definition,
% 0.19/0.46 ( powerset__Cong
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ( A = B )
% 0.19/0.46 => ( ( powerset @ A )
% 0.19/0.46 = ( powerset @ B ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setunion__Cong_type,type,
% 0.19/0.46 setunion__Cong: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setunion__Cong,definition,
% 0.19/0.46 ( setunion__Cong
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ( A = B )
% 0.19/0.46 => ( ( setunion @ A )
% 0.19/0.46 = ( setunion @ B ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(omega__Cong_type,type,
% 0.19/0.46 omega__Cong: $o ).
% 0.19/0.46
% 0.19/0.46 thf(omega__Cong,definition,
% 0.19/0.46 ( omega__Cong
% 0.19/0.46 = ( omega = omega ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(exuEu_type,type,
% 0.19/0.46 exuEu: $o ).
% 0.19/0.46
% 0.19/0.46 thf(exuEu,definition,
% 0.19/0.46 ( exuEu
% 0.19/0.46 = ( ! [Xphi: $i > $o] :
% 0.19/0.46 ( ( exu
% 0.19/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.46 => ! [Xx: $i,Xy: $i] :
% 0.19/0.46 ( ( Xphi @ Xx )
% 0.19/0.46 => ( ( Xphi @ Xy )
% 0.19/0.46 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(descr__Cong_type,type,
% 0.19/0.46 descr__Cong: $o ).
% 0.19/0.46
% 0.19/0.46 thf(descr__Cong,definition,
% 0.19/0.46 ( descr__Cong
% 0.19/0.46 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.19/0.46 ( ! [Xx: $i,Xy: $i] :
% 0.19/0.46 ( ( Xx = Xy )
% 0.19/0.46 => ( ( Xphi @ Xx )
% 0.19/0.46 <=> ( Xpsi @ Xy ) ) )
% 0.19/0.46 => ( ( exu
% 0.19/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.46 => ( ( exu
% 0.19/0.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.19/0.46 => ( ( descr
% 0.19/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.46 = ( descr
% 0.19/0.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(dsetconstr__Cong_type,type,
% 0.19/0.46 dsetconstr__Cong: $o ).
% 0.19/0.46
% 0.19/0.46 thf(dsetconstr__Cong,definition,
% 0.19/0.46 ( dsetconstr__Cong
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ( A = B )
% 0.19/0.46 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.19/0.46 ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ B )
% 0.19/0.46 => ( ( Xx = Xy )
% 0.19/0.46 => ( ( Xphi @ Xx )
% 0.19/0.46 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.19/0.46 => ( ( dsetconstr @ A
% 0.19/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.46 = ( dsetconstr @ B
% 0.19/0.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(subset_type,type,
% 0.19/0.46 subset: $i > $i > $o ).
% 0.19/0.46
% 0.19/0.46 thf(disjoint_type,type,
% 0.19/0.46 disjoint: $i > $i > $o ).
% 0.19/0.46
% 0.19/0.46 thf(setsmeet_type,type,
% 0.19/0.46 setsmeet: $i > $i > $o ).
% 0.19/0.46
% 0.19/0.46 thf(subsetI1_type,type,
% 0.19/0.46 subsetI1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(subsetI1,definition,
% 0.19/0.46 ( subsetI1
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( in @ Xx @ B ) )
% 0.19/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(eqimpsubset2_type,type,
% 0.19/0.46 eqimpsubset2: $o ).
% 0.19/0.46
% 0.19/0.46 thf(eqimpsubset2,definition,
% 0.19/0.46 ( eqimpsubset2
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ( A = B )
% 0.19/0.46 => ( subset @ B @ A ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(eqimpsubset1_type,type,
% 0.19/0.46 eqimpsubset1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(eqimpsubset1,definition,
% 0.19/0.46 ( eqimpsubset1
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ( A = B )
% 0.19/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(subsetI2_type,type,
% 0.19/0.46 subsetI2: $o ).
% 0.19/0.46
% 0.19/0.46 thf(subsetI2,definition,
% 0.19/0.46 ( subsetI2
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( in @ Xx @ B ) )
% 0.19/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(emptysetsubset_type,type,
% 0.19/0.46 emptysetsubset: $o ).
% 0.19/0.46
% 0.19/0.46 thf(emptysetsubset,definition,
% 0.19/0.46 ( emptysetsubset
% 0.19/0.46 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(subsetE_type,type,
% 0.19/0.46 subsetE: $o ).
% 0.19/0.46
% 0.19/0.46 thf(subsetE,definition,
% 0.19/0.46 ( subsetE
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( subset @ A @ B )
% 0.19/0.46 => ( ( in @ Xx @ A )
% 0.19/0.46 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(subsetE2_type,type,
% 0.19/0.46 subsetE2: $o ).
% 0.19/0.46
% 0.19/0.46 thf(subsetE2,definition,
% 0.19/0.46 ( subsetE2
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( subset @ A @ B )
% 0.19/0.46 => ( ~ ( in @ Xx @ B )
% 0.19/0.46 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(notsubsetI_type,type,
% 0.19/0.46 notsubsetI: $o ).
% 0.19/0.46
% 0.19/0.46 thf(notsubsetI,definition,
% 0.19/0.46 ( notsubsetI
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( ~ ( in @ Xx @ B )
% 0.19/0.46 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(notequalI1_type,type,
% 0.30/0.46 notequalI1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(notequalI1,definition,
% 0.30/0.46 ( notequalI1
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ~ ( subset @ A @ B )
% 0.30/0.46 => ( A != B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(notequalI2_type,type,
% 0.30/0.46 notequalI2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(notequalI2,definition,
% 0.30/0.46 ( notequalI2
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ~ ( in @ Xx @ B )
% 0.30/0.46 => ( A != B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(subsetRefl_type,type,
% 0.30/0.46 subsetRefl: $o ).
% 0.30/0.46
% 0.30/0.46 thf(subsetRefl,definition,
% 0.30/0.46 ( subsetRefl
% 0.30/0.46 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(subsetTrans_type,type,
% 0.30/0.46 subsetTrans: $o ).
% 0.30/0.46
% 0.30/0.46 thf(subsetTrans,definition,
% 0.30/0.46 ( subsetTrans
% 0.30/0.46 = ( ! [A: $i,B: $i,C: $i] :
% 0.30/0.46 ( ( subset @ A @ B )
% 0.30/0.46 => ( ( subset @ B @ C )
% 0.30/0.46 => ( subset @ A @ C ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinSub_type,type,
% 0.30/0.46 setadjoinSub: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinSub,definition,
% 0.30/0.46 ( setadjoinSub
% 0.30/0.46 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinSub2_type,type,
% 0.30/0.46 setadjoinSub2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinSub2,definition,
% 0.30/0.46 ( setadjoinSub2
% 0.30/0.46 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.30/0.46 ( ( subset @ A @ B )
% 0.30/0.46 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(subset2powerset_type,type,
% 0.30/0.46 subset2powerset: $o ).
% 0.30/0.46
% 0.30/0.46 thf(subset2powerset,definition,
% 0.30/0.46 ( subset2powerset
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( subset @ A @ B )
% 0.30/0.46 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setextsub_type,type,
% 0.30/0.46 setextsub: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setextsub,definition,
% 0.30/0.46 ( setextsub
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( subset @ A @ B )
% 0.30/0.46 => ( ( subset @ B @ A )
% 0.30/0.46 => ( A = B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(subsetemptysetimpeq_type,type,
% 0.30/0.46 subsetemptysetimpeq: $o ).
% 0.30/0.46
% 0.30/0.46 thf(subsetemptysetimpeq,definition,
% 0.30/0.46 ( subsetemptysetimpeq
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ( ( subset @ A @ emptyset )
% 0.30/0.46 => ( A = emptyset ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(powersetI1_type,type,
% 0.30/0.46 powersetI1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(powersetI1,definition,
% 0.30/0.46 ( powersetI1
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( subset @ B @ A )
% 0.30/0.46 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(powersetE1_type,type,
% 0.30/0.46 powersetE1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(powersetE1,definition,
% 0.30/0.46 ( powersetE1
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( in @ B @ ( powerset @ A ) )
% 0.30/0.46 => ( subset @ B @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(inPowerset_type,type,
% 0.30/0.46 inPowerset: $o ).
% 0.30/0.46
% 0.30/0.46 thf(inPowerset,definition,
% 0.30/0.46 ( inPowerset
% 0.30/0.46 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(powersetsubset_type,type,
% 0.30/0.46 powersetsubset: $o ).
% 0.30/0.46
% 0.30/0.46 thf(powersetsubset,definition,
% 0.30/0.46 ( powersetsubset
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( subset @ A @ B )
% 0.30/0.46 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(sepInPowerset_type,type,
% 0.30/0.46 sepInPowerset: $o ).
% 0.30/0.46
% 0.30/0.46 thf(sepInPowerset,definition,
% 0.30/0.46 ( sepInPowerset
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( in
% 0.30/0.46 @ ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 @ ( powerset @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(sepSubset_type,type,
% 0.30/0.46 sepSubset: $o ).
% 0.30/0.46
% 0.30/0.46 thf(sepSubset,definition,
% 0.30/0.46 ( sepSubset
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( subset
% 0.30/0.46 @ ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binunion_type,type,
% 0.30/0.46 binunion: $i > $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(binunionIL_type,type,
% 0.30/0.46 binunionIL: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binunionIL,definition,
% 0.30/0.46 ( binunionIL
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(upairset2IR_type,type,
% 0.30/0.46 upairset2IR: $o ).
% 0.30/0.46
% 0.30/0.46 thf(upairset2IR,definition,
% 0.30/0.46 ( upairset2IR
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binunionIR_type,type,
% 0.30/0.46 binunionIR: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binunionIR,definition,
% 0.30/0.46 ( binunionIR
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ B )
% 0.30/0.46 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binunionEcases_type,type,
% 0.30/0.46 binunionEcases: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binunionEcases,definition,
% 0.30/0.46 ( binunionEcases
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.30/0.46 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.30/0.46 => ( ( ( in @ Xx @ A )
% 0.30/0.46 => Xphi )
% 0.30/0.46 => ( ( ( in @ Xx @ B )
% 0.30/0.46 => Xphi )
% 0.30/0.46 => Xphi ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binunionE_type,type,
% 0.30/0.46 binunionE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binunionE,definition,
% 0.30/0.46 ( binunionE
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.30/0.46 => ( ( in @ Xx @ A )
% 0.30/0.46 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binunionLsub_type,type,
% 0.30/0.46 binunionLsub: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binunionLsub,definition,
% 0.30/0.46 ( binunionLsub
% 0.30/0.46 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binunionRsub_type,type,
% 0.30/0.46 binunionRsub: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binunionRsub,definition,
% 0.30/0.46 ( binunionRsub
% 0.30/0.46 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binintersect_type,type,
% 0.30/0.46 binintersect: $i > $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectI_type,type,
% 0.30/0.46 binintersectI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectI,definition,
% 0.30/0.46 ( binintersectI
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ B )
% 0.30/0.46 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectSubset5_type,type,
% 0.30/0.46 binintersectSubset5: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectSubset5,definition,
% 0.30/0.46 ( binintersectSubset5
% 0.30/0.46 = ( ! [A: $i,B: $i,C: $i] :
% 0.30/0.46 ( ( subset @ C @ A )
% 0.30/0.46 => ( ( subset @ C @ B )
% 0.30/0.46 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectEL_type,type,
% 0.30/0.46 binintersectEL: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectEL,definition,
% 0.30/0.46 ( binintersectEL
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.30/0.46 => ( in @ Xx @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectLsub_type,type,
% 0.30/0.46 binintersectLsub: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectLsub,definition,
% 0.30/0.46 ( binintersectLsub
% 0.30/0.46 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectSubset2_type,type,
% 0.30/0.46 binintersectSubset2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectSubset2,definition,
% 0.30/0.46 ( binintersectSubset2
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( subset @ A @ B )
% 0.30/0.46 => ( ( binintersect @ A @ B )
% 0.30/0.46 = A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectSubset3_type,type,
% 0.30/0.46 binintersectSubset3: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectSubset3,definition,
% 0.30/0.46 ( binintersectSubset3
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( ( binintersect @ A @ B )
% 0.30/0.46 = B )
% 0.30/0.46 => ( subset @ B @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectER_type,type,
% 0.30/0.46 binintersectER: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectER,definition,
% 0.30/0.46 ( binintersectER
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.30/0.46 => ( in @ Xx @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(disjointsetsI1_type,type,
% 0.30/0.46 disjointsetsI1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(disjointsetsI1,definition,
% 0.30/0.46 ( disjointsetsI1
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ~ ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & ( in @ Xx @ B ) )
% 0.30/0.46 => ( ( binintersect @ A @ B )
% 0.30/0.46 = emptyset ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectRsub_type,type,
% 0.30/0.46 binintersectRsub: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectRsub,definition,
% 0.30/0.46 ( binintersectRsub
% 0.30/0.46 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectSubset4_type,type,
% 0.30/0.46 binintersectSubset4: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectSubset4,definition,
% 0.30/0.46 ( binintersectSubset4
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( subset @ B @ A )
% 0.30/0.46 => ( ( binintersect @ A @ B )
% 0.30/0.46 = B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectSubset1_type,type,
% 0.30/0.46 binintersectSubset1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectSubset1,definition,
% 0.30/0.46 ( binintersectSubset1
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( ( binintersect @ A @ B )
% 0.30/0.46 = A )
% 0.30/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(bs114d_type,type,
% 0.30/0.46 bs114d: $o ).
% 0.30/0.46
% 0.30/0.46 thf(bs114d,definition,
% 0.30/0.46 ( bs114d
% 0.30/0.46 = ( ! [A: $i,B: $i,C: $i] :
% 0.30/0.46 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 0.30/0.46 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(regular_type,type,
% 0.30/0.46 regular: $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(setminus_type,type,
% 0.30/0.46 setminus: $i > $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(setminusI_type,type,
% 0.30/0.46 setminusI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setminusI,definition,
% 0.30/0.46 ( setminusI
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ~ ( in @ Xx @ B )
% 0.30/0.46 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setminusEL_type,type,
% 0.30/0.46 setminusEL: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setminusEL,definition,
% 0.30/0.46 ( setminusEL
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.30/0.46 => ( in @ Xx @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setminusER_type,type,
% 0.30/0.46 setminusER: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setminusER,definition,
% 0.30/0.46 ( setminusER
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.30/0.46 => ~ ( in @ Xx @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setminusSubset2_type,type,
% 0.30/0.46 setminusSubset2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setminusSubset2,definition,
% 0.30/0.46 ( setminusSubset2
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( subset @ A @ B )
% 0.30/0.46 => ( ( setminus @ A @ B )
% 0.30/0.46 = emptyset ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setminusERneg_type,type,
% 0.30/0.46 setminusERneg: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setminusERneg,definition,
% 0.30/0.46 ( setminusERneg
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.30/0.46 => ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setminusELneg_type,type,
% 0.30/0.46 setminusELneg: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setminusELneg,definition,
% 0.30/0.46 ( setminusELneg
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.30/0.46 => ( ~ ( in @ Xx @ B )
% 0.30/0.46 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setminusILneg_type,type,
% 0.30/0.46 setminusILneg: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setminusILneg,definition,
% 0.30/0.46 ( setminusILneg
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ~ ( in @ Xx @ A )
% 0.30/0.46 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setminusIRneg_type,type,
% 0.30/0.46 setminusIRneg: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setminusIRneg,definition,
% 0.30/0.46 ( setminusIRneg
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ B )
% 0.30/0.46 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setminusLsub_type,type,
% 0.30/0.46 setminusLsub: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setminusLsub,definition,
% 0.30/0.46 ( setminusLsub
% 0.30/0.46 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setminusSubset1_type,type,
% 0.30/0.46 setminusSubset1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setminusSubset1,definition,
% 0.30/0.46 ( setminusSubset1
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( ( setminus @ A @ B )
% 0.30/0.46 = emptyset )
% 0.30/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(symdiff_type,type,
% 0.30/0.46 symdiff: $i > $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(symdiffE_type,type,
% 0.30/0.46 symdiffE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(symdiffE,definition,
% 0.30/0.46 ( symdiffE
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 0.30/0.46 => ! [Xphi: $o] :
% 0.30/0.46 ( ( ( in @ Xx @ A )
% 0.30/0.46 => ( ~ ( in @ Xx @ B )
% 0.30/0.46 => Xphi ) )
% 0.30/0.46 => ( ( ~ ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ B )
% 0.30/0.46 => Xphi ) )
% 0.30/0.46 => Xphi ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(symdiffI1_type,type,
% 0.30/0.46 symdiffI1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(symdiffI1,definition,
% 0.30/0.46 ( symdiffI1
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ~ ( in @ Xx @ B )
% 0.30/0.46 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(symdiffI2_type,type,
% 0.30/0.46 symdiffI2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(symdiffI2,definition,
% 0.30/0.46 ( symdiffI2
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ~ ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ B )
% 0.30/0.46 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(symdiffIneg1_type,type,
% 0.30/0.46 symdiffIneg1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(symdiffIneg1,definition,
% 0.30/0.46 ( symdiffIneg1
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ B )
% 0.30/0.46 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(symdiffIneg2_type,type,
% 0.30/0.46 symdiffIneg2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(symdiffIneg2,definition,
% 0.30/0.46 ( symdiffIneg2
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ~ ( in @ Xx @ A )
% 0.30/0.46 => ( ~ ( in @ Xx @ B )
% 0.30/0.46 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(iskpair_type,type,
% 0.30/0.46 iskpair: $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(secondinupair_type,type,
% 0.30/0.46 secondinupair: $o ).
% 0.30/0.46
% 0.30/0.46 thf(secondinupair,definition,
% 0.30/0.46 ( secondinupair
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setukpairIL_type,type,
% 0.30/0.46 setukpairIL: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setukpairIL,definition,
% 0.30/0.46 ( setukpairIL
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setukpairIR_type,type,
% 0.30/0.46 setukpairIR: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setukpairIR,definition,
% 0.30/0.46 ( setukpairIR
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(kpairiskpair_type,type,
% 0.30/0.46 kpairiskpair: $o ).
% 0.30/0.46
% 0.30/0.46 thf(kpairiskpair,definition,
% 0.30/0.46 ( kpairiskpair
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(kpair_type,type,
% 0.30/0.46 kpair: $i > $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(kpairp_type,type,
% 0.30/0.46 kpairp: $o ).
% 0.30/0.46
% 0.30/0.46 thf(kpairp,definition,
% 0.30/0.46 ( kpairp
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(cartprod_type,type,
% 0.30/0.46 cartprod: $i > $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(singletonsubset_type,type,
% 0.30/0.46 singletonsubset: $o ).
% 0.30/0.46
% 0.30/0.46 thf(singletonsubset,definition,
% 0.30/0.46 ( singletonsubset
% 0.30/0.46 = ( ! [A: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(singletoninpowerset_type,type,
% 0.30/0.46 singletoninpowerset: $o ).
% 0.30/0.46
% 0.30/0.46 thf(singletoninpowerset,definition,
% 0.30/0.46 ( singletoninpowerset
% 0.30/0.46 = ( ! [A: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(singletoninpowunion_type,type,
% 0.30/0.46 singletoninpowunion: $o ).
% 0.30/0.46
% 0.30/0.46 thf(singletoninpowunion,definition,
% 0.30/0.46 ( singletoninpowunion
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(upairset2E_type,type,
% 0.30/0.46 upairset2E: $o ).
% 0.30/0.46
% 0.30/0.46 thf(upairset2E,definition,
% 0.30/0.46 ( upairset2E
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.30/0.46 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.30/0.46 => ( ( Xz = Xx )
% 0.30/0.46 | ( Xz = Xy ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(upairsubunion_type,type,
% 0.30/0.46 upairsubunion: $o ).
% 0.30/0.46
% 0.30/0.46 thf(upairsubunion,definition,
% 0.30/0.46 ( upairsubunion
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ B )
% 0.30/0.46 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(upairinpowunion_type,type,
% 0.30/0.46 upairinpowunion: $o ).
% 0.30/0.46
% 0.30/0.46 thf(upairinpowunion,definition,
% 0.30/0.46 ( upairinpowunion
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ B )
% 0.30/0.46 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(ubforcartprodlem1_type,type,
% 0.30/0.46 ubforcartprodlem1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(ubforcartprodlem1,definition,
% 0.30/0.46 ( ubforcartprodlem1
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ B )
% 0.30/0.46 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(ubforcartprodlem2_type,type,
% 0.30/0.46 ubforcartprodlem2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(ubforcartprodlem2,definition,
% 0.30/0.46 ( ubforcartprodlem2
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ B )
% 0.30/0.46 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(ubforcartprodlem3_type,type,
% 0.30/0.46 ubforcartprodlem3: $o ).
% 0.30/0.46
% 0.30/0.46 thf(ubforcartprodlem3,definition,
% 0.30/0.46 ( ubforcartprodlem3
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ B )
% 0.30/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(cartprodpairin_type,type,
% 0.30/0.46 cartprodpairin: $o ).
% 0.30/0.46
% 0.30/0.46 thf(cartprodpairin,definition,
% 0.30/0.46 ( cartprodpairin
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ B )
% 0.30/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(cartprodmempair1_type,type,
% 0.30/0.46 cartprodmempair1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(cartprodmempair1,definition,
% 0.30/0.46 ( cartprodmempair1
% 0.30/0.46 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.30/0.46 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.30/0.46 => ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & ? [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ B )
% 0.30/0.46 & ( Xu
% 0.30/0.46 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(cartprodmempair_type,type,
% 0.30/0.46 cartprodmempair: $o ).
% 0.30/0.46
% 0.30/0.46 thf(cartprodmempair,definition,
% 0.30/0.46 ( cartprodmempair
% 0.30/0.46 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.30/0.46 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.30/0.46 => ( iskpair @ Xu ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setunionE2_type,type,
% 0.30/0.46 setunionE2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setunionE2,definition,
% 0.30/0.46 ( setunionE2
% 0.30/0.46 = ( ! [A: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ ( setunion @ A ) )
% 0.30/0.46 => ? [X: $i] :
% 0.30/0.46 ( ( in @ X @ A )
% 0.30/0.46 & ( in @ Xx @ X ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setunionsingleton1_type,type,
% 0.30/0.46 setunionsingleton1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setunionsingleton1,definition,
% 0.30/0.46 ( setunionsingleton1
% 0.30/0.46 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setunionsingleton2_type,type,
% 0.30/0.46 setunionsingleton2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setunionsingleton2,definition,
% 0.30/0.46 ( setunionsingleton2
% 0.30/0.46 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setunionsingleton_type,type,
% 0.30/0.46 setunionsingleton: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setunionsingleton,definition,
% 0.30/0.46 ( setunionsingleton
% 0.30/0.46 = ( ! [Xx: $i] :
% 0.30/0.46 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 0.30/0.46 = Xx ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(singleton_type,type,
% 0.30/0.46 singleton: $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(singletonprop_type,type,
% 0.30/0.46 singletonprop: $o ).
% 0.30/0.46
% 0.30/0.46 thf(singletonprop,definition,
% 0.30/0.46 ( singletonprop
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( ( Xphi @ Xx )
% 0.30/0.46 => ( ( Xphi @ Xy )
% 0.30/0.46 => ( Xx = Xy ) ) ) ) )
% 0.30/0.46 => ( ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & ( Xphi @ Xx ) )
% 0.30/0.46 => ( singleton
% 0.30/0.46 @ ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(ex1_type,type,
% 0.30/0.46 ex1: $i > ( $i > $o ) > $o ).
% 0.30/0.46
% 0.30/0.46 thf(ex1E1_type,type,
% 0.30/0.46 ex1E1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(ex1E1,definition,
% 0.30/0.46 ( ex1E1
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( ( ex1 @ A
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 => ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(ex1I_type,type,
% 0.30/0.46 ex1I: $o ).
% 0.30/0.46
% 0.30/0.46 thf(ex1I,definition,
% 0.30/0.46 ( ex1I
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( Xphi @ Xx )
% 0.30/0.46 => ( ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( ( Xphi @ Xy )
% 0.30/0.46 => ( Xy = Xx ) ) )
% 0.30/0.46 => ( ex1 @ A
% 0.30/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(ex1I2_type,type,
% 0.30/0.46 ex1I2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(ex1I2,definition,
% 0.30/0.46 ( ex1I2
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( ( Xphi @ Xx )
% 0.30/0.46 => ( ( Xphi @ Xy )
% 0.30/0.46 => ( Xx = Xy ) ) ) ) )
% 0.30/0.46 => ( ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & ( Xphi @ Xx ) )
% 0.30/0.46 => ( ex1 @ A
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(singletonsuniq_type,type,
% 0.30/0.46 singletonsuniq: $o ).
% 0.30/0.46
% 0.30/0.46 thf(singletonsuniq,definition,
% 0.30/0.46 ( singletonsuniq
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( ( setadjoin @ Xx @ emptyset )
% 0.30/0.46 = ( setadjoin @ Xy @ emptyset ) )
% 0.30/0.46 => ( Xx = Xy ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(atmost1p_type,type,
% 0.30/0.46 atmost1p: $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(atleast2p_type,type,
% 0.30/0.46 atleast2p: $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(atmost2p_type,type,
% 0.30/0.46 atmost2p: $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(upairsetp_type,type,
% 0.30/0.46 upairsetp: $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(setukpairinjL1_type,type,
% 0.30/0.46 setukpairinjL1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setukpairinjL1,definition,
% 0.30/0.46 ( setukpairinjL1
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.30/0.46 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 0.30/0.46 => ( Xx = Xz ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(kfstsingleton_type,type,
% 0.30/0.46 kfstsingleton: $o ).
% 0.30/0.46
% 0.30/0.46 thf(kfstsingleton,definition,
% 0.30/0.46 ( kfstsingleton
% 0.30/0.46 = ( ! [Xu: $i] :
% 0.30/0.46 ( ( iskpair @ Xu )
% 0.30/0.46 => ( singleton
% 0.30/0.46 @ ( dsetconstr @ ( setunion @ Xu )
% 0.30/0.46 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(theprop_type,type,
% 0.30/0.46 theprop: $o ).
% 0.30/0.46
% 0.30/0.46 thf(theprop,definition,
% 0.30/0.46 ( theprop
% 0.30/0.46 = ( ! [X: $i] :
% 0.30/0.46 ( ( singleton @ X )
% 0.30/0.46 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(kfst_type,type,
% 0.30/0.46 kfst: $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(kfstpairEq_type,type,
% 0.30/0.46 kfstpairEq: $o ).
% 0.30/0.46
% 0.30/0.46 thf(kfstpairEq,definition,
% 0.30/0.46 ( kfstpairEq
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.30/0.46 = Xx ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(cartprodfstin_type,type,
% 0.30/0.46 cartprodfstin: $o ).
% 0.30/0.46
% 0.30/0.46 thf(cartprodfstin,definition,
% 0.30/0.46 ( cartprodfstin
% 0.30/0.46 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.30/0.46 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.30/0.46 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setukpairinjL2_type,type,
% 0.30/0.46 setukpairinjL2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setukpairinjL2,definition,
% 0.30/0.46 ( setukpairinjL2
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.30/0.46 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.30/0.46 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.30/0.46 => ( Xx = Xz ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setukpairinjL_type,type,
% 0.30/0.46 setukpairinjL: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setukpairinjL,definition,
% 0.30/0.46 ( setukpairinjL
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.30/0.46 ( ( ( kpair @ Xx @ Xy )
% 0.30/0.46 = ( kpair @ Xz @ Xu ) )
% 0.30/0.46 => ( Xx = Xz ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setukpairinjR11_type,type,
% 0.30/0.46 setukpairinjR11: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setukpairinjR11,definition,
% 0.30/0.46 ( setukpairinjR11
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xx = Xy )
% 0.30/0.46 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.30/0.46 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setukpairinjR12_type,type,
% 0.30/0.46 setukpairinjR12: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setukpairinjR12,definition,
% 0.30/0.46 ( setukpairinjR12
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xx = Xy )
% 0.30/0.46 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.30/0.46 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setukpairinjR1_type,type,
% 0.30/0.46 setukpairinjR1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setukpairinjR1,definition,
% 0.30/0.46 ( setukpairinjR1
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.30/0.46 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.30/0.46 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.30/0.46 => ( ( Xz = Xu )
% 0.30/0.46 => ( Xy = Xu ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(upairequniteq_type,type,
% 0.30/0.46 upairequniteq: $o ).
% 0.30/0.46
% 0.30/0.46 thf(upairequniteq,definition,
% 0.30/0.46 ( upairequniteq
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.30/0.46 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.30/0.46 = ( setadjoin @ Xz @ emptyset ) )
% 0.30/0.46 => ( Xx = Xy ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setukpairinjR2_type,type,
% 0.30/0.46 setukpairinjR2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setukpairinjR2,definition,
% 0.30/0.46 ( setukpairinjR2
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.30/0.46 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.30/0.46 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.30/0.46 => ( Xy = Xu ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setukpairinjR_type,type,
% 0.30/0.46 setukpairinjR: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setukpairinjR,definition,
% 0.30/0.46 ( setukpairinjR
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.30/0.46 ( ( ( kpair @ Xx @ Xy )
% 0.30/0.46 = ( kpair @ Xz @ Xu ) )
% 0.30/0.46 => ( Xy = Xu ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(ksndsingleton_type,type,
% 0.30/0.46 ksndsingleton: $o ).
% 0.30/0.46
% 0.30/0.46 thf(ksndsingleton,definition,
% 0.30/0.46 ( ksndsingleton
% 0.30/0.46 = ( ! [Xu: $i] :
% 0.30/0.46 ( ( iskpair @ Xu )
% 0.30/0.46 => ( singleton
% 0.30/0.46 @ ( dsetconstr @ ( setunion @ Xu )
% 0.30/0.46 @ ^ [Xx: $i] :
% 0.30/0.46 ( Xu
% 0.30/0.46 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(ksnd_type,type,
% 0.30/0.46 ksnd: $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(ksndpairEq_type,type,
% 0.30/0.46 ksndpairEq: $o ).
% 0.30/0.46
% 0.30/0.46 thf(ksndpairEq,definition,
% 0.30/0.46 ( ksndpairEq
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.30/0.46 = Xy ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(kpairsurjEq_type,type,
% 0.30/0.46 kpairsurjEq: $o ).
% 0.30/0.46
% 0.30/0.46 thf(kpairsurjEq,definition,
% 0.30/0.46 ( kpairsurjEq
% 0.30/0.46 = ( ! [Xu: $i] :
% 0.30/0.46 ( ( iskpair @ Xu )
% 0.30/0.46 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.30/0.46 = Xu ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(cartprodsndin_type,type,
% 0.30/0.46 cartprodsndin: $o ).
% 0.30/0.46
% 0.30/0.46 thf(cartprodsndin,definition,
% 0.30/0.46 ( cartprodsndin
% 0.30/0.46 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.30/0.46 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.30/0.46 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(cartprodpairmemEL_type,type,
% 0.30/0.46 cartprodpairmemEL: $o ).
% 0.30/0.46
% 0.30/0.46 thf(cartprodpairmemEL,definition,
% 0.30/0.46 ( cartprodpairmemEL
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.30/0.46 => ( in @ Xx @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(cartprodpairmemER_type,type,
% 0.30/0.46 cartprodpairmemER: $o ).
% 0.30/0.46
% 0.30/0.46 thf(cartprodpairmemER,definition,
% 0.30/0.46 ( cartprodpairmemER
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.30/0.46 => ( in @ Xy @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(cartprodmempaircEq_type,type,
% 0.30/0.46 cartprodmempaircEq: $o ).
% 0.30/0.46
% 0.30/0.46 thf(cartprodmempaircEq,definition,
% 0.30/0.46 ( cartprodmempaircEq
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ B )
% 0.30/0.46 => ( ( kpair @ Xx @ Xy )
% 0.30/0.46 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(cartprodfstpairEq_type,type,
% 0.30/0.46 cartprodfstpairEq: $o ).
% 0.30/0.46
% 0.30/0.46 thf(cartprodfstpairEq,definition,
% 0.30/0.46 ( cartprodfstpairEq
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ B )
% 0.30/0.46 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.30/0.46 = Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(cartprodsndpairEq_type,type,
% 0.30/0.46 cartprodsndpairEq: $o ).
% 0.30/0.46
% 0.30/0.46 thf(cartprodsndpairEq,definition,
% 0.30/0.46 ( cartprodsndpairEq
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ B )
% 0.30/0.46 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.30/0.46 = Xy ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(cartprodpairsurjEq_type,type,
% 0.30/0.46 cartprodpairsurjEq: $o ).
% 0.30/0.46
% 0.30/0.46 thf(cartprodpairsurjEq,definition,
% 0.30/0.46 ( cartprodpairsurjEq
% 0.30/0.46 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.30/0.46 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.30/0.46 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.30/0.46 = Xu ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(breln_type,type,
% 0.30/0.46 breln: $i > $i > $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(dpsetconstr_type,type,
% 0.30/0.46 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 0.30/0.46
% 0.30/0.46 thf(dpsetconstrI_type,type,
% 0.30/0.46 dpsetconstrI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(dpsetconstrI,definition,
% 0.30/0.46 ( dpsetconstrI
% 0.30/0.46 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ B )
% 0.30/0.46 => ( ( Xphi @ Xx @ Xy )
% 0.30/0.46 => ( in @ ( kpair @ Xx @ Xy )
% 0.30/0.46 @ ( dpsetconstr @ A @ B
% 0.30/0.46 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(dpsetconstrSub_type,type,
% 0.30/0.46 dpsetconstrSub: $o ).
% 0.30/0.46
% 0.30/0.46 thf(dpsetconstrSub,definition,
% 0.30/0.46 ( dpsetconstrSub
% 0.30/0.46 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.30/0.46 ( subset
% 0.30/0.46 @ ( dpsetconstr @ A @ B
% 0.30/0.46 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 0.30/0.46 @ ( cartprod @ A @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setOfPairsIsBReln_type,type,
% 0.30/0.46 setOfPairsIsBReln: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setOfPairsIsBReln,definition,
% 0.30/0.46 ( setOfPairsIsBReln
% 0.30/0.46 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.30/0.46 ( breln @ A @ B
% 0.30/0.46 @ ( dpsetconstr @ A @ B
% 0.30/0.46 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(dpsetconstrERa_type,type,
% 0.30/0.46 dpsetconstrERa: $o ).
% 0.30/0.46
% 0.30/0.46 thf(dpsetconstrERa,definition,
% 0.30/0.46 ( dpsetconstrERa
% 0.30/0.46 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ B )
% 0.30/0.46 => ( ( in @ ( kpair @ Xx @ Xy )
% 0.30/0.46 @ ( dpsetconstr @ A @ B
% 0.30/0.46 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.30/0.46 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(dpsetconstrEL1_type,type,
% 0.30/0.46 dpsetconstrEL1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(dpsetconstrEL1,definition,
% 0.30/0.46 ( dpsetconstrEL1
% 0.30/0.46 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( in @ ( kpair @ Xx @ Xy )
% 0.30/0.46 @ ( dpsetconstr @ A @ B
% 0.30/0.46 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.30/0.46 => ( in @ Xx @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(dpsetconstrEL2_type,type,
% 0.30/0.46 dpsetconstrEL2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(dpsetconstrEL2,definition,
% 0.30/0.46 ( dpsetconstrEL2
% 0.30/0.46 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( in @ ( kpair @ Xx @ Xy )
% 0.30/0.46 @ ( dpsetconstr @ A @ B
% 0.30/0.46 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.30/0.46 => ( in @ Xy @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(dpsetconstrER_type,type,
% 0.30/0.46 dpsetconstrER: $o ).
% 0.30/0.46
% 0.30/0.46 thf(dpsetconstrER,definition,
% 0.30/0.46 ( dpsetconstrER
% 0.30/0.46 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( in @ ( kpair @ Xx @ Xy )
% 0.30/0.46 @ ( dpsetconstr @ A @ B
% 0.30/0.46 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.30/0.46 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(func_type,type,
% 0.30/0.46 func: $i > $i > $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(funcSet_type,type,
% 0.30/0.46 funcSet: $i > $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(funcImageSingleton_type,type,
% 0.30/0.46 funcImageSingleton: $o ).
% 0.30/0.46
% 0.30/0.46 thf(funcImageSingleton,definition,
% 0.30/0.46 ( funcImageSingleton
% 0.30/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.46 ( ( func @ A @ B @ Xf )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( singleton
% 0.30/0.46 @ ( dsetconstr @ B
% 0.30/0.46 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(apProp_type,type,
% 0.30/0.46 apProp: $o ).
% 0.30/0.46
% 0.30/0.46 thf(apProp,definition,
% 0.30/0.46 ( apProp
% 0.30/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.46 ( ( func @ A @ B @ Xf )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( in
% 0.30/0.46 @ ( setunion
% 0.30/0.46 @ ( dsetconstr @ B
% 0.30/0.46 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 0.30/0.46 @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(ap_type,type,
% 0.30/0.46 ap: $i > $i > $i > $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(app_type,type,
% 0.30/0.46 app: $o ).
% 0.30/0.46
% 0.30/0.46 thf(app,definition,
% 0.30/0.46 ( app
% 0.30/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.46 ( ( func @ A @ B @ Xf )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(infuncsetfunc_type,type,
% 0.30/0.46 infuncsetfunc: $o ).
% 0.30/0.46
% 0.30/0.46 thf(infuncsetfunc,definition,
% 0.30/0.46 ( infuncsetfunc
% 0.30/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.46 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.30/0.46 => ( func @ A @ B @ Xf ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(ap2p_type,type,
% 0.30/0.46 ap2p: $o ).
% 0.30/0.46
% 0.30/0.46 thf(ap2p,definition,
% 0.30/0.46 ( ap2p
% 0.30/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.46 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(funcinfuncset_type,type,
% 0.30/0.46 funcinfuncset: $o ).
% 0.30/0.46
% 0.30/0.46 thf(funcinfuncset,definition,
% 0.30/0.46 ( funcinfuncset
% 0.30/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.46 ( ( func @ A @ B @ Xf )
% 0.30/0.46 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(lamProp_type,type,
% 0.30/0.46 lamProp: $o ).
% 0.30/0.46
% 0.30/0.46 thf(lamProp,definition,
% 0.30/0.46 ( lamProp
% 0.30/0.46 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.30/0.46 => ( func @ A @ B
% 0.30/0.46 @ ( dpsetconstr @ A @ B
% 0.30/0.46 @ ^ [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xf @ Xx )
% 0.30/0.46 = Xy ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(lam_type,type,
% 0.30/0.46 lam: $i > $i > ( $i > $i ) > $i ).
% 0.30/0.46
% 0.30/0.46 thf(lamp_type,type,
% 0.30/0.46 lamp: $o ).
% 0.30/0.46
% 0.30/0.46 thf(lamp,definition,
% 0.30/0.46 ( lamp
% 0.30/0.46 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.30/0.46 => ( func @ A @ B
% 0.30/0.46 @ ( lam @ A @ B
% 0.30/0.46 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(lam2p_type,type,
% 0.30/0.46 lam2p: $o ).
% 0.30/0.46
% 0.30/0.46 thf(lam2p,definition,
% 0.30/0.46 ( lam2p
% 0.30/0.46 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.30/0.46 => ( in
% 0.30/0.46 @ ( lam @ A @ B
% 0.30/0.46 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.30/0.46 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(brelnall1_type,type,
% 0.30/0.46 brelnall1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(brelnall1,definition,
% 0.30/0.46 ( brelnall1
% 0.30/0.46 = ( ! [A: $i,B: $i,R: $i] :
% 0.30/0.46 ( ( breln @ A @ B @ R )
% 0.30/0.46 => ! [Xphi: $i > $o] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ B )
% 0.30/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.30/0.46 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ R )
% 0.30/0.46 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(brelnall2_type,type,
% 0.30/0.46 brelnall2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(brelnall2,definition,
% 0.30/0.46 ( brelnall2
% 0.30/0.46 = ( ! [A: $i,B: $i,R: $i] :
% 0.30/0.46 ( ( breln @ A @ B @ R )
% 0.30/0.46 => ! [Xphi: $i > $o] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ B )
% 0.30/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.30/0.46 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ R )
% 0.30/0.46 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(ex1E2_type,type,
% 0.30/0.46 ex1E2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(ex1E2,definition,
% 0.30/0.46 ( ex1E2
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( ( ex1 @ A
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( ( Xphi @ Xx )
% 0.30/0.46 => ( ( Xphi @ Xy )
% 0.30/0.46 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(funcGraphProp1_type,type,
% 0.30/0.46 funcGraphProp1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(funcGraphProp1,definition,
% 0.30/0.46 ( funcGraphProp1
% 0.30/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.46 ( ( func @ A @ B @ Xf )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(funcGraphProp3_type,type,
% 0.30/0.46 funcGraphProp3: $o ).
% 0.30/0.46
% 0.30/0.46 thf(funcGraphProp3,definition,
% 0.30/0.46 ( funcGraphProp3
% 0.30/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.46 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(funcGraphProp2_type,type,
% 0.30/0.46 funcGraphProp2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(funcGraphProp2,definition,
% 0.30/0.46 ( funcGraphProp2
% 0.30/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.46 ( ( func @ A @ B @ Xf )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ B )
% 0.30/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.30/0.46 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.30/0.46 = Xy ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(funcextLem_type,type,
% 0.30/0.46 funcextLem: $o ).
% 0.30/0.46
% 0.30/0.46 thf(funcextLem,definition,
% 0.30/0.46 ( funcextLem
% 0.30/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.46 ( ( func @ A @ B @ Xf )
% 0.30/0.46 => ! [Xg: $i] :
% 0.30/0.46 ( ( func @ A @ B @ Xg )
% 0.30/0.46 => ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.30/0.46 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ B )
% 0.30/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 0.30/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(funcGraphProp4_type,type,
% 0.30/0.46 funcGraphProp4: $o ).
% 0.30/0.46
% 0.30/0.46 thf(funcGraphProp4,definition,
% 0.30/0.46 ( funcGraphProp4
% 0.30/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.46 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ B )
% 0.30/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.30/0.46 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.30/0.46 = Xy ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(subbreln_type,type,
% 0.30/0.46 subbreln: $o ).
% 0.30/0.46
% 0.30/0.46 thf(subbreln,definition,
% 0.30/0.46 ( subbreln
% 0.30/0.46 = ( ! [A: $i,B: $i,R: $i] :
% 0.30/0.46 ( ( breln @ A @ B @ R )
% 0.30/0.46 => ! [S: $i] :
% 0.30/0.46 ( ( breln @ A @ B @ S )
% 0.30/0.46 => ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ B )
% 0.30/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.30/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.30/0.46 => ( subset @ R @ S ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(eqbreln_type,type,
% 0.30/0.46 eqbreln: $o ).
% 0.30/0.46
% 0.30/0.46 thf(eqbreln,definition,
% 0.30/0.46 ( eqbreln
% 0.30/0.46 = ( ! [A: $i,B: $i,R: $i] :
% 0.30/0.46 ( ( breln @ A @ B @ R )
% 0.30/0.46 => ! [S: $i] :
% 0.30/0.46 ( ( breln @ A @ B @ S )
% 0.30/0.46 => ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ B )
% 0.30/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.30/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.30/0.46 => ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ B )
% 0.30/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.30/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 0.30/0.46 => ( R = S ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(funcext_type,type,
% 0.30/0.46 funcext: $o ).
% 0.30/0.46
% 0.30/0.46 thf(funcext,definition,
% 0.30/0.46 ( funcext
% 0.30/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.46 ( ( func @ A @ B @ Xf )
% 0.30/0.46 => ! [Xg: $i] :
% 0.30/0.46 ( ( func @ A @ B @ Xg )
% 0.30/0.46 => ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.30/0.46 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.30/0.46 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(funcext2_type,type,
% 0.30/0.46 funcext2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(funcext2,definition,
% 0.30/0.46 ( funcext2
% 0.30/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.46 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.30/0.46 => ! [Xg: $i] :
% 0.30/0.46 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 0.30/0.46 => ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.30/0.46 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.30/0.46 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(ap2apEq1_type,type,
% 0.30/0.46 ap2apEq1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(ap2apEq1,definition,
% 0.30/0.46 ( ap2apEq1
% 0.30/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.46 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.30/0.46 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(ap2apEq2_type,type,
% 0.30/0.46 ap2apEq2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(ap2apEq2,definition,
% 0.30/0.46 ( ap2apEq2
% 0.30/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.46 ( ( func @ A @ B @ Xf )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.30/0.46 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(beta1_type,type,
% 0.30/0.46 beta1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(beta1,definition,
% 0.30/0.46 ( beta1
% 0.30/0.46 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( ap @ A @ B
% 0.30/0.46 @ ( lam @ A @ B
% 0.30/0.46 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.30/0.46 @ Xx )
% 0.30/0.46 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(eta1_type,type,
% 0.30/0.46 eta1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(eta1,definition,
% 0.30/0.46 ( eta1
% 0.30/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.46 ( ( func @ A @ B @ Xf )
% 0.30/0.46 => ( ( lam @ A @ B
% 0.30/0.46 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.30/0.46 = Xf ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(lam2lamEq_type,type,
% 0.30/0.46 lam2lamEq: $o ).
% 0.30/0.46
% 0.30/0.46 thf(lam2lamEq,definition,
% 0.30/0.46 ( lam2lamEq
% 0.30/0.46 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.30/0.46 => ( ( lam @ A @ B
% 0.30/0.46 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.30/0.46 = ( lam @ A @ B
% 0.30/0.46 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(beta2_type,type,
% 0.30/0.46 beta2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(beta2,definition,
% 0.30/0.46 ( beta2
% 0.30/0.46 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( ap @ A @ B
% 0.30/0.46 @ ( lam @ A @ B
% 0.30/0.46 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.30/0.46 @ Xx )
% 0.30/0.46 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(eta2_type,type,
% 0.30/0.46 eta2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(eta2,definition,
% 0.30/0.46 ( eta2
% 0.30/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.46 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.30/0.46 => ( ( lam @ A @ B
% 0.30/0.46 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.30/0.46 = Xf ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(iffalseProp1_type,type,
% 0.30/0.46 iffalseProp1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(iffalseProp1,definition,
% 0.30/0.46 ( iffalseProp1
% 0.30/0.46 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( ~ Xphi
% 0.30/0.46 => ( in @ Xy
% 0.30/0.46 @ ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xz: $i] :
% 0.30/0.46 ( ( Xphi
% 0.30/0.46 & ( Xz = Xx ) )
% 0.30/0.46 | ( ~ Xphi
% 0.30/0.46 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(iffalseProp2_type,type,
% 0.30/0.46 iffalseProp2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(iffalseProp2,definition,
% 0.30/0.46 ( iffalseProp2
% 0.30/0.46 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( ~ Xphi
% 0.30/0.46 => ( ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xz: $i] :
% 0.30/0.46 ( ( Xphi
% 0.30/0.46 & ( Xz = Xx ) )
% 0.30/0.46 | ( ~ Xphi
% 0.30/0.46 & ( Xz = Xy ) ) ) )
% 0.30/0.46 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(iftrueProp1_type,type,
% 0.30/0.46 iftrueProp1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(iftrueProp1,definition,
% 0.30/0.46 ( iftrueProp1
% 0.30/0.46 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( Xphi
% 0.30/0.46 => ( in @ Xx
% 0.30/0.46 @ ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xz: $i] :
% 0.30/0.46 ( ( Xphi
% 0.30/0.46 & ( Xz = Xx ) )
% 0.30/0.46 | ( ~ Xphi
% 0.30/0.46 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(iftrueProp2_type,type,
% 0.30/0.46 iftrueProp2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(iftrueProp2,definition,
% 0.30/0.46 ( iftrueProp2
% 0.30/0.46 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( Xphi
% 0.30/0.46 => ( ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xz: $i] :
% 0.30/0.46 ( ( Xphi
% 0.30/0.46 & ( Xz = Xx ) )
% 0.30/0.46 | ( ~ Xphi
% 0.30/0.46 & ( Xz = Xy ) ) ) )
% 0.30/0.46 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(ifSingleton_type,type,
% 0.30/0.46 ifSingleton: $o ).
% 0.30/0.46
% 0.30/0.46 thf(ifSingleton,definition,
% 0.30/0.46 ( ifSingleton
% 0.30/0.46 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( singleton
% 0.30/0.46 @ ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xz: $i] :
% 0.30/0.46 ( ( Xphi
% 0.30/0.46 & ( Xz = Xx ) )
% 0.30/0.46 | ( ~ Xphi
% 0.30/0.46 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(if_type,type,
% 0.30/0.46 if: $i > $o > $i > $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(ifp_type,type,
% 0.30/0.46 ifp: $o ).
% 0.30/0.46
% 0.30/0.46 thf(ifp,definition,
% 0.30/0.46 ( ifp
% 0.30/0.46 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(theeq_type,type,
% 0.30/0.46 theeq: $o ).
% 0.30/0.46
% 0.30/0.46 thf(theeq,definition,
% 0.30/0.46 ( theeq
% 0.30/0.46 = ( ! [X: $i] :
% 0.30/0.46 ( ( singleton @ X )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ X )
% 0.30/0.46 => ( ( setunion @ X )
% 0.30/0.46 = Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(iftrue_type,type,
% 0.30/0.46 iftrue: $o ).
% 0.30/0.46
% 0.30/0.46 thf(iftrue,definition,
% 0.30/0.46 ( iftrue
% 0.30/0.46 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( Xphi
% 0.30/0.46 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.30/0.46 = Xx ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(iffalse_type,type,
% 0.30/0.46 iffalse: $o ).
% 0.30/0.46
% 0.30/0.46 thf(iffalse,definition,
% 0.30/0.46 ( iffalse
% 0.30/0.46 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( ~ Xphi
% 0.30/0.46 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.30/0.46 = Xy ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(iftrueorfalse_type,type,
% 0.30/0.46 iftrueorfalse: $o ).
% 0.30/0.46
% 0.30/0.46 thf(iftrueorfalse,definition,
% 0.30/0.46 ( iftrueorfalse
% 0.30/0.46 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectT_lem_type,type,
% 0.30/0.46 binintersectT_lem: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectT_lem,definition,
% 0.30/0.46 ( binintersectT_lem
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binunionT_lem_type,type,
% 0.30/0.46 binunionT_lem: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binunionT_lem,definition,
% 0.30/0.46 ( binunionT_lem
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(powersetT_lem_type,type,
% 0.30/0.46 powersetT_lem: $o ).
% 0.30/0.46
% 0.30/0.46 thf(powersetT_lem,definition,
% 0.30/0.46 ( powersetT_lem
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setminusT_lem_type,type,
% 0.30/0.46 setminusT_lem: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setminusT_lem,definition,
% 0.30/0.46 ( setminusT_lem
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ( in @ ( setminus @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(complementT_lem_type,type,
% 0.30/0.46 complementT_lem: $o ).
% 0.30/0.46
% 0.30/0.46 thf(complementT_lem,definition,
% 0.30/0.46 ( complementT_lem
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ( in @ ( setminus @ A @ X ) @ ( powerset @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setextT_type,type,
% 0.30/0.46 setextT: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setextT,definition,
% 0.30/0.46 ( setextT
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ X )
% 0.30/0.46 => ( in @ Xx @ Y ) ) )
% 0.30/0.46 => ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ Y )
% 0.30/0.46 => ( in @ Xx @ X ) ) )
% 0.30/0.46 => ( X = Y ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(subsetTI_type,type,
% 0.30/0.46 subsetTI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(subsetTI,definition,
% 0.30/0.46 ( subsetTI
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ X )
% 0.30/0.46 => ( in @ Xx @ Y ) ) )
% 0.30/0.46 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(powersetTI1_type,type,
% 0.30/0.46 powersetTI1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(powersetTI1,definition,
% 0.30/0.46 ( powersetTI1
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ X )
% 0.30/0.46 => ( in @ Xx @ Y ) ) )
% 0.30/0.46 => ( in @ X @ ( powerset @ Y ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(powersetTE1_type,type,
% 0.30/0.46 powersetTE1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(powersetTE1,definition,
% 0.30/0.46 ( powersetTE1
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ X @ ( powerset @ Y ) )
% 0.30/0.46 => ( ( in @ Xx @ X )
% 0.30/0.46 => ( in @ Xx @ Y ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(complementTI1_type,type,
% 0.30/0.46 complementTI1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(complementTI1,definition,
% 0.30/0.46 ( complementTI1
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ X )
% 0.30/0.46 => ~ ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(complementTE1_type,type,
% 0.30/0.46 complementTE1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(complementTE1,definition,
% 0.30/0.46 ( complementTE1
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ~ ( in @ Xx @ ( setminus @ A @ X ) )
% 0.30/0.46 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectTELcontra_type,type,
% 0.30/0.46 binintersectTELcontra: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectTELcontra,definition,
% 0.30/0.46 ( binintersectTELcontra
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ~ ( in @ Xx @ X )
% 0.30/0.46 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectTERcontra_type,type,
% 0.30/0.46 binintersectTERcontra: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectTERcontra,definition,
% 0.30/0.46 ( binintersectTERcontra
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ~ ( in @ Xx @ Y )
% 0.30/0.46 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(contrasubsetT_type,type,
% 0.30/0.46 contrasubsetT: $o ).
% 0.30/0.46
% 0.30/0.46 thf(contrasubsetT,definition,
% 0.30/0.46 ( contrasubsetT
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.30/0.46 => ( ( in @ Xx @ Y )
% 0.30/0.46 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(contrasubsetT1_type,type,
% 0.30/0.46 contrasubsetT1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(contrasubsetT1,definition,
% 0.30/0.46 ( contrasubsetT1
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( subset @ X @ Y )
% 0.30/0.46 => ( ~ ( in @ Xx @ Y )
% 0.30/0.46 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(contrasubsetT2_type,type,
% 0.30/0.46 contrasubsetT2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(contrasubsetT2,definition,
% 0.30/0.46 ( contrasubsetT2
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ( ( subset @ X @ Y )
% 0.30/0.46 => ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(contrasubsetT3_type,type,
% 0.30/0.46 contrasubsetT3: $o ).
% 0.30/0.46
% 0.30/0.46 thf(contrasubsetT3,definition,
% 0.30/0.46 ( contrasubsetT3
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ( ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) )
% 0.30/0.46 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(doubleComplementI1_type,type,
% 0.30/0.46 doubleComplementI1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(doubleComplementI1,definition,
% 0.30/0.46 ( doubleComplementI1
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ X )
% 0.30/0.46 => ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(doubleComplementE1_type,type,
% 0.30/0.46 doubleComplementE1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(doubleComplementE1,definition,
% 0.30/0.46 ( doubleComplementE1
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) )
% 0.30/0.46 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(doubleComplementSub1_type,type,
% 0.30/0.46 doubleComplementSub1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(doubleComplementSub1,definition,
% 0.30/0.46 ( doubleComplementSub1
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ( subset @ X @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(doubleComplementSub2_type,type,
% 0.30/0.46 doubleComplementSub2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(doubleComplementSub2,definition,
% 0.30/0.46 ( doubleComplementSub2
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ( subset @ ( setminus @ A @ ( setminus @ A @ X ) ) @ X ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(doubleComplementEq_type,type,
% 0.30/0.46 doubleComplementEq: $o ).
% 0.30/0.46
% 0.30/0.46 thf(doubleComplementEq,definition,
% 0.30/0.46 ( doubleComplementEq
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ( X
% 0.30/0.46 = ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(complementTnotintersectT_type,type,
% 0.30/0.46 complementTnotintersectT: $o ).
% 0.30/0.46
% 0.30/0.46 thf(complementTnotintersectT,definition,
% 0.30/0.46 ( complementTnotintersectT
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.30/0.46 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(complementImpComplementIntersect_type,type,
% 0.30/0.46 complementImpComplementIntersect: $o ).
% 0.30/0.46
% 0.30/0.46 thf(complementImpComplementIntersect,definition,
% 0.30/0.46 ( complementImpComplementIntersect
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.30/0.46 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(complementSubsetComplementIntersect_type,type,
% 0.30/0.46 complementSubsetComplementIntersect: $o ).
% 0.30/0.46
% 0.30/0.46 thf(complementSubsetComplementIntersect,definition,
% 0.30/0.46 ( complementSubsetComplementIntersect
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ( subset @ ( setminus @ A @ X ) @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(complementInPowersetComplementIntersect_type,type,
% 0.30/0.46 complementInPowersetComplementIntersect: $o ).
% 0.30/0.46
% 0.30/0.46 thf(complementInPowersetComplementIntersect,definition,
% 0.30/0.46 ( complementInPowersetComplementIntersect
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(contraSubsetComplement_type,type,
% 0.30/0.46 contraSubsetComplement: $o ).
% 0.30/0.46
% 0.30/0.46 thf(contraSubsetComplement,definition,
% 0.30/0.46 ( contraSubsetComplement
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ Y )
% 0.30/0.46 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(complementTcontraSubset_type,type,
% 0.30/0.46 complementTcontraSubset: $o ).
% 0.30/0.46
% 0.30/0.46 thf(complementTcontraSubset,definition,
% 0.30/0.46 ( complementTcontraSubset
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.30/0.46 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binunionTILcontra_type,type,
% 0.30/0.46 binunionTILcontra: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binunionTILcontra,definition,
% 0.30/0.46 ( binunionTILcontra
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.30/0.46 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binunionTIRcontra_type,type,
% 0.30/0.46 binunionTIRcontra: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binunionTIRcontra,definition,
% 0.30/0.46 ( binunionTIRcontra
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.30/0.46 => ~ ( in @ Xx @ Y ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(inIntersectImpInUnion_type,type,
% 0.30/0.46 inIntersectImpInUnion: $o ).
% 0.30/0.46
% 0.30/0.46 thf(inIntersectImpInUnion,definition,
% 0.30/0.46 ( inIntersectImpInUnion
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Z: $i] :
% 0.30/0.46 ( ( in @ Z @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.30/0.46 => ( in @ Xx @ ( binunion @ X @ Z ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(inIntersectImpInUnion2_type,type,
% 0.30/0.46 inIntersectImpInUnion2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(inIntersectImpInUnion2,definition,
% 0.30/0.46 ( inIntersectImpInUnion2
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Z: $i] :
% 0.30/0.46 ( ( in @ Z @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.30/0.46 => ( in @ Xx @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(inIntersectImpInIntersectUnions_type,type,
% 0.30/0.46 inIntersectImpInIntersectUnions: $o ).
% 0.30/0.46
% 0.30/0.46 thf(inIntersectImpInIntersectUnions,definition,
% 0.30/0.46 ( inIntersectImpInIntersectUnions
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Z: $i] :
% 0.30/0.46 ( ( in @ Z @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.30/0.46 => ( in @ Xx @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(intersectInPowersetIntersectUnions_type,type,
% 0.30/0.46 intersectInPowersetIntersectUnions: $o ).
% 0.30/0.46
% 0.30/0.46 thf(intersectInPowersetIntersectUnions,definition,
% 0.30/0.46 ( intersectInPowersetIntersectUnions
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Z: $i] :
% 0.30/0.46 ( ( in @ Z @ ( powerset @ A ) )
% 0.30/0.46 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(inComplementUnionImpNotIn1_type,type,
% 0.30/0.46 inComplementUnionImpNotIn1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(inComplementUnionImpNotIn1,definition,
% 0.30/0.46 ( inComplementUnionImpNotIn1
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.30/0.46 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(inComplementUnionImpInComplement1_type,type,
% 0.30/0.46 inComplementUnionImpInComplement1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(inComplementUnionImpInComplement1,definition,
% 0.30/0.46 ( inComplementUnionImpInComplement1
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.30/0.46 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binunionTE_type,type,
% 0.30/0.46 binunionTE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binunionTE,definition,
% 0.30/0.46 ( binunionTE
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xphi: $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.30/0.46 => ( ( ( in @ Xx @ X )
% 0.30/0.46 => Xphi )
% 0.30/0.46 => ( ( ( in @ Xx @ Y )
% 0.30/0.46 => Xphi )
% 0.30/0.46 => Xphi ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binunionTEcontra_type,type,
% 0.30/0.46 binunionTEcontra: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binunionTEcontra,definition,
% 0.30/0.46 ( binunionTEcontra
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ~ ( in @ Xx @ X )
% 0.30/0.46 => ( ~ ( in @ Xx @ Y )
% 0.30/0.46 => ~ ( in @ Xx @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(demorgan2a1_type,type,
% 0.30/0.46 demorgan2a1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(demorgan2a1,definition,
% 0.30/0.46 ( demorgan2a1
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.30/0.46 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(complementUnionInPowersetComplement_type,type,
% 0.30/0.46 complementUnionInPowersetComplement: $o ).
% 0.30/0.46
% 0.30/0.46 thf(complementUnionInPowersetComplement,definition,
% 0.30/0.46 ( complementUnionInPowersetComplement
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(demorgan2a2_type,type,
% 0.30/0.46 demorgan2a2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(demorgan2a2,definition,
% 0.30/0.46 ( demorgan2a2
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.30/0.46 => ( in @ Xx @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(demorgan1a_type,type,
% 0.30/0.46 demorgan1a: $o ).
% 0.30/0.46
% 0.30/0.46 thf(demorgan1a,definition,
% 0.30/0.46 ( demorgan1a
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) )
% 0.30/0.46 => ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(demorgan1b_type,type,
% 0.30/0.46 demorgan1b: $o ).
% 0.30/0.46
% 0.30/0.46 thf(demorgan1b,definition,
% 0.30/0.46 ( demorgan1b
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 0.30/0.46 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(demorgan1_type,type,
% 0.30/0.46 demorgan1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(demorgan1,definition,
% 0.30/0.46 ( demorgan1
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ( ( setminus @ A @ ( binintersect @ X @ Y ) )
% 0.30/0.46 = ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(demorgan2a_type,type,
% 0.30/0.46 demorgan2a: $o ).
% 0.30/0.46
% 0.30/0.46 thf(demorgan2a,definition,
% 0.30/0.46 ( demorgan2a
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.30/0.46 => ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(demorgan2b2_type,type,
% 0.30/0.46 demorgan2b2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(demorgan2b2,definition,
% 0.30/0.46 ( demorgan2b2
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.30/0.46 => ( ( in @ Xx @ ( setminus @ A @ Y ) )
% 0.30/0.46 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(demorgan2b_type,type,
% 0.30/0.46 demorgan2b: $o ).
% 0.30/0.46
% 0.30/0.46 thf(demorgan2b,definition,
% 0.30/0.46 ( demorgan2b
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 0.30/0.46 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(demorgan2_type,type,
% 0.30/0.46 demorgan2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(demorgan2,definition,
% 0.30/0.46 ( demorgan2
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ( ( setminus @ A @ ( binunion @ X @ Y ) )
% 0.30/0.46 = ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(woz13rule0_type,type,
% 0.30/0.46 woz13rule0: $o ).
% 0.30/0.46
% 0.30/0.46 thf(woz13rule0,definition,
% 0.30/0.46 ( woz13rule0
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.30/0.46 => ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(woz13rule1_type,type,
% 0.30/0.46 woz13rule1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(woz13rule1,definition,
% 0.30/0.46 ( woz13rule1
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Z: $i] :
% 0.30/0.46 ( ( in @ Z @ ( powerset @ A ) )
% 0.30/0.46 => ( ( subset @ X @ Z )
% 0.30/0.46 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(woz13rule2_type,type,
% 0.30/0.46 woz13rule2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(woz13rule2,definition,
% 0.30/0.46 ( woz13rule2
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Z: $i] :
% 0.30/0.46 ( ( in @ Z @ ( powerset @ A ) )
% 0.30/0.46 => ( ( subset @ Y @ Z )
% 0.30/0.46 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(woz13rule3_type,type,
% 0.30/0.46 woz13rule3: $o ).
% 0.30/0.46
% 0.30/0.46 thf(woz13rule3,definition,
% 0.30/0.46 ( woz13rule3
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Z: $i] :
% 0.30/0.46 ( ( in @ Z @ ( powerset @ A ) )
% 0.30/0.46 => ( ( subset @ X @ Y )
% 0.30/0.46 => ( ( subset @ X @ Z )
% 0.30/0.46 => ( subset @ X @ ( binintersect @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(woz13rule4_type,type,
% 0.30/0.46 woz13rule4: $o ).
% 0.30/0.46
% 0.30/0.46 thf(woz13rule4,definition,
% 0.30/0.46 ( woz13rule4
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Z: $i] :
% 0.30/0.46 ( ( in @ Z @ ( powerset @ A ) )
% 0.30/0.46 => ! [W: $i] :
% 0.30/0.46 ( ( in @ W @ ( powerset @ A ) )
% 0.30/0.46 => ( ( subset @ X @ Z )
% 0.30/0.46 => ( ( subset @ Y @ W )
% 0.30/0.46 => ( subset @ ( binintersect @ X @ Y ) @ ( binintersect @ Z @ W ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(woz1_1_type,type,
% 0.30/0.46 woz1_1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(woz1_1,definition,
% 0.30/0.46 ( woz1_1
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(woz1_2_type,type,
% 0.30/0.46 woz1_2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(woz1_2,definition,
% 0.30/0.46 ( woz1_2
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Z: $i] :
% 0.30/0.46 ( ( in @ Z @ ( powerset @ A ) )
% 0.30/0.46 => ! [W: $i] :
% 0.30/0.46 ( ( in @ W @ ( powerset @ A ) )
% 0.30/0.46 => ( ( setminus @ A @ ( binintersect @ ( binunion @ X @ Y ) @ ( binunion @ Z @ W ) ) )
% 0.30/0.46 = ( binunion @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) @ ( binintersect @ ( setminus @ A @ Z ) @ ( setminus @ A @ W ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(woz1_3_type,type,
% 0.30/0.46 woz1_3: $o ).
% 0.30/0.46
% 0.30/0.46 thf(woz1_3,definition,
% 0.30/0.46 ( woz1_3
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ! [Z: $i] :
% 0.30/0.46 ( ( in @ Z @ ( powerset @ A ) )
% 0.30/0.46 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(woz1_4_type,type,
% 0.30/0.46 woz1_4: $o ).
% 0.30/0.46
% 0.30/0.46 thf(woz1_4,definition,
% 0.30/0.46 ( woz1_4
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.30/0.46 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(woz1_5_type,type,
% 0.30/0.46 woz1_5: $o ).
% 0.30/0.46
% 0.30/0.46 thf(woz1_5,definition,
% 0.30/0.46 ( woz1_5
% 0.30/0.46 = ( ! [A: $i,X: $i] :
% 0.30/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.46 => ! [Y: $i] :
% 0.30/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.46 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(breln1_type,type,
% 0.30/0.46 breln1: $i > $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(breln1all2_type,type,
% 0.30/0.46 breln1all2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(breln1all2,definition,
% 0.30/0.46 ( breln1all2
% 0.30/0.46 = ( ! [A: $i,R: $i] :
% 0.30/0.46 ( ( breln1 @ A @ R )
% 0.30/0.46 => ! [Xphi: $i > $o] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.30/0.46 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ R )
% 0.30/0.46 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(breln1Set_type,type,
% 0.30/0.46 breln1Set: $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(breln1SetBreln1_type,type,
% 0.30/0.46 breln1SetBreln1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(breln1SetBreln1,definition,
% 0.30/0.46 ( breln1SetBreln1
% 0.30/0.46 = ( ! [A: $i,R: $i] :
% 0.30/0.46 ( ( in @ R @ ( breln1Set @ A ) )
% 0.30/0.46 => ( breln1 @ A @ R ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(transitive_type,type,
% 0.30/0.46 transitive: $i > $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(antisymmetric_type,type,
% 0.30/0.46 antisymmetric: $i > $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(reflexive_type,type,
% 0.30/0.46 reflexive: $i > $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(refltransitive_type,type,
% 0.30/0.46 refltransitive: $i > $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(refllinearorder_type,type,
% 0.30/0.46 refllinearorder: $i > $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(reflwellordering_type,type,
% 0.30/0.46 reflwellordering: $i > $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(choice2fnsingleton_type,type,
% 0.30/0.46 choice2fnsingleton: $o ).
% 0.30/0.46
% 0.30/0.46 thf(choice2fnsingleton,definition,
% 0.30/0.46 ( choice2fnsingleton
% 0.30/0.46 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ? [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ B )
% 0.30/0.46 & ( Xphi @ Xx @ Xy ) ) )
% 0.30/0.46 => ! [R: $i] :
% 0.30/0.46 ( ( in @ R @ ( breln1Set @ B ) )
% 0.30/0.46 => ( ( reflwellordering @ B @ R )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( singleton
% 0.30/0.46 @ ( dsetconstr @ B
% 0.30/0.46 @ ^ [Xy: $i] :
% 0.30/0.46 ( ( Xphi @ Xx @ Xy )
% 0.30/0.46 & ! [Xz: $i] :
% 0.30/0.46 ( ( in @ Xz @ B )
% 0.30/0.46 => ( ( Xphi @ Xx @ Xz )
% 0.30/0.46 => ( in @ ( kpair @ Xy @ Xz ) @ R ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setOfPairsIsBReln1_type,type,
% 0.30/0.46 setOfPairsIsBReln1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setOfPairsIsBReln1,definition,
% 0.30/0.46 ( setOfPairsIsBReln1
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $i > $o] :
% 0.30/0.46 ( breln1 @ A
% 0.30/0.46 @ ( dpsetconstr @ A @ A
% 0.30/0.46 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(breln1all1_type,type,
% 0.30/0.46 breln1all1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(breln1all1,definition,
% 0.30/0.46 ( breln1all1
% 0.30/0.46 = ( ! [A: $i,R: $i] :
% 0.30/0.46 ( ( breln1 @ A @ R )
% 0.30/0.46 => ! [Xphi: $i > $o] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.30/0.46 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ R )
% 0.30/0.46 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(subbreln1_type,type,
% 0.30/0.46 subbreln1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(subbreln1,definition,
% 0.30/0.46 ( subbreln1
% 0.30/0.46 = ( ! [A: $i,R: $i] :
% 0.30/0.46 ( ( breln1 @ A @ R )
% 0.30/0.46 => ! [S: $i] :
% 0.30/0.46 ( ( breln1 @ A @ S )
% 0.30/0.46 => ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.30/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.30/0.46 => ( subset @ R @ S ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(eqbreln1_type,type,
% 0.30/0.46 eqbreln1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(eqbreln1,definition,
% 0.30/0.46 ( eqbreln1
% 0.30/0.46 = ( ! [A: $i,R: $i] :
% 0.30/0.46 ( ( breln1 @ A @ R )
% 0.30/0.46 => ! [S: $i] :
% 0.30/0.46 ( ( breln1 @ A @ S )
% 0.30/0.46 => ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.30/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.30/0.46 => ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.30/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 0.30/0.46 => ( R = S ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(breln1invset_type,type,
% 0.30/0.46 breln1invset: $i > $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(breln1invprop_type,type,
% 0.30/0.46 breln1invprop: $o ).
% 0.30/0.46
% 0.30/0.46 thf(breln1invprop,definition,
% 0.30/0.46 ( breln1invprop
% 0.30/0.46 = ( ! [A: $i,R: $i] :
% 0.30/0.46 ( ( breln1 @ A @ R )
% 0.30/0.46 => ( breln1 @ A @ ( breln1invset @ A @ R ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(breln1invI_type,type,
% 0.30/0.46 breln1invI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(breln1invI,definition,
% 0.30/0.46 ( breln1invI
% 0.30/0.46 = ( ! [A: $i,R: $i] :
% 0.30/0.46 ( ( breln1 @ A @ R )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.30/0.46 => ( in @ ( kpair @ Xy @ Xx ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(breln1invE_type,type,
% 0.30/0.46 breln1invE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(breln1invE,definition,
% 0.30/0.46 ( breln1invE
% 0.30/0.46 = ( ! [A: $i,R: $i] :
% 0.30/0.46 ( ( breln1 @ A @ R )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( ( in @ ( kpair @ Xy @ Xx ) @ ( breln1invset @ A @ R ) )
% 0.30/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(breln1compset_type,type,
% 0.30/0.46 breln1compset: $i > $i > $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(breln1compprop_type,type,
% 0.30/0.46 breln1compprop: $o ).
% 0.30/0.46
% 0.30/0.46 thf(breln1compprop,definition,
% 0.30/0.46 ( breln1compprop
% 0.30/0.46 = ( ! [A: $i,R: $i] :
% 0.30/0.46 ( ( breln1 @ A @ R )
% 0.30/0.46 => ! [S: $i] :
% 0.30/0.46 ( ( breln1 @ A @ S )
% 0.30/0.46 => ( breln1 @ A @ ( breln1compset @ A @ R @ S ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(breln1compI_type,type,
% 0.30/0.46 breln1compI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(breln1compI,definition,
% 0.30/0.46 ( breln1compI
% 0.30/0.46 = ( ! [A: $i,R: $i] :
% 0.30/0.46 ( ( breln1 @ A @ R )
% 0.30/0.46 => ! [S: $i] :
% 0.30/0.46 ( ( breln1 @ A @ S )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ! [Xz: $i] :
% 0.30/0.46 ( ( in @ Xz @ A )
% 0.30/0.46 => ( ( in @ ( kpair @ Xx @ Xz ) @ R )
% 0.30/0.46 => ( ( in @ ( kpair @ Xz @ Xy ) @ S )
% 0.30/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(breln1compE_type,type,
% 0.30/0.46 breln1compE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(breln1compE,definition,
% 0.30/0.46 ( breln1compE
% 0.30/0.46 = ( ! [A: $i,R: $i] :
% 0.30/0.46 ( ( breln1 @ A @ R )
% 0.30/0.46 => ! [S: $i] :
% 0.30/0.46 ( ( breln1 @ A @ S )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) )
% 0.30/0.46 => ? [Xz: $i] :
% 0.30/0.46 ( ( in @ Xz @ A )
% 0.30/0.46 & ( in @ ( kpair @ Xx @ Xz ) @ R )
% 0.30/0.46 & ( in @ ( kpair @ Xz @ Xy ) @ S ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(breln1compEex_type,type,
% 0.30/0.46 breln1compEex: $o ).
% 0.30/0.46
% 0.30/0.46 thf(breln1compEex,definition,
% 0.30/0.46 ( breln1compEex
% 0.30/0.46 = ( ! [A: $i,R: $i] :
% 0.30/0.46 ( ( breln1 @ A @ R )
% 0.30/0.46 => ! [S: $i] :
% 0.30/0.46 ( ( breln1 @ A @ S )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) )
% 0.30/0.46 => ! [Xphi: $o] :
% 0.30/0.46 ( ! [Xz: $i] :
% 0.30/0.46 ( ( in @ Xz @ A )
% 0.30/0.46 => ( ( in @ ( kpair @ Xx @ Xz ) @ R )
% 0.30/0.46 => ( ( in @ ( kpair @ Xz @ Xy ) @ S )
% 0.30/0.46 => Xphi ) ) )
% 0.30/0.46 => Xphi ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(breln1unionprop_type,type,
% 0.30/0.46 breln1unionprop: $o ).
% 0.30/0.46
% 0.30/0.46 thf(breln1unionprop,definition,
% 0.30/0.46 ( breln1unionprop
% 0.30/0.46 = ( ! [A: $i,R: $i] :
% 0.30/0.46 ( ( breln1 @ A @ R )
% 0.30/0.46 => ! [S: $i] :
% 0.30/0.46 ( ( breln1 @ A @ S )
% 0.30/0.46 => ( breln1 @ A @ ( binunion @ R @ S ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(breln1unionIL_type,type,
% 0.30/0.46 breln1unionIL: $o ).
% 0.30/0.46
% 0.30/0.46 thf(breln1unionIL,definition,
% 0.30/0.46 ( breln1unionIL
% 0.30/0.46 = ( ! [A: $i,R: $i] :
% 0.30/0.46 ( ( breln1 @ A @ R )
% 0.30/0.46 => ! [S: $i] :
% 0.30/0.46 ( ( breln1 @ A @ S )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.30/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(breln1unionIR_type,type,
% 0.30/0.46 breln1unionIR: $o ).
% 0.30/0.46
% 0.30/0.46 thf(breln1unionIR,definition,
% 0.30/0.46 ( breln1unionIR
% 0.30/0.46 = ( ! [A: $i,R: $i] :
% 0.30/0.46 ( ( breln1 @ A @ R )
% 0.30/0.46 => ! [S: $i] :
% 0.30/0.46 ( ( breln1 @ A @ S )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.30/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(breln1unionI_type,type,
% 0.30/0.46 breln1unionI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(breln1unionI,definition,
% 0.30/0.46 ( breln1unionI
% 0.30/0.46 = ( ! [A: $i,R: $i] :
% 0.30/0.46 ( ( breln1 @ A @ R )
% 0.30/0.46 => ! [S: $i] :
% 0.30/0.46 ( ( breln1 @ A @ S )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.30/0.46 | ( in @ ( kpair @ Xx @ Xy ) @ S ) )
% 0.30/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(breln1unionE_type,type,
% 0.30/0.46 breln1unionE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(breln1unionE,definition,
% 0.30/0.46 ( breln1unionE
% 0.30/0.46 = ( ! [A: $i,R: $i] :
% 0.30/0.46 ( ( breln1 @ A @ R )
% 0.30/0.46 => ! [S: $i] :
% 0.30/0.46 ( ( breln1 @ A @ S )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) )
% 0.30/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.30/0.46 | ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(breln1unionEcases_type,type,
% 0.30/0.46 breln1unionEcases: $o ).
% 0.30/0.46
% 0.30/0.46 thf(breln1unionEcases,definition,
% 0.30/0.46 ( breln1unionEcases
% 0.30/0.46 = ( ! [A: $i,R: $i] :
% 0.30/0.46 ( ( breln1 @ A @ R )
% 0.30/0.46 => ! [S: $i] :
% 0.30/0.46 ( ( breln1 @ A @ S )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) )
% 0.30/0.46 => ! [Xphi: $o] :
% 0.30/0.46 ( ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.30/0.46 => Xphi )
% 0.30/0.46 => ( ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.30/0.46 => Xphi )
% 0.30/0.46 => Xphi ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(breln1unionCommutes_type,type,
% 0.30/0.46 breln1unionCommutes: $o ).
% 0.30/0.46
% 0.30/0.46 thf(breln1unionCommutes,definition,
% 0.30/0.46 ( breln1unionCommutes
% 0.30/0.46 = ( ! [A: $i,R: $i] :
% 0.30/0.46 ( ( breln1 @ A @ R )
% 0.30/0.46 => ! [S: $i] :
% 0.30/0.46 ( ( breln1 @ A @ S )
% 0.30/0.46 => ( ( binunion @ R @ S )
% 0.30/0.46 = ( binunion @ S @ R ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(woz2Ex_type,type,
% 0.30/0.46 woz2Ex: $o ).
% 0.30/0.46
% 0.30/0.46 thf(woz2Ex,definition,
% 0.30/0.46 ( woz2Ex
% 0.30/0.46 = ( ! [A: $i,R: $i] :
% 0.30/0.46 ( ( breln1 @ A @ R )
% 0.30/0.46 => ( R
% 0.30/0.46 = ( breln1invset @ A @ ( breln1invset @ A @ R ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(woz2W_type,type,
% 0.30/0.46 woz2W: $o ).
% 0.30/0.46
% 0.30/0.46 thf(woz2W,definition,
% 0.30/0.46 ( woz2W
% 0.30/0.46 = ( ! [A: $i,R: $i] :
% 0.30/0.46 ( ( breln1 @ A @ R )
% 0.30/0.46 => ! [S: $i] :
% 0.30/0.46 ( ( breln1 @ A @ S )
% 0.30/0.46 => ( ( breln1invset @ A @ ( breln1compset @ A @ R @ S ) )
% 0.30/0.46 = ( breln1compset @ A @ ( breln1invset @ A @ S ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(woz2A_type,type,
% 0.30/0.46 woz2A: $o ).
% 0.30/0.46
% 0.30/0.46 thf(woz2A,definition,
% 0.30/0.46 ( woz2A
% 0.30/0.46 = ( ! [A: $i,R: $i] :
% 0.30/0.46 ( ( breln1 @ A @ R )
% 0.30/0.46 => ! [S: $i] :
% 0.30/0.46 ( ( breln1 @ A @ S )
% 0.30/0.46 => ! [T: $i] :
% 0.30/0.46 ( ( breln1 @ A @ T )
% 0.30/0.46 => ( ( breln1compset @ A @ ( binunion @ R @ S ) @ T )
% 0.30/0.46 = ( binunion @ ( breln1compset @ A @ R @ T ) @ ( breln1compset @ A @ S @ T ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(woz2B_type,type,
% 0.30/0.46 woz2B: $o ).
% 0.30/0.46
% 0.30/0.46 thf(woz2B,definition,
% 0.30/0.46 ( woz2B
% 0.30/0.46 = ( ! [A: $i,R: $i] :
% 0.30/0.46 ( ( breln1 @ A @ R )
% 0.30/0.46 => ! [S: $i] :
% 0.30/0.46 ( ( breln1 @ A @ S )
% 0.30/0.46 => ! [T: $i] :
% 0.30/0.46 ( ( breln1 @ A @ T )
% 0.30/0.46 => ( ( breln1compset @ A @ ( binunion @ R @ S ) @ T )
% 0.30/0.46 = ( binunion @ ( breln1invset @ A @ ( breln1compset @ A @ ( breln1invset @ A @ T ) @ ( breln1invset @ A @ S ) ) ) @ ( breln1invset @ A @ ( breln1compset @ A @ ( breln1invset @ A @ T ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(image1Ex_type,type,
% 0.30/0.46 image1Ex: $o ).
% 0.30/0.46
% 0.30/0.46 thf(image1Ex,definition,
% 0.30/0.46 ( image1Ex
% 0.30/0.46 = ( ! [A: $i,Xf: $i > $i] :
% 0.30/0.46 ? [B: $i] :
% 0.30/0.46 ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ B )
% 0.30/0.46 <=> ? [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 & ( Xx
% 0.30/0.46 = ( Xf @ Xy ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(image1Ex1_type,type,
% 0.30/0.46 image1Ex1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(image1Ex1,definition,
% 0.30/0.46 ( image1Ex1
% 0.30/0.46 = ( ! [A: $i,Xf: $i > $i] :
% 0.30/0.46 ( exu
% 0.30/0.46 @ ^ [B: $i] :
% 0.30/0.46 ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ B )
% 0.30/0.46 <=> ? [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 & ( Xx
% 0.30/0.46 = ( Xf @ Xy ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(image1_type,type,
% 0.30/0.46 image1: $i > ( $i > $i ) > $i ).
% 0.30/0.46
% 0.30/0.46 thf(image1Equiv_type,type,
% 0.30/0.46 image1Equiv: $o ).
% 0.30/0.46
% 0.30/0.46 thf(image1Equiv,definition,
% 0.30/0.46 ( image1Equiv
% 0.30/0.46 = ( ! [A: $i,Xf: $i > $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx
% 0.30/0.46 @ ( image1 @ A
% 0.30/0.46 @ ^ [Xy: $i] : ( Xf @ Xy ) ) )
% 0.30/0.46 <=> ? [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 & ( Xx
% 0.30/0.46 = ( Xf @ Xy ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(image1E_type,type,
% 0.30/0.46 image1E: $o ).
% 0.30/0.46
% 0.30/0.46 thf(image1E,definition,
% 0.30/0.46 ( image1E
% 0.30/0.46 = ( ! [A: $i,Xf: $i > $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx
% 0.30/0.46 @ ( image1 @ A
% 0.30/0.46 @ ^ [Xy: $i] : ( Xf @ Xy ) ) )
% 0.30/0.46 => ? [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 & ( Xx
% 0.30/0.46 = ( Xf @ Xy ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(image1I_type,type,
% 0.30/0.46 image1I: $o ).
% 0.30/0.46
% 0.30/0.46 thf(image1I,definition,
% 0.30/0.46 ( image1I
% 0.30/0.46 = ( ! [A: $i,Xf: $i > $i,Xx: $i] :
% 0.30/0.46 ( ? [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 & ( Xx
% 0.30/0.46 = ( Xf @ Xy ) ) )
% 0.30/0.46 => ( in @ Xx
% 0.30/0.46 @ ( image1 @ A
% 0.30/0.46 @ ^ [Xy: $i] : ( Xf @ Xy ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(injective_type,type,
% 0.30/0.46 injective: $i > $i > $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(injFuncSet_type,type,
% 0.30/0.46 injFuncSet: $i > $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(injFuncSet,definition,
% 0.30/0.46 ( injFuncSet
% 0.30/0.46 = ( ^ [A: $i,B: $i] :
% 0.30/0.46 ( dsetconstr @ ( funcSet @ A @ B )
% 0.30/0.46 @ ^ [Xf: $i] : ( injective @ A @ B @ Xf ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(injFuncInInjFuncSet_type,type,
% 0.30/0.46 injFuncInInjFuncSet: $o ).
% 0.30/0.46
% 0.30/0.46 thf(injFuncInInjFuncSet,definition,
% 0.30/0.46 ( injFuncInInjFuncSet
% 0.30/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.46 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.30/0.46 => ( ( injective @ A @ B @ Xf )
% 0.30/0.46 => ( in @ Xf @ ( injFuncSet @ A @ B ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(injFuncSetFuncIn_type,type,
% 0.30/0.46 injFuncSetFuncIn: $o ).
% 0.30/0.46
% 0.30/0.46 thf(injFuncSetFuncIn,definition,
% 0.30/0.46 ( injFuncSetFuncIn
% 0.30/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.46 ( ( in @ Xf @ ( injFuncSet @ A @ B ) )
% 0.30/0.46 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(injFuncSetFuncInj,conjecture,
% 0.30/0.46 ( setextAx
% 0.30/0.46 => ( emptysetAx
% 0.30/0.46 => ( setadjoinAx
% 0.30/0.46 => ( powersetAx
% 0.30/0.46 => ( setunionAx
% 0.30/0.46 => ( omega0Ax
% 0.30/0.46 => ( omegaSAx
% 0.30/0.46 => ( omegaIndAx
% 0.30/0.46 => ( replAx
% 0.30/0.46 => ( foundationAx
% 0.30/0.46 => ( wellorderingAx
% 0.30/0.46 => ( descrp
% 0.30/0.46 => ( dsetconstrI
% 0.30/0.46 => ( dsetconstrEL
% 0.30/0.46 => ( dsetconstrER
% 0.30/0.46 => ( exuE1
% 0.30/0.46 => ( prop2setE
% 0.30/0.46 => ( emptysetE
% 0.30/0.46 => ( emptysetimpfalse
% 0.30/0.46 => ( notinemptyset
% 0.30/0.46 => ( exuE3e
% 0.30/0.46 => ( setext
% 0.30/0.46 => ( emptyI
% 0.30/0.46 => ( noeltsimpempty
% 0.30/0.46 => ( setbeta
% 0.30/0.46 => ( nonemptyE1
% 0.30/0.46 => ( nonemptyI
% 0.30/0.46 => ( nonemptyI1
% 0.30/0.46 => ( setadjoinIL
% 0.30/0.46 => ( emptyinunitempty
% 0.30/0.46 => ( setadjoinIR
% 0.30/0.46 => ( setadjoinE
% 0.30/0.46 => ( setadjoinOr
% 0.30/0.46 => ( setoftrueEq
% 0.30/0.46 => ( powersetI
% 0.30/0.46 => ( emptyinPowerset
% 0.30/0.46 => ( emptyInPowerset
% 0.30/0.46 => ( powersetE
% 0.30/0.46 => ( setunionI
% 0.30/0.46 => ( setunionE
% 0.30/0.46 => ( subPowSU
% 0.30/0.46 => ( exuE2
% 0.30/0.46 => ( nonemptyImpWitness
% 0.30/0.46 => ( uniqinunit
% 0.30/0.46 => ( notinsingleton
% 0.30/0.46 => ( eqinunit
% 0.30/0.46 => ( singletonsswitch
% 0.30/0.46 => ( upairsetE
% 0.30/0.46 => ( upairsetIL
% 0.30/0.46 => ( upairsetIR
% 0.30/0.46 => ( emptyE1
% 0.30/0.47 => ( vacuousDall
% 0.30/0.47 => ( quantDeMorgan1
% 0.30/0.47 => ( quantDeMorgan2
% 0.30/0.47 => ( quantDeMorgan3
% 0.30/0.47 => ( quantDeMorgan4
% 0.30/0.47 => ( prop2setI
% 0.30/0.47 => ( prop2set2propI
% 0.30/0.47 => ( notdexE
% 0.30/0.47 => ( notdallE
% 0.30/0.47 => ( exuI1
% 0.30/0.47 => ( exuI3
% 0.30/0.47 => ( exuI2
% 0.30/0.47 => ( inCongP
% 0.30/0.47 => ( in__Cong
% 0.30/0.47 => ( exuE3u
% 0.30/0.47 => ( exu__Cong
% 0.30/0.47 => ( emptyset__Cong
% 0.30/0.47 => ( setadjoin__Cong
% 0.30/0.47 => ( powerset__Cong
% 0.30/0.47 => ( setunion__Cong
% 0.30/0.47 => ( omega__Cong
% 0.30/0.47 => ( exuEu
% 0.30/0.47 => ( descr__Cong
% 0.30/0.47 => ( dsetconstr__Cong
% 0.30/0.47 => ( subsetI1
% 0.30/0.47 => ( eqimpsubset2
% 0.30/0.47 => ( eqimpsubset1
% 0.30/0.47 => ( subsetI2
% 0.30/0.47 => ( emptysetsubset
% 0.30/0.47 => ( subsetE
% 0.30/0.47 => ( subsetE2
% 0.30/0.47 => ( notsubsetI
% 0.30/0.47 => ( notequalI1
% 0.30/0.47 => ( notequalI2
% 0.30/0.47 => ( subsetRefl
% 0.30/0.47 => ( subsetTrans
% 0.30/0.47 => ( setadjoinSub
% 0.30/0.47 => ( setadjoinSub2
% 0.30/0.47 => ( subset2powerset
% 0.30/0.47 => ( setextsub
% 0.30/0.47 => ( subsetemptysetimpeq
% 0.30/0.47 => ( powersetI1
% 0.30/0.47 => ( powersetE1
% 0.30/0.47 => ( inPowerset
% 0.30/0.47 => ( powersetsubset
% 0.30/0.47 => ( sepInPowerset
% 0.30/0.47 => ( sepSubset
% 0.30/0.47 => ( binunionIL
% 0.30/0.47 => ( upairset2IR
% 0.30/0.47 => ( binunionIR
% 0.30/0.47 => ( binunionEcases
% 0.30/0.47 => ( binunionE
% 0.30/0.47 => ( binunionLsub
% 0.30/0.47 => ( binunionRsub
% 0.30/0.47 => ( binintersectI
% 0.30/0.47 => ( binintersectSubset5
% 0.30/0.47 => ( binintersectEL
% 0.30/0.47 => ( binintersectLsub
% 0.30/0.47 => ( binintersectSubset2
% 0.30/0.47 => ( binintersectSubset3
% 0.30/0.47 => ( binintersectER
% 0.30/0.47 => ( disjointsetsI1
% 0.30/0.47 => ( binintersectRsub
% 0.30/0.47 => ( binintersectSubset4
% 0.30/0.47 => ( binintersectSubset1
% 0.30/0.47 => ( bs114d
% 0.30/0.47 => ( setminusI
% 0.30/0.47 => ( setminusEL
% 0.30/0.47 => ( setminusER
% 0.30/0.47 => ( setminusSubset2
% 0.30/0.47 => ( setminusERneg
% 0.30/0.47 => ( setminusELneg
% 0.30/0.47 => ( setminusILneg
% 0.30/0.47 => ( setminusIRneg
% 0.30/0.47 => ( setminusLsub
% 0.30/0.47 => ( setminusSubset1
% 0.30/0.47 => ( symdiffE
% 0.30/0.47 => ( symdiffI1
% 0.30/0.47 => ( symdiffI2
% 0.30/0.47 => ( symdiffIneg1
% 0.30/0.47 => ( symdiffIneg2
% 0.30/0.47 => ( secondinupair
% 0.30/0.47 => ( setukpairIL
% 0.30/0.47 => ( setukpairIR
% 0.30/0.47 => ( kpairiskpair
% 0.30/0.47 => ( kpairp
% 0.30/0.47 => ( singletonsubset
% 0.30/0.47 => ( singletoninpowerset
% 0.30/0.47 => ( singletoninpowunion
% 0.30/0.47 => ( upairset2E
% 0.30/0.47 => ( upairsubunion
% 0.30/0.47 => ( upairinpowunion
% 0.30/0.47 => ( ubforcartprodlem1
% 0.30/0.47 => ( ubforcartprodlem2
% 0.30/0.47 => ( ubforcartprodlem3
% 0.30/0.47 => ( cartprodpairin
% 0.30/0.47 => ( cartprodmempair1
% 0.30/0.47 => ( cartprodmempair
% 0.30/0.47 => ( setunionE2
% 0.30/0.47 => ( setunionsingleton1
% 0.30/0.47 => ( setunionsingleton2
% 0.30/0.47 => ( setunionsingleton
% 0.30/0.47 => ( singletonprop
% 0.30/0.47 => ( ex1E1
% 0.30/0.47 => ( ex1I
% 0.30/0.47 => ( ex1I2
% 0.30/0.47 => ( singletonsuniq
% 0.30/0.47 => ( setukpairinjL1
% 0.30/0.47 => ( kfstsingleton
% 0.30/0.47 => ( theprop
% 0.30/0.47 => ( kfstpairEq
% 0.30/0.47 => ( cartprodfstin
% 0.30/0.47 => ( setukpairinjL2
% 0.30/0.47 => ( setukpairinjL
% 0.30/0.47 => ( setukpairinjR11
% 0.30/0.47 => ( setukpairinjR12
% 0.30/0.47 => ( setukpairinjR1
% 0.30/0.47 => ( upairequniteq
% 0.30/0.47 => ( setukpairinjR2
% 0.30/0.47 => ( setukpairinjR
% 0.30/0.47 => ( ksndsingleton
% 0.30/0.47 => ( ksndpairEq
% 0.30/0.47 => ( kpairsurjEq
% 0.30/0.47 => ( cartprodsndin
% 0.30/0.47 => ( cartprodpairmemEL
% 0.30/0.47 => ( cartprodpairmemER
% 0.30/0.47 => ( cartprodmempaircEq
% 0.30/0.47 => ( cartprodfstpairEq
% 0.30/0.47 => ( cartprodsndpairEq
% 0.30/0.47 => ( cartprodpairsurjEq
% 0.30/0.47 => ( dpsetconstrI
% 0.30/0.47 => ( dpsetconstrSub
% 0.30/0.47 => ( setOfPairsIsBReln
% 0.30/0.47 => ( dpsetconstrERa
% 0.30/0.47 => ( dpsetconstrEL1
% 0.30/0.47 => ( dpsetconstrEL2
% 0.30/0.47 => ( dpsetconstrER
% 0.30/0.47 => ( funcImageSingleton
% 0.30/0.47 => ( apProp
% 0.30/0.47 => ( app
% 0.30/0.47 => ( infuncsetfunc
% 0.30/0.47 => ( ap2p
% 0.30/0.47 => ( funcinfuncset
% 0.30/0.47 => ( lamProp
% 0.30/0.47 => ( lamp
% 0.30/0.47 => ( lam2p
% 0.30/0.47 => ( brelnall1
% 0.30/0.47 => ( brelnall2
% 0.30/0.47 => ( ex1E2
% 0.30/0.47 => ( funcGraphProp1
% 0.30/0.47 => ( funcGraphProp3
% 0.30/0.47 => ( funcGraphProp2
% 0.30/0.47 => ( funcextLem
% 0.30/0.47 => ( funcGraphProp4
% 0.30/0.47 => ( subbreln
% 0.30/0.47 => ( eqbreln
% 0.30/0.47 => ( funcext
% 0.30/0.47 => ( funcext2
% 0.30/0.47 => ( ap2apEq1
% 0.30/0.47 => ( ap2apEq2
% 0.30/0.47 => ( beta1
% 0.30/0.47 => ( eta1
% 0.30/0.47 => ( lam2lamEq
% 0.30/0.47 => ( beta2
% 0.30/0.47 => ( eta2
% 0.30/0.47 => ( iffalseProp1
% 0.30/0.47 => ( iffalseProp2
% 0.30/0.47 => ( iftrueProp1
% 0.30/0.47 => ( iftrueProp2
% 0.30/0.47 => ( ifSingleton
% 0.30/0.47 => ( ifp
% 0.30/0.47 => ( theeq
% 0.30/0.47 => ( iftrue
% 0.30/0.47 => ( iffalse
% 0.30/0.47 => ( iftrueorfalse
% 0.30/0.47 => ( binintersectT_lem
% 0.30/0.47 => ( binunionT_lem
% 0.30/0.47 => ( powersetT_lem
% 0.30/0.47 => ( setminusT_lem
% 0.30/0.47 => ( complementT_lem
% 0.30/0.47 => ( setextT
% 0.30/0.47 => ( subsetTI
% 0.30/0.47 => ( powersetTI1
% 0.30/0.47 => ( powersetTE1
% 0.30/0.47 => ( complementTI1
% 0.30/0.47 => ( complementTE1
% 0.30/0.47 => ( binintersectTELcontra
% 0.30/0.47 => ( binintersectTERcontra
% 0.30/0.47 => ( contrasubsetT
% 0.30/0.47 => ( contrasubsetT1
% 0.30/0.47 => ( contrasubsetT2
% 0.30/0.47 => ( contrasubsetT3
% 0.30/0.47 => ( doubleComplementI1
% 0.30/0.47 => ( doubleComplementE1
% 0.30/0.47 => ( doubleComplementSub1
% 0.30/0.47 => ( doubleComplementSub2
% 0.30/0.47 => ( doubleComplementEq
% 0.30/0.47 => ( complementTnotintersectT
% 0.30/0.47 => ( complementImpComplementIntersect
% 0.30/0.47 => ( complementSubsetComplementIntersect
% 0.30/0.47 => ( complementInPowersetComplementIntersect
% 0.30/0.47 => ( contraSubsetComplement
% 0.30/0.47 => ( complementTcontraSubset
% 0.30/0.47 => ( binunionTILcontra
% 0.30/0.47 => ( binunionTIRcontra
% 0.30/0.47 => ( inIntersectImpInUnion
% 0.30/0.47 => ( inIntersectImpInUnion2
% 0.30/0.47 => ( inIntersectImpInIntersectUnions
% 0.30/0.47 => ( intersectInPowersetIntersectUnions
% 0.30/0.47 => ( inComplementUnionImpNotIn1
% 0.30/0.47 => ( inComplementUnionImpInComplement1
% 0.30/0.47 => ( binunionTE
% 0.30/0.47 => ( binunionTEcontra
% 0.30/0.47 => ( demorgan2a1
% 0.30/0.47 => ( complementUnionInPowersetComplement
% 0.30/0.47 => ( demorgan2a2
% 0.30/0.47 => ( demorgan1a
% 0.30/0.47 => ( demorgan1b
% 0.30/0.47 => ( demorgan1
% 0.30/0.47 => ( demorgan2a
% 0.30/0.47 => ( demorgan2b2
% 0.30/0.47 => ( demorgan2b
% 0.30/0.47 => ( demorgan2
% 0.30/0.47 => ( woz13rule0
% 0.30/0.47 => ( woz13rule1
% 0.30/0.47 => ( woz13rule2
% 0.30/0.47 => ( woz13rule3
% 0.30/0.47 => ( woz13rule4
% 0.30/0.47 => ( woz1_1
% 0.30/0.47 => ( woz1_2
% 0.30/0.47 => ( woz1_3
% 0.30/0.47 => ( woz1_4
% 0.30/0.47 => ( woz1_5
% 0.30/0.47 => ( breln1all2
% 0.30/0.47 => ( breln1SetBreln1
% 0.30/0.47 => ( choice2fnsingleton
% 0.30/0.47 => ( setOfPairsIsBReln1
% 0.30/0.47 => ( breln1all1
% 0.30/0.47 => ( subbreln1
% 0.30/0.47 => ( eqbreln1
% 0.30/0.47 => ( breln1invprop
% 0.30/0.47 => ( breln1invI
% 0.30/0.47 => ( breln1invE
% 0.30/0.47 => ( breln1compprop
% 0.30/0.47 => ( breln1compI
% 0.30/0.47 => ( breln1compE
% 0.30/0.47 => ( breln1compEex
% 0.30/0.47 => ( breln1unionprop
% 0.30/0.47 => ( breln1unionIL
% 0.30/0.47 => ( breln1unionIR
% 0.30/0.47 => ( breln1unionI
% 0.30/0.47 => ( breln1unionE
% 0.30/0.47 => ( breln1unionEcases
% 0.30/0.47 => ( breln1unionCommutes
% 0.30/0.47 => ( woz2Ex
% 0.30/0.47 => ( woz2W
% 0.30/0.47 => ( woz2A
% 0.30/0.47 => ( woz2B
% 0.30/0.47 => ( image1Ex
% 0.30/0.47 => ( image1Ex1
% 0.30/0.47 => ( image1Equiv
% 0.30/0.47 => ( image1E
% 0.30/0.47 => ( image1I
% 0.30/0.47 => ( injFuncInInjFuncSet
% 0.30/0.47 => ( injFuncSetFuncIn
% 0.30/0.47 => ! [Xx: $i,Xy: $i,Xf: $i] :
% 0.30/0.47 ( ( in @ Xf @ ( injFuncSet @ Xx @ Xy ) )
% 0.30/0.47 => ( injective @ Xx @ Xy @ Xf ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 %------------------------------------------------------------------------------
% 0.30/0.47 ------------------------
% 3.02/1.65 ---- Embedded in THF ---
% 3.02/1.65 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 3.02/1.65 %%% Generated on Tue Feb 24 12:34:26 EST 2026
% 3.02/1.65 %%% using '$$dhol' embedding, version 1.3.0.
% 3.02/1.65 %%% Logic specification used:
% 3.02/1.65 %%% thf(spec, logic, $$dhol).
% 3.02/1.65
% 3.02/1.65 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.2k8MKHSw6S/DTF2DTF18170.p
% See solution above
% 3.02/1.65 ------------------------
% 3.02/1.69 ---- Cleaned THF ---
% 3.02/1.69 thf(in_type,type,
% 3.02/1.69 in: $i > $i > $o ).
% 3.02/1.69
% 3.02/1.69 thf(exu_type,type,
% 3.02/1.69 exu: ( $i > $o ) > $o ).
% 3.02/1.69
% 3.02/1.69 thf(exu,definition,
% 3.02/1.69 ( exu
% 3.02/1.69 = ( ^ [Xphi: $i > $o] :
% 3.02/1.69 ? [Xx: $i] :
% 3.02/1.69 ( ( Xphi @ Xx )
% 3.02/1.69 & ! [Xy: $i] :
% 3.02/1.69 ( ( Xphi @ Xy )
% 3.02/1.69 => ( Xx = Xy ) ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(setextAx_type,type,
% 3.02/1.69 setextAx: $o ).
% 3.02/1.69
% 3.02/1.69 thf(setextAx,definition,
% 3.02/1.69 ( setextAx
% 3.02/1.69 = ( ! [A: $i,B: $i] :
% 3.02/1.69 ( ! [Xx: $i] :
% 3.02/1.69 ( ( in @ Xx @ A )
% 3.02/1.69 <=> ( in @ Xx @ B ) )
% 3.02/1.69 => ( A = B ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(emptyset_type,type,
% 3.02/1.69 emptyset: $i ).
% 3.02/1.69
% 3.02/1.69 thf(emptysetAx_type,type,
% 3.02/1.69 emptysetAx: $o ).
% 3.02/1.69
% 3.02/1.69 thf(emptysetAx,definition,
% 3.02/1.69 ( emptysetAx
% 3.02/1.69 = ( ! [Xx: $i] :
% 3.02/1.69 ~ ( in @ Xx @ emptyset ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(setadjoin_type,type,
% 3.02/1.69 setadjoin: $i > $i > $i ).
% 3.02/1.69
% 3.02/1.69 thf(setadjoinAx_type,type,
% 3.02/1.69 setadjoinAx: $o ).
% 3.02/1.69
% 3.02/1.69 thf(setadjoinAx,definition,
% 3.02/1.69 ( setadjoinAx
% 3.02/1.69 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 3.02/1.69 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 3.02/1.69 <=> ( ( Xy = Xx )
% 3.02/1.69 | ( in @ Xy @ A ) ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(powerset_type,type,
% 3.02/1.69 powerset: $i > $i ).
% 3.02/1.69
% 3.02/1.69 thf(powersetAx_type,type,
% 3.02/1.69 powersetAx: $o ).
% 3.02/1.69
% 3.02/1.69 thf(powersetAx,definition,
% 3.02/1.69 ( powersetAx
% 3.02/1.69 = ( ! [A: $i,B: $i] :
% 3.02/1.69 ( ( in @ B @ ( powerset @ A ) )
% 3.02/1.69 <=> ! [Xx: $i] :
% 3.02/1.69 ( ( in @ Xx @ B )
% 3.02/1.69 => ( in @ Xx @ A ) ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(setunion_type,type,
% 3.02/1.69 setunion: $i > $i ).
% 3.02/1.69
% 3.02/1.69 thf(setunionAx_type,type,
% 3.02/1.69 setunionAx: $o ).
% 3.02/1.69
% 3.02/1.69 thf(setunionAx,definition,
% 3.02/1.69 ( setunionAx
% 3.02/1.69 = ( ! [A: $i,Xx: $i] :
% 3.02/1.69 ( ( in @ Xx @ ( setunion @ A ) )
% 3.02/1.69 <=> ? [B: $i] :
% 3.02/1.69 ( ( in @ Xx @ B )
% 3.02/1.69 & ( in @ B @ A ) ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(omega_type,type,
% 3.02/1.69 omega: $i ).
% 3.02/1.69
% 3.02/1.69 thf(omega0Ax_type,type,
% 3.02/1.69 omega0Ax: $o ).
% 3.02/1.69
% 3.02/1.69 thf(omega0Ax,definition,
% 3.02/1.69 ( omega0Ax
% 3.02/1.69 = ( in @ emptyset @ omega ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(omegaSAx_type,type,
% 3.02/1.69 omegaSAx: $o ).
% 3.02/1.69
% 3.02/1.69 thf(omegaSAx,definition,
% 3.02/1.69 ( omegaSAx
% 3.02/1.69 = ( ! [Xx: $i] :
% 3.02/1.69 ( ( in @ Xx @ omega )
% 3.02/1.69 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(omegaIndAx_type,type,
% 3.02/1.69 omegaIndAx: $o ).
% 3.02/1.69
% 3.02/1.69 thf(omegaIndAx,definition,
% 3.02/1.69 ( omegaIndAx
% 3.02/1.69 = ( ! [A: $i] :
% 3.02/1.69 ( ( ( in @ emptyset @ A )
% 3.02/1.69 & ! [Xx: $i] :
% 3.02/1.69 ( ( ( in @ Xx @ omega )
% 3.02/1.69 & ( in @ Xx @ A ) )
% 3.02/1.69 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 3.02/1.69 => ! [Xx: $i] :
% 3.02/1.69 ( ( in @ Xx @ omega )
% 3.02/1.69 => ( in @ Xx @ A ) ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(replAx_type,type,
% 3.02/1.69 replAx: $o ).
% 3.02/1.69
% 3.02/1.69 thf(replAx,definition,
% 3.02/1.69 ( replAx
% 3.02/1.69 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 3.02/1.69 ( ! [Xx: $i] :
% 3.02/1.69 ( ( in @ Xx @ A )
% 3.02/1.69 => ( exu
% 3.02/1.69 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 3.02/1.69 => ? [B: $i] :
% 3.02/1.69 ! [Xx: $i] :
% 3.02/1.69 ( ( in @ Xx @ B )
% 3.02/1.69 <=> ? [Xy: $i] :
% 3.02/1.69 ( ( in @ Xy @ A )
% 3.02/1.69 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(foundationAx_type,type,
% 3.02/1.69 foundationAx: $o ).
% 3.02/1.69
% 3.02/1.69 thf(foundationAx,definition,
% 3.02/1.69 ( foundationAx
% 3.02/1.69 = ( ! [A: $i] :
% 3.02/1.69 ( ? [Xx: $i] : ( in @ Xx @ A )
% 3.02/1.69 => ? [B: $i] :
% 3.02/1.69 ( ( in @ B @ A )
% 3.02/1.69 & ~ ? [Xx: $i] :
% 3.02/1.69 ( ( in @ Xx @ B )
% 3.02/1.69 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(wellorderingAx_type,type,
% 3.02/1.69 wellorderingAx: $o ).
% 3.02/1.69
% 3.02/1.69 thf(wellorderingAx,definition,
% 3.02/1.69 ( wellorderingAx
% 3.02/1.69 = ( ! [A: $i] :
% 3.02/1.69 ? [B: $i] :
% 3.02/1.69 ( ! [C: $i] :
% 3.02/1.69 ( ( in @ C @ B )
% 3.02/1.69 => ! [Xx: $i] :
% 3.02/1.69 ( ( in @ Xx @ C )
% 3.02/1.69 => ( in @ Xx @ A ) ) )
% 3.02/1.69 & ! [Xx: $i,Xy: $i] :
% 3.02/1.69 ( ( ( in @ Xx @ A )
% 3.02/1.69 & ( in @ Xy @ A ) )
% 3.02/1.69 => ( ! [C: $i] :
% 3.02/1.69 ( ( in @ C @ B )
% 3.02/1.69 => ( ( in @ Xx @ C )
% 3.02/1.69 <=> ( in @ Xy @ C ) ) )
% 3.02/1.69 => ( Xx = Xy ) ) )
% 3.02/1.69 & ! [C: $i,D: $i] :
% 3.02/1.69 ( ( ( in @ C @ B )
% 3.02/1.69 & ( in @ D @ B ) )
% 3.02/1.69 => ( ! [Xx: $i] :
% 3.02/1.69 ( ( in @ Xx @ C )
% 3.02/1.69 => ( in @ Xx @ D ) )
% 3.02/1.69 | ! [Xx: $i] :
% 3.02/1.69 ( ( in @ Xx @ D )
% 3.02/1.69 => ( in @ Xx @ C ) ) ) )
% 3.02/1.69 & ! [C: $i] :
% 3.02/1.69 ( ( ! [Xx: $i] :
% 3.02/1.69 ( ( in @ Xx @ C )
% 3.02/1.69 => ( in @ Xx @ A ) )
% 3.02/1.69 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 3.02/1.69 => ? [D: $i,Xx: $i] :
% 3.02/1.69 ( ( in @ D @ B )
% 3.02/1.69 & ( in @ Xx @ C )
% 3.02/1.69 & ~ ? [Xy: $i] :
% 3.02/1.69 ( ( in @ Xy @ D )
% 3.02/1.69 & ( in @ Xy @ C ) )
% 3.02/1.69 & ! [E: $i] :
% 3.02/1.69 ( ( in @ E @ B )
% 3.02/1.69 => ( ! [Xy: $i] :
% 3.02/1.69 ( ( in @ Xy @ E )
% 3.02/1.69 => ( in @ Xy @ D ) )
% 3.02/1.69 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(descr_type,type,
% 3.02/1.69 descr: ( $i > $o ) > $i ).
% 3.02/1.69
% 3.02/1.69 thf(descrp_type,type,
% 3.02/1.69 descrp: $o ).
% 3.02/1.69
% 3.02/1.69 thf(descrp,definition,
% 3.02/1.69 ( descrp
% 3.02/1.69 = ( ! [Xphi: $i > $o] :
% 3.02/1.69 ( ( exu
% 3.02/1.69 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.02/1.69 => ( Xphi
% 3.02/1.69 @ ( descr
% 3.02/1.69 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(dsetconstr_type,type,
% 3.02/1.69 dsetconstr: $i > ( $i > $o ) > $i ).
% 3.02/1.69
% 3.02/1.69 thf(dsetconstrI_type,type,
% 3.02/1.69 dsetconstrI: $o ).
% 3.02/1.69
% 3.02/1.69 thf(dsetconstrI,definition,
% 3.02/1.69 ( dsetconstrI
% 3.02/1.69 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.02/1.69 ( ( in @ Xx @ A )
% 3.02/1.69 => ( ( Xphi @ Xx )
% 3.02/1.69 => ( in @ Xx
% 3.02/1.69 @ ( dsetconstr @ A
% 3.02/1.69 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(dsetconstrEL_type,type,
% 3.02/1.69 dsetconstrEL: $o ).
% 3.02/1.69
% 3.02/1.69 thf(dsetconstrEL,definition,
% 3.02/1.69 ( dsetconstrEL
% 3.02/1.69 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.02/1.69 ( ( in @ Xx
% 3.02/1.69 @ ( dsetconstr @ A
% 3.02/1.69 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 3.02/1.69 => ( in @ Xx @ A ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(dsetconstrER_type,type,
% 3.02/1.69 dsetconstrER: $o ).
% 3.02/1.69
% 3.02/1.69 thf(dsetconstrER,definition,
% 3.02/1.69 ( dsetconstrER
% 3.02/1.69 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.02/1.69 ( ( in @ Xx
% 3.02/1.69 @ ( dsetconstr @ A
% 3.02/1.69 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 3.02/1.69 => ( Xphi @ Xx ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(exuE1_type,type,
% 3.02/1.69 exuE1: $o ).
% 3.02/1.69
% 3.02/1.69 thf(exuE1,definition,
% 3.02/1.69 ( exuE1
% 3.02/1.69 = ( ! [Xphi: $i > $o] :
% 3.02/1.69 ( ( exu
% 3.02/1.69 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.02/1.69 => ? [Xx: $i] :
% 3.02/1.69 ( ( Xphi @ Xx )
% 3.02/1.69 & ! [Xy: $i] :
% 3.02/1.69 ( ( Xphi @ Xy )
% 3.02/1.69 => ( Xx = Xy ) ) ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(prop2set_type,type,
% 3.02/1.69 prop2set: $o > $i ).
% 3.02/1.69
% 3.02/1.69 thf(prop2setE_type,type,
% 3.02/1.69 prop2setE: $o ).
% 3.02/1.69
% 3.02/1.69 thf(prop2setE,definition,
% 3.02/1.69 ( prop2setE
% 3.02/1.69 = ( ! [Xphi: $o,Xx: $i] :
% 3.02/1.69 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 3.02/1.69 => Xphi ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(emptysetE_type,type,
% 3.02/1.69 emptysetE: $o ).
% 3.02/1.69
% 3.02/1.69 thf(emptysetE,definition,
% 3.02/1.69 ( emptysetE
% 3.02/1.69 = ( ! [Xx: $i] :
% 3.02/1.69 ( ( in @ Xx @ emptyset )
% 3.02/1.69 => ! [Xphi: $o] : Xphi ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(emptysetimpfalse_type,type,
% 3.02/1.69 emptysetimpfalse: $o ).
% 3.02/1.69
% 3.02/1.69 thf(emptysetimpfalse,definition,
% 3.02/1.69 ( emptysetimpfalse
% 3.02/1.69 = ( ! [Xx: $i] :
% 3.02/1.69 ( ( in @ Xx @ emptyset )
% 3.02/1.69 => $false ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(notinemptyset_type,type,
% 3.02/1.69 notinemptyset: $o ).
% 3.02/1.69
% 3.02/1.69 thf(notinemptyset,definition,
% 3.02/1.69 ( notinemptyset
% 3.02/1.69 = ( ! [Xx: $i] :
% 3.02/1.69 ~ ( in @ Xx @ emptyset ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(exuE3e_type,type,
% 3.02/1.69 exuE3e: $o ).
% 3.02/1.69
% 3.02/1.69 thf(exuE3e,definition,
% 3.02/1.69 ( exuE3e
% 3.02/1.69 = ( ! [Xphi: $i > $o] :
% 3.02/1.69 ( ( exu
% 3.02/1.69 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.02/1.69 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(setext_type,type,
% 3.02/1.69 setext: $o ).
% 3.02/1.69
% 3.02/1.69 thf(setext,definition,
% 3.02/1.69 ( setext
% 3.02/1.69 = ( ! [A: $i,B: $i] :
% 3.02/1.69 ( ! [Xx: $i] :
% 3.02/1.69 ( ( in @ Xx @ A )
% 3.02/1.69 => ( in @ Xx @ B ) )
% 3.02/1.69 => ( ! [Xx: $i] :
% 3.02/1.69 ( ( in @ Xx @ B )
% 3.02/1.69 => ( in @ Xx @ A ) )
% 3.02/1.69 => ( A = B ) ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(emptyI_type,type,
% 3.02/1.69 emptyI: $o ).
% 3.02/1.69
% 3.02/1.69 thf(emptyI,definition,
% 3.02/1.69 ( emptyI
% 3.02/1.69 = ( ! [A: $i] :
% 3.02/1.69 ( ! [Xx: $i] :
% 3.02/1.69 ~ ( in @ Xx @ A )
% 3.02/1.69 => ( A = emptyset ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(noeltsimpempty_type,type,
% 3.02/1.69 noeltsimpempty: $o ).
% 3.02/1.69
% 3.02/1.69 thf(noeltsimpempty,definition,
% 3.02/1.69 ( noeltsimpempty
% 3.02/1.69 = ( ! [A: $i] :
% 3.02/1.69 ( ! [Xx: $i] :
% 3.02/1.69 ~ ( in @ Xx @ A )
% 3.02/1.69 => ( A = emptyset ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(setbeta_type,type,
% 3.02/1.69 setbeta: $o ).
% 3.02/1.69
% 3.02/1.69 thf(setbeta,definition,
% 3.02/1.69 ( setbeta
% 3.02/1.69 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.02/1.69 ( ( in @ Xx @ A )
% 3.02/1.69 => ( ( in @ Xx
% 3.02/1.69 @ ( dsetconstr @ A
% 3.02/1.69 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 3.02/1.69 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(nonempty_type,type,
% 3.02/1.69 nonempty: $i > $o ).
% 3.02/1.69
% 3.02/1.69 thf(nonempty,definition,
% 3.02/1.69 ( nonempty
% 3.02/1.69 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(nonemptyE1_type,type,
% 3.02/1.69 nonemptyE1: $o ).
% 3.02/1.69
% 3.02/1.69 thf(nonemptyE1,definition,
% 3.02/1.69 ( nonemptyE1
% 3.02/1.69 = ( ! [A: $i] :
% 3.02/1.69 ( ( nonempty @ A )
% 3.02/1.69 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(nonemptyI_type,type,
% 3.02/1.69 nonemptyI: $o ).
% 3.02/1.69
% 3.02/1.69 thf(nonemptyI,definition,
% 3.02/1.69 ( nonemptyI
% 3.02/1.69 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.02/1.69 ( ( in @ Xx @ A )
% 3.02/1.69 => ( ( Xphi @ Xx )
% 3.02/1.69 => ( nonempty
% 3.02/1.69 @ ( dsetconstr @ A
% 3.02/1.69 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(nonemptyI1_type,type,
% 3.02/1.69 nonemptyI1: $o ).
% 3.02/1.69
% 3.02/1.69 thf(nonemptyI1,definition,
% 3.02/1.69 ( nonemptyI1
% 3.02/1.69 = ( ! [A: $i] :
% 3.02/1.69 ( ? [Xx: $i] : ( in @ Xx @ A )
% 3.02/1.69 => ( nonempty @ A ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(setadjoinIL_type,type,
% 3.02/1.69 setadjoinIL: $o ).
% 3.02/1.69
% 3.02/1.69 thf(setadjoinIL,definition,
% 3.02/1.69 ( setadjoinIL
% 3.02/1.69 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(emptyinunitempty_type,type,
% 3.02/1.69 emptyinunitempty: $o ).
% 3.02/1.69
% 3.02/1.69 thf(emptyinunitempty,definition,
% 3.02/1.69 ( emptyinunitempty
% 3.02/1.69 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(setadjoinIR_type,type,
% 3.02/1.69 setadjoinIR: $o ).
% 3.02/1.69
% 3.02/1.69 thf(setadjoinIR,definition,
% 3.02/1.69 ( setadjoinIR
% 3.02/1.69 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 3.02/1.69 ( ( in @ Xy @ A )
% 3.02/1.69 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(setadjoinE_type,type,
% 3.02/1.69 setadjoinE: $o ).
% 3.02/1.69
% 3.02/1.69 thf(setadjoinE,definition,
% 3.02/1.69 ( setadjoinE
% 3.02/1.69 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 3.02/1.69 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 3.02/1.69 => ! [Xphi: $o] :
% 3.02/1.69 ( ( ( Xy = Xx )
% 3.02/1.69 => Xphi )
% 3.02/1.69 => ( ( ( in @ Xy @ A )
% 3.02/1.69 => Xphi )
% 3.02/1.69 => Xphi ) ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(setadjoinOr_type,type,
% 3.02/1.69 setadjoinOr: $o ).
% 3.02/1.69
% 3.02/1.69 thf(setadjoinOr,definition,
% 3.02/1.69 ( setadjoinOr
% 3.02/1.69 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 3.02/1.69 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 3.02/1.69 => ( ( Xy = Xx )
% 3.02/1.69 | ( in @ Xy @ A ) ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(setoftrueEq_type,type,
% 3.02/1.69 setoftrueEq: $o ).
% 3.02/1.69
% 3.02/1.69 thf(setoftrueEq,definition,
% 3.02/1.69 ( setoftrueEq
% 3.02/1.69 = ( ! [A: $i] :
% 3.02/1.69 ( ( dsetconstr @ A
% 3.02/1.69 @ ^ [Xx: $i] : $true )
% 3.02/1.69 = A ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(powersetI_type,type,
% 3.02/1.69 powersetI: $o ).
% 3.02/1.69
% 3.02/1.69 thf(powersetI,definition,
% 3.02/1.69 ( powersetI
% 3.02/1.69 = ( ! [A: $i,B: $i] :
% 3.02/1.69 ( ! [Xx: $i] :
% 3.02/1.69 ( ( in @ Xx @ B )
% 3.02/1.69 => ( in @ Xx @ A ) )
% 3.02/1.69 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(emptyinPowerset_type,type,
% 3.02/1.69 emptyinPowerset: $o ).
% 3.02/1.69
% 3.02/1.69 thf(emptyinPowerset,definition,
% 3.02/1.69 ( emptyinPowerset
% 3.02/1.69 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(emptyInPowerset_type,type,
% 3.02/1.69 emptyInPowerset: $o ).
% 3.02/1.69
% 3.02/1.69 thf(emptyInPowerset,definition,
% 3.02/1.69 ( emptyInPowerset
% 3.02/1.69 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(powersetE_type,type,
% 3.02/1.69 powersetE: $o ).
% 3.02/1.69
% 3.02/1.69 thf(powersetE,definition,
% 3.02/1.69 ( powersetE
% 3.02/1.69 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.02/1.69 ( ( in @ B @ ( powerset @ A ) )
% 3.02/1.69 => ( ( in @ Xx @ B )
% 3.02/1.69 => ( in @ Xx @ A ) ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(setunionI_type,type,
% 3.02/1.69 setunionI: $o ).
% 3.02/1.69
% 3.02/1.69 thf(setunionI,definition,
% 3.02/1.69 ( setunionI
% 3.02/1.69 = ( ! [A: $i,Xx: $i,B: $i] :
% 3.02/1.69 ( ( in @ Xx @ B )
% 3.02/1.69 => ( ( in @ B @ A )
% 3.02/1.69 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(setunionE_type,type,
% 3.02/1.69 setunionE: $o ).
% 3.02/1.69
% 3.02/1.69 thf(setunionE,definition,
% 3.02/1.69 ( setunionE
% 3.02/1.69 = ( ! [A: $i,Xx: $i] :
% 3.02/1.69 ( ( in @ Xx @ ( setunion @ A ) )
% 3.02/1.69 => ! [Xphi: $o] :
% 3.02/1.69 ( ! [B: $i] :
% 3.02/1.69 ( ( in @ Xx @ B )
% 3.02/1.69 => ( ( in @ B @ A )
% 3.02/1.69 => Xphi ) )
% 3.02/1.69 => Xphi ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(subPowSU_type,type,
% 3.02/1.69 subPowSU: $o ).
% 3.02/1.69
% 3.02/1.69 thf(subPowSU,definition,
% 3.02/1.69 ( subPowSU
% 3.02/1.69 = ( ! [A: $i,Xx: $i] :
% 3.02/1.69 ( ( in @ Xx @ A )
% 3.02/1.69 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(exuE2_type,type,
% 3.02/1.69 exuE2: $o ).
% 3.02/1.69
% 3.02/1.69 thf(exuE2,definition,
% 3.02/1.69 ( exuE2
% 3.02/1.69 = ( ! [Xphi: $i > $o] :
% 3.02/1.69 ( ( exu
% 3.02/1.69 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.02/1.69 => ? [Xx: $i] :
% 3.02/1.69 ! [Xy: $i] :
% 3.02/1.69 ( ( Xphi @ Xy )
% 3.02/1.69 <=> ( Xy = Xx ) ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(nonemptyImpWitness_type,type,
% 3.02/1.69 nonemptyImpWitness: $o ).
% 3.02/1.69
% 3.02/1.69 thf(nonemptyImpWitness,definition,
% 3.02/1.69 ( nonemptyImpWitness
% 3.02/1.69 = ( ! [A: $i] :
% 3.02/1.69 ( ( nonempty @ A )
% 3.02/1.69 => ? [Xx: $i] :
% 3.02/1.69 ( ( in @ Xx @ A )
% 3.02/1.69 & $true ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(uniqinunit_type,type,
% 3.02/1.69 uniqinunit: $o ).
% 3.02/1.69
% 3.02/1.69 thf(uniqinunit,definition,
% 3.02/1.69 ( uniqinunit
% 3.02/1.69 = ( ! [Xx: $i,Xy: $i] :
% 3.02/1.69 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 3.02/1.69 => ( Xx = Xy ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(notinsingleton_type,type,
% 3.02/1.69 notinsingleton: $o ).
% 3.02/1.69
% 3.02/1.69 thf(notinsingleton,definition,
% 3.02/1.69 ( notinsingleton
% 3.02/1.69 = ( ! [Xx: $i,Xy: $i] :
% 3.02/1.69 ( ( Xx != Xy )
% 3.02/1.69 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(eqinunit_type,type,
% 3.02/1.69 eqinunit: $o ).
% 3.02/1.69
% 3.02/1.69 thf(eqinunit,definition,
% 3.02/1.69 ( eqinunit
% 3.02/1.69 = ( ! [Xx: $i,Xy: $i] :
% 3.02/1.69 ( ( Xx = Xy )
% 3.02/1.69 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(singletonsswitch_type,type,
% 3.02/1.69 singletonsswitch: $o ).
% 3.02/1.69
% 3.02/1.69 thf(singletonsswitch,definition,
% 3.02/1.69 ( singletonsswitch
% 3.02/1.69 = ( ! [Xx: $i,Xy: $i] :
% 3.02/1.69 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 3.02/1.69 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(upairsetE_type,type,
% 3.02/1.69 upairsetE: $o ).
% 3.02/1.69
% 3.02/1.69 thf(upairsetE,definition,
% 3.02/1.69 ( upairsetE
% 3.02/1.69 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 3.02/1.69 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 3.02/1.69 => ( ( Xz = Xx )
% 3.02/1.69 | ( Xz = Xy ) ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(upairsetIL_type,type,
% 3.02/1.69 upairsetIL: $o ).
% 3.02/1.69
% 3.02/1.69 thf(upairsetIL,definition,
% 3.02/1.69 ( upairsetIL
% 3.02/1.69 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(upairsetIR_type,type,
% 3.02/1.69 upairsetIR: $o ).
% 3.02/1.69
% 3.02/1.69 thf(upairsetIR,definition,
% 3.02/1.69 ( upairsetIR
% 3.02/1.69 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(emptyE1_type,type,
% 3.02/1.69 emptyE1: $o ).
% 3.02/1.69
% 3.02/1.69 thf(emptyE1,definition,
% 3.02/1.69 ( emptyE1
% 3.02/1.69 = ( ! [A: $i,Xphi: $i > $o] :
% 3.02/1.69 ( ? [Xx: $i] :
% 3.02/1.69 ( ( in @ Xx @ A )
% 3.02/1.69 & ( Xphi @ Xx ) )
% 3.02/1.69 => ( ( ( dsetconstr @ A
% 3.02/1.69 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.02/1.69 = emptyset )
% 3.02/1.69 => $false ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(vacuousDall_type,type,
% 3.02/1.69 vacuousDall: $o ).
% 3.02/1.69
% 3.02/1.69 thf(vacuousDall,definition,
% 3.02/1.69 ( vacuousDall
% 3.02/1.69 = ( ! [Xphi: $i > $o,Xx: $i] :
% 3.02/1.69 ( ( in @ Xx @ emptyset )
% 3.02/1.69 => ( Xphi @ Xx ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(quantDeMorgan1_type,type,
% 3.02/1.69 quantDeMorgan1: $o ).
% 3.02/1.69
% 3.02/1.69 thf(quantDeMorgan1,definition,
% 3.02/1.69 ( quantDeMorgan1
% 3.02/1.69 = ( ! [A: $i,Xphi: $i > $o] :
% 3.02/1.69 ( ~ ! [Xx: $i] :
% 3.02/1.69 ( ( in @ Xx @ A )
% 3.02/1.69 => ( Xphi @ Xx ) )
% 3.02/1.69 => ? [Xx: $i] :
% 3.02/1.69 ( ( in @ Xx @ A )
% 3.02/1.69 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(quantDeMorgan2_type,type,
% 3.02/1.69 quantDeMorgan2: $o ).
% 3.02/1.69
% 3.02/1.69 thf(quantDeMorgan2,definition,
% 3.02/1.69 ( quantDeMorgan2
% 3.02/1.69 = ( ! [A: $i,Xphi: $i > $o] :
% 3.02/1.69 ( ! [Xx: $i] :
% 3.02/1.69 ( ( in @ Xx @ A )
% 3.02/1.69 => ~ ( Xphi @ Xx ) )
% 3.02/1.69 => ~ ? [Xx: $i] :
% 3.02/1.69 ( ( in @ Xx @ A )
% 3.02/1.69 & ( Xphi @ Xx ) ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(quantDeMorgan3_type,type,
% 3.02/1.69 quantDeMorgan3: $o ).
% 3.02/1.69
% 3.02/1.69 thf(quantDeMorgan3,definition,
% 3.02/1.69 ( quantDeMorgan3
% 3.02/1.69 = ( ! [A: $i,Xphi: $i > $o] :
% 3.02/1.69 ( ~ ? [Xx: $i] :
% 3.02/1.69 ( ( in @ Xx @ A )
% 3.02/1.69 & ( Xphi @ Xx ) )
% 3.02/1.69 => ! [Xx: $i] :
% 3.02/1.69 ( ( in @ Xx @ A )
% 3.02/1.69 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(quantDeMorgan4_type,type,
% 3.02/1.69 quantDeMorgan4: $o ).
% 3.02/1.69
% 3.02/1.69 thf(quantDeMorgan4,definition,
% 3.02/1.69 ( quantDeMorgan4
% 3.02/1.69 = ( ! [A: $i,Xphi: $i > $o] :
% 3.02/1.69 ( ? [Xx: $i] :
% 3.02/1.69 ( ( in @ Xx @ A )
% 3.02/1.69 & ~ ( Xphi @ Xx ) )
% 3.02/1.69 => ~ ! [Xx: $i] :
% 3.02/1.69 ( ( in @ Xx @ A )
% 3.02/1.69 => ( Xphi @ Xx ) ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(prop2setI_type,type,
% 3.02/1.69 prop2setI: $o ).
% 3.02/1.69
% 3.02/1.69 thf(prop2setI,definition,
% 3.02/1.69 ( prop2setI
% 3.02/1.69 = ( ! [Xphi: $o] :
% 3.02/1.69 ( Xphi
% 3.02/1.69 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(set2prop_type,type,
% 3.02/1.69 set2prop: $i > $o ).
% 3.02/1.69
% 3.02/1.69 thf(prop2set2propI_type,type,
% 3.02/1.69 prop2set2propI: $o ).
% 3.02/1.69
% 3.02/1.69 thf(prop2set2propI,definition,
% 3.02/1.69 ( prop2set2propI
% 3.02/1.69 = ( ! [Xphi: $o] :
% 3.02/1.69 ( Xphi
% 3.02/1.69 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(notdexE_type,type,
% 3.02/1.69 notdexE: $o ).
% 3.02/1.69
% 3.02/1.69 thf(notdexE,definition,
% 3.02/1.69 ( notdexE
% 3.02/1.69 = ( ! [A: $i,Xphi: $i > $o] :
% 3.02/1.69 ( ~ ? [Xx: $i] :
% 3.02/1.69 ( ( in @ Xx @ A )
% 3.02/1.69 & ( Xphi @ Xx ) )
% 3.02/1.69 => ! [Xx: $i] :
% 3.02/1.69 ( ( in @ Xx @ A )
% 3.02/1.69 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(notdallE_type,type,
% 3.02/1.69 notdallE: $o ).
% 3.02/1.69
% 3.02/1.69 thf(notdallE,definition,
% 3.02/1.69 ( notdallE
% 3.02/1.69 = ( ! [A: $i,Xphi: $i > $o] :
% 3.02/1.69 ( ~ ! [Xx: $i] :
% 3.02/1.69 ( ( in @ Xx @ A )
% 3.02/1.69 => ( Xphi @ Xx ) )
% 3.02/1.69 => ? [Xx: $i] :
% 3.02/1.69 ( ( in @ Xx @ A )
% 3.02/1.69 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(exuI1_type,type,
% 3.02/1.69 exuI1: $o ).
% 3.02/1.69
% 3.02/1.69 thf(exuI1,definition,
% 3.02/1.69 ( exuI1
% 3.02/1.69 = ( ! [Xphi: $i > $o] :
% 3.02/1.69 ( ? [Xx: $i] :
% 3.02/1.69 ( ( Xphi @ Xx )
% 3.02/1.69 & ! [Xy: $i] :
% 3.02/1.69 ( ( Xphi @ Xy )
% 3.02/1.69 => ( Xx = Xy ) ) )
% 3.02/1.69 => ( exu
% 3.02/1.69 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(exuI3_type,type,
% 3.02/1.69 exuI3: $o ).
% 3.02/1.69
% 3.02/1.69 thf(exuI3,definition,
% 3.02/1.69 ( exuI3
% 3.02/1.69 = ( ! [Xphi: $i > $o] :
% 3.02/1.69 ( ? [Xx: $i] : ( Xphi @ Xx )
% 3.02/1.69 => ( ! [Xx: $i,Xy: $i] :
% 3.02/1.69 ( ( Xphi @ Xx )
% 3.02/1.69 => ( ( Xphi @ Xy )
% 3.02/1.69 => ( Xx = Xy ) ) )
% 3.02/1.69 => ( exu
% 3.02/1.69 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(exuI2_type,type,
% 3.02/1.69 exuI2: $o ).
% 3.02/1.69
% 3.02/1.69 thf(exuI2,definition,
% 3.02/1.69 ( exuI2
% 3.02/1.69 = ( ! [Xphi: $i > $o] :
% 3.02/1.69 ( ? [Xx: $i] :
% 3.02/1.69 ! [Xy: $i] :
% 3.02/1.69 ( ( Xphi @ Xy )
% 3.02/1.69 <=> ( Xy = Xx ) )
% 3.02/1.69 => ( exu
% 3.02/1.69 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(inCongP_type,type,
% 3.02/1.69 inCongP: $o ).
% 3.02/1.69
% 3.02/1.69 thf(inCongP,definition,
% 3.02/1.69 ( inCongP
% 3.02/1.69 = ( ! [A: $i,B: $i] :
% 3.02/1.69 ( ( A = B )
% 3.02/1.69 => ! [Xx: $i,Xy: $i] :
% 3.02/1.69 ( ( Xx = Xy )
% 3.02/1.69 => ( ( in @ Xx @ A )
% 3.02/1.69 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(in__Cong_type,type,
% 3.02/1.69 in__Cong: $o ).
% 3.02/1.69
% 3.02/1.69 thf(in__Cong,definition,
% 3.02/1.69 ( in__Cong
% 3.02/1.69 = ( ! [A: $i,B: $i] :
% 3.02/1.69 ( ( A = B )
% 3.02/1.69 => ! [Xx: $i,Xy: $i] :
% 3.02/1.69 ( ( Xx = Xy )
% 3.02/1.69 => ( ( in @ Xx @ A )
% 3.02/1.69 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(exuE3u_type,type,
% 3.02/1.69 exuE3u: $o ).
% 3.02/1.69
% 3.02/1.69 thf(exuE3u,definition,
% 3.02/1.69 ( exuE3u
% 3.02/1.69 = ( ! [Xphi: $i > $o] :
% 3.02/1.69 ( ( exu
% 3.02/1.69 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.02/1.69 => ! [Xx: $i,Xy: $i] :
% 3.02/1.69 ( ( Xphi @ Xx )
% 3.02/1.69 => ( ( Xphi @ Xy )
% 3.02/1.69 => ( Xx = Xy ) ) ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(exu__Cong_type,type,
% 3.02/1.69 exu__Cong: $o ).
% 3.02/1.69
% 3.02/1.69 thf(exu__Cong,definition,
% 3.02/1.69 ( exu__Cong
% 3.02/1.69 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 3.02/1.69 ( ! [Xx: $i,Xy: $i] :
% 3.02/1.69 ( ( Xx = Xy )
% 3.02/1.69 => ( ( Xphi @ Xx )
% 3.02/1.69 <=> ( Xpsi @ Xy ) ) )
% 3.02/1.69 => ( ( exu
% 3.02/1.69 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.02/1.69 <=> ( exu
% 3.02/1.69 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(emptyset__Cong_type,type,
% 3.02/1.69 emptyset__Cong: $o ).
% 3.02/1.69
% 3.02/1.69 thf(emptyset__Cong,definition,
% 3.02/1.69 ( emptyset__Cong
% 3.02/1.69 = ( emptyset = emptyset ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(setadjoin__Cong_type,type,
% 3.02/1.69 setadjoin__Cong: $o ).
% 3.02/1.69
% 3.02/1.69 thf(setadjoin__Cong,definition,
% 3.02/1.69 ( setadjoin__Cong
% 3.02/1.69 = ( ! [Xx: $i,Xy: $i] :
% 3.02/1.69 ( ( Xx = Xy )
% 3.02/1.69 => ! [Xz: $i,Xu: $i] :
% 3.02/1.69 ( ( Xz = Xu )
% 3.02/1.69 => ( ( setadjoin @ Xx @ Xz )
% 3.02/1.69 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(powerset__Cong_type,type,
% 3.02/1.69 powerset__Cong: $o ).
% 3.02/1.69
% 3.02/1.69 thf(powerset__Cong,definition,
% 3.02/1.69 ( powerset__Cong
% 3.02/1.69 = ( ! [A: $i,B: $i] :
% 3.02/1.69 ( ( A = B )
% 3.02/1.69 => ( ( powerset @ A )
% 3.02/1.69 = ( powerset @ B ) ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(setunion__Cong_type,type,
% 3.02/1.69 setunion__Cong: $o ).
% 3.02/1.69
% 3.02/1.69 thf(setunion__Cong,definition,
% 3.02/1.69 ( setunion__Cong
% 3.02/1.69 = ( ! [A: $i,B: $i] :
% 3.02/1.69 ( ( A = B )
% 3.02/1.69 => ( ( setunion @ A )
% 3.02/1.69 = ( setunion @ B ) ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(omega__Cong_type,type,
% 3.02/1.69 omega__Cong: $o ).
% 3.02/1.69
% 3.02/1.69 thf(omega__Cong,definition,
% 3.02/1.69 ( omega__Cong
% 3.02/1.69 = ( omega = omega ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(exuEu_type,type,
% 3.02/1.69 exuEu: $o ).
% 3.02/1.69
% 3.02/1.69 thf(exuEu,definition,
% 3.02/1.69 ( exuEu
% 3.02/1.69 = ( ! [Xphi: $i > $o] :
% 3.02/1.69 ( ( exu
% 3.02/1.69 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.02/1.69 => ! [Xx: $i,Xy: $i] :
% 3.02/1.69 ( ( Xphi @ Xx )
% 3.02/1.69 => ( ( Xphi @ Xy )
% 3.02/1.69 => ( Xx = Xy ) ) ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(descr__Cong_type,type,
% 3.02/1.69 descr__Cong: $o ).
% 3.02/1.69
% 3.02/1.69 thf(descr__Cong,definition,
% 3.02/1.69 ( descr__Cong
% 3.02/1.69 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 3.02/1.69 ( ! [Xx: $i,Xy: $i] :
% 3.02/1.69 ( ( Xx = Xy )
% 3.02/1.69 => ( ( Xphi @ Xx )
% 3.02/1.69 <=> ( Xpsi @ Xy ) ) )
% 3.02/1.69 => ( ( exu
% 3.02/1.69 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.02/1.69 => ( ( exu
% 3.02/1.69 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 3.02/1.69 => ( ( descr
% 3.02/1.69 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.02/1.69 = ( descr
% 3.02/1.69 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(dsetconstr__Cong_type,type,
% 3.02/1.69 dsetconstr__Cong: $o ).
% 3.02/1.69
% 3.02/1.69 thf(dsetconstr__Cong,definition,
% 3.02/1.69 ( dsetconstr__Cong
% 3.02/1.69 = ( ! [A: $i,B: $i] :
% 3.02/1.69 ( ( A = B )
% 3.02/1.69 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 3.02/1.69 ( ! [Xx: $i] :
% 3.02/1.69 ( ( in @ Xx @ A )
% 3.02/1.69 => ! [Xy: $i] :
% 3.02/1.69 ( ( in @ Xy @ B )
% 3.02/1.69 => ( ( Xx = Xy )
% 3.02/1.69 => ( ( Xphi @ Xx )
% 3.02/1.69 <=> ( Xpsi @ Xy ) ) ) ) )
% 3.02/1.69 => ( ( dsetconstr @ A
% 3.02/1.69 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.02/1.69 = ( dsetconstr @ B
% 3.02/1.69 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(subset_type,type,
% 3.02/1.69 subset: $i > $i > $o ).
% 3.02/1.69
% 3.02/1.69 thf(disjoint_type,type,
% 3.02/1.69 disjoint: $i > $i > $o ).
% 3.02/1.69
% 3.02/1.69 thf(setsmeet_type,type,
% 3.02/1.69 setsmeet: $i > $i > $o ).
% 3.02/1.69
% 3.02/1.69 thf(subsetI1_type,type,
% 3.02/1.69 subsetI1: $o ).
% 3.02/1.69
% 3.02/1.69 thf(subsetI1,definition,
% 3.02/1.69 ( subsetI1
% 3.02/1.69 = ( ! [A: $i,B: $i] :
% 3.02/1.69 ( ! [Xx: $i] :
% 3.02/1.69 ( ( in @ Xx @ A )
% 3.02/1.69 => ( in @ Xx @ B ) )
% 3.02/1.69 => ( subset @ A @ B ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(eqimpsubset2_type,type,
% 3.02/1.69 eqimpsubset2: $o ).
% 3.02/1.69
% 3.02/1.69 thf(eqimpsubset2,definition,
% 3.02/1.69 ( eqimpsubset2
% 3.02/1.69 = ( ! [A: $i,B: $i] :
% 3.02/1.69 ( ( A = B )
% 3.02/1.69 => ( subset @ B @ A ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(eqimpsubset1_type,type,
% 3.02/1.69 eqimpsubset1: $o ).
% 3.02/1.69
% 3.02/1.69 thf(eqimpsubset1,definition,
% 3.02/1.69 ( eqimpsubset1
% 3.02/1.69 = ( ! [A: $i,B: $i] :
% 3.02/1.69 ( ( A = B )
% 3.02/1.69 => ( subset @ A @ B ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(subsetI2_type,type,
% 3.02/1.69 subsetI2: $o ).
% 3.02/1.69
% 3.02/1.69 thf(subsetI2,definition,
% 3.02/1.69 ( subsetI2
% 3.02/1.69 = ( ! [A: $i,B: $i] :
% 3.02/1.69 ( ! [Xx: $i] :
% 3.02/1.69 ( ( in @ Xx @ A )
% 3.02/1.69 => ( in @ Xx @ B ) )
% 3.02/1.69 => ( subset @ A @ B ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(emptysetsubset_type,type,
% 3.02/1.69 emptysetsubset: $o ).
% 3.02/1.69
% 3.02/1.69 thf(emptysetsubset,definition,
% 3.02/1.69 ( emptysetsubset
% 3.02/1.69 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(subsetE_type,type,
% 3.02/1.69 subsetE: $o ).
% 3.02/1.69
% 3.02/1.69 thf(subsetE,definition,
% 3.02/1.69 ( subsetE
% 3.02/1.69 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.02/1.69 ( ( subset @ A @ B )
% 3.02/1.69 => ( ( in @ Xx @ A )
% 3.02/1.69 => ( in @ Xx @ B ) ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(subsetE2_type,type,
% 3.02/1.69 subsetE2: $o ).
% 3.02/1.69
% 3.02/1.69 thf(subsetE2,definition,
% 3.02/1.69 ( subsetE2
% 3.02/1.69 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.02/1.69 ( ( subset @ A @ B )
% 3.02/1.69 => ( ~ ( in @ Xx @ B )
% 3.02/1.69 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(notsubsetI_type,type,
% 3.02/1.69 notsubsetI: $o ).
% 3.02/1.69
% 3.02/1.69 thf(notsubsetI,definition,
% 3.02/1.69 ( notsubsetI
% 3.02/1.69 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.02/1.69 ( ( in @ Xx @ A )
% 3.02/1.69 => ( ~ ( in @ Xx @ B )
% 3.02/1.69 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(notequalI1_type,type,
% 3.02/1.69 notequalI1: $o ).
% 3.02/1.69
% 3.02/1.69 thf(notequalI1,definition,
% 3.02/1.69 ( notequalI1
% 3.02/1.69 = ( ! [A: $i,B: $i] :
% 3.02/1.69 ( ~ ( subset @ A @ B )
% 3.02/1.69 => ( A != B ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(notequalI2_type,type,
% 3.02/1.69 notequalI2: $o ).
% 3.02/1.69
% 3.02/1.69 thf(notequalI2,definition,
% 3.02/1.69 ( notequalI2
% 3.02/1.69 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.02/1.69 ( ( in @ Xx @ A )
% 3.02/1.69 => ( ~ ( in @ Xx @ B )
% 3.02/1.69 => ( A != B ) ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(subsetRefl_type,type,
% 3.02/1.69 subsetRefl: $o ).
% 3.02/1.69
% 3.02/1.69 thf(subsetRefl,definition,
% 3.02/1.69 ( subsetRefl
% 3.02/1.69 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(subsetTrans_type,type,
% 3.02/1.69 subsetTrans: $o ).
% 3.02/1.69
% 3.02/1.69 thf(subsetTrans,definition,
% 3.02/1.69 ( subsetTrans
% 3.02/1.69 = ( ! [A: $i,B: $i,C: $i] :
% 3.02/1.69 ( ( subset @ A @ B )
% 3.02/1.69 => ( ( subset @ B @ C )
% 3.02/1.69 => ( subset @ A @ C ) ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(setadjoinSub_type,type,
% 3.02/1.69 setadjoinSub: $o ).
% 3.02/1.69
% 3.02/1.69 thf(setadjoinSub,definition,
% 3.02/1.69 ( setadjoinSub
% 3.02/1.69 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(setadjoinSub2_type,type,
% 3.02/1.69 setadjoinSub2: $o ).
% 3.02/1.69
% 3.02/1.69 thf(setadjoinSub2,definition,
% 3.02/1.69 ( setadjoinSub2
% 3.02/1.69 = ( ! [A: $i,Xx: $i,B: $i] :
% 3.02/1.69 ( ( subset @ A @ B )
% 3.02/1.69 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(subset2powerset_type,type,
% 3.02/1.69 subset2powerset: $o ).
% 3.02/1.69
% 3.02/1.69 thf(subset2powerset,definition,
% 3.02/1.69 ( subset2powerset
% 3.02/1.69 = ( ! [A: $i,B: $i] :
% 3.02/1.69 ( ( subset @ A @ B )
% 3.02/1.69 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(setextsub_type,type,
% 3.02/1.69 setextsub: $o ).
% 3.02/1.69
% 3.02/1.69 thf(setextsub,definition,
% 3.02/1.69 ( setextsub
% 3.02/1.69 = ( ! [A: $i,B: $i] :
% 3.02/1.69 ( ( subset @ A @ B )
% 3.02/1.69 => ( ( subset @ B @ A )
% 3.02/1.69 => ( A = B ) ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(subsetemptysetimpeq_type,type,
% 3.02/1.69 subsetemptysetimpeq: $o ).
% 3.02/1.69
% 3.02/1.69 thf(subsetemptysetimpeq,definition,
% 3.02/1.69 ( subsetemptysetimpeq
% 3.02/1.69 = ( ! [A: $i] :
% 3.02/1.69 ( ( subset @ A @ emptyset )
% 3.02/1.69 => ( A = emptyset ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(powersetI1_type,type,
% 3.02/1.69 powersetI1: $o ).
% 3.02/1.69
% 3.02/1.69 thf(powersetI1,definition,
% 3.02/1.69 ( powersetI1
% 3.02/1.69 = ( ! [A: $i,B: $i] :
% 3.02/1.69 ( ( subset @ B @ A )
% 3.02/1.69 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(powersetE1_type,type,
% 3.02/1.69 powersetE1: $o ).
% 3.02/1.69
% 3.02/1.69 thf(powersetE1,definition,
% 3.02/1.69 ( powersetE1
% 3.02/1.69 = ( ! [A: $i,B: $i] :
% 3.02/1.69 ( ( in @ B @ ( powerset @ A ) )
% 3.02/1.69 => ( subset @ B @ A ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(inPowerset_type,type,
% 3.02/1.69 inPowerset: $o ).
% 3.02/1.69
% 3.02/1.69 thf(inPowerset,definition,
% 3.02/1.69 ( inPowerset
% 3.02/1.69 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(powersetsubset_type,type,
% 3.02/1.69 powersetsubset: $o ).
% 3.02/1.69
% 3.02/1.69 thf(powersetsubset,definition,
% 3.02/1.69 ( powersetsubset
% 3.02/1.69 = ( ! [A: $i,B: $i] :
% 3.02/1.69 ( ( subset @ A @ B )
% 3.02/1.69 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(sepInPowerset_type,type,
% 3.02/1.69 sepInPowerset: $o ).
% 3.02/1.69
% 3.02/1.69 thf(sepInPowerset,definition,
% 3.02/1.69 ( sepInPowerset
% 3.02/1.69 = ( ! [A: $i,Xphi: $i > $o] :
% 3.02/1.69 ( in
% 3.02/1.69 @ ( dsetconstr @ A
% 3.02/1.69 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.02/1.69 @ ( powerset @ A ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(sepSubset_type,type,
% 3.02/1.69 sepSubset: $o ).
% 3.02/1.69
% 3.02/1.69 thf(sepSubset,definition,
% 3.02/1.69 ( sepSubset
% 3.02/1.69 = ( ! [A: $i,Xphi: $i > $o] :
% 3.02/1.69 ( subset
% 3.02/1.69 @ ( dsetconstr @ A
% 3.02/1.69 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.02/1.69 @ A ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(binunion_type,type,
% 3.02/1.69 binunion: $i > $i > $i ).
% 3.02/1.69
% 3.02/1.69 thf(binunionIL_type,type,
% 3.02/1.69 binunionIL: $o ).
% 3.02/1.69
% 3.02/1.69 thf(binunionIL,definition,
% 3.02/1.69 ( binunionIL
% 3.02/1.69 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.02/1.69 ( ( in @ Xx @ A )
% 3.02/1.69 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(upairset2IR_type,type,
% 3.02/1.69 upairset2IR: $o ).
% 3.02/1.69
% 3.02/1.69 thf(upairset2IR,definition,
% 3.02/1.69 ( upairset2IR
% 3.02/1.69 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(binunionIR_type,type,
% 3.02/1.69 binunionIR: $o ).
% 3.02/1.69
% 3.02/1.69 thf(binunionIR,definition,
% 3.02/1.69 ( binunionIR
% 3.02/1.69 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.02/1.69 ( ( in @ Xx @ B )
% 3.02/1.69 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(binunionEcases_type,type,
% 3.02/1.69 binunionEcases: $o ).
% 3.02/1.69
% 3.02/1.69 thf(binunionEcases,definition,
% 3.02/1.69 ( binunionEcases
% 3.02/1.69 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 3.02/1.69 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 3.02/1.69 => ( ( ( in @ Xx @ A )
% 3.02/1.69 => Xphi )
% 3.02/1.69 => ( ( ( in @ Xx @ B )
% 3.02/1.69 => Xphi )
% 3.02/1.69 => Xphi ) ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(binunionE_type,type,
% 3.02/1.69 binunionE: $o ).
% 3.02/1.69
% 3.02/1.69 thf(binunionE,definition,
% 3.02/1.69 ( binunionE
% 3.02/1.69 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.02/1.69 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 3.02/1.69 => ( ( in @ Xx @ A )
% 3.02/1.69 | ( in @ Xx @ B ) ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(binunionLsub_type,type,
% 3.02/1.69 binunionLsub: $o ).
% 3.02/1.69
% 3.02/1.69 thf(binunionLsub,definition,
% 3.02/1.69 ( binunionLsub
% 3.02/1.69 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(binunionRsub_type,type,
% 3.02/1.69 binunionRsub: $o ).
% 3.02/1.69
% 3.02/1.69 thf(binunionRsub,definition,
% 3.02/1.69 ( binunionRsub
% 3.02/1.69 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(binintersect_type,type,
% 3.02/1.69 binintersect: $i > $i > $i ).
% 3.02/1.69
% 3.02/1.69 thf(binintersectI_type,type,
% 3.02/1.69 binintersectI: $o ).
% 3.02/1.69
% 3.02/1.69 thf(binintersectI,definition,
% 3.02/1.69 ( binintersectI
% 3.02/1.69 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.02/1.69 ( ( in @ Xx @ A )
% 3.02/1.69 => ( ( in @ Xx @ B )
% 3.02/1.69 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(binintersectSubset5_type,type,
% 3.02/1.69 binintersectSubset5: $o ).
% 3.02/1.69
% 3.02/1.69 thf(binintersectSubset5,definition,
% 3.02/1.69 ( binintersectSubset5
% 3.02/1.69 = ( ! [A: $i,B: $i,C: $i] :
% 3.02/1.69 ( ( subset @ C @ A )
% 3.02/1.69 => ( ( subset @ C @ B )
% 3.02/1.69 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(binintersectEL_type,type,
% 3.02/1.69 binintersectEL: $o ).
% 3.02/1.69
% 3.02/1.69 thf(binintersectEL,definition,
% 3.02/1.69 ( binintersectEL
% 3.02/1.69 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.02/1.69 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 3.02/1.69 => ( in @ Xx @ A ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(binintersectLsub_type,type,
% 3.02/1.69 binintersectLsub: $o ).
% 3.02/1.69
% 3.02/1.69 thf(binintersectLsub,definition,
% 3.02/1.69 ( binintersectLsub
% 3.02/1.69 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(binintersectSubset2_type,type,
% 3.02/1.69 binintersectSubset2: $o ).
% 3.02/1.69
% 3.02/1.69 thf(binintersectSubset2,definition,
% 3.02/1.69 ( binintersectSubset2
% 3.02/1.69 = ( ! [A: $i,B: $i] :
% 3.02/1.69 ( ( subset @ A @ B )
% 3.02/1.69 => ( ( binintersect @ A @ B )
% 3.02/1.69 = A ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(binintersectSubset3_type,type,
% 3.02/1.69 binintersectSubset3: $o ).
% 3.02/1.69
% 3.02/1.69 thf(binintersectSubset3,definition,
% 3.02/1.69 ( binintersectSubset3
% 3.02/1.69 = ( ! [A: $i,B: $i] :
% 3.02/1.69 ( ( ( binintersect @ A @ B )
% 3.02/1.69 = B )
% 3.02/1.69 => ( subset @ B @ A ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(binintersectER_type,type,
% 3.02/1.69 binintersectER: $o ).
% 3.02/1.69
% 3.02/1.69 thf(binintersectER,definition,
% 3.02/1.69 ( binintersectER
% 3.02/1.69 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.02/1.69 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 3.02/1.69 => ( in @ Xx @ B ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(disjointsetsI1_type,type,
% 3.02/1.69 disjointsetsI1: $o ).
% 3.02/1.69
% 3.02/1.69 thf(disjointsetsI1,definition,
% 3.02/1.69 ( disjointsetsI1
% 3.02/1.69 = ( ! [A: $i,B: $i] :
% 3.02/1.69 ( ~ ? [Xx: $i] :
% 3.02/1.69 ( ( in @ Xx @ A )
% 3.02/1.69 & ( in @ Xx @ B ) )
% 3.02/1.69 => ( ( binintersect @ A @ B )
% 3.02/1.69 = emptyset ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(binintersectRsub_type,type,
% 3.02/1.69 binintersectRsub: $o ).
% 3.02/1.69
% 3.02/1.69 thf(binintersectRsub,definition,
% 3.02/1.69 ( binintersectRsub
% 3.02/1.69 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(binintersectSubset4_type,type,
% 3.02/1.69 binintersectSubset4: $o ).
% 3.02/1.69
% 3.02/1.69 thf(binintersectSubset4,definition,
% 3.02/1.69 ( binintersectSubset4
% 3.02/1.69 = ( ! [A: $i,B: $i] :
% 3.02/1.69 ( ( subset @ B @ A )
% 3.02/1.69 => ( ( binintersect @ A @ B )
% 3.02/1.69 = B ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(binintersectSubset1_type,type,
% 3.02/1.69 binintersectSubset1: $o ).
% 3.02/1.69
% 3.02/1.69 thf(binintersectSubset1,definition,
% 3.02/1.69 ( binintersectSubset1
% 3.02/1.69 = ( ! [A: $i,B: $i] :
% 3.02/1.69 ( ( ( binintersect @ A @ B )
% 3.02/1.69 = A )
% 3.02/1.69 => ( subset @ A @ B ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(bs114d_type,type,
% 3.02/1.69 bs114d: $o ).
% 3.02/1.69
% 3.02/1.69 thf(bs114d,definition,
% 3.02/1.69 ( bs114d
% 3.02/1.69 = ( ! [A: $i,B: $i,C: $i] :
% 3.02/1.69 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 3.02/1.69 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(regular_type,type,
% 3.02/1.69 regular: $i > $o ).
% 3.02/1.69
% 3.02/1.69 thf(setminus_type,type,
% 3.02/1.69 setminus: $i > $i > $i ).
% 3.02/1.69
% 3.02/1.69 thf(setminusI_type,type,
% 3.02/1.69 setminusI: $o ).
% 3.02/1.69
% 3.02/1.69 thf(setminusI,definition,
% 3.02/1.69 ( setminusI
% 3.02/1.69 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.02/1.69 ( ( in @ Xx @ A )
% 3.02/1.69 => ( ~ ( in @ Xx @ B )
% 3.02/1.69 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(setminusEL_type,type,
% 3.02/1.69 setminusEL: $o ).
% 3.02/1.69
% 3.02/1.69 thf(setminusEL,definition,
% 3.02/1.69 ( setminusEL
% 3.02/1.69 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.02/1.69 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 3.02/1.69 => ( in @ Xx @ A ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(setminusER_type,type,
% 3.02/1.69 setminusER: $o ).
% 3.02/1.69
% 3.02/1.69 thf(setminusER,definition,
% 3.02/1.69 ( setminusER
% 3.02/1.69 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.02/1.69 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 3.02/1.69 => ~ ( in @ Xx @ B ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(setminusSubset2_type,type,
% 3.02/1.69 setminusSubset2: $o ).
% 3.02/1.69
% 3.02/1.69 thf(setminusSubset2,definition,
% 3.02/1.69 ( setminusSubset2
% 3.02/1.69 = ( ! [A: $i,B: $i] :
% 3.02/1.69 ( ( subset @ A @ B )
% 3.02/1.69 => ( ( setminus @ A @ B )
% 3.02/1.69 = emptyset ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(setminusERneg_type,type,
% 3.02/1.69 setminusERneg: $o ).
% 3.02/1.69
% 3.02/1.69 thf(setminusERneg,definition,
% 3.02/1.69 ( setminusERneg
% 3.02/1.69 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.02/1.69 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 3.02/1.69 => ( ( in @ Xx @ A )
% 3.02/1.69 => ( in @ Xx @ B ) ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(setminusELneg_type,type,
% 3.02/1.69 setminusELneg: $o ).
% 3.02/1.69
% 3.02/1.69 thf(setminusELneg,definition,
% 3.02/1.69 ( setminusELneg
% 3.02/1.69 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.02/1.69 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 3.02/1.69 => ( ~ ( in @ Xx @ B )
% 3.02/1.69 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(setminusILneg_type,type,
% 3.02/1.69 setminusILneg: $o ).
% 3.02/1.69
% 3.02/1.69 thf(setminusILneg,definition,
% 3.02/1.69 ( setminusILneg
% 3.02/1.69 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.02/1.69 ( ~ ( in @ Xx @ A )
% 3.02/1.69 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(setminusIRneg_type,type,
% 3.02/1.69 setminusIRneg: $o ).
% 3.02/1.69
% 3.02/1.69 thf(setminusIRneg,definition,
% 3.02/1.69 ( setminusIRneg
% 3.02/1.69 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.02/1.69 ( ( in @ Xx @ B )
% 3.02/1.69 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(setminusLsub_type,type,
% 3.02/1.69 setminusLsub: $o ).
% 3.02/1.69
% 3.02/1.69 thf(setminusLsub,definition,
% 3.02/1.69 ( setminusLsub
% 3.02/1.69 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(setminusSubset1_type,type,
% 3.02/1.69 setminusSubset1: $o ).
% 3.02/1.69
% 3.02/1.69 thf(setminusSubset1,definition,
% 3.02/1.69 ( setminusSubset1
% 3.02/1.69 = ( ! [A: $i,B: $i] :
% 3.02/1.69 ( ( ( setminus @ A @ B )
% 3.02/1.69 = emptyset )
% 3.02/1.69 => ( subset @ A @ B ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(symdiff_type,type,
% 3.02/1.69 symdiff: $i > $i > $i ).
% 3.02/1.69
% 3.02/1.69 thf(symdiffE_type,type,
% 3.02/1.69 symdiffE: $o ).
% 3.02/1.69
% 3.02/1.69 thf(symdiffE,definition,
% 3.02/1.69 ( symdiffE
% 3.02/1.69 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.02/1.69 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 3.02/1.69 => ! [Xphi: $o] :
% 3.02/1.69 ( ( ( in @ Xx @ A )
% 3.02/1.69 => ( ~ ( in @ Xx @ B )
% 3.02/1.69 => Xphi ) )
% 3.02/1.69 => ( ( ~ ( in @ Xx @ A )
% 3.02/1.69 => ( ( in @ Xx @ B )
% 3.02/1.69 => Xphi ) )
% 3.02/1.69 => Xphi ) ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(symdiffI1_type,type,
% 3.02/1.69 symdiffI1: $o ).
% 3.02/1.69
% 3.02/1.69 thf(symdiffI1,definition,
% 3.02/1.69 ( symdiffI1
% 3.02/1.69 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.02/1.69 ( ( in @ Xx @ A )
% 3.02/1.69 => ( ~ ( in @ Xx @ B )
% 3.02/1.69 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(symdiffI2_type,type,
% 3.02/1.69 symdiffI2: $o ).
% 3.02/1.69
% 3.02/1.69 thf(symdiffI2,definition,
% 3.02/1.69 ( symdiffI2
% 3.02/1.69 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.02/1.69 ( ~ ( in @ Xx @ A )
% 3.02/1.69 => ( ( in @ Xx @ B )
% 3.02/1.69 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(symdiffIneg1_type,type,
% 3.02/1.69 symdiffIneg1: $o ).
% 3.02/1.69
% 3.02/1.69 thf(symdiffIneg1,definition,
% 3.02/1.69 ( symdiffIneg1
% 3.02/1.69 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.02/1.69 ( ( in @ Xx @ A )
% 3.02/1.69 => ( ( in @ Xx @ B )
% 3.02/1.69 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(symdiffIneg2_type,type,
% 3.02/1.69 symdiffIneg2: $o ).
% 3.02/1.69
% 3.02/1.69 thf(symdiffIneg2,definition,
% 3.02/1.69 ( symdiffIneg2
% 3.02/1.69 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.02/1.69 ( ~ ( in @ Xx @ A )
% 3.02/1.69 => ( ~ ( in @ Xx @ B )
% 3.02/1.69 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 3.02/1.69
% 3.02/1.69 thf(iskpair_type,type,
% 3.02/1.69 iskpair: $i > $o ).
% 3.02/1.69
% 3.02/1.69 thf(secondinupair_type,type,
% 3.02/1.69 secondinupair: $o ).
% 3.02/1.69
% 3.02/1.70 thf(secondinupair,definition,
% 3.02/1.70 ( secondinupair
% 3.02/1.70 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.02/1.70
% 3.02/1.70 thf(setukpairIL_type,type,
% 3.02/1.70 setukpairIL: $o ).
% 3.02/1.70
% 3.02/1.70 thf(setukpairIL,definition,
% 3.02/1.70 ( setukpairIL
% 3.02/1.70 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 3.02/1.70
% 3.02/1.70 thf(setukpairIR_type,type,
% 3.02/1.70 setukpairIR: $o ).
% 3.02/1.70
% 3.02/1.70 thf(setukpairIR,definition,
% 3.02/1.70 ( setukpairIR
% 3.02/1.70 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 3.02/1.70
% 3.02/1.70 thf(kpairiskpair_type,type,
% 3.02/1.70 kpairiskpair: $o ).
% 3.02/1.70
% 3.02/1.70 thf(kpairiskpair,definition,
% 3.02/1.70 ( kpairiskpair
% 3.02/1.70 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 3.02/1.70
% 3.02/1.70 thf(kpair_type,type,
% 3.02/1.70 kpair: $i > $i > $i ).
% 3.02/1.70
% 3.02/1.70 thf(kpairp_type,type,
% 3.02/1.70 kpairp: $o ).
% 3.02/1.70
% 3.02/1.70 thf(kpairp,definition,
% 3.02/1.70 ( kpairp
% 3.02/1.70 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 3.02/1.70
% 3.02/1.70 thf(cartprod_type,type,
% 3.02/1.70 cartprod: $i > $i > $i ).
% 3.02/1.70
% 3.02/1.70 thf(singletonsubset_type,type,
% 3.02/1.70 singletonsubset: $o ).
% 3.02/1.70
% 3.02/1.70 thf(singletonsubset,definition,
% 3.02/1.70 ( singletonsubset
% 3.02/1.70 = ( ! [A: $i,Xx: $i] :
% 3.02/1.70 ( ( in @ Xx @ A )
% 3.02/1.70 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 3.02/1.70
% 3.02/1.70 thf(singletoninpowerset_type,type,
% 3.02/1.70 singletoninpowerset: $o ).
% 3.02/1.70
% 3.02/1.70 thf(singletoninpowerset,definition,
% 3.02/1.70 ( singletoninpowerset
% 3.02/1.70 = ( ! [A: $i,Xx: $i] :
% 3.02/1.70 ( ( in @ Xx @ A )
% 3.02/1.70 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 3.02/1.70
% 3.02/1.70 thf(singletoninpowunion_type,type,
% 3.02/1.70 singletoninpowunion: $o ).
% 3.02/1.70
% 3.02/1.70 thf(singletoninpowunion,definition,
% 3.02/1.70 ( singletoninpowunion
% 3.02/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.02/1.70 ( ( in @ Xx @ A )
% 3.02/1.70 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 3.02/1.70
% 3.02/1.70 thf(upairset2E_type,type,
% 3.02/1.70 upairset2E: $o ).
% 3.02/1.70
% 3.02/1.70 thf(upairset2E,definition,
% 3.02/1.70 ( upairset2E
% 3.02/1.70 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 3.02/1.70 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 3.02/1.70 => ( ( Xz = Xx )
% 3.02/1.70 | ( Xz = Xy ) ) ) ) ) ).
% 3.02/1.70
% 3.02/1.70 thf(upairsubunion_type,type,
% 3.02/1.70 upairsubunion: $o ).
% 3.02/1.70
% 3.02/1.70 thf(upairsubunion,definition,
% 3.02/1.70 ( upairsubunion
% 3.02/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.02/1.70 ( ( in @ Xx @ A )
% 3.02/1.70 => ! [Xy: $i] :
% 3.02/1.70 ( ( in @ Xy @ B )
% 3.02/1.70 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 3.02/1.70
% 3.02/1.70 thf(upairinpowunion_type,type,
% 3.02/1.70 upairinpowunion: $o ).
% 3.02/1.70
% 3.02/1.70 thf(upairinpowunion,definition,
% 3.02/1.70 ( upairinpowunion
% 3.02/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.02/1.70 ( ( in @ Xx @ A )
% 3.02/1.70 => ! [Xy: $i] :
% 3.02/1.70 ( ( in @ Xy @ B )
% 3.02/1.70 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 3.02/1.70
% 3.02/1.70 thf(ubforcartprodlem1_type,type,
% 3.02/1.70 ubforcartprodlem1: $o ).
% 3.02/1.70
% 3.02/1.70 thf(ubforcartprodlem1,definition,
% 3.02/1.70 ( ubforcartprodlem1
% 3.02/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.02/1.70 ( ( in @ Xx @ A )
% 3.02/1.70 => ! [Xy: $i] :
% 3.02/1.70 ( ( in @ Xy @ B )
% 3.02/1.70 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 3.02/1.70
% 3.02/1.70 thf(ubforcartprodlem2_type,type,
% 3.02/1.70 ubforcartprodlem2: $o ).
% 3.02/1.70
% 3.02/1.70 thf(ubforcartprodlem2,definition,
% 3.02/1.70 ( ubforcartprodlem2
% 3.02/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.02/1.70 ( ( in @ Xx @ A )
% 3.02/1.70 => ! [Xy: $i] :
% 3.02/1.70 ( ( in @ Xy @ B )
% 3.02/1.70 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 3.02/1.70
% 3.02/1.70 thf(ubforcartprodlem3_type,type,
% 3.02/1.70 ubforcartprodlem3: $o ).
% 3.02/1.70
% 3.02/1.70 thf(ubforcartprodlem3,definition,
% 3.02/1.70 ( ubforcartprodlem3
% 3.02/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.02/1.70 ( ( in @ Xx @ A )
% 3.02/1.70 => ! [Xy: $i] :
% 3.02/1.70 ( ( in @ Xy @ B )
% 3.02/1.70 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 3.02/1.70
% 3.02/1.70 thf(cartprodpairin_type,type,
% 3.02/1.70 cartprodpairin: $o ).
% 3.02/1.70
% 3.02/1.70 thf(cartprodpairin,definition,
% 3.02/1.70 ( cartprodpairin
% 3.02/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.02/1.70 ( ( in @ Xx @ A )
% 3.02/1.70 => ! [Xy: $i] :
% 3.02/1.70 ( ( in @ Xy @ B )
% 3.02/1.70 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 3.02/1.70
% 3.02/1.70 thf(cartprodmempair1_type,type,
% 3.02/1.70 cartprodmempair1: $o ).
% 3.02/1.70
% 3.02/1.70 thf(cartprodmempair1,definition,
% 3.02/1.70 ( cartprodmempair1
% 3.02/1.70 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.02/1.70 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.02/1.70 => ? [Xx: $i] :
% 3.02/1.70 ( ( in @ Xx @ A )
% 3.02/1.70 & ? [Xy: $i] :
% 3.02/1.70 ( ( in @ Xy @ B )
% 3.02/1.70 & ( Xu
% 3.02/1.70 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 3.02/1.70
% 3.02/1.70 thf(cartprodmempair_type,type,
% 3.02/1.70 cartprodmempair: $o ).
% 3.02/1.70
% 3.02/1.70 thf(cartprodmempair,definition,
% 3.02/1.70 ( cartprodmempair
% 3.02/1.70 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.02/1.70 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.02/1.70 => ( iskpair @ Xu ) ) ) ) ).
% 3.02/1.70
% 3.02/1.70 thf(setunionE2_type,type,
% 3.02/1.70 setunionE2: $o ).
% 3.02/1.70
% 3.02/1.70 thf(setunionE2,definition,
% 3.02/1.70 ( setunionE2
% 3.02/1.70 = ( ! [A: $i,Xx: $i] :
% 3.02/1.70 ( ( in @ Xx @ ( setunion @ A ) )
% 3.02/1.70 => ? [X: $i] :
% 3.02/1.70 ( ( in @ X @ A )
% 3.02/1.70 & ( in @ Xx @ X ) ) ) ) ) ).
% 3.02/1.70
% 3.02/1.70 thf(setunionsingleton1_type,type,
% 3.02/1.70 setunionsingleton1: $o ).
% 3.02/1.70
% 3.02/1.70 thf(setunionsingleton1,definition,
% 3.02/1.70 ( setunionsingleton1
% 3.02/1.70 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 3.02/1.70
% 3.02/1.70 thf(setunionsingleton2_type,type,
% 3.02/1.70 setunionsingleton2: $o ).
% 3.02/1.70
% 3.02/1.70 thf(setunionsingleton2,definition,
% 3.02/1.70 ( setunionsingleton2
% 3.02/1.70 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 3.02/1.70
% 3.02/1.70 thf(setunionsingleton_type,type,
% 3.02/1.70 setunionsingleton: $o ).
% 3.02/1.70
% 3.02/1.70 thf(setunionsingleton,definition,
% 3.02/1.70 ( setunionsingleton
% 3.02/1.70 = ( ! [Xx: $i] :
% 3.02/1.70 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 3.02/1.70 = Xx ) ) ) ).
% 3.02/1.70
% 3.02/1.70 thf(singleton_type,type,
% 3.02/1.70 singleton: $i > $o ).
% 3.02/1.70
% 3.02/1.70 thf(singletonprop_type,type,
% 3.02/1.70 singletonprop: $o ).
% 3.02/1.70
% 3.02/1.70 thf(singletonprop,definition,
% 3.02/1.70 ( singletonprop
% 3.02/1.70 = ( ! [A: $i,Xphi: $i > $o] :
% 3.02/1.70 ( ! [Xx: $i] :
% 3.02/1.70 ( ( in @ Xx @ A )
% 3.02/1.70 => ! [Xy: $i] :
% 3.02/1.70 ( ( in @ Xy @ A )
% 3.02/1.70 => ( ( Xphi @ Xx )
% 3.02/1.70 => ( ( Xphi @ Xy )
% 3.02/1.70 => ( Xx = Xy ) ) ) ) )
% 3.02/1.70 => ( ? [Xx: $i] :
% 3.02/1.70 ( ( in @ Xx @ A )
% 3.02/1.70 & ( Xphi @ Xx ) )
% 3.02/1.70 => ( singleton
% 3.02/1.70 @ ( dsetconstr @ A
% 3.02/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 3.02/1.70
% 3.02/1.70 thf(ex1_type,type,
% 3.02/1.70 ex1: $i > ( $i > $o ) > $o ).
% 3.02/1.70
% 3.02/1.70 thf(ex1E1_type,type,
% 3.02/1.70 ex1E1: $o ).
% 3.02/1.70
% 3.02/1.70 thf(ex1E1,definition,
% 3.02/1.70 ( ex1E1
% 3.02/1.70 = ( ! [A: $i,Xphi: $i > $o] :
% 3.02/1.70 ( ( ex1 @ A
% 3.02/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.02/1.70 => ? [Xx: $i] :
% 3.02/1.70 ( ( in @ Xx @ A )
% 3.02/1.70 & ( Xphi @ Xx ) ) ) ) ) ).
% 3.02/1.70
% 3.02/1.70 thf(ex1I_type,type,
% 3.02/1.70 ex1I: $o ).
% 3.02/1.70
% 3.02/1.70 thf(ex1I,definition,
% 3.02/1.70 ( ex1I
% 3.02/1.70 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.02/1.70 ( ( in @ Xx @ A )
% 3.02/1.70 => ( ( Xphi @ Xx )
% 3.02/1.70 => ( ! [Xy: $i] :
% 3.02/1.70 ( ( in @ Xy @ A )
% 3.02/1.70 => ( ( Xphi @ Xy )
% 3.02/1.70 => ( Xy = Xx ) ) )
% 3.02/1.70 => ( ex1 @ A
% 3.02/1.70 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 3.02/1.70
% 3.02/1.70 thf(ex1I2_type,type,
% 3.02/1.70 ex1I2: $o ).
% 3.02/1.70
% 3.02/1.70 thf(ex1I2,definition,
% 3.02/1.70 ( ex1I2
% 3.02/1.70 = ( ! [A: $i,Xphi: $i > $o] :
% 3.02/1.70 ( ! [Xx: $i] :
% 3.02/1.70 ( ( in @ Xx @ A )
% 3.02/1.70 => ! [Xy: $i] :
% 3.02/1.70 ( ( in @ Xy @ A )
% 3.02/1.70 => ( ( Xphi @ Xx )
% 3.02/1.70 => ( ( Xphi @ Xy )
% 3.02/1.70 => ( Xx = Xy ) ) ) ) )
% 3.02/1.70 => ( ? [Xx: $i] :
% 3.02/1.70 ( ( in @ Xx @ A )
% 3.02/1.70 & ( Xphi @ Xx ) )
% 3.02/1.70 => ( ex1 @ A
% 3.02/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.02/1.70
% 3.02/1.70 thf(singletonsuniq_type,type,
% 3.02/1.70 singletonsuniq: $o ).
% 3.02/1.70
% 3.02/1.70 thf(singletonsuniq,definition,
% 3.02/1.70 ( singletonsuniq
% 3.02/1.70 = ( ! [Xx: $i,Xy: $i] :
% 3.02/1.70 ( ( ( setadjoin @ Xx @ emptyset )
% 3.02/1.70 = ( setadjoin @ Xy @ emptyset ) )
% 3.02/1.70 => ( Xx = Xy ) ) ) ) ).
% 3.02/1.70
% 3.02/1.70 thf(atmost1p_type,type,
% 3.02/1.70 atmost1p: $i > $o ).
% 3.02/1.70
% 3.02/1.70 thf(atleast2p_type,type,
% 3.02/1.70 atleast2p: $i > $o ).
% 3.02/1.70
% 3.02/1.70 thf(atmost2p_type,type,
% 3.02/1.70 atmost2p: $i > $o ).
% 3.02/1.70
% 3.02/1.70 thf(upairsetp_type,type,
% 3.02/1.70 upairsetp: $i > $o ).
% 3.02/1.70
% 3.02/1.70 thf(setukpairinjL1_type,type,
% 3.02/1.70 setukpairinjL1: $o ).
% 3.02/1.70
% 3.02/1.70 thf(setukpairinjL1,definition,
% 3.02/1.70 ( setukpairinjL1
% 3.02/1.70 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 3.02/1.70 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 3.02/1.70 => ( Xx = Xz ) ) ) ) ).
% 3.02/1.70
% 3.02/1.70 thf(kfstsingleton_type,type,
% 3.02/1.70 kfstsingleton: $o ).
% 3.02/1.70
% 3.02/1.70 thf(kfstsingleton,definition,
% 3.02/1.70 ( kfstsingleton
% 3.02/1.70 = ( ! [Xu: $i] :
% 3.02/1.70 ( ( iskpair @ Xu )
% 3.02/1.70 => ( singleton
% 3.02/1.70 @ ( dsetconstr @ ( setunion @ Xu )
% 3.02/1.70 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 3.02/1.70
% 3.02/1.70 thf(theprop_type,type,
% 3.02/1.70 theprop: $o ).
% 3.02/1.70
% 3.02/1.70 thf(theprop,definition,
% 3.02/1.70 ( theprop
% 3.02/1.70 = ( ! [X: $i] :
% 3.02/1.70 ( ( singleton @ X )
% 3.02/1.70 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 3.02/1.70
% 3.02/1.70 thf(kfst_type,type,
% 3.02/1.70 kfst: $i > $i ).
% 3.02/1.70
% 3.02/1.70 thf(kfstpairEq_type,type,
% 3.02/1.70 kfstpairEq: $o ).
% 3.02/1.70
% 3.02/1.70 thf(kfstpairEq,definition,
% 3.02/1.70 ( kfstpairEq
% 3.02/1.70 = ( ! [Xx: $i,Xy: $i] :
% 3.02/1.70 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 3.02/1.70 = Xx ) ) ) ).
% 3.02/1.70
% 3.02/1.70 thf(cartprodfstin_type,type,
% 3.02/1.70 cartprodfstin: $o ).
% 3.02/1.70
% 3.02/1.70 thf(cartprodfstin,definition,
% 3.02/1.70 ( cartprodfstin
% 3.02/1.70 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.02/1.70 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.02/1.70 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 3.02/1.70
% 3.02/1.70 thf(setukpairinjL2_type,type,
% 3.02/1.70 setukpairinjL2: $o ).
% 3.02/1.70
% 3.02/1.70 thf(setukpairinjL2,definition,
% 3.02/1.70 ( setukpairinjL2
% 3.02/1.70 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.02/1.70 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 3.02/1.70 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 3.02/1.70 => ( Xx = Xz ) ) ) ) ).
% 3.02/1.70
% 3.02/1.70 thf(setukpairinjL_type,type,
% 3.02/1.70 setukpairinjL: $o ).
% 3.02/1.70
% 3.02/1.70 thf(setukpairinjL,definition,
% 3.02/1.70 ( setukpairinjL
% 3.02/1.70 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.02/1.70 ( ( ( kpair @ Xx @ Xy )
% 3.02/1.70 = ( kpair @ Xz @ Xu ) )
% 3.02/1.70 => ( Xx = Xz ) ) ) ) ).
% 3.02/1.70
% 3.02/1.70 thf(setukpairinjR11_type,type,
% 3.02/1.70 setukpairinjR11: $o ).
% 3.02/1.70
% 3.02/1.70 thf(setukpairinjR11,definition,
% 3.02/1.70 ( setukpairinjR11
% 3.02/1.70 = ( ! [Xx: $i,Xy: $i] :
% 3.02/1.70 ( ( Xx = Xy )
% 3.02/1.70 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 3.02/1.70 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 3.02/1.70
% 3.02/1.70 thf(setukpairinjR12_type,type,
% 3.02/1.70 setukpairinjR12: $o ).
% 3.02/1.70
% 3.02/1.70 thf(setukpairinjR12,definition,
% 3.02/1.70 ( setukpairinjR12
% 3.02/1.70 = ( ! [Xx: $i,Xy: $i] :
% 3.02/1.70 ( ( Xx = Xy )
% 3.02/1.70 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 3.02/1.70 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 3.02/1.70
% 3.02/1.70 thf(setukpairinjR1_type,type,
% 3.02/1.70 setukpairinjR1: $o ).
% 3.02/1.70
% 3.02/1.70 thf(setukpairinjR1,definition,
% 3.02/1.70 ( setukpairinjR1
% 3.02/1.70 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.02/1.70 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 3.02/1.70 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 3.02/1.70 => ( ( Xz = Xu )
% 3.02/1.70 => ( Xy = Xu ) ) ) ) ) ).
% 3.02/1.70
% 3.02/1.70 thf(upairequniteq_type,type,
% 3.02/1.70 upairequniteq: $o ).
% 3.02/1.70
% 3.02/1.70 thf(upairequniteq,definition,
% 3.02/1.70 ( upairequniteq
% 3.02/1.70 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 3.02/1.70 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 3.02/1.70 = ( setadjoin @ Xz @ emptyset ) )
% 3.02/1.70 => ( Xx = Xy ) ) ) ) ).
% 3.02/1.70
% 3.02/1.70 thf(setukpairinjR2_type,type,
% 3.02/1.70 setukpairinjR2: $o ).
% 3.02/1.70
% 3.02/1.70 thf(setukpairinjR2,definition,
% 3.02/1.70 ( setukpairinjR2
% 3.02/1.70 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.02/1.70 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 3.02/1.70 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 3.02/1.70 => ( Xy = Xu ) ) ) ) ).
% 3.02/1.70
% 3.02/1.70 thf(setukpairinjR_type,type,
% 3.02/1.70 setukpairinjR: $o ).
% 3.02/1.70
% 3.02/1.70 thf(setukpairinjR,definition,
% 3.02/1.70 ( setukpairinjR
% 3.02/1.70 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.02/1.70 ( ( ( kpair @ Xx @ Xy )
% 3.02/1.70 = ( kpair @ Xz @ Xu ) )
% 3.02/1.70 => ( Xy = Xu ) ) ) ) ).
% 3.02/1.70
% 3.02/1.70 thf(ksndsingleton_type,type,
% 3.02/1.70 ksndsingleton: $o ).
% 3.02/1.70
% 3.02/1.70 thf(ksndsingleton,definition,
% 3.02/1.70 ( ksndsingleton
% 3.02/1.70 = ( ! [Xu: $i] :
% 3.02/1.70 ( ( iskpair @ Xu )
% 3.02/1.70 => ( singleton
% 3.02/1.70 @ ( dsetconstr @ ( setunion @ Xu )
% 3.02/1.70 @ ^ [Xx: $i] :
% 3.02/1.70 ( Xu
% 3.02/1.70 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 3.02/1.70
% 3.02/1.70 thf(ksnd_type,type,
% 3.02/1.70 ksnd: $i > $i ).
% 3.02/1.70
% 3.02/1.70 thf(ksndpairEq_type,type,
% 3.02/1.70 ksndpairEq: $o ).
% 3.02/1.70
% 3.02/1.70 thf(ksndpairEq,definition,
% 3.02/1.70 ( ksndpairEq
% 3.02/1.70 = ( ! [Xx: $i,Xy: $i] :
% 3.02/1.70 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 3.02/1.70 = Xy ) ) ) ).
% 3.02/1.70
% 3.02/1.70 thf(kpairsurjEq_type,type,
% 3.02/1.70 kpairsurjEq: $o ).
% 3.02/1.70
% 3.02/1.70 thf(kpairsurjEq,definition,
% 3.02/1.70 ( kpairsurjEq
% 3.02/1.70 = ( ! [Xu: $i] :
% 3.02/1.70 ( ( iskpair @ Xu )
% 3.02/1.70 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 3.02/1.70 = Xu ) ) ) ) ).
% 3.02/1.70
% 3.02/1.70 thf(cartprodsndin_type,type,
% 3.02/1.70 cartprodsndin: $o ).
% 3.02/1.70
% 3.02/1.70 thf(cartprodsndin,definition,
% 3.02/1.70 ( cartprodsndin
% 3.02/1.70 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.02/1.70 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.02/1.70 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 3.02/1.70
% 3.02/1.70 thf(cartprodpairmemEL_type,type,
% 3.02/1.70 cartprodpairmemEL: $o ).
% 3.02/1.70
% 3.02/1.70 thf(cartprodpairmemEL,definition,
% 3.02/1.70 ( cartprodpairmemEL
% 3.02/1.70 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 3.02/1.70 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 3.02/1.70 => ( in @ Xx @ A ) ) ) ) ).
% 3.02/1.70
% 3.02/1.70 thf(cartprodpairmemER_type,type,
% 3.02/1.70 cartprodpairmemER: $o ).
% 3.02/1.70
% 3.02/1.70 thf(cartprodpairmemER,definition,
% 3.02/1.70 ( cartprodpairmemER
% 3.02/1.70 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 3.02/1.70 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 3.02/1.70 => ( in @ Xy @ B ) ) ) ) ).
% 3.02/1.70
% 3.02/1.70 thf(cartprodmempaircEq_type,type,
% 3.02/1.70 cartprodmempaircEq: $o ).
% 3.02/1.70
% 3.02/1.70 thf(cartprodmempaircEq,definition,
% 3.17/1.70 ( cartprodmempaircEq
% 3.17/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ! [Xy: $i] :
% 3.17/1.70 ( ( in @ Xy @ B )
% 3.17/1.70 => ( ( kpair @ Xx @ Xy )
% 3.17/1.70 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(cartprodfstpairEq_type,type,
% 3.17/1.70 cartprodfstpairEq: $o ).
% 3.17/1.70
% 3.17/1.70 thf(cartprodfstpairEq,definition,
% 3.17/1.70 ( cartprodfstpairEq
% 3.17/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ! [Xy: $i] :
% 3.17/1.70 ( ( in @ Xy @ B )
% 3.17/1.70 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 3.17/1.70 = Xx ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(cartprodsndpairEq_type,type,
% 3.17/1.70 cartprodsndpairEq: $o ).
% 3.17/1.70
% 3.17/1.70 thf(cartprodsndpairEq,definition,
% 3.17/1.70 ( cartprodsndpairEq
% 3.17/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ! [Xy: $i] :
% 3.17/1.70 ( ( in @ Xy @ B )
% 3.17/1.70 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 3.17/1.70 = Xy ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(cartprodpairsurjEq_type,type,
% 3.17/1.70 cartprodpairsurjEq: $o ).
% 3.17/1.70
% 3.17/1.70 thf(cartprodpairsurjEq,definition,
% 3.17/1.70 ( cartprodpairsurjEq
% 3.17/1.70 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.17/1.70 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.17/1.70 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 3.17/1.70 = Xu ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(breln_type,type,
% 3.17/1.70 breln: $i > $i > $i > $o ).
% 3.17/1.70
% 3.17/1.70 thf(dpsetconstr_type,type,
% 3.17/1.70 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 3.17/1.70
% 3.17/1.70 thf(dpsetconstrI_type,type,
% 3.17/1.70 dpsetconstrI: $o ).
% 3.17/1.70
% 3.17/1.70 thf(dpsetconstrI,definition,
% 3.17/1.70 ( dpsetconstrI
% 3.17/1.70 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ! [Xy: $i] :
% 3.17/1.70 ( ( in @ Xy @ B )
% 3.17/1.70 => ( ( Xphi @ Xx @ Xy )
% 3.17/1.70 => ( in @ ( kpair @ Xx @ Xy )
% 3.17/1.70 @ ( dpsetconstr @ A @ B
% 3.17/1.70 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(dpsetconstrSub_type,type,
% 3.17/1.70 dpsetconstrSub: $o ).
% 3.17/1.70
% 3.17/1.70 thf(dpsetconstrSub,definition,
% 3.17/1.70 ( dpsetconstrSub
% 3.17/1.70 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 3.17/1.70 ( subset
% 3.17/1.70 @ ( dpsetconstr @ A @ B
% 3.17/1.70 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 3.17/1.70 @ ( cartprod @ A @ B ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(setOfPairsIsBReln_type,type,
% 3.17/1.70 setOfPairsIsBReln: $o ).
% 3.17/1.70
% 3.17/1.70 thf(setOfPairsIsBReln,definition,
% 3.17/1.70 ( setOfPairsIsBReln
% 3.17/1.70 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 3.17/1.70 ( breln @ A @ B
% 3.17/1.70 @ ( dpsetconstr @ A @ B
% 3.17/1.70 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(dpsetconstrERa_type,type,
% 3.17/1.70 dpsetconstrERa: $o ).
% 3.17/1.70
% 3.17/1.70 thf(dpsetconstrERa,definition,
% 3.17/1.70 ( dpsetconstrERa
% 3.17/1.70 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ! [Xy: $i] :
% 3.17/1.70 ( ( in @ Xy @ B )
% 3.17/1.70 => ( ( in @ ( kpair @ Xx @ Xy )
% 3.17/1.70 @ ( dpsetconstr @ A @ B
% 3.17/1.70 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 3.17/1.70 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(dpsetconstrEL1_type,type,
% 3.17/1.70 dpsetconstrEL1: $o ).
% 3.17/1.70
% 3.17/1.70 thf(dpsetconstrEL1,definition,
% 3.17/1.70 ( dpsetconstrEL1
% 3.17/1.70 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 3.17/1.70 ( ( in @ ( kpair @ Xx @ Xy )
% 3.17/1.70 @ ( dpsetconstr @ A @ B
% 3.17/1.70 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 3.17/1.70 => ( in @ Xx @ A ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(dpsetconstrEL2_type,type,
% 3.17/1.70 dpsetconstrEL2: $o ).
% 3.17/1.70
% 3.17/1.70 thf(dpsetconstrEL2,definition,
% 3.17/1.70 ( dpsetconstrEL2
% 3.17/1.70 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 3.17/1.70 ( ( in @ ( kpair @ Xx @ Xy )
% 3.17/1.70 @ ( dpsetconstr @ A @ B
% 3.17/1.70 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 3.17/1.70 => ( in @ Xy @ B ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(dpsetconstrER_type,type,
% 3.17/1.70 dpsetconstrER: $o ).
% 3.17/1.70
% 3.17/1.70 thf(dpsetconstrER,definition,
% 3.17/1.70 ( dpsetconstrER
% 3.17/1.70 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 3.17/1.70 ( ( in @ ( kpair @ Xx @ Xy )
% 3.17/1.70 @ ( dpsetconstr @ A @ B
% 3.17/1.70 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 3.17/1.70 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(func_type,type,
% 3.17/1.70 func: $i > $i > $i > $o ).
% 3.17/1.70
% 3.17/1.70 thf(funcSet_type,type,
% 3.17/1.70 funcSet: $i > $i > $i ).
% 3.17/1.70
% 3.17/1.70 thf(funcImageSingleton_type,type,
% 3.17/1.70 funcImageSingleton: $o ).
% 3.17/1.70
% 3.17/1.70 thf(funcImageSingleton,definition,
% 3.17/1.70 ( funcImageSingleton
% 3.17/1.70 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.17/1.70 ( ( func @ A @ B @ Xf )
% 3.17/1.70 => ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ( singleton
% 3.17/1.70 @ ( dsetconstr @ B
% 3.17/1.70 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(apProp_type,type,
% 3.17/1.70 apProp: $o ).
% 3.17/1.70
% 3.17/1.70 thf(apProp,definition,
% 3.17/1.70 ( apProp
% 3.17/1.70 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.17/1.70 ( ( func @ A @ B @ Xf )
% 3.17/1.70 => ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ( in
% 3.17/1.70 @ ( setunion
% 3.17/1.70 @ ( dsetconstr @ B
% 3.17/1.70 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 3.17/1.70 @ B ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(ap_type,type,
% 3.17/1.70 ap: $i > $i > $i > $i > $i ).
% 3.17/1.70
% 3.17/1.70 thf(app_type,type,
% 3.17/1.70 app: $o ).
% 3.17/1.70
% 3.17/1.70 thf(app,definition,
% 3.17/1.70 ( app
% 3.17/1.70 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.17/1.70 ( ( func @ A @ B @ Xf )
% 3.17/1.70 => ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(infuncsetfunc_type,type,
% 3.17/1.70 infuncsetfunc: $o ).
% 3.17/1.70
% 3.17/1.70 thf(infuncsetfunc,definition,
% 3.17/1.70 ( infuncsetfunc
% 3.17/1.70 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.17/1.70 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.17/1.70 => ( func @ A @ B @ Xf ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(ap2p_type,type,
% 3.17/1.70 ap2p: $o ).
% 3.17/1.70
% 3.17/1.70 thf(ap2p,definition,
% 3.17/1.70 ( ap2p
% 3.17/1.70 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.17/1.70 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.17/1.70 => ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(funcinfuncset_type,type,
% 3.17/1.70 funcinfuncset: $o ).
% 3.17/1.70
% 3.17/1.70 thf(funcinfuncset,definition,
% 3.17/1.70 ( funcinfuncset
% 3.17/1.70 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.17/1.70 ( ( func @ A @ B @ Xf )
% 3.17/1.70 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(lamProp_type,type,
% 3.17/1.70 lamProp: $o ).
% 3.17/1.70
% 3.17/1.70 thf(lamProp,definition,
% 3.17/1.70 ( lamProp
% 3.17/1.70 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.17/1.70 ( ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.17/1.70 => ( func @ A @ B
% 3.17/1.70 @ ( dpsetconstr @ A @ B
% 3.17/1.70 @ ^ [Xx: $i,Xy: $i] :
% 3.17/1.70 ( ( Xf @ Xx )
% 3.17/1.70 = Xy ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(lam_type,type,
% 3.17/1.70 lam: $i > $i > ( $i > $i ) > $i ).
% 3.17/1.70
% 3.17/1.70 thf(lamp_type,type,
% 3.17/1.70 lamp: $o ).
% 3.17/1.70
% 3.17/1.70 thf(lamp,definition,
% 3.17/1.70 ( lamp
% 3.17/1.70 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.17/1.70 ( ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.17/1.70 => ( func @ A @ B
% 3.17/1.70 @ ( lam @ A @ B
% 3.17/1.70 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(lam2p_type,type,
% 3.17/1.70 lam2p: $o ).
% 3.17/1.70
% 3.17/1.70 thf(lam2p,definition,
% 3.17/1.70 ( lam2p
% 3.17/1.70 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.17/1.70 ( ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.17/1.70 => ( in
% 3.17/1.70 @ ( lam @ A @ B
% 3.17/1.70 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 3.17/1.70 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(brelnall1_type,type,
% 3.17/1.70 brelnall1: $o ).
% 3.17/1.70
% 3.17/1.70 thf(brelnall1,definition,
% 3.17/1.70 ( brelnall1
% 3.17/1.70 = ( ! [A: $i,B: $i,R: $i] :
% 3.17/1.70 ( ( breln @ A @ B @ R )
% 3.17/1.70 => ! [Xphi: $i > $o] :
% 3.17/1.70 ( ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ! [Xy: $i] :
% 3.17/1.70 ( ( in @ Xy @ B )
% 3.17/1.70 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.17/1.70 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 3.17/1.70 => ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ R )
% 3.17/1.70 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(brelnall2_type,type,
% 3.17/1.70 brelnall2: $o ).
% 3.17/1.70
% 3.17/1.70 thf(brelnall2,definition,
% 3.17/1.70 ( brelnall2
% 3.17/1.70 = ( ! [A: $i,B: $i,R: $i] :
% 3.17/1.70 ( ( breln @ A @ B @ R )
% 3.17/1.70 => ! [Xphi: $i > $o] :
% 3.17/1.70 ( ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ! [Xy: $i] :
% 3.17/1.70 ( ( in @ Xy @ B )
% 3.17/1.70 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.17/1.70 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 3.17/1.70 => ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ R )
% 3.17/1.70 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(ex1E2_type,type,
% 3.17/1.70 ex1E2: $o ).
% 3.17/1.70
% 3.17/1.70 thf(ex1E2,definition,
% 3.17/1.70 ( ex1E2
% 3.17/1.70 = ( ! [A: $i,Xphi: $i > $o] :
% 3.17/1.70 ( ( ex1 @ A
% 3.17/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.17/1.70 => ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ! [Xy: $i] :
% 3.17/1.70 ( ( in @ Xy @ A )
% 3.17/1.70 => ( ( Xphi @ Xx )
% 3.17/1.70 => ( ( Xphi @ Xy )
% 3.17/1.70 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(funcGraphProp1_type,type,
% 3.17/1.70 funcGraphProp1: $o ).
% 3.17/1.70
% 3.17/1.70 thf(funcGraphProp1,definition,
% 3.17/1.70 ( funcGraphProp1
% 3.17/1.70 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.17/1.70 ( ( func @ A @ B @ Xf )
% 3.17/1.70 => ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(funcGraphProp3_type,type,
% 3.17/1.70 funcGraphProp3: $o ).
% 3.17/1.70
% 3.17/1.70 thf(funcGraphProp3,definition,
% 3.17/1.70 ( funcGraphProp3
% 3.17/1.70 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.17/1.70 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.17/1.70 => ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(funcGraphProp2_type,type,
% 3.17/1.70 funcGraphProp2: $o ).
% 3.17/1.70
% 3.17/1.70 thf(funcGraphProp2,definition,
% 3.17/1.70 ( funcGraphProp2
% 3.17/1.70 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.17/1.70 ( ( func @ A @ B @ Xf )
% 3.17/1.70 => ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ! [Xy: $i] :
% 3.17/1.70 ( ( in @ Xy @ B )
% 3.17/1.70 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 3.17/1.70 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.17/1.70 = Xy ) ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(funcextLem_type,type,
% 3.17/1.70 funcextLem: $o ).
% 3.17/1.70
% 3.17/1.70 thf(funcextLem,definition,
% 3.17/1.70 ( funcextLem
% 3.17/1.70 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.17/1.70 ( ( func @ A @ B @ Xf )
% 3.17/1.70 => ! [Xg: $i] :
% 3.17/1.70 ( ( func @ A @ B @ Xg )
% 3.17/1.70 => ( ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.17/1.70 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 3.17/1.70 => ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ! [Xy: $i] :
% 3.17/1.70 ( ( in @ Xy @ B )
% 3.17/1.70 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 3.17/1.70 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(funcGraphProp4_type,type,
% 3.17/1.70 funcGraphProp4: $o ).
% 3.17/1.70
% 3.17/1.70 thf(funcGraphProp4,definition,
% 3.17/1.70 ( funcGraphProp4
% 3.17/1.70 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.17/1.70 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.17/1.70 => ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ! [Xy: $i] :
% 3.17/1.70 ( ( in @ Xy @ B )
% 3.17/1.70 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 3.17/1.70 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.17/1.70 = Xy ) ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(subbreln_type,type,
% 3.17/1.70 subbreln: $o ).
% 3.17/1.70
% 3.17/1.70 thf(subbreln,definition,
% 3.17/1.70 ( subbreln
% 3.17/1.70 = ( ! [A: $i,B: $i,R: $i] :
% 3.17/1.70 ( ( breln @ A @ B @ R )
% 3.17/1.70 => ! [S: $i] :
% 3.17/1.70 ( ( breln @ A @ B @ S )
% 3.17/1.70 => ( ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ! [Xy: $i] :
% 3.17/1.70 ( ( in @ Xy @ B )
% 3.17/1.70 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.17/1.70 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 3.17/1.70 => ( subset @ R @ S ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(eqbreln_type,type,
% 3.17/1.70 eqbreln: $o ).
% 3.17/1.70
% 3.17/1.70 thf(eqbreln,definition,
% 3.17/1.70 ( eqbreln
% 3.17/1.70 = ( ! [A: $i,B: $i,R: $i] :
% 3.17/1.70 ( ( breln @ A @ B @ R )
% 3.17/1.70 => ! [S: $i] :
% 3.17/1.70 ( ( breln @ A @ B @ S )
% 3.17/1.70 => ( ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ! [Xy: $i] :
% 3.17/1.70 ( ( in @ Xy @ B )
% 3.17/1.70 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.17/1.70 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 3.17/1.70 => ( ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ! [Xy: $i] :
% 3.17/1.70 ( ( in @ Xy @ B )
% 3.17/1.70 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 3.17/1.70 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 3.17/1.70 => ( R = S ) ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(funcext_type,type,
% 3.17/1.70 funcext: $o ).
% 3.17/1.70
% 3.17/1.70 thf(funcext,definition,
% 3.17/1.70 ( funcext
% 3.17/1.70 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.17/1.70 ( ( func @ A @ B @ Xf )
% 3.17/1.70 => ! [Xg: $i] :
% 3.17/1.70 ( ( func @ A @ B @ Xg )
% 3.17/1.70 => ( ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.17/1.70 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 3.17/1.70 => ( Xf = Xg ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(funcext2_type,type,
% 3.17/1.70 funcext2: $o ).
% 3.17/1.70
% 3.17/1.70 thf(funcext2,definition,
% 3.17/1.70 ( funcext2
% 3.17/1.70 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.17/1.70 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.17/1.70 => ! [Xg: $i] :
% 3.17/1.70 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 3.17/1.70 => ( ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.17/1.70 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 3.17/1.70 => ( Xf = Xg ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(ap2apEq1_type,type,
% 3.17/1.70 ap2apEq1: $o ).
% 3.17/1.70
% 3.17/1.70 thf(ap2apEq1,definition,
% 3.17/1.70 ( ap2apEq1
% 3.17/1.70 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.17/1.70 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.17/1.70 => ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.17/1.70 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(ap2apEq2_type,type,
% 3.17/1.70 ap2apEq2: $o ).
% 3.17/1.70
% 3.17/1.70 thf(ap2apEq2,definition,
% 3.17/1.70 ( ap2apEq2
% 3.17/1.70 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.17/1.70 ( ( func @ A @ B @ Xf )
% 3.17/1.70 => ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.17/1.70 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(beta1_type,type,
% 3.17/1.70 beta1: $o ).
% 3.17/1.70
% 3.17/1.70 thf(beta1,definition,
% 3.17/1.70 ( beta1
% 3.17/1.70 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.17/1.70 ( ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.17/1.70 => ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ( ( ap @ A @ B
% 3.17/1.70 @ ( lam @ A @ B
% 3.17/1.70 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 3.17/1.70 @ Xx )
% 3.17/1.70 = ( Xf @ Xx ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(eta1_type,type,
% 3.17/1.70 eta1: $o ).
% 3.17/1.70
% 3.17/1.70 thf(eta1,definition,
% 3.17/1.70 ( eta1
% 3.17/1.70 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.17/1.70 ( ( func @ A @ B @ Xf )
% 3.17/1.70 => ( ( lam @ A @ B
% 3.17/1.70 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 3.17/1.70 = Xf ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(lam2lamEq_type,type,
% 3.17/1.70 lam2lamEq: $o ).
% 3.17/1.70
% 3.17/1.70 thf(lam2lamEq,definition,
% 3.17/1.70 ( lam2lamEq
% 3.17/1.70 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.17/1.70 ( ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.17/1.70 => ( ( lam @ A @ B
% 3.17/1.70 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 3.17/1.70 = ( lam @ A @ B
% 3.17/1.70 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(beta2_type,type,
% 3.17/1.70 beta2: $o ).
% 3.17/1.70
% 3.17/1.70 thf(beta2,definition,
% 3.17/1.70 ( beta2
% 3.17/1.70 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.17/1.70 ( ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.17/1.70 => ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ( ( ap @ A @ B
% 3.17/1.70 @ ( lam @ A @ B
% 3.17/1.70 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 3.17/1.70 @ Xx )
% 3.17/1.70 = ( Xf @ Xx ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(eta2_type,type,
% 3.17/1.70 eta2: $o ).
% 3.17/1.70
% 3.17/1.70 thf(eta2,definition,
% 3.17/1.70 ( eta2
% 3.17/1.70 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.17/1.70 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.17/1.70 => ( ( lam @ A @ B
% 3.17/1.70 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 3.17/1.70 = Xf ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(iffalseProp1_type,type,
% 3.17/1.70 iffalseProp1: $o ).
% 3.17/1.70
% 3.17/1.70 thf(iffalseProp1,definition,
% 3.17/1.70 ( iffalseProp1
% 3.17/1.70 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ! [Xy: $i] :
% 3.17/1.70 ( ( in @ Xy @ A )
% 3.17/1.70 => ( ~ Xphi
% 3.17/1.70 => ( in @ Xy
% 3.17/1.70 @ ( dsetconstr @ A
% 3.17/1.70 @ ^ [Xz: $i] :
% 3.17/1.70 ( ( Xphi
% 3.17/1.70 & ( Xz = Xx ) )
% 3.17/1.70 | ( ~ Xphi
% 3.17/1.70 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(iffalseProp2_type,type,
% 3.17/1.70 iffalseProp2: $o ).
% 3.17/1.70
% 3.17/1.70 thf(iffalseProp2,definition,
% 3.17/1.70 ( iffalseProp2
% 3.17/1.70 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ! [Xy: $i] :
% 3.17/1.70 ( ( in @ Xy @ A )
% 3.17/1.70 => ( ~ Xphi
% 3.17/1.70 => ( ( dsetconstr @ A
% 3.17/1.70 @ ^ [Xz: $i] :
% 3.17/1.70 ( ( Xphi
% 3.17/1.70 & ( Xz = Xx ) )
% 3.17/1.70 | ( ~ Xphi
% 3.17/1.70 & ( Xz = Xy ) ) ) )
% 3.17/1.70 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(iftrueProp1_type,type,
% 3.17/1.70 iftrueProp1: $o ).
% 3.17/1.70
% 3.17/1.70 thf(iftrueProp1,definition,
% 3.17/1.70 ( iftrueProp1
% 3.17/1.70 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ! [Xy: $i] :
% 3.17/1.70 ( ( in @ Xy @ A )
% 3.17/1.70 => ( Xphi
% 3.17/1.70 => ( in @ Xx
% 3.17/1.70 @ ( dsetconstr @ A
% 3.17/1.70 @ ^ [Xz: $i] :
% 3.17/1.70 ( ( Xphi
% 3.17/1.70 & ( Xz = Xx ) )
% 3.17/1.70 | ( ~ Xphi
% 3.17/1.70 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(iftrueProp2_type,type,
% 3.17/1.70 iftrueProp2: $o ).
% 3.17/1.70
% 3.17/1.70 thf(iftrueProp2,definition,
% 3.17/1.70 ( iftrueProp2
% 3.17/1.70 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ! [Xy: $i] :
% 3.17/1.70 ( ( in @ Xy @ A )
% 3.17/1.70 => ( Xphi
% 3.17/1.70 => ( ( dsetconstr @ A
% 3.17/1.70 @ ^ [Xz: $i] :
% 3.17/1.70 ( ( Xphi
% 3.17/1.70 & ( Xz = Xx ) )
% 3.17/1.70 | ( ~ Xphi
% 3.17/1.70 & ( Xz = Xy ) ) ) )
% 3.17/1.70 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(ifSingleton_type,type,
% 3.17/1.70 ifSingleton: $o ).
% 3.17/1.70
% 3.17/1.70 thf(ifSingleton,definition,
% 3.17/1.70 ( ifSingleton
% 3.17/1.70 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ! [Xy: $i] :
% 3.17/1.70 ( ( in @ Xy @ A )
% 3.17/1.70 => ( singleton
% 3.17/1.70 @ ( dsetconstr @ A
% 3.17/1.70 @ ^ [Xz: $i] :
% 3.17/1.70 ( ( Xphi
% 3.17/1.70 & ( Xz = Xx ) )
% 3.17/1.70 | ( ~ Xphi
% 3.17/1.70 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(if_type,type,
% 3.17/1.70 if: $i > $o > $i > $i > $i ).
% 3.17/1.70
% 3.17/1.70 thf(ifp_type,type,
% 3.17/1.70 ifp: $o ).
% 3.17/1.70
% 3.17/1.70 thf(ifp,definition,
% 3.17/1.70 ( ifp
% 3.17/1.70 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ! [Xy: $i] :
% 3.17/1.70 ( ( in @ Xy @ A )
% 3.17/1.70 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(theeq_type,type,
% 3.17/1.70 theeq: $o ).
% 3.17/1.70
% 3.17/1.70 thf(theeq,definition,
% 3.17/1.70 ( theeq
% 3.17/1.70 = ( ! [X: $i] :
% 3.17/1.70 ( ( singleton @ X )
% 3.17/1.70 => ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ X )
% 3.17/1.70 => ( ( setunion @ X )
% 3.17/1.70 = Xx ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(iftrue_type,type,
% 3.17/1.70 iftrue: $o ).
% 3.17/1.70
% 3.17/1.70 thf(iftrue,definition,
% 3.17/1.70 ( iftrue
% 3.17/1.70 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ! [Xy: $i] :
% 3.17/1.70 ( ( in @ Xy @ A )
% 3.17/1.70 => ( Xphi
% 3.17/1.70 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 3.17/1.70 = Xx ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(iffalse_type,type,
% 3.17/1.70 iffalse: $o ).
% 3.17/1.70
% 3.17/1.70 thf(iffalse,definition,
% 3.17/1.70 ( iffalse
% 3.17/1.70 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ! [Xy: $i] :
% 3.17/1.70 ( ( in @ Xy @ A )
% 3.17/1.70 => ( ~ Xphi
% 3.17/1.70 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 3.17/1.70 = Xy ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(iftrueorfalse_type,type,
% 3.17/1.70 iftrueorfalse: $o ).
% 3.17/1.70
% 3.17/1.70 thf(iftrueorfalse,definition,
% 3.17/1.70 ( iftrueorfalse
% 3.17/1.70 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ! [Xy: $i] :
% 3.17/1.70 ( ( in @ Xy @ A )
% 3.17/1.70 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(binintersectT_lem_type,type,
% 3.17/1.70 binintersectT_lem: $o ).
% 3.17/1.70
% 3.17/1.70 thf(binintersectT_lem,definition,
% 3.17/1.70 ( binintersectT_lem
% 3.17/1.70 = ( ! [A: $i,X: $i] :
% 3.17/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.70 => ! [Y: $i] :
% 3.17/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.70 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(binunionT_lem_type,type,
% 3.17/1.70 binunionT_lem: $o ).
% 3.17/1.70
% 3.17/1.70 thf(binunionT_lem,definition,
% 3.17/1.70 ( binunionT_lem
% 3.17/1.70 = ( ! [A: $i,X: $i] :
% 3.17/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.70 => ! [Y: $i] :
% 3.17/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.70 => ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(powersetT_lem_type,type,
% 3.17/1.70 powersetT_lem: $o ).
% 3.17/1.70
% 3.17/1.70 thf(powersetT_lem,definition,
% 3.17/1.70 ( powersetT_lem
% 3.17/1.70 = ( ! [A: $i,X: $i] :
% 3.17/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.70 => ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(setminusT_lem_type,type,
% 3.17/1.70 setminusT_lem: $o ).
% 3.17/1.70
% 3.17/1.70 thf(setminusT_lem,definition,
% 3.17/1.70 ( setminusT_lem
% 3.17/1.70 = ( ! [A: $i,X: $i] :
% 3.17/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.70 => ! [Y: $i] :
% 3.17/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.70 => ( in @ ( setminus @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(complementT_lem_type,type,
% 3.17/1.70 complementT_lem: $o ).
% 3.17/1.70
% 3.17/1.70 thf(complementT_lem,definition,
% 3.17/1.70 ( complementT_lem
% 3.17/1.70 = ( ! [A: $i,X: $i] :
% 3.17/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.70 => ( in @ ( setminus @ A @ X ) @ ( powerset @ A ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(setextT_type,type,
% 3.17/1.70 setextT: $o ).
% 3.17/1.70
% 3.17/1.70 thf(setextT,definition,
% 3.17/1.70 ( setextT
% 3.17/1.70 = ( ! [A: $i,X: $i] :
% 3.17/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.70 => ! [Y: $i] :
% 3.17/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.70 => ( ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ( ( in @ Xx @ X )
% 3.17/1.70 => ( in @ Xx @ Y ) ) )
% 3.17/1.70 => ( ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ( ( in @ Xx @ Y )
% 3.17/1.70 => ( in @ Xx @ X ) ) )
% 3.17/1.70 => ( X = Y ) ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(subsetTI_type,type,
% 3.17/1.70 subsetTI: $o ).
% 3.17/1.70
% 3.17/1.70 thf(subsetTI,definition,
% 3.17/1.70 ( subsetTI
% 3.17/1.70 = ( ! [A: $i,X: $i] :
% 3.17/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.70 => ! [Y: $i] :
% 3.17/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.70 => ( ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ( ( in @ Xx @ X )
% 3.17/1.70 => ( in @ Xx @ Y ) ) )
% 3.17/1.70 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(powersetTI1_type,type,
% 3.17/1.70 powersetTI1: $o ).
% 3.17/1.70
% 3.17/1.70 thf(powersetTI1,definition,
% 3.17/1.70 ( powersetTI1
% 3.17/1.70 = ( ! [A: $i,X: $i] :
% 3.17/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.70 => ! [Y: $i] :
% 3.17/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.70 => ( ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ( ( in @ Xx @ X )
% 3.17/1.70 => ( in @ Xx @ Y ) ) )
% 3.17/1.70 => ( in @ X @ ( powerset @ Y ) ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(powersetTE1_type,type,
% 3.17/1.70 powersetTE1: $o ).
% 3.17/1.70
% 3.17/1.70 thf(powersetTE1,definition,
% 3.17/1.70 ( powersetTE1
% 3.17/1.70 = ( ! [A: $i,X: $i] :
% 3.17/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.70 => ! [Y: $i] :
% 3.17/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.70 => ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ( ( in @ X @ ( powerset @ Y ) )
% 3.17/1.70 => ( ( in @ Xx @ X )
% 3.17/1.70 => ( in @ Xx @ Y ) ) ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(complementTI1_type,type,
% 3.17/1.70 complementTI1: $o ).
% 3.17/1.70
% 3.17/1.70 thf(complementTI1,definition,
% 3.17/1.70 ( complementTI1
% 3.17/1.70 = ( ! [A: $i,X: $i] :
% 3.17/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.70 => ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ( ( in @ Xx @ X )
% 3.17/1.70 => ~ ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(complementTE1_type,type,
% 3.17/1.70 complementTE1: $o ).
% 3.17/1.70
% 3.17/1.70 thf(complementTE1,definition,
% 3.17/1.70 ( complementTE1
% 3.17/1.70 = ( ! [A: $i,X: $i] :
% 3.17/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.70 => ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ( ~ ( in @ Xx @ ( setminus @ A @ X ) )
% 3.17/1.70 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(binintersectTELcontra_type,type,
% 3.17/1.70 binintersectTELcontra: $o ).
% 3.17/1.70
% 3.17/1.70 thf(binintersectTELcontra,definition,
% 3.17/1.70 ( binintersectTELcontra
% 3.17/1.70 = ( ! [A: $i,X: $i] :
% 3.17/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.70 => ! [Y: $i] :
% 3.17/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.70 => ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ( ~ ( in @ Xx @ X )
% 3.17/1.70 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(binintersectTERcontra_type,type,
% 3.17/1.70 binintersectTERcontra: $o ).
% 3.17/1.70
% 3.17/1.70 thf(binintersectTERcontra,definition,
% 3.17/1.70 ( binintersectTERcontra
% 3.17/1.70 = ( ! [A: $i,X: $i] :
% 3.17/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.70 => ! [Y: $i] :
% 3.17/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.70 => ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ( ~ ( in @ Xx @ Y )
% 3.17/1.70 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(contrasubsetT_type,type,
% 3.17/1.70 contrasubsetT: $o ).
% 3.17/1.70
% 3.17/1.70 thf(contrasubsetT,definition,
% 3.17/1.70 ( contrasubsetT
% 3.17/1.70 = ( ! [A: $i,X: $i] :
% 3.17/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.70 => ! [Y: $i] :
% 3.17/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.70 => ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 3.17/1.70 => ( ( in @ Xx @ Y )
% 3.17/1.70 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(contrasubsetT1_type,type,
% 3.17/1.70 contrasubsetT1: $o ).
% 3.17/1.70
% 3.17/1.70 thf(contrasubsetT1,definition,
% 3.17/1.70 ( contrasubsetT1
% 3.17/1.70 = ( ! [A: $i,X: $i] :
% 3.17/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.70 => ! [Y: $i] :
% 3.17/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.70 => ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ( ( subset @ X @ Y )
% 3.17/1.70 => ( ~ ( in @ Xx @ Y )
% 3.17/1.70 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(contrasubsetT2_type,type,
% 3.17/1.70 contrasubsetT2: $o ).
% 3.17/1.70
% 3.17/1.70 thf(contrasubsetT2,definition,
% 3.17/1.70 ( contrasubsetT2
% 3.17/1.70 = ( ! [A: $i,X: $i] :
% 3.17/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.70 => ! [Y: $i] :
% 3.17/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.70 => ( ( subset @ X @ Y )
% 3.17/1.70 => ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(contrasubsetT3_type,type,
% 3.17/1.70 contrasubsetT3: $o ).
% 3.17/1.70
% 3.17/1.70 thf(contrasubsetT3,definition,
% 3.17/1.70 ( contrasubsetT3
% 3.17/1.70 = ( ! [A: $i,X: $i] :
% 3.17/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.70 => ! [Y: $i] :
% 3.17/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.70 => ( ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) )
% 3.17/1.70 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(doubleComplementI1_type,type,
% 3.17/1.70 doubleComplementI1: $o ).
% 3.17/1.70
% 3.17/1.70 thf(doubleComplementI1,definition,
% 3.17/1.70 ( doubleComplementI1
% 3.17/1.70 = ( ! [A: $i,X: $i] :
% 3.17/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.70 => ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ( ( in @ Xx @ X )
% 3.17/1.70 => ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(doubleComplementE1_type,type,
% 3.17/1.70 doubleComplementE1: $o ).
% 3.17/1.70
% 3.17/1.70 thf(doubleComplementE1,definition,
% 3.17/1.70 ( doubleComplementE1
% 3.17/1.70 = ( ! [A: $i,X: $i] :
% 3.17/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.70 => ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ( ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) )
% 3.17/1.70 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(doubleComplementSub1_type,type,
% 3.17/1.70 doubleComplementSub1: $o ).
% 3.17/1.70
% 3.17/1.70 thf(doubleComplementSub1,definition,
% 3.17/1.70 ( doubleComplementSub1
% 3.17/1.70 = ( ! [A: $i,X: $i] :
% 3.17/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.70 => ( subset @ X @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(doubleComplementSub2_type,type,
% 3.17/1.70 doubleComplementSub2: $o ).
% 3.17/1.70
% 3.17/1.70 thf(doubleComplementSub2,definition,
% 3.17/1.70 ( doubleComplementSub2
% 3.17/1.70 = ( ! [A: $i,X: $i] :
% 3.17/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.70 => ( subset @ ( setminus @ A @ ( setminus @ A @ X ) ) @ X ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(doubleComplementEq_type,type,
% 3.17/1.70 doubleComplementEq: $o ).
% 3.17/1.70
% 3.17/1.70 thf(doubleComplementEq,definition,
% 3.17/1.70 ( doubleComplementEq
% 3.17/1.70 = ( ! [A: $i,X: $i] :
% 3.17/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.70 => ( X
% 3.17/1.70 = ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(complementTnotintersectT_type,type,
% 3.17/1.70 complementTnotintersectT: $o ).
% 3.17/1.70
% 3.17/1.70 thf(complementTnotintersectT,definition,
% 3.17/1.70 ( complementTnotintersectT
% 3.17/1.70 = ( ! [A: $i,X: $i] :
% 3.17/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.70 => ! [Y: $i] :
% 3.17/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.70 => ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 3.17/1.70 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(complementImpComplementIntersect_type,type,
% 3.17/1.70 complementImpComplementIntersect: $o ).
% 3.17/1.70
% 3.17/1.70 thf(complementImpComplementIntersect,definition,
% 3.17/1.70 ( complementImpComplementIntersect
% 3.17/1.70 = ( ! [A: $i,X: $i] :
% 3.17/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.70 => ! [Y: $i] :
% 3.17/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.70 => ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 3.17/1.70 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(complementSubsetComplementIntersect_type,type,
% 3.17/1.70 complementSubsetComplementIntersect: $o ).
% 3.17/1.70
% 3.17/1.70 thf(complementSubsetComplementIntersect,definition,
% 3.17/1.70 ( complementSubsetComplementIntersect
% 3.17/1.70 = ( ! [A: $i,X: $i] :
% 3.17/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.70 => ! [Y: $i] :
% 3.17/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.70 => ( subset @ ( setminus @ A @ X ) @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(complementInPowersetComplementIntersect_type,type,
% 3.17/1.70 complementInPowersetComplementIntersect: $o ).
% 3.17/1.70
% 3.17/1.70 thf(complementInPowersetComplementIntersect,definition,
% 3.17/1.70 ( complementInPowersetComplementIntersect
% 3.17/1.70 = ( ! [A: $i,X: $i] :
% 3.17/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.70 => ! [Y: $i] :
% 3.17/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.70 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(contraSubsetComplement_type,type,
% 3.17/1.70 contraSubsetComplement: $o ).
% 3.17/1.70
% 3.17/1.70 thf(contraSubsetComplement,definition,
% 3.17/1.70 ( contraSubsetComplement
% 3.17/1.70 = ( ! [A: $i,X: $i] :
% 3.17/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.70 => ! [Y: $i] :
% 3.17/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.70 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 3.17/1.70 => ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ( ( in @ Xx @ Y )
% 3.17/1.70 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(complementTcontraSubset_type,type,
% 3.17/1.70 complementTcontraSubset: $o ).
% 3.17/1.70
% 3.17/1.70 thf(complementTcontraSubset,definition,
% 3.17/1.70 ( complementTcontraSubset
% 3.17/1.70 = ( ! [A: $i,X: $i] :
% 3.17/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.70 => ! [Y: $i] :
% 3.17/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.70 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 3.17/1.70 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(binunionTILcontra_type,type,
% 3.17/1.70 binunionTILcontra: $o ).
% 3.17/1.70
% 3.17/1.70 thf(binunionTILcontra,definition,
% 3.17/1.70 ( binunionTILcontra
% 3.17/1.70 = ( ! [A: $i,X: $i] :
% 3.17/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.70 => ! [Y: $i] :
% 3.17/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.70 => ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 3.17/1.70 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(binunionTIRcontra_type,type,
% 3.17/1.70 binunionTIRcontra: $o ).
% 3.17/1.70
% 3.17/1.70 thf(binunionTIRcontra,definition,
% 3.17/1.70 ( binunionTIRcontra
% 3.17/1.70 = ( ! [A: $i,X: $i] :
% 3.17/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.70 => ! [Y: $i] :
% 3.17/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.70 => ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 3.17/1.70 => ~ ( in @ Xx @ Y ) ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(inIntersectImpInUnion_type,type,
% 3.17/1.70 inIntersectImpInUnion: $o ).
% 3.17/1.70
% 3.17/1.70 thf(inIntersectImpInUnion,definition,
% 3.17/1.70 ( inIntersectImpInUnion
% 3.17/1.70 = ( ! [A: $i,X: $i] :
% 3.17/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.70 => ! [Y: $i] :
% 3.17/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.70 => ! [Z: $i] :
% 3.17/1.70 ( ( in @ Z @ ( powerset @ A ) )
% 3.17/1.70 => ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 3.17/1.70 => ( in @ Xx @ ( binunion @ X @ Z ) ) ) ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(inIntersectImpInUnion2_type,type,
% 3.17/1.70 inIntersectImpInUnion2: $o ).
% 3.17/1.70
% 3.17/1.70 thf(inIntersectImpInUnion2,definition,
% 3.17/1.70 ( inIntersectImpInUnion2
% 3.17/1.70 = ( ! [A: $i,X: $i] :
% 3.17/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.70 => ! [Y: $i] :
% 3.17/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.70 => ! [Z: $i] :
% 3.17/1.70 ( ( in @ Z @ ( powerset @ A ) )
% 3.17/1.70 => ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 3.17/1.70 => ( in @ Xx @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(inIntersectImpInIntersectUnions_type,type,
% 3.17/1.70 inIntersectImpInIntersectUnions: $o ).
% 3.17/1.70
% 3.17/1.70 thf(inIntersectImpInIntersectUnions,definition,
% 3.17/1.70 ( inIntersectImpInIntersectUnions
% 3.17/1.70 = ( ! [A: $i,X: $i] :
% 3.17/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.70 => ! [Y: $i] :
% 3.17/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.70 => ! [Z: $i] :
% 3.17/1.70 ( ( in @ Z @ ( powerset @ A ) )
% 3.17/1.70 => ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 3.17/1.70 => ( in @ Xx @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(intersectInPowersetIntersectUnions_type,type,
% 3.17/1.70 intersectInPowersetIntersectUnions: $o ).
% 3.17/1.70
% 3.17/1.70 thf(intersectInPowersetIntersectUnions,definition,
% 3.17/1.70 ( intersectInPowersetIntersectUnions
% 3.17/1.70 = ( ! [A: $i,X: $i] :
% 3.17/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.70 => ! [Y: $i] :
% 3.17/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.70 => ! [Z: $i] :
% 3.17/1.70 ( ( in @ Z @ ( powerset @ A ) )
% 3.17/1.70 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(inComplementUnionImpNotIn1_type,type,
% 3.17/1.70 inComplementUnionImpNotIn1: $o ).
% 3.17/1.70
% 3.17/1.70 thf(inComplementUnionImpNotIn1,definition,
% 3.17/1.70 ( inComplementUnionImpNotIn1
% 3.17/1.70 = ( ! [A: $i,X: $i] :
% 3.17/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.70 => ! [Y: $i] :
% 3.17/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.70 => ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 3.17/1.70 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(inComplementUnionImpInComplement1_type,type,
% 3.17/1.70 inComplementUnionImpInComplement1: $o ).
% 3.17/1.70
% 3.17/1.70 thf(inComplementUnionImpInComplement1,definition,
% 3.17/1.70 ( inComplementUnionImpInComplement1
% 3.17/1.70 = ( ! [A: $i,X: $i] :
% 3.17/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.70 => ! [Y: $i] :
% 3.17/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.70 => ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 3.17/1.70 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(binunionTE_type,type,
% 3.17/1.70 binunionTE: $o ).
% 3.17/1.70
% 3.17/1.70 thf(binunionTE,definition,
% 3.17/1.70 ( binunionTE
% 3.17/1.70 = ( ! [A: $i,X: $i] :
% 3.17/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.70 => ! [Y: $i] :
% 3.17/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.70 => ! [Xphi: $o,Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ( ( in @ Xx @ ( binunion @ X @ Y ) )
% 3.17/1.70 => ( ( ( in @ Xx @ X )
% 3.17/1.70 => Xphi )
% 3.17/1.70 => ( ( ( in @ Xx @ Y )
% 3.17/1.70 => Xphi )
% 3.17/1.70 => Xphi ) ) ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(binunionTEcontra_type,type,
% 3.17/1.70 binunionTEcontra: $o ).
% 3.17/1.70
% 3.17/1.70 thf(binunionTEcontra,definition,
% 3.17/1.70 ( binunionTEcontra
% 3.17/1.70 = ( ! [A: $i,X: $i] :
% 3.17/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.70 => ! [Y: $i] :
% 3.17/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.70 => ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ( ~ ( in @ Xx @ X )
% 3.17/1.70 => ( ~ ( in @ Xx @ Y )
% 3.17/1.70 => ~ ( in @ Xx @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(demorgan2a1_type,type,
% 3.17/1.70 demorgan2a1: $o ).
% 3.17/1.70
% 3.17/1.70 thf(demorgan2a1,definition,
% 3.17/1.70 ( demorgan2a1
% 3.17/1.70 = ( ! [A: $i,X: $i] :
% 3.17/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.70 => ! [Y: $i] :
% 3.17/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.70 => ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 3.17/1.70 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(complementUnionInPowersetComplement_type,type,
% 3.17/1.70 complementUnionInPowersetComplement: $o ).
% 3.17/1.70
% 3.17/1.70 thf(complementUnionInPowersetComplement,definition,
% 3.17/1.70 ( complementUnionInPowersetComplement
% 3.17/1.70 = ( ! [A: $i,X: $i] :
% 3.17/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.70 => ! [Y: $i] :
% 3.17/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.70 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(demorgan2a2_type,type,
% 3.17/1.70 demorgan2a2: $o ).
% 3.17/1.70
% 3.17/1.70 thf(demorgan2a2,definition,
% 3.17/1.70 ( demorgan2a2
% 3.17/1.70 = ( ! [A: $i,X: $i] :
% 3.17/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.70 => ! [Y: $i] :
% 3.17/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.70 => ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 3.17/1.70 => ( in @ Xx @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(demorgan1a_type,type,
% 3.17/1.70 demorgan1a: $o ).
% 3.17/1.70
% 3.17/1.70 thf(demorgan1a,definition,
% 3.17/1.70 ( demorgan1a
% 3.17/1.70 = ( ! [A: $i,X: $i] :
% 3.17/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.70 => ! [Y: $i] :
% 3.17/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.70 => ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ( ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) )
% 3.17/1.70 => ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(demorgan1b_type,type,
% 3.17/1.70 demorgan1b: $o ).
% 3.17/1.70
% 3.17/1.70 thf(demorgan1b,definition,
% 3.17/1.70 ( demorgan1b
% 3.17/1.70 = ( ! [A: $i,X: $i] :
% 3.17/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.70 => ! [Y: $i] :
% 3.17/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.70 => ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ( ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 3.17/1.70 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(demorgan1_type,type,
% 3.17/1.70 demorgan1: $o ).
% 3.17/1.70
% 3.17/1.70 thf(demorgan1,definition,
% 3.17/1.70 ( demorgan1
% 3.17/1.70 = ( ! [A: $i,X: $i] :
% 3.17/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.70 => ! [Y: $i] :
% 3.17/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.70 => ( ( setminus @ A @ ( binintersect @ X @ Y ) )
% 3.17/1.70 = ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(demorgan2a_type,type,
% 3.17/1.70 demorgan2a: $o ).
% 3.17/1.70
% 3.17/1.70 thf(demorgan2a,definition,
% 3.17/1.70 ( demorgan2a
% 3.17/1.70 = ( ! [A: $i,X: $i] :
% 3.17/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.70 => ! [Y: $i] :
% 3.17/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.70 => ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 3.17/1.70 => ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(demorgan2b2_type,type,
% 3.17/1.70 demorgan2b2: $o ).
% 3.17/1.70
% 3.17/1.70 thf(demorgan2b2,definition,
% 3.17/1.70 ( demorgan2b2
% 3.17/1.70 = ( ! [A: $i,X: $i] :
% 3.17/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.70 => ! [Y: $i] :
% 3.17/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.70 => ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 3.17/1.70 => ( ( in @ Xx @ ( setminus @ A @ Y ) )
% 3.17/1.70 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(demorgan2b_type,type,
% 3.17/1.70 demorgan2b: $o ).
% 3.17/1.70
% 3.17/1.70 thf(demorgan2b,definition,
% 3.17/1.70 ( demorgan2b
% 3.17/1.70 = ( ! [A: $i,X: $i] :
% 3.17/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.70 => ! [Y: $i] :
% 3.17/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.70 => ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ( ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 3.17/1.70 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(demorgan2_type,type,
% 3.17/1.70 demorgan2: $o ).
% 3.17/1.70
% 3.17/1.70 thf(demorgan2,definition,
% 3.17/1.70 ( demorgan2
% 3.17/1.70 = ( ! [A: $i,X: $i] :
% 3.17/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.70 => ! [Y: $i] :
% 3.17/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.70 => ( ( setminus @ A @ ( binunion @ X @ Y ) )
% 3.17/1.70 = ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(woz13rule0_type,type,
% 3.17/1.70 woz13rule0: $o ).
% 3.17/1.70
% 3.17/1.70 thf(woz13rule0,definition,
% 3.17/1.70 ( woz13rule0
% 3.17/1.70 = ( ! [A: $i,X: $i] :
% 3.17/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.70 => ! [Y: $i] :
% 3.17/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.70 => ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 3.17/1.70 => ( in @ Xx @ A ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(woz13rule1_type,type,
% 3.17/1.70 woz13rule1: $o ).
% 3.17/1.70
% 3.17/1.70 thf(woz13rule1,definition,
% 3.17/1.70 ( woz13rule1
% 3.17/1.70 = ( ! [A: $i,X: $i] :
% 3.17/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.70 => ! [Y: $i] :
% 3.17/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.70 => ! [Z: $i] :
% 3.17/1.70 ( ( in @ Z @ ( powerset @ A ) )
% 3.17/1.70 => ( ( subset @ X @ Z )
% 3.17/1.70 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(woz13rule2_type,type,
% 3.17/1.70 woz13rule2: $o ).
% 3.17/1.70
% 3.17/1.70 thf(woz13rule2,definition,
% 3.17/1.70 ( woz13rule2
% 3.17/1.70 = ( ! [A: $i,X: $i] :
% 3.17/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.70 => ! [Y: $i] :
% 3.17/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.70 => ! [Z: $i] :
% 3.17/1.70 ( ( in @ Z @ ( powerset @ A ) )
% 3.17/1.70 => ( ( subset @ Y @ Z )
% 3.17/1.70 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(woz13rule3_type,type,
% 3.17/1.70 woz13rule3: $o ).
% 3.17/1.70
% 3.17/1.70 thf(woz13rule3,definition,
% 3.17/1.70 ( woz13rule3
% 3.17/1.70 = ( ! [A: $i,X: $i] :
% 3.17/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.70 => ! [Y: $i] :
% 3.17/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.70 => ! [Z: $i] :
% 3.17/1.70 ( ( in @ Z @ ( powerset @ A ) )
% 3.17/1.70 => ( ( subset @ X @ Y )
% 3.17/1.70 => ( ( subset @ X @ Z )
% 3.17/1.70 => ( subset @ X @ ( binintersect @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(woz13rule4_type,type,
% 3.17/1.70 woz13rule4: $o ).
% 3.17/1.70
% 3.17/1.70 thf(woz13rule4,definition,
% 3.17/1.70 ( woz13rule4
% 3.17/1.70 = ( ! [A: $i,X: $i] :
% 3.17/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.70 => ! [Y: $i] :
% 3.17/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.70 => ! [Z: $i] :
% 3.17/1.70 ( ( in @ Z @ ( powerset @ A ) )
% 3.17/1.70 => ! [W: $i] :
% 3.17/1.70 ( ( in @ W @ ( powerset @ A ) )
% 3.17/1.70 => ( ( subset @ X @ Z )
% 3.17/1.70 => ( ( subset @ Y @ W )
% 3.17/1.70 => ( subset @ ( binintersect @ X @ Y ) @ ( binintersect @ Z @ W ) ) ) ) ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(woz1_1_type,type,
% 3.17/1.70 woz1_1: $o ).
% 3.17/1.70
% 3.17/1.70 thf(woz1_1,definition,
% 3.17/1.70 ( woz1_1
% 3.17/1.70 = ( ! [A: $i,X: $i] :
% 3.17/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.70 => ! [Y: $i] :
% 3.17/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.70 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(woz1_2_type,type,
% 3.17/1.70 woz1_2: $o ).
% 3.17/1.70
% 3.17/1.70 thf(woz1_2,definition,
% 3.17/1.70 ( woz1_2
% 3.17/1.70 = ( ! [A: $i,X: $i] :
% 3.17/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.70 => ! [Y: $i] :
% 3.17/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.70 => ! [Z: $i] :
% 3.17/1.70 ( ( in @ Z @ ( powerset @ A ) )
% 3.17/1.70 => ! [W: $i] :
% 3.17/1.70 ( ( in @ W @ ( powerset @ A ) )
% 3.17/1.70 => ( ( setminus @ A @ ( binintersect @ ( binunion @ X @ Y ) @ ( binunion @ Z @ W ) ) )
% 3.17/1.70 = ( binunion @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) @ ( binintersect @ ( setminus @ A @ Z ) @ ( setminus @ A @ W ) ) ) ) ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(woz1_3_type,type,
% 3.17/1.70 woz1_3: $o ).
% 3.17/1.70
% 3.17/1.70 thf(woz1_3,definition,
% 3.17/1.70 ( woz1_3
% 3.17/1.70 = ( ! [A: $i,X: $i] :
% 3.17/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.70 => ! [Y: $i] :
% 3.17/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.70 => ! [Z: $i] :
% 3.17/1.70 ( ( in @ Z @ ( powerset @ A ) )
% 3.17/1.70 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(woz1_4_type,type,
% 3.17/1.70 woz1_4: $o ).
% 3.17/1.70
% 3.17/1.70 thf(woz1_4,definition,
% 3.17/1.70 ( woz1_4
% 3.17/1.70 = ( ! [A: $i,X: $i] :
% 3.17/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.70 => ! [Y: $i] :
% 3.17/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.70 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 3.17/1.70 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(woz1_5_type,type,
% 3.17/1.70 woz1_5: $o ).
% 3.17/1.70
% 3.17/1.70 thf(woz1_5,definition,
% 3.17/1.70 ( woz1_5
% 3.17/1.70 = ( ! [A: $i,X: $i] :
% 3.17/1.70 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.70 => ! [Y: $i] :
% 3.17/1.70 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.70 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(breln1_type,type,
% 3.17/1.70 breln1: $i > $i > $o ).
% 3.17/1.70
% 3.17/1.70 thf(breln1all2_type,type,
% 3.17/1.70 breln1all2: $o ).
% 3.17/1.70
% 3.17/1.70 thf(breln1all2,definition,
% 3.17/1.70 ( breln1all2
% 3.17/1.70 = ( ! [A: $i,R: $i] :
% 3.17/1.70 ( ( breln1 @ A @ R )
% 3.17/1.70 => ! [Xphi: $i > $o] :
% 3.17/1.70 ( ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ! [Xy: $i] :
% 3.17/1.70 ( ( in @ Xy @ A )
% 3.17/1.70 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.17/1.70 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 3.17/1.70 => ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ R )
% 3.17/1.70 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(breln1Set_type,type,
% 3.17/1.70 breln1Set: $i > $i ).
% 3.17/1.70
% 3.17/1.70 thf(breln1SetBreln1_type,type,
% 3.17/1.70 breln1SetBreln1: $o ).
% 3.17/1.70
% 3.17/1.70 thf(breln1SetBreln1,definition,
% 3.17/1.70 ( breln1SetBreln1
% 3.17/1.70 = ( ! [A: $i,R: $i] :
% 3.17/1.70 ( ( in @ R @ ( breln1Set @ A ) )
% 3.17/1.70 => ( breln1 @ A @ R ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(transitive_type,type,
% 3.17/1.70 transitive: $i > $i > $o ).
% 3.17/1.70
% 3.17/1.70 thf(antisymmetric_type,type,
% 3.17/1.70 antisymmetric: $i > $i > $o ).
% 3.17/1.70
% 3.17/1.70 thf(reflexive_type,type,
% 3.17/1.70 reflexive: $i > $i > $o ).
% 3.17/1.70
% 3.17/1.70 thf(refltransitive_type,type,
% 3.17/1.70 refltransitive: $i > $i > $o ).
% 3.17/1.70
% 3.17/1.70 thf(refllinearorder_type,type,
% 3.17/1.70 refllinearorder: $i > $i > $o ).
% 3.17/1.70
% 3.17/1.70 thf(reflwellordering_type,type,
% 3.17/1.70 reflwellordering: $i > $i > $o ).
% 3.17/1.70
% 3.17/1.70 thf(choice2fnsingleton_type,type,
% 3.17/1.70 choice2fnsingleton: $o ).
% 3.17/1.70
% 3.17/1.70 thf(choice2fnsingleton,definition,
% 3.17/1.70 ( choice2fnsingleton
% 3.17/1.70 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 3.17/1.70 ( ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ? [Xy: $i] :
% 3.17/1.70 ( ( in @ Xy @ B )
% 3.17/1.70 & ( Xphi @ Xx @ Xy ) ) )
% 3.17/1.70 => ! [R: $i] :
% 3.17/1.70 ( ( in @ R @ ( breln1Set @ B ) )
% 3.17/1.70 => ( ( reflwellordering @ B @ R )
% 3.17/1.70 => ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ( singleton
% 3.17/1.70 @ ( dsetconstr @ B
% 3.17/1.70 @ ^ [Xy: $i] :
% 3.17/1.70 ( ( Xphi @ Xx @ Xy )
% 3.17/1.70 & ! [Xz: $i] :
% 3.17/1.70 ( ( in @ Xz @ B )
% 3.17/1.70 => ( ( Xphi @ Xx @ Xz )
% 3.17/1.70 => ( in @ ( kpair @ Xy @ Xz ) @ R ) ) ) ) ) ) ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(setOfPairsIsBReln1_type,type,
% 3.17/1.70 setOfPairsIsBReln1: $o ).
% 3.17/1.70
% 3.17/1.70 thf(setOfPairsIsBReln1,definition,
% 3.17/1.70 ( setOfPairsIsBReln1
% 3.17/1.70 = ( ! [A: $i,Xphi: $i > $i > $o] :
% 3.17/1.70 ( breln1 @ A
% 3.17/1.70 @ ( dpsetconstr @ A @ A
% 3.17/1.70 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(breln1all1_type,type,
% 3.17/1.70 breln1all1: $o ).
% 3.17/1.70
% 3.17/1.70 thf(breln1all1,definition,
% 3.17/1.70 ( breln1all1
% 3.17/1.70 = ( ! [A: $i,R: $i] :
% 3.17/1.70 ( ( breln1 @ A @ R )
% 3.17/1.70 => ! [Xphi: $i > $o] :
% 3.17/1.70 ( ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ! [Xy: $i] :
% 3.17/1.70 ( ( in @ Xy @ A )
% 3.17/1.70 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.17/1.70 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 3.17/1.70 => ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ R )
% 3.17/1.70 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(subbreln1_type,type,
% 3.17/1.70 subbreln1: $o ).
% 3.17/1.70
% 3.17/1.70 thf(subbreln1,definition,
% 3.17/1.70 ( subbreln1
% 3.17/1.70 = ( ! [A: $i,R: $i] :
% 3.17/1.70 ( ( breln1 @ A @ R )
% 3.17/1.70 => ! [S: $i] :
% 3.17/1.70 ( ( breln1 @ A @ S )
% 3.17/1.70 => ( ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ! [Xy: $i] :
% 3.17/1.70 ( ( in @ Xy @ A )
% 3.17/1.70 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.17/1.70 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 3.17/1.70 => ( subset @ R @ S ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(eqbreln1_type,type,
% 3.17/1.70 eqbreln1: $o ).
% 3.17/1.70
% 3.17/1.70 thf(eqbreln1,definition,
% 3.17/1.70 ( eqbreln1
% 3.17/1.70 = ( ! [A: $i,R: $i] :
% 3.17/1.70 ( ( breln1 @ A @ R )
% 3.17/1.70 => ! [S: $i] :
% 3.17/1.70 ( ( breln1 @ A @ S )
% 3.17/1.70 => ( ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ! [Xy: $i] :
% 3.17/1.70 ( ( in @ Xy @ A )
% 3.17/1.70 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.17/1.70 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 3.17/1.70 => ( ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ! [Xy: $i] :
% 3.17/1.70 ( ( in @ Xy @ A )
% 3.17/1.70 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 3.17/1.70 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 3.17/1.70 => ( R = S ) ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(breln1invset_type,type,
% 3.17/1.70 breln1invset: $i > $i > $i ).
% 3.17/1.70
% 3.17/1.70 thf(breln1invprop_type,type,
% 3.17/1.70 breln1invprop: $o ).
% 3.17/1.70
% 3.17/1.70 thf(breln1invprop,definition,
% 3.17/1.70 ( breln1invprop
% 3.17/1.70 = ( ! [A: $i,R: $i] :
% 3.17/1.70 ( ( breln1 @ A @ R )
% 3.17/1.70 => ( breln1 @ A @ ( breln1invset @ A @ R ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(breln1invI_type,type,
% 3.17/1.70 breln1invI: $o ).
% 3.17/1.70
% 3.17/1.70 thf(breln1invI,definition,
% 3.17/1.70 ( breln1invI
% 3.17/1.70 = ( ! [A: $i,R: $i] :
% 3.17/1.70 ( ( breln1 @ A @ R )
% 3.17/1.70 => ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ! [Xy: $i] :
% 3.17/1.70 ( ( in @ Xy @ A )
% 3.17/1.70 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.17/1.70 => ( in @ ( kpair @ Xy @ Xx ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(breln1invE_type,type,
% 3.17/1.70 breln1invE: $o ).
% 3.17/1.70
% 3.17/1.70 thf(breln1invE,definition,
% 3.17/1.70 ( breln1invE
% 3.17/1.70 = ( ! [A: $i,R: $i] :
% 3.17/1.70 ( ( breln1 @ A @ R )
% 3.17/1.70 => ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ! [Xy: $i] :
% 3.17/1.70 ( ( in @ Xy @ A )
% 3.17/1.70 => ( ( in @ ( kpair @ Xy @ Xx ) @ ( breln1invset @ A @ R ) )
% 3.17/1.70 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(breln1compset_type,type,
% 3.17/1.70 breln1compset: $i > $i > $i > $i ).
% 3.17/1.70
% 3.17/1.70 thf(breln1compprop_type,type,
% 3.17/1.70 breln1compprop: $o ).
% 3.17/1.70
% 3.17/1.70 thf(breln1compprop,definition,
% 3.17/1.70 ( breln1compprop
% 3.17/1.70 = ( ! [A: $i,R: $i] :
% 3.17/1.70 ( ( breln1 @ A @ R )
% 3.17/1.70 => ! [S: $i] :
% 3.17/1.70 ( ( breln1 @ A @ S )
% 3.17/1.70 => ( breln1 @ A @ ( breln1compset @ A @ R @ S ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(breln1compI_type,type,
% 3.17/1.70 breln1compI: $o ).
% 3.17/1.70
% 3.17/1.70 thf(breln1compI,definition,
% 3.17/1.70 ( breln1compI
% 3.17/1.70 = ( ! [A: $i,R: $i] :
% 3.17/1.70 ( ( breln1 @ A @ R )
% 3.17/1.70 => ! [S: $i] :
% 3.17/1.70 ( ( breln1 @ A @ S )
% 3.17/1.70 => ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ! [Xy: $i] :
% 3.17/1.70 ( ( in @ Xy @ A )
% 3.17/1.70 => ! [Xz: $i] :
% 3.17/1.70 ( ( in @ Xz @ A )
% 3.17/1.70 => ( ( in @ ( kpair @ Xx @ Xz ) @ R )
% 3.17/1.70 => ( ( in @ ( kpair @ Xz @ Xy ) @ S )
% 3.17/1.70 => ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) ) ) ) ) ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(breln1compE_type,type,
% 3.17/1.70 breln1compE: $o ).
% 3.17/1.70
% 3.17/1.70 thf(breln1compE,definition,
% 3.17/1.70 ( breln1compE
% 3.17/1.70 = ( ! [A: $i,R: $i] :
% 3.17/1.70 ( ( breln1 @ A @ R )
% 3.17/1.70 => ! [S: $i] :
% 3.17/1.70 ( ( breln1 @ A @ S )
% 3.17/1.70 => ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ! [Xy: $i] :
% 3.17/1.70 ( ( in @ Xy @ A )
% 3.17/1.70 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) )
% 3.17/1.70 => ? [Xz: $i] :
% 3.17/1.70 ( ( in @ Xz @ A )
% 3.17/1.70 & ( in @ ( kpair @ Xx @ Xz ) @ R )
% 3.17/1.70 & ( in @ ( kpair @ Xz @ Xy ) @ S ) ) ) ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(breln1compEex_type,type,
% 3.17/1.70 breln1compEex: $o ).
% 3.17/1.70
% 3.17/1.70 thf(breln1compEex,definition,
% 3.17/1.70 ( breln1compEex
% 3.17/1.70 = ( ! [A: $i,R: $i] :
% 3.17/1.70 ( ( breln1 @ A @ R )
% 3.17/1.70 => ! [S: $i] :
% 3.17/1.70 ( ( breln1 @ A @ S )
% 3.17/1.70 => ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ! [Xy: $i] :
% 3.17/1.70 ( ( in @ Xy @ A )
% 3.17/1.70 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) )
% 3.17/1.70 => ! [Xphi: $o] :
% 3.17/1.70 ( ! [Xz: $i] :
% 3.17/1.70 ( ( in @ Xz @ A )
% 3.17/1.70 => ( ( in @ ( kpair @ Xx @ Xz ) @ R )
% 3.17/1.70 => ( ( in @ ( kpair @ Xz @ Xy ) @ S )
% 3.17/1.70 => Xphi ) ) )
% 3.17/1.70 => Xphi ) ) ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(breln1unionprop_type,type,
% 3.17/1.70 breln1unionprop: $o ).
% 3.17/1.70
% 3.17/1.70 thf(breln1unionprop,definition,
% 3.17/1.70 ( breln1unionprop
% 3.17/1.70 = ( ! [A: $i,R: $i] :
% 3.17/1.70 ( ( breln1 @ A @ R )
% 3.17/1.70 => ! [S: $i] :
% 3.17/1.70 ( ( breln1 @ A @ S )
% 3.17/1.70 => ( breln1 @ A @ ( binunion @ R @ S ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(breln1unionIL_type,type,
% 3.17/1.70 breln1unionIL: $o ).
% 3.17/1.70
% 3.17/1.70 thf(breln1unionIL,definition,
% 3.17/1.70 ( breln1unionIL
% 3.17/1.70 = ( ! [A: $i,R: $i] :
% 3.17/1.70 ( ( breln1 @ A @ R )
% 3.17/1.70 => ! [S: $i] :
% 3.17/1.70 ( ( breln1 @ A @ S )
% 3.17/1.70 => ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ! [Xy: $i] :
% 3.17/1.70 ( ( in @ Xy @ A )
% 3.17/1.70 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.17/1.70 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(breln1unionIR_type,type,
% 3.17/1.70 breln1unionIR: $o ).
% 3.17/1.70
% 3.17/1.70 thf(breln1unionIR,definition,
% 3.17/1.70 ( breln1unionIR
% 3.17/1.70 = ( ! [A: $i,R: $i] :
% 3.17/1.70 ( ( breln1 @ A @ R )
% 3.17/1.70 => ! [S: $i] :
% 3.17/1.70 ( ( breln1 @ A @ S )
% 3.17/1.70 => ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ! [Xy: $i] :
% 3.17/1.70 ( ( in @ Xy @ A )
% 3.17/1.70 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 3.17/1.70 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(breln1unionI_type,type,
% 3.17/1.70 breln1unionI: $o ).
% 3.17/1.70
% 3.17/1.70 thf(breln1unionI,definition,
% 3.17/1.70 ( breln1unionI
% 3.17/1.70 = ( ! [A: $i,R: $i] :
% 3.17/1.70 ( ( breln1 @ A @ R )
% 3.17/1.70 => ! [S: $i] :
% 3.17/1.70 ( ( breln1 @ A @ S )
% 3.17/1.70 => ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ! [Xy: $i] :
% 3.17/1.70 ( ( in @ Xy @ A )
% 3.17/1.70 => ( ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.17/1.70 | ( in @ ( kpair @ Xx @ Xy ) @ S ) )
% 3.17/1.70 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(breln1unionE_type,type,
% 3.17/1.70 breln1unionE: $o ).
% 3.17/1.70
% 3.17/1.70 thf(breln1unionE,definition,
% 3.17/1.70 ( breln1unionE
% 3.17/1.70 = ( ! [A: $i,R: $i] :
% 3.17/1.70 ( ( breln1 @ A @ R )
% 3.17/1.70 => ! [S: $i] :
% 3.17/1.70 ( ( breln1 @ A @ S )
% 3.17/1.70 => ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ! [Xy: $i] :
% 3.17/1.70 ( ( in @ Xy @ A )
% 3.17/1.70 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) )
% 3.17/1.70 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.17/1.70 | ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(breln1unionEcases_type,type,
% 3.17/1.70 breln1unionEcases: $o ).
% 3.17/1.70
% 3.17/1.70 thf(breln1unionEcases,definition,
% 3.17/1.70 ( breln1unionEcases
% 3.17/1.70 = ( ! [A: $i,R: $i] :
% 3.17/1.70 ( ( breln1 @ A @ R )
% 3.17/1.70 => ! [S: $i] :
% 3.17/1.70 ( ( breln1 @ A @ S )
% 3.17/1.70 => ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ A )
% 3.17/1.70 => ! [Xy: $i] :
% 3.17/1.70 ( ( in @ Xy @ A )
% 3.17/1.70 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) )
% 3.17/1.70 => ! [Xphi: $o] :
% 3.17/1.70 ( ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.17/1.70 => Xphi )
% 3.17/1.70 => ( ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 3.17/1.70 => Xphi )
% 3.17/1.70 => Xphi ) ) ) ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(breln1unionCommutes_type,type,
% 3.17/1.70 breln1unionCommutes: $o ).
% 3.17/1.70
% 3.17/1.70 thf(breln1unionCommutes,definition,
% 3.17/1.70 ( breln1unionCommutes
% 3.17/1.70 = ( ! [A: $i,R: $i] :
% 3.17/1.70 ( ( breln1 @ A @ R )
% 3.17/1.70 => ! [S: $i] :
% 3.17/1.70 ( ( breln1 @ A @ S )
% 3.17/1.70 => ( ( binunion @ R @ S )
% 3.17/1.70 = ( binunion @ S @ R ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(woz2Ex_type,type,
% 3.17/1.70 woz2Ex: $o ).
% 3.17/1.70
% 3.17/1.70 thf(woz2Ex,definition,
% 3.17/1.70 ( woz2Ex
% 3.17/1.70 = ( ! [A: $i,R: $i] :
% 3.17/1.70 ( ( breln1 @ A @ R )
% 3.17/1.70 => ( R
% 3.17/1.70 = ( breln1invset @ A @ ( breln1invset @ A @ R ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(woz2W_type,type,
% 3.17/1.70 woz2W: $o ).
% 3.17/1.70
% 3.17/1.70 thf(woz2W,definition,
% 3.17/1.70 ( woz2W
% 3.17/1.70 = ( ! [A: $i,R: $i] :
% 3.17/1.70 ( ( breln1 @ A @ R )
% 3.17/1.70 => ! [S: $i] :
% 3.17/1.70 ( ( breln1 @ A @ S )
% 3.17/1.70 => ( ( breln1invset @ A @ ( breln1compset @ A @ R @ S ) )
% 3.17/1.70 = ( breln1compset @ A @ ( breln1invset @ A @ S ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(woz2A_type,type,
% 3.17/1.70 woz2A: $o ).
% 3.17/1.70
% 3.17/1.70 thf(woz2A,definition,
% 3.17/1.70 ( woz2A
% 3.17/1.70 = ( ! [A: $i,R: $i] :
% 3.17/1.70 ( ( breln1 @ A @ R )
% 3.17/1.70 => ! [S: $i] :
% 3.17/1.70 ( ( breln1 @ A @ S )
% 3.17/1.70 => ! [T: $i] :
% 3.17/1.70 ( ( breln1 @ A @ T )
% 3.17/1.70 => ( ( breln1compset @ A @ ( binunion @ R @ S ) @ T )
% 3.17/1.70 = ( binunion @ ( breln1compset @ A @ R @ T ) @ ( breln1compset @ A @ S @ T ) ) ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(woz2B_type,type,
% 3.17/1.70 woz2B: $o ).
% 3.17/1.70
% 3.17/1.70 thf(woz2B,definition,
% 3.17/1.70 ( woz2B
% 3.17/1.70 = ( ! [A: $i,R: $i] :
% 3.17/1.70 ( ( breln1 @ A @ R )
% 3.17/1.70 => ! [S: $i] :
% 3.17/1.70 ( ( breln1 @ A @ S )
% 3.17/1.70 => ! [T: $i] :
% 3.17/1.70 ( ( breln1 @ A @ T )
% 3.17/1.70 => ( ( breln1compset @ A @ ( binunion @ R @ S ) @ T )
% 3.17/1.70 = ( binunion @ ( breln1invset @ A @ ( breln1compset @ A @ ( breln1invset @ A @ T ) @ ( breln1invset @ A @ S ) ) ) @ ( breln1invset @ A @ ( breln1compset @ A @ ( breln1invset @ A @ T ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(image1Ex_type,type,
% 3.17/1.70 image1Ex: $o ).
% 3.17/1.70
% 3.17/1.70 thf(image1Ex,definition,
% 3.17/1.70 ( image1Ex
% 3.17/1.70 = ( ! [A: $i,Xf: $i > $i] :
% 3.17/1.70 ? [B: $i] :
% 3.17/1.70 ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ B )
% 3.17/1.70 <=> ? [Xy: $i] :
% 3.17/1.70 ( ( in @ Xy @ A )
% 3.17/1.70 & ( Xx
% 3.17/1.70 = ( Xf @ Xy ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(image1Ex1_type,type,
% 3.17/1.70 image1Ex1: $o ).
% 3.17/1.70
% 3.17/1.70 thf(image1Ex1,definition,
% 3.17/1.70 ( image1Ex1
% 3.17/1.70 = ( ! [A: $i,Xf: $i > $i] :
% 3.17/1.70 ( exu
% 3.17/1.70 @ ^ [B: $i] :
% 3.17/1.70 ! [Xx: $i] :
% 3.17/1.70 ( ( in @ Xx @ B )
% 3.17/1.70 <=> ? [Xy: $i] :
% 3.17/1.70 ( ( in @ Xy @ A )
% 3.17/1.70 & ( Xx
% 3.17/1.70 = ( Xf @ Xy ) ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(image1_type,type,
% 3.17/1.70 image1: $i > ( $i > $i ) > $i ).
% 3.17/1.70
% 3.17/1.70 thf(image1Equiv_type,type,
% 3.17/1.70 image1Equiv: $o ).
% 3.17/1.70
% 3.17/1.70 thf(image1Equiv,definition,
% 3.17/1.70 ( image1Equiv
% 3.17/1.70 = ( ! [A: $i,Xf: $i > $i,Xx: $i] :
% 3.17/1.70 ( ( in @ Xx
% 3.17/1.70 @ ( image1 @ A
% 3.17/1.70 @ ^ [Xy: $i] : ( Xf @ Xy ) ) )
% 3.17/1.70 <=> ? [Xy: $i] :
% 3.17/1.70 ( ( in @ Xy @ A )
% 3.17/1.70 & ( Xx
% 3.17/1.70 = ( Xf @ Xy ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(image1E_type,type,
% 3.17/1.70 image1E: $o ).
% 3.17/1.70
% 3.17/1.70 thf(image1E,definition,
% 3.17/1.70 ( image1E
% 3.17/1.70 = ( ! [A: $i,Xf: $i > $i,Xx: $i] :
% 3.17/1.70 ( ( in @ Xx
% 3.17/1.70 @ ( image1 @ A
% 3.17/1.70 @ ^ [Xy: $i] : ( Xf @ Xy ) ) )
% 3.17/1.70 => ? [Xy: $i] :
% 3.17/1.70 ( ( in @ Xy @ A )
% 3.17/1.70 & ( Xx
% 3.17/1.70 = ( Xf @ Xy ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(image1I_type,type,
% 3.17/1.70 image1I: $o ).
% 3.17/1.70
% 3.17/1.70 thf(image1I,definition,
% 3.17/1.70 ( image1I
% 3.17/1.70 = ( ! [A: $i,Xf: $i > $i,Xx: $i] :
% 3.17/1.70 ( ? [Xy: $i] :
% 3.17/1.70 ( ( in @ Xy @ A )
% 3.17/1.70 & ( Xx
% 3.17/1.70 = ( Xf @ Xy ) ) )
% 3.17/1.70 => ( in @ Xx
% 3.17/1.70 @ ( image1 @ A
% 3.17/1.70 @ ^ [Xy: $i] : ( Xf @ Xy ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(injective_type,type,
% 3.17/1.70 injective: $i > $i > $i > $o ).
% 3.17/1.70
% 3.17/1.70 thf(injFuncSet_type,type,
% 3.17/1.70 injFuncSet: $i > $i > $i ).
% 3.17/1.70
% 3.17/1.70 thf(injFuncSet,definition,
% 3.17/1.70 ( injFuncSet
% 3.17/1.70 = ( ^ [A: $i,B: $i] :
% 3.17/1.70 ( dsetconstr @ ( funcSet @ A @ B )
% 3.17/1.70 @ ^ [Xf: $i] : ( injective @ A @ B @ Xf ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(injFuncInInjFuncSet_type,type,
% 3.17/1.70 injFuncInInjFuncSet: $o ).
% 3.17/1.70
% 3.17/1.70 thf(injFuncInInjFuncSet,definition,
% 3.17/1.70 ( injFuncInInjFuncSet
% 3.17/1.70 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.17/1.70 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.17/1.70 => ( ( injective @ A @ B @ Xf )
% 3.17/1.70 => ( in @ Xf @ ( injFuncSet @ A @ B ) ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(injFuncSetFuncIn_type,type,
% 3.17/1.70 injFuncSetFuncIn: $o ).
% 3.17/1.70
% 3.17/1.70 thf(injFuncSetFuncIn,definition,
% 3.17/1.70 ( injFuncSetFuncIn
% 3.17/1.70 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.17/1.70 ( ( in @ Xf @ ( injFuncSet @ A @ B ) )
% 3.17/1.70 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 3.17/1.70
% 3.17/1.70 thf(injFuncSetFuncInj,conjecture,
% 3.17/1.70 ( setextAx
% 3.17/1.70 => ( emptysetAx
% 3.17/1.70 => ( setadjoinAx
% 3.17/1.70 => ( powersetAx
% 3.17/1.70 => ( setunionAx
% 3.17/1.70 => ( omega0Ax
% 3.17/1.70 => ( omegaSAx
% 3.17/1.70 => ( omegaIndAx
% 3.17/1.70 => ( replAx
% 3.17/1.70 => ( foundationAx
% 3.17/1.70 => ( wellorderingAx
% 3.17/1.70 => ( descrp
% 3.17/1.70 => ( dsetconstrI
% 3.17/1.70 => ( dsetconstrEL
% 3.17/1.70 => ( dsetconstrER
% 3.17/1.70 => ( exuE1
% 3.17/1.70 => ( prop2setE
% 3.17/1.70 => ( emptysetE
% 3.17/1.70 => ( emptysetimpfalse
% 3.17/1.70 => ( notinemptyset
% 3.17/1.70 => ( exuE3e
% 3.17/1.70 => ( setext
% 3.17/1.70 => ( emptyI
% 3.17/1.70 => ( noeltsimpempty
% 3.17/1.70 => ( setbeta
% 3.17/1.70 => ( nonemptyE1
% 3.17/1.70 => ( nonemptyI
% 3.17/1.70 => ( nonemptyI1
% 3.17/1.70 => ( setadjoinIL
% 3.17/1.70 => ( emptyinunitempty
% 3.17/1.70 => ( setadjoinIR
% 3.17/1.70 => ( setadjoinE
% 3.17/1.70 => ( setadjoinOr
% 3.17/1.70 => ( setoftrueEq
% 3.17/1.70 => ( powersetI
% 3.17/1.70 => ( emptyinPowerset
% 3.17/1.70 => ( emptyInPowerset
% 3.17/1.70 => ( powersetE
% 3.17/1.70 => ( setunionI
% 3.17/1.70 => ( setunionE
% 3.17/1.70 => ( subPowSU
% 3.17/1.70 => ( exuE2
% 3.17/1.70 => ( nonemptyImpWitness
% 3.17/1.70 => ( uniqinunit
% 3.17/1.70 => ( notinsingleton
% 3.17/1.70 => ( eqinunit
% 3.17/1.70 => ( singletonsswitch
% 3.17/1.70 => ( upairsetE
% 3.17/1.70 => ( upairsetIL
% 3.17/1.70 => ( upairsetIR
% 3.17/1.70 => ( emptyE1
% 3.17/1.70 => ( vacuousDall
% 3.17/1.70 => ( quantDeMorgan1
% 3.17/1.70 => ( quantDeMorgan2
% 3.17/1.70 => ( quantDeMorgan3
% 3.17/1.70 => ( quantDeMorgan4
% 3.17/1.70 => ( prop2setI
% 3.17/1.70 => ( prop2set2propI
% 3.17/1.70 => ( notdexE
% 3.17/1.70 => ( notdallE
% 3.17/1.70 => ( exuI1
% 3.17/1.70 => ( exuI3
% 3.17/1.70 => ( exuI2
% 3.17/1.70 => ( inCongP
% 3.17/1.70 => ( in__Cong
% 3.17/1.70 => ( exuE3u
% 3.17/1.70 => ( exu__Cong
% 3.17/1.70 => ( emptyset__Cong
% 3.17/1.70 => ( setadjoin__Cong
% 3.17/1.70 => ( powerset__Cong
% 3.17/1.70 => ( setunion__Cong
% 3.17/1.70 => ( omega__Cong
% 3.17/1.70 => ( exuEu
% 3.17/1.70 => ( descr__Cong
% 3.17/1.70 => ( dsetconstr__Cong
% 3.17/1.70 => ( subsetI1
% 3.17/1.70 => ( eqimpsubset2
% 3.17/1.70 => ( eqimpsubset1
% 3.17/1.70 => ( subsetI2
% 3.17/1.70 => ( emptysetsubset
% 3.17/1.70 => ( subsetE
% 3.17/1.70 => ( subsetE2
% 3.17/1.70 => ( notsubsetI
% 3.17/1.70 => ( notequalI1
% 3.17/1.70 => ( notequalI2
% 3.17/1.70 => ( subsetRefl
% 3.17/1.70 => ( subsetTrans
% 3.17/1.70 => ( setadjoinSub
% 3.17/1.70 => ( setadjoinSub2
% 3.17/1.70 => ( subset2powerset
% 3.17/1.70 => ( setextsub
% 3.17/1.70 => ( subsetemptysetimpeq
% 3.17/1.70 => ( powersetI1
% 3.17/1.70 => ( powersetE1
% 3.17/1.70 => ( inPowerset
% 3.17/1.70 => ( powersetsubset
% 3.17/1.70 => ( sepInPowerset
% 3.17/1.70 => ( sepSubset
% 3.17/1.70 => ( binunionIL
% 3.17/1.70 => ( upairset2IR
% 3.17/1.70 => ( binunionIR
% 3.17/1.70 => ( binunionEcases
% 3.17/1.70 => ( binunionE
% 3.17/1.70 => ( binunionLsub
% 3.17/1.70 => ( binunionRsub
% 3.17/1.70 => ( binintersectI
% 3.17/1.70 => ( binintersectSubset5
% 3.17/1.70 => ( binintersectEL
% 3.17/1.70 => ( binintersectLsub
% 3.17/1.70 => ( binintersectSubset2
% 3.17/1.70 => ( binintersectSubset3
% 3.17/1.70 => ( binintersectER
% 3.17/1.70 => ( disjointsetsI1
% 3.17/1.70 => ( binintersectRsub
% 3.17/1.70 => ( binintersectSubset4
% 3.17/1.70 => ( binintersectSubset1
% 3.17/1.70 => ( bs114d
% 3.17/1.70 => ( setminusI
% 3.17/1.70 => ( setminusEL
% 3.17/1.70 => ( setminusER
% 3.17/1.70 => ( setminusSubset2
% 3.17/1.70 => ( setminusERneg
% 3.17/1.70 => ( setminusELneg
% 3.17/1.70 => ( setminusILneg
% 3.17/1.70 => ( setminusIRneg
% 3.17/1.70 => ( setminusLsub
% 3.17/1.70 => ( setminusSubset1
% 3.17/1.70 => ( symdiffE
% 3.17/1.70 => ( symdiffI1
% 3.17/1.70 => ( symdiffI2
% 3.17/1.70 => ( symdiffIneg1
% 3.17/1.70 => ( symdiffIneg2
% 3.17/1.70 => ( secondinupair
% 3.17/1.70 => ( setukpairIL
% 3.17/1.70 => ( setukpairIR
% 3.17/1.70 => ( kpairiskpair
% 3.17/1.70 => ( kpairp
% 3.17/1.70 => ( singletonsubset
% 3.17/1.70 => ( singletoninpowerset
% 3.17/1.70 => ( singletoninpowunion
% 3.17/1.70 => ( upairset2E
% 3.17/1.70 => ( upairsubunion
% 3.17/1.70 => ( upairinpowunion
% 3.17/1.70 => ( ubforcartprodlem1
% 3.17/1.70 => ( ubforcartprodlem2
% 3.17/1.70 => ( ubforcartprodlem3
% 3.17/1.70 => ( cartprodpairin
% 3.17/1.70 => ( cartprodmempair1
% 3.17/1.70 => ( cartprodmempair
% 3.17/1.70 => ( setunionE2
% 3.17/1.70 => ( setunionsingleton1
% 3.17/1.70 => ( setunionsingleton2
% 3.17/1.70 => ( setunionsingleton
% 3.17/1.70 => ( singletonprop
% 3.17/1.70 => ( ex1E1
% 3.17/1.70 => ( ex1I
% 3.17/1.70 => ( ex1I2
% 3.17/1.70 => ( singletonsuniq
% 3.17/1.70 => ( setukpairinjL1
% 3.17/1.70 => ( kfstsingleton
% 3.17/1.70 => ( theprop
% 3.17/1.70 => ( kfstpairEq
% 3.17/1.70 => ( cartprodfstin
% 3.17/1.70 => ( setukpairinjL2
% 3.17/1.70 => ( setukpairinjL
% 3.17/1.70 => ( setukpairinjR11
% 3.17/1.70 => ( setukpairinjR12
% 3.17/1.70 => ( setukpairinjR1
% 3.17/1.70 => ( upairequniteq
% 3.17/1.70 => ( setukpairinjR2
% 3.17/1.70 => ( setukpairinjR
% 3.17/1.70 => ( ksndsingleton
% 3.17/1.70 => ( ksndpairEq
% 3.17/1.70 => ( kpairsurjEq
% 3.17/1.70 => ( cartprodsndin
% 3.17/1.70 => ( cartprodpairmemEL
% 3.17/1.70 => ( cartprodpairmemER
% 3.17/1.70 => ( cartprodmempaircEq
% 3.17/1.70 => ( cartprodfstpairEq
% 3.17/1.70 => ( cartprodsndpairEq
% 3.17/1.70 => ( cartprodpairsurjEq
% 3.17/1.70 => ( dpsetconstrI
% 3.17/1.70 => ( dpsetconstrSub
% 3.17/1.70 => ( setOfPairsIsBReln
% 3.17/1.70 => ( dpsetconstrERa
% 3.17/1.70 => ( dpsetconstrEL1
% 3.17/1.70 => ( dpsetconstrEL2
% 3.17/1.70 => ( dpsetconstrER
% 3.17/1.70 => ( funcImageSingleton
% 3.17/1.70 => ( apProp
% 3.17/1.70 => ( app
% 3.17/1.70 => ( infuncsetfunc
% 3.17/1.70 => ( ap2p
% 3.17/1.70 => ( funcinfuncset
% 3.17/1.70 => ( lamProp
% 3.17/1.70 => ( lamp
% 3.17/1.70 => ( lam2p
% 3.17/1.70 => ( brelnall1
% 3.17/1.70 => ( brelnall2
% 3.17/1.70 => ( ex1E2
% 3.17/1.70 => ( funcGraphProp1
% 3.17/1.70 => ( funcGraphProp3
% 3.17/1.70 => ( funcGraphProp2
% 3.17/1.70 => ( funcextLem
% 3.17/1.70 => ( funcGraphProp4
% 3.17/1.70 => ( subbreln
% 3.17/1.70 => ( eqbreln
% 3.17/1.70 => ( funcext
% 3.17/1.70 => ( funcext2
% 3.17/1.70 => ( ap2apEq1
% 3.17/1.70 => ( ap2apEq2
% 3.17/1.70 => ( beta1
% 3.17/1.70 => ( eta1
% 3.17/1.70 => ( lam2lamEq
% 3.17/1.70 => ( beta2
% 3.17/1.70 => ( eta2
% 3.17/1.70 => ( iffalseProp1
% 3.17/1.70 => ( iffalseProp2
% 3.17/1.70 => ( iftrueProp1
% 3.17/1.70 => ( iftrueProp2
% 3.17/1.70 => ( ifSingleton
% 3.17/1.70 => ( ifp
% 3.17/1.70 => ( theeq
% 3.17/1.70 => ( iftrue
% 3.17/1.70 => ( iffalse
% 3.17/1.70 => ( iftrueorfalse
% 3.17/1.70 => ( binintersectT_lem
% 3.17/1.70 => ( binunionT_lem
% 3.17/1.70 => ( powersetT_lem
% 3.17/1.70 => ( setminusT_lem
% 3.17/1.70 => ( complementT_lem
% 3.17/1.70 => ( setextT
% 3.17/1.70 => ( subsetTI
% 3.17/1.70 => ( powersetTI1
% 3.17/1.70 => ( powersetTE1
% 3.17/1.70 => ( complementTI1
% 3.17/1.70 => ( complementTE1
% 3.17/1.70 => ( binintersectTELcontra
% 3.17/1.70 => ( binintersectTERcontra
% 3.17/1.70 => ( contrasubsetT
% 3.17/1.70 => ( contrasubsetT1
% 3.17/1.70 => ( contrasubsetT2
% 3.17/1.70 => ( contrasubsetT3
% 3.17/1.70 => ( doubleComplementI1
% 3.17/1.70 => ( doubleComplementE1
% 3.17/1.70 => ( doubleComplementSub1
% 3.17/1.70 => ( doubleComplementSub2
% 3.17/1.70 => ( doubleComplementEq
% 3.17/1.70 => ( complementTnotintersectT
% 3.17/1.70 => ( complementImpComplementIntersect
% 3.17/1.70 => ( complementSubsetComplementIntersect
% 3.17/1.70 => ( complementInPowersetComplementIntersect
% 3.17/1.70 => ( contraSubsetComplement
% 3.17/1.70 => ( complementTcontraSubset
% 3.17/1.70 => ( binunionTILcontra
% 3.17/1.70 => ( binunionTIRcontra
% 3.17/1.70 => ( inIntersectImpInUnion
% 3.17/1.70 => ( inIntersectImpInUnion2
% 3.17/1.70 => ( inIntersectImpInIntersectUnions
% 3.17/1.70 => ( intersectInPowersetIntersectUnions
% 3.17/1.70 => ( inComplementUnionImpNotIn1
% 3.17/1.70 => ( inComplementUnionImpInComplement1
% 3.17/1.70 => ( binunionTE
% 3.17/1.70 => ( binunionTEcontra
% 3.17/1.70 => ( demorgan2a1
% 3.17/1.70 => ( complementUnionInPowersetComplement
% 3.17/1.70 => ( demorgan2a2
% 3.17/1.70 => ( demorgan1a
% 3.17/1.70 => ( demorgan1b
% 3.17/1.70 => ( demorgan1
% 3.17/1.70 => ( demorgan2a
% 3.17/1.70 => ( demorgan2b2
% 3.17/1.70 => ( demorgan2b
% 3.17/1.70 => ( demorgan2
% 3.17/1.70 => ( woz13rule0
% 3.17/1.70 => ( woz13rule1
% 3.17/1.70 => ( woz13rule2
% 3.17/1.70 => ( woz13rule3
% 3.17/1.70 => ( woz13rule4
% 3.17/1.70 => ( woz1_1
% 3.17/1.70 => ( woz1_2
% 3.17/1.70 => ( woz1_3
% 3.17/1.70 => ( woz1_4
% 3.17/1.70 => ( woz1_5
% 3.17/1.70 => ( breln1all2
% 3.17/1.70 => ( breln1SetBreln1
% 3.17/1.70 => ( choice2fnsingleton
% 3.17/1.70 => ( setOfPairsIsBReln1
% 3.17/1.70 => ( breln1all1
% 3.17/1.70 => ( subbreln1
% 3.17/1.70 => ( eqbreln1
% 3.17/1.70 => ( breln1invprop
% 3.17/1.70 => ( breln1invI
% 3.17/1.70 => ( breln1invE
% 3.17/1.70 => ( breln1compprop
% 3.17/1.70 => ( breln1compI
% 3.17/1.70 => ( breln1compE
% 3.17/1.70 => ( breln1compEex
% 3.17/1.70 => ( breln1unionprop
% 3.17/1.70 => ( breln1unionIL
% 3.17/1.70 => ( breln1unionIR
% 3.17/1.70 => ( breln1unionI
% 3.17/1.70 => ( breln1unionE
% 3.17/1.70 => ( breln1unionEcases
% 3.17/1.70 => ( breln1unionCommutes
% 3.17/1.70 => ( woz2Ex
% 3.17/1.70 => ( woz2W
% 3.17/1.70 => ( woz2A
% 3.17/1.70 => ( woz2B
% 3.17/1.70 => ( image1Ex
% 3.17/1.70 => ( image1Ex1
% 3.17/1.70 => ( image1Equiv
% 3.17/1.70 => ( image1E
% 3.17/1.70 => ( image1I
% 3.17/1.70 => ( injFuncInInjFuncSet
% 3.17/1.70 => ( injFuncSetFuncIn
% 3.17/1.70 => ! [Xx: $i,Xy: $i,Xf: $i] :
% 3.17/1.70 ( ( in @ Xf @ ( injFuncSet @ Xx @ Xy ) )
% 3.17/1.70 => ( injective @ Xx @ Xy @ Xf ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 3.17/1.70 ------------------------
% 3.17/1.73 % (18363)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_18170 for (2999ds/183Mi)
% 3.17/1.74 % (18369)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_18170 for (2999ds/18Mi)
% 3.17/1.74 % (18366)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_18170 for (2999ds/2Mi)
% 3.17/1.74 % (18364)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_18170 for (2999ds/4Mi)
% 3.17/1.74 % (18365)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_18170 for (2999ds/27Mi)
% 3.17/1.74 % (18367)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_18170 for (2999ds/2Mi)
% 3.17/1.74 % (18368)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_18170 for (2999ds/275Mi)
% 3.17/1.74 % (18366)Instruction limit reached!
% 3.17/1.74 % (18366)------------------------------
% 3.17/1.74 % (18366)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.74 % (18366)Termination reason: Unknown
% 3.17/1.74 % (18367)Instruction limit reached!
% 3.17/1.74 % (18367)------------------------------
% 3.17/1.74 % (18367)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.74 % (18367)Termination reason: Unknown
% 3.17/1.74 % (18367)Termination phase: shuffling
% 3.17/1.74
% 3.17/1.74 % (18367)Memory used [KB]: 1663
% 3.17/1.74 % (18367)Time elapsed: 0.003 s
% 3.17/1.74 % (18367)Instructions burned: 2 (million)
% 3.17/1.74 % (18367)------------------------------
% 3.17/1.74 % (18367)------------------------------
% 3.17/1.74 % (18366)Termination phase: shuffling
% 3.17/1.74
% 3.17/1.74 % (18366)Memory used [KB]: 1663
% 3.17/1.74 % (18366)Time elapsed: 0.003 s
% 3.17/1.74 % (18366)Instructions burned: 2 (million)
% 3.17/1.74 % (18366)------------------------------
% 3.17/1.74 % (18366)------------------------------
% 3.17/1.74 % (18364)Instruction limit reached!
% 3.17/1.74 % (18364)------------------------------
% 3.17/1.74 % (18364)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.74 % (18364)Termination reason: Unknown
% 3.17/1.74 % (18364)Termination phase: shuffling
% 3.17/1.74
% 3.17/1.74 % (18364)Memory used [KB]: 1663
% 3.17/1.74 % (18364)Time elapsed: 0.004 s
% 3.17/1.74 % (18364)Instructions burned: 4 (million)
% 3.17/1.74 % (18364)------------------------------
% 3.17/1.74 % (18364)------------------------------
% 3.17/1.75 % (18369)Instruction limit reached!
% 3.17/1.75 % (18369)------------------------------
% 3.17/1.75 % (18369)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.75 % (18369)Termination reason: Unknown
% 3.17/1.75 % (18369)Termination phase: shuffling
% 3.17/1.75
% 3.17/1.75 % (18369)Memory used [KB]: 1918
% 3.17/1.75 % (18369)Time elapsed: 0.011 s
% 3.17/1.75 % (18369)Instructions burned: 19 (million)
% 3.17/1.75 % (18369)------------------------------
% 3.17/1.75 % (18369)------------------------------
% 3.17/1.75 % (18365)Instruction limit reached!
% 3.17/1.75 % (18365)------------------------------
% 3.17/1.75 % (18365)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.75 % (18365)Termination reason: Unknown
% 3.17/1.75 % (18365)Termination phase: shuffling
% 3.17/1.75
% 3.17/1.75 % (18365)Memory used [KB]: 2046
% 3.17/1.75 % (18365)Time elapsed: 0.015 s
% 3.17/1.75 % (18365)Instructions burned: 28 (million)
% 3.17/1.75 % (18365)------------------------------
% 3.17/1.75 % (18365)------------------------------
% 3.17/1.75 % (18370)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_18170 for (2999ds/3Mi)
% 3.17/1.75 % (18371)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_18170 for (2999ds/37Mi)
% 3.17/1.75 % (18372)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_18170 for (2999ds/15Mi)
% 3.17/1.76 % (18370)Instruction limit reached!
% 3.17/1.76 % (18370)------------------------------
% 3.17/1.76 % (18370)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.76 % (18370)Termination reason: Unknown
% 3.17/1.76 % (18370)Termination phase: shuffling
% 3.17/1.76
% 3.17/1.76 % (18370)Memory used [KB]: 1663
% 3.17/1.76 % (18370)Time elapsed: 0.004 s
% 3.17/1.76 % (18370)Instructions burned: 5 (million)
% 3.17/1.76 % (18370)------------------------------
% 3.17/1.76 % (18370)------------------------------
% 3.17/1.76 % (18372)Instruction limit reached!
% 3.17/1.76 % (18372)------------------------------
% 3.17/1.76 % (18372)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.76 % (18372)Termination reason: Unknown
% 3.17/1.76 % (18372)Termination phase: shuffling
% 3.17/1.76
% 3.17/1.76 % (18372)Memory used [KB]: 1791
% 3.17/1.76 % (18372)Time elapsed: 0.009 s
% 3.17/1.76 % (18372)Instructions burned: 15 (million)
% 3.17/1.76 % (18372)------------------------------
% 3.17/1.76 % (18372)------------------------------
% 3.17/1.76 % (18374)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_18170 for (2999ds/3Mi)
% 3.17/1.76 % (18374)Instruction limit reached!
% 3.17/1.76 % (18374)------------------------------
% 3.17/1.76 % (18374)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.76 % (18374)Termination reason: Unknown
% 3.17/1.76 % (18374)Termination phase: shuffling
% 3.17/1.76
% 3.17/1.76 % (18374)Memory used [KB]: 1663
% 3.17/1.76 % (18374)Time elapsed: 0.004 s
% 3.17/1.76 % (18374)Instructions burned: 4 (million)
% 3.17/1.76 % (18374)------------------------------
% 3.17/1.76 % (18374)------------------------------
% 3.17/1.77 % (18375)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_18170 for (2999ds/1041Mi)
% 3.17/1.77 % (18371)Instruction limit reached!
% 3.17/1.77 % (18371)------------------------------
% 3.17/1.77 % (18371)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.77 % (18371)Termination reason: Unknown
% 3.17/1.77 % (18371)Termination phase: shuffling
% 3.17/1.77
% 3.17/1.77 % (18371)Memory used [KB]: 2302
% 3.17/1.77 % (18371)Time elapsed: 0.018 s
% 3.17/1.77 % (18371)Instructions burned: 37 (million)
% 3.17/1.77 % (18371)------------------------------
% 3.17/1.77 % (18371)------------------------------
% 3.17/1.78 % (18378)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_18170 for (2999ds/16Mi)
% 3.17/1.78 % (18379)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_18170 for (2999ds/3Mi)
% 3.17/1.78 % (18379)Instruction limit reached!
% 3.17/1.78 % (18379)------------------------------
% 3.17/1.78 % (18379)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.78 % (18379)Termination reason: Unknown
% 3.17/1.78 % (18379)Termination phase: shuffling
% 3.17/1.78
% 3.17/1.78 % (18379)Memory used [KB]: 1663
% 3.17/1.78 % (18379)Time elapsed: 0.003 s
% 3.17/1.78 % (18379)Instructions burned: 3 (million)
% 3.17/1.78 % (18379)------------------------------
% 3.17/1.78 % (18379)------------------------------
% 3.17/1.79 % (18378)Instruction limit reached!
% 3.17/1.79 % (18378)------------------------------
% 3.17/1.79 % (18378)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.79 % (18378)Termination reason: Unknown
% 3.17/1.79 % (18378)Termination phase: shuffling
% 3.17/1.79
% 3.17/1.79 % (18378)Memory used [KB]: 1918
% 3.17/1.79 % (18378)Time elapsed: 0.009 s
% 3.17/1.79 % (18378)Instructions burned: 16 (million)
% 3.17/1.79 % (18378)------------------------------
% 3.17/1.79 % (18378)------------------------------
% 3.17/1.79 % (18380)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_18170 for (2999ds/3Mi)
% 3.17/1.79 % (18376)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_18170 for (2999ds/7Mi)
% 3.17/1.79 % (18380)Instruction limit reached!
% 3.17/1.79 % (18380)------------------------------
% 3.17/1.79 % (18380)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.79 % (18380)Termination reason: Unknown
% 3.17/1.79 % (18380)Termination phase: shuffling
% 3.17/1.79
% 3.17/1.79 % (18380)Memory used [KB]: 1663
% 3.17/1.79 % (18380)Time elapsed: 0.003 s
% 3.17/1.79 % (18380)Instructions burned: 3 (million)
% 3.17/1.79 % (18380)------------------------------
% 3.17/1.79 % (18380)------------------------------
% 3.17/1.79 % (18376)Instruction limit reached!
% 3.17/1.79 % (18376)------------------------------
% 3.17/1.79 % (18376)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.79 % (18376)Termination reason: Unknown
% 3.17/1.79 % (18376)Termination phase: shuffling
% 3.17/1.79
% 3.17/1.79 % (18376)Memory used [KB]: 1663
% 3.17/1.79 % (18376)Time elapsed: 0.005 s
% 3.17/1.79 % (18376)Instructions burned: 10 (million)
% 3.17/1.79 % (18376)------------------------------
% 3.17/1.79 % (18376)------------------------------
% 3.17/1.80 % (18382)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_18170 for (2999ds/7Mi)
% 3.17/1.80 % (18382)Instruction limit reached!
% 3.17/1.80 % (18382)------------------------------
% 3.17/1.80 % (18382)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.80 % (18382)Termination reason: Unknown
% 3.17/1.80 % (18382)Termination phase: shuffling
% 3.17/1.80
% 3.17/1.80 % (18382)Memory used [KB]: 1663
% 3.17/1.80 % (18382)Time elapsed: 0.005 s
% 3.17/1.80 % (18382)Instructions burned: 7 (million)
% 3.17/1.80 % (18382)------------------------------
% 3.17/1.80 % (18382)------------------------------
% 3.17/1.80 % (18383)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_18170 for (2999ds/3Mi)
% 3.17/1.80 % (18363)Instruction limit reached!
% 3.17/1.80 % (18363)------------------------------
% 3.17/1.80 % (18363)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.80 % (18363)Termination reason: Unknown
% 3.17/1.80 % (18363)Termination phase: Property scanning
% 3.17/1.80
% 3.17/1.80 % (18363)Memory used [KB]: 3070
% 3.17/1.80 % (18363)Time elapsed: 0.074 s
% 3.17/1.80 % (18363)Instructions burned: 183 (million)
% 3.17/1.80 % (18363)------------------------------
% 3.17/1.80 % (18363)------------------------------
% 3.17/1.80 % (18383)Instruction limit reached!
% 3.17/1.80 % (18383)------------------------------
% 3.17/1.80 % (18383)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.80 % (18383)Termination reason: Unknown
% 3.17/1.80 % (18383)Termination phase: shuffling
% 3.17/1.80
% 3.17/1.80 % (18383)Memory used [KB]: 1663
% 3.17/1.80 % (18383)Time elapsed: 0.004 s
% 3.17/1.80 % (18383)Instructions burned: 5 (million)
% 3.17/1.80 % (18383)------------------------------
% 3.17/1.80 % (18383)------------------------------
% 3.17/1.81 % (18385)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_18170 for (2999ds/4Mi)
% 3.17/1.81 % (18385)Instruction limit reached!
% 3.17/1.81 % (18385)------------------------------
% 3.17/1.81 % (18385)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.81 % (18385)Termination reason: Unknown
% 3.17/1.81 % (18385)Termination phase: shuffling
% 3.17/1.81
% 3.17/1.81 % (18385)Memory used [KB]: 1663
% 3.17/1.81 % (18385)Time elapsed: 0.003 s
% 3.17/1.81 % (18385)Instructions burned: 4 (million)
% 3.17/1.81 % (18385)------------------------------
% 3.17/1.81 % (18385)------------------------------
% 3.17/1.82 % (18390)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_18170 for (2999ds/21Mi)
% 3.17/1.82 % (18387)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_18170 for (2999ds/710Mi)
% 3.17/1.82 % (18386)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_18170 for (2999ds/18Mi)
% 3.17/1.82 % (18388)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_18170 for (2999ds/6Mi)
% 3.17/1.82 % (18389)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_18170 for (2999ds/902Mi)
% 3.17/1.82 % (18368)Instruction limit reached!
% 3.17/1.82 % (18368)------------------------------
% 3.17/1.82 % (18368)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.82 % (18368)Termination reason: Unknown
% 3.17/1.82 % (18368)Termination phase: Saturation
% 3.17/1.82
% 3.17/1.82 % (18368)Memory used [KB]: 5884
% 3.17/1.82 % (18368)Time elapsed: 0.086 s
% 3.17/1.82 % (18368)Instructions burned: 276 (million)
% 3.17/1.82 % (18368)------------------------------
% 3.17/1.82 % (18368)------------------------------
% 3.17/1.82 % (18388)Instruction limit reached!
% 3.17/1.82 % (18388)------------------------------
% 3.17/1.82 % (18388)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.82 % (18388)Termination reason: Unknown
% 3.17/1.82 % (18388)Termination phase: shuffling
% 3.17/1.82
% 3.17/1.82 % (18388)Memory used [KB]: 1663
% 3.17/1.82 % (18388)Time elapsed: 0.005 s
% 3.17/1.82 % (18388)Instructions burned: 7 (million)
% 3.17/1.82 % (18390)Instruction limit reached!
% 3.17/1.82 % (18390)------------------------------
% 3.17/1.82 % (18390)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.82 % (18388)------------------------------
% 3.17/1.82 % (18388)------------------------------
% 3.17/1.82 % (18390)Termination reason: Unknown
% 3.17/1.82 % (18390)Termination phase: shuffling
% 3.17/1.82
% 3.17/1.82 % (18390)Memory used [KB]: 1918
% 3.17/1.82 % (18390)Time elapsed: 0.007 s
% 3.17/1.82 % (18390)Instructions burned: 21 (million)
% 3.17/1.82 % (18390)------------------------------
% 3.17/1.82 % (18390)------------------------------
% 3.17/1.83 % (18386)Instruction limit reached!
% 3.17/1.83 % (18386)------------------------------
% 3.17/1.83 % (18386)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.83 % (18386)Termination reason: Unknown
% 3.17/1.83 % (18386)Termination phase: shuffling
% 3.17/1.83
% 3.17/1.83 % (18386)Memory used [KB]: 1918
% 3.17/1.83 % (18386)Time elapsed: 0.010 s
% 3.17/1.83 % (18386)Instructions burned: 18 (million)
% 3.17/1.83 % (18386)------------------------------
% 3.17/1.83 % (18386)------------------------------
% 3.17/1.83 % (18394)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_18170 for (2998ds/6Mi)
% 3.17/1.83 % (18394)Instruction limit reached!
% 3.17/1.83 % (18394)------------------------------
% 3.17/1.83 % (18394)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.83 % (18394)Termination reason: Unknown
% 3.17/1.83 % (18394)Termination phase: shuffling
% 3.17/1.83
% 3.17/1.83 % (18394)Memory used [KB]: 1663
% 3.17/1.83 % (18394)Time elapsed: 0.003 s
% 3.17/1.83 % (18394)Instructions burned: 6 (million)
% 3.17/1.83 % (18394)------------------------------
% 3.17/1.83 % (18394)------------------------------
% 3.17/1.84 % (18393)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_18170 for (2998ds/5Mi)
% 3.17/1.84 % (18395)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_18170 for (2998ds/377Mi)
% 3.17/1.84 % (18393)Instruction limit reached!
% 3.17/1.84 % (18393)------------------------------
% 3.17/1.84 % (18393)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.84 % (18393)Termination reason: Unknown
% 3.17/1.84 % (18393)Termination phase: shuffling
% 3.17/1.84
% 3.17/1.84 % (18393)Memory used [KB]: 1663
% 3.17/1.84 % (18393)Time elapsed: 0.005 s
% 3.17/1.84 % (18393)Instructions burned: 6 (million)
% 3.17/1.84 % (18393)------------------------------
% 3.17/1.84 % (18393)------------------------------
% 3.17/1.84 % (18396)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_18170 for (2998ds/779Mi)
% 3.17/1.84 % (18397)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_18170 for (2998ds/19Mi)
% 3.17/1.85 % (18397)Instruction limit reached!
% 3.17/1.85 % (18397)------------------------------
% 3.17/1.85 % (18397)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.85 % (18397)Termination reason: Unknown
% 3.17/1.85 % (18397)Termination phase: shuffling
% 3.17/1.85
% 3.17/1.85 % (18397)Memory used [KB]: 1918
% 3.17/1.85 % (18397)Time elapsed: 0.010 s
% 3.17/1.85 % (18397)Instructions burned: 20 (million)
% 3.17/1.85 % (18397)------------------------------
% 3.17/1.85 % (18397)------------------------------
% 3.17/1.86 % (18398)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_18170 for (2998ds/879Mi)
% 3.17/1.87 % (18401)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_18170 for (2998ds/17Mi)
% 3.17/1.88 % (18401)Instruction limit reached!
% 3.17/1.88 % (18401)------------------------------
% 3.17/1.88 % (18401)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.88 % (18401)Termination reason: Unknown
% 3.17/1.88 % (18401)Termination phase: shuffling
% 3.17/1.88
% 3.17/1.88 % (18401)Memory used [KB]: 1918
% 3.17/1.88 % (18401)Time elapsed: 0.010 s
% 3.17/1.88 % (18401)Instructions burned: 20 (million)
% 3.17/1.88 % (18401)------------------------------
% 3.17/1.88 % (18401)------------------------------
% 3.17/1.89 % (18405)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_18170 for (2998ds/3Mi)
% 3.17/1.89 % (18405)Instruction limit reached!
% 3.17/1.89 % (18405)------------------------------
% 3.17/1.89 % (18405)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.89 % (18405)Termination reason: Unknown
% 3.17/1.89 % (18405)Termination phase: shuffling
% 3.17/1.89
% 3.17/1.89 % (18405)Memory used [KB]: 1663
% 3.17/1.89 % (18405)Time elapsed: 0.003 s
% 3.17/1.89 % (18405)Instructions burned: 5 (million)
% 3.17/1.89 % (18405)------------------------------
% 3.17/1.89 % (18405)------------------------------
% 3.17/1.90 % (18406)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_18170 for (2998ds/30Mi)
% 3.17/1.91 % (18406)Instruction limit reached!
% 3.17/1.91 % (18406)------------------------------
% 3.17/1.91 % (18406)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.91 % (18406)Termination reason: Unknown
% 3.17/1.91 % (18406)Termination phase: shuffling
% 3.17/1.91
% 3.17/1.91 % (18406)Memory used [KB]: 2174
% 3.17/1.91 % (18406)Time elapsed: 0.010 s
% 3.17/1.91 % (18406)Instructions burned: 30 (million)
% 3.17/1.91 % (18406)------------------------------
% 3.17/1.91 % (18406)------------------------------
% 3.17/1.92 % (18409)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_18170 for (2997ds/127Mi)
% 3.17/1.97 % (18409)Instruction limit reached!
% 3.17/1.97 % (18409)------------------------------
% 3.17/1.97 % (18409)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.97 % (18409)Termination reason: Unknown
% 3.17/1.97 % (18409)Termination phase: Property scanning
% 3.17/1.97
% 3.17/1.97 % (18409)Memory used [KB]: 2942
% 3.17/1.97 % (18409)Time elapsed: 0.053 s
% 3.17/1.97 % (18409)Instructions burned: 127 (million)
% 3.17/1.97 % (18409)------------------------------
% 3.17/1.97 % (18409)------------------------------
% 3.17/1.99 % (18415)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_18170 for (2997ds/100Mi)
% 3.17/2.01 % (18395)Instruction limit reached!
% 3.17/2.01 % (18395)------------------------------
% 3.17/2.01 % (18395)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/2.01 % (18395)Termination reason: Unknown
% 3.17/2.01 % (18395)Termination phase: Saturation
% 3.17/2.01
% 3.17/2.01 % (18395)Memory used [KB]: 9466
% 3.17/2.01 % (18395)Time elapsed: 0.169 s
% 3.17/2.01 % (18395)Instructions burned: 379 (million)
% 3.17/2.01 % (18395)------------------------------
% 3.17/2.01 % (18395)------------------------------
% 3.17/2.02 % (18418)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_18170 for (2997ds/3Mi)
% 3.17/2.02 % (18418)Instruction limit reached!
% 3.17/2.02 % (18418)------------------------------
% 3.17/2.02 % (18418)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/2.02 % (18418)Termination reason: Unknown
% 3.17/2.02 % (18418)Termination phase: shuffling
% 3.17/2.02
% 3.17/2.02 % (18418)Memory used [KB]: 1663
% 3.17/2.02 % (18418)Time elapsed: 0.003 s
% 3.17/2.02 % (18418)Instructions burned: 4 (million)
% 3.17/2.02 % (18418)------------------------------
% 3.17/2.02 % (18418)------------------------------
% 5.17/2.03 % (18422)lrs+10_8:1_au=on:avsq=on:e2e=on:ins=3:s2a=on:s2at=3.0:ss=axioms:i=20:si=on:rtra=on_0 on DTF2THF_18170 for (2996ds/20Mi)
% 5.17/2.03 % (18415)Instruction limit reached!
% 5.17/2.03 % (18415)------------------------------
% 5.17/2.03 % (18415)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.17/2.03 % (18415)Termination reason: Unknown
% 5.17/2.03 % (18415)Termination phase: Property scanning
% 5.17/2.03
% 5.17/2.03 % (18415)Memory used [KB]: 2942
% 5.17/2.03 % (18415)Time elapsed: 0.041 s
% 5.17/2.03 % (18415)Instructions burned: 101 (million)
% 5.17/2.03 % (18415)------------------------------
% 5.17/2.03 % (18415)------------------------------
% 5.17/2.04 % (18422)Instruction limit reached!
% 5.17/2.04 % (18422)------------------------------
% 5.17/2.04 % (18422)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.17/2.04 % (18422)Termination reason: Unknown
% 5.17/2.04 % (18422)Termination phase: shuffling
% 5.17/2.04
% 5.17/2.04 % (18422)Memory used [KB]: 2046
% 5.17/2.04 % (18422)Time elapsed: 0.011 s
% 5.17/2.04 % (18422)Instructions burned: 21 (million)
% 5.17/2.04 % (18422)------------------------------
% 5.17/2.04 % (18422)------------------------------
% 5.17/2.04 % (18424)dis+1002_1:1_cbe=off:hud=5:nm=4:plsq=on:plsqr=7,1:prag=on:sp=const_max:tnu=1:i=86:si=on:rtra=on_0 on DTF2THF_18170 for (2996ds/86Mi)
% 5.17/2.05 % (18389)Refutation not found, incomplete strategy
% 5.17/2.05 % (18389)------------------------------
% 5.17/2.05 % (18389)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.17/2.05 % (18389)Termination reason: Refutation not found, incomplete strategy
% 5.17/2.05
% 5.17/2.05
% 5.17/2.05 % (18389)Memory used [KB]: 11513
% 5.17/2.05 % (18389)Time elapsed: 0.228 s
% 5.17/2.05 % (18389)Instructions burned: 472 (million)
% 5.17/2.05 % (18389)------------------------------
% 5.17/2.05 % (18389)------------------------------
% 5.17/2.06 % (18426)lrs+1010_1:1_au=on:cbe=off:nm=2:ntd=on:sd=2:ss=axioms:st=5.0:i=107:si=on:rtra=on_0 on DTF2THF_18170 for (2996ds/107Mi)
% 5.17/2.07 % (18427)lrs+2_1:1024_cnfonf=lazy_gen:fe=off:hud=15:plsq=on:plsqc=1:plsqr=32,1:i=39:si=on:rtra=on_0 on DTF2THF_18170 for (2996ds/39Mi)
% 5.17/2.08 % (18424)Instruction limit reached!
% 5.17/2.08 % (18424)------------------------------
% 5.17/2.08 % (18424)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.17/2.08 % (18424)Termination reason: Unknown
% 5.17/2.08 % (18424)Termination phase: Property scanning
% 5.17/2.08
% 5.17/2.08 % (18424)Memory used [KB]: 2814
% 5.17/2.08 % (18424)Time elapsed: 0.025 s
% 5.17/2.08 % (18424)Instructions burned: 86 (million)
% 5.17/2.08 % (18424)------------------------------
% 5.17/2.08 % (18424)------------------------------
% 5.17/2.09 % (18427)Instruction limit reached!
% 5.17/2.09 % (18427)------------------------------
% 5.17/2.09 % (18427)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.17/2.09 % (18427)Termination reason: Unknown
% 5.17/2.09 % (18427)Termination phase: shuffling
% 5.17/2.09
% 5.17/2.09 % (18427)Memory used [KB]: 2430
% 5.17/2.09 % (18427)Time elapsed: 0.013 s
% 5.17/2.09 % (18427)Instructions burned: 42 (million)
% 5.17/2.09 % (18427)------------------------------
% 5.17/2.09 % (18427)------------------------------
% 5.17/2.10 % (18429)dis+10_1:1_cnfonf=lazy_not_gen:fsr=off:kws=precedence:nwc=5.0:s2a=on:ss=axioms:st=1.5:i=448:si=on:rtra=on_0 on DTF2THF_18170 for (2996ds/448Mi)
% 5.17/2.10 % (18430)lrs+10_1:512_au=on:fde=unused:lma=on:nm=32:plsq=on:plsqc=1:plsqr=16121663,131072:sfv=off:sp=const_max:ss=axioms:st=3.0:tgt=full:i=46:si=on:rtra=on_0 on DTF2THF_18170 for (2996ds/46Mi)
% 5.17/2.10 % (18426)Instruction limit reached!
% 5.17/2.10 % (18426)------------------------------
% 5.17/2.10 % (18426)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.17/2.10 % (18426)Termination reason: Unknown
% 5.17/2.10 % (18426)Termination phase: SInE selection
% 5.17/2.10
% 5.17/2.10 % (18426)Memory used [KB]: 2814
% 5.17/2.10 % (18426)Time elapsed: 0.049 s
% 5.17/2.10 % (18426)Instructions burned: 108 (million)
% 5.17/2.10 % (18426)------------------------------
% 5.17/2.10 % (18426)------------------------------
% 5.17/2.12 % (18430)Instruction limit reached!
% 5.17/2.12 % (18430)------------------------------
% 5.17/2.12 % (18430)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.17/2.12 % (18430)Termination reason: Unknown
% 5.17/2.12 % (18430)Termination phase: shuffling
% 5.17/2.12
% 5.17/2.12 % (18430)Memory used [KB]: 2430
% 5.17/2.12 % (18430)Time elapsed: 0.019 s
% 5.17/2.12 % (18430)Instructions burned: 46 (million)
% 5.17/2.12 % (18430)------------------------------
% 5.17/2.12 % (18430)------------------------------
% 5.17/2.12 % (18433)lrs+10_1:10_au=on:av=off:cbe=off:cnfonf=lazy_pi_sigma_gen:ntd=on:plsq=on:plsqc=1:plsqr=32,1:i=98:si=on:rtra=on_0 on DTF2THF_18170 for (2996ds/98Mi)
% 5.80/2.13 % (18396)Instruction limit reached!
% 5.80/2.13 % (18396)------------------------------
% 5.80/2.13 % (18396)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.80/2.13 % (18396)Termination reason: Unknown
% 5.80/2.13 % (18396)Termination phase: Saturation
% 5.80/2.13
% 5.80/2.13 % (18396)Memory used [KB]: 7931
% 5.80/2.13 % (18396)Time elapsed: 0.284 s
% 5.80/2.13 % (18396)Instructions burned: 779 (million)
% 5.80/2.13 % (18396)------------------------------
% 5.80/2.13 % (18396)------------------------------
% 5.80/2.13 % (18434)ott+1002_1:1_apa=on:au=on:bd=off:cnfonf=off:fd=off:sos=on:sp=weighted_frequency:i=507:si=on:rtra=on_0 on DTF2THF_18170 for (2995ds/507Mi)
% 5.80/2.13 % (18387)Instruction limit reached!
% 5.80/2.13 % (18387)------------------------------
% 5.80/2.13 % (18387)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.80/2.13 % (18387)Termination reason: Unknown
% 5.80/2.13 % (18387)Termination phase: Saturation
% 5.80/2.13
% 5.80/2.13 % (18387)Memory used [KB]: 11769
% 5.80/2.13 % (18387)Time elapsed: 0.317 s
% 5.80/2.13 % (18387)Instructions burned: 712 (million)
% 5.80/2.13 % (18387)------------------------------
% 5.80/2.13 % (18387)------------------------------
% 5.80/2.14 % (18375)Instruction limit reached!
% 5.80/2.14 % (18375)------------------------------
% 5.80/2.14 % (18375)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.80/2.14 % (18375)Termination reason: Unknown
% 5.80/2.14 % (18375)Termination phase: Saturation
% 5.80/2.14
% 5.80/2.14 % (18375)Memory used [KB]: 18038
% 5.80/2.14 % (18375)Time elapsed: 0.372 s
% 5.80/2.14 % (18375)Instructions burned: 1043 (million)
% 5.80/2.14 % (18375)------------------------------
% 5.80/2.14 % (18375)------------------------------
% 5.80/2.14 % (18436)dis+1010_2:3_amm=off:fd=preordered:ixr=off:nm=0:pe=on:piset=equals:prag=on:sac=on:tgt=ground:i=149:si=on:rtra=on_0 on DTF2THF_18170 for (2995ds/149Mi)
% 5.80/2.14 % (18437)lrs+21_1:16_au=on:bd=off:piset=and:i=39:si=on:rtra=on_0 on DTF2THF_18170 for (2995ds/39Mi)
% 5.80/2.16 % (18437)Instruction limit reached!
% 5.80/2.16 % (18437)------------------------------
% 5.80/2.16 % (18437)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.80/2.16 % (18437)Termination reason: Unknown
% 5.80/2.16 % (18437)Termination phase: shuffling
% 5.80/2.16
% 5.80/2.16 % (18437)Memory used [KB]: 2430
% 5.80/2.16 % (18437)Time elapsed: 0.016 s
% 5.80/2.16 % (18437)Instructions burned: 40 (million)
% 5.80/2.16 % (18437)------------------------------
% 5.80/2.16 % (18437)------------------------------
% 5.80/2.16 % (18438)lrs+1004_1:1_cnfonf=lazy_pi_sigma_gen:fe=off:gs=on:hud=10:prag=on:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_18170 for (2995ds/6Mi)
% 5.80/2.16 % (18438)Instruction limit reached!
% 5.80/2.16 % (18438)------------------------------
% 5.80/2.16 % (18438)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.80/2.16 % (18438)Termination reason: Unknown
% 5.80/2.16 % (18438)Termination phase: shuffling
% 5.80/2.16
% 5.80/2.16 % (18438)Memory used [KB]: 1663
% 5.80/2.16 % (18438)Time elapsed: 0.005 s
% 5.80/2.16 % (18438)Instructions burned: 6 (million)
% 5.80/2.16 % (18438)------------------------------
% 5.80/2.16 % (18438)------------------------------
% 5.80/2.16 % (18433)Instruction limit reached!
% 5.80/2.16 % (18433)------------------------------
% 5.80/2.16 % (18433)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.80/2.16 % (18433)Termination reason: Unknown
% 5.80/2.16 % (18433)Termination phase: Property scanning
% 5.80/2.16
% 5.80/2.16 % (18433)Memory used [KB]: 2814
% 5.80/2.16 % (18433)Time elapsed: 0.046 s
% 5.80/2.16 % (18433)Instructions burned: 99 (million)
% 5.80/2.16 % (18433)------------------------------
% 5.80/2.16 % (18433)------------------------------
% 5.80/2.17 % (18443)lrs+1010_1:128_av=off:cbe=off:cnfonf=lazy_simp:fsr=off:ntd=on:ss=axioms:st=1.5:i=44:si=on:rtra=on_0 on DTF2THF_18170 for (2995ds/44Mi)
% 5.80/2.18 % (18440)dis+1002_1:1_aac=none:au=on:e2e=on:nm=2:nwc=10.0:sd=1:ss=axioms:st=3.0:i=1087:si=on:rtra=on_0 on DTF2THF_18170 for (2995ds/1087Mi)
% 5.80/2.18 % (18442)dis+10_1:1_aac=none:cs=on:i=47:si=on:rtra=on_0 on DTF2THF_18170 for (2995ds/47Mi)
% 5.80/2.19 % (18443)Instruction limit reached!
% 5.80/2.19 % (18443)------------------------------
% 5.80/2.19 % (18443)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.80/2.19 % (18443)Termination reason: Unknown
% 5.80/2.19 % (18443)Termination phase: shuffling
% 5.80/2.19
% 5.80/2.19 % (18443)Memory used [KB]: 2430
% 5.80/2.19 % (18443)Time elapsed: 0.014 s
% 5.80/2.19 % (18443)Instructions burned: 47 (million)
% 5.80/2.19 % (18443)------------------------------
% 5.80/2.19 % (18443)------------------------------
% 6.07/2.20 % (18436)Instruction limit reached!
% 6.07/2.20 % (18436)------------------------------
% 6.07/2.20 % (18436)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.07/2.20 % (18436)Termination reason: Unknown
% 6.07/2.20 % (18436)Termination phase: Function definition elimination
% 6.07/2.20
% 6.07/2.20 % (18436)Memory used [KB]: 3070
% 6.07/2.20 % (18436)Time elapsed: 0.056 s
% 6.07/2.20 % (18436)Instructions burned: 151 (million)
% 6.07/2.20 % (18436)------------------------------
% 6.07/2.20 % (18436)------------------------------
% 6.07/2.20 % (18445)dis+10_1:1_av=off:bd=off:cnfonf=lazy_gen:fe=off:hud=10:plsq=on:plsqc=1:plsqr=32,1:prag=on:tnu=1:i=432:si=on:rtra=on_0 on DTF2THF_18170 for (2995ds/432Mi)
% 6.07/2.20 % (18442)Instruction limit reached!
% 6.07/2.20 % (18442)------------------------------
% 6.07/2.20 % (18442)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.07/2.20 % (18442)Termination reason: Unknown
% 6.07/2.20 % (18442)Termination phase: shuffling
% 6.07/2.20
% 6.07/2.20 % (18442)Memory used [KB]: 2558
% 6.07/2.20 % (18442)Time elapsed: 0.023 s
% 6.07/2.20 % (18442)Instructions burned: 47 (million)
% 6.07/2.20 % (18442)------------------------------
% 6.07/2.20 % (18442)------------------------------
% 6.07/2.21 % (18447)lrs+1010_1:16_add=large:amm=off:anc=all_dependent:chr=on:fd=off:ins=2:sp=const_max:tnu=5:uhcvi=on:i=31:si=on:rtra=on_0 on DTF2THF_18170 for (2995ds/31Mi)
% 6.07/2.21 % (18446)lrs+10_7:1_cnfonf=lazy_gen:fe=off:nwc=5.0:plsq=on:i=62:si=on:rtra=on_0 on DTF2THF_18170 for (2995ds/62Mi)
% 6.07/2.22 % (18447)Instruction limit reached!
% 6.07/2.22 % (18447)------------------------------
% 6.07/2.22 % (18447)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.07/2.22 % (18447)Termination reason: Unknown
% 6.07/2.22 % (18447)Termination phase: shuffling
% 6.07/2.22
% 6.07/2.22 % (18447)Memory used [KB]: 2174
% 6.07/2.22 % (18447)Time elapsed: 0.014 s
% 6.07/2.22 % (18447)Instructions burned: 31 (million)
% 6.07/2.22 % (18447)------------------------------
% 6.07/2.22 % (18447)------------------------------
% 6.07/2.24 % (18446)Instruction limit reached!
% 6.07/2.24 % (18446)------------------------------
% 6.07/2.24 % (18446)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.07/2.24 % (18446)Termination reason: Unknown
% 6.07/2.24 % (18446)Termination phase: Property scanning
% 6.07/2.24
% 6.07/2.24 % (18446)Memory used [KB]: 2686
% 6.07/2.24 % (18446)Time elapsed: 0.030 s
% 6.07/2.24 % (18446)Instructions burned: 63 (million)
% 6.07/2.24 % (18446)------------------------------
% 6.07/2.24 % (18446)------------------------------
% 6.07/2.24 % (18450)dis+10_1:128_hud=1:i=96:si=on:rtra=on_0 on DTF2THF_18170 for (2994ds/96Mi)
% 6.07/2.26 % (18451)lrs+1002_1:4_au=on:cbe=off:cnfonf=lazy_pi_sigma_gen:hud=15:inj=on:nwc=4.0:s2a=on:sac=on:sp=reverse_arity:i=453:si=on:rtra=on_0 on DTF2THF_18170 for (2994ds/453Mi)
% 6.07/2.27 % (18398)Instruction limit reached!
% 6.07/2.27 % (18398)------------------------------
% 6.07/2.27 % (18398)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.07/2.27 % (18398)Termination reason: Unknown
% 6.07/2.27 % (18398)Termination phase: Saturation
% 6.07/2.27
% 6.07/2.27 % (18398)Memory used [KB]: 10746
% 6.07/2.27 % (18398)Time elapsed: 0.417 s
% 6.07/2.27 % (18398)Instructions burned: 880 (million)
% 6.07/2.27 % (18398)------------------------------
% 6.07/2.27 % (18398)------------------------------
% 6.07/2.28 % (18429)Instruction limit reached!
% 6.07/2.28 % (18429)------------------------------
% 6.07/2.28 % (18429)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.07/2.28 % (18429)Termination reason: Unknown
% 6.07/2.28 % (18429)Termination phase: Saturation
% 6.07/2.28
% 6.07/2.28 % (18429)Memory used [KB]: 7803
% 6.07/2.28 % (18429)Time elapsed: 0.184 s
% 6.07/2.28 % (18429)Instructions burned: 450 (million)
% 6.07/2.28 % (18429)------------------------------
% 6.07/2.28 % (18429)------------------------------
% 6.07/2.28 % (18450)Instruction limit reached!
% 6.07/2.28 % (18450)------------------------------
% 6.07/2.28 % (18450)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.07/2.28 % (18450)Termination reason: Unknown
% 6.07/2.28 % (18450)Termination phase: Property scanning
% 6.07/2.28
% 6.07/2.28 % (18450)Memory used [KB]: 2814
% 6.07/2.28 % (18450)Time elapsed: 0.045 s
% 6.07/2.28 % (18450)Instructions burned: 96 (million)
% 6.07/2.28 % (18450)------------------------------
% 6.07/2.28 % (18450)------------------------------
% 6.07/2.29 % (18453)dis+1002_1:1_bd=off:fsr=off:ntd=on:s2a=on:ss=axioms:i=83:si=on:rtra=on_0 on DTF2THF_18170 for (2994ds/83Mi)
% 6.07/2.29 % (18452)lrs-1004_1:1_au=on:bsd=on:cbe=off:cnfonf=off:fe=axiom:fsd=on:i=71:si=on:rtra=on_0 on DTF2THF_18170 for (2994ds/71Mi)
% 6.59/2.30 % (18457)lrs+10_1:128_cnfonf=lazy_not_gen_be_off:hud=1:ins=1:i=11:si=on:rtra=on_0 on DTF2THF_18170 for (2994ds/11Mi)
% 6.59/2.31 % (18457)Instruction limit reached!
% 6.59/2.31 % (18457)------------------------------
% 6.59/2.31 % (18457)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.59/2.31 % (18457)Termination reason: Unknown
% 6.59/2.31 % (18457)Termination phase: shuffling
% 6.59/2.31
% 6.59/2.31 % (18457)Memory used [KB]: 1918
% 6.59/2.31 % (18457)Time elapsed: 0.007 s
% 6.59/2.31 % (18457)Instructions burned: 12 (million)
% 6.59/2.31 % (18457)------------------------------
% 6.59/2.31 % (18457)------------------------------
% 6.59/2.31 % (18453)Instruction limit reached!
% 6.59/2.31 % (18453)------------------------------
% 6.59/2.31 % (18453)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.59/2.31 % (18453)Termination reason: Unknown
% 6.59/2.31 % (18453)Termination phase: Property scanning
% 6.59/2.31
% 6.59/2.31 % (18453)Memory used [KB]: 2814
% 6.59/2.31 % (18453)Time elapsed: 0.024 s
% 6.59/2.31 % (18453)Instructions burned: 85 (million)
% 6.59/2.31 % (18453)------------------------------
% 6.59/2.31 % (18453)------------------------------
% 6.59/2.32 % (18452)Instruction limit reached!
% 6.59/2.32 % (18452)------------------------------
% 6.59/2.32 % (18452)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.59/2.32 % (18452)Termination reason: Unknown
% 6.59/2.32 % (18452)Termination phase: Property scanning
% 6.59/2.32
% 6.59/2.32 % (18452)Memory used [KB]: 2814
% 6.59/2.32 % (18452)Time elapsed: 0.030 s
% 6.59/2.32 % (18452)Instructions burned: 74 (million)
% 6.59/2.32 % (18452)------------------------------
% 6.59/2.32 % (18452)------------------------------
% 6.59/2.32 % (18445)Instruction limit reached!
% 6.59/2.32 % (18445)------------------------------
% 6.59/2.32 % (18445)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.59/2.32 % (18445)Termination reason: Unknown
% 6.59/2.32 % (18445)Termination phase: Saturation
% 6.59/2.32
% 6.59/2.32 % (18445)Memory used [KB]: 3326
% 6.59/2.32 % (18445)Time elapsed: 0.124 s
% 6.59/2.32 % (18445)Instructions burned: 432 (million)
% 6.59/2.32 % (18445)------------------------------
% 6.59/2.32 % (18445)------------------------------
% 6.59/2.32 % (18459)lrs+1010_1:1_au=on:cbe=off:cnfonf=lazy_not_gen_be_off:fsr=off:spb=non_intro:i=41:si=on:rtra=on_0 on DTF2THF_18170 for (2993ds/41Mi)
% 6.59/2.32 % (18461)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_18170 for (2993ds/6Mi)
% 6.59/2.33 % (18461)Instruction limit reached!
% 6.59/2.33 % (18461)------------------------------
% 6.59/2.33 % (18461)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.59/2.33 % (18461)Termination reason: Unknown
% 6.59/2.33 % (18461)Termination phase: shuffling
% 6.59/2.33
% 6.59/2.33 % (18461)Memory used [KB]: 1663
% 6.59/2.33 % (18461)Time elapsed: 0.004 s
% 6.59/2.33 % (18461)Instructions burned: 9 (million)
% 6.59/2.33 % (18461)------------------------------
% 6.59/2.33 % (18461)------------------------------
% 6.59/2.33 % (18462)dis+10_1:1_avsq=on:cnfonf=off:e2e=on:nm=32:i=150:si=on:rtra=on_0 on DTF2THF_18170 for (2993ds/150Mi)
% 6.59/2.33 % (18463)ott+1010_1:1_au=on:s2a=on:sd=7:ss=axioms:st=1.5:i=343:si=on:rtra=on_0 on DTF2THF_18170 for (2993ds/343Mi)
% 6.59/2.33 % (18459)Instruction limit reached!
% 6.59/2.33 % (18459)------------------------------
% 6.59/2.33 % (18459)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.59/2.33 % (18459)Termination reason: Unknown
% 6.59/2.33 % (18459)Termination phase: shuffling
% 6.59/2.33
% 6.59/2.33 % (18459)Memory used [KB]: 2430
% 6.59/2.33 % (18459)Time elapsed: 0.017 s
% 6.59/2.33 % (18459)Instructions burned: 42 (million)
% 6.59/2.33 % (18459)------------------------------
% 6.59/2.33 % (18459)------------------------------
% 6.59/2.34 % (18464)ott+1010_1:4_cbe=off:chr=on:hfsq=on:hfsqr=1,16:hud=3:sac=on:sp=unary_frequency:ss=axioms:st=5.0:tnu=2:i=109:si=on:rtra=on_0 on DTF2THF_18170 for (2993ds/109Mi)
% 6.59/2.34 % (18465)dis+10_1:1_au=on:c=on:e2e=on:fsr=off:sos=all:sp=const_frequency:ss=axioms:i=253:si=on:rtra=on_0 on DTF2THF_18170 for (2993ds/253Mi)
% 6.59/2.35 % (18434)Instruction limit reached!
% 6.59/2.35 % (18434)------------------------------
% 6.59/2.35 % (18434)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.59/2.35 % (18434)Termination reason: Unknown
% 6.59/2.35 % (18434)Termination phase: Saturation
% 6.59/2.35
% 6.59/2.35 % (18434)Memory used [KB]: 8315
% 6.59/2.35 % (18434)Time elapsed: 0.215 s
% 6.59/2.35 % (18434)Instructions burned: 509 (million)
% 6.59/2.35 % (18434)------------------------------
% 6.59/2.35 % (18434)------------------------------
% 6.59/2.35 % (18466)lrs+10_1:1_au=on:cnfonf=off:sd=1:sgt=16:ss=axioms:i=140:si=on:rtra=on_0 on DTF2THF_18170 for (2993ds/140Mi)
% 6.59/2.36 % (18464)Instruction limit reached!
% 6.59/2.36 % (18464)------------------------------
% 6.59/2.36 % (18464)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.59/2.36 % (18464)Termination reason: Unknown
% 6.59/2.36 % (18464)Termination phase: SInE selection
% 6.59/2.36
% 6.59/2.36 % (18464)Memory used [KB]: 2814
% 6.59/2.36 % (18464)Time elapsed: 0.029 s
% 6.59/2.36 % (18464)Instructions burned: 111 (million)
% 6.59/2.36 % (18464)------------------------------
% 6.59/2.36 % (18464)------------------------------
% 6.59/2.37 % (18462)Instruction limit reached!
% 6.59/2.37 % (18462)------------------------------
% 6.59/2.37 % (18462)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.59/2.37 % (18462)Termination reason: Unknown
% 6.59/2.37 % (18462)Termination phase: Preprocessing 3
% 6.59/2.37
% 6.59/2.37 % (18462)Memory used [KB]: 5245
% 6.59/2.37 % (18462)Time elapsed: 0.045 s
% 6.59/2.37 % (18462)Instructions burned: 151 (million)
% 6.59/2.37 % (18462)------------------------------
% 6.59/2.37 % (18462)------------------------------
% 6.59/2.37 % (18467)dis+1010_16:1_acc=model:au=on:avsq=on:cond=fast:plsq=on:plsqc=2:plsqr=9267,262144:i=614:si=on:rtra=on_0 on DTF2THF_18170 for (2993ds/614Mi)
% 6.59/2.38 % (18468)lrs+1002_1:1_e2e=on:fde=unused:fe=axiom:prag=on:sd=1:sos=on:ss=axioms:st=2.0:i=202:si=on:rtra=on_0 on DTF2THF_18170 for (2993ds/202Mi)
% 6.59/2.39 % (18451)Instruction limit reached!
% 6.59/2.39 % (18451)------------------------------
% 6.59/2.39 % (18451)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.59/2.39 % (18451)Termination reason: Unknown
% 6.59/2.39 % (18451)Termination phase: Saturation
% 6.59/2.39
% 6.59/2.39 % (18451)Memory used [KB]: 7803
% 6.59/2.39 % (18451)Time elapsed: 0.133 s
% 6.59/2.39 % (18451)Instructions burned: 455 (million)
% 6.59/2.39 % (18451)------------------------------
% 6.59/2.39 % (18451)------------------------------
% 6.59/2.39 % (18466)Instruction limit reached!
% 6.59/2.39 % (18466)------------------------------
% 6.59/2.39 % (18466)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.59/2.39 % (18466)Termination reason: Unknown
% 6.59/2.39 % (18466)Termination phase: Property scanning
% 6.59/2.39
% 6.59/2.39 % (18466)Memory used [KB]: 2942
% 6.59/2.39 % (18466)Time elapsed: 0.037 s
% 6.59/2.39 % (18466)Instructions burned: 141 (million)
% 6.59/2.39 % (18466)------------------------------
% 6.59/2.39 % (18466)------------------------------
% 7.39/2.40 % (18475)dis+1002_1:1_au=on:cs=on:ins=3:kws=inv_frequency:nwc=10.0:ss=axioms:st=3.0:i=339:si=on:rtra=on_0 on DTF2THF_18170 for (2993ds/339Mi)
% 7.39/2.40 % (18477)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=283:si=on:rtra=on_0 on DTF2THF_18170 for (2993ds/283Mi)
% 7.39/2.42 % (18465)Instruction limit reached!
% 7.39/2.42 % (18465)------------------------------
% 7.39/2.42 % (18465)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.39/2.42 % (18465)Termination reason: Unknown
% 7.39/2.42 % (18465)Termination phase: Property scanning
% 7.39/2.42
% 7.39/2.42 % (18465)Memory used [KB]: 5884
% 7.39/2.42 % (18465)Time elapsed: 0.072 s
% 7.39/2.42 % (18465)Instructions burned: 256 (million)
% 7.39/2.42 % (18465)------------------------------
% 7.39/2.42 % (18465)------------------------------
% 7.39/2.42 % (18463)Instruction limit reached!
% 7.39/2.42 % (18463)------------------------------
% 7.39/2.42 % (18463)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.39/2.42 % (18463)Termination reason: Unknown
% 7.39/2.42 % (18463)Termination phase: Saturation
% 7.39/2.42
% 7.39/2.42 % (18463)Memory used [KB]: 8315
% 7.39/2.42 % (18463)Time elapsed: 0.089 s
% 7.39/2.42 % (18463)Instructions burned: 343 (million)
% 7.39/2.42 % (18463)------------------------------
% 7.39/2.42 % (18463)------------------------------
% 7.39/2.43 % (18488)dis+2_1:1_cha=on:hud=7:nwc=3.0:prag=on:tnu=2:i=134:si=on:rtra=on_0 on DTF2THF_18170 for (2992ds/134Mi)
% 7.39/2.43 % (18490)dis+1010_1:64_cnfonf=lazy_simp:fd=off:i=298:si=on:rtra=on_0 on DTF2THF_18170 for (2992ds/298Mi)
% 7.39/2.44 % (18468)Instruction limit reached!
% 7.39/2.44 % (18468)------------------------------
% 7.39/2.44 % (18468)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.39/2.44 % (18468)Termination reason: Unknown
% 7.39/2.44 % (18468)Termination phase: Property scanning
% 7.39/2.44
% 7.39/2.44 % (18468)Memory used [KB]: 5628
% 7.39/2.44 % (18468)Time elapsed: 0.061 s
% 7.39/2.44 % (18468)Instructions burned: 205 (million)
% 7.39/2.44 % (18468)------------------------------
% 7.39/2.44 % (18468)------------------------------
% 7.39/2.45 % (18500)lrs+2_1:8_erd=off:plsq=on:plsqr=32,1:sos=on:i=192:si=on:rtra=on_0 on DTF2THF_18170 for (2992ds/192Mi)
% 7.39/2.46 % (18488)Instruction limit reached!
% 7.39/2.46 % (18488)------------------------------
% 7.39/2.46 % (18488)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.39/2.46 % (18488)Termination reason: Unknown
% 7.39/2.46 % (18488)Termination phase: Property scanning
% 7.39/2.46
% 7.39/2.46 % (18488)Memory used [KB]: 2942
% 7.39/2.46 % (18488)Time elapsed: 0.036 s
% 7.39/2.46 % (18488)Instructions burned: 136 (million)
% 7.39/2.46 % (18488)------------------------------
% 7.39/2.46 % (18488)------------------------------
% 7.39/2.47 % (18510)lrs+10_1:1_bd=off:hud=1:nm=0:prag=on:sd=2:sgt=32:sos=all:ss=axioms:i=258:si=on:rtra=on_0 on DTF2THF_18170 for (2992ds/258Mi)
% 7.39/2.47 % (18477)Instruction limit reached!
% 7.39/2.47 % (18477)------------------------------
% 7.39/2.47 % (18477)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.39/2.47 % (18477)Termination reason: Unknown
% 7.39/2.47 % (18477)Termination phase: Saturation
% 7.39/2.47
% 7.39/2.47 % (18477)Memory used [KB]: 7675
% 7.39/2.47 % (18477)Time elapsed: 0.072 s
% 7.39/2.47 % (18477)Instructions burned: 283 (million)
% 7.39/2.47 % (18477)------------------------------
% 7.39/2.47 % (18477)------------------------------
% 7.39/2.48 % (18515)lrs+1002_1:1024_au=on:cond=fast:ntd=on:piset=and:plsq=on:plsqc=1:plsqr=183491,1048576:sac=on:tgt=ground:i=90:si=on:rtra=on_0 on DTF2THF_18170 for (2992ds/90Mi)
% 7.39/2.49 % (18475)Instruction limit reached!
% 7.39/2.49 % (18475)------------------------------
% 7.39/2.49 % (18475)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.39/2.49 % (18475)Termination reason: Unknown
% 7.39/2.49 % (18475)Termination phase: Saturation
% 7.39/2.49
% 7.39/2.49 % (18475)Memory used [KB]: 8059
% 7.39/2.49 % (18475)Time elapsed: 0.092 s
% 7.39/2.49 % (18475)Instructions burned: 340 (million)
% 7.39/2.49 % (18475)------------------------------
% 7.39/2.49 % (18475)------------------------------
% 7.86/2.50 % (18500)Instruction limit reached!
% 7.86/2.50 % (18500)------------------------------
% 7.86/2.50 % (18500)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.86/2.50 % (18500)Termination reason: Unknown
% 7.86/2.50 % (18500)Termination phase: Saturation
% 7.86/2.50
% 7.86/2.50 % (18500)Memory used [KB]: 3070
% 7.86/2.50 % (18500)Time elapsed: 0.050 s
% 7.86/2.50 % (18500)Instructions burned: 192 (million)
% 7.86/2.50 % (18500)------------------------------
% 7.86/2.50 % (18500)------------------------------
% 7.86/2.50 % (18523)dis+1004_16:1_avsq=on:avsqr=1,16:cnfonf=lazy_gen:cs=on:fe=off:tgt=ground:tnu=1:i=231:si=on:rtra=on_0 on DTF2THF_18170 for (2992ds/231Mi)
% 7.86/2.50 % (18490)Instruction limit reached!
% 7.86/2.50 % (18490)------------------------------
% 7.86/2.50 % (18490)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.86/2.50 % (18490)Termination reason: Unknown
% 7.86/2.50 % (18490)Termination phase: Saturation
% 7.86/2.50
% 7.86/2.50 % (18490)Memory used [KB]: 7675
% 7.86/2.50 % (18490)Time elapsed: 0.076 s
% 7.86/2.50 % (18490)Instructions burned: 300 (million)
% 7.86/2.50 % (18490)------------------------------
% 7.86/2.50 % (18490)------------------------------
% 7.86/2.50 % (18515)Instruction limit reached!
% 7.86/2.50 % (18515)------------------------------
% 7.86/2.50 % (18515)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.86/2.50 % (18515)Termination reason: Unknown
% 7.86/2.50 % (18515)Termination phase: Property scanning
% 7.86/2.50
% 7.86/2.50 % (18515)Memory used [KB]: 2814
% 7.86/2.50 % (18515)Time elapsed: 0.025 s
% 7.86/2.50 % (18515)Instructions burned: 92 (million)
% 7.86/2.50 % (18515)------------------------------
% 7.86/2.50 % (18515)------------------------------
% 7.86/2.51 % (18529)lrs+1002_1:1_au=on:s2a=on:sgt=32:slsq=on:slsqc=1:slsql=off:ss=axioms:i=458:si=on:rtra=on_0 on DTF2THF_18170 for (2992ds/458Mi)
% 7.86/2.51 % (18530)lrs+1010_1:16_add=large:amm=off:anc=all_dependent:chr=on:fd=off:ins=2:sp=const_max:tnu=5:uhcvi=on:i=31:si=on:rtra=on_0 on DTF2THF_18170 for (2992ds/31Mi)
% 7.86/2.51 % (18532)lrs+10_1:1_cnfonf=lazy_gen:kws=inv_precedence:nwc=10.0:s2a=on:sp=weighted_frequency:ss=axioms:i=297:si=on:rtra=on_0 on DTF2THF_18170 for (2992ds/297Mi)
% 7.86/2.52 % (18530)Instruction limit reached!
% 7.86/2.52 % (18530)------------------------------
% 7.86/2.52 % (18530)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.86/2.52 % (18530)Termination reason: Unknown
% 7.86/2.52 % (18530)Termination phase: shuffling
% 7.86/2.52
% 7.86/2.52 % (18530)Memory used [KB]: 2174
% 7.86/2.52 % (18530)Time elapsed: 0.010 s
% 7.86/2.52 % (18530)Instructions burned: 31 (million)
% 7.86/2.52 % (18530)------------------------------
% 7.86/2.52 % (18530)------------------------------
% 7.86/2.53 % (18537)dis+2_1:4_amm=off:au=on:cbe=off:cnfonf=lazy_gen:hfaw=0:hfsq=on:hfsql=off:plsq=on:plsqc=1:plsqr=35,4:sac=on:i=987:si=on:rtra=on_0 on DTF2THF_18170 for (2991ds/987Mi)
% 7.86/2.54 % (18510)Instruction limit reached!
% 7.86/2.54 % (18510)------------------------------
% 7.86/2.54 % (18510)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.86/2.54 % (18510)Termination reason: Unknown
% 7.86/2.54 % (18510)Termination phase: Saturation
% 7.86/2.54
% 7.86/2.54 % (18510)Memory used [KB]: 8059
% 7.86/2.54 % (18510)Time elapsed: 0.068 s
% 7.86/2.54 % (18510)Instructions burned: 259 (million)
% 7.86/2.54 % (18510)------------------------------
% 7.86/2.54 % (18510)------------------------------
% 7.86/2.55 % (18545)lrs+10_1:1024_avsq=on:avsqc=4:avsqr=584411,1048576:bd=off:bet=on:br=off:chr=on:fsd=on:fsdmm=1:hud=10:nm=0:ntd=on:nwc=9.0:piset=or:plsq=on:plsqc=2:plsql=on:plsqr=852219,1048576:rawr=on:sp=const_frequency:spb=units:i=182:si=on:rtra=on_0 on DTF2THF_18170 for (2991ds/182Mi)
% 7.86/2.55 % (18467)Instruction limit reached!
% 7.86/2.55 % (18467)------------------------------
% 7.86/2.55 % (18467)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.86/2.55 % (18467)Termination reason: Unknown
% 7.86/2.55 % (18467)Termination phase: Saturation
% 7.86/2.55
% 7.86/2.55 % (18467)Memory used [KB]: 10490
% 7.86/2.55 % (18467)Time elapsed: 0.176 s
% 7.86/2.55 % (18467)Instructions burned: 615 (million)
% 7.86/2.55 % (18467)------------------------------
% 7.86/2.55 % (18467)------------------------------
% 7.86/2.56 % (18548)dis+10_1:1_au=on:e2e=on:fde=none:nwc=10.0:s2a=on:s2at=2.0:i=182:si=on:rtra=on_0 on DTF2THF_18170 for (2991ds/182Mi)
% 7.86/2.56 % (18523)Instruction limit reached!
% 7.86/2.56 % (18523)------------------------------
% 7.86/2.56 % (18523)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.86/2.56 % (18523)Termination reason: Unknown
% 7.86/2.56 % (18523)Termination phase: Property scanning
% 7.86/2.56
% 7.86/2.56 % (18523)Memory used [KB]: 4093
% 7.86/2.56 % (18523)Time elapsed: 0.062 s
% 7.86/2.56 % (18523)Instructions burned: 233 (million)
% 7.86/2.56 % (18523)------------------------------
% 7.86/2.56 % (18523)------------------------------
% 7.86/2.57 % (18440)Instruction limit reached!
% 7.86/2.57 % (18440)------------------------------
% 7.86/2.57 % (18440)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.86/2.57 % (18440)Termination reason: Unknown
% 7.86/2.57 % (18440)Termination phase: Saturation
% 7.86/2.57
% 7.86/2.57 % (18440)Memory used [KB]: 13048
% 7.86/2.57 % (18440)Time elapsed: 0.394 s
% 7.86/2.57 % (18440)Instructions burned: 1087 (million)
% 7.86/2.57 % (18440)------------------------------
% 7.86/2.57 % (18440)------------------------------
% 7.86/2.57 % (18554)dis+10_1:8_cha=on:cnfonf=lazy_pi_sigma_gen:fe=axiom:hud=5:plsq=on:plsqr=32,1:prag=on:sos=all:i=154:si=on:rtra=on_0 on DTF2THF_18170 for (2991ds/154Mi)
% 7.86/2.58 % (18561)dis+1003_1:64_atotf=0.4:au=on:fsr=off:hflw=5:hfsq=on:hfsqc=20:hfsql=off:plsq=on:plsqc=1:plsqr=32,1:i=79:si=on:rtra=on_0 on DTF2THF_18170 for (2991ds/79Mi)
% 7.86/2.59 % (18532)Instruction limit reached!
% 7.86/2.59 % (18532)------------------------------
% 7.86/2.59 % (18532)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.86/2.59 % (18532)Termination reason: Unknown
% 7.86/2.59 % (18532)Termination phase: Saturation
% 7.86/2.59
% 7.86/2.59 % (18532)Memory used [KB]: 7675
% 7.86/2.59 % (18532)Time elapsed: 0.076 s
% 7.86/2.59 % (18532)Instructions burned: 299 (million)
% 7.86/2.59 % (18532)------------------------------
% 7.86/2.59 % (18532)------------------------------
% 7.86/2.59 % (18545)Instruction limit reached!
% 7.86/2.59 % (18545)------------------------------
% 7.86/2.59 % (18545)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.86/2.59 % (18545)Termination reason: Unknown
% 7.86/2.59 % (18545)Termination phase: Property scanning
% 7.86/2.59
% 7.86/2.59 % (18545)Memory used [KB]: 3070
% 7.86/2.59 % (18545)Time elapsed: 0.048 s
% 7.86/2.59 % (18545)Instructions burned: 182 (million)
% 7.86/2.59 % (18545)------------------------------
% 7.86/2.59 % (18545)------------------------------
% 8.93/2.60 % (18571)dis+1010_5:2_anc=all_dependent:awrs=converge:bd=off:bet=on:c=on:cha=on:cnfonf=lazy_not_gen_be_off:fe=axiom:hud=5:kws=precedence:piset=and:prag=on:s2a=on:sac=on:slsq=on:slsqc=1:sp=reverse_arity:ss=axioms:st=1.5:i=216:si=on:rtra=on_0 on DTF2THF_18170 for (2991ds/216Mi)
% 8.93/2.60 % (18561)Instruction limit reached!
% 8.93/2.60 % (18561)------------------------------
% 8.93/2.60 % (18561)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.93/2.60 % (18561)Termination reason: Unknown
% 8.93/2.60 % (18561)Termination phase: Property scanning
% 8.93/2.60
% 8.93/2.60 % (18561)Memory used [KB]: 2814
% 8.93/2.60 % (18561)Time elapsed: 0.023 s
% 8.93/2.60 % (18561)Instructions burned: 80 (million)
% 8.93/2.60 % (18561)------------------------------
% 8.93/2.60 % (18561)------------------------------
% 8.93/2.60 % (18572)lrs+1002_1:1_fe=off:hud=10:ins=2:sac=on:sp=const_frequency:i=114:si=on:rtra=on_0 on DTF2THF_18170 for (2991ds/114Mi)
% 8.93/2.61 % (18573)dis+1002_1:1_acc=on:au=on:bd=off:e2e=on:sgt=8:ss=axioms:i=922:si=on:rtra=on_0 on DTF2THF_18170 for (2991ds/922Mi)
% 8.93/2.61 % (18554)Instruction limit reached!
% 8.93/2.61 % (18554)------------------------------
% 8.93/2.61 % (18554)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.93/2.61 % (18554)Termination reason: Unknown
% 8.93/2.61 % (18554)Termination phase: Function definition elimination
% 8.93/2.61
% 8.93/2.61 % (18554)Memory used [KB]: 2942
% 8.93/2.61 % (18554)Time elapsed: 0.040 s
% 8.93/2.61 % (18554)Instructions burned: 154 (million)
% 8.93/2.61 % (18554)------------------------------
% 8.93/2.61 % (18554)------------------------------
% 8.93/2.61 % (18548)Instruction limit reached!
% 8.93/2.61 % (18548)------------------------------
% 8.93/2.61 % (18548)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.93/2.61 % (18548)Termination reason: Unknown
% 8.93/2.61 % (18548)Termination phase: Property scanning
% 8.93/2.61
% 8.93/2.61 % (18548)Memory used [KB]: 5628
% 8.93/2.61 % (18548)Time elapsed: 0.055 s
% 8.93/2.61 % (18548)Instructions burned: 184 (million)
% 8.93/2.61 % (18548)------------------------------
% 8.93/2.61 % (18548)------------------------------
% 8.93/2.62 % (18583)lrs+10_1:128_au=on:bd=off:cnfonf=lazy_not_gen_be_off:nwc=10.0:piset=and:i=182:si=on:rtra=on_0 on DTF2THF_18170 for (2991ds/182Mi)
% 8.93/2.62 % (18586)ott+1010_1:1024_au=on:av=off:c=on:cnfonf=conj_eager:piset=equals:s2a=on:sp=occurrence:spb=units:ss=axioms:i=164:si=on:rtra=on_0 on DTF2THF_18170 for (2991ds/164Mi)
% 8.93/2.63 % (18529)Instruction limit reached!
% 8.93/2.63 % (18529)------------------------------
% 8.93/2.63 % (18529)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.93/2.63 % (18529)Termination reason: Unknown
% 8.93/2.63 % (18529)Termination phase: Saturation
% 8.93/2.63
% 8.93/2.63 % (18529)Memory used [KB]: 9722
% 8.93/2.63 % (18529)Time elapsed: 0.122 s
% 8.93/2.63 % (18529)Instructions burned: 459 (million)
% 8.93/2.63 % (18529)------------------------------
% 8.93/2.63 % (18529)------------------------------
% 8.93/2.63 % (18572)Instruction limit reached!
% 8.93/2.63 % (18572)------------------------------
% 8.93/2.63 % (18572)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.93/2.63 % (18572)Termination reason: Unknown
% 8.93/2.63 % (18572)Termination phase: Property scanning
% 8.93/2.63
% 8.93/2.63 % (18572)Memory used [KB]: 2942
% 8.93/2.63 % (18572)Time elapsed: 0.031 s
% 8.93/2.63 % (18572)Instructions burned: 116 (million)
% 8.93/2.63 % (18572)------------------------------
% 8.93/2.63 % (18572)------------------------------
% 8.93/2.64 % (18598)lrs+1004_1:1_cnfonf=lazy_pi_sigma_gen:fe=off:gs=on:hud=10:prag=on:tnu=1:i=50:si=on:rtra=on_0 on DTF2THF_18170 for (2990ds/50Mi)
% 8.93/2.64 % (18601)dis-30_1:1_acc=on:bd=off:fsr=off:plsq=on:plsqr=32,1:i=368:si=on:rtra=on_0 on DTF2THF_18170 for (2990ds/368Mi)
% 8.93/2.65 % (18571)Instruction limit reached!
% 8.93/2.65 % (18571)------------------------------
% 8.93/2.65 % (18571)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.93/2.65 % (18571)Termination reason: Unknown
% 8.93/2.65 % (18571)Termination phase: Property scanning
% 8.93/2.65
% 8.93/2.65 % (18571)Memory used [KB]: 3198
% 8.93/2.65 % (18571)Time elapsed: 0.054 s
% 8.93/2.65 % (18571)Instructions burned: 218 (million)
% 8.93/2.65 % (18571)------------------------------
% 8.93/2.65 % (18571)------------------------------
% 8.93/2.65 % (18598)Instruction limit reached!
% 8.93/2.65 % (18598)------------------------------
% 8.93/2.65 % (18598)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.93/2.65 % (18598)Termination reason: Unknown
% 8.93/2.65 % (18598)Termination phase: shuffling
% 8.93/2.65
% 8.93/2.65 % (18598)Memory used [KB]: 2558
% 8.93/2.65 % (18598)Time elapsed: 0.015 s
% 8.93/2.65 % (18598)Instructions burned: 52 (million)
% 8.93/2.65 % (18598)------------------------------
% 8.93/2.65 % (18598)------------------------------
% 8.93/2.66 % (18604)dis+2_1:1_bd=off:e2e=on:hud=1:nm=3:piset=or:prag=on:s2a=on:sp=const_max:i=1021:si=on:rtra=on_0 on DTF2THF_18170 for (2990ds/1021Mi)
% 8.93/2.66 % (18586)Instruction limit reached!
% 8.93/2.66 % (18586)------------------------------
% 8.93/2.66 % (18586)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.93/2.66 % (18586)Termination reason: Unknown
% 8.93/2.66 % (18586)Termination phase: Property scanning
% 8.93/2.66
% 8.93/2.66 % (18586)Memory used [KB]: 2942
% 8.93/2.66 % (18586)Time elapsed: 0.042 s
% 8.93/2.66 % (18586)Instructions burned: 164 (million)
% 8.93/2.66 % (18586)------------------------------
% 8.93/2.66 % (18586)------------------------------
% 8.93/2.66 % (18583)Instruction limit reached!
% 8.93/2.66 % (18583)------------------------------
% 8.93/2.66 % (18583)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.93/2.66 % (18583)Termination reason: Unknown
% 8.93/2.66 % (18583)Termination phase: Property scanning
% 8.93/2.66
% 8.93/2.66 % (18583)Memory used [KB]: 3198
% 8.93/2.66 % (18583)Time elapsed: 0.046 s
% 8.93/2.66 % (18583)Instructions burned: 182 (million)
% 8.93/2.66 % (18583)------------------------------
% 8.93/2.66 % (18583)------------------------------
% 8.93/2.66 % (18608)dis+10_1:1_cnfonf=lazy_gen:fe=off:plsq=on:plsqc=1:plsqr=32,1:prag=on:i=267:si=on:rtra=on_0 on DTF2THF_18170 for (2990ds/267Mi)
% 8.93/2.67 % (18617)dis+21_1:5_au=on:ntd=on:nwc=5.0:s2a=on:ss=axioms:st=5.0:i=204:si=on:rtra=on_0 on DTF2THF_18170 for (2990ds/204Mi)
% 8.93/2.67 % (18619)dis+2_1:1024_atotf=0.0802881:au=on:bs=unit_only:cnfonf=lazy_not_gen_be_off:fde=none:nwc=10.0:prag=on:i=26:si=on:rtra=on_0 on DTF2THF_18170 for (2990ds/26Mi)
% 8.93/2.68 % (18619)Instruction limit reached!
% 8.93/2.68 % (18619)------------------------------
% 8.93/2.68 % (18619)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.93/2.68 % (18619)Termination reason: Unknown
% 8.93/2.68 % (18619)Termination phase: shuffling
% 8.93/2.68
% 8.93/2.68 % (18619)Memory used [KB]: 2174
% 8.93/2.68 % (18619)Time elapsed: 0.009 s
% 8.93/2.68 % (18619)Instructions burned: 29 (million)
% 8.93/2.68 % (18619)------------------------------
% 8.93/2.68 % (18619)------------------------------
% 8.93/2.69 % (18624)dis+1002_1:1_bd=preordered:c=on:cbe=off:cnfonf=lazy_not_gen_be_off:hud=10:ixr=off:prag=on:i=53:si=on:rtra=on_0 on DTF2THF_18170 for (2990ds/53Mi)
% 9.45/2.71 % (18624)Instruction limit reached!
% 9.45/2.71 % (18624)------------------------------
% 9.45/2.71 % (18624)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.45/2.71 % (18624)Termination reason: Unknown
% 9.45/2.71 % (18624)Termination phase: shuffling
% 9.45/2.71
% 9.45/2.71 % (18624)Memory used [KB]: 2686
% 9.45/2.71 % (18624)Time elapsed: 0.016 s
% 9.45/2.71 % (18624)Instructions burned: 56 (million)
% 9.45/2.71 % (18624)------------------------------
% 9.45/2.71 % (18624)------------------------------
% 9.45/2.72 % (18637)dis+1002_1:1_aac=none:au=on:e2e=on:nm=2:nwc=10.0:sd=1:ss=axioms:st=3.0:i=4436:si=on:rtra=on_0 on DTF2THF_18170 for (2990ds/4436Mi)
% 9.45/2.72 % (18617)Instruction limit reached!
% 9.45/2.72 % (18617)------------------------------
% 9.45/2.72 % (18617)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.45/2.72 % (18617)Termination reason: Unknown
% 9.45/2.72 % (18617)Termination phase: Property scanning
% 9.45/2.72
% 9.45/2.72 % (18617)Memory used [KB]: 3070
% 9.45/2.72 % (18617)Time elapsed: 0.053 s
% 9.45/2.72 % (18617)Instructions burned: 204 (million)
% 9.45/2.72 % (18617)------------------------------
% 9.45/2.72 % (18617)------------------------------
% 9.45/2.73 % (18608)Instruction limit reached!
% 9.45/2.73 % (18608)------------------------------
% 9.45/2.73 % (18608)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.45/2.73 % (18608)Termination reason: Unknown
% 9.45/2.73 % (18608)Termination phase: Saturation
% 9.45/2.73
% 9.45/2.73 % (18608)Memory used [KB]: 7675
% 9.45/2.73 % (18608)Time elapsed: 0.068 s
% 9.45/2.73 % (18608)Instructions burned: 270 (million)
% 9.45/2.73 % (18608)------------------------------
% 9.45/2.73 % (18608)------------------------------
% 9.45/2.73 % (18644)dis+1010_1:1_acc=model:avsq=on:ixr=off:prag=on:s2a=on:sac=on:i=1539:si=on:rtra=on_0 on DTF2THF_18170 for (2989ds/1539Mi)
% 9.45/2.74 % (18601)Instruction limit reached!
% 9.45/2.74 % (18601)------------------------------
% 9.45/2.74 % (18601)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.45/2.74 % (18601)Termination reason: Unknown
% 9.45/2.74 % (18601)Termination phase: Saturation
% 9.45/2.74
% 9.45/2.74 % (18601)Memory used [KB]: 9466
% 9.45/2.74 % (18601)Time elapsed: 0.098 s
% 9.45/2.74 % (18601)Instructions burned: 370 (million)
% 9.45/2.74 % (18601)------------------------------
% 9.45/2.74 % (18601)------------------------------
% 9.45/2.74 % (18647)lrs+2_1:1_bd=off:e2e=on:fde=none:kws=inv_precedence:plsq=on:plsqr=32,1:sos=on:i=408:si=on:rtra=on_0 on DTF2THF_18170 for (2989ds/408Mi)
% 9.45/2.75 % (18651)lrs+1002_1:1_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=4203469,65536:i=5205:si=on:rtra=on_0 on DTF2THF_18170 for (2989ds/5205Mi)
% 9.45/2.78 % (18537)Instruction limit reached!
% 9.45/2.78 % (18537)------------------------------
% 9.45/2.78 % (18537)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.45/2.78 % (18537)Termination reason: Unknown
% 9.45/2.78 % (18537)Termination phase: Saturation
% 9.45/2.78
% 9.45/2.78 % (18537)Memory used [KB]: 13048
% 9.45/2.78 % (18537)Time elapsed: 0.251 s
% 9.45/2.78 % (18537)Instructions burned: 989 (million)
% 9.45/2.78 % (18537)------------------------------
% 9.45/2.78 % (18537)------------------------------
% 9.45/2.79 % (18665)dis+1010_1:4_au=on:av=off:cbe=off:nwc=10.0:s2a=on:s2at=3.0:sd=2:ss=axioms:st=5.0:i=1913:si=on:rtra=on_0 on DTF2THF_18170 for (2989ds/1913Mi)
% 9.99/2.86 % (18647)Instruction limit reached!
% 9.99/2.86 % (18647)------------------------------
% 9.99/2.86 % (18647)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.99/2.86 % (18647)Termination reason: Unknown
% 9.99/2.86 % (18647)Termination phase: Saturation
% 9.99/2.86
% 9.99/2.86 % (18647)Memory used [KB]: 12025
% 9.99/2.86 % (18647)Time elapsed: 0.146 s
% 9.99/2.86 % (18647)Instructions burned: 410 (million)
% 9.99/2.86 % (18647)------------------------------
% 9.99/2.86 % (18647)------------------------------
% 9.99/2.87 % (18692)lrs+2_1:1_bd=off:cbe=off:piset=equals:sd=1:sgt=20:sos=all:ss=axioms:i=4350:si=on:rtra=on_0 on DTF2THF_18170 for (2988ds/4350Mi)
% 9.99/2.88 % (18604)First to succeed.
% 9.99/2.89 % (18604)Refutation found. Thanks to Tanya!
% 9.99/2.89 % SZS status Theorem for DTF2THF_18170
% 9.99/2.89 % SZS output start Proof for DTF2THF_18170
% 9.99/2.89 thf(func_def_0, type, in: $i > $i > $o).
% 9.99/2.89 thf(func_def_1, type, exu: ($i > $o) > $o).
% 9.99/2.89 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 9.99/2.89 thf(func_def_8, type, powerset: $i > $i).
% 9.99/2.89 thf(func_def_10, type, setunion: $i > $i).
% 9.99/2.89 thf(func_def_19, type, descr: ($i > $o) > $i).
% 9.99/2.89 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 9.99/2.89 thf(func_def_26, type, prop2set: $o > $i).
% 9.99/2.89 thf(func_def_36, type, nonempty: $i > $o).
% 9.99/2.89 thf(func_def_69, type, set2prop: $i > $o).
% 9.99/2.89 thf(func_def_88, type, subset: $i > $i > $o).
% 9.99/2.89 thf(func_def_89, type, disjoint: $i > $i > $o).
% 9.99/2.89 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 9.99/2.89 thf(func_def_114, type, binunion: $i > $i > $i).
% 9.99/2.89 thf(func_def_122, type, binintersect: $i > $i > $i).
% 9.99/2.89 thf(func_def_135, type, regular: $i > $o).
% 9.99/2.89 thf(func_def_136, type, setminus: $i > $i > $i).
% 9.99/2.89 thf(func_def_147, type, symdiff: $i > $i > $i).
% 9.99/2.89 thf(func_def_153, type, iskpair: $i > $o).
% 9.99/2.89 thf(func_def_158, type, kpair: $i > $i > $i).
% 9.99/2.89 thf(func_def_160, type, cartprod: $i > $i > $i).
% 9.99/2.89 thf(func_def_177, type, singleton: $i > $o).
% 9.99/2.89 thf(func_def_179, type, ex1: $i > ($i > $o) > $o).
% 9.99/2.89 thf(func_def_184, type, atmost1p: $i > $o).
% 9.99/2.89 thf(func_def_185, type, atleast2p: $i > $o).
% 9.99/2.89 thf(func_def_186, type, atmost2p: $i > $o).
% 9.99/2.89 thf(func_def_187, type, upairsetp: $i > $o).
% 9.99/2.89 thf(func_def_191, type, kfst: $i > $i).
% 9.99/2.89 thf(func_def_203, type, ksnd: $i > $i).
% 9.99/2.89 thf(func_def_213, type, breln: $i > $i > $i > $o).
% 9.99/2.89 thf(func_def_214, type, dpsetconstr: $i > $i > ($i > $i > $o) > $i).
% 9.99/2.89 thf(func_def_222, type, func: $i > $i > $i > $o).
% 9.99/2.89 thf(func_def_223, type, funcSet: $i > $i > $i).
% 9.99/2.89 thf(func_def_226, type, ap: $i > $i > $i > $i > $i).
% 9.99/2.89 thf(func_def_232, type, lam: $i > $i > ($i > $i) > $i).
% 9.99/2.89 thf(func_def_259, type, if: $i > $o > $i > $i > $i).
% 9.99/2.89 thf(func_def_323, type, breln1: $i > $i > $o).
% 9.99/2.89 thf(func_def_325, type, breln1Set: $i > $i).
% 9.99/2.89 thf(func_def_327, type, transitive: $i > $i > $o).
% 9.99/2.89 thf(func_def_328, type, antisymmetric: $i > $i > $o).
% 9.99/2.89 thf(func_def_329, type, reflexive: $i > $i > $o).
% 9.99/2.89 thf(func_def_330, type, refltransitive: $i > $i > $o).
% 9.99/2.89 thf(func_def_331, type, refllinearorder: $i > $i > $o).
% 9.99/2.89 thf(func_def_332, type, reflwellordering: $i > $i > $o).
% 9.99/2.89 thf(func_def_338, type, breln1invset: $i > $i > $i).
% 9.99/2.89 thf(func_def_342, type, breln1compset: $i > $i > $i > $i).
% 9.99/2.89 thf(func_def_360, type, image1: $i > ($i > $i) > $i).
% 9.99/2.89 thf(func_def_364, type, injective: $i > $i > $i > $o).
% 9.99/2.89 thf(func_def_365, type, injFuncSet: $i > $i > $i).
% 9.99/2.89 thf(func_def_382, type, sP0: $i > $i > $i > $o).
% 9.99/2.89 thf(func_def_383, type, sP1: $i > $i > $o).
% 9.99/2.89 thf(func_def_384, type, sP2: $i > $i > ($i > $o) > $i > $o).
% 9.99/2.89 thf(func_def_385, type, sP3: $o > $i > $i > $i > $o).
% 9.99/2.89 thf(func_def_386, type, sP4: $i > $i > $i > $i > $o).
% 9.99/2.89 thf(func_def_388, type, sP6: $i > $i > $i > $o).
% 9.99/2.89 thf(func_def_392, type, sP10: ($i > $i > $o) > $i > $i > $o).
% 9.99/2.89 thf(func_def_394, type, sP12: $i > $i > $i > $o).
% 9.99/2.89 thf(func_def_395, type, sP13: $o > $i > $i > $i > $o).
% 9.99/2.89 thf(func_def_396, type, sP14: $i > $i > $i > $o).
% 9.99/2.89 thf(func_def_397, type, sP15: $i > ($i > $o) > $o).
% 9.99/2.89 thf(func_def_398, type, sP16: ($i > $o) > $i > $o).
% 9.99/2.89 thf(func_def_400, type, sP18: $o > $i > $i > $o).
% 9.99/2.89 thf(func_def_401, type, sP19: $i > $i > $i > $o).
% 9.99/2.89 thf(func_def_402, type, sP20: $i > $i > $i > $o).
% 9.99/2.89 thf(func_def_403, type, sP21: $i > $i > $o).
% 9.99/2.89 thf(func_def_404, type, sP22: $i > $o).
% 9.99/2.89 thf(func_def_405, type, sP23: $i > $i > $o).
% 9.99/2.89 thf(func_def_406, type, sP24: $i > $i > $o).
% 9.99/2.89 thf(func_def_410, type, sP28: $i > $i > $i > $i > $o).
% 9.99/2.89 thf(func_def_412, type, sP30: $o > $i > $i > $o).
% 9.99/2.89 thf(func_def_416, type, sP34: $i > $i > $i > $i > $o).
% 9.99/2.89 thf(func_def_417, type, sP35: $i > $i > $i > $i > $o).
% 9.99/2.89 thf(func_def_418, type, sP36: $i > $i > $i > $i > $o).
% 9.99/2.89 thf(func_def_419, type, sP37: $i > $i > $i > $o).
% 9.99/2.89 thf(func_def_421, type, sP39: $i > $i > ($i > $o) > $i > $o).
% 9.99/2.89 thf(func_def_426, type, sP44: $i > ($i > $o) > $o).
% 9.99/2.89 thf(func_def_427, type, sP45: $i > ($i > $o) > $o).
% 9.99/2.89 thf(func_def_428, type, sP46: $i > $i > $o).
% 9.99/2.89 thf(func_def_433, type, sP51: $o > $i > $i > $o).
% 9.99/2.89 thf(func_def_434, type, sP52: $i > $i > $i > $o).
% 9.99/2.89 thf(func_def_435, type, sP53: $i > $i > $o).
% 9.99/2.89 thf(func_def_440, type, sP58: $i > ($i > $i > $o) > $o).
% 9.99/2.89 thf(func_def_445, type, sP63: $i > $i > $i > $o).
% 9.99/2.89 thf(func_def_446, type, sP64: $i > $i > $i > $o).
% 9.99/2.89 thf(func_def_447, type, sP65: $i > $i > $o).
% 9.99/2.89 thf(func_def_448, type, sP66: ($i > $o) > $i > $i > $o).
% 9.99/2.89 thf(func_def_452, type, sP70: $i > $i > $i > $o).
% 9.99/2.89 thf(func_def_456, type, sP74: $i > $i > $i > $i > $o).
% 9.99/2.89 thf(func_def_457, type, sP75: $o > $i > $i > $i > $i > $i > $o).
% 9.99/2.89 thf(func_def_459, type, sP77: $i > ($i > $o) > ($i > $o) > $i > $o).
% 9.99/2.89 thf(func_def_462, type, sP80: $i > $i > $i > $i > $o).
% 9.99/2.89 thf(func_def_463, type, sP81: ($i > $o) > $i > $i > $o).
% 9.99/2.89 thf(func_def_464, type, sP82: $i > $o).
% 9.99/2.89 thf(func_def_467, type, sK85: $i > $i > $i).
% 9.99/2.89 thf(func_def_468, type, sK86: $i > $i > $i > $i).
% 9.99/2.89 thf(func_def_469, type, sK87: $i > $i > $i > $i).
% 9.99/2.89 thf(func_def_476, type, sK94: $i > $i > $o).
% 9.99/2.89 thf(func_def_502, type, sK120: $i > $i > ($i > $o) > $i > $i).
% 9.99/2.89 thf(func_def_503, type, sK121: $i > $i > ($i > $o) > $i > $i).
% 9.99/2.89 thf(func_def_507, type, sK125: $i > $o).
% 9.99/2.89 thf(func_def_509, type, sK127: $i > $i > $i > $i > $o).
% 9.99/2.89 thf(func_def_522, type, sK140: $i > $o).
% 9.99/2.89 thf(func_def_525, type, sK143: $i > $i > ($i > $o) > $i).
% 9.99/2.89 thf(func_def_530, type, sK148: $i > $i > $i > $i).
% 9.99/2.89 thf(func_def_531, type, sK149: $i > $i > $i > $i > $i).
% 9.99/2.89 thf(func_def_549, type, sK167: $i > $i > $i).
% 9.99/2.89 thf(func_def_555, type, sK173: $i > $o).
% 9.99/2.89 thf(func_def_557, type, sK175: ($i > $o) > $i > $i).
% 9.99/2.89 thf(func_def_572, type, sK190: $i > $o).
% 9.99/2.89 thf(func_def_574, type, sK192: ($i > $o) > $i).
% 9.99/2.89 thf(func_def_575, type, sK193: ($i > $o) > $i).
% 9.99/2.89 thf(func_def_618, type, sK236: $i > $i > $o).
% 9.99/2.89 thf(func_def_638, type, sK256: $i > $i > $o).
% 9.99/2.89 thf(func_def_641, type, sK259: $i > $i).
% 9.99/2.89 thf(func_def_642, type, sK260: $i > $i > ($i > $i > $o) > $i).
% 9.99/2.89 thf(func_def_643, type, sK261: ($i > $i > $o) > $i > $i > $i).
% 9.99/2.89 thf(func_def_644, type, sK262: ($i > $i > $o) > $i > $i > $i).
% 9.99/2.89 thf(func_def_665, type, sK283: $i > $i > $i > $i).
% 9.99/2.89 thf(func_def_666, type, sK284: $i > $i > $i > $i > $i).
% 9.99/2.89 thf(func_def_681, type, sK299: $i > $i > $i > $o).
% 9.99/2.89 thf(func_def_689, type, sK307: ($i > $o) > $i > $i).
% 9.99/2.89 thf(func_def_690, type, sK308: ($i > $o) > $i > $i).
% 9.99/2.89 thf(func_def_691, type, sK309: $i > ($i > $o) > $i).
% 9.99/2.89 thf(func_def_693, type, sK311: $i > $o).
% 9.99/2.89 thf(func_def_697, type, sK315: $i > $i > $i).
% 9.99/2.89 thf(func_def_699, type, sK317: $i > $i).
% 9.99/2.89 thf(func_def_706, type, sK324: $i > $i > $o).
% 9.99/2.89 thf(func_def_708, type, sK326: $i > $i > $i > $i).
% 9.99/2.89 thf(func_def_709, type, sK327: $i > $i > $i > $o).
% 9.99/2.89 thf(func_def_718, type, sK336: ($i > $o) > $i).
% 9.99/2.89 thf(func_def_719, type, sK337: $i > $o).
% 9.99/2.89 thf(func_def_720, type, sK338: $i > $o).
% 9.99/2.89 thf(func_def_735, type, sK353: $i > $o).
% 9.99/2.89 thf(func_def_748, type, sK366: $i > $i).
% 9.99/2.89 thf(func_def_751, type, sK369: $i > $o).
% 9.99/2.89 thf(func_def_772, type, sK390: $i > $i).
% 9.99/2.89 thf(func_def_785, type, sK403: $i > $i).
% 9.99/2.89 thf(func_def_786, type, sK404: $i > $i).
% 9.99/2.89 thf(func_def_787, type, sK405: $i > $i).
% 9.99/2.89 thf(func_def_788, type, sK406: $i > $i > $i).
% 9.99/2.89 thf(func_def_789, type, sK407: $i > $i > $i).
% 9.99/2.89 thf(func_def_790, type, sK408: $i > $i > $i).
% 9.99/2.89 thf(func_def_791, type, sK409: $i > $i > $i).
% 9.99/2.89 thf(func_def_792, type, sK410: $i > $i > $i).
% 9.99/2.89 thf(func_def_793, type, sK411: $i > $i > $i > $i).
% 9.99/2.89 thf(func_def_794, type, sK412: $i > $i).
% 9.99/2.89 thf(func_def_795, type, sK413: $i > $i).
% 9.99/2.89 thf(func_def_796, type, sK414: $i > $i).
% 9.99/2.89 thf(func_def_797, type, sK415: $i > $i).
% 9.99/2.89 thf(func_def_798, type, sK416: $i > $i > $i).
% 9.99/2.89 thf(func_def_799, type, sK417: $i > $i > $i).
% 9.99/2.89 thf(func_def_800, type, sK418: $i > $i > $i).
% 9.99/2.89 thf(func_def_801, type, sK419: $i > $i > $i > $i).
% 9.99/2.89 thf(func_def_802, type, sK420: $i > $i > $i > $i).
% 9.99/2.89 thf(func_def_811, type, sK429: $i > $o).
% 9.99/2.89 thf(func_def_824, type, sK442: $i > $i).
% 9.99/2.89 thf(func_def_827, type, sK445: $i > ($i > $i) > $i > $i).
% 9.99/2.89 thf(func_def_830, type, sK448: $i > $i > $i).
% 9.99/2.89 thf(func_def_835, type, sK453: $i > $i > $i > $i > $i).
% 9.99/2.89 thf(func_def_836, type, sK454: $i > $i > $i > $i > $i > $i).
% 9.99/2.89 thf(func_def_852, type, sK470: $i > $i > $o).
% 9.99/2.89 thf(func_def_862, type, sK480: $i > $o).
% 9.99/2.89 thf(func_def_869, type, sK487: $i > $i).
% 9.99/2.89 thf(func_def_871, type, sK489: $i > ($i > $i) > $i > $i).
% 9.99/2.89 thf(func_def_881, type, sK499: $i > $i).
% 9.99/2.89 thf(func_def_884, type, sK502: $i > ($i > $i) > $i > $i).
% 9.99/2.89 thf(func_def_903, type, sK521: $i > $i > $i > $i > $i).
% 9.99/2.89 thf(func_def_904, type, sK522: $i > $i > $i > $i > $i).
% 9.99/2.89 thf(func_def_905, type, sK523: $i > $i > $i > $i > $i).
% 9.99/2.89 thf(func_def_906, type, sK524: $i > $i > $i > $i > $i).
% 9.99/2.89 thf(func_def_928, type, sK546: $i > $i > $i > $i).
% 9.99/2.89 thf(func_def_929, type, sK547: $i > $i > $i > $i > $i).
% 9.99/2.89 thf(func_def_930, type, sK548: $i > $i > $i > $i > $i).
% 9.99/2.89 thf(func_def_931, type, sK549: $i > $i > $i > $i > $i).
% 9.99/2.89 thf(func_def_938, type, sK556: $i > $i > $i > $i).
% 9.99/2.89 thf(func_def_939, type, sK557: $i > $i > $i > $i).
% 9.99/2.89 thf(func_def_951, type, sK569: $i > $o).
% 9.99/2.89 thf(func_def_958, type, sK576: $i > $i > $o).
% 9.99/2.89 thf(func_def_968, type, sK586: $i > $i > ($i > $o) > $i > $i).
% 9.99/2.89 thf(func_def_969, type, sK587: $i > $i > ($i > $o) > $i > $i).
% 9.99/2.89 thf(func_def_973, type, sK591: $i > $o).
% 9.99/2.89 thf(func_def_976, type, sK594: $i > $i).
% 9.99/2.89 thf(func_def_978, type, sK596: $i > $i).
% 9.99/2.89 thf(func_def_981, type, sK599: $i > $o).
% 9.99/2.89 thf(func_def_994, type, sK612: $i > $i > $i).
% 9.99/2.89 thf(func_def_1002, type, sK620: $i > $o).
% 9.99/2.89 thf(func_def_1004, type, sK622: $i > ($i > $o) > $i).
% 9.99/2.89 thf(func_def_1017, type, sK635: $i > $i).
% 9.99/2.89 thf(func_def_1020, type, sK638: $i > ($i > $i) > $i > $i).
% 9.99/2.89 thf(func_def_1052, type, sK670: $i > $o).
% 9.99/2.89 thf(func_def_1067, type, sK685: $i > ($i > $o) > $i).
% 9.99/2.89 thf(func_def_1068, type, sK686: $i > ($i > $o) > $i).
% 9.99/2.89 thf(func_def_1069, type, sK687: $i > ($i > $o) > $i).
% 9.99/2.89 thf(func_def_1071, type, sK689: $i > $o).
% 9.99/2.89 thf(func_def_1072, type, sK690: $i > $i).
% 9.99/2.89 thf(func_def_1098, type, sK716: $i > $i > $i).
% 9.99/2.89 thf(func_def_1099, type, sK717: $i > $i > $i > $i).
% 9.99/2.89 thf(func_def_1115, type, sK733: $o > $i > $i > $i).
% 9.99/2.89 thf(func_def_1118, type, sK736: $i > $i > $i).
% 9.99/2.89 thf(func_def_1141, type, sK759: $i > $o).
% 9.99/2.89 thf(func_def_1142, type, sK760: ($i > $o) > $i > $i).
% 9.99/2.89 thf(func_def_1143, type, sK761: $i > $i).
% 9.99/2.89 thf(func_def_1146, type, sK764: $i > $i > ($i > $i) > $i).
% 9.99/2.89 thf(func_def_1147, type, sK765: $i > $i > $i > $o).
% 9.99/2.89 thf(func_def_1163, type, sK781: $i > $i > $o).
% 9.99/2.89 thf(func_def_1172, type, sK790: $i > $i > $i).
% 9.99/2.89 thf(func_def_1173, type, sK791: $i > $i > $i).
% 9.99/2.89 thf(func_def_1174, type, sK792: $i > $o).
% 9.99/2.89 thf(func_def_1177, type, sK795: $i > ($i > $o) > $i).
% 9.99/2.89 thf(func_def_1179, type, sK797: $i > $i).
% 9.99/2.89 thf(func_def_1180, type, sK798: $i > $i).
% 9.99/2.89 thf(func_def_1181, type, sK799: $i > $i).
% 9.99/2.89 thf(func_def_1182, type, sK800: ($i > $i) > $i > $i).
% 9.99/2.89 thf(func_def_1183, type, sK801: $i > ($i > $i) > $i > $i).
% 9.99/2.89 thf(func_def_1196, type, sK814: $i > $o).
% 9.99/2.89 thf(func_def_1198, type, sK816: ($i > $o) > $i > $i).
% 9.99/2.89 thf(func_def_1212, type, sK830: $i > $i > $o).
% 9.99/2.89 thf(func_def_1213, type, sK831: ($i > $i > $o) > $i > $i).
% 9.99/2.89 thf(func_def_1214, type, sK832: $i > $i > ($i > $i > $o) > $i).
% 9.99/2.89 thf(func_def_1215, type, sK833: $i > $i > ($i > $i > $o) > $i).
% 9.99/2.89 thf(func_def_1216, type, sK834: $i > ($i > $i > $o) > $i).
% 9.99/2.89 thf(func_def_1217, type, sK835: $i > $i > ($i > $i > $o) > $i).
% 9.99/2.89 thf(func_def_1230, type, sK848: $i > $i > $i).
% 9.99/2.89 thf(func_def_1233, type, sK851: $i > $i > $i).
% 9.99/2.89 thf(func_def_1250, type, sK868: $i > $o).
% 9.99/2.89 thf(func_def_1251, type, sK869: $i > $i).
% 9.99/2.89 thf(func_def_1252, type, sK870: ($i > $o) > $i).
% 9.99/2.89 thf(func_def_1254, type, sK872: $i > $o).
% 9.99/2.89 thf(func_def_1267, type, sK885: $i > $i > $i > $i).
% 9.99/2.89 thf(func_def_1268, type, sK886: $i > $i > $i > $i).
% 9.99/2.89 thf(func_def_1269, type, sK887: $i > $i > $i > $i).
% 9.99/2.89 thf(func_def_1270, type, sK888: $i > $i > $i > $i).
% 9.99/2.89 thf(func_def_1278, type, sK896: $i > $i).
% 9.99/2.89 thf(func_def_1279, type, sK897: $i > $o).
% 9.99/2.89 thf(func_def_1282, type, sK900: $i > $i > $i).
% 9.99/2.89 thf(func_def_1283, type, sK901: $i > $i > $i > $i).
% 9.99/2.89 thf(func_def_1286, type, sK904: $i > $o).
% 9.99/2.89 thf(func_def_1310, type, sK928: ($i > $o) > $i > $i > $i).
% 9.99/2.89 thf(func_def_1311, type, sK929: ($i > $o) > $i > $i > $i).
% 9.99/2.89 thf(func_def_1314, type, sK932: $i > $o).
% 9.99/2.89 thf(func_def_1320, type, sK938: $i > $o).
% 9.99/2.89 thf(func_def_1323, type, sK941: $i > ($i > $o) > $i).
% 9.99/2.89 thf(func_def_1324, type, sK942: $i > $o).
% 9.99/2.89 thf(func_def_1325, type, sK943: $i > $o).
% 9.99/2.89 thf(func_def_1326, type, sK944: ($i > $o) > ($i > $o) > $i).
% 9.99/2.89 thf(func_def_1327, type, sK945: ($i > $o) > ($i > $o) > $i).
% 9.99/2.89 thf(func_def_1328, type, sK946: $i > $i).
% 9.99/2.89 thf(func_def_1331, type, sK949: $i > $i > ($i > $i) > $i).
% 9.99/2.89 thf(func_def_1350, type, sK968: $i > $o).
% 9.99/2.89 thf(func_def_1355, type, sK973: $i > $i > $i > $i).
% 9.99/2.89 thf(func_def_1356, type, sK974: $i > $i > $i > $i).
% 9.99/2.89 thf(func_def_1361, type, sK979: $i > $i).
% 9.99/2.89 thf(func_def_1370, type, sK988: $i > $i > $i).
% 9.99/2.89 thf(func_def_1371, type, sK989: $i > $o).
% 9.99/2.89 thf(func_def_1372, type, sK990: $i > $i).
% 9.99/2.89 thf(func_def_1373, type, sK991: ($i > $o) > $i).
% 9.99/2.89 thf(func_def_1380, type, sK998: $i > $i).
% 9.99/2.89 thf(func_def_1383, type, sK1001: $i > $i > ($i > $i) > $i).
% 9.99/2.89 thf(func_def_1398, type, sK1016: $i > $o).
% 9.99/2.89 thf(func_def_1399, type, sK1017: $i > $o).
% 9.99/2.89 thf(func_def_1400, type, sK1018: ($i > $o) > ($i > $o) > $i).
% 9.99/2.89 thf(func_def_1401, type, sK1019: ($i > $o) > ($i > $o) > $i).
% 9.99/2.89 thf(func_def_1405, type, sK1023: $i > $i > $i > $i > $i).
% 9.99/2.89 thf(func_def_1410, type, sK1028: $o > $i > $i > $i > $i > $i > $i).
% 9.99/2.89 thf(func_def_1418, type, sK1036: $i > $o).
% 9.99/2.89 thf(func_def_1420, type, sK1038: ($i > $o) > $i > $i).
% 9.99/2.89 thf(func_def_1427, type, sK1045: $i > ($i > $o) > ($i > $o) > $i > $i).
% 9.99/2.89 thf(func_def_1428, type, sK1046: $i > ($i > $o) > ($i > $o) > $i > $i).
% 9.99/2.89 thf(func_def_1431, type, sK1049: $i > $o).
% 9.99/2.89 thf(func_def_1432, type, sK1050: $i > $o).
% 9.99/2.89 thf(func_def_1447, type, sK1065: $i > $o).
% 9.99/2.89 thf(func_def_1450, type, sK1068: $i > ($i > $o) > $i).
% 9.99/2.89 thf(func_def_1460, type, sK1078: $i > $o).
% 9.99/2.89 thf(func_def_1462, type, sK1080: $i > ($i > $o) > $i).
% 9.99/2.89 thf(func_def_1463, type, sK1081: $i > $i > $i > $i > $i).
% 9.99/2.89 thf(func_def_1464, type, sK1082: $i > $i > $i > $i > $i).
% 9.99/2.89 thf(func_def_1476, type, sK1094: ($i > $o) > $i > $i > $i).
% 9.99/2.89 thf(func_def_1477, type, sK1095: ($i > $o) > $i > $i > $i).
% 9.99/2.89 thf(func_def_1480, type, sK1098: $i > $o).
% 9.99/2.89 thf(func_def_1482, type, sK1100: $i > $o).
% 9.99/2.89 thf(func_def_1485, type, sK1103: $i > $i > $o).
% 9.99/2.89 thf(func_def_1489, type, sK1107: $i > $i).
% 9.99/2.89 thf(func_def_1490, type, sK1108: $i > $i).
% 9.99/2.89 thf(func_def_1493, type, sK1111: $i > $i).
% 9.99/2.89 thf(func_def_1496, type, sK1114: $i > $i > ($i > $i) > $i).
% 9.99/2.89 thf(func_def_1514, type, sK1132: $i > $i > $o).
% 9.99/2.89 thf(func_def_1520, type, sK1138: $i > $i).
% 9.99/2.89 thf(func_def_1522, type, sK1140: ($i > $o) > $i).
% 9.99/2.89 thf(func_def_1523, type, sK1141: ($i > $o) > $i > $i).
% 9.99/2.89 thf(func_def_1524, type, sK1142: ($i > $o) > $i > $i).
% 9.99/2.89 thf(func_def_1525, type, sK1143: ($i > $o) > $i).
% 9.99/2.89 thf(func_def_1526, type, sK1144: ($i > $o) > $i > $i).
% 9.99/2.89 thf(func_def_1527, type, sK1145: ($i > $o) > $i > $i).
% 9.99/2.89 thf(func_def_1528, type, sK1146: ($i > $o) > $i > $i).
% 9.99/2.89 thf(func_def_1529, type, sK1147: $i > ($i > $o) > $i).
% 9.99/2.89 thf(func_def_1530, type, sK1148: ($i > $o) > $i).
% 9.99/2.89 thf(func_def_1531, type, sK1149: $i > $i).
% 9.99/2.89 thf(func_def_1532, type, sK1150: ($i > $o) > $i > $i).
% 9.99/2.89 thf(func_def_1533, type, sK1151: ($i > $o) > $i > $i).
% 9.99/2.89 thf(func_def_1534, type, sK1152: ($i > $o) > $i).
% 9.99/2.89 thf(func_def_1535, type, sK1153: $i > $i > ($i > $i > $o) > $i).
% 9.99/2.89 thf(func_def_1537, type, sK1155: ($i > $o) > $i > $i).
% 9.99/2.89 thf(func_def_1538, type, sK1156: $i > ($i > $o) > $i).
% 9.99/2.89 thf(func_def_1539, type, sK1157: $i > $i).
% 9.99/2.89 thf(func_def_1540, type, sK1158: $i > $i).
% 9.99/2.89 thf(func_def_1543, type, sK1161: ($i > $o) > $i).
% 9.99/2.89 thf(func_def_1544, type, sK1162: ($i > $o) > $i > $i).
% 9.99/2.89 thf(func_def_1545, type, sK1163: $i > ($i > $o) > $i).
% 9.99/2.89 thf(func_def_1546, type, sK1164: ($i > $o) > $i).
% 9.99/2.89 thf(func_def_1547, type, sK1165: ($i > $o) > $i > $i).
% 9.99/2.89 thf(func_def_1548, type, sK1166: $i > ($i > $o) > $i).
% 9.99/2.89 thf(func_def_1549, type, sK1167: ($i > $o) > $i).
% 9.99/2.89 thf(func_def_1550, type, sK1168: ($i > $o) > $i > $i).
% 9.99/2.89 thf(func_def_1551, type, sK1169: ($i > $o) > $i).
% 9.99/2.89 thf(func_def_1552, type, sK1170: ($i > $o) > $i > $i).
% 9.99/2.89 thf(func_def_1553, type, sK1171: ($i > $o) > $i > $i).
% 9.99/2.89 thf(func_def_1554, type, sK1172: $i > ($i > $o) > $i).
% 9.99/2.89 thf(func_def_1555, type, sK1173: $i > ($i > $o) > $i).
% 9.99/2.89 thf(func_def_1556, type, sK1174: ($i > $i) > $i > $i).
% 9.99/2.89 thf(func_def_1557, type, sK1175: ($i > $i) > $i > $i > $i).
% 9.99/2.89 thf(func_def_1558, type, sK1176: ($i > $i) > $i > $i > $i).
% 9.99/2.89 thf(func_def_1559, type, sK1177: ($i > $i) > $i > $i > $i).
% 9.99/2.89 thf(func_def_1560, type, sK1178: $i > $i).
% 9.99/2.89 thf(func_def_1561, type, sK1179: $i > ($i > $o) > $i).
% 9.99/2.89 thf(func_def_1562, type, sK1180: ($i > $o) > $i > $i).
% 9.99/2.89 thf(func_def_1563, type, sK1181: ($i > $o) > $i > $i).
% 9.99/2.89 thf(func_def_1564, type, sK1182: $i > ($i > $o) > $i).
% 9.99/2.89 thf(func_def_1565, type, sK1183: $i > ($i > $o) > $i).
% 9.99/2.89 thf(func_def_1567, type, sK1185: ($i > $o) > $i > $i).
% 9.99/2.89 thf(func_def_1568, type, sK1186: $i > $i).
% 9.99/2.89 thf(func_def_1570, type, sK1188: ($i > $o) > $i).
% 9.99/2.89 thf(func_def_1571, type, sK1189: ($i > $o) > $i).
% 9.99/2.89 thf(func_def_1572, type, sK1190: ($i > $o) > $i > $i).
% 9.99/2.89 thf(func_def_1573, type, sK1191: ($i > $o) > $i).
% 9.99/2.89 thf(func_def_1574, type, sK1192: ($i > $o) > $i).
% 9.99/2.89 thf(func_def_1575, type, sK1193: ($i > $o) > $i).
% 9.99/2.89 thf(func_def_1576, type, ph1194: !>[X0: $tType]:(X0)).
% 9.99/2.89 thf(f9824,plain,(
% 9.99/2.89 $false),
% 9.99/2.89 inference(subsumption_resolution,[status(thm)],[f9822,f3736])).
% 9.99/2.89 thf(f3736,plain,(
% 9.99/2.89 ($true != (injective @ sK228 @ sK229 @ sK230))),
% 9.99/2.89 inference(cnf_transformation,[status(thm)],[f1778])).
% 9.99/2.89 thf(f1778,plain,(
% 9.99/2.89 (upairsetIR = $true) & (upairsetE = $true) & (funcextLem = $true) & (cartprodsndin = $true) & (dsetconstr__Cong = $true) & (setadjoin__Cong = $true) & (cartprodfstpairEq = $true) & (cartprodpairsurjEq = $true) & (secondinupair = $true) & (funcGraphProp2 = $true) & (nonemptyE1 = $true) & (ap2apEq1 = $true) & (demorgan1a = $true) & (contraSubsetComplement = $true) & (foundationAx = $true) & (funcinfuncset = $true) & (emptysetE = $true) & (binunionIR = $true) & (exuE3e = $true) & (dpsetconstrER = $true) & (subsetI1 = $true) & (subsetE = $true) & (setminusIRneg = $true) & (exuE2 = $true) & (funcImageSingleton = $true) & (contrasubsetT = $true) & (doubleComplementEq = $true) & (setminusEL = $true) & (subsetTI = $true) & (setadjoinOr = $true) & (ex1E1 = $true) & (sepSubset = $true) & (dpsetconstrERa = $true) & (dsetconstrER = $true) & (cartprodsndpairEq = $true) & (wellorderingAx = $true) & (cartprodmempair1 = $true) & (emptyset__Cong = $true) & (descrp = $true) & (demorgan2a2 = $true) & (powersetsubset = $true) & (subsetTrans = $true) & (setoftrueEq = $true) & (cartprodpairmemEL = $true) & (doubleComplementSub1 = $true) & (kpairsurjEq = $true) & (powersetI = $true) & (powersetTE1 = $true) & (dpsetconstrEL2 = $true) & (breln1all2 = $true) & (binintersectTERcontra = $true) & (singletonsswitch = $true) & (doubleComplementE1 = $true) & (cartprodfstin = $true) & (subsetE2 = $true) & (binunionTE = $true) & (binunionIL = $true) & (emptyI = $true) & (notequalI1 = $true) & (exuI3 = $true) & (powerset__Cong = $true) & (ubforcartprodlem3 = $true) & (in__Cong = $true) & (kpairp = $true) & (complementTE1 = $true) & (woz1_5 = $true) & (apProp = $true) & (cartprodpairmemER = $true) & (bs114d = $true) & (setukpairinjR1 = $true) & (quantDeMorgan2 = $true) & (setunionI = $true) & (binintersectRsub = $true) & (binintersectTELcontra = $true) & (dsetconstrI = $true) & (image1Equiv = $true) & (contrasubsetT2 = $true) & (ex1E2 = $true) & (inComplementUnionImpNotIn1 = $true) & (upairequniteq = $true) & (breln1unionprop = $true) & (ex1I = $true) & (emptyInPowerset = $true) & (breln1unionIL = $true) & (inComplementUnionImpInComplement1 = $true) & (symdiffIneg1 = $true) & (funcGraphProp3 = $true) & (woz1_4 = $true) & (binunionE = $true) & (((in @ sK230 @ (injFuncSet @ sK228 @ sK229)) = $true) & ($true != (injective @ sK228 @ sK229 @ sK230))) & (subsetRefl = $true) & (lam2p = $true) & (setukpairinjR = $true) & (image1Ex1 = $true) & (setukpairIR = $true) & (image1Ex = $true) & (exuEu = $true) & (binunionT_lem = $true) & (powersetE = $true) & (eqbreln = $true) & (kfstsingleton = $true) & (setOfPairsIsBReln1 = $true) & (setminusELneg = $true) & (breln1invI = $true) & (omegaSAx = $true) & (iftrueProp2 = $true) & (cartprodmempaircEq = $true) & (binintersectSubset5 = $true) & (prop2setI = $true) & (woz13rule1 = $true) & (setadjoinIL = $true) & (sepInPowerset = $true) & (contrasubsetT3 = $true) & (setadjoinAx = $true) & (setunionsingleton1 = $true) & (singletoninpowerset = $true) & (complementInPowersetComplementIntersect = $true) & (demorgan2b2 = $true) & (setminusLsub = $true) & (notsubsetI = $true) & (emptysetimpfalse = $true) & (binintersectI = $true) & (subbreln = $true) & (ap2apEq2 = $true) & (descr__Cong = $true) & (inIntersectImpInUnion = $true) & (vacuousDall = $true) & (cartprodmempair = $true) & (breln1unionIR = $true) & (complementTI1 = $true) & (powersetT_lem = $true) & (ifp = $true) & (brelnall2 = $true) & (setminusERneg = $true) & (exuI1 = $true) & (binunionEcases = $true) & (breln1unionE = $true) & (woz2A = $true) & (setminusER = $true) & (complementTnotintersectT = $true) & (complementTcontraSubset = $true) & (setext = $true) & (ubforcartprodlem2 = $true) & (noeltsimpempty = $true) & (beta2 = $true) & (symdiffI1 = $true) & (binintersectSubset3 = $true) & (ex1I2 = $true) & (setukpairinjL2 = $true) & (singletoninpowunion = $true) & (upairset2E = $true) & (brelnall1 = $true) & (theeq = $true) & (breln1all1 = $true) & (binunionTILcontra = $true) & (singletonsubset = $true) & (dpsetconstrSub = $true) & (ap2p = $true) & (setukpairinjL = $true) & (setextT = $true) & (upairinpowunion = $true) & (eqimpsubset1 = $true) & (notdexE = $true) & (complementSubsetComplementIntersect = $true) & (setadjoinE = $true) & (breln1compI = $true) & (setextsub = $true) & (dsetconstrEL = $true) & (setminusI = $true) & (theprop = $true) & (setminusILneg = $true) & (eta2 = $true) & (exuI2 = $true) & (woz13rule0 = $true) & (setminusT_lem = $true) & (iffalseProp1 = $true) & (setunionE = $true) & (notdallE = $true) & (setminusSubset2 = $true) & (notinemptyset = $true) & (emptyinunitempty = $true) & (ksndsingleton = $true) & (woz1_2 = $true) & (setadjoinIR = $true) & (beta1 = $true) & (subbreln1 = $true) & (powersetAx = $true) & (breln1compprop = $true) & (upairsetIL = $true) & (demorgan1 = $true) & (woz1_1 = $true) & (iftrue = $true) & (quantDeMorgan1 = $true) & (setunionsingleton2 = $true) & (breln1compEex = $true) & (nonemptyI1 = $true) & (singletonprop = $true) & (iffalse = $true) & (funcGraphProp1 = $true) & (app = $true) & (symdiffI2 = $true) & (eqbreln1 = $true) & (subPowSU = $true) & (inPowerset = $true) & (binunionRsub = $true) & (demorgan2b = $true) & (emptyinPowerset = $true) & (binintersectSubset2 = $true) & (setukpairIL = $true) & (funcext = $true) & (binintersectSubset1 = $true) & (iffalseProp2 = $true) & (quantDeMorgan3 = $true) & (setminusSubset1 = $true) & (binunionTIRcontra = $true) & (breln1invE = $true) & (binunionTEcontra = $true) & (quantDeMorgan4 = $true) & (binintersectLsub = $true) & (setunion__Cong = $true) & (complementT_lem = $true) & (funcGraphProp4 = $true) & (lamp = $true) & (subset2powerset = $true) & (exuE1 = $true) & (setukpairinjL1 = $true) & (woz1_3 = $true) & (setadjoinSub = $true) & (inIntersectImpInUnion2 = $true) & (breln1unionEcases = $true) & (powersetI1 = $true) & (injFuncSetFuncIn = $true) & (complementUnionInPowersetComplement = $true) & (demorgan2 = $true) & (ubforcartprodlem1 = $true) & (nonemptyI = $true) & (binintersectER = $true) & (ifSingleton = $true) & (setunionsingleton = $true) & (emptysetsubset = $true) & (omegaIndAx = $true) & (woz13rule2 = $true) & (binintersectT_lem = $true) & (setextAx = $true) & (demorgan2a = $true) & (upairset2IR = $true) & (eqinunit = $true) & (emptyE1 = $true) & (breln1compE = $true) & (setadjoinSub2 = $true) & (infuncsetfunc = $true) & (choice2fnsingleton = $true) & (symdiffE = $true) & (lamProp = $true) & (nonemptyImpWitness = $true) & (demorgan2a1 = $true) & (binunionLsub = $true) & (notequalI2 = $true) & (kfstpairEq = $true) & (singletonsuniq = $true) & (contrasubsetT1 = $true) & (disjointsetsI1 = $true) & (upairsubunion = $true) & (replAx = $true) & (inIntersectImpInIntersectUnions = $true) & (setukpairinjR11 = $true) & (dpsetconstrEL1 = $true) & (cartprodpairin = $true) & (ksndpairEq = $true) & (setunionAx = $true) & (image1I = $true) & (woz2B = $true) & (uniqinunit = $true) & (exuE3u = $true) & (injFuncInInjFuncSet = $true) & (setbeta = $true) & (woz13rule3 = $true) & (setOfPairsIsBReln = $true) & (breln1SetBreln1 = $true) & (eqimpsubset2 = $true) & (emptysetAx = $true) & (binintersectSubset4 = $true) & (woz2Ex = $true) & (binintersectEL = $true) & (intersectInPowersetIntersectUnions = $true) & (doubleComplementI1 = $true) & (setukpairinjR2 = $true) & (powersetTI1 = $true) & (powersetE1 = $true) & (iftrueorfalse = $true) & (subsetemptysetimpeq = $true) & (setunionE2 = $true) & (omega0Ax = $true) & (breln1unionI = $true) & (inCongP = $true) & (prop2set2propI = $true) & (complementImpComplementIntersect = $true) & (woz2W = $true) & (lam2lamEq = $true) & (setukpairinjR12 = $true) & (breln1unionCommutes = $true) & (eta1 = $true) & (doubleComplementSub2 = $true) & (demorgan1b = $true) & (dpsetconstrI = $true) & (subsetI2 = $true) & (omega__Cong = $true) & (notinsingleton = $true) & (iftrueProp1 = $true) & (breln1invprop = $true) & (symdiffIneg2 = $true) & (funcext2 = $true) & (exu__Cong = $true) & (kpairiskpair = $true) & (woz13rule4 = $true) & (prop2setE = $true) & (image1E = $true)),
% 9.99/2.89 inference(skolemisation,[status(esa),new_symbols(skolem,[sK228,sK229,sK230])],[f1069,f1777])).
% 9.99/2.89 thf(f1777,plain,(
% 9.99/2.89 ? [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X2 @ (injFuncSet @ X0 @ X1))) & ($true != (injective @ X0 @ X1 @ X2))) => (((in @ sK230 @ (injFuncSet @ sK228 @ sK229)) = $true) & ($true != (injective @ sK228 @ sK229 @ sK230)))),
% 9.99/2.89 introduced(definition,[],[choice_axiom])).
% 9.99/2.89 thf(f1069,plain,(
% 9.99/2.89 (upairsetIR = $true) & (upairsetE = $true) & (funcextLem = $true) & (cartprodsndin = $true) & (dsetconstr__Cong = $true) & (setadjoin__Cong = $true) & (cartprodfstpairEq = $true) & (cartprodpairsurjEq = $true) & (secondinupair = $true) & (funcGraphProp2 = $true) & (nonemptyE1 = $true) & (ap2apEq1 = $true) & (demorgan1a = $true) & (contraSubsetComplement = $true) & (foundationAx = $true) & (funcinfuncset = $true) & (emptysetE = $true) & (binunionIR = $true) & (exuE3e = $true) & (dpsetconstrER = $true) & (subsetI1 = $true) & (subsetE = $true) & (setminusIRneg = $true) & (exuE2 = $true) & (funcImageSingleton = $true) & (contrasubsetT = $true) & (doubleComplementEq = $true) & (setminusEL = $true) & (subsetTI = $true) & (setadjoinOr = $true) & (ex1E1 = $true) & (sepSubset = $true) & (dpsetconstrERa = $true) & (dsetconstrER = $true) & (cartprodsndpairEq = $true) & (wellorderingAx = $true) & (cartprodmempair1 = $true) & (emptyset__Cong = $true) & (descrp = $true) & (demorgan2a2 = $true) & (powersetsubset = $true) & (subsetTrans = $true) & (setoftrueEq = $true) & (cartprodpairmemEL = $true) & (doubleComplementSub1 = $true) & (kpairsurjEq = $true) & (powersetI = $true) & (powersetTE1 = $true) & (dpsetconstrEL2 = $true) & (breln1all2 = $true) & (binintersectTERcontra = $true) & (singletonsswitch = $true) & (doubleComplementE1 = $true) & (cartprodfstin = $true) & (subsetE2 = $true) & (binunionTE = $true) & (binunionIL = $true) & (emptyI = $true) & (notequalI1 = $true) & (exuI3 = $true) & (powerset__Cong = $true) & (ubforcartprodlem3 = $true) & (in__Cong = $true) & (kpairp = $true) & (complementTE1 = $true) & (woz1_5 = $true) & (apProp = $true) & (cartprodpairmemER = $true) & (bs114d = $true) & (setukpairinjR1 = $true) & (quantDeMorgan2 = $true) & (setunionI = $true) & (binintersectRsub = $true) & (binintersectTELcontra = $true) & (dsetconstrI = $true) & (image1Equiv = $true) & (contrasubsetT2 = $true) & (ex1E2 = $true) & (inComplementUnionImpNotIn1 = $true) & (upairequniteq = $true) & (breln1unionprop = $true) & (ex1I = $true) & (emptyInPowerset = $true) & (breln1unionIL = $true) & (inComplementUnionImpInComplement1 = $true) & (symdiffIneg1 = $true) & (funcGraphProp3 = $true) & (woz1_4 = $true) & (binunionE = $true) & ? [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X2 @ (injFuncSet @ X0 @ X1))) & ($true != (injective @ X0 @ X1 @ X2))) & (subsetRefl = $true) & (lam2p = $true) & (setukpairinjR = $true) & (image1Ex1 = $true) & (setukpairIR = $true) & (image1Ex = $true) & (exuEu = $true) & (binunionT_lem = $true) & (powersetE = $true) & (eqbreln = $true) & (kfstsingleton = $true) & (setOfPairsIsBReln1 = $true) & (setminusELneg = $true) & (breln1invI = $true) & (omegaSAx = $true) & (iftrueProp2 = $true) & (cartprodmempaircEq = $true) & (binintersectSubset5 = $true) & (prop2setI = $true) & (woz13rule1 = $true) & (setadjoinIL = $true) & (sepInPowerset = $true) & (contrasubsetT3 = $true) & (setadjoinAx = $true) & (setunionsingleton1 = $true) & (singletoninpowerset = $true) & (complementInPowersetComplementIntersect = $true) & (demorgan2b2 = $true) & (setminusLsub = $true) & (notsubsetI = $true) & (emptysetimpfalse = $true) & (binintersectI = $true) & (subbreln = $true) & (ap2apEq2 = $true) & (descr__Cong = $true) & (inIntersectImpInUnion = $true) & (vacuousDall = $true) & (cartprodmempair = $true) & (breln1unionIR = $true) & (complementTI1 = $true) & (powersetT_lem = $true) & (ifp = $true) & (brelnall2 = $true) & (setminusERneg = $true) & (exuI1 = $true) & (binunionEcases = $true) & (breln1unionE = $true) & (woz2A = $true) & (setminusER = $true) & (complementTnotintersectT = $true) & (complementTcontraSubset = $true) & (setext = $true) & (ubforcartprodlem2 = $true) & (noeltsimpempty = $true) & (beta2 = $true) & (symdiffI1 = $true) & (binintersectSubset3 = $true) & (ex1I2 = $true) & (setukpairinjL2 = $true) & (singletoninpowunion = $true) & (upairset2E = $true) & (brelnall1 = $true) & (theeq = $true) & (breln1all1 = $true) & (binunionTILcontra = $true) & (singletonsubset = $true) & (dpsetconstrSub = $true) & (ap2p = $true) & (setukpairinjL = $true) & (setextT = $true) & (upairinpowunion = $true) & (eqimpsubset1 = $true) & (notdexE = $true) & (complementSubsetComplementIntersect = $true) & (setadjoinE = $true) & (breln1compI = $true) & (setextsub = $true) & (dsetconstrEL = $true) & (setminusI = $true) & (theprop = $true) & (setminusILneg = $true) & (eta2 = $true) & (exuI2 = $true) & (woz13rule0 = $true) & (setminusT_lem = $true) & (iffalseProp1 = $true) & (setunionE = $true) & (notdallE = $true) & (setminusSubset2 = $true) & (notinemptyset = $true) & (emptyinunitempty = $true) & (ksndsingleton = $true) & (woz1_2 = $true) & (setadjoinIR = $true) & (beta1 = $true) & (subbreln1 = $true) & (powersetAx = $true) & (breln1compprop = $true) & (upairsetIL = $true) & (demorgan1 = $true) & (woz1_1 = $true) & (iftrue = $true) & (quantDeMorgan1 = $true) & (setunionsingleton2 = $true) & (breln1compEex = $true) & (nonemptyI1 = $true) & (singletonprop = $true) & (iffalse = $true) & (funcGraphProp1 = $true) & (app = $true) & (symdiffI2 = $true) & (eqbreln1 = $true) & (subPowSU = $true) & (inPowerset = $true) & (binunionRsub = $true) & (demorgan2b = $true) & (emptyinPowerset = $true) & (binintersectSubset2 = $true) & (setukpairIL = $true) & (funcext = $true) & (binintersectSubset1 = $true) & (iffalseProp2 = $true) & (quantDeMorgan3 = $true) & (setminusSubset1 = $true) & (binunionTIRcontra = $true) & (breln1invE = $true) & (binunionTEcontra = $true) & (quantDeMorgan4 = $true) & (binintersectLsub = $true) & (setunion__Cong = $true) & (complementT_lem = $true) & (funcGraphProp4 = $true) & (lamp = $true) & (subset2powerset = $true) & (exuE1 = $true) & (setukpairinjL1 = $true) & (woz1_3 = $true) & (setadjoinSub = $true) & (inIntersectImpInUnion2 = $true) & (breln1unionEcases = $true) & (powersetI1 = $true) & (injFuncSetFuncIn = $true) & (complementUnionInPowersetComplement = $true) & (demorgan2 = $true) & (ubforcartprodlem1 = $true) & (nonemptyI = $true) & (binintersectER = $true) & (ifSingleton = $true) & (setunionsingleton = $true) & (emptysetsubset = $true) & (omegaIndAx = $true) & (woz13rule2 = $true) & (binintersectT_lem = $true) & (setextAx = $true) & (demorgan2a = $true) & (upairset2IR = $true) & (eqinunit = $true) & (emptyE1 = $true) & (breln1compE = $true) & (setadjoinSub2 = $true) & (infuncsetfunc = $true) & (choice2fnsingleton = $true) & (symdiffE = $true) & (lamProp = $true) & (nonemptyImpWitness = $true) & (demorgan2a1 = $true) & (binunionLsub = $true) & (notequalI2 = $true) & (kfstpairEq = $true) & (singletonsuniq = $true) & (contrasubsetT1 = $true) & (disjointsetsI1 = $true) & (upairsubunion = $true) & (replAx = $true) & (inIntersectImpInIntersectUnions = $true) & (setukpairinjR11 = $true) & (dpsetconstrEL1 = $true) & (cartprodpairin = $true) & (ksndpairEq = $true) & (setunionAx = $true) & (image1I = $true) & (woz2B = $true) & (uniqinunit = $true) & (exuE3u = $true) & (injFuncInInjFuncSet = $true) & (setbeta = $true) & (woz13rule3 = $true) & (setOfPairsIsBReln = $true) & (breln1SetBreln1 = $true) & (eqimpsubset2 = $true) & (emptysetAx = $true) & (binintersectSubset4 = $true) & (woz2Ex = $true) & (binintersectEL = $true) & (intersectInPowersetIntersectUnions = $true) & (doubleComplementI1 = $true) & (setukpairinjR2 = $true) & (powersetTI1 = $true) & (powersetE1 = $true) & (iftrueorfalse = $true) & (subsetemptysetimpeq = $true) & (setunionE2 = $true) & (omega0Ax = $true) & (breln1unionI = $true) & (inCongP = $true) & (prop2set2propI = $true) & (complementImpComplementIntersect = $true) & (woz2W = $true) & (lam2lamEq = $true) & (setukpairinjR12 = $true) & (breln1unionCommutes = $true) & (eta1 = $true) & (doubleComplementSub2 = $true) & (demorgan1b = $true) & (dpsetconstrI = $true) & (subsetI2 = $true) & (omega__Cong = $true) & (notinsingleton = $true) & (iftrueProp1 = $true) & (breln1invprop = $true) & (symdiffIneg2 = $true) & (funcext2 = $true) & (exu__Cong = $true) & (kpairiskpair = $true) & (woz13rule4 = $true) & (prop2setE = $true) & (image1E = $true)),
% 9.99/2.89 inference(flattening,[status(thm)],[f1068])).
% 9.99/2.89 thf(f1068,plain,(
% 9.99/2.89 (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X2 @ (injFuncSet @ X0 @ X1))) & ($true != (injective @ X0 @ X1 @ X2))) & (injFuncSetFuncIn = $true)) & (injFuncInInjFuncSet = $true)) & (image1I = $true)) & (image1E = $true)) & (image1Equiv = $true)) & (image1Ex1 = $true)) & (image1Ex = $true)) & (woz2B = $true)) & (woz2A = $true)) & (woz2W = $true)) & (woz2Ex = $true)) & (breln1unionCommutes = $true)) & (breln1unionEcases = $true)) & (breln1unionE = $true)) & (breln1unionI = $true)) & (breln1unionIR = $true)) & (breln1unionIL = $true)) & (breln1unionprop = $true)) & (breln1compEex = $true)) & (breln1compE = $true)) & (breln1compI = $true)) & (breln1compprop = $true)) & (breln1invE = $true)) & (breln1invI = $true)) & (breln1invprop = $true)) & (eqbreln1 = $true)) & (subbreln1 = $true)) & (breln1all1 = $true)) & (setOfPairsIsBReln1 = $true)) & (choice2fnsingleton = $true)) & (breln1SetBreln1 = $true)) & (breln1all2 = $true)) & (woz1_5 = $true)) & (woz1_4 = $true)) & (woz1_3 = $true)) & (woz1_2 = $true)) & (woz1_1 = $true)) & (woz13rule4 = $true)) & (woz13rule3 = $true)) & (woz13rule2 = $true)) & (woz13rule1 = $true)) & (woz13rule0 = $true)) & (demorgan2 = $true)) & (demorgan2b = $true)) & (demorgan2b2 = $true)) & (demorgan2a = $true)) & (demorgan1 = $true)) & (demorgan1b = $true)) & (demorgan1a = $true)) & (demorgan2a2 = $true)) & (complementUnionInPowersetComplement = $true)) & (demorgan2a1 = $true)) & (binunionTEcontra = $true)) & (binunionTE = $true)) & (inComplementUnionImpInComplement1 = $true)) & (inComplementUnionImpNotIn1 = $true)) & (intersectInPowersetIntersectUnions = $true)) & (inIntersectImpInIntersectUnions = $true)) & (inIntersectImpInUnion2 = $true)) & (inIntersectImpInUnion = $true)) & (binunionTIRcontra = $true)) & (binunionTILcontra = $true)) & (complementTcontraSubset = $true)) & (contraSubsetComplement = $true)) & (complementInPowersetComplementIntersect = $true)) & (complementSubsetComplementIntersect = $true)) & (complementImpComplementIntersect = $true)) & (complementTnotintersectT = $true)) & (doubleComplementEq = $true)) & (doubleComplementSub2 = $true)) & (doubleComplementSub1 = $true)) & (doubleComplementE1 = $true)) & (doubleComplementI1 = $true)) & (contrasubsetT3 = $true)) & (contrasubsetT2 = $true)) & (contrasubsetT1 = $true)) & (contrasubsetT = $true)) & (binintersectTERcontra = $true)) & (binintersectTELcontra = $true)) & (complementTE1 = $true)) & (complementTI1 = $true)) & (powersetTE1 = $true)) & (powersetTI1 = $true)) & (subsetTI = $true)) & (setextT = $true)) & (complementT_lem = $true)) & (setminusT_lem = $true)) & (powersetT_lem = $true)) & (binunionT_lem = $true)) & (binintersectT_lem = $true)) & (iftrueorfalse = $true)) & (iffalse = $true)) & (iftrue = $true)) & (theeq = $true)) & (ifp = $true)) & (ifSingleton = $true)) & (iftrueProp2 = $true)) & (iftrueProp1 = $true)) & (iffalseProp2 = $true)) & (iffalseProp1 = $true)) & (eta2 = $true)) & (beta2 = $true)) & (lam2lamEq = $true)) & (eta1 = $true)) & (beta1 = $true)) & (ap2apEq2 = $true)) & (ap2apEq1 = $true)) & (funcext2 = $true)) & (funcext = $true)) & (eqbreln = $true)) & (subbreln = $true)) & (funcGraphProp4 = $true)) & (funcextLem = $true)) & (funcGraphProp2 = $true)) & (funcGraphProp3 = $true)) & (funcGraphProp1 = $true)) & (ex1E2 = $true)) & (brelnall2 = $true)) & (brelnall1 = $true)) & (lam2p = $true)) & (lamp = $true)) & (lamProp = $true)) & (funcinfuncset = $true)) & (ap2p = $true)) & (infuncsetfunc = $true)) & (app = $true)) & (apProp = $true)) & (funcImageSingleton = $true)) & (dpsetconstrER = $true)) & (dpsetconstrEL2 = $true)) & (dpsetconstrEL1 = $true)) & (dpsetconstrERa = $true)) & (setOfPairsIsBReln = $true)) & (dpsetconstrSub = $true)) & (dpsetconstrI = $true)) & (cartprodpairsurjEq = $true)) & (cartprodsndpairEq = $true)) & (cartprodfstpairEq = $true)) & (cartprodmempaircEq = $true)) & (cartprodpairmemER = $true)) & (cartprodpairmemEL = $true)) & (cartprodsndin = $true)) & (kpairsurjEq = $true)) & (ksndpairEq = $true)) & (ksndsingleton = $true)) & (setukpairinjR = $true)) & (setukpairinjR2 = $true)) & (upairequniteq = $true)) & (setukpairinjR1 = $true)) & (setukpairinjR12 = $true)) & (setukpairinjR11 = $true)) & (setukpairinjL = $true)) & (setukpairinjL2 = $true)) & (cartprodfstin = $true)) & (kfstpairEq = $true)) & (theprop = $true)) & (kfstsingleton = $true)) & (setukpairinjL1 = $true)) & (singletonsuniq = $true)) & (ex1I2 = $true)) & (ex1I = $true)) & (ex1E1 = $true)) & (singletonprop = $true)) & (setunionsingleton = $true)) & (setunionsingleton2 = $true)) & (setunionsingleton1 = $true)) & (setunionE2 = $true)) & (cartprodmempair = $true)) & (cartprodmempair1 = $true)) & (cartprodpairin = $true)) & (ubforcartprodlem3 = $true)) & (ubforcartprodlem2 = $true)) & (ubforcartprodlem1 = $true)) & (upairinpowunion = $true)) & (upairsubunion = $true)) & (upairset2E = $true)) & (singletoninpowunion = $true)) & (singletoninpowerset = $true)) & (singletonsubset = $true)) & (kpairp = $true)) & (kpairiskpair = $true)) & (setukpairIR = $true)) & (setukpairIL = $true)) & (secondinupair = $true)) & (symdiffIneg2 = $true)) & (symdiffIneg1 = $true)) & (symdiffI2 = $true)) & (symdiffI1 = $true)) & (symdiffE = $true)) & (setminusSubset1 = $true)) & (setminusLsub = $true)) & (setminusIRneg = $true)) & (setminusILneg = $true)) & (setminusELneg = $true)) & (setminusERneg = $true)) & (setminusSubset2 = $true)) & (setminusER = $true)) & (setminusEL = $true)) & (setminusI = $true)) & (bs114d = $true)) & (binintersectSubset1 = $true)) & (binintersectSubset4 = $true)) & (binintersectRsub = $true)) & (disjointsetsI1 = $true)) & (binintersectER = $true)) & (binintersectSubset3 = $true)) & (binintersectSubset2 = $true)) & (binintersectLsub = $true)) & (binintersectEL = $true)) & (binintersectSubset5 = $true)) & (binintersectI = $true)) & (binunionRsub = $true)) & (binunionLsub = $true)) & (binunionE = $true)) & (binunionEcases = $true)) & (binunionIR = $true)) & (upairset2IR = $true)) & (binunionIL = $true)) & (sepSubset = $true)) & (sepInPowerset = $true)) & (powersetsubset = $true)) & (inPowerset = $true)) & (powersetE1 = $true)) & (powersetI1 = $true)) & (subsetemptysetimpeq = $true)) & (setextsub = $true)) & (subset2powerset = $true)) & (setadjoinSub2 = $true)) & (setadjoinSub = $true)) & (subsetTrans = $true)) & (subsetRefl = $true)) & (notequalI2 = $true)) & (notequalI1 = $true)) & (notsubsetI = $true)) & (subsetE2 = $true)) & (subsetE = $true)) & (emptysetsubset = $true)) & (subsetI2 = $true)) & (eqimpsubset1 = $true)) & (eqimpsubset2 = $true)) & (subsetI1 = $true)) & (dsetconstr__Cong = $true)) & (descr__Cong = $true)) & (exuEu = $true)) & (omega__Cong = $true)) & (setunion__Cong = $true)) & (powerset__Cong = $true)) & (setadjoin__Cong = $true)) & (emptyset__Cong = $true)) & (exu__Cong = $true)) & (exuE3u = $true)) & (in__Cong = $true)) & (inCongP = $true)) & (exuI2 = $true)) & (exuI3 = $true)) & (exuI1 = $true)) & (notdallE = $true)) & (notdexE = $true)) & (prop2set2propI = $true)) & (prop2setI = $true)) & (quantDeMorgan4 = $true)) & (quantDeMorgan3 = $true)) & (quantDeMorgan2 = $true)) & (quantDeMorgan1 = $true)) & (vacuousDall = $true)) & (emptyE1 = $true)) & (upairsetIR = $true)) & (upairsetIL = $true)) & (upairsetE = $true)) & (singletonsswitch = $true)) & (eqinunit = $true)) & (notinsingleton = $true)) & (uniqinunit = $true)) & (nonemptyImpWitness = $true)) & (exuE2 = $true)) & (subPowSU = $true)) & (setunionE = $true)) & (setunionI = $true)) & (powersetE = $true)) & (emptyInPowerset = $true)) & (emptyinPowerset = $true)) & (powersetI = $true)) & (setoftrueEq = $true)) & (setadjoinOr = $true)) & (setadjoinE = $true)) & (setadjoinIR = $true)) & (emptyinunitempty = $true)) & (setadjoinIL = $true)) & (nonemptyI1 = $true)) & (nonemptyI = $true)) & (nonemptyE1 = $true)) & (setbeta = $true)) & (noeltsimpempty = $true)) & (emptyI = $true)) & (setext = $true)) & (exuE3e = $true)) & (notinemptyset = $true)) & (emptysetimpfalse = $true)) & (emptysetE = $true)) & (prop2setE = $true)) & (exuE1 = $true)) & (dsetconstrER = $true)) & (dsetconstrEL = $true)) & (dsetconstrI = $true)) & (descrp = $true)) & (wellorderingAx = $true)) & (foundationAx = $true)) & (replAx = $true)) & (omegaIndAx = $true)) & (omegaSAx = $true)) & (omega0Ax = $true)) & (setunionAx = $true)) & (powersetAx = $true)) & (setadjoinAx = $true)) & (emptysetAx = $true)) & (setextAx = $true)),
% 9.99/2.89 inference(ennf_transformation,[status(thm)],[f718])).
% 9.99/2.89 thf(f718,plain,(
% 9.99/2.89 ~((setextAx = $true) => ((emptysetAx = $true) => ((setadjoinAx = $true) => ((powersetAx = $true) => ((setunionAx = $true) => ((omega0Ax = $true) => ((omegaSAx = $true) => ((omegaIndAx = $true) => ((replAx = $true) => ((foundationAx = $true) => ((wellorderingAx = $true) => ((descrp = $true) => ((dsetconstrI = $true) => ((dsetconstrEL = $true) => ((dsetconstrER = $true) => ((exuE1 = $true) => ((prop2setE = $true) => ((emptysetE = $true) => ((emptysetimpfalse = $true) => ((notinemptyset = $true) => ((exuE3e = $true) => ((setext = $true) => ((emptyI = $true) => ((noeltsimpempty = $true) => ((setbeta = $true) => ((nonemptyE1 = $true) => ((nonemptyI = $true) => ((nonemptyI1 = $true) => ((setadjoinIL = $true) => ((emptyinunitempty = $true) => ((setadjoinIR = $true) => ((setadjoinE = $true) => ((setadjoinOr = $true) => ((setoftrueEq = $true) => ((powersetI = $true) => ((emptyinPowerset = $true) => ((emptyInPowerset = $true) => ((powersetE = $true) => ((setunionI = $true) => ((setunionE = $true) => ((subPowSU = $true) => ((exuE2 = $true) => ((nonemptyImpWitness = $true) => ((uniqinunit = $true) => ((notinsingleton = $true) => ((eqinunit = $true) => ((singletonsswitch = $true) => ((upairsetE = $true) => ((upairsetIL = $true) => ((upairsetIR = $true) => ((emptyE1 = $true) => ((vacuousDall = $true) => ((quantDeMorgan1 = $true) => ((quantDeMorgan2 = $true) => ((quantDeMorgan3 = $true) => ((quantDeMorgan4 = $true) => ((prop2setI = $true) => ((prop2set2propI = $true) => ((notdexE = $true) => ((notdallE = $true) => ((exuI1 = $true) => ((exuI3 = $true) => ((exuI2 = $true) => ((inCongP = $true) => ((in__Cong = $true) => ((exuE3u = $true) => ((exu__Cong = $true) => ((emptyset__Cong = $true) => ((setadjoin__Cong = $true) => ((powerset__Cong = $true) => ((setunion__Cong = $true) => ((omega__Cong = $true) => ((exuEu = $true) => ((descr__Cong = $true) => ((dsetconstr__Cong = $true) => ((subsetI1 = $true) => ((eqimpsubset2 = $true) => ((eqimpsubset1 = $true) => ((subsetI2 = $true) => ((emptysetsubset = $true) => ((subsetE = $true) => ((subsetE2 = $true) => ((notsubsetI = $true) => ((notequalI1 = $true) => ((notequalI2 = $true) => ((subsetRefl = $true) => ((subsetTrans = $true) => ((setadjoinSub = $true) => ((setadjoinSub2 = $true) => ((subset2powerset = $true) => ((setextsub = $true) => ((subsetemptysetimpeq = $true) => ((powersetI1 = $true) => ((powersetE1 = $true) => ((inPowerset = $true) => ((powersetsubset = $true) => ((sepInPowerset = $true) => ((sepSubset = $true) => ((binunionIL = $true) => ((upairset2IR = $true) => ((binunionIR = $true) => ((binunionEcases = $true) => ((binunionE = $true) => ((binunionLsub = $true) => ((binunionRsub = $true) => ((binintersectI = $true) => ((binintersectSubset5 = $true) => ((binintersectEL = $true) => ((binintersectLsub = $true) => ((binintersectSubset2 = $true) => ((binintersectSubset3 = $true) => ((binintersectER = $true) => ((disjointsetsI1 = $true) => ((binintersectRsub = $true) => ((binintersectSubset4 = $true) => ((binintersectSubset1 = $true) => ((bs114d = $true) => ((setminusI = $true) => ((setminusEL = $true) => ((setminusER = $true) => ((setminusSubset2 = $true) => ((setminusERneg = $true) => ((setminusELneg = $true) => ((setminusILneg = $true) => ((setminusIRneg = $true) => ((setminusLsub = $true) => ((setminusSubset1 = $true) => ((symdiffE = $true) => ((symdiffI1 = $true) => ((symdiffI2 = $true) => ((symdiffIneg1 = $true) => ((symdiffIneg2 = $true) => ((secondinupair = $true) => ((setukpairIL = $true) => ((setukpairIR = $true) => ((kpairiskpair = $true) => ((kpairp = $true) => ((singletonsubset = $true) => ((singletoninpowerset = $true) => ((singletoninpowunion = $true) => ((upairset2E = $true) => ((upairsubunion = $true) => ((upairinpowunion = $true) => ((ubforcartprodlem1 = $true) => ((ubforcartprodlem2 = $true) => ((ubforcartprodlem3 = $true) => ((cartprodpairin = $true) => ((cartprodmempair1 = $true) => ((cartprodmempair = $true) => ((setunionE2 = $true) => ((setunionsingleton1 = $true) => ((setunionsingleton2 = $true) => ((setunionsingleton = $true) => ((singletonprop = $true) => ((ex1E1 = $true) => ((ex1I = $true) => ((ex1I2 = $true) => ((singletonsuniq = $true) => ((setukpairinjL1 = $true) => ((kfstsingleton = $true) => ((theprop = $true) => ((kfstpairEq = $true) => ((cartprodfstin = $true) => ((setukpairinjL2 = $true) => ((setukpairinjL = $true) => ((setukpairinjR11 = $true) => ((setukpairinjR12 = $true) => ((setukpairinjR1 = $true) => ((upairequniteq = $true) => ((setukpairinjR2 = $true) => ((setukpairinjR = $true) => ((ksndsingleton = $true) => ((ksndpairEq = $true) => ((kpairsurjEq = $true) => ((cartprodsndin = $true) => ((cartprodpairmemEL = $true) => ((cartprodpairmemER = $true) => ((cartprodmempaircEq = $true) => ((cartprodfstpairEq = $true) => ((cartprodsndpairEq = $true) => ((cartprodpairsurjEq = $true) => ((dpsetconstrI = $true) => ((dpsetconstrSub = $true) => ((setOfPairsIsBReln = $true) => ((dpsetconstrERa = $true) => ((dpsetconstrEL1 = $true) => ((dpsetconstrEL2 = $true) => ((dpsetconstrER = $true) => ((funcImageSingleton = $true) => ((apProp = $true) => ((app = $true) => ((infuncsetfunc = $true) => ((ap2p = $true) => ((funcinfuncset = $true) => ((lamProp = $true) => ((lamp = $true) => ((lam2p = $true) => ((brelnall1 = $true) => ((brelnall2 = $true) => ((ex1E2 = $true) => ((funcGraphProp1 = $true) => ((funcGraphProp3 = $true) => ((funcGraphProp2 = $true) => ((funcextLem = $true) => ((funcGraphProp4 = $true) => ((subbreln = $true) => ((eqbreln = $true) => ((funcext = $true) => ((funcext2 = $true) => ((ap2apEq1 = $true) => ((ap2apEq2 = $true) => ((beta1 = $true) => ((eta1 = $true) => ((lam2lamEq = $true) => ((beta2 = $true) => ((eta2 = $true) => ((iffalseProp1 = $true) => ((iffalseProp2 = $true) => ((iftrueProp1 = $true) => ((iftrueProp2 = $true) => ((ifSingleton = $true) => ((ifp = $true) => ((theeq = $true) => ((iftrue = $true) => ((iffalse = $true) => ((iftrueorfalse = $true) => ((binintersectT_lem = $true) => ((binunionT_lem = $true) => ((powersetT_lem = $true) => ((setminusT_lem = $true) => ((complementT_lem = $true) => ((setextT = $true) => ((subsetTI = $true) => ((powersetTI1 = $true) => ((powersetTE1 = $true) => ((complementTI1 = $true) => ((complementTE1 = $true) => ((binintersectTELcontra = $true) => ((binintersectTERcontra = $true) => ((contrasubsetT = $true) => ((contrasubsetT1 = $true) => ((contrasubsetT2 = $true) => ((contrasubsetT3 = $true) => ((doubleComplementI1 = $true) => ((doubleComplementE1 = $true) => ((doubleComplementSub1 = $true) => ((doubleComplementSub2 = $true) => ((doubleComplementEq = $true) => ((complementTnotintersectT = $true) => ((complementImpComplementIntersect = $true) => ((complementSubsetComplementIntersect = $true) => ((complementInPowersetComplementIntersect = $true) => ((contraSubsetComplement = $true) => ((complementTcontraSubset = $true) => ((binunionTILcontra = $true) => ((binunionTIRcontra = $true) => ((inIntersectImpInUnion = $true) => ((inIntersectImpInUnion2 = $true) => ((inIntersectImpInIntersectUnions = $true) => ((intersectInPowersetIntersectUnions = $true) => ((inComplementUnionImpNotIn1 = $true) => ((inComplementUnionImpInComplement1 = $true) => ((binunionTE = $true) => ((binunionTEcontra = $true) => ((demorgan2a1 = $true) => ((complementUnionInPowersetComplement = $true) => ((demorgan2a2 = $true) => ((demorgan1a = $true) => ((demorgan1b = $true) => ((demorgan1 = $true) => ((demorgan2a = $true) => ((demorgan2b2 = $true) => ((demorgan2b = $true) => ((demorgan2 = $true) => ((woz13rule0 = $true) => ((woz13rule1 = $true) => ((woz13rule2 = $true) => ((woz13rule3 = $true) => ((woz13rule4 = $true) => ((woz1_1 = $true) => ((woz1_2 = $true) => ((woz1_3 = $true) => ((woz1_4 = $true) => ((woz1_5 = $true) => ((breln1all2 = $true) => ((breln1SetBreln1 = $true) => ((choice2fnsingleton = $true) => ((setOfPairsIsBReln1 = $true) => ((breln1all1 = $true) => ((subbreln1 = $true) => ((eqbreln1 = $true) => ((breln1invprop = $true) => ((breln1invI = $true) => ((breln1invE = $true) => ((breln1compprop = $true) => ((breln1compI = $true) => ((breln1compE = $true) => ((breln1compEex = $true) => ((breln1unionprop = $true) => ((breln1unionIL = $true) => ((breln1unionIR = $true) => ((breln1unionI = $true) => ((breln1unionE = $true) => ((breln1unionEcases = $true) => ((breln1unionCommutes = $true) => ((woz2Ex = $true) => ((woz2W = $true) => ((woz2A = $true) => ((woz2B = $true) => ((image1Ex = $true) => ((image1Ex1 = $true) => ((image1Equiv = $true) => ((image1E = $true) => ((image1I = $true) => ((injFuncInInjFuncSet = $true) => ((injFuncSetFuncIn = $true) => ! [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X2 @ (injFuncSet @ X0 @ X1))) => ($true = (injective @ X0 @ X1 @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 9.99/2.89 inference(fool_elimination,[status(thm)],[f717])).
% 9.99/2.89 thf(f717,plain,(
% 9.99/2.89 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => (ubforcartprodlem2 => (ubforcartprodlem3 => (cartprodpairin => (cartprodmempair1 => (cartprodmempair => (setunionE2 => (setunionsingleton1 => (setunionsingleton2 => (setunionsingleton => (singletonprop => (ex1E1 => (ex1I => (ex1I2 => (singletonsuniq => (setukpairinjL1 => (kfstsingleton => (theprop => (kfstpairEq => (cartprodfstin => (setukpairinjL2 => (setukpairinjL => (setukpairinjR11 => (setukpairinjR12 => (setukpairinjR1 => (upairequniteq => (setukpairinjR2 => (setukpairinjR => (ksndsingleton => (ksndpairEq => (kpairsurjEq => (cartprodsndin => (cartprodpairmemEL => (cartprodpairmemER => (cartprodmempaircEq => (cartprodfstpairEq => (cartprodsndpairEq => (cartprodpairsurjEq => (dpsetconstrI => (dpsetconstrSub => (setOfPairsIsBReln => (dpsetconstrERa => (dpsetconstrEL1 => (dpsetconstrEL2 => (dpsetconstrER => (funcImageSingleton => (apProp => (app => (infuncsetfunc => (ap2p => (funcinfuncset => (lamProp => (lamp => (lam2p => (brelnall1 => (brelnall2 => (ex1E2 => (funcGraphProp1 => (funcGraphProp3 => (funcGraphProp2 => (funcextLem => (funcGraphProp4 => (subbreln => (eqbreln => (funcext => (funcext2 => (ap2apEq1 => (ap2apEq2 => (beta1 => (eta1 => (lam2lamEq => (beta2 => (eta2 => (iffalseProp1 => (iffalseProp2 => (iftrueProp1 => (iftrueProp2 => (ifSingleton => (ifp => (theeq => (iftrue => (iffalse => (iftrueorfalse => (binintersectT_lem => (binunionT_lem => (powersetT_lem => (setminusT_lem => (complementT_lem => (setextT => (subsetTI => (powersetTI1 => (powersetTE1 => (complementTI1 => (complementTE1 => (binintersectTELcontra => (binintersectTERcontra => (contrasubsetT => (contrasubsetT1 => (contrasubsetT2 => (contrasubsetT3 => (doubleComplementI1 => (doubleComplementE1 => (doubleComplementSub1 => (doubleComplementSub2 => (doubleComplementEq => (complementTnotintersectT => (complementImpComplementIntersect => (complementSubsetComplementIntersect => (complementInPowersetComplementIntersect => (contraSubsetComplement => (complementTcontraSubset => (binunionTILcontra => (binunionTIRcontra => (inIntersectImpInUnion => (inIntersectImpInUnion2 => (inIntersectImpInIntersectUnions => (intersectInPowersetIntersectUnions => (inComplementUnionImpNotIn1 => (inComplementUnionImpInComplement1 => (binunionTE => (binunionTEcontra => (demorgan2a1 => (complementUnionInPowersetComplement => (demorgan2a2 => (demorgan1a => (demorgan1b => (demorgan1 => (demorgan2a => (demorgan2b2 => (demorgan2b => (demorgan2 => (woz13rule0 => (woz13rule1 => (woz13rule2 => (woz13rule3 => (woz13rule4 => (woz1_1 => (woz1_2 => (woz1_3 => (woz1_4 => (woz1_5 => (breln1all2 => (breln1SetBreln1 => (choice2fnsingleton => (setOfPairsIsBReln1 => (breln1all1 => (subbreln1 => (eqbreln1 => (breln1invprop => (breln1invI => (breln1invE => (breln1compprop => (breln1compI => (breln1compE => (breln1compEex => (breln1unionprop => (breln1unionIL => (breln1unionIR => (breln1unionI => (breln1unionE => (breln1unionEcases => (breln1unionCommutes => (woz2Ex => (woz2W => (woz2A => (woz2B => (image1Ex => (image1Ex1 => (image1Equiv => (image1E => (image1I => (injFuncInInjFuncSet => (injFuncSetFuncIn => ! [X0 : $i,X1 : $i,X2 : $i] : ((in @ X2 @ (injFuncSet @ X0 @ X1)) => (injective @ X0 @ X1 @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 9.99/2.89 inference(rectify,[status(thm)],[f321])).
% 9.99/2.89 thf(f321,negated_conjecture,(
% 9.99/2.89 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => (ubforcartprodlem2 => (ubforcartprodlem3 => (cartprodpairin => (cartprodmempair1 => (cartprodmempair => (setunionE2 => (setunionsingleton1 => (setunionsingleton2 => (setunionsingleton => (singletonprop => (ex1E1 => (ex1I => (ex1I2 => (singletonsuniq => (setukpairinjL1 => (kfstsingleton => (theprop => (kfstpairEq => (cartprodfstin => (setukpairinjL2 => (setukpairinjL => (setukpairinjR11 => (setukpairinjR12 => (setukpairinjR1 => (upairequniteq => (setukpairinjR2 => (setukpairinjR => (ksndsingleton => (ksndpairEq => (kpairsurjEq => (cartprodsndin => (cartprodpairmemEL => (cartprodpairmemER => (cartprodmempaircEq => (cartprodfstpairEq => (cartprodsndpairEq => (cartprodpairsurjEq => (dpsetconstrI => (dpsetconstrSub => (setOfPairsIsBReln => (dpsetconstrERa => (dpsetconstrEL1 => (dpsetconstrEL2 => (dpsetconstrER => (funcImageSingleton => (apProp => (app => (infuncsetfunc => (ap2p => (funcinfuncset => (lamProp => (lamp => (lam2p => (brelnall1 => (brelnall2 => (ex1E2 => (funcGraphProp1 => (funcGraphProp3 => (funcGraphProp2 => (funcextLem => (funcGraphProp4 => (subbreln => (eqbreln => (funcext => (funcext2 => (ap2apEq1 => (ap2apEq2 => (beta1 => (eta1 => (lam2lamEq => (beta2 => (eta2 => (iffalseProp1 => (iffalseProp2 => (iftrueProp1 => (iftrueProp2 => (ifSingleton => (ifp => (theeq => (iftrue => (iffalse => (iftrueorfalse => (binintersectT_lem => (binunionT_lem => (powersetT_lem => (setminusT_lem => (complementT_lem => (setextT => (subsetTI => (powersetTI1 => (powersetTE1 => (complementTI1 => (complementTE1 => (binintersectTELcontra => (binintersectTERcontra => (contrasubsetT => (contrasubsetT1 => (contrasubsetT2 => (contrasubsetT3 => (doubleComplementI1 => (doubleComplementE1 => (doubleComplementSub1 => (doubleComplementSub2 => (doubleComplementEq => (complementTnotintersectT => (complementImpComplementIntersect => (complementSubsetComplementIntersect => (complementInPowersetComplementIntersect => (contraSubsetComplement => (complementTcontraSubset => (binunionTILcontra => (binunionTIRcontra => (inIntersectImpInUnion => (inIntersectImpInUnion2 => (inIntersectImpInIntersectUnions => (intersectInPowersetIntersectUnions => (inComplementUnionImpNotIn1 => (inComplementUnionImpInComplement1 => (binunionTE => (binunionTEcontra => (demorgan2a1 => (complementUnionInPowersetComplement => (demorgan2a2 => (demorgan1a => (demorgan1b => (demorgan1 => (demorgan2a => (demorgan2b2 => (demorgan2b => (demorgan2 => (woz13rule0 => (woz13rule1 => (woz13rule2 => (woz13rule3 => (woz13rule4 => (woz1_1 => (woz1_2 => (woz1_3 => (woz1_4 => (woz1_5 => (breln1all2 => (breln1SetBreln1 => (choice2fnsingleton => (setOfPairsIsBReln1 => (breln1all1 => (subbreln1 => (eqbreln1 => (breln1invprop => (breln1invI => (breln1invE => (breln1compprop => (breln1compI => (breln1compE => (breln1compEex => (breln1unionprop => (breln1unionIL => (breln1unionIR => (breln1unionI => (breln1unionE => (breln1unionEcases => (breln1unionCommutes => (woz2Ex => (woz2W => (woz2A => (woz2B => (image1Ex => (image1Ex1 => (image1Equiv => (image1E => (image1I => (injFuncInInjFuncSet => (injFuncSetFuncIn => ! [X1 : $i,X2 : $i,X12 : $i] : ((in @ X12 @ (injFuncSet @ X1 @ X2)) => (injective @ X1 @ X2 @ X12))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 9.99/2.89 inference(negated_conjecture,[status(cth)],[f320])).
% 9.99/2.89 thf(f320,conjecture,(
% 9.99/2.89 setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => (ubforcartprodlem2 => (ubforcartprodlem3 => (cartprodpairin => (cartprodmempair1 => (cartprodmempair => (setunionE2 => (setunionsingleton1 => (setunionsingleton2 => (setunionsingleton => (singletonprop => (ex1E1 => (ex1I => (ex1I2 => (singletonsuniq => (setukpairinjL1 => (kfstsingleton => (theprop => (kfstpairEq => (cartprodfstin => (setukpairinjL2 => (setukpairinjL => (setukpairinjR11 => (setukpairinjR12 => (setukpairinjR1 => (upairequniteq => (setukpairinjR2 => (setukpairinjR => (ksndsingleton => (ksndpairEq => (kpairsurjEq => (cartprodsndin => (cartprodpairmemEL => (cartprodpairmemER => (cartprodmempaircEq => (cartprodfstpairEq => (cartprodsndpairEq => (cartprodpairsurjEq => (dpsetconstrI => (dpsetconstrSub => (setOfPairsIsBReln => (dpsetconstrERa => (dpsetconstrEL1 => (dpsetconstrEL2 => (dpsetconstrER => (funcImageSingleton => (apProp => (app => (infuncsetfunc => (ap2p => (funcinfuncset => (lamProp => (lamp => (lam2p => (brelnall1 => (brelnall2 => (ex1E2 => (funcGraphProp1 => (funcGraphProp3 => (funcGraphProp2 => (funcextLem => (funcGraphProp4 => (subbreln => (eqbreln => (funcext => (funcext2 => (ap2apEq1 => (ap2apEq2 => (beta1 => (eta1 => (lam2lamEq => (beta2 => (eta2 => (iffalseProp1 => (iffalseProp2 => (iftrueProp1 => (iftrueProp2 => (ifSingleton => (ifp => (theeq => (iftrue => (iffalse => (iftrueorfalse => (binintersectT_lem => (binunionT_lem => (powersetT_lem => (setminusT_lem => (complementT_lem => (setextT => (subsetTI => (powersetTI1 => (powersetTE1 => (complementTI1 => (complementTE1 => (binintersectTELcontra => (binintersectTERcontra => (contrasubsetT => (contrasubsetT1 => (contrasubsetT2 => (contrasubsetT3 => (doubleComplementI1 => (doubleComplementE1 => (doubleComplementSub1 => (doubleComplementSub2 => (doubleComplementEq => (complementTnotintersectT => (complementImpComplementIntersect => (complementSubsetComplementIntersect => (complementInPowersetComplementIntersect => (contraSubsetComplement => (complementTcontraSubset => (binunionTILcontra => (binunionTIRcontra => (inIntersectImpInUnion => (inIntersectImpInUnion2 => (inIntersectImpInIntersectUnions => (intersectInPowersetIntersectUnions => (inComplementUnionImpNotIn1 => (inComplementUnionImpInComplement1 => (binunionTE => (binunionTEcontra => (demorgan2a1 => (complementUnionInPowersetComplement => (demorgan2a2 => (demorgan1a => (demorgan1b => (demorgan1 => (demorgan2a => (demorgan2b2 => (demorgan2b => (demorgan2 => (woz13rule0 => (woz13rule1 => (woz13rule2 => (woz13rule3 => (woz13rule4 => (woz1_1 => (woz1_2 => (woz1_3 => (woz1_4 => (woz1_5 => (breln1all2 => (breln1SetBreln1 => (choice2fnsingleton => (setOfPairsIsBReln1 => (breln1all1 => (subbreln1 => (eqbreln1 => (breln1invprop => (breln1invI => (breln1invE => (breln1compprop => (breln1compI => (breln1compE => (breln1compEex => (breln1unionprop => (breln1unionIL => (breln1unionIR => (breln1unionI => (breln1unionE => (breln1unionEcases => (breln1unionCommutes => (woz2Ex => (woz2W => (woz2A => (woz2B => (image1Ex => (image1Ex1 => (image1Equiv => (image1E => (image1I => (injFuncInInjFuncSet => (injFuncSetFuncIn => ! [X1 : $i,X2 : $i,X12 : $i] : ((in @ X12 @ (injFuncSet @ X1 @ X2)) => (injective @ X1 @ X2 @ X12)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 9.99/2.89 file('/export/starexec/sandbox2/tmp/tmp.2k8MKHSw6S/DTF2THF_18170.p',injFuncSetFuncInj)).
% 9.99/2.89 thf(f9822,plain,(
% 9.99/2.89 ($true = (injective @ sK228 @ sK229 @ sK230))),
% 9.99/2.89 inference(trivial_inequality_removal,[status(thm)],[f9815])).
% 9.99/2.89 thf(f9815,plain,(
% 9.99/2.89 ($true = (injective @ sK228 @ sK229 @ sK230)) | ($true != $true)),
% 9.99/2.89 inference(superposition,[status(thm)],[f6933,f7125])).
% 9.99/2.89 thf(f7125,plain,(
% 9.99/2.89 ($true = (in @ sK230 @ (dsetconstr @ (funcSet @ sK228 @ sK229) @ (injective @ sK228 @ sK229))))),
% 9.99/2.89 inference(beta_eta_normalization,[status(thm)],[f5349])).
% 9.99/2.89 thf(f5349,plain,(
% 9.99/2.89 ($true = (in @ sK230 @ ((^[Y0 : $i]: ((^[Y1 : $i]: (dsetconstr @ (funcSet @ Y0 @ Y1) @ (^[Y2 : $i]: (injective @ Y0 @ Y1 @ Y2)))))) @ sK228 @ sK229)))),
% 9.99/2.89 inference(definition_unfolding,[status(thm)],[f3737,f4974])).
% 9.99/2.89 thf(f4974,plain,(
% 9.99/2.89 (injFuncSet = (^[Y0 : $i]: ((^[Y1 : $i]: (dsetconstr @ (funcSet @ Y0 @ Y1) @ (^[Y2 : $i]: (injective @ Y0 @ Y1 @ Y2)))))))),
% 9.99/2.89 inference(cnf_transformation,[status(thm)],[f454])).
% 9.99/2.89 thf(f454,plain,(
% 9.99/2.89 (injFuncSet = (^[Y0 : $i]: ((^[Y1 : $i]: (dsetconstr @ (funcSet @ Y0 @ Y1) @ (^[Y2 : $i]: (injective @ Y0 @ Y1 @ Y2)))))))),
% 9.99/2.89 inference(fool_elimination,[status(thm)],[f453])).
% 9.99/2.89 thf(f453,plain,(
% 9.99/2.89 (injFuncSet = (^[X0 : $i, X1 : $i] : (dsetconstr @ (funcSet @ X0 @ X1) @ (^[X2 : $i] : (injective @ X0 @ X1 @ X2)))))),
% 9.99/2.89 inference(rectify,[status(thm)],[f317])).
% 9.99/2.89 thf(f317,axiom,(
% 9.99/2.89 (injFuncSet = (^[X3 : $i, X4 : $i] : (dsetconstr @ (funcSet @ X3 @ X4) @ (^[X12 : $i] : (injective @ X3 @ X4 @ X12)))))),
% 9.99/2.89 file('/export/starexec/sandbox2/tmp/tmp.2k8MKHSw6S/DTF2THF_18170.p',injFuncSet)).
% 9.99/2.89 thf(f3737,plain,(
% 9.99/2.89 ((in @ sK230 @ (injFuncSet @ sK228 @ sK229)) = $true)),
% 9.99/2.89 inference(cnf_transformation,[status(thm)],[f1778])).
% 9.99/2.89 thf(f6933,plain,(
% 9.99/2.89 ( ! [X2 : $i > $o,X0 : $i,X1 : $i] : (((in @ X1 @ (dsetconstr @ X0 @ X2)) != $true) | ($true = (X2 @ X1))) )),
% 9.99/2.89 inference(beta_eta_normalization,[status(thm)],[f6932])).
% 9.99/2.89 thf(f6932,plain,(
% 9.99/2.89 ( ! [X2 : $i > $o,X0 : $i,X1 : $i] : (($true != (in @ X1 @ (dsetconstr @ X0 @ (^[Y0 : $i]: (X2 @ Y0))))) | ($true = (X2 @ X1))) )),
% 9.99/2.89 inference(trivial_inequality_removal,[status(thm)],[f5813])).
% 9.99/2.89 thf(f5813,plain,(
% 9.99/2.89 ( ! [X2 : $i > $o,X0 : $i,X1 : $i] : (($true != $true) | ($true = (X2 @ X1)) | ($true != (in @ X1 @ (dsetconstr @ X0 @ (^[Y0 : $i]: (X2 @ Y0)))))) )),
% 9.99/2.89 inference(definition_unfolding,[status(thm)],[f4466,f3793])).
% 9.99/2.89 thf(f3793,plain,(
% 9.99/2.89 (dsetconstrER = $true)),
% 9.99/2.89 inference(cnf_transformation,[status(thm)],[f1778])).
% 9.99/2.89 thf(f4466,plain,(
% 9.99/2.89 ( ! [X2 : $i > $o,X0 : $i,X1 : $i] : (($true != (in @ X1 @ (dsetconstr @ X0 @ (^[Y0 : $i]: (X2 @ Y0))))) | ($true = (X2 @ X1)) | (dsetconstrER != $true)) )),
% 9.99/2.89 inference(cnf_transformation,[status(thm)],[f2495])).
% 9.99/2.89 thf(f2495,plain,(
% 9.99/2.89 (! [X0 : $i,X1 : $i,X2 : $i > $o] : (($true != (in @ X1 @ (dsetconstr @ X0 @ (^[Y0 : $i]: (X2 @ Y0))))) | ($true = (X2 @ X1))) | (dsetconstrER != $true)) & ((dsetconstrER = $true) | (($true = (in @ sK669 @ (dsetconstr @ sK668 @ (^[Y0 : $i]: (sK670 @ Y0))))) & ($true != (sK670 @ sK669))))),
% 9.99/2.89 inference(skolemisation,[status(esa),new_symbols(skolem,[sK668,sK669,sK670])],[f2493,f2494])).
% 9.99/2.89 thf(f2494,plain,(
% 9.99/2.89 ? [X3 : $i,X4 : $i,X5 : $i > $o] : (($true = (in @ X4 @ (dsetconstr @ X3 @ (^[Y0 : $i]: (X5 @ Y0))))) & ($true != (X5 @ X4))) => (($true = (in @ sK669 @ (dsetconstr @ sK668 @ (^[Y0 : $i]: (sK670 @ Y0))))) & ($true != (sK670 @ sK669)))),
% 9.99/2.89 introduced(definition,[],[choice_axiom])).
% 9.99/2.89 thf(f2493,plain,(
% 9.99/2.89 (! [X0 : $i,X1 : $i,X2 : $i > $o] : (($true != (in @ X1 @ (dsetconstr @ X0 @ (^[Y0 : $i]: (X2 @ Y0))))) | ($true = (X2 @ X1))) | (dsetconstrER != $true)) & ((dsetconstrER = $true) | ? [X3 : $i,X4 : $i,X5 : $i > $o] : (($true = (in @ X4 @ (dsetconstr @ X3 @ (^[Y0 : $i]: (X5 @ Y0))))) & ($true != (X5 @ X4))))),
% 9.99/2.89 inference(rectify,[status(thm)],[f2492])).
% 9.99/2.89 thf(f2492,plain,(
% 9.99/2.89 (! [X0 : $i,X1 : $i,X2 : $i > $o] : (($true != (in @ X1 @ (dsetconstr @ X0 @ (^[Y0 : $i]: (X2 @ Y0))))) | ($true = (X2 @ X1))) | (dsetconstrER != $true)) & ((dsetconstrER = $true) | ? [X0 : $i,X1 : $i,X2 : $i > $o] : (($true = (in @ X1 @ (dsetconstr @ X0 @ (^[Y0 : $i]: (X2 @ Y0))))) & ($true != (X2 @ X1))))),
% 9.99/2.89 inference(nnf_transformation,[status(thm)],[f1029])).
% 9.99/2.89 thf(f1029,plain,(
% 9.99/2.89 ! [X0 : $i,X1 : $i,X2 : $i > $o] : (($true != (in @ X1 @ (dsetconstr @ X0 @ (^[Y0 : $i]: (X2 @ Y0))))) | ($true = (X2 @ X1))) <=> (dsetconstrER = $true)),
% 9.99/2.89 inference(ennf_transformation,[status(thm)],[f894])).
% 9.99/2.89 thf(f894,plain,(
% 9.99/2.89 (dsetconstrER = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i > $o] : (($true = (in @ X1 @ (dsetconstr @ X0 @ (^[Y0 : $i]: (X2 @ Y0))))) => ($true = (X2 @ X1)))),
% 9.99/2.89 inference(fool_elimination,[status(thm)],[f893])).
% 9.99/2.89 thf(f893,plain,(
% 9.99/2.89 (dsetconstrER = ! [X0 : $i,X1 : $i,X2 : $i > $o] : ((in @ X1 @ (dsetconstr @ X0 @ (^[X3 : $i] : (X2 @ X3)))) => (X2 @ X1)))),
% 9.99/2.89 inference(rectify,[status(thm)],[f16])).
% 9.99/2.89 thf(f16,axiom,(
% 9.99/2.89 (dsetconstrER = ! [X3 : $i,X1 : $i,X0 : $i > $o] : ((in @ X1 @ (dsetconstr @ X3 @ (^[X2 : $i] : (X0 @ X2)))) => (X0 @ X1)))),
% 9.99/2.89 file('/export/starexec/sandbox2/tmp/tmp.2k8MKHSw6S/DTF2THF_18170.p',dsetconstrER)).
% 9.99/2.89 % SZS output end Proof for DTF2THF_18170
% 9.99/2.89 % (18604)------------------------------
% 9.99/2.89 % (18604)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.99/2.89 % (18604)Termination reason: Refutation
% 9.99/2.89
% 9.99/2.89 % (18604)Memory used [KB]: 12920
% 9.99/2.89 % (18604)Time elapsed: 0.227 s
% 9.99/2.89 % (18604)Instructions burned: 726 (million)
% 9.99/2.89 % (18604)------------------------------
% 9.99/2.89 % (18604)------------------------------
% 9.99/2.89 % (18361)Success in time 1.188 s
%------------------------------------------------------------------------------