%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU813^1 : TPTP v9.2.1. Released v3.7.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n019.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 09:37:25 AM UTC 2026
% Result : Theorem 8.47s 2.45s
% Output : Refutation 8.47s
% Verified :
% SZS Type : Refutation
% Derivation depth : 18
% Number of leaves : 12
% Syntax : Number of formulae : 91 ( 35 unt; 0 typ; 2 def)
% Number of atoms : 5728 (1520 equ; 0 cnn)
% Maximal formula atoms : 332 ( 62 avg)
% Number of connectives : 3075 ( 85 ~; 74 |;1028 &; 514 @)
% ( 9 <=>;1357 =>; 0 <=; 0 <~>)
% Maximal formula depth : 338 ( 29 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 686 ( 682 usr; 346 con; 0-6 aty)
% ( 8 !!; 0 ??; 0 @@+; 0 @@-)
% Number of variables : 162 ( 18 ^; 109 !; 35 ?; 162 :)
% Comments :
%------------------------------------------------------------------------------
thf(func_def_0,type,
in: $i > $i > $o ).
thf(func_def_1,type,
exu: ( $i > $o ) > $o ).
thf(func_def_6,type,
setadjoin: $i > $i > $i ).
thf(func_def_8,type,
powerset: $i > $i ).
thf(func_def_10,type,
setunion: $i > $i ).
thf(func_def_19,type,
descr: ( $i > $o ) > $i ).
thf(func_def_21,type,
dsetconstr: $i > ( $i > $o ) > $i ).
thf(func_def_26,type,
prop2set: $o > $i ).
thf(func_def_36,type,
nonempty: $i > $o ).
thf(func_def_69,type,
set2prop: $i > $o ).
thf(func_def_88,type,
subset: $i > $i > $o ).
thf(func_def_89,type,
disjoint: $i > $i > $o ).
thf(func_def_90,type,
setsmeet: $i > $i > $o ).
thf(func_def_114,type,
binunion: $i > $i > $i ).
thf(func_def_122,type,
binintersect: $i > $i > $i ).
thf(func_def_135,type,
regular: $i > $o ).
thf(func_def_136,type,
setminus: $i > $i > $i ).
thf(func_def_147,type,
symdiff: $i > $i > $i ).
thf(func_def_153,type,
iskpair: $i > $o ).
thf(func_def_158,type,
kpair: $i > $i > $i ).
thf(func_def_160,type,
cartprod: $i > $i > $i ).
thf(func_def_177,type,
singleton: $i > $o ).
thf(func_def_179,type,
ex1: $i > ( $i > $o ) > $o ).
thf(func_def_184,type,
atmost1p: $i > $o ).
thf(func_def_185,type,
atleast2p: $i > $o ).
thf(func_def_186,type,
atmost2p: $i > $o ).
thf(func_def_187,type,
upairsetp: $i > $o ).
thf(func_def_191,type,
kfst: $i > $i ).
thf(func_def_203,type,
ksnd: $i > $i ).
thf(func_def_213,type,
breln: $i > $i > $i > $o ).
thf(func_def_214,type,
dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_222,type,
func: $i > $i > $i > $o ).
thf(func_def_223,type,
funcSet: $i > $i > $i ).
thf(func_def_226,type,
ap: $i > $i > $i > $i > $i ).
thf(func_def_232,type,
lam: $i > $i > ( $i > $i ) > $i ).
thf(func_def_259,type,
if: $i > $o > $i > $i > $i ).
thf(func_def_323,type,
breln1: $i > $i > $o ).
thf(func_def_325,type,
breln1Set: $i > $i ).
thf(func_def_327,type,
transitive: $i > $i > $o ).
thf(func_def_328,type,
antisymmetric: $i > $i > $o ).
thf(func_def_329,type,
reflexive: $i > $i > $o ).
thf(func_def_330,type,
refltransitive: $i > $i > $o ).
thf(func_def_331,type,
refllinearorder: $i > $i > $o ).
thf(func_def_332,type,
reflwellordering: $i > $i > $o ).
thf(func_def_338,type,
breln1invset: $i > $i > $i ).
thf(func_def_342,type,
breln1compset: $i > $i > $i > $i ).
thf(func_def_360,type,
image1: $i > ( $i > $i ) > $i ).
thf(func_def_364,type,
injective: $i > $i > $i > $o ).
thf(func_def_365,type,
injFuncSet: $i > $i > $i ).
thf(func_def_369,type,
surjective: $i > $i > $i > $o ).
thf(func_def_370,type,
surjFuncSet: $i > $i > $i ).
thf(func_def_378,type,
omegaS: $i > $i ).
thf(func_def_382,type,
peano3: $i > $i > $i > $o ).
thf(func_def_383,type,
peano4: $i > $i > $i > $o ).
thf(func_def_384,type,
peano5: $i > $i > $i > $o ).
thf(func_def_386,type,
peano: $i > $i > $i > $o ).
thf(func_def_387,type,
transitiveset: $i > $o ).
thf(func_def_403,type,
sP0: $i > $i > $o ).
thf(func_def_405,type,
sP2: $i > $i > $i > $o ).
thf(func_def_406,type,
sP3: $i > $i > $o ).
thf(func_def_407,type,
sP4: $i > $o ).
thf(func_def_408,type,
sP5: $i > $i > $o ).
thf(func_def_409,type,
sP6: $i > $i > $o ).
thf(func_def_410,type,
sP7: $i > $i > $o ).
thf(func_def_412,type,
sP9: $i > ( $i > $i ) > $i > $o ).
thf(func_def_414,type,
sP11: $i > $i > $i > $i > $o ).
thf(func_def_415,type,
sP12: $i > $i > $i > $i > $o ).
thf(func_def_416,type,
sP13: ( $i > $o ) > $o ).
thf(func_def_417,type,
sP14: $i > ( $i > $o ) > $o ).
thf(func_def_418,type,
sP15: ( $i > $o ) > $i > $o ).
thf(func_def_419,type,
sP16: $i > $i > $i > $o ).
thf(func_def_423,type,
sP20: $i > $i > $o > $o ).
thf(func_def_424,type,
sP21: $i > $i > $i > $o ).
thf(func_def_426,type,
sP23: $o > $i > $i > $i > $o ).
thf(func_def_427,type,
sP24: $i > $i > $i > $o > $o ).
thf(func_def_428,type,
sP25: $i > $i > $i > $i > $o ).
thf(func_def_429,type,
sP26: $i > $i > $o ).
thf(func_def_431,type,
sP28: $i > ( $i > $o ) > $o ).
thf(func_def_433,type,
sP30: $i > ( $i > $o ) > $o ).
thf(func_def_435,type,
sP32: $i > ( $i > $o ) > $o ).
thf(func_def_436,type,
sP33: $i > ( $i > $o ) > $i > $o ).
thf(func_def_437,type,
sP34: $i > $i > $i > $i > $i > $o ).
thf(func_def_438,type,
sP35: $i > $i > $i > $o ).
thf(func_def_439,type,
sP36: $i > ( $i > $o ) > $o ).
thf(func_def_441,type,
sP38: ( $i > $i > $o ) > $i > $i > $o ).
thf(func_def_442,type,
sP39: ( $i > $i > $o ) > $i > $o ).
thf(func_def_444,type,
sP41: $i > $i > $i > $o ).
thf(func_def_445,type,
sP42: ( $i > $o ) > $i > $o ).
thf(func_def_446,type,
sP43: $i > ( $i > $o ) > $i > $o ).
thf(func_def_447,type,
sP44: ( $i > $o ) > $o ).
thf(func_def_448,type,
sP45: $o > $i > $i > $o ).
thf(func_def_450,type,
sP47: $i > $i > $i > $o ).
thf(func_def_451,type,
sP48: $i > $i > $i > $o ).
thf(func_def_452,type,
sP49: $i > ( $i > $o ) > ( $i > $o ) > $i > $o ).
thf(func_def_453,type,
sP50: $i > ( $i > $o ) > $i > $o ).
thf(func_def_455,type,
sP52: $i > $i > $i > $o > $i > $i > $o ).
thf(func_def_456,type,
sP53: $i > $i > $i > $o ).
thf(func_def_457,type,
sP54: $i > $i > $i > $o ).
thf(func_def_458,type,
sP55: $i > $i > $i > $o ).
thf(func_def_460,type,
sP57: ( $i > $o ) > $i > $o ).
thf(func_def_463,type,
sP60: $i > $i > $o ).
thf(func_def_465,type,
sP62: $i > $i > $i > $o ).
thf(func_def_467,type,
sP64: ( $i > $o ) > $o ).
thf(func_def_468,type,
sP65: $i > $o > $i > $o ).
thf(func_def_469,type,
sP66: $i > $i > $o > $o ).
thf(func_def_470,type,
sP67: $i > $i > $i > $o ).
thf(func_def_471,type,
sP68: ( $i > $o ) > $i > $o ).
thf(func_def_472,type,
sP69: $i > $i > $i > ( $i > $o ) > $o ).
thf(func_def_473,type,
sP70: ( $i > $o ) > $o ).
thf(func_def_475,type,
sP72: $i > $i > $i > $i > $o ).
thf(func_def_476,type,
sP73: $i > $i > $i > $o ).
thf(func_def_477,type,
sP74: ( $i > $i ) > $i > $i > $o ).
thf(func_def_479,type,
sP76: ( $i > $o ) > $o ).
thf(func_def_480,type,
sP77: $i > ( $i > $o ) > $o ).
thf(func_def_481,type,
sP78: ( $i > $o ) > $i > $o ).
thf(func_def_482,type,
sP79: $i > ( $i > $o ) > $o ).
thf(func_def_486,type,
sP83: $i > ( $i > $i ) > $i > $o ).
thf(func_def_489,type,
sP86: $i > $o ).
thf(func_def_490,type,
sP87: $i > ( $i > $i > $o ) > $i > $o ).
thf(func_def_491,type,
sP88: $i > ( $i > $i > $o ) > $i > $o ).
thf(func_def_492,type,
sP89: $i > ( $i > $i ) > $i > $o ).
thf(func_def_494,type,
sP91: $i > $i > $i > $o ).
thf(func_def_495,type,
sP92: $i > $o ).
thf(func_def_496,type,
sP93: $i > $o ).
thf(func_def_498,type,
sP95: $i > $i > $i > $o > $o ).
thf(func_def_499,type,
sP96: $i > $i > $i > $i > $o ).
thf(func_def_500,type,
sP97: ( $i > $i ) > $i > $i > $o ).
thf(func_def_502,type,
sP99: $i > $i > $i > $i > $o ).
thf(func_def_503,type,
sP100: $i > $i > $i > $o ).
thf(func_def_505,type,
sP102: ( $i > $o ) > $i > $o ).
thf(func_def_507,type,
sP104: ( $i > $o ) > $i > $o ).
thf(func_def_508,type,
sP105: ( $i > $o ) > $i > $i > $i > $o ).
thf(func_def_516,type,
sK113: $i > $i > $i ).
thf(func_def_521,type,
sK118: $i > $i > $i ).
thf(func_def_522,type,
sK119: $i > $i > $i ).
thf(func_def_523,type,
sK120: $i > $i > $i > $i ).
thf(func_def_524,type,
sK121: $i > $i > $i ).
thf(func_def_525,type,
sK122: $i > $i > $i ).
thf(func_def_526,type,
sK123: $i > $i > $i ).
thf(func_def_527,type,
sK124: $i > $i > $i ).
thf(func_def_528,type,
sK125: $i > $i > $i ).
thf(func_def_529,type,
sK126: $i > $i ).
thf(func_def_530,type,
sK127: $i > $i ).
thf(func_def_531,type,
sK128: $i > $i ).
thf(func_def_532,type,
sK129: $i > $i ).
thf(func_def_533,type,
sK130: $i > $i > $i ).
thf(func_def_534,type,
sK131: $i > $i > $i ).
thf(func_def_535,type,
sK132: $i > $i > $i ).
thf(func_def_536,type,
sK133: $i > $i > $i > $i ).
thf(func_def_537,type,
sK134: $i > $i > $i > $i ).
thf(func_def_538,type,
sK135: $i > $i ).
thf(func_def_561,type,
sK158: $i > $i > $i ).
thf(func_def_562,type,
sK159: $i > $i ).
thf(func_def_565,type,
sK162: $i > $i > ( $i > $i ) > $i ).
thf(func_def_566,type,
sK163: $i > ( $i > $i ) > $i > $i ).
thf(func_def_569,type,
sK166: $i > $i > $o ).
thf(func_def_570,type,
sK167: $i > $i > $i > $i > $i ).
thf(func_def_571,type,
sK168: $i > $i > $i > $i > $i ).
thf(func_def_572,type,
sK169: $i > $i > $i > $i > $i ).
thf(func_def_573,type,
sK170: $i > $i > $i > $i > $i ).
thf(func_def_585,type,
sK182: ( $i > $o ) > $i ).
thf(func_def_586,type,
sK183: $i > ( $i > $o ) > $i ).
thf(func_def_587,type,
sK184: $i > $o ).
thf(func_def_588,type,
sK185: ( $i > $o ) > $i > $i ).
thf(func_def_589,type,
sK186: ( $i > $o ) > $i > $i ).
thf(func_def_590,type,
sK187: $i > ( $i > $o ) > $i ).
thf(func_def_592,type,
sK189: $i > $o ).
thf(func_def_601,type,
sK198: $i > $o ).
thf(func_def_603,type,
sK200: $i > $i ).
thf(func_def_606,type,
sK203: $i > $i > ( $i > $i ) > $i ).
thf(func_def_613,type,
sK210: $i > $o ).
thf(func_def_633,type,
sK230: $i > $o ).
thf(func_def_634,type,
sK231: ( $i > $o ) > $i > $i ).
thf(func_def_635,type,
sK232: $i > $i > $i > $o ).
thf(func_def_636,type,
sK233: $i > $i > $i > $i ).
thf(func_def_656,type,
sK253: $i > $i > $i ).
thf(func_def_665,type,
sK262: $i > $i > $i > $i > $i ).
thf(func_def_666,type,
sK263: $i > $i > $i > $i > $i ).
thf(func_def_673,type,
sK270: $i > $i > $i ).
thf(func_def_674,type,
sK271: $i > $i > $i ).
thf(func_def_675,type,
sK272: $i > $o ).
thf(func_def_677,type,
sK274: $i > ( $i > $o ) > $i ).
thf(func_def_678,type,
sK275: $i > ( $i > $o ) > $i ).
thf(func_def_680,type,
sK277: $i > $o ).
thf(func_def_681,type,
sK278: ( $i > $o ) > $i > $i ).
thf(func_def_682,type,
sK279: $i > ( $i > $o ) > $i ).
thf(func_def_701,type,
sK298: $i > $i > $o ).
thf(func_def_716,type,
sK313: $i > ( $i > $o ) > $i > $i ).
thf(func_def_717,type,
sK314: $i > ( $i > $o ) > $i > $i ).
thf(func_def_718,type,
sK315: $i > ( $i > $o ) > $i ).
thf(func_def_721,type,
sK318: $i > $o ).
thf(func_def_732,type,
sK329: $i > $i > $i > $i > $i > $i ).
thf(func_def_742,type,
sK339: $i > $i > $i > $i ).
thf(func_def_746,type,
sK343: $i > $o ).
thf(func_def_748,type,
sK345: $i > ( $i > $o ) > $i ).
thf(func_def_749,type,
sK346: $i > ( $i > $o ) > $i ).
thf(func_def_773,type,
sK370: $i > $i > $o ).
thf(func_def_774,type,
sK371: ( $i > $i > $o ) > $i > $i ).
thf(func_def_775,type,
sK372: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_776,type,
sK373: ( $i > $i > $o ) > $i > $i ).
thf(func_def_777,type,
sK374: ( $i > $i > $o ) > $i > $i > $i ).
thf(func_def_780,type,
sK377: $i > $i > $i > $i ).
thf(func_def_781,type,
sK378: $i > $i > $i > $i > $i ).
thf(func_def_789,type,
sK386: $i > ( $i > $o ) > $i > $i ).
thf(func_def_790,type,
sK387: $i > ( $i > $o ) > $i > $i ).
thf(func_def_791,type,
sK388: ( $i > $o ) > $i > $i ).
thf(func_def_794,type,
sK391: $i > $o ).
thf(func_def_795,type,
sK392: $i > ( $i > $o ) > $i ).
thf(func_def_796,type,
sK393: ( $i > $o ) > $i ).
thf(func_def_797,type,
sK394: $i > $o ).
thf(func_def_806,type,
sK403: $i > $i > $i ).
thf(func_def_809,type,
sK406: $i > $i > $o ).
thf(func_def_833,type,
sK430: $i > $i ).
thf(func_def_835,type,
sK432: $i > $o ).
thf(func_def_840,type,
sK437: $i > $i > $i > $i ).
thf(func_def_841,type,
sK438: $i > $i > $i > $i ).
thf(func_def_874,type,
sK471: $i > $o ).
thf(func_def_875,type,
sK472: $i > ( $i > $o ) > ( $i > $o ) > $i > $i ).
thf(func_def_876,type,
sK473: $i > ( $i > $o ) > ( $i > $o ) > $i > $i ).
thf(func_def_879,type,
sK476: $i > $o ).
thf(func_def_880,type,
sK477: $i > $o ).
thf(func_def_881,type,
sK478: $i > ( $i > $o ) > $i > $i ).
thf(func_def_884,type,
sK481: $i > $o ).
thf(func_def_887,type,
sK484: $i > $i ).
thf(func_def_888,type,
sK485: ( $i > $i ) > $i > $i > $i ).
thf(func_def_889,type,
sK486: $i > $o ).
thf(func_def_890,type,
sK487: $i > $i > $i > $o > $i > $i > $i ).
thf(func_def_904,type,
sK501: $i > $o ).
thf(func_def_911,type,
sK508: $i > $i > $i > $i ).
thf(func_def_912,type,
sK509: $i > $i > $i > $i ).
thf(func_def_913,type,
sK510: $i > $i > $i > $i ).
thf(func_def_914,type,
sK511: $i > $i > $i > $i ).
thf(func_def_918,type,
sK515: $i > $i > $i > $i ).
thf(func_def_919,type,
sK516: $i > $i > $i > $i ).
thf(func_def_927,type,
sK524: $i > $i ).
thf(func_def_929,type,
sK526: $i > $o ).
thf(func_def_945,type,
sK542: $i > $i > $i ).
thf(func_def_949,type,
sK546: $i > $o ).
thf(func_def_951,type,
sK548: $i > ( $i > $o ) > $i ).
thf(func_def_952,type,
sK549: ( $i > $o ) > $i > $i ).
thf(func_def_968,type,
sK565: $i > $o ).
thf(func_def_969,type,
sK566: $i > $o ).
thf(func_def_970,type,
sK567: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_971,type,
sK568: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_986,type,
sK583: $i > $i > $i ).
thf(func_def_992,type,
sK589: $i > $i > $i > $i ).
thf(func_def_1006,type,
sK603: $i > $i ).
thf(func_def_1008,type,
sK605: $i > ( $i > $i ) > $i > $i ).
thf(func_def_1013,type,
sK610: $i > $o ).
thf(func_def_1021,type,
sK618: ( $i > $o ) > $i ).
thf(func_def_1022,type,
sK619: $i > $o ).
thf(func_def_1023,type,
sK620: ( $i > $o ) > $i ).
thf(func_def_1024,type,
sK621: $i > ( $i > $o ) > $i ).
thf(func_def_1025,type,
sK622: $i > $o ).
thf(func_def_1028,type,
sK625: $i > $i > $o ).
thf(func_def_1044,type,
sK641: $i > $o ).
thf(func_def_1047,type,
sK644: $i > $o ).
thf(func_def_1052,type,
sK649: $i > $o > $i > $i ).
thf(func_def_1058,type,
sK655: $i > $o ).
thf(func_def_1062,type,
sK659: $i > $i ).
thf(func_def_1071,type,
sK668: $i > $i ).
thf(func_def_1083,type,
sK680: $i > $i > $i > $o ).
thf(func_def_1097,type,
sK694: $i > $i > $i > ( $i > $o ) > $i ).
thf(func_def_1098,type,
sK695: $i > $i > $i > ( $i > $o ) > $i ).
thf(func_def_1099,type,
sK696: ( $i > $o ) > $i > $i ).
thf(func_def_1103,type,
sK700: $i > $o ).
thf(func_def_1106,type,
sK703: ( $i > $o ) > $i ).
thf(func_def_1107,type,
sK704: ( $i > $o ) > $i ).
thf(func_def_1108,type,
sK705: $i > $o ).
thf(func_def_1110,type,
sK707: $i > $o ).
thf(func_def_1111,type,
sK708: $i > $o ).
thf(func_def_1112,type,
sK709: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_1113,type,
sK710: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_1117,type,
sK714: $i > $i > $i > $i > $i ).
thf(func_def_1128,type,
sK725: $i > $i > $o ).
thf(func_def_1167,type,
sK764: $i > $i > $o ).
thf(func_def_1171,type,
sK768: $i > $i > $i > $i ).
thf(func_def_1172,type,
sK769: $i > $i > $i > $i > $i ).
thf(func_def_1193,type,
sK790: $i > $i ).
thf(func_def_1195,type,
sK792: ( $i > $i ) > $i > $i > $i ).
thf(func_def_1236,type,
sK833: $i > $i ).
thf(func_def_1245,type,
sK842: $i > ( $i > $o ) > $i ).
thf(func_def_1246,type,
sK843: ( $i > $o ) > $i ).
thf(func_def_1247,type,
sK844: $i > $o ).
thf(func_def_1266,type,
sK863: ( $i > $o ) > $i > $i ).
thf(func_def_1267,type,
sK864: ( $i > $o ) > $i > $i ).
thf(func_def_1268,type,
sK865: $i > ( $i > $o ) > $i ).
thf(func_def_1270,type,
sK867: $i > $o ).
thf(func_def_1284,type,
sK881: $i > $o ).
thf(func_def_1285,type,
sK882: ( $i > $o ) > $i > $i ).
thf(func_def_1286,type,
sK883: $i > ( $i > $o ) > $i ).
thf(func_def_1307,type,
sK904: $i > $i ).
thf(func_def_1308,type,
sK905: ( $i > $i ) > $i > $i > $i ).
thf(func_def_1347,type,
sK944: $i > $i > $o ).
thf(func_def_1365,type,
sK962: $i > $o ).
thf(func_def_1374,type,
sK971: $i > $i > $i ).
thf(func_def_1378,type,
sK975: $i > $i ).
thf(func_def_1379,type,
sK976: $i > $i ).
thf(func_def_1380,type,
sK977: ( $i > $i ) > $i > $i ).
thf(func_def_1381,type,
sK978: $i > ( $i > $i ) > $i > $i ).
thf(func_def_1382,type,
sK979: $i > $i ).
thf(func_def_1385,type,
sK982: $i > $i > ( $i > $i ) > $i ).
thf(func_def_1386,type,
sK983: $i > $i > $i ).
thf(func_def_1387,type,
sK984: $i > $i ).
thf(func_def_1390,type,
sK987: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_1391,type,
sK988: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_1392,type,
sK989: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_1393,type,
sK990: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_1396,type,
sK993: $i > $i > $o ).
thf(func_def_1413,type,
sK1010: $i > $i ).
thf(func_def_1414,type,
sK1011: ( $i > $i ) > $i > $i > $i ).
thf(func_def_1415,type,
sK1012: $i > ( $i > $i ) > $i > $i ).
thf(func_def_1420,type,
sK1017: $i > $i > $i > $i ).
thf(func_def_1421,type,
sK1018: $i > $i > $i > $i ).
thf(func_def_1439,type,
sK1036: $i > $i ).
thf(func_def_1440,type,
sK1037: $i > $i ).
thf(func_def_1447,type,
sK1044: $i > $i ).
thf(func_def_1448,type,
sK1045: $i > $i > $i > $i > $o ).
thf(func_def_1474,type,
sK1071: $i > $i ).
thf(func_def_1476,type,
sK1073: $i > ( $i > $i ) > $i > $i ).
thf(func_def_1477,type,
sK1074: ( $i > $i ) > $i > $i > $i ).
thf(func_def_1478,type,
sK1075: $i > ( $i > $i ) > $i > $i > $i ).
thf(func_def_1486,type,
sK1083: $i > $i ).
thf(func_def_1522,type,
sK1119: $i > $i > $o ).
thf(func_def_1525,type,
sK1122: $i > $o ).
thf(func_def_1530,type,
sK1127: $i > $i > $i > $i ).
thf(func_def_1531,type,
sK1128: $i > $i > $i > $i > $i ).
thf(func_def_1532,type,
sK1129: $i > $i > $i > $i > $i ).
thf(func_def_1533,type,
sK1130: $i > $i > $i > $i > $i ).
thf(func_def_1544,type,
sK1141: $i > $i > $i ).
thf(func_def_1557,type,
sK1154: $i > $o ).
thf(func_def_1558,type,
sK1155: ( $i > $o ) > $i > $i ).
thf(func_def_1559,type,
sK1156: ( $i > $o ) > $i > $i ).
thf(func_def_1568,type,
sK1165: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_1569,type,
sK1166: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_1570,type,
sK1167: ( $i > $o ) > $i > $i ).
thf(func_def_1574,type,
sK1171: $i > $o ).
thf(func_def_1585,type,
sK1182: $i > $i ).
thf(func_def_1587,type,
ph1184:
!>[X0: $tType] : X0 ).
thf(f9885,plain,
$false,
inference(avatar_sat_refutation,[status(thm)],[f9724,f9777,f9883]) ).
thf(f9883,plain,
spl1181_214,
inference(avatar_contradiction_clause,[status(thm)],[f9882]) ).
thf(f9882,plain,
( $false
| spl1181_214 ),
inference(trivial_inequality_removal,[status(thm)],[f9873]) ).
thf(f9873,plain,
( ( $false = $true )
| spl1181_214 ),
inference(backward_demodulation,[status(thm)],[f8737,f9854]) ).
thf(f9854,plain,
( ( $false
= ( in @ sK1183 @ sK955 ) )
| spl1181_214 ),
inference(trivial_inequality_removal,[status(thm)],[f9849]) ).
thf(f9849,plain,
( ( $true != $true )
| ( $false
= ( in @ sK1183 @ sK955 ) )
| spl1181_214 ),
inference(superposition,[status(thm)],[f9722,f7004]) ).
thf(f7004,plain,
! [X1: $i] :
( ( ( subset @ X1 @ sK955 )
= $true )
| ( ( in @ X1 @ sK955 )
= $false ) ),
inference(binary_proxy_clausification,[status(thm)],[f7003]) ).
thf(f7003,plain,
! [X1: $i] :
( ( ( in @ X1 @ sK955 )
=> ( subset @ X1 @ sK955 ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f7002]) ).
thf(f7002,plain,
! [X1: $i] :
( ( ^ [Y0: $i] :
( ( in @ Y0 @ sK955 )
=> ( subset @ Y0 @ sK955 ) )
@ X1 )
= $true ),
inference(pi_clausification,[status(thm)],[f7001]) ).
thf(f7001,plain,
( ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ sK955 )
=> ( subset @ Y0 @ sK955 ) ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f6464]) ).
thf(f6464,plain,
( $true
= ( ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ Y0 )
=> ( subset @ Y1 @ Y0 ) ) )
@ sK955 ) ),
inference(definition_unfolding,[status(thm)],[f4914,f3583]) ).
thf(f3583,plain,
( transitiveset
= ( ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ Y0 )
=> ( subset @ Y1 @ Y0 ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f869]) ).
thf(f869,plain,
( transitiveset
= ( ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ Y0 )
=> ( subset @ Y1 @ Y0 ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f868]) ).
thf(f868,plain,
( ( ^ [X0: $i] :
! [X1: $i] :
( ( in @ X1 @ X0 )
=> ( subset @ X1 @ X0 ) ) )
= transitiveset ),
inference(rectify,[status(thm)],[f331]) ).
thf(f331,axiom,
( ( ^ [X3: $i] :
! [X11: $i] :
( ( in @ X11 @ X3 )
=> ( subset @ X11 @ X3 ) ) )
= transitiveset ),
file('/export/starexec/sandbox2/tmp/tmp.H1wBmJGvZQ/DTF2THF_2486.p',transitiveset) ).
thf(f4914,plain,
( ( transitiveset @ sK955 )
= $true ),
inference(cnf_transformation,[status(thm)],[f3115]) ).
thf(f3115,plain,
( ( ubforcartprodlem2 = $true )
& ( funcGraphProp4 = $true )
& ( demorgan2a1 = $true )
& ( injFuncInInjFuncSet = $true )
& ( doubleComplementSub2 = $true )
& ( singletonprop = $true )
& ( symdiffE = $true )
& ( complementTE1 = $true )
& ( binunionTILcontra = $true )
& ( exuE1 = $true )
& ( woz1_2 = $true )
& ( subsetRefl = $true )
& ( notinemptyset = $true )
& ( demorgan2 = $true )
& ( ksndpairEq = $true )
& ( notinsingleton = $true )
& ( setminusERneg = $true )
& ( omegaIndAx = $true )
& ( setukpairinjL1 = $true )
& ( woz2B = $true )
& ( emptysetAx = $true )
& ( quantDeMorgan1 = $true )
& ( setminusLsub = $true )
& ( binintersectI = $true )
& ( funcinfuncset = $true )
& ( prop2set2propI = $true )
& ( breln1SetBreln1 = $true )
& ( emptyInPowerset = $true )
& ( setukpairinjR2 = $true )
& ( setadjoinAx = $true )
& ( funcGraphProp2 = $true )
& ( subsetemptysetimpeq = $true )
& ( powersetT_lem = $true )
& ( setoftrueEq = $true )
& ( theprop = $true )
& ( dsetconstr__Cong = $true )
& ( binintersectSubset3 = $true )
& ( iftrueorfalse = $true )
& ( binunionIR = $true )
& ( setukpairinjR1 = $true )
& ( demorgan2b = $true )
& ( vacuousDall = $true )
& ( dsetconstrEL = $true )
& ( binunionT_lem = $true )
& ( binintersectTERcontra = $true )
& ( setminusELneg = $true )
& ( setminusEL = $true )
& ( woz2Ex = $true )
& ( setminusIRneg = $true )
& ( breln1all1 = $true )
& ( woz13rule2 = $true )
& ( choice2fnsingleton = $true )
& ( sepSubset = $true )
& ( inCongP = $true )
& ( setunion__Cong = $true )
& ( setbeta = $true )
& ( iftrueProp1 = $true )
& ( powersetTI1 = $true )
& ( breln1compEex = $true )
& ( cartprodfstin = $true )
& ( ksndsingleton = $true )
& ( nonemptyI1 = $true )
& ( singletonsuniq = $true )
& ( subsetE2 = $true )
& ( breln1invprop = $true )
& ( demorgan1b = $true )
& ( breln1unionprop = $true )
& ( complementT_lem = $true )
& ( notdexE = $true )
& ( descr__Cong = $true )
& ( transitivesetOp1 = $true )
& ( contrasubsetT2 = $true )
& ( funcImageSingleton = $true )
& ( cartprodmempair1 = $true )
& ( image1Equiv = $true )
& ( app = $true )
& ( emptyset__Cong = $true )
& ( contraSubsetComplement = $true )
& ( complementImpComplementIntersect = $true )
& ( woz13rule3 = $true )
& ( setunionsingleton2 = $true )
& ( breln1invI = $true )
& ( surjFuncSetFuncIn = $true )
& ( binintersectTELcontra = $true )
& ( kpairiskpair = $true )
& ( binintersectSubset1 = $true )
& ( foundationAx = $true )
& ( image1E = $true )
& ( dpsetconstrEL2 = $true )
& ( prop2setE = $true )
& ( woz1_5 = $true )
& ( binunionTIRcontra = $true )
& ( setunionAx = $true )
& ( nonemptyImpWitness = $true )
& ( inComplementUnionImpNotIn1 = $true )
& ( ubforcartprodlem1 = $true )
& ( lam2p = $true )
& ( ap2apEq2 = $true )
& ( omegaSp = $true )
& ( symdiffI2 = $true )
& ( cartprodpairsurjEq = $true )
& ( kpairp = $true )
& ( in__Cong = $true )
& ( powersetI = $true )
& ( leftInvIsSurj = $true )
& ( setminusER = $true )
& ( upairsetIL = $true )
& ( cartprodfstpairEq = $true )
& ( subsetTI = $true )
& ( descrp = $true )
& ( demorgan1 = $true )
& ( lamp = $true )
& ( cartprodpairmemER = $true )
& ( powersetTE1 = $true )
& ( complementInPowersetComplementIntersect = $true )
& ( binintersectSubset4 = $true )
& ( subbreln1 = $true )
& ( apProp = $true )
& ( singletoninpowerset = $true )
& ( complementUnionInPowersetComplement = $true )
& ( powerset__Cong = $true )
& ( injFuncSetFuncInj = $true )
& ( binunionTEcontra = $true )
& ( theeq = $true )
& ( inIntersectImpInUnion = $true )
& ( subbreln = $true )
& ( beta1 = $true )
& ( setadjoinSub2 = $true )
& ( upairsetE = $true )
& ( breln1unionEcases = $true )
& ( emptyI = $true )
& ( eta2 = $true )
& ( replAx = $true )
& ( inIntersectImpInIntersectUnions = $true )
& ( demorgan1a = $true )
& ( surjFuncSetFuncSurj = $true )
& ( setunionsingleton = $true )
& ( beta2 = $true )
& ( binintersectLsub = $true )
& ( dpsetconstrSub = $true )
& ( upairset2E = $true )
& ( breln1unionE = $true )
& ( exuE3u = $true )
& ( setminusSubset1 = $true )
& ( funcGraphProp1 = $true )
& ( setadjoinIR = $true )
& ( ifSingleton = $true )
& ( demorgan2a2 = $true )
& ( woz1_4 = $true )
& ( lamProp = $true )
& ( cartprodpairmemEL = $true )
& ( dpsetconstrEL1 = $true )
& ( emptyinPowerset = $true )
& ( cartprodpairin = $true )
& ( secondinupair = $true )
& ( setukpairinjR11 = $true )
& ( omegaSAx = $true )
& ( breln1unionIR = $true )
& ( funcext2 = $true )
& ( ex1I = $true )
& ( upairset2IR = $true )
& ( symdiffIneg1 = $true )
& ( dpsetconstrER = $true )
& ( inIntersectImpInUnion2 = $true )
& ( emptyE1 = $true )
& ( subsetI2 = $true )
& ( binunionLsub = $true )
& ( binintersectSubset2 = $true )
& ( setOfPairsIsBReln = $true )
& ( emptyinunitempty = $true )
& ( dpsetconstrI = $true )
& ( eqbreln = $true )
& ( breln1all2 = $true )
& ( breln1unionIL = $true )
& ( setukpairinjR12 = $true )
& ( complementTnotintersectT = $true )
& ( breln1unionCommutes = $true )
& ( setextAx = $true )
& ( eqbreln1 = $true )
& ( image1Ex = $true )
& ( setadjoinE = $true )
& ( disjointsetsI1 = $true )
& ( uniqinunit = $true )
& ( exuE3e = $true )
& ( omega0Ax = $true )
& ( powersetE1 = $true )
& ( breln1compprop = $true )
& ( symdiffIneg2 = $true )
& ( setadjoinOr = $true )
& ( symdiffI1 = $true )
& ( doubleComplementI1 = $true )
& ( emptysetE = $true )
& ( contrasubsetT = $true )
& ( kpairsurjEq = $true )
& ( singletonsswitch = $true )
& ( upairsubunion = $true )
& ( infuncsetfunc = $true )
& ( setminusSubset2 = $true )
& ( sepInPowerset = $true )
& ( ap2p = $true )
& ( exu__Cong = $true )
& ( subset2powerset = $true )
& ( dsetconstrER = $true )
& ( singletonsubset = $true )
& ( notinself = $true )
& ( woz2W = $true )
& ( binunionEcases = $true )
& ( exuI1 = $true )
& ( upairinpowunion = $true )
& ( prop2setI = $true )
& ( powersetI1 = $true )
& ( funcextLem = $true )
& ( subsetI1 = $true )
& ( notequalI2 = $true )
& ( kfstpairEq = $true )
& ( subsetTrans = $true )
& ( emptysetsubset = $true )
& ( dsetconstrI = $true )
& ( doubleComplementEq = $true )
& ( intersectInPowersetIntersectUnions = $true )
& ( woz1_1 = $true )
& ( complementSubsetComplementIntersect = $true )
& ( ap2apEq1 = $true )
& ( breln1compI = $true )
& ( setminusT_lem = $true )
& ( doubleComplementSub1 = $true )
& ( wellorderingAx = $true )
& ( cartprodmempaircEq = $true )
& ( woz13rule4 = $true )
& ( iffalseProp1 = $true )
& ( inComplementUnionImpInComplement1 = $true )
& ( binunionRsub = $true )
& ( setukpairinjL = $true )
& ( iftrueProp2 = $true )
& ( iffalseProp2 = $true )
& ( powersetsubset = $true )
& ( doubleComplementE1 = $true )
& ( demorgan2b2 = $true )
& ( cartprodmempair = $true )
& ( injFuncSetFuncIn = $true )
& ( iffalse = $true )
& ( bs114d = $true )
& ( omegaSclos = $true )
& ( ifp = $true )
& ( ( transitiveset @ sK955 )
= $true )
& ( $true
!= ( transitiveset @ ( binintersect @ sK955 @ sK956 ) ) )
& ( $true
= ( transitiveset @ sK956 ) )
& ( setunionsingleton1 = $true )
& ( setOfPairsIsBReln1 = $true )
& ( powersetAx = $true )
& ( peanoSinj = $true )
& ( woz1_3 = $true )
& ( upairsetIR = $true )
& ( ex1E1 = $true )
& ( ubforcartprodlem3 = $true )
& ( notdallE = $true )
& ( exuI2 = $true )
& ( kfstsingleton = $true )
& ( setadjoinIL = $true )
& ( funcGraphProp3 = $true )
& ( notsubsetI = $true )
& ( upairequniteq = $true )
& ( notinself2 = $true )
& ( funcext = $true )
& ( complementTcontraSubset = $true )
& ( image1I = $true )
& ( binintersectEL = $true )
& ( woz13rule1 = $true )
& ( lam2lamEq = $true )
& ( setukpairinjR = $true )
& ( nonemptyE1 = $true )
& ( setukpairIL = $true )
& ( demorgan2a = $true )
& ( setukpairinjL2 = $true )
& ( image1Ex1 = $true )
& ( binintersectER = $true )
& ( nonemptyI = $true )
& ( iftrue = $true )
& ( breln1unionI = $true )
& ( inPowerset = $true )
& ( powersetE = $true )
& ( eta1 = $true )
& ( brelnall2 = $true )
& ( exuE2 = $true )
& ( setext = $true )
& ( quantDeMorgan4 = $true )
& ( subsetE = $true )
& ( woz13rule0 = $true )
& ( binintersectRsub = $true )
& ( woz2A = $true )
& ( setunionE2 = $true )
& ( ex1E2 = $true )
& ( omega__Cong = $true )
& ( breln1compE = $true )
& ( binunionE = $true )
& ( binunionIL = $true )
& ( eqinunit = $true )
& ( exuI3 = $true )
& ( binunionTE = $true )
& ( ex1I2 = $true )
& ( emptysetimpfalse = $true )
& ( quantDeMorgan3 = $true )
& ( binintersectSubset5 = $true )
& ( binintersectT_lem = $true )
& ( surjCantorThm = $true )
& ( setunionI = $true )
& ( breln1invE = $true )
& ( foundation2 = $true )
& ( setukpairIR = $true )
& ( setadjoinSub = $true )
& ( notequalI1 = $true )
& ( peano0notS = $true )
& ( quantDeMorgan2 = $true )
& ( subPowSU = $true )
& ( cartprodsndin = $true )
& ( eqimpsubset2 = $true )
& ( setminusI = $true )
& ( complementTI1 = $true )
& ( setunionE = $true )
& ( cartprodsndpairEq = $true )
& ( contrasubsetT1 = $true )
& ( dpsetconstrERa = $true )
& ( brelnall1 = $true )
& ( setminusILneg = $true )
& ( eqimpsubset1 = $true )
& ( setextT = $true )
& ( setadjoin__Cong = $true )
& ( contrasubsetT3 = $true )
& ( noeltsimpempty = $true )
& ( exuEu = $true )
& ( singletoninpowunion = $true )
& ( setextsub = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK955,sK956])],[f1132,f3114,f3113]) ).
thf(f3113,plain,
( ? [X0: $i] :
( ( ( transitiveset @ X0 )
= $true )
& ? [X1: $i] :
( ( ( transitiveset @ ( binintersect @ X0 @ X1 ) )
!= $true )
& ( ( transitiveset @ X1 )
= $true ) ) )
=> ( ( ( transitiveset @ sK955 )
= $true )
& ? [X1: $i] :
( ( ( transitiveset @ ( binintersect @ sK955 @ X1 ) )
!= $true )
& ( ( transitiveset @ X1 )
= $true ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f3114,plain,
( ? [X1: $i] :
( ( ( transitiveset @ ( binintersect @ sK955 @ X1 ) )
!= $true )
& ( ( transitiveset @ X1 )
= $true ) )
=> ( ( $true
!= ( transitiveset @ ( binintersect @ sK955 @ sK956 ) ) )
& ( $true
= ( transitiveset @ sK956 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1132,plain,
( ( ubforcartprodlem2 = $true )
& ( funcGraphProp4 = $true )
& ( demorgan2a1 = $true )
& ( injFuncInInjFuncSet = $true )
& ( doubleComplementSub2 = $true )
& ( singletonprop = $true )
& ( symdiffE = $true )
& ( complementTE1 = $true )
& ( binunionTILcontra = $true )
& ( exuE1 = $true )
& ( woz1_2 = $true )
& ( subsetRefl = $true )
& ( notinemptyset = $true )
& ( demorgan2 = $true )
& ( ksndpairEq = $true )
& ( notinsingleton = $true )
& ( setminusERneg = $true )
& ( omegaIndAx = $true )
& ( setukpairinjL1 = $true )
& ( woz2B = $true )
& ( emptysetAx = $true )
& ( quantDeMorgan1 = $true )
& ( setminusLsub = $true )
& ( binintersectI = $true )
& ( funcinfuncset = $true )
& ( prop2set2propI = $true )
& ( breln1SetBreln1 = $true )
& ( emptyInPowerset = $true )
& ( setukpairinjR2 = $true )
& ( setadjoinAx = $true )
& ( funcGraphProp2 = $true )
& ( subsetemptysetimpeq = $true )
& ( powersetT_lem = $true )
& ( setoftrueEq = $true )
& ( theprop = $true )
& ( dsetconstr__Cong = $true )
& ( binintersectSubset3 = $true )
& ( iftrueorfalse = $true )
& ( binunionIR = $true )
& ( setukpairinjR1 = $true )
& ( demorgan2b = $true )
& ( vacuousDall = $true )
& ( dsetconstrEL = $true )
& ( binunionT_lem = $true )
& ( binintersectTERcontra = $true )
& ( setminusELneg = $true )
& ( setminusEL = $true )
& ( woz2Ex = $true )
& ( setminusIRneg = $true )
& ( breln1all1 = $true )
& ( woz13rule2 = $true )
& ( choice2fnsingleton = $true )
& ( sepSubset = $true )
& ( inCongP = $true )
& ( setunion__Cong = $true )
& ( setbeta = $true )
& ( iftrueProp1 = $true )
& ( powersetTI1 = $true )
& ( breln1compEex = $true )
& ( cartprodfstin = $true )
& ( ksndsingleton = $true )
& ( nonemptyI1 = $true )
& ( singletonsuniq = $true )
& ( subsetE2 = $true )
& ( breln1invprop = $true )
& ( demorgan1b = $true )
& ( breln1unionprop = $true )
& ( complementT_lem = $true )
& ( notdexE = $true )
& ( descr__Cong = $true )
& ( transitivesetOp1 = $true )
& ( contrasubsetT2 = $true )
& ( funcImageSingleton = $true )
& ( cartprodmempair1 = $true )
& ( image1Equiv = $true )
& ( app = $true )
& ( emptyset__Cong = $true )
& ( contraSubsetComplement = $true )
& ( complementImpComplementIntersect = $true )
& ( woz13rule3 = $true )
& ( setunionsingleton2 = $true )
& ( breln1invI = $true )
& ( surjFuncSetFuncIn = $true )
& ( binintersectTELcontra = $true )
& ( kpairiskpair = $true )
& ( binintersectSubset1 = $true )
& ( foundationAx = $true )
& ( image1E = $true )
& ( dpsetconstrEL2 = $true )
& ( prop2setE = $true )
& ( woz1_5 = $true )
& ( binunionTIRcontra = $true )
& ( setunionAx = $true )
& ( nonemptyImpWitness = $true )
& ( inComplementUnionImpNotIn1 = $true )
& ( ubforcartprodlem1 = $true )
& ( lam2p = $true )
& ( ap2apEq2 = $true )
& ( omegaSp = $true )
& ( symdiffI2 = $true )
& ( cartprodpairsurjEq = $true )
& ( kpairp = $true )
& ( in__Cong = $true )
& ( powersetI = $true )
& ( leftInvIsSurj = $true )
& ( setminusER = $true )
& ( upairsetIL = $true )
& ( cartprodfstpairEq = $true )
& ( subsetTI = $true )
& ( descrp = $true )
& ( demorgan1 = $true )
& ( lamp = $true )
& ( cartprodpairmemER = $true )
& ( powersetTE1 = $true )
& ( complementInPowersetComplementIntersect = $true )
& ( binintersectSubset4 = $true )
& ( subbreln1 = $true )
& ( apProp = $true )
& ( singletoninpowerset = $true )
& ( complementUnionInPowersetComplement = $true )
& ( powerset__Cong = $true )
& ( injFuncSetFuncInj = $true )
& ( binunionTEcontra = $true )
& ( theeq = $true )
& ( inIntersectImpInUnion = $true )
& ( subbreln = $true )
& ( beta1 = $true )
& ( setadjoinSub2 = $true )
& ( upairsetE = $true )
& ( breln1unionEcases = $true )
& ( emptyI = $true )
& ( eta2 = $true )
& ( replAx = $true )
& ( inIntersectImpInIntersectUnions = $true )
& ( demorgan1a = $true )
& ( surjFuncSetFuncSurj = $true )
& ( setunionsingleton = $true )
& ( beta2 = $true )
& ( binintersectLsub = $true )
& ( dpsetconstrSub = $true )
& ( upairset2E = $true )
& ( breln1unionE = $true )
& ( exuE3u = $true )
& ( setminusSubset1 = $true )
& ( funcGraphProp1 = $true )
& ( setadjoinIR = $true )
& ( ifSingleton = $true )
& ( demorgan2a2 = $true )
& ( woz1_4 = $true )
& ( lamProp = $true )
& ( cartprodpairmemEL = $true )
& ( dpsetconstrEL1 = $true )
& ( emptyinPowerset = $true )
& ( cartprodpairin = $true )
& ( secondinupair = $true )
& ( setukpairinjR11 = $true )
& ( omegaSAx = $true )
& ( breln1unionIR = $true )
& ( funcext2 = $true )
& ( ex1I = $true )
& ( upairset2IR = $true )
& ( symdiffIneg1 = $true )
& ( dpsetconstrER = $true )
& ( inIntersectImpInUnion2 = $true )
& ( emptyE1 = $true )
& ( subsetI2 = $true )
& ( binunionLsub = $true )
& ( binintersectSubset2 = $true )
& ( setOfPairsIsBReln = $true )
& ( emptyinunitempty = $true )
& ( dpsetconstrI = $true )
& ( eqbreln = $true )
& ( breln1all2 = $true )
& ( breln1unionIL = $true )
& ( setukpairinjR12 = $true )
& ( complementTnotintersectT = $true )
& ( breln1unionCommutes = $true )
& ( setextAx = $true )
& ( eqbreln1 = $true )
& ( image1Ex = $true )
& ( setadjoinE = $true )
& ( disjointsetsI1 = $true )
& ( uniqinunit = $true )
& ( exuE3e = $true )
& ( omega0Ax = $true )
& ( powersetE1 = $true )
& ( breln1compprop = $true )
& ( symdiffIneg2 = $true )
& ( setadjoinOr = $true )
& ( symdiffI1 = $true )
& ( doubleComplementI1 = $true )
& ( emptysetE = $true )
& ( contrasubsetT = $true )
& ( kpairsurjEq = $true )
& ( singletonsswitch = $true )
& ( upairsubunion = $true )
& ( infuncsetfunc = $true )
& ( setminusSubset2 = $true )
& ( sepInPowerset = $true )
& ( ap2p = $true )
& ( exu__Cong = $true )
& ( subset2powerset = $true )
& ( dsetconstrER = $true )
& ( singletonsubset = $true )
& ( notinself = $true )
& ( woz2W = $true )
& ( binunionEcases = $true )
& ( exuI1 = $true )
& ( upairinpowunion = $true )
& ( prop2setI = $true )
& ( powersetI1 = $true )
& ( funcextLem = $true )
& ( subsetI1 = $true )
& ( notequalI2 = $true )
& ( kfstpairEq = $true )
& ( subsetTrans = $true )
& ( emptysetsubset = $true )
& ( dsetconstrI = $true )
& ( doubleComplementEq = $true )
& ( intersectInPowersetIntersectUnions = $true )
& ( woz1_1 = $true )
& ( complementSubsetComplementIntersect = $true )
& ( ap2apEq1 = $true )
& ( breln1compI = $true )
& ( setminusT_lem = $true )
& ( doubleComplementSub1 = $true )
& ( wellorderingAx = $true )
& ( cartprodmempaircEq = $true )
& ( woz13rule4 = $true )
& ( iffalseProp1 = $true )
& ( inComplementUnionImpInComplement1 = $true )
& ( binunionRsub = $true )
& ( setukpairinjL = $true )
& ( iftrueProp2 = $true )
& ( iffalseProp2 = $true )
& ( powersetsubset = $true )
& ( doubleComplementE1 = $true )
& ( demorgan2b2 = $true )
& ( cartprodmempair = $true )
& ( injFuncSetFuncIn = $true )
& ( iffalse = $true )
& ( bs114d = $true )
& ( omegaSclos = $true )
& ( ifp = $true )
& ? [X0: $i] :
( ( ( transitiveset @ X0 )
= $true )
& ? [X1: $i] :
( ( ( transitiveset @ ( binintersect @ X0 @ X1 ) )
!= $true )
& ( ( transitiveset @ X1 )
= $true ) ) )
& ( setunionsingleton1 = $true )
& ( setOfPairsIsBReln1 = $true )
& ( powersetAx = $true )
& ( peanoSinj = $true )
& ( woz1_3 = $true )
& ( upairsetIR = $true )
& ( ex1E1 = $true )
& ( ubforcartprodlem3 = $true )
& ( notdallE = $true )
& ( exuI2 = $true )
& ( kfstsingleton = $true )
& ( setadjoinIL = $true )
& ( funcGraphProp3 = $true )
& ( notsubsetI = $true )
& ( upairequniteq = $true )
& ( notinself2 = $true )
& ( funcext = $true )
& ( complementTcontraSubset = $true )
& ( image1I = $true )
& ( binintersectEL = $true )
& ( woz13rule1 = $true )
& ( lam2lamEq = $true )
& ( setukpairinjR = $true )
& ( nonemptyE1 = $true )
& ( setukpairIL = $true )
& ( demorgan2a = $true )
& ( setukpairinjL2 = $true )
& ( image1Ex1 = $true )
& ( binintersectER = $true )
& ( nonemptyI = $true )
& ( iftrue = $true )
& ( breln1unionI = $true )
& ( inPowerset = $true )
& ( powersetE = $true )
& ( eta1 = $true )
& ( brelnall2 = $true )
& ( exuE2 = $true )
& ( setext = $true )
& ( quantDeMorgan4 = $true )
& ( subsetE = $true )
& ( woz13rule0 = $true )
& ( binintersectRsub = $true )
& ( woz2A = $true )
& ( setunionE2 = $true )
& ( ex1E2 = $true )
& ( omega__Cong = $true )
& ( breln1compE = $true )
& ( binunionE = $true )
& ( binunionIL = $true )
& ( eqinunit = $true )
& ( exuI3 = $true )
& ( binunionTE = $true )
& ( ex1I2 = $true )
& ( emptysetimpfalse = $true )
& ( quantDeMorgan3 = $true )
& ( binintersectSubset5 = $true )
& ( binintersectT_lem = $true )
& ( surjCantorThm = $true )
& ( setunionI = $true )
& ( breln1invE = $true )
& ( foundation2 = $true )
& ( setukpairIR = $true )
& ( setadjoinSub = $true )
& ( notequalI1 = $true )
& ( peano0notS = $true )
& ( quantDeMorgan2 = $true )
& ( subPowSU = $true )
& ( cartprodsndin = $true )
& ( eqimpsubset2 = $true )
& ( setminusI = $true )
& ( complementTI1 = $true )
& ( setunionE = $true )
& ( cartprodsndpairEq = $true )
& ( contrasubsetT1 = $true )
& ( dpsetconstrERa = $true )
& ( brelnall1 = $true )
& ( setminusILneg = $true )
& ( eqimpsubset1 = $true )
& ( setextT = $true )
& ( setadjoin__Cong = $true )
& ( contrasubsetT3 = $true )
& ( noeltsimpempty = $true )
& ( exuEu = $true )
& ( singletoninpowunion = $true )
& ( setextsub = $true ) ),
inference(flattening,[status(thm)],[f1131]) ).
thf(f1131,plain,
( ? [X0: $i] :
( ( ( transitiveset @ X0 )
= $true )
& ? [X1: $i] :
( ( ( transitiveset @ ( binintersect @ X0 @ X1 ) )
!= $true )
& ( ( transitiveset @ X1 )
= $true ) ) )
& ( transitivesetOp1 = $true )
& ( peanoSinj = $true )
& ( peano0notS = $true )
& ( omegaSclos = $true )
& ( omegaSp = $true )
& ( notinself2 = $true )
& ( notinself = $true )
& ( foundation2 = $true )
& ( surjCantorThm = $true )
& ( leftInvIsSurj = $true )
& ( surjFuncSetFuncSurj = $true )
& ( surjFuncSetFuncIn = $true )
& ( injFuncSetFuncInj = $true )
& ( injFuncSetFuncIn = $true )
& ( injFuncInInjFuncSet = $true )
& ( image1I = $true )
& ( image1E = $true )
& ( image1Equiv = $true )
& ( image1Ex1 = $true )
& ( image1Ex = $true )
& ( woz2B = $true )
& ( woz2A = $true )
& ( woz2W = $true )
& ( woz2Ex = $true )
& ( breln1unionCommutes = $true )
& ( breln1unionEcases = $true )
& ( breln1unionE = $true )
& ( breln1unionI = $true )
& ( breln1unionIR = $true )
& ( breln1unionIL = $true )
& ( breln1unionprop = $true )
& ( breln1compEex = $true )
& ( breln1compE = $true )
& ( breln1compI = $true )
& ( breln1compprop = $true )
& ( breln1invE = $true )
& ( breln1invI = $true )
& ( breln1invprop = $true )
& ( eqbreln1 = $true )
& ( subbreln1 = $true )
& ( breln1all1 = $true )
& ( setOfPairsIsBReln1 = $true )
& ( choice2fnsingleton = $true )
& ( breln1SetBreln1 = $true )
& ( breln1all2 = $true )
& ( woz1_5 = $true )
& ( woz1_4 = $true )
& ( woz1_3 = $true )
& ( woz1_2 = $true )
& ( woz1_1 = $true )
& ( woz13rule4 = $true )
& ( woz13rule3 = $true )
& ( woz13rule2 = $true )
& ( woz13rule1 = $true )
& ( woz13rule0 = $true )
& ( demorgan2 = $true )
& ( demorgan2b = $true )
& ( demorgan2b2 = $true )
& ( demorgan2a = $true )
& ( demorgan1 = $true )
& ( demorgan1b = $true )
& ( demorgan1a = $true )
& ( demorgan2a2 = $true )
& ( complementUnionInPowersetComplement = $true )
& ( demorgan2a1 = $true )
& ( binunionTEcontra = $true )
& ( binunionTE = $true )
& ( inComplementUnionImpInComplement1 = $true )
& ( inComplementUnionImpNotIn1 = $true )
& ( intersectInPowersetIntersectUnions = $true )
& ( inIntersectImpInIntersectUnions = $true )
& ( inIntersectImpInUnion2 = $true )
& ( inIntersectImpInUnion = $true )
& ( binunionTIRcontra = $true )
& ( binunionTILcontra = $true )
& ( complementTcontraSubset = $true )
& ( contraSubsetComplement = $true )
& ( complementInPowersetComplementIntersect = $true )
& ( complementSubsetComplementIntersect = $true )
& ( complementImpComplementIntersect = $true )
& ( complementTnotintersectT = $true )
& ( doubleComplementEq = $true )
& ( doubleComplementSub2 = $true )
& ( doubleComplementSub1 = $true )
& ( doubleComplementE1 = $true )
& ( doubleComplementI1 = $true )
& ( contrasubsetT3 = $true )
& ( contrasubsetT2 = $true )
& ( contrasubsetT1 = $true )
& ( contrasubsetT = $true )
& ( binintersectTERcontra = $true )
& ( binintersectTELcontra = $true )
& ( complementTE1 = $true )
& ( complementTI1 = $true )
& ( powersetTE1 = $true )
& ( powersetTI1 = $true )
& ( subsetTI = $true )
& ( setextT = $true )
& ( complementT_lem = $true )
& ( setminusT_lem = $true )
& ( powersetT_lem = $true )
& ( binunionT_lem = $true )
& ( binintersectT_lem = $true )
& ( iftrueorfalse = $true )
& ( iffalse = $true )
& ( iftrue = $true )
& ( theeq = $true )
& ( ifp = $true )
& ( ifSingleton = $true )
& ( iftrueProp2 = $true )
& ( iftrueProp1 = $true )
& ( iffalseProp2 = $true )
& ( iffalseProp1 = $true )
& ( eta2 = $true )
& ( beta2 = $true )
& ( lam2lamEq = $true )
& ( eta1 = $true )
& ( beta1 = $true )
& ( ap2apEq2 = $true )
& ( ap2apEq1 = $true )
& ( funcext2 = $true )
& ( funcext = $true )
& ( eqbreln = $true )
& ( subbreln = $true )
& ( funcGraphProp4 = $true )
& ( funcextLem = $true )
& ( funcGraphProp2 = $true )
& ( funcGraphProp3 = $true )
& ( funcGraphProp1 = $true )
& ( ex1E2 = $true )
& ( brelnall2 = $true )
& ( brelnall1 = $true )
& ( lam2p = $true )
& ( lamp = $true )
& ( lamProp = $true )
& ( funcinfuncset = $true )
& ( ap2p = $true )
& ( infuncsetfunc = $true )
& ( app = $true )
& ( apProp = $true )
& ( funcImageSingleton = $true )
& ( dpsetconstrER = $true )
& ( dpsetconstrEL2 = $true )
& ( dpsetconstrEL1 = $true )
& ( dpsetconstrERa = $true )
& ( setOfPairsIsBReln = $true )
& ( dpsetconstrSub = $true )
& ( dpsetconstrI = $true )
& ( cartprodpairsurjEq = $true )
& ( cartprodsndpairEq = $true )
& ( cartprodfstpairEq = $true )
& ( cartprodmempaircEq = $true )
& ( cartprodpairmemER = $true )
& ( cartprodpairmemEL = $true )
& ( cartprodsndin = $true )
& ( kpairsurjEq = $true )
& ( ksndpairEq = $true )
& ( ksndsingleton = $true )
& ( setukpairinjR = $true )
& ( setukpairinjR2 = $true )
& ( upairequniteq = $true )
& ( setukpairinjR1 = $true )
& ( setukpairinjR12 = $true )
& ( setukpairinjR11 = $true )
& ( setukpairinjL = $true )
& ( setukpairinjL2 = $true )
& ( cartprodfstin = $true )
& ( kfstpairEq = $true )
& ( theprop = $true )
& ( kfstsingleton = $true )
& ( setukpairinjL1 = $true )
& ( singletonsuniq = $true )
& ( ex1I2 = $true )
& ( ex1I = $true )
& ( ex1E1 = $true )
& ( singletonprop = $true )
& ( setunionsingleton = $true )
& ( setunionsingleton2 = $true )
& ( setunionsingleton1 = $true )
& ( setunionE2 = $true )
& ( cartprodmempair = $true )
& ( cartprodmempair1 = $true )
& ( cartprodpairin = $true )
& ( ubforcartprodlem3 = $true )
& ( ubforcartprodlem2 = $true )
& ( ubforcartprodlem1 = $true )
& ( upairinpowunion = $true )
& ( upairsubunion = $true )
& ( upairset2E = $true )
& ( singletoninpowunion = $true )
& ( singletoninpowerset = $true )
& ( singletonsubset = $true )
& ( kpairp = $true )
& ( kpairiskpair = $true )
& ( setukpairIR = $true )
& ( setukpairIL = $true )
& ( secondinupair = $true )
& ( symdiffIneg2 = $true )
& ( symdiffIneg1 = $true )
& ( symdiffI2 = $true )
& ( symdiffI1 = $true )
& ( symdiffE = $true )
& ( setminusSubset1 = $true )
& ( setminusLsub = $true )
& ( setminusIRneg = $true )
& ( setminusILneg = $true )
& ( setminusELneg = $true )
& ( setminusERneg = $true )
& ( setminusSubset2 = $true )
& ( setminusER = $true )
& ( setminusEL = $true )
& ( setminusI = $true )
& ( bs114d = $true )
& ( binintersectSubset1 = $true )
& ( binintersectSubset4 = $true )
& ( binintersectRsub = $true )
& ( disjointsetsI1 = $true )
& ( binintersectER = $true )
& ( binintersectSubset3 = $true )
& ( binintersectSubset2 = $true )
& ( binintersectLsub = $true )
& ( binintersectEL = $true )
& ( binintersectSubset5 = $true )
& ( binintersectI = $true )
& ( binunionRsub = $true )
& ( binunionLsub = $true )
& ( binunionE = $true )
& ( binunionEcases = $true )
& ( binunionIR = $true )
& ( upairset2IR = $true )
& ( binunionIL = $true )
& ( sepSubset = $true )
& ( sepInPowerset = $true )
& ( powersetsubset = $true )
& ( inPowerset = $true )
& ( powersetE1 = $true )
& ( powersetI1 = $true )
& ( subsetemptysetimpeq = $true )
& ( setextsub = $true )
& ( subset2powerset = $true )
& ( setadjoinSub2 = $true )
& ( setadjoinSub = $true )
& ( subsetTrans = $true )
& ( subsetRefl = $true )
& ( notequalI2 = $true )
& ( notequalI1 = $true )
& ( notsubsetI = $true )
& ( subsetE2 = $true )
& ( subsetE = $true )
& ( emptysetsubset = $true )
& ( subsetI2 = $true )
& ( eqimpsubset1 = $true )
& ( eqimpsubset2 = $true )
& ( subsetI1 = $true )
& ( dsetconstr__Cong = $true )
& ( descr__Cong = $true )
& ( exuEu = $true )
& ( omega__Cong = $true )
& ( setunion__Cong = $true )
& ( powerset__Cong = $true )
& ( setadjoin__Cong = $true )
& ( emptyset__Cong = $true )
& ( exu__Cong = $true )
& ( exuE3u = $true )
& ( in__Cong = $true )
& ( inCongP = $true )
& ( exuI2 = $true )
& ( exuI3 = $true )
& ( exuI1 = $true )
& ( notdallE = $true )
& ( notdexE = $true )
& ( prop2set2propI = $true )
& ( prop2setI = $true )
& ( quantDeMorgan4 = $true )
& ( quantDeMorgan3 = $true )
& ( quantDeMorgan2 = $true )
& ( quantDeMorgan1 = $true )
& ( vacuousDall = $true )
& ( emptyE1 = $true )
& ( upairsetIR = $true )
& ( upairsetIL = $true )
& ( upairsetE = $true )
& ( singletonsswitch = $true )
& ( eqinunit = $true )
& ( notinsingleton = $true )
& ( uniqinunit = $true )
& ( nonemptyImpWitness = $true )
& ( exuE2 = $true )
& ( subPowSU = $true )
& ( setunionE = $true )
& ( setunionI = $true )
& ( powersetE = $true )
& ( emptyInPowerset = $true )
& ( emptyinPowerset = $true )
& ( powersetI = $true )
& ( setoftrueEq = $true )
& ( setadjoinOr = $true )
& ( setadjoinE = $true )
& ( setadjoinIR = $true )
& ( emptyinunitempty = $true )
& ( setadjoinIL = $true )
& ( nonemptyI1 = $true )
& ( nonemptyI = $true )
& ( nonemptyE1 = $true )
& ( setbeta = $true )
& ( noeltsimpempty = $true )
& ( emptyI = $true )
& ( setext = $true )
& ( exuE3e = $true )
& ( notinemptyset = $true )
& ( emptysetimpfalse = $true )
& ( emptysetE = $true )
& ( prop2setE = $true )
& ( exuE1 = $true )
& ( dsetconstrER = $true )
& ( dsetconstrEL = $true )
& ( dsetconstrI = $true )
& ( descrp = $true )
& ( wellorderingAx = $true )
& ( foundationAx = $true )
& ( replAx = $true )
& ( omegaIndAx = $true )
& ( omegaSAx = $true )
& ( omega0Ax = $true )
& ( setunionAx = $true )
& ( powersetAx = $true )
& ( setadjoinAx = $true )
& ( emptysetAx = $true )
& ( setextAx = $true ) ),
inference(ennf_transformation,[status(thm)],[f373]) ).
thf(f373,plain,
~ ( ( setextAx = $true )
=> ( ( emptysetAx = $true )
=> ( ( setadjoinAx = $true )
=> ( ( powersetAx = $true )
=> ( ( setunionAx = $true )
=> ( ( omega0Ax = $true )
=> ( ( omegaSAx = $true )
=> ( ( omegaIndAx = $true )
=> ( ( replAx = $true )
=> ( ( foundationAx = $true )
=> ( ( wellorderingAx = $true )
=> ( ( descrp = $true )
=> ( ( dsetconstrI = $true )
=> ( ( dsetconstrEL = $true )
=> ( ( dsetconstrER = $true )
=> ( ( exuE1 = $true )
=> ( ( prop2setE = $true )
=> ( ( emptysetE = $true )
=> ( ( emptysetimpfalse = $true )
=> ( ( notinemptyset = $true )
=> ( ( exuE3e = $true )
=> ( ( setext = $true )
=> ( ( emptyI = $true )
=> ( ( noeltsimpempty = $true )
=> ( ( setbeta = $true )
=> ( ( nonemptyE1 = $true )
=> ( ( nonemptyI = $true )
=> ( ( nonemptyI1 = $true )
=> ( ( setadjoinIL = $true )
=> ( ( emptyinunitempty = $true )
=> ( ( setadjoinIR = $true )
=> ( ( setadjoinE = $true )
=> ( ( setadjoinOr = $true )
=> ( ( setoftrueEq = $true )
=> ( ( powersetI = $true )
=> ( ( emptyinPowerset = $true )
=> ( ( emptyInPowerset = $true )
=> ( ( powersetE = $true )
=> ( ( setunionI = $true )
=> ( ( setunionE = $true )
=> ( ( subPowSU = $true )
=> ( ( exuE2 = $true )
=> ( ( nonemptyImpWitness = $true )
=> ( ( uniqinunit = $true )
=> ( ( notinsingleton = $true )
=> ( ( eqinunit = $true )
=> ( ( singletonsswitch = $true )
=> ( ( upairsetE = $true )
=> ( ( upairsetIL = $true )
=> ( ( upairsetIR = $true )
=> ( ( emptyE1 = $true )
=> ( ( vacuousDall = $true )
=> ( ( quantDeMorgan1 = $true )
=> ( ( quantDeMorgan2 = $true )
=> ( ( quantDeMorgan3 = $true )
=> ( ( quantDeMorgan4 = $true )
=> ( ( prop2setI = $true )
=> ( ( prop2set2propI = $true )
=> ( ( notdexE = $true )
=> ( ( notdallE = $true )
=> ( ( exuI1 = $true )
=> ( ( exuI3 = $true )
=> ( ( exuI2 = $true )
=> ( ( inCongP = $true )
=> ( ( in__Cong = $true )
=> ( ( exuE3u = $true )
=> ( ( exu__Cong = $true )
=> ( ( emptyset__Cong = $true )
=> ( ( setadjoin__Cong = $true )
=> ( ( powerset__Cong = $true )
=> ( ( setunion__Cong = $true )
=> ( ( omega__Cong = $true )
=> ( ( exuEu = $true )
=> ( ( descr__Cong = $true )
=> ( ( dsetconstr__Cong = $true )
=> ( ( subsetI1 = $true )
=> ( ( eqimpsubset2 = $true )
=> ( ( eqimpsubset1 = $true )
=> ( ( subsetI2 = $true )
=> ( ( emptysetsubset = $true )
=> ( ( subsetE = $true )
=> ( ( subsetE2 = $true )
=> ( ( notsubsetI = $true )
=> ( ( notequalI1 = $true )
=> ( ( notequalI2 = $true )
=> ( ( subsetRefl = $true )
=> ( ( subsetTrans = $true )
=> ( ( setadjoinSub = $true )
=> ( ( setadjoinSub2 = $true )
=> ( ( subset2powerset = $true )
=> ( ( setextsub = $true )
=> ( ( subsetemptysetimpeq = $true )
=> ( ( powersetI1 = $true )
=> ( ( powersetE1 = $true )
=> ( ( inPowerset = $true )
=> ( ( powersetsubset = $true )
=> ( ( sepInPowerset = $true )
=> ( ( sepSubset = $true )
=> ( ( binunionIL = $true )
=> ( ( upairset2IR = $true )
=> ( ( binunionIR = $true )
=> ( ( binunionEcases = $true )
=> ( ( binunionE = $true )
=> ( ( binunionLsub = $true )
=> ( ( binunionRsub = $true )
=> ( ( binintersectI = $true )
=> ( ( binintersectSubset5 = $true )
=> ( ( binintersectEL = $true )
=> ( ( binintersectLsub = $true )
=> ( ( binintersectSubset2 = $true )
=> ( ( binintersectSubset3 = $true )
=> ( ( binintersectER = $true )
=> ( ( disjointsetsI1 = $true )
=> ( ( binintersectRsub = $true )
=> ( ( binintersectSubset4 = $true )
=> ( ( binintersectSubset1 = $true )
=> ( ( bs114d = $true )
=> ( ( setminusI = $true )
=> ( ( setminusEL = $true )
=> ( ( setminusER = $true )
=> ( ( setminusSubset2 = $true )
=> ( ( setminusERneg = $true )
=> ( ( setminusELneg = $true )
=> ( ( setminusILneg = $true )
=> ( ( setminusIRneg = $true )
=> ( ( setminusLsub = $true )
=> ( ( setminusSubset1 = $true )
=> ( ( symdiffE = $true )
=> ( ( symdiffI1 = $true )
=> ( ( symdiffI2 = $true )
=> ( ( symdiffIneg1 = $true )
=> ( ( symdiffIneg2 = $true )
=> ( ( secondinupair = $true )
=> ( ( setukpairIL = $true )
=> ( ( setukpairIR = $true )
=> ( ( kpairiskpair = $true )
=> ( ( kpairp = $true )
=> ( ( singletonsubset = $true )
=> ( ( singletoninpowerset = $true )
=> ( ( singletoninpowunion = $true )
=> ( ( upairset2E = $true )
=> ( ( upairsubunion = $true )
=> ( ( upairinpowunion = $true )
=> ( ( ubforcartprodlem1 = $true )
=> ( ( ubforcartprodlem2 = $true )
=> ( ( ubforcartprodlem3 = $true )
=> ( ( cartprodpairin = $true )
=> ( ( cartprodmempair1 = $true )
=> ( ( cartprodmempair = $true )
=> ( ( setunionE2 = $true )
=> ( ( setunionsingleton1 = $true )
=> ( ( setunionsingleton2 = $true )
=> ( ( setunionsingleton = $true )
=> ( ( singletonprop = $true )
=> ( ( ex1E1 = $true )
=> ( ( ex1I = $true )
=> ( ( ex1I2 = $true )
=> ( ( singletonsuniq = $true )
=> ( ( setukpairinjL1 = $true )
=> ( ( kfstsingleton = $true )
=> ( ( theprop = $true )
=> ( ( kfstpairEq = $true )
=> ( ( cartprodfstin = $true )
=> ( ( setukpairinjL2 = $true )
=> ( ( setukpairinjL = $true )
=> ( ( setukpairinjR11 = $true )
=> ( ( setukpairinjR12 = $true )
=> ( ( setukpairinjR1 = $true )
=> ( ( upairequniteq = $true )
=> ( ( setukpairinjR2 = $true )
=> ( ( setukpairinjR = $true )
=> ( ( ksndsingleton = $true )
=> ( ( ksndpairEq = $true )
=> ( ( kpairsurjEq = $true )
=> ( ( cartprodsndin = $true )
=> ( ( cartprodpairmemEL = $true )
=> ( ( cartprodpairmemER = $true )
=> ( ( cartprodmempaircEq = $true )
=> ( ( cartprodfstpairEq = $true )
=> ( ( cartprodsndpairEq = $true )
=> ( ( cartprodpairsurjEq = $true )
=> ( ( dpsetconstrI = $true )
=> ( ( dpsetconstrSub = $true )
=> ( ( setOfPairsIsBReln = $true )
=> ( ( dpsetconstrERa = $true )
=> ( ( dpsetconstrEL1 = $true )
=> ( ( dpsetconstrEL2 = $true )
=> ( ( dpsetconstrER = $true )
=> ( ( funcImageSingleton = $true )
=> ( ( apProp = $true )
=> ( ( app = $true )
=> ( ( infuncsetfunc = $true )
=> ( ( ap2p = $true )
=> ( ( funcinfuncset = $true )
=> ( ( lamProp = $true )
=> ( ( lamp = $true )
=> ( ( lam2p = $true )
=> ( ( brelnall1 = $true )
=> ( ( brelnall2 = $true )
=> ( ( ex1E2 = $true )
=> ( ( funcGraphProp1 = $true )
=> ( ( funcGraphProp3 = $true )
=> ( ( funcGraphProp2 = $true )
=> ( ( funcextLem = $true )
=> ( ( funcGraphProp4 = $true )
=> ( ( subbreln = $true )
=> ( ( eqbreln = $true )
=> ( ( funcext = $true )
=> ( ( funcext2 = $true )
=> ( ( ap2apEq1 = $true )
=> ( ( ap2apEq2 = $true )
=> ( ( beta1 = $true )
=> ( ( eta1 = $true )
=> ( ( lam2lamEq = $true )
=> ( ( beta2 = $true )
=> ( ( eta2 = $true )
=> ( ( iffalseProp1 = $true )
=> ( ( iffalseProp2 = $true )
=> ( ( iftrueProp1 = $true )
=> ( ( iftrueProp2 = $true )
=> ( ( ifSingleton = $true )
=> ( ( ifp = $true )
=> ( ( theeq = $true )
=> ( ( iftrue = $true )
=> ( ( iffalse = $true )
=> ( ( iftrueorfalse = $true )
=> ( ( binintersectT_lem = $true )
=> ( ( binunionT_lem = $true )
=> ( ( powersetT_lem = $true )
=> ( ( setminusT_lem = $true )
=> ( ( complementT_lem = $true )
=> ( ( setextT = $true )
=> ( ( subsetTI = $true )
=> ( ( powersetTI1 = $true )
=> ( ( powersetTE1 = $true )
=> ( ( complementTI1 = $true )
=> ( ( complementTE1 = $true )
=> ( ( binintersectTELcontra = $true )
=> ( ( binintersectTERcontra = $true )
=> ( ( contrasubsetT = $true )
=> ( ( contrasubsetT1 = $true )
=> ( ( contrasubsetT2 = $true )
=> ( ( contrasubsetT3 = $true )
=> ( ( doubleComplementI1 = $true )
=> ( ( doubleComplementE1 = $true )
=> ( ( doubleComplementSub1 = $true )
=> ( ( doubleComplementSub2 = $true )
=> ( ( doubleComplementEq = $true )
=> ( ( complementTnotintersectT = $true )
=> ( ( complementImpComplementIntersect = $true )
=> ( ( complementSubsetComplementIntersect = $true )
=> ( ( complementInPowersetComplementIntersect = $true )
=> ( ( contraSubsetComplement = $true )
=> ( ( complementTcontraSubset = $true )
=> ( ( binunionTILcontra = $true )
=> ( ( binunionTIRcontra = $true )
=> ( ( inIntersectImpInUnion = $true )
=> ( ( inIntersectImpInUnion2 = $true )
=> ( ( inIntersectImpInIntersectUnions = $true )
=> ( ( intersectInPowersetIntersectUnions = $true )
=> ( ( inComplementUnionImpNotIn1 = $true )
=> ( ( inComplementUnionImpInComplement1 = $true )
=> ( ( binunionTE = $true )
=> ( ( binunionTEcontra = $true )
=> ( ( demorgan2a1 = $true )
=> ( ( complementUnionInPowersetComplement = $true )
=> ( ( demorgan2a2 = $true )
=> ( ( demorgan1a = $true )
=> ( ( demorgan1b = $true )
=> ( ( demorgan1 = $true )
=> ( ( demorgan2a = $true )
=> ( ( demorgan2b2 = $true )
=> ( ( demorgan2b = $true )
=> ( ( demorgan2 = $true )
=> ( ( woz13rule0 = $true )
=> ( ( woz13rule1 = $true )
=> ( ( woz13rule2 = $true )
=> ( ( woz13rule3 = $true )
=> ( ( woz13rule4 = $true )
=> ( ( woz1_1 = $true )
=> ( ( woz1_2 = $true )
=> ( ( woz1_3 = $true )
=> ( ( woz1_4 = $true )
=> ( ( woz1_5 = $true )
=> ( ( breln1all2 = $true )
=> ( ( breln1SetBreln1 = $true )
=> ( ( choice2fnsingleton = $true )
=> ( ( setOfPairsIsBReln1 = $true )
=> ( ( breln1all1 = $true )
=> ( ( subbreln1 = $true )
=> ( ( eqbreln1 = $true )
=> ( ( breln1invprop = $true )
=> ( ( breln1invI = $true )
=> ( ( breln1invE = $true )
=> ( ( breln1compprop = $true )
=> ( ( breln1compI = $true )
=> ( ( breln1compE = $true )
=> ( ( breln1compEex = $true )
=> ( ( breln1unionprop = $true )
=> ( ( breln1unionIL = $true )
=> ( ( breln1unionIR = $true )
=> ( ( breln1unionI = $true )
=> ( ( breln1unionE = $true )
=> ( ( breln1unionEcases = $true )
=> ( ( breln1unionCommutes = $true )
=> ( ( woz2Ex = $true )
=> ( ( woz2W = $true )
=> ( ( woz2A = $true )
=> ( ( woz2B = $true )
=> ( ( image1Ex = $true )
=> ( ( image1Ex1 = $true )
=> ( ( image1Equiv = $true )
=> ( ( image1E = $true )
=> ( ( image1I = $true )
=> ( ( injFuncInInjFuncSet = $true )
=> ( ( injFuncSetFuncIn = $true )
=> ( ( injFuncSetFuncInj = $true )
=> ( ( surjFuncSetFuncIn = $true )
=> ( ( surjFuncSetFuncSurj = $true )
=> ( ( leftInvIsSurj = $true )
=> ( ( surjCantorThm = $true )
=> ( ( foundation2 = $true )
=> ( ( notinself = $true )
=> ( ( notinself2 = $true )
=> ( ( omegaSp = $true )
=> ( ( omegaSclos = $true )
=> ( ( peano0notS = $true )
=> ( ( peanoSinj = $true )
=> ( ( transitivesetOp1 = $true )
=> ! [X0: $i] :
( ( ( transitiveset @ X0 )
= $true )
=> ! [X1: $i] :
( ( ( transitiveset @ X1 )
= $true )
=> ( ( transitiveset @ ( binintersect @ X0 @ X1 ) )
= $true ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f372]) ).
thf(f372,plain,
~ ( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> ( setunionE2
=> ( setunionsingleton1
=> ( setunionsingleton2
=> ( setunionsingleton
=> ( singletonprop
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> ( setukpairinjL1
=> ( kfstsingleton
=> ( theprop
=> ( kfstpairEq
=> ( cartprodfstin
=> ( setukpairinjL2
=> ( setukpairinjL
=> ( setukpairinjR11
=> ( setukpairinjR12
=> ( setukpairinjR1
=> ( upairequniteq
=> ( setukpairinjR2
=> ( setukpairinjR
=> ( ksndsingleton
=> ( ksndpairEq
=> ( kpairsurjEq
=> ( cartprodsndin
=> ( cartprodpairmemEL
=> ( cartprodpairmemER
=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ( cartprodpairsurjEq
=> ( dpsetconstrI
=> ( dpsetconstrSub
=> ( setOfPairsIsBReln
=> ( dpsetconstrERa
=> ( dpsetconstrEL1
=> ( dpsetconstrEL2
=> ( dpsetconstrER
=> ( funcImageSingleton
=> ( apProp
=> ( app
=> ( infuncsetfunc
=> ( ap2p
=> ( funcinfuncset
=> ( lamProp
=> ( lamp
=> ( lam2p
=> ( brelnall1
=> ( brelnall2
=> ( ex1E2
=> ( funcGraphProp1
=> ( funcGraphProp3
=> ( funcGraphProp2
=> ( funcextLem
=> ( funcGraphProp4
=> ( subbreln
=> ( eqbreln
=> ( funcext
=> ( funcext2
=> ( ap2apEq1
=> ( ap2apEq2
=> ( beta1
=> ( eta1
=> ( lam2lamEq
=> ( beta2
=> ( eta2
=> ( iffalseProp1
=> ( iffalseProp2
=> ( iftrueProp1
=> ( iftrueProp2
=> ( ifSingleton
=> ( ifp
=> ( theeq
=> ( iftrue
=> ( iffalse
=> ( iftrueorfalse
=> ( binintersectT_lem
=> ( binunionT_lem
=> ( powersetT_lem
=> ( setminusT_lem
=> ( complementT_lem
=> ( setextT
=> ( subsetTI
=> ( powersetTI1
=> ( powersetTE1
=> ( complementTI1
=> ( complementTE1
=> ( binintersectTELcontra
=> ( binintersectTERcontra
=> ( contrasubsetT
=> ( contrasubsetT1
=> ( contrasubsetT2
=> ( contrasubsetT3
=> ( doubleComplementI1
=> ( doubleComplementE1
=> ( doubleComplementSub1
=> ( doubleComplementSub2
=> ( doubleComplementEq
=> ( complementTnotintersectT
=> ( complementImpComplementIntersect
=> ( complementSubsetComplementIntersect
=> ( complementInPowersetComplementIntersect
=> ( contraSubsetComplement
=> ( complementTcontraSubset
=> ( binunionTILcontra
=> ( binunionTIRcontra
=> ( inIntersectImpInUnion
=> ( inIntersectImpInUnion2
=> ( inIntersectImpInIntersectUnions
=> ( intersectInPowersetIntersectUnions
=> ( inComplementUnionImpNotIn1
=> ( inComplementUnionImpInComplement1
=> ( binunionTE
=> ( binunionTEcontra
=> ( demorgan2a1
=> ( complementUnionInPowersetComplement
=> ( demorgan2a2
=> ( demorgan1a
=> ( demorgan1b
=> ( demorgan1
=> ( demorgan2a
=> ( demorgan2b2
=> ( demorgan2b
=> ( demorgan2
=> ( woz13rule0
=> ( woz13rule1
=> ( woz13rule2
=> ( woz13rule3
=> ( woz13rule4
=> ( woz1_1
=> ( woz1_2
=> ( woz1_3
=> ( woz1_4
=> ( woz1_5
=> ( breln1all2
=> ( breln1SetBreln1
=> ( choice2fnsingleton
=> ( setOfPairsIsBReln1
=> ( breln1all1
=> ( subbreln1
=> ( eqbreln1
=> ( breln1invprop
=> ( breln1invI
=> ( breln1invE
=> ( breln1compprop
=> ( breln1compI
=> ( breln1compE
=> ( breln1compEex
=> ( breln1unionprop
=> ( breln1unionIL
=> ( breln1unionIR
=> ( breln1unionI
=> ( breln1unionE
=> ( breln1unionEcases
=> ( breln1unionCommutes
=> ( woz2Ex
=> ( woz2W
=> ( woz2A
=> ( woz2B
=> ( image1Ex
=> ( image1Ex1
=> ( image1Equiv
=> ( image1E
=> ( image1I
=> ( injFuncInInjFuncSet
=> ( injFuncSetFuncIn
=> ( injFuncSetFuncInj
=> ( surjFuncSetFuncIn
=> ( surjFuncSetFuncSurj
=> ( leftInvIsSurj
=> ( surjCantorThm
=> ( foundation2
=> ( notinself
=> ( notinself2
=> ( omegaSp
=> ( omegaSclos
=> ( peano0notS
=> ( peanoSinj
=> ( transitivesetOp1
=> ! [X0: $i] :
( ( transitiveset @ X0 )
=> ! [X1: $i] :
( ( transitiveset @ X1 )
=> ( transitiveset @ ( binintersect @ X0 @ X1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f334]) ).
thf(f334,negated_conjecture,
~ ( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> ( setunionE2
=> ( setunionsingleton1
=> ( setunionsingleton2
=> ( setunionsingleton
=> ( singletonprop
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> ( setukpairinjL1
=> ( kfstsingleton
=> ( theprop
=> ( kfstpairEq
=> ( cartprodfstin
=> ( setukpairinjL2
=> ( setukpairinjL
=> ( setukpairinjR11
=> ( setukpairinjR12
=> ( setukpairinjR1
=> ( upairequniteq
=> ( setukpairinjR2
=> ( setukpairinjR
=> ( ksndsingleton
=> ( ksndpairEq
=> ( kpairsurjEq
=> ( cartprodsndin
=> ( cartprodpairmemEL
=> ( cartprodpairmemER
=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ( cartprodpairsurjEq
=> ( dpsetconstrI
=> ( dpsetconstrSub
=> ( setOfPairsIsBReln
=> ( dpsetconstrERa
=> ( dpsetconstrEL1
=> ( dpsetconstrEL2
=> ( dpsetconstrER
=> ( funcImageSingleton
=> ( apProp
=> ( app
=> ( infuncsetfunc
=> ( ap2p
=> ( funcinfuncset
=> ( lamProp
=> ( lamp
=> ( lam2p
=> ( brelnall1
=> ( brelnall2
=> ( ex1E2
=> ( funcGraphProp1
=> ( funcGraphProp3
=> ( funcGraphProp2
=> ( funcextLem
=> ( funcGraphProp4
=> ( subbreln
=> ( eqbreln
=> ( funcext
=> ( funcext2
=> ( ap2apEq1
=> ( ap2apEq2
=> ( beta1
=> ( eta1
=> ( lam2lamEq
=> ( beta2
=> ( eta2
=> ( iffalseProp1
=> ( iffalseProp2
=> ( iftrueProp1
=> ( iftrueProp2
=> ( ifSingleton
=> ( ifp
=> ( theeq
=> ( iftrue
=> ( iffalse
=> ( iftrueorfalse
=> ( binintersectT_lem
=> ( binunionT_lem
=> ( powersetT_lem
=> ( setminusT_lem
=> ( complementT_lem
=> ( setextT
=> ( subsetTI
=> ( powersetTI1
=> ( powersetTE1
=> ( complementTI1
=> ( complementTE1
=> ( binintersectTELcontra
=> ( binintersectTERcontra
=> ( contrasubsetT
=> ( contrasubsetT1
=> ( contrasubsetT2
=> ( contrasubsetT3
=> ( doubleComplementI1
=> ( doubleComplementE1
=> ( doubleComplementSub1
=> ( doubleComplementSub2
=> ( doubleComplementEq
=> ( complementTnotintersectT
=> ( complementImpComplementIntersect
=> ( complementSubsetComplementIntersect
=> ( complementInPowersetComplementIntersect
=> ( contraSubsetComplement
=> ( complementTcontraSubset
=> ( binunionTILcontra
=> ( binunionTIRcontra
=> ( inIntersectImpInUnion
=> ( inIntersectImpInUnion2
=> ( inIntersectImpInIntersectUnions
=> ( intersectInPowersetIntersectUnions
=> ( inComplementUnionImpNotIn1
=> ( inComplementUnionImpInComplement1
=> ( binunionTE
=> ( binunionTEcontra
=> ( demorgan2a1
=> ( complementUnionInPowersetComplement
=> ( demorgan2a2
=> ( demorgan1a
=> ( demorgan1b
=> ( demorgan1
=> ( demorgan2a
=> ( demorgan2b2
=> ( demorgan2b
=> ( demorgan2
=> ( woz13rule0
=> ( woz13rule1
=> ( woz13rule2
=> ( woz13rule3
=> ( woz13rule4
=> ( woz1_1
=> ( woz1_2
=> ( woz1_3
=> ( woz1_4
=> ( woz1_5
=> ( breln1all2
=> ( breln1SetBreln1
=> ( choice2fnsingleton
=> ( setOfPairsIsBReln1
=> ( breln1all1
=> ( subbreln1
=> ( eqbreln1
=> ( breln1invprop
=> ( breln1invI
=> ( breln1invE
=> ( breln1compprop
=> ( breln1compI
=> ( breln1compE
=> ( breln1compEex
=> ( breln1unionprop
=> ( breln1unionIL
=> ( breln1unionIR
=> ( breln1unionI
=> ( breln1unionE
=> ( breln1unionEcases
=> ( breln1unionCommutes
=> ( woz2Ex
=> ( woz2W
=> ( woz2A
=> ( woz2B
=> ( image1Ex
=> ( image1Ex1
=> ( image1Equiv
=> ( image1E
=> ( image1I
=> ( injFuncInInjFuncSet
=> ( injFuncSetFuncIn
=> ( injFuncSetFuncInj
=> ( surjFuncSetFuncIn
=> ( surjFuncSetFuncSurj
=> ( leftInvIsSurj
=> ( surjCantorThm
=> ( foundation2
=> ( notinself
=> ( notinself2
=> ( omegaSp
=> ( omegaSclos
=> ( peano0notS
=> ( peanoSinj
=> ( transitivesetOp1
=> ! [X11: $i] :
( ( transitiveset @ X11 )
=> ! [X16: $i] :
( ( transitiveset @ X16 )
=> ( transitiveset @ ( binintersect @ X11 @ X16 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f333]) ).
thf(f333,conjecture,
( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> ( setunionE2
=> ( setunionsingleton1
=> ( setunionsingleton2
=> ( setunionsingleton
=> ( singletonprop
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> ( setukpairinjL1
=> ( kfstsingleton
=> ( theprop
=> ( kfstpairEq
=> ( cartprodfstin
=> ( setukpairinjL2
=> ( setukpairinjL
=> ( setukpairinjR11
=> ( setukpairinjR12
=> ( setukpairinjR1
=> ( upairequniteq
=> ( setukpairinjR2
=> ( setukpairinjR
=> ( ksndsingleton
=> ( ksndpairEq
=> ( kpairsurjEq
=> ( cartprodsndin
=> ( cartprodpairmemEL
=> ( cartprodpairmemER
=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ( cartprodpairsurjEq
=> ( dpsetconstrI
=> ( dpsetconstrSub
=> ( setOfPairsIsBReln
=> ( dpsetconstrERa
=> ( dpsetconstrEL1
=> ( dpsetconstrEL2
=> ( dpsetconstrER
=> ( funcImageSingleton
=> ( apProp
=> ( app
=> ( infuncsetfunc
=> ( ap2p
=> ( funcinfuncset
=> ( lamProp
=> ( lamp
=> ( lam2p
=> ( brelnall1
=> ( brelnall2
=> ( ex1E2
=> ( funcGraphProp1
=> ( funcGraphProp3
=> ( funcGraphProp2
=> ( funcextLem
=> ( funcGraphProp4
=> ( subbreln
=> ( eqbreln
=> ( funcext
=> ( funcext2
=> ( ap2apEq1
=> ( ap2apEq2
=> ( beta1
=> ( eta1
=> ( lam2lamEq
=> ( beta2
=> ( eta2
=> ( iffalseProp1
=> ( iffalseProp2
=> ( iftrueProp1
=> ( iftrueProp2
=> ( ifSingleton
=> ( ifp
=> ( theeq
=> ( iftrue
=> ( iffalse
=> ( iftrueorfalse
=> ( binintersectT_lem
=> ( binunionT_lem
=> ( powersetT_lem
=> ( setminusT_lem
=> ( complementT_lem
=> ( setextT
=> ( subsetTI
=> ( powersetTI1
=> ( powersetTE1
=> ( complementTI1
=> ( complementTE1
=> ( binintersectTELcontra
=> ( binintersectTERcontra
=> ( contrasubsetT
=> ( contrasubsetT1
=> ( contrasubsetT2
=> ( contrasubsetT3
=> ( doubleComplementI1
=> ( doubleComplementE1
=> ( doubleComplementSub1
=> ( doubleComplementSub2
=> ( doubleComplementEq
=> ( complementTnotintersectT
=> ( complementImpComplementIntersect
=> ( complementSubsetComplementIntersect
=> ( complementInPowersetComplementIntersect
=> ( contraSubsetComplement
=> ( complementTcontraSubset
=> ( binunionTILcontra
=> ( binunionTIRcontra
=> ( inIntersectImpInUnion
=> ( inIntersectImpInUnion2
=> ( inIntersectImpInIntersectUnions
=> ( intersectInPowersetIntersectUnions
=> ( inComplementUnionImpNotIn1
=> ( inComplementUnionImpInComplement1
=> ( binunionTE
=> ( binunionTEcontra
=> ( demorgan2a1
=> ( complementUnionInPowersetComplement
=> ( demorgan2a2
=> ( demorgan1a
=> ( demorgan1b
=> ( demorgan1
=> ( demorgan2a
=> ( demorgan2b2
=> ( demorgan2b
=> ( demorgan2
=> ( woz13rule0
=> ( woz13rule1
=> ( woz13rule2
=> ( woz13rule3
=> ( woz13rule4
=> ( woz1_1
=> ( woz1_2
=> ( woz1_3
=> ( woz1_4
=> ( woz1_5
=> ( breln1all2
=> ( breln1SetBreln1
=> ( choice2fnsingleton
=> ( setOfPairsIsBReln1
=> ( breln1all1
=> ( subbreln1
=> ( eqbreln1
=> ( breln1invprop
=> ( breln1invI
=> ( breln1invE
=> ( breln1compprop
=> ( breln1compI
=> ( breln1compE
=> ( breln1compEex
=> ( breln1unionprop
=> ( breln1unionIL
=> ( breln1unionIR
=> ( breln1unionI
=> ( breln1unionE
=> ( breln1unionEcases
=> ( breln1unionCommutes
=> ( woz2Ex
=> ( woz2W
=> ( woz2A
=> ( woz2B
=> ( image1Ex
=> ( image1Ex1
=> ( image1Equiv
=> ( image1E
=> ( image1I
=> ( injFuncInInjFuncSet
=> ( injFuncSetFuncIn
=> ( injFuncSetFuncInj
=> ( surjFuncSetFuncIn
=> ( surjFuncSetFuncSurj
=> ( leftInvIsSurj
=> ( surjCantorThm
=> ( foundation2
=> ( notinself
=> ( notinself2
=> ( omegaSp
=> ( omegaSclos
=> ( peano0notS
=> ( peanoSinj
=> ( transitivesetOp1
=> ! [X11: $i] :
( ( transitiveset @ X11 )
=> ! [X16: $i] :
( ( transitiveset @ X16 )
=> ( transitiveset @ ( binintersect @ X11 @ X16 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.H1wBmJGvZQ/DTF2THF_2486.p',binintTransitive) ).
thf(f9722,plain,
( ( ( subset @ sK1183 @ sK955 )
!= $true )
| spl1181_214 ),
inference(avatar_component_clause,[status(thm)],[f9721]) ).
thf(f9721,definition,
( spl1181_214
<=> ( ( subset @ sK1183 @ sK955 )
= $true ) ),
introduced(definition,[new_symbols(naming,[spl1181_214])],[avatar_definition]) ).
thf(f8737,plain,
( $true
= ( in @ sK1183 @ sK955 ) ),
inference(trivial_inequality_removal,[status(thm)],[f8730]) ).
thf(f8730,plain,
( ( $true != $true )
| ( $true
= ( in @ sK1183 @ sK955 ) ) ),
inference(superposition,[status(thm)],[f6995,f6850]) ).
thf(f6850,plain,
( ( in @ sK1183 @ ( binintersect @ sK955 @ sK956 ) )
= $true ),
inference(binary_proxy_clausification,[status(thm)],[f6848]) ).
thf(f6848,plain,
( ( ( in @ sK1183 @ ( binintersect @ sK955 @ sK956 ) )
=> ( subset @ sK1183 @ ( binintersect @ sK955 @ sK956 ) ) )
= $false ),
inference(beta_eta_normalization,[status(thm)],[f6847]) ).
thf(f6847,plain,
( ( ^ [Y0: $i] :
( ( in @ Y0 @ ( binintersect @ sK955 @ sK956 ) )
=> ( subset @ Y0 @ ( binintersect @ sK955 @ sK956 ) ) )
@ sK1183 )
= $false ),
inference(sigma_clausification,[status(thm)],[f6846]) ).
thf(f6846,plain,
( ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ ( binintersect @ sK955 @ sK956 ) )
=> ( subset @ Y0 @ ( binintersect @ sK955 @ sK956 ) ) ) )
!= $true ),
inference(beta_eta_normalization,[status(thm)],[f6465]) ).
thf(f6465,plain,
( ( ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ Y0 )
=> ( subset @ Y1 @ Y0 ) ) )
@ ( binintersect @ sK955 @ sK956 ) )
!= $true ),
inference(definition_unfolding,[status(thm)],[f4913,f3583]) ).
thf(f4913,plain,
( $true
!= ( transitiveset @ ( binintersect @ sK955 @ sK956 ) ) ),
inference(cnf_transformation,[status(thm)],[f3115]) ).
thf(f6995,plain,
! [X3: $i,X4: $i,X5: $i] :
( ( ( in @ X4 @ ( binintersect @ X3 @ X5 ) )
!= $true )
| ( ( in @ X4 @ X3 )
= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f6227]) ).
thf(f6227,plain,
! [X3: $i,X4: $i,X5: $i] :
( ( ( in @ X4 @ ( binintersect @ X3 @ X5 ) )
!= $true )
| ( ( in @ X4 @ X3 )
= $true )
| ( $true != $true ) ),
inference(definition_unfolding,[status(thm)],[f4554,f4892]) ).
thf(f4892,plain,
binintersectEL = $true,
inference(cnf_transformation,[status(thm)],[f3115]) ).
thf(f4554,plain,
! [X3: $i,X4: $i,X5: $i] :
( ( ( in @ X4 @ X3 )
= $true )
| ( ( in @ X4 @ ( binintersect @ X3 @ X5 ) )
!= $true )
| ( binintersectEL != $true ) ),
inference(cnf_transformation,[status(thm)],[f2806]) ).
thf(f2806,plain,
( ( ( binintersectEL = $true )
| ( ( ( in @ sK761 @ sK760 )
!= $true )
& ( $true
= ( in @ sK761 @ ( binintersect @ sK760 @ sK762 ) ) ) ) )
& ( ! [X3: $i,X4: $i,X5: $i] :
( ( ( in @ X4 @ X3 )
= $true )
| ( ( in @ X4 @ ( binintersect @ X3 @ X5 ) )
!= $true ) )
| ( binintersectEL != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK760,sK761,sK762])],[f2804,f2805]) ).
thf(f2805,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ( $true
!= ( in @ X1 @ X0 ) )
& ( $true
= ( in @ X1 @ ( binintersect @ X0 @ X2 ) ) ) )
=> ( ( ( in @ sK761 @ sK760 )
!= $true )
& ( $true
= ( in @ sK761 @ ( binintersect @ sK760 @ sK762 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f2804,plain,
( ( ( binintersectEL = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( $true
!= ( in @ X1 @ X0 ) )
& ( $true
= ( in @ X1 @ ( binintersect @ X0 @ X2 ) ) ) ) )
& ( ! [X3: $i,X4: $i,X5: $i] :
( ( ( in @ X4 @ X3 )
= $true )
| ( ( in @ X4 @ ( binintersect @ X3 @ X5 ) )
!= $true ) )
| ( binintersectEL != $true ) ) ),
inference(rectify,[status(thm)],[f2803]) ).
thf(f2803,plain,
( ( ( binintersectEL = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( $true
!= ( in @ X1 @ X0 ) )
& ( $true
= ( in @ X1 @ ( binintersect @ X0 @ X2 ) ) ) ) )
& ( ! [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X1 @ X0 ) )
| ( $true
!= ( in @ X1 @ ( binintersect @ X0 @ X2 ) ) ) )
| ( binintersectEL != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f1077]) ).
thf(f1077,plain,
( ( binintersectEL = $true )
<=> ! [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X1 @ X0 ) )
| ( $true
!= ( in @ X1 @ ( binintersect @ X0 @ X2 ) ) ) ) ),
inference(ennf_transformation,[status(thm)],[f929]) ).
thf(f929,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X1 @ ( binintersect @ X0 @ X2 ) ) )
=> ( $true
= ( in @ X1 @ X0 ) ) )
<=> ( binintersectEL = $true ) ),
inference(fool_elimination,[status(thm)],[f928]) ).
thf(f928,plain,
( binintersectEL
= ( ! [X0: $i,X1: $i,X2: $i] :
( ( in @ X1 @ ( binintersect @ X0 @ X2 ) )
=> ( in @ X1 @ X0 ) ) ) ),
inference(rectify,[status(thm)],[f110]) ).
thf(f110,axiom,
( binintersectEL
= ( ! [X3: $i,X1: $i,X4: $i] :
( ( in @ X1 @ ( binintersect @ X3 @ X4 ) )
=> ( in @ X1 @ X3 ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.H1wBmJGvZQ/DTF2THF_2486.p',binintersectEL) ).
thf(f9777,plain,
spl1181_213,
inference(avatar_contradiction_clause,[status(thm)],[f9776]) ).
thf(f9776,plain,
( $false
| spl1181_213 ),
inference(trivial_inequality_removal,[status(thm)],[f9767]) ).
thf(f9767,plain,
( ( $false = $true )
| spl1181_213 ),
inference(backward_demodulation,[status(thm)],[f8639,f9733]) ).
thf(f9733,plain,
( ( ( in @ sK1183 @ sK956 )
= $false )
| spl1181_213 ),
inference(trivial_inequality_removal,[status(thm)],[f9725]) ).
thf(f9725,plain,
( ( ( in @ sK1183 @ sK956 )
= $false )
| ( $true != $true )
| spl1181_213 ),
inference(superposition,[status(thm)],[f9719,f6778]) ).
thf(f6778,plain,
! [X1: $i] :
( ( ( subset @ X1 @ sK956 )
= $true )
| ( ( in @ X1 @ sK956 )
= $false ) ),
inference(binary_proxy_clausification,[status(thm)],[f6777]) ).
thf(f6777,plain,
! [X1: $i] :
( $true
= ( ( in @ X1 @ sK956 )
=> ( subset @ X1 @ sK956 ) ) ),
inference(beta_eta_normalization,[status(thm)],[f6776]) ).
thf(f6776,plain,
! [X1: $i] :
( ( ^ [Y0: $i] :
( ( in @ Y0 @ sK956 )
=> ( subset @ Y0 @ sK956 ) )
@ X1 )
= $true ),
inference(pi_clausification,[status(thm)],[f6775]) ).
thf(f6775,plain,
( ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ sK956 )
=> ( subset @ Y0 @ sK956 ) ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f6466]) ).
thf(f6466,plain,
( $true
= ( ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ Y0 )
=> ( subset @ Y1 @ Y0 ) ) )
@ sK956 ) ),
inference(definition_unfolding,[status(thm)],[f4912,f3583]) ).
thf(f4912,plain,
( $true
= ( transitiveset @ sK956 ) ),
inference(cnf_transformation,[status(thm)],[f3115]) ).
thf(f9719,plain,
( ( ( subset @ sK1183 @ sK956 )
!= $true )
| spl1181_213 ),
inference(avatar_component_clause,[status(thm)],[f9718]) ).
thf(f9718,definition,
( spl1181_213
<=> ( ( subset @ sK1183 @ sK956 )
= $true ) ),
introduced(definition,[new_symbols(naming,[spl1181_213])],[avatar_definition]) ).
thf(f8639,plain,
( ( in @ sK1183 @ sK956 )
= $true ),
inference(trivial_inequality_removal,[status(thm)],[f8631]) ).
thf(f8631,plain,
( ( $true != $true )
| ( ( in @ sK1183 @ sK956 )
= $true ) ),
inference(superposition,[status(thm)],[f6761,f6850]) ).
thf(f6761,plain,
! [X2: $i,X0: $i,X1: $i] :
( ( $true
!= ( in @ X2 @ ( binintersect @ X0 @ X1 ) ) )
| ( ( in @ X2 @ X1 )
= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f6288]) ).
thf(f6288,plain,
! [X2: $i,X0: $i,X1: $i] :
( ( $true != $true )
| ( $true
!= ( in @ X2 @ ( binintersect @ X0 @ X1 ) ) )
| ( ( in @ X2 @ X1 )
= $true ) ),
inference(definition_unfolding,[status(thm)],[f4631,f4883]) ).
thf(f4883,plain,
binintersectER = $true,
inference(cnf_transformation,[status(thm)],[f3115]) ).
thf(f4631,plain,
! [X2: $i,X0: $i,X1: $i] :
( ( ( in @ X2 @ X1 )
= $true )
| ( $true
!= ( in @ X2 @ ( binintersect @ X0 @ X1 ) ) )
| ( binintersectER != $true ) ),
inference(cnf_transformation,[status(thm)],[f2887]) ).
thf(f2887,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X2 @ X1 )
= $true )
| ( $true
!= ( in @ X2 @ ( binintersect @ X0 @ X1 ) ) ) )
| ( binintersectER != $true ) )
& ( ( binintersectER = $true )
| ( ( ( in @ sK819 @ sK818 )
!= $true )
& ( ( in @ sK819 @ ( binintersect @ sK817 @ sK818 ) )
= $true ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK817,sK818,sK819])],[f2885,f2886]) ).
thf(f2886,plain,
( ? [X3: $i,X4: $i,X5: $i] :
( ( ( in @ X5 @ X4 )
!= $true )
& ( ( in @ X5 @ ( binintersect @ X3 @ X4 ) )
= $true ) )
=> ( ( ( in @ sK819 @ sK818 )
!= $true )
& ( ( in @ sK819 @ ( binintersect @ sK817 @ sK818 ) )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f2885,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X2 @ X1 )
= $true )
| ( $true
!= ( in @ X2 @ ( binintersect @ X0 @ X1 ) ) ) )
| ( binintersectER != $true ) )
& ( ( binintersectER = $true )
| ? [X3: $i,X4: $i,X5: $i] :
( ( ( in @ X5 @ X4 )
!= $true )
& ( ( in @ X5 @ ( binintersect @ X3 @ X4 ) )
= $true ) ) ) ),
inference(rectify,[status(thm)],[f2884]) ).
thf(f2884,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X2 @ X1 )
= $true )
| ( $true
!= ( in @ X2 @ ( binintersect @ X0 @ X1 ) ) ) )
| ( binintersectER != $true ) )
& ( ( binintersectER = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X2 @ X1 )
!= $true )
& ( $true
= ( in @ X2 @ ( binintersect @ X0 @ X1 ) ) ) ) ) ),
inference(nnf_transformation,[status(thm)],[f1087]) ).
thf(f1087,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X2 @ X1 )
= $true )
| ( $true
!= ( in @ X2 @ ( binintersect @ X0 @ X1 ) ) ) )
<=> ( binintersectER = $true ) ),
inference(ennf_transformation,[status(thm)],[f421]) ).
thf(f421,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X2 @ ( binintersect @ X0 @ X1 ) ) )
=> ( ( in @ X2 @ X1 )
= $true ) )
<=> ( binintersectER = $true ) ),
inference(fool_elimination,[status(thm)],[f420]) ).
thf(f420,plain,
( binintersectER
= ( ! [X0: $i,X1: $i,X2: $i] :
( ( in @ X2 @ ( binintersect @ X0 @ X1 ) )
=> ( in @ X2 @ X1 ) ) ) ),
inference(rectify,[status(thm)],[f114]) ).
thf(f114,axiom,
( binintersectER
= ( ! [X3: $i,X4: $i,X1: $i] :
( ( in @ X1 @ ( binintersect @ X3 @ X4 ) )
=> ( in @ X1 @ X4 ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.H1wBmJGvZQ/DTF2THF_2486.p',binintersectER) ).
thf(f9724,plain,
( ~ spl1181_213
| ~ spl1181_214 ),
inference(avatar_split_clause,[status(thm)],[f9714,f9721,f9718]) ).
thf(f9714,plain,
( ( ( subset @ sK1183 @ sK955 )
!= $true )
| ( ( subset @ sK1183 @ sK956 )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f9708]) ).
thf(f9708,plain,
( ( ( subset @ sK1183 @ sK956 )
!= $true )
| ( $false = $true )
| ( ( subset @ sK1183 @ sK955 )
!= $true ) ),
inference(superposition,[status(thm)],[f6849,f7005]) ).
thf(f7005,plain,
! [X3: $i,X4: $i,X5: $i] :
( ( ( subset @ X4 @ ( binintersect @ X3 @ X5 ) )
= $true )
| ( ( subset @ X4 @ X5 )
!= $true )
| ( ( subset @ X4 @ X3 )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f6169]) ).
thf(f6169,plain,
! [X3: $i,X4: $i,X5: $i] :
( ( ( subset @ X4 @ X5 )
!= $true )
| ( ( subset @ X4 @ ( binintersect @ X3 @ X5 ) )
= $true )
| ( ( subset @ X4 @ X3 )
!= $true )
| ( $true != $true ) ),
inference(definition_unfolding,[status(thm)],[f4489,f4856]) ).
thf(f4856,plain,
binintersectSubset5 = $true,
inference(cnf_transformation,[status(thm)],[f3115]) ).
thf(f4489,plain,
! [X3: $i,X4: $i,X5: $i] :
( ( ( subset @ X4 @ ( binintersect @ X3 @ X5 ) )
= $true )
| ( ( subset @ X4 @ X3 )
!= $true )
| ( ( subset @ X4 @ X5 )
!= $true )
| ( binintersectSubset5 != $true ) ),
inference(cnf_transformation,[status(thm)],[f2732]) ).
thf(f2732,plain,
( ( ( binintersectSubset5 = $true )
| ( ( ( subset @ sK712 @ ( binintersect @ sK711 @ sK713 ) )
!= $true )
& ( ( subset @ sK712 @ sK711 )
= $true )
& ( ( subset @ sK712 @ sK713 )
= $true ) ) )
& ( ! [X3: $i,X4: $i,X5: $i] :
( ( ( subset @ X4 @ ( binintersect @ X3 @ X5 ) )
= $true )
| ( ( subset @ X4 @ X3 )
!= $true )
| ( ( subset @ X4 @ X5 )
!= $true ) )
| ( binintersectSubset5 != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK711,sK712,sK713])],[f2730,f2731]) ).
thf(f2731,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ( ( subset @ X1 @ ( binintersect @ X0 @ X2 ) )
!= $true )
& ( ( subset @ X1 @ X0 )
= $true )
& ( ( subset @ X1 @ X2 )
= $true ) )
=> ( ( ( subset @ sK712 @ ( binintersect @ sK711 @ sK713 ) )
!= $true )
& ( ( subset @ sK712 @ sK711 )
= $true )
& ( ( subset @ sK712 @ sK713 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f2730,plain,
( ( ( binintersectSubset5 = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( ( subset @ X1 @ ( binintersect @ X0 @ X2 ) )
!= $true )
& ( ( subset @ X1 @ X0 )
= $true )
& ( ( subset @ X1 @ X2 )
= $true ) ) )
& ( ! [X3: $i,X4: $i,X5: $i] :
( ( ( subset @ X4 @ ( binintersect @ X3 @ X5 ) )
= $true )
| ( ( subset @ X4 @ X3 )
!= $true )
| ( ( subset @ X4 @ X5 )
!= $true ) )
| ( binintersectSubset5 != $true ) ) ),
inference(rectify,[status(thm)],[f2729]) ).
thf(f2729,plain,
( ( ( binintersectSubset5 = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( ( subset @ X1 @ ( binintersect @ X0 @ X2 ) )
!= $true )
& ( ( subset @ X1 @ X0 )
= $true )
& ( ( subset @ X1 @ X2 )
= $true ) ) )
& ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( subset @ X1 @ ( binintersect @ X0 @ X2 ) )
= $true )
| ( ( subset @ X1 @ X0 )
!= $true )
| ( ( subset @ X1 @ X2 )
!= $true ) )
| ( binintersectSubset5 != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f1383]) ).
thf(f1383,plain,
( ( binintersectSubset5 = $true )
<=> ! [X0: $i,X1: $i,X2: $i] :
( ( ( subset @ X1 @ ( binintersect @ X0 @ X2 ) )
= $true )
| ( ( subset @ X1 @ X0 )
!= $true )
| ( ( subset @ X1 @ X2 )
!= $true ) ) ),
inference(flattening,[status(thm)],[f1382]) ).
thf(f1382,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( ( subset @ X1 @ ( binintersect @ X0 @ X2 ) )
= $true )
| ( ( subset @ X1 @ X2 )
!= $true )
| ( ( subset @ X1 @ X0 )
!= $true ) )
<=> ( binintersectSubset5 = $true ) ),
inference(ennf_transformation,[status(thm)],[f488]) ).
thf(f488,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( ( subset @ X1 @ X0 )
= $true )
=> ( ( ( subset @ X1 @ X2 )
= $true )
=> ( ( subset @ X1 @ ( binintersect @ X0 @ X2 ) )
= $true ) ) )
<=> ( binintersectSubset5 = $true ) ),
inference(fool_elimination,[status(thm)],[f487]) ).
thf(f487,plain,
( binintersectSubset5
= ( ! [X0: $i,X1: $i,X2: $i] :
( ( subset @ X1 @ X0 )
=> ( ( subset @ X1 @ X2 )
=> ( subset @ X1 @ ( binintersect @ X0 @ X2 ) ) ) ) ) ),
inference(rectify,[status(thm)],[f109]) ).
thf(f109,axiom,
( binintersectSubset5
= ( ! [X3: $i,X5: $i,X4: $i] :
( ( subset @ X5 @ X3 )
=> ( ( subset @ X5 @ X4 )
=> ( subset @ X5 @ ( binintersect @ X3 @ X4 ) ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.H1wBmJGvZQ/DTF2THF_2486.p',binintersectSubset5) ).
thf(f6849,plain,
( ( subset @ sK1183 @ ( binintersect @ sK955 @ sK956 ) )
= $false ),
inference(binary_proxy_clausification,[status(thm)],[f6848]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : SEU813^1 : TPTP v9.2.1. Released v3.7.0.
% 0.00/0.12 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.13/0.31 % Computer : n019.cluster.edu
% 0.13/0.31 % Model : x86_64 x86_64
% 0.13/0.31 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.31 % Memory : 8042.1875MB
% 0.13/0.31 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.31 % CPULimit : 300
% 0.13/0.31 % WCLimit : 300
% 0.13/0.31 % DateTime : Tue Feb 24 12:34:54 EST 2026
% 0.13/0.31 % CPUTime :
% 0.13/0.31 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.16/0.41 ---- Original DTF file ---
% 0.16/0.41 thf(spec,logic,$$dhol).
% 0.16/0.41 %------------------------------------------------------------------------------
% 0.16/0.41 % File : SEU813^1 : TPTP v9.2.1. Released v3.7.0.
% 0.16/0.41 % Domain : Set Theory
% 0.16/0.41 % Problem : Transitive Sets
% 0.16/0.41 % Version : Especial.
% 0.16/0.41 % English : (! X:i.transitiveset X -> (! Y:i.transitiveset Y ->
% 0.16/0.41 % transitiveset (binintersect X Y)))
% 0.16/0.41
% 0.16/0.41 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.16/0.41 % Source : [Bro08]
% 0.16/0.41 % Names : ZFC315g [Bro08]
% 0.16/0.41
% 0.16/0.41 % Status : Theorem
% 0.16/0.41 % Rating : 0.22 v9.1.0, 0.25 v9.0.0, 0.30 v8.2.0, 0.38 v8.1.0, 0.27 v7.5.0, 0.29 v7.4.0, 0.56 v7.2.0, 0.50 v7.1.0, 0.62 v7.0.0, 0.57 v6.4.0, 0.50 v6.3.0, 0.60 v6.2.0, 0.57 v6.1.0, 0.71 v5.5.0, 0.83 v5.4.0, 0.60 v5.2.0, 1.00 v3.7.0
% 0.16/0.41 % Syntax : Number of formulae : 721 ( 332 unt; 388 typ; 332 def)
% 0.16/0.41 % Number of atoms : 2041 ( 477 equ; 0 cnn)
% 0.16/0.41 % Maximal formula atoms : 332 ( 6 avg)
% 0.16/0.41 % Number of connectives : 4446 ( 77 ~; 14 |; 54 &;3232 @)
% 0.16/0.41 % ( 17 <=>;1052 =>; 0 <=; 0 <~>)
% 0.16/0.41 % Maximal formula depth : 337 ( 2 avg)
% 0.16/0.41 % Number of types : 2 ( 0 usr)
% 0.16/0.41 % Number of type conns : 190 ( 190 >; 0 *; 0 +; 0 <<)
% 0.16/0.41 % Number of symbols : 391 ( 388 usr; 333 con; 0-4 aty)
% 0.16/0.41 % Number of variables : 1157 ( 76 ^;1037 !; 44 ?;1157 :)
% 0.16/0.41 % SPC : TH0_THM_EQU_NAR
% 0.16/0.41
% 0.16/0.41 % Comments : http://mathgate.info/detsetitem.php?id=501
% 0.16/0.41 %------------------------------------------------------------------------------
% 0.16/0.41 thf(in_type,type,
% 0.16/0.41 in: $i > $i > $o ).
% 0.16/0.41
% 0.16/0.41 thf(exu_type,type,
% 0.16/0.41 exu: ( $i > $o ) > $o ).
% 0.16/0.41
% 0.16/0.41 thf(exu,definition,
% 0.16/0.41 ( exu
% 0.16/0.41 = ( ^ [Xphi: $i > $o] :
% 0.16/0.41 ? [Xx: $i] :
% 0.16/0.41 ( ( Xphi @ Xx )
% 0.16/0.41 & ! [Xy: $i] :
% 0.16/0.41 ( ( Xphi @ Xy )
% 0.16/0.41 => ( Xx = Xy ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(setextAx_type,type,
% 0.16/0.41 setextAx: $o ).
% 0.16/0.41
% 0.16/0.41 thf(setextAx,definition,
% 0.16/0.41 ( setextAx
% 0.16/0.41 = ( ! [A: $i,B: $i] :
% 0.16/0.41 ( ! [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 <=> ( in @ Xx @ B ) )
% 0.16/0.41 => ( A = B ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(emptyset_type,type,
% 0.16/0.41 emptyset: $i ).
% 0.16/0.41
% 0.16/0.41 thf(emptysetAx_type,type,
% 0.16/0.41 emptysetAx: $o ).
% 0.16/0.41
% 0.16/0.41 thf(emptysetAx,definition,
% 0.16/0.41 ( emptysetAx
% 0.16/0.41 = ( ! [Xx: $i] :
% 0.16/0.41 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(setadjoin_type,type,
% 0.16/0.41 setadjoin: $i > $i > $i ).
% 0.16/0.41
% 0.16/0.41 thf(setadjoinAx_type,type,
% 0.16/0.41 setadjoinAx: $o ).
% 0.16/0.41
% 0.16/0.41 thf(setadjoinAx,definition,
% 0.16/0.41 ( setadjoinAx
% 0.16/0.41 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.16/0.41 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.16/0.41 <=> ( ( Xy = Xx )
% 0.16/0.41 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(powerset_type,type,
% 0.16/0.41 powerset: $i > $i ).
% 0.16/0.41
% 0.16/0.41 thf(powersetAx_type,type,
% 0.16/0.41 powersetAx: $o ).
% 0.16/0.41
% 0.16/0.41 thf(powersetAx,definition,
% 0.16/0.41 ( powersetAx
% 0.16/0.41 = ( ! [A: $i,B: $i] :
% 0.16/0.41 ( ( in @ B @ ( powerset @ A ) )
% 0.16/0.41 <=> ! [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ B )
% 0.16/0.41 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(setunion_type,type,
% 0.16/0.41 setunion: $i > $i ).
% 0.16/0.41
% 0.16/0.41 thf(setunionAx_type,type,
% 0.16/0.41 setunionAx: $o ).
% 0.16/0.41
% 0.16/0.41 thf(setunionAx,definition,
% 0.16/0.41 ( setunionAx
% 0.16/0.41 = ( ! [A: $i,Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ ( setunion @ A ) )
% 0.16/0.41 <=> ? [B: $i] :
% 0.16/0.41 ( ( in @ Xx @ B )
% 0.16/0.41 & ( in @ B @ A ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(omega_type,type,
% 0.16/0.41 omega: $i ).
% 0.16/0.41
% 0.16/0.41 thf(omega0Ax_type,type,
% 0.16/0.41 omega0Ax: $o ).
% 0.16/0.41
% 0.16/0.41 thf(omega0Ax,definition,
% 0.16/0.41 ( omega0Ax
% 0.16/0.41 = ( in @ emptyset @ omega ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(omegaSAx_type,type,
% 0.16/0.41 omegaSAx: $o ).
% 0.16/0.41
% 0.16/0.41 thf(omegaSAx,definition,
% 0.16/0.41 ( omegaSAx
% 0.16/0.41 = ( ! [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ omega )
% 0.16/0.41 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(omegaIndAx_type,type,
% 0.16/0.41 omegaIndAx: $o ).
% 0.16/0.41
% 0.16/0.41 thf(omegaIndAx,definition,
% 0.16/0.41 ( omegaIndAx
% 0.16/0.41 = ( ! [A: $i] :
% 0.16/0.41 ( ( ( in @ emptyset @ A )
% 0.16/0.41 & ! [Xx: $i] :
% 0.16/0.41 ( ( ( in @ Xx @ omega )
% 0.16/0.41 & ( in @ Xx @ A ) )
% 0.16/0.41 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.16/0.41 => ! [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ omega )
% 0.16/0.41 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(replAx_type,type,
% 0.16/0.41 replAx: $o ).
% 0.16/0.41
% 0.16/0.41 thf(replAx,definition,
% 0.16/0.41 ( replAx
% 0.16/0.41 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.16/0.41 ( ! [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 => ( exu
% 0.16/0.41 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.16/0.41 => ? [B: $i] :
% 0.16/0.41 ! [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ B )
% 0.16/0.41 <=> ? [Xy: $i] :
% 0.16/0.41 ( ( in @ Xy @ A )
% 0.16/0.41 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(foundationAx_type,type,
% 0.16/0.41 foundationAx: $o ).
% 0.16/0.41
% 0.16/0.41 thf(foundationAx,definition,
% 0.16/0.41 ( foundationAx
% 0.16/0.41 = ( ! [A: $i] :
% 0.16/0.41 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.16/0.41 => ? [B: $i] :
% 0.16/0.41 ( ( in @ B @ A )
% 0.16/0.41 & ~ ? [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ B )
% 0.16/0.41 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(wellorderingAx_type,type,
% 0.16/0.41 wellorderingAx: $o ).
% 0.16/0.41
% 0.16/0.41 thf(wellorderingAx,definition,
% 0.16/0.41 ( wellorderingAx
% 0.16/0.41 = ( ! [A: $i] :
% 0.16/0.41 ? [B: $i] :
% 0.16/0.41 ( ! [C: $i] :
% 0.16/0.41 ( ( in @ C @ B )
% 0.16/0.41 => ! [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ C )
% 0.16/0.41 => ( in @ Xx @ A ) ) )
% 0.16/0.41 & ! [Xx: $i,Xy: $i] :
% 0.16/0.41 ( ( ( in @ Xx @ A )
% 0.16/0.41 & ( in @ Xy @ A ) )
% 0.16/0.41 => ( ! [C: $i] :
% 0.16/0.41 ( ( in @ C @ B )
% 0.16/0.41 => ( ( in @ Xx @ C )
% 0.16/0.41 <=> ( in @ Xy @ C ) ) )
% 0.16/0.41 => ( Xx = Xy ) ) )
% 0.16/0.41 & ! [C: $i,D: $i] :
% 0.16/0.41 ( ( ( in @ C @ B )
% 0.16/0.41 & ( in @ D @ B ) )
% 0.16/0.41 => ( ! [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ C )
% 0.16/0.41 => ( in @ Xx @ D ) )
% 0.16/0.41 | ! [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ D )
% 0.16/0.41 => ( in @ Xx @ C ) ) ) )
% 0.16/0.41 & ! [C: $i] :
% 0.16/0.41 ( ( ! [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ C )
% 0.16/0.41 => ( in @ Xx @ A ) )
% 0.16/0.41 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.16/0.41 => ? [D: $i,Xx: $i] :
% 0.16/0.41 ( ( in @ D @ B )
% 0.16/0.41 & ( in @ Xx @ C )
% 0.16/0.41 & ~ ? [Xy: $i] :
% 0.16/0.41 ( ( in @ Xy @ D )
% 0.16/0.41 & ( in @ Xy @ C ) )
% 0.16/0.41 & ! [E: $i] :
% 0.16/0.41 ( ( in @ E @ B )
% 0.16/0.41 => ( ! [Xy: $i] :
% 0.16/0.41 ( ( in @ Xy @ E )
% 0.16/0.41 => ( in @ Xy @ D ) )
% 0.16/0.41 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(descr_type,type,
% 0.16/0.41 descr: ( $i > $o ) > $i ).
% 0.16/0.41
% 0.16/0.41 thf(descrp_type,type,
% 0.16/0.41 descrp: $o ).
% 0.16/0.41
% 0.16/0.41 thf(descrp,definition,
% 0.16/0.41 ( descrp
% 0.16/0.41 = ( ! [Xphi: $i > $o] :
% 0.16/0.41 ( ( exu
% 0.16/0.41 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.41 => ( Xphi
% 0.16/0.41 @ ( descr
% 0.16/0.41 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(dsetconstr_type,type,
% 0.16/0.41 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.16/0.41
% 0.16/0.41 thf(dsetconstrI_type,type,
% 0.16/0.41 dsetconstrI: $o ).
% 0.16/0.41
% 0.16/0.41 thf(dsetconstrI,definition,
% 0.16/0.41 ( dsetconstrI
% 0.16/0.41 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 => ( ( Xphi @ Xx )
% 0.16/0.41 => ( in @ Xx
% 0.16/0.41 @ ( dsetconstr @ A
% 0.16/0.41 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(dsetconstrEL_type,type,
% 0.16/0.41 dsetconstrEL: $o ).
% 0.16/0.41
% 0.16/0.41 thf(dsetconstrEL,definition,
% 0.16/0.41 ( dsetconstrEL
% 0.16/0.41 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.16/0.41 ( ( in @ Xx
% 0.16/0.41 @ ( dsetconstr @ A
% 0.16/0.41 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.16/0.41 => ( in @ Xx @ A ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(dsetconstrER_type,type,
% 0.16/0.41 dsetconstrER: $o ).
% 0.16/0.41
% 0.16/0.41 thf(dsetconstrER,definition,
% 0.16/0.41 ( dsetconstrER
% 0.16/0.41 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.16/0.41 ( ( in @ Xx
% 0.16/0.41 @ ( dsetconstr @ A
% 0.16/0.41 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.16/0.41 => ( Xphi @ Xx ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(exuE1_type,type,
% 0.16/0.41 exuE1: $o ).
% 0.16/0.41
% 0.16/0.41 thf(exuE1,definition,
% 0.16/0.41 ( exuE1
% 0.16/0.41 = ( ! [Xphi: $i > $o] :
% 0.16/0.41 ( ( exu
% 0.16/0.41 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.41 => ? [Xx: $i] :
% 0.16/0.41 ( ( Xphi @ Xx )
% 0.16/0.41 & ! [Xy: $i] :
% 0.16/0.41 ( ( Xphi @ Xy )
% 0.16/0.41 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(prop2set_type,type,
% 0.16/0.41 prop2set: $o > $i ).
% 0.16/0.41
% 0.16/0.41 thf(prop2setE_type,type,
% 0.16/0.41 prop2setE: $o ).
% 0.16/0.41
% 0.16/0.41 thf(prop2setE,definition,
% 0.16/0.41 ( prop2setE
% 0.16/0.41 = ( ! [Xphi: $o,Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.16/0.41 => Xphi ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(emptysetE_type,type,
% 0.16/0.41 emptysetE: $o ).
% 0.16/0.41
% 0.16/0.41 thf(emptysetE,definition,
% 0.16/0.41 ( emptysetE
% 0.16/0.41 = ( ! [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ emptyset )
% 0.16/0.41 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(emptysetimpfalse_type,type,
% 0.16/0.41 emptysetimpfalse: $o ).
% 0.16/0.41
% 0.16/0.41 thf(emptysetimpfalse,definition,
% 0.16/0.41 ( emptysetimpfalse
% 0.16/0.41 = ( ! [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ emptyset )
% 0.16/0.41 => $false ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(notinemptyset_type,type,
% 0.16/0.41 notinemptyset: $o ).
% 0.16/0.41
% 0.16/0.41 thf(notinemptyset,definition,
% 0.16/0.41 ( notinemptyset
% 0.16/0.41 = ( ! [Xx: $i] :
% 0.16/0.41 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(exuE3e_type,type,
% 0.16/0.41 exuE3e: $o ).
% 0.16/0.41
% 0.16/0.41 thf(exuE3e,definition,
% 0.16/0.41 ( exuE3e
% 0.16/0.41 = ( ! [Xphi: $i > $o] :
% 0.16/0.41 ( ( exu
% 0.16/0.41 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.41 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(setext_type,type,
% 0.16/0.41 setext: $o ).
% 0.16/0.41
% 0.16/0.41 thf(setext,definition,
% 0.16/0.41 ( setext
% 0.16/0.41 = ( ! [A: $i,B: $i] :
% 0.16/0.41 ( ! [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 => ( in @ Xx @ B ) )
% 0.16/0.41 => ( ! [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ B )
% 0.16/0.41 => ( in @ Xx @ A ) )
% 0.16/0.41 => ( A = B ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(emptyI_type,type,
% 0.16/0.41 emptyI: $o ).
% 0.16/0.41
% 0.16/0.41 thf(emptyI,definition,
% 0.16/0.41 ( emptyI
% 0.16/0.41 = ( ! [A: $i] :
% 0.16/0.41 ( ! [Xx: $i] :
% 0.16/0.41 ~ ( in @ Xx @ A )
% 0.16/0.41 => ( A = emptyset ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(noeltsimpempty_type,type,
% 0.16/0.41 noeltsimpempty: $o ).
% 0.16/0.41
% 0.16/0.41 thf(noeltsimpempty,definition,
% 0.16/0.41 ( noeltsimpempty
% 0.16/0.41 = ( ! [A: $i] :
% 0.16/0.41 ( ! [Xx: $i] :
% 0.16/0.41 ~ ( in @ Xx @ A )
% 0.16/0.41 => ( A = emptyset ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(setbeta_type,type,
% 0.16/0.41 setbeta: $o ).
% 0.16/0.41
% 0.16/0.41 thf(setbeta,definition,
% 0.16/0.41 ( setbeta
% 0.16/0.41 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 => ( ( in @ Xx
% 0.16/0.41 @ ( dsetconstr @ A
% 0.16/0.41 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.16/0.41 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(nonempty_type,type,
% 0.16/0.41 nonempty: $i > $o ).
% 0.16/0.41
% 0.16/0.41 thf(nonempty,definition,
% 0.16/0.41 ( nonempty
% 0.16/0.41 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(nonemptyE1_type,type,
% 0.16/0.41 nonemptyE1: $o ).
% 0.16/0.41
% 0.16/0.41 thf(nonemptyE1,definition,
% 0.16/0.41 ( nonemptyE1
% 0.16/0.41 = ( ! [A: $i] :
% 0.16/0.41 ( ( nonempty @ A )
% 0.16/0.41 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(nonemptyI_type,type,
% 0.16/0.41 nonemptyI: $o ).
% 0.16/0.41
% 0.16/0.41 thf(nonemptyI,definition,
% 0.16/0.41 ( nonemptyI
% 0.16/0.41 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 => ( ( Xphi @ Xx )
% 0.16/0.41 => ( nonempty
% 0.16/0.41 @ ( dsetconstr @ A
% 0.16/0.41 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(nonemptyI1_type,type,
% 0.16/0.41 nonemptyI1: $o ).
% 0.16/0.41
% 0.16/0.41 thf(nonemptyI1,definition,
% 0.16/0.41 ( nonemptyI1
% 0.16/0.41 = ( ! [A: $i] :
% 0.16/0.41 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.16/0.41 => ( nonempty @ A ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(setadjoinIL_type,type,
% 0.16/0.41 setadjoinIL: $o ).
% 0.16/0.41
% 0.16/0.41 thf(setadjoinIL,definition,
% 0.16/0.41 ( setadjoinIL
% 0.16/0.41 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(emptyinunitempty_type,type,
% 0.16/0.41 emptyinunitempty: $o ).
% 0.16/0.41
% 0.16/0.41 thf(emptyinunitempty,definition,
% 0.16/0.41 ( emptyinunitempty
% 0.16/0.41 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(setadjoinIR_type,type,
% 0.16/0.41 setadjoinIR: $o ).
% 0.16/0.41
% 0.16/0.41 thf(setadjoinIR,definition,
% 0.16/0.41 ( setadjoinIR
% 0.16/0.41 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.16/0.41 ( ( in @ Xy @ A )
% 0.16/0.41 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(setadjoinE_type,type,
% 0.16/0.41 setadjoinE: $o ).
% 0.16/0.41
% 0.16/0.41 thf(setadjoinE,definition,
% 0.16/0.41 ( setadjoinE
% 0.16/0.41 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.16/0.41 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.16/0.41 => ! [Xphi: $o] :
% 0.16/0.41 ( ( ( Xy = Xx )
% 0.16/0.41 => Xphi )
% 0.16/0.41 => ( ( ( in @ Xy @ A )
% 0.16/0.41 => Xphi )
% 0.16/0.41 => Xphi ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(setadjoinOr_type,type,
% 0.16/0.41 setadjoinOr: $o ).
% 0.16/0.41
% 0.16/0.41 thf(setadjoinOr,definition,
% 0.16/0.41 ( setadjoinOr
% 0.16/0.41 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.16/0.41 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.16/0.41 => ( ( Xy = Xx )
% 0.16/0.41 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(setoftrueEq_type,type,
% 0.16/0.41 setoftrueEq: $o ).
% 0.16/0.41
% 0.16/0.41 thf(setoftrueEq,definition,
% 0.16/0.41 ( setoftrueEq
% 0.16/0.41 = ( ! [A: $i] :
% 0.16/0.41 ( ( dsetconstr @ A
% 0.16/0.41 @ ^ [Xx: $i] : $true )
% 0.16/0.41 = A ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(powersetI_type,type,
% 0.16/0.41 powersetI: $o ).
% 0.16/0.41
% 0.16/0.41 thf(powersetI,definition,
% 0.16/0.41 ( powersetI
% 0.16/0.41 = ( ! [A: $i,B: $i] :
% 0.16/0.41 ( ! [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ B )
% 0.16/0.41 => ( in @ Xx @ A ) )
% 0.16/0.41 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(emptyinPowerset_type,type,
% 0.16/0.41 emptyinPowerset: $o ).
% 0.16/0.41
% 0.16/0.41 thf(emptyinPowerset,definition,
% 0.16/0.41 ( emptyinPowerset
% 0.16/0.41 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(emptyInPowerset_type,type,
% 0.16/0.41 emptyInPowerset: $o ).
% 0.16/0.41
% 0.16/0.41 thf(emptyInPowerset,definition,
% 0.16/0.41 ( emptyInPowerset
% 0.16/0.41 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(powersetE_type,type,
% 0.16/0.41 powersetE: $o ).
% 0.16/0.41
% 0.16/0.41 thf(powersetE,definition,
% 0.16/0.41 ( powersetE
% 0.16/0.41 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.41 ( ( in @ B @ ( powerset @ A ) )
% 0.16/0.41 => ( ( in @ Xx @ B )
% 0.16/0.41 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(setunionI_type,type,
% 0.16/0.41 setunionI: $o ).
% 0.16/0.41
% 0.16/0.41 thf(setunionI,definition,
% 0.16/0.41 ( setunionI
% 0.16/0.41 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.16/0.41 ( ( in @ Xx @ B )
% 0.16/0.41 => ( ( in @ B @ A )
% 0.16/0.41 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(setunionE_type,type,
% 0.16/0.41 setunionE: $o ).
% 0.16/0.41
% 0.16/0.41 thf(setunionE,definition,
% 0.16/0.41 ( setunionE
% 0.16/0.41 = ( ! [A: $i,Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ ( setunion @ A ) )
% 0.16/0.41 => ! [Xphi: $o] :
% 0.16/0.41 ( ! [B: $i] :
% 0.16/0.41 ( ( in @ Xx @ B )
% 0.16/0.41 => ( ( in @ B @ A )
% 0.16/0.41 => Xphi ) )
% 0.16/0.41 => Xphi ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(subPowSU_type,type,
% 0.16/0.41 subPowSU: $o ).
% 0.16/0.41
% 0.16/0.41 thf(subPowSU,definition,
% 0.16/0.41 ( subPowSU
% 0.16/0.41 = ( ! [A: $i,Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(exuE2_type,type,
% 0.16/0.41 exuE2: $o ).
% 0.16/0.41
% 0.16/0.41 thf(exuE2,definition,
% 0.16/0.41 ( exuE2
% 0.16/0.41 = ( ! [Xphi: $i > $o] :
% 0.16/0.41 ( ( exu
% 0.16/0.41 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.41 => ? [Xx: $i] :
% 0.16/0.41 ! [Xy: $i] :
% 0.16/0.41 ( ( Xphi @ Xy )
% 0.16/0.41 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(nonemptyImpWitness_type,type,
% 0.16/0.41 nonemptyImpWitness: $o ).
% 0.16/0.41
% 0.16/0.41 thf(nonemptyImpWitness,definition,
% 0.16/0.41 ( nonemptyImpWitness
% 0.16/0.41 = ( ! [A: $i] :
% 0.16/0.41 ( ( nonempty @ A )
% 0.16/0.41 => ? [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 & $true ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(uniqinunit_type,type,
% 0.16/0.41 uniqinunit: $o ).
% 0.16/0.41
% 0.16/0.41 thf(uniqinunit,definition,
% 0.16/0.41 ( uniqinunit
% 0.16/0.41 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.41 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.16/0.41 => ( Xx = Xy ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(notinsingleton_type,type,
% 0.16/0.41 notinsingleton: $o ).
% 0.16/0.41
% 0.16/0.41 thf(notinsingleton,definition,
% 0.16/0.41 ( notinsingleton
% 0.16/0.41 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.41 ( ( Xx != Xy )
% 0.16/0.41 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(eqinunit_type,type,
% 0.16/0.41 eqinunit: $o ).
% 0.16/0.41
% 0.16/0.41 thf(eqinunit,definition,
% 0.16/0.41 ( eqinunit
% 0.16/0.41 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.41 ( ( Xx = Xy )
% 0.16/0.41 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(singletonsswitch_type,type,
% 0.16/0.41 singletonsswitch: $o ).
% 0.16/0.41
% 0.16/0.41 thf(singletonsswitch,definition,
% 0.16/0.41 ( singletonsswitch
% 0.16/0.41 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.41 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.16/0.41 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(upairsetE_type,type,
% 0.16/0.41 upairsetE: $o ).
% 0.16/0.41
% 0.16/0.41 thf(upairsetE,definition,
% 0.16/0.41 ( upairsetE
% 0.16/0.41 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.16/0.41 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.16/0.41 => ( ( Xz = Xx )
% 0.16/0.41 | ( Xz = Xy ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(upairsetIL_type,type,
% 0.16/0.41 upairsetIL: $o ).
% 0.16/0.41
% 0.16/0.41 thf(upairsetIL,definition,
% 0.16/0.41 ( upairsetIL
% 0.16/0.41 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(upairsetIR_type,type,
% 0.16/0.41 upairsetIR: $o ).
% 0.16/0.41
% 0.16/0.41 thf(upairsetIR,definition,
% 0.16/0.41 ( upairsetIR
% 0.16/0.41 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(emptyE1_type,type,
% 0.16/0.41 emptyE1: $o ).
% 0.16/0.41
% 0.16/0.41 thf(emptyE1,definition,
% 0.16/0.41 ( emptyE1
% 0.16/0.41 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.41 ( ? [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 & ( Xphi @ Xx ) )
% 0.16/0.41 => ( ( ( dsetconstr @ A
% 0.16/0.41 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.41 = emptyset )
% 0.16/0.41 => $false ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(vacuousDall_type,type,
% 0.16/0.41 vacuousDall: $o ).
% 0.16/0.41
% 0.16/0.41 thf(vacuousDall,definition,
% 0.16/0.41 ( vacuousDall
% 0.16/0.41 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ emptyset )
% 0.16/0.41 => ( Xphi @ Xx ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(quantDeMorgan1_type,type,
% 0.16/0.41 quantDeMorgan1: $o ).
% 0.16/0.41
% 0.16/0.41 thf(quantDeMorgan1,definition,
% 0.16/0.41 ( quantDeMorgan1
% 0.16/0.41 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.41 ( ~ ! [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 => ( Xphi @ Xx ) )
% 0.16/0.41 => ? [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(quantDeMorgan2_type,type,
% 0.16/0.41 quantDeMorgan2: $o ).
% 0.16/0.41
% 0.16/0.41 thf(quantDeMorgan2,definition,
% 0.16/0.41 ( quantDeMorgan2
% 0.16/0.41 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.41 ( ! [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 => ~ ( Xphi @ Xx ) )
% 0.16/0.41 => ~ ? [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(quantDeMorgan3_type,type,
% 0.16/0.41 quantDeMorgan3: $o ).
% 0.16/0.41
% 0.16/0.41 thf(quantDeMorgan3,definition,
% 0.16/0.41 ( quantDeMorgan3
% 0.16/0.41 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.41 ( ~ ? [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 & ( Xphi @ Xx ) )
% 0.16/0.41 => ! [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(quantDeMorgan4_type,type,
% 0.16/0.41 quantDeMorgan4: $o ).
% 0.16/0.41
% 0.16/0.41 thf(quantDeMorgan4,definition,
% 0.16/0.41 ( quantDeMorgan4
% 0.16/0.41 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.41 ( ? [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 & ~ ( Xphi @ Xx ) )
% 0.16/0.41 => ~ ! [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(prop2setI_type,type,
% 0.16/0.41 prop2setI: $o ).
% 0.16/0.41
% 0.16/0.41 thf(prop2setI,definition,
% 0.16/0.41 ( prop2setI
% 0.16/0.41 = ( ! [Xphi: $o] :
% 0.16/0.41 ( Xphi
% 0.16/0.41 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(set2prop_type,type,
% 0.16/0.41 set2prop: $i > $o ).
% 0.16/0.41
% 0.16/0.41 thf(prop2set2propI_type,type,
% 0.16/0.41 prop2set2propI: $o ).
% 0.16/0.41
% 0.16/0.41 thf(prop2set2propI,definition,
% 0.16/0.41 ( prop2set2propI
% 0.16/0.41 = ( ! [Xphi: $o] :
% 0.16/0.41 ( Xphi
% 0.16/0.41 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(notdexE_type,type,
% 0.16/0.41 notdexE: $o ).
% 0.16/0.41
% 0.16/0.41 thf(notdexE,definition,
% 0.16/0.41 ( notdexE
% 0.16/0.41 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.41 ( ~ ? [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 & ( Xphi @ Xx ) )
% 0.16/0.41 => ! [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(notdallE_type,type,
% 0.16/0.41 notdallE: $o ).
% 0.16/0.41
% 0.16/0.41 thf(notdallE,definition,
% 0.16/0.41 ( notdallE
% 0.16/0.41 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.41 ( ~ ! [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 => ( Xphi @ Xx ) )
% 0.16/0.41 => ? [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(exuI1_type,type,
% 0.16/0.41 exuI1: $o ).
% 0.16/0.41
% 0.16/0.41 thf(exuI1,definition,
% 0.16/0.41 ( exuI1
% 0.16/0.41 = ( ! [Xphi: $i > $o] :
% 0.16/0.41 ( ? [Xx: $i] :
% 0.16/0.41 ( ( Xphi @ Xx )
% 0.16/0.41 & ! [Xy: $i] :
% 0.16/0.41 ( ( Xphi @ Xy )
% 0.16/0.41 => ( Xx = Xy ) ) )
% 0.16/0.41 => ( exu
% 0.16/0.41 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(exuI3_type,type,
% 0.16/0.41 exuI3: $o ).
% 0.16/0.41
% 0.16/0.41 thf(exuI3,definition,
% 0.16/0.41 ( exuI3
% 0.16/0.41 = ( ! [Xphi: $i > $o] :
% 0.16/0.41 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.16/0.41 => ( ! [Xx: $i,Xy: $i] :
% 0.16/0.41 ( ( Xphi @ Xx )
% 0.16/0.41 => ( ( Xphi @ Xy )
% 0.16/0.41 => ( Xx = Xy ) ) )
% 0.16/0.41 => ( exu
% 0.16/0.41 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(exuI2_type,type,
% 0.16/0.41 exuI2: $o ).
% 0.16/0.41
% 0.16/0.41 thf(exuI2,definition,
% 0.16/0.41 ( exuI2
% 0.16/0.41 = ( ! [Xphi: $i > $o] :
% 0.16/0.41 ( ? [Xx: $i] :
% 0.16/0.41 ! [Xy: $i] :
% 0.16/0.41 ( ( Xphi @ Xy )
% 0.16/0.41 <=> ( Xy = Xx ) )
% 0.16/0.41 => ( exu
% 0.16/0.41 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(inCongP_type,type,
% 0.16/0.41 inCongP: $o ).
% 0.16/0.41
% 0.16/0.41 thf(inCongP,definition,
% 0.16/0.41 ( inCongP
% 0.16/0.41 = ( ! [A: $i,B: $i] :
% 0.16/0.41 ( ( A = B )
% 0.16/0.41 => ! [Xx: $i,Xy: $i] :
% 0.16/0.41 ( ( Xx = Xy )
% 0.16/0.41 => ( ( in @ Xx @ A )
% 0.16/0.41 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(in__Cong_type,type,
% 0.16/0.41 in__Cong: $o ).
% 0.16/0.41
% 0.16/0.41 thf(in__Cong,definition,
% 0.16/0.41 ( in__Cong
% 0.16/0.41 = ( ! [A: $i,B: $i] :
% 0.16/0.41 ( ( A = B )
% 0.16/0.41 => ! [Xx: $i,Xy: $i] :
% 0.16/0.41 ( ( Xx = Xy )
% 0.16/0.41 => ( ( in @ Xx @ A )
% 0.16/0.41 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(exuE3u_type,type,
% 0.16/0.41 exuE3u: $o ).
% 0.16/0.41
% 0.16/0.41 thf(exuE3u,definition,
% 0.16/0.41 ( exuE3u
% 0.16/0.41 = ( ! [Xphi: $i > $o] :
% 0.16/0.41 ( ( exu
% 0.16/0.41 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.41 => ! [Xx: $i,Xy: $i] :
% 0.16/0.41 ( ( Xphi @ Xx )
% 0.16/0.41 => ( ( Xphi @ Xy )
% 0.16/0.41 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(exu__Cong_type,type,
% 0.16/0.41 exu__Cong: $o ).
% 0.16/0.41
% 0.16/0.41 thf(exu__Cong,definition,
% 0.16/0.41 ( exu__Cong
% 0.16/0.41 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.16/0.41 ( ! [Xx: $i,Xy: $i] :
% 0.16/0.41 ( ( Xx = Xy )
% 0.16/0.41 => ( ( Xphi @ Xx )
% 0.16/0.41 <=> ( Xpsi @ Xy ) ) )
% 0.16/0.41 => ( ( exu
% 0.16/0.41 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.41 <=> ( exu
% 0.16/0.41 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(emptyset__Cong_type,type,
% 0.16/0.41 emptyset__Cong: $o ).
% 0.16/0.41
% 0.16/0.41 thf(emptyset__Cong,definition,
% 0.16/0.41 ( emptyset__Cong
% 0.16/0.41 = ( emptyset = emptyset ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(setadjoin__Cong_type,type,
% 0.16/0.41 setadjoin__Cong: $o ).
% 0.16/0.41
% 0.16/0.41 thf(setadjoin__Cong,definition,
% 0.16/0.41 ( setadjoin__Cong
% 0.16/0.41 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.41 ( ( Xx = Xy )
% 0.16/0.41 => ! [Xz: $i,Xu: $i] :
% 0.16/0.41 ( ( Xz = Xu )
% 0.16/0.41 => ( ( setadjoin @ Xx @ Xz )
% 0.16/0.41 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(powerset__Cong_type,type,
% 0.16/0.41 powerset__Cong: $o ).
% 0.16/0.41
% 0.16/0.41 thf(powerset__Cong,definition,
% 0.16/0.41 ( powerset__Cong
% 0.16/0.41 = ( ! [A: $i,B: $i] :
% 0.16/0.41 ( ( A = B )
% 0.16/0.41 => ( ( powerset @ A )
% 0.16/0.41 = ( powerset @ B ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(setunion__Cong_type,type,
% 0.16/0.41 setunion__Cong: $o ).
% 0.16/0.41
% 0.16/0.41 thf(setunion__Cong,definition,
% 0.16/0.41 ( setunion__Cong
% 0.16/0.41 = ( ! [A: $i,B: $i] :
% 0.16/0.41 ( ( A = B )
% 0.16/0.41 => ( ( setunion @ A )
% 0.16/0.41 = ( setunion @ B ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(omega__Cong_type,type,
% 0.16/0.41 omega__Cong: $o ).
% 0.16/0.41
% 0.16/0.41 thf(omega__Cong,definition,
% 0.16/0.41 ( omega__Cong
% 0.16/0.41 = ( omega = omega ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(exuEu_type,type,
% 0.16/0.41 exuEu: $o ).
% 0.16/0.41
% 0.16/0.41 thf(exuEu,definition,
% 0.16/0.41 ( exuEu
% 0.16/0.41 = ( ! [Xphi: $i > $o] :
% 0.16/0.41 ( ( exu
% 0.16/0.41 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.41 => ! [Xx: $i,Xy: $i] :
% 0.16/0.41 ( ( Xphi @ Xx )
% 0.16/0.41 => ( ( Xphi @ Xy )
% 0.16/0.41 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(descr__Cong_type,type,
% 0.16/0.41 descr__Cong: $o ).
% 0.16/0.41
% 0.16/0.41 thf(descr__Cong,definition,
% 0.16/0.41 ( descr__Cong
% 0.16/0.41 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.16/0.41 ( ! [Xx: $i,Xy: $i] :
% 0.16/0.41 ( ( Xx = Xy )
% 0.16/0.41 => ( ( Xphi @ Xx )
% 0.16/0.41 <=> ( Xpsi @ Xy ) ) )
% 0.16/0.41 => ( ( exu
% 0.16/0.41 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.41 => ( ( exu
% 0.16/0.41 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.16/0.41 => ( ( descr
% 0.16/0.41 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.41 = ( descr
% 0.16/0.41 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(dsetconstr__Cong_type,type,
% 0.16/0.41 dsetconstr__Cong: $o ).
% 0.16/0.41
% 0.16/0.41 thf(dsetconstr__Cong,definition,
% 0.16/0.41 ( dsetconstr__Cong
% 0.16/0.41 = ( ! [A: $i,B: $i] :
% 0.16/0.41 ( ( A = B )
% 0.16/0.41 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.16/0.41 ( ! [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 => ! [Xy: $i] :
% 0.16/0.41 ( ( in @ Xy @ B )
% 0.16/0.41 => ( ( Xx = Xy )
% 0.16/0.41 => ( ( Xphi @ Xx )
% 0.16/0.41 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.16/0.41 => ( ( dsetconstr @ A
% 0.16/0.41 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.41 = ( dsetconstr @ B
% 0.16/0.41 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(subset_type,type,
% 0.16/0.41 subset: $i > $i > $o ).
% 0.16/0.41
% 0.16/0.41 thf(disjoint_type,type,
% 0.16/0.41 disjoint: $i > $i > $o ).
% 0.16/0.41
% 0.16/0.41 thf(setsmeet_type,type,
% 0.16/0.41 setsmeet: $i > $i > $o ).
% 0.16/0.41
% 0.16/0.41 thf(subsetI1_type,type,
% 0.16/0.41 subsetI1: $o ).
% 0.16/0.41
% 0.16/0.41 thf(subsetI1,definition,
% 0.16/0.41 ( subsetI1
% 0.16/0.41 = ( ! [A: $i,B: $i] :
% 0.16/0.41 ( ! [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 => ( in @ Xx @ B ) )
% 0.16/0.41 => ( subset @ A @ B ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(eqimpsubset2_type,type,
% 0.16/0.41 eqimpsubset2: $o ).
% 0.16/0.41
% 0.16/0.41 thf(eqimpsubset2,definition,
% 0.16/0.41 ( eqimpsubset2
% 0.16/0.41 = ( ! [A: $i,B: $i] :
% 0.16/0.41 ( ( A = B )
% 0.16/0.41 => ( subset @ B @ A ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(eqimpsubset1_type,type,
% 0.16/0.41 eqimpsubset1: $o ).
% 0.16/0.41
% 0.16/0.41 thf(eqimpsubset1,definition,
% 0.16/0.41 ( eqimpsubset1
% 0.16/0.41 = ( ! [A: $i,B: $i] :
% 0.16/0.41 ( ( A = B )
% 0.16/0.41 => ( subset @ A @ B ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(subsetI2_type,type,
% 0.16/0.41 subsetI2: $o ).
% 0.16/0.41
% 0.16/0.41 thf(subsetI2,definition,
% 0.16/0.41 ( subsetI2
% 0.16/0.41 = ( ! [A: $i,B: $i] :
% 0.16/0.41 ( ! [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 => ( in @ Xx @ B ) )
% 0.16/0.41 => ( subset @ A @ B ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(emptysetsubset_type,type,
% 0.16/0.41 emptysetsubset: $o ).
% 0.16/0.41
% 0.16/0.41 thf(emptysetsubset,definition,
% 0.16/0.41 ( emptysetsubset
% 0.16/0.41 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(subsetE_type,type,
% 0.16/0.41 subsetE: $o ).
% 0.16/0.41
% 0.16/0.41 thf(subsetE,definition,
% 0.16/0.41 ( subsetE
% 0.16/0.41 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.41 ( ( subset @ A @ B )
% 0.16/0.41 => ( ( in @ Xx @ A )
% 0.16/0.41 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(subsetE2_type,type,
% 0.16/0.41 subsetE2: $o ).
% 0.16/0.41
% 0.16/0.41 thf(subsetE2,definition,
% 0.16/0.41 ( subsetE2
% 0.16/0.41 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.41 ( ( subset @ A @ B )
% 0.16/0.41 => ( ~ ( in @ Xx @ B )
% 0.16/0.41 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(notsubsetI_type,type,
% 0.16/0.41 notsubsetI: $o ).
% 0.16/0.41
% 0.16/0.41 thf(notsubsetI,definition,
% 0.16/0.41 ( notsubsetI
% 0.16/0.41 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 => ( ~ ( in @ Xx @ B )
% 0.16/0.41 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(notequalI1_type,type,
% 0.16/0.41 notequalI1: $o ).
% 0.16/0.41
% 0.16/0.41 thf(notequalI1,definition,
% 0.16/0.41 ( notequalI1
% 0.16/0.41 = ( ! [A: $i,B: $i] :
% 0.16/0.41 ( ~ ( subset @ A @ B )
% 0.16/0.41 => ( A != B ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(notequalI2_type,type,
% 0.16/0.41 notequalI2: $o ).
% 0.16/0.41
% 0.16/0.41 thf(notequalI2,definition,
% 0.16/0.41 ( notequalI2
% 0.16/0.41 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 => ( ~ ( in @ Xx @ B )
% 0.16/0.41 => ( A != B ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(subsetRefl_type,type,
% 0.16/0.41 subsetRefl: $o ).
% 0.16/0.41
% 0.16/0.41 thf(subsetRefl,definition,
% 0.16/0.41 ( subsetRefl
% 0.16/0.41 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(subsetTrans_type,type,
% 0.16/0.41 subsetTrans: $o ).
% 0.16/0.41
% 0.16/0.41 thf(subsetTrans,definition,
% 0.16/0.41 ( subsetTrans
% 0.16/0.41 = ( ! [A: $i,B: $i,C: $i] :
% 0.16/0.41 ( ( subset @ A @ B )
% 0.16/0.41 => ( ( subset @ B @ C )
% 0.16/0.41 => ( subset @ A @ C ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(setadjoinSub_type,type,
% 0.16/0.41 setadjoinSub: $o ).
% 0.16/0.41
% 0.16/0.41 thf(setadjoinSub,definition,
% 0.16/0.41 ( setadjoinSub
% 0.16/0.41 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(setadjoinSub2_type,type,
% 0.16/0.41 setadjoinSub2: $o ).
% 0.16/0.41
% 0.16/0.41 thf(setadjoinSub2,definition,
% 0.16/0.41 ( setadjoinSub2
% 0.16/0.41 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.16/0.41 ( ( subset @ A @ B )
% 0.16/0.41 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(subset2powerset_type,type,
% 0.16/0.41 subset2powerset: $o ).
% 0.16/0.41
% 0.16/0.41 thf(subset2powerset,definition,
% 0.16/0.41 ( subset2powerset
% 0.16/0.41 = ( ! [A: $i,B: $i] :
% 0.16/0.41 ( ( subset @ A @ B )
% 0.16/0.41 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(setextsub_type,type,
% 0.16/0.41 setextsub: $o ).
% 0.16/0.41
% 0.16/0.41 thf(setextsub,definition,
% 0.16/0.41 ( setextsub
% 0.16/0.41 = ( ! [A: $i,B: $i] :
% 0.16/0.41 ( ( subset @ A @ B )
% 0.16/0.41 => ( ( subset @ B @ A )
% 0.16/0.41 => ( A = B ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(subsetemptysetimpeq_type,type,
% 0.16/0.41 subsetemptysetimpeq: $o ).
% 0.16/0.41
% 0.16/0.41 thf(subsetemptysetimpeq,definition,
% 0.16/0.41 ( subsetemptysetimpeq
% 0.16/0.41 = ( ! [A: $i] :
% 0.16/0.41 ( ( subset @ A @ emptyset )
% 0.16/0.41 => ( A = emptyset ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(powersetI1_type,type,
% 0.16/0.41 powersetI1: $o ).
% 0.16/0.41
% 0.16/0.41 thf(powersetI1,definition,
% 0.16/0.41 ( powersetI1
% 0.16/0.41 = ( ! [A: $i,B: $i] :
% 0.16/0.41 ( ( subset @ B @ A )
% 0.16/0.41 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(powersetE1_type,type,
% 0.16/0.41 powersetE1: $o ).
% 0.16/0.41
% 0.16/0.41 thf(powersetE1,definition,
% 0.16/0.41 ( powersetE1
% 0.16/0.41 = ( ! [A: $i,B: $i] :
% 0.16/0.41 ( ( in @ B @ ( powerset @ A ) )
% 0.16/0.41 => ( subset @ B @ A ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(inPowerset_type,type,
% 0.16/0.41 inPowerset: $o ).
% 0.16/0.41
% 0.16/0.41 thf(inPowerset,definition,
% 0.16/0.41 ( inPowerset
% 0.16/0.41 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(powersetsubset_type,type,
% 0.16/0.41 powersetsubset: $o ).
% 0.16/0.41
% 0.16/0.41 thf(powersetsubset,definition,
% 0.16/0.41 ( powersetsubset
% 0.16/0.41 = ( ! [A: $i,B: $i] :
% 0.16/0.41 ( ( subset @ A @ B )
% 0.16/0.41 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(sepInPowerset_type,type,
% 0.16/0.41 sepInPowerset: $o ).
% 0.16/0.41
% 0.16/0.41 thf(sepInPowerset,definition,
% 0.16/0.41 ( sepInPowerset
% 0.16/0.41 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.41 ( in
% 0.16/0.41 @ ( dsetconstr @ A
% 0.16/0.41 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.41 @ ( powerset @ A ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(sepSubset_type,type,
% 0.16/0.41 sepSubset: $o ).
% 0.16/0.41
% 0.16/0.41 thf(sepSubset,definition,
% 0.16/0.41 ( sepSubset
% 0.16/0.41 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.41 ( subset
% 0.16/0.41 @ ( dsetconstr @ A
% 0.16/0.41 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.41 @ A ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(binunion_type,type,
% 0.16/0.41 binunion: $i > $i > $i ).
% 0.16/0.41
% 0.16/0.41 thf(binunionIL_type,type,
% 0.16/0.41 binunionIL: $o ).
% 0.16/0.41
% 0.16/0.41 thf(binunionIL,definition,
% 0.16/0.41 ( binunionIL
% 0.16/0.41 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(upairset2IR_type,type,
% 0.16/0.41 upairset2IR: $o ).
% 0.16/0.41
% 0.16/0.41 thf(upairset2IR,definition,
% 0.16/0.41 ( upairset2IR
% 0.16/0.41 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(binunionIR_type,type,
% 0.16/0.41 binunionIR: $o ).
% 0.16/0.41
% 0.16/0.41 thf(binunionIR,definition,
% 0.16/0.41 ( binunionIR
% 0.16/0.41 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ B )
% 0.16/0.41 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(binunionEcases_type,type,
% 0.16/0.41 binunionEcases: $o ).
% 0.16/0.41
% 0.16/0.41 thf(binunionEcases,definition,
% 0.16/0.41 ( binunionEcases
% 0.16/0.41 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.16/0.41 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.16/0.41 => ( ( ( in @ Xx @ A )
% 0.16/0.41 => Xphi )
% 0.16/0.41 => ( ( ( in @ Xx @ B )
% 0.16/0.41 => Xphi )
% 0.16/0.41 => Xphi ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(binunionE_type,type,
% 0.16/0.41 binunionE: $o ).
% 0.16/0.41
% 0.16/0.41 thf(binunionE,definition,
% 0.16/0.41 ( binunionE
% 0.16/0.41 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.16/0.41 => ( ( in @ Xx @ A )
% 0.16/0.41 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(binunionLsub_type,type,
% 0.16/0.41 binunionLsub: $o ).
% 0.16/0.41
% 0.16/0.41 thf(binunionLsub,definition,
% 0.16/0.41 ( binunionLsub
% 0.16/0.41 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(binunionRsub_type,type,
% 0.16/0.41 binunionRsub: $o ).
% 0.16/0.41
% 0.16/0.41 thf(binunionRsub,definition,
% 0.16/0.41 ( binunionRsub
% 0.16/0.41 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(binintersect_type,type,
% 0.16/0.41 binintersect: $i > $i > $i ).
% 0.16/0.41
% 0.16/0.41 thf(binintersectI_type,type,
% 0.16/0.41 binintersectI: $o ).
% 0.16/0.41
% 0.16/0.41 thf(binintersectI,definition,
% 0.16/0.41 ( binintersectI
% 0.16/0.41 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 => ( ( in @ Xx @ B )
% 0.16/0.41 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(binintersectSubset5_type,type,
% 0.16/0.41 binintersectSubset5: $o ).
% 0.16/0.41
% 0.16/0.41 thf(binintersectSubset5,definition,
% 0.16/0.41 ( binintersectSubset5
% 0.16/0.41 = ( ! [A: $i,B: $i,C: $i] :
% 0.16/0.41 ( ( subset @ C @ A )
% 0.16/0.41 => ( ( subset @ C @ B )
% 0.16/0.41 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(binintersectEL_type,type,
% 0.16/0.41 binintersectEL: $o ).
% 0.16/0.41
% 0.16/0.41 thf(binintersectEL,definition,
% 0.16/0.41 ( binintersectEL
% 0.16/0.41 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.16/0.41 => ( in @ Xx @ A ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(binintersectLsub_type,type,
% 0.16/0.41 binintersectLsub: $o ).
% 0.16/0.41
% 0.16/0.41 thf(binintersectLsub,definition,
% 0.16/0.41 ( binintersectLsub
% 0.16/0.41 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(binintersectSubset2_type,type,
% 0.16/0.41 binintersectSubset2: $o ).
% 0.16/0.41
% 0.16/0.41 thf(binintersectSubset2,definition,
% 0.16/0.41 ( binintersectSubset2
% 0.16/0.41 = ( ! [A: $i,B: $i] :
% 0.16/0.41 ( ( subset @ A @ B )
% 0.16/0.41 => ( ( binintersect @ A @ B )
% 0.16/0.41 = A ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(binintersectSubset3_type,type,
% 0.16/0.41 binintersectSubset3: $o ).
% 0.16/0.41
% 0.16/0.41 thf(binintersectSubset3,definition,
% 0.16/0.41 ( binintersectSubset3
% 0.16/0.41 = ( ! [A: $i,B: $i] :
% 0.16/0.41 ( ( ( binintersect @ A @ B )
% 0.16/0.41 = B )
% 0.16/0.41 => ( subset @ B @ A ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(binintersectER_type,type,
% 0.16/0.41 binintersectER: $o ).
% 0.16/0.41
% 0.16/0.41 thf(binintersectER,definition,
% 0.16/0.41 ( binintersectER
% 0.16/0.41 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.16/0.41 => ( in @ Xx @ B ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(disjointsetsI1_type,type,
% 0.16/0.41 disjointsetsI1: $o ).
% 0.16/0.41
% 0.16/0.41 thf(disjointsetsI1,definition,
% 0.16/0.41 ( disjointsetsI1
% 0.16/0.41 = ( ! [A: $i,B: $i] :
% 0.16/0.41 ( ~ ? [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 & ( in @ Xx @ B ) )
% 0.16/0.41 => ( ( binintersect @ A @ B )
% 0.16/0.41 = emptyset ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(binintersectRsub_type,type,
% 0.16/0.41 binintersectRsub: $o ).
% 0.16/0.41
% 0.16/0.41 thf(binintersectRsub,definition,
% 0.16/0.41 ( binintersectRsub
% 0.16/0.41 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(binintersectSubset4_type,type,
% 0.16/0.41 binintersectSubset4: $o ).
% 0.16/0.41
% 0.16/0.41 thf(binintersectSubset4,definition,
% 0.16/0.41 ( binintersectSubset4
% 0.16/0.41 = ( ! [A: $i,B: $i] :
% 0.16/0.41 ( ( subset @ B @ A )
% 0.16/0.41 => ( ( binintersect @ A @ B )
% 0.16/0.41 = B ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(binintersectSubset1_type,type,
% 0.16/0.41 binintersectSubset1: $o ).
% 0.16/0.41
% 0.16/0.41 thf(binintersectSubset1,definition,
% 0.16/0.41 ( binintersectSubset1
% 0.16/0.41 = ( ! [A: $i,B: $i] :
% 0.16/0.41 ( ( ( binintersect @ A @ B )
% 0.16/0.41 = A )
% 0.16/0.41 => ( subset @ A @ B ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(bs114d_type,type,
% 0.16/0.41 bs114d: $o ).
% 0.16/0.41
% 0.16/0.41 thf(bs114d,definition,
% 0.16/0.41 ( bs114d
% 0.16/0.41 = ( ! [A: $i,B: $i,C: $i] :
% 0.16/0.41 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 0.16/0.41 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(regular_type,type,
% 0.16/0.41 regular: $i > $o ).
% 0.16/0.41
% 0.16/0.41 thf(setminus_type,type,
% 0.16/0.41 setminus: $i > $i > $i ).
% 0.16/0.41
% 0.16/0.41 thf(setminusI_type,type,
% 0.16/0.41 setminusI: $o ).
% 0.16/0.41
% 0.16/0.41 thf(setminusI,definition,
% 0.16/0.41 ( setminusI
% 0.16/0.41 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 => ( ~ ( in @ Xx @ B )
% 0.16/0.41 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(setminusEL_type,type,
% 0.16/0.41 setminusEL: $o ).
% 0.16/0.41
% 0.16/0.41 thf(setminusEL,definition,
% 0.16/0.41 ( setminusEL
% 0.16/0.41 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.16/0.41 => ( in @ Xx @ A ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(setminusER_type,type,
% 0.16/0.41 setminusER: $o ).
% 0.16/0.41
% 0.16/0.41 thf(setminusER,definition,
% 0.16/0.41 ( setminusER
% 0.16/0.41 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.16/0.41 => ~ ( in @ Xx @ B ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(setminusSubset2_type,type,
% 0.16/0.41 setminusSubset2: $o ).
% 0.16/0.41
% 0.16/0.41 thf(setminusSubset2,definition,
% 0.16/0.41 ( setminusSubset2
% 0.16/0.41 = ( ! [A: $i,B: $i] :
% 0.16/0.41 ( ( subset @ A @ B )
% 0.16/0.41 => ( ( setminus @ A @ B )
% 0.16/0.41 = emptyset ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(setminusERneg_type,type,
% 0.16/0.41 setminusERneg: $o ).
% 0.16/0.41
% 0.16/0.41 thf(setminusERneg,definition,
% 0.16/0.41 ( setminusERneg
% 0.16/0.41 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.41 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.16/0.41 => ( ( in @ Xx @ A )
% 0.16/0.41 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(setminusELneg_type,type,
% 0.16/0.41 setminusELneg: $o ).
% 0.16/0.41
% 0.16/0.41 thf(setminusELneg,definition,
% 0.16/0.41 ( setminusELneg
% 0.16/0.41 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.41 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.16/0.41 => ( ~ ( in @ Xx @ B )
% 0.16/0.41 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(setminusILneg_type,type,
% 0.16/0.41 setminusILneg: $o ).
% 0.16/0.41
% 0.16/0.41 thf(setminusILneg,definition,
% 0.16/0.41 ( setminusILneg
% 0.16/0.41 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.41 ( ~ ( in @ Xx @ A )
% 0.16/0.41 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(setminusIRneg_type,type,
% 0.16/0.41 setminusIRneg: $o ).
% 0.16/0.41
% 0.16/0.41 thf(setminusIRneg,definition,
% 0.16/0.41 ( setminusIRneg
% 0.16/0.41 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ B )
% 0.16/0.41 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(setminusLsub_type,type,
% 0.16/0.41 setminusLsub: $o ).
% 0.16/0.41
% 0.16/0.41 thf(setminusLsub,definition,
% 0.16/0.41 ( setminusLsub
% 0.16/0.41 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(setminusSubset1_type,type,
% 0.16/0.41 setminusSubset1: $o ).
% 0.16/0.41
% 0.16/0.41 thf(setminusSubset1,definition,
% 0.16/0.41 ( setminusSubset1
% 0.16/0.41 = ( ! [A: $i,B: $i] :
% 0.16/0.41 ( ( ( setminus @ A @ B )
% 0.16/0.41 = emptyset )
% 0.16/0.41 => ( subset @ A @ B ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(symdiff_type,type,
% 0.16/0.41 symdiff: $i > $i > $i ).
% 0.16/0.41
% 0.16/0.41 thf(symdiffE_type,type,
% 0.16/0.41 symdiffE: $o ).
% 0.16/0.41
% 0.16/0.41 thf(symdiffE,definition,
% 0.16/0.41 ( symdiffE
% 0.16/0.41 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 0.16/0.41 => ! [Xphi: $o] :
% 0.16/0.41 ( ( ( in @ Xx @ A )
% 0.16/0.41 => ( ~ ( in @ Xx @ B )
% 0.16/0.41 => Xphi ) )
% 0.16/0.41 => ( ( ~ ( in @ Xx @ A )
% 0.16/0.41 => ( ( in @ Xx @ B )
% 0.16/0.41 => Xphi ) )
% 0.16/0.41 => Xphi ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(symdiffI1_type,type,
% 0.16/0.41 symdiffI1: $o ).
% 0.16/0.41
% 0.16/0.41 thf(symdiffI1,definition,
% 0.16/0.41 ( symdiffI1
% 0.16/0.41 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 => ( ~ ( in @ Xx @ B )
% 0.16/0.41 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(symdiffI2_type,type,
% 0.16/0.41 symdiffI2: $o ).
% 0.16/0.41
% 0.16/0.41 thf(symdiffI2,definition,
% 0.16/0.41 ( symdiffI2
% 0.16/0.41 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.41 ( ~ ( in @ Xx @ A )
% 0.16/0.41 => ( ( in @ Xx @ B )
% 0.16/0.41 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(symdiffIneg1_type,type,
% 0.16/0.41 symdiffIneg1: $o ).
% 0.16/0.41
% 0.16/0.41 thf(symdiffIneg1,definition,
% 0.16/0.41 ( symdiffIneg1
% 0.16/0.41 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 => ( ( in @ Xx @ B )
% 0.16/0.41 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(symdiffIneg2_type,type,
% 0.16/0.41 symdiffIneg2: $o ).
% 0.16/0.41
% 0.16/0.41 thf(symdiffIneg2,definition,
% 0.16/0.41 ( symdiffIneg2
% 0.16/0.41 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.41 ( ~ ( in @ Xx @ A )
% 0.16/0.41 => ( ~ ( in @ Xx @ B )
% 0.16/0.41 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(iskpair_type,type,
% 0.16/0.41 iskpair: $i > $o ).
% 0.16/0.41
% 0.16/0.41 thf(secondinupair_type,type,
% 0.16/0.41 secondinupair: $o ).
% 0.16/0.41
% 0.16/0.41 thf(secondinupair,definition,
% 0.16/0.41 ( secondinupair
% 0.16/0.41 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(setukpairIL_type,type,
% 0.16/0.41 setukpairIL: $o ).
% 0.16/0.41
% 0.16/0.41 thf(setukpairIL,definition,
% 0.16/0.41 ( setukpairIL
% 0.16/0.41 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(setukpairIR_type,type,
% 0.16/0.41 setukpairIR: $o ).
% 0.16/0.41
% 0.16/0.41 thf(setukpairIR,definition,
% 0.16/0.41 ( setukpairIR
% 0.16/0.41 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(kpairiskpair_type,type,
% 0.16/0.41 kpairiskpair: $o ).
% 0.16/0.41
% 0.16/0.41 thf(kpairiskpair,definition,
% 0.16/0.41 ( kpairiskpair
% 0.16/0.41 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(kpair_type,type,
% 0.16/0.41 kpair: $i > $i > $i ).
% 0.16/0.41
% 0.16/0.41 thf(kpairp_type,type,
% 0.16/0.41 kpairp: $o ).
% 0.16/0.41
% 0.16/0.41 thf(kpairp,definition,
% 0.16/0.41 ( kpairp
% 0.16/0.41 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(cartprod_type,type,
% 0.16/0.41 cartprod: $i > $i > $i ).
% 0.16/0.41
% 0.16/0.41 thf(singletonsubset_type,type,
% 0.16/0.41 singletonsubset: $o ).
% 0.16/0.41
% 0.16/0.41 thf(singletonsubset,definition,
% 0.16/0.41 ( singletonsubset
% 0.16/0.41 = ( ! [A: $i,Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(singletoninpowerset_type,type,
% 0.16/0.41 singletoninpowerset: $o ).
% 0.16/0.41
% 0.16/0.41 thf(singletoninpowerset,definition,
% 0.16/0.41 ( singletoninpowerset
% 0.16/0.41 = ( ! [A: $i,Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(singletoninpowunion_type,type,
% 0.16/0.41 singletoninpowunion: $o ).
% 0.16/0.41
% 0.16/0.41 thf(singletoninpowunion,definition,
% 0.16/0.41 ( singletoninpowunion
% 0.16/0.41 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(upairset2E_type,type,
% 0.16/0.41 upairset2E: $o ).
% 0.16/0.41
% 0.16/0.41 thf(upairset2E,definition,
% 0.16/0.41 ( upairset2E
% 0.16/0.41 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.16/0.41 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.16/0.41 => ( ( Xz = Xx )
% 0.16/0.41 | ( Xz = Xy ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(upairsubunion_type,type,
% 0.16/0.41 upairsubunion: $o ).
% 0.16/0.41
% 0.16/0.41 thf(upairsubunion,definition,
% 0.16/0.41 ( upairsubunion
% 0.16/0.41 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 => ! [Xy: $i] :
% 0.16/0.41 ( ( in @ Xy @ B )
% 0.16/0.41 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(upairinpowunion_type,type,
% 0.16/0.41 upairinpowunion: $o ).
% 0.16/0.41
% 0.16/0.41 thf(upairinpowunion,definition,
% 0.16/0.41 ( upairinpowunion
% 0.16/0.41 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 => ! [Xy: $i] :
% 0.16/0.41 ( ( in @ Xy @ B )
% 0.16/0.41 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(ubforcartprodlem1_type,type,
% 0.16/0.41 ubforcartprodlem1: $o ).
% 0.16/0.41
% 0.16/0.41 thf(ubforcartprodlem1,definition,
% 0.16/0.41 ( ubforcartprodlem1
% 0.16/0.41 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 => ! [Xy: $i] :
% 0.16/0.41 ( ( in @ Xy @ B )
% 0.16/0.41 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(ubforcartprodlem2_type,type,
% 0.16/0.41 ubforcartprodlem2: $o ).
% 0.16/0.41
% 0.16/0.41 thf(ubforcartprodlem2,definition,
% 0.16/0.41 ( ubforcartprodlem2
% 0.16/0.41 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 => ! [Xy: $i] :
% 0.16/0.41 ( ( in @ Xy @ B )
% 0.16/0.41 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(ubforcartprodlem3_type,type,
% 0.16/0.41 ubforcartprodlem3: $o ).
% 0.16/0.41
% 0.16/0.41 thf(ubforcartprodlem3,definition,
% 0.16/0.41 ( ubforcartprodlem3
% 0.16/0.41 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 => ! [Xy: $i] :
% 0.16/0.41 ( ( in @ Xy @ B )
% 0.16/0.41 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(cartprodpairin_type,type,
% 0.16/0.41 cartprodpairin: $o ).
% 0.16/0.41
% 0.16/0.41 thf(cartprodpairin,definition,
% 0.16/0.41 ( cartprodpairin
% 0.16/0.41 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 => ! [Xy: $i] :
% 0.16/0.41 ( ( in @ Xy @ B )
% 0.16/0.41 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(cartprodmempair1_type,type,
% 0.16/0.41 cartprodmempair1: $o ).
% 0.16/0.41
% 0.16/0.41 thf(cartprodmempair1,definition,
% 0.16/0.41 ( cartprodmempair1
% 0.16/0.41 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.16/0.41 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.16/0.41 => ? [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 & ? [Xy: $i] :
% 0.16/0.41 ( ( in @ Xy @ B )
% 0.16/0.41 & ( Xu
% 0.16/0.41 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(cartprodmempair_type,type,
% 0.16/0.41 cartprodmempair: $o ).
% 0.16/0.41
% 0.16/0.41 thf(cartprodmempair,definition,
% 0.16/0.41 ( cartprodmempair
% 0.16/0.41 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.16/0.41 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.16/0.41 => ( iskpair @ Xu ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(setunionE2_type,type,
% 0.16/0.41 setunionE2: $o ).
% 0.16/0.41
% 0.16/0.41 thf(setunionE2,definition,
% 0.16/0.41 ( setunionE2
% 0.16/0.41 = ( ! [A: $i,Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ ( setunion @ A ) )
% 0.16/0.41 => ? [X: $i] :
% 0.16/0.41 ( ( in @ X @ A )
% 0.16/0.41 & ( in @ Xx @ X ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(setunionsingleton1_type,type,
% 0.16/0.41 setunionsingleton1: $o ).
% 0.16/0.41
% 0.16/0.41 thf(setunionsingleton1,definition,
% 0.16/0.41 ( setunionsingleton1
% 0.16/0.41 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(setunionsingleton2_type,type,
% 0.16/0.41 setunionsingleton2: $o ).
% 0.16/0.41
% 0.16/0.41 thf(setunionsingleton2,definition,
% 0.16/0.41 ( setunionsingleton2
% 0.16/0.41 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(setunionsingleton_type,type,
% 0.16/0.41 setunionsingleton: $o ).
% 0.16/0.41
% 0.16/0.41 thf(setunionsingleton,definition,
% 0.16/0.41 ( setunionsingleton
% 0.16/0.41 = ( ! [Xx: $i] :
% 0.16/0.41 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 0.16/0.41 = Xx ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(singleton_type,type,
% 0.16/0.41 singleton: $i > $o ).
% 0.16/0.41
% 0.16/0.41 thf(singletonprop_type,type,
% 0.16/0.41 singletonprop: $o ).
% 0.16/0.41
% 0.16/0.41 thf(singletonprop,definition,
% 0.16/0.41 ( singletonprop
% 0.16/0.41 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.41 ( ! [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 => ! [Xy: $i] :
% 0.16/0.41 ( ( in @ Xy @ A )
% 0.16/0.41 => ( ( Xphi @ Xx )
% 0.16/0.41 => ( ( Xphi @ Xy )
% 0.16/0.41 => ( Xx = Xy ) ) ) ) )
% 0.16/0.41 => ( ? [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 & ( Xphi @ Xx ) )
% 0.16/0.41 => ( singleton
% 0.16/0.41 @ ( dsetconstr @ A
% 0.16/0.41 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(ex1_type,type,
% 0.16/0.41 ex1: $i > ( $i > $o ) > $o ).
% 0.16/0.41
% 0.16/0.41 thf(ex1E1_type,type,
% 0.16/0.41 ex1E1: $o ).
% 0.16/0.41
% 0.16/0.41 thf(ex1E1,definition,
% 0.16/0.41 ( ex1E1
% 0.16/0.41 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.41 ( ( ex1 @ A
% 0.16/0.41 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.41 => ? [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(ex1I_type,type,
% 0.16/0.41 ex1I: $o ).
% 0.16/0.41
% 0.16/0.41 thf(ex1I,definition,
% 0.16/0.41 ( ex1I
% 0.16/0.41 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 => ( ( Xphi @ Xx )
% 0.16/0.41 => ( ! [Xy: $i] :
% 0.16/0.41 ( ( in @ Xy @ A )
% 0.16/0.41 => ( ( Xphi @ Xy )
% 0.16/0.41 => ( Xy = Xx ) ) )
% 0.16/0.41 => ( ex1 @ A
% 0.16/0.41 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(ex1I2_type,type,
% 0.16/0.41 ex1I2: $o ).
% 0.16/0.41
% 0.16/0.41 thf(ex1I2,definition,
% 0.16/0.41 ( ex1I2
% 0.16/0.41 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.41 ( ! [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ A )
% 0.16/0.42 => ( ( Xphi @ Xx )
% 0.16/0.42 => ( ( Xphi @ Xy )
% 0.16/0.42 => ( Xx = Xy ) ) ) ) )
% 0.16/0.42 => ( ? [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 & ( Xphi @ Xx ) )
% 0.16/0.42 => ( ex1 @ A
% 0.16/0.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(singletonsuniq_type,type,
% 0.16/0.42 singletonsuniq: $o ).
% 0.16/0.42
% 0.16/0.42 thf(singletonsuniq,definition,
% 0.16/0.42 ( singletonsuniq
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.42 ( ( ( setadjoin @ Xx @ emptyset )
% 0.16/0.42 = ( setadjoin @ Xy @ emptyset ) )
% 0.16/0.42 => ( Xx = Xy ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(atmost1p_type,type,
% 0.16/0.42 atmost1p: $i > $o ).
% 0.16/0.42
% 0.16/0.42 thf(atleast2p_type,type,
% 0.16/0.42 atleast2p: $i > $o ).
% 0.16/0.42
% 0.16/0.42 thf(atmost2p_type,type,
% 0.16/0.42 atmost2p: $i > $o ).
% 0.16/0.42
% 0.16/0.42 thf(upairsetp_type,type,
% 0.16/0.42 upairsetp: $i > $o ).
% 0.16/0.42
% 0.16/0.42 thf(setukpairinjL1_type,type,
% 0.16/0.42 setukpairinjL1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setukpairinjL1,definition,
% 0.16/0.42 ( setukpairinjL1
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.16/0.42 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 0.16/0.42 => ( Xx = Xz ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(kfstsingleton_type,type,
% 0.16/0.42 kfstsingleton: $o ).
% 0.16/0.42
% 0.16/0.42 thf(kfstsingleton,definition,
% 0.16/0.42 ( kfstsingleton
% 0.16/0.42 = ( ! [Xu: $i] :
% 0.16/0.42 ( ( iskpair @ Xu )
% 0.16/0.42 => ( singleton
% 0.16/0.42 @ ( dsetconstr @ ( setunion @ Xu )
% 0.16/0.42 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(theprop_type,type,
% 0.16/0.42 theprop: $o ).
% 0.16/0.42
% 0.16/0.42 thf(theprop,definition,
% 0.16/0.42 ( theprop
% 0.16/0.42 = ( ! [X: $i] :
% 0.16/0.42 ( ( singleton @ X )
% 0.16/0.42 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(kfst_type,type,
% 0.16/0.42 kfst: $i > $i ).
% 0.16/0.42
% 0.16/0.42 thf(kfstpairEq_type,type,
% 0.16/0.42 kfstpairEq: $o ).
% 0.16/0.42
% 0.16/0.42 thf(kfstpairEq,definition,
% 0.16/0.42 ( kfstpairEq
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.42 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.16/0.42 = Xx ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(cartprodfstin_type,type,
% 0.16/0.42 cartprodfstin: $o ).
% 0.16/0.42
% 0.16/0.42 thf(cartprodfstin,definition,
% 0.16/0.42 ( cartprodfstin
% 0.16/0.42 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.16/0.42 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.16/0.42 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setukpairinjL2_type,type,
% 0.16/0.42 setukpairinjL2: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setukpairinjL2,definition,
% 0.16/0.42 ( setukpairinjL2
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.16/0.42 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.16/0.42 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.16/0.42 => ( Xx = Xz ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setukpairinjL_type,type,
% 0.16/0.42 setukpairinjL: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setukpairinjL,definition,
% 0.16/0.42 ( setukpairinjL
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.16/0.42 ( ( ( kpair @ Xx @ Xy )
% 0.16/0.42 = ( kpair @ Xz @ Xu ) )
% 0.16/0.42 => ( Xx = Xz ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setukpairinjR11_type,type,
% 0.16/0.42 setukpairinjR11: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setukpairinjR11,definition,
% 0.16/0.42 ( setukpairinjR11
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.42 ( ( Xx = Xy )
% 0.16/0.42 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.16/0.42 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setukpairinjR12_type,type,
% 0.16/0.42 setukpairinjR12: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setukpairinjR12,definition,
% 0.16/0.42 ( setukpairinjR12
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.42 ( ( Xx = Xy )
% 0.16/0.42 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.16/0.42 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setukpairinjR1_type,type,
% 0.16/0.42 setukpairinjR1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setukpairinjR1,definition,
% 0.16/0.42 ( setukpairinjR1
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.16/0.42 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.16/0.42 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.16/0.42 => ( ( Xz = Xu )
% 0.16/0.42 => ( Xy = Xu ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(upairequniteq_type,type,
% 0.16/0.42 upairequniteq: $o ).
% 0.16/0.42
% 0.16/0.42 thf(upairequniteq,definition,
% 0.16/0.42 ( upairequniteq
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.16/0.42 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.16/0.42 = ( setadjoin @ Xz @ emptyset ) )
% 0.16/0.42 => ( Xx = Xy ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setukpairinjR2_type,type,
% 0.16/0.42 setukpairinjR2: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setukpairinjR2,definition,
% 0.16/0.42 ( setukpairinjR2
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.16/0.42 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.16/0.42 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.16/0.42 => ( Xy = Xu ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setukpairinjR_type,type,
% 0.16/0.42 setukpairinjR: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setukpairinjR,definition,
% 0.16/0.42 ( setukpairinjR
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.16/0.42 ( ( ( kpair @ Xx @ Xy )
% 0.16/0.42 = ( kpair @ Xz @ Xu ) )
% 0.16/0.42 => ( Xy = Xu ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(ksndsingleton_type,type,
% 0.16/0.42 ksndsingleton: $o ).
% 0.16/0.42
% 0.16/0.42 thf(ksndsingleton,definition,
% 0.16/0.42 ( ksndsingleton
% 0.16/0.42 = ( ! [Xu: $i] :
% 0.16/0.42 ( ( iskpair @ Xu )
% 0.16/0.42 => ( singleton
% 0.16/0.42 @ ( dsetconstr @ ( setunion @ Xu )
% 0.16/0.42 @ ^ [Xx: $i] :
% 0.16/0.42 ( Xu
% 0.16/0.42 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(ksnd_type,type,
% 0.16/0.42 ksnd: $i > $i ).
% 0.16/0.42
% 0.16/0.42 thf(ksndpairEq_type,type,
% 0.16/0.42 ksndpairEq: $o ).
% 0.16/0.42
% 0.16/0.42 thf(ksndpairEq,definition,
% 0.16/0.42 ( ksndpairEq
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.42 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.16/0.42 = Xy ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(kpairsurjEq_type,type,
% 0.16/0.42 kpairsurjEq: $o ).
% 0.16/0.42
% 0.16/0.42 thf(kpairsurjEq,definition,
% 0.16/0.42 ( kpairsurjEq
% 0.16/0.42 = ( ! [Xu: $i] :
% 0.16/0.42 ( ( iskpair @ Xu )
% 0.16/0.42 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.16/0.42 = Xu ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(cartprodsndin_type,type,
% 0.16/0.42 cartprodsndin: $o ).
% 0.16/0.42
% 0.16/0.42 thf(cartprodsndin,definition,
% 0.16/0.42 ( cartprodsndin
% 0.16/0.42 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.16/0.42 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.16/0.42 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(cartprodpairmemEL_type,type,
% 0.16/0.42 cartprodpairmemEL: $o ).
% 0.16/0.42
% 0.16/0.42 thf(cartprodpairmemEL,definition,
% 0.16/0.42 ( cartprodpairmemEL
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.16/0.42 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.16/0.42 => ( in @ Xx @ A ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(cartprodpairmemER_type,type,
% 0.16/0.42 cartprodpairmemER: $o ).
% 0.16/0.42
% 0.16/0.42 thf(cartprodpairmemER,definition,
% 0.16/0.42 ( cartprodpairmemER
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.16/0.42 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.16/0.42 => ( in @ Xy @ B ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(cartprodmempaircEq_type,type,
% 0.16/0.42 cartprodmempaircEq: $o ).
% 0.16/0.42
% 0.16/0.42 thf(cartprodmempaircEq,definition,
% 0.16/0.42 ( cartprodmempaircEq
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ B )
% 0.16/0.42 => ( ( kpair @ Xx @ Xy )
% 0.16/0.42 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(cartprodfstpairEq_type,type,
% 0.16/0.42 cartprodfstpairEq: $o ).
% 0.16/0.42
% 0.16/0.42 thf(cartprodfstpairEq,definition,
% 0.16/0.42 ( cartprodfstpairEq
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ B )
% 0.16/0.42 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.16/0.42 = Xx ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(cartprodsndpairEq_type,type,
% 0.16/0.42 cartprodsndpairEq: $o ).
% 0.16/0.42
% 0.16/0.42 thf(cartprodsndpairEq,definition,
% 0.16/0.42 ( cartprodsndpairEq
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ B )
% 0.16/0.42 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.16/0.42 = Xy ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(cartprodpairsurjEq_type,type,
% 0.16/0.42 cartprodpairsurjEq: $o ).
% 0.16/0.42
% 0.16/0.42 thf(cartprodpairsurjEq,definition,
% 0.16/0.42 ( cartprodpairsurjEq
% 0.16/0.42 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.16/0.42 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.16/0.42 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.16/0.42 = Xu ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(breln_type,type,
% 0.16/0.42 breln: $i > $i > $i > $o ).
% 0.16/0.42
% 0.16/0.42 thf(dpsetconstr_type,type,
% 0.16/0.42 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 0.16/0.42
% 0.16/0.42 thf(dpsetconstrI_type,type,
% 0.16/0.42 dpsetconstrI: $o ).
% 0.16/0.42
% 0.16/0.42 thf(dpsetconstrI,definition,
% 0.16/0.42 ( dpsetconstrI
% 0.16/0.42 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ B )
% 0.16/0.42 => ( ( Xphi @ Xx @ Xy )
% 0.16/0.42 => ( in @ ( kpair @ Xx @ Xy )
% 0.16/0.42 @ ( dpsetconstr @ A @ B
% 0.16/0.42 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(dpsetconstrSub_type,type,
% 0.16/0.42 dpsetconstrSub: $o ).
% 0.16/0.42
% 0.16/0.42 thf(dpsetconstrSub,definition,
% 0.16/0.42 ( dpsetconstrSub
% 0.16/0.42 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.16/0.42 ( subset
% 0.16/0.42 @ ( dpsetconstr @ A @ B
% 0.16/0.42 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 0.16/0.42 @ ( cartprod @ A @ B ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setOfPairsIsBReln_type,type,
% 0.16/0.42 setOfPairsIsBReln: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setOfPairsIsBReln,definition,
% 0.16/0.42 ( setOfPairsIsBReln
% 0.16/0.42 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.16/0.42 ( breln @ A @ B
% 0.16/0.42 @ ( dpsetconstr @ A @ B
% 0.16/0.42 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(dpsetconstrERa_type,type,
% 0.16/0.42 dpsetconstrERa: $o ).
% 0.16/0.42
% 0.16/0.42 thf(dpsetconstrERa,definition,
% 0.16/0.42 ( dpsetconstrERa
% 0.16/0.42 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ B )
% 0.16/0.42 => ( ( in @ ( kpair @ Xx @ Xy )
% 0.16/0.42 @ ( dpsetconstr @ A @ B
% 0.16/0.42 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.16/0.42 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(dpsetconstrEL1_type,type,
% 0.16/0.42 dpsetconstrEL1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(dpsetconstrEL1,definition,
% 0.16/0.42 ( dpsetconstrEL1
% 0.16/0.42 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.16/0.42 ( ( in @ ( kpair @ Xx @ Xy )
% 0.16/0.42 @ ( dpsetconstr @ A @ B
% 0.16/0.42 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.16/0.42 => ( in @ Xx @ A ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(dpsetconstrEL2_type,type,
% 0.16/0.42 dpsetconstrEL2: $o ).
% 0.16/0.42
% 0.16/0.42 thf(dpsetconstrEL2,definition,
% 0.16/0.42 ( dpsetconstrEL2
% 0.16/0.42 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.16/0.42 ( ( in @ ( kpair @ Xx @ Xy )
% 0.16/0.42 @ ( dpsetconstr @ A @ B
% 0.16/0.42 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.16/0.42 => ( in @ Xy @ B ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(dpsetconstrER_type,type,
% 0.16/0.42 dpsetconstrER: $o ).
% 0.16/0.42
% 0.16/0.42 thf(dpsetconstrER,definition,
% 0.16/0.42 ( dpsetconstrER
% 0.16/0.42 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.16/0.42 ( ( in @ ( kpair @ Xx @ Xy )
% 0.16/0.42 @ ( dpsetconstr @ A @ B
% 0.16/0.42 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.16/0.42 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(func_type,type,
% 0.16/0.42 func: $i > $i > $i > $o ).
% 0.16/0.42
% 0.16/0.42 thf(funcSet_type,type,
% 0.16/0.42 funcSet: $i > $i > $i ).
% 0.16/0.42
% 0.16/0.42 thf(funcImageSingleton_type,type,
% 0.16/0.42 funcImageSingleton: $o ).
% 0.16/0.42
% 0.16/0.42 thf(funcImageSingleton,definition,
% 0.16/0.42 ( funcImageSingleton
% 0.16/0.42 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.42 ( ( func @ A @ B @ Xf )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( singleton
% 0.16/0.42 @ ( dsetconstr @ B
% 0.16/0.42 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(apProp_type,type,
% 0.16/0.42 apProp: $o ).
% 0.16/0.42
% 0.16/0.42 thf(apProp,definition,
% 0.16/0.42 ( apProp
% 0.16/0.42 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.42 ( ( func @ A @ B @ Xf )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( in
% 0.16/0.42 @ ( setunion
% 0.16/0.42 @ ( dsetconstr @ B
% 0.16/0.42 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 0.16/0.42 @ B ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(ap_type,type,
% 0.16/0.42 ap: $i > $i > $i > $i > $i ).
% 0.16/0.42
% 0.16/0.42 thf(app_type,type,
% 0.16/0.42 app: $o ).
% 0.16/0.42
% 0.16/0.42 thf(app,definition,
% 0.16/0.42 ( app
% 0.16/0.42 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.42 ( ( func @ A @ B @ Xf )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(infuncsetfunc_type,type,
% 0.16/0.42 infuncsetfunc: $o ).
% 0.16/0.42
% 0.16/0.42 thf(infuncsetfunc,definition,
% 0.16/0.42 ( infuncsetfunc
% 0.16/0.42 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.42 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.16/0.42 => ( func @ A @ B @ Xf ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(ap2p_type,type,
% 0.16/0.42 ap2p: $o ).
% 0.16/0.42
% 0.16/0.42 thf(ap2p,definition,
% 0.16/0.42 ( ap2p
% 0.16/0.42 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.42 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(funcinfuncset_type,type,
% 0.16/0.42 funcinfuncset: $o ).
% 0.16/0.42
% 0.16/0.42 thf(funcinfuncset,definition,
% 0.16/0.42 ( funcinfuncset
% 0.16/0.42 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.42 ( ( func @ A @ B @ Xf )
% 0.16/0.42 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(lamProp_type,type,
% 0.16/0.42 lamProp: $o ).
% 0.16/0.42
% 0.16/0.42 thf(lamProp,definition,
% 0.16/0.42 ( lamProp
% 0.16/0.42 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.16/0.42 ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.16/0.42 => ( func @ A @ B
% 0.16/0.42 @ ( dpsetconstr @ A @ B
% 0.16/0.42 @ ^ [Xx: $i,Xy: $i] :
% 0.16/0.42 ( ( Xf @ Xx )
% 0.16/0.42 = Xy ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(lam_type,type,
% 0.16/0.42 lam: $i > $i > ( $i > $i ) > $i ).
% 0.16/0.42
% 0.16/0.42 thf(lamp_type,type,
% 0.16/0.42 lamp: $o ).
% 0.16/0.42
% 0.16/0.42 thf(lamp,definition,
% 0.16/0.42 ( lamp
% 0.16/0.42 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.16/0.42 ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.16/0.42 => ( func @ A @ B
% 0.16/0.42 @ ( lam @ A @ B
% 0.16/0.42 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(lam2p_type,type,
% 0.16/0.42 lam2p: $o ).
% 0.16/0.42
% 0.16/0.42 thf(lam2p,definition,
% 0.16/0.42 ( lam2p
% 0.16/0.42 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.16/0.42 ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.16/0.42 => ( in
% 0.16/0.42 @ ( lam @ A @ B
% 0.16/0.42 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.16/0.42 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(brelnall1_type,type,
% 0.16/0.42 brelnall1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(brelnall1,definition,
% 0.16/0.42 ( brelnall1
% 0.16/0.42 = ( ! [A: $i,B: $i,R: $i] :
% 0.16/0.42 ( ( breln @ A @ B @ R )
% 0.16/0.42 => ! [Xphi: $i > $o] :
% 0.16/0.42 ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ B )
% 0.16/0.42 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.16/0.42 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ R )
% 0.16/0.42 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(brelnall2_type,type,
% 0.16/0.42 brelnall2: $o ).
% 0.16/0.42
% 0.16/0.42 thf(brelnall2,definition,
% 0.16/0.42 ( brelnall2
% 0.16/0.42 = ( ! [A: $i,B: $i,R: $i] :
% 0.16/0.42 ( ( breln @ A @ B @ R )
% 0.16/0.42 => ! [Xphi: $i > $o] :
% 0.16/0.42 ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ B )
% 0.16/0.42 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.16/0.42 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ R )
% 0.16/0.42 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(ex1E2_type,type,
% 0.16/0.42 ex1E2: $o ).
% 0.16/0.42
% 0.16/0.42 thf(ex1E2,definition,
% 0.16/0.42 ( ex1E2
% 0.16/0.42 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.42 ( ( ex1 @ A
% 0.16/0.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ A )
% 0.16/0.42 => ( ( Xphi @ Xx )
% 0.16/0.42 => ( ( Xphi @ Xy )
% 0.16/0.42 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(funcGraphProp1_type,type,
% 0.16/0.42 funcGraphProp1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(funcGraphProp1,definition,
% 0.16/0.42 ( funcGraphProp1
% 0.16/0.42 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.42 ( ( func @ A @ B @ Xf )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(funcGraphProp3_type,type,
% 0.16/0.42 funcGraphProp3: $o ).
% 0.16/0.42
% 0.16/0.42 thf(funcGraphProp3,definition,
% 0.16/0.42 ( funcGraphProp3
% 0.16/0.42 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.42 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(funcGraphProp2_type,type,
% 0.16/0.42 funcGraphProp2: $o ).
% 0.16/0.42
% 0.16/0.42 thf(funcGraphProp2,definition,
% 0.16/0.42 ( funcGraphProp2
% 0.16/0.42 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.42 ( ( func @ A @ B @ Xf )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ B )
% 0.16/0.42 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.16/0.42 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.16/0.42 = Xy ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(funcextLem_type,type,
% 0.16/0.42 funcextLem: $o ).
% 0.16/0.42
% 0.16/0.42 thf(funcextLem,definition,
% 0.16/0.42 ( funcextLem
% 0.16/0.42 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.42 ( ( func @ A @ B @ Xf )
% 0.16/0.42 => ! [Xg: $i] :
% 0.16/0.42 ( ( func @ A @ B @ Xg )
% 0.16/0.42 => ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.16/0.42 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ B )
% 0.16/0.42 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 0.16/0.42 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(funcGraphProp4_type,type,
% 0.16/0.42 funcGraphProp4: $o ).
% 0.16/0.42
% 0.16/0.42 thf(funcGraphProp4,definition,
% 0.16/0.42 ( funcGraphProp4
% 0.16/0.42 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.42 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ B )
% 0.16/0.42 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.16/0.42 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.16/0.42 = Xy ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(subbreln_type,type,
% 0.16/0.42 subbreln: $o ).
% 0.16/0.42
% 0.16/0.42 thf(subbreln,definition,
% 0.16/0.42 ( subbreln
% 0.16/0.42 = ( ! [A: $i,B: $i,R: $i] :
% 0.16/0.42 ( ( breln @ A @ B @ R )
% 0.16/0.42 => ! [S: $i] :
% 0.16/0.42 ( ( breln @ A @ B @ S )
% 0.16/0.42 => ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ B )
% 0.16/0.42 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.16/0.42 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.16/0.42 => ( subset @ R @ S ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(eqbreln_type,type,
% 0.16/0.42 eqbreln: $o ).
% 0.16/0.42
% 0.16/0.42 thf(eqbreln,definition,
% 0.16/0.42 ( eqbreln
% 0.16/0.42 = ( ! [A: $i,B: $i,R: $i] :
% 0.16/0.42 ( ( breln @ A @ B @ R )
% 0.16/0.42 => ! [S: $i] :
% 0.16/0.42 ( ( breln @ A @ B @ S )
% 0.16/0.42 => ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ B )
% 0.16/0.42 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.16/0.42 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.16/0.42 => ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ B )
% 0.16/0.42 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.16/0.42 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 0.16/0.42 => ( R = S ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(funcext_type,type,
% 0.16/0.42 funcext: $o ).
% 0.16/0.42
% 0.16/0.42 thf(funcext,definition,
% 0.16/0.42 ( funcext
% 0.16/0.42 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.42 ( ( func @ A @ B @ Xf )
% 0.16/0.42 => ! [Xg: $i] :
% 0.16/0.42 ( ( func @ A @ B @ Xg )
% 0.16/0.42 => ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.16/0.42 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.16/0.42 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(funcext2_type,type,
% 0.16/0.42 funcext2: $o ).
% 0.16/0.42
% 0.16/0.42 thf(funcext2,definition,
% 0.16/0.42 ( funcext2
% 0.16/0.42 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.42 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.16/0.42 => ! [Xg: $i] :
% 0.16/0.42 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 0.16/0.42 => ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.16/0.42 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.16/0.42 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(ap2apEq1_type,type,
% 0.16/0.42 ap2apEq1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(ap2apEq1,definition,
% 0.16/0.42 ( ap2apEq1
% 0.16/0.42 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.42 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.16/0.42 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(ap2apEq2_type,type,
% 0.16/0.42 ap2apEq2: $o ).
% 0.16/0.42
% 0.16/0.42 thf(ap2apEq2,definition,
% 0.16/0.42 ( ap2apEq2
% 0.16/0.42 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.42 ( ( func @ A @ B @ Xf )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.16/0.42 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(beta1_type,type,
% 0.16/0.42 beta1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(beta1,definition,
% 0.16/0.42 ( beta1
% 0.16/0.42 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.16/0.42 ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( ( ap @ A @ B
% 0.16/0.42 @ ( lam @ A @ B
% 0.16/0.42 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.16/0.42 @ Xx )
% 0.16/0.42 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(eta1_type,type,
% 0.16/0.42 eta1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(eta1,definition,
% 0.16/0.42 ( eta1
% 0.16/0.42 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.42 ( ( func @ A @ B @ Xf )
% 0.16/0.42 => ( ( lam @ A @ B
% 0.16/0.42 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.16/0.42 = Xf ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(lam2lamEq_type,type,
% 0.16/0.42 lam2lamEq: $o ).
% 0.16/0.42
% 0.16/0.42 thf(lam2lamEq,definition,
% 0.16/0.42 ( lam2lamEq
% 0.16/0.42 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.16/0.42 ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.16/0.42 => ( ( lam @ A @ B
% 0.16/0.42 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.16/0.42 = ( lam @ A @ B
% 0.16/0.42 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(beta2_type,type,
% 0.16/0.42 beta2: $o ).
% 0.16/0.42
% 0.16/0.42 thf(beta2,definition,
% 0.16/0.42 ( beta2
% 0.16/0.42 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.16/0.42 ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( ( ap @ A @ B
% 0.16/0.42 @ ( lam @ A @ B
% 0.16/0.42 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.16/0.42 @ Xx )
% 0.16/0.42 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(eta2_type,type,
% 0.16/0.42 eta2: $o ).
% 0.16/0.42
% 0.16/0.42 thf(eta2,definition,
% 0.16/0.42 ( eta2
% 0.16/0.42 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.42 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.16/0.42 => ( ( lam @ A @ B
% 0.16/0.42 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.16/0.42 = Xf ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(iffalseProp1_type,type,
% 0.16/0.42 iffalseProp1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(iffalseProp1,definition,
% 0.16/0.42 ( iffalseProp1
% 0.16/0.42 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ A )
% 0.16/0.42 => ( ~ Xphi
% 0.16/0.42 => ( in @ Xy
% 0.16/0.42 @ ( dsetconstr @ A
% 0.16/0.42 @ ^ [Xz: $i] :
% 0.16/0.42 ( ( Xphi
% 0.16/0.42 & ( Xz = Xx ) )
% 0.16/0.42 | ( ~ Xphi
% 0.16/0.42 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(iffalseProp2_type,type,
% 0.16/0.42 iffalseProp2: $o ).
% 0.16/0.42
% 0.16/0.42 thf(iffalseProp2,definition,
% 0.16/0.42 ( iffalseProp2
% 0.16/0.42 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ A )
% 0.16/0.42 => ( ~ Xphi
% 0.16/0.42 => ( ( dsetconstr @ A
% 0.16/0.42 @ ^ [Xz: $i] :
% 0.16/0.42 ( ( Xphi
% 0.16/0.42 & ( Xz = Xx ) )
% 0.16/0.42 | ( ~ Xphi
% 0.16/0.42 & ( Xz = Xy ) ) ) )
% 0.16/0.42 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(iftrueProp1_type,type,
% 0.16/0.42 iftrueProp1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(iftrueProp1,definition,
% 0.16/0.42 ( iftrueProp1
% 0.16/0.42 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ A )
% 0.16/0.42 => ( Xphi
% 0.16/0.42 => ( in @ Xx
% 0.16/0.42 @ ( dsetconstr @ A
% 0.16/0.42 @ ^ [Xz: $i] :
% 0.16/0.42 ( ( Xphi
% 0.16/0.42 & ( Xz = Xx ) )
% 0.16/0.42 | ( ~ Xphi
% 0.16/0.42 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(iftrueProp2_type,type,
% 0.16/0.42 iftrueProp2: $o ).
% 0.16/0.42
% 0.16/0.42 thf(iftrueProp2,definition,
% 0.16/0.42 ( iftrueProp2
% 0.16/0.42 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ A )
% 0.16/0.42 => ( Xphi
% 0.16/0.42 => ( ( dsetconstr @ A
% 0.16/0.42 @ ^ [Xz: $i] :
% 0.16/0.42 ( ( Xphi
% 0.16/0.42 & ( Xz = Xx ) )
% 0.16/0.42 | ( ~ Xphi
% 0.16/0.42 & ( Xz = Xy ) ) ) )
% 0.16/0.42 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(ifSingleton_type,type,
% 0.16/0.42 ifSingleton: $o ).
% 0.16/0.42
% 0.16/0.42 thf(ifSingleton,definition,
% 0.16/0.42 ( ifSingleton
% 0.16/0.42 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ A )
% 0.16/0.42 => ( singleton
% 0.16/0.42 @ ( dsetconstr @ A
% 0.16/0.42 @ ^ [Xz: $i] :
% 0.16/0.42 ( ( Xphi
% 0.16/0.42 & ( Xz = Xx ) )
% 0.16/0.42 | ( ~ Xphi
% 0.16/0.42 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(if_type,type,
% 0.16/0.42 if: $i > $o > $i > $i > $i ).
% 0.16/0.42
% 0.16/0.42 thf(ifp_type,type,
% 0.16/0.42 ifp: $o ).
% 0.16/0.42
% 0.16/0.42 thf(ifp,definition,
% 0.16/0.42 ( ifp
% 0.16/0.42 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ A )
% 0.16/0.42 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(theeq_type,type,
% 0.16/0.42 theeq: $o ).
% 0.16/0.42
% 0.16/0.42 thf(theeq,definition,
% 0.16/0.42 ( theeq
% 0.16/0.42 = ( ! [X: $i] :
% 0.16/0.42 ( ( singleton @ X )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ X )
% 0.16/0.42 => ( ( setunion @ X )
% 0.16/0.42 = Xx ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(iftrue_type,type,
% 0.16/0.42 iftrue: $o ).
% 0.16/0.42
% 0.16/0.42 thf(iftrue,definition,
% 0.16/0.42 ( iftrue
% 0.16/0.42 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ A )
% 0.16/0.42 => ( Xphi
% 0.16/0.42 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.16/0.42 = Xx ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(iffalse_type,type,
% 0.16/0.42 iffalse: $o ).
% 0.16/0.42
% 0.16/0.42 thf(iffalse,definition,
% 0.16/0.42 ( iffalse
% 0.16/0.42 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ A )
% 0.16/0.42 => ( ~ Xphi
% 0.16/0.42 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.16/0.42 = Xy ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(iftrueorfalse_type,type,
% 0.16/0.42 iftrueorfalse: $o ).
% 0.16/0.42
% 0.16/0.42 thf(iftrueorfalse,definition,
% 0.16/0.42 ( iftrueorfalse
% 0.16/0.42 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ A )
% 0.16/0.42 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(binintersectT_lem_type,type,
% 0.16/0.42 binintersectT_lem: $o ).
% 0.16/0.42
% 0.16/0.42 thf(binintersectT_lem,definition,
% 0.16/0.42 ( binintersectT_lem
% 0.16/0.42 = ( ! [A: $i,X: $i] :
% 0.16/0.42 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.42 => ! [Y: $i] :
% 0.16/0.42 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.42 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(binunionT_lem_type,type,
% 0.16/0.42 binunionT_lem: $o ).
% 0.16/0.42
% 0.16/0.42 thf(binunionT_lem,definition,
% 0.16/0.42 ( binunionT_lem
% 0.16/0.42 = ( ! [A: $i,X: $i] :
% 0.16/0.42 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.42 => ! [Y: $i] :
% 0.16/0.42 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.42 => ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(powersetT_lem_type,type,
% 0.16/0.42 powersetT_lem: $o ).
% 0.16/0.42
% 0.16/0.42 thf(powersetT_lem,definition,
% 0.16/0.42 ( powersetT_lem
% 0.16/0.42 = ( ! [A: $i,X: $i] :
% 0.16/0.42 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.42 => ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setminusT_lem_type,type,
% 0.16/0.42 setminusT_lem: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setminusT_lem,definition,
% 0.16/0.42 ( setminusT_lem
% 0.16/0.42 = ( ! [A: $i,X: $i] :
% 0.16/0.42 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.42 => ! [Y: $i] :
% 0.16/0.42 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.42 => ( in @ ( setminus @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(complementT_lem_type,type,
% 0.16/0.42 complementT_lem: $o ).
% 0.16/0.42
% 0.16/0.42 thf(complementT_lem,definition,
% 0.16/0.42 ( complementT_lem
% 0.16/0.42 = ( ! [A: $i,X: $i] :
% 0.16/0.42 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.42 => ( in @ ( setminus @ A @ X ) @ ( powerset @ A ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setextT_type,type,
% 0.16/0.42 setextT: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setextT,definition,
% 0.16/0.42 ( setextT
% 0.16/0.42 = ( ! [A: $i,X: $i] :
% 0.16/0.42 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.42 => ! [Y: $i] :
% 0.16/0.42 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.42 => ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( ( in @ Xx @ X )
% 0.16/0.42 => ( in @ Xx @ Y ) ) )
% 0.16/0.42 => ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( ( in @ Xx @ Y )
% 0.16/0.42 => ( in @ Xx @ X ) ) )
% 0.16/0.42 => ( X = Y ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(subsetTI_type,type,
% 0.16/0.42 subsetTI: $o ).
% 0.16/0.42
% 0.16/0.42 thf(subsetTI,definition,
% 0.16/0.42 ( subsetTI
% 0.16/0.42 = ( ! [A: $i,X: $i] :
% 0.16/0.42 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.42 => ! [Y: $i] :
% 0.16/0.42 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.42 => ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( ( in @ Xx @ X )
% 0.16/0.42 => ( in @ Xx @ Y ) ) )
% 0.16/0.42 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(powersetTI1_type,type,
% 0.16/0.42 powersetTI1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(powersetTI1,definition,
% 0.16/0.42 ( powersetTI1
% 0.16/0.42 = ( ! [A: $i,X: $i] :
% 0.16/0.42 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.42 => ! [Y: $i] :
% 0.16/0.42 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.42 => ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( ( in @ Xx @ X )
% 0.16/0.42 => ( in @ Xx @ Y ) ) )
% 0.16/0.42 => ( in @ X @ ( powerset @ Y ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(powersetTE1_type,type,
% 0.16/0.42 powersetTE1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(powersetTE1,definition,
% 0.16/0.42 ( powersetTE1
% 0.16/0.42 = ( ! [A: $i,X: $i] :
% 0.16/0.42 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.42 => ! [Y: $i] :
% 0.16/0.42 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( ( in @ X @ ( powerset @ Y ) )
% 0.16/0.42 => ( ( in @ Xx @ X )
% 0.16/0.42 => ( in @ Xx @ Y ) ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(complementTI1_type,type,
% 0.16/0.42 complementTI1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(complementTI1,definition,
% 0.16/0.42 ( complementTI1
% 0.16/0.42 = ( ! [A: $i,X: $i] :
% 0.16/0.42 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( ( in @ Xx @ X )
% 0.16/0.42 => ~ ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(complementTE1_type,type,
% 0.16/0.42 complementTE1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(complementTE1,definition,
% 0.16/0.42 ( complementTE1
% 0.16/0.42 = ( ! [A: $i,X: $i] :
% 0.16/0.42 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( ~ ( in @ Xx @ ( setminus @ A @ X ) )
% 0.16/0.42 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(binintersectTELcontra_type,type,
% 0.16/0.42 binintersectTELcontra: $o ).
% 0.16/0.42
% 0.16/0.42 thf(binintersectTELcontra,definition,
% 0.16/0.42 ( binintersectTELcontra
% 0.16/0.42 = ( ! [A: $i,X: $i] :
% 0.16/0.42 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.42 => ! [Y: $i] :
% 0.16/0.42 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( ~ ( in @ Xx @ X )
% 0.16/0.42 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(binintersectTERcontra_type,type,
% 0.16/0.42 binintersectTERcontra: $o ).
% 0.16/0.42
% 0.16/0.42 thf(binintersectTERcontra,definition,
% 0.16/0.42 ( binintersectTERcontra
% 0.16/0.42 = ( ! [A: $i,X: $i] :
% 0.16/0.42 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.42 => ! [Y: $i] :
% 0.16/0.42 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( ~ ( in @ Xx @ Y )
% 0.16/0.42 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(contrasubsetT_type,type,
% 0.16/0.42 contrasubsetT: $o ).
% 0.16/0.42
% 0.16/0.42 thf(contrasubsetT,definition,
% 0.16/0.42 ( contrasubsetT
% 0.16/0.42 = ( ! [A: $i,X: $i] :
% 0.16/0.42 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.42 => ! [Y: $i] :
% 0.16/0.42 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.16/0.42 => ( ( in @ Xx @ Y )
% 0.16/0.42 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(contrasubsetT1_type,type,
% 0.16/0.42 contrasubsetT1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(contrasubsetT1,definition,
% 0.16/0.42 ( contrasubsetT1
% 0.16/0.42 = ( ! [A: $i,X: $i] :
% 0.16/0.42 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.42 => ! [Y: $i] :
% 0.16/0.42 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( ( subset @ X @ Y )
% 0.16/0.42 => ( ~ ( in @ Xx @ Y )
% 0.16/0.42 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(contrasubsetT2_type,type,
% 0.16/0.42 contrasubsetT2: $o ).
% 0.16/0.42
% 0.16/0.42 thf(contrasubsetT2,definition,
% 0.16/0.42 ( contrasubsetT2
% 0.16/0.42 = ( ! [A: $i,X: $i] :
% 0.16/0.42 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.42 => ! [Y: $i] :
% 0.16/0.42 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.42 => ( ( subset @ X @ Y )
% 0.16/0.42 => ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(contrasubsetT3_type,type,
% 0.16/0.42 contrasubsetT3: $o ).
% 0.16/0.42
% 0.16/0.42 thf(contrasubsetT3,definition,
% 0.16/0.42 ( contrasubsetT3
% 0.16/0.42 = ( ! [A: $i,X: $i] :
% 0.16/0.42 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.42 => ! [Y: $i] :
% 0.16/0.42 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.42 => ( ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) )
% 0.16/0.42 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(doubleComplementI1_type,type,
% 0.16/0.42 doubleComplementI1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(doubleComplementI1,definition,
% 0.16/0.42 ( doubleComplementI1
% 0.16/0.42 = ( ! [A: $i,X: $i] :
% 0.16/0.42 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( ( in @ Xx @ X )
% 0.16/0.42 => ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(doubleComplementE1_type,type,
% 0.16/0.42 doubleComplementE1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(doubleComplementE1,definition,
% 0.16/0.42 ( doubleComplementE1
% 0.16/0.42 = ( ! [A: $i,X: $i] :
% 0.16/0.42 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) )
% 0.16/0.42 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(doubleComplementSub1_type,type,
% 0.16/0.42 doubleComplementSub1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(doubleComplementSub1,definition,
% 0.16/0.42 ( doubleComplementSub1
% 0.16/0.42 = ( ! [A: $i,X: $i] :
% 0.16/0.42 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.42 => ( subset @ X @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(doubleComplementSub2_type,type,
% 0.16/0.42 doubleComplementSub2: $o ).
% 0.16/0.42
% 0.16/0.42 thf(doubleComplementSub2,definition,
% 0.16/0.42 ( doubleComplementSub2
% 0.16/0.42 = ( ! [A: $i,X: $i] :
% 0.16/0.42 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.42 => ( subset @ ( setminus @ A @ ( setminus @ A @ X ) ) @ X ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(doubleComplementEq_type,type,
% 0.16/0.42 doubleComplementEq: $o ).
% 0.16/0.42
% 0.16/0.42 thf(doubleComplementEq,definition,
% 0.16/0.42 ( doubleComplementEq
% 0.16/0.42 = ( ! [A: $i,X: $i] :
% 0.16/0.42 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.42 => ( X
% 0.16/0.42 = ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(complementTnotintersectT_type,type,
% 0.16/0.42 complementTnotintersectT: $o ).
% 0.16/0.42
% 0.16/0.42 thf(complementTnotintersectT,definition,
% 0.16/0.42 ( complementTnotintersectT
% 0.16/0.42 = ( ! [A: $i,X: $i] :
% 0.16/0.42 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.42 => ! [Y: $i] :
% 0.16/0.42 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.16/0.42 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(complementImpComplementIntersect_type,type,
% 0.16/0.42 complementImpComplementIntersect: $o ).
% 0.16/0.42
% 0.16/0.42 thf(complementImpComplementIntersect,definition,
% 0.16/0.42 ( complementImpComplementIntersect
% 0.16/0.42 = ( ! [A: $i,X: $i] :
% 0.16/0.42 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.42 => ! [Y: $i] :
% 0.16/0.42 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.16/0.42 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(complementSubsetComplementIntersect_type,type,
% 0.16/0.42 complementSubsetComplementIntersect: $o ).
% 0.16/0.42
% 0.16/0.42 thf(complementSubsetComplementIntersect,definition,
% 0.16/0.42 ( complementSubsetComplementIntersect
% 0.16/0.42 = ( ! [A: $i,X: $i] :
% 0.16/0.42 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.42 => ! [Y: $i] :
% 0.16/0.42 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.42 => ( subset @ ( setminus @ A @ X ) @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(complementInPowersetComplementIntersect_type,type,
% 0.16/0.42 complementInPowersetComplementIntersect: $o ).
% 0.16/0.42
% 0.16/0.42 thf(complementInPowersetComplementIntersect,definition,
% 0.16/0.42 ( complementInPowersetComplementIntersect
% 0.16/0.42 = ( ! [A: $i,X: $i] :
% 0.16/0.42 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.42 => ! [Y: $i] :
% 0.16/0.42 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.42 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(contraSubsetComplement_type,type,
% 0.16/0.42 contraSubsetComplement: $o ).
% 0.16/0.42
% 0.16/0.42 thf(contraSubsetComplement,definition,
% 0.16/0.42 ( contraSubsetComplement
% 0.16/0.42 = ( ! [A: $i,X: $i] :
% 0.16/0.42 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.42 => ! [Y: $i] :
% 0.16/0.42 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.42 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( ( in @ Xx @ Y )
% 0.16/0.42 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(complementTcontraSubset_type,type,
% 0.16/0.42 complementTcontraSubset: $o ).
% 0.16/0.42
% 0.16/0.42 thf(complementTcontraSubset,definition,
% 0.16/0.42 ( complementTcontraSubset
% 0.16/0.42 = ( ! [A: $i,X: $i] :
% 0.16/0.42 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.42 => ! [Y: $i] :
% 0.16/0.42 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.42 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.16/0.42 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(binunionTILcontra_type,type,
% 0.16/0.42 binunionTILcontra: $o ).
% 0.16/0.42
% 0.16/0.42 thf(binunionTILcontra,definition,
% 0.16/0.42 ( binunionTILcontra
% 0.16/0.42 = ( ! [A: $i,X: $i] :
% 0.16/0.42 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.42 => ! [Y: $i] :
% 0.16/0.42 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.16/0.42 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(binunionTIRcontra_type,type,
% 0.16/0.42 binunionTIRcontra: $o ).
% 0.16/0.42
% 0.16/0.42 thf(binunionTIRcontra,definition,
% 0.16/0.42 ( binunionTIRcontra
% 0.16/0.42 = ( ! [A: $i,X: $i] :
% 0.16/0.42 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.42 => ! [Y: $i] :
% 0.16/0.42 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.16/0.42 => ~ ( in @ Xx @ Y ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(inIntersectImpInUnion_type,type,
% 0.16/0.42 inIntersectImpInUnion: $o ).
% 0.16/0.42
% 0.16/0.42 thf(inIntersectImpInUnion,definition,
% 0.16/0.42 ( inIntersectImpInUnion
% 0.16/0.42 = ( ! [A: $i,X: $i] :
% 0.16/0.42 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.42 => ! [Y: $i] :
% 0.16/0.42 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.42 => ! [Z: $i] :
% 0.16/0.42 ( ( in @ Z @ ( powerset @ A ) )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.16/0.42 => ( in @ Xx @ ( binunion @ X @ Z ) ) ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(inIntersectImpInUnion2_type,type,
% 0.16/0.42 inIntersectImpInUnion2: $o ).
% 0.16/0.42
% 0.16/0.42 thf(inIntersectImpInUnion2,definition,
% 0.16/0.42 ( inIntersectImpInUnion2
% 0.16/0.42 = ( ! [A: $i,X: $i] :
% 0.16/0.42 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.42 => ! [Y: $i] :
% 0.16/0.42 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.42 => ! [Z: $i] :
% 0.16/0.42 ( ( in @ Z @ ( powerset @ A ) )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.16/0.42 => ( in @ Xx @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(inIntersectImpInIntersectUnions_type,type,
% 0.16/0.42 inIntersectImpInIntersectUnions: $o ).
% 0.16/0.42
% 0.16/0.42 thf(inIntersectImpInIntersectUnions,definition,
% 0.16/0.42 ( inIntersectImpInIntersectUnions
% 0.16/0.42 = ( ! [A: $i,X: $i] :
% 0.16/0.42 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.42 => ! [Y: $i] :
% 0.16/0.42 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.42 => ! [Z: $i] :
% 0.16/0.42 ( ( in @ Z @ ( powerset @ A ) )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.16/0.42 => ( in @ Xx @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(intersectInPowersetIntersectUnions_type,type,
% 0.16/0.42 intersectInPowersetIntersectUnions: $o ).
% 0.16/0.42
% 0.16/0.42 thf(intersectInPowersetIntersectUnions,definition,
% 0.16/0.42 ( intersectInPowersetIntersectUnions
% 0.16/0.42 = ( ! [A: $i,X: $i] :
% 0.16/0.42 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.42 => ! [Y: $i] :
% 0.16/0.42 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.42 => ! [Z: $i] :
% 0.16/0.42 ( ( in @ Z @ ( powerset @ A ) )
% 0.16/0.42 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(inComplementUnionImpNotIn1_type,type,
% 0.16/0.42 inComplementUnionImpNotIn1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(inComplementUnionImpNotIn1,definition,
% 0.16/0.42 ( inComplementUnionImpNotIn1
% 0.16/0.42 = ( ! [A: $i,X: $i] :
% 0.16/0.42 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.42 => ! [Y: $i] :
% 0.16/0.42 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.16/0.42 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(inComplementUnionImpInComplement1_type,type,
% 0.16/0.42 inComplementUnionImpInComplement1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(inComplementUnionImpInComplement1,definition,
% 0.16/0.42 ( inComplementUnionImpInComplement1
% 0.16/0.42 = ( ! [A: $i,X: $i] :
% 0.16/0.42 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.42 => ! [Y: $i] :
% 0.16/0.42 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.16/0.42 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(binunionTE_type,type,
% 0.16/0.42 binunionTE: $o ).
% 0.16/0.42
% 0.16/0.42 thf(binunionTE,definition,
% 0.16/0.42 ( binunionTE
% 0.16/0.42 = ( ! [A: $i,X: $i] :
% 0.16/0.42 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.42 => ! [Y: $i] :
% 0.16/0.42 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.42 => ! [Xphi: $o,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.16/0.42 => ( ( ( in @ Xx @ X )
% 0.16/0.42 => Xphi )
% 0.16/0.42 => ( ( ( in @ Xx @ Y )
% 0.16/0.42 => Xphi )
% 0.16/0.42 => Xphi ) ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(binunionTEcontra_type,type,
% 0.16/0.42 binunionTEcontra: $o ).
% 0.16/0.42
% 0.16/0.42 thf(binunionTEcontra,definition,
% 0.16/0.42 ( binunionTEcontra
% 0.16/0.42 = ( ! [A: $i,X: $i] :
% 0.16/0.42 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.42 => ! [Y: $i] :
% 0.16/0.42 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( ~ ( in @ Xx @ X )
% 0.16/0.42 => ( ~ ( in @ Xx @ Y )
% 0.16/0.42 => ~ ( in @ Xx @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(demorgan2a1_type,type,
% 0.16/0.42 demorgan2a1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(demorgan2a1,definition,
% 0.16/0.42 ( demorgan2a1
% 0.16/0.42 = ( ! [A: $i,X: $i] :
% 0.16/0.42 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.42 => ! [Y: $i] :
% 0.16/0.42 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.16/0.42 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(complementUnionInPowersetComplement_type,type,
% 0.16/0.42 complementUnionInPowersetComplement: $o ).
% 0.16/0.42
% 0.16/0.42 thf(complementUnionInPowersetComplement,definition,
% 0.16/0.42 ( complementUnionInPowersetComplement
% 0.16/0.42 = ( ! [A: $i,X: $i] :
% 0.16/0.42 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.42 => ! [Y: $i] :
% 0.16/0.42 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.42 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(demorgan2a2_type,type,
% 0.16/0.42 demorgan2a2: $o ).
% 0.16/0.42
% 0.16/0.42 thf(demorgan2a2,definition,
% 0.16/0.42 ( demorgan2a2
% 0.16/0.42 = ( ! [A: $i,X: $i] :
% 0.16/0.42 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.42 => ! [Y: $i] :
% 0.16/0.42 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.16/0.42 => ( in @ Xx @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(demorgan1a_type,type,
% 0.16/0.42 demorgan1a: $o ).
% 0.16/0.42
% 0.16/0.42 thf(demorgan1a,definition,
% 0.16/0.42 ( demorgan1a
% 0.16/0.42 = ( ! [A: $i,X: $i] :
% 0.16/0.42 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.42 => ! [Y: $i] :
% 0.16/0.42 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) )
% 0.16/0.42 => ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(demorgan1b_type,type,
% 0.16/0.42 demorgan1b: $o ).
% 0.16/0.42
% 0.16/0.42 thf(demorgan1b,definition,
% 0.16/0.42 ( demorgan1b
% 0.16/0.42 = ( ! [A: $i,X: $i] :
% 0.16/0.42 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.42 => ! [Y: $i] :
% 0.16/0.42 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 0.16/0.42 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(demorgan1_type,type,
% 0.16/0.42 demorgan1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(demorgan1,definition,
% 0.16/0.42 ( demorgan1
% 0.16/0.42 = ( ! [A: $i,X: $i] :
% 0.16/0.42 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.42 => ! [Y: $i] :
% 0.16/0.42 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.42 => ( ( setminus @ A @ ( binintersect @ X @ Y ) )
% 0.16/0.42 = ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(demorgan2a_type,type,
% 0.16/0.42 demorgan2a: $o ).
% 0.16/0.42
% 0.16/0.42 thf(demorgan2a,definition,
% 0.16/0.42 ( demorgan2a
% 0.16/0.42 = ( ! [A: $i,X: $i] :
% 0.16/0.42 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.42 => ! [Y: $i] :
% 0.16/0.42 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.16/0.42 => ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(demorgan2b2_type,type,
% 0.16/0.42 demorgan2b2: $o ).
% 0.16/0.42
% 0.16/0.42 thf(demorgan2b2,definition,
% 0.16/0.42 ( demorgan2b2
% 0.16/0.42 = ( ! [A: $i,X: $i] :
% 0.16/0.42 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.42 => ! [Y: $i] :
% 0.16/0.42 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.16/0.42 => ( ( in @ Xx @ ( setminus @ A @ Y ) )
% 0.16/0.42 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(demorgan2b_type,type,
% 0.16/0.42 demorgan2b: $o ).
% 0.16/0.42
% 0.16/0.42 thf(demorgan2b,definition,
% 0.16/0.42 ( demorgan2b
% 0.16/0.42 = ( ! [A: $i,X: $i] :
% 0.16/0.42 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.42 => ! [Y: $i] :
% 0.16/0.42 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 0.16/0.42 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(demorgan2_type,type,
% 0.16/0.42 demorgan2: $o ).
% 0.16/0.42
% 0.16/0.42 thf(demorgan2,definition,
% 0.16/0.42 ( demorgan2
% 0.16/0.42 = ( ! [A: $i,X: $i] :
% 0.16/0.42 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.42 => ! [Y: $i] :
% 0.16/0.42 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.42 => ( ( setminus @ A @ ( binunion @ X @ Y ) )
% 0.16/0.42 = ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(woz13rule0_type,type,
% 0.16/0.42 woz13rule0: $o ).
% 0.16/0.42
% 0.16/0.42 thf(woz13rule0,definition,
% 0.16/0.42 ( woz13rule0
% 0.16/0.42 = ( ! [A: $i,X: $i] :
% 0.16/0.42 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.42 => ! [Y: $i] :
% 0.16/0.42 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.16/0.42 => ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(woz13rule1_type,type,
% 0.16/0.42 woz13rule1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(woz13rule1,definition,
% 0.16/0.42 ( woz13rule1
% 0.16/0.42 = ( ! [A: $i,X: $i] :
% 0.16/0.42 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.42 => ! [Y: $i] :
% 0.16/0.42 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.42 => ! [Z: $i] :
% 0.16/0.42 ( ( in @ Z @ ( powerset @ A ) )
% 0.16/0.42 => ( ( subset @ X @ Z )
% 0.16/0.42 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(woz13rule2_type,type,
% 0.16/0.42 woz13rule2: $o ).
% 0.16/0.42
% 0.16/0.42 thf(woz13rule2,definition,
% 0.16/0.42 ( woz13rule2
% 0.16/0.42 = ( ! [A: $i,X: $i] :
% 0.16/0.42 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.42 => ! [Y: $i] :
% 0.16/0.42 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.42 => ! [Z: $i] :
% 0.16/0.42 ( ( in @ Z @ ( powerset @ A ) )
% 0.16/0.42 => ( ( subset @ Y @ Z )
% 0.16/0.42 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(woz13rule3_type,type,
% 0.16/0.42 woz13rule3: $o ).
% 0.16/0.42
% 0.16/0.42 thf(woz13rule3,definition,
% 0.16/0.42 ( woz13rule3
% 0.16/0.42 = ( ! [A: $i,X: $i] :
% 0.16/0.42 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.42 => ! [Y: $i] :
% 0.16/0.42 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.42 => ! [Z: $i] :
% 0.16/0.42 ( ( in @ Z @ ( powerset @ A ) )
% 0.16/0.42 => ( ( subset @ X @ Y )
% 0.16/0.42 => ( ( subset @ X @ Z )
% 0.16/0.42 => ( subset @ X @ ( binintersect @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(woz13rule4_type,type,
% 0.16/0.42 woz13rule4: $o ).
% 0.16/0.42
% 0.16/0.42 thf(woz13rule4,definition,
% 0.16/0.42 ( woz13rule4
% 0.16/0.42 = ( ! [A: $i,X: $i] :
% 0.16/0.42 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.42 => ! [Y: $i] :
% 0.16/0.42 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.42 => ! [Z: $i] :
% 0.16/0.42 ( ( in @ Z @ ( powerset @ A ) )
% 0.16/0.42 => ! [W: $i] :
% 0.16/0.42 ( ( in @ W @ ( powerset @ A ) )
% 0.16/0.42 => ( ( subset @ X @ Z )
% 0.16/0.42 => ( ( subset @ Y @ W )
% 0.16/0.42 => ( subset @ ( binintersect @ X @ Y ) @ ( binintersect @ Z @ W ) ) ) ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(woz1_1_type,type,
% 0.16/0.42 woz1_1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(woz1_1,definition,
% 0.16/0.42 ( woz1_1
% 0.16/0.42 = ( ! [A: $i,X: $i] :
% 0.16/0.42 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.42 => ! [Y: $i] :
% 0.16/0.42 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.42 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(woz1_2_type,type,
% 0.16/0.42 woz1_2: $o ).
% 0.16/0.42
% 0.16/0.42 thf(woz1_2,definition,
% 0.16/0.42 ( woz1_2
% 0.16/0.42 = ( ! [A: $i,X: $i] :
% 0.16/0.42 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.42 => ! [Y: $i] :
% 0.16/0.42 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.42 => ! [Z: $i] :
% 0.16/0.42 ( ( in @ Z @ ( powerset @ A ) )
% 0.16/0.42 => ! [W: $i] :
% 0.16/0.42 ( ( in @ W @ ( powerset @ A ) )
% 0.16/0.42 => ( ( setminus @ A @ ( binintersect @ ( binunion @ X @ Y ) @ ( binunion @ Z @ W ) ) )
% 0.16/0.42 = ( binunion @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) @ ( binintersect @ ( setminus @ A @ Z ) @ ( setminus @ A @ W ) ) ) ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(woz1_3_type,type,
% 0.16/0.42 woz1_3: $o ).
% 0.16/0.42
% 0.16/0.42 thf(woz1_3,definition,
% 0.16/0.42 ( woz1_3
% 0.16/0.42 = ( ! [A: $i,X: $i] :
% 0.16/0.42 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.42 => ! [Y: $i] :
% 0.16/0.42 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.42 => ! [Z: $i] :
% 0.16/0.42 ( ( in @ Z @ ( powerset @ A ) )
% 0.16/0.42 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(woz1_4_type,type,
% 0.16/0.42 woz1_4: $o ).
% 0.16/0.42
% 0.16/0.42 thf(woz1_4,definition,
% 0.16/0.42 ( woz1_4
% 0.16/0.42 = ( ! [A: $i,X: $i] :
% 0.16/0.42 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.42 => ! [Y: $i] :
% 0.16/0.42 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.42 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.16/0.42 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(woz1_5_type,type,
% 0.16/0.42 woz1_5: $o ).
% 0.16/0.42
% 0.16/0.42 thf(woz1_5,definition,
% 0.16/0.42 ( woz1_5
% 0.16/0.42 = ( ! [A: $i,X: $i] :
% 0.16/0.42 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.42 => ! [Y: $i] :
% 0.16/0.42 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.42 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(breln1_type,type,
% 0.16/0.42 breln1: $i > $i > $o ).
% 0.16/0.42
% 0.16/0.42 thf(breln1all2_type,type,
% 0.16/0.42 breln1all2: $o ).
% 0.16/0.42
% 0.16/0.42 thf(breln1all2,definition,
% 0.16/0.42 ( breln1all2
% 0.16/0.42 = ( ! [A: $i,R: $i] :
% 0.16/0.42 ( ( breln1 @ A @ R )
% 0.16/0.42 => ! [Xphi: $i > $o] :
% 0.16/0.42 ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ A )
% 0.16/0.42 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.16/0.42 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ R )
% 0.16/0.42 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(breln1Set_type,type,
% 0.16/0.42 breln1Set: $i > $i ).
% 0.16/0.42
% 0.16/0.42 thf(breln1SetBreln1_type,type,
% 0.16/0.42 breln1SetBreln1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(breln1SetBreln1,definition,
% 0.16/0.42 ( breln1SetBreln1
% 0.16/0.42 = ( ! [A: $i,R: $i] :
% 0.16/0.42 ( ( in @ R @ ( breln1Set @ A ) )
% 0.16/0.42 => ( breln1 @ A @ R ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(transitive_type,type,
% 0.16/0.42 transitive: $i > $i > $o ).
% 0.16/0.42
% 0.16/0.42 thf(antisymmetric_type,type,
% 0.16/0.42 antisymmetric: $i > $i > $o ).
% 0.16/0.42
% 0.16/0.42 thf(reflexive_type,type,
% 0.16/0.42 reflexive: $i > $i > $o ).
% 0.16/0.42
% 0.16/0.42 thf(refltransitive_type,type,
% 0.16/0.42 refltransitive: $i > $i > $o ).
% 0.16/0.42
% 0.16/0.42 thf(refllinearorder_type,type,
% 0.16/0.42 refllinearorder: $i > $i > $o ).
% 0.16/0.42
% 0.16/0.42 thf(reflwellordering_type,type,
% 0.16/0.42 reflwellordering: $i > $i > $o ).
% 0.16/0.42
% 0.16/0.42 thf(choice2fnsingleton_type,type,
% 0.16/0.42 choice2fnsingleton: $o ).
% 0.16/0.42
% 0.16/0.42 thf(choice2fnsingleton,definition,
% 0.16/0.42 ( choice2fnsingleton
% 0.16/0.42 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.16/0.42 ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ? [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ B )
% 0.16/0.42 & ( Xphi @ Xx @ Xy ) ) )
% 0.16/0.42 => ! [R: $i] :
% 0.16/0.42 ( ( in @ R @ ( breln1Set @ B ) )
% 0.16/0.42 => ( ( reflwellordering @ B @ R )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( singleton
% 0.16/0.42 @ ( dsetconstr @ B
% 0.16/0.42 @ ^ [Xy: $i] :
% 0.16/0.42 ( ( Xphi @ Xx @ Xy )
% 0.16/0.42 & ! [Xz: $i] :
% 0.16/0.42 ( ( in @ Xz @ B )
% 0.16/0.42 => ( ( Xphi @ Xx @ Xz )
% 0.16/0.42 => ( in @ ( kpair @ Xy @ Xz ) @ R ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setOfPairsIsBReln1_type,type,
% 0.16/0.42 setOfPairsIsBReln1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setOfPairsIsBReln1,definition,
% 0.16/0.42 ( setOfPairsIsBReln1
% 0.16/0.42 = ( ! [A: $i,Xphi: $i > $i > $o] :
% 0.16/0.42 ( breln1 @ A
% 0.16/0.42 @ ( dpsetconstr @ A @ A
% 0.16/0.42 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(breln1all1_type,type,
% 0.16/0.42 breln1all1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(breln1all1,definition,
% 0.16/0.42 ( breln1all1
% 0.16/0.42 = ( ! [A: $i,R: $i] :
% 0.16/0.42 ( ( breln1 @ A @ R )
% 0.16/0.42 => ! [Xphi: $i > $o] :
% 0.16/0.42 ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ A )
% 0.16/0.42 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.16/0.42 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ R )
% 0.16/0.42 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(subbreln1_type,type,
% 0.16/0.42 subbreln1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(subbreln1,definition,
% 0.16/0.42 ( subbreln1
% 0.16/0.42 = ( ! [A: $i,R: $i] :
% 0.16/0.42 ( ( breln1 @ A @ R )
% 0.16/0.42 => ! [S: $i] :
% 0.16/0.42 ( ( breln1 @ A @ S )
% 0.16/0.42 => ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ A )
% 0.16/0.42 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.16/0.42 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.16/0.42 => ( subset @ R @ S ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(eqbreln1_type,type,
% 0.16/0.42 eqbreln1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(eqbreln1,definition,
% 0.16/0.42 ( eqbreln1
% 0.16/0.42 = ( ! [A: $i,R: $i] :
% 0.16/0.42 ( ( breln1 @ A @ R )
% 0.16/0.42 => ! [S: $i] :
% 0.16/0.42 ( ( breln1 @ A @ S )
% 0.16/0.42 => ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ A )
% 0.16/0.42 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.16/0.42 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.16/0.42 => ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ A )
% 0.16/0.42 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.16/0.42 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 0.16/0.42 => ( R = S ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(breln1invset_type,type,
% 0.16/0.42 breln1invset: $i > $i > $i ).
% 0.16/0.42
% 0.16/0.42 thf(breln1invprop_type,type,
% 0.16/0.42 breln1invprop: $o ).
% 0.16/0.42
% 0.16/0.42 thf(breln1invprop,definition,
% 0.16/0.42 ( breln1invprop
% 0.16/0.42 = ( ! [A: $i,R: $i] :
% 0.16/0.42 ( ( breln1 @ A @ R )
% 0.16/0.42 => ( breln1 @ A @ ( breln1invset @ A @ R ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(breln1invI_type,type,
% 0.16/0.42 breln1invI: $o ).
% 0.16/0.42
% 0.16/0.42 thf(breln1invI,definition,
% 0.16/0.42 ( breln1invI
% 0.16/0.42 = ( ! [A: $i,R: $i] :
% 0.16/0.42 ( ( breln1 @ A @ R )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ A )
% 0.16/0.42 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.16/0.42 => ( in @ ( kpair @ Xy @ Xx ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(breln1invE_type,type,
% 0.16/0.42 breln1invE: $o ).
% 0.16/0.42
% 0.16/0.42 thf(breln1invE,definition,
% 0.16/0.42 ( breln1invE
% 0.16/0.42 = ( ! [A: $i,R: $i] :
% 0.16/0.42 ( ( breln1 @ A @ R )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ A )
% 0.16/0.42 => ( ( in @ ( kpair @ Xy @ Xx ) @ ( breln1invset @ A @ R ) )
% 0.16/0.42 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(breln1compset_type,type,
% 0.16/0.42 breln1compset: $i > $i > $i > $i ).
% 0.16/0.42
% 0.16/0.42 thf(breln1compprop_type,type,
% 0.16/0.42 breln1compprop: $o ).
% 0.16/0.42
% 0.16/0.42 thf(breln1compprop,definition,
% 0.16/0.42 ( breln1compprop
% 0.16/0.42 = ( ! [A: $i,R: $i] :
% 0.16/0.42 ( ( breln1 @ A @ R )
% 0.16/0.42 => ! [S: $i] :
% 0.16/0.42 ( ( breln1 @ A @ S )
% 0.16/0.42 => ( breln1 @ A @ ( breln1compset @ A @ R @ S ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(breln1compI_type,type,
% 0.16/0.42 breln1compI: $o ).
% 0.16/0.42
% 0.16/0.42 thf(breln1compI,definition,
% 0.16/0.42 ( breln1compI
% 0.16/0.42 = ( ! [A: $i,R: $i] :
% 0.16/0.42 ( ( breln1 @ A @ R )
% 0.16/0.42 => ! [S: $i] :
% 0.16/0.42 ( ( breln1 @ A @ S )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ A )
% 0.16/0.42 => ! [Xz: $i] :
% 0.16/0.42 ( ( in @ Xz @ A )
% 0.16/0.42 => ( ( in @ ( kpair @ Xx @ Xz ) @ R )
% 0.16/0.42 => ( ( in @ ( kpair @ Xz @ Xy ) @ S )
% 0.16/0.42 => ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) ) ) ) ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(breln1compE_type,type,
% 0.16/0.42 breln1compE: $o ).
% 0.16/0.42
% 0.16/0.42 thf(breln1compE,definition,
% 0.16/0.42 ( breln1compE
% 0.16/0.42 = ( ! [A: $i,R: $i] :
% 0.16/0.42 ( ( breln1 @ A @ R )
% 0.16/0.42 => ! [S: $i] :
% 0.16/0.42 ( ( breln1 @ A @ S )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ A )
% 0.16/0.42 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) )
% 0.16/0.42 => ? [Xz: $i] :
% 0.16/0.42 ( ( in @ Xz @ A )
% 0.16/0.42 & ( in @ ( kpair @ Xx @ Xz ) @ R )
% 0.16/0.42 & ( in @ ( kpair @ Xz @ Xy ) @ S ) ) ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(breln1compEex_type,type,
% 0.16/0.42 breln1compEex: $o ).
% 0.16/0.42
% 0.16/0.42 thf(breln1compEex,definition,
% 0.16/0.42 ( breln1compEex
% 0.16/0.42 = ( ! [A: $i,R: $i] :
% 0.16/0.42 ( ( breln1 @ A @ R )
% 0.16/0.42 => ! [S: $i] :
% 0.16/0.42 ( ( breln1 @ A @ S )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ A )
% 0.16/0.42 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) )
% 0.16/0.42 => ! [Xphi: $o] :
% 0.16/0.42 ( ! [Xz: $i] :
% 0.16/0.42 ( ( in @ Xz @ A )
% 0.16/0.42 => ( ( in @ ( kpair @ Xx @ Xz ) @ R )
% 0.16/0.42 => ( ( in @ ( kpair @ Xz @ Xy ) @ S )
% 0.16/0.42 => Xphi ) ) )
% 0.16/0.42 => Xphi ) ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(breln1unionprop_type,type,
% 0.16/0.42 breln1unionprop: $o ).
% 0.16/0.42
% 0.16/0.42 thf(breln1unionprop,definition,
% 0.16/0.42 ( breln1unionprop
% 0.16/0.42 = ( ! [A: $i,R: $i] :
% 0.16/0.42 ( ( breln1 @ A @ R )
% 0.16/0.42 => ! [S: $i] :
% 0.16/0.42 ( ( breln1 @ A @ S )
% 0.16/0.42 => ( breln1 @ A @ ( binunion @ R @ S ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(breln1unionIL_type,type,
% 0.16/0.42 breln1unionIL: $o ).
% 0.16/0.42
% 0.16/0.42 thf(breln1unionIL,definition,
% 0.16/0.42 ( breln1unionIL
% 0.16/0.42 = ( ! [A: $i,R: $i] :
% 0.16/0.42 ( ( breln1 @ A @ R )
% 0.16/0.42 => ! [S: $i] :
% 0.16/0.42 ( ( breln1 @ A @ S )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ A )
% 0.16/0.42 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.16/0.42 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(breln1unionIR_type,type,
% 0.16/0.42 breln1unionIR: $o ).
% 0.16/0.42
% 0.16/0.42 thf(breln1unionIR,definition,
% 0.16/0.42 ( breln1unionIR
% 0.16/0.42 = ( ! [A: $i,R: $i] :
% 0.16/0.42 ( ( breln1 @ A @ R )
% 0.16/0.42 => ! [S: $i] :
% 0.16/0.42 ( ( breln1 @ A @ S )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ A )
% 0.16/0.42 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.16/0.42 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(breln1unionI_type,type,
% 0.16/0.42 breln1unionI: $o ).
% 0.16/0.42
% 0.16/0.42 thf(breln1unionI,definition,
% 0.16/0.42 ( breln1unionI
% 0.16/0.42 = ( ! [A: $i,R: $i] :
% 0.16/0.42 ( ( breln1 @ A @ R )
% 0.16/0.42 => ! [S: $i] :
% 0.16/0.42 ( ( breln1 @ A @ S )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ A )
% 0.16/0.42 => ( ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.16/0.42 | ( in @ ( kpair @ Xx @ Xy ) @ S ) )
% 0.16/0.42 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(breln1unionE_type,type,
% 0.16/0.42 breln1unionE: $o ).
% 0.16/0.42
% 0.16/0.42 thf(breln1unionE,definition,
% 0.16/0.42 ( breln1unionE
% 0.16/0.42 = ( ! [A: $i,R: $i] :
% 0.16/0.42 ( ( breln1 @ A @ R )
% 0.16/0.42 => ! [S: $i] :
% 0.16/0.42 ( ( breln1 @ A @ S )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ A )
% 0.16/0.42 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) )
% 0.16/0.42 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.16/0.42 | ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(breln1unionEcases_type,type,
% 0.16/0.42 breln1unionEcases: $o ).
% 0.16/0.42
% 0.16/0.42 thf(breln1unionEcases,definition,
% 0.16/0.42 ( breln1unionEcases
% 0.16/0.42 = ( ! [A: $i,R: $i] :
% 0.16/0.42 ( ( breln1 @ A @ R )
% 0.16/0.42 => ! [S: $i] :
% 0.16/0.42 ( ( breln1 @ A @ S )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ A )
% 0.16/0.42 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) )
% 0.16/0.42 => ! [Xphi: $o] :
% 0.16/0.42 ( ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.16/0.42 => Xphi )
% 0.16/0.42 => ( ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.16/0.42 => Xphi )
% 0.16/0.42 => Xphi ) ) ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(breln1unionCommutes_type,type,
% 0.16/0.42 breln1unionCommutes: $o ).
% 0.16/0.42
% 0.16/0.42 thf(breln1unionCommutes,definition,
% 0.16/0.42 ( breln1unionCommutes
% 0.16/0.42 = ( ! [A: $i,R: $i] :
% 0.16/0.42 ( ( breln1 @ A @ R )
% 0.16/0.42 => ! [S: $i] :
% 0.16/0.42 ( ( breln1 @ A @ S )
% 0.16/0.42 => ( ( binunion @ R @ S )
% 0.16/0.42 = ( binunion @ S @ R ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(woz2Ex_type,type,
% 0.16/0.42 woz2Ex: $o ).
% 0.16/0.42
% 0.16/0.42 thf(woz2Ex,definition,
% 0.16/0.42 ( woz2Ex
% 0.16/0.42 = ( ! [A: $i,R: $i] :
% 0.16/0.42 ( ( breln1 @ A @ R )
% 0.16/0.42 => ( R
% 0.16/0.42 = ( breln1invset @ A @ ( breln1invset @ A @ R ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(woz2W_type,type,
% 0.16/0.42 woz2W: $o ).
% 0.16/0.42
% 0.16/0.42 thf(woz2W,definition,
% 0.16/0.42 ( woz2W
% 0.16/0.42 = ( ! [A: $i,R: $i] :
% 0.16/0.42 ( ( breln1 @ A @ R )
% 0.16/0.42 => ! [S: $i] :
% 0.16/0.42 ( ( breln1 @ A @ S )
% 0.16/0.42 => ( ( breln1invset @ A @ ( breln1compset @ A @ R @ S ) )
% 0.16/0.42 = ( breln1compset @ A @ ( breln1invset @ A @ S ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(woz2A_type,type,
% 0.16/0.42 woz2A: $o ).
% 0.16/0.42
% 0.16/0.42 thf(woz2A,definition,
% 0.16/0.42 ( woz2A
% 0.16/0.42 = ( ! [A: $i,R: $i] :
% 0.16/0.42 ( ( breln1 @ A @ R )
% 0.16/0.42 => ! [S: $i] :
% 0.16/0.42 ( ( breln1 @ A @ S )
% 0.16/0.42 => ! [T: $i] :
% 0.16/0.42 ( ( breln1 @ A @ T )
% 0.16/0.42 => ( ( breln1compset @ A @ ( binunion @ R @ S ) @ T )
% 0.16/0.42 = ( binunion @ ( breln1compset @ A @ R @ T ) @ ( breln1compset @ A @ S @ T ) ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(woz2B_type,type,
% 0.16/0.42 woz2B: $o ).
% 0.16/0.42
% 0.16/0.42 thf(woz2B,definition,
% 0.16/0.42 ( woz2B
% 0.16/0.42 = ( ! [A: $i,R: $i] :
% 0.16/0.42 ( ( breln1 @ A @ R )
% 0.16/0.42 => ! [S: $i] :
% 0.16/0.42 ( ( breln1 @ A @ S )
% 0.16/0.42 => ! [T: $i] :
% 0.16/0.42 ( ( breln1 @ A @ T )
% 0.16/0.42 => ( ( breln1compset @ A @ ( binunion @ R @ S ) @ T )
% 0.16/0.42 = ( binunion @ ( breln1invset @ A @ ( breln1compset @ A @ ( breln1invset @ A @ T ) @ ( breln1invset @ A @ S ) ) ) @ ( breln1invset @ A @ ( breln1compset @ A @ ( breln1invset @ A @ T ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(image1Ex_type,type,
% 0.16/0.42 image1Ex: $o ).
% 0.16/0.42
% 0.16/0.42 thf(image1Ex,definition,
% 0.16/0.42 ( image1Ex
% 0.16/0.42 = ( ! [A: $i,Xf: $i > $i] :
% 0.16/0.42 ? [B: $i] :
% 0.16/0.42 ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ B )
% 0.16/0.42 <=> ? [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ A )
% 0.16/0.42 & ( Xx
% 0.16/0.42 = ( Xf @ Xy ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(image1Ex1_type,type,
% 0.16/0.42 image1Ex1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(image1Ex1,definition,
% 0.16/0.42 ( image1Ex1
% 0.16/0.42 = ( ! [A: $i,Xf: $i > $i] :
% 0.16/0.42 ( exu
% 0.16/0.42 @ ^ [B: $i] :
% 0.16/0.42 ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ B )
% 0.16/0.42 <=> ? [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ A )
% 0.16/0.42 & ( Xx
% 0.16/0.42 = ( Xf @ Xy ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(image1_type,type,
% 0.16/0.42 image1: $i > ( $i > $i ) > $i ).
% 0.16/0.42
% 0.16/0.42 thf(image1Equiv_type,type,
% 0.16/0.42 image1Equiv: $o ).
% 0.16/0.42
% 0.16/0.42 thf(image1Equiv,definition,
% 0.16/0.42 ( image1Equiv
% 0.16/0.42 = ( ! [A: $i,Xf: $i > $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx
% 0.16/0.42 @ ( image1 @ A
% 0.16/0.42 @ ^ [Xy: $i] : ( Xf @ Xy ) ) )
% 0.16/0.42 <=> ? [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ A )
% 0.16/0.42 & ( Xx
% 0.16/0.42 = ( Xf @ Xy ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(image1E_type,type,
% 0.16/0.42 image1E: $o ).
% 0.16/0.42
% 0.16/0.42 thf(image1E,definition,
% 0.16/0.42 ( image1E
% 0.16/0.42 = ( ! [A: $i,Xf: $i > $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx
% 0.16/0.42 @ ( image1 @ A
% 0.16/0.42 @ ^ [Xy: $i] : ( Xf @ Xy ) ) )
% 0.16/0.42 => ? [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ A )
% 0.16/0.42 & ( Xx
% 0.16/0.42 = ( Xf @ Xy ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(image1I_type,type,
% 0.16/0.42 image1I: $o ).
% 0.16/0.42
% 0.16/0.42 thf(image1I,definition,
% 0.16/0.42 ( image1I
% 0.16/0.42 = ( ! [A: $i,Xf: $i > $i,Xx: $i] :
% 0.16/0.42 ( ? [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ A )
% 0.16/0.42 & ( Xx
% 0.16/0.42 = ( Xf @ Xy ) ) )
% 0.16/0.42 => ( in @ Xx
% 0.16/0.42 @ ( image1 @ A
% 0.16/0.42 @ ^ [Xy: $i] : ( Xf @ Xy ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(injective_type,type,
% 0.16/0.42 injective: $i > $i > $i > $o ).
% 0.16/0.42
% 0.16/0.42 thf(injFuncSet_type,type,
% 0.16/0.42 injFuncSet: $i > $i > $i ).
% 0.16/0.42
% 0.16/0.42 thf(injFuncInInjFuncSet_type,type,
% 0.16/0.43 injFuncInInjFuncSet: $o ).
% 0.16/0.43
% 0.16/0.43 thf(injFuncInInjFuncSet,definition,
% 0.16/0.43 ( injFuncInInjFuncSet
% 0.16/0.43 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.43 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.16/0.43 => ( ( injective @ A @ B @ Xf )
% 0.16/0.43 => ( in @ Xf @ ( injFuncSet @ A @ B ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(injFuncSetFuncIn_type,type,
% 0.16/0.43 injFuncSetFuncIn: $o ).
% 0.16/0.43
% 0.16/0.43 thf(injFuncSetFuncIn,definition,
% 0.16/0.43 ( injFuncSetFuncIn
% 0.16/0.43 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.43 ( ( in @ Xf @ ( injFuncSet @ A @ B ) )
% 0.16/0.43 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(injFuncSetFuncInj_type,type,
% 0.16/0.43 injFuncSetFuncInj: $o ).
% 0.16/0.43
% 0.16/0.43 thf(injFuncSetFuncInj,definition,
% 0.16/0.43 ( injFuncSetFuncInj
% 0.16/0.43 = ( ! [Xx: $i,Xy: $i,Xf: $i] :
% 0.16/0.43 ( ( in @ Xf @ ( injFuncSet @ Xx @ Xy ) )
% 0.16/0.43 => ( injective @ Xx @ Xy @ Xf ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(surjective_type,type,
% 0.16/0.43 surjective: $i > $i > $i > $o ).
% 0.16/0.43
% 0.16/0.43 thf(surjFuncSet_type,type,
% 0.16/0.43 surjFuncSet: $i > $i > $i ).
% 0.16/0.43
% 0.16/0.43 thf(surjFuncSetFuncIn_type,type,
% 0.16/0.43 surjFuncSetFuncIn: $o ).
% 0.16/0.43
% 0.16/0.43 thf(surjFuncSetFuncIn,definition,
% 0.16/0.43 ( surjFuncSetFuncIn
% 0.16/0.43 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.43 ( ( in @ Xf @ ( surjFuncSet @ A @ B ) )
% 0.16/0.43 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(surjFuncSetFuncSurj_type,type,
% 0.16/0.43 surjFuncSetFuncSurj: $o ).
% 0.16/0.43
% 0.16/0.43 thf(surjFuncSetFuncSurj,definition,
% 0.16/0.43 ( surjFuncSetFuncSurj
% 0.16/0.43 = ( ! [Xx: $i,Xy: $i,Xf: $i] :
% 0.16/0.43 ( ( in @ Xf @ ( surjFuncSet @ Xx @ Xy ) )
% 0.16/0.43 => ( surjective @ Xx @ Xy @ Xf ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(leftInvIsSurj_type,type,
% 0.16/0.43 leftInvIsSurj: $o ).
% 0.16/0.43
% 0.16/0.43 thf(leftInvIsSurj,definition,
% 0.16/0.43 ( leftInvIsSurj
% 0.16/0.43 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.16/0.43 ( ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.16/0.43 => ! [Xg: $i] :
% 0.16/0.43 ( ( in @ Xg @ ( funcSet @ B @ A ) )
% 0.16/0.43 => ( ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ( ( ap @ B @ A @ Xg @ ( Xf @ Xx ) )
% 0.16/0.43 = Xx ) )
% 0.16/0.43 => ( surjective @ B @ A @ Xg ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(surjCantorThm_type,type,
% 0.16/0.43 surjCantorThm: $o ).
% 0.16/0.43
% 0.16/0.43 thf(surjCantorThm,definition,
% 0.16/0.43 ( surjCantorThm
% 0.16/0.43 = ( ! [A: $i,Xf: $i] :
% 0.16/0.43 ( ( in @ Xf @ ( funcSet @ A @ ( powerset @ A ) ) )
% 0.16/0.43 => ~ ( surjective @ A @ ( powerset @ A ) @ Xf ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(foundation2_type,type,
% 0.16/0.43 foundation2: $o ).
% 0.16/0.43
% 0.16/0.43 thf(foundation2,definition,
% 0.16/0.43 ( foundation2
% 0.16/0.43 = ( ! [A: $i] :
% 0.16/0.43 ( ( nonempty @ A )
% 0.16/0.43 => ? [X: $i] :
% 0.16/0.43 ( ( in @ X @ A )
% 0.16/0.43 & ( ( binintersect @ X @ A )
% 0.16/0.43 = emptyset ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(notinself_type,type,
% 0.16/0.43 notinself: $o ).
% 0.16/0.43
% 0.16/0.43 thf(notinself,definition,
% 0.16/0.43 ( notinself
% 0.16/0.43 = ( ! [A: $i] :
% 0.16/0.43 ~ ( in @ A @ A ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(notinself2_type,type,
% 0.16/0.43 notinself2: $o ).
% 0.16/0.43
% 0.16/0.43 thf(notinself2,definition,
% 0.16/0.43 ( notinself2
% 0.16/0.43 = ( ! [A: $i,B: $i] :
% 0.16/0.43 ( ( in @ A @ B )
% 0.16/0.43 => ~ ( in @ B @ A ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(omegaS_type,type,
% 0.16/0.43 omegaS: $i > $i ).
% 0.16/0.43
% 0.16/0.43 thf(omegaSp_type,type,
% 0.16/0.43 omegaSp: $o ).
% 0.16/0.43
% 0.16/0.43 thf(omegaSp,definition,
% 0.16/0.43 ( omegaSp
% 0.16/0.43 = ( ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ omega )
% 0.16/0.43 => ( in @ ( omegaS @ Xx ) @ omega ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(omegaSclos_type,type,
% 0.16/0.43 omegaSclos: $o ).
% 0.16/0.43
% 0.16/0.43 thf(omegaSclos,definition,
% 0.16/0.43 ( omegaSclos
% 0.16/0.43 = ( ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ omega )
% 0.16/0.43 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(peano0notS_type,type,
% 0.16/0.43 peano0notS: $o ).
% 0.16/0.43
% 0.16/0.43 thf(peano0notS,definition,
% 0.16/0.43 ( peano0notS
% 0.16/0.43 = ( ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ omega )
% 0.16/0.43 => ( ( omegaS @ Xx )
% 0.16/0.43 != emptyset ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(peano3_type,type,
% 0.16/0.43 peano3: $i > $i > $i > $o ).
% 0.16/0.43
% 0.16/0.43 thf(peano4_type,type,
% 0.16/0.43 peano4: $i > $i > $i > $o ).
% 0.16/0.43
% 0.16/0.43 thf(peano5_type,type,
% 0.16/0.43 peano5: $i > $i > $i > $o ).
% 0.16/0.43
% 0.16/0.43 thf(peanoSinj_type,type,
% 0.16/0.43 peanoSinj: $o ).
% 0.16/0.43
% 0.16/0.43 thf(peanoSinj,definition,
% 0.16/0.43 ( peanoSinj
% 0.16/0.43 = ( ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ omega )
% 0.16/0.43 => ! [Xy: $i] :
% 0.16/0.43 ( ( in @ Xy @ omega )
% 0.16/0.43 => ( ( ( omegaS @ Xx )
% 0.16/0.43 = ( omegaS @ Xy ) )
% 0.16/0.43 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(peano_type,type,
% 0.16/0.43 peano: $i > $i > $i > $o ).
% 0.16/0.43
% 0.16/0.43 thf(transitiveset_type,type,
% 0.16/0.43 transitiveset: $i > $o ).
% 0.16/0.43
% 0.16/0.43 thf(transitiveset,definition,
% 0.16/0.43 ( transitiveset
% 0.16/0.43 = ( ^ [A: $i] :
% 0.16/0.43 ! [X: $i] :
% 0.16/0.43 ( ( in @ X @ A )
% 0.16/0.43 => ( subset @ X @ A ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(transitivesetOp1_type,type,
% 0.16/0.43 transitivesetOp1: $o ).
% 0.16/0.43
% 0.16/0.43 thf(transitivesetOp1,definition,
% 0.16/0.43 ( transitivesetOp1
% 0.16/0.43 = ( ! [X: $i] :
% 0.16/0.43 ( ( transitiveset @ X )
% 0.16/0.43 => ! [A: $i] :
% 0.16/0.43 ( ( in @ A @ X )
% 0.16/0.43 => ( subset @ A @ X ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(binintTransitive,conjecture,
% 0.16/0.43 ( setextAx
% 0.16/0.43 => ( emptysetAx
% 0.16/0.43 => ( setadjoinAx
% 0.16/0.43 => ( powersetAx
% 0.16/0.43 => ( setunionAx
% 0.16/0.43 => ( omega0Ax
% 0.16/0.43 => ( omegaSAx
% 0.16/0.43 => ( omegaIndAx
% 0.16/0.43 => ( replAx
% 0.16/0.43 => ( foundationAx
% 0.16/0.43 => ( wellorderingAx
% 0.16/0.43 => ( descrp
% 0.16/0.43 => ( dsetconstrI
% 0.16/0.43 => ( dsetconstrEL
% 0.16/0.43 => ( dsetconstrER
% 0.16/0.43 => ( exuE1
% 0.16/0.43 => ( prop2setE
% 0.16/0.43 => ( emptysetE
% 0.16/0.43 => ( emptysetimpfalse
% 0.16/0.43 => ( notinemptyset
% 0.16/0.43 => ( exuE3e
% 0.16/0.43 => ( setext
% 0.16/0.43 => ( emptyI
% 0.16/0.43 => ( noeltsimpempty
% 0.16/0.43 => ( setbeta
% 0.16/0.43 => ( nonemptyE1
% 0.16/0.43 => ( nonemptyI
% 0.16/0.43 => ( nonemptyI1
% 0.16/0.43 => ( setadjoinIL
% 0.16/0.43 => ( emptyinunitempty
% 0.16/0.43 => ( setadjoinIR
% 0.16/0.43 => ( setadjoinE
% 0.16/0.43 => ( setadjoinOr
% 0.16/0.43 => ( setoftrueEq
% 0.16/0.43 => ( powersetI
% 0.16/0.43 => ( emptyinPowerset
% 0.16/0.43 => ( emptyInPowerset
% 0.16/0.43 => ( powersetE
% 0.16/0.43 => ( setunionI
% 0.16/0.43 => ( setunionE
% 0.16/0.43 => ( subPowSU
% 0.16/0.43 => ( exuE2
% 0.16/0.43 => ( nonemptyImpWitness
% 0.16/0.43 => ( uniqinunit
% 0.16/0.43 => ( notinsingleton
% 0.16/0.43 => ( eqinunit
% 0.16/0.43 => ( singletonsswitch
% 0.16/0.43 => ( upairsetE
% 0.16/0.43 => ( upairsetIL
% 0.16/0.43 => ( upairsetIR
% 0.16/0.43 => ( emptyE1
% 0.16/0.43 => ( vacuousDall
% 0.16/0.43 => ( quantDeMorgan1
% 0.16/0.43 => ( quantDeMorgan2
% 0.16/0.43 => ( quantDeMorgan3
% 0.16/0.43 => ( quantDeMorgan4
% 0.16/0.43 => ( prop2setI
% 0.16/0.43 => ( prop2set2propI
% 0.16/0.43 => ( notdexE
% 0.16/0.43 => ( notdallE
% 0.16/0.43 => ( exuI1
% 0.16/0.43 => ( exuI3
% 0.16/0.43 => ( exuI2
% 0.16/0.43 => ( inCongP
% 0.16/0.43 => ( in__Cong
% 0.16/0.43 => ( exuE3u
% 0.16/0.43 => ( exu__Cong
% 0.16/0.43 => ( emptyset__Cong
% 0.16/0.43 => ( setadjoin__Cong
% 0.16/0.43 => ( powerset__Cong
% 0.16/0.43 => ( setunion__Cong
% 0.16/0.43 => ( omega__Cong
% 0.16/0.43 => ( exuEu
% 0.16/0.43 => ( descr__Cong
% 0.16/0.43 => ( dsetconstr__Cong
% 0.16/0.43 => ( subsetI1
% 0.16/0.43 => ( eqimpsubset2
% 0.16/0.43 => ( eqimpsubset1
% 0.16/0.43 => ( subsetI2
% 0.16/0.43 => ( emptysetsubset
% 0.16/0.43 => ( subsetE
% 0.16/0.43 => ( subsetE2
% 0.16/0.43 => ( notsubsetI
% 0.16/0.43 => ( notequalI1
% 0.16/0.43 => ( notequalI2
% 0.16/0.43 => ( subsetRefl
% 0.16/0.43 => ( subsetTrans
% 0.16/0.43 => ( setadjoinSub
% 0.16/0.43 => ( setadjoinSub2
% 0.16/0.43 => ( subset2powerset
% 0.16/0.43 => ( setextsub
% 0.16/0.43 => ( subsetemptysetimpeq
% 0.16/0.43 => ( powersetI1
% 0.16/0.43 => ( powersetE1
% 0.16/0.43 => ( inPowerset
% 0.16/0.43 => ( powersetsubset
% 0.16/0.43 => ( sepInPowerset
% 0.16/0.43 => ( sepSubset
% 0.16/0.43 => ( binunionIL
% 0.16/0.43 => ( upairset2IR
% 0.16/0.43 => ( binunionIR
% 0.16/0.43 => ( binunionEcases
% 0.16/0.43 => ( binunionE
% 0.16/0.43 => ( binunionLsub
% 0.16/0.43 => ( binunionRsub
% 0.16/0.43 => ( binintersectI
% 0.16/0.43 => ( binintersectSubset5
% 0.16/0.43 => ( binintersectEL
% 0.16/0.43 => ( binintersectLsub
% 0.16/0.43 => ( binintersectSubset2
% 0.16/0.43 => ( binintersectSubset3
% 0.16/0.43 => ( binintersectER
% 0.16/0.43 => ( disjointsetsI1
% 0.16/0.43 => ( binintersectRsub
% 0.16/0.43 => ( binintersectSubset4
% 0.16/0.43 => ( binintersectSubset1
% 0.16/0.43 => ( bs114d
% 0.16/0.43 => ( setminusI
% 0.16/0.43 => ( setminusEL
% 0.16/0.43 => ( setminusER
% 0.16/0.43 => ( setminusSubset2
% 0.16/0.43 => ( setminusERneg
% 0.16/0.43 => ( setminusELneg
% 0.16/0.43 => ( setminusILneg
% 0.16/0.43 => ( setminusIRneg
% 0.16/0.43 => ( setminusLsub
% 0.16/0.43 => ( setminusSubset1
% 0.16/0.43 => ( symdiffE
% 0.16/0.43 => ( symdiffI1
% 0.16/0.43 => ( symdiffI2
% 0.16/0.43 => ( symdiffIneg1
% 0.16/0.43 => ( symdiffIneg2
% 0.16/0.43 => ( secondinupair
% 0.16/0.43 => ( setukpairIL
% 0.16/0.43 => ( setukpairIR
% 0.16/0.43 => ( kpairiskpair
% 0.16/0.43 => ( kpairp
% 0.16/0.43 => ( singletonsubset
% 0.16/0.43 => ( singletoninpowerset
% 0.16/0.43 => ( singletoninpowunion
% 0.16/0.43 => ( upairset2E
% 0.16/0.43 => ( upairsubunion
% 0.16/0.43 => ( upairinpowunion
% 0.16/0.43 => ( ubforcartprodlem1
% 0.16/0.43 => ( ubforcartprodlem2
% 0.16/0.43 => ( ubforcartprodlem3
% 0.16/0.43 => ( cartprodpairin
% 0.16/0.43 => ( cartprodmempair1
% 0.16/0.43 => ( cartprodmempair
% 0.16/0.43 => ( setunionE2
% 0.16/0.43 => ( setunionsingleton1
% 0.16/0.43 => ( setunionsingleton2
% 0.16/0.43 => ( setunionsingleton
% 0.16/0.43 => ( singletonprop
% 0.16/0.43 => ( ex1E1
% 0.16/0.43 => ( ex1I
% 0.16/0.43 => ( ex1I2
% 0.16/0.43 => ( singletonsuniq
% 0.16/0.43 => ( setukpairinjL1
% 0.16/0.43 => ( kfstsingleton
% 0.16/0.43 => ( theprop
% 0.16/0.43 => ( kfstpairEq
% 0.16/0.43 => ( cartprodfstin
% 0.16/0.43 => ( setukpairinjL2
% 0.16/0.43 => ( setukpairinjL
% 0.16/0.43 => ( setukpairinjR11
% 0.16/0.43 => ( setukpairinjR12
% 0.16/0.43 => ( setukpairinjR1
% 0.16/0.43 => ( upairequniteq
% 0.16/0.43 => ( setukpairinjR2
% 0.16/0.43 => ( setukpairinjR
% 0.16/0.43 => ( ksndsingleton
% 0.16/0.43 => ( ksndpairEq
% 0.16/0.43 => ( kpairsurjEq
% 0.16/0.43 => ( cartprodsndin
% 0.16/0.43 => ( cartprodpairmemEL
% 0.16/0.43 => ( cartprodpairmemER
% 0.16/0.43 => ( cartprodmempaircEq
% 0.16/0.43 => ( cartprodfstpairEq
% 0.16/0.43 => ( cartprodsndpairEq
% 0.16/0.43 => ( cartprodpairsurjEq
% 0.16/0.43 => ( dpsetconstrI
% 0.16/0.43 => ( dpsetconstrSub
% 0.16/0.43 => ( setOfPairsIsBReln
% 0.16/0.43 => ( dpsetconstrERa
% 0.16/0.43 => ( dpsetconstrEL1
% 0.16/0.43 => ( dpsetconstrEL2
% 0.16/0.43 => ( dpsetconstrER
% 0.16/0.43 => ( funcImageSingleton
% 0.16/0.43 => ( apProp
% 0.16/0.43 => ( app
% 0.16/0.43 => ( infuncsetfunc
% 0.16/0.43 => ( ap2p
% 0.16/0.43 => ( funcinfuncset
% 0.16/0.43 => ( lamProp
% 0.16/0.43 => ( lamp
% 0.16/0.43 => ( lam2p
% 0.16/0.43 => ( brelnall1
% 0.16/0.43 => ( brelnall2
% 0.16/0.43 => ( ex1E2
% 0.16/0.43 => ( funcGraphProp1
% 0.16/0.43 => ( funcGraphProp3
% 0.16/0.43 => ( funcGraphProp2
% 0.16/0.43 => ( funcextLem
% 0.16/0.43 => ( funcGraphProp4
% 0.16/0.43 => ( subbreln
% 0.16/0.43 => ( eqbreln
% 0.16/0.43 => ( funcext
% 0.16/0.43 => ( funcext2
% 0.16/0.43 => ( ap2apEq1
% 0.16/0.43 => ( ap2apEq2
% 0.16/0.43 => ( beta1
% 0.16/0.43 => ( eta1
% 0.16/0.43 => ( lam2lamEq
% 0.16/0.43 => ( beta2
% 0.16/0.43 => ( eta2
% 0.16/0.43 => ( iffalseProp1
% 0.16/0.43 => ( iffalseProp2
% 0.16/0.43 => ( iftrueProp1
% 0.16/0.43 => ( iftrueProp2
% 0.16/0.43 => ( ifSingleton
% 0.16/0.43 => ( ifp
% 0.16/0.43 => ( theeq
% 0.16/0.43 => ( iftrue
% 0.16/0.43 => ( iffalse
% 0.16/0.43 => ( iftrueorfalse
% 0.16/0.43 => ( binintersectT_lem
% 0.16/0.43 => ( binunionT_lem
% 0.16/0.43 => ( powersetT_lem
% 0.16/0.43 => ( setminusT_lem
% 0.16/0.43 => ( complementT_lem
% 0.16/0.43 => ( setextT
% 0.16/0.43 => ( subsetTI
% 0.16/0.43 => ( powersetTI1
% 0.16/0.43 => ( powersetTE1
% 0.16/0.43 => ( complementTI1
% 0.16/0.43 => ( complementTE1
% 0.16/0.43 => ( binintersectTELcontra
% 0.16/0.43 => ( binintersectTERcontra
% 0.16/0.43 => ( contrasubsetT
% 0.16/0.43 => ( contrasubsetT1
% 0.16/0.43 => ( contrasubsetT2
% 0.16/0.43 => ( contrasubsetT3
% 0.16/0.43 => ( doubleComplementI1
% 0.16/0.43 => ( doubleComplementE1
% 0.16/0.43 => ( doubleComplementSub1
% 0.16/0.43 => ( doubleComplementSub2
% 0.16/0.43 => ( doubleComplementEq
% 0.16/0.43 => ( complementTnotintersectT
% 0.16/0.43 => ( complementImpComplementIntersect
% 0.16/0.43 => ( complementSubsetComplementIntersect
% 0.16/0.43 => ( complementInPowersetComplementIntersect
% 0.16/0.43 => ( contraSubsetComplement
% 0.16/0.43 => ( complementTcontraSubset
% 0.16/0.43 => ( binunionTILcontra
% 0.16/0.43 => ( binunionTIRcontra
% 0.16/0.43 => ( inIntersectImpInUnion
% 0.16/0.43 => ( inIntersectImpInUnion2
% 0.16/0.43 => ( inIntersectImpInIntersectUnions
% 0.16/0.43 => ( intersectInPowersetIntersectUnions
% 0.16/0.43 => ( inComplementUnionImpNotIn1
% 0.16/0.43 => ( inComplementUnionImpInComplement1
% 0.16/0.43 => ( binunionTE
% 0.16/0.43 => ( binunionTEcontra
% 0.16/0.43 => ( demorgan2a1
% 0.16/0.43 => ( complementUnionInPowersetComplement
% 0.16/0.43 => ( demorgan2a2
% 0.16/0.43 => ( demorgan1a
% 0.16/0.43 => ( demorgan1b
% 0.16/0.43 => ( demorgan1
% 0.16/0.43 => ( demorgan2a
% 0.16/0.43 => ( demorgan2b2
% 0.16/0.43 => ( demorgan2b
% 0.16/0.43 => ( demorgan2
% 0.16/0.43 => ( woz13rule0
% 0.16/0.43 => ( woz13rule1
% 0.16/0.43 => ( woz13rule2
% 0.16/0.43 => ( woz13rule3
% 0.16/0.43 => ( woz13rule4
% 0.16/0.43 => ( woz1_1
% 0.16/0.43 => ( woz1_2
% 0.16/0.43 => ( woz1_3
% 0.16/0.43 => ( woz1_4
% 0.16/0.43 => ( woz1_5
% 0.16/0.43 => ( breln1all2
% 0.16/0.43 => ( breln1SetBreln1
% 0.16/0.43 => ( choice2fnsingleton
% 0.16/0.43 => ( setOfPairsIsBReln1
% 0.16/0.43 => ( breln1all1
% 0.16/0.43 => ( subbreln1
% 0.16/0.43 => ( eqbreln1
% 0.16/0.43 => ( breln1invprop
% 0.16/0.43 => ( breln1invI
% 0.16/0.43 => ( breln1invE
% 0.16/0.43 => ( breln1compprop
% 0.16/0.43 => ( breln1compI
% 0.16/0.43 => ( breln1compE
% 0.16/0.43 => ( breln1compEex
% 0.16/0.43 => ( breln1unionprop
% 0.16/0.43 => ( breln1unionIL
% 0.16/0.43 => ( breln1unionIR
% 0.16/0.43 => ( breln1unionI
% 0.16/0.43 => ( breln1unionE
% 0.16/0.43 => ( breln1unionEcases
% 0.16/0.43 => ( breln1unionCommutes
% 0.16/0.43 => ( woz2Ex
% 0.16/0.43 => ( woz2W
% 0.16/0.43 => ( woz2A
% 0.16/0.43 => ( woz2B
% 0.16/0.43 => ( image1Ex
% 0.16/0.43 => ( image1Ex1
% 0.16/0.43 => ( image1Equiv
% 0.16/0.43 => ( image1E
% 0.16/0.43 => ( image1I
% 0.16/0.43 => ( injFuncInInjFuncSet
% 0.16/0.43 => ( injFuncSetFuncIn
% 0.16/0.43 => ( injFuncSetFuncInj
% 0.16/0.43 => ( surjFuncSetFuncIn
% 0.16/0.43 => ( surjFuncSetFuncSurj
% 0.16/0.43 => ( leftInvIsSurj
% 0.16/0.43 => ( surjCantorThm
% 0.16/0.43 => ( foundation2
% 0.16/0.43 => ( notinself
% 0.16/0.43 => ( notinself2
% 0.16/0.43 => ( omegaSp
% 0.16/0.43 => ( omegaSclos
% 0.16/0.43 => ( peano0notS
% 0.16/0.43 => ( peanoSinj
% 0.16/0.43 => ( transitivesetOp1
% 0.16/0.43 => ! [X: $i] :
% 0.16/0.43 ( ( transitiveset @ X )
% 0.16/0.43 => ! [Y: $i] :
% 0.16/0.43 ( ( transitiveset @ Y )
% 0.16/0.43 => ( transitiveset @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 %------------------------------------------------------------------------------
% 0.16/0.43 ------------------------
% 2.85/1.46 ---- Embedded in THF ---
% 2.85/1.46 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.85/1.46 %%% Generated on Tue Feb 24 12:34:55 EST 2026
% 2.85/1.46 %%% using '$$dhol' embedding, version 1.3.0.
% 2.85/1.46 %%% Logic specification used:
% 2.85/1.46 %%% thf(spec, logic, $$dhol).
% 2.85/1.46
% 2.85/1.46 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.H1wBmJGvZQ/DTF2DTF2486.p
% See solution above
% 2.85/1.46 ------------------------
% 3.05/1.50 ---- Cleaned THF ---
% 3.05/1.50 thf(in_type,type,
% 3.05/1.50 in: $i > $i > $o ).
% 3.05/1.50
% 3.05/1.50 thf(exu_type,type,
% 3.05/1.50 exu: ( $i > $o ) > $o ).
% 3.05/1.50
% 3.05/1.50 thf(exu,definition,
% 3.05/1.50 ( exu
% 3.05/1.50 = ( ^ [Xphi: $i > $o] :
% 3.05/1.50 ? [Xx: $i] :
% 3.05/1.50 ( ( Xphi @ Xx )
% 3.05/1.50 & ! [Xy: $i] :
% 3.05/1.50 ( ( Xphi @ Xy )
% 3.05/1.50 => ( Xx = Xy ) ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(setextAx_type,type,
% 3.05/1.50 setextAx: $o ).
% 3.05/1.50
% 3.05/1.50 thf(setextAx,definition,
% 3.05/1.50 ( setextAx
% 3.05/1.50 = ( ! [A: $i,B: $i] :
% 3.05/1.50 ( ! [Xx: $i] :
% 3.05/1.50 ( ( in @ Xx @ A )
% 3.05/1.50 <=> ( in @ Xx @ B ) )
% 3.05/1.50 => ( A = B ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(emptyset_type,type,
% 3.05/1.50 emptyset: $i ).
% 3.05/1.50
% 3.05/1.50 thf(emptysetAx_type,type,
% 3.05/1.50 emptysetAx: $o ).
% 3.05/1.50
% 3.05/1.50 thf(emptysetAx,definition,
% 3.05/1.50 ( emptysetAx
% 3.05/1.50 = ( ! [Xx: $i] :
% 3.05/1.50 ~ ( in @ Xx @ emptyset ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(setadjoin_type,type,
% 3.05/1.50 setadjoin: $i > $i > $i ).
% 3.05/1.50
% 3.05/1.50 thf(setadjoinAx_type,type,
% 3.05/1.50 setadjoinAx: $o ).
% 3.05/1.50
% 3.05/1.50 thf(setadjoinAx,definition,
% 3.05/1.50 ( setadjoinAx
% 3.05/1.50 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 3.05/1.50 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 3.05/1.50 <=> ( ( Xy = Xx )
% 3.05/1.50 | ( in @ Xy @ A ) ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(powerset_type,type,
% 3.05/1.50 powerset: $i > $i ).
% 3.05/1.50
% 3.05/1.50 thf(powersetAx_type,type,
% 3.05/1.50 powersetAx: $o ).
% 3.05/1.50
% 3.05/1.50 thf(powersetAx,definition,
% 3.05/1.50 ( powersetAx
% 3.05/1.50 = ( ! [A: $i,B: $i] :
% 3.05/1.50 ( ( in @ B @ ( powerset @ A ) )
% 3.05/1.50 <=> ! [Xx: $i] :
% 3.05/1.50 ( ( in @ Xx @ B )
% 3.05/1.50 => ( in @ Xx @ A ) ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(setunion_type,type,
% 3.05/1.50 setunion: $i > $i ).
% 3.05/1.50
% 3.05/1.50 thf(setunionAx_type,type,
% 3.05/1.50 setunionAx: $o ).
% 3.05/1.50
% 3.05/1.50 thf(setunionAx,definition,
% 3.05/1.50 ( setunionAx
% 3.05/1.50 = ( ! [A: $i,Xx: $i] :
% 3.05/1.50 ( ( in @ Xx @ ( setunion @ A ) )
% 3.05/1.50 <=> ? [B: $i] :
% 3.05/1.50 ( ( in @ Xx @ B )
% 3.05/1.50 & ( in @ B @ A ) ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(omega_type,type,
% 3.05/1.50 omega: $i ).
% 3.05/1.50
% 3.05/1.50 thf(omega0Ax_type,type,
% 3.05/1.50 omega0Ax: $o ).
% 3.05/1.50
% 3.05/1.50 thf(omega0Ax,definition,
% 3.05/1.50 ( omega0Ax
% 3.05/1.50 = ( in @ emptyset @ omega ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(omegaSAx_type,type,
% 3.05/1.50 omegaSAx: $o ).
% 3.05/1.50
% 3.05/1.50 thf(omegaSAx,definition,
% 3.05/1.50 ( omegaSAx
% 3.05/1.50 = ( ! [Xx: $i] :
% 3.05/1.50 ( ( in @ Xx @ omega )
% 3.05/1.50 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(omegaIndAx_type,type,
% 3.05/1.50 omegaIndAx: $o ).
% 3.05/1.50
% 3.05/1.50 thf(omegaIndAx,definition,
% 3.05/1.50 ( omegaIndAx
% 3.05/1.50 = ( ! [A: $i] :
% 3.05/1.50 ( ( ( in @ emptyset @ A )
% 3.05/1.50 & ! [Xx: $i] :
% 3.05/1.50 ( ( ( in @ Xx @ omega )
% 3.05/1.50 & ( in @ Xx @ A ) )
% 3.05/1.50 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 3.05/1.50 => ! [Xx: $i] :
% 3.05/1.50 ( ( in @ Xx @ omega )
% 3.05/1.50 => ( in @ Xx @ A ) ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(replAx_type,type,
% 3.05/1.50 replAx: $o ).
% 3.05/1.50
% 3.05/1.50 thf(replAx,definition,
% 3.05/1.50 ( replAx
% 3.05/1.50 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 3.05/1.50 ( ! [Xx: $i] :
% 3.05/1.50 ( ( in @ Xx @ A )
% 3.05/1.50 => ( exu
% 3.05/1.50 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 3.05/1.50 => ? [B: $i] :
% 3.05/1.50 ! [Xx: $i] :
% 3.05/1.50 ( ( in @ Xx @ B )
% 3.05/1.50 <=> ? [Xy: $i] :
% 3.05/1.50 ( ( in @ Xy @ A )
% 3.05/1.50 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(foundationAx_type,type,
% 3.05/1.50 foundationAx: $o ).
% 3.05/1.50
% 3.05/1.50 thf(foundationAx,definition,
% 3.05/1.50 ( foundationAx
% 3.05/1.50 = ( ! [A: $i] :
% 3.05/1.50 ( ? [Xx: $i] : ( in @ Xx @ A )
% 3.05/1.50 => ? [B: $i] :
% 3.05/1.50 ( ( in @ B @ A )
% 3.05/1.50 & ~ ? [Xx: $i] :
% 3.05/1.50 ( ( in @ Xx @ B )
% 3.05/1.50 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(wellorderingAx_type,type,
% 3.05/1.50 wellorderingAx: $o ).
% 3.05/1.50
% 3.05/1.50 thf(wellorderingAx,definition,
% 3.05/1.50 ( wellorderingAx
% 3.05/1.50 = ( ! [A: $i] :
% 3.05/1.50 ? [B: $i] :
% 3.05/1.50 ( ! [C: $i] :
% 3.05/1.50 ( ( in @ C @ B )
% 3.05/1.50 => ! [Xx: $i] :
% 3.05/1.50 ( ( in @ Xx @ C )
% 3.05/1.50 => ( in @ Xx @ A ) ) )
% 3.05/1.50 & ! [Xx: $i,Xy: $i] :
% 3.05/1.50 ( ( ( in @ Xx @ A )
% 3.05/1.50 & ( in @ Xy @ A ) )
% 3.05/1.50 => ( ! [C: $i] :
% 3.05/1.50 ( ( in @ C @ B )
% 3.05/1.50 => ( ( in @ Xx @ C )
% 3.05/1.50 <=> ( in @ Xy @ C ) ) )
% 3.05/1.50 => ( Xx = Xy ) ) )
% 3.05/1.50 & ! [C: $i,D: $i] :
% 3.05/1.50 ( ( ( in @ C @ B )
% 3.05/1.50 & ( in @ D @ B ) )
% 3.05/1.50 => ( ! [Xx: $i] :
% 3.05/1.50 ( ( in @ Xx @ C )
% 3.05/1.50 => ( in @ Xx @ D ) )
% 3.05/1.50 | ! [Xx: $i] :
% 3.05/1.50 ( ( in @ Xx @ D )
% 3.05/1.50 => ( in @ Xx @ C ) ) ) )
% 3.05/1.50 & ! [C: $i] :
% 3.05/1.50 ( ( ! [Xx: $i] :
% 3.05/1.50 ( ( in @ Xx @ C )
% 3.05/1.50 => ( in @ Xx @ A ) )
% 3.05/1.50 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 3.05/1.50 => ? [D: $i,Xx: $i] :
% 3.05/1.50 ( ( in @ D @ B )
% 3.05/1.50 & ( in @ Xx @ C )
% 3.05/1.50 & ~ ? [Xy: $i] :
% 3.05/1.50 ( ( in @ Xy @ D )
% 3.05/1.50 & ( in @ Xy @ C ) )
% 3.05/1.50 & ! [E: $i] :
% 3.05/1.50 ( ( in @ E @ B )
% 3.05/1.50 => ( ! [Xy: $i] :
% 3.05/1.50 ( ( in @ Xy @ E )
% 3.05/1.50 => ( in @ Xy @ D ) )
% 3.05/1.50 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(descr_type,type,
% 3.05/1.50 descr: ( $i > $o ) > $i ).
% 3.05/1.50
% 3.05/1.50 thf(descrp_type,type,
% 3.05/1.50 descrp: $o ).
% 3.05/1.50
% 3.05/1.50 thf(descrp,definition,
% 3.05/1.50 ( descrp
% 3.05/1.50 = ( ! [Xphi: $i > $o] :
% 3.05/1.50 ( ( exu
% 3.05/1.50 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.05/1.50 => ( Xphi
% 3.05/1.50 @ ( descr
% 3.05/1.50 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(dsetconstr_type,type,
% 3.05/1.50 dsetconstr: $i > ( $i > $o ) > $i ).
% 3.05/1.50
% 3.05/1.50 thf(dsetconstrI_type,type,
% 3.05/1.50 dsetconstrI: $o ).
% 3.05/1.50
% 3.05/1.50 thf(dsetconstrI,definition,
% 3.05/1.50 ( dsetconstrI
% 3.05/1.50 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.05/1.50 ( ( in @ Xx @ A )
% 3.05/1.50 => ( ( Xphi @ Xx )
% 3.05/1.50 => ( in @ Xx
% 3.05/1.50 @ ( dsetconstr @ A
% 3.05/1.50 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(dsetconstrEL_type,type,
% 3.05/1.50 dsetconstrEL: $o ).
% 3.05/1.50
% 3.05/1.50 thf(dsetconstrEL,definition,
% 3.05/1.50 ( dsetconstrEL
% 3.05/1.50 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.05/1.50 ( ( in @ Xx
% 3.05/1.50 @ ( dsetconstr @ A
% 3.05/1.50 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 3.05/1.50 => ( in @ Xx @ A ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(dsetconstrER_type,type,
% 3.05/1.50 dsetconstrER: $o ).
% 3.05/1.50
% 3.05/1.50 thf(dsetconstrER,definition,
% 3.05/1.50 ( dsetconstrER
% 3.05/1.50 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.05/1.50 ( ( in @ Xx
% 3.05/1.50 @ ( dsetconstr @ A
% 3.05/1.50 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 3.05/1.50 => ( Xphi @ Xx ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(exuE1_type,type,
% 3.05/1.50 exuE1: $o ).
% 3.05/1.50
% 3.05/1.50 thf(exuE1,definition,
% 3.05/1.50 ( exuE1
% 3.05/1.50 = ( ! [Xphi: $i > $o] :
% 3.05/1.50 ( ( exu
% 3.05/1.50 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.05/1.50 => ? [Xx: $i] :
% 3.05/1.50 ( ( Xphi @ Xx )
% 3.05/1.50 & ! [Xy: $i] :
% 3.05/1.50 ( ( Xphi @ Xy )
% 3.05/1.50 => ( Xx = Xy ) ) ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(prop2set_type,type,
% 3.05/1.50 prop2set: $o > $i ).
% 3.05/1.50
% 3.05/1.50 thf(prop2setE_type,type,
% 3.05/1.50 prop2setE: $o ).
% 3.05/1.50
% 3.05/1.50 thf(prop2setE,definition,
% 3.05/1.50 ( prop2setE
% 3.05/1.50 = ( ! [Xphi: $o,Xx: $i] :
% 3.05/1.50 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 3.05/1.50 => Xphi ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(emptysetE_type,type,
% 3.05/1.50 emptysetE: $o ).
% 3.05/1.50
% 3.05/1.50 thf(emptysetE,definition,
% 3.05/1.50 ( emptysetE
% 3.05/1.50 = ( ! [Xx: $i] :
% 3.05/1.50 ( ( in @ Xx @ emptyset )
% 3.05/1.50 => ! [Xphi: $o] : Xphi ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(emptysetimpfalse_type,type,
% 3.05/1.50 emptysetimpfalse: $o ).
% 3.05/1.50
% 3.05/1.50 thf(emptysetimpfalse,definition,
% 3.05/1.50 ( emptysetimpfalse
% 3.05/1.50 = ( ! [Xx: $i] :
% 3.05/1.50 ( ( in @ Xx @ emptyset )
% 3.05/1.50 => $false ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(notinemptyset_type,type,
% 3.05/1.50 notinemptyset: $o ).
% 3.05/1.50
% 3.05/1.50 thf(notinemptyset,definition,
% 3.05/1.50 ( notinemptyset
% 3.05/1.50 = ( ! [Xx: $i] :
% 3.05/1.50 ~ ( in @ Xx @ emptyset ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(exuE3e_type,type,
% 3.05/1.50 exuE3e: $o ).
% 3.05/1.50
% 3.05/1.50 thf(exuE3e,definition,
% 3.05/1.50 ( exuE3e
% 3.05/1.50 = ( ! [Xphi: $i > $o] :
% 3.05/1.50 ( ( exu
% 3.05/1.50 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.05/1.50 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(setext_type,type,
% 3.05/1.50 setext: $o ).
% 3.05/1.50
% 3.05/1.50 thf(setext,definition,
% 3.05/1.50 ( setext
% 3.05/1.50 = ( ! [A: $i,B: $i] :
% 3.05/1.50 ( ! [Xx: $i] :
% 3.05/1.50 ( ( in @ Xx @ A )
% 3.05/1.50 => ( in @ Xx @ B ) )
% 3.05/1.50 => ( ! [Xx: $i] :
% 3.05/1.50 ( ( in @ Xx @ B )
% 3.05/1.50 => ( in @ Xx @ A ) )
% 3.05/1.50 => ( A = B ) ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(emptyI_type,type,
% 3.05/1.50 emptyI: $o ).
% 3.05/1.50
% 3.05/1.50 thf(emptyI,definition,
% 3.05/1.50 ( emptyI
% 3.05/1.50 = ( ! [A: $i] :
% 3.05/1.50 ( ! [Xx: $i] :
% 3.05/1.50 ~ ( in @ Xx @ A )
% 3.05/1.50 => ( A = emptyset ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(noeltsimpempty_type,type,
% 3.05/1.50 noeltsimpempty: $o ).
% 3.05/1.50
% 3.05/1.50 thf(noeltsimpempty,definition,
% 3.05/1.50 ( noeltsimpempty
% 3.05/1.50 = ( ! [A: $i] :
% 3.05/1.50 ( ! [Xx: $i] :
% 3.05/1.50 ~ ( in @ Xx @ A )
% 3.05/1.50 => ( A = emptyset ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(setbeta_type,type,
% 3.05/1.50 setbeta: $o ).
% 3.05/1.50
% 3.05/1.50 thf(setbeta,definition,
% 3.05/1.50 ( setbeta
% 3.05/1.50 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.05/1.50 ( ( in @ Xx @ A )
% 3.05/1.50 => ( ( in @ Xx
% 3.05/1.50 @ ( dsetconstr @ A
% 3.05/1.50 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 3.05/1.50 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(nonempty_type,type,
% 3.05/1.50 nonempty: $i > $o ).
% 3.05/1.50
% 3.05/1.50 thf(nonempty,definition,
% 3.05/1.50 ( nonempty
% 3.05/1.50 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(nonemptyE1_type,type,
% 3.05/1.50 nonemptyE1: $o ).
% 3.05/1.50
% 3.05/1.50 thf(nonemptyE1,definition,
% 3.05/1.50 ( nonemptyE1
% 3.05/1.50 = ( ! [A: $i] :
% 3.05/1.50 ( ( nonempty @ A )
% 3.05/1.50 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(nonemptyI_type,type,
% 3.05/1.50 nonemptyI: $o ).
% 3.05/1.50
% 3.05/1.50 thf(nonemptyI,definition,
% 3.05/1.50 ( nonemptyI
% 3.05/1.50 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.05/1.50 ( ( in @ Xx @ A )
% 3.05/1.50 => ( ( Xphi @ Xx )
% 3.05/1.50 => ( nonempty
% 3.05/1.50 @ ( dsetconstr @ A
% 3.05/1.50 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(nonemptyI1_type,type,
% 3.05/1.50 nonemptyI1: $o ).
% 3.05/1.50
% 3.05/1.50 thf(nonemptyI1,definition,
% 3.05/1.50 ( nonemptyI1
% 3.05/1.50 = ( ! [A: $i] :
% 3.05/1.50 ( ? [Xx: $i] : ( in @ Xx @ A )
% 3.05/1.50 => ( nonempty @ A ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(setadjoinIL_type,type,
% 3.05/1.50 setadjoinIL: $o ).
% 3.05/1.50
% 3.05/1.50 thf(setadjoinIL,definition,
% 3.05/1.50 ( setadjoinIL
% 3.05/1.50 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(emptyinunitempty_type,type,
% 3.05/1.50 emptyinunitempty: $o ).
% 3.05/1.50
% 3.05/1.50 thf(emptyinunitempty,definition,
% 3.05/1.50 ( emptyinunitempty
% 3.05/1.50 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(setadjoinIR_type,type,
% 3.05/1.50 setadjoinIR: $o ).
% 3.05/1.50
% 3.05/1.50 thf(setadjoinIR,definition,
% 3.05/1.50 ( setadjoinIR
% 3.05/1.50 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 3.05/1.50 ( ( in @ Xy @ A )
% 3.05/1.50 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(setadjoinE_type,type,
% 3.05/1.50 setadjoinE: $o ).
% 3.05/1.50
% 3.05/1.50 thf(setadjoinE,definition,
% 3.05/1.50 ( setadjoinE
% 3.05/1.50 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 3.05/1.50 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 3.05/1.50 => ! [Xphi: $o] :
% 3.05/1.50 ( ( ( Xy = Xx )
% 3.05/1.50 => Xphi )
% 3.05/1.50 => ( ( ( in @ Xy @ A )
% 3.05/1.50 => Xphi )
% 3.05/1.50 => Xphi ) ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(setadjoinOr_type,type,
% 3.05/1.50 setadjoinOr: $o ).
% 3.05/1.50
% 3.05/1.50 thf(setadjoinOr,definition,
% 3.05/1.50 ( setadjoinOr
% 3.05/1.50 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 3.05/1.50 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 3.05/1.50 => ( ( Xy = Xx )
% 3.05/1.50 | ( in @ Xy @ A ) ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(setoftrueEq_type,type,
% 3.05/1.50 setoftrueEq: $o ).
% 3.05/1.50
% 3.05/1.50 thf(setoftrueEq,definition,
% 3.05/1.50 ( setoftrueEq
% 3.05/1.50 = ( ! [A: $i] :
% 3.05/1.50 ( ( dsetconstr @ A
% 3.05/1.50 @ ^ [Xx: $i] : $true )
% 3.05/1.50 = A ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(powersetI_type,type,
% 3.05/1.50 powersetI: $o ).
% 3.05/1.50
% 3.05/1.50 thf(powersetI,definition,
% 3.05/1.50 ( powersetI
% 3.05/1.50 = ( ! [A: $i,B: $i] :
% 3.05/1.50 ( ! [Xx: $i] :
% 3.05/1.50 ( ( in @ Xx @ B )
% 3.05/1.50 => ( in @ Xx @ A ) )
% 3.05/1.50 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(emptyinPowerset_type,type,
% 3.05/1.50 emptyinPowerset: $o ).
% 3.05/1.50
% 3.05/1.50 thf(emptyinPowerset,definition,
% 3.05/1.50 ( emptyinPowerset
% 3.05/1.50 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(emptyInPowerset_type,type,
% 3.05/1.50 emptyInPowerset: $o ).
% 3.05/1.50
% 3.05/1.50 thf(emptyInPowerset,definition,
% 3.05/1.50 ( emptyInPowerset
% 3.05/1.50 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(powersetE_type,type,
% 3.05/1.50 powersetE: $o ).
% 3.05/1.50
% 3.05/1.50 thf(powersetE,definition,
% 3.05/1.50 ( powersetE
% 3.05/1.50 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.05/1.50 ( ( in @ B @ ( powerset @ A ) )
% 3.05/1.50 => ( ( in @ Xx @ B )
% 3.05/1.50 => ( in @ Xx @ A ) ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(setunionI_type,type,
% 3.05/1.50 setunionI: $o ).
% 3.05/1.50
% 3.05/1.50 thf(setunionI,definition,
% 3.05/1.50 ( setunionI
% 3.05/1.50 = ( ! [A: $i,Xx: $i,B: $i] :
% 3.05/1.50 ( ( in @ Xx @ B )
% 3.05/1.50 => ( ( in @ B @ A )
% 3.05/1.50 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(setunionE_type,type,
% 3.05/1.50 setunionE: $o ).
% 3.05/1.50
% 3.05/1.50 thf(setunionE,definition,
% 3.05/1.50 ( setunionE
% 3.05/1.50 = ( ! [A: $i,Xx: $i] :
% 3.05/1.50 ( ( in @ Xx @ ( setunion @ A ) )
% 3.05/1.50 => ! [Xphi: $o] :
% 3.05/1.50 ( ! [B: $i] :
% 3.05/1.50 ( ( in @ Xx @ B )
% 3.05/1.50 => ( ( in @ B @ A )
% 3.05/1.50 => Xphi ) )
% 3.05/1.50 => Xphi ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(subPowSU_type,type,
% 3.05/1.50 subPowSU: $o ).
% 3.05/1.50
% 3.05/1.50 thf(subPowSU,definition,
% 3.05/1.50 ( subPowSU
% 3.05/1.50 = ( ! [A: $i,Xx: $i] :
% 3.05/1.50 ( ( in @ Xx @ A )
% 3.05/1.50 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(exuE2_type,type,
% 3.05/1.50 exuE2: $o ).
% 3.05/1.50
% 3.05/1.50 thf(exuE2,definition,
% 3.05/1.50 ( exuE2
% 3.05/1.50 = ( ! [Xphi: $i > $o] :
% 3.05/1.50 ( ( exu
% 3.05/1.50 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.05/1.50 => ? [Xx: $i] :
% 3.05/1.50 ! [Xy: $i] :
% 3.05/1.50 ( ( Xphi @ Xy )
% 3.05/1.50 <=> ( Xy = Xx ) ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(nonemptyImpWitness_type,type,
% 3.05/1.50 nonemptyImpWitness: $o ).
% 3.05/1.50
% 3.05/1.50 thf(nonemptyImpWitness,definition,
% 3.05/1.50 ( nonemptyImpWitness
% 3.05/1.50 = ( ! [A: $i] :
% 3.05/1.50 ( ( nonempty @ A )
% 3.05/1.50 => ? [Xx: $i] :
% 3.05/1.50 ( ( in @ Xx @ A )
% 3.05/1.50 & $true ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(uniqinunit_type,type,
% 3.05/1.50 uniqinunit: $o ).
% 3.05/1.50
% 3.05/1.50 thf(uniqinunit,definition,
% 3.05/1.50 ( uniqinunit
% 3.05/1.50 = ( ! [Xx: $i,Xy: $i] :
% 3.05/1.50 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 3.05/1.50 => ( Xx = Xy ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(notinsingleton_type,type,
% 3.05/1.50 notinsingleton: $o ).
% 3.05/1.50
% 3.05/1.50 thf(notinsingleton,definition,
% 3.05/1.50 ( notinsingleton
% 3.05/1.50 = ( ! [Xx: $i,Xy: $i] :
% 3.05/1.50 ( ( Xx != Xy )
% 3.05/1.50 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(eqinunit_type,type,
% 3.05/1.50 eqinunit: $o ).
% 3.05/1.50
% 3.05/1.50 thf(eqinunit,definition,
% 3.05/1.50 ( eqinunit
% 3.05/1.50 = ( ! [Xx: $i,Xy: $i] :
% 3.05/1.50 ( ( Xx = Xy )
% 3.05/1.50 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(singletonsswitch_type,type,
% 3.05/1.50 singletonsswitch: $o ).
% 3.05/1.50
% 3.05/1.50 thf(singletonsswitch,definition,
% 3.05/1.50 ( singletonsswitch
% 3.05/1.50 = ( ! [Xx: $i,Xy: $i] :
% 3.05/1.50 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 3.05/1.50 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(upairsetE_type,type,
% 3.05/1.50 upairsetE: $o ).
% 3.05/1.50
% 3.05/1.50 thf(upairsetE,definition,
% 3.05/1.50 ( upairsetE
% 3.05/1.50 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 3.05/1.50 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 3.05/1.50 => ( ( Xz = Xx )
% 3.05/1.50 | ( Xz = Xy ) ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(upairsetIL_type,type,
% 3.05/1.50 upairsetIL: $o ).
% 3.05/1.50
% 3.05/1.50 thf(upairsetIL,definition,
% 3.05/1.50 ( upairsetIL
% 3.05/1.50 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(upairsetIR_type,type,
% 3.05/1.50 upairsetIR: $o ).
% 3.05/1.50
% 3.05/1.50 thf(upairsetIR,definition,
% 3.05/1.50 ( upairsetIR
% 3.05/1.50 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(emptyE1_type,type,
% 3.05/1.50 emptyE1: $o ).
% 3.05/1.50
% 3.05/1.50 thf(emptyE1,definition,
% 3.05/1.50 ( emptyE1
% 3.05/1.50 = ( ! [A: $i,Xphi: $i > $o] :
% 3.05/1.50 ( ? [Xx: $i] :
% 3.05/1.50 ( ( in @ Xx @ A )
% 3.05/1.50 & ( Xphi @ Xx ) )
% 3.05/1.50 => ( ( ( dsetconstr @ A
% 3.05/1.50 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.05/1.50 = emptyset )
% 3.05/1.50 => $false ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(vacuousDall_type,type,
% 3.05/1.50 vacuousDall: $o ).
% 3.05/1.50
% 3.05/1.50 thf(vacuousDall,definition,
% 3.05/1.50 ( vacuousDall
% 3.05/1.50 = ( ! [Xphi: $i > $o,Xx: $i] :
% 3.05/1.50 ( ( in @ Xx @ emptyset )
% 3.05/1.50 => ( Xphi @ Xx ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(quantDeMorgan1_type,type,
% 3.05/1.50 quantDeMorgan1: $o ).
% 3.05/1.50
% 3.05/1.50 thf(quantDeMorgan1,definition,
% 3.05/1.50 ( quantDeMorgan1
% 3.05/1.50 = ( ! [A: $i,Xphi: $i > $o] :
% 3.05/1.50 ( ~ ! [Xx: $i] :
% 3.05/1.50 ( ( in @ Xx @ A )
% 3.05/1.50 => ( Xphi @ Xx ) )
% 3.05/1.50 => ? [Xx: $i] :
% 3.05/1.50 ( ( in @ Xx @ A )
% 3.05/1.50 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(quantDeMorgan2_type,type,
% 3.05/1.50 quantDeMorgan2: $o ).
% 3.05/1.50
% 3.05/1.50 thf(quantDeMorgan2,definition,
% 3.05/1.50 ( quantDeMorgan2
% 3.05/1.50 = ( ! [A: $i,Xphi: $i > $o] :
% 3.05/1.50 ( ! [Xx: $i] :
% 3.05/1.50 ( ( in @ Xx @ A )
% 3.05/1.50 => ~ ( Xphi @ Xx ) )
% 3.05/1.50 => ~ ? [Xx: $i] :
% 3.05/1.50 ( ( in @ Xx @ A )
% 3.05/1.50 & ( Xphi @ Xx ) ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(quantDeMorgan3_type,type,
% 3.05/1.50 quantDeMorgan3: $o ).
% 3.05/1.50
% 3.05/1.50 thf(quantDeMorgan3,definition,
% 3.05/1.50 ( quantDeMorgan3
% 3.05/1.50 = ( ! [A: $i,Xphi: $i > $o] :
% 3.05/1.50 ( ~ ? [Xx: $i] :
% 3.05/1.50 ( ( in @ Xx @ A )
% 3.05/1.50 & ( Xphi @ Xx ) )
% 3.05/1.50 => ! [Xx: $i] :
% 3.05/1.50 ( ( in @ Xx @ A )
% 3.05/1.50 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(quantDeMorgan4_type,type,
% 3.05/1.50 quantDeMorgan4: $o ).
% 3.05/1.50
% 3.05/1.50 thf(quantDeMorgan4,definition,
% 3.05/1.50 ( quantDeMorgan4
% 3.05/1.50 = ( ! [A: $i,Xphi: $i > $o] :
% 3.05/1.50 ( ? [Xx: $i] :
% 3.05/1.50 ( ( in @ Xx @ A )
% 3.05/1.50 & ~ ( Xphi @ Xx ) )
% 3.05/1.50 => ~ ! [Xx: $i] :
% 3.05/1.50 ( ( in @ Xx @ A )
% 3.05/1.50 => ( Xphi @ Xx ) ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(prop2setI_type,type,
% 3.05/1.50 prop2setI: $o ).
% 3.05/1.50
% 3.05/1.50 thf(prop2setI,definition,
% 3.05/1.50 ( prop2setI
% 3.05/1.50 = ( ! [Xphi: $o] :
% 3.05/1.50 ( Xphi
% 3.05/1.50 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(set2prop_type,type,
% 3.05/1.50 set2prop: $i > $o ).
% 3.05/1.50
% 3.05/1.50 thf(prop2set2propI_type,type,
% 3.05/1.50 prop2set2propI: $o ).
% 3.05/1.50
% 3.05/1.50 thf(prop2set2propI,definition,
% 3.05/1.50 ( prop2set2propI
% 3.05/1.50 = ( ! [Xphi: $o] :
% 3.05/1.50 ( Xphi
% 3.05/1.50 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(notdexE_type,type,
% 3.05/1.50 notdexE: $o ).
% 3.05/1.50
% 3.05/1.50 thf(notdexE,definition,
% 3.05/1.50 ( notdexE
% 3.05/1.50 = ( ! [A: $i,Xphi: $i > $o] :
% 3.05/1.50 ( ~ ? [Xx: $i] :
% 3.05/1.50 ( ( in @ Xx @ A )
% 3.05/1.50 & ( Xphi @ Xx ) )
% 3.05/1.50 => ! [Xx: $i] :
% 3.05/1.50 ( ( in @ Xx @ A )
% 3.05/1.50 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(notdallE_type,type,
% 3.05/1.50 notdallE: $o ).
% 3.05/1.50
% 3.05/1.50 thf(notdallE,definition,
% 3.05/1.50 ( notdallE
% 3.05/1.50 = ( ! [A: $i,Xphi: $i > $o] :
% 3.05/1.50 ( ~ ! [Xx: $i] :
% 3.05/1.50 ( ( in @ Xx @ A )
% 3.05/1.50 => ( Xphi @ Xx ) )
% 3.05/1.50 => ? [Xx: $i] :
% 3.05/1.50 ( ( in @ Xx @ A )
% 3.05/1.50 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(exuI1_type,type,
% 3.05/1.50 exuI1: $o ).
% 3.05/1.50
% 3.05/1.50 thf(exuI1,definition,
% 3.05/1.50 ( exuI1
% 3.05/1.50 = ( ! [Xphi: $i > $o] :
% 3.05/1.50 ( ? [Xx: $i] :
% 3.05/1.50 ( ( Xphi @ Xx )
% 3.05/1.50 & ! [Xy: $i] :
% 3.05/1.50 ( ( Xphi @ Xy )
% 3.05/1.50 => ( Xx = Xy ) ) )
% 3.05/1.50 => ( exu
% 3.05/1.50 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(exuI3_type,type,
% 3.05/1.50 exuI3: $o ).
% 3.05/1.50
% 3.05/1.50 thf(exuI3,definition,
% 3.05/1.50 ( exuI3
% 3.05/1.50 = ( ! [Xphi: $i > $o] :
% 3.05/1.50 ( ? [Xx: $i] : ( Xphi @ Xx )
% 3.05/1.50 => ( ! [Xx: $i,Xy: $i] :
% 3.05/1.50 ( ( Xphi @ Xx )
% 3.05/1.50 => ( ( Xphi @ Xy )
% 3.05/1.50 => ( Xx = Xy ) ) )
% 3.05/1.50 => ( exu
% 3.05/1.50 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(exuI2_type,type,
% 3.05/1.50 exuI2: $o ).
% 3.05/1.50
% 3.05/1.50 thf(exuI2,definition,
% 3.05/1.50 ( exuI2
% 3.05/1.50 = ( ! [Xphi: $i > $o] :
% 3.05/1.50 ( ? [Xx: $i] :
% 3.05/1.50 ! [Xy: $i] :
% 3.05/1.50 ( ( Xphi @ Xy )
% 3.05/1.50 <=> ( Xy = Xx ) )
% 3.05/1.50 => ( exu
% 3.05/1.50 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(inCongP_type,type,
% 3.05/1.50 inCongP: $o ).
% 3.05/1.50
% 3.05/1.50 thf(inCongP,definition,
% 3.05/1.50 ( inCongP
% 3.05/1.50 = ( ! [A: $i,B: $i] :
% 3.05/1.50 ( ( A = B )
% 3.05/1.50 => ! [Xx: $i,Xy: $i] :
% 3.05/1.50 ( ( Xx = Xy )
% 3.05/1.50 => ( ( in @ Xx @ A )
% 3.05/1.50 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(in__Cong_type,type,
% 3.05/1.50 in__Cong: $o ).
% 3.05/1.50
% 3.05/1.50 thf(in__Cong,definition,
% 3.05/1.50 ( in__Cong
% 3.05/1.50 = ( ! [A: $i,B: $i] :
% 3.05/1.50 ( ( A = B )
% 3.05/1.50 => ! [Xx: $i,Xy: $i] :
% 3.05/1.50 ( ( Xx = Xy )
% 3.05/1.50 => ( ( in @ Xx @ A )
% 3.05/1.50 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(exuE3u_type,type,
% 3.05/1.50 exuE3u: $o ).
% 3.05/1.50
% 3.05/1.50 thf(exuE3u,definition,
% 3.05/1.50 ( exuE3u
% 3.05/1.50 = ( ! [Xphi: $i > $o] :
% 3.05/1.50 ( ( exu
% 3.05/1.50 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.05/1.50 => ! [Xx: $i,Xy: $i] :
% 3.05/1.50 ( ( Xphi @ Xx )
% 3.05/1.50 => ( ( Xphi @ Xy )
% 3.05/1.50 => ( Xx = Xy ) ) ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(exu__Cong_type,type,
% 3.05/1.50 exu__Cong: $o ).
% 3.05/1.50
% 3.05/1.50 thf(exu__Cong,definition,
% 3.05/1.50 ( exu__Cong
% 3.05/1.50 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 3.05/1.50 ( ! [Xx: $i,Xy: $i] :
% 3.05/1.50 ( ( Xx = Xy )
% 3.05/1.50 => ( ( Xphi @ Xx )
% 3.05/1.50 <=> ( Xpsi @ Xy ) ) )
% 3.05/1.50 => ( ( exu
% 3.05/1.50 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.05/1.50 <=> ( exu
% 3.05/1.50 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(emptyset__Cong_type,type,
% 3.05/1.50 emptyset__Cong: $o ).
% 3.05/1.50
% 3.05/1.50 thf(emptyset__Cong,definition,
% 3.05/1.50 ( emptyset__Cong
% 3.05/1.50 = ( emptyset = emptyset ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(setadjoin__Cong_type,type,
% 3.05/1.50 setadjoin__Cong: $o ).
% 3.05/1.50
% 3.05/1.50 thf(setadjoin__Cong,definition,
% 3.05/1.50 ( setadjoin__Cong
% 3.05/1.50 = ( ! [Xx: $i,Xy: $i] :
% 3.05/1.50 ( ( Xx = Xy )
% 3.05/1.50 => ! [Xz: $i,Xu: $i] :
% 3.05/1.50 ( ( Xz = Xu )
% 3.05/1.50 => ( ( setadjoin @ Xx @ Xz )
% 3.05/1.50 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(powerset__Cong_type,type,
% 3.05/1.50 powerset__Cong: $o ).
% 3.05/1.50
% 3.05/1.50 thf(powerset__Cong,definition,
% 3.05/1.50 ( powerset__Cong
% 3.05/1.50 = ( ! [A: $i,B: $i] :
% 3.05/1.50 ( ( A = B )
% 3.05/1.50 => ( ( powerset @ A )
% 3.05/1.50 = ( powerset @ B ) ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(setunion__Cong_type,type,
% 3.05/1.50 setunion__Cong: $o ).
% 3.05/1.50
% 3.05/1.50 thf(setunion__Cong,definition,
% 3.05/1.50 ( setunion__Cong
% 3.05/1.50 = ( ! [A: $i,B: $i] :
% 3.05/1.50 ( ( A = B )
% 3.05/1.50 => ( ( setunion @ A )
% 3.05/1.50 = ( setunion @ B ) ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(omega__Cong_type,type,
% 3.05/1.50 omega__Cong: $o ).
% 3.05/1.50
% 3.05/1.50 thf(omega__Cong,definition,
% 3.05/1.50 ( omega__Cong
% 3.05/1.50 = ( omega = omega ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(exuEu_type,type,
% 3.05/1.50 exuEu: $o ).
% 3.05/1.50
% 3.05/1.50 thf(exuEu,definition,
% 3.05/1.50 ( exuEu
% 3.05/1.50 = ( ! [Xphi: $i > $o] :
% 3.05/1.50 ( ( exu
% 3.05/1.50 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.05/1.50 => ! [Xx: $i,Xy: $i] :
% 3.05/1.50 ( ( Xphi @ Xx )
% 3.05/1.50 => ( ( Xphi @ Xy )
% 3.05/1.50 => ( Xx = Xy ) ) ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(descr__Cong_type,type,
% 3.05/1.50 descr__Cong: $o ).
% 3.05/1.50
% 3.05/1.50 thf(descr__Cong,definition,
% 3.05/1.50 ( descr__Cong
% 3.05/1.50 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 3.05/1.50 ( ! [Xx: $i,Xy: $i] :
% 3.05/1.50 ( ( Xx = Xy )
% 3.05/1.50 => ( ( Xphi @ Xx )
% 3.05/1.50 <=> ( Xpsi @ Xy ) ) )
% 3.05/1.50 => ( ( exu
% 3.05/1.50 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.05/1.50 => ( ( exu
% 3.05/1.50 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 3.05/1.50 => ( ( descr
% 3.05/1.50 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.05/1.50 = ( descr
% 3.05/1.50 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(dsetconstr__Cong_type,type,
% 3.05/1.50 dsetconstr__Cong: $o ).
% 3.05/1.50
% 3.05/1.50 thf(dsetconstr__Cong,definition,
% 3.05/1.50 ( dsetconstr__Cong
% 3.05/1.50 = ( ! [A: $i,B: $i] :
% 3.05/1.50 ( ( A = B )
% 3.05/1.50 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 3.05/1.50 ( ! [Xx: $i] :
% 3.05/1.50 ( ( in @ Xx @ A )
% 3.05/1.50 => ! [Xy: $i] :
% 3.05/1.50 ( ( in @ Xy @ B )
% 3.05/1.50 => ( ( Xx = Xy )
% 3.05/1.50 => ( ( Xphi @ Xx )
% 3.05/1.50 <=> ( Xpsi @ Xy ) ) ) ) )
% 3.05/1.50 => ( ( dsetconstr @ A
% 3.05/1.50 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.05/1.50 = ( dsetconstr @ B
% 3.05/1.50 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(subset_type,type,
% 3.05/1.50 subset: $i > $i > $o ).
% 3.05/1.50
% 3.05/1.50 thf(disjoint_type,type,
% 3.05/1.50 disjoint: $i > $i > $o ).
% 3.05/1.50
% 3.05/1.50 thf(setsmeet_type,type,
% 3.05/1.50 setsmeet: $i > $i > $o ).
% 3.05/1.50
% 3.05/1.50 thf(subsetI1_type,type,
% 3.05/1.50 subsetI1: $o ).
% 3.05/1.50
% 3.05/1.50 thf(subsetI1,definition,
% 3.05/1.50 ( subsetI1
% 3.05/1.50 = ( ! [A: $i,B: $i] :
% 3.05/1.50 ( ! [Xx: $i] :
% 3.05/1.50 ( ( in @ Xx @ A )
% 3.05/1.50 => ( in @ Xx @ B ) )
% 3.05/1.50 => ( subset @ A @ B ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(eqimpsubset2_type,type,
% 3.05/1.50 eqimpsubset2: $o ).
% 3.05/1.50
% 3.05/1.50 thf(eqimpsubset2,definition,
% 3.05/1.50 ( eqimpsubset2
% 3.05/1.50 = ( ! [A: $i,B: $i] :
% 3.05/1.50 ( ( A = B )
% 3.05/1.50 => ( subset @ B @ A ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(eqimpsubset1_type,type,
% 3.05/1.50 eqimpsubset1: $o ).
% 3.05/1.50
% 3.05/1.50 thf(eqimpsubset1,definition,
% 3.05/1.50 ( eqimpsubset1
% 3.05/1.50 = ( ! [A: $i,B: $i] :
% 3.05/1.50 ( ( A = B )
% 3.05/1.50 => ( subset @ A @ B ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(subsetI2_type,type,
% 3.05/1.50 subsetI2: $o ).
% 3.05/1.50
% 3.05/1.50 thf(subsetI2,definition,
% 3.05/1.50 ( subsetI2
% 3.05/1.50 = ( ! [A: $i,B: $i] :
% 3.05/1.50 ( ! [Xx: $i] :
% 3.05/1.50 ( ( in @ Xx @ A )
% 3.05/1.50 => ( in @ Xx @ B ) )
% 3.05/1.50 => ( subset @ A @ B ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(emptysetsubset_type,type,
% 3.05/1.50 emptysetsubset: $o ).
% 3.05/1.50
% 3.05/1.50 thf(emptysetsubset,definition,
% 3.05/1.50 ( emptysetsubset
% 3.05/1.50 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(subsetE_type,type,
% 3.05/1.50 subsetE: $o ).
% 3.05/1.50
% 3.05/1.50 thf(subsetE,definition,
% 3.05/1.50 ( subsetE
% 3.05/1.50 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.05/1.50 ( ( subset @ A @ B )
% 3.05/1.50 => ( ( in @ Xx @ A )
% 3.05/1.50 => ( in @ Xx @ B ) ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(subsetE2_type,type,
% 3.05/1.50 subsetE2: $o ).
% 3.05/1.50
% 3.05/1.50 thf(subsetE2,definition,
% 3.05/1.50 ( subsetE2
% 3.05/1.50 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.05/1.50 ( ( subset @ A @ B )
% 3.05/1.50 => ( ~ ( in @ Xx @ B )
% 3.05/1.50 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(notsubsetI_type,type,
% 3.05/1.50 notsubsetI: $o ).
% 3.05/1.50
% 3.05/1.50 thf(notsubsetI,definition,
% 3.05/1.50 ( notsubsetI
% 3.05/1.50 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.05/1.50 ( ( in @ Xx @ A )
% 3.05/1.50 => ( ~ ( in @ Xx @ B )
% 3.05/1.50 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(notequalI1_type,type,
% 3.05/1.50 notequalI1: $o ).
% 3.05/1.50
% 3.05/1.50 thf(notequalI1,definition,
% 3.05/1.50 ( notequalI1
% 3.05/1.50 = ( ! [A: $i,B: $i] :
% 3.05/1.50 ( ~ ( subset @ A @ B )
% 3.05/1.50 => ( A != B ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(notequalI2_type,type,
% 3.05/1.50 notequalI2: $o ).
% 3.05/1.50
% 3.05/1.50 thf(notequalI2,definition,
% 3.05/1.50 ( notequalI2
% 3.05/1.50 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.05/1.50 ( ( in @ Xx @ A )
% 3.05/1.50 => ( ~ ( in @ Xx @ B )
% 3.05/1.50 => ( A != B ) ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(subsetRefl_type,type,
% 3.05/1.50 subsetRefl: $o ).
% 3.05/1.50
% 3.05/1.50 thf(subsetRefl,definition,
% 3.05/1.50 ( subsetRefl
% 3.05/1.50 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(subsetTrans_type,type,
% 3.05/1.50 subsetTrans: $o ).
% 3.05/1.50
% 3.05/1.50 thf(subsetTrans,definition,
% 3.05/1.50 ( subsetTrans
% 3.05/1.50 = ( ! [A: $i,B: $i,C: $i] :
% 3.05/1.50 ( ( subset @ A @ B )
% 3.05/1.50 => ( ( subset @ B @ C )
% 3.05/1.50 => ( subset @ A @ C ) ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(setadjoinSub_type,type,
% 3.05/1.50 setadjoinSub: $o ).
% 3.05/1.50
% 3.05/1.50 thf(setadjoinSub,definition,
% 3.05/1.50 ( setadjoinSub
% 3.05/1.50 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(setadjoinSub2_type,type,
% 3.05/1.50 setadjoinSub2: $o ).
% 3.05/1.50
% 3.05/1.50 thf(setadjoinSub2,definition,
% 3.05/1.50 ( setadjoinSub2
% 3.05/1.50 = ( ! [A: $i,Xx: $i,B: $i] :
% 3.05/1.50 ( ( subset @ A @ B )
% 3.05/1.50 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(subset2powerset_type,type,
% 3.05/1.50 subset2powerset: $o ).
% 3.05/1.50
% 3.05/1.50 thf(subset2powerset,definition,
% 3.05/1.50 ( subset2powerset
% 3.05/1.50 = ( ! [A: $i,B: $i] :
% 3.05/1.50 ( ( subset @ A @ B )
% 3.05/1.50 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(setextsub_type,type,
% 3.05/1.50 setextsub: $o ).
% 3.05/1.50
% 3.05/1.50 thf(setextsub,definition,
% 3.05/1.50 ( setextsub
% 3.05/1.50 = ( ! [A: $i,B: $i] :
% 3.05/1.50 ( ( subset @ A @ B )
% 3.05/1.50 => ( ( subset @ B @ A )
% 3.05/1.50 => ( A = B ) ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(subsetemptysetimpeq_type,type,
% 3.05/1.50 subsetemptysetimpeq: $o ).
% 3.05/1.50
% 3.05/1.50 thf(subsetemptysetimpeq,definition,
% 3.05/1.50 ( subsetemptysetimpeq
% 3.05/1.50 = ( ! [A: $i] :
% 3.05/1.50 ( ( subset @ A @ emptyset )
% 3.05/1.50 => ( A = emptyset ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(powersetI1_type,type,
% 3.05/1.50 powersetI1: $o ).
% 3.05/1.50
% 3.05/1.50 thf(powersetI1,definition,
% 3.05/1.50 ( powersetI1
% 3.05/1.50 = ( ! [A: $i,B: $i] :
% 3.05/1.50 ( ( subset @ B @ A )
% 3.05/1.50 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(powersetE1_type,type,
% 3.05/1.50 powersetE1: $o ).
% 3.05/1.50
% 3.05/1.50 thf(powersetE1,definition,
% 3.05/1.50 ( powersetE1
% 3.05/1.50 = ( ! [A: $i,B: $i] :
% 3.05/1.50 ( ( in @ B @ ( powerset @ A ) )
% 3.05/1.50 => ( subset @ B @ A ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(inPowerset_type,type,
% 3.05/1.50 inPowerset: $o ).
% 3.05/1.50
% 3.05/1.50 thf(inPowerset,definition,
% 3.05/1.50 ( inPowerset
% 3.05/1.50 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(powersetsubset_type,type,
% 3.05/1.50 powersetsubset: $o ).
% 3.05/1.50
% 3.05/1.50 thf(powersetsubset,definition,
% 3.05/1.50 ( powersetsubset
% 3.05/1.50 = ( ! [A: $i,B: $i] :
% 3.05/1.50 ( ( subset @ A @ B )
% 3.05/1.50 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(sepInPowerset_type,type,
% 3.05/1.50 sepInPowerset: $o ).
% 3.05/1.50
% 3.05/1.50 thf(sepInPowerset,definition,
% 3.05/1.50 ( sepInPowerset
% 3.05/1.50 = ( ! [A: $i,Xphi: $i > $o] :
% 3.05/1.50 ( in
% 3.05/1.50 @ ( dsetconstr @ A
% 3.05/1.50 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.05/1.50 @ ( powerset @ A ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(sepSubset_type,type,
% 3.05/1.50 sepSubset: $o ).
% 3.05/1.50
% 3.05/1.50 thf(sepSubset,definition,
% 3.05/1.50 ( sepSubset
% 3.05/1.50 = ( ! [A: $i,Xphi: $i > $o] :
% 3.05/1.50 ( subset
% 3.05/1.50 @ ( dsetconstr @ A
% 3.05/1.50 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.05/1.50 @ A ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(binunion_type,type,
% 3.05/1.50 binunion: $i > $i > $i ).
% 3.05/1.50
% 3.05/1.50 thf(binunionIL_type,type,
% 3.05/1.50 binunionIL: $o ).
% 3.05/1.50
% 3.05/1.50 thf(binunionIL,definition,
% 3.05/1.50 ( binunionIL
% 3.05/1.50 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.05/1.50 ( ( in @ Xx @ A )
% 3.05/1.50 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(upairset2IR_type,type,
% 3.05/1.50 upairset2IR: $o ).
% 3.05/1.50
% 3.05/1.50 thf(upairset2IR,definition,
% 3.05/1.50 ( upairset2IR
% 3.05/1.50 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(binunionIR_type,type,
% 3.05/1.50 binunionIR: $o ).
% 3.05/1.50
% 3.05/1.50 thf(binunionIR,definition,
% 3.05/1.50 ( binunionIR
% 3.05/1.50 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.05/1.50 ( ( in @ Xx @ B )
% 3.05/1.50 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(binunionEcases_type,type,
% 3.05/1.50 binunionEcases: $o ).
% 3.05/1.50
% 3.05/1.50 thf(binunionEcases,definition,
% 3.05/1.50 ( binunionEcases
% 3.05/1.50 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 3.05/1.50 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 3.05/1.50 => ( ( ( in @ Xx @ A )
% 3.05/1.50 => Xphi )
% 3.05/1.50 => ( ( ( in @ Xx @ B )
% 3.05/1.50 => Xphi )
% 3.05/1.50 => Xphi ) ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(binunionE_type,type,
% 3.05/1.50 binunionE: $o ).
% 3.05/1.50
% 3.05/1.50 thf(binunionE,definition,
% 3.05/1.50 ( binunionE
% 3.05/1.50 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.05/1.50 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 3.05/1.50 => ( ( in @ Xx @ A )
% 3.05/1.50 | ( in @ Xx @ B ) ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(binunionLsub_type,type,
% 3.05/1.50 binunionLsub: $o ).
% 3.05/1.50
% 3.05/1.50 thf(binunionLsub,definition,
% 3.05/1.50 ( binunionLsub
% 3.05/1.50 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(binunionRsub_type,type,
% 3.05/1.50 binunionRsub: $o ).
% 3.05/1.50
% 3.05/1.50 thf(binunionRsub,definition,
% 3.05/1.50 ( binunionRsub
% 3.05/1.50 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(binintersect_type,type,
% 3.05/1.50 binintersect: $i > $i > $i ).
% 3.05/1.50
% 3.05/1.50 thf(binintersectI_type,type,
% 3.05/1.50 binintersectI: $o ).
% 3.05/1.50
% 3.05/1.50 thf(binintersectI,definition,
% 3.05/1.50 ( binintersectI
% 3.05/1.50 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.05/1.50 ( ( in @ Xx @ A )
% 3.05/1.50 => ( ( in @ Xx @ B )
% 3.05/1.50 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(binintersectSubset5_type,type,
% 3.05/1.50 binintersectSubset5: $o ).
% 3.05/1.50
% 3.05/1.50 thf(binintersectSubset5,definition,
% 3.05/1.50 ( binintersectSubset5
% 3.05/1.50 = ( ! [A: $i,B: $i,C: $i] :
% 3.05/1.50 ( ( subset @ C @ A )
% 3.05/1.50 => ( ( subset @ C @ B )
% 3.05/1.50 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(binintersectEL_type,type,
% 3.05/1.50 binintersectEL: $o ).
% 3.05/1.50
% 3.05/1.50 thf(binintersectEL,definition,
% 3.05/1.50 ( binintersectEL
% 3.05/1.50 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.05/1.50 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 3.05/1.50 => ( in @ Xx @ A ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(binintersectLsub_type,type,
% 3.05/1.50 binintersectLsub: $o ).
% 3.05/1.50
% 3.05/1.50 thf(binintersectLsub,definition,
% 3.05/1.50 ( binintersectLsub
% 3.05/1.50 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(binintersectSubset2_type,type,
% 3.05/1.50 binintersectSubset2: $o ).
% 3.05/1.50
% 3.05/1.50 thf(binintersectSubset2,definition,
% 3.05/1.50 ( binintersectSubset2
% 3.05/1.50 = ( ! [A: $i,B: $i] :
% 3.05/1.50 ( ( subset @ A @ B )
% 3.05/1.50 => ( ( binintersect @ A @ B )
% 3.05/1.50 = A ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(binintersectSubset3_type,type,
% 3.05/1.50 binintersectSubset3: $o ).
% 3.05/1.50
% 3.05/1.50 thf(binintersectSubset3,definition,
% 3.05/1.50 ( binintersectSubset3
% 3.05/1.50 = ( ! [A: $i,B: $i] :
% 3.05/1.50 ( ( ( binintersect @ A @ B )
% 3.05/1.50 = B )
% 3.05/1.50 => ( subset @ B @ A ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(binintersectER_type,type,
% 3.05/1.50 binintersectER: $o ).
% 3.05/1.50
% 3.05/1.50 thf(binintersectER,definition,
% 3.05/1.50 ( binintersectER
% 3.05/1.50 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.05/1.50 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 3.05/1.50 => ( in @ Xx @ B ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(disjointsetsI1_type,type,
% 3.05/1.50 disjointsetsI1: $o ).
% 3.05/1.50
% 3.05/1.50 thf(disjointsetsI1,definition,
% 3.05/1.50 ( disjointsetsI1
% 3.05/1.50 = ( ! [A: $i,B: $i] :
% 3.05/1.50 ( ~ ? [Xx: $i] :
% 3.05/1.50 ( ( in @ Xx @ A )
% 3.05/1.50 & ( in @ Xx @ B ) )
% 3.05/1.50 => ( ( binintersect @ A @ B )
% 3.05/1.50 = emptyset ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(binintersectRsub_type,type,
% 3.05/1.50 binintersectRsub: $o ).
% 3.05/1.50
% 3.05/1.50 thf(binintersectRsub,definition,
% 3.05/1.50 ( binintersectRsub
% 3.05/1.50 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(binintersectSubset4_type,type,
% 3.05/1.50 binintersectSubset4: $o ).
% 3.05/1.50
% 3.05/1.50 thf(binintersectSubset4,definition,
% 3.05/1.50 ( binintersectSubset4
% 3.05/1.50 = ( ! [A: $i,B: $i] :
% 3.05/1.50 ( ( subset @ B @ A )
% 3.05/1.50 => ( ( binintersect @ A @ B )
% 3.05/1.50 = B ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(binintersectSubset1_type,type,
% 3.05/1.50 binintersectSubset1: $o ).
% 3.05/1.50
% 3.05/1.50 thf(binintersectSubset1,definition,
% 3.05/1.50 ( binintersectSubset1
% 3.05/1.50 = ( ! [A: $i,B: $i] :
% 3.05/1.50 ( ( ( binintersect @ A @ B )
% 3.05/1.50 = A )
% 3.05/1.50 => ( subset @ A @ B ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(bs114d_type,type,
% 3.05/1.50 bs114d: $o ).
% 3.05/1.50
% 3.05/1.50 thf(bs114d,definition,
% 3.05/1.50 ( bs114d
% 3.05/1.50 = ( ! [A: $i,B: $i,C: $i] :
% 3.05/1.50 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 3.05/1.50 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(regular_type,type,
% 3.05/1.50 regular: $i > $o ).
% 3.05/1.50
% 3.05/1.50 thf(setminus_type,type,
% 3.05/1.50 setminus: $i > $i > $i ).
% 3.05/1.50
% 3.05/1.50 thf(setminusI_type,type,
% 3.05/1.50 setminusI: $o ).
% 3.05/1.50
% 3.05/1.50 thf(setminusI,definition,
% 3.05/1.50 ( setminusI
% 3.05/1.50 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.05/1.50 ( ( in @ Xx @ A )
% 3.05/1.50 => ( ~ ( in @ Xx @ B )
% 3.05/1.50 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(setminusEL_type,type,
% 3.05/1.50 setminusEL: $o ).
% 3.05/1.50
% 3.05/1.50 thf(setminusEL,definition,
% 3.05/1.50 ( setminusEL
% 3.05/1.50 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.05/1.50 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 3.05/1.50 => ( in @ Xx @ A ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(setminusER_type,type,
% 3.05/1.50 setminusER: $o ).
% 3.05/1.50
% 3.05/1.50 thf(setminusER,definition,
% 3.05/1.50 ( setminusER
% 3.05/1.50 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.05/1.50 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 3.05/1.50 => ~ ( in @ Xx @ B ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(setminusSubset2_type,type,
% 3.05/1.50 setminusSubset2: $o ).
% 3.05/1.50
% 3.05/1.50 thf(setminusSubset2,definition,
% 3.05/1.50 ( setminusSubset2
% 3.05/1.50 = ( ! [A: $i,B: $i] :
% 3.05/1.50 ( ( subset @ A @ B )
% 3.05/1.50 => ( ( setminus @ A @ B )
% 3.05/1.50 = emptyset ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(setminusERneg_type,type,
% 3.05/1.50 setminusERneg: $o ).
% 3.05/1.50
% 3.05/1.50 thf(setminusERneg,definition,
% 3.05/1.50 ( setminusERneg
% 3.05/1.50 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.05/1.50 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 3.05/1.50 => ( ( in @ Xx @ A )
% 3.05/1.50 => ( in @ Xx @ B ) ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(setminusELneg_type,type,
% 3.05/1.50 setminusELneg: $o ).
% 3.05/1.50
% 3.05/1.50 thf(setminusELneg,definition,
% 3.05/1.50 ( setminusELneg
% 3.05/1.50 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.05/1.50 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 3.05/1.50 => ( ~ ( in @ Xx @ B )
% 3.05/1.50 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(setminusILneg_type,type,
% 3.05/1.50 setminusILneg: $o ).
% 3.05/1.50
% 3.05/1.50 thf(setminusILneg,definition,
% 3.05/1.50 ( setminusILneg
% 3.05/1.50 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.05/1.50 ( ~ ( in @ Xx @ A )
% 3.05/1.50 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(setminusIRneg_type,type,
% 3.05/1.50 setminusIRneg: $o ).
% 3.05/1.50
% 3.05/1.50 thf(setminusIRneg,definition,
% 3.05/1.50 ( setminusIRneg
% 3.05/1.50 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.05/1.50 ( ( in @ Xx @ B )
% 3.05/1.50 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(setminusLsub_type,type,
% 3.05/1.50 setminusLsub: $o ).
% 3.05/1.50
% 3.05/1.50 thf(setminusLsub,definition,
% 3.05/1.50 ( setminusLsub
% 3.05/1.50 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(setminusSubset1_type,type,
% 3.05/1.50 setminusSubset1: $o ).
% 3.05/1.50
% 3.05/1.50 thf(setminusSubset1,definition,
% 3.05/1.50 ( setminusSubset1
% 3.05/1.50 = ( ! [A: $i,B: $i] :
% 3.05/1.50 ( ( ( setminus @ A @ B )
% 3.05/1.50 = emptyset )
% 3.05/1.50 => ( subset @ A @ B ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(symdiff_type,type,
% 3.05/1.50 symdiff: $i > $i > $i ).
% 3.05/1.50
% 3.05/1.50 thf(symdiffE_type,type,
% 3.05/1.50 symdiffE: $o ).
% 3.05/1.50
% 3.05/1.50 thf(symdiffE,definition,
% 3.05/1.50 ( symdiffE
% 3.05/1.50 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.05/1.50 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 3.05/1.50 => ! [Xphi: $o] :
% 3.05/1.50 ( ( ( in @ Xx @ A )
% 3.05/1.50 => ( ~ ( in @ Xx @ B )
% 3.05/1.50 => Xphi ) )
% 3.05/1.50 => ( ( ~ ( in @ Xx @ A )
% 3.05/1.50 => ( ( in @ Xx @ B )
% 3.05/1.50 => Xphi ) )
% 3.05/1.50 => Xphi ) ) ) ) ) ).
% 3.05/1.50
% 3.05/1.50 thf(symdiffI1_type,type,
% 3.05/1.50 symdiffI1: $o ).
% 3.05/1.50
% 3.05/1.50 thf(symdiffI1,definition,
% 3.05/1.50 ( symdiffI1
% 3.05/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ( ~ ( in @ Xx @ B )
% 3.05/1.51 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(symdiffI2_type,type,
% 3.05/1.51 symdiffI2: $o ).
% 3.05/1.51
% 3.05/1.51 thf(symdiffI2,definition,
% 3.05/1.51 ( symdiffI2
% 3.05/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.05/1.51 ( ~ ( in @ Xx @ A )
% 3.05/1.51 => ( ( in @ Xx @ B )
% 3.05/1.51 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(symdiffIneg1_type,type,
% 3.05/1.51 symdiffIneg1: $o ).
% 3.05/1.51
% 3.05/1.51 thf(symdiffIneg1,definition,
% 3.05/1.51 ( symdiffIneg1
% 3.05/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ( ( in @ Xx @ B )
% 3.05/1.51 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(symdiffIneg2_type,type,
% 3.05/1.51 symdiffIneg2: $o ).
% 3.05/1.51
% 3.05/1.51 thf(symdiffIneg2,definition,
% 3.05/1.51 ( symdiffIneg2
% 3.05/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.05/1.51 ( ~ ( in @ Xx @ A )
% 3.05/1.51 => ( ~ ( in @ Xx @ B )
% 3.05/1.51 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(iskpair_type,type,
% 3.05/1.51 iskpair: $i > $o ).
% 3.05/1.51
% 3.05/1.51 thf(secondinupair_type,type,
% 3.05/1.51 secondinupair: $o ).
% 3.05/1.51
% 3.05/1.51 thf(secondinupair,definition,
% 3.05/1.51 ( secondinupair
% 3.05/1.51 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(setukpairIL_type,type,
% 3.05/1.51 setukpairIL: $o ).
% 3.05/1.51
% 3.05/1.51 thf(setukpairIL,definition,
% 3.05/1.51 ( setukpairIL
% 3.05/1.51 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(setukpairIR_type,type,
% 3.05/1.51 setukpairIR: $o ).
% 3.05/1.51
% 3.05/1.51 thf(setukpairIR,definition,
% 3.05/1.51 ( setukpairIR
% 3.05/1.51 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(kpairiskpair_type,type,
% 3.05/1.51 kpairiskpair: $o ).
% 3.05/1.51
% 3.05/1.51 thf(kpairiskpair,definition,
% 3.05/1.51 ( kpairiskpair
% 3.05/1.51 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(kpair_type,type,
% 3.05/1.51 kpair: $i > $i > $i ).
% 3.05/1.51
% 3.05/1.51 thf(kpairp_type,type,
% 3.05/1.51 kpairp: $o ).
% 3.05/1.51
% 3.05/1.51 thf(kpairp,definition,
% 3.05/1.51 ( kpairp
% 3.05/1.51 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(cartprod_type,type,
% 3.05/1.51 cartprod: $i > $i > $i ).
% 3.05/1.51
% 3.05/1.51 thf(singletonsubset_type,type,
% 3.05/1.51 singletonsubset: $o ).
% 3.05/1.51
% 3.05/1.51 thf(singletonsubset,definition,
% 3.05/1.51 ( singletonsubset
% 3.05/1.51 = ( ! [A: $i,Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(singletoninpowerset_type,type,
% 3.05/1.51 singletoninpowerset: $o ).
% 3.05/1.51
% 3.05/1.51 thf(singletoninpowerset,definition,
% 3.05/1.51 ( singletoninpowerset
% 3.05/1.51 = ( ! [A: $i,Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(singletoninpowunion_type,type,
% 3.05/1.51 singletoninpowunion: $o ).
% 3.05/1.51
% 3.05/1.51 thf(singletoninpowunion,definition,
% 3.05/1.51 ( singletoninpowunion
% 3.05/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(upairset2E_type,type,
% 3.05/1.51 upairset2E: $o ).
% 3.05/1.51
% 3.05/1.51 thf(upairset2E,definition,
% 3.05/1.51 ( upairset2E
% 3.05/1.51 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 3.05/1.51 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 3.05/1.51 => ( ( Xz = Xx )
% 3.05/1.51 | ( Xz = Xy ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(upairsubunion_type,type,
% 3.05/1.51 upairsubunion: $o ).
% 3.05/1.51
% 3.05/1.51 thf(upairsubunion,definition,
% 3.05/1.51 ( upairsubunion
% 3.05/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ! [Xy: $i] :
% 3.05/1.51 ( ( in @ Xy @ B )
% 3.05/1.51 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(upairinpowunion_type,type,
% 3.05/1.51 upairinpowunion: $o ).
% 3.05/1.51
% 3.05/1.51 thf(upairinpowunion,definition,
% 3.05/1.51 ( upairinpowunion
% 3.05/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ! [Xy: $i] :
% 3.05/1.51 ( ( in @ Xy @ B )
% 3.05/1.51 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(ubforcartprodlem1_type,type,
% 3.05/1.51 ubforcartprodlem1: $o ).
% 3.05/1.51
% 3.05/1.51 thf(ubforcartprodlem1,definition,
% 3.05/1.51 ( ubforcartprodlem1
% 3.05/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ! [Xy: $i] :
% 3.05/1.51 ( ( in @ Xy @ B )
% 3.05/1.51 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(ubforcartprodlem2_type,type,
% 3.05/1.51 ubforcartprodlem2: $o ).
% 3.05/1.51
% 3.05/1.51 thf(ubforcartprodlem2,definition,
% 3.05/1.51 ( ubforcartprodlem2
% 3.05/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ! [Xy: $i] :
% 3.05/1.51 ( ( in @ Xy @ B )
% 3.05/1.51 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(ubforcartprodlem3_type,type,
% 3.05/1.51 ubforcartprodlem3: $o ).
% 3.05/1.51
% 3.05/1.51 thf(ubforcartprodlem3,definition,
% 3.05/1.51 ( ubforcartprodlem3
% 3.05/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ! [Xy: $i] :
% 3.05/1.51 ( ( in @ Xy @ B )
% 3.05/1.51 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(cartprodpairin_type,type,
% 3.05/1.51 cartprodpairin: $o ).
% 3.05/1.51
% 3.05/1.51 thf(cartprodpairin,definition,
% 3.05/1.51 ( cartprodpairin
% 3.05/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ! [Xy: $i] :
% 3.05/1.51 ( ( in @ Xy @ B )
% 3.05/1.51 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(cartprodmempair1_type,type,
% 3.05/1.51 cartprodmempair1: $o ).
% 3.05/1.51
% 3.05/1.51 thf(cartprodmempair1,definition,
% 3.05/1.51 ( cartprodmempair1
% 3.05/1.51 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.05/1.51 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.05/1.51 => ? [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 & ? [Xy: $i] :
% 3.05/1.51 ( ( in @ Xy @ B )
% 3.05/1.51 & ( Xu
% 3.05/1.51 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(cartprodmempair_type,type,
% 3.05/1.51 cartprodmempair: $o ).
% 3.05/1.51
% 3.05/1.51 thf(cartprodmempair,definition,
% 3.05/1.51 ( cartprodmempair
% 3.05/1.51 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.05/1.51 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.05/1.51 => ( iskpair @ Xu ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(setunionE2_type,type,
% 3.05/1.51 setunionE2: $o ).
% 3.05/1.51
% 3.05/1.51 thf(setunionE2,definition,
% 3.05/1.51 ( setunionE2
% 3.05/1.51 = ( ! [A: $i,Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ ( setunion @ A ) )
% 3.05/1.51 => ? [X: $i] :
% 3.05/1.51 ( ( in @ X @ A )
% 3.05/1.51 & ( in @ Xx @ X ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(setunionsingleton1_type,type,
% 3.05/1.51 setunionsingleton1: $o ).
% 3.05/1.51
% 3.05/1.51 thf(setunionsingleton1,definition,
% 3.05/1.51 ( setunionsingleton1
% 3.05/1.51 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(setunionsingleton2_type,type,
% 3.05/1.51 setunionsingleton2: $o ).
% 3.05/1.51
% 3.05/1.51 thf(setunionsingleton2,definition,
% 3.05/1.51 ( setunionsingleton2
% 3.05/1.51 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(setunionsingleton_type,type,
% 3.05/1.51 setunionsingleton: $o ).
% 3.05/1.51
% 3.05/1.51 thf(setunionsingleton,definition,
% 3.05/1.51 ( setunionsingleton
% 3.05/1.51 = ( ! [Xx: $i] :
% 3.05/1.51 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 3.05/1.51 = Xx ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(singleton_type,type,
% 3.05/1.51 singleton: $i > $o ).
% 3.05/1.51
% 3.05/1.51 thf(singletonprop_type,type,
% 3.05/1.51 singletonprop: $o ).
% 3.05/1.51
% 3.05/1.51 thf(singletonprop,definition,
% 3.05/1.51 ( singletonprop
% 3.05/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 3.05/1.51 ( ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ! [Xy: $i] :
% 3.05/1.51 ( ( in @ Xy @ A )
% 3.05/1.51 => ( ( Xphi @ Xx )
% 3.05/1.51 => ( ( Xphi @ Xy )
% 3.05/1.51 => ( Xx = Xy ) ) ) ) )
% 3.05/1.51 => ( ? [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 & ( Xphi @ Xx ) )
% 3.05/1.51 => ( singleton
% 3.05/1.51 @ ( dsetconstr @ A
% 3.05/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(ex1_type,type,
% 3.05/1.51 ex1: $i > ( $i > $o ) > $o ).
% 3.05/1.51
% 3.05/1.51 thf(ex1E1_type,type,
% 3.05/1.51 ex1E1: $o ).
% 3.05/1.51
% 3.05/1.51 thf(ex1E1,definition,
% 3.05/1.51 ( ex1E1
% 3.05/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 3.05/1.51 ( ( ex1 @ A
% 3.05/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.05/1.51 => ? [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 & ( Xphi @ Xx ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(ex1I_type,type,
% 3.05/1.51 ex1I: $o ).
% 3.05/1.51
% 3.05/1.51 thf(ex1I,definition,
% 3.05/1.51 ( ex1I
% 3.05/1.51 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ( ( Xphi @ Xx )
% 3.05/1.51 => ( ! [Xy: $i] :
% 3.05/1.51 ( ( in @ Xy @ A )
% 3.05/1.51 => ( ( Xphi @ Xy )
% 3.05/1.51 => ( Xy = Xx ) ) )
% 3.05/1.51 => ( ex1 @ A
% 3.05/1.51 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(ex1I2_type,type,
% 3.05/1.51 ex1I2: $o ).
% 3.05/1.51
% 3.05/1.51 thf(ex1I2,definition,
% 3.05/1.51 ( ex1I2
% 3.05/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 3.05/1.51 ( ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ! [Xy: $i] :
% 3.05/1.51 ( ( in @ Xy @ A )
% 3.05/1.51 => ( ( Xphi @ Xx )
% 3.05/1.51 => ( ( Xphi @ Xy )
% 3.05/1.51 => ( Xx = Xy ) ) ) ) )
% 3.05/1.51 => ( ? [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 & ( Xphi @ Xx ) )
% 3.05/1.51 => ( ex1 @ A
% 3.05/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(singletonsuniq_type,type,
% 3.05/1.51 singletonsuniq: $o ).
% 3.05/1.51
% 3.05/1.51 thf(singletonsuniq,definition,
% 3.05/1.51 ( singletonsuniq
% 3.05/1.51 = ( ! [Xx: $i,Xy: $i] :
% 3.05/1.51 ( ( ( setadjoin @ Xx @ emptyset )
% 3.05/1.51 = ( setadjoin @ Xy @ emptyset ) )
% 3.05/1.51 => ( Xx = Xy ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(atmost1p_type,type,
% 3.05/1.51 atmost1p: $i > $o ).
% 3.05/1.51
% 3.05/1.51 thf(atleast2p_type,type,
% 3.05/1.51 atleast2p: $i > $o ).
% 3.05/1.51
% 3.05/1.51 thf(atmost2p_type,type,
% 3.05/1.51 atmost2p: $i > $o ).
% 3.05/1.51
% 3.05/1.51 thf(upairsetp_type,type,
% 3.05/1.51 upairsetp: $i > $o ).
% 3.05/1.51
% 3.05/1.51 thf(setukpairinjL1_type,type,
% 3.05/1.51 setukpairinjL1: $o ).
% 3.05/1.51
% 3.05/1.51 thf(setukpairinjL1,definition,
% 3.05/1.51 ( setukpairinjL1
% 3.05/1.51 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 3.05/1.51 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 3.05/1.51 => ( Xx = Xz ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(kfstsingleton_type,type,
% 3.05/1.51 kfstsingleton: $o ).
% 3.05/1.51
% 3.05/1.51 thf(kfstsingleton,definition,
% 3.05/1.51 ( kfstsingleton
% 3.05/1.51 = ( ! [Xu: $i] :
% 3.05/1.51 ( ( iskpair @ Xu )
% 3.05/1.51 => ( singleton
% 3.05/1.51 @ ( dsetconstr @ ( setunion @ Xu )
% 3.05/1.51 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(theprop_type,type,
% 3.05/1.51 theprop: $o ).
% 3.05/1.51
% 3.05/1.51 thf(theprop,definition,
% 3.05/1.51 ( theprop
% 3.05/1.51 = ( ! [X: $i] :
% 3.05/1.51 ( ( singleton @ X )
% 3.05/1.51 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(kfst_type,type,
% 3.05/1.51 kfst: $i > $i ).
% 3.05/1.51
% 3.05/1.51 thf(kfstpairEq_type,type,
% 3.05/1.51 kfstpairEq: $o ).
% 3.05/1.51
% 3.05/1.51 thf(kfstpairEq,definition,
% 3.05/1.51 ( kfstpairEq
% 3.05/1.51 = ( ! [Xx: $i,Xy: $i] :
% 3.05/1.51 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 3.05/1.51 = Xx ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(cartprodfstin_type,type,
% 3.05/1.51 cartprodfstin: $o ).
% 3.05/1.51
% 3.05/1.51 thf(cartprodfstin,definition,
% 3.05/1.51 ( cartprodfstin
% 3.05/1.51 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.05/1.51 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.05/1.51 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(setukpairinjL2_type,type,
% 3.05/1.51 setukpairinjL2: $o ).
% 3.05/1.51
% 3.05/1.51 thf(setukpairinjL2,definition,
% 3.05/1.51 ( setukpairinjL2
% 3.05/1.51 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.05/1.51 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 3.05/1.51 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 3.05/1.51 => ( Xx = Xz ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(setukpairinjL_type,type,
% 3.05/1.51 setukpairinjL: $o ).
% 3.05/1.51
% 3.05/1.51 thf(setukpairinjL,definition,
% 3.05/1.51 ( setukpairinjL
% 3.05/1.51 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.05/1.51 ( ( ( kpair @ Xx @ Xy )
% 3.05/1.51 = ( kpair @ Xz @ Xu ) )
% 3.05/1.51 => ( Xx = Xz ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(setukpairinjR11_type,type,
% 3.05/1.51 setukpairinjR11: $o ).
% 3.05/1.51
% 3.05/1.51 thf(setukpairinjR11,definition,
% 3.05/1.51 ( setukpairinjR11
% 3.05/1.51 = ( ! [Xx: $i,Xy: $i] :
% 3.05/1.51 ( ( Xx = Xy )
% 3.05/1.51 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 3.05/1.51 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(setukpairinjR12_type,type,
% 3.05/1.51 setukpairinjR12: $o ).
% 3.05/1.51
% 3.05/1.51 thf(setukpairinjR12,definition,
% 3.05/1.51 ( setukpairinjR12
% 3.05/1.51 = ( ! [Xx: $i,Xy: $i] :
% 3.05/1.51 ( ( Xx = Xy )
% 3.05/1.51 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 3.05/1.51 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(setukpairinjR1_type,type,
% 3.05/1.51 setukpairinjR1: $o ).
% 3.05/1.51
% 3.05/1.51 thf(setukpairinjR1,definition,
% 3.05/1.51 ( setukpairinjR1
% 3.05/1.51 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.05/1.51 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 3.05/1.51 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 3.05/1.51 => ( ( Xz = Xu )
% 3.05/1.51 => ( Xy = Xu ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(upairequniteq_type,type,
% 3.05/1.51 upairequniteq: $o ).
% 3.05/1.51
% 3.05/1.51 thf(upairequniteq,definition,
% 3.05/1.51 ( upairequniteq
% 3.05/1.51 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 3.05/1.51 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 3.05/1.51 = ( setadjoin @ Xz @ emptyset ) )
% 3.05/1.51 => ( Xx = Xy ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(setukpairinjR2_type,type,
% 3.05/1.51 setukpairinjR2: $o ).
% 3.05/1.51
% 3.05/1.51 thf(setukpairinjR2,definition,
% 3.05/1.51 ( setukpairinjR2
% 3.05/1.51 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.05/1.51 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 3.05/1.51 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 3.05/1.51 => ( Xy = Xu ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(setukpairinjR_type,type,
% 3.05/1.51 setukpairinjR: $o ).
% 3.05/1.51
% 3.05/1.51 thf(setukpairinjR,definition,
% 3.05/1.51 ( setukpairinjR
% 3.05/1.51 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.05/1.51 ( ( ( kpair @ Xx @ Xy )
% 3.05/1.51 = ( kpair @ Xz @ Xu ) )
% 3.05/1.51 => ( Xy = Xu ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(ksndsingleton_type,type,
% 3.05/1.51 ksndsingleton: $o ).
% 3.05/1.51
% 3.05/1.51 thf(ksndsingleton,definition,
% 3.05/1.51 ( ksndsingleton
% 3.05/1.51 = ( ! [Xu: $i] :
% 3.05/1.51 ( ( iskpair @ Xu )
% 3.05/1.51 => ( singleton
% 3.05/1.51 @ ( dsetconstr @ ( setunion @ Xu )
% 3.05/1.51 @ ^ [Xx: $i] :
% 3.05/1.51 ( Xu
% 3.05/1.51 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(ksnd_type,type,
% 3.05/1.51 ksnd: $i > $i ).
% 3.05/1.51
% 3.05/1.51 thf(ksndpairEq_type,type,
% 3.05/1.51 ksndpairEq: $o ).
% 3.05/1.51
% 3.05/1.51 thf(ksndpairEq,definition,
% 3.05/1.51 ( ksndpairEq
% 3.05/1.51 = ( ! [Xx: $i,Xy: $i] :
% 3.05/1.51 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 3.05/1.51 = Xy ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(kpairsurjEq_type,type,
% 3.05/1.51 kpairsurjEq: $o ).
% 3.05/1.51
% 3.05/1.51 thf(kpairsurjEq,definition,
% 3.05/1.51 ( kpairsurjEq
% 3.05/1.51 = ( ! [Xu: $i] :
% 3.05/1.51 ( ( iskpair @ Xu )
% 3.05/1.51 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 3.05/1.51 = Xu ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(cartprodsndin_type,type,
% 3.05/1.51 cartprodsndin: $o ).
% 3.05/1.51
% 3.05/1.51 thf(cartprodsndin,definition,
% 3.05/1.51 ( cartprodsndin
% 3.05/1.51 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.05/1.51 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.05/1.51 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(cartprodpairmemEL_type,type,
% 3.05/1.51 cartprodpairmemEL: $o ).
% 3.05/1.51
% 3.05/1.51 thf(cartprodpairmemEL,definition,
% 3.05/1.51 ( cartprodpairmemEL
% 3.05/1.51 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 3.05/1.51 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 3.05/1.51 => ( in @ Xx @ A ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(cartprodpairmemER_type,type,
% 3.05/1.51 cartprodpairmemER: $o ).
% 3.05/1.51
% 3.05/1.51 thf(cartprodpairmemER,definition,
% 3.05/1.51 ( cartprodpairmemER
% 3.05/1.51 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 3.05/1.51 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 3.05/1.51 => ( in @ Xy @ B ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(cartprodmempaircEq_type,type,
% 3.05/1.51 cartprodmempaircEq: $o ).
% 3.05/1.51
% 3.05/1.51 thf(cartprodmempaircEq,definition,
% 3.05/1.51 ( cartprodmempaircEq
% 3.05/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ! [Xy: $i] :
% 3.05/1.51 ( ( in @ Xy @ B )
% 3.05/1.51 => ( ( kpair @ Xx @ Xy )
% 3.05/1.51 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(cartprodfstpairEq_type,type,
% 3.05/1.51 cartprodfstpairEq: $o ).
% 3.05/1.51
% 3.05/1.51 thf(cartprodfstpairEq,definition,
% 3.05/1.51 ( cartprodfstpairEq
% 3.05/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ! [Xy: $i] :
% 3.05/1.51 ( ( in @ Xy @ B )
% 3.05/1.51 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 3.05/1.51 = Xx ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(cartprodsndpairEq_type,type,
% 3.05/1.51 cartprodsndpairEq: $o ).
% 3.05/1.51
% 3.05/1.51 thf(cartprodsndpairEq,definition,
% 3.05/1.51 ( cartprodsndpairEq
% 3.05/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ! [Xy: $i] :
% 3.05/1.51 ( ( in @ Xy @ B )
% 3.05/1.51 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 3.05/1.51 = Xy ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(cartprodpairsurjEq_type,type,
% 3.05/1.51 cartprodpairsurjEq: $o ).
% 3.05/1.51
% 3.05/1.51 thf(cartprodpairsurjEq,definition,
% 3.05/1.51 ( cartprodpairsurjEq
% 3.05/1.51 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.05/1.51 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.05/1.51 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 3.05/1.51 = Xu ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(breln_type,type,
% 3.05/1.51 breln: $i > $i > $i > $o ).
% 3.05/1.51
% 3.05/1.51 thf(dpsetconstr_type,type,
% 3.05/1.51 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 3.05/1.51
% 3.05/1.51 thf(dpsetconstrI_type,type,
% 3.05/1.51 dpsetconstrI: $o ).
% 3.05/1.51
% 3.05/1.51 thf(dpsetconstrI,definition,
% 3.05/1.51 ( dpsetconstrI
% 3.05/1.51 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ! [Xy: $i] :
% 3.05/1.51 ( ( in @ Xy @ B )
% 3.05/1.51 => ( ( Xphi @ Xx @ Xy )
% 3.05/1.51 => ( in @ ( kpair @ Xx @ Xy )
% 3.05/1.51 @ ( dpsetconstr @ A @ B
% 3.05/1.51 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(dpsetconstrSub_type,type,
% 3.05/1.51 dpsetconstrSub: $o ).
% 3.05/1.51
% 3.05/1.51 thf(dpsetconstrSub,definition,
% 3.05/1.51 ( dpsetconstrSub
% 3.05/1.51 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 3.05/1.51 ( subset
% 3.05/1.51 @ ( dpsetconstr @ A @ B
% 3.05/1.51 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 3.05/1.51 @ ( cartprod @ A @ B ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(setOfPairsIsBReln_type,type,
% 3.05/1.51 setOfPairsIsBReln: $o ).
% 3.05/1.51
% 3.05/1.51 thf(setOfPairsIsBReln,definition,
% 3.05/1.51 ( setOfPairsIsBReln
% 3.05/1.51 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 3.05/1.51 ( breln @ A @ B
% 3.05/1.51 @ ( dpsetconstr @ A @ B
% 3.05/1.51 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(dpsetconstrERa_type,type,
% 3.05/1.51 dpsetconstrERa: $o ).
% 3.05/1.51
% 3.05/1.51 thf(dpsetconstrERa,definition,
% 3.05/1.51 ( dpsetconstrERa
% 3.05/1.51 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ! [Xy: $i] :
% 3.05/1.51 ( ( in @ Xy @ B )
% 3.05/1.51 => ( ( in @ ( kpair @ Xx @ Xy )
% 3.05/1.51 @ ( dpsetconstr @ A @ B
% 3.05/1.51 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 3.05/1.51 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(dpsetconstrEL1_type,type,
% 3.05/1.51 dpsetconstrEL1: $o ).
% 3.05/1.51
% 3.05/1.51 thf(dpsetconstrEL1,definition,
% 3.05/1.51 ( dpsetconstrEL1
% 3.05/1.51 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 3.05/1.51 ( ( in @ ( kpair @ Xx @ Xy )
% 3.05/1.51 @ ( dpsetconstr @ A @ B
% 3.05/1.51 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 3.05/1.51 => ( in @ Xx @ A ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(dpsetconstrEL2_type,type,
% 3.05/1.51 dpsetconstrEL2: $o ).
% 3.05/1.51
% 3.05/1.51 thf(dpsetconstrEL2,definition,
% 3.05/1.51 ( dpsetconstrEL2
% 3.05/1.51 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 3.05/1.51 ( ( in @ ( kpair @ Xx @ Xy )
% 3.05/1.51 @ ( dpsetconstr @ A @ B
% 3.05/1.51 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 3.05/1.51 => ( in @ Xy @ B ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(dpsetconstrER_type,type,
% 3.05/1.51 dpsetconstrER: $o ).
% 3.05/1.51
% 3.05/1.51 thf(dpsetconstrER,definition,
% 3.05/1.51 ( dpsetconstrER
% 3.05/1.51 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 3.05/1.51 ( ( in @ ( kpair @ Xx @ Xy )
% 3.05/1.51 @ ( dpsetconstr @ A @ B
% 3.05/1.51 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 3.05/1.51 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(func_type,type,
% 3.05/1.51 func: $i > $i > $i > $o ).
% 3.05/1.51
% 3.05/1.51 thf(funcSet_type,type,
% 3.05/1.51 funcSet: $i > $i > $i ).
% 3.05/1.51
% 3.05/1.51 thf(funcImageSingleton_type,type,
% 3.05/1.51 funcImageSingleton: $o ).
% 3.05/1.51
% 3.05/1.51 thf(funcImageSingleton,definition,
% 3.05/1.51 ( funcImageSingleton
% 3.05/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.05/1.51 ( ( func @ A @ B @ Xf )
% 3.05/1.51 => ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ( singleton
% 3.05/1.51 @ ( dsetconstr @ B
% 3.05/1.51 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(apProp_type,type,
% 3.05/1.51 apProp: $o ).
% 3.05/1.51
% 3.05/1.51 thf(apProp,definition,
% 3.05/1.51 ( apProp
% 3.05/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.05/1.51 ( ( func @ A @ B @ Xf )
% 3.05/1.51 => ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ( in
% 3.05/1.51 @ ( setunion
% 3.05/1.51 @ ( dsetconstr @ B
% 3.05/1.51 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 3.05/1.51 @ B ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(ap_type,type,
% 3.05/1.51 ap: $i > $i > $i > $i > $i ).
% 3.05/1.51
% 3.05/1.51 thf(app_type,type,
% 3.05/1.51 app: $o ).
% 3.05/1.51
% 3.05/1.51 thf(app,definition,
% 3.05/1.51 ( app
% 3.05/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.05/1.51 ( ( func @ A @ B @ Xf )
% 3.05/1.51 => ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(infuncsetfunc_type,type,
% 3.05/1.51 infuncsetfunc: $o ).
% 3.05/1.51
% 3.05/1.51 thf(infuncsetfunc,definition,
% 3.05/1.51 ( infuncsetfunc
% 3.05/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.05/1.51 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.05/1.51 => ( func @ A @ B @ Xf ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(ap2p_type,type,
% 3.05/1.51 ap2p: $o ).
% 3.05/1.51
% 3.05/1.51 thf(ap2p,definition,
% 3.05/1.51 ( ap2p
% 3.05/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.05/1.51 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.05/1.51 => ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(funcinfuncset_type,type,
% 3.05/1.51 funcinfuncset: $o ).
% 3.05/1.51
% 3.05/1.51 thf(funcinfuncset,definition,
% 3.05/1.51 ( funcinfuncset
% 3.05/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.05/1.51 ( ( func @ A @ B @ Xf )
% 3.05/1.51 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(lamProp_type,type,
% 3.05/1.51 lamProp: $o ).
% 3.05/1.51
% 3.05/1.51 thf(lamProp,definition,
% 3.05/1.51 ( lamProp
% 3.05/1.51 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.05/1.51 ( ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.05/1.51 => ( func @ A @ B
% 3.05/1.51 @ ( dpsetconstr @ A @ B
% 3.05/1.51 @ ^ [Xx: $i,Xy: $i] :
% 3.05/1.51 ( ( Xf @ Xx )
% 3.05/1.51 = Xy ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(lam_type,type,
% 3.05/1.51 lam: $i > $i > ( $i > $i ) > $i ).
% 3.05/1.51
% 3.05/1.51 thf(lamp_type,type,
% 3.05/1.51 lamp: $o ).
% 3.05/1.51
% 3.05/1.51 thf(lamp,definition,
% 3.05/1.51 ( lamp
% 3.05/1.51 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.05/1.51 ( ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.05/1.51 => ( func @ A @ B
% 3.05/1.51 @ ( lam @ A @ B
% 3.05/1.51 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(lam2p_type,type,
% 3.05/1.51 lam2p: $o ).
% 3.05/1.51
% 3.05/1.51 thf(lam2p,definition,
% 3.05/1.51 ( lam2p
% 3.05/1.51 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.05/1.51 ( ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.05/1.51 => ( in
% 3.05/1.51 @ ( lam @ A @ B
% 3.05/1.51 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 3.05/1.51 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(brelnall1_type,type,
% 3.05/1.51 brelnall1: $o ).
% 3.05/1.51
% 3.05/1.51 thf(brelnall1,definition,
% 3.05/1.51 ( brelnall1
% 3.05/1.51 = ( ! [A: $i,B: $i,R: $i] :
% 3.05/1.51 ( ( breln @ A @ B @ R )
% 3.05/1.51 => ! [Xphi: $i > $o] :
% 3.05/1.51 ( ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ! [Xy: $i] :
% 3.05/1.51 ( ( in @ Xy @ B )
% 3.05/1.51 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.05/1.51 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 3.05/1.51 => ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ R )
% 3.05/1.51 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(brelnall2_type,type,
% 3.05/1.51 brelnall2: $o ).
% 3.05/1.51
% 3.05/1.51 thf(brelnall2,definition,
% 3.05/1.51 ( brelnall2
% 3.05/1.51 = ( ! [A: $i,B: $i,R: $i] :
% 3.05/1.51 ( ( breln @ A @ B @ R )
% 3.05/1.51 => ! [Xphi: $i > $o] :
% 3.05/1.51 ( ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ! [Xy: $i] :
% 3.05/1.51 ( ( in @ Xy @ B )
% 3.05/1.51 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.05/1.51 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 3.05/1.51 => ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ R )
% 3.05/1.51 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(ex1E2_type,type,
% 3.05/1.51 ex1E2: $o ).
% 3.05/1.51
% 3.05/1.51 thf(ex1E2,definition,
% 3.05/1.51 ( ex1E2
% 3.05/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 3.05/1.51 ( ( ex1 @ A
% 3.05/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.05/1.51 => ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ! [Xy: $i] :
% 3.05/1.51 ( ( in @ Xy @ A )
% 3.05/1.51 => ( ( Xphi @ Xx )
% 3.05/1.51 => ( ( Xphi @ Xy )
% 3.05/1.51 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(funcGraphProp1_type,type,
% 3.05/1.51 funcGraphProp1: $o ).
% 3.05/1.51
% 3.05/1.51 thf(funcGraphProp1,definition,
% 3.05/1.51 ( funcGraphProp1
% 3.05/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.05/1.51 ( ( func @ A @ B @ Xf )
% 3.05/1.51 => ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(funcGraphProp3_type,type,
% 3.05/1.51 funcGraphProp3: $o ).
% 3.05/1.51
% 3.05/1.51 thf(funcGraphProp3,definition,
% 3.05/1.51 ( funcGraphProp3
% 3.05/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.05/1.51 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.05/1.51 => ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(funcGraphProp2_type,type,
% 3.05/1.51 funcGraphProp2: $o ).
% 3.05/1.51
% 3.05/1.51 thf(funcGraphProp2,definition,
% 3.05/1.51 ( funcGraphProp2
% 3.05/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.05/1.51 ( ( func @ A @ B @ Xf )
% 3.05/1.51 => ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ! [Xy: $i] :
% 3.05/1.51 ( ( in @ Xy @ B )
% 3.05/1.51 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 3.05/1.51 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.05/1.51 = Xy ) ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(funcextLem_type,type,
% 3.05/1.51 funcextLem: $o ).
% 3.05/1.51
% 3.05/1.51 thf(funcextLem,definition,
% 3.05/1.51 ( funcextLem
% 3.05/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.05/1.51 ( ( func @ A @ B @ Xf )
% 3.05/1.51 => ! [Xg: $i] :
% 3.05/1.51 ( ( func @ A @ B @ Xg )
% 3.05/1.51 => ( ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.05/1.51 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 3.05/1.51 => ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ! [Xy: $i] :
% 3.05/1.51 ( ( in @ Xy @ B )
% 3.05/1.51 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 3.05/1.51 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(funcGraphProp4_type,type,
% 3.05/1.51 funcGraphProp4: $o ).
% 3.05/1.51
% 3.05/1.51 thf(funcGraphProp4,definition,
% 3.05/1.51 ( funcGraphProp4
% 3.05/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.05/1.51 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.05/1.51 => ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ! [Xy: $i] :
% 3.05/1.51 ( ( in @ Xy @ B )
% 3.05/1.51 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 3.05/1.51 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.05/1.51 = Xy ) ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(subbreln_type,type,
% 3.05/1.51 subbreln: $o ).
% 3.05/1.51
% 3.05/1.51 thf(subbreln,definition,
% 3.05/1.51 ( subbreln
% 3.05/1.51 = ( ! [A: $i,B: $i,R: $i] :
% 3.05/1.51 ( ( breln @ A @ B @ R )
% 3.05/1.51 => ! [S: $i] :
% 3.05/1.51 ( ( breln @ A @ B @ S )
% 3.05/1.51 => ( ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ! [Xy: $i] :
% 3.05/1.51 ( ( in @ Xy @ B )
% 3.05/1.51 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.05/1.51 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 3.05/1.51 => ( subset @ R @ S ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(eqbreln_type,type,
% 3.05/1.51 eqbreln: $o ).
% 3.05/1.51
% 3.05/1.51 thf(eqbreln,definition,
% 3.05/1.51 ( eqbreln
% 3.05/1.51 = ( ! [A: $i,B: $i,R: $i] :
% 3.05/1.51 ( ( breln @ A @ B @ R )
% 3.05/1.51 => ! [S: $i] :
% 3.05/1.51 ( ( breln @ A @ B @ S )
% 3.05/1.51 => ( ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ! [Xy: $i] :
% 3.05/1.51 ( ( in @ Xy @ B )
% 3.05/1.51 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.05/1.51 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 3.05/1.51 => ( ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ! [Xy: $i] :
% 3.05/1.51 ( ( in @ Xy @ B )
% 3.05/1.51 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 3.05/1.51 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 3.05/1.51 => ( R = S ) ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(funcext_type,type,
% 3.05/1.51 funcext: $o ).
% 3.05/1.51
% 3.05/1.51 thf(funcext,definition,
% 3.05/1.51 ( funcext
% 3.05/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.05/1.51 ( ( func @ A @ B @ Xf )
% 3.05/1.51 => ! [Xg: $i] :
% 3.05/1.51 ( ( func @ A @ B @ Xg )
% 3.05/1.51 => ( ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.05/1.51 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 3.05/1.51 => ( Xf = Xg ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(funcext2_type,type,
% 3.05/1.51 funcext2: $o ).
% 3.05/1.51
% 3.05/1.51 thf(funcext2,definition,
% 3.05/1.51 ( funcext2
% 3.05/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.05/1.51 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.05/1.51 => ! [Xg: $i] :
% 3.05/1.51 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 3.05/1.51 => ( ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.05/1.51 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 3.05/1.51 => ( Xf = Xg ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(ap2apEq1_type,type,
% 3.05/1.51 ap2apEq1: $o ).
% 3.05/1.51
% 3.05/1.51 thf(ap2apEq1,definition,
% 3.05/1.51 ( ap2apEq1
% 3.05/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.05/1.51 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.05/1.51 => ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.05/1.51 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(ap2apEq2_type,type,
% 3.05/1.51 ap2apEq2: $o ).
% 3.05/1.51
% 3.05/1.51 thf(ap2apEq2,definition,
% 3.05/1.51 ( ap2apEq2
% 3.05/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.05/1.51 ( ( func @ A @ B @ Xf )
% 3.05/1.51 => ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.05/1.51 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(beta1_type,type,
% 3.05/1.51 beta1: $o ).
% 3.05/1.51
% 3.05/1.51 thf(beta1,definition,
% 3.05/1.51 ( beta1
% 3.05/1.51 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.05/1.51 ( ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.05/1.51 => ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ( ( ap @ A @ B
% 3.05/1.51 @ ( lam @ A @ B
% 3.05/1.51 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 3.05/1.51 @ Xx )
% 3.05/1.51 = ( Xf @ Xx ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(eta1_type,type,
% 3.05/1.51 eta1: $o ).
% 3.05/1.51
% 3.05/1.51 thf(eta1,definition,
% 3.05/1.51 ( eta1
% 3.05/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.05/1.51 ( ( func @ A @ B @ Xf )
% 3.05/1.51 => ( ( lam @ A @ B
% 3.05/1.51 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 3.05/1.51 = Xf ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(lam2lamEq_type,type,
% 3.05/1.51 lam2lamEq: $o ).
% 3.05/1.51
% 3.05/1.51 thf(lam2lamEq,definition,
% 3.05/1.51 ( lam2lamEq
% 3.05/1.51 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.05/1.51 ( ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.05/1.51 => ( ( lam @ A @ B
% 3.05/1.51 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 3.05/1.51 = ( lam @ A @ B
% 3.05/1.51 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(beta2_type,type,
% 3.05/1.51 beta2: $o ).
% 3.05/1.51
% 3.05/1.51 thf(beta2,definition,
% 3.05/1.51 ( beta2
% 3.05/1.51 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.05/1.51 ( ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.05/1.51 => ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ( ( ap @ A @ B
% 3.05/1.51 @ ( lam @ A @ B
% 3.05/1.51 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 3.05/1.51 @ Xx )
% 3.05/1.51 = ( Xf @ Xx ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(eta2_type,type,
% 3.05/1.51 eta2: $o ).
% 3.05/1.51
% 3.05/1.51 thf(eta2,definition,
% 3.05/1.51 ( eta2
% 3.05/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.05/1.51 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.05/1.51 => ( ( lam @ A @ B
% 3.05/1.51 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 3.05/1.51 = Xf ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(iffalseProp1_type,type,
% 3.05/1.51 iffalseProp1: $o ).
% 3.05/1.51
% 3.05/1.51 thf(iffalseProp1,definition,
% 3.05/1.51 ( iffalseProp1
% 3.05/1.51 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ! [Xy: $i] :
% 3.05/1.51 ( ( in @ Xy @ A )
% 3.05/1.51 => ( ~ Xphi
% 3.05/1.51 => ( in @ Xy
% 3.05/1.51 @ ( dsetconstr @ A
% 3.05/1.51 @ ^ [Xz: $i] :
% 3.05/1.51 ( ( Xphi
% 3.05/1.51 & ( Xz = Xx ) )
% 3.05/1.51 | ( ~ Xphi
% 3.05/1.51 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(iffalseProp2_type,type,
% 3.05/1.51 iffalseProp2: $o ).
% 3.05/1.51
% 3.05/1.51 thf(iffalseProp2,definition,
% 3.05/1.51 ( iffalseProp2
% 3.05/1.51 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ! [Xy: $i] :
% 3.05/1.51 ( ( in @ Xy @ A )
% 3.05/1.51 => ( ~ Xphi
% 3.05/1.51 => ( ( dsetconstr @ A
% 3.05/1.51 @ ^ [Xz: $i] :
% 3.05/1.51 ( ( Xphi
% 3.05/1.51 & ( Xz = Xx ) )
% 3.05/1.51 | ( ~ Xphi
% 3.05/1.51 & ( Xz = Xy ) ) ) )
% 3.05/1.51 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(iftrueProp1_type,type,
% 3.05/1.51 iftrueProp1: $o ).
% 3.05/1.51
% 3.05/1.51 thf(iftrueProp1,definition,
% 3.05/1.51 ( iftrueProp1
% 3.05/1.51 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ! [Xy: $i] :
% 3.05/1.51 ( ( in @ Xy @ A )
% 3.05/1.51 => ( Xphi
% 3.05/1.51 => ( in @ Xx
% 3.05/1.51 @ ( dsetconstr @ A
% 3.05/1.51 @ ^ [Xz: $i] :
% 3.05/1.51 ( ( Xphi
% 3.05/1.51 & ( Xz = Xx ) )
% 3.05/1.51 | ( ~ Xphi
% 3.05/1.51 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(iftrueProp2_type,type,
% 3.05/1.51 iftrueProp2: $o ).
% 3.05/1.51
% 3.05/1.51 thf(iftrueProp2,definition,
% 3.05/1.51 ( iftrueProp2
% 3.05/1.51 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ! [Xy: $i] :
% 3.05/1.51 ( ( in @ Xy @ A )
% 3.05/1.51 => ( Xphi
% 3.05/1.51 => ( ( dsetconstr @ A
% 3.05/1.51 @ ^ [Xz: $i] :
% 3.05/1.51 ( ( Xphi
% 3.05/1.51 & ( Xz = Xx ) )
% 3.05/1.51 | ( ~ Xphi
% 3.05/1.51 & ( Xz = Xy ) ) ) )
% 3.05/1.51 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(ifSingleton_type,type,
% 3.05/1.51 ifSingleton: $o ).
% 3.05/1.51
% 3.05/1.51 thf(ifSingleton,definition,
% 3.05/1.51 ( ifSingleton
% 3.05/1.51 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ! [Xy: $i] :
% 3.05/1.51 ( ( in @ Xy @ A )
% 3.05/1.51 => ( singleton
% 3.05/1.51 @ ( dsetconstr @ A
% 3.05/1.51 @ ^ [Xz: $i] :
% 3.05/1.51 ( ( Xphi
% 3.05/1.51 & ( Xz = Xx ) )
% 3.05/1.51 | ( ~ Xphi
% 3.05/1.51 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(if_type,type,
% 3.05/1.51 if: $i > $o > $i > $i > $i ).
% 3.05/1.51
% 3.05/1.51 thf(ifp_type,type,
% 3.05/1.51 ifp: $o ).
% 3.05/1.51
% 3.05/1.51 thf(ifp,definition,
% 3.05/1.51 ( ifp
% 3.05/1.51 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ! [Xy: $i] :
% 3.05/1.51 ( ( in @ Xy @ A )
% 3.05/1.51 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(theeq_type,type,
% 3.05/1.51 theeq: $o ).
% 3.05/1.51
% 3.05/1.51 thf(theeq,definition,
% 3.05/1.51 ( theeq
% 3.05/1.51 = ( ! [X: $i] :
% 3.05/1.51 ( ( singleton @ X )
% 3.05/1.51 => ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ X )
% 3.05/1.51 => ( ( setunion @ X )
% 3.05/1.51 = Xx ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(iftrue_type,type,
% 3.05/1.51 iftrue: $o ).
% 3.05/1.51
% 3.05/1.51 thf(iftrue,definition,
% 3.05/1.51 ( iftrue
% 3.05/1.51 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ! [Xy: $i] :
% 3.05/1.51 ( ( in @ Xy @ A )
% 3.05/1.51 => ( Xphi
% 3.05/1.51 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 3.05/1.51 = Xx ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(iffalse_type,type,
% 3.05/1.51 iffalse: $o ).
% 3.05/1.51
% 3.05/1.51 thf(iffalse,definition,
% 3.05/1.51 ( iffalse
% 3.05/1.51 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ! [Xy: $i] :
% 3.05/1.51 ( ( in @ Xy @ A )
% 3.05/1.51 => ( ~ Xphi
% 3.05/1.51 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 3.05/1.51 = Xy ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(iftrueorfalse_type,type,
% 3.05/1.51 iftrueorfalse: $o ).
% 3.05/1.51
% 3.05/1.51 thf(iftrueorfalse,definition,
% 3.05/1.51 ( iftrueorfalse
% 3.05/1.51 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ! [Xy: $i] :
% 3.05/1.51 ( ( in @ Xy @ A )
% 3.05/1.51 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(binintersectT_lem_type,type,
% 3.05/1.51 binintersectT_lem: $o ).
% 3.05/1.51
% 3.05/1.51 thf(binintersectT_lem,definition,
% 3.05/1.51 ( binintersectT_lem
% 3.05/1.51 = ( ! [A: $i,X: $i] :
% 3.05/1.51 ( ( in @ X @ ( powerset @ A ) )
% 3.05/1.51 => ! [Y: $i] :
% 3.05/1.51 ( ( in @ Y @ ( powerset @ A ) )
% 3.05/1.51 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(binunionT_lem_type,type,
% 3.05/1.51 binunionT_lem: $o ).
% 3.05/1.51
% 3.05/1.51 thf(binunionT_lem,definition,
% 3.05/1.51 ( binunionT_lem
% 3.05/1.51 = ( ! [A: $i,X: $i] :
% 3.05/1.51 ( ( in @ X @ ( powerset @ A ) )
% 3.05/1.51 => ! [Y: $i] :
% 3.05/1.51 ( ( in @ Y @ ( powerset @ A ) )
% 3.05/1.51 => ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(powersetT_lem_type,type,
% 3.05/1.51 powersetT_lem: $o ).
% 3.05/1.51
% 3.05/1.51 thf(powersetT_lem,definition,
% 3.05/1.51 ( powersetT_lem
% 3.05/1.51 = ( ! [A: $i,X: $i] :
% 3.05/1.51 ( ( in @ X @ ( powerset @ A ) )
% 3.05/1.51 => ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(setminusT_lem_type,type,
% 3.05/1.51 setminusT_lem: $o ).
% 3.05/1.51
% 3.05/1.51 thf(setminusT_lem,definition,
% 3.05/1.51 ( setminusT_lem
% 3.05/1.51 = ( ! [A: $i,X: $i] :
% 3.05/1.51 ( ( in @ X @ ( powerset @ A ) )
% 3.05/1.51 => ! [Y: $i] :
% 3.05/1.51 ( ( in @ Y @ ( powerset @ A ) )
% 3.05/1.51 => ( in @ ( setminus @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(complementT_lem_type,type,
% 3.05/1.51 complementT_lem: $o ).
% 3.05/1.51
% 3.05/1.51 thf(complementT_lem,definition,
% 3.05/1.51 ( complementT_lem
% 3.05/1.51 = ( ! [A: $i,X: $i] :
% 3.05/1.51 ( ( in @ X @ ( powerset @ A ) )
% 3.05/1.51 => ( in @ ( setminus @ A @ X ) @ ( powerset @ A ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(setextT_type,type,
% 3.05/1.51 setextT: $o ).
% 3.05/1.51
% 3.05/1.51 thf(setextT,definition,
% 3.05/1.51 ( setextT
% 3.05/1.51 = ( ! [A: $i,X: $i] :
% 3.05/1.51 ( ( in @ X @ ( powerset @ A ) )
% 3.05/1.51 => ! [Y: $i] :
% 3.05/1.51 ( ( in @ Y @ ( powerset @ A ) )
% 3.05/1.51 => ( ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ( ( in @ Xx @ X )
% 3.05/1.51 => ( in @ Xx @ Y ) ) )
% 3.05/1.51 => ( ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ( ( in @ Xx @ Y )
% 3.05/1.51 => ( in @ Xx @ X ) ) )
% 3.05/1.51 => ( X = Y ) ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(subsetTI_type,type,
% 3.05/1.51 subsetTI: $o ).
% 3.05/1.51
% 3.05/1.51 thf(subsetTI,definition,
% 3.05/1.51 ( subsetTI
% 3.05/1.51 = ( ! [A: $i,X: $i] :
% 3.05/1.51 ( ( in @ X @ ( powerset @ A ) )
% 3.05/1.51 => ! [Y: $i] :
% 3.05/1.51 ( ( in @ Y @ ( powerset @ A ) )
% 3.05/1.51 => ( ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ( ( in @ Xx @ X )
% 3.05/1.51 => ( in @ Xx @ Y ) ) )
% 3.05/1.51 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(powersetTI1_type,type,
% 3.05/1.51 powersetTI1: $o ).
% 3.05/1.51
% 3.05/1.51 thf(powersetTI1,definition,
% 3.05/1.51 ( powersetTI1
% 3.05/1.51 = ( ! [A: $i,X: $i] :
% 3.05/1.51 ( ( in @ X @ ( powerset @ A ) )
% 3.05/1.51 => ! [Y: $i] :
% 3.05/1.51 ( ( in @ Y @ ( powerset @ A ) )
% 3.05/1.51 => ( ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ( ( in @ Xx @ X )
% 3.05/1.51 => ( in @ Xx @ Y ) ) )
% 3.05/1.51 => ( in @ X @ ( powerset @ Y ) ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(powersetTE1_type,type,
% 3.05/1.51 powersetTE1: $o ).
% 3.05/1.51
% 3.05/1.51 thf(powersetTE1,definition,
% 3.05/1.51 ( powersetTE1
% 3.05/1.51 = ( ! [A: $i,X: $i] :
% 3.05/1.51 ( ( in @ X @ ( powerset @ A ) )
% 3.05/1.51 => ! [Y: $i] :
% 3.05/1.51 ( ( in @ Y @ ( powerset @ A ) )
% 3.05/1.51 => ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ( ( in @ X @ ( powerset @ Y ) )
% 3.05/1.51 => ( ( in @ Xx @ X )
% 3.05/1.51 => ( in @ Xx @ Y ) ) ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(complementTI1_type,type,
% 3.05/1.51 complementTI1: $o ).
% 3.05/1.51
% 3.05/1.51 thf(complementTI1,definition,
% 3.05/1.51 ( complementTI1
% 3.05/1.51 = ( ! [A: $i,X: $i] :
% 3.05/1.51 ( ( in @ X @ ( powerset @ A ) )
% 3.05/1.51 => ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ( ( in @ Xx @ X )
% 3.05/1.51 => ~ ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(complementTE1_type,type,
% 3.05/1.51 complementTE1: $o ).
% 3.05/1.51
% 3.05/1.51 thf(complementTE1,definition,
% 3.05/1.51 ( complementTE1
% 3.05/1.51 = ( ! [A: $i,X: $i] :
% 3.05/1.51 ( ( in @ X @ ( powerset @ A ) )
% 3.05/1.51 => ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ( ~ ( in @ Xx @ ( setminus @ A @ X ) )
% 3.05/1.51 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(binintersectTELcontra_type,type,
% 3.05/1.51 binintersectTELcontra: $o ).
% 3.05/1.51
% 3.05/1.51 thf(binintersectTELcontra,definition,
% 3.05/1.51 ( binintersectTELcontra
% 3.05/1.51 = ( ! [A: $i,X: $i] :
% 3.05/1.51 ( ( in @ X @ ( powerset @ A ) )
% 3.05/1.51 => ! [Y: $i] :
% 3.05/1.51 ( ( in @ Y @ ( powerset @ A ) )
% 3.05/1.51 => ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ( ~ ( in @ Xx @ X )
% 3.05/1.51 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(binintersectTERcontra_type,type,
% 3.05/1.51 binintersectTERcontra: $o ).
% 3.05/1.51
% 3.05/1.51 thf(binintersectTERcontra,definition,
% 3.05/1.51 ( binintersectTERcontra
% 3.05/1.51 = ( ! [A: $i,X: $i] :
% 3.05/1.51 ( ( in @ X @ ( powerset @ A ) )
% 3.05/1.51 => ! [Y: $i] :
% 3.05/1.51 ( ( in @ Y @ ( powerset @ A ) )
% 3.05/1.51 => ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ( ~ ( in @ Xx @ Y )
% 3.05/1.51 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(contrasubsetT_type,type,
% 3.05/1.51 contrasubsetT: $o ).
% 3.05/1.51
% 3.05/1.51 thf(contrasubsetT,definition,
% 3.05/1.51 ( contrasubsetT
% 3.05/1.51 = ( ! [A: $i,X: $i] :
% 3.05/1.51 ( ( in @ X @ ( powerset @ A ) )
% 3.05/1.51 => ! [Y: $i] :
% 3.05/1.51 ( ( in @ Y @ ( powerset @ A ) )
% 3.05/1.51 => ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 3.05/1.51 => ( ( in @ Xx @ Y )
% 3.05/1.51 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(contrasubsetT1_type,type,
% 3.05/1.51 contrasubsetT1: $o ).
% 3.05/1.51
% 3.05/1.51 thf(contrasubsetT1,definition,
% 3.05/1.51 ( contrasubsetT1
% 3.05/1.51 = ( ! [A: $i,X: $i] :
% 3.05/1.51 ( ( in @ X @ ( powerset @ A ) )
% 3.05/1.51 => ! [Y: $i] :
% 3.05/1.51 ( ( in @ Y @ ( powerset @ A ) )
% 3.05/1.51 => ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ( ( subset @ X @ Y )
% 3.05/1.51 => ( ~ ( in @ Xx @ Y )
% 3.05/1.51 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(contrasubsetT2_type,type,
% 3.05/1.51 contrasubsetT2: $o ).
% 3.05/1.51
% 3.05/1.51 thf(contrasubsetT2,definition,
% 3.05/1.51 ( contrasubsetT2
% 3.05/1.51 = ( ! [A: $i,X: $i] :
% 3.05/1.51 ( ( in @ X @ ( powerset @ A ) )
% 3.05/1.51 => ! [Y: $i] :
% 3.05/1.51 ( ( in @ Y @ ( powerset @ A ) )
% 3.05/1.51 => ( ( subset @ X @ Y )
% 3.05/1.51 => ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(contrasubsetT3_type,type,
% 3.05/1.51 contrasubsetT3: $o ).
% 3.05/1.51
% 3.05/1.51 thf(contrasubsetT3,definition,
% 3.05/1.51 ( contrasubsetT3
% 3.05/1.51 = ( ! [A: $i,X: $i] :
% 3.05/1.51 ( ( in @ X @ ( powerset @ A ) )
% 3.05/1.51 => ! [Y: $i] :
% 3.05/1.51 ( ( in @ Y @ ( powerset @ A ) )
% 3.05/1.51 => ( ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) )
% 3.05/1.51 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(doubleComplementI1_type,type,
% 3.05/1.51 doubleComplementI1: $o ).
% 3.05/1.51
% 3.05/1.51 thf(doubleComplementI1,definition,
% 3.05/1.51 ( doubleComplementI1
% 3.05/1.51 = ( ! [A: $i,X: $i] :
% 3.05/1.51 ( ( in @ X @ ( powerset @ A ) )
% 3.05/1.51 => ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ( ( in @ Xx @ X )
% 3.05/1.51 => ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(doubleComplementE1_type,type,
% 3.05/1.51 doubleComplementE1: $o ).
% 3.05/1.51
% 3.05/1.51 thf(doubleComplementE1,definition,
% 3.05/1.51 ( doubleComplementE1
% 3.05/1.51 = ( ! [A: $i,X: $i] :
% 3.05/1.51 ( ( in @ X @ ( powerset @ A ) )
% 3.05/1.51 => ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ( ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) )
% 3.05/1.51 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(doubleComplementSub1_type,type,
% 3.05/1.51 doubleComplementSub1: $o ).
% 3.05/1.51
% 3.05/1.51 thf(doubleComplementSub1,definition,
% 3.05/1.51 ( doubleComplementSub1
% 3.05/1.51 = ( ! [A: $i,X: $i] :
% 3.05/1.51 ( ( in @ X @ ( powerset @ A ) )
% 3.05/1.51 => ( subset @ X @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(doubleComplementSub2_type,type,
% 3.05/1.51 doubleComplementSub2: $o ).
% 3.05/1.51
% 3.05/1.51 thf(doubleComplementSub2,definition,
% 3.05/1.51 ( doubleComplementSub2
% 3.05/1.51 = ( ! [A: $i,X: $i] :
% 3.05/1.51 ( ( in @ X @ ( powerset @ A ) )
% 3.05/1.51 => ( subset @ ( setminus @ A @ ( setminus @ A @ X ) ) @ X ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(doubleComplementEq_type,type,
% 3.05/1.51 doubleComplementEq: $o ).
% 3.05/1.51
% 3.05/1.51 thf(doubleComplementEq,definition,
% 3.05/1.51 ( doubleComplementEq
% 3.05/1.51 = ( ! [A: $i,X: $i] :
% 3.05/1.51 ( ( in @ X @ ( powerset @ A ) )
% 3.05/1.51 => ( X
% 3.05/1.51 = ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(complementTnotintersectT_type,type,
% 3.05/1.51 complementTnotintersectT: $o ).
% 3.05/1.51
% 3.05/1.51 thf(complementTnotintersectT,definition,
% 3.05/1.51 ( complementTnotintersectT
% 3.05/1.51 = ( ! [A: $i,X: $i] :
% 3.05/1.51 ( ( in @ X @ ( powerset @ A ) )
% 3.05/1.51 => ! [Y: $i] :
% 3.05/1.51 ( ( in @ Y @ ( powerset @ A ) )
% 3.05/1.51 => ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 3.05/1.51 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(complementImpComplementIntersect_type,type,
% 3.05/1.51 complementImpComplementIntersect: $o ).
% 3.05/1.51
% 3.05/1.51 thf(complementImpComplementIntersect,definition,
% 3.05/1.51 ( complementImpComplementIntersect
% 3.05/1.51 = ( ! [A: $i,X: $i] :
% 3.05/1.51 ( ( in @ X @ ( powerset @ A ) )
% 3.05/1.51 => ! [Y: $i] :
% 3.05/1.51 ( ( in @ Y @ ( powerset @ A ) )
% 3.05/1.51 => ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 3.05/1.51 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(complementSubsetComplementIntersect_type,type,
% 3.05/1.51 complementSubsetComplementIntersect: $o ).
% 3.05/1.51
% 3.05/1.51 thf(complementSubsetComplementIntersect,definition,
% 3.05/1.51 ( complementSubsetComplementIntersect
% 3.05/1.51 = ( ! [A: $i,X: $i] :
% 3.05/1.51 ( ( in @ X @ ( powerset @ A ) )
% 3.05/1.51 => ! [Y: $i] :
% 3.05/1.51 ( ( in @ Y @ ( powerset @ A ) )
% 3.05/1.51 => ( subset @ ( setminus @ A @ X ) @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(complementInPowersetComplementIntersect_type,type,
% 3.05/1.51 complementInPowersetComplementIntersect: $o ).
% 3.05/1.51
% 3.05/1.51 thf(complementInPowersetComplementIntersect,definition,
% 3.05/1.51 ( complementInPowersetComplementIntersect
% 3.05/1.51 = ( ! [A: $i,X: $i] :
% 3.05/1.51 ( ( in @ X @ ( powerset @ A ) )
% 3.05/1.51 => ! [Y: $i] :
% 3.05/1.51 ( ( in @ Y @ ( powerset @ A ) )
% 3.05/1.51 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(contraSubsetComplement_type,type,
% 3.05/1.51 contraSubsetComplement: $o ).
% 3.05/1.51
% 3.05/1.51 thf(contraSubsetComplement,definition,
% 3.05/1.51 ( contraSubsetComplement
% 3.05/1.51 = ( ! [A: $i,X: $i] :
% 3.05/1.51 ( ( in @ X @ ( powerset @ A ) )
% 3.05/1.51 => ! [Y: $i] :
% 3.05/1.51 ( ( in @ Y @ ( powerset @ A ) )
% 3.05/1.51 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 3.05/1.51 => ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ( ( in @ Xx @ Y )
% 3.05/1.51 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(complementTcontraSubset_type,type,
% 3.05/1.51 complementTcontraSubset: $o ).
% 3.05/1.51
% 3.05/1.51 thf(complementTcontraSubset,definition,
% 3.05/1.51 ( complementTcontraSubset
% 3.05/1.51 = ( ! [A: $i,X: $i] :
% 3.05/1.51 ( ( in @ X @ ( powerset @ A ) )
% 3.05/1.51 => ! [Y: $i] :
% 3.05/1.51 ( ( in @ Y @ ( powerset @ A ) )
% 3.05/1.51 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 3.05/1.51 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(binunionTILcontra_type,type,
% 3.05/1.51 binunionTILcontra: $o ).
% 3.05/1.51
% 3.05/1.51 thf(binunionTILcontra,definition,
% 3.05/1.51 ( binunionTILcontra
% 3.05/1.51 = ( ! [A: $i,X: $i] :
% 3.05/1.51 ( ( in @ X @ ( powerset @ A ) )
% 3.05/1.51 => ! [Y: $i] :
% 3.05/1.51 ( ( in @ Y @ ( powerset @ A ) )
% 3.05/1.51 => ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 3.05/1.51 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(binunionTIRcontra_type,type,
% 3.05/1.51 binunionTIRcontra: $o ).
% 3.05/1.51
% 3.05/1.51 thf(binunionTIRcontra,definition,
% 3.05/1.51 ( binunionTIRcontra
% 3.05/1.51 = ( ! [A: $i,X: $i] :
% 3.05/1.51 ( ( in @ X @ ( powerset @ A ) )
% 3.05/1.51 => ! [Y: $i] :
% 3.05/1.51 ( ( in @ Y @ ( powerset @ A ) )
% 3.05/1.51 => ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 3.05/1.51 => ~ ( in @ Xx @ Y ) ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(inIntersectImpInUnion_type,type,
% 3.05/1.51 inIntersectImpInUnion: $o ).
% 3.05/1.51
% 3.05/1.51 thf(inIntersectImpInUnion,definition,
% 3.05/1.51 ( inIntersectImpInUnion
% 3.05/1.51 = ( ! [A: $i,X: $i] :
% 3.05/1.51 ( ( in @ X @ ( powerset @ A ) )
% 3.05/1.51 => ! [Y: $i] :
% 3.05/1.51 ( ( in @ Y @ ( powerset @ A ) )
% 3.05/1.51 => ! [Z: $i] :
% 3.05/1.51 ( ( in @ Z @ ( powerset @ A ) )
% 3.05/1.51 => ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 3.05/1.51 => ( in @ Xx @ ( binunion @ X @ Z ) ) ) ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(inIntersectImpInUnion2_type,type,
% 3.05/1.51 inIntersectImpInUnion2: $o ).
% 3.05/1.51
% 3.05/1.51 thf(inIntersectImpInUnion2,definition,
% 3.05/1.51 ( inIntersectImpInUnion2
% 3.05/1.51 = ( ! [A: $i,X: $i] :
% 3.05/1.51 ( ( in @ X @ ( powerset @ A ) )
% 3.05/1.51 => ! [Y: $i] :
% 3.05/1.51 ( ( in @ Y @ ( powerset @ A ) )
% 3.05/1.51 => ! [Z: $i] :
% 3.05/1.51 ( ( in @ Z @ ( powerset @ A ) )
% 3.05/1.51 => ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 3.05/1.51 => ( in @ Xx @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(inIntersectImpInIntersectUnions_type,type,
% 3.05/1.51 inIntersectImpInIntersectUnions: $o ).
% 3.05/1.51
% 3.05/1.51 thf(inIntersectImpInIntersectUnions,definition,
% 3.05/1.51 ( inIntersectImpInIntersectUnions
% 3.05/1.51 = ( ! [A: $i,X: $i] :
% 3.05/1.51 ( ( in @ X @ ( powerset @ A ) )
% 3.05/1.51 => ! [Y: $i] :
% 3.05/1.51 ( ( in @ Y @ ( powerset @ A ) )
% 3.05/1.51 => ! [Z: $i] :
% 3.05/1.51 ( ( in @ Z @ ( powerset @ A ) )
% 3.05/1.51 => ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 3.05/1.51 => ( in @ Xx @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(intersectInPowersetIntersectUnions_type,type,
% 3.05/1.51 intersectInPowersetIntersectUnions: $o ).
% 3.05/1.51
% 3.05/1.51 thf(intersectInPowersetIntersectUnions,definition,
% 3.05/1.51 ( intersectInPowersetIntersectUnions
% 3.05/1.51 = ( ! [A: $i,X: $i] :
% 3.05/1.51 ( ( in @ X @ ( powerset @ A ) )
% 3.05/1.51 => ! [Y: $i] :
% 3.05/1.51 ( ( in @ Y @ ( powerset @ A ) )
% 3.05/1.51 => ! [Z: $i] :
% 3.05/1.51 ( ( in @ Z @ ( powerset @ A ) )
% 3.05/1.51 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(inComplementUnionImpNotIn1_type,type,
% 3.05/1.51 inComplementUnionImpNotIn1: $o ).
% 3.05/1.51
% 3.05/1.51 thf(inComplementUnionImpNotIn1,definition,
% 3.05/1.51 ( inComplementUnionImpNotIn1
% 3.05/1.51 = ( ! [A: $i,X: $i] :
% 3.05/1.51 ( ( in @ X @ ( powerset @ A ) )
% 3.05/1.51 => ! [Y: $i] :
% 3.05/1.51 ( ( in @ Y @ ( powerset @ A ) )
% 3.05/1.51 => ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 3.05/1.51 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(inComplementUnionImpInComplement1_type,type,
% 3.05/1.51 inComplementUnionImpInComplement1: $o ).
% 3.05/1.51
% 3.05/1.51 thf(inComplementUnionImpInComplement1,definition,
% 3.05/1.51 ( inComplementUnionImpInComplement1
% 3.05/1.51 = ( ! [A: $i,X: $i] :
% 3.05/1.51 ( ( in @ X @ ( powerset @ A ) )
% 3.05/1.51 => ! [Y: $i] :
% 3.05/1.51 ( ( in @ Y @ ( powerset @ A ) )
% 3.05/1.51 => ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 3.05/1.51 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(binunionTE_type,type,
% 3.05/1.51 binunionTE: $o ).
% 3.05/1.51
% 3.05/1.51 thf(binunionTE,definition,
% 3.05/1.51 ( binunionTE
% 3.05/1.51 = ( ! [A: $i,X: $i] :
% 3.05/1.51 ( ( in @ X @ ( powerset @ A ) )
% 3.05/1.51 => ! [Y: $i] :
% 3.05/1.51 ( ( in @ Y @ ( powerset @ A ) )
% 3.05/1.51 => ! [Xphi: $o,Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ( ( in @ Xx @ ( binunion @ X @ Y ) )
% 3.05/1.51 => ( ( ( in @ Xx @ X )
% 3.05/1.51 => Xphi )
% 3.05/1.51 => ( ( ( in @ Xx @ Y )
% 3.05/1.51 => Xphi )
% 3.05/1.51 => Xphi ) ) ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(binunionTEcontra_type,type,
% 3.05/1.51 binunionTEcontra: $o ).
% 3.05/1.51
% 3.05/1.51 thf(binunionTEcontra,definition,
% 3.05/1.51 ( binunionTEcontra
% 3.05/1.51 = ( ! [A: $i,X: $i] :
% 3.05/1.51 ( ( in @ X @ ( powerset @ A ) )
% 3.05/1.51 => ! [Y: $i] :
% 3.05/1.51 ( ( in @ Y @ ( powerset @ A ) )
% 3.05/1.51 => ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ( ~ ( in @ Xx @ X )
% 3.05/1.51 => ( ~ ( in @ Xx @ Y )
% 3.05/1.51 => ~ ( in @ Xx @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(demorgan2a1_type,type,
% 3.05/1.51 demorgan2a1: $o ).
% 3.05/1.51
% 3.05/1.51 thf(demorgan2a1,definition,
% 3.05/1.51 ( demorgan2a1
% 3.05/1.51 = ( ! [A: $i,X: $i] :
% 3.05/1.51 ( ( in @ X @ ( powerset @ A ) )
% 3.05/1.51 => ! [Y: $i] :
% 3.05/1.51 ( ( in @ Y @ ( powerset @ A ) )
% 3.05/1.51 => ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 3.05/1.51 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(complementUnionInPowersetComplement_type,type,
% 3.05/1.51 complementUnionInPowersetComplement: $o ).
% 3.05/1.51
% 3.05/1.51 thf(complementUnionInPowersetComplement,definition,
% 3.05/1.51 ( complementUnionInPowersetComplement
% 3.05/1.51 = ( ! [A: $i,X: $i] :
% 3.05/1.51 ( ( in @ X @ ( powerset @ A ) )
% 3.05/1.51 => ! [Y: $i] :
% 3.05/1.51 ( ( in @ Y @ ( powerset @ A ) )
% 3.05/1.51 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(demorgan2a2_type,type,
% 3.05/1.51 demorgan2a2: $o ).
% 3.05/1.51
% 3.05/1.51 thf(demorgan2a2,definition,
% 3.05/1.51 ( demorgan2a2
% 3.05/1.51 = ( ! [A: $i,X: $i] :
% 3.05/1.51 ( ( in @ X @ ( powerset @ A ) )
% 3.05/1.51 => ! [Y: $i] :
% 3.05/1.51 ( ( in @ Y @ ( powerset @ A ) )
% 3.05/1.51 => ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 3.05/1.51 => ( in @ Xx @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(demorgan1a_type,type,
% 3.05/1.51 demorgan1a: $o ).
% 3.05/1.51
% 3.05/1.51 thf(demorgan1a,definition,
% 3.05/1.51 ( demorgan1a
% 3.05/1.51 = ( ! [A: $i,X: $i] :
% 3.05/1.51 ( ( in @ X @ ( powerset @ A ) )
% 3.05/1.51 => ! [Y: $i] :
% 3.05/1.51 ( ( in @ Y @ ( powerset @ A ) )
% 3.05/1.51 => ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ( ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) )
% 3.05/1.51 => ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(demorgan1b_type,type,
% 3.05/1.51 demorgan1b: $o ).
% 3.05/1.51
% 3.05/1.51 thf(demorgan1b,definition,
% 3.05/1.51 ( demorgan1b
% 3.05/1.51 = ( ! [A: $i,X: $i] :
% 3.05/1.51 ( ( in @ X @ ( powerset @ A ) )
% 3.05/1.51 => ! [Y: $i] :
% 3.05/1.51 ( ( in @ Y @ ( powerset @ A ) )
% 3.05/1.51 => ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ( ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 3.05/1.51 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(demorgan1_type,type,
% 3.05/1.51 demorgan1: $o ).
% 3.05/1.51
% 3.05/1.51 thf(demorgan1,definition,
% 3.05/1.51 ( demorgan1
% 3.05/1.51 = ( ! [A: $i,X: $i] :
% 3.05/1.51 ( ( in @ X @ ( powerset @ A ) )
% 3.05/1.51 => ! [Y: $i] :
% 3.05/1.51 ( ( in @ Y @ ( powerset @ A ) )
% 3.05/1.51 => ( ( setminus @ A @ ( binintersect @ X @ Y ) )
% 3.05/1.51 = ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(demorgan2a_type,type,
% 3.05/1.51 demorgan2a: $o ).
% 3.05/1.51
% 3.05/1.51 thf(demorgan2a,definition,
% 3.05/1.51 ( demorgan2a
% 3.05/1.51 = ( ! [A: $i,X: $i] :
% 3.05/1.51 ( ( in @ X @ ( powerset @ A ) )
% 3.05/1.51 => ! [Y: $i] :
% 3.05/1.51 ( ( in @ Y @ ( powerset @ A ) )
% 3.05/1.51 => ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 3.05/1.51 => ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(demorgan2b2_type,type,
% 3.05/1.51 demorgan2b2: $o ).
% 3.05/1.51
% 3.05/1.51 thf(demorgan2b2,definition,
% 3.05/1.51 ( demorgan2b2
% 3.05/1.51 = ( ! [A: $i,X: $i] :
% 3.05/1.51 ( ( in @ X @ ( powerset @ A ) )
% 3.05/1.51 => ! [Y: $i] :
% 3.05/1.51 ( ( in @ Y @ ( powerset @ A ) )
% 3.05/1.51 => ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 3.05/1.51 => ( ( in @ Xx @ ( setminus @ A @ Y ) )
% 3.05/1.51 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(demorgan2b_type,type,
% 3.05/1.51 demorgan2b: $o ).
% 3.05/1.51
% 3.05/1.51 thf(demorgan2b,definition,
% 3.05/1.51 ( demorgan2b
% 3.05/1.51 = ( ! [A: $i,X: $i] :
% 3.05/1.51 ( ( in @ X @ ( powerset @ A ) )
% 3.05/1.51 => ! [Y: $i] :
% 3.05/1.51 ( ( in @ Y @ ( powerset @ A ) )
% 3.05/1.51 => ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ( ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 3.05/1.51 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(demorgan2_type,type,
% 3.05/1.51 demorgan2: $o ).
% 3.05/1.51
% 3.05/1.51 thf(demorgan2,definition,
% 3.05/1.51 ( demorgan2
% 3.05/1.51 = ( ! [A: $i,X: $i] :
% 3.05/1.51 ( ( in @ X @ ( powerset @ A ) )
% 3.05/1.51 => ! [Y: $i] :
% 3.05/1.51 ( ( in @ Y @ ( powerset @ A ) )
% 3.05/1.51 => ( ( setminus @ A @ ( binunion @ X @ Y ) )
% 3.05/1.51 = ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(woz13rule0_type,type,
% 3.05/1.51 woz13rule0: $o ).
% 3.05/1.51
% 3.05/1.51 thf(woz13rule0,definition,
% 3.05/1.51 ( woz13rule0
% 3.05/1.51 = ( ! [A: $i,X: $i] :
% 3.05/1.51 ( ( in @ X @ ( powerset @ A ) )
% 3.05/1.51 => ! [Y: $i] :
% 3.05/1.51 ( ( in @ Y @ ( powerset @ A ) )
% 3.05/1.51 => ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 3.05/1.51 => ( in @ Xx @ A ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(woz13rule1_type,type,
% 3.05/1.51 woz13rule1: $o ).
% 3.05/1.51
% 3.05/1.51 thf(woz13rule1,definition,
% 3.05/1.51 ( woz13rule1
% 3.05/1.51 = ( ! [A: $i,X: $i] :
% 3.05/1.51 ( ( in @ X @ ( powerset @ A ) )
% 3.05/1.51 => ! [Y: $i] :
% 3.05/1.51 ( ( in @ Y @ ( powerset @ A ) )
% 3.05/1.51 => ! [Z: $i] :
% 3.05/1.51 ( ( in @ Z @ ( powerset @ A ) )
% 3.05/1.51 => ( ( subset @ X @ Z )
% 3.05/1.51 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(woz13rule2_type,type,
% 3.05/1.51 woz13rule2: $o ).
% 3.05/1.51
% 3.05/1.51 thf(woz13rule2,definition,
% 3.05/1.51 ( woz13rule2
% 3.05/1.51 = ( ! [A: $i,X: $i] :
% 3.05/1.51 ( ( in @ X @ ( powerset @ A ) )
% 3.05/1.51 => ! [Y: $i] :
% 3.05/1.51 ( ( in @ Y @ ( powerset @ A ) )
% 3.05/1.51 => ! [Z: $i] :
% 3.05/1.51 ( ( in @ Z @ ( powerset @ A ) )
% 3.05/1.51 => ( ( subset @ Y @ Z )
% 3.05/1.51 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(woz13rule3_type,type,
% 3.05/1.51 woz13rule3: $o ).
% 3.05/1.51
% 3.05/1.51 thf(woz13rule3,definition,
% 3.05/1.51 ( woz13rule3
% 3.05/1.51 = ( ! [A: $i,X: $i] :
% 3.05/1.51 ( ( in @ X @ ( powerset @ A ) )
% 3.05/1.51 => ! [Y: $i] :
% 3.05/1.51 ( ( in @ Y @ ( powerset @ A ) )
% 3.05/1.51 => ! [Z: $i] :
% 3.05/1.51 ( ( in @ Z @ ( powerset @ A ) )
% 3.05/1.51 => ( ( subset @ X @ Y )
% 3.05/1.51 => ( ( subset @ X @ Z )
% 3.05/1.51 => ( subset @ X @ ( binintersect @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(woz13rule4_type,type,
% 3.05/1.51 woz13rule4: $o ).
% 3.05/1.51
% 3.05/1.51 thf(woz13rule4,definition,
% 3.05/1.51 ( woz13rule4
% 3.05/1.51 = ( ! [A: $i,X: $i] :
% 3.05/1.51 ( ( in @ X @ ( powerset @ A ) )
% 3.05/1.51 => ! [Y: $i] :
% 3.05/1.51 ( ( in @ Y @ ( powerset @ A ) )
% 3.05/1.51 => ! [Z: $i] :
% 3.05/1.51 ( ( in @ Z @ ( powerset @ A ) )
% 3.05/1.51 => ! [W: $i] :
% 3.05/1.51 ( ( in @ W @ ( powerset @ A ) )
% 3.05/1.51 => ( ( subset @ X @ Z )
% 3.05/1.51 => ( ( subset @ Y @ W )
% 3.05/1.51 => ( subset @ ( binintersect @ X @ Y ) @ ( binintersect @ Z @ W ) ) ) ) ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(woz1_1_type,type,
% 3.05/1.51 woz1_1: $o ).
% 3.05/1.51
% 3.05/1.51 thf(woz1_1,definition,
% 3.05/1.51 ( woz1_1
% 3.05/1.51 = ( ! [A: $i,X: $i] :
% 3.05/1.51 ( ( in @ X @ ( powerset @ A ) )
% 3.05/1.51 => ! [Y: $i] :
% 3.05/1.51 ( ( in @ Y @ ( powerset @ A ) )
% 3.05/1.51 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(woz1_2_type,type,
% 3.05/1.51 woz1_2: $o ).
% 3.05/1.51
% 3.05/1.51 thf(woz1_2,definition,
% 3.05/1.51 ( woz1_2
% 3.05/1.51 = ( ! [A: $i,X: $i] :
% 3.05/1.51 ( ( in @ X @ ( powerset @ A ) )
% 3.05/1.51 => ! [Y: $i] :
% 3.05/1.51 ( ( in @ Y @ ( powerset @ A ) )
% 3.05/1.51 => ! [Z: $i] :
% 3.05/1.51 ( ( in @ Z @ ( powerset @ A ) )
% 3.05/1.51 => ! [W: $i] :
% 3.05/1.51 ( ( in @ W @ ( powerset @ A ) )
% 3.05/1.51 => ( ( setminus @ A @ ( binintersect @ ( binunion @ X @ Y ) @ ( binunion @ Z @ W ) ) )
% 3.05/1.51 = ( binunion @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) @ ( binintersect @ ( setminus @ A @ Z ) @ ( setminus @ A @ W ) ) ) ) ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(woz1_3_type,type,
% 3.05/1.51 woz1_3: $o ).
% 3.05/1.51
% 3.05/1.51 thf(woz1_3,definition,
% 3.05/1.51 ( woz1_3
% 3.05/1.51 = ( ! [A: $i,X: $i] :
% 3.05/1.51 ( ( in @ X @ ( powerset @ A ) )
% 3.05/1.51 => ! [Y: $i] :
% 3.05/1.51 ( ( in @ Y @ ( powerset @ A ) )
% 3.05/1.51 => ! [Z: $i] :
% 3.05/1.51 ( ( in @ Z @ ( powerset @ A ) )
% 3.05/1.51 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(woz1_4_type,type,
% 3.05/1.51 woz1_4: $o ).
% 3.05/1.51
% 3.05/1.51 thf(woz1_4,definition,
% 3.05/1.51 ( woz1_4
% 3.05/1.51 = ( ! [A: $i,X: $i] :
% 3.05/1.51 ( ( in @ X @ ( powerset @ A ) )
% 3.05/1.51 => ! [Y: $i] :
% 3.05/1.51 ( ( in @ Y @ ( powerset @ A ) )
% 3.05/1.51 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 3.05/1.51 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(woz1_5_type,type,
% 3.05/1.51 woz1_5: $o ).
% 3.05/1.51
% 3.05/1.51 thf(woz1_5,definition,
% 3.05/1.51 ( woz1_5
% 3.05/1.51 = ( ! [A: $i,X: $i] :
% 3.05/1.51 ( ( in @ X @ ( powerset @ A ) )
% 3.05/1.51 => ! [Y: $i] :
% 3.05/1.51 ( ( in @ Y @ ( powerset @ A ) )
% 3.05/1.51 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(breln1_type,type,
% 3.05/1.51 breln1: $i > $i > $o ).
% 3.05/1.51
% 3.05/1.51 thf(breln1all2_type,type,
% 3.05/1.51 breln1all2: $o ).
% 3.05/1.51
% 3.05/1.51 thf(breln1all2,definition,
% 3.05/1.51 ( breln1all2
% 3.05/1.51 = ( ! [A: $i,R: $i] :
% 3.05/1.51 ( ( breln1 @ A @ R )
% 3.05/1.51 => ! [Xphi: $i > $o] :
% 3.05/1.51 ( ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ! [Xy: $i] :
% 3.05/1.51 ( ( in @ Xy @ A )
% 3.05/1.51 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.05/1.51 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 3.05/1.51 => ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ R )
% 3.05/1.51 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(breln1Set_type,type,
% 3.05/1.51 breln1Set: $i > $i ).
% 3.05/1.51
% 3.05/1.51 thf(breln1SetBreln1_type,type,
% 3.05/1.51 breln1SetBreln1: $o ).
% 3.05/1.51
% 3.05/1.51 thf(breln1SetBreln1,definition,
% 3.05/1.51 ( breln1SetBreln1
% 3.05/1.51 = ( ! [A: $i,R: $i] :
% 3.05/1.51 ( ( in @ R @ ( breln1Set @ A ) )
% 3.05/1.51 => ( breln1 @ A @ R ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(transitive_type,type,
% 3.05/1.51 transitive: $i > $i > $o ).
% 3.05/1.51
% 3.05/1.51 thf(antisymmetric_type,type,
% 3.05/1.51 antisymmetric: $i > $i > $o ).
% 3.05/1.51
% 3.05/1.51 thf(reflexive_type,type,
% 3.05/1.51 reflexive: $i > $i > $o ).
% 3.05/1.51
% 3.05/1.51 thf(refltransitive_type,type,
% 3.05/1.51 refltransitive: $i > $i > $o ).
% 3.05/1.51
% 3.05/1.51 thf(refllinearorder_type,type,
% 3.05/1.51 refllinearorder: $i > $i > $o ).
% 3.05/1.51
% 3.05/1.51 thf(reflwellordering_type,type,
% 3.05/1.51 reflwellordering: $i > $i > $o ).
% 3.05/1.51
% 3.05/1.51 thf(choice2fnsingleton_type,type,
% 3.05/1.51 choice2fnsingleton: $o ).
% 3.05/1.51
% 3.05/1.51 thf(choice2fnsingleton,definition,
% 3.05/1.51 ( choice2fnsingleton
% 3.05/1.51 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 3.05/1.51 ( ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ? [Xy: $i] :
% 3.05/1.51 ( ( in @ Xy @ B )
% 3.05/1.51 & ( Xphi @ Xx @ Xy ) ) )
% 3.05/1.51 => ! [R: $i] :
% 3.05/1.51 ( ( in @ R @ ( breln1Set @ B ) )
% 3.05/1.51 => ( ( reflwellordering @ B @ R )
% 3.05/1.51 => ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ( singleton
% 3.05/1.51 @ ( dsetconstr @ B
% 3.05/1.51 @ ^ [Xy: $i] :
% 3.05/1.51 ( ( Xphi @ Xx @ Xy )
% 3.05/1.51 & ! [Xz: $i] :
% 3.05/1.51 ( ( in @ Xz @ B )
% 3.05/1.51 => ( ( Xphi @ Xx @ Xz )
% 3.05/1.51 => ( in @ ( kpair @ Xy @ Xz ) @ R ) ) ) ) ) ) ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(setOfPairsIsBReln1_type,type,
% 3.05/1.51 setOfPairsIsBReln1: $o ).
% 3.05/1.51
% 3.05/1.51 thf(setOfPairsIsBReln1,definition,
% 3.05/1.51 ( setOfPairsIsBReln1
% 3.05/1.51 = ( ! [A: $i,Xphi: $i > $i > $o] :
% 3.05/1.51 ( breln1 @ A
% 3.05/1.51 @ ( dpsetconstr @ A @ A
% 3.05/1.51 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(breln1all1_type,type,
% 3.05/1.51 breln1all1: $o ).
% 3.05/1.51
% 3.05/1.51 thf(breln1all1,definition,
% 3.05/1.51 ( breln1all1
% 3.05/1.51 = ( ! [A: $i,R: $i] :
% 3.05/1.51 ( ( breln1 @ A @ R )
% 3.05/1.51 => ! [Xphi: $i > $o] :
% 3.05/1.51 ( ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ! [Xy: $i] :
% 3.05/1.51 ( ( in @ Xy @ A )
% 3.05/1.51 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.05/1.51 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 3.05/1.51 => ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ R )
% 3.05/1.51 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(subbreln1_type,type,
% 3.05/1.51 subbreln1: $o ).
% 3.05/1.51
% 3.05/1.51 thf(subbreln1,definition,
% 3.05/1.51 ( subbreln1
% 3.05/1.51 = ( ! [A: $i,R: $i] :
% 3.05/1.51 ( ( breln1 @ A @ R )
% 3.05/1.51 => ! [S: $i] :
% 3.05/1.51 ( ( breln1 @ A @ S )
% 3.05/1.51 => ( ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ! [Xy: $i] :
% 3.05/1.51 ( ( in @ Xy @ A )
% 3.05/1.51 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.05/1.51 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 3.05/1.51 => ( subset @ R @ S ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(eqbreln1_type,type,
% 3.05/1.51 eqbreln1: $o ).
% 3.05/1.51
% 3.05/1.51 thf(eqbreln1,definition,
% 3.05/1.51 ( eqbreln1
% 3.05/1.51 = ( ! [A: $i,R: $i] :
% 3.05/1.51 ( ( breln1 @ A @ R )
% 3.05/1.51 => ! [S: $i] :
% 3.05/1.51 ( ( breln1 @ A @ S )
% 3.05/1.51 => ( ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ! [Xy: $i] :
% 3.05/1.51 ( ( in @ Xy @ A )
% 3.05/1.51 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.05/1.51 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 3.05/1.51 => ( ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ! [Xy: $i] :
% 3.05/1.51 ( ( in @ Xy @ A )
% 3.05/1.51 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 3.05/1.51 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 3.05/1.51 => ( R = S ) ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(breln1invset_type,type,
% 3.05/1.51 breln1invset: $i > $i > $i ).
% 3.05/1.51
% 3.05/1.51 thf(breln1invprop_type,type,
% 3.05/1.51 breln1invprop: $o ).
% 3.05/1.51
% 3.05/1.51 thf(breln1invprop,definition,
% 3.05/1.51 ( breln1invprop
% 3.05/1.51 = ( ! [A: $i,R: $i] :
% 3.05/1.51 ( ( breln1 @ A @ R )
% 3.05/1.51 => ( breln1 @ A @ ( breln1invset @ A @ R ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(breln1invI_type,type,
% 3.05/1.51 breln1invI: $o ).
% 3.05/1.51
% 3.05/1.51 thf(breln1invI,definition,
% 3.05/1.51 ( breln1invI
% 3.05/1.51 = ( ! [A: $i,R: $i] :
% 3.05/1.51 ( ( breln1 @ A @ R )
% 3.05/1.51 => ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ! [Xy: $i] :
% 3.05/1.51 ( ( in @ Xy @ A )
% 3.05/1.51 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.05/1.51 => ( in @ ( kpair @ Xy @ Xx ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(breln1invE_type,type,
% 3.05/1.51 breln1invE: $o ).
% 3.05/1.51
% 3.05/1.51 thf(breln1invE,definition,
% 3.05/1.51 ( breln1invE
% 3.05/1.51 = ( ! [A: $i,R: $i] :
% 3.05/1.51 ( ( breln1 @ A @ R )
% 3.05/1.51 => ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ! [Xy: $i] :
% 3.05/1.51 ( ( in @ Xy @ A )
% 3.05/1.51 => ( ( in @ ( kpair @ Xy @ Xx ) @ ( breln1invset @ A @ R ) )
% 3.05/1.51 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(breln1compset_type,type,
% 3.05/1.51 breln1compset: $i > $i > $i > $i ).
% 3.05/1.51
% 3.05/1.51 thf(breln1compprop_type,type,
% 3.05/1.51 breln1compprop: $o ).
% 3.05/1.51
% 3.05/1.51 thf(breln1compprop,definition,
% 3.05/1.51 ( breln1compprop
% 3.05/1.51 = ( ! [A: $i,R: $i] :
% 3.05/1.51 ( ( breln1 @ A @ R )
% 3.05/1.51 => ! [S: $i] :
% 3.05/1.51 ( ( breln1 @ A @ S )
% 3.05/1.51 => ( breln1 @ A @ ( breln1compset @ A @ R @ S ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(breln1compI_type,type,
% 3.05/1.51 breln1compI: $o ).
% 3.05/1.51
% 3.05/1.51 thf(breln1compI,definition,
% 3.05/1.51 ( breln1compI
% 3.05/1.51 = ( ! [A: $i,R: $i] :
% 3.05/1.51 ( ( breln1 @ A @ R )
% 3.05/1.51 => ! [S: $i] :
% 3.05/1.51 ( ( breln1 @ A @ S )
% 3.05/1.51 => ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ! [Xy: $i] :
% 3.05/1.51 ( ( in @ Xy @ A )
% 3.05/1.51 => ! [Xz: $i] :
% 3.05/1.51 ( ( in @ Xz @ A )
% 3.05/1.51 => ( ( in @ ( kpair @ Xx @ Xz ) @ R )
% 3.05/1.51 => ( ( in @ ( kpair @ Xz @ Xy ) @ S )
% 3.05/1.51 => ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) ) ) ) ) ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(breln1compE_type,type,
% 3.05/1.51 breln1compE: $o ).
% 3.05/1.51
% 3.05/1.51 thf(breln1compE,definition,
% 3.05/1.51 ( breln1compE
% 3.05/1.51 = ( ! [A: $i,R: $i] :
% 3.05/1.51 ( ( breln1 @ A @ R )
% 3.05/1.51 => ! [S: $i] :
% 3.05/1.51 ( ( breln1 @ A @ S )
% 3.05/1.51 => ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ! [Xy: $i] :
% 3.05/1.51 ( ( in @ Xy @ A )
% 3.05/1.51 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) )
% 3.05/1.51 => ? [Xz: $i] :
% 3.05/1.51 ( ( in @ Xz @ A )
% 3.05/1.51 & ( in @ ( kpair @ Xx @ Xz ) @ R )
% 3.05/1.51 & ( in @ ( kpair @ Xz @ Xy ) @ S ) ) ) ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(breln1compEex_type,type,
% 3.05/1.51 breln1compEex: $o ).
% 3.05/1.51
% 3.05/1.51 thf(breln1compEex,definition,
% 3.05/1.51 ( breln1compEex
% 3.05/1.51 = ( ! [A: $i,R: $i] :
% 3.05/1.51 ( ( breln1 @ A @ R )
% 3.05/1.51 => ! [S: $i] :
% 3.05/1.51 ( ( breln1 @ A @ S )
% 3.05/1.51 => ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ! [Xy: $i] :
% 3.05/1.51 ( ( in @ Xy @ A )
% 3.05/1.51 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) )
% 3.05/1.51 => ! [Xphi: $o] :
% 3.05/1.51 ( ! [Xz: $i] :
% 3.05/1.51 ( ( in @ Xz @ A )
% 3.05/1.51 => ( ( in @ ( kpair @ Xx @ Xz ) @ R )
% 3.05/1.51 => ( ( in @ ( kpair @ Xz @ Xy ) @ S )
% 3.05/1.51 => Xphi ) ) )
% 3.05/1.51 => Xphi ) ) ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(breln1unionprop_type,type,
% 3.05/1.51 breln1unionprop: $o ).
% 3.05/1.51
% 3.05/1.51 thf(breln1unionprop,definition,
% 3.05/1.51 ( breln1unionprop
% 3.05/1.51 = ( ! [A: $i,R: $i] :
% 3.05/1.51 ( ( breln1 @ A @ R )
% 3.05/1.51 => ! [S: $i] :
% 3.05/1.51 ( ( breln1 @ A @ S )
% 3.05/1.51 => ( breln1 @ A @ ( binunion @ R @ S ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(breln1unionIL_type,type,
% 3.05/1.51 breln1unionIL: $o ).
% 3.05/1.51
% 3.05/1.51 thf(breln1unionIL,definition,
% 3.05/1.51 ( breln1unionIL
% 3.05/1.51 = ( ! [A: $i,R: $i] :
% 3.05/1.51 ( ( breln1 @ A @ R )
% 3.05/1.51 => ! [S: $i] :
% 3.05/1.51 ( ( breln1 @ A @ S )
% 3.05/1.51 => ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ! [Xy: $i] :
% 3.05/1.51 ( ( in @ Xy @ A )
% 3.05/1.51 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.05/1.51 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(breln1unionIR_type,type,
% 3.05/1.51 breln1unionIR: $o ).
% 3.05/1.51
% 3.05/1.51 thf(breln1unionIR,definition,
% 3.05/1.51 ( breln1unionIR
% 3.05/1.51 = ( ! [A: $i,R: $i] :
% 3.05/1.51 ( ( breln1 @ A @ R )
% 3.05/1.51 => ! [S: $i] :
% 3.05/1.51 ( ( breln1 @ A @ S )
% 3.05/1.51 => ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ! [Xy: $i] :
% 3.05/1.51 ( ( in @ Xy @ A )
% 3.05/1.51 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 3.05/1.51 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(breln1unionI_type,type,
% 3.05/1.51 breln1unionI: $o ).
% 3.05/1.51
% 3.05/1.51 thf(breln1unionI,definition,
% 3.05/1.51 ( breln1unionI
% 3.05/1.51 = ( ! [A: $i,R: $i] :
% 3.05/1.51 ( ( breln1 @ A @ R )
% 3.05/1.51 => ! [S: $i] :
% 3.05/1.51 ( ( breln1 @ A @ S )
% 3.05/1.51 => ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ! [Xy: $i] :
% 3.05/1.51 ( ( in @ Xy @ A )
% 3.05/1.51 => ( ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.05/1.51 | ( in @ ( kpair @ Xx @ Xy ) @ S ) )
% 3.05/1.51 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(breln1unionE_type,type,
% 3.05/1.51 breln1unionE: $o ).
% 3.05/1.51
% 3.05/1.51 thf(breln1unionE,definition,
% 3.05/1.51 ( breln1unionE
% 3.05/1.51 = ( ! [A: $i,R: $i] :
% 3.05/1.51 ( ( breln1 @ A @ R )
% 3.05/1.51 => ! [S: $i] :
% 3.05/1.51 ( ( breln1 @ A @ S )
% 3.05/1.51 => ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ! [Xy: $i] :
% 3.05/1.51 ( ( in @ Xy @ A )
% 3.05/1.51 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) )
% 3.05/1.51 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.05/1.51 | ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(breln1unionEcases_type,type,
% 3.05/1.51 breln1unionEcases: $o ).
% 3.05/1.51
% 3.05/1.51 thf(breln1unionEcases,definition,
% 3.05/1.51 ( breln1unionEcases
% 3.05/1.51 = ( ! [A: $i,R: $i] :
% 3.05/1.51 ( ( breln1 @ A @ R )
% 3.05/1.51 => ! [S: $i] :
% 3.05/1.51 ( ( breln1 @ A @ S )
% 3.05/1.51 => ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ! [Xy: $i] :
% 3.05/1.51 ( ( in @ Xy @ A )
% 3.05/1.51 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) )
% 3.05/1.51 => ! [Xphi: $o] :
% 3.05/1.51 ( ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.05/1.51 => Xphi )
% 3.05/1.51 => ( ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 3.05/1.51 => Xphi )
% 3.05/1.51 => Xphi ) ) ) ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(breln1unionCommutes_type,type,
% 3.05/1.51 breln1unionCommutes: $o ).
% 3.05/1.51
% 3.05/1.51 thf(breln1unionCommutes,definition,
% 3.05/1.51 ( breln1unionCommutes
% 3.05/1.51 = ( ! [A: $i,R: $i] :
% 3.05/1.51 ( ( breln1 @ A @ R )
% 3.05/1.51 => ! [S: $i] :
% 3.05/1.51 ( ( breln1 @ A @ S )
% 3.05/1.51 => ( ( binunion @ R @ S )
% 3.05/1.51 = ( binunion @ S @ R ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(woz2Ex_type,type,
% 3.05/1.51 woz2Ex: $o ).
% 3.05/1.51
% 3.05/1.51 thf(woz2Ex,definition,
% 3.05/1.51 ( woz2Ex
% 3.05/1.51 = ( ! [A: $i,R: $i] :
% 3.05/1.51 ( ( breln1 @ A @ R )
% 3.05/1.51 => ( R
% 3.05/1.51 = ( breln1invset @ A @ ( breln1invset @ A @ R ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(woz2W_type,type,
% 3.05/1.51 woz2W: $o ).
% 3.05/1.51
% 3.05/1.51 thf(woz2W,definition,
% 3.05/1.51 ( woz2W
% 3.05/1.51 = ( ! [A: $i,R: $i] :
% 3.05/1.51 ( ( breln1 @ A @ R )
% 3.05/1.51 => ! [S: $i] :
% 3.05/1.51 ( ( breln1 @ A @ S )
% 3.05/1.51 => ( ( breln1invset @ A @ ( breln1compset @ A @ R @ S ) )
% 3.05/1.51 = ( breln1compset @ A @ ( breln1invset @ A @ S ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(woz2A_type,type,
% 3.05/1.51 woz2A: $o ).
% 3.05/1.51
% 3.05/1.51 thf(woz2A,definition,
% 3.05/1.51 ( woz2A
% 3.05/1.51 = ( ! [A: $i,R: $i] :
% 3.05/1.51 ( ( breln1 @ A @ R )
% 3.05/1.51 => ! [S: $i] :
% 3.05/1.51 ( ( breln1 @ A @ S )
% 3.05/1.51 => ! [T: $i] :
% 3.05/1.51 ( ( breln1 @ A @ T )
% 3.05/1.51 => ( ( breln1compset @ A @ ( binunion @ R @ S ) @ T )
% 3.05/1.51 = ( binunion @ ( breln1compset @ A @ R @ T ) @ ( breln1compset @ A @ S @ T ) ) ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(woz2B_type,type,
% 3.05/1.51 woz2B: $o ).
% 3.05/1.51
% 3.05/1.51 thf(woz2B,definition,
% 3.05/1.51 ( woz2B
% 3.05/1.51 = ( ! [A: $i,R: $i] :
% 3.05/1.51 ( ( breln1 @ A @ R )
% 3.05/1.51 => ! [S: $i] :
% 3.05/1.51 ( ( breln1 @ A @ S )
% 3.05/1.51 => ! [T: $i] :
% 3.05/1.51 ( ( breln1 @ A @ T )
% 3.05/1.51 => ( ( breln1compset @ A @ ( binunion @ R @ S ) @ T )
% 3.05/1.51 = ( binunion @ ( breln1invset @ A @ ( breln1compset @ A @ ( breln1invset @ A @ T ) @ ( breln1invset @ A @ S ) ) ) @ ( breln1invset @ A @ ( breln1compset @ A @ ( breln1invset @ A @ T ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(image1Ex_type,type,
% 3.05/1.51 image1Ex: $o ).
% 3.05/1.51
% 3.05/1.51 thf(image1Ex,definition,
% 3.05/1.51 ( image1Ex
% 3.05/1.51 = ( ! [A: $i,Xf: $i > $i] :
% 3.05/1.51 ? [B: $i] :
% 3.05/1.51 ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ B )
% 3.05/1.51 <=> ? [Xy: $i] :
% 3.05/1.51 ( ( in @ Xy @ A )
% 3.05/1.51 & ( Xx
% 3.05/1.51 = ( Xf @ Xy ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(image1Ex1_type,type,
% 3.05/1.51 image1Ex1: $o ).
% 3.05/1.51
% 3.05/1.51 thf(image1Ex1,definition,
% 3.05/1.51 ( image1Ex1
% 3.05/1.51 = ( ! [A: $i,Xf: $i > $i] :
% 3.05/1.51 ( exu
% 3.05/1.51 @ ^ [B: $i] :
% 3.05/1.51 ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ B )
% 3.05/1.51 <=> ? [Xy: $i] :
% 3.05/1.51 ( ( in @ Xy @ A )
% 3.05/1.51 & ( Xx
% 3.05/1.51 = ( Xf @ Xy ) ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(image1_type,type,
% 3.05/1.51 image1: $i > ( $i > $i ) > $i ).
% 3.05/1.51
% 3.05/1.51 thf(image1Equiv_type,type,
% 3.05/1.51 image1Equiv: $o ).
% 3.05/1.51
% 3.05/1.51 thf(image1Equiv,definition,
% 3.05/1.51 ( image1Equiv
% 3.05/1.51 = ( ! [A: $i,Xf: $i > $i,Xx: $i] :
% 3.05/1.51 ( ( in @ Xx
% 3.05/1.51 @ ( image1 @ A
% 3.05/1.51 @ ^ [Xy: $i] : ( Xf @ Xy ) ) )
% 3.05/1.51 <=> ? [Xy: $i] :
% 3.05/1.51 ( ( in @ Xy @ A )
% 3.05/1.51 & ( Xx
% 3.05/1.51 = ( Xf @ Xy ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(image1E_type,type,
% 3.05/1.51 image1E: $o ).
% 3.05/1.51
% 3.05/1.51 thf(image1E,definition,
% 3.05/1.51 ( image1E
% 3.05/1.51 = ( ! [A: $i,Xf: $i > $i,Xx: $i] :
% 3.05/1.51 ( ( in @ Xx
% 3.05/1.51 @ ( image1 @ A
% 3.05/1.51 @ ^ [Xy: $i] : ( Xf @ Xy ) ) )
% 3.05/1.51 => ? [Xy: $i] :
% 3.05/1.51 ( ( in @ Xy @ A )
% 3.05/1.51 & ( Xx
% 3.05/1.51 = ( Xf @ Xy ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(image1I_type,type,
% 3.05/1.51 image1I: $o ).
% 3.05/1.51
% 3.05/1.51 thf(image1I,definition,
% 3.05/1.51 ( image1I
% 3.05/1.51 = ( ! [A: $i,Xf: $i > $i,Xx: $i] :
% 3.05/1.51 ( ? [Xy: $i] :
% 3.05/1.51 ( ( in @ Xy @ A )
% 3.05/1.51 & ( Xx
% 3.05/1.51 = ( Xf @ Xy ) ) )
% 3.05/1.51 => ( in @ Xx
% 3.05/1.51 @ ( image1 @ A
% 3.05/1.51 @ ^ [Xy: $i] : ( Xf @ Xy ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(injective_type,type,
% 3.05/1.51 injective: $i > $i > $i > $o ).
% 3.05/1.51
% 3.05/1.51 thf(injFuncSet_type,type,
% 3.05/1.51 injFuncSet: $i > $i > $i ).
% 3.05/1.51
% 3.05/1.51 thf(injFuncInInjFuncSet_type,type,
% 3.05/1.51 injFuncInInjFuncSet: $o ).
% 3.05/1.51
% 3.05/1.51 thf(injFuncInInjFuncSet,definition,
% 3.05/1.51 ( injFuncInInjFuncSet
% 3.05/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.05/1.51 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.05/1.51 => ( ( injective @ A @ B @ Xf )
% 3.05/1.51 => ( in @ Xf @ ( injFuncSet @ A @ B ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(injFuncSetFuncIn_type,type,
% 3.05/1.51 injFuncSetFuncIn: $o ).
% 3.05/1.51
% 3.05/1.51 thf(injFuncSetFuncIn,definition,
% 3.05/1.51 ( injFuncSetFuncIn
% 3.05/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.05/1.51 ( ( in @ Xf @ ( injFuncSet @ A @ B ) )
% 3.05/1.51 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(injFuncSetFuncInj_type,type,
% 3.05/1.51 injFuncSetFuncInj: $o ).
% 3.05/1.51
% 3.05/1.51 thf(injFuncSetFuncInj,definition,
% 3.05/1.51 ( injFuncSetFuncInj
% 3.05/1.51 = ( ! [Xx: $i,Xy: $i,Xf: $i] :
% 3.05/1.51 ( ( in @ Xf @ ( injFuncSet @ Xx @ Xy ) )
% 3.05/1.51 => ( injective @ Xx @ Xy @ Xf ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(surjective_type,type,
% 3.05/1.51 surjective: $i > $i > $i > $o ).
% 3.05/1.51
% 3.05/1.51 thf(surjFuncSet_type,type,
% 3.05/1.51 surjFuncSet: $i > $i > $i ).
% 3.05/1.51
% 3.05/1.51 thf(surjFuncSetFuncIn_type,type,
% 3.05/1.51 surjFuncSetFuncIn: $o ).
% 3.05/1.51
% 3.05/1.51 thf(surjFuncSetFuncIn,definition,
% 3.05/1.51 ( surjFuncSetFuncIn
% 3.05/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.05/1.51 ( ( in @ Xf @ ( surjFuncSet @ A @ B ) )
% 3.05/1.51 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(surjFuncSetFuncSurj_type,type,
% 3.05/1.51 surjFuncSetFuncSurj: $o ).
% 3.05/1.51
% 3.05/1.51 thf(surjFuncSetFuncSurj,definition,
% 3.05/1.51 ( surjFuncSetFuncSurj
% 3.05/1.51 = ( ! [Xx: $i,Xy: $i,Xf: $i] :
% 3.05/1.51 ( ( in @ Xf @ ( surjFuncSet @ Xx @ Xy ) )
% 3.05/1.51 => ( surjective @ Xx @ Xy @ Xf ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(leftInvIsSurj_type,type,
% 3.05/1.51 leftInvIsSurj: $o ).
% 3.05/1.51
% 3.05/1.51 thf(leftInvIsSurj,definition,
% 3.05/1.51 ( leftInvIsSurj
% 3.05/1.51 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.05/1.51 ( ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.05/1.51 => ! [Xg: $i] :
% 3.05/1.51 ( ( in @ Xg @ ( funcSet @ B @ A ) )
% 3.05/1.51 => ( ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ A )
% 3.05/1.51 => ( ( ap @ B @ A @ Xg @ ( Xf @ Xx ) )
% 3.05/1.51 = Xx ) )
% 3.05/1.51 => ( surjective @ B @ A @ Xg ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(surjCantorThm_type,type,
% 3.05/1.51 surjCantorThm: $o ).
% 3.05/1.51
% 3.05/1.51 thf(surjCantorThm,definition,
% 3.05/1.51 ( surjCantorThm
% 3.05/1.51 = ( ! [A: $i,Xf: $i] :
% 3.05/1.51 ( ( in @ Xf @ ( funcSet @ A @ ( powerset @ A ) ) )
% 3.05/1.51 => ~ ( surjective @ A @ ( powerset @ A ) @ Xf ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(foundation2_type,type,
% 3.05/1.51 foundation2: $o ).
% 3.05/1.51
% 3.05/1.51 thf(foundation2,definition,
% 3.05/1.51 ( foundation2
% 3.05/1.51 = ( ! [A: $i] :
% 3.05/1.51 ( ( nonempty @ A )
% 3.05/1.51 => ? [X: $i] :
% 3.05/1.51 ( ( in @ X @ A )
% 3.05/1.51 & ( ( binintersect @ X @ A )
% 3.05/1.51 = emptyset ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(notinself_type,type,
% 3.05/1.51 notinself: $o ).
% 3.05/1.51
% 3.05/1.51 thf(notinself,definition,
% 3.05/1.51 ( notinself
% 3.05/1.51 = ( ! [A: $i] :
% 3.05/1.51 ~ ( in @ A @ A ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(notinself2_type,type,
% 3.05/1.51 notinself2: $o ).
% 3.05/1.51
% 3.05/1.51 thf(notinself2,definition,
% 3.05/1.51 ( notinself2
% 3.05/1.51 = ( ! [A: $i,B: $i] :
% 3.05/1.51 ( ( in @ A @ B )
% 3.05/1.51 => ~ ( in @ B @ A ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(omegaS_type,type,
% 3.05/1.51 omegaS: $i > $i ).
% 3.05/1.51
% 3.05/1.51 thf(omegaSp_type,type,
% 3.05/1.51 omegaSp: $o ).
% 3.05/1.51
% 3.05/1.51 thf(omegaSp,definition,
% 3.05/1.51 ( omegaSp
% 3.05/1.51 = ( ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ omega )
% 3.05/1.51 => ( in @ ( omegaS @ Xx ) @ omega ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(omegaSclos_type,type,
% 3.05/1.51 omegaSclos: $o ).
% 3.05/1.51
% 3.05/1.51 thf(omegaSclos,definition,
% 3.05/1.51 ( omegaSclos
% 3.05/1.51 = ( ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ omega )
% 3.05/1.51 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(peano0notS_type,type,
% 3.05/1.51 peano0notS: $o ).
% 3.05/1.51
% 3.05/1.51 thf(peano0notS,definition,
% 3.05/1.51 ( peano0notS
% 3.05/1.51 = ( ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ omega )
% 3.05/1.51 => ( ( omegaS @ Xx )
% 3.05/1.51 != emptyset ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(peano3_type,type,
% 3.05/1.51 peano3: $i > $i > $i > $o ).
% 3.05/1.51
% 3.05/1.51 thf(peano4_type,type,
% 3.05/1.51 peano4: $i > $i > $i > $o ).
% 3.05/1.51
% 3.05/1.51 thf(peano5_type,type,
% 3.05/1.51 peano5: $i > $i > $i > $o ).
% 3.05/1.51
% 3.05/1.51 thf(peanoSinj_type,type,
% 3.05/1.51 peanoSinj: $o ).
% 3.05/1.51
% 3.05/1.51 thf(peanoSinj,definition,
% 3.05/1.51 ( peanoSinj
% 3.05/1.51 = ( ! [Xx: $i] :
% 3.05/1.51 ( ( in @ Xx @ omega )
% 3.05/1.51 => ! [Xy: $i] :
% 3.05/1.51 ( ( in @ Xy @ omega )
% 3.05/1.51 => ( ( ( omegaS @ Xx )
% 3.05/1.51 = ( omegaS @ Xy ) )
% 3.05/1.51 => ( Xx = Xy ) ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(peano_type,type,
% 3.05/1.51 peano: $i > $i > $i > $o ).
% 3.05/1.51
% 3.05/1.51 thf(transitiveset_type,type,
% 3.05/1.51 transitiveset: $i > $o ).
% 3.05/1.51
% 3.05/1.51 thf(transitiveset,definition,
% 3.05/1.51 ( transitiveset
% 3.05/1.51 = ( ^ [A: $i] :
% 3.05/1.51 ! [X: $i] :
% 3.05/1.51 ( ( in @ X @ A )
% 3.05/1.51 => ( subset @ X @ A ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(transitivesetOp1_type,type,
% 3.05/1.51 transitivesetOp1: $o ).
% 3.05/1.51
% 3.05/1.51 thf(transitivesetOp1,definition,
% 3.05/1.51 ( transitivesetOp1
% 3.05/1.51 = ( ! [X: $i] :
% 3.05/1.51 ( ( transitiveset @ X )
% 3.05/1.51 => ! [A: $i] :
% 3.05/1.51 ( ( in @ A @ X )
% 3.05/1.51 => ( subset @ A @ X ) ) ) ) ) ).
% 3.05/1.51
% 3.05/1.51 thf(binintTransitive,conjecture,
% 3.05/1.51 ( setextAx
% 3.05/1.51 => ( emptysetAx
% 3.05/1.51 => ( setadjoinAx
% 3.05/1.51 => ( powersetAx
% 3.05/1.51 => ( setunionAx
% 3.05/1.51 => ( omega0Ax
% 3.05/1.51 => ( omegaSAx
% 3.05/1.51 => ( omegaIndAx
% 3.05/1.51 => ( replAx
% 3.05/1.51 => ( foundationAx
% 3.05/1.51 => ( wellorderingAx
% 3.05/1.51 => ( descrp
% 3.05/1.51 => ( dsetconstrI
% 3.05/1.51 => ( dsetconstrEL
% 3.05/1.51 => ( dsetconstrER
% 3.05/1.51 => ( exuE1
% 3.05/1.51 => ( prop2setE
% 3.05/1.51 => ( emptysetE
% 3.05/1.51 => ( emptysetimpfalse
% 3.05/1.51 => ( notinemptyset
% 3.05/1.51 => ( exuE3e
% 3.05/1.51 => ( setext
% 3.05/1.51 => ( emptyI
% 3.05/1.51 => ( noeltsimpempty
% 3.05/1.51 => ( setbeta
% 3.05/1.51 => ( nonemptyE1
% 3.05/1.51 => ( nonemptyI
% 3.05/1.51 => ( nonemptyI1
% 3.05/1.51 => ( setadjoinIL
% 3.05/1.51 => ( emptyinunitempty
% 3.05/1.51 => ( setadjoinIR
% 3.05/1.51 => ( setadjoinE
% 3.05/1.51 => ( setadjoinOr
% 3.05/1.51 => ( setoftrueEq
% 3.05/1.51 => ( powersetI
% 3.05/1.51 => ( emptyinPowerset
% 3.05/1.51 => ( emptyInPowerset
% 3.05/1.51 => ( powersetE
% 3.05/1.51 => ( setunionI
% 3.05/1.51 => ( setunionE
% 3.05/1.51 => ( subPowSU
% 3.05/1.51 => ( exuE2
% 3.05/1.51 => ( nonemptyImpWitness
% 3.05/1.51 => ( uniqinunit
% 3.05/1.51 => ( notinsingleton
% 3.05/1.51 => ( eqinunit
% 3.05/1.51 => ( singletonsswitch
% 3.05/1.51 => ( upairsetE
% 3.05/1.51 => ( upairsetIL
% 3.05/1.51 => ( upairsetIR
% 3.05/1.51 => ( emptyE1
% 3.05/1.51 => ( vacuousDall
% 3.05/1.51 => ( quantDeMorgan1
% 3.05/1.51 => ( quantDeMorgan2
% 3.05/1.51 => ( quantDeMorgan3
% 3.05/1.51 => ( quantDeMorgan4
% 3.05/1.51 => ( prop2setI
% 3.05/1.51 => ( prop2set2propI
% 3.05/1.51 => ( notdexE
% 3.05/1.51 => ( notdallE
% 3.05/1.51 => ( exuI1
% 3.05/1.51 => ( exuI3
% 3.05/1.51 => ( exuI2
% 3.05/1.51 => ( inCongP
% 3.05/1.51 => ( in__Cong
% 3.05/1.51 => ( exuE3u
% 3.05/1.51 => ( exu__Cong
% 3.05/1.51 => ( emptyset__Cong
% 3.05/1.51 => ( setadjoin__Cong
% 3.05/1.51 => ( powerset__Cong
% 3.05/1.51 => ( setunion__Cong
% 3.05/1.51 => ( omega__Cong
% 3.05/1.51 => ( exuEu
% 3.05/1.51 => ( descr__Cong
% 3.05/1.51 => ( dsetconstr__Cong
% 3.05/1.51 => ( subsetI1
% 3.05/1.51 => ( eqimpsubset2
% 3.05/1.51 => ( eqimpsubset1
% 3.05/1.51 => ( subsetI2
% 3.05/1.51 => ( emptysetsubset
% 3.05/1.51 => ( subsetE
% 3.05/1.51 => ( subsetE2
% 3.05/1.51 => ( notsubsetI
% 3.05/1.51 => ( notequalI1
% 3.05/1.51 => ( notequalI2
% 3.05/1.51 => ( subsetRefl
% 3.05/1.51 => ( subsetTrans
% 3.05/1.51 => ( setadjoinSub
% 3.05/1.51 => ( setadjoinSub2
% 3.05/1.51 => ( subset2powerset
% 3.05/1.51 => ( setextsub
% 3.05/1.51 => ( subsetemptysetimpeq
% 3.05/1.51 => ( powersetI1
% 3.05/1.51 => ( powersetE1
% 3.05/1.51 => ( inPowerset
% 3.05/1.51 => ( powersetsubset
% 3.05/1.51 => ( sepInPowerset
% 3.05/1.51 => ( sepSubset
% 3.05/1.51 => ( binunionIL
% 3.05/1.51 => ( upairset2IR
% 3.05/1.51 => ( binunionIR
% 3.05/1.51 => ( binunionEcases
% 3.05/1.51 => ( binunionE
% 3.05/1.51 => ( binunionLsub
% 3.05/1.51 => ( binunionRsub
% 3.05/1.51 => ( binintersectI
% 3.05/1.51 => ( binintersectSubset5
% 3.05/1.51 => ( binintersectEL
% 3.05/1.51 => ( binintersectLsub
% 3.05/1.51 => ( binintersectSubset2
% 3.05/1.51 => ( binintersectSubset3
% 3.05/1.51 => ( binintersectER
% 3.05/1.51 => ( disjointsetsI1
% 3.05/1.51 => ( binintersectRsub
% 3.05/1.51 => ( binintersectSubset4
% 3.05/1.51 => ( binintersectSubset1
% 3.05/1.51 => ( bs114d
% 3.05/1.51 => ( setminusI
% 3.05/1.51 => ( setminusEL
% 3.05/1.51 => ( setminusER
% 3.05/1.51 => ( setminusSubset2
% 3.05/1.51 => ( setminusERneg
% 3.05/1.51 => ( setminusELneg
% 3.05/1.51 => ( setminusILneg
% 3.05/1.51 => ( setminusIRneg
% 3.05/1.51 => ( setminusLsub
% 3.05/1.51 => ( setminusSubset1
% 3.05/1.51 => ( symdiffE
% 3.05/1.51 => ( symdiffI1
% 3.05/1.51 => ( symdiffI2
% 3.05/1.51 => ( symdiffIneg1
% 3.05/1.51 => ( symdiffIneg2
% 3.05/1.51 => ( secondinupair
% 3.05/1.51 => ( setukpairIL
% 3.05/1.51 => ( setukpairIR
% 3.05/1.51 => ( kpairiskpair
% 3.05/1.51 => ( kpairp
% 3.05/1.51 => ( singletonsubset
% 3.05/1.51 => ( singletoninpowerset
% 3.05/1.51 => ( singletoninpowunion
% 3.05/1.51 => ( upairset2E
% 3.05/1.51 => ( upairsubunion
% 3.05/1.51 => ( upairinpowunion
% 3.05/1.51 => ( ubforcartprodlem1
% 3.05/1.51 => ( ubforcartprodlem2
% 3.05/1.51 => ( ubforcartprodlem3
% 3.05/1.51 => ( cartprodpairin
% 3.05/1.51 => ( cartprodmempair1
% 3.05/1.51 => ( cartprodmempair
% 3.05/1.51 => ( setunionE2
% 3.05/1.51 => ( setunionsingleton1
% 3.05/1.51 => ( setunionsingleton2
% 3.05/1.51 => ( setunionsingleton
% 3.05/1.51 => ( singletonprop
% 3.05/1.51 => ( ex1E1
% 3.05/1.51 => ( ex1I
% 3.05/1.51 => ( ex1I2
% 3.05/1.51 => ( singletonsuniq
% 3.05/1.51 => ( setukpairinjL1
% 3.05/1.51 => ( kfstsingleton
% 3.05/1.51 => ( theprop
% 3.05/1.51 => ( kfstpairEq
% 3.05/1.51 => ( cartprodfstin
% 3.05/1.51 => ( setukpairinjL2
% 3.05/1.51 => ( setukpairinjL
% 3.05/1.51 => ( setukpairinjR11
% 3.05/1.51 => ( setukpairinjR12
% 3.05/1.51 => ( setukpairinjR1
% 3.05/1.51 => ( upairequniteq
% 3.05/1.51 => ( setukpairinjR2
% 3.05/1.51 => ( setukpairinjR
% 3.05/1.51 => ( ksndsingleton
% 3.05/1.51 => ( ksndpairEq
% 3.05/1.51 => ( kpairsurjEq
% 3.05/1.51 => ( cartprodsndin
% 3.05/1.51 => ( cartprodpairmemEL
% 3.05/1.51 => ( cartprodpairmemER
% 3.05/1.51 => ( cartprodmempaircEq
% 3.05/1.51 => ( cartprodfstpairEq
% 3.05/1.51 => ( cartprodsndpairEq
% 3.05/1.51 => ( cartprodpairsurjEq
% 3.05/1.51 => ( dpsetconstrI
% 3.05/1.51 => ( dpsetconstrSub
% 3.05/1.51 => ( setOfPairsIsBReln
% 3.05/1.51 => ( dpsetconstrERa
% 3.05/1.51 => ( dpsetconstrEL1
% 3.05/1.51 => ( dpsetconstrEL2
% 3.05/1.51 => ( dpsetconstrER
% 3.05/1.51 => ( funcImageSingleton
% 3.05/1.51 => ( apProp
% 3.05/1.51 => ( app
% 3.05/1.51 => ( infuncsetfunc
% 3.05/1.51 => ( ap2p
% 3.05/1.51 => ( funcinfuncset
% 3.05/1.51 => ( lamProp
% 3.05/1.51 => ( lamp
% 3.05/1.51 => ( lam2p
% 3.05/1.51 => ( brelnall1
% 3.05/1.51 => ( brelnall2
% 3.05/1.51 => ( ex1E2
% 3.05/1.51 => ( funcGraphProp1
% 3.05/1.51 => ( funcGraphProp3
% 3.05/1.51 => ( funcGraphProp2
% 3.05/1.51 => ( funcextLem
% 3.05/1.51 => ( funcGraphProp4
% 3.05/1.51 => ( subbreln
% 3.05/1.51 => ( eqbreln
% 3.05/1.51 => ( funcext
% 3.05/1.51 => ( funcext2
% 3.05/1.51 => ( ap2apEq1
% 3.05/1.51 => ( ap2apEq2
% 3.05/1.51 => ( beta1
% 3.05/1.51 => ( eta1
% 3.05/1.51 => ( lam2lamEq
% 3.05/1.51 => ( beta2
% 3.05/1.51 => ( eta2
% 3.05/1.51 => ( iffalseProp1
% 3.05/1.51 => ( iffalseProp2
% 3.05/1.51 => ( iftrueProp1
% 3.05/1.51 => ( iftrueProp2
% 3.05/1.51 => ( ifSingleton
% 3.05/1.51 => ( ifp
% 3.05/1.51 => ( theeq
% 3.05/1.51 => ( iftrue
% 3.05/1.51 => ( iffalse
% 3.05/1.51 => ( iftrueorfalse
% 3.05/1.51 => ( binintersectT_lem
% 3.05/1.51 => ( binunionT_lem
% 3.05/1.51 => ( powersetT_lem
% 3.05/1.51 => ( setminusT_lem
% 3.05/1.51 => ( complementT_lem
% 3.05/1.51 => ( setextT
% 3.05/1.51 => ( subsetTI
% 3.05/1.51 => ( powersetTI1
% 3.05/1.51 => ( powersetTE1
% 3.05/1.51 => ( complementTI1
% 3.05/1.51 => ( complementTE1
% 3.05/1.51 => ( binintersectTELcontra
% 3.05/1.51 => ( binintersectTERcontra
% 3.05/1.51 => ( contrasubsetT
% 3.05/1.51 => ( contrasubsetT1
% 3.05/1.51 => ( contrasubsetT2
% 3.05/1.51 => ( contrasubsetT3
% 3.05/1.51 => ( doubleComplementI1
% 3.05/1.51 => ( doubleComplementE1
% 3.05/1.51 => ( doubleComplementSub1
% 3.05/1.51 => ( doubleComplementSub2
% 3.05/1.51 => ( doubleComplementEq
% 3.05/1.51 => ( complementTnotintersectT
% 3.05/1.51 => ( complementImpComplementIntersect
% 3.05/1.51 => ( complementSubsetComplementIntersect
% 3.05/1.51 => ( complementInPowersetComplementIntersect
% 3.05/1.51 => ( contraSubsetComplement
% 3.05/1.51 => ( complementTcontraSubset
% 3.05/1.51 => ( binunionTILcontra
% 3.05/1.51 => ( binunionTIRcontra
% 3.05/1.51 => ( inIntersectImpInUnion
% 3.05/1.51 => ( inIntersectImpInUnion2
% 3.05/1.51 => ( inIntersectImpInIntersectUnions
% 3.05/1.51 => ( intersectInPowersetIntersectUnions
% 3.05/1.51 => ( inComplementUnionImpNotIn1
% 3.05/1.51 => ( inComplementUnionImpInComplement1
% 3.05/1.51 => ( binunionTE
% 3.05/1.51 => ( binunionTEcontra
% 3.05/1.51 => ( demorgan2a1
% 3.05/1.51 => ( complementUnionInPowersetComplement
% 3.05/1.51 => ( demorgan2a2
% 3.05/1.51 => ( demorgan1a
% 3.05/1.51 => ( demorgan1b
% 3.05/1.51 => ( demorgan1
% 3.05/1.51 => ( demorgan2a
% 3.05/1.51 => ( demorgan2b2
% 3.05/1.51 => ( demorgan2b
% 3.05/1.51 => ( demorgan2
% 3.05/1.51 => ( woz13rule0
% 3.05/1.51 => ( woz13rule1
% 3.05/1.51 => ( woz13rule2
% 3.05/1.51 => ( woz13rule3
% 3.05/1.51 => ( woz13rule4
% 3.05/1.51 => ( woz1_1
% 3.05/1.51 => ( woz1_2
% 3.05/1.51 => ( woz1_3
% 3.05/1.51 => ( woz1_4
% 3.05/1.51 => ( woz1_5
% 3.05/1.51 => ( breln1all2
% 3.05/1.51 => ( breln1SetBreln1
% 3.05/1.51 => ( choice2fnsingleton
% 3.05/1.51 => ( setOfPairsIsBReln1
% 3.05/1.51 => ( breln1all1
% 3.05/1.51 => ( subbreln1
% 3.05/1.51 => ( eqbreln1
% 3.05/1.51 => ( breln1invprop
% 3.05/1.51 => ( breln1invI
% 3.05/1.51 => ( breln1invE
% 3.05/1.51 => ( breln1compprop
% 3.05/1.51 => ( breln1compI
% 3.05/1.51 => ( breln1compE
% 3.05/1.51 => ( breln1compEex
% 3.05/1.51 => ( breln1unionprop
% 3.05/1.51 => ( breln1unionIL
% 3.05/1.51 => ( breln1unionIR
% 3.05/1.51 => ( breln1unionI
% 3.05/1.51 => ( breln1unionE
% 3.05/1.51 => ( breln1unionEcases
% 3.05/1.51 => ( breln1unionCommutes
% 3.05/1.51 => ( woz2Ex
% 3.05/1.51 => ( woz2W
% 3.05/1.51 => ( woz2A
% 3.05/1.51 => ( woz2B
% 3.05/1.51 => ( image1Ex
% 3.05/1.51 => ( image1Ex1
% 3.05/1.51 => ( image1Equiv
% 3.05/1.51 => ( image1E
% 3.05/1.51 => ( image1I
% 3.05/1.51 => ( injFuncInInjFuncSet
% 3.05/1.51 => ( injFuncSetFuncIn
% 3.05/1.51 => ( injFuncSetFuncInj
% 3.05/1.51 => ( surjFuncSetFuncIn
% 3.05/1.51 => ( surjFuncSetFuncSurj
% 3.05/1.51 => ( leftInvIsSurj
% 3.05/1.51 => ( surjCantorThm
% 3.05/1.51 => ( foundation2
% 3.05/1.51 => ( notinself
% 3.05/1.51 => ( notinself2
% 3.05/1.51 => ( omegaSp
% 3.05/1.51 => ( omegaSclos
% 3.05/1.51 => ( peano0notS
% 3.05/1.51 => ( peanoSinj
% 3.05/1.51 => ( transitivesetOp1
% 3.05/1.51 => ! [X: $i] :
% 3.05/1.51 ( ( transitiveset @ X )
% 3.05/1.51 => ! [Y: $i] :
% 3.05/1.51 ( ( transitiveset @ Y )
% 3.05/1.51 => ( transitiveset @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 3.05/1.51 ------------------------
% 3.05/1.54 % (2592)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_2486 for (2999ds/183Mi)
% 3.05/1.55 % (2593)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_2486 for (2999ds/4Mi)
% 3.05/1.55 % (2597)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_2486 for (2999ds/275Mi)
% 3.05/1.55 % (2595)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_2486 for (2999ds/2Mi)
% 3.05/1.55 % (2594)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_2486 for (2999ds/27Mi)
% 3.05/1.55 % (2596)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_2486 for (2999ds/2Mi)
% 3.05/1.55 % (2598)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_2486 for (2999ds/18Mi)
% 3.05/1.55 % (2595)Instruction limit reached!
% 3.05/1.55 % (2595)------------------------------
% 3.05/1.55 % (2595)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.55 % (2595)Termination reason: Unknown
% 3.05/1.55 % (2595)Termination phase: shuffling
% 3.05/1.55 % (2596)Instruction limit reached!
% 3.05/1.55 % (2596)------------------------------
% 3.05/1.55 % (2596)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.55 % (2596)Termination reason: Unknown
% 3.05/1.55 % (2596)Termination phase: shuffling
% 3.05/1.55
% 3.05/1.55 % (2596)Memory used [KB]: 1663
% 3.05/1.55 % (2596)Time elapsed: 0.003 s
% 3.05/1.55 % (2596)Instructions burned: 2 (million)
% 3.05/1.55 % (2596)------------------------------
% 3.05/1.55 % (2596)------------------------------
% 3.05/1.55
% 3.05/1.55 % (2595)Memory used [KB]: 1663
% 3.05/1.55 % (2595)Time elapsed: 0.003 s
% 3.05/1.55 % (2595)Instructions burned: 3 (million)
% 3.05/1.55 % (2595)------------------------------
% 3.05/1.55 % (2595)------------------------------
% 3.05/1.55 % (2593)Instruction limit reached!
% 3.05/1.55 % (2593)------------------------------
% 3.05/1.55 % (2593)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.55 % (2593)Termination reason: Unknown
% 3.05/1.55 % (2593)Termination phase: shuffling
% 3.05/1.55
% 3.05/1.55 % (2593)Memory used [KB]: 1663
% 3.05/1.55 % (2593)Time elapsed: 0.004 s
% 3.05/1.55 % (2593)Instructions burned: 5 (million)
% 3.05/1.55 % (2593)------------------------------
% 3.05/1.55 % (2593)------------------------------
% 3.05/1.56 % (2598)Instruction limit reached!
% 3.05/1.56 % (2598)------------------------------
% 3.05/1.56 % (2598)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.56 % (2598)Termination reason: Unknown
% 3.05/1.56 % (2598)Termination phase: shuffling
% 3.05/1.56
% 3.05/1.56 % (2598)Memory used [KB]: 1918
% 3.05/1.56 % (2598)Time elapsed: 0.011 s
% 3.05/1.56 % (2598)Instructions burned: 18 (million)
% 3.05/1.56 % (2598)------------------------------
% 3.05/1.56 % (2598)------------------------------
% 3.05/1.56 % (2594)Instruction limit reached!
% 3.05/1.56 % (2594)------------------------------
% 3.05/1.56 % (2594)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.56 % (2594)Termination reason: Unknown
% 3.05/1.56 % (2594)Termination phase: shuffling
% 3.05/1.56
% 3.05/1.56 % (2594)Memory used [KB]: 2174
% 3.05/1.56 % (2594)Time elapsed: 0.015 s
% 3.05/1.56 % (2594)Instructions burned: 29 (million)
% 3.05/1.56 % (2594)------------------------------
% 3.05/1.56 % (2594)------------------------------
% 3.05/1.57 % (2599)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_2486 for (2999ds/3Mi)
% 3.05/1.57 % (2600)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_2486 for (2999ds/37Mi)
% 3.05/1.57 % (2599)Instruction limit reached!
% 3.05/1.57 % (2599)------------------------------
% 3.05/1.57 % (2599)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.57 % (2599)Termination reason: Unknown
% 3.05/1.57 % (2599)Termination phase: shuffling
% 3.05/1.57
% 3.05/1.57 % (2599)Memory used [KB]: 1663
% 3.05/1.57 % (2599)Time elapsed: 0.003 s
% 3.05/1.57 % (2599)Instructions burned: 3 (million)
% 3.05/1.57 % (2599)------------------------------
% 3.05/1.57 % (2599)------------------------------
% 3.05/1.57 % (2601)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_2486 for (2999ds/15Mi)
% 3.05/1.58 % (2601)Instruction limit reached!
% 3.05/1.58 % (2601)------------------------------
% 3.05/1.58 % (2601)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.58 % (2601)Termination reason: Unknown
% 3.05/1.58 % (2601)Termination phase: shuffling
% 3.05/1.58
% 3.05/1.58 % (2601)Memory used [KB]: 1918
% 3.05/1.58 % (2601)Time elapsed: 0.009 s
% 3.05/1.58 % (2601)Instructions burned: 16 (million)
% 3.05/1.58 % (2601)------------------------------
% 3.05/1.58 % (2601)------------------------------
% 3.05/1.58 % (2603)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_2486 for (2999ds/1041Mi)
% 3.05/1.58 % (2602)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_2486 for (2999ds/3Mi)
% 3.05/1.58 % (2602)Instruction limit reached!
% 3.05/1.58 % (2602)------------------------------
% 3.05/1.58 % (2602)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.58 % (2602)Termination reason: Unknown
% 3.05/1.58 % (2602)Termination phase: shuffling
% 3.05/1.58
% 3.05/1.58 % (2602)Memory used [KB]: 1663
% 3.05/1.58 % (2602)Time elapsed: 0.003 s
% 3.05/1.58 % (2602)Instructions burned: 3 (million)
% 3.05/1.58 % (2602)------------------------------
% 3.05/1.58 % (2602)------------------------------
% 3.05/1.58 % (2600)Instruction limit reached!
% 3.05/1.58 % (2600)------------------------------
% 3.05/1.58 % (2600)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.58 % (2600)Termination reason: Unknown
% 3.05/1.58 % (2600)Termination phase: shuffling
% 3.05/1.58
% 3.05/1.58 % (2600)Memory used [KB]: 2430
% 3.05/1.58 % (2600)Time elapsed: 0.019 s
% 3.05/1.58 % (2600)Instructions burned: 39 (million)
% 3.05/1.58 % (2600)------------------------------
% 3.05/1.58 % (2600)------------------------------
% 3.05/1.58 % (2604)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_2486 for (2999ds/7Mi)
% 3.05/1.59 % (2604)Instruction limit reached!
% 3.05/1.59 % (2604)------------------------------
% 3.05/1.59 % (2604)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.59 % (2604)Termination reason: Unknown
% 3.05/1.59 % (2604)Termination phase: shuffling
% 3.05/1.59
% 3.05/1.59 % (2604)Memory used [KB]: 1663
% 3.05/1.59 % (2604)Time elapsed: 0.005 s
% 3.05/1.59 % (2604)Instructions burned: 8 (million)
% 3.05/1.59 % (2604)------------------------------
% 3.05/1.59 % (2604)------------------------------
% 3.05/1.59 % (2592)Instruction limit reached!
% 3.05/1.59 % (2592)------------------------------
% 3.05/1.59 % (2592)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.05/1.59 % (2592)Termination reason: Unknown
% 3.05/1.59 % (2592)Termination phase: Property scanning
% 3.05/1.59
% 3.05/1.59 % (2592)Memory used [KB]: 3070
% 3.05/1.59 % (2592)Time elapsed: 0.049 s
% 3.05/1.59 % (2592)Instructions burned: 187 (million)
% 3.05/1.59 % (2592)------------------------------
% 3.05/1.59 % (2592)------------------------------
% 3.08/1.60 % (2606)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_2486 for (2999ds/3Mi)
% 3.08/1.60 % (2605)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_2486 for (2999ds/16Mi)
% 3.08/1.60 % (2606)Instruction limit reached!
% 3.08/1.60 % (2606)------------------------------
% 3.08/1.60 % (2606)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.60 % (2606)Termination reason: Unknown
% 3.08/1.60 % (2606)Termination phase: shuffling
% 3.08/1.60
% 3.08/1.60 % (2606)Memory used [KB]: 1663
% 3.08/1.60 % (2606)Time elapsed: 0.003 s
% 3.08/1.60 % (2606)Instructions burned: 3 (million)
% 3.08/1.60 % (2606)------------------------------
% 3.08/1.60 % (2606)------------------------------
% 3.08/1.60 % (2609)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_2486 for (2999ds/3Mi)
% 3.08/1.60 % (2607)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_2486 for (2999ds/3Mi)
% 3.08/1.60 % (2609)Instruction limit reached!
% 3.08/1.60 % (2609)------------------------------
% 3.08/1.60 % (2609)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.60 % (2609)Termination reason: Unknown
% 3.08/1.60 % (2609)Termination phase: shuffling
% 3.08/1.60
% 3.08/1.60 % (2609)Memory used [KB]: 1663
% 3.08/1.60 % (2609)Time elapsed: 0.002 s
% 3.08/1.60 % (2609)Instructions burned: 3 (million)
% 3.08/1.60 % (2609)------------------------------
% 3.08/1.60 % (2609)------------------------------
% 3.08/1.60 % (2607)Instruction limit reached!
% 3.08/1.60 % (2607)------------------------------
% 3.08/1.60 % (2607)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.60 % (2607)Termination reason: Unknown
% 3.08/1.60 % (2607)Termination phase: shuffling
% 3.08/1.60
% 3.08/1.60 % (2607)Memory used [KB]: 1663
% 3.08/1.60 % (2607)Time elapsed: 0.004 s
% 3.08/1.60 % (2607)Instructions burned: 4 (million)
% 3.08/1.60 % (2607)------------------------------
% 3.08/1.60 % (2607)------------------------------
% 3.08/1.60 % (2608)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_2486 for (2999ds/7Mi)
% 3.08/1.60 % (2605)Instruction limit reached!
% 3.08/1.60 % (2605)------------------------------
% 3.08/1.60 % (2605)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.60 % (2605)Termination reason: Unknown
% 3.08/1.60 % (2605)Termination phase: shuffling
% 3.08/1.60
% 3.08/1.60 % (2605)Memory used [KB]: 1918
% 3.08/1.60 % (2605)Time elapsed: 0.009 s
% 3.08/1.60 % (2605)Instructions burned: 17 (million)
% 3.08/1.60 % (2605)------------------------------
% 3.08/1.60 % (2605)------------------------------
% 3.08/1.61 % (2608)Instruction limit reached!
% 3.08/1.61 % (2608)------------------------------
% 3.08/1.61 % (2608)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.61 % (2608)Termination reason: Unknown
% 3.08/1.61 % (2608)Termination phase: shuffling
% 3.08/1.61
% 3.08/1.61 % (2608)Memory used [KB]: 1663
% 3.08/1.61 % (2608)Time elapsed: 0.005 s
% 3.08/1.61 % (2608)Instructions burned: 8 (million)
% 3.08/1.61 % (2608)------------------------------
% 3.08/1.61 % (2608)------------------------------
% 3.08/1.61 % (2611)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_2486 for (2999ds/18Mi)
% 3.08/1.61 % (2610)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_2486 for (2999ds/4Mi)
% 3.08/1.61 % (2611)Instruction limit reached!
% 3.08/1.61 % (2611)------------------------------
% 3.08/1.61 % (2611)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.61 % (2611)Termination reason: Unknown
% 3.08/1.61 % (2611)Termination phase: shuffling
% 3.08/1.61
% 3.08/1.61 % (2611)Memory used [KB]: 1918
% 3.08/1.61 % (2611)Time elapsed: 0.006 s
% 3.08/1.61 % (2611)Instructions burned: 18 (million)
% 3.08/1.61 % (2611)------------------------------
% 3.08/1.61 % (2611)------------------------------
% 3.08/1.62 % (2610)Instruction limit reached!
% 3.08/1.62 % (2610)------------------------------
% 3.08/1.62 % (2610)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.62 % (2610)Termination reason: Unknown
% 3.08/1.62 % (2610)Termination phase: shuffling
% 3.08/1.62
% 3.08/1.62 % (2610)Memory used [KB]: 1663
% 3.08/1.62 % (2610)Time elapsed: 0.004 s
% 3.08/1.62 % (2610)Instructions burned: 5 (million)
% 3.08/1.62 % (2610)------------------------------
% 3.08/1.62 % (2610)------------------------------
% 3.08/1.62 % (2612)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_2486 for (2999ds/710Mi)
% 3.08/1.62 % (2613)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_2486 for (2999ds/6Mi)
% 3.08/1.62 % (2613)Instruction limit reached!
% 3.08/1.62 % (2613)------------------------------
% 3.08/1.62 % (2613)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.62 % (2613)Termination reason: Unknown
% 3.08/1.62 % (2613)Termination phase: shuffling
% 3.08/1.62
% 3.08/1.62 % (2613)Memory used [KB]: 1663
% 3.08/1.62 % (2613)Time elapsed: 0.004 s
% 3.08/1.62 % (2613)Instructions burned: 6 (million)
% 3.08/1.62 % (2613)------------------------------
% 3.08/1.62 % (2613)------------------------------
% 3.08/1.62 % (2614)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_2486 for (2999ds/902Mi)
% 3.08/1.62 % (2615)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_2486 for (2999ds/21Mi)
% 3.08/1.63 % (2615)Instruction limit reached!
% 3.08/1.63 % (2615)------------------------------
% 3.08/1.63 % (2615)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.63 % (2615)Termination reason: Unknown
% 3.08/1.63 % (2615)Termination phase: shuffling
% 3.08/1.63
% 3.08/1.63 % (2615)Memory used [KB]: 2046
% 3.08/1.63 % (2615)Time elapsed: 0.007 s
% 3.08/1.63 % (2615)Instructions burned: 22 (million)
% 3.08/1.63 % (2615)------------------------------
% 3.08/1.63 % (2615)------------------------------
% 3.08/1.63 % (2616)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_2486 for (2999ds/5Mi)
% 3.08/1.63 % (2616)Instruction limit reached!
% 3.08/1.63 % (2616)------------------------------
% 3.08/1.63 % (2616)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.63 % (2616)Termination reason: Unknown
% 3.08/1.63 % (2616)Termination phase: shuffling
% 3.08/1.63
% 3.08/1.63 % (2616)Memory used [KB]: 1663
% 3.08/1.63 % (2616)Time elapsed: 0.004 s
% 3.08/1.63 % (2616)Instructions burned: 6 (million)
% 3.08/1.63 % (2616)------------------------------
% 3.08/1.63 % (2616)------------------------------
% 3.08/1.64 % (2617)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_2486 for (2999ds/6Mi)
% 3.08/1.64 % (2617)Instruction limit reached!
% 3.08/1.64 % (2617)------------------------------
% 3.08/1.64 % (2617)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.64 % (2617)Termination reason: Unknown
% 3.08/1.64 % (2617)Termination phase: shuffling
% 3.08/1.64
% 3.08/1.64 % (2617)Memory used [KB]: 1663
% 3.08/1.64 % (2617)Time elapsed: 0.004 s
% 3.08/1.64 % (2617)Instructions burned: 6 (million)
% 3.08/1.64 % (2617)------------------------------
% 3.08/1.64 % (2617)------------------------------
% 3.08/1.65 % (2618)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_2486 for (2998ds/377Mi)
% 3.08/1.65 % (2619)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_2486 for (2998ds/779Mi)
% 3.08/1.66 % (2620)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_2486 for (2998ds/19Mi)
% 3.08/1.67 % (2620)Instruction limit reached!
% 3.08/1.67 % (2620)------------------------------
% 3.08/1.67 % (2620)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.67 % (2620)Termination reason: Unknown
% 3.08/1.67 % (2620)Termination phase: shuffling
% 3.08/1.67
% 3.08/1.67 % (2620)Memory used [KB]: 1918
% 3.08/1.67 % (2620)Time elapsed: 0.010 s
% 3.08/1.67 % (2620)Instructions burned: 19 (million)
% 3.08/1.67 % (2620)------------------------------
% 3.08/1.67 % (2620)------------------------------
% 3.08/1.67 % (2597)Instruction limit reached!
% 3.08/1.67 % (2597)------------------------------
% 3.08/1.67 % (2597)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.67 % (2597)Termination reason: Unknown
% 3.08/1.67 % (2597)Termination phase: Property scanning
% 3.08/1.67
% 3.08/1.67 % (2597)Memory used [KB]: 5884
% 3.08/1.67 % (2597)Time elapsed: 0.126 s
% 3.08/1.67 % (2597)Instructions burned: 277 (million)
% 3.08/1.67 % (2597)------------------------------
% 3.08/1.67 % (2597)------------------------------
% 3.08/1.68 % (2621)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_2486 for (2998ds/879Mi)
% 3.08/1.69 % (2622)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_2486 for (2998ds/17Mi)
% 3.08/1.70 % (2622)Instruction limit reached!
% 3.08/1.70 % (2622)------------------------------
% 3.08/1.70 % (2622)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.70 % (2622)Termination reason: Unknown
% 3.08/1.70 % (2622)Termination phase: shuffling
% 3.08/1.70
% 3.08/1.70 % (2622)Memory used [KB]: 1918
% 3.08/1.70 % (2622)Time elapsed: 0.009 s
% 3.08/1.70 % (2622)Instructions burned: 17 (million)
% 3.08/1.70 % (2622)------------------------------
% 3.08/1.70 % (2622)------------------------------
% 3.08/1.71 % (2623)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_2486 for (2998ds/3Mi)
% 3.08/1.72 % (2623)Instruction limit reached!
% 3.08/1.72 % (2623)------------------------------
% 3.08/1.72 % (2623)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.72 % (2623)Termination reason: Unknown
% 3.08/1.72 % (2623)Termination phase: shuffling
% 3.08/1.72
% 3.08/1.72 % (2623)Memory used [KB]: 1663
% 3.08/1.72 % (2623)Time elapsed: 0.004 s
% 3.08/1.72 % (2623)Instructions burned: 4 (million)
% 3.08/1.72 % (2623)------------------------------
% 3.08/1.72 % (2623)------------------------------
% 3.08/1.73 % (2624)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_2486 for (2997ds/30Mi)
% 3.08/1.75 % (2624)Instruction limit reached!
% 3.08/1.75 % (2624)------------------------------
% 3.08/1.75 % (2624)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.75 % (2624)Termination reason: Unknown
% 3.08/1.75 % (2624)Termination phase: shuffling
% 3.08/1.75
% 3.08/1.75 % (2624)Memory used [KB]: 2302
% 3.08/1.75 % (2624)Time elapsed: 0.016 s
% 3.08/1.75 % (2624)Instructions burned: 31 (million)
% 3.08/1.75 % (2624)------------------------------
% 3.08/1.75 % (2624)------------------------------
% 3.08/1.76 % (2618)Instruction limit reached!
% 3.08/1.76 % (2618)------------------------------
% 3.08/1.76 % (2618)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.76 % (2618)Termination reason: Unknown
% 3.08/1.76 % (2618)Termination phase: Saturation
% 3.08/1.76
% 3.08/1.76 % (2618)Memory used [KB]: 9594
% 3.08/1.76 % (2618)Time elapsed: 0.115 s
% 3.08/1.76 % (2618)Instructions burned: 378 (million)
% 3.08/1.76 % (2618)------------------------------
% 3.08/1.76 % (2618)------------------------------
% 3.08/1.76 % (2625)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_2486 for (2997ds/127Mi)
% 3.08/1.77 % (2626)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_2486 for (2997ds/100Mi)
% 3.08/1.80 % (2626)Instruction limit reached!
% 3.08/1.80 % (2626)------------------------------
% 3.08/1.80 % (2626)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.80 % (2626)Termination reason: Unknown
% 3.08/1.80 % (2626)Termination phase: Preprocessing 1
% 3.08/1.80
% 3.08/1.80 % (2626)Memory used [KB]: 2814
% 3.08/1.80 % (2626)Time elapsed: 0.037 s
% 3.08/1.80 % (2626)Instructions burned: 101 (million)
% 3.08/1.80 % (2626)------------------------------
% 3.08/1.80 % (2626)------------------------------
% 3.08/1.82 % (2627)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_2486 for (2997ds/3Mi)
% 3.08/1.82 % (2627)Instruction limit reached!
% 3.08/1.82 % (2627)------------------------------
% 3.08/1.82 % (2627)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.82 % (2627)Termination reason: Unknown
% 3.08/1.82 % (2627)Termination phase: shuffling
% 3.08/1.82
% 3.08/1.82 % (2627)Memory used [KB]: 1663
% 3.08/1.82 % (2627)Time elapsed: 0.003 s
% 3.08/1.82 % (2627)Instructions burned: 4 (million)
% 3.08/1.82 % (2627)------------------------------
% 3.08/1.82 % (2627)------------------------------
% 3.08/1.82 % (2625)Instruction limit reached!
% 3.08/1.82 % (2625)------------------------------
% 3.08/1.82 % (2625)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.82 % (2625)Termination reason: Unknown
% 3.08/1.82 % (2625)Termination phase: Property scanning
% 3.08/1.82
% 3.08/1.82 % (2625)Memory used [KB]: 3070
% 3.08/1.82 % (2625)Time elapsed: 0.057 s
% 3.08/1.82 % (2625)Instructions burned: 128 (million)
% 3.08/1.82 % (2625)------------------------------
% 3.08/1.82 % (2625)------------------------------
% 3.08/1.83 % (2628)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_2486 for (2996ds/20Mi)
% 3.08/1.83 % (2628)Instruction limit reached!
% 3.08/1.83 % (2628)------------------------------
% 3.08/1.83 % (2628)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.83 % (2628)Termination reason: Unknown
% 3.08/1.83 % (2628)Termination phase: shuffling
% 3.08/1.83
% 3.08/1.83 % (2628)Memory used [KB]: 2046
% 3.08/1.83 % (2628)Time elapsed: 0.008 s
% 3.08/1.83 % (2628)Instructions burned: 21 (million)
% 3.08/1.83 % (2628)------------------------------
% 3.08/1.83 % (2628)------------------------------
% 3.08/1.83 % (2629)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_2486 for (2996ds/86Mi)
% 3.08/1.84 % (2630)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_2486 for (2996ds/107Mi)
% 3.08/1.87 % (2630)Instruction limit reached!
% 3.08/1.87 % (2630)------------------------------
% 3.08/1.87 % (2630)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.87 % (2630)Termination reason: Unknown
% 3.08/1.87 % (2630)Termination phase: SInE selection
% 3.08/1.87
% 3.08/1.87 % (2630)Memory used [KB]: 2814
% 3.08/1.87 % (2630)Time elapsed: 0.029 s
% 3.08/1.87 % (2630)Instructions burned: 108 (million)
% 3.08/1.87 % (2630)------------------------------
% 3.08/1.87 % (2630)------------------------------
% 3.08/1.87 % (2629)Instruction limit reached!
% 3.08/1.87 % (2629)------------------------------
% 3.08/1.87 % (2629)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.87 % (2629)Termination reason: Unknown
% 3.08/1.87 % (2629)Termination phase: Property scanning
% 3.08/1.87
% 3.08/1.87 % (2629)Memory used [KB]: 2814
% 3.08/1.87 % (2629)Time elapsed: 0.040 s
% 3.08/1.87 % (2629)Instructions burned: 88 (million)
% 3.08/1.87 % (2629)------------------------------
% 3.08/1.87 % (2629)------------------------------
% 3.08/1.88 % (2631)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_2486 for (2996ds/39Mi)
% 3.08/1.89 % (2632)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_2486 for (2996ds/448Mi)
% 3.08/1.89 % (2631)Instruction limit reached!
% 3.08/1.89 % (2631)------------------------------
% 3.08/1.89 % (2631)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.89 % (2631)Termination reason: Unknown
% 3.08/1.89 % (2631)Termination phase: shuffling
% 3.08/1.89
% 3.08/1.89 % (2631)Memory used [KB]: 2430
% 3.08/1.89 % (2631)Time elapsed: 0.012 s
% 3.08/1.89 % (2631)Instructions burned: 39 (million)
% 3.08/1.89 % (2631)------------------------------
% 3.08/1.89 % (2631)------------------------------
% 3.08/1.90 % (2633)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_2486 for (2996ds/46Mi)
% 3.08/1.91 % (2633)Instruction limit reached!
% 3.08/1.91 % (2633)------------------------------
% 3.08/1.91 % (2633)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.91 % (2633)Termination reason: Unknown
% 3.08/1.91 % (2633)Termination phase: shuffling
% 3.08/1.91
% 3.08/1.91 % (2633)Memory used [KB]: 2558
% 3.08/1.91 % (2633)Time elapsed: 0.014 s
% 3.08/1.91 % (2633)Instructions burned: 47 (million)
% 3.08/1.91 % (2633)------------------------------
% 3.08/1.91 % (2633)------------------------------
% 3.08/1.92 % (2634)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_2486 for (2995ds/98Mi)
% 3.08/1.95 % (2603)Instruction limit reached!
% 3.08/1.95 % (2603)------------------------------
% 3.08/1.95 % (2603)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.95 % (2603)Termination reason: Unknown
% 3.08/1.95 % (2603)Termination phase: Saturation
% 3.08/1.95
% 3.08/1.95 % (2603)Memory used [KB]: 18933
% 3.08/1.95 % (2603)Time elapsed: 0.372 s
% 3.08/1.95 % (2603)Instructions burned: 1043 (million)
% 3.08/1.95 % (2603)------------------------------
% 3.08/1.95 % (2603)------------------------------
% 3.08/1.95 % (2634)Instruction limit reached!
% 3.08/1.95 % (2634)------------------------------
% 3.08/1.95 % (2634)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.95 % (2634)Termination reason: Unknown
% 3.08/1.95 % (2634)Termination phase: Property scanning
% 3.08/1.95
% 3.08/1.95 % (2634)Memory used [KB]: 2814
% 3.08/1.95 % (2634)Time elapsed: 0.027 s
% 3.08/1.95 % (2634)Instructions burned: 100 (million)
% 3.08/1.95 % (2634)------------------------------
% 3.08/1.95 % (2634)------------------------------
% 3.08/1.96 % (2619)Instruction limit reached!
% 3.08/1.96 % (2619)------------------------------
% 3.08/1.96 % (2619)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.96 % (2619)Termination reason: Unknown
% 3.08/1.96 % (2619)Termination phase: Saturation
% 3.08/1.96
% 3.08/1.96 % (2619)Memory used [KB]: 7931
% 3.08/1.96 % (2619)Time elapsed: 0.310 s
% 3.08/1.96 % (2619)Instructions burned: 779 (million)
% 3.08/1.96 % (2619)------------------------------
% 3.08/1.96 % (2619)------------------------------
% 3.08/1.96 % (2635)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_2486 for (2995ds/507Mi)
% 3.08/1.96 % (2612)Instruction limit reached!
% 3.08/1.96 % (2612)------------------------------
% 3.08/1.96 % (2612)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.96 % (2612)Termination reason: Unknown
% 3.08/1.96 % (2612)Termination phase: Saturation
% 3.08/1.96
% 3.08/1.96 % (2612)Memory used [KB]: 11769
% 3.08/1.96 % (2612)Time elapsed: 0.348 s
% 3.08/1.96 % (2612)Instructions burned: 711 (million)
% 3.08/1.96 % (2612)------------------------------
% 3.08/1.96 % (2612)------------------------------
% 3.08/1.97 % (2636)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_2486 for (2995ds/149Mi)
% 3.08/1.98 % (2637)lrs+21_1:16_au=on:bd=off:piset=and:i=39:si=on:rtra=on_0 on DTF2THF_2486 for (2995ds/39Mi)
% 3.08/1.98 % (2638)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_2486 for (2995ds/6Mi)
% 3.08/1.98 % (2638)Instruction limit reached!
% 3.08/1.98 % (2638)------------------------------
% 3.08/1.98 % (2638)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.98 % (2638)Termination reason: Unknown
% 3.08/1.98 % (2638)Termination phase: shuffling
% 3.08/1.98
% 3.08/1.98 % (2638)Memory used [KB]: 1663
% 3.08/1.98 % (2638)Time elapsed: 0.005 s
% 3.08/1.98 % (2638)Instructions burned: 8 (million)
% 3.08/1.98 % (2638)------------------------------
% 3.08/1.98 % (2638)------------------------------
% 3.08/2.00 % (2637)Instruction limit reached!
% 3.08/2.00 % (2637)------------------------------
% 3.08/2.00 % (2637)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/2.00 % (2637)Termination reason: Unknown
% 3.08/2.00 % (2637)Termination phase: shuffling
% 3.08/2.00
% 3.08/2.00 % (2637)Memory used [KB]: 2430
% 3.08/2.00 % (2637)Time elapsed: 0.020 s
% 3.08/2.00 % (2637)Instructions burned: 40 (million)
% 3.08/2.00 % (2637)------------------------------
% 3.08/2.00 % (2637)------------------------------
% 3.08/2.00 % (2639)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_2486 for (2995ds/1087Mi)
% 3.08/2.02 % (2640)dis+10_1:1_aac=none:cs=on:i=47:si=on:rtra=on_0 on DTF2THF_2486 for (2995ds/47Mi)
% 6.40/2.03 % (2640)Instruction limit reached!
% 6.40/2.03 % (2640)------------------------------
% 6.40/2.03 % (2640)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.40/2.03 % (2640)Termination reason: Unknown
% 6.40/2.03 % (2640)Termination phase: shuffling
% 6.40/2.03
% 6.40/2.03 % (2640)Memory used [KB]: 2558
% 6.40/2.03 % (2640)Time elapsed: 0.016 s
% 6.40/2.03 % (2640)Instructions burned: 47 (million)
% 6.40/2.03 % (2640)------------------------------
% 6.40/2.03 % (2640)------------------------------
% 6.40/2.03 % (2636)Instruction limit reached!
% 6.40/2.03 % (2636)------------------------------
% 6.40/2.03 % (2636)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.40/2.03 % (2636)Termination reason: Unknown
% 6.40/2.03 % (2636)Termination phase: Function definition elimination
% 6.40/2.03
% 6.40/2.03 % (2636)Memory used [KB]: 2942
% 6.40/2.03 % (2636)Time elapsed: 0.067 s
% 6.40/2.03 % (2636)Instructions burned: 149 (million)
% 6.40/2.03 % (2636)------------------------------
% 6.40/2.03 % (2636)------------------------------
% 6.40/2.04 % (2641)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_2486 for (2994ds/44Mi)
% 6.40/2.05 % (2614)Instruction limit reached!
% 6.40/2.05 % (2614)------------------------------
% 6.40/2.05 % (2614)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.40/2.05 % (2614)Termination reason: Unknown
% 6.40/2.05 % (2614)Termination phase: Saturation
% 6.40/2.05
% 6.40/2.05 % (2614)Memory used [KB]: 13432
% 6.40/2.05 % (2614)Time elapsed: 0.424 s
% 6.40/2.05 % (2614)Instructions burned: 902 (million)
% 6.40/2.05 % (2614)------------------------------
% 6.40/2.05 % (2614)------------------------------
% 6.40/2.05 % (2642)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_2486 for (2994ds/432Mi)
% 6.40/2.06 % (2643)lrs+10_7:1_cnfonf=lazy_gen:fe=off:nwc=5.0:plsq=on:i=62:si=on:rtra=on_0 on DTF2THF_2486 for (2994ds/62Mi)
% 6.40/2.06 % (2641)Instruction limit reached!
% 6.40/2.06 % (2641)------------------------------
% 6.40/2.06 % (2641)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.40/2.06 % (2641)Termination reason: Unknown
% 6.40/2.06 % (2641)Termination phase: shuffling
% 6.40/2.06
% 6.40/2.06 % (2641)Memory used [KB]: 2430
% 6.40/2.06 % (2641)Time elapsed: 0.014 s
% 6.40/2.06 % (2641)Instructions burned: 44 (million)
% 6.40/2.06 % (2641)------------------------------
% 6.40/2.06 % (2641)------------------------------
% 6.40/2.07 % (2632)Instruction limit reached!
% 6.40/2.07 % (2632)------------------------------
% 6.40/2.07 % (2632)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.40/2.07 % (2632)Termination reason: Unknown
% 6.40/2.07 % (2632)Termination phase: Saturation
% 6.40/2.07
% 6.40/2.07 % (2632)Memory used [KB]: 7803
% 6.40/2.07 % (2632)Time elapsed: 0.181 s
% 6.40/2.07 % (2632)Instructions burned: 450 (million)
% 6.40/2.07 % (2632)------------------------------
% 6.40/2.07 % (2632)------------------------------
% 6.40/2.07 % (2644)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_2486 for (2994ds/31Mi)
% 6.40/2.09 % (2645)dis+10_1:128_hud=1:i=96:si=on:rtra=on_0 on DTF2THF_2486 for (2994ds/96Mi)
% 6.40/2.09 % (2621)Instruction limit reached!
% 6.40/2.09 % (2621)------------------------------
% 6.40/2.09 % (2621)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.40/2.09 % (2621)Termination reason: Unknown
% 6.40/2.09 % (2621)Termination phase: Saturation
% 6.40/2.09
% 6.40/2.09 % (2621)Memory used [KB]: 10746
% 6.40/2.09 % (2621)Time elapsed: 0.434 s
% 6.40/2.09 % (2621)Instructions burned: 879 (million)
% 6.40/2.09 % (2621)------------------------------
% 6.40/2.09 % (2621)------------------------------
% 6.40/2.09 % (2643)Instruction limit reached!
% 6.40/2.09 % (2643)------------------------------
% 6.40/2.09 % (2643)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.40/2.09 % (2643)Termination reason: Unknown
% 6.40/2.09 % (2643)Termination phase: shuffling
% 6.40/2.09
% 6.40/2.09 % (2643)Memory used [KB]: 2814
% 6.40/2.09 % (2643)Time elapsed: 0.030 s
% 6.40/2.09 % (2643)Instructions burned: 63 (million)
% 6.40/2.09 % (2643)------------------------------
% 6.40/2.09 % (2643)------------------------------
% 6.40/2.10 % (2644)Instruction limit reached!
% 6.40/2.10 % (2644)------------------------------
% 6.40/2.10 % (2644)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.40/2.10 % (2644)Termination reason: Unknown
% 6.40/2.10 % (2644)Termination phase: shuffling
% 6.40/2.10
% 6.40/2.10 % (2644)Memory used [KB]: 2302
% 6.40/2.10 % (2644)Time elapsed: 0.012 s
% 6.40/2.10 % (2644)Instructions burned: 31 (million)
% 6.40/2.10 % (2644)------------------------------
% 6.40/2.10 % (2644)------------------------------
% 6.40/2.11 % (2646)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_2486 for (2994ds/453Mi)
% 6.40/2.11 % (2647)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_2486 for (2994ds/71Mi)
% 6.40/2.11 % (2648)dis+1002_1:1_bd=off:fsr=off:ntd=on:s2a=on:ss=axioms:i=83:si=on:rtra=on_0 on DTF2THF_2486 for (2994ds/83Mi)
% 6.40/2.11 % (2645)Instruction limit reached!
% 6.40/2.11 % (2645)------------------------------
% 6.40/2.11 % (2645)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.40/2.11 % (2645)Termination reason: Unknown
% 6.40/2.11 % (2645)Termination phase: Property scanning
% 6.40/2.11
% 6.40/2.11 % (2645)Memory used [KB]: 2814
% 6.40/2.11 % (2645)Time elapsed: 0.026 s
% 6.40/2.11 % (2645)Instructions burned: 96 (million)
% 6.40/2.11 % (2645)------------------------------
% 6.40/2.11 % (2645)------------------------------
% 6.95/2.12 % (2649)lrs+10_1:128_cnfonf=lazy_not_gen_be_off:hud=1:ins=1:i=11:si=on:rtra=on_0 on DTF2THF_2486 for (2994ds/11Mi)
% 6.95/2.13 % (2649)Instruction limit reached!
% 6.95/2.13 % (2649)------------------------------
% 6.95/2.13 % (2649)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.95/2.13 % (2649)Termination reason: Unknown
% 6.95/2.13 % (2649)Termination phase: shuffling
% 6.95/2.13
% 6.95/2.13 % (2649)Memory used [KB]: 1918
% 6.95/2.13 % (2649)Time elapsed: 0.005 s
% 6.95/2.13 % (2649)Instructions burned: 14 (million)
% 6.95/2.13 % (2649)------------------------------
% 6.95/2.13 % (2649)------------------------------
% 6.95/2.14 % (2650)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_2486 for (2993ds/41Mi)
% 6.95/2.14 % (2647)Instruction limit reached!
% 6.95/2.14 % (2647)------------------------------
% 6.95/2.14 % (2647)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.95/2.14 % (2647)Termination reason: Unknown
% 6.95/2.14 % (2647)Termination phase: Property scanning
% 6.95/2.14
% 6.95/2.14 % (2647)Memory used [KB]: 2814
% 6.95/2.14 % (2647)Time elapsed: 0.034 s
% 6.95/2.14 % (2647)Instructions burned: 72 (million)
% 6.95/2.14 % (2647)------------------------------
% 6.95/2.14 % (2647)------------------------------
% 6.95/2.15 % (2650)Instruction limit reached!
% 6.95/2.15 % (2650)------------------------------
% 6.95/2.15 % (2650)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.95/2.15 % (2650)Termination reason: Unknown
% 6.95/2.15 % (2650)Termination phase: shuffling
% 6.95/2.15
% 6.95/2.15 % (2650)Memory used [KB]: 2430
% 6.95/2.15 % (2650)Time elapsed: 0.012 s
% 6.95/2.15 % (2650)Instructions burned: 41 (million)
% 6.95/2.15 % (2650)------------------------------
% 6.95/2.15 % (2650)------------------------------
% 6.95/2.15 % (2648)Instruction limit reached!
% 6.95/2.15 % (2648)------------------------------
% 6.95/2.15 % (2648)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.95/2.15 % (2648)Termination reason: Unknown
% 6.95/2.15 % (2648)Termination phase: Property scanning
% 6.95/2.15
% 6.95/2.15 % (2648)Memory used [KB]: 2814
% 6.95/2.15 % (2648)Time elapsed: 0.039 s
% 6.95/2.15 % (2648)Instructions burned: 85 (million)
% 6.95/2.15 % (2648)------------------------------
% 6.95/2.15 % (2648)------------------------------
% 6.95/2.16 % (2652)dis+10_1:1_avsq=on:cnfonf=off:e2e=on:nm=32:i=150:si=on:rtra=on_0 on DTF2THF_2486 for (2993ds/150Mi)
% 6.95/2.16 % (2651)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_2486 for (2993ds/6Mi)
% 6.95/2.16 % (2651)Instruction limit reached!
% 6.95/2.16 % (2651)------------------------------
% 6.95/2.16 % (2651)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.95/2.16 % (2651)Termination reason: Unknown
% 6.95/2.16 % (2651)Termination phase: shuffling
% 6.95/2.16
% 6.95/2.16 % (2651)Memory used [KB]: 1663
% 6.95/2.16 % (2651)Time elapsed: 0.004 s
% 6.95/2.16 % (2651)Instructions burned: 6 (million)
% 6.95/2.16 % (2651)------------------------------
% 6.95/2.16 % (2651)------------------------------
% 6.95/2.17 % (2653)ott+1010_1:1_au=on:s2a=on:sd=7:ss=axioms:st=1.5:i=343:si=on:rtra=on_0 on DTF2THF_2486 for (2993ds/343Mi)
% 6.95/2.18 % (2635)Instruction limit reached!
% 6.95/2.18 % (2635)------------------------------
% 6.95/2.18 % (2635)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.95/2.18 % (2635)Termination reason: Unknown
% 6.95/2.18 % (2635)Termination phase: Saturation
% 6.95/2.18
% 6.95/2.18 % (2635)Memory used [KB]: 8443
% 6.95/2.18 % (2635)Time elapsed: 0.219 s
% 6.95/2.18 % (2635)Instructions burned: 508 (million)
% 6.95/2.18 % (2635)------------------------------
% 6.95/2.18 % (2635)------------------------------
% 6.95/2.18 % (2654)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_2486 for (2993ds/109Mi)
% 6.95/2.19 % (2655)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_2486 for (2993ds/253Mi)
% 7.24/2.22 % (2652)Instruction limit reached!
% 7.24/2.22 % (2652)------------------------------
% 7.24/2.22 % (2652)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.24/2.22 % (2652)Termination reason: Unknown
% 7.24/2.22 % (2652)Termination phase: Preprocessing 3
% 7.24/2.22
% 7.24/2.22 % (2652)Memory used [KB]: 5245
% 7.24/2.22 % (2652)Time elapsed: 0.059 s
% 7.24/2.22 % (2652)Instructions burned: 151 (million)
% 7.24/2.22 % (2652)------------------------------
% 7.24/2.22 % (2652)------------------------------
% 7.24/2.22 % (2642)Instruction limit reached!
% 7.24/2.22 % (2642)------------------------------
% 7.24/2.22 % (2642)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.24/2.22 % (2642)Termination reason: Unknown
% 7.24/2.22 % (2642)Termination phase: Saturation
% 7.24/2.22
% 7.24/2.22 % (2642)Memory used [KB]: 3326
% 7.24/2.22 % (2642)Time elapsed: 0.170 s
% 7.24/2.22 % (2642)Instructions burned: 433 (million)
% 7.24/2.22 % (2642)------------------------------
% 7.24/2.22 % (2642)------------------------------
% 7.24/2.23 % (2654)Instruction limit reached!
% 7.24/2.23 % (2654)------------------------------
% 7.24/2.23 % (2654)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.24/2.23 % (2654)Termination reason: Unknown
% 7.24/2.23 % (2654)Termination phase: SInE selection
% 7.24/2.23
% 7.24/2.23 % (2654)Memory used [KB]: 2814
% 7.24/2.23 % (2654)Time elapsed: 0.049 s
% 7.24/2.23 % (2654)Instructions burned: 109 (million)
% 7.24/2.23 % (2654)------------------------------
% 7.24/2.23 % (2654)------------------------------
% 7.24/2.23 % (2656)lrs+10_1:1_au=on:cnfonf=off:sd=1:sgt=16:ss=axioms:i=140:si=on:rtra=on_0 on DTF2THF_2486 for (2992ds/140Mi)
% 7.24/2.23 % (2657)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_2486 for (2992ds/614Mi)
% 7.24/2.24 % (2658)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_2486 for (2992ds/202Mi)
% 7.24/2.28 % (2653)Instruction limit reached!
% 7.24/2.28 % (2653)------------------------------
% 7.24/2.28 % (2653)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.24/2.28 % (2653)Termination reason: Unknown
% 7.24/2.28 % (2653)Termination phase: Saturation
% 7.24/2.28
% 7.24/2.28 % (2653)Memory used [KB]: 8443
% 7.24/2.28 % (2653)Time elapsed: 0.116 s
% 7.24/2.28 % (2653)Instructions burned: 346 (million)
% 7.24/2.28 % (2653)------------------------------
% 7.24/2.28 % (2653)------------------------------
% 7.24/2.29 % (2659)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_2486 for (2992ds/339Mi)
% 7.24/2.29 % (2646)Instruction limit reached!
% 7.24/2.29 % (2646)------------------------------
% 7.24/2.29 % (2646)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.24/2.29 % (2646)Termination reason: Unknown
% 7.24/2.29 % (2646)Termination phase: Saturation
% 7.24/2.29
% 7.24/2.29 % (2646)Memory used [KB]: 7803
% 7.24/2.29 % (2646)Time elapsed: 0.187 s
% 7.24/2.29 % (2646)Instructions burned: 455 (million)
% 7.24/2.29 % (2646)------------------------------
% 7.24/2.29 % (2646)------------------------------
% 7.24/2.29 % (2656)Instruction limit reached!
% 7.24/2.29 % (2656)------------------------------
% 7.24/2.29 % (2656)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.24/2.29 % (2656)Termination reason: Unknown
% 7.24/2.29 % (2656)Termination phase: Property scanning
% 7.24/2.29
% 7.24/2.29 % (2656)Memory used [KB]: 3070
% 7.24/2.29 % (2656)Time elapsed: 0.063 s
% 7.24/2.29 % (2656)Instructions burned: 141 (million)
% 7.24/2.29 % (2656)------------------------------
% 7.24/2.29 % (2656)------------------------------
% 8.06/2.30 % (2655)Instruction limit reached!
% 8.06/2.30 % (2655)------------------------------
% 8.06/2.30 % (2655)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.06/2.30 % (2655)Termination reason: Unknown
% 8.06/2.30 % (2655)Termination phase: Property scanning
% 8.06/2.30
% 8.06/2.30 % (2655)Memory used [KB]: 5884
% 8.06/2.30 % (2655)Time elapsed: 0.108 s
% 8.06/2.30 % (2655)Instructions burned: 253 (million)
% 8.06/2.30 % (2655)------------------------------
% 8.06/2.30 % (2655)------------------------------
% 8.06/2.31 % (2660)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_2486 for (2992ds/283Mi)
% 8.06/2.31 % (2661)dis+2_1:1_cha=on:hud=7:nwc=3.0:prag=on:tnu=2:i=134:si=on:rtra=on_0 on DTF2THF_2486 for (2992ds/134Mi)
% 8.06/2.33 % (2662)dis+1010_1:64_cnfonf=lazy_simp:fd=off:i=298:si=on:rtra=on_0 on DTF2THF_2486 for (2992ds/298Mi)
% 8.06/2.34 % (2658)Instruction limit reached!
% 8.06/2.34 % (2658)------------------------------
% 8.06/2.34 % (2658)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.06/2.34 % (2658)Termination reason: Unknown
% 8.06/2.34 % (2658)Termination phase: Property scanning
% 8.06/2.34
% 8.06/2.34 % (2658)Memory used [KB]: 5756
% 8.06/2.34 % (2658)Time elapsed: 0.096 s
% 8.06/2.34 % (2658)Instructions burned: 203 (million)
% 8.06/2.34 % (2658)------------------------------
% 8.06/2.34 % (2658)------------------------------
% 8.06/2.36 % (2663)lrs+2_1:8_erd=off:plsq=on:plsqr=32,1:sos=on:i=192:si=on:rtra=on_0 on DTF2THF_2486 for (2991ds/192Mi)
% 8.06/2.37 % (2661)Instruction limit reached!
% 8.06/2.37 % (2661)------------------------------
% 8.06/2.37 % (2661)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.06/2.37 % (2661)Termination reason: Unknown
% 8.06/2.37 % (2661)Termination phase: Property scanning
% 8.06/2.37
% 8.06/2.37 % (2661)Memory used [KB]: 3070
% 8.06/2.37 % (2661)Time elapsed: 0.061 s
% 8.06/2.37 % (2661)Instructions burned: 135 (million)
% 8.06/2.37 % (2661)------------------------------
% 8.06/2.37 % (2661)------------------------------
% 8.06/2.39 % (2664)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_2486 for (2991ds/258Mi)
% 8.47/2.43 % (2660)Instruction limit reached!
% 8.47/2.43 % (2660)------------------------------
% 8.47/2.43 % (2660)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.47/2.43 % (2660)Termination reason: Unknown
% 8.47/2.43 % (2660)Termination phase: Saturation
% 8.47/2.43
% 8.47/2.43 % (2660)Memory used [KB]: 7675
% 8.47/2.43 % (2660)Time elapsed: 0.119 s
% 8.47/2.43 % (2660)Instructions burned: 284 (million)
% 8.47/2.43 % (2660)------------------------------
% 8.47/2.43 % (2660)------------------------------
% 8.47/2.44 % (2663)Instruction limit reached!
% 8.47/2.44 % (2663)------------------------------
% 8.47/2.44 % (2663)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.47/2.44 % (2663)Termination reason: Unknown
% 8.47/2.44 % (2663)Termination phase: Property scanning
% 8.47/2.44
% 8.47/2.44 % (2663)Memory used [KB]: 3070
% 8.47/2.44 % (2663)Time elapsed: 0.084 s
% 8.47/2.44 % (2663)Instructions burned: 193 (million)
% 8.47/2.44 % (2663)------------------------------
% 8.47/2.44 % (2663)------------------------------
% 8.47/2.44 % (2659)Instruction limit reached!
% 8.47/2.44 % (2659)------------------------------
% 8.47/2.44 % (2659)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.47/2.44 % (2659)Termination reason: Unknown
% 8.47/2.44 % (2659)Termination phase: Saturation
% 8.47/2.44
% 8.47/2.44 % (2659)Memory used [KB]: 8315
% 8.47/2.44 % (2659)Time elapsed: 0.102 s
% 8.47/2.44 % (2659)Instructions burned: 339 (million)
% 8.47/2.44 % (2659)------------------------------
% 8.47/2.44 % (2659)------------------------------
% 8.47/2.44 % (2639)First to succeed.
% 8.47/2.44 % (2665)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_2486 for (2990ds/90Mi)
% 8.47/2.45 % (2639)Refutation found. Thanks to Tanya!
% 8.47/2.45 % SZS status Theorem for DTF2THF_2486
% 8.47/2.45 % SZS output start Proof for DTF2THF_2486
% 8.47/2.45 thf(func_def_0, type, in: $i > $i > $o).
% 8.47/2.45 thf(func_def_1, type, exu: ($i > $o) > $o).
% 8.47/2.45 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 8.47/2.45 thf(func_def_8, type, powerset: $i > $i).
% 8.47/2.45 thf(func_def_10, type, setunion: $i > $i).
% 8.47/2.45 thf(func_def_19, type, descr: ($i > $o) > $i).
% 8.47/2.45 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 8.47/2.45 thf(func_def_26, type, prop2set: $o > $i).
% 8.47/2.45 thf(func_def_36, type, nonempty: $i > $o).
% 8.47/2.45 thf(func_def_69, type, set2prop: $i > $o).
% 8.47/2.45 thf(func_def_88, type, subset: $i > $i > $o).
% 8.47/2.45 thf(func_def_89, type, disjoint: $i > $i > $o).
% 8.47/2.45 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 8.47/2.45 thf(func_def_114, type, binunion: $i > $i > $i).
% 8.47/2.45 thf(func_def_122, type, binintersect: $i > $i > $i).
% 8.47/2.45 thf(func_def_135, type, regular: $i > $o).
% 8.47/2.45 thf(func_def_136, type, setminus: $i > $i > $i).
% 8.47/2.45 thf(func_def_147, type, symdiff: $i > $i > $i).
% 8.47/2.45 thf(func_def_153, type, iskpair: $i > $o).
% 8.47/2.45 thf(func_def_158, type, kpair: $i > $i > $i).
% 8.47/2.45 thf(func_def_160, type, cartprod: $i > $i > $i).
% 8.47/2.45 thf(func_def_177, type, singleton: $i > $o).
% 8.47/2.45 thf(func_def_179, type, ex1: $i > ($i > $o) > $o).
% 8.47/2.45 thf(func_def_184, type, atmost1p: $i > $o).
% 8.47/2.45 thf(func_def_185, type, atleast2p: $i > $o).
% 8.47/2.45 thf(func_def_186, type, atmost2p: $i > $o).
% 8.47/2.45 thf(func_def_187, type, upairsetp: $i > $o).
% 8.47/2.45 thf(func_def_191, type, kfst: $i > $i).
% 8.47/2.45 thf(func_def_203, type, ksnd: $i > $i).
% 8.47/2.45 thf(func_def_213, type, breln: $i > $i > $i > $o).
% 8.47/2.45 thf(func_def_214, type, dpsetconstr: $i > $i > ($i > $i > $o) > $i).
% 8.47/2.45 thf(func_def_222, type, func: $i > $i > $i > $o).
% 8.47/2.45 thf(func_def_223, type, funcSet: $i > $i > $i).
% 8.47/2.45 thf(func_def_226, type, ap: $i > $i > $i > $i > $i).
% 8.47/2.45 thf(func_def_232, type, lam: $i > $i > ($i > $i) > $i).
% 8.47/2.45 thf(func_def_259, type, if: $i > $o > $i > $i > $i).
% 8.47/2.45 thf(func_def_323, type, breln1: $i > $i > $o).
% 8.47/2.45 thf(func_def_325, type, breln1Set: $i > $i).
% 8.47/2.45 thf(func_def_327, type, transitive: $i > $i > $o).
% 8.47/2.45 thf(func_def_328, type, antisymmetric: $i > $i > $o).
% 8.47/2.45 thf(func_def_329, type, reflexive: $i > $i > $o).
% 8.47/2.45 thf(func_def_330, type, refltransitive: $i > $i > $o).
% 8.47/2.45 thf(func_def_331, type, refllinearorder: $i > $i > $o).
% 8.47/2.45 thf(func_def_332, type, reflwellordering: $i > $i > $o).
% 8.47/2.45 thf(func_def_338, type, breln1invset: $i > $i > $i).
% 8.47/2.45 thf(func_def_342, type, breln1compset: $i > $i > $i > $i).
% 8.47/2.45 thf(func_def_360, type, image1: $i > ($i > $i) > $i).
% 8.47/2.45 thf(func_def_364, type, injective: $i > $i > $i > $o).
% 8.47/2.45 thf(func_def_365, type, injFuncSet: $i > $i > $i).
% 8.47/2.45 thf(func_def_369, type, surjective: $i > $i > $i > $o).
% 8.47/2.45 thf(func_def_370, type, surjFuncSet: $i > $i > $i).
% 8.47/2.45 thf(func_def_378, type, omegaS: $i > $i).
% 8.47/2.45 thf(func_def_382, type, peano3: $i > $i > $i > $o).
% 8.47/2.45 thf(func_def_383, type, peano4: $i > $i > $i > $o).
% 8.47/2.45 thf(func_def_384, type, peano5: $i > $i > $i > $o).
% 8.47/2.45 thf(func_def_386, type, peano: $i > $i > $i > $o).
% 8.47/2.45 thf(func_def_387, type, transitiveset: $i > $o).
% 8.47/2.45 thf(func_def_403, type, sP0: $i > $i > $o).
% 8.47/2.45 thf(func_def_405, type, sP2: $i > $i > $i > $o).
% 8.47/2.45 thf(func_def_406, type, sP3: $i > $i > $o).
% 8.47/2.45 thf(func_def_407, type, sP4: $i > $o).
% 8.47/2.45 thf(func_def_408, type, sP5: $i > $i > $o).
% 8.47/2.45 thf(func_def_409, type, sP6: $i > $i > $o).
% 8.47/2.45 thf(func_def_410, type, sP7: $i > $i > $o).
% 8.47/2.45 thf(func_def_412, type, sP9: $i > ($i > $i) > $i > $o).
% 8.47/2.45 thf(func_def_414, type, sP11: $i > $i > $i > $i > $o).
% 8.47/2.45 thf(func_def_415, type, sP12: $i > $i > $i > $i > $o).
% 8.47/2.45 thf(func_def_416, type, sP13: ($i > $o) > $o).
% 8.47/2.45 thf(func_def_417, type, sP14: $i > ($i > $o) > $o).
% 8.47/2.45 thf(func_def_418, type, sP15: ($i > $o) > $i > $o).
% 8.47/2.45 thf(func_def_419, type, sP16: $i > $i > $i > $o).
% 8.47/2.45 thf(func_def_423, type, sP20: $i > $i > $o > $o).
% 8.47/2.45 thf(func_def_424, type, sP21: $i > $i > $i > $o).
% 8.47/2.45 thf(func_def_426, type, sP23: $o > $i > $i > $i > $o).
% 8.47/2.45 thf(func_def_427, type, sP24: $i > $i > $i > $o > $o).
% 8.47/2.45 thf(func_def_428, type, sP25: $i > $i > $i > $i > $o).
% 8.47/2.45 thf(func_def_429, type, sP26: $i > $i > $o).
% 8.47/2.45 thf(func_def_431, type, sP28: $i > ($i > $o) > $o).
% 8.47/2.45 thf(func_def_433, type, sP30: $i > ($i > $o) > $o).
% 8.47/2.45 thf(func_def_435, type, sP32: $i > ($i > $o) > $o).
% 8.47/2.45 thf(func_def_436, type, sP33: $i > ($i > $o) > $i > $o).
% 8.47/2.45 thf(func_def_437, type, sP34: $i > $i > $i > $i > $i > $o).
% 8.47/2.45 thf(func_def_438, type, sP35: $i > $i > $i > $o).
% 8.47/2.45 thf(func_def_439, type, sP36: $i > ($i > $o) > $o).
% 8.47/2.45 thf(func_def_441, type, sP38: ($i > $i > $o) > $i > $i > $o).
% 8.47/2.45 thf(func_def_442, type, sP39: ($i > $i > $o) > $i > $o).
% 8.47/2.45 thf(func_def_444, type, sP41: $i > $i > $i > $o).
% 8.47/2.45 thf(func_def_445, type, sP42: ($i > $o) > $i > $o).
% 8.47/2.45 thf(func_def_446, type, sP43: $i > ($i > $o) > $i > $o).
% 8.47/2.45 thf(func_def_447, type, sP44: ($i > $o) > $o).
% 8.47/2.45 thf(func_def_448, type, sP45: $o > $i > $i > $o).
% 8.47/2.45 thf(func_def_450, type, sP47: $i > $i > $i > $o).
% 8.47/2.45 thf(func_def_451, type, sP48: $i > $i > $i > $o).
% 8.47/2.45 thf(func_def_452, type, sP49: $i > ($i > $o) > ($i > $o) > $i > $o).
% 8.47/2.45 thf(func_def_453, type, sP50: $i > ($i > $o) > $i > $o).
% 8.47/2.45 thf(func_def_455, type, sP52: $i > $i > $i > $o > $i > $i > $o).
% 8.47/2.45 thf(func_def_456, type, sP53: $i > $i > $i > $o).
% 8.47/2.45 thf(func_def_457, type, sP54: $i > $i > $i > $o).
% 8.47/2.45 thf(func_def_458, type, sP55: $i > $i > $i > $o).
% 8.47/2.45 thf(func_def_460, type, sP57: ($i > $o) > $i > $o).
% 8.47/2.45 thf(func_def_463, type, sP60: $i > $i > $o).
% 8.47/2.45 thf(func_def_465, type, sP62: $i > $i > $i > $o).
% 8.47/2.45 thf(func_def_467, type, sP64: ($i > $o) > $o).
% 8.47/2.45 thf(func_def_468, type, sP65: $i > $o > $i > $o).
% 8.47/2.45 thf(func_def_469, type, sP66: $i > $i > $o > $o).
% 8.47/2.45 thf(func_def_470, type, sP67: $i > $i > $i > $o).
% 8.47/2.45 thf(func_def_471, type, sP68: ($i > $o) > $i > $o).
% 8.47/2.45 thf(func_def_472, type, sP69: $i > $i > $i > ($i > $o) > $o).
% 8.47/2.45 thf(func_def_473, type, sP70: ($i > $o) > $o).
% 8.47/2.45 thf(func_def_475, type, sP72: $i > $i > $i > $i > $o).
% 8.47/2.45 thf(func_def_476, type, sP73: $i > $i > $i > $o).
% 8.47/2.45 thf(func_def_477, type, sP74: ($i > $i) > $i > $i > $o).
% 8.47/2.45 thf(func_def_479, type, sP76: ($i > $o) > $o).
% 8.47/2.45 thf(func_def_480, type, sP77: $i > ($i > $o) > $o).
% 8.47/2.45 thf(func_def_481, type, sP78: ($i > $o) > $i > $o).
% 8.47/2.45 thf(func_def_482, type, sP79: $i > ($i > $o) > $o).
% 8.47/2.45 thf(func_def_486, type, sP83: $i > ($i > $i) > $i > $o).
% 8.47/2.45 thf(func_def_489, type, sP86: $i > $o).
% 8.47/2.45 thf(func_def_490, type, sP87: $i > ($i > $i > $o) > $i > $o).
% 8.47/2.45 thf(func_def_491, type, sP88: $i > ($i > $i > $o) > $i > $o).
% 8.47/2.45 thf(func_def_492, type, sP89: $i > ($i > $i) > $i > $o).
% 8.47/2.45 thf(func_def_494, type, sP91: $i > $i > $i > $o).
% 8.47/2.45 thf(func_def_495, type, sP92: $i > $o).
% 8.47/2.45 thf(func_def_496, type, sP93: $i > $o).
% 8.47/2.45 thf(func_def_498, type, sP95: $i > $i > $i > $o > $o).
% 8.47/2.45 thf(func_def_499, type, sP96: $i > $i > $i > $i > $o).
% 8.47/2.45 thf(func_def_500, type, sP97: ($i > $i) > $i > $i > $o).
% 8.47/2.45 thf(func_def_502, type, sP99: $i > $i > $i > $i > $o).
% 8.47/2.45 thf(func_def_503, type, sP100: $i > $i > $i > $o).
% 8.47/2.45 thf(func_def_505, type, sP102: ($i > $o) > $i > $o).
% 8.47/2.45 thf(func_def_507, type, sP104: ($i > $o) > $i > $o).
% 8.47/2.45 thf(func_def_508, type, sP105: ($i > $o) > $i > $i > $i > $o).
% 8.47/2.45 thf(func_def_516, type, sK113: $i > $i > $i).
% 8.47/2.45 thf(func_def_521, type, sK118: $i > $i > $i).
% 8.47/2.45 thf(func_def_522, type, sK119: $i > $i > $i).
% 8.47/2.45 thf(func_def_523, type, sK120: $i > $i > $i > $i).
% 8.47/2.45 thf(func_def_524, type, sK121: $i > $i > $i).
% 8.47/2.45 thf(func_def_525, type, sK122: $i > $i > $i).
% 8.47/2.45 thf(func_def_526, type, sK123: $i > $i > $i).
% 8.47/2.45 thf(func_def_527, type, sK124: $i > $i > $i).
% 8.47/2.45 thf(func_def_528, type, sK125: $i > $i > $i).
% 8.47/2.45 thf(func_def_529, type, sK126: $i > $i).
% 8.47/2.45 thf(func_def_530, type, sK127: $i > $i).
% 8.47/2.45 thf(func_def_531, type, sK128: $i > $i).
% 8.47/2.45 thf(func_def_532, type, sK129: $i > $i).
% 8.47/2.45 thf(func_def_533, type, sK130: $i > $i > $i).
% 8.47/2.45 thf(func_def_534, type, sK131: $i > $i > $i).
% 8.47/2.45 thf(func_def_535, type, sK132: $i > $i > $i).
% 8.47/2.45 thf(func_def_536, type, sK133: $i > $i > $i > $i).
% 8.47/2.45 thf(func_def_537, type, sK134: $i > $i > $i > $i).
% 8.47/2.45 thf(func_def_538, type, sK135: $i > $i).
% 8.47/2.45 thf(func_def_561, type, sK158: $i > $i > $i).
% 8.47/2.45 thf(func_def_562, type, sK159: $i > $i).
% 8.47/2.45 thf(func_def_565, type, sK162: $i > $i > ($i > $i) > $i).
% 8.47/2.45 thf(func_def_566, type, sK163: $i > ($i > $i) > $i > $i).
% 8.47/2.45 thf(func_def_569, type, sK166: $i > $i > $o).
% 8.47/2.45 thf(func_def_570, type, sK167: $i > $i > $i > $i > $i).
% 8.47/2.45 thf(func_def_571, type, sK168: $i > $i > $i > $i > $i).
% 8.47/2.45 thf(func_def_572, type, sK169: $i > $i > $i > $i > $i).
% 8.47/2.45 thf(func_def_573, type, sK170: $i > $i > $i > $i > $i).
% 8.47/2.45 thf(func_def_585, type, sK182: ($i > $o) > $i).
% 8.47/2.45 thf(func_def_586, type, sK183: $i > ($i > $o) > $i).
% 8.47/2.45 thf(func_def_587, type, sK184: $i > $o).
% 8.47/2.45 thf(func_def_588, type, sK185: ($i > $o) > $i > $i).
% 8.47/2.45 thf(func_def_589, type, sK186: ($i > $o) > $i > $i).
% 8.47/2.45 thf(func_def_590, type, sK187: $i > ($i > $o) > $i).
% 8.47/2.45 thf(func_def_592, type, sK189: $i > $o).
% 8.47/2.45 thf(func_def_601, type, sK198: $i > $o).
% 8.47/2.45 thf(func_def_603, type, sK200: $i > $i).
% 8.47/2.45 thf(func_def_606, type, sK203: $i > $i > ($i > $i) > $i).
% 8.47/2.45 thf(func_def_613, type, sK210: $i > $o).
% 8.47/2.45 thf(func_def_633, type, sK230: $i > $o).
% 8.47/2.45 thf(func_def_634, type, sK231: ($i > $o) > $i > $i).
% 8.47/2.45 thf(func_def_635, type, sK232: $i > $i > $i > $o).
% 8.47/2.45 thf(func_def_636, type, sK233: $i > $i > $i > $i).
% 8.47/2.45 thf(func_def_656, type, sK253: $i > $i > $i).
% 8.47/2.45 thf(func_def_665, type, sK262: $i > $i > $i > $i > $i).
% 8.47/2.45 thf(func_def_666, type, sK263: $i > $i > $i > $i > $i).
% 8.47/2.45 thf(func_def_673, type, sK270: $i > $i > $i).
% 8.47/2.45 thf(func_def_674, type, sK271: $i > $i > $i).
% 8.47/2.45 thf(func_def_675, type, sK272: $i > $o).
% 8.47/2.45 thf(func_def_677, type, sK274: $i > ($i > $o) > $i).
% 8.47/2.45 thf(func_def_678, type, sK275: $i > ($i > $o) > $i).
% 8.47/2.45 thf(func_def_680, type, sK277: $i > $o).
% 8.47/2.45 thf(func_def_681, type, sK278: ($i > $o) > $i > $i).
% 8.47/2.45 thf(func_def_682, type, sK279: $i > ($i > $o) > $i).
% 8.47/2.45 thf(func_def_701, type, sK298: $i > $i > $o).
% 8.47/2.45 thf(func_def_716, type, sK313: $i > ($i > $o) > $i > $i).
% 8.47/2.45 thf(func_def_717, type, sK314: $i > ($i > $o) > $i > $i).
% 8.47/2.45 thf(func_def_718, type, sK315: $i > ($i > $o) > $i).
% 8.47/2.45 thf(func_def_721, type, sK318: $i > $o).
% 8.47/2.45 thf(func_def_732, type, sK329: $i > $i > $i > $i > $i > $i).
% 8.47/2.45 thf(func_def_742, type, sK339: $i > $i > $i > $i).
% 8.47/2.45 thf(func_def_746, type, sK343: $i > $o).
% 8.47/2.45 thf(func_def_748, type, sK345: $i > ($i > $o) > $i).
% 8.47/2.45 thf(func_def_749, type, sK346: $i > ($i > $o) > $i).
% 8.47/2.45 thf(func_def_773, type, sK370: $i > $i > $o).
% 8.47/2.45 thf(func_def_774, type, sK371: ($i > $i > $o) > $i > $i).
% 8.47/2.45 thf(func_def_775, type, sK372: $i > ($i > $i > $o) > $i > $i).
% 8.47/2.45 thf(func_def_776, type, sK373: ($i > $i > $o) > $i > $i).
% 8.47/2.45 thf(func_def_777, type, sK374: ($i > $i > $o) > $i > $i > $i).
% 8.47/2.45 thf(func_def_780, type, sK377: $i > $i > $i > $i).
% 8.47/2.45 thf(func_def_781, type, sK378: $i > $i > $i > $i > $i).
% 8.47/2.45 thf(func_def_789, type, sK386: $i > ($i > $o) > $i > $i).
% 8.47/2.45 thf(func_def_790, type, sK387: $i > ($i > $o) > $i > $i).
% 8.47/2.45 thf(func_def_791, type, sK388: ($i > $o) > $i > $i).
% 8.47/2.45 thf(func_def_794, type, sK391: $i > $o).
% 8.47/2.45 thf(func_def_795, type, sK392: $i > ($i > $o) > $i).
% 8.47/2.45 thf(func_def_796, type, sK393: ($i > $o) > $i).
% 8.47/2.45 thf(func_def_797, type, sK394: $i > $o).
% 8.47/2.45 thf(func_def_806, type, sK403: $i > $i > $i).
% 8.47/2.45 thf(func_def_809, type, sK406: $i > $i > $o).
% 8.47/2.45 thf(func_def_833, type, sK430: $i > $i).
% 8.47/2.45 thf(func_def_835, type, sK432: $i > $o).
% 8.47/2.45 thf(func_def_840, type, sK437: $i > $i > $i > $i).
% 8.47/2.45 thf(func_def_841, type, sK438: $i > $i > $i > $i).
% 8.47/2.45 thf(func_def_874, type, sK471: $i > $o).
% 8.47/2.45 thf(func_def_875, type, sK472: $i > ($i > $o) > ($i > $o) > $i > $i).
% 8.47/2.45 thf(func_def_876, type, sK473: $i > ($i > $o) > ($i > $o) > $i > $i).
% 8.47/2.45 thf(func_def_879, type, sK476: $i > $o).
% 8.47/2.45 thf(func_def_880, type, sK477: $i > $o).
% 8.47/2.45 thf(func_def_881, type, sK478: $i > ($i > $o) > $i > $i).
% 8.47/2.45 thf(func_def_884, type, sK481: $i > $o).
% 8.47/2.45 thf(func_def_887, type, sK484: $i > $i).
% 8.47/2.45 thf(func_def_888, type, sK485: ($i > $i) > $i > $i > $i).
% 8.47/2.45 thf(func_def_889, type, sK486: $i > $o).
% 8.47/2.45 thf(func_def_890, type, sK487: $i > $i > $i > $o > $i > $i > $i).
% 8.47/2.45 thf(func_def_904, type, sK501: $i > $o).
% 8.47/2.45 thf(func_def_911, type, sK508: $i > $i > $i > $i).
% 8.47/2.45 thf(func_def_912, type, sK509: $i > $i > $i > $i).
% 8.47/2.45 thf(func_def_913, type, sK510: $i > $i > $i > $i).
% 8.47/2.45 thf(func_def_914, type, sK511: $i > $i > $i > $i).
% 8.47/2.45 thf(func_def_918, type, sK515: $i > $i > $i > $i).
% 8.47/2.45 thf(func_def_919, type, sK516: $i > $i > $i > $i).
% 8.47/2.45 thf(func_def_927, type, sK524: $i > $i).
% 8.47/2.45 thf(func_def_929, type, sK526: $i > $o).
% 8.47/2.45 thf(func_def_945, type, sK542: $i > $i > $i).
% 8.47/2.45 thf(func_def_949, type, sK546: $i > $o).
% 8.47/2.45 thf(func_def_951, type, sK548: $i > ($i > $o) > $i).
% 8.47/2.45 thf(func_def_952, type, sK549: ($i > $o) > $i > $i).
% 8.47/2.45 thf(func_def_968, type, sK565: $i > $o).
% 8.47/2.45 thf(func_def_969, type, sK566: $i > $o).
% 8.47/2.45 thf(func_def_970, type, sK567: ($i > $o) > ($i > $o) > $i).
% 8.47/2.45 thf(func_def_971, type, sK568: ($i > $o) > ($i > $o) > $i).
% 8.47/2.45 thf(func_def_986, type, sK583: $i > $i > $i).
% 8.47/2.45 thf(func_def_992, type, sK589: $i > $i > $i > $i).
% 8.47/2.45 thf(func_def_1006, type, sK603: $i > $i).
% 8.47/2.45 thf(func_def_1008, type, sK605: $i > ($i > $i) > $i > $i).
% 8.47/2.45 thf(func_def_1013, type, sK610: $i > $o).
% 8.47/2.45 thf(func_def_1021, type, sK618: ($i > $o) > $i).
% 8.47/2.45 thf(func_def_1022, type, sK619: $i > $o).
% 8.47/2.45 thf(func_def_1023, type, sK620: ($i > $o) > $i).
% 8.47/2.45 thf(func_def_1024, type, sK621: $i > ($i > $o) > $i).
% 8.47/2.45 thf(func_def_1025, type, sK622: $i > $o).
% 8.47/2.45 thf(func_def_1028, type, sK625: $i > $i > $o).
% 8.47/2.45 thf(func_def_1044, type, sK641: $i > $o).
% 8.47/2.45 thf(func_def_1047, type, sK644: $i > $o).
% 8.47/2.45 thf(func_def_1052, type, sK649: $i > $o > $i > $i).
% 8.47/2.45 thf(func_def_1058, type, sK655: $i > $o).
% 8.47/2.45 thf(func_def_1062, type, sK659: $i > $i).
% 8.47/2.45 thf(func_def_1071, type, sK668: $i > $i).
% 8.47/2.45 thf(func_def_1083, type, sK680: $i > $i > $i > $o).
% 8.47/2.45 thf(func_def_1097, type, sK694: $i > $i > $i > ($i > $o) > $i).
% 8.47/2.45 thf(func_def_1098, type, sK695: $i > $i > $i > ($i > $o) > $i).
% 8.47/2.45 thf(func_def_1099, type, sK696: ($i > $o) > $i > $i).
% 8.47/2.45 thf(func_def_1103, type, sK700: $i > $o).
% 8.47/2.45 thf(func_def_1106, type, sK703: ($i > $o) > $i).
% 8.47/2.45 thf(func_def_1107, type, sK704: ($i > $o) > $i).
% 8.47/2.45 thf(func_def_1108, type, sK705: $i > $o).
% 8.47/2.45 thf(func_def_1110, type, sK707: $i > $o).
% 8.47/2.45 thf(func_def_1111, type, sK708: $i > $o).
% 8.47/2.45 thf(func_def_1112, type, sK709: ($i > $o) > ($i > $o) > $i).
% 8.47/2.45 thf(func_def_1113, type, sK710: ($i > $o) > ($i > $o) > $i).
% 8.47/2.45 thf(func_def_1117, type, sK714: $i > $i > $i > $i > $i).
% 8.47/2.45 thf(func_def_1128, type, sK725: $i > $i > $o).
% 8.47/2.45 thf(func_def_1167, type, sK764: $i > $i > $o).
% 8.47/2.45 thf(func_def_1171, type, sK768: $i > $i > $i > $i).
% 8.47/2.45 thf(func_def_1172, type, sK769: $i > $i > $i > $i > $i).
% 8.47/2.45 thf(func_def_1193, type, sK790: $i > $i).
% 8.47/2.45 thf(func_def_1195, type, sK792: ($i > $i) > $i > $i > $i).
% 8.47/2.45 thf(func_def_1236, type, sK833: $i > $i).
% 8.47/2.45 thf(func_def_1245, type, sK842: $i > ($i > $o) > $i).
% 8.47/2.45 thf(func_def_1246, type, sK843: ($i > $o) > $i).
% 8.47/2.45 thf(func_def_1247, type, sK844: $i > $o).
% 8.47/2.45 thf(func_def_1266, type, sK863: ($i > $o) > $i > $i).
% 8.47/2.45 thf(func_def_1267, type, sK864: ($i > $o) > $i > $i).
% 8.47/2.45 thf(func_def_1268, type, sK865: $i > ($i > $o) > $i).
% 8.47/2.45 thf(func_def_1270, type, sK867: $i > $o).
% 8.47/2.45 thf(func_def_1284, type, sK881: $i > $o).
% 8.47/2.45 thf(func_def_1285, type, sK882: ($i > $o) > $i > $i).
% 8.47/2.45 thf(func_def_1286, type, sK883: $i > ($i > $o) > $i).
% 8.47/2.45 thf(func_def_1307, type, sK904: $i > $i).
% 8.47/2.45 thf(func_def_1308, type, sK905: ($i > $i) > $i > $i > $i).
% 8.47/2.45 thf(func_def_1347, type, sK944: $i > $i > $o).
% 8.47/2.45 thf(func_def_1365, type, sK962: $i > $o).
% 8.47/2.45 thf(func_def_1374, type, sK971: $i > $i > $i).
% 8.47/2.45 thf(func_def_1378, type, sK975: $i > $i).
% 8.47/2.45 thf(func_def_1379, type, sK976: $i > $i).
% 8.47/2.45 thf(func_def_1380, type, sK977: ($i > $i) > $i > $i).
% 8.47/2.45 thf(func_def_1381, type, sK978: $i > ($i > $i) > $i > $i).
% 8.47/2.45 thf(func_def_1382, type, sK979: $i > $i).
% 8.47/2.45 thf(func_def_1385, type, sK982: $i > $i > ($i > $i) > $i).
% 8.47/2.45 thf(func_def_1386, type, sK983: $i > $i > $i).
% 8.47/2.45 thf(func_def_1387, type, sK984: $i > $i).
% 8.47/2.45 thf(func_def_1390, type, sK987: $i > $i > ($i > $i > $o) > $i).
% 8.47/2.45 thf(func_def_1391, type, sK988: $i > ($i > $i > $o) > $i > $i).
% 8.47/2.45 thf(func_def_1392, type, sK989: $i > ($i > $i > $o) > $i > $i).
% 8.47/2.45 thf(func_def_1393, type, sK990: $i > ($i > $i > $o) > $i > $i).
% 8.47/2.45 thf(func_def_1396, type, sK993: $i > $i > $o).
% 8.47/2.45 thf(func_def_1413, type, sK1010: $i > $i).
% 8.47/2.45 thf(func_def_1414, type, sK1011: ($i > $i) > $i > $i > $i).
% 8.47/2.45 thf(func_def_1415, type, sK1012: $i > ($i > $i) > $i > $i).
% 8.47/2.45 thf(func_def_1420, type, sK1017: $i > $i > $i > $i).
% 8.47/2.45 thf(func_def_1421, type, sK1018: $i > $i > $i > $i).
% 8.47/2.45 thf(func_def_1439, type, sK1036: $i > $i).
% 8.47/2.45 thf(func_def_1440, type, sK1037: $i > $i).
% 8.47/2.45 thf(func_def_1447, type, sK1044: $i > $i).
% 8.47/2.45 thf(func_def_1448, type, sK1045: $i > $i > $i > $i > $o).
% 8.47/2.45 thf(func_def_1474, type, sK1071: $i > $i).
% 8.47/2.45 thf(func_def_1476, type, sK1073: $i > ($i > $i) > $i > $i).
% 8.47/2.45 thf(func_def_1477, type, sK1074: ($i > $i) > $i > $i > $i).
% 8.47/2.45 thf(func_def_1478, type, sK1075: $i > ($i > $i) > $i > $i > $i).
% 8.47/2.45 thf(func_def_1486, type, sK1083: $i > $i).
% 8.47/2.45 thf(func_def_1522, type, sK1119: $i > $i > $o).
% 8.47/2.45 thf(func_def_1525, type, sK1122: $i > $o).
% 8.47/2.45 thf(func_def_1530, type, sK1127: $i > $i > $i > $i).
% 8.47/2.45 thf(func_def_1531, type, sK1128: $i > $i > $i > $i > $i).
% 8.47/2.45 thf(func_def_1532, type, sK1129: $i > $i > $i > $i > $i).
% 8.47/2.45 thf(func_def_1533, type, sK1130: $i > $i > $i > $i > $i).
% 8.47/2.45 thf(func_def_1544, type, sK1141: $i > $i > $i).
% 8.47/2.45 thf(func_def_1557, type, sK1154: $i > $o).
% 8.47/2.45 thf(func_def_1558, type, sK1155: ($i > $o) > $i > $i).
% 8.47/2.45 thf(func_def_1559, type, sK1156: ($i > $o) > $i > $i).
% 8.47/2.45 thf(func_def_1568, type, sK1165: ($i > $o) > $i > $i > $i > $i).
% 8.47/2.45 thf(func_def_1569, type, sK1166: ($i > $o) > $i > $i > $i > $i).
% 8.47/2.45 thf(func_def_1570, type, sK1167: ($i > $o) > $i > $i).
% 8.47/2.45 thf(func_def_1574, type, sK1171: $i > $o).
% 8.47/2.45 thf(func_def_1585, type, sK1182: $i > $i).
% 8.47/2.45 thf(func_def_1587, type, ph1184: !>[X0: $tType]:(X0)).
% 8.47/2.45 thf(f9885,plain,(
% 8.47/2.45 $false),
% 8.47/2.45 inference(avatar_sat_refutation,[status(thm)],[f9724,f9777,f9883])).
% 8.47/2.45 thf(f9883,plain,(
% 8.47/2.45 spl1181_214),
% 8.47/2.45 inference(avatar_contradiction_clause,[status(thm)],[f9882])).
% 8.47/2.45 thf(f9882,plain,(
% 8.47/2.45 $false | spl1181_214),
% 8.47/2.45 inference(trivial_inequality_removal,[status(thm)],[f9873])).
% 8.47/2.45 thf(f9873,plain,(
% 8.47/2.45 ($false = $true) | spl1181_214),
% 8.47/2.45 inference(backward_demodulation,[status(thm)],[f8737,f9854])).
% 8.47/2.45 thf(f9854,plain,(
% 8.47/2.45 ($false = (in @ sK1183 @ sK955)) | spl1181_214),
% 8.47/2.45 inference(trivial_inequality_removal,[status(thm)],[f9849])).
% 8.47/2.45 thf(f9849,plain,(
% 8.47/2.45 ($true != $true) | ($false = (in @ sK1183 @ sK955)) | spl1181_214),
% 8.47/2.45 inference(superposition,[status(thm)],[f9722,f7004])).
% 8.47/2.45 thf(f7004,plain,(
% 8.47/2.45 ( ! [X1 : $i] : (((subset @ X1 @ sK955) = $true) | ((in @ X1 @ sK955) = $false)) )),
% 8.47/2.45 inference(binary_proxy_clausification,[status(thm)],[f7003])).
% 8.47/2.45 thf(f7003,plain,(
% 8.47/2.45 ( ! [X1 : $i] : ((((in @ X1 @ sK955) => (subset @ X1 @ sK955)) = $true)) )),
% 8.47/2.45 inference(beta_eta_normalization,[status(thm)],[f7002])).
% 8.47/2.45 thf(f7002,plain,(
% 8.47/2.45 ( ! [X1 : $i] : ((((^[Y0 : $i]: ((in @ Y0 @ sK955) => (subset @ Y0 @ sK955))) @ X1) = $true)) )),
% 8.47/2.45 inference(pi_clausification,[status(thm)],[f7001])).
% 8.47/2.45 thf(f7001,plain,(
% 8.47/2.45 ((!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ sK955) => (subset @ Y0 @ sK955)))) = $true)),
% 8.47/2.45 inference(beta_eta_normalization,[status(thm)],[f6464])).
% 8.47/2.45 thf(f6464,plain,(
% 8.47/2.45 ($true = ((^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ Y0) => (subset @ Y1 @ Y0))))) @ sK955))),
% 8.47/2.45 inference(definition_unfolding,[status(thm)],[f4914,f3583])).
% 8.47/2.45 thf(f3583,plain,(
% 8.47/2.45 (transitiveset = (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ Y0) => (subset @ Y1 @ Y0))))))),
% 8.47/2.45 inference(cnf_transformation,[status(thm)],[f869])).
% 8.47/2.45 thf(f869,plain,(
% 8.47/2.45 (transitiveset = (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ Y0) => (subset @ Y1 @ Y0))))))),
% 8.47/2.45 inference(fool_elimination,[status(thm)],[f868])).
% 8.47/2.45 thf(f868,plain,(
% 8.47/2.45 ((^[X0 : $i] : (! [X1 : $i] : ((in @ X1 @ X0) => (subset @ X1 @ X0)))) = transitiveset)),
% 8.47/2.45 inference(rectify,[status(thm)],[f331])).
% 8.47/2.45 thf(f331,axiom,(
% 8.47/2.45 ((^[X3 : $i] : (! [X11 : $i] : ((in @ X11 @ X3) => (subset @ X11 @ X3)))) = transitiveset)),
% 8.47/2.45 file('/export/starexec/sandbox2/tmp/tmp.H1wBmJGvZQ/DTF2THF_2486.p',transitiveset)).
% 8.47/2.45 thf(f4914,plain,(
% 8.47/2.45 ((transitiveset @ sK955) = $true)),
% 8.47/2.45 inference(cnf_transformation,[status(thm)],[f3115])).
% 8.47/2.45 thf(f3115,plain,(
% 8.47/2.45 (ubforcartprodlem2 = $true) & (funcGraphProp4 = $true) & (demorgan2a1 = $true) & (injFuncInInjFuncSet = $true) & (doubleComplementSub2 = $true) & (singletonprop = $true) & (symdiffE = $true) & (complementTE1 = $true) & (binunionTILcontra = $true) & (exuE1 = $true) & (woz1_2 = $true) & (subsetRefl = $true) & (notinemptyset = $true) & (demorgan2 = $true) & (ksndpairEq = $true) & (notinsingleton = $true) & (setminusERneg = $true) & (omegaIndAx = $true) & (setukpairinjL1 = $true) & (woz2B = $true) & (emptysetAx = $true) & (quantDeMorgan1 = $true) & (setminusLsub = $true) & (binintersectI = $true) & (funcinfuncset = $true) & (prop2set2propI = $true) & (breln1SetBreln1 = $true) & (emptyInPowerset = $true) & (setukpairinjR2 = $true) & (setadjoinAx = $true) & (funcGraphProp2 = $true) & (subsetemptysetimpeq = $true) & (powersetT_lem = $true) & (setoftrueEq = $true) & (theprop = $true) & (dsetconstr__Cong = $true) & (binintersectSubset3 = $true) & (iftrueorfalse = $true) & (binunionIR = $true) & (setukpairinjR1 = $true) & (demorgan2b = $true) & (vacuousDall = $true) & (dsetconstrEL = $true) & (binunionT_lem = $true) & (binintersectTERcontra = $true) & (setminusELneg = $true) & (setminusEL = $true) & (woz2Ex = $true) & (setminusIRneg = $true) & (breln1all1 = $true) & (woz13rule2 = $true) & (choice2fnsingleton = $true) & (sepSubset = $true) & (inCongP = $true) & (setunion__Cong = $true) & (setbeta = $true) & (iftrueProp1 = $true) & (powersetTI1 = $true) & (breln1compEex = $true) & (cartprodfstin = $true) & (ksndsingleton = $true) & (nonemptyI1 = $true) & (singletonsuniq = $true) & (subsetE2 = $true) & (breln1invprop = $true) & (demorgan1b = $true) & (breln1unionprop = $true) & (complementT_lem = $true) & (notdexE = $true) & (descr__Cong = $true) & (transitivesetOp1 = $true) & (contrasubsetT2 = $true) & (funcImageSingleton = $true) & (cartprodmempair1 = $true) & (image1Equiv = $true) & (app = $true) & (emptyset__Cong = $true) & (contraSubsetComplement = $true) & (complementImpComplementIntersect = $true) & (woz13rule3 = $true) & (setunionsingleton2 = $true) & (breln1invI = $true) & (surjFuncSetFuncIn = $true) & (binintersectTELcontra = $true) & (kpairiskpair = $true) & (binintersectSubset1 = $true) & (foundationAx = $true) & (image1E = $true) & (dpsetconstrEL2 = $true) & (prop2setE = $true) & (woz1_5 = $true) & (binunionTIRcontra = $true) & (setunionAx = $true) & (nonemptyImpWitness = $true) & (inComplementUnionImpNotIn1 = $true) & (ubforcartprodlem1 = $true) & (lam2p = $true) & (ap2apEq2 = $true) & (omegaSp = $true) & (symdiffI2 = $true) & (cartprodpairsurjEq = $true) & (kpairp = $true) & (in__Cong = $true) & (powersetI = $true) & (leftInvIsSurj = $true) & (setminusER = $true) & (upairsetIL = $true) & (cartprodfstpairEq = $true) & (subsetTI = $true) & (descrp = $true) & (demorgan1 = $true) & (lamp = $true) & (cartprodpairmemER = $true) & (powersetTE1 = $true) & (complementInPowersetComplementIntersect = $true) & (binintersectSubset4 = $true) & (subbreln1 = $true) & (apProp = $true) & (singletoninpowerset = $true) & (complementUnionInPowersetComplement = $true) & (powerset__Cong = $true) & (injFuncSetFuncInj = $true) & (binunionTEcontra = $true) & (theeq = $true) & (inIntersectImpInUnion = $true) & (subbreln = $true) & (beta1 = $true) & (setadjoinSub2 = $true) & (upairsetE = $true) & (breln1unionEcases = $true) & (emptyI = $true) & (eta2 = $true) & (replAx = $true) & (inIntersectImpInIntersectUnions = $true) & (demorgan1a = $true) & (surjFuncSetFuncSurj = $true) & (setunionsingleton = $true) & (beta2 = $true) & (binintersectLsub = $true) & (dpsetconstrSub = $true) & (upairset2E = $true) & (breln1unionE = $true) & (exuE3u = $true) & (setminusSubset1 = $true) & (funcGraphProp1 = $true) & (setadjoinIR = $true) & (ifSingleton = $true) & (demorgan2a2 = $true) & (woz1_4 = $true) & (lamProp = $true) & (cartprodpairmemEL = $true) & (dpsetconstrEL1 = $true) & (emptyinPowerset = $true) & (cartprodpairin = $true) & (secondinupair = $true) & (setukpairinjR11 = $true) & (omegaSAx = $true) & (breln1unionIR = $true) & (funcext2 = $true) & (ex1I = $true) & (upairset2IR = $true) & (symdiffIneg1 = $true) & (dpsetconstrER = $true) & (inIntersectImpInUnion2 = $true) & (emptyE1 = $true) & (subsetI2 = $true) & (binunionLsub = $true) & (binintersectSubset2 = $true) & (setOfPairsIsBReln = $true) & (emptyinunitempty = $true) & (dpsetconstrI = $true) & (eqbreln = $true) & (breln1all2 = $true) & (breln1unionIL = $true) & (setukpairinjR12 = $true) & (complementTnotintersectT = $true) & (breln1unionCommutes = $true) & (setextAx = $true) & (eqbreln1 = $true) & (image1Ex = $true) & (setadjoinE = $true) & (disjointsetsI1 = $true) & (uniqinunit = $true) & (exuE3e = $true) & (omega0Ax = $true) & (powersetE1 = $true) & (breln1compprop = $true) & (symdiffIneg2 = $true) & (setadjoinOr = $true) & (symdiffI1 = $true) & (doubleComplementI1 = $true) & (emptysetE = $true) & (contrasubsetT = $true) & (kpairsurjEq = $true) & (singletonsswitch = $true) & (upairsubunion = $true) & (infuncsetfunc = $true) & (setminusSubset2 = $true) & (sepInPowerset = $true) & (ap2p = $true) & (exu__Cong = $true) & (subset2powerset = $true) & (dsetconstrER = $true) & (singletonsubset = $true) & (notinself = $true) & (woz2W = $true) & (binunionEcases = $true) & (exuI1 = $true) & (upairinpowunion = $true) & (prop2setI = $true) & (powersetI1 = $true) & (funcextLem = $true) & (subsetI1 = $true) & (notequalI2 = $true) & (kfstpairEq = $true) & (subsetTrans = $true) & (emptysetsubset = $true) & (dsetconstrI = $true) & (doubleComplementEq = $true) & (intersectInPowersetIntersectUnions = $true) & (woz1_1 = $true) & (complementSubsetComplementIntersect = $true) & (ap2apEq1 = $true) & (breln1compI = $true) & (setminusT_lem = $true) & (doubleComplementSub1 = $true) & (wellorderingAx = $true) & (cartprodmempaircEq = $true) & (woz13rule4 = $true) & (iffalseProp1 = $true) & (inComplementUnionImpInComplement1 = $true) & (binunionRsub = $true) & (setukpairinjL = $true) & (iftrueProp2 = $true) & (iffalseProp2 = $true) & (powersetsubset = $true) & (doubleComplementE1 = $true) & (demorgan2b2 = $true) & (cartprodmempair = $true) & (injFuncSetFuncIn = $true) & (iffalse = $true) & (bs114d = $true) & (omegaSclos = $true) & (ifp = $true) & (((transitiveset @ sK955) = $true) & (($true != (transitiveset @ (binintersect @ sK955 @ sK956))) & ($true = (transitiveset @ sK956)))) & (setunionsingleton1 = $true) & (setOfPairsIsBReln1 = $true) & (powersetAx = $true) & (peanoSinj = $true) & (woz1_3 = $true) & (upairsetIR = $true) & (ex1E1 = $true) & (ubforcartprodlem3 = $true) & (notdallE = $true) & (exuI2 = $true) & (kfstsingleton = $true) & (setadjoinIL = $true) & (funcGraphProp3 = $true) & (notsubsetI = $true) & (upairequniteq = $true) & (notinself2 = $true) & (funcext = $true) & (complementTcontraSubset = $true) & (image1I = $true) & (binintersectEL = $true) & (woz13rule1 = $true) & (lam2lamEq = $true) & (setukpairinjR = $true) & (nonemptyE1 = $true) & (setukpairIL = $true) & (demorgan2a = $true) & (setukpairinjL2 = $true) & (image1Ex1 = $true) & (binintersectER = $true) & (nonemptyI = $true) & (iftrue = $true) & (breln1unionI = $true) & (inPowerset = $true) & (powersetE = $true) & (eta1 = $true) & (brelnall2 = $true) & (exuE2 = $true) & (setext = $true) & (quantDeMorgan4 = $true) & (subsetE = $true) & (woz13rule0 = $true) & (binintersectRsub = $true) & (woz2A = $true) & (setunionE2 = $true) & (ex1E2 = $true) & (omega__Cong = $true) & (breln1compE = $true) & (binunionE = $true) & (binunionIL = $true) & (eqinunit = $true) & (exuI3 = $true) & (binunionTE = $true) & (ex1I2 = $true) & (emptysetimpfalse = $true) & (quantDeMorgan3 = $true) & (binintersectSubset5 = $true) & (binintersectT_lem = $true) & (surjCantorThm = $true) & (setunionI = $true) & (breln1invE = $true) & (foundation2 = $true) & (setukpairIR = $true) & (setadjoinSub = $true) & (notequalI1 = $true) & (peano0notS = $true) & (quantDeMorgan2 = $true) & (subPowSU = $true) & (cartprodsndin = $true) & (eqimpsubset2 = $true) & (setminusI = $true) & (complementTI1 = $true) & (setunionE = $true) & (cartprodsndpairEq = $true) & (contrasubsetT1 = $true) & (dpsetconstrERa = $true) & (brelnall1 = $true) & (setminusILneg = $true) & (eqimpsubset1 = $true) & (setextT = $true) & (setadjoin__Cong = $true) & (contrasubsetT3 = $true) & (noeltsimpempty = $true) & (exuEu = $true) & (singletoninpowunion = $true) & (setextsub = $true)),
% 8.47/2.45 inference(skolemisation,[status(esa),new_symbols(skolem,[sK955,sK956])],[f1132,f3114,f3113])).
% 8.47/2.45 thf(f3113,plain,(
% 8.47/2.45 ? [X0 : $i] : (((transitiveset @ X0) = $true) & ? [X1 : $i] : (((transitiveset @ (binintersect @ X0 @ X1)) != $true) & ((transitiveset @ X1) = $true))) => (((transitiveset @ sK955) = $true) & ? [X1 : $i] : (((transitiveset @ (binintersect @ sK955 @ X1)) != $true) & ((transitiveset @ X1) = $true)))),
% 8.47/2.45 introduced(definition,[],[choice_axiom])).
% 8.47/2.45 thf(f3114,plain,(
% 8.47/2.45 ? [X1 : $i] : (((transitiveset @ (binintersect @ sK955 @ X1)) != $true) & ((transitiveset @ X1) = $true)) => (($true != (transitiveset @ (binintersect @ sK955 @ sK956))) & ($true = (transitiveset @ sK956)))),
% 8.47/2.45 introduced(definition,[],[choice_axiom])).
% 8.47/2.45 thf(f1132,plain,(
% 8.47/2.45 (ubforcartprodlem2 = $true) & (funcGraphProp4 = $true) & (demorgan2a1 = $true) & (injFuncInInjFuncSet = $true) & (doubleComplementSub2 = $true) & (singletonprop = $true) & (symdiffE = $true) & (complementTE1 = $true) & (binunionTILcontra = $true) & (exuE1 = $true) & (woz1_2 = $true) & (subsetRefl = $true) & (notinemptyset = $true) & (demorgan2 = $true) & (ksndpairEq = $true) & (notinsingleton = $true) & (setminusERneg = $true) & (omegaIndAx = $true) & (setukpairinjL1 = $true) & (woz2B = $true) & (emptysetAx = $true) & (quantDeMorgan1 = $true) & (setminusLsub = $true) & (binintersectI = $true) & (funcinfuncset = $true) & (prop2set2propI = $true) & (breln1SetBreln1 = $true) & (emptyInPowerset = $true) & (setukpairinjR2 = $true) & (setadjoinAx = $true) & (funcGraphProp2 = $true) & (subsetemptysetimpeq = $true) & (powersetT_lem = $true) & (setoftrueEq = $true) & (theprop = $true) & (dsetconstr__Cong = $true) & (binintersectSubset3 = $true) & (iftrueorfalse = $true) & (binunionIR = $true) & (setukpairinjR1 = $true) & (demorgan2b = $true) & (vacuousDall = $true) & (dsetconstrEL = $true) & (binunionT_lem = $true) & (binintersectTERcontra = $true) & (setminusELneg = $true) & (setminusEL = $true) & (woz2Ex = $true) & (setminusIRneg = $true) & (breln1all1 = $true) & (woz13rule2 = $true) & (choice2fnsingleton = $true) & (sepSubset = $true) & (inCongP = $true) & (setunion__Cong = $true) & (setbeta = $true) & (iftrueProp1 = $true) & (powersetTI1 = $true) & (breln1compEex = $true) & (cartprodfstin = $true) & (ksndsingleton = $true) & (nonemptyI1 = $true) & (singletonsuniq = $true) & (subsetE2 = $true) & (breln1invprop = $true) & (demorgan1b = $true) & (breln1unionprop = $true) & (complementT_lem = $true) & (notdexE = $true) & (descr__Cong = $true) & (transitivesetOp1 = $true) & (contrasubsetT2 = $true) & (funcImageSingleton = $true) & (cartprodmempair1 = $true) & (image1Equiv = $true) & (app = $true) & (emptyset__Cong = $true) & (contraSubsetComplement = $true) & (complementImpComplementIntersect = $true) & (woz13rule3 = $true) & (setunionsingleton2 = $true) & (breln1invI = $true) & (surjFuncSetFuncIn = $true) & (binintersectTELcontra = $true) & (kpairiskpair = $true) & (binintersectSubset1 = $true) & (foundationAx = $true) & (image1E = $true) & (dpsetconstrEL2 = $true) & (prop2setE = $true) & (woz1_5 = $true) & (binunionTIRcontra = $true) & (setunionAx = $true) & (nonemptyImpWitness = $true) & (inComplementUnionImpNotIn1 = $true) & (ubforcartprodlem1 = $true) & (lam2p = $true) & (ap2apEq2 = $true) & (omegaSp = $true) & (symdiffI2 = $true) & (cartprodpairsurjEq = $true) & (kpairp = $true) & (in__Cong = $true) & (powersetI = $true) & (leftInvIsSurj = $true) & (setminusER = $true) & (upairsetIL = $true) & (cartprodfstpairEq = $true) & (subsetTI = $true) & (descrp = $true) & (demorgan1 = $true) & (lamp = $true) & (cartprodpairmemER = $true) & (powersetTE1 = $true) & (complementInPowersetComplementIntersect = $true) & (binintersectSubset4 = $true) & (subbreln1 = $true) & (apProp = $true) & (singletoninpowerset = $true) & (complementUnionInPowersetComplement = $true) & (powerset__Cong = $true) & (injFuncSetFuncInj = $true) & (binunionTEcontra = $true) & (theeq = $true) & (inIntersectImpInUnion = $true) & (subbreln = $true) & (beta1 = $true) & (setadjoinSub2 = $true) & (upairsetE = $true) & (breln1unionEcases = $true) & (emptyI = $true) & (eta2 = $true) & (replAx = $true) & (inIntersectImpInIntersectUnions = $true) & (demorgan1a = $true) & (surjFuncSetFuncSurj = $true) & (setunionsingleton = $true) & (beta2 = $true) & (binintersectLsub = $true) & (dpsetconstrSub = $true) & (upairset2E = $true) & (breln1unionE = $true) & (exuE3u = $true) & (setminusSubset1 = $true) & (funcGraphProp1 = $true) & (setadjoinIR = $true) & (ifSingleton = $true) & (demorgan2a2 = $true) & (woz1_4 = $true) & (lamProp = $true) & (cartprodpairmemEL = $true) & (dpsetconstrEL1 = $true) & (emptyinPowerset = $true) & (cartprodpairin = $true) & (secondinupair = $true) & (setukpairinjR11 = $true) & (omegaSAx = $true) & (breln1unionIR = $true) & (funcext2 = $true) & (ex1I = $true) & (upairset2IR = $true) & (symdiffIneg1 = $true) & (dpsetconstrER = $true) & (inIntersectImpInUnion2 = $true) & (emptyE1 = $true) & (subsetI2 = $true) & (binunionLsub = $true) & (binintersectSubset2 = $true) & (setOfPairsIsBReln = $true) & (emptyinunitempty = $true) & (dpsetconstrI = $true) & (eqbreln = $true) & (breln1all2 = $true) & (breln1unionIL = $true) & (setukpairinjR12 = $true) & (complementTnotintersectT = $true) & (breln1unionCommutes = $true) & (setextAx = $true) & (eqbreln1 = $true) & (image1Ex = $true) & (setadjoinE = $true) & (disjointsetsI1 = $true) & (uniqinunit = $true) & (exuE3e = $true) & (omega0Ax = $true) & (powersetE1 = $true) & (breln1compprop = $true) & (symdiffIneg2 = $true) & (setadjoinOr = $true) & (symdiffI1 = $true) & (doubleComplementI1 = $true) & (emptysetE = $true) & (contrasubsetT = $true) & (kpairsurjEq = $true) & (singletonsswitch = $true) & (upairsubunion = $true) & (infuncsetfunc = $true) & (setminusSubset2 = $true) & (sepInPowerset = $true) & (ap2p = $true) & (exu__Cong = $true) & (subset2powerset = $true) & (dsetconstrER = $true) & (singletonsubset = $true) & (notinself = $true) & (woz2W = $true) & (binunionEcases = $true) & (exuI1 = $true) & (upairinpowunion = $true) & (prop2setI = $true) & (powersetI1 = $true) & (funcextLem = $true) & (subsetI1 = $true) & (notequalI2 = $true) & (kfstpairEq = $true) & (subsetTrans = $true) & (emptysetsubset = $true) & (dsetconstrI = $true) & (doubleComplementEq = $true) & (intersectInPowersetIntersectUnions = $true) & (woz1_1 = $true) & (complementSubsetComplementIntersect = $true) & (ap2apEq1 = $true) & (breln1compI = $true) & (setminusT_lem = $true) & (doubleComplementSub1 = $true) & (wellorderingAx = $true) & (cartprodmempaircEq = $true) & (woz13rule4 = $true) & (iffalseProp1 = $true) & (inComplementUnionImpInComplement1 = $true) & (binunionRsub = $true) & (setukpairinjL = $true) & (iftrueProp2 = $true) & (iffalseProp2 = $true) & (powersetsubset = $true) & (doubleComplementE1 = $true) & (demorgan2b2 = $true) & (cartprodmempair = $true) & (injFuncSetFuncIn = $true) & (iffalse = $true) & (bs114d = $true) & (omegaSclos = $true) & (ifp = $true) & ? [X0 : $i] : (((transitiveset @ X0) = $true) & ? [X1 : $i] : (((transitiveset @ (binintersect @ X0 @ X1)) != $true) & ((transitiveset @ X1) = $true))) & (setunionsingleton1 = $true) & (setOfPairsIsBReln1 = $true) & (powersetAx = $true) & (peanoSinj = $true) & (woz1_3 = $true) & (upairsetIR = $true) & (ex1E1 = $true) & (ubforcartprodlem3 = $true) & (notdallE = $true) & (exuI2 = $true) & (kfstsingleton = $true) & (setadjoinIL = $true) & (funcGraphProp3 = $true) & (notsubsetI = $true) & (upairequniteq = $true) & (notinself2 = $true) & (funcext = $true) & (complementTcontraSubset = $true) & (image1I = $true) & (binintersectEL = $true) & (woz13rule1 = $true) & (lam2lamEq = $true) & (setukpairinjR = $true) & (nonemptyE1 = $true) & (setukpairIL = $true) & (demorgan2a = $true) & (setukpairinjL2 = $true) & (image1Ex1 = $true) & (binintersectER = $true) & (nonemptyI = $true) & (iftrue = $true) & (breln1unionI = $true) & (inPowerset = $true) & (powersetE = $true) & (eta1 = $true) & (brelnall2 = $true) & (exuE2 = $true) & (setext = $true) & (quantDeMorgan4 = $true) & (subsetE = $true) & (woz13rule0 = $true) & (binintersectRsub = $true) & (woz2A = $true) & (setunionE2 = $true) & (ex1E2 = $true) & (omega__Cong = $true) & (breln1compE = $true) & (binunionE = $true) & (binunionIL = $true) & (eqinunit = $true) & (exuI3 = $true) & (binunionTE = $true) & (ex1I2 = $true) & (emptysetimpfalse = $true) & (quantDeMorgan3 = $true) & (binintersectSubset5 = $true) & (binintersectT_lem = $true) & (surjCantorThm = $true) & (setunionI = $true) & (breln1invE = $true) & (foundation2 = $true) & (setukpairIR = $true) & (setadjoinSub = $true) & (notequalI1 = $true) & (peano0notS = $true) & (quantDeMorgan2 = $true) & (subPowSU = $true) & (cartprodsndin = $true) & (eqimpsubset2 = $true) & (setminusI = $true) & (complementTI1 = $true) & (setunionE = $true) & (cartprodsndpairEq = $true) & (contrasubsetT1 = $true) & (dpsetconstrERa = $true) & (brelnall1 = $true) & (setminusILneg = $true) & (eqimpsubset1 = $true) & (setextT = $true) & (setadjoin__Cong = $true) & (contrasubsetT3 = $true) & (noeltsimpempty = $true) & (exuEu = $true) & (singletoninpowunion = $true) & (setextsub = $true)),
% 8.47/2.45 inference(flattening,[status(thm)],[f1131])).
% 8.47/2.45 thf(f1131,plain,(
% 8.47/2.45 ((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i] : (((transitiveset @ X0) = $true) & ? [X1 : $i] : (((transitiveset @ (binintersect @ X0 @ X1)) != $true) & ((transitiveset @ X1) = $true))) & (transitivesetOp1 = $true)) & (peanoSinj = $true)) & (peano0notS = $true)) & (omegaSclos = $true)) & (omegaSp = $true)) & (notinself2 = $true)) & (notinself = $true)) & (foundation2 = $true)) & (surjCantorThm = $true)) & (leftInvIsSurj = $true)) & (surjFuncSetFuncSurj = $true)) & (surjFuncSetFuncIn = $true)) & (injFuncSetFuncInj = $true)) & (injFuncSetFuncIn = $true)) & (injFuncInInjFuncSet = $true)) & (image1I = $true)) & (image1E = $true)) & (image1Equiv = $true)) & (image1Ex1 = $true)) & (image1Ex = $true)) & (woz2B = $true)) & (woz2A = $true)) & (woz2W = $true)) & (woz2Ex = $true)) & (breln1unionCommutes = $true)) & (breln1unionEcases = $true)) & (breln1unionE = $true)) & (breln1unionI = $true)) & (breln1unionIR = $true)) & (breln1unionIL = $true)) & (breln1unionprop = $true)) & (breln1compEex = $true)) & (breln1compE = $true)) & (breln1compI = $true)) & (breln1compprop = $true)) & (breln1invE = $true)) & (breln1invI = $true)) & (breln1invprop = $true)) & (eqbreln1 = $true)) & (subbreln1 = $true)) & (breln1all1 = $true)) & (setOfPairsIsBReln1 = $true)) & (choice2fnsingleton = $true)) & (breln1SetBreln1 = $true)) & (breln1all2 = $true)) & (woz1_5 = $true)) & (woz1_4 = $true)) & (woz1_3 = $true)) & (woz1_2 = $true)) & (woz1_1 = $true)) & (woz13rule4 = $true)) & (woz13rule3 = $true)) & (woz13rule2 = $true)) & (woz13rule1 = $true)) & (woz13rule0 = $true)) & (demorgan2 = $true)) & (demorgan2b = $true)) & (demorgan2b2 = $true)) & (demorgan2a = $true)) & (demorgan1 = $true)) & (demorgan1b = $true)) & (demorgan1a = $true)) & (demorgan2a2 = $true)) & (complementUnionInPowersetComplement = $true)) & (demorgan2a1 = $true)) & (binunionTEcontra = $true)) & (binunionTE = $true)) & (inComplementUnionImpInComplement1 = $true)) & (inComplementUnionImpNotIn1 = $true)) & (intersectInPowersetIntersectUnions = $true)) & (inIntersectImpInIntersectUnions = $true)) & (inIntersectImpInUnion2 = $true)) & (inIntersectImpInUnion = $true)) & (binunionTIRcontra = $true)) & (binunionTILcontra = $true)) & (complementTcontraSubset = $true)) & (contraSubsetComplement = $true)) & (complementInPowersetComplementIntersect = $true)) & (complementSubsetComplementIntersect = $true)) & (complementImpComplementIntersect = $true)) & (complementTnotintersectT = $true)) & (doubleComplementEq = $true)) & (doubleComplementSub2 = $true)) & (doubleComplementSub1 = $true)) & (doubleComplementE1 = $true)) & (doubleComplementI1 = $true)) & (contrasubsetT3 = $true)) & (contrasubsetT2 = $true)) & (contrasubsetT1 = $true)) & (contrasubsetT = $true)) & (binintersectTERcontra = $true)) & (binintersectTELcontra = $true)) & (complementTE1 = $true)) & (complementTI1 = $true)) & (powersetTE1 = $true)) & (powersetTI1 = $true)) & (subsetTI = $true)) & (setextT = $true)) & (complementT_lem = $true)) & (setminusT_lem = $true)) & (powersetT_lem = $true)) & (binunionT_lem = $true)) & (binintersectT_lem = $true)) & (iftrueorfalse = $true)) & (iffalse = $true)) & (iftrue = $true)) & (theeq = $true)) & (ifp = $true)) & (ifSingleton = $true)) & (iftrueProp2 = $true)) & (iftrueProp1 = $true)) & (iffalseProp2 = $true)) & (iffalseProp1 = $true)) & (eta2 = $true)) & (beta2 = $true)) & (lam2lamEq = $true)) & (eta1 = $true)) & (beta1 = $true)) & (ap2apEq2 = $true)) & (ap2apEq1 = $true)) & (funcext2 = $true)) & (funcext = $true)) & (eqbreln = $true)) & (subbreln = $true)) & (funcGraphProp4 = $true)) & (funcextLem = $true)) & (funcGraphProp2 = $true)) & (funcGraphProp3 = $true)) & (funcGraphProp1 = $true)) & (ex1E2 = $true)) & (brelnall2 = $true)) & (brelnall1 = $true)) & (lam2p = $true)) & (lamp = $true)) & (lamProp = $true)) & (funcinfuncset = $true)) & (ap2p = $true)) & (infuncsetfunc = $true)) & (app = $true)) & (apProp = $true)) & (funcImageSingleton = $true)) & (dpsetconstrER = $true)) & (dpsetconstrEL2 = $true)) & (dpsetconstrEL1 = $true)) & (dpsetconstrERa = $true)) & (setOfPairsIsBReln = $true)) & (dpsetconstrSub = $true)) & (dpsetconstrI = $true)) & (cartprodpairsurjEq = $true)) & (cartprodsndpairEq = $true)) & (cartprodfstpairEq = $true)) & (cartprodmempaircEq = $true)) & (cartprodpairmemER = $true)) & (cartprodpairmemEL = $true)) & (cartprodsndin = $true)) & (kpairsurjEq = $true)) & (ksndpairEq = $true)) & (ksndsingleton = $true)) & (setukpairinjR = $true)) & (setukpairinjR2 = $true)) & (upairequniteq = $true)) & (setukpairinjR1 = $true)) & (setukpairinjR12 = $true)) & (setukpairinjR11 = $true)) & (setukpairinjL = $true)) & (setukpairinjL2 = $true)) & (cartprodfstin = $true)) & (kfstpairEq = $true)) & (theprop = $true)) & (kfstsingleton = $true)) & (setukpairinjL1 = $true)) & (singletonsuniq = $true)) & (ex1I2 = $true)) & (ex1I = $true)) & (ex1E1 = $true)) & (singletonprop = $true)) & (setunionsingleton = $true)) & (setunionsingleton2 = $true)) & (setunionsingleton1 = $true)) & (setunionE2 = $true)) & (cartprodmempair = $true)) & (cartprodmempair1 = $true)) & (cartprodpairin = $true)) & (ubforcartprodlem3 = $true)) & (ubforcartprodlem2 = $true)) & (ubforcartprodlem1 = $true)) & (upairinpowunion = $true)) & (upairsubunion = $true)) & (upairset2E = $true)) & (singletoninpowunion = $true)) & (singletoninpowerset = $true)) & (singletonsubset = $true)) & (kpairp = $true)) & (kpairiskpair = $true)) & (setukpairIR = $true)) & (setukpairIL = $true)) & (secondinupair = $true)) & (symdiffIneg2 = $true)) & (symdiffIneg1 = $true)) & (symdiffI2 = $true)) & (symdiffI1 = $true)) & (symdiffE = $true)) & (setminusSubset1 = $true)) & (setminusLsub = $true)) & (setminusIRneg = $true)) & (setminusILneg = $true)) & (setminusELneg = $true)) & (setminusERneg = $true)) & (setminusSubset2 = $true)) & (setminusER = $true)) & (setminusEL = $true)) & (setminusI = $true)) & (bs114d = $true)) & (binintersectSubset1 = $true)) & (binintersectSubset4 = $true)) & (binintersectRsub = $true)) & (disjointsetsI1 = $true)) & (binintersectER = $true)) & (binintersectSubset3 = $true)) & (binintersectSubset2 = $true)) & (binintersectLsub = $true)) & (binintersectEL = $true)) & (binintersectSubset5 = $true)) & (binintersectI = $true)) & (binunionRsub = $true)) & (binunionLsub = $true)) & (binunionE = $true)) & (binunionEcases = $true)) & (binunionIR = $true)) & (upairset2IR = $true)) & (binunionIL = $true)) & (sepSubset = $true)) & (sepInPowerset = $true)) & (powersetsubset = $true)) & (inPowerset = $true)) & (powersetE1 = $true)) & (powersetI1 = $true)) & (subsetemptysetimpeq = $true)) & (setextsub = $true)) & (subset2powerset = $true)) & (setadjoinSub2 = $true)) & (setadjoinSub = $true)) & (subsetTrans = $true)) & (subsetRefl = $true)) & (notequalI2 = $true)) & (notequalI1 = $true)) & (notsubsetI = $true)) & (subsetE2 = $true)) & (subsetE = $true)) & (emptysetsubset = $true)) & (subsetI2 = $true)) & (eqimpsubset1 = $true)) & (eqimpsubset2 = $true)) & (subsetI1 = $true)) & (dsetconstr__Cong = $true)) & (descr__Cong = $true)) & (exuEu = $true)) & (omega__Cong = $true)) & (setunion__Cong = $true)) & (powerset__Cong = $true)) & (setadjoin__Cong = $true)) & (emptyset__Cong = $true)) & (exu__Cong = $true)) & (exuE3u = $true)) & (in__Cong = $true)) & (inCongP = $true)) & (exuI2 = $true)) & (exuI3 = $true)) & (exuI1 = $true)) & (notdallE = $true)) & (notdexE = $true)) & (prop2set2propI = $true)) & (prop2setI = $true)) & (quantDeMorgan4 = $true)) & (quantDeMorgan3 = $true)) & (quantDeMorgan2 = $true)) & (quantDeMorgan1 = $true)) & (vacuousDall = $true)) & (emptyE1 = $true)) & (upairsetIR = $true)) & (upairsetIL = $true)) & (upairsetE = $true)) & (singletonsswitch = $true)) & (eqinunit = $true)) & (notinsingleton = $true)) & (uniqinunit = $true)) & (nonemptyImpWitness = $true)) & (exuE2 = $true)) & (subPowSU = $true)) & (setunionE = $true)) & (setunionI = $true)) & (powersetE = $true)) & (emptyInPowerset = $true)) & (emptyinPowerset = $true)) & (powersetI = $true)) & (setoftrueEq = $true)) & (setadjoinOr = $true)) & (setadjoinE = $true)) & (setadjoinIR = $true)) & (emptyinunitempty = $true)) & (setadjoinIL = $true)) & (nonemptyI1 = $true)) & (nonemptyI = $true)) & (nonemptyE1 = $true)) & (setbeta = $true)) & (noeltsimpempty = $true)) & (emptyI = $true)) & (setext = $true)) & (exuE3e = $true)) & (notinemptyset = $true)) & (emptysetimpfalse = $true)) & (emptysetE = $true)) & (prop2setE = $true)) & (exuE1 = $true)) & (dsetconstrER = $true)) & (dsetconstrEL = $true)) & (dsetconstrI = $true)) & (descrp = $true)) & (wellorderingAx = $true)) & (foundationAx = $true)) & (replAx = $true)) & (omegaIndAx = $true)) & (omegaSAx = $true)) & (omega0Ax = $true)) & (setunionAx = $true)) & (powersetAx = $true)) & (setadjoinAx = $true)) & (emptysetAx = $true)) & (setextAx = $true)),
% 8.47/2.45 inference(ennf_transformation,[status(thm)],[f373])).
% 8.47/2.45 thf(f373,plain,(
% 8.47/2.45 ~((setextAx = $true) => ((emptysetAx = $true) => ((setadjoinAx = $true) => ((powersetAx = $true) => ((setunionAx = $true) => ((omega0Ax = $true) => ((omegaSAx = $true) => ((omegaIndAx = $true) => ((replAx = $true) => ((foundationAx = $true) => ((wellorderingAx = $true) => ((descrp = $true) => ((dsetconstrI = $true) => ((dsetconstrEL = $true) => ((dsetconstrER = $true) => ((exuE1 = $true) => ((prop2setE = $true) => ((emptysetE = $true) => ((emptysetimpfalse = $true) => ((notinemptyset = $true) => ((exuE3e = $true) => ((setext = $true) => ((emptyI = $true) => ((noeltsimpempty = $true) => ((setbeta = $true) => ((nonemptyE1 = $true) => ((nonemptyI = $true) => ((nonemptyI1 = $true) => ((setadjoinIL = $true) => ((emptyinunitempty = $true) => ((setadjoinIR = $true) => ((setadjoinE = $true) => ((setadjoinOr = $true) => ((setoftrueEq = $true) => ((powersetI = $true) => ((emptyinPowerset = $true) => ((emptyInPowerset = $true) => ((powersetE = $true) => ((setunionI = $true) => ((setunionE = $true) => ((subPowSU = $true) => ((exuE2 = $true) => ((nonemptyImpWitness = $true) => ((uniqinunit = $true) => ((notinsingleton = $true) => ((eqinunit = $true) => ((singletonsswitch = $true) => ((upairsetE = $true) => ((upairsetIL = $true) => ((upairsetIR = $true) => ((emptyE1 = $true) => ((vacuousDall = $true) => ((quantDeMorgan1 = $true) => ((quantDeMorgan2 = $true) => ((quantDeMorgan3 = $true) => ((quantDeMorgan4 = $true) => ((prop2setI = $true) => ((prop2set2propI = $true) => ((notdexE = $true) => ((notdallE = $true) => ((exuI1 = $true) => ((exuI3 = $true) => ((exuI2 = $true) => ((inCongP = $true) => ((in__Cong = $true) => ((exuE3u = $true) => ((exu__Cong = $true) => ((emptyset__Cong = $true) => ((setadjoin__Cong = $true) => ((powerset__Cong = $true) => ((setunion__Cong = $true) => ((omega__Cong = $true) => ((exuEu = $true) => ((descr__Cong = $true) => ((dsetconstr__Cong = $true) => ((subsetI1 = $true) => ((eqimpsubset2 = $true) => ((eqimpsubset1 = $true) => ((subsetI2 = $true) => ((emptysetsubset = $true) => ((subsetE = $true) => ((subsetE2 = $true) => ((notsubsetI = $true) => ((notequalI1 = $true) => ((notequalI2 = $true) => ((subsetRefl = $true) => ((subsetTrans = $true) => ((setadjoinSub = $true) => ((setadjoinSub2 = $true) => ((subset2powerset = $true) => ((setextsub = $true) => ((subsetemptysetimpeq = $true) => ((powersetI1 = $true) => ((powersetE1 = $true) => ((inPowerset = $true) => ((powersetsubset = $true) => ((sepInPowerset = $true) => ((sepSubset = $true) => ((binunionIL = $true) => ((upairset2IR = $true) => ((binunionIR = $true) => ((binunionEcases = $true) => ((binunionE = $true) => ((binunionLsub = $true) => ((binunionRsub = $true) => ((binintersectI = $true) => ((binintersectSubset5 = $true) => ((binintersectEL = $true) => ((binintersectLsub = $true) => ((binintersectSubset2 = $true) => ((binintersectSubset3 = $true) => ((binintersectER = $true) => ((disjointsetsI1 = $true) => ((binintersectRsub = $true) => ((binintersectSubset4 = $true) => ((binintersectSubset1 = $true) => ((bs114d = $true) => ((setminusI = $true) => ((setminusEL = $true) => ((setminusER = $true) => ((setminusSubset2 = $true) => ((setminusERneg = $true) => ((setminusELneg = $true) => ((setminusILneg = $true) => ((setminusIRneg = $true) => ((setminusLsub = $true) => ((setminusSubset1 = $true) => ((symdiffE = $true) => ((symdiffI1 = $true) => ((symdiffI2 = $true) => ((symdiffIneg1 = $true) => ((symdiffIneg2 = $true) => ((secondinupair = $true) => ((setukpairIL = $true) => ((setukpairIR = $true) => ((kpairiskpair = $true) => ((kpairp = $true) => ((singletonsubset = $true) => ((singletoninpowerset = $true) => ((singletoninpowunion = $true) => ((upairset2E = $true) => ((upairsubunion = $true) => ((upairinpowunion = $true) => ((ubforcartprodlem1 = $true) => ((ubforcartprodlem2 = $true) => ((ubforcartprodlem3 = $true) => ((cartprodpairin = $true) => ((cartprodmempair1 = $true) => ((cartprodmempair = $true) => ((setunionE2 = $true) => ((setunionsingleton1 = $true) => ((setunionsingleton2 = $true) => ((setunionsingleton = $true) => ((singletonprop = $true) => ((ex1E1 = $true) => ((ex1I = $true) => ((ex1I2 = $true) => ((singletonsuniq = $true) => ((setukpairinjL1 = $true) => ((kfstsingleton = $true) => ((theprop = $true) => ((kfstpairEq = $true) => ((cartprodfstin = $true) => ((setukpairinjL2 = $true) => ((setukpairinjL = $true) => ((setukpairinjR11 = $true) => ((setukpairinjR12 = $true) => ((setukpairinjR1 = $true) => ((upairequniteq = $true) => ((setukpairinjR2 = $true) => ((setukpairinjR = $true) => ((ksndsingleton = $true) => ((ksndpairEq = $true) => ((kpairsurjEq = $true) => ((cartprodsndin = $true) => ((cartprodpairmemEL = $true) => ((cartprodpairmemER = $true) => ((cartprodmempaircEq = $true) => ((cartprodfstpairEq = $true) => ((cartprodsndpairEq = $true) => ((cartprodpairsurjEq = $true) => ((dpsetconstrI = $true) => ((dpsetconstrSub = $true) => ((setOfPairsIsBReln = $true) => ((dpsetconstrERa = $true) => ((dpsetconstrEL1 = $true) => ((dpsetconstrEL2 = $true) => ((dpsetconstrER = $true) => ((funcImageSingleton = $true) => ((apProp = $true) => ((app = $true) => ((infuncsetfunc = $true) => ((ap2p = $true) => ((funcinfuncset = $true) => ((lamProp = $true) => ((lamp = $true) => ((lam2p = $true) => ((brelnall1 = $true) => ((brelnall2 = $true) => ((ex1E2 = $true) => ((funcGraphProp1 = $true) => ((funcGraphProp3 = $true) => ((funcGraphProp2 = $true) => ((funcextLem = $true) => ((funcGraphProp4 = $true) => ((subbreln = $true) => ((eqbreln = $true) => ((funcext = $true) => ((funcext2 = $true) => ((ap2apEq1 = $true) => ((ap2apEq2 = $true) => ((beta1 = $true) => ((eta1 = $true) => ((lam2lamEq = $true) => ((beta2 = $true) => ((eta2 = $true) => ((iffalseProp1 = $true) => ((iffalseProp2 = $true) => ((iftrueProp1 = $true) => ((iftrueProp2 = $true) => ((ifSingleton = $true) => ((ifp = $true) => ((theeq = $true) => ((iftrue = $true) => ((iffalse = $true) => ((iftrueorfalse = $true) => ((binintersectT_lem = $true) => ((binunionT_lem = $true) => ((powersetT_lem = $true) => ((setminusT_lem = $true) => ((complementT_lem = $true) => ((setextT = $true) => ((subsetTI = $true) => ((powersetTI1 = $true) => ((powersetTE1 = $true) => ((complementTI1 = $true) => ((complementTE1 = $true) => ((binintersectTELcontra = $true) => ((binintersectTERcontra = $true) => ((contrasubsetT = $true) => ((contrasubsetT1 = $true) => ((contrasubsetT2 = $true) => ((contrasubsetT3 = $true) => ((doubleComplementI1 = $true) => ((doubleComplementE1 = $true) => ((doubleComplementSub1 = $true) => ((doubleComplementSub2 = $true) => ((doubleComplementEq = $true) => ((complementTnotintersectT = $true) => ((complementImpComplementIntersect = $true) => ((complementSubsetComplementIntersect = $true) => ((complementInPowersetComplementIntersect = $true) => ((contraSubsetComplement = $true) => ((complementTcontraSubset = $true) => ((binunionTILcontra = $true) => ((binunionTIRcontra = $true) => ((inIntersectImpInUnion = $true) => ((inIntersectImpInUnion2 = $true) => ((inIntersectImpInIntersectUnions = $true) => ((intersectInPowersetIntersectUnions = $true) => ((inComplementUnionImpNotIn1 = $true) => ((inComplementUnionImpInComplement1 = $true) => ((binunionTE = $true) => ((binunionTEcontra = $true) => ((demorgan2a1 = $true) => ((complementUnionInPowersetComplement = $true) => ((demorgan2a2 = $true) => ((demorgan1a = $true) => ((demorgan1b = $true) => ((demorgan1 = $true) => ((demorgan2a = $true) => ((demorgan2b2 = $true) => ((demorgan2b = $true) => ((demorgan2 = $true) => ((woz13rule0 = $true) => ((woz13rule1 = $true) => ((woz13rule2 = $true) => ((woz13rule3 = $true) => ((woz13rule4 = $true) => ((woz1_1 = $true) => ((woz1_2 = $true) => ((woz1_3 = $true) => ((woz1_4 = $true) => ((woz1_5 = $true) => ((breln1all2 = $true) => ((breln1SetBreln1 = $true) => ((choice2fnsingleton = $true) => ((setOfPairsIsBReln1 = $true) => ((breln1all1 = $true) => ((subbreln1 = $true) => ((eqbreln1 = $true) => ((breln1invprop = $true) => ((breln1invI = $true) => ((breln1invE = $true) => ((breln1compprop = $true) => ((breln1compI = $true) => ((breln1compE = $true) => ((breln1compEex = $true) => ((breln1unionprop = $true) => ((breln1unionIL = $true) => ((breln1unionIR = $true) => ((breln1unionI = $true) => ((breln1unionE = $true) => ((breln1unionEcases = $true) => ((breln1unionCommutes = $true) => ((woz2Ex = $true) => ((woz2W = $true) => ((woz2A = $true) => ((woz2B = $true) => ((image1Ex = $true) => ((image1Ex1 = $true) => ((image1Equiv = $true) => ((image1E = $true) => ((image1I = $true) => ((injFuncInInjFuncSet = $true) => ((injFuncSetFuncIn = $true) => ((injFuncSetFuncInj = $true) => ((surjFuncSetFuncIn = $true) => ((surjFuncSetFuncSurj = $true) => ((leftInvIsSurj = $true) => ((surjCantorThm = $true) => ((foundation2 = $true) => ((notinself = $true) => ((notinself2 = $true) => ((omegaSp = $true) => ((omegaSclos = $true) => ((peano0notS = $true) => ((peanoSinj = $true) => ((transitivesetOp1 = $true) => ! [X0 : $i] : (((transitiveset @ X0) = $true) => ! [X1 : $i] : (((transitiveset @ X1) = $true) => ((transitiveset @ (binintersect @ X0 @ X1)) = $true))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 8.47/2.45 inference(fool_elimination,[status(thm)],[f372])).
% 8.47/2.45 thf(f372,plain,(
% 8.47/2.45 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => (ubforcartprodlem2 => (ubforcartprodlem3 => (cartprodpairin => (cartprodmempair1 => (cartprodmempair => (setunionE2 => (setunionsingleton1 => (setunionsingleton2 => (setunionsingleton => (singletonprop => (ex1E1 => (ex1I => (ex1I2 => (singletonsuniq => (setukpairinjL1 => (kfstsingleton => (theprop => (kfstpairEq => (cartprodfstin => (setukpairinjL2 => (setukpairinjL => (setukpairinjR11 => (setukpairinjR12 => (setukpairinjR1 => (upairequniteq => (setukpairinjR2 => (setukpairinjR => (ksndsingleton => (ksndpairEq => (kpairsurjEq => (cartprodsndin => (cartprodpairmemEL => (cartprodpairmemER => (cartprodmempaircEq => (cartprodfstpairEq => (cartprodsndpairEq => (cartprodpairsurjEq => (dpsetconstrI => (dpsetconstrSub => (setOfPairsIsBReln => (dpsetconstrERa => (dpsetconstrEL1 => (dpsetconstrEL2 => (dpsetconstrER => (funcImageSingleton => (apProp => (app => (infuncsetfunc => (ap2p => (funcinfuncset => (lamProp => (lamp => (lam2p => (brelnall1 => (brelnall2 => (ex1E2 => (funcGraphProp1 => (funcGraphProp3 => (funcGraphProp2 => (funcextLem => (funcGraphProp4 => (subbreln => (eqbreln => (funcext => (funcext2 => (ap2apEq1 => (ap2apEq2 => (beta1 => (eta1 => (lam2lamEq => (beta2 => (eta2 => (iffalseProp1 => (iffalseProp2 => (iftrueProp1 => (iftrueProp2 => (ifSingleton => (ifp => (theeq => (iftrue => (iffalse => (iftrueorfalse => (binintersectT_lem => (binunionT_lem => (powersetT_lem => (setminusT_lem => (complementT_lem => (setextT => (subsetTI => (powersetTI1 => (powersetTE1 => (complementTI1 => (complementTE1 => (binintersectTELcontra => (binintersectTERcontra => (contrasubsetT => (contrasubsetT1 => (contrasubsetT2 => (contrasubsetT3 => (doubleComplementI1 => (doubleComplementE1 => (doubleComplementSub1 => (doubleComplementSub2 => (doubleComplementEq => (complementTnotintersectT => (complementImpComplementIntersect => (complementSubsetComplementIntersect => (complementInPowersetComplementIntersect => (contraSubsetComplement => (complementTcontraSubset => (binunionTILcontra => (binunionTIRcontra => (inIntersectImpInUnion => (inIntersectImpInUnion2 => (inIntersectImpInIntersectUnions => (intersectInPowersetIntersectUnions => (inComplementUnionImpNotIn1 => (inComplementUnionImpInComplement1 => (binunionTE => (binunionTEcontra => (demorgan2a1 => (complementUnionInPowersetComplement => (demorgan2a2 => (demorgan1a => (demorgan1b => (demorgan1 => (demorgan2a => (demorgan2b2 => (demorgan2b => (demorgan2 => (woz13rule0 => (woz13rule1 => (woz13rule2 => (woz13rule3 => (woz13rule4 => (woz1_1 => (woz1_2 => (woz1_3 => (woz1_4 => (woz1_5 => (breln1all2 => (breln1SetBreln1 => (choice2fnsingleton => (setOfPairsIsBReln1 => (breln1all1 => (subbreln1 => (eqbreln1 => (breln1invprop => (breln1invI => (breln1invE => (breln1compprop => (breln1compI => (breln1compE => (breln1compEex => (breln1unionprop => (breln1unionIL => (breln1unionIR => (breln1unionI => (breln1unionE => (breln1unionEcases => (breln1unionCommutes => (woz2Ex => (woz2W => (woz2A => (woz2B => (image1Ex => (image1Ex1 => (image1Equiv => (image1E => (image1I => (injFuncInInjFuncSet => (injFuncSetFuncIn => (injFuncSetFuncInj => (surjFuncSetFuncIn => (surjFuncSetFuncSurj => (leftInvIsSurj => (surjCantorThm => (foundation2 => (notinself => (notinself2 => (omegaSp => (omegaSclos => (peano0notS => (peanoSinj => (transitivesetOp1 => ! [X0 : $i] : ((transitiveset @ X0) => ! [X1 : $i] : ((transitiveset @ X1) => (transitiveset @ (binintersect @ X0 @ X1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 8.47/2.46 inference(rectify,[status(thm)],[f334])).
% 8.47/2.46 thf(f334,negated_conjecture,(
% 8.47/2.46 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => (ubforcartprodlem2 => (ubforcartprodlem3 => (cartprodpairin => (cartprodmempair1 => (cartprodmempair => (setunionE2 => (setunionsingleton1 => (setunionsingleton2 => (setunionsingleton => (singletonprop => (ex1E1 => (ex1I => (ex1I2 => (singletonsuniq => (setukpairinjL1 => (kfstsingleton => (theprop => (kfstpairEq => (cartprodfstin => (setukpairinjL2 => (setukpairinjL => (setukpairinjR11 => (setukpairinjR12 => (setukpairinjR1 => (upairequniteq => (setukpairinjR2 => (setukpairinjR => (ksndsingleton => (ksndpairEq => (kpairsurjEq => (cartprodsndin => (cartprodpairmemEL => (cartprodpairmemER => (cartprodmempaircEq => (cartprodfstpairEq => (cartprodsndpairEq => (cartprodpairsurjEq => (dpsetconstrI => (dpsetconstrSub => (setOfPairsIsBReln => (dpsetconstrERa => (dpsetconstrEL1 => (dpsetconstrEL2 => (dpsetconstrER => (funcImageSingleton => (apProp => (app => (infuncsetfunc => (ap2p => (funcinfuncset => (lamProp => (lamp => (lam2p => (brelnall1 => (brelnall2 => (ex1E2 => (funcGraphProp1 => (funcGraphProp3 => (funcGraphProp2 => (funcextLem => (funcGraphProp4 => (subbreln => (eqbreln => (funcext => (funcext2 => (ap2apEq1 => (ap2apEq2 => (beta1 => (eta1 => (lam2lamEq => (beta2 => (eta2 => (iffalseProp1 => (iffalseProp2 => (iftrueProp1 => (iftrueProp2 => (ifSingleton => (ifp => (theeq => (iftrue => (iffalse => (iftrueorfalse => (binintersectT_lem => (binunionT_lem => (powersetT_lem => (setminusT_lem => (complementT_lem => (setextT => (subsetTI => (powersetTI1 => (powersetTE1 => (complementTI1 => (complementTE1 => (binintersectTELcontra => (binintersectTERcontra => (contrasubsetT => (contrasubsetT1 => (contrasubsetT2 => (contrasubsetT3 => (doubleComplementI1 => (doubleComplementE1 => (doubleComplementSub1 => (doubleComplementSub2 => (doubleComplementEq => (complementTnotintersectT => (complementImpComplementIntersect => (complementSubsetComplementIntersect => (complementInPowersetComplementIntersect => (contraSubsetComplement => (complementTcontraSubset => (binunionTILcontra => (binunionTIRcontra => (inIntersectImpInUnion => (inIntersectImpInUnion2 => (inIntersectImpInIntersectUnions => (intersectInPowersetIntersectUnions => (inComplementUnionImpNotIn1 => (inComplementUnionImpInComplement1 => (binunionTE => (binunionTEcontra => (demorgan2a1 => (complementUnionInPowersetComplement => (demorgan2a2 => (demorgan1a => (demorgan1b => (demorgan1 => (demorgan2a => (demorgan2b2 => (demorgan2b => (demorgan2 => (woz13rule0 => (woz13rule1 => (woz13rule2 => (woz13rule3 => (woz13rule4 => (woz1_1 => (woz1_2 => (woz1_3 => (woz1_4 => (woz1_5 => (breln1all2 => (breln1SetBreln1 => (choice2fnsingleton => (setOfPairsIsBReln1 => (breln1all1 => (subbreln1 => (eqbreln1 => (breln1invprop => (breln1invI => (breln1invE => (breln1compprop => (breln1compI => (breln1compE => (breln1compEex => (breln1unionprop => (breln1unionIL => (breln1unionIR => (breln1unionI => (breln1unionE => (breln1unionEcases => (breln1unionCommutes => (woz2Ex => (woz2W => (woz2A => (woz2B => (image1Ex => (image1Ex1 => (image1Equiv => (image1E => (image1I => (injFuncInInjFuncSet => (injFuncSetFuncIn => (injFuncSetFuncInj => (surjFuncSetFuncIn => (surjFuncSetFuncSurj => (leftInvIsSurj => (surjCantorThm => (foundation2 => (notinself => (notinself2 => (omegaSp => (omegaSclos => (peano0notS => (peanoSinj => (transitivesetOp1 => ! [X11 : $i] : ((transitiveset @ X11) => ! [X16 : $i] : ((transitiveset @ X16) => (transitiveset @ (binintersect @ X11 @ X16)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 8.47/2.46 inference(negated_conjecture,[status(cth)],[f333])).
% 8.47/2.46 thf(f333,conjecture,(
% 8.47/2.46 setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => (ubforcartprodlem2 => (ubforcartprodlem3 => (cartprodpairin => (cartprodmempair1 => (cartprodmempair => (setunionE2 => (setunionsingleton1 => (setunionsingleton2 => (setunionsingleton => (singletonprop => (ex1E1 => (ex1I => (ex1I2 => (singletonsuniq => (setukpairinjL1 => (kfstsingleton => (theprop => (kfstpairEq => (cartprodfstin => (setukpairinjL2 => (setukpairinjL => (setukpairinjR11 => (setukpairinjR12 => (setukpairinjR1 => (upairequniteq => (setukpairinjR2 => (setukpairinjR => (ksndsingleton => (ksndpairEq => (kpairsurjEq => (cartprodsndin => (cartprodpairmemEL => (cartprodpairmemER => (cartprodmempaircEq => (cartprodfstpairEq => (cartprodsndpairEq => (cartprodpairsurjEq => (dpsetconstrI => (dpsetconstrSub => (setOfPairsIsBReln => (dpsetconstrERa => (dpsetconstrEL1 => (dpsetconstrEL2 => (dpsetconstrER => (funcImageSingleton => (apProp => (app => (infuncsetfunc => (ap2p => (funcinfuncset => (lamProp => (lamp => (lam2p => (brelnall1 => (brelnall2 => (ex1E2 => (funcGraphProp1 => (funcGraphProp3 => (funcGraphProp2 => (funcextLem => (funcGraphProp4 => (subbreln => (eqbreln => (funcext => (funcext2 => (ap2apEq1 => (ap2apEq2 => (beta1 => (eta1 => (lam2lamEq => (beta2 => (eta2 => (iffalseProp1 => (iffalseProp2 => (iftrueProp1 => (iftrueProp2 => (ifSingleton => (ifp => (theeq => (iftrue => (iffalse => (iftrueorfalse => (binintersectT_lem => (binunionT_lem => (powersetT_lem => (setminusT_lem => (complementT_lem => (setextT => (subsetTI => (powersetTI1 => (powersetTE1 => (complementTI1 => (complementTE1 => (binintersectTELcontra => (binintersectTERcontra => (contrasubsetT => (contrasubsetT1 => (contrasubsetT2 => (contrasubsetT3 => (doubleComplementI1 => (doubleComplementE1 => (doubleComplementSub1 => (doubleComplementSub2 => (doubleComplementEq => (complementTnotintersectT => (complementImpComplementIntersect => (complementSubsetComplementIntersect => (complementInPowersetComplementIntersect => (contraSubsetComplement => (complementTcontraSubset => (binunionTILcontra => (binunionTIRcontra => (inIntersectImpInUnion => (inIntersectImpInUnion2 => (inIntersectImpInIntersectUnions => (intersectInPowersetIntersectUnions => (inComplementUnionImpNotIn1 => (inComplementUnionImpInComplement1 => (binunionTE => (binunionTEcontra => (demorgan2a1 => (complementUnionInPowersetComplement => (demorgan2a2 => (demorgan1a => (demorgan1b => (demorgan1 => (demorgan2a => (demorgan2b2 => (demorgan2b => (demorgan2 => (woz13rule0 => (woz13rule1 => (woz13rule2 => (woz13rule3 => (woz13rule4 => (woz1_1 => (woz1_2 => (woz1_3 => (woz1_4 => (woz1_5 => (breln1all2 => (breln1SetBreln1 => (choice2fnsingleton => (setOfPairsIsBReln1 => (breln1all1 => (subbreln1 => (eqbreln1 => (breln1invprop => (breln1invI => (breln1invE => (breln1compprop => (breln1compI => (breln1compE => (breln1compEex => (breln1unionprop => (breln1unionIL => (breln1unionIR => (breln1unionI => (breln1unionE => (breln1unionEcases => (breln1unionCommutes => (woz2Ex => (woz2W => (woz2A => (woz2B => (image1Ex => (image1Ex1 => (image1Equiv => (image1E => (image1I => (injFuncInInjFuncSet => (injFuncSetFuncIn => (injFuncSetFuncInj => (surjFuncSetFuncIn => (surjFuncSetFuncSurj => (leftInvIsSurj => (surjCantorThm => (foundation2 => (notinself => (notinself2 => (omegaSp => (omegaSclos => (peano0notS => (peanoSinj => (transitivesetOp1 => ! [X11 : $i] : ((transitiveset @ X11) => ! [X16 : $i] : ((transitiveset @ X16) => (transitiveset @ (binintersect @ X11 @ X16))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 8.47/2.46 file('/export/starexec/sandbox2/tmp/tmp.H1wBmJGvZQ/DTF2THF_2486.p',binintTransitive)).
% 8.47/2.46 thf(f9722,plain,(
% 8.47/2.46 ((subset @ sK1183 @ sK955) != $true) | spl1181_214),
% 8.47/2.46 inference(avatar_component_clause,[status(thm)],[f9721])).
% 8.47/2.46 thf(f9721,definition,(
% 8.47/2.46 spl1181_214 <=> ((subset @ sK1183 @ sK955) = $true)),
% 8.47/2.46 introduced(definition,[new_symbols(naming,[spl1181_214])],[avatar_definition])).
% 8.47/2.46 thf(f8737,plain,(
% 8.47/2.46 ($true = (in @ sK1183 @ sK955))),
% 8.47/2.46 inference(trivial_inequality_removal,[status(thm)],[f8730])).
% 8.47/2.46 thf(f8730,plain,(
% 8.47/2.46 ($true != $true) | ($true = (in @ sK1183 @ sK955))),
% 8.47/2.46 inference(superposition,[status(thm)],[f6995,f6850])).
% 8.47/2.46 thf(f6850,plain,(
% 8.47/2.46 ((in @ sK1183 @ (binintersect @ sK955 @ sK956)) = $true)),
% 8.47/2.46 inference(binary_proxy_clausification,[status(thm)],[f6848])).
% 8.47/2.46 thf(f6848,plain,(
% 8.47/2.46 (((in @ sK1183 @ (binintersect @ sK955 @ sK956)) => (subset @ sK1183 @ (binintersect @ sK955 @ sK956))) = $false)),
% 8.47/2.46 inference(beta_eta_normalization,[status(thm)],[f6847])).
% 8.47/2.46 thf(f6847,plain,(
% 8.47/2.46 (((^[Y0 : $i]: ((in @ Y0 @ (binintersect @ sK955 @ sK956)) => (subset @ Y0 @ (binintersect @ sK955 @ sK956)))) @ sK1183) = $false)),
% 8.47/2.46 inference(sigma_clausification,[status(thm)],[f6846])).
% 8.47/2.46 thf(f6846,plain,(
% 8.47/2.46 ((!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ (binintersect @ sK955 @ sK956)) => (subset @ Y0 @ (binintersect @ sK955 @ sK956))))) != $true)),
% 8.47/2.46 inference(beta_eta_normalization,[status(thm)],[f6465])).
% 8.47/2.46 thf(f6465,plain,(
% 8.47/2.46 (((^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ Y0) => (subset @ Y1 @ Y0))))) @ (binintersect @ sK955 @ sK956)) != $true)),
% 8.47/2.46 inference(definition_unfolding,[status(thm)],[f4913,f3583])).
% 8.47/2.46 thf(f4913,plain,(
% 8.47/2.46 ($true != (transitiveset @ (binintersect @ sK955 @ sK956)))),
% 8.47/2.46 inference(cnf_transformation,[status(thm)],[f3115])).
% 8.47/2.46 thf(f6995,plain,(
% 8.47/2.46 ( ! [X3 : $i,X4 : $i,X5 : $i] : (((in @ X4 @ (binintersect @ X3 @ X5)) != $true) | ((in @ X4 @ X3) = $true)) )),
% 8.47/2.46 inference(trivial_inequality_removal,[status(thm)],[f6227])).
% 8.47/2.46 thf(f6227,plain,(
% 8.47/2.46 ( ! [X3 : $i,X4 : $i,X5 : $i] : (((in @ X4 @ (binintersect @ X3 @ X5)) != $true) | ((in @ X4 @ X3) = $true) | ($true != $true)) )),
% 8.47/2.46 inference(definition_unfolding,[status(thm)],[f4554,f4892])).
% 8.47/2.46 thf(f4892,plain,(
% 8.47/2.46 (binintersectEL = $true)),
% 8.47/2.46 inference(cnf_transformation,[status(thm)],[f3115])).
% 8.47/2.46 thf(f4554,plain,(
% 8.47/2.46 ( ! [X3 : $i,X4 : $i,X5 : $i] : (((in @ X4 @ X3) = $true) | ((in @ X4 @ (binintersect @ X3 @ X5)) != $true) | (binintersectEL != $true)) )),
% 8.47/2.46 inference(cnf_transformation,[status(thm)],[f2806])).
% 8.47/2.46 thf(f2806,plain,(
% 8.47/2.46 ((binintersectEL = $true) | (((in @ sK761 @ sK760) != $true) & ($true = (in @ sK761 @ (binintersect @ sK760 @ sK762))))) & (! [X3 : $i,X4 : $i,X5 : $i] : (((in @ X4 @ X3) = $true) | ((in @ X4 @ (binintersect @ X3 @ X5)) != $true)) | (binintersectEL != $true))),
% 8.47/2.46 inference(skolemisation,[status(esa),new_symbols(skolem,[sK760,sK761,sK762])],[f2804,f2805])).
% 8.47/2.46 thf(f2805,plain,(
% 8.47/2.46 ? [X0 : $i,X1 : $i,X2 : $i] : (($true != (in @ X1 @ X0)) & ($true = (in @ X1 @ (binintersect @ X0 @ X2)))) => (((in @ sK761 @ sK760) != $true) & ($true = (in @ sK761 @ (binintersect @ sK760 @ sK762))))),
% 8.47/2.46 introduced(definition,[],[choice_axiom])).
% 8.47/2.46 thf(f2804,plain,(
% 8.47/2.46 ((binintersectEL = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (($true != (in @ X1 @ X0)) & ($true = (in @ X1 @ (binintersect @ X0 @ X2))))) & (! [X3 : $i,X4 : $i,X5 : $i] : (((in @ X4 @ X3) = $true) | ((in @ X4 @ (binintersect @ X3 @ X5)) != $true)) | (binintersectEL != $true))),
% 8.47/2.46 inference(rectify,[status(thm)],[f2803])).
% 8.47/2.46 thf(f2803,plain,(
% 8.47/2.46 ((binintersectEL = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (($true != (in @ X1 @ X0)) & ($true = (in @ X1 @ (binintersect @ X0 @ X2))))) & (! [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X1 @ X0)) | ($true != (in @ X1 @ (binintersect @ X0 @ X2)))) | (binintersectEL != $true))),
% 8.47/2.46 inference(nnf_transformation,[status(thm)],[f1077])).
% 8.47/2.46 thf(f1077,plain,(
% 8.47/2.46 (binintersectEL = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X1 @ X0)) | ($true != (in @ X1 @ (binintersect @ X0 @ X2))))),
% 8.47/2.46 inference(ennf_transformation,[status(thm)],[f929])).
% 8.47/2.46 thf(f929,plain,(
% 8.47/2.46 ! [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X1 @ (binintersect @ X0 @ X2))) => ($true = (in @ X1 @ X0))) <=> (binintersectEL = $true)),
% 8.47/2.46 inference(fool_elimination,[status(thm)],[f928])).
% 8.47/2.46 thf(f928,plain,(
% 8.47/2.46 (binintersectEL = ! [X0 : $i,X1 : $i,X2 : $i] : ((in @ X1 @ (binintersect @ X0 @ X2)) => (in @ X1 @ X0)))),
% 8.47/2.46 inference(rectify,[status(thm)],[f110])).
% 8.47/2.46 thf(f110,axiom,(
% 8.47/2.46 (binintersectEL = ! [X3 : $i,X1 : $i,X4 : $i] : ((in @ X1 @ (binintersect @ X3 @ X4)) => (in @ X1 @ X3)))),
% 8.47/2.46 file('/export/starexec/sandbox2/tmp/tmp.H1wBmJGvZQ/DTF2THF_2486.p',binintersectEL)).
% 8.47/2.46 thf(f9777,plain,(
% 8.47/2.46 spl1181_213),
% 8.47/2.46 inference(avatar_contradiction_clause,[status(thm)],[f9776])).
% 8.47/2.46 thf(f9776,plain,(
% 8.47/2.46 $false | spl1181_213),
% 8.47/2.46 inference(trivial_inequality_removal,[status(thm)],[f9767])).
% 8.47/2.46 thf(f9767,plain,(
% 8.47/2.46 ($false = $true) | spl1181_213),
% 8.47/2.46 inference(backward_demodulation,[status(thm)],[f8639,f9733])).
% 8.47/2.46 thf(f9733,plain,(
% 8.47/2.46 ((in @ sK1183 @ sK956) = $false) | spl1181_213),
% 8.47/2.46 inference(trivial_inequality_removal,[status(thm)],[f9725])).
% 8.47/2.46 thf(f9725,plain,(
% 8.47/2.46 ((in @ sK1183 @ sK956) = $false) | ($true != $true) | spl1181_213),
% 8.47/2.46 inference(superposition,[status(thm)],[f9719,f6778])).
% 8.47/2.46 thf(f6778,plain,(
% 8.47/2.46 ( ! [X1 : $i] : (((subset @ X1 @ sK956) = $true) | ((in @ X1 @ sK956) = $false)) )),
% 8.47/2.46 inference(binary_proxy_clausification,[status(thm)],[f6777])).
% 8.47/2.46 thf(f6777,plain,(
% 8.47/2.46 ( ! [X1 : $i] : (($true = ((in @ X1 @ sK956) => (subset @ X1 @ sK956)))) )),
% 8.47/2.46 inference(beta_eta_normalization,[status(thm)],[f6776])).
% 8.47/2.46 thf(f6776,plain,(
% 8.47/2.46 ( ! [X1 : $i] : ((((^[Y0 : $i]: ((in @ Y0 @ sK956) => (subset @ Y0 @ sK956))) @ X1) = $true)) )),
% 8.47/2.46 inference(pi_clausification,[status(thm)],[f6775])).
% 8.47/2.46 thf(f6775,plain,(
% 8.47/2.46 ((!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ sK956) => (subset @ Y0 @ sK956)))) = $true)),
% 8.47/2.46 inference(beta_eta_normalization,[status(thm)],[f6466])).
% 8.47/2.46 thf(f6466,plain,(
% 8.47/2.46 ($true = ((^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ Y0) => (subset @ Y1 @ Y0))))) @ sK956))),
% 8.47/2.46 inference(definition_unfolding,[status(thm)],[f4912,f3583])).
% 8.47/2.46 thf(f4912,plain,(
% 8.47/2.46 ($true = (transitiveset @ sK956))),
% 8.47/2.46 inference(cnf_transformation,[status(thm)],[f3115])).
% 8.47/2.46 thf(f9719,plain,(
% 8.47/2.46 ((subset @ sK1183 @ sK956) != $true) | spl1181_213),
% 8.47/2.46 inference(avatar_component_clause,[status(thm)],[f9718])).
% 8.47/2.46 thf(f9718,definition,(
% 8.47/2.46 spl1181_213 <=> ((subset @ sK1183 @ sK956) = $true)),
% 8.47/2.46 introduced(definition,[new_symbols(naming,[spl1181_213])],[avatar_definition])).
% 8.47/2.46 thf(f8639,plain,(
% 8.47/2.46 ((in @ sK1183 @ sK956) = $true)),
% 8.47/2.46 inference(trivial_inequality_removal,[status(thm)],[f8631])).
% 8.47/2.46 thf(f8631,plain,(
% 8.47/2.46 ($true != $true) | ((in @ sK1183 @ sK956) = $true)),
% 8.47/2.46 inference(superposition,[status(thm)],[f6761,f6850])).
% 8.47/2.46 thf(f6761,plain,(
% 8.47/2.46 ( ! [X2 : $i,X0 : $i,X1 : $i] : (($true != (in @ X2 @ (binintersect @ X0 @ X1))) | ((in @ X2 @ X1) = $true)) )),
% 8.47/2.46 inference(trivial_inequality_removal,[status(thm)],[f6288])).
% 8.47/2.46 thf(f6288,plain,(
% 8.47/2.46 ( ! [X2 : $i,X0 : $i,X1 : $i] : (($true != $true) | ($true != (in @ X2 @ (binintersect @ X0 @ X1))) | ((in @ X2 @ X1) = $true)) )),
% 8.47/2.46 inference(definition_unfolding,[status(thm)],[f4631,f4883])).
% 8.47/2.46 thf(f4883,plain,(
% 8.47/2.46 (binintersectER = $true)),
% 8.47/2.46 inference(cnf_transformation,[status(thm)],[f3115])).
% 8.47/2.46 thf(f4631,plain,(
% 8.47/2.46 ( ! [X2 : $i,X0 : $i,X1 : $i] : (((in @ X2 @ X1) = $true) | ($true != (in @ X2 @ (binintersect @ X0 @ X1))) | (binintersectER != $true)) )),
% 8.47/2.46 inference(cnf_transformation,[status(thm)],[f2887])).
% 8.47/2.46 thf(f2887,plain,(
% 8.47/2.46 (! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X2 @ X1) = $true) | ($true != (in @ X2 @ (binintersect @ X0 @ X1)))) | (binintersectER != $true)) & ((binintersectER = $true) | (((in @ sK819 @ sK818) != $true) & ((in @ sK819 @ (binintersect @ sK817 @ sK818)) = $true)))),
% 8.47/2.46 inference(skolemisation,[status(esa),new_symbols(skolem,[sK817,sK818,sK819])],[f2885,f2886])).
% 8.47/2.46 thf(f2886,plain,(
% 8.47/2.46 ? [X3 : $i,X4 : $i,X5 : $i] : (((in @ X5 @ X4) != $true) & ((in @ X5 @ (binintersect @ X3 @ X4)) = $true)) => (((in @ sK819 @ sK818) != $true) & ((in @ sK819 @ (binintersect @ sK817 @ sK818)) = $true))),
% 8.47/2.46 introduced(definition,[],[choice_axiom])).
% 8.47/2.46 thf(f2885,plain,(
% 8.47/2.46 (! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X2 @ X1) = $true) | ($true != (in @ X2 @ (binintersect @ X0 @ X1)))) | (binintersectER != $true)) & ((binintersectER = $true) | ? [X3 : $i,X4 : $i,X5 : $i] : (((in @ X5 @ X4) != $true) & ((in @ X5 @ (binintersect @ X3 @ X4)) = $true)))),
% 8.47/2.46 inference(rectify,[status(thm)],[f2884])).
% 8.47/2.46 thf(f2884,plain,(
% 8.47/2.46 (! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X2 @ X1) = $true) | ($true != (in @ X2 @ (binintersect @ X0 @ X1)))) | (binintersectER != $true)) & ((binintersectER = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (((in @ X2 @ X1) != $true) & ($true = (in @ X2 @ (binintersect @ X0 @ X1)))))),
% 8.47/2.46 inference(nnf_transformation,[status(thm)],[f1087])).
% 8.47/2.46 thf(f1087,plain,(
% 8.47/2.46 ! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X2 @ X1) = $true) | ($true != (in @ X2 @ (binintersect @ X0 @ X1)))) <=> (binintersectER = $true)),
% 8.47/2.46 inference(ennf_transformation,[status(thm)],[f421])).
% 8.47/2.46 thf(f421,plain,(
% 8.47/2.46 ! [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X2 @ (binintersect @ X0 @ X1))) => ((in @ X2 @ X1) = $true)) <=> (binintersectER = $true)),
% 8.47/2.46 inference(fool_elimination,[status(thm)],[f420])).
% 8.47/2.46 thf(f420,plain,(
% 8.47/2.46 (binintersectER = ! [X0 : $i,X1 : $i,X2 : $i] : ((in @ X2 @ (binintersect @ X0 @ X1)) => (in @ X2 @ X1)))),
% 8.47/2.46 inference(rectify,[status(thm)],[f114])).
% 8.47/2.46 thf(f114,axiom,(
% 8.47/2.46 (binintersectER = ! [X3 : $i,X4 : $i,X1 : $i] : ((in @ X1 @ (binintersect @ X3 @ X4)) => (in @ X1 @ X4)))),
% 8.47/2.46 file('/export/starexec/sandbox2/tmp/tmp.H1wBmJGvZQ/DTF2THF_2486.p',binintersectER)).
% 8.47/2.46 thf(f9724,plain,(
% 8.47/2.46 ~spl1181_213 | ~spl1181_214),
% 8.47/2.46 inference(avatar_split_clause,[status(thm)],[f9714,f9721,f9718])).
% 8.47/2.46 thf(f9714,plain,(
% 8.47/2.46 ((subset @ sK1183 @ sK955) != $true) | ((subset @ sK1183 @ sK956) != $true)),
% 8.47/2.46 inference(trivial_inequality_removal,[status(thm)],[f9708])).
% 8.47/2.46 thf(f9708,plain,(
% 8.47/2.46 ((subset @ sK1183 @ sK956) != $true) | ($false = $true) | ((subset @ sK1183 @ sK955) != $true)),
% 8.47/2.46 inference(superposition,[status(thm)],[f6849,f7005])).
% 8.47/2.46 thf(f7005,plain,(
% 8.47/2.46 ( ! [X3 : $i,X4 : $i,X5 : $i] : (((subset @ X4 @ (binintersect @ X3 @ X5)) = $true) | ((subset @ X4 @ X5) != $true) | ((subset @ X4 @ X3) != $true)) )),
% 8.47/2.46 inference(trivial_inequality_removal,[status(thm)],[f6169])).
% 8.47/2.46 thf(f6169,plain,(
% 8.47/2.46 ( ! [X3 : $i,X4 : $i,X5 : $i] : (((subset @ X4 @ X5) != $true) | ((subset @ X4 @ (binintersect @ X3 @ X5)) = $true) | ((subset @ X4 @ X3) != $true) | ($true != $true)) )),
% 8.47/2.46 inference(definition_unfolding,[status(thm)],[f4489,f4856])).
% 8.47/2.46 thf(f4856,plain,(
% 8.47/2.46 (binintersectSubset5 = $true)),
% 8.47/2.46 inference(cnf_transformation,[status(thm)],[f3115])).
% 8.47/2.46 thf(f4489,plain,(
% 8.47/2.46 ( ! [X3 : $i,X4 : $i,X5 : $i] : (((subset @ X4 @ (binintersect @ X3 @ X5)) = $true) | ((subset @ X4 @ X3) != $true) | ((subset @ X4 @ X5) != $true) | (binintersectSubset5 != $true)) )),
% 8.47/2.46 inference(cnf_transformation,[status(thm)],[f2732])).
% 8.47/2.46 thf(f2732,plain,(
% 8.47/2.46 ((binintersectSubset5 = $true) | (((subset @ sK712 @ (binintersect @ sK711 @ sK713)) != $true) & ((subset @ sK712 @ sK711) = $true) & ((subset @ sK712 @ sK713) = $true))) & (! [X3 : $i,X4 : $i,X5 : $i] : (((subset @ X4 @ (binintersect @ X3 @ X5)) = $true) | ((subset @ X4 @ X3) != $true) | ((subset @ X4 @ X5) != $true)) | (binintersectSubset5 != $true))),
% 8.47/2.46 inference(skolemisation,[status(esa),new_symbols(skolem,[sK711,sK712,sK713])],[f2730,f2731])).
% 8.47/2.46 thf(f2731,plain,(
% 8.47/2.46 ? [X0 : $i,X1 : $i,X2 : $i] : (((subset @ X1 @ (binintersect @ X0 @ X2)) != $true) & ((subset @ X1 @ X0) = $true) & ((subset @ X1 @ X2) = $true)) => (((subset @ sK712 @ (binintersect @ sK711 @ sK713)) != $true) & ((subset @ sK712 @ sK711) = $true) & ((subset @ sK712 @ sK713) = $true))),
% 8.47/2.46 introduced(definition,[],[choice_axiom])).
% 8.47/2.46 thf(f2730,plain,(
% 8.47/2.46 ((binintersectSubset5 = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (((subset @ X1 @ (binintersect @ X0 @ X2)) != $true) & ((subset @ X1 @ X0) = $true) & ((subset @ X1 @ X2) = $true))) & (! [X3 : $i,X4 : $i,X5 : $i] : (((subset @ X4 @ (binintersect @ X3 @ X5)) = $true) | ((subset @ X4 @ X3) != $true) | ((subset @ X4 @ X5) != $true)) | (binintersectSubset5 != $true))),
% 8.47/2.46 inference(rectify,[status(thm)],[f2729])).
% 8.47/2.46 thf(f2729,plain,(
% 8.47/2.46 ((binintersectSubset5 = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (((subset @ X1 @ (binintersect @ X0 @ X2)) != $true) & ((subset @ X1 @ X0) = $true) & ((subset @ X1 @ X2) = $true))) & (! [X0 : $i,X1 : $i,X2 : $i] : (((subset @ X1 @ (binintersect @ X0 @ X2)) = $true) | ((subset @ X1 @ X0) != $true) | ((subset @ X1 @ X2) != $true)) | (binintersectSubset5 != $true))),
% 8.47/2.46 inference(nnf_transformation,[status(thm)],[f1383])).
% 8.47/2.46 thf(f1383,plain,(
% 8.47/2.46 (binintersectSubset5 = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i] : (((subset @ X1 @ (binintersect @ X0 @ X2)) = $true) | ((subset @ X1 @ X0) != $true) | ((subset @ X1 @ X2) != $true))),
% 8.47/2.46 inference(flattening,[status(thm)],[f1382])).
% 8.47/2.46 thf(f1382,plain,(
% 8.47/2.46 ! [X0 : $i,X1 : $i,X2 : $i] : ((((subset @ X1 @ (binintersect @ X0 @ X2)) = $true) | ((subset @ X1 @ X2) != $true)) | ((subset @ X1 @ X0) != $true)) <=> (binintersectSubset5 = $true)),
% 8.47/2.46 inference(ennf_transformation,[status(thm)],[f488])).
% 8.47/2.46 thf(f488,plain,(
% 8.47/2.46 ! [X0 : $i,X1 : $i,X2 : $i] : (((subset @ X1 @ X0) = $true) => (((subset @ X1 @ X2) = $true) => ((subset @ X1 @ (binintersect @ X0 @ X2)) = $true))) <=> (binintersectSubset5 = $true)),
% 8.47/2.46 inference(fool_elimination,[status(thm)],[f487])).
% 8.47/2.46 thf(f487,plain,(
% 8.47/2.46 (binintersectSubset5 = ! [X0 : $i,X1 : $i,X2 : $i] : ((subset @ X1 @ X0) => ((subset @ X1 @ X2) => (subset @ X1 @ (binintersect @ X0 @ X2)))))),
% 8.47/2.46 inference(rectify,[status(thm)],[f109])).
% 8.47/2.46 thf(f109,axiom,(
% 8.47/2.46 (binintersectSubset5 = ! [X3 : $i,X5 : $i,X4 : $i] : ((subset @ X5 @ X3) => ((subset @ X5 @ X4) => (subset @ X5 @ (binintersect @ X3 @ X4)))))),
% 8.47/2.46 file('/export/starexec/sandbox2/tmp/tmp.H1wBmJGvZQ/DTF2THF_2486.p',binintersectSubset5)).
% 8.47/2.46 thf(f6849,plain,(
% 8.47/2.46 ((subset @ sK1183 @ (binintersect @ sK955 @ sK956)) = $false)),
% 8.47/2.46 inference(binary_proxy_clausification,[status(thm)],[f6848])).
% 8.47/2.46 % SZS output end Proof for DTF2THF_2486
% 8.47/2.46 % (2639)------------------------------
% 8.47/2.46 % (2639)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.47/2.46 % (2639)Termination reason: Refutation
% 8.47/2.46
% 8.47/2.46 % (2639)Memory used [KB]: 13432
% 8.47/2.46 % (2639)Time elapsed: 0.452 s
% 8.47/2.46 % (2639)Instructions burned: 912 (million)
% 8.47/2.46 % (2639)------------------------------
% 8.47/2.46 % (2639)------------------------------
% 8.47/2.46 % (2591)Success in time 0.939 s
%------------------------------------------------------------------------------