%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU798^1 : TPTP v9.2.1. Released v3.7.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n024.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 09:37:17 AM UTC 2026
% Result : Theorem 2.19s 1.79s
% Output : Refutation 2.19s
% Verified :
% SZS Type : Refutation
% Derivation depth : 15
% Number of leaves : 5
% Syntax : Number of formulae : 35 ( 14 unt; 0 typ; 0 def)
% Number of atoms : 5045 (1348 equ; 0 cnn)
% Maximal formula atoms : 317 ( 144 avg)
% Number of connectives : 2624 ( 45 ~; 29 |; 965 &; 310 @)
% ( 3 <=>;1272 =>; 0 <=; 0 <~>)
% Maximal formula depth : 324 ( 67 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 15 ( 15 >; 0 *; 0 +; 0 <<)
% Number of symbols : 646 ( 643 usr; 321 con; 0-6 aty)
% Number of variables : 97 ( 31 ^; 48 !; 18 ?; 97 :)
% 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_380,type,
sP0: $i > $i > $o ).
thf(func_def_382,type,
sP2: $i > $i > $i > $o ).
thf(func_def_383,type,
sP3: $i > $i > $i > $o ).
thf(func_def_384,type,
sP4: ( $i > $o ) > $o ).
thf(func_def_385,type,
sP5: ( $i > $o ) > $i > $i > $o ).
thf(func_def_386,type,
sP6: ( $i > $o ) > $i > $o ).
thf(func_def_388,type,
sP8: $i > $i > $i > $o ).
thf(func_def_389,type,
sP9: $i > $i > $o > $o ).
thf(func_def_391,type,
sP11: ( $i > $o ) > $i > $o ).
thf(func_def_393,type,
sP13: ( $i > $o ) > $i > $o ).
thf(func_def_394,type,
sP14: $i > $i > ( $i > $o ) > $o ).
thf(func_def_395,type,
sP15: $i > ( $i > $o ) > $o ).
thf(func_def_396,type,
sP16: $i > $i > ( $i > $o ) > $i > $o ).
thf(func_def_397,type,
sP17: $i > ( $i > $o ) > $o ).
thf(func_def_399,type,
sP19: $o > $i > $i > $o ).
thf(func_def_401,type,
sP21: ( $i > $i ) > $i > $i > $o ).
thf(func_def_403,type,
sP23: $i > $i > $i > $i > $o ).
thf(func_def_404,type,
sP24: $i > $i > $i > $o ).
thf(func_def_405,type,
sP25: $i > $i > $i > $o ).
thf(func_def_406,type,
sP26: $i > $i > $i > $o ).
thf(func_def_407,type,
sP27: $i > $i > $o ).
thf(func_def_409,type,
sP29: ( $i > $o ) > $o ).
thf(func_def_410,type,
sP30: $o > $i > $i > $i > $o ).
thf(func_def_411,type,
sP31: $i > $i > $i > $i > $o ).
thf(func_def_414,type,
sP34: $i > $i > $i > $o ).
thf(func_def_416,type,
sP36: $i > $i > $i > $o ).
thf(func_def_417,type,
sP37: ( $i > $i > $o ) > $i > $i > $o ).
thf(func_def_418,type,
sP38: ( $i > $i > $o ) > $i > $o ).
thf(func_def_420,type,
sP40: ( $i > $o ) > $i > $o ).
thf(func_def_421,type,
sP41: ( $i > $o ) > $i > $o ).
thf(func_def_422,type,
sP42: $i > $i > $o ).
thf(func_def_425,type,
sP45: ( $i > $o ) > $i > $o ).
thf(func_def_429,type,
sP49: $i > $i > $o > $o ).
thf(func_def_430,type,
sP50: $i > $i > $i > $o ).
thf(func_def_431,type,
sP51: $i > $i > $i > $o ).
thf(func_def_433,type,
sP53: $i > ( $i > $o ) > $o ).
thf(func_def_436,type,
sP56: $i > ( $i > $o ) > $o ).
thf(func_def_437,type,
sP57: $i > $i > ( $i > $o ) > $o ).
thf(func_def_439,type,
sP59: $i > $i > $i > $i > $o ).
thf(func_def_440,type,
sP60: $i > $i > $i > $o ).
thf(func_def_441,type,
sP61: ( $i > $o ) > $o ).
thf(func_def_442,type,
sP62: $i > $i > $i > $i > $o ).
thf(func_def_443,type,
sP63: $i > $i > $i > $i > $i > $o ).
thf(func_def_445,type,
sP65: $i > $o > $i > $i > $i > $i > $o ).
thf(func_def_446,type,
sP66: $o > $i > $i > $i > $o ).
thf(func_def_447,type,
sP67: $i > $o > $i > $i > $o ).
thf(func_def_448,type,
sP68: ( $i > $o ) > ( $i > $o ) > $i > $i > $o ).
thf(func_def_449,type,
sP69: $i > $o ).
thf(func_def_450,type,
sP70: $i > $o ).
thf(func_def_451,type,
sP71: ( $i > $o ) > $o ).
thf(func_def_453,type,
sP73: ( $i > $o ) > $i > $o ).
thf(func_def_454,type,
sP74: $i > ( $i > $o ) > $o ).
thf(func_def_455,type,
sP75: $i > $i > $i > $o ).
thf(func_def_456,type,
sP76: $i > $i > $o ).
thf(func_def_457,type,
sP77: $i > $i > $o ).
thf(func_def_458,type,
sP78: $i > $i > $o ).
thf(func_def_459,type,
sP79: $i > $i > $o ).
thf(func_def_460,type,
sP80: $i > $o ).
thf(func_def_461,type,
sP81: ( $i > $o ) > $o ).
thf(func_def_462,type,
sP82: ( $i > $o ) > $i > $o ).
thf(func_def_464,type,
sP84: $i > $o ).
thf(func_def_465,type,
sP85: $i > $i > ( $i > $i > $o ) > $o ).
thf(func_def_466,type,
sP86: $i > ( $i > $i > $o ) > $i > $o ).
thf(func_def_467,type,
sP87: $i > $i > $i > $o ).
thf(func_def_469,type,
sP89: $i > $i > $o > $o ).
thf(func_def_470,type,
sP90: $i > $i > $i > $o ).
thf(func_def_471,type,
sP91: $i > $i > $i > $o ).
thf(func_def_472,type,
sP92: ( $i > $i ) > $i > $i > $o ).
thf(func_def_475,type,
sP95: $i > $i > ( $i > $i ) > $o ).
thf(func_def_477,type,
sP97: ( $i > $o ) > $i > $o ).
thf(func_def_478,type,
sP98: $i > $i > $i > ( $i > $o ) > $o ).
thf(func_def_479,type,
sP99: $i > $i > $i > $i > $o ).
thf(func_def_480,type,
sP100: $i > $i > $i > $i > $o ).
thf(func_def_481,type,
sP101: $i > $i > ( $i > $i ) > $o ).
thf(func_def_489,type,
sK109: $i > $i > $i ).
thf(func_def_490,type,
sK110: $i > $i > $i ).
thf(func_def_491,type,
sK111: $i > $i > $o ).
thf(func_def_521,type,
sK141: $i > $i > $i > $i ).
thf(func_def_522,type,
sK142: $i > $i > $i > $i ).
thf(func_def_523,type,
sK143: $i > $i > $i > $i ).
thf(func_def_524,type,
sK144: $i > $i > $i > $i ).
thf(func_def_535,type,
sK155: ( $i > $o ) > $i ).
thf(func_def_536,type,
sK156: $i > ( $i > $o ) > $i ).
thf(func_def_537,type,
sK157: $i > $o ).
thf(func_def_541,type,
sK161: ( $i > $o ) > $i > $i > $i ).
thf(func_def_542,type,
sK162: $i > $o ).
thf(func_def_566,type,
sK186: $i > $o ).
thf(func_def_568,type,
sK188: $i > ( $i > $o ) > $i ).
thf(func_def_569,type,
sK189: ( $i > $o ) > $i > $i ).
thf(func_def_585,type,
sK205: $i > $i > $i > $i ).
thf(func_def_586,type,
sK206: $i > $i > $i > $i > $i ).
thf(func_def_614,type,
sK234: $i > $i > $o > $i ).
thf(func_def_620,type,
sK240: $i > $i ).
thf(func_def_621,type,
sK241: ( $i > $i ) > $i > $i > $i ).
thf(func_def_640,type,
sK260: $i > $o ).
thf(func_def_642,type,
sK262: $i > ( $i > $o ) > $i ).
thf(func_def_643,type,
sK263: ( $i > $o ) > $i > $i ).
thf(func_def_676,type,
sK296: $i > $i > ( $i > $o ) > $i ).
thf(func_def_677,type,
sK297: $i > $i > ( $i > $o ) > $i ).
thf(func_def_678,type,
sK298: ( $i > $o ) > $i > $i ).
thf(func_def_681,type,
sK301: $i > $o ).
thf(func_def_691,type,
sK311: $i > $i > ( $i > $o ) > $i > $i ).
thf(func_def_692,type,
sK312: $i > $i > ( $i > $o ) > $i > $i ).
thf(func_def_693,type,
sK313: $i > ( $i > $o ) > $i ).
thf(func_def_697,type,
sK317: $i > $o ).
thf(func_def_714,type,
sK334: $i > $o ).
thf(func_def_715,type,
sK335: ( $i > $o ) > $i > $i ).
thf(func_def_716,type,
sK336: $i > ( $i > $o ) > $i ).
thf(func_def_737,type,
sK357: $i > $i ).
thf(func_def_738,type,
sK358: $i > $i ).
thf(func_def_739,type,
sK359: ( $i > $i ) > $i > $i ).
thf(func_def_740,type,
sK360: ( $i > $i ) > $i > $i > $i ).
thf(func_def_741,type,
sK361: $i > $i ).
thf(func_def_743,type,
sK363: $i > $i > $i > $i ).
thf(func_def_744,type,
sK364: $i > $i > $i > $i > $i ).
thf(func_def_745,type,
sK365: $i > $i > $i > $i > $i ).
thf(func_def_746,type,
sK366: $i > $i > $i > $i > $i ).
thf(func_def_772,type,
sK392: $i > $i > $i > $i ).
thf(func_def_773,type,
sK393: $i > $i > $i > $i ).
thf(func_def_786,type,
sK406: $i > $i > $i ).
thf(func_def_792,type,
sK412: $i > $i > $i ).
thf(func_def_800,type,
sK420: ( $i > $o ) > $i ).
thf(func_def_801,type,
sK421: ( $i > $o ) > $i ).
thf(func_def_802,type,
sK422: $i > $o ).
thf(func_def_804,type,
sK424: $i > $i > $i > $i > $o ).
thf(func_def_832,type,
sK452: $i > $i > $i ).
thf(func_def_837,type,
sK457: $i > $o ).
thf(func_def_840,type,
sK460: $i > $i ).
thf(func_def_842,type,
sK462: $i > ( $i > $i ) > $i > $i ).
thf(func_def_856,type,
sK476: $i > $o ).
thf(func_def_859,type,
sK479: $i > $i > $i > $i ).
thf(func_def_860,type,
sK480: $i > $i > $i > $i ).
thf(func_def_868,type,
sK488: $i > $i > $o ).
thf(func_def_869,type,
sK489: ( $i > $i > $o ) > $i > $i ).
thf(func_def_870,type,
sK490: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_871,type,
sK491: ( $i > $i > $o ) > $i > $i ).
thf(func_def_872,type,
sK492: ( $i > $i > $o ) > $i > $i > $i ).
thf(func_def_873,type,
sK493: ( $i > $o ) > $i > $i ).
thf(func_def_874,type,
sK494: ( $i > $o ) > $i > $i ).
thf(func_def_875,type,
sK495: ( $i > $o ) > $i > $i ).
thf(func_def_877,type,
sK497: $i > $o ).
thf(func_def_884,type,
sK504: $i > $i > $i ).
thf(func_def_887,type,
sK507: $i > $i ).
thf(func_def_891,type,
sK511: $i > $i > $o ).
thf(func_def_896,type,
sK516: $i > $o ).
thf(func_def_900,type,
sK520: $i > $i > $i ).
thf(func_def_904,type,
sK524: $i > $o ).
thf(func_def_905,type,
sK525: ( $i > $o ) > $i > $i ).
thf(func_def_906,type,
sK526: ( $i > $o ) > $i > $i ).
thf(func_def_921,type,
sK541: $i > $o ).
thf(func_def_922,type,
sK542: ( $i > $o ) > $i > $i ).
thf(func_def_949,type,
sK569: $i > $i ).
thf(func_def_952,type,
sK572: $i > $i > ( $i > $i ) > $i ).
thf(func_def_957,type,
sK577: $i > $o ).
thf(func_def_988,type,
sK608: $i > $i > $i > $o ).
thf(func_def_995,type,
sK615: $i > $i > $i > $i ).
thf(func_def_996,type,
sK616: $i > $i > $i > $i > $i ).
thf(func_def_1000,type,
sK620: $i > $o ).
thf(func_def_1005,type,
sK625: $i > $i > $o ).
thf(func_def_1016,type,
sK636: $i > $i ).
thf(func_def_1017,type,
sK637: $i > $i > $o ).
thf(func_def_1029,type,
sK649: $i > $i > $i ).
thf(func_def_1030,type,
sK650: $i > $o ).
thf(func_def_1031,type,
sK651: ( $i > $o ) > $i ).
thf(func_def_1041,type,
sK661: $i > $o ).
thf(func_def_1042,type,
sK662: ( $i > $o ) > $i > $i ).
thf(func_def_1043,type,
sK663: $i > ( $i > $o ) > $i ).
thf(func_def_1044,type,
sK664: $i > $o ).
thf(func_def_1062,type,
sK682: $i > $i ).
thf(func_def_1065,type,
sK685: $i > $i > ( $i > $i ) > $i ).
thf(func_def_1066,type,
sK686: $i > $i > ( $i > $o ) > $i ).
thf(func_def_1067,type,
sK687: $i > $i > ( $i > $o ) > $i ).
thf(func_def_1068,type,
sK688: $i > ( $i > $o ) > $i ).
thf(func_def_1071,type,
sK691: $i > $o ).
thf(func_def_1074,type,
sK694: $i > $i > $i ).
thf(func_def_1088,type,
sK708: $i > $i ).
thf(func_def_1092,type,
sK712: $i > $i > $i > $i > $i ).
thf(func_def_1101,type,
sK721: $i > $i > $i > $i ).
thf(func_def_1102,type,
sK722: $i > $i > $i > $i ).
thf(func_def_1113,type,
sK733: $i > ( $i > $o ) > $i ).
thf(func_def_1114,type,
sK734: ( $i > $o ) > $i ).
thf(func_def_1115,type,
sK735: $i > $o ).
thf(func_def_1116,type,
sK736: $i > $i > $i > $i > $i ).
thf(func_def_1117,type,
sK737: $i > $i > $i > $i > $i ).
thf(func_def_1122,type,
sK742: $i > $i > $i > $i > $i > $i ).
thf(func_def_1150,type,
sK770: $i > $i > $i ).
thf(func_def_1160,type,
sK780: $i > $i > $o ).
thf(func_def_1173,type,
sK793: $i > $o > $i > $i > $i > $i > $i ).
thf(func_def_1209,type,
sK829: $i > $i > $o ).
thf(func_def_1231,type,
sK851: $i > $o ).
thf(func_def_1238,type,
sK858: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_1239,type,
sK859: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_1242,type,
sK862: $i > $o ).
thf(func_def_1243,type,
sK863: $i > $o ).
thf(func_def_1244,type,
sK864: $i > $i ).
thf(func_def_1245,type,
sK865: $i > $i ).
thf(func_def_1247,type,
sK867: $i > $o ).
thf(func_def_1251,type,
sK871: ( $i > $o ) > $i ).
thf(func_def_1252,type,
sK872: $i > ( $i > $o ) > $i ).
thf(func_def_1253,type,
sK873: $i > $o ).
thf(func_def_1284,type,
sK904: $i > $i ).
thf(func_def_1287,type,
sK907: $i > $i > ( $i > $i ) > $i ).
thf(func_def_1293,type,
sK913: $i > $i ).
thf(func_def_1304,type,
sK924: $i > ( $i > $o ) > $i ).
thf(func_def_1305,type,
sK925: $i > ( $i > $o ) > $i ).
thf(func_def_1306,type,
sK926: ( $i > $o ) > $i > $i ).
thf(func_def_1308,type,
sK928: $i > $o ).
thf(func_def_1317,type,
sK937: $i > $o ).
thf(func_def_1318,type,
sK938: $i > $i ).
thf(func_def_1319,type,
sK939: $i > $i ).
thf(func_def_1320,type,
sK940: $i > $i ).
thf(func_def_1321,type,
sK941: $i > $i ).
thf(func_def_1322,type,
sK942: $i > $i > $i ).
thf(func_def_1323,type,
sK943: $i > $i > $i ).
thf(func_def_1324,type,
sK944: $i > $i > $i ).
thf(func_def_1325,type,
sK945: $i > $i > $i ).
thf(func_def_1326,type,
sK946: $i > $i > $i ).
thf(func_def_1327,type,
sK947: $i > $i > $i > $i ).
thf(func_def_1328,type,
sK948: $i > $i > $i ).
thf(func_def_1329,type,
sK949: $i > $i > $i ).
thf(func_def_1330,type,
sK950: $i > $i > $i ).
thf(func_def_1331,type,
sK951: $i > $i > $i ).
thf(func_def_1332,type,
sK952: $i > $i > $i ).
thf(func_def_1333,type,
sK953: $i > $i > $i > $i ).
thf(func_def_1334,type,
sK954: $i > $i > $i > $i ).
thf(func_def_1335,type,
sK955: $i > $i ).
thf(func_def_1337,type,
sK957: $i > ( $i > $o ) > $i ).
thf(func_def_1338,type,
sK958: ( $i > $o ) > $i ).
thf(func_def_1339,type,
sK959: $i > $o ).
thf(func_def_1341,type,
sK961: $i > $o ).
thf(func_def_1342,type,
sK962: ( $i > $o ) > $i > $i ).
thf(func_def_1343,type,
sK963: ( $i > $o ) > $i > $i ).
thf(func_def_1350,type,
sK970: $i > $i > $i ).
thf(func_def_1351,type,
sK971: $i > $i ).
thf(func_def_1354,type,
sK974: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_1355,type,
sK975: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_1356,type,
sK976: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_1357,type,
sK977: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_1358,type,
sK978: $i > $i > $o ).
thf(func_def_1382,type,
sK1002: $i > $o ).
thf(func_def_1383,type,
sK1003: $i > $i > $i > $i ).
thf(func_def_1387,type,
sK1007: $i > $o ).
thf(func_def_1388,type,
sK1008: $i > $o ).
thf(func_def_1389,type,
sK1009: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_1390,type,
sK1010: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_1401,type,
sK1021: $i > $i > $i > $i ).
thf(func_def_1402,type,
sK1022: $i > $i > $i > $o ).
thf(func_def_1406,type,
sK1026: $i > $i > $i > $i ).
thf(func_def_1410,type,
sK1030: $i > $i ).
thf(func_def_1416,type,
sK1036: $i > $i > $o ).
thf(func_def_1421,type,
sK1041: $i > $i ).
thf(func_def_1423,type,
sK1043: ( $i > $i ) > $i > $i > $i ).
thf(func_def_1433,type,
sK1053: $i > $o ).
thf(func_def_1434,type,
sK1054: $i > $o ).
thf(func_def_1435,type,
sK1055: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_1436,type,
sK1056: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_1444,type,
sK1064: $i > $i > $o ).
thf(func_def_1450,type,
sK1070: $i > $o ).
thf(func_def_1470,type,
sK1090: $i > $i ).
thf(func_def_1473,type,
sK1093: $i > $i > ( $i > $i ) > $i ).
thf(func_def_1474,type,
sK1094: $i > $i > ( $i > $i ) > $i ).
thf(func_def_1478,type,
sK1098: $i > $i > $i > ( $i > $o ) > $i ).
thf(func_def_1479,type,
sK1099: $i > $i > $i > ( $i > $o ) > $i ).
thf(func_def_1480,type,
sK1100: ( $i > $o ) > $i > $i ).
thf(func_def_1484,type,
sK1104: $i > $o ).
thf(func_def_1489,type,
sK1109: $i > $i > $i > $i > $i ).
thf(func_def_1490,type,
sK1110: $i > $i > $i > $i > $i ).
thf(func_def_1491,type,
sK1111: $i > $i > $i > $i > $i ).
thf(func_def_1492,type,
sK1112: $i > $i > $i > $i > $i ).
thf(func_def_1504,type,
sK1124: $i > $o ).
thf(func_def_1508,type,
sK1128: $i > $i ).
thf(func_def_1509,type,
sK1129: ( $i > $i ) > $i > $i > $i ).
thf(func_def_1510,type,
sK1130: $i > $i > ( $i > $i ) > $i ).
thf(func_def_1516,type,
sK1136: $i > $o ).
thf(func_def_1525,type,
ph1145:
!>[X0: $tType] : X0 ).
thf(f9357,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f9356,f4973]) ).
thf(f4973,plain,
( ( in @ sK1117 @ ( funcSet @ sK1119 @ sK1118 ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f3328]) ).
thf(f3328,plain,
( ( demorgan2a = $true )
& ( breln1unionprop = $true )
& ( notdallE = $true )
& ( setukpairinjL = $true )
& ( setunion__Cong = $true )
& ( doubleComplementSub2 = $true )
& ( binintersectEL = $true )
& ( prop2setI = $true )
& ( inPowerset = $true )
& ( subsetTI = $true )
& ( binunionTIRcontra = $true )
& ( eqbreln = $true )
& ( ex1I = $true )
& ( complementT_lem = $true )
& ( singletoninpowunion = $true )
& ( setadjoinAx = $true )
& ( contrasubsetT = $true )
& ( inIntersectImpInIntersectUnions = $true )
& ( singletoninpowerset = $true )
& ( breln1invI = $true )
& ( binunionT_lem = $true )
& ( woz2W = $true )
& ( funcGraphProp3 = $true )
& ( setext = $true )
& ( setextsub = $true )
& ( noeltsimpempty = $true )
& ( symdiffI1 = $true )
& ( setminusER = $true )
& ( nonemptyI1 = $true )
& ( kpairiskpair = $true )
& ( exuE1 = $true )
& ( powersetT_lem = $true )
& ( doubleComplementEq = $true )
& ( beta2 = $true )
& ( beta1 = $true )
& ( uniqinunit = $true )
& ( exu__Cong = $true )
& ( exuI2 = $true )
& ( exuI3 = $true )
& ( binunionRsub = $true )
& ( binunionTEcontra = $true )
& ( woz1_4 = $true )
& ( emptysetimpfalse = $true )
& ( funcImageSingleton = $true )
& ( upairinpowunion = $true )
& ( setminusLsub = $true )
& ( binunionIR = $true )
& ( demorgan2b2 = $true )
& ( binintersectI = $true )
& ( ifSingleton = $true )
& ( iffalse = $true )
& ( ex1E1 = $true )
& ( powersetsubset = $true )
& ( dpsetconstrERa = $true )
& ( intersectInPowersetIntersectUnions = $true )
& ( binintersectSubset3 = $true )
& ( setukpairinjR2 = $true )
& ( complementSubsetComplementIntersect = $true )
& ( binintersectSubset4 = $true )
& ( cartprodpairmemER = $true )
& ( ubforcartprodlem1 = $true )
& ( exuI1 = $true )
& ( sepSubset = $true )
& ( subsetI1 = $true )
& ( breln1all1 = $true )
& ( kpairsurjEq = $true )
& ( powersetTI1 = $true )
& ( inIntersectImpInUnion2 = $true )
& ( doubleComplementI1 = $true )
& ( nonemptyE1 = $true )
& ( quantDeMorgan4 = $true )
& ( woz13rule2 = $true )
& ( woz1_3 = $true )
& ( dsetconstr__Cong = $true )
& ( omegaIndAx = $true )
& ( binunionTILcontra = $true )
& ( complementTnotintersectT = $true )
& ( inComplementUnionImpNotIn1 = $true )
& ( setminusIRneg = $true )
& ( kfstpairEq = $true )
& ( cartprodsndin = $true )
& ( theprop = $true )
& ( breln1compE = $true )
& ( cartprodsndpairEq = $true )
& ( notinsingleton = $true )
& ( brelnall1 = $true )
& ( ap2apEq2 = $true )
& ( upairequniteq = $true )
& ( setadjoinSub2 = $true )
& ( funcext2 = $true )
& ( omegaSAx = $true )
& ( emptyI = $true )
& ( exuE2 = $true )
& ( powersetTE1 = $true )
& ( descrp = $true )
& ( setminusELneg = $true )
& ( eqinunit = $true )
& ( funcext = $true )
& ( lamp = $true )
& ( inCongP = $true )
& ( notinemptyset = $true )
& ( subPowSU = $true )
& ( infuncsetfunc = $true )
& ( ap2apEq1 = $true )
& ( breln1unionIL = $true )
& ( descr__Cong = $true )
& ( woz13rule3 = $true )
& ( ifp = $true )
& ( breln1unionEcases = $true )
& ( image1I = $true )
& ( ap2p = $true )
& ( emptyinPowerset = $true )
& ( powersetI1 = $true )
& ( doubleComplementE1 = $true )
& ( iffalseProp1 = $true )
& ( contraSubsetComplement = $true )
& ( subsetTrans = $true )
& ( omega0Ax = $true )
& ( vacuousDall = $true )
& ( quantDeMorgan2 = $true )
& ( breln1compI = $true )
& ( prop2setE = $true )
& ( setukpairinjR12 = $true )
& ( lam2lamEq = $true )
& ( complementInPowersetComplementIntersect = $true )
& ( breln1unionI = $true )
& ( binintersectSubset2 = $true )
& ( contrasubsetT1 = $true )
& ( iffalseProp2 = $true )
& ( lam2p = $true )
& ( brelnall2 = $true )
& ( notequalI1 = $true )
& ( upairsetE = $true )
& ( powersetI = $true )
& ( cartprodmempaircEq = $true )
& ( dpsetconstrSub = $true )
& ( symdiffE = $true )
& ( bs114d = $true )
& ( foundationAx = $true )
& ( demorgan1b = $true )
& ( setukpairIL = $true )
& ( in__Cong = $true )
& ( setukpairIR = $true )
& ( ubforcartprodlem3 = $true )
& ( demorgan2a2 = $true )
& ( singletonprop = $true )
& ( eqimpsubset2 = $true )
& ( notsubsetI = $true )
& ( cartprodpairin = $true )
& ( setadjoin__Cong = $true )
& ( subsetI2 = $true )
& ( subsetE = $true )
& ( binunionTE = $true )
& ( notdexE = $true )
& ( setunionE = $true )
& ( woz1_2 = $true )
& ( iftrueProp1 = $true )
& ( dpsetconstrEL1 = $true )
& ( demorgan1a = $true )
& ( eta2 = $true )
& ( dpsetconstrEL2 = $true )
& ( setextT = $true )
& ( setminusSubset1 = $true )
& ( cartprodfstin = $true )
& ( woz1_1 = $true )
& ( woz2A = $true )
& ( funcinfuncset = $true )
& ( subset2powerset = $true )
& ( cartprodpairsurjEq = $true )
& ( breln1SetBreln1 = $true )
& ( binintersectTELcontra = $true )
& ( cartprodpairmemEL = $true )
& ( setukpairinjR1 = $true )
& ( woz13rule1 = $true )
& ( setunionsingleton1 = $true )
& ( funcextLem = $true )
& ( setukpairinjR = $true )
& ( setextAx = $true )
& ( setoftrueEq = $true )
& ( image1Equiv = $true )
& ( kfstsingleton = $true )
& ( cartprodmempair = $true )
& ( complementUnionInPowersetComplement = $true )
& ( eqimpsubset1 = $true )
& ( setunionsingleton = $true )
& ( secondinupair = $true )
& ( woz1_5 = $true )
& ( powersetE = $true )
& ( subsetRefl = $true )
& ( woz13rule4 = $true )
& ( setadjoinSub = $true )
& ( binunionE = $true )
& ( quantDeMorgan1 = $true )
& ( complementTI1 = $true )
& ( funcGraphProp2 = $true )
& ( setunionAx = $true )
& ( breln1invE = $true )
& ( powersetE1 = $true )
& ( image1Ex = $true )
& ( ubforcartprodlem2 = $true )
& ( eqbreln1 = $true )
& ( doubleComplementSub1 = $true )
& ( contrasubsetT2 = $true )
& ( setminusI = $true )
& ( binintersectT_lem = $true )
& ( wellorderingAx = $true )
& ( nonemptyI = $true )
& ( setminusILneg = $true )
& ( demorgan2 = $true )
& ( eta1 = $true )
& ( breln1compprop = $true )
& ( singletonsswitch = $true )
& ( apProp = $true )
& ( funcGraphProp4 = $true )
& ( setminusSubset2 = $true )
& ( binintersectSubset5 = $true )
& ( emptysetE = $true )
& ( breln1invprop = $true )
& ( kpairp = $true )
& ( woz2B = $true )
& ( exuE3e = $true )
& ( choice2fnsingleton = $true )
& ( theeq = $true )
& ( emptysetAx = $true )
& ( setadjoinE = $true )
& ( complementTE1 = $true )
& ( image1Ex1 = $true )
& ( nonemptyImpWitness = $true )
& ( inComplementUnionImpInComplement1 = $true )
& ( symdiffI2 = $true )
& ( binintersectTERcontra = $true )
& ( setunionI = $true )
& ( omega__Cong = $true )
& ( setbeta = $true )
& ( powersetAx = $true )
& ( exuEu = $true )
& ( dsetconstrEL = $true )
& ( notequalI2 = $true )
& ( singletonsuniq = $true )
& ( upairset2IR = $true )
& ( setunionE2 = $true )
& ( setminusT_lem = $true )
& ( binintersectSubset1 = $true )
& ( demorgan2a1 = $true )
& ( iftrueorfalse = $true )
& ( inIntersectImpInUnion = $true )
& ( ksndpairEq = $true )
& ( setukpairinjL2 = $true )
& ( breln1compEex = $true )
& ( prop2set2propI = $true )
& ( upairsetIR = $true )
& ( powerset__Cong = $true )
& ( emptyinunitempty = $true )
& ( setukpairinjR11 = $true )
& ( emptysetsubset = $true )
& ( ksndsingleton = $true )
& ( emptyset__Cong = $true )
& ( dpsetconstrER = $true )
& ( ex1E2 = $true )
& ( upairset2E = $true )
& ( demorgan1 = $true )
& ( breln1unionCommutes = $true )
& ( setunionsingleton2 = $true )
& ( setOfPairsIsBReln = $true )
& ( singletonsubset = $true )
& ( subbreln1 = $true )
& ( binintersectRsub = $true )
& ( woz13rule0 = $true )
& ( ( in @ sK1117 @ ( funcSet @ sK1119 @ sK1118 ) )
= $true )
& ( $true
= ( injective @ sK1119 @ sK1118 @ sK1117 ) )
& ( ( in @ sK1117 @ ( injFuncSet @ sK1119 @ sK1118 ) )
!= $true )
& ( iftrueProp2 = $true )
& ( symdiffIneg1 = $true )
& ( setOfPairsIsBReln1 = $true )
& ( lamProp = $true )
& ( binunionEcases = $true )
& ( dsetconstrER = $true )
& ( sepInPowerset = $true )
& ( setminusERneg = $true )
& ( woz2Ex = $true )
& ( cartprodfstpairEq = $true )
& ( subsetE2 = $true )
& ( cartprodmempair1 = $true )
& ( complementImpComplementIntersect = $true )
& ( binunionIL = $true )
& ( setadjoinOr = $true )
& ( disjointsetsI1 = $true )
& ( breln1all2 = $true )
& ( ex1I2 = $true )
& ( subsetemptysetimpeq = $true )
& ( upairsetIL = $true )
& ( binintersectER = $true )
& ( binintersectLsub = $true )
& ( dsetconstrI = $true )
& ( dpsetconstrI = $true )
& ( complementTcontraSubset = $true )
& ( setminusEL = $true )
& ( replAx = $true )
& ( subbreln = $true )
& ( quantDeMorgan3 = $true )
& ( exuE3u = $true )
& ( binunionLsub = $true )
& ( demorgan2b = $true )
& ( emptyInPowerset = $true )
& ( app = $true )
& ( symdiffIneg2 = $true )
& ( setadjoinIL = $true )
& ( contrasubsetT3 = $true )
& ( iftrue = $true )
& ( funcGraphProp1 = $true )
& ( breln1unionIR = $true )
& ( breln1unionE = $true )
& ( setadjoinIR = $true )
& ( setukpairinjL1 = $true )
& ( upairsubunion = $true )
& ( image1E = $true )
& ( emptyE1 = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK1117,sK1118,sK1119])],[f1345,f3327]) ).
thf(f3327,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X0 @ ( funcSet @ X2 @ X1 ) ) )
& ( ( injective @ X2 @ X1 @ X0 )
= $true )
& ( ( in @ X0 @ ( injFuncSet @ X2 @ X1 ) )
!= $true ) )
=> ( ( ( in @ sK1117 @ ( funcSet @ sK1119 @ sK1118 ) )
= $true )
& ( $true
= ( injective @ sK1119 @ sK1118 @ sK1117 ) )
& ( ( in @ sK1117 @ ( injFuncSet @ sK1119 @ sK1118 ) )
!= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1345,plain,
( ( demorgan2a = $true )
& ( breln1unionprop = $true )
& ( notdallE = $true )
& ( setukpairinjL = $true )
& ( setunion__Cong = $true )
& ( doubleComplementSub2 = $true )
& ( binintersectEL = $true )
& ( prop2setI = $true )
& ( inPowerset = $true )
& ( subsetTI = $true )
& ( binunionTIRcontra = $true )
& ( eqbreln = $true )
& ( ex1I = $true )
& ( complementT_lem = $true )
& ( singletoninpowunion = $true )
& ( setadjoinAx = $true )
& ( contrasubsetT = $true )
& ( inIntersectImpInIntersectUnions = $true )
& ( singletoninpowerset = $true )
& ( breln1invI = $true )
& ( binunionT_lem = $true )
& ( woz2W = $true )
& ( funcGraphProp3 = $true )
& ( setext = $true )
& ( setextsub = $true )
& ( noeltsimpempty = $true )
& ( symdiffI1 = $true )
& ( setminusER = $true )
& ( nonemptyI1 = $true )
& ( kpairiskpair = $true )
& ( exuE1 = $true )
& ( powersetT_lem = $true )
& ( doubleComplementEq = $true )
& ( beta2 = $true )
& ( beta1 = $true )
& ( uniqinunit = $true )
& ( exu__Cong = $true )
& ( exuI2 = $true )
& ( exuI3 = $true )
& ( binunionRsub = $true )
& ( binunionTEcontra = $true )
& ( woz1_4 = $true )
& ( emptysetimpfalse = $true )
& ( funcImageSingleton = $true )
& ( upairinpowunion = $true )
& ( setminusLsub = $true )
& ( binunionIR = $true )
& ( demorgan2b2 = $true )
& ( binintersectI = $true )
& ( ifSingleton = $true )
& ( iffalse = $true )
& ( ex1E1 = $true )
& ( powersetsubset = $true )
& ( dpsetconstrERa = $true )
& ( intersectInPowersetIntersectUnions = $true )
& ( binintersectSubset3 = $true )
& ( setukpairinjR2 = $true )
& ( complementSubsetComplementIntersect = $true )
& ( binintersectSubset4 = $true )
& ( cartprodpairmemER = $true )
& ( ubforcartprodlem1 = $true )
& ( exuI1 = $true )
& ( sepSubset = $true )
& ( subsetI1 = $true )
& ( breln1all1 = $true )
& ( kpairsurjEq = $true )
& ( powersetTI1 = $true )
& ( inIntersectImpInUnion2 = $true )
& ( doubleComplementI1 = $true )
& ( nonemptyE1 = $true )
& ( quantDeMorgan4 = $true )
& ( woz13rule2 = $true )
& ( woz1_3 = $true )
& ( dsetconstr__Cong = $true )
& ( omegaIndAx = $true )
& ( binunionTILcontra = $true )
& ( complementTnotintersectT = $true )
& ( inComplementUnionImpNotIn1 = $true )
& ( setminusIRneg = $true )
& ( kfstpairEq = $true )
& ( cartprodsndin = $true )
& ( theprop = $true )
& ( breln1compE = $true )
& ( cartprodsndpairEq = $true )
& ( notinsingleton = $true )
& ( brelnall1 = $true )
& ( ap2apEq2 = $true )
& ( upairequniteq = $true )
& ( setadjoinSub2 = $true )
& ( funcext2 = $true )
& ( omegaSAx = $true )
& ( emptyI = $true )
& ( exuE2 = $true )
& ( powersetTE1 = $true )
& ( descrp = $true )
& ( setminusELneg = $true )
& ( eqinunit = $true )
& ( funcext = $true )
& ( lamp = $true )
& ( inCongP = $true )
& ( notinemptyset = $true )
& ( subPowSU = $true )
& ( infuncsetfunc = $true )
& ( ap2apEq1 = $true )
& ( breln1unionIL = $true )
& ( descr__Cong = $true )
& ( woz13rule3 = $true )
& ( ifp = $true )
& ( breln1unionEcases = $true )
& ( image1I = $true )
& ( ap2p = $true )
& ( emptyinPowerset = $true )
& ( powersetI1 = $true )
& ( doubleComplementE1 = $true )
& ( iffalseProp1 = $true )
& ( contraSubsetComplement = $true )
& ( subsetTrans = $true )
& ( omega0Ax = $true )
& ( vacuousDall = $true )
& ( quantDeMorgan2 = $true )
& ( breln1compI = $true )
& ( prop2setE = $true )
& ( setukpairinjR12 = $true )
& ( lam2lamEq = $true )
& ( complementInPowersetComplementIntersect = $true )
& ( breln1unionI = $true )
& ( binintersectSubset2 = $true )
& ( contrasubsetT1 = $true )
& ( iffalseProp2 = $true )
& ( lam2p = $true )
& ( brelnall2 = $true )
& ( notequalI1 = $true )
& ( upairsetE = $true )
& ( powersetI = $true )
& ( cartprodmempaircEq = $true )
& ( dpsetconstrSub = $true )
& ( symdiffE = $true )
& ( bs114d = $true )
& ( foundationAx = $true )
& ( demorgan1b = $true )
& ( setukpairIL = $true )
& ( in__Cong = $true )
& ( setukpairIR = $true )
& ( ubforcartprodlem3 = $true )
& ( demorgan2a2 = $true )
& ( singletonprop = $true )
& ( eqimpsubset2 = $true )
& ( notsubsetI = $true )
& ( cartprodpairin = $true )
& ( setadjoin__Cong = $true )
& ( subsetI2 = $true )
& ( subsetE = $true )
& ( binunionTE = $true )
& ( notdexE = $true )
& ( setunionE = $true )
& ( woz1_2 = $true )
& ( iftrueProp1 = $true )
& ( dpsetconstrEL1 = $true )
& ( demorgan1a = $true )
& ( eta2 = $true )
& ( dpsetconstrEL2 = $true )
& ( setextT = $true )
& ( setminusSubset1 = $true )
& ( cartprodfstin = $true )
& ( woz1_1 = $true )
& ( woz2A = $true )
& ( funcinfuncset = $true )
& ( subset2powerset = $true )
& ( cartprodpairsurjEq = $true )
& ( breln1SetBreln1 = $true )
& ( binintersectTELcontra = $true )
& ( cartprodpairmemEL = $true )
& ( setukpairinjR1 = $true )
& ( woz13rule1 = $true )
& ( setunionsingleton1 = $true )
& ( funcextLem = $true )
& ( setukpairinjR = $true )
& ( setextAx = $true )
& ( setoftrueEq = $true )
& ( image1Equiv = $true )
& ( kfstsingleton = $true )
& ( cartprodmempair = $true )
& ( complementUnionInPowersetComplement = $true )
& ( eqimpsubset1 = $true )
& ( setunionsingleton = $true )
& ( secondinupair = $true )
& ( woz1_5 = $true )
& ( powersetE = $true )
& ( subsetRefl = $true )
& ( woz13rule4 = $true )
& ( setadjoinSub = $true )
& ( binunionE = $true )
& ( quantDeMorgan1 = $true )
& ( complementTI1 = $true )
& ( funcGraphProp2 = $true )
& ( setunionAx = $true )
& ( breln1invE = $true )
& ( powersetE1 = $true )
& ( image1Ex = $true )
& ( ubforcartprodlem2 = $true )
& ( eqbreln1 = $true )
& ( doubleComplementSub1 = $true )
& ( contrasubsetT2 = $true )
& ( setminusI = $true )
& ( binintersectT_lem = $true )
& ( wellorderingAx = $true )
& ( nonemptyI = $true )
& ( setminusILneg = $true )
& ( demorgan2 = $true )
& ( eta1 = $true )
& ( breln1compprop = $true )
& ( singletonsswitch = $true )
& ( apProp = $true )
& ( funcGraphProp4 = $true )
& ( setminusSubset2 = $true )
& ( binintersectSubset5 = $true )
& ( emptysetE = $true )
& ( breln1invprop = $true )
& ( kpairp = $true )
& ( woz2B = $true )
& ( exuE3e = $true )
& ( choice2fnsingleton = $true )
& ( theeq = $true )
& ( emptysetAx = $true )
& ( setadjoinE = $true )
& ( complementTE1 = $true )
& ( image1Ex1 = $true )
& ( nonemptyImpWitness = $true )
& ( inComplementUnionImpInComplement1 = $true )
& ( symdiffI2 = $true )
& ( binintersectTERcontra = $true )
& ( setunionI = $true )
& ( omega__Cong = $true )
& ( setbeta = $true )
& ( powersetAx = $true )
& ( exuEu = $true )
& ( dsetconstrEL = $true )
& ( notequalI2 = $true )
& ( singletonsuniq = $true )
& ( upairset2IR = $true )
& ( setunionE2 = $true )
& ( setminusT_lem = $true )
& ( binintersectSubset1 = $true )
& ( demorgan2a1 = $true )
& ( iftrueorfalse = $true )
& ( inIntersectImpInUnion = $true )
& ( ksndpairEq = $true )
& ( setukpairinjL2 = $true )
& ( breln1compEex = $true )
& ( prop2set2propI = $true )
& ( upairsetIR = $true )
& ( powerset__Cong = $true )
& ( emptyinunitempty = $true )
& ( setukpairinjR11 = $true )
& ( emptysetsubset = $true )
& ( ksndsingleton = $true )
& ( emptyset__Cong = $true )
& ( dpsetconstrER = $true )
& ( ex1E2 = $true )
& ( upairset2E = $true )
& ( demorgan1 = $true )
& ( breln1unionCommutes = $true )
& ( setunionsingleton2 = $true )
& ( setOfPairsIsBReln = $true )
& ( singletonsubset = $true )
& ( subbreln1 = $true )
& ( binintersectRsub = $true )
& ( woz13rule0 = $true )
& ? [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X0 @ ( funcSet @ X2 @ X1 ) ) )
& ( ( injective @ X2 @ X1 @ X0 )
= $true )
& ( ( in @ X0 @ ( injFuncSet @ X2 @ X1 ) )
!= $true ) )
& ( iftrueProp2 = $true )
& ( symdiffIneg1 = $true )
& ( setOfPairsIsBReln1 = $true )
& ( lamProp = $true )
& ( binunionEcases = $true )
& ( dsetconstrER = $true )
& ( sepInPowerset = $true )
& ( setminusERneg = $true )
& ( woz2Ex = $true )
& ( cartprodfstpairEq = $true )
& ( subsetE2 = $true )
& ( cartprodmempair1 = $true )
& ( complementImpComplementIntersect = $true )
& ( binunionIL = $true )
& ( setadjoinOr = $true )
& ( disjointsetsI1 = $true )
& ( breln1all2 = $true )
& ( ex1I2 = $true )
& ( subsetemptysetimpeq = $true )
& ( upairsetIL = $true )
& ( binintersectER = $true )
& ( binintersectLsub = $true )
& ( dsetconstrI = $true )
& ( dpsetconstrI = $true )
& ( complementTcontraSubset = $true )
& ( setminusEL = $true )
& ( replAx = $true )
& ( subbreln = $true )
& ( quantDeMorgan3 = $true )
& ( exuE3u = $true )
& ( binunionLsub = $true )
& ( demorgan2b = $true )
& ( emptyInPowerset = $true )
& ( app = $true )
& ( symdiffIneg2 = $true )
& ( setadjoinIL = $true )
& ( contrasubsetT3 = $true )
& ( iftrue = $true )
& ( funcGraphProp1 = $true )
& ( breln1unionIR = $true )
& ( breln1unionE = $true )
& ( setadjoinIR = $true )
& ( setukpairinjL1 = $true )
& ( upairsubunion = $true )
& ( image1E = $true )
& ( emptyE1 = $true ) ),
inference(flattening,[status(thm)],[f1344]) ).
thf(f1344,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X0 @ ( injFuncSet @ X2 @ X1 ) )
!= $true )
& ( ( injective @ X2 @ X1 @ X0 )
= $true )
& ( $true
= ( in @ X0 @ ( funcSet @ X2 @ X1 ) ) ) )
& ( 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)],[f443]) ).
thf(f443,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 )
=> ! [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X0 @ ( funcSet @ X2 @ X1 ) ) )
=> ( ( ( injective @ X2 @ X1 @ X0 )
= $true )
=> ( ( in @ X0 @ ( injFuncSet @ X2 @ X1 ) )
= $true ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f442]) ).
thf(f442,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
=> ! [X0: $i,X1: $i,X2: $i] :
( ( in @ X0 @ ( funcSet @ X2 @ X1 ) )
=> ( ( injective @ X2 @ X1 @ X0 )
=> ( in @ X0 @ ( injFuncSet @ X2 @ X1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f319]) ).
thf(f319,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
=> ! [X12: $i,X4: $i,X3: $i] :
( ( in @ X12 @ ( funcSet @ X3 @ X4 ) )
=> ( ( injective @ X3 @ X4 @ X12 )
=> ( in @ X12 @ ( injFuncSet @ X3 @ X4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f318]) ).
thf(f318,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
=> ! [X12: $i,X4: $i,X3: $i] :
( ( in @ X12 @ ( funcSet @ X3 @ X4 ) )
=> ( ( injective @ X3 @ X4 @ X12 )
=> ( in @ X12 @ ( injFuncSet @ X3 @ X4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.IMUvashtid/DTF2THF_11553.p',injFuncInInjFuncSet) ).
thf(f9356,plain,
( ( in @ sK1117 @ ( funcSet @ sK1119 @ sK1118 ) )
!= $true ),
inference(subsumption_resolution,[status(thm)],[f9355,f4972]) ).
thf(f4972,plain,
( $true
= ( injective @ sK1119 @ sK1118 @ sK1117 ) ),
inference(cnf_transformation,[status(thm)],[f3328]) ).
thf(f9355,plain,
( ( $true
!= ( injective @ sK1119 @ sK1118 @ sK1117 ) )
| ( ( in @ sK1117 @ ( funcSet @ sK1119 @ sK1118 ) )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f9352]) ).
thf(f9352,plain,
( ( ( in @ sK1117 @ ( funcSet @ sK1119 @ sK1118 ) )
!= $true )
| ( $true
!= ( injective @ sK1119 @ sK1118 @ sK1117 ) )
| ( $true != $true ) ),
inference(superposition,[status(thm)],[f6706,f6531]) ).
thf(f6531,plain,
! [X3: $i > $o,X4: $i,X5: $i] :
( ( $true
= ( in @ X5 @ ( dsetconstr @ X4 @ X3 ) ) )
| ( $true
!= ( X3 @ X5 ) )
| ( ( in @ X5 @ X4 )
!= $true ) ),
inference(beta_eta_normalization,[status(thm)],[f6530]) ).
thf(f6530,plain,
! [X3: $i > $o,X4: $i,X5: $i] :
( ( $true
= ( in @ X5
@ ( dsetconstr @ X4
@ ^ [Y0: $i] : ( X3 @ Y0 ) ) ) )
| ( ( in @ X5 @ X4 )
!= $true )
| ( $true
!= ( X3 @ X5 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f6136]) ).
thf(f6136,plain,
! [X3: $i > $o,X4: $i,X5: $i] :
( ( $true != $true )
| ( $true
= ( in @ X5
@ ( dsetconstr @ X4
@ ^ [Y0: $i] : ( X3 @ Y0 ) ) ) )
| ( ( in @ X5 @ X4 )
!= $true )
| ( $true
!= ( X3 @ X5 ) ) ),
inference(definition_unfolding,[status(thm)],[f4503,f4948]) ).
thf(f4948,plain,
dsetconstrI = $true,
inference(cnf_transformation,[status(thm)],[f3328]) ).
thf(f4503,plain,
! [X3: $i > $o,X4: $i,X5: $i] :
( ( $true
!= ( X3 @ X5 ) )
| ( ( in @ X5 @ X4 )
!= $true )
| ( $true
= ( in @ X5
@ ( dsetconstr @ X4
@ ^ [Y0: $i] : ( X3 @ Y0 ) ) ) )
| ( dsetconstrI != $true ) ),
inference(cnf_transformation,[status(thm)],[f2848]) ).
thf(f2848,plain,
( ( ( dsetconstrI = $true )
| ( ( $true
= ( sK851 @ sK853 ) )
& ( ( in @ sK853 @ sK852 )
= $true )
& ( ( in @ sK853
@ ( dsetconstr @ sK852
@ ^ [Y0: $i] : ( sK851 @ Y0 ) ) )
!= $true ) ) )
& ( ! [X3: $i > $o,X4: $i,X5: $i] :
( ( $true
!= ( X3 @ X5 ) )
| ( ( in @ X5 @ X4 )
!= $true )
| ( $true
= ( in @ X5
@ ( dsetconstr @ X4
@ ^ [Y0: $i] : ( X3 @ Y0 ) ) ) ) )
| ( dsetconstrI != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK851,sK852,sK853])],[f2846,f2847]) ).
thf(f2847,plain,
( ? [X0: $i > $o,X1: $i,X2: $i] :
( ( ( X0 @ X2 )
= $true )
& ( $true
= ( in @ X2 @ X1 ) )
& ( ( in @ X2
@ ( dsetconstr @ X1
@ ^ [Y0: $i] : ( X0 @ Y0 ) ) )
!= $true ) )
=> ( ( $true
= ( sK851 @ sK853 ) )
& ( ( in @ sK853 @ sK852 )
= $true )
& ( ( in @ sK853
@ ( dsetconstr @ sK852
@ ^ [Y0: $i] : ( sK851 @ Y0 ) ) )
!= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f2846,plain,
( ( ( dsetconstrI = $true )
| ? [X0: $i > $o,X1: $i,X2: $i] :
( ( ( X0 @ X2 )
= $true )
& ( $true
= ( in @ X2 @ X1 ) )
& ( ( in @ X2
@ ( dsetconstr @ X1
@ ^ [Y0: $i] : ( X0 @ Y0 ) ) )
!= $true ) ) )
& ( ! [X3: $i > $o,X4: $i,X5: $i] :
( ( $true
!= ( X3 @ X5 ) )
| ( ( in @ X5 @ X4 )
!= $true )
| ( $true
= ( in @ X5
@ ( dsetconstr @ X4
@ ^ [Y0: $i] : ( X3 @ Y0 ) ) ) ) )
| ( dsetconstrI != $true ) ) ),
inference(rectify,[status(thm)],[f2845]) ).
thf(f2845,plain,
( ( ( dsetconstrI = $true )
| ? [X0: $i > $o,X1: $i,X2: $i] :
( ( ( X0 @ X2 )
= $true )
& ( $true
= ( in @ X2 @ X1 ) )
& ( ( in @ X2
@ ( dsetconstr @ X1
@ ^ [Y0: $i] : ( X0 @ Y0 ) ) )
!= $true ) ) )
& ( ! [X0: $i > $o,X1: $i,X2: $i] :
( ( ( X0 @ X2 )
!= $true )
| ( $true
!= ( in @ X2 @ X1 ) )
| ( ( in @ X2
@ ( dsetconstr @ X1
@ ^ [Y0: $i] : ( X0 @ Y0 ) ) )
= $true ) )
| ( dsetconstrI != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f1180]) ).
thf(f1180,plain,
( ( dsetconstrI = $true )
<=> ! [X0: $i > $o,X1: $i,X2: $i] :
( ( ( X0 @ X2 )
!= $true )
| ( $true
!= ( in @ X2 @ X1 ) )
| ( ( in @ X2
@ ( dsetconstr @ X1
@ ^ [Y0: $i] : ( X0 @ Y0 ) ) )
= $true ) ) ),
inference(flattening,[status(thm)],[f1179]) ).
thf(f1179,plain,
( ! [X0: $i > $o,X1: $i,X2: $i] :
( ( ( in @ X2
@ ( dsetconstr @ X1
@ ^ [Y0: $i] : ( X0 @ Y0 ) ) )
= $true )
| ( ( X0 @ X2 )
!= $true )
| ( $true
!= ( in @ X2 @ X1 ) ) )
<=> ( dsetconstrI = $true ) ),
inference(ennf_transformation,[status(thm)],[f600]) ).
thf(f600,plain,
( ! [X0: $i > $o,X1: $i,X2: $i] :
( ( $true
= ( in @ X2 @ X1 ) )
=> ( ( ( X0 @ X2 )
= $true )
=> ( ( in @ X2
@ ( dsetconstr @ X1
@ ^ [Y0: $i] : ( X0 @ Y0 ) ) )
= $true ) ) )
<=> ( dsetconstrI = $true ) ),
inference(fool_elimination,[status(thm)],[f599]) ).
thf(f599,plain,
( dsetconstrI
= ( ! [X0: $i > $o,X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( X0 @ X2 )
=> ( in @ X2
@ ( dsetconstr @ X1
@ ^ [X3: $i] : ( X0 @ X3 ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f14]) ).
thf(f14,axiom,
( dsetconstrI
= ( ! [X0: $i > $o,X3: $i,X1: $i] :
( ( in @ X1 @ X3 )
=> ( ( X0 @ X1 )
=> ( in @ X1
@ ( dsetconstr @ X3
@ ^ [X2: $i] : ( X0 @ X2 ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.IMUvashtid/DTF2THF_11553.p',dsetconstrI) ).
thf(f6706,plain,
( $true
!= ( in @ sK1117 @ ( dsetconstr @ ( funcSet @ sK1119 @ sK1118 ) @ ( injective @ sK1119 @ sK1118 ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f6411]) ).
thf(f6411,plain,
( ( in @ sK1117
@ ( ^ [Y0: $i,Y1: $i] :
( dsetconstr @ ( funcSet @ Y0 @ Y1 )
@ ^ [Y2: $i] : ( injective @ Y0 @ Y1 @ Y2 ) )
@ sK1119
@ sK1118 ) )
!= $true ),
inference(definition_unfolding,[status(thm)],[f4971,f4448]) ).
thf(f4448,plain,
( injFuncSet
= ( ^ [Y0: $i,Y1: $i] :
( dsetconstr @ ( funcSet @ Y0 @ Y1 )
@ ^ [Y2: $i] : ( injective @ Y0 @ Y1 @ Y2 ) ) ) ),
inference(cnf_transformation,[status(thm)],[f892]) ).
thf(f892,plain,
( injFuncSet
= ( ^ [Y0: $i,Y1: $i] :
( dsetconstr @ ( funcSet @ Y0 @ Y1 )
@ ^ [Y2: $i] : ( injective @ Y0 @ Y1 @ Y2 ) ) ) ),
inference(fool_elimination,[status(thm)],[f891]) ).
thf(f891,plain,
( ( ^ [X0: $i,X1: $i] :
( dsetconstr @ ( funcSet @ X0 @ X1 )
@ ^ [X2: $i] : ( injective @ X0 @ X1 @ X2 ) ) )
= injFuncSet ),
inference(rectify,[status(thm)],[f317]) ).
thf(f317,axiom,
( ( ^ [X3: $i,X4: $i] :
( dsetconstr @ ( funcSet @ X3 @ X4 )
@ ^ [X12: $i] : ( injective @ X3 @ X4 @ X12 ) ) )
= injFuncSet ),
file('/export/starexec/sandbox/tmp/tmp.IMUvashtid/DTF2THF_11553.p',injFuncSet) ).
thf(f4971,plain,
( ( in @ sK1117 @ ( injFuncSet @ sK1119 @ sK1118 ) )
!= $true ),
inference(cnf_transformation,[status(thm)],[f3328]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.06 % Problem : SEU798^1 : TPTP v9.2.1. Released v3.7.0.
% 0.00/0.07 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.06/0.25 % Computer : n024.cluster.edu
% 0.06/0.25 % Model : x86_64 x86_64
% 0.06/0.25 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.06/0.25 % Memory : 8042.1875MB
% 0.06/0.25 % OS : Linux 3.10.0-693.el7.x86_64
% 0.06/0.25 % CPULimit : 300
% 0.06/0.25 % WCLimit : 300
% 0.06/0.25 % DateTime : Tue Feb 24 07:34:10 EST 2026
% 0.06/0.25 % CPUTime :
% 0.06/0.25 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.06/0.31 ---- Original DTF file ---
% 0.06/0.32 thf(spec,logic,$$dhol).
% 0.06/0.32 %------------------------------------------------------------------------------
% 0.06/0.32 % File : SEU798^1 : TPTP v9.2.1. Released v3.7.0.
% 0.06/0.32 % Domain : Set Theory
% 0.06/0.32 % Problem : More about Functions - Injective Functions
% 0.06/0.32 % Version : Especial.
% 0.06/0.32 % English : (! A:i.! B:i.! f:i.in f (funcSet A B) -> injective A B f ->
% 0.06/0.32 % in f (injFuncSet A B))
% 0.06/0.32
% 0.06/0.32 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.06/0.32 % Source : [Bro08]
% 0.06/0.32 % Names : ZFC300g [Bro08]
% 0.06/0.32
% 0.06/0.32 % Status : Theorem
% 0.06/0.32 % 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, 0.71 v5.5.0, 0.83 v5.4.0, 0.60 v5.2.0, 1.00 v3.7.0
% 0.06/0.32 % Syntax : Number of formulae : 683 ( 317 unt; 365 typ; 317 def)
% 0.06/0.32 % Number of atoms : 1956 ( 457 equ; 0 cnn)
% 0.06/0.32 % Maximal formula atoms : 317 ( 6 avg)
% 0.06/0.32 % Number of connectives : 4307 ( 73 ~; 14 |; 53 &;3136 @)
% 0.06/0.32 % ( 17 <=>;1014 =>; 0 <=; 0 <~>)
% 0.06/0.32 % Maximal formula depth : 323 ( 2 avg)
% 0.06/0.32 % Number of types : 2 ( 0 usr)
% 0.06/0.32 % Number of type conns : 170 ( 170 >; 0 *; 0 +; 0 <<)
% 0.06/0.32 % Number of symbols : 368 ( 365 usr; 318 con; 0-4 aty)
% 0.06/0.32 % Number of variables : 1124 ( 78 ^;1003 !; 43 ?;1124 :)
% 0.06/0.32 % SPC : TH0_THM_EQU_NAR
% 0.06/0.32
% 0.06/0.32 % Comments : http://mathgate.info/detsetitem.php?id=423
% 0.06/0.32 %------------------------------------------------------------------------------
% 0.06/0.32 thf(in_type,type,
% 0.06/0.32 in: $i > $i > $o ).
% 0.06/0.32
% 0.06/0.32 thf(exu_type,type,
% 0.06/0.32 exu: ( $i > $o ) > $o ).
% 0.06/0.32
% 0.06/0.32 thf(exu,definition,
% 0.06/0.32 ( exu
% 0.06/0.32 = ( ^ [Xphi: $i > $o] :
% 0.06/0.32 ? [Xx: $i] :
% 0.06/0.32 ( ( Xphi @ Xx )
% 0.06/0.32 & ! [Xy: $i] :
% 0.06/0.32 ( ( Xphi @ Xy )
% 0.06/0.32 => ( Xx = Xy ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(setextAx_type,type,
% 0.06/0.32 setextAx: $o ).
% 0.06/0.32
% 0.06/0.32 thf(setextAx,definition,
% 0.06/0.32 ( setextAx
% 0.06/0.32 = ( ! [A: $i,B: $i] :
% 0.06/0.32 ( ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 <=> ( in @ Xx @ B ) )
% 0.06/0.32 => ( A = B ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(emptyset_type,type,
% 0.06/0.32 emptyset: $i ).
% 0.06/0.32
% 0.06/0.32 thf(emptysetAx_type,type,
% 0.06/0.32 emptysetAx: $o ).
% 0.06/0.32
% 0.06/0.32 thf(emptysetAx,definition,
% 0.06/0.32 ( emptysetAx
% 0.06/0.32 = ( ! [Xx: $i] :
% 0.06/0.32 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(setadjoin_type,type,
% 0.06/0.32 setadjoin: $i > $i > $i ).
% 0.06/0.32
% 0.06/0.32 thf(setadjoinAx_type,type,
% 0.06/0.32 setadjoinAx: $o ).
% 0.06/0.32
% 0.06/0.32 thf(setadjoinAx,definition,
% 0.06/0.32 ( setadjoinAx
% 0.06/0.32 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.06/0.32 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.06/0.32 <=> ( ( Xy = Xx )
% 0.06/0.32 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(powerset_type,type,
% 0.06/0.32 powerset: $i > $i ).
% 0.06/0.32
% 0.06/0.32 thf(powersetAx_type,type,
% 0.06/0.32 powersetAx: $o ).
% 0.06/0.32
% 0.06/0.32 thf(powersetAx,definition,
% 0.06/0.32 ( powersetAx
% 0.06/0.32 = ( ! [A: $i,B: $i] :
% 0.06/0.32 ( ( in @ B @ ( powerset @ A ) )
% 0.06/0.32 <=> ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ B )
% 0.06/0.32 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(setunion_type,type,
% 0.06/0.32 setunion: $i > $i ).
% 0.06/0.32
% 0.06/0.32 thf(setunionAx_type,type,
% 0.06/0.32 setunionAx: $o ).
% 0.06/0.32
% 0.06/0.32 thf(setunionAx,definition,
% 0.06/0.32 ( setunionAx
% 0.06/0.32 = ( ! [A: $i,Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ ( setunion @ A ) )
% 0.06/0.32 <=> ? [B: $i] :
% 0.06/0.32 ( ( in @ Xx @ B )
% 0.06/0.32 & ( in @ B @ A ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(omega_type,type,
% 0.06/0.32 omega: $i ).
% 0.06/0.32
% 0.06/0.32 thf(omega0Ax_type,type,
% 0.06/0.32 omega0Ax: $o ).
% 0.06/0.32
% 0.06/0.32 thf(omega0Ax,definition,
% 0.06/0.32 ( omega0Ax
% 0.06/0.32 = ( in @ emptyset @ omega ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(omegaSAx_type,type,
% 0.06/0.32 omegaSAx: $o ).
% 0.06/0.32
% 0.06/0.32 thf(omegaSAx,definition,
% 0.06/0.32 ( omegaSAx
% 0.06/0.32 = ( ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ omega )
% 0.06/0.32 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(omegaIndAx_type,type,
% 0.06/0.32 omegaIndAx: $o ).
% 0.06/0.32
% 0.06/0.32 thf(omegaIndAx,definition,
% 0.06/0.32 ( omegaIndAx
% 0.06/0.32 = ( ! [A: $i] :
% 0.06/0.32 ( ( ( in @ emptyset @ A )
% 0.06/0.32 & ! [Xx: $i] :
% 0.06/0.32 ( ( ( in @ Xx @ omega )
% 0.06/0.32 & ( in @ Xx @ A ) )
% 0.06/0.32 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.06/0.32 => ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ omega )
% 0.06/0.32 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(replAx_type,type,
% 0.06/0.32 replAx: $o ).
% 0.06/0.32
% 0.06/0.32 thf(replAx,definition,
% 0.06/0.32 ( replAx
% 0.06/0.32 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.06/0.32 ( ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ( exu
% 0.06/0.32 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.06/0.32 => ? [B: $i] :
% 0.06/0.32 ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ B )
% 0.06/0.32 <=> ? [Xy: $i] :
% 0.06/0.32 ( ( in @ Xy @ A )
% 0.06/0.32 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(foundationAx_type,type,
% 0.06/0.32 foundationAx: $o ).
% 0.06/0.32
% 0.06/0.32 thf(foundationAx,definition,
% 0.06/0.32 ( foundationAx
% 0.06/0.32 = ( ! [A: $i] :
% 0.06/0.32 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.06/0.32 => ? [B: $i] :
% 0.06/0.32 ( ( in @ B @ A )
% 0.06/0.32 & ~ ? [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ B )
% 0.06/0.32 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(wellorderingAx_type,type,
% 0.06/0.32 wellorderingAx: $o ).
% 0.06/0.32
% 0.06/0.32 thf(wellorderingAx,definition,
% 0.06/0.32 ( wellorderingAx
% 0.06/0.32 = ( ! [A: $i] :
% 0.06/0.32 ? [B: $i] :
% 0.06/0.32 ( ! [C: $i] :
% 0.06/0.32 ( ( in @ C @ B )
% 0.06/0.32 => ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ C )
% 0.06/0.32 => ( in @ Xx @ A ) ) )
% 0.06/0.32 & ! [Xx: $i,Xy: $i] :
% 0.06/0.32 ( ( ( in @ Xx @ A )
% 0.06/0.32 & ( in @ Xy @ A ) )
% 0.06/0.32 => ( ! [C: $i] :
% 0.06/0.32 ( ( in @ C @ B )
% 0.06/0.32 => ( ( in @ Xx @ C )
% 0.06/0.32 <=> ( in @ Xy @ C ) ) )
% 0.06/0.32 => ( Xx = Xy ) ) )
% 0.06/0.32 & ! [C: $i,D: $i] :
% 0.06/0.32 ( ( ( in @ C @ B )
% 0.06/0.32 & ( in @ D @ B ) )
% 0.06/0.32 => ( ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ C )
% 0.06/0.32 => ( in @ Xx @ D ) )
% 0.06/0.32 | ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ D )
% 0.06/0.32 => ( in @ Xx @ C ) ) ) )
% 0.06/0.32 & ! [C: $i] :
% 0.06/0.32 ( ( ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ C )
% 0.06/0.32 => ( in @ Xx @ A ) )
% 0.06/0.32 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.06/0.32 => ? [D: $i,Xx: $i] :
% 0.06/0.32 ( ( in @ D @ B )
% 0.06/0.32 & ( in @ Xx @ C )
% 0.06/0.32 & ~ ? [Xy: $i] :
% 0.06/0.32 ( ( in @ Xy @ D )
% 0.06/0.32 & ( in @ Xy @ C ) )
% 0.06/0.32 & ! [E: $i] :
% 0.06/0.32 ( ( in @ E @ B )
% 0.06/0.32 => ( ! [Xy: $i] :
% 0.06/0.32 ( ( in @ Xy @ E )
% 0.06/0.32 => ( in @ Xy @ D ) )
% 0.06/0.32 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(descr_type,type,
% 0.06/0.32 descr: ( $i > $o ) > $i ).
% 0.06/0.32
% 0.06/0.32 thf(descrp_type,type,
% 0.06/0.32 descrp: $o ).
% 0.06/0.32
% 0.06/0.32 thf(descrp,definition,
% 0.06/0.32 ( descrp
% 0.06/0.32 = ( ! [Xphi: $i > $o] :
% 0.06/0.32 ( ( exu
% 0.06/0.32 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.06/0.32 => ( Xphi
% 0.06/0.32 @ ( descr
% 0.06/0.32 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(dsetconstr_type,type,
% 0.06/0.32 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.06/0.32
% 0.06/0.32 thf(dsetconstrI_type,type,
% 0.06/0.32 dsetconstrI: $o ).
% 0.06/0.32
% 0.06/0.32 thf(dsetconstrI,definition,
% 0.06/0.32 ( dsetconstrI
% 0.06/0.32 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ( ( Xphi @ Xx )
% 0.06/0.32 => ( in @ Xx
% 0.06/0.32 @ ( dsetconstr @ A
% 0.06/0.32 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(dsetconstrEL_type,type,
% 0.06/0.32 dsetconstrEL: $o ).
% 0.06/0.32
% 0.06/0.32 thf(dsetconstrEL,definition,
% 0.06/0.32 ( dsetconstrEL
% 0.06/0.32 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.06/0.32 ( ( in @ Xx
% 0.06/0.32 @ ( dsetconstr @ A
% 0.06/0.32 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.06/0.32 => ( in @ Xx @ A ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(dsetconstrER_type,type,
% 0.06/0.32 dsetconstrER: $o ).
% 0.06/0.32
% 0.06/0.32 thf(dsetconstrER,definition,
% 0.06/0.32 ( dsetconstrER
% 0.06/0.32 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.06/0.32 ( ( in @ Xx
% 0.06/0.32 @ ( dsetconstr @ A
% 0.06/0.32 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.06/0.32 => ( Xphi @ Xx ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(exuE1_type,type,
% 0.06/0.32 exuE1: $o ).
% 0.06/0.32
% 0.06/0.32 thf(exuE1,definition,
% 0.06/0.32 ( exuE1
% 0.06/0.32 = ( ! [Xphi: $i > $o] :
% 0.06/0.32 ( ( exu
% 0.06/0.32 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.06/0.32 => ? [Xx: $i] :
% 0.06/0.32 ( ( Xphi @ Xx )
% 0.06/0.32 & ! [Xy: $i] :
% 0.06/0.32 ( ( Xphi @ Xy )
% 0.06/0.32 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(prop2set_type,type,
% 0.06/0.32 prop2set: $o > $i ).
% 0.06/0.32
% 0.06/0.32 thf(prop2setE_type,type,
% 0.06/0.32 prop2setE: $o ).
% 0.06/0.32
% 0.06/0.32 thf(prop2setE,definition,
% 0.06/0.32 ( prop2setE
% 0.06/0.32 = ( ! [Xphi: $o,Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.06/0.32 => Xphi ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(emptysetE_type,type,
% 0.06/0.32 emptysetE: $o ).
% 0.06/0.32
% 0.06/0.32 thf(emptysetE,definition,
% 0.06/0.32 ( emptysetE
% 0.06/0.32 = ( ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ emptyset )
% 0.06/0.32 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(emptysetimpfalse_type,type,
% 0.06/0.32 emptysetimpfalse: $o ).
% 0.06/0.32
% 0.06/0.32 thf(emptysetimpfalse,definition,
% 0.06/0.32 ( emptysetimpfalse
% 0.06/0.32 = ( ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ emptyset )
% 0.06/0.32 => $false ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(notinemptyset_type,type,
% 0.06/0.32 notinemptyset: $o ).
% 0.06/0.32
% 0.06/0.32 thf(notinemptyset,definition,
% 0.06/0.32 ( notinemptyset
% 0.06/0.32 = ( ! [Xx: $i] :
% 0.06/0.32 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(exuE3e_type,type,
% 0.06/0.32 exuE3e: $o ).
% 0.06/0.32
% 0.06/0.32 thf(exuE3e,definition,
% 0.06/0.32 ( exuE3e
% 0.06/0.32 = ( ! [Xphi: $i > $o] :
% 0.06/0.32 ( ( exu
% 0.06/0.32 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.06/0.32 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(setext_type,type,
% 0.06/0.32 setext: $o ).
% 0.06/0.32
% 0.06/0.32 thf(setext,definition,
% 0.06/0.32 ( setext
% 0.06/0.32 = ( ! [A: $i,B: $i] :
% 0.06/0.32 ( ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ( in @ Xx @ B ) )
% 0.06/0.32 => ( ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ B )
% 0.06/0.32 => ( in @ Xx @ A ) )
% 0.06/0.32 => ( A = B ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(emptyI_type,type,
% 0.06/0.32 emptyI: $o ).
% 0.06/0.32
% 0.06/0.32 thf(emptyI,definition,
% 0.06/0.32 ( emptyI
% 0.06/0.32 = ( ! [A: $i] :
% 0.06/0.32 ( ! [Xx: $i] :
% 0.06/0.32 ~ ( in @ Xx @ A )
% 0.06/0.32 => ( A = emptyset ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(noeltsimpempty_type,type,
% 0.06/0.32 noeltsimpempty: $o ).
% 0.06/0.32
% 0.06/0.32 thf(noeltsimpempty,definition,
% 0.06/0.32 ( noeltsimpempty
% 0.06/0.32 = ( ! [A: $i] :
% 0.06/0.32 ( ! [Xx: $i] :
% 0.06/0.32 ~ ( in @ Xx @ A )
% 0.06/0.32 => ( A = emptyset ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(setbeta_type,type,
% 0.06/0.32 setbeta: $o ).
% 0.06/0.32
% 0.06/0.32 thf(setbeta,definition,
% 0.06/0.32 ( setbeta
% 0.06/0.32 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ( ( in @ Xx
% 0.06/0.32 @ ( dsetconstr @ A
% 0.06/0.32 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.06/0.32 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(nonempty_type,type,
% 0.06/0.32 nonempty: $i > $o ).
% 0.06/0.32
% 0.06/0.32 thf(nonempty,definition,
% 0.06/0.32 ( nonempty
% 0.06/0.32 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(nonemptyE1_type,type,
% 0.06/0.32 nonemptyE1: $o ).
% 0.06/0.32
% 0.06/0.32 thf(nonemptyE1,definition,
% 0.06/0.32 ( nonemptyE1
% 0.06/0.32 = ( ! [A: $i] :
% 0.06/0.32 ( ( nonempty @ A )
% 0.06/0.32 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(nonemptyI_type,type,
% 0.06/0.32 nonemptyI: $o ).
% 0.06/0.32
% 0.06/0.32 thf(nonemptyI,definition,
% 0.06/0.32 ( nonemptyI
% 0.06/0.32 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ( ( Xphi @ Xx )
% 0.06/0.32 => ( nonempty
% 0.06/0.32 @ ( dsetconstr @ A
% 0.06/0.32 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(nonemptyI1_type,type,
% 0.06/0.32 nonemptyI1: $o ).
% 0.06/0.32
% 0.06/0.32 thf(nonemptyI1,definition,
% 0.06/0.32 ( nonemptyI1
% 0.06/0.32 = ( ! [A: $i] :
% 0.06/0.32 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.06/0.32 => ( nonempty @ A ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(setadjoinIL_type,type,
% 0.06/0.32 setadjoinIL: $o ).
% 0.06/0.32
% 0.06/0.32 thf(setadjoinIL,definition,
% 0.06/0.32 ( setadjoinIL
% 0.06/0.32 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(emptyinunitempty_type,type,
% 0.06/0.32 emptyinunitempty: $o ).
% 0.06/0.32
% 0.06/0.32 thf(emptyinunitempty,definition,
% 0.06/0.32 ( emptyinunitempty
% 0.06/0.32 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(setadjoinIR_type,type,
% 0.06/0.32 setadjoinIR: $o ).
% 0.06/0.32
% 0.06/0.32 thf(setadjoinIR,definition,
% 0.06/0.32 ( setadjoinIR
% 0.06/0.32 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.06/0.32 ( ( in @ Xy @ A )
% 0.06/0.32 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(setadjoinE_type,type,
% 0.06/0.32 setadjoinE: $o ).
% 0.06/0.32
% 0.06/0.32 thf(setadjoinE,definition,
% 0.06/0.32 ( setadjoinE
% 0.06/0.32 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.06/0.32 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.06/0.32 => ! [Xphi: $o] :
% 0.06/0.32 ( ( ( Xy = Xx )
% 0.06/0.32 => Xphi )
% 0.06/0.32 => ( ( ( in @ Xy @ A )
% 0.06/0.32 => Xphi )
% 0.06/0.32 => Xphi ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(setadjoinOr_type,type,
% 0.06/0.32 setadjoinOr: $o ).
% 0.06/0.32
% 0.06/0.32 thf(setadjoinOr,definition,
% 0.06/0.32 ( setadjoinOr
% 0.06/0.32 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.06/0.32 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.06/0.32 => ( ( Xy = Xx )
% 0.06/0.32 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(setoftrueEq_type,type,
% 0.06/0.32 setoftrueEq: $o ).
% 0.06/0.32
% 0.06/0.32 thf(setoftrueEq,definition,
% 0.06/0.32 ( setoftrueEq
% 0.06/0.32 = ( ! [A: $i] :
% 0.06/0.32 ( ( dsetconstr @ A
% 0.06/0.32 @ ^ [Xx: $i] : $true )
% 0.06/0.32 = A ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(powersetI_type,type,
% 0.06/0.32 powersetI: $o ).
% 0.06/0.32
% 0.06/0.32 thf(powersetI,definition,
% 0.06/0.32 ( powersetI
% 0.06/0.32 = ( ! [A: $i,B: $i] :
% 0.06/0.32 ( ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ B )
% 0.06/0.32 => ( in @ Xx @ A ) )
% 0.06/0.32 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(emptyinPowerset_type,type,
% 0.06/0.32 emptyinPowerset: $o ).
% 0.06/0.32
% 0.06/0.32 thf(emptyinPowerset,definition,
% 0.06/0.32 ( emptyinPowerset
% 0.06/0.32 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(emptyInPowerset_type,type,
% 0.06/0.32 emptyInPowerset: $o ).
% 0.06/0.32
% 0.06/0.32 thf(emptyInPowerset,definition,
% 0.06/0.32 ( emptyInPowerset
% 0.06/0.32 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(powersetE_type,type,
% 0.06/0.32 powersetE: $o ).
% 0.06/0.32
% 0.06/0.32 thf(powersetE,definition,
% 0.06/0.32 ( powersetE
% 0.06/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.06/0.32 ( ( in @ B @ ( powerset @ A ) )
% 0.06/0.32 => ( ( in @ Xx @ B )
% 0.06/0.32 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(setunionI_type,type,
% 0.06/0.32 setunionI: $o ).
% 0.06/0.32
% 0.06/0.32 thf(setunionI,definition,
% 0.06/0.32 ( setunionI
% 0.06/0.32 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.06/0.32 ( ( in @ Xx @ B )
% 0.06/0.32 => ( ( in @ B @ A )
% 0.06/0.32 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(setunionE_type,type,
% 0.06/0.32 setunionE: $o ).
% 0.06/0.32
% 0.06/0.32 thf(setunionE,definition,
% 0.06/0.32 ( setunionE
% 0.06/0.32 = ( ! [A: $i,Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ ( setunion @ A ) )
% 0.06/0.32 => ! [Xphi: $o] :
% 0.06/0.32 ( ! [B: $i] :
% 0.06/0.32 ( ( in @ Xx @ B )
% 0.06/0.32 => ( ( in @ B @ A )
% 0.06/0.32 => Xphi ) )
% 0.06/0.32 => Xphi ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(subPowSU_type,type,
% 0.06/0.32 subPowSU: $o ).
% 0.06/0.32
% 0.06/0.32 thf(subPowSU,definition,
% 0.06/0.32 ( subPowSU
% 0.06/0.32 = ( ! [A: $i,Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(exuE2_type,type,
% 0.06/0.32 exuE2: $o ).
% 0.06/0.32
% 0.06/0.32 thf(exuE2,definition,
% 0.06/0.32 ( exuE2
% 0.06/0.32 = ( ! [Xphi: $i > $o] :
% 0.06/0.32 ( ( exu
% 0.06/0.32 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.06/0.32 => ? [Xx: $i] :
% 0.06/0.32 ! [Xy: $i] :
% 0.06/0.32 ( ( Xphi @ Xy )
% 0.06/0.32 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(nonemptyImpWitness_type,type,
% 0.06/0.32 nonemptyImpWitness: $o ).
% 0.06/0.32
% 0.06/0.32 thf(nonemptyImpWitness,definition,
% 0.06/0.32 ( nonemptyImpWitness
% 0.06/0.32 = ( ! [A: $i] :
% 0.06/0.32 ( ( nonempty @ A )
% 0.06/0.32 => ? [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 & $true ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(uniqinunit_type,type,
% 0.06/0.32 uniqinunit: $o ).
% 0.06/0.32
% 0.06/0.32 thf(uniqinunit,definition,
% 0.06/0.32 ( uniqinunit
% 0.06/0.32 = ( ! [Xx: $i,Xy: $i] :
% 0.06/0.32 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.06/0.32 => ( Xx = Xy ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(notinsingleton_type,type,
% 0.06/0.32 notinsingleton: $o ).
% 0.06/0.32
% 0.06/0.32 thf(notinsingleton,definition,
% 0.06/0.32 ( notinsingleton
% 0.06/0.32 = ( ! [Xx: $i,Xy: $i] :
% 0.06/0.32 ( ( Xx != Xy )
% 0.06/0.32 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(eqinunit_type,type,
% 0.06/0.32 eqinunit: $o ).
% 0.06/0.32
% 0.06/0.32 thf(eqinunit,definition,
% 0.06/0.32 ( eqinunit
% 0.06/0.32 = ( ! [Xx: $i,Xy: $i] :
% 0.06/0.32 ( ( Xx = Xy )
% 0.06/0.32 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(singletonsswitch_type,type,
% 0.06/0.32 singletonsswitch: $o ).
% 0.06/0.32
% 0.06/0.32 thf(singletonsswitch,definition,
% 0.06/0.32 ( singletonsswitch
% 0.06/0.32 = ( ! [Xx: $i,Xy: $i] :
% 0.06/0.32 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.06/0.32 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(upairsetE_type,type,
% 0.06/0.32 upairsetE: $o ).
% 0.06/0.32
% 0.06/0.32 thf(upairsetE,definition,
% 0.06/0.32 ( upairsetE
% 0.06/0.32 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.06/0.32 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.06/0.32 => ( ( Xz = Xx )
% 0.06/0.32 | ( Xz = Xy ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(upairsetIL_type,type,
% 0.06/0.32 upairsetIL: $o ).
% 0.06/0.32
% 0.06/0.32 thf(upairsetIL,definition,
% 0.06/0.32 ( upairsetIL
% 0.06/0.32 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(upairsetIR_type,type,
% 0.06/0.32 upairsetIR: $o ).
% 0.06/0.32
% 0.06/0.32 thf(upairsetIR,definition,
% 0.06/0.32 ( upairsetIR
% 0.06/0.32 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(emptyE1_type,type,
% 0.06/0.32 emptyE1: $o ).
% 0.06/0.32
% 0.06/0.32 thf(emptyE1,definition,
% 0.06/0.32 ( emptyE1
% 0.06/0.32 = ( ! [A: $i,Xphi: $i > $o] :
% 0.06/0.32 ( ? [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 & ( Xphi @ Xx ) )
% 0.06/0.32 => ( ( ( dsetconstr @ A
% 0.06/0.32 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.06/0.32 = emptyset )
% 0.06/0.32 => $false ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(vacuousDall_type,type,
% 0.06/0.32 vacuousDall: $o ).
% 0.06/0.32
% 0.06/0.32 thf(vacuousDall,definition,
% 0.06/0.32 ( vacuousDall
% 0.06/0.32 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ emptyset )
% 0.06/0.32 => ( Xphi @ Xx ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(quantDeMorgan1_type,type,
% 0.06/0.32 quantDeMorgan1: $o ).
% 0.06/0.32
% 0.06/0.32 thf(quantDeMorgan1,definition,
% 0.06/0.32 ( quantDeMorgan1
% 0.06/0.32 = ( ! [A: $i,Xphi: $i > $o] :
% 0.06/0.32 ( ~ ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ( Xphi @ Xx ) )
% 0.06/0.32 => ? [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(quantDeMorgan2_type,type,
% 0.06/0.32 quantDeMorgan2: $o ).
% 0.06/0.32
% 0.06/0.32 thf(quantDeMorgan2,definition,
% 0.06/0.32 ( quantDeMorgan2
% 0.06/0.32 = ( ! [A: $i,Xphi: $i > $o] :
% 0.06/0.32 ( ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ~ ( Xphi @ Xx ) )
% 0.06/0.32 => ~ ? [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(quantDeMorgan3_type,type,
% 0.06/0.32 quantDeMorgan3: $o ).
% 0.06/0.32
% 0.06/0.32 thf(quantDeMorgan3,definition,
% 0.06/0.32 ( quantDeMorgan3
% 0.06/0.32 = ( ! [A: $i,Xphi: $i > $o] :
% 0.06/0.32 ( ~ ? [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 & ( Xphi @ Xx ) )
% 0.06/0.32 => ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(quantDeMorgan4_type,type,
% 0.06/0.32 quantDeMorgan4: $o ).
% 0.06/0.32
% 0.06/0.32 thf(quantDeMorgan4,definition,
% 0.06/0.32 ( quantDeMorgan4
% 0.06/0.32 = ( ! [A: $i,Xphi: $i > $o] :
% 0.06/0.32 ( ? [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 & ~ ( Xphi @ Xx ) )
% 0.06/0.32 => ~ ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(prop2setI_type,type,
% 0.06/0.32 prop2setI: $o ).
% 0.06/0.32
% 0.06/0.32 thf(prop2setI,definition,
% 0.06/0.32 ( prop2setI
% 0.06/0.32 = ( ! [Xphi: $o] :
% 0.06/0.32 ( Xphi
% 0.06/0.32 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(set2prop_type,type,
% 0.06/0.32 set2prop: $i > $o ).
% 0.06/0.32
% 0.06/0.32 thf(prop2set2propI_type,type,
% 0.06/0.32 prop2set2propI: $o ).
% 0.06/0.32
% 0.06/0.32 thf(prop2set2propI,definition,
% 0.06/0.32 ( prop2set2propI
% 0.06/0.32 = ( ! [Xphi: $o] :
% 0.06/0.32 ( Xphi
% 0.06/0.32 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(notdexE_type,type,
% 0.06/0.32 notdexE: $o ).
% 0.06/0.32
% 0.06/0.32 thf(notdexE,definition,
% 0.06/0.32 ( notdexE
% 0.06/0.32 = ( ! [A: $i,Xphi: $i > $o] :
% 0.06/0.32 ( ~ ? [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 & ( Xphi @ Xx ) )
% 0.06/0.32 => ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(notdallE_type,type,
% 0.06/0.32 notdallE: $o ).
% 0.06/0.32
% 0.06/0.32 thf(notdallE,definition,
% 0.06/0.32 ( notdallE
% 0.06/0.32 = ( ! [A: $i,Xphi: $i > $o] :
% 0.06/0.32 ( ~ ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ( Xphi @ Xx ) )
% 0.06/0.32 => ? [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(exuI1_type,type,
% 0.06/0.32 exuI1: $o ).
% 0.06/0.32
% 0.06/0.32 thf(exuI1,definition,
% 0.06/0.32 ( exuI1
% 0.06/0.32 = ( ! [Xphi: $i > $o] :
% 0.06/0.32 ( ? [Xx: $i] :
% 0.06/0.32 ( ( Xphi @ Xx )
% 0.06/0.32 & ! [Xy: $i] :
% 0.06/0.32 ( ( Xphi @ Xy )
% 0.06/0.32 => ( Xx = Xy ) ) )
% 0.06/0.32 => ( exu
% 0.06/0.32 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(exuI3_type,type,
% 0.06/0.32 exuI3: $o ).
% 0.06/0.32
% 0.06/0.32 thf(exuI3,definition,
% 0.06/0.32 ( exuI3
% 0.06/0.32 = ( ! [Xphi: $i > $o] :
% 0.06/0.32 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.06/0.32 => ( ! [Xx: $i,Xy: $i] :
% 0.06/0.32 ( ( Xphi @ Xx )
% 0.06/0.32 => ( ( Xphi @ Xy )
% 0.06/0.32 => ( Xx = Xy ) ) )
% 0.06/0.32 => ( exu
% 0.06/0.32 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(exuI2_type,type,
% 0.06/0.32 exuI2: $o ).
% 0.06/0.32
% 0.06/0.32 thf(exuI2,definition,
% 0.06/0.32 ( exuI2
% 0.06/0.32 = ( ! [Xphi: $i > $o] :
% 0.06/0.32 ( ? [Xx: $i] :
% 0.06/0.32 ! [Xy: $i] :
% 0.06/0.32 ( ( Xphi @ Xy )
% 0.06/0.32 <=> ( Xy = Xx ) )
% 0.06/0.32 => ( exu
% 0.06/0.32 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(inCongP_type,type,
% 0.06/0.32 inCongP: $o ).
% 0.06/0.32
% 0.06/0.32 thf(inCongP,definition,
% 0.06/0.32 ( inCongP
% 0.06/0.32 = ( ! [A: $i,B: $i] :
% 0.06/0.32 ( ( A = B )
% 0.06/0.32 => ! [Xx: $i,Xy: $i] :
% 0.06/0.32 ( ( Xx = Xy )
% 0.06/0.32 => ( ( in @ Xx @ A )
% 0.06/0.32 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(in__Cong_type,type,
% 0.06/0.32 in__Cong: $o ).
% 0.06/0.32
% 0.06/0.32 thf(in__Cong,definition,
% 0.06/0.32 ( in__Cong
% 0.06/0.32 = ( ! [A: $i,B: $i] :
% 0.06/0.32 ( ( A = B )
% 0.06/0.32 => ! [Xx: $i,Xy: $i] :
% 0.06/0.32 ( ( Xx = Xy )
% 0.06/0.32 => ( ( in @ Xx @ A )
% 0.06/0.32 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(exuE3u_type,type,
% 0.06/0.32 exuE3u: $o ).
% 0.06/0.32
% 0.06/0.32 thf(exuE3u,definition,
% 0.06/0.32 ( exuE3u
% 0.06/0.32 = ( ! [Xphi: $i > $o] :
% 0.06/0.32 ( ( exu
% 0.06/0.32 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.06/0.32 => ! [Xx: $i,Xy: $i] :
% 0.06/0.32 ( ( Xphi @ Xx )
% 0.06/0.32 => ( ( Xphi @ Xy )
% 0.06/0.32 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(exu__Cong_type,type,
% 0.06/0.32 exu__Cong: $o ).
% 0.06/0.32
% 0.06/0.32 thf(exu__Cong,definition,
% 0.06/0.32 ( exu__Cong
% 0.06/0.32 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.06/0.32 ( ! [Xx: $i,Xy: $i] :
% 0.06/0.32 ( ( Xx = Xy )
% 0.06/0.32 => ( ( Xphi @ Xx )
% 0.06/0.32 <=> ( Xpsi @ Xy ) ) )
% 0.06/0.32 => ( ( exu
% 0.06/0.32 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.06/0.32 <=> ( exu
% 0.06/0.32 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(emptyset__Cong_type,type,
% 0.06/0.32 emptyset__Cong: $o ).
% 0.06/0.32
% 0.06/0.32 thf(emptyset__Cong,definition,
% 0.06/0.32 ( emptyset__Cong
% 0.06/0.32 = ( emptyset = emptyset ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(setadjoin__Cong_type,type,
% 0.06/0.32 setadjoin__Cong: $o ).
% 0.06/0.32
% 0.06/0.32 thf(setadjoin__Cong,definition,
% 0.06/0.32 ( setadjoin__Cong
% 0.06/0.32 = ( ! [Xx: $i,Xy: $i] :
% 0.06/0.32 ( ( Xx = Xy )
% 0.06/0.32 => ! [Xz: $i,Xu: $i] :
% 0.06/0.32 ( ( Xz = Xu )
% 0.06/0.32 => ( ( setadjoin @ Xx @ Xz )
% 0.06/0.32 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(powerset__Cong_type,type,
% 0.06/0.32 powerset__Cong: $o ).
% 0.06/0.32
% 0.06/0.32 thf(powerset__Cong,definition,
% 0.06/0.32 ( powerset__Cong
% 0.06/0.32 = ( ! [A: $i,B: $i] :
% 0.06/0.32 ( ( A = B )
% 0.06/0.32 => ( ( powerset @ A )
% 0.06/0.32 = ( powerset @ B ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(setunion__Cong_type,type,
% 0.06/0.32 setunion__Cong: $o ).
% 0.06/0.32
% 0.06/0.32 thf(setunion__Cong,definition,
% 0.06/0.32 ( setunion__Cong
% 0.06/0.32 = ( ! [A: $i,B: $i] :
% 0.06/0.32 ( ( A = B )
% 0.06/0.32 => ( ( setunion @ A )
% 0.06/0.32 = ( setunion @ B ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(omega__Cong_type,type,
% 0.06/0.32 omega__Cong: $o ).
% 0.06/0.32
% 0.06/0.32 thf(omega__Cong,definition,
% 0.06/0.32 ( omega__Cong
% 0.06/0.32 = ( omega = omega ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(exuEu_type,type,
% 0.06/0.32 exuEu: $o ).
% 0.06/0.32
% 0.06/0.32 thf(exuEu,definition,
% 0.06/0.32 ( exuEu
% 0.06/0.32 = ( ! [Xphi: $i > $o] :
% 0.06/0.32 ( ( exu
% 0.06/0.32 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.06/0.32 => ! [Xx: $i,Xy: $i] :
% 0.06/0.32 ( ( Xphi @ Xx )
% 0.06/0.32 => ( ( Xphi @ Xy )
% 0.06/0.32 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(descr__Cong_type,type,
% 0.06/0.32 descr__Cong: $o ).
% 0.06/0.32
% 0.06/0.32 thf(descr__Cong,definition,
% 0.06/0.32 ( descr__Cong
% 0.06/0.32 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.06/0.32 ( ! [Xx: $i,Xy: $i] :
% 0.06/0.32 ( ( Xx = Xy )
% 0.06/0.32 => ( ( Xphi @ Xx )
% 0.06/0.32 <=> ( Xpsi @ Xy ) ) )
% 0.06/0.32 => ( ( exu
% 0.06/0.32 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.06/0.32 => ( ( exu
% 0.06/0.32 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.06/0.32 => ( ( descr
% 0.06/0.32 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.06/0.32 = ( descr
% 0.06/0.32 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(dsetconstr__Cong_type,type,
% 0.06/0.32 dsetconstr__Cong: $o ).
% 0.06/0.32
% 0.06/0.32 thf(dsetconstr__Cong,definition,
% 0.06/0.32 ( dsetconstr__Cong
% 0.06/0.32 = ( ! [A: $i,B: $i] :
% 0.06/0.32 ( ( A = B )
% 0.06/0.32 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.06/0.32 ( ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ! [Xy: $i] :
% 0.06/0.32 ( ( in @ Xy @ B )
% 0.06/0.32 => ( ( Xx = Xy )
% 0.06/0.32 => ( ( Xphi @ Xx )
% 0.06/0.32 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.06/0.32 => ( ( dsetconstr @ A
% 0.06/0.32 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.06/0.32 = ( dsetconstr @ B
% 0.06/0.32 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(subset_type,type,
% 0.06/0.32 subset: $i > $i > $o ).
% 0.06/0.32
% 0.06/0.32 thf(disjoint_type,type,
% 0.06/0.32 disjoint: $i > $i > $o ).
% 0.06/0.32
% 0.06/0.32 thf(setsmeet_type,type,
% 0.06/0.32 setsmeet: $i > $i > $o ).
% 0.06/0.32
% 0.06/0.32 thf(subsetI1_type,type,
% 0.06/0.32 subsetI1: $o ).
% 0.06/0.32
% 0.06/0.32 thf(subsetI1,definition,
% 0.06/0.32 ( subsetI1
% 0.06/0.32 = ( ! [A: $i,B: $i] :
% 0.06/0.32 ( ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ( in @ Xx @ B ) )
% 0.06/0.32 => ( subset @ A @ B ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(eqimpsubset2_type,type,
% 0.06/0.32 eqimpsubset2: $o ).
% 0.06/0.32
% 0.06/0.32 thf(eqimpsubset2,definition,
% 0.06/0.32 ( eqimpsubset2
% 0.06/0.32 = ( ! [A: $i,B: $i] :
% 0.06/0.32 ( ( A = B )
% 0.06/0.32 => ( subset @ B @ A ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(eqimpsubset1_type,type,
% 0.06/0.32 eqimpsubset1: $o ).
% 0.06/0.32
% 0.06/0.32 thf(eqimpsubset1,definition,
% 0.06/0.32 ( eqimpsubset1
% 0.06/0.32 = ( ! [A: $i,B: $i] :
% 0.06/0.32 ( ( A = B )
% 0.06/0.32 => ( subset @ A @ B ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(subsetI2_type,type,
% 0.06/0.32 subsetI2: $o ).
% 0.06/0.32
% 0.06/0.32 thf(subsetI2,definition,
% 0.06/0.32 ( subsetI2
% 0.06/0.32 = ( ! [A: $i,B: $i] :
% 0.06/0.32 ( ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ( in @ Xx @ B ) )
% 0.06/0.32 => ( subset @ A @ B ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(emptysetsubset_type,type,
% 0.06/0.32 emptysetsubset: $o ).
% 0.06/0.32
% 0.06/0.32 thf(emptysetsubset,definition,
% 0.06/0.32 ( emptysetsubset
% 0.06/0.32 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(subsetE_type,type,
% 0.06/0.32 subsetE: $o ).
% 0.06/0.32
% 0.06/0.32 thf(subsetE,definition,
% 0.06/0.32 ( subsetE
% 0.06/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.06/0.32 ( ( subset @ A @ B )
% 0.06/0.32 => ( ( in @ Xx @ A )
% 0.06/0.32 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(subsetE2_type,type,
% 0.06/0.32 subsetE2: $o ).
% 0.06/0.32
% 0.06/0.32 thf(subsetE2,definition,
% 0.06/0.32 ( subsetE2
% 0.06/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.06/0.32 ( ( subset @ A @ B )
% 0.06/0.32 => ( ~ ( in @ Xx @ B )
% 0.06/0.32 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(notsubsetI_type,type,
% 0.06/0.32 notsubsetI: $o ).
% 0.06/0.32
% 0.06/0.32 thf(notsubsetI,definition,
% 0.06/0.32 ( notsubsetI
% 0.06/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ( ~ ( in @ Xx @ B )
% 0.06/0.32 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(notequalI1_type,type,
% 0.06/0.32 notequalI1: $o ).
% 0.06/0.32
% 0.06/0.32 thf(notequalI1,definition,
% 0.06/0.32 ( notequalI1
% 0.06/0.32 = ( ! [A: $i,B: $i] :
% 0.06/0.32 ( ~ ( subset @ A @ B )
% 0.06/0.32 => ( A != B ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(notequalI2_type,type,
% 0.06/0.32 notequalI2: $o ).
% 0.06/0.32
% 0.06/0.32 thf(notequalI2,definition,
% 0.06/0.32 ( notequalI2
% 0.06/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ( ~ ( in @ Xx @ B )
% 0.06/0.32 => ( A != B ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(subsetRefl_type,type,
% 0.06/0.32 subsetRefl: $o ).
% 0.06/0.32
% 0.06/0.32 thf(subsetRefl,definition,
% 0.06/0.32 ( subsetRefl
% 0.06/0.32 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(subsetTrans_type,type,
% 0.06/0.32 subsetTrans: $o ).
% 0.06/0.32
% 0.06/0.32 thf(subsetTrans,definition,
% 0.06/0.32 ( subsetTrans
% 0.06/0.32 = ( ! [A: $i,B: $i,C: $i] :
% 0.06/0.32 ( ( subset @ A @ B )
% 0.06/0.32 => ( ( subset @ B @ C )
% 0.06/0.32 => ( subset @ A @ C ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(setadjoinSub_type,type,
% 0.06/0.32 setadjoinSub: $o ).
% 0.06/0.32
% 0.06/0.32 thf(setadjoinSub,definition,
% 0.06/0.32 ( setadjoinSub
% 0.06/0.32 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(setadjoinSub2_type,type,
% 0.06/0.32 setadjoinSub2: $o ).
% 0.06/0.32
% 0.06/0.32 thf(setadjoinSub2,definition,
% 0.06/0.32 ( setadjoinSub2
% 0.06/0.32 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.06/0.32 ( ( subset @ A @ B )
% 0.06/0.32 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(subset2powerset_type,type,
% 0.06/0.32 subset2powerset: $o ).
% 0.06/0.32
% 0.06/0.32 thf(subset2powerset,definition,
% 0.06/0.32 ( subset2powerset
% 0.06/0.32 = ( ! [A: $i,B: $i] :
% 0.06/0.32 ( ( subset @ A @ B )
% 0.06/0.32 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(setextsub_type,type,
% 0.06/0.32 setextsub: $o ).
% 0.06/0.32
% 0.06/0.32 thf(setextsub,definition,
% 0.06/0.32 ( setextsub
% 0.06/0.32 = ( ! [A: $i,B: $i] :
% 0.06/0.32 ( ( subset @ A @ B )
% 0.06/0.32 => ( ( subset @ B @ A )
% 0.06/0.32 => ( A = B ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(subsetemptysetimpeq_type,type,
% 0.06/0.32 subsetemptysetimpeq: $o ).
% 0.06/0.32
% 0.06/0.32 thf(subsetemptysetimpeq,definition,
% 0.06/0.32 ( subsetemptysetimpeq
% 0.06/0.32 = ( ! [A: $i] :
% 0.06/0.32 ( ( subset @ A @ emptyset )
% 0.06/0.32 => ( A = emptyset ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(powersetI1_type,type,
% 0.06/0.32 powersetI1: $o ).
% 0.06/0.32
% 0.06/0.32 thf(powersetI1,definition,
% 0.06/0.32 ( powersetI1
% 0.06/0.32 = ( ! [A: $i,B: $i] :
% 0.06/0.32 ( ( subset @ B @ A )
% 0.06/0.32 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(powersetE1_type,type,
% 0.06/0.32 powersetE1: $o ).
% 0.06/0.32
% 0.06/0.32 thf(powersetE1,definition,
% 0.06/0.32 ( powersetE1
% 0.06/0.32 = ( ! [A: $i,B: $i] :
% 0.06/0.32 ( ( in @ B @ ( powerset @ A ) )
% 0.06/0.32 => ( subset @ B @ A ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(inPowerset_type,type,
% 0.06/0.32 inPowerset: $o ).
% 0.06/0.32
% 0.06/0.32 thf(inPowerset,definition,
% 0.06/0.32 ( inPowerset
% 0.06/0.32 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(powersetsubset_type,type,
% 0.06/0.32 powersetsubset: $o ).
% 0.06/0.32
% 0.06/0.32 thf(powersetsubset,definition,
% 0.06/0.32 ( powersetsubset
% 0.06/0.32 = ( ! [A: $i,B: $i] :
% 0.06/0.32 ( ( subset @ A @ B )
% 0.06/0.32 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(sepInPowerset_type,type,
% 0.06/0.32 sepInPowerset: $o ).
% 0.06/0.32
% 0.06/0.32 thf(sepInPowerset,definition,
% 0.06/0.32 ( sepInPowerset
% 0.06/0.32 = ( ! [A: $i,Xphi: $i > $o] :
% 0.06/0.32 ( in
% 0.06/0.32 @ ( dsetconstr @ A
% 0.06/0.32 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.06/0.32 @ ( powerset @ A ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(sepSubset_type,type,
% 0.06/0.32 sepSubset: $o ).
% 0.06/0.32
% 0.06/0.32 thf(sepSubset,definition,
% 0.06/0.32 ( sepSubset
% 0.06/0.32 = ( ! [A: $i,Xphi: $i > $o] :
% 0.06/0.32 ( subset
% 0.06/0.32 @ ( dsetconstr @ A
% 0.06/0.32 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.06/0.32 @ A ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(binunion_type,type,
% 0.06/0.32 binunion: $i > $i > $i ).
% 0.06/0.32
% 0.06/0.32 thf(binunionIL_type,type,
% 0.06/0.32 binunionIL: $o ).
% 0.06/0.32
% 0.06/0.32 thf(binunionIL,definition,
% 0.06/0.32 ( binunionIL
% 0.06/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(upairset2IR_type,type,
% 0.06/0.32 upairset2IR: $o ).
% 0.06/0.32
% 0.06/0.32 thf(upairset2IR,definition,
% 0.06/0.32 ( upairset2IR
% 0.06/0.32 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(binunionIR_type,type,
% 0.06/0.32 binunionIR: $o ).
% 0.06/0.32
% 0.06/0.32 thf(binunionIR,definition,
% 0.06/0.32 ( binunionIR
% 0.06/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ B )
% 0.06/0.32 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(binunionEcases_type,type,
% 0.06/0.32 binunionEcases: $o ).
% 0.06/0.32
% 0.06/0.32 thf(binunionEcases,definition,
% 0.06/0.32 ( binunionEcases
% 0.06/0.32 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.06/0.32 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.06/0.32 => ( ( ( in @ Xx @ A )
% 0.06/0.32 => Xphi )
% 0.06/0.32 => ( ( ( in @ Xx @ B )
% 0.06/0.32 => Xphi )
% 0.06/0.32 => Xphi ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(binunionE_type,type,
% 0.06/0.32 binunionE: $o ).
% 0.06/0.32
% 0.06/0.32 thf(binunionE,definition,
% 0.06/0.32 ( binunionE
% 0.06/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.06/0.32 => ( ( in @ Xx @ A )
% 0.06/0.32 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(binunionLsub_type,type,
% 0.06/0.32 binunionLsub: $o ).
% 0.06/0.32
% 0.06/0.32 thf(binunionLsub,definition,
% 0.06/0.32 ( binunionLsub
% 0.06/0.32 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(binunionRsub_type,type,
% 0.06/0.32 binunionRsub: $o ).
% 0.06/0.32
% 0.06/0.32 thf(binunionRsub,definition,
% 0.06/0.32 ( binunionRsub
% 0.06/0.32 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(binintersect_type,type,
% 0.06/0.32 binintersect: $i > $i > $i ).
% 0.06/0.32
% 0.06/0.32 thf(binintersectI_type,type,
% 0.06/0.32 binintersectI: $o ).
% 0.06/0.32
% 0.06/0.32 thf(binintersectI,definition,
% 0.06/0.32 ( binintersectI
% 0.06/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ( ( in @ Xx @ B )
% 0.06/0.32 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(binintersectSubset5_type,type,
% 0.06/0.32 binintersectSubset5: $o ).
% 0.06/0.32
% 0.06/0.32 thf(binintersectSubset5,definition,
% 0.06/0.32 ( binintersectSubset5
% 0.06/0.32 = ( ! [A: $i,B: $i,C: $i] :
% 0.06/0.32 ( ( subset @ C @ A )
% 0.06/0.32 => ( ( subset @ C @ B )
% 0.06/0.32 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(binintersectEL_type,type,
% 0.06/0.32 binintersectEL: $o ).
% 0.06/0.32
% 0.06/0.32 thf(binintersectEL,definition,
% 0.06/0.32 ( binintersectEL
% 0.06/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.06/0.32 => ( in @ Xx @ A ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(binintersectLsub_type,type,
% 0.06/0.32 binintersectLsub: $o ).
% 0.06/0.32
% 0.06/0.32 thf(binintersectLsub,definition,
% 0.06/0.32 ( binintersectLsub
% 0.06/0.32 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(binintersectSubset2_type,type,
% 0.06/0.32 binintersectSubset2: $o ).
% 0.06/0.32
% 0.06/0.32 thf(binintersectSubset2,definition,
% 0.06/0.32 ( binintersectSubset2
% 0.06/0.32 = ( ! [A: $i,B: $i] :
% 0.06/0.32 ( ( subset @ A @ B )
% 0.06/0.32 => ( ( binintersect @ A @ B )
% 0.06/0.32 = A ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(binintersectSubset3_type,type,
% 0.06/0.32 binintersectSubset3: $o ).
% 0.06/0.32
% 0.06/0.32 thf(binintersectSubset3,definition,
% 0.06/0.32 ( binintersectSubset3
% 0.06/0.32 = ( ! [A: $i,B: $i] :
% 0.06/0.32 ( ( ( binintersect @ A @ B )
% 0.06/0.32 = B )
% 0.06/0.32 => ( subset @ B @ A ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(binintersectER_type,type,
% 0.06/0.32 binintersectER: $o ).
% 0.06/0.32
% 0.06/0.32 thf(binintersectER,definition,
% 0.06/0.32 ( binintersectER
% 0.06/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.06/0.32 => ( in @ Xx @ B ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(disjointsetsI1_type,type,
% 0.06/0.32 disjointsetsI1: $o ).
% 0.06/0.32
% 0.06/0.32 thf(disjointsetsI1,definition,
% 0.06/0.32 ( disjointsetsI1
% 0.06/0.32 = ( ! [A: $i,B: $i] :
% 0.06/0.32 ( ~ ? [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 & ( in @ Xx @ B ) )
% 0.06/0.32 => ( ( binintersect @ A @ B )
% 0.06/0.32 = emptyset ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(binintersectRsub_type,type,
% 0.06/0.32 binintersectRsub: $o ).
% 0.06/0.32
% 0.06/0.32 thf(binintersectRsub,definition,
% 0.06/0.32 ( binintersectRsub
% 0.06/0.32 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(binintersectSubset4_type,type,
% 0.06/0.32 binintersectSubset4: $o ).
% 0.06/0.32
% 0.06/0.32 thf(binintersectSubset4,definition,
% 0.06/0.32 ( binintersectSubset4
% 0.06/0.32 = ( ! [A: $i,B: $i] :
% 0.06/0.32 ( ( subset @ B @ A )
% 0.06/0.32 => ( ( binintersect @ A @ B )
% 0.06/0.32 = B ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(binintersectSubset1_type,type,
% 0.06/0.32 binintersectSubset1: $o ).
% 0.06/0.32
% 0.06/0.32 thf(binintersectSubset1,definition,
% 0.06/0.32 ( binintersectSubset1
% 0.06/0.32 = ( ! [A: $i,B: $i] :
% 0.06/0.32 ( ( ( binintersect @ A @ B )
% 0.06/0.32 = A )
% 0.06/0.32 => ( subset @ A @ B ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(bs114d_type,type,
% 0.06/0.32 bs114d: $o ).
% 0.06/0.32
% 0.06/0.32 thf(bs114d,definition,
% 0.06/0.32 ( bs114d
% 0.06/0.32 = ( ! [A: $i,B: $i,C: $i] :
% 0.06/0.32 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 0.06/0.32 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(regular_type,type,
% 0.06/0.32 regular: $i > $o ).
% 0.06/0.32
% 0.06/0.32 thf(setminus_type,type,
% 0.06/0.32 setminus: $i > $i > $i ).
% 0.06/0.32
% 0.06/0.32 thf(setminusI_type,type,
% 0.06/0.32 setminusI: $o ).
% 0.06/0.32
% 0.06/0.32 thf(setminusI,definition,
% 0.06/0.32 ( setminusI
% 0.06/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ( ~ ( in @ Xx @ B )
% 0.06/0.32 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(setminusEL_type,type,
% 0.06/0.32 setminusEL: $o ).
% 0.06/0.32
% 0.06/0.32 thf(setminusEL,definition,
% 0.06/0.32 ( setminusEL
% 0.06/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.06/0.32 => ( in @ Xx @ A ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(setminusER_type,type,
% 0.06/0.32 setminusER: $o ).
% 0.06/0.32
% 0.06/0.32 thf(setminusER,definition,
% 0.06/0.32 ( setminusER
% 0.06/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.06/0.32 => ~ ( in @ Xx @ B ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(setminusSubset2_type,type,
% 0.06/0.32 setminusSubset2: $o ).
% 0.06/0.32
% 0.06/0.32 thf(setminusSubset2,definition,
% 0.06/0.32 ( setminusSubset2
% 0.06/0.32 = ( ! [A: $i,B: $i] :
% 0.06/0.32 ( ( subset @ A @ B )
% 0.06/0.32 => ( ( setminus @ A @ B )
% 0.06/0.32 = emptyset ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(setminusERneg_type,type,
% 0.06/0.32 setminusERneg: $o ).
% 0.06/0.32
% 0.06/0.32 thf(setminusERneg,definition,
% 0.06/0.32 ( setminusERneg
% 0.06/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.06/0.32 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.06/0.32 => ( ( in @ Xx @ A )
% 0.06/0.32 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(setminusELneg_type,type,
% 0.06/0.32 setminusELneg: $o ).
% 0.06/0.32
% 0.06/0.32 thf(setminusELneg,definition,
% 0.06/0.32 ( setminusELneg
% 0.06/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.06/0.32 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.06/0.32 => ( ~ ( in @ Xx @ B )
% 0.06/0.32 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(setminusILneg_type,type,
% 0.06/0.32 setminusILneg: $o ).
% 0.06/0.32
% 0.06/0.32 thf(setminusILneg,definition,
% 0.06/0.32 ( setminusILneg
% 0.06/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.06/0.32 ( ~ ( in @ Xx @ A )
% 0.06/0.32 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(setminusIRneg_type,type,
% 0.06/0.32 setminusIRneg: $o ).
% 0.06/0.32
% 0.06/0.32 thf(setminusIRneg,definition,
% 0.06/0.32 ( setminusIRneg
% 0.06/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ B )
% 0.06/0.32 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(setminusLsub_type,type,
% 0.06/0.32 setminusLsub: $o ).
% 0.06/0.32
% 0.06/0.32 thf(setminusLsub,definition,
% 0.06/0.32 ( setminusLsub
% 0.06/0.32 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(setminusSubset1_type,type,
% 0.06/0.32 setminusSubset1: $o ).
% 0.06/0.32
% 0.06/0.32 thf(setminusSubset1,definition,
% 0.06/0.32 ( setminusSubset1
% 0.06/0.32 = ( ! [A: $i,B: $i] :
% 0.06/0.32 ( ( ( setminus @ A @ B )
% 0.06/0.32 = emptyset )
% 0.06/0.32 => ( subset @ A @ B ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(symdiff_type,type,
% 0.06/0.32 symdiff: $i > $i > $i ).
% 0.06/0.32
% 0.06/0.32 thf(symdiffE_type,type,
% 0.06/0.32 symdiffE: $o ).
% 0.06/0.32
% 0.06/0.32 thf(symdiffE,definition,
% 0.06/0.32 ( symdiffE
% 0.06/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 0.06/0.32 => ! [Xphi: $o] :
% 0.06/0.32 ( ( ( in @ Xx @ A )
% 0.06/0.32 => ( ~ ( in @ Xx @ B )
% 0.06/0.32 => Xphi ) )
% 0.06/0.32 => ( ( ~ ( in @ Xx @ A )
% 0.06/0.32 => ( ( in @ Xx @ B )
% 0.06/0.32 => Xphi ) )
% 0.06/0.32 => Xphi ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(symdiffI1_type,type,
% 0.06/0.32 symdiffI1: $o ).
% 0.06/0.32
% 0.06/0.32 thf(symdiffI1,definition,
% 0.06/0.32 ( symdiffI1
% 0.06/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ( ~ ( in @ Xx @ B )
% 0.06/0.32 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(symdiffI2_type,type,
% 0.06/0.32 symdiffI2: $o ).
% 0.06/0.32
% 0.06/0.32 thf(symdiffI2,definition,
% 0.06/0.32 ( symdiffI2
% 0.06/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.06/0.32 ( ~ ( in @ Xx @ A )
% 0.06/0.32 => ( ( in @ Xx @ B )
% 0.06/0.32 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(symdiffIneg1_type,type,
% 0.06/0.32 symdiffIneg1: $o ).
% 0.06/0.32
% 0.06/0.32 thf(symdiffIneg1,definition,
% 0.06/0.32 ( symdiffIneg1
% 0.06/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ( ( in @ Xx @ B )
% 0.06/0.32 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(symdiffIneg2_type,type,
% 0.06/0.32 symdiffIneg2: $o ).
% 0.06/0.32
% 0.06/0.32 thf(symdiffIneg2,definition,
% 0.06/0.32 ( symdiffIneg2
% 0.06/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.06/0.32 ( ~ ( in @ Xx @ A )
% 0.06/0.32 => ( ~ ( in @ Xx @ B )
% 0.06/0.32 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(iskpair_type,type,
% 0.06/0.32 iskpair: $i > $o ).
% 0.06/0.32
% 0.06/0.32 thf(secondinupair_type,type,
% 0.06/0.32 secondinupair: $o ).
% 0.06/0.32
% 0.06/0.32 thf(secondinupair,definition,
% 0.06/0.32 ( secondinupair
% 0.06/0.32 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(setukpairIL_type,type,
% 0.06/0.32 setukpairIL: $o ).
% 0.06/0.32
% 0.06/0.32 thf(setukpairIL,definition,
% 0.06/0.32 ( setukpairIL
% 0.06/0.32 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(setukpairIR_type,type,
% 0.06/0.32 setukpairIR: $o ).
% 0.06/0.32
% 0.06/0.32 thf(setukpairIR,definition,
% 0.06/0.32 ( setukpairIR
% 0.06/0.32 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(kpairiskpair_type,type,
% 0.06/0.32 kpairiskpair: $o ).
% 0.06/0.32
% 0.06/0.32 thf(kpairiskpair,definition,
% 0.06/0.32 ( kpairiskpair
% 0.06/0.32 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(kpair_type,type,
% 0.06/0.32 kpair: $i > $i > $i ).
% 0.06/0.32
% 0.06/0.32 thf(kpairp_type,type,
% 0.06/0.32 kpairp: $o ).
% 0.06/0.32
% 0.06/0.32 thf(kpairp,definition,
% 0.06/0.32 ( kpairp
% 0.06/0.32 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(cartprod_type,type,
% 0.06/0.32 cartprod: $i > $i > $i ).
% 0.06/0.32
% 0.06/0.32 thf(singletonsubset_type,type,
% 0.06/0.32 singletonsubset: $o ).
% 0.06/0.32
% 0.06/0.32 thf(singletonsubset,definition,
% 0.06/0.32 ( singletonsubset
% 0.06/0.32 = ( ! [A: $i,Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(singletoninpowerset_type,type,
% 0.06/0.32 singletoninpowerset: $o ).
% 0.06/0.32
% 0.06/0.32 thf(singletoninpowerset,definition,
% 0.06/0.32 ( singletoninpowerset
% 0.06/0.32 = ( ! [A: $i,Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(singletoninpowunion_type,type,
% 0.06/0.32 singletoninpowunion: $o ).
% 0.06/0.32
% 0.06/0.32 thf(singletoninpowunion,definition,
% 0.06/0.32 ( singletoninpowunion
% 0.06/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(upairset2E_type,type,
% 0.06/0.32 upairset2E: $o ).
% 0.06/0.32
% 0.06/0.32 thf(upairset2E,definition,
% 0.06/0.32 ( upairset2E
% 0.06/0.32 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.06/0.32 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.06/0.32 => ( ( Xz = Xx )
% 0.06/0.32 | ( Xz = Xy ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(upairsubunion_type,type,
% 0.06/0.32 upairsubunion: $o ).
% 0.06/0.32
% 0.06/0.32 thf(upairsubunion,definition,
% 0.06/0.32 ( upairsubunion
% 0.06/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ! [Xy: $i] :
% 0.06/0.32 ( ( in @ Xy @ B )
% 0.06/0.32 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(upairinpowunion_type,type,
% 0.06/0.32 upairinpowunion: $o ).
% 0.06/0.32
% 0.06/0.32 thf(upairinpowunion,definition,
% 0.06/0.32 ( upairinpowunion
% 0.06/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ! [Xy: $i] :
% 0.06/0.32 ( ( in @ Xy @ B )
% 0.06/0.32 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(ubforcartprodlem1_type,type,
% 0.06/0.32 ubforcartprodlem1: $o ).
% 0.06/0.32
% 0.06/0.32 thf(ubforcartprodlem1,definition,
% 0.06/0.32 ( ubforcartprodlem1
% 0.06/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ! [Xy: $i] :
% 0.06/0.32 ( ( in @ Xy @ B )
% 0.06/0.32 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(ubforcartprodlem2_type,type,
% 0.06/0.32 ubforcartprodlem2: $o ).
% 0.06/0.32
% 0.06/0.32 thf(ubforcartprodlem2,definition,
% 0.06/0.32 ( ubforcartprodlem2
% 0.06/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ! [Xy: $i] :
% 0.06/0.32 ( ( in @ Xy @ B )
% 0.06/0.32 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(ubforcartprodlem3_type,type,
% 0.06/0.32 ubforcartprodlem3: $o ).
% 0.06/0.32
% 0.06/0.32 thf(ubforcartprodlem3,definition,
% 0.06/0.32 ( ubforcartprodlem3
% 0.06/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ! [Xy: $i] :
% 0.06/0.32 ( ( in @ Xy @ B )
% 0.06/0.32 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(cartprodpairin_type,type,
% 0.06/0.32 cartprodpairin: $o ).
% 0.06/0.32
% 0.06/0.32 thf(cartprodpairin,definition,
% 0.06/0.32 ( cartprodpairin
% 0.06/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ! [Xy: $i] :
% 0.06/0.32 ( ( in @ Xy @ B )
% 0.06/0.32 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(cartprodmempair1_type,type,
% 0.06/0.32 cartprodmempair1: $o ).
% 0.06/0.32
% 0.06/0.32 thf(cartprodmempair1,definition,
% 0.06/0.32 ( cartprodmempair1
% 0.06/0.32 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.06/0.32 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.06/0.32 => ? [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 & ? [Xy: $i] :
% 0.06/0.32 ( ( in @ Xy @ B )
% 0.06/0.32 & ( Xu
% 0.06/0.32 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(cartprodmempair_type,type,
% 0.06/0.32 cartprodmempair: $o ).
% 0.06/0.32
% 0.06/0.32 thf(cartprodmempair,definition,
% 0.06/0.32 ( cartprodmempair
% 0.06/0.32 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.06/0.32 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.06/0.32 => ( iskpair @ Xu ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(setunionE2_type,type,
% 0.06/0.32 setunionE2: $o ).
% 0.06/0.32
% 0.06/0.32 thf(setunionE2,definition,
% 0.06/0.32 ( setunionE2
% 0.06/0.32 = ( ! [A: $i,Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ ( setunion @ A ) )
% 0.06/0.32 => ? [X: $i] :
% 0.06/0.32 ( ( in @ X @ A )
% 0.06/0.32 & ( in @ Xx @ X ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(setunionsingleton1_type,type,
% 0.06/0.32 setunionsingleton1: $o ).
% 0.06/0.32
% 0.06/0.32 thf(setunionsingleton1,definition,
% 0.06/0.32 ( setunionsingleton1
% 0.06/0.32 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(setunionsingleton2_type,type,
% 0.06/0.32 setunionsingleton2: $o ).
% 0.06/0.32
% 0.06/0.32 thf(setunionsingleton2,definition,
% 0.06/0.32 ( setunionsingleton2
% 0.06/0.32 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(setunionsingleton_type,type,
% 0.06/0.32 setunionsingleton: $o ).
% 0.06/0.32
% 0.06/0.32 thf(setunionsingleton,definition,
% 0.06/0.32 ( setunionsingleton
% 0.06/0.32 = ( ! [Xx: $i] :
% 0.06/0.32 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 0.06/0.32 = Xx ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(singleton_type,type,
% 0.06/0.32 singleton: $i > $o ).
% 0.06/0.32
% 0.06/0.32 thf(singletonprop_type,type,
% 0.06/0.32 singletonprop: $o ).
% 0.06/0.32
% 0.06/0.32 thf(singletonprop,definition,
% 0.06/0.32 ( singletonprop
% 0.06/0.32 = ( ! [A: $i,Xphi: $i > $o] :
% 0.06/0.32 ( ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ! [Xy: $i] :
% 0.06/0.32 ( ( in @ Xy @ A )
% 0.06/0.32 => ( ( Xphi @ Xx )
% 0.06/0.32 => ( ( Xphi @ Xy )
% 0.06/0.32 => ( Xx = Xy ) ) ) ) )
% 0.06/0.32 => ( ? [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 & ( Xphi @ Xx ) )
% 0.06/0.32 => ( singleton
% 0.06/0.32 @ ( dsetconstr @ A
% 0.06/0.32 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(ex1_type,type,
% 0.06/0.32 ex1: $i > ( $i > $o ) > $o ).
% 0.06/0.32
% 0.06/0.32 thf(ex1E1_type,type,
% 0.06/0.32 ex1E1: $o ).
% 0.06/0.32
% 0.06/0.32 thf(ex1E1,definition,
% 0.06/0.32 ( ex1E1
% 0.06/0.32 = ( ! [A: $i,Xphi: $i > $o] :
% 0.06/0.32 ( ( ex1 @ A
% 0.06/0.32 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.06/0.32 => ? [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(ex1I_type,type,
% 0.06/0.32 ex1I: $o ).
% 0.06/0.32
% 0.06/0.32 thf(ex1I,definition,
% 0.06/0.32 ( ex1I
% 0.06/0.32 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ( ( Xphi @ Xx )
% 0.06/0.32 => ( ! [Xy: $i] :
% 0.06/0.32 ( ( in @ Xy @ A )
% 0.06/0.32 => ( ( Xphi @ Xy )
% 0.06/0.32 => ( Xy = Xx ) ) )
% 0.06/0.32 => ( ex1 @ A
% 0.06/0.32 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(ex1I2_type,type,
% 0.06/0.32 ex1I2: $o ).
% 0.06/0.32
% 0.06/0.32 thf(ex1I2,definition,
% 0.06/0.32 ( ex1I2
% 0.06/0.32 = ( ! [A: $i,Xphi: $i > $o] :
% 0.06/0.32 ( ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ! [Xy: $i] :
% 0.06/0.32 ( ( in @ Xy @ A )
% 0.06/0.32 => ( ( Xphi @ Xx )
% 0.06/0.32 => ( ( Xphi @ Xy )
% 0.06/0.32 => ( Xx = Xy ) ) ) ) )
% 0.06/0.32 => ( ? [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 & ( Xphi @ Xx ) )
% 0.06/0.32 => ( ex1 @ A
% 0.06/0.32 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(singletonsuniq_type,type,
% 0.06/0.32 singletonsuniq: $o ).
% 0.06/0.32
% 0.06/0.32 thf(singletonsuniq,definition,
% 0.06/0.32 ( singletonsuniq
% 0.06/0.32 = ( ! [Xx: $i,Xy: $i] :
% 0.06/0.32 ( ( ( setadjoin @ Xx @ emptyset )
% 0.06/0.32 = ( setadjoin @ Xy @ emptyset ) )
% 0.06/0.32 => ( Xx = Xy ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(atmost1p_type,type,
% 0.06/0.32 atmost1p: $i > $o ).
% 0.06/0.32
% 0.06/0.32 thf(atleast2p_type,type,
% 0.06/0.32 atleast2p: $i > $o ).
% 0.06/0.32
% 0.06/0.32 thf(atmost2p_type,type,
% 0.06/0.32 atmost2p: $i > $o ).
% 0.06/0.32
% 0.06/0.32 thf(upairsetp_type,type,
% 0.06/0.32 upairsetp: $i > $o ).
% 0.06/0.32
% 0.06/0.32 thf(setukpairinjL1_type,type,
% 0.06/0.32 setukpairinjL1: $o ).
% 0.06/0.32
% 0.06/0.32 thf(setukpairinjL1,definition,
% 0.06/0.32 ( setukpairinjL1
% 0.06/0.32 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.06/0.32 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 0.06/0.32 => ( Xx = Xz ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(kfstsingleton_type,type,
% 0.06/0.32 kfstsingleton: $o ).
% 0.06/0.32
% 0.06/0.32 thf(kfstsingleton,definition,
% 0.06/0.32 ( kfstsingleton
% 0.06/0.32 = ( ! [Xu: $i] :
% 0.06/0.32 ( ( iskpair @ Xu )
% 0.06/0.32 => ( singleton
% 0.06/0.32 @ ( dsetconstr @ ( setunion @ Xu )
% 0.06/0.32 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(theprop_type,type,
% 0.06/0.32 theprop: $o ).
% 0.06/0.32
% 0.06/0.32 thf(theprop,definition,
% 0.06/0.32 ( theprop
% 0.06/0.32 = ( ! [X: $i] :
% 0.06/0.32 ( ( singleton @ X )
% 0.06/0.32 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(kfst_type,type,
% 0.06/0.32 kfst: $i > $i ).
% 0.06/0.32
% 0.06/0.32 thf(kfstpairEq_type,type,
% 0.06/0.32 kfstpairEq: $o ).
% 0.06/0.32
% 0.06/0.32 thf(kfstpairEq,definition,
% 0.06/0.32 ( kfstpairEq
% 0.06/0.32 = ( ! [Xx: $i,Xy: $i] :
% 0.06/0.32 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.06/0.32 = Xx ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(cartprodfstin_type,type,
% 0.06/0.32 cartprodfstin: $o ).
% 0.06/0.32
% 0.06/0.32 thf(cartprodfstin,definition,
% 0.06/0.32 ( cartprodfstin
% 0.06/0.32 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.06/0.32 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.06/0.32 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(setukpairinjL2_type,type,
% 0.06/0.32 setukpairinjL2: $o ).
% 0.06/0.32
% 0.06/0.32 thf(setukpairinjL2,definition,
% 0.06/0.32 ( setukpairinjL2
% 0.06/0.32 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.06/0.32 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.06/0.32 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.06/0.32 => ( Xx = Xz ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(setukpairinjL_type,type,
% 0.06/0.32 setukpairinjL: $o ).
% 0.06/0.32
% 0.06/0.32 thf(setukpairinjL,definition,
% 0.06/0.32 ( setukpairinjL
% 0.06/0.32 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.06/0.32 ( ( ( kpair @ Xx @ Xy )
% 0.06/0.32 = ( kpair @ Xz @ Xu ) )
% 0.06/0.32 => ( Xx = Xz ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(setukpairinjR11_type,type,
% 0.06/0.32 setukpairinjR11: $o ).
% 0.06/0.32
% 0.06/0.32 thf(setukpairinjR11,definition,
% 0.06/0.32 ( setukpairinjR11
% 0.06/0.32 = ( ! [Xx: $i,Xy: $i] :
% 0.06/0.32 ( ( Xx = Xy )
% 0.06/0.32 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.06/0.32 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(setukpairinjR12_type,type,
% 0.06/0.32 setukpairinjR12: $o ).
% 0.06/0.32
% 0.06/0.32 thf(setukpairinjR12,definition,
% 0.06/0.32 ( setukpairinjR12
% 0.06/0.32 = ( ! [Xx: $i,Xy: $i] :
% 0.06/0.32 ( ( Xx = Xy )
% 0.06/0.32 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.06/0.32 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(setukpairinjR1_type,type,
% 0.06/0.32 setukpairinjR1: $o ).
% 0.06/0.32
% 0.06/0.32 thf(setukpairinjR1,definition,
% 0.06/0.32 ( setukpairinjR1
% 0.06/0.32 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.06/0.32 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.06/0.32 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.06/0.32 => ( ( Xz = Xu )
% 0.06/0.32 => ( Xy = Xu ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(upairequniteq_type,type,
% 0.06/0.32 upairequniteq: $o ).
% 0.06/0.32
% 0.06/0.32 thf(upairequniteq,definition,
% 0.06/0.32 ( upairequniteq
% 0.06/0.32 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.06/0.32 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.06/0.32 = ( setadjoin @ Xz @ emptyset ) )
% 0.06/0.32 => ( Xx = Xy ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(setukpairinjR2_type,type,
% 0.06/0.32 setukpairinjR2: $o ).
% 0.06/0.32
% 0.06/0.32 thf(setukpairinjR2,definition,
% 0.06/0.32 ( setukpairinjR2
% 0.06/0.32 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.06/0.32 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.06/0.32 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.06/0.32 => ( Xy = Xu ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(setukpairinjR_type,type,
% 0.06/0.32 setukpairinjR: $o ).
% 0.06/0.32
% 0.06/0.32 thf(setukpairinjR,definition,
% 0.06/0.32 ( setukpairinjR
% 0.06/0.32 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.06/0.32 ( ( ( kpair @ Xx @ Xy )
% 0.06/0.32 = ( kpair @ Xz @ Xu ) )
% 0.06/0.32 => ( Xy = Xu ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(ksndsingleton_type,type,
% 0.06/0.32 ksndsingleton: $o ).
% 0.06/0.32
% 0.06/0.32 thf(ksndsingleton,definition,
% 0.06/0.32 ( ksndsingleton
% 0.06/0.32 = ( ! [Xu: $i] :
% 0.06/0.32 ( ( iskpair @ Xu )
% 0.06/0.32 => ( singleton
% 0.06/0.32 @ ( dsetconstr @ ( setunion @ Xu )
% 0.06/0.32 @ ^ [Xx: $i] :
% 0.06/0.32 ( Xu
% 0.06/0.32 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(ksnd_type,type,
% 0.06/0.32 ksnd: $i > $i ).
% 0.06/0.32
% 0.06/0.32 thf(ksndpairEq_type,type,
% 0.06/0.32 ksndpairEq: $o ).
% 0.06/0.32
% 0.06/0.32 thf(ksndpairEq,definition,
% 0.06/0.32 ( ksndpairEq
% 0.06/0.32 = ( ! [Xx: $i,Xy: $i] :
% 0.06/0.32 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.06/0.32 = Xy ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(kpairsurjEq_type,type,
% 0.06/0.32 kpairsurjEq: $o ).
% 0.06/0.32
% 0.06/0.32 thf(kpairsurjEq,definition,
% 0.06/0.32 ( kpairsurjEq
% 0.06/0.32 = ( ! [Xu: $i] :
% 0.06/0.32 ( ( iskpair @ Xu )
% 0.06/0.32 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.06/0.32 = Xu ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(cartprodsndin_type,type,
% 0.06/0.32 cartprodsndin: $o ).
% 0.06/0.32
% 0.06/0.32 thf(cartprodsndin,definition,
% 0.06/0.32 ( cartprodsndin
% 0.06/0.32 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.06/0.32 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.06/0.32 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(cartprodpairmemEL_type,type,
% 0.06/0.32 cartprodpairmemEL: $o ).
% 0.06/0.32
% 0.06/0.32 thf(cartprodpairmemEL,definition,
% 0.06/0.32 ( cartprodpairmemEL
% 0.06/0.32 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.06/0.32 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.06/0.32 => ( in @ Xx @ A ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(cartprodpairmemER_type,type,
% 0.06/0.32 cartprodpairmemER: $o ).
% 0.06/0.32
% 0.06/0.32 thf(cartprodpairmemER,definition,
% 0.06/0.32 ( cartprodpairmemER
% 0.06/0.32 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.06/0.32 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.06/0.32 => ( in @ Xy @ B ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(cartprodmempaircEq_type,type,
% 0.06/0.32 cartprodmempaircEq: $o ).
% 0.06/0.32
% 0.06/0.32 thf(cartprodmempaircEq,definition,
% 0.06/0.32 ( cartprodmempaircEq
% 0.06/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ! [Xy: $i] :
% 0.06/0.32 ( ( in @ Xy @ B )
% 0.06/0.32 => ( ( kpair @ Xx @ Xy )
% 0.06/0.32 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(cartprodfstpairEq_type,type,
% 0.06/0.32 cartprodfstpairEq: $o ).
% 0.06/0.32
% 0.06/0.32 thf(cartprodfstpairEq,definition,
% 0.06/0.32 ( cartprodfstpairEq
% 0.06/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ! [Xy: $i] :
% 0.06/0.32 ( ( in @ Xy @ B )
% 0.06/0.32 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.06/0.32 = Xx ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(cartprodsndpairEq_type,type,
% 0.06/0.32 cartprodsndpairEq: $o ).
% 0.06/0.32
% 0.06/0.32 thf(cartprodsndpairEq,definition,
% 0.06/0.32 ( cartprodsndpairEq
% 0.06/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ! [Xy: $i] :
% 0.06/0.32 ( ( in @ Xy @ B )
% 0.06/0.32 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.06/0.32 = Xy ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(cartprodpairsurjEq_type,type,
% 0.06/0.32 cartprodpairsurjEq: $o ).
% 0.06/0.32
% 0.06/0.32 thf(cartprodpairsurjEq,definition,
% 0.06/0.32 ( cartprodpairsurjEq
% 0.06/0.32 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.06/0.32 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.06/0.32 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.06/0.32 = Xu ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(breln_type,type,
% 0.06/0.32 breln: $i > $i > $i > $o ).
% 0.06/0.32
% 0.06/0.32 thf(dpsetconstr_type,type,
% 0.06/0.32 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 0.06/0.32
% 0.06/0.32 thf(dpsetconstrI_type,type,
% 0.06/0.32 dpsetconstrI: $o ).
% 0.06/0.32
% 0.06/0.32 thf(dpsetconstrI,definition,
% 0.06/0.32 ( dpsetconstrI
% 0.06/0.32 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ! [Xy: $i] :
% 0.06/0.32 ( ( in @ Xy @ B )
% 0.06/0.32 => ( ( Xphi @ Xx @ Xy )
% 0.06/0.32 => ( in @ ( kpair @ Xx @ Xy )
% 0.06/0.32 @ ( dpsetconstr @ A @ B
% 0.06/0.32 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(dpsetconstrSub_type,type,
% 0.06/0.32 dpsetconstrSub: $o ).
% 0.06/0.32
% 0.06/0.32 thf(dpsetconstrSub,definition,
% 0.06/0.32 ( dpsetconstrSub
% 0.06/0.32 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.06/0.32 ( subset
% 0.06/0.32 @ ( dpsetconstr @ A @ B
% 0.06/0.32 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 0.06/0.32 @ ( cartprod @ A @ B ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(setOfPairsIsBReln_type,type,
% 0.06/0.32 setOfPairsIsBReln: $o ).
% 0.06/0.32
% 0.06/0.32 thf(setOfPairsIsBReln,definition,
% 0.06/0.32 ( setOfPairsIsBReln
% 0.06/0.32 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.06/0.32 ( breln @ A @ B
% 0.06/0.32 @ ( dpsetconstr @ A @ B
% 0.06/0.32 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(dpsetconstrERa_type,type,
% 0.06/0.32 dpsetconstrERa: $o ).
% 0.06/0.32
% 0.06/0.32 thf(dpsetconstrERa,definition,
% 0.06/0.32 ( dpsetconstrERa
% 0.06/0.32 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ! [Xy: $i] :
% 0.06/0.32 ( ( in @ Xy @ B )
% 0.06/0.32 => ( ( in @ ( kpair @ Xx @ Xy )
% 0.06/0.32 @ ( dpsetconstr @ A @ B
% 0.06/0.32 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.06/0.32 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(dpsetconstrEL1_type,type,
% 0.06/0.32 dpsetconstrEL1: $o ).
% 0.06/0.32
% 0.06/0.32 thf(dpsetconstrEL1,definition,
% 0.06/0.32 ( dpsetconstrEL1
% 0.06/0.32 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.06/0.32 ( ( in @ ( kpair @ Xx @ Xy )
% 0.06/0.32 @ ( dpsetconstr @ A @ B
% 0.06/0.32 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.06/0.32 => ( in @ Xx @ A ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(dpsetconstrEL2_type,type,
% 0.06/0.32 dpsetconstrEL2: $o ).
% 0.06/0.32
% 0.06/0.32 thf(dpsetconstrEL2,definition,
% 0.06/0.32 ( dpsetconstrEL2
% 0.06/0.32 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.06/0.32 ( ( in @ ( kpair @ Xx @ Xy )
% 0.06/0.32 @ ( dpsetconstr @ A @ B
% 0.06/0.32 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.06/0.32 => ( in @ Xy @ B ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(dpsetconstrER_type,type,
% 0.06/0.32 dpsetconstrER: $o ).
% 0.06/0.32
% 0.06/0.32 thf(dpsetconstrER,definition,
% 0.06/0.32 ( dpsetconstrER
% 0.06/0.32 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.06/0.32 ( ( in @ ( kpair @ Xx @ Xy )
% 0.06/0.32 @ ( dpsetconstr @ A @ B
% 0.06/0.32 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.06/0.32 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(func_type,type,
% 0.06/0.32 func: $i > $i > $i > $o ).
% 0.06/0.32
% 0.06/0.32 thf(funcSet_type,type,
% 0.06/0.32 funcSet: $i > $i > $i ).
% 0.06/0.32
% 0.06/0.32 thf(funcImageSingleton_type,type,
% 0.06/0.32 funcImageSingleton: $o ).
% 0.06/0.32
% 0.06/0.32 thf(funcImageSingleton,definition,
% 0.06/0.32 ( funcImageSingleton
% 0.06/0.32 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.06/0.32 ( ( func @ A @ B @ Xf )
% 0.06/0.32 => ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ( singleton
% 0.06/0.32 @ ( dsetconstr @ B
% 0.06/0.32 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(apProp_type,type,
% 0.06/0.32 apProp: $o ).
% 0.06/0.32
% 0.06/0.32 thf(apProp,definition,
% 0.06/0.32 ( apProp
% 0.06/0.32 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.06/0.32 ( ( func @ A @ B @ Xf )
% 0.06/0.32 => ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ( in
% 0.06/0.32 @ ( setunion
% 0.06/0.32 @ ( dsetconstr @ B
% 0.06/0.32 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 0.06/0.32 @ B ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(ap_type,type,
% 0.06/0.32 ap: $i > $i > $i > $i > $i ).
% 0.06/0.32
% 0.06/0.32 thf(app_type,type,
% 0.06/0.32 app: $o ).
% 0.06/0.32
% 0.06/0.32 thf(app,definition,
% 0.06/0.32 ( app
% 0.06/0.32 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.06/0.32 ( ( func @ A @ B @ Xf )
% 0.06/0.32 => ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(infuncsetfunc_type,type,
% 0.06/0.32 infuncsetfunc: $o ).
% 0.06/0.32
% 0.06/0.32 thf(infuncsetfunc,definition,
% 0.06/0.32 ( infuncsetfunc
% 0.06/0.32 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.06/0.32 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.06/0.32 => ( func @ A @ B @ Xf ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(ap2p_type,type,
% 0.06/0.32 ap2p: $o ).
% 0.06/0.32
% 0.06/0.32 thf(ap2p,definition,
% 0.06/0.32 ( ap2p
% 0.06/0.32 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.06/0.32 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.06/0.32 => ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(funcinfuncset_type,type,
% 0.06/0.32 funcinfuncset: $o ).
% 0.06/0.32
% 0.06/0.32 thf(funcinfuncset,definition,
% 0.06/0.32 ( funcinfuncset
% 0.06/0.32 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.06/0.32 ( ( func @ A @ B @ Xf )
% 0.06/0.32 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(lamProp_type,type,
% 0.06/0.32 lamProp: $o ).
% 0.06/0.32
% 0.06/0.32 thf(lamProp,definition,
% 0.06/0.32 ( lamProp
% 0.06/0.32 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.06/0.32 ( ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.06/0.32 => ( func @ A @ B
% 0.06/0.32 @ ( dpsetconstr @ A @ B
% 0.06/0.32 @ ^ [Xx: $i,Xy: $i] :
% 0.06/0.32 ( ( Xf @ Xx )
% 0.06/0.32 = Xy ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(lam_type,type,
% 0.06/0.32 lam: $i > $i > ( $i > $i ) > $i ).
% 0.06/0.32
% 0.06/0.32 thf(lamp_type,type,
% 0.06/0.32 lamp: $o ).
% 0.06/0.32
% 0.06/0.32 thf(lamp,definition,
% 0.06/0.32 ( lamp
% 0.06/0.32 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.06/0.32 ( ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.06/0.32 => ( func @ A @ B
% 0.06/0.32 @ ( lam @ A @ B
% 0.06/0.32 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(lam2p_type,type,
% 0.06/0.32 lam2p: $o ).
% 0.06/0.32
% 0.06/0.32 thf(lam2p,definition,
% 0.06/0.32 ( lam2p
% 0.06/0.32 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.06/0.32 ( ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.06/0.32 => ( in
% 0.06/0.32 @ ( lam @ A @ B
% 0.06/0.32 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.06/0.32 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(brelnall1_type,type,
% 0.06/0.32 brelnall1: $o ).
% 0.06/0.32
% 0.06/0.32 thf(brelnall1,definition,
% 0.06/0.32 ( brelnall1
% 0.06/0.32 = ( ! [A: $i,B: $i,R: $i] :
% 0.06/0.32 ( ( breln @ A @ B @ R )
% 0.06/0.32 => ! [Xphi: $i > $o] :
% 0.06/0.32 ( ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ! [Xy: $i] :
% 0.06/0.32 ( ( in @ Xy @ B )
% 0.06/0.32 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.06/0.32 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.06/0.32 => ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ R )
% 0.06/0.32 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(brelnall2_type,type,
% 0.06/0.32 brelnall2: $o ).
% 0.06/0.32
% 0.06/0.32 thf(brelnall2,definition,
% 0.06/0.32 ( brelnall2
% 0.06/0.32 = ( ! [A: $i,B: $i,R: $i] :
% 0.06/0.32 ( ( breln @ A @ B @ R )
% 0.06/0.32 => ! [Xphi: $i > $o] :
% 0.06/0.32 ( ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ! [Xy: $i] :
% 0.06/0.32 ( ( in @ Xy @ B )
% 0.06/0.32 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.06/0.32 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.06/0.32 => ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ R )
% 0.06/0.32 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(ex1E2_type,type,
% 0.06/0.32 ex1E2: $o ).
% 0.06/0.32
% 0.06/0.32 thf(ex1E2,definition,
% 0.06/0.32 ( ex1E2
% 0.06/0.32 = ( ! [A: $i,Xphi: $i > $o] :
% 0.06/0.32 ( ( ex1 @ A
% 0.06/0.32 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.06/0.32 => ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ! [Xy: $i] :
% 0.06/0.32 ( ( in @ Xy @ A )
% 0.06/0.32 => ( ( Xphi @ Xx )
% 0.06/0.32 => ( ( Xphi @ Xy )
% 0.06/0.32 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(funcGraphProp1_type,type,
% 0.06/0.32 funcGraphProp1: $o ).
% 0.06/0.32
% 0.06/0.32 thf(funcGraphProp1,definition,
% 0.06/0.32 ( funcGraphProp1
% 0.06/0.32 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.06/0.32 ( ( func @ A @ B @ Xf )
% 0.06/0.32 => ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(funcGraphProp3_type,type,
% 0.06/0.32 funcGraphProp3: $o ).
% 0.06/0.32
% 0.06/0.32 thf(funcGraphProp3,definition,
% 0.06/0.32 ( funcGraphProp3
% 0.06/0.32 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.06/0.32 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.06/0.32 => ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(funcGraphProp2_type,type,
% 0.06/0.32 funcGraphProp2: $o ).
% 0.06/0.32
% 0.06/0.32 thf(funcGraphProp2,definition,
% 0.06/0.32 ( funcGraphProp2
% 0.06/0.32 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.06/0.32 ( ( func @ A @ B @ Xf )
% 0.06/0.32 => ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ! [Xy: $i] :
% 0.06/0.32 ( ( in @ Xy @ B )
% 0.06/0.32 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.06/0.32 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.06/0.32 = Xy ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(funcextLem_type,type,
% 0.06/0.32 funcextLem: $o ).
% 0.06/0.32
% 0.06/0.32 thf(funcextLem,definition,
% 0.06/0.32 ( funcextLem
% 0.06/0.32 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.06/0.32 ( ( func @ A @ B @ Xf )
% 0.06/0.32 => ! [Xg: $i] :
% 0.06/0.32 ( ( func @ A @ B @ Xg )
% 0.06/0.32 => ( ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.06/0.32 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.06/0.32 => ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ! [Xy: $i] :
% 0.06/0.32 ( ( in @ Xy @ B )
% 0.06/0.32 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 0.06/0.32 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(funcGraphProp4_type,type,
% 0.06/0.32 funcGraphProp4: $o ).
% 0.06/0.32
% 0.06/0.32 thf(funcGraphProp4,definition,
% 0.06/0.32 ( funcGraphProp4
% 0.06/0.32 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.06/0.32 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.06/0.32 => ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ! [Xy: $i] :
% 0.06/0.32 ( ( in @ Xy @ B )
% 0.06/0.32 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.06/0.32 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.06/0.32 = Xy ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(subbreln_type,type,
% 0.06/0.32 subbreln: $o ).
% 0.06/0.32
% 0.06/0.32 thf(subbreln,definition,
% 0.06/0.32 ( subbreln
% 0.06/0.32 = ( ! [A: $i,B: $i,R: $i] :
% 0.06/0.32 ( ( breln @ A @ B @ R )
% 0.06/0.32 => ! [S: $i] :
% 0.06/0.32 ( ( breln @ A @ B @ S )
% 0.06/0.32 => ( ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ! [Xy: $i] :
% 0.06/0.32 ( ( in @ Xy @ B )
% 0.06/0.32 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.06/0.32 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.06/0.32 => ( subset @ R @ S ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(eqbreln_type,type,
% 0.06/0.32 eqbreln: $o ).
% 0.06/0.32
% 0.06/0.32 thf(eqbreln,definition,
% 0.06/0.32 ( eqbreln
% 0.06/0.32 = ( ! [A: $i,B: $i,R: $i] :
% 0.06/0.32 ( ( breln @ A @ B @ R )
% 0.06/0.32 => ! [S: $i] :
% 0.06/0.32 ( ( breln @ A @ B @ S )
% 0.06/0.32 => ( ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ! [Xy: $i] :
% 0.06/0.32 ( ( in @ Xy @ B )
% 0.06/0.32 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.06/0.32 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.06/0.32 => ( ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ! [Xy: $i] :
% 0.06/0.32 ( ( in @ Xy @ B )
% 0.06/0.32 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.06/0.32 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 0.06/0.32 => ( R = S ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(funcext_type,type,
% 0.06/0.32 funcext: $o ).
% 0.06/0.32
% 0.06/0.32 thf(funcext,definition,
% 0.06/0.32 ( funcext
% 0.06/0.32 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.06/0.32 ( ( func @ A @ B @ Xf )
% 0.06/0.32 => ! [Xg: $i] :
% 0.06/0.32 ( ( func @ A @ B @ Xg )
% 0.06/0.32 => ( ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.06/0.32 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.06/0.32 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(funcext2_type,type,
% 0.06/0.32 funcext2: $o ).
% 0.06/0.32
% 0.06/0.32 thf(funcext2,definition,
% 0.06/0.32 ( funcext2
% 0.06/0.32 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.06/0.32 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.06/0.32 => ! [Xg: $i] :
% 0.06/0.32 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 0.06/0.32 => ( ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.06/0.32 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.06/0.32 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(ap2apEq1_type,type,
% 0.06/0.32 ap2apEq1: $o ).
% 0.06/0.32
% 0.06/0.32 thf(ap2apEq1,definition,
% 0.06/0.32 ( ap2apEq1
% 0.06/0.32 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.06/0.32 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.06/0.32 => ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.06/0.32 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(ap2apEq2_type,type,
% 0.06/0.32 ap2apEq2: $o ).
% 0.06/0.32
% 0.06/0.32 thf(ap2apEq2,definition,
% 0.06/0.32 ( ap2apEq2
% 0.06/0.32 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.06/0.32 ( ( func @ A @ B @ Xf )
% 0.06/0.32 => ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.06/0.32 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(beta1_type,type,
% 0.06/0.32 beta1: $o ).
% 0.06/0.32
% 0.06/0.32 thf(beta1,definition,
% 0.06/0.32 ( beta1
% 0.06/0.32 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.06/0.32 ( ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.06/0.32 => ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ( ( ap @ A @ B
% 0.06/0.32 @ ( lam @ A @ B
% 0.06/0.32 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.06/0.32 @ Xx )
% 0.06/0.32 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(eta1_type,type,
% 0.06/0.32 eta1: $o ).
% 0.06/0.32
% 0.06/0.32 thf(eta1,definition,
% 0.06/0.32 ( eta1
% 0.06/0.32 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.06/0.32 ( ( func @ A @ B @ Xf )
% 0.06/0.32 => ( ( lam @ A @ B
% 0.06/0.32 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.06/0.32 = Xf ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(lam2lamEq_type,type,
% 0.06/0.32 lam2lamEq: $o ).
% 0.06/0.32
% 0.06/0.32 thf(lam2lamEq,definition,
% 0.06/0.32 ( lam2lamEq
% 0.06/0.32 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.06/0.32 ( ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.06/0.32 => ( ( lam @ A @ B
% 0.06/0.32 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.06/0.32 = ( lam @ A @ B
% 0.06/0.32 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(beta2_type,type,
% 0.06/0.32 beta2: $o ).
% 0.06/0.32
% 0.06/0.32 thf(beta2,definition,
% 0.06/0.32 ( beta2
% 0.06/0.32 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.06/0.32 ( ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.06/0.32 => ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ( ( ap @ A @ B
% 0.06/0.32 @ ( lam @ A @ B
% 0.06/0.32 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.06/0.32 @ Xx )
% 0.06/0.32 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(eta2_type,type,
% 0.06/0.32 eta2: $o ).
% 0.06/0.32
% 0.06/0.32 thf(eta2,definition,
% 0.06/0.32 ( eta2
% 0.06/0.32 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.06/0.32 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.06/0.32 => ( ( lam @ A @ B
% 0.06/0.32 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.06/0.32 = Xf ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(iffalseProp1_type,type,
% 0.06/0.32 iffalseProp1: $o ).
% 0.06/0.32
% 0.06/0.32 thf(iffalseProp1,definition,
% 0.06/0.32 ( iffalseProp1
% 0.06/0.32 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ! [Xy: $i] :
% 0.06/0.32 ( ( in @ Xy @ A )
% 0.06/0.32 => ( ~ Xphi
% 0.06/0.32 => ( in @ Xy
% 0.06/0.32 @ ( dsetconstr @ A
% 0.06/0.32 @ ^ [Xz: $i] :
% 0.06/0.32 ( ( Xphi
% 0.06/0.32 & ( Xz = Xx ) )
% 0.06/0.32 | ( ~ Xphi
% 0.06/0.32 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(iffalseProp2_type,type,
% 0.06/0.32 iffalseProp2: $o ).
% 0.06/0.32
% 0.06/0.32 thf(iffalseProp2,definition,
% 0.06/0.32 ( iffalseProp2
% 0.06/0.32 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ! [Xy: $i] :
% 0.06/0.32 ( ( in @ Xy @ A )
% 0.06/0.32 => ( ~ Xphi
% 0.06/0.32 => ( ( dsetconstr @ A
% 0.06/0.32 @ ^ [Xz: $i] :
% 0.06/0.32 ( ( Xphi
% 0.06/0.32 & ( Xz = Xx ) )
% 0.06/0.32 | ( ~ Xphi
% 0.06/0.32 & ( Xz = Xy ) ) ) )
% 0.06/0.32 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(iftrueProp1_type,type,
% 0.06/0.32 iftrueProp1: $o ).
% 0.06/0.32
% 0.06/0.32 thf(iftrueProp1,definition,
% 0.06/0.32 ( iftrueProp1
% 0.06/0.32 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ! [Xy: $i] :
% 0.06/0.32 ( ( in @ Xy @ A )
% 0.06/0.32 => ( Xphi
% 0.06/0.32 => ( in @ Xx
% 0.06/0.32 @ ( dsetconstr @ A
% 0.06/0.32 @ ^ [Xz: $i] :
% 0.06/0.32 ( ( Xphi
% 0.06/0.32 & ( Xz = Xx ) )
% 0.06/0.32 | ( ~ Xphi
% 0.06/0.32 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(iftrueProp2_type,type,
% 0.06/0.32 iftrueProp2: $o ).
% 0.06/0.32
% 0.06/0.32 thf(iftrueProp2,definition,
% 0.06/0.32 ( iftrueProp2
% 0.06/0.32 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ! [Xy: $i] :
% 0.06/0.32 ( ( in @ Xy @ A )
% 0.06/0.32 => ( Xphi
% 0.06/0.32 => ( ( dsetconstr @ A
% 0.06/0.32 @ ^ [Xz: $i] :
% 0.06/0.32 ( ( Xphi
% 0.06/0.32 & ( Xz = Xx ) )
% 0.06/0.32 | ( ~ Xphi
% 0.06/0.32 & ( Xz = Xy ) ) ) )
% 0.06/0.32 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(ifSingleton_type,type,
% 0.06/0.32 ifSingleton: $o ).
% 0.06/0.32
% 0.06/0.32 thf(ifSingleton,definition,
% 0.06/0.32 ( ifSingleton
% 0.06/0.32 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ! [Xy: $i] :
% 0.06/0.32 ( ( in @ Xy @ A )
% 0.06/0.32 => ( singleton
% 0.06/0.32 @ ( dsetconstr @ A
% 0.06/0.32 @ ^ [Xz: $i] :
% 0.06/0.32 ( ( Xphi
% 0.06/0.32 & ( Xz = Xx ) )
% 0.06/0.32 | ( ~ Xphi
% 0.06/0.32 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(if_type,type,
% 0.06/0.32 if: $i > $o > $i > $i > $i ).
% 0.06/0.32
% 0.06/0.32 thf(ifp_type,type,
% 0.06/0.32 ifp: $o ).
% 0.06/0.32
% 0.06/0.32 thf(ifp,definition,
% 0.06/0.32 ( ifp
% 0.06/0.32 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ! [Xy: $i] :
% 0.06/0.32 ( ( in @ Xy @ A )
% 0.06/0.32 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(theeq_type,type,
% 0.06/0.32 theeq: $o ).
% 0.06/0.32
% 0.06/0.32 thf(theeq,definition,
% 0.06/0.32 ( theeq
% 0.06/0.32 = ( ! [X: $i] :
% 0.06/0.32 ( ( singleton @ X )
% 0.06/0.32 => ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ X )
% 0.06/0.32 => ( ( setunion @ X )
% 0.06/0.32 = Xx ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(iftrue_type,type,
% 0.06/0.32 iftrue: $o ).
% 0.06/0.32
% 0.06/0.32 thf(iftrue,definition,
% 0.06/0.32 ( iftrue
% 0.06/0.32 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ! [Xy: $i] :
% 0.06/0.32 ( ( in @ Xy @ A )
% 0.06/0.32 => ( Xphi
% 0.06/0.32 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.06/0.32 = Xx ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(iffalse_type,type,
% 0.06/0.32 iffalse: $o ).
% 0.06/0.32
% 0.06/0.32 thf(iffalse,definition,
% 0.06/0.32 ( iffalse
% 0.06/0.32 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ! [Xy: $i] :
% 0.06/0.32 ( ( in @ Xy @ A )
% 0.06/0.32 => ( ~ Xphi
% 0.06/0.32 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.06/0.32 = Xy ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(iftrueorfalse_type,type,
% 0.06/0.32 iftrueorfalse: $o ).
% 0.06/0.32
% 0.06/0.32 thf(iftrueorfalse,definition,
% 0.06/0.32 ( iftrueorfalse
% 0.06/0.32 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ! [Xy: $i] :
% 0.06/0.32 ( ( in @ Xy @ A )
% 0.06/0.32 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(binintersectT_lem_type,type,
% 0.06/0.32 binintersectT_lem: $o ).
% 0.06/0.32
% 0.06/0.32 thf(binintersectT_lem,definition,
% 0.06/0.32 ( binintersectT_lem
% 0.06/0.32 = ( ! [A: $i,X: $i] :
% 0.06/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.06/0.32 => ! [Y: $i] :
% 0.06/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.06/0.32 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(binunionT_lem_type,type,
% 0.06/0.32 binunionT_lem: $o ).
% 0.06/0.32
% 0.06/0.32 thf(binunionT_lem,definition,
% 0.06/0.32 ( binunionT_lem
% 0.06/0.32 = ( ! [A: $i,X: $i] :
% 0.06/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.06/0.32 => ! [Y: $i] :
% 0.06/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.06/0.32 => ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(powersetT_lem_type,type,
% 0.06/0.32 powersetT_lem: $o ).
% 0.06/0.32
% 0.06/0.32 thf(powersetT_lem,definition,
% 0.06/0.32 ( powersetT_lem
% 0.06/0.32 = ( ! [A: $i,X: $i] :
% 0.06/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.06/0.32 => ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(setminusT_lem_type,type,
% 0.06/0.32 setminusT_lem: $o ).
% 0.06/0.32
% 0.06/0.32 thf(setminusT_lem,definition,
% 0.06/0.32 ( setminusT_lem
% 0.06/0.32 = ( ! [A: $i,X: $i] :
% 0.06/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.06/0.32 => ! [Y: $i] :
% 0.06/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.06/0.32 => ( in @ ( setminus @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(complementT_lem_type,type,
% 0.06/0.32 complementT_lem: $o ).
% 0.06/0.32
% 0.06/0.32 thf(complementT_lem,definition,
% 0.06/0.32 ( complementT_lem
% 0.06/0.32 = ( ! [A: $i,X: $i] :
% 0.06/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.06/0.32 => ( in @ ( setminus @ A @ X ) @ ( powerset @ A ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(setextT_type,type,
% 0.06/0.32 setextT: $o ).
% 0.06/0.32
% 0.06/0.32 thf(setextT,definition,
% 0.06/0.32 ( setextT
% 0.06/0.32 = ( ! [A: $i,X: $i] :
% 0.06/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.06/0.32 => ! [Y: $i] :
% 0.06/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.06/0.32 => ( ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ( ( in @ Xx @ X )
% 0.06/0.32 => ( in @ Xx @ Y ) ) )
% 0.06/0.32 => ( ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ( ( in @ Xx @ Y )
% 0.06/0.32 => ( in @ Xx @ X ) ) )
% 0.06/0.32 => ( X = Y ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(subsetTI_type,type,
% 0.06/0.32 subsetTI: $o ).
% 0.06/0.32
% 0.06/0.32 thf(subsetTI,definition,
% 0.06/0.32 ( subsetTI
% 0.06/0.32 = ( ! [A: $i,X: $i] :
% 0.06/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.06/0.32 => ! [Y: $i] :
% 0.06/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.06/0.32 => ( ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ( ( in @ Xx @ X )
% 0.06/0.32 => ( in @ Xx @ Y ) ) )
% 0.06/0.32 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(powersetTI1_type,type,
% 0.06/0.32 powersetTI1: $o ).
% 0.06/0.32
% 0.06/0.32 thf(powersetTI1,definition,
% 0.06/0.32 ( powersetTI1
% 0.06/0.32 = ( ! [A: $i,X: $i] :
% 0.06/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.06/0.32 => ! [Y: $i] :
% 0.06/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.06/0.32 => ( ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ( ( in @ Xx @ X )
% 0.06/0.32 => ( in @ Xx @ Y ) ) )
% 0.06/0.32 => ( in @ X @ ( powerset @ Y ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(powersetTE1_type,type,
% 0.06/0.32 powersetTE1: $o ).
% 0.06/0.32
% 0.06/0.32 thf(powersetTE1,definition,
% 0.06/0.32 ( powersetTE1
% 0.06/0.32 = ( ! [A: $i,X: $i] :
% 0.06/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.06/0.32 => ! [Y: $i] :
% 0.06/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.06/0.32 => ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ( ( in @ X @ ( powerset @ Y ) )
% 0.06/0.32 => ( ( in @ Xx @ X )
% 0.06/0.32 => ( in @ Xx @ Y ) ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(complementTI1_type,type,
% 0.06/0.32 complementTI1: $o ).
% 0.06/0.32
% 0.06/0.32 thf(complementTI1,definition,
% 0.06/0.32 ( complementTI1
% 0.06/0.32 = ( ! [A: $i,X: $i] :
% 0.06/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.06/0.32 => ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ( ( in @ Xx @ X )
% 0.06/0.32 => ~ ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(complementTE1_type,type,
% 0.06/0.32 complementTE1: $o ).
% 0.06/0.32
% 0.06/0.32 thf(complementTE1,definition,
% 0.06/0.32 ( complementTE1
% 0.06/0.32 = ( ! [A: $i,X: $i] :
% 0.06/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.06/0.32 => ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ( ~ ( in @ Xx @ ( setminus @ A @ X ) )
% 0.06/0.32 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(binintersectTELcontra_type,type,
% 0.06/0.32 binintersectTELcontra: $o ).
% 0.06/0.32
% 0.06/0.32 thf(binintersectTELcontra,definition,
% 0.06/0.32 ( binintersectTELcontra
% 0.06/0.32 = ( ! [A: $i,X: $i] :
% 0.06/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.06/0.32 => ! [Y: $i] :
% 0.06/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.06/0.32 => ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ( ~ ( in @ Xx @ X )
% 0.06/0.32 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(binintersectTERcontra_type,type,
% 0.06/0.32 binintersectTERcontra: $o ).
% 0.06/0.32
% 0.06/0.32 thf(binintersectTERcontra,definition,
% 0.06/0.32 ( binintersectTERcontra
% 0.06/0.32 = ( ! [A: $i,X: $i] :
% 0.06/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.06/0.32 => ! [Y: $i] :
% 0.06/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.06/0.32 => ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ( ~ ( in @ Xx @ Y )
% 0.06/0.32 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(contrasubsetT_type,type,
% 0.06/0.32 contrasubsetT: $o ).
% 0.06/0.32
% 0.06/0.32 thf(contrasubsetT,definition,
% 0.06/0.32 ( contrasubsetT
% 0.06/0.32 = ( ! [A: $i,X: $i] :
% 0.06/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.06/0.32 => ! [Y: $i] :
% 0.06/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.06/0.32 => ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.06/0.32 => ( ( in @ Xx @ Y )
% 0.06/0.32 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(contrasubsetT1_type,type,
% 0.06/0.32 contrasubsetT1: $o ).
% 0.06/0.32
% 0.06/0.32 thf(contrasubsetT1,definition,
% 0.06/0.32 ( contrasubsetT1
% 0.06/0.32 = ( ! [A: $i,X: $i] :
% 0.06/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.06/0.32 => ! [Y: $i] :
% 0.06/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.06/0.32 => ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ( ( subset @ X @ Y )
% 0.06/0.32 => ( ~ ( in @ Xx @ Y )
% 0.06/0.32 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(contrasubsetT2_type,type,
% 0.06/0.32 contrasubsetT2: $o ).
% 0.06/0.32
% 0.06/0.32 thf(contrasubsetT2,definition,
% 0.06/0.32 ( contrasubsetT2
% 0.06/0.32 = ( ! [A: $i,X: $i] :
% 0.06/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.06/0.32 => ! [Y: $i] :
% 0.06/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.06/0.32 => ( ( subset @ X @ Y )
% 0.06/0.32 => ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(contrasubsetT3_type,type,
% 0.06/0.32 contrasubsetT3: $o ).
% 0.06/0.32
% 0.06/0.32 thf(contrasubsetT3,definition,
% 0.06/0.32 ( contrasubsetT3
% 0.06/0.32 = ( ! [A: $i,X: $i] :
% 0.06/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.06/0.32 => ! [Y: $i] :
% 0.06/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.06/0.32 => ( ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) )
% 0.06/0.32 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(doubleComplementI1_type,type,
% 0.06/0.32 doubleComplementI1: $o ).
% 0.06/0.32
% 0.06/0.32 thf(doubleComplementI1,definition,
% 0.06/0.32 ( doubleComplementI1
% 0.06/0.32 = ( ! [A: $i,X: $i] :
% 0.06/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.06/0.32 => ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ( ( in @ Xx @ X )
% 0.06/0.32 => ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(doubleComplementE1_type,type,
% 0.06/0.32 doubleComplementE1: $o ).
% 0.06/0.32
% 0.06/0.32 thf(doubleComplementE1,definition,
% 0.06/0.32 ( doubleComplementE1
% 0.06/0.32 = ( ! [A: $i,X: $i] :
% 0.06/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.06/0.32 => ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ( ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) )
% 0.06/0.32 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(doubleComplementSub1_type,type,
% 0.06/0.32 doubleComplementSub1: $o ).
% 0.06/0.32
% 0.06/0.32 thf(doubleComplementSub1,definition,
% 0.06/0.32 ( doubleComplementSub1
% 0.06/0.32 = ( ! [A: $i,X: $i] :
% 0.06/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.06/0.32 => ( subset @ X @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(doubleComplementSub2_type,type,
% 0.06/0.32 doubleComplementSub2: $o ).
% 0.06/0.32
% 0.06/0.32 thf(doubleComplementSub2,definition,
% 0.06/0.32 ( doubleComplementSub2
% 0.06/0.32 = ( ! [A: $i,X: $i] :
% 0.06/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.06/0.32 => ( subset @ ( setminus @ A @ ( setminus @ A @ X ) ) @ X ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(doubleComplementEq_type,type,
% 0.06/0.32 doubleComplementEq: $o ).
% 0.06/0.32
% 0.06/0.32 thf(doubleComplementEq,definition,
% 0.06/0.32 ( doubleComplementEq
% 0.06/0.32 = ( ! [A: $i,X: $i] :
% 0.06/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.06/0.32 => ( X
% 0.06/0.32 = ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(complementTnotintersectT_type,type,
% 0.06/0.32 complementTnotintersectT: $o ).
% 0.06/0.32
% 0.06/0.32 thf(complementTnotintersectT,definition,
% 0.06/0.32 ( complementTnotintersectT
% 0.06/0.32 = ( ! [A: $i,X: $i] :
% 0.06/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.06/0.32 => ! [Y: $i] :
% 0.06/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.06/0.32 => ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.06/0.32 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(complementImpComplementIntersect_type,type,
% 0.06/0.32 complementImpComplementIntersect: $o ).
% 0.06/0.32
% 0.06/0.32 thf(complementImpComplementIntersect,definition,
% 0.06/0.32 ( complementImpComplementIntersect
% 0.06/0.32 = ( ! [A: $i,X: $i] :
% 0.06/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.06/0.32 => ! [Y: $i] :
% 0.06/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.06/0.32 => ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.06/0.32 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(complementSubsetComplementIntersect_type,type,
% 0.06/0.32 complementSubsetComplementIntersect: $o ).
% 0.06/0.32
% 0.06/0.32 thf(complementSubsetComplementIntersect,definition,
% 0.06/0.32 ( complementSubsetComplementIntersect
% 0.06/0.32 = ( ! [A: $i,X: $i] :
% 0.06/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.06/0.32 => ! [Y: $i] :
% 0.06/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.06/0.32 => ( subset @ ( setminus @ A @ X ) @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(complementInPowersetComplementIntersect_type,type,
% 0.06/0.32 complementInPowersetComplementIntersect: $o ).
% 0.06/0.32
% 0.06/0.32 thf(complementInPowersetComplementIntersect,definition,
% 0.06/0.32 ( complementInPowersetComplementIntersect
% 0.06/0.32 = ( ! [A: $i,X: $i] :
% 0.06/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.06/0.32 => ! [Y: $i] :
% 0.06/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.06/0.32 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(contraSubsetComplement_type,type,
% 0.06/0.32 contraSubsetComplement: $o ).
% 0.06/0.32
% 0.06/0.32 thf(contraSubsetComplement,definition,
% 0.06/0.32 ( contraSubsetComplement
% 0.06/0.32 = ( ! [A: $i,X: $i] :
% 0.06/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.06/0.32 => ! [Y: $i] :
% 0.06/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.06/0.32 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.06/0.32 => ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ( ( in @ Xx @ Y )
% 0.06/0.32 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(complementTcontraSubset_type,type,
% 0.06/0.32 complementTcontraSubset: $o ).
% 0.06/0.32
% 0.06/0.32 thf(complementTcontraSubset,definition,
% 0.06/0.32 ( complementTcontraSubset
% 0.06/0.32 = ( ! [A: $i,X: $i] :
% 0.06/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.06/0.32 => ! [Y: $i] :
% 0.06/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.06/0.32 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.06/0.32 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(binunionTILcontra_type,type,
% 0.06/0.32 binunionTILcontra: $o ).
% 0.06/0.32
% 0.06/0.32 thf(binunionTILcontra,definition,
% 0.06/0.32 ( binunionTILcontra
% 0.06/0.32 = ( ! [A: $i,X: $i] :
% 0.06/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.06/0.32 => ! [Y: $i] :
% 0.06/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.06/0.32 => ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.06/0.32 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(binunionTIRcontra_type,type,
% 0.06/0.32 binunionTIRcontra: $o ).
% 0.06/0.32
% 0.06/0.32 thf(binunionTIRcontra,definition,
% 0.06/0.32 ( binunionTIRcontra
% 0.06/0.32 = ( ! [A: $i,X: $i] :
% 0.06/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.06/0.32 => ! [Y: $i] :
% 0.06/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.06/0.32 => ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.06/0.32 => ~ ( in @ Xx @ Y ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(inIntersectImpInUnion_type,type,
% 0.06/0.32 inIntersectImpInUnion: $o ).
% 0.06/0.32
% 0.06/0.32 thf(inIntersectImpInUnion,definition,
% 0.06/0.32 ( inIntersectImpInUnion
% 0.06/0.32 = ( ! [A: $i,X: $i] :
% 0.06/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.06/0.32 => ! [Y: $i] :
% 0.06/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.06/0.32 => ! [Z: $i] :
% 0.06/0.32 ( ( in @ Z @ ( powerset @ A ) )
% 0.06/0.32 => ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.06/0.32 => ( in @ Xx @ ( binunion @ X @ Z ) ) ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(inIntersectImpInUnion2_type,type,
% 0.06/0.32 inIntersectImpInUnion2: $o ).
% 0.06/0.32
% 0.06/0.32 thf(inIntersectImpInUnion2,definition,
% 0.06/0.32 ( inIntersectImpInUnion2
% 0.06/0.32 = ( ! [A: $i,X: $i] :
% 0.06/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.06/0.32 => ! [Y: $i] :
% 0.06/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.06/0.32 => ! [Z: $i] :
% 0.06/0.32 ( ( in @ Z @ ( powerset @ A ) )
% 0.06/0.32 => ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.06/0.32 => ( in @ Xx @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(inIntersectImpInIntersectUnions_type,type,
% 0.06/0.32 inIntersectImpInIntersectUnions: $o ).
% 0.06/0.32
% 0.06/0.32 thf(inIntersectImpInIntersectUnions,definition,
% 0.06/0.32 ( inIntersectImpInIntersectUnions
% 0.06/0.32 = ( ! [A: $i,X: $i] :
% 0.06/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.06/0.32 => ! [Y: $i] :
% 0.06/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.06/0.32 => ! [Z: $i] :
% 0.06/0.32 ( ( in @ Z @ ( powerset @ A ) )
% 0.06/0.32 => ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.06/0.32 => ( in @ Xx @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(intersectInPowersetIntersectUnions_type,type,
% 0.06/0.32 intersectInPowersetIntersectUnions: $o ).
% 0.06/0.32
% 0.06/0.32 thf(intersectInPowersetIntersectUnions,definition,
% 0.06/0.32 ( intersectInPowersetIntersectUnions
% 0.06/0.32 = ( ! [A: $i,X: $i] :
% 0.06/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.06/0.32 => ! [Y: $i] :
% 0.06/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.06/0.32 => ! [Z: $i] :
% 0.06/0.32 ( ( in @ Z @ ( powerset @ A ) )
% 0.06/0.32 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(inComplementUnionImpNotIn1_type,type,
% 0.06/0.32 inComplementUnionImpNotIn1: $o ).
% 0.06/0.32
% 0.06/0.32 thf(inComplementUnionImpNotIn1,definition,
% 0.06/0.32 ( inComplementUnionImpNotIn1
% 0.06/0.32 = ( ! [A: $i,X: $i] :
% 0.06/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.06/0.32 => ! [Y: $i] :
% 0.06/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.06/0.32 => ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.06/0.32 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(inComplementUnionImpInComplement1_type,type,
% 0.06/0.32 inComplementUnionImpInComplement1: $o ).
% 0.06/0.32
% 0.06/0.32 thf(inComplementUnionImpInComplement1,definition,
% 0.06/0.32 ( inComplementUnionImpInComplement1
% 0.06/0.32 = ( ! [A: $i,X: $i] :
% 0.06/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.06/0.32 => ! [Y: $i] :
% 0.06/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.06/0.32 => ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.06/0.32 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(binunionTE_type,type,
% 0.06/0.32 binunionTE: $o ).
% 0.06/0.32
% 0.06/0.32 thf(binunionTE,definition,
% 0.06/0.32 ( binunionTE
% 0.06/0.32 = ( ! [A: $i,X: $i] :
% 0.06/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.06/0.32 => ! [Y: $i] :
% 0.06/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.06/0.32 => ! [Xphi: $o,Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ( ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.06/0.32 => ( ( ( in @ Xx @ X )
% 0.06/0.32 => Xphi )
% 0.06/0.32 => ( ( ( in @ Xx @ Y )
% 0.06/0.32 => Xphi )
% 0.06/0.32 => Xphi ) ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(binunionTEcontra_type,type,
% 0.06/0.32 binunionTEcontra: $o ).
% 0.06/0.32
% 0.06/0.32 thf(binunionTEcontra,definition,
% 0.06/0.32 ( binunionTEcontra
% 0.06/0.32 = ( ! [A: $i,X: $i] :
% 0.06/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.06/0.32 => ! [Y: $i] :
% 0.06/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.06/0.32 => ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ( ~ ( in @ Xx @ X )
% 0.06/0.32 => ( ~ ( in @ Xx @ Y )
% 0.06/0.32 => ~ ( in @ Xx @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(demorgan2a1_type,type,
% 0.06/0.32 demorgan2a1: $o ).
% 0.06/0.32
% 0.06/0.32 thf(demorgan2a1,definition,
% 0.06/0.32 ( demorgan2a1
% 0.06/0.32 = ( ! [A: $i,X: $i] :
% 0.06/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.06/0.32 => ! [Y: $i] :
% 0.06/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.06/0.32 => ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.06/0.32 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(complementUnionInPowersetComplement_type,type,
% 0.06/0.32 complementUnionInPowersetComplement: $o ).
% 0.06/0.32
% 0.06/0.32 thf(complementUnionInPowersetComplement,definition,
% 0.06/0.32 ( complementUnionInPowersetComplement
% 0.06/0.32 = ( ! [A: $i,X: $i] :
% 0.06/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.06/0.32 => ! [Y: $i] :
% 0.06/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.06/0.32 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(demorgan2a2_type,type,
% 0.06/0.32 demorgan2a2: $o ).
% 0.06/0.32
% 0.06/0.32 thf(demorgan2a2,definition,
% 0.06/0.32 ( demorgan2a2
% 0.06/0.32 = ( ! [A: $i,X: $i] :
% 0.06/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.06/0.32 => ! [Y: $i] :
% 0.06/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.06/0.32 => ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.06/0.32 => ( in @ Xx @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(demorgan1a_type,type,
% 0.06/0.32 demorgan1a: $o ).
% 0.06/0.32
% 0.06/0.32 thf(demorgan1a,definition,
% 0.06/0.32 ( demorgan1a
% 0.06/0.32 = ( ! [A: $i,X: $i] :
% 0.06/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.06/0.32 => ! [Y: $i] :
% 0.06/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.06/0.32 => ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ( ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) )
% 0.06/0.32 => ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(demorgan1b_type,type,
% 0.06/0.32 demorgan1b: $o ).
% 0.06/0.32
% 0.06/0.32 thf(demorgan1b,definition,
% 0.06/0.32 ( demorgan1b
% 0.06/0.32 = ( ! [A: $i,X: $i] :
% 0.06/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.06/0.32 => ! [Y: $i] :
% 0.06/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.06/0.32 => ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ( ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 0.06/0.32 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(demorgan1_type,type,
% 0.06/0.32 demorgan1: $o ).
% 0.06/0.32
% 0.06/0.32 thf(demorgan1,definition,
% 0.06/0.32 ( demorgan1
% 0.06/0.32 = ( ! [A: $i,X: $i] :
% 0.06/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.06/0.32 => ! [Y: $i] :
% 0.06/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.06/0.32 => ( ( setminus @ A @ ( binintersect @ X @ Y ) )
% 0.06/0.32 = ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(demorgan2a_type,type,
% 0.06/0.32 demorgan2a: $o ).
% 0.06/0.32
% 0.06/0.32 thf(demorgan2a,definition,
% 0.06/0.32 ( demorgan2a
% 0.06/0.32 = ( ! [A: $i,X: $i] :
% 0.06/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.06/0.32 => ! [Y: $i] :
% 0.06/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.06/0.32 => ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.06/0.32 => ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(demorgan2b2_type,type,
% 0.06/0.32 demorgan2b2: $o ).
% 0.06/0.32
% 0.06/0.32 thf(demorgan2b2,definition,
% 0.06/0.32 ( demorgan2b2
% 0.06/0.32 = ( ! [A: $i,X: $i] :
% 0.06/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.06/0.32 => ! [Y: $i] :
% 0.06/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.06/0.32 => ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.06/0.32 => ( ( in @ Xx @ ( setminus @ A @ Y ) )
% 0.06/0.32 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(demorgan2b_type,type,
% 0.06/0.32 demorgan2b: $o ).
% 0.06/0.32
% 0.06/0.32 thf(demorgan2b,definition,
% 0.06/0.32 ( demorgan2b
% 0.06/0.32 = ( ! [A: $i,X: $i] :
% 0.06/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.06/0.32 => ! [Y: $i] :
% 0.06/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.06/0.32 => ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ( ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 0.06/0.32 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(demorgan2_type,type,
% 0.06/0.32 demorgan2: $o ).
% 0.06/0.32
% 0.06/0.32 thf(demorgan2,definition,
% 0.06/0.32 ( demorgan2
% 0.06/0.32 = ( ! [A: $i,X: $i] :
% 0.06/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.06/0.32 => ! [Y: $i] :
% 0.06/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.06/0.32 => ( ( setminus @ A @ ( binunion @ X @ Y ) )
% 0.06/0.32 = ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(woz13rule0_type,type,
% 0.06/0.32 woz13rule0: $o ).
% 0.06/0.32
% 0.06/0.32 thf(woz13rule0,definition,
% 0.06/0.32 ( woz13rule0
% 0.06/0.32 = ( ! [A: $i,X: $i] :
% 0.06/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.06/0.32 => ! [Y: $i] :
% 0.06/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.06/0.32 => ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.06/0.32 => ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(woz13rule1_type,type,
% 0.06/0.32 woz13rule1: $o ).
% 0.06/0.32
% 0.06/0.32 thf(woz13rule1,definition,
% 0.06/0.32 ( woz13rule1
% 0.06/0.32 = ( ! [A: $i,X: $i] :
% 0.06/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.06/0.32 => ! [Y: $i] :
% 0.06/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.06/0.32 => ! [Z: $i] :
% 0.06/0.32 ( ( in @ Z @ ( powerset @ A ) )
% 0.06/0.32 => ( ( subset @ X @ Z )
% 0.06/0.32 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(woz13rule2_type,type,
% 0.06/0.32 woz13rule2: $o ).
% 0.06/0.32
% 0.06/0.32 thf(woz13rule2,definition,
% 0.06/0.32 ( woz13rule2
% 0.06/0.32 = ( ! [A: $i,X: $i] :
% 0.06/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.06/0.32 => ! [Y: $i] :
% 0.06/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.06/0.32 => ! [Z: $i] :
% 0.06/0.32 ( ( in @ Z @ ( powerset @ A ) )
% 0.06/0.32 => ( ( subset @ Y @ Z )
% 0.06/0.32 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(woz13rule3_type,type,
% 0.06/0.32 woz13rule3: $o ).
% 0.06/0.32
% 0.06/0.32 thf(woz13rule3,definition,
% 0.06/0.32 ( woz13rule3
% 0.06/0.32 = ( ! [A: $i,X: $i] :
% 0.06/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.06/0.32 => ! [Y: $i] :
% 0.06/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.06/0.32 => ! [Z: $i] :
% 0.06/0.32 ( ( in @ Z @ ( powerset @ A ) )
% 0.06/0.32 => ( ( subset @ X @ Y )
% 0.06/0.32 => ( ( subset @ X @ Z )
% 0.06/0.32 => ( subset @ X @ ( binintersect @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(woz13rule4_type,type,
% 0.06/0.32 woz13rule4: $o ).
% 0.06/0.32
% 0.06/0.32 thf(woz13rule4,definition,
% 0.06/0.32 ( woz13rule4
% 0.06/0.32 = ( ! [A: $i,X: $i] :
% 0.06/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.06/0.32 => ! [Y: $i] :
% 0.06/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.06/0.32 => ! [Z: $i] :
% 0.06/0.32 ( ( in @ Z @ ( powerset @ A ) )
% 0.06/0.32 => ! [W: $i] :
% 0.06/0.32 ( ( in @ W @ ( powerset @ A ) )
% 0.06/0.32 => ( ( subset @ X @ Z )
% 0.06/0.32 => ( ( subset @ Y @ W )
% 0.06/0.32 => ( subset @ ( binintersect @ X @ Y ) @ ( binintersect @ Z @ W ) ) ) ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(woz1_1_type,type,
% 0.06/0.32 woz1_1: $o ).
% 0.06/0.32
% 0.06/0.32 thf(woz1_1,definition,
% 0.06/0.32 ( woz1_1
% 0.06/0.32 = ( ! [A: $i,X: $i] :
% 0.06/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.06/0.32 => ! [Y: $i] :
% 0.06/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.06/0.32 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(woz1_2_type,type,
% 0.06/0.32 woz1_2: $o ).
% 0.06/0.32
% 0.06/0.32 thf(woz1_2,definition,
% 0.06/0.32 ( woz1_2
% 0.06/0.32 = ( ! [A: $i,X: $i] :
% 0.06/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.06/0.32 => ! [Y: $i] :
% 0.06/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.06/0.32 => ! [Z: $i] :
% 0.06/0.32 ( ( in @ Z @ ( powerset @ A ) )
% 0.06/0.32 => ! [W: $i] :
% 0.06/0.32 ( ( in @ W @ ( powerset @ A ) )
% 0.06/0.32 => ( ( setminus @ A @ ( binintersect @ ( binunion @ X @ Y ) @ ( binunion @ Z @ W ) ) )
% 0.06/0.32 = ( binunion @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) @ ( binintersect @ ( setminus @ A @ Z ) @ ( setminus @ A @ W ) ) ) ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(woz1_3_type,type,
% 0.06/0.32 woz1_3: $o ).
% 0.06/0.32
% 0.06/0.32 thf(woz1_3,definition,
% 0.06/0.32 ( woz1_3
% 0.06/0.32 = ( ! [A: $i,X: $i] :
% 0.06/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.06/0.32 => ! [Y: $i] :
% 0.06/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.06/0.32 => ! [Z: $i] :
% 0.06/0.32 ( ( in @ Z @ ( powerset @ A ) )
% 0.06/0.32 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(woz1_4_type,type,
% 0.06/0.32 woz1_4: $o ).
% 0.06/0.32
% 0.06/0.32 thf(woz1_4,definition,
% 0.06/0.32 ( woz1_4
% 0.06/0.32 = ( ! [A: $i,X: $i] :
% 0.06/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.06/0.32 => ! [Y: $i] :
% 0.06/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.06/0.32 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.06/0.32 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(woz1_5_type,type,
% 0.06/0.32 woz1_5: $o ).
% 0.06/0.32
% 0.06/0.32 thf(woz1_5,definition,
% 0.06/0.32 ( woz1_5
% 0.06/0.32 = ( ! [A: $i,X: $i] :
% 0.06/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.06/0.32 => ! [Y: $i] :
% 0.06/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.06/0.32 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(breln1_type,type,
% 0.06/0.32 breln1: $i > $i > $o ).
% 0.06/0.32
% 0.06/0.32 thf(breln1all2_type,type,
% 0.06/0.32 breln1all2: $o ).
% 0.06/0.32
% 0.06/0.32 thf(breln1all2,definition,
% 0.06/0.32 ( breln1all2
% 0.06/0.32 = ( ! [A: $i,R: $i] :
% 0.06/0.32 ( ( breln1 @ A @ R )
% 0.06/0.32 => ! [Xphi: $i > $o] :
% 0.06/0.32 ( ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ! [Xy: $i] :
% 0.06/0.32 ( ( in @ Xy @ A )
% 0.06/0.32 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.06/0.32 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.06/0.32 => ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ R )
% 0.06/0.32 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(breln1Set_type,type,
% 0.06/0.32 breln1Set: $i > $i ).
% 0.06/0.32
% 0.06/0.32 thf(breln1SetBreln1_type,type,
% 0.06/0.32 breln1SetBreln1: $o ).
% 0.06/0.32
% 0.06/0.32 thf(breln1SetBreln1,definition,
% 0.06/0.32 ( breln1SetBreln1
% 0.06/0.32 = ( ! [A: $i,R: $i] :
% 0.06/0.32 ( ( in @ R @ ( breln1Set @ A ) )
% 0.06/0.32 => ( breln1 @ A @ R ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(transitive_type,type,
% 0.06/0.32 transitive: $i > $i > $o ).
% 0.06/0.32
% 0.06/0.32 thf(antisymmetric_type,type,
% 0.06/0.32 antisymmetric: $i > $i > $o ).
% 0.06/0.32
% 0.06/0.32 thf(reflexive_type,type,
% 0.06/0.32 reflexive: $i > $i > $o ).
% 0.06/0.32
% 0.06/0.32 thf(refltransitive_type,type,
% 0.06/0.32 refltransitive: $i > $i > $o ).
% 0.06/0.32
% 0.06/0.32 thf(refllinearorder_type,type,
% 0.06/0.32 refllinearorder: $i > $i > $o ).
% 0.06/0.32
% 0.06/0.32 thf(reflwellordering_type,type,
% 0.06/0.32 reflwellordering: $i > $i > $o ).
% 0.06/0.32
% 0.06/0.32 thf(choice2fnsingleton_type,type,
% 0.06/0.32 choice2fnsingleton: $o ).
% 0.06/0.32
% 0.06/0.32 thf(choice2fnsingleton,definition,
% 0.06/0.32 ( choice2fnsingleton
% 0.06/0.32 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.06/0.32 ( ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ? [Xy: $i] :
% 0.06/0.32 ( ( in @ Xy @ B )
% 0.06/0.32 & ( Xphi @ Xx @ Xy ) ) )
% 0.06/0.32 => ! [R: $i] :
% 0.06/0.32 ( ( in @ R @ ( breln1Set @ B ) )
% 0.06/0.32 => ( ( reflwellordering @ B @ R )
% 0.06/0.32 => ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ( singleton
% 0.06/0.32 @ ( dsetconstr @ B
% 0.06/0.32 @ ^ [Xy: $i] :
% 0.06/0.32 ( ( Xphi @ Xx @ Xy )
% 0.06/0.32 & ! [Xz: $i] :
% 0.06/0.32 ( ( in @ Xz @ B )
% 0.06/0.32 => ( ( Xphi @ Xx @ Xz )
% 0.06/0.32 => ( in @ ( kpair @ Xy @ Xz ) @ R ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(setOfPairsIsBReln1_type,type,
% 0.06/0.32 setOfPairsIsBReln1: $o ).
% 0.06/0.32
% 0.06/0.32 thf(setOfPairsIsBReln1,definition,
% 0.06/0.32 ( setOfPairsIsBReln1
% 0.06/0.32 = ( ! [A: $i,Xphi: $i > $i > $o] :
% 0.06/0.32 ( breln1 @ A
% 0.06/0.32 @ ( dpsetconstr @ A @ A
% 0.06/0.32 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(breln1all1_type,type,
% 0.06/0.32 breln1all1: $o ).
% 0.06/0.32
% 0.06/0.32 thf(breln1all1,definition,
% 0.06/0.32 ( breln1all1
% 0.06/0.32 = ( ! [A: $i,R: $i] :
% 0.06/0.32 ( ( breln1 @ A @ R )
% 0.06/0.32 => ! [Xphi: $i > $o] :
% 0.06/0.32 ( ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ! [Xy: $i] :
% 0.06/0.32 ( ( in @ Xy @ A )
% 0.06/0.32 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.06/0.32 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.06/0.32 => ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ R )
% 0.06/0.32 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(subbreln1_type,type,
% 0.06/0.32 subbreln1: $o ).
% 0.06/0.32
% 0.06/0.32 thf(subbreln1,definition,
% 0.06/0.32 ( subbreln1
% 0.06/0.32 = ( ! [A: $i,R: $i] :
% 0.06/0.32 ( ( breln1 @ A @ R )
% 0.06/0.32 => ! [S: $i] :
% 0.06/0.32 ( ( breln1 @ A @ S )
% 0.06/0.32 => ( ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ! [Xy: $i] :
% 0.06/0.32 ( ( in @ Xy @ A )
% 0.06/0.32 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.06/0.32 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.06/0.32 => ( subset @ R @ S ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(eqbreln1_type,type,
% 0.06/0.32 eqbreln1: $o ).
% 0.06/0.32
% 0.06/0.32 thf(eqbreln1,definition,
% 0.06/0.32 ( eqbreln1
% 0.06/0.32 = ( ! [A: $i,R: $i] :
% 0.06/0.32 ( ( breln1 @ A @ R )
% 0.06/0.32 => ! [S: $i] :
% 0.06/0.32 ( ( breln1 @ A @ S )
% 0.06/0.32 => ( ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ! [Xy: $i] :
% 0.06/0.32 ( ( in @ Xy @ A )
% 0.06/0.32 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.06/0.32 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.06/0.32 => ( ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ! [Xy: $i] :
% 0.06/0.32 ( ( in @ Xy @ A )
% 0.06/0.32 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.06/0.32 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 0.06/0.32 => ( R = S ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(breln1invset_type,type,
% 0.06/0.32 breln1invset: $i > $i > $i ).
% 0.06/0.32
% 0.06/0.32 thf(breln1invprop_type,type,
% 0.06/0.32 breln1invprop: $o ).
% 0.06/0.32
% 0.06/0.32 thf(breln1invprop,definition,
% 0.06/0.32 ( breln1invprop
% 0.06/0.32 = ( ! [A: $i,R: $i] :
% 0.06/0.32 ( ( breln1 @ A @ R )
% 0.06/0.32 => ( breln1 @ A @ ( breln1invset @ A @ R ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(breln1invI_type,type,
% 0.06/0.32 breln1invI: $o ).
% 0.06/0.32
% 0.06/0.32 thf(breln1invI,definition,
% 0.06/0.32 ( breln1invI
% 0.06/0.32 = ( ! [A: $i,R: $i] :
% 0.06/0.32 ( ( breln1 @ A @ R )
% 0.06/0.32 => ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ! [Xy: $i] :
% 0.06/0.32 ( ( in @ Xy @ A )
% 0.06/0.32 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.06/0.32 => ( in @ ( kpair @ Xy @ Xx ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(breln1invE_type,type,
% 0.06/0.32 breln1invE: $o ).
% 0.06/0.32
% 0.06/0.32 thf(breln1invE,definition,
% 0.06/0.32 ( breln1invE
% 0.06/0.32 = ( ! [A: $i,R: $i] :
% 0.06/0.32 ( ( breln1 @ A @ R )
% 0.06/0.32 => ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ! [Xy: $i] :
% 0.06/0.32 ( ( in @ Xy @ A )
% 0.06/0.32 => ( ( in @ ( kpair @ Xy @ Xx ) @ ( breln1invset @ A @ R ) )
% 0.06/0.32 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(breln1compset_type,type,
% 0.06/0.32 breln1compset: $i > $i > $i > $i ).
% 0.06/0.32
% 0.06/0.32 thf(breln1compprop_type,type,
% 0.06/0.32 breln1compprop: $o ).
% 0.06/0.32
% 0.06/0.32 thf(breln1compprop,definition,
% 0.06/0.32 ( breln1compprop
% 0.06/0.32 = ( ! [A: $i,R: $i] :
% 0.06/0.32 ( ( breln1 @ A @ R )
% 0.06/0.32 => ! [S: $i] :
% 0.06/0.32 ( ( breln1 @ A @ S )
% 0.06/0.32 => ( breln1 @ A @ ( breln1compset @ A @ R @ S ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(breln1compI_type,type,
% 0.06/0.32 breln1compI: $o ).
% 0.06/0.32
% 0.06/0.32 thf(breln1compI,definition,
% 0.06/0.32 ( breln1compI
% 0.06/0.32 = ( ! [A: $i,R: $i] :
% 0.06/0.32 ( ( breln1 @ A @ R )
% 0.06/0.32 => ! [S: $i] :
% 0.06/0.32 ( ( breln1 @ A @ S )
% 0.06/0.32 => ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ! [Xy: $i] :
% 0.06/0.32 ( ( in @ Xy @ A )
% 0.06/0.32 => ! [Xz: $i] :
% 0.06/0.32 ( ( in @ Xz @ A )
% 0.06/0.32 => ( ( in @ ( kpair @ Xx @ Xz ) @ R )
% 0.06/0.32 => ( ( in @ ( kpair @ Xz @ Xy ) @ S )
% 0.06/0.32 => ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) ) ) ) ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(breln1compE_type,type,
% 0.06/0.32 breln1compE: $o ).
% 0.06/0.32
% 0.06/0.32 thf(breln1compE,definition,
% 0.06/0.32 ( breln1compE
% 0.06/0.32 = ( ! [A: $i,R: $i] :
% 0.06/0.32 ( ( breln1 @ A @ R )
% 0.06/0.32 => ! [S: $i] :
% 0.06/0.32 ( ( breln1 @ A @ S )
% 0.06/0.32 => ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ! [Xy: $i] :
% 0.06/0.32 ( ( in @ Xy @ A )
% 0.06/0.32 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) )
% 0.06/0.32 => ? [Xz: $i] :
% 0.06/0.32 ( ( in @ Xz @ A )
% 0.06/0.32 & ( in @ ( kpair @ Xx @ Xz ) @ R )
% 0.06/0.32 & ( in @ ( kpair @ Xz @ Xy ) @ S ) ) ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(breln1compEex_type,type,
% 0.06/0.32 breln1compEex: $o ).
% 0.06/0.32
% 0.06/0.32 thf(breln1compEex,definition,
% 0.06/0.32 ( breln1compEex
% 0.06/0.32 = ( ! [A: $i,R: $i] :
% 0.06/0.32 ( ( breln1 @ A @ R )
% 0.06/0.32 => ! [S: $i] :
% 0.06/0.32 ( ( breln1 @ A @ S )
% 0.06/0.32 => ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ! [Xy: $i] :
% 0.06/0.32 ( ( in @ Xy @ A )
% 0.06/0.32 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) )
% 0.06/0.32 => ! [Xphi: $o] :
% 0.06/0.32 ( ! [Xz: $i] :
% 0.06/0.32 ( ( in @ Xz @ A )
% 0.06/0.32 => ( ( in @ ( kpair @ Xx @ Xz ) @ R )
% 0.06/0.32 => ( ( in @ ( kpair @ Xz @ Xy ) @ S )
% 0.06/0.32 => Xphi ) ) )
% 0.06/0.32 => Xphi ) ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(breln1unionprop_type,type,
% 0.06/0.32 breln1unionprop: $o ).
% 0.06/0.32
% 0.06/0.32 thf(breln1unionprop,definition,
% 0.06/0.32 ( breln1unionprop
% 0.06/0.32 = ( ! [A: $i,R: $i] :
% 0.06/0.32 ( ( breln1 @ A @ R )
% 0.06/0.32 => ! [S: $i] :
% 0.06/0.32 ( ( breln1 @ A @ S )
% 0.06/0.32 => ( breln1 @ A @ ( binunion @ R @ S ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(breln1unionIL_type,type,
% 0.06/0.32 breln1unionIL: $o ).
% 0.06/0.32
% 0.06/0.32 thf(breln1unionIL,definition,
% 0.06/0.32 ( breln1unionIL
% 0.06/0.32 = ( ! [A: $i,R: $i] :
% 0.06/0.32 ( ( breln1 @ A @ R )
% 0.06/0.32 => ! [S: $i] :
% 0.06/0.32 ( ( breln1 @ A @ S )
% 0.06/0.32 => ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ! [Xy: $i] :
% 0.06/0.32 ( ( in @ Xy @ A )
% 0.06/0.32 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.06/0.32 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(breln1unionIR_type,type,
% 0.06/0.32 breln1unionIR: $o ).
% 0.06/0.32
% 0.06/0.32 thf(breln1unionIR,definition,
% 0.06/0.32 ( breln1unionIR
% 0.06/0.32 = ( ! [A: $i,R: $i] :
% 0.06/0.32 ( ( breln1 @ A @ R )
% 0.06/0.32 => ! [S: $i] :
% 0.06/0.32 ( ( breln1 @ A @ S )
% 0.06/0.32 => ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ! [Xy: $i] :
% 0.06/0.32 ( ( in @ Xy @ A )
% 0.06/0.32 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.06/0.32 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(breln1unionI_type,type,
% 0.06/0.32 breln1unionI: $o ).
% 0.06/0.32
% 0.06/0.32 thf(breln1unionI,definition,
% 0.06/0.32 ( breln1unionI
% 0.06/0.32 = ( ! [A: $i,R: $i] :
% 0.06/0.32 ( ( breln1 @ A @ R )
% 0.06/0.32 => ! [S: $i] :
% 0.06/0.32 ( ( breln1 @ A @ S )
% 0.06/0.32 => ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ! [Xy: $i] :
% 0.06/0.32 ( ( in @ Xy @ A )
% 0.06/0.32 => ( ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.06/0.32 | ( in @ ( kpair @ Xx @ Xy ) @ S ) )
% 0.06/0.32 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(breln1unionE_type,type,
% 0.06/0.32 breln1unionE: $o ).
% 0.06/0.32
% 0.06/0.32 thf(breln1unionE,definition,
% 0.06/0.32 ( breln1unionE
% 0.06/0.32 = ( ! [A: $i,R: $i] :
% 0.06/0.32 ( ( breln1 @ A @ R )
% 0.06/0.32 => ! [S: $i] :
% 0.06/0.32 ( ( breln1 @ A @ S )
% 0.06/0.32 => ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ! [Xy: $i] :
% 0.06/0.32 ( ( in @ Xy @ A )
% 0.06/0.32 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) )
% 0.06/0.32 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.06/0.32 | ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(breln1unionEcases_type,type,
% 0.06/0.32 breln1unionEcases: $o ).
% 0.06/0.32
% 0.06/0.32 thf(breln1unionEcases,definition,
% 0.06/0.32 ( breln1unionEcases
% 0.06/0.32 = ( ! [A: $i,R: $i] :
% 0.06/0.32 ( ( breln1 @ A @ R )
% 0.06/0.32 => ! [S: $i] :
% 0.06/0.32 ( ( breln1 @ A @ S )
% 0.06/0.32 => ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ A )
% 0.06/0.32 => ! [Xy: $i] :
% 0.06/0.32 ( ( in @ Xy @ A )
% 0.06/0.32 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) )
% 0.06/0.32 => ! [Xphi: $o] :
% 0.06/0.32 ( ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.06/0.32 => Xphi )
% 0.06/0.32 => ( ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.06/0.32 => Xphi )
% 0.06/0.32 => Xphi ) ) ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(breln1unionCommutes_type,type,
% 0.06/0.32 breln1unionCommutes: $o ).
% 0.06/0.32
% 0.06/0.32 thf(breln1unionCommutes,definition,
% 0.06/0.32 ( breln1unionCommutes
% 0.06/0.32 = ( ! [A: $i,R: $i] :
% 0.06/0.32 ( ( breln1 @ A @ R )
% 0.06/0.32 => ! [S: $i] :
% 0.06/0.32 ( ( breln1 @ A @ S )
% 0.06/0.32 => ( ( binunion @ R @ S )
% 0.06/0.32 = ( binunion @ S @ R ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(woz2Ex_type,type,
% 0.06/0.32 woz2Ex: $o ).
% 0.06/0.32
% 0.06/0.32 thf(woz2Ex,definition,
% 0.06/0.32 ( woz2Ex
% 0.06/0.32 = ( ! [A: $i,R: $i] :
% 0.06/0.32 ( ( breln1 @ A @ R )
% 0.06/0.32 => ( R
% 0.06/0.32 = ( breln1invset @ A @ ( breln1invset @ A @ R ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(woz2W_type,type,
% 0.06/0.32 woz2W: $o ).
% 0.06/0.32
% 0.06/0.32 thf(woz2W,definition,
% 0.06/0.32 ( woz2W
% 0.06/0.32 = ( ! [A: $i,R: $i] :
% 0.06/0.32 ( ( breln1 @ A @ R )
% 0.06/0.32 => ! [S: $i] :
% 0.06/0.32 ( ( breln1 @ A @ S )
% 0.06/0.32 => ( ( breln1invset @ A @ ( breln1compset @ A @ R @ S ) )
% 0.06/0.32 = ( breln1compset @ A @ ( breln1invset @ A @ S ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(woz2A_type,type,
% 0.06/0.32 woz2A: $o ).
% 0.06/0.32
% 0.06/0.32 thf(woz2A,definition,
% 0.06/0.32 ( woz2A
% 0.06/0.32 = ( ! [A: $i,R: $i] :
% 0.06/0.32 ( ( breln1 @ A @ R )
% 0.06/0.32 => ! [S: $i] :
% 0.06/0.32 ( ( breln1 @ A @ S )
% 0.06/0.32 => ! [T: $i] :
% 0.06/0.32 ( ( breln1 @ A @ T )
% 0.06/0.32 => ( ( breln1compset @ A @ ( binunion @ R @ S ) @ T )
% 0.06/0.32 = ( binunion @ ( breln1compset @ A @ R @ T ) @ ( breln1compset @ A @ S @ T ) ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(woz2B_type,type,
% 0.06/0.32 woz2B: $o ).
% 0.06/0.32
% 0.06/0.32 thf(woz2B,definition,
% 0.06/0.32 ( woz2B
% 0.06/0.32 = ( ! [A: $i,R: $i] :
% 0.06/0.32 ( ( breln1 @ A @ R )
% 0.06/0.32 => ! [S: $i] :
% 0.06/0.32 ( ( breln1 @ A @ S )
% 0.06/0.32 => ! [T: $i] :
% 0.06/0.32 ( ( breln1 @ A @ T )
% 0.06/0.32 => ( ( breln1compset @ A @ ( binunion @ R @ S ) @ T )
% 0.06/0.32 = ( binunion @ ( breln1invset @ A @ ( breln1compset @ A @ ( breln1invset @ A @ T ) @ ( breln1invset @ A @ S ) ) ) @ ( breln1invset @ A @ ( breln1compset @ A @ ( breln1invset @ A @ T ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(image1Ex_type,type,
% 0.06/0.32 image1Ex: $o ).
% 0.06/0.32
% 0.06/0.32 thf(image1Ex,definition,
% 0.06/0.32 ( image1Ex
% 0.06/0.32 = ( ! [A: $i,Xf: $i > $i] :
% 0.06/0.32 ? [B: $i] :
% 0.06/0.32 ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ B )
% 0.06/0.32 <=> ? [Xy: $i] :
% 0.06/0.32 ( ( in @ Xy @ A )
% 0.06/0.32 & ( Xx
% 0.06/0.32 = ( Xf @ Xy ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(image1Ex1_type,type,
% 0.06/0.32 image1Ex1: $o ).
% 0.06/0.32
% 0.06/0.32 thf(image1Ex1,definition,
% 0.06/0.32 ( image1Ex1
% 0.06/0.32 = ( ! [A: $i,Xf: $i > $i] :
% 0.06/0.32 ( exu
% 0.06/0.32 @ ^ [B: $i] :
% 0.06/0.32 ! [Xx: $i] :
% 0.06/0.32 ( ( in @ Xx @ B )
% 0.06/0.32 <=> ? [Xy: $i] :
% 0.06/0.32 ( ( in @ Xy @ A )
% 0.06/0.32 & ( Xx
% 0.06/0.32 = ( Xf @ Xy ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(image1_type,type,
% 0.06/0.32 image1: $i > ( $i > $i ) > $i ).
% 0.06/0.32
% 0.06/0.32 thf(image1Equiv_type,type,
% 0.06/0.32 image1Equiv: $o ).
% 0.06/0.32
% 0.06/0.32 thf(image1Equiv,definition,
% 0.06/0.32 ( image1Equiv
% 0.06/0.32 = ( ! [A: $i,Xf: $i > $i,Xx: $i] :
% 0.06/0.32 ( ( in @ Xx
% 0.06/0.32 @ ( image1 @ A
% 0.06/0.32 @ ^ [Xy: $i] : ( Xf @ Xy ) ) )
% 0.06/0.32 <=> ? [Xy: $i] :
% 0.06/0.32 ( ( in @ Xy @ A )
% 0.06/0.32 & ( Xx
% 0.06/0.32 = ( Xf @ Xy ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(image1E_type,type,
% 0.06/0.32 image1E: $o ).
% 0.06/0.32
% 0.06/0.32 thf(image1E,definition,
% 0.06/0.32 ( image1E
% 0.06/0.32 = ( ! [A: $i,Xf: $i > $i,Xx: $i] :
% 0.06/0.32 ( ( in @ Xx
% 0.06/0.32 @ ( image1 @ A
% 0.06/0.32 @ ^ [Xy: $i] : ( Xf @ Xy ) ) )
% 0.06/0.32 => ? [Xy: $i] :
% 0.06/0.32 ( ( in @ Xy @ A )
% 0.06/0.32 & ( Xx
% 0.06/0.32 = ( Xf @ Xy ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(image1I_type,type,
% 0.06/0.32 image1I: $o ).
% 0.06/0.32
% 0.06/0.32 thf(image1I,definition,
% 0.06/0.32 ( image1I
% 0.06/0.32 = ( ! [A: $i,Xf: $i > $i,Xx: $i] :
% 0.06/0.32 ( ? [Xy: $i] :
% 0.06/0.32 ( ( in @ Xy @ A )
% 0.06/0.32 & ( Xx
% 0.06/0.32 = ( Xf @ Xy ) ) )
% 0.06/0.32 => ( in @ Xx
% 0.06/0.32 @ ( image1 @ A
% 0.06/0.32 @ ^ [Xy: $i] : ( Xf @ Xy ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(injective_type,type,
% 0.06/0.32 injective: $i > $i > $i > $o ).
% 0.06/0.32
% 0.06/0.32 thf(injFuncSet_type,type,
% 0.06/0.32 injFuncSet: $i > $i > $i ).
% 0.06/0.32
% 0.06/0.32 thf(injFuncSet,definition,
% 0.06/0.32 ( injFuncSet
% 0.06/0.32 = ( ^ [A: $i,B: $i] :
% 0.06/0.32 ( dsetconstr @ ( funcSet @ A @ B )
% 0.06/0.32 @ ^ [Xf: $i] : ( injective @ A @ B @ Xf ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(injFuncInInjFuncSet,conjecture,
% 0.06/0.32 ( setextAx
% 0.06/0.32 => ( emptysetAx
% 0.06/0.32 => ( setadjoinAx
% 0.06/0.32 => ( powersetAx
% 0.06/0.32 => ( setunionAx
% 0.06/0.32 => ( omega0Ax
% 0.06/0.32 => ( omegaSAx
% 0.06/0.32 => ( omegaIndAx
% 0.06/0.32 => ( replAx
% 0.06/0.32 => ( foundationAx
% 0.06/0.32 => ( wellorderingAx
% 0.06/0.32 => ( descrp
% 0.06/0.32 => ( dsetconstrI
% 0.06/0.32 => ( dsetconstrEL
% 0.06/0.32 => ( dsetconstrER
% 0.06/0.32 => ( exuE1
% 0.06/0.32 => ( prop2setE
% 0.06/0.32 => ( emptysetE
% 0.06/0.32 => ( emptysetimpfalse
% 0.06/0.32 => ( notinemptyset
% 0.06/0.32 => ( exuE3e
% 0.06/0.32 => ( setext
% 0.06/0.32 => ( emptyI
% 0.06/0.32 => ( noeltsimpempty
% 0.06/0.32 => ( setbeta
% 0.06/0.32 => ( nonemptyE1
% 0.06/0.32 => ( nonemptyI
% 0.06/0.32 => ( nonemptyI1
% 0.06/0.32 => ( setadjoinIL
% 0.06/0.32 => ( emptyinunitempty
% 0.06/0.32 => ( setadjoinIR
% 0.06/0.32 => ( setadjoinE
% 0.06/0.32 => ( setadjoinOr
% 0.06/0.32 => ( setoftrueEq
% 0.06/0.32 => ( powersetI
% 0.06/0.32 => ( emptyinPowerset
% 0.06/0.32 => ( emptyInPowerset
% 0.06/0.32 => ( powersetE
% 0.06/0.32 => ( setunionI
% 0.06/0.32 => ( setunionE
% 0.06/0.32 => ( subPowSU
% 0.06/0.32 => ( exuE2
% 0.06/0.32 => ( nonemptyImpWitness
% 0.06/0.32 => ( uniqinunit
% 0.06/0.32 => ( notinsingleton
% 0.06/0.32 => ( eqinunit
% 0.06/0.32 => ( singletonsswitch
% 0.06/0.32 => ( upairsetE
% 0.06/0.32 => ( upairsetIL
% 0.06/0.32 => ( upairsetIR
% 0.06/0.32 => ( emptyE1
% 0.06/0.32 => ( vacuousDall
% 0.06/0.32 => ( quantDeMorgan1
% 0.06/0.32 => ( quantDeMorgan2
% 0.06/0.32 => ( quantDeMorgan3
% 0.06/0.32 => ( quantDeMorgan4
% 0.06/0.32 => ( prop2setI
% 0.06/0.32 => ( prop2set2propI
% 0.06/0.32 => ( notdexE
% 0.06/0.32 => ( notdallE
% 0.06/0.32 => ( exuI1
% 0.06/0.32 => ( exuI3
% 0.06/0.32 => ( exuI2
% 0.06/0.32 => ( inCongP
% 0.06/0.32 => ( in__Cong
% 0.06/0.32 => ( exuE3u
% 0.06/0.32 => ( exu__Cong
% 0.06/0.32 => ( emptyset__Cong
% 0.06/0.32 => ( setadjoin__Cong
% 0.06/0.32 => ( powerset__Cong
% 0.06/0.32 => ( setunion__Cong
% 0.06/0.32 => ( omega__Cong
% 0.06/0.32 => ( exuEu
% 0.06/0.32 => ( descr__Cong
% 0.06/0.32 => ( dsetconstr__Cong
% 0.06/0.32 => ( subsetI1
% 0.06/0.32 => ( eqimpsubset2
% 0.06/0.32 => ( eqimpsubset1
% 0.06/0.32 => ( subsetI2
% 0.06/0.32 => ( emptysetsubset
% 0.06/0.32 => ( subsetE
% 0.06/0.32 => ( subsetE2
% 0.06/0.32 => ( notsubsetI
% 0.06/0.32 => ( notequalI1
% 0.06/0.32 => ( notequalI2
% 0.06/0.32 => ( subsetRefl
% 0.06/0.32 => ( subsetTrans
% 0.06/0.32 => ( setadjoinSub
% 0.06/0.32 => ( setadjoinSub2
% 0.06/0.32 => ( subset2powerset
% 0.06/0.32 => ( setextsub
% 0.06/0.32 => ( subsetemptysetimpeq
% 0.06/0.32 => ( powersetI1
% 0.06/0.32 => ( powersetE1
% 0.06/0.32 => ( inPowerset
% 0.06/0.32 => ( powersetsubset
% 0.06/0.32 => ( sepInPowerset
% 0.06/0.32 => ( sepSubset
% 0.06/0.32 => ( binunionIL
% 0.06/0.32 => ( upairset2IR
% 0.06/0.32 => ( binunionIR
% 0.06/0.32 => ( binunionEcases
% 0.06/0.32 => ( binunionE
% 0.06/0.32 => ( binunionLsub
% 0.06/0.32 => ( binunionRsub
% 0.06/0.32 => ( binintersectI
% 0.06/0.32 => ( binintersectSubset5
% 0.06/0.32 => ( binintersectEL
% 0.06/0.32 => ( binintersectLsub
% 0.06/0.32 => ( binintersectSubset2
% 0.06/0.32 => ( binintersectSubset3
% 0.06/0.32 => ( binintersectER
% 0.06/0.32 => ( disjointsetsI1
% 0.06/0.32 => ( binintersectRsub
% 0.06/0.32 => ( binintersectSubset4
% 0.06/0.32 => ( binintersectSubset1
% 0.06/0.32 => ( bs114d
% 0.06/0.32 => ( setminusI
% 0.06/0.32 => ( setminusEL
% 0.06/0.32 => ( setminusER
% 0.06/0.32 => ( setminusSubset2
% 0.06/0.32 => ( setminusERneg
% 0.06/0.32 => ( setminusELneg
% 0.06/0.32 => ( setminusILneg
% 0.06/0.32 => ( setminusIRneg
% 0.06/0.32 => ( setminusLsub
% 0.06/0.32 => ( setminusSubset1
% 0.06/0.32 => ( symdiffE
% 0.06/0.32 => ( symdiffI1
% 0.06/0.32 => ( symdiffI2
% 0.06/0.32 => ( symdiffIneg1
% 0.06/0.32 => ( symdiffIneg2
% 0.06/0.32 => ( secondinupair
% 0.06/0.32 => ( setukpairIL
% 0.06/0.32 => ( setukpairIR
% 0.06/0.32 => ( kpairiskpair
% 0.06/0.32 => ( kpairp
% 0.06/0.32 => ( singletonsubset
% 0.06/0.32 => ( singletoninpowerset
% 0.06/0.32 => ( singletoninpowunion
% 0.06/0.32 => ( upairset2E
% 0.06/0.32 => ( upairsubunion
% 0.06/0.32 => ( upairinpowunion
% 0.06/0.32 => ( ubforcartprodlem1
% 0.06/0.32 => ( ubforcartprodlem2
% 0.06/0.32 => ( ubforcartprodlem3
% 0.06/0.32 => ( cartprodpairin
% 0.06/0.32 => ( cartprodmempair1
% 0.06/0.32 => ( cartprodmempair
% 0.06/0.32 => ( setunionE2
% 0.06/0.32 => ( setunionsingleton1
% 0.06/0.32 => ( setunionsingleton2
% 0.06/0.32 => ( setunionsingleton
% 0.06/0.32 => ( singletonprop
% 0.06/0.32 => ( ex1E1
% 0.06/0.32 => ( ex1I
% 0.06/0.32 => ( ex1I2
% 0.06/0.32 => ( singletonsuniq
% 0.06/0.32 => ( setukpairinjL1
% 0.06/0.32 => ( kfstsingleton
% 0.06/0.32 => ( theprop
% 0.06/0.32 => ( kfstpairEq
% 0.06/0.32 => ( cartprodfstin
% 0.06/0.32 => ( setukpairinjL2
% 0.06/0.32 => ( setukpairinjL
% 0.06/0.32 => ( setukpairinjR11
% 0.06/0.32 => ( setukpairinjR12
% 0.06/0.32 => ( setukpairinjR1
% 0.06/0.32 => ( upairequniteq
% 0.06/0.32 => ( setukpairinjR2
% 0.06/0.32 => ( setukpairinjR
% 0.06/0.32 => ( ksndsingleton
% 0.06/0.32 => ( ksndpairEq
% 0.06/0.32 => ( kpairsurjEq
% 0.06/0.32 => ( cartprodsndin
% 0.06/0.32 => ( cartprodpairmemEL
% 0.06/0.32 => ( cartprodpairmemER
% 0.06/0.32 => ( cartprodmempaircEq
% 0.06/0.32 => ( cartprodfstpairEq
% 0.06/0.32 => ( cartprodsndpairEq
% 0.06/0.32 => ( cartprodpairsurjEq
% 0.06/0.32 => ( dpsetconstrI
% 0.06/0.32 => ( dpsetconstrSub
% 0.06/0.32 => ( setOfPairsIsBReln
% 0.06/0.32 => ( dpsetconstrERa
% 0.06/0.32 => ( dpsetconstrEL1
% 0.06/0.32 => ( dpsetconstrEL2
% 0.06/0.32 => ( dpsetconstrER
% 0.06/0.32 => ( funcImageSingleton
% 0.06/0.32 => ( apProp
% 0.06/0.32 => ( app
% 0.06/0.32 => ( infuncsetfunc
% 0.06/0.32 => ( ap2p
% 0.06/0.32 => ( funcinfuncset
% 0.06/0.32 => ( lamProp
% 0.06/0.32 => ( lamp
% 0.06/0.32 => ( lam2p
% 0.06/0.32 => ( brelnall1
% 0.06/0.32 => ( brelnall2
% 0.06/0.32 => ( ex1E2
% 0.06/0.32 => ( funcGraphProp1
% 0.06/0.32 => ( funcGraphProp3
% 0.06/0.32 => ( funcGraphProp2
% 0.06/0.32 => ( funcextLem
% 0.06/0.32 => ( funcGraphProp4
% 0.06/0.32 => ( subbreln
% 0.06/0.32 => ( eqbreln
% 0.06/0.32 => ( funcext
% 0.06/0.32 => ( funcext2
% 0.06/0.32 => ( ap2apEq1
% 0.06/0.32 => ( ap2apEq2
% 0.06/0.32 => ( beta1
% 0.06/0.32 => ( eta1
% 0.06/0.32 => ( lam2lamEq
% 0.06/0.32 => ( beta2
% 0.06/0.32 => ( eta2
% 0.06/0.32 => ( iffalseProp1
% 0.06/0.32 => ( iffalseProp2
% 0.06/0.32 => ( iftrueProp1
% 0.06/0.32 => ( iftrueProp2
% 0.06/0.32 => ( ifSingleton
% 0.06/0.32 => ( ifp
% 0.06/0.32 => ( theeq
% 0.06/0.32 => ( iftrue
% 0.06/0.32 => ( iffalse
% 0.06/0.32 => ( iftrueorfalse
% 0.06/0.32 => ( binintersectT_lem
% 0.06/0.32 => ( binunionT_lem
% 0.06/0.32 => ( powersetT_lem
% 0.06/0.32 => ( setminusT_lem
% 0.06/0.32 => ( complementT_lem
% 0.06/0.32 => ( setextT
% 0.06/0.32 => ( subsetTI
% 0.06/0.32 => ( powersetTI1
% 0.06/0.32 => ( powersetTE1
% 0.06/0.32 => ( complementTI1
% 0.06/0.32 => ( complementTE1
% 0.06/0.32 => ( binintersectTELcontra
% 0.06/0.32 => ( binintersectTERcontra
% 0.06/0.32 => ( contrasubsetT
% 0.06/0.32 => ( contrasubsetT1
% 0.06/0.32 => ( contrasubsetT2
% 0.06/0.32 => ( contrasubsetT3
% 0.06/0.32 => ( doubleComplementI1
% 0.06/0.32 => ( doubleComplementE1
% 0.06/0.32 => ( doubleComplementSub1
% 0.06/0.32 => ( doubleComplementSub2
% 0.06/0.32 => ( doubleComplementEq
% 0.06/0.32 => ( complementTnotintersectT
% 0.06/0.32 => ( complementImpComplementIntersect
% 0.06/0.32 => ( complementSubsetComplementIntersect
% 0.06/0.32 => ( complementInPowersetComplementIntersect
% 0.06/0.32 => ( contraSubsetComplement
% 0.06/0.32 => ( complementTcontraSubset
% 0.06/0.32 => ( binunionTILcontra
% 0.06/0.32 => ( binunionTIRcontra
% 0.06/0.32 => ( inIntersectImpInUnion
% 0.06/0.32 => ( inIntersectImpInUnion2
% 0.06/0.32 => ( inIntersectImpInIntersectUnions
% 0.06/0.32 => ( intersectInPowersetIntersectUnions
% 0.06/0.32 => ( inComplementUnionImpNotIn1
% 0.06/0.32 => ( inComplementUnionImpInComplement1
% 0.06/0.32 => ( binunionTE
% 0.06/0.32 => ( binunionTEcontra
% 0.06/0.32 => ( demorgan2a1
% 0.06/0.32 => ( complementUnionInPowersetComplement
% 0.06/0.32 => ( demorgan2a2
% 0.06/0.32 => ( demorgan1a
% 0.06/0.32 => ( demorgan1b
% 0.06/0.32 => ( demorgan1
% 0.06/0.32 => ( demorgan2a
% 0.06/0.32 => ( demorgan2b2
% 0.06/0.32 => ( demorgan2b
% 0.06/0.32 => ( demorgan2
% 0.06/0.32 => ( woz13rule0
% 0.06/0.32 => ( woz13rule1
% 0.06/0.32 => ( woz13rule2
% 0.06/0.32 => ( woz13rule3
% 0.06/0.32 => ( woz13rule4
% 0.06/0.32 => ( woz1_1
% 0.06/0.32 => ( woz1_2
% 0.06/0.32 => ( woz1_3
% 0.06/0.32 => ( woz1_4
% 0.06/0.32 => ( woz1_5
% 0.06/0.32 => ( breln1all2
% 0.06/0.32 => ( breln1SetBreln1
% 0.06/0.32 => ( choice2fnsingleton
% 0.06/0.32 => ( setOfPairsIsBReln1
% 0.06/0.32 => ( breln1all1
% 0.06/0.32 => ( subbreln1
% 0.06/0.32 => ( eqbreln1
% 0.06/0.32 => ( breln1invprop
% 0.06/0.32 => ( breln1invI
% 0.06/0.32 => ( breln1invE
% 0.06/0.32 => ( breln1compprop
% 0.06/0.32 => ( breln1compI
% 0.06/0.32 => ( breln1compE
% 0.06/0.32 => ( breln1compEex
% 0.06/0.32 => ( breln1unionprop
% 0.06/0.32 => ( breln1unionIL
% 0.06/0.32 => ( breln1unionIR
% 0.06/0.32 => ( breln1unionI
% 0.06/0.32 => ( breln1unionE
% 0.06/0.32 => ( breln1unionEcases
% 0.06/0.32 => ( breln1unionCommutes
% 0.06/0.32 => ( woz2Ex
% 0.06/0.32 => ( woz2W
% 0.06/0.32 => ( woz2A
% 0.06/0.32 => ( woz2B
% 0.06/0.32 => ( image1Ex
% 0.06/0.32 => ( image1Ex1
% 0.06/0.32 => ( image1Equiv
% 0.06/0.32 => ( image1E
% 0.06/0.32 => ( image1I
% 0.06/0.32 => ! [A: $i,B: $i,Xf: $i] :
% 0.06/0.32 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.06/0.32 => ( ( injective @ A @ B @ Xf )
% 0.06/0.32 => ( in @ Xf @ ( injFuncSet @ A @ B ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 %------------------------------------------------------------------------------
% 0.06/0.32 ------------------------
% 2.12/1.13 ---- Embedded in THF ---
% 2.12/1.13 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.12/1.13 %%% Generated on Tue Feb 24 07:34:11 EST 2026
% 2.12/1.13 %%% using '$$dhol' embedding, version 1.3.0.
% 2.12/1.13 %%% Logic specification used:
% 2.12/1.13 %%% thf(spec, logic, $$dhol).
% 2.12/1.13
% 2.12/1.13 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.IMUvashtid/DTF2DTF11553.p
% See solution above
% 2.12/1.13 ------------------------
% 2.12/1.17 ---- Cleaned THF ---
% 2.12/1.17 thf(in_type,type,
% 2.12/1.17 in: $i > $i > $o ).
% 2.12/1.17
% 2.12/1.17 thf(exu_type,type,
% 2.12/1.17 exu: ( $i > $o ) > $o ).
% 2.12/1.17
% 2.12/1.17 thf(exu,definition,
% 2.12/1.17 ( exu
% 2.12/1.17 = ( ^ [Xphi: $i > $o] :
% 2.12/1.17 ? [Xx: $i] :
% 2.12/1.17 ( ( Xphi @ Xx )
% 2.12/1.17 & ! [Xy: $i] :
% 2.12/1.17 ( ( Xphi @ Xy )
% 2.12/1.17 => ( Xx = Xy ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(setextAx_type,type,
% 2.12/1.17 setextAx: $o ).
% 2.12/1.17
% 2.12/1.17 thf(setextAx,definition,
% 2.12/1.17 ( setextAx
% 2.12/1.17 = ( ! [A: $i,B: $i] :
% 2.12/1.17 ( ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 <=> ( in @ Xx @ B ) )
% 2.12/1.17 => ( A = B ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(emptyset_type,type,
% 2.12/1.17 emptyset: $i ).
% 2.12/1.17
% 2.12/1.17 thf(emptysetAx_type,type,
% 2.12/1.17 emptysetAx: $o ).
% 2.12/1.17
% 2.12/1.17 thf(emptysetAx,definition,
% 2.12/1.17 ( emptysetAx
% 2.12/1.17 = ( ! [Xx: $i] :
% 2.12/1.17 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(setadjoin_type,type,
% 2.12/1.17 setadjoin: $i > $i > $i ).
% 2.12/1.17
% 2.12/1.17 thf(setadjoinAx_type,type,
% 2.12/1.17 setadjoinAx: $o ).
% 2.12/1.17
% 2.12/1.17 thf(setadjoinAx,definition,
% 2.12/1.17 ( setadjoinAx
% 2.12/1.17 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.12/1.17 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.12/1.17 <=> ( ( Xy = Xx )
% 2.12/1.17 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(powerset_type,type,
% 2.12/1.17 powerset: $i > $i ).
% 2.12/1.17
% 2.12/1.17 thf(powersetAx_type,type,
% 2.12/1.17 powersetAx: $o ).
% 2.12/1.17
% 2.12/1.17 thf(powersetAx,definition,
% 2.12/1.17 ( powersetAx
% 2.12/1.17 = ( ! [A: $i,B: $i] :
% 2.12/1.17 ( ( in @ B @ ( powerset @ A ) )
% 2.12/1.17 <=> ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ B )
% 2.12/1.17 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(setunion_type,type,
% 2.12/1.17 setunion: $i > $i ).
% 2.12/1.17
% 2.12/1.17 thf(setunionAx_type,type,
% 2.12/1.17 setunionAx: $o ).
% 2.12/1.17
% 2.12/1.17 thf(setunionAx,definition,
% 2.12/1.17 ( setunionAx
% 2.12/1.17 = ( ! [A: $i,Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ ( setunion @ A ) )
% 2.12/1.17 <=> ? [B: $i] :
% 2.12/1.17 ( ( in @ Xx @ B )
% 2.12/1.17 & ( in @ B @ A ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(omega_type,type,
% 2.12/1.17 omega: $i ).
% 2.12/1.17
% 2.12/1.17 thf(omega0Ax_type,type,
% 2.12/1.17 omega0Ax: $o ).
% 2.12/1.17
% 2.12/1.17 thf(omega0Ax,definition,
% 2.12/1.17 ( omega0Ax
% 2.12/1.17 = ( in @ emptyset @ omega ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(omegaSAx_type,type,
% 2.12/1.17 omegaSAx: $o ).
% 2.12/1.17
% 2.12/1.17 thf(omegaSAx,definition,
% 2.12/1.17 ( omegaSAx
% 2.12/1.17 = ( ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ omega )
% 2.12/1.17 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(omegaIndAx_type,type,
% 2.12/1.17 omegaIndAx: $o ).
% 2.12/1.17
% 2.12/1.17 thf(omegaIndAx,definition,
% 2.12/1.17 ( omegaIndAx
% 2.12/1.17 = ( ! [A: $i] :
% 2.12/1.17 ( ( ( in @ emptyset @ A )
% 2.12/1.17 & ! [Xx: $i] :
% 2.12/1.17 ( ( ( in @ Xx @ omega )
% 2.12/1.17 & ( in @ Xx @ A ) )
% 2.12/1.17 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 2.12/1.17 => ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ omega )
% 2.12/1.17 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(replAx_type,type,
% 2.12/1.17 replAx: $o ).
% 2.12/1.17
% 2.12/1.17 thf(replAx,definition,
% 2.12/1.17 ( replAx
% 2.12/1.17 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 2.12/1.17 ( ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ( exu
% 2.12/1.17 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 2.12/1.17 => ? [B: $i] :
% 2.12/1.17 ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ B )
% 2.12/1.17 <=> ? [Xy: $i] :
% 2.12/1.17 ( ( in @ Xy @ A )
% 2.12/1.17 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(foundationAx_type,type,
% 2.12/1.17 foundationAx: $o ).
% 2.12/1.17
% 2.12/1.17 thf(foundationAx,definition,
% 2.12/1.17 ( foundationAx
% 2.12/1.17 = ( ! [A: $i] :
% 2.12/1.17 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.12/1.17 => ? [B: $i] :
% 2.12/1.17 ( ( in @ B @ A )
% 2.12/1.17 & ~ ? [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ B )
% 2.12/1.17 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(wellorderingAx_type,type,
% 2.12/1.17 wellorderingAx: $o ).
% 2.12/1.17
% 2.12/1.17 thf(wellorderingAx,definition,
% 2.12/1.17 ( wellorderingAx
% 2.12/1.17 = ( ! [A: $i] :
% 2.12/1.17 ? [B: $i] :
% 2.12/1.17 ( ! [C: $i] :
% 2.12/1.17 ( ( in @ C @ B )
% 2.12/1.17 => ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ C )
% 2.12/1.17 => ( in @ Xx @ A ) ) )
% 2.12/1.17 & ! [Xx: $i,Xy: $i] :
% 2.12/1.17 ( ( ( in @ Xx @ A )
% 2.12/1.17 & ( in @ Xy @ A ) )
% 2.12/1.17 => ( ! [C: $i] :
% 2.12/1.17 ( ( in @ C @ B )
% 2.12/1.17 => ( ( in @ Xx @ C )
% 2.12/1.17 <=> ( in @ Xy @ C ) ) )
% 2.12/1.17 => ( Xx = Xy ) ) )
% 2.12/1.17 & ! [C: $i,D: $i] :
% 2.12/1.17 ( ( ( in @ C @ B )
% 2.12/1.17 & ( in @ D @ B ) )
% 2.12/1.17 => ( ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ C )
% 2.12/1.17 => ( in @ Xx @ D ) )
% 2.12/1.17 | ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ D )
% 2.12/1.17 => ( in @ Xx @ C ) ) ) )
% 2.12/1.17 & ! [C: $i] :
% 2.12/1.17 ( ( ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ C )
% 2.12/1.17 => ( in @ Xx @ A ) )
% 2.12/1.17 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 2.12/1.17 => ? [D: $i,Xx: $i] :
% 2.12/1.17 ( ( in @ D @ B )
% 2.12/1.17 & ( in @ Xx @ C )
% 2.12/1.17 & ~ ? [Xy: $i] :
% 2.12/1.17 ( ( in @ Xy @ D )
% 2.12/1.17 & ( in @ Xy @ C ) )
% 2.12/1.17 & ! [E: $i] :
% 2.12/1.17 ( ( in @ E @ B )
% 2.12/1.17 => ( ! [Xy: $i] :
% 2.12/1.17 ( ( in @ Xy @ E )
% 2.12/1.17 => ( in @ Xy @ D ) )
% 2.12/1.17 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(descr_type,type,
% 2.12/1.17 descr: ( $i > $o ) > $i ).
% 2.12/1.17
% 2.12/1.17 thf(descrp_type,type,
% 2.12/1.17 descrp: $o ).
% 2.12/1.17
% 2.12/1.17 thf(descrp,definition,
% 2.12/1.17 ( descrp
% 2.12/1.17 = ( ! [Xphi: $i > $o] :
% 2.12/1.17 ( ( exu
% 2.12/1.17 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.12/1.17 => ( Xphi
% 2.12/1.17 @ ( descr
% 2.12/1.17 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(dsetconstr_type,type,
% 2.12/1.17 dsetconstr: $i > ( $i > $o ) > $i ).
% 2.12/1.17
% 2.12/1.17 thf(dsetconstrI_type,type,
% 2.12/1.17 dsetconstrI: $o ).
% 2.12/1.17
% 2.12/1.17 thf(dsetconstrI,definition,
% 2.12/1.17 ( dsetconstrI
% 2.12/1.17 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ( ( Xphi @ Xx )
% 2.12/1.17 => ( in @ Xx
% 2.12/1.17 @ ( dsetconstr @ A
% 2.12/1.17 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(dsetconstrEL_type,type,
% 2.12/1.17 dsetconstrEL: $o ).
% 2.12/1.17
% 2.12/1.17 thf(dsetconstrEL,definition,
% 2.12/1.17 ( dsetconstrEL
% 2.12/1.17 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.12/1.17 ( ( in @ Xx
% 2.12/1.17 @ ( dsetconstr @ A
% 2.12/1.17 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.12/1.17 => ( in @ Xx @ A ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(dsetconstrER_type,type,
% 2.12/1.17 dsetconstrER: $o ).
% 2.12/1.17
% 2.12/1.17 thf(dsetconstrER,definition,
% 2.12/1.17 ( dsetconstrER
% 2.12/1.17 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.12/1.17 ( ( in @ Xx
% 2.12/1.17 @ ( dsetconstr @ A
% 2.12/1.17 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.12/1.17 => ( Xphi @ Xx ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(exuE1_type,type,
% 2.12/1.17 exuE1: $o ).
% 2.12/1.17
% 2.12/1.17 thf(exuE1,definition,
% 2.12/1.17 ( exuE1
% 2.12/1.17 = ( ! [Xphi: $i > $o] :
% 2.12/1.17 ( ( exu
% 2.12/1.17 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.12/1.17 => ? [Xx: $i] :
% 2.12/1.17 ( ( Xphi @ Xx )
% 2.12/1.17 & ! [Xy: $i] :
% 2.12/1.17 ( ( Xphi @ Xy )
% 2.12/1.17 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(prop2set_type,type,
% 2.12/1.17 prop2set: $o > $i ).
% 2.12/1.17
% 2.12/1.17 thf(prop2setE_type,type,
% 2.12/1.17 prop2setE: $o ).
% 2.12/1.17
% 2.12/1.17 thf(prop2setE,definition,
% 2.12/1.17 ( prop2setE
% 2.12/1.17 = ( ! [Xphi: $o,Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 2.12/1.17 => Xphi ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(emptysetE_type,type,
% 2.12/1.17 emptysetE: $o ).
% 2.12/1.17
% 2.12/1.17 thf(emptysetE,definition,
% 2.12/1.17 ( emptysetE
% 2.12/1.17 = ( ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ emptyset )
% 2.12/1.17 => ! [Xphi: $o] : Xphi ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(emptysetimpfalse_type,type,
% 2.12/1.17 emptysetimpfalse: $o ).
% 2.12/1.17
% 2.12/1.17 thf(emptysetimpfalse,definition,
% 2.12/1.17 ( emptysetimpfalse
% 2.12/1.17 = ( ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ emptyset )
% 2.12/1.17 => $false ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(notinemptyset_type,type,
% 2.12/1.17 notinemptyset: $o ).
% 2.12/1.17
% 2.12/1.17 thf(notinemptyset,definition,
% 2.12/1.17 ( notinemptyset
% 2.12/1.17 = ( ! [Xx: $i] :
% 2.12/1.17 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(exuE3e_type,type,
% 2.12/1.17 exuE3e: $o ).
% 2.12/1.17
% 2.12/1.17 thf(exuE3e,definition,
% 2.12/1.17 ( exuE3e
% 2.12/1.17 = ( ! [Xphi: $i > $o] :
% 2.12/1.17 ( ( exu
% 2.12/1.17 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.12/1.17 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(setext_type,type,
% 2.12/1.17 setext: $o ).
% 2.12/1.17
% 2.12/1.17 thf(setext,definition,
% 2.12/1.17 ( setext
% 2.12/1.17 = ( ! [A: $i,B: $i] :
% 2.12/1.17 ( ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ( in @ Xx @ B ) )
% 2.12/1.17 => ( ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ B )
% 2.12/1.17 => ( in @ Xx @ A ) )
% 2.12/1.17 => ( A = B ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(emptyI_type,type,
% 2.12/1.17 emptyI: $o ).
% 2.12/1.17
% 2.12/1.17 thf(emptyI,definition,
% 2.12/1.17 ( emptyI
% 2.12/1.17 = ( ! [A: $i] :
% 2.12/1.17 ( ! [Xx: $i] :
% 2.12/1.17 ~ ( in @ Xx @ A )
% 2.12/1.17 => ( A = emptyset ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(noeltsimpempty_type,type,
% 2.12/1.17 noeltsimpempty: $o ).
% 2.12/1.17
% 2.12/1.17 thf(noeltsimpempty,definition,
% 2.12/1.17 ( noeltsimpempty
% 2.12/1.17 = ( ! [A: $i] :
% 2.12/1.17 ( ! [Xx: $i] :
% 2.12/1.17 ~ ( in @ Xx @ A )
% 2.12/1.17 => ( A = emptyset ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(setbeta_type,type,
% 2.12/1.17 setbeta: $o ).
% 2.12/1.17
% 2.12/1.17 thf(setbeta,definition,
% 2.12/1.17 ( setbeta
% 2.12/1.17 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ( ( in @ Xx
% 2.12/1.17 @ ( dsetconstr @ A
% 2.12/1.17 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.12/1.17 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(nonempty_type,type,
% 2.12/1.17 nonempty: $i > $o ).
% 2.12/1.17
% 2.12/1.17 thf(nonempty,definition,
% 2.12/1.17 ( nonempty
% 2.12/1.17 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(nonemptyE1_type,type,
% 2.12/1.17 nonemptyE1: $o ).
% 2.12/1.17
% 2.12/1.17 thf(nonemptyE1,definition,
% 2.12/1.17 ( nonemptyE1
% 2.12/1.17 = ( ! [A: $i] :
% 2.12/1.17 ( ( nonempty @ A )
% 2.12/1.17 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(nonemptyI_type,type,
% 2.12/1.17 nonemptyI: $o ).
% 2.12/1.17
% 2.12/1.17 thf(nonemptyI,definition,
% 2.12/1.17 ( nonemptyI
% 2.12/1.17 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ( ( Xphi @ Xx )
% 2.12/1.17 => ( nonempty
% 2.12/1.17 @ ( dsetconstr @ A
% 2.12/1.17 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(nonemptyI1_type,type,
% 2.12/1.17 nonemptyI1: $o ).
% 2.12/1.17
% 2.12/1.17 thf(nonemptyI1,definition,
% 2.12/1.17 ( nonemptyI1
% 2.12/1.17 = ( ! [A: $i] :
% 2.12/1.17 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.12/1.17 => ( nonempty @ A ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(setadjoinIL_type,type,
% 2.12/1.17 setadjoinIL: $o ).
% 2.12/1.17
% 2.12/1.17 thf(setadjoinIL,definition,
% 2.12/1.17 ( setadjoinIL
% 2.12/1.17 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(emptyinunitempty_type,type,
% 2.12/1.17 emptyinunitempty: $o ).
% 2.12/1.17
% 2.12/1.17 thf(emptyinunitempty,definition,
% 2.12/1.17 ( emptyinunitempty
% 2.12/1.17 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(setadjoinIR_type,type,
% 2.12/1.17 setadjoinIR: $o ).
% 2.12/1.17
% 2.12/1.17 thf(setadjoinIR,definition,
% 2.12/1.17 ( setadjoinIR
% 2.12/1.17 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.12/1.17 ( ( in @ Xy @ A )
% 2.12/1.17 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(setadjoinE_type,type,
% 2.12/1.17 setadjoinE: $o ).
% 2.12/1.17
% 2.12/1.17 thf(setadjoinE,definition,
% 2.12/1.17 ( setadjoinE
% 2.12/1.17 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.12/1.17 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.12/1.17 => ! [Xphi: $o] :
% 2.12/1.17 ( ( ( Xy = Xx )
% 2.12/1.17 => Xphi )
% 2.12/1.17 => ( ( ( in @ Xy @ A )
% 2.12/1.17 => Xphi )
% 2.12/1.17 => Xphi ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(setadjoinOr_type,type,
% 2.12/1.17 setadjoinOr: $o ).
% 2.12/1.17
% 2.12/1.17 thf(setadjoinOr,definition,
% 2.12/1.17 ( setadjoinOr
% 2.12/1.17 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.12/1.17 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.12/1.17 => ( ( Xy = Xx )
% 2.12/1.17 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(setoftrueEq_type,type,
% 2.12/1.17 setoftrueEq: $o ).
% 2.12/1.17
% 2.12/1.17 thf(setoftrueEq,definition,
% 2.12/1.17 ( setoftrueEq
% 2.12/1.17 = ( ! [A: $i] :
% 2.12/1.17 ( ( dsetconstr @ A
% 2.12/1.17 @ ^ [Xx: $i] : $true )
% 2.12/1.17 = A ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(powersetI_type,type,
% 2.12/1.17 powersetI: $o ).
% 2.12/1.17
% 2.12/1.17 thf(powersetI,definition,
% 2.12/1.17 ( powersetI
% 2.12/1.17 = ( ! [A: $i,B: $i] :
% 2.12/1.17 ( ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ B )
% 2.12/1.17 => ( in @ Xx @ A ) )
% 2.12/1.17 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(emptyinPowerset_type,type,
% 2.12/1.17 emptyinPowerset: $o ).
% 2.12/1.17
% 2.12/1.17 thf(emptyinPowerset,definition,
% 2.12/1.17 ( emptyinPowerset
% 2.12/1.17 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(emptyInPowerset_type,type,
% 2.12/1.17 emptyInPowerset: $o ).
% 2.12/1.17
% 2.12/1.17 thf(emptyInPowerset,definition,
% 2.12/1.17 ( emptyInPowerset
% 2.12/1.17 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(powersetE_type,type,
% 2.12/1.17 powersetE: $o ).
% 2.12/1.17
% 2.12/1.17 thf(powersetE,definition,
% 2.12/1.17 ( powersetE
% 2.12/1.17 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.12/1.17 ( ( in @ B @ ( powerset @ A ) )
% 2.12/1.17 => ( ( in @ Xx @ B )
% 2.12/1.17 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(setunionI_type,type,
% 2.12/1.17 setunionI: $o ).
% 2.12/1.17
% 2.12/1.17 thf(setunionI,definition,
% 2.12/1.17 ( setunionI
% 2.12/1.17 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.12/1.17 ( ( in @ Xx @ B )
% 2.12/1.17 => ( ( in @ B @ A )
% 2.12/1.17 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(setunionE_type,type,
% 2.12/1.17 setunionE: $o ).
% 2.12/1.17
% 2.12/1.17 thf(setunionE,definition,
% 2.12/1.17 ( setunionE
% 2.12/1.17 = ( ! [A: $i,Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ ( setunion @ A ) )
% 2.12/1.17 => ! [Xphi: $o] :
% 2.12/1.17 ( ! [B: $i] :
% 2.12/1.17 ( ( in @ Xx @ B )
% 2.12/1.17 => ( ( in @ B @ A )
% 2.12/1.17 => Xphi ) )
% 2.12/1.17 => Xphi ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(subPowSU_type,type,
% 2.12/1.17 subPowSU: $o ).
% 2.12/1.17
% 2.12/1.17 thf(subPowSU,definition,
% 2.12/1.17 ( subPowSU
% 2.12/1.17 = ( ! [A: $i,Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(exuE2_type,type,
% 2.12/1.17 exuE2: $o ).
% 2.12/1.17
% 2.12/1.17 thf(exuE2,definition,
% 2.12/1.17 ( exuE2
% 2.12/1.17 = ( ! [Xphi: $i > $o] :
% 2.12/1.17 ( ( exu
% 2.12/1.17 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.12/1.17 => ? [Xx: $i] :
% 2.12/1.17 ! [Xy: $i] :
% 2.12/1.17 ( ( Xphi @ Xy )
% 2.12/1.17 <=> ( Xy = Xx ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(nonemptyImpWitness_type,type,
% 2.12/1.17 nonemptyImpWitness: $o ).
% 2.12/1.17
% 2.12/1.17 thf(nonemptyImpWitness,definition,
% 2.12/1.17 ( nonemptyImpWitness
% 2.12/1.17 = ( ! [A: $i] :
% 2.12/1.17 ( ( nonempty @ A )
% 2.12/1.17 => ? [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 & $true ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(uniqinunit_type,type,
% 2.12/1.17 uniqinunit: $o ).
% 2.12/1.17
% 2.12/1.17 thf(uniqinunit,definition,
% 2.12/1.17 ( uniqinunit
% 2.12/1.17 = ( ! [Xx: $i,Xy: $i] :
% 2.12/1.17 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.12/1.17 => ( Xx = Xy ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(notinsingleton_type,type,
% 2.12/1.17 notinsingleton: $o ).
% 2.12/1.17
% 2.12/1.17 thf(notinsingleton,definition,
% 2.12/1.17 ( notinsingleton
% 2.12/1.17 = ( ! [Xx: $i,Xy: $i] :
% 2.12/1.17 ( ( Xx != Xy )
% 2.12/1.17 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(eqinunit_type,type,
% 2.12/1.17 eqinunit: $o ).
% 2.12/1.17
% 2.12/1.17 thf(eqinunit,definition,
% 2.12/1.17 ( eqinunit
% 2.12/1.17 = ( ! [Xx: $i,Xy: $i] :
% 2.12/1.17 ( ( Xx = Xy )
% 2.12/1.17 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(singletonsswitch_type,type,
% 2.12/1.17 singletonsswitch: $o ).
% 2.12/1.17
% 2.12/1.17 thf(singletonsswitch,definition,
% 2.12/1.17 ( singletonsswitch
% 2.12/1.17 = ( ! [Xx: $i,Xy: $i] :
% 2.12/1.17 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.12/1.17 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(upairsetE_type,type,
% 2.12/1.17 upairsetE: $o ).
% 2.12/1.17
% 2.12/1.17 thf(upairsetE,definition,
% 2.12/1.17 ( upairsetE
% 2.12/1.17 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.12/1.17 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.12/1.17 => ( ( Xz = Xx )
% 2.12/1.17 | ( Xz = Xy ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(upairsetIL_type,type,
% 2.12/1.17 upairsetIL: $o ).
% 2.12/1.17
% 2.12/1.17 thf(upairsetIL,definition,
% 2.12/1.17 ( upairsetIL
% 2.12/1.17 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(upairsetIR_type,type,
% 2.12/1.17 upairsetIR: $o ).
% 2.12/1.17
% 2.12/1.17 thf(upairsetIR,definition,
% 2.12/1.17 ( upairsetIR
% 2.12/1.17 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(emptyE1_type,type,
% 2.12/1.17 emptyE1: $o ).
% 2.12/1.17
% 2.12/1.17 thf(emptyE1,definition,
% 2.12/1.17 ( emptyE1
% 2.12/1.17 = ( ! [A: $i,Xphi: $i > $o] :
% 2.12/1.17 ( ? [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 & ( Xphi @ Xx ) )
% 2.12/1.17 => ( ( ( dsetconstr @ A
% 2.12/1.17 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.12/1.17 = emptyset )
% 2.12/1.17 => $false ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(vacuousDall_type,type,
% 2.12/1.17 vacuousDall: $o ).
% 2.12/1.17
% 2.12/1.17 thf(vacuousDall,definition,
% 2.12/1.17 ( vacuousDall
% 2.12/1.17 = ( ! [Xphi: $i > $o,Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ emptyset )
% 2.12/1.17 => ( Xphi @ Xx ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(quantDeMorgan1_type,type,
% 2.12/1.17 quantDeMorgan1: $o ).
% 2.12/1.17
% 2.12/1.17 thf(quantDeMorgan1,definition,
% 2.12/1.17 ( quantDeMorgan1
% 2.12/1.17 = ( ! [A: $i,Xphi: $i > $o] :
% 2.12/1.17 ( ~ ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ( Xphi @ Xx ) )
% 2.12/1.17 => ? [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(quantDeMorgan2_type,type,
% 2.12/1.17 quantDeMorgan2: $o ).
% 2.12/1.17
% 2.12/1.17 thf(quantDeMorgan2,definition,
% 2.12/1.17 ( quantDeMorgan2
% 2.12/1.17 = ( ! [A: $i,Xphi: $i > $o] :
% 2.12/1.17 ( ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ~ ( Xphi @ Xx ) )
% 2.12/1.17 => ~ ? [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(quantDeMorgan3_type,type,
% 2.12/1.17 quantDeMorgan3: $o ).
% 2.12/1.17
% 2.12/1.17 thf(quantDeMorgan3,definition,
% 2.12/1.17 ( quantDeMorgan3
% 2.12/1.17 = ( ! [A: $i,Xphi: $i > $o] :
% 2.12/1.17 ( ~ ? [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 & ( Xphi @ Xx ) )
% 2.12/1.17 => ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(quantDeMorgan4_type,type,
% 2.12/1.17 quantDeMorgan4: $o ).
% 2.12/1.17
% 2.12/1.17 thf(quantDeMorgan4,definition,
% 2.12/1.17 ( quantDeMorgan4
% 2.12/1.17 = ( ! [A: $i,Xphi: $i > $o] :
% 2.12/1.17 ( ? [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 & ~ ( Xphi @ Xx ) )
% 2.12/1.17 => ~ ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ( Xphi @ Xx ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(prop2setI_type,type,
% 2.12/1.17 prop2setI: $o ).
% 2.12/1.17
% 2.12/1.17 thf(prop2setI,definition,
% 2.12/1.17 ( prop2setI
% 2.12/1.17 = ( ! [Xphi: $o] :
% 2.12/1.17 ( Xphi
% 2.12/1.17 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(set2prop_type,type,
% 2.12/1.17 set2prop: $i > $o ).
% 2.12/1.17
% 2.12/1.17 thf(prop2set2propI_type,type,
% 2.12/1.17 prop2set2propI: $o ).
% 2.12/1.17
% 2.12/1.17 thf(prop2set2propI,definition,
% 2.12/1.17 ( prop2set2propI
% 2.12/1.17 = ( ! [Xphi: $o] :
% 2.12/1.17 ( Xphi
% 2.12/1.17 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(notdexE_type,type,
% 2.12/1.17 notdexE: $o ).
% 2.12/1.17
% 2.12/1.17 thf(notdexE,definition,
% 2.12/1.17 ( notdexE
% 2.12/1.17 = ( ! [A: $i,Xphi: $i > $o] :
% 2.12/1.17 ( ~ ? [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 & ( Xphi @ Xx ) )
% 2.12/1.17 => ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(notdallE_type,type,
% 2.12/1.17 notdallE: $o ).
% 2.12/1.17
% 2.12/1.17 thf(notdallE,definition,
% 2.12/1.17 ( notdallE
% 2.12/1.17 = ( ! [A: $i,Xphi: $i > $o] :
% 2.12/1.17 ( ~ ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ( Xphi @ Xx ) )
% 2.12/1.17 => ? [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(exuI1_type,type,
% 2.12/1.17 exuI1: $o ).
% 2.12/1.17
% 2.12/1.17 thf(exuI1,definition,
% 2.12/1.17 ( exuI1
% 2.12/1.17 = ( ! [Xphi: $i > $o] :
% 2.12/1.17 ( ? [Xx: $i] :
% 2.12/1.17 ( ( Xphi @ Xx )
% 2.12/1.17 & ! [Xy: $i] :
% 2.12/1.17 ( ( Xphi @ Xy )
% 2.12/1.17 => ( Xx = Xy ) ) )
% 2.12/1.17 => ( exu
% 2.12/1.17 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(exuI3_type,type,
% 2.12/1.17 exuI3: $o ).
% 2.12/1.17
% 2.12/1.17 thf(exuI3,definition,
% 2.12/1.17 ( exuI3
% 2.12/1.17 = ( ! [Xphi: $i > $o] :
% 2.12/1.17 ( ? [Xx: $i] : ( Xphi @ Xx )
% 2.12/1.17 => ( ! [Xx: $i,Xy: $i] :
% 2.12/1.17 ( ( Xphi @ Xx )
% 2.12/1.17 => ( ( Xphi @ Xy )
% 2.12/1.17 => ( Xx = Xy ) ) )
% 2.12/1.17 => ( exu
% 2.12/1.17 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(exuI2_type,type,
% 2.12/1.17 exuI2: $o ).
% 2.12/1.17
% 2.12/1.17 thf(exuI2,definition,
% 2.12/1.17 ( exuI2
% 2.12/1.17 = ( ! [Xphi: $i > $o] :
% 2.12/1.17 ( ? [Xx: $i] :
% 2.12/1.17 ! [Xy: $i] :
% 2.12/1.17 ( ( Xphi @ Xy )
% 2.12/1.17 <=> ( Xy = Xx ) )
% 2.12/1.17 => ( exu
% 2.12/1.17 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(inCongP_type,type,
% 2.12/1.17 inCongP: $o ).
% 2.12/1.17
% 2.12/1.17 thf(inCongP,definition,
% 2.12/1.17 ( inCongP
% 2.12/1.17 = ( ! [A: $i,B: $i] :
% 2.12/1.17 ( ( A = B )
% 2.12/1.17 => ! [Xx: $i,Xy: $i] :
% 2.12/1.17 ( ( Xx = Xy )
% 2.12/1.17 => ( ( in @ Xx @ A )
% 2.12/1.17 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(in__Cong_type,type,
% 2.12/1.17 in__Cong: $o ).
% 2.12/1.17
% 2.12/1.17 thf(in__Cong,definition,
% 2.12/1.17 ( in__Cong
% 2.12/1.17 = ( ! [A: $i,B: $i] :
% 2.12/1.17 ( ( A = B )
% 2.12/1.17 => ! [Xx: $i,Xy: $i] :
% 2.12/1.17 ( ( Xx = Xy )
% 2.12/1.17 => ( ( in @ Xx @ A )
% 2.12/1.17 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(exuE3u_type,type,
% 2.12/1.17 exuE3u: $o ).
% 2.12/1.17
% 2.12/1.17 thf(exuE3u,definition,
% 2.12/1.17 ( exuE3u
% 2.12/1.17 = ( ! [Xphi: $i > $o] :
% 2.12/1.17 ( ( exu
% 2.12/1.17 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.12/1.17 => ! [Xx: $i,Xy: $i] :
% 2.12/1.17 ( ( Xphi @ Xx )
% 2.12/1.17 => ( ( Xphi @ Xy )
% 2.12/1.17 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(exu__Cong_type,type,
% 2.12/1.17 exu__Cong: $o ).
% 2.12/1.17
% 2.12/1.17 thf(exu__Cong,definition,
% 2.12/1.17 ( exu__Cong
% 2.12/1.17 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.12/1.17 ( ! [Xx: $i,Xy: $i] :
% 2.12/1.17 ( ( Xx = Xy )
% 2.12/1.17 => ( ( Xphi @ Xx )
% 2.12/1.17 <=> ( Xpsi @ Xy ) ) )
% 2.12/1.17 => ( ( exu
% 2.12/1.17 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.12/1.17 <=> ( exu
% 2.12/1.17 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(emptyset__Cong_type,type,
% 2.12/1.17 emptyset__Cong: $o ).
% 2.12/1.17
% 2.12/1.17 thf(emptyset__Cong,definition,
% 2.12/1.17 ( emptyset__Cong
% 2.12/1.17 = ( emptyset = emptyset ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(setadjoin__Cong_type,type,
% 2.12/1.17 setadjoin__Cong: $o ).
% 2.12/1.17
% 2.12/1.17 thf(setadjoin__Cong,definition,
% 2.12/1.17 ( setadjoin__Cong
% 2.12/1.17 = ( ! [Xx: $i,Xy: $i] :
% 2.12/1.17 ( ( Xx = Xy )
% 2.12/1.17 => ! [Xz: $i,Xu: $i] :
% 2.12/1.17 ( ( Xz = Xu )
% 2.12/1.17 => ( ( setadjoin @ Xx @ Xz )
% 2.12/1.17 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(powerset__Cong_type,type,
% 2.12/1.17 powerset__Cong: $o ).
% 2.12/1.17
% 2.12/1.17 thf(powerset__Cong,definition,
% 2.12/1.17 ( powerset__Cong
% 2.12/1.17 = ( ! [A: $i,B: $i] :
% 2.12/1.17 ( ( A = B )
% 2.12/1.17 => ( ( powerset @ A )
% 2.12/1.17 = ( powerset @ B ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(setunion__Cong_type,type,
% 2.12/1.17 setunion__Cong: $o ).
% 2.12/1.17
% 2.12/1.17 thf(setunion__Cong,definition,
% 2.12/1.17 ( setunion__Cong
% 2.12/1.17 = ( ! [A: $i,B: $i] :
% 2.12/1.17 ( ( A = B )
% 2.12/1.17 => ( ( setunion @ A )
% 2.12/1.17 = ( setunion @ B ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(omega__Cong_type,type,
% 2.12/1.17 omega__Cong: $o ).
% 2.12/1.17
% 2.12/1.17 thf(omega__Cong,definition,
% 2.12/1.17 ( omega__Cong
% 2.12/1.17 = ( omega = omega ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(exuEu_type,type,
% 2.12/1.17 exuEu: $o ).
% 2.12/1.17
% 2.12/1.17 thf(exuEu,definition,
% 2.12/1.17 ( exuEu
% 2.12/1.17 = ( ! [Xphi: $i > $o] :
% 2.12/1.17 ( ( exu
% 2.12/1.17 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.12/1.17 => ! [Xx: $i,Xy: $i] :
% 2.12/1.17 ( ( Xphi @ Xx )
% 2.12/1.17 => ( ( Xphi @ Xy )
% 2.12/1.17 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(descr__Cong_type,type,
% 2.12/1.17 descr__Cong: $o ).
% 2.12/1.17
% 2.12/1.17 thf(descr__Cong,definition,
% 2.12/1.17 ( descr__Cong
% 2.12/1.17 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.12/1.17 ( ! [Xx: $i,Xy: $i] :
% 2.12/1.17 ( ( Xx = Xy )
% 2.12/1.17 => ( ( Xphi @ Xx )
% 2.12/1.17 <=> ( Xpsi @ Xy ) ) )
% 2.12/1.17 => ( ( exu
% 2.12/1.17 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.12/1.17 => ( ( exu
% 2.12/1.17 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 2.12/1.17 => ( ( descr
% 2.12/1.17 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.12/1.17 = ( descr
% 2.12/1.17 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(dsetconstr__Cong_type,type,
% 2.12/1.17 dsetconstr__Cong: $o ).
% 2.12/1.17
% 2.12/1.17 thf(dsetconstr__Cong,definition,
% 2.12/1.17 ( dsetconstr__Cong
% 2.12/1.17 = ( ! [A: $i,B: $i] :
% 2.12/1.17 ( ( A = B )
% 2.12/1.17 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.12/1.17 ( ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ! [Xy: $i] :
% 2.12/1.17 ( ( in @ Xy @ B )
% 2.12/1.17 => ( ( Xx = Xy )
% 2.12/1.17 => ( ( Xphi @ Xx )
% 2.12/1.17 <=> ( Xpsi @ Xy ) ) ) ) )
% 2.12/1.17 => ( ( dsetconstr @ A
% 2.12/1.17 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.12/1.17 = ( dsetconstr @ B
% 2.12/1.17 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(subset_type,type,
% 2.12/1.17 subset: $i > $i > $o ).
% 2.12/1.17
% 2.12/1.17 thf(disjoint_type,type,
% 2.12/1.17 disjoint: $i > $i > $o ).
% 2.12/1.17
% 2.12/1.17 thf(setsmeet_type,type,
% 2.12/1.17 setsmeet: $i > $i > $o ).
% 2.12/1.17
% 2.12/1.17 thf(subsetI1_type,type,
% 2.12/1.17 subsetI1: $o ).
% 2.12/1.17
% 2.12/1.17 thf(subsetI1,definition,
% 2.12/1.17 ( subsetI1
% 2.12/1.17 = ( ! [A: $i,B: $i] :
% 2.12/1.17 ( ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ( in @ Xx @ B ) )
% 2.12/1.17 => ( subset @ A @ B ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(eqimpsubset2_type,type,
% 2.12/1.17 eqimpsubset2: $o ).
% 2.12/1.17
% 2.12/1.17 thf(eqimpsubset2,definition,
% 2.12/1.17 ( eqimpsubset2
% 2.12/1.17 = ( ! [A: $i,B: $i] :
% 2.12/1.17 ( ( A = B )
% 2.12/1.17 => ( subset @ B @ A ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(eqimpsubset1_type,type,
% 2.12/1.17 eqimpsubset1: $o ).
% 2.12/1.17
% 2.12/1.17 thf(eqimpsubset1,definition,
% 2.12/1.17 ( eqimpsubset1
% 2.12/1.17 = ( ! [A: $i,B: $i] :
% 2.12/1.17 ( ( A = B )
% 2.12/1.17 => ( subset @ A @ B ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(subsetI2_type,type,
% 2.12/1.17 subsetI2: $o ).
% 2.12/1.17
% 2.12/1.17 thf(subsetI2,definition,
% 2.12/1.17 ( subsetI2
% 2.12/1.17 = ( ! [A: $i,B: $i] :
% 2.12/1.17 ( ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ( in @ Xx @ B ) )
% 2.12/1.17 => ( subset @ A @ B ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(emptysetsubset_type,type,
% 2.12/1.17 emptysetsubset: $o ).
% 2.12/1.17
% 2.12/1.17 thf(emptysetsubset,definition,
% 2.12/1.17 ( emptysetsubset
% 2.12/1.17 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(subsetE_type,type,
% 2.12/1.17 subsetE: $o ).
% 2.12/1.17
% 2.12/1.17 thf(subsetE,definition,
% 2.12/1.17 ( subsetE
% 2.12/1.17 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.12/1.17 ( ( subset @ A @ B )
% 2.12/1.17 => ( ( in @ Xx @ A )
% 2.12/1.17 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(subsetE2_type,type,
% 2.12/1.17 subsetE2: $o ).
% 2.12/1.17
% 2.12/1.17 thf(subsetE2,definition,
% 2.12/1.17 ( subsetE2
% 2.12/1.17 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.12/1.17 ( ( subset @ A @ B )
% 2.12/1.17 => ( ~ ( in @ Xx @ B )
% 2.12/1.17 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(notsubsetI_type,type,
% 2.12/1.17 notsubsetI: $o ).
% 2.12/1.17
% 2.12/1.17 thf(notsubsetI,definition,
% 2.12/1.17 ( notsubsetI
% 2.12/1.17 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ( ~ ( in @ Xx @ B )
% 2.12/1.17 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(notequalI1_type,type,
% 2.12/1.17 notequalI1: $o ).
% 2.12/1.17
% 2.12/1.17 thf(notequalI1,definition,
% 2.12/1.17 ( notequalI1
% 2.12/1.17 = ( ! [A: $i,B: $i] :
% 2.12/1.17 ( ~ ( subset @ A @ B )
% 2.12/1.17 => ( A != B ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(notequalI2_type,type,
% 2.12/1.17 notequalI2: $o ).
% 2.12/1.17
% 2.12/1.17 thf(notequalI2,definition,
% 2.12/1.17 ( notequalI2
% 2.12/1.17 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ( ~ ( in @ Xx @ B )
% 2.12/1.17 => ( A != B ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(subsetRefl_type,type,
% 2.12/1.17 subsetRefl: $o ).
% 2.12/1.17
% 2.12/1.17 thf(subsetRefl,definition,
% 2.12/1.17 ( subsetRefl
% 2.12/1.17 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(subsetTrans_type,type,
% 2.12/1.17 subsetTrans: $o ).
% 2.12/1.17
% 2.12/1.17 thf(subsetTrans,definition,
% 2.12/1.17 ( subsetTrans
% 2.12/1.17 = ( ! [A: $i,B: $i,C: $i] :
% 2.12/1.17 ( ( subset @ A @ B )
% 2.12/1.17 => ( ( subset @ B @ C )
% 2.12/1.17 => ( subset @ A @ C ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(setadjoinSub_type,type,
% 2.12/1.17 setadjoinSub: $o ).
% 2.12/1.17
% 2.12/1.17 thf(setadjoinSub,definition,
% 2.12/1.17 ( setadjoinSub
% 2.12/1.17 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(setadjoinSub2_type,type,
% 2.12/1.17 setadjoinSub2: $o ).
% 2.12/1.17
% 2.12/1.17 thf(setadjoinSub2,definition,
% 2.12/1.17 ( setadjoinSub2
% 2.12/1.17 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.12/1.17 ( ( subset @ A @ B )
% 2.12/1.17 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(subset2powerset_type,type,
% 2.12/1.17 subset2powerset: $o ).
% 2.12/1.17
% 2.12/1.17 thf(subset2powerset,definition,
% 2.12/1.17 ( subset2powerset
% 2.12/1.17 = ( ! [A: $i,B: $i] :
% 2.12/1.17 ( ( subset @ A @ B )
% 2.12/1.17 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(setextsub_type,type,
% 2.12/1.17 setextsub: $o ).
% 2.12/1.17
% 2.12/1.17 thf(setextsub,definition,
% 2.12/1.17 ( setextsub
% 2.12/1.17 = ( ! [A: $i,B: $i] :
% 2.12/1.17 ( ( subset @ A @ B )
% 2.12/1.17 => ( ( subset @ B @ A )
% 2.12/1.17 => ( A = B ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(subsetemptysetimpeq_type,type,
% 2.12/1.17 subsetemptysetimpeq: $o ).
% 2.12/1.17
% 2.12/1.17 thf(subsetemptysetimpeq,definition,
% 2.12/1.17 ( subsetemptysetimpeq
% 2.12/1.17 = ( ! [A: $i] :
% 2.12/1.17 ( ( subset @ A @ emptyset )
% 2.12/1.17 => ( A = emptyset ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(powersetI1_type,type,
% 2.12/1.17 powersetI1: $o ).
% 2.12/1.17
% 2.12/1.17 thf(powersetI1,definition,
% 2.12/1.17 ( powersetI1
% 2.12/1.17 = ( ! [A: $i,B: $i] :
% 2.12/1.17 ( ( subset @ B @ A )
% 2.12/1.17 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(powersetE1_type,type,
% 2.12/1.17 powersetE1: $o ).
% 2.12/1.17
% 2.12/1.17 thf(powersetE1,definition,
% 2.12/1.17 ( powersetE1
% 2.12/1.17 = ( ! [A: $i,B: $i] :
% 2.12/1.17 ( ( in @ B @ ( powerset @ A ) )
% 2.12/1.17 => ( subset @ B @ A ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(inPowerset_type,type,
% 2.12/1.17 inPowerset: $o ).
% 2.12/1.17
% 2.12/1.17 thf(inPowerset,definition,
% 2.12/1.17 ( inPowerset
% 2.12/1.17 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(powersetsubset_type,type,
% 2.12/1.17 powersetsubset: $o ).
% 2.12/1.17
% 2.12/1.17 thf(powersetsubset,definition,
% 2.12/1.17 ( powersetsubset
% 2.12/1.17 = ( ! [A: $i,B: $i] :
% 2.12/1.17 ( ( subset @ A @ B )
% 2.12/1.17 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(sepInPowerset_type,type,
% 2.12/1.17 sepInPowerset: $o ).
% 2.12/1.17
% 2.12/1.17 thf(sepInPowerset,definition,
% 2.12/1.17 ( sepInPowerset
% 2.12/1.17 = ( ! [A: $i,Xphi: $i > $o] :
% 2.12/1.17 ( in
% 2.12/1.17 @ ( dsetconstr @ A
% 2.12/1.17 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.12/1.17 @ ( powerset @ A ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(sepSubset_type,type,
% 2.12/1.17 sepSubset: $o ).
% 2.12/1.17
% 2.12/1.17 thf(sepSubset,definition,
% 2.12/1.17 ( sepSubset
% 2.12/1.17 = ( ! [A: $i,Xphi: $i > $o] :
% 2.12/1.17 ( subset
% 2.12/1.17 @ ( dsetconstr @ A
% 2.12/1.17 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.12/1.17 @ A ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(binunion_type,type,
% 2.12/1.17 binunion: $i > $i > $i ).
% 2.12/1.17
% 2.12/1.17 thf(binunionIL_type,type,
% 2.12/1.17 binunionIL: $o ).
% 2.12/1.17
% 2.12/1.17 thf(binunionIL,definition,
% 2.12/1.17 ( binunionIL
% 2.12/1.17 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(upairset2IR_type,type,
% 2.12/1.17 upairset2IR: $o ).
% 2.12/1.17
% 2.12/1.17 thf(upairset2IR,definition,
% 2.12/1.17 ( upairset2IR
% 2.12/1.17 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(binunionIR_type,type,
% 2.12/1.17 binunionIR: $o ).
% 2.12/1.17
% 2.12/1.17 thf(binunionIR,definition,
% 2.12/1.17 ( binunionIR
% 2.12/1.17 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ B )
% 2.12/1.17 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(binunionEcases_type,type,
% 2.12/1.17 binunionEcases: $o ).
% 2.12/1.17
% 2.12/1.17 thf(binunionEcases,definition,
% 2.12/1.17 ( binunionEcases
% 2.12/1.17 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 2.12/1.17 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.12/1.17 => ( ( ( in @ Xx @ A )
% 2.12/1.17 => Xphi )
% 2.12/1.17 => ( ( ( in @ Xx @ B )
% 2.12/1.17 => Xphi )
% 2.12/1.17 => Xphi ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(binunionE_type,type,
% 2.12/1.17 binunionE: $o ).
% 2.12/1.17
% 2.12/1.17 thf(binunionE,definition,
% 2.12/1.17 ( binunionE
% 2.12/1.17 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.12/1.17 => ( ( in @ Xx @ A )
% 2.12/1.17 | ( in @ Xx @ B ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(binunionLsub_type,type,
% 2.12/1.17 binunionLsub: $o ).
% 2.12/1.17
% 2.12/1.17 thf(binunionLsub,definition,
% 2.12/1.17 ( binunionLsub
% 2.12/1.17 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(binunionRsub_type,type,
% 2.12/1.17 binunionRsub: $o ).
% 2.12/1.17
% 2.12/1.17 thf(binunionRsub,definition,
% 2.12/1.17 ( binunionRsub
% 2.12/1.17 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(binintersect_type,type,
% 2.12/1.17 binintersect: $i > $i > $i ).
% 2.12/1.17
% 2.12/1.17 thf(binintersectI_type,type,
% 2.12/1.17 binintersectI: $o ).
% 2.12/1.17
% 2.12/1.17 thf(binintersectI,definition,
% 2.12/1.17 ( binintersectI
% 2.12/1.17 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ( ( in @ Xx @ B )
% 2.12/1.17 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(binintersectSubset5_type,type,
% 2.12/1.17 binintersectSubset5: $o ).
% 2.12/1.17
% 2.12/1.17 thf(binintersectSubset5,definition,
% 2.12/1.17 ( binintersectSubset5
% 2.12/1.17 = ( ! [A: $i,B: $i,C: $i] :
% 2.12/1.17 ( ( subset @ C @ A )
% 2.12/1.17 => ( ( subset @ C @ B )
% 2.12/1.17 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(binintersectEL_type,type,
% 2.12/1.17 binintersectEL: $o ).
% 2.12/1.17
% 2.12/1.17 thf(binintersectEL,definition,
% 2.12/1.17 ( binintersectEL
% 2.12/1.17 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.12/1.17 => ( in @ Xx @ A ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(binintersectLsub_type,type,
% 2.12/1.17 binintersectLsub: $o ).
% 2.12/1.17
% 2.12/1.17 thf(binintersectLsub,definition,
% 2.12/1.17 ( binintersectLsub
% 2.12/1.17 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(binintersectSubset2_type,type,
% 2.12/1.17 binintersectSubset2: $o ).
% 2.12/1.17
% 2.12/1.17 thf(binintersectSubset2,definition,
% 2.12/1.17 ( binintersectSubset2
% 2.12/1.17 = ( ! [A: $i,B: $i] :
% 2.12/1.17 ( ( subset @ A @ B )
% 2.12/1.17 => ( ( binintersect @ A @ B )
% 2.12/1.17 = A ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(binintersectSubset3_type,type,
% 2.12/1.17 binintersectSubset3: $o ).
% 2.12/1.17
% 2.12/1.17 thf(binintersectSubset3,definition,
% 2.12/1.17 ( binintersectSubset3
% 2.12/1.17 = ( ! [A: $i,B: $i] :
% 2.12/1.17 ( ( ( binintersect @ A @ B )
% 2.12/1.17 = B )
% 2.12/1.17 => ( subset @ B @ A ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(binintersectER_type,type,
% 2.12/1.17 binintersectER: $o ).
% 2.12/1.17
% 2.12/1.17 thf(binintersectER,definition,
% 2.12/1.17 ( binintersectER
% 2.12/1.17 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.12/1.17 => ( in @ Xx @ B ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(disjointsetsI1_type,type,
% 2.12/1.17 disjointsetsI1: $o ).
% 2.12/1.17
% 2.12/1.17 thf(disjointsetsI1,definition,
% 2.12/1.17 ( disjointsetsI1
% 2.12/1.17 = ( ! [A: $i,B: $i] :
% 2.12/1.17 ( ~ ? [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 & ( in @ Xx @ B ) )
% 2.12/1.17 => ( ( binintersect @ A @ B )
% 2.12/1.17 = emptyset ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(binintersectRsub_type,type,
% 2.12/1.17 binintersectRsub: $o ).
% 2.12/1.17
% 2.12/1.17 thf(binintersectRsub,definition,
% 2.12/1.17 ( binintersectRsub
% 2.12/1.17 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(binintersectSubset4_type,type,
% 2.12/1.17 binintersectSubset4: $o ).
% 2.12/1.17
% 2.12/1.17 thf(binintersectSubset4,definition,
% 2.12/1.17 ( binintersectSubset4
% 2.12/1.17 = ( ! [A: $i,B: $i] :
% 2.12/1.17 ( ( subset @ B @ A )
% 2.12/1.17 => ( ( binintersect @ A @ B )
% 2.12/1.17 = B ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(binintersectSubset1_type,type,
% 2.12/1.17 binintersectSubset1: $o ).
% 2.12/1.17
% 2.12/1.17 thf(binintersectSubset1,definition,
% 2.12/1.17 ( binintersectSubset1
% 2.12/1.17 = ( ! [A: $i,B: $i] :
% 2.12/1.17 ( ( ( binintersect @ A @ B )
% 2.12/1.17 = A )
% 2.12/1.17 => ( subset @ A @ B ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(bs114d_type,type,
% 2.12/1.17 bs114d: $o ).
% 2.12/1.17
% 2.12/1.17 thf(bs114d,definition,
% 2.12/1.17 ( bs114d
% 2.12/1.17 = ( ! [A: $i,B: $i,C: $i] :
% 2.12/1.17 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 2.12/1.17 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(regular_type,type,
% 2.12/1.17 regular: $i > $o ).
% 2.12/1.17
% 2.12/1.17 thf(setminus_type,type,
% 2.12/1.17 setminus: $i > $i > $i ).
% 2.12/1.17
% 2.12/1.17 thf(setminusI_type,type,
% 2.12/1.17 setminusI: $o ).
% 2.12/1.17
% 2.12/1.17 thf(setminusI,definition,
% 2.12/1.17 ( setminusI
% 2.12/1.17 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ( ~ ( in @ Xx @ B )
% 2.12/1.17 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(setminusEL_type,type,
% 2.12/1.17 setminusEL: $o ).
% 2.12/1.17
% 2.12/1.17 thf(setminusEL,definition,
% 2.12/1.17 ( setminusEL
% 2.12/1.17 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.12/1.17 => ( in @ Xx @ A ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(setminusER_type,type,
% 2.12/1.17 setminusER: $o ).
% 2.12/1.17
% 2.12/1.17 thf(setminusER,definition,
% 2.12/1.17 ( setminusER
% 2.12/1.17 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.12/1.17 => ~ ( in @ Xx @ B ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(setminusSubset2_type,type,
% 2.12/1.17 setminusSubset2: $o ).
% 2.12/1.17
% 2.12/1.17 thf(setminusSubset2,definition,
% 2.12/1.17 ( setminusSubset2
% 2.12/1.17 = ( ! [A: $i,B: $i] :
% 2.12/1.17 ( ( subset @ A @ B )
% 2.12/1.17 => ( ( setminus @ A @ B )
% 2.12/1.17 = emptyset ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(setminusERneg_type,type,
% 2.12/1.17 setminusERneg: $o ).
% 2.12/1.17
% 2.12/1.17 thf(setminusERneg,definition,
% 2.12/1.17 ( setminusERneg
% 2.12/1.17 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.12/1.17 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.12/1.17 => ( ( in @ Xx @ A )
% 2.12/1.17 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(setminusELneg_type,type,
% 2.12/1.17 setminusELneg: $o ).
% 2.12/1.17
% 2.12/1.17 thf(setminusELneg,definition,
% 2.12/1.17 ( setminusELneg
% 2.12/1.17 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.12/1.17 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.12/1.17 => ( ~ ( in @ Xx @ B )
% 2.12/1.17 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(setminusILneg_type,type,
% 2.12/1.17 setminusILneg: $o ).
% 2.12/1.17
% 2.12/1.17 thf(setminusILneg,definition,
% 2.12/1.17 ( setminusILneg
% 2.12/1.17 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.12/1.17 ( ~ ( in @ Xx @ A )
% 2.12/1.17 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(setminusIRneg_type,type,
% 2.12/1.17 setminusIRneg: $o ).
% 2.12/1.17
% 2.12/1.17 thf(setminusIRneg,definition,
% 2.12/1.17 ( setminusIRneg
% 2.12/1.17 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ B )
% 2.12/1.17 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(setminusLsub_type,type,
% 2.12/1.17 setminusLsub: $o ).
% 2.12/1.17
% 2.12/1.17 thf(setminusLsub,definition,
% 2.12/1.17 ( setminusLsub
% 2.12/1.17 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(setminusSubset1_type,type,
% 2.12/1.17 setminusSubset1: $o ).
% 2.12/1.17
% 2.12/1.17 thf(setminusSubset1,definition,
% 2.12/1.17 ( setminusSubset1
% 2.12/1.17 = ( ! [A: $i,B: $i] :
% 2.12/1.17 ( ( ( setminus @ A @ B )
% 2.12/1.17 = emptyset )
% 2.12/1.17 => ( subset @ A @ B ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(symdiff_type,type,
% 2.12/1.17 symdiff: $i > $i > $i ).
% 2.12/1.17
% 2.12/1.17 thf(symdiffE_type,type,
% 2.12/1.17 symdiffE: $o ).
% 2.12/1.17
% 2.12/1.17 thf(symdiffE,definition,
% 2.12/1.17 ( symdiffE
% 2.12/1.17 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 2.12/1.17 => ! [Xphi: $o] :
% 2.12/1.17 ( ( ( in @ Xx @ A )
% 2.12/1.17 => ( ~ ( in @ Xx @ B )
% 2.12/1.17 => Xphi ) )
% 2.12/1.17 => ( ( ~ ( in @ Xx @ A )
% 2.12/1.17 => ( ( in @ Xx @ B )
% 2.12/1.17 => Xphi ) )
% 2.12/1.17 => Xphi ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(symdiffI1_type,type,
% 2.12/1.17 symdiffI1: $o ).
% 2.12/1.17
% 2.12/1.17 thf(symdiffI1,definition,
% 2.12/1.17 ( symdiffI1
% 2.12/1.17 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ( ~ ( in @ Xx @ B )
% 2.12/1.17 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(symdiffI2_type,type,
% 2.12/1.17 symdiffI2: $o ).
% 2.12/1.17
% 2.12/1.17 thf(symdiffI2,definition,
% 2.12/1.17 ( symdiffI2
% 2.12/1.17 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.12/1.17 ( ~ ( in @ Xx @ A )
% 2.12/1.17 => ( ( in @ Xx @ B )
% 2.12/1.17 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(symdiffIneg1_type,type,
% 2.12/1.17 symdiffIneg1: $o ).
% 2.12/1.17
% 2.12/1.17 thf(symdiffIneg1,definition,
% 2.12/1.17 ( symdiffIneg1
% 2.12/1.17 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ( ( in @ Xx @ B )
% 2.12/1.17 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(symdiffIneg2_type,type,
% 2.12/1.17 symdiffIneg2: $o ).
% 2.12/1.17
% 2.12/1.17 thf(symdiffIneg2,definition,
% 2.12/1.17 ( symdiffIneg2
% 2.12/1.17 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.12/1.17 ( ~ ( in @ Xx @ A )
% 2.12/1.17 => ( ~ ( in @ Xx @ B )
% 2.12/1.17 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(iskpair_type,type,
% 2.12/1.17 iskpair: $i > $o ).
% 2.12/1.17
% 2.12/1.17 thf(secondinupair_type,type,
% 2.12/1.17 secondinupair: $o ).
% 2.12/1.17
% 2.12/1.17 thf(secondinupair,definition,
% 2.12/1.17 ( secondinupair
% 2.12/1.17 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(setukpairIL_type,type,
% 2.12/1.17 setukpairIL: $o ).
% 2.12/1.17
% 2.12/1.17 thf(setukpairIL,definition,
% 2.12/1.17 ( setukpairIL
% 2.12/1.17 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(setukpairIR_type,type,
% 2.12/1.17 setukpairIR: $o ).
% 2.12/1.17
% 2.12/1.17 thf(setukpairIR,definition,
% 2.12/1.17 ( setukpairIR
% 2.12/1.17 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(kpairiskpair_type,type,
% 2.12/1.17 kpairiskpair: $o ).
% 2.12/1.17
% 2.12/1.17 thf(kpairiskpair,definition,
% 2.12/1.17 ( kpairiskpair
% 2.12/1.17 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(kpair_type,type,
% 2.12/1.17 kpair: $i > $i > $i ).
% 2.12/1.17
% 2.12/1.17 thf(kpairp_type,type,
% 2.12/1.17 kpairp: $o ).
% 2.12/1.17
% 2.12/1.17 thf(kpairp,definition,
% 2.12/1.17 ( kpairp
% 2.12/1.17 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(cartprod_type,type,
% 2.12/1.17 cartprod: $i > $i > $i ).
% 2.12/1.17
% 2.12/1.17 thf(singletonsubset_type,type,
% 2.12/1.17 singletonsubset: $o ).
% 2.12/1.17
% 2.12/1.17 thf(singletonsubset,definition,
% 2.12/1.17 ( singletonsubset
% 2.12/1.17 = ( ! [A: $i,Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(singletoninpowerset_type,type,
% 2.12/1.17 singletoninpowerset: $o ).
% 2.12/1.17
% 2.12/1.17 thf(singletoninpowerset,definition,
% 2.12/1.17 ( singletoninpowerset
% 2.12/1.17 = ( ! [A: $i,Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(singletoninpowunion_type,type,
% 2.12/1.17 singletoninpowunion: $o ).
% 2.12/1.17
% 2.12/1.17 thf(singletoninpowunion,definition,
% 2.12/1.17 ( singletoninpowunion
% 2.12/1.17 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(upairset2E_type,type,
% 2.12/1.17 upairset2E: $o ).
% 2.12/1.17
% 2.12/1.17 thf(upairset2E,definition,
% 2.12/1.17 ( upairset2E
% 2.12/1.17 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.12/1.17 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.12/1.17 => ( ( Xz = Xx )
% 2.12/1.17 | ( Xz = Xy ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(upairsubunion_type,type,
% 2.12/1.17 upairsubunion: $o ).
% 2.12/1.17
% 2.12/1.17 thf(upairsubunion,definition,
% 2.12/1.17 ( upairsubunion
% 2.12/1.17 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ! [Xy: $i] :
% 2.12/1.17 ( ( in @ Xy @ B )
% 2.12/1.17 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(upairinpowunion_type,type,
% 2.12/1.17 upairinpowunion: $o ).
% 2.12/1.17
% 2.12/1.17 thf(upairinpowunion,definition,
% 2.12/1.17 ( upairinpowunion
% 2.12/1.17 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ! [Xy: $i] :
% 2.12/1.17 ( ( in @ Xy @ B )
% 2.12/1.17 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(ubforcartprodlem1_type,type,
% 2.12/1.17 ubforcartprodlem1: $o ).
% 2.12/1.17
% 2.12/1.17 thf(ubforcartprodlem1,definition,
% 2.12/1.17 ( ubforcartprodlem1
% 2.12/1.17 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ! [Xy: $i] :
% 2.12/1.17 ( ( in @ Xy @ B )
% 2.12/1.17 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(ubforcartprodlem2_type,type,
% 2.12/1.17 ubforcartprodlem2: $o ).
% 2.12/1.17
% 2.12/1.17 thf(ubforcartprodlem2,definition,
% 2.12/1.17 ( ubforcartprodlem2
% 2.12/1.17 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ! [Xy: $i] :
% 2.12/1.17 ( ( in @ Xy @ B )
% 2.12/1.17 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(ubforcartprodlem3_type,type,
% 2.12/1.17 ubforcartprodlem3: $o ).
% 2.12/1.17
% 2.12/1.17 thf(ubforcartprodlem3,definition,
% 2.12/1.17 ( ubforcartprodlem3
% 2.12/1.17 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ! [Xy: $i] :
% 2.12/1.17 ( ( in @ Xy @ B )
% 2.12/1.17 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(cartprodpairin_type,type,
% 2.12/1.17 cartprodpairin: $o ).
% 2.12/1.17
% 2.12/1.17 thf(cartprodpairin,definition,
% 2.12/1.17 ( cartprodpairin
% 2.12/1.17 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ! [Xy: $i] :
% 2.12/1.17 ( ( in @ Xy @ B )
% 2.12/1.17 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(cartprodmempair1_type,type,
% 2.12/1.17 cartprodmempair1: $o ).
% 2.12/1.17
% 2.12/1.17 thf(cartprodmempair1,definition,
% 2.12/1.17 ( cartprodmempair1
% 2.12/1.17 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.12/1.17 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.12/1.17 => ? [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 & ? [Xy: $i] :
% 2.12/1.17 ( ( in @ Xy @ B )
% 2.12/1.17 & ( Xu
% 2.12/1.17 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(cartprodmempair_type,type,
% 2.12/1.17 cartprodmempair: $o ).
% 2.12/1.17
% 2.12/1.17 thf(cartprodmempair,definition,
% 2.12/1.17 ( cartprodmempair
% 2.12/1.17 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.12/1.17 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.12/1.17 => ( iskpair @ Xu ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(setunionE2_type,type,
% 2.12/1.17 setunionE2: $o ).
% 2.12/1.17
% 2.12/1.17 thf(setunionE2,definition,
% 2.12/1.17 ( setunionE2
% 2.12/1.17 = ( ! [A: $i,Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ ( setunion @ A ) )
% 2.12/1.17 => ? [X: $i] :
% 2.12/1.17 ( ( in @ X @ A )
% 2.12/1.17 & ( in @ Xx @ X ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(setunionsingleton1_type,type,
% 2.12/1.17 setunionsingleton1: $o ).
% 2.12/1.17
% 2.12/1.17 thf(setunionsingleton1,definition,
% 2.12/1.17 ( setunionsingleton1
% 2.12/1.17 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(setunionsingleton2_type,type,
% 2.12/1.17 setunionsingleton2: $o ).
% 2.12/1.17
% 2.12/1.17 thf(setunionsingleton2,definition,
% 2.12/1.17 ( setunionsingleton2
% 2.12/1.17 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(setunionsingleton_type,type,
% 2.12/1.17 setunionsingleton: $o ).
% 2.12/1.17
% 2.12/1.17 thf(setunionsingleton,definition,
% 2.12/1.17 ( setunionsingleton
% 2.12/1.17 = ( ! [Xx: $i] :
% 2.12/1.17 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 2.12/1.17 = Xx ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(singleton_type,type,
% 2.12/1.17 singleton: $i > $o ).
% 2.12/1.17
% 2.12/1.17 thf(singletonprop_type,type,
% 2.12/1.17 singletonprop: $o ).
% 2.12/1.17
% 2.12/1.17 thf(singletonprop,definition,
% 2.12/1.17 ( singletonprop
% 2.12/1.17 = ( ! [A: $i,Xphi: $i > $o] :
% 2.12/1.17 ( ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ! [Xy: $i] :
% 2.12/1.17 ( ( in @ Xy @ A )
% 2.12/1.17 => ( ( Xphi @ Xx )
% 2.12/1.17 => ( ( Xphi @ Xy )
% 2.12/1.17 => ( Xx = Xy ) ) ) ) )
% 2.12/1.17 => ( ? [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 & ( Xphi @ Xx ) )
% 2.12/1.17 => ( singleton
% 2.12/1.17 @ ( dsetconstr @ A
% 2.12/1.17 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(ex1_type,type,
% 2.12/1.17 ex1: $i > ( $i > $o ) > $o ).
% 2.12/1.17
% 2.12/1.17 thf(ex1E1_type,type,
% 2.12/1.17 ex1E1: $o ).
% 2.12/1.17
% 2.12/1.17 thf(ex1E1,definition,
% 2.12/1.17 ( ex1E1
% 2.12/1.17 = ( ! [A: $i,Xphi: $i > $o] :
% 2.12/1.17 ( ( ex1 @ A
% 2.12/1.17 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.12/1.17 => ? [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(ex1I_type,type,
% 2.12/1.17 ex1I: $o ).
% 2.12/1.17
% 2.12/1.17 thf(ex1I,definition,
% 2.12/1.17 ( ex1I
% 2.12/1.17 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ( ( Xphi @ Xx )
% 2.12/1.17 => ( ! [Xy: $i] :
% 2.12/1.17 ( ( in @ Xy @ A )
% 2.12/1.17 => ( ( Xphi @ Xy )
% 2.12/1.17 => ( Xy = Xx ) ) )
% 2.12/1.17 => ( ex1 @ A
% 2.12/1.17 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(ex1I2_type,type,
% 2.12/1.17 ex1I2: $o ).
% 2.12/1.17
% 2.12/1.17 thf(ex1I2,definition,
% 2.12/1.17 ( ex1I2
% 2.12/1.17 = ( ! [A: $i,Xphi: $i > $o] :
% 2.12/1.17 ( ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ! [Xy: $i] :
% 2.12/1.17 ( ( in @ Xy @ A )
% 2.12/1.17 => ( ( Xphi @ Xx )
% 2.12/1.17 => ( ( Xphi @ Xy )
% 2.12/1.17 => ( Xx = Xy ) ) ) ) )
% 2.12/1.17 => ( ? [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 & ( Xphi @ Xx ) )
% 2.12/1.17 => ( ex1 @ A
% 2.12/1.17 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(singletonsuniq_type,type,
% 2.12/1.17 singletonsuniq: $o ).
% 2.12/1.17
% 2.12/1.17 thf(singletonsuniq,definition,
% 2.12/1.17 ( singletonsuniq
% 2.12/1.17 = ( ! [Xx: $i,Xy: $i] :
% 2.12/1.17 ( ( ( setadjoin @ Xx @ emptyset )
% 2.12/1.17 = ( setadjoin @ Xy @ emptyset ) )
% 2.12/1.17 => ( Xx = Xy ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(atmost1p_type,type,
% 2.12/1.17 atmost1p: $i > $o ).
% 2.12/1.17
% 2.12/1.17 thf(atleast2p_type,type,
% 2.12/1.17 atleast2p: $i > $o ).
% 2.12/1.17
% 2.12/1.17 thf(atmost2p_type,type,
% 2.12/1.17 atmost2p: $i > $o ).
% 2.12/1.17
% 2.12/1.17 thf(upairsetp_type,type,
% 2.12/1.17 upairsetp: $i > $o ).
% 2.12/1.17
% 2.12/1.17 thf(setukpairinjL1_type,type,
% 2.12/1.17 setukpairinjL1: $o ).
% 2.12/1.17
% 2.12/1.17 thf(setukpairinjL1,definition,
% 2.12/1.17 ( setukpairinjL1
% 2.12/1.17 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.12/1.17 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 2.12/1.17 => ( Xx = Xz ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(kfstsingleton_type,type,
% 2.12/1.17 kfstsingleton: $o ).
% 2.12/1.17
% 2.12/1.17 thf(kfstsingleton,definition,
% 2.12/1.17 ( kfstsingleton
% 2.12/1.17 = ( ! [Xu: $i] :
% 2.12/1.17 ( ( iskpair @ Xu )
% 2.12/1.17 => ( singleton
% 2.12/1.17 @ ( dsetconstr @ ( setunion @ Xu )
% 2.12/1.17 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(theprop_type,type,
% 2.12/1.17 theprop: $o ).
% 2.12/1.17
% 2.12/1.17 thf(theprop,definition,
% 2.12/1.17 ( theprop
% 2.12/1.17 = ( ! [X: $i] :
% 2.12/1.17 ( ( singleton @ X )
% 2.12/1.17 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(kfst_type,type,
% 2.12/1.17 kfst: $i > $i ).
% 2.12/1.17
% 2.12/1.17 thf(kfstpairEq_type,type,
% 2.12/1.17 kfstpairEq: $o ).
% 2.12/1.17
% 2.12/1.17 thf(kfstpairEq,definition,
% 2.12/1.17 ( kfstpairEq
% 2.12/1.17 = ( ! [Xx: $i,Xy: $i] :
% 2.12/1.17 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 2.12/1.17 = Xx ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(cartprodfstin_type,type,
% 2.12/1.17 cartprodfstin: $o ).
% 2.12/1.17
% 2.12/1.17 thf(cartprodfstin,definition,
% 2.12/1.17 ( cartprodfstin
% 2.12/1.17 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.12/1.17 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.12/1.17 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(setukpairinjL2_type,type,
% 2.12/1.17 setukpairinjL2: $o ).
% 2.12/1.17
% 2.12/1.17 thf(setukpairinjL2,definition,
% 2.12/1.17 ( setukpairinjL2
% 2.12/1.17 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.12/1.17 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.12/1.17 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.12/1.17 => ( Xx = Xz ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(setukpairinjL_type,type,
% 2.12/1.17 setukpairinjL: $o ).
% 2.12/1.17
% 2.12/1.17 thf(setukpairinjL,definition,
% 2.12/1.17 ( setukpairinjL
% 2.12/1.17 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.12/1.17 ( ( ( kpair @ Xx @ Xy )
% 2.12/1.17 = ( kpair @ Xz @ Xu ) )
% 2.12/1.17 => ( Xx = Xz ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(setukpairinjR11_type,type,
% 2.12/1.17 setukpairinjR11: $o ).
% 2.12/1.17
% 2.12/1.17 thf(setukpairinjR11,definition,
% 2.12/1.17 ( setukpairinjR11
% 2.12/1.17 = ( ! [Xx: $i,Xy: $i] :
% 2.12/1.17 ( ( Xx = Xy )
% 2.12/1.17 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.12/1.17 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(setukpairinjR12_type,type,
% 2.12/1.17 setukpairinjR12: $o ).
% 2.12/1.17
% 2.12/1.17 thf(setukpairinjR12,definition,
% 2.12/1.17 ( setukpairinjR12
% 2.12/1.17 = ( ! [Xx: $i,Xy: $i] :
% 2.12/1.17 ( ( Xx = Xy )
% 2.12/1.17 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.12/1.17 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(setukpairinjR1_type,type,
% 2.12/1.17 setukpairinjR1: $o ).
% 2.12/1.17
% 2.12/1.17 thf(setukpairinjR1,definition,
% 2.12/1.17 ( setukpairinjR1
% 2.12/1.17 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.12/1.17 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.12/1.17 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.12/1.17 => ( ( Xz = Xu )
% 2.12/1.17 => ( Xy = Xu ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(upairequniteq_type,type,
% 2.12/1.17 upairequniteq: $o ).
% 2.12/1.17
% 2.12/1.17 thf(upairequniteq,definition,
% 2.12/1.17 ( upairequniteq
% 2.12/1.17 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.12/1.17 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.12/1.17 = ( setadjoin @ Xz @ emptyset ) )
% 2.12/1.17 => ( Xx = Xy ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(setukpairinjR2_type,type,
% 2.12/1.17 setukpairinjR2: $o ).
% 2.12/1.17
% 2.12/1.17 thf(setukpairinjR2,definition,
% 2.12/1.17 ( setukpairinjR2
% 2.12/1.17 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.12/1.17 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.12/1.17 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.12/1.17 => ( Xy = Xu ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(setukpairinjR_type,type,
% 2.12/1.17 setukpairinjR: $o ).
% 2.12/1.17
% 2.12/1.17 thf(setukpairinjR,definition,
% 2.12/1.17 ( setukpairinjR
% 2.12/1.17 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.12/1.17 ( ( ( kpair @ Xx @ Xy )
% 2.12/1.17 = ( kpair @ Xz @ Xu ) )
% 2.12/1.17 => ( Xy = Xu ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(ksndsingleton_type,type,
% 2.12/1.17 ksndsingleton: $o ).
% 2.12/1.17
% 2.12/1.17 thf(ksndsingleton,definition,
% 2.12/1.17 ( ksndsingleton
% 2.12/1.17 = ( ! [Xu: $i] :
% 2.12/1.17 ( ( iskpair @ Xu )
% 2.12/1.17 => ( singleton
% 2.12/1.17 @ ( dsetconstr @ ( setunion @ Xu )
% 2.12/1.17 @ ^ [Xx: $i] :
% 2.12/1.17 ( Xu
% 2.12/1.17 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(ksnd_type,type,
% 2.12/1.17 ksnd: $i > $i ).
% 2.12/1.17
% 2.12/1.17 thf(ksndpairEq_type,type,
% 2.12/1.17 ksndpairEq: $o ).
% 2.12/1.17
% 2.12/1.17 thf(ksndpairEq,definition,
% 2.12/1.17 ( ksndpairEq
% 2.12/1.17 = ( ! [Xx: $i,Xy: $i] :
% 2.12/1.17 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 2.12/1.17 = Xy ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(kpairsurjEq_type,type,
% 2.12/1.17 kpairsurjEq: $o ).
% 2.12/1.17
% 2.12/1.17 thf(kpairsurjEq,definition,
% 2.12/1.17 ( kpairsurjEq
% 2.12/1.17 = ( ! [Xu: $i] :
% 2.12/1.17 ( ( iskpair @ Xu )
% 2.12/1.17 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 2.12/1.17 = Xu ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(cartprodsndin_type,type,
% 2.12/1.17 cartprodsndin: $o ).
% 2.12/1.17
% 2.12/1.17 thf(cartprodsndin,definition,
% 2.12/1.17 ( cartprodsndin
% 2.12/1.17 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.12/1.17 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.12/1.17 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(cartprodpairmemEL_type,type,
% 2.12/1.17 cartprodpairmemEL: $o ).
% 2.12/1.17
% 2.12/1.17 thf(cartprodpairmemEL,definition,
% 2.12/1.17 ( cartprodpairmemEL
% 2.12/1.17 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 2.12/1.17 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 2.12/1.17 => ( in @ Xx @ A ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(cartprodpairmemER_type,type,
% 2.12/1.17 cartprodpairmemER: $o ).
% 2.12/1.17
% 2.12/1.17 thf(cartprodpairmemER,definition,
% 2.12/1.17 ( cartprodpairmemER
% 2.12/1.17 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 2.12/1.17 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 2.12/1.17 => ( in @ Xy @ B ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(cartprodmempaircEq_type,type,
% 2.12/1.17 cartprodmempaircEq: $o ).
% 2.12/1.17
% 2.12/1.17 thf(cartprodmempaircEq,definition,
% 2.12/1.17 ( cartprodmempaircEq
% 2.12/1.17 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ! [Xy: $i] :
% 2.12/1.17 ( ( in @ Xy @ B )
% 2.12/1.17 => ( ( kpair @ Xx @ Xy )
% 2.12/1.17 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(cartprodfstpairEq_type,type,
% 2.12/1.17 cartprodfstpairEq: $o ).
% 2.12/1.17
% 2.12/1.17 thf(cartprodfstpairEq,definition,
% 2.12/1.17 ( cartprodfstpairEq
% 2.12/1.17 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ! [Xy: $i] :
% 2.12/1.17 ( ( in @ Xy @ B )
% 2.12/1.17 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 2.12/1.17 = Xx ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(cartprodsndpairEq_type,type,
% 2.12/1.17 cartprodsndpairEq: $o ).
% 2.12/1.17
% 2.12/1.17 thf(cartprodsndpairEq,definition,
% 2.12/1.17 ( cartprodsndpairEq
% 2.12/1.17 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ! [Xy: $i] :
% 2.12/1.17 ( ( in @ Xy @ B )
% 2.12/1.17 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 2.12/1.17 = Xy ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(cartprodpairsurjEq_type,type,
% 2.12/1.17 cartprodpairsurjEq: $o ).
% 2.12/1.17
% 2.12/1.17 thf(cartprodpairsurjEq,definition,
% 2.12/1.17 ( cartprodpairsurjEq
% 2.12/1.17 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.12/1.17 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.12/1.17 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 2.12/1.17 = Xu ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(breln_type,type,
% 2.12/1.17 breln: $i > $i > $i > $o ).
% 2.12/1.17
% 2.12/1.17 thf(dpsetconstr_type,type,
% 2.12/1.17 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 2.12/1.17
% 2.12/1.17 thf(dpsetconstrI_type,type,
% 2.12/1.17 dpsetconstrI: $o ).
% 2.12/1.17
% 2.12/1.17 thf(dpsetconstrI,definition,
% 2.12/1.17 ( dpsetconstrI
% 2.12/1.17 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ! [Xy: $i] :
% 2.12/1.17 ( ( in @ Xy @ B )
% 2.12/1.17 => ( ( Xphi @ Xx @ Xy )
% 2.12/1.17 => ( in @ ( kpair @ Xx @ Xy )
% 2.12/1.17 @ ( dpsetconstr @ A @ B
% 2.12/1.17 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(dpsetconstrSub_type,type,
% 2.12/1.17 dpsetconstrSub: $o ).
% 2.12/1.17
% 2.12/1.17 thf(dpsetconstrSub,definition,
% 2.12/1.17 ( dpsetconstrSub
% 2.12/1.17 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 2.12/1.17 ( subset
% 2.12/1.17 @ ( dpsetconstr @ A @ B
% 2.12/1.17 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 2.12/1.17 @ ( cartprod @ A @ B ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(setOfPairsIsBReln_type,type,
% 2.12/1.17 setOfPairsIsBReln: $o ).
% 2.12/1.17
% 2.12/1.17 thf(setOfPairsIsBReln,definition,
% 2.12/1.17 ( setOfPairsIsBReln
% 2.12/1.17 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 2.12/1.17 ( breln @ A @ B
% 2.12/1.17 @ ( dpsetconstr @ A @ B
% 2.12/1.17 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(dpsetconstrERa_type,type,
% 2.12/1.17 dpsetconstrERa: $o ).
% 2.12/1.17
% 2.12/1.17 thf(dpsetconstrERa,definition,
% 2.12/1.17 ( dpsetconstrERa
% 2.12/1.17 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ! [Xy: $i] :
% 2.12/1.17 ( ( in @ Xy @ B )
% 2.12/1.17 => ( ( in @ ( kpair @ Xx @ Xy )
% 2.12/1.17 @ ( dpsetconstr @ A @ B
% 2.12/1.17 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.12/1.17 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(dpsetconstrEL1_type,type,
% 2.12/1.17 dpsetconstrEL1: $o ).
% 2.12/1.17
% 2.12/1.17 thf(dpsetconstrEL1,definition,
% 2.12/1.17 ( dpsetconstrEL1
% 2.12/1.17 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.12/1.17 ( ( in @ ( kpair @ Xx @ Xy )
% 2.12/1.17 @ ( dpsetconstr @ A @ B
% 2.12/1.17 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.12/1.17 => ( in @ Xx @ A ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(dpsetconstrEL2_type,type,
% 2.12/1.17 dpsetconstrEL2: $o ).
% 2.12/1.17
% 2.12/1.17 thf(dpsetconstrEL2,definition,
% 2.12/1.17 ( dpsetconstrEL2
% 2.12/1.17 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.12/1.17 ( ( in @ ( kpair @ Xx @ Xy )
% 2.12/1.17 @ ( dpsetconstr @ A @ B
% 2.12/1.17 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.12/1.17 => ( in @ Xy @ B ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(dpsetconstrER_type,type,
% 2.12/1.17 dpsetconstrER: $o ).
% 2.12/1.17
% 2.12/1.17 thf(dpsetconstrER,definition,
% 2.12/1.17 ( dpsetconstrER
% 2.12/1.17 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.12/1.17 ( ( in @ ( kpair @ Xx @ Xy )
% 2.12/1.17 @ ( dpsetconstr @ A @ B
% 2.12/1.17 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.12/1.17 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(func_type,type,
% 2.12/1.17 func: $i > $i > $i > $o ).
% 2.12/1.17
% 2.12/1.17 thf(funcSet_type,type,
% 2.12/1.17 funcSet: $i > $i > $i ).
% 2.12/1.17
% 2.12/1.17 thf(funcImageSingleton_type,type,
% 2.12/1.17 funcImageSingleton: $o ).
% 2.12/1.17
% 2.12/1.17 thf(funcImageSingleton,definition,
% 2.12/1.17 ( funcImageSingleton
% 2.12/1.17 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.12/1.17 ( ( func @ A @ B @ Xf )
% 2.12/1.17 => ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ( singleton
% 2.12/1.17 @ ( dsetconstr @ B
% 2.12/1.17 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(apProp_type,type,
% 2.12/1.17 apProp: $o ).
% 2.12/1.17
% 2.12/1.17 thf(apProp,definition,
% 2.12/1.17 ( apProp
% 2.12/1.17 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.12/1.17 ( ( func @ A @ B @ Xf )
% 2.12/1.17 => ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ( in
% 2.12/1.17 @ ( setunion
% 2.12/1.17 @ ( dsetconstr @ B
% 2.12/1.17 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 2.12/1.17 @ B ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(ap_type,type,
% 2.12/1.17 ap: $i > $i > $i > $i > $i ).
% 2.12/1.17
% 2.12/1.17 thf(app_type,type,
% 2.12/1.17 app: $o ).
% 2.12/1.17
% 2.12/1.17 thf(app,definition,
% 2.12/1.17 ( app
% 2.12/1.17 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.12/1.17 ( ( func @ A @ B @ Xf )
% 2.12/1.17 => ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(infuncsetfunc_type,type,
% 2.12/1.17 infuncsetfunc: $o ).
% 2.12/1.17
% 2.12/1.17 thf(infuncsetfunc,definition,
% 2.12/1.17 ( infuncsetfunc
% 2.12/1.17 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.12/1.17 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.12/1.17 => ( func @ A @ B @ Xf ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(ap2p_type,type,
% 2.12/1.17 ap2p: $o ).
% 2.12/1.17
% 2.12/1.17 thf(ap2p,definition,
% 2.12/1.17 ( ap2p
% 2.12/1.17 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.12/1.17 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.12/1.17 => ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(funcinfuncset_type,type,
% 2.12/1.17 funcinfuncset: $o ).
% 2.12/1.17
% 2.12/1.17 thf(funcinfuncset,definition,
% 2.12/1.17 ( funcinfuncset
% 2.12/1.17 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.12/1.17 ( ( func @ A @ B @ Xf )
% 2.12/1.17 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(lamProp_type,type,
% 2.12/1.17 lamProp: $o ).
% 2.12/1.17
% 2.12/1.17 thf(lamProp,definition,
% 2.12/1.17 ( lamProp
% 2.12/1.17 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.12/1.17 ( ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.12/1.17 => ( func @ A @ B
% 2.12/1.17 @ ( dpsetconstr @ A @ B
% 2.12/1.17 @ ^ [Xx: $i,Xy: $i] :
% 2.12/1.17 ( ( Xf @ Xx )
% 2.12/1.17 = Xy ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(lam_type,type,
% 2.12/1.17 lam: $i > $i > ( $i > $i ) > $i ).
% 2.12/1.17
% 2.12/1.17 thf(lamp_type,type,
% 2.12/1.17 lamp: $o ).
% 2.12/1.17
% 2.12/1.17 thf(lamp,definition,
% 2.12/1.17 ( lamp
% 2.12/1.17 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.12/1.17 ( ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.12/1.17 => ( func @ A @ B
% 2.12/1.17 @ ( lam @ A @ B
% 2.12/1.17 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(lam2p_type,type,
% 2.12/1.17 lam2p: $o ).
% 2.12/1.17
% 2.12/1.17 thf(lam2p,definition,
% 2.12/1.17 ( lam2p
% 2.12/1.17 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.12/1.17 ( ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.12/1.17 => ( in
% 2.12/1.17 @ ( lam @ A @ B
% 2.12/1.17 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 2.12/1.17 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(brelnall1_type,type,
% 2.12/1.17 brelnall1: $o ).
% 2.12/1.17
% 2.12/1.17 thf(brelnall1,definition,
% 2.12/1.17 ( brelnall1
% 2.12/1.17 = ( ! [A: $i,B: $i,R: $i] :
% 2.12/1.17 ( ( breln @ A @ B @ R )
% 2.12/1.17 => ! [Xphi: $i > $o] :
% 2.12/1.17 ( ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ! [Xy: $i] :
% 2.12/1.17 ( ( in @ Xy @ B )
% 2.12/1.17 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.12/1.17 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 2.12/1.17 => ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ R )
% 2.12/1.17 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(brelnall2_type,type,
% 2.12/1.17 brelnall2: $o ).
% 2.12/1.17
% 2.12/1.17 thf(brelnall2,definition,
% 2.12/1.17 ( brelnall2
% 2.12/1.17 = ( ! [A: $i,B: $i,R: $i] :
% 2.12/1.17 ( ( breln @ A @ B @ R )
% 2.12/1.17 => ! [Xphi: $i > $o] :
% 2.12/1.17 ( ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ! [Xy: $i] :
% 2.12/1.17 ( ( in @ Xy @ B )
% 2.12/1.17 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.12/1.17 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 2.12/1.17 => ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ R )
% 2.12/1.17 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(ex1E2_type,type,
% 2.12/1.17 ex1E2: $o ).
% 2.12/1.17
% 2.12/1.17 thf(ex1E2,definition,
% 2.12/1.17 ( ex1E2
% 2.12/1.17 = ( ! [A: $i,Xphi: $i > $o] :
% 2.12/1.17 ( ( ex1 @ A
% 2.12/1.17 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.12/1.17 => ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ! [Xy: $i] :
% 2.12/1.17 ( ( in @ Xy @ A )
% 2.12/1.17 => ( ( Xphi @ Xx )
% 2.12/1.17 => ( ( Xphi @ Xy )
% 2.12/1.17 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(funcGraphProp1_type,type,
% 2.12/1.17 funcGraphProp1: $o ).
% 2.12/1.17
% 2.12/1.17 thf(funcGraphProp1,definition,
% 2.12/1.17 ( funcGraphProp1
% 2.12/1.17 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.12/1.17 ( ( func @ A @ B @ Xf )
% 2.12/1.17 => ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(funcGraphProp3_type,type,
% 2.12/1.17 funcGraphProp3: $o ).
% 2.12/1.17
% 2.12/1.17 thf(funcGraphProp3,definition,
% 2.12/1.17 ( funcGraphProp3
% 2.12/1.17 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.12/1.17 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.12/1.17 => ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(funcGraphProp2_type,type,
% 2.12/1.17 funcGraphProp2: $o ).
% 2.12/1.17
% 2.12/1.17 thf(funcGraphProp2,definition,
% 2.12/1.17 ( funcGraphProp2
% 2.12/1.17 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.12/1.17 ( ( func @ A @ B @ Xf )
% 2.12/1.17 => ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ! [Xy: $i] :
% 2.12/1.17 ( ( in @ Xy @ B )
% 2.12/1.17 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 2.12/1.17 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.12/1.17 = Xy ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(funcextLem_type,type,
% 2.12/1.17 funcextLem: $o ).
% 2.12/1.17
% 2.12/1.17 thf(funcextLem,definition,
% 2.12/1.17 ( funcextLem
% 2.12/1.17 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.12/1.17 ( ( func @ A @ B @ Xf )
% 2.12/1.17 => ! [Xg: $i] :
% 2.12/1.17 ( ( func @ A @ B @ Xg )
% 2.12/1.17 => ( ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.12/1.17 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.12/1.17 => ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ! [Xy: $i] :
% 2.12/1.17 ( ( in @ Xy @ B )
% 2.12/1.17 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 2.12/1.17 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(funcGraphProp4_type,type,
% 2.12/1.17 funcGraphProp4: $o ).
% 2.12/1.17
% 2.12/1.17 thf(funcGraphProp4,definition,
% 2.12/1.17 ( funcGraphProp4
% 2.12/1.17 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.12/1.17 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.12/1.17 => ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ! [Xy: $i] :
% 2.12/1.17 ( ( in @ Xy @ B )
% 2.12/1.17 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 2.12/1.17 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.12/1.17 = Xy ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(subbreln_type,type,
% 2.12/1.17 subbreln: $o ).
% 2.12/1.17
% 2.12/1.17 thf(subbreln,definition,
% 2.12/1.17 ( subbreln
% 2.12/1.17 = ( ! [A: $i,B: $i,R: $i] :
% 2.12/1.17 ( ( breln @ A @ B @ R )
% 2.12/1.17 => ! [S: $i] :
% 2.12/1.17 ( ( breln @ A @ B @ S )
% 2.12/1.17 => ( ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ! [Xy: $i] :
% 2.12/1.17 ( ( in @ Xy @ B )
% 2.12/1.17 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.12/1.17 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 2.12/1.17 => ( subset @ R @ S ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(eqbreln_type,type,
% 2.12/1.17 eqbreln: $o ).
% 2.12/1.17
% 2.12/1.17 thf(eqbreln,definition,
% 2.12/1.17 ( eqbreln
% 2.12/1.17 = ( ! [A: $i,B: $i,R: $i] :
% 2.12/1.17 ( ( breln @ A @ B @ R )
% 2.12/1.17 => ! [S: $i] :
% 2.12/1.17 ( ( breln @ A @ B @ S )
% 2.12/1.17 => ( ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ! [Xy: $i] :
% 2.12/1.17 ( ( in @ Xy @ B )
% 2.12/1.17 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.12/1.17 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 2.12/1.17 => ( ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ! [Xy: $i] :
% 2.12/1.17 ( ( in @ Xy @ B )
% 2.12/1.17 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 2.12/1.17 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 2.12/1.17 => ( R = S ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(funcext_type,type,
% 2.12/1.17 funcext: $o ).
% 2.12/1.17
% 2.12/1.17 thf(funcext,definition,
% 2.12/1.17 ( funcext
% 2.12/1.17 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.12/1.17 ( ( func @ A @ B @ Xf )
% 2.12/1.17 => ! [Xg: $i] :
% 2.12/1.17 ( ( func @ A @ B @ Xg )
% 2.12/1.17 => ( ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.12/1.17 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.12/1.17 => ( Xf = Xg ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(funcext2_type,type,
% 2.12/1.17 funcext2: $o ).
% 2.12/1.17
% 2.12/1.17 thf(funcext2,definition,
% 2.12/1.17 ( funcext2
% 2.12/1.17 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.12/1.17 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.12/1.17 => ! [Xg: $i] :
% 2.12/1.17 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 2.12/1.17 => ( ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.12/1.17 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.12/1.17 => ( Xf = Xg ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(ap2apEq1_type,type,
% 2.12/1.17 ap2apEq1: $o ).
% 2.12/1.17
% 2.12/1.17 thf(ap2apEq1,definition,
% 2.12/1.17 ( ap2apEq1
% 2.12/1.17 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.12/1.17 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.12/1.17 => ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.12/1.17 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(ap2apEq2_type,type,
% 2.12/1.17 ap2apEq2: $o ).
% 2.12/1.17
% 2.12/1.17 thf(ap2apEq2,definition,
% 2.12/1.17 ( ap2apEq2
% 2.12/1.17 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.12/1.17 ( ( func @ A @ B @ Xf )
% 2.12/1.17 => ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.12/1.17 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(beta1_type,type,
% 2.12/1.17 beta1: $o ).
% 2.12/1.17
% 2.12/1.17 thf(beta1,definition,
% 2.12/1.17 ( beta1
% 2.12/1.17 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.12/1.17 ( ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.12/1.17 => ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ( ( ap @ A @ B
% 2.12/1.17 @ ( lam @ A @ B
% 2.12/1.17 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 2.12/1.17 @ Xx )
% 2.12/1.17 = ( Xf @ Xx ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(eta1_type,type,
% 2.12/1.17 eta1: $o ).
% 2.12/1.17
% 2.12/1.17 thf(eta1,definition,
% 2.12/1.17 ( eta1
% 2.12/1.17 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.12/1.17 ( ( func @ A @ B @ Xf )
% 2.12/1.17 => ( ( lam @ A @ B
% 2.12/1.17 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 2.12/1.17 = Xf ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(lam2lamEq_type,type,
% 2.12/1.17 lam2lamEq: $o ).
% 2.12/1.17
% 2.12/1.17 thf(lam2lamEq,definition,
% 2.12/1.17 ( lam2lamEq
% 2.12/1.17 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.12/1.17 ( ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.12/1.17 => ( ( lam @ A @ B
% 2.12/1.17 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 2.12/1.17 = ( lam @ A @ B
% 2.12/1.17 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(beta2_type,type,
% 2.12/1.17 beta2: $o ).
% 2.12/1.17
% 2.12/1.17 thf(beta2,definition,
% 2.12/1.17 ( beta2
% 2.12/1.17 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.12/1.17 ( ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.12/1.17 => ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ( ( ap @ A @ B
% 2.12/1.17 @ ( lam @ A @ B
% 2.12/1.17 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 2.12/1.17 @ Xx )
% 2.12/1.17 = ( Xf @ Xx ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(eta2_type,type,
% 2.12/1.17 eta2: $o ).
% 2.12/1.17
% 2.12/1.17 thf(eta2,definition,
% 2.12/1.17 ( eta2
% 2.12/1.17 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.12/1.17 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.12/1.17 => ( ( lam @ A @ B
% 2.12/1.17 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 2.12/1.17 = Xf ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(iffalseProp1_type,type,
% 2.12/1.17 iffalseProp1: $o ).
% 2.12/1.17
% 2.12/1.17 thf(iffalseProp1,definition,
% 2.12/1.17 ( iffalseProp1
% 2.12/1.17 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ! [Xy: $i] :
% 2.12/1.17 ( ( in @ Xy @ A )
% 2.12/1.17 => ( ~ Xphi
% 2.12/1.17 => ( in @ Xy
% 2.12/1.17 @ ( dsetconstr @ A
% 2.12/1.17 @ ^ [Xz: $i] :
% 2.12/1.17 ( ( Xphi
% 2.12/1.17 & ( Xz = Xx ) )
% 2.12/1.17 | ( ~ Xphi
% 2.12/1.17 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(iffalseProp2_type,type,
% 2.12/1.17 iffalseProp2: $o ).
% 2.12/1.17
% 2.12/1.17 thf(iffalseProp2,definition,
% 2.12/1.17 ( iffalseProp2
% 2.12/1.17 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ! [Xy: $i] :
% 2.12/1.17 ( ( in @ Xy @ A )
% 2.12/1.17 => ( ~ Xphi
% 2.12/1.17 => ( ( dsetconstr @ A
% 2.12/1.17 @ ^ [Xz: $i] :
% 2.12/1.17 ( ( Xphi
% 2.12/1.17 & ( Xz = Xx ) )
% 2.12/1.17 | ( ~ Xphi
% 2.12/1.17 & ( Xz = Xy ) ) ) )
% 2.12/1.17 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(iftrueProp1_type,type,
% 2.12/1.17 iftrueProp1: $o ).
% 2.12/1.17
% 2.12/1.17 thf(iftrueProp1,definition,
% 2.12/1.17 ( iftrueProp1
% 2.12/1.17 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ! [Xy: $i] :
% 2.12/1.17 ( ( in @ Xy @ A )
% 2.12/1.17 => ( Xphi
% 2.12/1.17 => ( in @ Xx
% 2.12/1.17 @ ( dsetconstr @ A
% 2.12/1.17 @ ^ [Xz: $i] :
% 2.12/1.17 ( ( Xphi
% 2.12/1.17 & ( Xz = Xx ) )
% 2.12/1.17 | ( ~ Xphi
% 2.12/1.17 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(iftrueProp2_type,type,
% 2.12/1.17 iftrueProp2: $o ).
% 2.12/1.17
% 2.12/1.17 thf(iftrueProp2,definition,
% 2.12/1.17 ( iftrueProp2
% 2.12/1.17 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ! [Xy: $i] :
% 2.12/1.17 ( ( in @ Xy @ A )
% 2.12/1.17 => ( Xphi
% 2.12/1.17 => ( ( dsetconstr @ A
% 2.12/1.17 @ ^ [Xz: $i] :
% 2.12/1.17 ( ( Xphi
% 2.12/1.17 & ( Xz = Xx ) )
% 2.12/1.17 | ( ~ Xphi
% 2.12/1.17 & ( Xz = Xy ) ) ) )
% 2.12/1.17 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(ifSingleton_type,type,
% 2.12/1.17 ifSingleton: $o ).
% 2.12/1.17
% 2.12/1.17 thf(ifSingleton,definition,
% 2.12/1.17 ( ifSingleton
% 2.12/1.17 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ! [Xy: $i] :
% 2.12/1.17 ( ( in @ Xy @ A )
% 2.12/1.17 => ( singleton
% 2.12/1.17 @ ( dsetconstr @ A
% 2.12/1.17 @ ^ [Xz: $i] :
% 2.12/1.17 ( ( Xphi
% 2.12/1.17 & ( Xz = Xx ) )
% 2.12/1.17 | ( ~ Xphi
% 2.12/1.17 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(if_type,type,
% 2.12/1.17 if: $i > $o > $i > $i > $i ).
% 2.12/1.17
% 2.12/1.17 thf(ifp_type,type,
% 2.12/1.17 ifp: $o ).
% 2.12/1.17
% 2.12/1.17 thf(ifp,definition,
% 2.12/1.17 ( ifp
% 2.12/1.17 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ! [Xy: $i] :
% 2.12/1.17 ( ( in @ Xy @ A )
% 2.12/1.17 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(theeq_type,type,
% 2.12/1.17 theeq: $o ).
% 2.12/1.17
% 2.12/1.17 thf(theeq,definition,
% 2.12/1.17 ( theeq
% 2.12/1.17 = ( ! [X: $i] :
% 2.12/1.17 ( ( singleton @ X )
% 2.12/1.17 => ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ X )
% 2.12/1.17 => ( ( setunion @ X )
% 2.12/1.17 = Xx ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(iftrue_type,type,
% 2.12/1.17 iftrue: $o ).
% 2.12/1.17
% 2.12/1.17 thf(iftrue,definition,
% 2.12/1.17 ( iftrue
% 2.12/1.17 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ! [Xy: $i] :
% 2.12/1.17 ( ( in @ Xy @ A )
% 2.12/1.17 => ( Xphi
% 2.12/1.17 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 2.12/1.17 = Xx ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(iffalse_type,type,
% 2.12/1.17 iffalse: $o ).
% 2.12/1.17
% 2.12/1.17 thf(iffalse,definition,
% 2.12/1.17 ( iffalse
% 2.12/1.17 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ! [Xy: $i] :
% 2.12/1.17 ( ( in @ Xy @ A )
% 2.12/1.17 => ( ~ Xphi
% 2.12/1.17 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 2.12/1.17 = Xy ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(iftrueorfalse_type,type,
% 2.12/1.17 iftrueorfalse: $o ).
% 2.12/1.17
% 2.12/1.17 thf(iftrueorfalse,definition,
% 2.12/1.17 ( iftrueorfalse
% 2.12/1.17 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ! [Xy: $i] :
% 2.12/1.17 ( ( in @ Xy @ A )
% 2.12/1.17 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(binintersectT_lem_type,type,
% 2.12/1.17 binintersectT_lem: $o ).
% 2.12/1.17
% 2.12/1.17 thf(binintersectT_lem,definition,
% 2.12/1.17 ( binintersectT_lem
% 2.12/1.17 = ( ! [A: $i,X: $i] :
% 2.12/1.17 ( ( in @ X @ ( powerset @ A ) )
% 2.12/1.17 => ! [Y: $i] :
% 2.12/1.17 ( ( in @ Y @ ( powerset @ A ) )
% 2.12/1.17 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(binunionT_lem_type,type,
% 2.12/1.17 binunionT_lem: $o ).
% 2.12/1.17
% 2.12/1.17 thf(binunionT_lem,definition,
% 2.12/1.17 ( binunionT_lem
% 2.12/1.17 = ( ! [A: $i,X: $i] :
% 2.12/1.17 ( ( in @ X @ ( powerset @ A ) )
% 2.12/1.17 => ! [Y: $i] :
% 2.12/1.17 ( ( in @ Y @ ( powerset @ A ) )
% 2.12/1.17 => ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(powersetT_lem_type,type,
% 2.12/1.17 powersetT_lem: $o ).
% 2.12/1.17
% 2.12/1.17 thf(powersetT_lem,definition,
% 2.12/1.17 ( powersetT_lem
% 2.12/1.17 = ( ! [A: $i,X: $i] :
% 2.12/1.17 ( ( in @ X @ ( powerset @ A ) )
% 2.12/1.17 => ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(setminusT_lem_type,type,
% 2.12/1.17 setminusT_lem: $o ).
% 2.12/1.17
% 2.12/1.17 thf(setminusT_lem,definition,
% 2.12/1.17 ( setminusT_lem
% 2.12/1.17 = ( ! [A: $i,X: $i] :
% 2.12/1.17 ( ( in @ X @ ( powerset @ A ) )
% 2.12/1.17 => ! [Y: $i] :
% 2.12/1.17 ( ( in @ Y @ ( powerset @ A ) )
% 2.12/1.17 => ( in @ ( setminus @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(complementT_lem_type,type,
% 2.12/1.17 complementT_lem: $o ).
% 2.12/1.17
% 2.12/1.17 thf(complementT_lem,definition,
% 2.12/1.17 ( complementT_lem
% 2.12/1.17 = ( ! [A: $i,X: $i] :
% 2.12/1.17 ( ( in @ X @ ( powerset @ A ) )
% 2.12/1.17 => ( in @ ( setminus @ A @ X ) @ ( powerset @ A ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(setextT_type,type,
% 2.12/1.17 setextT: $o ).
% 2.12/1.17
% 2.12/1.17 thf(setextT,definition,
% 2.12/1.17 ( setextT
% 2.12/1.17 = ( ! [A: $i,X: $i] :
% 2.12/1.17 ( ( in @ X @ ( powerset @ A ) )
% 2.12/1.17 => ! [Y: $i] :
% 2.12/1.17 ( ( in @ Y @ ( powerset @ A ) )
% 2.12/1.17 => ( ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ( ( in @ Xx @ X )
% 2.12/1.17 => ( in @ Xx @ Y ) ) )
% 2.12/1.17 => ( ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ( ( in @ Xx @ Y )
% 2.12/1.17 => ( in @ Xx @ X ) ) )
% 2.12/1.17 => ( X = Y ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(subsetTI_type,type,
% 2.12/1.17 subsetTI: $o ).
% 2.12/1.17
% 2.12/1.17 thf(subsetTI,definition,
% 2.12/1.17 ( subsetTI
% 2.12/1.17 = ( ! [A: $i,X: $i] :
% 2.12/1.17 ( ( in @ X @ ( powerset @ A ) )
% 2.12/1.17 => ! [Y: $i] :
% 2.12/1.17 ( ( in @ Y @ ( powerset @ A ) )
% 2.12/1.17 => ( ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ( ( in @ Xx @ X )
% 2.12/1.17 => ( in @ Xx @ Y ) ) )
% 2.12/1.17 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(powersetTI1_type,type,
% 2.12/1.17 powersetTI1: $o ).
% 2.12/1.17
% 2.12/1.17 thf(powersetTI1,definition,
% 2.12/1.17 ( powersetTI1
% 2.12/1.17 = ( ! [A: $i,X: $i] :
% 2.12/1.17 ( ( in @ X @ ( powerset @ A ) )
% 2.12/1.17 => ! [Y: $i] :
% 2.12/1.17 ( ( in @ Y @ ( powerset @ A ) )
% 2.12/1.17 => ( ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ( ( in @ Xx @ X )
% 2.12/1.17 => ( in @ Xx @ Y ) ) )
% 2.12/1.17 => ( in @ X @ ( powerset @ Y ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(powersetTE1_type,type,
% 2.12/1.17 powersetTE1: $o ).
% 2.12/1.17
% 2.12/1.17 thf(powersetTE1,definition,
% 2.12/1.17 ( powersetTE1
% 2.12/1.17 = ( ! [A: $i,X: $i] :
% 2.12/1.17 ( ( in @ X @ ( powerset @ A ) )
% 2.12/1.17 => ! [Y: $i] :
% 2.12/1.17 ( ( in @ Y @ ( powerset @ A ) )
% 2.12/1.17 => ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ( ( in @ X @ ( powerset @ Y ) )
% 2.12/1.17 => ( ( in @ Xx @ X )
% 2.12/1.17 => ( in @ Xx @ Y ) ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(complementTI1_type,type,
% 2.12/1.17 complementTI1: $o ).
% 2.12/1.17
% 2.12/1.17 thf(complementTI1,definition,
% 2.12/1.17 ( complementTI1
% 2.12/1.17 = ( ! [A: $i,X: $i] :
% 2.12/1.17 ( ( in @ X @ ( powerset @ A ) )
% 2.12/1.17 => ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ( ( in @ Xx @ X )
% 2.12/1.17 => ~ ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(complementTE1_type,type,
% 2.12/1.17 complementTE1: $o ).
% 2.12/1.17
% 2.12/1.17 thf(complementTE1,definition,
% 2.12/1.17 ( complementTE1
% 2.12/1.17 = ( ! [A: $i,X: $i] :
% 2.12/1.17 ( ( in @ X @ ( powerset @ A ) )
% 2.12/1.17 => ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ( ~ ( in @ Xx @ ( setminus @ A @ X ) )
% 2.12/1.17 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(binintersectTELcontra_type,type,
% 2.12/1.17 binintersectTELcontra: $o ).
% 2.12/1.17
% 2.12/1.17 thf(binintersectTELcontra,definition,
% 2.12/1.17 ( binintersectTELcontra
% 2.12/1.17 = ( ! [A: $i,X: $i] :
% 2.12/1.17 ( ( in @ X @ ( powerset @ A ) )
% 2.12/1.17 => ! [Y: $i] :
% 2.12/1.17 ( ( in @ Y @ ( powerset @ A ) )
% 2.12/1.17 => ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ( ~ ( in @ Xx @ X )
% 2.12/1.17 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(binintersectTERcontra_type,type,
% 2.12/1.17 binintersectTERcontra: $o ).
% 2.12/1.17
% 2.12/1.17 thf(binintersectTERcontra,definition,
% 2.12/1.17 ( binintersectTERcontra
% 2.12/1.17 = ( ! [A: $i,X: $i] :
% 2.12/1.17 ( ( in @ X @ ( powerset @ A ) )
% 2.12/1.17 => ! [Y: $i] :
% 2.12/1.17 ( ( in @ Y @ ( powerset @ A ) )
% 2.12/1.17 => ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ( ~ ( in @ Xx @ Y )
% 2.12/1.17 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(contrasubsetT_type,type,
% 2.12/1.17 contrasubsetT: $o ).
% 2.12/1.17
% 2.12/1.17 thf(contrasubsetT,definition,
% 2.12/1.17 ( contrasubsetT
% 2.12/1.17 = ( ! [A: $i,X: $i] :
% 2.12/1.17 ( ( in @ X @ ( powerset @ A ) )
% 2.12/1.17 => ! [Y: $i] :
% 2.12/1.17 ( ( in @ Y @ ( powerset @ A ) )
% 2.12/1.17 => ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 2.12/1.17 => ( ( in @ Xx @ Y )
% 2.12/1.17 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(contrasubsetT1_type,type,
% 2.12/1.17 contrasubsetT1: $o ).
% 2.12/1.17
% 2.12/1.17 thf(contrasubsetT1,definition,
% 2.12/1.17 ( contrasubsetT1
% 2.12/1.17 = ( ! [A: $i,X: $i] :
% 2.12/1.17 ( ( in @ X @ ( powerset @ A ) )
% 2.12/1.17 => ! [Y: $i] :
% 2.12/1.17 ( ( in @ Y @ ( powerset @ A ) )
% 2.12/1.17 => ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ( ( subset @ X @ Y )
% 2.12/1.17 => ( ~ ( in @ Xx @ Y )
% 2.12/1.17 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(contrasubsetT2_type,type,
% 2.12/1.17 contrasubsetT2: $o ).
% 2.12/1.17
% 2.12/1.17 thf(contrasubsetT2,definition,
% 2.12/1.17 ( contrasubsetT2
% 2.12/1.17 = ( ! [A: $i,X: $i] :
% 2.12/1.17 ( ( in @ X @ ( powerset @ A ) )
% 2.12/1.17 => ! [Y: $i] :
% 2.12/1.17 ( ( in @ Y @ ( powerset @ A ) )
% 2.12/1.17 => ( ( subset @ X @ Y )
% 2.12/1.17 => ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(contrasubsetT3_type,type,
% 2.12/1.17 contrasubsetT3: $o ).
% 2.12/1.17
% 2.12/1.17 thf(contrasubsetT3,definition,
% 2.12/1.17 ( contrasubsetT3
% 2.12/1.17 = ( ! [A: $i,X: $i] :
% 2.12/1.17 ( ( in @ X @ ( powerset @ A ) )
% 2.12/1.17 => ! [Y: $i] :
% 2.12/1.17 ( ( in @ Y @ ( powerset @ A ) )
% 2.12/1.17 => ( ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) )
% 2.12/1.17 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(doubleComplementI1_type,type,
% 2.12/1.17 doubleComplementI1: $o ).
% 2.12/1.17
% 2.12/1.17 thf(doubleComplementI1,definition,
% 2.12/1.17 ( doubleComplementI1
% 2.12/1.17 = ( ! [A: $i,X: $i] :
% 2.12/1.17 ( ( in @ X @ ( powerset @ A ) )
% 2.12/1.17 => ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ( ( in @ Xx @ X )
% 2.12/1.17 => ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(doubleComplementE1_type,type,
% 2.12/1.17 doubleComplementE1: $o ).
% 2.12/1.17
% 2.12/1.17 thf(doubleComplementE1,definition,
% 2.12/1.17 ( doubleComplementE1
% 2.12/1.17 = ( ! [A: $i,X: $i] :
% 2.12/1.17 ( ( in @ X @ ( powerset @ A ) )
% 2.12/1.17 => ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ( ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) )
% 2.12/1.17 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(doubleComplementSub1_type,type,
% 2.12/1.17 doubleComplementSub1: $o ).
% 2.12/1.17
% 2.12/1.17 thf(doubleComplementSub1,definition,
% 2.12/1.17 ( doubleComplementSub1
% 2.12/1.17 = ( ! [A: $i,X: $i] :
% 2.12/1.17 ( ( in @ X @ ( powerset @ A ) )
% 2.12/1.17 => ( subset @ X @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(doubleComplementSub2_type,type,
% 2.12/1.17 doubleComplementSub2: $o ).
% 2.12/1.17
% 2.12/1.17 thf(doubleComplementSub2,definition,
% 2.12/1.17 ( doubleComplementSub2
% 2.12/1.17 = ( ! [A: $i,X: $i] :
% 2.12/1.17 ( ( in @ X @ ( powerset @ A ) )
% 2.12/1.17 => ( subset @ ( setminus @ A @ ( setminus @ A @ X ) ) @ X ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(doubleComplementEq_type,type,
% 2.12/1.17 doubleComplementEq: $o ).
% 2.12/1.17
% 2.12/1.17 thf(doubleComplementEq,definition,
% 2.12/1.17 ( doubleComplementEq
% 2.12/1.17 = ( ! [A: $i,X: $i] :
% 2.12/1.17 ( ( in @ X @ ( powerset @ A ) )
% 2.12/1.17 => ( X
% 2.12/1.17 = ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(complementTnotintersectT_type,type,
% 2.12/1.17 complementTnotintersectT: $o ).
% 2.12/1.17
% 2.12/1.17 thf(complementTnotintersectT,definition,
% 2.12/1.17 ( complementTnotintersectT
% 2.12/1.17 = ( ! [A: $i,X: $i] :
% 2.12/1.17 ( ( in @ X @ ( powerset @ A ) )
% 2.12/1.17 => ! [Y: $i] :
% 2.12/1.17 ( ( in @ Y @ ( powerset @ A ) )
% 2.12/1.17 => ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 2.12/1.17 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(complementImpComplementIntersect_type,type,
% 2.12/1.17 complementImpComplementIntersect: $o ).
% 2.12/1.17
% 2.12/1.17 thf(complementImpComplementIntersect,definition,
% 2.12/1.17 ( complementImpComplementIntersect
% 2.12/1.17 = ( ! [A: $i,X: $i] :
% 2.12/1.17 ( ( in @ X @ ( powerset @ A ) )
% 2.12/1.17 => ! [Y: $i] :
% 2.12/1.17 ( ( in @ Y @ ( powerset @ A ) )
% 2.12/1.17 => ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 2.12/1.17 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(complementSubsetComplementIntersect_type,type,
% 2.12/1.17 complementSubsetComplementIntersect: $o ).
% 2.12/1.17
% 2.12/1.17 thf(complementSubsetComplementIntersect,definition,
% 2.12/1.17 ( complementSubsetComplementIntersect
% 2.12/1.17 = ( ! [A: $i,X: $i] :
% 2.12/1.17 ( ( in @ X @ ( powerset @ A ) )
% 2.12/1.17 => ! [Y: $i] :
% 2.12/1.17 ( ( in @ Y @ ( powerset @ A ) )
% 2.12/1.17 => ( subset @ ( setminus @ A @ X ) @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(complementInPowersetComplementIntersect_type,type,
% 2.12/1.17 complementInPowersetComplementIntersect: $o ).
% 2.12/1.17
% 2.12/1.17 thf(complementInPowersetComplementIntersect,definition,
% 2.12/1.17 ( complementInPowersetComplementIntersect
% 2.12/1.17 = ( ! [A: $i,X: $i] :
% 2.12/1.17 ( ( in @ X @ ( powerset @ A ) )
% 2.12/1.17 => ! [Y: $i] :
% 2.12/1.17 ( ( in @ Y @ ( powerset @ A ) )
% 2.12/1.17 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(contraSubsetComplement_type,type,
% 2.12/1.17 contraSubsetComplement: $o ).
% 2.12/1.17
% 2.12/1.17 thf(contraSubsetComplement,definition,
% 2.12/1.17 ( contraSubsetComplement
% 2.12/1.17 = ( ! [A: $i,X: $i] :
% 2.12/1.17 ( ( in @ X @ ( powerset @ A ) )
% 2.12/1.17 => ! [Y: $i] :
% 2.12/1.17 ( ( in @ Y @ ( powerset @ A ) )
% 2.12/1.17 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 2.12/1.17 => ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ( ( in @ Xx @ Y )
% 2.12/1.17 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(complementTcontraSubset_type,type,
% 2.12/1.17 complementTcontraSubset: $o ).
% 2.12/1.17
% 2.12/1.17 thf(complementTcontraSubset,definition,
% 2.12/1.17 ( complementTcontraSubset
% 2.12/1.17 = ( ! [A: $i,X: $i] :
% 2.12/1.17 ( ( in @ X @ ( powerset @ A ) )
% 2.12/1.17 => ! [Y: $i] :
% 2.12/1.17 ( ( in @ Y @ ( powerset @ A ) )
% 2.12/1.17 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 2.12/1.17 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(binunionTILcontra_type,type,
% 2.12/1.17 binunionTILcontra: $o ).
% 2.12/1.17
% 2.12/1.17 thf(binunionTILcontra,definition,
% 2.12/1.17 ( binunionTILcontra
% 2.12/1.17 = ( ! [A: $i,X: $i] :
% 2.12/1.17 ( ( in @ X @ ( powerset @ A ) )
% 2.12/1.17 => ! [Y: $i] :
% 2.12/1.17 ( ( in @ Y @ ( powerset @ A ) )
% 2.12/1.17 => ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 2.12/1.17 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(binunionTIRcontra_type,type,
% 2.12/1.17 binunionTIRcontra: $o ).
% 2.12/1.17
% 2.12/1.17 thf(binunionTIRcontra,definition,
% 2.12/1.17 ( binunionTIRcontra
% 2.12/1.17 = ( ! [A: $i,X: $i] :
% 2.12/1.17 ( ( in @ X @ ( powerset @ A ) )
% 2.12/1.17 => ! [Y: $i] :
% 2.12/1.17 ( ( in @ Y @ ( powerset @ A ) )
% 2.12/1.17 => ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 2.12/1.17 => ~ ( in @ Xx @ Y ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(inIntersectImpInUnion_type,type,
% 2.12/1.17 inIntersectImpInUnion: $o ).
% 2.12/1.17
% 2.12/1.17 thf(inIntersectImpInUnion,definition,
% 2.12/1.17 ( inIntersectImpInUnion
% 2.12/1.17 = ( ! [A: $i,X: $i] :
% 2.12/1.17 ( ( in @ X @ ( powerset @ A ) )
% 2.12/1.17 => ! [Y: $i] :
% 2.12/1.17 ( ( in @ Y @ ( powerset @ A ) )
% 2.12/1.17 => ! [Z: $i] :
% 2.12/1.17 ( ( in @ Z @ ( powerset @ A ) )
% 2.12/1.17 => ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 2.12/1.17 => ( in @ Xx @ ( binunion @ X @ Z ) ) ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(inIntersectImpInUnion2_type,type,
% 2.12/1.17 inIntersectImpInUnion2: $o ).
% 2.12/1.17
% 2.12/1.17 thf(inIntersectImpInUnion2,definition,
% 2.12/1.17 ( inIntersectImpInUnion2
% 2.12/1.17 = ( ! [A: $i,X: $i] :
% 2.12/1.17 ( ( in @ X @ ( powerset @ A ) )
% 2.12/1.17 => ! [Y: $i] :
% 2.12/1.17 ( ( in @ Y @ ( powerset @ A ) )
% 2.12/1.17 => ! [Z: $i] :
% 2.12/1.17 ( ( in @ Z @ ( powerset @ A ) )
% 2.12/1.17 => ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 2.12/1.17 => ( in @ Xx @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(inIntersectImpInIntersectUnions_type,type,
% 2.12/1.17 inIntersectImpInIntersectUnions: $o ).
% 2.12/1.17
% 2.12/1.17 thf(inIntersectImpInIntersectUnions,definition,
% 2.12/1.17 ( inIntersectImpInIntersectUnions
% 2.12/1.17 = ( ! [A: $i,X: $i] :
% 2.12/1.17 ( ( in @ X @ ( powerset @ A ) )
% 2.12/1.17 => ! [Y: $i] :
% 2.12/1.17 ( ( in @ Y @ ( powerset @ A ) )
% 2.12/1.17 => ! [Z: $i] :
% 2.12/1.17 ( ( in @ Z @ ( powerset @ A ) )
% 2.12/1.17 => ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 2.12/1.17 => ( in @ Xx @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(intersectInPowersetIntersectUnions_type,type,
% 2.12/1.17 intersectInPowersetIntersectUnions: $o ).
% 2.12/1.17
% 2.12/1.17 thf(intersectInPowersetIntersectUnions,definition,
% 2.12/1.17 ( intersectInPowersetIntersectUnions
% 2.12/1.17 = ( ! [A: $i,X: $i] :
% 2.12/1.17 ( ( in @ X @ ( powerset @ A ) )
% 2.12/1.17 => ! [Y: $i] :
% 2.12/1.17 ( ( in @ Y @ ( powerset @ A ) )
% 2.12/1.17 => ! [Z: $i] :
% 2.12/1.17 ( ( in @ Z @ ( powerset @ A ) )
% 2.12/1.17 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(inComplementUnionImpNotIn1_type,type,
% 2.12/1.17 inComplementUnionImpNotIn1: $o ).
% 2.12/1.17
% 2.12/1.17 thf(inComplementUnionImpNotIn1,definition,
% 2.12/1.17 ( inComplementUnionImpNotIn1
% 2.12/1.17 = ( ! [A: $i,X: $i] :
% 2.12/1.17 ( ( in @ X @ ( powerset @ A ) )
% 2.12/1.17 => ! [Y: $i] :
% 2.12/1.17 ( ( in @ Y @ ( powerset @ A ) )
% 2.12/1.17 => ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 2.12/1.17 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(inComplementUnionImpInComplement1_type,type,
% 2.12/1.17 inComplementUnionImpInComplement1: $o ).
% 2.12/1.17
% 2.12/1.17 thf(inComplementUnionImpInComplement1,definition,
% 2.12/1.17 ( inComplementUnionImpInComplement1
% 2.12/1.17 = ( ! [A: $i,X: $i] :
% 2.12/1.17 ( ( in @ X @ ( powerset @ A ) )
% 2.12/1.17 => ! [Y: $i] :
% 2.12/1.17 ( ( in @ Y @ ( powerset @ A ) )
% 2.12/1.17 => ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 2.12/1.17 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(binunionTE_type,type,
% 2.12/1.17 binunionTE: $o ).
% 2.12/1.17
% 2.12/1.17 thf(binunionTE,definition,
% 2.12/1.17 ( binunionTE
% 2.12/1.17 = ( ! [A: $i,X: $i] :
% 2.12/1.17 ( ( in @ X @ ( powerset @ A ) )
% 2.12/1.17 => ! [Y: $i] :
% 2.12/1.17 ( ( in @ Y @ ( powerset @ A ) )
% 2.12/1.17 => ! [Xphi: $o,Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ( ( in @ Xx @ ( binunion @ X @ Y ) )
% 2.12/1.17 => ( ( ( in @ Xx @ X )
% 2.12/1.17 => Xphi )
% 2.12/1.17 => ( ( ( in @ Xx @ Y )
% 2.12/1.17 => Xphi )
% 2.12/1.17 => Xphi ) ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(binunionTEcontra_type,type,
% 2.12/1.17 binunionTEcontra: $o ).
% 2.12/1.17
% 2.12/1.17 thf(binunionTEcontra,definition,
% 2.12/1.17 ( binunionTEcontra
% 2.12/1.17 = ( ! [A: $i,X: $i] :
% 2.12/1.17 ( ( in @ X @ ( powerset @ A ) )
% 2.12/1.17 => ! [Y: $i] :
% 2.12/1.17 ( ( in @ Y @ ( powerset @ A ) )
% 2.12/1.17 => ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ( ~ ( in @ Xx @ X )
% 2.12/1.17 => ( ~ ( in @ Xx @ Y )
% 2.12/1.17 => ~ ( in @ Xx @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(demorgan2a1_type,type,
% 2.12/1.17 demorgan2a1: $o ).
% 2.12/1.17
% 2.12/1.17 thf(demorgan2a1,definition,
% 2.12/1.17 ( demorgan2a1
% 2.12/1.17 = ( ! [A: $i,X: $i] :
% 2.12/1.17 ( ( in @ X @ ( powerset @ A ) )
% 2.12/1.17 => ! [Y: $i] :
% 2.12/1.17 ( ( in @ Y @ ( powerset @ A ) )
% 2.12/1.17 => ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 2.12/1.17 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(complementUnionInPowersetComplement_type,type,
% 2.12/1.17 complementUnionInPowersetComplement: $o ).
% 2.12/1.17
% 2.12/1.17 thf(complementUnionInPowersetComplement,definition,
% 2.12/1.17 ( complementUnionInPowersetComplement
% 2.12/1.17 = ( ! [A: $i,X: $i] :
% 2.12/1.17 ( ( in @ X @ ( powerset @ A ) )
% 2.12/1.17 => ! [Y: $i] :
% 2.12/1.17 ( ( in @ Y @ ( powerset @ A ) )
% 2.12/1.17 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(demorgan2a2_type,type,
% 2.12/1.17 demorgan2a2: $o ).
% 2.12/1.17
% 2.12/1.17 thf(demorgan2a2,definition,
% 2.12/1.17 ( demorgan2a2
% 2.12/1.17 = ( ! [A: $i,X: $i] :
% 2.12/1.17 ( ( in @ X @ ( powerset @ A ) )
% 2.12/1.17 => ! [Y: $i] :
% 2.12/1.17 ( ( in @ Y @ ( powerset @ A ) )
% 2.12/1.17 => ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 2.12/1.17 => ( in @ Xx @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(demorgan1a_type,type,
% 2.12/1.17 demorgan1a: $o ).
% 2.12/1.17
% 2.12/1.17 thf(demorgan1a,definition,
% 2.12/1.17 ( demorgan1a
% 2.12/1.17 = ( ! [A: $i,X: $i] :
% 2.12/1.17 ( ( in @ X @ ( powerset @ A ) )
% 2.12/1.17 => ! [Y: $i] :
% 2.12/1.17 ( ( in @ Y @ ( powerset @ A ) )
% 2.12/1.17 => ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ( ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) )
% 2.12/1.17 => ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(demorgan1b_type,type,
% 2.12/1.17 demorgan1b: $o ).
% 2.12/1.17
% 2.12/1.17 thf(demorgan1b,definition,
% 2.12/1.17 ( demorgan1b
% 2.12/1.17 = ( ! [A: $i,X: $i] :
% 2.12/1.17 ( ( in @ X @ ( powerset @ A ) )
% 2.12/1.17 => ! [Y: $i] :
% 2.12/1.17 ( ( in @ Y @ ( powerset @ A ) )
% 2.12/1.17 => ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ( ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 2.12/1.17 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(demorgan1_type,type,
% 2.12/1.17 demorgan1: $o ).
% 2.12/1.17
% 2.12/1.17 thf(demorgan1,definition,
% 2.12/1.17 ( demorgan1
% 2.12/1.17 = ( ! [A: $i,X: $i] :
% 2.12/1.17 ( ( in @ X @ ( powerset @ A ) )
% 2.12/1.17 => ! [Y: $i] :
% 2.12/1.17 ( ( in @ Y @ ( powerset @ A ) )
% 2.12/1.17 => ( ( setminus @ A @ ( binintersect @ X @ Y ) )
% 2.12/1.17 = ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(demorgan2a_type,type,
% 2.12/1.17 demorgan2a: $o ).
% 2.12/1.17
% 2.12/1.17 thf(demorgan2a,definition,
% 2.12/1.17 ( demorgan2a
% 2.12/1.17 = ( ! [A: $i,X: $i] :
% 2.12/1.17 ( ( in @ X @ ( powerset @ A ) )
% 2.12/1.17 => ! [Y: $i] :
% 2.12/1.17 ( ( in @ Y @ ( powerset @ A ) )
% 2.12/1.17 => ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 2.12/1.17 => ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(demorgan2b2_type,type,
% 2.12/1.17 demorgan2b2: $o ).
% 2.12/1.17
% 2.12/1.17 thf(demorgan2b2,definition,
% 2.12/1.17 ( demorgan2b2
% 2.12/1.17 = ( ! [A: $i,X: $i] :
% 2.12/1.17 ( ( in @ X @ ( powerset @ A ) )
% 2.12/1.17 => ! [Y: $i] :
% 2.12/1.17 ( ( in @ Y @ ( powerset @ A ) )
% 2.12/1.17 => ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 2.12/1.17 => ( ( in @ Xx @ ( setminus @ A @ Y ) )
% 2.12/1.17 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(demorgan2b_type,type,
% 2.12/1.17 demorgan2b: $o ).
% 2.12/1.17
% 2.12/1.17 thf(demorgan2b,definition,
% 2.12/1.17 ( demorgan2b
% 2.12/1.17 = ( ! [A: $i,X: $i] :
% 2.12/1.17 ( ( in @ X @ ( powerset @ A ) )
% 2.12/1.17 => ! [Y: $i] :
% 2.12/1.17 ( ( in @ Y @ ( powerset @ A ) )
% 2.12/1.17 => ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ( ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 2.12/1.17 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(demorgan2_type,type,
% 2.12/1.17 demorgan2: $o ).
% 2.12/1.17
% 2.12/1.17 thf(demorgan2,definition,
% 2.12/1.17 ( demorgan2
% 2.12/1.17 = ( ! [A: $i,X: $i] :
% 2.12/1.17 ( ( in @ X @ ( powerset @ A ) )
% 2.12/1.17 => ! [Y: $i] :
% 2.12/1.17 ( ( in @ Y @ ( powerset @ A ) )
% 2.12/1.17 => ( ( setminus @ A @ ( binunion @ X @ Y ) )
% 2.12/1.17 = ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(woz13rule0_type,type,
% 2.12/1.17 woz13rule0: $o ).
% 2.12/1.17
% 2.12/1.17 thf(woz13rule0,definition,
% 2.12/1.17 ( woz13rule0
% 2.12/1.17 = ( ! [A: $i,X: $i] :
% 2.12/1.17 ( ( in @ X @ ( powerset @ A ) )
% 2.12/1.17 => ! [Y: $i] :
% 2.12/1.17 ( ( in @ Y @ ( powerset @ A ) )
% 2.12/1.17 => ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 2.12/1.17 => ( in @ Xx @ A ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(woz13rule1_type,type,
% 2.12/1.17 woz13rule1: $o ).
% 2.12/1.17
% 2.12/1.17 thf(woz13rule1,definition,
% 2.12/1.17 ( woz13rule1
% 2.12/1.17 = ( ! [A: $i,X: $i] :
% 2.12/1.17 ( ( in @ X @ ( powerset @ A ) )
% 2.12/1.17 => ! [Y: $i] :
% 2.12/1.17 ( ( in @ Y @ ( powerset @ A ) )
% 2.12/1.17 => ! [Z: $i] :
% 2.12/1.17 ( ( in @ Z @ ( powerset @ A ) )
% 2.12/1.17 => ( ( subset @ X @ Z )
% 2.12/1.17 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(woz13rule2_type,type,
% 2.12/1.17 woz13rule2: $o ).
% 2.12/1.17
% 2.12/1.17 thf(woz13rule2,definition,
% 2.12/1.17 ( woz13rule2
% 2.12/1.17 = ( ! [A: $i,X: $i] :
% 2.12/1.17 ( ( in @ X @ ( powerset @ A ) )
% 2.12/1.17 => ! [Y: $i] :
% 2.12/1.17 ( ( in @ Y @ ( powerset @ A ) )
% 2.12/1.17 => ! [Z: $i] :
% 2.12/1.17 ( ( in @ Z @ ( powerset @ A ) )
% 2.12/1.17 => ( ( subset @ Y @ Z )
% 2.12/1.17 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(woz13rule3_type,type,
% 2.12/1.17 woz13rule3: $o ).
% 2.12/1.17
% 2.12/1.17 thf(woz13rule3,definition,
% 2.12/1.17 ( woz13rule3
% 2.12/1.17 = ( ! [A: $i,X: $i] :
% 2.12/1.17 ( ( in @ X @ ( powerset @ A ) )
% 2.12/1.17 => ! [Y: $i] :
% 2.12/1.17 ( ( in @ Y @ ( powerset @ A ) )
% 2.12/1.17 => ! [Z: $i] :
% 2.12/1.17 ( ( in @ Z @ ( powerset @ A ) )
% 2.12/1.17 => ( ( subset @ X @ Y )
% 2.12/1.17 => ( ( subset @ X @ Z )
% 2.12/1.17 => ( subset @ X @ ( binintersect @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(woz13rule4_type,type,
% 2.12/1.17 woz13rule4: $o ).
% 2.12/1.17
% 2.12/1.17 thf(woz13rule4,definition,
% 2.12/1.17 ( woz13rule4
% 2.12/1.17 = ( ! [A: $i,X: $i] :
% 2.12/1.17 ( ( in @ X @ ( powerset @ A ) )
% 2.12/1.17 => ! [Y: $i] :
% 2.12/1.17 ( ( in @ Y @ ( powerset @ A ) )
% 2.12/1.17 => ! [Z: $i] :
% 2.12/1.17 ( ( in @ Z @ ( powerset @ A ) )
% 2.12/1.17 => ! [W: $i] :
% 2.12/1.17 ( ( in @ W @ ( powerset @ A ) )
% 2.12/1.17 => ( ( subset @ X @ Z )
% 2.12/1.17 => ( ( subset @ Y @ W )
% 2.12/1.17 => ( subset @ ( binintersect @ X @ Y ) @ ( binintersect @ Z @ W ) ) ) ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(woz1_1_type,type,
% 2.12/1.17 woz1_1: $o ).
% 2.12/1.17
% 2.12/1.17 thf(woz1_1,definition,
% 2.12/1.17 ( woz1_1
% 2.12/1.17 = ( ! [A: $i,X: $i] :
% 2.12/1.17 ( ( in @ X @ ( powerset @ A ) )
% 2.12/1.17 => ! [Y: $i] :
% 2.12/1.17 ( ( in @ Y @ ( powerset @ A ) )
% 2.12/1.17 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(woz1_2_type,type,
% 2.12/1.17 woz1_2: $o ).
% 2.12/1.17
% 2.12/1.17 thf(woz1_2,definition,
% 2.12/1.17 ( woz1_2
% 2.12/1.17 = ( ! [A: $i,X: $i] :
% 2.12/1.17 ( ( in @ X @ ( powerset @ A ) )
% 2.12/1.17 => ! [Y: $i] :
% 2.12/1.17 ( ( in @ Y @ ( powerset @ A ) )
% 2.12/1.17 => ! [Z: $i] :
% 2.12/1.17 ( ( in @ Z @ ( powerset @ A ) )
% 2.12/1.17 => ! [W: $i] :
% 2.12/1.17 ( ( in @ W @ ( powerset @ A ) )
% 2.12/1.17 => ( ( setminus @ A @ ( binintersect @ ( binunion @ X @ Y ) @ ( binunion @ Z @ W ) ) )
% 2.12/1.17 = ( binunion @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) @ ( binintersect @ ( setminus @ A @ Z ) @ ( setminus @ A @ W ) ) ) ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(woz1_3_type,type,
% 2.12/1.17 woz1_3: $o ).
% 2.12/1.17
% 2.12/1.17 thf(woz1_3,definition,
% 2.12/1.17 ( woz1_3
% 2.12/1.17 = ( ! [A: $i,X: $i] :
% 2.12/1.17 ( ( in @ X @ ( powerset @ A ) )
% 2.12/1.17 => ! [Y: $i] :
% 2.12/1.17 ( ( in @ Y @ ( powerset @ A ) )
% 2.12/1.17 => ! [Z: $i] :
% 2.12/1.17 ( ( in @ Z @ ( powerset @ A ) )
% 2.12/1.17 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(woz1_4_type,type,
% 2.12/1.17 woz1_4: $o ).
% 2.12/1.17
% 2.12/1.17 thf(woz1_4,definition,
% 2.12/1.17 ( woz1_4
% 2.12/1.17 = ( ! [A: $i,X: $i] :
% 2.12/1.17 ( ( in @ X @ ( powerset @ A ) )
% 2.12/1.17 => ! [Y: $i] :
% 2.12/1.17 ( ( in @ Y @ ( powerset @ A ) )
% 2.12/1.17 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 2.12/1.17 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(woz1_5_type,type,
% 2.12/1.17 woz1_5: $o ).
% 2.12/1.17
% 2.12/1.17 thf(woz1_5,definition,
% 2.12/1.17 ( woz1_5
% 2.12/1.17 = ( ! [A: $i,X: $i] :
% 2.12/1.17 ( ( in @ X @ ( powerset @ A ) )
% 2.12/1.17 => ! [Y: $i] :
% 2.12/1.17 ( ( in @ Y @ ( powerset @ A ) )
% 2.12/1.17 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(breln1_type,type,
% 2.12/1.17 breln1: $i > $i > $o ).
% 2.12/1.17
% 2.12/1.17 thf(breln1all2_type,type,
% 2.12/1.17 breln1all2: $o ).
% 2.12/1.17
% 2.12/1.17 thf(breln1all2,definition,
% 2.12/1.17 ( breln1all2
% 2.12/1.17 = ( ! [A: $i,R: $i] :
% 2.12/1.17 ( ( breln1 @ A @ R )
% 2.12/1.17 => ! [Xphi: $i > $o] :
% 2.12/1.17 ( ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ! [Xy: $i] :
% 2.12/1.17 ( ( in @ Xy @ A )
% 2.12/1.17 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.12/1.17 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 2.12/1.17 => ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ R )
% 2.12/1.17 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(breln1Set_type,type,
% 2.12/1.17 breln1Set: $i > $i ).
% 2.12/1.17
% 2.12/1.17 thf(breln1SetBreln1_type,type,
% 2.12/1.17 breln1SetBreln1: $o ).
% 2.12/1.17
% 2.12/1.17 thf(breln1SetBreln1,definition,
% 2.12/1.17 ( breln1SetBreln1
% 2.12/1.17 = ( ! [A: $i,R: $i] :
% 2.12/1.17 ( ( in @ R @ ( breln1Set @ A ) )
% 2.12/1.17 => ( breln1 @ A @ R ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(transitive_type,type,
% 2.12/1.17 transitive: $i > $i > $o ).
% 2.12/1.17
% 2.12/1.17 thf(antisymmetric_type,type,
% 2.12/1.17 antisymmetric: $i > $i > $o ).
% 2.12/1.17
% 2.12/1.17 thf(reflexive_type,type,
% 2.12/1.17 reflexive: $i > $i > $o ).
% 2.12/1.17
% 2.12/1.17 thf(refltransitive_type,type,
% 2.12/1.17 refltransitive: $i > $i > $o ).
% 2.12/1.17
% 2.12/1.17 thf(refllinearorder_type,type,
% 2.12/1.17 refllinearorder: $i > $i > $o ).
% 2.12/1.17
% 2.12/1.17 thf(reflwellordering_type,type,
% 2.12/1.17 reflwellordering: $i > $i > $o ).
% 2.12/1.17
% 2.12/1.17 thf(choice2fnsingleton_type,type,
% 2.12/1.17 choice2fnsingleton: $o ).
% 2.12/1.17
% 2.12/1.17 thf(choice2fnsingleton,definition,
% 2.12/1.17 ( choice2fnsingleton
% 2.12/1.17 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 2.12/1.17 ( ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ? [Xy: $i] :
% 2.12/1.17 ( ( in @ Xy @ B )
% 2.12/1.17 & ( Xphi @ Xx @ Xy ) ) )
% 2.12/1.17 => ! [R: $i] :
% 2.12/1.17 ( ( in @ R @ ( breln1Set @ B ) )
% 2.12/1.17 => ( ( reflwellordering @ B @ R )
% 2.12/1.17 => ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ( singleton
% 2.12/1.17 @ ( dsetconstr @ B
% 2.12/1.17 @ ^ [Xy: $i] :
% 2.12/1.17 ( ( Xphi @ Xx @ Xy )
% 2.12/1.17 & ! [Xz: $i] :
% 2.12/1.17 ( ( in @ Xz @ B )
% 2.12/1.17 => ( ( Xphi @ Xx @ Xz )
% 2.12/1.17 => ( in @ ( kpair @ Xy @ Xz ) @ R ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(setOfPairsIsBReln1_type,type,
% 2.12/1.17 setOfPairsIsBReln1: $o ).
% 2.12/1.17
% 2.12/1.17 thf(setOfPairsIsBReln1,definition,
% 2.12/1.17 ( setOfPairsIsBReln1
% 2.12/1.17 = ( ! [A: $i,Xphi: $i > $i > $o] :
% 2.12/1.17 ( breln1 @ A
% 2.12/1.17 @ ( dpsetconstr @ A @ A
% 2.12/1.17 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(breln1all1_type,type,
% 2.12/1.17 breln1all1: $o ).
% 2.12/1.17
% 2.12/1.17 thf(breln1all1,definition,
% 2.12/1.17 ( breln1all1
% 2.12/1.17 = ( ! [A: $i,R: $i] :
% 2.12/1.17 ( ( breln1 @ A @ R )
% 2.12/1.17 => ! [Xphi: $i > $o] :
% 2.12/1.17 ( ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ! [Xy: $i] :
% 2.12/1.17 ( ( in @ Xy @ A )
% 2.12/1.17 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.12/1.17 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 2.12/1.17 => ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ R )
% 2.12/1.17 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(subbreln1_type,type,
% 2.12/1.17 subbreln1: $o ).
% 2.12/1.17
% 2.12/1.17 thf(subbreln1,definition,
% 2.12/1.17 ( subbreln1
% 2.12/1.17 = ( ! [A: $i,R: $i] :
% 2.12/1.17 ( ( breln1 @ A @ R )
% 2.12/1.17 => ! [S: $i] :
% 2.12/1.17 ( ( breln1 @ A @ S )
% 2.12/1.17 => ( ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ! [Xy: $i] :
% 2.12/1.17 ( ( in @ Xy @ A )
% 2.12/1.17 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.12/1.17 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 2.12/1.17 => ( subset @ R @ S ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(eqbreln1_type,type,
% 2.12/1.17 eqbreln1: $o ).
% 2.12/1.17
% 2.12/1.17 thf(eqbreln1,definition,
% 2.12/1.17 ( eqbreln1
% 2.12/1.17 = ( ! [A: $i,R: $i] :
% 2.12/1.17 ( ( breln1 @ A @ R )
% 2.12/1.17 => ! [S: $i] :
% 2.12/1.17 ( ( breln1 @ A @ S )
% 2.12/1.17 => ( ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ! [Xy: $i] :
% 2.12/1.17 ( ( in @ Xy @ A )
% 2.12/1.17 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.12/1.17 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 2.12/1.17 => ( ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ! [Xy: $i] :
% 2.12/1.17 ( ( in @ Xy @ A )
% 2.12/1.17 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 2.12/1.17 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 2.12/1.17 => ( R = S ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(breln1invset_type,type,
% 2.12/1.17 breln1invset: $i > $i > $i ).
% 2.12/1.17
% 2.12/1.17 thf(breln1invprop_type,type,
% 2.12/1.17 breln1invprop: $o ).
% 2.12/1.17
% 2.12/1.17 thf(breln1invprop,definition,
% 2.12/1.17 ( breln1invprop
% 2.12/1.17 = ( ! [A: $i,R: $i] :
% 2.12/1.17 ( ( breln1 @ A @ R )
% 2.12/1.17 => ( breln1 @ A @ ( breln1invset @ A @ R ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(breln1invI_type,type,
% 2.12/1.17 breln1invI: $o ).
% 2.12/1.17
% 2.12/1.17 thf(breln1invI,definition,
% 2.12/1.17 ( breln1invI
% 2.12/1.17 = ( ! [A: $i,R: $i] :
% 2.12/1.17 ( ( breln1 @ A @ R )
% 2.12/1.17 => ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ! [Xy: $i] :
% 2.12/1.17 ( ( in @ Xy @ A )
% 2.12/1.17 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.12/1.17 => ( in @ ( kpair @ Xy @ Xx ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(breln1invE_type,type,
% 2.12/1.17 breln1invE: $o ).
% 2.12/1.17
% 2.12/1.17 thf(breln1invE,definition,
% 2.12/1.17 ( breln1invE
% 2.12/1.17 = ( ! [A: $i,R: $i] :
% 2.12/1.17 ( ( breln1 @ A @ R )
% 2.12/1.17 => ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ! [Xy: $i] :
% 2.12/1.17 ( ( in @ Xy @ A )
% 2.12/1.17 => ( ( in @ ( kpair @ Xy @ Xx ) @ ( breln1invset @ A @ R ) )
% 2.12/1.17 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(breln1compset_type,type,
% 2.12/1.17 breln1compset: $i > $i > $i > $i ).
% 2.12/1.17
% 2.12/1.17 thf(breln1compprop_type,type,
% 2.12/1.17 breln1compprop: $o ).
% 2.12/1.17
% 2.12/1.17 thf(breln1compprop,definition,
% 2.12/1.17 ( breln1compprop
% 2.12/1.17 = ( ! [A: $i,R: $i] :
% 2.12/1.17 ( ( breln1 @ A @ R )
% 2.12/1.17 => ! [S: $i] :
% 2.12/1.17 ( ( breln1 @ A @ S )
% 2.12/1.17 => ( breln1 @ A @ ( breln1compset @ A @ R @ S ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(breln1compI_type,type,
% 2.12/1.17 breln1compI: $o ).
% 2.12/1.17
% 2.12/1.17 thf(breln1compI,definition,
% 2.12/1.17 ( breln1compI
% 2.12/1.17 = ( ! [A: $i,R: $i] :
% 2.12/1.17 ( ( breln1 @ A @ R )
% 2.12/1.17 => ! [S: $i] :
% 2.12/1.17 ( ( breln1 @ A @ S )
% 2.12/1.17 => ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ! [Xy: $i] :
% 2.12/1.17 ( ( in @ Xy @ A )
% 2.12/1.17 => ! [Xz: $i] :
% 2.12/1.17 ( ( in @ Xz @ A )
% 2.12/1.17 => ( ( in @ ( kpair @ Xx @ Xz ) @ R )
% 2.12/1.17 => ( ( in @ ( kpair @ Xz @ Xy ) @ S )
% 2.12/1.17 => ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) ) ) ) ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(breln1compE_type,type,
% 2.12/1.17 breln1compE: $o ).
% 2.12/1.17
% 2.12/1.17 thf(breln1compE,definition,
% 2.12/1.17 ( breln1compE
% 2.12/1.17 = ( ! [A: $i,R: $i] :
% 2.12/1.17 ( ( breln1 @ A @ R )
% 2.12/1.17 => ! [S: $i] :
% 2.12/1.17 ( ( breln1 @ A @ S )
% 2.12/1.17 => ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ! [Xy: $i] :
% 2.12/1.17 ( ( in @ Xy @ A )
% 2.12/1.17 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) )
% 2.12/1.17 => ? [Xz: $i] :
% 2.12/1.17 ( ( in @ Xz @ A )
% 2.12/1.17 & ( in @ ( kpair @ Xx @ Xz ) @ R )
% 2.12/1.17 & ( in @ ( kpair @ Xz @ Xy ) @ S ) ) ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(breln1compEex_type,type,
% 2.12/1.17 breln1compEex: $o ).
% 2.12/1.17
% 2.12/1.17 thf(breln1compEex,definition,
% 2.12/1.17 ( breln1compEex
% 2.12/1.17 = ( ! [A: $i,R: $i] :
% 2.12/1.17 ( ( breln1 @ A @ R )
% 2.12/1.17 => ! [S: $i] :
% 2.12/1.17 ( ( breln1 @ A @ S )
% 2.12/1.17 => ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ! [Xy: $i] :
% 2.12/1.17 ( ( in @ Xy @ A )
% 2.12/1.17 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) )
% 2.12/1.17 => ! [Xphi: $o] :
% 2.12/1.17 ( ! [Xz: $i] :
% 2.12/1.17 ( ( in @ Xz @ A )
% 2.12/1.17 => ( ( in @ ( kpair @ Xx @ Xz ) @ R )
% 2.12/1.17 => ( ( in @ ( kpair @ Xz @ Xy ) @ S )
% 2.12/1.17 => Xphi ) ) )
% 2.12/1.17 => Xphi ) ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(breln1unionprop_type,type,
% 2.12/1.17 breln1unionprop: $o ).
% 2.12/1.17
% 2.12/1.17 thf(breln1unionprop,definition,
% 2.12/1.17 ( breln1unionprop
% 2.12/1.17 = ( ! [A: $i,R: $i] :
% 2.12/1.17 ( ( breln1 @ A @ R )
% 2.12/1.17 => ! [S: $i] :
% 2.12/1.17 ( ( breln1 @ A @ S )
% 2.12/1.17 => ( breln1 @ A @ ( binunion @ R @ S ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(breln1unionIL_type,type,
% 2.12/1.17 breln1unionIL: $o ).
% 2.12/1.17
% 2.12/1.17 thf(breln1unionIL,definition,
% 2.12/1.17 ( breln1unionIL
% 2.12/1.17 = ( ! [A: $i,R: $i] :
% 2.12/1.17 ( ( breln1 @ A @ R )
% 2.12/1.17 => ! [S: $i] :
% 2.12/1.17 ( ( breln1 @ A @ S )
% 2.12/1.17 => ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ! [Xy: $i] :
% 2.12/1.17 ( ( in @ Xy @ A )
% 2.12/1.17 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.12/1.17 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(breln1unionIR_type,type,
% 2.12/1.17 breln1unionIR: $o ).
% 2.12/1.17
% 2.12/1.17 thf(breln1unionIR,definition,
% 2.12/1.17 ( breln1unionIR
% 2.12/1.17 = ( ! [A: $i,R: $i] :
% 2.12/1.17 ( ( breln1 @ A @ R )
% 2.12/1.17 => ! [S: $i] :
% 2.12/1.17 ( ( breln1 @ A @ S )
% 2.12/1.17 => ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ! [Xy: $i] :
% 2.12/1.17 ( ( in @ Xy @ A )
% 2.12/1.17 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 2.12/1.17 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(breln1unionI_type,type,
% 2.12/1.17 breln1unionI: $o ).
% 2.12/1.17
% 2.12/1.17 thf(breln1unionI,definition,
% 2.12/1.17 ( breln1unionI
% 2.12/1.17 = ( ! [A: $i,R: $i] :
% 2.12/1.17 ( ( breln1 @ A @ R )
% 2.12/1.17 => ! [S: $i] :
% 2.12/1.17 ( ( breln1 @ A @ S )
% 2.12/1.17 => ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ! [Xy: $i] :
% 2.12/1.17 ( ( in @ Xy @ A )
% 2.12/1.17 => ( ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.12/1.17 | ( in @ ( kpair @ Xx @ Xy ) @ S ) )
% 2.12/1.17 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(breln1unionE_type,type,
% 2.12/1.17 breln1unionE: $o ).
% 2.12/1.17
% 2.12/1.17 thf(breln1unionE,definition,
% 2.12/1.17 ( breln1unionE
% 2.12/1.17 = ( ! [A: $i,R: $i] :
% 2.12/1.17 ( ( breln1 @ A @ R )
% 2.12/1.17 => ! [S: $i] :
% 2.12/1.17 ( ( breln1 @ A @ S )
% 2.12/1.17 => ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ! [Xy: $i] :
% 2.12/1.17 ( ( in @ Xy @ A )
% 2.12/1.17 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) )
% 2.12/1.17 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.12/1.17 | ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(breln1unionEcases_type,type,
% 2.12/1.17 breln1unionEcases: $o ).
% 2.12/1.17
% 2.12/1.17 thf(breln1unionEcases,definition,
% 2.12/1.17 ( breln1unionEcases
% 2.12/1.17 = ( ! [A: $i,R: $i] :
% 2.12/1.17 ( ( breln1 @ A @ R )
% 2.12/1.17 => ! [S: $i] :
% 2.12/1.17 ( ( breln1 @ A @ S )
% 2.12/1.17 => ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ A )
% 2.12/1.17 => ! [Xy: $i] :
% 2.12/1.17 ( ( in @ Xy @ A )
% 2.12/1.17 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) )
% 2.12/1.17 => ! [Xphi: $o] :
% 2.12/1.17 ( ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.12/1.17 => Xphi )
% 2.12/1.17 => ( ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 2.12/1.17 => Xphi )
% 2.12/1.17 => Xphi ) ) ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(breln1unionCommutes_type,type,
% 2.12/1.17 breln1unionCommutes: $o ).
% 2.12/1.17
% 2.12/1.17 thf(breln1unionCommutes,definition,
% 2.12/1.17 ( breln1unionCommutes
% 2.12/1.17 = ( ! [A: $i,R: $i] :
% 2.12/1.17 ( ( breln1 @ A @ R )
% 2.12/1.17 => ! [S: $i] :
% 2.12/1.17 ( ( breln1 @ A @ S )
% 2.12/1.17 => ( ( binunion @ R @ S )
% 2.12/1.17 = ( binunion @ S @ R ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(woz2Ex_type,type,
% 2.12/1.17 woz2Ex: $o ).
% 2.12/1.17
% 2.12/1.17 thf(woz2Ex,definition,
% 2.12/1.17 ( woz2Ex
% 2.12/1.17 = ( ! [A: $i,R: $i] :
% 2.12/1.17 ( ( breln1 @ A @ R )
% 2.12/1.17 => ( R
% 2.12/1.17 = ( breln1invset @ A @ ( breln1invset @ A @ R ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(woz2W_type,type,
% 2.12/1.17 woz2W: $o ).
% 2.12/1.17
% 2.12/1.17 thf(woz2W,definition,
% 2.12/1.17 ( woz2W
% 2.12/1.17 = ( ! [A: $i,R: $i] :
% 2.12/1.17 ( ( breln1 @ A @ R )
% 2.12/1.17 => ! [S: $i] :
% 2.12/1.17 ( ( breln1 @ A @ S )
% 2.12/1.17 => ( ( breln1invset @ A @ ( breln1compset @ A @ R @ S ) )
% 2.12/1.17 = ( breln1compset @ A @ ( breln1invset @ A @ S ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(woz2A_type,type,
% 2.12/1.17 woz2A: $o ).
% 2.12/1.17
% 2.12/1.17 thf(woz2A,definition,
% 2.12/1.17 ( woz2A
% 2.12/1.17 = ( ! [A: $i,R: $i] :
% 2.12/1.17 ( ( breln1 @ A @ R )
% 2.12/1.17 => ! [S: $i] :
% 2.12/1.17 ( ( breln1 @ A @ S )
% 2.12/1.17 => ! [T: $i] :
% 2.12/1.17 ( ( breln1 @ A @ T )
% 2.12/1.17 => ( ( breln1compset @ A @ ( binunion @ R @ S ) @ T )
% 2.12/1.17 = ( binunion @ ( breln1compset @ A @ R @ T ) @ ( breln1compset @ A @ S @ T ) ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(woz2B_type,type,
% 2.12/1.17 woz2B: $o ).
% 2.12/1.17
% 2.12/1.17 thf(woz2B,definition,
% 2.12/1.17 ( woz2B
% 2.12/1.17 = ( ! [A: $i,R: $i] :
% 2.12/1.17 ( ( breln1 @ A @ R )
% 2.12/1.17 => ! [S: $i] :
% 2.12/1.17 ( ( breln1 @ A @ S )
% 2.12/1.17 => ! [T: $i] :
% 2.12/1.17 ( ( breln1 @ A @ T )
% 2.12/1.17 => ( ( breln1compset @ A @ ( binunion @ R @ S ) @ T )
% 2.12/1.17 = ( binunion @ ( breln1invset @ A @ ( breln1compset @ A @ ( breln1invset @ A @ T ) @ ( breln1invset @ A @ S ) ) ) @ ( breln1invset @ A @ ( breln1compset @ A @ ( breln1invset @ A @ T ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(image1Ex_type,type,
% 2.12/1.17 image1Ex: $o ).
% 2.12/1.17
% 2.12/1.17 thf(image1Ex,definition,
% 2.12/1.17 ( image1Ex
% 2.12/1.17 = ( ! [A: $i,Xf: $i > $i] :
% 2.12/1.17 ? [B: $i] :
% 2.12/1.17 ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ B )
% 2.12/1.17 <=> ? [Xy: $i] :
% 2.12/1.17 ( ( in @ Xy @ A )
% 2.12/1.17 & ( Xx
% 2.12/1.17 = ( Xf @ Xy ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(image1Ex1_type,type,
% 2.12/1.17 image1Ex1: $o ).
% 2.12/1.17
% 2.12/1.17 thf(image1Ex1,definition,
% 2.12/1.17 ( image1Ex1
% 2.12/1.17 = ( ! [A: $i,Xf: $i > $i] :
% 2.12/1.17 ( exu
% 2.12/1.17 @ ^ [B: $i] :
% 2.12/1.17 ! [Xx: $i] :
% 2.12/1.17 ( ( in @ Xx @ B )
% 2.12/1.17 <=> ? [Xy: $i] :
% 2.12/1.17 ( ( in @ Xy @ A )
% 2.12/1.17 & ( Xx
% 2.12/1.17 = ( Xf @ Xy ) ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(image1_type,type,
% 2.12/1.17 image1: $i > ( $i > $i ) > $i ).
% 2.12/1.17
% 2.12/1.17 thf(image1Equiv_type,type,
% 2.12/1.17 image1Equiv: $o ).
% 2.12/1.17
% 2.12/1.17 thf(image1Equiv,definition,
% 2.12/1.17 ( image1Equiv
% 2.12/1.17 = ( ! [A: $i,Xf: $i > $i,Xx: $i] :
% 2.12/1.17 ( ( in @ Xx
% 2.12/1.17 @ ( image1 @ A
% 2.12/1.17 @ ^ [Xy: $i] : ( Xf @ Xy ) ) )
% 2.12/1.17 <=> ? [Xy: $i] :
% 2.12/1.17 ( ( in @ Xy @ A )
% 2.12/1.17 & ( Xx
% 2.12/1.17 = ( Xf @ Xy ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(image1E_type,type,
% 2.12/1.17 image1E: $o ).
% 2.12/1.17
% 2.12/1.17 thf(image1E,definition,
% 2.12/1.17 ( image1E
% 2.12/1.17 = ( ! [A: $i,Xf: $i > $i,Xx: $i] :
% 2.12/1.17 ( ( in @ Xx
% 2.12/1.17 @ ( image1 @ A
% 2.12/1.17 @ ^ [Xy: $i] : ( Xf @ Xy ) ) )
% 2.12/1.17 => ? [Xy: $i] :
% 2.12/1.17 ( ( in @ Xy @ A )
% 2.12/1.17 & ( Xx
% 2.12/1.17 = ( Xf @ Xy ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(image1I_type,type,
% 2.12/1.17 image1I: $o ).
% 2.12/1.17
% 2.12/1.17 thf(image1I,definition,
% 2.12/1.17 ( image1I
% 2.12/1.17 = ( ! [A: $i,Xf: $i > $i,Xx: $i] :
% 2.12/1.17 ( ? [Xy: $i] :
% 2.12/1.17 ( ( in @ Xy @ A )
% 2.12/1.17 & ( Xx
% 2.12/1.17 = ( Xf @ Xy ) ) )
% 2.12/1.17 => ( in @ Xx
% 2.12/1.17 @ ( image1 @ A
% 2.12/1.17 @ ^ [Xy: $i] : ( Xf @ Xy ) ) ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(injective_type,type,
% 2.12/1.17 injective: $i > $i > $i > $o ).
% 2.12/1.17
% 2.12/1.17 thf(injFuncSet_type,type,
% 2.12/1.17 injFuncSet: $i > $i > $i ).
% 2.12/1.17
% 2.12/1.17 thf(injFuncSet,definition,
% 2.12/1.17 ( injFuncSet
% 2.12/1.17 = ( ^ [A: $i,B: $i] :
% 2.12/1.17 ( dsetconstr @ ( funcSet @ A @ B )
% 2.12/1.17 @ ^ [Xf: $i] : ( injective @ A @ B @ Xf ) ) ) ) ).
% 2.12/1.17
% 2.12/1.17 thf(injFuncInInjFuncSet,conjecture,
% 2.12/1.17 ( setextAx
% 2.12/1.17 => ( emptysetAx
% 2.12/1.17 => ( setadjoinAx
% 2.12/1.17 => ( powersetAx
% 2.12/1.17 => ( setunionAx
% 2.12/1.17 => ( omega0Ax
% 2.12/1.17 => ( omegaSAx
% 2.12/1.17 => ( omegaIndAx
% 2.12/1.17 => ( replAx
% 2.12/1.17 => ( foundationAx
% 2.12/1.17 => ( wellorderingAx
% 2.12/1.17 => ( descrp
% 2.12/1.17 => ( dsetconstrI
% 2.12/1.17 => ( dsetconstrEL
% 2.12/1.17 => ( dsetconstrER
% 2.12/1.17 => ( exuE1
% 2.12/1.17 => ( prop2setE
% 2.12/1.17 => ( emptysetE
% 2.12/1.17 => ( emptysetimpfalse
% 2.12/1.17 => ( notinemptyset
% 2.12/1.17 => ( exuE3e
% 2.12/1.17 => ( setext
% 2.12/1.17 => ( emptyI
% 2.12/1.17 => ( noeltsimpempty
% 2.12/1.17 => ( setbeta
% 2.12/1.17 => ( nonemptyE1
% 2.12/1.17 => ( nonemptyI
% 2.12/1.17 => ( nonemptyI1
% 2.12/1.17 => ( setadjoinIL
% 2.12/1.17 => ( emptyinunitempty
% 2.12/1.17 => ( setadjoinIR
% 2.12/1.17 => ( setadjoinE
% 2.12/1.17 => ( setadjoinOr
% 2.12/1.17 => ( setoftrueEq
% 2.12/1.17 => ( powersetI
% 2.12/1.17 => ( emptyinPowerset
% 2.12/1.17 => ( emptyInPowerset
% 2.12/1.17 => ( powersetE
% 2.12/1.17 => ( setunionI
% 2.12/1.17 => ( setunionE
% 2.12/1.17 => ( subPowSU
% 2.12/1.17 => ( exuE2
% 2.12/1.17 => ( nonemptyImpWitness
% 2.12/1.17 => ( uniqinunit
% 2.12/1.17 => ( notinsingleton
% 2.12/1.17 => ( eqinunit
% 2.12/1.17 => ( singletonsswitch
% 2.12/1.17 => ( upairsetE
% 2.12/1.17 => ( upairsetIL
% 2.12/1.17 => ( upairsetIR
% 2.12/1.17 => ( emptyE1
% 2.12/1.17 => ( vacuousDall
% 2.12/1.17 => ( quantDeMorgan1
% 2.12/1.17 => ( quantDeMorgan2
% 2.12/1.17 => ( quantDeMorgan3
% 2.12/1.17 => ( quantDeMorgan4
% 2.12/1.17 => ( prop2setI
% 2.12/1.17 => ( prop2set2propI
% 2.12/1.17 => ( notdexE
% 2.12/1.17 => ( notdallE
% 2.12/1.17 => ( exuI1
% 2.12/1.17 => ( exuI3
% 2.12/1.17 => ( exuI2
% 2.12/1.17 => ( inCongP
% 2.12/1.17 => ( in__Cong
% 2.12/1.17 => ( exuE3u
% 2.12/1.17 => ( exu__Cong
% 2.12/1.17 => ( emptyset__Cong
% 2.12/1.17 => ( setadjoin__Cong
% 2.12/1.17 => ( powerset__Cong
% 2.12/1.17 => ( setunion__Cong
% 2.12/1.17 => ( omega__Cong
% 2.12/1.17 => ( exuEu
% 2.12/1.17 => ( descr__Cong
% 2.12/1.17 => ( dsetconstr__Cong
% 2.12/1.17 => ( subsetI1
% 2.12/1.17 => ( eqimpsubset2
% 2.12/1.17 => ( eqimpsubset1
% 2.12/1.17 => ( subsetI2
% 2.12/1.17 => ( emptysetsubset
% 2.12/1.17 => ( subsetE
% 2.12/1.17 => ( subsetE2
% 2.12/1.17 => ( notsubsetI
% 2.12/1.17 => ( notequalI1
% 2.12/1.17 => ( notequalI2
% 2.12/1.17 => ( subsetRefl
% 2.12/1.17 => ( subsetTrans
% 2.12/1.17 => ( setadjoinSub
% 2.12/1.17 => ( setadjoinSub2
% 2.12/1.17 => ( subset2powerset
% 2.12/1.17 => ( setextsub
% 2.12/1.17 => ( subsetemptysetimpeq
% 2.12/1.17 => ( powersetI1
% 2.12/1.17 => ( powersetE1
% 2.12/1.17 => ( inPowerset
% 2.12/1.17 => ( powersetsubset
% 2.12/1.17 => ( sepInPowerset
% 2.12/1.17 => ( sepSubset
% 2.12/1.17 => ( binunionIL
% 2.12/1.17 => ( upairset2IR
% 2.12/1.17 => ( binunionIR
% 2.12/1.17 => ( binunionEcases
% 2.12/1.17 => ( binunionE
% 2.12/1.17 => ( binunionLsub
% 2.12/1.17 => ( binunionRsub
% 2.12/1.17 => ( binintersectI
% 2.12/1.17 => ( binintersectSubset5
% 2.12/1.17 => ( binintersectEL
% 2.12/1.17 => ( binintersectLsub
% 2.12/1.17 => ( binintersectSubset2
% 2.12/1.17 => ( binintersectSubset3
% 2.12/1.17 => ( binintersectER
% 2.12/1.17 => ( disjointsetsI1
% 2.12/1.17 => ( binintersectRsub
% 2.12/1.17 => ( binintersectSubset4
% 2.12/1.17 => ( binintersectSubset1
% 2.12/1.17 => ( bs114d
% 2.12/1.17 => ( setminusI
% 2.12/1.17 => ( setminusEL
% 2.12/1.17 => ( setminusER
% 2.12/1.17 => ( setminusSubset2
% 2.12/1.17 => ( setminusERneg
% 2.12/1.17 => ( setminusELneg
% 2.12/1.17 => ( setminusILneg
% 2.12/1.17 => ( setminusIRneg
% 2.12/1.17 => ( setminusLsub
% 2.12/1.17 => ( setminusSubset1
% 2.12/1.17 => ( symdiffE
% 2.12/1.17 => ( symdiffI1
% 2.12/1.17 => ( symdiffI2
% 2.12/1.17 => ( symdiffIneg1
% 2.12/1.17 => ( symdiffIneg2
% 2.12/1.17 => ( secondinupair
% 2.12/1.17 => ( setukpairIL
% 2.12/1.17 => ( setukpairIR
% 2.12/1.17 => ( kpairiskpair
% 2.12/1.17 => ( kpairp
% 2.12/1.17 => ( singletonsubset
% 2.12/1.17 => ( singletoninpowerset
% 2.12/1.17 => ( singletoninpowunion
% 2.12/1.17 => ( upairset2E
% 2.12/1.17 => ( upairsubunion
% 2.12/1.17 => ( upairinpowunion
% 2.12/1.17 => ( ubforcartprodlem1
% 2.12/1.17 => ( ubforcartprodlem2
% 2.12/1.17 => ( ubforcartprodlem3
% 2.12/1.17 => ( cartprodpairin
% 2.12/1.17 => ( cartprodmempair1
% 2.12/1.17 => ( cartprodmempair
% 2.12/1.17 => ( setunionE2
% 2.12/1.17 => ( setunionsingleton1
% 2.12/1.17 => ( setunionsingleton2
% 2.12/1.17 => ( setunionsingleton
% 2.12/1.17 => ( singletonprop
% 2.12/1.17 => ( ex1E1
% 2.12/1.17 => ( ex1I
% 2.12/1.17 => ( ex1I2
% 2.12/1.17 => ( singletonsuniq
% 2.12/1.17 => ( setukpairinjL1
% 2.12/1.17 => ( kfstsingleton
% 2.12/1.17 => ( theprop
% 2.12/1.17 => ( kfstpairEq
% 2.12/1.17 => ( cartprodfstin
% 2.12/1.17 => ( setukpairinjL2
% 2.12/1.17 => ( setukpairinjL
% 2.12/1.17 => ( setukpairinjR11
% 2.12/1.17 => ( setukpairinjR12
% 2.12/1.17 => ( setukpairinjR1
% 2.12/1.17 => ( upairequniteq
% 2.12/1.17 => ( setukpairinjR2
% 2.12/1.17 => ( setukpairinjR
% 2.12/1.17 => ( ksndsingleton
% 2.12/1.17 => ( ksndpairEq
% 2.12/1.17 => ( kpairsurjEq
% 2.12/1.17 => ( cartprodsndin
% 2.12/1.17 => ( cartprodpairmemEL
% 2.12/1.17 => ( cartprodpairmemER
% 2.12/1.17 => ( cartprodmempaircEq
% 2.12/1.17 => ( cartprodfstpairEq
% 2.12/1.17 => ( cartprodsndpairEq
% 2.12/1.17 => ( cartprodpairsurjEq
% 2.12/1.17 => ( dpsetconstrI
% 2.12/1.17 => ( dpsetconstrSub
% 2.12/1.17 => ( setOfPairsIsBReln
% 2.12/1.17 => ( dpsetconstrERa
% 2.12/1.17 => ( dpsetconstrEL1
% 2.12/1.17 => ( dpsetconstrEL2
% 2.12/1.17 => ( dpsetconstrER
% 2.12/1.17 => ( funcImageSingleton
% 2.12/1.17 => ( apProp
% 2.12/1.17 => ( app
% 2.12/1.17 => ( infuncsetfunc
% 2.12/1.17 => ( ap2p
% 2.12/1.17 => ( funcinfuncset
% 2.12/1.17 => ( lamProp
% 2.12/1.17 => ( lamp
% 2.12/1.17 => ( lam2p
% 2.12/1.17 => ( brelnall1
% 2.12/1.17 => ( brelnall2
% 2.12/1.17 => ( ex1E2
% 2.12/1.17 => ( funcGraphProp1
% 2.12/1.17 => ( funcGraphProp3
% 2.12/1.17 => ( funcGraphProp2
% 2.12/1.17 => ( funcextLem
% 2.12/1.17 => ( funcGraphProp4
% 2.12/1.17 => ( subbreln
% 2.12/1.17 => ( eqbreln
% 2.12/1.17 => ( funcext
% 2.12/1.17 => ( funcext2
% 2.12/1.17 => ( ap2apEq1
% 2.12/1.17 => ( ap2apEq2
% 2.12/1.17 => ( beta1
% 2.12/1.17 => ( eta1
% 2.12/1.17 => ( lam2lamEq
% 2.12/1.17 => ( beta2
% 2.12/1.17 => ( eta2
% 2.12/1.17 => ( iffalseProp1
% 2.12/1.17 => ( iffalseProp2
% 2.12/1.17 => ( iftrueProp1
% 2.12/1.17 => ( iftrueProp2
% 2.12/1.17 => ( ifSingleton
% 2.12/1.17 => ( ifp
% 2.12/1.17 => ( theeq
% 2.12/1.17 => ( iftrue
% 2.12/1.17 => ( iffalse
% 2.12/1.17 => ( iftrueorfalse
% 2.12/1.17 => ( binintersectT_lem
% 2.12/1.17 => ( binunionT_lem
% 2.12/1.17 => ( powersetT_lem
% 2.12/1.17 => ( setminusT_lem
% 2.12/1.17 => ( complementT_lem
% 2.12/1.17 => ( setextT
% 2.12/1.17 => ( subsetTI
% 2.12/1.17 => ( powersetTI1
% 2.12/1.17 => ( powersetTE1
% 2.12/1.17 => ( complementTI1
% 2.12/1.17 => ( complementTE1
% 2.12/1.17 => ( binintersectTELcontra
% 2.12/1.17 => ( binintersectTERcontra
% 2.12/1.17 => ( contrasubsetT
% 2.12/1.17 => ( contrasubsetT1
% 2.12/1.17 => ( contrasubsetT2
% 2.12/1.17 => ( contrasubsetT3
% 2.12/1.17 => ( doubleComplementI1
% 2.12/1.17 => ( doubleComplementE1
% 2.12/1.17 => ( doubleComplementSub1
% 2.12/1.17 => ( doubleComplementSub2
% 2.12/1.17 => ( doubleComplementEq
% 2.12/1.17 => ( complementTnotintersectT
% 2.12/1.17 => ( complementImpComplementIntersect
% 2.12/1.17 => ( complementSubsetComplementIntersect
% 2.12/1.17 => ( complementInPowersetComplementIntersect
% 2.12/1.17 => ( contraSubsetComplement
% 2.12/1.17 => ( complementTcontraSubset
% 2.12/1.17 => ( binunionTILcontra
% 2.12/1.17 => ( binunionTIRcontra
% 2.12/1.17 => ( inIntersectImpInUnion
% 2.12/1.17 => ( inIntersectImpInUnion2
% 2.12/1.17 => ( inIntersectImpInIntersectUnions
% 2.12/1.17 => ( intersectInPowersetIntersectUnions
% 2.12/1.17 => ( inComplementUnionImpNotIn1
% 2.12/1.17 => ( inComplementUnionImpInComplement1
% 2.12/1.17 => ( binunionTE
% 2.12/1.17 => ( binunionTEcontra
% 2.12/1.17 => ( demorgan2a1
% 2.12/1.17 => ( complementUnionInPowersetComplement
% 2.12/1.17 => ( demorgan2a2
% 2.12/1.17 => ( demorgan1a
% 2.12/1.17 => ( demorgan1b
% 2.12/1.17 => ( demorgan1
% 2.12/1.17 => ( demorgan2a
% 2.12/1.17 => ( demorgan2b2
% 2.12/1.17 => ( demorgan2b
% 2.12/1.17 => ( demorgan2
% 2.12/1.17 => ( woz13rule0
% 2.12/1.17 => ( woz13rule1
% 2.12/1.17 => ( woz13rule2
% 2.12/1.17 => ( woz13rule3
% 2.12/1.17 => ( woz13rule4
% 2.12/1.17 => ( woz1_1
% 2.12/1.17 => ( woz1_2
% 2.12/1.17 => ( woz1_3
% 2.12/1.17 => ( woz1_4
% 2.12/1.17 => ( woz1_5
% 2.12/1.17 => ( breln1all2
% 2.12/1.17 => ( breln1SetBreln1
% 2.12/1.17 => ( choice2fnsingleton
% 2.12/1.17 => ( setOfPairsIsBReln1
% 2.12/1.17 => ( breln1all1
% 2.12/1.17 => ( subbreln1
% 2.12/1.17 => ( eqbreln1
% 2.12/1.17 => ( breln1invprop
% 2.12/1.17 => ( breln1invI
% 2.12/1.17 => ( breln1invE
% 2.12/1.17 => ( breln1compprop
% 2.12/1.17 => ( breln1compI
% 2.12/1.17 => ( breln1compE
% 2.12/1.17 => ( breln1compEex
% 2.12/1.17 => ( breln1unionprop
% 2.12/1.17 => ( breln1unionIL
% 2.12/1.17 => ( breln1unionIR
% 2.12/1.17 => ( breln1unionI
% 2.12/1.17 => ( breln1unionE
% 2.12/1.17 => ( breln1unionEcases
% 2.12/1.17 => ( breln1unionCommutes
% 2.12/1.17 => ( woz2Ex
% 2.12/1.17 => ( woz2W
% 2.12/1.17 => ( woz2A
% 2.12/1.17 => ( woz2B
% 2.12/1.17 => ( image1Ex
% 2.12/1.17 => ( image1Ex1
% 2.12/1.17 => ( image1Equiv
% 2.12/1.17 => ( image1E
% 2.12/1.17 => ( image1I
% 2.12/1.17 => ! [A: $i,B: $i,Xf: $i] :
% 2.12/1.17 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.12/1.17 => ( ( injective @ A @ B @ Xf )
% 2.12/1.17 => ( in @ Xf @ ( injFuncSet @ A @ B ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.12/1.17 ------------------------
% 2.19/1.20 % (11816)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_11553 for (2999ds/183Mi)
% 2.19/1.20 % (11817)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_11553 for (2999ds/4Mi)
% 2.19/1.20 % (11818)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_11553 for (2999ds/27Mi)
% 2.19/1.20 % (11819)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_11553 for (2999ds/2Mi)
% 2.19/1.20 % (11820)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_11553 for (2999ds/2Mi)
% 2.19/1.20 % (11821)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_11553 for (2999ds/275Mi)
% 2.19/1.20 % (11822)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_11553 for (2999ds/18Mi)
% 2.19/1.20 % (11819)Instruction limit reached!
% 2.19/1.20 % (11819)------------------------------
% 2.19/1.20 % (11819)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.19/1.20 % (11819)Termination reason: Unknown
% 2.19/1.20 % (11819)Termination phase: shuffling
% 2.19/1.20
% 2.19/1.20 % (11819)Memory used [KB]: 1663
% 2.19/1.20 % (11819)Time elapsed: 0.002 s
% 2.19/1.20 % (11819)Instructions burned: 3 (million)
% 2.19/1.20 % (11819)------------------------------
% 2.19/1.20 % (11819)------------------------------
% 2.19/1.20 % (11820)Instruction limit reached!
% 2.19/1.20 % (11820)------------------------------
% 2.19/1.20 % (11820)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.19/1.20 % (11820)Termination reason: Unknown
% 2.19/1.20 % (11820)Termination phase: shuffling
% 2.19/1.20
% 2.19/1.20 % (11820)Memory used [KB]: 1663
% 2.19/1.20 % (11820)Time elapsed: 0.002 s
% 2.19/1.20 % (11820)Instructions burned: 3 (million)
% 2.19/1.20 % (11820)------------------------------
% 2.19/1.20 % (11820)------------------------------
% 2.19/1.20 % (11817)Instruction limit reached!
% 2.19/1.20 % (11817)------------------------------
% 2.19/1.20 % (11817)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.19/1.20 % (11817)Termination reason: Unknown
% 2.19/1.20 % (11817)Termination phase: shuffling
% 2.19/1.20
% 2.19/1.20 % (11817)Memory used [KB]: 1663
% 2.19/1.20 % (11817)Time elapsed: 0.003 s
% 2.19/1.20 % (11817)Instructions burned: 6 (million)
% 2.19/1.20 % (11817)------------------------------
% 2.19/1.20 % (11817)------------------------------
% 2.19/1.21 % (11822)Instruction limit reached!
% 2.19/1.21 % (11822)------------------------------
% 2.19/1.21 % (11822)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.19/1.21 % (11822)Termination reason: Unknown
% 2.19/1.21 % (11822)Termination phase: shuffling
% 2.19/1.21
% 2.19/1.21 % (11822)Memory used [KB]: 1918
% 2.19/1.21 % (11822)Time elapsed: 0.007 s
% 2.19/1.21 % (11822)Instructions burned: 18 (million)
% 2.19/1.21 % (11822)------------------------------
% 2.19/1.21 % (11822)------------------------------
% 2.19/1.21 % (11818)Instruction limit reached!
% 2.19/1.21 % (11818)------------------------------
% 2.19/1.21 % (11818)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.19/1.21 % (11818)Termination reason: Unknown
% 2.19/1.21 % (11818)Termination phase: shuffling
% 2.19/1.21
% 2.19/1.21 % (11818)Memory used [KB]: 2046
% 2.19/1.21 % (11818)Time elapsed: 0.009 s
% 2.19/1.21 % (11818)Instructions burned: 29 (million)
% 2.19/1.21 % (11818)------------------------------
% 2.19/1.21 % (11818)------------------------------
% 2.19/1.21 % (11823)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_11553 for (2999ds/3Mi)
% 2.19/1.21 % (11824)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_11553 for (2999ds/37Mi)
% 2.19/1.21 % (11825)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_11553 for (2999ds/15Mi)
% 2.19/1.21 % (11823)Instruction limit reached!
% 2.19/1.21 % (11823)------------------------------
% 2.19/1.21 % (11823)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.19/1.21 % (11823)Termination reason: Unknown
% 2.19/1.21 % (11823)Termination phase: shuffling
% 2.19/1.21
% 2.19/1.21 % (11823)Memory used [KB]: 1663
% 2.19/1.21 % (11823)Time elapsed: 0.002 s
% 2.19/1.21 % (11823)Instructions burned: 3 (million)
% 2.19/1.21 % (11823)------------------------------
% 2.19/1.21 % (11823)------------------------------
% 2.19/1.22 % (11826)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_11553 for (2999ds/3Mi)
% 2.19/1.22 % (11825)Instruction limit reached!
% 2.19/1.22 % (11825)------------------------------
% 2.19/1.22 % (11825)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.19/1.22 % (11825)Termination reason: Unknown
% 2.19/1.22 % (11825)Termination phase: shuffling
% 2.19/1.22
% 2.19/1.22 % (11825)Memory used [KB]: 1791
% 2.19/1.22 % (11825)Time elapsed: 0.006 s
% 2.19/1.22 % (11825)Instructions burned: 16 (million)
% 2.19/1.22 % (11825)------------------------------
% 2.19/1.22 % (11825)------------------------------
% 2.19/1.22 % (11826)Instruction limit reached!
% 2.19/1.22 % (11826)------------------------------
% 2.19/1.22 % (11826)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.19/1.22 % (11826)Termination reason: Unknown
% 2.19/1.22 % (11826)Termination phase: shuffling
% 2.19/1.22
% 2.19/1.22 % (11826)Memory used [KB]: 1663
% 2.19/1.22 % (11826)Time elapsed: 0.002 s
% 2.19/1.22 % (11826)Instructions burned: 3 (million)
% 2.19/1.22 % (11826)------------------------------
% 2.19/1.22 % (11826)------------------------------
% 2.19/1.22 % (11827)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_11553 for (2999ds/1041Mi)
% 2.19/1.22 % (11824)Instruction limit reached!
% 2.19/1.22 % (11824)------------------------------
% 2.19/1.22 % (11824)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.19/1.22 % (11824)Termination reason: Unknown
% 2.19/1.22 % (11824)Termination phase: shuffling
% 2.19/1.22
% 2.19/1.22 % (11824)Memory used [KB]: 2430
% 2.19/1.22 % (11824)Time elapsed: 0.011 s
% 2.19/1.22 % (11824)Instructions burned: 38 (million)
% 2.19/1.22 % (11824)------------------------------
% 2.19/1.22 % (11824)------------------------------
% 2.19/1.22 % (11828)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_11553 for (2999ds/7Mi)
% 2.19/1.22 % (11828)Instruction limit reached!
% 2.19/1.22 % (11828)------------------------------
% 2.19/1.22 % (11828)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.19/1.22 % (11828)Termination reason: Unknown
% 2.19/1.22 % (11828)Termination phase: shuffling
% 2.19/1.22
% 2.19/1.22 % (11828)Memory used [KB]: 1663
% 2.19/1.22 % (11828)Time elapsed: 0.003 s
% 2.19/1.22 % (11828)Instructions burned: 7 (million)
% 2.19/1.22 % (11828)------------------------------
% 2.19/1.22 % (11828)------------------------------
% 2.19/1.23 % (11829)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_11553 for (2999ds/16Mi)
% 2.19/1.23 % (11830)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_11553 for (2999ds/3Mi)
% 2.19/1.23 % (11830)Instruction limit reached!
% 2.19/1.23 % (11830)------------------------------
% 2.19/1.23 % (11830)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.19/1.23 % (11830)Termination reason: Unknown
% 2.19/1.23 % (11830)Termination phase: shuffling
% 2.19/1.23
% 2.19/1.23 % (11830)Memory used [KB]: 1663
% 2.19/1.23 % (11830)Time elapsed: 0.002 s
% 2.19/1.23 % (11830)Instructions burned: 3 (million)
% 2.19/1.23 % (11830)------------------------------
% 2.19/1.23 % (11830)------------------------------
% 2.19/1.23 % (11831)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_11553 for (2999ds/3Mi)
% 2.19/1.23 % (11829)Instruction limit reached!
% 2.19/1.23 % (11829)------------------------------
% 2.19/1.23 % (11829)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.19/1.23 % (11829)Termination reason: Unknown
% 2.19/1.23 % (11829)Termination phase: shuffling
% 2.19/1.23
% 2.19/1.23 % (11829)Memory used [KB]: 1918
% 2.19/1.23 % (11829)Time elapsed: 0.006 s
% 2.19/1.23 % (11829)Instructions burned: 18 (million)
% 2.19/1.23 % (11829)------------------------------
% 2.19/1.23 % (11829)------------------------------
% 2.19/1.23 % (11831)Instruction limit reached!
% 2.19/1.23 % (11831)------------------------------
% 2.19/1.23 % (11831)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.19/1.23 % (11831)Termination reason: Unknown
% 2.19/1.23 % (11831)Termination phase: shuffling
% 2.19/1.23
% 2.19/1.23 % (11831)Memory used [KB]: 1663
% 2.19/1.23 % (11831)Time elapsed: 0.002 s
% 2.19/1.23 % (11831)Instructions burned: 3 (million)
% 2.19/1.23 % (11831)------------------------------
% 2.19/1.23 % (11831)------------------------------
% 2.19/1.23 % (11832)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_11553 for (2999ds/7Mi)
% 2.19/1.23 % (11832)Instruction limit reached!
% 2.19/1.23 % (11832)------------------------------
% 2.19/1.23 % (11832)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.19/1.23 % (11832)Termination reason: Unknown
% 2.19/1.23 % (11832)Termination phase: shuffling
% 2.19/1.23
% 2.19/1.23 % (11832)Memory used [KB]: 1663
% 2.19/1.23 % (11832)Time elapsed: 0.003 s
% 2.19/1.23 % (11832)Instructions burned: 7 (million)
% 2.19/1.23 % (11832)------------------------------
% 2.19/1.23 % (11832)------------------------------
% 2.19/1.24 % (11833)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_11553 for (2999ds/3Mi)
% 2.19/1.24 % (11833)Instruction limit reached!
% 2.19/1.24 % (11833)------------------------------
% 2.19/1.24 % (11833)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.19/1.24 % (11833)Termination reason: Unknown
% 2.19/1.24 % (11833)Termination phase: shuffling
% 2.19/1.24
% 2.19/1.24 % (11833)Memory used [KB]: 1663
% 2.19/1.24 % (11833)Time elapsed: 0.002 s
% 2.19/1.24 % (11833)Instructions burned: 4 (million)
% 2.19/1.24 % (11833)------------------------------
% 2.19/1.24 % (11833)------------------------------
% 2.19/1.24 % (11834)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_11553 for (2999ds/4Mi)
% 2.19/1.24 % (11835)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_11553 for (2999ds/18Mi)
% 2.19/1.24 % (11834)Instruction limit reached!
% 2.19/1.24 % (11834)------------------------------
% 2.19/1.24 % (11834)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.19/1.24 % (11834)Termination reason: Unknown
% 2.19/1.24 % (11834)Termination phase: shuffling
% 2.19/1.24
% 2.19/1.24 % (11834)Memory used [KB]: 1663
% 2.19/1.24 % (11834)Time elapsed: 0.002 s
% 2.19/1.24 % (11834)Instructions burned: 4 (million)
% 2.19/1.24 % (11834)------------------------------
% 2.19/1.24 % (11834)------------------------------
% 2.19/1.24 % (11836)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_11553 for (2999ds/710Mi)
% 2.19/1.25 % (11835)Instruction limit reached!
% 2.19/1.25 % (11835)------------------------------
% 2.19/1.25 % (11835)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.19/1.25 % (11835)Termination reason: Unknown
% 2.19/1.25 % (11835)Termination phase: shuffling
% 2.19/1.25
% 2.19/1.25 % (11835)Memory used [KB]: 1918
% 2.19/1.25 % (11835)Time elapsed: 0.006 s
% 2.19/1.25 % (11835)Instructions burned: 19 (million)
% 2.19/1.25 % (11835)------------------------------
% 2.19/1.25 % (11835)------------------------------
% 2.19/1.25 % (11816)Instruction limit reached!
% 2.19/1.25 % (11816)------------------------------
% 2.19/1.25 % (11816)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.19/1.25 % (11816)Termination reason: Unknown
% 2.19/1.25 % (11816)Termination phase: Property scanning
% 2.19/1.25
% 2.19/1.25 % (11816)Memory used [KB]: 3070
% 2.19/1.25 % (11816)Time elapsed: 0.047 s
% 2.19/1.25 % (11816)Instructions burned: 183 (million)
% 2.19/1.25 % (11816)------------------------------
% 2.19/1.25 % (11816)------------------------------
% 2.19/1.25 % (11837)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_11553 for (2999ds/6Mi)
% 2.19/1.25 % (11837)Instruction limit reached!
% 2.19/1.25 % (11837)------------------------------
% 2.19/1.25 % (11837)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.19/1.25 % (11837)Termination reason: Unknown
% 2.19/1.25 % (11837)Termination phase: shuffling
% 2.19/1.25
% 2.19/1.25 % (11837)Memory used [KB]: 1663
% 2.19/1.25 % (11837)Time elapsed: 0.003 s
% 2.19/1.25 % (11837)Instructions burned: 8 (million)
% 2.19/1.25 % (11837)------------------------------
% 2.19/1.25 % (11837)------------------------------
% 2.19/1.25 % (11838)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_11553 for (2999ds/902Mi)
% 2.19/1.25 % (11839)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_11553 for (2999ds/21Mi)
% 2.19/1.26 % (11840)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_11553 for (2999ds/5Mi)
% 2.19/1.26 % (11840)Instruction limit reached!
% 2.19/1.26 % (11840)------------------------------
% 2.19/1.26 % (11840)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.19/1.26 % (11840)Termination reason: Unknown
% 2.19/1.26 % (11840)Termination phase: shuffling
% 2.19/1.26
% 2.19/1.26 % (11840)Memory used [KB]: 1663
% 2.19/1.26 % (11840)Time elapsed: 0.003 s
% 2.19/1.26 % (11840)Instructions burned: 7 (million)
% 2.19/1.26 % (11840)------------------------------
% 2.19/1.26 % (11840)------------------------------
% 2.19/1.26 % (11841)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_11553 for (2999ds/6Mi)
% 2.19/1.26 % (11841)Instruction limit reached!
% 2.19/1.26 % (11841)------------------------------
% 2.19/1.26 % (11841)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.19/1.26 % (11841)Termination reason: Unknown
% 2.19/1.26 % (11841)Termination phase: shuffling
% 2.19/1.26
% 2.19/1.26 % (11841)Memory used [KB]: 1663
% 2.19/1.26 % (11841)Time elapsed: 0.003 s
% 2.19/1.26 % (11841)Instructions burned: 8 (million)
% 2.19/1.26 % (11841)------------------------------
% 2.19/1.26 % (11841)------------------------------
% 2.19/1.26 % (11839)Instruction limit reached!
% 2.19/1.26 % (11839)------------------------------
% 2.19/1.26 % (11839)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.19/1.26 % (11839)Termination reason: Unknown
% 2.19/1.26 % (11839)Termination phase: shuffling
% 2.19/1.26
% 2.19/1.26 % (11839)Memory used [KB]: 1918
% 2.19/1.26 % (11839)Time elapsed: 0.007 s
% 2.19/1.26 % (11839)Instructions burned: 23 (million)
% 2.19/1.26 % (11839)------------------------------
% 2.19/1.26 % (11839)------------------------------
% 2.19/1.27 % (11842)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_11553 for (2999ds/377Mi)
% 2.19/1.27 % (11843)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_11553 for (2999ds/779Mi)
% 2.19/1.27 % (11844)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_11553 for (2999ds/19Mi)
% 2.19/1.28 % (11844)Instruction limit reached!
% 2.19/1.28 % (11844)------------------------------
% 2.19/1.28 % (11844)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.19/1.28 % (11844)Termination reason: Unknown
% 2.19/1.28 % (11844)Termination phase: shuffling
% 2.19/1.28
% 2.19/1.28 % (11844)Memory used [KB]: 1918
% 2.19/1.28 % (11844)Time elapsed: 0.006 s
% 2.19/1.28 % (11844)Instructions burned: 19 (million)
% 2.19/1.28 % (11844)------------------------------
% 2.19/1.28 % (11844)------------------------------
% 2.19/1.28 % (11821)Instruction limit reached!
% 2.19/1.28 % (11821)------------------------------
% 2.19/1.28 % (11821)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.19/1.28 % (11821)Termination reason: Unknown
% 2.19/1.28 % (11821)Termination phase: Saturation
% 2.19/1.28
% 2.19/1.28 % (11821)Memory used [KB]: 5884
% 2.19/1.28 % (11821)Time elapsed: 0.076 s
% 2.19/1.28 % (11821)Instructions burned: 277 (million)
% 2.19/1.28 % (11821)------------------------------
% 2.19/1.28 % (11821)------------------------------
% 2.19/1.28 % (11845)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_11553 for (2999ds/879Mi)
% 2.19/1.29 % (11846)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_11553 for (2999ds/17Mi)
% 2.19/1.29 % (11846)Instruction limit reached!
% 2.19/1.29 % (11846)------------------------------
% 2.19/1.29 % (11846)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.19/1.29 % (11846)Termination reason: Unknown
% 2.19/1.29 % (11846)Termination phase: shuffling
% 2.19/1.29
% 2.19/1.29 % (11846)Memory used [KB]: 2046
% 2.19/1.29 % (11846)Time elapsed: 0.006 s
% 2.19/1.29 % (11846)Instructions burned: 18 (million)
% 2.19/1.29 % (11846)------------------------------
% 2.19/1.29 % (11846)------------------------------
% 2.19/1.30 % (11847)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_11553 for (2998ds/3Mi)
% 2.19/1.30 % (11847)Instruction limit reached!
% 2.19/1.30 % (11847)------------------------------
% 2.19/1.30 % (11847)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.19/1.30 % (11847)Termination reason: Unknown
% 2.19/1.30 % (11847)Termination phase: shuffling
% 2.19/1.30
% 2.19/1.30 % (11847)Memory used [KB]: 1663
% 2.19/1.30 % (11847)Time elapsed: 0.002 s
% 2.19/1.30 % (11847)Instructions burned: 3 (million)
% 2.19/1.30 % (11847)------------------------------
% 2.19/1.30 % (11847)------------------------------
% 2.19/1.31 % (11848)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_11553 for (2998ds/30Mi)
% 2.19/1.32 % (11848)Instruction limit reached!
% 2.19/1.32 % (11848)------------------------------
% 2.19/1.32 % (11848)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.19/1.32 % (11848)Termination reason: Unknown
% 2.19/1.32 % (11848)Termination phase: shuffling
% 2.19/1.32
% 2.19/1.32 % (11848)Memory used [KB]: 2174
% 2.19/1.32 % (11848)Time elapsed: 0.009 s
% 2.19/1.32 % (11848)Instructions burned: 30 (million)
% 2.19/1.32 % (11848)------------------------------
% 2.19/1.32 % (11848)------------------------------
% 2.19/1.33 % (11849)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_11553 for (2998ds/127Mi)
% 2.19/1.36 % (11849)Instruction limit reached!
% 2.19/1.36 % (11849)------------------------------
% 2.19/1.36 % (11849)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.19/1.36 % (11849)Termination reason: Unknown
% 2.19/1.36 % (11849)Termination phase: Property scanning
% 2.19/1.36
% 2.19/1.36 % (11849)Memory used [KB]: 2942
% 2.19/1.36 % (11849)Time elapsed: 0.033 s
% 2.19/1.36 % (11849)Instructions burned: 128 (million)
% 2.19/1.36 % (11849)------------------------------
% 2.19/1.36 % (11849)------------------------------
% 2.19/1.37 % (11842)Instruction limit reached!
% 2.19/1.37 % (11842)------------------------------
% 2.19/1.37 % (11842)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.19/1.37 % (11842)Termination reason: Unknown
% 2.19/1.37 % (11842)Termination phase: Saturation
% 2.19/1.37
% 2.19/1.37 % (11842)Memory used [KB]: 9466
% 2.19/1.37 % (11842)Time elapsed: 0.099 s
% 2.19/1.37 % (11842)Instructions burned: 380 (million)
% 2.19/1.37 % (11842)------------------------------
% 2.19/1.37 % (11842)------------------------------
% 2.19/1.37 % (11850)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_11553 for (2998ds/100Mi)
% 2.19/1.37 % (11851)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_11553 for (2998ds/3Mi)
% 2.19/1.38 % (11851)Instruction limit reached!
% 2.19/1.38 % (11851)------------------------------
% 2.19/1.38 % (11851)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.19/1.38 % (11851)Termination reason: Unknown
% 2.19/1.38 % (11851)Termination phase: shuffling
% 2.19/1.38
% 2.19/1.38 % (11851)Memory used [KB]: 1663
% 2.19/1.38 % (11851)Time elapsed: 0.002 s
% 2.19/1.38 % (11851)Instructions burned: 3 (million)
% 2.19/1.38 % (11851)------------------------------
% 2.19/1.38 % (11851)------------------------------
% 2.19/1.39 % (11852)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_11553 for (2998ds/20Mi)
% 2.19/1.39 % (11852)Instruction limit reached!
% 2.19/1.39 % (11852)------------------------------
% 2.19/1.39 % (11852)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.19/1.39 % (11852)Termination reason: Unknown
% 2.19/1.39 % (11852)Termination phase: shuffling
% 2.19/1.39
% 2.19/1.39 % (11852)Memory used [KB]: 1918
% 2.19/1.39 % (11852)Time elapsed: 0.008 s
% 2.19/1.39 % (11852)Instructions burned: 22 (million)
% 2.19/1.39 % (11852)------------------------------
% 2.19/1.39 % (11852)------------------------------
% 2.19/1.39 % (11850)Instruction limit reached!
% 2.19/1.39 % (11850)------------------------------
% 2.19/1.39 % (11850)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.19/1.39 % (11850)Termination reason: Unknown
% 2.19/1.39 % (11850)Termination phase: Property scanning
% 2.19/1.39
% 2.19/1.39 % (11850)Memory used [KB]: 2942
% 2.19/1.39 % (11850)Time elapsed: 0.027 s
% 2.19/1.39 % (11850)Instructions burned: 101 (million)
% 2.19/1.39 % (11850)------------------------------
% 2.19/1.39 % (11850)------------------------------
% 2.19/1.40 % (11853)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_11553 for (2997ds/86Mi)
% 2.19/1.40 % (11854)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_11553 for (2997ds/107Mi)
% 2.19/1.42 % (11853)Instruction limit reached!
% 2.19/1.42 % (11853)------------------------------
% 2.19/1.42 % (11853)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.19/1.42 % (11853)Termination reason: Unknown
% 2.19/1.42 % (11853)Termination phase: Property scanning
% 2.19/1.42
% 2.19/1.42 % (11853)Memory used [KB]: 2814
% 2.19/1.42 % (11853)Time elapsed: 0.024 s
% 2.19/1.42 % (11853)Instructions burned: 88 (million)
% 2.19/1.42 % (11853)------------------------------
% 2.19/1.42 % (11853)------------------------------
% 2.19/1.43 % (11854)Instruction limit reached!
% 2.19/1.43 % (11854)------------------------------
% 2.19/1.43 % (11854)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.19/1.43 % (11854)Termination reason: Unknown
% 2.19/1.43 % (11854)Termination phase: SInE selection
% 2.19/1.43
% 2.19/1.43 % (11854)Memory used [KB]: 2814
% 2.19/1.43 % (11854)Time elapsed: 0.029 s
% 2.19/1.43 % (11854)Instructions burned: 110 (million)
% 2.19/1.43 % (11854)------------------------------
% 2.19/1.43 % (11854)------------------------------
% 2.19/1.43 % (11855)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_11553 for (2997ds/39Mi)
% 2.19/1.44 % (11856)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_11553 for (2997ds/448Mi)
% 2.19/1.44 % (11855)Instruction limit reached!
% 2.19/1.44 % (11855)------------------------------
% 2.19/1.44 % (11855)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.19/1.44 % (11855)Termination reason: Unknown
% 2.19/1.44 % (11855)Termination phase: shuffling
% 2.19/1.44
% 2.19/1.44 % (11855)Memory used [KB]: 2430
% 2.19/1.44 % (11855)Time elapsed: 0.012 s
% 2.19/1.44 % (11855)Instructions burned: 40 (million)
% 2.19/1.44 % (11855)------------------------------
% 2.19/1.44 % (11855)------------------------------
% 2.19/1.45 % (11857)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_11553 for (2997ds/46Mi)
% 2.19/1.46 % (11836)Instruction limit reached!
% 2.19/1.46 % (11836)------------------------------
% 2.19/1.46 % (11836)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.19/1.46 % (11836)Termination reason: Unknown
% 2.19/1.46 % (11836)Termination phase: Saturation
% 2.19/1.46
% 2.19/1.46 % (11836)Memory used [KB]: 11769
% 2.19/1.46 % (11836)Time elapsed: 0.216 s
% 2.19/1.46 % (11836)Instructions burned: 710 (million)
% 2.19/1.46 % (11836)------------------------------
% 2.19/1.46 % (11836)------------------------------
% 2.19/1.46 % (11843)Instruction limit reached!
% 2.19/1.46 % (11843)------------------------------
% 2.19/1.46 % (11843)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.19/1.46 % (11843)Termination reason: Unknown
% 2.19/1.46 % (11843)Termination phase: Saturation
% 2.19/1.46
% 2.19/1.46 % (11843)Memory used [KB]: 7803
% 2.19/1.46 % (11843)Time elapsed: 0.191 s
% 2.19/1.46 % (11843)Instructions burned: 779 (million)
% 2.19/1.46 % (11843)------------------------------
% 2.19/1.46 % (11843)------------------------------
% 2.19/1.47 % (11857)Instruction limit reached!
% 2.19/1.47 % (11857)------------------------------
% 2.19/1.47 % (11857)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.19/1.47 % (11857)Termination reason: Unknown
% 2.19/1.47 % (11857)Termination phase: shuffling
% 2.19/1.47
% 2.19/1.47 % (11857)Memory used [KB]: 2430
% 2.19/1.47 % (11857)Time elapsed: 0.014 s
% 2.19/1.47 % (11857)Instructions burned: 48 (million)
% 2.19/1.47 % (11857)------------------------------
% 2.19/1.47 % (11857)------------------------------
% 2.19/1.47 % (11858)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_11553 for (2997ds/98Mi)
% 2.19/1.47 % (11859)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_11553 for (2997ds/507Mi)
% 2.19/1.48 % (11827)Instruction limit reached!
% 2.19/1.48 % (11827)------------------------------
% 2.19/1.48 % (11827)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.19/1.48 % (11827)Termination reason: Unknown
% 2.19/1.48 % (11827)Termination phase: Saturation
% 2.19/1.48
% 2.19/1.48 % (11827)Memory used [KB]: 18038
% 2.19/1.48 % (11827)Time elapsed: 0.260 s
% 2.19/1.48 % (11827)Instructions burned: 1042 (million)
% 2.19/1.48 % (11827)------------------------------
% 2.19/1.48 % (11827)------------------------------
% 2.19/1.48 % (11860)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_11553 for (2997ds/149Mi)
% 2.19/1.49 % (11861)lrs+21_1:16_au=on:bd=off:piset=and:i=39:si=on:rtra=on_0 on DTF2THF_11553 for (2997ds/39Mi)
% 2.19/1.49 % (11858)Instruction limit reached!
% 2.19/1.49 % (11858)------------------------------
% 2.19/1.49 % (11858)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.19/1.49 % (11858)Termination reason: Unknown
% 2.19/1.49 % (11858)Termination phase: Property scanning
% 2.19/1.49
% 2.19/1.49 % (11858)Memory used [KB]: 2814
% 2.19/1.49 % (11858)Time elapsed: 0.027 s
% 2.19/1.49 % (11858)Instructions burned: 101 (million)
% 2.19/1.49 % (11858)------------------------------
% 2.19/1.49 % (11858)------------------------------
% 2.19/1.50 % (11861)Instruction limit reached!
% 2.19/1.50 % (11861)------------------------------
% 2.19/1.50 % (11861)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.19/1.50 % (11861)Termination reason: Unknown
% 2.19/1.50 % (11861)Termination phase: shuffling
% 2.19/1.50
% 2.19/1.50 % (11861)Memory used [KB]: 2430
% 2.19/1.50 % (11861)Time elapsed: 0.012 s
% 2.19/1.50 % (11861)Instructions burned: 40 (million)
% 2.19/1.50 % (11861)------------------------------
% 2.19/1.50 % (11861)------------------------------
% 2.19/1.50 % (11862)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_11553 for (2997ds/6Mi)
% 2.19/1.51 % (11862)Instruction limit reached!
% 2.19/1.51 % (11862)------------------------------
% 2.19/1.51 % (11862)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.19/1.51 % (11862)Termination reason: Unknown
% 2.19/1.51 % (11862)Termination phase: shuffling
% 2.19/1.51
% 2.19/1.51 % (11862)Memory used [KB]: 1663
% 2.19/1.51 % (11862)Time elapsed: 0.003 s
% 2.19/1.51 % (11862)Instructions burned: 6 (million)
% 2.19/1.51 % (11862)------------------------------
% 2.19/1.51 % (11862)------------------------------
% 2.19/1.51 % (11863)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_11553 for (2996ds/1087Mi)
% 2.19/1.51 % (11860)Instruction limit reached!
% 2.19/1.51 % (11860)------------------------------
% 2.19/1.51 % (11860)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.19/1.51 % (11860)Termination reason: Unknown
% 2.19/1.51 % (11860)Termination phase: Function definition elimination
% 2.19/1.51
% 2.19/1.51 % (11860)Memory used [KB]: 2942
% 2.19/1.51 % (11860)Time elapsed: 0.039 s
% 2.19/1.51 % (11860)Instructions burned: 149 (million)
% 2.19/1.51 % (11860)------------------------------
% 2.19/1.51 % (11860)------------------------------
% 2.19/1.52 % (11864)dis+10_1:1_aac=none:cs=on:i=47:si=on:rtra=on_0 on DTF2THF_11553 for (2996ds/47Mi)
% 2.19/1.52 % (11838)Instruction limit reached!
% 2.19/1.52 % (11838)------------------------------
% 2.19/1.52 % (11838)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.19/1.52 % (11838)Termination reason: Unknown
% 2.19/1.52 % (11838)Termination phase: Saturation
% 2.19/1.52
% 2.19/1.52 % (11838)Memory used [KB]: 13048
% 2.19/1.52 % (11838)Time elapsed: 0.269 s
% 2.19/1.52 % (11838)Instructions burned: 903 (million)
% 2.19/1.52 % (11838)------------------------------
% 2.19/1.52 % (11838)------------------------------
% 2.19/1.52 % (11865)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_11553 for (2996ds/44Mi)
% 2.19/1.53 % (11864)Instruction limit reached!
% 2.19/1.53 % (11864)------------------------------
% 2.19/1.53 % (11864)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.19/1.53 % (11864)Termination reason: Unknown
% 2.19/1.53 % (11864)Termination phase: shuffling
% 2.19/1.53
% 2.19/1.53 % (11864)Memory used [KB]: 2558
% 2.19/1.53 % (11864)Time elapsed: 0.014 s
% 2.19/1.53 % (11864)Instructions burned: 47 (million)
% 2.19/1.53 % (11864)------------------------------
% 2.19/1.53 % (11864)------------------------------
% 2.19/1.53 % (11866)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_11553 for (2996ds/432Mi)
% 2.19/1.54 % (11845)Instruction limit reached!
% 2.19/1.54 % (11845)------------------------------
% 2.19/1.54 % (11845)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.19/1.54 % (11845)Termination reason: Unknown
% 2.19/1.54 % (11845)Termination phase: Saturation
% 2.19/1.54
% 2.19/1.54 % (11845)Memory used [KB]: 10618
% 2.19/1.54 % (11845)Time elapsed: 0.252 s
% 2.19/1.54 % (11845)Instructions burned: 881 (million)
% 2.19/1.54 % (11845)------------------------------
% 2.19/1.54 % (11845)------------------------------
% 2.19/1.54 % (11865)Instruction limit reached!
% 2.19/1.54 % (11865)------------------------------
% 2.19/1.54 % (11865)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.19/1.54 % (11865)Termination reason: Unknown
% 2.19/1.54 % (11865)Termination phase: shuffling
% 2.19/1.54
% 2.19/1.54 % (11865)Memory used [KB]: 2430
% 2.19/1.54 % (11865)Time elapsed: 0.013 s
% 2.19/1.54 % (11865)Instructions burned: 45 (million)
% 2.19/1.54 % (11865)------------------------------
% 2.19/1.54 % (11865)------------------------------
% 2.19/1.54 % (11867)lrs+10_7:1_cnfonf=lazy_gen:fe=off:nwc=5.0:plsq=on:i=62:si=on:rtra=on_0 on DTF2THF_11553 for (2996ds/62Mi)
% 2.19/1.55 % (11868)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_11553 for (2996ds/31Mi)
% 2.19/1.55 % (11869)dis+10_1:128_hud=1:i=96:si=on:rtra=on_0 on DTF2THF_11553 for (2996ds/96Mi)
% 2.19/1.55 % (11856)Instruction limit reached!
% 2.19/1.55 % (11856)------------------------------
% 2.19/1.55 % (11856)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.19/1.55 % (11856)Termination reason: Unknown
% 2.19/1.55 % (11856)Termination phase: Saturation
% 2.19/1.55
% 2.19/1.55 % (11856)Memory used [KB]: 7803
% 2.19/1.55 % (11856)Time elapsed: 0.112 s
% 2.19/1.55 % (11856)Instructions burned: 449 (million)
% 2.19/1.55 % (11856)------------------------------
% 2.19/1.55 % (11856)------------------------------
% 2.19/1.55 % (11867)Instruction limit reached!
% 2.19/1.55 % (11867)------------------------------
% 2.19/1.55 % (11867)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.19/1.55 % (11867)Termination reason: Unknown
% 2.19/1.55 % (11867)Termination phase: shuffling
% 2.19/1.55
% 2.19/1.55 % (11867)Memory used [KB]: 2686
% 2.19/1.55 % (11867)Time elapsed: 0.018 s
% 2.19/1.55 % (11867)Instructions burned: 62 (million)
% 2.19/1.55 % (11867)------------------------------
% 2.19/1.55 % (11867)------------------------------
% 2.19/1.56 % (11868)Instruction limit reached!
% 2.19/1.56 % (11868)------------------------------
% 2.19/1.56 % (11868)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.19/1.56 % (11868)Termination reason: Unknown
% 2.19/1.56 % (11868)Termination phase: shuffling
% 2.19/1.56
% 2.19/1.56 % (11868)Memory used [KB]: 2430
% 2.19/1.56 % (11868)Time elapsed: 0.011 s
% 2.19/1.56 % (11868)Instructions burned: 34 (million)
% 2.19/1.56 % (11868)------------------------------
% 2.19/1.56 % (11868)------------------------------
% 2.19/1.56 % (11870)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_11553 for (2996ds/453Mi)
% 2.19/1.56 % (11871)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_11553 for (2996ds/71Mi)
% 2.19/1.56 % (11872)dis+1002_1:1_bd=off:fsr=off:ntd=on:s2a=on:ss=axioms:i=83:si=on:rtra=on_0 on DTF2THF_11553 for (2996ds/83Mi)
% 2.19/1.57 % (11869)Instruction limit reached!
% 2.19/1.57 % (11869)------------------------------
% 2.19/1.57 % (11869)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.19/1.57 % (11869)Termination reason: Unknown
% 2.19/1.57 % (11869)Termination phase: Property scanning
% 2.19/1.57
% 2.19/1.57 % (11869)Memory used [KB]: 2814
% 2.19/1.57 % (11869)Time elapsed: 0.027 s
% 2.19/1.57 % (11869)Instructions burned: 98 (million)
% 2.19/1.57 % (11869)------------------------------
% 2.19/1.57 % (11869)------------------------------
% 2.19/1.58 % (11873)lrs+10_1:128_cnfonf=lazy_not_gen_be_off:hud=1:ins=1:i=11:si=on:rtra=on_0 on DTF2THF_11553 for (2996ds/11Mi)
% 2.19/1.58 % (11871)Instruction limit reached!
% 2.19/1.58 % (11871)------------------------------
% 2.19/1.58 % (11871)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.19/1.58 % (11871)Termination reason: Unknown
% 2.19/1.58 % (11871)Termination phase: Property scanning
% 2.19/1.58
% 2.19/1.58 % (11871)Memory used [KB]: 2814
% 2.19/1.58 % (11871)Time elapsed: 0.020 s
% 2.19/1.58 % (11871)Instructions burned: 72 (million)
% 2.19/1.58 % (11871)------------------------------
% 2.19/1.58 % (11871)------------------------------
% 2.19/1.58 % (11873)Instruction limit reached!
% 2.19/1.58 % (11873)------------------------------
% 2.19/1.58 % (11873)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.19/1.58 % (11873)Termination reason: Unknown
% 2.19/1.58 % (11873)Termination phase: shuffling
% 2.19/1.58
% 2.19/1.58 % (11873)Memory used [KB]: 1791
% 2.19/1.58 % (11873)Time elapsed: 0.004 s
% 2.19/1.58 % (11873)Instructions burned: 11 (million)
% 2.19/1.58 % (11873)------------------------------
% 2.19/1.58 % (11873)------------------------------
% 2.19/1.59 % (11872)Instruction limit reached!
% 2.19/1.59 % (11872)------------------------------
% 2.19/1.59 % (11872)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.19/1.59 % (11872)Termination reason: Unknown
% 2.19/1.59 % (11872)Termination phase: Property scanning
% 2.19/1.59
% 2.19/1.59 % (11872)Memory used [KB]: 2814
% 2.19/1.59 % (11872)Time elapsed: 0.023 s
% 2.19/1.59 % (11872)Instructions burned: 84 (million)
% 2.19/1.59 % (11872)------------------------------
% 2.19/1.59 % (11872)------------------------------
% 2.19/1.59 % (11874)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_11553 for (2995ds/41Mi)
% 2.19/1.59 % (11875)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_11553 for (2995ds/6Mi)
% 2.19/1.60 % (11875)Instruction limit reached!
% 2.19/1.60 % (11875)------------------------------
% 2.19/1.60 % (11875)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.19/1.60 % (11875)Termination reason: Unknown
% 2.19/1.60 % (11875)Termination phase: shuffling
% 2.19/1.60
% 2.19/1.60 % (11875)Memory used [KB]: 1663
% 2.19/1.60 % (11875)Time elapsed: 0.003 s
% 2.19/1.60 % (11875)Instructions burned: 7 (million)
% 2.19/1.60 % (11875)------------------------------
% 2.19/1.60 % (11875)------------------------------
% 2.19/1.60 % (11876)dis+10_1:1_avsq=on:cnfonf=off:e2e=on:nm=32:i=150:si=on:rtra=on_0 on DTF2THF_11553 for (2995ds/150Mi)
% 2.19/1.60 % (11874)Instruction limit reached!
% 2.19/1.60 % (11874)------------------------------
% 2.19/1.60 % (11874)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.19/1.60 % (11874)Termination reason: Unknown
% 2.19/1.60 % (11874)Termination phase: shuffling
% 2.19/1.60
% 2.19/1.60 % (11874)Memory used [KB]: 2430
% 2.19/1.60 % (11874)Time elapsed: 0.012 s
% 2.19/1.60 % (11874)Instructions burned: 41 (million)
% 2.19/1.60 % (11874)------------------------------
% 2.19/1.60 % (11874)------------------------------
% 2.19/1.60 % (11877)ott+1010_1:1_au=on:s2a=on:sd=7:ss=axioms:st=1.5:i=343:si=on:rtra=on_0 on DTF2THF_11553 for (2995ds/343Mi)
% 2.19/1.61 % (11859)Instruction limit reached!
% 2.19/1.61 % (11859)------------------------------
% 2.19/1.61 % (11859)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.19/1.61 % (11859)Termination reason: Unknown
% 2.19/1.61 % (11859)Termination phase: Saturation
% 2.19/1.61
% 2.19/1.61 % (11859)Memory used [KB]: 8315
% 2.19/1.61 % (11859)Time elapsed: 0.166 s
% 2.19/1.61 % (11859)Instructions burned: 509 (million)
% 2.19/1.61 % (11859)------------------------------
% 2.19/1.61 % (11859)------------------------------
% 2.19/1.61 % (11878)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_11553 for (2995ds/109Mi)
% 2.19/1.62 % (11879)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_11553 for (2995ds/253Mi)
% 2.19/1.64 % (11866)Instruction limit reached!
% 2.19/1.64 % (11866)------------------------------
% 2.19/1.64 % (11866)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.19/1.64 % (11866)Termination reason: Unknown
% 2.19/1.64 % (11866)Termination phase: Saturation
% 2.19/1.64
% 2.19/1.64 % (11866)Memory used [KB]: 3326
% 2.19/1.64 % (11866)Time elapsed: 0.110 s
% 2.19/1.64 % (11866)Instructions burned: 434 (million)
% 2.19/1.64 % (11866)------------------------------
% 2.19/1.64 % (11866)------------------------------
% 2.19/1.64 % (11876)Instruction limit reached!
% 2.19/1.64 % (11876)------------------------------
% 2.19/1.64 % (11876)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.19/1.64 % (11876)Termination reason: Unknown
% 2.19/1.64 % (11876)Termination phase: Preprocessing 3
% 2.19/1.64
% 2.19/1.64 % (11876)Memory used [KB]: 5245
% 2.19/1.64 % (11876)Time elapsed: 0.046 s
% 2.19/1.64 % (11876)Instructions burned: 151 (million)
% 2.19/1.64 % (11876)------------------------------
% 2.19/1.64 % (11876)------------------------------
% 2.19/1.64 % (11878)Instruction limit reached!
% 2.19/1.64 % (11878)------------------------------
% 2.19/1.64 % (11878)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.19/1.64 % (11878)Termination reason: Unknown
% 2.19/1.64 % (11878)Termination phase: SInE selection
% 2.19/1.64
% 2.19/1.64 % (11878)Memory used [KB]: 2814
% 2.19/1.64 % (11878)Time elapsed: 0.030 s
% 2.19/1.64 % (11878)Instructions burned: 110 (million)
% 2.19/1.64 % (11878)------------------------------
% 2.19/1.64 % (11878)------------------------------
% 2.19/1.65 % (11880)lrs+10_1:1_au=on:cnfonf=off:sd=1:sgt=16:ss=axioms:i=140:si=on:rtra=on_0 on DTF2THF_11553 for (2995ds/140Mi)
% 2.19/1.65 % (11881)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_11553 for (2995ds/614Mi)
% 2.19/1.65 % (11882)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_11553 for (2995ds/202Mi)
% 2.19/1.68 % (11870)Instruction limit reached!
% 2.19/1.68 % (11870)------------------------------
% 2.19/1.68 % (11870)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.19/1.68 % (11870)Termination reason: Unknown
% 2.19/1.68 % (11870)Termination phase: Saturation
% 2.19/1.68
% 2.19/1.68 % (11870)Memory used [KB]: 7803
% 2.19/1.68 % (11870)Time elapsed: 0.115 s
% 2.19/1.68 % (11870)Instructions burned: 456 (million)
% 2.19/1.68 % (11870)------------------------------
% 2.19/1.68 % (11870)------------------------------
% 2.19/1.68 % (11880)Instruction limit reached!
% 2.19/1.68 % (11880)------------------------------
% 2.19/1.68 % (11880)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.19/1.68 % (11880)Termination reason: Unknown
% 2.19/1.68 % (11880)Termination phase: Property scanning
% 2.19/1.68
% 2.19/1.68 % (11880)Memory used [KB]: 3070
% 2.19/1.68 % (11880)Time elapsed: 0.038 s
% 2.19/1.68 % (11880)Instructions burned: 141 (million)
% 2.19/1.68 % (11880)------------------------------
% 2.19/1.68 % (11880)------------------------------
% 2.19/1.69 % (11883)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_11553 for (2995ds/339Mi)
% 2.19/1.69 % (11877)Instruction limit reached!
% 2.19/1.69 % (11877)------------------------------
% 2.19/1.69 % (11877)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.19/1.69 % (11877)Termination reason: Unknown
% 2.19/1.69 % (11877)Termination phase: Saturation
% 2.19/1.69
% 2.19/1.69 % (11877)Memory used [KB]: 8315
% 2.19/1.69 % (11877)Time elapsed: 0.090 s
% 2.19/1.69 % (11877)Instructions burned: 343 (million)
% 2.19/1.69 % (11877)------------------------------
% 2.19/1.69 % (11877)------------------------------
% 2.19/1.69 % (11884)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_11553 for (2995ds/283Mi)
% 2.19/1.70 % (11879)Instruction limit reached!
% 2.19/1.70 % (11879)------------------------------
% 2.19/1.70 % (11879)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.19/1.70 % (11879)Termination reason: Unknown
% 2.19/1.70 % (11879)Termination phase: Property scanning
% 2.19/1.70
% 2.19/1.70 % (11879)Memory used [KB]: 5884
% 2.19/1.70 % (11879)Time elapsed: 0.074 s
% 2.19/1.70 % (11879)Instructions burned: 256 (million)
% 2.19/1.70 % (11879)------------------------------
% 2.19/1.70 % (11879)------------------------------
% 2.19/1.70 % (11885)dis+2_1:1_cha=on:hud=7:nwc=3.0:prag=on:tnu=2:i=134:si=on:rtra=on_0 on DTF2THF_11553 for (2994ds/134Mi)
% 2.19/1.71 % (11886)dis+1010_1:64_cnfonf=lazy_simp:fd=off:i=298:si=on:rtra=on_0 on DTF2THF_11553 for (2994ds/298Mi)
% 2.19/1.71 % (11882)Instruction limit reached!
% 2.19/1.71 % (11882)------------------------------
% 2.19/1.71 % (11882)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.19/1.71 % (11882)Termination reason: Unknown
% 2.19/1.71 % (11882)Termination phase: Property scanning
% 2.19/1.71
% 2.19/1.71 % (11882)Memory used [KB]: 5500
% 2.19/1.71 % (11882)Time elapsed: 0.059 s
% 2.19/1.71 % (11882)Instructions burned: 202 (million)
% 2.19/1.71 % (11882)------------------------------
% 2.19/1.71 % (11882)------------------------------
% 2.19/1.72 % (11887)lrs+2_1:8_erd=off:plsq=on:plsqr=32,1:sos=on:i=192:si=on:rtra=on_0 on DTF2THF_11553 for (2994ds/192Mi)
% 2.19/1.74 % (11885)Instruction limit reached!
% 2.19/1.74 % (11885)------------------------------
% 2.19/1.74 % (11885)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.19/1.74 % (11885)Termination reason: Unknown
% 2.19/1.74 % (11885)Termination phase: Property scanning
% 2.19/1.74
% 2.19/1.74 % (11885)Memory used [KB]: 2942
% 2.19/1.74 % (11885)Time elapsed: 0.036 s
% 2.19/1.74 % (11885)Instructions burned: 137 (million)
% 2.19/1.74 % (11885)------------------------------
% 2.19/1.74 % (11885)------------------------------
% 2.19/1.75 % (11888)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_11553 for (2994ds/258Mi)
% 2.19/1.77 % (11884)Instruction limit reached!
% 2.19/1.77 % (11884)------------------------------
% 2.19/1.77 % (11884)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.19/1.77 % (11884)Termination reason: Unknown
% 2.19/1.77 % (11884)Termination phase: Saturation
% 2.19/1.77
% 2.19/1.77 % (11884)Memory used [KB]: 7675
% 2.19/1.77 % (11884)Time elapsed: 0.073 s
% 2.19/1.77 % (11884)Instructions burned: 286 (million)
% 2.19/1.77 % (11884)------------------------------
% 2.19/1.77 % (11884)------------------------------
% 2.19/1.77 % (11887)Instruction limit reached!
% 2.19/1.77 % (11887)------------------------------
% 2.19/1.77 % (11887)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.19/1.77 % (11887)Termination reason: Unknown
% 2.19/1.77 % (11887)Termination phase: Saturation
% 2.19/1.77
% 2.19/1.77 % (11887)Memory used [KB]: 7547
% 2.19/1.77 % (11887)Time elapsed: 0.051 s
% 2.19/1.77 % (11887)Instructions burned: 194 (million)
% 2.19/1.77 % (11887)------------------------------
% 2.19/1.77 % (11887)------------------------------
% 2.19/1.78 % (11889)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_11553 for (2994ds/90Mi)
% 2.19/1.78 % (11883)Instruction limit reached!
% 2.19/1.78 % (11883)------------------------------
% 2.19/1.78 % (11883)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.19/1.78 % (11883)Termination reason: Unknown
% 2.19/1.78 % (11883)Termination phase: Saturation
% 2.19/1.78
% 2.19/1.78 % (11883)Memory used [KB]: 8187
% 2.19/1.78 % (11883)Time elapsed: 0.093 s
% 2.19/1.78 % (11883)Instructions burned: 342 (million)
% 2.19/1.78 % (11883)------------------------------
% 2.19/1.78 % (11883)------------------------------
% 2.19/1.78 % (11890)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_11553 for (2994ds/231Mi)
% 2.19/1.78 % (11886)Instruction limit reached!
% 2.19/1.78 % (11886)------------------------------
% 2.19/1.78 % (11886)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.19/1.78 % (11886)Termination reason: Unknown
% 2.19/1.78 % (11886)Termination phase: Saturation
% 2.19/1.78
% 2.19/1.78 % (11886)Memory used [KB]: 7675
% 2.19/1.78 % (11886)Time elapsed: 0.076 s
% 2.19/1.78 % (11886)Instructions burned: 300 (million)
% 2.19/1.78 % (11886)------------------------------
% 2.19/1.78 % (11886)------------------------------
% 2.19/1.78 % (11863)First to succeed.
% 2.19/1.79 % (11863)Refutation found. Thanks to Tanya!
% 2.19/1.79 % SZS status Theorem for DTF2THF_11553
% 2.19/1.79 % SZS output start Proof for DTF2THF_11553
% 2.19/1.79 thf(func_def_0, type, in: $i > $i > $o).
% 2.19/1.79 thf(func_def_1, type, exu: ($i > $o) > $o).
% 2.19/1.79 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 2.19/1.79 thf(func_def_8, type, powerset: $i > $i).
% 2.19/1.79 thf(func_def_10, type, setunion: $i > $i).
% 2.19/1.79 thf(func_def_19, type, descr: ($i > $o) > $i).
% 2.19/1.79 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 2.19/1.79 thf(func_def_26, type, prop2set: $o > $i).
% 2.19/1.79 thf(func_def_36, type, nonempty: $i > $o).
% 2.19/1.79 thf(func_def_69, type, set2prop: $i > $o).
% 2.19/1.79 thf(func_def_88, type, subset: $i > $i > $o).
% 2.19/1.79 thf(func_def_89, type, disjoint: $i > $i > $o).
% 2.19/1.79 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 2.19/1.79 thf(func_def_114, type, binunion: $i > $i > $i).
% 2.19/1.79 thf(func_def_122, type, binintersect: $i > $i > $i).
% 2.19/1.79 thf(func_def_135, type, regular: $i > $o).
% 2.19/1.79 thf(func_def_136, type, setminus: $i > $i > $i).
% 2.19/1.79 thf(func_def_147, type, symdiff: $i > $i > $i).
% 2.19/1.79 thf(func_def_153, type, iskpair: $i > $o).
% 2.19/1.79 thf(func_def_158, type, kpair: $i > $i > $i).
% 2.19/1.79 thf(func_def_160, type, cartprod: $i > $i > $i).
% 2.19/1.79 thf(func_def_177, type, singleton: $i > $o).
% 2.19/1.79 thf(func_def_179, type, ex1: $i > ($i > $o) > $o).
% 2.19/1.79 thf(func_def_184, type, atmost1p: $i > $o).
% 2.19/1.79 thf(func_def_185, type, atleast2p: $i > $o).
% 2.19/1.79 thf(func_def_186, type, atmost2p: $i > $o).
% 2.19/1.79 thf(func_def_187, type, upairsetp: $i > $o).
% 2.19/1.79 thf(func_def_191, type, kfst: $i > $i).
% 2.19/1.79 thf(func_def_203, type, ksnd: $i > $i).
% 2.19/1.79 thf(func_def_213, type, breln: $i > $i > $i > $o).
% 2.19/1.79 thf(func_def_214, type, dpsetconstr: $i > $i > ($i > $i > $o) > $i).
% 2.19/1.79 thf(func_def_222, type, func: $i > $i > $i > $o).
% 2.19/1.79 thf(func_def_223, type, funcSet: $i > $i > $i).
% 2.19/1.79 thf(func_def_226, type, ap: $i > $i > $i > $i > $i).
% 2.19/1.79 thf(func_def_232, type, lam: $i > $i > ($i > $i) > $i).
% 2.19/1.79 thf(func_def_259, type, if: $i > $o > $i > $i > $i).
% 2.19/1.79 thf(func_def_323, type, breln1: $i > $i > $o).
% 2.19/1.79 thf(func_def_325, type, breln1Set: $i > $i).
% 2.19/1.79 thf(func_def_327, type, transitive: $i > $i > $o).
% 2.19/1.79 thf(func_def_328, type, antisymmetric: $i > $i > $o).
% 2.19/1.79 thf(func_def_329, type, reflexive: $i > $i > $o).
% 2.19/1.79 thf(func_def_330, type, refltransitive: $i > $i > $o).
% 2.19/1.79 thf(func_def_331, type, refllinearorder: $i > $i > $o).
% 2.19/1.79 thf(func_def_332, type, reflwellordering: $i > $i > $o).
% 2.19/1.79 thf(func_def_338, type, breln1invset: $i > $i > $i).
% 2.19/1.79 thf(func_def_342, type, breln1compset: $i > $i > $i > $i).
% 2.19/1.79 thf(func_def_360, type, image1: $i > ($i > $i) > $i).
% 2.19/1.79 thf(func_def_364, type, injective: $i > $i > $i > $o).
% 2.19/1.79 thf(func_def_365, type, injFuncSet: $i > $i > $i).
% 2.19/1.79 thf(func_def_380, type, sP0: $i > $i > $o).
% 2.19/1.79 thf(func_def_382, type, sP2: $i > $i > $i > $o).
% 2.19/1.79 thf(func_def_383, type, sP3: $i > $i > $i > $o).
% 2.19/1.79 thf(func_def_384, type, sP4: ($i > $o) > $o).
% 2.19/1.79 thf(func_def_385, type, sP5: ($i > $o) > $i > $i > $o).
% 2.19/1.79 thf(func_def_386, type, sP6: ($i > $o) > $i > $o).
% 2.19/1.79 thf(func_def_388, type, sP8: $i > $i > $i > $o).
% 2.19/1.79 thf(func_def_389, type, sP9: $i > $i > $o > $o).
% 2.19/1.79 thf(func_def_391, type, sP11: ($i > $o) > $i > $o).
% 2.19/1.79 thf(func_def_393, type, sP13: ($i > $o) > $i > $o).
% 2.19/1.79 thf(func_def_394, type, sP14: $i > $i > ($i > $o) > $o).
% 2.19/1.79 thf(func_def_395, type, sP15: $i > ($i > $o) > $o).
% 2.19/1.79 thf(func_def_396, type, sP16: $i > $i > ($i > $o) > $i > $o).
% 2.19/1.79 thf(func_def_397, type, sP17: $i > ($i > $o) > $o).
% 2.19/1.79 thf(func_def_399, type, sP19: $o > $i > $i > $o).
% 2.19/1.79 thf(func_def_401, type, sP21: ($i > $i) > $i > $i > $o).
% 2.19/1.79 thf(func_def_403, type, sP23: $i > $i > $i > $i > $o).
% 2.19/1.79 thf(func_def_404, type, sP24: $i > $i > $i > $o).
% 2.19/1.79 thf(func_def_405, type, sP25: $i > $i > $i > $o).
% 2.19/1.79 thf(func_def_406, type, sP26: $i > $i > $i > $o).
% 2.19/1.79 thf(func_def_407, type, sP27: $i > $i > $o).
% 2.19/1.79 thf(func_def_409, type, sP29: ($i > $o) > $o).
% 2.19/1.79 thf(func_def_410, type, sP30: $o > $i > $i > $i > $o).
% 2.19/1.79 thf(func_def_411, type, sP31: $i > $i > $i > $i > $o).
% 2.19/1.79 thf(func_def_414, type, sP34: $i > $i > $i > $o).
% 2.19/1.79 thf(func_def_416, type, sP36: $i > $i > $i > $o).
% 2.19/1.79 thf(func_def_417, type, sP37: ($i > $i > $o) > $i > $i > $o).
% 2.19/1.79 thf(func_def_418, type, sP38: ($i > $i > $o) > $i > $o).
% 2.19/1.79 thf(func_def_420, type, sP40: ($i > $o) > $i > $o).
% 2.19/1.79 thf(func_def_421, type, sP41: ($i > $o) > $i > $o).
% 2.19/1.79 thf(func_def_422, type, sP42: $i > $i > $o).
% 2.19/1.79 thf(func_def_425, type, sP45: ($i > $o) > $i > $o).
% 2.19/1.79 thf(func_def_429, type, sP49: $i > $i > $o > $o).
% 2.19/1.79 thf(func_def_430, type, sP50: $i > $i > $i > $o).
% 2.19/1.79 thf(func_def_431, type, sP51: $i > $i > $i > $o).
% 2.19/1.79 thf(func_def_433, type, sP53: $i > ($i > $o) > $o).
% 2.19/1.79 thf(func_def_436, type, sP56: $i > ($i > $o) > $o).
% 2.19/1.79 thf(func_def_437, type, sP57: $i > $i > ($i > $o) > $o).
% 2.19/1.79 thf(func_def_439, type, sP59: $i > $i > $i > $i > $o).
% 2.19/1.79 thf(func_def_440, type, sP60: $i > $i > $i > $o).
% 2.19/1.79 thf(func_def_441, type, sP61: ($i > $o) > $o).
% 2.19/1.79 thf(func_def_442, type, sP62: $i > $i > $i > $i > $o).
% 2.19/1.79 thf(func_def_443, type, sP63: $i > $i > $i > $i > $i > $o).
% 2.19/1.79 thf(func_def_445, type, sP65: $i > $o > $i > $i > $i > $i > $o).
% 2.19/1.79 thf(func_def_446, type, sP66: $o > $i > $i > $i > $o).
% 2.19/1.79 thf(func_def_447, type, sP67: $i > $o > $i > $i > $o).
% 2.19/1.79 thf(func_def_448, type, sP68: ($i > $o) > ($i > $o) > $i > $i > $o).
% 2.19/1.79 thf(func_def_449, type, sP69: $i > $o).
% 2.19/1.79 thf(func_def_450, type, sP70: $i > $o).
% 2.19/1.79 thf(func_def_451, type, sP71: ($i > $o) > $o).
% 2.19/1.79 thf(func_def_453, type, sP73: ($i > $o) > $i > $o).
% 2.19/1.79 thf(func_def_454, type, sP74: $i > ($i > $o) > $o).
% 2.19/1.79 thf(func_def_455, type, sP75: $i > $i > $i > $o).
% 2.19/1.79 thf(func_def_456, type, sP76: $i > $i > $o).
% 2.19/1.79 thf(func_def_457, type, sP77: $i > $i > $o).
% 2.19/1.79 thf(func_def_458, type, sP78: $i > $i > $o).
% 2.19/1.79 thf(func_def_459, type, sP79: $i > $i > $o).
% 2.19/1.79 thf(func_def_460, type, sP80: $i > $o).
% 2.19/1.79 thf(func_def_461, type, sP81: ($i > $o) > $o).
% 2.19/1.79 thf(func_def_462, type, sP82: ($i > $o) > $i > $o).
% 2.19/1.79 thf(func_def_464, type, sP84: $i > $o).
% 2.19/1.79 thf(func_def_465, type, sP85: $i > $i > ($i > $i > $o) > $o).
% 2.19/1.79 thf(func_def_466, type, sP86: $i > ($i > $i > $o) > $i > $o).
% 2.19/1.79 thf(func_def_467, type, sP87: $i > $i > $i > $o).
% 2.19/1.79 thf(func_def_469, type, sP89: $i > $i > $o > $o).
% 2.19/1.79 thf(func_def_470, type, sP90: $i > $i > $i > $o).
% 2.19/1.79 thf(func_def_471, type, sP91: $i > $i > $i > $o).
% 2.19/1.79 thf(func_def_472, type, sP92: ($i > $i) > $i > $i > $o).
% 2.19/1.79 thf(func_def_475, type, sP95: $i > $i > ($i > $i) > $o).
% 2.19/1.79 thf(func_def_477, type, sP97: ($i > $o) > $i > $o).
% 2.19/1.79 thf(func_def_478, type, sP98: $i > $i > $i > ($i > $o) > $o).
% 2.19/1.79 thf(func_def_479, type, sP99: $i > $i > $i > $i > $o).
% 2.19/1.79 thf(func_def_480, type, sP100: $i > $i > $i > $i > $o).
% 2.19/1.79 thf(func_def_481, type, sP101: $i > $i > ($i > $i) > $o).
% 2.19/1.79 thf(func_def_489, type, sK109: $i > $i > $i).
% 2.19/1.79 thf(func_def_490, type, sK110: $i > $i > $i).
% 2.19/1.79 thf(func_def_491, type, sK111: $i > $i > $o).
% 2.19/1.79 thf(func_def_521, type, sK141: $i > $i > $i > $i).
% 2.19/1.79 thf(func_def_522, type, sK142: $i > $i > $i > $i).
% 2.19/1.79 thf(func_def_523, type, sK143: $i > $i > $i > $i).
% 2.19/1.79 thf(func_def_524, type, sK144: $i > $i > $i > $i).
% 2.19/1.79 thf(func_def_535, type, sK155: ($i > $o) > $i).
% 2.19/1.79 thf(func_def_536, type, sK156: $i > ($i > $o) > $i).
% 2.19/1.79 thf(func_def_537, type, sK157: $i > $o).
% 2.19/1.79 thf(func_def_541, type, sK161: ($i > $o) > $i > $i > $i).
% 2.19/1.79 thf(func_def_542, type, sK162: $i > $o).
% 2.19/1.79 thf(func_def_566, type, sK186: $i > $o).
% 2.19/1.79 thf(func_def_568, type, sK188: $i > ($i > $o) > $i).
% 2.19/1.79 thf(func_def_569, type, sK189: ($i > $o) > $i > $i).
% 2.19/1.79 thf(func_def_585, type, sK205: $i > $i > $i > $i).
% 2.19/1.79 thf(func_def_586, type, sK206: $i > $i > $i > $i > $i).
% 2.19/1.79 thf(func_def_614, type, sK234: $i > $i > $o > $i).
% 2.19/1.79 thf(func_def_620, type, sK240: $i > $i).
% 2.19/1.79 thf(func_def_621, type, sK241: ($i > $i) > $i > $i > $i).
% 2.19/1.79 thf(func_def_640, type, sK260: $i > $o).
% 2.19/1.79 thf(func_def_642, type, sK262: $i > ($i > $o) > $i).
% 2.19/1.79 thf(func_def_643, type, sK263: ($i > $o) > $i > $i).
% 2.19/1.79 thf(func_def_676, type, sK296: $i > $i > ($i > $o) > $i).
% 2.19/1.79 thf(func_def_677, type, sK297: $i > $i > ($i > $o) > $i).
% 2.19/1.79 thf(func_def_678, type, sK298: ($i > $o) > $i > $i).
% 2.19/1.79 thf(func_def_681, type, sK301: $i > $o).
% 2.19/1.79 thf(func_def_691, type, sK311: $i > $i > ($i > $o) > $i > $i).
% 2.19/1.79 thf(func_def_692, type, sK312: $i > $i > ($i > $o) > $i > $i).
% 2.19/1.79 thf(func_def_693, type, sK313: $i > ($i > $o) > $i).
% 2.19/1.79 thf(func_def_697, type, sK317: $i > $o).
% 2.19/1.79 thf(func_def_714, type, sK334: $i > $o).
% 2.19/1.79 thf(func_def_715, type, sK335: ($i > $o) > $i > $i).
% 2.19/1.79 thf(func_def_716, type, sK336: $i > ($i > $o) > $i).
% 2.19/1.79 thf(func_def_737, type, sK357: $i > $i).
% 2.19/1.79 thf(func_def_738, type, sK358: $i > $i).
% 2.19/1.79 thf(func_def_739, type, sK359: ($i > $i) > $i > $i).
% 2.19/1.79 thf(func_def_740, type, sK360: ($i > $i) > $i > $i > $i).
% 2.19/1.79 thf(func_def_741, type, sK361: $i > $i).
% 2.19/1.79 thf(func_def_743, type, sK363: $i > $i > $i > $i).
% 2.19/1.79 thf(func_def_744, type, sK364: $i > $i > $i > $i > $i).
% 2.19/1.79 thf(func_def_745, type, sK365: $i > $i > $i > $i > $i).
% 2.19/1.79 thf(func_def_746, type, sK366: $i > $i > $i > $i > $i).
% 2.19/1.79 thf(func_def_772, type, sK392: $i > $i > $i > $i).
% 2.19/1.79 thf(func_def_773, type, sK393: $i > $i > $i > $i).
% 2.19/1.79 thf(func_def_786, type, sK406: $i > $i > $i).
% 2.19/1.79 thf(func_def_792, type, sK412: $i > $i > $i).
% 2.19/1.79 thf(func_def_800, type, sK420: ($i > $o) > $i).
% 2.19/1.79 thf(func_def_801, type, sK421: ($i > $o) > $i).
% 2.19/1.79 thf(func_def_802, type, sK422: $i > $o).
% 2.19/1.79 thf(func_def_804, type, sK424: $i > $i > $i > $i > $o).
% 2.19/1.79 thf(func_def_832, type, sK452: $i > $i > $i).
% 2.19/1.79 thf(func_def_837, type, sK457: $i > $o).
% 2.19/1.79 thf(func_def_840, type, sK460: $i > $i).
% 2.19/1.79 thf(func_def_842, type, sK462: $i > ($i > $i) > $i > $i).
% 2.19/1.79 thf(func_def_856, type, sK476: $i > $o).
% 2.19/1.79 thf(func_def_859, type, sK479: $i > $i > $i > $i).
% 2.19/1.79 thf(func_def_860, type, sK480: $i > $i > $i > $i).
% 2.19/1.79 thf(func_def_868, type, sK488: $i > $i > $o).
% 2.19/1.79 thf(func_def_869, type, sK489: ($i > $i > $o) > $i > $i).
% 2.19/1.79 thf(func_def_870, type, sK490: $i > ($i > $i > $o) > $i > $i).
% 2.19/1.79 thf(func_def_871, type, sK491: ($i > $i > $o) > $i > $i).
% 2.19/1.79 thf(func_def_872, type, sK492: ($i > $i > $o) > $i > $i > $i).
% 2.19/1.79 thf(func_def_873, type, sK493: ($i > $o) > $i > $i).
% 2.19/1.79 thf(func_def_874, type, sK494: ($i > $o) > $i > $i).
% 2.19/1.79 thf(func_def_875, type, sK495: ($i > $o) > $i > $i).
% 2.19/1.79 thf(func_def_877, type, sK497: $i > $o).
% 2.19/1.79 thf(func_def_884, type, sK504: $i > $i > $i).
% 2.19/1.79 thf(func_def_887, type, sK507: $i > $i).
% 2.19/1.79 thf(func_def_891, type, sK511: $i > $i > $o).
% 2.19/1.79 thf(func_def_896, type, sK516: $i > $o).
% 2.19/1.79 thf(func_def_900, type, sK520: $i > $i > $i).
% 2.19/1.79 thf(func_def_904, type, sK524: $i > $o).
% 2.19/1.79 thf(func_def_905, type, sK525: ($i > $o) > $i > $i).
% 2.19/1.79 thf(func_def_906, type, sK526: ($i > $o) > $i > $i).
% 2.19/1.79 thf(func_def_921, type, sK541: $i > $o).
% 2.19/1.79 thf(func_def_922, type, sK542: ($i > $o) > $i > $i).
% 2.19/1.79 thf(func_def_949, type, sK569: $i > $i).
% 2.19/1.79 thf(func_def_952, type, sK572: $i > $i > ($i > $i) > $i).
% 2.19/1.79 thf(func_def_957, type, sK577: $i > $o).
% 2.19/1.79 thf(func_def_988, type, sK608: $i > $i > $i > $o).
% 2.19/1.79 thf(func_def_995, type, sK615: $i > $i > $i > $i).
% 2.19/1.79 thf(func_def_996, type, sK616: $i > $i > $i > $i > $i).
% 2.19/1.79 thf(func_def_1000, type, sK620: $i > $o).
% 2.19/1.79 thf(func_def_1005, type, sK625: $i > $i > $o).
% 2.19/1.79 thf(func_def_1016, type, sK636: $i > $i).
% 2.19/1.79 thf(func_def_1017, type, sK637: $i > $i > $o).
% 2.19/1.79 thf(func_def_1029, type, sK649: $i > $i > $i).
% 2.19/1.79 thf(func_def_1030, type, sK650: $i > $o).
% 2.19/1.79 thf(func_def_1031, type, sK651: ($i > $o) > $i).
% 2.19/1.79 thf(func_def_1041, type, sK661: $i > $o).
% 2.19/1.79 thf(func_def_1042, type, sK662: ($i > $o) > $i > $i).
% 2.19/1.79 thf(func_def_1043, type, sK663: $i > ($i > $o) > $i).
% 2.19/1.79 thf(func_def_1044, type, sK664: $i > $o).
% 2.19/1.79 thf(func_def_1062, type, sK682: $i > $i).
% 2.19/1.79 thf(func_def_1065, type, sK685: $i > $i > ($i > $i) > $i).
% 2.19/1.79 thf(func_def_1066, type, sK686: $i > $i > ($i > $o) > $i).
% 2.19/1.79 thf(func_def_1067, type, sK687: $i > $i > ($i > $o) > $i).
% 2.19/1.79 thf(func_def_1068, type, sK688: $i > ($i > $o) > $i).
% 2.19/1.79 thf(func_def_1071, type, sK691: $i > $o).
% 2.19/1.79 thf(func_def_1074, type, sK694: $i > $i > $i).
% 2.19/1.79 thf(func_def_1088, type, sK708: $i > $i).
% 2.19/1.79 thf(func_def_1092, type, sK712: $i > $i > $i > $i > $i).
% 2.19/1.79 thf(func_def_1101, type, sK721: $i > $i > $i > $i).
% 2.19/1.79 thf(func_def_1102, type, sK722: $i > $i > $i > $i).
% 2.19/1.79 thf(func_def_1113, type, sK733: $i > ($i > $o) > $i).
% 2.19/1.79 thf(func_def_1114, type, sK734: ($i > $o) > $i).
% 2.19/1.79 thf(func_def_1115, type, sK735: $i > $o).
% 2.19/1.79 thf(func_def_1116, type, sK736: $i > $i > $i > $i > $i).
% 2.19/1.79 thf(func_def_1117, type, sK737: $i > $i > $i > $i > $i).
% 2.19/1.79 thf(func_def_1122, type, sK742: $i > $i > $i > $i > $i > $i).
% 2.19/1.79 thf(func_def_1150, type, sK770: $i > $i > $i).
% 2.19/1.79 thf(func_def_1160, type, sK780: $i > $i > $o).
% 2.19/1.79 thf(func_def_1173, type, sK793: $i > $o > $i > $i > $i > $i > $i).
% 2.19/1.79 thf(func_def_1209, type, sK829: $i > $i > $o).
% 2.19/1.79 thf(func_def_1231, type, sK851: $i > $o).
% 2.19/1.79 thf(func_def_1238, type, sK858: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.19/1.79 thf(func_def_1239, type, sK859: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.19/1.79 thf(func_def_1242, type, sK862: $i > $o).
% 2.19/1.79 thf(func_def_1243, type, sK863: $i > $o).
% 2.19/1.79 thf(func_def_1244, type, sK864: $i > $i).
% 2.19/1.79 thf(func_def_1245, type, sK865: $i > $i).
% 2.19/1.79 thf(func_def_1247, type, sK867: $i > $o).
% 2.19/1.79 thf(func_def_1251, type, sK871: ($i > $o) > $i).
% 2.19/1.79 thf(func_def_1252, type, sK872: $i > ($i > $o) > $i).
% 2.19/1.79 thf(func_def_1253, type, sK873: $i > $o).
% 2.19/1.79 thf(func_def_1284, type, sK904: $i > $i).
% 2.19/1.79 thf(func_def_1287, type, sK907: $i > $i > ($i > $i) > $i).
% 2.19/1.79 thf(func_def_1293, type, sK913: $i > $i).
% 2.19/1.79 thf(func_def_1304, type, sK924: $i > ($i > $o) > $i).
% 2.19/1.79 thf(func_def_1305, type, sK925: $i > ($i > $o) > $i).
% 2.19/1.79 thf(func_def_1306, type, sK926: ($i > $o) > $i > $i).
% 2.19/1.79 thf(func_def_1308, type, sK928: $i > $o).
% 2.19/1.79 thf(func_def_1317, type, sK937: $i > $o).
% 2.19/1.79 thf(func_def_1318, type, sK938: $i > $i).
% 2.19/1.79 thf(func_def_1319, type, sK939: $i > $i).
% 2.19/1.79 thf(func_def_1320, type, sK940: $i > $i).
% 2.19/1.79 thf(func_def_1321, type, sK941: $i > $i).
% 2.19/1.79 thf(func_def_1322, type, sK942: $i > $i > $i).
% 2.19/1.79 thf(func_def_1323, type, sK943: $i > $i > $i).
% 2.19/1.79 thf(func_def_1324, type, sK944: $i > $i > $i).
% 2.19/1.79 thf(func_def_1325, type, sK945: $i > $i > $i).
% 2.19/1.79 thf(func_def_1326, type, sK946: $i > $i > $i).
% 2.19/1.79 thf(func_def_1327, type, sK947: $i > $i > $i > $i).
% 2.19/1.79 thf(func_def_1328, type, sK948: $i > $i > $i).
% 2.19/1.79 thf(func_def_1329, type, sK949: $i > $i > $i).
% 2.19/1.79 thf(func_def_1330, type, sK950: $i > $i > $i).
% 2.19/1.79 thf(func_def_1331, type, sK951: $i > $i > $i).
% 2.19/1.79 thf(func_def_1332, type, sK952: $i > $i > $i).
% 2.19/1.79 thf(func_def_1333, type, sK953: $i > $i > $i > $i).
% 2.19/1.79 thf(func_def_1334, type, sK954: $i > $i > $i > $i).
% 2.19/1.79 thf(func_def_1335, type, sK955: $i > $i).
% 2.19/1.79 thf(func_def_1337, type, sK957: $i > ($i > $o) > $i).
% 2.19/1.79 thf(func_def_1338, type, sK958: ($i > $o) > $i).
% 2.19/1.79 thf(func_def_1339, type, sK959: $i > $o).
% 2.19/1.79 thf(func_def_1341, type, sK961: $i > $o).
% 2.19/1.79 thf(func_def_1342, type, sK962: ($i > $o) > $i > $i).
% 2.19/1.79 thf(func_def_1343, type, sK963: ($i > $o) > $i > $i).
% 2.19/1.79 thf(func_def_1350, type, sK970: $i > $i > $i).
% 2.19/1.79 thf(func_def_1351, type, sK971: $i > $i).
% 2.19/1.79 thf(func_def_1354, type, sK974: $i > $i > ($i > $i > $o) > $i).
% 2.19/1.79 thf(func_def_1355, type, sK975: $i > ($i > $i > $o) > $i > $i).
% 2.19/1.79 thf(func_def_1356, type, sK976: $i > $i > ($i > $i > $o) > $i).
% 2.19/1.79 thf(func_def_1357, type, sK977: $i > $i > ($i > $i > $o) > $i).
% 2.19/1.79 thf(func_def_1358, type, sK978: $i > $i > $o).
% 2.19/1.79 thf(func_def_1382, type, sK1002: $i > $o).
% 2.19/1.79 thf(func_def_1383, type, sK1003: $i > $i > $i > $i).
% 2.19/1.79 thf(func_def_1387, type, sK1007: $i > $o).
% 2.19/1.79 thf(func_def_1388, type, sK1008: $i > $o).
% 2.19/1.79 thf(func_def_1389, type, sK1009: ($i > $o) > ($i > $o) > $i).
% 2.19/1.79 thf(func_def_1390, type, sK1010: ($i > $o) > ($i > $o) > $i).
% 2.19/1.79 thf(func_def_1401, type, sK1021: $i > $i > $i > $i).
% 2.19/1.79 thf(func_def_1402, type, sK1022: $i > $i > $i > $o).
% 2.19/1.79 thf(func_def_1406, type, sK1026: $i > $i > $i > $i).
% 2.19/1.79 thf(func_def_1410, type, sK1030: $i > $i).
% 2.19/1.79 thf(func_def_1416, type, sK1036: $i > $i > $o).
% 2.19/1.79 thf(func_def_1421, type, sK1041: $i > $i).
% 2.19/1.79 thf(func_def_1423, type, sK1043: ($i > $i) > $i > $i > $i).
% 2.19/1.79 thf(func_def_1433, type, sK1053: $i > $o).
% 2.19/1.79 thf(func_def_1434, type, sK1054: $i > $o).
% 2.19/1.79 thf(func_def_1435, type, sK1055: ($i > $o) > ($i > $o) > $i).
% 2.19/1.79 thf(func_def_1436, type, sK1056: ($i > $o) > ($i > $o) > $i).
% 2.19/1.79 thf(func_def_1444, type, sK1064: $i > $i > $o).
% 2.19/1.79 thf(func_def_1450, type, sK1070: $i > $o).
% 2.19/1.79 thf(func_def_1470, type, sK1090: $i > $i).
% 2.19/1.79 thf(func_def_1473, type, sK1093: $i > $i > ($i > $i) > $i).
% 2.19/1.79 thf(func_def_1474, type, sK1094: $i > $i > ($i > $i) > $i).
% 2.19/1.79 thf(func_def_1478, type, sK1098: $i > $i > $i > ($i > $o) > $i).
% 2.19/1.79 thf(func_def_1479, type, sK1099: $i > $i > $i > ($i > $o) > $i).
% 2.19/1.79 thf(func_def_1480, type, sK1100: ($i > $o) > $i > $i).
% 2.19/1.79 thf(func_def_1484, type, sK1104: $i > $o).
% 2.19/1.79 thf(func_def_1489, type, sK1109: $i > $i > $i > $i > $i).
% 2.19/1.79 thf(func_def_1490, type, sK1110: $i > $i > $i > $i > $i).
% 2.19/1.79 thf(func_def_1491, type, sK1111: $i > $i > $i > $i > $i).
% 2.19/1.79 thf(func_def_1492, type, sK1112: $i > $i > $i > $i > $i).
% 2.19/1.79 thf(func_def_1504, type, sK1124: $i > $o).
% 2.19/1.79 thf(func_def_1508, type, sK1128: $i > $i).
% 2.19/1.79 thf(func_def_1509, type, sK1129: ($i > $i) > $i > $i > $i).
% 2.19/1.79 thf(func_def_1510, type, sK1130: $i > $i > ($i > $i) > $i).
% 2.19/1.79 thf(func_def_1516, type, sK1136: $i > $o).
% 2.19/1.79 thf(func_def_1525, type, ph1145: !>[X0: $tType]:(X0)).
% 2.19/1.79 thf(f9357,plain,(
% 2.19/1.79 $false),
% 2.19/1.79 inference(subsumption_resolution,[status(thm)],[f9356,f4973])).
% 2.19/1.79 thf(f4973,plain,(
% 2.19/1.79 ((in @ sK1117 @ (funcSet @ sK1119 @ sK1118)) = $true)),
% 2.19/1.79 inference(cnf_transformation,[status(thm)],[f3328])).
% 2.19/1.79 thf(f3328,plain,(
% 2.19/1.79 (demorgan2a = $true) & (breln1unionprop = $true) & (notdallE = $true) & (setukpairinjL = $true) & (setunion__Cong = $true) & (doubleComplementSub2 = $true) & (binintersectEL = $true) & (prop2setI = $true) & (inPowerset = $true) & (subsetTI = $true) & (binunionTIRcontra = $true) & (eqbreln = $true) & (ex1I = $true) & (complementT_lem = $true) & (singletoninpowunion = $true) & (setadjoinAx = $true) & (contrasubsetT = $true) & (inIntersectImpInIntersectUnions = $true) & (singletoninpowerset = $true) & (breln1invI = $true) & (binunionT_lem = $true) & (woz2W = $true) & (funcGraphProp3 = $true) & (setext = $true) & (setextsub = $true) & (noeltsimpempty = $true) & (symdiffI1 = $true) & (setminusER = $true) & (nonemptyI1 = $true) & (kpairiskpair = $true) & (exuE1 = $true) & (powersetT_lem = $true) & (doubleComplementEq = $true) & (beta2 = $true) & (beta1 = $true) & (uniqinunit = $true) & (exu__Cong = $true) & (exuI2 = $true) & (exuI3 = $true) & (binunionRsub = $true) & (binunionTEcontra = $true) & (woz1_4 = $true) & (emptysetimpfalse = $true) & (funcImageSingleton = $true) & (upairinpowunion = $true) & (setminusLsub = $true) & (binunionIR = $true) & (demorgan2b2 = $true) & (binintersectI = $true) & (ifSingleton = $true) & (iffalse = $true) & (ex1E1 = $true) & (powersetsubset = $true) & (dpsetconstrERa = $true) & (intersectInPowersetIntersectUnions = $true) & (binintersectSubset3 = $true) & (setukpairinjR2 = $true) & (complementSubsetComplementIntersect = $true) & (binintersectSubset4 = $true) & (cartprodpairmemER = $true) & (ubforcartprodlem1 = $true) & (exuI1 = $true) & (sepSubset = $true) & (subsetI1 = $true) & (breln1all1 = $true) & (kpairsurjEq = $true) & (powersetTI1 = $true) & (inIntersectImpInUnion2 = $true) & (doubleComplementI1 = $true) & (nonemptyE1 = $true) & (quantDeMorgan4 = $true) & (woz13rule2 = $true) & (woz1_3 = $true) & (dsetconstr__Cong = $true) & (omegaIndAx = $true) & (binunionTILcontra = $true) & (complementTnotintersectT = $true) & (inComplementUnionImpNotIn1 = $true) & (setminusIRneg = $true) & (kfstpairEq = $true) & (cartprodsndin = $true) & (theprop = $true) & (breln1compE = $true) & (cartprodsndpairEq = $true) & (notinsingleton = $true) & (brelnall1 = $true) & (ap2apEq2 = $true) & (upairequniteq = $true) & (setadjoinSub2 = $true) & (funcext2 = $true) & (omegaSAx = $true) & (emptyI = $true) & (exuE2 = $true) & (powersetTE1 = $true) & (descrp = $true) & (setminusELneg = $true) & (eqinunit = $true) & (funcext = $true) & (lamp = $true) & (inCongP = $true) & (notinemptyset = $true) & (subPowSU = $true) & (infuncsetfunc = $true) & (ap2apEq1 = $true) & (breln1unionIL = $true) & (descr__Cong = $true) & (woz13rule3 = $true) & (ifp = $true) & (breln1unionEcases = $true) & (image1I = $true) & (ap2p = $true) & (emptyinPowerset = $true) & (powersetI1 = $true) & (doubleComplementE1 = $true) & (iffalseProp1 = $true) & (contraSubsetComplement = $true) & (subsetTrans = $true) & (omega0Ax = $true) & (vacuousDall = $true) & (quantDeMorgan2 = $true) & (breln1compI = $true) & (prop2setE = $true) & (setukpairinjR12 = $true) & (lam2lamEq = $true) & (complementInPowersetComplementIntersect = $true) & (breln1unionI = $true) & (binintersectSubset2 = $true) & (contrasubsetT1 = $true) & (iffalseProp2 = $true) & (lam2p = $true) & (brelnall2 = $true) & (notequalI1 = $true) & (upairsetE = $true) & (powersetI = $true) & (cartprodmempaircEq = $true) & (dpsetconstrSub = $true) & (symdiffE = $true) & (bs114d = $true) & (foundationAx = $true) & (demorgan1b = $true) & (setukpairIL = $true) & (in__Cong = $true) & (setukpairIR = $true) & (ubforcartprodlem3 = $true) & (demorgan2a2 = $true) & (singletonprop = $true) & (eqimpsubset2 = $true) & (notsubsetI = $true) & (cartprodpairin = $true) & (setadjoin__Cong = $true) & (subsetI2 = $true) & (subsetE = $true) & (binunionTE = $true) & (notdexE = $true) & (setunionE = $true) & (woz1_2 = $true) & (iftrueProp1 = $true) & (dpsetconstrEL1 = $true) & (demorgan1a = $true) & (eta2 = $true) & (dpsetconstrEL2 = $true) & (setextT = $true) & (setminusSubset1 = $true) & (cartprodfstin = $true) & (woz1_1 = $true) & (woz2A = $true) & (funcinfuncset = $true) & (subset2powerset = $true) & (cartprodpairsurjEq = $true) & (breln1SetBreln1 = $true) & (binintersectTELcontra = $true) & (cartprodpairmemEL = $true) & (setukpairinjR1 = $true) & (woz13rule1 = $true) & (setunionsingleton1 = $true) & (funcextLem = $true) & (setukpairinjR = $true) & (setextAx = $true) & (setoftrueEq = $true) & (image1Equiv = $true) & (kfstsingleton = $true) & (cartprodmempair = $true) & (complementUnionInPowersetComplement = $true) & (eqimpsubset1 = $true) & (setunionsingleton = $true) & (secondinupair = $true) & (woz1_5 = $true) & (powersetE = $true) & (subsetRefl = $true) & (woz13rule4 = $true) & (setadjoinSub = $true) & (binunionE = $true) & (quantDeMorgan1 = $true) & (complementTI1 = $true) & (funcGraphProp2 = $true) & (setunionAx = $true) & (breln1invE = $true) & (powersetE1 = $true) & (image1Ex = $true) & (ubforcartprodlem2 = $true) & (eqbreln1 = $true) & (doubleComplementSub1 = $true) & (contrasubsetT2 = $true) & (setminusI = $true) & (binintersectT_lem = $true) & (wellorderingAx = $true) & (nonemptyI = $true) & (setminusILneg = $true) & (demorgan2 = $true) & (eta1 = $true) & (breln1compprop = $true) & (singletonsswitch = $true) & (apProp = $true) & (funcGraphProp4 = $true) & (setminusSubset2 = $true) & (binintersectSubset5 = $true) & (emptysetE = $true) & (breln1invprop = $true) & (kpairp = $true) & (woz2B = $true) & (exuE3e = $true) & (choice2fnsingleton = $true) & (theeq = $true) & (emptysetAx = $true) & (setadjoinE = $true) & (complementTE1 = $true) & (image1Ex1 = $true) & (nonemptyImpWitness = $true) & (inComplementUnionImpInComplement1 = $true) & (symdiffI2 = $true) & (binintersectTERcontra = $true) & (setunionI = $true) & (omega__Cong = $true) & (setbeta = $true) & (powersetAx = $true) & (exuEu = $true) & (dsetconstrEL = $true) & (notequalI2 = $true) & (singletonsuniq = $true) & (upairset2IR = $true) & (setunionE2 = $true) & (setminusT_lem = $true) & (binintersectSubset1 = $true) & (demorgan2a1 = $true) & (iftrueorfalse = $true) & (inIntersectImpInUnion = $true) & (ksndpairEq = $true) & (setukpairinjL2 = $true) & (breln1compEex = $true) & (prop2set2propI = $true) & (upairsetIR = $true) & (powerset__Cong = $true) & (emptyinunitempty = $true) & (setukpairinjR11 = $true) & (emptysetsubset = $true) & (ksndsingleton = $true) & (emptyset__Cong = $true) & (dpsetconstrER = $true) & (ex1E2 = $true) & (upairset2E = $true) & (demorgan1 = $true) & (breln1unionCommutes = $true) & (setunionsingleton2 = $true) & (setOfPairsIsBReln = $true) & (singletonsubset = $true) & (subbreln1 = $true) & (binintersectRsub = $true) & (woz13rule0 = $true) & (((in @ sK1117 @ (funcSet @ sK1119 @ sK1118)) = $true) & ($true = (injective @ sK1119 @ sK1118 @ sK1117)) & ((in @ sK1117 @ (injFuncSet @ sK1119 @ sK1118)) != $true)) & (iftrueProp2 = $true) & (symdiffIneg1 = $true) & (setOfPairsIsBReln1 = $true) & (lamProp = $true) & (binunionEcases = $true) & (dsetconstrER = $true) & (sepInPowerset = $true) & (setminusERneg = $true) & (woz2Ex = $true) & (cartprodfstpairEq = $true) & (subsetE2 = $true) & (cartprodmempair1 = $true) & (complementImpComplementIntersect = $true) & (binunionIL = $true) & (setadjoinOr = $true) & (disjointsetsI1 = $true) & (breln1all2 = $true) & (ex1I2 = $true) & (subsetemptysetimpeq = $true) & (upairsetIL = $true) & (binintersectER = $true) & (binintersectLsub = $true) & (dsetconstrI = $true) & (dpsetconstrI = $true) & (complementTcontraSubset = $true) & (setminusEL = $true) & (replAx = $true) & (subbreln = $true) & (quantDeMorgan3 = $true) & (exuE3u = $true) & (binunionLsub = $true) & (demorgan2b = $true) & (emptyInPowerset = $true) & (app = $true) & (symdiffIneg2 = $true) & (setadjoinIL = $true) & (contrasubsetT3 = $true) & (iftrue = $true) & (funcGraphProp1 = $true) & (breln1unionIR = $true) & (breln1unionE = $true) & (setadjoinIR = $true) & (setukpairinjL1 = $true) & (upairsubunion = $true) & (image1E = $true) & (emptyE1 = $true)),
% 2.19/1.79 inference(skolemisation,[status(esa),new_symbols(skolem,[sK1117,sK1118,sK1119])],[f1345,f3327])).
% 2.19/1.79 thf(f3327,plain,(
% 2.19/1.79 ? [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X0 @ (funcSet @ X2 @ X1))) & ((injective @ X2 @ X1 @ X0) = $true) & ((in @ X0 @ (injFuncSet @ X2 @ X1)) != $true)) => (((in @ sK1117 @ (funcSet @ sK1119 @ sK1118)) = $true) & ($true = (injective @ sK1119 @ sK1118 @ sK1117)) & ((in @ sK1117 @ (injFuncSet @ sK1119 @ sK1118)) != $true))),
% 2.19/1.79 introduced(definition,[],[choice_axiom])).
% 2.19/1.79 thf(f1345,plain,(
% 2.19/1.79 (demorgan2a = $true) & (breln1unionprop = $true) & (notdallE = $true) & (setukpairinjL = $true) & (setunion__Cong = $true) & (doubleComplementSub2 = $true) & (binintersectEL = $true) & (prop2setI = $true) & (inPowerset = $true) & (subsetTI = $true) & (binunionTIRcontra = $true) & (eqbreln = $true) & (ex1I = $true) & (complementT_lem = $true) & (singletoninpowunion = $true) & (setadjoinAx = $true) & (contrasubsetT = $true) & (inIntersectImpInIntersectUnions = $true) & (singletoninpowerset = $true) & (breln1invI = $true) & (binunionT_lem = $true) & (woz2W = $true) & (funcGraphProp3 = $true) & (setext = $true) & (setextsub = $true) & (noeltsimpempty = $true) & (symdiffI1 = $true) & (setminusER = $true) & (nonemptyI1 = $true) & (kpairiskpair = $true) & (exuE1 = $true) & (powersetT_lem = $true) & (doubleComplementEq = $true) & (beta2 = $true) & (beta1 = $true) & (uniqinunit = $true) & (exu__Cong = $true) & (exuI2 = $true) & (exuI3 = $true) & (binunionRsub = $true) & (binunionTEcontra = $true) & (woz1_4 = $true) & (emptysetimpfalse = $true) & (funcImageSingleton = $true) & (upairinpowunion = $true) & (setminusLsub = $true) & (binunionIR = $true) & (demorgan2b2 = $true) & (binintersectI = $true) & (ifSingleton = $true) & (iffalse = $true) & (ex1E1 = $true) & (powersetsubset = $true) & (dpsetconstrERa = $true) & (intersectInPowersetIntersectUnions = $true) & (binintersectSubset3 = $true) & (setukpairinjR2 = $true) & (complementSubsetComplementIntersect = $true) & (binintersectSubset4 = $true) & (cartprodpairmemER = $true) & (ubforcartprodlem1 = $true) & (exuI1 = $true) & (sepSubset = $true) & (subsetI1 = $true) & (breln1all1 = $true) & (kpairsurjEq = $true) & (powersetTI1 = $true) & (inIntersectImpInUnion2 = $true) & (doubleComplementI1 = $true) & (nonemptyE1 = $true) & (quantDeMorgan4 = $true) & (woz13rule2 = $true) & (woz1_3 = $true) & (dsetconstr__Cong = $true) & (omegaIndAx = $true) & (binunionTILcontra = $true) & (complementTnotintersectT = $true) & (inComplementUnionImpNotIn1 = $true) & (setminusIRneg = $true) & (kfstpairEq = $true) & (cartprodsndin = $true) & (theprop = $true) & (breln1compE = $true) & (cartprodsndpairEq = $true) & (notinsingleton = $true) & (brelnall1 = $true) & (ap2apEq2 = $true) & (upairequniteq = $true) & (setadjoinSub2 = $true) & (funcext2 = $true) & (omegaSAx = $true) & (emptyI = $true) & (exuE2 = $true) & (powersetTE1 = $true) & (descrp = $true) & (setminusELneg = $true) & (eqinunit = $true) & (funcext = $true) & (lamp = $true) & (inCongP = $true) & (notinemptyset = $true) & (subPowSU = $true) & (infuncsetfunc = $true) & (ap2apEq1 = $true) & (breln1unionIL = $true) & (descr__Cong = $true) & (woz13rule3 = $true) & (ifp = $true) & (breln1unionEcases = $true) & (image1I = $true) & (ap2p = $true) & (emptyinPowerset = $true) & (powersetI1 = $true) & (doubleComplementE1 = $true) & (iffalseProp1 = $true) & (contraSubsetComplement = $true) & (subsetTrans = $true) & (omega0Ax = $true) & (vacuousDall = $true) & (quantDeMorgan2 = $true) & (breln1compI = $true) & (prop2setE = $true) & (setukpairinjR12 = $true) & (lam2lamEq = $true) & (complementInPowersetComplementIntersect = $true) & (breln1unionI = $true) & (binintersectSubset2 = $true) & (contrasubsetT1 = $true) & (iffalseProp2 = $true) & (lam2p = $true) & (brelnall2 = $true) & (notequalI1 = $true) & (upairsetE = $true) & (powersetI = $true) & (cartprodmempaircEq = $true) & (dpsetconstrSub = $true) & (symdiffE = $true) & (bs114d = $true) & (foundationAx = $true) & (demorgan1b = $true) & (setukpairIL = $true) & (in__Cong = $true) & (setukpairIR = $true) & (ubforcartprodlem3 = $true) & (demorgan2a2 = $true) & (singletonprop = $true) & (eqimpsubset2 = $true) & (notsubsetI = $true) & (cartprodpairin = $true) & (setadjoin__Cong = $true) & (subsetI2 = $true) & (subsetE = $true) & (binunionTE = $true) & (notdexE = $true) & (setunionE = $true) & (woz1_2 = $true) & (iftrueProp1 = $true) & (dpsetconstrEL1 = $true) & (demorgan1a = $true) & (eta2 = $true) & (dpsetconstrEL2 = $true) & (setextT = $true) & (setminusSubset1 = $true) & (cartprodfstin = $true) & (woz1_1 = $true) & (woz2A = $true) & (funcinfuncset = $true) & (subset2powerset = $true) & (cartprodpairsurjEq = $true) & (breln1SetBreln1 = $true) & (binintersectTELcontra = $true) & (cartprodpairmemEL = $true) & (setukpairinjR1 = $true) & (woz13rule1 = $true) & (setunionsingleton1 = $true) & (funcextLem = $true) & (setukpairinjR = $true) & (setextAx = $true) & (setoftrueEq = $true) & (image1Equiv = $true) & (kfstsingleton = $true) & (cartprodmempair = $true) & (complementUnionInPowersetComplement = $true) & (eqimpsubset1 = $true) & (setunionsingleton = $true) & (secondinupair = $true) & (woz1_5 = $true) & (powersetE = $true) & (subsetRefl = $true) & (woz13rule4 = $true) & (setadjoinSub = $true) & (binunionE = $true) & (quantDeMorgan1 = $true) & (complementTI1 = $true) & (funcGraphProp2 = $true) & (setunionAx = $true) & (breln1invE = $true) & (powersetE1 = $true) & (image1Ex = $true) & (ubforcartprodlem2 = $true) & (eqbreln1 = $true) & (doubleComplementSub1 = $true) & (contrasubsetT2 = $true) & (setminusI = $true) & (binintersectT_lem = $true) & (wellorderingAx = $true) & (nonemptyI = $true) & (setminusILneg = $true) & (demorgan2 = $true) & (eta1 = $true) & (breln1compprop = $true) & (singletonsswitch = $true) & (apProp = $true) & (funcGraphProp4 = $true) & (setminusSubset2 = $true) & (binintersectSubset5 = $true) & (emptysetE = $true) & (breln1invprop = $true) & (kpairp = $true) & (woz2B = $true) & (exuE3e = $true) & (choice2fnsingleton = $true) & (theeq = $true) & (emptysetAx = $true) & (setadjoinE = $true) & (complementTE1 = $true) & (image1Ex1 = $true) & (nonemptyImpWitness = $true) & (inComplementUnionImpInComplement1 = $true) & (symdiffI2 = $true) & (binintersectTERcontra = $true) & (setunionI = $true) & (omega__Cong = $true) & (setbeta = $true) & (powersetAx = $true) & (exuEu = $true) & (dsetconstrEL = $true) & (notequalI2 = $true) & (singletonsuniq = $true) & (upairset2IR = $true) & (setunionE2 = $true) & (setminusT_lem = $true) & (binintersectSubset1 = $true) & (demorgan2a1 = $true) & (iftrueorfalse = $true) & (inIntersectImpInUnion = $true) & (ksndpairEq = $true) & (setukpairinjL2 = $true) & (breln1compEex = $true) & (prop2set2propI = $true) & (upairsetIR = $true) & (powerset__Cong = $true) & (emptyinunitempty = $true) & (setukpairinjR11 = $true) & (emptysetsubset = $true) & (ksndsingleton = $true) & (emptyset__Cong = $true) & (dpsetconstrER = $true) & (ex1E2 = $true) & (upairset2E = $true) & (demorgan1 = $true) & (breln1unionCommutes = $true) & (setunionsingleton2 = $true) & (setOfPairsIsBReln = $true) & (singletonsubset = $true) & (subbreln1 = $true) & (binintersectRsub = $true) & (woz13rule0 = $true) & ? [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X0 @ (funcSet @ X2 @ X1))) & ((injective @ X2 @ X1 @ X0) = $true) & ((in @ X0 @ (injFuncSet @ X2 @ X1)) != $true)) & (iftrueProp2 = $true) & (symdiffIneg1 = $true) & (setOfPairsIsBReln1 = $true) & (lamProp = $true) & (binunionEcases = $true) & (dsetconstrER = $true) & (sepInPowerset = $true) & (setminusERneg = $true) & (woz2Ex = $true) & (cartprodfstpairEq = $true) & (subsetE2 = $true) & (cartprodmempair1 = $true) & (complementImpComplementIntersect = $true) & (binunionIL = $true) & (setadjoinOr = $true) & (disjointsetsI1 = $true) & (breln1all2 = $true) & (ex1I2 = $true) & (subsetemptysetimpeq = $true) & (upairsetIL = $true) & (binintersectER = $true) & (binintersectLsub = $true) & (dsetconstrI = $true) & (dpsetconstrI = $true) & (complementTcontraSubset = $true) & (setminusEL = $true) & (replAx = $true) & (subbreln = $true) & (quantDeMorgan3 = $true) & (exuE3u = $true) & (binunionLsub = $true) & (demorgan2b = $true) & (emptyInPowerset = $true) & (app = $true) & (symdiffIneg2 = $true) & (setadjoinIL = $true) & (contrasubsetT3 = $true) & (iftrue = $true) & (funcGraphProp1 = $true) & (breln1unionIR = $true) & (breln1unionE = $true) & (setadjoinIR = $true) & (setukpairinjL1 = $true) & (upairsubunion = $true) & (image1E = $true) & (emptyE1 = $true)),
% 2.19/1.79 inference(flattening,[status(thm)],[f1344])).
% 2.19/1.79 thf(f1344,plain,(
% 2.19/1.79 (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i,X1 : $i,X2 : $i] : ((((in @ X0 @ (injFuncSet @ X2 @ X1)) != $true) & ((injective @ X2 @ X1 @ X0) = $true)) & ($true = (in @ X0 @ (funcSet @ X2 @ X1)))) & (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)),
% 2.19/1.79 inference(ennf_transformation,[status(thm)],[f443])).
% 2.19/1.79 thf(f443,plain,(
% 2.19/1.79 ~((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) => ! [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X0 @ (funcSet @ X2 @ X1))) => (((injective @ X2 @ X1 @ X0) = $true) => ((in @ X0 @ (injFuncSet @ X2 @ X1)) = $true)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.19/1.79 inference(fool_elimination,[status(thm)],[f442])).
% 2.19/1.79 thf(f442,plain,(
% 2.19/1.79 ~(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 => ! [X0 : $i,X1 : $i,X2 : $i] : ((in @ X0 @ (funcSet @ X2 @ X1)) => ((injective @ X2 @ X1 @ X0) => (in @ X0 @ (injFuncSet @ X2 @ X1))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.19/1.79 inference(rectify,[status(thm)],[f319])).
% 2.19/1.79 thf(f319,negated_conjecture,(
% 2.19/1.79 ~(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 => ! [X12 : $i,X4 : $i,X3 : $i] : ((in @ X12 @ (funcSet @ X3 @ X4)) => ((injective @ X3 @ X4 @ X12) => (in @ X12 @ (injFuncSet @ X3 @ X4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.19/1.79 inference(negated_conjecture,[status(cth)],[f318])).
% 2.19/1.79 thf(f318,conjecture,(
% 2.19/1.79 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 => ! [X12 : $i,X4 : $i,X3 : $i] : ((in @ X12 @ (funcSet @ X3 @ X4)) => ((injective @ X3 @ X4 @ X12) => (in @ X12 @ (injFuncSet @ X3 @ X4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.19/1.79 file('/export/starexec/sandbox/tmp/tmp.IMUvashtid/DTF2THF_11553.p',injFuncInInjFuncSet)).
% 2.19/1.79 thf(f9356,plain,(
% 2.19/1.79 ((in @ sK1117 @ (funcSet @ sK1119 @ sK1118)) != $true)),
% 2.19/1.79 inference(subsumption_resolution,[status(thm)],[f9355,f4972])).
% 2.19/1.79 thf(f4972,plain,(
% 2.19/1.79 ($true = (injective @ sK1119 @ sK1118 @ sK1117))),
% 2.19/1.79 inference(cnf_transformation,[status(thm)],[f3328])).
% 2.19/1.79 thf(f9355,plain,(
% 2.19/1.79 ($true != (injective @ sK1119 @ sK1118 @ sK1117)) | ((in @ sK1117 @ (funcSet @ sK1119 @ sK1118)) != $true)),
% 2.19/1.79 inference(trivial_inequality_removal,[status(thm)],[f9352])).
% 2.19/1.79 thf(f9352,plain,(
% 2.19/1.79 ((in @ sK1117 @ (funcSet @ sK1119 @ sK1118)) != $true) | ($true != (injective @ sK1119 @ sK1118 @ sK1117)) | ($true != $true)),
% 2.19/1.79 inference(superposition,[status(thm)],[f6706,f6531])).
% 2.19/1.79 thf(f6531,plain,(
% 2.19/1.79 ( ! [X3 : $i > $o,X4 : $i,X5 : $i] : (($true = (in @ X5 @ (dsetconstr @ X4 @ X3))) | ($true != (X3 @ X5)) | ((in @ X5 @ X4) != $true)) )),
% 2.19/1.79 inference(beta_eta_normalization,[status(thm)],[f6530])).
% 2.19/1.79 thf(f6530,plain,(
% 2.19/1.79 ( ! [X3 : $i > $o,X4 : $i,X5 : $i] : (($true = (in @ X5 @ (dsetconstr @ X4 @ (^[Y0 : $i]: (X3 @ Y0))))) | ((in @ X5 @ X4) != $true) | ($true != (X3 @ X5))) )),
% 2.19/1.79 inference(trivial_inequality_removal,[status(thm)],[f6136])).
% 2.19/1.79 thf(f6136,plain,(
% 2.19/1.79 ( ! [X3 : $i > $o,X4 : $i,X5 : $i] : (($true != $true) | ($true = (in @ X5 @ (dsetconstr @ X4 @ (^[Y0 : $i]: (X3 @ Y0))))) | ((in @ X5 @ X4) != $true) | ($true != (X3 @ X5))) )),
% 2.19/1.79 inference(definition_unfolding,[status(thm)],[f4503,f4948])).
% 2.19/1.79 thf(f4948,plain,(
% 2.19/1.79 (dsetconstrI = $true)),
% 2.19/1.79 inference(cnf_transformation,[status(thm)],[f3328])).
% 2.19/1.79 thf(f4503,plain,(
% 2.19/1.79 ( ! [X3 : $i > $o,X4 : $i,X5 : $i] : (($true != (X3 @ X5)) | ((in @ X5 @ X4) != $true) | ($true = (in @ X5 @ (dsetconstr @ X4 @ (^[Y0 : $i]: (X3 @ Y0))))) | (dsetconstrI != $true)) )),
% 2.19/1.79 inference(cnf_transformation,[status(thm)],[f2848])).
% 2.19/1.79 thf(f2848,plain,(
% 2.19/1.79 ((dsetconstrI = $true) | (($true = (sK851 @ sK853)) & ((in @ sK853 @ sK852) = $true) & ((in @ sK853 @ (dsetconstr @ sK852 @ (^[Y0 : $i]: (sK851 @ Y0)))) != $true))) & (! [X3 : $i > $o,X4 : $i,X5 : $i] : (($true != (X3 @ X5)) | ((in @ X5 @ X4) != $true) | ($true = (in @ X5 @ (dsetconstr @ X4 @ (^[Y0 : $i]: (X3 @ Y0)))))) | (dsetconstrI != $true))),
% 2.19/1.79 inference(skolemisation,[status(esa),new_symbols(skolem,[sK851,sK852,sK853])],[f2846,f2847])).
% 2.19/1.79 thf(f2847,plain,(
% 2.19/1.79 ? [X0 : $i > $o,X1 : $i,X2 : $i] : (((X0 @ X2) = $true) & ($true = (in @ X2 @ X1)) & ((in @ X2 @ (dsetconstr @ X1 @ (^[Y0 : $i]: (X0 @ Y0)))) != $true)) => (($true = (sK851 @ sK853)) & ((in @ sK853 @ sK852) = $true) & ((in @ sK853 @ (dsetconstr @ sK852 @ (^[Y0 : $i]: (sK851 @ Y0)))) != $true))),
% 2.19/1.79 introduced(definition,[],[choice_axiom])).
% 2.19/1.79 thf(f2846,plain,(
% 2.19/1.79 ((dsetconstrI = $true) | ? [X0 : $i > $o,X1 : $i,X2 : $i] : (((X0 @ X2) = $true) & ($true = (in @ X2 @ X1)) & ((in @ X2 @ (dsetconstr @ X1 @ (^[Y0 : $i]: (X0 @ Y0)))) != $true))) & (! [X3 : $i > $o,X4 : $i,X5 : $i] : (($true != (X3 @ X5)) | ((in @ X5 @ X4) != $true) | ($true = (in @ X5 @ (dsetconstr @ X4 @ (^[Y0 : $i]: (X3 @ Y0)))))) | (dsetconstrI != $true))),
% 2.19/1.79 inference(rectify,[status(thm)],[f2845])).
% 2.19/1.79 thf(f2845,plain,(
% 2.19/1.79 ((dsetconstrI = $true) | ? [X0 : $i > $o,X1 : $i,X2 : $i] : (((X0 @ X2) = $true) & ($true = (in @ X2 @ X1)) & ((in @ X2 @ (dsetconstr @ X1 @ (^[Y0 : $i]: (X0 @ Y0)))) != $true))) & (! [X0 : $i > $o,X1 : $i,X2 : $i] : (((X0 @ X2) != $true) | ($true != (in @ X2 @ X1)) | ((in @ X2 @ (dsetconstr @ X1 @ (^[Y0 : $i]: (X0 @ Y0)))) = $true)) | (dsetconstrI != $true))),
% 2.19/1.79 inference(nnf_transformation,[status(thm)],[f1180])).
% 2.19/1.79 thf(f1180,plain,(
% 2.19/1.79 (dsetconstrI = $true) <=> ! [X0 : $i > $o,X1 : $i,X2 : $i] : (((X0 @ X2) != $true) | ($true != (in @ X2 @ X1)) | ((in @ X2 @ (dsetconstr @ X1 @ (^[Y0 : $i]: (X0 @ Y0)))) = $true))),
% 2.19/1.79 inference(flattening,[status(thm)],[f1179])).
% 2.19/1.79 thf(f1179,plain,(
% 2.19/1.79 ! [X0 : $i > $o,X1 : $i,X2 : $i] : ((((in @ X2 @ (dsetconstr @ X1 @ (^[Y0 : $i]: (X0 @ Y0)))) = $true) | ((X0 @ X2) != $true)) | ($true != (in @ X2 @ X1))) <=> (dsetconstrI = $true)),
% 2.19/1.79 inference(ennf_transformation,[status(thm)],[f600])).
% 2.19/1.79 thf(f600,plain,(
% 2.19/1.79 ! [X0 : $i > $o,X1 : $i,X2 : $i] : (($true = (in @ X2 @ X1)) => (((X0 @ X2) = $true) => ((in @ X2 @ (dsetconstr @ X1 @ (^[Y0 : $i]: (X0 @ Y0)))) = $true))) <=> (dsetconstrI = $true)),
% 2.19/1.79 inference(fool_elimination,[status(thm)],[f599])).
% 2.19/1.79 thf(f599,plain,(
% 2.19/1.79 (dsetconstrI = ! [X0 : $i > $o,X1 : $i,X2 : $i] : ((in @ X2 @ X1) => ((X0 @ X2) => (in @ X2 @ (dsetconstr @ X1 @ (^[X3 : $i] : (X0 @ X3)))))))),
% 2.19/1.79 inference(rectify,[status(thm)],[f14])).
% 2.19/1.79 thf(f14,axiom,(
% 2.19/1.79 (dsetconstrI = ! [X0 : $i > $o,X3 : $i,X1 : $i] : ((in @ X1 @ X3) => ((X0 @ X1) => (in @ X1 @ (dsetconstr @ X3 @ (^[X2 : $i] : (X0 @ X2)))))))),
% 2.19/1.79 file('/export/starexec/sandbox/tmp/tmp.IMUvashtid/DTF2THF_11553.p',dsetconstrI)).
% 2.19/1.79 thf(f6706,plain,(
% 2.19/1.79 ($true != (in @ sK1117 @ (dsetconstr @ (funcSet @ sK1119 @ sK1118) @ (injective @ sK1119 @ sK1118))))),
% 2.19/1.79 inference(beta_eta_normalization,[status(thm)],[f6411])).
% 2.19/1.79 thf(f6411,plain,(
% 2.19/1.79 ((in @ sK1117 @ ((^[Y0 : $i]: ((^[Y1 : $i]: (dsetconstr @ (funcSet @ Y0 @ Y1) @ (^[Y2 : $i]: (injective @ Y0 @ Y1 @ Y2)))))) @ sK1119 @ sK1118)) != $true)),
% 2.19/1.79 inference(definition_unfolding,[status(thm)],[f4971,f4448])).
% 2.19/1.79 thf(f4448,plain,(
% 2.19/1.79 (injFuncSet = (^[Y0 : $i]: ((^[Y1 : $i]: (dsetconstr @ (funcSet @ Y0 @ Y1) @ (^[Y2 : $i]: (injective @ Y0 @ Y1 @ Y2)))))))),
% 2.19/1.79 inference(cnf_transformation,[status(thm)],[f892])).
% 2.19/1.79 thf(f892,plain,(
% 2.19/1.79 (injFuncSet = (^[Y0 : $i]: ((^[Y1 : $i]: (dsetconstr @ (funcSet @ Y0 @ Y1) @ (^[Y2 : $i]: (injective @ Y0 @ Y1 @ Y2)))))))),
% 2.19/1.79 inference(fool_elimination,[status(thm)],[f891])).
% 2.19/1.79 thf(f891,plain,(
% 2.19/1.79 ((^[X0 : $i, X1 : $i] : (dsetconstr @ (funcSet @ X0 @ X1) @ (^[X2 : $i] : (injective @ X0 @ X1 @ X2)))) = injFuncSet)),
% 2.19/1.79 inference(rectify,[status(thm)],[f317])).
% 2.19/1.79 thf(f317,axiom,(
% 2.19/1.79 ((^[X3 : $i, X4 : $i] : (dsetconstr @ (funcSet @ X3 @ X4) @ (^[X12 : $i] : (injective @ X3 @ X4 @ X12)))) = injFuncSet)),
% 2.19/1.79 file('/export/starexec/sandbox/tmp/tmp.IMUvashtid/DTF2THF_11553.p',injFuncSet)).
% 2.19/1.79 thf(f4971,plain,(
% 2.19/1.79 ((in @ sK1117 @ (injFuncSet @ sK1119 @ sK1118)) != $true)),
% 2.19/1.79 inference(cnf_transformation,[status(thm)],[f3328])).
% 2.19/1.79 % SZS output end Proof for DTF2THF_11553
% 2.19/1.79 % (11863)------------------------------
% 2.19/1.79 % (11863)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.19/1.79 % (11863)Termination reason: Refutation
% 2.19/1.79
% 2.19/1.79 % (11863)Memory used [KB]: 13048
% 2.19/1.79 % (11863)Time elapsed: 0.301 s
% 2.19/1.79 % (11863)Instructions burned: 871 (million)
% 2.19/1.79 % (11863)------------------------------
% 2.19/1.79 % (11863)------------------------------
% 2.19/1.79 % (11815)Success in time 0.598 s
%------------------------------------------------------------------------------